PHYSICAL REVIEW VOLUME 166, NUMBER 5 25 FEBRUARY 1968 Conspiracy and Evasion: Property of Regge Poles* Er LIOT LEADER't Canendhsh Laboratory, Cambridge, England (Received 19 June 1967) It is shown that if standard methods are used to apply the Regge-pole theory to relativistic problems in which the external particles have nonzero spin, then there exist constraint equations which enforce relation- ships between the residue and trajectory functions of the participating poles in the region t 0. The con- straint equations follow directly from general quantum-mechanical principles and it is therefore essential to satisfy them. Moreover, the number of constraint equationsincreases roughly as the fourth power of the spin of the external particles. The structure of the constraint equations also diGers radically according to whether the t-channel process has equal-mass particles in both the initial and the final state or unequal-mass particles in both the initial and the final state. A separate treatment of the various situations is given. Several examples are worked out in detail: mE — + mX, xE* ~ 7rE*, m. p ~ ~p, EÃ — + EE, and pp ~ pp. The discussion of the general case in which the external particles have arbitrary spins requires a slight extension of previously given methods for the Reggeization of processes with spin. A very simple, and completely general scheme for the Reggeization, and for the classification of the Regge poles involved, is given. Finally, a discussion is given of the fundamental underlying group-theoretical origin of the con- straint equations, and it is suggested that the necessity to satisfy them artificially represents a weakness in our present methods of applying Regge-pole theory to relativistic processes. I. INTRODUCTION ' 'N the past year it has begun to become rather ap- ~ ~ parent that our present methods of applying Regge's original ideas, developed within the context of potential scattering, to more realistic situations in- volving the relativistic scattering of particles with arbitrary masses and spins, seers from a certain naivete. The appearance of several papers' pointing out paradoxes, pseudoparadoxes and downright idio- syncrasies in the predictions of and requirements on the Regge-pole theory has borne witness to this situation. In the present paper, we shall show that there is a class of constraints, the existence of which follows directly from the general principles of quantum me- chanics, which are amtomaticu/ly satisfied in any "decent" theory but which appear in the Regge theory in a very complicated and restrictive guise. These constraints enforce relationships among the residues and/or the trajectories of quite dissimilar Regge poles, and lead to very powerful experimental predictions. It was our original motivation to examine and explore the phenomenological consequences of these constraints but our present feeling is that the requirements are so arti6cal, and even arbitrary, that it is much more likely that their existence is simply a manifestation of a weak- ness in our standard method of Reggeizing relativistic problems. Two classic examples of these constraint conditions have already appeared in the literature. * Supported in part by the Air Force Once of Scientific Re- search OAR, under Grant No. AF EOAR 65-36, with the European Once of Aerospace Research, United States Air Force. t Present address: Department oi Physics, Kest6eld College, Kidderpore Avenue, London, N. W.3, England. M. L. Goldberger and C. Edward Jones, Phys. Rev. Letters 17, 105 (1966); D. Z. Freedman, C. E. Jones, and J. M. Wang, Phys. Rev. 155, 1645 (1967). (i) In backward z-X scattering, in which the ex- changed Regge poles are fermions, Gribov et al. ' showed that the Regge poles have to occur in pairs of opposite parity, with trajectory functions cr~(N) which intersect at @=0. (ii) In the theory of nucleon-nucleon scattering it has been known for a long time' that some of the t-channel helicity amplitudes are related to each other as t ~ 0, namely, in the notation of Ref. 3, fr s~f4 fs— ~t a— s t ~ 0. Since the f; receive contributions from various types of Regge pole, Eq. (1) enforces a relationship amongst the trajectory functions and residue functions of dissimilar Regge poles. The consequences of Eq. (1) were partially analyzed by Volkov and Gribov4 some time ago, but for inexplicable reasons their work seems to have passed unnoticed. More recently, Durand' has reexamined Eq. (1) and discussed alternative methods of satisfying it. In both the above examples, the results quoted are reached by Ineans of a study of the relationship be- tween the Regge-pole trajectory and residue functions, and the invariant functions (A, 8 in nN; Fs, F v, Fr, . F~, Fr in AVIV) used in the epxression for the scattering matrix of the process concerned. It is the assumed analyticity of these invariant functions, i.e. , their non- singular behavior at t=0 or N=O, ' which leads to the above results. 'V. N. Gribov, L. Okun', and I. Ya. Pomeranchuk, Zh. Eksperim. i Teor. Fiz. 45, 1114 (1963) LEnglish transl. : Soviet Phys. — JETP 18, 769 (1964)j. 'M. L. Goldberger, M. T. Grisaru, S. W. MacDowell, and D. Y. Wong, Phys. Rev. 120, 2250 (1960). 4 D. V. Volkov and V. N. Gribov, Zh. Eksperim. i Teor. Fiz. 44, 1068 (1963) I English transl. : Soviet Phys. — JETP 17, 720 (1963) g. 6 Loyal Durand, III, Phys. Rev. Letters 18, 58 (196'7). ' In a theory with exchange of mass-zero particles, this would not be true. 166 1599
Transcript
PHYSICAL REVIEW VOLUME 166, NUMBER 5 25 FEBRUARY 1968
Conspiracy and Evasion: Property of Regge Poles*
Er LIOT LEADER't
Canendhsh Laboratory, Cambridge, England
(Received 19 June 1967)
It is shown that if standard methods are used to apply the Regge-pole theory to relativistic problems inwhich the external particles have nonzero spin, then there exist constraint equations which enforce relation-ships between the residue and trajectory functions of the participating poles in the region t 0. The con-straint equations follow directly from general quantum-mechanical principles and it is therefore essentialto satisfy them. Moreover, the number of constraint equationsincreases roughly as the fourth power of thespin of the external particles. The structure of the constraint equations also diGers radically according towhether the t-channel process has equal-mass particles in both the initial and the final state or unequal-massparticles in both the initial and the final state. A separate treatment of the various situations is given.Several examples are worked out in detail: mE —+ mX, xE*~ 7rE*, m.p ~ ~p, EÃ —+ EE, and pp ~ pp.The discussion of the general case in which the external particles have arbitrary spins requires a slightextension of previously given methods for the Reggeization of processes with spin. A very simple, andcompletely general scheme for the Reggeization, and for the classification of the Regge poles involved,is given. Finally, a discussion is given of the fundamental underlying group-theoretical origin of the con-straint equations, and it is suggested that the necessity to satisfy them artificially represents a weaknessin our present methods of applying Regge-pole theory to relativistic processes.
I. INTRODUCTION' 'N the past year it has begun to become rather ap-~ ~ parent that our present methods of applyingRegge's original ideas, developed within the context ofpotential scattering, to more realistic situations in-volving the relativistic scattering of particles witharbitrary masses and spins, seers from a certainnaivete. The appearance of several papers' pointingout paradoxes, pseudoparadoxes and downright idio-syncrasies in the predictions of and requirements onthe Regge-pole theory has borne witness to thissituation.
In the present paper, we shall show that there is aclass of constraints, the existence of which followsdirectly from the general principles of quantum me-chanics, which are amtomaticu/ly satisfied in any"decent" theory but which appear in the Regge theoryin a very complicated and restrictive guise. Theseconstraints enforce relationships among the residuesand/or the trajectories of quite dissimilar Regge poles,and lead to very powerful experimental predictions. Itwas our original motivation to examine and explore thephenomenological consequences of these constraints butour present feeling is that the requirements are soarti6cal, and even arbitrary, that it is much more likelythat their existence is simply a manifestation of a weak-ness in our standard method of Reggeizing relativisticproblems.
Two classic examples of these constraint conditionshave already appeared in the literature.
* Supported in part by the Air Force Once of Scientific Re-search OAR, under Grant No. AF EOAR 65-36, with the EuropeanOnce of Aerospace Research, United States Air Force.
t Present address: Department oi Physics, Kest6eld College,Kidderpore Avenue, London, N.W.3, England.
M. L. Goldberger and C. Edward Jones, Phys. Rev. Letters17, 105 (1966); D. Z. Freedman, C. E. Jones, and J. M. Wang,Phys. Rev. 155, 1645 (1967).
(i) In backward z-X scattering, in which the ex-changed Regge poles are fermions, Gribov et al. ' showedthat the Regge poles have to occur in pairs of oppositeparity, with trajectory functions cr~(N) which intersectat @=0.
(ii) In the theory of nucleon-nucleon scattering ithas been known for a long time' that some of thet-channel helicity amplitudes are related to each otheras t ~ 0, namely, in the notation of Ref. 3,
fr s~f4 fs—~t a—s t ~ 0.Since the f; receive contributions from various types ofRegge pole, Eq. (1) enforces a relationship amongst thetrajectory functions and residue functions of dissimilarRegge poles. The consequences of Eq. (1) were partiallyanalyzed by Volkov and Gribov4 some time ago, but forinexplicable reasons their work seems to have passedunnoticed. More recently, Durand' has reexaminedEq. (1) and discussed alternative methods of satisfyingit.
In both the above examples, the results quoted arereached by Ineans of a study of the relationship be-tween the Regge-pole trajectory and residue functions,and the invariant functions (A, 8 in nN; Fs, Fv, Fr, .F~, Fr in AVIV) used in the epxression for the scatteringmatrix of the process concerned. It is the assumedanalyticity of these invariant functions, i.e., their non-singular behavior at t=0 or N=O, ' which leads to theabove results.
'V. N. Gribov, L. Okun', and I. Ya. Pomeranchuk, Zh.Eksperim. i Teor. Fiz. 45, 1114 (1963) LEnglish transl. : SovietPhys. —JETP 18, 769 (1964)j.
'M. L. Goldberger, M. T. Grisaru, S. W. MacDowell, andD. Y. Wong, Phys. Rev. 120, 2250 (1960).
4 D. V. Volkov and V. N. Gribov, Zh. Eksperim. i Teor. Fiz. 44,1068 (1963) I English transl. : Soviet Phys. —JETP 17, 720(1963)g.
6 Loyal Durand, III, Phys. Rev. Letters 18, 58 (196'7).' In a theory with exchange of mass-zero particles, this would
not be true.
166 1599
1600
Since the Regge poles which contribute to the aboveprocess also contribute to a vast number of other pro-cesses, it is of great importance to inquire as to theexistence of analogous equations of constrain. t in otherprocesses, and as to whether the conditions imposed onthe Regge-pole functions in satisfying, say Eq. (1), aresufhcient for, or even compatible with, the conditionsdemanded by these further constraint equations.
