$\pi N\to \eta N$ process in a $\chi$QM approach

A chiral quark model approach is used to investigate the $\pi^{-}p \to \eta n$ process at low energies. The roles of the most relevant nucleon resonances in $n\leq 2$ shells are briefly discussed.


Introduction
The π − p → ηn reaction provides a suitable probe to investigate the structure of low-lying nucleon resonances as well as the ηN interaction.
Recent high precision data released by the BNL Crystal Ball Collaboration [1] has revived the interest in that process.The impact of those data on the meson-baryon interactions has been emphasized by the SAID Group [2].Extensive theoretical efforts are also being deployed via coupled-channel formalisms, such as the K-matrix approach [3], meson-exchange model [4], chiral model [5], T-matrix [6], and dynamical formalism [7].
We have extended to the πN → ηN process a comprehensive and unified approach [8] to the meson photoproduction, based on the low energy QCD Lagrangian in terms of quarks degrees of freedom.This latter formalism has been developed and proven [9] to be successful in investigating γp → ηp, K + Λ and γN → πN reactions.In this approach, only a few parameters are required.In particular, only one parameter is needed for the nucleon resonances to be coupled to the pseudoscalar mesons.All the resonances can be treated consistently in the quark model.

Theoretical frame
In the chiral quark model, the low energy quark-meson interactions are described by the effective Lagrangian where vector (V µ ) and axial (A µ ) currents read with ξ = exp (iφ m /f m ), where f m is the meson decay constant.ψ and φ m are the pion and quark fields, respectively.The η meson production amplitude can be expressed in terms of Mandelstam variables, The s-and u-channel transitions are given by: where ω π and ω η are the energies of the incoming π-meson and outgoing ηmeson, respectively.H π and H η are the standard quark-meson couplings at tree level.|N i , |N j , and |N f stand for the initial, intermediate, and final state baryons, respectively, and their corresponding kinetic energies are E i , E j , and E f .Given that the a 0 meson decay is dominated by πη channel [11], we consider the a 0 exchange as the prominent contribution to the t-channel, where m a 0 is the mass of the a 0 meson.With above effective Lagrangian and following the procedures used in Ref. [8], we obtain the amplitude in the harmonic oscillator basis [10].

Results and discussion
Using the formalism sketched above, we have investigated the cross-section for the π − p → ηn process.In our model, non-resonant components include nucleon pole term, u-channel contributions (treated as degenerate to the harmonic oscillator shell n), and t-channel contributions due to the a 0 -exchange.
Here we use the Breit-Wigner masses and widths given in the PDG [11].For meson-nucleon-nucleon couplings we adopt g πN N =13.48 and g ηN N =0.81.
Our results for the differential cross-section are depicted in Fig. [1] for pion incident momenta P lab π = 0.718, 0.850, and 1.005 GeV, corresponding to the total centre-of-mass energies W = 1.507, 1.576, and 1.674 GeV, respectively.We get a good agreement with the data at those energies (full curves).In order to single out the importance of various resonances, at each energy we show results while one significant resonance is switched off.The S 11 (1535) plays a crucial role in this energy range.At the lowest energies it has a constructive effect, while at the highest one its contribution becomes destructive.The S 11 (1650) has a (much) smaller and destructive effect.The role of the D 13 (1520), shown at W=1.576 GeV, is merely to produce the right curvature.At the highest energy, although the overall contribution from n=2 shell is rather small, the P 11 (1710) produces significant effects.This point was emphasized in our recent work [10], and led us to adopt here a reversed sign for that resonance from the beginning.That sign change for the P 11 (1710) could be an indication, e.g. for the breakdown of the non-relativistic constituent quark model or for unconventional configurations inside that resonance.More investigation is needed to underpin the origins of this novelty.