P H Y S I C A L R E V I E W D V O L U M E 1 2 , N U M B E R 9 1 N O V E M B E R 1 9 7 5

Mass formulas, strong gravity, and scalar-tensor universality

Kashyap V. Vasavada Department of Physics, Indiana-Purdue University. I~~dianapol i s , Indiana 46205

(Received 27 May 1975)

We show that tensor dominance of the stress tensor and a scalar-tensor universality postulate lead to values of the D / F ratio and the coupling constants of the f meson and Pomeron to nucleons in very good agreement with experiments.

In analogy with the vector dominance of tile electromagnetic current , tensor dominance of the energy-momentum s t r e s s t ensor has been con- siderably explored in recent years . ' Extensive tensor-field-theoretic treatments213 of the so- called "strong gravity" have been given. Yet, s o m e simple considerations given here have not appeared in the l i terature. It i s known that, if only the f andf ' meson contributions to the pole- dominated mat r ix elements a r e kept, cer tain (meson-baryon) universality type relat ions (e.g., GfNN/Gfnn = 1 ) a r e obtained.'"' These a r e in con- flict with experiments by a factor of 2 o r 3. In Ref. 4 we introduced an explicit Pomeron contr i - bution to remove this d i ~ c r e p a n c y . ~ Some in- terest ing bounds w e r e obtxined but t h e r e were too many unknowns to make a unique comparison with experiments. In the present note we study sys tem- atically the symmetry breaking i n m a s s e s and use a recently proposed sca la r - tensor universality p ~ s t u l a t e . ~ We will be mainly concerned with spin-$ baryons.

The matr ix elements of the energy-momentum tensor between baryon octet s ta tes a r e given by

6'. the average m a s s of the baryon octet, is intro- duced to make the couplings d i m e n ~ i o n 1 e s s . W ~ i s the m a s s of the tensor meson.

As discussed p r e v i ~ u s l y , ~ it is necessary t o in- troduce s o m e contribution in addition to that of f and f ' mesons. We identify this with the Pomeron te rm. It i s t reated a s a factorizable SU(3)-singlet Regge pole with slope a$ and intercept 1. The recently observed logarithmic r i s e in total c r o s s sect ions and the considerable difference between nN and KN total c r o s s sect ions have cas t s o m e doubt on this s imple picture. On the other hand. these effects and violation of factorization do not s e e m to be m o r e than 10 o r 20%. So these a s - sumptions a r e s t i l l approximately valid. The Pomeron-tensor contribution i s taken into account by introducing a spin-2 part ic le with m a s s Mp =l/6;. This can be regarded a s a convenient s imple way of paramet r iz ing the Pomeron contr i - bution and the actual existence of such a par t i c le i s not really crucial.

The singlet and octet components off and f ' a r e given by

P p v The m a s s formula fo r a tensor nonet gives the = ~ @ ' ) [ ( y , ~ + Fz(q2) mixing angle 8 =31°* 2'. The so-called ideal m h -

ing angle i s given by c o s f ? = ~ ( B = 3 5 . 3 " ) . The

+ 9'g.u - quqv Fdq2)] u ~ ) . (1) couplings t o the s t r e s s tensor a r e denoted by MB g,, g,, and gp. The f o r m factor F , ( q 2 ) gets contr i -

Here i denotes various SU(3) members . P = p +p' , butions f r o m spin-2 mesons. Saturating it with

q = P -p ' , andM, i s the baryon mass . The tensor - P? f I ? a n d f 6 at q 2 = O we get (Ref. 4,

meson couplings t o the s t r e s s tensor and baryon -

s ta tes a r e defined by 1% = & . ~ ? P N N ~ + g l ~ l ~ ~ + ~ 8 g , w M , (5)

(TI @p 10) = W3g+ ,v , (2) 2 ( F - 1 ) = K ~ ~ P A w M + ~ Y ~ ~ , x v R + +F - z~gga:8NNM, (6)

Pi @')lTlB, @)) 2 ( F - 1 )

M~ =gpgP.WM +glglWB -4F_1 K8gWNiCi9 (7)

2 F + 1 - M~ 'gPgPXV6' +g1glhWa - m&!ag8NNM.

( ) B @ ( ) H e r e g,,, and g.,, a r e singlet and octet tensor + 4MTM, J

12 - MASS F O R M U L A S , S T R O N G G R A V I T Y , AND S C A L A R - T E N S O R . . . 2919

(f,, j , ) coupling constants to the baryons. g,, a re assumed to be SU(3) symmetric with F +D =l.

From Eqs. (5)-(8) it can be readily seen that the baryons obey the Gell-Mann-Okubo mass formula

This i s not completely unexpected since we allowed only singlet and octet intermediate states, but it does provide insight into the dynamics of symmetry breakinge7

Now, taking differences between the masses we obtain

independent of any values of the coupling constants. Hence

This gives D/'F = - 0.3 in excellent agreement with the values (- 0.2 to - 0.3) determined by Barger, Olsson, and ~ e e d e r " and Berger and Fox." These authors fitted tensor trajectory contributions to meson-baryon and baryon-baryon reactions. Note that, in contrast, the SU(6) value of $ is completely off.