There is little hope of discovering or studying con-straint equations in general by the methods used inthe ~Ã and XE cases since to begin rvith one has littleidea of how to go about formulating the decompositionof the scattering matrix into invariant functions. How-ever, the clue to an alternative method emerges whenone realizes that the particular combination of t-channelamp]itudes occurring in Eq. (1) has a very simplesignificance. It is just that combination of t-channelamplitudes which for t=0 is equal to one of the s-channelFlV —+ XX hehcity-Qip amplitudes, i.e.,
whereft z~f4 —fz=&—4 for 1=0,
4
The constraint (1) is thus an immediate consequence ofthe conservation of angular momentum and of the factthat for processes of the type mt+m~~ mt+m2, 8 isproportional to t for small 8. In this form the generaliza-tion to processes involving particles of arbitrary spin isfairly straightforward, and will be dealt with fully inSec. IV.~ We mention here only that one finds a vastnumber of constraint equations; their number goingup roughly as the fourth power of the spin, for ex-
ample, in processes like fermion-fermion scattering.In processes analogous to backward (u small) Tr/
scattering, which we prefer to describe as forward (1small) processes of the type mt+ m2 —' m&+mt,(mrWmu), the point 8=0 does not coincide, at finiteenergies, with the point t=0, so that the above methodis not directly applicable. Nevertheless, one Ands thata very similar approach, using the properties of thecrossing matrix, suQices. One also Ands constraintsanalogous to (1)at t 0 in all processes of type mt+m2 —&
m2+mr, mt/m2 even when the exchanged Regge poleis bosonic. The precise description of this situation andthe diGerences between the fermionic and bosonic casesare dealt with in Sec. III.
~ In the course of preparation of this manuscript me received acopy of a paper by E. Abers and V. Teplitz LPhys. Rev. 158,1365 (196/)g which applies rather similar methods to the questionof the Reggeization of field theories.
in the notation of Ref. 3. The behavior demanded inEq. (1) is then just a consequence of the kinematicalrequirement that for 8 —+ 0, (where 0 is the s-channelc.m. scattering angle)
~ sin'(rz8)
oc t.
Since the same Re~«ge poles which contribute to pro-cesses like mt+md~ mt+m~ will also contribute toprocesses of the type mt+m2~ m2+mt one is againfaced with the question of compatability: Are the pro-perties enforced on the Regge-pole functions by con-straints arising in the first type of process automaticallycompatible with the properties enforced in the lattertype of process? The answer to this question dependsto some extent on the fact that the constraint equationsdo not lead to unique statements about the Regge tra-jectories and residues, and there is an element ofarbitrariness involved in our choice as to what pro-perties we shall consider acceptable.
Basically we shall divide the solutions of the con-straint equations into three main types.
(a) Conspiratorial: If the constraint equations aresatisfied by demanding a relationship between thetrujeciort'es of different Regge poles, (i.e., poles withdifferent intern. al quantum numbers), we shall saythere is a conspiracy among them.
(b) Evasive: If the constraint equations can besatisfied without demanding a relationship amongsttrajectories, but simply by enforcing certain conditionson the res~'die functions, then we shall say that a con-spiracy is evaded, and shall refer to the situation asevasion.
(c) Daughterlike: If the constraint equations aresatisfied by demanding the existence of sequences ofRegge poles with the same internal quantum numbersbut di6erent trajectory functions then the solution willbe called daughterlike. 8
There always exists the possibility that the Reggepoles can satisfy the constraint equations by decouplingthemselves completely from the process at t=0. Thisis 3, rather unacceptable situation which we shall referto as trivial evasion.
In this paper we show the following:
(i) For bosonic poles or poles of even fermionnumber it is never accessary to have a conspiracy. Theconditions, which lead to the necessity for the existenceof pairs of opposite parity fermion trajectories, areautomatically satisfied by the requirements of thefactorization theorem when the poles are bosonic or ofeven fermion number (Sec. III).
(ii) The constraint conditions analogous to (2), forthe case of arbitrary spin, do not require a conspiracyand can always be satisfied by a nontrivial evasivesolution. This result is valid to u/l orders in s, and leadsto s-channel amplitudes whose contribution from each
Regge pole has a behavior in t as t ~ 0 which is factoriz-able for all s (Sec. IV).
(iii) In any process the leading term as s-+ oo of thecontribution of each Regge pole to the s-channel
In a conspiratorial or daughterlike situation it may also benecessary to enforce some conditions on the residue functions.
PROPERTY OF RE GGE POLES 160i
helicity amplitudes has the behavior
f,s. s(*) octa(l' 'I+I( dli as t~0in the absence of conspiracy.
Although conspiracy is not necessary, there is noobvious reason why it might not nevertheless occur.We have been unable to solve the general problem ofcompatibility for conspiratorial solutions. It does seem,however, from looking at examples, that the standardconspiratorial solution used in the EA problem4 willnot work in other cases. This, we feel, is very reasonablesince if there are conspiracies the 1VX process ought notto be a good place to find them since only three of thefour possible types of Regge pole can occur in nucleon-nucleon scattering. In this paper we have looked onlyat conspiracies of the parity doublet type, and therelevant behavior of f,s ,b
' is g.iven in Eq. (68).The question of daughter trajectories is not dealt
with in detail here, and a brief discussion of their role inequal-mass constraints, where they can provide analternative to the evasive solution, is given in Sec. VI. Acritique of the group-theoretic approach' to the con-straints at t= 0 is given. The proof of the uniqueness ofthe I orentz-pole hypothesis' is questioned, and thegeneral problem of group theory for unequal-massprocesses is touched upon.
From the various solutions there follow very strikingexperimental consequences. For the evasive type, largenumbers of amplitudes vanish or get related to eachother for 3=0. For the conspiratorial and daughtertypes, one is predicting the existence of hitherto un-identified Regge trajectories which ought to manifestthemselves as particles and resonances. Thus if onecontinues to take the Regge-pole model seriously, onecan hope to distinguish between the types of solutionexperimentally.
In what follows, we shall make use of the conceptof the kinematically normal behavior (k.n.b.) of anamplitude. It is de6ned as the most singular behavioran amplitude may have consistent with the 6niteness ofall experimental parameters. It is this behavior whichis yielded for example, byWang s analysis of singularity-free helicity amplitudes. " In practice, however, in anydynamical theory the actual behavior of amplitudesmay diRer from their k.n.b. (For example, k.n.b. is notconsistent with the factorization theorem. ) Wheneverthis is so, there will always be some experimental con-sequence. Some analysis in this direction has beencarried out in Wang's later paper. "
In Sec. II we discuss the kinematics and crossingproperties of an arbitrary mass scattering process. With
'A. Sciarrino and M. Toiler, University of Rome, Internalnote No. 108, 1966 (unpublished); M. Toiler, Nuovo Cimento 37,8631 (1965); D. Z. Freedman and J. M. Wang, Phys. Rev.Letters 18, 863 (1967).' J.Finkelstein and J.M. Wang, University of California Radia-tion Laboratory Report No. UCRL-17500, 1967 (unpublished).
only minor modifications, we shall follow the notationof Ref. 11.
In Secs. III—V we study the constraint equations forprocesses of the type mt+ms-+ ms+m4 with miWms,ms/m4, mt+ms ~ mt+ms and mt+ms ~ mt+ms,m2/F3, respectively. Several examples are worked outin detail.
In Sec. VI we consider the group-theoretic approachto the constraints at t=0 and in the final section (VII)we attempt to discuss at a more fundamental level theorigin and cause of the constraints and their implicationsfor the Regge theory as presently interpreted. We com-pare the situation at 1=0 in the Regge-pole theory withthe simpler case of elementary particle exchange. Someremarks are made which may be relevant to recentattempts to generalize the concept of Regge poles. '
with the definitionss (pA+ pB)
t = (p~ —pc)'.
The erst particle on either side of a reaction formula isto be treated as "particle number 1," in the sense ofJacob and Wick, "when defining helicity amplitudes.
Following Wang" we define s-channel helicity ampli-tudes f,s,s'&(s, t), which are related to the Jacob-Wick
"M. Jacob and G. C. Wiclt, Ann. Phys. (N. Y.) 7, 404 (1959).
II. KINEMATICS, CROSSING,AND REQGEIZATION
A. Kinematics
We consider a scattering process
3+8 —+ C+D
involving particles of arbitrary mass m~, re~,arbitrary spin s~, sn, , and four-momenta p~, pn,The physical process takes place in the s channel (seeFig. 1) and the t channel is defined as the process
1602 ELLIOT LEADER 166
andS '= [s—(m,—m, )'j[s—(m;+m;)']
=mA +mB +mc +mD
We also have for the c.m. momenta
P,ts=(1/4s) S@', ji=AB or CD. (9)
The physical region for all three channels is given by
P(s, t) &0, (10)where
(t)(s, t) = st( P m' —s—t)—s(mB' —mD') (mA' —mc')
t(mA mB ) (mc' mD') —(mA m—D mB'mc')
X (mA'+mD' mB' m—c') . —(11)
In terms of it we have
sin8c=2[sit (s,t)]I"/ SAB ScD, 0(8c&tr. (12)
In the c.m. system of the t channel, with scatteringangle 0& defined as the angle between D and C we have
helicity amplitudes as follows:
f (c)(S t) 2~(Sp /p ) llsf (Jacob W-iok) (5)
Subscript a, b, . refer, of course, to the helicities ofparticles A, B, , etc., and pAB and pcD are themagnitudes of the c.m. momenta of particles A, 8, and
C, D.In an analogous way we define t-channel helicity
amplitudes f„,gb(c)-(t, s), where c,, d are the helicities ofA. , D the antiparticles of A and D.
In the c.m. system of the s channel, with scatteringangle 0„defined as the angle between A and C, we have
COS8c= (1/ SAB Scl))[2st+s sp m-
+(mA' —mB')(mc' —mD')) (6)with
C. Reggeization
The Reggeization of arbitrary spin processes hasalready been discussed in the literature. ""Ke shallessentially follow the method of Gell-Mann et a/. , '4
though we shall make some minor generalizations totheir formalism. To begin with, in order to classify theRegge poles, we introduce a new operator
We shall be interested in the behavior of 0&, 0, and theX; as t —+ 0. It is clear from the above equations thatthis behavior will be a sensitive function of the massesinvolved, and we shall therefore handle the analysis inthree distinct sections for the cases m~/mg, m~@m~,.mg=ssg, fpsgg=mD, ' and mg/mg, mg=m~.
where I';; is the exchange operator of Jacob and Wick"and p; is the intrinsic parity factor for particle j, andconstruct normalized eigenstates of angular momentumJ, parity p', E;;, and helicity:
cos8, = (1/9"AcV'~D) [2st+ t' t p m'—+ (mD' —mB') (mc' —mA') j, (13)
withV',ts= [t—(m;+m, )'][t—(m;—m;)']
p;,'= (1/4t) v';,', ij =AC or BD.
si» =2[t4(s, t)j'"/&Ac&BD.
