Sum rules based on current algebra are obtained for the currents evaluated between states of arbitrary spin and mass. Both the infinite-momentum limit and the dispersion method are shown to yield the same result. These sum rules are given explicitly for the crossed-channel amplitudes. © 1967 The American Physical Society

Bander, M

Physical Review

Sum rules based on current algebra are obtained for the currents evaluated between states of arbitrary spin and mass. Both the infinite-momentum limit and the dispersion method are shown to yield the same result. These sum rules are given explicitly for the crossed-channel amplitudes

Bander, Myron

eScholarship - University of California

UC IrvineUC Irvine Previously Published WorksTitleCurrent-Algebra Sum Rules for States of Arbitrary Mass and SpinPermalinkhttps://escholarship.org/uc/item/6q04f19zJournalPhysical Review, 160(5)ISSN0031-899XAuthorBander, MyronPublication Date1967-08-25DOI10.1103/physrev.160.1416Copyright InformationThis work is made available under the terms of a Creative Commons Attribution License, availalbe at https://creativecommons.org/licenses/by/4.0/ Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaPII VS ICAL REVIEW VOLUME 16 0, NUMBER 5 25 AUGUST 1967Current-Algebra Sum Rules for States of Arbitrary Mass and SpinMYRON BANDERDepartment of Physics, University of California, Iroine, California(Received 24 February 1967)Sum rules based orj current algebra are obtained fear the currents evaluated between states of arbitraryspin and mass. Both the infinite-momentum limit and the dispersion method are shown to yield the sameresult. These sum rules are given explicitly for the crossed-channel amplitudes.I. INTRODUCTIONS UM rules from current algebra have been obtainedprincipally using two methods. One is to evaluatethe commutators in the infinite-total-momentum limit.The other is through the use of dispersion relations. 'Explicit sum rules have been written for the case ofscalar particles' ' and for equal-mass spin-one-halfstates. ' 4 The other sum rules have usually involved theevaluation of the commutators between states withequal momentum.In this article sum rules for the commutator betweenarbitrary spin and mass states are obtained. Bothmethods have been shown to give the same results. Thecommutation relations are'B(»)Pyo' (x) Jy"' ~(0)5=ic»5'(x) Jy" &(0),'&( )LJ ' ( ) J "' '(o)5= "&'(*)J "' '(0), (1 )8(xs)P " (x), J~" ~(0)5=ic ~ 8'(x)Jy"'»(0),which we summarize as8(xs)P' (x) J"' t'(0)5=if »6'(x)J"' &(0). (1b)In Eq. (1a) ct,P,y run through the eight SU(3) indicesand the subscript V,A denotes vector or axial-vectorcurrents. In Eq. (1b) the indices run through theSU(3)XSU(3) group. The c», foe& are structure con-stants of SU(3) and SU(3)OXSU(3), respectively.The above corrunutation relations are presumedcorrect for v=o. For v=1,2,3 there appear additionalgradient Schwinger terms. ' The existence of these termsdoes not come into the discussions below.In Sec. II a general representation for these commu-tators, evaluated between helicity states of arbitrarymass and spin, is developed in terms of helicity ampli-tudes in the crossed channel. Dispersion relations forkinematically free amplitudes are stated.In Sec. III the syrnrnetry properties due to parityconservation for both the amplitudes and vertex func-tions are presented.*Partially supported by the National Science Foundation.'R. Dashen and M. Gell-Mann, in I'roceedings of the ThirdCoral Gables Conference on Symmetry Principles at IIigh Energy(W. H. Freeman and Company, San Francisco, California, 1966).' S. Fubini and G. Furlan, Physics 1, 229 (1965); S. Fubini,Nuovo Cimento 4BA, 475 (1966).'