Multiperipheral approach to photon-meson-Pomeron couplings

P H Y S I C A L R E V I E W D V O L U M E 1 8 , N U M B E R 1 1 J U L Y 1 9 7 8

Multiperipheral approach to photon-meson-Pomeron couplings

Paul M. Fishbane Department of Physics, University of Virginia, Charlottesville, Va. 22903

Jeremiah D. Sullivan Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

(Received 20 September 1977)

A multiperipheral model is used to calculate the full spin dependence of the photon-p-Pomeron Regge vertex function. Inclusion of both natural-parity ( w ) and unnatural-parity (w) intermediate states at the top of the ladder leads to a pattern of approximate helicity conservation. The model also predicts an anomalous refractive term. The predictions of the model for the differential cross section are too small when compared to experiment, however. Photoproduction of 6.1, 4, and I) are briefly considered as well.

I. INTRODUCTION

It i s universally agreed that diffractive process- e s play a central role in high-energy scattering including photon-induced processes; the experimental evidence i s overwhelming. In spite of this there does not yet exist a realistic, quantitative theory of diffraction scattering.

In this paper we use the multiperipheral model to make calculations of some photon-meson-Pomeron couplings. Our approach i s motivated by and i s a logical continuation of the now classic work of Berman and Drell' who used the multiperipheral model in a qualitative way to predict and describe diffractive photoproduction of vector m e ~ o n s . ~

The f i rs t part of this work i s devoted to a quantitative study of the model of Berman and Drell for Y N - ~ O N . We make a full calculation of the spin structure of the photon and vector-meson lines in order to compare the model to the vector-meson density-matrix elements now measured in polarized photoproduction. We find, remarkably, that when both n and w intermediate states a r e included the model tends toward approximate helicity conservation for y N - p O N a s observed in experiments. This comes about through an approximate cancellation of n and w (opposite parity) contributions to the helicity-nonconserving amplitudes.

The overall amplitude, however, i s too small by about an order of magnitude. This i s a common failure of the multiperipheral model when it i s re- str icted to pion exchanges along the sides of the t-channel ladder. It occurs also, for example, in nN o r nn ~ c a t t e r i n g . ~ We believe that the inclusion of higher-lying Regge trajectory exchanges may correct the normalization discrepancy without los- ing the tendency towards helicity conservation &nd the other key results which we find. Calculations of the nucleon-nucleon-Pomeron vertex have been car r ied out by Wingate4 using the model developed

here. He found, using only pion exchange, good agreement with experiment for both the spin dependence and the overall magnitude. We do not un- derstand at this time why pion exchange works well for the baryon case but not the meson case.

The model used in our work i s also related to that used by Kane and P ~ m p l i n ~ ~ in their study of the large- impact-parameter contributions to diffraction scattering. Although these authors dealt only with hadronic projectiles and did not explicit- ly discuss photoproduction, their findings and ours a r e for the most part in general agreement. One a r e a of partial disagreement i s the issue of s- channel helicity conservation. The pattern we find in p photoproduction and Wingate4 finds in ?TN and NN scattering of a cancellation between the contributions to helicity flip amplitudes coming from natural- and unnatural-parity intermediate states i s not brought out by the authors of Ref. 4(a). They point out only that specific individual intermediate state contributions do not separately conserve s- channel helicity, a result we find a s well.

Care must be taken, however, to distinguish the c lass of diagonal and diagonal-like processes reported here and in Ref. 4 for which the phases of coupling constants are tightly constrained and the c lass of off-diagonal processes such a s n N - A , N which were the main interest in the work of Kane and Pumplin. The relative phases of the couplings for this lat ter class a r e poorly constrained and largely unknown. Hence, approximate helicity conservation for these processes, while possible, i s not compelling.

In addition to approximate helicity conservation in yN-pON we find that the model automatically in- corporates in a natural way the S8ding-Pumplin- Bauer mechanism5" which yields the skewing of the p0 mass distribution observed in experiment. Our model also predicts an anomalous "refractive" (e.g., real) part of the diffractive amplitude whose

136 P A U L M . F I S H B A N E A N D J E R E M I A H D . S U L L I V A N - 18

existence was f i r s t pointed out by BauerO7 These topics have been discussed a l so by Spital and Yennie,' and Yennieg f r o m a different but physical- l y equivalent point of view. The anomalous r e a l p a r t s associated with diffractive production of un- s table part ic les a r e an important feature of the Pomeranchuk contribution. We a l so d i scuss yN - WN in the s a m e sp i r i t a s the photoproduction of pO. T h e prediction f o r the overal l s i z e of the c r o s s sect ion again t u r n s out low when only pion exchange is included.

Another par t of th i s work is devoted to a study of the photoproduction of m o r e mass ive mesons. F o r the most par t we concern ourselves with the over- a l l s i ze of the differential c r o s s section, particu- l a r l y a t t = 0. One could, of course, work out the full spin s t ruc ture a s we have done f o r the and W.

We take up in turn yN - $N and yN- $No F o r the l a t t e r we employ D, F, D* and F* intermediate mesons, charmed s ta tes which a r e predicted by the SU(4) model of hadrons.1° The resu l t s differ f r o m experimental resu l t s i n the s a m e way a s our o ther calculations. Tha t is, the prediction f o r the absolute normalization is low. However, if one cons iders only residue rat ios the resu l t s a r e more successful.

II. BASIC EQUATIONS

We begin by set t ing down the bas ic multiperiph- e r a 1 equations and philosophy which we use throughout th i s paper. F igure 1 shows the multiper ipheral "Pomeron" which descr ibes the dif- f ractive process a + c -a' + c. In i t s original form"*12 i t was hoped that the mult iper ipheral lad- d e r would b e dominated by the exchange (s ides of the ladder) of the lightest m a s s par t i c les consis- tent with the quantum number constraints imposed

FIG. 1. Schematic representation of the multiperiph- era1 Pomeron. The rungs of the ladder are various meson states, R . Pions (or the lightest possible particles) form the sides of the ladder.

b y the external par t ic les; normally the exchanged part ic les could b e pions. As a resul t of extensive s tudies of the mult iper ipheral model i t has become c lear3 that the exchange of more massive part ic les (p,w, etc.), indeed ent i re Regge famil ies , must a l s o b e included. The reason f o r this is the fact that these more mass ive mesons, while generating t-channel s ingular i t ies more distant than those generated by the pion, have Regge t ra jec tory in- t e r c e p t s @(O)- i which l ie above the pion intercept, a, (9) " 0.