B. Crossing
and I~'l'»'p p)=2[(1+»;,~;)(1+&~;,-~;)3 '"X( I &; &;), )+ppI J; —l(;—l~;)+ptt;rt;( —1)'"+'
X[I';—~;—~,)+ppIs;~;~, )j}, (22)Finally,
(16) wherev=0 for J integral= ~~for J half-integral,
p= &1, p= &1.These states are correctly normalized to one, and have
the property that(17)cd;ab — ~ ~~& ca;db ' J c'a';8'b'al bl clgl I Is;x;~;; p,p)=p( 1)J Ix;~,~, ; p,,)—-
where
The helicity amplitudes of the s and t channels are andrelated to each other via the crossing matrix of Truemanand Wick" as follows:
'c T. L. Trneman and G. C. Wick, Ann. Phys. (N. V.) 26, 322(1964),
~'t I~»'»" pP) = P( 1)' "I&; &;&,", p, p—). -(24)
4 M. Gell-Mann, M. L. Go/dberger, F. E. I.ow, E. Marx, andF. Zachariasen, Phys. Rev. 133, 8145 {1964).' F. Calogero, J. Charap, and E. Squires, Ann. Phys. {N. Y.)25, 325 (j.963).
166 P ROPERTY OF REGGE POLES 1603
TAnr. z I. Allowed states~J; A;X, ; P, tt) for FF, BB, and Ft systems. Note that (i) X;X; means X;WX, and X,W —X,',
(ii) states with X;=A;=0 are shown separately.
X;Xf
X;XJ
G(—1)r= (—1)~
G(—1)r= —(—1)d
);X~X;X;'A; —);00
T+J even
T+J odd );X~X;).gX);X;X; —34
T+7 even
T+J odd
Note that when m;= m; we have
R;;=P;; for BB and FF systems= —P,; for FIf' systems.
Also we have that R;;=G(—1)~ or C(—1)~ whenthese are applicable. Thus R;; is a conserved operator.Since J is conserved we see that the quantum numbers
p, and p are individually conserved.(If C or G are not applicable, e.g. , for fermionic poles,
or unequal-mass cases, we simply ignore p and deal withstates obtained from (22) by formally putting p=0).
The quantum numbers p, p together with baryonnumber, isotopic spin and signature (r) provide a con-venient labelling system for arbitrary Regge poles. InTable I we list the allowed helicity states for variousvalues of p and p for the cases of FF, BB, and FF.Thesituation for other systems, e.g., F&F2, where F& and F2are different fermions, is easily deduced. The restric-tions arising from the generalized Pauli exclusionprinciple would no longer apply.
%e now introduce modi6ed t-channel helicityamplitudes"
f — (t = (~2 COSt8 ) [A(db)+A(ca) [
X (~2 Sinl g )—
[ A(d b) A(ca) [f —— (t) (25)
f;db("(p, p),=
I (1+4b)(1+4 b)(1+8.-)(1+8, -)] "'X(fca;t[b+ppf a c;db+p —C t—)At()1)&« 1)'"')+"--If-.='. +P.f='. », -(26)
where» =max(IA(db) I; IA(ca) I }.
These amplitudes have a partial-wave expansioninvolving the functions B~+(0) introduced in Ref. 13.
f . adb((pt)p) = p(27+1)L(«I T~(ptp) Idb)
(PDBPA C)
XBA(d», A(.;)'+(gt)+(«I 2"'( P p) Idb)——
where A(ca) = c—a, etc. , and take combinations of themto form "parity symmetry conserving amplitudes. "
and are thus dominated by the Regge poles in T~(p,p)as s&= cose& —+~.
Equation (27), because of its good analytic proper-ties, is the most suitable starting point for the Sommer-feld-Watson transform and the Reggeization procedure;and this is carried out exactly as described in Ref. 14so we shall not discuss it any further here.
However, it is sometimes more convenient to dealdirectly with the unmodified t-channel helicity ampli-tudes, and we shall note here a very useful formula forthe partial-wave expansion of certain linear combina-tions of them. For arbitrary coe%cients A, B, C, D wehave
XBA(sb) A(, —,)d (8,)], (27) where the sum runs over all Regge poles. Here t, is the
1604 ELLIOT LEADER 166
usual signature factor
1+~~—ic [a (@pc,) , c—[
much less singular than
~$t a{as)—x(»:a) )
2 sin~I n(p, p, t-) —vj(30) or
~- $[ x(8s)+s (cm) ~
The residue functions r(p, p, r; ») are the unmodifiedresidues of T~(p, p) at the relevant poles in the I plane.
III. PROCESSES WITH mgWmc, ' maWmD
Here the»-channel process is of the type (unequal-mass pair) ~ (unequal-mass pair) and we shall thusdenote it as a UU process. YVe wish to investigate thebehavior of the modified»-channel amplitudes f&'& ast ~ 0 in order to deduce properties of the Regge polesin this region.
It is assumed that the f&t)(»,s) are functions whoseanalyticity in s is comparable with the Lehmann ellipseanalyticity of the spinless case. If we define analogousmodified helicity functions in the s channel as
since there could be subtle cancellations in the summa-tion involved in (32). However, their behavior catbtbo»
be more siegllur. %e shall refer to this most singularallowed behavior as the kinematically normal be-havior (k..n.b.) of the amplitude.
From Eq. (26) we have for the parity-symmetry con-serving amplitudes that
f —db"'(p) =p(+ peeve( —1)"+'" 'y ( 1 )b(ab)+Xmp2 (3g)
and
f db(t) ( p)—pq pt)ct)~( 1)cc+td c—
)('( 1)A.(ab)+Amp (39)where for case (i)
(8& —yg COSt g ) [tt(ab)+—b(«& [
)((~2 s)nt. g )—(tt(ab) —it(cd)()(f b(c) (31)
then the f"(s,») will have a region. of analyticity in»comparable with the Lehmann ellipse.
Using (25), (31), and the inverse of (17) we canrelate the f&') to the f'& getting
f d(t) (g2 COS g ) (b(db)+b(ca)l (~2 Sin). g ) Ib(db) b&ca) I
&&3II,,„..,,, 'd '(V2 COS-'g, )[b&a'b')+
&& (v2 sin~~gc)(b&a'b') b&""')(f;d ~ &' (32)
Now as» ~ 0 we see from (13) and (16) that
p O(» tt[tt(db)—b(ca)[)—
p =0(»-kit[(db)+b(ca) I)
and for case (ii),
Pz —0(»—bib(db)+b(«) [)
p2 —Q(»—k(b(db) —b(ca) I)
It is thus clear that unless
dl. (db) =X(«)=O,
we will always have
f--; "'(P)=~f--; '"(—P)
(4o)
(41)
(42)
(43)
The result can be stated more precisely as follows.De6ne
so that0, ~ ~&~2,
cosy'] ~ 1
sin-', 8& ~ 1.
sjn~~g, o: ]~~~
(ii) If md)t)t&;, m)t())td) or vice versa, then
0)—x ~ t'")so that
cos&g& ~ ~l/2
(33)
(34)
(35)
s„=sgnt (m~ —mc) (m)) —mn) j,» = sgn(l~(df) —~(«) I
—I~(df )+»l(«) I),
dl =f Idl(df) —dl(«)
I
—I&(df)+d[(«) I I
=2min{ IX(df) I; IX(«) I}.Then
f.', "'(P)= --f-;'"( P)L1+-o(»"—)j,and the k.n.b. is
f „(t)(p) tt- »--$&[b(db)( [ (+c )t[()a
(44)
(45)
(46)
All the other functions occurring in (32) tend to un-exceptional limits as» +0. We thus 6n-d that for case (i)
f — (t) —Q(»-gtt(db) b(ca)[)—and for case (ii)
Let us now examine the effect of this behavior on theparameters of the Regge poles. From (27) and theknown properties of the e~~ functions, we deduce thatthe k.n.b. for the residues of Td(p, p) is
db(p»)) at »bct a(P, P)»—tt([b(db)(+(b(ca)[+1)
as» —b 0. (47)f db(t) —0(»-bl tt(db)+b(«) [) (37)
LThere is, of course, the well-known problem of how toThe behavior of the f&') as» —) 0 could of course be use Regge theory at »=0 in UU-type processes, but all
PROPERTY OI' REGLE POLES j.605
known methods, daughters or no daughters, lead toEq. (47).]
Now suppose that as s-+~, f&'&(p) is dominated bysome Regge pole with trajectory function n(p, p). ThenEq. (45) demands either of the following:
(6) There is a conspiracy, i.e., there exists a secondRegge pole with quantum number —p (and thereforewith opposite rP), whose trajectory is such that
and~ y—I—2n
c1a1,c1a1rc2a2', c2a2 (53)
2 tx t I ~(clal) I+I A(c2a2) I+1)t2[maxf [ A(cla1) I I A(c2a2) I} al ~
c1a1,c2a2 )
To see what the most singular behavior is compatiblewith factorization, ve consider the simple case c~ ——d~,Gy= Aj., C2=dg, Qg=bg.
For the terms in (51) we now have
a(p) =a(—p) at t=0,and whose residues satisfy
(4S)and since this has to hold for all c~a~, c~a~ we see thatthe simplest consistent solution is to take
l~(&b) —~(«)I
= It (&b)+~(«) I (50)
i.e., states for which h. (db) and/or A(ca) =0.Hence if there are no states satisfying (50), then a
Regge pole seeking evasion would completely decoupleitself from the process at t=0. But this is precisely thesituation when the t channel has odd fermion number,e.g. , is of the type 8+8 -+ 8+F.Thus fermionic poles,i.e., poles with odd fermion number, must conspire ifthey are to avoid total decoupling. This is just theresult of Gribov et cl.' in a more general guise.
Putting aside the extremely unpalatable alternativeof total decoupling, we thus conclude that (a) fermionicpoles must conspire and their residues can have k.n.b. ;(b) bosonic (and even-fermion number") poles havethe choice of conspiring or avoiding conspiracy at theexpense of decoupling partially from the process att= 0.
