Kinematic Singularities
PDF · 153 pages · 27.0 MB
Open PDF file
Mostly handwritten notes in Phil's hand, from his Berkeley years (1970-1977). The legible opening sections cover 2-to-2 reaction kinematics: the Breit (brick wall) frame, its construction by rotating and boosting, the pion-nucleon equal-mass case, the expressions for s and t, scattering angles and cos theta, and the link to Toller frames. Later pages are largely illegible OCR of algebra on kinematic singularities and physical regions.
AI-written summary; may contain errors. This description is approximate.
Extracted text (machine-read; may contain errors)
Vinemafic Singelarcties
PhilLucht
;
|
|
|
i
Cons/wus/lab
|
|
~QMSKINEMATICS s=Cartes fsE\-sey——7
b=Ges =afsa
hawtetomt a7(a,mind) A=aarly he om
pteeoewitamt *\2- ae ayt= p=Ae“eerie: =m)
pre|LerCurny |]a-Caos]
oResssecTLONS, (Tesksmmneune) aieSux<a Sn #_>Garhwh) Qipay ry =Me,Ph=iA=Sr
agi= ot/[CowFze] 7t
sde=atlTats)"Scoat)[rm |
de mh ek/ * * ee ah. 4 2gACA,Hyet) anyovaa~#RZETES)/
. =|:fs fe-tePr ="arya
de 1 an rr . .arrerrescalia *64-2ge\mly
—1—R--~Ty owMe=ieBn.$00) :~
;;SE(pe=Ue)& Se
| .eSlom,5
Jk
—ShaegineeeeSHh oluxiya. posilete bolidoSaou viudachsRtki= Oo.
jum: fhe sumd Tass put. beThal Swe
ns
orm: Qho vactn -9, =(Ei, oh os. soacebler.
—__\ so,Gaoa_yon savas _*oom mk"Gao2—mm.
nn Gn".
Ae nS !
>
a
4
1
1
SS
ft
é TheBreitFrame(alias,thebrickwallframe)
|aIna2-to-2reaction wherep)+ppgoes.top,+p4,itisalwayspossible tofindaLorentz frameinwhiehthemomentum ofoneofthefinalparticles, say
Pz,isequal andopposite themomentum ofoneoftheinitials, saypj.Insuch
_aframe,particle oneappears tobouncestraight backandbecomes particle 3. |The reason suck aframe ispossible isthis: Start with:
A‘
rn Fi=(a,»,¢,4) ps=Cefiqsh)
These are arbitrary but timelike 4-veotors. Now gotorest frame ofp;:
lada Ja ~ | B=(B.9.0,9) j Be=©F5,8)
Next, rotate sothat p;hasonly az-momentum. p,isunaffected:
aoe,
x = ia P=(G0, 0,0) QS=(€,0,0, h)
Nowboost inthezdirection (passive boost). This causes p3,todecrease to
@smaller positive value (assuming hispositive) andcauses pj,togonegative. Stop boosting when these momenta are equal and oppésite, You have
arrived inthe Breit frame onthe saxis:
Bow i et=(B,0,058) = (Eym9,+2)
@ Byassuming thathwaspositive, wenowhavezpositive. Ofcourse, attheh stage wecould have rotated 180 degrees around y,say, toget -hinstead ofh,
soinfact, 2has nosign restriction.
Forsomeweirdmassconfiguration, then,theBreitframegives:
! 4
.
1 oR 3
}
Ingeneral, 2does notbounee offthebriek wall inavery nicewaytobecome 4.
However, ifthemass strueture islike PI-Norequal mass, then:
=a2 a . ae MEM MyM Pe-B >BRE 9EE, &\l=lh]
ond (RU) CRD =PP=QR)~, . ‘Therequirement thatp4and-p2haveeqhallength aZe2b and opequal transverse momenta restricts these
vectorstohavetipsonthedottedline,Thus: |Pea
;/Giventhemisses,thetwodegrees e@' offreedom (sand t)are now
" \y .Nt thelengthofp,andtheangle.Ke? : ofpot :eo: Ht He. /yt 2
Nowwehave/a true "briek wall". ' ow .
* i
e -2- 5
Se Thesesimplefactsconcerning theBreitframearetobenoted(hereweareagain inthe PI-N mass struoture orequati masses only):
fh—y—> 3iA
y P=(-Pose, p,smo pe pa=Vfl
2‘ 1
:¥aR=CA,8 ' 2,0ot a3 L\: Pceso se, |BrP.=—pprese =~¢
‘The explicit ‘expressions forsandtare:
ae 2. a,gt * aS=(te) =mame +2|meept|me Pi+e] =S(P,9)cos"
= oa yhKeepd=-49 ;
Unlike the oms expressions for sand t,here the angle appears only ins-Thus,
afixed-t process with svarying (energy oms varying) has asimple picture
imtheBreit frame. Changing smerely changes the angle 0,Nowtherangeof
©is0to90°. For «given t,the physical region starts atthe value of8
obtained with ©=0,Then as=gets infinite, 9approaches 90°.
1
s Aconvenient aspeet oftheBreit frame (again, inspecial masscase) isthatthe momentum transfer has vanishing energy component:
Gehef =(0,6,0)-2—) +
Asdiscussed inAFFR, this makes some daluluations easier, Inpassing, Inotiee
that inthe following special ease, the Breit frame coincides with the frames
used with theBCPvariables ("Standard Toller frame!’)
nr<VT :Inotherwiras,ifyoudoaTollerdecomposition of22-to-2amplitude, thenthe Toller frames (marked with x's) coineide with Breit frames.
= 4 \_——>é ; Bytheway,inageneral <~ G masssituation, the .
A a! 2=2 momentum transfer is
tag notalwaysspacelike. o> fy SeeMandlestanm diagranseS inKibble, Ones Froissart.
oe References: . 1( AFFR page 400 ,use ofBreit frame
2( MSpage 210, problem 6onBreit frame
3(Omes Froissart, p68and70,‘why +notalways spacelike
‘
1
AZ
t
1
|
About the two different z's which you cah choose for _z,
t
@ Consider thissituation: bey i=O, F ts Ams | §&Ze=Arts { f
coms A
}Comppitinrndcomport. QuikMinskLook: 2Lie,AK Obviously, SerOi=T ' ‘ *
:
3Ze=ewsei =cos(rx) =|cose=-Ze
i
Iseemtoforget thatthissimple factisindependent ofallthemasses, etc.
Justforfun,letscompute themandcompapre:
eS=Crepe)=atemt-2g-Re]
=owraant —ALE faze] |
i sektus 2
ASarg +RELY 22 !
Ret |
i Re EL= Sewhiwd Co=deem Pr>NCAme) 2A‘
UR~
ae
=\Cyatontfs‘atax) ‘
t
So Be=Geert) AE2 ACeewhmttdons)FEpantome)SCRAEoe
{
ala= FaRs-2)+ (ok(dad) flowdoZery
e Geyhwk)ACEvadbo)'
.|
ae=Kak Qu-2)&Cwmty ois) Lee]
WEwhat) KORaay’wm)
:
-Quest: DoceR<o odoaye Ualinside ALO? 2.
Doko,daw<a=Aso? ..LehKeen—amet-: |
TO SL TO x aEPae 2BaCo8©,=eelke) WAVse, --ODfale: aa.AENS:FaFee .
—-stte. 2 ook eee ee_—OY) Ba=cosGas =SeslthF%)+ae----.
4h bene .opFeed fo
ads Cem omimt) is whewawsywaAatCnt=)on-mt)1.Es WEEESEK_ eeee ee ao -oeQe=JAS)/25sB=qosOg=Sslek-sZ)+0 -Gso.~\SEme Me)IS” tT asenq_- ge-=OG LS Cone
S81. 0ea7 _. ee! steAe ESfetchone+mstmys. -
©.@®¥so.@ FRA Uae} goan
~.(Sasthoalea) 4(sesgicwen) y(Ses@e-aea)
|SOR \os"956.9x. »IORI
=(Sesfrhra]sa)GSsireyray Gestet-ey ta) go,
. 2 BUSTSeqhah :
~DiyaNiaWd MtblyShsghgdar - ae
LdaN+Lqie+XzNXINNNIs =RGyH<0 Teovisakug Ny=28k+AcMSsoSosa"he. m4Qhows cee jtt one ~
ue. aad . Oe(asked qh(Laat) +96(2ahee CLESANAA 2aA)
a ——lesesexay SO Roawe BL,
lite_—Reeggs). weeeQadeve:AOR SayGe=G6)—aft Ses _.€
_==AGatgaan=fpGOA I>To ee eae SOnsinidasacleFrmoueh.2MeeEe ad Sdas
YuakleeQack.ayia<bWhey Oe
7 iisan
Yeo,ws.erradroaia.qoms bsSaaxlaonaallMaso;e
oS SL se
. AE MSS ee eeeee
_ - Oo<aech. : an .
So. lBGatYeyGosscdeGaOa a
TA heBAABEAD shag Sy SO |
Ruste voieg ESB=(Bt IRQ
ee oe. 2Myre how 8. -oe. .-
Alidate) 4.(20k 2TH) ~4hbk
_ WCB Bek+6,BeBGK +BBG) wee_~Gediahess ek) el eCZevgKiByDWKBrtZeHKB) Sayin,+GogBeaBEols:BE)Ue-<O° 2=Selle S7Y- et -
.Oo
1.
1
<, t
e;Sonethawedajowackion SORA).
>dain.SGoAaa)=$\—aGet) || ~ H Utrte,
| ick, <0=A270)aoR (vb)kit, DO andi A rn<O. _
|| i _|bbQos:.LLPoandAAtAL) <0,Non
:a a)<\bycamSax=Swe©quadsankA. || OSS=~r-k| k
yl SUK=J=OG840/ 2Wot..d\voscmaldposonah_sMees fossrk.
.ewe bok)21[Dd)__Qasassa=WNW“a . :4 Ms Mk fe)
ag= \e=Ai=k ZN ~tt
‘edohsss:a(t=+PoGede)2 r ) te °
écau: (araJiwpos.onlinshLivroAYO. |b)Thom. 9% => “geal 20
p= Ya204:
Abe4jaae-Apbi-Qeg— Ade PFaohwrie! saw|||agno= oe
1 ronmled sets. anne ae itakSES)- ales)SO ge dette 7 _
jay! satu)||Pyne|Aitcapy,s) —#(Srb-e)+ FG)
+i+1 : @ deat| oe AG@rals fs angryeHs 1||
HtlL}tblte4Lraceet/a\ |Cusemer:Afouba)Maepp
*Yai voabIy abt)ech=labsa)ihsortspeelLeRect =(ern)leet TT |!
—:\
1
Pcllongtes Enc i Book d @ Sak PD, ond 0. Du uw tnaAi2 POCA anynurte "
1 ANTS
ME)
@Yoe_pluysical sqion jerthrvorialedeas —pi*andf°iotre
ees
ea Cerneeee {\:H
.Cnofrorwd.ad,[eevarralae, p+Papcow Binseno fu
8i
i
~ emmy
ee
.
Pop=SibFp =IENGI- BR>leie)—lele
3pop. +Ceteme lorem) ~Wau
Rat, Tied=p+$m/p+benehoualsage) ee
OM comida —nneflnonasibss aMgpO
$Cype)=[Gama =e LE=£ f
& lick toca Quin Lancto
Lonk Dike? | Q
@)ao Lng ao “th,#0 O,lawn ia
posskud dehinsle | | fe
4 ° ~2- ‘N
DueStan)=9[Yau —pl=\alf,—>£2. oe ro 2
ites omDar=o Q,=0boundcmee iooakinfe nd
w= OC=fhm) =8an \
af = O= acme) =
oe J Qh, peau :
—_
i jl PaPlCQetme: aOeeee
FCfirRG),fe)=mm NY
d;PLY
= =Rife _oh=fh(erste) -0. _Sp=SR
O,py=am,lpm7RY gest awe)
Que =O urn R= ez
iy -3-1
@_ So,idWooly 9,9. one tnkue, nif bothore pavk, Gross uso
_ asa Mok 2p. Wye Or Daber woos dmSaad Glia
a)Oonm Oot eilles%Yom, on_S~"YH
o $
=: =lelel+ép. 7IeBl-ff =f 20.
i (hue 9:0: £09pp,<rymq
kodak, ep = Bef =eA
9 <- E1\ 0Pe
< — Cp,ou
BkFemm, so—F<=-mim, =>0-0,S-mw, 7|
H
Aluka, Cag) Oy SAwee
ACnCasy GON 20 |ZA
an i.
i
QOS sous duced orm (Aad Vea), 2S, on tuo Argon, 82docs
re t °
SydConyhe) == SS<> +
QQ. >TA, ~:~. S~WM
ZT ya7
; a
SR 0 OQ i
— gunele“ow
roe ‘N
(10)eeQaidadpaQawnAslousdAnneaoAA‘\wz=0
- \ fF :
suk ont20 UKSw _ __.2Sant ee
Onan fo7iolan;Aeyakenasasnda AhesDSuaface wil,s=0"plame- 10 wtYours=o E(w \s\e0-
A sete
Pas wbevsoedun ofAC SlcMSleQs]pone >
_ A Sepak aseh
(A LATN\
Kx 3
_ )=dG,get)=owe
wil Sorsded odora ulead onsen doafowkww Ro A) : 6 ‘ —
Nanl
ee] CREE nn
a EIR SCOOP CICS CUE CCS)
27 Bjiae Ge
~ G[Gepaithoa ava)aedTee2
a|TOOSPPaCC Cesere eco
a :
.Fodsobat Bo Ln
~-\yegte 2 be
.- -VORaan - weeee
1 setarts nteat-2(6 f=ppanen) |oR=sombamis—Uebelanti)4pACupsi@ Lone. ose ae NEE8Seamed —.deCeestang) betsamt)+ZeZeMk,1dMey)
=Spsent EES=BOCA BeBe(G8)NaN)_
SD tkeOeDkKESada)-CONS RAN
a . elo . a
San sks +e 1 .
oe Be
Wu giafetetet} jas Otay) ooLL eee
®Sssal.2Be=K+Ks~8)+ComesOey) aae eyeoe Rn) MG]Quagine2s) 202ek aNautafourwordhYKPsPskphs. Veer_- .
© aS RI eeAke =shyge +ez) .
4pits a pits .
=S/appee AChwosea. \: — :
QuéieQememo2EAL f-Doe _
-@Wake SRA? STAs Qe(ed.
gO Wk SEA eet Lk
H eu =PoMSEN Bens peste sEaaliscelaio.kus “fee eaCo ee
Y
|Sowseaneowgens' : -.eBux KrkQa+e K So oa ESS oe
joaKak@ed- a|
SS.-ABr2)HO =EKCa- P+a/- ye
=Ky- =O ASSAJFARKO LLL a - a
©QueSatta bea sakoke fee
8 2het BELO Kesenicso/ | -
@ads rados,Wee,aoefasmeGr
«(WE -Eake-o 2s =sieho 7L
_2Nae) RZEHO= & : .
2h .
_Vonpy stats ay§aeo| re
©DBgowndeacker B=S/Oppeed :
ss Gam
—.sbat\= G?-3)-2pfeo 7
-@Ya, Cat) A.Wer-KakomaAun ak. :
. - - oe ce wee
Qua. wenskr€l sq... 2 Lee ee
©.stele Fskez- 4a| |
N . . -
0 a
z3 <—s + _To >aPE Veee. temu Uphllere —eensapBeeteae =affsesso] —
fp —-Pp aaa.ea!RxPsomecdsnsbhia |ae aioe eee ee
we a a) AIagtma)
= ib 22obeaeadea, Tacdhueniaedeelyalae be wattleoeyenaLLt opaana Joe --3 } Shalt NauaQu=dbhaidy
' 1 i
OTe Gr) megingsayhiaaaegleetomaBreycheMraghovh.. bn2baal yon(0)eoeires oeaa |;|booe~ LI|bo .
-i.e. ce >
Chee 3=Shaped) ne Garmin)
arentesOearpensanLoe.ooLap ar~My, §lnnioler 060nndoarantaol Sheanauth tar .
a ee cee
-_ weanACR =adsaed) me ato. . a a
ate Epetn) ly ene ee“OTT iy!a+ LnaBakionlet| age20%tess ATA
a
_ mpamd we.pour oA_(i/s\.
ey. le
ae
appeno ure Sand Orch: ° ‘
3 CX)=9,G)+b GQ)seay ee
Qhos! et pe a aMile fassurseptastessesy
4
a
=(aa)S_+(abrbads +G@oqrcarxhbe)S +:
2=(safpe-slsre\(faads +[abbals +lacscatbhs S|
A & cS
= (s+ [et-slsre)(ASa@SeCS)
= sfs+ FS+Gs«--+
sant R= A
R= ACt-2)+ 8.
= A: a+AM-B+eC
Ware! A=aa, t
R= abe ba i
CQ =ac +caetbb, |
a =| .a, eeChaTa)". [stareig=Gay|
(=2mm) Ss Guy ST
,”
-a- q
agy a
[ixaxsbe* |=|-sCoxsbe) +C)fe)z lant) 4
os?et cole tbteSANKis iseos ~20m ae
=(mb-9 a
ae
f - (ha Ou =_S|LEGS 493Cine) —ECEWES |
=_S[lsGibews)S +Fenton) +datasfs|
=S| TSGrbewy) Se Comer +Geto) S|
@!Yawn, crwslovne durch free:
whe . =\
Qu) x)=ST) (oivwe) s+GitesGuia)S[T+(ase) @+Chew)Naty) S
a
= eS &Creme) Ss+Comteamd+Gavtane)S
(was at a,=
bisXottome |by=amya _
Q. = 1
bi=onstWMe
__ On=abe +NON a
SE
Awd =(any Q bb; = +0,+bb.‘
= yam) tone.
z wey Heeee Cone Oke
i
neh ame msMP|e sekek = (h-~ ~My’ MeMY| 30 ~
= Cay — Goto H
a_Cony=Menv Cmteatongag’ 3
@Veo B=A= :
= b(h-2)+e = {
= LA+F(t-2)3c!
@_ Ta mgsdia auwplticohom :i pat.
Ree =CMC ae) —(newaisaMe) +O
=O=One =bei) =2Cole)OSM SH) WTAE) nse’) Oma Cw |
=Contam Ombre) —OwWe) (NiWY) Dantas aLopswy
== Devens—2etbandawie athe
=aCatay comme —tt was) =26.
eee) Onejas|+QE)S +Qk8+2E)SeT
eres eeevg =_ott ee !
’
=antearntome—thug= 3
SvE-uw
@ od wa -©.dweokshu, {apical Sashonmd quence: _ Lo,DabtaedatesSoaksvnsechanba -\Ge]
Quick &.&Soman Tddae a—WXconsodYawayrug dokao ee . , |-. 09hdasqualmoed,ROR, pot eee
-@You,monUSge,We.dadaHUougoofeewongboeo|w==NakUNLdegAMabrDeaeverduabic loomadaw. Cosiat A),2-“yyWALsar: weae‘oe --
ie we 222ite _
re. \ . afl
-©Gaytodaudole ocluslvaolS-va a oesaw oy -. -.to —ReGagsowsawhALE-RELL- A - a ror, ateye YeeetyVyuegt
sane | oe oe
~—— Aun osemankBE:9\s0come,faswoad a oe
Ree=ovbcut_= Se|Gotntleemi-an) =Seating)olermmT, t
_.oeJangSLenkwasBlapssssanteee |HLA=Feesched aceerciey oaseted) eetatedty)
_. Ce ee wae 3TORO) LL
Qa a a a _ ae
"S,Jxconcg RASae]abner onsen] PS
@Qeuatues,Pextemuralworpe magabia ,(aeskew
@YouViasihuokee:ihoingoardenLoow
Quocrodeor Namwdise &f lookQa skia.\s-
presod.Darah cotlenGatad Quce,
: — LO ®
ee . Se
oo, ~~ 8
anoleonavinsakaudi. dualssshanakiow.
@©Crd,rarchiny FesdaawoLendoak
3) Se 1aad @-05Kehoe=-3S,
_-:“Soquam.34828semkrat1K<nC0.
@BacurSadlernuedsle,puenows ' ;
-Veampg(ttute)ontexonsfoshite!ontrong. Mun,
JasWL2rwadlampbitmas |
OW Vavousdse B=SmhaQagapent:
e BoC/ste\= yayyesYor
S 7S503 ‘Dualroad=agpuy.(“E*) —ImCls3e),==Yuu
Qngeeapeswvaiten Vous)nanan «
\bah=om ‘ .
