On Pion Nucleon Resonances Citation Marx-Oberländer, Egon (1963) On Pion Nucleon Resonances. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Q819-RH19. https://resolver.caltech.edu/CaltechTHESIS:04232012-142537704 Abstract This thesis shows a method for setting up a numerical computation to find resonances in the pion-nucleon system. It takes into account inelastic reactions through a generalized N/D method, and problems introduced by unstable particles are analyzed. General formulae are calculated for the kinematics of reactions involving the πN, ρN, ωN and πN* channels, relating sets of Lorents invariant amplitudes (free of kinematical singularities) to helicity amplitudes in a form that exhibits the reflection symmetry when the total energy W is replaced by - W, and a simple form is found for the partial wave expansions. A detailed discussion of the isospin part of the equations is included. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) ; helicity amplitudes, isospin, partial wave expansions, particle physics, Regge poles, pion nucleon resonances Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Gell-Mann, Murray (advisor) Zachariasen, Fredrik (co-advisor) Thesis Committee: Unknown, Unknown Defense Date: 1 January 1963 Record Number: CaltechTHESIS:04232012-142537704 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04232012-142537704 DOI: 10.7907/Q819-RH19 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 6960 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 23 Apr 2012 22:41 Last Modified: 04 Jan 2024 19:26 Thesis Files Preview PDF - Final Version See Usage Policy. 28MB Repository Staff Only: item control page

ON PION NUCLEON RE SONANCES T he•i• by Egon Man·Oberllndel' In Partial Fulflllment of the Requirement• For the Degree of Doctor of Philosophy California Inatitute of Technology Pa•adena, Call:fornia 1963 ACKN OWLEDGMENTS It is my pleasure t o thank Dr. Murray Gell-Mann and Dr. Fredrik Zachari asen f or gui d a n ce in the work conducted in this thesis; and Dr. Jon Mathe w s and Dr. William G. Wagner for much helpful advice. " w a s h e lpful with mathematical problems. Dr. A. Erdelyi I would also like t o e xp r e s s my g ratitude to the Agency for International Development, the C ali f ornia Institute of Technology and the University of Chile f o r financ ial suppo rt . Finally, I wish to thank M rs . Alra e Tingley for expertly typing this thesis. .ABSTRACT Thia theaia ahowa a method for aetting up a numerical computatlon to find. reaonancea in the pion·nucl~on ayatem. It takea into account inelaatic reactlona through a generalised N/D method, and problema introduced by unatable particle• are analysed. General formulae are calculated for the kinematics of reaction• involving • channels, relating aeta of Lorent11 invariant the wN, pN, CI)N and wN &Jnplitudea (free of kinematical aingularltlea) to helicity amplitude• in a form that exhlblta the reflection aymmetry when the total energy W la replaced by • W, and a almple form \a found. for the partial wave expanalona. \a included. A det&Uec:l dlacuaalon of the \aoepin part of the equation• TABLE OJ' CONTENTS Section Paae I INTRODUCTION 1 u I<INI:M.A TICS 4 Dl ISOSPIN "KINSMA TICS" 142 IV DYN.AWCS 149 AI!PeDclicee A SYM.WETR Y OPERA TtONS 161 • DETERWJN.ATION or HELICITY STATES FOR PAll TICLES OF SPIN 1 /2 .AND 1 181 c ~ FUNCTIONS 185 D DETERMINATION Or PROJECTION OPERATORS 191 E ., LINEAR RELATIONS BETWEEN INVARIANTS zos SPIN 3/Z J'OitM.ALISW 219 Q N/D METHOD ZZ9 H COMPLEX SINGULARITIES 240 I NOTATIONS .AND CONVENTIONS 280 -1- I. INTRODUCTION In the laat few yeare aeve:ral reaonancea have been diacovered in the pion-nucleon ayatem, and attempt• have been made to predict their p:ropertiea. t The general pu:rpoae o! thb theala la to provide the f r ame- work for a calculation to determine euch reaonancea. The uae of a generalized N/D method, for a problem with several channela, ia proposed. to take \nto acco\Ult the etlect of tnelaatle acatterlng, related to the main p:roceea of pion-nucleon ecatterlng through Wlltarlty. The channela with eeveral piona are approximated by combln\ng aeveral particle• in a reaonant etate or unatable particle. SpecUically, the p and ~ a nucleon. Another reaonance wlth a relatively low maaa ta the J • 3/2, vector meaone are included in our formulae together with • I= l/2 p-wave pion nucleon reeonance, denoted by N • Thla ahoulcl be obtained ln a calculation where it is included ln a croaaed channel, ln a aelf conalatent way. The N* ean alao be put into the calculation aa an external unatable particle in a wN • channel. There la no theoretleal reaaon to exclude tbia channel, but uncertalntiea in a theory wltb apin 3/2 partlclea and the corroaponding experimental data do not permit an unambtauoua approach to ita tnteractlona, ao it mlaht be convenient to try a tirat calculation at leaat without lt. The exact problem obvlouely baa an lnfinlte number ot charmele, and the u.ual procedure t.e to keep only thoee with the lowe•t maeeee. tn part n the aeneral kinematic• for the reaction• under conalder• atlon l• developed. The mam reaulta are the partial wave ezpana lona of f See. for iutance, reference• 16, 18, 19 and ll; further reference• are given ln theae. -Zcertain amplitudea, related to the beHc\ty amplltudea but coming !rom atatea with definite parity, and their connection to Lorents invariant amplitude •· Their form ia auc:h that they c:an be aclvantageoualy uaed in calculation a with Regge pole a. In part m a •ktnematlc:a• ta developed !or the laoapin part of the matrlx element a, that waa not included ln part II. It la patterned tn a !orm aimilar to that of ordinary klnematica. In part IV, the problema related to the dynamic a o1 the reactlona are diacueaed. .Both the uae of the generalised N/D method, with a poaelble set of approxbnatiooa, and the evaluation of aimple dlagrama that would be uaed aa input data, are examined. The appencltcea form &lao an important part of thla theab, and many aeneral reaulta and formulae uaed ln the main bocly are derived there. They are more or leaa ael!-contained unite, azul a moderate familiarity with the matter• dlacuaaed there might facUltate the reading o i the theaia itaelt. In appendix A we examine the e!fecta of lnvariance of the atrona interaction• under aeveral aymmetry operationa. Special attentlon ta devoted to the reflection aymmetry that relate• amplitude• at poalttve and negative energiea. In appendix B, the determination of atate vectora with de!tnlte helicttlea la ahown. J In appendix C, t&blea !or the rotation matricea ct).p(8) for am.all value• of ~ ancl f.' are ,tven, and proofa of two aeural eapanatoaa, uaina either Legendre polynomial• or theb derlvatlYea, are ahown. In appendix D, two methode to get proje ction operatol"e for partial wave amplitude• are dlecu•eed ln detail, and a aeneral ozpreeelon for amplitudes llke thoee ueed ln part II le obtained. m appendix E, a peeudo•vector alaebra !or !our-vectore, dmtlar to the one for ordinary three -vector a, h developed. It le uaed then to filld a way of cbooelna Lorentz invariant am.plttudee for the p + N - p +N reaction without lntroduc:lni kinematical elqulal'ltiee. In appendix F, a field theory for epin l /'Z partlelee ie pal'tially developed ln Ol'der to find etate vectore, propagator, etc. In appendix G, the N/D method for eeveral channele te explained, together with a dbcue•icm of the dlfflculttee arleina when tlme revereal lnvariance (ezpreaeed by the symmetry of the scattering matrtx) le to be built into an approximate eolution besides unltarity. In appendix H, a dlecueaton of the problema preeented by unetable external particle• and the related complex eingularitiee le alven. One way to take them into account ie ehown ln detail. Finally, in appendix t, eome ol the. notatiolU u.ed in thie thNla are etated, specially when booke aDd papere on modern phyelce ehow a areat variety in their choice of conventlone and detinitlone. -·II. KINEMATICS In thte part, all particle a wlll be aa aumed to have zero t.otoptc eptn. The leotoptc eptn part ol the atnplltudee will be left for part UL a) • + N- • + N Four-moment\1111 coneervatlon alvee the relation p + q c p' + q' (1) that reduce• the number of independent momenta to three. We will uee the combtnattone p • i (p + p') 0 = i(q + q•) A • i(q- q') = ~(p' - p) (Z) q'-- --.,.. -- ~ Fil• 1. Gene r&l d!aaram for the reaction I"rom the three vector• we can form eix ecalare. tr +N - tr +N Four of them are fixed. correepondlng to the external maeeee p z • p' z • M z q z = q' 2 .... l (3) The ee relatione glve P·A•O·A= O (5) -5One poaaible choice of independent scalar variable• ia • =(p + q) 2 z (P + Q) 2 (6a) t • (p' - p) z • 4A 2 (6b) We can alao defl11e u • (p' - q) that l• related to • 2 = (P - C) 2 (6c} and t by • + t + u • 2(M 2 + v. 2 ) (6d) Any matrix element for thi• reaction can be written in the form (7) where the a1 are Lorenta invariant •calar amplitude•, the M 1 are in- variant• characterlatic of the reaction and u( p) la a apinor for a nucleon - of momentum p. The invariant• are acalar• in •pace-time and 4 x 4 matricee in apin apace, anct lt le convenient to buUd lnto them the eymmetriea of the problem. Lorentz acalare can be fonned with the momenta and y • eplnore u( p ), u( p1) obey The "' (rJ- M)u(p) • 0 (Sa) ('I' - M)u( p) .. 0 (Sb) u( p)(rJ' - W) a 0 (8c) anc1 from 8b Hence the poeaible factor• -;, j,. in invariant• can be eliminated by movlna f to the riabt and f' to the left. J"actora like P • 0 and ~ 2 are in- -6eluded in the ai. Parity conaervation under atrong interaction• eliminate• lnvarianu like y 5 and y 5 ~, which chana• aign under the parity operation, etnce the interaction• are euch that the matrbt element la a true ecalar ancl not a paeuclo·acalar. tn appendix E it ia ehown how invariant. containing the antleymmetric tenaor ~ ~~"P can be expreaaecS in term• of other• that do not uee lt. lt la now eaay to convince O\&raelvea that all term• in a matrix ampllt\lde ean be written uatna the two invariant• 1 and f/l, and T ia of the aeneral form (9) T • A+ Bf/l It might be dealrable to write a in term• of lnvarianta conatructed from Pauli apin m&tricee and l·vectora. In o r der to do thla we chooae the center-of-mae• ayatem, where -- -· -· p+qzp +q •0 (10) p q Fig. 2. Incoming and outgoing l-momenta -7- We call t p 0 • p 0 • nucleon eneray (11) p 0 + ~ • W • total ene:rav (12) x • coe 8 • p• p • q. q (U) ..., ..., ... • ( E zM +M ) u(p) i - ... a.•Jg ) uo u•a) (1 + E r (l.b) where t u • (It>) 0 (lJa, b) 0 Ii >, It> beina the initial and final 2-component epinore, aDd - Q. - .. ( 0 (16) - J"rom equation• lS and 16 it ie obv\oue that relatione Uke ,,.. a... .. .. u • < tier·.. -a ..... .. f > u0 O.• a.• b b 0 l (17) -· bold. and that any term with &D odd number of o • le aero. Hence u(p')Tu(;) • ~ u~t(1 + .. -rni )ts(A , • < fl ~zW' A+ (W-~E+M) + (- .... +B~0 • 0)) (1 m)u 0 ;'. + 0 B) !jG A+ (W+~l\E-M)B);. q' ;. ci It> {18) -8wbe re equation• 10 to 1Z and 1 -, o=z<q+q) -z =q-•z ==E-M z z p-z a: p-·z czq - (19) (ZO) -· q'a~~ (11) The relation between the matrix elemeDt ancl the cro•• •ection i8 a• follows:* (22) then the trauition probability 1• wfl• Uv) 4 4 2 M 4E q (23) • (25) a (p +q .. P r • q ' > z z Iaj 2 0 aDd the cl1f1erentlal cro•• section We have to uee the fonn\lla (26a, b) *See Ref. 1 ••ction l4d or Ref. 2 chaptel' 15. - 9we aet I I 0 0 8(p + q - p - q ) 0 0 (27) ejQ'I Abo (28) (29) We can define new amplitu.clea ~i by (lO) ancl comb\nlng equatioM 18, 29, and 30 we obtain 3 1a ~2 a E +M A + ( W - M) B i tr (lla) E - M -A + t~ + M)B (31b) fW 2W 4. aa \n Ref. 3. Formwae 31 exhibit the eymmetry polnted out ln Ref. 4, that ia (32) (eee appencllx A). If a eum over final epin and averaae over initial •pin i• pedormecl, we get -10- I ~ .i l<tl3li >1 2 a pine • i- ~ <fl3li><d:Jflf> epine (l3) when we have written -.... ci • 3 • 31 + cr· q W• 2 (34) aAd the eompleteneee relation . I ~~><).1·1 (35) epin ba8 been ueecl. Stnce Tr 1• Z .. -- -.. .. ... Tr (0'• a ft• b) • Za• b (36a) (l6b) (37) an4 the total croea section 1• cr r tn = c!S) •Z• .)_~ c!x{l•1 1z+ l31 12 +2xRe(•1•;)) (38) -11lf we define a "scalar product" of 2-component vectora f' IE ( f'l ) (39) "z by rl ~ o"' • \ . ., -1 where dx rp f C~$~ (40) ~ 1 G• ( (41) X wa have Here we have written 8 !or (43) It la eeaentially equivalent to the 3 defined in equation 34, ancllt ahou14 be evident lrom the context which one la uaed. Comparing the two expreaalone for the croea aection, equationa 30 and A-6, we obtain for the heli.city amplitude• (44a) (44b) -· where the flrat aplnor refer• to the outgoing nucleon of momentum p •• in the direction p (8, f') , and the eecond to the incoming nucleon of momenA tum p- i in the direction p(O, 0). * Thle G diffe ra from the one in equation D7 by a factor of 2. -12Following the pbaee convention• of Ref. 5, we determine in appendix B coe t ''· t'it> • ( ein I6 e i fl ) je.....;> • ( -t· - eln y8 e..ft "" coe y ) (41b) E lther by ue ing ... •• ... .., V• q ( coe 8 •• q a eln Be - ein Be - • ) • .. "" (46) coe 8 .., . or noticing that, eince q • - p and q • - p , l1 ... cr• q"I i > a - ).a I i > (47&) 1 cr• ... q•• • - l.c < f I < fI1 (47b) we aet ..li'Om equation• 44 1ioato • co• 4<•1 + 'z> 1 - to;itl • - ein (48&) t• it'( 3 1 .. 3 2) (48b) The element• of the eubmatrix TJ(W) defined in Ref. 5 ancl appendix A can be written in the fonn final --J- TJ • initial IP .. I io io o.J 1 -to Q.J z -uwbel'e the partial wave hellcity amplitudes are (49b) (49c) The flret equalltiee ln equation• 49b and .f9e come from parity coneerva- tion, expreaeed for tbeae partial wave helidty amplitude• by RS-43: (JO) where the "t and •t are the lntrlnalc paritlea and aplne of the ~OUI' panicle• a, b. c , del cL , g •1 For the reaction in thle aection (SOa) Ueing equaUone .A-5. C-9, C-11 anel C-6 we aet the following partial wave expanaiona (Sla) (Jib) Hence (Sla) -14- :Jl .. :,2 =I o~(P;+l +P;) (52b) Jrd+t (52c) (52d) (5.2e) L 3z • (fJ +1,-P; +i - '1 +P; > (52f) J•l+i where the 11 • are 4eftnecS aa uaual in terOUJ of phaae ahifta for atate• 1 of parity (-1) , J • l • t 1 11 * • -:::- e ill• eln 61 • I PI (53) ancl wo can write the formulae in the form 31 = I .p; (fl +1 - fJ_P; -1> (S4a) l (54b) Obvioualy, 1.t + • 1 J J 2 (cl + a.z ) (55a, b) -15- Proceeding aa indicated \n appendix D, we obtain from formulae 54 ' ,1 .. • ( -Pl-1) p' (56a, b) I. Tben we have from D-15 (57) Actually we will \lee a a lightly cllfferent notation, and the procedu.re to get the partlal wav~ expanaion and projection ope ..atora, aa well aa the equation• connect\na A and B to T 1 and T z will be ahown ln detail for thta aimple caae& the mo ..e complicated ones to follow beina atraiaht· forward generaltaatione of thia. Following the proceclu..e in appendix D, accordina to equation D-23 we define (58&) (58 b) X z: COS 6 J The minu.a aign in the denominator i.8 there becauae dj., _ ~ (8) haa one. In aenerallt will be a factor of (-l)).+tJ., where ). and ~ refer to the flnt term, i.e., ie then t and • in thia caae. The partial wave expanaion -16- ~ [~-few) +cai(w))P;+l(x) T 1(W,x) • J•t+i (59&) T 2 cw, x) • L [(a.f(w) - ca~(w)) P; +l(x) J•l.+i - (•few) + •~(W)) P;(x) J ~cw> • .few)+ ·~Cw) ~~(W) = of (W) - (59b) a~(W) (60&, b) We notlce that ~ ancl ,~ COJ'I'eeponcl to tr&Deltion• between •tate• of CS.flnlte parity. In fact, from equatiOil 1\5-~: P IJM;"tl.z> • "t"zC-1) J-·~·· 2 fJ, Mz·"t· -~ 2 > (61) where 'li, •t are tho illtrillelc parity ancl •pin of the particle•, ancl the cle!lnition•: (62) we get P IJMJ* >. •c-u1 1JMpl: > •inc• J • t + t .•1 • 9, •z • o, "; • 1, 'lz • -1. Recalllna •9a, we fine! that (61) -17- < JM; + I i I =,~ (6U.) -r 1JMJ - > = ~ (64b) TJ JM; + > < J.Ma - 1 ~ < JM; + I !. p TJ IJM; - > • 0 (64c) r 1JMa • > • o (64d) < JMa- 1 !. p Equatiol18 Me and 64d are ol couree expected from parity conae rvation• .Abo from A-14 we have ofc-w) • - oi<w) (65a, b) and hence (65c, d) th1a laet form of re1lec:tion aymmetry beina obvioua from rewriting 59 u (66&) (66b) To obtain the projection opentore, we continue aa ln appendix D. Rewritina equation• 51 aa t (W, x, •) • 1 2: (J +t)oi(W) d4 (x) (67a) Jal+-9 tz(W, x. t>) • I J•t+i (J +t) o.~ ( W)e i•ctf. - i (x.) (67b) -18we get from D- S (68a) (68b) From equation• 58, (69b) U•ing thia time equation• ClO, Cll and C6, from 68 and 69 we get (TOa) (70b) and from 60 (71a) (71b) which are the required projecti-on formulae. From equation• 48 and 58 we derive (7?&, b) and combining the•• with 31, (7lb) In general, equat\on.e 72 wlll not be tl'lvtal a• in thls caee. A final remark about time revereal: for -:rN ecattertng , parity couervatton already demand• the •ymmet:ry of the •catterlng matrix, and no new relattonl are impo1ed by ttme rever1al. b) p + N - 1r + N Momentum con.ervation gives in thle ca1e p+k:zp ' +q (74) We tal< e ae independent momenta k, q and P = ; (p + p 1 ) (75) Here we have p Z • p t2 • M2 k ? =m 1 q ?. = ~ 2 (76) We can cbooee ae independent 1calar variable• I =(P + i k + i q) Z 1 2 t • 'i (k - q) -Fia. 3. q _.,... ..,. General diag ram for the reaction p + N- w + N (77a, b) -ZO- E quation• 8 are etUl valid: .. (-j - M)u( p) • 0 -. -u( p )(,1' • M) • 0 Lorent• ecalars that can be formed ars •• t. f, ~. 1. -r, k• «, P. (78a, b) ~, q• c. The lut two can be included tn the invariant amplttudee (eee 7). Equation 74 gi.vee rl ='I - I + ll (7 9) and equatlone 78 allow lt to reduce to a combhlatton of '!. and 1. abo allow the elimination of They 1 from a matrix element. FoJ' vector flelda, it te neceaeary to tmpoee a eubeldiary condi tion, eorreepondlng to the continuity equation for the 4-current, R9-4. 3, 8A ~.o (80) ~x v. For plane wavee A ... = « e -lk• X (81) =0 (82) " and 80 becomes k· t' which ellmlnatee anotheJ' eealar. The relation ,. . =m 2 "'2 (81) ahowa that the matrix element can contain only )l to the power• 0 and 1. A final condition is that it hae to be linear and homoseneoue tn t'. To !tnd the num ber of independent lnvariante, we apply the - 21appropriate symm etry conditions to the T matrix (ln thle caae, parity conservation only): 1 J --::.-- T = IPrl ft.nal to -~ ~ J J C11 ~0 J Cll C!l4 t -1 0.6 initial ~1 -tl -+<> -t -1 J J •CI6 J ·CI·J -o.? J J Cl6 J -0.5 X Hlql/lk1)1/l. (84) -fl1 -c;J Notice that in thta caee, for equation 50, 'l a C'l c a1 'lb IZ 'ld IZ •1 and hence '1 g = -1 (8 S) There are, then, •lx independent amplltudea, and the correapondtng invariant• can be choaen (86) The y S are included due to the pseudoecalar nature of the 'II' meaonJ the matrix element should be a true ecalar aa tn eectton a). -7?- ln analogy to equations 30 and )4, we expect to write the erose section in the form (87a) (87b) The N 1 are invariants involving Paull matrtceea "Ne ehooae N N -- =l v• ~ 1 .......... 2 N .,. ........ =:iq· c v• q 4 =l ' · ; ;. ' .,. ...... N =i c v· q 6 • lv· q v. c .,. K N5 (88) lit• The polarl11atlon vectors (89a) have to be normalized eo that • c • ~ = -1 (89b) From equation 82 we derive -- k •C c 0 = -r 0 (8 9c) and introducing this ln 90b we get and lf we delln.e the direction of € - relative to k by (89cl) we obtain z - Z 1•1 = f k0 - ko z-z (8 9e) &k anclln particular for the dUferent heUeity etatea, 2 ,;(1) 1 • 1 kz 2 ,;(0) 1 • _!2 m 1«'<-1) 12 c 1 (89f) Equation 24 become• ( 90&) (90b) Now p 0 and k are fixed, and 0 p'2•Mz+qz (9la) ~ = .,.z + qz ( 91b) 0 (92) (91) (94) - ... .. We have to calculate next the equation. co~mecttng the u(p )M u(p) with 1 the Ni. -24- Wtth the convention• u.ed here, "s .: ( o t ) l 0 "s turn• out to be (95) and hence (96) We abo "•• 89c. \i(p')Mlu(p) a~ u~'(t+i;is;)~s~«o·;.;)(t+i~L)uo ~~ir .-~ (E·~~r ·Ml uo. <,I , (1 -i:&)( -t!_;..-, (1- ~)( 1; - It te obviou. that only odd power• of a. wtU g\ve non•cero term•. >) Other invariant• are l'educed •lmUarly U•lng (97a) (97b) we get -• - = <tjiE+M)i(E'+M)t N + lE·M)t(E '·M)t N ?.M 1 ?M 7 \i(p )~u(p) + (E+M),~E'+M).(E ·M), N + {E-M)i(E'·M).(E+M) N It> Mk0 s 2Jdli0 6 (98a) -25- .. -u(p •)M u(p) - =- <f I (E+M)~(E'+M)+(E-M+k) iM 2 o 2M"k N~ (E ·M) t(!:'-M) 9(E+M)(E-M+k ) o ZMk N5+ 0 o ?M 1 (E+M) t(E'+M) t(E•M)(E+M+k ) - (E·M)t(E'·M)t(E+M+k) o N - 0 I N6 t > (98b) _( ... ')M (-) <fj (E·M)t(E'·M)t(E'+M) N + (E+M)t(E'+M)i(E'·M) N ta P lu P • • iM 3 iM 4 (E+M) t(E'+M) t(E·M)(E+E'+k ) - 0 4Mli 0 (E-M) ~(E'·M) t(E+M)(E+E'+k0 ) 4Mk N5+ 0 N6ji > (98c:) .. .. -u(p •)M4 u(p) =< f I (E·M)t(E'·M)i(E'+M)(E+M+k ) o N iM 1 (E+M)i(E+M)t(E'·M)(E-M+k ) +- 0 'IM N4 (E+M) t(E'+M) 'E·M)(E+M+k )(lt+E'+k ) 4Mk + 0 0 0 N5 (E ·M)y(E'·M)t(E+M)(E-M+k )(E+E'+k ) + 4Mk o o N61t > (98d) 0 -(-')M (-) < ,, SE·M},(E'·M)i(E'+M) N (E+M),(E'+M),(E'·M) N uP 5u P =lM 3• 2M 4 (E+M) t(E 1 +M) i(E·M)<lc, lMk . 0 (E-M) i(E'·M) t(E+M)fio Ns+ 7.Mk 0 N61 t > ( 9Se) - 76- -u(p-· ) M 6u (p) - • <II • (E - M ) ~(E'·M) t(E'+M)(E+M+k ) o N 2M (E+M);(E '+M ) t (E'·M)(E - M+k ) 0 ?M 1 {E+M) t (E ' +M) t (E· M)(E+M+k )fio ?Mk N4+ 0 N5 0 (98f) and hence ~(E+M)(E '+M) [a._ + ( W ·M)Q ) (99a) (a • ( W+M)G ] z• ~(E ·M)(E'·M) 8wW ~~ 'l (99b) 31. 8 & StrW ? 8 • ~(E ·M~E '-M) ( E '+M ! [-a3 + cw+M)a4 + 2a5 - 7(W+M)a61 ( 99c:) 1 wW J • ~(E+M)i¥'+M) (E-M) [?a.- ?(W+M)Q • ( W+E')G + ( W+E ')( W+M)Q f&,..k & ., l .. 3 5 0 (99e) 3 6 • ~(E ·MljE '·M) (E+M) l ?a..+ ?(W·M)Q + ( W+E ')Q 6wWk · """1 7 3 0 + ( W+E ')( W - M ) a + ?q a + ,q (w-M)a } 4 0 5 0 6 Here the reflection aymmetry t. evtdent: -Z7The unpolariaed croea aection ia equal to: (101) ln addition to the relation• in 36& and 36b. we have .. .. .. .. ... ... ... .. .. ........ - ....... - .... + Tr(cr• a tr• b cr• c fl• d) • Z(a• b C• d - a• c b• d a• d b· c) (102) and theae allow ua to calculate (103) - where we have aasumed « to be real and alao (104a) -l8We write ln what foUowe '· q It & • co• e (104b) The •wn over polariaatlona la done by ualna .I pol. (105&) I ;. ; b· ; • ;. b + K '· & ,. ; (105b) pol. &Ad atvea • • z • + l(l+IC)x Re (:J 3' ) + 2(1+«) Re (3 ; ) + 2(l+«x )x Re (:1 3 ) 3 5 2 6 3 4 ... 2 _. 2 • + 2(l+K)x Re (3 .. ) + 2(l+«)x Re (\J :J ) + 2(1+«)x Re (3_.:f ) 3 5 3 6 5 (106a) which can be expre•eed, according to D-7, by -Z9· l+Kx 2 l+K (l+K)x (l+K)x (l+K)x l+IC l+ICX 2 (l+Kx2 )x (l+K)x (l+K)x2 (l+K)x (l+Kx 2 )x l+a 2 (l+tc)x 2 (l+K)x l+K (1+/C)x (l+K)x (l+IC)Xz l+K (l+K)x l+IC (l+K)x 2 (l+K)x (l+A")x l+tc (-l+te)x (1 +tc)x (-l+K)x 3+« l+Kx (l+(C)x l+Kx2 l+«x2 G• 2 whe11e IC 2 (106b) l+IC ia defined in equatlon 105a (106c) We next have to date rmine the quantitiea and we uee equation• B-7, B·ll (with B • , • 0) and the onea correaponding to 47: (107a) ;. <i I t > • • ~c I t > (107b) (108) (108b) (I Ole) (1084) (lOSe) (lOSf) Uetna AS and 84, (109a) (109b) (109c) (109d) . • ,fz (109e) • pl +1+ pl .J l (l +Z) 1 ,- -• . 1 _____ ,fz l (109!) -31By inverting these linear equations we obtain th~ partial wave expansion• 3 c _!_ [e1 + "z+ (1+x)Cl + (l-x)G ) 1 5 6 1 z..fz (llOa) = ..L. [ a 1- ~ 2 + (1 +x)a. 5 - (1-x)n 6 ] (llOb) 3 2 z..fz . •3= - ..L [ a.s- a.6] (llOc) 1 w4 c ·7z [11 5 + ca 6] (llOd) ..fz :J c _!