Pion Nucleon

Pion Nuci€on | Phil Lucht | PION NUCLEON NOTES MAY 1974 . D 1.Miso. a.Polarization inhelicity amplitudes. 1 b. Gasiorowics ABnormalization 2 o.Status ofamplitude (full) relations. 3 4.library info hunting 4 2.Crossing. . + a,SU crossing with neutrad pion only 5 b.8-7 crossing 6 ce.T,4 (t-channel!) interms ofAandB 7 “ 4.8-0 crossing with isospin 10 .Time reversal with isospin. lL f+Crossing Relations with Aand B. 12 -comments and adding isospin indices. 13- Gioter ellthisstuffwasmotivated bydesire tofindt-channel helicity amplitudes interms ofAand Bbycrossing right atthestart. Belowthisisdonewithexcruciating painviathehelicity _crossing relations. See part (¢)above.) - \3s Isospin and the t-channel . a.discussion ofantiparticle isospin states. 15 b.crossing statement with isospin 5 c.the general t-channel isospin expansion 16 a.PI-N case: expansion with charged pions p5. 29 e.PI-Ncase:expansion withcartesian pionsp6. 20 @ £.Relation of7(0) andT(1) toF(+) andF(-) al g-PI-N isospin amplitude crossing relations. 20 h.charged versus cartesian ‘pions p8. 22 4.Full amplitude interrelations. a.between f1,f2 and fyyyfp— — 23 b. note on rotation of spin states. 24 c.between f),f2 and the various pairs f,g 25 a.between f),f> and4,BofCGLN. 26 e.between 4,3 and fy4 orTy, at 35. Catalog ofPartial Wave and related amplitdes a.submatrix diagonalization with parity eigenstates 28 : bd, justification for phase shift parametrization 30 .c.various partial wave expénsions 32 a.Goldberger Watson projection operator method footnote 34 - 6.The DCS equation. (in terms ofA'and 3B) a.through s~channel (via part 4eabove) 35 b.between A;B-end Tj, (t-channel) via helicity crossings 37 ©.DCS through t-channel 38 (Hote: these are ofcourse unpolarized DCS) _1+Optical theorem, polarisation intermsofAYandBo |__ a) 4 Uy m5:TERS,BaF*(pseryABAQuer.-ABApyse) Ke4AYU WAY hdbY,GHSS sof Gas: MESRRADADBorel SHPED PHrhGarOoKSaySe”FeG- . = adn: BUGq,SezMpbbbot wah:AptGh)a,b(=hh) a” 1 Reduction forPI-Nscattering. Polarization, \ | 1 =@ The full equations are: 1 ‘ o) Oo Be de=ZiFaomtennSt gh=£(\+RG)|| . * @ cumle , 0 ;aneZovongCAMLNLEMD TrasFinget’Sand | - Obviously theonlysurviving polarization tensor istheL=1,sincethefinal |“ beam3isspin}. ThePyforthisfinalbeamisgivenby, H * * 'saleZaSeta) kal}FamFacumt~FaginFm’(. Wecan seethat only m3=+4contributes inthis sum, forwhich the SQRT isunity mes ae oy el. i) , : ; AER =2,a[SomCelene] Ofcourse aslong asthe initial beam has some obscure polarization, Pyand Py are also presnet. These explicit formulas are still abit messy. e . a} *HlAl= 2Za, Sabrn,Pi For anunpolarized input beam, these equations reduce to:| ee. ie +_ Re canelf <4ZUbe ElLimGnd=£2LRTfatal) eyPyfe “* | s&sal>libelt|fut+4,-£_] a QeanlA)=A“ms|asa Ss= ae sebeis Watts FTP ee eee Finally, since parity isconserved insuch astrong interaction, weget: eu >“ a5={led1h ; . ie* | vo ge fee Befe ge. at |ZEB=~dw|+fttes)=dm[Sa--feF-]| a|=28m[te#2]a Thesearecorrect.See(5.84)5.| — .c : . Gasiorowics PI-N normalizations. — Looking atp364 and p370 and p34ofGas, wecan figure out that: 1a — T=jarsELA+ AB) T= 4 « f= -s6rim T vs on However, after converting tothe BD metric, wefind that CGLN defined their Aand Baccording to: We.=aLA-BAlu --Kustw T= WeSuds 7 Putting these facts together, (fidthesame inboth cases) wefind: ces e A= WA \ BY=-aGB Ow ~ a4 : Status oftheVarious Pion-Nucleon Amplitudes soFar. (Excluding partial waves.) @ Here,anarrowmeanswehavean.explicitequationforxintermsofy: K—yY Thus, here iswhere westand: : TRE7 fi(OIG ST ~~) , CABAY, Gh), Ca), Byefe-), Te,7) f # “Ithink thatnowIcangetthepolarisdtion ofrtheDCSinteras ofany : amplitudes with complete accuracy. Algo, can relate any amplitudes to anyotheramplitudes.Nowampreparedtoplay. |'. . | | | . i ©‘J i 1 * 1 Soosedlaierienenlialialict memeneemennennnmmenma meetin Spm ee Sano na aa 4 eeeoer he = 1... ..|Information Hunting. _ : . ae B \ | hoy.7 —tonverning thenuciesriscience-abstracts: thelibrary staff-has-gathered =~ :6_--together allthefiveyearindices, tHe’last5-yearperiodbeing1967-1971. ."Porallyearscoferedbyhensinddeng,thesomal‘authorapdsubjectindices "+ —~~——+have-been disoarded:—Thue;they-have -only--these-annmual—indices~for-1972;-19733~ |2. fiheseSpyearindices’ areextrenely thorough. Theyhavebothsubjectand authorindices: thisiasurely theplace tolookforanything ifyoudqnot -——+—*+~have-a-ghean-journa} reference. oo omnes -- me a ! wed oepe ee .- ~-. xampie? Iwas Isoking forgnARCrepdrt titled LAMS-2740 byAVK.Kerman. There ‘,— -—+as-no-jourtialreference for-this.—I-slould-have then looked upKerman~in- the a appropriate 5-year index. Thatwouldhavegivenme:the publication reference _ . Tifthereeverwasone,Nowitturnedgatdnthisasethatsincethething “i__-was se-long-thet~it-wa neven-published. Then-what- do-you-do?—+ ~ -- ~~" ~"“Ghenyoua>CheckoneFPthefilesinijthe looked PoomtoseeiFUCBhasa~~—+—- ‘oopy-sitherup-here,—in thedppository inDoe,oron-microcards. ‘Thereare~—two kinda Ofmicrocards. ‘Thepositives donotmake copies, butfromthe a“~nfcrofiche youcanwakedopies directly fromtheminthelibrary. =~ i . \ ; : ‘ j i 1aera a pal --— pe ee pe -aeee >. on or Le Fp - [ete eto : ee i t ee i -OF ee ee ee te a4 re ee =--| ne ee 7Fe ee icine ies tiem eae od -Salt eae ee!Fe a ee oi Pe ee Pe - if. Lo -- nee poa : t ey a oe ee 1da a — rn a — ae eeeCe ToA re :a — — a re ob bee ~ee ee { + pt 1 re fg a ae + - i “ t “ Crossing: 8,U with neutral pion only. es Thisisthecasethatisconsidered inallthebooks.Forexample, see dD Gasiorowics page 363f. Using thereduction formaliam ofLSZ(asdiscussed - inB+D volume 2,say) veknow that wecan write the S-matrix amplitude for the e-channel process tweet © inthefollowing way: <P'a/}eq> =MsCeq's 2)=Ta(PFP19) <(er Vora, ee. Segy !“(Voy SRceit(dagenin% : Here weonly need extract the pion féelde from the assymptotic states, since . - the mucleon fields will not beneeded‘in the comparison that istofollow. - Byasimilar application ofthe reduction formulae, wecari write the S-matrix amplitude fortheu-ohamnel process ay) ¢har @ ,: inthefollowing way: KXpQD=Ma(Pa'<— pQ)=Ta(pp) Qa). ot xia-y <<.) iGex=(iS SP ne eS geen T ESGe)Sey ToITIP SHIDOKSO Bysimply comparing thesetwoexpressions, wecanquickly conclude that: -,;|ratat ol 'een’| Ta(e2-4,PhQE-q)= Te(PCa 0-9)=TA 9’)mer @ Thissaysthattheamplitude forthephysical u-ohannel process (where-q :and=q"arepositive!) isgivenbythee-channel. amplituse contizued to :aparticular unphysical point. This point isunphysical forthes-channel i because the qand q'appearing inside 7,are both negative. : Wow, inthe special case ofneutral pion scattering, the s~channel is the same channel asthe u-channel, because achannel isdefined bythe typeofparticles involved. Thus,inthiscase(only) wecansay: \ TN et? ‘ =a! o!_ ote, = TaCPinaiind)=Te(eraieina) =TeCeaeg’) : ganat chev Ifthe 7,function iswritten ont inthe standard parity-invariant way, : f er~TeCAUPY) =TENLAGH) +LG+A) BisJace) : Then weseethat the identity ofthe q-and u-channels implies that: - oe aTCP-4,Pig)=ROLLACS)-+4)B(u9)]a)| (Again, thisisonlytrue forAandBdescribing theneutral pion-nucleonamplitude.) 