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THE ROLE OF THE ROPER RESONANCE IN NUCLEI

Bertrand Desplanques Division de Physique Théorique} Institut de Physique Nucléaire, F-91406 Orsay Cedex

Invited Contribution lr>t European Workshop on Hadronic Physics in the 1990'a with multi-GeV électrons SeïWac June 27 - July I, 1988

IPNO/TH 88-51 t September 1988

'Laboratoire awocié au C.N.R.S. ABSTRACT In a mean field approximation, the nucléon in nuclei may be considered as the superposition of a nucléon with the structure of the free one and its excitations, the Roper resonance in particular. Estimates of this admixture are given. Consequences for the saturation properties of nuclear matter and for the swelling of nucléons in nuclei are considered.

1 INTRODUCTION

It is now well known that the excitation in nuclei of the A resonance at 1230 MeV (simply denoted A in the following) plays a non negligible role for any observable concerned with spin-isospin type excitations. This can be easily understood in a quark model where this resonance appears as a spin-isoapin excitation of the nucléon constituenta themselves. Similarly, it may be thought that the radial excitation of the nucléon will have some role in scalar type excitations, such as intrinsic radial densities of the nucléon inside the nucleus. It is generally believed that the first radial excitation of the nucléon is the Roper resonance (denoted N* in the following). While it is not presently clear which degree of freedom is involved in this excitation, two features indicate that the Roper resonance may be the first next resonance to give relatively large contributions in nuclei : its relatively low energy excitation with respect to the nucléon (500 MeV instead of 300 MeV for the A resonance) and its coupling to Nn and Air channels (1/2 of those in the nucléon case). In the following, we implicitly assume that the nucléon and Roper resonance can be identified to the first and second states, with full spatial symmetry, of a model of constituent quarks moving in an harmonic oscillator confining potential. Such a model, as any other one, does not completely account for the properties of the Roper resonance and neighbouring baryons. This uncertainty may obviously affect results presented in the following. In the best case, where the states of the above model represent the dominant components of the physical states, some renormalization of the results should be performed. In a mean field approach, the Roper resonance may be admixed to the nucléon in nuclei. This has two immediate consequences : the nucléon to which the Roper resonance has been admixed has a size and an energy different from those for the free nucléon. These effects can be respectively related to the question of the swelling of nucléons in nuclei and of the saturation properties of nuclear matter. The relationship to the swelling can be understood as follows. A nucléon made of 3 constituents quarks with a larger radius can be analyzed in terms of free baryons and considered as a superposition of the free nucléon and its radial excitations. If the swelling is not too large, the only important excitation will be the Roper resonance. The relevance to the saturation properties of nuclear matter stems from the fact that it is one of the candidates (if not the only one) to provide the extra binding energy which remains unexplained in approaches based on non relativistic 2- body forces. Relativistic corrections of dynamical origin are rather repulsive and only reinforce the need for further binding energy. Those of kinematical origin are likely to be incorporated in these approaches, implicitly (through fit to NN scattering data) or explicitly (Bonn and Paris potentials for instance). In the next sections, we will successively describe the mechanisms leading to the (coherent) exci- tation of the Roper resonance in nuclei, provide some numerical estimates and discuss its relevance for saturation properties of nuclear matter and for the swelling of nucléons in nuclei.

2 MECHANISMS FOR THE EXCITATION OF THE ROPER RESONANCE IN NUCLEI

2.1 Coupling to the "

The simplest model to excite the Roper resonance in nuclei is inspired by relativistic approaches

I O Q based on "a" and w exchanges, but with the "ff" meson coupled to quarks instead of nucléons 1^'0. The process is depicted in fig. Ia.

N .—N . N » I N, A I N

Tt TC

a) b)

Figure 1

Diagrams leading to the excitation of the Roper resonance. From now on, two features appear to be important. In nuclear matter, all the nucléons can coherently contribute to the excitation of the Roper resonance so that the probability to excite this particle from some nucléon is proportional to the square of the number of nucléons ourrounding that nucléon. In order to make some comparison, it should be noticed that for the A resonance quoted above, this probability is only proportional to the number of those nucléons. The second point is dealing with the JV* —» Nc coupling. In a static non relativistic model and at zero momentum transfer, this coupling vanishes due to the orthogonality of the wave functions. This is not any more true if one goes beyond thià approximation and introduce kinematical corrections arising from the small components of the Dirac spinors of quarks, so that the coupling, N* —+ ./Va, involves the internal momentum of quarks in baryons. That such contributions are not negligible is emphasized by the large N" -» Nit to N -* Nn ratio (^ 1/2) where the same kind of argument applies. At the same order, however, the N* —> JVo* coupling remains equal to zero. This makes an esaential difference with the NN -+ NN interaction since the cancellation there between a and w exchange contributions is absent here for the NN -> NN* interaction.