_[ -~- ca 2 + ..fz.o.3 + {2Q4 • (1-x)o. 5 - (l+x)G 6 ] (UOe) 3 ..!. [ -o1+ ca 2 + .fa o 3 - .fz o 4 - (1-x)o 5 + (l+x)a6 ] z.fz (UOf) 5 6 z 2'12 From the•e equation• wo derive the J .1 = pl' +1+ .,,' z..fz t 1 1 1 0 0 -1 -1 J •z = 0 0 t p.t +1 .. pI J "J Ill z •t (equation D·ll) 0 0 m ~ -1 P 1' +1- P 1' 0 0 -1 1 z..fz 0 t 0 ' pl +1+ pl J 2 • • 1: 1 m k0 1 0 0 1 -1 1-x J .pl" +1+ pl" J .6 ·~:a z.Ju (t +z) P 1" +l+ P 1" = 2/U(l+Z) -ltx 2 -2 -1-x l+x and using 106b we get the X t (equation D-16) (111) -322 2 2 l+x l+x J Xz::: 0 0 -ko 1 1 J x. • X rn X -z pl. +1 + PI -ltx z.fz -l+x 0 0 .pt +f+- p.f ko :z m - 1 ·1 X •a 1 1 1 -1 0 (U2) 0 ·1 -1 0 0 0 0 1 -1 0 0 From appendix C aection b we have \laecl the relatione obtained by eq\lating both expreaaions for the ct{tl- to eliminate the derlvallvea of the Legendre J polynomial• in X l • .Aleo in x: and x~ we have •ned the definition of K that gives -z (1 + K) t; • (1 + ~ ) f!kz • mzo kmo ko c- m We alao define ~~~~ Jsojll t, • -icail J ~~~~ • le '+o,+t 1 1 Tl•[ etn 4e • • + -ioatl ] 1 coe 8 i 1 T • [ 4 in 1 .. 1' 1 coaJ i (lU) H~~~~ H~1!1 J .} J ~z • ~+ 0 4 (U4) J J ,sJ6 • a J5 - a J6 J ,.4 • G.). • o,z From tbeae we flncl.by \UiiDJ A·5 and 84, the partial wave expanat.on of (ll5) T l, 6 • L J•l~ p " ~ ltJ I ( 3, 6 .Jl (l +!) .. " p.C+l ) 6, l .fJ (l +Z) Comblnlna wlth equation• 108, we obtain T 1• z../z rll + .fz. X :yl + .fz. :14 T•• z..fz. :J2 + .fz. :r3 + .fzxrl4 T 2• ko m (2~+ Zx~ + 23 ) 3 5 (U6) ko m (2?2+ Zx3.. + z:J6 ) T 5• T l • - .fz !J4 T 6 • - .f2 3 3 and uaing equation• 99: Tl= -/ (E+M)(E'+M) ( 4C\ + 4(W -W)Oz + (E'·M){a +(W -M)a -za -Z( W -M)a }] 3 4 5 6 .w .rz + x -/(E-M)(E'-M) (E'+M) [-a +(w+M)a +za -zcw+M)a ] a..fz .w l 4 s 6 T 2 a -/(E +!fJ!!'+M) [ Z(W -M)~+Zmza +(E-M){ -(W+E')a +(W+E')(W+M)0 3 8 4 2 -Zq 0 a5 + Za(W+M)G }]+ x v(E-M)(E'·WK~'+M)~[ .a +(W+M)a ~ 6 s.wm 3 • (U7) T • {(E+M)(E'+M) (E'·M)[ .. 0 .. (W-M)G + 20 + 2(W-M)Q) 1 e./z .w 1 • s 6 - .35 - TS • -/(~;tJ!lE'-M)[ Z(W+M)G:t· zm 20 2 + (E+M){(W+E')Cl1 + (W+E')(W·M)a4 + z~a 5 ., 2q (W-M)a6 }) + x -I(E+1);~·:)(E-M)ko[ a 3 + (W-M)G4 0 - 20 - 2( W -M)G ) 5 6 (117) The reflection aymmetry ie now expressed by 'i J J (-W) 111 'l(i)~l+l (W) l • 1, 2, 3 (118a) T 1( - W) c 'l(l)Ti+J(W) i • 1, 2, 3 (U8b) where "t • 1 i a: 1, 3 i :a 2 (118c:) Thia la ln agreement with equation A-14. The projection operator• that expreaa the ts1J ln term• of the Ti. are (119) Calculations aimUar to thoae in thia aect\on are aho-.n in detaU in aection Ita, and a aeneral c:aae ia •tudied in appendix D. -36Time revereal invariance relate• the matrix elemente of the l'eaction p + N - w + N with thoee of • + N - p + N. It ie •xpr••sed by R5-S5 (120) Motel For a photon (maas aero), tiM invariants M1 appear only in certain combinations d.ue to gau.ae invarianoe (eee Ret. 10, for instance). We also have -;. ' 1fl 0 J are T 2 azul T 5 (J'2 azad tbe amplit'lc:les 3 5 and 3 6 are absenta so J J J anci '•· or ca3 and a 4 ). c:) p + N - p + N Momentum conservation p + k. p' + k' (12la) allow• us to choose aa independent variable• p • i (p + p') K • i (k + k') A • i (k • k') = ~ (p'- p) (12lb) aatidying relations P · A c K· A • 0 (12lc:, d, e) that leave two independent ac:alar variablee, which can be chosen., for inetance s • (P + 1<)2 (12lf, g) We also have in thia caee two equation• like 8Z: k •« • k '· •' • 0 that are equivalent to (122) -37- A·« ' = K· ~ · Abo (1Z3a, b) -- permit ua to eliminate ; and u (p ')(p'- M) c 0 uz•a. b) fl. from invarlante. The matrix element «'. abo hae to be linear and hom ogeneoue in both « an4 Flg. 4. General diagram for the reaction p + N - p + N In thia case, both time revereal invariance and parity coneervation reduce the number of lnvarlante, according to the rulee derived in appendb: A. To find the number of independent invariant., we look at the matrix -Lr. IP'rl ~1 +.o t -1 -il -~o -t-1 t!J z J o.J J a.. JJ tO ato.J 0.8 t!J o.J 6 J t -1 o.J -+ 1 J J G9 C15 ~0 c;z J a9 Cls ClJ final initial .1 -t 0 -i -1 2 3 ••J J C16 t17 a.J 8 J 3 J J c;l J 4 ~ J 9 ';o J c;z J ';} o.J J 9 J Cl8 J C18 Cl7 ClJ Cl2 3 J 0.~ J Cl. J ell J Clz a; (lZS) ·l8where relation• A-35 anc:l A-47 have been uaed. There are 12 independent elementa. If we then try to write clown the 12 invarianta, we come up rather with 1• that obey all the hnpoeec:l condition•. ~-, M 2a ,. M 8 • (P•« K•«' + P·«' K•E)Jt t ~ t• M Ml • P• « t' + P· «1 t M.. • 9 a K• « t + K· «' ;. M 10 • K· c t''f.- K· «' ;. 1;. i p. «(t' ~ • f. t') •i P• E'(tJt• ;.t) ~~· K· c :K• «' .... M 1z• K· « K• «' Jt M5 • p. ~ P• «' ~)· M6. p. « p. «I 1:. ~.· «• •• 1:. M 7 (126) =p. « K· «' + P• «' 1<• c Theee invariant• are apparently all independent. However lt ia poaalble to find two linear relatlona that they aatiefy and thta ia related to the fact that no more than four vectors can be linearly independent in a fourc:limenaional vector apace. In appendix E one way of gettln& theee relatione ie ahown. They aaoe the following equalitiea betwf!len spinora: z 2 2 MK• P ~+ P M 2 + MM4 - M:8 + (K• P- P )M - MK· P ~ 3 + P ~.· 0 9 (127a) 2 2 2 ( K A 2 + (K• P) ] ~+ MK• P M 2 + K• PM.. + K M 7 + M(K 2 - K• P)M - 4 ~ 2 2 0 - 2M~ 2 - ( A K + (K• P) ) Mu+ MK· P M,_ 4 • 0 2 2 Theee allow u• to eliminate M 8 and M 12 (or introducing kinematical alnaularitlea. M• 9 (1Z7b) anc:l ~ 2 ) without Thb ie dl•cu••ed further in the -39appendix. (See also Ref. ll. ) We can write, as in equations 30 and 34 ~·l<tl3l1>1 2 (1Z8) ~ :.IN; II. (1Z9) i•l Time reve real lnvart..ance aaaln impo••• llmitationa on the form of the N ; the rule ia analoao~o~a to that for the 1 w 1: il initial and final momenta and polarisation vector• are interchanged, and the order of the matrlcea reveraect, the Ni should remain invariant. .Asaln there are two Unear relatione amoni the lf invariant•, derived alao in appendix E. • - -1 Nl • «• « ........ ,.... ,. I •t .. - - N 2 • fl· « ,. • N 3 • «• « I f1• ~ cr• a -t . . ,;.t ,. N~ •· f' 1t • « . "'' ... ... . ' -- ....... ,. ""' -. -,. -... - ..... - t N 5t • ' • f'- ....... x• f' + k • f' .K • · N6. X • f' .... It• f' N 1 • f' u• « v• • I ,;. - t ,.. t AI .. t - N 8 • ll• f' f1' • - - - .. • - ,. .. -·I.......... - , •• + k· f' ,. I r • Nlz • 13 cr• c v• € • • x • I( • & 1• f -· &• f' (T• K fT• ~ -I .;. + ,. &• « tr• Jr. t~• f' I .. 'I--- ..... ' . ' -,.I -I .. ,. - ' ' ,. -,. -·-,. -,. NU • ,. N - tr• & f1• f' + , . • f'.,. t -ff' , . ... f' -1 - NlO • k • « ,. f' N .... fT• K N9 • .K • ~ (I• f' - +a •« fl• ,.. " ' ,.. , . x' rr• a -· - ' • (a· ,. f'.. ,.•· « + ,., ,., -,. a • ~... ,., • • f' ) , • .It f1• .It 14 (UO) -40· (lJla) (Ulb) where ~ ... X • COli (J • &• k In a manner almilar to equation• 18 ancl 98, we derive: 1, J • 1, 2, ••• ,14 The coefficient• (UZ) a lJ are given in Table 1. From theae equat1ona we set i, J • 1, 2, •••• 14 correaponcllna to equation.• 31 an.d 99. vention (n.ot for the 1 in (U3) We have uaed the aummation con- It>, of courae); the table for the coefficient• in equation. 132 ia the tran.apoae of Table 1. There la not much evidence of the rellection symmetry, nor ie any expected. It la now convenient to eliminate the two euperlluoua invariant• and amplitudes. We chooae to eliminate ~l and ~ 3 (tor no real aood reaeon; according to the problem it mi~ht be better to take any other two that can be dete rmtnec! from equatlorw 131, or a linear combination. Equa• tiona D-18 fl. might be ueeful if a cban ~e l• made. ) lt ahould be remembered that the aete of a1 and ~; are n.ot u.aique for a given tran•ltion amplitude, due to the auper1luoua invar\ante, but U they are only an intermediate etep, lt might be convenient to uae &1114. -·1TABLE 1 Nl N! Nl E+M ·-m- .. l,E+MY:,_W -M! Mz 2 .. M) .. zMkz0 k2 (W+E)(W+k0 ) . • . • - • .. .. k2 (W+k0 ) • .. zmo 2M • kz(:&+W) IU k-z(W·M)JE+M) AU E+M --m • - E·M -m ... (Jf•M)bW+M) 2 kZ(E+M) ZWkz 0 kZ(w ·M)(E+M) 2Mk 2 0 -•zTABLE 1 (Cofttlnued) N5 .. ~ M4 Ms N7 N8 k2 ~2 2int l'Dko k2 (E+M+2k0 ) Mz Ml N6 0 k2(k0 -M) .. !Mlio 2140 k2(E+M+Zk0 ) • <iMlio ~iA+E) 0 k2( w2 +EM+k02 ) imt0 k2 (E+M) 4M i' w~+E~ - 4 0 k2i W +JlVE+M~ k2,E+M) .. 8 0 !mo k2 (W+E~ 4Mllt0 2 u 0 kl(k +M) k'zW+E! - " 0 i" 1w ·M)iE+M~ .u, i' iW+E~W-MUE+M~ 8 k 8U 2 2 0 M7 Me k2 E ~E+M) • 4Mk0 - k2E~W ·MUE+M~ Mk0 k2 (E+.U+Zk0 ) M9 4M1lo ~0 ... Mu ... k2(E+M) 41( -z k .m ... k2 (E·M+Zk0 ) _k2jE+M) kziE+Ml kl(E+M) 'IM IM .. i' 1W·MUE+M~ ~2 IU 2 ~) • ~. • - 2 i' iW·MUE+M! .. 4M :u I Ill 2 i' !w -MUE +Ml k'2 w - -u;r -z nr k2 W ... "lJr k .. - 8M • - - - ..u .. TABLE 1 (Conelnued) N9 NlO lt•M -n:r • Wz k2 ·m kZ m ~1 Nu fE•M)m+M! ... ' iE·W~ 2 aMkz0 k2(E·M)(E+M-k ) -· 2Mlt! 0 0 Ml '2 iii k~p~:: .. wuw+E~ ZN.ki k2 ·-m 0 M4 Ms ,zw :"'2 . ·-u;.r ~w ·-m- • .. .. .. • kz!W+E)ZiE·Ml • • -a z k IW+E~ IW+MUE·W~ ... - k IEitWW+E) 0 8Mk1 0 w, M7 Me ~ ~0 • ·m i 2w -m- ~1 0 2 ... k (w+mE·N) • " "'· k ~E·M~ • kzlE•Ml • .. . i fE·W~ liA • k2iW+~E·W! • • k2iE•M) ... • ,a aWJ~tz kz m k'Zw "'W' ~2 - • ~3 • • 0 :"'2 k IW+E~W+MUE·M~ 2 lfbio 4U 2 • ZMll2 0 ~. • .. :-2 k lW+NUE•M! zWk02 ..... TABLE 1 (ConttauecS) Nu ~• • • Mz • k2(E•M) .. zmo Ml • ki~E-M) 11llJiO • iZE&·M) M4 • £Z1i;M) Ms -k IU • ;ziw+~;~JC-Ma k2 lW+~iE•Ml k2iW+El(W+MlfE·M) k2 (E•M) k2EJi•M) " 2 M6 ~ Ms (E•M) 8 4EI - kZiW+~E·Ml 4 • ~0 Mu ~2 0 ID .0 8140 " 0 • [ZElWg~iE·Ml 4 - 0 k (E·M) - 411&0 k2(E·M) k2(E·M) -k k 2 4SZ 2 2 (E•M) IU iW+MUE·W! 8Ja ~3 • ~. • m k (E·MJ IJJ _.k'2cw•msE-w) .. - 4 5- We define N a N' 1 1 1 a 1, Z, ••• , 10 ' Nu• "tz (U4) Theee equa.tlone, together with 131, give ue the l'elatlone i • 1, z••..• 14 J • 1, Z, ••• , U (U5) and lrom I I :JJNJ • 8 1N 1 ' • :JlbljNJ we eet (136) A lengthy calcula.tlon, aimllar to the one that alvea equation 106, permlte u• to lind the metric G for thi• c:aee. l. e. when It<< 1, we have In the non-relativletic llmlt, G-=2 / · 2 lx x z -Zx -z -1+2x2 zx 2x 2 2 \ 2x x zx -z 2 x2 x2 2x z 2 3 2x -z+bz 3 2 2x x 2 Sz 2x -z+tox2 8x 8 lxz fx z -2+b2 -Zx Zx 2 2x+2x 2x z+zx b a.2 2x zx Z+Zx2 2x 2 Zx2 1 2x 2+2x2 Zx ,.z zx b 2 Zx -Z+b2 x x Zx x2 1 x z x x2 x Z x 2x -Zx Z 2 ,.z Zx "' J 3x "' lx 9 l lx l 3 4z z z+b2 ax 8 8x Zx 2+2x2 zx 2x -2 Z 2+2z 2x ax 8 8x 2 -2+10x2 2 -b zxZ 2 -Zx Zx Z zx 3 4x 1 2.x 2 zx 2 zx2 8 8x Z+6x2 8x Zx 3 I. I 3 (137) z+a2 Zx b Z+2x2 o6z -z+6x2 ,, Zxz x3 zx 2x+2.x a2 \ zxz \ 2 2x \ 2+2x2 2 I Zx \ "' x -1+2x x zx Zx -z 2 ·•7· We calculate next ''-c'-41~.~· ~ <~c~41Ntl~.~ > •t (ll8) l•l ulDI the •tat. vee•~"• from B-7, B·ll (complex coDjuaatecl far the tta&l atate). We aet: (139) .. ~ . _.,_ (139) We follow the aame atepa 1ea4taa to equU•n• U1 an4 U2 to ••• the partial wav. expaaaton of the 31 au the pnJectton ope~to.-a (t!Mae &Jala ln a non•welattvlatte approartmatton). =\cw.aJ U we wtite •l ~(JC)nftaJa.fcw, 1•1+~ t we AM that ~ nf aa.cl ~ aaoe alven by equatioaa 140a &IUl Table 2. J 1 • • n ... i<J c-- ··P.~+P.) .. f(l+2) .. -.-.. .. (140a) J 1 • • n 9 ~:~ •zJ21(.1+2) CPJ +1+ P .... > .. . . . W.u• l(.U+l)PI+z• ll(Z1+3)Pl+l+ l(.C +l)(U+l)P.t- (1+2)(2-t+l)P.t-J 41 (f +Z)(.U +1U21 +l) nJ • 'cu +t)P;+ 2 + l.f cu +3)P:+1 + ' ' ' +2)(21 +l)P; + Cf +2)(U +3)P;_1 12 "'(l +2)(21 +lUU +l) -so. "if ...• c .r 1 ::,.,.. :r ~N ' --' ' 1 ~ fll"t N • 1 e 1 1 1 • • c N "" ' ' ~ M - fll"t ..,r i ... cJ .• .lJ• ..."• "" T • ' ' - ... ' N ' • tMaO" • ~I' N ~ < toe ....,. ~ff· • 1 • • J• N ...... • • ' ~ • i .... • N N "ii • • • • ... • ... • M• .... .... N • • ic -• N -• ' ... " '= -• .t! 1 = ...."• ' .... ...."• .,~ N .... ....t J1 -r N M - • • • ~ ..... t -.,• N d N • -... - ... • • • • • ... ¥ .... M M N ....1'4 ...• • ' I -; ' • N N I 2 N 1 ~ N 1f 1 • • • 1 c = i • it 1 • • ~ c ... • i • t I ""o~~'~ fll"t ~ ~ ~r ~ N .... • " N 1l =•• ....• .... t .... • .... • -r .... • N • • ... • -r -r ... • .. ..=rr 0 rr l 'N •""' ro• •"' ,.• .... •• ,.o- r.r N $c -51· J"nm D·lZ we have then (1-61) wbeH the p&l'entbeaie indicate• that there l• eo eum over l. The •~• of CO\ll'ae the cohamna in lfOb. xt ·xfub., xf an4 b d are given by eqGatlone lf!a ancl Table l. X: • t (P.t +1+ p wbel'e the 11 J) J 1 Kz • ~-PI.+l+ PJ) x~. "•r•zt (~. ~. ~. ;:+/) K~ • yfJ'i+! ) J K5 • (;m +~ ~ =;:;t) 1 2'/z (PJ +l+ P.e) ~ • t<· pl+l+ Pl.) - • •.t 1+x x(l+x) -2 2 -(1-x ) - 6 7 1-x 1+x - 12 l•x 1 - - - - - - n - - 2K -(1 +x) -f(1 +z) z -2 1-x 4(1-x) - 2x Zx 10 ·4U+x) 1•3x 2 2 l+lx -4(1-x) 2 9 - - - 2 ·2 z - 2 X 2 -(1-x ) 2x 1 X - .. z -x(l-x) - - 1-x -1 X~ X 1 2 X 2 Zx 1 -1 -1 -1 - - -1 -1 -2x -1 2x -Zx - - -z - - -x -1 1 - - 1+2x X Go -x 2 1 xJ9 -1 -1 -(1-lx) -2 X 7 XJ 8 - - l+x -0-a) - .. - - - • s 1-x 1 1 l+s l+x l Z(l+z) 2x -2 2 2x ·(1-x) -U-x) -2(1-x) -1 2 1 l+x xi X~ l+x • XJ 1 X~ X~ 1 XJ TABLE 3 .. -(1-x) 1 2 1 xit l+x -1 2 1 J Xtz (142b) • I N u. -SlNow, according to D·Zl and D·ZS (aee alao 113 and U4), we define: T 1• 7 • 1~1atl at: 1 ·i-ta;t aln coe { 4 e· 1• 1t •J:i ·l T 2, a• Z coal T l, 9• T 4, 10 • 1 4 • ·2 ainl t .Jt., -it i.:.1 fto;tg • cos "! '·i~J~. (141) • aln 1 e ---'r-i·,.l_.J~--1....,.._...... * . 2 a in Z 4 4 zt., coa e• 2 e in '·tl;~ 4 coa { • {f' J J J ~•• 10 • Cl) ••• (l.U) It aboulcl be remembered that the ~: correapon4 to tranaitiona between atatee ot a definite parity (aee equation• 61 through M). In the preaent caee we have P IJM;~~2 > • (·l)ljJMJ-~·"z > jJMa>,"z • >• J; (jJMJ~>-.2 > • jJM&·>,·"z (14Sa) >) (14Sb) Uelna A·5 and 1Z5 we find the pal'tial wave expanelon for the T "' P.c -1 + J(ZJ +i) 1 )] Uetna equatlou U9 we gett (147) -55- TlO* ~+ Z8z+ x:J6 .. 2~0 2 Ts• (.f2x:J1+ z.fz x32 + .f_z:J3 +.fz x':J5 +.fz x '16 - l:{z:J7 + .fz3 9 (147) ko - .fz ·~o + .fz 3tz) iii Tu• (.fz 3 + .fzx3 +.fz :J5 +.fZx:J + ?...fl. 38 + 3 6 1 •o l.fzxr~9 + .fz 310 +.fZx~12)~ T6 • (• .fz "3 • .f2 .,9 • .fz "t2) -;n Tl2 • (•.fZ ~•• z.fz 3 2 • .fz 's· .fz x:J 6 • /'z 310) ko m From equationa lll and 136 we have (148a) ancl uaing equation• 147, we calculate (148b) It ia toUDd that we can write the ell tn the form C The coeltloienta reapectlvely. tl • c (O) + c (1) X + e (Z) x 2 u u (148c) tl c~O), c~P, c 1~'1.) are abown ln Table a 4, 5 and 6 The reftection aymmetry ie now clearly exhibited, and lt is expreaaec! by J J ~~ c- w) • 'lcu-'t+,cw) i • 1, 2, •••• 6 (149a) Ti(•W) * 'l(l) Tl+6(W) l • 1. 2, .••• 6 (149b) "t• 1 i • s. 6 "t• -1 l • 1, 2, 3,. (149c) -56The projection operator• for the ~~ are (eoe D-28): ~.7 • iS_1 [Tl,7Pt + T7,1Pttl)u J l rt [ llPt+l+ (1+2)P1 _1 1 ~'z,a• l'J_ Tz,s 1 Zl+t + Ta.z tP1 +Z+ l(l +2)P1 21+3 J dx 1rl J 9. "!{' fl,, £ . -1 ~ lltu• ii: [ J l TS,llpt. Tll,Spl+l]dx rt [ ~6, 12• 'I J •l T 6, 12 v'l(l+Z)(P1 t 2 -P1 ) Zl +3 v't(J+2)(PJ+l-PJ-l)] + T 12, 6 Zl +l For a ma•• zero vector particle we are led to the case of Compton acatteri.ng. Gauge invariance demand• then that only certain combination• of the invariant• M 1 appear; and the matrix element• of TJ for which "-l, or >.d are sero are abaeot. dx -57TABLE 4 Oz (E •M)(E +M .. k <i.w 0 ) (E+M)w·M~ lt+M .. r.w- "· 2 ( E-M)[ (E+M) ·Zk (E+M)-k!J 2 m (E·M)(E+M-k ) 4wwm 2 4wwm 2 0 0 (E-M)k 0 ••• w -z k2(E+M-k0 ) - z/z wWm 4..f2 wWm k k (E·M) 1 (E-.M)[ m +k (W+M:)] 2..f2 wWm- 4.f2 wWm 0 0 (E+M)(E-M·k ) E+M 0 r.w- 'iww E·M ~E-M~W+M) r.w- •• (E+M)[ (E·M) -Zk (E-M)·lt ) 2 m (E+M)(E·M·k ) ZwWm fwWm 2 2 2 0 E+M • 4wW 0 0 (E+M)k 4ww 0 :--2 k (E·M·k0 ) 2/2 wWm k (E+M) 4..f2 wWm 2 (E+M)( m +k (W ·M)] z..fz wWm 4..fz wWm 0 0 ·58TABLE 4 (Continued) a• 2 k (lE-M+4k0 ) 16ww k~i~wM) k2 (W+E)(W+M) 4wwm 2 k k (W+E)(E•M+Zk ) 0 2 0 awwm 2 16wwm2 k2(E-M+2k0 ) k2 k2(E+W) 31wW 16wW 1WV k'2(5E-M+lk0 ) &.fa wWm • k2 (W•M) k2[ W 2 +E(2E+M+lk0 )] 8/'2 wWm 2 2 i [ w -E(k +M)} 16.fz wWm 84"2wWm 2 k (3E-+M +4k ) k2 ·r.v • • k2(E·M) 0 16wW .. ~(W+E)(E-N) 16V2 wWm k2(W+EUE·N) 0 8/'2 wWm k2(W+E) 2 (E+M) Jz.. w k2tE+M) • i'ww 4wWm k k2 (W+E)(E+M+2k ) z • 8wwm r.w- 16ww k2(5E+M+3k ) 0 • k2 (W+MJ a/z wWm 2 0 k 2 (E+M+2k ) kz &,fz wWm 0 0 k2( w 2 +E(2E-M+3k0 )) ~ k2(W+E) 2 (l~;-MI 16wwm 2 2 .. k ~E·W) l ..w k2(W+E)(E+M! . 16.fz wWm &.fz wWm k' [ w 2 ·E(k0 -M)] .. k2 (W+E)(&+M) a.fz wWm 16/z wWm 2 TABLE 4 (Continued) a, _ kz(E+~(W·MI lZwW a7 k2tE+M) 1g,w 16.. k2 (W +E)z(E+M)(W -M) &wwm 2 16wwm 2 k0 ki W +E)(E +M)( W -M) 8wwm 2 k2 (E+M) • k2(E+M)tW·M.) k2(W +E)(E-M)(W +M) k2 E(E-M) 16.fz wWm &..fz wWm .. ~E(E·M)(W+M) 8.f2 1rWm k2(E+M~W •M) Jz, - 16ww -z 16wW k2 ~E-M)(W+M) • k2(W+E)(E-M)(W+M) _ k E(E-M) 16/Z wWm 8..f2 vWm .. kz(~-r.~W+M) .. k:!wM' k2 (E·M~W+M) 16w • k2(E+M)tW-M) k2JE+M) k2 (E+M~W-M) 32wW k2( W+E) 2 (E ·M)( W +M) t,;.w k 0 16w k2 (W+E)(E·M) 8wwm2 16wwm2 8 Z wWm k0k~W+E)(E·M)(W+M) 8wwm 2 . k2(E~M).W+M) k2jE·M) .. k2 (E-M)w+M) k2 (W+E)(E+M)(W-M) .. k2 E(E+M) .. ~E(~+M)(W •M) 16.f2 wWm &..fz wWm 1g,..w 16w 8/2 wWm • k2 (W+E)(E+M)(W-M) k2 E(E+M) k2 E(E+M)(W-M) 16.f2 wWm 8.f2 wWm 8.f2 wWm -60TABLE 4 (Continued) ~0 -z r.wk k2(lll:-M+4k0 ) • 16wW • k2 llS-M) lliwW 2 2 k k (E-M+Zk ) -0 0 8wwm 2 '2 • 1WW"' 16wWmz 2 k (E•M+2k0 ) k2 ~E+M) 16ww .w j k2 (E•W:•k0 ) k k 2(2E+.M+3k ) 8.f2 wWm 8f2 wWm 16.f2wWm kl)k2 (E·M) k2 2 k k (k +M) 0 0 a.fz vWm 2 k (3E+M+4k ) ii'W'" 16wW k2(W•Ml a.fzwwm 0 0 0 '~r.t.M> k kz(W-M) 0 4wWm 2 '2 ·r.v 2CE+M+2k ) kzk 0 0 8wwm 2 k2 (E+M+2k0 ) • --.1""6w'""'!w:=o---- k2(E+M•k ) k0k2(zE-W+Jk0) &.f2 wWm 8.f2 wWm ·----0- k2 (W+M) ko'z(ko·M) &.f2 wWm 8..f2 wWm k 0 k2(E·M) 16.f2 wWm k2(E·M) 32wW • k2lE+M) zwW ~k2 (E-M) 16wwm2 2 k • J JE·M) .w 16..f2 wWm ·61· TABLE 4 (Continued) ~3 C1u Tl Tz 2 k • ~E+M~W•M~ Jz, k2 (E·~l~(W+M) • k Tl lZw 2 2 k (E+M)(W-M) 0 16wwm 2 k iE+M~W·M~ 2 T4 32w k T6 • Ta • k2 lE+MUW·M~ 4wWm'2 E•M 1iW 4wWm z .1E·M~W+N~ •• 16.fz wWm 4.fZ wWm 4.f2 wWm k (E •M)(W +M) 0 k (E·M) k (E•M)(W+.M) 4.f2 wWm 4i'{'l. wWm 0 • 0 k2 (E•Ml\:+M~ JZw • "liW' k2 iE+t.f~W ·M~ Jz.. 11rV • • kl(E·M~ k2 fE·MHW+MJ T9 16wwm2 E+M E·M .. ~E+M~W·M~ •• (E·M~W+M! lw 4-wWm 4wWm z • E+M rww- . {Ei!'~(W -MI (E+M)(W·M) .. k k16.fz wWm - .. Jc (E+W)(W·M) k0~(E+M)(W ·M) k (E+M) k (E+M)(W-M) 16/2 wWm 4.f2 wWm 4.fz wWm 2 k ~E-M~W+W~ TlO 32w 2 T12 k2(E+M) k (E·M)(W+M) 2 k!k (E •M)( W +M) Tu • (E+M)~W·M) 8w k 0 (E•M) 16/l wWm T7 lwW E+M • iwW 2 0 ~ (E·MUW+M~ k kZ(lt-M)(W+M) 0 T5 E·M TwW ~. 0 k (E+M) 0 4.f2 wWm 0 0 4.f2 wWm 0 -62TABLE 5 Tl ~ az E+M (E+M)(E-M·k ) ...w .TwW" Tz ,z 0 <iww • k 2 (E+M) Tl al • 0 k (E +M ) 2 k k (W+M) 2... w 4wWm 0 z 2wWm 0 • T4 k (E+M) 0 Ts 1:lzwwm - (E+M)[ k (E·M·k )•mz) 0 0 'k2 (E+M+lk0 ) •.fz ... wm &.fz 'I'Wm • T6 ...z (E·M)(E+M·k ) T7 E·M TwW" 4 ...w • k 0 1;'\\r - Ta T9 z k!(lt•M) k (E·M) z... wm2 2vw 0 k 0 k2(W·M) +trWm z ,··· TlO k (E·M) Tll T1z 0 z..fz wWm • (E·M)[ k 0 (E+M·k0 )•m 2) 4,fz •Wm • k2(E·M+lk0 ) 8.f2 wWm • -63T .ABLE 5 (Continued) k2 T,; .. rziE+M) k 2(E+M) &.. k 0 w k2[ w2 -E(k0 -M)) ••wm 2 k2 (E +M)( W •Ml Jzww l6wW k 0 k2(W+E)(E+M) 8wWm k 2 · ;;zlf;~) 0 k 2 ( W +E)(E +M)( W ·M) 8wwm2 k2 (E·M)~W+M) 3!w k2[ W(W+M)+zk!J k2 (E 2 +MW) kZr k (2EW+EM-M 2)-m2E ) 8-/2 • Wm 8-/2 wWm &..fz wWm k (E·M) 0 a.fzwwm 0 k (E-M)(W+M) k (E·M) 0 16-/2 wWm • 0 16..f2 wWm [2 ·r. - ':~!il'' ki w2 ·E(k0 +W)] . z k 0 • k2jE·M) Jzww J2ww k k2 ( W+E)(E·M) k 0 k2(w+E)(E-M)(W tM) 8wwm 2 8wWm 0 ••Wm z p I k2 (E+W) k2 (E +M)~W -M~ klr W(W -M)+2k!J • k2 (E 2 ·MW) &.fz wWm &..fz wWm k2[ k 0 (ZEW-EM-M 2 )-m 2 E) a..fz wW'm i 2(E+M) 16wW JzwW 12w k (E+M) k (E+M) k (E+M)(W-M) 8.f2wWm "' 16..f2 wWm 16/2 wWm 0 0 0 TABLE 5 (Coutluued.) • 2 k k E(E+M)(W -M) 0 4wwm 2 • ~:~:wM) • k2(E•M)(W+W) k2l E (k0 •E)•MW) a.fz wWm k2( (E • .M) W ( W +M) +1.MEk0 3 16wW • 8.f2 wWm k.,(E-M) k (E•W)(W+W) a.fz wWm &.fz wWm • 8.f2 wWm 0 • • • k1 tJt•M) I'•W .. ' 2 (E·Mj(W+M) l&wW 2 k k lt(E•M) 2 k k E(E·M)('W+U) 2 k k (W-W) uwm2 • • 4wwm 2 4'1'Wm 2 ,zjE+M) k2(E+M)~V ·M! .. 16wW 0 • l'ww 0 k'Zr (E +M)W(W•U)·2lak0 ) ~Zr E(k0 ·E)+MW) .. &..fz wWm &{1. wWm • k (E+M) k (E+U)(W-M) &..fz wWm a/z wWm 0 0 0 s/z wWm -65· TABLE 5 (Coatin'led.) Cit -;-z .. k 2 (E+M) • 3!.. 16wW •!rz<ko·M) 4wwm 1 • • 8wWm 2 k 2k?'(E +W)( W ·M) - 0. 8wW'mJ i~l!wM) a0 k2(E+3k0 ) a..fz wWm k (E·M) -S..fz 0 wWm • a/z wWm • a/z wWm k (E•M) . ---:::----16..f2 0 16..f2 wWm k (E·M)(W+M) 0 wWm k2 r. .. k~~~vr, 32w k~2 (E-M)(W+M) k 2k2(k +M) 0 0 z 8wWmz k2jE+M) k2JE+M) 4t•Wm - u;.w J!wW 2 k 0 k (E+lk0 ) S..fz wWm k &..fz wWm •2... Zw k-11 0 8..f2 wWm k (Et M) • 0 16..fz wWm 2 k2 (E +M~W·M) 32.. k2E 8/z wWm k (E+M) 0 0 8wwm • k (E+M)(W•M) 0 .16..f2 wWm -66TABLE 5 (Continued) ~3 .E+M Tl ·r.w Tz • Tl .. ~4 • iE+.MUW·M! BwW k 2 (E+M) k!(E+M)(W·M) 4wWmz 4wWm 2 0 • T4 k (E+M) Ts 0 • 41'2 wWm T6 • T1 rwv k (E+M)( W •M) • E•M • 0 41'2 wWm !E·M~W+M! 8w .. Ta k!(E-M)(W+M) k!(E·M) T9 4wWm 2 • • Tto k (E-M)(W+M) k (E·M) Tu Tu .,._wm2 0 41'2 wWm • 0 4..f2 wWm ·67· TABLE 6 az (ll a• as k2 (E+U) i6ww k2iE+M) - • Tl • - • Tz • ... • Tl • T. • Ts • lwWm .. z • • .. T7 • • .. • 0 a,fz wWm • 2 [ ~E·ld) • 1 .w • • 2 2 T9 • TlO .. Tu • Tu .. • .. • 16wWmz • 2 k k (E+M) • 2 k~ (E+M) 0 ... T6 Ts a02k2 (E+M) Jz.w· • • 0 awwm 2 ... • • & k (E•M) • k 0 k2(E+M) 161'2wWm • k2(E·M~ • Jz.. w - &!' • 2 (E•M) 16wWm 2 - 2 k 0 k (E·Mt ko£2(1t·M) s,fz •Wm 16.f2 trWm • -68TABLE 6 (Continued) a, Tl .. k2iE+M~W •Ml ,~ • a!k2(E +W)( ·M) W Tl 16wWm z • T4 2 k k (E +M)( W ·W) Ts 0 16.f2 wWm Ts a, k2iE+N) k2 lE+~lW·M~ 16. • 6wW • • 2 k~ (E+M) 0 8wWm z k2 !E·N~W+N! lZw (E+M)(W•M) 8wWm 2 • 2 k (E+M) 0 8.f2 wWm k2~E·M) I .w • k2(E+M)(W-M) a/z wWm I • • 2 - • k lE-Mw+M! 16· .. 2 k!' (E•M) T9 2 16wWm z 8wWm TlO • T12 2 a!k • • k0 ' • • 2 k!k (:tz:-M)(W +M) Tu I .. T6 T7 Oa . • Tz 07 • ~t!'2 CE-U)(W +M) 8wWm 2 • • k 0 k2 (E-M)CW+M) k 0 k2 (E·M) 16.f2 wWm a,.fz wWm - • .. • a £2(E -u)( w +M) • 0 a.fz wWm -69TABLE 6 (Continued) C1u ~z k2~E+Ml k2iE +MU w-u~ ~0 Tl 2 k ~E+M) • 1 wW zww 2 k~ (E+N) 0 awwm 2 - 2 k~ (E+M) 0 16wWm 2 k k (E+M) .. aOJz wWm 0 0 k'2 ~E-M~ i&ww • .. k , 0 • i 2(E·M) J!wW 2 (E +M)(W ·M) .. 16/z wWm • • g ... - • k2 iE ·M~W +M) Jz., • s.. wm z 2 k!k (E ·W)( W +M) 16wwm2 16vWm • - - k k2(E-M) 0 8.f2 wWm • - k 0 k2(E·M) 16.f2wWm • . • k~(E•M) TlO TlZ k~z(E+M)(W' •M) 0 16wWm2 k 2k2(E·M) Tu - 1612 •Wm Ta T9 z k ,z(E+M) • T6 T7 J2wW • T4 Ts C\4 • Tz Tl <\3 k 0 z k2 (E -M)( W +M) t6.fz wWm - .. - - • • • 4l) p+N-w+N The klnematlca of thia reaction ia very almUar to that of pN ae&,tterlng, the dU'ference arlalna from the fact that time reveraal no longer red'ICea the zuamber of independent amplltudea, alnce the p and w meaona are different parttclea. The maaaea &l'e alao alightly clilfer• ent, ao that aome of the equatlone 121 change. p + k . p' + k' p 2 • p •2 ,...2 • - k (149&) 2 am 2 k ,z •m •2 K • ~ (k + k ) 1 (149c) (149d) P·6 • 0 1 2 K·6 •l'Cm • m ,z ) (149e) Pz+ Az • wz K2+ 62 • \ Fla. 5. (1491) Cmz + m'2) a s (P + K) 2 (149h) t . 46 2 (149l) General cllaaram for the reaction p + N - w +N Equation• 1Z2, 123 and 1Z4 are .till valid.: k•c •k'·•' •0 (150&) /}.• c • • K• « (150b) A• c' • (150c) (p·M)~(p) • 0 (151&) u(p 'Hp' • M) • 0 (lSlb) we now aet zo invariant.: ~. «•« • Mu· p •• ,. M2 -= c•c'Jt Ml• JJ' ),44. tJtt' M 5 • K• c't ~2- p ••;·;. ~ 3 : K• c K• t:' ~.· K· t: K· t: Jt 1 ~s· p. « P· « M6• K•t:'JJt I ~ 6 ~~~ P•c P·c'Jt M 7 • K• t: t' M a P•t: K•c • 17 Ma• K· t: t'Jt ~8· p. « K• c'~ M9• p. c't ~ •K·«P•t: M1o• p. «'t~ M 20 • K· « P· 9 (lSZ) I c'Jt Two equation• •imUar to 127 can be found1 the procedure ie the eame, a• indicated in appendix E. The important difference ie that K• A ie no lon&er zero, and thie will probably introduce a&l1Uonal terD'le • .Again we can wl'lte (153) (154) . - -· Nl a C• C I N2 ~ =a• c a· c .,. _,.I - t I.,.-.,.-· Nl • &• « &• c I ~I - I ,., - ' • - .,., .,. 0'• A 0'• • ./01 - I N4•tt·«•·« N5 • k • c -1 - N 6 Ill • • • N7 a A• C I "' • c... , - 1 ~I ... /'f - ./0 K• C G"• It fT• K • . .,., -"'. I ./0 - N 8 • a• c II> c v· , , ,.. v• a # -. &• -·-~·-,. 