REFERENCEIC/72/32

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

PROPERTIES OF REAL PHOTON

AKD PHOTO-INDUCED PSEUDOSCALAR MESON

INCLUSIVE REACTION

N. Murai

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL.

SCIENTIFICAND CULTURALORGANIZATION 1972 MIRAMARE-TRIESTE

IC/72/32

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

PROPERTIES OF REAL PHOTON

AND PHOTO-INDUCED PSEUDOSCALAR MESON INCLUSIVE REACTION *

N. Murai

International Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE - TRIESTE

May 1972

To "be submitted for publication.

41I

ABSTRACT

The inclusive reaction, y + p •* c + (anything) (c = ir~ , K~) is dis-

cussed in association with the properties of the real photon. Assuming the

fixed-pole behaviour of the partial amplitude for y + (a vector meson) -*

(a meson) + c at high energies in the framework of a multiperipheral model,

the contribution to the structure function, integrated over the transverse

momentum of c , is shown to "be appreciably large, in addition to the one

coming from the diagram which has "been expected to contribute mainly in the

framework of this model. From the viewpoint of experiments we expect from

our simple calculations that the structure function at the point of the small

longitudinal momentum of c in the centre-of-mass system decreases to a con-

stant like s » if the fixed-pole behaviour exists. In the case of TT~

the absolute value of the contribution is too small to tell us anything definixe

about the properties of the real photon because even the main term, connected

to the multiplicity, is smaller than the experimental value, as recently pre-

dicted. There is a possibility of obtaining, information on the real photon

from the photo-induced kaon inclusive reaction. This proposal Is related to

the validity of the multiperipheral model for the kaon reaction.

-1-

I. INTRODUCTION

Several years ago, Gribov, Ioffe and Fomeranchuk raised the question of

the order of longitudinal distances which are important in the processes of

scattering at high energies . Ioffe subsequently showed, from some features

of experimental results in deep inelastic e-p scattering, that the large longi-

tudinal distances play an important role in the processes with small photon2)

mass . Many predictions have been made concerning the electromagnetic inter-

actions of hadrons by using the vector meson dominance model (VMD) whichattributes the same properties to the real photon as to hadrons. However, the

3)validity of VMD has recently been questionedh)

The experimental data on multibody photoproduction and the inclusive

reaction y + p -+• (a pseudoscalar meson) + (anything) have gradually9)accumulated. These data, especially on the y fragmentation region of

the photo-induced inclusive reaction, would provide us with information on the

properties of the real photon. The results obtained from their analysis

are the following:

a) When interpreted in a Regge-pole framework , the t-dependence

of the structure function of the inclusive reaction leads to a trajec-

tory associated with the y -*• TT~ vertex,with a(o) w 0.0 and a slope

of«1 GeV"2 at E =9-3 GeV. T'

if" production cross-section is

y-2 7)

ofttl GeV at E = 9.3 GeV. The corresponding expression of the

°dk3 R

where s = , (P X + p 2 )2 , s" = ^ + p 2 - k )

2 , t = (P 1 - k)

a{t) is the exchanged Regge trajectory which is coupled to the photonTOT i

with a residue G(t) and <J_. (s ,t) is the total cross-section forn

the reggeon-proton process.

b) Risk et al. compared inclusive photoproduction data in the

energy range E = 9.0 - l8.0 GeV wift a multi-Regge model with the usual

slope a' = 1 GeV"2 of the trajectory aft) in |t|< 0.6 . They

obtained a good agreement with the experimental data.

12)

c) Satz et al. applied the dual resonance model for diffractive

dissociation to two-pion and KK photoproduction and obtained a good

agreement with the experiments.

-2-

—" Ir• t —"t-IT— -j mtUl i r i i ilhin " ....... ... „. , ... .^i*-..,, . „ : , J

The consensus of these authors is that the real photon behaves like a

hadron. However, the following remarks on these results may be noted:

a) Unfortunately, the data can be available only at E = 9-3 GeV

because of the A peak in the region of the Y fragmentation. In

determining a(t) by experimental distribution, Cs'/s) ~ is

fitted for finite t-intervals. The contribution of the residue G(t)

is not clear.

b) The Y fragmentation region near the kinematic boundary is a

little underestimated in the longitudinal momentum distributions, though

their fits are globally in good agreement with experimental data..

c) The p meson strongly dominates the final two-pion spectrum and

we cannot obtain clear information on the properties of the y-~n vertex

at high energies. Further studies" would be necessary to Qive a de-

finite answer.

