susy SuperSymmetry Primer

hep-ph/9709356 v3 7 Apr 1999 hep/-ph///9/7/0/9/3/5/6v/3 April /7/, /1/9/9/9A SUPERSYMMETR Y PRIMERSTEPHEN P /. MAR TIN yR andal l Physics L ab or atory/, University of MichiganA nn A rb or MI /4/8/1/0/9/-/1/1/2/0 USAIp r o vide a p edagogical in tro duction to sup ersymmetry /. The lev el of dis/-cussion is aimed at readers who are familiar with the Standard Mo del andquan tum /eld theory /, but who ha v e little or no prior exp osure to sup ersym/-metry /.T opics co v ered include/: motiv ations for sup ersymmetry/;; the con/-struction of sup ersymmetric Lagrangians/;; sup ersymmetry/-breakin g in terac/-tions/;; the Minimal Sup ersymmetric Standard Mo del /(MSSM/)/;; R /-parit ya n dits consequences/;; the origins of sup ersymmetry breaking/;; the mass sp ectrumof the MSSM/;; deca ys of sup ersymmetric particles/;; exp erimen tal signals forsup ersymmetry/;; and some extensions of the minimal framew ork/. This is anextended v ersion of a con tribution to the b o ok Persp e ctives on Sup ersym/-metry /, edited b yG /.L /. K a n e/( W orld Scien ti/c/, Singap ore /1/9/9/8/)/.Con ten ts/1 In tro duction /2/2 In terlude/: Notations and Con v en tions /1/2/3 Sup ersymmetric lagrangians /1/5/3/./1 The simplest sup ersymmetric mo del/: a free c hiral sup erm ultiplet /././. /././. /. /1/6/3/./2 In teractions of c hiral sup erm ultiplets /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /2/0/3/./3 Lagrangians for gauge sup erm ultiplets /. /././. /./. /././. /././. /././. /././. /././. /. /2/3/3/./4 Sup ersymmetric gauge in t e r a c t i o n s /././. /././. /./. /././. /././. /././. /././. /././. /. /2/4/3/./5 Summary/: Ho w to build a sup ersymmetric mo del /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /2/6/4 Soft sup ersymmetry breaking in teractions /2/8/5 The Minimal Sup ersymmetric Standard Mo del /3/0/5/./1 The sup erp oten tial and sup ersymmetric in teractions /. /././. /././. /././. /././. /. /3/0/5/./2 R /-parit y /(also kno wn as matter parit y/) and its consequences /. /. /. /. /. /. /. /. /. /. /3/4/5/./3 Soft sup ersymmetry breaking in the MSSM /. /./. /././. /././. /././. /././. /././. /. /3/6/5/./4 Hin ts of an Organizing Principle /. /././. /././. /./. /././. /././. /././. /././. /././. /. /3/7/6 Origins of sup ersymmetry breaking /4/1/6/./1 General considerations for sup ersymmetry breaking /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /4/1/6/./2 The goldstino and the gra v i t i n o /./. /././. /././. /./. /././. /././. /././. /././. /././. /. /4/5/6/./3 Gra vit y/-mediated sup ersymmetry breaking mo dels /./. /././. /././. /././. /././. /. /4/9/6/./4 Gauge/-mediated sup ersymmetry breaking mo dels /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /5/0 ySince Octob er /1/, /1/9/9/8/: Departmen to f P h ysics/, Northern Illinois Univ ersit y /, DeKalb IL /6/0/1/1/5/, andTheoretical Ph ysics/, F ermi National Accelerator Lab oratory /, Bata via IL /6/0/5/1/0/. email/: spmartin/@f nal /.go v/1 /7 The mass sp ectrum of the MSSM /5/5/7/./1 Renormalization Group Equations /././. /././. /./. /././. /././. /././. /././. /././. /. /5/6/7/./2 Electro w eak symmetry breaking and the Higgs b osons /. /. /./. /. /./. /. /./. /. /./. /6/0/7/./3 Neutralinos and c harginos /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /6/5/7/./4 T h e g l u i n o /././. /./. /././. /././. /././. /././. /././. /./. /././. /././. /././. /././. /././. /. /6/8/7/./5 The squark and slepton mass sp ectrum /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /6/8/7/./6 Summary/: the MSSM sparticle sp ectrum /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /7/2/8 Sparticle deca ys /7/5/8/./1 Deca ys of neutralinos and c harginos /./. /././. /./. /././. /././. /././. /././. /././. /. /7/5/8/./2 Slepton deca ys /./. /././. /././. /././. /././. /././. /./. /././. /././. /././. /././. /././. /. /7/6/8/./3 Squark deca y s /. /./. /././. /././. /././. /././. /././. /./. /././. /././. /././. /././. /././. /. /7/7/8/./4 Gluino deca y s /. /./. /././. /././. /././. /././. /././. /./. /././. /././. /././. /././. /././. /. /7/7/8/./5 Deca ys to the gra vitino//goldstino /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /7/7/9 Exp erimen tal signals for sup ersymmetry /7/9/9/./1 Signals at e /+e /;colliders /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /7/9/9/./2 Signals at hadron colliders /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /8/2/9/./3 Dark matter detection /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /8/4/1/0 Some miscellaneous v ariations /8/5/1/0/./1 Mo dels with R /-parit y violation/. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /8/5/1/0/./2 The next/-to/-minimal sup ersymmetric standard mo del /. /. /./. /. /./. /. /./. /. /./. /8/7/1/0/./3 Extra D /-term con tributions to scalar masses /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /. /8/7/1/1 Concluding remarks /9/0App endix/: Nonrenormalizable sup ersymmetric lagrangians /9/0References /9/5/1 In tro ductionThe Standard Mo del of high energy ph ysics pro vides a remark ably successful descriptionof presen tly kno wn phenomena/. The exp erimen tal high/-energy fron tier has adv anced in tothe h undreds of GeV range with no con/rmed deviations from Standard Mo del predictionsand few unam biguous hin ts of additional structure/. Still/, it seems quite clear that theStandard Mo del is a w ork in progress and will ha v e to b e extended to describ e ph ysics atarbitrarily high energies/. Certainly a new framew ork will b e required at the reduced Planc kscale MP /=/( /8 /GNewton /) /; /1 /= /2/=/2 /: /4 / /1/0 /1/8GeV/, where quan tum gra vitational e/ects b ecomeimp ortan t/. Based only on a prop er resp ect for the p o w er of Nature to surprise us/, it seemsnearly as ob vious that new ph ysics exists in the /1/6 orders of magnitude in energy b et w eenthe presen tly explored territory and the Planc k scale/.The mere fact that the ratio MP /= MW is so h uge is already a p o w erful clue to the c har/-acter of ph ysics b ey ond the Standard Mo del/, b ecause of the infamous /\hierarc h y problem/"/. /1This is not really a di/cult y with the Standard Mo del itself/, but rather a disturbing sensitiv/-it y of the Higgs p oten tial to new ph ysics in almost an y imaginable extension of the StandardMo del/. The electrically neutral part of the Standard Mo del Higgs /eld is a complex scalar/2 (a)Hf (b)HSFigure /1/: Quan tum corrections to the Higgs /(mass/) /2/.H with a classical p oten tial giv en b yV /= m /2H j H j /2/+ / j H j /4/: /(/1/./1/)The Standard Mo del requires a non/-v anishing v acuum exp ectation v alue /(VEV/) for H atthe minim um of the p oten tial/. This will o ccur if m /2H /< /0/, resulting in h H i /= q /; m /2H /= /2 / /.Since w e kno w exp erimen tally that h H i /= /1/7/4 GeV from measuremen ts of the prop erties ofthe w eak in teractions/, it m ust b e that m /2H is v ery roughly of order /; /(/1/0/0 GeV/) /2/.H o w ev er/,m /2H receiv es enormous quan tum corrections from the virtual e/ects of ev ery particle whic hcouples/, directly or indirectly /, to the Higgs /eld/.F or example/, in Fig/. /1a w eh a v e a correction to m /2H from a lo op con taining a Diracfermion f with mass mf /. If the Higgs /eld couples to f with a term in the lagrangian/; /f H ff /,t h e nt h eF eynman diagram in Fig/. /1a yields a correction/
m /2H /= j /f j /2 /1/6 / /2 h/; /2/ /2UV /+/6 m /2f ln /(/UV /=mf /)/+ /:/:/: i/: /(/1/./2/)Here /UV is an ultra violet momen tum cuto/ used to regulate the lo op in tegral/;; it should b ein terpreted as the energy scale at whic h new ph ysics en ters to alter the high/-energy b eha viorof the theory /. The ellipses represen t terms whic h dep end on the precise manner in whic h themomen tum cuto/ is applied/, and whic h do not get large as /UV do es/. Eac h of the leptonsand quarks of the Standard Mo del can pla y the role of f /;; for quarks/, eq/. /(/1/./2/) should b em ultiplied b y /3 to accoun t for color/. The largest correction comes when f is the top quarkwith /f / /1/. The problem is that if /UV is of order MP /,s a y /, then this quan tum correctionto m /2H is some /3/0 orders of magnitude larger than the aimed/-for v alue of m /2H //; /(/1/0/0GeV/) /2/. This is only directly a problem for corrections to the Higgs scalar b oson /(mass/) /2/,b ecause quan tum corrections to fermion and gauge b oson masses do not ha v e the quadraticsensitivit yt o /UV found in eq/. /(/1/./2/)/. Ho w ev er/, the quarks and leptons and the electro w eakgauge b osons Z /0/, W /of the Standard Mo del all o w e their masses to h H i /, so that the en tiremass sp ectrum of the Standard Mo del is directly or indirectly sensitiv e to the cuto/ /UV /.One could imagine that the solution is to simply pic k an ultra violet cuto/ /UV whic hi snot to o large/. Ho w ev er/, one still has to conco ct some new ph ysics at the scale /UV whic hnot only alters the propagators in the lo op/, but actually cuts o/ the lo op in tegral/. This isnot easy to do in a theory whose lagrangian do es not con tain more than t w o deriv ativ es/, andhigher deriv ativ e theories generally su/er from a loss of unitarit y /. In string theories/, lo opin tegrals are cut o/ at high Euclidean momen tum p b y factors e /; p /2/= / /2UV/, but then /UV isa string scale whic h is usually though tt o b e n o t v ery far b elo w MP /.F urthermore/, there isa con tribution similar to eq/. /(/1/./2/) from the virtual e/ects of an y arbitrarily hea vy particleswhic h migh t exist/. F or example/, supp ose there exists a hea vy complex scalar particle Swith mass mS whic h couples to the Higgs with a lagrangian term /; /S j H j /2j S j /2/. Then the/3 (b)HF (a)HFFigure /2/: Tw o/-lo op corrections to the Higgs /(mass/) /2d u e t oah e a vy fermion/.F eynman diagram in Fig/. /1b giv es a correction/
m /2H /= /S /1/6 / /2 h/ /2UV /; /2 m /2S ln /(/UV /=mS /)/+ /:/:/: i/: /(/1/./3/)If one rejects a ph ysical in terpretation of /UV and uses dimensional regularization on thelo op in tegral instead of a momen tum cuto//, then there will b e no / /2UV piece/. Ho w ev er/, ev enthen the term prop ortional to m /2S cannot b e eliminated without the ph ysically unjusti/abletuning of a coun ter/-term sp eci/cally for that purp ose/. So m /2H is sensitiv e to the massesof the he aviest particles that H couples to/;; if mS is v ery large/, its e/ects on the StandardMo del do not decouple/, but instead mak ei t v ery di/cult to understand wh y m /2H is so small/.This problem arises ev en if there is no direct coupling b et w een the Standard Mo delHiggs b oson and the unkno wn hea vy particles/. F or example/, supp ose that there exists ahea vy fermion F whic h/, unlik e the quarks and leptons of the Standard Mo del/, has v ector/-lik e quan tum n um b ers and therefore gets a large mass mF without coupling to the Higgs/eld/. /[In other w ords/, an arbitrarily large mass term of the form mF FF is not forbiddenb ya n y symmetry /, including SU /(/2/)L /./] In that case/, no diagram lik e Fig/. /1a exists for F /.Nev ertheless there will b e a correction to m /2H as long as F shares some gauge in teractionswith the Standard Mo del Higgs /eld/;; these ma y b e the familiar electro w eak in teractions/,or some unkno wn gauge forces whic h are brok en at a v ery high energy scale inaccessible toexp erimen t/. In an y case/, the t w o/-lo op F eynman diagrams in Fig/. /2 yield a correction/
m /2H /= x / g /2 /1/6 / /2 /!/2ha / /2UV /+/4 /8 m /2F ln /(/UV /=mF /)/+ /:/:/: i/;; /(/1/./4/)where g is the gauge coupling in question/, and x is a group theory factor of order /1/. /(Sp ecif/-ically /, x is the pro duct of the quadratic Casimir in v arian to f H and the Dynkin index of Ffor the gauge group in question/./) The co e/cien t a dep ends on the precise metho d of cuttingo/ the momen tum in tegrals/. It do es not arise at all if one rejects the p ossibilit yo f a p h ys/-ical in terpretation for /UV and uses dimensional regularization/, but the m /2F con tributionis alw a ys presen t/. The n umerical factor /( g /2/= /1/6 / /2/) /2ma y b e quite small /(of order /1/0 /; /5forelectro w eak in teractions/)/, but the imp ortan tp o i n t is that these con tributions to /
m /2H aresensitiv e to the largest masses and//or ultra violet cuto/ in the theory /, presumably of orderMP /. The /\natural/" /(mass/) /2of a fundamen tal Higgs scalar/, includin g quan tum corrections/,seems to b e more lik e M /2P than the exp erimen tally fa v ored v alue/! Ev en v ery indirect con/-tributions from F eynman diagrams with three or more lo ops can giv e unacceptably largecon tributions to /
m /2H /. If the Higgs b oson is a fundamen tal particle/, w eh a v et w o options/:either w em ust mak e the rather bizarre assumption that there do not exist any hea vy par/-ticles whic hc o u p l e/( e v en indirectly or extremely w eakly/) to the Higgs scalar /eld/, or somerather striking cancellation is needed b et w een the v arious con tributions to /
m /2H /./4 The systematic cancellation of the dangerous con tributions to /
m /2H can only b e brough tab out b y the t yp e of conspiracy whic h is b etter kno wn to ph ysicists as a symmetry /.I t i sapparen t from comparing eqs/. /(/1/./2/)/, /(/1/./3/) that the new symmetry ough t to relate fermionsand b osons/, b ecause of the relativ em i n us sign b et w een fermion lo op and b oson lo op con tri/-butions to /
m /2H /. /(Note that /S m ust b e p ositiv e if the scalar p oten tial is to b e b oundedfrom b elo w/./) If eac h of the quarks and leptons of the Standard Mo del is accompanied b yt w o complex scalars with /S /= j /f j /2/, then the / /2UV con tributions of Figs/. /1a and /1b willneatly cancel/. /2Clearly /, more restrictions on the theory will b e necessary to ensure thatthis success p ersists to higher orders/, so that/, for example/, the con tributions in Fig/. /2 andeq/. /(/1/./4/) from a v ery hea vy fermion are cancelled b y the t w o/-lo op e/ects of some v ery hea vyb osons/. F ortunately /, conditions for cancelling all suc hc o n tributions to scalar masses arenot only p ossible/, but are actually una v oidable once w e merely assume that a symmetryrelating fermions and b osons/, called a sup ersymmetry /, should exist/.A sup ersymmetry transformation turns a b osonic state in to a fermionic state/, and vicev ersa/. The op erator Q whic h generates suc h transformations m ust b e an an ticomm utingspinor/, withQ j Boson i /= j F ermion i /;; Q j F ermion i /= j Boson i /: /(/1/./5/)Spinors are in trinsically complex ob jects/, so Q y/(the hermitian conjugate of Q /) is also asymmetry generator/. Because Q and Q yare fermionic op erators/, they carry spin angularmomen tum /1///2/, so it is clear that sup ersymmetry m ust b e a spacetime symmetry /. The p os/-sible forms for suc h symmetries in an in teracting quan tum /eld theory are highly restrictedb y the Haag/-Lopuszanski/-Sohnius extension of the Coleman/-Mandula theorem/. /3F or realistictheories whic h/, lik e the Standard Mo del/, ha v ec hiral fermions /(i/.e/./, fermions whose left/- andrigh t/-handed pieces transform di/eren tly under the gauge group/) and th us the p ossibilit yof parit y/-violating in teractions/, this theorem implies that the generators Q and Q ym ustsatisfy an algebra of an ticomm utation and comm utation relations with the sc hematic formf Q/;; Q yg /= P //(/1/./6/)f Q/;; Q g /= f Q y/;;Q yg /=/0 /(/1/./7/)/[ P //;;Q /]/= /[ P //;;Q y/]/= /0 /(/1/./8/)where P /is the momen tum generator of spacetime translations/. Here w eh a v e ruthlesslysuppressed the spinor indices on Q and Q y/;; after dev eloping some notation w e will/, in section/3/./1/, deriv e the precise v ersion of eqs/. /(/1/./6/)/-/(/1/./8/) with indices restored/. In the mean time/, w esimply note that the app earance of P /on the righ t/-hand side of eq/. /(/1/./6/) is unsurprising/,since it transforms under Loren tz b o osts and rotations as a spin/-/1 ob ject while Q and Q yon the left/-hand side eac h transform as spin/-/1///2 ob jects/.The single/-particle states of a sup ersymmetric theory fall naturally in to irreducible rep/-resen tations of the sup ersymmetry algebra whic h are called sup ermultiplets /. Eac h sup erm ul/-tiplet con tains b oth fermion and b oson states/, whic h are commonly kno wn as sup erp artnersof eac h other/. By de/nition/, if j / i and j / /0i are mem b ers of the same sup erm ultiplet/, thenj / /0i is prop ortional to some com bination of Q and Q yop erators acting on j / i /,u p t oaspacetime translation or rotation/. The /(mass/) /2op erator /; P /2comm utes with the op eratorsQ /, Q y/, and with all spacetime rotation and translation op erators/, so it follo ws immediatelythat particles whic h inhabit the same irreducible sup erm ultiplet m ust ha v e equal eigen v aluesof /; P /2/, and therefore equal masses/./5 The sup ersymmetry generators Q/;; Q yalso comm ute with the generators of gauge trans/-formations/. Therefore particles in the same sup erm ultiplet m ust also b e in the same repre/-sen tation of the gauge group/, and so m ust ha v e the same electric c harges/, w eak isospin/, andcolor degrees of freedom/.Eac h sup erm ultiplet con tains an equal n um b er of fermion and b oson degrees of freedom/.T o pro v e this/, consider the op erator /( /; /1/) /2 swhere s is the spin angular momen tum/. Bythe spin/-statistics theorem/, this op erator has eigen v alue /+/1 acting on a b osonic state andeigen v alue /; /1 acting on a fermionic state/. An y fermionic op erator will turn a b osonicstate in to a fermionic state and vice v ersa/. Therefore /( /; /1/) /2 sm ust an ticomm ute with ev eryfermionic op erator in the theory /, and in particular with Q and Q y/. No w consider thesubspace of states j i i in a sup erm ultiplet whic hh a v e the same eigen v alue p /of the four/-momen tum op erator P //. In view of eq/. /(/1/./8/)/, an yc o m bination of Q or Q yacting on j i i willgiv e another state j i /0i whic h has the same four/-momen tum eigen v alue/. Therefore one has acompleteness relation Pi j i ih i j /= /1 within this subspace of states/. No w one can tak e a traceo v er all suc h states of the op erator /( /; /1/) /2 sP //(including eac h spin helicit y state separately/)/:Xi h i j /( /; /1/) /2 sP /j i i /= Xi h i j /( /; /1/) /2 sQQ yj i i /+ Xi h i j /( /; /1/) /2 sQ yQ j i i/= Xi h i j /( /; /1/) /2 sQQ yj i i /+ Xi Xj h i j /( /; /1/) /2 sQ yj j ih j j Q j i i/= Xi h i j /( /; /1/) /2 sQQ yj i i /+ Xj h j j Q /( /; /1/) /2 sQ yj j i/= Xi h i j /( /; /1/) /2 sQQ yj i i/; Xj h j j /( /; /1/) /2 sQQ yj j i/= /0 /: /(/1/./9/)The /rst equalit yf o l l o ws from the sup ersymmetry algebra relation eq/. /(/1/./6/)/;; the second andthird from use of the completeness relation/;; and the fourth from the fact that /( /; /1/) /2 sm ustan ticomm ute with Q /.N o w Pi h i j /( /; /1/) /2 sP /j i i /= p /T r/[/( /; /1/) /2 s/] is just prop ortional to then um b er of b osonic degrees of freedom nB min us the n um b er of fermionic degrees of freedomnF in the trace/, so thatnB /= nF /(/1/./1/0/)m ust hold for a giv en p //6/= /0 in eac h sup erm ultiplet/.The simplest p ossibilit y for a sup erm ultiplet whic h is consisten t with eq/. /(/1/./1/0/) has asingle W eyl fermion /(with t w o helicit y states/, so nF /= /2/) and t w o real scalars /(eac h withnB /= /1/)/. It is natural to assem ble the t w o real scalar degrees of freedom in to a complex scalar/eld/;; as w e will see b elo w this pro vides for con v enien tf o r m ulation of the sup ersymmetryalgebra/, F eynman rules/, sup ersymmetry violating e/ects/, etc/. This com bination of a t w o/-comp onen tW eyl fermion and a complex scalar /eld is called a chir al or matter or sc alarsup erm ultiplet/.The next simplest p ossibilit y for a sup erm ultiplet con tains a spin/-/1 v ector b oson/. If thetheory is to b e renormalizable this m ust b e a gauge b oson whic h is massless/, at least b eforethe gauge symmetry is sp on taneously brok en/. A massless spin/-/1 b oson has t w o helicit ystates/, so the n um b er of b osonic degrees of freedom is nB /= /2/. Its sup erpartner is thereforea massless spin/-/1///2 W eyl fermion/, again with t w o helicit y states/, so nF /= /2/. /(If one triedinstead to use a massless spin/-/3///2 fermion/, the theory w ould not b e renormalizable/./) Gaugeb osons m ust transform as the adjoin t represen tation of the gauge group/, so their fermionic/6 partners/, called gauginos /,m ust also/. Since the adjoin t represen tation of a gauge group isalw a ys its o wn conjugate/, this means in particular that these fermions m ust ha v e the samegauge transformation prop erties for left/-handed and for righ t/-handed comp onen ts/. Suc ha com bination of spin/-/1///2 gauginos and spin/-/1 gauge b osons is called a gauge or ve ctorsup erm ultiplet/.There are other p ossible com binations of particles with spins whic h can satisfy eq/. /(/1/./1/0/)/.Ho w ev er/, these are alw a ys reducible to com binations of c hiral and gauge sup erm ultiplets ifthey ha v e renormalizable in teractions/, except in certain theories with /\extended/" sup er/-symmetry /. Theories with extended sup ersymmetry ha v e more than one distinct cop y of thesup ersymmetry generators Q/;; Q y/. Suc h theories are mathematically am using/, but eviden tlydo not ha v ea n y phenomenological prosp ects/. The reason is that extended sup ersymmetryin four/-dimensional /eld theories cannot allo w for c hiral fermions or parit y violation as ob/-serv ed in the Standard Mo del/. So w e will not discuss suc h p ossibiliti es further/, althoughextended sup ersymmetry in higher dimensional /eld theories migh t describ e the real w orldif the extra dimensions are compacti/ed/, and extended sup ersymmetry in four dimensionspro vides in teresting to ym o d e l s /. The ordinary /, non/-extended/, phenomenologically/-vi ablet yp e of sup ersymmetric mo del is sometimes called N /= /1 sup ersymmetry /, with N referringto the n um b er of sup ersymmetries /(the n um b er of distinct copies of Q/;; Q y/)/.In a sup ersymmetric extension of the Standard Mo del/, /4 /;; /5 /;; /6eac h of the kno wn funda/-men tal particles m ust therefore b e in either a c hiral or gauge sup erm ultiplet and ha v easup erpartner with spin di/ering b y /1///2 unit/. The /rst step in understanding the excitingphenomenological consequences of this prediction is to decide ho w the kno wn particles /tin to sup erm ultiplets/, and to giv e them appropriate names/. A crucial observ ation here isthat only c hiral sup erm ultiplets can con tain fermions whose left/-handed parts transformdi/eren tly under the gauge group than their righ t/-handed parts/. All of the Standard Mo delfermions /(the kno wn quarks and leptons/) ha v e this prop ert y /, so they m ust b e mem b ers ofc hiral sup erm ultiplets/. yThe names for the spin/-/0 partners of the quarks and leptons areconstructed b y prep ending an /\s/"/, whic h is short for scalar/. Th us generically they are calledsquarks and sleptons /(short for /\scalar quark/" and /\scalar lepton/"/)/. The left/-handed andrigh t/-handed pieces of the quarks and leptons are separate t w o/-comp onen tW eyl fermionswith di/eren t gauge transformation prop erties in the Standard Mo del/, so eac hm ust ha v ei t so wn complex scalar partner/. The sym b ols for the squarks and sleptons are the same as forthe corresp onding fermion/, but with a tilde used to denote the sup erpartner of a StandardMo del particle/. F or example/, the sup erpartners of the left/-handed and righ t/-handed partsof the electron Dirac /eld are called left/- and righ t/-handed selectrons/, and are denoted eeLand eeR /. It is imp ortan tt ok eep in mind that the /\handedness/" here do es not refer to thehelicit y of the selectrons /(they are spin/-/0 particles/) but to that of their sup erpartners/. Asimilar nomenclature applies for sm uons and staus/: e/L /, e/R /, e/L /, e/R /. In the Standard Mo delthe neutrinos are alw a ys left/-handed/, so the sneutrinos are denoted generically b y e/ /, with ap ossible subscript indicating whic hl e p t o n/ a v or they carry/: e/e /, e// /, e// /. Finally /, a completelist of the squarks is eqL /, eqR with q /= u/;; d/;; s/;; c/;; b/;; t /. The gauge in teractions of eac h of thesesquark and slepton /eld are the same as for the corresp onding Standard Mo del fermion/;; forinstance/, a left/-handed squark lik e euL will couple to the W b oson while euR will not/.It seems clear that the Higgs scalar b oson m u s tr e s i d e i nac hiral sup erm ultiplet/, sinceit has spin /0/. Actually /, it turns out that one c hiral sup erm ultiplet is not enough/. One w a yto see this is to note that if there w ere only one Higgs c hiral sup erm ultiplet/, the electro w eak yIn particular/, one cannot attempt to mak e a spin/-/1///2 neutrino b e the sup erpartner of the spin/-/1 photon/;;the neutrino is in a doublet/, and the photon neutral/, under w eak isospin/./7 T able /1/: Chiral sup erm ultiplets in the Minimal Sup ersymmetric Standard Mo del/. Names spin /0 spin /1///2 SU /(/3/)C /;;S U /(/2/)L /;;U /(/1/)Y squarks/, quarks Q /( euL edL /) /( uL dL /) /( /3 /;; /2 /;; /1 /6 /) /( / /3 families/) u eu /R u yR /( /3 /;; /1 /;; /; /2 /3 /) d ed /R d yR /( /3 /;; /1 /;; /1 /3 /) sleptons/, leptons L /( e/ eeL /) /( / eL /) /( /1 /;; /2 /;; /; /1 /2 /) /( / /3 families/) e ee /R e yR /( /1 /;; /1 /;; /1/) Higgs/, higgsinos Hu /( H /+u H /0u /) /( eH /+u eH /0u /) /( /1 /;; /2 /;; /+ /1 /2 /) Hd /( H /0d H /;d /) /( eH /0d eH /;d /) /( /1 /;; /2 /;; /; /1 /2 /) gauge symmetry w ould su/er a triangle gauge anomaly /, and w ould b e inconsisten ta saquan tum theory /. This is b ecause the conditions for cancellation of gauge anomalies includeT r/[ Y /3/]/= T r/[ T /2/3 Y /]/= /0 /;; where T/3 and Y are the third comp onen to f w eak isospin and thew eak h yp erc harge/, resp ectiv ely /, in a normalization where the ordinary electric c harge isQEM /= T/3 /+ Y /. The traces run o v er all of the left/-handed W eyl fermionic degrees of freedomin the theory /. In the Standard Mo del/, these conditions are already satis/ed/, somewhatmiraculously /,b y the kno wn quarks and leptons/. No w/, a fermionic partner of a Higgs c hiralsup erm ultiplet m ust b e a w eak iso doublet with w eak h yp erc harge Y /=/1 /= /2o r Y /= /; /1 /= /2/. Ineither case alone/, suc h a fermion will mak e a non/-zero con tribution to the traces and sp oilthe anomaly cancellation/. This can b e a v oided if there are t w o Higgs sup erm ultiplets/, onewith eac ho f Y /= / /1 /= /2/. In that case the total con tribution to the anomaly traces from thet w o fermionic mem b ers of the Higgs c hiral sup erm ultiplets will v anish/. As w e will see insection /5/./1/, b oth of these are also necessary for another completely di/eren t reason/: b ecauseof the structure of sup ersymmetric theories/, only a Y /=/+ /1 /= /2 Higgs c hiral sup erm ultipletcan ha v e the Y uk a w a couplings necessary to giv em a s s e s t o c harge /+/2 /= /3 up/-t yp e quarks /(up/,c harm/, top/)/, and only a Y /= /; /1 /= /2 Higgs can ha v e the Y uk a w a couplings necessary to giv emasses to c harge /; /1 /= /3d o wn/-t yp e quarks /(do wn/, strange/, b ottom/) and to c harged leptons/.W e will call the SU /(/2/)L /-doublet complex scalar /elds corresp onding to these t w o cases Huand Hd resp ectiv ely /. zThe w eak isospin comp onen ts of Hu with T/3 /=/( /+ /1 /= /2/, /; /1 /= /2/) ha v eelectric c harges /1/, /0 resp ectiv ely /, and are denoted /( H /+u /, H /0u /)/. Similarly /,t h e SU /(/2/)L /-doubletcomplex scalar Hd has T/3 /=/( /+ /1 /= /2/, /; /1 /= /2/) comp onen ts /( H /0d /, H /;d /)/. The neutral scalar thatcorresp onds to the ph ysical Standard Mo del Higgs b oson is in a linear com bination of H /0uand H /0d /;;w e will discuss this further in section /7/./2/. The generic nomenclature for a spin/-/1///2 sup erpartner is to app end /\/-ino/" to the name of the Standard Mo del particle/, so thefermionic partners of the Higgs scalars are called higgsinos/. They are denoted b y eHu /, eHdfor the SU /(/2/)L /-doublet left/-handed W eyl spinor /elds/, with w eak isospin comp onen ts eH /+u /,eH /0u and eH /0d /, eH /;d /.W eh a v en o w found all of the c hiral sup erm ultiplets of a minimal phenomenologi/-cally viable extension of the Standard Mo del/. They are summarized in T able /1/, classi/-/ed according to their transformation prop erties under the Standard Mo del gauge group zOther notations whic h are p opular in the literature ha v e Hd /;;Hu /! H/1 /;;H/2 or H/;; H /. The one used here hasthe virtue of making it easy to remem b er whic h Higgs is resp onsible for giving masses to whic h quarks/./8 T able /2/: Gauge sup erm ultiplets in the Minimal Sup ersymmetric Standard Mo del/. Names spin /1///2 spin /1 SU /(/3/)C /;;S U /(/2/)L /;;U /(/1/)Y gluino/, gluon eg g /( /8 /;; /1 /;; /0/) winos/, W b osons fW /fW /0 W /W /0 /( /1 /;; /3 /;; /0/) bino/, B b oson eB /0 B /0 /( /1 /;; /1 /;; /0/) SU /(/3/)C / SU /(/2/)L / U /(/1/)Y /, whic h com bines uL /;;dL and //;; eL degrees of freedom in to SU /(/2/)Ldoublets/. Here w eh a v e follo w ed the standard con v en tion that all c hiral sup erm ultiplets arede/ned in terms of left/-handed W eyl spinors/, so that the c onjugates of the righ t/-handedquarks and leptons /(and their sup erpartners/) app ear in T able /1/. This proto col for de/n/-ing c hiral sup erm ultiplets turns out to b e v ery useful for constructing sup ersymmetric la/-grangians/, as w e will see in section /3/. It is useful also to ha v eas y m b ol for eac ho ft h ec hiral sup erm ultiplets as a whole/;; these are indicated in the second column of T able /1/. Th usfor example Q stands for the SU /(/2/)L /-doublet c hiral sup erm ultiplet con taining euL /;;uL /(withw eak isospin comp onen t T/3 /=/+ /1 /= /2/)/, and edL /;;dL /(with T/3 /= /; /1 /= /2/)/, while u stands for theSU /(/2/)L /-singlet sup erm ultiplet con taining eu /R /;;u yR /. There are three families for eac h of thequark and lepton sup erm ultiplets/, but w eh a v e used /rst/-family represen tativ es in T able/1/. Belo w/, a family index i /=/1 /;; /2 /;; /3 will b e a/xed to the c hiral sup erm ultiplet names /( Qi /, ui /;;/:/:/: /) when needed/, e/.g/. /( e/1 /;; e/2 /;; e/3 /)/=/( e /;; //;; / /)/. The bar on u /, d /, e /elds is part of the name/,and do es not denote an y kind of conjugation/.It is in teresting to note that the Higgs c hiral sup erm ultiplet Hd /(con taining H /0d /, H /;d /, eH /0d /,eH /;d /) has exactly the same Standard Mo del gauge quan tum n um b ers as the left/-handed slep/-tons and leptons Li /,e /. g /. /( e/ /, eeL /, / /, eL /)/. Naiv ely one migh t therefore supp ose that w e couldha v e b een more economical in our assignmen tb y taking a neutrino and a Higgs scalar to b esup erpartners/, instead of putting them in separate sup erm ultiplets/. This w ould amoun tt othe prop osal that the Higgs b oson and a sneutrino should b e the same particle/. This is anice try whic h pla y ed a k ey role in some of the /rst attempts to connect sup ersymmetry tophenomenology /, /4but it is no w kno wn not to w ork/. Ev en ignoring the anomaly cancellationproblem men tioned ab o v e/, man y insoluble phenomenological problems w ould result/, includ/-ing lepton n um b er violation and a mass for at least one of the neutrinos in gross violationof exp erimen tal b ounds/. Therefore/, all of the sup erpartners of Standard Mo del particles arereally new particles/, and cannot b e iden ti/ed with some other Standard Mo del state/.The v ector b osons of the Standard Mo del clearly m ust reside in gauge sup erm ultiplets/.Their fermionic sup erpartners are generically referred to as gauginos/. The SU /(/3/)C colorgauge in teractions of QCD are mediated b y the gluon/, whose spin/-/1///2 color/-o ctet sup er/-symmetric partner is the gluino/. As usual/, a tilde is used to denote the sup ersymmetricpartner of a Standard Mo del state/, so the sym b ols for the gluon and gluino are g and egresp ectiv ely /. The electro w eak gauge symmetry SU /(/2/)L / U /(/1/)Y has asso ciated with it spin/-/1 gauge b osons W /+/;;W /0/;;W /;and B /0/, with spin/-/1///2 sup erpartners fW /+/;; fW /0/;; fW /;and eB /0/,called winos and bino /. After electro w eak symmetry breaking/, the W /0/, B /0gauge eigenstatesmix to giv e mass eigenstates Z /0and / /. The corresp onding gaugino mixtures of fW /0andeB /0are called zino /( eZ /0/) and photino /( e/ /)/;; if sup ersymmetry w ere un brok en/, they w ould b emass eigenstates with masses mZ and /0/. T able /2 summarizes the gauge sup erm ultiplets of/9 a minimal sup ersymmetric extension of the Standard Mo del/.The c hiral and gauge sup erm ultiplets in T ables /1 and /2 mak e up the particle con ten to fthe Minimal Sup ersymmetric Standard Mo del /(MSSM/)/. The most ob vious and in terestingfeature of this theory is that none of the sup erpartners of the Standard Mo del particleshas b een disco v ered as of this writing/. If sup ersymmetry w ere un brok en/, then there w ouldha v e to b e selectrons eeL and eeR with masses exactly equal to me /=/0 /: /5/1/1 /:/:/: MeV/. A similarstatemen t applies to eac h of the other sleptons and squarks/, and there w ould also ha v et ob e a massless gluino and photino/. These particles w ould ha v e b een extraordinarily easyto detect long ago/. Clearly /, therefore/, sup ersymmetry is a br oken symmetry in the v acuumstate c hosen b y nature/.Av ery imp ortan t clue as to the nature of sup ersymmetry breaking can b e obtainedb y returning to the motiv ation pro vided b y the hierarc h y problem/. Sup ersymmetry forcedus to in tro duce t w o complex scalar /elds for eac h Standard Mo del Dirac fermion/, whic hi sjust what is needed to enable a cancellation of the quadratically div ergen t/( / /2UV /) pieces ofeqs/. /(/1/./2/) and /(/1/./3/)/. This sort of cancellation also requires that the asso ciated dimensionlesscouplings should b e related /(e/.g/. /S /= j /f j /2/)/. The necessary relationships b et w een couplingsindeed o ccur in un brok en sup ersymmetry /,a s w e will see in section /3/. In fact/, un brok ensup ersymmetry guaran tees that the quadratic div ergences in scalar squared masses m ustv anish to all orders in p erturbation theory /. xNo w/, if brok en sup ersymmetry is still to pro videa solution to the hierarc h y problem/, then the relationships b et w een dimensionless couplingswhic h hold in an un brok en sup ersymmetric theory m ust b e main tained/. Otherwise/, therew ould b e quadratically div ergen t radiativ e corrections to the Higgs scalar masses of the form/
m /2H /= /1 /8 / /2 /( /S /;j /f j /2/)/ /2UV /+ /:/:/: /: /(/1/./1/1/)W e are therefore led to consider /\soft/" sup ersymmetry breaking/. This means that thee/ectiv e lagrangian of the MSSM can b e written in the formL /= LSUSY /+ Lsoft /;; /(/1/./1/2/)where LSUSY preserv es sup ersymmetry in v ariance/, and Lsoft violates sup ersymmetry butcon tains only mass terms and couplings with p ositive mass dimension/. Without furtherjusti/cation/, soft sup ersymmetry breaking migh ts e e ml i k e a rather arbitrary requiremen t/.F ortunately /,w e will see in section /6 that theoretical mo dels for sup ersymmetry breaking canindeed yield e/ectiv e lagrangians with just suc h terms for Lsoft /. If the largest mass scaleasso ciated with the soft terms is denoted msoft /, then the additional non/-sup ersymmetriccorrections to the Higgs scalar /(mass/) /2m ust v anish in the msoft /! /0 limit/, so b y dimensionalanalysis they cannot b e prop ortional to / /2UV /. More generally /, these mo dels main tain thecancellation of quadratically div ergen t terms in the radiativ e corrections of all scalar masses/,to all orders in p erturbation theory /. The corrections also cannot go lik e/
m /2H / msoft /UV /,b ecause in general the lo op momen tum in tegrals alw a ys div erge either quadratically orlogarithmically /, not linearly /,a s /UV /!/1 /.S o t h e y m ust b e of the form/
m /2H /= m /2soft // /1/6 / /2 ln /(/UV /=msoft /)/+ /:/:/: //: /(/1/./1/3/) xA simple w a y to understand this is to note that un brok en sup ersymmetry requires the degeneracy of scalarand fermion masses/. Radiativ e corrections to fermion masses are kno wn to div erge at most logarithmical l y /,so the same m ust b e true for scalar masses in un brok en sup ersymmetry /./1/0 Here / is sc hematic for v arious dimensionless couplings/, and the ellipses stand b oth forterms whic h are indep enden to f /UV and for higher lo op corrections /(whic h dep end on /UVthrough p o w ers of logarithms/)/.Since the mass splittings b et w een the kno wn Standard Mo del particles and their sup er/-partners are just determined b y the parameters msoft app earing in Lsoft /, eq/. /(/1/./1/3/) tells usthat the sup erpartner masses cannot b e to o h uge/. Otherwise/, w ew ould lose our successfulcure for the hierarc h y problem since the m /2soft corrections to the Higgs scalar /(mass/) /2w ouldb e unnaturally large compared to the electro w eak breaking scale of /1/7/4 GeV/. The top andb ottom squarks and the winos and bino giv e esp ecially large con tributions to /
m /2Hu and/