9 • & • c; .K • c v• a v· a I ,, ./' - t .. , . ... I" N • • c K• c ,.. v· a 10 N I 'I,..,- - -II> N :a • c •· c tr• a (155) 16 . , ---, . - .., - - --· -.. -1 ,., N ' a .,. &• c ff• 1t ,.. c 17 Nu•· . .... • v· • ... I -- N19= r • N 20 = ('f· k - -1 ,. c ('f• c v· c v• k The two llnear relatione t hat can be uaed to eliminate two invariant• are: (156a) (156b) and they are obtained the aame way equation• 1.31 were, as ahown in appendix E . The relation between t he two seta of invariant• ia i, J • 1, 2 •••• ' 20 (157) The coefficients a 1j are given in Table 7 w here we have aet .a* • .f (E*M)(E '*M) (157a) ZM ~W·MtR+ 0 (E+Ml!K+M)R. - M7 4111&0 ~ iW+EUE+WUE'+M~R- - .. ~0 4Mii0 ( W+EUE+M)v='+M)R. M<J Ma 0 D k~'R+ 0 4Uk0 4Mii0 ,za+ k'za+ - 0 • k'J;;,E)R + • 0 -1M" - 41401 - 4Mit0 k~'R+ 0 - I 0 M6 Ms k-·~oR + - ~R+ 0 - it:ot'0 - mro (E+M)~E'+M)a· 4 ZMk0 k'0 k' 1 a+ - - kZR+ N. Nl -4Wii - 0 (E+M)(E'+M)(E·W·k' )R• 0 !E+~'+M)R- ,. ziliii0 ''0 ! W·MUE+M)(E'+M}R. Z 0k~ (E+.Mtw'+M.)a· Nz (E+Mm'+M)R• Ml M4 .. - Mz ~ ·m R+ ~ TA.BLE 7 (E+M~(E'+M.)R• -g - iM (E+M)(.E '+M)R• - ... ... Ns • • ~ w • ,, 0 0 .. .. ... - Ma M9 ~0 - 0 0 " 0 (W+EUE•tWE'·M)R+ 4 I w+E)(Eur'~'-M)R + • 4Ml0 0 ~·a· • 1m': k2R• •m;, klc•a· 0 mo - (E·M)JE'·M)R + -4D IE-!!HE '-MlR + 0 (E·MJt[;M)R + ... z Mli'0 k lW+E)R .. • - - iilk'0 k' 2a· 0 mr• 0 ,.~a· 0 k2R-'ill£ 0 -fiA' .. ,zR- -zm. - Ms 2MJi0i'0 k - ~- - NCJ - M7 0 (E-M}(E'-M)(E+M·k' )R + 0 . (E·MB'-M)R a £' + • (W+M)WdMl\E'·MlR + 0 146 ~W+M)R .. - 2M - 1 {E·Mtt'·M)R + - a· m N8 - ),{. ).(3 .M2 ~ NT N6 TABLE 7 (Continued) 4JJ (E - WUE '·M)R + .. - .. f!:·Mllf '·MJR + - NlO -..I .•• 0 - - - - - - ~0 m M9 W!E '+MlR .. 0 •m. k' W(E+W)a· lE'+MlR .. - 4M 0 4Mii0 0 0 - 4Mii k' W(E •W)R+ 4Mk0 Ma M7 k' (E·M)lt+ - - IE'+M~·- - 2M - f\3 k' (E+M)a· - M6 0 - - )d5 (k~·M)(E+M)R• - lE+lfla· z HfDio 0 (k' +M.)(E•M)R + 0 (E~R+ zJAO l - Nll M4 M Mz ~ Nn - - !U lE'·M)R+ - WlE '·M)R+ - m • (E'irJR+ - - ~. TABLE 7 (Continued) 0 0 illk'0 (W+E~(E'·M)R + 4M£'0 iMJi'0 WiW+E)(E '·M)R + - 0 k W(E '-M)R+ 0 - 'IMit' k (E'-W)R+ 0 zm• (k +M)(E'·M)R + 0 '~Ia+ "ts "'' • -.a -76· z2 ·"'~ ·~ • .Z:• +"'a ·r • • ~ I j .;. -,._ .F. ·~ ea:t ·~ ·r ~ ~ cQ < f-4 • ~ ., ,._ -r• • •cr: - .i•"' 0~..!:3. + ·~J - i .;. . ~· • N l1 .., ~ ~ •• ' • •cr: 2 • • ll 0 • &i4 Ja1 ~ Mo Mo • • .... l1 ~ :a ,._ ~ :f • 2 ' 0 '£ - • ~6 - •. ~9 M 20 - ~~ ~7 - • - ~s ~ ~l .. ~2 Mu ~ 0 ' 0 0 0 8 0 (\V+E)(W -M)&+M)(E'+M)~- 8 (W+E)(E+~E'+M)R- 8 (W+E')(W·NltE+M)(E'+M)R- I (W+E')(E~(E'+M)R• Mkolto • 0 k~'tW-M)R+ 0 amo 0 8)4 8 k~'R+ .. kl(w+~(W-M)R + 8Mit0 k2 (W+E')R+ 0 ~(...V+EJiW•M)Rt lllk0 (W+Et(W+E'~(W·Mt(E+M)(E'+M)R­ lmt0 k0 ( W+E)( W+E ')(E +rz9(E' +M)R • lJU&o k~'(~·M)R+ 0 -~ k~'Rt Nl k2(W+E')R+ BM {W ·M)(E+M)(E'+M)R- M I 0 (E+M),E'+N)R- 0 'iilot (W+E')(E+M)lE'+M)B• N2 TABLE 7 (Contillllecl) + • .. 0 2 8 0 k' ( -Nt_21W -M)R + 0 81&' k' 2 ( W+E)R + 8Mk" 0 8Mk01 k'~ (W•M.)R+ 0 k'~ R+ o k'fw +E~(W·M)R + 8 ,. 0 + ~ k'f&~)R ... 2 ·~(W·M)R+ k'~ 0 k~R 0 ... z.. . ~ -•2Jl + It N4 • _.., • _.., Mzo - - ---.9 8M ( W- M)(E+M)(E '+M)R - (E+M)( E '+M)R - AM ----.8 - M. M.. _ ( W- M)(E+M)(E '+M)R. 8M - SM (E+M)( E '+M)R- SM ( W- M)( E+M)(E '+M)R - 8M (E+M)(E'+M)R - AM ( W -M)(E+M)(E'+M)R- 8M: (E+M)(E'+M)R - - - ---.7 M.. ----.6 M_ - ---.s M... ----.4 M. -u M.. ~2 M11 NS - - _ - _ - - _ - N6 8M (E - M)(E'- M)R + 0 0 8Mk k1 (W+E)(W+E ')(E- M)(E '- M).R + 8Mk10 0 8M1t 0 8 k ( W+E)(W+1(E-M)(E'-M)R + - (W+E)(E-M)(E'-M)R+ 0 8M ' (W+E ')(W+M)LE - M)( E '- M)R + - (W+E')(E- M)(E '-M)R + ( W+E)( W+E'){ W+M)~ E- M)(E'- M)R+ SMk0 k0 - AM ( W +M)(E-M)(E'-M)R + - ( W+E ')( E - r~'E '- M)R + 'IMit0 0 N7 TABLE 7 (Continued) 0 8Mk kl(W+E')R- 0 8Mk klk~(W+M)R- 8 k k~R- ~ 0 0 8:Mk :-2 0 k k~(W +M)R- --a-M - k2 (W+E~W+M)R- AMli0 k2{W+E')R- 8Mit0 kl(W+E')(W+M)R- - - ~~R8Mlt0 N 8 -I ~ 0 M 20 U.. -~9 "''a :U.... ~ 0 am• .. 8 • 0 k 12(W+WW+N)a• lUll'0 k12(WtE)R- It -•z(W+fMW+M)R.. £6 1 ' -•2.. 0 It I t . ~7 -•2.. 0 k -.. (W+M)a· -•2 • It (W+EtR• ~5 llldi0 IMii' 0 lt I t (W+U)a· -·2...- 8~ k'~0 a· 0 mr• k'2a· ~. - ~l ~2 Mu NCJ u .. lil (W+W)(E-W)(E'·M)R+ SE -W)'E. -M) R + lid (W+WUE·M)(E'·M)R + ill (E-.,E'-W)R+ IU (W+M)(E·MHE'·M)R + (E•Milf'·!:QR + (W+W)(EjrJ(E'-M)R + (E·M)fti·M)R + NlO "u " 0 _ W(W+Ear-M)R + 0 (W+EllfkW)R+ TABLE 1 ( Coatinued) 4 0 W(W+Eav:fW)R .. MJio (W+E'~E+M)R­ ~2 I • -.1 ..0 -80· • • t- ~ • ~ • " i i: -4 -"'w"' • u ~ (Q < .... "' -;:- • ··:. ~ ... Jal 0 N ~ -81Since we have, aa in equation 93, (158) we derive, correapoo.cUng to equation 131: i, j • 1, 2, ••• t 20 Theae (159) •J' a how the reflection symmetry, ae ia eaay to ••e from Table 7. Thia come• from the fact that the lae~r ,.. k ia alway• on t he right, and o-• k' on the left. We will eliminate the invariante N and N , and we aet 20 19 i • 1, 2, ••• ,18 (160a) which toaether with eqi.Ultlone 156 give i • 1. 2, •••• 20 J • 1, 2, ••• ,18 (160b) and hence (160c) We aext calcwate the matrix G that give• the unpolarised. eroae aection, and we get Table 8 of \ Gij (with K << 1). a xz lx 1 X 18 1 17 xz X X 1 xz X 16 a a "1 z x2 xz 1 1 z ••lS az X X ll X X 1 X X 12 "2 1 "a 1 1 ,l· X "2 "3 X X xz 3 xz 11 10 xl "2 X 9 X X 8 xz xz 7 6 az X x2 X 5 a x2 X • 1 "2 1 1 a 1 l X X 1 X 2 1 1 X l 1 4 l 2 1 X X 1 1 x2 1 a X 1 1 z X 1 1 X l zz X X l xz a X 1 zz X a 1 X xl xz 1 1 xz 1 xz z " zz X X l " X "az X X 1 xz X xz 1 a "2 X xl xz X 9 1 X 1 xl x3 X xz x2, x2 X 8 a 7 x1. X X 3x x2. 6 X X 1 xz X " xz xz xl zz z 1 X x2 X 5 1 X X xz 1 X X xz 1 X X x2 X X .z .z xl zz 10 TABLE 8 1 " X 1 lx -xz 3 -x X X 1 1 I 1 X -a l lx z -x ' X lx z -x -x X 1 xz X x2 "1 X 1 1 xz 1 X X "2 X 13 X X l X 12 3 xz X X 1 z X 1 "1. X 1 u xz 1 z X 1 l -xz -x lx l 1 X X lx •X 1 " xz X -zz 1 X X ' 1 X X 1 x2. X l 15 X X 1 .z X 1 14 -x lx l -x X 1 1 X X 1 xz X 2. z "1 X X "1 16 (161) -a 3 l -x 1 z z 1 1 Z " xz X X "1. 1 X 1 18 .• CD N lx •Xz -zz lx " 1 1 z X 1 x2 X 1 1 X X xz X 17 -82-a The matrix ~ TJ now has the form final _L_TJ• initial IPrl fl to t-1 -~1 "' ·iD -~-1 tl io t-1 ·tl -to -t-1 atClJ J Gz o,J o,J o.J (15 7 8 4 3 J .fo ~J O:s a,f, ~7 GJ 9 . J Cl6 J ~z J at a:. J J ~8 afs rat? c;, GJ.s of, ofl ~I O:o ClJ9 ,,.J8 o,J7 rafz aft o,J 6 o,J 5 o,J o.J 4 3 o,J 2 (162) .-: In the •am• way equation• 139 were obtained, we get l+x l•x 'itrn· co• "!6 [ -r~- 2 l+x l-x 2 z- 3s+-r 3,- - r ~o- (l•a)3u· (1-x)314 • (l·x)a. 7 - (l•x)J18] (163) -83- 8 ·l• [ '·~-ll~lz •lll-,; e - 1-x z 1-x 2 l•a ~- - r ~s•-r 1~x 2 •6•--r- ~o- (l+x):Ju + (l+x)~.- (l+x)l17+ (l+x)1Y18] (163) -··· (163) A• ln equatln 140., we write lk(W,JI)• 2 ~(x)U~(x)a:(W) UW) l•l+l l wbel'e (164a) ·85- nuJ (l64a) . . . . . .. n J _____________ ........,...._l(l+2XU+l)P ____.....,;1______ ...;_.,;;.. • (l+2)(U+l)Pl·l l(U+l)Pf+z• Jl(.U+l)Pl+l+ 15 J 4.1 (l +2)(21 +1)(21 +3)(l•x) "' . l(.U+l)Pl+Z+ 3l(U+3)Pf+l+ 3(1+2)(2l+l)P1 + (l+2)(21+3)PJ•l nl,•-------------.-..-(-l-+Z-)-(U--+-1)-(U __+_l_)(_l_+x_)_______________ and t he ~l are given tn T able 9 for IC << 1. 0.2 J 43 J 4. J •s J ., ~ J "a J J Cl9 TABLE 9 J J ~0 ~ J J ~z ~3 J ~. J ~s 2 -:62 2 J ~1 J ~8 -1-z -z 317 X 3 t6 X X X X X ~ X . 1 1 -1 1 15 3 '· '1 3 '11 X 1 X 1 X X 1 1 1-x X ~ l·x 32 -x -1 1 1 1 -2 -2 ~ l X 1 -2 X I fox -2 2 2 1-x -2 2 z -1 X -2 2 z l·x - 1-xa -1-x-z -1- -1-x -1- - -l+x- -1-x- 1- -1- 1-xz -l+x- 1-x 3z-x2 3z+x 2 -1-x -2+2z -2-h: '3 - - -2 z - .. - - -2 2 - - -2- --2 ·Z+Zx "•Js - -- 1- 1- -- -- -- -- -- -- -- 1 - -3+x ·Z·Zx l+x - 1 l+x l+x - . ,6 1-x . - - 1-x - 1-x -1+z2 - l+x 1 -x -1 1 -l+x l+x -1 z z -1+x la-x -lx-x l+x -x 2+2.x - - - - -Z+Zx 'a - - -z -2 - - - -Z+Zx Z+Zx •"'• 39 - - - --1 - - - - - - 1 -1 - - - 1+x- -l+x- - - - -3+x2 -3-xz - -1 • to -1 -x -x-x -z+x -x - •z z 1 -x l+z 1•x -z x-x -- 1 'z:Ju ·1 - -1 -1 - -1 - - - - - - 1 -- -x-z l+s -1+x 1 1+x 1-z -1 - -1 - - - - z z -l+z a l+z •z -z -1 z-x -x-x z 1 -z -x-z2 -x+x -x -x - - - - - . - - - - -1 - l+z ·l+z - -1 1 1 -1 -1 -1 - 1 - - - - - - -1 - l+x 1-x - 1 ~8 -1 - 1 J ~ -87From D12, we have then (165) and by D-16 (166) where the J.l!.l are 1 J I<} = 4 (P1 +1 + p 1 ) J 1 K2 =z:~2<- P.t+1+ P,> K~ = ~lj1 +2) ( KJ 4 J ;.m _~ _~ ~ + ~1(1+2) I pl +Z + p1+1 = 4 K7 • ' 1'l"+l ]'l"lJ" - pl ) pl-1 ) urr - 1l1T 1 21?. (- pl+1+ P.f) J 1 K8 = ~ (Pl +1 + p l) KJ ="1(1+2) 1 pl+2 + pl+1 _ 9 KJ 10 2.f2 \ U+3 -u;r pl _ pl-1 ) "Ul! U+r v't;,;+2) I Pl+2 pl+1 pl + Pl-1) 'lTi1' ~ n+J 2 = KUJ = '!1 <- P 1 +1 + P 1 > mr -88J _v't(t+2) l_pt+2 K 13 "4 " \ Zl +3 p.f+1 pt - 'Il'11 - 1T+J + pt-1) -mr (166a) and. the ~~ are given in Table 10 for K << 1. Wo define now the following amplitudes: T 1,10 =1f-l;tl * f-i-1;f1 8 . 8 cos 'I stn'! e -{q~ (16 7) 1;. 1 ;i-1 T 4, 13 f -t-1;!-1 = Z dn 2 8 coe 28 e 2{9' :t Z sin 28 cos Z z8 e i9' 2 T s. 14 = f~-1;t 1 28 e 2 sin 'I cos l e f -·l;tl zt., * Z ein '!8 coe z'Ie e t• J J x4 x5 -1 -1+x 1+x - Zx Zx 6 1+x 8 9 - 17 -Z+Zx 18 1 - 1 J x7 1+x X X J Xs -x X X TABLE 10 J J J J X 9 X10 X11 X12 -1 1 X -l+x 1+x J xl3 1 x+x 1+x - -l+x 1-x - 1 X x X -1 1-x -x - - -1 -1 -x+x X X 1 - - 1 - - 1 -1 1 -l+x - 1+x - 2 -2 2 - - 2 - 2 - -1+x J x6 -1 -x -1 J X14 -1 1-x -(1-x) - 2 J X1s 1-x -1 X (l+x)2 -1-x 1 - 1 1-x J J X17 X18 1 -1 - J X16 1+x -2+2x 2 z 2+Zx 1+x -2 2x -Zx -2.-Zx l+x z X X X X X X X -1 -1 1 X -1 -x X -1+x 1-x (166b) -1 -1 -x 1-x l+x - - - 1+x - - 1-x2 - 2 - 2 2 - - -x2 - 2 - -1 - 2 - 2 -1 2 l+x -x -(1-x) -(l+x) -x -1+x 1-x x+x -x -x -x x-x 10 -1+x2 x+x 1+x -1 +x x-x 11 - - - - - - 2 1 - - -1 -z2 12 - - - - - 2 1 - - 1 13 -2+2x 1+x - -1 2. - - 1-x -2-Zx 2x - - -x -Zx z -1 2x 2 -1 14 -2+2x 1+x - - 1 -2 - - -1+x 2+2x Zx 2 15 - - -Z - - 1 - - -1 16 2 - - 2 - - 1 - - 1 - - 1 1+x 1-x -x+x --1+x2 1+x 2 - - -l+x 2 x+x 7 J xl 1 - - - - - - - l+x J Xz 5 4 2 3 - 1 J x1 1+x I "' • C» -90- ft-1:!0 T 1 16 • 6 • - 2 ain ~ cos T h ,.1. l!o ,1!"1 8 17 = 8 • ein 1 e f -t1iiO z~6 e -·tcps 2 ein z ~(J cos ~lJ e zt., ! =* .. -.Po: ~1 -i· -..., cos z (167) f :t J J ~2. u ~~~ «;s J * c;6 2 ein J -j0;~-1 ~ coa;9 e 2~ 2 lJ J J ~6. 1s • o.7 * c;2 (168) J J J ~•• u = ~3 * «;a J J J ,9,18 = "14 :t c; 1 The partial wave expansion of the T 1'• ia then (169) )] -91- T 4,13 a T s, 14 \ l J 1'4, 13 /..; J•! +~ \ l- J =/..; t3s, 14 J=l +i " pl J1 (1 +2) J + t13, 4 p~· J " p l +1 .J1 (1 +2) J p; +1 ] J1 (l +2) + ~'14, s It (l +2) (169) T I++ l 7,16 :Ill Jd IJJ p " l 7,16 -/1(1+2) " J p 1 +1 ] - Jll6, 1 .fi. (1 +2) p; J +1 - t318, 9 ..;, (1 +2) J Using equations 163 we get: (170) 2 k k 1 T 3 • (2x:J + 2:J 2 + a:J + Zx3 + Zx 3 + 23 + Zx:J + z3 + Zx~ ~ 1 3 4 5 12 14 16 mm 8 k k' T = (2x:J6 + 237 + Zx38 + 2x39 + 2x2 :J10 + 2311 + 2x313 + 2315 + 2x~17 ) : ; 12 1 -9Z- Tl4• ~+ x35+ 310+ 2318 2 T • (.fZx"t +.fz x"J3 +.fz x 3~,+.fz :J +.fz 3 8 +.fZx:J10 + z.fz <J12 +Z.f2 x3 6 14 6 + .fz 317+ .fz x:J18) ko m T15• (.fz ~ + .fz 33 + .fz x:J5t·.fz x'36 +.fz x:J8 +.fz x 2 w10 + z.fz l\t ko + 'l.fz a::Ju + .fz x317 + .fz 318) m T 7• (.fz ko 't+ .fz 33+ .fz •'s+ .fz 'ta) Iii (170) ko Tt6• c.fz 36+ .fz 3a+ .f2 ·~o+ .fz ~7) iii T 8 a (.f2 a:J + .fz xl4 + .fz x'~-1 + .fz • + .fz 3 + .fz x:J10 + .fz a 9 1 6 13 5 k' + .fz x31,,.. z./2 \fl6 + 2-fz X~S) ~ T 17 10 a (.fz a,_+ .fz :J + .fz a3 + .fz x3 + .fz x3 + .f2 x'~-3 + .fz xJ 4 5 6 13 9 k' + .fz 314 + 'l.fz. 315+ z.fz x317) ~ T • (-.fz 3 - .fz 3 • .fz 6 9 9 k' xf\o • .fz 313);;. k' T18• (-.fz W,.· .f21. • .fz x35 - .fz w,_ 4 );;. P"l"Om equatf.ona 159, 160c ancl 170 we c:an flncl ctn) -93· wheJ'e we can wl'ite (17la) The coeUic\eate c~). eg) a.ncl cl:) are alven ln table• 11. 12 aDdU reepectlvely where we have aaain eet (17lb) i • 1, z••••• 9 (172&) l • 1, z••••• 9 (172b) t • 6, 1, a. 9 The pnjection ope ratol'e for ~.10 • ~ J Cl72c) IJ~ an: r.: ( T1,10pl + T10,1Pl +1 1 slIT 15 z.. u. 1 ·1 \ 2, u )c1x 3.f :Pl +l+(J+Z)PJ.l Zl +i ( . (173) ~.lZ • ~ r.~ T 3,1zl'l + TlZ, 3Pl+1) dx 1 rt /. .fl (J +Z) (P.f +2- p.f) J '•· u • 'I J •1 \T-6,. U Zl +3 - T 13 • 4 /1 (J +2) (P1 +l• P.t _1)) ZJ+t dz . -94- 1sl <. J 1Ss,l4 • 'I -l T 5, H ~.f (t +2) (PJ+z• P 1 ) Zl+3 ~.f(l+2) (PJ+l .. pl-1)) - Tl4,5 Zl+l dx 11!,15 • ~S~1 (T6,15Pt• T15, 6PH1) dx J 1 s -/1 (J +2) . (PJ +2- p l) 1 ( P7, 16 • 'I ..1 T 7, 16 1'~,17 • ~ Zl +3 (173) s.: ( T8,17 pl. T17, 8PJ+1) dx J {t(l +2) (PI +z• PI) 1 sl ( IS9,18 • 'I _ T9, 18 Zi +3 1 Remark• etmUar to thoee at t he end of eectton b apply ln caae one of the parttelee haa aero mae a. It would correapond, for lnetance, to photoproductlon of 2 or 3 plone in a reaonant etate. T9 Ta T7 T6 Ts T4 Tl Tz Tl wWm +fz wWm' - 4/Z wWm' 0 k' (W+M)R- 0 k 1 R- 0 4{2 GWm 1 0 k' (W+M)R- 4{2 v Wm 0 4-/Z wWm 0 k (W-M)R+ - k (W+M)R.. - 4/2 wWm' k'R• 0 -4/2 k a+ 4/2 •Wm 0 k R .. •• • (W+~R- R- iww (W+M)R• • 8ww mm' lW-MUE+~!E'+MlR'lw lw .. (W.,.A+ •• .. (W+,R- a2 '~itt a· 4w mm !E+Ml~'+M.~R- -lri' R+ GVt a· ~ al 1 0 0 - 2-lz •Wm 1 l:{z •Wm 1 Jr.' a· .. (E'+.M)R- wWm 0 - 2/z k R+ z.lz •Wm _ (E+M)R- r.r I "' R.. ..0 • a· ~ GWmm' 0 ( (E+M)(E'+M-k )•k (W+M)) a· -~ R+ R. -lW'Vi TABLE U T5 T9 Ta T7 0 0 0 IIK'w (E-M)(E'·M)R + 16WW a/z wWm' a/z vWm' 4/2 wWm' lt (E'Hd)(W+M+lt' )Ro 0 a.fz wWm "'a:"•2lt0 R - 0 0 (E -U)[ lt~(.E +M)- Wk ) R + +12 wWm' + 8{2 wWm 4/2 wWm' ( m' 2 +k1 (W+M)) R. k (E'+M)R- S.f2 wWm (E+M)(E+M-k0 )R a7z wWm o (E+M)( k~(E ·M)- Wkc) B- 161FW o o o (E+MHE-M-k )R- IwW (E+M)R- (E +M)(W-M+k' )R- l .w (E+M)lE'+M)R- (E'+M)(E+M-k )R- 4.fz wWm ( m +k (W -M)] R + 2 •72 wWm (E+M)(E'+M-k' )R• k R0 r.W" o k'R- r.w T6 - a.. w (E+M)(W•M+k' )R. a, [m2(E'+M)+(W+M)(k'W·Mk )+2Mk k'1R- k (E'+M)(E+M-k )R- k (E'+M)[k'(E+M)·Wk )R. o o oO" o o o o o - 81rWmm· 8wWmm' 4•Wmm· T4 T3 (W-M)R+ Tz Gw r.w (E+W)R- (E+M-k' )Bo . Tl as a,. TABLE 11 (Continued) I 1 o- ...0 0 - a.fz iW·MUE'+~Ra/z wWm' Ta T9 •Wm 1 (W -MUE'+W)R- - .a.f2 wWm k'(E+M)R- - !E'+M)R'"' lww 0 - 8wWmm 1 k' (W +M)(E+M)R'"' - iE'+MlR• lww T7 T6 Ts T4 Tl T2 Tl Q7 8wmm' 0 - 0 0 - a.fz wm' k' (E'+M)R .. s.fz 1r111' k' (E'+M)R'"' - .alz wm k' (E+M)R'"' ~11' (E'+Mlll'"' 0 k' 2(E+M)R'"' (E'+MlR'"' - 111' - Oa •• 0 0 0 - ... (W+E)(E'+M)R4-.fZ wWm' &.fz wWm 0 - a.fz wWm' a/z wWm' t' 2(W+E)R- - 0 (W+E)(E'+M)(W+M+k' )R'"' a.fz wWm 0 (E-M)( k~(E+M)-Wk J R + 0 a/2 •Wm (E-M)(E+M-k )R + a/z wWm 0 (E+M)(k'(E-M)-Wk )R- 16wW 0 (E+Y)(W-M+k' )R. - (E +M)(E'+M)R'"' u.w 8wWmm1 0 (W+E)(E'+M)(k'(E+M)-Wk )R 0 t&WW (E+M)(W -M+k' )R'"' . . (E·M~~E'·M)R ,..w + - ~0 (E+M)(E-M-k )R- - (E+M~R- .. - lwwmm' (W+EXE'+M)f£+M-k )R- - (E+lfJR'"' ~ TABLE 11 (Continued) I I ~ -D -· ·~ lw - Ts T9 Ta 8t2 wWm' • (W ·M)(E•+MtR• a/z wwm• 11'111 1 0 811. 11'111' k' (E'+M)ll• - HZ • 161'2 wWm' 0 k~ R .. 1612 wWm' .- k' (E'+M)a· (W -W)(E'+M)a· 0 ~··· -1il/z wWm "~0 0 - ~a· zwW - T7 (W -M.){E+M~~'+M)R.. .... 16/zwwm• ~ \,(vi_i:M).lt- 0 - u~/z wWm' k ~ (W+M)R• 1 0 - lWZ wWm .. k~16/z(W-M)R+ wWm k~~(W+W)R• . 3z.. J 11' J!1+M~E'+ Wl.a· sw lW·!!}(E+~E'+!!!R• iGWmm' 'E+M~E'+W)R• IGWmm' 0 0 k k' (W-M)(E+W)(E'+M)R• 0 k. k 1 (E+W)(E'+M)Jl• 0 • (W+M)'fj!fJ(E•-M)a+ z... ~-M~~E'·~B + J!. • (W•M)(E+!'fPE'+M)R- .. iE+M~~E'+!!lll wW ~. Gu 16/z ... wm lW+E'llE+M~a· • .. (E~~- 0 k' ( W+E .. E+M)R• .'nDm. •• tE'+Mla- O,.z +lzsm 4/z wWm T6 1W+E'UE+~R· •• (E'+t'ja· wWmm' T4 - (E'+M·R· Tl .. fW +E-'!'W +MUE+M! R- Tz Tl On TABLE U (Contin•d) I • ao ..0 lE-M~E'-MIR.. Tz .w (E+M)~E'+W)R• .. k2(W+E')R.. lW -MUE+Wl\f '+Mia· ~(W+E')(WW)R.. IE+~E'+MlR- .. klW+E')a· 16 Z wWm kt;+E')R t 1 2wWm Ts T6 T7 T 9 T8 tlr/2 wWm' .. r' 2(W+E)R.. 1612 ..wm• k 2(W+E)R.. z..w 1612 wWm 16{2 wWm 1612 wWm' k'~0 a· • k' 2(W+E)(W+M)R- 16/z wWm' t6.f2 •Wm' 0 - k' 2l W+E)!W+M!R16/z wWm' k'~ a· .. k2(W+E')R + 16/Zw'Nm zww k2(W+E'UW·M)R+ t6fz 11Wm lZw j • IGwmm' .. (E+M~~:ww,a· Wmm iW·MUE+aE'+!!!R- j .. w l 0 k (W+E')(E+M)(E'+M)R.. 3 .. (E·M)iE'·M)R + (E+MyE 1 +M)ll- T4 T3 16.. mm IW +MUE-M~E'-M!R + Jz.. 3 .w CE+M)jE'+M)R- ~7 lW+EUW+E'UW·N~E+~IE'+!:!}R- .. • (W-W)(~z!*E'+MJR.. ~6 lW+EUW+E~+M),E'+M)ll .. zww 3 ..w Tl .. IE+M~IE'+M~R- ~ TABLE U (Continu.d.) • ..0 t ..0 .. SE+M~E'+MlRj _ lW -Ml~E+Wll.E'+M~R- !. (W +E')(W +M)ll.. t 216/2 wWm T5 T6 T9 T8 T7 3 .w - k'Z.:0 (W+M)R.. tltlz ..wm' lMZ wWm' k ~ CW+M)a· · o 1 - 16/2 wWm' k' 2(W+E)R.. uJz.wm' ...z k (W +E)R. 1612 wWm lf¥(2 wWm - k~'0 Jl+ k2(W+E')(W -W}R+ -16{2wWm kz.:'0 a· .w - SE+M~E'+M!R- IGWmm' .. SW ·MUE+M~E'+M!R .. . Jzw 16trWmm' T4 T3 . l&rwmm' I6{z.wm• k' 2(W+E)(W+M)R.. 2(W+E)(W +M)R• -k• uJz.wm.t Jhf2.wm ~k'(W-M)R+ 0 16/'l.wWm ~'(W+M)R• 0 .. (W•M)(E+M~E'+M)R .. Jzw izw .. (W-M)(E+~(E'+M)R .. 0 lt' (W+E)(W•M)(E+M)(E'+M)R- z.w 0 It' (W +E)(E-+M)(E'+M)R• k (W+E')(W •M)(E+M)CE'+M)a· 0 iW+M)(Eit!lhE;'•M)R t - SE-M!lE'·M~R + j ww SW+~r=·MU!'-MlR t T2 2ww {W•!')(E+M»E'+J.(}R• 12w s l!+!!lJE'+M)R.. azo lW -~~E~SE'+MIR.. (\, Tl '\a TA5LE U tContlnued) f • 0 0 a· k'(W+M)R+ It'll+ +fz wWm' Tl8 0 wWm' +lz wWm' 0 0 4(1 k'CW+W)R+ 0 k (W+M)R- ~ '.t~z wWm +fz wWm' 0 Tl7 k'R+ 4(ZwWm k0 .- ... HZwWm 0 - wWm -+fz k •• -r.vr ... T16 TIS Tl' 0 • (Wi:1JR+ a• k (W-M)ll+ _(W•!fiB.+ lw -r.r a• iW tM~E-M!!E•·MIR + wWmm* • !E-M~rt-~R+ iw · mm Tll Tu - (W+~a· J;W"' •• - r.r Tu TlO Gz - (W-M)R+ 8wW a• '\ Gl 0 2:/z wWm' 0 k'R+ z{z vWm' (E'·N)R i: z{z wWm 0 It a· Z ZwWm ~-M)R+ ·r.w B.+ :a• .. r.r GW'mm' 0 [ (E-:W)(E'-M·k' )·k (W ·W)] R + 1'iW" It- a• 4;W TABLE 11 ( CODtiuud) 0 ......•• Tl8 Tl7 Tl6 TlS Tl4 Tl3 Tl2 Til TlO 0 0 0 0 0 0 4{2 wWm 0 1 4{2 1rWm 1 (m 12 tk 1 (W-M)]R+ (E 1 -M)(E-M-k )R + 4{2 wWm o [m 2 +k (W+M)]R- 4{2 wWm (E-M)(E 1 -M-k 1 )R + 41rW Sw 0 + o 0 4{2 wWm' k (E 1 -M)R+ 8{2 wWm (E+M)(E-M-k )R- 8{2 wWm (E-M)( E+M-k )R (E-1R+ 0 0 0 0 0 0 0 8{2 wWm 1 0 8{2 vWm' 12 k k R+ k (E'-M)(W-M+k )R + a.fz wWm (E+M)[ k~(E-M)-Wk ) R- 8{2 wWm 0 0 (E-M)[ k' (E +M)- Wk 1R + I6iW 6.w SwWmm' (E-M)(W +M+k' )R + 8wWmm' k (E 1 -M)(k 1 (E-M)-Wk 1R+ 16vw k R+ 4iW 0 0 0 (E+M)(E '+M)R- 16wW (E-M)(W+M+k 1 )R + a6 (E-M)~E '-M)R + 0 0 k (E'-M)(E-M-k )R + s.w (E-M)R + . as k 1 R+ '11rWmm' 2 [ m (E 1 -M)+(W -M)(Wk 1 +Mk )-ZMk k 1 1R + 4•W (W+M)R- 4trW 0 (E -M-k 1 )R + a4 TABLE 11 (Continued) I I 0 N .... Tl8 Tl1 Tl6 TIS Tl4 Tll TlZ Tu TlO 0 0 8{Z wWm 1 (W+M)(E'-M)R+ 8f2 wWm' (W+M)(E'-~+ 47Z wWm k' (E-M)R + SwW (E'-M)R+ s-wwmni' k' (W -M)(E-M)R + 0 0 8..fZwWm' 8{2 wWm' k' (E'-M)R + 0 k'(E'-M)R+ 4{2 wWm k'(E-M)R + Sw (E'-M)R + Bmnm"' 0 k' 2 (E-M)R+ 0 0 (W+E)(E'-M)R + 4/z wWm 1 8{2 wWm o (E+M)(E-M-k )R- 8{2 wWm (E-M)(E+M-k )R + Sw (E-~R+ BwWmm' (W+E)(E'-M)(E-M-k )R + 0 0 0 0 &{z wWm' k 2(W+E)R+ a{z wWm' I (W+E)(E'-M)(W -M+k' )R + 8{2 •Wm (E+M)[ k~(E-M)- Wk0 ] R- 8..fl wWm 0 (E-M)(k1 (E+M)-Wk ]Rt lOiW (E-M)(W+M+k' )R + .w (E-MllE'-M}R + 8wWmm' 0 ( W+E)(E'·M)[ k' (E-M)- Wk ] R + (E+M)(E'+M)R16wW lfnrW 8ww 8w SwW 0 (E+M)(W +M+k' )R + ~0 (E-M)R+ a9 (E'-M)R+ a8 (E'·M)R- a7 TABLE 11 (Continued) I u. I .... 0 (E '·M)R + (E'-1R+ Tl6 Tl8 Tl7 - Tl5 8{2 wWm ' _ ( W+M)( E '-M)R + 8{z wWm' ( W+M)(E '-M)R+ ( W+E ')(E -M)R + 4/2 wWm Tl4 - 0 0 - a{z 1r111' k' (E '-M)R + s.fz vm' k 1 (E '-M)R + - _ ( W +E')(E-M)R + 4/2 vm 8,.. Smnm 1 - 811' ' k' ( W+E ')(E -M) R + ( W+E '~(W -M)(E- M )R + wWmm 1 - Tl3 Tl2 - - Tu 0 a. _ (E '-M)R+ - (E'-M)R+ Sv W ~2 TlO <lu 0 16.f2 flWm' 0 ,..wm' 2 k' k R+ - t6.fz 0 16..fz wWm k-·~ R + 0 k~'R- -16/z •Wm £~'R+ 2vW wWm ' 16.fz •Wm' 0 k--~ ( W- M)R + 0 - 16/z k~ (W-M)R+ 0 - 16/z vWm k~'( W -M)R + 0 - 16..f2 flWm k~' (W-M)R+ 32vW I61r Wmm 1 {W+M)( E -M)*E ' -M)R + JZ1r ' 0 _ (E -M)JE '-M)R- 0 ! W+MUE -MU E '-MlR + 16vWmm 1 k k' ( W+M)(E -M)( E '-M)R + - iE-M)(E'·M~R32,w 0 z,w k k' (E -M)(E '-M)R + 0 _ (W- M)(E+M)~E'+M}R3Zv _ iE+M~(E'+M~R- 3lYW _ ( W+M)(E -M)(E '-M)R + Jlv W '\. ~E -MH E '-M!R + ~3 T .ABLE 11 (Continued) I I ~ 0 ~W+M)(E-M)(E'·M~R+ 3ZYW ( W+M)(E -M)(E'-M)R + + - (E-M)~E'-M)R 3z.w _ (E·M~(E'-M)R+ TlZ T13 T18 T17 T16 k2(W +E')R + T15 16{z •Wm' _k' 2(W+E)R + 16/z wWm ' k 2(W+E)R + 16{2 fiWm 16{2 ftWm _k2(W+E')R- 3 tr W T14 16wWmm 1 - 16{2 YWm' 0 - 16.fz y'lfm' k'~ R+ 16/z wWm' 16,fz wWm' "z(W+E)(W -M)R + 0 16/2 •Wm k' 2k R + -z + • k (W+E')R 16/z •Wm k2(W+E ')R- 3211'W (E-M)(E '-M)R + k' 2(W+E)(W-M)R + 16{2 wWm k2(W ~E')(W+M)R- 16.f2 •Wm k2(W +E')(W -M)R + 3ltrW 0 ~E-M~(E'-M}R + 3lwW - (W+E)(W+E')~Yi +M)(E-M)(E'-M)R + 6wWmm 1 _ (W+E)(W+E'~E-M)(E '-M)R+ 16w mm 1 k {W+E')(E-M)(E'-M)R+ (E+M)(E '+M)R3lwW 32.. 3z... w {E-M)(E'-M)R + _ ~ W -MUE+M~E' +M ~R- 3 .w 32wW _ !E+M~E'+M~R- Tll TlO - + ~7 - ~W+M)(E-M)(E'·M~R 32wW '\6 (E-M)(E'·M~R + ~5 TABLE 11 (Contin11ed) I "' I ....0 0 0 T18 T17 Tl6 TIS k 2( W+E)R+ 16/z wWm ' kk0 ( W- M)R+ 16/z .. wm' 16/2 •Wm' 16.fz .. wm' 16fz.wm k 0 2k' a· 16{z .. wm t' 2( W+E)R+ 0 16{2 vWm 0 k~'R+ k ~ ( W-M)R+ 1 t 2 ( W+E')( W+M)R- 16/z .. wm t 2( W+E')( W-M)R+ 32.. w 12.. w (E-M)(E'-M)R + ( W+M)( E -M)(E'-M)R + Tl4 {E ·M~(E'-M)R + 3 wW 3z.. w {W +M)(E -M)( E '-M)R + Tu l61rWmm' k' ( W+E)(E-M)(E ' -M)R + 3lwW (E+M)(E '+M)R- z.. w (E-M~lE'-M)R + 16vWmiil' k ( W+E')( W+M)(E-M)(E'-M)R+ 3Z•W ( W-M)(E+M)(E '+M)R- 32.. w ( W+M)(E-M)(E'-M)R + ~9 TlZ Tll TlO ~8 TABLE 11 (Continued) 32.. (W+M)( E -MME '-M)R+ 32.. w ( W+M)(E -M)(E '-M )R + 16.. Wumi' + 16.f2 1rWm' k 2( W+E)(W - M) R + 16.f2 •Wm' k 2 ( W+E)(W -M)R + 1612 •Wm 0 kit' ( W+M)R -2... 16.f2 •Wm k~'0 ( W-M) R+ 0 k' ( W+E)(W+M)( E -M)(E '-M) R + ( W -M)( E+Mi(E '+M)Rj .. w (W+M)( E -M)(E '-M)R + 32•W 0 zo I t 0 0" ~ T9 Ta T7 T6 T5 T4 Tl Tl Tl 0 -4/z •Wm' 0 k'R+ 4{Z trWm - 0 k'Jl+ .. iwWmm 1 0 k k 1 R+ .. 'li'W R+ ~ :t.. wmm' 0 .. 4{Z •Wm' 0 k'(W-M)R+ k {W- M)R+ 0 0 k k' ( W- M)R+ Sw - Z:lz wWm' k' R+ 0 -2:/z vWm 0 k R+ - z... wmm' 0 0 k k' R+ .. 4i"W R+ ( W·M~R+ - +/2 wWm . • al az 4wW 0 (E-M·k' )R a,. 2 - - •lz •Wm' £'' 2+k'(E'·M-Zk ))R+ 0 0 •lz.wm [k2 +k0 (E·M-lk'0 )) R + i1rWmm* ( m 2k'o +m'~o )R + - TABLE lZ 0 0 0 i/z wWm 4/Z 1rWm1 k' (E·M)R+ - (E-M)(E+M-k )R + 0 )ll + .. k' (E•M)(E+M-k 8w'Nuun* a• (E·Mw+ as I -..1 0 ...• n T9 Ts T7 0 0 a.fz vWm 0 0 8.. wmrn 1 0 8.f2 'II'Wm' k' (E-M)(E'·M)R + 0 8/z vWm' 0 s/zwWm' - 4{2 •Wm • (W+M)iE'·M~R- 0 0 k (W+M)(E'·M)R'• 8ww iE'·M)R+ .. k (E '-M)R+ .. - Q7 (k' ( W+M+k' )(E·M)·k~ ] R"' a.fz vWm k (E+M)(E'+M)R- 0 (It (k' +M)·k' (E+M)) (E·M)R+ _(E-Ml!;~·M)R~ Ts t6.. W (E·M)(E'·W)R+ .. T6 2 0 8wWmm1 00 [ W(E-M)k k'·k~' -k'la~R+ 16iw (E+M)(E'+MJR .. 8. (E·M)R+ T4 T3 T2 Tl Q6 ~ - 0 - 4{2 1l'm 8{2 •m' k' (E '·M)R• - 0 fhrmm' 0 k (E'-M)R+ 0 k k' (E '-M)R + - {E'·M)R+ 8. TABLE 12 (Continued) 0 - 4{2 vWm' k' (E·M)R- - a..fz wWm 0 8wWmm1 0 (E·M)(E+M)·k )R + 0 k' (E -M)(E +M-k )R + .. (E~~~R+ 09 I I Q:l 0 - T9 Tg T7 T6 - 0 0 a/z vWm 0 + .. It' (E·M)(E'~M)R SJz wWm' &/2 wWm' 1 (k 2(W+E)+k0 (E-M)(W+M+k'0 ))R+ - lt (E+M)(E'+M)R- &.fz wWm 0 ( 1t (k' +M)·k' (E+M)} (E - M) R + 0 a/2 1 (W+M}(E'·.M)R+ wWm' - +fz wWm k (E'-W)R+ Ts 0 .. (E · Id\(E ' ·.M)ll+ 6wW - lwWmm 1 w 0 lt (W+M)(E'·M)R+ •• a.. 0 - lwmm1 0 s/z wm' k' (E'·M)R + 4{2 tim Jr. (E'·M)R+ 0 0 lt It' {E'•MtR+ Uf'•W)R + (E'-~R+ (E·M~'-M)R+ {lt~(E-M)( lt~(E+M)-WkJ ..k'~0(W +E)}R + 8,..Wmm 1 I wW • {E+Ml(E '+M)R. a• .. lE-M)R+ ~z flu T4 Tl Tz Tl ~0 TABLE lZ (Couttn\led) t ....0• -o T9 T8 T7 T6 Ts T4 16/2 •Wm' 0 1612 vWm' k~(E-M)(E'-M)R + 0 ( k (E '+M)+k' (E-M)) (E '- M)R + k (E+M)( E'+M)Ro 161'2 wWm 16'12 wWm 0 16'12 .,wm 0 0 0 16.fz wWm' t6.fz .,wm' k' (W+M)(E-M)(E'-M)R + 0 [ k ( W -U)(E'+M)·k' (W+M)(E-M)) (E'·M)R + 16.fz •Wm k (W-M)(E+.M)(E'+M)Ro 0 [ k (W+M)(E'·M)-k' ( W-M)(E+M)) (E-M)R + 0 [k (E'·M)+k' (E+M)) (E·M)R + 3zvW (W+M)(E- M)(E'-M)R + llvW (E·M)(E'·M ) R+ 3ZwW 3 .. w (E-M)~E'·M)R + (W+M)(E-M)(E'-M)R + 16vWmm• 0 16w Wmm' 0 T3 3 l'fl (W -M)( E+M)~E' +M)R- (k~'0 2 +k'la02 )(W ·M) R + .w (E+M)V:'tM)Rj - ~. (k'~' z+k'~Z)R + Tz Tl f\3 TABLE 12 (Continued) t • 0 ~ ~ T9 T8 T7 T6 - 0 0 16/z 1rWm' k~(E-M)(E'-M)R + 16/2 wWm 1 [ ( W+E)(E'+M)·k' ( E - M))E '-M ) R + 16/z yWm k (E+M)(E '+M)R - 16/z wWm 0 16.fz flWm 0 - 0 16..fz wWm' k ' (W+M)(E-M)(E'-M)R + lWZ wWm' [ ( W+E)(W - M)(E ' +M)+k' ( W+M)(E- M)) (E '-M)R + 16/z 1rWm k ( W -M)(E+M)(E '+M)R- 0 [ k (W +M)(E '·M)+( W +E ')( W - M)) (E - M)R + 0 [ k (E '·M)·( W+E ')(E+M))CE - M)R + 3 'flw (W+M)(E ·M~(E' -M) R t l .w w _ (E · M)!E '-M)R + Ts 3 ( W +M)( E - MtPE '-M)R + lZw _ (E ·M~E '-M)R + T4 0 16.. Wmm' 16wwmm ' 0 [klk'(W+E')+k'~ (W+E)](W-M)R+ 0 [k~' ( W+E ') +t ' 2 ( W+E)) R + 3z. ( W- W)( E+~i E '+M) R- ( E +M)(E ' +M)R- 3z.w ~6 Tl Tz Tl ats TABLE lZ (Contim1ed) • t t: - T9 T8 T7 T6 Ts T4 T3 Tz T1 utlz .. wm 0 0 16/z vWm' 16/z •Wm' k' ( W+M)(E- M)(E'-M)R + 0 k' (E-N)(E '-M)R+ - 0 [ k (W-W)(E'+:M)-k' ( W+M)(E-M)](E'·M)R + 1612 wWm' 0 lll.fz wWm lll/2 11'Wm' 0 0 k (W -M)(E+M.)fE'+M)R"' 0 k (E+M)(E'+M)R"' [ k (E'+M)+k' (E-.M))CE' -M)R + ... 