This article is devoted to the photo-induced reaction Y + P "*" c + (any-

thing) , where c = 7T or K . As a test, ve shall examine the graph

shown in Fig.2 within the framework of the multiperipheral model. A fixed

pole' ot(t) = 0 is assumed to dominate the amplitude for y + V -+• d + c at

high energies *), where V is a vector meson and d is another meson produced.

This fixed-pole behaviour describes the properties of the real photon, as we

shall remark below in A) and B). In practice, we shall calculate the struc-. CM

ture function at the point of small **) longitudinal momentum k|( of the

particle c in the centre-of-mass system. In the high-energy limit the

dominant structure function integrated over the transverse momentum is a con-

stant. This ter

the multiplicity.

stant. This term has been expected to give the scaling law * as well as

The graph shown in Fig.2 gives an additive term which decreases like s

because a(0) » 0 . The fixed-pole behaviour leads to the appreciably large

coefficient of s in the structure function. We can distinguish the term

*) In the low-energy region the production of resonances would dominate the

amplitude.

**->The elastic p° (yp -«-.TrVp , M + _ < 1.0 GeV) are small at kJM =• 0 in

YP -• IT + (anything).7' It is not necessary to take account of the effect

from the reaction.

from the main constant term by the energy dependence of the structure function.

The ratio of the fixed-pole term to the main term is recognized over the usual

statistical error (about 10j0 in our simple calculations.

Before proceeding with our calculations we note the following:

A) In pseudoscalar meson photoproduction on the nucleon the differential

cross-sections can be explained effectively with a fixed pole whose intercept

is a(0) = 0 and slope a' =s 0 up to the region |tj 2, 1 (GeV/c) .*) ' Can

we find this fixed-pole behaviour only in the pseudoscalar meson photoproduction on

the nucleon? Some insight into this question can be obtained from the dual15)

hadrodynamics with the electromagnetic interactions . If charge distri-

bution is pointlike in the internal space (i.e. the "name" space of vertices

in the so-called fish-net diagrams , the amplitude of photoproduction on

the meson has Begge behaviour at high energies. If charge distribution is

uniform, a fixed-pole behaviour is shown. We thus see that the properties of

the real photon strongly depend on charge distribution in the fish-net diagrams

exemplifying hadronic matter.

B) VMD seems to be a powerful tool in analyses of forward scattering and

total cross-sections. However, it has turned out that a systematic dis-

crepancy is found in Compton scattering of a proton between the predictions of

VMD and large-angle differential cross-sections . VMD usually underestimates

the experimental data there.**)

In this article we also assume VMD for total cross-sections.***^ -

*) This problem has been subjected to much phenomenological work over the past

decade. Here we limit ourselves to showing reviews (Ref.lU).

**)Brodsky and others have suggested,from the viewpoint of the parton

model, that the discrepancy is associated only with the two-photon processes.

But in this article we consider a test on a one-photon process.

***)Brodsky and others have also suggested that in the forward Compton scat-

tering of a proton there is a discrepancy between the prediction of VMD

and experimental data and that the discrepancy should be connected with

nuclear y absorption cross-section. However, we assume it in our

approximat ion.

II. THE MODEL

First we consider the case c = TT" for simplicity of description. Tiie

results for c = .K** can be obtained by modifying the kinematics and changing

the Regge trajectories exchanged.

Within the framework of the multiperipheral model, the diagram in

Fig.3a contributes to the production cross-section in the region x s* 0 where

x - 2kn /Ss t and k(| is the longitudinal momentum of IT in the

centre-of-mass system. In fact, many successful predictions have recently

been made in the measured secondary spectra in hadronic inclusive reactions,

assuming the same type of diagram as shown, in Fig.3a in the central region#^17^ T ft)

x z 0 . 'Here we use the generalized discontinuity theorem and assume

that the contributions of "anything" can be replaced by total cross-sections.