2.2 2TT exchange

While the reference above to the "o" exchange makes possible some relation to nuclear relativistic approaches, it is well known that the "a" particle is an effective one, which rather accounts for 2ir exchange in a relative 5 state. A representative contribution is shown in fig.Ib. Apart for the fact that it is more realistic, this approach makes possible some relation with calculations of the 2JT exchange contribution to the NN interaction. Indeed the main difference resides in coupling constants, N" -» NJT or N* -> ATT instead of JV -> NJT or N -> AJT. Similarly in a mean field approximation, one may think of some relation between the N —> N* transition potential and the 2 JT exchange contribution to the nucleon-nucleus potential. Single JT and p exchanges can also contribute to the excitation of the Roper resonance. Due to the coherence of the different contributions involved in fig.Ib, their contribution is likely to be smaller than the one due to 2ir exchange, as for the NN interaction. In any case, it would rather give an enhancement of the contribution considered here.

3 QUANTITATIVE ESTIMATES

The quantity of interest here is the Roper resonance admixture amplitude to the nucléon, denoted a. As details about its estimates have been given elsewhere 4, we only reproduce here the main results. 3.1 'V exchange

As previously mentioned, the advantage of the approach based on the 1V exchange is the possibility to refer to nuclear relativistic approaches. In this case, the expression of a is given by :

where VJ is the a exchange potential acting on the quarks of the baryon under consideration. It incorporates terms of the order p2/m£ and its strength is fixed by the relation :

V* '(P = O) = \V? , (2) with V? = 400 MeV. To calculate the numerator in (1), one also needa a quark description of the nucléon. It is taken from RefA The main parameter in this model, &§ = 0.22 fm2, has a value which gives rise to a mean square radius which is 1/3 of the charge one, while the spacing between the nucléon and the state which might correspond to the Roper resonance is somewhat too large (770 MeV instead of 500 MeV which is used throughout this paper). However, the predicted energy spacing between the nucléon and the first negative parity states is correct. With the above values, one then gets a = 0.4, which corresponds to a probability for the Roper resonance admixture to the nucléon in nuclei of 16%.

3.2 2 TT exchange

Calculating from first principles the contribution of diagrams shown in fig. Ib with a reasonable accuracy is presently hopeless. They depend on hadronic form factors, short range correlations or also on the p exchange contribution which for some part plays the role of a cut-off. Aa a guide, we may use the example of the NN interaction which is sensitive to the same uncertainties but with the advantage that there exist a large amount of data. However, since in this case the calculation of the 2* exchange to the NN — » JVJV* amplitude essentially repeats the one which is done for NN — » NN, the only difference arising from coupling constants, we believe it is simpler to scale by appropriate factors present contributions to this interaction, or to the nucleon-nucleus interaction. From the JV* — > Nn decay, it is known that the ratio of the N" Nx and N N TT coupling constants is 1/2. As far as it is known about the N' — > &ir decay, this value is compatible with the ratio of the N* Air and TVA* coupling constants. This equality is consistent with quark model predictions which, furthermore, allow to determine the relative signs. It is thus reasonable that the overall scaling factor be close to 1/2. The 2* exchange contribution to the TV-nucleus interaction might be of the order of 150 MeV, which is the value necessary to cancel the repulsive contribution due to the w exchange, the actual attraction of 50 MeV being roughly provided by single TT and p exchange. Accounting for differences in the effect of short range correlations and for the scaling factor 1/2, one then gets a transition potential W —» N" of 50 MeV. Using this value leads to an admixture amplitude of 0.1. This value, which corresponds to a 1% admixture of Roper resonance per nucléon, is much smaller than the previous estimate. We however do not believe that there is contradiction. In our estimate of the 2 JT exchange contribution, we implicitly included some irp exchange contribution which has a repulsive character and for which the overall factor 1/2 may not apply. If this contribution is assumed to behave like an w exchange, then, the second estimate would become closer to the first one. In the following, we will concentrate on the lower estimate, a = 0.1, keeping in mind that it may be an underestimate with some respects.