%. Weisberger, Phys. Rev. 143, 1302 (1966); S. Adler, ibid.143, 1144 (1966); I. Muzinich, ibid. 151, 1206 (1966); D. Amati,R. Jengo, and E. Remiddi (unpublished).4 J. Bjorken, Phys. Rev. 148, 1467 (1966).~ M. Gell-Mann, Physics 1, 63 (1962).e J. Schwinger, Phys. Rev. Letters 3, 296 (1959).In Sec. IV sum rules are obtained and summarizedin Sec. V for the cases of spin zero and one-half.The use of the crossed-channel amplitudes directlywill facilitate discussion of convergence based onRegge-pole arguments.II. KINEMATIC ANALYSIS ANDDISPERSION RELATIONSWe are interested in studying the matrix element ofthe commutator of two currents between arbitraryhadron states. We choose to label these states by theirthree-momenta, helicities, and a myriad of otherquantum numbers which do not change under Lorentztransformations. These other quantum numbers willnot be explicitly shown. Sum rules may be obtainedfrom either of the following expressions':» ) ""' (&&Q)= (27r)s(4cotcos)t&s dex eccl *g(xs)orx(u; ) I P": (*),J"' '(o)5l u; ~ ), (2)(& ~Q)(2sr)s(4co &o )'" d'x e'&& *2jx(Pt; ) tIP" (x), Jr"1' '(0)5I ps; )ts), (3)which is related to the absorptive part of S.In the aboveI'=Pt+Ps ~=pt Ps Q=e+—vswith qt+pt —ps+ps and &o;= (p,'+m )"', where m; isthe invariant mass of state i. We likewise introduce theinvariant variablesy=I' Q,Under the Lorentz group, S transforms as follows:S., " -e(~,~,Q)=~",~""Dx, x,*'~"(ll (Pt ~))» ~ ' "(~'(Ps ~))XS„,,„,»'" -&(Xr,ft~, ftQ), (4)with an analogous expression for E.. In the above A isan arbitrary I.orentz transformation and D), ,),(~)(IV(p,A)) is the spin-J representation of the rotationgroup for the Wigner rotation determined by p and A.'S&" is not quite a covariant amplitude. There exists non-covariant parts which are related to the Schwinger terms. For afull discussion see Ref. 4.160 1416160 CURRENT —ALGEBRA SUM RULESSmay be considered a scattering amplitude for a currentof momentum q~ impinging on a state with momentumpi and going to a current of momentum qs and a statewith momentum ps.We shall obtain a representation for S and R in theBreit coordinate system, P=O, 4 along the s direction,and Q in the x-s plane. The form in any other frame ofreference may be obtained by means of Eq. (4).Quanti-ties in the Breit system will be denoted by the sub-script B.P&»= ((2mP+2ms' —t)'i' 0 0; 0),Ai)»= ((m '—ms')/g(P') T(m)/Q(P'); 0; 0),QB = (v/Q(p'); [ql q2 +v(ml m2 )/P $ (5)XV'(p')/T(m) ' {("/P')(q—P qss—+v (m P ms—s)/P')'XP'/T'(m) —Q'}'~' 0),withT(m) = (P 2t(mP+—ms')+ (m '—ms')'j'tIt is straightforward to show that the Lorentz trans-formation needed to take us from the Breit system to anarbitrary one specified by P,A,Q is given byA"0=P"/P)30,A». = (A» —A»0A, 0)/A&, (6)A», = (Q» —A»0Q&' —A. sQ&')/Q~',A;=".s,P-»Q"/V'(e),with P= LQs*T(m) O'. It is convenient to write Q» asQ» A»0Qi)0+. A»—Qsz+.Kwith K P=K tI(=0 and K'= —(Qe*)' A, and A „may then be rewrittenA», =K»/Q&*,(6')A»v = 0».»P.AsK~/Q(p)which shows their independence of v.The convenience of the Breit system is that theamplitude 5& is simply related to the crossed-channelamplitude for the reaction (pi,Xi)+ (ps, )(s) ~ (qi, p, ', n)+(qs, v; P), where the bars denote antiparticles and udenotes the complex-conjugate current. If S denotesthe crossed-channel amplitude, then following the argu-ments of Trueman and Wick, ' we find that Sp equalsS in the system pi= ps with center-of-mass