Socnn | UgeeYon Lows
+lo i~O
+) 1~Ye |40 bay —Shab beeew
~\173 iy . |-t {wld
©@Ow, adaapsrivks Vein a pdotsdbde.Que OOSeASrealeschubAugQalaa,anangenfeteedgeeeM >OF a
t
628 +
AonstrKinematias Qustalew + - . a
- -— _ we ee - -
@“Brqpwalomas cass,aA. Comeate)COM,MTSo.WSK.S=Congety”«aaaphysical Ourashadd unOeesectbrndd. OKOiaQuast, uk
| wolusaoundeten toyCom at!" _i
ee _ bo. SK iechapowk,
H
i
Wecomonnndian + i
- - - _ ee
Bostowdony: +TAGminAG,whnh)=Sase-z)+nme|S1HS)
Baalboundany: ~a . >“
- ! Loe eee
Qk.s=Guraty, AGmi}=0, Jud,Volsquatassitedckbae sawsWma),80aaven,boselvaSaKt: .
Dom Capencreelak-2)+commons)
AeCwstMay =ACsewed\ —Cyrwd—aant)Cws=ey)
- - i . rn oe
| DAR S—Cevady—Girma ondrnd Ss.
Gg) .
- iB=adenylyl yh~mv,—|On—mboHg=md) ' Cty+m)
-_7(egrsrmYComrals~2erts)—CoEGrae/Cstme) =Jdass.
|=[omenk=Uy+myeng -yiavi,WaaameVege)
>Larsonsen2g~2-msmWns, =Leahwrngnel any=neyYe
© =4Loins ens) =r {
-oole Crista) ;_ - - 3k= hacake) +vtyCootH5)
Vy
Se, =ms(memd) +wn(whns) =Sa.
_ —COMM)
: t
Neonat 5Ws-S~KAZ =~sCnyewd) —Onseme] —aAnstun)
_ msty
Ve.Cogts)~rtvee)—ey(mt) —rites cme?)(igty)
=Chcontny)Lngeny Gnswkswf,eva]eee
===Bribe —Bmand —wy—womsmoteMya4vtyWE mate My~ME=WEhy—wbany —WHEny~WgoMy —Yity>
Loe a4 QrakeodLSQASAL CoA8At “i vAasi a3 ©s4lmwowwe3 HHO. Y 1SN
@TNanazON,=T= agam, K=OVA 6) ASD.
t= Thy=M.
@Wadee aoees ee MLEm, LehAEM.S
t=aaa wt)=&(mye) 4 :ame _
Drieagrtanuinte wieG88,©.
\ .
(of i 4 ‘
sn otbein Lonoraac soa :
©Wwgatinal squationss uilshchrdaleronins (haphysical so9in.
'
'
b As, 2O Kiltble Vonnctnon Ase WS. 9. \IS~
Jf
@Verona, vseMavi A [2°]<1on!dodenowtmiass) mtu the s-olrasanal
7 " <
~~ i
.. fl
ine unRACY=tasint+20,0BT
\ @c@. 16). H
an uanor pose. Owa: ~mtomt =- BESy +20
aA ZepS *
;H
oo: WEN]Sy Lae FRET Ye aA
7
dncomme dedocd :
aanh—_ay® ty als, mat) a(3omat
.
a gual. We bron od $L-T. uroguadit, aks Gxof
Dronannsla xenMejia Ton, uralea S&F Mm.(rt8,
wisi ‘neCheons candi jrok urilea24s0canabstae—> Grom urnbeod askKhodebe thPind Trot 21S iaEAU
S53 bit_fIZ1 ceKb Se “s<ot >
Ruwonle ov le) <i
— - —
axAhanooxpdrsitliy compwbed oud OVS wise e
= OD, Que be omnos*
N\<\p hans d}=4 ———__—.
OVomtocpbrcdy compen (|S, oarqake
(2s(Fw Ze) gp2s (29% o@)VY*
18 23(Eoooh coms) Gtwheme)(SMSlata ocala]
QaLavaoa.YiuareoatNoobuoosss MaasoQuod 5,Qasdoflertnetd
AMSonnegdoce ad\wilh 2A Sta
See
Ady=[5=Carma"][s=CurvedSS=GayeresT[S=Gretol a
LaQingas(sat CO
bolita Vo Vnang, Ad =|. Ono ocnme Irae condel oomneasad
____diwasd so.
doson: dnanyplysical ssquiin, 1Od|=-a.GaCicer consth ce ule Mag ih acerand dveeD opeh:
| sQG.t) So.
-zmsso} 6CV
.
po
‘
@_Vn_ormokr od conedox Mw cosa altace, we sanATM
aT ee a.YsESSE Sea ELON
era ZO ee TitwR
_os PX NEE!
a AN
rmanteQua2=~\eunur . ee ok Dwusaotd vafas @“Masoapesla ohflorgamesol) dragas
Mramondolvislersa. Go) noeaanues atolivous Quin voclovious, because aaywololic, dp s=0 ,t=0.
yslick (as sows bowin CaQue channel with Cal|
: Dusted cola f|
—fid-e Troumits(hoWighodkDWwodaldhon-emdiadyusd Auosynglola Coroobsyy Se
_ aMa ederollthal GnMecbespote Zi
Vas nada afaVeMs ola daOrnealorae ALS2Q)
ee®Example: gncomovauea rrddace OSM\~12) widSea) o_o ~126), Menau
0 oe coma coat omens Gro Marolor
A exonplo, suppsor usotcoas of owe. linod USO.
i
oo -3-
Me workdobw (he value cl8unre Ghaiolusicol s-d
ed = \ +
aSie 1<\ sos. oe’ Se
s ae
ae BudodyS7SaauntsyomsahQlunsséol -Te — :
gy EE\e"\-\) <O = S~Sa(S- St)%O =
— 1
lee w[s=mrom 42GETgy
WOOFGVAG2lot BA AuancawSo }
> 2
Ok, Clete 266,—20.9,)ASSteERSA |Se eC
e S%Sa SESM. sa “Toke s
: + os a a C)be) G)a tg =EME EZpp [seeReatoanlSawdon
a<r =<>or
Masaya Asi_Oad|S>
a)QePy=WV Cod==TEPa— phpOay|=Gawnnt)/2
sy) Pa==(OAGPS)=~LAE -PEPsea[=Gen me7
: Game TO +DEM Ey-
J ay= Srar
I Sk =GT
i
a
. -4-
,
Chcournn Oy) <> (lalsy
dM gamsral: 1
Sreeee fa,=[Seawemy eee TTfoepM a a
Noro, (2)aachonwwsl AW. And ii dorole whida ween
iadp lagloobinson. Aud ge Qe9s) All amamneute
Qoust on9 annmat \roune Si,=0cA_ornunad
f
92Nk Nisodanaicall s-ohawnl. .
AdaWiAadnmsnd 2 p
t
Qeyom =>S% mem _@GCN)SOeke.DyPy >“tH Sy wa A
00s<=my=3 tSCm~nmy™ \L“|
pots<=Wi, Sa <Genoa <n
; es
1
@OsyorWtQaa= @Oho2distaibuins
DatA(reamannta AS
pecaneoprunphoy PZ KN Seeom)» : SN
a Oe
'
GD0Wequeued preted. -m CNre _.____Nesisle&e ksennea se. wie £ ——__w= Owgrenade Fal ZL. -BAR. 2CORA, WAara__ - toeme ee me SONY cant!
.---SuiceWok Oa Be—-larsause shud, dead “WEO,K0.Dx=22,GKoka Qwiapowk,s=2,s0. x24 LeC2-O): emt enk =Vet _
==) es ee _
“©Waean axgread SaukavSeas. i - .
Sat3S_sexties 2 . ~eAestu= ZZ) _ws-x tks . _. —..! .
_8 _- -: . ~fo “aeae ahue,OreLimeSHOnmuthbe:REZ ve[Sopa =2.— weMintaso.omwek her he-2x+Z ee
-@SesngenialSoman apnesayoyACS)=Obegk Apps! SokAas doopal Gutd}=o, lenaed.
i
@Graminghe:IkLedwhetmyeatinWN.QhZealqeeewhe Re0)WhWoke Byerh? _— .
AHL Seana =Beker) k=,
>S-4sG@)+ohwls ko| a3 s[s-24k}+ =0 -poe Teg as
eo (KrOYsagaaseo |9XzA)«Bro
gas Gees 2adetAPare) ete keteee,Rasy adofWe,3c0adhuro——Wypreete, orymetoiabeant i erate
.Oo, Xeo BSks£28 seYawSoa,gosro Nook lesle2,se- _ . ;Oden Sves0 SUXS THO, sommmstbdSEQAunO. e Od, ke=xasl.Go Stated Jape
wo ee eR
@Yiw weed: clued 2a Doh Grane: KWAE DO
—Oe 2 Lee = : -e nee -
re aes) XPQo. .
Givaains owksids OneWaypeitotn,
-:
©GkWarame danwewek Bycl=m> t<o.... @
®SoyDatwnagolond ohSol® adun mu)WeaYan Onait QaesaSchennai ee ee
|@®Wa2542OWvequasar,“z=coudlenh? -
OO "re bastt aD oSeats GL
—Wetual Stst= x=aeye a
Se oma (BERS OTR
oe C4CRA 4EE =ondehd cone,
sont+Gxt +FY <t[eal +(e3) @
Ocai, couduadsh.x=0,k=t amdsolaledunleas&=lA. Sow=amdonety. eat
_ _2- |!
1®Foewocek =OrDA+WH.WelkeundOkB=0_k Ore5mOedutle, SoKESales,ondsaber)... 6. Quite,,WeGercavenmuilareOmDestShpoutaale
QD aggpia?Gasae, S29sofOFGYM 7. ~s(\-De1k=0_9. reS@)/a .
wo BRAKeon. 2 Robe! UectMoe
t
-a\+paaeENGHe _SO
- => (en\q-th-4e)= th9KeixQx-£) 7<=25.
©
Soast&Keo5eel 3azsekerUxtZ Quro
~~ragnarok ,Zacoksaashock Quis|es
Byeer ueaketx-de .oo
— - ' <a = «=> = ~
Na BAP, RAV, sv.kKx2), - ee ee t
Yes2,da-3,s0 e~Sxade,
.Wears, a=+T,s0=IpoFR a ed +0
Ji ye
STA”(/ivee
~ " vapid’ - ~
& pent
7 oe -
|
Pusetonsslfaa i
Do Webron 1ipam oaOG8)=0 ,baaden
—@ Trequaskion MhodcnraraieMoenstulara|2>\equaldofe— Qwusicod sro, and ude co (21S cased dokh?
s ;
‘ Mgt. H) buck nailonystad" -4 t —$
wat Qt 7
wl OQ oe doVihitethsoil
eet @= foON 2 LZduwug ha. |
_ weet Q+ /Uy XN
eA Q -
: p= sheGuedaath
.Se AVsVVNR
Z ‘
——@thenagemingsgistDaesigonAAAS =AA.Dowuae
. . Q_l#1St =, 868 20 x|
GS) =) ZO
nd sonenns Smsso- bP
——©Wacinadedaagintoiiagen0SOaaaimcache,adgiesoun.tagnA$+Ohuaim2ackeagainwas arrows Ai)vnhsarpqanparale “aspraanasatin"oAfeICLbl _——rodatuin OFesunat nootwuasgn(sive), (nowanabe
a d
=O Sondots),thewcrammul docequitnutesyousnd S=0,Qonak2)>soba QSpuiccat u- i
tJ i
es
i
ed >
~ ee sassasesresoessssssoasssessassstsitasstesessesoaseesesas,HistsssseeesseeseeeteeetEeCSSISEREEEIES. hed 7
ge cainAaealytandy,aeet 4 panne Sleen rts Mee SS Sent ahpdrneeae eenela . sperms eet fymune ee road 4 ErgeSeieatthSelmetartarsektstris fengegen ‘ STS ies sean eine aeon: SEE Mien aie tteaneea iEis matarcbarecnanas Sultana geeonera ae { Slutcoantees eatin ntschantteas etgeo,teams ig !+Betsyeeatiacta inhceoelayas : |HickGamlaidktatenaied atingcertesetjocnr jTeoindie hemeangnOEerarmonetuntershih fe { TimiaorSebelSeeotgrinnl,nn hh : seertoesyiGielen em : ;acterisRSSemen ibepeesoyiyrt . : 1 shockpres eecoegomant.However,iour‘es.seman 4‘etraionsaponintadtheesatheaessFawbewerguarnoms . a hiuc Regattas GMC heme ttepeda Eichten at baspetantlechaa : LE | Scieiregechaabrmaeie Popetenwetted Wennpre rayeieatsaviee tosousos,wowea ramones & : siimenrcre acpeiFinessnepanmeteenthStone a“ “1 : vk renedWendonotenementees ate Rinewaten SertoyngPacertadGOT Seaeyteiciateet mgcteatiees auevoaatt EGeneralSeatingroetdheNesRep nis i Sinem ewctnanas spicuteetereed Sen !Eitamara eaeFete Freeeta7aPipedfeyramegpore ? :Sieeyege Seieiany Telarc ems nei aes a :RetSee ea eeeneni neue 4 ite Reet ets tie Biogeniayotere |Biersstognenuwemtengetenmreeeeneaue ¥ TealiomAnandteneothepeanesr"uniteecalhepetig SSSinaahacne eterna at . SEINP itisehAechere ! rereheogreeaoaeee iEeEbceilionmalieseitiede Sisachicuacietan ced SSciti aaeaeee : RijoemmonermnarestedenenaeReywheter | IER ooasamarinnerrhseseAliasneTeeeaefoe — Teliieesremac'nfartinyneweebeeimeartesh ‘rooacron ‘here aia frtateeter(Sna0anygenenn M2221s7it0hrweypy«rg btsaaate ‘‘Stineeispnvoyaneaeguamematiateadde ‘etalonatec a;3d e |See RR cerslsHagmetiroane sree sotenmneyohnnestete 2MEBs , econtecutoffthealnorptive parsAyAyanaAyarthwildoorsbesekeedaceheeyei ‘sumptionsbootkabehavierts2haolgieens Aet . ‘sighemerson,heareleatingthesngslaiesoftautanfortheapprostionthetUneSonat ‘twoarabe,beeegytadamenaZt=AMGIOYThcampeonfanndtpipawlbe {Belowangularmonenorattentennerin.sreapretnaionikdonnehcpearl Hehesaotenaletothom?aremaremee]eteAndatheparesibetadoC *; HowerthecoollyspedaieveteSietefoneiespasm teoes roctatherepraetntonSeideduyReloces ope ifermeceedeichnesteiedbretfase meeire Eoheromintanci ttt“Eatwereyore ic: Enola,eeniinapresSoperacidenate iponcnteesaengetlieear ‘ =Remain TilethsteneatycocheforthePaesot SUSSOMfethesepeeritionSotawhotoeaeetees : , "Thcpptimatncrobereneeebetofa :SLinlayraion!inetropercesorsuneafeforprThemeetsRee : ' ‘Seeantabutewhepcrfecal ayyswnaide maetinheetewionofhisprocs.tPAreposarthenboundedbynadie etlweainaymachcnnhatte relBtacterbl 'fresetptirptefoeaaik 4 :cotterSevemopnain tinesCsrrROot Taept,tpnnereneeeFs Scat ESSENSE. SESS |eee maemesonlays 1+ pmtiabaStigeemcg iSamat rearsanCae O TES Seaime carmelSioeSles i 2|*S:Mandewam,PapeRev.118183{/839) WV:toed4see 2
NS ee ad
Ve e €i S wantsssosasananagananagggggggggausagngege_
1aasadesassesonsgsessssssestessssssussanessssssesnaenl Lf }100 - en TidUponowingttepimaybstin, 5 H AuepolTemoerteT ciespregcetany perthacetonscaepaarrrfrete CEGReetergrtyLaneso Pre Sveputneacrefareea ome pean ‘eteeoheedatyoann netsHat vanecreetnvatnecnegeycon 1 SatayweyaraSenetaig tneea : Seteateueyelayeem feiiyohohepplsmnwhtomaay : iEMainepacedtart 2 predicttscaster fe
.SOM meshes iene flwigoemaT thcece Adenohaer*ferensins—mDnnnge my)”i) SCoutincaneo wedsaltsnce2 a
:Fothncvmemrinmmetet Sieftdelem 9 mtoccmtn OOaintah-nitesoy H PihafemTatamafeegwhere Bhpbel oschgratythheren1YallowedtaswheeAydyandertothepcs . : icfeehsaane arcspoeye bin“oPBeblon SESS clSetbeat wheeuddane j‘abbhingofhedevine lbhbehPP onevethestrip(10)everep,attheregionTWTeUhm344, Ix. i eeneeanee “Taneybeenahogsnyt Sx iIBoRS K Bers o 4MarogisraxsermasanrArionamp BeHa esaaehtemaemootedits) Eaa sept,Taerere Farce, iyTheisadawinSe b { eesti meme huesettrgeinSESEtt stayteneti.0Rectngercnvaniartetnteee, ReseticisearsSSRISENSpremuetons : manzeme! Rening iemaanietderenSoeaeatteBemep : i.Daze frcekicamacrrarcoran, BeiReedy rinema10Etstenegiin. opGarytemoc is aactestn,@Rcmntoarenartarontoniy ecmechasiimcn adenPattieSyreeceereoo }3Spee eR aelnswi SUSSeamagenceafeeSaindrane ui i(althoughinthatcaseweassumethatithasvanishing. sencouscosetinatesa'splaneio‘theline ‘Banaofthecaideandlvesein Iorl-aedepdH, iuy).Heallchetheramen othe eager commieriesfoee IstemedatesterntheprocesLeapecyceMH a aEien agYenoterandnym4-6Tn(©bbl.thssaneusuaptingweaeeeeee 2thwe4] 5 inanamnivenemaral SSeRGTidenteGeeaBS gyweHascateSega4sororrlsBETa 5 i wpe Sipe : isteoratearsemaresche
: ath
euareth atid, ° Bratann,tedelaytak Inerhbe iTerkenaeden tonehiewaar SoRedameRecman deonoceo ; ViTessieem Eimbeknthrirepein DRS aceaseaa ss iSehnremieEEcenresee SReee ete,Ottoton ; ‘ retntritanetpak, ‘Sirdarasebeonan Reteinntemetinewsaih eeeThawigdedleiimecteinencesed + teOtAmine(meRRewee “yAiontinindshagemt 4 }Ty ttm=ae ryeilypositive,« minemtmttmtntymtmees fyathe on Oo sanemioi seasonalictame,SeeBEEsemenoo P ‘ revtteKomtimttnttnd, 0)fayersteAp1agse ESEBASISweftiy“iat
- taaicotratnamgenbrpce BiartaraientetosoBenaar~ThemanIaeASTWS 7 . tmentaareoe(Ever=(Eaeosen(~Eu, KA, Eenpret88igskesemeoftenbycedeagegeasee 5protBad.tieeta8eeeesr WZn'y iiaelem aodThs.tential ioseatiaOO a jGES I% ertwetiaSeerioma : : PAWWeEtEemBebb ey aera Autad=(0/t09fernat;s0asfancinnywthavetnbasineat)oece 2 .Mempindadiitmetanndnaly —casonccsef Neder: acta,i)SemenMataaeesOOiy deghelaceLoCo, _LX tenbeneanteGOGacehoweelecisteake . PoeGeS Gl _ Rete”MUNrerawhaetePlater,(9) Fete ele MIN; sapphicamentestttobeahepplegertrmy . 1shia : ODNe ESSEpangsRimtvaremdesoatiaa f :rerteeretnntednale oyanaa Risen Sees: ia iwhereermnicenane Papandan tio ogSenal f-
4 4H
& i
aweBGCKT fanash P |
aka
"7ye .
OSrowle yapromos aaeFrou, :
EN nie D= A Ask Sined Sy a
—________@) __duspsmas cachAokdia5 ee
Rakes jaesVE ETSW
ens tyete=tehae, haeao¢bicitt:de . { :
ahe
adere =}
a oe We DeckSo.passiehe obsuaLigedRasgisu, susstanlar(rose
— Pruloin (pans Wr i
—sie=o. be duéclog =Ye.Ywe, akKable
FXO =[Eteehh= [or |="
ss
KGS =Goes 2k(2exth +S
K=O (he £—(URESO Ly,| Stes |Rems SE NCY=VGey VE
—OWomindeWokTipStakofOSachowkG)eennea e gostundnAteti= 212dDeas” nnaa oA,ns (19008VO?| —
—ke0 |F277SEE? _
eg——©Dinosptebans_Ge-ondQs Ss
— @Auswoded Asgrpiars UaeLouseoxac ASGUOLAD 5rouneoxackJanta &Qe. 0 (6)
|
dur, On=9(este) I a
——@ Invendd tapimebls doDueSo,Tosbodyalos Pumesgla doTieSaTanbodsasks,hen a
a e
,é7:
—
, yooeGe PEt SeEe
1 ——EE
VAS Ahe im _
z ro lraadSook AEcpeea doTE 2 the, thts branches, cosad =
[y 4)A eeTua: g(A)=_—t— ta|$I
a , C) OL
eg pe Fe
)fai
__ Fa}
_
3as.Om|et4= c
Qhom She Ost slerba ok Aa sree flor 0.
:
=:
GeAaM
“4 . 4
LnQmuteiitJoniom.s2> +Rymemotica) 7
JON mvanaoaike Guem.o astofY-vwma pip... a’. ulaataro
Mo Cowadtauckikels wvourpadd ?
