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arXiv:1706.06382 · Physical review. D/Physical review. D.

Beyond integrability: Baryon-baryon backward scattering in the massive Gross-Neveu model

MICHAEL F. THIES

Published Oct 23, 2017• 3 citations
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Abstract

Due to integrability, baryon-baryon scattering in the massless Gross-Neveu model at large $N$ features only forward elastic scattering. A bare mass term breaks integrability and is therefore expected to induce backward elastic scattering as well as inelastic reactions. We confirm these expectations by a study of baryon-baryon scattering in the massive Gross-Neveu model near the nonrelativistic limit. This restriction enables us to solve the time-dependent Hartree-Fock equations with controlled approximations, using a combination of analytical methods from an effective field theory and the numerical solution of partial differential equations.

DOI
10.1103/physrevd.96.076012

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Due to integrability, baryon-baryon scattering in the massless Gross-Neveu model at large NNN features only forward elastic scattering. A bare mass term breaks integrability and is therefore expected to induce backward elastic scattering as well as inelastic reactions. We confirm these expectations by a study of baryon-baryon scattering in the massive Gross-Neveu model near the nonrelativistic limit. This restriction enables us to solve the time-dependent Hartree-Fock equations with controlled approximations, using a combination of analytical methods from an effective field theory and the numerical solution of partial differential equations.
The subject of this work time dependent solutions of the massless Gross-Neveu model. The Gross-Neveu model is a 1+1 dimensional field theory of N flavours of fermions with four-fermion interaction. In the limit of large flavour numbers N , this theory can be solved exactly by a mean field approximation equivalent to the Hartree-Fock method of relativistic many-particle physics. Nearly all solutions that have been found until now, like “kinks” and “DHN-baryons”, are time independent, since there exists a systematic way to generate such solutions. The only time dependent solution, the breather, has been found by guessing. In this thesis we present new time dependent solutions. Firstly, we show that a special case, the scattering of an arbitrary number of kinks, can be mapped onto an already solved problem. Secondly, we use the fact that Hartree-Fock potential and wave functions of the already known solutions can be written as fractions of polynomials of exponential functions. We make an ansatz of new polynomials, determine their coefficients via the Dirac equation and confirm the self consistency. That way we find analytic solutions for the scattering of up to four DHN-baryons or breathers and can even give a, yet unproven, formula for the scattering of an arbitrary number of such objects.
The two-dimensional Gross-Neveu model is anticipated to undergo a crystalline phase transition at high baryon charge densities. This conclusion is drawn from the mean-field approximation, which closely resembles models of Peierls instability. We demonstrate that this transition indeed occurs when both the rank of the symmetry group and the dimension of the particle representation contributing to the baryon density are large (the large N limit). We derive this result through the exact solution of the model, developing the large N limit of the Bethe ansatz. Our analytical construction of the large-N solution of the Bethe ansatz equations aligns perfectly with the periodic (finite-gap) solution of the Korteweg-de Vries (KdV) of the mean-field analysis.
We propose a model that describes approximately the strong-coupling limit of interacting electrons and holes in a one-dimensional quantum wire. An exact solution for this model at the SU ( n ) symmetric point of equal electron and hole masses and for a particular ratio of the forward- ( f ) to backward-scattering ( g ) is obtained by means of the Bethe ansatz. The model discussed is a hybrid between the chiral invariant Gross-Neveu model of quantum chromodynamics and the massive Thirring model. While the eigenstates exhibit a huge degeneracy in colour (spin) space as in the U → ∞ Hubbard model, the charge excitation spectrum comprizes excitons, bi-excitons and other electron-hole complexes (droplets) for not too repulsive interactions ( − 2/√3 ⩽ g ⩽ − 1). In the regime of strong repulsion ( g < − 2/√3), the bi-excitons and large droplets disappear from the spectrum.