Nevertheless, i n the bulk of this paper we will r e s t r i c t our discussion to the c a s e of un-Regge- i zed pion exchange along the s ides of the multiper ipheral ladder. This is interesting because: (i) Pion exchange while not dominant i s nonethe- l e s s an important component in the smal l t region; (ii) we a r e rea l ly only using pion exchange f o r the topmost s ides of the ladder in Fig. 1 since we a r e only calculating the Pomeron residue functions and not the t rajectory function itself; and (iii) the heli- c i ty s t ruc ture and other fea tures which we find a r e physically interest ing i n the i r own right and need not b e obscured by detai ls which a r e special to Regge exchanges.

The rungs of the multiperipheral ladder in Fig. 1 a r e low-mass meson s ta tes (sometimes re fe r red to a s clusters) . We will make the common approximation that the rungs may be taken to b e a s table hadron o r a low-mass resonance. The fact that the rungs of the ladder a r e dominated b y low-mass s y s t e m s is now well established by experiment. Indeed, the study of the multiparticle final s ta tes which dominate the c r o s s section a t high energ ies h a s confirmed the bas ic validity of the multiperipheral approach and has begun to determine the b a s i c input parameters.13

Le t u s f i r s t consider the c a s e where the external par t ic les a r e chosen according to a = a' = pion and c = nucleon. Choosing the s ides of the ladder t o b e pions we may pick R =p(770) [R = ~ ( 7 0 0 ) , f(1200), g(1680), . . . a r e a l so possible but give nu- mer ica l ly s m a l l e r answers]. In th i s f o r m the model i s nothing m o r e than an integral equation f o r (off shell) 7TN scat ter ing a s i l lustrated in Fig. 2,

A s is well known3 the solution of th i s integral equation gives a I ~ N total c r o s s section which fa l l s

FIG. 2. Integral equation nN elastic scattering, sometimes referred to as the multiperipheral bootstrap.

18 - M U L T I P E R I P H E R A L A P P R O A C H T O P H O T O N - M E S O N - P O M E R O N ... 137

with energy. In other words, the Pomeron intercept when self-consistently determined a s in Fig. 2 lies below 1.

We, however, may regard Fig. 2 in a different sense. It may be interpreted a s a relation between on-mass-shell rN scattering and an integral over off-mass-shell nN scattering folded with the (p- dominated) nn- nn amplitude. We will insert a phenomenological model for the off-mass-shell nN amplitude which has a physically correct Pomeron trajectory, a (0 ) = 1. The output on-mass-shell amplitude will enjoy the same correct energy dependence with a coefficient, the pion-pion-Pomer- on (nnP) residue function, which i s the object of interest. Of course we will only generate a finite fraction of this residue function since, a s de- scribed above, we a re restricting the uppermost (only) sides of the ladder to be pions. Before pre- senting numerical results for the special case of nN scattering we need to establish some notation.

Fo r the general case a + c -a' + c we choose momenta and invariant variables a s shown in Fig. 3. Namely,

We a r e interested in the leading behavior in the limit s d m , t fixed. It i s convenient to express al l vectors in t e rms of their "plus, transverse, and minus" components14 in the overall center-of- mass system,

FIG. 3. Kinematics of the loop integral for the process a + c - a ' i c .

The variable x i s the fraction of the large plus momentum which flows through the lines q, and q,. It i s useful to scale the two-dimensional transverse momenta according to

Note a t high energies

and

It i s interesting to note that a t t = 0, (2.6a) and (2.6b) tell us that

42' -qi2 =x(pz2 - 1112)* 0 , (2.7)

even a s s - m when pZ2 - p12# 0. Thus, processes involving large mass changes a t the external lines cannot have both the internal pions simultaneously near the mass shell.

In t e rms of these variables the loop integration in Fig. 3 takes the form

P A U L M . F I S H B A N E A N D J E R E M I A H D . S U L L I V A N 18 -

FIG. 4. The absorptive part of the graph in Fig. 3. The vertic a1 line indicates the unitarity cut.

where M denotes the invariant amplitude. (We temporarily assume that a, a', and R a r e spinless.) Rather than ca r ry out the four-dimensional integral in Eq. (2.8), i t i s eas ier to take the s- channel discontinuity of Eq. (2.8) and reduce i t to a three-dimensional integration relating absoptive parts. After carrying this out Eq. (2.8) simplifies to (Fig. 4)

(2.9)

and in Eqs. (2.2)- (2.6) we se t 12= pR2. Equation (2.9) is identical to an equation derived by different techniques some time ago by Bertocchi, Fubini, and Tonin.l2

As a phenomenological model for off-shell .irN scattering we choose

where ~ ( t ) i s the Pomeron trajectory

a ( t ) = 1 + (0.3 GeV2) t , (2.11)

so = 1 GeV2,

and P and b a r e taken from experiment

b = 4.7 GeV-'. (2.13)

Note that P is the product a t t = 0 of the nrrP and N N P residue functions,

P = Y,, ( 'arm (0) . (2.14)

All of the off -mass-shell dependence in Eq. (2.10) i s contained in the factor C$ which we assume can

b e represented a s the product of form faetors

whose form we discuss below. Once we have calculated A,,,;,, f rom Eq. (2.9) we can reconstruct the full amplitude f rom

Substituting Eq. (2.10) into Eq. (2.9) one sees im- mediately

where the reduced residue function Te,,(t) i s given by the integral

Another feature of general interest i s the phase of the scattering amplitude. Fo r the elast ic scattering of stable particles the phase i s given entire- ly by the Regge signature factor a s written in Eq. (2.16). The reduced residue functions, Eq. (2.18), a r e real for t 6 0.

When dealing with unstable particles, however, the residue functions a r e not always real; this follows directly from Eq. (2.18). If during the integration over x and : one should cross the pole a t qZ2 = mr2(or q12 = mu2) the proper prescription i s

The 6-function t e rm contributes a complex part to F, and hence the overall amplitude, Eq. (2.16), picks up an energy -independent phase in addition to the usual signature phase. Fo r the case of diffraction scattering a t t = 0 where the amplitude i s normally pure imaginary (diffractive) the 8-function te rm in Eq. (2.19) contributes a rea l (refractive) part to the amplitude. It i s easy to establish the conditions under which ? becomes complex. F rom Eq. (2.6b), qZ2 = mu2 corresponds to

We may regard Eq. (2.20) a s a relation among the masses. As a function of x and 5 the left-hand side i s maximized a t xo = 1 - pR/p2 and the right-hand side i s minimized at c=c. Hence the criterion for

18 - M U L T I P E R I P H E R A L A P P R O A C H T O P H O T O N - M E S O N - P O M E R O N . . . 139

a complex residue function i s valid even when the absorptive part itself becomes

