P H Y S I C A L R E V I E W D V O L U M E 9 , N U M B E R 9 1 M A Y 1 9 7 4

Role of the Pomeron in tensor-meson dominance*

L. R. Ram oha ant Department of Physics, Purdue University, Lafayette, Indiana 47907

Kashyap V. Vasavada Department of Physics, Indiana-Purdue University, Indianapolis, Indiana 46205

(Received 15 October 1973)

We include the Pomeron contribution in the tensor-meson dominance relations and find that (i) the previous discrepancies are resolved, (ii) the mass radii satisfy the Gell-Mann-Okubo type relations, and (iii) some bounds on the Pomeron slope and the tensor F / D ratio, con- sistent with experiments, are obtained.

The success of the vector-meson dominance of el without a P ~ i 0 r i restrictions based on the no- the electromagnetic current has led to efforts tions of duality, strong exchange degeneracy, uni- towards a study of tensor-meson dominance of the versality, etc. Also we consider the matrix ele- energy-momentum tensor.' Several authors have ments of the energy-momentum tensor for the investigated saturation schemes with known tensor pseudoscalar meson and the baryon octets. mesons.213 If only the f and f ' meson contribu- We begin by defining the couplings and the form tions a re retained, certain universality-type re- factors. The matrix elements of the energy-mo- lations for their couplings a re obtained which a re mentum tensor B,, for the mesons Mi are given by in conflict with the available experimental infor- m a t i ~ n . ~ This has led to the inclusion of subtrac- ( M i ( p ' )I Q,,/ Mi(P)) P,F?(q2)

tion c o n s t a r ~ t s , ~ * ~ . ~ the existence of exotic SU(3) + (q2g,, - qPqu)F'!(q2), singlet tensor meson^,^ etc. Although it has been suggested that the Pomeron also contributes to (1)

the matrix elements of the energy-momentum ten- where P = (p +p ' ) and q =p -p ', and the states a re s ~ r , ~ . ~ an explicit calculation taking its contribu- normalized such that tion into account has not been carried out. In the present work we investigate the role of the Pomer- (Mi(P ' )I Mi(p) ) = 2 ~ , ( 2 n ) ~ 6 ~ ( C - G ' ) . (2 )

on in tensor dominance at a phenomenological lev- For the baryons we have I

1 1 (@(P')B,.iBi(P)) = a p t )[(Y,P, + ~ , P , ) ~ F ~ c ' ) + P , P , ~ F , B ( ~ ~ ) + ( ~ ~ ~ , . - U , ~ . ) ~ F , B ( ~ ~ ) ] u(P), (3)

with the same normalization a s for mesons. At zero momentum transfer the form factors a re normalized to

F ~ ( O ) = 1, Ff(0) = 1, F,B(O) = O . (4)

The form factors allow us to define the tensor- mass radii to be

where ~ ' ( 0 ) denotes dF(q2)/dq 1,2,, . The tensor- meson couplings to the s t ress tensor a re given by

(TI QpuIO) = ~ T ~ ~ T E , , , (6)

and the couplings of the tensor mesons to the pseudoscalar mesons and the baryons a re

(8) Here i = 1. . -8 is the SU(3) index.

Now the nature of the Pomeron singularity has not been resolved as yet. For our purpose we shall treat it a s a factorizable linear Regge t ra- jectory with intercept unity and slope a;. The Pomeron contribution will be taken into account by introducing a spin 2 + particle of mass Mp2 = l/ak, with the Pomeron couplings given by (6)) (7), and (8) with the replacement of T by the Pomeron (P). We consider this a s simply a convenient way of parametrizing the Pomeron contribution and the

L. R . RAM MOHAN A N D K A S H Y A P V. VASAVADA

actual existence of such a particle is not crucial. yM2 = 6 gfGfMi~i + g f r G f ' ~ . ~ ~ ~ G P M ~ M ~ Moreover, the Pomeron will be regarded a s an i rnf2 2: f + m; ) ' (12)

SU(3) singlet. This seems to be in agreement with high energy experiments to within 10-20%.