P-P:.includes omassem. uy flswsan,“dm Lrog GrenGrsroax
aca=\) dwvonola AQlaia Sri oxDartinne Amos
2 s
9 . 5:
Qadas ornlesin Dares pain wivarg, : °
:Q Q,-+P-0 =490- ae c
so:_had ncaa frost signaArt
eAalana eH
Pr,fa= 02Ox"Oe =_scalar alan.
n 6 . et Y
Debut te,9°0%Py Kepee ASOS =eelam Sanaa P<carKasda
Louw: : . .ig. ley=
)One adne@angl eeae
ea
—~2- :
Soa_43.4), ae as 7=>
a OS
=ak(PARA) aggq) =ak(OTNN
=aahote NGetl=aadfae \PU et 7
8
—_
ECputefogs)=[PP peMT=earPXphPsPe |a
F
v 8 D|
— Sa Q\cotxQuekofdreQanuwi agegosP\OOWU)oamboos—— Gus
¢
Q.a.a.9,\ = — BGs) Ver:
=—Ak(4Megrque)
wh $0aAtgAs Noowly —nsed one Slaw dokinma Bho stqree
.338 ~_ .
fo) nsonyshuctiou dl!Oneonomela +Cheolusicol Argus _
A= Cer Te=2,0 )
Py =LpPeMppoePohDO)_ate
Yo —'3- 4
mon: Men — & q
N= dtxe GO)=akiopi= &wolQQ= le |
ssvepuke,Ps.= P.= (8.9!0.9
_1
5p=esp =\-aQ ate =Tear ome) 2\—4a)| i
Q DeTyeps GPMhyOre=6@y=UC
Wg =GGG) pf|b)”. gy
™i ayo PfseI|a =leestals| C= eS eae7ee
anew-<4)) = o()68) —cyo@.|
=-GG8 :
A=GCG
; beQa-Ph|.= soG\- 6G)6G,
1
JonpainoufaonGlafapwilhteedeneCae(ert, |
cuovire O52 Deemudhleetrok og= ae Sus
Si
p:Oe = AD: 1
t Psoe @P\a,=(pba Pe pp un
ae. i _
4
v ~4~ ~4]
hf we a easily
‘Sak De.
SO
eee :
GQ=Gl 6as yiloudhnakcaAwad“nero
ee=haa “ 4:f7::—
aposvl%e witatRoaskmaa £0,ohVook2urlast D
eS Zaphp =o 27@PAa=O al.5:
_ MOVey jyof= _H -
__
5iL20SyanikofinsponduckUscdina,Dasa yeu saan ZaBe Memtolgosq
Wain=>aM—— anbsad. dkffi=O.Que: ee
-s-york 43
dsj 26CEH)=o tule)=Oob. . |(auig)=o [OE] obs po
. Lae
_
_ 2 G@J>o a GER) So __~ . : _ All Yoleda one| ay aSZoaI) << Adfeeed= ao | _oe) GUE) =0 i _i
‘
OR OT TO
E01 eT Owain Gla Saws ST
oo _ me Q. 2 Q2, Ps
‘ Vow J=O =|e ——
73{
=MieMinMee MaelON SS
Ss aeRO NTNCNGCi‘Ba=o —
&=psseet (QebP)pop)flaca F
/ —6- -
9).o5deonmuincule Dahir ayy=dyes Ui=UN.
WOOMan, dali digas peWanoovabnar:
Tou Aye ae :
:
_.
aTilda ds dss ee
_ _
Qo Vedsde _=CNT YeteaeeSthsdon _I D2
Ny (i
__ Ve? at i— VEdatseoo|=dales|KX\“erT\\\oO
\_£
ee10MuZiddy|
_ Vide Ydsdo|1disAdOdy _\dy dy O : a
——MkGagaSey AWNisfuck od oo
_ a a eee ee~|odyfaiamdeydnd,dniI dy dy| |deddydesde day=GV] dyda(Gada tseda dy de df
moa a ©Ydandy =(.-) -G= 0p —29-0,~2fype-
-
\-se0 *y ¥
.do=RQ UAfe=dytidy ChPe|“Windle Satter %
“ p22.5oe4
. 3On_rstacimo MraCroweonadar\or,elenrmimawts 4
@ ToaW=2 ng areuitk bo pare Maok :
a .=SOT °Q e-P. pt
en oeee i(ep Otoe pC pt of| col Pi
re Cate
~@ Basa:teaGlromaoun ice?CoadtrealesnoeBe
ea)Dadinn mood-doaeillaan24add Cil*lastondto“oak
_ oCPR) =: Sey a PArn pt
en 3 Bt °
-
= OTE OBtee
_ Ew (_pt-2p. ~Pe
Thnk { ee tee ex||
a 4— a—_@ NaasJako_=9Fx\stcoadddoarcond .
es ~20ipa may|PP] aTs Tan gl] Lea
. a”
.
Aad wih le 42) +3
- ° i)a Goi)OG) ES)
Ann(nebundakArd end tae s\rollacl upa=
_ fA Q {
Poet
CN ° he .
ea 70)
_ t CBi-P) PwPed ux$a)* ott
_ 1 pf pe Q a
Tole) Souteoodddoald)Denaexcep Sok
__ oo oo. _S tl_
{4 a eSiSe I po~20-P PS=2pef : : .
| P=26ity pt=2py6 ae| _ 4 pe Fa a to|
—tasd iaooad, —9tkld thudecef
ny ~2ppe _
a cd 2 Nvar ——
[| Reps afy tlee BE ~2pe{o} O_. ic] _
7Fat, GuiamannhaciaeSER
en ye '
oe
__
: :
_
Cau\ei =G2)-Gy *Crown eo
Se
~ranye K
‘e ‘ 42
QoGranReagorins Quapre :
=[Aaeispeh TPP| eeBae
1
| sj] fp pape =GCP EY a
f
i
@ Tay onsact Nasphata scorer: ‘ y
pnt [eReSH UGpeped=lempanes ‘Gasp
3
;2
|
a
3 $ Hi
[op Se SS Sw=Sep SaSSSSw| E00 0 0a
[sySg
j
i
7 .etod .
3)Yawtoler ~a*Ishceaaddnsoomd:
[RO\hse\[9Se| (Ca)|Safi|e[SaSe [=[S.-Sie |[Ss |[SisSin[[sySee]\Swhf \SwSin}\Si:Sea
ed feSee fu:Ses Q:Sao
Sach SgSea Sis Sig:Sew
wf SiwSa SwSey SynSed
©Wandake CavJuak A+odddyaxcord
:
Ea ‘3 S See:
+ : Sat BY Sia:Sew
_ = Sn SaSq SeSra SuSw) @
Lay
Qe |7Se Ss Seo[S04 Sr\% SaSesSeuSew_||Sio-9 SiySer___Sta SaSaSew|
|syy-See_ ||Spots JSy.SyJSenSe anSo
h $$
“
|SuS. SwoSa SeSeapS Seo|Piss sat “saS SeSee)Sua: SwiSe StoSis:SauSia} Wee aapdrcn) |
= Su:Sn__SuSee_ SuSs Sus Sas
[SaSy SuSe eS |SaSw||SusSvSis'See Sus:Sas_SsSw|[Sens SwSteSysSySwSw|t
do BS un Dat Uno ” '
t :
= SwSi SwSee SvSx SvSan
Se:SvSaSu SeSw]SeSapSy-Su____ SaSu SaySysifSeSewS & Ssu-Sei San-Sow
Okie Werpanera unlmtatev Qerlea Dee
| SSn Su SeSs SieSy Soo[OT SesSt SaSu SuiSs Bw Se
a aceni__Sen* Ga: i Stee [OT
H
| Lndsxback srgnr olstron ah) olamor, (ron ol? (de) =
soo mot"orb 2of =>? Shedogu
1‘
@Wade: Moro Aspva condd mastloopulled padior |
6 — i
@Soykoamu susheau alsigna.(ainMasrome caher. 10 0 | iy a
be Q.=CApe) Qs=GR+A+s) a4 al +.
‘
“Ne oe ——-©Siyuhodmoe ofCanOsdanigGroen +Gas/Cayer:___
Sate eee
FE S _
Sune an a pe (on7 dam de, (#R+A.—APN=Fo LakNey.= + oe
yy > dyaaey © Sasy
sy eg
oestoeoaoe Wedye> {
—Begunntie ncnBhs BSSoe
(We 3bowdanie a:2,0 SySu,Sar _
Ee3 —_ - 1osO dy dyede |}SsBy© _ \ _
_ tS Se Ss_ 9
_
!_™T-wt#NNTuwuswvawvwre’"Sy Sy SS
ss —s- 4
-
;43
.
0: @SoOabadnawd annedyesob.QaliBon Butaasunphie :
a Qa
oS
a Sen >fo]
—
~
6
=
Ts 5 —
4> ° NS S33 SO a
es a) _ (SS. SaSiwOl
a — 5 ____@Fhopus 4 ee
_ LSaSO
__
1SaSySyOO = USusSurSuSsO [|
_ 1S Sy =:Saxe D
i
fe) theadouaeQomamSnETE _
q:
—
;©
___
Cs 0 —— +t
=|oot! =a |sss)50 5S3501
ets atSSa
:
Anat foe damasdalce postiune amooaes Aron ddes4 of
, ~6- ws
<
o)N= 2 Y\<o = JoistaL 3
+ ss ~
Ua: oe = oT eee -
——
|ss Se8
= (8 GS =CNG =eGe=So) __
_ VES) hePpl JOCSS) =252.|Piss seI ss,
= Gy PEs) Bes Se
_
=GUESS. =Gas
= S44, =2US. = _ 7 /
_
= A(Si,5.S j
: [or Qoroauth WresttaMok —_ACSS., Sa)>0,1 emmyLZ ‘ :a2 Ont ont Oo. YJ. ‘
Gm 7) ‘[i aanne) |\__ aen | ‘ _
PoAs et .7 7,7 ny,~ Mm .
SsCasta7OsSa{panes“|‘Meas“hasya6 ; peecoat beSoALIA ArartroleLe
Vw olanse heoll+oomukim .2
Ce
a aA 43
1Waleoxaug connaabery (heWeniowd qpams immisalibia caeOduye imdthowde ey t ‘
;
CGel= GRe)dy
H
Quo, Shr_wnrvqualtics oseMeeaime InonyJaleoMi3 4
(Se IS
PSAS fy 0 (a
@Nawtsromad dnamoGot kaoanQasasquamdoteorditeseguehomadoled viato S~cordate, Sheromble ay |
Jao <T. _OnA Su Sean pA SU/wa war ust (lao ono
aslo €hera Qeboy
Lnomy oosr. “nia ta
lismdaQasqualima maled byNEqnen | awe, A Vhs squakdma imal sdlov oY
Quix hwo ews: A ‘
J; ACue it, (+9))>Oatm me,Oey >o
PS
(a
PEASAAVVANRXV Et YR
a,
. HInaduno\ponesodaclee me.sedDaire ragoas (OAP=Ss.
Msiatursgqeim AQronCOA)=S>Comet) >RP,>Bukf.-0.>oniis<>C~pr\t <afone $Tetaaraies AD|0 RevA ‘ Se —
, ie
~g- .
0 ah thos :
BG WSSNS Oeeg
GAG wsS)=(SaGuawa([S=Gr=m)|. _
_ ance “~— a _
a = FeComeha
Yodo ueAudeack skmam) hhieeB<MWis,— Fea =Po _
Q;eyEhE>0 sseast boveGeetacum, 0@
bean owm<0
—@ASwowoy: jd)lod2onoonsama sido (aos asunisle
Mala ®tp =htP)amd onndarkery eiSeCmameS
4) Wand Vaneance, sider,Vowsider,VaomArmatile yoraadele
_ i =Ch=PS)and onachedet ioKExComnm
GWE ha2wasaali, one, SySO
= =Guus S=Cues |>O-| £ =St=s[2Caisn)l (mm) >oC
Xess GOW)= Yoient)—Ga_ kVSs ott =[ante =Ce
Pa Owequectmcostvancho . oy “a1"anodeswankee 43
@So, Vue W= now aimed) (he sSNendard oodatrmna on
5Cen pe OJ aS |ee ee ;
— =awaeéaera 9]2f)
esTeGren . To a
a en a ee k= he osntfs=PSo Sted
Jf,suk Pu0.3AesAL sf =(oP _
liters btsdasenqaach) tosGonShiota
a ) 3ajj>0 _ Vow 2 = =
2 A
K ‘
_‘
GS s-CMI. Sy
he AKAE)= AL) de aks Kid
ooWak 208)Va Att, 4>t)>o 9 en a
_ _
aOY eS
© MaybeJsddsrdonwe (roaysditboopwns_€COP)SucksA
—_p* =LeeVege 0,0,02 Sand aWdnning spacaliler 1
p*= SGovo, (pys-p) PKs eTsBS 0IO = oO,0,PRA NN
~l0- oy
DOC, gueMing akadime MP... fsoneaalorhany fmol
\ineQshe nookra DL)(aeee
ps (EQpe,©0,0) . __
P= pt=(hee@Pep 89)
|
oea and=aBEic) _
o-Ps=fedsLye]tofeA)=Ph
__2,=(PP.—Par}—_§_!——. ee
TF LO =>-(e)-( . — OW
—Sipascagt_C4:3.12) andvincorkagin Tayawed _
TT
=frtr fta
GCs (i: a
Chuckussim
oo diaemoaed Deep syNN
ie Aha |7 4d
\
QRooadd\Qroaeturn WekanTesuasOustA OLD _somabike (Suda).
Bes anomie Wea 6 JeCun,=0. Fy
Oo_UADANAwe.okNGSKidAo:77[DlNS RitXteASOinch alincamtkoh
ZX
ZN
ES \ LK
wieinOPnviwuit VuktnoeAuntpane’ omBuaud, Mak2,>0
—_\2__}iifad.han. -TeGtro-taccat sngadAWasnoir:a
pAstafse =A:Sp95
0) =
é . =)4- “
8
ao 7:
_
, a
ISa3Oo- [A
a oe es
:
AAS oa
—
ot otLit a
1Om tkmt .
©
Oc 0©Os
Qindeinen fabeapluysised areco fr.Sb: .(ulhast : ‘ . 8 *_o Oo ie) i
ee Re
ies sas
[{i een aw nf |
ee 21 1 Ss
_ay :
__ sat(\ [=e nme = | _
a=w=Stmye-2s Be)
ee —y'~ GMS —2myb
- oun fe) fe) _ _
==FECA) (Ysa=(s-mF) )+(sempem")(
)fa 43
FO Neko:Ouienulajst vioenigrinal isBhaosswrasas,Onoxtionse ~&Wound
. . 6) iN}is .
lg CO, 3,S-%
AG Yadapoduuoaddaca bakit barcan
SG: alela, Meefplats-G@ yLeal<Waeldhais) ee eto Sx;
-Orr qussby Siete 2.2=G9)/€ ardCrew
=3)\-3-k =“@-kyo =AGEL) SO
QnQaida Aagusens gza +a, cmush' les spacelike, ie aon
=Vira <p<=[rene)
© Gall4ERI, Dee le,A=pr camSleonck onoaryMea
_____ Na,22=somon@ Seah AS2th yO
@Petine: te iy~ae | 4NOT} Gall, Sells spacllien
s ; t\ 2
: fn>A
7|\ear, BaspsQauds
| ,
3
© Polmah viforprotetena+ . Ga Whae(oo aeSE
Ti dn Dae lant, enquasibdrDak
iOe
ee VEN—“tLOKAADEMANorg a, vr ae
e eee),
_ Se
Gon: ew,9,=O,0,0,% E ,|_ =(€,o,0 \. XZ _secs:OO SEE -_
Diy deanGoaowsclasBookcoudian SLAcanyg
— AeRAS —
— horeia_Sno ee A —
Pa : 43
Baohic Unnihanitiy -dre Quodlursicad nso oyGow"
@Wad consider, a9 tolseacellee dvagemoral .Le, uoillemad
Din XO HAE
ttn AADLSePabASeancaeUagaating5ve ——
A. : [tT
i a Oe te; Qa a - -
1
ce
a AT2G) RE SE 2a
Nes =|AKARACADT =4 _
BAT BH BC" BOF _
_ CAT CBF Cer C-Oe]
__ DAT DRT O-ct 0-0* 7
=|Geenp Pot) O-B)psOA)@-Hl |L(=@)- (fe) Chea)(faefe) —C@e03)Ps(Per (Qf)|I]
a
-2- Q=(0,0,0) =
a Ode Q,- XS) _——_—a y
_
“Repo
Wa, Dacndihnn @SObieman’
SaVOCEWeWeEUVeaORWnveerfs wrrhomts obepel
Sas SG) Seas TsfaleGay
SVahdida dt6COSET EBCasad) Sat -
finewagsDabs,
SN OIC NC eedSale
~@Dux sadnspallposesondovafuer tQ=ER =ERi Drooae aond Tun. RQ=E 8=ER
SEG ee tle sy2 fog|preySe Qt?)CHP) P
ene (Ee AAASZoeA cgi OEPRTCE]
’ v/
‘
3 aa 43
A Yc7
Gal hfe a Pups Ye :
_ ese Rafe rat gle _
_ Se Pe Ya Ss. __
| =u |-m@ep~ee __ __.2 Ufa ~Uwb =Bef 4s _
=2pPy-Ue Ut Ls _
: SS
a) Wot JS7
i Va, sided do«
-~ 4
_
. =v AQe APs =Ye _
= Qo fePw =Afy Ha |. _
: ~Untr -Uefs Uk fe
_-s -3 —3 1 3s
oe + = Sad ao
Pe aaek __ oe FaeEe eeee —UlJu|>_ ahh hp he-te le |e :
— Oooo tes
Veo dasa ckpbsGadeYes2C=20']
se: =Be=eke =KAT ~
ai 4 ~ SorZerker Ae PAT AZOOL [eeBeeve|=Sap KTkateh datept —epht ae }
808 BgSARETESTE) 2pGeELLY_=See\§ Reestk 2p"(bEthetite)Hol(katate) =ox< 1 = 2 ude\_|
= = BONS So: tel ges
=e)—6p"phAK=Et YTpeer t)TY
+ (Eek) Pg! 6© SIT PRche=IPAth a — 2 A00.)
( a mn
@QReoth ssomaAnbon:
S=G(RRe)=o[accor Se] —
Yas, &boatalasodn wadB=+i8/yche,sosmug
Tin. SG
= Sahatdear00)8 _ a adon
Ne Ste =) SCox) tn BV eee Bis e 48
=e =
Maa: AASSag 4=GAG)
—Didsagth CaisagathdefolfbgI.scanTaa“ARdhd
yi toasdDapsc angieadeeally daleddosre Sashanmninsaiy ohodsYea OOoP
AQ,jel.u.
tieonsnestingchaseszacefommlas4
tosy
1,Consider this situatii : Wp
ag"=acea\. OF[vsed, .
f
4Wes AL BCRrctebendTELead]
GQ2 Shee AVEWass
NSB,dayeie Ge)Sogn onBeQe
* aQhr @,a(RotePyMe)xXLead}. x[saleBGr-s)h
Saee =GO-\ba x
e AY |\8@ =GQO\asy .Yaa, Saa
Saw(Smee)=aySase)3,(5Sis)
“\’ N0G.Geemend)=WO)ShsehydeemS)AQ(SH) .i
Integrate cms over the last two particles ',and 5, then doa4-body phase space,
finally integrate overmass6,5. HereTpmshowing howyouexpose justthelast
ems phase space. Obviously Icould continue this iteration some more:
j
av¢ e\aQGrates ss)=Ores):Kass4s(syS58)HSQASeeNSSe
v 4“aSSaaalesantaes S30)=OSViseSao.snls)40.60#5)
S SexSowaSuid: a
®
SAGs (SHwhrelya2)
ye52 =Bel ©\tsedssde
%AQ,(Sas HEoe)
*AQSap:aH,Sas)
AQ SarmdSs)
yAQ.055 ISar)
@
e
—-2e -
hyomaorcad Geaapes ca©BeaBaahin,snicanity, Qhiaeglansapresegpiaweaio
®torea=SeGhmyaSGewySGree8) etnWisahhwaeSKdemeaFe
a ee Sos. FS tM eeOE san £Scpeasy =OCP)SG) ae . Behe 7- Eepee ~ - a oe Beike nee et
@ComadanoOakinomarauyslewdLOanyemaays hanes ean
dasFam whegeSr) Reoncomshawa=-- aoe,
:we . us.aO.=dhtte SE “eek TTTahadade an n=ahd” Beggs FRESE
@KewcomSends- : uh RGmad)=Seated-Laem! _oeos
AGTRBah eeeeSobows):one.PasPuQeantesaagachine, owedrmamsney eq),aemmm*
«Nea dysomgiao dypact4 . pe=Ayn)(aT“LandySots, TeaSmYe ee ee eeeeef 7ID. a:te aah r ne eet@CressQuek«SeTMDowerBOYNalgene Beeyomaa“Ai,oSeatMermanwlanSomPo
rot Rg 2 ~-> Cr)* wee Constante SRR YreGo) ATAlesdQdesea:Zeee I
rua 7 * of mea OA ySom,a ae
: daerend5Bay>ake @®\am squab,sed~--
-mm Tome RY 1~
souw) VOCUspun oohMW) we Ka, Trad eeka)
@Weowingu: pelosSika,COP,COeikshe,owlsJomobenCA ne
AnkndkionAAyABal,VomA%anoaaunnsdsathsdaniieal rwoarsa)squabonwi,5 eeSEOSDcoatapeceGangomenrah. ,.@Waster *.4oeamtoo°
Wye) ~4SSGA, . fo
Alajbye) =tae HLeeVeeLae RG)=48 ) nea)
:.= pak, Cat. FaSons3, day dhdaBETey
‘ \
ne eeeyeeeereapneinmate pipeaeyapp;
Bola de aScpysCte pes) _Fis. Wsdod
andl cesrsons veniam BAS.Cyrsalamokeds ounomapao. vedmea elMaei),so£70Jorauts80.sawallan 184:- 3& Loge Nghe pak eS-__. dy.Sas.Veepaevae, cieeee
©DMA, eresangende,“SatBshagSogghi a _... . ree ,-oeer t+t=Nan:reFOG)Wena,(sas r2hA)Le@)eea)}
-FaQvalening,DakeInetSELES) ahonMetwe 7-
__®Mat, Bs FE-WRVPIE, soaRowe: fees
.* DMSaE,EFCAo(ls tw)Seas-WErele)eEle(%e) a. see a ~ - :
DedusJAE,wspidleup*vee E oS
es ae _ es
—sea emt echletmia) sees). CEseal
duhmoor-B=SSP Remyb 07,a
@NeswrentsordSeeSn +QedeLeAddoh
Pig ggET aaae bot _ we SbheeZee) LOA ©ogyectine@nerets!
