Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 026, 18 pages      hep-ph/0602165

Functional Integral Approaches to the Bosonization of Effective Multi-Quark Interactions with UA(1) Breaking

Brigitte Hiller a, Alexander A. Osipov b, Véronique Bernard c and Alex H. Blin a
a) Centro de Física Teórica, Departamento de Física da Universidade de Coimbra, 3004-516 Coimbra, Portugal
b) Laboratory of Nuclear Problems, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia
c) Laboratoire de Physique Théorique 3-5, Université Louis Pasteur, rue de l'Université, F-67084 Strasbourg, France

Received October 27, 2005, in final form February 13, 2006; Published online February 23, 2006

Low energy hadron phenomenology involving the (u,d,s) quarks is often approached through effective multi-quark Lagrangians with the symmetries of QCD. A very successful approach consists in taking the four-quark Nambu-Jona-Lasinio Lagrangian with the chiral UL(3) × UR(3) symmetry in the massless limit, combined with the UA(1) breaking six-quark flavour determinant interaction of 't Hooft. We review the present status and some very recent developments related to the functional integration over the cubic term in auxiliary mesonic variables that one introduces to bosonize the system. Various approaches for handling this functional, which cannot be integrated exactly, are discussed: the stationary phase approximation, the perturbative expansion, the loop expansion, their interrelation and importance for the evaluation of the effective action. The intricate group structure rules out the method of Airy's integral. The problem of the instability of the vacuum is stated and a solution given by including eight-quark interactions.

Key words: field theory; functional integral methods; stationary phase method; 't Hooft interactions; semiclassical corrections; effective action.

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