Neutral kaon mixing beyond the Standard Model with n f = 2 + 1 chiral fermions. Part 2: non perturbative renormalisation of the ΔF = 2 four-quark operators

Abstract

We compute the renormalisation factors (Z-matrices) of the ∆F = 2 fourquark operators needed for Beyond the Standard Model (BSM) kaon mixing. We work with n f = 2+1 flavours of Domain-Wall fermions whose chiral-flavour properties are essential to maintain a continuum-like mixing pattern. We introduce new RI-SMOM renormalisation schemes, which we argue are better behaved compared to the commonly-used corresponding RI-MOM one. We find that, once converted to MS, the Z-factors computed through these RI-SMOM schemes are in good agreement but differ significantly from the ones computed through the RI-MOM scheme. The RI-SMOM Z-factors presented here have been used to compute the BSM neutral kaon mixing matrix elements in the companion paper [1]. We argue that the renormalisation procedure is responsible for the discrepancies observed by different collaborations, we will investigate and elucidate the origin of these differences throughout this work.

BSM Kaon Mixing at the Physical Point

“…We use the non-perturbative Rome-Southampton method [17] with non-exceptional kinematics (RI-SMOM) [18]. The RI-SMOM scheme for the SM four-quark operator is described in [19], and the extension to the full SUSY basis in [20]. The renormalised matrix elements can be expressed as,…”

“…Details on the renormalisation procedure and the definition of these projectors can be found in [20].…”

Fierz transformations and renormalization schemes for fourquark operators

“…Therefore, the γ µ and / q-projectors defined above (and in [13]) lead to independent renormalisation schemes.…”

“…For the operators Q 3,4 = S S ∓ PP, where no γ µ structure is present, the trick is to use a Fierz identity [13] …”

“…The latter are also compatible with SWME [5,6] where the renormalisation is performed at 1-loop in perturbation theory. In [13], we studied in details the RI-MOM and SMOM schemes, and pointed out potential issues with the RI-MOM procedure with exceptional kinematics. Using non-diagonal Fierz transformation, I show here that the SMOM schemes called (γ µ , γ µ ) and (/ q, / q) are mathematically different.…”