On saturating the form factors F: and F:, F: YB 2=6[pzii~$ +gf tG? ' )~ i~ j + g p ~ $ ~ i B i ) + ( l - 211 - mft2 with the f and f ' mesons and the Pomeron poles mp

we obtain from the equations (4) and (5) the rela- tions

(13)

The octet and singlet components off and f ' a r e l=gfGfwpi + B ~ ~ G ~ ( M ~ M ~ + ~ P G P Y ~ M~ (9) given by

1 =g f ~ ( 1 ) ~ B ~ B ~ + ~ ~ ' G ~ I B , B ~ (1) + ~ P G P B ~ B ~ (1) (10) fo=f cos8- f 's ine,

o =g r ~ ( 2 ) ~ B ~ B ~ + g f : f ' G f ~ B ~ ~ ~ (2 , +gpG$iiBi (11) f8=f s ine+f ' C O S ~ , (14)

and and the SU(3)-symmetric tensor couplings a r e

GTMiMi M ~ M ~ T = G ~ . . { f [(;.;I+ (cos28- a sin28)(2RK) + (cos28- sin28)q0q,]

+ft[(-2 s i n ~ c o s 8 q 0 q 0 + (-4) sin8cos8 ( ~ R K ) ] ) (15) and

GTBiBi& B j T = G f N N { ~ [ ( ~ F - D ) ~ N + ( ~ F ( c o s ~ ~ - s ~ ~ ~ ~ ) - D ) Ez (SF - D) + (D(2 sin28 - cos28) + 3 cos28 F ) ~ C + (D(-2 sin28 - cos28) +3Fcos28) XA]

+ (-3 sin8 cos 8 ) f ' [ (2~ )% + (F - D)% + (F + ~ D ) E A ] } , (16)

where F +D = 1. In agreement with experiment f ' has been decoupled from sa and FN in the above.

In order that the normalization conditions (9), (lo), and (11) be satisfied for the entire multiplet i t i s clearly necessary that the octet parts of f and f' cancel in these equations. From (15) and (16) i t i s seen that this can be achieved by taking

From mass formulas for the tensor nonet 6 r30.5". Then al l of the equations (9), (lo), and (11) s im- plify to the following three relations:

Since the Pomeron exchange in elastic scattering i s known to be consistent with s -channel helicity conservation7 we expect GFiB=O, and this leads to G Y ~ , =O. This is consistent with one of the de- terminations of the f couplings.' If in future, ex- periments indicate that G Y ~ , # O but ~ $ 2 ~ is st i l l zero then (20) will have to be modified either by a subtraction constant o r extra contributions. For the present we assume that

Now we use Eqs. (18)-(20) to eliminate gf and gp in the expressions for tensor mass radii. De- fining

and using a; mp2 = 1, mf '/wf, =0.7, we have

(Y- 1) y: =6[- f fb - (Y(_";;f2] 9

and

Tensor-mass radii represent the distribution of the hadronic matter in particles. Here we have expressed these radii in t e rms of experimentally determinable quantities by using tensor-meson dominance. While a direct observation of these radii requires probing particles by the graviton, in- formation about hadronic distribution of matter is

9 - R O L E O F T H E P O M E R O N IN T E N S O R - M E S O N DOMINANCE 2629

beginning to be available from various other theo- retical approaches also. For example a simple connection betweenpp scattering and the proton electromagnetic form factor suggests some equiv- alence of matter density and charge densityeg Also Y. S. Kim and one of us (K. V.) have considered inverse scattering formalism to obtain the nucleon and the p-meson strong-interaction radii from the S-matrix approach using experimental information on phase shifts and bound-state parameters."

Quite independent of any numerical values of the parameters in Eqs. (23)-(29) we see that the (radius)' satisfy Gell-Mann-Okubo type relations. Thus

and

Note that the saturation of baryon and meson ma- t r ix elements of the commutators of the conformal generator KO with the vector o r axial-vector cur- rents j, and their divergences a, j,, in the infi- nite-momentum limit, leads to the result that the radii within the meson o r baryon multiplets are e q ~ a l . ~ . ~ However, here we shall not invoke prop- ert ies of the currents under the conformal trans- formations.

If we assume that the mass radii within the same multiplets increase as the masses increase, then we at once obtain restrictions on the coupling con- stants. Thus r; > vK2 > Y: requires that

In the following we will show that experiments in- dicate X> 1; hence we have Y> X > 1. If we re- qure v,' > v: > vA2 > v: we obtain a constraint on the F/D ratio in G : j iBi couplings. In particular, it i s found that F> 1, o r F/D i s negative. This is in excellent agreement with the Regge pole fits of Barger et al." and Berger and A typical value of F = 1.25 to 1.4 is obtained in Refs. 11 and 12.

Further constraints can be obtained by demand- ing that al l the r 2 a re positive since the matter density is expected to be positive. The only ad- ditional restrictions arise from Y: > 0 and rN2> 0 and we have

cub> (X- l)/m, ' ( Y - 1) (33)

and

respectively. Since Y> X as shown above the con- dition (34) i s the more restrictive one.