16-fz 11'Wm lfdZ •Wm 0 [k (W+M)(E'·M)+(W+E')(W-M))(E -M)R + 0 ( k (E'·M)-(W+E ')(E+W)] (E·M)R + Jz. J2., ( W+M)(E -MUE '·M)R+ 3 .w z..w (E·M)jE'-M)R + 16.. Wmm' ( W +l.4)(E-M~E '·M)R + 16.-wmm* [k~'0 (W+E')·k'~02 ) (W-M)R+ lZ1F (W-M)(E+MijE'+M)R- ~ (E-M)1E'·M)R + - w ( k~~(W+E')-k ~~~ 1l + 3 _ (E+M!!E'+M)R"' ~7 TABLE 12 (Continued) • • N - 0 0 0 [ ( W+E)(E '+M)·k' (E · M))E '·M)R + - 16tf2 . wm k (E +M )( E ' +M)R- 1~/z • Wm T8 16-/2 16{2 wWm' k~(E-M)(E'-M) R + T9 vwm• T7 T6 0 [ k (E t· M)+k' (E +M)](E - M ) R + .w (E - MYE '-W)R+ Ts j i2.W (E - M )( E '-M !R + JSW T 16/z 1rWm 0 16,(2 • Wm' 0 16{2 1rWm' k' ( W+M)(E - M)(E '-M)R + - 0 [ ( W+E )( W- 1\t)(E '+M)+k' ( W+M)(E · M)) (E '-M)R + - 1W2 ..wm k ( W- M)4E +M)(E '+M) R. - 0 ( k (W+M)(E 1 ·M)-k~( W · M)(E +M)) (E - l.4)R + l!w .. ( W +M)( E-M~E'·M)R + - 0 16iwmm1 (W+M)(E - M)(E '-M) R + 0 [k2k' 2 -k'~ ( W+E )](W- M)R + [ k~'02 -t' 2k 0 ( W+E)] R + 16vwmm 1 _ ( W- M)(E +MUE ' +M)R.. 12. 0 zo ' +M)R- (E+M)(E 32.w T4 T3 Tz Tl ~9 TA BL E 12 (Continued) I w. I - Tl& Tl1 T16 TlS 41z wWm' 0 k'R .. 4/2 wWm 0 • .. 0 4{2 •Wm' 0 k' ( W+M)R- 4{2 wWm k (W+M).R.. - - Tl4 k R .. - - Tu 0 2/2 wWm 0 k'a· liz •Wm k a· - - zwWmm' 4wWmm1 0 0 k k'R .. 0 1 0 .. - 4-fz wWm' 0 0 .... z+lt'(E'+M·2k ))a· (k 4.f2 1rWm [k~+k0(E+M·Zk;,l a· .. .. Gwmm' (m~~ +m 1 ~0)R .. - GW (E+W-k' )R .. n· 4w'W'" a• Gl k k' (W+M)R.. 0 • a.w 'lw'IOmm' 0 0 k k'R- hW" ( W+M~R - Ql Tl2 Tll TlO a· ~ T .ABLE ll (Continued) • 0 8wWmm1 0 +fz wWm' 0 k' (E+M)R• (E+M)(E-M-k )R- 0 - H2 11'Wm - k' (E+M)(E-M·k )R .. (Ei~~lt- «s • • -..... - TlZ 0 2 2 0 0 8. 0 - - Tl6 Tl7 0 afz •Wm 0 0 - 0 afz •Wm' k' (E+M)(E'+M)R- 8/2 •Wm' 0 [k 1 (W- M+k' )-k' 2tc) R. a{z .wm k (E-M)(E'·M)R + 0 wWm - ... fW-M)(E'+M)R ... 8t2 wWm' - - 4/z 0 ... 8{2 wm' k' (E '+M)R• - -i{z wm 0 k (E'+M)R. k (E'+M)R- 0 [k (k' -M)·k' (E -M)}( E+M)R'"' 0 .. - &.w 6wW iE+M)lE ' +M)R ... Svmm 1 k k~(E'+M)(.R. • !E ' tM~R- ... .. k (W-M)(E'+M)R'"' s... wmm 1 - §, (E'+M~R- Ga - - T18 0 A,wmm' 0 - a7 iE+W)~E'+M)R- TlS Tl4 (E -Ml(E'·M)R+ I wW ( W(E+M)k k' -k~' -k'~ ] a· - Tl3 iE+M~R- - s. Tll TlO a6 TABLE 12 (Conti.Dued) 0 8vwmm' 0 0 - 4.fz.wm' k' (E+M)R- - 8{2 wWm (E+M)(E-M-k )R'"' 0 k' (E+M)(E-M·k )R- .~ _ (ES~R- a9 I I \.n ~ ~ T18 Tl7 T16 TIS T14 Tu T12 Tu TlO - 0 0 - 1 ffW (E +M)lE'+M)R- 1 .w 0 0 812 .. wm• k' (E+M)(E '+M)R- 8{2 wWm' [ k' 2( W+E )+k~(E+M)( W-M+k;)l R- • s/2 'flWm k (E-M)(E'-M).R + s/2 'fl Wm 0 [ k (k' - M)-k1 (E-M)) (E+M.)R- 0 s.. wmm' _ (E+M)AE'+M)R- 0 {k' (E+M)[ k' (E - M)-WkJ .. ,·~ cw+E)} a· 1 'flw (E • .Mt(E'-M).R.+ a. (E+M)R- ~0 - 8wWmm 1 8{2 .. wm' (W-M)(E '+M)R- 4..f2 wWm k (E'+M)R0 0 k ( 'N·MHE'+M)R- 8wW' (E '+M)R- Gu TABLE 12 (ContiD\led) 0 0 - 811'DUn' 0 0 - e/z t~Wm' k' (E'+M)R- •..f2 wm k (E '+M)R- - - k k' (E'+M)R- _ (E~:M)R- ~2 • 0' I t: 16 Tl8 Tl7 T Tl5 Tl4 Tl3 T12 Tu TlO t6.fzwwm 0 o 161'2 wWm' k' (E+M)(E'+M)R- [It (E'-M)+k' (E +M)) (E'+M)Ro 0 1612 wWm' 161'2 wWm k (E-:W)(E'·M)R - _o;;..... _ _ _ __+ 0 [k (E'+M)+k' (E-M)) (E+M)R .. w o 0 0 o 161'2 vWm' k' (W-M)(E+M)(E'+M)R- t6.fz t~Wm' [ko(W+M)(E'-M)·k'0 (W-M)(E+M)) (E+M)R- 16/i. t~Wm k (W+M)(E-M)(E'-M)R+ 16-12 wWm ( k (W ·MUE'+M)-k' (W+M)(E-M)] (E+M)R- Jzy {W -M)(E+M~E•+M)R­ !E+M~E'+W)Rj .. 3 .w (W -M)(E+MW:'+M)Rllw n ... wmm• (k~'o2 +k'~02 )(W+M)R- 32. (W +M)(E·M~E' -M)R + ~. (E +MJ:(E'+M)R• t&.wmm' ci~·0 2 =tk'~02 )a· 3 ww _ (E-.M,(E'·M)R + a.3 TABLE 12 (Continued) • • t= ..., 16trWmm 1 Tl4 - - - Tl6 Tl7 T18 TlS (E+M~(E '+M)R3 vW 16vWmm 1 0 0 0 1612 •Wm' k' (E+M)(E'+M)R- 16/2 wWm' [ ( W+E)(E '-M)-k 1 (E+M)] (E ' +M) R- 16/2 •Wm 0 - 0 16{2 •Wm' k' ( W-M)(E+M)(E '+M) R - 16{2 'fiWm' [ ( W+E)( W+M)(E '-M)+k' ( W-M)(E+M)] (E '+M)R - 16{2 •Wm k ( W+M)( E -M)( E '-M)R + k (E-M)(E '-M)R + 0 ( k ( W - M)( E '+M)~ W +E')( W+M)(E -M)] (E+M)R- 32vv ( W-M)(E+M)~::!:: ' +M) R - 3itrW 16.fz •Wm 0 1 ( W-M)( E+M)(E '+M)R- 1 [ k~0 ( W +E ')+k ~0 ( W+E )]( W+M)R- 3lw \ ( W+M)(E - M~(E' -M)R + ~6 16/2 •Wm 0 [ k (E '+M)-( W+E ')(E -M)] (E+M)R- 3 wW ( E +M~lE '+M)R- - 1 [ k~0 ( W+E ')+£' 2k 0 ( W+E)] R- '-M)R+ - (E -M)(E Jl1rW Tl3 T12 Tu TlO ~5 TABLE l l (Continued) I I :::00 Tl8 16{2 wWm [ k (E'+M)-( W+E ')(E -M)) (E+M)R- 0 16{2 wWm' 0 ( k (E'-M)+k' ( E+M)) (E '+M) R- 16{2 'fiWm 0 16{2 wWm' k' (E+M)(E '+M)R- - 0 k (E -M)(E '-M)R + - 0 0 0 16.fz wWm 0 k ( W+M)(E -M)(E '-M)R + 16{2 wWm ( k ( W-M)(E 1 +M)+( W+E')( W+M)(E - M)) (E+M)R- 0 16.f2 trWm' k' (W-M)(E+M)(E 1 +M)R- 16.fz trWm 1 ( k ( W+M)(E '-M)-k' (W - M)(E+M)) (E'+M)R- - - 32w 3zvw 0 _ (W-M)(E+M~E'+M)R- (E+M)(E '+M)R- l6vWmm' 32ww - 12 _ ( W-M)(E +M)(E '+M)R- 16vWmm' 1 [ k~'0 (W+E )-k k 20 ] ( W+M)R- 3 wW Tll T17 2 I - M) R + - ( w +M)(E-M){E 3ZvW ~8 _ (E+M~E '+M)R- - T16 1 32wW (E -M)(E 1 -M)R + [ klx0 ( W+E 1 )-k'~0 ) R- Tl. Tu T12 Tu TlO ~7 T .AIIJt.atLEZ ~~Cbtiad.~d) I I .... .... ..0 Tl4 TIS Tl7 Tl6 0 16/2 trWm 0 [k (E '+M)+k'(E -Mi(E +M)R- 32.W 16.fz tr Wm 0 16.fz tr Wm' 16.fz v Wm' k~( E +M)( E '+M)R- [ ( W+E )(E '-M)-k' (E +M}iE ' +M) R- 0 k (E -M)(E '-M) R + - (E +M)(E '+M) R32vW Tl5 - (E +M)(E '+M)R- Tl3 0 16.f2 'fiWm o 1 16{2 wWm ' k'0 ( W- M)(E +M)(E '+M)R· 16{2 v Wm ' [ ( W+E )( W+M)(E '-M)+k' ( W- M)(E +M)) (E +M)R- 16..fz .,wm k ( W+M)(E · M )(E '-M)R + 0 [ k ( W-M)(E '+M)·k~( W+M)(E · M)] (E+M) R- ( W- M)(E +MME '+M)R32w JZ•W ( W- M)(E +M)(E '+M) R- 1611'Wmm' 16.wmm' - [ k2k'o2 -k'~0 ( W+E )] ( W+M) R- 32.w (W+M)(E - M)(E '-M)R + G20 ( k2k'0 2-k'Zk0 ( W+E )) R- (E · M)lE '· M)R + 3 vW TlZ Tll TlO G19 TA BL E 12 (Conti nued) ' 0 I .... N -121TABLE 13 ~ az a3 a4 a7 a6 as iE+M~iE'+M~R- T1 6wW Tz k k' (E+M)(E'+M)R0 0 T3 8wWmm' T4 Ts k (E+M)(E'+M)R 0 T6 - a/z wWm T7 k' (E+M)(E'+M)R 0 Ta - afz wWm' T9 _ ~E-M~~E'·M!R + .w TlO Tll k k'(E-M}(E'·M)R+ 0 T12 Tl3 0 SwWrnrn' • Tl4 k (E-M)(E '·M)R + Tl5 0 a/z YWm T16 k' (E-M)(E'·M)R+ Tl1 TlS 0 a/z wWm' -122TA B L E 13 (Continued) ~0 _ ( E +M)(E '+M)R- l6trW k k' (E +M)(E'+M)R- o 0 8wWmm 1 k (E +M)(E'+M)R- o 8.f2 trWm k 1 (E +M)( E '+M)R. 0 S.f2 wWm 1 6.W k k' (E -M(E '-M)R+ 0 0 8wWmm 1 k (E -M)(E'·M)R + 0 s/'2 wwm k~(E-M)(E' ·M)R+ a..fz •Wm' ~1 T .ABLE U ( Coatlnu.d) Tl fE+W~~E'+M~R- 2wW - Tz k k' (E+M)(E'+M) R• 0 'l'l 0 16wWmm' 0 k' (E+W)(E'+M) a· 0 k ( W·M)(E+N)(E'+M) a· 0 - k' (W•W)(E+M)(E'+M.)R.. 0 • • IE·M~E'•MIR+ 2wW (W+M)(E-~(E·M')R + .w • • k k' (E·M)(E'•M)R.+ • ~ Tu Tl" 0 16wWmm 1 .. i&wWmm' • • 16/2 wWm - T16 0 0 • 0 Tl5 k k' (W+M)(E•W)(E'·M)R+ k (Jt•M)(E'•W)R + Tl8 i6trWmm1 1W'2 wWm' Tu T17 0 16/z vWm T9 T12 0 • k (E+M)(E '+N) a· Ta TlO • k k' ( W• U)(E +M)(E '+M) R • .. T6 T7 lZw - T. Ts (W •W)(E+M!!f'+M}(R• k' (E•M)(E '·M)R + 0 16.f2 wWm' • -1Z4TABL E 13 (Continued) ( E+M)~E'+M)R"' ( W -M)(E+M~E ' +M)R­ k k' (E+M)(E '+M)R- k k' ( W- M)(E+M)(E '+M )R .. 16,..Wmm' 16.Wmm1 3 .w o 0 l~11' 0 0 • • • k (E +.M)(E'+M).Ro k ( W •.M)(E+M)(E '+M)R"' 0 16.fz wWm k' (E+M)(E '+M)R- o k' ( W·M)(E+M)(E '+M)R .. 0 t61'z •Wm' T9 TlO .. Tu T12 k k' (E-M)(E '•M)R+ k k' (W+M)(E -M)(E '·M)R+ 16wWmm 1 i&wwmm' 0 0 0 0 Tu • • Tl-4 • .. kc,(E ·M)(E '•M)R + T15 161z wWm 16-/2 •Wm Tl6 k' (E-M.)(E '·M)R + 0 k' ( W+M)(E -M)(E '·M)R + 0 t6.fz •Wm' • TABLE U (Continued) _ {W- W)(Jt +M)(E '+W)lf" T1 JzwW .. Tz Tl 0 k k~(E+l4)(E '+W)R .. • 16wW'mm1 T4 • T5 - k (E+W)(E'+M)R• 0 T6 16wWmm9 .. • k (W •M)(E+M)(E'+M)R0 16J2 wWm k~(E+M)(E'+M)Jl• ----~--- 164'2 wWrn' 16/z wWm - • T7 Ta • k k 1 (W•M)(E+M)(E'+M)a2 0 k~( W·M)(E +M)(E '+M) R- ------------------16.f2 wwm• T9 TlO Tu Tl2 • k . k' (E·M)(E'•M)R + k k' (W+M)(E•M)(E'•W)R+ 16wWmm' -------·t~&.~w~-nun~•~---- 0 0 Tu 0 0 • • Tl4 ._.o _________________ k (W+M)(E·MHE'·M)R+ Tl5 161"! wWm 164'2 wWm • Tl6 k' (E·M)(E'·M)R.t Tl7 T18 0 tWz wWm' - ·126TABLE ll (Continued) 0 zo .. (W -M)(E+1(E'+M).R'" 3~. k k' (E+M)(E'+N.)R- o 0 16wWmm 1 k k' (W-M)(E+M)(E'+M)R'" 0 0 9--.. -------..,.l&......~w~mm-.. - k (E+M)(E'+M)R. ·---..----0 16.f2 YWm .. . ..-o-~---16.'"2 wWm' k' (lt+W)(E'+M)I\ .. .. k ( W -W)(E+M)(E'+M)a· .--o----~~-------1W2 wWm .. k~( W ·M)(lt+MHE '+M)R- ·------~---------u,.fz wWm' .. • T9 TlO Tu • 16wwmm1 T12 Tu • Tlf .. • k k' (W+M)(E•.M)(E'·M).R+ 0 0 l6wWmm* • k (E-M)(E'•M)R + • k ( W +M)(E -M)(E 1 ·M) R + 16/2 wWm 16/2 wWm k' (E•M)(E'-M)R+ k' (W+M)(E-M)(E'·M)R + 0 9 16/2 trWm' .. 0 0 --------.---------16.f2 wWm' - -127- • channel e) The wN Ae ha• already been indicated ln the introduction, the lnelu•ion • channel in a flr•t calculation \e unlikely, eo we will not of the wN develop the aeneral ldnematlce for proceeeee involvlna it. Nevel'tbeleae, ln appendix F we have obtained all the neceeeary fonnwae for the calculation of matrix element• for thie channel. With the propaaator aiven by F•19 and the vertlcea cliecu•eecl in pal't IV the ampUtu.clee correeponcling to araphe involving internal N• can be cleterminecl, anclif one or two external N •• • an involved, the .tate vector• given in .1'·15 together with formulae B -7 and B -13 permit ua to calculate the hellcity amplitude• * f~ ~ ~~ c d 1._. Then equation D·23 a·'b define• the T). ~ ,~ ~ that have a definite parity, and the correeponcling c cl a·'"b partial wave expanelona and projection ~rator• are alven by D-26 and D•28 reepecUvely, aD4 thu.e the matrix element• that are needed can be calculated. Ae an example, we will calculate the neceeeary formulae for the • kinematic• of the l'eaction w + N - • , +N • I --~ .,. --- na. 6. __ ..r q --- ., • General diaal'am fol' the reaction ,.. + N - w+ N • -1Z8· p + q. ,. + ,. (174&) Q.-\ (q + q') p.-\ (p + p') A.. i- (q-q').-\ (p'- p) (174c) (174d) (174e) wheaoe M 1 le the ma•• of the N •. The matrix element Q le now of the form Q • \iv(p)Tv.,_tt.,.(P, (175a) T "P. • a1M1, "t' fl75b) The nwnber ol independent lnvarl&nu le 6, ae can be • een from 3 0 ,..!_ TJ • inltlal 'I y1 0 1 ·1'0 3 ·yO fprj l 3 0 J ~ J a.z J 4} a..J "% 10 J a.z a.s J a.,J J Cll 1 ·yO cil a.J a.J s fllz ·y3 0 ~ ~ ,lz ~ final 6 (176) where parity con•ervation and time reversallnvariance have been taken into account. We will contT&ct the two vector lndlcee ,.,., v with two •polaris&• • tlon vector•• «..,_. c.., to bave the equatlone in a more familiar fonn. ·lZ9· '¥ and lj. can be ellmlnated from the invariant•, equation• l'·l4b, 4, that J and (', operatina to Equation• l"·l•a, c indicate that the right and left re•pectlvely, atve aero, and from I"·Zb we can derive two further relation•. All the• e can be written 1• M' A·~ • p. ~ (177a) 7'•0 (177b) A· •' • .p. c' (1.7 7c) The invariant• that we obtain, after taktna into account all obvlou• conatralnta, are M 5 • o. ~ O· c • M6 • Q. c 0· c'll M3 • p. c p. c • (178) M7• p. c Q. c • + Q. c p • c ' M_. • p. ~ p. c'~ M • (P. c 0• c'+ 0• c p. ~·)~ 8 There are two •qerfluou• lnvartant1, anclln the way lncUcated ln appendix E, wtth the help of equation• 117 we find ' 2M MP·~·P • ' 2 +6M.·MM 7 • 0 (179&) [A aa 0 + (P· Q) z1~- M •P· OM 2 + lQ aM 3 • 2A zM 5 • 2P· OM7 + M'M8 • 0 The•e ahow that M• &D4 M 8 , or ~ ancl w 8 , can be eliminated without lnti'CKiucing kinematical alnaularitiea. U we cleltne (180) f 3 1 ·y 0& 1' 0 • I 2 8 ZifJ Z co•t•in y • J J J ~ ••• •t ::1: • • J J ~z. 5 • (15 * (16J (181) J J J ~3. 6. 4 3 . 4 4 (18Z) ancl the projection operator• -131C!..J, .. • ,...1 "If 1 1 (' [T l.tP.t+l+(.t+Z)P1 _1 'I J -1 1, " Zl +i (181) If we uae the atate vector from .1'·15 in equatioaa 175, we can lind the c: cl 1 f~ ~ ~ ~ ao that appear in equation• 180, and from them we aet the relatione (184) (185) The coeUlclenta c(O) c(l) and c(ljZ) aJ'e abown ln table• 1•, 15 ancll6 lJ • lj reapectlvely. FI'Om theae table• we eee that the reflection aymmetry in tbla cue le expreeae4 by Ti(•W) • 'l(i)Ti+l(W) i =1, Z, l (186&) J J 1 • 1, 2, l (186b) i • 3 l • 1, 2 (186c:) ~t<-w> • "cu~'t+lcw> T6 Ts T4 Tl Tz Tl E ·M' Z4www• 2 .. --------;;.;;.. ZE 2+EM'+w' 2 4{3 wWM ' 2 • (E·M')(4E -4EM'-sWZ) "'T.W E+M' 4{3wWM' 2 2 ZE ·EM'+M' 24wwM' 2 12 2 (E+M*)(4E +4EM'·SM ) TYW (\ s..· (Wi'M~E-M') Gz a• 4{3 wWM' • ZE( W-M')(E+M•)+M'( W+M'}(E-M') 2 ( Yl+W'UE·M')(4E ·.U:M'·SM'j 241rWM'z ( W-M'~E+.W') 4{3 wWM' ZE(W +W'UE-M')- M'( W.. w'UE+M') 2 ( W-M')(E+M ')(4E +4EM'·SM'i 2"-WM'z TABLE 14 N ""• ...• T6 Ts T4 T3 Tz Tl 2 16{3 wWN' 2 p2(zE 2 +EM'+M' ) 96www' p2(E -M')(4E 2 -4EM' ·M' 2) 16{3 wWM' .. PlZE{W+M')(E-M')·M'(W -M')(E+.M')] pl(zE2· EM'+M'J 16.. ~ I6.f3 wWM' • p2[ ZE(W-M')(E+M'}+M'(W+M')(E-M')] 96wWM Z 1 p2(W+M')(E·M')(4E 2-4EM'·M'Z) p2( W·M'),E+M') 16{3 wWM' 96wwM' 2 p2(W · M ')(E+M'(4E 2 +4EM '·M ' 2) 1 p2 ( W+M~(E-M) 16w a4 p2(E+M')(4E 2 +4EM'·M 2) 96wwM' 2 al 1'.A.BLE 14 (Continued) • ~,- ""' I ~ 3 T6 Ts T4 T Tl Tl t6.f3 .wu• pZr 2(W~)(E+M')+M'(E·M')) 96•WM'z p2(E-U')( .f(W+q0 ) 2--iM'(W+q0 )-M'1 lfr/3 .ww• pl[ Z(W ~)(E-M')·M'(E+W')) 96.ww• 2 p2(E+M')[ 4(W,) 2-4M'(W,)·M' 2) as a, 1 2 2 16..-w 0 ) 0 1W3 .ww• Pi 2(W+q )(W-M')(E+M')·M'( W+M')CE-M')] 961fWM' p2(W+M')(E-M')[ .f(W+q 2+4M'(W,)·M' 2) p2( W· M ')(E+M ') 16{3 'I' WM' pi 2(W~)(W+M ..E-M')+M'( W-M')(E+W')] 1 96•WM z pl(W-M')CE+M')( f(W ,) -4M'(W~)-M' ) 16.. w .. p2(W+M )(E-M') TABLE 14 (Contin~ed) "•" • -. T6 Ts T4 T3 Tz Tl -z 0 1 2 s... w 1 a.w pl( W-M')(E+M ') 8{3 wWM' p2[ Zq0 ( W+M')(E-M')+M'( W-M')(E+M ')) 48trWM p~ (W -M')(E+M'){ 4E(W+tto)+M' 2 } +4EM'(W+M')(E-M')) p2 ( W+M ')( ~·M') aa 1 8/3 wWM' PZ-r 2q0 (E+M')+M'(E-M 1 )] 48wWM 8{3 wWM' p2[ Z~(W-M')(E+M') - M'(W+M')(E-M')] 48wWM' p2[ (E-M'){4E(W+C~c,)+M' } +4EM'(E+M')] p2[ ( W+M')(E-M'){ 4E(W+q0)+M'~·4EM 1 (W -M')(E+M')) 8..f3 wWM' P [ Zq (E-M')-M'(E+M'}] 48wWM 0 p2[ ( E+M '){ 4E(W+q )+M' 2} -4EM'(E-M')) a7 TABLE 14 (Continued) U1 -"••"' T6 T5 T4 T3 Tz Tl 1 z 4Ez(E·M')·M'(4E-M 1 )(E+M') . 241fWM 1 Z41fWM 1 l 2 .. 4E ( W+M')(E-M ')+M'(.f.E-M')( W -M')(E+M') Z4wWM Z Z41fWM 1 z 2 4E ( w- M ')(E +M ') -M I ( .f.E+M ')( w +M ')(E -M') az 4E 2(E+M')+M'(4E+M')(E-M') ~ TABLE 15 1 o- "" I ... T4 T6 ·r 5 - ;z~E-M') 6.w TJ 16{3 wWM'z 22i2E2+EM'+M'2! -12•WM' 2 -z 3 p E - EM'+M '2} - e2!2E216.f3 vWM' Tz -z t&.w 3 P E 2 12www' Tl p2(E+M') (l3 16. ll•WM'z E 2 (W ·M')(E+M')·«~oM' 2E) 1613 wWM' pj ZE(W ·M')(E+M')+M'(W+M'lE-M')) lZ•WM'Z p2[ E 2(W+M')(E-M')·fioM 12E) p2 (W+M'~E-M') 16. lfrl3 •WM' 2~ 2E(W+M'UE·M'l·M'(W-MlE+M'U p, p2(w -M'UE+M') a" TABLE 15 (Continued) I ~ ' ~ \N T6 Ts T4 T3 Tz Tl 16.. 1 16/3 .. wM'z 2 p2(2E 2+EM'+M' ) tz.. wM' 2 E( W~)(E-M')-M'(C~c,E+M'W)) tz.. wM' E(W+Cic,)(W+M')(E-U')+M'(M'ct!-w 2E+WM' 2)] 16{3 .. wM' elzE(W ·M')(E+M')+M'(W+M')(E·W')] p, t(w+aE-M') 16{3 .. ww• 121rWM 16-fl wWM' p, 1 E(W~)(W-M')(E+W')+M'(M'~+W 2E·WM ;) pj 2E(W+M')(E-M')-M'(W ·M')(E+M')] p, p2(w -M'UE+M') a, p2(2E 2 -EM'+M' 2 ) 1ZwWM 1 l p2{ E(W ~)(E+M')+M'(CfoE-M' W)) as TABLE 15 (Continued) • 0) • ..... ~ T6 Ts T4 T3 Tz Tl 2 8/1 .. wM' 2 p2( ZE z+EM' +M' '1 lZw WM'z p2( ZWE(E-M')·M'E(E-q0 )-M' 2 w) 8{3 wWM' p2 (zE 2 -EM'+M' 2) 1ZwWM z 1 p2( ZWE(E+M')+M'E(E-~)-M' w] a7 8.. as 0 1 s.. 0 0 lZwWM'z ZWE(W+M')(E-M')+M' 2 q (E-q )+M'W(W E - M S.fl wWM 1 p2( ZE(W -M')(E+M')+M'(W+M')(E-M')) p, p2(W+M')W:-M') 8{3 1rWM 1 p2( ZE( W +M')(E -M')-M'( W-M')(E+M')) lZ1rWM' 12 )] pe[ ZWE(W- M ')(E+M') +M' zq (E -flc,)-M' W (WE- M 2)] _p ( W -M~E+M') 2 TABLE 15 (Continued) I • -c "" - T6 Ts T4 Tl Tz Tl T6 Ts T4 T3 Tz Tl az Ci3 as 96wWM' 2 1 1 96wWM' 2 p2[ (E-M')(4E 2 +M' 2)-4M'E(E+M ')] 96wWM' 96wWM'l p2[ (W- M ')(E+M')(4E 2+M ' 2)-4M ' E ( W+M ')(E - M ')] Ci6 96wWM' 2 96wwM'z p2[ (W-M')(E+M')(4El +M' 2)-4M'E(W +M')(E-M')] a• 0 I .-•. p2[ (E-M')(4E 2 +M' 2 )-4M ' E (E+M ')) p2[ (W+M')(E-M'}{-4E 2 +M 12)+4M E(W - M ')(E+M')) 96wWM ?£ E+M ')(4E 2+M ')+4M ' E(E-M')] p2[ (E+M')(4E 2 +M ' 21+4M'E(E-M ')] ~ TABLE 16 T6 T5 T4 T3 Tz Tl -48wWM1 l - 1 p2(4E 2+4EM •+M' 2) 2 2 UE+M'! -fz~4E -4EM'+M' 48wWM z 0.7 as 1 48wWM' _p2 (4E 2 +4EM'+M'z~( W+M ')(E-M') - _p2(4E 2-4EM'+M' 2)(w -M')(E+M') 48wWM z TABLE 16 (Continued) • I •........ -142Ill. ISOSPIN "KINEMATICS" In part II we have developed the kinematics for the different proeeaaea assuming that all particles have iaoap\n zero. The effect of iaoapin will be to increase the multiplicity of each atate, but it does not interfere wtth the k inematics and ao it can be superimposed on it. a) w + N - w+N Piona have iaoepin 1 and nucleons 1 'I , ao that the 3 1 can be either in a state with I a 'I or I • 1 . wN ayatem Since only atrong inter• action• are included, charge independence ia aaawned and it ia aufficien t to apeci!y the magnitude of the iaospin. The iaospin apace for the wN aystem will be a direct product of the apaces for the , and N, and the atate vector• lor states with a definite total isoapin are given by the corresponding Clebach- Gordan coefflcienta, i. e. I1 ,l15 > • l C(Il' 12 , hi• -t 2•• 12•) It1• 1• -x 2• > lt 2 • 1215 > (1) 12• (See ref. 6, aection 10, for inatance.) Aa an example we will give 12 3 1 I1 1 11 I1 1 11·1>=~3 11• 0 > 1•1>+~y 11• 1 > 1··1> or 3 1 f2 °) 1 f1 ·l ) 0 I 1 • -z > • \1 "J t ~ <o , +~ 3 ( 01 <1 ) (la) (lb) We have taken for apin 1 the usual representation described in ref. 1 -143eection 7h, where 0 0 0 ) (2a) Tl• ( 0 0 -i 0 i 0 or (2b) .. + • 11. 1 > • +(=~ ) v2 For epln call them Ti. (l) 0 ~ we get the uaual Pauli apin matrices, and we wtll We have then (4) where each term contain• lmpllcitly a unit operator that acta on the other etate vector of the direct product. In general (Ia) where '• on the left hand aide la an operator, ancl on the rig ht a num• ber, ancl 12 ji, I > • l(I+l) II, 1 > • In • (5b) wN acatterlng , we can alway• write the amplitude• in the form (6) where the atate vector• correspond to the iaoapln of the particle•. and l • 1. z (7a) (7b) Theae two are the only "acalar•" (acalara wuter rotation• t.n a three• climenaional laoepln apace) that can be formed. U we want the matrix element• for atatea with a definite total laoapln. we calculate (8) To calculate < tl Mz It > we can write iz • (T + \ ;,z l(I+l) • T(T+l) + T·; + \ ,.(.,+1) -u T· ., • 1(1+1) • '"t (9) and equation 8 glvea ~> • ~+ az (lOa) aci> • ~- za2 (lOb) U the croaa aectlon la neeclecl lor a panlculal' reaction, we can !lnd lt fl'om equation 6, or by ualng the Q(t) of equation 10 ancl tM tnverae of equation 1 (aee R6•3. 8): ~~~. > lt 2 • I.-~.> • LC(~t2t:~• .rz·~.) Itt• t &Ad remembering that total taoapin l• conaervecl. > (U) ·145- b) Reaction• involving vector meaons The p-me•on has boapln 1 and the pN channel baa the •am• •tructure aa the 1rN channel. The w•me•on h&8 l•o•pln 0, and hence the wN channel ha• 1 alway• I • '! . W• have for the w + N - w + N reaction (12a) alnce the clepencience on the w le trivial. It can al•o be written (12b) where .A(w) • ~M(•) (Ua) - .. (Ub) M(•) • •· ~ - We have wri"•n here w for the l•o•pln etate vector of the pion. Obviouely we have For the reaction w + N - w + N we flnci (15a) (15b) (15c) (15cl) -146· Equation 15c exproaaea the fact that the w meaon b neutral and the charge of the nu.cleon has to 'be coneerved• c) Reaction• lnvolvin& the wN • channel • We can conatru.ct the atatea lor the N , that la, for a particle of iaoapln 4, out of thoae for iaoapin and ~ , by ualn& the correapondlna 1 Clebach·Gordan coefllclenta. 8 We write thle atate Ne(I.), or ~(1 ) ll • we want to make explicit the three•vector nature ol N (Ish then each • Nk(ts), la a two•component aplnor. k • 1, Z, J, We have, then, 1 1 1 0 • ~( 0) '"'76(1) 1 ·rz<o, l 0 - 76(1 ) 0 ~ (~) i (16) 1 0 l 0 Tz ( 1) • 1 • N (• y) • 3 N(·y)• ·rz<t > 0 It l• eaay to check that all four .tatea obey the equation (17) in analoay to equation F·lb tor the wave function of a apln • For • + N - • +N ~ particle. • we can write then (18) i • 1, 2, l (19) -147 There are only throe inclepen4ent invariant• becauae the total laoepln can 5 ~ , talte the valuea 1 '23 and 'l ancl the invariant• are aecond rank ten1or1 in the fo.-m we have written them. We can chooae (20a) (20b) (20c) OU:lel' lnvariant• a:ro .-eclucecl to comblnatlona of the•e with the help of the :relation• (21a) (Zlb) (21c) o:r are ae:ro between •tate vecto.