On the $b$-quark running mass in QCD and the SM

TL;DR

This paper analyzes electroweak corrections to relate the $b$-quark's running mass in QCD to its Standard Model parameters, proposing a gauge-independent method to extract the Yukawa coupling at the electroweak scale.

Contribution

It introduces a gauge-independent prescription for relating the $b$-quark's running mass to its Yukawa coupling, enabling systematic resummation of large logarithms.

Findings

Electroweak corrections are small compared to $y_b$ and $M_b$ relations.

The proposed relation allows direct determination of the $b$-quark Yukawa coupling.

The method resums large logarithms systematically.

Abstract

We consider electroweak corrections to the relation between the running $\overline{\mathrm{MS}}$ mass $m_b$ of the $b$ quark in the five-flavor QCD$\times$QED effective theory and its counterpart in the Standard Model (SM). As a bridge between the two parameters, we use the pole mass $M_b$ of the $b$ quark, which can be calculated in both models. The running mass is not a fundamental parameter of the SM Lagrangian, but the product of the running Yukawa coupling $y_b$ and the Higgs vacuum expectation value. Since there exist different prescriptions to define the latter, the relations considered in the paper involve a certain amount of freedom. All the definitions can be related to each other in perturbation theory. Nevertheless, we argue in favor of a certain gauge-independent prescription and provide a relation which can be directly used to deduce the value of the Yukawa coupling of the $b$ quark at the electroweak scale from its effective QCD running mass. This approach allows one to resum large logarithms $\ln(m_b/M_t)$ systematically. Numerical analysis shows that, indeed, the corrections to the proposed relation are much smaller than those between $y_b$ and $M_b$.