Relativistic NJL model with light and heavy quarks

Abstract

Abstract. We study the Nambu-Jona-Lasinio model with light and heavy quarks in a relativistic approach. We emphasize relevant regularization issues as well as the transition from light to heavy quarks. The approach of the electromagnetic meson form factor to the Isgur-Wise function in the heavy quark limit is also discussed. 11.30.Rd,12.39.Fe The physics of light and heavy quarks and their corresponding effective field theories cannot be more disparate even though a smooth transition between both limits is expected. In the case of light up, down and strange quarks the spontaneous breaking of chiral symmetry is the dominant feature which explains the mass gap between pions and kaons and the rest of the hadronic spectrum enabling the use of Chiral Perturbation Theory (ChPT) for energies much smaller than the mass gap [1,2]. In the opposite limit of heavy charm, bottom and top quarks, spin symmetry largely explains the degeneracy between hadronic states which differ only in the spin of the heavy quark like e.g. B(5280) vs B * (5325), or D(1870) vs D * (2010) and a systematic Heavy Quark Effective Theory (HQET) [3,4,5] can be designed for masses much larger than the mass gap. Besides these two fairly known extreme limits, the understanding of the transition from light to heavy quarks is not only of theoretical interest, but may also provide some insight into lattice simulations where the putative light quarks are most frequently artificially heavy. Unfortunately, there is no general framework describing the heavy-light transition in a model independent way, even though ChPT and HQET describe the extreme cases. PACS.In HQET the heavy quark limit is taken before implementing dimensional regularization because as is well known heavy particles do not decouple in this regularization scheme. For a heavy quark the relevant degrees of freedom are given by where Ψ (x) is the heavy quark spinor, m 0 is the heavy quark mass, and v µ is a quadrivector where the spacial components v corresponds to the velocity of the heavy quark and the time component is chosen in order to have v 2 = 1. After integrating out the irrelevant degrees of freedom, the resulting effective Lagrangian is expanded in 1/m 0 , and the propagator for the heavy quark effective field, in leading order, is given byk µ being the residual momentum of a heavy quark with total momentum k µ + m 0 v µ . In this work we discuss the heavy-light transition with the guidance of the Nambu-Jona-Lasinio (NJL) model for quarks (for reviews see e.g. [6,7,8,10]). The corresponding Lagrangian readswhere λ are the N 2 f − 1 flavour SU (N f ) Gell-Mann matrices andm 0 = diag(m u0 , m d0 , m s0 , ..., m n0 ) is a diagonal current mass matrix which explicitly breaks chiral invariance. With the exception of the mass term all flavours are treated on the same footing and Lagrangian (3) is invariant under the SU R (N f ) ⊗ SU L (N f ) chiral group and also under SU (N c ) global transformations. Summation over color and flavour indexes is implicit. As it is well known the NJL mode...

Relativistic NJL model with light and heavy quarks

“…It is possible, however, to overcome this problem in a way that the treatment of light and heavy mesons is as symmetric as possible, so that at any stage the heavy pseudoscalars would become Goldstone bosons if the quarks were light. Further details will be further elaborated elsewhere [21].…”