Then the TT production cross-section is given as

, V 8M a - f r &k Sl kpkq W ^C° dk3" (27r)3s J ^ ? £tl - y

2)2(t2 - n2/

where M , y and m are the masses of the proton, the pion and the p2

meson, respectively, s = (p - q)

the coupling constant of "mrp , and

2 2 2meson, respectively, s = (p - q) , t^ = q , t ~ (k + q) , f is

Wpa = j d\e i < 1 X <P2iJP(x) Ja(0)|p2>

with J (x) the source of the p meson. We have employed approximations

of s » y and s » M , and have assumed that the p meson is coupled

to the conserved current.

Under the assumption of the small cross-section of the longitudinal p

meson off the proton, we have

pa ^2_ TOT*D CT M apP

*) To cite all references is beyond this short article. See, for example,

Ref.17.

-5-

with

TC = •• ^ <(p. 2<1 J

1 -

where B 2 = (p2 - k - q) . Following Ref.19, we make a simple reggeization

for the p meson by the replacement

K2~ctp

e x P [ A2 V

mp (2.1)

with a (t )= J + a , s » (p? - k) and where A is a positive constant.

We do not reggeize the IT meson and consider the off-mass shell correction

After some manipulations, following Refs.19 and 20, in the approximation

of high energies *), we obtain

~ - TOT TOT 2 -P*

F?(0,s) = dkfJVCM ,2,dkl( d k

Yff PP _TT (2.2)

where k^ is the transverse momentum, kfi is the energy of TT in the centre-

of-mass system, I is an integer introduced to take care of analogous dia-

grams, and

X {(.

with

and

C - 1 + — ^ - (1 - An z)1-2

.00 _t

|-dt .

•) In integration over the five-body phase space we have put s » W , y

in the centre-of-mass system.

-6-

We obtain J™ = 0.0U3 with the choice of f2/l*TT = 2 , and A. * A = 1a 1 2

from the experimental observations of the k^ distribution.

Now we are in a position to estimate the contribution of the diagram

Fig.3b. First we regard the reggeon a like the ̂ meson and obtain

ovtp, + q) - m,)JL Q

ef )d TT f d%

TT)3 S ' T^r(Sir)- (tn - , 2 ) 2 (t9 - m 2 ) 2

where mfl is the mass of the particle d , and e is the polarization four-

vector of the photon.

Averaging over the polarization of the photon, we reggeize the ir

meson by the following replacement:

ef(2q

2,- V

5lll 2 0 t ( t l ) r. ,. .2exp[A1(t1- u

2)] , (2.3)

to pwhere a(t) « a (t - u ) , i, • (p, - k) and srt is a constant used in

1 1 0the conventional Regge pole model. We have normalized the Regge-pole residue

2near t̂. = y after considering the contributions of natural parity exchange(see Appendix).

Our reggeization may be too simple to discuss.the phase of the production

amplitudes from the recent observations of the inclusive reaction for polarized7)

photons up to E = 9*3 GeV. In the Regge-pole framework, phases from the

signature terms of Regge poles would play an important role in the discus-22)

sions . However, we now concentrate only on the fixed-pole behaviour.

Therefore, we assume our simple reggeization which would give effectively the

differential cross-section of YP "*" dir" at high energies.

In order to perform our calculation analytically, we do not reggeize the

p meson and assume the following replacement:

m2

£-2-2-* B — exp[A2(t2 -m2)] . (2.U)

In this replacement our requirement is that the right-hand side of (2.M should

be almost equal to that of (2.1) at t« w 0 . Our final result for ratios

does not change if these factors are of the same order.

Thus we obtain the following contribution from Fig.3b to F (x,s),1 ~* rtassuming a = 0 :

2 2 fTT0T m2 j"

fc ' " S ^ ^ 2 - m2l -̂ , (2.5)p S

where L has the same significance as I , and

The result is not sensitive to the mass m ; therefore, our high-energy

approximation is consistent in the centre-of-mass system. A numerical cal-

culation gives J. = 0.1*0 . The factor s~ in (2,5) comes from the zero

intercept a(0) = -a y S o of the effective trajectory.