2 Dd >AG)= Alas) ~yo . &G) =-8(4,5) : -- . : | - -- ny — - : : o Crossing. S,Tcase. = Forthes-channel process: 4N<—wt © . wri: heS=mat: Litud t 1 me _ wewritetheS-natrix amp! feas“ised ea)=<eh'led=Gs)FC,ry) A\i=i 5iq ighy— nd ~lle)raySOGStTeasnyGrreeca payer Forthet-channel process: ww <—NN @® mewritetheS-matrix amplitude as:‘Tite(Gu’<pL)=Caq|pLp= VESle(p£,Q,9 ~iPyi)(4 & _ > tax“ADM, Sies} aireora py2 Now acomparison ofthe objects Tabove shows that: Wes’—e¢)=VERS)Tee,#1-0.9) Telerei-aig) =-EQs9, Ba!) = Preromani, dingoTnnahSatonn Thislastlineisthecrossing statement. Notethatitapplies totheamplitude withoneofthespinors pulled off.There isnoneedtomakeany assumptions like, _ awC-e) =Vep) wirich really isnot true. . Ifweexpress the s-channel amplitude inthe usual way, ToC ria)=LAGs.t) +Chea’) B54) ucps) * Thenwecanwrite thet-channel amplitude likeso: = -—ro Py*,; eyCate#-F')=-Tes) |AGA+Eea)BOS]ucas) ae: __i? Vawe Again, weshould keepinmindthat(here)qandp'arenegative quantities. Theylie inthe past light cone. Here, tispositive and snegative. The idea of continuation isclear, 7. Now, ufilike thé S-U-srossing cass, the Sand Tchannels are never the r same(asdefined here), 0thereisnoequation analogous toA(s,u)#-A(u,8). a PI-N t-channel helicity amplitudes interms ofAand Bvia direct crossing.PioWtrohannel helicity amplitudes intermsofAandBviadirect crossing. _ - Keepinginmindthatp'andqarenegative fourvectors, weknowfromstudying crossing thattheinvariant (Jackson) amplitude forthe Aephysicalt-channelprocesswreNN is*givenby:wl wht,= Ne Th-aq'= pf)=—Wees) -A-@el ured) ="OG jtSCH yt w ’v . v=-\-ehm(TeamEn-s) (op)\E+ whore asbefore weareusing: egve =, °<- QE(q+q)Wu=2m=~W, rh: TheAterm maybetackled first. Itgives: ¢ 7 = 3 a at Ams cacTemem) (vfGe) -(ERS) Since weare inthe cms, weofcourse know afew helpful things: BS 3 =CeE=% andCF)=-C8) oe Withthese aide, theA-term immediately reduces tot Alo=—7A(*cnx) | Mext, the Qterm, This term vaniskes fortvo different reasons, oneof which isthat the two terms which added above now cancel. Also, itturns out that @,itself is0because: :Q=tl-Car) =£L%-AQ=0- Brtews: GoalF-Can= sle-Ca- @ ALL that remains ie the momentum Bterm. We find: = 3 a3 i (—o a (ete ~Eom=~Cempeancdy(af EB)2ED¥he) : =-28 (WPERR -ENED) 7e+m : Some algebra isrequired to simplify thig, sohere are some.of the steps: 2 a22s.2)\2 5¢ea =pa=-Pa (an ZaBa)a+BE)4yuoPASgeose “Ft a7,2meelel70 svt [C228mgiese) Ge+(r28EiSN) GJa Now wehave assuméd that>isalongthe¢axis(thenucleon)andn=y.Notice —-@vythe way that the parity-allowed terms inthe t-channel \look like --insome sense ~~the parity-violating terms inthe e-channel._/ a) ot oe :w=(55) x=(40) * oeabel . ': + -2- => Beforegoingontocalculate thet-channel helicity amplitudes, itisnecessary tounderstand clearly the spinor vofDirac- theory. First, consider aparticle (ferméon) (not anantifermion) travelling inthe +zdirection with z-component ofspin equal to+4. Inother words, this particle has helicity +. The question is: which spinor describes such aparticle. Aseverybody knows, the answer is: U(RS)=UCpepeSateinsasthowd)(3)\e w(9 + 1S)= = ate = Sad =F\oy BPs(1) 7 . .erm . : Somehow, everyone knows that (1,0) represents anelectron orfermion with spin ‘ +%along the zaxis. This isthewayd,issetup. 1 Now, consider ananti-fermion travelling inthe +sdiréction with spin i inthe plus 2direction. Thie isapositive helicity anti-fermion. Everything isthe same asbefore, except this isanantifermion. Question: which spinor 1 represents such aparticle? The answer igopposite what myintuition suggests. -Itist - } Jefe) 0)1 aA VO).=vV(p=pe,S=+2ssopana)=|e+m(8 ve©) This rather subtle point isdiscussed ~ °inBidpage68;Thepointissimply (5)this: this state v(p,s) represents . anegative-energy fermion with spin down(along~z).Butinholetheory, ananti-fermion istheabsence ofa e negative energy fermion. Itisahole inthe sea, The absence ofsomething with spin down isahole with spin up, just asinsolid state physics. Inourpresent application, wehave our anti-nucleon travelling inthe _-> =direction, Ifthis anti-nucleon has +'helicity, then ithas spin inthe ~2direction. Thus, for the +helicity antinucleon, one should use v=(1,0). Naively,-this iswhat one might expect inthe first place, but wehave seen that itisreally acase ofadouble-negative. Now, recalling the factor relating the Jackson (Q8D) Mamplitude to the MShelicity amplitudes 7,wehave: N= an+)au Sn)%}=-(4t)cog=(-2Ap+2Bong]cose)VY)%UC) + ry try :«+ (+2BEig)sine)uy)O%u(y) where, asdiscussed above, =( =(2 Hay = =w=() ue=@) Wate w= CO Thus, inparticular wefind: ‘ ay, =&thtToys =+BEEIgelsuRe _EG, a=k} =@ ~"WToojer =~2AQ+2.Brmajge|oose H‘ : 4 -3-. Inordertocomparethesewiththeamplitudes quotedbyMSpages and 456 4wehave todoalittle time-reversal. According toMS(5.51) p232 we know that: (perl)~(Haat) Sposipom =CY Tip G=_ tte >S00;wy=G) Syl;00 >foo;++=Face;09 1 =)foo;wR fi,90 ' Then our final results should be: a +t); |—4n"T+tyoo=-2AR+2Brw|q{eose !2 Ul . ; j49Trjoo=BIEiqdsue | There remains aslight ambiguity concerning the meaning of9.The way I have calculated things above, Ihave been using ©ae‘the angle between the incident nucleon pand the final pion ofmomentum ~q: Of, an2% nae -@rw- A mn Ne B==) =—Pigjcose HG~&-(-Q)=-Gqlwose -o%Ig|sime Suppose instead that wehad used asour scattering angle the angle 9'defined asthe angle tothe other pion ,ie, the one with momentum q'. Then we'd have: ar 1K / : ne) Ne Ba=Pa=polese’ _ can=Bq eqcose+GQSway , , .. Buk@owde’onsuppSamande :cose’=-c089 andswe=SUS Interms ofthis other angle, all the sin and cos terms inthe helicity , amplitudes would have their signes changéd. Sothe question that remains isthis: doMSusetheangle from ptothecrossed pion, ordothey usetheangle to the pion that was inthe final state inthe s-channel aswell. Myinclination isthat they use the former (called O=above). This isbecause : S a = (p+q) = -)) =s=(pra)=CPO) Te)compare, - e . De @ Cp-p) =ts Thus, the formulae above stand asthey are. The lower one disagrees byasign. "from HS's. - ' : al | SUCrossingwithIsospin. i sand 4 | Again,goslowandstayclean.Startwith: 4 lye % ; ~igx >SW)» ,, sIIF oe)Wicd ps)~Samay CPIM GFHageallo(E7kK YP : Now rewrite shuffling stars and using fact that: igo}=gon Thus get, igx>Sam, aysOp) =Saray(CR G9Cytrbgicatendp>GCKe] Now, reverse positions ofthetwo(...) objects, switch xandy,switch . iand j. Result is: - a igh:AM), aysAw) -Viidy(OR$f)cerittacogatld (€77%ge) Since the fields are inside aTOP, their order does not matter. Comparison ofthislastexpressionwiththefirstshowsthat: ad ws det,"1 Se ey=TECinq ea) Here Ihave been thinking interms of.charged pions. Ofcourse the thing also goes through with cartesian pions. For cartesian pions, however, (a) =(a) sowewould forget the stars inthe last line above. Ae)ay ,. A . q=|4 Jocordesiansprsia+wuhastai=Sie. —® a - ! = Time Reversal with Isospin. | ; For some reason this keeps goofing me,up. Goin slow motion: axXO,, ayighyY Taipan) ~Saray(ERKJCerrFscranny (49%, 7) : » signAIF,, * -69! LTPdina» Saeay(EPREN eerriQianoyi® GPE&7 ‘The matrix elements yields itself with-p and p'reversed,so get: migxAwe (KOS of ~Sardy (EPLAPYceiriqinracghr(ORE* Nowsimulataneously switch the(...) projectors andalso switch xandy ' andalso swithh iatidj: + i} ~ighx >H(A) (BOE >HI~Sandy(EME424T(gerdgayr (8% =@Above weused the fact the the fields are hermitian and also the fact that - the ordering offields isunimportant inside atime ordered product. So, comparing this last expression with the first we find that: . ' [Wereiea)] =Tea tad Thisresult isvalid fortothcharged andcarteisn pions. = - 74 : | Crossing Relations fortheAandBamplitudes withIsospin. | The following statements areallmade interms ofcartesian ° pion isospin labels only. The general form ofthe PI-N amplitude may be written as: pe ox uior v. Welt nay=ze)|A,Gu)+4+4)CoD) uP) All isospin indices are shown explicitly. For the carteisian labele . wehave shown that each amplitude can bewritten as: «pO. | ‘C A=SMSA9*tis.AP Sp C) -) BE=Sa, C4chsh. 6° From crossing wefound that, red ae :ThCea 4)=TTColas 8) ; and from time reversal we found that, : Peoy Bes ogy]* e Tea pa=[T(Gnea)| Combining these two relations, wecan show that: a ¥ - THCla'pa)=[7Ppa; pia)| This isequation (23.16) ofGas. Gas ‘called thie relation "crossing" but : itseems tometobeacombination ofcrossing and time-reversal.. Inorder * to apply this relation tothe form for Tabove, we first need alittle Dirac algebra: * * ** Lice cay] =[utcorgeaagwlDg)”=EeWMia Ua st=MMs CUpeUe=ae)Heuy’) ; ~[ncays ace] . Thus, the relation may be written as: .