m /2Hd /, but the gluino mass and all the other squark and slepton masses also feed in indi/-rectly /, through radiativ e corrections to the top and b ottom squark masses/. F urthermore/, inmost viable mo dels of sup ersymmetry breaking that are not unduly con triv ed/, the sup er/-partner masses do not di/er from eac ho t h e r b y more than ab out an order of magnitude/.Using /UV / MP and / / /1 in eq/. /(/1/./1/3/)/, one /nds that roughly sp eaking msoft /,a n dtherefore the masses of at least the ligh test few sup erpartners/, should b e at the most ab out/1T eV or so/, in order for the MSSM scalar p oten tial to pro vide a Higgs VEV resulting inmW /;;mZ /= /8/0/./4/, /9/1/./2 GeV without miraculous cancellations/. This is the b est reason forthe optimism among man y theorists that sup ersymmetry will b e disco v ered at LEP/2/, theT ev atron/, the LHC/, or a next generation lepton linear collider/.Ho w ev er/, it is useful to k eep in mind that the hierarc h y problem w as not the historicalmotiv ation for the dev elopmen t of sup ersymmetry in the early /1/9/7/0/'s/. The sup ersymmetryalgebra and sup ersymmetric /eld theories w ere originally conco cted indep enden tly in v ariousdisguises /7 /;; /8 /;; /9 /;; /1/0whic h b ear little resem blance to the MSSM/. It is quite impressiv e that atheory whic hw as dev elop ed for quite di/eren t reasons/, including purely aesthetic ones/, canlater b e found to pro vide a solution for the hierarc h y problem/.One migh ta l s o w onder if there is an y go o d reason wh y all of the sup erpartners of theStandard Mo del particles should b e hea vy enough to ha v ea v oided disco v ery so far/. Thereis/. All of the particles in the MSSM whic hh a v e b een disco v ered so far ha v e somethingin common/;; they w ould necessarily b e massless in the absence of electro w eak symmetrybreaking/. In particular/, the masses of the W //;;Z /0b osons and all quarks and leptons areequal to dimensionless coupling constan ts times the Higgs VEV / /1/7/4 GeV/, while thephoton and gluon are required to b e massless b y electromagnetic and QCD gauge in v ariance/.Con v ersely /, all of the undisco v ered particles in the MSSM ha v e exactly the opp osite prop ert y /,since eac h o ft h e mc a nh a v e a lagrangian mass term in the absence of electro w eak symmetrybreaking/. F or the squarks/, sleptons/, and Higgs scalars this follo ws from a general prop ert yof complex scalar /elds that a mass term m /2j / j /2is alw a ys allo w ed b y all gauge symmetries/.F or the higgsinos and gauginos/, it follo ws from the fact that they are fermions in a realrepresen tation of the gauge group/. So/, from the p oin t of view of the MSSM/, the disco v ery ofthe top quark in /1/9/9/5 mark ed a quite natural milestone/;; the already/-disco v ered particles areprecisely those whic h had to b e ligh t/, based on the principle of electro w eak gauge symmetry /.There is a single exception/: one neutral Higgs scalar b oson should b e ligh ter than ab out/1/5/0 GeV if sup ersymmetry is correct/, for reasons to b e discussed in section /7/./2/.Av ery imp ortan t feature of the MSSM is that the sup erpartners listed in T ables /1 and/2 are not necessarily the mass eigenstates of the theory /. This is b ecause after electro w eaksymmetry breaking and sup ersymmetry breaking e/ects are included/, there can b e mixingbe t w een the electro w eak gauginos and the higgsinos/, and within the v arious sets of squarksand sleptons and Higgs scalars whic hh a v e the same electric c harge/. The lone exceptionis the gluino/, whic h is a color o ctet fermion and therefore do es not ha v e the appropriate/1/1 quan tum n um b ers to mix with an y other particle/. The masses and mixings of the sup er/-partners are ob viously of paramoun t imp ortance to exp erimen talists/. It is p erhaps sligh tlyless ob vious that these phenomenological issues are all quite directly related to one cen tralquestion whic h is also the fo cus of m uc h of the theoretical w ork in sup ersymmetry/: /\Ho wis sup ersymmetry brok en/?/" The reason for this is that most of what w e do not alreadykno w ab out the MSSM has to do with Lsoft /. The structure of sup ersymmetric lagrangiansallo ws v ery little arbitrariness/, as w e will see in section /3/. In fact/, all of the dimensionlesscouplings and all but one mass term in the sup ersymmetric part of the MSSM lagrangiancorresp ond directly to some parameter in the ordinary Standard Mo del whic h has alreadyb een measured b y exp erimen t/. F or example/, w e will /nd out that the sup ersymmetric cou/-pling of a gluino to a squark and a quark is determined b y the QCD coupling constan t/S /.I n c o n trast/, the sup ersymmetry/-breaking part of the lagrangian apparen tly con tainsman y unkno wn parameters and a considerable amoun t of arbitrariness/. Eac h of the masssplittings b et w een Standard Mo del particles and their sup erpartners corresp ond to terms inthe MSSM lagrangian whic h are purely sup ersymmetry/-breaking in their origin and e/ect/.These soft sup ersymmetry/-breaking terms can also in tro duce a large n um b er of mixing an/-gles and CP/-violating phases not found in the Standard Mo del/. F ortunately /,a s w e will see insection /5/./4/, there is already rather strong evidence that the sup ersymmetry/-breaking termsin the MSSM are actually not arbitrary at all/. F urthermore/, the additional parameters willb e measured and constrained as the sup erpartners are detected/. F rom a theoretical p er/-sp ectiv e/, the c hallenge is to explain all of these parameters with a mo del for sup ersymmetrybreaking/.The rest of our discussion is organized as follo ws/. Section /2 pro vides a list of imp ortan tnotations/. In section /3/, w e will learn ho w to construct lagrangians for sup ersymmetric /eldtheories/. Soft sup ersymmetry/-breaking couplings are describ ed in section /4/. In section /5/,w e will apply the preceding general results to the sp ecial case of the MSSM/, in tro duce theconcept of R /-parit y /, and emphasize the imp ortance of the structure of the soft terms/. Section/6 outlines some considerations for understanding the origin of sup ersymmetry breaking/, andthe consequences of v arious prop osals/. In section /7/, w e will study the mass and mixing anglepatterns of the new particles predicted b y the MSSM/. Their deca y mo des are considered insection /8/, and some of the qualitativ e features of exp erimen tal signals for sup ersymmetry arereview ed in section /9/. Section /1/0 describ es some sample v ariations on the standard MSSMpicture/. The discussion will b e lac king in historical accuracy or p ersp ectiv e/, for whic h theauthor ap ologizes in adv ance/. The reader is encouraged to consult the man y outstandingtextb o oks/, /1/1 /; /1/8review articles/, /1/9 /; /3/4and the reprin tv olume/, /3/5whic hc o n tain a m uc h moreconsisten t guide to the original literature/./2 In terlude/: Notations and Con v en tionsBefore pro ceeding to discuss the construction of sup ersymmetric lagrangians/, w en e e d t osp ecify our notations/. It is o v erwhelmingly con v enien tt o e m p l o yt w o/-comp onen tW eyl no/-tation for fermions/, rather than four comp onen t Dirac or Ma jorana spinors/. The lagrangianof the Standard Mo del /(and sup ersymmetric extensions of it/) violates parit y/;; eac h Diracfermion has left/-handed and righ t/-handed parts with completely di/eren te l e c t r o w eak gaugein teractions/. If one used four/-comp onen t notation/, one w ould therefore ha v e to includeclumsy left/- and righ t/-handed pro jection op eratorsPL/;;R /=/( /1 / / /5 /) /= /2 /(/2/./1/)/1/2 all o v er the place/. The t w o/-comp onen tW eyl fermion notation has the adv an tage of treatingfermionic degrees of freedom with di/eren t gauge quan tum n um b ers separately from thestart /(as Nature in tended for us to do/)/. But an ev en b etter reason for using t w o/-comp onen tnotation here is that in sup ersymmetric mo dels the minimal building blo c ks of matter arec hiral sup erm ultiplets/, eac h of whic hc o n tains a single t w o/-comp onen tW eyl fermion/.Since t w o/-comp onen t fermion notation ma y b e unfamiliar to some readers/, w e will sp ecifyour con v en tions yb y sho wing ho w they corresp ond to the four/-comp onen t fermion language/.A four/-comp onen t Dirac fermion / D with mass M is describ ed b y the lagrangianLDirac /= /; i / D / //@/ / D /; M / D / D /: /(/2/./2/)W e use a spacetime metric /// /= diag/( /; /1 /;; /1 /;; /1 /;; /1/)/. F or our purp oses it is con v enien t to usethe sp eci/c represen tation of the /4 / /4 gamma matrices giv en in /2 / /2 blo c ks b y/ / /= //0 // // /0 //;; / /5 /= //1 /0/0 /; /1 //;; /(/2/./3/)where//0 /= //0 /= //1 /0/0 /1 //;; //1 /= /; //1 /= //0 /1/1 /0 //;;//2 /= /; //2 /= //0 /; ii /0 //;; //3 /= /; //3 /= //1 /0/0 /; /1 //: /(/2/./4/)In this basis/, a four comp onen t Dirac spinor is written in terms of /2 t w o/-comp onen t/, complex/,an ticomm uting ob jects /( / /)/ with / /=/1 /;; /2 and /( / y/) /_ /with /_ / /=/1 /;; /2/:/ D /= //// y /_ / //;; / D /=/(/ // y/_ / /) /: /(/2/./5/)The undotted /(dotted/) indices are used for the /rst /(last/) t w o comp onen ts of a Dirac spinor/.The heigh ts of these indices are imp ortan t/;; for example/, comparing eqs/. /(/2/./2/)/-/(/2/./5/)/, w eobserv e that the matrices /( / //)/ /_ / and /( / //) /_ //de/ned b y eq/. /(/2/./4/) carry indices with theheigh ts as indicated/. The spinor indices are raised and lo w ered using the an tisymmetricsym bo l / /1/2/= /; / /2/1/= //2/1 /= /; //1/2 /=/1 /;; //1/1 /= //2/2 /= / /1/1/= / /2/2/= /0/, according to// /= /// / //;; / //= / //// /;; / y/_ / /= //_ / /_/ / y /_//;; / y /_ //= / /_ / /_// y/_/ /: /(/2/./6/)This is consisten t since /// / // /= / / //// /= / / / and //_ / /_/ / /_/ /_ / /= / /_ / /_///_/ /_ / /= / /_ / /_ / /. The /eld / iscalled a /\left/-handed W eyl spinor/" and / yis a /\righ t/-handed W eyl spinor/"/. The names /t/,b ecausePL / D /= ////0 //;; PR / D /= //0/ y /_ / //: /(/2/./7/)The hermitian conjugate of a left/-handed W eyl spinor is a righ t/-handed W eyl spinor /( / / /) y/=/( / y/)/_ / and vice v ersa /( / y /_ //) y/= / //. Therefore an y particular fermionic degrees of freedom canb e describ ed equally w ell using a W eyl spinor whic h is left/-handed /(with an undotted index/)or b y one whic hi s r i g h t/-handed /(with a dotted index/)/. By con v en tion/, all names of fermion yThe con v en tions used here are the same as in Ref/. /1/1/,e x c e p tt h a tw e use a dagger rather than a bar toindicate hermitian conjugation for W eyl spinors/./1/3 /elds are c hosen so that left/-handed W eyl spinors do not carry daggers and righ t/-handedW eyl spinors do carry daggers/, as in eq/. /(/2/./5/)/.It is useful to abbreviate expressions with t w o spinor /elds b y suppressing undottedindices con tracted lik e // and dotted indices con tracted lik e/_ / /_ //. In particular/,// / / /// /= / //// / //= /; / //// / //= / //// / //= / /// / // /(/2/./8/)with/, con v enien tly /, no min us sign in the end/. /[A min us sign app eared in eq/. /(/2/./8/) fromexc hanging the order of an ticomm uting spinors/, but it disapp eared due to the an tisymmetryof the / sym b ol/./] Lik ewise/, / y/ yand / y/ yare equiv alen t abbreviations for / y/_ / / y /_ //=/( // /) //,the complex conjugate of // /. In a similar w a y /,/ y / // /= /; // // y/=/( / y / // /) //= /; /( // // y/) //(/2/./9/)stands for / y/_ / /( / //) /_ / /// /, etc/. With these con v en tions/, the Dirac lagrangian eq/. /(/2/./2/) can no wb e rewritten/:LDirac /= /; i / D / //@/ / D /; M / D / D /(/2/./1/0/)/= /; i/ y / //@/ / /; i/ y / //@/ / /; M /( // /+ / y/ y/) /(/2/./1/1/)where w eh a v e dropp ed a total deriv ativ e piece i/@/ /( / y / // /) whic h do es not a/ect the action/.A four/-comp onen t Ma jorana spinor can b e obtained from the Dirac spinor of eq/. /(/2/./5/)b y imp osing the constrain t / /= / /, so that/ M /= //// y /_ / //;; / M /=/(/ // y/_ / /) /: /(/2/./1/2/)The lagrangian for a Ma jorana fermion with mass MLMa jorana /= /; i /2 / M / //@/ / M /; /1 /2 M / M / M /(/2/./1/3/)in the four/-comp onen t Ma jorana spinor form can therefore b e rewrittenLMa jorana /= /; i/ y / //@/ / /; /1 /2 M /( // /+ / y/ y/) /(/2/./1/4/)in the more economical t w o/-comp onen tW eyl spinor represen tation/. /[Note that ev en though// is an ticomm uting/, // and its complex conjugate / y/ ydo not v anish/, b ecause of thesuppressed / sym b ol/, see eq/. /(/2/./8/)/./]More generally /,a n y theory in v olving spin/-/1///2 fermions can always b e written do wn interms of a collection of left/-handed W eyl spinors / i withL /= /; i/ y i / //@/ / i /+ /:/:/: /(/2/./1/5/)where the ellipses represen t p ossible mass terms/, gauge in teractions/, and Y uk a w ai n teractionswith scalar /elds/. Here the index i runs o v er the appropriate gauge and / a v or indices ofthe fermions/;; it is raised or lo w ered b y hermitian conjugation/. There is a di/eren t / i forthe left/-handed piece and for the hermitian conjugate of the righ t/-handed piece of a Diracfermion/. If one has an y expression in v olving bilinears in four/-comp onen t spinors/ /1 /= ///1/ y/1 /and / /2 /= ///2/ y/2 //;; /(/2/./1/6/)/1/4 then one can translate in to t w o/-comp onen tW eyl spinor language /(or vice v ersa/) using thedictionary/: / /1 PL / /2 /= //1 //2 /;; / /1 PR / /2 /= / y/1 / y/2 /;; /(/2/./1/7/) / /1 / /PL / /2 /= / y/1 / ///2 /;; / /1 / /PR / /2 /= //1 / // y/2 /(/2/./1/8/)etc/. W ew i l l i n tro duce a few other W eyl spinor iden tities in the follo wing as they are needed/.Let us no w see ho w the Standard Mo del quarks and leptons are describ ed in this nota/-tion/. The complete list of left/-handed W eyl spinors can b e giv en names corresp onding tothe c hiral sup erm ultiplets in T able /1/:Qi /= /( ud /) /;; /( cs /) /;; /( tb /) /(/2/./1/9/) ui /= u/;; c/;; t di /= d/;; s/;; b /(/2/./2/0/)Li /= /( /e e /) /;; /( // / /) /;; /( // / /) /(/2/./2/1/) ei /= e/;; //;; //: /(/2/./2/2/)Here i /=/1 /;; /2 /;; /3 is a family index/. The bars on these /elds are part of the names of the /elds/,and do not denote an y kind of conjugation/. Rather/, the un barred /elds are the left/-handedpieces of a Dirac spinor/, while the barred /elds are the names giv en to the conjugates ofthe righ t/-handed piece of a Dirac spinor/. F or example/, e is the same thing as eL in T able /1/,and e is the same as e yR /.T ogether they form a Dirac spinor/:/e e y // /eLeR //(/2/./2/3/)with similar equations for all of the other quark and c harged lepton Dirac spinors/. /(Theneutrinos of the Standard Mo del are not part of a Dirac spinor/./) The /elds Qi and Li arew eak iso doublets whic h alw a ys go together when one is constructing in teractions in v arian tunder the full Standard Mo del gauge group SU /(/3/)C / SU /(/2/)L / U /(/1/)Y /. Suppressing all colorand w eak isospin indices/, the purely kinetic part of the Standard Mo del fermion lagrangiandensit yi st h e nL /= /; iQ y i / //@/ Qi /; i u y i / //@/ ui /; i d y i / //@/ di /; iL y i / //@/ Li /; i e y i / //@/ ei /(/2/./2/4/)with the family index i /=/1 /;; /2 /;; /3 summed o v er/./3 Sup ersymmetric lagrangiansIn this section w e will describ e the construction of sup ersymmetric lagrangians/. Our aimis to arriv e at a sort of recip e whic h will allo w us to write do wn the allo w ed in teractionsand mass terms of a general sup ersymmetric theory /, so that later w e can apply the resultsto the sp ecial case of the MSSM/. W e will not use the sup er/eld language/, /3/6whic h is oftenmore elegan t and e/cien t for those who kno w it/, but whic h migh t seem rather cabalisticto some readers/. Our approac h is therefore in tended to b e rather complemen tary to thesup er/eld deriv ations giv en in Refs/. /1/1 /; /1/8W e b egin b y considering the simplest example of asup ersymmetric theory in four dimensions/./1/5 /3/./1 The simplest sup ersymmetric mo del/: a fr e ec h i r al sup ermultipletThe minim um fermion con ten to fa n y theory in four dimensions consists of a single left/-handed t w o/-comp onen tW eyl fermion / /. Since this is an in trinsically complex ob ject/, itseems sensible to c ho ose as its sup erpartner a complex scalar /eld / /. The simplest actionw e can write do wn for these /elds just consists of kinetic energy terms for eac h/:S /= Zd /4x /( Lscalar /+ Lfermion /) /(/3/./1/)Lscalar /= /; /@ // //@/ / /;; Lfermion /= /; i/ y / //@/ / /: /(/3/./2/)This is called the massless/, non/-in teracting Wess/-Zumino mo del /, /9and it corresp onds to asingle c hiral sup erm ultiplet as discussed in the In tro duction/.A sup ersymmetry transformation should turn the scalar b oson / in to something in/-v olving the fermion / / /. The simplest p ossibilit y for the transformation of the scalar /eldis// /= // /;; // //= / y/ y/(/3/./3/)where / /is an in/nitesimal/, an ticomm uting/, t w o/-comp onen tW eyl fermion ob ject whic hparameterizes the sup ersymmetry transformation/. Un til section /6/./2/, w e will b e discussingglobal sup ersymmetry /, whic h means that / /is a constan t/, satisfying /@/ / //= /0/. Since / hasdimensions of /(mass/) /3 /= /2and / has dimensions of /(mass/)/, it m ust b e that / has dimensionsof /(mass/) /; /1 /= /2/. Using eq/. /(/3/./3/)/, w e /nd that the scalar part of the lagrangian transforms as/ Lscalar /= /; //@ // /@/ / //; / y/@ // y/@/ //: /(/3/./4/)W ew ould lik e for this to b e cancelled b y / Lfermion /, at least up to a total deriv ativ e/, so thatthe action will b e in v arian t under the sup ersymmetry transformation/. Comparing eq/. /(/3/./4/)with Lfermion /,w e see that for this to ha v ea n yc hance of happ ening/, // should b e linear in/ yand in / and con tain one spacetime deriv ativ e/. Up to a m ultiplicativ e constan t/, there isonly one p ossibilit y to try/:// / /= i /( / // y/)/ /@/ / /;; // y/_ / /= /; i /( // //)/_ / /@/ / //: /(/3/./5/)With this guess/, one immediately obtains/ Lfermion /= /; // / / //@/ / /@/ / //+ / y / // // y/@/ /@/ //: /(/3/./6/)This can b e put in a sligh tly more useful form b y emplo ying the P auli matrix iden tities/[ / / / //+ / / / //] // /= /; /2 / /// // /;; /[ / // //+ / // //] /_//_ / /= /; /2 / /// /_//_ / /(/3/./7/)and using the fact that partial deriv ativ es comm ute /( /@/ /@/ /= /@/ /@/ /)/. Equation /(/3/./6/) thenb ecomes/ Lfermion /= //@ // /@/ / //+ / y/@ // y/@/ //; /@/ /// / / // /@/ / //+ // /@ // //+ / y/ y/@ // //: /(/3/./8/)The /rst t w o terms here just cancel against / Lscalar /, while the remaining con tribution is atotal deriv ativ e/. So w e arriv ea t/S /= Zd /4x /( / Lscalar /+ / Lfermion /)/=/0 /;; /(/3/./9/)/1/6 justifying our guess of the n umerical m ultiplicativ e factor made in eq/. /(/3/./5/)/.W e are not quite /nished in demonstrating that the theory describ ed b y eq/. /(/3/./1/) issup ersymmetric/. W em ust also sho w that the sup ersymmetry algebra closes/;; in other w ords/,that the comm utator of t w o sup ersymmetry transformations is another symmetry of thetheory /. Using eq/. /(/3/./5/) in eq/. /(/3/./3/)/, one /nds/( ///2 ///1 /; ///1 ///2 /) / /= i /( //1 / // y/2 /; //2 / // y/1 /) /@/ //: /(/3/./1/0/)This is a remark able result/;; in w ords/, w eh a v e found that the comm utator of t w o sup er/-symmetry transformations giv es us bac k the deriv ativ e of the original /eld/. Since /@/ justcorresp onds to the generator of spacetime translations P/ /, eq/. /(/3/./1/0/) implies the form of thesup ersymmetry algebra whic hw as foreshado w ed in eq/. /(/1/./6/) of the In tro duction/. /(W e willmak e this statemen t more explicit b efore the end of this section/./)All of this will b e for naugh ti fw e do not /nd the same result for the fermion / /,h o w ev er/.Using eq/. /(/3/./3/) in eq/. /(/3/./5/)/, w e /nd/( ///2 ///1 /; ///1 ///2 /) / / /= i /( / // y/1 /)/ //2 /@/ / /; i /( / // y/2 /)/ //1 /@/ / /: /(/3/./1/1/)W e can put this in to a more useful form b y applying the Fierz iden tit y// /( // /)/= /; // /( // /) /; // /( // /) /(/3/./1/2/)with / /= / // y/1 /, / /= //2 /, / /= /@/ / /, and again with / /= / // y/2 /, / /= //1 /, / /= /@/ / /, follo w ed in eac hcase b y an application of the iden tit y eq/. /(/2/./9/)/. The result is/( ///2 ///1 /; ///1 ///2 /) / / /= i /( //1 / // y/2 /; //2 / // y/1 /) /@/ / //; i//1 / / y/2 / //@/ / /+ i//2 / / y/1 / //@/ / /: /(/3/./1/3/)The last t w o terms in /(/3/./1/3/) v anish on/-shell/;; that is/, if the equation of motion / //@/ / /=/0follo wing from the action is enforced/. The remaining piece is exactly the same spacetimetranslation that w e found for the scalar /eld/.The fact that the sup ersymmetry algebra only closes on/-shell /(when the classical equa/-tions of motion are satis/ed/) migh t b e somewhat w orrisome/, since w ew ould lik e the sym/-metry to hold ev en quan tum mec hanically /. This can b e /xed b y a tric k/. W ei n v en t a newcomplex scalar /eld F whic hd o e sn o t h a v e a kinetic term/. Suc h /elds are called auxiliary /,and they are really just b o ok/-k eeping devices whic h allo w the symmetry algebra to closeo//-shell/. The lagrangian densit yf o r F and its complex conjugate is justLauxiliary /= F /F/: /(/3/./1/4/)The dimensions of F are /(mass/) /2/, unlik e an ordinary scalar /eld whic h has dimensions of/(mass/)/. Equation /(/3/./1/4/) leads to the not/-v ery/-exciting equations of motion F /= F //=/0/. Ho w ev er/, w e can use the auxiliary /elds to our adv an tage b y including them in thesup ersymmetry transformation rules/. In view of eq/. /(/3/./1/3/)/, a plausible thing to do is tomak e F transform in t oam ultiple of the equation of motion for / /:/F /= i/ y / //@/ / /;; /F //= /; i/@/ / y / ///: /(/3/./1/5/)Once again w eh a v ec hosen the o v erall factor on the righ t hand side b y virtue of foresigh t/.No w the auxiliary part of the lagrangian densit y transforms as/ Lauxiliary /= i/ y / //@/ / F //; i/@/ / y / //F /(/3/./1/6/)/1/7 whic hv anishes on/-shell/, but not for arbitrary o//-shell /eld con/gurations/. It is easy to seethat b y adding an extra term to the transformation la wf o r / and / y/:// / /= i /( / // y/)/ /@/ / /+ // F /;; // y/_ / /= /; i /( // //)/_ / /@/ / //+ / y/_ / F //(/3/./1/7/)one obtains an additional con tribution to / Lfermion whic h just cancels with / Lauxiliary /,u pt oa total deriv ativ e term/. So our /\mo di/ed/" theory with L /= Lscalar /+ Lfermion /+ Lauxiliary isstill in v arian t under sup ersymmetry transformations/. Pro ceeding as b efore/, one no w obtainsfor eac h of the /elds X /= //;; / //;;/ /;;/ y/;;F /;;F //,/( ///2 ///1 /; ///1 ///2 /) X /= i /( //1 / // y/2 /; //2 / // y/1 /) /@/ X /(/3/./1/8/)using eqs/. /(/3/./3/)/, /(/3/./1/5/)/, and /(/3/./1/7/)/, but without resorting to an y of the equations of motion/.So w eh a v e succeeded in sho wing that sup ersymmetry is a v alid symmetry of the lagrangiano//-shell/.In retrosp ect/, one can see wh yw e needed to in tro duce the auxiliary /eld F in order toget the sup ersymmetry algebra to w ork o//-shell/. On/-shell/, the complex scalar /eld / hast w o real propagating degrees of freedom/, whic hm a t c h with the t w o spin p olarization statesof / /. O//-shell/, ho w ev er/, the W eyl fermion / is a complex t w o/-comp onen t ob ject/, so it hasfour real degrees of freedom/. /(Going on/-shell eliminates half of the propagating degrees offreedom for / /, b ecause the lagrangian is linear in time deriv ativ es/, so that the canonicalmomen ta can b e reexpressed in terms of the con/guration v ariables without time deriv ativ esand are not indep enden t phase space co ordinates/./) T om a k et h e n um b ers of b osonic andfermionic degrees of freedom matc h o//-shell as w ell as on/-shell/, w e had to in tro duce t w o morereal scalar degrees of freedom in the complex /eld F /, whic h are eliminated when one go eson/-shell/. The auxiliary /eld form ulation is esp ecially useful when discussing sp on taneoussup ersymmetry breaking/, as w e will see in section /6/.In v ariance of the action under a symmetry transformation alw a ys implies the existenceof a conserv ed curren t/, and sup ersymmetry is no exception/. The sup er curr ent J // is anan ticomm uting four/-v ector whic h also carries a spinor index/, as b e/ts the curren t asso ciatedwith a symmetry with fermionic generators/. /3/7By the usual No ether pro cedure/, one /ndsfor the sup ercurren t /(and its hermitian conjugate/) in terms of the v ariations of the /eldsX /= //;; / //;;/ /;;/ y/;;F /;;F //:/J/ /+ / yJ y/ / XX /X / L / /( /@ /X /) /; K/ /;; /(/3/./1/9/)where K/ is the ob ject whose div ergence is the v ariation of the lagrangian densit y underthe sup ersymmetry transformation/, /@ /K/ /= / L /. A little w ork rev eals thatJ // /=/( / / / // /)/ /@/ / //;; J y //_ / /=/( / y / // //)/_ / /@/ //: /(/3/./2/0/)The sup ercurren t and its hermitian conjugate are separately conserv ed/:/@/ J // /=/0 /;; /@/ J y //_ / /=/0 /(/3/./2/1/)a s c a nb ev eri/ed b y use of the equations of motion/. F rom these curren ts one constructs theconserv ed c hargesQ/ /= p /2 Zd /3xJ /0/ /;; Q y/_ / /= p /2 Zd /3xJ y /0/_ / /(/3/./2/2/)/1/8 whic h are the generators of sup ersymmetry transformations/. /(The factor of p /2 normaliza/-tion is included to agree with an arbitrary historical con v en tion/./) As quan tum mec hanicalop erators/, they satisfyh/Q /+ / yQ y/;;X i/= /; i p /2 /X /(/3/./2/3/)for an y/ e l d X /, up to terms whic hv anish on/-shell/. This can b e v eri/ed explicitly b y usingthe canonical equal/-time comm utation and an ticomm utation relations/[ / /( x /) /;;/ /( y /)/] /= /[ / //( x /) /;;/ //( y /)/] /= i/ /(/3/)/( x /; y /)/;; /(/3/./2/4/)f / y/_ / /( x /) /;;/ / /( y /) g /= /; / /0/ /_ / / /(/3/)/( x /; y /) /(/3/./2/5/)deriv ed from the free /eld theory lagrangian eq/. /(/3/./1/)/. Here / /= /@/0 / /and / //= /@/0 / are themomen ta conjugate to / and / /resp ectiv ely /.N o w the con ten t of eq/. /(/3/./1/8/) can b e expressedin terms of canonical comm utators ash//2 Q /+ / y/2 Q y/;; /[ //1 Q /+ / y/1 Q y/;;X /] i/; h//1 Q /+ / y/1 Q y/;; /[ //2 Q /+ / y/2 Q y/;;X /] i/=/2/( //2 / // y/1 /; //1 / // y/2 /) i/@/ X /(/3/./2/6/)up to terms whic hv anish on/-shell/. The spacetime momen tum op erator P /is giv en in termsof the canonical v ariables b y P /0/= // //+ /@j //@ j/ //+ i/ y / j/@j / and P j/= /; //@ j/ /; / //@ j/ //+i/ y / /0/@ j/ /, where j is the spacetime v ector index restricted to the three spatial dimensions/.It generates spacetime translations on the /elds X according to/[ P/ /;;X /]/= i/@/ X/: /(/3/./2/7/)By rearranging the terms in eq/. /(/3/./2/6/) using the Jacobi iden tit y /,w e therefore ha v eh/[ //2 Q /+ / y/2 Q y/;;//1 Q /+ / y/1 Q y/] /;;X i/=/2 /( //2 / // y/1 /; //1 / // y/2 /)/[ P/ /;;X /] /;; /(/3/./2/8/)for an y X /, s oi tm ust b e that/[ //2 Q /+ / y/2 Q y/;;//1 Q /+ / y/1 Q y/]/= /2 /( //2 / // y/1 /; //1 / // y/2 /) P/ /(/3/./2/9/)up to terms whic hv anish on/-shell/. No wb y expanding out eq/. /(/3/./2/9/)/, one obtains the non/-sc hematic form of the sup ersymmetry algebra relationsf Q/ /;;Q y/_ / g /=/2 / // /_ / P/ /;; /(/3/./3/0/)f Q/ /;;Q/ g /= f Q y/_ / /;;Q y/_/ g /=/0 /(/3/./3/1/)as promised in the In tro duction/. /[The comm utator in eq/. /(/3/./2/9/) turns in to an ticomm utatorsin eqs/. /(/3/./3/0/) and /(/3/./3/1/) in the pro cess of extracting the an ticomm uting spinors //1 and //2 /./]The results /[ Q/ /;;P/ /] /= /0 and /[ Q y/_ / /;;P/ /]/= /0 f o l l o w immediately from eq/. /(/3/./2/7/) and the factthat the sup ersymmetry transformations are global /(indep enden t of p osition in spacetime/)/.This demonstration of the sup ersymmetry algebra in terms of the canonical generators Qand Q yrequires the use of the Hamiltonian equations of motion/, but the symmetry itself isv alid o//-shell at the lev el of the lagrangian/, as w eh a v e already sho wn/./1/9 /3/./2 Inter actions of chir al sup ermultipletsIn a realistic theory lik e the MSSM/, there are man yc hiral sup erm ultiplets whic hh a v e b othgauge and non/-gauge in teractions/. In this subsection/, our task is to construct the mostgeneral p ossible theory of masses and non/-gauge in teractions for particles that liv ei nc hi/-ral sup erm ultiplets/. In the MSSM these are the quarks/, squarks/, leptons/, sleptons/, Higgsscalars and higgsino fermions/. W e will /nd that the form of the non/-gauge couplings/, includ/-ing mass terms/, is highly restricted b y the requiremen t that the action is in v arian t undersup ersymmetry transformations/. /(Gauge in teractions will b e dealt with in the follo wingsubsections/./)Our starting p oin t is the lagrangian densit y for a collection of free c hiral sup erm ultipletslab elled b y an index i whic h runs o v er all gauge and / a v or degrees of freedom/. Since w ewill w an t to construct an in teracting theory with sup ersymmetry closing o//-shell/, eac hsup erm ultiplet con tains a complex scalar /i and a left/-handed W eyl fermion / i as ph ysicaldegrees of freedom/, plus a complex auxiliary /eld Fi whic h do es not propagate/. The resultsof the previous subsection tell us that the free part of the Lagrangian isLfree /= /; /@ // / i/@/ /i /; i/ y i / //@/ / i /+ F / iFi /(/3/./3/2/)where w e sum o v er rep eated indices i /(not to b e confused with the suppressed spinor indices/)/,with the con v en tion that /elds /i and / i alw a ys carry lo w ered indices/, while their conjugatesalw a ys carry raised indices/. It is in v arian t under the sup ersymmetry transformation//i /= // i // / i/= / y/ y i/(/3/./3/3/)/ /( / i /)/ /= i /( / // y/)/ /@/ /i /+ // Fi / /( / y i/)/_ / /= /; i /( // //)/_ / /@/ / / i/+ / y/_ / F / i/(/3/./3/4/)/Fi /= i/ y / //@/ / i /F / i/= /; i/@/ / y i / ///: /(/3/./3/5/)As w ew i l l n o w argue/, the most general set of renormalizable in teractions for these /eldscan b e written in the simple formLin t /= /; /1 /2 W ij/ i / j /+ W iFi /+c /: c /: /;; /(/3/./3/6/)where W ijand W iare some functions of the b osonic /elds with dimensions of /(mass/) and/(mass/) /2resp ectiv ely /, and /\c /: c /: /" henceforth stands for complex conjugate/. A t this p oin t/,w e are not assuming that W ijand W iare related to eac h other in an yw a y whatso ev er/.Ho w ev er/, so on w e will /nd out that they ar e related/, whic hi sw h yw eh a v ec hosen thesame letter for them/. Notice that eq/. /(/2/./8/) tells us that W ijis symmetric under i /$ j /.No w/, let us require the lagrangian to b e renormalizable b yp o w er coun ting/, so that eac hterm has /eld con ten t with mass dimension / /4/. It follo ws immediately that w ed on o tneed to consider the p ossibilit yo f W ijor W ib eing functions of the fermionic or auxiliary/elds/. F or the same reason/, w e can tak e W ito b e at most a quadratic p olynomial/, and W ijlinear/, in the /elds /i and / / i/. Also/, w e do not need to consider including in Lin t an y termwhic h is a function of the scalar /elds /i /;;/ / ionly /. If there w ere suc h a term/, then undera sup ersymmetry transformation eq/. /(/3/./3/3/) it w ould go in to another function of the scalar/elds only /,m ultiplied b y // i or / y/ y i/, and with no spacetime deriv ativ es or Fi /, F / i/elds/.It is easy to see from eqs/. /(/3/./3/3/)/-/(/3/./3/6/) that nothing of this form can p ossibly b e cancelledb y the sup ersymmetry transformation of an y other term in the lagrangian/. So eq/. /(/3/./3/6/) isindeed the most general p ossibilit y/!/2/0 W em ust no w require that Lin t is in v arian t under the sup ersymmetry transformations/,since Lfree w as already in v arian tb y itself/. It is easiest to divide the v ariation of Lin t in tosev eral parts whic hm ust cancel separately /. First/, w e consider the part whic hc o n tains fourspinors/:/ Lin t j/4 /; spinor /= /; /1 /2 /W ij //k /( // k /)/( / i / j /) /; /1 /2 /W ij // / k /( / y/ y k/)/( / i / j /)/+c /: c /: /(/3/./3/7/)The term prop ortional to /( // k /)/( / i / j /) cannot cancel against an y other term/. F ortunately /,ho w ev er/, the Fierz iden tit y eq/. /(/3/./1/2/) implies/( // i /)/( / j / k /)/+/( // j /)/( / k / i /)/+/( // k /)/( / i / j /)/=/0 /;; /(/3/./3/8/)whic h allo ws this con tribution to / Lin t to v anish iden tically if and only if /W ij/=/ /k istotally symmetric under in terc hange of i/;; j/;; k /. There is no suc h iden tit ya v ailable for theterm prop ortional to /( / y/ y k/)/( / i / j /)/. Since it cannot cancel with an y other term/, requiringit to b e absen t just tells us that W ijcannot con tain / / k/. In other w ords/, W ijis analytic/(or holomorphic /) in the complex /elds /k /.So far/, what w eh a v e learned is that w e can writeW ij/= M ij/+ y ij k/k /(/3/./3/9/)where M ijis a symmetric mass matrix for the fermion /elds/, and y ij kis a Y uk a w a couplingof a scalar /k and t w o fermions / i / j whic hm ust b e totally symmetric under in terc hange ofi/;; j/;; k /. It is con v enien t to writeW ij/= / /2 //i //j W /(/3/./4/0/)where w eh a v ei n tro duced a v ery useful ob jectW /= /1 /2 M ij/i /j /+ /1 /6 y ij k/i /j /k /(/3/./4/1/)whic h is called the sup erp otential /. This is not a scalar p oten tial in the ordinary sense/;; infact/, it is not ev en real/. It is instead an analytic function of the scalar /elds /i treated ascomplex v ariables/.Con tin uing on our v aun ted quest/, w e next consider the parts of / Lin t whic hc o n tain aspacetime deriv ativ e/:/ Lin t j/@ /= /; iW ij/@/ /j / i / // y/; iW i/@/ / i / // y/+c /: c /: /(/3/./4/2/)Here w eh a v e used the iden tit y eq/. /(/2/./9/) on the second term/, whic h came from /( /Fi /) W i/.No ww e can use eq/. /(/3/./4/0/) to observ e thatW ij/@/ /j /= /@/ //W //i //: /(/3/./4/3/)Then it is clear that eq/. /(/3/./4/2/) will b e a total deriv ativ e if and only ifW i/= /W //i /= M ij/j /+ /1 /2 y ij k/j /k /;; /(/3/./4/4/)/2/1 whic h explains wh yw ec hose its name the w a yw e did/. The remaining terms in / Lin t areall linear in Fi or F / i/, and it is easy to sho w that they cancel/, giv en the results for W iandW ijthat w eh a v e already found/.T o recap/, w eh a v e found that the most general non/-gauge in teractions for c hiral su/-p erm ultiplets are determined b y a single analytic function of the complex scalar /elds/,the sup erp oten tial W /. The auxiliary /elds Fi and F / ican b e eliminated using their clas/-sical equations of motion/. The part of Lfree /+ Lin t that con tains the auxiliary /elds isFi F / i/+ W iFi /+ W /i F / i/, leading to the equations of motionFi /= /; W /i /;; F / i/= /; W i/: /(/3/./4/5/)Th us the auxiliary /elds are expressible algebraically /(without an yd e r i v ativ es/) in terms ofthe scalar /elds/. After making the replacemen t eq/. /(/3/./4/5/) in Lfree /+ Lin t /,w e obtain thelagrangian densit yL /= /; /@ // / i/@/ /i /; i/ y i / //@/ / i /; /1 /2 /W ij/ i / j /+ W / ij/ y i/ y j //; W iW /i /: /(/3/./4/6/)/(Since Fi and F / iapp ear only quadratically in the action/, the result of instead doing afunctional in tegral o v er them at the quan tum lev el has precisely the same e/ect/./) No wthat the non/-propagating /elds Fi /;;F / iha v e b een eliminated/, it is clear from eq/. /(/3/./4/6/) thatthe scalar p oten tial for the theory is just giv en in terms of the sup erp oten tial b y /(recall Lcon tains /; V /)/:V /( //;; / //)/= W iW /i /= Fi F / i/= M /ik M kj/ / i/j /(/3/./4/7/)/+ /1 /2 M iny /jk n /i / / j/ / k/+ /1 /2 M /in y jk n/ / i/j /k /+ /1 /4 y ij ny /kl n /i /j / / k/ / l/:This scalar p oten tial is automatically b ounded from b elo w/;; in fact/, since it is a sum of squaresof absolute v alues /(of the W i/)/, it is alw a ys non/-negativ e/. If w e substitute the general formfor the sup erp oten tial eq/. /(/3/./4/1/) in to eq/. /(/3/./4/6/)/, w e obtain for the full lagrangian densit yL /= /; /@ // / i/@/ /i /; i/ y i / //@/ / i/; /1 /2 M ij/ i / j /; /1 /2 M /ij / y i/ y j/; V /( //;; / //)/; /1 /2 y ij k/i / j / k /; /1 /2 y /ij k / / i/ y j/ y k/: /(/3/./4/8/)No ww e can compare the masses of the fermions and scalars b y lo oking at the linearizedequations of motion/:/@ //@/ /i /= M /ik M kj/j /+ /:/:/: /;; /(/3/./4/9/)/; i / //@/ / i /= M /ij / y j/+ /:/:/: /;; /; i/ //@/ / y i/= M ij/ j /+ /:/:/: /: /(/3/./5/0/)One can eliminate / in terms of / yand vice v ersa in eq/. /(/3/./5/0/)/, obtaining /[after use of theiden tit y eq/. /(/3/./7/)/]/@ //@/ / i /= M /ik M kj/ j /+ /:/:/: /;; /@ //@/ / y j/= / y iM /ik M kj/+ /:/:/: /: /(/3/./5/1/)Therefore/, the fermions and the b osons satisfy the same w a v e equation with exactly thesame /(mass/) /2matrix with real non/-negativ e eigen v alues/, namely /( M /2/)i j/= M /ik M kj/. Itfollo ws that diagonalizing this matrix giv es a collection of c hiral sup erm ultiplets eac ho fwhic h con tains a mass/-degenerate complex scalar and W eyl fermion/, in agreemen t with thegeneral argumen t in the In tro duction/./2/2 /3/./3 L agr angians for gauge sup ermultipletsThe propagating degrees of freedom in a gauge sup erm ultiplet are a massless gauge b oson/eld A a/ and a t w o/-comp onen tW eyl fermion gaugino / a/. The index a here runs o v er theadjoin t represen tation of the gauge group /( a /=/1 /:/:/: /8f o r SU /(/3/)C color gluons and gluinos/;;a /=/1 /;; /2 /;; /3f o r SU /(/2/)L w eak isospin/;; a /=/1 f o r U /(/1/)Y w eak h yp erc harge/)/. The gaugetransformations of the v ector sup erm ultiplet /elds are then/gauge A a/ /= /; /@/ / a/+ gf abcA b/ / c/(/3/./5/2/)/gauge / a/= gf abc/ b/ c/(/3/./5/3/)where / ais an in/nitesimal gauge transformation parameter/, g is the gauge coupling/, andf abcare the totally an tisymmetric structure constan ts whic h de/ne the gauge group/. /(Thesp ecial case of an ab elian group lik e U /(/1/)Y is obtained b y just setting f abc/= /0/;; in particularthe corresp onding gaugino is a gauge singlet in that case/./)The on/-shell degrees of freedom for A a/ and / a/ amoun tt ot w o b osonic and t w o fermionichelicit y states /(for eac h a /)/, as required b y sup ersymmetry /.H o w ev er/, o//-shell / a/ consistsof t w o complex/, or four real/, fermionic degrees of freedom/, while A a/ only has three realb osonic degrees of freedom/;; one degree of freedom is remo v ed b y the inhomogeneous gaugetransformation eq/. /(/3/./5/2/)/. So/, w e will need one real b osonic auxiliary /eld/, traditionallycalled D a/, in order for sup ersymmetry to b e consisten t o//-shell/. This /eld also transformsas an adjoin t of the gauge group /[i/.e/./, lik e eq/. /(/3/./5/3/) with / /! D /] and satis/es /( D a/) //= D a/.Lik e the c hiral auxiliary /elds Fi /, it has dimensions of /(mass/) /2and th us no kinetic term/, sothat it can b e eliminated on/-shell using its algebraic equation of motion/.Therefore/, the lagrangian densit y for a gauge sup erm ultiplet ough tt o b eLgauge /= /; /1 /4 F a// F // a/; i/ y a / /D/ / a/+ /1 /2 D aD a/(/3/./5/4/)whereF a// /= /@/ A a/ /; /@/ A a/ /; gf abcA b/ A c/ /(/3/./5/5/)is the usual Y ang/-Mills /eld strength/, andD/ / a/= /@/ / a/; gf abcA b/ / c/(/3/./5/6/)is the co v arian td e r i v ativ e of the gaugino /eld/. One can infer the appropriate form forthe sup ersymmetry transformation of the /elds/, up to m ultiplicativ e constan ts/, from therequiremen ts that they should b e linear in the in/nitesimal parameters //;; / ywith dimensionsof /(mass/) /; /1 /= /2/;; that /A a/ is real/;; and that /D ashould b e real and prop ortional to the /eldequations for the gaugino/, in analogy with the role of the auxiliary /eld F in the c hiralsup erm ultiplet case/. Th us one can guess/, up to m ultiplicativ e factors/,/A a/ /= /; /1 p /2 h/ y // / a/+ / y a // / i/(/3/./5/7/)// a/ /= /; i /2 p /2 /( / / / // /)/ F a// /+ /1 p /2 // D a/(/3/./5/8/)/D a/= i p /2 h/ y / /D/ / a/; D/ / y a / // i/: /(/3/./5/9/)/2/3 The factors of p /2 are c hosen yso that the action obtained b yi n tegrating Lgauge is in v arian t/.It is no w a little bit tedious/, but straigh tforw ard/, to c hec k that eq/. /(/3/./1/8/) is mo di/ed to/( ///2 ///1 /; ///1 ///2 /) X /= i /( //1 / // y/2 /; //2 / // y/1 /) D/ X /(/3/./6/0/)for X equal to an y of the gauge/-co v arian t/ e l d s F a// /, / a/, / y a/, D a/,a sw ell as arbitrary co v ari/-an t deriv ativ es acting on them/. This ensures that the sup ersymmetry algebra eqs/. /(/3/./3/0/)/-/(/3/./3/1/) is realized on gauge/-in v arian tc o m binations of /elds in gauge sup erm ultiplets/, as theyw ere on the c hiral sup erm ultiplets/. zThese calculations require the use of iden tities// / / // /= // / / // /=/( / y / // // y/) //=/( / y / // // y/) //;; /(/3/./6/1/) / // / / //= / // / //; / // / //; / // / //; i/ // // // /;; /(/3/./6/2/)/ // /_ / / /_/// /= /; /2 / // / /_//_ / /: /(/3/./6/3/)If w e had not included the auxiliary /eld D a/, then the sup ersymmetry algebra eq/. /(/3/./6/0/)w ould hold only after using the equations of motion for / aand / y a/. The auxiliary /elds justsatisfy the equations of motion D a/= /0/, but this is no longer true if one couples the gaugesup erm ultiplets to c hiral sup erm ultiplets/, as w en o w do/./3/./4 Sup ersymmetric gauge inter actionsFinally w e are ready to consider a general lagrangian densit y for a sup ersymmetric theorywith b oth c hiral and gauge sup erm ultiplets/. Supp ose that the c hiral sup erm ultiplets trans/-form under the gauge group in a represen tation with hermitian matrices /( T a/)i jsatisfying/[ T a/;;T b/]/= if abcT c/./[ F or example/, if the gauge group is SU /(/2/)/, then f abc/= / abc/, and the T aare /1 /= /2 times the P auli matrices for a c hiral sup erm ultiplet transforming in the fundamen talrepresen tation/./] Th us/gauge Xi /= ig / a/( T aX /)i /(/3/./6/4/)for Xi /= /i /;;/ i /;;Fi /;; since sup ersymmetry and gauge transformations comm ute/, the scalar/,fermion/, and auxiliary /elds m ust b e in the same represen tation of the gauge group/. T oha v e a gauge/-in v arian t lagrangian/, w e need to turn the ordinary deriv ativ es in eq/. /(/3/./3/2/)in to co v arian t deriv ativ es/:/@/ /i /! D/ /i /= /@/ /i /+ ig A a/ /( T a/ /)i /(/3/./6/5/)/@/ / / i/! D/ / / i/= /@/ / / i/; ig A a/ /( / /T a/) i/(/3/./6/6/)/@/ / i /! D/ / i /= /@/ / i /+ ig A a/ /( T a/ /)i /: /(/3/./6/7/)Naiv ely /, this simple pro cedure ac hiev es the goal of coupling the v ector b osons in the gaugesup erm ultiplet to the scalars and fermions in the c hiral sup erm ultiplets/. Ho w ev er/, w e alsoha v e to consider whether there are an y other in teractions allo w ed b y gauge in v ariance in v olv/-ing the gaugino and D a/elds whic h migh th a v e to b e included to mak e a sup ersymmetriclagrangian/. yF or future con v enience in treating the MSSM/, w eh a v ec hosen complex phases so that our / a/, / y aare equalto /; i /, i times the gaugino spinors in Ref/. /1/1zThe sup ersymmetry transformations eqs/. /(/3/./5/7/)/-/(/3/./5/9/) are non/-linear for non/-ab elian gauge symmetries/,b ecause of the gauge /elds con tained in the co v arian t deriv ativ es acting on the gaugino /elds and in the/eld strength F a// /. By adding ev en more auxiliary /elds b esides D a/, one can mak e the sup ersymmetrytransformations linear in the /elds/. The v ersion giv en here in whic h those extra auxiliary /elds ha v e b eenremo v ed b y gauge transformations is called /\W ess/-Zumino gauge/"/. /3/8/2/4 In fact/, there are three suc h p ossibilities whic h are renormalizable /(of mass dimension/ /4/)/, namely/( / /T a/ /) / a/;; / y a/( / yT a/ /) and /( / /T a/ /) D a/: /(/3/./6/8/)No w one can add them/, with arbitrary dimensionless coupling co e/cien ts/, to the lagrangiansfor the c hiral and gauge sup erm ultiplets and demand that the whole mess b e real and in/-v arian t under sup ersymmetry transformations/, up to a total deriv ativ e/. Not surprisingly /,this is p ossible only if one mo di/es the sup ersymmetry transformation la ws for the mat/-ter /elds to include gauge/-co v arian t rather than ordinary deriv ativ es /(and to include onestrategically/-c hosen extra term in /Fi /)/://i /= // i /(/3/./6/9/)/ /( / i /)/ /= i /( / // y/)/ D/ /i /+ // Fi /(/3/./7/0/)/Fi /= i/ y / /D/ / i /+ p /2 g /( T a/ /)i / y/ y a/: /(/3/./7/1/)After some algebra one can no w /x the co e/cien ts for the terms in eq/. /(/3/./6/8/)/, so that thefull lagrangian densit y for a renormalizable sup ersymmetric theory isL /= Lgauge /+ Lc hiral/; p /2 g h/( / /T a/ /) / a/+ / y a/( / yT a/ /) i/+ g /( / /T a/ /) D a/: /(/3/./7/2/)Here Lc hiral means the c hiral sup erm ultiplet lagrangian found in section /3/./2 /[e/.g/./, eq/. /(/3/./4/6/)or /(/3/./4/8/)/]/, but with ordinary deriv ativ es replaced ev erywhere b y gauge/-co v arian t deriv a/-tiv es/, and Lgauge w as giv en in eq/. /(/3/./5/4/)/. T o pro v e that eq/. /(/3/./7/2/) is in v arian t under thesup ersymmetry transformations/, one m ust use the iden tit yW i/( T a/) ji /j /=/0 /: /(/3/./7/3/)This is precisely the condition that m ust b e satis/ed an yw a y in order for the sup erp oten tial/(and th us Lc hiral /)t ob e g a u g e i n v arian t/, since the left side is prop ortional to /gauge W /.The last t w o lines in eq/. /(/3/./7/2/) are in teractions whose strengths are /xed to b e gaugecouplings b y the requiremen ts of sup ersymmetry /,e v en though they are not gauge in terac/-tions from the p oin t of view of an ordinary /eld theory /. The second line is a direct couplingof gauginos to matter /elds whic h is the /\sup ersymmetrization/" of the usual gauge b osoncoupling to matter /elds/. The last line com bines with the /(/1 /= /2/) D aD aterm in Lgauge topro vide an equation of motionD a/= /; g /( / /T a/ /) /: /(/3/./7/4/)Lik e the auxiliary /elds Fi and F / i/,t h e D aare expressible purely algebraically in terms ofthe scalar /elds/. Replacing the auxiliary /elds in eq/. /(/3/./7/2/) using eq/. /(/3/./7/4/)/, one /nds thatthe complete scalar p oten tial is /(recall L/ /; V /)/:V /( //;; / //)/= F / iFi /+ /1 /2 Xa D aD a/= W /i W i/+ /1 /2 Xa g /2a /( / /T a/ /) /2/: /(/3/./7/5/)The t w ot yp es of terms in this expression are called /\ F /-term/" and /\ D /-term/" con tributions/,resp ectiv ely /. In the second term in eq/. /(/3/./7/5/)/, w eh a v en o w written an explicit sum Pa to/2/5 co v er the case that the gauge group has sev eral distinct factors with di/eren t gauge couplingsga /./[ F or instance/, in the MSSM the three factors SU /(/3/)C /, SU /(/2/)L and U /(/1/)Y ha v e di/eren tgauge couplings g/3 /, g and g /0/./] Since V /( //;; / //) is a sum of squares/, it is alw a ys greater thanor equal to zero for ev ery /eld con/guration/. It is a v ery in teresting and unique featureof sup ersymmetric theories that the scalar p oten tial is completely determined b y the otherin teractions in the theory /. The F /-terms are /xed b yY uk a w a couplings and fermion massterms/, and the D /-terms are /xed b y the gauge in teractions/.By using No ether/'s pro cedure /[see eq/. /(/3/./1/9/)/]/, one /nds the conserv ed sup ercurren tJ // /=/( / / / // i /)/ D/ / / i/; i /( / // y i/)/ W /i/; /1 /2 p /2 /( / / / // // y a/)/ F a// /; i p /2 g/ /T a/ /( / // y a/)/ /;; /(/3/./7/6/)generalizing the expression giv en in eq/. /(/3/./2/0/) for the W ess/-Zumino mo del/. This expressionwill b e useful when w e discuss certain asp ects of sp on taneous sup ersymmetry breaking insection /6/./2/./3/./5 Summary/: How to build a sup ersymmetric mo delIn a renormalizable sup ersymmetric /eld theory /,t h e i n teractions and masses of all particlesare determined just b y their gauge transformation prop erties and b y the sup erp oten tialW /. By construction/, w e found that W had to b e an analytic function of the complexscalar /elds /i /, whic ha r ea l w a ys de/ned to transform under sup ersymmetry in to left /-handedW eyl fermions/. W e should men tion that in an equiv alen t language/, W is said to b e afunction of c hiral sup er/elds /. /3/6A sup er/eld is a single ob ject whic hc o n tains as comp onen tsall of the b osonic/, fermionic/, and auxiliary /elds within the corresp onding sup erm ultiplet/,e/.g/. /i / /( /i /;;/ i /;;Fi /)/. /(This is analogous to the w a y in whic h one often describ es a w eakisospin doublet or color triplet b yam ulticomp onen t /eld/./) The gauge quan tum n um b ersand mass dimension of a c hiral sup er/eld are the same as that of its scalar comp onen t/. Inthe sup er/eld form ulation/, one writes instead of eq/. /(/3/./4/1/)W /= /1 /2 M ij/i /j /+ /1 /6 y ij k/i /j /k /(/3/./7/7/)whic h means exactly the same thing/. While this en tails no di/erence in practical results/,the fancier v ersion eq/. /(/3/./7/7/) at least serv es to remind us that W determines not only thescalar in teractions in the theory /, but the fermion masses and Y uk a w a couplings as w ell/. Thederiv ation of all of our preceding results can b e obtained somewhat more elegan tly usingsup er/eld metho ds/, whic hh a v e the adv an tage of making in v ariance under sup ersymmetrytransformations manifest/. W eh a v ea v oided this extra la y er of notation on purp ose/, in fa v orof the more p edestrian but hop efully more familiar comp onen t /eld approac h/. The latteris at least more appropriate for making con tact with phenomenology in a univ erse withsup ersymmetry breaking/. The only /(o ccasional/) use w e will mak e of sup er/eld notationis the purely cosmetic one of follo wing the common practice of sp ecifying sup erp oten tialslik e eq/. /(/3/./7/7/) rather than /(/3/./4/1/)/. The sp eci/cation of the sup erp oten tial is really a co defor the terms that it implies in the lagrangian/, so the reader ma y feel free to think of thesup erp oten tial either as a function of the scalar /elds /i or as the same function of thesup er/elds /i whic h con tain them/.Giv en the sup erm ultiplet con ten t of the theory /, the form of the sup erp oten tial is re/-stricted b y gauge in v ariance/. In an yg i v en theory /, only a subset of the couplings M ijand/2/6 i kj (a) (b)ji lkFigure /3/: The dimensionl ess non/-gauge in teraction v ertices in a sup ersymmetric theory/: /(a/) scalar/-fermion/-fermion Y uk a w ai n teraction y ij k/, /(b/) quartic scalar in teraction y ij ny /kl n /. i kj (a) (b)ij (c)ijFigure /4/: Sup ersymmetric dimensionful couplings/: /(a/) /(scalar/) /3in teraction v ertex M /in y jkn/, /(b/) fermionmass term M ij/, /(c/) scalar /(mass/) /2term M /ik M kj/.y ij kwill b e allo w ed to b e non/-zero/. The en tries of the mass matrix M ijcan only b e non/-zerofor i and j suc h that the sup erm ultiplets /i and /j transform under the gauge group inrepresen tations whic h are conjugates of eac h other/. /(In fact/, in the MSSM there is only onesuc h term/, as w e will see/./) Lik ewise/, the Y uk a w a couplings y ij kcan only b e non/-zero when/i /,/j /, and /k transform in represen tations whic h can com bine to form a singlet/.The in teractions implied b y the sup erp oten tial eq/. /(/3/./7/7/) are sho wn xin Figs/. /3 and/4/. Those in Fig/. /3 are all determined b y the dimensionless parameters y ij k/. The Y uk a w ain teraction in Fig/. /3a corresp onds to the next/-to/-last term in eq/. /(/3/./4/8/)/. F or eac h particularY uk a w a coupling of /i / j / k with strength y ij k/, there m ust b e equal couplings of /j / i / k and/k / i / j /, since y ij kis completely symmetric under in terc hange of an yt w o of its indices assho wn in section /3/./2/. There is also a dimensionless coupling for /i /j / / k/ / l/, with strengthy ij ny /kl n as required b y sup ersymmetry /[see the last term in eq/. /(/3/./4/7/)/]/. The arro ws on b oththe fermion and scalar lines follo w the c hiralit y/;; i/.e/./, one direction for propagation of / and / and the other for the propagation of / /and / y/.T h us there is a v ertex corresp onding to theone in Fig/. /3a but with all arro ws rev ersed/, corresp onding to the complex conjugate /[the lastterm in eq/. /(/3/./4/8/)/]/. The relationship b et w een the in teractions in Figs/. /3a and /3b is exactlyof the sp ecial t yp e needed to cancel the quadratic div ergences in quan tum corrections toscalar masses/, as discussed in the In tro duction /[compare Fig/. /1/]/.In Fig/. /4/, w es h o w the only in teractions corresp onding to renormalizable and sup ersym/-metric v ertices with dimensions of /(mass/) and /(mass/) /2/. First/, there are /(scalar/) /3couplingswhic h are en tirely determined b y the sup erp oten tial mass parameters M ijand Y uk a w a cou/-plings y ij k/, as indicated b y the second and third terms in eq/. /(/3/./4/7/)/. The propagators ofthe fermions and scalars in the theory are constructed in the usual w a y using the fermionmass M ijand scalar /(mass/) /2M /in M nj/. Of particular in terest is the fact that the fermionmass term M ijl e a d s t oac hiralit y/-c hanging insertion in the fermion propagator/;; note thedirections of the arro ws in Fig/. /4b/. There is no suc ha r r o w/-rev ersal for a scalar propagatorin a theory with exact sup ersymmetry/;; as sho wn in Fig/. /4c/, if one treats the scalar /(mass/) /2 xHere/, the auxiliary /elds ha v e b een eliminated using their equations of motion /(/\in tegrated out/"/) as ineq/. /(/3/./4/8/)/. It is quite p ossible instead to giv eF eynman rules whic h include the auxiliary /elds/, although thistends to b e less useful in phenomenologi cal applicatio ns/./2/7 (a) (b) (c) (d) (e) (f) (g) (h)Figure /5/: Sup ersymmetric gauge in teraction v ertices/.term as an insertion in the propagator/, the arro w direction is preserv ed/. Again/, for eac ho fFigures /4a and /4b there is an in teraction with all arro ws rev ersed/.In Fig/. /5 w es h o w in a similar manner the gauge in teractions in a sup ersymmetric theory /.Figures /5a/,b/,c o ccur only when the gauge group is non/-ab elian /(e/.g/. for SU /(/3/)C color andSU /(/2/)L w eak isospin in the MSSM/)/. Figures /5a and /5b are the in teractions of gauge b osonswhic h deriv e from the /rst term in eq/. /(/3/./5/4/)/. In the MSSM these are exactly the same asthe w ell/-kno wn QCD gluon and electro w eak gauge b oson v ertices of the Standard Mo del/./(W e do not sho w the in teractions of ghost /elds/, whic h are necessary only for consisten tl o o pamplitudes/./) Figures /5c/,d/,e/,f are just the standard in teractions b et w een gauge b osons andfermion and scalar /elds whic hm ust o ccur in an y gauge theory b ecause of the form of theco v arian td e r i v ativ e/;; they come from eqs/. /(/3/./5/6/) and /(/3/./6/5/)/-/(/3/./6/7/) inserted in the kinetic partof the lagrangian/. Figure /5c sho ws the coupling of a gaugino to a gauge b oson/;; the gauginoline in a F eynman diagram is traditionally dra wn as a solid fermion line sup erimp osed ona gauge b oson squiggly line/. In Fig/. /5g w eh a v e the coupling of a gaugino to a c hiralfermion and a complex scalar /[the /rst term in the second line in eq/. /(/3/./7/2/)/]/. One canthink of this as the /\sup ersymmetrization/" of Figure /5e or /5f/;; an y of these three v erticesma y b e obtained from an y other /(up to a factor of p /2 /)b y replacing t w o of the particles b ytheir sup ersymmetric partners/. There is also an in teraction lik e Fig/. /5g but with all arro wsrev ersed/, corresp onding to the complex conjugate term in the lagrangian /[the second termin the second line in eq/. /(/3/./7/2/)/]/. Finally in Fig/. /5h w eh a v e a scalar quartic in teractionv ertex /[the last term in eq/. /(/3/./7/5/)/] whic h is also determined b y the gauge coupling/.The results of this section can b e used as a recip e for constructing the sup ersymmetricin teractions for an y mo del/. In the case of the MSSM/, w e already kno w the gauge group/,particle con ten t and the gauge transformation prop erties/, so it only remains to decide onthe sup erp oten tial/. This w e will do in section /5/./1/./4 Soft sup ersymmetry breaking in teractionsA realistic phenomenological mo del m ust con tain sup ersymmetry breaking/. F rom a theo/-retical p ersp ectiv e/, w e exp ect that sup ersymmetry /, if it exists at all/, should b e an exactsymmetry whic hi s s p o n taneously brok en/. In other w ords/, the ultimate mo del should ha v ea lagrangian densit y whic hi s i n v arian t under sup ersymmetry /, but a v acuum state whic hi snot/. In this w a y /, sup ersymmetry is hidden at lo w energies in a manner exactly analogousto the fate of the electro w eak symmetry in the ordinary Standard Mo del/./2/8 Man y mo dels of sp on taneous symmetry breaking ha v e indeed b een prop osed and w e willmen tion the basic ideas of some of them in section /6/. These alw a ys in v olv e extending theMSSM to include new particles and in teractions at v ery high mass scales/, and there is noconsensus on exactly ho w this should b e done/. Ho w ev er/, from a practical p oin t of view/, itis extremely useful to simply parameterize our ignorance of these issues b y just in tro ducingextra terms whic h break sup ersymmetry explicitly in the e/ectiv e MSSM lagrangian/. Asw as argued in the In tro duction/, the extra sup ersymmetry/-breaking couplings should b e soft/(of p ositiv e mass dimension/) in order to b e able to naturally main tain a hierarc h yb e t w eenthe electro w eak scale and the Planc k /(or some other v ery large/) mass scale/. This means inparticular that w e should not consider an y dimensionless sup ersymmetry/-breaking couplings/.In the con text of a general renormalizable theory /, the p ossible soft sup ersymmetry/-breaking terms in the lagrangian areLsoft /= /; /1 /2 /( M/ / a/ a/+c /: c /: /) /; /( m /2/) ij / j //i/; //1 /2 b ij/i /j /+ /1 /6 a ij k/i /j /k /+c /: c /: //;; /(/4/./1/)Lma yb e soft /= /; /1 /2 c jki / / i/j /k /+c /: c /: /(/4/./2/)They consist of gaugino masses M/ for eac h gauge group/, scalar /(mass/) /2terms /( m /2/) ji andb ij/, and /(scalar/) /3couplings a ij kand c jki /. One migh tw onder wh yw eh a v e not includedp ossible soft mass terms for the c hiral sup erm ultiplet fermions/. The reason is that includingsuc h terms w ould b e redundan t/;; they can alw a ys b e absorb ed in to a rede/nition of the su/-p erp oten tial and the terms /( m /2/) ji and c jki /. It has b een sho wn rigorously that a softly/-brok ensup ersymmetric theory with Lsoft as giv en b y eq/. /(/4/./1/) is indeed free of quadratic div er/-gences in quan tum corrections to scalar masses/, to all orders in p erturbation theory /. /3/9Thesituation is sligh tly more subtle if one tries to include the non/-analytic /(scalar/) /3couplingsin Lma yb e soft /.I f a n y of the c hiral sup erm ultiplets in the theory are completely unc hargedunder all gauge symmetries/, then non/-zero c jki terms can lead to quadratic div ergences/, de/-spite the fact that they are formally soft/. No w/, this constrain t need not apply to the MSSM/,whic hd o e sn o t h a v ea n y gauge/-singlet c hiral sup erm ultiplets/. Nev ertheless/, the p ossibilit yof c jki terms is nearly alw a ys neglected/. /4/0The real reason for this is that it is extremelydi/cult to construct an y mo del of sp on taneous sup ersymmetry breaking in whic ht h e c jkiare not utterly negligibly small/. Equation /(/4/./1/) is therefore usually tak en to b e the mostgeneral soft sup ersymmetry/-breaking lagrangian/.It should b e clear that Lsoft indeed breaks sup ersymmetry /, s i n c ei ti n v olv es only scalarsand gauginos/, and not their resp ectiv e sup erpartners/. In fact/, the soft terms in Lsoft arecapable of giving masses to all of the scalars and gauginos in a theory /,e v en if the gaugeb osons and fermions in c hiral sup erm ultiplets are massless /(or relativ ely ligh t/)/. The gauginomasses M/ are alw a ys allo w ed b y gauge symmetry /. The /( m /2/) ij terms are allo w ed for i/;; jsuc h that /i /, / j /transform in complex conjugate represen tations of eac h other under allgauge symmetries/;; in particular this is true of course when i /= j /,s o e v ery scalar is eligibleto get a mass in this w a y if sup ersymmetry is brok en/. The remaining soft terms ma yo rma y not b e allo w ed b y the symmetries/. In this regard it is useful to note that the b ijanda ij kterms ha v e the same form as the M ijand y ij kterms in the sup erp oten tial /[compareeq/. /(/4/./1/) to eq/. /(/3/./4/1/) or eq/. /(/3/./7/7/)/]/, so they will b e allo w ed b y gauge in v ariance if andonly if a corresp onding sup erp oten tial term is allo w ed/. The F eynman diagram in teractionscorresp onding to the allo w ed soft terms in eq/. /(/4/./1/) are sho wn in Fig/. /6/. As b efore/, for eac h/2/9 (a) (b)ij (c)ij (d)i jkFigure /6/: Soft sup ersymmetry/-breakin g terms/: /(a/) Gaugino mass insertion M/ /;; /(b/) non/-analytic scalar/(mass/) /2/( m /2/) ij /;; /(c/) analytic scalar /(mass/) /2b ij/;; /(d/) /(scalar/) /3coupling a ij k/.of the in teractions in Figs/. /6a/,c/,d there is one with all arro ws rev ersed/, corresp onding tothe complex conjugate term in the lagrangian/. W e will apply these general results to thesp eci/c case of the MSSM in the next section/./5 The Minimal Sup ersymmetric Standard Mo delIn sections /3 and /4/, w eh a v e found a general recip e for constructing lagrangians for softlybrok en sup ersymmetric theories/. W ea r e n o w ready to apply these general results to theMSSM/. The particle con ten t for the MSSM w as describ ed in the In tro duction/. In this sectionw e will complete the mo del b y sp ecifying the sup erp oten tial and the soft/-breaking terms/./5/./1 The sup erp otential and sup ersymmetric inter actionsThe sup erp oten tial for the MSSM is giv en b yWMSSM /= u yu QHu /; d yd QHd /; e ye LHd /+ /Hu Hd /: /(/5/./1/)The ob jects Hu /, Hd /, Q /, L /, u /, d /, e app earing in eq/. /(/5/./1/) are c hiral sup er/elds corresp ondingto the c hiral sup erm ultiplets in T able /1/. /(Alternativ ely /, they can b e just though to fa sthe corresp onding scalar /elds/, as w as done in section /3/, but w e prefer not to put thetildes on Q /, L /, u /, d /, e in order to reduce clutter/./) The dimensionless Y uk a w a couplingparameters yu /;; yd /;; ye are /3 / /3 matrices in family space/. Here w eh a v e suppressed all ofthe gauge /[ SU /(/3/)C color and SU /(/2/)L w eak isospin/] and family indices/. The /\ / term/"/, asit is traditionally called/, can b e written out as / /( Hu /)/ /( Hd /)/ / ///, where / //is used to tietogether SU /(/2/)L w eak isospin indices //;; / /=/1 /;; /2 in a gauge/-in v arian tw a y /.L i k ewise/, theterm u yu QHu can b e written out as u ia /( yu /)i jQ aj/ /( Hu /)/ / ///, where i /=/1 /;; /2 /;; /3 is a familyindex/, and a /=/1 /;; /2 /;; /3 is a color index whic hi sr a i s e d/( l o w ered/) in the /3 /( /3 /) represen tationof SU /(/3/)C /.The / term in eq/. /(/5/./1/) is the sup ersymmetric v ersion of the Higgs b oson mass inthe Standard Mo del/. It is unique/, b ecause terms H /u Hu or H /d Hd are forbidden in thesup erp oten tial/, since it m ust b e analytic in the c hiral sup er/elds /(or equiv alen tly in thescalar /elds/) treated as complex v ariables/, as sho wn in section /3/./2/. W e can also see from theform of eq/. /(/5/./1/) wh yb o t h Hu and Hd are needed in order to giv eY uk a w a couplings/, andth us masses/, to all of the quarks and leptons/. Since the sup erp oten tial m ust b e analytic/,the uQHu Y uk a w a terms cannot b e replaced b y something lik e uQH /d /. Similarly /, the dQHdand eL Hd terms cannot b e replaced b y something lik e d QH /u and eLH /u /. The analogousY uk a w a couplings w ould b e allo w ed in a general non/-sup ersymmetric t w o Higgs doubletmo del/, but are forbidden b y the structure of sup ersymmetry /.S o w e need b oth Hu and Hd /,ev en without in v oking the argumen t based on anomaly cancellation whic hw as men tionedin the In tro duction/./3/0 Hu0 tLtR† (a)Hu0 tLtR† (a)Hu0 tLtR* (c)Figure /7/: The top/-quark Y uk a w a coupling /(a/) and its sup ersymmetrization s /(b/)/,/(c/)/, all of strength yt /.The Y uk a w a matrices determine the masses and CKM mixing angles of the ordinaryquarks and leptons/, after the neutral scalar comp onen ts of Hu and Hd get VEVs/. Since thetop quark/, b ottom quark and tau lepton are the hea viest fermions in the Standard Mo del/,it is often useful to mak e an appro ximation that only the /(/3 /;; /3/) family comp onen ts of eac hof yu /, yd and ye are imp ortan t/:yu / /0/@ /0 /0 /0/0 /0 /0/0 /0 yt /1A/;; yd / /0/@ /0 /0 /0/0 /0 /0/0 /0 yb /1A/;; ye / /0/@ /0 /0 /0/0 /0 /0/0 /0 y/ /1A/: /(/5/./2/)In this limit/, only the third family and Higgs /elds con tribute to the MSSM sup erp oten tial/.It is instructiv e to write the sup erp oten tial in terms of the separate SU /(/2/)L w eak isospincomp onen ts /[ Q/3 /=/( tb /)/;; L/3 /=/( // / /)/;; Hu /=/( H /+u H /0u /)/;; Hd /=/( H /0d H /;d /)/;; u/3 /= t /;; d/3 /= b /;; e/3 /= / /]/, so/:WMSSM / yt /( ttH /0u /; tbH /+u /) /; yb /( b tH /;d /; b bH /0d /) /; y/ /( /// H /;d /; //H /0d /)/+ / /( H /+u H /;d /; H /0u H /0d /) /: /(/5/./3/)The min us signs inside the paren theses app ear b ecause of the an tisymmetry of the / //sym b ol used to tie up the SU /(/2/)L indices/. The min us signs in eq/. /(/5/./1/) w ere c hosen so thatthe terms yt ttH /0u /, yb bbH /0d /,a n d y/ //H /0d /, whic h will b ecome the top/, b ottom and tau masseswhen H /0u and H /0d get VEVs/, ha v e p ositiv e signs in eq/. /(/5/./3/)/.Since the Y uk a w ai n teractions y ij kin a general sup ersymmetric theory m ust b e com/-pletely symmetric under in terc hange of i/;; j/;; k /,w ek n o w that yu /, yd and ye imply not onlyHiggs/-quark/-quark and Higgs/-lepton/-lepton couplings as in the Standard Mo del/, but alsosquark/-Higgsino/-quark and slepton/-Higgsino/-lepton in teractions/. T o illustrate this/, w es h o win Figs/. /7a/,b/,c some of the in teractions whic hi n v olv e the top/-quark Y uk a w a coupling yt /.Figure /7a is the Standard Mo del/-lik e coupling of the top quark to the neutral complexscalar Higgs b oson/, whic h follo ws from the /rst term in eq/. /(/5/./3/)/. F or v ariet y /,w eh a v eused tL and t yR in place of their synon yms t and t in Fig/. /7/;; see the discussion in the /nalparagraph in section /2/. In Fig/. /7b/, w eh a v e the coupling of the left/-handed top squark etL tothe neutral higgsino /eld eH /0u and righ t/-handed top quark/, while in Fig/. /7c the righ t/-handedtop/-squark /eld /(kno wn either as e t or et /R dep ending on taste/) couples to eH /0u and tL /.F oreac h of the three in teractions/, there is another with H /0u /! H /+u and tL /!/; bL /(with tildeswhere appropriate/)/, corresp onding to the second part of the /rst term in eq/. /(/5/./3/)/. All ofthese in teractions are required b y sup ersymmetry to ha v e the same strength yt /. This is alsoan incon tro v ertible prediction of softly/-brok en sup ersymmetry at tree/-lev el/, since these in/-teractions are dimensionless and can b e mo di/ed b y the in tro duction of soft sup ersymmetrybreaking only through /nite /(and small/) radiativ e corrections/. A useful mnemonic is thateac h of Figs/. /7a/,b/,c can b e obtained from an y of the others b yc hanging t w o of the particlesin to their sup erpartners/./3/1 (a)tR*tRtL tL* (b)Hu0Hu0*tL tL* (c)Hu0Hu0*tR* tRFigure /8/: Some of the /(scalar/) /4in teractions with strength prop ortional to y /2t /. q gq (a)qL, lL, Hu, Hd WqL, lL, Hu, Hd (b)q, l, H u, Hd Bq, l, Hu, Hd (c)Figure /9/: Couplings of the gluino/, wino/, and bino to MSSM /(scalar/, fermion/) pairs/.There are also scalar quartic in teractions with strength prop ortional to y /2t /, as can b eseen e/.g/. from Fig/. /3b or the last term in eq/. /(/3/./4/7/)/. Three of them are sho wn in Fig/. /8/. Thereader is in vited to c hec k/, using eq/. /(/3/./4/7/) and eq/. /(/5/./3/)/, that there are nine more/, whic hcan b e obtained b y replacing etL /! ebL and//or H /0u /! H /+u in eac hv ertex/. This illustratesthe remark able econom y of sup ersymmetry/;; there are man yi n teractions determined b yonly a single parameter/! In a similar w a y /, the existence of all the other quark and leptonY uk a w a couplings in the sup erp oten tial eq/. /(/5/./1/) leads not only to Higgs/-quark/-quark andHiggs/-lepton/-lepton lagrangian terms as in the ordinary Standard Mo del/, but also to squark/-higgsino/-quark and slepton/-higgsino/-lepton terms/, and scalar quartic couplings /[/(squark/) /4/,/(slepton/) /4/, /(squark/) /2/(slepton/) /2/, /(squark/) /2/(Higgs/) /2/, and /(slepton/) /2/(Higgs/) /2/]/. If needed/, thesecan all b e obtained in terms of the Y uk a w a matrices yu /, yd /,a n d ye as outlined ab o v e/.Ho w ev er/, it is useful to note that the dimensionless in teractions determined b y thesup erp oten tial are often not the most imp ortan t ones of direct in terest for phenomenology /.This is b ecause the Y uk a w a couplings are already kno wn to b e v ery small/, except for those ofthe third family /(top/, b ottom/, tau/)/. Instead/, deca y and esp ecially pro duction pro cesses forsup erpartners in the MSSM are t ypically dominated b y the sup ersymmetric in teractions ofgauge/-coupling strength/, as w e will explore in more detail in sections /8 and /9/. The couplingsof the Standard Mo del gauge b osons /(photon/, W //, Z /0and gluons/) to the MSSM particlesare determined completely b y the gauge in v ariance of the kinetic terms in the lagrangian/.The gauginos also couple to /(squark/, quark/) and /(slepton/, lepton/) and /(Higgs/, higgsino/)pairs as illustrated in the general case in Fig/. /5g and the second line in eq/. /(/3/./7/2/)/. F orinstance/, eac h of the squark/-quark/-gluino couplings is giv en b y p /2 g/3 /( eqT aq eg /+c /: c /: /) whereT a/( a /=/1 /:/:/: /8/) are the Gell/-Mann matrices for SU /(/3/)C /. The F eynman diagram for thisin teraction is sho wn in Fig/. /9a/. In Figs/. /9b/,c w es h o w in a similar w a y the couplings of/(squark/, quark/)/, /(lepton/, slepton/) and /(Higgs/, higgsino/) pairs to the winos and bino/, withstrengths prop ortional to the electro w eak gauge couplings g and g /0resp ectiv ely /. The winosonly couple to the left/-handed squarks and sleptons/, and the /(lepton/, slepton/) and /(Higgs/,higgsino/) pairs of course do not couple to the gluino/. The bino couplings for eac h /(scalar/,fermion/) pair are also prop ortional to the w eak h yp erc harges Y as giv en in T able /1/. Thein teractions sho wn in Fig/. /9 pro vide for deca ys eq /! q eg and eq /! fWq /0and eq /! eBq when the/3/2 /nal states are kinematically allo w ed to b e on/-shell/. Ho w ev er/, a complication is that thefW and eB states are not mass eigenstates/, b ecause of mixing due to electro w eak symmetrybreaking/, as w e will see in section /7/./3/.There are also v arious scalar quartic in teractions in the MSSM whic h are uniquelydetermined b y gauge in v ariance and sup ersymmetry /, according to the last term in eq/. /(/3/./7/5/)illustrated in Fig/. /5h/. Among them are /(Higgs/) /4terms prop ortional to g /2and g /0 /2in thescalar p oten tial/. These are the direct generalization of the last term in the Standard Mo delHiggs p oten tial/, eq/. /(/1/./1/)/, to the case of the MSSM/. W ew i l l h a v e o ccasion to iden tify themexplicitly when w e discuss the minimization of the MSSM Higgs p oten tial in section /7/./2/.The dimensionful terms in the sup ersymmetric part of the MSSM lagrangian are alldep enden to n / /. F ollo wing the general result of eq/. /(/3/./4/8/)/, w e /nd that / pro vides forhiggsino fermion mass termsL//; / /( eH /+u eH /;d /; eH /0u eH /0d /)/+ c /: c /: /;; /(/5/./4/)as w ell as Higgs /(mass/) /2terms in the scalar p oten tial/;L / V /j / j /2/( j H /0u j /2/+ j H /+u j /2/+ j H /0d j /2/+ j H /;d j /2/) /: /(/5/./5/)Since eq/. /(/5/./5/) is p ositiv e/-de/nite/, it is clear that w e cannot understand electro w eak sym/-metry breaking without including sup ersymmetry/-breaking /(mass/) /2soft terms for the Higgsscalars/, whic h can b e negativ e/. An explicit treatmen t of the Higgs scalar p oten tial willtherefore ha v et ow ait un til w eh a v ei n tro duced the soft terms for the MSSM/. Ho w ev er/, w ecan already see a puzzle/: w e exp ect that / should b e roughly of order /1/0 /2or /1/0 /3GeV/, inorder to allo w a Higgs VEV of order /1/7/4 GeV without to o m uc h miraculous cancellation b e/-t w een j / j /2and the negativ e soft /(mass/) /2terms that w eh a v e not written do wn y et/. But wh yshould / b e so small compared to/, sa y /, MP /, and in particular wh y should it b e roughly ofthe same order as msoft /? The scalar p oten tial of the MSSM seems to dep end on t w ot yp es ofdimensionful parameters whic h are conceptually quite distinct/, namely the sup ersymmetry/-resp ecting mass / and the sup ersymmetry/-breaking soft mass terms/. Y et the observ ed v aluefor the electro w eak breaking scale suggests that without miraculous cancellations/, b oth ofthese apparen tly unrelated mass scales should b e within an order of magnitude or so of /1/0/0GeV/. This puzzle is called /\the / problem/"/. Sev eral di/eren t solutions to the / problemha v e b een prop osed/, in v olving extensions of the MSSM of v arying in tricacy /. They all w orkin roughly the same w a y/;; the parameter / is required or assumed to b e completely absen tat tree/-lev el/, and is to b e replaced b y the VEV/(s/) of some new /eld/(s/)/. The latter are inturn determined b y minimizing a p oten tial whic h dep ends on soft sup ersymmetry/-breakingterms/. In this w a y /, the v alue of the e/ectiv e parameter / is no longer conceptually distinctfrom the mec hanism of sup ersymmetry breaking/;; if w e can explain wh y msoft / MP /,w ewill also b e able to understand wh y / is of the same order/. In section /1/0/./2 w e will de/-scrib e one suc h mec hanism/. Some other attractiv e solutions for the / problem are prop osedin Refs/. /4/1 /;; /4/2 /;; /4/3F rom the p oin t of view of the MSSM/, ho w ev er/, w e can just treat / as anindep endent parameter/.The / /-term and the Y uk a w a couplings in the sup erp oten tial eq/. /(/5/./1/) com bine to yield/(scalar/) /3couplings /[see the second and third terms on the righ t/-hand side of eq/. /(/3/./4/7/)/] ofthe formL / / //( e u yu euH /0 /d /+ e d yd edH /0 /u /+ e e ye eeH /0 /u/+ e u yu ed H /;/d /+ e d yd euH /+ /u /+ e e ye e/H /+ /u /)/+c /: c /: /(/5/./6/)/3/3 Hd0* tR*tL (a)Hu0* bR*bL (b)Hu0* τR*τL (c)Figure /1/0/: Some of the sup ersymmetric /(scalar/) /3couplings prop ortional to / /yt /, / /yb /, and / /y/ /. s or bd u u uL Qλ′′ λ′Figure /1/1/: Squarks can mediate disastrously rapid proton deca yi f R /-parit y is violated/.In Fig/. /1/0 w e sho w some of these couplings whic h are prop ortional to / /yt /, / /yb /,a n d / /y/resp ectiv ely /. These pla y an imp ortan t role in determining the mixing of top squarks/, b ottomsquarks/, and tau sleptons/, as w e will see in section /7/./5/./5/./2 R /-p arity /(also known as matter p arity/) and its c onse quenc esThe sup erp oten tial eq/. /(/5/./1/) is minimal in the sense that it is su/cien t to pro duce a phe/-nomenologically viable mo del/. Ho w ev er/, there are other terms that one could write do wnwhic h are gauge/-in v arian t and analytic in the c hiral sup er/elds/, but are not included in theMSSM b ecause they violate either bary on n um b er /(B/) or total lepton n um b er /(L/)/. The mostgeneral gauge/-in v arian t and renormalizable sup erp oten tial w ould include not only eq/. /(/5/./1/)/,but also the termsW/
L/=/1 /= /1 /2 / ij kLi Lj ek /+ / /0 ij kLi Qj dk /+ / /0 iLi Hu /(/5/./7/)W/
B/=/1 /= /1 /2 / /0/0 ij k ui dj dk /(/5/./8/)where w eh a v e restored family indices i /=/1 /;; /2 /;; /3/. The c hiral sup erm ultiplets carry bary onn um b er assignmen ts B /= /+/1 /= /3f o r Qi /;;B /= /; /1 /= /3f o r ui /;; di /;; and B /= /0 for all others/. Thetotal lepton n um b er assignmen ts are L /= /+/1 for Li /,L/= /; /1f o r ei /, and L /= /0 for allothers/. Therefore/, the terms in eq/. /(/5/./7/) violate total lepton n um be r b y /1 unit /(as w ell asthe individual lepton / a v ors/) and those in eq/. /(/5/./8/) violate bary on n um be r b y /1 unit/.The p ossible existence of suc h terms migh t seem rather disturbing/, since B/- and L/-violating pro cesses ha v en e v er b een seen exp erimen tally /. The most ob vious exp erimen talconstrain t comes from the non/-observ ation of proton deca y /, whic hw ould violate b oth Band L b y /1 unit/. If b oth / /0and / /0/0couplings w ere presen t and of order unit y /, then thelifetime of the proton w ould b e measured in min utes or hours/! F or example/, the F eynmangraph in Fig/. /1/1 w ould lead to p /+/! e /+/ /0or e /+K /0or / /+/ /0or / /+K /0or // /+or /K /+etc/. dep ending on whic h comp onen ts of / /0are largest/, and these pro cesses w ould seem to b ecompletely unsuppressed since the necessary couplings are all renormalizable/. /(The coupling/ /0/0m ust b e an tisymmetric in its last t w o/ a v or indices/, since the color indices are con tractedan tisymmetrically /. That is wh y the squark in Fig/. /1/1 is e s or e b but not e d /,f o r u/;; d quarks in/3/4 the initial state/./) In con trast/, the deca y time of the proton in to these mo des is measuredto b e in excess of /1/0 /3/2y ears/. Man y other pro cesses also giv ev ery signi/can t constrain ts onthe violation of lepton and bary on n um b ers/;; these are review ed in Ref/. /4/4One could simply try to tak eBa n dLc o n s e r v ation as a p ostulate in the MSSM/. Ho w/-ev er/, this is clearly a step bac kw ards from the situation in the Standard Mo del/, where theconserv ation of these quan tum n um b ers is not assumed/, but is rather a pleasan tly /\acci/-den tal/" consequence of the fact that there are no p ossible renormalizable lagrangian termswhic h violate B or L/. F urthermore/, there is a quite general obstacle to treating B and Las fundamen tal symmetries of nature/, since they are kno wn to b e necessarily violated b ynon/-p erturbativ e electro w eak e/ects /(ev en though those e/ects are calculably negligible forexp erimen ts at ordinary energies/)/. Therefore/, in the MSSM one adds a new symmetry whic hhas the e/ect of eliminating the p ossibilit y of B and L violating terms in the renormalizablesup erp oten tial/, while allo wing the go o d terms in eq/. /(/5/./1/)/. This new symmetry is called/\ R /-parit y/" /6or equiv alen tly /\matter parit y/"/. /4/5Matter parit yi s am ultiplicativ ely conserv ed quan tum n um b er de/ned asPM /=/( /; /1/) /3/(B /; L/)/(/5/./9/)for eac h particle in the theory /. I ti s e a s yt oc hec k that the quark and lepton sup erm ultipletsall ha v e PM /= /; /1/, while the Higgs sup erm ultiplets Hu and Hd ha v e PM /= /+/1/. The gaugeb osons and gauginos of course do not carry bary on n um b er or lepton n um b er/, so they areassigned matter parit y PM /= /+/1/. The symmetry principle to b e enforced is that a termin the Lagrangian /(or in the sup erp oten t i a l /)i sa l l o w ed only if the pro duct of PM for allof the /elds in it is /+/1/. It is easy to see that eac h of the terms in eq/. /(/5/./7/) and /(/5/./8/) isth us forbidden/, while the go o d and necessary terms in eq/. /(/5/./1/) are allo w ed/. This discretesymmetry comm utes with sup ersymmetry /, as all mem b ers of a giv en sup erm ultiplet ha v ethe same matter parit y /. The adv an tage of matter parit y is that it can in principle b ean exact and fundamen tal symmetry /, whic h B and L themselv es cannot/, since they arekno wn to b e violated b y non/-p erturbativ e electro w eak e/ects/. So ev en with exact matterparit y conserv ation in the MSSM/, one exp ects that bary on n um b er and total lepton n um be rviolation will o ccur in v ery tin y amoun ts/, due to nonrenormalizable terms in the Lagrangian/.Ho w ev er/, the MSSM do es not ha v e renormalizable in teractions that violate B or L/, with thestandard assumption of matter parit yc o n s e r v ation/.It is sometimes useful to recast matter parit yi nt e r m s o f R /-parit y /, de/ned for eac hparticle asPR /=/( /; /1/) /3/(B /; L/)/+/2 s/(/5/./1/0/)where s is the spin of the particle/. No w/, matter parit yc o n s e r v ation and R /-parit y conser/-v ation are precisely equiv alen t/, since the pro duct of /( /; /1/) /2 sis of course equal to /+/1 for theparticles in v olv ed in an yi n teraction v ertex in a theory that conserv es angular momen tum/.Ho w ev er/, particles within the same sup erm ultiplet do not ha v e the same R /-parit y /. In gen/-eral/, symmetries with the prop ert y that particles within the same m ultiplet ha v e di/eren tc harges are called R symmetries/;; they do not comm ute with sup ersymmetry /.C o n tin uousU /(/1/) R symmetries are often encoun tered in the mo del/-buildin g literature/;; they should notb e confused with R /-parit y /, whic h is a discrete Z/2 symmetry /. In fact/, the matter parit yv er/-sion of R /-parit ym a k es clear that there is really nothing in trinsically /\ R /" ab out it/;; in otherw ords it secretly do es comm ute with sup ersymmetry /, so its name is somewhat sub optimal/.Nev ertheless/, the R /-parit y assignmen ti sv ery useful for phenomenology b ecause all of the/3/5 Standard Mo del particles and the Higgs b osons ha v ee v en R /-parit y/( PR /= /+/1/)/, while all ofthe squarks/, sleptons/, gauginos/, and higgsinos ha v eo d d R /-parit y/( PR /= /; /1/)/.The R /-parit y o dd particles are kno wn as /\sup ersymmetric particles/" or /\sparticles/" forshort/, and they are distinguished b y a tilde /(see T ables /1 and /2/)/. If R /-parit y is exactlyconserv ed/, then there can b e no mixing b et w een the sparticles and the PR /= /+/1 particles/.F urthermore/, ev ery in teraction v ertex in the theory con tains an ev en n um be r o f PR /= /; /1sparticles/. This has three extremely imp ortan t phenomenological consequences/:/ The ligh test sparticle with PR /= /; /1/, called the /\ligh test sup ersymmetric particle/"or LSP /,m ust b e absolutely stable/. If the LSP is electrically neutral/, it in teractsonly w eakly with ordinary matter/, and so can mak e an attractiv e candidate /4/6for thenon/-bary onic dark matter whic h seems to b e required b y cosmology /./ Eac h sparticle other than the LSP m ust ev en tually deca yi n to a state whic hc o n tainsan o dd n um b er of LSPs /(usually just one/)/./ In collider exp erimen ts/, sparticles can only b e pro duced in ev en n um b ers /(usuallyt w o/-at/-a/-time/)/.W e de/ne the MSSM to conserv e R /-parit y or equiv alen tly matter parit y /. While thisdecision seems to b e w ell/-motiv ated phenomenologically b y proton deca y constrain ts and thehop e that the LSP will pro vide a go o d dark matter candidate/, it migh t app ear somewhatad ho c from a theoretical p oin t of view/. After all/, the MSSM w ould not su/er an yi n ternalinconsistency if w e did not imp ose matter parit y conserv ation/. F urthermore/, it is fair to askwh y matter parit y should b e exactly conserv ed/, giv en that the kno wn discrete symmetriesin the Standard Mo del /(ordinary parit y P /,c harge conjugation C /, time rev ersal T /, etc/./) areall kno wn to b e inexact symmetries/. F ortunately /,i t is sensible to form ulate matter parit ya sa discrete symmetry whic h is exactly conserv ed/. In general/, exactly conserv ed/, or /\gauged/"discrete symmetries /4/7can exist pro vided that they satisfy certain anomaly cancellationconditions /4/8/(m uc h lik ec o n tin uous gauged symmetries/)/. One particularly attractiv ew a y thiscould o ccur is if B /; Li sa c o n tin uous U /(/1/) gauge symmetry whic hi s s p o n taneously brok enat some v ery high energy scale/. F rom eq/. /(/5/./9/)/, w e observ e that PM is actually a discretesubgroup of the con tin uous U /(/1/)B /; L group/. Therefore/, if gauged U /(/1/)B /; L is brok en b y scalarVEVs /(or other order parameters/) whic h carry only ev en in teger v alues of /3/(B /; L/)/, then PMwill automatically surviv e as an exactly conserv ed remnan t/. A v ariet y of extensions of theMSSM in whic h exact R /-parit y arises in just this w a yh a v e b een prop osed/. /4/9 /;; /5/0It ma y alsob e p ossible to ha v e gauged discrete symmetries whic hd o n o t o w e their exact conserv ationto an underlying con tin uous gauged symmetry /, but rather to some other structure suc ha scan o ccur in string theory /. It is also p ossible that R /-parit yi sb r o k en/, or is replaced b y somealternativ e discrete symmetry /.W e will brie/ y consider these as v ariations on the MSSM insection /1/0/./1/./5/./3 Soft sup ersymmetry br e aking in the MSSMT o complete the description of the MSSM/, w e need to sp ecify the soft sup ersymmetrybreaking terms/. In section /4/, w e learned ho w to write do wn the most general set of suc hterms in an y sup ersymmetric theory /. Applying this recip e to the MSSM/, w eh a v e/:L MSSMsoft /= /; /1 /2 /M/3 eg eg /+ M/2 fW fW /+ M/1 eB eB //+c /: c /:/3/6 /; /e u au eQHu /; e d ad eQHd /; e e ae eLHd //+c /: c /:/; eQ ym /2Q eQ /; eL ym /2L eL /; e u m /2 u e u y/; e d m /2 d e d y/; e e m /2 e e e y/; m /2Hu H /u Hu /; m /2Hd H /d Hd /; /( bHu Hd /+c /: c /: /) /: /(/5/./1/1/)In eq/. /(/5/./1/1/)/, M/3 /, M/2 /, and M/1 are the gluino/, wino/, and bino mass terms/. Here/, and fromno w on/, w e suppress the adjoin t represen tation gauge indices on the wino and gluino /elds/,and the gauge indices on all of the c hiral sup erm ultiplet /elds/. The second line in eq/. /(/5/./1/1/)con tains the /(scalar/) /3couplings /[of the t yp e a ij kin eq/. /(/4/./1/)/]/. Eac ho f au /, ad /, ae is acomplex /3 / /3 matrix in family space/, with dimensions of /(mass/)/. They are in one/-to/-onecorresp ondence with the Y uk a w a coupling matrices in the sup erp oten tial/. The third line ofeq/. /(/5/./1/1/) consists of squark and slepton mass terms of the /( m /2/) ji t yp e in eq/. /(/4/./1/)/. Eac ho fm /2Q /, m /2 u /, m /2 d /, m /2L /, m /2 e i sa/3 / /3 matrix in family space whic h can ha v e complex en tries/, butthey m ust b e hermitian so that the lagrangian is real/. /(T oa v oid clutter/, w e do not put tildeson the Q in m /2Q /, etc/./) Finally /, in the last line of eq/. /(/5/./1/1/) w eh a v e sup ersymmetry/-breakingcon tributions to the Higgs p oten tial/;; m /2Hu and m /2Hd are /(mass/) /2terms of the /( m /2/) ji t yp e/,while b is the only /(mass/) /2term of the t yp e b ijin eq/. /(/4/./1/) whic h can o ccur in the MSSM/. ySc hematically /,w e can writeM/1 /;;M/2 /;;M/3 /;; au /;; ad /;; ae / msoft /;; /(/5/./1/2/)m /2Q /;; m /2L /;; m /2 u /;; m /2 d /;; m /2 e /;;m /2Hu /;;m /2Hd /;;b / m /2soft /(/5/./1/3/)with a c haracteristic mass scale msoft whic hi s n o t m uc h larger than /1/0 /3GeV/, as arguedin the In tro duction/. The expression eq/. /(/5/./1/1/) is the most general soft sup ersymmetry/-breaking Lagrangian of the form eq/. /(/4/./1/) whic h is compatible with gauge in v ariance andmatter parit y conserv ation/.Unlik e the sup ersymmetry/-preserving part of the lagrangian/, L MSSMsoft in tro duces man ynew parameters whic hw ere not presen t in the ordinary Standard Mo del/. A careful coun t /5/1rev eals that there are /1/0/5 masses/, phases and mixing angles in the MSSM lagrangian whic hcannot b e rotated a w a yb y rede/ning the phases and / a v or basis for the quark and leptonsup erm ultiplets/, and whic hh a v e no coun terpart in the ordinary Standard Mo del/. Th us/, inprinciple/, sup ersymmetry /(or more precisely /, sup ersymmetry br e aking /) app ears to in tro ducea tremendous arbitrariness in the lagrangian/./5/./4 Hints of an Or ganizing PrincipleF ortunately /, there is already go o d exp erimen tal evidence that some sort of p o w erful /\orga/-nizing principle/" m ust go v ern the soft terms/. This is b ecause most of the new parametersin eq/. /(/5/./1/1/) in v olv e/ a v or mixing or CP violation of the t yp e whic h is already sev erely re/-stricted b y exp erimen t/. /5/2F or example/, supp ose that m /2 e is not diagonal in a basis /( eeR /;; e/R /;; e/R /)of sleptons whose sup erpartners are the righ t/-handed pieces of the Standard Mo del masseigenstates e/;; //;; / /. In that case slepton mixing o ccurs/, and the individual lepton n um b erswill not b e conserv ed/. This is true ev en for pro cesses whic h only in v olv e the sleptons asvirtual particles/. A particularly strong limit on this p ossibilit y comes from the exp erimen talconstrain to n / /! e/ /, /5/3whic h can o ccur via the one/-lo op diagram in Fig/. /1/2a featuringa virtual bino and slepton/. The cross represen ts an insertion of L MSSMsoft //; /( m /2 e /)/2/1 eeR e/ /R /,and the slepton/-bino v ertices are determined b y the w eak h yp erc harge gauge coupling /[see yThe parameter w e call b is often seen in the literature as m /2/1/2 or m /2/3 or B/ /./3/7 (a)µ eγ µeB (b)dssd g g ds sdFigure /1/2/: Diagrams whic h cause / a v or violation in mo dels with arbitrary soft masses/.Fig/. /5g and eq/. /(/3/./7/2/)/]/. There are similar diagrams if the left/-handed slepton mass matrixm /2L has arbitrary o//-diagonal en tries/. If m /2L or m /2 e w ere /\random/"/, with all en tries ofcomparable size/, then the con tributions to BR/( / /! e/ /)w ould b e ab out /5 or /6 orders ofmagnitude larger than the curren t exp erimen tal upp er limit of /5 / /1/0 /; /1/1/,e v en if the sleptonsare as hea vy as /1 T eV/. Therefore the form of the slepton mass matrices m ust b e sev erelyconstrained/.There are also imp ortan t exp erimen tal constrain ts on the squark /(mass/) /2matrices/. Thestrongest of these come from the neutral k aon system/. The e/ectiv e hamiltonian for K /0/$ K /0mixing gets con tributions from the diagram in Fig/. /1/2b/, among others/, if L MSSMsoft con tains/(mass/) /2terms whic h mix do wn squarks and strange squarks/. The gluino/-squark/-quarkv ertices in Fig/. /1/2b are all /xed b y sup ersymmetry to b e of strong in teraction strength/;;there are similar diagrams in whic h the bino and winos are exc hanged/. /5/4If the squark andgaugino masses are of order /1 T eV or less/, one /nds that limits on the parameters /