energy gtand momentum transfer —', (v+mp+ms'+qp+qs t)To relate Ss to S in the crossed-channel center-of-masssystem we must perform a Lorentz transformationalong the s direction with velocity T(m)/(mp —mes). Asthis is a Lorentz transformation along the s directionthe helicities can either remain the same or changesigns. It turns out that ) 2 changes signs. ' If I- is theabove-mentioned transformation, thenSs)„y ""' s= I» I"„8), ), (7)Each current may be considered a superposition of avector and a scalar "particle. " (For conserved currentsthe scalar part is missing from the absorptive part. ) Wewish to decompose the amplitude 8 into helicity arnpli-tudes. The reason is that we are easily able to removekinematic v singularities from S helicity amplitudes. ""We define the crossed-channel amplitudes by8. )„,)„»" t'= P P t"(q; li, ~i)t"(qs; ls, ~s)(lI,oI) (L2(r2)X+ (4,»r), ((2, n); ig, x2(v, t) 1 (8)where the summation is over (l,a) = (0,0), (1, +),(1,0), a,nd (1, —), and t»(q; l, 0) may be determined asfollows: Let t», (q) be the Lorentz transformation whichtakes the vector (g(qs); 0; 0; 0) to q». For q spacelike,t»„(q) is complex. The transformation is further specifiedby requiring that it be a boost in the z direction followedby a rotation in the q—s plane. For q in the x-s planev=0C(q')' —q'ji"3.0 I =0q*q'/L(q')' —q'3"'t".(q) =Q(q2) q» q»q0/L(q0)2 q211/20—q'v'(q')/E(q')' —q')"'q'v'(q')/L(q')' —q't'"1. 3Then for Eq. (8)t»(q; 0, 0)=t», (q),t"(q; 1, +)= —Lt"*(q)+ t".(q)j/v2,t"(q; 1, o)= t"./(q),t»(q; 1, —) = Lt». (q) —tt»„(q) j/V2.(10)Combining Eqs. (4), (5), (8), and (10) we find a repre-sentation for S in an arbitrary coordinate system in'T. Trueman and G. Wick, Ann. Phys. (N.Y.) 26, 322 (1966);I. Mnsinich, J. Math. Phys. 5, 1418 (1964).terms of the crossed-channel helicity amplitudes:S..." ' -s(P,A,Q)=D), ), '(~»(W(p A))D), ), (~»(W(ps, A)Z ('qii l4 +&)Z (qsi lsi +0)(~1 &1) (~2 4'2)X+ ' (li, »i)(lm, 2); xr', »xs'(v&t) ~ (11)The easiest way to see this is to take the crossing relations de-rived in Ref. g for direct-channel backward scattering where theBreit and center-of-mass systems coincide."L.C. Wang, Phys. Rev. 142, 1187 (1966)."Such a dispersion relation may be proved for Q2(0 using theJost-Lehman-Dyson representation.M YRON BANDER 160An elementary but tedious calculation yieldsZ"o( )=(~—Q)"/2V'( ')Z"*(qt)= L(~—Q)"(&+qi' —qs') —4qP~"j/2V'(qP) T(q)T(q) 1Z, (q,) = — {(5 Q)"(m '—m s—v)T(m) 4qPQ)s+2Zv, g(qp) (v(&+qp —qs')—(mP mP)—(f 3qi' —qP))—/T(q) —4qP~"},Zv„(qi) =Av„,with Qe* defined in Eq. (5), andT(q) = l:&'—2&(qp+q ')+ (qi' —q ')j"'.(12)Zv„(qs) is obtained from Zv„(qt) by interchanging qPand qP, letting v~ —v, and Q)& ~—Q»*. A similaranalysis may be carried out for E. and we write the resultR)„&,v"' &s(P,a,Q)= D„)„* (»()W(p A&) )Dg, g, (») (W(p, A) )X Q 2 Z (ql ii&1)Z"(q2 i2 o 2)where Zv(q; l, o.) is related to Zv„(q) in the same way ast" (q; i, a) is related to tv„(q) Lcf. Eq. (10)], andz „(q)=lAL,f(q)j, .then for any Lorentz transformationr„,„,v &(I', i4&) =i&&. „.D),;&„'(»)(W(p,h))XD. ..