Structure of the Meson – Nucleon – N*(1535) Vertices M. Dillig * ** and C. V. Z. Vasconcellos* * Instituto di Fisica Universidade do Rio Grande do Sul, Porto Alegre RS, Brazil ** Institute for Theoretical Physics III University Erlangen – Nürnberg, Erlangen, Germany PACS: Keywords: Abstract We investigate the strength of the coupling of , , , , and mesons to the nucleon – N* S (1535) vertex. Starting from effective meson – baryon Lagrangians we calculate the influence of open pion, eta and two-pion – nucleon channels, which give rise to complex off shell corrections (form factors). We find a significant renormalization of the meson – NN* coupling constants at the threshold. Email: mdillig@theorie3.physik.uni-erlangen.de Email: Supported in part by the Kernforschungszentrum KFZ, Jülich, Germany In the last decade, particularly with the advent of modern sychrotrons with a phase space cooling of the beams (IUCF, CELSIUS, COSY), there has been an increasing experimental activity on the exclusive production of mesons in proton-proton collisions (and on light nuclei) near the meson thresholds (1). Characteristic for these reactions is the very large momentum transfer of 0.5 – 1.5 GeV/c – corresponding to a typical scale of less than ½ fm in coordinate space - already at threshold. Thus such processes test the dynamics of meson – exchange currents and baryon resonances and ultimately might bridge the gap to a more fundamental description in QCD degrees of freedom, i. e. quarks and gluons. In fact, a variety of specific meson channels, ranging from pion production up to the -meson, have been investigated (a recent survey on the meson production at COSY is given in ref. 2). In this note we focus on the exclusive production of the eta meson right at it s threshold at a CMS energy of s = 2.426 GeV, corresponding to a beam momentum of p = 0.768 GeV/c. The – meson production is very interesting from various reasons: among others it is the isoscalar SU(3) partner of the pion and thus acts as an isospin filter and it s cross section at excess energies a few MeV above threshold is one order of magnitude larger than the corresponding cross section for neutral pion production (3) (inspite of the much larger momentum transfer). This and other unique feature have triggered an intensive experimental program in the past years (4). In parallel there have been an increasing number of teoretical models for the production mechanism in the framework of effective meson – baryon Lagrangians: models with the excitation of specific baryon resonances (5,..,8), coupled channel models (9) or attempts to relate the production to the excitation of othere baryonic or more exotic mechanisms (10). There is concensus in the meson- exchange (MEC) models that threshold eta production is dominated by the negative parity S – resonance N*(1535) resonance. It has been found that the exchange of mesons (135), (548), (770), (783), (550) and (980) have a significant influence on the absolute magnitude of the total cross section (5,..8; presently only data on the total cross section are available). One serious problem of such caclulations is that the microscopic nature of the coupling constants entering in these models is not well understood: the relative phases between the various meson exchanges vary from calculation to calculation (such a problem is typical for most meson production reactions dominated by intermediate baryon resonances, to mention only as an example the open strangeness production in pp – pYK processes (11)). In this note we investigate the influence of open decay channels of the N*(1535) at the threshold on the meson NN* vertices. Starting from effective meson – baryon Lagrangians we caclulate imaginary corrections to the standard real coupling constants in the dominant one and two loop approximation with N, N and N intermediate states (Fig. 1). The corresponding fractional N* -- N decay rates are given as (12): N* N 35% – 55%, N 30% – 55%, N 1% – 10%; the decay rate into other channels is below 5%. We remark that for the kinematics at eta threshold the coupling to the channel in the and propagator is strictly forbidden (compare Fig. 2 for energy- momentum flow at the eta threshold). Fig. 1: Leading loop corrections from the N*(1535) coupling to the open meson – nucleon channels: one and and two loop vertex corrections (a,b), coupling to the meson cloud (c). __________ Fig. 2: Kinematics and notation for the one loop vertex correction at the threshold. Within standard Feynman rules the calculation of the various diagrams is straightforward. We examplify the main steps explicitly for the one-loop contri-bution. With the notation fig. 2 the resulting coupling constants are given as (note p = With ( = and ) (and correspondingly for the intermediate state). Above denote the various meson-baryon-N* and meson-meson vertices (13). With the identity (14) we reduce the corresponding relativistic NB Lagrangians to their leading (static) nonrelativistic limit and evaluate the diagrams in time-ordered perturbation theory, suppressing