Probing the chiral weak Hamiltonian at finite volumes

Abstract

Non-leptonic kaon decays are often described through an effective chiral weak Hamiltonian, whose couplings ("low-energy constants") encode all non-perturbative QCD physics. It has recently been suggested that these low-energy constants could be determined at finite volumes by matching the non-perturbatively measured three-point correlation functions between the weak Hamiltonian and two left-handed flavour currents, to analytic predictions following from chiral perturbation theory. Here we complete the analytic side in two respects: by inspecting how small ("ǫ-regime") and intermediate or large ("p-regime") quark masses connect to each other, and by including in the discussion the two leading ∆I = 1/2 operators. We show that the ǫ-regime offers a straightforward strategy for disentangling the coefficients of the ∆I = 1/2 operators, and that in the p-regime finite-volume effects are significant in these observables once the pseudoscalar mass M and the box length L are in the regime M L < ∼ 5.0. July 20061 pilar.hernandez@ific.uv.es 2

Exploring the role of the charm quark in theΔI=1/2rule

“…As is well-known, qChiPT displays unphysical artifacts; in particular, in the context of K → ππ transitions GoltermanPallante ambiguities make the matching of QCD to SU(3) qChiPT ill-defined [50,51]. This is however not the case for SU (4), where the ratios of correlation functions we will deal with (see below) present no ambiguities in the quenched approximation, as discussed in [28,33]. Quenched results are not worked out explicitly in [33] for SU(3) ChiPT.…”

“…This is however not the case for SU (4), where the ratios of correlation functions we will deal with (see below) present no ambiguities in the quenched approximation, as discussed in [28,33]. Quenched results are not worked out explicitly in [33] for SU(3) ChiPT. As can be seen in the formulae gathered in Appendix A, while the -regime formulae are essentially insensitive to quenching, the NLO prediction p-regime predictions for the relevant correlation functions in the octet channel displays 1/N f factors, that signal the need to take into account non-decoupled singlet contributions to repeat the computation in the quenched case.…”

“…(2.1) rewritten in terms of operators that transform in irreducible representations of the flavour group SU(3) L × SU(3) R spanned by the light u, d, s quarks. The outcome of this exercise is [33] …”

“…The construction of the relevant chiral effective weak Hamiltonians has been reviewed in [33]. Given a leading-order chiral Lagrangian of the form (either for 15) where M is the mass matrix and θ the vacuum angle, the leading-order SU(4) Hamiltonian reads 6…”

“…A variant of the above procedure consists of computing the correlation functions D ± 1,2 with -regime kinematics for the light quarks, as proposed in [33]. In that case the computation is carried out at fixed value of the topological charge ν, and parity is not preserved; as a result, for a given value of ν the contribution to D…”