P H Y S I C A L R E V I E W V O L U M E 1 3 8 , N U M B E R 6 B 21 J U N E 1 9 6 5

Vector Mesons and Regge Trajectories* K. AHMED

Physics Department, Imperial College, London, England

(Received 28 December 1964; revised manuscript received 18 February 1965)

Freund and Oehme's note concerning Reggeization of massive vector mesons in field theory is re-examined in the light of Polkinghorne's work; it is found that the Regge-pole approximation for the vector meson is not entirely justified.

1 INTRODUCTION

THIS note is based on a paper by Freud and Oehme1

in which they have studied the possibility of Reggeization of elementary massive neutral vector mesons in field theory, and wherein they consider all the relevant sixth-order graphs with the exchange of odd number of vector mesons in the s channel (assuming charge-conjugation invariance) and their corresponding crossed graphs. They show that the least-order non-vanishing coefficient a^ (s) in the perturbation expansion of the Regge-pole trajectory a(s) does not have an imaginary part in the range of s permitted by elastic unitarity. The graphs considered by them are not all planar, while the rules they applied for determining the asymptotic behavior2-4 of the graphs are essentially those of the planar graphs. Their procedure is not mathematically well defined and the discrepancy in­volved is discussed in Sec. III. How one can avoid this is shown in Sec. IV by applying Polkinghorne's5

prescription to evaluate the asymptotic limit of non-planar graphs which have contour pinches in some of the Feynman parametric planes. This then leads to the generation of an imaginary part in the sum over all graphs for azis) which is forbidden by elastic unitarity and the assumption of a simple Regge pole in the complex angular-momentum plane.

Therefore one arrives at the conclusion that at best there is a Reggeized particle plus some other equally important singularity—which is what one might a priori expect from the rather general set of diagrams considered by Freund and Oehme.

The asymptotic limit and other notations adopted here are the same as in Ref. 1, and the implicit assump­tion as usual is that of retaining only the leading term in each Feynman integral.

Some further remarks are made in Sec. V.

* The research reported in this document has been sponsored in part by the U. S. Air Force Office of Scientific Research OAR through the European Office Aerospace Research U. S. Air Force.

1 P. G. O. Freund and R. Oehme, Phys. Letters 5,362 (1963); and also see lectures by R. Oehme, in Strong Interactions and High Energy Physics Scottish Universities Summer School 1963, edited by R. G. Moorhouse (Oliver and Boyd, Edinburgh, 1964).

2 P. G. Federbush and M. T. Grisaru, Ann. Phys. (N.Y.) 22, 263 (1963) (I); 22, 299 (1963) (II); J. C. Polkinghorne, J. Math. Phys. 4, 503 (1963).

3 G. Tiktopoulos, Phys. Rev. 131, 480 (1963); for a definition on nonplanarity and other related properties see ibid. 131, 2373 (1963).

4 I . G. Halliday, Nuovo Cimento 30, 177 (1963). 5 J. C. Polkinghorne, J. Math. Phys. 4, 1396 (1963).

II. PRELIMINARIES

A. Result of Unitarity

Oehme and Freund1 consider the diagrams (a), (b), and (c) of Fig. 1 and their corresponding crossed dia­grams in sixth order of renormalized perturbation theory for Tr+if~ scattering mediated in the s channel by neutral massive vector mesons in the asymptotic limit when t—>oo. In the graphs (a), (b), and (c) we denote the vector mesons by coq.

Oehme and Freund suppose the existence of a leading Regge trajectory a-(s) (odd signature) associated with a vector meson of mass m with fx^m<2 y, where fj, is the pion mass. Then the function F(s,\) which uniquely interpolates the physical partial-wave amplitude Fi(s) for sufficiently large I becomes

F(s,\) = [«(*)-\'T1Ks)+R(s,X) , (1)

£-+

CO '

or

******* W » * * * * M

i 4 0

MM*m***M******i

S - * <*>'

(J

(C) FIG. 1. The relevant sixth-order Feynman graphs with the

exchange of an odd number of vector mesons.