-a clue to equation 17. Next we can calculate the a(I) by u•lna any of the atate• with total l101pln I. W• aet Q( i) • <\+ az (2Ja) Q( i )• ~· j az + i al (22b) 4a2 + j a3 (22c) a( ~ ) • ~. For 1r +N • - w + N we have (2la) i • 1, z (Zlb) In thia caae the two invariant• are vectors ln leoapln apace. ~.k•lzT~c (Z4a) M2, k • T•,. Tk (24b) -- ror a{ l' 3 ) • • ~'J ~. ~ a2 (25a) ac "!1 > • $ !i - (25b) • , +N - 'J 2 * 'J a2 w + N there ie only one invariant& a • < N I ~C•> IN:> (26) ~(•) • ~~. k(•) (27a) ~. kC•> • 1 z•~t (27b) ac i) • . .fz 0, (Z7c) Thla la all the i•o•pln •kinematic•" that i• needed for the fo\lr cliUerent channels. Matrix elements •here totall•o•pln and it1 &•component are not both conaerved are aero for etrong lnteractlona. Finally we would like to point O\lt that thla •ection •how• only one way of writing down the different ampllt\lcleaa and other equivalent way• are poa•lble. In particular, U the analogy to part 11 l• carried one atep further. the laovectora correeponcllng to particle• of laoapin 1 can be included ln the invariants. For ln•tance, the invarlm b for • + N - • + N can be choaen (28a, b) -149IV. DYNAMICS a) General approach The aolutione to problema in relativhtic quantum mechanics propoeed ao far all demand that aome approximation be uaed in order to get a practical calculation. It ia deelrable to buUd. into t¥ approximate aolution ae many propertlee that the exact aolution le known to have u le poaelble without getting into undue compllcatlona. One euch aol11tlon that incorporate• unitarf.ty into the acattering matrix la the N/D method. preaented in appencllx G. Eaaentlally, the aubmatrlx of the acattering matrix correaponcling to a (conaerved) total ang\llar momentum J when a partial wave expanaion ia made ia eeen to obey the equation SJ~t =1 aa ln reference 5. (1) There a matrix TJ la defined by (2) and hence r obey• 21m TJ • TJ.TJ U (3) TJ l8 aymmetrtc (aa ia demanded by time reveraal invariance). Scattering amplltudea, in a theory where the S•matrix ia unitary, obey diaperalon relatione that make explicit certain cute in the comples energy (or energy aqurec!) plane. \lDitarUy cute. Each T 3 Ia then written in the fol'm Theae are called ·150- (4) eo that DJ hae all the unltarlty cute and no other•, and ~ the :re• matnlng eingularitiee of TJ. Fl'om equation• 3 and 4 we can find relatione for 1m~ and 1m DJ llke thoee tn Q ..14a, b, which can be written ae integral equatlona. Two of the poeelble app:roachee to a numerical computation will be briefly cllecu•eed. It i• poeeible to aeeume a certain eet of 8bljularttiee given for T , not lnclucSlng the unltarity cute, euch ae cut• arielng from the exch ange of a certaiD particle ln a c:roeeed channel. The lntea:ral equation• for N and D can then be eolvecl ae le clone in :reference• 16 and 23 for a very •lmple choice of the •lngularltle•, or the eolution can be calculated on a computer. It l• abo poeeible to chooee an approximate eolutton for N, from the Born approximation for inetance, and then calculate D by performing the inte g ral• in equation 0·14c. The inelutlc chalulell are related to plon nucleon •catterlng by the unUarlty equation•, and theb pn•ence can p1"o4uce :re•onancee a• ia cUecuaeec! in :reference• 16, 18 and 23. It t• then deelrable to lnc:lucle theee channele, at leaat in an approximate way. Thh le done by con• eldering eeve:ral particle• repreeented by one unetable particle or leobar when there l• a :reeonance b1 the co:rreeponcUng ey•tem. U tM in!onnatlon we want to uee ae input for the eolution ie fairly exteneive, it look• advieable to uee the eecond euggeeted approximation for the N/D method, •tnce involve• only the calculatioo of integral• ·151· and not the D\Unerical aolution of integral equation•. Thia method hae the diaadvant&ge ot giving a T-matrix that ia not aymmetric, a• required by time reveraal tnvariance. and hence untt&rity la abo violated even in the approximation of conatdering only a lew cbanneb (aee the dbcuaeion in appendix 0). The J"eynrnan diagram• of the Born approximation, including the nucleon pole in the main cbanael and other polea in the croeaed cb.anneb are ahown in aection IVb, and a few example• are worked out in aection IVd. laoapin t.a conaerved by atrong lnteractiou. and hence the atatea with different total iaoepin can be treated aeparately. The way to project out the correapondiag amplitude• ta abown in part UL Total angular momentun) la abo conaervedr thla leac!a to the uae ol a partial wave expanaton. Moreover, parity la abo coneerved by atrong interactional thta permit• one to eeparate the eete ol amplituclee correaponcllng to a given J in two groupe, one for traneitione between et&tee of parity + aDd the other for parity •• matrix element• between at&tea of dUlerent parity are alwaye aero. Thh eeparatlon ie buUt J into the amplitude• called "' ln part U, and the eimple relation between etatee of definite parity and thoee of definite hellcity i.e ehown for inatance in equation n-6z. The eete of amplitude• or matrix elem ente are related by an equation of the form (5a) For lnetance, for wN ec:atterina alone, ·152J J 0 "1 + "2 1 • Tz 1 1 1 72 Tz 1 1 ·rz Tz 0 Tht.e doea not change equation l. (Sb) Tz (It ahould be noticed that -\ TJ and i T' 1 are the matrtc:ea that obey equation 0-12. Tht.e agree• with equation• 11·49, ll-52 and 11•53 for wN acatterlng.) Slngularitlea like branch cute coming from parti.al wavea expan•iona and denaity of atate• factor• cu be elitninatec:l from the equ.atiolUI by redefining the T-matri.x aa ia ahown for iutance in refel'ence 19. One obvtoua change ia the choice of the total energy W iuteact of •• w2 . aa the lnclepencient variable. Other change• lUte thoae incticatect in R19·28 to 32 have the purpo•e of alvina the amplitude• the right threahold behavior, when the •\lbatractton point for D le taken at threabold, u in equation 0·144. The tnclualon in the problem of un•table external particle• and the reaulting complex alngularitle• atve riae to the need of modifying the unltarlty relationa. . Thia. la ahoWB in 4etaU in appendix H for the ca•e dlacuaaed in reference 16 and a c!lacua•ion of the •ppllcatlon to the problem in conaidel'ation ia abo included. Reference• 18 andl9 alao deal with theae difflcultiea, in a different way. The matrix elementa can then be calculatecl for wN acatterina !or amall value• of J and the two value• of parity and laoaplnJ and from the phaae ahlfta (equatlona Il·5l) the poa•ible reaonance• can be found. -151b) .Feynman diagram• U we exclude external N • iaobare. we have to con•lder the pole term• indicated in figure• 1 to 6. To write down the amplitude• corre•ponding to the different graph8 we follow Feynman'• rule• a• de•cribed in reference 1 section t•c, for tn•tance. that faetore like .,l,.. an4 Yl.wk ~ E(p) M,. It ehould be noticed are not lndudotlln later when the cro•• eection i• computed. a but a4cled Neither are the power• of 2w, and momentum con•ervatlon at each vertex can be taken into account by choo•ing adequately the momenta of internal particle•. The propagator• for particle• of eptn 0, -\ and 1 are well known, and for epln 4 it l• given by equation F·l9. The form of the Interaction• l• more u ncertain, and •ome principle of minimal com• plication is u•ually invoked to rule out certain term• and leave other•. The final Ju•tlltcatton come• of cour•e from the check with expel'lment, but cletaUecl data and a aood theory to make calculation• are required to reach eouncl conclueione. The value• of all the effective coupllna con•tant• must al•o be cletennined from experiment. Wlle•• a theory like one of higher •ymmetrle• of •trong interaction• t. u•ed. The verticee involved ln figure• 1 to 6 will be taken a• follow• c V(wNN) • gwNNY5 • V(wNN ) • i NNe(p + q ) 1r ~ V(pNN) • ipNN'Vv t V(pww) • lpww(q"' + qv.) ~ (eee ftg. 7a) (6) (eee fla. ?b) (7) Ceee tia. 7c) (8) (see fta. 7d) (9) -154- ' .,. .,.~ " / .,. .,. N -~ N -~ .,. / ~ .,. -~- -~- p .,. -~- (a) .,. ~- N Graphs for w + N - 1r N (d) (c) (b) Fig. 1. N +N P N N .,. :JN~-(a) (b) p p _ .,_.__.....N ..,. - - (c) Fig. Z. .,. ~-...-- 1 1.,. ...-I~ N N (d) Graphs for p + N- 1r + N -155- N --+___.l.- - .... - N (b) (a) 7T N (c) N Graphs for w + N- 1r + N Fig. 3. p N ~ N p p (cf N N N p N N (b) (a) Fig. 4. (d) Graph• for p + N - p + N p N w ~ I 17T N I • N N (b) (a) Fig. 5. Graphs for w + N - I~ N (c) p +N N N N (b) (a) Fig. 6. w Graphs for w + N - w +N - 156 - 1T l q·p-p' k I N·J* l I p=p-q k=p-p U,~(p') E}J- I I N ~ I N ,..N , p p 7Tk ~ - p (b) (0) kz:rq-q' (C) I Nj* p-p-k O.p Cp') Pt Ef-17T·J - ~ - q (d) 7Tk q' - ~ - N p ·k, E 1l 'e) N P (f) p' Fig. 7. p~ p q Int e r a cti o n v e rticea , -157V(pNN, c g p NN y y (p ..,. + k ... h s <••• llg. 7e) (10) V(-..NN) c lwNN"~ V (eee flg. 7f) (11) V(•pw) II '•P«r.)..f,l1'Tk>.k~ (aee fta. 7g) (lZ) <••e na. 7h) (B) V(ppp) =glp (:k't6V" + kV 6v>.+ k~6~) The •NN vertex ie the uaual choice la rather arl>ltrary. v5 coupllnl• In tho •NN* vertex. the It correeponda to tM interaction La1ranalan given in reference Zl, anclla the only term that la non•aero when the N• la on the ma•• abe.U. A general vertex co\.lld be written The term• with y 5 are ruled out becauae the N • la known to be a p•wave reaonanee o! a (paeudoaealar) plon and a nucleon, eo that u " trana!orma Uke a paeuc!ovector with reepect to the lnclez J.li and eo doe• the combination of the paeucloacalar pion an4 the vector• ln equation 14. It la euy to aee from equation• F·lb and F-2b v u (p) • 0 - (15a) p u (p) • 0 (15b) " " ... .... " that all term• but the firet 1ive zero when actina on an external N•. But it ie quite poe aible that they atve important contrlbutlona when the ·158N • la an tnt. .-mediate panicle. The pNN Yel'tex la the one that le found tn reference 20 and COI'reeponcla to the coupUna ol the p meeon to the nucleon cu•rent. The aame can be add about the p1Mr Yel"tex. • wnex the•• te aleo a more or 1••• arbitrary For the pNN cholce to be made. In equatloll 10 we ban taken a comblnatloll of 7 and bema actded by pal'lty coneide ratlona. A a in equation 14, 5 there &l'e maay Othel' ten•ora that can be formed with the momenta 8, the "Y uad "Y matricee, ~•t of which alv. aero contribution with the particle• oa the mae• abell. Eq\aatlon U ia almtlar to 8. The .. ,.., vertex 4oea not appear ln the La&l'&nliall cliecuaaed in l'efe renee !0, but lt doe• not violate the known conae rYatlon law• of atrona hueractlona. Parity con•ervatlon demand• that a paeudoacalar formed with the momenta and polariaatton v.ctora of the me•one be the form of the lnteractlon& the Ollly ODe that CaD be formed ia «~"Pk~k~«v«~ where «~vp la the antlaymmetric (Levt.-Ct.vUa) tenaor. In the 3p vertex we ban taken what reeulta from the unitary aymmetry olre!el'ence 10J lt abowcl be remembered that~ p m ..oa couple• to the laoapln current, of which it la al•o a aource. Othe:r verticea are forblclden by coneervatlon1awa of atrona lnteractlODI, like O.parity, l•o•pln, parity• It tbe wN • channel l• abo t.nc1u4e4 t.n the problem, many other arapha with eltbel' one or two external Ne'• haYe to be coaalc!erecl. Al•o new venice•, •laown ln fta. 8, lAtrocluc:e new coupllDI conatanta tnto the problem. -159- "k I ~q=P-P I * Ni~ I N j >-I>- , 31' N·I p p' p UJL (p) U11 (-p') uJL<i>> u.,liS'> (c) (b) Fla. 8. p' Venice• with two ~·. There are many poeatble forma for the tnteractlona; a dmple choice could be V(1rN•N*) • 11fN•N•~s6 v 11 V(p~N*) • gpN*N*~~a..,.., (aee fta. Sa) (16) C••• ttg. 8b) (17) V(wN*N*) :a gwN*N*~~6 ., (eee fig. Sc) (18) 11 Equation• 17 and 18 aaree with the propoeltion that the vector meaona are coupled to the current of the N*, equation F-20c. c) Iaoapin dynamic• In aection IVb we have not written down the iaoaptn part ln ver• ttcea and propagator•, and we propoae to do eo here. remembered that the • and p have I. :a 1 (vector• in iaoapln apace), the cw baa I • 0 (acalar), the N baa I • and the N • baa I • It abould be i (two component aplnor) j (direct product ol a vector and a aptnor). By conaiderina the poaelble expreaatona of adequate rank ln boapin epace, weflAcl V(wNN) • Tk (19) V( •NN•) :c 6Jklz (20) V(pNN) • Tk (21) V(p1Mr) • •it:ijk (ZZ) V(pNN•) • 6jklz (ZJ) V(QINN) • 1 2 V(•pw) • 6jk (25) V(ppp) • • t•tjk (26) Vhr~N., • 6lj"k (27) V(p~~ • 6lJTk (28) V(wN*~ • 'tiz (29) • Equation• W•21a. b, and W•l7 when applied to the N , are uaecl to ncll.lce the number Gf po11l'ble term1. The propaaator• for the iao1pln pan of the wave function• are cletenntned by 1ummb~a ctvel' the l101pin •tate• of the particle P•L 1~ > <~1 (30) ~ where f).> l• the atate vector lor Is• ~. We aet P(w) • 6jk: 13 (ll) P(p) • 6jk: 13 (32) P(w) • 1 (33) ·161(l4) P(N) • 1 2 P(N, • f 6Jklz• (35) d) Example • We will a how how the determination of the amplitude a corre • apondlng to the graph• in figure• 1 to 6 t. carried out. we will take the araph from lb (aee lia. 9). follow•~ For inatance, The iaoapin part goea ae wtth the notation of equation It-6. and ualng equation• 19 and 34, (36) Notice the importance of chooaina the lndlcea in a correct way. Nc - -':'Qb - --.---.-P ...:;'~N Na p-q _ _ _...,_ _ ___..._ - p Fta. 9. I rrd- #1'- q' Nucleon exchanae diagram Equation 17 atvea (37) .Accordingly, (l8a, b) ·16Z· For the re•t of the amplit\Ule, we have (39a) Between •plnore, So, ln equation II-9 (39b) ,AaO ta,NN s •. (Zw)t zw2 w•.z.,.1wz .. (Mz ..... z,z+[w4.z(Mz+p.z)wz+cMz •.,.z)2]x (39c) We write B in the fo nn ()94) U we waat to 1\nd reeonancee ln the pl/Z channel, t. e. 3 J • 'I poaitive parity (•ee equatlona U·61 and 63), we have to determice 1'~/Z (W). th\a effect, we ct.term.lne T 1 and. T z from equation U·73 To -163(40a) T 2 • (E·M)\r+.M) a+1bx 4w (40b) We wt11 al•o do tbe ealeulatton for the 1raph lzt. fiaure 2e (eee .,. fta. to). - ~ -- q k'=p-p . I p p' na. 10 w meeozt. exchanae cllaaram (42) He ace .. ~· 1 (4la) a2 • o o4) • 1 0{ i) • 1 (4lb) Space pan: a :a Between eplaol'a, We have uaecl equation• E-18 and k• c • o. 5 v ";.~; • v5"1.CP +-;. • ,.,, 1 • ul·y 5; • 2p • k y 5 ; + My5 ;-;_ + Zp• • YJ Jl + My5 1Jl 2 • (m -ap' ·kh5 i + ZMy 5 /li + zp. •v 5 Jt + q• • v 5 tl HeQc:e y• k A q" • • (m2 • Jp'· k • k· <lhst + aMy5;)l + 2'V5JiP. • ~· (m2 - Zp'• k • k· q)O (45) a.4 • ZO (46) where -165· G(W ,x) • (211') 6r (p-p ,) z-m,z1 (46a) 3 If we are atW lntereeted ln J • 'I. poaitt.ve parity, we have to deter• mine IS3/2 , pJ/2 and ~l/1. 3 • Tbie can be done if we ua e equatio~ 2 1 II-U7 and U•119 after expaoeaalna the at.'• in terms of W and x. It should alao be remembered t hat the o.f's in this caae are the elements ol • 1 ~ ~ TJ and not of TJ itseU. \J Tkf lPrl e) Dlacuaalon of the input data To cleflDe the lett haDd cute in our approximation of the N/D metboct, the dUferent maaaes aad coupUna conatanta have to be deter• mined from experiment or by eome theoretical calculation. The maaaes of the stable particles (under atrong interacttona), the plona aaclnucleona, have been kDown for a lona time, aa baa the The unatable particles, • p, "' &Acl N , have been founcl aa naonancea ln 2w, lw and wN ayatema in acattertng experim.ent1. The eaergy at which the re•onAAce la obaerved give• the :maaa of the correaponc:llng particle. to the •NN The wiclth of the N • t.a relatecl • couplina conataat, and that of the p to the pww coqpltna conatant (••• reference 21). II we conalder that tiM p meaon te coupled to the iaoapln cur• rent with a. unlveraal couplt.na conatant, -.. have a relation between the p11'W, pNN, lp and PN.N. coup1'dll conatanta. Th • aame '•• true "or I the "' meaon ln relation to the hype reba rae current, relatina the •• coupling conata.nta for wNN and wN N • ·166· lf tlt.e meeona and nacleona obey approximately a higher aymmetJ>y, like unitary symmetry (see reference ZO), the coupling conatante lnvolvina the p or th w are ea aeDtially the eame. Further iQforma• tion about the pNN couplina coaetant can be extracted from wN ec.attering data. and about the wNN one, from the iaoecalar nucleon form. factor or n\lcleon nucleon ecatterina 4ata. A cUacuaalcm about the c.twp interaction CaD be found in reference zz. Tbie ..owd leave without even an approximate dete rmlnation the pNN • and wN•N• couplina conatanu, ancl they ahould be conelcMred aa free parameter• to be adjueted eo ... to stve the beet &Jreement of the reaulta of tho calculatlon with experimental clata about the wN reeo• n.ancee. O.he r data tbat are rather Wl4l'e Hain eu alao be varied around the ueumed value. -167- APPENDIX A SYMMETRY OPERATIONS a) Reflection aymmetry We will make a general aaalys\8 of the aymmetry pointed out by MacDowell in :reference 4, where he finda the tranaformation p:rope:rtiea of the transition amplitudes ln the w + N - 1F + N reaction when W - • W, that ia, under a change of alp ln the total energy. Our definition• of poaltive and negative eneray epinore for particle• will be the following: u.c;.~, .JJ:zH (1 •z'ii: J( IPo D) (la) uJP.~I •j-~ir (1 +.i.\t )( IPO!. >) where the two-component apino:ra IP•).. > aatiafy 11 .... .. p ,. p, ). > • ). I* p, ).. > (Z) ao that ).. ia the hellcity of the state. Alao (l) - Fol' p non•sero, we have E > M and it ahould be realised that a c:hana• of E into ·E ca.rries u+ into u We next compute and u: '- into -u • -168- 1-E+M (·1 + \J~ rE+M"-a.• pi ":\-o.p•(·lp.~ 0 • \J-m"" > )" • ~-\;r (1 + Jk )( z~~.~ >) Hence (4b) For acatterin1 of meeone and nucleona, the fo1lowin1 helictty amplt• tudea are defined a• in reference J: Fig. 1. General diagram -169where (5a) (Sb) - .. (5c) - p la the relative momentum of the incomina partlclea. The ctlfferential cl'Oa • aectlon la (6) Comparina with the expre•elona of the cro•• •ec:tlon in term• of Lorenta• bavarlant amplitude• (II·Z9, 0·94, etc.) IP"al IPb I ~c~ct'~a~ (7) where (8) From (9) etc. , lt can be •een that the tranalormatlon of W alan of the ene rgie• of all the particle•. • W chana•• the - From equatlo• B·ll we ••• that «(0) chana•• algn when W - ... • W, but not c(•l). In addition we have -... .. .. aY (•a y • a• y) • (a y • a• y)y ..y Y~o oo oo ~o (10) -170 (-W) f: 0 - - =«{·•kW)· k = _ (·l)l+~f: 0 (W) (11) 0 and, remembering that the amplitude• ai are alwaya !unctions of • = W 2 , it ia eaay to prove that for the reactiona with nuc::leone and meaona (12) where the T'e are defined in part II. Combining equations 4, 7, 8, 12 and remembering the remark after equation 3 we get I.Pdl _ - - ~ u_(pc• ~c)T(• W , ~ ~d)u_(pa' ~a) 1;d1 _ _ .. lPbl u +(Pc• ~c)YsYoT(- W , ~ ~d)yoy5u+(Pa• ~a) (U) U4) U p&J'ity la not c:oDaerved, then T haa factor• of the form a + by 5 , and (15) ancl equation 12 ia no longel" true. -171- B ut it ia atill poaeible to relate arnplitudea at negative ener gy with thoae at poaitive enereY• inatead of y y • 0 Thia ia accompli.ahed by uaing only y 5 We have, for instance, that 5 ·~(t+~~~M ;.p);.p( IPLJ • 2).~ (t + ~~~;~ WM-21~ 1-P;, ->.>) Hence (16a) (l6 b) where the relation • .. ,.> ,. Ip,• ~ > a 2~e 21~ I •p, (17a) can be derived from equation• B-7 by aubetltutlng ' - , .. ' · ff - ff + •. From equation B-12 we get • ,. €(K,~) • (•1) 1+~ e -21~. .. €(•k,-~) (17b) and alao -. ~(It, ~) • e ·2i~- ,. ~( •K, -~) (17c) Other needed relation• are (18) ~(-k,-~)•e 2i~ «(k.~) (19) -172If we write {p } to denote the eet of the four-momenta of the 1 particle•, we can verily for the different reactions that where ., i• the lntrineic parity factor (eee equation ll-SO) and t'0 8 arbitrary (the final reeult will be lndepende nt of t' 0 ). ie Equation• 7, 8, 16 and 20 give Actually the function on the left-hand elde hae a different initial etate aleo, corr&eponding to a relative momentum in the direction (w, •o +..), and equation A-5 can no longer be uaed.. We have to calculate the equivalent of R5 ·l0, which become• c l < ..-e,.,.,,.~c• .).41JM;-).c.).d>< JM:-).c-~dl s(- W) IJ'M'J·).a·~> JJ', MM' < J'M';-).a .l.... lw. t' o +..;-).a • -l.... ·'"b ·o > (22) (23a) -173- (23b) f." = ).c • ).d < JM;-l.c• -l.d IS(· W) I J ' M'J·l.a -~ > • <JM;·).c.).dls (·W) IJM;-).a·~>6JJ ' 6MM ' 1 (2Jc) and s11batltutlng in 22: = L i! ei(l.+~)(•+•)d~ 2 1 • .,_(, .. 9)< JM:·).c• -).d I SJ (- W) IJMJ-}.a -~ > J (24) and aince (ZS) we obtain for the modll\ed f(·~i (Z6) where C-4 halil been uaed. Equation l1 then becomea: -174- and hence (Z7) If parity ie coaee rvecl, we have (Z8) and we get back equation 14. In thle eecond form, the symmetry correoponda to a apace-time reflection, i. •·, a change of clirectt.on of all four axee in epace•thne. Energy and momentum change elgn, eince they are component• of a four• vector. Spin can be c:onelclered either a• a peeudovector or components of a second rank teneol', and 4oee not change eign. Hence the belicity, that le, the projection of the epln on the momentwn of a panicle, cloee change elgn. We notice, finally, that the exponential in the etata vector for plane wave• le not changed by thle operation, eince the coordinate four• vector aleo changes elgn: e -ip• x •e -l(-p)• (•x) (29) 1n the firet part of the diecueeion of the refiection eyrrunetry we have combined thle operation wlth the parity operation (to be diecuaeed. -175briefly in aection b of thla appendlx), the combination of both bein1 obvtoualy equivalent to time reflection (not to be conluaec5 with time revereal). Sometbne• in reaction• like p + p - N + R we have matrix amplitude• of the form - -.. -. a c u(p)Tv(p) (30) where v(p) la a apinor for the antiparticle; and the effect of the aymmetry operator• can be aeen from (ll) · ~ <1 - .ii><-z. r,.... ,) • • n~ -~Kr (1 + ~~~) Cz,._~21~ 1-P• ._ >) Hence (lla) (3Zb) Notice the difference in a\gn of the phaae factors when compared to thoae in equation• 16. -176b) Parity Thia operation COI'reeponda to a change of direction of all thl'ee apace axee; poattion veetora and momenta change algn, epin doe• not, and, hence, helicity alao cbangea sign. The operator for aptnore la v0 : '~o"+IP'.~I • '~•jEJr (1 + ik)( IP.o~ >) • ~ Eztf (1 • iia)( IP.~>) • zuzt~~ EJ¥ (1 + iJif'')(I·Prv) (ll) A• to the effect on the matrix ln apln apace T , lt can be eeen from -.. .. .. .. v0 (a0v0 - a•v) • (a0'Y0 • (•a)•vh 0 (34&) a b - a•b • a b (l4b) 0 0 -- 0 0 • (•a)·(·b) (34c) It become• obvtoue that U parity te to be coneervect, the matrtx T ebould not contain terma like that in equation 15. The effect on helicity amplitude• h atuclied in refel'ence 5 an4 la a\lmmarlaed by equation RS-43. (35) -177c) Time revereal Thie eymmetry operation, more aptly callect "revereal of the dlrectlon of motion• by Wlgner on p. 325 of reference 11, correeponcle to the operation of time inversion tn cluetcal mechanice. Here , together with the 4lrectton of time, the elpe of momenta, epine and q\&&lltitlee wboee claeeical expreeelon contain• the time to an odd power, are ebanged. Consequently, helicUy b unallecte4. ID thla caee, care hae to be taken with the exponeatial in the plane waYe, eo we will write - .. .. +(x, t;p, ~) • u+(p. ~)• •lp• X (36) The operator for tlme rever•al le antiunltary (eee reference U, page l26 tL ), anct we have for aplnore - - .- ,+(a, •ta-p. ~) • ~o~iv 5 K+(x. tap.).) (36a) where K •tand• for the ope ration of complex conJu&ation. lncteec:l we have j -nT' EZM (t i:J)( . lp,). >*/ ~+M (1 i;J )( lp,). >*/ • Gz j -nr + • l YoYz i + 0 0 ,\ ,\ e e l(Et-p• ~) -l{ E( •t)•(•p)•;) ~ .. .. .. "" • '-J-nf" (t • y.m )( lfT 20,,.' ~· ) e·l[ E (•t)•(•p)•XJ -178- • e ·Z1~ - .... "'(x. •ta •p, M Hence the p~e fac:tor 1J t. cletenninedl 'l = e ·21""-'r (36b) We have ueed nlatf.one auch a• (l?a) (l?b) tcr 2 lP•'- > • e A • •21,_., A -p,l. > (37c) which are eaey to check. U we call (38a) we have u' • u ·1 • 'Ys'Yi"to (l8b) u• * u (38c) Uu+(p,).) • e 21~u+( •p,).) (39a) -u+(p,l.)U ·1 • e ·20.~·· .... u+(-p,l.) (l 9b) a le a nw:nbor, we have: Then, rememberina that --. - a c: u(p , ).') Tu(p, l.) u(p •. ). u·1uTu·1uu(p. ).) 1 ID ) -179- _.< -p.. ,, ""')uTu·1u •c •p, .. ") " • e ztCk-~')" u = \lt (•p.., k)(UTU •1 ) T y 0 u(•p-, , ).•) (4 0) With our pantcwar choice tor y , we obvioualy have Jl m c 0, 2, 5 (41a) m •1, l (41b) an.d hence (42a) m c 1, l,l, J (42b) (42d) Alao from B•12 w. aet ;(,.~, • -e -zt~;-·c-~~) (43a) c 0 (k.~) • . -zt~c•c-k.k) 0 (43b) ancl hence • • ·21~ • - <·'- ).)) ..... [ p 0 c 0 (-k, ).) - (•p)• c E q,\\&tiona 4U and 43c •bow what the operator doe• to Ts lt ••••ntlally l'ever••• the orclel' of the factor• and chan1•• the •lgna of the three- -180- momenta. The effect of the change o! sign in t i8 to interchange the initial ancl fill&l atateaJ ebb abowe already for the aplnora in equation 40 and the complex conjugate of the polarisation vector in 4lc. II a theory la invariant \11\cler time reveraal, the mlltrix element Q aho\114 at moat be multiplied by a pbaee factor whan initial and final atatea are t.nterchaoged aad the alan of all momenta changecl, leavlna the hellcitlea Wl&ffected. U we expreae the reaQlta of our calculatlona where the eubacrlpt R ie a reminder of the l'eVel'ae ordel' of the factor a. Invarlaace thea demande that (45) (46) abould be choaen ao that an exchanae of the tnltlal and final momenta and polarisation vector, toaether with a reveraalln the OJ'cler of the matricea involved. leavea them unchanaecl. Notice that the phaae facton ln equa.tlona 19 and 43 wUl produce a factor e 21Cl.•J1)• n.eceaaary to a•t the rtaht • 5 whea ~ dependence in equation ancl fl have been lntercbanacsd. The effect on the partial wave hellcity ampUtudea ia atven in ll5·55 (41) -181APPENDIX B DETERMINATION OF HELICITY STATES .FOR PARTICLES OF SPIN t AND 1 a) Spin ~ We follow the conventions ln reference 5, and compute fbat the * J'Otation matrtx R.,,e, 0 R~~tv•e Jk -t~J •e •l,SJ ·lyJ Ye • (1) =i crk (Z) Remombe ring that z (3) ..k •1 we get by aerioa expanaloft ·l ~Clerk e ~ 4 • eoa l' • lcrk aln 'I (4) Hence .,,,,0 • R • (•tnz ~·: l 9 -t-I coal e 0 et 9 •aiD 1 cl alft! e tj coa -'1 6 • aln2 coa ~ ~ e tl (5) The hollclty atatea are j9, f';"- > • R.,,e, -~+>.. • eU•tJRtp,B, o+>.. (6) I9, ,, t > • e1 • 41Rt' 8 0 ( 1 ) • ( co a9 ! 1., ) (7a) • ' ¥See refe renee 6 aectiOD 13. 0 alft I e -182- ~) . (} - sin 'I e ..... coe ~ (7b) b) Spin 1 The vector meson• will be labeled aa particle• b and d, and the X).. a (•1) •->..e ·hrJY "').. (8) have to be u•ed (see RS·U). Since we are dsaling here with ordinary three•vectora, the rotation matrice• are well Jmown; but they are abo eaey to determine hom equattOft 1: J J J • y • a (9a) • (9b) .