(2.21) complex.1 Let u s now return to the nN scattering example,

This relation i s simply the condition which states Fig. 2, and for simplicity specialize to t = 0. The that the decay p2 - R + m, can physically occur, product of the two coupling constants which appear i.e., ,u2 i s unstable. [Note that the signature factor in Eq. (2.18) needs only to be replaced by pnn ver- prescription given in Eq. (2.16) for constructing the tices (see Appendix A) and the p propagator. full amplitude from the absorptive part remains Namely,

where we use k , = k , and q, =q,. The overall factor of [2] in Eq. (2.22) i s a Clebsch-Gordon coefficient coming from the pnn vertices. Thus we have for the snP reduced residue function,

It i s now necessary to specify the form factor sides of the ladder we must choose F , accordingly. F,(q2). If we simply se t F, = 1 the transverse-mo- The elementary divergence (which gets worse for mentum integration i s divergent. Since the basic the cases involving higher spin which we discuss spir i t of the multiperipheral model i s a restriction below) i s a signal that al l results will be somewhat to the region of low momentum transfers along the sensitive to the choice and scale of cutoff.

FIG. 5 . Dependence of the reduced residue function, Eq. (2.23), on the cutoff mass ~ " n the off-shell form factor for various choices of this form factor.


We have investigated two choices: (i) square cutoffs

~ , ( q ~ ) = 0(A2- 1q2 - m : / ) (2.24)

and (ii) inverse -power cutoffs

where A2 s e t s the scale of the f o r m factor. More complicated choices, e.g., exponential, Dur r -P i l - kuhn, l5 etc., could b e made but have no special .

mer i t , a t l eas t in our case. In Fig, 5 we i l lustrate the sensitivity of the quan-

tity YTj(O), Eq. (2.23), to the cutoff parameter A2 f o r the above cutoff choices. Resul ts were obtained b y numerical integration on the computer. Note that i f the s imple mult iper ipheral model of Fig. 2 were exact, the resu l t would b e Trn(t) = 1 f o r a l l t. It i s c lea r that unless A' is chosen ridiculously l a r g e y,,(O) i s about a n o r d e r of magnitude too small . The methods and approximations of Ref. 4 a indicate a deficiency of approximately a fac tor of 1/30 f o r this process . F o r definiteness we will work with a dipole fo rm fac tor and A2 = 1.0 GeV2 in o u r subsequent work. Doubling the value of A2 would roughly double the resul ts .

111. p PHOTOPRODUCTION

In th i s section we apply the multiperipheral model , a s developed in the previous section, to make detailed calculations f o r photoproduction of mesons. Extensive experimental data f o r the reaction exists.16 In part icular we will calculate the t dependence and the helicity dependence of the ypP residue functions a s well a s d i scuss the mech- a n i s m which causes the m a s s skewing in p photoproduction. These calculations extend the pioneer- ing work of Berman and Drell' who abstracted f r o m the multiperipheral model the key prediction that vec tor mesons should be produced diffractively but otherwise used it only f o r order-of-magnitude calculations.

In o r d e r to calculate helicity amplitudes we need explicit helicity wave functions f o r the incoming y and the final Using the (V', V x , Vy, V - ) notation of Eq. (2,2) and the s a m e choice of coordinate sys- t e m we have

Note that in writing Eqs. (3.1) and (3.2) we have

kept only f i r s t -o rder t e r m s in the scat ter ing angle

which is proper i n the W-w, fixed-t l imit in which we work. Since our calculations have nothing to do with the nucleon vertex i t is not necessary to specify nucleon sp inors o r nucleon helicity indicies. In effect the nucleons may be t reated a s spinless.

We denote the corresponding hel<city amplitudes F ,p ,S(~ , t ) by

F1,l = F - l , - l = F l ,

FO,, = -F 0,-1 = F z , (3.3)

F-l , l =Fl,y-l =F3 . The normalization is specified by

In addition, helicity mat r ix elements a r e defined a s i n Ref. 16, F o r our case of pure natural-pari ty exchange one h a s

P ; O = IFJ2 /D, (3.5a)

Re(&) =Re[$@: - F,*)F,]/D, (3.5b)

p:,., = R~[F:F,I/D , ( 3 . 5 ~ )

Im(pfo) = Re[$(A: +A,*)A2]/D, (3,6a)

and

1mCo,2,,,) 1.4, l 2 - I A ~ 12]/0, (3.6b)

where

D = IF,^^+ l ~ ( ~ 1 ~ + IF^]^.

The quantities (3.5a)- ( 3 . 5 ~ ) can be measured with unpolarized photon beams. The quantities (3.6a) and (3.6b) require plane-polarized photons.

A. o contribution

We begin with the c a s e considered by Berman and Drel l , R = w(783), Fig. 6(a). The ylrw and pnw

( a ) ( b )

FIG. 6. The two graphs for p photoproduction which we consider in this paper.

18 - M U L T I Q E R I P H E R A L A P P R O A C H T O P H O T O N - M E S O N - P O M E R O N . . . 141

vertices and couplings a r e specified in Appendix A. Note that they automatically satisfy gauge invariance. The corresponding ywP residue functions a r e given by Eq. (2.18) after the replacement

- t [ ( G ~ v / c )']

1 I I I l l I I T 1

FIG. 7. Reduced residue functions plotted vs t for p photoproduction. Superscripts refer to the particle com- posing the topmost rung of the multiperipheral ladder: (a) A,=+l-A,=+l; (b) & , = l - A , = O ; and (c) & , = l - A , =- 1. Also shown in Fig. 7 (a) i s the point estimated in Ref. 1.


Substituting helicity wave functions, Eqs. (3-1) and (3.2), into Eq. (3-7) we obtain the expressions f o r M,,,(x,, h,, = 1), h, =+ 1, 0 which a r e recorded in appendix B. Substituting these in turn into Eq. (2.18) and performing a t r iple numerical integration we find the reduced residue functions yIW)(t). T h e resu l t s f o r a dipole f o r m factor , cutoff A2= 1.0 (GeV/c)', a r e shorn i n Figs. 7(a)-7(c). Using these one can construct the full amplitudes f rom Eqs. (2.16) and (2-17).

It i s interesting to compare with the est imate of B e r m a n and Drell. They worked only a t t = 0 which means only M,, (1,l) is nonzero, They assumed the integration in Eq. (2.18) was dominated by the region q,2 =mn2 and x = 1; effectively,

where the las t s tep is obtained by using Eq. (2-7) and neglecting the pion m a s s squared, Hence the i r approximations lead to

- 0) - w '1' 4 n 167 ' (3.9)

This is a l so shown i n Fig. 7(a).