(")(W(p.,~))r, , "' (»,AQ), (16)and it suffices to know I' in the Breit system, where wedenote it by r),i, &„»' 'r(f). From Ref. (12) we know thatdue to rotational invariancer», „""(f)=8», ,-,r&, ,&," '(f),r&, i,gs ' (f) Sing, —,&sag, x 4 (/) irx, ,x,+' ~(f) =8),—&,par&, x,+' ~(f),With r+= —(r *+irv)/V2 and r-= (r*—ir v)/K2.III. SYMMETRY PROPERTIESUnless the states under consideration are identicaland eigenstates of the scattering-matrix, time-reversalinvariance does not lead to any interesting restrictions.The only useful relations among the 5's, E's, and I"'scome from parity invariance.Let qi, q2 be the intrinsic parities of the states; g,qpthe parities of the vector parts of the currents (i.e.,minus for vector current and plus for pseudovectorcurrents). Then(ll &1) ()2 N2)X1 X2 (v)f)rt crP$1/2(—1)»+»+(i+&4 (—1) &+&'+~+~'—(ll, (rl) (l2, &2)X(r ' (ly, ri) (&,rs); Xi', —A4' ( & ) ( )XA (lf, —IT) (&9, —Its); —ij, —Xs(v)f) ' (17)From Ref. 10 we know thatA '~«„,) «, „),&„&„(v,t)/Leos-', 8&~sj'"+ 'Lsin —,'8(]'" 'ihas no kinematic v singularities. In the above X='A~ —) ~and O-=a.i—~~ and 0& is the f-channel scattering angle.The expression for the angle may be obtained fromcos8, (v) = Pvf+ (qP —qP) (mP —mP)g/IX( )T(q)l, (14)sin8, (v) = Qr)'Qt/T(q) . —We assume that the A's satisfy the following dispersionrelation, which we write in unsubtracted form:(ll el), (l2, ~2); Xl, is (v&f)gnPLeos —8 (v)gi" +iLsi 'n8 (v)g'~ 'The above expression takes into account the fact thatif 2 is a Fermi state, then in the crossed channel, thecharge-conjugate state has opposite parity. An analog-ous expression holds for the a' s.For the vertex function, parity conservation implies$1/2r ~ o, *, z; &'(])— ( 1)Ji+»—&i—&sro, *, z; r(f) (1g)with an extra minus sign for I'&.In the subsequent discussion it will be convenient tointroduce "parity eigenstates""lp. ~~)=(1/v2)(lp; l )~lp; —l )) (»)for X=0,1 -.J, and to express the amplitudes andvertex in terms of these states.(4&&1) ~ (i2t&2) l ~lt~24 ) )lcoss8i(v'))i""'iLsms8i(v')]i" —'i(v' —v)A representation for the vertex function will also berequired. This has been given by Durand, DeCelles,and Marr. "Letr.. ." '(&,~)=(2 )'(4 )'"(p; & l~" "(o)lp"L. Durand, D. De Celles, and R. Marr, Phys. Rev. 126, 1882(1962),IV. SUM RULESA. P~~ MethodUsing' translation invariance we rewrite Eq. (3) asP'& Q)(2~)s(4(o o),)'" d4x e-"@ *2ix(p; z ILJ" (-', a), &" '(——;a)lp;l() (2o)"These are really eigenstates of l =Pe'~~&, (See Ref. 12.)160 CURRENT —ALGEBRA SU M RULESand note that using the commutation relations, Eq. (1), B. Dispersion-Relation MethodStarting with Eq. (2) we evaluate, using integrationby parts,(22r) 4= (4(ol(o2)'" d4x e'o '"b(xo)(q,)„S„,„"'- (P,~,Q)2(22r)3(4(r)1(d2)1/2 d4e42; re(2; )Q2~ Q2"- P ~'&'/v'(2) p "v'( t)/T( ).—(22)The crucial assumption is that for the leading terms inPo, Pv the limit is uniform. (This is equivalent to theassumption that certain terms in the expansion of St""obey unsubtracted dispersion relations). CombiningEqs. (11), (12), (16), and (22) we obtain two sets ofsum rules:T'(m){+ (1, y), (1, —); lr, —13(vr tr (tl rIt2 )ap f . 2 2hsin'8, (v)++ (1, —) (1, +); Xr, —12(Vr tj(tlr(t2))r„,„o' (t)(2ml'+ 2m2' —t)'"andPl y —m2I'1, , )„*' '(t)( ){A (1, —), (1, +); Xr, —12(v t (ll g2)sin83 (v)(1, —1), (1, +1); lr, —12 (Vr t j (tl r Ir(2 ))=v2~f "Ll'l, , l,+ "(t)+1'l, ,l, ' '(t)]. (24)If these sum rules are taken between parity statesLcf. Eq. (19)], then for each choice of l(1,X2 either (23)or (24) is trivially satisfied by having both sides vanish.X (pl, l(1 I LJ' (*/2), J" ~(—~/2)] I p; &2&=2~f-»ro' ~(P,d). (21)Since Q is fixed in the above, both F12 and (t22 vary as weintegrate along Qp. To get sum rules for fixed F12 and q2'we choose a particular frame. First perform an accelera-tion along the z direction to eliminate the time com-ponent of 6& and then accelerate to +~ along the ydirection. Note that any Wigner rotations associatedwith the foregoing transformations are the same onboth sides of Eq. (21) and hence may be ignored.Changing