Semileptonic weak Hamiltonian to $$ \mathcal{O} $$(ααs) in momentum-space subtraction schemes

Abstract

The CKM unitarity precision test of the Standard Model requires a systematic treatment of electromagnetic and strong corrections for semi-leptonic decays. Electromagnetic corrections require the renormalization of a semileptonic four-fermion operator. In this work we calculate the $$ \mathcal{O} $$ O (ααs) perturbative scheme conversion between the $$ \overline{\textrm{MS}} $$ MS ¯ scheme and several momentum-space subtraction schemes, which can also be implemented on the lattice. We consider schemes defined by MOM and SMOM kinematics and emphasize the importance of the choice of projector for each case. The conventional projector, that has been used in the literature for MOM kinematics, generates QCD corrections to the conversion factor that do not vanish for α = 0 and which generate an artificial dependence on the lattice matching scale that would only disappear after summing all orders of perturbation theory. This can be traced to the violation of a Ward identity that holds in the α = 0 limit. We show how to remedy this by judicious choices of projector, and define two new schemes $$ \overline{\textrm{RI}} $$ RI ¯ -MOM and $$ \overline{\textrm{RI}} $$ RI ¯ -SMOM. We prove that the Wilson coefficients in the new schemes are free from pure QCD contributions, and find that the Wilson coefficients (and operator matrix elements) have greatly reduced scale dependence. Our choice of the $$ \overline{\textrm{MS}} $$ MS ¯ scheme over the traditional W-mass scheme is motivated by the fact that, besides being more tractable at higher orders, unlike the latter it allows for a transparent separation of scales. We exploit this to obtain renormalization-group-improved leading-log and next-to-leading-log strong corrections to the electromagnetic contributions and study the (QED-induced) dependence on the lattice matching scale.

Semileptonic weak Hamiltonian to in momentum-space subtraction schemes

“…For MOM kinematics we derive the operator conversion factor using both the conventional projector, comparing against the literature and finding a total agreement with the previous works at O(α) [22], and our new RI-MOM projector described in [24]. W on the next-to-leading-log contribution: the upper limit is obtained with γ…”

“…In our work [24], we show that the conventional definition of the projector P [29] entering the condition (5) on the renormalized four-point function leads, in contrast to what happens in the MS, to a finite renormalization of the semi-leptonic operator even in the case QED is neglected. This can be traced to the violation of the QCD Ward identity (WI) for the quark current and it reflects in a scale dependence of the conversion factor.…”

“…For all the details on the technicalities of our calculations, we refer the reader to Ref. [24]. The respective conversion factors between the MS and the RI-MOM and RI-SMOM schemes exhibit a scale dependence that mirrors the scale dependence of the relevant semi-leptonic Wilson coefficient.…”

“…For the determination of the operator conversion factor in the SMOM kinematics we only consider our newly define scheme RI-SMOM [24]. Again, as in the MOM case, all the pure-QCD corrections cancel as expected.…”

“…[22], including the one-loop perturbative matching to the W -mass scheme. In our work [24], we perform the perturbative matching at two-loop level for different momentumspace subtraction schemes. These schemes are regulator-independent (RI) and are defined through a condition on a projected renormalized Green's function for a particular off-shell momentum configuration.…”