However, this is not the final picture since we have upto now ignored the consequences of the factorizationtheorem. If we concentrate on a given process thenthe factorization theorem imposes relationships amongstthe various residues of a single Regge pole, in the form
cl 1'~1~1 2 2'~2~2 1 1'~2~2 c2 2'~1~1 &(51')
r;, a&(p) = s s.zr„,s&( p-) a—t t= 0. (49)
(8) A conspiracy is evaded, i.e., there is no need forthe second Regge pole, but in order to satisfy (45) theresidues have to behave less singularly than their k.n.b.by a factor t&~. In this case the Regge pole effectivelydecouples from all states with A/0. It thus remainscoupled to states which have
r„si~t-&I~~. ~«"+'~&s'&~ ' '~~ for bosonic poles (54)
andr„a.dg t ')&k.n.b.) for bosons
r« , et, ~ t~ 't ' .'Xk.n.b. , for fermions. (56b)
But Eq. (56a) gives precisely the behavior specified in hin order to evade a conspiracy. Thus for bosonic polesthe above simple choice of residue behavior guaranteescompatibility with evasion. 8"e thus corIclude that thereis no need for conspiracy in the case of bosonic or even-
fermion number potes.On the other hand, for fermion poles the behavior
demanded in (56b) is not strong enough to satisfy therequirements for evasion. We thus reinforce our earlierconclusion that fermiol poLes must conspire; or totallydecouple from the process at t=0.
The above behavior leads to interesting experimentalconsequences.
To see these we consider what effect the behavior(54) and (55) has on the s-channel amplitudes. We havenot been able to find any simple result for arbitrary s,but if we keep only the Leadieg term in s, then from (19)
cosx~= ~1~t/lm, '—m~ I,cosxo =a 1.+t/
Imc' —ma'
I
(57)
r«;gbcc t)(l+(«)I+I&(~&)l—&—&~) for fermionic poles. (55)
I et us now compare this behavior with the k.n.b.given in (47). We have
where c~, c~, , etc., refer to the diferent helicitystates of particles C, , etc.
If we substitute into (51) the k.n.b. for the residuesat t=0 we see that it is not satisfied in general. For wewould require, e.g. ,
max(l~(dibi) I ltl(ciai) I }+m»(l~(drab. ) I, I~(ciao) I }=max{Is( , c)al, IA(dibs)l}
+max( I «dibi) I I ~(ciao) I } (52)
which is not true in general. Thus the k.n.b. as given
by Eq. (47) is not compatible with factorisation.16In what follows, bosonic will also include even-fermion
number.
according as m|.-'—mg'~~0; and
cosxe =&1+t/ I mo —rND
cosxo=~l~t! Im, ' moil, —
according as mg' —@AD'~~0. Thus if mg' —no~'~~0,
xg= (2t/Imo' —m~'I)'"
xo= (—2t/Imo' m~'I)'"—or
xz=ir (2t/lmo' —ma I)xo=ir —(—2t/I mo mg'I )"—
(58)
(59)
1606 EI.I. IOT LEADER
and )f mg2 —sgD2~~0
x))= (—2t/[ms) ygDI
)i/2
xD = (2t/ I ms) ' —ms)'I ) '"
orxs) =m —(—2t/Im~' —ms)'I)"',xs) =x—(2t/ I
~s)'—~s)'I ) '~'
(6o)
simple factorizable behaviors for the residues which dorequire conspiracy. For example, we could take for theresidues of each of two conspiring poles of opposite (p)
(x t$( jh(ca) I+jh(8b) j—2M—1—2al
IA(«) I»d IA(db) I &~,
r ;—,.0- t~«'('. )[-'-'.[ for IA(ca) I&M& [A(db) I,
Now
dr
d~.(""(0)= d~.'(~)I 0.
d6"(62)
In what follows, it makes no diQerence whether theX' approach zero or m, so for simphcity, we shall con-sider the case with all X' —& 0 as t —+ 0.
Consider now d),„s(s) as &~0. Expanding aboutt.=0, we have
where M is a positive integer, and where the residuesmust satisfy (49).
This leads to the following behavior for the leadingterm as s —+~ of the contribution of a pair of con-spiring Regge poles:
Let )(, =max{[A(ac) I; [A(db) [}and
X; =min{ IA(ac) I; IA. (db) I }.d)„'(0)= b)„
and by the definition of the dq„~ functions,
d(r) d"d..'(~) = V,~lc-' "l~,.)
d6 d6
(63)
(68)~ ~t, ~ ~. +a s + )««—)[s for(») o- t»[)t(ao)+A(st)[+k)«)» for y &~&),
f (e) ~ t-', [A(cc)+A(sb)[+1[s—»(x~»—x«;») f()r y (slrl
So,= (Xl(—iJ„)"c—"s~lp).
d~. '""(0)=( I(—~~.)'lt )=o if I) —ul &r.
Now substituting (54) into (29) we have, for t=o,for bosonic poles
This has the remarkable property that the no@ spiv gip-amplitude is suppressed by a factor tM, whereas theamplitudes with X;„=Mor 0 and ) „&3fare allowedto have their normal behavior.
This should be contrasted with (66a) where only thenon-spin-Rip amplitude avoids suppression. The caseM =1 has been applied to several processes in Ref. 17.
(t) (x ts( jh(ca) I+jh(8b) jlJ ca, db (65) IV. PROCESSES VGTH yng ——gnat) gnat——ynD
and putting this result into (17) and expanding each ofthe d functions in the crossing matrix as in (61), andusing (64), we see that
The t-channel process is now of the type (equal-masspair) ~ (equal-mass pair) and we denote it as anEE process.
From Eqs. (11) and (12) we note that as t~ 0
for no conspiracy.For the fermionic poles we get
sin0, ~ t'",and it then follows from Eq. (31) that
(69)
f s;gQ(' ~ t&[[~(~'[+[~('")[ '[ (ferrnionic poles). (66b)
I Equation (66) is true only for the leading term in sas s-+00.j
Thus the Regge-pole theory leads to a highly re-stricted spin structure as t ~ 0 in processes dominatedby bosonic or fermionic poles, and this results in manyinteresting experimental consequences. '~ The theory alsodemands the existence of parity doublets for fermionicpoles and this too should manifest itself in remark-able experimental consequences.
Although the simple solution (54) for the residue be-havior eliminates the necessity for conspiracy in thecase of bosonic Regge poles, it is possible to hand less
"T.W. Rogers and G. C. Fox (to be published).
as t~0J cd, ab (70)
The behavior implied by (69) is of fundamentalsigniicance. It simply expresses the fact that in theforward direction (8,=0 implies t=o for EE processes)no net helicity Qip is allowed if angular momentum is tobe conserved. It therefore arises purely as a result of therotational invariance of the theory.
Since each amplitude f') is expressed in terms of thef") via the crossing relation (17) we see that for everys-channel amplitude with nonzero net helicity Qipthere will exist a linear combination of t-channelamplitudes which has to vanish as t —+ 0 at a prescribedrate. This will then imply constraints and relationshipsamong the t-channel amplitudes as t ~ 0 and will, upon
PROPERTY OF RE GGE POLES
Reggeization, lead to relationships among the tra-jectories and residues of the participating Regge poles.
In order to study these conditions quantitatively, itwill prove convenient to deal directly with the Ne-
modrfted helicity amplitudes f,d, b(') and f;,,db(') and to
use Eq. (17) itself rather than its inverse which we havebeen using up to now.
From (69) and (17) we thus have that wheneveriA((rb) —A(cd) t WOt
We shall refer to these as e(12t(rtiotbs of cotbstr(bi/tt.
The equations of constraint are obviously not all in-dependent since parity conservation, time-reversalinvariance, etc. , reduce the number of independents-channel amplitudes, and constraint equations arisingfrom related s-channel amplitudes will clearly notgive independent information. Therefore, one wouldconclude that there are as many constraint equationsas there are independent s-channel amplitudes. Strictlyspeaking, in order to get the exact behavior of thet-channel amplitudes one should then solve the wholeset of simultaneous equation (71) in the region t=OHowever, it will prove more convenient to introduce firstan approximate type of k.n.b. for the t-channel ampli-tudes which gives their correct dependence on t ast ~ 0, but which does not guarantee that the Eq. (71)are satisfied. It is thus a necessary but not sufficientspecification of the behavior as t ~ 0.
Ke obtain this behavior by studying the inverse of(17) and picking out in the expression for f;,, db") themost singular term as t —+ 0. We shall not go into thedetails here since this is essentially the procedureadopted by Wang" in studying the behavior of f(') att=0. However, it was not stressed by Wang that herconditions, while necessary, are insuQicient to guaranteeaccord with the fundamental requirements of (70) and(71). As before, we shall refer to this approximate, andmost singular behavior as the kinema, tically normalbehavior.
In studying the constraint equation (71) it will some-times happen that the k.n.b. specified in (73) is sufficientto satisfy one or more of the constraint equations. If thisoccurs, the relevant constraint equations do not carryany further information.
There is another mechanism which complicates thequestion as to which and how many of the equationsare information carrying, and this is connected with
parity conservation. To see this we rewrite the left-hand side of (71) as
I Q ~ clal, dfblf (t)a'b/, c/8/
1 p {~ „e'e', d' b'f, , d, b, (t)a/ b/ c/(
+M„db "=' "' 'f ce .., d—b.'"}. (74)
Using parity conservation,
f, , „-, ~ ( 1)te(d'b') tt(c'ee')-f, , d, , (73)
whereQAQO
( 1)sC'+sA sD sB- —'}(IBg D
Eq. (74) becomes
I —1 P {1' c' 'de'V+~ ( 1)A(d'b') A(c'e')—e/ b', c'd'
e' a', d' b—') f—. .—. —, (t) (76)Also since
d- ..'( )=(-1) "~,.'( -X),we have from (18) that
(77)
with
where
x =-'~—'t~f27
2[A= rto=/2AB(1+VABt+ ' ' ')
r)B= —r[D= —tbBA(1+2)BAt+ ' ' '),
[[tAB = (tNB tNA —S)/2im—A»B
(79)
(80)
andr)AB 12 ((1/212A2) —(»/ »B2)1.
Thus from (78) and (79)
„b c' e'; d' b' —(
—1)s—A+s—B+sO+sD e b c d- ———
ead bt-+0
Xlim M, dbce 'd", (82)t~0
and depending on the helicities involved, the two termsin the parentheses of (76) can either reinforce eachother or cancel out. If they cancel then the leading termfor small t goes at least like t'" and this is sometimessufficient to satisfy the constraint equation. (See theexamples 2' —+ tr1[7 and EX—+ 1V1V below).
