Perturbative and non-perturbative renormalization results of the Chromomagnetic Operator on the Lattice

TL;DR

This paper investigates the renormalization and operator mixing of the Chromomagnetic Operator on the lattice, providing both perturbative and non-perturbative results to understand its behavior in quantum chromodynamics.

Contribution

It identifies the mixing pattern of the CMO with other operators in both dimensional regularization and lattice regularization, including non-perturbative matrix element measurements.

Findings

CMO mixes with 9 operators in dimensional regularization.

On the lattice, 3 additional operators with lower dimension also mix.

Non-perturbative measurements of the $K- o\pi$ matrix element are underway.

Abstract

The Chromomagnetic operator (CMO) mixes with a large number of operators under renormalization. We identify which operators can mix with the CMO, at the quantum level. Even in dimensional regularization (DR), which has the simplest mixing pattern, the CMO mixes with a total of 9 other operators, forming a basis of dimension-five, Lorentz scalar operators with the same flavor content as the CMO. Among them, there are also gauge noninvariant operators; these are BRST invariant and vanish by the equations of motion, as required by renormalization theory. On the other hand using a lattice regularization further operators with $d \leq 5$ will mix; choosing the lattice action in a manner as to preserve certain discrete symmetries, a minimul set of 3 additional operators (all with $d<5$) will appear. In order to compute all relevant mixing coefficients, we calculate the quark-antiquark (2-pt) and the quark-antiquark-gluon (3-pt) Green's functions of the CMO at nonzero quark masses. These calculations were performed in the continuum (dimensional regularization) and on the lattice using the maximally twisted mass fermion action and the Symanzik improved gluon action. In parallel, non-perturbative measurements of the $K-\pi$ matrix element are being performed in simulations with 4 dynamical ($N_f = 2+1+1$) twisted mass fermions and the Iwasaki improved gluon action.