Maatna tatrfowlgghone2yaaaBeASoG). S'S.QopreondeKeevet; Gul2 SwWOK Tee!“gk Va ee Le eee-@ =SGieites theeSrd not,
yeSroma.. Se BSA)It) ~\e a Saeyie, ——(eee\aUpba
Yr
Boa: , aet
prackunt ELEGY. Bubmn Pikpromt, EtEL=O
- .. dhe (84=9.Ane
ee ea .ead Y
- oe Hep he — ay ares
a ne --\e-€ —~wselato 2LL _.
. yee tl ee rras : ; dasaEY=eonsen.MejisBarthWatcanst and&#
GEssasp SANBAG so YE Cs-scssebee cee eo+Sas,
Qo Fay =GSosetSe) —AssyAr a
t
=Pl SAFHSLR UHWIAUS,“USO
--QWscowmNanangdoe!2!mepeeeeeeee
OW, wdAGIS) 29;Ban.&Asalod.fs20.Soe SeniASSate |)D2,win&sealVa,O69.nus
@inatnasveentoarmdhere?!Tfyoutry,toqvalutate &@pinanarbitrary framesitcanbe done. Even ifs.[f.0, you can still doitif.you drop thepositive energy conditions.
(Keeping these conditions .forces-dQ>-= 0,that-is for sure because-you-can-go toaBreit -
frame and show it.) Most interesting'is this: ifs.LT.0, you can still define a2;as
thephysical angle “ofvector pr. However,itturnsoutthatthis.variable tstotally irrelevant. Thisisclearly indicated bythemessyfunctional, formof21thatappears 4nourdz,dzzintegral inaddition totheK-function. Infact, when youaredoing
Breit kinematics,‘itis.the,"angleof theversor:that dsthesignificant variable, chg, not the angle-of the vector. Itisthis versor angle which isrelated to‘aphysical
vector angle inthecms frame’ connected tothis Breit frame by’the operator o,which
werecallwas(-i)tingsanimaginary‘2-bpost. :oar e
coe ee ey ae ee
‘ - >-adBene , ee
,
ia
0 %
Sn _
él ‘
s {le oama Jeroranns
@ Yowwen, om som altro uaity Pate:
to
dQ,=(4AESmice 1
¢
© Vgw suppose Usdinas antadrouas lao Tee1422 3e4soohening
{ rn 8
® Zo
g ar
i
> olpastes mo,brwwillens ST 3 ‘ f
eee Rei
-Coons wn-laody,adaastSpace. foe
Tadgs spaces aeScgs0)Saqe8) rT ee ae
ee SECs) -_
DY se£>0,bromocuetnawuetriste,andwrconOrrgo_gaNeaeenston £=0.Vora,poe eeee
ee ~ Hee eeeae ee t =sewn oe -
~~ BBL=de,Sees) SCUR-pV#-8) - .
~@SiwFeHERD, weomtuompwaedheded=Tey camdoWalemreilee rornydfaramy)Srabisinn oaota,Que-iwdowdbanSeiteyocslanae #Pr=IBlsvie,eesd, ok\laasdonEuabidson geom.
Syne” daceGapSetiySan.SGOT) YY
GO&dagiudqlab WETDare: wte@eaya i&
a t
eo =S385DRTCR)-Sa28(s2886)+8,-50.
a :
HoDekkEL)AQEMMISCR):teoO Be@ a2,
Pie\4G58 7)SASSSE weamor(hie[AGEs — oe
Oeparler, onSCE)aanaraid@® =OUn sdoy.
QinWad>0,Wrens TE=iatstad>Smstrand.
e i
/+
}‘
Wie}partxcte Pa22800.522Ad
-pantechg
i
dQ = 4%. do, 3S, ‘ \aoan p=(P-p,ee T
o a i
@_ dQ,=AXX: Saplode\ih. dyed
_=Qua, Sah, |SarSia S|
EA EHRESSet
ai aa
B® Ourwien satoconpuhs jana Piso ews Poms:-—— st L=Sac.) fap. Gamer) 3CE=ve=my
Sa SrpaSEa)SEPe)OE-)
, ;
a b Y= =
_ So: 2 B= \et-me
SESS BSCEESCaio) CEB) es.
db: SCE 26.6,4mt— mt
a * 4 __=+8 oS =
to — .
= :
ee
i .
aaalboxed Owasd: Lt
Jimask!) nami(Ge) ff
ang rack thrd Unpure
1
+
WEG jwise cmoama the PT
dolla Tot bok Vous: .
bo Shoryy=SVG) OKeR) |REZ *
1
:°
,
g aayTONRte! pee oscttiumstl (ly mloannod Slalo. Foxerp arTH8\ UMowkeals row, |ada+doRUdows U2)uoctenis
@_BnSeleuiteqrademaydrbowMolto QeMoteSeberTho| &oul (hows wwot fro sind og} eCA).
i t >telAte ee ee Oe TS
/ 4
|SlashicMidonitin ;onwvension doGiagstony
——— : BD ahaa dQ,=Sed) Yds Wid'aS
a sedalg aR
@_ Conver toww augie usronotano veyVand
-_
-=(Git). fas 2Ke)SOE AQ,=Skepts 4.40 5 af
eei
PTR =Te =CNa0, TAT a
$$$ $$ # a
Mole:dnsquad anossoe isACswlomt =+)\s-40eaTdgu, Gz oh: bets (ny? =Cm s
one mot Sou: +
i
ae18]=an FeNag ee
'
—@ Brgpanal caning, howmdsimeomadybeta, watqk
_
{Fi due) =Ge). agDatTeEs
ee RO | a
—————== -SQhoss aie(eeefooapicidedtical om, tolemnseaLyant
— ‘ e iD —
ld)dnl =FyqeVADTeTe: jo
TE
/
Cons St =
D_Goriag ana urd, sd ooasS04! 1 (a
fi 4,dQ$=(G2) Vissi tgFS ET
flaw odd: ‘ &
Da
Sda-f =(GX): Sahat, FGSSSG)
|J
=(AX) \okdh,§QG Autes eet nn Oa
a @Asdenmakurcl necomwhkosk wil:
nd “4s
2 GrredFe.SahdheF
t
ee —
Q ; rare fat!
fa lbind-= aes
i
a1
‘
Pima: 4* aAf
4> GD Ree
RPaNSPe Er aSs T :
S=Sado,Saz,[k=wt=ms+LEER~2ppal 8k.=(pj-p.)°
=ever dp,S\4.-t= 4DEEa!~20,00Ba
2.72B+Versialates coyd~) (ee
Gs FeeNhSkink 4Rees—eq2H3UppaHrela]
=..pe LHD SinescoscadetevaluedRP ERTL SRTSHOP Gane)
Ae Zomd dio Qa“Wkvasa! dA2,and&
Naowdty :SeiJas-sin(G4)t=[20-y-eelsDegas at
‘ ,
=|petR-eevean] =[+rates —teaa) |Ro
=_\-k@aag daw ko4RGRe)=SFB QeEy ,a
¥bd KE,Sea)a
:2
A
ska) =USAGA pe=2sia
=asd) OO; SsRiesike)f-
- '
|a
|
Al_omuy among is. baconnsa.?ee a
a RL A3ES GT
A =o, po f and =k— QI-4/s_, so
G
; i~Ce, GE)
t
"AEE =dieGO 5OVag Te1
faa =iWGEe EH Teo
$
| ®Sromap ds: dow/HY)Socks, S:4XSy=Cry=Uy
Alan, disc,TRS=QiAe? nod oanodylic..
$lbowl=apie Jpe aGANT GA) GIL|
| i
t
i
‘ 3
y
emonal YEfasts aikonatySSEee
Qnorchne: : i :
OTE =daTes iyaa ee =Dw Te .
aieCRUG: PEAT EETGS Gt)
ZsS$+3(%k-2)+8 4%ape d4593 <u
= dada, =dhdhi- Soe
2 dkdt= PrPad dad y
A
we
ne Yl |«as HG) =iOe SENSS GaGa)|
|
@Trompe: inWh, WOE =HC) od pd 2
1Ou.TeGal=pet ee TTT
pee fae
(hue:dad Skydet(se Se
ee
¢.
,
(hedouble valusdren arid” |
or :
iad
R(eosf =) =—'—SC45") +SE8g~ia5%) Sugt MAD
= de \S(d-v5\ +3Co- ;
Qu = wt As
— '
t
i
tT
‘
}
ee
.
,'
i
cr
° °° *
“
Sas) = S@-e) +8-2) |v
t ° ov
—___ i= fey sith =eG yOa;
- * 2a 2\-e%y
NOKaSore OS CESRESCEO)
oe = 7
= \oddq. =. :Bla-4) = -° ve 4 ARPT t
a i ca . = A
= .{
=ETETS1SyTPGptg SZ
y “4
AYNoppsadss (rok: x
=a: de 4,02) Axd=dade
_
~~ 26.4) :
x aw, |S,
“x= (peg) =ptah+2pq casoj=(Q2g)=pag’=204ce
ey Ot .XU= pra =(Wisasks9.Ys
= = oq cose Sua
Lyvalues18. D|
_ few 9.4) =volumerdain =[amy
QC) AG9)
Bub: Gay [aa af eya@ad =Jay3a[|fathead Gea |fae gee
=r}~a@yo~iw =\dAA
=lor en qenphtaiaph =gbaph _\pe = ote
GRY Ga=PY Cay at apt)= .pape etyaftlaph=easytp)=leay-2)
=\*ter(Gua: T=apuyyare =JyTe a
= reps) gy[ES=
alee west(f=Fagg} Pca) id
. = ee
PeBeat RSSST aR IRIEGa I
.es q . a. ae
en
A Ba a!a sothat
a ee (218) HG,By8)=HOHE BEL+2+ORE i
He IVErepresentsthe,arcaoftheplanetriangle,whosesidesareVE,VE,VE. iz os. Inthefollowing: weshallusethenotation . ci H4 4
‘es, eu HEE8)_ante6,69 |
‘*
Inthe variables: ,,&04.(2.11)becomes 1
has % 1Ma. dat at ae.w 218)(atNe+1)BO,tt)FTE ,
\ [. aa: 890 Sq
; \ | -fosadteafasfaciae-2me etoR(wisrmer M2,ferme). gy
“ y \ Onecaneasily ‘verify thefollowing identity, which willoftenbeuseful
*®Q 36) 116,§4.£)=2[840((@+9-8)Ki—9B, :
3 é
ol 5 ‘ wherepisatwo-dimensional vectorsuchthar De—12).a ( Using (2.16), eq,(2.15) canbeexpressed intheweetul form
' "
ae. \ 1
. ' C217) mbBNE HBO, fya) =EE
i ' .: ; Bg Sgti| ar.asieaefahra(so,Oe ee .)
at [ ‘Aninteresting pointconcerning theasymptotic limitistoconsiderwhat; happens inthe forward direction where weknow that thekinematical condi-
ee r ' i tions impose that Y=%4—%,=0. Ifweconsider thekinematics oftheprob-
oF a 1 lemmore closely, wescethat |I’|<Val. This means that ifweset#00
"iie1 before ¢->0 theintegral equation weareconsidering nolonger contains this
42 {| limitation. |yk i '1 Infactifwetaketholimitfort-+0ofourasymptotic equation (2.17)
vO QuaLeSe8GQo) ==a . |' poe Wy. ee) ; .@alone =o. SOye.
PPcseT Raat eameeeREE WT Sean ER re hieTeeredasSrMeg asTReereals SethOEEaisPRCeIMTSaBahae Bewth osMe SPESFhhMMBERpeaksIalgeEs
=a RET
s, in-
,/
¢ i
AG=duel = OxX dk,dk a Ss
a crea
i
@®Moa doled do consort bp 5
‘ '
Qe: AG Ak)=At ee DKG
3 G
i} od .
AB=OWGG e\dx\dyCUSFES TSGewTsay) |as a eee
5 : x
@Consuten :\dyOES \=VayFO =HEeSo-#50).
wo 9 Vetee® 2 *yen
;©
Av=tG¥GG% VaxTGXOT Ged\S ray
Quik =so> tet = So K= 2Komd dx=24k
sa::i) “ z
SS q \ geeTS Es eT
i
1
i Woks 7eae
Bk gemnal, aShaasksZeBO
On:
ol Hk Bs
a
z=e\+ Bo Mans A=ACs,me,ms) —
« %
;
| Mai
OD ©Om AS audusame so: Ot Ass a. -
p= (1+. :
_ R= (\+oak) +(itat) +Cirati) -1-2(leat C+au)(rat)
le £4On ot UM) Vf=Bateh
~ H=Mad (Hy +tter tt) ~!203 £44,
;
_
=o Alby) =AeA, 1
_ =ot & ~—Qaktke
Qhessfou: R@a2 =FS)1)AGt,4) -SBkha 7 ~
© Nai_gp totllsquadnnonslyA>AGntm*)= s(s—Yeme° { = Use
©Nde —
= —
\_ as \-a+(sh ——
dnsquad nao Ws2Na A8(3Ut 4 GA)— "PF as
t =e —
| aeartyonan {
SF_6Shoanguior says.thesasme.jai,ges lsae(s(s-4d"=55S=Fa
— "=E€gghak.VY Se)—Chasy—— ~
PAs =Ala) +OSA)ki
2 2 ee
ee
—_—— 3 9
hou peAsn my/2fBSo. ence_omnt: wae SSesAtSy
Sn a
ne
~ woiti.W(a22) osm S|
t ——
= \+QpyS +CkKE*2G)S + “20 Kung
7 =(l+axsbe |=lean wars Kathe AQabx
GS ESRESQsSY 7 at OCS=ei+Ubs +4fkZ* G\S"wee
\
@“flow, roweed 2. i
Grade Kat
a OY SS
+
= Ls (Bb+HHW)S
2 +CARIN UGAREFGAUG_ —_ESTSrsO
GS WUAZ+EAt TetCG6+Utht-.) S
ar |
— ——_ +0hd +GEtesb)S)
ao A s
— — & =D=A =TRC EAE CACO COG CHY)S|re eee=lee oS +08 eb 4eUs"atkt) TT aT a a= = Ce, A a
|
ch
Ag DUNT|Q 00 Lee Wa Mshy YuhALD WG AN_O WADO, CVI AD OM M3Wy
S 19why im(no _onoxitordn AS) akwast!d _
See
ee OC Teorevod leeaap davwanin of——wiiegastian Sinoobse,umcheb$a pry
arepDinde, vowuidaa, “oo3-90JeKO.
ny ; -Onjock,QeRomesgeha,(roonerettydannaunwi,)KroS OF
a
do...
n
y NORMALIZATION DicTLONARY.
=44.G7 Qete :Sokea? :aw @ster=ceesteepy (~acc beyse)abit
TeLOTpd,CHIR=CyS-B), OyeA,Cn" l= IRDOlu? =03a)
p>=TEatt ineiHeex=Se40,4 Cy)ThSent) Tes <PeI TPB>As=GSient)- TTXsdy Xeety+RRAONTHeee 2Sut=Wepre abetcbsmel.55lil.Gee) Sy CatiaA(Tae . + 2% =Yeftae|SeTRESS A(TE-Te) =QZaTHT oAA oeAWSats[eta a =QomTe WwTeste
s 5 « Satie y ".¥)(8:gdee@deGEh)Te aSa={acutacedbateaanee CAN,ae75Cee oh Fle
Fiddtun: =Seelacscte"] orelane,bavetl=is© FAy . 6 de Celn>RuaT=am/le Coe1,me,2E ealemalworehncbeaiy wtu =viv =26/6 oa
ap=\BesLeue”+ dtve"],cosVRE VyY=Jes"uy
Noles <pLPD=<aLatin 9Tach] acea?(ep)
Tastes] aute STIR? ee
| “Coneninas Mee +GitQt ae a |
Sakewrat Ms ~GxGave VER Year: Aw
| 1Kallen sind +Qa@ukve(Feem| RjokauBal Wee —Cw YeYPREGAR flees me
'Yoder aed ~Qa@he RE fer |
| {|Howes The ©Ge@lke YRE~Yom?
| *Glapilalsn —Sse -Gr!yefe 3
Raa TF+mt@a Yeon? Ris1bese ®CivarSpian Gis) GN Re Ye =&few
ive. Sms 4 3 Wiliams Tse +(onGry 7rh Ymvolumesofbot Gasinowens =Tee -@ & \ Va
WoxtrisSpomen <Msid)+\\t ( ;
a 1
‘osksar. Nona Frnnuale ikQs+1SS_——S—SS—_—_ +
20a SD
=3s ae Sd
opkcal » RwT=A=VA-of
a)plawasspaces: dQ,=mba.) TTSFpeSGy-wHh & ary
a Exc ees Vac
do=—\__p dk lorsA
@Highauosan: QkAG—S. !{ $
@ Wwpores. cme prone paca?
dp,Siem) dlp,Segiewt) Sle) =Fg. S
Aer Ady ushooratum. Vaasis b= S-
$ 6 y
fi
'>
T
a
fi
a
P.w A.
--Vuvirfomousd. Jo_)Dasdnam calewotin -
esOMpastaindSinrdattfadOneslebee\ Dare
OO et
We.rasa: KN?=Q\e>- <x"). a
—-a ABC SA LECT SR) _
- ~-=Bile). <eer SO
Rife ooPath, Pf =Pe.
ae T= 2068) 2 Sadea DL#Pae-
ExeSeay - -
e2Seomaegh= JSCP-LS ce-w) .
7 Danuta: 2A726 Cn. Sle-e)
- —=&SCR-R KP cheche”
@Meoinamsory wai.WeaSraadigkanadJeotderulrane hak; /
VatSGaly.Seetmg) set)=AOaota 4fs
©
@ToaYoke.Ge)ondcomulhbotasrchen oydp,Shag)dlp,S-mn'} Quequae: - _
| ES26CedlpBMSGA)AMATCm)SHB
Gy Bhat)cale’>ap,FOEwad)dpapdms a
Dor doeB ilaspote onVathsdeatoge —
tray 7 EGE OSSERR) SoianeiieeiielES og)aedSSCCDSea: ei) Sea a
r.4
—Aaae eee __
yaoi. & aa vs]
Qhuschon: w>=dis S(R=)
a =20<>
Nelay=aglietcehittleet sacarb TS
~
~_1ORERED ¥.SbOg )_-—_
SONNE HED Neg Ne .
=WearscraTNScanaaatigs eeap =_“Js y=.) __Pt fi .
