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Vol 2 7 H» 1 7 [2] CA9600449 TIU-PP-94-90 I. INTRODUCTION Oct 1994 Nuclcon-nucleon brcntsstrahlung has been extensively investigated both experimentally and theoretically during the past 30 years. For pp —> ppj the experimental data available A-Excitation and Exchange Corrections for NN-Bremsstrahlung covers an energy domain between 42 MeV [I] and 730 MeV [2], the most rrccnl results being obtained at incident proton energies around the pion production threshold [3-5]. Up to the pion production threshold, the NPf-j potential model using realistic NN-potcntials M. Jetter and H. W. Fearing for the description of the nuclear force plus first order approximation for the electromagnetic interaction gives the most successful description. TRWMF, W0\ Wesbrook Mail, Vancouver, B. G, A comparison of different NN-interactions yields widely equivalent WJVir-results in the whole kinematic range [6], so that bremsstrahlung provides a sensitive test for the dynam- Canada V6T SAS ical model used to describe the photon emission, lu the framework of the potential model, however, there is little controversy about the basic features and the correction terms to be used. In particular, relativistic spin corrections as derived in [7,8] and rescattering contri- butions [9] are used in the most recent analyses [6,10], see also [11,12]. Moreover, the role of Coulomb corrections in ppy [13] and exchange currents in the np-j cross section have been discussed [9,14]. Abstract The main theoretical problems left are presumably related to shortcomings in principle of the potential model description. For example, Lorcntz invariance can only be accounted The role of the relativistic amplitudes for a number of O(k) processes usually neglected for approximately by a covariant treatment of the kinematic transformations, inclusion of in potential model calculations of NN-bremsstrahlung is investigated. In particular, we higher order terms in (p/m) in the electromagnetic operator (relativistic spin corrections) consider the A-excitation pole contributions related to the one-pion and one-rho exchange and an appropriate description of the NN-intcracti»n. Further common features of existing and in addition include the exchange contributions induced by the radiative w, p -* -x-j potential model calculations are the absence of dynamical baryou resonances and the neglect decays. The contributions are calculated from relativislic Born amplitudes fitted to A- of two-body currents beyond the O(k°) terms given by the soft photon approximation (SPA). production and absorption data in the energy range up to 1 GeV and then used to supplement The role of the A-excitation was studied in the 1970*s in two different approaches. The potential model and soft photon calculations for nucleon-nucleon brctnsstrahlung. The authors of [15] used a dispersion analysis in order to correct the one-pion exchange neutron- effects on JV/V7-observables, although moderate in general, are found to be important in proton electromagnetic current. The effect of the A-resonance on the npy cïîss section at 200 some kinematic domains. MeV turned out to be small. In [16] and [17], the A-excitalion part of the ppj amplitude was derived from phenomcnological Lagrangians and combined with soft photon or one-boson (submitted to Physical Review C) exchange Horn calculations for the radiative background. A more sophisticated potential approach including effects of the A based on a coupled PACS: 13.75.Cs, 25.20.-x, 25.40.-h channel calculation of the half off-shell NN- and N A T-matrices together with a phenonieno- logical JVA7-vcrtcx has been published recently [18,19]. In contrast to the njry-results of [15], the authors find an appreciable A contribution to the 280 MeV ppj observables. For higher energies where the potential model is inappropriate, the closely related prob- lem of dileplon production has recently been studied in the framework of an effective one- boson exchange plus A-excitation Born approximation [20]. As mentioned above, a rigorous derivation of induced two-body currents is not possible to all orders. The reason is that, unlike e. g. for a one-pion exchange potential [21], the gauge invariant replacement VN(P) -» VN (p*— e/l) in the argument of a general NN-potential Vtf [9,14] leads to au unique expression for the induced current only in SPA. To this order, the induced current approximates contributions due to the exchange of charged mesons and vanishes for ppi- The radiative decay processes we are considering here are thus not accounted for in the conventional potential model. So far, only the radiative ut —* x°7 decay for ppy has been considered as an example of such an O(Jt) internal radiation process [22|. The purpose of the