‘ ya x AMS)+3(keg)BM)=[AMCs] ETaeq 1BMaoy x x eh AMG=[ABGs]* oneBMG=-[BMuD]™ )Ifweuse(b*)notation ononesideoftheseequations, theybecomealso =@@& trueforcharged pions. Keeping withthecartesians, weseethat: i(3) *Th: wltisindependent ofthe"type"of e\AGu)=1[A® w,s)] pionsused.itisreally@statement about fVv @ _ cy 4t-channel isospin amplitudes, eg, :=OeeT eo, tos, - I ' Comment onCrossing. Ihavebeenusingthereduction formulae ofaxiomatic field theory toprove these crossing relations. Field theory happens to ‘beamodel inwhich crossing istrues But really crossing ismore general. Itisthe analyticity statement that the 3channel amplitudes are just 3boundary values of one hugongous analytic function] Inthis general frame ofreference, all these crossing relations are trivial : and infact belong on the same level askinematics. For example, toshow S-U crossing relations, wewould simply observe that T(s,t,u) issupposed togiven the scattering amplitudes . inall three channels, Here the various spin, helicity, isospin ete indices are suppressed. Now, ifthe sand the uchannel are the same channel, then itmust betrue that T(s,t,u) =T(u,t,s). Itcould be noother way. Then all the other details follow like B(s,u) =-B(u,s) simply from the definition ofB.Ifweadd isospin, the statement is . alittle more complicated, but the results are all the same (seeMartin Spearman equation (5.100) for‘acomplete statement ofcrossing with isospin shown explicitly. See equation (7.34) for acomplete statement ofcrossing with helicity shown explicitly. ) About Aand B. Whenthesefunctions appearwithoutisospinindices, onesaysthethe —@isdealingwithaparticular processwithaknown,given setofparticles anfCisospia labels). Itmst beunderstood that when onedoes acrossing of some sort that the isospin labels ofthe crossed process are determined dythe lab{és ofthe uncrossed process. For example, back onthe neutral s-u page wehad: a aTq) VAP) AG,4) onl stayTalons] ~ROYACu,S) The indices must betacked onintelligently, not hapha,ardly: Ae ho5)ght ry xi)ate Te(ia). ~2) ACS) TPG )~ae)Ahus) Similarly, inthe s-t crossing situation wehad: . ~~.): : Te--) ~Te|AGH:-- co! . Tol) ~KP) TAG). - Again, wecan add isospin inddces: ee peTre TEC) AxaGt} . (Be 7-9 pet —® Teas~VCH) ABSGH) . The meaning here of(b*) isalittle complicated. The isospin ofacrossed ~ nucleot isumusally.taken tobeopposite the uncrossed, so(b*)=(-b). This ! allows CGmanipulation, ete etc. we———Qressing:titeommgdouonadSadia!=smne.dirswoydoexprose)tne—+-showasl_Larrn8i]. dnstieitiy omptioedsa dndensofAomd8, dome.—— HB ekg prspeatnnens wohidly deearlar Sachets eressiapin henna ofRemsen re _- --ee —-—BaapaoVAXpi igenadGreSe2uwereatng)amdhindAMGW)---ce ania tenalunnag!onmbeska GalMoms(= -omnchGatdlenagen (1954)4PaceAMO.Rfdauber Cif.p26F.Bukmouae|HCP)wane, ads(p25)MayamgaecktrotMat)msdeeT,iasmfect hraoeinn058thaws yvacsiyaivas Ws§-heanfestacs\ecia I Saadostcterstrcn Sowa"elStacks Actas,fey.toondseyvidaspr_~Chewtiddowitw Ei of 8. eee Gil WennGata, CandieRectealn)QIaunancno Snares seaain; lit | n==Onlypradnangromreutionr Direolsnodleinp O0AfroCASS),22penneott _--_-rewodtisa). YYont_tonddSere indereofTO)audaS.Tarra Sete,OrawinkeddmgetsTHYnekToSCoieraglana AGreenable cmathrad (Momradsto.dialidespin ap) te)Ore re wef oe ee set -——Countiwrd (beckon_aan ).Vertraaicailypviews Dacecw- ——-4stenting gorink osetwerga, somatliry dle.cdlMs}ab).= Su(blMalad) =omdvermmcandy CodiMslae= (EALMielez)Fst_okor fouprone.2=Oaquedinbetertaeabncheek,V8)smsjoerCalera)trouSpd. ahedag —ecbBikatooveatealeCagle" noewey Bemid == etreeon.gagemaaaeaaingJoySoUagurteuronppsench, Dwd. a set eee Re ee ne Shoeee ee :we ee= — -4 a 2| ee ~- oe ee — —oeH —_——-- ; ‘ - 4 Hl oS—: Isospinandthet-channel. —>FOomaFOeke, | =, Carefully definethes-andt-channels asfollows: . @ateoerd ®Gtrorer+e* The crossing statement with isospin isthis: ) @, :7 Seem; FarmalTO|Tarma;Trowy=<Teme;(Tata)|T|(Tamd)¢ iLema where (Iu7i)c means. C\Tdwd> ,thecharge conjugated state (antiparticle). =[Td . Unfortunately, these physical antiparticle isospin states donot transform according ththe usual angular: momentum (isospin) algebra. This . isSimply due tothe fact that Cdoes not commute with I.Infact, ecly=-1¢ crh,=+he ch=-he Observing that (1,G) =0,HSsuggest defining anew kind ofantiparticle isospin state: el >kw, ald\Imy =%GlIinw? =UC € VImp> =We(-) (E-my, : These barred isospin states docouple inthe usual way, and they are related © tothe physical antiparticle states asshown above orbelow: . = p Ctorend : Vimy = CNP™(Ime [Dindge=MECI™[E-my : e@ Theideaistodoallmanipulations withthebarred states, then return to ' the physical states atthe end ofacalculation. ' ' 4sanexample ofthe meaning ofthese states, wecan write the isospin 1 state ofaproton asi |=144 =\E-8> Theanti-proton state physical is: Op= |b. Oper le-De ' H +, Butthestate thatwewilluseinallthealgebra is: (P=|E2).= MC)larzy You see that the I;components change sigm inthese —ibarred states, and-sone ofthestates thenselvea Qit= LEE,pick upanoverall minus sign. Itwould benice ifwecould avoid these extra overall minus signs. Recalling that the charge parity ofacharged state isarbitrary (asopposed : tothe charge parity ofaneutral state), wecan avoid the above minus signs byincluding them inthe definition of|byand|#) Inother words, choose: : Upr=+lp7? 7- Ipp= +le-2> - Cly=-le> p= lepat+lED Uy=-l Thus, wechoose the phase sothat the states come out with +signe. Now the only confusion isthereversal oftheI,sign. 4similar analysis canbedoneforthepionstates.Here,wehavesome . = conditions imposed. First, weknow that the neutral pion has C=+.Second, since the pion isself conjugate, wewould like tohave the barred states identical to the unbarred. Then: 7 ty <== = = * OursYet, City> FR =>WFP=-ots-1h ° | wee | MN«—«—2"overthispiononcemore.lechoosethevariousphasesendcharge | parities so that: . ' Wer=leyay ! RTS (41 =hei?) This isall convention. However, since the antipluspion isaminuspion, we . can add to the bottom line above as hhown, With all these preliminaries out ofthe way, wenow write the t-channel scattering amplitude with isospin shown explicitly and with helicity indices . suppressed along with the invariants sand ¢: @ me =ta Cremelat)elThCamaDaw=CC semeEeeTTDd ®= a+aed+b @= exdearb Nowdrop thebars ontheaand4isospin states, since they arestandard states. We-will keep inmind that a-and denter as antiparticles. Now wecan expand ' _ interms ofisospin amplitudes T(I) using usual CGcoefficients: = me Tad . (J=P MRM ESCrmremlEMrtmtama(tHTa), eLet's apply this immediately tothe PI-N t-channel. Wehave: 1 "«Tet ov vyYew w ' @= eta d+b “@= exdHarb : Now leta,b,%,(@ bedefined astheisospin labels inthe s-channel (not in -the j-channel). That is, ' MWa=% My= m=&md=b . Interms ofthese labels, wewould write asthe s-channel amplitude: SPTPtag> ‘ ‘Using these same labels, the t-channel amplitude is: ‘ xa wn” teeny 4) y=" 4+ , . d. eal (By=Ci)Ci) ZCheyxlIHYK48)alTHY T(z) Cae; C) =o)"ay®«|<ipi-aloaree-b ee|oopT(x29)+Cipinalldy dabalinyTe21} Wedonot include Im2 because the 4CGvanishes. From either the s-channel : 7 or the g=channel one can see that: - ©mere=smth ==©]Marysverel . Qh.Ara=~b+a 2dta= P+b _ * .-3- . niedNow lets return tothe general t-channel formalism where wehad, Zam Tard w HermesTeed|TOYTamasTomy=GV Cry ZT)-¢emTemaNYGemtTameTAY Ifweuse the properties ofthe GCcoefficients given inMSappendix oranywhere else, wecan show that: SereTamaEMD=CrysetTemeE-N\Tanto? ‘ * Tad O-Ta-ICab-maTyoml InY=GYRE. (a) <Tau TALDome> “Recalling the definition ofthe multipole, tensors used inpolarization, wecan rewrite these as: sew,(azar apn - SremeTamaIH)=Ci)\es<teme\T[Tented sem Soa an De-Ta-T GaedeTH)=GPMVEER<aT|Teme>GA) epoet ss INserting these into the above expansion wefind: . LATO|o>=ciyettat mend—t—TON ar\eren) xt ) Cc T-T4-E v zm =A xZTezy-@) “(aaei) »ZGren PViamaydTinllT Hawi Wecan see, for example, that unless I,=Ic, there will benoI=-0 "wave" inthe t-channel, which ofcourse makes sense. Wecan write the above expansion inadirect product sense, where the first operator operates inthe c-a mainfold and the second operates inthe d-b manifold: 0 =. SAT? aby=LSE) LAT VedCalbd $. Oy Se zy TetTat Te Zhml(ZTo 1 oO)= n VL= ‘The object_within the parehthesés isthe direct product oftwo irreducible tensor operators. Weknow how tocombine such things using more CGcoéfficitnts. Weget: eCa+")asoy, xo a ET eT)=ZZVecxeleutey =ATA grolr-wre]nc)Sa 2yet . Tegee . -~he ByUsing the CGsymmetry properties, wecan find arelation between Iand I': yzeteenh7nett GryRetten Zo".AMGrn m=ZSr'oITwin=Z“cayrb]TmTw) ot oh™ im TetTan qyenent 1 Ee 'oy phar 2etUa shar' oye) =tl U >ONPS4G peed=crewvitigen. . iN"42S te) , = = vee J =UT=02,4 aL4DT=wt, >galoF- 1 .T=13,5,.. 