~~@Gono) sand: — sce | —
___ URE2partantestake. |
. 3
_.Gheeoflmaim.oredireof<elsM> LoL
“L171 @ipd=SEsco)» ‘a~Re:a““ENC OT
_.@a?= 5s@-3') ae— ___? oe C3 a _ _
@ Weolse ual,bower, Mook —— - -
gesldey =BQ) Le oS
\ed> =dos (Ge-d)\am>-4 a
;oene °own tee—-@Souporwion slraue God,a a _
~. i=kleod|Dex BO :
<eig>=38Ciims =2S ae Asoy —_
a= > ajo
©Vader,
eo= KASD I | —
-- Sml@e ks bd =kG _ _
oe 1Seefeld « fe .____2e|b=aBeeea-i=,feet
@hus. ~ : -7Sa: Bet>+
TO-HotEES|=-
ae So 3
—@ Shiu j ;
———- fa ToeMea Tal39
EE err OaOW“Sgn dichuingnssTe Xe
—________-_ wpe ——— _ _a see=_crioa) TOE vs -_
_fe)MinddcatashanQak: A=230 [geosRaw dslonemy - a
\dQi \te{aa
—Qa, d-omaar, Ties GoCa. LeTTj —
m4 i :
x4 as
Tide=Can YdySarleoremlTIpypES “ae
——— Ste A ee
Es =\e>=Whe! sho "TWslain oA} ;—__ Dat_ounas Vu Noaneided Mar {dow, CsMraSideyla_cnee Qacheneaa
a . ™~
a13
<;oeSAXeS _
—[QmarlTs l=1Sanh Thyp2arl +Key Su,
— = Th WS. Bidinsc ee /
EES wo
|Gosling farguar
"
5— Re
Sux * Tr ~
rn =
— =Tr (i as 4 —_
—@ Miwa, DeoOom aa
ae) =a ss Fe Taal]a
a Ipoene cmalea freronal 509rsod vaula? _
.Li_ Alien pels 2lds}sa=(Sal=>Ws min T “ H :
“OA \
About fassstaheef
ne A2<g.m,42mnaude Pur42,de; saon> ~
=SOMA Ary! ©SEEM @.©BCGicps)DeSAMSanh
fw Sina PI)=HCCC) unbem
Tt"
r
: =SoyV1S8fejzed Ge
J 1 NY BySyy yy:
. . i
2Yl 4-Ja=a‘Ce-@) ainaeiq Uebell
:cec eh ObrEs: a
AQKinda,dS=By1PRSOR) Saw _= TI
0 i is
30, Zahn, SS=! Sa Sale
DD Wencrnballassceaty|R'™NY cookoust4Xtandan!
OAC
-2= 4
_ @Was AD.Oa%esonotun2 (Vico pooss: co™KW.Rack
sh samablacore ° :
tr» UV
Marky eit fatheons connect ommrOolanse sQatin »
Que tition nora Uumoddl let+
, a
roJ =+i
0. Cate\Sgla=<Hbleai) +idsCela:
}
fe)
THE Mes
. m=baby*
3 i
|
GDGhos Qno drial cra ochre a”
eee BR
ao ~35 4
O_S Wets Soaeees
a
CS
—___@_U_SS=sS's1, ondAS=iTGlowyorof
at 2imeeNT
—;
2 dnS<ind| =1 a
—
728 ited—
Ea
o ~~
Se leme - _ a
oe = A a ! VK Wh, 2d;
oe =A GG belTelriaMSye
Dhan, ®becomes
7 Vines. QhokGaianooaoguct wdkses =
OT
,
“ 5
.0 Uwde ‘
|yt
. [|
acts ace
lear, Gro slate TMG Ws wsk Pro prmectrm oA
mwSs)=SS
Nokoo Gaack IMva 1 .
fovea aquol dpa=U~Me = Grusw slateeg=oo.
@ Nad Qeke on badex oyamune umitondy dA Sb wOo "eg : "NOAA Ane! @\
“ T 7 a
Que: _ a
Fyjg) +[xem2a\Selrwmasl=t7| cing: Se
. ee@Puting ododyde | =2eld=-c|me" I
Og=oh(ae)ns) Z :eS.
| { :
ic
_
. (yuu: Cin= sey UeHe,
{ +e
” U-' 5
Portia! unecurclivaie H
i. @)SpjarVrouw Draws Mes vnoun pouk oaendta : ‘ ¢
eS
Te=we(ores) \invgl |eos
t= OeFQCi- nyes(2ss TTbe S ,
@Drouin, Qhatk rae Worx ==ieSclash — i
Sb: '
:
.;
5
@ \bwrtariti, roctichuna on pbuckrl-amo cas trrchos |:
No aso Mork Wy ofudiata 1Qneonsen ofro LowtreMasnuaLornmit Ad
:1SEO,Heatavsapies
ue 7 a
irO=>W=T FZQyn) jp
Tm== FYOy) |
Tet =Br (av+1)
: pee +
- ad
- —3T 5
patized urane comosi :
4 @) Owlonpast Miiacoault :
Neg eeose) P|Se re) |es keS
=Te29. Sotohe wiQoted.SoaMeetotal| Cowohoar,
.
§ Q
-
n= 2asaj.sivR oe=A&R
Bit,unavods, Oiadiscsroeashodnd. (hatGraded |
Seaslenix passat: don oatot aomceldatim wills She,
ola. '
ecs (un: SmSO
GM==I Timea@ijal wee j -
[oy=SEZ(vu) sw =gf
: :
aay y
TS YeeamosOnneuebeigCinSPge)alsette2!
[oeGrow)=a(a5)J :a <S
Sy=@90", Qian:,:.3
: Los (wee) =Ur(2541 |
[SieCaer)=YorGS).
)
’
|:
; = =:
: 24 ne—
=i =BE..
;.
a
: |
a
i
ee
nv vei
Oe
i
. {
t
nN
coe
:a
- 76> 6
Ss= => =Tt=-3(7-1
H
stalsa oso camasard so Gack |
aon : dp: +
\atas AQ. = sext). [LSCeo- mt) Node Guck Glaose
io_oon CaS tmQe Ancram oh Vy oss rca _(2r\3_ inn
- Ngsake vvonawa .
H
@Ghus, oemsal onaubarctiy a
4a ; Ka TRS=-2|Td —Git|.| F 7 :
AQue someon S
=
1
@_\ usrnsatusk 1aamd be teheVopiin2on shade stedon
Teas =+2DanLeeitiayl |
ey rorlonduy= only, Ardirarcked mikoni jafasS~Dacrsmoll
g39
a
>
4 7
|oo -~7- 6
Aacducrion,fogasgnlan wibsgaad
@oSojoe Moe
“at =RG aeTS — deoS 2Sand GeSETEOTELa ee ¥dp.apy—
oka Set) SCR ferpt)Okanoruaeaadanes.bo:
mLDeets=tZSap,Slgems ene FEA VEL)”
~ muck... an eeht ere
eee ~~
—owSaspedataLem] e8-25-etsar]
_2Rak 2%Ad, 8le-28h aeae wl] ©
“ee %=Te Med gk
= peRRQa -_= —-=:
SaALe_coneonuatuan. So. ee
—©GoadiffeSeonmuyCHS.andabyGeesepaced Gx.
=ePeeekerhs eas=
Tn eS SEVLTESTE
c —8- 6;
Sedu paste) waieyanihontty WS
amrea=o =\ag\a(ead) (3,6)1(s,6" ~
xads
Se
‘S2)=ja 323)mestsG __ee Sn,
x 8
oS ee a itasawak
QM JoSak a2 i aah KA. r
tre a) = el=0d bnls=2bab.
a's. cy T o)
a
- se P
~@]&SawnWS.(0)2oomB=AMSmardereeownDek
Tt)=TearGat)SL =eeKegcATelooab)
@FiomMS.(iu!)aalsefom.~3~useManseDaleropptasion:
.Lagsd\Bloat =GelSPE 2ssi)BeCoyCoal[a6
——@Bathing Deandagen, weSi -
ast)
=f& OETAOsaAre) Toa(S| > GH se¢e‘asCoss)dyeco)“Taaau(9)oo
_ @Fastest fiahoon
a Tet)=f&{Sass Pleo) Ts)vy.— (Giggs a
—--@WienonSe=FFTaye ahead Onteeaegck:Copiers)
a SQ9= seeZsryPeles RI. C4an\.
~QkitSmoshoes,OerSewdoad 2tackn.cethapaatsal
--+GyhwoAsoeeme SL _ =
a A)vasaTsodofan= 180CHzg)s 2a0sloudandiyed nap.
oe 8)uaeTao TSaceoeheuniteuicle. .
OB ee
tT rot “ye . get es Fe
CE RRC ae
ao oT oeensaesnn
- a -Hat [aecgaia@oae@\—
Stat ROMS
-MadeOaat.phsicod:anpgeo,dmb)=O,Dhsuikanitnikagd |ee cSoO_----Mea,jpaty2,aedC38)ofoeeonlenaQaik—
ON aROPA) =SolSERe)
_dee ESBatWatJI-BF008(4-@.), ChuaaayHin,
TEE TT a)0|aibyouoblewaeG=lrg oarsa
wee ' peeeeogFygh—_ BakeBELato TTiilgdoGSatedCStie?Niroaliayonekbe
2Ceotl)Rano Orn.oupirakinn tha(Beg)ad Heong AaB.akat $e
t——eCTEDionanteA=Re?faBe
_ag Rae ak fe
D8 oy ft poh weeee
a“ae9Btedkslige. cobalt,ahanitdDoe
Ml C=Io,okLaasshane,¢esemallWatbakafogenoiae..
See foot weaie SRO
Qala SayetatamaetonYadREL. eeachat)
Gt nena Ansbn
SS—_f~sathifookMG)auOeonde.sedg aesac“aaa=Tato Ba)_.&S=aang=[¢6%oa fo-
Sapte enytede
~ ster:Thaaiae
i cedTf———-a
Hana 64
hs : apthrHysICanReviEWVOLUME136,NUitDERop26OCTOBERi964|iJAnalyticityPropertiesofHelicityAmplitudesandConstructionofKinematical : Fa Singularity-Free AmplitudésforAnySpin* :(
YeHat CaliforniatnsituteofTeebntory, Pasadena,California 5(Received 11June1964) eh1 ify
“Theanalytcitypropertiesofhelicityamplitudesforbinary[tonofparticwitharbitraryapnaae H athesearecorrectlypredictedbyperturhationtheory;(ii)thecrassingrelationsofhelicityamplitudesneat, | if poliohesubdairofcetteatodindsNoepease i shrepro treamoliudeforanyspinarecontractedbylmodivieghlicigaha,eel | aeheigkaeaealiedtoarbitrarypin,Heltyamplesareprovedfoateeaeeeeel i ¥atthehigh-energylimit H | 2‘| 1.INTRODUCTION Forni!scattering®®. | i” "HES-matrix theoryofstronginteractionsisbased=i ey i 4TOntheanalyticitypropertiesofsatteringample_f+fsDL2MA+ WMy9B, w+tudes.Amongvariouskindsofamplitudes, thehelicity©f,,_=:in(0/2)CWMA (11) . {amplitude introduced byJacobandWick?ismost +P30+.)MBWA, t convenient forpractical applications. Untilnow,ifone a :Santed toknowtheanalyticity properties ofhelicity orWYscatteringt | !amplitudes,onehadtolookforthelinearlv.independent- WetaSe)=BG2p'Getm'G,, { :Lorentz-invariant scalarsbuiltupfromthefom{ect )=(BGmG)e—ps |“momentaandspinparameters oftheexternalparticles, fretfir)=(BrPG,
!:thecoefficientsofwhicharefreefromkinematical A+fa(If, =p, {‘singularitiesandsatisfy‘theMandelstamepresene | (=coss),(1.2)i tation. (Here, thespinparameters include Diracede toa Gh BG. ! ? ‘matrices,polarizationvectors,andfermionspinors)3L(+Sma(2)Jam'Gy Ge, i; "NPThen,onehadtoknowtherelationbetweenhelicity CU)fora—-WEG+G,), (y=sind). 3 . amplitudes andthesecoefficients. Thishasbeendone i: | j :forwz,aN,andNNscatterings*4andtheircrossedForthepr—+NWprocess! |{ - reactions.‘ Formorecomplicated scattering problems, a| : 1] 4prescription forfinding kinematial singeketeene1feat294+2mghcod], ;; :amplitudeshasbeengiven.However,itisnoteasyto San.2EQBsind, (1.3)|.followtheprescription. Forexample, itwasdificult
,
5 "1 evenfor179scattering,‘Therefore,thisindirectmetliodReadersraefromEas.(1)to(1.3)thatthe!3 1]willnotbeusedinthefollowing, Instead, wewill™0di icityamplitude,H; ‘Investigatetheanalyticitypropertiesofhelicityampli-Inaarerses[o0s(0/2)-™Hl[sin(9/2)]--a1 Hi .‘tudesfromthebeginning. t xseyp, :z 1}ASwillbeshowninthefollowing,theanalyticityDahan, neNods,40fort a91: {1Propertiesofhelicityamplitudesarenot80complicated TheMayBs" -i [f(rsimplescatteringproblems? jfMdActasiseven(orodd), |g fo—— satisfiestheMandelstam’representation ifweneglect Ié Fcombetk,supportedinpartby,theU.S,AtomicEnergyPossible(kinematical) polesats=0andp=0.Wecan:: ‘mmission, ; ib ed ibe HEQndueofabsencefromPhynicsDepartment,TokyoUni-getridofthesebymultiplying Eq,(1.4)bystandp* 1venityofBaueationToker‘apesDeva veGandZganbepredicted). ©pmzesicee BEMBIAG:FCheaCompany,LTRS,WORe~Theaniplitude(1-4)doéenotsatisfytheMandelstam % ey Seale pany,Tae, *representhtion ingeneral.However,itwillbeshown
, 1)2bfoeebandG.C.Wick,Ann.Phys.(N.¥.)7,404(1959).thatwecanmodifyhelicityamplitudes foranyspinin
£viHayeew106eTEEBBTowand'Y.Namba,ohaYaythatthemodifiedamplitudessatisfy a5HEYCottverseeME,Gris,8.wr.MacDowellandMandelstamrepresentations. Alistofkinematical :: OKMingPsTeYayaaliey an(ivy,Singularity-free amplitudeswillbegiveninSeeVILL RAC ateaen,NinoCimaneaeae Makinguseofthisresult,ageneralizedMacDowell
. nic slees alopp oedeteoreciprocitywillbeprovedinSec.V.Helicityamplitudes @saicyamidesfeefayGOeoneeforanyspinwillbeprovedtosatisfytheFroissartlimitpresentation, athighenergies(Sec.Vi). : 4.
BSO7 | EE
H1 iF L
hy ; \;
i
' 8508 YASUO HARA
i It.ANALYTICITY INcosoANDIN¢! systemoftheschannel, 0,isthescattering angleinthe”
a Inthefollowing, weassume thattheanalyticity ‘tterof-mass system oftheschannel, and
i properties ofscattering amplitudes arecorrectly pre: = (s+mge—meie dictedbyperturbation theory.Thenithasbeenshownt pam(otme—me)/26),iq thattheamplitudes ofbinaryreactionse+b4>c+d 4Pa[s—(matin)Ls—(memY(As),(25)
:canbewrittenast j i; THEBANG), —4Qa)HORLALE LEmet(mdm(me—me)] :
ig;1 xspp. E
a where Tisrelatedtothecorresponding Stmatri> a {F element throughwe corresponding Simatsis yy(2.3),0(or1)ifnoanene=1(or—1)duetothe aePinvariance ofthestrong interactions* 3
‘a (ebalS—A1| paps)=(2x)8'(d.t PaPaBr)/ Nextletusconsiderthefollowingamplitude: aai XCespeveErAT] 2), CDeo0(0/2)H™MLsin(O/)H¥-PTra adds iB BA(s4.u) isananalytic function of, and and. Nedende andpeededa). (2.6)4 safestheMande reprtniation A(v@)0 gandwedehd. (26) ig polynomial inthefourmomenta oftheexterrial par- (Chisamplitude hasbeenshown" todenend oncos6,a ticlesp;andspinparameters B;whichinclude Dirac butnoton_cos(0/2) andsin(0/2). Then,wecanwrite“a matrices, fermion spinors, andpolarization vectors. itas.
“A ‘The expectation valueof(2.1)betweenhelicitystates + we : dinthecenter-of-mass systemoftheschannel (helicity Thaaren’=E By(s,h#)[Polynomial inp22
;| amplitudes) canbewrittenas 1iE /2,pp’,andcos.p*orpt :TrasseseE BAstalEPolmnoialin 82, persodextn#54 . 7 \ xTLCoetmds, (27)_‘i 2%,pp’,sin(0/2)andc0s(0/2)IC9" of9° fed Tisansh q Hence, wehavefound thatfor: reals,7”isanalytic. ,HLCorte"! (23)“inthecosdplanewithcutsontherealaxis,SincecosdA | isTinearin,wehavealsofoundthat7”isanalyticin‘A Inderiving (2.3)wehavemadeuséofthefactthatthethecutplane(forfixeds)_withcuts!(twin) and?+H helicitystatesarelinearcombinations ofdirect products(—@,Dim?—s—tais). Therefore, wehaveonlyto78 ofaf() and1(p)(\stands forhelicity angyisastudytheanalyticity properties ofhelicity amplitudesi Lorentzindex),where. Hasfunctionsofs. 4
‘it e1(9)=(05cosei,~sinf)NP,{:TILANALYTICITYIN5;GENERALCASEa4 i c=5Psin0,Pcosh/m, ‘Thusfar,wehaveonlyconsideredthehelicityampli- a 1*(p)= (0;—c0s8, i,sind)/V2, { tudeforthesreaction (Taay.a.,")- Ashasbeenshown
oH yya(@)=L (0+)05(0/2),(P+)sin(0/2), ¢ intheprevious section,thehelicityamplitudes forthe FEi pcos (0/2), sin(0/2)}L2m(ge-+m)yy",-Lteaction(Taya)ssnokinematicalsingularities— itAre,insifitisdividedby ; i waa(p)=L— (p+) sin(@/2), (p+) cos(0/2), « :oo psin(0/2},—p cos(/2)]— st Leos(o/yyrrnifsin( O/TA.:} X[2m(P+m pi) Crossingrelationsbetweenhelicityamplitudesinthe 2 1(9,sin Forexam ‘city3channel,Tsy.s.',andhelicityamplitudes inthe¢
fl ‘Thebectyamplitude, Tapencnyin(23)isrlated884Wick"andbyMuzinich* According toTrueman¢ totheconventionalhelicity‘amplitudeof‘absandWick, tH Wick?through —oa Hi *91isdefinedinRef.} TracrasmDCHfroaroade/a-a| 19}24) gtaSFGadus£53Low,©Maryaad ; Tn(2.3), pois theenergyoftheparticleiinthécenterASRS PekardECintes(tobepublished); [: of-masssystemoftheschannel,p(p")isthemoientusn CalogeroandJ.Charap,Ana.Phys(N¥.)26,44(1964), it oftheinitial(final)particlesintheonerltanues ClaspTrenandG:C.Wick,Ann.PAS¥)a2y TeGekpo (repos—telbe-pdte(n— pdandTaeooRRS reatbe r i, 83(be=(Detpad’,(=(de—Pa)=(popa)and ®, onMuzinich's relations can giveninthe
; PTRESTess aia somaiey. a i
Wang 66
1
_ Phil's comments onKinematic Singularities March 11, 1977.
N
@ Temthinksne intermsofLing-Lie Wangatthemonent. Normally, youwrite
; thet-channel helicity amplitude asaJWexpansion inthe t-channel, andyou
___ observe that itignice. to"pull" thehalf-dngle factor. Thus: _
~ oy B -8. ae=Leong) fe)"at(a) gheyRx(eosSe) SE ee ae
. See (TKews) _- L-'zk~ =7
Thefunction f?“defined inthiswayhason}ydynamical cutsinthez,plane,the planeyouuseforgoingovertoeFGprojection. Thissamefunction, st) 38cor ~~thesamereason analytic ins. ~ —
Hereisanimportant point, however. Indefining thefunction £''*)(s,t), we
haveexplicitly removed afactorwhichyoucanseeissingular atthet-channel @errerhoras. Ie, atsuch thresholds, the p's inthe denominator gotozero sothe
factor which wasremoved blows up. BUT—-nevertheless —-the"cleansed" amplitude
~ ~ltS45stillcontains tilet-chennel threbhold singularities. |~ 7
-i“apyother words,byexposingthehalf-angle actorasabove,youhavenotexposedythe t-channel threshold singularities ofthe,t-channel helicity anplitudes,although thehalf-ange factorpulledoutdoe’have,t~-channel threshold singularities.~TheGésired ”singularites areactually contained inthecrossing matricesaa wal,
ay -o- kg -ee Ga =MoySadcad kex2)
; . _Yvte oralenLnnn. cmala\eioa GEL.DosesSpans waded bogk
- 2.Thefact thet yduhave toinclude not“only theh(@y) butalso thed.did.d factors
~explaind“\ihy myhaive FFoissaré-Gribov-Projection with spin Method didnotgive the
. cdrect answer!Tf “YouStartoffwithf-channel JWexpansion} youpulioffthehalf-@- selefactor thereby getting analyticity-in a,which ablows theFOP.But,youneverexpose fully thet~channel threshold sings in'the discontinuity you areintegrating,
80you donot "Learn" the threshold behavior inthis manner.