variables from Qp to v=QpPp —Q P we findthat in the infinite-momentum limit (keeping v fixed)"x(p; l, ILD-(*),~: (o)]lp, ; l,)—f »Tl, , l, "' &(t)+Schwinger terms (25)(see Ref. 6), where D~(x) =B„t'&' ~(x). The first term onthe right in Eq. (25) could likewise be analyzed by themethods discussed previously; however, the only as-sumption we need make are that terms proportional toP" and, e" t)„P B,~Q'r/g(p) vanish as v ~+~. Keepingterms proportional to the above we And two limitingrelations:T(V)lim (Q((tl )A ~(o,o), ; 1, —1 (v, t' gi, I((2 )T(m)Q))~Pv1'1,),." "(t)—(2m12+ 2m 22—t)'"—(m12—m22)T(m)andXl'1, ,)„' '(t)+ 1')., 1,*' '(t), (26)T(m)lim (gl')'"A'&(p, p) V 1, , )„(Vrtj Vl r 92 )tr ~goo=f.»r„&,v' ~(t), (27)where the indices x,y denote the combinationsL(—)—(+)]/K2 and L(+)+ (—)]/i', respectively.To obtain sum rules we must require that certainterms in the expansion of S satisfy unsubstracted dis-persion relation, i.e., vanish as v —+&~. This require-ment, applied to the coefficients of (6 Q) "e" t),P tr. ~Q r/—g(t2) (where we treat Z&p and Z", as independentvectors), yieldsT(q) 1lim A (p, p), xaP"-+"T(m) Qs'T'(q) 1~PT'(m) (Qi)*)' 2q12X{A z, r, ; )ql2+A v, v; lr12) rlim A &(p, p), y, )„, )„v ~goovT(q) 1 A"'. ,; 1,&„. (2g)T(m) Qi)*'4 pre must remember that in this method () is 6xeg and un Similar relations may be obtained by consid i g therelated to v. quantity S""(q2)„and combining the two sets of limiting1420 M YRON BANDER 160relations we find Spin zero:lim"~"T'(m) sin'81(v)X '(+ (1, +), (1—); 4i —4(v& ti ql 1 qs )+~ (1—) (1 +)' 4 —4(v& ti ql 1 q2 )}r4, ," (t)(2mls+2m 22 t)"—(ml' —ms')F4 )„' (t), (29)T(m)T'(m)(2 (1, +), (1, ) (V, t; ql, q2 )Sin281(v)(2m12+ 2m22 —t) ' 'ml' —m 'X rs' (t)— r*. (t)T(m)=rrf »F,&(t), (31)lim X""+"T(m) sin8, (v)X(&'(1,—), (1, +); 4, —4(v, t; ql', q2)(1, +)(1,—); 4, —4(v) ti ql p q2)}= —v2f. )2 I r„4+' (t)+r„4— (t)7. (30)where Fl'(t) is the form factor appearing in the tensordecomposition of the current,(p I J"' 'I p )= (2 )'(~ .)'"I:F '(t)F"+F"(t)&"7.Spin one-half:Any relations other than the above which might havebeen obtained before combining expressions from(ql)„5&" and 5&"(q2) „can be seen to be trivially satisfiedbecause of parity invariance.Equations (29) and (30) are not yet singularity free;however, noting that if the limit of the following ex-pression exists,v+F(v) (1 v'.() dv'= const. ,( '—)LF ( ')7'"llmp ~+00(where F(v) is a polynonual in v), then1n(v')dv'= const.Using the dispersion relations I Eq. (15)7 and the abovelemma we recover the sum rules of Eqs. (23) and (24).Besides the sum rules there also occur many relationswith zero on the right-hand side. These are of the super-convergence type" and will not be further discussed.V. SUMMARYWe restate the sum rules derived for the two practicalcases of spin-zero and spin —one-half states:"V. de Alfaro, S. Fubini, G. Furland, and G, Rosetti, Phys.Letters 21, 576 (1966).{(2 (1, +), (1, —); 1/2, 1/2aPT'(m) sin'8, (v)X (v t ql q2 )+(2 (1, —), (1, +); 1/2, 1/2(v t ql, q2 )}andm$ mQ1' ~(t)— f'* '(t),(2m12+2m 22 —t) '" T(m)(32)j~ (1, —) (1, +); +1/2, —1/2A aPsin8& (v)X(v) ti ql & q2) + (1, —)(1, +);+1/2, —1/2(vt tj ql ) q2)}~f-' Pr„„„,+' ~(t)+r„„„~(t)7. (33)For equal mass we may identify the F's with theusual form factors. For vector current,1' /, 1/ ' (t) = 2mG (t)P+1/2, 1/2 ' +1+1/2, 1/2 ' (g )GZOther cases may be obtained using parity-transforma-tion properties.For higher spin states, sum rules for X~& —X2 andX~& —X~&1 have zero on the right-hand side and areof the superconvergence type.