mK and/K app earing in the neutral k aon system e/ectiv e hamiltonian sev erely restrict the amoun tof do wn/-strange squark mixing and CP/-violating complex phases that one can tolerate inthe soft parameters/. /5/5Considerably w eak er/, but still in teresting/, constrain ts come fromthe D /0/;; D /0and B /0/;; B /0neutral meson systems/, and the deca y b /! s/ /. /5/6After the Higgsscalar /elds get VEVs/, the au /, ad /, ae matrices con tribute o//-diagonal squark and slepton/(mass/) /2terms /[for example/, e d ad eQHd /+c /: c /: /! /( ad /)/1/2 h H /0d i esL ed /R /+c /: c /: /, etc/./]/, so their formis also strongly constrained b y/ a v or/-c hanging neutral curren t/( F CNC/) limits/. There areother signi/can t constrain ts on CP/-violating phases in the gaugino masses and /(scalar/) /3softcouplings follo wing from limits on the electric dip ole momen ts of the neutron and electron/. /5/7All of these p oten tially dangerous F CNC and CP/-violating e/ects in the MSSM can b eev aded if one assumes /(or can explain/!/) that sup ersymmetry breaking should b e suitably/\univ ersal/"/. In particular/, one can supp ose that the squark and slepton /(mass/) /2matricesare / a v or/-blind/. This means that they should eac h b e prop ortional to the /3 / /3 iden tit ymatrix in family space/:m /2Q /= m /2Q /1 /;; m /2 u /= m /2 u /1 /;; m /2 d /= m /2 d /1 /;; m /2L /= m /2L /1 /;; m /2 e /= m /2 e /1 /: /(/5/./1/4/)If so/, then all squark and slepton mixing angles are rendered trivial/, b ecause squarks andsleptons with the same electro w eak quan tum n um b ers will b e degenerate in mass and canb e rotated in to eac h other at will/. Sup ersymmetric con tributions to F CNC pro cesses willtherefore b e v ery small in suc h an idealized limit/, mo dulo the mixing due to au /, ad /, ae /.One can mak e the further assumption that the /(scalar/) /3couplings are eac h prop ortional tothe corresp onding Y uk a w a coupling matrix/:au /= Au /0 yu /;; ad /= Ad /0 yd /;; ae /= Ae /0 ye /: /(/5/./1/5/)/3/8 This ensures that only the squarks and sleptons of the third family can ha v e large /(scalar/) /3couplings/. Finally /, one can a v oid disastrously large CP/-violating e/ects with the assumptionthat the soft parameters do not in tro duce new complex phases/. This is automatic for m /2Huand m /2Hd /, and for m /2Q /, m /2 u etc/. if eq/. /(/5/./1/4/) is assumed/;; if they w ere not real n um b ers/, thelagrangian w ould not b e real/. One can also /x / in the sup erp oten tial and b in eq/. /(/5/./1/1/)to b e real/, b y an appropriate phase rotation of Hu and Hd /. If one then assumes thatarg/( M/1 /) /;; arg/( M/2 /) /;; arg/( M/3 /) /;; arg/( Au /0 /) /;; arg/( Ad /0 /) /;; arg/( Ae /0 /)/= /0 o r //;; /(/5/./1/6/)then the only CP/-violating phase in the theory will b e the ordinary CKM phase found in theordinary Y uk a w a couplings/. T ogether/, the conditions eqs/. /(/5/./1/4/)/-/(/5/./1/6 /)m a k e up a ratherw eak v ersion of what is often called the assumption of soft/-br e aking universality /.The soft/-breaking univ ersalit y relations eqs/. /(/5/./1/4/)/-/(/5/./1/6/) /(or stronger v ersions of them/)are presumed to b e the result of some sp eci/c mo del for the origin of sup ersymmetry break/-ing/, ev en though there is considerable disagreemen t among theorists as to what the sp eci/cmo del should actually b e/. In an y case/, they are indicativ e of an underlying simplicit yo rsymmetry of the lagrangian at some v ery high energy scale Q/0 /, whic hw e will call the /\inputscale/"/. If w e use this lagrangian to compute masses and cross/-sections and deca y rates forexp erimen ts at ordinary energies near the electro w eak scale/, the results will in v olv e largelogarithms of order ln/( Q/0 /=mZ /) coming from lo op diagrams/. As is usual in quan tum /eld the/-ory /, the large logarithms can b e con v enien tly resummed using renormalization group /(R G/)equations/, b y treating the couplings and masses app earing in the lagrangian as /\running/"parameters/. Therefore/, eqs/. /(/5/./1/4/)/-/(/5/./1/6/) should b e in terpreted as b oundary conditions onthe running soft parameters at the R Gs c a l e Q/0 whic hi sv ery far remo v ed from direct ex/-p erimen tal prob es/. W em ust then R G/-ev olv e all of the soft parameters/, the sup erp oten tialparameters/, and the gauge couplings do wn to the electro w eak scale or comparable scaleswhere h umans p erform exp erimen ts/.A t the electro w eak scale/, eqs/. /(/5/./1/4/) and /(/5/./1/5/) will no longer hold/. Ho w ev er/, R G cor/-rections due to gauge in teractions will resp ect eqs/. /(/5/./1/4/) and /(/5/./1/5/)/, while R G correctionsdue to Y uk a w ai n teractions are quite small except for couplings in v olving the top squarks/(stops/) and p ossibly the b ottom squarks /(sb ottoms/) and tau sleptons /(staus/)/. In particu/-lar/, the /(scalar/) /3couplings should b e quite negligible for the squarks and sleptons of the/rst t w o families/. F urthermore/, R Ge v olution do es not in tro duce new CP/-violating phases/.Therefore/, if univ ersalit y can b e arranged to hold at the input scale/, sup ersymmetric con/-tributions to F CNC and CP/-violating observ ables can b e acceptably small in comparison topresen t limits /(although quite p ossibly measurable in future exp erimen ts/)/.One go o d reason to b e optimistic that suc h a program can succeed is the celebratedapparen t uni/cation of gauge couplings in the MSSM/. /5/8The /1/-lo op R G equations for theStandard Mo del gauge couplings g/1 /;;g/2 /;;g/3 are giv en b yd dt ga /= /1 /1/6 / /2 ba g /3a /) d dt / /; /1a /= /; ba /2 / /( a /=/1 /;; /2 /;; /3/) /(/5/./1/7/)where t /=l n /( Q/=Q/0 /)w i t h Q the R G scale/. In the Standard Mo del/, b SMa /=/( /4 /1 /= /1/0 /;; /; /1/9 /= /6 /;;/; /7/)/, while in the MSSM one /nds instead b MSSMa /= /(/3/3 /= /5 /;; /1 /;; /; /3/)/. The latter set of co ef/-/cien ts are larger b ecause of the virtual e/ects of the extra MSSM particles in lo ops/. Thenormalization for g/1 here is c hosen to agree with the canonical co v arian t deriv ativ e for granduni/cation of the gauge group SU /(/3/)C / SU /(/2/)L / U /(/1/)Y in to SU /(/5/) or SO /(/1/0/)/. Th us interms of the con v en tional electro w eak gauge couplings g and g /0with e /= g sin /W /= g /0cos /W /,/3/9 2 4 6 8 1 01 21 41 61 8 Log10(Q/1 GeV)0102030405060α−1 α1−1 α2−1 α3−1Figure /1/3/: R Ge v olution of the in v erse gauge couplings / /; /1a /( Q /) in the Standard Mo del /(dashed lines/) andthe MSSM /(solid lines/)/. In the MSSM case/, //3 /( mZ /)i s v aried b et w een /0 /: /1/1/3 and /0 /: /1/2/3/, and the sparticlemass thresholds b et w een /2/5/0 GeV and /1 T eV/. Tw o/-lo op e/ects are included/.one has g/2 /= g and g/1 /= p /5 /= /3 g /0/.T h e q u a n tities /a /= g /2a /= /4 / ha v e the nice prop ert y thattheir recipro cals run linearly with R G scale at one/-lo op order/. In Fig/. /1/3 w e compare theR Ge v olution of the / /; /1a /, including t w o/-lo op e/ects/, in the Standard Mo del /(dashed lines/)and the MSSM /(solid lines/)/. Unlik e the Standard Mo del/, the MSSM includes just the righ tparticle con ten t to ensure that the gauge couplings can unify /, at a scale MU / /2 / /1/0 /1/6GeV/. While the apparen t uni/cation of gauge couplings at MU could b e just an acciden t/, itma y also b e tak en as a strong hin ti n f a v or of a grand uni/ed theory /(GUT/) or sup erstringmo dels/, b oth of whic h indeed predict gauge coupling uni/cation b elo w MP /.F urthermore/, ifw e tak e this hin t seriously /, then it means that w e can reasonably exp ect to apply a similarR G analysis to the other MSSM couplings and soft masses as w ell/.W em ust men tion that there are t w o other p ossible t yp es of explanations for the suppres/-sion of F CNCs in the MSSM/, whic h could replace the univ ersalit yh yp othesis of eqs/. /(/5/./1/4/)/-/(/5/./1/6/)/. One migh t refer to them as /\irrelev ancy/" and /\alignmen t/" of the soft masses/. The/\irrelev ancy/" idea is that the sparticles masses are simply extr emely hea vy /, so that their con/-tributions to F CNC and CP/-violating diagrams lik e Figs/. /1/2a/,b are highly suppressed/. Inpractice/, ho w ev er/, the degree of suppression needed often requires msoft / /1T eV for at leastsome of the scalar masses/;; this seems to go directly against the motiv ation for sup ersymme/-try as a cure for the hierarc h y problem as discussed in the In tro duction/. Nev ertheless/, it isp ossible to arrange a sc heme where this can w ork in a sensible w a y /. /5/9The /\alignmen t/" ideais that the squark /(mass/) /2matrices do not ha v e the / a v or/-blindness indicated in eq/. /(/5/./1/4/)/,but are arranged in / a v or space to b e aligned with the relev an tY uk a w a matrices in just suc haw a ya st oa v oid large F CNC e/ects/. /4/0 /;; /6/0The alignmen tm o d e l s t ypically require rathersp ecial / a v or symmetries/. In an y case/, w e will not discuss these p ossibiliti es further/.In practice/, a giv en mo del for the origin of sup ersymmetry breaking ma ym a k e predic/-tions for the MSSM soft terms that are ev en stronger than eqs/. /(/5/./1/4/)/-/(/5/./1/6/)/. In the nextsection w e will discuss the ideas that go in to making suc h predictions/, b efore turning totheir implications for the MSSM sp ectrum in section /7/./4/0 /6 Origins of sup ersymmetry breaking/6/./1 Gener al c onsider ations for sup ersymmetry br e akingIn the MSSM/, sup ersymmetry breaking is simply in tro duced explicitly /.H o w ev er/, w eh a v eseen that the soft parameters cannot b e arbitrary /. In order to understand ho w patternslik e eqs/. /(/5/./1/4/)/, /(/5/./1/5/) and /(/5/./1/6/) can emerge/, it is necessary to consider mo dels in whic hsup ersymmetry is sp on taneously brok en/. By de/nition/, this means that the v acuum state j /0 iis not in v arian t under sup ersymmetry transformations/, so Q/ j /0 i /6/=/0 a n d Q y/_ / j /0 i /6/=/0 /. N o w/,in global sup ersymmetry /, the Hamiltonian op erator H can b e related to the sup ersymmetrygenerators through the algebra eq/. /(/3/./3/0/)/:H /= P /0/= /1 /4 /( Q/1 Q y/1 /+ Q y/1 Q/1 /+ Q/2 Q y/2 /+ Q y/2 Q/2 /) /: /(/6/./1/)If sup ersymmetry is un brok en in the v acuum state/, it follo ws that H j /0 i /= /0 and the v acuumhas zero energy /. Con v ersely /, if sup ersymmetry is sp on taneously brok en in the v acuum state/,then the v acuum m ust ha v ep o s i t i v e energy /, sinceh /0 j H j /0 i /= /1 /4 /k Q/1 j /0 ik /2/+ k Q y/1 j /0 ik /2/+ k Q/2 j /0 ik /2/+ k Q y/2 j /0 ik /2 //> /0 /(/6/./2/)if the Hilb ert space is to ha v e p ositiv e norm/. If spacetime/-dep enden t e/ects and fermioncondensates can b e neglected/, then h /0 j H j /0 i /= h /0 j V j /0 i /,w h e r e V is the scalar p oten tial ineq/. /(/3/./7/5/)/. Therefore sup ersymmetry will b e sp on taneously brok en if Fi and//or D ado es notv anish in the ground state/. Note that if an y state exists in whic ha l l Fi and D av anish/, thenit will ha v e zero energy /, implying that sup ersymmetry cannot b e sp on taneously brok en inthe true ground state/. Therefore the w a yt o a c hiev es p o n taneous sup ersymmetry breakingis to lo ok for mo dels in whic h the equations Fi /= /0 and D a/= /0 cannot b e sim ultaneouslysatis/ed for any v alues of the /elds/.Sup ersymmetry breaking with non/-zero D /-terms can b e ac hiev ed through the F a y et/-Iliop oulos mec hanism/. /6/1If the gauge symmetry includes a U /(/1/) factor/, then one can in tro ducea term linear in the corresp onding auxiliary /eld of the gauge sup erm ultiplet/:LF a y et /; Iliop oulos /= /D /(/6/./3/)where / is a constan t parameter with dimensions of /(mass/) /2/. This term is gauge/-in v arian tand sup ersymmetric b y itself/. /[Note that the sup ersymmetry transformation /D in eq/. /(/3/./5/9/)is a total deriv ativ ef o r a U /(/1/) gauge symmetry /./] If w e include it in the lagrangian/, then Dma y get a non/-zero VEV/, dep ending on the other in teractions of the scalar /elds that arec harged under the U /(/1/)/. T o see this/, w e can write the relev an t part of the scalar p oten tialusing eqs/. /(/3/./5/4/) and /(/3/./7/2/) asV /= /1 /2 D /2/; /D /+ gD Xi qi / / i/i /(/6/./4/)where the qi are the c harges of the scalar /elds /i under the U /(/1/) gauge group in question/.The presence of the F a y et/-Iliop oulos term mo di/es the equation of motion eq/. /(/3/./7/4/) toD /= / /; g Xi qi / / i/i /: /(/6/./5/)No w supp ose that the scalar /elds /i ha v e other in teractions /(suc h as large sup erp oten tialmass terms/) whic hp r e v en t them from getting VEVs/. Then the auxiliary /eld D will b e/4/1 forced to get a VEV equal to / /, and sup ersymmetry will b e brok en/. This mec hanismcannot w ork for non/-ab elian gauge groups/, ho w ev er/, since the analog of eq/. /(/6/./3/) w ould notb e gauge/-in v arian t/.In the MSSM/, one can imagine that the D term for U /(/1/)Y has a F a y et/-Iliop oulos termwhic h is the principal source of sup ersymmetry breaking/. Unfortunately /,t h i s w ould b e animmediate disaster/, b ecause at least some of the squarks and sleptons w ould just get non/-zeroVEVs /(breaking color/, electromagnetism/, and//or lepton n um b er/, but not sup ersymmetry/) inorder to satisfy eq/. /(/6/./5/)/, b ecause they do not ha v e sup erp oten tial mass terms/. This meansthat a F a y et/-Iliop oulos term for U /(/1/)Y m ust b e sub dominan t compared to other sources ofsup ersymmetry breaking in the MSSM/, if not absen t altogether/. One could also attemptto trigger sup ersymmetry breaking with a F a y et/-Iliop oulos term for some other U /(/1/) gaugesymmetry whic hi s a s y et unkno wn b ecause it is sp on taneously brok en at a v ery high massscale or b ecause it do es not couple to the Standard Mo del particles/. Ho w ev er/, if this is theultimate source for sup ersymmetry breaking/, it pro v es di/cult to giv e appropriate masses toall of the MSSM particles/, esp ecially the gauginos/. In an y case/, w e will not discuss D /-termbreaking as the ultimate origin of sup ersymmetry violation an y further/, although it ma ynot b e ruled out/. /6/2Mo dels where sup ersymmetry breaking is due to non/-zero F /-terms/, called O/'Raifear/-taigh mo dels/, /6/3ma yh a v eb r i g h ter phenomenological prosp ects/. The idea is to pic k a setof c hiral sup erm ultiplets /i / /( /i /;;/ i /;;Fi /) and a sup erp oten tial W in suc ha w a y that theequations Fi /= /; /W //=/ / / i/=/0 h a v en o s i m ultaneous solution/. Then V /= Pi j Fi j /2willha v e to b e p ositiv e at its minim um/, ensuring that sup ersymmetry is brok en/. The simplestexample whic h do es this has three c hiral sup erm ultiplets withW /= /; k //1 /+ m //2 //3 /+ y /2 //1 / /2/3 /: /(/6/./6/)Note that W con tains a linear term/, with k ha ving dimensions of /(mass/) /2/. This is onlyp ossible if //1 is a gauge singlet/. In section /3 w ec heated and did not men tion suc haterm/, b ecause w e knew that the MSSM con tains no suc h singlet c hiral sup erm ultiplet/.Nev ertheless/, it should b e clear from retracing the deriv ation in section /3/./2 that suc haterm is allo w ed if a gauge/-singlet c hiral sup erm ultiplet is added to the theory /. In fact/, alinear term is absolutely necessary to ac hiev e F /-term breaking/, since otherwise setting all/i /= /0 will alw a ys giv e a sup ersymmetric global minim um with all Fi /= /0/. Without loss ofgeneralit y /,w e can c ho ose k /, m /, and y to b e real and p ositiv e/( b y a phase rotation of the/elds/)/. The scalar p oten tial follo wing from eq/. /(/6/./6/) isV /= j F/1 j /2/+ j F/2 j /2/+ j F/3 j /2/;; /(/6/./7/)F/1 /= k /; y /2 / / /2/3 /;; F/2 /= /; m/ //3 /;; F/3 /= /; m/ //2 /; y/ //1 / //3 /: /(/6/./8/)Clearly /, F/1 /= /0 and F/2 /= /0 are not compatible/, so sup ersymmetry m ust indeed b e brok en/.If m /2/>y k /(whic hw e assume from no w on/)/, then it is easy to sho w that the absoluteminim um of the p oten tial is at //2 /= //3 /=/0w i t h //1 undetermined/, so F/1 /= k and V /= k /2at the minim um of the p oten tial/. The fact that //1 is undetermined is an example of a /\/ atdirection/" in the scalar p oten tial/;; this is a common feature of sup ersymmetric mo dels/. y yMore generally /, /\/ at directions/" are non/-compact lines and surfaces in the space of scalar /elds along whic hthe scalar p oten tial v anishes/. The classical scalar p oten tial of the MSSM w ould ha v e man y / at directions ifsup ersymmetry w ere not brok en/./4/2 If w e prescien tly c ho ose to expand V around //1 /= /0/, the mass sp ectrum of the theoryconsists of /6 real scalars with tree/-lev el squared masses/0 /;; /0 /;;m /2/;;m /2/;;m /2/; yk /;; m /2/+ yk /: /(/6/./9/)Mean while/, there are /3 W eyl fermions with masses/0 /;;m /;;m /: /(/6/./1/0/)The non/-degeneracy of scalars and fermions is a clear sign that sup ersymmetry has b eensp on taneously brok en/. The /0 eigen v alues in eqs/. /(/6/./9/) and /(/6/./1/0/) corresp ond to the complexscalar //1 and its fermionic partner / /1 /.H o w ev er/, //1 and / /1 ha v e di/eren t reasons for b eingmassless/. The masslessness of //1 corresp onds to the existence of the / at direction/, since an yv alue of //1 giv es the same energy at tree/-lev el/. This / at direction is an acciden tal feature ofthe classical scalar p oten tial/, and in this case it is remo v ed /(/\lifted/"/) b y quan tum corrections/.This can b e seen b y computing the Coleman/-W ein b erg one/-lo op e/ectiv ep o t e n tial/. /6/4Aftersome calculation/, one /nds the result that the global minim um is indeed /xed at //1 /= //2 /=//3 /= /0/, with the complex scalar //1 receiving a small p ositiv e/-de/nite /(mass/) /2equal tom /2//1 /= /1 /3/2 / /2 /"/ym /4 k /+ y /3k /ln /m /2/+ yk m /2/; yk //+/2 y /2m /2 /ln /[/1 /; y /2k /2 m /4 /] /; /1 / /#/: /(/6/./1/1/)/[In the limit yk / m /2/, this reduces to m /2//1 /= y /4k /2/= /(/4/8 / /2m /2/)/./] In con trast/, the W eyl fermion/ /1 remains exactly massless b ecause of a general feature of all mo dels with sp on taneouslybrok en sup ersymmetry /. T o understand this/, recall that the sp on taneous breaking of an yglobal symmetry alw a ys giv es rise to a massless Nam bu/-Goldstone mo de with the samequan tum n um b ers as the brok en symmetry generator/. In the case of sup ersymmetry /, thebrok en generator is the fermionic c harge Q/ /, so the Nam bu/-Goldstone particle m ust b e amassless neutral W eyl fermion called the goldstino /. In the O/'Raifeartaigh mo del example/,/ /1 is the goldstino b ecause it is the fermionic partner of the auxiliary /eld F/1 whic h got aVEV/. /(W e will pro v e these statemen ts in a more general con text in section /6/./2/./)The O/'Raifeartaigh sup erp oten tial determines the mass scale of sup ersymmetry breakingp F/1 in terms of a dimensionful parameter k whic h is put in b y hand/. This is somewhat adho c /, since p k will ha v et o b em uc h less than MP in order to giv e the righ t order of magnitudefor the MSSM soft terms/. W ew ould lik et o h a v e a mec hanism whic h can instead generatesuc h scales naturally /. This can b e done in mo dels of dynamical sup ersymmetry breaking/.In suc h theories/, the small /(compared to MP /) mass scales asso ciated with sup ersymmetrybreaking arise b y dimensional transm utation/. In other w ords/, they generally feature a newasymptotically/-free non/-Ab elian gauge symmetry with a gauge coupling g whic h is p ertur/-bativ ea t MP and whic h gets strong in the infrared at some smaller scale / / e /; /8 / /2/= j b j g /2/0MP /,where g/0 is the running gauge coupling at MP with b eta function /;j b j g /3/= /1/6 / /2/. Just as inQCD/, it is p erfectly natural for / to b e man y orders of magnitude b elo w the Planc k scale/.Sup ersymmetry breaking ma y then b e b est describ ed in terms of the e/ectiv e dynamics ofthe strongly coupled theory /. One p ossibilit y is that the auxiliary F /eld for a comp ositec hiral sup erm ultiplet /(built out of the fundamen tal /elds whic h transform under the newstrongly/-coupled gauge group/) obtains a VEV/. Constructing mo dels whic h actually breaksup ersymmetry in an acceptable w a y is a highly non/-trivial business/;; for more informationw e refer the reader to Ref/. /6/5The one thing that is no w clear ab out sp on taneous sup ersymmetry breaking /(dynamicalor not/) is that it requires us to extend the MSSM/. The ultimate sup ersymmetry/-breaking/4/3 (Hidden sector)(Visible sector)Supersymmetry breaking origin MSSMFlavor-blind interactionsFigure /1/4/: The presumed sc hematic structure for sup ersymmetry breaking/.order parameter cannot b elong to an yo f t h es u p e r m ultiplets of the MSSM/;; a D /-term VEVfor U /(/1/)Y do es not lead to an acceptable sp ectrum/, and there is no candidate gauge/-singletwhose F /-term could dev elop a VEV/. Therefore one m ust ask what e/ects ar e resp onsiblefor sp on taneous sup ersymmetry breaking/, and ho w sup ersymmetry breakdo wn is /\comm u/-nicated/" to the MSSM particles/. It is v ery di/cult to ac hiev e the latter in a phenomeno/-logically viable w a yw orking only with renormalizable in teractions at tree/-lev el/. First/, it isproblematic to giv e masses to the MSSM gauginos/, b ecause sup ersymmetry do es not allo w/(scalar/)/-/(gaugino/)/-/(gaugino/) couplings whic h could turn in to gaugino mass terms when thescalar gets a VEV/. Second/, at least some of the MSSM squarks and sleptons w ould ha v et ob e unacceptably ligh t/, and should ha v e b een disco v ered already /. This can b e understo o d ina general w a y from the existence of a sum rule whic hg o v erns the tree/-lev el squared massesof scalars and c hiral fermions in theories with sp on taneous sup ersymmetry breaking/:T r/[ M /2real scalars /]/= /2 T r/[ M /2c hiral fermions /] /: /(/6/./1/2/)If sup ersymmetry w ere not brok en/, then eq/. /(/6/./1/2/) w ould follo w immediately from thedegeneracy of complex scalars /[with t w o real scalar comp onen ts/, hence the factor of /2/]and their W eyl fermion sup erpartners/. Ho w ev er/, eq/. /(/6/./1/2/) still holds at tree/-lev el whensup ersymmetry is brok en sp on taneously b y F /-terms and D /-terms/, as one can v erify ingeneral b y explicitly computing the /(mass/) /2matrices for arbitrary v alues of the /elds/. zOne can easily see/, for example/, that with the O/'Raifeartaigh sp ectrum of eqs/. /(/6/./9/) and/(/6/./1/0/)/, the sum rule eq/. /(/6/./1/2/) is indeed satis/ed/. This sum rule seems to b e bad news fora phenomenologically viable mo del/, b ecause the masses of all of the MSSM c hiral fermionsare already kno wn to b e small /(except for the top quark and the higgsinos/)/. Ev en if w ecould succeed in ev ading this/, there is no reason wh y the resulting MSSM soft terms in thist yp e of mo del should satisfy conditions lik e eqs/. /(/5/./1/4/) or /(/5/./1/5/)/.F or these reasons/, w e exp ect that the MSSM soft terms arise indirectly or radiativ ely /,rather than from tree/-lev el renormalizable couplings to the sup ersymmetry/-breaking orderparameters/. Sup ersymmetry breaking eviden tly o ccurs in a /\hidden sector/" of particleswhic hh a v e no /(or only v ery small/) direct couplings to the /\visible sector/" c hiral sup er/-m ultiplets of the MSSM/. Ho w ev er/, the t w o sectors do share some in teractions whic h areresp onsible for mediating sup ersymmetry breaking from the hidden sector to the visible sec/-tor/, where they app ear as calculable soft terms/. /(See Fig/. /1/4/./) In this scenario/, the tree/-lev elsum rule eq/. /(/6/./1/2/) need not hold for the visible sector /elds/, so that a phenomenologicallyviable sup erpartner mass sp ectrum is in principle ac hiev able/. As a b on us/, if the mediatingin teractions are / a v or/-blind/, then the soft terms app earing in the MSSM ma y automaticallyob ey conditions lik e eqs/. /(/5/./1/4/)/, /(/5/./1/5/) and /(/5/./1/6/)/.There are t w o main comp eting prop osals for what the mediating in teractions migh tb e /.The /rst /(and historically the more p opular/) is that they are gra vitational/. More precisely /, zThis assumes only that the trace of the U /(/1/) c harges o v er all c hiral sup erm ultipl ets in the theory v anishes/(T r/[ T a/] /= /0/)/. This holds for U /(/1/)Y in the MSSM and more generally for an y non/-anomalous gauge symmetry /./4/4 they are asso ciated with the new ph ysics/, including gra vit y /, whic he n ters at the Planc k scale/.In this gr avity/-me diate ds u p ersymmetry br e aking scenario/, if sup ersymmetry is brok en in thehidden sector b y a VEV h F i /, then the soft terms in the visible sector should b e roughly ofordermsoft / h F i MP /;; /(/6/./1/3/)b y dimensional analysis/. This is b ecause w e kno w that msoft m ust v anish in the limith F i/! /0 where sup ersymmetry is un brok en/, and also in the limit MP /!/1 /(corresp onding toGNewton /! /0/) in whic hg r a vit y b ecomes irrelev an t/. F or msoft of order a few h undred GeV/, onew ould therefore exp ect that the scale asso ciated with the origin of sup ersymmetry breakingin the hidden sector should b e roughly p h F i/ /1/0 /1/0or /1/0 /1/1GeV/. Another p ossibilit y is thatthe sup ersymmetry breaking order parameter is a gaugino condensate h /0 j / a/ bj /0 i /= / ab/ /3/6/=/0/. If the comp osite /eld / a/ bis part of an auxiliary /eld F for some /(p erhaps comp osite/)c hiral sup er/eld/, then b y dimensional analysis w e exp ect sup ersymmetry breaking soft termsof ordermsoft / / /3 M /2P /;; /(/6/./1/4/)with/, e/ectiv ely /, h F i/ / /3/= MP /. In that case/, the scale asso ciated with dynamical sup er/-symmetry breaking should b e more lik e/ / /1/0 /1/3GeV/.The second main p ossibilit y is that the / a v or/-blind mediating in teractions for sup ersym/-metry breaking are the ordinary electro w eak and QCD gauge in teractions/. In this gauge/-me diate d sup ersymmetry br e aking scenario/, the MSSM soft terms arise from lo op diagramsin v olving some messenger particles/. The messengers couple to a sup ersymmetry/-breakingVEV h F i /, and also ha v e SU /(/3/)C / SU /(/2/)L / U /(/1/)Y in teractions whic hp r o vide a link to theMSSM/. Then/, using dimensional analysis/, one estimates for the MSSM soft termsmsoft / /a /4 / h F i Mmess /(/6/./1/5/)where the /a /= /4 / is a lo op factor for F eynman diagrams in v olving gauge in teractions/, andMmess is a c haracteristic scale of the masses of the messenger /elds/. So if Mmess and p h F iare roughly comparable/, then the scale of sup ersymmetry breaking can b e as lo w as ab outp h F i/ /1/0 /4or /1/0 /5GeV /(m uc hl o w er than in the gra vit y/-mediated case/!/) to giv e msoft ofthe righ t order of magnitude/./6/./2 The goldstino and the gr avitinoAs explained in the previous section/, the sp on taneous breaking of global sup ersymmetryimplies the existence of a massless W eyl fermion/, the goldstino/. In the particular case of theO/'Raifeartaigh mo del/, the goldstino w as iden ti/ed to b e / /1 /. More generally /,w em i g h t exp ectthat in the case of F /-term or D /-term breaking/, the goldstino is the fermionic comp onen to fthe sup erm ultiplet whose auxiliary /eld obtains a VEV/.Let us mak e this more precise b y actually pro ving that the goldstino exists and/, in thepro cess/, iden tifying it/. This is actually rather easy /. Consider a general sup ersymmetricmo del with b oth gauge and c hiral sup erm ultiplets as in section /3/. The fermionic degrees of/4/5 freedom consist of gauginos /( / a/) and c hiral fermions /( / i /)/. After some of the scalar /elds inthe theory obtain VEVs/, the fermion mass matrix will ha v e the form/:Mfermion /= //0 p /2 ga /( h / /i T a/) ip /2 ga /( h / /i T a/) jh W iji //(/6/./1/6/)in the /( / a/;;/ i /) basis/. /[The o//-diagonal en tries in this matrix come from the second line ineq/. /(/3/./7/2/)/, and the lo w er righ te n try can b e seen in eq/. /(/3/./4/6/)/./] No ww e simply note thatMfermion annihilates the v ectoreG /= /h D ai /= p /2h Fi i //: /(/6/./1/7/)The /rst ro wo f Mfermion annihilates eG b y virtue of the requiremen t eq/. /(/3/./7/3/) that thesup erp oten tial is gauge in v arian t/, and the second ro w annihilates eG b ecause of the conditionh /@ V /=/@ /i i /= /0 whic hm ust b e satis/ed at the minim um of the scalar p oten tial/. Eq/. /(/6/./1/7/)is prop ortional to the goldstino w a v efunction/;; it is non/-trivial if and only if at least one ofthe auxiliary /elds has a VEV/, breaking sup ersymmetry /.S o w eh a v e pro v en that if globalsup ersymmetry is sp on taneously brok en/, then the goldstino exists and has zero mass/, andthat its comp onen ts among the v arious fermions in the theory are just prop ortional to thecorresp onding auxiliary /eld VEVs/.W e can deriv e another v ery imp ortan tp r o p e r t y of the goldstino b y considering the formof the conserv ed sup ercurren t eq/. /(/3/./7/6/)/. Supp ose for simplicit y xthat the non/-v anishingauxiliary /eld VEV is h F i and that its goldstino sup erpartner is eG /. Then the sup ercurren tconserv ation equation tells us that/0/= /@/ J // /= i h F i /( / //@/ eG y/)/ /+ /@/ j // /+ /:/:/: /(/6/./1/8/)where j // is the part of the sup ercurren t whic hi n v olv es all of the other sup erm ultiplets/, andthe ellipses represen t other con tributions of the goldstino sup erm ultiplet to /@/ J // whic hw ecan ignore/. /[The /rst term in eq/. /(/6/./1/8/) comes from the second term in eq/. /(/3/./7/6/)/, using the equation of motion Fi /= /; W /i for the goldstino/'s auxiliary /eld/./] This equation of motionfor the goldstino /eld allo ws us to write an e/ectiv e lagrangianLgoldstino /= /; i eG y / //@/ eG /; /1 h F i /( eG /@/ j //+c /: c /: /) /(/6/./1/9/)whic h describ es the in teractions of the goldstino with all of the other fermion/-b oson pairs/. /6/6In particular/, since j // /=/( / / / // i /)/ /@/ / / i/; /(/1 /= /2 p /2 /) / / / // // y aF a// /+ /:/:/: /, there are goldstino/-scalar/-c hiral fermion and goldstino/-gaugino/-gauge b oson v ertices as sho wn in Fig/. /1/5/. Sincethis deriv ation dep ends only on sup ercurren t conserv ation/, eq/. /(/6/./1/9/) holds indep enden tly ofthe details of ho w sup ersymmetry breaking is comm unicated from h F i to the MSSM sector/elds /( /i /;;/ i /) and /( / a/;;A a/)/. It ma y app ear strange at /rst that the in teraction terms ineq/. /(/6/./1/9/) get larger as h F i go es to zero/. Ho w ev er/, the in teraction term eG/@/ j /con tains t w oderiv ativ es whic h turn out to alw a ys giv e a kinematic factor prop ortional to the /(mass/) /2di/erence of the sup erpartners when they are on/-shell/, i/.e/. m /2/i /; m /2/ i and m /2/ /; m /2A forFigs/. /1/5a and /1/5b resp ectiv ely /. These can b e non/-zero only b y virtue of sup ersymmetrybreaking/, so they m ust also v anish as h F i/! /0/, and the in teraction is w ell/-de/ned in that xMore generally /, if sup ersymmetry is sp on taneously brok en b y VEVs for sev eral auxiliary /elds Fi and D a/,then one should mak e the replacemen t h F i/! /( Pi jh Fi ij /2/+ /1 /2 Pa h D ai /2/) /1 /= /2ev erywhere in the follo wing/./4/6 φ Gψ (a)λ GA (b)Figure /1/5/: Goldstino//gra vi tin o in teractions with sup erpartner pairs /( //;; / /)a n d /( / a/;;A a/)/.limit/. Nev ertheless/, for /xed v alues of m /2/i /; m /2/ i and m /2/ /; m /2A /, the in teraction term ineq/. /(/6/./1/9/) can b e phenomenologically imp ortan ti f h F i is not to o large/. /6/6 /;; /6/7 /;; /6/8 /;; /6/9The ab o v e remarks apply to the breaking of global sup ersymmetry /.H o w ev er/, when onetak es in to accoun tg r a vit y /, sup ersymmetry m ust b e a lo cal symmetry /. This means thatthe spinor parameter / /whic h /rst app eared in section /3/./1 is no longer a constan t/, butcan v ary from p oin tt o p o i n t in spacetime/. The resulting lo cally sup ersymmetric theory iscalled sup er gr avity /. /7/0 /;; /7/1It necessarily uni/es the spacetime symmetries of ordinary generalrelativit y with lo cal sup ersymmetry transformations/. In sup ergra vit y /, the spin/-/2 gra vitonhas a spin/-/3///2 fermion sup erpartner called the gra vitino/, whic hw e will denote e/ // /. Thegra vitino has o dd R /-parit y/( PR /= /; /1/)/, as can b e seen from the de/nition eq/. /(/5/./1/0/)/. Itcarries b oth a v ector index /( / /) and a spinor index /( / /)/, and transforms inhomogeneouslyunder lo cal sup ersymmetry transformations/:/ e/ // /= /; /@/ / //+ /:/:/: /(/6/./2/0/)Th us the gra vitino should b e though t of as the /\gauge/" particle of lo cal sup ersymmetrytransformations /[compare eq/. /(/3/./5/2/)/]/. As long as sup ersymmetry is un brok en/, the gra vitonand the gra vitino are b oth massless/, eac h with t w o spin helicit y states/. Once sup ersymmetryis sp on taneously brok en/, the gra vitino acquires a mass b y absorbing /(/\eating/"/) the goldstino/,whic h b ecomes its longitudinal /(helicit y / /1 /= /2/) comp onen ts/. This is called the sup er/-Higgsmec hanism/. It is en tirely analogous to the ordinary Higgs mec hanism for gauge theories/,b y whic ht h e W /and Z /0gauge b osons in the Standard Mo del gain mass b y absorbingthe Nam bu/-Goldstone b osons asso ciated with the sp on taneously brok en electro w eak gaugein v ariance/. The coun ting w orks/, b ecause the massiv e spin/-/3///2 gra vitino no w has four helicit ystates/, of whic ht w ow ere originally assigned to the w ould/-b e goldstino/. The gra vitino massis traditionally called m/3 /= /2 /, and in the case of F /-term breaking can b e estimated as /7/2m/3 /= /2 / h F i MP /;; /(/6/./2/1/)This follo ws simply from dimensional analysis/, since m/3 /= /2 m ust v anish in the limits thatsup ersymmetry is restored /( h F i/! /0/) and that gra vit y is turned o/ /( MP /!/1 /)/. Equa/-tion /(/6/./2/1/) means that one has v ery di/eren t exp ectations for the mass of the gra vitino ingra vit y/-mediated and in gauge/-mediated mo dels/, b ecause they usually mak ev ery di/eren tpredictions for h F i /.In the gra vit y/-mediated sup ersymmetry breaking case/, the gra vitino mass is comparableto the masses of the MSSM sparticles /[compare eqs/. /(/6/./1/3/) and /(/6/./2/1/)/]/. Therefore m/3 /= /2 isexp ected to b e at least /1/0/0 GeV or so/. Its in teractions will b e of gra vitational strength/,so the gra vitino will not pla ya n y role in collider ph ysics/, but it can b e a v ery imp ortan tconsideration in cosmology /. /7/3If it is the LSP /, then it is stable and its primordial densit ycould easily exceed the critical densit y /, causing the univ erse to b ecome matter/-dominated/4/7 to o early /.E v en if it is not the LSP /, the gra vitino can cause problems unless its densit yi sdiluted b y in/ ation at late times/, or it deca ys su/cien tly rapidly /.In con trast/, gauge/-mediated sup ersymmetry breaking mo dels predict that the gra vitinois m uc h ligh ter than the MSSM sparticles as long as Mmess / MP /. This can b e seenb y comparing eqs/. /(/6/./1/5/) and /(/6/./2/1/)/. The gra vitino is almost certainly the LSP in thiscase/, and all of the MSSM sparticles will ev en tually deca yi n to /nal states that includeit/. Naiv ely /, one migh t exp ect that these deca ys are extremely slo w/. Ho w ev er/, this is notnecessarily true/, b ecause the gra vitino inherits the non /-gra vitational in teractions of thegoldstino it has absorb ed/. This means that the gra vitino/, or more precisely its longitudinal/(goldstino/) comp onen ts/, can pla y an imp ortan t role in collider ph ysics exp erimen ts/. Themass of the gra vitino can generally b e ignored for kinematic purp oses/, as can its transv erse/(helicit y / /3 /= /2/) comp onen ts whic h really do ha v e only gra vitational in teractions/. Thereforein collider phenomenology discussions one ma yi n terc hangeably use the same sym bo l eG forthe goldstino and for the gra vitino of whic h it is the longitudinal /(helicit y / /1 /= /2/) part/. Byusing the e/ectiv e lagrangian eq/. /(/6/./1/9/)/, one can compute that the deca yr a t e o f a n y sparticleeX in to its Standard Mo del partner X plus a gra vitino//goldstino eG is giv en b y/;/( eX /! X eG /)/= m /5eX /1/6 / h F i /2 / /1 /; m /2X m /2eX /!/4/: /(/6/./2/2/)This corresp onds to either Fig/. /1/5a or /1/5b/, with /( eX/;; X /)/= /( //;; / /)o r /( //;; A /) resp ectiv ely /.One factor /(/1 /; m /2X /=m /2eX /) /2came from the deriv ativ es in the in teraction term in eq/. /(/6/./1/9/)ev aluated for on/-shell /nal states/, and another suc h factor comes from the kinematic phasespace in tegral with m/3 /= /2 / meX /;;mX /.If the sup erm ultiplet con taining the goldstino and h F i has canonically/-normalized kineticterms/, and one requires the tree/-lev el v acuum energy to v anish/, then the estimate eq/. /(/6/./2/1/)ma y b e sharp ened tom/3 /= /2 /= h F i p /3 MP /: /(/6/./2/3/)In that case/, one can rewrite eq/. /(/6/./2/2/) as/;/( eX /! X eG /)/= m /5eX /4/8 /M /2P m /2/3 /= /2 / /1 /; m /2X m /2eX /!