• (9c) from R6-5. 4Z; the cor:re•pondlng "'>.. a:re the u.ual polarisation vector• glveD in R6-5. 44. (10) - 183- R.,,e.o s -atn • (coo. atn coa ff 0 IZ :) ff 0 (coo 8 coo" cos 6 etn - aln • coa f11 f' - atn () 0 ( coo 8 •• , : 8 1 otn08 ) 0 coa 8 0 oln B coot') aln 9 ain • coa 8 (U) We then calculate 1 c(+l) • e •if1R 6 ..!.. ( -t J • • · 0 .fz 0 coa 6 coa fl# + l sln ") coa 8 aln • - l coa • ( • •in () e -if~ ·. fz (lZa) ~J i(O) • Rl!, t'o O ( . (::: ::: :) (llb) cos 8 «< -u • e 1"Re 0 ..!.. ( ·•· .fz :~ J 0 8 C08 tp + 1 eln " ) • co• 8 a1n • • l coa ff • C08 ( (llc) aln () aAc:l hence, ualng U-891 to obtain the correct nonnallsatlon for the four• vector ~. coa () coa " + l at.n " ) coa 8 •in • • 1 coa f' ( • ain 8 (13a) ·184· k c(o) • - sin 6 coe 0 m .~ i(·l). - .fz ( fJ) eln 8 stu <1 (Ub) coe 9 - co• 9 coe fJ + i alD. • ) • coe 8 eln tp • l cos • ( etn 9 (Uc:) It ehoulc:l be remembered that the hellcltlee correepond to partlclee - moving in the direction of ... p • and that 6 and of - + p. tp deltne the dlrec:tlon -185APPENDIX C dJ ~ a) FUNCTIONS IJaeful aymmetriea: <..,.(6) =(-1)~-..,.d~~{ B) (1) 3 • .,.d~ ( 2) z:: ( -1) _,, .,.{•-6) (3) {4) (5) (6) (7) b) Half integer indice• (table) 6 J :r l + ~ 6 coa a l +i l [ P.f +l(x) - P.f (x)] (8a, b) X a COli {9) (10) J (11) d 1 1 (6) ·z·z = 1 8 [ .pt +1 (x) + p .f (x)] 2 coat" 8 ain 8 = (12) 2 aln -y 1' l [ .p• (x) + p• (x)) l (t +l)V't (l +2) l +l V't(l+2) [Pt+z(x) = 4 co• z28 etn . lJ 2 21 +3 + (13) Pl+l(x) Pt (x) 21 +t - zn:J - Pl -l(x) ·Ju +1 Cl 4 ) e = = ze 2 coa 2 ain "! [ P" (x) + P" (x)] (t +1)-/t (t +2) l +1 r n +3 .J.t (l +Z) 2 coa 3 ! r .t P~ +z(x) 4 COB j pt (x) - l1 +3 21 +l pt-1(x) + 21 +1 -~21r7-+,...,3;___ J (t +2)P;'_ 1(x)- (17) 21 +1 3lPl+l(x) 2f +) + tf fl 3(1+2)Pt(x) + (x) ] Zl +3 , (.f +2)P _ 1 1 3 (18) i1 +1 T Z ain J P;\1(x) 3(t +2)P;' (x) + Z1 +1 rlPJ.+Z(x) ! 3l 2l +3 1 - (16) = l (I +1)(1 +z) 1 = pt +l(x) p l +2(x) e 28 4 coa z sin "i (1~) t ~ l (1 +1)(1 +i) II: (19) (20) = 4 coa . 3 8 2 6 Z aln 2 {l +l)v'(J. -l)t (t +2)(t +3) = .f(.t -1~1. ~~ +2)\.f +;> 8 cos l sin [ P ~ (x) - P"' (x)] +1 (21) 1 p.t+3(x) pt-+ Z(x) 2.Pl+l(x) [ (21 +5)(21 +J) + (21 +3 )(2l +1)- (21 +5)(21 +1) 1 2P (x) 1 - (21 +3}(21-1} + Pt -l(x) Pt -z(x) (21 H)(ll +1) + (21 +i)(H -1) ~j (22) -187- dJ 5 1 (8) - 1' · 2 = 4 co a 2 !!. a in 3 ~ 2 2 (.t +l)~(.t-1)1 ( f +2)(1 +3) - ~(1-1)1 (.t +2)(1 +3) 3 6 8 coa - aln - r -28 2 [ P .f"'+l(x) + P 1"' (x)] p.t +3 (x) (23) p.f +2(x) _ - (u +5)(21 +3) + (21 +J)(zi +l) 2 ZP.t +l(x) 2P1 (x) P _ (x) 1 1 pl-2(x) + (21 +s)(u +1) .. (21 +3)(21-1}- (21 +3)(21 +1) + (21 +1)(!1-1) -; J (24) = 4 6 6 4 coa ~sin l l (.t +l)(.t +2)(1 -1)(1 +3) r U "h (4) tP1 +z(x) - (4) ll P 1 +l(x) + 4 3 (1 +2)P~ ) (x) (l. +2)P~ ~ (x) - 4 21 +J II ~(l -l){l +3) " ~ 8 8 co• 1 J (ZS) .t P 1 +3 31 P 1 +Z 2(1 +S)P l +1 (21 +s)(u +3) + (21 +l)(u +1) t ( 21 +s)(?J +1) 2 aln '! 3(1 +2)P _ 1 1 .t t (t +t)(t +z)J(.t -l)(t +3) + r 2(1 -3)P (x) 1 + (21 +3){21-1) (.l +2)P _ ] 1 2 (26) - (Zl +3 )(21 +1) - (Zl +1)(21-1) 4 coa ~ aln4 ~ ~(1-1)(1+3~ =8 coa 'I8 aln4 'I ll +1 + r 2(.l-3)P 1 - (21 +3)(il-1) c U +1 r +3 l.f p ( 4 ) (x) 3(1 +2)P( 4 ) + 1 +1 + l -1 4 (I +2)P(t -1 ) -J l p ( 4+)z(x) 1 21 zl +t 21 H (27) 21 +t lPl+l(x) (21 +5)(21 +l) 3.tP1 + (x) 2 2(1 +5)P1 + (x) 1 (21+3)(21 +1) + (21 +S){ii+t) 3(.t+Z)P _ (x) 1 1 - (21 +J )(Zl +1) (l+2)P _ ] 1 2 + (21 +1)(21-1) (28) - 138 r(l-l)l P~~l (x) (1-1)1 (l +1)(1 +i)(l +3) ~ (21 +5)(21 +3) 4 coe = 5 {il +5)(21 +1) (2.1 +J)(if-1) 1 (.t +2)(1 +3)P~ _> (x) 5 (Z1 +3)(21 +1) + 5 C08 J d 5 5 6 10(.f -l)(t +3 )Pl + (x) 1 (2l +5)(21 +1) S(l+2)(.t+l)P _ (x) 1 1 (zl+J)(21+t) '"!•2 5 =(1-1}1 U+1)(1 +z)(l +i) .. 1 ( 21 +5)( 21 +1) S(l-l)tP1 +Z(x) (21 +1)(21-1) + (1+2)(1+3)P1 (x)] + (21+i)(z1-1S (5) (f-l)tP 1 +3(x) 10(1 -l)(t +3)P~ } tx) 5 + +5)(21 +3) + (zl +3)(21 +I) lO(.t ·1)(1 +3)P~S)(x) + (21 +3)(Z1-l) 2 r (30) (5) 5(1-l)fP1 +Z 5(1 +Z)(I +3)Pl:) (x) (I +2)(1 H)P~:) (x) -J 1 + (zl +3)(21 +1) + (21 +1)(21-1} 1 (29) 10(1 -1)(1 +3 )PI (x) r (u ze l + (21 +3}(21 +I) (il +5){21 +3) 4 sin 5(8) ( 21 +t)(il -1) - r z 2 5 (1-l)lPl+J(x) 1 S 2 (x) 121 +3)(21 +1) 10(1 -l)(.t +3)P~ ) (x) S(t +2)(1 +J)P~ ~ (x) + .. 1 + + S(l·l).l P~ l 10(1-l)(l +3)P~ } tx) 5 21 5 ~ {l-1)tP1 +3 (x) (ll) 5(1-l)lPl+Z(x) =-Do -• tn-s~e- (21 +s)(21 +3) + (21 +3)(21 +U 'I . 10(1-1)(1 +3)P + (x) 10(1-1)(1 +3)P (x) 1 1 1 - (21 +s)(u +t) + (u +1)(21-1) 5(1 +Z)(l +3)Pl·l(x) (21 +3}(21 +1) + (I +Z)(l +3)Pt _ 2 (x) ] (ZI +1)(21-1) (3 2) -189c:) Integer indices (table) J a I. (.H) ( 34) 9 9 -2 c:oe '! •in '! J ciio<8) • • I ..flAI. +1) (35) P (x) 1 v't(l+l) [Pl.21 +1 - 21 +1 +l(x) 6 2 c:os '! 8in "! Pl. -l(x)] (36) ( 37) J c!il(B) • 2 6 1" 1(1+1) 2 co• 1 il +1 2 8 = 1 {1 +1) r u +t 1 [I. P 1 +1 2 c:o• d .. uHn ri.PI.+l(x) 28 = J " • rl. PI.21+1 +l(x) " (1. +l)Pl-l(x) -J -Pl. (x) + if+l 'I · 2 sin Z 11 • 1 P 1 +1(x) 28 2 sin ! - ze + p l (x) + " + P.t (x) + (x) 21 +1 - pI. (x) + (l+l)P1 _1(x) -J 21 +t " (I. +l)P l-l(x) 1 (I. +l)P1 ..1(x) -J il +t 21 +I (38) (39) (40) (41) 2 8 J 4 co• '! •ln 1' " d (8) • P (x) 20 v'(l.-1)1. (1 +1)(1. +2) I. (42) (44) -190- d J Zl (8) • - 4 cos 38 8 ?. sin 'I [ ~ l (I +1)4(1 -1)(1 +Z) - r v'(l-y(l+l~ 1 \ 4 c:oa 2 • 1n '! '" IPI+l(x) 2.1. +l '" (l+l)P1 _1(x)J + P (x) l Z1 +l m + pl+l(x) lPl(x) 21 +1 + (ll +i)(D .1) 1Pl+2(x) (21 +3)(H +1) P 1 _1(x) - Zl +t J d.zt( 9) • - r 8 J6 4c:oayain 1 '" lPl+l(x) 21 +1 1 (1 +l)v'(.t -1)(1 +Z) v'(.t-l)(l+Zb f (t +l)Pt .. zhd - (21 +t)(Zl ·l) tPli2(x) ! J (46) "' "' (l+l)P.t-l(x)] + P 1 (x) + Z1 +1 (47) lPl(x) pt+l(x) e , 3 1 - (D +3)(21 +t) + 21 +1 4 coa 'I • n 'I .. (45) - (21 +3)(21-t) (t +l)P _ (x)1 2 P _ (x) 1 1 21 +l + (21 +!)(Zl-1) ! J (48) 4 4 4 4 cos f(.t -1)1 P\ 2 (x) Z(l -l)P~ ! (x) dzz< 9) • {1-1)1(1+1)(1+2) 1 (:U+JJ(Zl+l) !f+t J 4 6(t -l)(t +2)P~ >(x) + (il+J)(zl-1) 1 1 4 2(1 +2)P1~ (x) - 21+1 (.1 +1)(1 +2)P~ ! 2 <x) J (ll+l)(!f-1) + (49) 1 r(l -1)1 PI +z(x) u1 nun +IJ + =4 coe " '!e + J cl.zzC 8 > = 2(1 +2)Pt _ (x) 1 21 +1 + 2(1 -1)Pl +l (x) 21 +t 6(1-l)(l +2)Pl (x) + rn +3)(!1-U (.t +1)(1 +2)P1 _ 2 (x) ·] (SO) (21 +t)(zl-U r P!~2 (x) 2(l-1)P~ ]1 (x) (1-1)1 +1)(1 +2) _ (21 +1)(21 +1) u ll 4 sin 4 ~ 4 (1-l)l + (1 2(1 +2)P~~ (x) 6(1·1)(1 +l)P(;)(x) + (il +3)(2l-1J + 11 +1 1 {l +l)(l +Z)P~~ (x) + 2 (D +1)(21-1) J(51) -191- 1 II [(l-l).tP1 +2 (x) 2(1-l)P.f+l(x) 6(1-1)(1+2)P1 (x) 4 8 - (fi +3)(21 tl) • 2l +1 + (21 +3)(Z1-l) 4 eln 2 .. 2(1 +2)P _ (x) 1 1 2l +t + (l +1)(1 +2)P.f _ (x) 2 -j ( 21 +1)(21-1) (52) d) Recurelon relatione and general formulae One convenient way of calculating one of the•e functlona is to atart with (53) for an inte ger 1 • 1 d~ • J.l +l(6) • d~ ( 8) can be dete rmlned by repeated uae of 1 .J(t +~·. +1)(1 •f1) ( - 8i'{ii""'g ~ + J.l cot 9 - D d) d~1 .. (6) n (54) ""'" For a half inte ger J = l + .;. one can atart !rom equatlon 53, then uae and equation 54. Other recurQiOn relatione that mig ht be useful, especially if only ernall value• of J are needed, are (56) .JtJ+vHJ~-1) d~Un • {(J+~)(J+~-1) (1+ co• B)d~-!1 • J.l_1(8) +2 J 2 2 ein 8 J ·1 J -~ v J -1 J.l•l( Ll) d ~ J.l_ 1(8) + (J -~)(J-~-1) (1· coa 8)d~+l, u (57) -192For proofs and further relations, see references S and 6. We will prove next that the forms of the expansions in sections b and c are general . ct1mm in terms of hyper• R6-4.14 expresses the 1 geometric functions, related by R7-10. 8.16 to Jacobi polynomials: j ctm'm< 9> 8 • [ (j-m} l(j+m '21 ]~ (cos 'I) {]+Dlj IO-m'J r Zj+m-m' m '•m (-sin 2) (m'·m) I X F(m'-J, -m-j, m'-m+l;-tan m'~ m ~ 0 8 2 !) (59) COS 6 =X (m' a: m 2:: 0 le a trivial limitation; relations 1 through 7 can be used. ) ct' <> [ m'm x a H-m) 'H+m'~ .-J' +m) I •m I 1 J+m-m' m '•m 2 <-ltx) 2J(m'-m) I (ltx) 2 :+t ) r = X F(m'-J, -m-j, m '-m+l; m'+m m 1 -m (J-m) tU+m ') ,-J~ (1+x) Z (-l+x) 2 {J+m) I{J· m ') I zm'(m'-m) I X (J-m )-1 p ( rr_:;;,m• m+m (x) I J·m' m'-m 2 :zt(•l) I j ) m'+m -it r(J+m')I(J·m')I -J~ {]+til) l(f-m) I ~~~ (l+x) 2 2 m X p(m'•m, m'+m)(x) J-m' =c(l+x) 2 (60a) m '-m m'+m 2 (1-x) m'•m (1-x) 2 (m 1 •Dl,m' +m) ( ) p j -m I X (60b) -1 CJlwhere m'-m c: i [<i+ m'J IH·m') 1-j 2 -m' ]tm I -m)1 =(-1) 2 (60c:) W e will determine the c:oe.Uic:iente in the following expansion in Legf:lndre polynomiala: m'+m -i d"m'm(x) (l+x) 2 m'•m 2 (1-x) -~k=O (61a) From RS-6. 3. 42 ~a -z sl 2k+l J ·l d m 'm(x) (l+x) m'+na 2 m'•m (1-x) 2 Pk(x) dx (6lb) and u.aing equation 60b Rodrigue•' formulae for Jaeobl and Legendre polynomials, R7·10. 8.10 and R7-10.10. 7, are (62b) and hence (63&) m'+m elk18 (-1) j+k- z r u+m '>. 2 -J-k [ (]+rn) 1(]-m) I(J-m') I JkT1 (lc:t i ) (63b) -194Integrating by parts, we get the followine; two expreaalona: (64a) (64b) b of degree 2kJ hence ~ •0 for k > J + m' (6Sa) for k<j .. m' (65b) J+m' ~m'm(x) = - - m....1+"'"m_....l......_m_*..-m-. (l+x) Z l ~Pk(x) (66) Z k•J-m' (1-x) This reawt agrees with thoae in aectiona b and c. In reference 7, page 164, (i) it 18 atated that derivatives of orthogonal polynomlala abo form a ayetem of orthogonal polynomiab. G We will uae the Gegenbauer polynomtah T 1 , aerivatlvea of Legendre polynomials: d j , m m (x) (l+x) .. 1 rn'+m 2 (1-x) - m •m >. Z = defined ln term• of m+m' h. Tk '-' -k •m-m , (x) (67a) k•O where (67b) ancl, using a formula in reference 8 page 783, -195- b k= 2k +1 2 ( 67c) Uainb equation• 60 , 62a and 62b, we get dk+m+m ' X r 2 k -J dxk+m+m' . (1-x ) ( 68a) dx ( l)j+k-(m ' +m) / 2 2 - j - kr dj+m') c2k = • . O+m)l • m) f{J-m 1) I X ~k+m+m 1'J I k·m - m I (k+t) -Jt ( 68b) and, integrating by parts , bk =<-uk+m+m ' czk ( 69a) ( 69b) Hence ~ =0 +m { 70a) for k < j • m (70b) for k > j -196 ~'m(x) • (l+x) f+m m+m' z (1-x) m'-m 2 h. T ,(x) m'+rn -k k-m-m (71) k=j-m alao ln agreement wlth result• ln aectton• b and c:. The tntegrab in equ.attona 64 and 69 can be reduced to beta functions B(m, m) (see R8-4. 5. 54) to determine ak and bk; ln practice, though, lt might be preferable to u•e recurdon relation.. -197APPENDIX D DETERMINATION OF PROJECTION OPERATORS One way to determine the projection operator• for a partial wave expanaion ll ahown tn the flret part of thll appendix. By aum• mlng over final heltc:itiea and averaging over initial onee, equation A-6 givea ~ (1) Ualng A·S (~)unpol• (2• a +ln2.b +u (~) I I<J+tHJ'+!) JJ' J )( J' 0.~ ~ ;~ ~ (o.~ ~ ·~ ~) c d a·o c d' a·'b • J J' d~..(8)d~..(8) (2) Hence the total cro11 aection ie S~dO cr • J dddx) cr• 2w 1 dx ·1 X • COl 6 i 21 +1 i l to be replaced by 2 tf the mae• of the correepondtng particle ie mero. R6·4. 59 glvea (5) Carrying out the integration and awnmation over J' in 4, t1' 1:1 LI Zw (2a +t)(Zab +t) a (~) z.. c where <2•a +t)(z.b +t) (J+t) lo.f~, ,z J L t ) ICl: 12 (J + (6) ni J,l o.f are the independent element• of ,;, TJ and n 1 the number of tlmea o 1 appear• in it. On the other hand, ll 3i ia a aet of amplitu.dea like thoae in U-30 for inatance, we can write ( ~)unpo1• (2•a +\nz.b+1) ljl :r 7°t/'J 1 t • (2•a +1)(2ab Ofl) 3 G:J where Gij (7) = Gjl• ia a •metric• determined by •wnmlng over aplna and polarizatlona. Thia h how equation II-37 w aa derived. (Notice that :J la uaec:l both for a column vector of :. •e aa in equation 7 and for a 1 Zx 2 matrix aa in 11-34; they are eaaenttally the same entity.) The :rk are next expanded in partial waves, giving :Jk • L of Ai!t (8a) J,i or, uaing matrix notation (8b) -199Sllb•tltllting 8b into 7, • cz. a +tn1 2• b +t) J,'LJ' c Jt A Jt cA J' o J' (9) z.. +J) 6"' o Jt ( .)r 1 dx A Jt GA 1 •)u1 , (10) ( dcr) unpol an cr = (2a +1)(2a a b JJ' -1 .By comparing thia eqllation with 6, we obtain (11) I! we define (12) and a generalization of the "acalar product" of D--4:0 (13) we get from U: (14) -zoo. or (15) We can calculate the vector• x:. 1 GpJ nt(J+i) l (16) and 15 can be rewritten . ~. r>~t &Dei we can call the :Jclx (17) xf projection vector•. U a dUferent eet of amplitude• 3' • B:J (18) b then c:hoaen, we have from 7 dcr ) 1 ,t . ( Cln a (21 +1)(21 +1) 3 0 <P~ UDpol a b and (19) From 8b, and A 1J • BAJ •J "t J J • BA. i • Bftt (20) (Zl) -ZOl· (2Z) Check: which b equation 17 Sometime• it i l &a•y to compute the projection operator• directly • For inatance, if we define the amplitude• • T (M (Z3) Jl. a ).c- ).d Then, ualns equation• A-5, C-71 and C-4, we can expr••• them in term• of Gegenbauer polynomiah (derivative• of Leaendre polynomial•) (Z4) -202- or, 11 we define rl* ~c~d~~a~ • <-u~~a. 1 • a.1 ~c~d 1 ~a~ -~c-~d~~.~ (ZS) fl. ~ ~d ~~ ~ • {z 2:J (J~) ~~·~ ~~ ~ ! \~zm(J, "'p.)Ti_i'.zp.+Zm(xl) c a ·"b c cl a·"b m•O where J'o and J'~ are either Ji and Ji•l, or both Ji•+, whichever are tnte1er•. From equation A-5, u•ln1 D-5, we 1et (27) We wtU u•e next the other expr•••t.on for the polynomtal8, equation C-66. d~ tn term• of Leaendre (28) 0 and ).~ are either ). and )..1, or both ).. +• where ~ Equation 28 exhibit• the projection operator• • showing that for theae amplitude• only 2). + 1 Legendre polynomial• (and no derivative• or powers of z) are involved. From tho prope rtie • of the helict.ty amplitude a and hellc.lty etatea under the parity trandormatlon lt te eaey to see that the ampll• J± tudee ~ ( ~) correapond to tranaitions between states of definite parity. Coneequently, their coefficients in the partial wave expansion (equation 26) are either even or odd functions of x. J± The properties of the IS(~) under the reflection c:liacus•ed in seetion a) of appendix A can be derived using equation A·l4: -204- J± . ~~X ·~~ ( .. \ v ) c d' a o ~+v. J =Q~J c ~d'·~a·-b ~ (· W) :t: (·1) a .. ~ .. ~·~'"- (- W) c: d' a·o (Z9a) and from equation Z6 • T ~ ~ • ~ ,.__ ( • W, x) ;:a ( ·1) c d• a o ~+~ ~ T ~ ~ • ~ l... ( W, x) c d ' a ·o (Z9b) .. zos.APPENDIX E LINEAR RELATIONS BETWEEN INVARIANTS Ae wae pointed out lor equation• U-126 and D·UO, ln the more complicated caaea there are r e latione among invariant. that are not obvtoue and that are related to the lact that no more than n vectora can be linearly independent in an n•dimenatonal vector apace . The atratght!orward algebraic atatement of thta fact involve• the anthym• metric tenaor of the correapondlng rank, and la not immediately ueeful to find theae relattona. In thb appendix a way of finding them will be abown, and the problem of the introduction of kinematical aingularltiea for the lnvarla.nt amplltw!ea will be dlacueeed Firat we wlU ahow how equation• U·Ul are obtained for the three•dimenaional caae, aince the other caae ia a direct generaltzation of thb one. Incidentally, we will abo derive aome uaeful relation• for reducing term• conta ining the antlayrrunetric tenaore to a combina• tlon of the lnvarlanta ueed here . by " • .... The vector product will be indicated .. .. ... By expanding (a" b) " (c " d) in two different waya , or other· wiae, we get --- where [a b c) ----... ...,_...... ...... ... __ .... [ a b c) d • [ d a b) c + [ d c a] b + [ d b c) a (1) i• the "box product" of the three vectora, that la ~.... .. .... [ a b c] = a A b• c (Za) or l.j, k • 1, z. 3 (Zb) ·206· By making the eubetltutiona t.n equation 1, we lind alter some alaebraic redu.;tlou. .................. _._._ - --- .. [a b c) 4 • d• a b 1\ c + 4• b c 1\ a + 4• c a 1\ b (3) From the property of the Pauli epin matrices vl "J • 6lJ+ iclJk17 k (4a) (eee Rl-1.101) . we get .... -.. .. .. ... .... rr• a v• b • a• b + lrr• a 1\ b (4b) .......... ____ .. -a + v• v• a • (4c) ff'• 0'• b b la• b By repeated I.Uie of theee eq\l&tlon•, we have (Sa) where x K , . It' • co• 6 On the othez: hand, and ueing equation l, we have -,.. --,. ,. t~ -,. • t' 'c 1\ ac + t: ' x' 1\ x • 1(. x' c 1\ • ' • x' • c « ' 1\ a «• • ' and hence, uetng equation 4b aaain (Sb) ... -Z.01- (Sc) Combining Sa and Sc, we get xNi • xNi • N l + lN.& + Nf, • N7 - N9 + Nh • 0 which le equation Il•llla. (6) ( The Ni are defined in 11-UO.) 1f the eame procedure ia applied to a eltghtly different expree• •ion like <'A ;") A (k A ~) + (k' A i) A (k' A if), we get the eame equation 6. ( The combination of terms hae to be cho•en •o that time tnvariance b •atiefied. ) We try a more complicated expree•lon ne:xt. By expanding (=" b) " [ (~" d) " (d" a] in two ctilferent way•. we get: ................ _ ... + (a•----...... .. (a• c a• c)e e c e a. c)d l .. .. .. .. -.... d)f -- A ...c + (o•,. ..£ ..a• .. .. -.... -A ..e + b• b• d • - d b• (a• e b• d • b· c a• ~ A f b• b· c .. a• f A c)d 1f we •et now ; • k, !r • ''· '; • ;, d • ;., ; • '· 7 • ''· multiply by i .. and u•e 4b, after aome algebraic reduction• we obtain: (7b) which i• equation ll·Ul b. A ueefu.l relation obtained from equtione 4a, 4c and 4b i• -zos- ---- ............ -.................... .. l{ a b c] a rr• a rr• b <r• c - a. b rr· e + C:• a rr• b • b• c: "• a We will adapt now theee c:aleulatione to four•veetore. (8) It ehould be remembered that we have to uee the Minkoweki metric: when a eu.m• mation over an index whose range ie 0, 1, Z, l inetanc:e te performed. . , ,v. For 0,1, 2,3 (9a) where the metric: teneor le given by i 00 • • i 11 • •g 22 a •g 33 =1 (9b) We aho uee (9c:) We define a "triple vector product" (lOa) by the eq uaUon (lOb) where ~)...,.vp ta the antieymmetrte teneor and ~OlZl • l (lOc:) Obvtouely, the triple vector product ie sero when any two fac:tore are equal, and c:hangee eign when two of the vector• are interchanged, while a eyc:lic: permutation doe• not change lt. We deline the "box product" for four four•vec:tore by (lla) -209- Hence [abed) -= (a 1\ b 1\ c)• d ,. d· (a 1\ b 1\ c) (llb) (The parenthe•e• around the vector product are actually auperfluoue.) [abed] • • ao ~ •z al bo ~ bz b3 co cl cz c3 do ~ dz dl (llc) We would like to emphaai&e the different behavior of box product• of three•vectora and four•vectora under a cyclic permutation: the latter change •ign, and hence [ abed] • • a• b 1\ c 1\ d (lld) eo that the position of the dot te not irrelevant after a certain definition baa been choaen. The box product of four-vectors la ce ro when any two vectora are equal, or, more generally, when the four vectora are linearly dependent. An equation analogous to 1 can be obtained. Since any ftve four• vector• are linearly clependent, we can write "tal+ o.2a 2 + o.lal + o4a.f.. + osas =- o Multtplicattona by a 1 A a 2 1\ a 3 , a 1 /\ a 2 /\ a 4 , a 1 /\a 3 /\a gtve 4 3 4 and a 2/\a Aa o.4[ ala 2a3a4) + o.S[ ala2a3a5) • 0 o.l[ala2a4a3] + o.5[ala2&•as] c 0 o.z( alala•az] + o.s[ a1a3a4•51 c o ~[a 2 a 3 a4a1) +4 5[a 2 a 3 a•a5) cO and hence [a1a2a3a4)a5 + [a5a1a2a3)a4 + [a4a5ala2)a3 + [a3a4a5a1]a2 + [a 2 a 3 a 4 a 5 ]a1 • 0 (12) We get the equation equivalent to 3 by the analOiOUa eubetitutioneJ tbeae give [a1a2ala4]a5. a•.•s a1Aa2Aa3 _ a3•as a"Aa1Aa2 + a 2• a 5 a 3A a 4 A a 1 - a 1• a 5 a 2A a 3 A a 4 (U) 3 4 1 2 3 4 4 1 3 2 3 We have •ubetituted a 1Aa Aa - (a Aa Aa )A(a Aa Aa )A(a Aa Aa ), etc. By direct computation we can ahow that ·~~vp«GJ'yp • • 6).o~1'ts6"Y• 6~~6t'Y6" 0- 6~Y6t'n6vf' + 6~0 6..Y6"#S+ 6~,6~0 6"Y + &~Y65-LI'6" 6 (14) &JUt by its uee 1 z 3 4 5) 1 2 3 • 5 [(a A a A a )A a A a y • E~"Pa,.a.._aPEP 4~& 6&,s 1 2 3 • 5 • • «~yp«G~Pa~af'aPao.a,S • al··· az•• s .3 + .t.•s a2a3··· + -~az·a" a3 •• 5 y y y 1 4 23 5 1 52 4 J 12 5 3 4 •&•a aa•a -a•a a•a a y •aa•a a•a y y -211or z 5 •a•a 1 5 az· a•)al (a1A a zA a 3 ) A a 4 A a 5 a (a1 • a4 a•a 14 3 5 1534Z • (a • a a • a - a • a a ·a )a + (a2 • a 4 a 3 • a s .. a z• a 5 a3 • a 41 )a (15) Equation 15 can abo be proven by noticing that the vector product l• perpendicular to all three vectors, and hence (a1A a 2A a 3 ) 1\ a 4 A a 5 t. perpendicular to a1 ~t. a 2A a 3 and llea in the "plane" of a 1, a 2 , a 3 • We can then write 1 z 3 4 5 1 2 l (a A a 1t. a ) A a A a • ~· + e~ 2a + e~3 a and multiplyina by a 1 O•~a·a 4 4 and a z +4 2a·a 5 4 3 4 +e~ a•a 1 o • ~al ••s + e~2a2· a' + e~lal• a5 and hence ~ oe a 24 ·a a 3 •as • a 2 •a5 a3 •a4 1 s 3 4 1 4 3 5 a 2 oe a • a a•a •a•a a•a 2 5 1 5 2 4 a •a .. a •a a •a The conatant of proportionality can be found from a particular caee (with unit vectore, for inetance). Another uaeful relation l8 obtained in the following way: alA a2A .3.a4A .s,.. a6 • .. [a''A a5A a6 ala2a3] • . (a"A asA a'i A alA a2• a3 and ueing 15 we get -ZlZ- 1 l 3 4 56 14 z 53 6 1 5 2 6 3 .. al\al\a•aAaAa •·a·a a•a a•a -a•a a•a a•a 1 • a •a 6 a l •a• a 3 •a 5 +a1•a4 a 2•a 6 a 3•a5 (16) Or lt can be 4erlved directly from equation 14. By dbect comptltation we can ahow that (17) We uae it and 14to derive the exact relation: a,. b c -v "'-tJVp "'- f,l V p ~·a A b A c • €,. =1b €l.pv p a}.Itp CVE a.fS'rp~ 0.'J ~'J 'r'J 5 = t ~ 5 (6~a.6~1''v-r + 6AIS6f.l.,6va. + ~,.6~ o.6,,s· 6l.o.&~.,.6v~ Us) U•lng 18 we prove the foUowtn&: [ abed] c a A b A c• d • ~ ~· a A b A c - + •i y i i, y• a b c A A 5 {" ~ ~ • b• c I. + c• a )I - a• b ~)­ • ~(1. )$ f. - be c I. + C• a )I ... a• b I)} and after •ome reductlcm.a and permutatlou [ abed] • y 5 (I. )l t/. - .. c• d '- )$ + be d I. t/. • be c f. ~ • a• d ~ I - a• b tJ ~ + a• b c• d - a• c be d + a• d b• c) + a• c ~ ~ (19) -zult t. ea•y to prove equation• RZ•.Al8: y y • 4 (20a) y ,.,. - y ... • -1.;. (ZOb) y .,.,. )I y .,. • 4&• b (ZOe) .... ,.,. (20d) Ualng the flrat three and 14 and 16, we lind (U) Uaingl? we have: z .. 6 Ys'VAaA ~A.p.vpaA'V,.,.'YvYp • • 6 'Vs;. (22) EqWLtiou 18, 19, 21 and 22 show that any te:rma containing the antbymmetrtc tenaor can be eliminated from matrix elements. Ou:r next taek ie to fln4 the linear relatione among the U•1Z6, following the method• that p:roduce(ll eq\l&tf.one 6 and 7b. ~ of Uee of rather abnple expree.toae Uke ( c AKA P) A c' A K + (c' "K 1\ P) Ac A K produce only ldentltie •. Non-trivial reaulta are obtained aa followa. By repeated uee of 18, 16, D·121, D·lll, D•l24 and :rearranging of facw:re, we get 'Y" (E A ~ A P) A (« 1 A ~ A K) A (K A ~ A P) • 'Ysh• « A A A P 'Y• «' A~ A K 'Y• K A A A p • c A A A P• c' A ~ A K 'Y• K A A A P + « A A A P • K A A A P 'Y" «'A ~ A K .. c' A A A K• K A A A P 'Y• « A A A P) II - <t /1.; - ~· « ; + p. « 4Ht'~ II. - A• •'It + K· «'iaH~tiJ/ + K· p j.) + (•E• c' A 21<• P + K• « A 2P· c' + A• « A• c' K• P)(JI,4(J1 + K• P 4) 2 2 2 • (·K• c .A P + P• c A K· P)(,''4{ Jt- A• «'Jt + K• c'4Q{) + (•K• c'A 2 K• P + P• c'A 2K 2 )(t /. 1'. • ~. c 1 + p. « ~) a 2 K• P A 2 {MK• P~+ P M 2 + MM4 - M + (K• p .. P 2 )M 8 9 .. MK• PMU + P 2 ~ 4} (2la) On the other hand, t.l8lng 15 and D-124 we get 'Y• (c A A A P) A (c' A A A K) A (K A A A P) c [ c' AKP)[ KAPc)fi c 0 (2lb) and hence z 2 MK• P~ + P M + MM • M + (K• p ... p )M .. MK• PMU 4 2 8 9 + p2~4 c 0 In a aim.Uar way (23c) -215- cy• (c A K A ~) A (c' A K A P) A (A A K A P) .. c A K A A• c' A K A P cy• A AKA P + c A K A A• ~ A K A P cy• c 1 A K A P • c' A K A P• A A K A P cy• c A K 1\ A) c 2 2 2 2 2 2 2 2 2 A {(K•A -K P } H 2 - MK A M 4 - A K M 6 • MK K·P~ 2 2 2 1 2 + {A K• P+A 2 K +(K• P) }M + {K P 2K• P+P 2 K A 2 -A 2(K• P) 2 8 • (K•P) 1 }~· MA 2 K·P~ 0 + 2M(K•P) 2 ~1 + {P 2K 2 - ZA 2 K·P • P 2A 2 • ZP 2 K·P· (K•P) 2}~ 2 (Z4a) v· (c A K A.. A) A (c' A K A P) A (A A K A P) • [ cAKP)[ Ac'KP) ¥ • ( c 'KPA] y• (K2 P A c A A • K• P c A A A K + K• « A A K A P) a v• {K2 P A c A (• A • c' KAPAA+ A 2c• AKA P) 2 • P• Kc A (•A• c' K A P A A +A c• A K A P) A K + K· C(•A• .