III. RESULTS

We do not include the reaction from two-body on each mass shell to

stable one-body on the mass shell in the total cross-section. We have re-

placed the contribution of (anything) by ' the total cross-section for two

particles on the mass shell. Then the contribution of the diagram in Fig.3b

leads to an additional term to that of Fig.3a,which is expected to be the

principal one at high energies. We can distinguish the two terms from their

energy dependence. First we examine their ratios,

The ratip of F (0,B) to F (0,s) follows from (2.2) and (2.5):u a

F (0 s) ll I 2 . P Jr = . — .

a a i M a,

TOTIf we assume the factorization for the residues of the pomeron, a is. , TOT TOT TOT / TOT ™ Vgiven by o = a _ a /a . Then we have

rU * 1.3

putting I • I. . The ratio r is 0.68 even at E y = 9-3 GeV.ft D m

-8-

The integrals in J^ and J' contain 2a' logCl + f a /s |) + A and

2a log( l + |S / s Q | ) + i^ tbefore putting a * 0 ) , respectively. We can see

that J is large if 2ot log(l + li^/s j) can "be a A, at high energies;

and so on. When a a 0 this situation can be realized. But it is not

the case with j" "because of a » 1 (for a S o , J^/J77 = 1.? GeV2). Thisa p t p b a

provides a mathematical explanation of the large value J,/J w 10 GeV , thatoa

is, the integral values of J and J depend sensitively on the slope of theEL D

trajectories.

As for the photo-induced kaon inclusive reaction, we consider graphs

similar to Fig.3a and Fig.3b obtained by replacing the p meson by the K*

meson. Making some numerical adjustments in the preceding calculations we

obtain

K _ 7.2i> _

s

and rK = 0.U2 at E = 9.3 GeV.

From studies of our simple model, we thus have the following conjecture.

Within the framework of the multiperipheral model, the contribution from Fig.3b

gives the appreciably large term, though decreasing like s , in addition

to that of Fig.3a expected^dominant at k(, = 0 , if the fixed pole a(t) = 0

dominates the amplitude for y + V -*• d + c . From the viewpoint of experi-CM —1

nents we expect that the structure function at k,, = 0 decreases like s

to a constant as s increases. If the fixed pole does not exist, we cannot

find such a feature. The slope of our effective trajectory a{t ) in

Y + V -*• d + c might be related to the charge distribution of hadronic matter

as remarked at A) in Sec.I.Our next step is to examine the absolute values of F(0,s) . If the

multiperipheral mechanism works well in the central region x a 0 , to re-23)

produce it the multiplicity <^n~y must be given by

with

TOTAs we have remarked already, from VMD the total cross-section a is given

TOT P P TOTby O* = (f /e ) a p . Thus we have a small multiplicity c^ = 0.019

with I = 1+ , while the experimental value is c" = O.UU ± O.k . Thea 10) TT

multiplicity predicted is very small . This difficulty of the multi-

peripheral model prevents us from making a definite statement on the proper-

ties of the real photon from available experimental data for TT~—Q-

The ratio of the production cross-section of K to fr (K/ff ratio) is

small compared with our knowledge of the total cross-sections and the low-_ _ 3 3

energy phenomena. The K /u ratios of k_ d Of/dk near the point x = 0at E = 18 GeV ; are 0.09 at k = 0.5 GeV/c , 0.07 at k = 1.0 GeV/cand 0.08 at k =1.6 GeV/c. If we put the K /IT ratio equal to 0.1 for any

2k)1value of k, * the coefficient c in the multiplicity for the kaon is

-v 0.0U . *

Unfortunately, we have less experimental information on the kaon

than on the pion. It is plausible to assume that the multiplied factor for

J in F (0,s) is of the same order as in the case of the pion, since this& £L

factor includes only the total cross-sections and the coupling constants.Then we have cv ..^ ^ 0.01 predicted in our model. Thus we plausibly assumeK.,M-rthat the multiplicity of the kaon is roughly equal to that predicted with the

multiperipheral model. In fact, we have also found that the kaon multiplicity19)is roughly consistent with the multiperipheral model in hadronic processes

As for the pion, we might have a speculative picture; the soft-pion

emission may exist only for the pion and the mechanism gives a sufficient19)

number of pions . If the multiperipheral mechanism works well in the photo-

induced kaon reaction, as argued above, we can obtain information on the real

photon from the reaction by examining the energy dependence of the structure

function at x = 0 . Unfortunately, at present we have experimental data

for the kaon reaction at high energies only at E = l8 GeV . •

IV. FINAL REMARKS

In conclusion we may make the following remarks:

a) Near the kinematic boundary of the y fragmentation, the contribution

from Fig.3b to the production cross-section is naturally finite with thei- *)reggeized p-meson because of the high increase of |s ] . This contribution

does not correspond to the small slope of a(t ) ; actually it corresponds

to the graph shown in Fig.U. In this region we must take into account the

dominant production of vector mesons.

b) A large value of YP distribution at large k x and small k,, is- T)favourable to the fixed-pole behaviour. Experimental data for IT do not

show such a feature.