B. Pion contribution

A second and important contribution to p photoproduction i s the choice R = f , Fig. 6(b). This w a s not calculated by Berman and Dre l l i n the i r or iginal work. It tu rns out to be much more complicated than the w intermediate s ta te but much r i c h e r in i t s physical content. Numerically i t i s about the s a m e size.

T o calculate the n contribution to the ypP coupling we make the replacement in Eq. (2.18),

where the overal l f ac tor of [2] counts both R = 7' intermediate s tates . If we make the replacement ~ ( k , , h) - k, in Eq. (3.10) the resul t i s nonzero except a t the mass -she l l point q12 = p2. Similarly, the p meson fac- t o r i s not gauge invariant. Th is is a problem which frequently occurs when one-pion-exchange approximations a r e made in photoproduction. We cure i t in a s tandard way, a s follows:

Replace the photon coupling in Eq. (3.10) by

where V i s some four vector. The right-hand s ide of (3.11) is gauge invariant f o r any choice of V , In our work the choice V = p , is part icular ly at t ract ive f o r two reasons. F i r s t we note that the fac tor of k L 0 V i n the donominator introduces a spurious singularity into the amplitude a t the point k, V = 0. The choice V=p,

' h a s the vir tue that k, .pl - s /2 in the l imit of interest , Hence the spurious singularity is a t a point f a r f r o m the region in which we a r e applying the model and should not be ser ious. Second we note the additional t e r m on the right-hand s ide of Eq. (3.11) i s negligible compared to the f i r s t f o r h , =i 1. It only contributes t o the helicity s tate X, =O (which is not p resen t f o r r e a l photons but possible f o r vir tual photons).

We make a s imi la r change in the p coupling except we choose V=p2. Thus the final, gauge-invariant f o r m we use in place of Eq. (3-10) is

Evaluating Eq. (3.12) f o r explicit helicity s ta tes gives the expressions recorded in Appendix B. We then use Eq. (2.18) to calculate the pion contribution to the ypP residue functions.

When car ry ing out the integrations we encounter the phenomenon discussed in Sec. 11. Namely, because the 'decay p - nn is allowed physically, the pole a t qZ2 = mT2 occurs in the integration region,

and the ypP residue function picks up an imaginary part . (Equivalently, the yN - pN amplitude picks up a r e a l par t which does not vanish a s s -m.)

T h i s was f i r s t pointed out by Bauer7 who used a different fo rmal i sm but an equivalent physical approach. Note that this anomalous behavior does not occur f o r the w intermediate s ta te s ince p - nw is a closed channel.

M U L T I P E R I P H E R A L A P P R O A C H T O P H O T O N - M E S O N - P O M E R O N.... 143

FIG. 8. Various contributions to two-pion photoproduction. Particle lines marked with a double bar are on shell. The contribution from (a), the anamalous or refractive term, is automatically included in our calculation of Fig. 6 (b). The contribution (b) is the no-scattering term and combines naturally with the result from (a). The contribution (c) is the so-called proper p-photoproduction amplitude. In our model this consists of Fig. 6(a) and the parts of Fig. 6(b) in which the p is connected by at least one off-shell pion.

This same anomalous t e rm i s intimately related to another well-known phenomenon of p photoproduction, the mass skewing of the p resonance peak. The &function t e rm associated with the propagator qzz - mr2 corresponds physically to the exchanged pion propagating on shell. The situation i s illus- t rated in Fig. 8(a) where the double ba r s denote on- shell particles. We have also indicated the final decay of the p meson. The upper-right vertex in Fig. 8(a) where the two pions combine to forrn a p which subsequently decays back into two pions i s merely a representation of n- n final-state scattering. We recognize one t e rm which i s not yet included, the no scattering case, Fig. 803).

The contributions from Fig. 8(a) and 8(b) combine to a simple result. Namely, let Mb(l1, 1,) be the invariant amplitude (we suppress al l but the pion momenta) for Fig. 8(b) and Mbl the piece which i s a p wave in the two pions. The contribution M,(l1, I,) from Fig. 8(a) is , of course, pure p wave. Combining, we have

where 6 i s the p-wave n-n scattering phase shift.

Equation (3,13) was f i r s t obtained by Pumplin6 who argued from the physical point of view that the or - iginal %ding skewing mechanism,' which corresponds to using Mbl without modification, plus the direct graph Fig. 8(c), was deficient in that i t did not ca r ry the phase of n-n elastic scattering a s required by unitarity and moreover, involved some double counting. It i s pleasing to see that the multiperipheral approach we a r e using leads in a natural and automatic way to the same conclusions. F o r the details of fitting the Sijding-Pumplin-Bauer model to actual data we refer the reader to Refs. 8 and 16.

We finally remark that the so-called "proper" o r "direct" amplitude, usually represented a s in Fig. 8(c), i s to be identified a s the sum of al l contributions to yN-pON except those where the p line i s connected to the nucleons via two on-the-mass- shell pions. This i s also the philosophy of B a ~ e r . ~ Note, therefore, that contributions from our calculation of Fig. 6(b) where the p0 i s coupled via one (or two) off-shell pions a r e to be identified a s part of the "proper" amplitude. Actually in practice this lat ter te rm should probably be split off and treated separately since i t can be rapidly varying a s a function of m,,. We have calculated i t only at the point m,, = m,.

Figures 7(a)-7(c) show the contribution of the pion intermediate state to the various reduced residues, including the "anomalous " imaginary par t corresponding to the physical n-n state. Also shown i s the sum of the w and pion intermediate- s tate contributions.

To illustrate further the special features which a r e associated with the open channel kinematics p -- nn we show in Figs. 7(a)-7(c) the residue functions (labeled T*) calculated for the diagram in Fig. 6(b) where the mass of'the pseudoscalar particle in the top rung i s increased to m,. The exchanged lines remain pions with no mass change and the couplings a r e unchanged from the physical pion calculation.

C. Combined results and discussion

In this part we consider in detail the results from the previous subsections and make compari- sons to experimental data. F i r s t consider the differential c ross section.

At t = 0 the combined result of the n and w intermediate-state contributions for the s -m asymptot- i c c ross section for yN-pON with the coupling constants of Appendix B i s

dB1 = 0.77 Lb/(GeV/c)2 (3.14a) dt ,o,e,

and


(0.60 -' m,, 0.88 GeV) . (3.14b)

As explained above, the "proper" c r o s s sect ion i s calculated f r o m ReFn and FW whereas the coherent "background" c r o s s sect ion [Fig. 8(a) and 8(b)] i s calculated f r o m ImTn. These values a r e to be compared to the experimental value a t 9.3 GeV/c (phenomenological Sijding nzethod of Ref. 16 ad- justed to the value I?, = 130 MeV used in this pap- e r ) ,

The experimental c r o s s sect ion decreases slightly with energy because of contributions f r o m non- leading t ra jec tor ies , but any reasonabie extrapol- ation to high energies would give a p roper forward c r o s s section f o r the Pomeron contributions alone of (751 5) CLb/(GeV/c)z. Thus we s e e that our model calculation f o r this piece of the amplitude i s s m a l l by an o r d e r of magnitude.