/4/;; /(/6/./2/4/)and this is ho w the form ula is sometimes presen ted b y those who prefer to tak e eq/. /(/6/./2/3/)seriously /. Note that the deca y width is larger for smaller h F i /, or equiv alen tly for smallerm/3 /= /2 /, if the other masses are /xed/. If eX is a mixture of sup erpartners of di/eren t StandardMo del particles X /, then eq/. /(/6/./2/2/) should b e m ultiplied b y a suppression factor equal to thesquare of the cosine of the appropriate mixing angle/. If meX is of order /1/0/0 GeV or more/,and p h F i /</ few / /1/0 /6GeV /[corresp onding to m/3 /= /2 less than roughly /1 k eV according toeq/. /(/6/./2/3/)/]/, then the deca y eX /! X eG can o ccur quic kly enough to b e observ ed in a mo derncollider detector/. This giv es rise to some v ery in teresting phenomenological signatures/,whic hw e will discuss further in sections /8/./5 and /9/.W en o w turn to a sligh tly more systematic analysis of the w a y in whic h the MSSM softterms arise/, considering in turn the gra vit y/-mediated and gauge/-mediated scenarios/./4/8 /6/./3 Gr avity/-me diate ds u p ersymmetry br e aking mo delsThe de/ning feature of these mo dels is that the hidden sector of the theory comm unicateswith our MSSM only /(or dominan tly/) through gra vitational/-strength in teractions/. In ane/ectiv e /eld theory format/, this means that the sup ergra vit y lagrangian con tains nonren/-ormalizable terms whic h comm unicate b et w een the t w o sectors and whic h are suppressed b ypo w ers of the Planc k mass/, since the gra vitational coupling is prop ortional to /1 /= MP /. Thesewill includeLNR /= /; /1 MP FX Xa /1 /2 fa / a/ a/+c /: c /:/; /1 M /2P FX F /X k ij /i / / j/; /1 MP FX /( /1 /6 y /0 ij k/i /j /k /+ /1 /2 / /0 ij/i /j /)/+ c /: c /: /(/6/./2/5/)where FX is the auxiliary /eld for a c hiral sup erm ultiplet X in the hidden sector/, and /i and/ aare the scalar and gaugino /elds in the MSSM/. By themselv es/, the terms in eq/. /(/6/./2/5/)are not sup ersymmetric/, but it is p ossible to sho w that they are part of a nonrenormalizablesup ersymmetric lagrangian /(see App endix/) whic hc o n tains other terms that w em a y ignore/.No w if one assumes that h FX i/ /1/0 /1/0or /1/0 /1/1GeV/, then LNR will giv e us nothing other thana lagrangian of the form Lsoft in eq/. /(/4/./1/)/, with MSSM soft terms of order a few h undredGeV/. /[Note that terms of the form Lma yb e soft in eq/. /(/4/./2/) do not arise/./]The dimensionless parameters fa /, k ij /, y /0 ij kand / /0 ijin LNR are to b e determined b y theunderlying theory /. This is a di/cult en terprise in general/, but a dramatic simpli/cationo ccurs if one assumes a /\minimal/" form for the normalization of kinetic terms and gaugein teractions in the full/, nonrenormalizable sup ergra vit y lagrangian /(see App endix/)/. In thatcase/, one /nds that there is a common fa /= f for the three gauginos/;; k ij /= k/ ij is the samefor all scalars/;; and the other couplings are prop ortional to the corresp onding sup erp oten tialparameters/, so that y /0 ij k/= /y ij kand / /0 ij/= // ijwith univ ersal dimensionless constan ts /and / /. Then one /nds that the soft terms in L MSSMsoft can all b e written in terms of just fourparameters/:m/1 /= /2 /= f h FX i MP /;; m /2/0 /= k jh FX ij /2 M /2P /;; A/0 /= / h FX i MP /;; B/0 /= / h FX i MP /: /(/6/./2/6/)In terms of these/, one can write for the parameters app earing in eq/. /(/5/./1/1/)/:M/3 /= M/2 /= M/1 /= m/1 /= /2 /;; /(/6/./2/7/)m /2Q /= m /2 u /= m /2 d /= m /2L /= m /2 e /= m /2/0 /1 /;; m /2Hu /= m /2Hd /= m /2/0 /;; /(/6/./2/8/)au /= A/0 yu /;; ad /= A/0 yd /;; ae /= A/0 ye /;; /(/6/./2/9/)b /= B/0 //: /(/6/./3/0/)It is a matter of some con tro v ersy whether the assumptions going in to this parameterizationare completely w ell/-motiv ated on purely theoretical grounds/, /{but from a phenomenologicalp ersp ectiv e they are clearly v ery nice/. This framew ork successfully ev ades the most danger/-ous t yp es of F CNC and CP/-violation as discussed in section /5/./4/. In particular/, eqs/. /(/6/./2/8/) /{The familiar / a v or/-blindnes s of gra vitational in teractions expressed in Einstein/'s equiv alence principle do esnot/, b y itself/, tell us an ything ab out the form of eq/. /(/6/./2/5/)/./4/9 and /(/6/./2/9/) are just stronger v ersions of eqs/. /(/5/./1/4/) and /(/5/./1/5/)/, resp ectiv ely /.I f m/1 /= /2 /, A/0 andB/0 all ha v e the same complex phase/, then eq/. /(/5/./1/6/) will also b e satis/ed/.Equations /(/6/./2/7/)/-/(/6/./3/0/) also ha v e the virtue of b eing highly predictiv e/. /[Of course/,eq/. /(/6/./3/0/) is con ten t/-free unless one can relate B/0 to the other parameters in some non/-trivial w a y /./] As discussed in section /5/./4/, they should b e applied as R G b oundary conditionsat the scale MP /. The R Ge v olution of the soft parameters do wn to the electro w eak scalewill then allo w us to predict the en tire MSSM sp ectrum in terms of just /v e parametersm/1 /= /2 /, m /2/0 /, A/0 /, B/0 /,a n d / /(plus the already/-measured gauge and Y uk a w a couplings of theMSSM/)/. In practice/, the appro ximation is usually made of starting this R G running fromthe uni/cation scale MU / /2 / /1/0 /1/6GeV instead of MP /. The reason for this is that theapparen t uni/cation of gauge couplings giv es us a strong hin t that w e kno w somethingab out ho w the R G equations are b eha ving up to MU /, but giv es us little guidance ab outwhat to exp ect at scales b et w een MU and MP /. The error made in neglecting these e/ects isprop ortional to a lo op suppression factor times ln/( MP /= MU /) and can b e partially absorb edin to a rede/nition of m /2/0 /, m/1 /= /2 /, A/0 and B/0 /, but in some cases can lead to imp ortan t e/ects/. /7/4The framew ork describ ed in the ab o v e few paragraphs has b een the sub ject of the bulk ofphenomenological studies of sup ersymmetry /. It is sometimes referred to as the minimalsup er gr avity or sup er gr avity/-inspir e d scenario for the soft terms/. A few examples of theman y useful n umerical R G studies of the MSSM sp ectrum whic hh a v e b een p erformed inthis framew ork can b e found in Ref/. /7/5P articular mo dels of gra vit y/-mediated sup ersymmetry breaking can b e ev en more pre/-dictiv e/, relating some of the parameters m/1 /= /2 /, m /2/0 /, A/0 and B/0 to eac h other and to the massof the gra vitino m/3 /= /2 /.F or example/, three p opular kinds of mo dels for the soft terms are/:/ Dilaton/-dominated/: /7/6m /2/0 /= m /2/3 /= /2 /;; m/1 /= /2 /= /; A/0 /= p /3 m/3 /= /2 /./ P olon yi/: /7/7m /2/0 /= m /2/3 /= /2 /;; A/0 /=/( /3 /; p /3 /) m/3 /= /2 /;; m/1 /= /2 /= O /( m/3 /= /2 /)/./ /\No/-scale/"/: /7/8m/1 /= /2 / m/0 /;;A/0 /;;m/3 /= /2 /.The dilaton/-dominated scenario arises in a particular limit of sup erstring theory /. Whileit app ears to b e highly predictiv e/, it can easily b e generalized in other limits/. /7/9The P olon yimo del has the adv an tage of b eing the simplest p ossible mo del for sup ersymmetry breakingin the hidden sector/, but it is rather ad ho c a n dd o e s n o ts e e m t oh a v e a sp ecial place ingrander sc hemes lik e sup erstrings/. The /\no/-scale/" limit ma y arise in a lo w/-energy limitof sup erstrings in whic h the gra vitino mass scale is undetermined at tree/-lev el /(hence thename/)/. It implies that only the gaugino masses are appreciable at MP /.A s w e will see insection /7/./1/, R Ge v olution feeds m/1 /= /2 in to the squark/, slepton and Higgs /(mass/) /2parameterswith su/cien t magnitude to giv e acceptable phenomenology at the electro w eak scale/. Morerecen tv ersions of the no/-scale scenario/, ho w ev er/, also can giv e signi/can t A/0 and m /2/0 atMP /. In man y cases B/0 can also b e predicted in terms of the other parameters/, but thisis quite sensitiv e to mo del assumptions/. F or phenomenological studies/, m/1 /= /2 /, m /2/0 /, A/0 andB/0 are usually just tak en to b e con v enien t indep endent parameters of our ignorance of thesup ersymmetry breaking mec hanism/./6/./4 Gauge/-me diate ds u p ersymmetry br e aking mo delsA strong alternativ e to the scenario describ ed in the previous section is pro vided b y thegauge/-mediated sup ersymmetry breaking prop osal/. /8/0 /;; /8/1The basic idea is to in tro duce somenew c hiral sup erm ultiplets/, called messengers/, whic h couple to the ultimate source of sup er/-symmetry breaking/, and whic h also couple indirectly to the /(s/)quarks and /(s/)leptons and/5/0 Higgs/(inos/) of the MSSM through the ordinary SU /(/3/)C / SU /(/2/)L / U /(/1/)Y gauge b osonand gaugino in teractions/. In this w a y /, the ordinary gauge in teractions/, rather than gra vit y /,are resp onsible for the app earance of soft terms in the MSSM/. There is still gra vitationalcomm unication b et w een the MSSM and the source of sup ersymmetry breaking/, of course/,but that e/ect is no w relativ ely unimp ortan t compared to the gauge in teraction e/ects/.In the simplest suc h mo del/, the messenger /elds are a set of c hiral sup erm ultiplets q /, q /,/` /, /` whic h transform under SU /(/3/)C / SU /(/2/)L / U /(/1/)Y asq / /( /3 /;; /1 /;; /; /1 /3 /)/;; q / /( /3 /;; /1 /;; /1 /3 /)/;; /` / /( /1 /;; /2 /;; /1 /2 /)/;; /` / /( /1 /;; /2 /;; /; /1 /2 /) /: /(/6/./3/1/)These sup erm ultiplets con tain messenger quarks / q /;;/ q and scalar quarks q/;; q and messengerleptons / /` /;;/ /` and scalar leptons /`/;; /` /. All of these particles m ust get v ery large masses so asnot to ha v e b een disco v ered already /. T h e y m a n a g et od o s ob y coupling to a gauge/-singletc hiral sup erm ultiplet S through a sup erp oten tial/:Wmess /= y/2 S/` /` /+ y/3 Sq q/: /(/6/./3/2/)The scalar comp onen to f S and its auxiliary /( F /-term/) comp onen ta r ee a c h supp osed to ac/-quire VEVs/, denoted h S i and h FS i resp ectiv ely /. This can b e accomplished either b y puttingS in to an O/'Raifeartaigh/-t yp e mo del/, /8/0or b y a dynamical mec hanism/. /8/1Exactly ho w thishapp ens is a v ery in teresting and imp ortan t question/. Here/, w e will simply parameter/-ize our ignorance of the precise mec hanism of sup ersymmetry breaking b y asserting thatS participates in another part of the sup erp oten tial/, call it Wbreaking /, whic h pro vides forsup ersymmetry breakdo wn/.Let us no w consider the mass sp ectrum of the messenger fermions and b osons/. Themessenger part of the sup erp oten tial no w e/ectiv ely b ecomes Wmess /= y/2 h S i /` /` /+ y/3 h S i q q /.So/, the fermionic messenger /elds pair up to get mass terms/:L /= /; /( y/2 h S i / /` / /` /+ y/3 h S i / q / q /+c /: c /: /) /(/6/./3/3/)as in eq/. /(/3/./4/8/)/. Mean while/, their scalar messenger partners /`/;; /` and q q ha v e a scalar p oten tialgiv en b y /(neglecting D /-term con tributions/, whic h do not a/ect the follo wing discussion/)/:V /= /// / /Wmess //` /// / /2/+ /// / /Wmess / /` /// / /2/+ /// / /Wmess /q /// / /2/+ /// / /Wmess / q /// / /2/+ /// / /Wmess /S /+ /Wbreaking /S /// / /2/(/6/./3/4/)as in eq/. /(/3/./4/7/)/. No w/, using the supp osition thath /Wbreaking /=/ S i /= /;h F /S i /(/6/./3/5/)/(with h /Wmess /=/ S i /= /0/)/, and replacing S and FS b y their VEVs/, one /nds quadratic massterms in the p oten tial for the messenger scalar leptons/:V /= j y/2 h S ij /2 /j /` j /2/+ j /` j /2 //+ j y/3 h S ij /2 /j q j /2/+ j q j /2 //; /y/2 h FS i /` /` /+ y/3 h FS i q q /+c /: c /: //+ quartic terms /: /(/6/./3/6/)The /rst line in eq/. /(/6/./3/6/) represen ts sup ersymmetric mass terms that go along with eq/. /(/6/./3/3/)/,while the second line consists of soft sup ersymmetry/-breaking masses/. The complex scalarmessengers /`/;; /` th us obtain a /(mass/) /2matrix equal to/:/j y/2 h S ij /2/; y //2 h F /S i/; y/2 h FS i j y/2 h S ij /2 //(/6/./3/7/)/5/1 〈 S 〉〈 FS 〉 B, W, gFigure /1/6/: Con tributions to the MSSM gaugino masses in gauge/-mediated sup ersymmetry breaking mo delsarise from one/-lo op graphs in v olving virtual messenger particles/.with squared mass eigen v alues j y/2 h S ij /2/j y/2 h FS ij /. In just the same w a y /, the scalars q/;; q getsquared masses j y/3 h S ij /2/j y/3 h FS ij /.So far/, w eh a v e found that the e/ect of sup ersymmetry breaking is to split eac h messengersup erm ultiplet pair apart/:/`/;; /` /: m /2fermions /= j y/2 h S ij /2/;; m /2scalars /= j y/2 h S ij /2/j y/2 h FS ij /;; /(/6/./3/8/)q/;; q /: m /2fermions /= j y/3 h S ij /2/;; m /2scalars /= j y/3 h S ij /2/j y/3 h FS ij /: /(/6/./3/9/)The sup ersymmetry violation apparen t in this messenger sp ectrum for h FS i /6/= /0 is comm uni/-cated to the MSSM sparticles through radiativ e quan tum corrections/. The MSSM gauginosobtain masses from the /1/-lo op graph sho wn in Fig/. /1/6/. The scalar and fermion lines inthe lo op are messenger /elds/. Recall that the in teraction v ertices in Fig/. /1/6 are of gaugecoupling strength ev en though they do not in v olv e gauge b osons/;; compare Fig/. /5g/. In thisw a y /, gauge/-mediation pro vides that q/;; q messenger lo ops giv e masses to the gluino and thebino/, and /`/;; /` messenger lo ops giv e masses to the wino and bino /elds/. By computing the/1/-lo op diagrams one /nds /8/1that the resulting MSSM gaugino masses are giv en b yMa /= /a /4 / / /;; /( a /=/1 /;; /2 /;; /3/) /;; /(/6/./4/0/)/(in the normalization discussed in section /5/./4/) where w eh a v ei n tro duced a mass parameter/ /h FS i /= h S i /: /(/6/./4/1/)/(Note that if h FS i w ere /0/, then / /= /0 and the messenger scalars w ould b e degenerate withtheir fermionic sup erpartners and there w ould b e no con tribution to the MSSM gauginomasses/./) In con trast/, the corresp onding MSSM gauge b osons cannot get a corresp ondingmass shift/, since they are protected b yg a u g ei n v ariance/. So sup ersymmetry breaking hasb een successfully comm unicated to the MSSM /(/\visible sector/"/)/. T o a go o d appro ximation/,eq/. /(/6/./4/0/) holds for the running gaugino masses at an R Gs c a l e Q/0 corresp onding to thea v erage c haracteristic mass of the hea vy messenger particles/, roughly of order Mmess / yi h S i /.The running mass parameters can then b e R G/-ev olv ed do wn to the electro w eak scale topredict the ph ysical masses to b e measured b y future exp erimen ts/.The scalars of the MSSM do not get an y radiativ e corrections to their masses at one/-lo oporder/. The leading con tribution to their masses comes from the t w o/-lo op graphs sho wn inFig/. /1/7/, with the messenger fermions /(hea vy solid lines/) and messenger scalars /(hea vy dashedlines/) and ordinary gauge b osons and gauginos running around the lo ops/. By computingthese graphs/, one /nds that eac h MSSM scalar / gets a /(mass/) /2giv en b y/:m /2/ /=/2 / /2 /"///3 /4 / //2C //3 /+ ///2 /4 / //2C //2 /+ ///1 /4 / //2C //1 /#/: /(/6/./4/2/)/5/2 Figure /1/7/: Con tributions to MSSM scalar squared masses in gauge/-mediated sup ersymmetry breaking mo delsarise in leading order from these t w o/-lo op F eynman graphs/.Here C /a are the quadratic Casimir group theory in v arian ts for the scalar / for eac h gaugegroup/. They are de/ned b y C /a / ji /=/( T aT a/) ji where the T aare the group generators whic hact on the scalar / /. Explicitly /, they are/:C //3 /= /8/>/</>/: /4 /= /3 for / /= eQi /;; e ui /;; e di /;;/0 for / /= eLi /;; e ei /;;Hu /;;Hd /(/6/./4/3/)C //2 /= /8/>/</>/: /3 /= /4 for / /= eQi /;; eLi /;;Hu /;;Hd /;;/0 for / /= e ui /;; e di /;; e ei /(/6/./4/4/)C //1 /=/3 Y /2/ /= /5 for eac h / with w eak h yp erc harge Y/ /: /(/6/./4/5/)The squared masses in eq/. /(/6/./4/2/) are p ositiv e /(fortunately/!/)/.The terms au /, ad /, ae arise /rst at t w o/-lo op order/, and are suppressed b y an extra factorof /a /= /(/4 / /) compared to the gaugino masses/. So/, to a v ery go o d appro ximation one has/, atthe messenger scale/,au /= ad /= ae /=/0 /;; /(/6/./4/6/)a signi/can tly stronger condition than eq/. /(/5/./1/5/)/. Again/, eqs/. /(/6/./4/2/) and /(/6/./4/6/) should b eapplied at an R G scale equal to the a v erage mass of the messenger /elds running in thelo ops/. Ho w ev er/, after ev olving the R G equations do wn to the electro w eak scale/, non/-zeroau /, ad and ae are generated prop ortional to the corresp onding Y uk a w a matrices and thenon/-zero gaugino masses/, as w e will see in section /7/./1/. These will only b e large for thethird family squarks and sleptons/, in the appro ximation of eq/. /(/5/./2/)/. The parameter b ma yalso b e tak en to v anish near the messenger scale/, but this is quite mo del/-dep enden t/, andin an y case b will b e non/-zero when it is R G/-ev olv ed to the electro w eak scale/. In practice/,b is determined b y the requiremen t of correct electro w eak symmetry breaking/, as discussedb elo w in section /7/./2/.Because the gaugino masses arise at one /-lo op order and the scalar /(mass/) /2con tributionsapp ear at two /-lo op order/, b oth eq/. /(/6/./4/0/) and /(/6/./4/2/) corresp ond to the estimate eq/. /(/6/./1/5/)for msoft /, with Mmess / yi h S i /. Equations /(/6/./4/0/) and /(/6/./4/2/) hold in the limit of smallh FS i /=yi h S i /2/, corresp onding to mass splittings within eac h messenger sup erm ultiplet thatare small compared to the o v erall messenger mass scale/. The subleading corrections in anexpansion in h FS i /=yi h S i /2turn out /8/2to b e quite small unless there are v ery large hierarc hiesin the messenger sector/.The mo del w eh a v e describ ed so far is often called the minimal mo del of gauge/-mediatedsup ersymmetry breaking/. Let us no w generalize it to a more complicated messenger sec/-tor/. Supp ose that q/;; q and /`/;; /` are replaced b y a collection of messengers /i /;; /i with a/5/3 sup erp oten tialWmess /= Xi yi S /i /i /: /(/6/./4/7/)The bar means that the c hiral sup er/elds /i transform as the complex conjugate represen/-tations of the /i c hiral sup er/elds/. T ogether they are said to form a /\v ector/-lik e/" /(real/)represen tation of the Standard Mo del gauge group/. As b efore/, the fermionic comp onen ts ofeac h pair /i and /i pair up to get squared masses yi h S i and their scalar partners mix to getsquared masses j yi h S ij /2/j yi h FS ij /. The MSSM gaugino mass parameters induced are no wMa /= /a /4 / / Xi na /( i /) /( a /=/1 /;; /2 /;; /3/) /(/6/./4/8/)where na /( i /) is the Dynkin index for eac h/i /+ /i /, in a normalization where n/3 /=/1 f o r a /3 /+ /3of SU /(/3/)C and n/2 /= /1 for a pair of doublets of SU /(/2/)L /.F or U /(/1/)Y /, one has n/1 /=/6 Y /2/= /5for eac h messenger pair with w eak h yp erc harges / Y /. In computing n/1 one m ust remem be rto add up the con tributions for eac h comp onen to f a n SU /(/3/)C or SU /(/2/)L m ultiplet/. So/, forexample/, /( n/1 /;;n/2 /;;n/3 /)/= /( /2 /= /5 /;; /0 /;; /1 /)f o r q /+ q and /( n/1 /;;n/2 /;;n/3 /)/=/( /3 /= /5 /;; /1 /;; /0/) for /` /+ /` /.T h us thetotal is Pi /( n/1 /;;n/2 /;;n/3 /)/=/( /1 /;; /1 /;; /1/) for the minimal mo del/, so that eq/. /(/6/./4/8/) is in agreemen twith eq/. /(/6/./4/0/)/. On general group/-theoretic grounds/, n/2 and n/3 m ust b e in tegers/, and n/1 isalw a ys an in teger m ultiple of /1 /= /5 if fractional electric c harges are con/ned/.The MSSM scalar masses in this generalized gauge/-mediation framew ork are no w/:m /2/ /=/2 / /2 /"///3 /4 / //2C //3 Xi n/3 /( i /)/+ ///2 /4 / //2C //2 Xi n/2 /( i /)/+ ///1 /4 / //2C //1 Xi n/1 /( i /) /#/: /(/6/./4/9/)In writing eqs/. /(/6/./4/8/) and /(/6/./4/9/) as simple sums/, w eh a v e implicitl y assumed that the mes/-sengers are all appro ximately equal in mass/, withMmess / yi h S i /: /(/6/./5/0/)This is a go o d appro ximation if the yi are not to o di/eren t from eac h other/, b ecause the de/-p endence of the MSSM mass sp ectrum on the yi is only logarithmic /(due to R G running/) for/xed //. Ho w ev er/, if large hierarc hies in the messenger masses are presen t/, then the additiv econ tributions to the gaugino and scalar masses from eac h individual messenger m ultiplet ishould really instead b e incorp orated at the mass scale of that messenger m ultiplet/. ThenR Ge v olution is used to run these v arious con tributions do wn to the electro w eak or T eVscale/;; the individual messenger con tributions to scalar and gaugino masses as indicatedab o v e can b e though t of as threshold corrections to this R G running/.Messengers with masses far b elo w the GUT scale will a/ect the running of gauge cou/-plings and migh t therefore b e exp ected to ruin the apparen t uni/cation sho wn in Fig/. /1/3/.Ho w ev er/, if the messengers come in complete m ultiplets of the SU /(/5/) global symmetry kthat con tains the Standard Mo del gauge group and are not v ery di/eren t in mass/, thenappro ximate uni/cation of gauge couplings will still o ccur when they are extrap olated upto the same scale MU /(but with a larger uni/ed v alue for the gauge couplings at that scale/)/. kThis SU /(/5/) symmetry ma yo rm a y not b e promoted to a lo cal gauge symmetry at the GUT scale/. F orour presen t purp oses/, it is used simply as a classi/cation sc heme/, since the global SU /(/5/) symmetry is onlyappro ximate b elo w the GUT scale at the messenger mass scale where gauge mediation tak es place/./5/4 F or this reason/, a p opular class of mo dels is obtained b y taking the messengers to consistof N/5 copies of the /5 /+ /5 of SU /(/5/)/, resulting inN/5 /= Xi n/1 /( i /)/= Xi n/2 /( i /)/= Xi n/3 /( i /) /: /(/6/./5/1/)In terms of this in teger parameter N/5 /, eqs/. /(/6/./4/8/) and /(/6/./4/9/) reduce toMa /= /a /4 / / N/5 /(/6/./5/2/)m /2/ /=/2 / /2N/5 /3Xa /=/1 C /a //a /4 / //2/;; /(/6/./5/3/)since no w there are N/5 copies of the minimal messenger sector particles running aroundthe lo ops/. F or example/, the minimal mo del in eq/. /(/6/./3/1/) corresp onds to N/5 /= /1/. A singlecop yo f /1/0 /+ /1/0 of SU /(/5/) has Dynkin indices Pi na /( i /) /= /3/, and so can b e substituted for /3copies of /5 /+ /5 /. /(Other com binations of messenger m ultiplets can also preserv e the apparen tuni/cation of gauge couplings/./) Note that the gaugino masses scale lik e N/5 /, while the scalarmasses scale lik e p N/5 /. This means that sleptons and squarks will tend to b e relativ elyligh ter for larger v alues of N/5 in non/-minimal mo dels/. Ho w ev er/, if N/5 is to o large/, then therunning gauge couplings will div erge b efore they can unify at MU /.F or messenger masses oforder /1/0 /6GeV or less/, for example/, one needs N/5 / /4/.There are man y other p ossible generalizations of the basic gauge/-mediation scenario asdescrib ed ab o v e/. An imp ortan t general exp ectation in these mo dels is that the strongly/-in teracting sparticles /(squarks/, gluino/) should b e hea vier than w eakly/-in teracting sparticles/(sleptons/, bino/, winos/, higgsinos/) simply b ecause of the hierarc h y of gauge couplings //3 />//2 />//1 /. The common feature whic hm a k es all of these mo dels v ery attractiv e is that themasses of the squarks and sleptons dep end only on their gauge quan tum n um b ers/, leadingautomatically to the degeneracy of squark and slepton masses needed for suppression ofF CNC e/ects/. But the most distinctiv e phenomenological prediction of gauge/-mediatedmo dels ma y b e the fact that the gra vitino is the LSP /. This can ha v e crucial consequencesfor b oth cosmology and collider ph ysics/, as w e will discuss further in sections /8/./5 and /9/./7 The mass sp ectrum of the MSSMIn this section/, w e will study the sparticle and Higgs mass sp ectrum of the MSSM/. W e willpa y sp ecial atten tion to the general classes of mo dels whic h/ t i n to the minimal sup ergra v/-it y eqs/. /(/6/./2/7/)/-/(/6/./2/9/) or gauge/-mediated eqs/. /(/6/./4/0/)/-/(/6/./4/6 /) b oundary conditions for the softterms/. As w eh a v e already discussed in section /5/./4/, the renormalization group /(R G/) equa/-tions are a crucial to ol in determining the lagrangian at the electro w eak scale/, giv en a setof b oundary conditions on the theory at the /(m uc h higher/) input scale/. Therefore/, w e willb egin b y lo oking at the R G equations for the parameters of the mo del/, in section /7/./1/. Ofcourse/, the b oundary conditions on soft parameters are quite mo del/-dep endent ev en withinthe minimal sup ergra vit y and gauge/-mediated framew orks/, but there are some imp ortan tgeneral lessons to b e learned from the form of the R G equations/. Once the R G equationsha v e b een used to determine the e/ectiv e lagrangian at the electro w eak scale/, one can usethe results of the earlier sections to predict the mass sp ectrum/, mixing angles/, and in ter/-actions of all of the new particles in the mo del/. In section /7/./2 w e will discuss electro w eaksymmetry breaking and the Higgs scalars/. Sections /7/./3/, /7/./4/, /7/./5 are dev oted to the sparticlemasses and mixings/. Finally in section /7/./6 w e will summarize some of the general featuresand exp ectations for the MSSM sp ectrum/./5/5 /7/./1 R enormalization Gr oup EquationsIn order to translate a set of predictions at the input scale in to ph ysically meaningfulquan tities whic h describ e ph ysics at the electro w eak scale/, it is necessary to ev olv e thegauge couplings/, sup erp oten tial parameters/, and soft terms using the R G equations/. As atec hnical aside/, w e note that when computing R G e/ects and other radiativ e correctionsin sup ersymmetry /, it is imp ortan tt oc ho ose regularization and renormalization sc hemesthat do not violate sup ersymmetry /. The most p opular regularization metho d for discussingradiativ e corrections within the Standard Mo del is dimensional regularization /(DREG/)/, inwhic h the n um b er of spacetime dimensions is con tin ued to d /=/4 /; /2 / /. Unfortunately /, DREGviolates sup ersymmetry explicitly b ecause it in tro duces a mismatc hb e t w een the n um b ers ofgauge b oson degrees of freedom and the gaugino degrees of freedom o//-shell/. This mismatc his only /2 / /, but can b e m ultiplied b y factors up to /1 /=/ nin an n /-lo op calculation/. In DREG/,sup ersymmetric relations b et w een dimensionless coupling constan ts /(/\sup ersymmetric W ardiden tities/"/) are therefore disresp ected b y radiativ e corrections in v olving the /nite parts ofone/-lo op graphs and b y the div ergen t parts of t w o/-lo op graphs/. Instead/, one ma y usethe sligh tly di/eren ts c heme kno wn as regularization b y dimensional reduction/, or DRED/,whic h do es resp ect sup ersymmetry /. /8/3In the DRED metho d/, all momen tum in tegrals arestill p erformed in d /=/4 /; /2 / dimensions/, but the v ector index / on the gauge b oson/elds A a/ no w runs o v er all /4 dimensions/. Running couplings are then renormalized usingDRED with mo di/ed minimal subtraction /( DR/) rather than the usual DREG with mo di/edminimal subtraction /( MS/)/. In particular/, the b oundary conditions at the input scale shouldb e applied in the sup ersymmetry/-preserving DR sc heme/. /(See Ref/. /8/4for an alternativ esup ersymmetric sc heme/./) One lo op / /-functions are alw a ys the same in the t w os c hemes/,but it is imp ortan t to realize that the MS sc heme do es violate sup ersymmetry /, so that DRis preferred yfrom that p oin t of view/. /(It is also p ossible to w ork consisten tly within the MS sc heme/, as long as one is careful to correctly translate all DR couplings and masses in totheir MS coun terparts/. /8/8 /;; /8/9/)The MSSM R G equations in the DR sc heme are giv en in Refs/. /9/0 /; /9/3/;; they are no wk n o wnfor the gauge couplings and sup erp oten tial parameters up to /3/-lo op order/, and for the softparameters at /2/-lo op order/. Ho w ev er/, for man y purp oses including p edagogical ones itsu/ces to w ork in the /1/-lo op appro ximation/. Here/, w e will also use the appro ximation thatonly the third family Y uk a w a couplings are signi/can t/;; see eq/. /(/5/./2/)/. Then the sup erp oten tialparameters run with scale according to/:d dt yt /= yt /1/6 / /2 h/6 j yt j /2/+ j yb j /2/; /1/6 /3 g /2/3 /; /3 g /2/2 /; /1/3 /1/5 g /2/1 i/;; /(/7/./1/)d dt yb /= yb /1/6 / /2 h/6 j yb j /2/+ j yt j /2/+ j y/ j /2/; /1/6 /3 g /2/3 /; /3 g /2/2 /; /7 /1/5 g /2/1 i/;; /(/7/./2/)d dt y/ /= y/ /1/6 / /2 h/4 j y/ j /2/+/3 j yb j /2/; /3 g /2/2 /; /9 /5 g /2/1 i/;; /(/7/./3/)d dt / /= / /1/6 / /2 h/3 j yt j /2/+/3 j yb j /2/+ j y/ j /2/; /3 g /2/2 /; /3 /5 g /2/1 i/: /(/7/./4/)The one/-lo op R G equations for the gauge couplings g/1 /;;g/2 /;;g/3 ha v e already b een listed ineq/. /(/5/./1/7/)/. Note that the / /-functions /(the quan tities on the righ t side of eac h equation/) for yEv en the DRED sc heme ma y not pro vide a sup ersymmetric regulator/, b ecause of am biguities whic h app earat /v e/-lo op order at the latest/. /8/5F ortunately /, this do es not seem to cause an y practical di/culties/. /8/6See alsoRef/. /8/7for a promising prop osal whic ha v oids doing violence to the n um b er of spacetime dimensions/./5/6 eac h sup ersymmetric parameter are prop ortional to the parameter itself/. This is actually aconsequence of a general and p o w erful result kno wn as the sup ersymmetric nonr enormaliza/-tion the or em /. /9/4This theorem implies that the logarithmically div ergen t con tributions to agiv en pro cess can alw a ys b e written in the form of a w a v e/-function renormalization/, withoutan yv ertex renormalization/. zIt is true for an y sup ersymmetric theory /, not just the MSSM/,and holds to all orders in p erturbation theory /. It can b e pro v ed most easily using sup er/eldtec hniques/. In particular/, it means that once w eh a v e a theory whic h can explain wh y /is of order /1/0 /2or /1/0 /3GeV at tree/-lev el/, w e do not ha v et o w orry ab out / b eing infected/(made v ery large/) b y radiativ e corrections in v olving the masses of some v ery hea vy unkno wnparticles/;; all suc hR G corrections to / will b e directly prop ortional to / itself/.The one/-lo op R G equations for the three gaugino mass parameters in the MSSM aredetermined b y the same quan tities b MSSMa whic h app ear in the gauge coupling R G eqs/. /(/5/./1/7/)/:d dt Ma /= /1 /8 / /2 ba g /2a Ma /( ba /=/3 /3 /= /5 /;; /1 /;; /; /3/) /(/7/./5/)for a /=/1 /;; /2 /;; /3/. It is therefore easy to sho w that the three ratios Ma /=g /2a are eac h constan t/(R G/-scale indep enden t/) up to small t w o/-lo op corrections/. In minimal sup ergra vit y mo dels/,w e can therefore writeMa /( Q /)/= g /2a /( Q /) g /2a /( Q/0 /) m/1 /= /2 /( a /=/1 /;; /2 /;; /3/) /(/7/./6/)at an yR G scale Q/<Q/0 /,w h e r e Q/0 is the input scale whic h is presumably nearly equal toMP /. Since the gauge couplings are observ ed to unify at MU / /0 /: /0/1 MP /, one exp ects xthatg /2/1 /( Q/0 /) / g /2/2 /( Q/0 /) / g /2/3 /( Q/0 /)/. Therefore/, one /nds thatM/1 g /2/1 /= M/2 g /2/2 /= M/3 g /2/3 /(/7/./7/)at an yR G scale/, up to small t w o/-lo op e/ects and p ossibly larger threshold e/ects near MUand MP /. The common v alue in eq/. /(/7/./7/) is also equal to m/1 /= /2 /=g /2U in minimal sup ergra vit ymo dels/, where gU is the uni/ed gauge coupling at the input scale where m/1 /= /2 is the commongaugino mass/. In terestingly /, eq/. /(/7/./7/) is also the solution to the one/-lo op R G equations inthe case of the gauge/-mediated b oundary conditions eq/. /(/6/./4/0/) applied at the messengermass scale/. This is true ev en though there is no suc h thing as a uni/ed gaugino mass m/1 /= /2in the gauge/-mediated case/, b ecause of the fact that the gaugino masses are prop ortionalto the g /2a times a constan t/. So eq/. /(/7/./7/) is theoretically w ell/-motiv ated /(but certainly notinevitable/) in b oth framew orks/. The prediction eq/. /(/7/./7/) is particularly useful since thegauge couplings g /2/1 /, g /2/2 /,a n d g /2/3 are already quite w ell kno wn at the electro w eak scale fromexp erimen t/. Therefore they can b e extrap olated up to at least MU /, assuming that theapparen t uni/cation of gauge couplings is not a fak e/. The gaugino mass parameters feedin to the R G equations for all of the other soft terms/, as w e will see/.Next w e consider the /1/-lo op R G equations for the analytic soft parameters au /, ad /, ae /.In mo dels ob eying eq/. /(/5/./1/5/)/, these matrices start o/ prop ortional to the corresp onding zActually /,t h e r e is v ertex renormalizatio n in the /eld theory in whic h auxiliary /elds ha v e b een in tegratedout/, but the sum of div ergen t con tributions for a giv en pro cess alw a ys has the form of w a v e/-function renor/-malization/. See Ref/. /2/3for a discussion of this p oin t/.xIn a GUT mo del/, it is automatic that the gauge couplings and gaugino masses are uni/ed at all scalesQ/> MU and in particular at Q / MP /, b ecause in the uni/ed theory the gauginos all liv e in the samerepresen tation of the uni/ed gauge group/. In man y sup erstring mo dels/, this is also kno w n t o be a g oodappro ximation /./5/7 Y uk a w a couplings at the input scale/, and the R Ge v olution resp ects this prop ert y /. With theappro ximation of eq/. /(/5/./2/)/, one can therefore also write/, at an yR G scale/,au / /0/@ /0 /0 /0/0 /0 /0/0 /0 at /1A/;; ad / /0/@ /0 /0 /0/0 /0 /0/0 /0 ab /1A/;; ae / /0/@ /0 /0 /0/0 /0 /0/0 /0 a/ /1A/;; /(/7/./8/)whic h de/nes /{running parameters at /, ab /,a n d a/ /. The R G equations for these parametersand b are giv en b y/1/6 / /2 d dt at /= at h/1/8 j yt j /2/+ j yb j /2/; /1/6 /3 g /2/3 /; /3 g /2/2 /; /1/3 /1/5 g /2/1 i/+/2 ab y /b yt /+ yt h/3/2 /3 g /2/3 M/3 /+/6 g /2/2 M/2 /+ /2/6 /1/5 g /2/1 M/1 i/;; /(/7/./9/)/1/6 / /2 d dt ab /= ab h/1/8 j yb j /2/+ j yt j /2/+ j y/ j /2/; /1/6 /3 g /2/3 /; /3 g /2/2 /; /7 /1/5 g /2/1 i/+/2 at y /t yb /+/2 a/ y // yb /+ yb h/3/2 /3 g /2/3 M/3 /+/6 g /2/2 M/2 /+ /1/4 /1/5 g /2/1 M/1 i/;; /(/7/./1/0/)/1/6 / /2 d dt a/ /= a/ h/1/2 j y/ j /2/+/3 j yb j /2/; /3 g /2/2 /; /9 /5 g /2/1 i/+/6 ab y /b y/ /+ y/ h/6 g /2/2 M/2 /+ /1/8 /5 g /2/1 M/1 i/;; /(/7/./1/1/)/1/6 / /2 d dt b /= b h/3 j yt j /2/+/3 j yb j /2/+ j y/ j /2/; /3 g /2/2 /; /3 /5 g /2/1 i/+ / h/6 at y /t /+/6 ab y /b /+/2 a/ y // /+/6 g /2/2 M/2 /+ /6 /5 g /2/1 M/1 i/(/7/./1/2/)in this appro ximation/. The / /-function for eac h of these soft parameters is not prop ortionalto the parameter itself/;; this mak es sense b ecause couplings whic h violate sup ersymmetryare not protected b y the sup ersymmetric nonrenormalization theorem/. In particular/, ev en ifA/0 and B/0 app earing in eqs/. /(/6/./2/9/) and /(/6/./3/0/) v anish at the input scale/, the R G correctionsprop ortional to gaugino masses app earing in eqs/. /(/7/./9/)/-/(/7/./1/2/) ensure that at /, ab /, a/ and bwill still b e non/-zero at the electro w eak scale/.Next let us consider the R G equations for the scalar masses in the MSSM/. In the ap/-pro ximation of eqs/. /(/5/./2/) and /(/7/./8/)/, the squarks and sleptons of the /rst t w o families ha v eonly gauge in teractions/. This means that if the scalar masses satisfy a b oundary conditionlik e eq/. /(/5/./1/4/) at an input R G scale/, then when renormalized to an yo t h e rR G scale/, theywill still b e almost diagonal/, with the appro ximate formm /2Q / /0B/@ m /2Q/1 /0 /0/0 m /2Q/1 /0/0 /0 m /2Q/3 /1CA /;; m /2 u / /0/@ m /2 u/1 /0 /0/0 m /2 u/1 /0/0 /0 m /2 u/3 /1A/;; /(/7/./1/3/)etc/. The /rst and second family squarks and sleptons with giv en gauge quan tum n um b ersremain v ery nearly degenerate/, but the third family squarks and sleptons feel the e/ects ofthe larger Y uk a w a couplings and so get renormalized di/eren tly /. The one/-lo op R G equations /{W em ust w arn the reader that rescaled soft parameters At /= at /=yt /, Ab /= ab /=yb /,a n d A/ /= a/ /=y/ arecommonly used in the literature/. W e do not follo w this notation/, b ecause it cannot b e generalized b ey ondthe appro ximation of eqs/. /(/5/./2/)/, /(/7/./8/) without in tro ducing horrible complications suc h as non/-p olynomi alR G equations/, and b ecause at /, ab and a/ are the couplings that actually app ear in the lagrangian an yw a y /./5/8 for the /rst and second family squark and slepton squared masses can b e written as k/1/6 / /2 d dt m /2/ /= /; Xa /=/1 /;; /2 /;; /3 /8 g /2a C /a j Ma j /2/(/7/./1/4/)for eac h scalar / /, where the Pa is o v er the three gauge groups U /(/1/)Y /, SU /(/2/)L and SU /(/3/)C /;;Ma are the corresp onding running gaugino mass parameters whic h are kno wn from eq/. /(/7/./7/)/;;and the constan ts C /a are the same quadratic Casimir in v arian ts whic h app eared in eqs/. /(/6/./4/3/)/-/(/6/./4/5/)/. An imp ortan t feature of eq/. /(/7/./1/4/) is that the righ t/-hand sides are strictly negativ e/,so that the scalar /(mass/) /2parameters gr ow as they are R G/-ev olv ed from the input scaledo wn to the electro w eak scale/. Ev en if the scalars ha v ez e r oo rv ery small masses at theinput scale/, as in the /\no/-scale/" b oundary condition limit m /2/0 /= /0/, they will obtain largep ositiv e squared masses at the electro w eak scale/, thanks to the e/ects of the gaugino masses/.The R G equations for the /(mass/) /2parameters of the Higgs scalars and third familysquarks and sleptons get the same gauge con tributions as in eq/. /(/7/./1/4/)/, but they also ha v econ tributions due to the large Y uk a w a/( yt/;;b/;;/ /) and soft /( at/;;b/;;/ /) couplings/. A t one/-lo op order/,these only app ear in three com binations/:Xt /=/2 j yt j /2/( m /2Hu /+ m /2Q/3 /+ m /2 u/3 /)/+ /2 j at j /2/;; /(/7/./1/5/)Xb /=/2 j yb j /2/( m /2Hd /+ m /2Q/3 /+ m /2 d/3 /)/+/2 j ab j /2/;; /(/7/./1/6/)X/ /=/2 j y/ j /2/( m /2Hd /+ m /2L/3 /+ m /2 e/3 /)/+/2 j a/ j /2/: /(/7/./1/7/)In terms of these quan tities/, the R G equations for the soft Higgs /(mass/) /2parameters m /2Huand m /2Hd are/1/6 / /2 d dt m /2Hu /=/3 Xt /; /6 g /2/2 j M/2 j /2/; /6 /5 g /2/1 j M/1 j /2/;; /(/7/./1/8/)/1/6 / /2 d dt m /2Hd /=/3 Xb /+ X/ /; /6 g /2/2 j M/2 j /2/; /6 /5 g /2/1 j M/1 j /2/: /(/7/./1/9/)Note that Xt /, Xb /,a n d X/ are p ositiv e/, so their e/ect is alw a ys to de cr e ase the Higgs massesas one ev olv es the R G equations do wn w ard from the input scale to the electro w eak scale/.Since yt is the largest of the Y uk a w a couplings b ecause of the exp erimen tal fact that thetop quark is hea vy /, Xt is t ypically exp ected to b e larger than Xb and X/ /. This can causethe R G/-ev olv ed m /2Hu to run negativ e near the electro w eak scale/, helping to destabilize thepo i n t Hu /=/0 a n d s o p r o v oking a Higgs VEV whic h is just what w ew an t/. //Th us a largetop Y uk a w a coupling fa v ors the breakdo wn of the electro w eak symmetry breaking b ecauseit induces negativ e radiativ e corrections to the Higgs /(mass/) /2/.The third family squark and slepton /(mass/) /2parameters also get con tributions whic hdep end on Xt /, Xb and X/ /. Their R G equations are giv en b y/1/6 / /2 d dt m /2Q/3 /= Xt /+ Xb /; /3/2 /3 g /2/3 j M/3 j /2/; /6 g /2/2 j M/2 j /2/; /2 /1/5 g /2/1 j M/1 j /2/(/7/./2/0/) kThere are also terms in the scalar /(mass/) /2R G equations whic h are prop ortional to T r/[ Ym /2/] /(the sum of thew eak h yp erc harge times the soft /(mass/) /2for all scalars in the theory/)/. Ho w ev er/, these con tributions v anishin b oth the cases of minimal sup ergra vit y and gauge/-mediated b oundary conditions for the soft terms/, asone can see b y explicitl y calculating T r/[ Ym /2/]i n e a c h case/. If T r/[ Ym /2/] is zero at the input scale/, then it willremain zero under R Ge v olution/. Therefore w e neglect suc h terms in our discussion/, although they can ha v ean imp ortan t e/ect in more general situations/.