200YoDahYuctyn \ The point ofall this detail isthat the direct product ofoperators shown inthe parenthesis onthe preceeding page isnot necessarily anisotopic scalar operator asIonce thought. IfIisaninteger, then itisanisoscalar operator plus the M=0 piece ofanI=2 operator, plus soon. Inthe case that I=1 wehave inparticular: on cot esAne 2,0 0 20 @(21%"0 TH)=fzCooltomrmyyT”+f$Zeadhvmiay T” Ce Lapel Re re? “=DANCE) TS Coyaxplieckcake.) Inthe application tothe PI-N t-channel, this second term above will give nothing because the NNsystem cannot yield Ie2. Inthis special case only, the sun shown turns out to be an isoscalar. Application: Now consider again the t-channel ofPI-N scattering. The setups are described back onpage 2.Wefind: _ole =ee Co} 5) ‘2,cy3-$-L b ita = C1 ——, rc deelTOley=onA, AMayey™Gay) . a a. xAcielPlayColey 1) om — Thesecond matrix element (remember, these things arejust CCcoefficients) shows that Iisrestricted to0and 1.First, wecan calculate the Imo term: t= a) a) zeodemm=GOTFeTO)<ipliddeey =FTO)EpSve. eThe I=1 term isabit more complicated: . et (8), pm Amy etdem=(ATH. TM) CI) FZcigTraycert) aq ’™ -? 4 > ~~ —-7 : \ -5- : ced Wehaveherethedotproductoftwosphericalvectors.Thiscanbeconvertedintoauseful cartesian vector dot product byFecalling the relation tothe angular momentum operators: - ot —4,- .To=FeawTe Ta2FhreyLD] . Here, Irefers tothe isospin ofthe manifold inwhich you are representing the operator T.Forthepions, I=1andforthenucleons Img.Then: me , Os veCU a x>ZoltZTWOT ey=saeHea C,MMS + van ; gCail DeetsOUTTT+CKEarTydgorr BIEamet me=| =frpaDeelteley +<ataleydertala) +<Altylardeltleyt LF ==7 (ARTERBED+1c01F47-duE1073BB @Owatlast wecan connect tothe Pauli matrices. The full I=1 term isnow: (4) 4£ = = e-=GPMorCay.(eGo oRDoentio 29]: — © QblElay=$<blSled> =+Tey <eltiay =Lew \ ._— wane )yaa> oYAc armny=GHD |HethGeya) PUttingthesetwotermstogether weget: eT ‘) L744 @ wTy=éay,GUTPacy= KTOSS,. BMYTcofterSYWD)C All indices here are physical, isospin indices. Ie, this isthe answer for aphysical scattering process. Thematrix t/is theoneGasiorowics calls t'. ‘Theeigenvectors of£3arethephysical pions like (100).Notice that thefirst term (the I<0term)@ does notcontaim allthe"diegonal" ‘amplitude. (By diagonal ismeant a= (6anda pb-) 6).‘‘(4)wv> Tatlay=eMosyh. -eM)BSL -6-., —) However,ifwewanted,wecouldperformaunitarytransformation ontheseisospin states ofthepiononlytoget,veartesian’ 1eospin-states. These 'unphysical pion states are labelled with indices 1,2,3. Such atransformation : takes the matrices t' into new matrices t. These new tmatrices are characterized bythefactthatthey have nodiagonal elements anywhere (noteven in3). So, for the 123 type pions which Iindicate byijindices wehave: ayy ey) 4 rcsSUTey=EMO Se ETO Ge | I : e) aos Hune Cj=iEsc Se:Girdas 6GuSh | B-4[3,,3¢JoevriThus,using123typepionstates,ourfinglaxeweris: pore 1yt) « C) PPA,[eprrPicy =WDBde HeTHYBS AM Aadsothelongsoughtafterquestion isakswered: Yes,theamplitudes F(+) |andF(-)usedbyGas.areproportional tothet-channel isospin amplitudes. | e Infact,directlycomparing thiswithGas(23.20): | : .” Co) CG)[Strl> =Sgr7+egadFal wemay conclude that: a r 1 Oo (2), G)_P=aETheo TesSERS | +o=Fo ' eS ) FOs =TE . Onthe following sheets weshowed that: y ™; Fe=s Jar’ st] ~e=FeeFO? ) y - :Fee slrh—p) Th=pt4oro . Based onthe above findings, we would prédict thet: : Co) ) on }, yo. ETM TMs ETP ea 0) q He) i mp o-3T"-31 TH=19 ~479 These agree perfectly with (5.116) and (5.117) ofKS(the isospin crossing Hmatrices.) ~ \ . . | . -1- @ Let's nowreexamine thecharged pionresults. Onpage5wefound that: )=Fa or FF (<ee\T \tay=FSpidbe-Fe?toa°Sun| . where FtandF~were just defined onthe last page, andwhere t'istheusual angular momentum 3~dimensional representation (see p247 Gas. orp203 Schiff). Inother words, these t'matrices satisfy the required commutation relations . andt§haseigenvectors corresponding tothecharged,physical pions. However, weknow another expansion for the above amplitude: indirect s-channel isospin amplitudes. The result is: \ . Gpoitday=Cerm\sabarcem|ipayT” +Cam|iezarcamlyeroy Toderive the relations quoted onpage 6,welook attwo terms inboth expansions. Namely, eCiS ns date POF =TH! a \273 LumMoye FI<0=STH ETH r)Thuswehavederived : (C) Me »Le! Yo 2Pee ateeoTl ang FUeLTH TA oa - . “a! -8- 1 . Charged_vs. neutral pions. : e Supposeone-has«natrixelementofanoperator takenbetweenchargedpionstates and<.Then,iffthesetofstates %ifformsanorthonormal =-set, weoan resxpress that matrix element. ' LAT =BZCAPATIGILD | a3 For the pion system, the set of9(complex) numbers (i\x) forms aunitary : transformation connecting two representations. Sofar, wehave not specified 1 what representation \iy is. Let's define the representation |i)such that theunitary transformation ¢<\4) performs thefollowing services i :37 a Tee=2<p ty<lay : Here, tistheisospin operator. Weknovthattgqisthestandard isospin ' representation, andwewant t;;tobey | fon 3 - _ot ig=TAExe 1 ‘Thie them determines the transformation, We find that: : .yeMeoy pw ‘ derely-mofes =Qe : t) oON \izs : act aa? 1 Interms ofthis"unit veotor notation” wecansay? ' ' 'eu) Via o@|e a(t) $ i=|yh Qos |i 3= i\ ' °° , Hote that these things are vectors inthe cartesian space, not inthe charged pion space. Inother words, the unit vectors are: : a $b wn ° nw fo}fie (3)=(8)=(8) Since the matrices t dorepresent isospin inthis 1234 representation, we expect andfindthats oho") otoy : Pity =t2 =+OO!=ayyyi é : Also, wecan show that: = yNOMEN) tas id's ma . a 7 Connection between f,fpandfy4fy. = Thetrickhereisknowing howtomanipulate helicity statesandspinstates inthe same breath. When one rotates ahelicity state, the helicity stays the same, but when one rotates aspin state, the rotation matrices must becalled in, The page following shows exactly how spin states gounder ' @rotation, along with some irrelevant gimmickry. Inwhat floows (follows) Iwill use agreek ufor helicity indices, and mfor spin labels. Ofcourse atsero angle, both types ofstates coincide (afact wemake use of!) . So, start off with helicity states all round and make use ofthe helicity projection operators: Ayn= Komp WFloowy =Copull STAT SRF Roomy =Cody) I+ By!MeLlogy ‘ Wecannot just close down the states ‘pecause a!isquantized insome wierd . direction. Wecan close after inserting aunity anthe left, however: Twp=Zcoplopimpdog tmK+Yup!Soota> =Lgogp'lodimy Say.|S88)+Yup!Sele9)] . That dude onthe left may becaloulated byextracting arotation operator fromtheleftsideandapplying ittotherightside,whereitrotates @those spin states: . aot s\¢y : Kodplod im)=Coop'|Rleg tw=Zghcde-4}<oon'|com> i =p@= De Thus, the helicity amplitudes are, . <l * Aye’)= ‘]= = Se- Byey il=Wwhl- eodeh 1. fas=cost[E44] =feQ5 _.. sig 24 pam: . Tae=see[4-S] € =-6*7L, => - - \ i - te - - - 1 ' Rotationofspinstates, 4 1 =@ Exercise: toillustrate themanipulation ofnon-helicity plane-wave states |with spin. . What istlie effect ofarotation operator onthe state shown: ! @PARS? \RSy> =\keg SoD . Naively wewould say itwas simply: a. BIRSY> =18RD =BO, [RKSyd ; Butthis‘car’beachieved bythefollowing setupmanipulations: :(allrepeated indices aretobesummed) ' . WRSe%_= ROIS =|LmrdnlBPOS=<ai> \msy> =Cul? LTMP<TM\ Lasd> . ., BIRSS =LOmRDDeaLMCTeaLesy? ; =dmR>LOM'>CTHQmse> @ = BD |swyComeWedy> =2OAl® (yee) =Trreth) Raw __=1%)@18> =(2S : Here, atwiddle onanystate means that state rotated byoperator R. 1 Infact, using this twiddle notation, we‘can dothe whole thing without . ever using the rotation matrices explicitly: - oo : IRSyp =Lf [omydswnlLs? . : . RIRSID =<MEY|TayGomnlLaas ; . . = SulBD17GRIDus) etusor) _ =LEED \rmyomDa? —_=IBD-FSD e@ Weareusingtheseshorthand factstarpughout: CANBD=caeee=A : A=Yomrdom| =RPL “4 — [Riy<om] ad=_RiwMdeamlay =Alay=[Spam Rey=loca | Relations betweenf)and£2andthefurictions f,getc. S _‘Thef1andfpfunctions aredefined thesamewaybyeveryone, Ithink, Theyareusedwithspinstates, nothelicity states. Namely: = 1Ca at .Hho=Alomar -2ZGimipy : ; ims STsh FR RRS =Ae } Therelation ofthese tothesetoffunctions (f,g) ofMSp251is: : 7mM=fI+igeala $3 due A=ae=Rxswe| $2fsfuse ! 3=~fsme - :Our old friend Williams defines some functions (g,h) according to: : JM=9-ehe-e} =gh [guafas,thw=goa] |qe4S+4,0086 ' =e dus+fisine . : Goldberger and Watson define athird setoffunctions (f,g) by: I ' ars : Til)f+igsmet 23=83 Laswe =ns. |1 FeS.