4 RELATION TO THE SATURATION PROPERTIES OF NU- CLEAR MATTER

In the mean field approximation used here, the extra binding energy due to the Roper resonance admixture is roughly given by :

S(E/'A) ••= -az(M* - M) = -5 MeV , (a ~ 0.1) . (3)

This contribution, calculated at the nuclear matter density of 0.17 fm~3, may seem larger than what is presently necessary. We do not believe it is so. Indeed, the present mechanism does not improve the saturation density which is predicted too high. Some repulsive contribution, looking like the one provided by relativistic dynamical effects, which have a strong density dependence, is desirable to lower the saturation density". That repulsive contribution should be compensated for. This picture s* represents a realization of what was guessed few years ago , namely two different mechanisms beyond 2--body forces are involved to get the right saturation properties. Furthermore, the treatment of the A excitation as an active degree of freedom seems to give some further repulsion , which has therefore also to be compensated for. The effect discussed here is absent in usual approaches, where the Roper resonance is not ex- plicitly incorporated. To account for it, 3-body forces have to be considered. As it can be seen in fig.2, where the whole effect is depicted, they should include nucléons in the final or initial states, but also A resonances. This goes beyond usual 3-body forces. In fact, most of the effect is due to these extra-pieces of the 3-body force. Due to the difficulty to deal with the loops in fig.2, it may be appropriate to replace each of them by an effective a exchange. Using such an approximation, we got an extra binding energy of 0.1 MeV in the 3-nucleon system (perturbative calculation).

5 RELATION TO THE SWELLING OF NUCLEONS IN NU- CLEI

The nucléon to which the Roper resonance has been admixed has properties different from those of the free nucléon. In particular, there will be changes of its various radii (charge, matter,...). Quite generally, this change for the proton charge square radius is given by :

2 l c h) = -«À (IdS < N\jQ(x)e* \N* > x2o) (4) O?* W / a^-tO

N N1A N I ITT TT

N ; N1A N* N1A N i i ITT i \! * N I N;A » N

Figure 2

Diagrammatic representation of S-body contributions

For the quark model already mentioned, the relative modification takes the value :

(5)

This estimate is strongly sensitive to different ingredients. It depends on uncertainties in a, already discussed, but also on the transition charge form factor, N —» N". With this respect, it should be reminded that the charge radius of the nucléon is not fully understood in the model used for its description. On the other hand, while the above estimate is much smaller than the one which has been claimed to be necessary to account for various effects (EMC, "missing charge" in inelastic electron scattering, • • •), it should be mentioned that there are other contributions which are likely to make the effective charge radius of the proton in the nuclear medium larger, such as mesonic exchange currents ^ or nuclear correlations 10-H. 6 CONCLUDING REMARKS

It is not necessary to insist on the speculative character of a large part of the present paper. While the role attributed to the Roper resonance in saturation properties of nuclear matter or in the swelling of nucléons in nuclei is not unreasonable and rather welcome, it remains to be confirmed by appropriate experiments. In our mind, however, a better understanding of what is the Roper resonance itself is first required. This implies studies tending to produce the Roper resonance in elementary, hadronic or electromagnetic, processes. The charge excitation should deserve a particular attention since it is likely to be a sensitive test of the idea that the Roper resonance is a collective radial excitation of the nucléon. These studies, which have already been performed but should be more extensive, should help to narrow the range of predictions concerning the role of the Roper resonance in both the saturation properties of nuclear matter and the swelling of nucléons in nuclei.

References

1) L.S. Celenza, A. Rosenthal and C.M. Shakin, Phys. Rev. C31 (1985) 232

2) P. Guichon, Phys. Lett. 200 (1988) 235

3) P. Grange, A. Lejeune, M. Martzolffand J.F. Mathiot (in preparation)

4) B. Desplanques, Z. Phys. A - Atomic Nuclei 330 (1988) 331

5) B. Silvestre-Brac and C. Gignoux, Phys. Rev. D32 (1985) 743

6) A.D. Jackson, M. Rho and E. Krotscheck, Nucl. Phys. A407 (1983) 495

7) B. ter Haar and R. Malfliet, Phys. Rep. 149 (1987) 207

8) B. Desplanques, Few-Body Systems, Suppl. 2 (1987) 370

9) B. Desplanques, Perspectives in Nuclear Physics at Intermediate Energies, eds S. Boffi, C. Cioffi degli Atti and M.M. Giannini (World Scientific, 1987) p.173

10) M. Ericson and M. Rosa Clot, Z. Phys. A - Atomic Nuclei, 324 (1986) 373

11) B. Desplanques, preprint IPNO/TH 88-32