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20Octf9w the ROLE of the ROPER RESONANCE in NUCLEI 20OCTf9W 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 "<T" meson 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.
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