+See. :qenh owl . B SuisJ"=9" :: ' . t ! | 1 -_ . = —ote eeee ee Cee - 1 Pauli Reduction ofthe A'Bamplitudes. Relating AandBtof)andfo.SanhsReduotion oftheMAamplitudes: _,Relating 4andBtofandfor Z@ Fromtheoriginal paperofCGLN,wehadtheuseofthePauligammamatrices and the Pauli metric. Converting tothe BDmetric and gammas, Ifind that CGLE's expression for the Jackson “invariant amplitude" tobe: ~ 7sdeahoeVadbnvebonwetf. W=GIy;=%[-A-AB)ue Q=tra we MR adEBT a3, 0/4Sh ABET geal ceeee ‘ ‘The Aterm is simply: ry py) EAM _t Caranems cui-TSR. where iand fare standard spin states, not helicity states. Wecan see clearly the Acontribution tof]and fp. The second term must besplit into two parts. The energy part gives: : E4m Ripe TPe-. =tfa= 0 ~p)EA™ “ol v Q 0,Ge)BPFla+TEEE\ , whilethemomentumpertis =@CRC) EEye, SHFEH+EHNA / Algebra suggests: ' (B-FVE Fe)+(FFE B)=CHAEP) +(FPG) 7 =PIBPREPRYG=-K Sosumeudun porkai a(-8) fl +CFME PIO -agc® <FlP PID . Now weoan read off the expressions for f,and fpwhere, : ° E. whaA,=~SeLA+eW] yecel EMEP BRO Note:tointerpretMasJackson'sM, vi)wemust use Gu=2m, Since CGLN use tu=1,Moe7iG[A+a(we)| wecannotusetheirTforoubM,Thus,ra2G1.->allmy2m'sshouldnotbethere. => Togetthereal fyand f,wehave toconvert from M-type tof-type amplitudes: fegat 47TERA |bene=> Sw; ge+)asetwend] |HemelMere. Relating Aand Btof,, and Ty, . = Sofar,thisiswhatweknow: ,bode CUd : fi.=costo (Sef) Fy=singe (F-) pao acefeaw[ASB fn=Taa f=SsLase(wen)] Lets make sure weknow.exactly what all these amplitudes are. The f,, babiesaredefined onp247ofMSsothattheir’squaregivestheDCS.Theixtypearedefined with the same normalisation. TheAandBareexactly those ofCGLN where: ~ ~ = se LON. ra S=Be—LOWECee),\— Mb “= We[-A- taAe]8]a:Tus2m —@ MN=~BeF Itisthen asimple matter toshow that: fin=este|gts}(2mA+B(S-m= 2) $4.=sivxe(Fon)(ATs+tn8)+amBLSmth) /ps ,ITe=costs|paar\C >Tr=ste[gin1¢ VAR deeswadqueaguawieWS4(91), =@ --t A Pion Nucleon Catalog ofPartial Wave Amplitudes.PionNucleon Catalog ofPartial WaveAmplitudes. =o Inwhatfollows wewillbeusingthefollowing symbols: ns x(TeTH) Tas) (SteSe-)(Mae,Sy,Me,Ib-) soy(RS) sefs-) All ofthese notations refer topartial wave amplitudes. The host ofnotations referring tothe full PI-N amplitudes isdiscussed elsewhere. For example, Ont Th THe o) © cTATTHTH,FORO, ALBAYFAL,See,Ter,Ag . One possible source of confusion isthe reation between fand Tamplitudes. Forthefullamplitudes, therelation is: - ©Forthepartial wave amplitudes, sos TV,thefamplitudes aredefinied feeaisottdifferently (and arbitrarily) as: x Lt -edt fata qeneg : The phase, shifts and inelasticities are used torepresent S.Itoan be shown that (S;{21 soitisappropriate to,use: 2S35 C) Sn=Nn© The connection between these Syz amplitudes and the Ty, amplitudes isthe MS standard relation: ; _, ~~2185s Selects = T= Me —4 1= .>2:Sse The amplitudes 1,,. are defined by: Sa HHP - vO =i '(e ! ymBie Soa) Tyedye!(CO)—rk ' Thus, ifwecan define what wemean bythe amplitudes Ty, ,then the meaning of all the other amplitudes follows. The motivation for defining the amplitudes Tf: isasfollows. Consider . the partial wave $matrix elements: v 7 : ty j = vuSymteeTe=SmxlSISKpy For aparticular J,weare showing the matrix elements ofSinside asort ~-of helicity manifold; The idea is#imply- this: since Sisunitary within thismanifold, weknowthatifitwerealsodiagonal, thenwecouldrepresent =>each diagonal element as aphase. This theorem isproved insection "group". Thus, wevant toavoid representations inwhich Sisnon-diagonal. But 7 a . : -2- MIMD=«—=fortunately Sinnon-diagonel intheusualhelicity representation. Infactvhatwehaveist v y vs sj !ay. [WF WLR ¥PR yr Pa s vTs Te Te Tae ‘The last step was done with the help ofparity. Recall that ingeneral J-S,-S Plymye? =10,C-) [Smtpm? | andinourspecial caseofPI-Nwehave | V3 | Plympyy =-Cd*[smaD | which means that I rep ' LempTboyy=EGY Cowpe\TSey !vy S |Tye =+Tle i ThetaskisnowtofindsomestatesinwhichTigdisgonal. Thenthe | diagonal elements ofthecorresponding SJcanbewrittenasphaseshifts e (or, when the possibility ofother channels istaken into account, asphase shifts and inelasticities). Toward this end, let usdefine two explicit states aand bas: .> -+) =+ aC tse)=[pmey-lym-ay] &jayeEi +t9] frume>=aPomray-[may] bY>Af?-] |Here the $'s refer ofcourse tohelicity (not "spin" wrt fixed amis, eto). It iseasily shown that: these etates areparity eigenstates: | | Sei,wk H . Play=a "isy Ple>= Gi) “ly ' Ifone is+,then the other is=,but wecannot say which iswhich unless we . know the value ofJ.Earlier, this caused mesome confusion, but now Iforget, why. Whatarethematrix elenents ofTinthisparity eigenstate representation? | Straightforwardcomputation revedls: | 1 5 x so Weseethattheoff-diagonal <o\T\az=Tee.+h =TR elements vanish. Weknowthis, v without anycalculation becaus T\by=To-T= theoperator Tisparity ®COLTbp=TasTe-Wee— invaFiant. It‘cannotconneet| =° = states ofopposite parity. | <al1\o>=<bT\ayp= Oo Naturally, that-iewhywe|. chose such etates in the : first place. ee : a ra -3- ' —) Nowthatwehaveadiagonal T,wecangoaheadanddefinetheinelasticitiesand phase shifts. This really completes the “catalog” ofpartial wave i amplitudes, but there are afew more details tobeworked out. H First, let's show explicitly why adiagonal Sallows for aphase. Here Iwill use only the elastic channel sothere will benoinclasticity. Start with: SS=4 > <oppg looney =edus|S'S\oouiy ; Zn Logit)S12fapPCOnfoujelS|OO}’> Rather than laboriously convert this tothe JHrepresentation byineerting unity four tines, lets just start over inthe JM: S321 Compalomasr =CompehS'soinar = ct ;=%Snpe]ST)SpeyCHMpn|S|TD Wehave used the fact that Sconserves J(but does not "conserve" helicity). Although Ishow only one helicity index} we can imagine each as afull set for the two particles inthe channel. Continuing: Teoy eShay=4%ShsShigns=SpsyucIfthese Smatrices are diagonal wecan add:ye | Te s =S = ,:= | Buape=SeerSowsSees&—L=|Spal pe. ‘Thus, each diagonal element is just apliase. Next, let's repeat the argument including channels other than the elastic channel. Ineach state,then, wewill add achannel index. Weconsider . scattering that begins and ends inthe game channel (channel e), but weallow for lots ofintermediate channels. : | Sortse[SMey=ZKonuselS [Myo>Mpae1S]Tyacey i iw ol\+92SonselSST Mads|S[Typed : Picking offthediagohal “term(i=f): 1 ; Cs end|"33|way =ZA + 4=ZVSil+SZ"Say _ e@If3isdiagonal intheelastic channel, wecansay: i { — om .AsSyal”= 1-25 |=ji- 1 oe SBE is . : -4e HEBD=—«-8,theclenents ofthediagonal elastic submatrix maybewritten as: : S_ ide : “Siye =AW 0°97) ae OKAEL Inour particular case wewrite (the 2isarbitrary): e = 2cSysSyz =Ue: © Next, toexplain tyeorigin ofthenotation Sy, .Weknowthatparity invariance tells us that: . a Cian : .a pionondumscc andangular monentun says: =Se ForagivenJ,thepossibilities forthejnitial andfinalLare: | hee VES leeVEX | ‘They oan bethe eame, orthey can differ byone. Ifthey differed byone, parity would beviolated, thus they are the same: Inpion-nucleon scattering then we have avery simple situation: VAT =3 khelbel= vetNatt : Stth, @_teshowedthatthestate"a"hadparity: lays1)"“jap | Butwaknowthatitsparityoanaleobewritten:———.._(_\)"jay |>LETHE &isllen YY TE | awdL=T+hHSsdateh ~sTy+- | Thus, the subscript +or-isareminder totell uswhat value Ltakes for the state in question. Next, wetackbe the problem ofwrititig the standard partiel wave expansion interms ofthese parity eigenstate partial wave amplitudes. The partial wave amplitude appearing onpage 1maybeexpanded as: H x: m= Spt =fy os 1 : THRE GATVY=Sparen larayay+dullddelTO<b> Tho elements of the transformation matrix are: ay\ =e= SlyenLosyea] =A=cab | 5g,: fd .' Silby=ALG) —GAY) =CY —aedhebh. : =&(-\Fte te i —@ Y= ROI =tC) . Thus, wefind: : _ ao- a Whe=2[Te+¢))Ter| 1 -5- eNext,stuffthisintothepartialwaveexpansion forthwith:edyen) xe aa SpoyUPACome [Tee EAC) +TeCr)4yrl)] SexZEon) 4d3+@)[Te+Tk (=) +22STN) 3+4()|Ts=Tee Changing totheevamplitudes justforfunthisbecomes: . TvSus=Fase) a[Sstf] dee) . Nowwelook>somenasty properties (representations)of thedmatricesxm, namely: vO 1 uw ’det@)=FADcosto]RagCos)—Pog(eae)Tv . aAa@)= 1sureilease)+Rez(cone\] With these, wemay show the mysterious formlae: / ' @fee=ZS feeease[Pre-Ped =(irh)eosts ;.: ‘5 -seg f=z[Ss--Sse]sing[Pos+Peale? =G-h)sute € Those primes are derivatives with respect