The problem with the proper c r o s s section i s the s a m e one we encountered i n the model calculation we made in Sec. I1 f o r nN- nN scattering. There we found that the model, with reasonable choices f o r the form factors , gave resu l t s f o r da/dt which w e r e smal l compared to experiment by a factor -100 (dipole f o r m factor) , -1O(monopole fo rm factor). (Recall do/dt i s proportional to the square of the quantity plotted in Fig. 5.)

In the original Berman-Drel l calculation this shortfal l of the model was not a problem since they worked exclusively with ratios. In effect the i r calculation corresponds to taking the ra t io of our re - s u l t s i n th i s section f o r yN - pON (w only) to our resu l t s in Sec. I1 f o r nN - nN. This rat io com- p a r e s favorably with the corresponding experimental ratio.

One can ask a t this point about the mechanisms which could br ing the magnitude of the amplitude t o the observed value, Among other mechanisms one could ei ther (a) choose the cutoff m a s s in the f o r m factors much l a r g e r than A Z = 1 (GeV/c)' o r (b) include other intermediate s ta tes on the top rung of the exchanged ladder (analogs to w and n) a s well a s along the s ides of the ladder (analogs of the n). Method (a) would sca le up a l l the r e a l res - idues in Fig. 7. The anomalous imaginary par t s of the residue functions would a l s o sca le up but a t a somewhat s lower ra te s ince i t i s only F,(qI2) which en te rs [F,(qZa = m r 2 ) = 11. Similarly, method (b) would increase the r e a l p a r t s of the residue functions and again have a s m a l l e r effect on the imaginary p a r t s s ince p - 2n is the only decay channel physically open to the p, and hence only

the subset consisting of subsititutions on the q , l ine alone will make fur ther contributions to the anomalous piece.

T r i a l calculations indicate that i f we increase the form-factor cutoff A [method (a)] the ypP residue functions y , ( t ) a l l sca le up a t about the s a m e rate. Using dipole form factors i t would take, however, a value of Az i n excess of 5 (GeV/cI2 to generate the full experimental c r o s s section i n this manner (cf. Fig. 5). Such a high value f o r A2 s e e m s unlikely.

Since other intermediate s ta tes a r e cer tainly possible one expects some relief a t l eas t f rom method (b). We approach this question by imag- ining a spectrum of higher m a s s vector s t a t e s i n place of the w in Fig. 6(a). As Appendix B shows, these s tates , like the w, have a n-p vertex which requi res a sca le fac tor with the dimensions of in- v e r s e mass . The contributions of the intermediate s t a t e s depend on how we t rea t this scale m a s s m,. The most reasonable option, i t s e e m s t o us , would b e to take m, = kR, the m a s s of the vector intermediate state, while a t the s a m e t ime taking the overal l dimensionless coupling constant to b e the s a m e o r d e r of magnitude a s g,,,. The curve (a ) in Fig. 9 shows that in this scheme du/dt drops very rapidly with increasing pR. If on the other hand we take a constant sca le factor m, = m, instead and keep the dimensionless coupling the s a m e a s g,,, we get the curve (b) shown i n Fig. 9. In this l a t t e r

FIG. 9. The contribution of higher-mass intermediate vector states (analogous to the w) in p photoproduction. Curve (a) shows the output when the scale mass in the coupling is proportional to the intermediate state mass. Curve (b) shows the output for a constant scale mass.

18 - M U L T I P E R I P H E R A L A P P R O A C H T O P H O T O N - M E S O N - P O M E R O N ... 145

scheme do/dt does not d rop a s rapidly with pR; nevertheless i t s t i l l s e e m s improbable that there a r e enough natural-pari ty intermediate s t a t e s to build up the fac tor of 10 required to explain the experimental amplitude.

Similarly, consider heavier pseudoscalar intermediate s t a t e s in place of the n (top rung only) i n Fig. 6(b). We r e f e r to these a s n*. Here the coupling constants a r e naturally dimensionless s o the question of sca le factor does not a r i s e ; we keep the couplings fixed. As Fig. 7(a) shows, f o r the c a s e m,a =m, the contribution a t t = 0 of such a s ta te i s only about half of the n contribution itself. [ ~ o t e that n* s t a t e s do not contribute any anomalous (imaginary) par t to 7 since the decay p - nn* is ener- getically forbidden.] Higher values of m,* would give s t i l l s m a l l e r contributions. Again it s e e m s unlikely that a sufficient spectrum of unnatural- par i ty s ta tes ex i s t s to build up the required factor of 10 in the amplitude.

Finally consider the exchange (s ides of the lad- d e r ) of heavier m a s s pseudoscalar s ta tes . As the calculation of K exchange in Sec. IV shows, these contributions a r e s imi la r ly small. It is our belief, therefore, that the missing contributions to the amplitude come largely f rom replacing the pions b y o ther meson (Regge) exchanges (p, w, etc.) on the s ides of the ladder. F o r a n analysis of the experimental evidence f rom multiparticle production which indicates direct ly that o ther exchanges in addition to pion exchange a r e required, s e e Ref. 13.

As remarked above, the model predictions f o r the yN - pON and nN - nN amplitudes a r e equally low in comparison to experiment when just pion exchanges a r e included on the s ides of the diagram in Fig. 4. We find roughly the s a m e situation in Sec. IV f o r w photoproduction. One can conjecture that this is not a n accident but r a t h e r has s o m e deeper significance. This would imply that the our model could be t rusted f o r amplitude rat ios .