Expanding the terms in parentheses in (76) aboutt=O, and collecting together the results of (76)—(82),we arrive at the final form of the constraint equations:
c'e';d'b'f, , &, , (t)a'b'c' d'& [) 1+piet
cc ts[tt(ab) —tt(cd) [ (83)
c' e';—d—' b' —(—
1)s A+sB+sO+sD e b e—d-——~ dbcad b
X dc~as "(7r XA)d—y bsB(tr XB)d—c.c«(m XO)—Xdd.;D(~ »). (78)—
Now from (19) as t ~ 0 all the X —+ 12tr. More preciselyfor small t one has
1608 ELLIOT LEADER
with and where we have used the shortened notation
] &20( ~c'a'd'b')~( —) c'a';d'b' (84)
db&t db&t (0 2~) (87)
where
~(+) &&c'a', 8'b' d, s&d&, &sBd, sCd~, &sD
+&'2&A[2&B~- '"d ~ "(t-'lb a'Bdd d'D
~d'd db'b )+t&B~c'c da'a
X (~d d'Ddb b'B &—b b'Bdd b'D)
—2&A(~- ' ~ ' db'a' dd d'
+hb'b Ad d Dd Ad -c)]+I (&A 2&B
sA+b, bsBQ, sC+d'dsD+. . . ~ (8$)
d~btt dbtt (S) ~
0= /2 ~
do
Tables of these functions are given in the Appendix Ifor several spin values.
Finally,
2)cad b (2)A'g C/'gB'QD)
X ( 1)2(sA+sB) a b —e —d+—b(—d'b') &t(c'—a') (88)
(86)
X (+b'b dd'd &-sd'd db b)'+f(&A(&B[Q ~t Cgd~d D((&A+a~ Ad@ b
B
+2&B+~a Bd, A) Q, Agba B
X (2&A~c e'Cdd d'D
+(0B&-td'd d ' )]+' ' '
In practice, if we have mg=mg and m~ ——mD we in-evitably have that A=C or A=C and B=D or B=D,and thus s~ = sg, s~= s~. In this case many terms in thesum in (83) are related to each other by parity conserva-tion, conservation of symmetry, or G-parity conserva-tion. The result, although it looks extremely forbidding,is much simpler to use in practical calculation than (83).
One gets
1 1
a')e'&0 b' d'&0 1++ 0 1+()Bsp 1+hciai a'& c'&0 b—' d'&0 1+$ai—(~..db"" "'f -'d b
"'
+gg dac' a'; —'b'f— , I';d'b' +~ dba'c'; d' b'f ' ';d'b' +~ db
a c'd' b' f— ,, ';d'b' )—
1+- ZZZ Z +ZZZ Z2 a')0 b' c'&0 d' 0 1+gaip a'=0 b' &0ed' 0
If the t channel is elastic scattering, then the number of terms in the sum of (89) is further reduced by time-reversalinvariance, but the writing down of the summation is so complicated that it is not worth reproducing. It is simplerjust to remember the rule
f, ;. „b('&= (—1)b(".-a') b("")fd a, e ."—& (elastic t c—hannel only). (90)
Equations (89) or (83) give the constraints in terms of the full helicity amplitudes f;, &b "&. For t—h.e study of theconstraints on the Regge pa, rameters it is convenient to rewrite (89) in terms of the parity-symmetry-conservingpartial-wave amplitudes Td(p, p). Using (28) and (89), we get
There are two ways of studying the effect of (91) on the Regge parameters. The best method is to do an angularintegration with a suitably chosen projection function and to obtain constraint equations for the Ts(p, p) directly,and these can then be analytically continued to complex J. However we have been unable to carry out this pro-jection in the gerberal case, although it has been possible for the examples below. The next best thing to do is toeffectively apply the Sommerfeld-Watson transformation to (91), which amounts to replacing the f&') in (89) bytheir asymptotic form for large s. In doing this one may lose a certain amount of information if too few terms aretaken in the asymptotic expansion. One gets then relations among the trajectories and residues of the form
[2(r(p P r)+1]f.Z ( Z Z [(1+()ao)(1+&)s o)(1+t).;)] '[m,.db"""'+p)bcA'+..db"" "]p)p~& b' a'&c'&0 d'&0
X«d .'a ('a(v(P p r)+ 2 (1+~. "—), '[~:db"""'+pp'(t(7:db=" ""']r;«-',a b (p,p, r)a'&—cl&0 8'&0
+ P Q (1+)-, )—I~ a'o;a'b'l, a, , (P p r)) ~ t1(( A(ab) d(e —d)]—i} (94)
c'(0 8'& 0
where
~lal Ql blcadb
= [(1+t)a b )(1+t)a b )(1+!)-...)(1+5;. ;)] 'X [~eadb ' dbid'b'), b(o'd') —' ' ( cosHt)
+plbCA~cadb ' db(d' )b(bc'W) ' ' ( cost)i)] ~
(95)
%hen only two of the particles have nonzero spin(say 8 and D) the above equations simplify enormously.Only the quantum numbers p= p=+1 are allowed, andone gets for (88),
be shown that there exists a purely evasive solutionsatisfying the exact constraint equations in which thebehavior of the residues is drastically diferent fromthe behavior arrived at by studying only the asymptoticlimit.
A. Existence of an Evasive Solution in theAsymptotic Limit
Let f,,,—,a, b&')" be the leading terri as s (or si) ~x) in
f, —, , a. b&') arising from the rbth Regge pole. From (29)
and the properties of d),„Jfunctions, one has that
r,—, db~ 3 ~'2 if ~csab +1
r,—,.db ~ const if(97)
which is not generally compatible with the factorization.requirement (51).
We wish now to study what effect the factorizationrequirement has on the residues, and to see how thisinfluences the behavior of the s-channel amplitudes.
It will be instructive to tackle this problem from twodifferent angles. Firstly we shall study the constraintequations in the asymptotic limit and derive the neces-sary and suQicient behavior of the residues and s-channelamplitudes. We shall then approach the problem using apartial-wave expansion and show that this behavior isno longer sufhcient to satisfy the constraint equationsif conspiracies and daughters are excluded. It will then
[(1+sa.b.)(1+ha b, )]i)oZ(»+1) ZJ 8') 0 1+t)a o
Xdb &a b ),o (cos8() et: tb()" b) ') (96)
»d a similar simplification for (94).In order to study the consequences of the constraint
equations for the Regge-pole parameters we first con-sider the eGect of the factorization theorem. If we focusour attention on a given EE process, we see that thek.n.b. of the residues implied by (73) is not compatiblewith the factorization theorem. For from (73) we wouldconclude that
—»2(2J)!X
P+ l&(d'~') I)'(~—l~(d'~') I)!—(99)
is the function introduced by Fox and Leader" whichexpresses the factorizability of the dz„~ functions in theasymptotic limit.
Since the residues satisfy the factorization theoremwe may put
and then write
, a, , (n) f), , (a)(t)f)a, b, (n)(t) (1oo)
whereca, 8 b gca 1'b' )
g, .&')-= [(1+~.-...)(1+~;.-.)(2~.+1)f.]'t'2&pAo
Xt). -"'"'@A(.s ) "(si) (1O2)
"G. C. P'ox and E. Leader, Phys. Rev. Letters 18, 628 (1967).
f. . . , (t)n ( 1)b(d'b')
4(PAOPDB)
X[(1+~.-.)(1+~; .)(1+~a b)(1+~a b)]"X(2 .+1)f.r. -.', a b
&")
X (4(a b ) -(sb) ab(c s')""(«)&
(98)where
+i(a'b()'(«) = (—1)"""2)
E LL I OT LEA D ER 166
and~~I/4
/sa b'""= (—1)""&&(1+4b)(1+4-b)2 Dg
X(241„+1)t„]'/2bS b &"&Q,s(~ b &"(sc) . (103)
demanded by (97) is, in this process, consistent with
(109).Thus by the original argument which led to (97) we
get that(s)n fx ('[A()ip, ) jg) 7
If we putfa b"=2 f~ b"" (104)
when the b», „'" are given by (108).From (106) it now follows that
f &, is&a cs-fsil&&ac&I+lb(&b&l& 21= (+ +) (113)
we can put
where
(s) n (s)nh (s) n
(s) n ~ c'a', , (t) n5 ca ca gc'a'
(106)
(107)
In what follows we shall use g),„(') and g), „'"genericallyfor the g and h functions of (107). Now the minimummodification we can make to the behavior (97) in orderto satisfy the factorization theorem, is to take"
bb„'"& ~ t&/4 if 1/"a= -+1,(n) ~ ]I/O if ~'Ap—
This then implies that
(108)
~ ]—1/2
~ const
~ ]1/2
if 1/sb =ca+ 1 and 2/ ca+1
jf peat&ib— (109)
if 2/""b=+1 but 2&"=—1
and make use of the fact that the crossing matrix is aproduct of functions, e.g. ,
M„„"""'=M,:"'M„", (10S)
when the residues r;, , sb(+,+) have the behavior givenin (109). Equa, tion (113) should be contrasted with theoriginal behavior (70):
(s) ~ ~$]h. (ab)—A(cd) [ (g A(ac)+h. (db) )J cd, ab 7
enforced by conservation of angular momentum. %esee, therefore, that the factorization theorem leads tothe existence of an evasive solution (at least in theasymptotic limit) in which the spin dependence in theforward direction is drastically modified whenever bothparticles in the reaction have nonzero spin.
The above argument breaks down when the pole nis not of the (+,+) type, since then the Eq. (110) is
analogous to 6ctitious processes, for example, like
L+2r -+ L+rr',
where ~' has the same mass as but opposite parity tothe m. As listed in Table I, some helicity states are nowforbidden. Nevertheless the arguments leading to (97)require only minor modification and lead to the follow-
ing results:For poles of the (—,—) variety
and, of course, factorization is satiated. LWe shall showlater that actually (109) applies only to poles of the
(+,+) variety. ) Consider now the generic equation
( ) oc f1/2
~ constfX ]—1/2
~cadb
~caleb
~caleb + 1 but
&& +ca1
(114)gca—
(s) (n) ~ )i p p r(t)n (110)
Provided that 22 is a pole of the (+,+) variety, thestructure of (110) is identical with the constraint equa-tions which would arise in the scattering process
L+2r -+ L+2r,
Equa, tion (113) is again valid.For poles of the (+,—) type
(y ) cc fs/2 if r/ca/tb —+1if q' ~b= —1
~ fl/2 if ~cash—+1
and r/ca=+1(11S)
but
where the I particles have spin sl„and helicities X, p,except that b), „(n' would be replaced by bQQ" (")b), „(n'.But
(111)
by (108).Thus we may associate
g1 '&" 4—& fop. p s&"(L2r-b L2r)
gb' "~fpp;&'/s
' "(LL~2r2r) ~
Moreover, the behavior
if q"'»'=+1~ const if g~ p =—1,
and (113) holds.Finally for poles of the (—,+) variety
r..—;~b(—,+)oc ]~ t'/2 if
~caleb + 1
~caleb
~ca//b —+1
and r/"=+1(116)
but gca—
and again (113) holds. It is worth noting that theleading term in the contributions to the s-channelamplitudes coming from poles of a gi2/erb tyPe possessspecial symmetry properties in the asymptotic limit.Thus,
f is& i/s, p& —pr/&2/&( 1)2s 4+a cf &
(s& ip, p& —(1 17)
"Iam indebted to G. C. Fox for enlightenment on this point. fas&;cb' "' = p/&( 1) 'fcs;ab '
PROPERTY OF REGGE POLES 16ii
and
f &. ,s(P) (P P) p—2/&&&( 1)2Pxf, & s(.P) (P P)
Taking J=n+1, however, we get
(119) r 1-'»(+ +)—~2roo":"(++)and similar results when b and d are manipulated.Equation (118) is actually true also in the nonasymp-totic region.