B1470

VECTOR MESONS AND REGGE TRAJECTORIES B1471

where we have dropped the signature subscript "—" which is always implied. The function R(s,\) is non-singular on the surface \ = a(s). Then with the help of the continued elastic unitarity condition6 in the s channel, they get the following vital equations for Reggeization of the massive vector meson when per­turbation expansions for the relevant functions7 are also considered:

for 0 = Ima2 (s) = Im/33 (s) = Imbs (s) (2)

where Si= (W+2JLI)2 is the first inelastic threshold. The function G:2(S) was defined in the introduction, $3(s) is the coefficient of g6 in the perturbation expansion of f3 (s) (the residue function of the Regge pole) in powers of g2 (coupling constant squared), and b%(s) is the coef­ficient of g6 in the perturbation expansion for b(s), which in turn is related to the high-energy behavior of the /-channel absorptive part as

At(s,t)~b(s)t°<*\

B. Planar Graph

The leading high-energy behavior of the graph (a) when t —» 00—ie (and e —> 0) is calculated by the usual methods for planar graphs with spin2 n»8 and it turns out to be9

Xln[X2-2(Xi+X2) ( X a + X s ) ] ^ - 1 ^ ; s,m2), (3)

where U(kj) sJm

2) = \{K2^zS—m2<p(Kj),

<P(\) =XlX2+X2X3+X3Xi , (3')

and where the constant G6 is proportional to g6 (g being the coupling constant) and is the same for all the sixth-order graphs considered here.

III. NONPLANAR GRAPHS

Now we go over to the discussion of the nonplanar graphs (b) and (c). Their contributions being the same, we need calculate only one of them. In the discussion of these two graphs our method differs essentially from that given in Ref. 1.

The numerator contribution2™-8 in the Feynman in­tegral for the graph (b) contributes a power tz asymp-

6 See Eq. (45) in Lectures by R. Oehme, (Ref. 1). 7 See Eqs. (6) and (7) of the paper in Ref. 1 and the correspond­

ing expansion of b(s) as in Eq. (231) of R. Oehme's lectures (Ref. 1).

8 J. C. Polkinghorne, J. Math. Phys. 5, 1491 (1964). M. Gell-Mann et al.f Phys. Rev. 133, B145 (1964).

9 A comparison of Eq. (3) with a part of Eq. (14) of Ref. 1 im­mediately shows that we have performed integrations over four of the parameters occurring there and have reproduced the result in the form (3) here.

totically and hence one writes

I^(s,t;m2)^(-tYG6 [ dav • -dX8«(ai+- • -+X«-1 ) Jo

XC(aA\)tg(a&\)t+d(a&\; 5,m2)]"3, (4) where

g M , X ) =ai&(02+a 2+X 2+X 3 ) +aip2^z+a2piM—«2#2Xi, (5)

C(afi,\) = p(X)+0(«AX), < ^ > X ? S>m^ = U(\;s,in2)+0'(aA\), (6)

U(\;s,m2) and <p(k) being given by Eq. (3'). The terms involving (afi) or (0:1,0:2; ^1,^2) grouped under the symbol 0 (a,0,X) and O' {afiy\) do not involve the leading terms when t —» 00 and hence may be neglected.

Now we first illustrate the discrepancy involved in Ref. 1 in finding the asymptotic behavior of the integral (4). When t —><*>, one must look for the zeros of g(ai,l3i,\j) given in (5) and integrate near that region to get the leading term. Oehme and Freund found this term by integrating near the end points (0:1,02)= ( M ) and (ftjft) = (€,€) (e is an arbitrarily small number) and obtained, by the usual procedure of "scaling" for planar graphs first introduced by Federbush and Grisaru,2 the following:

I^(s,t;m2)^( Jo

daidazdfiidfizb (a i+a 2— 1)

X8(jSi+j52 - l ) r1 3 re

Jo 3=1 Jo dp\dp2

X-

P i P 2 5 [ 0 ( P i ) + E X y - l ]

(7)

<(-t\nt) I day ' ' Jo

dX35(ai+a2— 1)

X*(&+ft- 1)«(Z Xy-m&AWTHd)-1- (7')

Now the transition from (7) to (7') is defined mathe­matically only when g as a function of the transformed variables a,-, ft, Xy (i=l, 2; j=l, 2, 3) does not vanish within the domain of integration. However, this is not the case. A glance at the function g in (5) tells that it vanishes inside the domain and consequently its trans­form g vanishes without all of its terms having to vanish separately. The above procedure of integration near the edge of the hypercontour, if blindly carried through, introduces an unnecessary complication,10 namely,

10 This difficulty of having to distort the contour and then the possible subsequent cancellations was in fact communicated to the author by Professor Oehme in private correspondence, and neces­sitated looking at the problem in a different way.

