Abstract
The Gross–Neveu model with UL(N f) × UR(N f) chiral symmetry is reconsidered in the large N c limit. The known analytical solution for the time dependent interaction of any number of twisted kinks and breathers is cast into a more revealing form. The (x, t)-dependent factors are isolated from constant coefficients and twist matrices. These latter generalize the twist phases of the single flavor model. The crucial tool is an identity for the inverse of a sum of two square matrices, derived from the known formula for the determinant of such a sum.
Highlights
Solvable model problems play a central role in teaching as well as in “intellectual body building” (JohnNegele)
The chiral GN model with UL(Nf ) × UR(Nf ) symmetry is probably one of the most complicated quantum field theories that one can still solve analytically, at least in the large Nc limit where semiclassical methods become exact. This has incited us to reconsider the question of soliton dynamics in this model
The main idea was to manipulate the formal expression for the mean field in such a way that the (x, t) dependent factors are neatly separated from constant coefficients and twist matrices
Summary
Solvable model problems play a central role in teaching as well as in “intellectual body building” Time dependent problems like hadron scattering can only indirectly be dealt with, for instance by calculating scattering lengths with Luscher’s method [4] These limitations make it desirable to gain some experience with interacting relativistic bound states of massless fermions in a reliable way, complementary to numerical studies of lattice QCD. The twisted kink in the one-flavor chiral GN model was discovered by Shei using inverse scattering theory [9] It is a “chord soliton” in the sense that its mean field traces out a straight line between two different vacua on the chiral circle. Compact analytical formulas are given for any number and complexity of solitons or breathers These works have been further extended to the multiflavor case, first by using methods akin to inverse scattering theory in condensed matter physics [12]. The proof of a mathematical identity is relegated to the appendix
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