B. Existence of an Evasive Solution in theExact Case
We shall now show that the above behavior is quiteinsuKcient to satisfy the constraint equations in thenonasymptotic limit. To do this we shall study severalexamples, and we shall learn from these that an evasivesolution can still be found, but with a more drasticbehavior for the residues than (109), (114)—(116).Weshall then give a proof of the existence of an evasivesolution in the general case.
rp p., pp(+ +) oo t—1/2 (120)
There are, of course, no constraint equations.(b) srp~1rp. There are tWO COnStraint equatiOnS
arising from the amplitudes fp., l(') and f 1,1('. Theformer carries no information, and the latter leads,using (96), to the equation
Equation (127) has precisely the same structure as (121)and leads to the requirements
rpI/II/N~Np(y y) (SIIt-1/2+g(tl/2)
r, I/I/l N*N*(+ +)—+„t—1/2+g(tl/2)
r] &//I/'NpNp(+ +)~ tl/2
(128)
Using (127) and (128) we see that (126) requires
Z(2J+ 1)([(42)Tp'+3T~- ]dip'(8/)—T;s'dso'(«)) t. (1—29)
Projecting with
n(n —1)I/p;pp(+ +) IX tl/2 (124)
(n+1)(n+2)
Thus we must have, if we avoid daughterlike sequences,
I/Il 'pp(+ +) OO tl/'2
T),„(srsriT happ, X,/s). ——
Projecting with Ps+1(8&)—Ps 1(8/), we get after somemanipulation we get
(J—2)(J—3) '/'dope+'(8 ) — — dsos '(81),
(J+3)(J+4)
(J+2)(J+3) '"J+1 v2 T s+1 J+1
J(J+1)(J—1)(J—2) "'
T llew-1 v2 Tpps —1 Tl—ls—1
(J—1)J(J+1)(J+2) "'[(V'2) (T~~'+' —TW' ')
(J+3)(J+4)+3(T~y '—Ty y )]+Tg p +'—T~// '~t. (130)
J(J+1) By similar methods to those used following (122)tl/2 (122) we deduce that
We continue this equation to complex J and assumethat there is a pole at J=n. (It has of course, p =p=+.)
Taking J=n 1 in (122) we get —for the residues
&OO"'"
and
II/I/;N~NP(+ +) ~ t
rI~I/II;NpNp(+ +)—/2„+ g(t)
', N N (+ +)—1/6(s, ~+Q(t)
(n+1)(n+2) "'rl 1"'»(+ +) t'/ . (123)
n(n —1)
In all the above examples, the t channel involvedcoupling to the mx system so that only one type ofRegge pole, with p= p=+1, was allowed. Conspiracy
1612 ELLIOT LEADER
is therefore not really a possibility, and the above ex-ercise simply shows that an evasive solution as againsta daughterlike solution, exists.
We now look at more complicated examples, involv-ing a mixture of Regge poles.
(d) EX +PE—. There are two amplitudes f «« ««&'.
and f y, ««&'& which could give rise to constraintequations. The 6rst of these turns out not to carry in-formation and the second leads to the requirement
Projecting with Pz~l(8t) P—z 1(8,) yields
7+2 J—1f J+1 f J I —foJ+1+f J 1-
J+1 J2J+1
f Jtt tl/2 (134)J(7+1)
For an evasive solution the terms belonging to eachtype of Regge pole must separately satisfy (134).Thuswe get
Note that the connection between our general nota-tion and the more usual one for the SS problem is asfollows:
T». gy~( —+)= fo~
yt~( ——) = f1+
y» t«NN;NN( +) gyt tl/2
(e) pp—y pp. In this example we shall see the vital
role played by the factorization theorem. The processis described by seventeen independent amplitudes ofwhich ten give rise to constraint equations. Of these,four yield no information and we are left with theconstraints arising from fll, l 1 ', foo, l 1 ', flo. ol ',flo; —lo", fol;1—1", and f 11., 1 1'. They are, in theabove order:
Since, for the moment, we are looking for an evasive from which we deduce thattype solution, the above equations have to be satisfiedseparately for each type of Regge pole.
Consider first the (+,—) type. From Eq. (141) wesee that
so that the above behavior for ryp 10 is not acceptableLooking at the role of rpl; 1—1 in (141), and using (158)and the factorization theorem, we conclude that
and10;10PP'PP( +)~ t3/2
rll 10 ' ( +)~t ~
(148) rolllPP PP(+ &+,)'(149) Similarly, we deduce that
(161)
All the other equations are now satisied.Consider now Regge poles of type (—,—). From
Z(2&+1){T—11;1—1 (—,—)[duu (83)+du —2 (83)j Using the results (156)—(165), we see that Eqs. (136)Tlp;10 ( )[dll (83)+dl—1 (8l)g ~ t'", (151) and (140) are satisfied.
1614 ELL IOT LEAD ER 166
dn
(166) Z ~(p) 1."'"'=rii. 00»'»(+ +)=v2app't '"+O(t'")
and now Eq. (139) is also satisfied. The cancellation in(141) is a more subtle matter. All the terms automatic-ally satisfy the equation except for the coefFicient ofPq(0, ) We. must therefore show that
using (156) and (166).Thus all the constraint equationsare satisfied.
Looking at the results of these examples we see P A(s, V)A(s, X'+1) A(s, X'+n 1)—emerging a definite pattern of behavior for the residuesin an evasive solution. Namely, that
y —,00(P p) cc tk( l~(~P)l+l~(00) I 1}
and
(p ) &z t&(l+(ep)l+l&(pp)i+i&
fol
for
p~ +1 (168) (We take n&0. If n(0 a similar result follows. )Proof: Since
From (174) the left-hand side of (176) at any e is
() lei(c/2 —c)JcJ' ne—i(c/2+c J„lp)
(rl= Z I 1( 1(-2~.) e"""""~+"(-2J.) "
M),„(")"'n'=(1/r!)(d"/de")Mb„"'&'(e) I, (). (183)
Thus (176) becomes
2sA 2sB
f.a:.b(c)"'~'= Z Z Z Z ~~"ear'n —2sA m 2sB c', 5' r, r'
Thus at &=0,Xe '('/+') nl/1). 2sA 2sB
n=2sA m=2sBtbll nl+Iml!L „(ts)
A (s,X')A (s, 'A'+1)) I
X g ti(+"')v& (t)(D"'(t)r, r -0
A(s, ) '+n —1)Mb„"'+""'(.) I. ,/'r l
IP, I(—2J2) LJ,+Ub]"(—2~v) "lt » (177)-o 2/2)
If we now take
c', 5'(r) c' c'+nM (r')b'+m V
XR;;+b+„b," -+'(t) . (184)
which is clearly zero if I)i—/tbl )rb+rLenzmu 4:
dr
p ()1'+222)A (s,)i')A (s, 7+1) A (s, )i'+I—1)d6"
R;. Sb»~'(t) =A(S~—.y;n )XA (s&, y. —+1) A (s&, x, ,——1)XA(sD, ys. b.)A(sD, ys. b+1) A(s)), xs b.—1)
XC(t; P,p, h(c'a'))C(t; P,p;h(d'b')), (185)
XMb "'+""'(e) I, 2=0 if 1)i—/il) rb+r+1. (178)
This follows by similar methods upon replacing X'+21rb
by (J,—222) when acting on the state l)i'+2b).The contribution of a single Regge pole to f(') is
given by (29) as
y, ,—.=min(c'; a'},x, .—.=max{c'; a'}, etc.
lim C(t; p,p; h.(c',a')) = C(p,p; h(c'a'))
(186)
(187)
=(-1)1+™ L(1+b-.)(1+b--.")(1+b-.)4(pczpDB)
X(1+8m 2b.)]' 'L2n(p, p, r)+1]err;;+n. b+m b (p,p, )
(n, c. )( si)—=(+t)r;;+„.b+ b, (P,p, r)[(1+8„,)(1+bn 2c)
X (1+bmO)(1+ 8m —2b )1'/21.mn(t, S), (179)
say, where L „(t,s) is finite at t=p.
exists and is 6nite, and where
C(p,p;h(a'c')) =pibc~C(p, p; h(c'a')), (188)
and use the results of the Lemmas, we see that the sumover b' and c' in (184) is zero if
Ic—al ) I
I I+r and/orld —bl & l~l+".
Thus every term on the right-hand side of (184) oforder $&~ ~nl+I~I+~"'} contributes only to s-channel ampli-tudes with
Ic—al+ I
d bI&
Irb I+ I irbI +r+—r'. Alterna-
tively, f,z, , ' hb(a)s in it only powers of t greater than orequal to
Ic—al+ I
d—b I, i.e.,
~(s) (up~ j.) 0(- tie I ~(«) [+ I &(&5) ) }J cd, c5
1616 ELLIOT LEADER 166
R;.—..s.b»—'(i) = (c'+a') (d'+b')XA(sg, yc;) . A(sg) x, .——1)XA(s/1, ys b ) ~ A(s/1, xs. b
—1)XC(t; p,p; A(c'a'))C(t; p,p; A(d'b')). (193)
Applying the same arguments as used in the pp=+1case, we see that a term on the right. -hand side of (184)with given value of ~e~+ ~m~+r+r' can in this casecontribute to [c—a)+ (d—b( & )/b)+ )m)I+r+r'+2.Hence the necessity of the extra power of i in (192).Thus (192) and (193) provide an evasive solution forwhich the s-channel helicity amplitudes have thebehavior
(s)yp 1 ~ $$( ) A(ca) )1 t A(db) ( }g cd;ab
with, as required by (118),
(194)
Thus we have the result that (185) and (180) providean evasive solution to the constraint equations, forwhich the s-channel amplitudes possess a factorizablet dependemce, even though the amplitudes themselvesare not actually factorizable in the nonasymptotic limit.