B1472 K. A H M E D

having to distort the contours of integration away from the fixed double pole appearing in (7'). Clearly, then there is no well-defined procedure for avoiding this fixed double pole.

We can avoid all this by adopting a technique given by Polkinghorne5 which appropriately takes into ac­count the pinch contribution.

The fact that the function g as given in Eq. (5) does vanish within the integration domain suggests the possi­bility of generation of a pinch in the asymptotic limit, in the light of Polkinghorne's above-mentioned work. Although we do not show it explicitly here, one could verify it trivially by Polkinghorne's procedure. Adopting the notation in Ref. 5, we may rewrite g as follows:

+ (C+p1p2xr1), (8) where

Pi=ai(j8i+X8), P*=/3i(ai+X8), (9)

<2 = ai/3i(X2+\3), (10)

and the aiy 02 pinches occur at PiXf"1, i^Xf1, respectively, corresponding to posi- (10') tive values of «2, 02.

Then, following Polkinghorne, the pinch contribution can be obtained by multiplying the integrand in (4) with the factor \-ld(a2—PiXf ^(ft—P2Xf 0 and inte­grating over «2 and 02. (In the Appendix we give a short derivation of the pinch contribution for our case.)

2wi r1 3 f1

/ ( & ) ( ^ ; m 2 ) - ( - 0 3 ^ 6 — / UdXj daidfiiXXr1

S(ai+/31+X1-1P1+X1-1P2+ £ Xy- l)C(ahfih\j) j —1

(ID where

The residual coefficient (Q+P1P2X1-1) in the pinch contribution (11) vanishes if.ai=0 or 0i=O [see Eqs. (9) and (10)]. Hence, integrating near these end points, we get logarithmic enhancement in the asymptotic behavior as follows:

IW(s,t; m2)^ (-v*Gt)t \nt [ J[ dXMT, Xy-1)

XU-^s,™2), (12)

where U(\j;s,m2) is given by Eq. (3'). Hence, adding the graphs (a), (b), (c), and the corresponding crossed ones, we obtain the asymptotic behavior as

FW(s,t; m2)^(-G&)t(2 lnt+iic)bz(s; m2), (13)

where /*! 3 3

b,(s;m2) = / II dXMZ Ay-l)^*/;^m2) J0 j=l j=l

X[ln{X2-2(Xi+X2)(X2+X3)}+27ri] (14)

and is the same as the function bs(s) defined earlier.

Since there is no elastic branch line1 in the functions defined by

[ n^ii^-Dt/-1 Jo j = 1 i^1

or by

f I I fajt(t Xi-l)^-1Xln{X2-2(Xi+X2)(X2+X3)},

Jo 3=1 J - l

the expression bz(s; m2) in Eq. (14) has a pure imaginary part in the second term; this contradicts the vital con­dition (2), which, as mentioned earlier, has been derived from unitarity. Hence, the assumption (1) is not entirely justified and one concludes that the vector meson may not Reggeize.11 In fact, it appears that there is a Regge pole which corresponds to the planar graphs of the type (a) and its suitable iterations in the s channel, plus some other equally important singularity corresponding to the nonplanar graphs of the type (b) and its suitable iterations. This would then lend some further truth to the already existing belief that Regge poles are associ­ated with end-point effects and correspond to potential scattering, while in a fully relativistic model in field theory where diagrams possess all three double spectral functions, the Regge-pole approximation is not possible.

In what follows we now make a few remarks related to our calculation.