tozofusual Legendre functions. Note once sgéin that the full helicity flip amplitude vanishes inthe forwarddirection. (itdoes) Theother twoamplitudes canbeobtained fromparity inthe usual way. Wecould ifwewanted rewrite the above asLsums, but that does not seem toouseful. Instead, lets convert tothetyandfpamplitdes: 7 , : £a)[fsPofeRal “1u a ‘These are basically Singh's equations (2.5). Since weare. usingparity invariance, there isno need for 3labels oneach amplitude asused by Singh. There ssanother common form for these expansions. Using the MSnotationwecanexpress the"f"and"g"fullamplitudes as: @ fahtheose -Aleli- ee1-2Selie-28] qaSiswe =4+ZVy] sine a 1 Here we have used the rather obvious definitions: teS23,4 ‘ sind f55=fy =Ffsegay =Seq | SomeLegendre identities arehelpful (seeJackson p59eg): : +o FlaPia=eryPy; i' ' 4 / ' 1fueR=Wane S>My-eh=Ai-2e! ~Ctryee =-2% 1 | spf__ot- i singAf=+fsanGRgloos 3-733.) | : Thus, wemayrestate theexpansions as: 1 feZz[Werf +2fe}Conse) <a iaS _ cy ow+, = ~ ((cose| ‘SeZGWtions >¢ade=Zi[ferpe] te(eoss) Sometiges this thing iswritten asatwo-by-two matrix inspin space ast i | ares M=1)+ign(F) | When expressed asatwo-by-two matrix, the result isindependent ofthe way inwhich fand gwere defined (and onthese definitions the authors. vary alot). So we write: «a, 0! , =—sine8 -A8 <4 Aa a4.FeZ,UGe) feeTr+it &AleHye]Rt ;Asing=Raby|- t A aie Rd . | F=Zdelfeted +eRakeZBYeh] W | This last line agrees exactly with GW(p463 #76). Itisthe same asGas. (23.28){ except Gashasthewrong sign onthe last term. Itlooke like Williams (3.20) f and(3.21) exact. Martin Spearman (5.91) exact. ' Thislastresult ismuchmore elegantly arrived atbythemethod of fprojection operators, This idea isdiscussed inGas. p369 and inmore detail byGipage361ff. Itis@child oftheLSformaliem. Theprojection operators |are inamixed representation: the spin part isinthe angmomrep, but the anbital part isinthe coordinate rep. Ihave appended apage ofnotes from 1 GWwhich shows how itall works. Below isthe result. You can see inaway : how these operators project out ofthe "state" anthe right the piece with Jaled and Jel}, ' fo=dl2(a+)[ded-+Jods|Ra(cose)wy R2Akh Tenens “Pea.ee: —@ aa hat + HRLEV] 1 +1 I ‘ | oo. a .1 Serb yatongwnityerougehan: (ounSin!) Se SyISIR SID=Za CeSybaSvPLSVISXOKISIEMD* oToe aE V82100)«Rema Ora[Someones LTMAS>. WwnanLeekModio 2. 2Sensi =gTHtels WTA=ByyhaeSSIS =.Orane082WeideAruen.te CrSeu eawation CB) ge ee |HSRQD= 2*<iqSvWalsyaie x (8) Fe 8 RL me egNe ee ~ oseos eaSash ComdaseyrosuBs)_ _.= SoSewotoaendFoonSeat! LocvacapuisstedéIS,P2ad-irksinstok Qhwlo2ee, waea 2NodSakeyuan deosstualan cham,CoLecl drole wouls,dda©]inadevagsDercoarimelmateadettering arkgarciaicesermed, Ca.| LBD castbeOesame, mastator bo24...dadanparbslwortJy. _PagcootwosAalfangone,LfLeTe4,L25-a)Dea,fad. — cheagisell.Macnee _waings2hso: Fo TORTS ekSulcoe(SUEDE eS Begin 5S)teeeaeporters,Ss5SteeSFr.Onge.juan qin2,DrtarontwoS'SDheatuonk: Sand SEdram, eat, —-pooheonnteeok (Sta) <p ee -ao FRIES SEED WSne DSBRES RAD=ZoGms vile) PL-Be eee ME st - ee ee eee eee - nen |Ce eeee MerayurewankUSD) =Lee.Arpad wroargorosugha(2s): _ a nnereeee-5-—V-—.<-- Py<aSVISIRS =AECRERYSuisTH<Tm)Muy+SygeBed _@- Dad BraeGoaease Tyee -- -- pererotptyboya yeYe Save) =ZoWORN RFCaTDCdn» Se— BR ---7 a Rone taleSeoefotos) -~—Fda sds =-Maitiwa sripatsdeDosspinatelicoadgaia eadsepseOants ~gala =CinAanCislny=ZZCWI omlbe eet eansanSlretny amdaddy FiocdtaecmtedMeaiassMest LL Sih ZS ASSSa -Dae,wrthewe ee RNCNONSIZad)OlEDaeMino landolaccesaybaWu ZTSpekgy etstaadSADEERSSCA)=ECB)MaanmeYOdelbemglsk: A=ONAL tateSerta Anti « Be 278s2H)Seaanstp .TrTvene Rak Tasntaycheepossilede SSfeseucls&sp: _..._ OFackwsannieokGoamawier “ouwal” (73.25) 938. Pion Hucleon DCS expressions. = Weoanstartwithanexpression weknow,whichis: — a ()a (3); « T+-|geHeLesaf We kn ‘thi fs) *aiseCsmYTHe=costoC2mA+B(s-mbdya) /am™ i Tae =suivte(A[stopt)anisemen]) Jyie re ‘InPI-N kinematics, the angles are given by: Le & ~Bele ae suite=&(\-2)= 2/4 costo=blite)= (1+Hag) Substituting andgrouping termswefiretget: | Sie - 2 (m7 vatsted} Ll(3 AYPARREsiCes #-(@)i lalaeSee)(tiaCe+(ashes)(teBY Here, 0 isthe lab pion energy and we‘have used:. weSontyt orm samt Lssowtep’ Wed tm um Nowbysomenearmiracle,thecoefficient of/&/*reducesto(1-t/4n®).This e@istrue atany energy, not just inthe H.E, limit. Atthis point, itseems fore profitable totry towork backwards from Singh's result which is: _ eS te 24ay ee amet cottivn(gtsgray41wetlonF1g)?4.t(5—Conta$=(Feu)[GrSe)[var ESR O08)4ET!aS~Pee)lol mereal AYGestlin yt(5_(eal =aSaSCHoa+e[8+hSea)aehsh))wstlon/2 Itisonly after some phenomenal algebra that weshow that the first and third coefficients match, ie, that: Ge 27Wy1 Yo) SPGG=Oe) MeV +Ce)RBIs (=(w+Ym) Finally, after awhole page ofalgebra, wecan show that the coefficients of 73/2 also match. Thus, wehave verified theformula ofSingh, which Inow. - rewrite using his Aamplitude: e L nel et fine one eefa.~b.0 +2As =\8)» (aap)[CaeIAT+gelS1eit : w+t/4m : 7=A+(Tf) ® Conclusion: these are indeed the exact Aand Bof CGLN. There must be an easier Pion Nucleon, The t-channel amplitudes in terme ofAand B,Parity. j 1 —@ First, let usstate all weknow about the parity relations. Assume f=0 sowecan use MS(5.37). Then, : AU=TMM (-V)**84~ SH vl,Schhavavnah,: ce)BA 6 te : Spyudofe =CCA Septeesa PePerpd p>parj . ‘H ' 1 . (-}+(0-0) BeBe. 'Slee Fusouo=oFPeosta°Cy : ' he 8) eo) 3) ey :>See fl ond|PL -S 4 5 KeChal: Sagoo=anoyFopccudeeQOOG) i TTNI a) Hy a @) .2 fase fl ad She -he H Itturns out that the parity relations are the same ineither channel. There are several tricky minus signs here, one ofwhich isdue tothe fact that the in< trinsic parity ofananti-proton isminus. Phe helicity crossing relations are (MSp342): . e) (t) y e TH=sexTH-eesToe eosX=—HS+vitp"), () 4) KyTH=cxTH+wxTY . at . sakea suy=ale (*= §) iS) 3 . Tae=sm%o+esxapt) ’ ze a . ityTre =ws%al-sy1) Iwas planning toactually execute this helicity crossing operation, but now itlooks too painful. For examgle, Ihave tounload those s-charinel scattering * angle sin andcosfactors present inthes-channel helicity amplitudes. Atthe * same time, Ihave touse messy expressions for X,the crossing angle. Itseems like aton ofalgebra, I'll bet there isamore direot approach. ‘ Well, except for anoverall phase, Ithink itwill work. First, lets write down’ all we know: al2TSCpFTIS~Cmapye] =S-2s(riinyt) +th=postivetmSandkoh! B=R= L(A-Wn®) =positue OeSond£ch. mp 3p kG-Y) ila 2p=3. ‘ZhEq=SuBente ~-E cuudloow momo yk 4e=pimomemunnuntaue Untoh. ; oe7 5) t Weknowstillmore: : d=t[se-(wp}] =4sofsite, =Ytpoetsintoe (puns) =k[-st-F1 VYttm. =pariswillchawnale, i Those s-channel half angles expressions are (from p366): j eshte, =\-9/t'/ol site.=)-stVol , Finally, thes-channel helicity amplitudes are, asfunctions of4andB: : 4TS=costo.ArmA+(sep 8ard) 7 SatTs=suites |Gutyc)A+(stand+ye) 8]AIS Putting all these things together into the crossing equations, weget: APSE ITTE=ATMapEGY +BERg(S~bsph)erat”StnpwhE-wah eUsing the following identity which brings incos ©, (Ers~ib-po) =2peycosee=£(S-4) one isable to show after mucho algebra that: P 2 LSFEuPTE=ALSS 2IepF+OPACIEC29:cose)? >(dort =QeeA—2g,eosOeB Ifwe set that phase factor onthe left equal to +1, then thie result isexactly that ofMS(7.96). The fact that this did work out, uptoaphase, tends to support the feeling that MShave asign error intheir (7.91) equations for | the s-channel amplitudes. ‘The second t-chennel helicity amplitude isgiven by: .: atl .Jeane -alt+ ef47 =seVFive : >(ianTY=~Bakes . * ~=~BaeltSwe, = :Tosummarize, hereiswhatwefound(tobecompared with(7.96): 1 tit a ~the main point isthat| ;=SFatTS=-2peAe2mge BooseBETSN|iS . outright. Thenrephaso:_\~SutTh=leBsinee - - ~_ Derivation ofSingh's DCS equation from t-channel amplitudes, Wehave already done this using the s-channel amplitudes, and after moh pain wegot the right answer. I+should bemuch cleaner here. Asnoted on . p341 NS, the helicity crossing matrix is orthogonal soaDCS can bewritten ineither sortchannel amplitudes. Ofcourse itisalways thes-channel DCS. |So,westart asbefore: " !