From (177), (180), and (185) the behavior of theactual residues is
For the case Pp= —1, the above solution is unaccept-able since one must have
fcc; Bb (P&p) =Ppf c c;Sb —(P—&p) r
which for Pp= —1 is not satisied by (185).Using Lemma 4 we construct a solution with the
correct symmetry (191)by taking
((1+b-.)(1+b-.)(1+b--.")X(1+i/m —2b~) $ r c'+cn; b'+m b'(PP = 1)
= ]kflcl+l~l+1)g, , „.b. b, »—1(/), (192)
and then choosing
V. PROCESSES WITH m~/mg, ' m~=mD
The t-channel process is of the type (unequal-masspair) —+ (equal-mass pair) and we denote it as a UEprocess. "
As in Sec. III, we study the behavior of f&'&(t,s) ast —+ 0 using Eq. (32).
It is seen from (13) and (16) that
cos8&~ t'~' sin8& —& 1 as t —+ 0, (197)
so that the factors involving cos-,'0& and sin~o, areinnocuous in (32).
However, the rotation angles X,& and Xz for the equal-mass pair are very sensitive to the order in which thelimits s~~, t —+0 are taken. Thus
cos+g =- Lt(s+mg' —m/1')g~ (1/2(] 4mA2) 1/2
+2m@'(m/1' —mn )g (198)
It is worth noting that the helicity dependence of theresidue behavior (186) deduced in the EE case bysatisfying the equal-mass constraint equations to allorders in s, coincides with the behavior (54) obtainedby satisfying the unequal-unequal (UU) constraintsto leading order only in s. The reason for this is discussedin Sec. VI.
As was done in the UU case we can also considerconspiratorial solutions involving poles of opposite p.%e then get for the leading term in s the same behavioras given in Eq. (68). The residues behave as in (67)but there is no longer a simple condition like (49) relat-ing the residues of opposite p. Instead one has
Thus the behavior conjectured on the basis of theexamples is shown to be true in general, i.e., there doesexist an evasive solution and it has the remarkableproperty that it leads to s™channel amplitudes whosemain l behavior, as $ —+ 0, is factorizable.
costly~ V ~ (2oo)
d„„~(x)~ (coax) ~
as cosX-+ca, we are led in (32) to the result
(201)
f, &b(c) cc t—k(cX+cC) (2o2)
independently of the helicities. One then fi.nds" from
'0 Much of the argument in this section is due to T. W. Rogers.
The limits of the other angles are unexceptional, sothat using the property
PROPERTY OF REGGE POLES 1617
for~ tkp r/s
—&(P P)]t—k(ssA+~C)
p= ance( —1)"'
(204)
respectively.But this behavior is totally at variance with the
behavior deduced for EE and UU processes, since if weconsider four processes, say
(26) that
f q s~(0(p p) ~ t-k(~a+ac) for p o ( 1)a/2
(z g-f(ek+ec —1) for p o ( 1)/(/2 (203)where
ec~=+))~/))c if A and C are bosons=—))~/))c if A and C are fermions,
and therefore, naively, that the residues ought to havek.n.b.
&&(p &) o.-///() m ~(y, n)l] //(~a+~—o+&)
or
and the residues can be evaluated from
(EE)r».&(uv) jl/2 (21o)
VL GROUP THEOR& AT ]=0A. Grouy-Theory Ayyroach
Consider the s-channel process
using (54) with either (109), (114), (115), and (116) or(190) and (196), according to whether the coupling tothe equal-mass pair is evasive or not.
It is well known that the limit t —+ 0 is subtle whenunequal masses are present. " The correct way toapproach t=0 is probably through a 6xed-s dispersionrelation and this would correspond to the order of limitsused in deriving (209) and (210). Again in this casethere can be conspiratorial solutions involving poles ofopposite p and the behavior (209) is then replaced bythat given in (68).
A+8 -+ A+8,G+G~ H+B,A+6) A+K,G+8 -+ H+8,
A+8 —+ C+D
&=p~-pc, Q= p~-pr)(205)
and le't
(211)
be the momentum-transfer four-vectors, so that fromEq. (4)where m«mII, so that the processes are, respectively,
of the type EE, UU, UK, and UE, then the residuesmust satisfy the factorization requirement
and. (204) is quite incompatible with (206), (54), and(109) or (185).
To get the correct behavior we consider only theleading term for large s in (198) then
cosXg~ t'" as t —+ 0,
and (213)pc=((y"+mc')'", p').
Then if m~=~c) it is easy to show that at t=0 in
(207every Lorentz frame, E is a null vector i.e.) p ~ op
and, similarly,E= (o,o,o,o). (214)
(208)cosmic t'". That is, t=0 implies forward scattering in al]. referencesystems.
However, if ns~&~c and t=0 is a physical point forthe process, then if we go to the s-channe]. c.m. framewith the z axis along, or antiparallel, to y—y', thenat t=0
In fact, the behavior of X~ and Xc in this case is ex-actly the same as in the EE case, (79) and (80), pro-vided we take only the leading term in s in the latterequation. Moreover, for the leadir/g term ir/, s in Xoand X& we have from (19) precisely the same behavioras in the UU case (58).
Thus so long as we deal with only the leading termin s, the equal-mass pair and the unequal-mass pair inthe UE process behave exactly as they would in EE-and UU-type processes, respectively. It is now clearthat the behavior of the leading term in the s-channelamplitude can be deduced directly from the result (106)since in the proof of (106) no reference was made to themass relations and since (106) was deduced for arbi-trary t; i.e., corresponding to taking limits in the orders large then t small.
Thus we have for the leading term, in the absence ofconspiracy,
mx' —mc'+mD' —ma'-(1,0,0,1), (215)
i.e., E is a lightlike four-vector. It follows thatis a lightlike vector in all frames which can be reachefrom the s-channel c.m. frame by arbitrary Lorentztransformations.
Let us now go to the c.m. frame of the t channel. Put
p~=(—(p +m&2))/2 p)
pc=((p'+mc')'", y),and (216)
(8) ~ t)/(A(ac)(+]A(bd)t)~ cd, ab
"D. Z. Freedman and J. M. Wang, phys. Re&. ]p3 ~596(209) (i9m&.
ELLIOT LEADER.
assumption that if
( (p2+~~) I /2 (p2+ygo2) 1 /2 0 0 0)=(—gt, o, o, o). (217)
thenf( )=S(~(*))
lim f(y(x)) = f(lim y(x)),
Then in the limit t~0, E becomes the null vectorE= (0,0,0,0) indepeldeetly of any relation amongst themasses.
Comparing (214), (215), and (217) it is clear that ingoing to the limit 3=0 while in the t-channel c.m. framewe must be doing something quite drastic. This is alsoshown by the fact that at t=o, p—= ~p~ =im in thecase m~=mo —=m, whereas we need p=i~ if m~Wmc.
From (214) and (215) one would argue that at t=othe relevant symmetry groups of the physical scatteringamplitudes are O(3,1) in the case un~ mc, m——s——mn anda group G isomorphic to T.XO(2) (the group oftranslations and rotations in 2 dimensions) for the caseof unequal masses. On the other hand, from (217) it isargued' that the relevant group is O(3,1) Lor O(4) if s isbelow threshold) in all cases, independently of themasses; and this leads to a classification of Regge polesin terms of "Lorentz" poles. Since Regge poles pre-sumably have an independent existence it seems mostunlikely that there is any fundamental preference forO(3,1), since if one treats the general mass case, steeringclear of 3=0 in the t-channel c.m. frame, one is simplynever led to O(3,1) at 6nite s.
This dHBculty at t=O in the t-channel c.m. frame isintimately bound up with the troubles which were dis-covered even in the spinless case. ' " In the usual dis-cussion of these diKculties one is always thinking ofs —+~. Ke wish to stress that the difhculties at 3=0have little to do with s~~.
Consider for simplicity a spinless process. Let A(s, t)be the invariant scattering amplitude. Let s lie in aclosed domain X).
Then certainly the limit t —+ 0 exists and
a result which is only true if the mapping x ~ y(g) jsnonsingular at x=O, which is not true for the mapping
(s, t) ~ (t,s,) a,t t=O.
Thus irrespective of whether s is large or not, onecuemot take the limit t ~ 0 if f~" regarded as a func-tion of t and st. Or, if one does take the limit, it may havenothing to do with the physics at t=0. Now we claimthat the fallacy which leads to the spurious results, that0(3,1) holds independently of the masses, is preciselyof this nature. For in order to deduce the symmetry it isnecessary to regard fq„~"{s,t) as a function of thevectors E and p= pg+ pc, p'= p~+ pD, i.e.,
f ."'(~,t) =A.(E,p,p'),
and it is not true, in the unequal-mass case, that
is singular at t= 0. Thus the group O(3,1) is not relevantfor the unequal-mass case in general. However, let usnow see what happens as s~~. Ke assume that forthe unequal-mass case there exists an asymptotic ex-pansion of the form
contradicting (220). The fallacy, of course, lies in the
However, we have, for the unequal-mass case, that
lim s~= 1 for all s&X),t~0
and therefore, naieely,
+ G~&(P~ErQ)s d{ijs)
+''' (223)
Then the term lim, „G»,(P,E,Q) has in it lim, „E= (0,0,0,0), and therefore, for the teadheg terna, and onlyfor the leading term, the relevant symmetry groupbecomes O(3,1).Thus, in summary, we expect that
(224)
where'„x(s, t) =O(s &" ') and where the relevant groupswould be O(3,1) for p„q(t) and T2XO(2) for g„q(s,t).This result explains why we found the same residuebehavior when treating the EE case to all orders in s
RE GGE POLES 1619
and the UU case only to leading order in s. (See end ofSec. IV.) It also suggests that the role of daughters inthe UU case is simply to eliminate a spurious 0(3,1)symmetry from terms of order s &') ' and lower,that develops as a result of our present methods ofReggeization.
B. Uniqueness of the Lorentz-Pole Hyyothesis
It has been claimed" that it is not possible for anumber of Lorentz poles to conspire in such a way as togive a result equivalent to having one single non-conspiring Regge pole. The proof given results from astudy of m.p~ xp. Ke wish to show that this claim isnot justi6ed.
In the first place, our general evasive solution is acounter-example to this claim. The evasive solution iseliminated in Ref. 10 by the implicit assumption thatrr, i » does not vanish at t=O Lsee Eq. (125)g.
In the second. place, even assuming that r~, ~&0 att=0, a proof of the existence of an infinite sequence ofRegge poles using only the process xp —+ xp certainlydoes not say anything about the question of conspiracy,since in this process true conspiracy is in any case im-
possible since only one type of Regge pole can con-tribute to it. If one assumes ri, ~&0 in the constraintequation. (122), then one is certainly forced into aninfinite sequence of Regge poles. On the other hand, inthe case XX~EF, we know that the constraint equa-tion can be satisfied by a finite conspiring sequence ofRegge poles. ' But, as we mentioned in Sec. I, this Gnitesequence might well be incompatible in other processes.
