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SIGMA 2 (2006), 026, 18 pages hep-ph/0602165
https://doi.org/10.3842/SIGMA.2006.026
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
Abstract
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.
pdf (308 kb)
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