-• K A A A P +A 2c• A K A P) AKA P} = -K 2A 2 M 6 + A 2K• PM8 + {K2 P 2 .. (JC• P) 2 - A 2P 2} ~z + A 2{K2P 2.. (K•P) 2 }~ 4 (24b) whel'e U and 15 have been ueed to get Z4b. From theee two: 2 2 2 2 2 2 2 A 2{K2P 2 ·(K•P) }Mz + MK A M + MK K·PM.,• {A K + (K•P) }.M 4 8 + (A 2+K•P){(K•P) - K 2P }Mc,+ MA K·P~0 + ZM K·P~z 2 2 2 2 (24c) -216- Also <y• (~ A K A P) A (t:' A A 1\ P) 1\ (K A A 1\ P) 2 2 2 2 2 • . ~ {K P • (K• P) }~+ MA K· PM + MA 2 K 2 M 4 5 2 2 2 2 2 2 2 + M (K P -A K• P)M.r - M K · PM8 - M {K P - (K• P ) 2}M 2 2 2 2 2 ... MP A ~ + M{A P +(K• P) 2 -K 2P }M + ZM 2P 2 ~ 11 0 9 2 (2Sa) y• (c A K A P) 1\ (~' A A 1\ P) A (K A A A P) a • [ «KAP)[ Ke'AP) c J' 2 2 2 2 2 MK 2A M • MA K·P~ + M{P A -K P 2 + (K•P) 2 }~ 5 ... MA 2 {K P -(K•P) }~ 2 2 2 1 3 (25b) 2 2 2 2 2 2 2 A {K P -(K• P) }~· A K• PM • K P 4 M.r+ MK• PM8 + M{K 2P 2 -(K•P) 2}M9 • P 2 A 2 ~ 0 ... ZMP 2 ~ 2 / 2 2 2 2 - A {K P - (K•P) }M c: 0 13 (25c) One more l'elatlon that can be derived from Uc: and the rule• for multtpllc:atlon of determinant• l• 1 ~ -~ ·•z 1 •a3 •az -a3 2 [ala2a3a"Ha5a6a7a8] • ... al• a s s a •a 2 =- a 3 •a s a •a 5 • •o 3 2 -~ 3 2 7 l ·•z a•a 2 7 a •a al. a7 4 7 a ·• 4 l ••3 a•a a 2• a 8 .3 •• 8 a " •a 8 8 5 .3 a6 7 3 7 a3 8 8 .3 -·3 ao ai a2 •az "1 8 ~ •t •z 1 7 •o -~ 4 4 •o -a,. al• a6 2 6 a •a .3 ••6 4 6 a •a 2 s 5 5 •o a1 •z 6 6 •o· a61 a2 (26) ·217Thb give• an alternative, and maybe eaaler, way of finding equations Z4b ancl lSb. The three equations ~3c, 2.4 and 25c are not independent, aa ahould be expected. Thb then eervea aa a check. since they were obtained independently. A little further manipulation gives equation. It muet be remembered that equation• U•l2-' were uaed repeatedly ancl hence theee eqwlttea are oDly true when multiplied on both ddea -- - by the col"reaponding epinora u(p') and u(p). Thia la a baalc dUfer• once between them and equation• U-Ul. Theae calc:ulatione, unfortunately, do not ehow a eyatematic way of obtalnln& relatione between invariant• that can be shown to work in all caaea. The general idea la to obtain noa•trivial reaulta by two cllfi'ereDt expa118lou of a awlldently complicated expreaalon. The method outlined by Hearn in reference 1Z doea not eeem to lead to the right number of invariant• in an obvioua way, and further relatione like equatlona 11·127 are neceaeary. We wtU now brle1ly c!tac:aaa the problem of kinematical alngularlttea. In abort, they can be defined aa thoae aingularitlea not preaeat in the reaction• with eplnleaa pa:rttclea. In the aame reference 12 it la pointed out that, ln perturbation theory, the only difference between matrix elements for aplnleaa reaettona &Del the corl'eepondtng onea for reactions involving particle• with apln llea ln the adclitt.onal factor• that appear in the numerator. Tbeee have to be reduced to the choeen eet of tnvarlanta without intro• ductng ef.ngularltiee. -218· It l8 obvious that no denominators are involved ln the reduction o£ the matrix element to a linea r combination of invariants like thoae 14 in U-126. Any terms involving the antisymmetric tensor can be reduced by using equations 16, 19, 21 and 22. In the particular case of pN elaatlc: scattering, equatlona IJ•\27 ahow that M 8 and ~Z ' introducing new singularities. for instance, can be eliminated without Thla proves, to all orders in perturb&• tion theory, that the remaining lZ invariants are asaoclatt.d with amplitudes free of kinematical alngularltlel. Thie gives no way of telling if a set of amplitudes free of kinematical elngularitlea can be found until the actual relations are calculated: but it la presumed that thle is always poaalble. -2.19APPENDIX F SPIN j FORMALISM This problem hae been treated in eeveral different waye; we will follow more or leee reference U. See also reference 14 and other reference• quoted there. 3 The wave function• for epin I particle• will be written ae a eet of four four-component epinore a teneor index in epace•tlme. "'v.. fl. a: 0, 1, Z, 3; the index f.1 i.e From the general theory o! particle• with spin we can derive RU-4. 56a: (la) (lb) where pv ie the uaual differential operator when actin, on a function of x • f.1 p .,. 8 = -i 8 fJ = -l 8x' f.1 The eub•idlary condition lb eervea to eliminate the spin (2) i part in the Multiplying la on the left by p + M, we get (pz- Mz)"' =o (Za) "' Multiplying it on the left by y tJ. and ueing lb and the commutation rulee for y matricee, p ~ )J. tJ. • 0 (Zb) -220Both equations la and lb can be incl •.1ded in one of the form A J..l" where A (p)+ " .,v ie given =o ( 3a) by RU-4. S6b: A!JV (p) :a (p- M) &!JV - .;_(y p +y p ) +~ y (p + M)y ~ J.l V V 1J .) 1J V This can be checked by m ultiplying 3a on the left by y IJ (lb) or p ; we get 1.1 respectively ( ~ Pv +~My")+"= 0 2 (- Mp v + -3 (lc) PP v + ~.) Mpy v >+v =0 {3d) and by combining these with la we get back equations 1. Equation la can be obtained from a Lagrangian density by the usual equation• ( R9·2 . 4) (4) Fro m Rll -7 .124 (Sa) where the arrow indicates the direction in which A !J.V operates. Or, to bring it into a form cloaer to R9-7. 27: (5b) -ZZlFrom the Lag rangian density we can obtain the energy mornen• tum tenaor and ang ular momentum tensor. U in R-Z. 20 ( 6a) we s um over the splnor index implicitly. we get (6b) We carry out theee operations on Sb and uae equations h T Cl~ is re- duced to (6c) It should be noticed that the adjoint state vector 7;;.,. aatiefiea the equations if; ()1 + M ) =0 .... (7a) ... .... = 0 (7b) ~ y which are directly derived from la and lb: and that for functions that obey these field equa tiona the Lagrangian denaity reduces to zero. Equation R 9 •2. 27 for the spin tensor (Sa) can be written The Aij. ~p are defined in terms of inflniteahnal rotation• &.J~p by -Z2l(9a) ~IJ bae actually two indicea: tl • which ie a vector index, and an impllcit aplnor index. We determine then, by ualng R9-2. Z4 for a vector field and the equation• above R9-7. 30 for a apinor field, for butance, that (9b) (9c) where (9d) Carrying out t ne calculation• in 8b and uaing equation• 1, we obtain sK.l.p :11 -l<'lil.y ""' P .. l Py K"'l.) - \ ili14(y Kal.p + ul.py ">"'14 +t;aiCYl.+p• ~Kyp~Ji). +\Jipyl.+K- ;).yp+; (10) Of epectal interest will be t he z•component of spin 3 We wUl atudy next the plane wave aoluttona of tne apln 'I equation. We write ·2Z3\(1 (x) "" =rru (p- , ~ )e •lp• X (lZ) ... where rr le a normalisation factor such that if the u u u = -1 p are normalized to (13a) v "' (llb) the integration being carried out over a volume of normalization V. Hence 1 M I, l z =v·'E where E u l.l (Uc) =p 0 is the energy of the particle. We notice that u 1.1 and obey the equation• (p • M)u,...(p) =0 (14a) y u (p) c 0 (14b) ... 1£ 1.1 u1.1 (-j • M) = 0 (14c) u y ,.,. =0 (1~) ~- where pl.l ie the momentum four-vector and no longer an operator. We will prove next that the following a tate vector a correspond to different hellcity etatea for the apin ~ particle: (l!Sa) (lSb) (lSc) (15d) ·224- - The momentum p has been omitted, and the labeh correepond to f. epln 1 and epln ~ etat••· heUcitte• of the correepondlng epin 3 The epln Z atatee have been conlidered a direct product of epin 1 and epln i •tatea, and the correaponding C1ebach- Gord.an coefficients The u (~) eattefy trivially equatton la, elnce the Dirac have been used. tJ aplnore u(Jo.') do. - In what followa we wUl chooee the z•axle in the direction of p , to avoid unneceeaary algebra. •(1) it 0 -1 ·l 0 7z = Then ~(0) : 0 0 E ~(-1) c :Q 1 Tz (-~ (16a) (1:\> where 1-~>· ~1 I~ >= ( ~ Equation lb for J u~ ( 1) (16b) give• -v..,utJ<t> = Ta ~-v 1 + i'Y 2H1 + .Jt~'v1 6 3 ) - &<cr1+ tcr 2 ) cr1+ icr 2 - (o-1 + lcrz) El~ («rl +icr 2) • • 0 alnce (o-1 + lcrz) I \- > = • 0 \~ :I l 1 0 0 -225- IPI f+M 2 0 ~J~l I~ I •0 and ln a similar way we can prove that the other two atatea abo aathfy equation lb. We can check the helicities of the etatea by aubatttuting 12 into 11, and after performing a h'ivlal integration we get s3 c!) = ~ {- t(i l u 1 (t)u(i) + ~ tut(i)u(~))· ~ u1 {.)« :(l)cru«.,_(l)u(~)J = ~ (1 + +.> !h l =1 atnce ·ZZ60 and 1 ~ 3 u(lt: 1f ) • * ; u (* it)- .Aleo the u()..) are normalised eo that + u u = l, and hence u t u ~ m i(. t u(t)ylu(•t)+ t u(to)yzu(-~) + t u(-;)ylu(i) + \ u( -l)y zu(t))] M • F . f'J'tt·lt"J!?+Ti+Ti-'J~)ti -z z 1 E 1 1 Z 1 1 ZE E } ~ The third term in s1 aaain gives no contribution, aa can be aeen from the computation: •0 and almUarly for the other products. -227- The hellcitiee of the other two etate vectore can be checked in the eame way. AU the reeulte obtained eo far are of couree un- changed when the llelde are quantlaed. Our next taek le to determine the propagator for a apln particle. To thia e:£fect. we have to find an operator i dl~-1 (p) euch that (17) It le etraightforward to check that (18a) (Thle operator la incorrectly stven by Rll-4. 56~ It can aleo be written in a eUehtly more aymmetric fonn ~p 1 pl p C\f.l(p) • (f+M)(6l.,.• ~ )+! (yl + JXHf•M)(v tl + jf> M (18b) The fbet term ia, of courae, the direct product of the correapondlng operator for a pin ~ and epln 1 particle•. The propagator la then, with the convention• ueed in reference 1 page 478 and h a c • 1 (19) where p la the four•momentum. The mlnue alp la related to the normallaatlon of the uf.l, equation lla. -ZZ8- Another quantity of interest la the current vector, given by R9-Z. 31 J Jl :: (ZOa) Written in terms of aplnors , lt becomes 8l - l {p iii ) ~v v v (lOb) U we take the Lagrangian den.lty given by Sb, ancl use equations 1 and 1 in the reduction ol terma. we get (ZOe) for the current o£ spin 4 particle •. .APPENDIX G N/D METHOD In •ection IVa we point out that the T ::r T matrix can be written N 15 (1) in order to •eparato it ln two part• auch that D ha• only the ri~~:bt­ bancl cut•, arieing through u.nitarity, and N the re•t of the cut•; the equation• they obey will inaure that unitarlty be •atiefled. In thl• appendix we will not be concerned with the complication• arblna born a.nomalou• threeholda and complex •lngularitie•: the•e are left to appendix H. The T matrix l• defined from the •catterlng matrix by (Z) where P f.l and P~ are the initial and final total four-momenta. It can be •hown that the S matrls l• unitary for phy•lcal value• of the variable• (•ee reference 1, eection llct, for potential ecattering, reference 9, •ection 17. 4, for a general ca•e), t. e. s' s • 1 (la) ss' • 1 (lb) (For infinite and contlnuou• matrlce• both equation• have to be proven ••paratelyJ one i• not a coneequence of the other.) Relation 3a can be written L<b ls' lc ><ciS Ia> ::r < bla > c (lc) -230- where the awn over intermediate •tate• become• an inte 6 ratlon for contlnuoua spectra. Subatituting 2 into 3c we get c valid when (4a) For the caae of •catterlng of two •plnl••• particle• in the center of rna•• •y•tem, equation 4 become• - -· where q, q q 0 , q~, are the three-momenta of the intermediate particle•: their energiee, and W, the total energy. 4 T-Tt :a (2w) i r q-zc!lqldn· J 6 (2w) 9 6[ q ( 0 jql) +q~(jqj) • W) Tt T and uelng It-25 T · T t •2wi f . t dfl•p(W)TT q (5) where (6) -z +m 2 q 2 :aq 0 -z + m ,z q'0 = q (6a, b, c) We write now a partial wave expansion for T T(W, x) • Zw~(W) L l a11d •ubetitutint:> into equation 5 (21 +l)T1 (W)P1 (x) (7) -231- where the angle• e• are ahown in figure 1, and obey the 8, 8' and equation COS 0• "= COS 6 COiil 9 1 + ain 8 ain 9 1 COS f'' ~ z:. Cfi. y i'lg. 1. Direction of momenta Reference 8 on page 1327 give• the addition theorem for Legendre polynomial• l Pl(x•) • P (x)P1 (x1 ) + 1 zi ~ Pf(x)P;n(x')coa m11' m:al and ualng thh and the orthogonality ol Legendre polynomial• we get finally (Sa) or 2 (8b) ain 61 (8c) ImT, = IT, I t6, T 1 cr e •ZlZ· We will drop the aubacrtpt in what followa. We can uae equation 1 and get D * lm N • N • Im D c IN Iz and along the right•hand cut where lrn N ~ 0 we have (9a) and along the left-hand cut where Im D =0, we hav~ 1m N • D 1m T (9b) Now we can uae equation• 9 to write dbperaton relatione. For tnetance, from 9a we get •·•o f D a l--;-- J where a 11: wZ; a a 0 1 ClO •• N(a ')de' (a*-• H•'·a-{EJ (9c) 0 correaponda to the threahold for the reaction, and l8 uaually taken equal to a 1 aince thb would. give the right threahold behavior. Equation 8a ia eaatly generalized to the caae of several channela of two particle• with apln: (10 ) where the T;r are now matricea with element• correaponding to the hellcity amplitude• in the different channel-. for lnatance. The auper index J wUl be dropped in what follow•. Time reveraal lnvariance require• that (11) -233- eo that equation 10 become• (lZ) We cao write equatioa 1 now in the form (U) and we get the equation• correepondlng to 9a and 9b: lm D • -N (14a) 1m N • (lm T)D (14b) where the element. involved are now matrlcea. From 14& we get again (14c) L where 8(•'-•t) lea dlagoo.al matrix whose lth element le 8(•'-•t)• the etep function with •i equal to the threahold of the correeponding reaction. Thle give• the actual lower limite of integration; and we obeene that ® doe• not commute with N. We can aleo define the aubetractione at different poln.ta for dWerent channele: (l<kt) (DO eum over t) Wllh the definition of D in 14c with •o ~:: 0 we can eaeUy find a connection with the form factor for the aimpllfled caee we are aeaumlng (see reference 15). F ~===Fig. z. Diagram for form factor• Unttarity applied. to graph. like that of fig\lre Z glvea the follow· tng equation for the column vector F of form factor• for the different channels: lm :F(a) = Tt (s) F(a) (15) We alao know that (16) J'(O) • Q where 0 ta the "charge vector." We will prove that (17) sattafies both equation•. trivially. From 1•c we get D(O) =1, ao that 16 i8 obtained. Abo from 17 hn F • 1m (D T)-lQ • ·(Dt)-1(hn DT)(DT)-10 (ualng 14a) and aince N ie real tn the region under conalderatlon, we get back equation 15. -235- Alao in reference 15 lt ia proved that equation• 14a and b are solved aa integral equation• wtth a •ymmetric Im T at the left-hand cut, then T l8 •ymmetric. unlta:rtty cut. The only cut T-TT could have 18 the But in thi• region If we take the complex conjugate of thb equation, 1m ( T • T T ) = T T T • • T t T = - Im ( T- T T ) and hence T-T T ha• no cut• (nor other •ingularltie•), and alnce it goea to aero at buinity, it le aero everywhere and T l• aymmetric. Unfortunately, lf the determlnantal approximation te uaed, taklDg N equal to the Born approximation for lnetance, T turns out not to be aymmetric and, although equation 12 ia aatt•fted, unltarity a• exprea ••d by equation 10 ie not. Several waya to ln•ure the aymmetry of the T matrix have been tried. The more obvioua poaalbllity of writlna (18) doe a not lead to a tractable expree aton for 1m T. Next tt baa been •uaae•ted to uee T • Z( DN-1+ (NT)-lDT]-1 (19) lrnT• ~T.hn[DN-l+(NT)-lDT) Along the right-hand cut we have 1m N = 0 (ZOa) Im D • -N (ZOb) Hence Im T = • i T-r (hn D)N-l+ (NT)-llm oY] T :: •i T-r ·NN·l ... (NT)-lNT) T and equation 12 is aatialled by thla symmetric T. Nevertheleaa, there ia a aerioua difficulty that ahowa up when a Umt.tina proceaa ia carried out to reduce a two chaanel problem to one chalUlel by lettlns a coupling constant go to zero. Aasume for inatanae that we atart with a aymmetrlc N ol the lorm (21) t:b(a) aa tn a caae Uke that in figure 3. .............. ............. "~~~ t ~-v I I I I / 0 "-. --Fig. 3 . Two channel problem I I -237- We will aimpUfy the notation by writing equation He as D =1 • y ®N (22) Equation 19 can be written T =N{ ~ D + \ N·lo TN) ·l :: N( 1-~j ®N - i N•l(.) N® )N) -1 s 1~ 8 U T ie calculated we get ~ + o2(d 1 T. A where and when ~- 0 we get additional term• ln the denominator for the acattering in the firat channel. A elight variation where T N =N T , = l(ND-1+ o·1N)-l D =1 - i J®N - preeente the same problem. is N® ·Z38· Thf.s problem l• not pre•ent when we write T • N;D..lN! (23) D •1 .. (24) r Nt®Nt •o that if N l• aym.metric, •o are D and T, but there are problem• of a different kind. Ill the lir•t place, along the uaitarlty cut where N i• real, it h not clear lf all lt• el1envaluee, and hence N;,, are real. to Be •Ide• the signa of the element• of the diagonal matrix •lmUar N~ are not determined, and ln general a different choice wlll pro- duce a different T matrix. One way of obtaining a aymmetric T matrix without •olving integral equation• ia the following procedure by iterattc:m. We aet, for lnataace, with NB equal to the Born approximation, 1 ~ (N + N T (25a) ) • NB (25b) (25c) D •1 • .\ ®N U we define X by (Z6a) N • N 2 +X £rom 2Sa we get, rememberlag that NB ia aymmetric XT • .. X We want to lmpo•e the condition that T (26a) is symmetric, t. e., (Z?a) and uaing equation• 26a and b, we get -239- (Z7b) The iteration procedure would be to uae (Z8a) (l8b) Then xi la determined from 27'b and we take (28c) Equation l7b lnaurea that X ia anttaymmetric but in general tt will not be real. U the final X haa an appreciable imaginary part, then T wUl not be symmetric and the procedure la not of much uae. Since the whole method le generally only an approximation, lt might be beat *o glve op the exact fulfillment of time reveraal invariance and unitarity and uae equation 14dl or to aolve equatlon a 14a and 14b with aome appropriate input for lm T at the left-hand cut aa lntearal -240APPENDIX H COMPLEX SINGULARITIES It is well known that certain graphs involving unstable particles or anomalous thresholds have complex aingularitiee, and that these demand great care ln the uae of concepts like unitarity and a straightforward application of the N/D method of appendix G le lmpoeelble. We wUl follow closely the procedure developed by Ball, Frazer and Nauenberg in reference 16, although we will chang e some definitions of branch cuts, thoug h presumably their choice ie arbitrary. approach ie to consider the inelastic reaction 11' Their + N - tr + w + N and approximate the two pion system by a resonance or unstable particle. We will work with the amplitudes ,.. + N - 1r T 11 + ,.. + N and T T 11 for tr + N -n-+N, 22 for w + • + N - w + w + N. b a function of the usual variables a a function of •• t T 21 for and t, T 21 will be and w, the square of the energy of the two pions in their own center of mass system ; the other two variable• necessary to define the relative motion of the two pions are eliminated by the assumption that the piona are in a reaonant state or unstable particle. we will have two additional variable•, invariance demands that T metric i.n "1 and w 2 • 12 '-'! and w • 2 be equal to T 21 In T 22 Time reveraal • and that T 22 be sym- The particles will be considered to be ap\nlesa, and t·he two pion system, in an S •wave resonance. We will alao use the conventions of reference 16, where the T matrix obeys the unitarity relatione ae formulated by Blankenbecler- in refe renee 15. ·Z41- In the a channel, for all variables havin,~ physical values z 8 > (M + ~ ) 2) we have (t < 0, w, w , w > 4~ , 2 1 (la) (lb) where the primee indicate interxnediate vari.ablee, and the aponde to an integ ral over the four momenta ~ L corre- of the intermediate particle a, (2) where P ia the total ·four-momentum. of the eyatem. The inte g ration indicated in equation 2 differ• from that in G -4b by factors of 2q 01 resulting from the 6p(q~- m~ h thl• ia due to a different definition of the T matrix that la Lorent• invariant. Abo •• stand• for a & i~, etc., and the traneition amplitude• •atidy T ij (x, y, ••• ) = T 1Jcx•, y•, ••• ) ( 3) E quations 1, in dispersion g raph lang uage, correspond to the relation• in fi gure 1. disc s disc~ o.,w I / I ~~~ I Fig. 1. Dispersion graph• From equation 3 we derive obviously (4a) (4b) • T zz<• .. •t, ""1-· ""z) (4c) For our calculations we would like to know the diacontlnuitlea in • for constant w, w1, ""z• for inatance , -243- The dhcontinuity in w of T Zl for constant •• t can be derived from the equation corresponding to figure 2 Dhpereion graph• for the w-reactlon Flg. 2. whel'e T ie the acatterlng amplitude for ,..., scattering. Unitarity applied to pion-pion acattering gives for the S-wave acattering amplitude f (w) 1 e i6(w) sin 6 (w) = -:':'T.":'l p,w, p(w) • (7) J J. "'"!P z (8) The derivation of equation 7 goea as followe (9) Uaing equation 2 we can write, for the C. M . ayetem: 1 ) Im T (w, x) :a Z • ~ ct'q~ 2,6 (q Z • ..._ Z)Zw6 (q,z • f.L Z(21r) 4 6 3 (q+ - q -, ) ~ (Z1r) ( 2v) P P Proceeding in a similar form to the derivation of equation G- 5, we get -244T (w, x)= p(w) \d!l • T*{w, x ') T {w,x") 4W J q Then we write the partial wave T(w,x) = Z. expaneion ( 21 +l)f {w)P (x) 1 1 t and Uke equation G - 8 we get {10) and equation 7 follow• immediately, for l a 0. U we take the two plone ln equation 6 to be in an S ·wave etate, we project out, like above, T 21 (8, t, w+) - T 21 (e, t, w_) l6(w+) • Zte etn 6(w+)T 21 (e, t, w) (11) We define T f 2 ae the contribution to T due to the diecon• 22 nected proceas of figure 3, we have then D l 3 .. , T zz • U•) 6 (p • p )l(w) ... -· where p and p are the initial and final nucleon three-momenta. Fig. 3. Dieconnected graph (lZ) -245- E quation 6 can t h!3n abo be written T 21(e,t,w+)- T 21 (a,t,wJ = 21L Trz<•.t',w+,w:>T 21 h.t",w: > (ll) where now the nucleon l• abo included in the intermediate state. Subetltutlng this expreeelon and equation lb in equation 5 we obtain T 21 (•+• t, w) • T Zl(• .• t, w) s ztL {T 21 <•+•t',w)T11(a_,t") :> + T l~ ( e +' t', w, w T Zl (a., t ", t~.t_)} (14) where (14a) A •lmilar reaeontng applied to T 22 gtvee (1Sa) T zz<•· t, ~· wz+> • T zz<•· t, "\• (&)z) i6(w c Zle 2 ) • •ln 6(w 2 _)T zz<•• t, W,• w 2 _) (15b) and the equivalent to equation 14 T~z<• •• t.Wj_.wz)- Tiz<• •• t•'";•wz) c zt L {T 21 (e+,t'.w,_)T 21<• •• t",w2) (16) -246- 1£ we introduce the function• (17a) (17b) (17c) we can prove that M 21 and M 22 have no diacontinultiea in w, ~· w 2 for these variables greater than 0. For lnatanee, and ualng equation U, 10 and 7 MZl(a, t, w+)- M 21 Ca, t, wJ a: Zlf(w+)p(w+)T Zl(a, t, w ) f(w+) - Zlp(w +)l(wJf(w +) - f(w+)f(w_) - Tzl(a,t,w_) =0 In our approximation we will not have to conalde r the branch c:uta in t. It will be shown that, to aatidy the equations (18&) ciiacw M 22 ca , t, ~· w 2 ) 1 =o (18b) Dlagrama like thoee in figure 4 have to be included, giving contributiona -247- where the cr' • --""" --- --- correspond to the !our-momentum squared of the nucleon propagator. " -~ / Flg. 4. -- ---- ------- Additional diagram• U we now make a partial wave expansion (20) From equation• la, 14 and 16 we get, alter integr ating over aome ot the variable• in L (lla) + + L•~ 2<•+.w.w')lf(w')I M'21 <•.•w') 2 • L ~z(•+· "\• 1 w') j.t'(w ) all valid for s i:!:: (M + ..,.) 2 ; where L' 1 2 ~z<• .• Wz· w') now stands for (Zlb) (2lc) (22) (23a) (Zlb) (23c) (23d) p(w) being d•fined in equation 8. We will indicate •ome •tep• in the integration over a three• particle intermediate atate. x Tll(a,t',w')T~(a,t",w') We ct.fine k :: q' + q" IC • q' .. q" -249- k =0 . Then k - K = 0 2 2 2 2 2 impltea K =0 and from k +K -4..,z= 0 we J et 'K = k -4., 0 We will take momentarily a ayetem where x 6 (2.k K )8(P •k )8(k +K )6(k •K ) T 00 0 0 0 0 0 0 We notice that t 1 ta a function of • a and x" 21 (a, t', w')T and x 1 •21(e, t", w') =qi · q, t" of =<it q, q.. =P... - ..k , and w' =k z, so T 21 does not depend -250- on K. We choose now the C. M . I •= T 21 TZ1 r - system , where. P = 0 - dO ... 