Thus we believe that the question of group-theoreticsolutions to the constraint equations is still open. Itmay turn out that only two extreme alternatives arepossible: evasion or 0(3,1) conspiracy, but the proof islacking.
VII. DISCUSSION AND CONCLUSION
We have seen above that the constraint equationsimpose very serious restrictions on the trajectories andresidues of Regge poles in the neighborhood of t= 0. Wehave analyzed svme possible methods of satisfying theseconstraints in the general case of the scattering ofparticles of arbitrary spin and in several examples, inparticular the construction of an evasive solltioe, andhave indicated the experimental consequences. It, isclear, however, that these conditions, fundamentally,have little to do with the Regge model itself. Any modelwhich uses as input the helicity amplitudes in thecrossed channel will run into the same diQiculties. 22 Thereason for this can be seen group-theoretically. Thereduction of Poincare group down to the little groupproceeds quite di8erently according to whether thetotal four-momentum is a timelike, spacelike, or null
"It is possible that Weinberg's approach to Feynman diagramsfPhys. Rev. 133,81318 (1964)]may also suBer from this difHculty.
vector, i.e., according as 1~~0 or 1=0. The crossingmatrix, which relates the regions t&0 and t&0, has asimple structure because the "amount of symmetry" inthe two regions is the same and therefore the number ofindependent helicity amplitudes needed in each region isthe same. On the other hand, at t=0 there is greatersymmetry, hence fewer independent amplitudes arerequired to describe the s-matrix, and therefore theconstraints. The attempts to study the situation at1=0 by group-theoretic methods are fraught withdiKculty. The relevant groups for processes withequal-mass particles or unequal-mass particles arequite di6erent. The symmetries hold only at one pointt=o. The 0(3,1) or 0(4) partial-wave amplitudes arediagonal only at 3=0, so away from I,=O it might benecessary to have new poles appearing in the non-diagonal amplitudes.
It is instructive to consider the situation at t=0 insome less complicated theories than the Regge model.For example, in single elementary-particle exchangemodels, one always describes the exchanged particle as arepresentation of the Lorentz group. Thus a p meson istreated as a four-vector, an A~ meson as an axial four-vector, and so on. This is so in both the Feynrnan-diagram approach and in dispersion theory. Alterna-tively, one could imagine describing p or A~ mesonexchange by means of a Breit-Wigner resonance in therelevant crossed-channel partial-wave helicity arnpli-tude, and these two methods are not at all equivalent,except in special circumstances.
Consider, for example, nucleon-nucleon scattering. Inthe latter approach the A~ would appear as a resonancein the 1-channel amplitude fr~(t) at J=1, t=m~, '. Itwould thus contribute only to the amplitude fs in thenotation of Ref. (3). On the other hand, Ai exchangecalculated using dispersion-theoretic methods leads to acontribution to Eg, the invariant function which is thecoeKcient of the axial coupling (ps'„). (ps'„), andfrom this one finds a contribution to both fs and fr In.fact one gets
which is just right to satisfy (1) as t +0 (which im-—plies p'~ —rN'). The reason for this discrepancy isclear. The axial four-vector behaves like an axialthree-vector at, and only at the point t=tn&, '. Every-where else it behaves like a mixture of an axial three-vector and a pseudoscalar. Only the axial three-vectorpart resonates and, close to 1=m~, ', one could neglectthe pseudoscalar part. However at t= 0 it is essential tohave both parts in precisely the right proportions.
For p-meson exchange there is no discrepancy. Onewould expect contributions to fs, f4, and fs by thepartial-wave approach, and this is what one finds ontreating the p as a four-vector with just electric coupling.Moreover, the contribution to f4 has an explicit factorof t in it guaranteeing that (1) is satisfied. The differencebetween the p and A~ cases lies in the fact that the p
1620 ELLIOT LEADER 166
is coupled to a conserved current whereas the A~ isnot 23
Another example is nucleon exchange in the directchannel of xE —+xE. Naively one might have triedto describe the nucleon as a pole in the p-wave ampli-tude with zero contribution to the s-wave. It is wellknown that nucleon exchange as normally treated con-tributes to both s and p waves and it is essential to keepboth contributions as s ~ 0.
In all these cases, in which the elementary-particleexchange is fed into the invariant amplitudes of theproblem, there is never any trouble at /=0. This is be-cause the decomposition into invariants does notutilize the little group and is therefore a global process.Thus 3=0 is simply not a special point. Thus parametriz-ing a theory which is to be useful globally, i.e., for all t,in terms of the amplitudes of a single channel, does notseem to be a very transparent or logical scheme. In-deed Regge theory seems to be the only case where thisis done, and this, presumably, because of too naive ageneralization of the potential-theory results to therelativistic situation. .
All this leads one to suspect, therefore, that itought to be possible to formulate Regge-pole theory in acovariant form which would be a more natural ex-tension of the potential situation to the relativistic one,and in which the question of constraints at t=0 wouldnot arise. It would, of course, be possible to reexpressthe content of- such a theory in terms of Regge poles aspresently used, presumably in a unique way, and thiswould then be equivalent to giving a unique prescrip-tion for the solution of the constraint equations .
X242
J—2'
J 3e2'
X242
p, ~ 3
'1 —v3 v3
32
32
0 —1
2 —2 0 2 —2
—2
2 2 0 —2 —2
3
12
1~3232
32
-,'K3 —-,'VS
—-',v3
0 g6 0 —2 0
ACKNOWLEDGMENTS
The author is enormously indebted to Professor M.Gell-Mann for many stimulating ideas and discussionsduring the early stages of this work. He is also gratefulto T. W. Rogers, G. C. Fox, Dr. R. J. N. Phillips, andDr. H. Hogaasen for enlightening comments.
APPENDIX I
32
32
1 —1
1 L
1~32
0 —1
0
0 —1.
32
For convenience we list here some relevant values ofthe functions
d),„~(8=n./2) .
2'These matters have been discussed in more detail by L.Durand, III, (see Ref. 5) and J. C. Taylor, Clarendon LaboratoryReport, 1967 (unpubhshed).
APPENDIX II%e list here an enumeration of the independent
amplitudes for some processes with arbitrary spin.(i) Elastic fermion-thermion scattering:
fs'&0: (a) fs' fs w——ith —fs'&(fr= fr')(fs and with
f,'& —f»0,(b) fs' —fs——with fs'&—(fr= —fr')& fs',
(c) fs'&f»0with fr=&fr'and with ft'Wfr,
f,' and fr of same sign, and —fs'&fr'&fs',
(d) fs'& —fs&0; »i fr, fr'
with helicity transitions (fr'fs~ f1f2). Independent
elements:
Number of elements =~sELS (R'+1)—E(2R—1)g, where
R= S+~s.(iii) Elastic fermion-boson scattering:
P+8 ~ Ii+8
with helicity transitions (b'f '~ 0 f). Independent elements
f'&0: (a) b'=0; all stand f,(b) b', bW0 and same elements as for
I'r+&s ~ &r+Fs
Number of elements=srM(&+2), where
M = (2ss+1)(2stt+1) .
PHYSICAL REVIEW VOLUME 166, NUM B ER 5 25 FEBRUARY 1968
Regge-Pole Couplings to Nucleons in a Field-Theory Model
ARTHUR R. SWITTt*
Department of pttysics, University of Wisconsin, Madison, Wisconsin
(Received 15 September 1967)
perturbatjon theoly is used to consider the coupling of both normal- and abnormal-parity Regge polesto nucleons. The kinematic dependence of the residue functions on the trajectory function is obtained. Allspin-pip amplitudes have fixed poles in both the angular momentum and energy variables. One of theabnormal parity trajectories satis6es a conspiracy condition corresponding to Freedman and Wang'sclass II. The daughter pole in this case moves exactly parallel to the leading trajectory, unlike the daughterarising from unequal-mass kinematics.
I. INTRODUCTION' gERTURIiATION —THEORY models of Regge poles
have been useful as a technique for testing various
conjectured properties of Reggeized scattering ampli-
tudes. ' On the other hand, all of the work on summing
infinite classes of perturbation-theory diagrams has
involved spinless particles; while the results so obtained
are interesting, they are not particularly useful for thephenomeno1ogist who attempts to 6t experimentally
observed high-energy cross sections with Regge poles.Particles with spin are always involved. The problem
of summing diagrams with internal particles having
spin has not been solved. "The difBculties due to the
necessity of including more than just simple ladder
diagrams have made the analysis of eighth-order dia-
t Present address: Physics Department, University of Massa-chusetts, Amherst, Mass.
* Work supported in part by the University of Wisconsin Re-search Committee with funds granted by the Wisconsin Alumni
Research Foundation, and in part by the U. S. Atomic EnergyCommission under Contract No. AT(11-1)—881, g C00-881-119.
' For a complete discussion of the techniques and justidcationof high-energy perturbation theory, as well as references to theproblems to which it has been applied, see R. Eden, P. Landshoff,D. Olive, and J. C. Polkinghorne, The Analytic 5 Matrig (Cam-bridge University Press, New York, 1966), Chap. 3.
' J. C. Polkinghorne, J. Math. Phys. S, 1491 (1964); J. V.Greenman, ibid. 7, 1782 I,'1966); 8, 26 (1967).
grams prohibitive; summation of higher-order diagramsis out of the question. In this paper we consider themore modest problem of coupling previously developedRegge poles to external states involving particles withspin. Only the external states and at most one internalline involve spin. The motion of the Regge poles in thecomplex / plane is assumed to be determined entirelyby the coupling to the lowest-mass intermediate stateswhich are composed of spinless scalar and pseudoscalarparticles. If real Regge trajectories are dominated bynearby singularities, the model is a quite reasonable onefor determining the coupling to higher spin states.
We analyze the coupling of boson trajectories ofboth parities to nucleon-antinucleon states. The methodis applicable to states containing particles of any spin,but the nucleon system is the most interesting one fromthe experimental point of view. We obtain expressionsfor the asymptotic form of the t-channe1 invarianthelicity amplitudes for the processes 3f3EI' —+ MM',g9f —&3fM', and EN —+PE; M denotes a spinlessmeson whose parity depends on whether we are consider-ing normal or abnormal parity trajectories. The kine-matic dependence of the Regge residues on the trajectoryfunction n(t) is such that the coeflicient of the leadingpower of (—s/s )isoI'(—cr) for every amplitude con-
