Quenched chiral perturbation theory to one loop

Abstract

We calculate the divergences of the generating functional of quenched Chiral Perturbation Theory at one loop, and renormalize the theory by an appropriate definition of the counterterms. We show that the quenched chiral logarithms can be accounted for by defining a renormalized B 0 parameter which, at lowest order, is proportional to the vacuum expectation value of the scalar quark density. Finally, we calculate several quantities at one loop to better analyze the modifications induced by quenching in the ultraviolet finite part of the one-loop corrections. We point out that some of the finite loop corrections may diverge in the chiral limit. PACS: 12.39.Fe, 12.38.Gc.

On the determination of nonleptonic kaon decays from K→π matrix elements

“…The (partially) quenched expansion we consider in this paper is systematic if we take M 2 to be of order p 2 , in other words, if we take the parameter m 2 0 to be of the same order as the quark mass, just as in the case of quenched ChPT [16,20,24].…”

Lattice extraction ofK→ππamplitudes to next-to-leading order in partially quenched and in full chiral perturbation theory

“…Also in the partially quenched case, the coefficient of the counterterm divergence depends on the number of sea quarks, N [23]. The N dependence of the necessary coefficients for the (8, 1)'s was calculated following [28,33], and the results are presented in Table 3.…”

“…The N dependence of the necessary coefficients for the (8, 1)'s was calculated following [28,33], and the results are presented in Table 3. This paper uses a different basis from [23] for the (8, 1)'s, and also several more LEC's appear here, so the calculation was redone for this work. The usual method was employed, expanding the action around the classical solution (background field method) and using a heat kernel expansion.…”

“…The counterterm expressions needed to construct this physical amplitude are given in [13] Eqs (21)(22)(23)(24), and the finite logarithmic contributions are given there in Appendix C. Counterterms needed to construct the above K → ππ amplitude can be obtained from K 0 → K 0 ; K + → π + , ∆I = 3/2 (non-degenerate quarks); and K → ππ, ∆I = 3/2 at only one value of the unphysical kinematics (e.g. UK1 3 ).…”

“…Therefore, the PQS method is used to compute the NLO amplitudes in this paper. Finally, it should be mentioned that the ∆I = 1/2, K → ππ amplitudes receive enhanced finite volume contributions in the partially quenched theory [22,23]. However, when m sea = m u = m d , the infra-red divergences in the K → ππ amplitudes (at UK1 and UK2) vanish in PQChPT in the infinite volume Minkowski space amplitudes.…”