//One should think of /\ m /2Hu /" as a parameter un to itself/, and not as the square of some m ythical real n um be rmHu /.T h us there is nothing strange ab out ha ving m /2Hu /< /0/. Ho w ev er/, strictly sp eaking m /2Hu /< /0 is neithernecessary nor su/cien t for electro w eak symmetry breaking/;; see section /7/./2/./5/9 /1/6 / /2 d dt m /2 u/3 /=/2 Xt /; /3/2 /3 g /2/3 j M/3 j /2/; /3/2 /1/5 g /2/1 j M/1 j /2/(/7/./2/1/)/1/6 / /2 d dt m /2 d/3 /=/2 Xb /; /3/2 /3 g /2/3 j M/3 j /2/; /8 /1/5 g /2/1 j M/1 j /2/(/7/./2/2/)/1/6 / /2 d dt m /2L/3 /= X/ /; /6 g /2/2 j M/2 j /2/; /3 /5 g /2/1 j M/1 j /2/(/7/./2/3/)/1/6 / /2 d dt m /2 e/3 /=/2 X/ /; /2/4 /5 g /2/1 j M/1 j /2/: /(/7/./2/4/)In eqs/. /(/7/./1/8/)/-/(/7/./2/4 /)/, the terms prop ortional to j M/3 j /2/, j M/2 j /2and j M/1 j /2are just the sameones as in eq/. /(/7/./1/4/)/. Note that the terms prop ortional to Xt app ear with smaller n umericalco e/cien ts in the m /2Q/3 and m /2 u/3 R G equations than they did for the Higgs scalars/, andthey do not app ear at all in the m /2 d/3 /, m /2L/3 and m /2 e/3 R G equations/. F urthermore/, the third/-family squark /(mass/) /2get a large p ositiv ec o n tribution prop ortional to j M/3 j /2from the R Gev olution/, whic h the Higgs scalars do not get/. These facts mak e it easy to understand wh ythe Higgs scalars in the MSSM can get VEVs/, but the squarks and sleptons/, ha ving largep ositiv e /(mass/) /2/,d o n o t /. An examination of the R G equations /(/7/./9/)/-/(/7/./1/2/) /, /(/7/./1/4/)/, and/(/7/./1/8/)/-/(/7/./2/4/) rev eals that if the gaugino mass parameters M/1 /, M/2 /, and M/3 are non/-zero atthe input scale/, then all of the other soft terms will b e generated/. This is wh y the /\no/-scale/"limit with m/1 /= /2 / m/0 /;;A/0 /;;B/0 can b e phenomenologically viable ev en though the squarksand sleptons are massless at tree/-lev el/. On the other hand/, if the gaugino masses w ere tov anish at tree/-lev el/, then they w ould not get an yc o n tributions to their masses at one/-lo oporder/;; in that case M/1 /, M/2 /, and M/3 w ould b e extremely small/.No w that w eh a v e review ed the e/ects of R Ge v olution from the input scale do wn tothe electro w e a ko rT eV scale/, w e are ready to w ork out the exp ected features of the MSSMsp ectrum in some detail/. W e will b egin with the Higgs sector in the next section/./7/./2 Ele ctr owe ak symmetry br e aking and the Higgs b osonsIn the MSSM/, the description of electro w eak symmetry breaking is sligh tly complicated b ythe fact that there are t w o complex Higgs doublets Hu /=/( H /+u /;;H /0u /) and Hd /=/( H /0d /;;H /;d /)rather than just one in the ordinary Standard Mo del/. The classical scalar p oten tial for theHiggs scalar /elds in the MSSM is giv en b yV /= /( j / j /2/+ m /2Hu /)/( j H /0u j /2/+ j H /+u j /2/)/+/( j / j /2/+ m /2Hd /)/( j H /0d j /2/+ j H /;d j /2/)/+ b /( H /+u H /;d /; H /0u H /0d /)/+ c /: c /:/+ /1 /8 /( g /2/+ g /0 /2/)/( j H /0u j /2/+ j H /+u j /2/;j H /0d j /2/;j H /;d j /2/) /2/+ /1 /2 g /2j H /+u H /0 /d /+ H /0u H /;/d j /2/: /(/7/./2/5/)The terms prop ortional to j / j /2come from F /-terms /[see the /rst term on the righ t/-handside of eq/. /(/5/./5/)/]/. The terms prop ortional to m /2Hu /, m /2Hd and b are nothing but a rewritingof the last three terms of eq/. /(/5/./1/1/)/. Finally /, the terms prop ortional to g /2and g /0 /2are theD /-term con tributions whic hm a yb e d e r i v ed from the general form ula eq/. /(/3/./7/5/)/, after somerearranging/. The full scalar p oten tial of the theory will also include man y terms in v olvingthe squark and slepton /elds that w e can ignore here/, since they do not get VEVs b ecausethey ha v e large p ositiv e /(mass/) /2/.W en o wh a v e to demand that the minim um of this p oten tial should break electro w eaksymmetry do wn to electromagnetism SU /(/2/)L / U /(/1/)Y /! U /(/1/)EM /, in accord with exp erimen t/./6/0 W e can use the freedom to mak e gauge transformations to simplify this analysis/. First/, thefreedom to mak e SU /(/2/)L gauge transformations allo ws us to rotate a w a y a p ossible VEV forone of the w eak isospin comp onen ts of one of the scalar /elds/;; so without loss of generalit yw e can tak e H /+u /= /0 at the minim um of the p oten tial/. Then one /nds that a minim um of thep oten tial satisfying /@ V /=/@ H /+u /=/0 m ust also ha v e H /;d /= /0/. This is go o d/, b ecause it meansthat at the minim um of the p oten tial electromagnetism is necessarily un brok en/, since thec harged comp onen ts of the Higgs scalars cannot get VEVs/. So after setting H /+u /= H /;d /=/0w e are left to consider the scalar p oten tialV /= /( j / j /2/+ m /2Hu /) j H /0u j /2/+/( j / j /2/+ m /2Hd /) j H /0d j /2/; /( bH /0u H /0d /+c /: c /: /)/+ /1 /8 /( g /2/+ g /0 /2/)/( j H /0u j /2/;j H /0d j /2/) /2/: /(/7/./2/6/)The only term in this p oten tial whic h dep ends on the phases of the /elds is the b /-term/.Therefore a rede/nition of the phases of Hu and Hd can absorb an y phase in b /,s ow ec a ntak e b to b e real and p ositiv e/. Then it is clear that a minim um of the p oten tial V requiresthat H /0u H /0d is also real and p ositiv e/, so h H /0u i and h H /0d i m ust ha v e opp osite phases/. W ec a ntherefore use a U /(/1/)Y gauge transformation to mak e them b oth b e real and p ositiv e withoutloss of generalit y /, since Hu and Hd ha v e opp osite w eak h yp erc harges /( / /1 /= /2/)/. It follo ws thatCP cannot b e sp on taneously brok en b y the Higgs scalar p oten tial/, since all of the VEVsand couplings can b e sim ultaneously c hosen to b e real/. This means that the Higgs scalarmass eigenstates can b e assigned w ell/-de/ned eigen v alues of CP /.Note that the b /-term alw a ys fa v ors electro w eak symmetry breaking/. The com binationof the b term and the terms m /2Hu and m /2Hd can allo w for one linear com bination of H /0u andH /0d to ha v e a negativ e /(mass/) /2near H /0u /= H /0d /= /0/. This requires thatb /2/> /( j / j /2/+ m /2Hu /)/( j / j /2/+ m /2Hd /) /: /(/7/./2/7/)If this inequalit y is not satis/ed/, then H /0u /= H /0d /= /0 will b e a stable minim um of thep oten tial/, and electro w eak symmetry breaking will not o ccur/. A negativ ev alue for j / j /2/+ m /2Huwill help eq/. /(/7/./2/7/) to b e satis/ed/, but it is not necessary /.F urthermore/, ev en if m /2Hu /< /0/,there ma y b e no electro w eak symmetry breaking if j / j is to o large or if b is to o small/.Still/, the large negativ ec o n tributions to m /2Hu from the R G equation /(/7/./1/8/) discussed in theprevious section are an imp ortan t factor in ensuring that electro w eak symmetry breakingcan o ccur in mo dels with minimal sup ergra vit y or gauge/-mediated b oundary conditions forthe soft terms/.In order for the MSSM scalar p oten tial to b e viable/, it is not enough that the p oin tH /0u /= H /0d /= /0 is destabilized b y a negativ e /(mass/) /2direction/;; w em ust also mak e surethat the p oten tial is b ounded from b elo w for arbitrarily large v alues of the scalar /elds/,so that V will really ha v e a minim um/. /(Recall from the discussion in sections /3/./2 and /3/./4that scalar p oten tials in purely sup ersymmetric theories are automatically p ositiv e and soclearly b ounded from b elo w/. But/, no w that w eh a v ei n tro duced sup ersymmetry breaking/,w em ust b e careful/./) The scalar quartic in teractions in V will stabilize the p oten tial foralmost all arbitrarily large v alues of H /0u and H /0d /.H o w ev er/, there are sp ecial directions in/eld space with j H /0u j /= j H /0d j /, along whic h the quartic con tributions to V /[the second linein eq/. /(/7/./2/6/)/] are iden tically zero/. Suc h directions in /eld space are called D /-/ at directions/,b ecause along them the part of the scalar p oten tial coming from D /-terms v anishes/. Inorder for the p oten tial to b e b ounded from b elo w/, w e need the quadratic part of the scalarp oten tial to b e p ositiv e along the D /-/ at directions/. This requiremen t amoun ts to/2 b/< /2 j / j /2/+ m /2Hu /+ m /2Hd /: /(/7/./2/8/)/6/1 In terestingly /,i f m /2Hu /= m /2Hd /, the constrain ts eqs/. /(/7/./2/7/) and /(/7/./2/8/) cannot b oth b e satis/ed/.In mo dels deriv ed from the minimal sup ergra vit y or gauge/-mediated b oundary conditions/,m /2Hu /= m /2Hd holds at tree/-lev el at the input scale/, but the Xt con tribution to the R G equationfor m /2Hu naturally pushes it to negativ e or small v alues m /2Hu /<m /2Hd at the electro w eakscale/, as w es a w in section /7/./1/. Unless this e/ect is large/, the parameter space in whic hthe electro w eak symmetry is brok en w ould b e quite small/. So in these mo dels electro w eaksymmetry breaking is actually driv en purely b y quan tum corrections/;; this mec hanism istherefore kno wn as r adiative ele ctr owe ak symmetry br e aking /. The realization that this w orksmost naturally with a large top/-quark Y uk a w a coupling pro vides additional motiv ation forthese mo dels/. /9/0 /;; /9/5Ha ving established the conditions necessary for H /0u and H /0d to get non/-zero VEVs/, w ecan no w require that they are compatible with the observ ed phenomenology of electro w eaksymmetry breaking SU /(/2/)L / U /(/1/)Y /! U /(/1/)EM /. Let us write h H /0u i /= vu and h H /0d i /= vd forthe VEVs at the minim um of the p oten tial/. These VEVs can b e connected to the kno wnmass of the Z /0b oson and the electro w eak gauge couplings/:v /2u /+ v /2d /= v /2/=/2 m /2Z /= /( g /2/+ g /0 /2/) / /(/1/7/4 GeV /) /2/: /(/7/./2/9/)The ratio of the t w o VEVs is traditionally written astan / / vu /=vd /: /(/7/./3/0/)The v alue of tan / is not /xed b y presen t exp erimen ts/, but it dep ends on the lagrangianparameters of the MSSM in a calculable w a y /. Since vu /= v sin / and vd /= v cos / w ere tak ento b e real and p ositiv e/, w eh a v e/0 /</ /< / /= /2/, a requiremen t that will b e sharp ened b elo w/.No w one can write do wn the conditions /@V /= /@H /0u /= /@ V /=/@ H /0d /= /0 under whic h the p oten tialeq/. /(/7/./2/6/) will ha v e a minim um satisfying eqs/. /(/7/./2/9/) and /(/7/./3/0/)/:j / j /2/+ m /2Hd /= b tan / /; /( m /2Z /= /2/) cos /2 / /;; /(/7/./3/1/)j / j /2/+ m /2Hu /= b cot / /+/( m /2Z /= /2/) cos /2 //: /(/7/./3/2/)It is easy to c hec k that these equations indeed satisfy the necessary conditions eqs/. /(/7/./2/7/)and /(/7/./2/8/)/. They allo w us to eliminate t w o of the lagrangian parameters b and j / j in fa v orof tan / /, but do not determine the phase of / /.As an aside/, w e note that eqs/. /(/7/./3/1/) and /(/7/./3/2/) highligh t the /\ / problem/" already men/-tioned in section /5/./1/. If w e view j / j /2/, b /, m /2Hu and m /2Hd as input parameters/, and m /2Z andtan / as output parameters obtained b y solving these t w o equations/, then without miracu/-lous cancellations w e exp ect that all of the input parameters ough t to b e within an orderof magnitude or t w oo f m /2Z /.H o w ev er/, in the MSSM/, / is a sup ersymmetry/-resp ecting pa/-rameter app earing in the sup erp oten tial/, while b /, m /2Hu /, m /2Hd are sup ersymmetry/-breakingparameters/. This has lead to a widespread b elief that the MSSM m ust b e extended atv ery high energies to include a mec hanism whic h relates the e/ectiv ev alue of / to thesup ersymmetry/-breaking mec hanism in some w a y/;; see section /1/0/./2 and Refs/. /4/1 /;; /4/2 /;; /4/3for ex/-amples/.The Higgs scalar /elds in the MSSM consist of t w o complex SU /(/2/)L /-doublet/, or eigh treal/, scalar degrees of freedom/. When the electro w eak symmetry is brok en/, three of them arethe w ould/-b e Nam bu/-Goldstone b osons G /0/, G /whic h b ecome the longitudinal mo des of theZ /0and W /massiv ev ector b osons/. The remaining /v e Higgs scalar mass eigenstates consistof one CP/-o dd neutral scalar A /0/,a c harge /+/1 scalar H /+and its conjugate c harge /; /1 scalar/6/2 H /;/, and t w o CP/-ev en neutral scalars h /0and H /0/. In terms of the original gauge/-eigenstate/elds/, the mass eigenstates and w ould/-b e Nam bu/-Goldstone b osons are giv en b y/G /0A /0 //= p /2 /sin / /; cos /cos / sin / //Im /[ H /0u /]Im /[ H /0d /] //;; /(/7/./3/3/)/G /+H /+ //= /sin / /; cos /cos / sin / //H /+uH /;/d //;; /(/7/./3/4/)with G /;/= G /+ /and H /;/= H /+ //,a n d/h /0H /0 //= p /2 /cos / /; sin /sin / cos / //Re /[ H /0u /] /; vuRe /[ H /0d /] /; vd //: /(/7/./3/5/)whic h de/nes a mixing angle / /. The tree/-lev el masses of these /elds can b e found b yexpanding the scalar p oten tial around the minim um/. One obtainsm /2A /0 /=/2 b/= sin /2 / /(/7/./3/6/)m /2H / /= m /2A /0 /+ m /2W /(/7/./3/7/)m /2h /0/;;H /0 /= /1 /2 /m /2A /0 /+ m /2Z / q /( m /2A /0 /+ m /2Z /) /2/; /4 m /2Z m /2A /0 cos /2/2 / //: /(/7/./3/8/)In terms of these masses/, the mixing angle / app earing in eq/. /(/7/./3/5/) is determined at tree/-lev el b ysin /2 / sin /2 / /= /; m /2A /0 /+ m /2Z m /2H /0 /; m /2h /0 /;; cos /2 / cos /2 / /= /; m /2A /0 /; m /2Z m /2H /0 /; m /2h /0 /: /(/7/./3/9/)The F eynman rules for couplings of the mass eigenstate Higgs scalars to the Standard Mo delquarks and leptons and the electro w eak v ector b osons/, as w ell as to the v arious sparticles/,ha v e b een w ork ed out in detail in Ref/. /9/6 /;; /9/7The masses of A /0/, H /0and H /can in principle b e arbitrarily large since they all gro wwith b/= sin /2 / /. In con trast/, the mass of h /0is b ounded from ab o v e/. It is not hard to sho wfrom eq/. /(/7/./3/8/) thatmh /0 /< j cos /2 / j mZ /(/7/./4/0/)at tree/-lev el/. /9/8If this inequalit yw ere robust/, it w ould guaran tee that the ligh test Higgs b osonof the MSSM w ould b e kinematically accessible to LEP/2/, with large regions of parameterspace already ruled out/. Ho w ev er/, the tree/-lev el mass form ulas giv en ab o v e for the Higgsmass eigenstates are sub ject to quite signi/can t quan tum corrections whic h are esp eciallyimp ortan tt o t a k ei n to accoun t in the case of h /0/. The largest suc hc o n tributions t ypicallycome from top/-stop lo op corrections to the terms in the scalar p oten tial/. In the limit ofstop squark masses met/1 /, met/2 m uc h greater than the top quark mass mt /, one /nds a one/-lo opradiativ e correction to eq/. /(/7/./3/8/)/:/
/( m /2h /0 /)/= /3 /4 / /2 v /2y /4t sin /4/ ln /met/1 met/2 m /2t //: /(/7/./4/1/)Including this and other corrections/, /9/9 /;; /1/0/0one can obtain only a considerably w eak er/, butstill v ery in teresting/, b oundmh /0 /</ /1/3/0 GeV /(/7/./4/2/)/6/3 in the MSSM/. This assumes that all of the sparticles that can con tribute to /
/( m /2h /0 /) in lo opsha v e masses that do not exceed /1 T eV/. By adding extra sup erm ultiplets to the MSSM/, thisb ound can b e made ev en w eak er/. Ho w ev er/, assuming that none of the MSSM sparticles ha v emasses exceeding /1 T eV and that all of the couplings in the theory remain p erturbativ eu pto the uni/cation scale/, one still /nds /1/0/1mh /0 /</ /1/5/0 GeV /: /(/7/./4/3/)This b ound is also w eak ened if/, for example/, the top squarks are hea vier than /1 T eV/, but theupp er b ound rises only logarithmically with the soft masses/, as can b e seen from eq/. /(/7/./4/1/)/.Th us it is a fairly robust prediction of sup ersymmetry at the electro w eak scale that at leastone of the Higgs scalar b osons m ust b e ligh t/.An in teresting limit o ccurs when mA /0 / mZ /. In that case/, mh /0 can saturate the upp erb ound just men tioned with mh /0 / mZ j cos /2 / j at tree/-lev el/, but sub ject to large p ositiv equan tum corrections/. The particles A /0/, H /0/, and H /are m uc hh e a vier and nearly degenerate/,forming an isospin doublet whic h decouples from su/cien tly lo w/-energy exp erimen ts/. Theangle / is /xed to b e appro ximately / /; //= /2/. In this limit/, h /0has the same couplings toquarks and leptons and electro w eak gauge b osons as w ould the ph ysical Higgs b oson of theordinary Standard Mo del without sup ersymmetry /. Indeed/, mo del/-buildi ng exp eriences ha v esho wn that it is quite common for h /0to b eha v ei n a w a y nearly indistinguishable from aStandard Mo del/-lik e Higgs b oson/, ev en if mA /0 is not to o h uge/. On the other hand/, it isimp ortan tt o k eep in mind that the couplings of h /0migh t turn out to deviate in imp ortan tw a ys from those of a Standard Mo del Higgs b oson/. F o rag i v en set of mo del parameters/,it is v ery imp ortan tt ot a k ei n to accoun t the complete set of one/-lo op corrections and ev enthe dominan tt w o/-lo op e/ects in a leading logarithm appro ximation in order to get accuratepredictions for the Higgs masses and mixing angles/. /9/9 /;; /1/0/0In the MSSM/, the masses and CKM mixing angles of the quarks and leptons are de/-termined b y the Y uk a w a couplings of the sup erp oten tial and the parameter tan / /. Thisis b ecause the top/, c harm and up quarks get masses prop ortional to vu /= v sin / andthe b ottom/, strange/, and do wn quarks and the c harge leptons get masses prop ortional tovd /= v cos / /. Therefore one /nds at tree/-lev elyt /= gmt p /2 mW sin / /;; yb /= gmb p /2 mW cos / /;; y/ /= gm/ p /2 mW cos / /: /(/7/./4/4/)These relations hold for the running masses of t/;; b/;; / rather than the ph ysical p ole masseswhic h are signi/can tly larger/. /1/0/2Including those corrections/, one can relate the Y uk a w acouplings to tan / and the kno wn fermion masses and CKM mixing angles/. It is no w clearwh yw eh a v e not neglected yb and y/ /,e v en though mb /;;m/ / mt /.T o a /rst appro ximation/,yb /=yt /=/( mb /=mt /)t a n / and y/ /=yt /=/( m/ /=mt /)t a n / /, so that yb and y/ cannot b e neglected iftan / is m uc h larger than /1/. In fact/, there are go o d theoretical motiv ations for consideringmo dels with large tan / /.F or example/, mo dels based on the GUT gauge group SO /(/1/0/) /(orcertain of its subgroups/) can unify the running top/, b ottom and tau Y uk a w a couplings atthe uni/cation scale/;; this requires tan / to b e v ery roughly of order mt /=mb /. /1/0/3 /;; /1/0/4Note that if one tries to mak es i n / to o small/, yt will b ecome nonp erturbativ ely large/.Requiring that yt do es not blo wu pa b o v e the electro w eak scale/, one /nds that tan / />/ /1 /: /2or so/, dep ending on the mass of the top quark/, the QCD coupling/, and other /ne details/.In principle/, one can also determine a lo w er b ound on cos / and th us an upp er b ound ontan / b y requiring that yb and y/ are not nonp erturbativ ely large/. This giv es a rough upp erb ound of tan / /</ /6/5/. Ho w ev er/, this is complicated sligh tly b y the fact that the b ottom/6/4 quark mass gets signi/can t one/-lo op corrections in the large tan / limit/. /1/0/4One can obtain asligh tly stronger upp er b ound on tan / in mo dels where m /2Hu /= m /2Hd at the input scale/, b yrequiring that yb do es not signi/can tly exceed yt /. /[Otherwise/, Xb w ould b e larger than Xtin eqs/. /(/7/./1/8/) and /(/7/./1/9/)/, so one w ould /nd m /2Hd /<m /2Hu at the electro w eak scale/, and theminim um of the p oten tial w ould ha v et o b ea t h H /0d i /> h H /0u i whic hw ould b e a con tradictionwith the supp osition that tan / is large/./] In the follo wing/, w e will see that the parametertan / has an imp ortan t e/ect on the masses and mixings of the MSSM sparticles/./7/./3 Neutr alinos and char ginosThe higgsinos and electro w eak gauginos mix with eac h other b ecause of the e/ects of elec/-tro w eak symmetry breaking/. The neutral higgsinos /( eH /0u and eH /0d /) and the neutral gauginos/( eB /, fW /0/) com bine to form four neutral mass eigenstates called neutr alinos /. The c hargedhiggsinos /( eH /+u and eH /;d /) and winos /( fW /+and fW /;/) mix to form t w o mass eigenstates withc harge / /1 called char ginos /.W e will denote yythe neutralino and c hargino mass eigenstatesb y eNi /( i /=/1 /;; /2 /;; /3 /;; /4/) and eC /i /( i /=/1 /;; /2/)/. By con v en tion/, these are lab elled in ascending order/,so that meN/1 /<meN/2 /<meN/3 /<meN/4 and meC/1 /<meC/2 /.T h e l i g h test neutralino/, eN/1 /, is usuallyassumed to b e the LSP /, unless there is a ligh ter gra vitino or unless R /-parit y is not conserv ed/,b ecause it is the only MSSM particle whic h can mak e a go o d cold dark matter candidate/.In this subsection/, w e will describ e the mass sp ectrum and mixing of the neutralinos andc harginos in the MSSM/.In the gauge/-eigenstate basis / /0/=/( eB/;; fW /0/;; eH /0d /;; eH /0u /)/, the neutralino mass terms in thelagrangian areL//; /1 /2 /( / /0/) TMeN / /0/+c /: c /: /(/7/./4/5/)whereMeN /= /0BB/@ M/1 /0 /; c/ sW mZ s/ sW mZ/0 M/2 c/ cW mZ /; s/ cW mZ/; c/ sW mZ c/ cW mZ /0 /; /s/ sW mZ /; s/ cW mZ /; / /0 /1CCA /: /(/7/./4/6/)Here w eh a v ei n tro duced abbreviations s/ /=s i n / /, c/ /=c o s / /, sW /=s i n /W /, and cW /=cos /W /. The en tries M/1 and M/2 in this matrix come directly from the MSSM soft Lagrangian/[see eq/. /(/5/./1/1/)/] while the en tries /; / are the sup ersymmetric higgsino mass terms /[seeeq/. /(/5/./4/)/]/. The terms prop ortional to mZ are the result of Higgs/-higgsino/-gaugino couplings/[see eq/. /(/3/./7/2/) and Fig/. /5g/]/, with the Higgs scalars getting their VEVs /[eqs/. /(/7/./2/9/)/,/(/7/./3/0/) /]/.The mass matrix MeN can b e diagonalized b y a unitary matrix N with eNi /= Nij / /0j /, so thatM diageN /= N /MeN N /; /1/(/7/./4/7/)has p ositiv er e a le n tries meN/1 /, meN/2 /, meN/3 /, meN/4 on the diagonal/. These are the absolute v aluesof the eigen v alues of MeN /,o r e q u i v alen tly the square ro ots of the eigen v alues of M yeN MeN /.The indices /( i/;; j /)o n Nij are /(mass/, gauge/) eigenstate lab els/. The mass eigen v alues and themixing matrix Nij c a nb eg i v en in closed form in terms of the parameters M/1 /, M/2 /, / andtan / /, but the results are v ery complicated and not v ery illuminating/. yyOther common notations use e / /0i or eZi for neutralinos/, and e / /i or fW /i for c harginos/./6/5 In general/, the parameters M/1 /, M/2 /,a n d / can ha v e arbitrary complex phases/. Inthe broad class of minimal sup ergra vit y or gauge/-mediated mo dels satisfying the gauginouni/cation conditions eq/. /(/6/./2/7/) or /(/6/./4/0/)/, M/2 and M/1 will ha v e the same complex phasewhic h is preserv ed b yR Ge v olution eq/. /(/7/./5/)/. In that case/, a rede/nition of the phases of eBand fW allo ws us to mak e M/1 and M/2 b oth real and p ositiv e/. The phase of / is then reallyap h ysical parameter whic h cannot b e rotated a w a y /./[ W eh a v e already used up the freedomto rede/ne the phases of the Higgs /elds/, since w eh a v ep i c k ed b and h H /0u i and h H /0d i to b ereal and p ositiv e/, to guaran tee that the o//-diagonal en tries in eq/. /(/7/./4/6/) prop ortional to mZare real/./] Ho w ev er/, if / is not real/, then there can b e p oten tially disastrous CP/-violatinge/ects in lo w/-energy ph ysics/, including electric dip ole momen ts for b oth the electron andthe neutron/. Therefore/, it is usual /(although not mandatory b ecause of the p ossibilit yo fnon trivial cancellations/) to assume that / is real in the same set of phase con v en tions whic hmak e M/1 /, M/2 /, b /, h H /0u i and h H /0d i real and p ositiv e/. The sign of / is still undetermined b ythis constrain t/.In mo dels whic h satisfy eq/. /(/7/./7/)/, one has the nice predictionM/1 / /5 /3 tan /2/W M/2 / /0 /: /5 M/2 /(/7/./4/8/)at the electro w eak scale/. If so/, then the neutralino masses and mixing angles dep end on onlythree unkno wn parameters/. This assumption is su/cien tly theoretically comp elling that ithas b een made in almost all phenomenological studies/;; nev ertheless it should b e recognizedas an assumption/, to b e tested someda yb y exp erimen t/.Sp ecializi ng further/, there is an in teresting and not unlik ely limit in whic h electro w eaksymmetry breaking e/ects can b e view ed as a small p erturbation on the neutralino massmatrix/. IfmZ /j / / M/1 j /;; j / / M/2 j /(/7/./4/9/)then the neutralino mass eigenstates are v ery nearly eN/1 / eB /;; eN/2 / fW /0/;; eN/3 /;; eN/4 / /( eH /0u /eH /0d /) /= p /2/, with mass eigen v alues/:meN/1 /= M/1 /; m /2Z s /2W /( M/1 /+ / sin /2 / /) / /2/; M /2/1 /+ /:/:/: /(/7/./5/0/)meN/2 /= M/2 /; m /2W /( M/2 /+ / sin /2 / /) / /2/; M /2/2 /+ /:/:/: /(/7/./5/1/)meN/3 /;;meN/4 /= j / j /+ m /2Z /(/1 /; / sin /2 / /)/( j / j /+ M/1 c /2W /+ M/2 s /2W /) /2/( j / j /+ M/1 /)/( j / j /+ M/2 /) /+ /:/:/: /;; /(/7/./5/2/)j / j /+ m /2Z /(/1 /+ / sin /2 / /)/( j / j/; M/1 c /2W /; M/2 s /2W /) /2/( j / j/; M/1 /)/( j / j/; M/2 /) /+ /:/:/: /(/7/./5/3/)where w eh a v e assumed / is real with sign / /= / /1/. The lab eling of the mass eigenstates eN/1and eN/2 assumes M/1 /<M/2 /< j / j /;; otherwise the subscripts ma y need to b e rearranged/. Itturns out that a /\bino/-lik e/" LSP eN/1 can v ery easily ha v et h e r i g h t cosmological abundanceto mak e a go o d dark matter candidate/, so the large j / j limit ma y b e preferred from thatpo i n t of view/. In addition/, this limit tends to emerge from minimal sup ergra vit y b oundaryconditions on the soft parameters/, whic h often require j / j to b e larger than M/1 and M/2 inorder to get correct electro w eak symmetry breaking/./6/6 The c hargino sp ectrum can b e analyzed in a similar w a y /. In the gauge/-eigenstate basis/ //=/( fW /+/;; eH /+u /;; fW /;/;; eH /;d /)/, the c hargino mass terms in the lagrangian areL//; /1 /2 /( / //) TMeC / //+c /: c /: /(/7/./5/4/)where/, in /2 / /2 blo c k form/,MeC /= //0 X TX /0 //;; X /= /M/2 p /2 s/ mWp /2 c/ mW / //: /(/7/./5/5/)The mass eigenstates are related to the gauge eigenstates b yt w o unitary /2 / /2 matrices Uand V according to/eC /+/1eC /+/2 //= V /fW /+eH /+u //;; /eC /;/1eC /;/2 //= U /fW /;eH /;d //: /(/7/./5/6/)Note that there are di/eren t mixing matrices for the p ositiv ely c harged states and for thenegativ ely c harged states/. They are to b e c hosen so thatU /XV /; /1/= /meC/1 /0/0 meC/2 //: /(/7/./5/7/)Because these are only /2 / /2 matrices/, it is not hard to solv e for the masses explicitly/:m /2eC/1 /;;m /2eC/2 /= /1 /2 h/( j M/2 j /2/+ j / j /2/+/2 m /2W /)/ q /( j M/2 j /2/+ j / j /2/+/2 m /2W /) /2/; /4 j /M/2 /; m /2W sin /2 / j /2 i/: /(/7/./5/8/)It should b e noted that these are the /(doubly degenerate/) eigen v alues of the /4 / /4 matrixM yeC MeC /, or equiv alen tly the eigen v alues of X yX /, but they are not the squares of the eigen/-v alues of X /. In the limit of eq/. /(/7/./4/9/) with real M/2 and / /, one /nds that the c harginos masseigenstates consist of a wino/-lik e eC //1 and and a higgsino/-lik e eC //2 /, with massesmeC/1 /= M/2 /; m /2W /( M/2 /+ / sin /2 / /) / /2/; M /2/2 /+ /:/:/: /(/7/./5/9/)meC/2 /= j / j /+ m /2W /( j / j /+ /M/2 sin /2 / /) / /2/; M /2/2 /+ /:/:/: /: /(/7/./6/0/)Here again the lab eling assumes M/2 /< j / j /, and / is the sign of / /.A m usingly /, the ligh terc hargino eC/1 is nearly degenerate with the second ligh test neutralino eN/2 in this limit/, but thisis not an exact result/. Their higgsino/-lik e colleagues eN/3 /, eN/4 and eC/2 ha v e masses of order j / j /.The case of M/1 / /0 /: /5 M/2 /j / j is not uncommonly found in viable mo dels follo wing fromthe b oundary conditions in section /6/, and it has b een elev ated to the status of a b enc hmarkscenario in man y phenomenological studies/. Ho w ev er it cannot b e o v eremphasized that suc hexp ectations are not mandatory /.In practice/, the masses and mixing angles for the neutralinos and c harginos are b estcomputed n umerically /. The corresp onding F eynman rules ma y b e inferred in terms of N /,U and V from the MSSM lagrangian as discussed ab o v e/;; they are collected in Refs/. /1/9 /;; /9/6/6/7 /7/./4 The gluinoThe gluino is a color o ctet fermion/, so it cannot mix with an y other particle in the MSSM/,ev en if R /-parit y is violated/. In this regard/, it is unique among all of the MSSM sparticles/.In the mo dels follo wing from minimal sup ergra vit y or gauge/-mediated b oundary conditions/,the gluino mass parameter M/3 is related to the bino and wino mass parameters M/1 and M/2b y eq/. /(/7/./7/)M/3 /= /S / sin /2/W M/2 /= /3 /5 /S / cos /2/W M/1 /(/7/./6/1/)at an yR G scale/, up to small t w o/-lo op corrections/. If w e use v alues /S /=/0 /: /1/1/8/, / /=/1 /= /1/2/8/,sin /2/W /=/0 /: /2/3/, then one /nds the rough predictionM/3 /: M/2 /: M/1 / /7/: /2/:/1 /(/7/./6/2/)at the electro w eak scale/. In particular/, w e susp ect that the gluino should b e m uc h hea vierthan the ligh ter neutralinos and c harginos/.F or more precise estimates/, one m ust tak ei n to accoun t the fact that the parameter M/3is really a running mass whic h has an implicit dep endence on the R G scale Q /. Because thegluino is a strongly in teracting particle/, M/3 runs rather quic kly with Q /[see eq/. /(/7/./5/)/]/. A moreuseful quan tit yp h ysically is the R G scale/-indep en den tm a s s me g at whic h the renormalizedgluino propagator has a p ole/. Including one/-lo op corrections to the gluino propagator dueto gluon exc hange and quark/-squark lo ops/, one /nds that the p ole mass is giv en in terms ofthe running mass in the DR sc heme b y /8/8me g /= M/3 /( Q /) //1/+ /S /4 / /[ /1 /5/+ /6l n /( Q/= M/3 /)/+ XAeq /] //(/7/./6/3/)whereAeq /= Z/1/0 dx x ln/[ xm /2eq /= M /2/3 /+/( /1 /; x /) m /2q /= M /2/3 /; x /(/1 /; x /)/] /: /(/7/./6/4/)The sum in eq/. /(/7/./6/3/) is o v er all /1/2 squark/-quark sup erm ultiplets/, and w eh a v e neglectedsmall e/ects due to squark mixing/. It is easy to c hec k that requiring me g to b e indep enden to fQ in eq/. /(/7/./6/3/) repro duces the one/-lo op R G equation for M/3 /( Q /) in eq/. /(/7/./5/)/. The correctionterms prop ortional to /S in eq/. /(/7/./6/3/) can b e quite signi/can t/, so that me g /= M/3 /( M/3 /)c a nexceed unit yb y /2/5/% or more/. The reasons for this are that the gluino is strongly in teracting/,with a large group theory factor /[the /1/5 in eq/. /(/7/./6/3/)/] due to its color o ctet nature/, and thatit couples to all the squark/-quark pairs/. Of course/, there are similar corrections whic h relatethe running masses of all the other MSSM particles to their ph ysical masses/. These ha v eb een systematically ev aluated at one/-lo op order in Ref/. /1/0/5They are more complicated inform and usually n umerically smaller than for the gluino/, but in some cases they could b equite imp ortan t in future e/orts to connect a giv en candidate mo del for the soft terms toexp erimen tally measured masses and mixing angles of the MSSM particles/./7/./5 The squark and slepton mass sp e ctrumIn principle/, an y scalars with the same electric c harge/, R /-parit y /, and color quan tum n um b erscan mix with eac h other/. This means that with completely arbitrary soft terms/, the masseigenstates of the squarks and sleptons of the MSSM should b e obtained b y diagonalizing/6/8 three /6 / /6 /(mass/) /2matrices for up/-t yp e squarks /( euL /, ecL /, etL /, euR /, ecR /, etR /)/, do wn/-t yp e squarks/( edL /, esL /, ebL /, edR /, esR /, ebR /)/, and c harged sleptons /( eeL /, e/L /, e/L /, eeR /, e/R /, e/R /)/, and one /3 / /3matrix for sneutrinos /( e/e /, e// /, e// /)/. F ortunately /, the general h yp othesis of / a v or/-blind softparameters eqs/. /(/5/./1/4/) and /(/5/./1/5/) predicts that most of these mixing angles are v ery small/.The third/-family squarks and sleptons can ha v ev ery di/eren t masses compared to their/rst/- and second/-family coun terparts/, b ecause of the e/ects of large Y uk a w a/( yt /, yb /, y/ /)and soft /( at /, ab /, a/ /) couplings in the R G equations /(/7/./2/0/)/-/(/7/./2/4 /)/. F urthermore/, they canha v e substan tial mixing in pairs /( etL /, etR /)/, /( ebL /, ebR /) and /( e/L /, e/R /)/. In con trast/, the /rst/-and second/-family squarks and sleptons ha v e negligible Y uk a w a couplings/, so they end upin /7 v ery nearly degenerate/, unmixed pairs /( eeR /;; e/R /)/, /( e/e /;; e// /)/, /( eeL /;; e/L /)/, /( euR /;; ecR /)/, /( edR /;; esR /)/,/( euL /;; ecL /)/, /( edL /;; esL /)/. As w eh a v e already discussed in section /5/./4/, this a v oids the problem ofdisastrously large virtual sparticle con tributions to F CNC pro cesses/.Let us /rst consider the sp ectrum of /rst/- and second/-family squarks and sleptons/. Inmo dels /tting in to b oth of the broad categories of minimal sup ergra vit y /[eq/. /(/6/./2/8/)/] orgauge/-mediated /[eq/. /(/6/./4/2/)/] b oundary conditions/, their running masses can b e con v enien tlyparameterized in the follo wing w a y/:m /2Q/1 /= m /2Q/2 /= m /2/0 /+ K/3 /+ K/2 /+ /1 /3/6 K/1 /;; /(/7/./6/5/)m /2 u/1 /= m /2 u/2 /= m /2/0 /+ K/3 /+ /4 /9 K/1 /;; /(/7/./6/6/)m /2 d/1 /= m /2 d/2 /= m /2/0 /+ K/3 /+ /1 /9 K/1 /;; /(/7/./6/7/)m /2L/1 /= m /2L/2 /= m /2/0 /+ K/2 /+ /1 /4 K/1 /;; /(/7/./6/8/)m /2 e/1 /= m /2 e/2 /= m /2/0 /+ K/1 /: /(/7/./6/9/)In minimal sup ergra vit y mo dels/, m /2/0 is the common scalar /(mass/) /2whic h app ears in eq/. /(/6/./2/8/)/.It can b e /0 in the /\no/-scale/" limit/, but it could also b e the dominan t source of the scalarmasses/. The con tributions K/3 /, K/2 and K/1 are due to the R G running prop ortional to thegaugino masses/;; see eq/. /(/7/./1/4/)/. They are strictly p ositiv e/. A k ey p oin t is that the same K/3 /,K/2 and K/1 app ear ev erywhere in eqs/. /(/7/./6/5/)/-/(/7/./6/9/)/, since all of the c hiral sup erm ultipletscouple to the same gauginos with the same gauge couplings/. The di/eren t co e/cien ts infron to f K/1 just corresp ond to the v arious v alues of w eak h yp erc harge squared for eac hscalar/. The quan tities K/1 /, K/2 /, K/3 dep end on the R G scale Q at whic h they are ev aluated/.Explicitly /, they are found b y solving eq/. /(/7/./1/4/)/:Ka /( Q /)/= /8/</: /3 /= /5/3 /= /4/4 /= /3 /9/=/;; / /1 /2 / /2 Zln Q/0ln Q dt g /2a /( t /) j Ma /( t /) j /2/( a /=/1 /;; /2 /;; /3/) /: /(/7/./7/0/)Here Q/0 is the input R G scale at whic h the b oundary condition eq/. /(/6/./2/8/) is applied/, andQ should b e tak en to b e ev aluated near the squark and slepton mass under consideration/,presumably less than ab out /1 T eV or so/. The v alues of the running parameters ga /( Q /)a n dMa /( Q /) can b e found using eqs/. /(/5/./1/7/) and /(/7/./7/)/. If the input scale is appro ximated b ythe apparen t scale of gauge coupling uni/cation Q/0 /= MU / /2 / /1/0 /1/6GeV/, one /nds thatn umericallyK/1 / /0 /: /1/5 m /2/1 /= /2 /;; K/2 / /0 /: /5 m /2/1 /= /2 /;; K/3 / /(/4 /: /5t o/6 /: /5/) m /2/1 /= /2 /: /(/7/./7/1/)/6/9 for Q near /1 T eV/. Here m/1 /= /2 is the common gaugino mass parameter at the uni/cation scale/.Note that K/3 / K/2 / K/1 /;; this is a direct consequence of the relativ e sizes of the gaugecouplings g/3 /, g/2 /, and g/1 /. The large uncertain t yi n K/3 is due in part to the exp erimen taluncertain t y in the QCD coupling constan t/, and in part to the uncertain t y in where to c ho oseQ /, since K/3 runs rather quic kly b elo w/1 T eV/. If the gauge couplings and gaugino massesare uni/ed b et w een MU and MP /,a s w ould o ccur in a GUT mo del/, then the e/ect of R Grunning for MU /<Q/<MP can b e absorb ed in to a rede/nition of m /2/0 /. Otherwise/, it addsa further uncertain t y whic h is roughly prop ortional to ln/( MP /= MU /)/, compared to the largercon tributions in eq/. /(/7/./7/0/) whic h go roughly lik el n /( MU /= /1T eV/)/.In gauge/-mediated mo dels/, the same parameterization eqs/. /(/7/./6/5/)/-/(/7/./6/9 /) holds/, but m /2/0 isalw a ys /0/. A t the input scale Q/0 /, eac h MSSM scalar gets con tributions to its /(mass/) /2whic hdep end only on its gauge in teractions/, as in eq/. /(/6/./4/2/)/. It is not hard to see that in generalthese con tribute in exactly the same pattern as K/1 /, K/2 /,a n d K/3 in eq/. /(/7/./6/5/)/-/(/7/./6/9/)/. Thesubsequen te v olution of the scalar squared masses do wn to the electro w eak scale again justyields more con tributions to the K/1 /, K/2 /,a n d K/3 parameters/. It is somewhat more di/cultto giv e meaningful n umerical estimates for these parameters in gauge/-mediated mo dels thanin the minimal sup ergra vit y mo dels/, b ecause of uncertain ties in the messenger mass scale/(s/)and in the m ultiplicities of the messenger /elds/. Ho w ev er/, in the gauge/-mediated case onequite generally exp ects that the n umerical v alues of the ratios K/3 /=K/2 /, K/3 /=K/1 and K/2 /=K/1should b e ev en larger than in eq/. /(/7/./7/1/)/. There are t w o reasons for this/. First/, the runningsquark squared masses start o/ larger than slepton squared masses already at the inputscale in gauge/-mediated mo dels/, rather than ha ving a common v alue m /2/0 /.F urthermore/, inthe gauge/-mediated case/, the input scale Q/0 is t ypically m uc hl o w er than MP or MU /, so thatthe R Ge v olution giv es relativ ely more w eigh t to smaller R G scales where the hierarc hiesg/3 />g/2 />g/1 and M/3 />M/2 />M/1 are already in e/ect/.In general/, one therefore exp ects that the squarks should b e considerably hea vier thanthe sleptons/, with the e/ect b eing more pronounced in gauge/-mediated sup ersymmetry breaking mo dels than in minimal sup ergra vit y mo dels/. F or an y sp eci/c c hoice of mo del/, thise/ect can b e easily quan ti/ed with an R G analysis/. The hierarc h y msquark />mslepton tendsto hold ev en in mo dels whic h do not really /t in to an y of the categories outlined in section/6/, b ecause the R G con tributions to squark masses from the gluino are alw a ys presen ta n dusually quite large/, since QCD has a larger gauge coupling than the electro w eak in teractions/.There is also a /\h yp er/ne/" splitting in the squark and slepton mass sp ectrum pro ducedb y electro w eak symmetry breaking/. Eac h squark and slepton / will get a con tribution /
/to its /(mass/) /2/, coming from the SU /(/2/)L and U /(/1/)Y D /-term quartic in teractions /[see the lastterm in eq/. /(/3/./7/5/)/] of the form /(squark/) /2/(Higgs/) /2and /(slepton/) /2/(Higgs/) /2/, when the neutralHiggs scalars H /0u and H /0d get VEVs/. They are mo del/-indep e ndent for a giv en v alue of tan / /,and are giv en b y/
/ /=/( T //3 /; Q /EM sin /2/W /)c o s /2 /m /2Z /;; /(/7/./7/2/)where T //3 and Q /EM are the third comp onen to f w eak isospin and the electric c harge of thec hiral sup erm ultiplet to whic h / b elongs/. /[F or example/, /
u /=/( /1 /2 /; /2 /3 sin /2/W /)c o s /2 /m /2Zand /
u /=/( /2 /3 sin /2/W /)c o s /2 /m /2Z /]/. These D /-term con tributions are t ypically smaller thanthe m /2/0 and K/1 /, K/2 /, K/3 con tributions/, but should not b e neglected/. They split apart thecomp onen ts of the SU /(/2/)L /-doublet sleptons and squarks L/1 /=/( e/e /;; eeL /)/, etc/. Including them/,the /rst/-family squark and slepton masses are no w giv en b y/:m /2edL /= m /2/0 /+ K/3 /+ K/2 /+ /1 /3/6 K/1 /+/
d /;; /(/7/./7/3/)/7/0 m /2euL /= m /2/0 /+ K/3 /+ K/2 /+ /1 /3/6 K/1 /+/
u /;; /(/7/./7/4/)m /2euR /= m /2/0 /+ K/3 /+ /4 /9 K/1 /+/
u /;; /(/7/./7/5/)m /2edR /= m /2/0 /+ K/3 /+ /1 /9 K/1 /+/
d /;; /(/7/./7/6/)m /2e eL /= m /2/0 /+ K/2 /+ /1 /4 K/1 /+/
e /;; /(/7/./7/7/)m /2e / /= m /2/0 /+ K/2 /+ /1 /4 K/1 /+/
/ /;; /(/7/./7/8/)m /2e eR /= m /2/0 /+ K/1 /+/
e /;; /(/7/./7/9/)with iden tical form ulas for the second/-family squarks and sleptons/. The mass splittings forthe left/-handed squarks and sleptons are go v erned b y mo del/-indep en den t sum rulesm /2e eL /; m /2e /e /= m /2edL /; m /2euL /= /; cos /2 /m /2W /: /(/7/./8/0/)Since cos /2 //< /0 in the allo w ed range tan //> /1/, it follo ws that me eL />me /e and medL />meuL /,with the magnitude of the splittings constrained b y electro w eak symmetry breaking/.Let us next consider the masses of the top squarks/, for whic h there are sev eral non/-negligible con tributions/. First/, there are /(mass/) /2terms for et /L etL and et /R etR whic h are justequal to m /2Q/3 /+/
u and m /2 u/3 /+/
u /, resp ectiv ely /, just as for the /rst/- and second/-familysquarks/. Second/, there are con tributions equal to m /2t for eac ho f et /L etL and et /R etR /. Thesecome from F /-terms in the scalar p oten tial of the form y /2t H /0 /u H /0u et /L etL and y /2t H /0 /u H /0u et /R etR/(see Figs/. /8b and /8c/)/, with the Higgs /elds replaced b y their VEVs/. These con tributionsare of course presen t for all of the squarks and sleptons/, but they are m uc h to o small tow orry ab out except in the case of the top squarks/. Third/, there are con tributions to thescalar p oten tial from F /-terms of the form /; /yt e t etH /0 /d /+c /: c /: /;; see eqs/. /(/5/./6/) and Fig/. /1/0a/.These b ecome /; /v yt cos / et /R etL /+c /: c /: when H /0d is replaced b y its VEV/. Finally /, there arecon tributions to the scalar p oten tial from the soft /(scalar/) /3couplings at e t eQ/3 H /0u /+c /: c /: /[see the/rst term of the second line of eq/. /(/5/./1/1/) and eq/. /(/7/./8/)/]/, whic h b ecome at v sin / etL et /R /+c /: c /:when H /0u is replaced b y its VEV/. Putting these all together/, w eh a v ea/( m a s s /) /2matrix forthe top squarks/, whic h in the gauge/-eigenstate basis /( etL /, etR /)i s g i v en b y/;L / /( et /L et /R /) m /2et /etLetR //(/7/./8/1/)wherem /2et /= /m /2Q/3 /+ m /2t /+/
u v /( at sin / /; /yt cos / /)v /( at sin / /; /yt cos / /) m /2 u/3 /+ m /2t /+/
u //: /(/7/./8/2/)This matrix can b e diagonalized to giv e mass eigenstates/et/1et/2 //= /cos /et sin /et/; sin /et cos /et //etLetR //(/7/./8/3/)with m /2et/1 /<m /2et/2 b eing the eigen v alues of eq/. /(/7/./8/2/) and /0 / /et / / /. Because of the largeR G e/ects prop ortional to Xt in eq/. /(/7/./2/0/) and eq/. /(/7/./2/1/)/, at the electro w eak scale one/nds that m /2 u/3 /<m /2Q/3 /, and b oth of these quan tities are usually signi/can tly smaller thanthe squark squared masses for the /rst t w o families/. The diagonal terms m /2t in eq/. /(/7/./8/2/)/7/1 tend to mitigate this e/ect somewhat/, but the o//-diagonal en tries will t ypically induce asigni/can t mixing whic ha l w a ys reduces the ligh ter top/-squark /(mass/) /2eigen v alue/. F or thisreason/, it is often found in mo dels that et/1 is the ligh test squark of all/.Av ery similar analysis can b e p erformed for the b ottom squarks and c harged tau slep/-tons/, whic h in their resp ectiv e gauge/-eigenstate bases /( ebL /, ebR /) and /( e/L /, e/R /)h a v e /(mass/) /2matrices/:m /2eb /= / m /2Q/3 /+/
d v /( ab cos / /; /yb sin / /)v /( ab cos / /; /yb sin / /) m /2 d/3 /+/
d /!/;; /(/7/./8/4/)m /2e / /= /m /2L/3 /+/
e v /( a/ cos / /; /y/ sin / /)v /( a/ cos / /; /y/ sin / /) m /2 e/3 /+/