If one adopts this point of view, consistency with the above opinion that the missing contributions to the amplitude come f r o m Regge exchanges in place of pion exchanges i n Fig. 4 implies that there is a n approximate equality i n the ra t io of pion and individual Reggeon couplings to yw and np, Namely,

where the subscript a, denotes a Reggeon coupling.'5a Similarly, our picture that s-channel helicity conservation comes about f r o m cancellations between natural-pari ty (represented by w) and unnatural-pari ty (represented by n) intermediate s t a t e s is equivalent to the s tatement

The question becomes, therefore, whether re la -

tions such a s (3.16a) and (3.16b) a r e a t a l l credi- ble. Recalling the many approximate pat terns which have been observed among Regge couplings (exchange degeneracy, f coupled Pomeron, etc. ), one cannot help but speculate that fu r ther rela- tions, such a s the above, remain to b e discovered. Presumably the ultimate dynamical b a s i s f o r such relat ions res ides in the fact that the 9, p, o, f , etc., a r e a l l built f r o m a common quark-anti-quark pair. (We have no idea how to formulate a n actual proof.) The second of the relat ions given above, Eq. (3.16 b) , s e e m s part icular ly at t ract ive if one accepts the conventional wisdom that the Reggeized multiper ipheral model is a good f i r s t approximation to multiparticle production a t high energy and the ex- per imenta l fact of s-channel helicity conservation f o r diagonal and diagonal-like diffractive processes . Namely, a s observed h e r e and in Ref. 4(a), individual intermediate-s tate contributions do not conserve s-channel helicity, hence some kind of cancellation must in fact b e the case. The picture that cancellation in helicity-flip amplitudes comes about t rajectory by t ra jec tory a s s tated by Eq. (3.16b) s e e m s m o r e natural than the only al terna- tive which would have the sum of a l l natural-pari ty contributions to a given helicity-flip amplitude for - tuitously cancel against the s u m of a l l unnatural- par i ty contributions. Again, however, we can offer no actual proof a t this time.

Next l e t u s consider the model predictions f o r the density matr ices . The existing experimental resu l t s (Ref. 16) f o r the p photoproduction density mat r ices include (unavoidably) both the "proper" and "coherent background" contributions. One mus t calculate the corresponding theoret ical com- bination.

Let F ( s , t ) be the proper amplitude and iR(s , t ) be the anomalous contribution calculated above a t m,, = m,. The qunatities 7; and Rey'; determine F; Im% de te rmines R. The discussion presented a- bove showed that the full Feynman amplitude f o r yN - nnlV is

~ ( ~ , t , m , , , e,, $,)=x s X ( s , t , mr,)Mx(er, +,)

where IM'(B,, $,) is the invariant mat r ix element describing the decay p - ~ ' (1 , ) + n-(l,), h is the heli- c i ty of the p, and O,, I$, specify the direct ion of n* i n the p r e s t f rame. ( F o r convenience we suppress the helicity label f o r the incoming photon.)

The full differential c r o s s sect ion is accordingly

146 P A U L M . F I S H B A N E A N D J E R E M I A H D . S U L L I V A N

where c$, is the two-body phase space f o r p decay. Note

and / d m 2 [MA 1 2 = 2 m P r p . (3.19) 2 2

ei8cosg= mrr - m ~ mr," mp2 + imprp (3.21)

We use Breit-Wigner f o r m s f o r the phase shifts; namely, with a constant width, r, = 130 MeV.

FIG. 10. Density-matrix elements as defined by Eqs. (3.5) and (3.6) in the text. The curves marked w (rr) are the predictions if only the w (rr) intermediate state is included-see Fig. 6 . The curves marked w + a are the predictions for the coherent sum. Note the cancellations between terms coming from a and w intermediate states. The experimental data are from Ref. 16.


When one squares the amplitude 5 to calculate cos26 the c ro s s section and the density-matrix elements,

(3.23)

one encounters sin26, cos26, and sin6 cos6 terms. The results reported in Ref. 16 correspond to the sin6 cos6

= 0.06. m a s s range O.GO~m, ,~ 0.88 GeV. F o r this range

(3.24)

Thus the experiment density-matrix elements (3'22) should be compared to simple generalizations of

FIG. 11. Density-matrix elements as in Fig. 1 0 except that a heavy-mass unnatural-parity a* contribution has been included. The theoretical predictions are those of the model after the o and n* amplitudes have been multiplied by 1 0 and the n amplitudes by 1 for the reasons discussed in the text.

148 P A U L M . F I S H R A N E A N D J E R E M I A H D . S U L L I V A N

Eqs. (3.5)-(3.6),

and s imi la r ly fo r the other elements. The model predictions a r e shown in Fig. 10 along

with the 9.3 GeV/c data f rom Ref. 16. The s tr iking fea ture i s the substantial cancellations between the natural-pari ty (w) and unnatural-parity (8) contributions in the denisty-matrix elements which correspond to k,+ k,.

Since the proper amplitude generated by our mo- de l i s roughly an order of magnitude s m a l l e r than the experimentally observed proper amplitude it is difficult to draw meaningful conclusions f rom Fig. 10. (The coherent background is about the c o r r e c t s ize , however.) As mentioned e a r l i e r i t i s quite possible that when additional contributions to the p roper amplitude a r e found the anomalous par t s will not be changed much f rom the values shown in Fig. 7. To i l lustrate the effect on the density-ma- t r ix elements we have recalculated them af te r mul- tiplying the reduced residue functions 77 and ?;* i n Figs. 7(a)-7(c) by 10 and the pion contributions, Re?; and Im?; by unity. The resu l t s a r e shown in Fig. 11. Again one s e e s a s t rong pat tern of cancellation between natural-parity and unnatural-parity contributions in helicity-nonconserving te rms . (This feature i s a l so observed in Ref. 4 f o r the nucleon-nucleon- Pomeron vertex.) One notes that the elements involving z e r o helicity have the wrong sign compared to experiment. This could easi ly b e changed by a slight var iat ion in the enhancement fac tors f o r natural- and unnatural-parity intermediate states. However, our purpose h e r e i s only to emphasize the smal lness of these elements. We believe th i s cancellation i s general property and offers an explanation of the approximate heli- c i ty conservation observed in Pomeron dominated two-body processes .

This pattern is , of course, a l ready evident in Figs. 7(b) and 7(c). Note that the n* and w helicity flip contributions have everywhere the opposite signs. It is interesting to note, however, that f o r It I s 0.5 GeV2 this is not the case f o r the s and w contributions. This situation ia another conse- quence of the open-channel kinematics of p-nn . F o r It 1 5 0.5 GeV2, the dominate contribution t o ReF;,, comes f rom integration regions in which the propagator qZ2 - mr2 takes on positive (timelike) values. F o r l a r g e r t, negative (spacelike) values dominate. Note that It 1 "0.5 GeV2 is a l so the point a t which I m z , , cease to grow. When the top rung is n* in place of n, 9,'-m,2 is always negative.

FIG. 12. The only low-mass intermediate-state contribution to w photoproduction.