6[ (v•-k r -2 - M 2 ] 6(P ·k ) dk Kr-oZ d jk ) 2 -k 1 i 8( 2•) I 0 0 .K 411' 0 L (21 +l)P1 0 (x)M~(e, w') 1 -z ~ 0, q-•z ~ 0, q-"z i2: 0 we get: Va =q + q' + q" 2:: M + 2tJ, From q 0 - hence the 6[ e-(M +2Jt)] 2 fro m k +K 2 • 4..,. 2 2 factor. Vs > M + Vw' Also We have seen that K we !conclude k 2 ~ 4tJ 2 • lrom or 0 Abo 0 2 s 0, and k2 2: 0 impliee, ·251- ~ T 21 1' ;1 c: ). (21 +l)Pl (x) ~ 1 f....J (.fa · M ) 2 . . fJ- 2 d w' 1 j 16(2'1') - ~ 1;.1 \]~- jf(w') I2 • M'21(•.w')~l ( a. w') and equati on 23 b i e obta ined. Since j£(w) I ie large only in the n e i ghbor- hood of the re•onance at w M 22 =m 2p' we can uae t he val ues of M 21 and at thi• point in the integrals. We get then (24c) where (..fe -M) 2 Pz(•) • li[ • • ( M +2p.) ) r z J 4Jl 2 2 dw' pz(• , w') jf(~') 1 (25) E quations 25 have t he •ame form a• the partial wave unitarlty relatione fo~t channeb with two •table particlee; the cUfference ie all contained ln t:be generalized denat.ty of etatee of equation 26. W e follow reference 16 ln the analyeie of the behavior of th e ·252- alngularltles when w l8 varied. The one pion exchange of figure 5 wi.ll glve a pole in the t variable, with realdue gf(w) where g i.e the pion nucleon coupling conatant and f(w) the ,....... acatterlng ampll• tude. Fig. 5. One pion exchange graph We will atudy the s-wave projection of thla pole: T 0 21 (Z6) (a, w) a f(w)B(a, w) B (a, w) = o.(a, w) lo ~~·· w) + o.~a, w~ w g - ( •, w) + a, w CL (27) (Z8a) CL(a, w) c ~(a, w) = 2 2 ••~ Z a • a(ZM +w.. . ,_ ) + ( M .. ..,_ 1x M Z•w) (28b) From equation 27 we aee that B (a, w) haa aingularlttea -.here the logarithm ia either zero or infinite, that la, where (I!S(x,w)] The eolutiona are 2 = [o.(a,w)] z (29) ·Z53- (lOa) (lOb) (31) r 2 and (M • vw) r 2• The branch points in e are (M+p) 2 , (M•tJ.) 2 , (M + vw) The branch cute will be choeen a• in llgul'e 6: thl• impliee • • a. (• , w) • • a.(e, w) (32) We abo chooee a. > 0 above the right-hand cut. We next give w a emall poeitlve imaginary part, and take the branch points • • 0 and • • oo below and above the left•hand cut. Hence for • • 0 the numel'• ator- of the logarithm ie zero, and for s =co, the denominator, and we can join both branch poinu by a branch cut. It ehould be remembered that the Riemann surface for the logarithmic function bae an infinite number of eheete, and not only two as the aquare I'OOt, aDd care bae to be exercleed to compute the die• continultlee aero•• two branch eute running together. For emall 6 > 0, and s:t: (w) real ds:t(w) 1m s:t: (w + 16) • 6 1 c a;,;; ~ 4M 2 -p 2 r1' * 2J! i z11 2 -<A1 ~ 2 \Jw(4~ -w) (ll) -254- r , (M+r-)z (nr-fW)l 0 t ;> s_ r~ 1) (M-.JW)"2- s > r+ )( )(+ .---------- ( (M+y)'2 I (M -p.)t I (M+JWY lb) 0 i .,.. I+ lj lM-~)~ ) I (M-r)'t '- - - - - - . - " " I (M+.Jwt I+ r - - _ _, ) (M~i \ _/ Fig. 6. } / s (M y--)L ( J) .,-----I ( M ·~,44f s-+ - + (M-rwY - -- ~j ()( (c) 0 , .... S)(. s+ Singularities of B(e, w) / (t--.rft:J)l -255We notice that 1m a(w) changes algn at the same time ddlw) doe a. It l8 easy to check that for lncreaalng w wo flnd the altuation ln ftgu.rea 6a, b, c. The turning pointe are reached 1m. 8 - c 0 (34a) ... a (M•jj) a+ 2 (34b) =(M+JL) ... tt 2 3 (l4c) Im .+ =0 (a .Sa) 2 a+ = (M + ~) 2 3 a .. (M·~.) + 1;r = It can be seen that both a. and ~ (35b) (35c) change algn when Im •:t change• algn, ao that we alwaya have o(a+' w) (36) l'(a+' w) < 0 and theae lnaqualltlea ehow that this branch cut ln B(a , w) can be llmlted to the aegment between "+ and •.· In the phyalcal region, z w > 4tJ , •:t: become• complex. U the branch cute of ~w(41J. Z•w) are chosen to run from 4!-1 Z to ao and from 0 to • m, and its value ln the region from 0 to 4~ Z la choaen posttlve, then (37) and the branch cute of B(a , ~) appear aa shown ln flgure 6d. Abo from equation 27 we eee that the discontinuity of B(a, w) acrose the cute of a.(a, w) la sero. -256A convenient repreaentatlon lor B(a, w) la B(a. w+l6) • !. r , Jr + a(a ', w+i. 6 ) da I (38) •'-• alnce (38a) diac B(a, w) • Zlo(a, w) acroaa the branch c:uta of the loaaritbm that form the path r + and are lAdicated lD flaure 6. :For real w we have a•c. w).!. r . .. lr+ • _l r c•(a',w+l6) .. J "+ ..... • ·• - • • c:la'• o(a', w+t6) da' ' • .... 6-0 (equation 32 t. uaed) • !. r C1(1',w+i6) da' .. lr+ • ' - s • • • B (a , w) 2 aDd lor real a, B(a, w) la real even for w > 4v • U the oDly lntenc:tlon we conalder ta the one pion exchange (eee figure 5) and U there are no anomaloua threaholde, or complex alngularitlea, equation• 24 and 26 give for the S•wave amplitude• M~ (written without the auperacrlpt ln what follows): Mu.C•) • 1 f Ql) MuC•~)pl(a~)Mu(•~) ;; J(M+p)Z da' •' .. 8 f e» c!s' M21(a~, m!)Pz(•~)Mll(e~, m!) 2 • J(M+2~) a' • • + !_ (39a) Mzz<•· w.' wz) 1 ~~~ -1 'If S Mzt(•~· c,;>Pt<•~>Mzlh~. wz) z da' _....._ __ .._....__ _....__ _ 00 •'· a (M .... ) da' M22(a~, "\'m!)pz(a~)Mzz(a~, w2: + !. f CD , J(M+4J.) 2 Sillce for 'm!) a'• • ~ ~· Ca)z > 2~ 2 (1 + m) (39c) anomaloua threaholde appear, we should write clown equation• 39 for w, ~· w2 leea than thb valae aDd then continue the oquationa analytically to the value of intereet, m!. But, aa l8 pointed out in reference 16, thla procedure leada to the wrong cUac:ontinulty equation• in ~ and the contribution• of the gra.pM in ftaure 4 Indicated ln equation• 19 have to be included. have to project out the contribution to s-wave acattering. According to R16·5. Z this pvea M..l(•) -w~ ~wftw)p(w) r•_(w) p (a')o.(e',w) .+ w - 8 d•'--~--------( ) •' -Fig. 7. -- ql -- ------- / ~ ;;:Tq" / Additional contribution to T 21 We -258- From equation 19 sTu<•· t) T' =--..;;.;.-...r---r 21 Cp+p'·q')z-M2 g Tll We c:hooee the plon C. M. eyetem, and we deltne k • q 1 + q". then .. k•O 1 ~ lq-· I = lq- · I = 'I\JI.I)•4v- Po e-M +w Po= r ZVIAl .. -· Wo c:hooee p + p ' z + , T Zl (p+p') z a (k+q) 2 ae the polar axle, thea gTu = e·Zq' (p +p' )+~ 2·M z+2 ,q.. ,liP- +p... , lx 0 0 0 = Taldng the S -wave projection of the two pion eyetem, =e We have (259) We next notice that the branch pointe from the logarltm are preciaely ••(w) aa given in equation lOb, and hence and next we can take the S·wave projection in the C. M. ayatem and uae 17b to arrive at the deaired result. between (M·tt) 2 The branch cut in p1(a) waa choaen 2 and (M+tt) • Since the phyelcal cuta are already included in the integral• in equation 39b, we want only the contrib~.&tion of the left•hand aingularitiea tn e, which la clone by f.ncluding Mu ander the integral. Similar terma are added to equation l9c. Equation• 39 become (40a) (40b) -260- \ •_(~) pl(e')Cl(e', c.;)Mll(a', wz) J.+("l) 1 + s•Jwz)d 'pl(a')Cl(l',wz)MZl(a',c.;) •f(wzP(wz) 1 + - • + l l • -· e+(wz) • CIO (M+tJ.)z da' a -• Mzl(e~, ~)pl(a~)MZl(a:, wz) • -• de' Mzz<•~·~·m!)Pz(•~)Mzz<•:,m;,wz) \go • J (M+Zl.\) 2 (40c) •'·• EquatiODa 40 have to be modified when the alnaulartty • +(w) of B (a, w) in M 21 reacbea the point • begiDDiAg at • :a =(M~) 2 and turns arowu:l. The integral• (M+Z~) 2 are ~.m.affected by thia aingularlty, and hence equation 40a l1 ~.m.cbanged, and there are no anomaloue thre a holds or complex eingularitlee in the • + N - • + N reaction, at least ln thb approximation. We wtll continue equation 40b both for w with a emall poaltlve ancl negative imaginary part, and we wlll ebow that M 21 ~· no dia• continuity in w. Firat for w - w + 16, we define the continuation of M 21 to the raOD•phyalcal aheet reached golna downwards through the W\ltarlty cut for (M-+,..) 2 ~ • < (M+z.-) 2 , by €- 0 Unharity gives for Mjl' ( 41) j • 1, Z (-i2a) -2.61- M (a·i•) 1 Mjl(a+l•) • l·Zlp ia-l«)Mu(•-l•} 1 and hence we en write 2 We continue next to Re w > 4t-' , noting that we get into an unphyateal n. Mu· .-· [ clhc M~(a', w+)) p (a')M (a') 1 11 (45) whore dlac Mg (•', w,.> le the diacontinutty of thta fWlctlon aero• • the cut of .B(a, w+) with the direction ahown in figure 6. we get, remembering that dlac ~l • 0, From equation 45 -262- and from equation 43 11 dt.c M21(•. w+) • diac M 21 (a, w+) l . 2tpl(•)MU(•) (46b) (46c) and aubetltutlng thle into equation 45, and almpltfytng with the ald of equation 43, Unttarity applied to W•'l' ac:atterlng gtvee f(w+i€)-f(w-i€) • Ztp(w+if:)f(w•lt')f(w+if:) (48b) 1 1 ~) = g + 2lp(w) (48c:) -263- and we get Equtlon 7 can be written l(w)p(w) • e t&(w) •in 6(w) For w • m! there b a l'eaoD&nce, t. e. &(m!) • J , anc! hence and from equation 48c, (50b) &D.d tatrocluclD.g equatlOD• 50 ta "''· ·Z64· 1• 2 2 i ( r(M+J.t)Z _(~+)., cla'\ o.(a '' m z)pl(e')Mll(a ') M., (a, m ) rz B(a, m ) + de'_ ___.P;.-.1 .::....--.:.:~8 ~1 p P ,.. a+(w+) (M+Jt)~ I ·• 1 + -W + !. w s 01) (M+v) ..a ' Pt(a~)M21(•~· m!)Mu(• _) 2Q8 ~----Tf~----L------- a •8 Sco de' Mzz<•~· m;, m!>Pzh~)Mzt<• :. m!) (M+ZJ.l)Z (51) •'·• We wtU repeat thh continuation, but now with w - ~ • 10. From equation 33 we aee that (52) which la true abo lor Re w > 4.,. Z according to the choice of branch cute mentioned above eq\l&tlon 37. Tbia lmpllea that the contour la deformed now into the upper half-plane, and we have to define the analytic continuation• of Mjl going upward through the cut III Mjl (a) • M~l(e) I+Zip a)M)j(a) (51) 1 By taking the complex conjugate of equation 48c and replacing w by w •, we get, uaing eqWltlon 48b 1 ~ f~(w) 1 .,~tp • ~.1\WI •cw., (S4a) and hence (54b) eo that the cute for p(w) baYe to be taken from •01) to 0 and from _.., co, which agreea with the choice of cute for •::tr(w). 2 to -265- Contlnulng equation 4-0b to (55) Taking the dtacontinultlea of equation 55 acroaa the path from ••(w_) to (M+.,.) 2 : + 21[ diac M 21 (a, w_)] p1 (a)M~(•) (56a) p 1 (a)~1(a) anct hence 1 + ltw )p(w ) diac: M 21 (e, w.) • - Zio.(a, w_) Tor Re w > 4f.l 2 , - rn- 1-~(s) Mu (56 b) (a) p(w) ta real and equation• 48b aDd 54b give (57 a) -266- (57 b) We abo have (57 c) r _(w+) da 8 J(M+J1)2 I Cl(8 It W+)pl (a I)~~ (a I) a -a (58) Thia equation •how• that the diacontlnuity in w of M 21 i• aero, aa requlr•d by equation 18a. The equation for M 22 ie handled in much the aame way. We wUl continue first in "\• giving it a emall po•ltive imaginary part, and 2 keeping w fixed at a value lea a than 21-l (1 + 2 -iiA> + 1 wltwz)p(w2) 1 f(M+p.) r•Jwz)cl 'pl(s')o.(s',wz)M21(•'•Co.)+) J a+(wz) • e ·• 2 [cUec M~(e'•"1+)]p~e 1 )M 21 (•'•wz) de ------------~~~------~~ 11' . • +("1 ·' • -· 1 + - ) 1 r- m Mzt<e+• "1+)pl(e~M21(•:. c.>z) +- \ 4•'------_.~----------~ • Jc~~>z • -• an4 ueing oquatione 43. "6c and 48c, we get 1 Mzz(t '"\+• w2) c: trf4("l+)p(~+) + + s(M+p.)z I Cl(e'. ~+)pt(•')M21(e'. wz) e+("i.+)ct. • ·• 1 r•Jwl+)d, o.(e'•"l+)pl(e')M21(e',wz) wl("l+)p("l+) .)(M+v)2 e e -e 1 se_(w2) Pt(•')Cl(s',wz)M21(•'•"1+) de' -----....,.;;-..=..-_.;;;.... ri(wz)p(wz) •+<wz> • ·• 1 + -, 1 + • s M2l(e~, '"l+)p1(e~)M21(e~, wz) de' ~~--~------~~---(M+,.t)2 e •e 00 I 00 (M+Zv)z de' Mzz<·~· "1+· m!>PzCe~MzzCe~. 1 e -• m!· wz) - ( 60) -268- We continue now in w2 , keeping w2 :S ~· The third integral in equation 60 h affected, and the path from a+(w ) to (M....,.) 2 ln the region II defined by the cut ln 2 Ilea ~~~ and another deut la made lu the third iutegral ln equation <&Oc (aee figure 8). Equation 60 then (61) Uatng 46b and e we can flnd diae M 21' and theu 48c allow• aome almpllftcatlona; we get and ualng 43 aud 62 we c:an write 61 in the form (6.3) ( M-JA-)z. 0 Fig. 8. Continuation of M 22 U we want to expreee all lntegrale in terme of M2, (e+' ~+). we have to find the dlec: o! thla ftmc:tlon both aero•• figure 8. c1 1 and c 2 ln The firet can be found from 46b and c:, and the eec:ond directly from equation 49. we aet (64a) -270- (64b) and equation 63 becomea M l 1 (Mi)a)Z a(a', ..,.+)pl(•')Mzt(•~· wl+) (a, w. , w ) • da' ------..;;._-...:;;.::-....;.......;.._ 22 1+ Z+ trf*("l+)p(~+) a+(..,.+) a ·• + + l 'd(~+)p(~+) r·J~+) o.(a',wl+lpl(a')MZl(a~,wz+) J(M+p)l da' • ·• (W+~)z 1 (' da' A trli(w2+)p(wz+) J•+(wz+) •'·• where pl(a ')Mu(• ') AaC1(a',w2-+)p 1(a') [Mu(•~~+)· ll4(a',~) ( 1 ·M~+)p("i+) ) and A' ie obtained from A by replacing f* by f. U we take now '-1 • w 2 • mp2 + l6 and uae equation• 50, J (65a) ·211· 2 2 M2Z(a, mp' m.p) II l!.. r (.M+.,.) : . 'o.,.t• ', zn!>Pt<• ')[ Mzt<• ~ m~)-lo(a', m!Hl-tpl(a ')Mu<• ')}] • J • +(mp2) • ·• 2 21 ·- la.Jm.) r~t 2 o(e', m,)pt(a')[ M 21 _ 2 z <•+· m,)·lo.(a', mp){l+tp 1(a')M11(a'H) w (M+)1)2 I 9 _ _ •a +.!.roo cia' Mu(•~· m;)pl(a~U·'zt(a~, m!) • Jcw+.,_)z •'-• CD cia' M.z2(•+• m;, m!)~z<•+)Mzz(•~• m:, m!) , J(M+lP.)z • -• + !. r (66) In equationa 61 ancl 66 we notice the additional te nna dt&e to the complex alD&\llarltlea, th&t la, the lntearala fro.m •t<m!) to 2 2 (M+v-) to a (m ). (M~) 2 and lrom . , StW loUowtna reference 16, we write (67a) N • (nlj) (67b) D a: (6tj + cllj) (67c) Thea, from 67a, we caa Wl"lte (674) N • MD z and U we .-eatrict ouraelvea to w • "\ • w2 c: mp and we dl'Op them aa variablea, tbla ta written O\lt (68a) ·272· ';.z<•) .. ~~(•)c;.z<•) + Mlz[ 1 + dzz<•>l (68b) nzt<•) a Mzt<•)[ 1 + c!u<•)) + MzzC•)dzt<•) (68c) nzz<•> .. Mnh>c;_zC•) + Mzz<s)[ 1 + dzz<•)) (68d) In reference 16 they next ellmbtate the Mlj and lind the relatione between the n 1j and cStj by uetng equation• 68 and the integral equatlone 40&, 51 and 66 for the Mlf ( "'--( •) • .. -1 -u s s•. 1t (M+p.) • 2 (m2) + p --sao 1 V' We will merely quote the reeulte: • z <mp) (M~)2 1 p1(•')o(e )dn(•') ) de' ...__,..__..-....._ Pth~)Au(•') (M+f,t.)2 4•·--~-----1'•1 • ·• (69a) (69b) (69c) (69cS) 1 r a(•')du(•') nut•> • i Jr d•' •'·• (70a) (70b) -2731 r o(•')du(•') n 21(a) • e(a) + ;Jr da' •'-• l t ( +.. (M+)t) a+(m!) 2 - l• 2 _(m ) p2 (M+)t) pl(s')o.(•')n..l(s') )d•' ----~~ .'. . . . • ·• (70c) (70d) '\t and '\z have cut• along (C1) + (C 2) and frozn We ob•erve that (M+,I.) 2 to 001 ct 21 and d 22 only from (.M+Z1-1) r. have only cut• along 2 to ao; and the nlj It b then •tratghtforward to prove that the right equation• for the dhcontinultlea are obtained. For instance, from 68c: and d n21(8)( 1 + dzz(e)) • nzz(8)dZ1(8) Mu(•) • l 1 + dlih)) l I + dza(•)] • dlz(•)dzl(a) we wtU take the dhconttnulty aero•• dl•c M 21 • (71a) c2 diec [ n 21 c1 + d 22 > - n 22 d 21 ] :&(.+) [ n21(l + dzz) • n 22d 21 - dlac {n 21 (1 +dzz)• n 22 d 22 }) di8C 6 A{.+)a(a_) where A • (1 + c!uHl + d 22 > • ~ 2 4 21 • From equation• 69 and 70 we get diec {n 21 (1 + d 22 ) - n d } zz (2lo + Zlodu- 2p1o.nu)(l + c~ 22 > 22 21 - (Zlo~z • Zpt=tz>dzl -274- aDCI hence dlec M 21 a zto[ (l+cSuHl+dzz)·c;zdzll - Zplo~l(l+dzz)+Zpl~zdu ----=-----------...-;;.-,....--.....---.__.;;.;;.._...;;....;;.;;;-..;;;;.. Thb aareee with what b obtalnecS from equation 51. We obeerve that the elniularltlee of the n 1J and ct1J are no longer eeparate, that la, there no lonaer l• a wdtal'lty right-hand cut, and left•hand cute. There are now two poealbUltlea: to aolve the integral equatiODe, ae lt ia clone in reference 16, or to make an approximation ln the per• turbatlon theory aplrit. B(a) la of firat order tn g, and ao ta a(a), hence, 11 we keep only g to the fbat order, we are left wtth (7Za) (72b) (7Zc) (724) nu<•) =o 2 (73&) ~ (e) • B(a) (73b) n = B(•) (73c) 21 (•) (73d) n22(•) • 0 The only cllfference with the equations for the caee without anomaloue thre•hold• i• the additional term in equation 72b. The analy•la of the problem of complex eingularitie• ha• been re•tricted •o far to a •imple example, although it probably ha• all, or at lea.•t moet of the feature•• of the real problem. Of couree we could try to repeat the analy•i• in the general ca•e of particle• with epin, but the amount of algebra involved l• rather forbidding. We will take then the alternative of a••wning that no fundamental change• are introduced by the different compllcatione. One generalisation wU1 have to be the inclueion of other than the S•wave component• of the Mlj" The analytic propertie• of the inter• action term can be deduced from the propertle• of the Legendre function• of the •econd kind, since R8·6. 3. 44 give• a•x and the •arne reference. Oil p. 1328 give• •o that the branch pointe coming from the log are the •arne lor all partial wave•. (74) -276- On the other hand, (76) and the dtacontinuitlea of the B ta not se ro for odd I. l (a.~) acroaa the branch cute of n(a, ~) Tbh can be taken care of by defining (77) for odd l (aee equation 57c). The incluaion of particle• with epin hae the effect of increaaing the number of partial wave tranaltlon amplitude• for a given proceaa, but all projection operator• have the form of equation 74 aa can be eeen in part U and appendix D. Moreover. for the amplitude• that correapond to atatee of definite parity only every other P 1 appear•. and equation 77 can etlU be uaed where neceaeary. Thua. the analytic propel'tiee of the partial wave amplitude• for the eame reaction aJ'e the aame fol' each aubmatrix characterised by lta total J and parity, ancl pl'eaumably obey the aame equationa. We alao obae rve that equation• 69 and 70 al'e linear ln the ~J and nlj' eo that contribution to N coming from other graph• can be included without dlaturbing the equatlona. It ebould be l'emembered that n ta the dlacontlnuity of B aero•• ita branch cute. eo. Wlleee the bl'anch cuta colnclde, the new gl'aph will not contribute to the integral involving Cl(a, w). Other extraneoua cute coming from the kinematic• can be ell.Jninated by changing the deflnltlona of the Mlj, ae ta pointed out in part IV. A mon clelleate matter le the lnclu•lon of eeveral UD.8table parttc:lee. For lnetance, the N • l8 a p•wave reeonance ln the •N eyetem, eo that the • + • + N etate etuclied eo far can aleo be approxi• • matecl by w + N , ae te clone in reference 18. In thle article aho a dU'fereftt treatment of unetable external pal'ttclee ie used, wblch might be \lMiw. • +N ODe reaeoD&ble aeeumptloft ie that the phaae epacee for the • and the p + N do not ove •lap to &Dy elgnlflcant extent, eo that we are not counttna etatee twice when we conelder both. . The analyete made ln thle appendix can t hen be repeated by COD• eiderillg the 1t' N ' channel llletead of or in aclctltton to the pN channel. Thle le however DOt exempt of new probleme. If YJ le the energy of the trN eyatem tn lte c:eDter of ma•• eqU&I'ed, the diecontlnulty equatioae for 11 are aeeoclate4 with diagram• like that in fla~re 9, wbic:h havolvee the wN ecattering amplttude, iteeU a part of the pi'Oblem, ud the Juetlfication for introcluctng experimental data or reeulte of an independent calculation. •• could be doae with the ww ecattering ampli• tucle f(w), h clu'blou. One poeatbUlty te maktas a calc:ulatton to deter• mine the wN ecatterlng tn aome approximation, and. feeding thia back into the appropriate llltegrale. I I I I Ftg. 9. Dlagram for the N* -218- The graph correeponding to the interaction at.ngularitiea ie ahown ln figure 10, it wUl give a pole in the t variable with reetdue gT(rr) where T(rr) ie tn the wN ecattering amplitude. - - - - - -....------------,. Flg. 10. Nucleon exchange g raph Taking the S·wave projection of thta pole, we have (7 8) B(e, YJ) • .. 4 (e, n) log j(s, n) + o.(s, n) - (•• ')) +a(e. '1) ,.. (79) (SO a) .A(. •·ll) • s z-•<'1-M z+z"'••s 2 z 1 2 ) - (M ·~-t Hrr-v ) (SOb) and we can dete rmlne z z 2 •.<n> =T(l + M~) + (1 - 2M ~ )YJ • Jt Y•,_u-:---.. -2 J£ n-CM-p.)z] [ (M~)z·'71 2M (81) -1.79- A• expected, these branch points of B(s, n) become complex • 2 when the N becomes unstable, i.e., when l'l > (M + tJ) • If both unatable particles are conaldered aimultaneo\.lsly, one baa to worry abo\.lt two variable ma••e•, and diagrams with two un•table vertices OCC\.lr, like that in Fig. 11. rr --- p I I ITT N* Fig. 11. N Diagram with two \.lnatable ve rticea The other \.lnatable meaon conaldered, the w, baa only one mode of decay, into three plona. B\lt theae diagram• have been neglected, and the wp-.r vertex la certainly not \.lnatable alnce the maaaes of the two vector meaona are almoat eq\.lal. Moreover, the width corre- aponding to the w meaon ia •mall, and hence it can be treated •• a atable particle without making any obvlo\1• big errors. -280· APPENDIX I NOTATIONS AND CONVENTIONS a) Summation convention When an index l• repeated in a monomial expre•aion, a matton over the full range of the index le implied. •wn· For instance l = 1, 2, ••• , n (la) me ana (lb) This te not so when the lDdex l• a number, and abo when parenthesea are put around one of the indices. That 18, we would write . and no summation over l la implied. There la also an important variation of the conventionJ when we uae a greek lnux whoae range h 0,1, 2, l, indicating the component. of a vector or tenaor in Lorentz space, we write -- = p• k c p"k"'• Poko .. p • k P z Poko -p~~-Pzkz•P3k3 z -z • ptlptJ. • Po - P Correeponc:Ungly, 6v• (la) (3b) la not the ordinary Kronecker delta, but the metric tensor ta a apace with a Mlnkoweki metric , tbat la (4) 6 tJ." • 0 It ta atralghtlorward to check that the ordinary rulea for auma carry through wttho\lt change. For lnatance p 6 k ::& p k • p• k P P" " ~ .,a Thla ia the acalar product of four -vectors aa defined ln equation 3a, anclln other reference• uae le made of the metric ten1or gt-1" that la the •arne aa our 6p.v' and upper and lower inclic~• are u.aecl aa in teaaor calculua. The antiaymmetrlc or Levl•Civtta tenaor •...,_,, la defined ln equation ElO, and tbb moclUled aummation convention appllea to itt~. A vector pl'ocluct la indicated by the aymbol "· b) y matricea In our convention, the y matricea obey the antlcommutatton relation• (Sa) Notice that 2 Y0 • • 2 2 2 'V1 • • Yz • • y 3 =1 - (5b) They are related to the matricea G, 15 by (6a) - l • 1, 2, l The matrlcea fS, G aJ'e Hermitean, and hence y 0 (6b) - la Hermltean and y anttbe rmltean. Thia can be expreaaed by (7) . . zaz .. U A le a matrix, A• la lte complex conjugate, AT ite tran•poee and At lte Hermltean conjugate, that ie (8) When an expllclt repreaentation la needed for the y matrtcea, we uee y who ra the tr 1 0 - • (9a) are the Pauli aptn matrices -;) (9b) We &leo deflao (lOa) &Ad ln the rep re • entation of 9a Ys • (: : ! (lOb) Abo (lOc) eo we can give to tJ the value 5 ln 4J(luation 1, since y 5 anti commute• with "o• In fact, it te eaay to eee that "' • 0,1. 2, l (lOd) l 0 1. z. l (lOe) The unlt matrix b more often than not omitted, but lta preaence and rankle eaaily deduced from the context. For lnatance, when we ·283write p y tJ. IJ + M, the M ts mwtiplyt.na a unit matrix ol fourth rank (1_..). .. and tn the expreast.on for y 0 t.n equation 9a the 1 etanda for a eecond rank unit matrix (1 ). 2 A alaeh will be \Ued bL connection wt&h a .four vector to mean (U) c) General conventions ltquatt.ona and fiaurea are numbered conaecutively ln each part or appencltx. When a reference t.a made, the part or appendix t.a 1Acllcated unleaa t.t b an eq~o~&tton tn the aame part. When an equation ll"Om an outalde reference l• uaod, lta number ie t.ncllcated &lao. So R9•22. U me ana equation 22.13 ln reference 9. Convention• that have not been mentioned in thie appendix are either the u•ual one• or they are explalneclill the text. -284REFERENCES 1. s. s. Schweber, An Introduction to Relatlviaitc Quantum Field Theory. (Row, Peteraon and Co., 1961). 2. F. MancU, Introduction to Quantum Field Theory. 3. O. F. Chew, M. L. Goldberger, F. E . Low, and Y. Nambu, Phya . 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