"
: 12, FRONSDAL'S MODELrorLEPTONIC INTERACTIONS 1187 E:FQ}|/Theeresultsteadustothefollowingconclusions : 3feaet aeFAbdld Econcerning thelowest onlerradiative corrections to Fi€.S.Diagrams peo- i ,portbnal to Che P iymuon-decay: . oeFaybagtbataw EFH. (1)Thereisnoinfiniterenormalization ofthedirect 1 :i:term(oroportionaltoD).Thereisnorenormalization Whenwehaveonlytodealwiththedzectfour 4 iCe fermioninteraction,thelowestorderradiativecorrec. b {(2)Thecoupling constant Cysuffersaninfinite ‘tiongivesadivergent renormalization ofthecoupling§ ;irenormalization, constaht.Also,afterintroduction ofthechargedscalar 3ae intermediate boson,wegetinaddition toaninfiniteneFheceaacmparethemodelproposedbyFronsdalintermbdiate, bosor§aninfiniterenormalization ofCs, E*H andtheconventional approach, wescethatthesitua-
contrary tothestatement made byFronsdal. Sointhis TE pacttioninFronsdal’smodelasfarasudecayisconcerned. caseFronsdal’smodeldoesnotgiveanimprovement of.
4:iworsethanintheconventionaltreatment,because cofventionalapproacheither.Thisresultcontra- ES .oftheakeaen® OFamInfinitecoupling-constant 5%Fyonsdal’sstatementthatthecouplingconstant 4 4 ;rn
renormalization isfiniteinthiscase.. F i Mi.§DECAY OFNEUTRON
3 ;fWeconsidertheradiativecorrectionsinlowestorder 1ACKNOWLEDGMENTS
. ::
totheprocess ThefuthorisespeciallyindebtedtoProfessorD.R. a »nophet, Yennie;whodrewhisattentiontoFronsdal’smodel,for 4 otWeproceed inexactly thesamewayasinthecaseofmanyétimulating andenlightening discussions. ‘The .a fithedecay.Westatetheresultswithoutgoingintotheassistance ofRaeinthechecking ofsomeofthecalew, FE +|,detailsofthecalculation, lationsjsgratefullyacknowledged, -TE 1i —. Bi:
|MYSTICAL REVIEW vouume142,wusoee« FEBRUARY 1966 :"] GeneralMethodofConstructing HelicityAmplitudes Freefrom: P Kinematic Singularities andZeros
a anLane-Lan CuavWax, . 7 “RGLawrence Radistion Laboratory, Univerity ofCalifornie, Berkeley, California2 all:4 (Recved 16July1965f » TE ‘
Asimy.
andstraightstforward method toidentify and ‘thekinematic singularities ofhelicity . ‘a i amp cee ened matbdtodentroasngvelalon, Aastofssploncefeotanea” : matic sgularteandzerosisobainedfortwoparticleewopgicereactionsanyepeeooden, a‘exceptthatforboson-fermion interactions withgeneral mass‘assignments thereisstillakinematic‘singularity lefeintheamplitude. ‘
. q '
1.INTRODUCTION amplitudes andthenanalytically continue theirpartial q 2Samet . . ‘ waveheligityamplitudes withdefiniteparitynthetotal a 1JNdynamicalcalculations ofscatteringamplitudes, ngularomentumplane’‘Thereskinematic-singy- ; : riceaeeeea, 0,weKoematiesinglarin-feejoeyomentumplan!ThereforeMoemati a iGrblitudes,whichhaveonlysingularitiesofdynamical—latitv-ftehelicity.amplitudesarenotonlysuitablefor_ iriginandsatisfytheMandelstam representation. D.N.2calsalultcabataliafulluble-lo- Renee {+Williamshassucceededinconstructing acompletesetee einotvaticityamplivaenene Se a f “invariant scalaramplitudes freeofkinematic singu- we yamplitudes.frites andsuitablefordynamicalcalculations. How._,Recently,¥,Harahasproposedamethodtoremove tnesandsuitablefordynami ization,theKinematicsingularitiesofhelicityamplitudesby a {_Ser,hisamplitudes arenotsuitable forReggeization, the kinematic si
P . 5a‘nematic SiNgperturbation fieldtheory, withemphasis on JReaseiae,firstwehavetoremoveallthekinematic thresholdbehaviorofpartial-wave amplitudes and :oo 1 SRilesteomthesocaleparty-conservinghelcty {esoliPlationsJnthspaperwedencenant E ionyanCogstruton offavariantSealc Amplitudes“GM.GallMann, M.Goliberger, F.Low,E.Marg,andPe = ©] ccd GitesCouuinotoSealeAmplitudesZachariasentVie.ev.133BLAS(anode©MAP f gsdealighSetterProneLawrenceKatonLabora:*¥HartWigsRet16HOY190Fareompasonof * fryReportUCRL-TIS- (unpublished), resultsandours,seeRel.9,
¥
¥ x
[email protected],discussed byWang.Consider HE)64),where.thetwiddle -= means that thefamous factor ofGs hasbeen removed. This function isanalytic
- intexcept, for usual dynamical cuts. However, itdoes have threshold and other
_. Signularities ins. aa se as
__-A.similer statenent canthenbemadejfort+)(s,t).. -
. —- Theuntwiddled amplitudes are connected bytheTrueman Wick crossing relations, -
-.80thatWang's equation (III.4) really saysitall.Allthekinematic singularities—ofFS) mustbesitting inthecrossing d-functiong inequation (III.4).
_ 2.Looking naively atthese.crossing functions, thedesired factors. seemtoappear —-—- An.the, denominators ifyouthink ofdé)=(osx)* .so.you immediately pickup
-.things like s,45. What youaredeveloping here isJackson Hite formula (26).
Jackson's formila igmuchmoreefficient, Jthink,. Obviously, sinceWangdoes2lot
ofwork, _youcannot. justthinkofthecosines. ..-. _. :
3.Notice that Ling-Lie does notparticularly mention any"partial wave threshold
behavrio" whereas Jackson. andHite.use.this astheir.physical..starting-point.~ Both
emethodshaveecertainelégance, butthere“$8.0deaying:--Jacksons is.simpler,
keIwould._note that,Heradoesusethethreshold behavior asastarting point,but..Somehow thewholepaperasobscure tomebecause, thereare.nod-functions ofJacob _-
andWick. lWnag claimes that Hara isbased somehow, on,perturbative field theory, a
thoughIdidnt“hoticy thatwhenIperised Hera'spaper.: --
5.Colling andSquirés basically quotes Wang'sregult's andthusendorees Wangsmethod. .—
_.Jackson andHitewas"not,done,butdoesgettentioned inCSnoteadded. --
! -
..-6+ZfindtheWangsolution rathercomplicated, justhardtoevenreadthe -—.
-=Tegults.IpreferJacksonandHitepresentation. a ~
: - i oboe - -
— oo . wee eee - oa ee
-@- —eee a -
!
i
‘
i.
.
.
Mek C8
a iJOURNAL OFMATMEMATICAL PHYSICS VOLUME 9,NUMBER I}NOVEMBER 1968
Kinematic Singularities ofHelicity andffransyersity Amplitudes and wedeAsymptotic Regge-Pole Contributions
{ Department ofAppliedMathematics andTheoretical Physic,UniversityofCambridge, England The
(Received24January1968) . ‘group ‘Thecrossingrelationfortransversity amplitudes isusedin'asimpleproofofthekinematic-singularitsrsctueothcamplitudes, both pentane angirene ose eyParson withearlierresultsismade,Thekinematic properties ofhelicityandtransverity amplludes areEmployed inthederivation ofanasymptotic expression [ofthecontribution ofReggepolestotheamplitude. ;
3 i - Two¥ 1.INTRODUCTION transversity amplitudesmaymakeforasimpler frame4 Thekinematic-singularity structure ofhelicity Phenomenological analysis, inspiteofthevarious singleyamplitudes hasbeengivenbyHaratandbyWang.helicity-dependent kinematicfactors.Sincethispart:Boththeproofsandthefinalresultsare,however,ofthepaperisforapplicationtoinelasticprocesses, th} rathercomplicated. Atthresholds andpseudothresh- Where'dataareatpresentratherinadequate, no whereéoldsthebehaviorofhelicityamplitudes maybeapologyismadeforthesomewhatcrudeapproxima- netrelatedtothatofpartial-wave amplitudes, andtionssuggested. . a omerFrautschi andJones*haveshownhow,atleastinTheAppendix isdevotedtoshowingtheequivalence moms ,certaincases,thisrelationship maybeusedtoobtain oftheresultsofSec.2withthoseofWang** andto actsacs. thepowersofthesingularities verysimply.Weshowthederivation ofthekinematic-singularity structure Pipval: thatthisresultisquitegeneral. ofthefelicityamplitudes invariousparticular-mass ._4Cohen-Tannoudji erai.alsoderivedthekinematic.cases.i ad, “cohen 7 ema
a 'singularitystructureofhelicityamplitudes,thistime)memaric.“SINGULARITY STRUCTURE OF 1an !fromtheproperties ofinvariant amplitudes, without s“HELICITY AMPLITUDES. ¢usingcrossing, andStappobtained similar results vi . - reaction .from basic analyticity properties. Reference 4also,Wereviewbrieflytheproperties ofhelicitystates ‘employsthetransversity amplitudes ofKotariski® ina_i?ordettoestablish somenotations andconventions. ‘nthewploy: yampli afKotariski i .
i derivationofkinematic constraints onthehelicity Single-Rarticle reststatesofaparticle ofmassmand thatthe ‘4amplitudes. Section 2ofthepresent paperisdevoted SPiNstransform underrotations according tolargelytothepresentation ofaverysimpleproofofthe RIB)=XsR)lie),B=(m,0). (1) do kinematic-singularitystructure ofhelicity amplitudes. 1 * dt i‘Themethodemploysthesamephysicalassumptions Weusetheconventions ofEdmonds" fortherotation where asdoesWang,? butbyusing transversity amplitudes* matrices.
: :andtheirkinematic propertiest greatsimplification isStatesofmomentum pwithp*=m*aredefined by 8obtained.- Ip2)=H(p)pa), 2) Parityo InSec.3some results onthehigh-energy limitofawhere ’Regge-pole contribution’ tothehelicityamplitude H(p)=RZ), @) Se :arederived(forgeneralmass),andquestions aboutand,for! inagreedaughter contributions, conspiracy, andevasion are) 7 inh2(si :considered. InSee.4itisshownhowrecourseto?=!mGosh£,msinhZ(sinDosg, ia_ sin0sinp,cos'0)},(4) (Weassi ‘a *Present address: Department ofApplied Mathematics, The wehaveUniversity, Liverpool 3,Englond., Trans 32Y.Hara,Phys.Rev.136,BSO7(1964). Z(p)=exp(—ifK,), ro} helicity2 :SLL Wang, Phys, Rev. 142, 1187 (966)
: ; . 4 28” FranchiandL.Jones,Phys.Rev.164,1918(1967). Rp)&exp(—ips,)exp(—I04,)exp(—ipd,).(6) Tan +G.Cohen-Tannoudji, A.Morel,andH.Navelet,Ano.Phys, fi 5 oe YBBa48ge Inpartichlar,for : . £14,P.Bap,Phys.Rev. 180-4251(1967), t whereTACTGREL AEEPoeaan99io, 4%=(7,0,0,9), 0) 1M.GellMunn, ger,F.E.Low,E.Marg,a ; 4]Zachariasen, Phy,Rev138,BasC1968), #9=(72,009, 920, ® 4‘LL.Wang,Phys.Rev.Letters,16,756(1966). = 5 2LoL,Wang,Phys:Rev.183,1664(196) 5A.R:Edmonds,AngularMomentuminQuantumMeckanles — i@"A.MeKerreland L.Serio,NuovoCimentoS2A,1223(1967), (Princeton UniversiyPress,Princston,Nox19o0)snaeat MtJac
Bi 18246 ‘
i
i
1
aris - ionatéc sings -
- - ~ t - tee
e NoKerrel achieves «simplification overHangbyusingthetransversity amplitudes.
Hedoes not explain thesé objects, but itlooks like ithas something, todawith
quantizing spintransversely tathescattering pldne,maybe, Cohen- etc.intheir_hugongous 80pagepaperonthis.subject usesuchamplitudes, andtheywereinvented
- — dy Kotanski in196. 2 eee ee -
sees Therelation oftransversity amplitudes to_the usualJWhelicity amplitudes -—
.----is given.in (21).Basically, thematrix uis.u (R,(90°)). Asyoumight guess, .
.convoluting in.thése u-matrices asin (21)causes thehelicity crossing matrices
togoaway. and.you getasimple phase xelation, 1gather that equation (27). really .
- -links the.tzchannel.cms transversity_amplitudes tothes-channel cmstrensyersity _
angles. .It1léoka likeyougettheusual angies_in there,.‘byt theyappéar just
~~asphases. eee eee - —-
. The. general philosohpy_of identifying the Ininematic_sings isthe same as. -
that.of Wang. You_assume. theyareall_given.by thefamous balf-angle factor, and..
-=~thenyou.usecrossing. Itsjust.thathere, trossing isso simple. For.the. =—
-trensverity. amplitudes, the4threshold conditions are.given in(49) through. _
(52).Thenallyouhavetodoisconyebt.theseback,intéthehelcity.amplitudes._ - e@The full answer.is stated. in.(7Q), like Jackson Hite (26). The thetas indicate.
the, parity step-ups which, may, beneeded. we ee eee
Section 3of.thepaperdigals_with suggested formsforreggefitting, and__-Section ,déalsgenerally. withphenomonology: usingtransvertity, amplitudes, Inote
. ~.that hedoes. showthe "pole.factors" inthefitting formulas. Iamnot sure __
--whether McKerrell hasfully considered. theconstraints that.goalong withthe._
-kinesings.Jackson andHite_showed thatthape_dn_fact removye_the poles, we.Inthe,appendix, results ane.shown.toagreewiththoseofWangatBerkeley. -
: ~Jackson Hite. isnotedinaNotedAddedinProof. eee eeee
-oe to ape eee -
_- St Fee Bobet Le - --
‘
-- <a --
~- a . wane ee - +
@. ~ —— amme «
2
Jack Hite(C2)
'
:
Reprinted from Ture Putysicat, Review, Vol. 169, No. 5,1248-1274, 25May 1968
Pant iU8.
1
t
'
Kinematic Singularities and Threshold Relations for
‘Helicity Amplifudes*
J.D.Jacesoks
Department ofPhysics ondLawrence Radiation Loboraory, University ofCalifornie, Berkey, Cekfornia 94720
a0
6.=amt
DeportesofPhi,Universope,Urtoe,Minas6101 ®(Received4December1967)
‘The kinematic singularities oftwo-body helicity amplitudes atthresholds andtheconcomitant relations
among these araplitudes arediscussed inadirect andelenientary way, without recourse tothesingularitystructureofthecrossingmatrix.Thetoolsarethoseofnonrelativistic quantummechanics,asbetsa ’situation where p—-0, with spins combined into channel spins SandRuseell-Saunders coupling of
L48=J. The kinemati singularitiesareshowntofollowfromamismatchbetweenJandZforeachterm inthepattialwave series, ‘The method isapplicable atpeeudothresholds (on—m)* aswellasnotmal
‘thresholds (m:-+-m)¥ with twoformal changes invelvingapintrinsicpartyandabelicty-dependent phase. ‘Therelations among thedifferent helicity amplitudes atthethresholdsareshowntoresultfromthepresence atthresholdoffewerRussell-Saunders amplitudesthanthereareindependent helicityaroplitudes. Thewse ‘ofinvariant amplitudesisshowntobeanalternative whichautomatically yieldsthekinematicsingularities andalsthethresholdrelationsamongthehelicityamplitudes. Adiscuaionisgveaofdynamicalexceptions tothethresholdconstraints, resultingfromlesssingularthgnstandardbehavioratathreshold.Thethreshold relationsare-important constraints ontheamplitudes, and must besatisGed byanyrealistic model. Inthe
useofchannel amplitudesforperipheralprocessesinthe#channel,teexplicitimpositionafaltherlations atthetchannel thresholds isnecessary inorder toassbre adiferential cross section without spurious,
Polelike singularities in¢whoeevariationcouldinsomecircumstances completelycontrolthe/dependence ‘The reactions #WKYand+N—»x’areusedasillstyations, Thelatterprocesisexpeciallyilluminating becauseits-channel amplitudes haveapote(ratherthanasimpleinverse-square-root singularity) atthe ‘apseudotheshold, £=0.09 (GeV/c).Thepraximityofthispointtothephysicalregionofthe¢channel medinsthatthethresholdrelationsthereareoferucalimportance. Theconsequences oftheseconstraints ontheerosscction anddecaydensitymatrixofthe8arddiscussedwithintheirameworkofthe Reggepole model.Comparison withexperiment implies thatthedynamics maketheamplitudes forwx’—>4have less-thanthestandard kinematic singularity at41pseudathreshold andsoavoid almost allthethreshold
constraints. Examplesarecitedfromtheliteraturewhere'useofRegge-poleformulaspossessingthespurious kinematicfactorsbasledtoincorrectinferencesconcemingthedynamicbehaviorofReggeresidues.
1.INTRODUCTION theexternal masses, hasreceived considerable attention
“TE suestion ofkinematicsnguate ofSmatix_ 3hepastfewyears,Historia, theutoffvarantclements, thatis,singularities associated withndbythéUnitedStatesOficeofNavalResearchunderContract thethreshold values ofs,4,and, andsodepending'on No.NONR 1834(05).
— }Presentaddress!DepartmentofPhysics,Universityof :*Supported inpart bytheU.S.Atomic Energy Cotumission Oregon, Eugene, Ore.
1
. . \
1249 KINEMATIC SINGULARITIES AND THRESHOLD RELATIONS 169
amplitudes incombination withexplicit kinematic G a ;factorsmadeupfromthemomenta andthespinor e
Dirac operators automatically took into account thekinematic singularities oftheproblemClassinexamples.aretheAandBamplitudes inpion-nucleon scattering Fis.1.Diagramforthe=*—*‘andthefourinvariant amplitudes Ay,---,Asinpion procemeth~so4d,photoproduction. ‘The existence and construction of
invariant amplitudes freeofkinematic singularities forageneralprocesshasbeendiscussed byHepp?Williams .t>andmorerecently byFox.‘Butwiththeconsideration .
iseofhelichy amplitudes* became prevalent, chiefly Butthestructure oftheformulas ofRef.12hasbeen
because (a)theformalism iscompletely generat, (b) questioned, with special reference tothepoint /=0
theangular momentum andparityexpansions arebyLin'*andongeneralgroundsbyStack.’* straightforward, andfinally (c)thehelicity amplitudes Another aspect ofthisgeneral problem, recognized
satisfy elegant crossing relations.*? Thework ofHara during thepastyear, istheexistence ofrelationships
andWang? solved, apart fromafewdetails, theproblem between various helicity amplitudes atthekinematic
ofdetermining thekinematic singularities ofhelicity thresholds. These_threshold_ conditions _or_hinematic
amplitudes. Wang made extensive useofthecrossing constrainis arediscussed byJones" interms ofpartial-syBatti,whleHarausedpartialvayethresholdbehaviorWaveexpansionsandorbitalangularmomentum fortheandlthecrossingmairis.Since then,otherdiscussions normalthresholds, byDiuandLeBellac"* intermsof‘ofthekinematic-singularity structure ofheligity ampli theconnection between invariant andhelicity ampli-
tudes have been given from other points ofview. tudes, with special emphasis on¢=0, andalsobyInperipheral, reactions, the(channel amplitudes Cohen-Tannoudji, MorelandNavelet)® andFox.”Ia,oftenposseéskinematic singularities thatare'sometimes Regge-pole theorywithtwoparticlesofequalmassclosetothephysicalregionoflargesandsmall(nega-TEELNN=>sp)theappropriatepeeudothreshold tive) 1.For example,intheprocessab—>cd,illustfated movestoT=0. weproblemofkinematiccon- inFig.1,the¢-channel helicity amplitudes mayhave Straits 1solved by“conspiracy” or“evasion,”™inverse-square-root (orworse) singularities atoneordepending onwiWhetherGFnot8nae heedsmoreofthepoints(=(matm.)), (mo-tm4)* andtheassistanceofanothertrajectoryinordertosatisfy@
{=Gne-m,), (omy—mig)', thenorgnal thresholds and theconditions inanontrivial fashion,pseudothresholds, respectively. Thepseudothresholds —Themainpurposes ofthepresentpaperare(1)to.‘canlieconsiderably closertore-physteal vchannel presentauniiedand_straightforwacd treatmentoffegion thandynamic singularities, suchasAchannel_ thekinematic singularitiesandthreshold conditis
oles. Consequently, itséems imporlanttotakeproper”Telictyamplitudesusingorbitalangularmomentum, account ofsuch Kinematic singularitiesinatheoretical. toshowwithintheframeworkoftheRegge- ‘model thatistobeconfronted with experiment, An’ polemodel howtoincorporate properly thekinematic__attemptwasmadefortheRegge-pole modeltodothis’structure intothecrosssectionsanddensitymatrices.byexhibiting inthes-chdniel crosssection alltheWeshowthatthegeneralresultsofRefs,9and10arct-channel kinematic singularities, leaving supposedly obtainable byconsiderationsofthethresholdsalone, smoothly varying residue functions forphenomeno- Without reference tothecrossing matrix, Ouruseof
logical fitting. Thiscompendium offormulas formany orbital angular momentum parallels theoriginal workdifferentreactionswasthen'tobeviewedastheultimate ofHara,*butwearecarefultodistinguish betweeninRegge-pole phenomenology. Some analysis ofdata normal thresholds andpseudothresholds. Frautschi andonthebasisoftheseforrnulashasalreadybeendone.#4Jones!havealsousedorbitalangularmomentum araargumentstoverifyandinterpretthesingularitystruc- yeEaChaeMidGaltier:FBLow,and¥.Nembsyureinanumberofspecificexamples. TE.argte.PhysAca3,551968) Theendresultsoftheproperincorporation oftheroeNiinns,LawrenceRalston LaboratoryReportNo.Kinematicstructureintothecrosssectionanddensity 4G,C.Fox,Phys.Rev.157,1493967), matricesarephenomenological formulasverydifferent 1MfeepbindCMe,annPsGx)7,404ps9),fromthoseofWang™inthattheyconformtothe ghTrosmanandGC.Wick,Anu.Pye.(N.Y)26,422requizementsofLin!*andStack'*andpossessno TJMusil, J‘Math, Phys.5,1481(1964), +-channel kinematic-singularity factors. Thesomewhat
TELEEhanWangPionRevs1,1870969 TF,Li,Ps.Rev,15,1515(1961 «gtGpeengsis AMorchaNoveAn,Pon,inSackeesti (N,¥)45,259(1968). . RHEE.Joes,NuovgCimento80,S1496D, «NHCP, Stapp, Phys: Rev.160,1251(1967), 4B.DiuandM.LeBellac, Nuovo itoSSA,158(1968).TEEREoeegogtae BoE Se. QB2A RenabRor,CeoS049(96. bi, WS,Frautachi andL,Jones, Phys. Rev. 164,918 (1967). ‘SE,Leader, Phys. Rev. 166,1599 (1968).
|_| eckson andHite --- et - .