(2 ye : geeSTAYTeETE i . ri ae ~ 1 at us : | va eT _ / _ Boesel“WTas =~2pA+2mqeBoosee =-2pA ube A=(A- nue0s9 a . {“WtT=qelEBane i Atthis point wecanrewrite A'several ways using theidentity wederived | earlier: - : , Apgecora=tG-u) > A=A+ m(s-wB (Hat -t] Adding uinthere doesn't really help much. Interms ofthe pion lab momentum, aye 1 +h e some t _ (co+t/4mn)B w=s-wst+4md =A+ =. 2thC4-t/4m*) Regardless ofhow you write A', the DCS is: = 2 > .(evAT=(2): i[-2peA'l+\qedéBsiueel'f an 4s Var From this, Singh's equation follows atonce. Onemust becareful touse/p,2/ forexample because inthee-channel py?isnegative, asist.Theresult ist os2 4 ut Cant) a ==(He ~Ye)(ata, s———-.)18) se(Sn) \Het)[Aan [e/a = 2ams. (ay[some|aw lst =' dk FAL WS Ags ' Using thetrivially derived relation kj?=(mw)? —8,wecanfinally cast this thing into the exact HSform onpage 456: 22 ee RSH Ye z z=1(2) (1-4) LA)fe(oa x)18 aay,s TS\49s. sam m : —_Notice that these are still the original CGLN Aand Bamplitudes. Ihave not i dealt with any other Aand Bfunctions inallthese sheets. Remember that it ' isthese AandBfunctions thatarefreeofallkinematic singularities and=j constraints, They are where you should inject Regge poles, Isaspect. : PI-N: optical theorem and polarization interme ofAand B. =@; _We have already derived anexpression for the DCS interms ofA‘and B, the Singh equation. Avariation ofthis equation (the same variation that appears inMS) appears inthe Pit ofRarita, Riddel tal1967. The optical theorem refers tothe optical theorem averaged oninitial spins, In general one would say? . » Cab)at a™s ‘ ;= nu4 ' Gy©)=FgMeLTaat] Fels)=EBascaygarny &Ta Ga=TE =meet Inthe case of pion -nucleon wehave merely, a 258)=Todnle(e+T]]=AydwTeGt=-a] pe|Fromthe"notes"xeknowthat: “we ;:edTaGet=)= tpl2ma+6 Oe]=Ae\A+8]=Safle: which follows fromthedefinition of4’.Thus, , %_ 2:a2 1 _ = 7 eT=2BenA’ [Re=(Re)dnlatse=0 | Next, the polarization. Start from helicity amplitudes: SER=2Omoarte|. , * He=2(Ap)sitocateBro|(Ztrls-ab8)) (AGwept)rabGwe!=sine, bx|ncAB"nbpt)(5-54 w)RAY|eursts ro=Sous Lo]B[A@?] =xsingHast”bm[AB] 64nts. Cha sds . rk tot ,=(Sue f i =md ‘.(76mc-)BA] #PP.it '=-sme_Am(A8*) [4e5167NS¢ : =9 - . | TTON mise. | | Roais.dovmudos bee0=N e a Ipoysays,ofl=4Bow|There00] de=a|Teosal+[\*) eeAr=- toro) +|Tro-0 Tao-o =.-Tros0oea{~” n=%=4; Page=2abw|ThoseHerel P=4BalTrow Topol| FUTyopol” +(Tre-6l*& wt —L '——tas irlyokKalen,lioowtcowactDumwaeea— ebeni1.eisTAYSeahlbadag2 pot teoSEE ha Se OOtt seb}t—t -| Cae tr we_aE SaBOM»ontIRaa a aDPSS a oe EpaTa) adePD ne ottot i t —_— La a en|Rinadadewaiteomcmeplettide, JnaEQ,entedTH(d).nein on tIS. aha at iapt eee nenafh:ie ere pata —— aoad =AT tekOO Hy,Gore aaChay |eebboy :ti onSa,SpaspoleoudppAccamtlagufla— dane yt. ~ _ ——— =~i———_-- '—1Ri aA AT een : wo a ne eo Se anwuiing baGoHSacephbale pakkee al hes f i vi ey YY £1“eae tsrolaes Eaee porea as ST etept [ieCae(a yeEben Teli) RelyisiSect ot)peat od ptesfpotfp webb SEES SemabBaeSeJ ee=oe ee eee dace — aes a Seri ore ~~ -ee — — a"4 i -egetea eo eene an, a TSCafe BCFA= etrs ——phim. HOsantbeniie, ————ianh - eo oleh) -}eS: TENS (BpHTGHQo —ATG aelh Cie)Bpne()):CA) Tie lyesa, pigs,Ssiaoganduck. woeee Se aenercanrOna ee See - So a ET anne OnncnnennaSeanee anpt Co ee oe accra TY PE, es fracche etcoe an) cn ne + “+s .RonieSonrmudos fanWoW Re(D.=4dew|TroeCONN == :a . a ae=ae{|Troael+[Tes.20.f) Trepo=-Troe sf . Mma d P= 2MlTrorotroro] FATpesel +4Teo;-0l* § \ ry Rosse: : 2 ' = (p+Ques9SiRdaagedmmanming)1eet 6 2)nedLaine,WlecwnaeAas astqreyY ™ 2)olmamta oneofoundes D)ABP ete as ! | ¢ ‘ /4 : “aq -t TNO TF7 VaPress): r} < «ofwi ect e “yO, f NeWoT { t j & ais H aya :.! 1 _—<~' wW— tH wa >A H oe beended !Ve,OneSTaudUprocseare, | | |Ranchoou:spiteNee TY]FerNaS| e|a} eid | Qudoud8deputeont: ' |Wemmee Bue olesalen =[SHCmysts~Cmeany”weenye Sarhas£[e-M] | Coyihe &\gdonduont: 4=0 we2AM sk p=&[sem(arya). Cus aoondadow apr, do: AinteWaauJe hema=(Step) Et AnWe=creed eos,=seme) |QrnesoanvaneQeWemTeWe.AtUneK.Osa: e|WeZdle)dorTL Marthe duuotktyepdiattly, Fasranpte: | Ee Grete Crem TE!+neat +Cnet!) . wna:Creyeet)=ccasinh©4bsiaX CNG =SESS ute Crea) =aosF:OHH cose GHGs) =suk:es=SAS Jaring(F32)Yorportamavomanes ¢ | | Tae To TeeteTet | e@|MP0(one)HY~Go)EY | ' way, LZ. : PiON NUCLEON KINEMATICS | ' . s “Nt fu . S=ane THwh _. k=(4) awh>NN ar . ir=(qe) aNoan ;‘*84 Atk+uU =aM Ue : —ne XG) =Qncyo= DeSE aTHe parla toSouunddunrahnd: . . . “* deoeSve 2atu, StuAh42elee t 2m 26 .pleclabenugy. , p=afOe . eeby,Position Acult:S=(Maingv=sp!: Posdhinehgles gt gtant-#,-(G8) @a De 9USMa,|Poe Qua,prrbure Looks Siler waite: poeitp ‘1ge\a =e = Neo duetwosapanate, duepersLon psSoshiee.BudAit aMcpaipia) =BEN AGA+£C4+4)86.4)| W@) ©Opfawoand Qimikinolepomsprbddonve,ofcastbejreafd,So wg, UlenMue [Liaw cael ntandy Enea lo:ey (Bustrwd) P| Qua:AM(MQ;Ka)=ZepLAGat2B]al): eQepmes —pat(y)=AG+85,2)=” seeEt, p ~ deeFaLow it,sah. fhe .g®asadscoldtgs pinky8 wa a aay natPG waht IE=trems btsamom«Gta(My (eT vemi¥e. 2 ;epoé[ifp|OrdaStachohTes) s et. ya ihoe é ‘ ; set%40‘ws SSR Sey. No t Seeore(asi) Set(nay=L1bGe S=OS. (users) a, LHe|anatupiaStachine4105.9) . ffreo“*-TE,°)=AG)+d&G2).go fFSs .A a get!ead and :ee ane . v[LTTE EID S|EEE | StseB =m=BO } e@ae(uy=bl ©Quisatsomeaula, | Sez(ny=mT 1Rode'swnadigia\noflposite|eat’ ‘ , : Sf CROSSING STATE MENT ; eWebundtat:amy) _aluPeasy_ame5,8)=aanep™cas)|MS Qwe: +,(o=naka)?; canalQ)<Sine} Defame,Cressmureranod)curawngold oun: z7 ix To=To To] -4[T@=s Vea, QernSsWg,say! * ay TO) ==TO). 4 . t et 1 1 | ! . : ‘ @ i Kinematically Clean Amplitudes forWw NN. AndtheRegge CEXDCS. __fi Thisisjustanexercice inusingthechartonpage364ofKS,Thestarting point for an annihilation process is: Gr Stag Flean) =(Se)* (Jeope)” (ean) yHere,Tisobtained fromTbydividing bythesinandcosfactors. Wewould | dothis division even ifthose factors are unphysical. Iamthinking here ' ofcomputing at-channel clean amplitude, butinthes-channel highenergy |region. Inother words, large 24. The+and=superscripts aretelling usthatwearedealing with i special"parity conserving" linearcombinations ofhelicity amplitudes. 7 Itisnecessary tousesuchlincombs because theyhavefewer singularities | forustoworryabout.Thisarisesbecauseofthe"kinematic constraints” onlincombs backinchapter 7ofUS. | Ax A wets. A- i T= Tye #COS Feat | Bythe way, atlast wesee why MSdid their spin-regge theory with such wierd amplitudes. You want "parity conserving" amplitudes tobe chucked out byyour regge machine; then you_can suck out ofthese the properkinematic singularities. "6 @ However,inthecasehere,axba0,ead,Bq=Bp=0sowesimply | have: A A A A :+ - Buse varrk ITar=TTthj00 Ve=Tyejoo ° 1 Nowwehavethree numbers tocompute (n,5,,+)thenwecan | compute the divide-bys for the hat-T's, 1 atbeced : ! =RGNMC) anh 8=\ DMZ 1 na e=1) n=) | 2) TV otoxced xd .&B=EAOWHE)C) =-Cl\) ett 1 Tet = 8=CH) Sty =+] DPnso- : : ak Te98SO)PalSarl2aJEJackyust. \Compute Ss: Kmysarnt. ! , 1SFB nanan Toleoitsd=€). 7 | +) =OBK /.QaSetS4—woe(= 2|p.a) . i_~~foTs,peo,ahCe=gtlaunergoddeasodsh t a - -2- me... SeeOege4-mon(W)s4- L020FoandRyoenfaTe. Conclusion :Te©(eae) Tew4 Compute Pt: pesp?,p=IM.FatTas,leosop20soBe=0 | B=PLAS FaTeylebandpenlas p=ptteosopec/ So far then we seem to have found that: 4 oA Me (elene) EGY Te . R ° ar ’RO. (EPUR ae’ tue") Te ow : Next, weexpose thekinematic factorg which makethehat-T: ,| A DAL al | Tye=be@®Tye was P(e) =(Suze) (ceste) ! peed dsabeo. by=(inte)dose)=Fxsaey4 | —@ Tage =Taio ; . a Tey =+singe Tej00 . Now let's write out the Regge assymptotic forgs for the hat amplitudes: A yt?)old tap [fue=Al-eree) BesosgMoyC8) {.Ndi Sor(xs)Jooanenmn | Ne)! GDS! Kar}! Specifieally inour ease we get:a « |vebaeyBde ANTS@o!(2) i Fojeo=Blaren ba hywa ] ' deh a=1 A _nmWadeyT GheeCaaylC%) ' fue=%[areSeayneer ! Now rearrange some ofthe details: 1 1 siebelesel]=sin(ne*ym)=Cay"singta)fpsuttepeapa =@ >sir@-)]=-siume - | Als: Jean! =Seabee! =oe) .|a 2 FF@n =Meare cal =coSS=o(A)OH :- ak -3- => LookfirstatTy,UsingMSdefinition ofFIfind: ‘Qe -4 A ee OB(,(8% |Tigo=eeCN1f.):(©+FQ) | where, oa _ mo - I!re@ay!eye] FQ)= at Sine . 1 Several comments here. First, that factor (-1)* came from the(~z). IfIwere!morecareful with-themeaning of-2,Imight beableto-make thatfactor I goaway. Second, also from zweare picking upsome threshold factors (pq). However, we-already_kmow explicitly the singularity structure ofthis amplitude, soweshould absorb that apparant factor and exhibit the ' real factors, ' A sot st: forsoo=(355BYFY ¢got ) | SoNNet A y se |+ Ss >Tea-tteFH)(SE),LL | —@Jt4 VowdoTyr.Asbefore, werearrange things eowecanextract thefunction F(a] A Cs Nau Toa 7 '~Tye=Cy)FR):te:CG).