The effective action of four-quark operators and non-leptonic weak transitions

Abstract

My talk at the Montpellier Workshop was a summary of recent work which has now appeared in print in quite a detailed form: [1] and [2]. Reference [1] describes the derivation, under a well defined set of approximations, of a low-energy effective chiral Lagrangian of the strong interactions from QCD. The effective Lagrangian includes only the Nambu-Goldstone modes (i.e., the octet of pseudoscalar particles) in the presence of external sources, with inclusion of the explicit chiral symmetry breaking due to the light quark masses M. The coupling constantsof the O( p 4) terms of the chiral Lagrangian are calculated first to leading order in the 1/ N c expansion but with O( α s ) terms and higher orders neglected. Next, the corrections O( α s ) to the couplings which remain in the chiral limit M → 0 are also taken into account. In reference [2], the matrix elements of four-quark operators of the effective weak Hamiltonian for ΔS = 1 and ΔS = 2 transitions are studied with two approaches. First inclusively : two-point functions of four-quark operators are studied. Their corresponding spectral functions at very short-distances are computed using perturbative QCD with inclusion of O( α s ) corrections; and at very long-distances, using the chiral effective realization of QCD in terms of Goldstoneparticles. A qualitative picture emerges which requires, for consistency between the two extreme behaviours, a large coupling constant for ΔI = 1 2 transitions. The other more direct approach described in ref. [2], consists in calculating the effective action of four-quark operators by integrating at the quark fields in a gluonic background and in the presence of a source term which triggers spontaneous chiral symmetry breaking. The procedure follows the method elaborated in ref. [1]. This way one can compute the coupling constants of the lowest order effective chiral Lagrangian with ΔS = 1(ΔI = 1 2 and 3 2 ) and ΔS = 2 (the so called B ̂ - parameter ). The calculations include the well known leading O( N c 2) contributions as well as the subleading O( N c ) with inclusion of the O( α s N c ) terms. The picture which emerges is very encouraging. Comparison with related work can also be found in the lists of references in [1] and [2].