Four quark operators for kaon bag parameter with gradient flow

Abstract

To study the CP-violation using the K 0 −K 0 oscillation, we need the kaon bag parameter which represents QCD corrections in the leading Feynman diagrams. The lattice QCD provides us with the only way to evaluate the kaon bag parameter directly from the first principles of QCD. However, a calculation of relevant four quark operators with theoretically sound Wilson-type lattice quarks had to carry a numerically big burden of extra renormalizations and resolution of extra mixings due to the explicit chiral violation. Recently, the small flow-time expansion (SFtX) method was proposed as a general method based on the gradient flow to correctly calculate any renormalized observables on the lattice, irrespective of the explicit violations of related symmetries on the lattice. To apply the SFtX method, we need matching coefficients, which relate finite operators at small flow times in the gradient flow scheme to renormalized observables in conventional renormalization schemes. In this paper, we calculate the matching coefficients for four quark operators and quark bilinear operators, relevant to the kaon bag parameter.

The effective electroweak Hamiltonian in the gradient-flow formalism

“…3 An immediate comparison of this result to the NLO expression of Ref. [10] is not possible, because the latter is obtained in the DR scheme. 4 Since the non-mixing basis in Ref.…”

The electroweak Hamiltonian in the gradient flow formalism

“…Our complete results as well as more details on our calculation and a comparison to the NLO results of of Ref. [12] will be presented elsewhere.…”

Flavor-diagonal CP violation: the electric dipole moment

“…The renormalization pattern of local operators can be ameliorated at finite flow time. The connection to QCD at can be done either perturbatively in the continuum [ 127 – 134 ], or non-perturbatively using Ward identities [ 88 , 135 , 136 ]. Below we discuss the matching to QCD in the context of the short flow-time expansion (SFTE).…”

“…a bilinear or a 4-fermion operator (see Ref. [ 134 ] for a study of ), is still greatly simplified because it renormalizes multiplicatively depending on the number of fermion fields, i.e. with and its powers we renormalize any fermion local operator at [ 88 ].…”