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Current-Algebra Sum Rules for States of Arbitrary Mass and Spin

Myron Bander

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UC IrvineUC Irvine Previously Published WorksTitleCurrent-Algebra Sum Rules for States of Arbitrary Mass and SpinPermalinkhttps://escholarship.org/uc/item/5xn421nbJournalPhysical Review, 168(5)ISSN0031-899XAuthorBander, MyronPublication Date1968-04-25DOI10.1103/physrev.168.1926.2Copyright InformationThis work is made available under the terms of a Creative Commons Attribution License, availalbe at https://creativecommons.org/licenses/by/4.0/ Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaPHYSICAL REVIEW VOLUME 168, NUMBER 5 25 APRIL 1968 Errata Current-Algebra Sum Rules for States of Arbitrary Mass and Spin, MYRON BANDER [Phys. Rev. 160, 1416 (1967)]. In deriving the sum rules by the P --+ oo method, a Wigner rotation has been left out. This changes Eqs. (23) and (24) to (1+cos0t(i,))X1+x2 } - , ) a"ii/J(l,-),(l,+);X1.->.2(v,t)Sl110t(V (23) =Y27r Ja/J-r[r>.1 ,>./; 'Y (t)+ rx1 ,>.2-;-r(t) J. (24) The right-hand side of Eq. (29) should read 168 I would like to thank Dr. David Gordon for communications bringing these errors to my atten­tion, and for sending me a report of his work with H. G. Dosch. 1 1 H. G. Dosch and D. Gordon, this issue, Phys. Rev. 168, 1817 (1968). 1926 

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Current-algebra sum rules for states of arbitrary mass and spin

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