—Table ofContents: =~ - - 4 - edit
-@-.. Introduction . 7~ /
It.Notation andBasic Concepts - , 7 . -
. _
A. Notation~ ~ ~ ~ a
~ 7 7BL.NoSpins ‘ 7 77 ~ .
a ~¢.Gutine oftheMethod =~~ i ee ~~
"7" Dy"Helicity amplitudes andTheir Partial WaveExpansions ~~
TT "BE,NormalThresholds andPseudoThresholds =”
oO
ITI. Kinematic Singularities” . ~ ~
7
ALPEPE toN DELTA 7 ~ - -
~ "~~
B,BehavioratPhysicalBoundary:powersoftTee _. “G.thegeneral result 77 too. _~
IV. Threshold Relations
“TSS SS" BEBE toNDELTA oT we
a“~"",dnvariant’ amplitudes forPIPIto:NDELTA Te
“~~~G,dynamical exceptions ~~~aenas
D.s-channel threshold relations for'PI NtoPIDELTA
e E.invariant amplitudes inthes-chahnel - -.-y F,general remarks
~~“~¥.CrossSections andDecayDensity Matriceb es ——
”. ~~ Ayabsence oft-channel kinsingsinthe cross section ~~~~ a
B.cross Sections anddecey correls forPEN toPYDELTA ©~~~ ~°~~
- ~
C.other reactions ~ 7 . ~ 7s 7
-”WE.summaryandConcitisions ~~~~~ To -~~"
a,dande-fiiietions, Clebshes ~~ TT oS
~—""~g, ‘Super example: PINtokKY ~ -: a 7 .—
~~" @,more threshold relations~ po — ——.
~"7D, Regge Theory - . 7 TT ~~
OT "1, Sing Structure andResidue Behavipr ~ 7 7 _
Ta Toma t . yo eee
~ “~~~3,choosing nonsense — ' ——
note added inproof. ~~ -
oe a_-- ---4- wee
-@- -- ee -- -
. - ~ -- - 1 - ~ --. -
wee Se ee tt
{
. - oe H oe. —
-Jackson andHite. . . ReadMarch11,1977
- I..Introduction. Work of.other people isreviewed briefly: .e Hepp,Williams, Fox: construction ofinvariant amplitudes inspecial cases- Jatob, “Wick :tHegeneral theory ‘ . TF a
—Wang: used croséing.matrices |_ eee ee
Hara: partial wave threshold behavior’ andcrossing matrices
-Jones: threshold conditions =“kinematic constraints “(Note!)~ -~~
on Diu and IeBeIlac: kc... .- — -
Cohen-T, Morel, Navelet: Fox: more onk.c. ;
——" +McKerrwit: ‘Hasapaper on“thts whole subject, note addedcin proof, — -
a Barmawi:-‘regge.theor'yinLScoupling. weeek Since pseudothresholds lie very clése tophysical scattering region, they should be
taken into account? They‘will obtein previdus-results without going tocrossing
matrices... Sop eee ee ee
Il.Notation andBasic Concepts. They will#base their arguments onnon-rel QMconceptswhichere-valid atthreshold etoz‘Theyaretiefirstto-show-that thesé“Simple ideas- .are enough toexplain.everything.. Since regge theory intachannel igdf.interest, that
is their choice of channels here.
--—A,Notation. Eigure2sayet' all.Panticld helicities. drenotgiven..Tyeteare clearly, defined." Certain nice’ kinematic relations are noted.
- B.Nospins. The'angular momentum behavior -is‘taken’ asa"law. ofnature™. They sayit
can beshown from FGP, but they will later not show this inthe spin case, but rather
willagaintake“itds‘Tawofnature.Fornovspiss, they”show'fowthfeshola behavior @=Wctly compenstates Py,function togive.nekine’singsinspinless scattering. . 7
~G.Outline-of-Method. They-will do‘standard-non-rel 1Scoupling. Ie, ndneed Tor the
complicated relativistic exactness ofMcKerrellsL$states... Everything willboil downtosor t.of@counting problem.
D.Helicity Amplitudes artiPilexpansions. ieopen with theusual JWexpansion-and
show thekine factor inthed-functions. Partial wave amplitudes ofdefinite natruality
- argdefined intheusual-way (see ms)inthehelicity formatism. Putting this back into
~ theJWexpansion, you getcan define full amplitudes of_definite naturality. These -full amplitudes appearing in(15)havethekinsingfactor cleaned out;ie,these amp.ituee
-->are-clear in-z. ~Equation-(16} shows-the factors - soe
-_—-— Inorder toexpress the full naturelity amps intermsofpartialwavenaturality amps,youhavetoshuffle around thed-fynctions. Thisgivesthee-functions ofof GGLMZ; Nobig deal. Since these functions are shuffled d-functiions withthe
_kinfactorremoved, theyhavelargez-dependence asshownin(18),Fine.Nocomplaints.
— E, Pseudothresholds are anlittle-different..s Consider-theusual-two-body helicity state. Asyougodéwntothepseudothrehold, theState ofthelighter particle m,hasenergy E=-m,which is.certainly abitstrange: Ig,youneedacomplex LTtotakeyour
~single particle~hebicity state tothispoint. Thisthing isjustaphasg. So,whynotredefing amplitude toincporporate thatphaseasin(19).Therayisjustarbitrary, keepsexponentintegral. Called,apoeudoanpl:itise.Second-compliéation is-mtneparityreationofamptitudes whenyouareat —— peeuddthrehold. I_am not. surprised tofind that the lighter_perticle hasan "effective"
parity=paritytimes(-1)28.°These,thep,arethetwoimportant changes. eoInplications: (1)"different threshold powerofTpversus Tyforfermion amps. Ingeneral, this affects the contraints too ‘for botb bosons and fermions.
Threshold behavior ofpartial wave amps ofdefinite naturality isthen
|reduced toe"counting problen" im(20).Recall: lawofnature.
| :
co
r
IIL. Kinematic Singularities.
A._An example PIPItoNDBETA: ‘Hereisanexample where you’sit dovnand @‘compute uptheVarious miniimm values ofLallowed by,arigulak momentum' apdparity considerations. For the fernion side ofthe system, thélightér ferminn
has its intrinsic parity effectively reversed atthe pseudothreshold, dothe
entries inthetable arenotthe same forNvérsug P.. -
Ihave dorie dut this example fully onanattached sheet. Looks good.
B.Physical Boundary: powers oft.° The “half-anglé factor" given in(16). has
certainsingularities att=O."Thesearéshownaboveequation{243s.Forall masses different, thehelicity amplitudes (full, left side ‘of(10))" donothave
anysuchsingularities, sincéinphysical region .Thérefore,. from&),‘the definite naturality full amplitudes doHave t=0 sings. These are sfiown in(24).Ontheotherhand,youcanshowthatthedefinite-naturality functions,(15) have nosingularities atKibble 9=0,andthatherefore the regular helicity
amplitudes 10,dohave singularitigs atfs0. Thus, towrite acorrect regular
helicity amplitude, you should include this fact tomake the thing smobth atg-0,
asdone in(25). .
Bothe these effects have acommon cause: the half-angle factor.
.the general forma. For low values ofJ,aswehave seen, ther'é. is"erratic
behavior" ofthe mingmum Lvalues. When Jislerger than all channel spins, a
standard behavior sets in. For these low Jvalues, the difference between the
general form and the exact, result is full zeros. Thus; the singularities are,
correctly, given bythe ‘general forma, See footnote 30, ,
The,general formula isgiven in(26).. Clearly, the minimuni value ofLgoes
withthemaximum value ofthechannel spinwhich is51¢21forexemple. However, @ parity mayincrease theminimum Lbyoneunit, aswehavegeenintheexample. This
factisshown ,inequation (27)andthe preceeding equation. -:
Note: general formula (26) isnot same thing as(20). Equatich (20) ‘isshowing
the threshold behavior ofthe definite-naturality partial wave amplitudes. Ithink
the general form of(20) inthe, general (large J)case isthis:
:
Seb 4 y=Years +Anan GOSedaCXEGE bpmTose +EOWC =Bein lisTrey +ELC, CCV .“28aLeHtWS,Ste SeLei GN LT egy
: . ‘ wes ol daw
These are the general minimum Lvalues above the erratic region. Note that the. pseudothreshold intrinsic effect isalready builtintotheseformules. Noticethat these things have avery intuitive form, aside from the parity factor which
may step up'L4,. byone.unit. The,totel J.minum the maximum spin. What could be
more easy $oremember! !!1! as + wo ot -
Equation: (26)is+showing notthepartial wavethresli¢ld behavior; ‘ut‘rather thekinsings inthefuli-(bpt definite naturality) amplitudes. Agnoted'alreddy,
theisisduetothe"Mismatch". iisyoucangeéfromequation’*(17) and(18),the 7GGLMe-functions have-as*their Yedding* power zJ-” where misusual‘max ofIsNF.
| The"mismatch" isbetweenthispower(jim)andtheminimumL'listed above. @ Thus: ’
-Q~
F endp se -\4\ t=40eande¢ ya ley Frawiste a)=ORR CyO®CTL Cy
- -tanes ue ~
-ae: oy=(=mw- ba)_=Frmeatmathely. wale~ee wfhe =Gem -teh 2 FL : ~ :
« Neee eee Xd2(oom ysa m=emnageCatt eeeeee CCE Ge Oe tee a vey
.Comments: _Noticethatthefhresbold behavior ofthepartial waveamplitudes daesnot: depend onthe helicity. ofthe particles involved, only ontheir spin. However,~one‘couldviewthtsas~ai.z- s4i+maxinum -hebicitiess ,-modwhothe-parity-factorr -which.might step you upone unit, ‘and mucdulo the ‘low Jerratic. beliavior.
Incontrast, the:"cinematic singulapitiest: ‘ofthefullamplitude dodepénd - omthe value mermax (/A7/,/X /Jetr> =boomneee ee
|~Comparison ath”crossirig method-results: —4putthesecomments-sorieshere"elew: --oeee wo ee
; IV.Threshold Relations (also called Kinematic Constraints) Atthet-channel threbholds, ~~
thet-chamnél helicityamplitudes havetheKinematic singularities, wehavealready @_noticedsuetothemibmatch ofthnehtold tebavdor_of thepartialwaveamplitudes, versus the behavior ofthe group functions.
Thet-channiel helicity amplitudes atthet~channel-threshoids, havealso~ certain linesr‘relations called kinematic constraints, anexample ofwhichis._. shownin(40). Often, both concepts must bédealt with together. Toller hasshown~howthetwoideasarequitetrivially connected asaninverston-problem ©—~ -- ... Notice that the t-channel helicity amplitudes donot have s-channel, __
thresholds, etcandetc. Ie,thet-channel helicity amplitudes donothave s-channel
kinematic singularities, HowdoIknowthis?| L4ugink tbareasonisThISTtwo | Bethet-channe} oiexpansron tetImee=tepinnde pehedrsinsidecos@=z.Bub :frow|5), zs\linear in’s. EachtermintheJW_ehynsignje-poignonial inz,there!ialThushe\t~channel peFOtryamplitudestreaplyticin-se-~~ -esAetesLictdpprocslywoswhere+ft~channDiexnansion, __ cohvet’g¢gJunich isonly nagatafe s.Howevs ‘ink the argument coulthe
extendfUyreplacing theJWwithaSirepresentation. Then-agaim, I'mnotSwreoftb“| Wont-RNOW Howonéknows this. _..ave _sicipped reading, this section for’ now.
: — + SO. ee
¥.Cross Sections and Decays. _ to - -
Aswewellknow,inthelargesphysical s-channel, youcanwritetheDCS interms ofthet-channel helicity emplitudés, duetotheorthogonality ofthe
helicity crossing matrices inthephysical s-channel. Butthese t-channel helicity -amplitudes havethekinematic singularities already noted.Thus,thesekinesingsshould -betaken into accouht when writing regge formulas, eg,asinequation (57).
Te,you will_see theusual powers oftand "pole-like factors"inthedenominator. eThaveseenthismanytimes inregge fitting formilas. Fine.
BAL (over), :
ig
First, consider this fact: the t-channel helicity amplitudes have kinematic
singularities atthresholds inthe variables t(tOchannel thresholds), because
thearguments inthehalf-angle factor goinfinite atthesepoints. Going infinite @ takes you tothe branch point inthese functions.
However, afinite value ofthe variables isnever going tomake one of
the half-angle factor arguemtns infinite.
Therefore, thet-channel helicity amplitudes donot have thes-channel
threshold singularities, and vice versa. The t-charinel helicity amplitudes have
only the t-channel threshold singularities.
Knowin gthis fact, howdowejustify the equation (55)? Thé answer seems to
bethat this replacement isonly valid inthe physical s-channel, so,you would
never "hit" upon the t-channel threshold singularities which are present inthe
right side of(55) outside the physical s-channel.
Ifyou trytomake amore general version of(55) which ‘isvalid everlythere, it
would have toshow how the singularities work. ‘The cfossing.matrices would have
toexhibit singularities inthe right way tomake things work.
_So.this is.why.it ispermissable tohave."polelike"factors 'asinequation (57). You would never beallowed tocontinue this equation tdpositive t,because
those poles donot exist: inthet-channel. Equations,like (57) are only. applicable
atnegative t,Ofcourse, that iswhere they are intended.
Itturns out that the pole-like factors will beeliminated ifyou incorporate
the kine constrains aswell asthe sings. Stack thought this up.
Stop reading atthis point, ontoconclusions.
Vi.Summary “atid“Cénclusiviis. -A_good “summary. ,Hote:onsome‘@ynamical exceptions, eg,ifsomechahniel spinisexcluded‘by.isospinorwomething LiKe"that. ~ TheydonétcYeiintodothisforthefirst,time.Claimis‘that“theuse|e ofnon-re] QMigbetter waytogothan viathe,crossing matrices. |,
Acomment rear’the,éndabout,Wetting Yeing,ricer’,but,nomerition of Tollerat‘all.’The‘obviotis Yeason4sthatto)lgr's fi¥stmetinion ofthesefungtions islaterin1966intHé4-0paper, aridCozensa Papers.) ~ |a5sathtt ere weed 8 faut Appendix D:Regge theory"consissbitig altgrthesbovs,‘ilat”shouldgTegge.pole term look 1ike?, They show theKirlematic singularities, factor called K(t) here,andtheyassume “shooses sénse" which Iwould always do. Endregult is(D.7).
Atain effect facto? ismentioned which does ‘something at,very smallt.
Finally, the"chooses nonsesne" mechahism (¢alled GeIl-Mann) ‘isdefined and
they show what happens'if youusejt. oo .
to :
roo e
iN
-—-LadinTILA2Nenomngta, a- -eae
i Jak. Pace a_PeGy2roe >»yet -
re oe Qe.
;PeTLSLO a2] aonoeao EO on
=[!Tae DR,Ta -|.
a7
- ~MO ; .-& -7.ey oO)
r
Inthe chart Ihave circled the "winners", You consider all possible Ssystems, and
pick thelowest Lyougetthroughout. Fine, Iagree, allworks good,
‘
DesiM-Semphing 22 eae ee
@OTidendieynnsgine— [Asem =Mimbo
—_ - 2 eee eee -! — === ~ ~ -
©Saesakawecuegor WeUSWA =Lunesa.
yw =ELCastes VN.LSM) 7THM, . ! _
-©Condens VASLegsPegi woe ee -
trmid=ZeSa,Smutseronreslaner-Lenn). MFOGwragpisy! Rca antawh©hawCONVemrenheavensfinewee a
t.. —_
i
. ..
jee
we ee ee a mee ee
oe eee ee -—| . Se ee
. a ns Cee
. _ _-- ' a -
@ i
i
-
pms 4
—QipainsopOwed)ithe1Ssewing) wished ASallis Sel 2
r)fo)RioniasieSE ese .-=.— »“aX “He ~ a
==WhckGads0-pare dela LS? oweus;ated .a a 2eee eee — Comes wml
_. . oe i en
eee —- “+wean ee -
eo ee
-- - - } --
|
i
Woe ee
_ --oeee -oei-
- a a 1‘
-@ { .
.
-i
. . i
-
i
H|
.-Tadsa oeasSuge Cee
]OMLa Vapanengonsin teOsKeto!) Geb
©HYodesonoyss: AGO) LGN AWRew
-Bawe(unQuALGGPOALOoaatfT-—:PrCeosoe)=Gosee)”+onanpruee
Won NGO)=Ze ACKopCemseed. Lt0Costa
—-@Logqene weBodhCop) aswe) pp
-—wh,SomcingokwsSeEL -—
ONaaeSAT,coat: Co ee~_BonGved~ZLGreylomeraiary dy(6)~
en ee
7 22BN DdPTD ©OEGALEAAD :
[email protected],Drs_Sbagk : :
eBak(8)=206)Rasa(oy)+“racedJone. _
meatybybreag\d”22h
Partial WaveAnalysis with1S.coupling. i eweeeeoe -
-
1.Ihaveneverreallydonethisbefore.ReadaboutitinWerle.Ithinkwiththe @ ititeriar ofvexerret1”i sould beable toquickly run through ‘the theory. I'am interested 1
. in-thresholds in this scheme. : -
- 2°How Should westart off? Perhaps weshould start with aiS-matrix elemént inthe ~
two particle canonical states: -- Sieecelied
_ — -Srl -4 ~ae ee -
eee ee S995 ge A
"iets saythatinitialandfinalarecmssystems. Perhapsa)andq,aredirected along .thezaxis,buttheothersystemisrotated atsomeangle.Sopulftherotation toget
tee eee eee -- 2 wee ,
a. AWM. LMG Me ==H -- ~
= = Cs _ 4mae AO =ReLQavsgude Le Le
eee a A Logg _—— -
~wtog:- S4e%aeslRoTala he -——— ---4
Nowtheideaissimplytoexpandthetwostitesystems intoHeKerreil's Isstates. ~
. - Lone 4 a —_..©SteGus),o_-—t--.wee ae 4
x, ’_& Vat “Ot um)\P Mess. —-——.\fmLogis =&—aaCUsgtessm) 17”MUssse)_- wee
eBSELL Cone, rn
ee, Re ep ee Qe ee eei,* Ow) %. TT Waray =ALS ‘ae Q(s,84sa4). DasslOee)+--+ ~ cee ae -——-— —
1 ee- -4 ~\-gly-- ANvoa eo. --Claseswves) RGRn#,9)lamas .
ue. we oe eee ae
-olyGhvadhead! agate,atkUQ\Y4, .p=G-Mow
._ . §$ a rr -
y
y : Ayx VQaewdeFeLeman ealGgDe acee ee
-e-- 286ATED. aySea 2.Oks,\WGene 3-ae akaK*_JFE.awSea)zAsytassmn) Doaylasg)C(31823YUe3s
- . oe ee a ee 4 - = eeeee
-2-‘ ,
oe -Petes .- - eo | DeCOO) PayHam
--Hm Hantriodsspom_— oe -WwwadMako TLLG. Tk. 7
_sO
_ QsMrCssesvyss) -
eo. Fl VeepCUS C&sustvansh)
J. TLD. etoyptel dagets 7
TL Bhagat Cea VETpmbeady .