*) This result is the same with the elementary p meson.

-10-

c) Two-particle correlation in the photo-induced reaction would be larger

than in purely hadronlc processes on the assumption of uniform charge dis-

tribution in hadronic matter, (See A} in Sec.I.) In the Regge-pole frame-

work, the uniform charge distribution would be connected to the fixed-pole

behaviour.

ACKNOWLEDGMENTS

The author is grateful to Professor Abdus Salam, Professor P. Budini,

the International Atomic Energy Agency and UNESCO for hospitality at the

International Centre for Theoretical Physics, Trieste. He would like to

thank Professor L. Bertocchi and Dr. M. Rafique for valuable comments, a

critical reading of the manuscript and scientific information. The early

stage of this work was stimulated "by discussions with Mr. Tateaki Sasaki at

the University of Tokyo.

-11-

• APPENDIX

In this appendix we note the kinematics for the reaction YP "*" 7I"ir in

order to clarify assumptions in the reggeization in the graph shovn in Fig.3b.

We have assumed that the longitudinal spin component of the p meson

off the nucleon is very small. • Therefore, with the reaction yy •*• Tnt our

kinematics is almost the same. The t-channel parity-conserving helicity25)

amplitudes free from all kinematical singularities are the following :

01;01 0 -l;01

(t - y2) ({t - (m - y)2}{t - (m + y)2}}1

ft t01;01 | 0 -l;1 + z, 1 -

P P

The differential cross-section in the s channel in terms of the

helicity amplitude s is given "by

dadft

1 + z.

8TT2 (B - m2Q]

+ ( t - y 2 ) 2 ( t - (mp - y ) 2 ) ( t - (mp

+ - g - I - t) [{t - (mp - u.)2Ht - (mp

where

2 s t - t(m2 2u2).-

Z* (u2 - t ) ({ t - (mp - u) 2 Ht - (mp + y)

In the conventional coupling schemes at t - 0 we require that F.

and F should be proportional to t . Then the differential cross-section2 22 2y2

at t = 0 decreases more rapidly than that in the region of st » m y

s increases (z •*• 1 as t ••• 0)t •

as

We see a dip in the forward cross-

section at high energies.

In order to elimin

require the following relation as another choice:

In order to eliminate the apparent pole in f0 1. 0 1

a t t we can

-12-

The above relation implies that the, differential cross-section receives the

contributions of both unnatural and natural parity exchanged at t • 0 .

We have assumed in Sec,II that the total contribution from unnatural and natural

parity exchange gives effectively the fixed-pole behaviour to the differential

cross-section, as we see in the pseudoscalar meson photoproduction on the

nucleon.

The left-hand side of (2.3) summed over polarization of the photon and

averaged over the direction of q. is (ef) |t-|/{3(t - u ) } at high

energies. After considering the contribution of natural parity exchange (we

neglect the effects from the mass of the p meson), we have normalized the2

squared Regge-pole amplitude near the point of t = y , because the contri-

bution of unnatural parity exchange dominates there. Then we again normalize2 —2

it at t. = 0 in order to get rid of the pole behaviour (t - y ) and to

include this factor in the Eegge residue. Another factor 1/2 comes from

averaging over the polarization of the photon.

-13-

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-15-

G(t)

Pi ^

Fig.l

Fiff. 2

-it,-

g.4f mm-

Fig. 3a

k , A v

q. -fc-q

/\

Fig. 3b

-17-

Fig. 4

Fig.l

Fie.2

Fig. 3

Fig.U

FIGURE CAPTIONS

Dominant diagram expected to contribute to IT production near the

kinematic boundary for beam associated IT" .

Diagram including a partial amplitude dominated by a fixed pole.

a) Dominant diagram expected to contribute to ir~ production near

the central region kn'ft* 0 in the multiperipheral framework.

b) Diagram vith a fixed-pole behaviour of the amplitude for

"YP + (a meson) + ir" . .

Diagram of Fig.3b near the kinematic boundary for beam associated IT"

- I S - 1 7 MAG 1972

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