With the above value of F, we have only two independent equations :

At this stage we use the scalar-tensor universal- ity postulate recently given by Nath, Arnowitt, and Friedmana3 This postulate, which would be analo- gous to the well-known KSRFIKawarabayashi- Suzuki-Riazuddin-Fayyazuddin) relation, states that the tensor and scalar couplings to the s t r e s s tensor are universal. This gives (in our notation)

where the scalar coupling Fa i s defined by

A value of F,(=F,) =97 MeV is found to be con- sistent with the electron-positron annihilation data.3 A comparable value is obtained by Carruth- e r s in Ref. 9. With this value of F,, we find

/gf 1 =0.076, jgf 1 1 =0.064,

and (1 6)

Igp/ = 0 . 0 9 7 6 ~ r .

Now it is well established that f' is decoupled from NH. This implies, from Eq. (4), that

If we choose g,:,,, g,, > 0, we need the relative signs g,< 0, g, > 0, and gp < 0. This can be achieved by taking gi > O , gfl < 0 , and gp<O.

Then Eqs. (12) and (13) give unique values of gf, and gp, for a given mixing angle 8 . The r e - sults a r e sensitive to the value of 8. We find

and 21.6

8 = 35.3'(ideal)-gf, = 38.4, g,, =- . (19) JgP

For comparison we note that the meson-baryon tensor universality relations obtained by saturating nucleon and pion matrix elements with the j meson implylg4 gf, - 9.2, in complete disagreement with the values given above.

There a r e several dispersion relation estimatesr2 for gfNN. The values are: Engels (30.3i2.9); Schaile (20.4); Strauss (32.6 i 9.7); Goldberg (17.2); Liu and McGee (20.3-24.2). These vary considerably but none of them agrees with the meson-baryon universality value. The experi- mental value of gp, can be obtained by comparison with the high-energy proton-proton scattering fits, e.g., those of Barger et a1 . I 0 The spin-averaged proton-proton scattering amplitude A(s, t ) is normalized such that

1 a,,, =- ImA(s, t =0 ) . (20)

For the spin-2 P exchange we find

Barger e t a1 . take

where the residue factor /3 ( t ) / P (0) has been in- serted by us. The data are fitted at t = 0.

Extrapolating Eq. (22) to the pole at t =Mp2 we obtain

Now the Pomeron residues a r e usually consistent with being structureless. Hence /3 (1~1,~)//3 (0) - 1. Then from Ref. 10 we have

2920 K A S H Y A P V . VASAVADA 1 2 -

It is interesting to note that, even if we take /3 ( t ) = l / r ( c u ( t ) ) given by the Gell-Mann mechanism for ghost elimination or the Veneziano model, p ( M p 2 ) / p ( 0 ) = 1. Equation ( 2 4 ) agrees with our prediction Eq. ( 1 8 ) for = 0 . 4 4 GeV-2 and with E q . ( 1 9 ) for a& = 0 . 0 8 GeV2. These values a r e within the generally accepted rangeL3: 0 < a & < 0 . 4 5 .

Fits of Ref. 10 can also be used to find the ex- perimental value of g f , by using Eq. ( 2 3 ) with f replacing P Cf is identified with P' trajectory). Unlike the Pomeron case, however, p ( t ) could change rapidly here. In particular, a right signa- ture pole at t = - 0.5 GeV2 should be avoided by factors like ci(t), l / r ( c u ( t )) etc. The former gives P ( M f 2 ) / p ( 0 ) = 2 , whereas the latter gives 1.33 for a P' trajectory a&(t) = 0 .5 + t . From the value given in Ref. 10, we find gf, = 17.0 and 11.3 in the two cases. In addition to this, because of the possible structures involved in the residue func- tion, there could be considerable variation in the extrapolation from t = 0 to t = Mf ' . At any rate, these values a re not too far off from the dispersion relation values which a r e more reliable since, in principle at least, no extrapolation i s involved. The value of gp,, however, essentially comes from the asymptotic value of the total c ross sec- tion and hence i s more reliable.

Thus we have shown that the dominance of the s t r e s s tensor by Pomeron, f, and f ' mesons gives the Gell-Mann-Okubo mass formula for baryons, the D / F ratio, and PNN and f NN couplings in good agreement with the experiments. The latter re- sults follow from the scalar-tensor universality which i s thus seen to be in better agreement with experiments than the meson-baryon universality.

It is a straightforward matter to extend these considerations to other multiplets. Unfortunately the large amount of symmetry breaking in the pseudoscalar meson octet masses does lead to considerable difficulty. For this reason we have decided to consider the baryon case separately. In addition to mass formulas, one can readily find expressions for tensor radii of particles and obtain more relations. These and other matters will be presented elsewhere.