present work is to provide an estimate of the rote of baryon reso- nances and internal radiation processes reliable up to photon energies of about 300 MeV corres|>oruling to the highest photon energy possible in the 730 MeV [ijrf-cxperiulcnt of VOL 2 7 H» 1 7 [2]. From a comparison with recent analyses of pion-photoprodiiclion [23], we expect the involve only on-shell nucléons, pseudovector coupling for the pion instead of equation (3) A(]232)-resonancc and the radiative decay of the u) and p to be the leading corrections. would not change any of our results. From eq. (3) - (6), the vertex functions read A coupled channel calculation for the NN — NA system omitting the contribution of the AA-states [24] is typically represented by a set of Lippmann-Schwinger equations = -igpHtl \-f - ÏAJV = VAN "I" V&NGNTNN + *^Û«A-^A/V» (2) Inclusion of the A thus modifies the NN-amplitude TNf/ and yields an additional ampli- and one derives the transition amplitudes for diagram (a) in Fig. 1 [17]: tude TAN. The pure NN-interaction below the pion production threshold is known to be well described by phenomenological and/or meson theoretic potentials so that we feel safe Pi - Jt)^r£tt(<fc,Pl - Jb)u(Pl) identifying TNN of equation (1) with a pure NN-T-malrix calculated from a realistic nuclear potential in effect absorbing the Vn&G&TfrH piece. By then adding the A-excitation Born terms, we basically neglect the modifications of the amplitude T^fj (equation (2)) induced by the iterative terms. Empirically, i. c. from a comparison of the Born amplitude V&N with (8) coupled channel A-absorplion predictions for lab energies of 50 —300 MeV [24], and exper- imental A-produclion data at 800 MeV [25] and 970 MeV [26], it turns out that the effect and equivalently for diagram (b) of the iteration can be simulated by a simple energy-dependent reseating of the Born ampli- tude in good approximation. We are then left basically with possible errors originating from the superposition of a (necessarily real and therefore not unitary) Born amplitude with the iterated potential model and SPA amplitudes. Our approach for the A is thus complemen- tary to a full coupled channel calculation [18] which systematically avoids the shortcomings mentioned above but cannot be extended easily to processes such as the internal radiative decays and - at least in the NNy-case - cannot be applied at higher energies. In the following two sections, a description of the model together with a discussion of Here, the energies and momenta are constrained according to pt + pj = P3+pt + k,q ~p^—p•t the parameters will be given. Results for ppy and njrj arc presented in section IV. We have is the four momentum transferred by the meson, t,, the photon unit polarization vector, u, « taken care to test thoroughly the influence of the experimental and theoretical uncertainties denote free nucléon Dirac spinors and P the appropriate propagators for the mesons and for the various ingredients of the amplitude that can not be fixed to accurate data. The the A. With the tilde in equation (9) we indicate that the vertex function has to be taken results might be useful to estimate the reliability not only of this but of any comparable from the Hcrmitian conjugate of the corresponding Lagrangian. calculation. Associated with eqs. (8) and (9) are isospin factors, e. g. in case of proton-proton brcmsstrahlung: I T3 I Xy)(Xy I r3 \ Xr) = | II. INCLUSION OF THE A The full A excitation amplitude including the isospin factors and a relative minus for graphs involving interchange of the final state particles is then For the derivation of the A-excitation terms depicted in Fig. 1 we start from the inter- action Lagrangians for the isovector mesons x and p (sec J27J) in the notation of [28]: (3) 3 T (4) (5) (6) l;p3,p4))}. (10) For the electromagnetic NA current of diagram (a) we use the parametrization [29] where V'A is the spin-3/2 Rarila-Schwinger field and f stands for the isospin operator for the transition of an isospin-1/2 and isospin-1 to an isospin-3/2 particle [27]. We have used the pscudscalar jrWiV-coupliiig for convenience. As the irAW-vcrticcs in our calculations ^-i-ff- W) + 2{{jrv - k • pg" s f-,#,»(<:,;») > equal to ryM^{k,p) up to a relative minus sign in the term proportional to G\. In writing down the Lorcnlz invariant propagators ( 15) and ( 16) we have dropped the static 2 1 The couplings G'i and G2 arc experimentally determined by a fit to the MI+/EI+ multipole approximation q* —• — q of the potential models A and B. As we an far from the pole in all pion-pholoproduclion cross section. Values for Gt and Gj range from G\ = 2.0, Gj = 0, our geometries and rescale our final amplitudes this has very little rlfccl on the end results.
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