A note on the power divergence in lattice calculations of ∆ I = 1 / 2 K → ππ amplitudes at M K = M π

In this note, we clarify a point concerning the power divergence in lattice calculations of ∆I = 1/2 K → ππ decay amplitudes. There have been worries that this divergence might show up in the Minkowski amplitudes at MK = Mπ with all the mesons at rest. Here we demonstrate, via an explicit calculation in leading-order Chiral Perturbation Theory, that the power divergence is absent at the above kinematic point, as predicted by CPS symmetry. PACS numbers: 11.15.Ha,12.38.Gc,12.15Ff The subtraction of a power divergence, which arises via the mixing of dimension-six four-fermion operators with those of lower dimension has been one of the central issues in lattice calculations of ∆I = 1/2 K → ππ amplitudes. This power divergence is of course unphysical, and can be related to a shift of the vacuum due to the inclusion of the weak interaction in Chiral Perturbation Theory (χPT) [1, 2, 3, 4]. It results in the so-called tadpole operators, which contribute to the processes K → |0〉 and K̄ → |0〉, in χPT with weak interactions. As argued in Ref. [5], this power divergence should be absent for K → ππ amplitudes when ms = md = mu (mu,d,s are the masses of u, d and s quarks), due to the exact CPS symmetry [1] of the fourfermion operators that mediate K → ππ decays. In Ref. [6], it was argued that the power divergence indeed does disappear in Euclidean space at MK = Mπ. However, a naive calculation in Minkowski space suggests that this power divergence might still be present at MK = Mπ when all mesons are at rest. The issue is relevant, as it has been proposed that this unphysical kinematic point can be used to extract the low-energy constants relevant for ∆I = 1/2 K → ππ to order p in χPT [7, 8]. In this note, we show, via an explicit calculation in χPT, that also in Minkowski space the power divergence is not present in ∆I = 1/2 K → ππ amplitudes at MK = Mπ, with all mesons at rest. Since it has already been argued in Ref. [9] that the ∆I = 1/2 K → ππ amplitudes in partially quenched χPT at the kinematic point MK = Mπ suffer from problems related to the lack of unitarity [6, 10, 11], we concentrate here on full QCD. Our conclusions on the power divergence will however not change in the (partially) quenched case. To simplify the discussion, we only consider weak operators in the (8, 1) irrep of SU(3)L × SU(3)R. The weak mass operator in this irrep at O(p2) in the chiral expansion is O 2 = α2 × {