e //: /(/7/./8/5/)These can b e diagonalized to giv e mass eigenstates eb/1 /;; eb/2 and e//1 /;; e//2 in exact analogy witheq/. /(/7/./8/3/)/.The magnitude and imp ortance of mixing in the sb ottom and stau sectors dep ends onho w large tan / is/. If tan / is not to o large /(in practice/, this usually means less than ab out/1/0 or so/, dep ending on the situation under study/)/, the sb ottoms and staus do not get a v erylarge e/ect from the mixing terms and the R G e/ects due to Xb and X/ /, b ecause yb /;;y/ / ytfrom eq/. /(/7/./4/4/)/. In that case the mass eigenstates are v ery nearly the same as the gaugeeigenstates ebL /, ebR /, e/L and e/R /. The latter three/, and /~ // /, will b e nearly degenerate with their/rst/- and second/-family coun terparts with the same SU /(/3/)C / SU /(/2/)L / U /(/1/)Y quan tumn um b ers/. Ho w ev er/, ev en in the case of small tan / /, ebL will feel the e/ects of the large topY uk a w a coupling b ecause it is part of the doublet eQ/3 whic hc o n tains etL /. In particular/, fromeq/. /(/7/./2/0/) w e see that Xt acts to decrease m /2eQ/3 as it is R G/-ev olv ed do wn from the inputscale to the electro w eak scale/. Therefore the mass of ebL can b e signi/can tly less than themasses of edL and esL /.F or larger v alues of tan / /, the mixing in eqs/. /(/7/./8/4/) and /(/7/./8/5/) can b e quite signi/can t/,b ecause yb /, y/ and ab /, a/ are non/-negligible/. Just as in the case of the top squarks/, theligh ter sb ottom and stau mass eigenstates /(denoted eb/1 and e//1 /) can b e signi/can tly ligh terthan their /rst/- and second/-family coun terparts/. F urthermore/, e// can b e signi/can tly ligh terthan the nearly degenerate e/e /, e// /.The requiremen t that the third/-family squarks and sleptons should all ha v e p ositiv e/(mass/) /2implies limits on the sizes of at sin / /; /yt cos / /, ab cos / /; /yb sin / /, and a/ cos / /;/y/ sin / /. If they are to o large/, the smaller eigen v alue of eq/. /(/7/./8/2/)/, /(/7/./8/4/) or /(/7/./8/5/) will b edriv en negativ e/, implying that a squark or c harged slepton gets a VEV/, breaking SU /(/3/)C orelectromagnetism/. Since this is clearly unacceptable/, one can put b ounds on the /(scalar/) /3couplings/, or equiv alen tly on the parameter A/0 in minimal sup ergra vit y mo dels/. Ev en if allof the /(mass/) /2eigen v alues are p ositiv e/, the presence of large /(scalar/) /3couplings can yieldglobal minima of the scalar p oten tial with non/-zero squark and//or c harged slepton VEVswhic h are disconnected from the v acuum whic hc o n s e r v es SU /(/3/)C and electromagnetism/. /1/0/6Ho w ev er/, it is not alw a ys clear whether the non/-existence of suc h disconnected global minimashould really b e tak en as a constrain t/, b ecause the tunneling rate from our /\go o d/" v acuumto the /\bad/" v acua can easily b e m uc h longer than the age of the univ erse/. /1/0/7/7/./6 Summary/: the MSSM sp article sp e ctrumIn the MSSM there are /3/2 distinct masses corresp onding to undisco v ered particles/, notincludin g the gra vitino/. In this section w eh a v e explained ho w the masses and mixing/7/2 T able /3/: Undisco v ered particles in the Minimal Sup ersymmetric Standard Mo del Names Spin PR Mass Eigenstates Gauge Eigenstates Higgs b osons /0 /+/1 h /0H /0A /0H / H /0u H /0d H /+u H /;d euL euR edL edR /\/" squarks /0 /; /1 esL esR ecL ecR /\/" et/1 et/2 eb/1 eb/2 etL etR ebL ebR eeL eeR e/e /\/" sleptons /0 /; /1 e/L e/R e// /\/" e//1 e//2 e// e/L e/R e// neutralinos /1 /= /2 /; /1 eN/1 eN/2 eN/3 eN/4 eB /0fW /0eH /0u eH /0d c harginos /1 /= /2 /; /1 eC //1 eC //2 fW /eH /+u eH /;d gluino /1 /= /2 /; /1 eg /\/" gra vitino /=goldstino /3 /= /2 /; /1 eG /\/" angles for these particles can b e computed/, giv en an underlying mo del for the soft terms atsome input scale/. Assuming only that the mixing of /rst/- and second/-family squarks andsleptons is negligible/, the mass eigenstates of the MSSM are listed in T able /3/. A completeset of F eynman rules for the in teractions of these particles with eac h other and with theStandard Mo del quarks/, leptons/, and gauge b osons can b e found in Refs/. /1/9 /;; /9/6Sp eci/c mo delsfor the soft terms t ypically predict the masses and the mixing angles angles for the MSSMin terms of far few er parameters/. F or example/, in the minimal sup ergra vit y mo dels/, one hasonly the parameters m /2/0 /, m/1 /= /2 /, A/0 /, / /,a n d b whic h are not already measured b y exp erimen t/.On the other hand/, in gauge/-mediated sup ersymmetry breaking mo dels/, the free parametersinclude at least the scale //, the t ypical messenger mass scale Mmess /, the in teger n um be rN/5 of copies of the minimal messengers/, the goldstino deca y constan t h F i /, and the Higgsmass parameters / and b /. After R Ge v olving the soft terms do wn to the electro w eak scale/,one can imp ose that the scalar p oten tial giv es correct electro w eak symmetry breaking/. Thisallo ws us to trade j / j and b /(or B/0 /) for one parameter tan / /, as in eqs/. /(/7/./3/1/)/-/(/7/./3/2/)/. So/,to a reasonable appro ximation/, the en tire mass sp ectrum in minimal sup ergra vit y mo delsis determined b yo n l y / v e unkno wn parameters/: m /2/0 /, m/1 /= /2 /, A/0 /, tan / /,a n dA r g /( / /)/, while inthe simplest gauge/-mediated sup ersymmetry breaking mo dels one can pic k parameters //,Mmess /, N/5 /, h F i /, tan / /, and Arg/( / /)/. Both framew orks are highly predictiv e/. Of course/, itis easy to imagine that the essen tial ph ysics of sup ersymmetry breaking is not captured b yeither of these t w o scenarios in their minimal forms/.While it w ould b e a mistak e to underestimate the uncertain ties in the MSSM massand mixing sp ectrum/, it is also useful to k eep in mind some general lessons that recur inv arious di/eren t scenarios/. Indeed/, there has emerged a sort of folklore concerning lik elyfeatures of the MSSM sp ectrum/, whic h is partly based on theoretical bias and partly on theconstrain ts inheren ti na n y sup ersymmetric theory /.W e remark on these features mainlyb ecause they represen t the prev ailing prejudice among sup ersymmetry theorists/, whic hi s/7/3 certainly a useful thing for the reader to kno we v en if he or she wisely decides to remainsk eptical/. F or example/, it is p erhaps not unlik ely that/:/ The LSP is the ligh test neutralino eN/1 /, unless the gra vitino is ligh ter or R /-parit y is notconserv ed/. If //> M/1 /;;M/2 /, then eN/1 is lik ely to b e bino/-lik e/, with a mass roughly /0/./5times the masses of eN/2 and eC/1 /. In the opp osite case //<M/1 /;;M/2 /, then eN/1 has a largehiggsino con ten ta n d eN/2 and eC/1 are not m uc h hea vier/./ The gluino will b e m uc h hea vier than the ligh ter neutralinos and c harginos/. Thisis certainly true in the case of the /\standard/" gaugino mass relation eq/. /(/7/./7/)/;; moregenerally /, the running gluino mass parameter gro ws relativ ely quic kly as it is R G/-ev olv ed in to the infrared b ecause the QCD coupling is larger than the electro w eakgauge couplings/. So ev en if there are big corrections to the gaugino mass b oundaryconditions eqs/. /(/6/./2/7/) or /(/6/./4/0/)/, the gluino mass parameter M/3 is lik ely to come outlarger than M/1 and M/2 /./ The squarks of the /rst and second families are nearly degenerate and m uc h hea vierthan the sleptons/. This is b ecause eac h squark mass gets the same large p ositiv e/-de/nite radiativ e corrections from lo ops in v olving the gluino/. The left/-handed squarkseuL /, edL /, esL and ecL are lik ely to b e hea vier than their righ t/-handed coun terparts euR /,edR /, esR and ecR /, b ecause of the e/ect of K/2 in eqs/. /(/7/./7/3/)/-/(/7/./7/9/) /./ The squarks of the /rst t w o families cannot b e ligh ter than ab out /0/./8 times the mass ofthe gluino in minimal sup ergra vit y mo dels/, and ab out /0/./6 times the mass of the gluinoin the simplest gauge/-mediated mo dels as discussed in section /6/./4 if the n um be r o fmessenger squark pairs is N/5 / /4/. In the minimal sup ergra vit y case this is b ecause thegluino mass feeds in to the squark masses through R Ge v olution/;; in the gauge/-mediatedcase it is b ecause the gluino and squark masses are tied together b y eqs/. /(/6/./4/0/) and/(/6/./4/2/) /[m ultiplied b y N/5 /, as explained at the end of section /6/./4/]/./ The ligh ter stop et/1 and the ligh ter sb ottom eb/1 are probably the ligh test squarks/. Thisis b ecause stop and sb ottom mixing e/ects and the e/ects of Xt and Xb in eqs/. /(/7/./2/0/)/-/(/7/./2/2/) b oth tend to decrease the ligh ter stop and sb ottom masses/./ The ligh test c harged slepton is probably a stau e//1 /. The mass di/erence me eR /; me //1 islik ely to b e signi/can ti f t a n / is large/, b ecause of the e/ects of a large tau Y uk a w acoupling/. F or smaller tan / /, e//1 is predominan tly e/R and it is not so m uc h ligh ter thaneeR /, e/R /./ The left/-handed c harged sleptons eeL and e/L are lik ely to b e hea vier than their righ t/-handed coun terparts eeR and e/R /. This is b ecause of the e/ect of K/2 in eq/. /(/7/./7/7/)/./(Note also that /
e /; /
e is p ositiv e but v ery small b ecause of the n umerical acciden tsin /2/W / /1 /= /4/./)/ The ligh test neutral Higgs b oson h /0should b e ligh ter than ab out /1/5/0 GeV/, and ma ybe m uc h ligh ter than the other Higgs scalar mass eigenstates A /0/, H //, H /0/.In Figure /1/8 w es h o w a qualitativ es k etc h of a sample MSSM mass sp ectrum whic h illustratesthese features/. V ariations in the mo del parameters can ha v e imp ortan t and predictablee/ects/. F or example/, taking larger /(smaller/) m /2/0 in minimal sup ergra vit y mo dels will tendto mo v e the en tire sp ectrum of squarks/, sleptons and the Higgs scalars A /0/, H //, H /0higher/7/4 N1N2 C1N3, N4 C2g eRνe, eL µRνµ, µL τ2, ντdR, uRuL, dL sR, cRcL, sL τ1t1b1b2, t2 h0A0, H0, H+MassFigure /1/8/: A sc hematic sample sp ectrum for the undisco v ered particles in the MSSM/. This sp ectrum ispresen ted for en tertainmen t purp oses only /.N o w arran t y /, expressed or implied/, guaran tees that this sp ectrumlo oks an ything lik e the real w orld/./(lo w er/) compared to the neutralinos/, c harginos and gluino/;; taking larger v alues of tan / withother mo del parameters held /xed will usually tend to lo w er eb/1 and e//1 masses compared tothose of the other sparticles/, etc/. The imp ortan tp o i n t is that b y measuring the masses andmixing angles of the MSSM particles w e will b e able to gain a great deal of information whic hcan rule out or b olster evidence for comp eting prop osals for the origin of sup ersymmetrybreaking/. T esting the v arious p ossible organizing principles will pro vide the high/-energyph ysicists of the next millennium with an exciting c hallenge/./8 Sparticle deca ysIn this section w e will giv e a brief qualitativ eo v erview of the deca y patterns of sparticlesin the MSSM/, assuming that R /-parit y is exactly conserv ed/. W e will consider in turn thep ossible deca ys of neutralinos/, c harginos/, sleptons/, squarks/, and the gluino/. If/, as is mostoften assumed/, the ligh test neutralino eN/1 is the LSP /, then all deca yc hains will end upcon taining it in the /nal state/. In section /8/./5 w e consider the alternativ e p ossibilit y thatthe gra vitino//goldstino eG is the LSP /./8/./1 De c ays of neutr alinos and char ginosLet us /rst consider the p ossible t w o/-b o dy deca ys/. Eac h neutralino and c hargino con tainsat least a small admixture of the electro w eak gauginos eB /, fW /0or fW //,a s w es a w in section/7/./3/. So eNi and eCi inherit couplings of w eak in teraction strength to /(scalar/, fermion/) pairs/,as sho wn in Fig/. /9b/,c/. If sleptons or squarks are su/cien tly ligh t/, a neutralino or c harginocan therefore deca yi n to lepton/+slepton or quark/+squark/. /(W e will often not distinguishbe t w een particle and an tiparticle names and lab els in this section/./) Since sleptons areprobably ligh ter than squarks/, the lepton/+slepton /nal states are more lik ely to b e op en/.A neutralino or c hargino ma y also deca yi n to an y ligh ter neutralino or c hargino plus aHiggs scalar or an electro w eak gauge b oson/, b ecause they inherit the gaugino/-higgsino/-Higgs /(see Fig/. /9b/,c/) and SU /(/2/)L gaugino/-gaugino/-v ector b oson /(see Fig/. /5c/) couplings oftheir comp onen ts/. So/, the p ossible t w o/-b o dy deca y mo des for neutralinos and c harginos in/7/5 the MSSM are/:eNi /! Z eNj /;; W eCj /;; h /0eNj /;; /` e/` /;; / e//;; /[ A /0eNj /;; H /0eNj /;; H /eC /j /;; q eq /]/;; /(/8/./1/)eCi /! W eNj /;; Z eC/1 /;; h /0eC/1 /;; /` e//;; / e/`/;; /[ A /0eC/1 /;; H /0eC/1 /;; H /eNj /;; q eq /0/] /;; /(/8/./2/)using a generic notation / /, /` /, q for neutrinos/, c harged leptons/, and quarks/. The /nal statesin brac k ets are the more kinematically/-implausi bl e ones/. /(Since h /0is required to b e ligh t/, itis the most lik ely of the Higgs scalars to app ear in these deca ys/./) F or the hea vier neutralinosand c hargino /( eN/3 /, eN/4 and eC/2 /)/, one or more of the deca ys in eqs/. /(/8/./1/) and /(/8/./2/) is lik elyto b e kinematically allo w ed/. Ho w ev er/, it ma y b e that all of these t w o/-b o dy mo des arekinematically forbidden for a giv en c hargino or neutralino/, esp ecially in the case of eC/1 andeN/2 deca ys/. If so/, then one has three/-b o dy deca yseNi /! ff eNj /;; eNi /! ff /0eCj /;; eCi /! ff /0eNj /;; and eC/2 /! ff /0eC/1 /;; /(/8/./3/)through the same /(but no w o//-shell/) gauge b osons/, Higgs scalars/, sleptons/, and squarksthat app eared in the t w o/-b o dy deca ys eqs/. /(/8/./1/) and /(/8/./2/)/. Here f is generic notation for alepton or quark/, with f and f /0b elonging to the same SU /(/2/)L m ultiplet/. The c hargino andneutralino deca y widths in to the v arious /nal states can b e found in Ref/. /1/0/8 /;; /1/0/9The deca yseC //1 /! /` // eN/1 /;; eN/2 /! /` /+/` /;eN/1 /(/8/./4/)can b e particularly imp ortan t for phenomenology /, b ecause the leptons in the /nal stateoften will result in clean signals/. In certain regions of parameter space/, the ab o v e deca yscan b e suppressed b y kinematics or b y coupling/, and one/-lo op deca ys /(notably eN/2 /! / eN/1 /)migh t pla y an imp ortan t role/. /1/1/0/8/./2 Slepton de c aysSleptons ha v et w o/-b o dy deca ys in to a lepton and a c hargino or neutralino/, b ecause of thegaugino admixture of the latter/, as can b e seen directly from the couplings in Figs/. /9b/,c/.The t w o/-b o dy deca yse/` /! /` eNi /;; e/` /! / eCi /;; e/ /! / eNi /;; e/ /! /` eCi /(/8/./5/)are therefore of w eak in teraction strength/. In particular/, the direct deca yse/` /! /` eN/1 and e/ /! / eN/1 /(/8/./6/)are /(essen tially y/) alw a ys kinematically allo w ed if eN/1 is the LSP /.H o w ev er/, if the sleptons aresu/cien tly hea vy /, then the t w o/-b o dy deca ys to c harginos and hea vier neutralinos can b eimp ortan t/, esp eciallye/` /! / eC/1 /;; e/` /! /` eN/2 /;; and e/ /! /` eC/1 /: /(/8/./7/)The righ t/-handed sleptons do not ha v e a coupling to the SU /(/2/)L gauginos/, so they t ypicallyprefer the direct deca y e/`R /! /` eN/1 /,i f eN/1 is bino/-lik e/. In con trast/, the left/-handed sleptonsma y prefer to deca y as in eq/. /(/8/./7/) rather than the direct deca ys to the LSP as in eq/. /(/8/./6/)/,if the former is kinematically op en and if eC/1 and eN/2 are mostly wino/. This is b ecause theslepton/-lepton/-wino in teractions in Fig/. /9b are prop ortional to the SU /(/2/)L gauge coupling g /,whereas the slepton/-lepton/-bino in teractions in Fig/. /9c are prop ortional to the m uc h smallerU /(/1/)Y coupling g /0/. General results for these deca y widths can b e found in Ref/. /1/0/9 yAn exception o ccurs if the mass di/erence me //1 /; me N/1 is less than m/ /./7/6 /8/./3 Squark de c aysIf the deca y eq /! q eg is kinematically allo w ed/, it will alw a ys dominate/, b ecause the quark/-squark/-gluino v ertex in Fig/. /9a has QCD strength/. Otherwise/, the squarks can deca yi n to aquark plus neutralino or c hargino/: eq /! q eNi or q /0eCi /. The direct deca y to the LSP eq /! q eN/1is alw a ys kinematically fa v ored/, and for righ t/-handed squarks it can dominate b ecause eN/1is mostly bino/. Ho w ev er/, the left/-handed squarks ma y strongly prefer to deca yi n to hea vierc harginos or neutralinos instead/, for example eq /! q eN/2 or q /0eC/1 /, b ecause the relev an t squark/-quark/-wino couplings are m uc h bigger than the squark/-quark/-bino couplings/. Squark deca ysto higgsino/-lik ec harginos and neutralinos are less imp ortan t/, except in the cases of stopsand sb ottoms whic hh a v e sizeable Y uk a w a couplings/. The gluino/, c hargino or neutralinoresulting from the squark deca y will in turn deca y /, and so on/, un til a /nal state con tainingeN/1 is reac hed/. This can result in v ery n umerous and complicated deca yc hain p ossibilitie scalled cascade deca ys/. /1/1/1Sp ecial atten tion m ust b e pa y ed to the top squark/, b ecause it isp ossible that the deca ys et/1 /! t eg and et/1 /! t eN/1 are b oth kinematically forbidden/. If so/, thenthe stop ma y deca y only in to c harginos/, b y et/1 /! b eC/1 /.I f e v en this deca y is kinematicallyclosed/, then the stop has only the / a v or/-suppressed deca yt o ac harm quark/: et/1 /! c eN/1 /. Thisdeca y can b e v ery slo w/, /1/1/2so that the ligh test stop can b e quasi/-stable on the time scalerelev an t for collider ph ysics/, and can hadronize and form b ound states inside the detector/./8/./4 Gluino de c aysThe deca y of the gluino can only pro ceed through an on/-shell or a virtual squark/. If t w o/-b o dydeca ys eg /! q eq are op en/, they will dominate/, again b ecause the relev an t gluino/-quark/-squarkcoupling in Fig/. /9a has QCD strength/. Since the top and b ottom squarks can easily b e m uc hligh ter than all of the other squarks/, it is quite p ossible that eg /! t et/1 and//or eg /! b eb/1 arethe only a v ailable t w o/-b o dy deca y mo de/(s/) for the gluino/, in whic h case they will dominateo v er all others/. If instead all of the squarks are hea vier than the gluino/, the gluino willdeca y only through o//-shell squarks/, so eg /! qq /0eNi and qq /0eCi /. The squarks/, neutralinosand c harginos in these /nal states will then deca y as discussed ab o v e/, so there can b e v eryman y comp eting gluino deca yc hains/. These cascade deca ys can ha v e /nal/-state branc hingfractions that are individuall y small and quite sensitiv e to the parameters of the mo del/./8/./5 De c ays to the gr avitino//goldstinoMost phenomenological studies of sup ersymmetry assume explicitly or implicitl y that theligh test neutralino is the LSP /.T h i s i s t ypically the case in gra vit y/-mediated mo dels forthe soft terms/. Ho w ev er/, in gauge/-mediated mo dels /(and in /\no/-scale/" mo dels/)/, the LSPis instead the gra vitino/. As w es a w in section /6/./2/, a v ery ligh tg r a vitino ma yb e r e l e v an tfor collider phenomenology /, b ecause it con tains as its longitudinal comp onen t the goldstino/,whic h has a non/-gra vitational coupling to all sparticle/-particle pairs /( eX/;; X /)/. The deca yrate found in eq/. /(/6/./2/2/) for eX /! X eG is usually not fast enough to comp ete with the otherdeca ys of sparticles eX as men tioned ab o v e/, exc ept in the case that eX is the next/-to/-ligh testsup ersymmetric particle /(NLSP/)/. Since the NLSP has no comp eting deca ys/, it should alw a ysdeca yi n to its sup erpartner and the LSP gra vitino/.In principle/, an y of the MSSM sup erpartners could b e the NLSP in mo dels with a ligh tgoldstino/, but most mo dels with gauge/-mediation of sup ersymmetry breaking ha v e eithera neutralino or a c harged lepton pla ying this role/. The argumen t for this can b e seenimmediately from eqs/. /(/6/./4/8/) and /(/6/./4/9/)/;; since //1 /<//2 /;;//3 /, those sup erpartners whic hh a v e/7/7 only U /(/1/)Y in teractions will tend to get the smallest masses/. The gauge/-eigenstate sparticleswith this prop ert y are the bino and the righ t/-handed sleptons eeR /, e/R /, e/R /, so the appropriatecorresp onding mass eigenstates should b e plausible candidates for the NLSP /.First supp ose that eN/1 is the NLSP in ligh t goldstino mo dels/. Since eN/1 con tains an ad/-mixture of the photino /(the linear com bination of bino and neutral wino whose sup erpartneris the photon/)/, from eq/. /(/6/./2/2/) it should then deca yi n to photon /+ goldstino//gra vitino witha width giv en b y/;/( eN/1 /! / eG /)/= /2 / /1/0 /; /3//1 / /meN/1 /1/0/0 GeV //5 / p h F i /1/0/0 T eV /!/; /4eV /: /(/8/./8/)Here //1 / /j N/1/1 cos /W /+ N/1/2 sin /W j /2is the /\photino con ten t/" of eN/1 /, in terms of theneutralino mixing matrix Nij de/ned b y eq/. /(/7/./4/7/)/. W eh a v e normalized meN/1 and p h F ito /(v ery roughly/) minim um exp ected v alues in gauge/-mediated mo dels/. This width is m uc hsmaller than for a t ypical / a v or/-unsuppressed w eak in teraction deca y /, but it is still largeenough to allo w eN/1 to deca y b efore it has left a collider detector/, if p h F i is less than a fewthousand T eV in gauge/-mediated mo dels/, or equiv alen tly if m/3 /= /2 is less than a k eV or sowhen eq/. /(/6/./2/1/) holds/. In fact/, from eq/. /(/8/./8/)/, the mean deca y length of an eN/1 with energyE in the lab frame isd /=/9 /: /9 / /1/0 /; /3 /1 //1 / /( E /2/=m /2eN/1 /; /1/) /1 /= /2 /meN/1 /1/0/0 GeV //; /5 / p h F i /1/0/0 T eV /!/4cm /;; /(/8/./9/)whic h could b e an ywhere from sub/-micron to m ulti/-kilometer dep ending on the scale ofsup ersymmetry breaking p h F i /. /(In other mo dels with a gra vitino LSP whic h are not de/-scrib ed b y F /-term breaking of global sup ersymmetry /, including certain /\no/-scale/" mo dels/, /1/1/3the same form ulas ma y b e applied with h F i/! p /3 m/3 /= /2 MP /./)Of course/, eN/1 is not a pure photino/, but con tains also admixtures of the sup erpartnerof the Z b oson and the neutral Higgs scalars/. So/, one can also ha v e /6/8eN/1 /! Z eG /, h /0eG /,A /0eG /,o r H /0eG /, with deca y widths giv en in Ref/. /6/9Of these deca ys/, the last t w o are unlik elyto b e kinematically allo w ed/, and only the eN/1 /! / eG mo de is guaran teed to b e kinematicallyallo w ed for a gra vitino LSP /.F urthermore/, ev en if they are op en/, the deca ys eN/1 /! Z eG andeN/1 /! h /0eG are sub ject to strong kinematic suppressions prop ortional to /(/1 /; m /2Z /=m /2eN/1 /) /4and/(/1 /; m /2h /0 /=m /2eN/1 /) /4/, resp ectiv ely /, in view of eq/. /(/6/./2/2/)/. Still/, these deca ys ma y pla y an imp ortan trole in phenomenology if p h F i is not to o large/, eN/1 has a sizeable zino or higgsino con ten t/,and meN/1 is signi/can tly greater than mZ or mh /0 /.Ac harged slepton mak es another lik ely candidate for the NLSP /. Actually /, it is imp ortan tto note that more than one slepton can act e/ectiv ely as the NLSP /,e v en though one of themis sligh tly ligh ter/, if they are su/cien tly degenerate in mass so that eac h has no kinematicallyallo w ed deca ys except to the goldstino/. In GMSB mo dels/, the squared masses obtained b yeeR /, e/R and e/R are equal b ecause of the / a v or/-blindness of the gauge couplings/. Ho w ev er/,this is not the whole story /, b ecause one m ust tak ei n to accoun t mixing with eeL /, e/L /,a n de/L and renormalization group running/. These e/ects are v ery small for eeR and e/R b ecauseof the tin y electron and m uon Y uk a w a couplings/, so w e can quite generally treat them asdegenerate/, unmixed mass eigenstates/. In con trast/, e/R usually has a quite signi/can t mixingwith e/L /, prop ortional to the tau Y uk a w a coupling/. This means that the ligh ter stau masseigenstate e//1 is pushed lo w er in mass than eeR or e/R /,b y an amoun t that dep ends moststrongly on tan / /.I f t a n / is not to o large then the stau mixing e/ect lea v es the slepton/7/8 mass eigenstates eeR /, e/R /,a n d e//1 degenerate to within less than m/ / /1 /: /8 GeV/, so theyact e/ectiv ely as co/-NLSPs/. In particular/, this means that ev en though the stau is sligh tlyligh ter/, the three/-b o dy slepton deca ys eeR /! e/ /e/ //1 and e/R /! // /e/ //1 are not kinematicallyallo w ed/;; the only allo w ed deca ys for the three ligh test sleptons are eeR /! e eG and e/R /! / eGand e//1 /! / eG /. This situation is called the /\slepton co/-NLSP/" scenario/.F or larger v alues of tan / /, the ligh ter stau eigenstate e//1 is more than /1 /: /8 GeV ligh ter thaneeR and e/R and eN/1 /. This means that the deca ys eN/1 /! / e//1 and eeR /! e/ e//1 and e/R /! // e//1are op en/. Then e//1 is the sole NLSP /, with all other MSSM sup ersymmetric particles ha vingkinematically allo w ed deca ys in to it/. This is called the /\stau NLSP/" scenario/.In an y case/, a slepton NLSP can deca yl i k e e/` /! /` eG according to eq/. /(/6/./2/2/)/, with a widthand deca y length just giv en b y eqs/. /(/8/./8/) and /(/8/./9/) with the replacemen ts //1 / /! /1a n dmeN/1 /! me/` /. So/, just as for the neutralino NLSP case/, the deca y e/` /! /` eG can b e either fastor v ery slo w/, dep ending on the scale of sup ersymmetry breaking/.If p h F i is larger than roughly /1/0 /3T eV /(or the gra vitino is hea vier than a k eV or so/)/,then the NLSP is so long/-liv ed that it will usually escap e a t ypical collider detector/. If eN/1is the NLSP /, then/, it migh ta sw ell b e the LSP from the p oin t of view of collider ph ysics/.Ho w ev er/, the deca yo f eN/1 in to the gra vitino is ob viously still crucial for cosmology /, since anunstable eN/1 is clearly not a go o d dark matter candidate while the gra vitino LSP conceiv ablycould b e/. On the other hand/, if the NLSP is a long/-liv ed c harged slepton/, then one can seeits trac ks /(or p ossibly deca y kinks/) inside a collider detector/. /6/8The presence of a massiv ec harged NLSP can b e established b y measuring its anomalously high ionization rate or itstime/-of/-/ igh t in the detector/./9 Exp erimen tal signals for sup ersymmetrySo far/, the exp erimen tal study of sup ersymmetry has unfortunately b een con/ned to settinglimits/. As w eh a v e already remark ed in section /5/./4/, there can b e indirect signals for sup er/-symmetry from pro cesses that are rare or forbidden in the Standard Mo del but can ha v econ tributions from lo ops in v olving virtual sparticles/. These include / /! e/ /, b /! s/ /, neu/-tral meson mixing/, electric dip ole momen ts for the neutron and the electron/, etc/. There arealso virtual sparticle e/ects on Standard Mo del predictions lik e Rb /(the fraction of b b pairsin hadronic Z deca ys/)/. /1/1/4Extensions of the MSSM /(GUT and otherwise/) can quite easilypredict proton deca y and neutron/-an tineutron oscillations at lo w but observ able rates/, ev enif R /-parit y is exactly conserv ed/. Ho w ev er/, it w ould b e quite di/cult to ascrib e a p ositiv eresult for an y of these pro cesses to sup ersymmetry in an unam biguous w a y /. There is nosubstitute for the direct detection of sparticles/. In this section w e will giv e an incompleteand en tirely qualitativ e review of some of the p ossible signals for direct detection of sup er/-symmetry /. The reader is encouraged to consult Refs/. /2/7 /;; /3/4 /;; /1/1/5for recen t reviews whic hc o v erthe sub ject more systematically /./9/./1 Signals at e /+e /;c ol lidersA t e /+e /;colliders/, sparticles /(other than the gluino/) can b e pair/-pro duced through tree/-lev elpro cesses/:e /+e /;/! eC /+i eC /;j /;; eNi eNj /;; e/` e/`/;; e/ e//;; eq eq/: /(/9/./1/)with cross/-sections determined just b y the electro w eak gauge couplings and the sparticlemixings/. All of the pro cesses in eq/. /(/9/./1/) get con tributions from the s /-c hannel exc hange of/7/9 e−e+e+, µ+, τ+ e−, µ−, τ−γ, Z (a)e−e+e+ e−Ni (b)Figure /1/9/: Diagrams con tributing to slepton pair/-pro duction at e /+e /;colliders/.the Z b oson and /(for c harged sparticle pairs/) of the photon/. In the cases of eC /+i eC /;j /, eNi eNj /,eeR eeR /, eeL eeL and e/e e/e pro duction/, there are also t /-c hannel con tributions from the exc hangesof a virtual sneutrino/, selectron/, neutralino/, neutralino and c hargino/, resp ectiv ely /. The t /-c hannel con tributions are quite signi/can t if the exc hanged sparticle is not to o hea vy /,a n din terference b et w een the s /- and t /-c hannel con tributions can b e either destructiv e or construc/-tiv e/. F or example/, the pro duction of wino/-lik e eC /+/1 eC /;/1 pairs t ypically su/ers a destructiv ein terference b et w een the s /-c hannel graphs with / /;; Z exc hange and the t /-c hannel graphs withe/e exc hange/, if the sneutrinos are not to o hea vy /. In the case of sleptons/, the pair/-pro ductionof sm uons and staus pro ceeds only through the s /-c hannel diagrams of Fig/. /1/9a/, while selec/-tron pro duction also has a con tribution from the t /-c hannel exc hanges of the neutralinos/, assho wn in Fig/. /1/9b/. /[W eh a v e dra wn the neutralino line as if it w ere a pure gaugino/, sincethe gaugino comp onen ts of eNi are resp onsible for the coupling to electron/-selectron/./] F orthis reason/, selectron pro duction ma y b e signi/can tly larger than sm uon or stau pro ductionat e /+e /;colliders/. The imp ortan ti n teractions for sparticle pro duction pro cesses are alw a ysof electro w eak in teraction strength/, namely the ones sho wn in Figs/. /9b/,c and the ordinarygauge in teractions/. The cross sections are to o complicated to b e listed here/, but can b efound in Ref/. /1/0/9The pair/-pro duced sparticles will deca y as discussed in section /8/. If the LSP is the ligh t/-est neutralino/, it will alw a ys escap e the detector b ecause it has no strong or electromagneticin teractions/. Therefore ev ery ev en t will ha v et w o LSPs lea ving the detector/, so there will b eat least /2 meN/1 of missing energy /( // E /)/. F or example/, in the case of eC /+/1 eC /;/1 pro duction/, the p os/-sible signals include a pair of acollinear leptons plus // E /, one lepton and a pair of jets plus // E /,and m ultiple jets plus // E /. The relativ e imp ortance of these signals dep ends on the branc hingfraction of the c hargino in to the comp eting c hannels eC/1 /! /`/ eN/1 and qq /0eN/1 /. In the case ofslepton pair/-pro duction/, the signal should b e t w o energetic/, acollinear/, same/-/ a v or leptonsplus // E /. It is not di/cult to construct the other p ossible signatures for sparticle pairs/, whic hcan b ecome quite complicated for the hea vier c harginos/, neutralinos and squarks/.A t the CERN LEP e /+e /;collider/, one has a reasonable p ossibilit y of seeing neutralino/,c hargino/, c harged slepton/, sneutrino/, or top/-squark pairs/. In the LEP/1 runs at p s /= mZ /, themeasuremen to ft h e i n visible deca y width of the Z b oson placed a lo w er b ound on sneutrinomasses of ab out /4/0 GeV/, ev en though they can deca y completely in visibly lik e e/ /! / eN/1 /.Similarly /, the con tribution of eN/1 eN/1 to the in visible width of the Z rules out a signi/can tregion of parameter space/, with a lo w er b ound on meN/1 whic h unfortunately dep ends stronglyon the other parameters/. Mo del/-indep endent lo w er b ounds ha v e b een set on the c hargedsparticle masses of roughly mZ /= /2/. A t this writing/, LEP/2 upgrades at p s /= /1/3/0/-/1/4/0/, /1/6/1/,/1/7/2/, /1/8/3/, /1/8/9 GeV and b ey ond are con tin uing to raise the lo w er b ounds on the ligh test/\visible/" sparticles/. It is w orth noting that in all future e /+e /;collider searc hes/, there will b ea large bac kground for the acollinear leptons plus // E and the lepton plus jets plus // E signals/8/0 from W /+W /;pro duction with one or b oth of the W b osons deca ying leptonically /.H o w ev er/,these and other Standard Mo del bac kgrounds can b e k ept under con trol with clev er cuts/.It should also b e men tioned that LEP/2 is conducting a promising searc h for the ligh testHiggs b oson of sup ersymmetry through e /+e /;/! h /0Z or p erhaps e /+e /;/! h /0A /0/.H o w ev er/,observ ation of the Higgs at LEP/2 w ould b e only a p o w erful clue that w e are on the righ ttrac k in pursuing sup ersymmetry /, and not a pro of/. Con v ersely /, the non/-observ ation of h /0at LEP/2 should not b e construed as evidence against sup ersymmetry /, in view of eqs/. /(/7/./4/2/)and /(/7/./4/3/)/.A t a future linear e /+e /;collider with p s /=af e w h undred GeV to /1/./5 T eV/, the pro/-cesses in eq/. /(/9/./1/) should b e prob ed close to the kinematic limit/, giv en su/cien ti n tegratedluminosit y /. /1/1/6In the case of e/ e/ pro duction/, this assumes that some of the deca ys are visi/-ble/, rather than just e/ /! / eN/1 /. In the cases of the hea vier sparticles/, the cascade deca ysmen tioned in the previous section will pro vide a ric h set of signals to study /. By making useof p olarized b eams and the relativ ely clean e /+e /;collider en vironmen t/, one can disen tanglethe sparticle sp ectrum/. F or example/, measuring the maxim um and minim um energy end/-po i n ts of the leptons pro duced in e /+e /;/! e/`R e/`R with e/`R /! /` eN/1 /, one can precisely determinebo t h me/`R and meN/1 /.B y v arying the p olarization of the electron b eam/, one can con trol theW /+W /;bac kground and sim ultaneously c hec k that the con tributions to the sparticle pro/-duction cross/-sections ha v e the correct magnitude and v ary in the righ tw a y /. This will allo wone to c hec k the spin and the /\handedness/" of the pro duced squarks and sleptons/. Similarprecision studies of c hargino and neutralino pro duction can also b e p erformed/. In general/,a high/-energy linear lepton collider will pro vide an excellen tw a y of testing sup ersymmetricrelations/. It is also w orth noting that searc hes for e /+e /;/! h /0Z /, h /0A /0/, H /0Z /, H /0A /0andH /+H /;should b e able to de/nitiv ely test the Higgs sector of the MSSM/.If the gra vitino is the LSP as in gauge/-mediated mo dels/, then one m ust tak ei n to accoun tthe p ossibilitie s men tioned in section /8/./5/. If the ligh test neutralino is the NLSP and thedeca y eN/1 /! / eG o ccurs within the detector/, then ev en the pro cess e /+e /;/! eN/1 eN/1 leads toa dramatic signal of t w o energetic photons plus missing energy /. /6/7 /;; /6/8 /;; /6/9There are signi/can tbac kgrounds to the / / // E signal/, but they are easily remo v ed b y cuts/. Eac h of the othersparticle pair/-pro duction mo des eq/. /(/9/./1/) will lead to the same signals as in the neutralinoLSP case/, but no w with t w o additional energetic photons whic h should mak e the exp eri/-men talists/'tasks quite easy /. If the deca y length for eN/1 /! / eG is m uc h larger than the sizeof a detector/, then the signals rev ert bac k to those found in the neutralino LSP scenario/.In an in termediate regime for the eN/1 /! / eG deca y length/, one ma y see ev en ts with one orb oth photons displaced from the ev en tv ertex b y a macroscopic distance/.If the NLSP is a c harged slepton e/` /, then e /+e /;/! e/` /+e/` /;follo w ed b y prompt deca yse/` /! /` eG will yield t w o energetic same/-/ a v or leptons in ev ery ev en t/, and with a di/eren tenergy distribution than the acollinear leptons that w ould follo w from either eC /+/1 eC /;/1 or e/` /+e/` /;pro duction in the neutralino LSP scenario/. The W /+W /;bac kground can b e a problem here/,but can b e defeated with angular cuts at LEP/2 or p olarized b eams at future e /+e /;colliders/.P air/-pro duction of non/-NLSP sparticles will yield unmistak able signals whic h are the sameas those found in the neutralino NLSP case but with t w o additional energetic leptons /(notnecessarily of the same / a v or/)/. A p erhaps ev en more exciting p ossibilit y is that the NLSPis a slepton whic h deca ys v ery slo wly /. /6/8If the slepton NLSP is so long/-liv ed that it deca ysoutside the detector/, then slepton pair/-pro duction will lead to ev en ts featuring a pair ofc harged particle trac ks with a high ionization rate whic h b etra ys their v ery large mass/. Ifthe sleptons deca y within the detector/, then one can lo ok for kinks in the c harged particletrac ks/, or a macroscopic impact parameter/. The pair/-pro duction of an y of the other hea vy/8/1 c harged sparticles will also yield hea vy c harged particle trac ks or deca y kinks/, plus leptonsand//or jets/, but no // E unless the deca yc hains happ en to include neutrinos/. It ma ya l s o b ep ossible to iden tify the presence of a hea vy c harged NLSP b y measuring its anomalouslylong time/-of/-/ igh t through the detector/./9/./2 Signals at hadr on c ol lidersA t hadron colliders/, the most imp ortan tc hannels for sparticle pro duction are t ypicallyexp ected to b eeC /+i eC /;j /;; eNi eC /j /;; eNi eNj /;; and /(/9/./2/)eg eg/;; eg eq/;; eq eq/: /(/9/./3/)A t the F ermilab T ev atron p p collider with p s /=/2T eV/, the c hargino and neutralino pro/-duction pro cesses /(through v alence quark annihilation in to virtual w eak b osons/) tend toha v e the larger cross/-sections/, unless the squarks or gluino are rather ligh t /(less than /3/0/0GeV or so/)/. In a t ypical scenario where eC/1 and eN/2 are mostly SU /(/2/)L gauginos and eN/1 ismostly bino/, the largest pro duction cross/-sections in eq/. /(/9/./2/) b elong to the eC/1 eC/1 and eN/2 eC/1c hannels/, b ecause they ha v e signi/can t couplings to W and / /;; Z b osons/, resp ectiv ely /.A tthe future CERN LHC pp collider with p s / /1/4 T eV/, the situation is t ypically rev ersed/,with pro duction of gluinos and squarks b y gluon fusion and gluon/-quark fusion usually dom/-inating/, unless the gluino and squarks are hea vier than /1 T eV or so/. A t b oth colliders/, onecan also ha v e asso ciated pro duction of a c hargino or neutralino together with a squark orgluino/, but the cross/-sections for suc h pro cesses are probably signi/can tly lo w er than for theones in eqs/. /(/9/./2/) and /(/9/./3/)/. Slepton pair pro duction ma y b e rather small at the T ev atron/,but migh t b e observ able there or at the LHC/. /1/1/7Cross/-sections for sparticle pro duction athadron colliders can b e found in Ref/. /1/1/8The deca ys of the pro duced sparticles result in /nal states with t w o neutralino LSPswhic h escap e the detector/. The LSPs again carry a w a y at least /2 meN/1 of missing energy /, butat hadron colliders only the comp onen t of the missing energy whic h is manifest in momen tatransv erse to the colliding b eams /(denoted // ET /) is observ able/. Therefore in general theobserv able signals for sup ersymmetry at hadron colliders are n leptons /+ m jets /+ // ET /,where either n or m migh t b e /0/. There are imp ortan t Standard Mo del bac kgrounds to man yof these signals/, esp ecially from pro cesses in v olving pro duction of W and Z b osons whic hcan deca y to neutrinos/, yielding // ET /. Therefore it is imp ortan t to iden tify sp eci/c signalsfor whic h the bac kgrounds can b e reduced/. Of course/, this dep ends on whic h sparticlesare b eing pro duced and ho w they are deca ying/. F or example/, the /\classic/" // ET signal forsup ersymmetry at hadron colliders is ev en ts with jets and // ET but no energetic isolatedleptons/. The latter requiremen t reduces bac kgrounds from Standard Mo del pro cesses withleptonic W deca ys/, and is ob viously most e/ectiv e if the relev an t sparticle deca ys ha v esizeable branc hing fractions in to c hannels with no leptons in the /nal state/.Another t yp e of signal arises if the gluino deca ys with a signi/can tb r a n c hing fractionto hadrons plus a c hargino/, whic h can subsequen tly deca yi n to a /nal state with a c hargedlepton/, a neutrino/, and eN/1 /. Since the gluino do esn/'t kno wa n ything ab out electric c harge/,the single c harged lepton pro duced from eac h gluino deca yc a n h a v e either sign with equalprobabilit y /. This means that gluino pair pro duction will often lead to ev en ts with t w oleptons with the same c harge /(but p ossibly di/eren t/ a v ors/) plus jets and // ET /. This signalcan also arise from eq eq and eq eg pro duction/, e/.g/. if the squarks deca yl i k e eq /! q eg /. Thissame/-sign dilepton signal /1/1/9has small ph ysics bac kgrounds from the Standard Mo del b oth/8/2 at the T ev atron and the LHC/. The reason is that the largest bac kground sources for isolatedlepton pairs/, namely W /+W /;/, Drell/-Y an and t t pro duction/, can only yield opp osite/-c hargedileptons/.Despite the bac kgrounds just men tioned/, opp osite/-c harge dilepton signals/, e/.g/. fromslepton pair pro duction with subsequen td e c a ys e/` /! /` eN/1 /, can giv e an observ able signalesp ecially at the LHC/.Another useful p ossibilit y is the trilepton signal/, /1/2/0whic h features three leptons plus// ET and p ossibly jets/. This can come ab out from eC/1 eN/2 pro duction follo w ed b y the deca ysindicated in eq/. /(/8/./4/)/, in whic h case one exp ects little hadronic activit y in the ev en t/. Itcould also come from eg eg /, eq eg /,o r eq eq pro duction/, with one of the gluinos or squarks deca yingthrough a eC/1 and the other through a eN/2 /. In that case/, there will b e jets from the deca ys/, inaddition to the three leptons and // ET /. These signatures rely on the eN/2 ha ving a signi/can tbranc hing fraction for the three/-b o dy deca y to leptons in eq/. /(/8/./4/)/. F or this reason/, thet w o/-b o dy deca y mo des in eq/. /(/8/./1/) are sometimes called /\sp oiler/" mo des/, since if they arekinematically allo w ed they can dominate/, sp oiling the trilepton signal/. This is b ecause if theeN/2 deca y is through an on/-shell h /0/, then the /nal state will v ery lik ely include b ottom/-quarkjets rather than isolated leptons/, while if the deca y is through an on/-shell Z /, then there canstill b e t w o leptons but there are Standard Mo del bac kgrounds with unfortunately similarkinematics from pro cesses in v olving Z /! /` /+/` /;/. Either w a y /, the trilepton signal can b esp oiled/, but other leptons /+ jets /+ // ET signals ma y b e observ able ab o v e Standard Mo delbac kgrounds/, esp ecially if b ottom quark jets can b e tagged with high e/ciency /.The single lepton plus jets plus // ET signal /1/2/1has large Standard Mo del bac kgrounds frompro cesses with W /! /`/ /.H o w ev er/, it also can ha v e a large rate from v arious sup erpartnerpro duction mo des/, and ma y still giv e the b est signal at the LHC/. One should also b e a w areof v ery in teresting signals whic h can arise for particular ranges of parameters/. F or example/,in a scenario studied in Ref/. /1/1/5/, the only t w o/-b o dy deca yc hannel for the gluino is eg /! b eb/1 /,with subsequen t deca ys eb/1 /! b eN/2 and eN/2 /! /` /+/` /;eN/1 or eN/2 /! qq eN/1 /. In that case/, gluinopair pro duction giv es a sp ectacular signal of four b ottom jets plus up to four leptons plus// ET /. In general/, pro duction of relativ ely ligh t et/1 and eb/1 can giv e hadron collider signals ric hin b ottom jets/, either through direct pro duction or cascade deca ys/.If the gra vitino is the LSP /, these signals can b e signi/can tly mo di/ed/. If the NLSP isa neutralino with a prompt deca y eN/1 /! / eG /, then one exp ects ev en ts with t w o energetic/,isolated photons plus // ET from the escaping gra vitinos/, rather than just // ET /. So at a hadroncollider the signal is / / /+ X /+// ET where X is an y collection of leptons plus jets/. The StandardMo del bac kgrounds relev an t for suc he v en ts are quite small/. If the eN/1 deca yl e n g t hi sl o n genough/, then it ma y b e measurable b ecause the photons will not p oin tb a c k to the ev en tv ertex/. If the eN/1 deca y is outside of the detector/, then one just has the usual leptons /+ jets/+// ET signals as discussed ab o v e in the neutralino LSP scenario/.In the case that the NLSP is a c harged slepton/, then the deca y e/` /! /` eG can pro videt w o extra leptons in eac he v en t/, compared to the signals with a neutralino LSP /.I ft h e e//1is su/cien tly ligh ter than the other c harged sleptons eeR /, e/R /, and so is e/ectiv ely the soleNLSP /, then ev en ts will alw a ys ha v e a pair of taus/. If the slepton NLSP is long/-liv ed/, onecan lo ok for ev en ts with a pair of v ery hea vy c harged particle trac ks or a long time/-of/-/ igh tin the detector/. Since slepton pair/-pro duction usually has a m uc h smaller cross/-section thanthe pro cesses in eq/. /(/9/./2/) and /(/9/./3/)/, this will t ypically b e accompanied b y leptons and//or jetsfrom the same ev en tv ertex/, whic hm a y b e of crucial help in iden tifying candidate ev en ts/.It is also quite p ossible that the deca y length of e/` /! /` eG is measurable within the detector/,seen as a macroscopic kink in the c harged particle trac k/./8/3 /9/./3 Dark matter dete ctionOne of the ma jor successes of sup ersymmetry with exact R /-parit y conserv ation is that anelectrically neutral LSP can b e a go o d candidate for the dark matter/. There are threeob vious candidates/: the gra vitino/, the ligh test sneutrino/, and the ligh test neutralino/. Ifthe