Analytic inclusion of the scale dependence of the anomalous dimension matrix in Standard Model Effective Theory

TL;DR

This paper derives analytic solutions for the RGEs in Standard Model effective theory that incorporate the scale dependence of the anomalous dimension matrix, improving accuracy over traditional methods.

Contribution

It introduces a method to analytically solve RGEs considering the dominant scale dependence of the anomalous dimension matrix, including complex multi-operator mixing scenarios.

Findings

Analytic solutions account for running of gauge and Yukawa couplings.

Solutions applicable to direct and two-step operator mixing.

Generalized to multiple operators with complex mixing.

Abstract

The renormalization group equations (RGEs) in Standard Model effective theory are usually either solved analytically, neglecting the scale dependence of gauge and Yukawa couplings, or numerically without such approximations. We present analytic solutions of RGEs that take into account the dominant scale dependence of the anomalous-dimension matrix due to the running of the QCD coupling and of the top-Yukawa coupling. We consider first the case for which a given operator is generated directly through mixing with the parent operator whose Wilson coefficient is non-vanishing at the new physics scale. Subsequently we consider the case of two-step running, in which two operators do not mix directly, but only via a third mediator operator. We generalize these solutions to an arbitrary number of operators and show how even in this case analytic solutions can be obtained.