IV. w PHOTOPRODUCTION

Figure 12 shows the diagram which i s analogous to tine w intermediate s tate in photoproduction. It has a n isospin weight of t h r e e compared to the corresponding amplitude for p0 photoproduction. Otherwise the calculation i s precisely that of p0 photoproduction with a change of coupling constant. Namely,

Talring the yN;;pON amplitudes froill Fig. 7 we find a t t = 0 and w c r o s s section by this channel, do/dt = 0.28 pb/(GeV/c)', compared to the high-energy experimental value16 of -12 pb/(GeV/c)', typical disagreement fo r the model,

The intermediate s ta te analogous to the n in p0 photoproduction would be a n isospin-1, G-parity- even, pseudoscalar meson. A low-lying s ta te with these quantum numbers does not exist, hence the density-matrix elements of w photoproduction by the above channels will be those corresponding to the w intermediate s ta te in Fig. 10. Presumably higher-mass contributions would r e d r e s s the bal- ance and reestabl ish approximate helicity conservation.

We can a l so t e s t the effect of higher-mass intermediate s ta tes a s well a s s ta tes on the s i d e s of the ladder in this process . F igure 13(a) and 13(b) show

FIG. 13. Some higher-mass contributions to w photoproduction.

18 .- l l 1 U L T I P E R I P N E R A L A P P R O A C H T O P H O T O N - M E S O N - P O M E R O N . . .

two more contributions to w photoproduction, analogous to Fig. 6(b) and 6(a), respectively. Using SU(3) values f o r g,,, ( see Appendix A) we find, f o r example, that if we take the cutoff parameter a s 1 GeV2 in a dipole fo rm factor , d r~ /d t (yNF wN) -0.001 p b / ( G e V / ~ ) ~ . The s h a r p dropoff with mass - e s of intermediate part ic les i s typical. Computa.- tion of yN;*wN gives a s imi la r , v e r y s m a l l resul t . A m o r e proper calculation would t r e a t the s ides of the ladder a s Regge exchanges. Those with inter- cep ts g r e a t e r than z e r o a r e undoubtedly the important ones.

V. @I AND I) PHOTOPRODUCTION

The case of @ photoproduction rnay present an especial ly interest ing t e s t because the reaction i s thought t o proceed purely by Pomeron exchange; Fig. 14 shows th ree contributions to Q photoproduction which a r e sizable.

We study only the s ize of the forward c r o s s section here , and give the value of this quantity cor - responding to the th ree graphs separately, r a t h e r than the coherent sum. We use values of the coupling constants given in Appendix A. Corresponding to graph (a) we find dcr/dt = 2 ~ l O - ~ p b / ( G e V / c ) ~ (a low value which resu l t s f rom the very smal l 4 - np branching ratio); to graph (b) do/dt = 0.03 p b / ( ~ e ~ / c)'; and to graph (c ) do/dt = 4 ~ l O " ' p b / ( G e V / c ) ~ (in this c a s e the low value i s associated with the la rge K* mass) . The major contribution thus comes f rom Fig. 14(b); i t i s the typical value of -100 s m a l l compared to the experimental value f o r Q photoprod~ct ion '~ of da/dt - 2 pb/(GeV/c)'.

F igure 15 shows a s imi la r s e t of graphs f o r the photoproduction of the 4 meson. Because of the l a r g e D, D*, F , and F* masses , both the contributions f rom Figs 15(a) and (c) a r e s t rongly suppressed. Using a form factor cutoff A2 = 1 (GeV/ c2)' a s in a l l our previous work, du/dt f o r Figs.

FIG. 14. Contributions to d, photoproduction.

FIG. 15. Contributions.to + photoproduction.

15(a ) and 15(c) a r e each "lo-' yb/(GeV/c)'. One might argue that f o r such s t a t e s we should u s e a l a r g e r cutoff value. However, increasing this value to A2 = 4 ( G ~ v / c ~ ) ~ increases the c r o s s sect ion only by a factor of -6 x10+3, ent i rely inadequate. Figure 15(b) i s not dynamically suppressed a s s t rongly and i s dominant. However, the s m a l l + - 2n branching ra t io has i t s effect, s o that the contribution of Fig. 15(b) i s do/dt " 2.5 x 1W4 pb/ (GeV)'. One can compare this to the experimental value17 of 50X10-3 ~ b / ( G e v ) ' ; again the model re - su l t i s a fac tor of "200 small.

VI. SUMMARY AND CONCLUSIONS

We have shown that a s imple mult iper ipheral model of diffractive p photoproduction with pion exchange contains many of the fea tures observed experimentally: approximate helicity conservation, skewing of the p-meson m a s s distribution (SEding- Pumplin-Bauer m e ~ h a n s i m ) ~ ~ ~ and a n anomalous r e a l par t of the yN -pN diffractive amplitude (Bauer effect),7 However, a s in the original work of Dre l l and Berman' the absolute normalization of the amplitude is low compared to experiment by a factor -6-10. If one wants to re ta in the concept of mult iper ipheral ism a t a l l , the missing contributions presumably must come f r o m multiperipheral-type d iagrams where the exchanged pions a r e supplemented by the exchange of Regge t ra jec - t o r i e s corresponding to the vector and tensor mesons.

Generalization of the model to w , @, and photoproduction show a s i m i l a r shortfal l of the theoret- i ca l predictions compared to experiment. F u r t h e r calculations using the complete s e t of posslble Regge exchanges i n addition to the pion exchanges considered in this work would be v e r y interesting.

150 P A U L M . F I S H B A N E A N D J E R E M I A H D . S U L L I V A N

ACKNOWLEDGMENT c This work was supported in par t by the National

Science Foundation under Grants NSF PHYS77- 00151 and NSF PHYS75-21590.

APPENDIX A

In this Appendix we compile fo r convenience va- r ious coupling constants used in this work. A larg- e r review is provided by Ref. 18. In some c a s e s the couplings a r e direct ly measured (in magnitude), while in o thers they mus t be deduced f rom other data and theoretical input. Those couplings which can be obtained f rom decay widths a r e given in Table I without comment. We d i scuss the others below. In each c a s e the choice of momentum is that indicated in Fig. 16.

The coupling g,,, can be obtained f rom the measured value for g,,, by a s imple vector-dominance argumentLg

where we have used fQ2/4a = 2.56 f rom e+e'- p0 data and gYTw2/4a = 0.029 f r o m the par t i a l width for the decay w-aO+y.

Recently the p a r t i a l width f o r p' - a- +y has been measuredz0 by means of the Primakoff effect. The resu l t s of this experiment give gmY2/4n = 0.0013

FIG. 16. Momentum and particle labelings for the calculation of coupling constants.

which, when combined with vector dominance and the value fw2/4a = 18.4 f r o m e'e+ - w , implies

This i s about a factor of three s m a l l e r than the above est imate, ( ~ 1 ) . It is not possible to say a t this t ime whether this indicates a substant ialbreak- down of vector dominance o r a n experimentalprob- lem.