gra vitino is the LSP /, as in gauge/-mediated mo dels/, then relic gra vitinos left o v er fromthe early univ erse w ould b e essen tially imp ossible to detect ev en if they can b e arrangedto ha v e the righ t cosmological densit yt o d a y /. The p ossibilit y of a sneutrino LSP makingup the dark matter with a cosmologically in teresting densit yh a s n o w b een ruled out b ydirect searc hes/. /1/2/2The most attractiv e prosp ects for direct detection of sup ersymmetricdark matter/, therefore/, are based on the idea that the ligh test neutralino eN/1 is the LSP /,a shapp ens quite naturally in the minimal sup ergra vit y mo dels/.In the early univ erse/, sparticles existed in thermal equilibrium with the ordinary Stan/-dard Mo del particles/. As the univ erse co oled and expanded/, the sparticles could no longerb e pro duced and they all annihilated or deca y ed in to eN/1 /. The remaining eN/1 can annihi/-late through pro cesses eN/1 eN/1 /! f f with t /-c hannel exc hange of squarks and sleptons or thes /-c hannel exc hange of Higgs scalars or a Z b oson/. Dep ending on the mass of eN/1 /, otherpro cesses lik e eN/1 eN/1 /! W /+W /;/, ZZ /, Zh /0/, h /0h /0or ev en W /H //, ZA /0/, h /0A /0/, h /0H /0/, H /0A /0/,H /0H /0/, A /0A /0/,o r H /+H /;could also ha v e b een imp ortan t/. Ev en tually /, as the densit y of LSPsdecreased/, the annihilation rate b ecame v ery small/, and the eN/1 relic densit y is determinedb y this small rate and the subsequen t dilution due to the expansion of the univ erse/.It is a remark able coincidence that the predicted densit y of a bino/-lik e /(or p erhapshiggsino/-lik e/) neutralino LSP obtained b y doing these calculations carefully can b e in therigh t range to mak e up a signi/can t fraction of the critical densit y of the univ erse/, andp erhaps to explain the rotation curv es of galaxies/. /1/2/3/(A wino/-lik e eN/1 w ould ha v e only atin y relic densit y /, but giv en the gaugino mass hierarc h y eq/. /(/7/./4/8/)/, there is little motiv ationfor suc h a thing an yw a y /./) It is also necessary to require that the densit y of surviving LSPsnot b e to o large/, so that the univ erse could ha v er e a c hed its presen t size and age of atleast /1/0 /1/0y ears/. This tends to put an upp er limit on the LSP mass/, but unfortunatelyit is di/cult to mak e a general/, parameter/-indep enden t b ound out of this b ecause if themasses are arranged just righ t/, the LSP ma y happ en to annihilate v ery e/cien tly through aresonance/. If neutralino LSPs really mak e up the cold dark matter/, then their mass densit yin our neigh b orho o d ough t to b e at least ab out /0/./1 GeV//cm /3in order to explain the rotationcurv es of galaxies/. In principle/, they should b e detectable through their w eak in teractionswith ordinary matter/, or b y their ongoing annihilations/.The direct detection of eN/1 dep ends on their elastic scattering o/ of hea vy n uclei ina detector/. A t a fundamen tal lev el/, eN/1 can scatter o/ of a quark b y virtual exc hange ofsquarks/, a Z b oson/, or Higgs scalars/, or can scatter o/ of gluons through one/-lo op diagrams/.The energy transferred to the n ucleus in these collisions is t ypically of order tens of k eV/.Ho w ev er/, there are imp ortan t bac kgrounds from radioactivit y and cosmic ra ys/. The optimaldetector material /(e/.g/. germanium/, silicon/, or niobium/) dep ends on the details of the eN/1 /-n ucleus in teraction/. Presen t detectors are still not sensitiv e to most regions of parameterspace/, but there is hop e that this can c hange in the future/.Another/, more indirect/, w a y to detect neutralino LSPs is through ongoing annihilations/.This can o ccur in regions of space where the densit y is greatly enhanced compared toour o wn neigh b orho o d/. This can o ccur if the LSPs lose energy b y rep eated scattering o/of n uclei/, ev en tually b ecoming concen trated inside massiv e astronomical b o dies lik e theEarth or the Sun/. In this case the annihilation of neutralino pairs in to neutrinos is the/8/4 most imp ortan t pro cess/, since no other particles can escap e from the cen ter of the ob jectwhere the annihilation is going on/. In particular/, m uon neutrinos and an tineutrinos fromeN/1 eN/1 /! // // will tra v el large distances/, /nally undergoing a c harged/-curren ti n teractionleading to energetic m uons p oin ting bac k to the cen ter of the Earth or Sun/. There are alsoin teresting p ossible signatures from neutralino LSP annihilation in the galactic halo whic hmigh t pro duce detectable quan tities of high/-energy photons/, p ositrons/, and an tiprotons/. /1/2/3/1/0 Some miscellaneous v ariationsIn this section w e will brie/ y consider a few v ariations on the simple picture of the MSSMthat has b een outlined ab o v e/. First/, w e will consider the p ossibilit yo f R /-parit y violationin section /1/0/./1/. Another ob vious w a y to extend the MSSM is to in tro duce new c hiralsup erm ultiplets/, corresp onding to scalars and fermions that are all su/cien tly hea vy to ha v ea v oided disco v ery so far/. In general/, this requires that the new c hiral sup erm ultiplets m ustform a real represen tation of the Standard Mo del gauge group/. The simplest suc h p ossibilit yis that the new particles liv e in just one gauge/-singlet c hiral sup erm ultiplet/;; this p ossibilit yis discussed in section /1/0/./2/. One can also extend the MSSM b yi n tro ducing new gaugein teractions whic ha r e s p o n taneously brok en at v ery high energies/. The p ossibiliti es hereinclude GUT mo dels lik e SU /(/5/) and SO /(/1/0/) and E/6 whic h unify the Standard Mo del gaugein teractions/, with imp ortan t implications for rare pro cesses lik e proton deca y /. Sup erstringmo dels also quite generically imply that the Standard Mo del gauge group should b e extendedat high energies/. There is a v ast literature on these p ossibilities/, but w e will concen trateinstead on the implications of just adding a single additional ab elian factor to the gaugegroup/, in section /1/0/./3/./1/0/./1 Mo dels with R /-p arity violation/.So far w eh a v e assumed that R /-parit y /(or equiv alen tly matter parit y/) is an exact symmetryof the MSSM/. This assumption precludes renormalizable proton deca y and predicts that theLSP should b e stable/, but despite these virtues R /-parit y is not inevitable/. Because of thethreat of proton deca y /,w e exp ect that if R /-parit y is violated/, then in the renormalizablelagrangian either B/-violating or L/-violating couplings are allo w ed/, but not b oth/, as explainedin section /5/./2/.One prop osal is that matter parit y can b e replaced b y an alternativ e discrete symmetrywhic h still manages to forbid proton deca y at the lev el of the renormalizable lagrangian/.The p ossibilities ha v e b een cataloged in Ref/. /1/2/4/, where it w as found that pro vided no newparticles are to b e added to the MSSM/, that the discrete symmetry is family/-indep enden t/,and that it can b e de/ned at the lev el of the sup erp oten tial/, there is only one other candidatediscrete symmetry b esides matter parit y /. That other p ossibilit yi s a Z/3 discrete symmetry/1/2/4whic hw as originally called /\bary on parit y/"/, but is more appropriately referred to as/\bary on trialit y/"/. The bary on trialit yo f a n y particle with bary on n um be r B a n d w eakh yp erc harge Y is de/ned to b eZ B/3 /=e x p //2 /i /3 /[B /; /2 Y /] //: /(/1/0/./1/)It is easy to c hec k that this is alw a ys a cub e ro ot of unit y for the MSSM particles/, sinceB /; /2 Y is alw a ys an in teger/. The symmetry principle to b e enforced is that the pro duct of thebary on trialities of the particles in an y term in the lagrangian /(or sup erp oten tial/) m ust b e /1/.This symmetry conserv es bary on n um b er at the renormalizable lev el while allo wing lepton/8/5 (a)N1l ll ν λ (b)N1l lq q′ λ′ (c)N1q qq′ q′′ λ′′Figure /2/0/: Deca ys of the eN/1 LSP in mo dels with R /-parit y violation /[see eqs/. /(/5/./7/) and /(/5/./8/)/]/.n um b er violation/;; in other w ords/, it allo ws the sup erp oten tial terms in eq/. /(/5/./7/) but forbidsthose in eq/. /(/5/./8/)/. In fact/, bary on trialit y conserv ation has the remark able prop ert y that itabsolutely forbids proton deca y /. /1/2/5The reason for this is simply that bary on trialit y requiresthat B can only b e violated in m ultiples of /3 units /(ev en in nonrenormalizable in teractions/)/,while an y kind of proton deca yw ould ha v e to violate B b y /1 unit/. So it is eminen tlyfalsi/able/. Similarly /, bary on trialit y conserv ation predicts that exp erimen tal searc hes forneutron/-an tineutron oscillations will b e negativ e/, since they w ould violate bary on n um be r b y/2 units/. Ho w ev er/, bary on trialit y conserv ation do es allo w the LSP to deca y /. If one adds somenew c hiral sup erm ultiplets to the MSSM /(corresp onding to particles whic h are presumablyv ery hea vy/)/, one can conco ct a v ariet y of new candidate discrete symmetries b esides matterparit y and bary on trialit y /. Some of these will allo w B violation in the sup erp oten tial/, whileforbidding the lepton n um b er violating sup erp oten tial terms in eq/. /(/5/./7/)/.Another idea is that matter parit y is an exact symmetry of the underlying sup erp oten tial/,but it is sp on taneously brok en b y the VEV of a scalar with PR /= /; /1/. One p ossibilit yi sthat an MSSM sneutrino gets a VEV/, /1/2/6since sneutrinos are scalars carrying L/=/1/. Ho w ev er/,there are strong b ounds /1/2/7on SU /(/2/)L /-doublet sneutrino VEVs h e/ i/ mZ coming from therequiremen t that the corresp onding neutrinos do not ha v e large masses/. It is somewhatdi/cult to understand wh ys u c h a small VEV should o ccur/, since the scalar p oten tial whic hpro duces it m ust include soft sneutrino /(mass/) /2terms of order m /2soft /. One can get aroundthis b y instead in tro ducing a new gauge/-singlet c hiral sup erm ultiplet with L/= /; /1/. Thescalar comp onen t can get a large VEV/, whic h can induce L/-violating terms /(and in generalB/-violating terms also/) in the lo w/-energy e/ectiv e sup erp oten tial of the MSSM/. /1/2/7In an y case/, if R /-parit y is violated/, then the LSP will deca y /, completely altering thesignals for sup ersymmetry /. The t yp e of signal to lo ok for dep ends on the form of R /-parit yviolation/. If there are L/-violating terms of the t yp e / and//or / /0as in eq/. /(/5/./7/)/, then the /nalstates from eN/1 deca y will alw a ys in v olv eac harged lepton or a neutrino plus either a pairof additional c harged leptons or a pair of jets/. Tw o suc h deca ys are sho wn in Fig/. /2/0a/,b/,but there are others/. Signals for sup ersymmetry will therefore alw a ys include leptons orlarge missing energy /, or b oth/. On the other hand/, if terms of the form / /0/0in eq/. /(/5/./8/) arepresen t instead/, then there are bary on/-n um b er violating deca ys eN/1 /! qq /0q /0/0from graphslik e the one sho wn in Fig/. /2/0c/. In that case/, sup ersymmetric ev en ts will alw a ys ha v e lotsof hadronic activit y /, and will only ha v e missing energy signatures when the other parts ofthe deca yc hains happ en to include neutrinos/. This could mak e the disco v ery and studyof sup ersymmetry v ery di/cult/. There are other p ossibilities/, to o/, b ecause if R /-parit yi sviolated/, then the deca ying LSP need not b e eN/1 /, and sparticles whic h are not the LSP canin principle deca y directly to Standard Mo dels quarks and leptons/. If / /0is non/-zero/, thensquarks can b e pro duced as resonances at the e /p collider at HERA/. A complete surv ey ofthe p ossibilities w ould b e far to o complicated to presen t here/./8/6 /1/0/./2 The next/-to/-minimal sup ersymmetric standar dm o delThe simplest p ossible extension of the particle con ten t of the MSSM is to add a new gauge/-singlet c hiral sup erm ultiplet/. The resulting mo del is often called the next/-to/-minimal sup er/-symmetric standard mo del /(NMSSM/)/. /1/2/8The most general p ossible sup erp oten tial for thismo del is giv en b yWNMSSM /= /1 /6 kS /3/+ /1 /2 /S S /2/+ /S Hu Hd /+ WMSSM /;; /(/1/0/./2/)where S stands for b oth the new c hiral sup erm ultiplet and its scalar comp onen t/. /(Therecould also b e a term linear in S in WNMSSM /, but this can alw a ys b e remo v ed b y rede/ningS b y a constan t shift/./)One of the virtues of the NMSSM is that it can pro vide a solution to the / problemmen tioned in sections /5/./1 and /7/./2/. T o understand this/, supp ose w es e t /S /= / /=/0 s othat there are no mass terms or dimensionful parameters in the sup erp oten tial at all/. Thenan e/ectiv e / /-term for Hu Hd will still arise from the third term in eq/. /(/1/0/./2/) if S gets aVEV/, with / /= / h S i /. The absence of dimensionful terms in WNMSSM can b e enforced b yin tro ducing a new symmetry /(in v arious di/eren tw a ys/)/. The soft terms in the lagrangiangiv ea c o n tribution to the scalar p oten tial whic h can b e written asV NMSSMsoft /=/( /1 /6 ak S /3/+ a/ SHu Hd /+c /: c /: /)/+ m /2S j S j /2/+ V MSSMsoft /;; /(/1/0/./3/)where ak and a/ ha v e dimensions of mass/. One ma yn o ws e t b /=/0 i n V MSSMsoft /, b ecause ane/ectiv ev alue for b will b e generated/, equal to a/ h S i /. If the new parameters k /, / /, ak and a/are c hosen correctly /, then phenomenologically acceptable VEVs will b e induced for S /, H /0u /,and H /0d /. A correct treatmen t of this requires the inclusion of one/-lo op radiativ e corrections/.But the imp ortan t p oin t is that the scale of the VEV h S i /, and therefore the e/ectiv ev alueof / /, is then determined b y the soft terms of order msoft /, instead of b eing a free parameterwhic h is conceptually indep enden t of sup ersymmetry breaking/.The NMSSM con tains/, b esides the particles of the MSSM/, a real PR /= /+/1 scalar/, a realPR /= /+/1 pseudoscalar/, and a PR /= /; /1W eyl fermion /\singlino/"/. These /elds ha v e no gaugecouplings of their o wn/, so they can only in teract with Standard Mo del particles b y mixingwith the neutral MSSM /elds with the same spin and c harge/. The real scalar mixes withthe MSSM particles h /0and H /0/, and the pseudo/-scalar mixes with A /0/. One of the e/ects ofreplacing the / term b y the dynamical /eld S is to raise the upp er b ound on the ligh testHiggs mass/, for a giv en set of the other parameters in the theory /.H o w ev er/, the b ound ineq/. /(/7/./4/3/) is still resp ected in the NMSSM /(and an y other p erturbativ e extension of theMSSM/)/, pro vided only that the sparticles that con tribute in lo ops to the Higgs mass areligh ter than /1 T eV or so/. The o dd R /-parit y singlino mixes with the four MSSM neutralinos/,so there are really /v e neutralinos no w/. In man y regions of parameter space/, mixing e/ectsin v olving the singlet /elds are small/, and they essen tially just decouple/. In that case/, thephenomenology of the NMSSM is nearly indistinguishabl e from that of the MSSM/. Ho w ev er/,if an y of the /v e NMSSM neutralinos /(and esp ecially the LSP/) has a large mixing b et w eenthe singlino and the usual gauginos and higgsinos/, then the signatures for sparticles can b ealtered in imp ortan tw a ys/. /1/2/9/1/0/./3 Extr a D /-term c ontributions to sc alar massesAnother w a y to generalize the MSSM is to include additional gauge in teractions/. The sim/-plest gauge extension of the MSSM in tro duces just an additional ab elian gauge symmetry /,/8/7 whic hw e can call U /(/1/)X /. As long as U /(/1/)X is brok en at a v ery high mass scale/, then thecorresp onding v ector gauge b oson and gaugino fermion will b e v ery hea vy and will decouplefrom ph ysics at the T eV scale and b elo w/. If so/, one migh t supp ose that all e/ects follo wingfrom the existence of U /(/1/)X will b e completely negligible for collider exp erimen ts in theforeseeable future/. Ho w ev er/, this is not necessarily so/, b ecause as long as the MSSM /eldscarry U /(/1/)X c harges/, the breaking of U /(/1/)X at a v ery high energy scale can lea v e its imprin ton the soft terms of the MSSM/. /1/3/0T o see ho w this w orks/, let us consider the scalar p oten tial for a mo del in whic h U /(/1/)X isbrok en/. Supp ose that the MSSM scalar /elds/, denoted generically b y /i /, carry U /(/1/)X c hargesxi /. In order to break U /(/1/)X /,w ea l s oi n tro duce a pair of c hiral sup erm ultiplets with U /(/1/)Xc harges / /1/, denoted S/+ and S/; /. These /elds are singlets under the Standard Mo del gaugegroup SU /(/3/)C / SU /(/2/)L / U /(/1/)Y /, so that when they get VEVs/, they will just break U /(/1/)X /.An ob vious guess for the sup erp oten tial con taining S/+ and S/; is W /= MS/+ S/; /, where M isa sup ersymmetric mass/. Ho w ev er/, unless M v anishes or is v ery small/, it will yield p ositiv e/-semide/nite quadratic terms in the scalar p oten tial of the form V /= j M j /2/( j S/+ j /2/+ j S/; j /2/)whic h will force the minim um to b e at S/+ /= S/; /= /0/. Since w ew an t S/+ and S/; to obtainVEVs/, this is unacceptable/. Therefore w e assume that M is /0 /(or v ery small/) and that theleading con tribution to the sup erp oten tial comes instead from a nonrenormalizable term/,sa y/:W /= / /2 MP S /2/+ S /2/; /: /(/1/0/./4/)/(Non/-renormalizable terms in the sup erp oten tial ob ey the same rules as w e found b efore/;; inparticular they m ust b e analytic functions of the c hiral sup er/elds/. See the App endix formore details on non/-renormalizable lagrangians in sup ersymmetric theories/./) The equationsof motion for the auxiliary /elds are then F /S/+ /= /; /@W /= /@S/+ /= /; /( //= MP /) S/+ S /2/; and F /S/; /=/; /@ W /=/@ S/; /= /; /( //= MP /) S/; S /2/+ /, and the corresp onding con tribution to the scalar p oten tial isVF /= j FS/+ j /2/+ j FS/; j /2/+ /:/:/: /= j / j /2 M /2P /j S/+ j /4j S/; j /2/+ j S/+ j /2j S/; j /4 //+ /:/:/: /: /(/1/0/./5/)Here the ellipses represen t other terms that are higher order in /1 /= MP /(see App endix/)/, whic hw e can safely ignore/. In addition/, there are soft terms whic hm ust b e tak en in to accoun t/:Vsoft /= m /2/+ j S/+ j /2/+ m /2/; j S/; j /2/; /a /2 MP S /2/+ S /2/; /+c /: c /: //: /(/1/0/./6/)The terms with m /2/+ and m /2/; are soft masses for S/+ and S/; /. W e can assume that theycome from a minimal sup ergra vit y framew ork at the Planc k scale/, but in general they willb e renormalized di/eren tly /, due to di/eren ti n teractions for S/+ and S/; whic hw eh a v e notb othered to write do wn in eq/. /(/1/0/./4/) b ecause they in v olv e /elds that will not get VEVs/.The last term is a /\soft/" term exactly analogous to the ones app earing in the second line ofeq/. /(/4/./1/)/, with a of order msoft /. The coupling a/= /2 MP is actually dimensionless/, but shouldb e treated as soft b ecause of its origin and its tin y magnitude/. Suc h terms arise from thesup ergra vit y lagrangian in an exactly analogous w a y to the usual soft terms/. Usually onecan just ignore them/, but this one pla ys a crucial role in the gauge symmetry breakingmec hanism/. The scalar p oten tial for terms con taining S/+ and S/; is no w/:V /= /1 /2 g /2X /j S/+ j /2/;j S/; j /2/+ Xi xi j /i j /2 //2/+ VF /+ Vsoft /: /(/1/0/./7/)/8/8 The /rst term is the square of the U /(/1/)X D /-term /[see eqs/. /(/3/./7/4/) and /(/3/./7/5/)/]/, and gX is theU /(/1/)X gauge coupling/. The scalar p oten tial eq/. /(/1/0/./7/) has a nearly D /-/ at direction/, b ecausethe D /-term part v anishes for /i /= /0 and an y j S/+ j /= j S/; j /. Therefore/, if m /2/+ /+ m /2/; /< /0/, thepo i n t S/+ /= S/; /= /0 will b e destabilized and S/+ and S/; can obtain large VEVs/. Without lossof generalit y /,w e can tak e a and / to b oth b e real and p ositiv e for purp oses of minimizingthe scalar p oten tial/. As long as a /2/; /8 / /2/( m /2/+ /+ m /2/; /) /> /0/, the global minim um of the p oten tialo ccurs forh S/+ i /2/h S/; i /2/ aMP /6 / /2 h/1/+ q /1 /; /6 / /2/( m /2/+ /+ m /2/; /) /=a /2 i/(/1/0/./8/)/(with h /i i /= /0/)/, so h S/+ i/ h S/; i/O /( p msoft MP /)/. The VF con tribution is what stabilizes thescalar p oten tial at v ery large /eld strengths/. The VEVs of S/+ and S/; will b e m uc h largerthan /1 T eV as long as a is not to o small/. Therefore the U /(/1/)X gauge b oson and gauginocan b e v ery hea vy /, with masses of order gX h S/ i /, and pla y no role in collider ph ysics/.Ho w ev er/, there is also a small deviation from h S/+ i /= h S/; i /,a s l o n ga s m /2/+ /6/= m /2/; /.A tthe minim um of the p oten tial with /@V /= /@S/+ /= /@ V /=/@ S/; /= /0/, the leading order di/erence inthe VEVs is giv en b yh S/+ i /2/;h S/; i /2/= /; /1 gX h DX i/ /1 /2 g /2X /( m /2/; /; m /2/+ /) /(/1/0/./9/)assuming that h S/+ i and h S/; i are m uc h larger than their di/erence/. After in tegrating outthe S/+ and S/; b y replacing them with their equations of motion expanded around theminim um of the p oten tial/, one /nds that the MSSM scalars /i eac h receiv e a correction totheir /(mass/) /2giv en b y/
m /2i /= /; xi gX h DX i /;; /(/1/0/./1/0/)in addition to the usual soft terms deriv ed from the minimal sup ergra vit y b oundary condi/-tions and R G equations/. The D /-term corrections eq/. /(/1/0/./1/0/) can b e roughly of the order ofm /2soft at most/, since they are all prop ortional to m /2/; /; m /2/+ /. Note that the result eq/. /(/1/0/./1/0/)do es not actually dep end on our c hoice of the nonrenormalizable sup erp oten tial/, as longas it pro duces the required symmetry breaking with large VEVs/;; this is a general feature/.In a sense/, the soft sup ersymmetry/-breaking terms m /2/+ and m /2/; ha v e b een recycled in to anon/-zero D /-term for U /(/1/)X /, whic h then lea v es its /\/ngerprin t/" on the sp ectrum of MSSMscalar masses/. The most imp ortan t feature of the correction eq/. /(/1/0/./1/0/) is that eac h MSSMscalar /(mass/) /2obtains a correction just prop ortional to its c harge xi under the sp on taneouslybrok en gauge group/, with a univ ersal factor gX h DX i /.F rom the p oin t of view of T eV scaleph ysics/, the quan tit y gX h DX i can simply b e tak en to parameterize our ignorance of ho wU /(/1/)X got brok en/. T ypically /, the c harges xi are rational n um b ers and do not all ha v e thesame sign/, so that a particular candidate U /(/1/)X can lea v e a quite distinctiv e pattern of masssplittings on the squark and slepton sp ectrum/.The additional gauge symmetry U /(/1/)X in the ab o v e discussion can stand alone/, orma y p erhaps b e em b edded in a larger non/-ab elian gauge group/. If the gauge group forthe underlying theory at the Planc k scale con tains more than one new U /(/1/) factor/, theneac h suc h factor can mak eac o n tribution exactly analogous to eq/. /(/1/0/./1/0/)/. Additional U /(/1/)gauge groups are quite common in sup erstring mo dels/, so from that p oin t of view one ma yb eoptimistic ab out the existence of the corresp onding D /-term corrections/. Once one merelyassumes the existence of additional U /(/1/) gauge groups at v ery high energies/, it is quite/8/9 unnatural to assume that suc h D /-term con tributions to the MSSM scalar masses shouldv anish/, unless there is an exact symmetry whic h will enforce m /2/+ /= m /2/; /. The only questionis whether or not the magnitude of the D /-term con tributions is signi/can t compared to theusual minimal sup ergra vit y and R Gc o n tributions/;; it ma yv ery w ell not b e/. Note also thatas long as the c harges xi are family/-indep ende n t/, then from eq/. /(/1/0/./1/0/) the squarks andsleptons with the same electro w eak quan tum n um b ers remain degenerate/, main taining thenatural suppression of F CNC e/ects/. So it is quite p ossible that e/orts to understand thesparticle sp ectrum of the MSSM will need to tak ei n to accoun t the p ossibilit yo f D /-termsfrom additional gauge groups/./1/1 Concluding remarksIn this primer/, I ha v e attempted to con v ey some of the more essen tial features of sup ersym/-metry as it is kno wn so far/. One of the most amazing qualities of sup ersymmetry is thatso m uc hi s k n o wn ab out it already /, despite the presen tl a c k of direct exp erimen tal data/.Ev en the terms and stak es of man y of the imp ortan t outstanding questions/, esp ecially theparamoun t issue /\Ho w is sup ersymmetry brok en/?/"/, are already rather clear/. That this canb e so is a testamen t to the unreasonably predictiv e qualit y of the symmetry itself/.W eh a v e seen that sensible and economical mo dels for sup ersymmetry at the T eV scalecan b e used as con v enien t templates for exp erimen tal searc hes/. As summarized in section/7/./6/, t w o of the simplest p ossibilitie s are the /\minimal sup ergra vit y/" scenario with newparameters m /2/0 /, m/1 /= /2 /, A/0 /, tan / and Arg/( / /)/, and the /\gauge/-mediated/" scenario with newparameters //, Mmess /, N/5 /, h F i /, tan / /, and Arg/( / /)/. Ho w ev er/, one should not lose sigh to ft h efact that the only indisp ensable idea of sup ersymmetry is simply that of a symmetry b et w eenfermions and b osons/. Nature ma yo rm a y not b e kind enough to realize this b eautiful ideawithin one of the sp eci/c framew orks that ha v e already b een explored w ell b y theorists/.The exp erimen tal v eri/cation of sup ersymmetry will not b e an end/, but rather a rev o/-lution in high energy ph ysics/. It seems lik ely to presen t us with questions and c hallengeswhic hw e can only guess at presen tly /. The measuremen t of sparticle masses/, pro ductioncross/-sections/, and deca ys mo des will rule out some mo dels for sup ersymmetry breakingand lend credence to others/. W e will b e able to test the principle of R /-parit y conserv ation/,the idea that sup ersymmetry has something to do with the dark matter/, and p ossibly mak econnections to other asp ects of cosmology including bary ogenesis and in/ ation/. Other fun/-damen tal questions/, lik e the origin of the / parameter and the rather p eculiar hierarc hicalstructure of the Y uk a w a couplings ma y b e brough ti n to sharp er fo cus with the disco v eryof the MSSM sp ectrum/. Understanding the precise connection of sup ersymmetry to theelectro w eak scale will surely op en the windo wt oe v en deep er lev els of fundamen tal ph ysics/.Ac kno wledgmen tsI am grateful to G/. Kane and J/. W ells for man y helpful commen ts on this primer/. I am alsoindebted to m y other collab orators on sup ersymmetric matters/, S/. Am brosanio/, N/. Ark ani/-Hamed/, D/. Casta /~ no/, M/. Dine/, T/. Gherghetta/, I/. Jac k/, D/.R/.T/. Jones/, C/. Kolda/, G/. Kribs/,S/. Mrenna/, M/. V aughn/, Y/. Y amada and esp ecially P /. Ramond/, for coun tless illuminatingand inspiring con v ersations on the sub jects discussed here/. I thank the Asp en Cen ter forPh ysics and the Stanford Linear Accelerator Cen ter for their hospitalit y /. This w ork w assupp orted in part b y the U/.S/. Departmen t of Energy /./9/0 App endix/: Nonrenormalizable sup ersymmetric lagrangiansIn section /3/, w e discussed only renormalizable sup ersymmetric lagrangians/. Ho w ev er/, lik e allkno wn theories that include general relativit y /, sup ergra vit y is nonrenormalizable as a quan/-tum /eld theory /. It is therefore clear that nonrenormalizable in teractions m ust b e presen tin an yl o w/-energy e/ectiv e description of the MSSM/. F ortunately /, these can b e neglected formost phenomenological purp oses/, b ecause nonrenormalizable in teractions ha v e couplings ofnegativ e mass dimension/, prop ortional to p o w ers of /1 /= MP /(or p erhaps /1 /= /UV /, where /UVis some other cuto/ scale asso ciated with new ph ysics/)/. This means that their e/ects atordinary energy scales E accessible to exp erimen t are t ypically suppressed b yp o w ers ofE/= MP /(or b yp o w ers of E/= /UV /)/. F or energies E /</ /1T eV/, the e/ects of nonrenormalizablein teractions are therefore usually to o small to b e in teresting/.Still/, there are sev eral reasons wh y one migh tb e i n terested in nonrenormalizable con tri/-butions to sup ersymmetric lagrangians/. First/, some v ery rare pro cesses /(lik e proton deca y/)can only b e describ ed using an e/ectiv e MSSM lagrangian whic h includes nonrenormalizableterms/. Second/, one ma yb e i n terested in understanding ph ysics at v ery high energy scaleswhere the suppression asso ciated with nonrenormalizable terms is not enough to stop them from b eing imp ortan t/. F or example/, this could b e the case in the study of the v ery earlyuniv erse/, or in understanding ho w additional gauge symmetries get brok en/. Third/, the non/-renormalizable in teractions ma yp l a y a crucial role in understanding how sup ersymmetrybreaking is transmitted to the MSSM/. Finally /, it is sometimes useful to treat strongly/-coupled sup ersymmetric gauge theories using nonrenormalizable e/ectiv e lagrangians/, inthe same w a y that c hiral e/ectiv e lagrangians are used to study hadron ph ysics in QCD/.Unfortunately /,w e will not b e able to treat these rather complicated sub jects in an y sort ofsystematic w a y /. Instead/, w e will merely sk etc h for the reader a few of the k ey elemen ts thatgo in to de/ning a nonrenormalizable sup ersymmetric lagrangian/, so that they ma y hop efullyseem sligh tly less m ysterious when encoun tered in other w orks/. More detailed treatmen tsma y b e found in Refs/. /1/7 /;; /2/0Let us consider a sup ersymmetric theory con taining gauge and c hiral sup erm ultipletswhose lagrangian ma yc o n tain terms that are nonrenormalizable/. It turns out that the partof the lagrangian con taining terms up to t w o spacetime deriv ativ es is completely determinedb y sp ecifying three indep enden t functions of the scalar /elds /(or equiv alen tly /, yof the c hiralsup er/elds/)/. They are/:/ The sup erp oten tial W /( /i /)/, whic hw eh a v e already encoun tered in the case of renormal/-izable sup ersymmetric lagrangians/. It m ust b e an analytic function of the sup er/eldstreated as complex v ariables/;; in other w ords it dep ends only on the /i and not on the/ / i/. It has dimensions of /(mass/) /3/./ The K/ ahler p otential K /( /i /;;/ / i/)/. Unlik e the sup erp oten tial/, the K/ ahler p oten tial isa function of b oth /i and / / i/. It is real/, and has dimensions of /(mass/) /2/. In thesp ecial case of renormalizable theories/, w ed i d n o th a v e to discuss the K/ ahler p oten tialexplicitly /, b ecause at tree/-lev el there is only one p ossibilit y for it/: K /= / i //i /(withthe index i summed o v er as usual/)/./ The gauge kinetic function fab /( /i /)/. Lik e the sup erp oten tial/, this is an analytic func/-tion of the /i treated as complex v ariables/. It is dimensionless and symmetric under yThe reader will lose nothing here b y considering them as functions of the scalar /elds/;; ho w ev er/, in a moresophisticated treatmen ts o m e v alue w ould b e lost/./9/1 in terc hange of its t w o indices a/;; b /, whic h run o v er the adjoin t represen tations of thegauge groups of the mo del/. In the sp ecial case of renormalizable sup ersymmetric la/-grangians/, it is just a constan t /(indep enden to f t h e /i /)/, and is equal to the iden tit ymatrix divided b y the gauge coupling squared/: fab /= /ab /=g /2a /. More generally /, it alsodetermines the nonrenormalizable couplings of the gauge sup erm ultiplets/.The whole lagrangian with up to t w od e r i v ativ es can no w b e written do wn in terms of thesefunctions/. This is a non/-trivial consequence of sup ersymmetry /, b ecause man y di/eren tindividual couplings in the lagrangian are determined b y the same three functions/. Thisapplies not only to theories with an ultra violet cuto/ lik e sup ergra vit y /, but also to e/ectiv etheories where one has in tegrated out ultra violet degrees of freedom/.F or example/, in sup ergra vit y mo dels the part of the scalar p oten tial whic h do es notdep end on the gauge kinetic function can b e found as follo ws/. First/, one ma y de/ne thereal/, dimensionless /\K/ ahler function/"/:G /= K M /2P /+l n W M /3P /+l n W / M /3P /: /(A/./1/)/(Just to maximize the confusion/, G is also sometimes referred to as the K/ ahler p oten tial/.Also/, man y authors w ork in units with MP /=/1 /,w h i c h simpli/es the expressions but cansligh tly obscure the corresp ondence with the global sup erymmetry limit of large MP /./) F romG /, one can construct its deriv ativ es with resp ect to the scalar /elds and their complexconjugates/: G i/= /G /= //i /;; Gi /= /G /= // / i/;;a n d G ji /= / /2G/=/ / / i//j /. Note that G ji really onlydep ends on K /. So using the same con v en tion in whic h raised /(lo w ered/) indices i corresp ondto deriv ativ es with resp ect to /i /( / / i/)/, w eh a v e G ji /= K ji /= M /2P /, whic h is sometimes called theK/ ahler metric/. The in v erse of this matrix is denoted /( G /; /1/) ji /, or equiv alen tly M /2P /( K /; /1/) ji /,s othat /( G /; /1/) ki G jk /=/( G /; /1/) jk G ki /= / ji /. In terms of these ob jects/, the direct generalization of theF /-term con tribution to the scalar p oten tial in ordinary renormalizable global sup ersymmetryturns out to b e/, after a complicated deriv ation/: /7/0 /;; /7/1V /= M /4P e G hG i/( G /; /1/) ji Gj /; /3 i/(A/./2/)in sup ergra vit y /. It can b e rewritten in a sligh tly less compact form/:V /= e K/= M /2P /"/( K /; /1/) ij /W i/+ /1 M /2P WK i //W /j /+ /1 M /2P W /Kj //; /3 M /2P WW / /#/(A/./3/)where K i/= / K /=/ /i and Kj /= / K /=/ / / j/. The order parameters for sup ersymmetry breaking/(analogous to the auxiliary /elds in the renormalizable/, global sup ersymmetry case/) turnout to b eFi /= /; M /2P e G/= /2/( G /; /1/) ji Gj /= /; e K/= /2 M /2P/( K /; /1/) ji /W /j /+ /1 M /2P W /Kj //(A/./4/)in sup ergra vit y /. In other w ords/, lo cal sup ersymmetry will b e brok en if one or more of theFi obtain a VEV/. The gra vitino then absorbs the w ould/-b e goldstino and obtains a massgiv en b ym /2/3 /= /2 /= /1 /3 M /2P h K ij Fi F / ji /: /(A/./5/)/9/2 No w if one assumes a /\minimal/" K/ ahler p oten tial K /= / / i/i /, then K ji /=/( K /; /1/) ji /= / ji /,s othat expanding eqs/. /(A/./3/) and /(A/./4/) to lo w est order in /1 /= MP just repro duces the resultsFi /= /; W /i and V /= W iW /i whic hw ere found in section /3/./2 for renormalizable global sup er/-symmetric theories /[see eqs/. /(/3/./4/5/)/-/(/3/./4/7 /)/]/. Equation /(A/./5/) also repro duces the expressionfor the gra vitino mass that w as quoted in eq/. /(/6/./2/1/)/.The scalar p oten tial eq/. /(A/./2/) do es not y et include con tributions from gauge in teractions/.The D /-term con tributions to the scalar p oten tial are giv en b yV /= /1 /2 Re f /; /1ab bD abD b/;; bD a/= /; K i/( T a/) ji /j /;; /(A/./6/)where Re f /; /1ab is the in v erse of the real part of the gauge kinetic function matrix/. In thecase that fab /= /ab /=g /2a and K i/= / / i/, this just repro duces the result of section /3/./4 forthe renormalizable global sup ersymmetry scalar p oten tial/, with bD a/= D a/=g abe i n g t h e D /-term order parameter for sup ersymmetry breaking/. If sup ersymmetry breaking tak es placethrough F /-term breaking/, then it is often not necessary to include the sup ergra vit y e/ectson the D /-terms explicitly /. There are also man yc o n tributions to the lagrangian other thanthe scalar p oten tial whic h dep end on the three functions W /, K and fab /, whic h can b e foundin Ref/. /7/1It should b e noted that unlik e in the case of global sup ersymmetry /, the scalar p oten tialin sup ergra vit yi s not necessarily non/-negativ e/, b ecause of the /; /3 term in eq/. /(A/./2/)/. Thismeans that in principle/, one can ha v e sup ersymmetry breaking with a p ositiv e/, negativ e/,or zero v acuum energy /. The last option migh t seem to b e preferred phenomenologically b ythe absence of a cosmological constan t/, although it is not clear wh y the terms in the scalarp oten tial should conspire to ha v e h V i /= /0 at the minim um/. F urthermore/, it is not at all clearthat h V i /= /0 really corresp onds to the requiremen to f a v anishing observ able/, quan tum/-corrected cosmological constan t/. /1/3/1In an y case/, with h V i /= /0 imp osed as a constrain t/, zeqs/. /(A/./3/)/-/(A/./5/) tell us that h K ij Fi F / ji /=/3 M /4P e h G i/=/3 e h K i /= M /2Pjh W ij /2/= M /2P /, and an equiv alen tform ula for the gra vitino mass is therefore m/3 /= /2 /= e h G i /= /2MP /.An instructiv e sp ecial case arises if w e assume a /\minimal/" K/ ahler p oten tial and dividethe /elds /i in to a visible sector including the MSSM /elds /'i and a hidden sector con taininga /eld X whic h breaks sup ersymmetry for us /(and other /elds that w e need not treatexplicitly/)/. In other w ords/, supp ose that the sup erp oten tial and the K/ ahler p oten tial ha v ethe formW /= Wvis /( /'i /)/+ Whid /( X /)/;; /(A/./7/)K /= /' / i/'i /+ X /X/: /(A/./8/)No w let us further assume that the dynamics of the hidden sector /elds giv es rise to non/-zeroVEVsh X i /= xMP /;; h Whid i /= wM /2P /;; h /Whid /=/ X i /= w /0MP /: /(A/./9/)whic h de/nes a dimensionless quan tit y x and w /, w /0with dimensions of /(mass/)/. Requiringh V i /=/0 y i e l d s j w /0/+ x /w j /2/=/3 j w j /2/,a n dm/3 /= /2 /= jh FX ij p /3 MP /= e j x j /2/= /2j w j /: /(A/./1/0/) zW e do this only to follo w a p opular example/;; as just noted w e cannot endorse this imp osition /./9/3 No ww e supp ose that it is v alid to expand the scalar p oten tial in p o w ers of the dimensionlessquan tities w/= MP /, w /0/= MP /, /'i /= MP /, etc/./, k eeping only terms that dep end on the visible sector/elds /'i /. It is not a di/cult exercise to sho w that in leading order the result is/:V /= /( W /vis /)i /( Wvis /) i/+ m /2/3 /= /2 /' / i/'i/+ e j x j /2/= /2 hw //'i /( Wvis /) i/+/( x /w /0//+ j x j /2w //; /3 w //) Wvis /+c /: c /: i/: /(A/./1/1/)A tric ky p oin t here is that w eh a v e rescaled the visible sector sup erp oten tial Wvis /!e /;j x j /2/= /2Wvis ev erywhere/, in order that the /rst term in eq/. /(A/./1/1/) is the usual/, prop erlynormalized/, F /-term con tribution in global sup ersymmetry /. The next term is a univ ersalsoft scalar /(mass/) /2of the form eq/. /(/6/./2/8/) withm /2/0 /= jh FX ij /2 /3 M /2P /= m /2/3 /= /2 /: /(A/./1/2/)The second line of eq/. /(A/./1/1/) just yields soft /(scalar/) /3and /(scalar/) /2analytic couplings ofthe form eqs/. /(/6/./2/9/) and /(/6/./3/0/)/, withA/0 /= /; h FX i MP x //;; B/0 /= h FX i MP //; x //+ /1 x /+ w /0//=w / //(A/./1/3/)since /'i /( Wvis /) iis equal to /3 Wvis for the cubic part of Wvis /, and to /2 Wvis for the quadraticpart/. /[If the complex phases of x /, w /, w /0can b e rotated a w a y /, then eq/. /(A/./1/3/) impliesB/0 /= A/0 /; m/3 /= /2 /, but there are man y e/ects whic h can ruin this prediction/./] The P olon yimo del men tioned in section /6/./3 is just the sp ecial case of this exercise in whic h Whid isassumed to b e linear in X /.Ho w ev er/, there is no particular reason wh y W and K m ust ha v e the simple form eq/. /(A/./7/)and eq/. /(A/./8/)/. F urthermore/, w eh a v en o ty et explained ho w gaugino masses arise fromnonrenormalizable terms/. This requires a non/-minimal gauge kinetic function fab /. If thegauge kinetic function can b e expanded in p o w ers of /1 /= MP asfab /= /ab h/1 g /2a /+ /1 MP f ia /i /+ /:/:/: i/;; /(A/./1/4/)then it is p ossible to sho w that the gaugino mass induced b y sup ersymmetry breaking ism/ a/= /1 /2 MP Re/[ f ia /] h Fi i /: /(A/./1/5/)The assumption of univ ersal gaugino masses therefore follo ws if the dimensionless quan titiesf ia are the same for eac h of the three MSSM gauge groups/;; this can b e automatic in certainGUT and sup erstring mo dels/. Similarly /, the sup erp oten tial can b e expanded with thesc hematic formW /= Wren /+ /1 MP / /4/+ /1 M /2P / /5/+ /:/:/: /(A/./1/6/)where Wren is the renormalizable sup erp oten tial with terms up to / /3/.I t m a y also b e p ossibleto expand the K/ ahler p oten tial lik eK /= /i / / i/+ /1 MP /( / / /3/+ / / /2/ /+ / // /2/+ / /3/)/+ /:/:/: /;; /(A/./1/7/)/9/4 If one no w plugs eqs/. /(A/./1/6/) and /(A/./1/7/) with arbitrary hidden sector /elds and VEVs in toeq/. /(A/./2/)/, one obtains a general form lik e eq/. /(/6/./2/5/) for the soft terms/. It is only whensp ecial assumptions are made /[lik e eqs/. /(A/./7/)/,/(A/./8/)/] that one gets the phenomenologicallydesirable results in eqs/. /(/6/./2/6/)/-/(/6/./3/0/)/. This is wh y it is often said that sup ergra vit yb yitself do es not guaran tee univ ersalit y of the soft terms/. F urthermore/, there is no guaran teethat expansions in /1 /= MP of the form giv en ab o v ea r ev alid or appropriate/. In sup erstringmo dels/, the /\dilaton/" and /\mo duli/" /elds ha v eK / ahler p oten tial terms prop ortional toM /2P ln /[/( / /+ / //) /= MP /]/. /(The mo duli are massless /elds whic h do not app ear in the tree/-lev elp erturbativ e sup erp oten tial/. The dilaton is a sp ecial mo dulus /eld whose VEV determinesthe gauge couplings in the theory /./)Finally /, let us men tion ho w gaugino condensates can giv e rise to sup ersymmetry breakingin sup ergra vit y mo dels/. This requires that the gauge kinetic function has a non/-trivialdep endence on the scalar /elds/, as in eq/. /(A/./1/4/)/. Then eq/. /(A/./4/) is mo di/ed toFi /= /; M /2P e G/= /2/( G /; /1/) ji Gj /; /1 /4 /( K /; /1/) ji /@fab /@/j / a/ b/+ /:/:/: /: /(A/./1/8/)No w if there is a gaugino condensate h / a/ bi /= / ab/ /3and h /( K /; /1/) ji /@fab /=/@ /j i/ /1 /= MP /, thenh Fi i/ / /3/= MP /. 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