Now we consider some typical numerical esti-

mates. The Pomeron couplings can be obtained by comparing the contribution of tensor meson (2') pole on the P trajectory with the factorizable Regge pole contribution at t = O which is simply re- lated to the total cross sections. An elementary calculation then gives

where C = ~ T ( ( Y ~ ) ~ s , , So being the usual factor for Regge contributions. Thus if cub is known, G,,,, ~ 2 2 , can be determined from the asymptotic cross sections. Since we regard the Pomeron a s a factorizable Regge pole with cu,(O) = 1, possible logarithmic rise of asymptotic cross sections is outside the scope of our work. At any rate this determination should be approximately valid. For the present, however, we will just consider the ratios of the coupling constants. From (351, (36), and (37) we have

The quark model predicts the last ratio to be $ and this is consistent with experiment to within 15%. (Thus X > 1.) The ratio Y is more difficult to ex- tract. If we make a stmilar comparison with the Regge pole (P' = f ) exchange we find

2.05-2.43 (Ref. 11) Y = G ~ ~ ~ / G ~ , ~ = 1.9 (Ref. 12). (39)

In our normalization Gf,, is related to the f- 2n decay width by

leading to G, ,, = 10.8. Dispersion relation esti- mates for Y vary considerably. Engels8 finds Y=3.44 i0.33 and G Y ~ , = 0, whereas Schlaile and Strauss13 find nonzero values for G:;, and Y=2.31 and Y = 3.7 i 1.1, respectively. Goldberg14 obtains Y = 1.95 using fixed-u dispersion relations and Liu and McGee15 have Y = 2.3-2.75 from continuous moment sum rules with nonzero G Y ~ , having large uncertainties. Note that in the absence of Pomeron contributions Eqs. (18) and (19) lead to a unique value2r3 Y= 1. On the other hand Freund6 has sug- gested that strong exchange degeneracy and uni- versality of the vector-meson couplings induces a universality into the tensor couplings leading to Y = 2. In spite of the variations in experimental numbers it appears that Y= 1 or $ are not consis- tent with experiment.

With X = $ the following lower bounds a re ob- tained for cub i.e.,

L . R . R A M MOHAN AND K A S H Y A P V. VASAVADA 9 - ity, and str ict duality were imposed and the mass

0.43 GeV-2 radii were not considered. One result was that (41) X = Y = g and hence the Pomeron contribution could

not resolve the discrepancies off couplings.

Alternately, for a given value of af, we can obtain bounds on Y (or on GYJ,). Recent fitsl"ased on Serpukhov and CERN ISR data indicate that the Pomeron slope l ies in the range 0 < a&< 0.45.

Now in Ref. 10 i t was found that the strong in- teraction radius Y, of the nucleon i s in the range of 0.15 to 0.50 fermi. The bootstrap hypothesis a s - cribes essentially the whole mass of the particles to self-consistent strong interactions. In accor- dance with this if we assume that this strong in- teraction radius i s the same a s the tensor mass radius found here we obtain 0.3 < af,< 0.9 GeV-2 for values of Y between 2 and 4. Also i t i s c lear that Y, cannot be arbitrari ly large unless Y i s close to g since there is an upper limit on af, (=. 1 Gev-'). An interesting point i s that in Regge- pole potential scattering theory the radius R of a particle i s given by

This gives a typical value of R ~ 0 . 6 fermi. The proportionality of R2 to a' is consistent with our formulation. Thus in spite of the current experi- mental inaccessibility of the mass radii, a con- sistent interesting picture emerges.

As we have already mentioned, ~ r e u n d % a s ex- amined the question of tensor dominance in a broad framework covering several ideas. In that work, however, strong exchange degeneracy, universal-

Large contributions of some exotic multi-quark SU(3) singlet tensor mesons were required. It re- mains fo r the future experiments to discover such exotic mesons. In the meantime it appears that many of the above-mentioned theoretically attrac- tive concepts a r e found to be broken in various contexts. In addition the naturk of Pomeron itself is not clear . It has not yet been possible to gener- ate the Pomeron through dual quark models. This has motivated our reexamination of this question on a phenomenological basis, although we can not rule out contributions from exotic mesons. We can, however, remove the discrepancies by using the experimental information without imposing the duality, exchange degeneracy, and universality constraints. Apart from resolving the inconsis- tency arising from saturation of tensor dominance relations by f and f ' mesons only, we a r e able to obtain various bounds on the Pomeron slope and the F / D ratio for tensor-meson couplings. In addition the mass radii have been shown to obey the Gell-Mann-Okubo type relations.

We wish to thank the Purdue Part icle Physics Group for discussions and Professor Rosen, Pro- fessor Sugawara, and Professor Loeffler for hos- pitality during the summer of 1973. One of us (K. V.) wishes to aknowledge the award of an I. U. Summer Research Fellowship.

*Work supported in par t by the U. S. Atomic Energy Commission.

tP re sen t address: Department of Physics, University of Delhi, Delhi 7, India.

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