(*5)J .: Apbefore, weredefine the residue function toabsorb the apparent thresholdsingularity andtodisplaytheknownsingularities. Atthesametime, | wewant toarrange tohave the residue contain certain factors ofatokill.thosesquarerootsof.a..Thesesquare rootscreateuawanted cuts.» 1=4Thus (and itcertainly seems weak), i - . . * - — - 1 “ s\' Vx sect Hhe=2.+(5a)*jaa”Fe-(7)-..3 eee me wer “yy tee eeee a -ya-JbE) =te ge)! SRTee . ot al ' AAS) >toe teewe(KR).¢FO)G5) ' = * ~ 4 - 7 ~ - 4 ThepowerofSQRT(+)iswronghere,‘butIhavefollowedcarefullytherules =@given anKSp363. They donotallow n=~—1. Lets compare toCSrules: ; _ ons ‘es4 .soWa Vo. .- > (&) b=Mor, lolput=comnragaring =Y=+4, -4- _| vol.Yes, MShave anerror intheir use ofn.Asshown onp266 ofAnnals 7 ofPhysics (long and complete article byCohen etal1968 about these . kinematic singularities), MSshould really have n=O orn=~1, not +1.Sohereletsrestatetheioamplitudes withthiscorrection: | A Bat 7sy 7 |Veoo =>See FX) (SE i >St-9m|(8) Ly yG-a s\* ty=Dibe-- ~Yee): -&) Taso =244- Jk R@®)- GY Back onpage 2weshowed the connection between the hat and no-hats, so: a A | Tet00(pst)=TtoTao(hil)=EsinogTeo|“ > ° Sse dow,Sinoe=Uta"=J=o>GE). ; Lo When this factor ofcos@ isinoluded, wesee that both full helicity amplitudes have the same regge power behavior. This isthe general situation, ofcourse. All helicity amplitudes have the same regge poles 7 ~and thesame power behavior. Itisonlytheresidues thataredifferent.—Also, ofcourse, the kinematic singularities hidden within the residues - -are different, aswejust-found-explicitly here inthe PI-N case. Also, — the ALPHA dependence isdifferent aswecan see olearly from the general Regge formla backonpage2._ :an ;hoes While weare here, why not write out the DCS that results from theabove amplitudes. Notice that here weavoid having todotheexplicit j crossing tothes-channel amplitudes because oforthogonality! | The DCS is: - 2 ( . 1 ae( !=tex} @)no)¥ _.2eG-domd!) =Gag¢TerlROyy Fy Crs oe wokys —___. -4,2) Moi, y ; #=ES) rolFo9Fat+E(E-Yene-ynYsiite| 56)ld Now get rid ofthat sine: ' - us S +x| = . —sote,42?=[Spel awEB : =i \pel= 2)t—der - —— , 2 > 2 : id=EGE meBG=Pil yestbbal} ~5- a} Sothehighenergys-channel PI-Ndosshouldbe: 2a-2, s i 2. a\ anor; ge=FSS(BSreal”piesFibatal— Netsttttolf ~“Here wehave used only one pole for the amplitudes, sothis is -most appropriate tothe CEX reaction where oltly the RHO oan occur, Isuppose the pomeron could beused along and then this would work for the elastic case. Two things tonote: 1)att=O the second term vanishes sothere will deadip att=0 inthe DCS. There isnomystery about this. Everybody knows that the helicity flip amplitude vanishes att=0. 2)attsuch that a(t)=0 thereshould alsobeadipbecause againthesecond termvanishes.This isthe famous dip inthe CEX DCS and Isuppose this should beagood victory for aregge fit with only one pole, However, the arguments which lead tothat power ofALPHA inthere are alittle weak. But even ifthat-isnot the"right" powerofALPHA, itseemspossible thatsomepowerofalpha will enter whe second term via the rotation functions, and the dip atte-,6 still supports the whole idea. }$ Formoredetails see: 7 1)Hohler etal, Phys Letters 20,79 (1966) 2)Arbab aidChit, “PhysRev147,1045(1966) -- -—— | i i ' - = - -- -. - - - al - zy "Choosing sense and nonsense: PI-N scattering.” ne Herewearediscussing thepossible presance ofzeroesintheresidue functions atplaces whore the trajectory hits some integer. 1 Inthe case ofpiontnucleon scattering, wemust deal with the Regge ! expansion inthe t-channel toget out the high energy PI-N behavior. The t-channel process ist + @, eh Tr>NN Ted =Tees, Ase-d pero. For the T,, and T__ amplitudes, J-0 isasense-sense point, sothere are nomysterious factors yet (but wait till below). For the amplitude Ty- wehave \=1.For this amplitude, Je0 issense-nonsense. Thus, for sure thie amplitude has asquare root, zero: pe- »Wa=oy Forthis same amplitude, J--1 isnondense~nonsense, since =| andp20. 7 However, T,=-T,_ and for TL, itturns out that Je-l isa sense-nonsenge because "d=-\ and so .Thus, wemust include asquare root zero at J==1 also: pr-~Nea Ja-ay =Jaen) ‘The business of "sense-nonsense” choosing does not come into play , inour application here because the initial and final channels are different. —@ Consider JaQ, Thenwecandrawthree pictures: v yy . to hasnh Altr>BY)=sae~Vy * aS ¥ vB - ved 2 = :ney rlplo) =(Ly,) 1H a Se ary peoSBepee b(rrstm).= Ss ass Here wehave ass,en, and mnresidue. Factorization here does not lead ustothe conclusion that ssornnhas azero because these three ! residues donot apply tothe same process. Inthe example given inMS H P448with NNscattering, there allresidues doapply tothesame |vertices. Then, factorization forces: either osormntopick upa factor J,Ifnnpicks upthe zero, then you have “chosen sense". You have decoupled from the’ nonsense at j=0. Note: another source ofconfusion. HSseem todefine their residue Bto |‘exclude the various rotation function square roots, whereas here Iam } thinking ofthesesquarerocts asbeingincludey in(« : =@Thus, MS in effect mxetmin expose these ' factors twice. ee !. TEACH | oe nee - . - - i ' - . |: c|Beltioersion pelgtion, | e |1,Thisisoneofthosethingswhérethereimsomuchtechnical detail that every time you makeamistakeyouhavetostartover.Thereare several subtle details which must bekept inmind: i 2.Here are some of the details: ' i a)thesymbol “s"always stands fortheenergy ineither the PI+-PorthePl—pprocess.Whenyouwritethedispersion |integral for one process, the other gets mixed indue tothe everpresent lefthandcut. b)both ofthese scattering amplitudes have onepole. Inthe full ” analytic function that describes both processes itisofcourse H the same pole, but when’ you look atthe separate scattering| amplitudes thispoleappears atdifferent places inthev-plane.i Thecomments made tothis regard inPI-N Kinematics are correct.| *_ e)Thedisprels that Ihave derived aretwice-subtracted. One subtraction pointis4u,theotheris-u.Thereshouldreally |betwo subtraction constants ineach equation but the crossing H property relates these two constants sothere isonly one constent | ineachequation. 1 4)However, these subtraction constants donot occur inthe same wayinbothequations. his justcomes outinthealgebra. !e)theBorn terms also ardnotthesame; inparticular, they do | notcancleinthe(~)equation asyoumightthink. f)infolding over the left hand cut integral onto theright, \ thereisatrickyextréminussignthatarisesingoingfrom @\ T(v+ie) toT(-v-ie) then to-T(-v+ie). This causes thebare . ,vtobeextracted from-the (-) integral, nof the (+). |3,Itseems that youcarinot really doadisperéion relation without subtractions and then just "add" the subtractions. You have to ifelude them from the start téget itall right. 4.Eventually Ishould compare mfresult withtheresult inJacksons notes for 227 which Ican trust iscorrect. There certainly are other options for chodsing the subtraction points. Also, usually one does not need any subtzactions atall forthe (-)disprel. Whereas, usually you need“notonebuttwosubtractions fortheGone.Thereason forthislies inthecrossingproperty. Thiscrossingpropertycausesthe(-) [disprel toconverge faster byanextra power. Moreover, usually you saythatthecrosssection difference goestozero,sothatgainsyou || ‘another power ofconvergence!, Thismeanstwopowers worthofdifference. | i5.IntheformIhaveused,theldiprel isindependent of'thescaleof the amplitude used, except for the Born constant. However, once you ‘ use the optical theorem toget cross sections inthere, you have to 1makeachoiceofscale. i! | i @ |4 1 :' { i 1 ' i | | @-saietsaes i . i * 2 |©ed ;|®», - [u ! ‘ Rasy GE+ape -agSewa ‘eevis wh,aio)! _| a <;|=-cyaTIGasweSoya!+aNaut Tuy: $= TO) atakTO)=Ow.L O-yy-v) | YmVins H ei"oddGee)mE5yeDikearubhaimpeoe} a yy Toso" TO);artyeY=ay‘5So~Yad—_— wy! dy!8sus2&¥Wi2WH, |*zarsSir we) LaeSalleywi)orwww=aay gay TETO fe OA LareS@)=HV)CAM)fomCAH7CAP)eH) a *\du! 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