In this note and in Ref. 4 we have treated the Pomeron on a completely phenomenological level, without any restrictions based on the notions of str ict two-component duality. Also, unlike the so- called tensor Cf -f ' ) dominated Pomeron model,14 no attempt has been made to relate the Pomeron contribution to the tensor-meson contributions. Only the satisfaction of mass relations has been required. Thus in general different results will be obtained in the two approaches. A drawback of our approach will be that, in order to explain SU(3) breaking of the high-energy total cross sec- tions (which seems to be actually less than 20%), we will have to introduce a small octet component of the Pomeron with f r ee parameters, This i s in contrast to the model of Ref. 14 where such param- e ters a r e determined. On the other hand, as we have already remarked previously,* the str ict duality concept does run into various problems. Furthermore, it has not been possible to generate a realistic Pomeron within the framework of dual quark models. Hence, such an alternative approach seems to be worth considering.

The author wishes to thank Dr. F. T. Meiere for a discussion.

IP . G. 0. Freund, Phys. Lett. 2, 136 (1962) ; R . Del- bourgo, A. Salam, and J. Strathdee, Nuovo Cimento *, 593 (1967); K. Raman, Phys. Rev. D 3, 2900 (1971) ; B . Renner, Phys. Lett. @, 599 (1970); H . Kleinert. L . P. Staunton. and P . H . Weisz, Nucl. Phys. E, 87 (1972) ; H. Pagels, Phys. Rev. 144, 1250 (1966) . . ,

'c. J. Isham, A . Salam, and J . Strathdee, Phys. Rev. D 3, 867 (1971) ; K . Raman, ibid. 2, 1577 q 9 7 0 ) ; B . Zumino, in Lectures on Elementary Particles and Quantum Field Theory, 1970 Brandeis Summer Institute in Theoretical Physics, edited by S . Deser, M. Grisaru, and H. Pendleton (MIT P r e s s , Cambridge, Mass ., 1970) .

3 ~ . Nath, R. Arnowitt, and M. H. Friedman, Phys. Rev. D 5, 1572 (1972) ; Phys. Lett. G, 361 (1972) .

4 ~ . R. Ram Mohan and K. V. Vasavada, Phys. Rev. D 2, 2627 (1974) .

5 ~ n extra I = 0 piece in the trace of the s t r e s s tensor has been also previously suggested. See, for example,

M . Gell-Mann, in Proceedings of the Third Hawaii Topical Conference on Particle Physics (Western Periodicals, North Hollywood, Calif ., 1969) ; P . Car- ru thers , Phys. Rev. D 2, 2265 (1970) .

ru his definition of the coupling constant gTBB is more convenient f o r our purpose he re than the one used in Ref. 4 ( G mB) which differs by the factor ( Z / M ~ ) . Of course, when we assume SU(3) symmetry fo r these, different results a r e obtained. Only g$ lB i s used in the following, and the superscript i s dropped.

he fact that we obtained l inear mass formulas can be, of course, traced to the definitions of the SU(3) sym- metric coupling in Eq. (3). Contrary to the claims which a r e sometimes made in l i terature, this cannot be determined a priori without appealing to the experi- ments. We could have obtained quadratic mass formu- l a s in exactly the same manner.

'while this work was being completed, we became aware of an unpublished report by H. Pagels [University of

12 - M A S S F O R M U L A S , S T R O N G G R A V I T Y , A N D S C A L A R - T E N S O R . . . 2921

North Carolina technical report (unpublished)] and the work by P . Carruthers quoted in Ref. 9 . Pagels ob- tains our Eqs. (9) and (10). However, he deals with singlet and octet contributions in a general way and does not consider measurable coupling constants a s we have done here . The Pomeron i s not included there and all the other results of our work a r e not obtained. I wish to thank Dr . H. Pagels for sending a copy of his unpublished report .

$In a scheme with saturation of the t race of s t r e s s tensor by sca lar mesons, the same relation for the D//F ratio would be found. This result has been previously ob- tained by P . Carruthers, Phys. Rev. D 3, 959 (1971). Of course, the available information on a sca lar nonet and i ts couplings i s nowhere near the corresponding

case of the tensor nonet. I%. Barger , M. Olsson, and D. Reeder, Nucl. Phys.

B5, 411 (1968). " r ~ e r g e r and G. Fox, Phys. Rev. 188, 2120 (1969). 125. Engels, Nucl. Phys. E, 141 (1971); H. Schlaile,

Ph.D . thesis, Karlsruhe, 1970 (unpublished); R . Strauss, Ph.D. thesis, Karlsruhe, 1970 (unpub- lished); G. Ebel et a1 ., Nucl. Phys. B33 (1971); H . Goldberg, Phys. Rev. 171, 1485 (1968); Y. C . Liu and I. J. McGee, Phys. Rev. D 3, 183 (1971).

13v. Barger , K. Geer, and F. Halzen, Nucl. Phys. B49, 302 (1972).

for example, R. Carlitz, M . B. Green, and A . Zee, Phys. Rev. D 4, 3439 (1971) and references quoted therein.