antiparticle excitation in the loop. With the substitution NN to the NN* vertices (6,7,15), we obtain in an obvious notation L= with L = L = L = from the L = Then for the pion loop with an intermediate N we obtain with E(k ) = and (k ) = . Next we perform two steps: - we isolate the imaginary part of the coupling constant from with - we choose the momentum of the incoming proton and the external meson p along the z-axis and perform the angular integration analytically. Combing all factors the final result is given as with the appreviations A = ,B= ;y = with y = y and from -y . In the contributions from the - isobar intermediate state exchange the overall structure is preserved (except of the coupling constants and masses only the spin and isospin factors change). The contributions from the loop and the . triangle are of similar structure and given as: - - 2 loop: triangle: All the other NN* coupling constants are calculated in the same spirit (the final results are detailed and summarized in ref. 16). With the imaginary component at hand, we determine the full structure of g** . In a first step we assume, that the magnitude of the imaginary component is small compared to the corresponding real part, and we derive the full imaginary part of each coupling constant as the sum of the various individual contributions (a more consistent determination of the coupling constants has to start from a selfconsistent inclusion of the full coupling constant, including the imaginary part at the vertices NN* in the loop and triangle diagrams itself, resulting in a set of coupled equations for tlhe coupling constants). In the representation with the mixing angle , we fix the absolute value from independent information on the squared coupling constants, such as from the corresponding N* decay or from selective scattering processes estimates in the literature. We collect the various coupling constants as input for our calculation from various sources in the literature. Concerning the NN coupling constants we follow the values of the Bonn group extracted from NN phase shifts (17). The NN*, NN* and NN* strengths are taken from the corresponding partial decay rates of the N* (1535) (12); for the , , and coupling we are guided by the recent analysis in refs. (6,7), based on universality arguments for the NN and the additional coupling constants we take N and N SU(6) relations (18) from s-wave pi N scattering (19), g NN* coupling constants. For from the and f width (12) or from the decay widths into the channel or related models (20,21); finally we fix g investigations on the from the width squared (except of the decay (22) or QCD sum rules (23), whereas g from recent is taken ( 12). We should remark, that practically all coupling constants NN coupling), show significant ambiguities up to a factor 2. A representative set of coupling constants is summarized in Tab. 1. Table 1: Compliation of coupling constants for our calculation. We present a few characteristic results of our calculation in Fig. 3 for the complex angle . For NN* the dominant loop correction from fig. 1 is the triangle diagram, which dominates the other diagrams by a roughly a factor 3 (Fig. 3(a)). Among the vertex corrections there is a cancellation between the N and the intermediate state in the exchange, resulting from the opposite sign in the isospin factors. From f* = 2 f the dominates the N contribution; and exchange are of similar importance (the small NN coupling is compensated by the large g** coupling; note, however, the large uncertainty in f (17); finally the ontribution is suppressed both by the weak coupling strength and the large energy denominators. Fig. 3: (a) Contribution of the various intermediate states (comp. Fig. 1) to the complex angle of the NN* coupling constant at the N threshold; (b) relative phases of the NN* coupling constants. Summing all contributions, we find for the various mesons the characteristic pattern for the complex angle in the NN* coupling constants as shown in Fig. 3(b) with a typical ambiguity up to 50% both in the absolute magnitude and in the mixing angle. Summarizing, we find a substantial inflluence of the open N, N and N channels on the microscopic structure of the meson – nucleon – N*(1535) coupling constants. Though the calculation of the imaginary component in the N* coupling is well under control due to the presence of effectively only three open N* decay channels, the results show still large uncertainties reflecting the poor knowledge of the absolute values of the various coupling constants in the model. Quantiatively the results will further change for a selfconsistent evaluation of the couplings (which, however, then requires a model for details of the full microscopic derivation from the underlying meson-quark coupling constants, a task, which is much harder to formulate and evaluate selfsonsistently). We expect that even our preliminary findings have a significant influence on a quantiative comparison of model calculations with the experiment. 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