The aO lifetime provides a useful consistency check. If vector dominance i s a ~ s u m e d , ' ~

TABLE I . Definition of the vertices and numerical values for the couplings used in this work.

ac2 a b c Coupling ra-bc (MeV) 471

no,r ~ ~ l l V y 6 ~ P ( ~ ) ~ u ~ ~ ( k 2 ) k $ Eq. (A11 10.2 mu Eq. (A21 3.3

Eq. (A31 13.3

@ K+ K - ~ @ K K ~ ~ ( P ) ( ~ I - k 2 ) P 1 . 9 + 0 .1 1.5

# nt nr g # r ~ ~ ~ ( P ) ( k l - k2)11 2 0.08

a Not used in calculations. Instead, the vector-dominance value 4.0 x 1 0 - ~ i s adopted. See text for discussion.

Used in calculations.


Another estimate has been given by Renard21 gd.; (g*) =gd =4.0 x f rom a fit to data for the process e*em- w0 + s o . He 41r 4lr obtains

However, his analysis needs modification if the p(1250) resonance suggested in Ref. 22 i s confirmed.

As a working value we adopt in this paper the value of g,, given by Eq. ( ~ 1 ) and listed in Table I. Moreover, we adopt the phase implied by vector-meson-dominance, namely, g,&$-,,, i s positive. This a s su re s that the amplitude for yp-pop has a positive imaginary par t a t high energies and t=O.

Another group of coupling constants involves kaons a s intermediate states. From Su(3) symmetry, we have

The experimental value of g,KoK*02 i s roughly 4 the ~ ~ ( 3 1 value given above; the former is the value used in the text.

We also require the couplings of the $/J particle to the charmed mesons. S U ( ~ ) symmetry gives

Here F, D a r e the 0- and F*,D* a r e the 1- charmed mesons defined in Ref. 10.

APPENDIX B

Denote by Mp,,(Xp, x , ) the numerator factors fo r p photoproduction with an w(783) a s the top rung of the multiperipheral ladder. Direct evaluation of Eq. (3.7) gives

- : a ' ( q x ~ + ~ z m ) [( mp2 +a t , z - - (qx)2 - (qy)2+qX(- z ( ~ 2 ) and

( - + ( a 2 - a 3 ) q x ( - z t ) 1 1 2 +$(a2+a3)z t -$a& -$a,{[(qx)2- (qY)2]z ++z2 t } ) , (B3)

where z = l -x and from Eq. (3.12) a r e

4a4 = - 4mw2, (B7) and

4a5= - 2(mp2 - t ) . (B8) M:A(-I, 1 ) =[21eg,r(- 2z[(qx)2 - ( q y ) ~ ] -

The numerator factors for p photoproduction with a pion a s the top rung of the ladder a s calculated ( ~ 1 1 )

152 P A U L M . F I S H B A N E A N D J E R E M I A H D . S U L L I V A N 18 -

's. Berman and S. Drell, Phys. Rev. 133, B791 (1964). 2 ~ o r other recent attempts to calculate Pomeron resi-

dues in the spiri t of multiperipheralism see S. T . Jones, Phys. Rev. D g , 1496 (1976); and J. W. Dash and S. T. Jones, ibid. 2, 1817 (1975). These authors were particularly interested in the t dependence of Regge residues. See also Ref. 4a.

3 ~ . Chew, T. Rogers, and D. Snider, Phys. Rev. D 2, 765 (1970); M. L. Goldberger, in Developments in High Energy Physics, Proceedings of the International School of Physics "Enrico Fermi", Course LIV, 1971, edited by R. Gatto (Academic, New York, 1972), p. 1; D. Tow, Phys. Rev. D 2, 154 (1970).

4 ~ . Wingate, Phys. Rev. D 2, 2565 (1977). 4 a ~ . Pumplin and G. L. Kane, Phys. Rev. D 11, 1183

(1975). 5 ~ . SFding, Phys. Rev. Lett. 19, 702 (1966). See also

A. Krass, Phys. Rev. 159, 1496 (1967). 6 ~ . Pumplin, Phys. Rev. D 2, 1859 (1970). 7 ~ . Bauer, Phys. Rev. D 3, 2671 (1971); Phys. Rev. Lett.

25, 485 (1970); 25, 704(E) (1970). 8 ~ T ~ p i t a l and D. R. Yennie, Phys. Rev. D 9, 126 (1974). 'D. R. Yennie, Rev. Mod. Phys. 47, 311 (1975). 'Osee, e.g., M. K. Gaillard, B. W. Lee, and J. L. Rosner,

Rev. Mod. Phys. 47, 277 (1975). Indeed the D and D* have recently been discovered: G. Goldhaber, et a l . , Phys. Rev. Lett. 37, 255 (1976).

I'D. Amati, A. Stanghellini, and S. Fubini, Nuovo Cimen- t o 2, 896 (1962).

1 2 ~ . Bertocchi, S. Fubini, and M. Tonin, Nuovo Cimento

25, 626 (1962). 1 3 F Piri lz, G. Thomas, and C. Quigg, Phys. Rev. D 12,

92 (1975); S. Pinsky and G. Thomas, ibid. 2, 1350 (1974).

I 4 ~ h e s e variables a r e defined byp,=po i p s ; the transverse components a r e untransformed. We write p , = (p+, Jr, p-). These variables are closely related to the Sudalcov variables. See, e.g., S. J. Chang and S.-K. Ma, Phys. Rev. 180, 1506 (1969).

l5H. P. Diirr and H. Pilkuhn, Nuovo Cimento 40, 899 (1965).

15a~quations (3.16a) and (3.16b) a r e t o be understood in a symbolic sense. Namely the coupling of a Reggeon to particles with spin will in general involve more than one coupling parameter as well as the usual nonsense factors, etc.

Ballam et al . , Phys. Rev. D 5 , 545 (1972); ibid., 7, 3150 (1973).

I7B. linapp et al . , Phys. Rev. Lett. 3, 1040 (1975); W. Y. Lee, in Proceedings of the 1975 International Symposium on Lepton and Photon Interactions a t High Energies, Stanford, Cal$omia, edited by W. T. Kirk (Stanford Linear Accelerator Center, Stanford, 1976), p. 225.

"M. Nagels et al . , Nucl. Phys. B s , 1 (1976). '$M. Gell-Mann, D. H. Sharp, and W. Wagner, Phys. Rev.

Lett. 8, 261 (1962). 'OB. Gobbi et a l . , Phys. Rev. Lett. 3, 1450 (1974). "F. Renard, Nuovo Cimento 6 2 , 979 (1969). "M. Conversi et al., Phys. Lett. 5 3 , 493 (1974).

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Multiperipheral approach to photon-meson-Pomeron couplings