J7Slce BTU Gs
Do wk\PeTAS= PeaGR)Veowahis):
CS r=FGSohAe)CeataryalTete)Oe - Sati S53 .cawDain RYSYCowoiTgoy.
@ARALek _
see — _‘
- - - ~ en
—34.
Sr oo oe
-@ SoYagavs. \Tlavawer -- - -.
FL \stFRYORR |sewste- -. Bs --b - foes
- -2aeTy CUsihasve) Ussgyvsy 2 ee .meee ROOMS Msht) CGrasam) Le
- ee eee eee -_- 2+DanEYGiLSE)TTGotCae -
aoe\.-TeAasPers®ona4Ogesdy lo‘yeytable
7 Herets"the-Ls ‘expansion: Foreachvalue otJ,you"must sufioverall Yeasonable Laids: 7
.—- A.dont understand why there are &.delta functions_on the right. - —
— —— am ee eee ee ee woe
e@ So rn ~
— — see.
-_— ———T meee ee —_— weoe
a .
ee SG Ce
eee ee
. eek a -
- . \ -
ee —— _ a‘
-
H
mek (2)
e 6
-
"
’ *J 1
'
S 4i : ‘
ILNUOVO CIMENTO: Vou.LVIA,Nb 1°Luglio 1963
|
ait 1
: efOntheElimination of¢ChannelThresholdSingularities 7 h: frémtheDifferential Cross-Section. aHfi q if8A.sexe: 7 iDepartment ofAppliedMathematics andheoretfent Physics, q iUniversityofCambridge-| _(eiovato il28Maggio 1908) |
‘
f
} Tswaspointed outbyJacxsox andIre(!)thatthekinematio singularities (+4)iA ofchannel helicityamplitudes atthresholds andpseudo-thitsholds in¢douotlead! tocorresponding singularities inthedifferential éross-scetion.| Inref.(1)certainreac: lionsarestudied imdetailtodemonstrato thecancellation ofthepole,andnowfor. j4 swulaeazogivonforthedifferential eross-scetion toreplaco thoseofref.(). .cans Jnthislettertherelation between helicity andtransversity amplitudes (7)isem-oaployedtoprove,inthecaseofgeneralmassandarbitraryathatthosquareofthe t modulus ofatransvorsity ainplitude, withthecorrectintexptotation ofitscontinua,an tiontothoneighbourhood ofthresholds andsendo-threshold} int,hasnosingularityi ‘there.Explicit formulao aregivenwhichallowaparametrizatjon trulyindependent of tho kinematics,
q JWoshallusomuchofthenotationandsomeoftheresultofret.)which,with ; i {tsnumbered equations, willboreferred tointhofollowing asM.Anyexplanations. |necessary foranunderstanding ofthoargument will,however, bogiven, “| ‘Thotchannel process considered is+b-+e+d, anditwillboassumed thatthis alchannelhasintegralspin.ItfollowsfromtheresultsofMthgtthocontribution ofa ii Keggepoleoftrajectory «tothetchannel helicity amplituje fornegative tands § :aaymptotically largemaybewritten '
a) Sel KeaalOYeaalt\R(0,0). 4
. :—— Hl a :(0)1.B.3nexson andG.3Laem:Ktnematte singularities andthereieatonsforhtelyewplte : iules,Woskoleypreprint,UCILA17959, Novenber 1967;Phym.lee.(bobppubllsned), . i |YY,Mata:Pye.stee.,188,33507(1008), i : ! ()Yesta, Wax? Pina, tons 142, 1487 (1000),
: it(9i.Courae-taxswunay, A.Monks, andIf,Nave: Kinemallal Hvmntarilce, evssing mainte :findincmatice constratnts fortwo-body hellily amplitudes, Haclsyprepnt, Apel196TseeofPie t j(Govopune. Hl t - i ()AyMokimiurts: Kinematic sinndaritin ofhelicly onatransernith amplitudes ondangie :Reapemircontrintions, Caniuridge prevrint, DAMTY 68/1,Nuvenver IMJourn,MaleThente i veabled),
. (9banWana:Dae.He,188,1685(1902), \ } ' ©A.Roxas detePapasPetr, 29,009(1900). '
q
‘
y
a f
Stapp ‘67
‘
1252 HENRY P.STAPP ye 10
question,ThemethodofWangmakesessentialuseof_partilejisexpressedintermsofitscoyariantvelocity e anextraassumption, Thisassumption isthatifcertain’a=9y/m,by asingular Kinematic functions withzerosaredivided out BaD" BE24),ofthehelicityamplitude, thentheresultingfunction nMbymultiplication|tcom eGRFor hasnoKinematic singularities incertainvariables.AnyDSiEieassociated withalower]dottedindex’the _such singularities necessaily arse-from afaire ofa*Paftcle lower| dotted index ‘; ot isgivenbythesamefunction ofitsvelocityacting generalized Legendre expansion toconverge, anditis bythe n ' end "venditisonMibymultiplication fromtheriglt.Foranyunitary assertedthetthisisadynamicalquestion,Whilethis9%MpPYmulllication fromtheHigHasttheQU-41), seemsreasonable, itisnotabsolutely convincing; since5°),sreaeirirthat “astheDotation specif Tee)notyetfullyunderstand. thedynaraics of4)7ssche(2)-+1)-dimensional irreducible representa-elementary-particle systems. Thusitisnotabsolutely DY447 y A :elements stems.“husitis tion6ftherotationgroup.Thematrixelementsofthe inconceivable thatakinematic singularity could cause trixel bs. " 4 " DTA] arehomogeneous polynomials inthematrix theseriestodiverge.InanycasethequestionarisesgueentsofA,andDLA]forgeneralAisdelinedby ‘whetherthisassumptionisadynamicalassumption Clementsof4,andDvie thatgoesbeyondthebasicanalyticitypropertiesused“02°1_sathespi byHeppandWillams: WeBndthatthisextraassump Lopiiier_Saat, svatem_somaltinn afsne_ssin3_By notreallyneeded. fatteenlonenee ‘Anassumption essentially equivalent totheextra ‘K€theform assumptionofWang ismade alsobyHara, whorelies 1BD UL(-2)7] *heavily onperturbation theory. :AsintheworkofHaraandWang,onlyfour-particle_*=(aya) v _Heactions areconsidered, Jtisfurtherassumedthatthe =(o/+1-vye) ye ‘woinitialparticleshaveunequalmasses,-and thatthe.(Pt1—vy9)/(20?+2) \/ twofinalparticles haveunequal masses.Thepassage | 1 toequal-mass limitshasbeendiscussed byWang. Halee) D0]. (2.5)
_itSINGULARITIES AT¢=0 ‘herhationmatiRjiseements
A“heheftyazuplitudeisgivenby |Breede, 12)@<r HERS, (21)wheretnesignisminusfor'—n=—1,andplusother- pikeSistheSmatrixandRisaproductofrotationwise, :‘QperaiorsRyoneforeachfinalpatie.Theeenter-o- \Thelbasicanalyticityassumption isthatthe ‘massframeisusedandthesaxisistakentoliealong functions areanalytic functions ofthecom ntsof thedirection ofoneoftheincoming particles, ‘The“themomeatan-watom exeptafdunembal singehelicityA;ofthisparticleisjustthescomponent ofits_Tarities.* Itfollowsfromthis,andLorentz invariance,spin.Theother incoming particle hashelicity Xs,which “thatAf{can bewritten in,theform?
isminusthecomponent ofitsspin.Thetwofinal r owearticles moviinthe«-splane,thefistmoving inthe Pasretbeeetaeetdsrow sere, (27)“rection @,thesecond intheopposite direction. where thecoefficients ,5,¢anddaremeromorphic‘Thetworotations Ryactonthespinspacesofthe_functioiis'of thescalarjnvatiants with,atmost,SlletwoinalParticles,andeachgivesarotationthrough"polesat6=0.Here¢isgivenby™ angle8.Speci it clej,one gle6.Specifically,forciterfinalparticlej,onehas 1ge(shi—sol—Gt-uct+2abe), es) .
.Rexiin), (2.2)here474,andwaretheMandelstamvariablesand whereJyithecomponent ofthespinvectorJthat "| ttn ‘etsinthespinspace ofthefinal particle j.'(rie tamat—aat— amet)» ‘TheSmatrixisrelatedtotheMffunction by* a’b=hmm mtn),
SeBM, 29 Pocodme—m—m). whereBisaproductofboosts,oneforeachparticles!Thesurface¢=0,whichisthesetofpointswherethe ‘Menorindhespesenttian oharealidaerankofthegramdetermingnt islesthanthree,includes sitherlowerdottedorlowerundotted. Foraparticle theboundaries ofthephysical regions. Thepossibility of associatedwithalowerundottedindextheboostsfor“Tpwitner,GroupTheoryandItsApplicationtotheQuantum =Mehaviclof“AlomieSpectra(Academie:PressTne,NewYork, ,,{HenryP.Stapp,Phys.Rev.125,2139(1962);alsoinProceed. 1989), ¥ .Bedierts2ihEnergyRlesicsandElementarywieILMuzinich,andD.N.Williams,Phys.Rev.130,e@ries, ternal , naVienne, 968)phe400 BeyMamAleedce5,218"'
'.
i .
on ::
: +Reprinted fromTnePrtvstcat Revtew, Vol.160,No.5,1251-1256, 25August‘1967 /< ested taU.S. 49
1
Analyticity Properties ofHelicity Amplitudes* 5= Pa
‘Henay'P,Starr caeaoe LawrenceRadiationLabrtery,UnveryofCaforna,Birkny,C2ijernia’ ade(Received 3April1967) ‘ tee
‘Theanalytic structureofhelicityamplitudesfsdetivedfrombaieanslyticityproperties,Previous derivationsrelied operosing properties andextra assumptions, :
ey @I.INTRODUCTION momentum vectors become functions ofscalar invari-
:jts.ants,However,thefunctions’thatexpressthesecom- “TiSproblem ofexpressing catering amplitudes ever, ns thiinterms offunctions ofscalar invariants without Ponents.intermsoftheinvariantshaye.numerous introducingextrasingularitieshasbeensolvedby‘inematicsingularities; whichtheamplitude,itselfjs Hepp? andWilliims.? Their solution hasaformthat ¢Pected toinherit. Also, thevarious rotations.ndisnotconvenient, however,formanypracticalpurposes, boostsneededtodefinethehelicityamplitudes haveThe ieoatDecors ifinvolves pveduetioa'et theXibematic singularities. Phustheanalvtic structure ofamplitudetoitsiereducibleonfonente? Thoughsun,ibe-belicityamplitudes; consideredasfunctionsofthefareduction isinprinciple straightforward, itisin-2-aatinvatiants. wouldbe¢faa practice cumbersome. Moreover, theirreducible com-~<ted.Tttimsout,however,that.most ofthesingponents, thoughthenaturalmathematical quantities, {=ztie¥cancel.leaving the helicityamplinades warenotnicephysically: Forexample,theirreducible Tessonably_ simpleanalvticity properties, ThePurposecomponents mixdifferentparityeigenstates. ThisThevendt faceaemkswowbestained vfreedteatsthattheconditionofinvarianceunderspace,THeTullisnotnew,havingbeenobtaiaedalreadyreflection doesnotleadtoanysimplereduction inthePYHara’andWang. theirmetedasseve numberofirreducible components. Itleadsratherto*Cuitous. Ratherthanstartingdirectlyfromthebasiccomplicated relations between different irreducible mO™entum-spaceanalyticityproperties,orequivalently ‘components, Forthisreason, among others, theelegant {romtheanalyticity properties deduced-by HeppandresultsofHeppandWilliams havehadTittleorno,Willams, they‘basetheirconclusions consistencyfeaultsofHeppandWillamshavehadTitleornowithwell-known crossingrelationsforhelicityampli-Tarmayparponsthemostconvesient formoftheTw4es-Sincethecrossing.felations_are_themsey
scattering amplitudeisintermsofhelicityamplitudes, etived_from_the basicmomentum-space_analyticity. /, Tiehelityamplitudes, likeanyothers,becomefunc.—BEQBErtes. thelrprocedureis,evidentlypermissibi. tionsofscalarinvariantswhehevaluatedinthecenter-Butitisroundabout, Onewolildexpectittobe:simplerof-massframe.Thisisbecausethecomponents ofthe{0orkdirectlywiththebasicproperties, andthis ofmassfr i indeedthecase.napognaas undertheauscesoftheU.&:Atomic "Thereis,asecondteasorr.forzrgéonsidering” the"TiltasHepp,Helv.Phys:Acta37,55(1968). =, Ate eae ae eeTesortaReport”+YanusHae,PrysReg136,507Peeiscoo> No.UERE-I1113, 1968(anpublished). ‘Ling-Lie Chau Wang, Pays, Rev.142,11871966),'*~*
OEE
read July 29, 1977
Analytic Properties ofHelicity Amplitudes
e1,Henryobservesthatthefancyinvariants} schemeofWillidmsandeverybody simplydoesnothelpmuchinparactice, people are/still going tousethehelicity amplitudes.
What singularities dothey have? Hara andWang answered this using crossing stuff. Since
crossing follows from p-analyticity, Henry thinks there should beafaster way to
show this. Heseems toelimitate some assumption used byWang andHara.
2.The main content ofthe paper istoexamine the various sing's bycomparing helicity
amplitudes tothe M-functioris. Henry only ¢onsiders 4-particle amplitudes inthe xz
plane. Unequal masses allround areassumed. Asusual, Henry treats general spin
particle ascombination ofspin-+ with Clebsh's.
3.Singscomeimtheusualvarieties: |
A.s=0. Turns outthings areanalytid asJongs asf40.5B.threshold andpseudothreshold appeat inthispaper asv[=41sings.C.B=0.(Kade) {
4.Ihave notread this paper somuch fortheresults asforstudying alittle Henry's
M-function notation, since weareabout todiscuss this. So,forthis purpose only the
first section ofthepaper ierAlevant. Wedtudy thevarious equations:
:
(2.1). H=RS. Ibelieve that Henry alwayd uses Swhen hespeake of"canonical
amplitudes" ofMcKerrelt, Then theRhere aretheregular rotations which convert
youtohelicity amplitudes. SoIwould sayy S=T, andH=Ty. “4i e(2.3) S=BM.HeretheBarejusttheL(p)ofMcKerrell which convert Tgtom. Isee noproblem here. H
(2.5)gavemeaproblembecauseIdidalgebrawrong,nowOK.Pointisthesquarerootisgone. {
(2.7) says that forthesimple case ofspin-O andspin-$ youcanwrite amplitude
interms of4scaler functions. This isexplained insome other peper. Ithink
probably parity would reduce-this to2functions. Not imortant now.
1 Soreally thereisnotmuchherethatishelpful tomeinunderstanding henry's notation . Ithink Idounderstand italready. File the paper and store for later use.
i
1
e@ i
i
'
‘
1
Stapp‘68
TT x
2092 HENRY P.STAPP m4
onlythrough thefactorsA(s—1)'* andX(#—1)!, theexplicitdefinition ao)
respectively, whereas Ria(@) isknown. yIfonechodsestheframewhereo4=03.and0,=—o [| seseaneejeanents thenthethemetsementsofHaethe“transverse LQOM#[™2,a(eersearsseaneanedamplitudes” ofKotanski.® Then therotation matrix
‘RO isdiagonal, instead ofBG?) andB(6%2), This Keerepresentation ofHisdenotedbyHar. XTL@Bi(02sees)WeertRi(P)Bi(09y6e1)Note thatif(®—1) [or(#=1)] iszero,thenthe oo
boost factor B(0.,4) [orB(8,2)] becomes unity. Then eathedependence ofHonAortfhonKor#7)ideter. xTOBi(odseu)W iar1Ry(0)B,(vv',¢x3). (2.90)mined bythematrix elements oftheknown rotation
operator R(@).Thisiinmediately: givesthekinematic Thesummation ontherightisoverthe4¥combinations constraints, asweshallseeinthenextsection. ofsigns ofthevarious gai,where aa, b,c,ord.Itwas
Processes withhigher spinsaredealtwithbycon- shown inRef.4thatthe.coefficients a(¢:;s,f) canbestructing their amplitudes {rom tensor productsofspin-$madefunctionsoftheinvariantssandfthatarefreeof amplitudes. For thepurpose ofthis (purely mathe- kinematic singularities at4x40.
matical) construction onecanconsideraparticleofspinOnecanwritetheequation.analogousto(2.9a)for Jandvelocity »tobeacomposite system (inapurely either theMfunction ortheSmatrix bysimply
mathematical sense) ofn>2/spin-t particlesofvelocity replacingtheHfandH;eitherbyMandMyorbyS »,Letthelabels ontheparticles ofo+b—>ctd beandS;,respectively. Theconversions between thethreethosen sothatJa=Je(mod}) andJe=Ju(modl). Letformsgothrough because boththeboosts andthe
Nae=max(2J.,2J.) and Now=max(2Jy2J4).. Then rotations areconverted inpassage through the$C,itnagineaProcesswithMachNumspin-particles,9theformappropriatetothespact‘ontheotherside.*
Sianelocty-os dndJenvewithvelocityngLet@,£449"oveobtainsfrom(2.9)theanalogof(28);represent the.setofClebsch-Gordan operators that rrndet©Fradeaat® a(2,108)combinethelast27oftheWsparticlesconstituting Mra EFrasa.tRraaat, (2400)particle@intoaparticlewithspinJe,Let@s,@.,and .)
€4besimilarly defined, Let8,bétheoperator that where thesumover7isasumarisingfromthelineat projectse1ch’ofthefirst.(@.<—J)pairs’ofparticlescombinations [email protected] fromthesctofNgparticlesconstituting particleaontoRiaaa\(@) isalinearcombination ofproductsofNV aspin-0 system, ‘Thatis,8isatensor product ofelementary rotation operator matrix elements Ryn)
@N..—J.) singlet projection operators, acting onthese andRi,.c(9), which mustsatisfy
QNee~J) pairs ofparticles. And let8s,8,and$4besimilarlydefined.ThenZfor,thecomposite systemis Daido (2.10b)
veritten as,
.‘Thefunction F,aap.” is@linear combination ofprod-
=uctsofNfactorslikeFuypayandFayigeof(2.80)which H=[6.8C)8 (8:8C2)]ITOH: alsomustsatisfy(2.10b).Inparticular, wehave
XLSO6)8 (XE). (2.90)Prova?=ILe+", t (xen) J Das(voP—1)'";544;WI],(2.100) This equation isschematic, foritdoes not“make . de eatexplicittheparticular waythattheW’variables fortheWheretheAesontheright-hand sidesatisfy(2.10b).operatorinthecenterareseparated intothefour‘Therepresentation (2.10)isdiscussedindetailinthe spacesoftheouteroperators. Butthisseparation hasAppendix. Themainpertinent features arethatthealready beenexplained. (See,also,theAppendix.) Also, dependence ofFray.’ upondsentersonly-through(2.94)doesnotconveytheinformation thatinformingthefactorsNes(te'—1)'", andRraans(@)isolinear thetensorproduct [I]©2//}eachofthefourterms combination ofproducts ofmatrix elements.of WVele-corresponding tothefourdifferentpossiblevaluesofmentaryrotations(2.5)havingthecorrecttotalhelici-(Ged)inHyistobecombined independently witheachtiesNwasspecified by(2.10b)..ach “Thearguments thatfollowholdforeachtermofthe ofthefourtermsofeachoftheotherHj,togive rn ‘Thustheinde ‘ll ited,altogether 4¥terms,whichhaveindependent. cocffici- SW"in(2-108), Thustheindexywillbeomjtted entsa(@s,*+-205 ¢x**-eyr5f). Thisfactisexhibitedin———— -——. ngsefiPrecis oeTemata Sev ~~ fihetnergy Pipsiesond.Elementary Portides,Trieste "aA.Rota, AtiPhy,Poon,29,69(1966);30,629(196).feppsteal AtomicBrergyAgency,Vienna,1565),Bq2.29)
1
. Read July 29, 1977
Kinematic Constraints onHelicity Amplitudes
@imsingsofhelicity amplitudes wasahotsubjectatthetime1968.Betweenthetine
ofthis and Henry's prededing paper, Jacksof and Hite came out with their version basedonpartialwaveanalysis. Here,Henrycae[eohisearlierpaper,There,kelocated
thesingulardity branchpoints,but{ignoredthepolesandzeros.Herehefollowsthe others bycomputing the strengths ofthe various pole powers and zeros. Asusual,
heisforced into all kinds ofmessy stuff toexplain the general spin case. Not very
degant, alsothenotation ismesgy. IseethyJackson andHiteeaughtonsowell
compared tothis,say,although inprinciplg thisisamorebasicderivation.
Ihave notread this paper, just aquik scan. Sotre forlater usemaybe.
{
|
1
t
|
i
e!
1
|
|
1
1.
i
‘
|
i
‘
:
1
e |!
t