(1) We may remark here on the apparent simplicity of the pinch arising in our graph [see Eq. (8)] as com­pared to the pinch in the graph discussed by Polking­horne in Sec. 4 of Ref. S. In our case the pinches move towards the end points when we integrate near the end points ai=0 and /3i=0; whereas in the more complicated (and higher order) graph discussed in Ref. 5, one requires at least n+S parameters to be set equal to zero [see Eq. (29) of Ref. 5] to move the pinches to the origin. But this is not at all important in relation to what we discuss here. What is important is that asymptotically there do arise pinches inside the physical contours of integration of parameters a2 and /32 [see (10')] which have to be appropriately accounted for before one goes on to integrate near the edges for logarithmic enhance­ment in the asymptotic behavior [Eqs. (11) and (12)]. Also, since the behavior so obtained turns out to be of

11 We may remind the reader here once again that this may be true only when the leading terms alone are kept. The result may change when the other terms are also included.

IV. IMAGINARY TERM V. CONCLUSIONS AND REMARKS

VECTOR MESONS AND REGGE TRAJECTORIES B1473

the same order (~t Int) as that of the planar graph [Eq. (3)3, one cannot simply ignore it. Hence the imaginary asymptotic behavior is associated with the occurrence at t= co—ie of a singularity given by pinches in the x and y integrations and end points in the fi integrations. This contradicts elastic unitarity if the pure Regge-pole assumption is retained, which is undesirable.

(2) If the pion line is replaced by the nucleon line in the graphs (a), (b), and (c) the situation does not appear to change.

(3) We hope to discuss in a future publication the nature of the other singularity, which appears to arise through a suitable iteration in the s channel of the sixth-order graph (b) discussed here. The presence of pinch in the graph (b) points, at least in the light of Polking-horne's work, to the generation of a Regge cut12"14

when the graph is iterated and summed.

ACKNOWLEDGMENTS

I am greatly indebted to Professor R. Oehme for suggesting this problem to me and to Professor P. T. Matthews for his critical remarks and encouragement. I am also grateful to Dr. J. C. Polkinghorne for his very useful comments on this note.

Finally, I also wish to express my acknowledgment to the Pakistan Atomic Energy Commission and the British Department of Technical Co-operation for a financial grant under the Colombo Plan.

APPENDIX

The pinch calculation for the integral (4) can be illustrated by calculating the pinch contribution of the following integral:

pP2 rx2 rvz

1= dp dx dy Xr^ixy+flt+dlr*, (Al) J 0 J x\ J vi

where

ff=Xi[—a2+Xr1aiO?i+X8)] y

y = 0 2 - A r l f t ( a i + X O , (A2)

/3=aift(X2+X3)+Xr1Q;ift(igi+X3) (ai+Xj). 12 D. Amati, S. Fubini, and A. Stanghellini, Nuovo Cimento 26,

896(1962). 13 J. C. Polkinghorne, Phys. Rev. 128, 2459 (1962). 14 S. Mandelstam, Nuovo Cimento 30, 1127, 1148, and 1113

(1963).

The factor Xf-1 in (Al) comes from the Jacobean

I J I • I d(a2,02,ai,0i,Xi>X2,X3) |

Also dp in (Al) is a combined notation for the integra­tions over ai, ft, Xi, X2 and X3. There is of course an over-all 8 function which in fact serves to determine the limits of integration.

Performing the y and x integrations in (Al) one easily obtains the following:

/=— f dp\A(pt+d)-* 2t Jo L

Xln \+R(P,t) ,(A3) \L(xiyi+P)t+d2£(x2y2+p)t+d-]\ J

where R(p}t) is a rational function of t which is not im­portant here for our consideration. From Eq. (A2) it is clear that the lower limits of the (x,y) integrations in (Al) may become negative (for positive Feynman parameters) whereas the upper limits must be positive. In such a case the (x,y) integrations pass through their origin. So when xh y\ are less than zero (and 0 is suf­ficiently small) and /—» + 00— 6̂, the two factors in the numerator of the logarithm both tend to — 00+f €, and those in the denominator both tend to + 00—ie. Hence the correct logarithmic branch is then defined by

lnl=—2ir£.

Hence the pinch contribution to (Al) is given by

2wi r^ — / dp\rl(pt+dY\ (A4) It Jo

It is clear from the above procedure that the pinch con­tribution to an integral of the type (Al) comes from the region inside the domain of integration where both x and y vanish. Thus it follows that this contribution can be easily evaluated by multiplying the integrand by i(x)»(y) or by X1-15[a2-Xr1(^i+X3)«i]5[ft-Xr1

X (ai+X2)ft] and integrating over QJ2 and ft.