Relating bottom quark mass in DR-bar and MS-bar regularization schemes

TL;DR

This paper tackles the problem of obtaining the bottom-quark mass in the DR-bar scheme at the $Z$-boson scale, $m_b^{\overline{\rm DR}}(M_Z)$, from experimental data typically quoted in the MS-bar scheme, $m_b^{\overline{\rm MS}}(m_b)$. The authors present a two-step method: (i) evolve $m_b^{\overline{\rm MS}}(m_b)$ to $M_Z$ using NNLO renormalization-group running, and (ii) convert the result to the DR-bar scheme at the same scale via a two-loop MS–DR relation, avoiding the ambiguities associated with the pole mass. They show that the MS–DR conversion exhibits excellent perturbative convergence and provide explicit formulas, including $m_b^{\overline{\rm DR}}(\mu) = m_b^{\overline{\rm MS}}(\mu)\left( 1 - \frac{1}{3}a_s(\mu) - \frac{29}{72}a_s(\mu)^2 + O(a_s^3) \right)$. Numerically, they find $m_b^{\overline{\rm DR}}(M_Z) = 2.83 \pm 0.11$ GeV for $m_b^{\overline{\rm MS}}(m_b) = 4.2 \pm 0.1$ GeV and $\alpha_s(M_Z)_{\overline{\rm MS}} = 0.1172$, with a broader range $2.65$–$3.03$ GeV corresponding to $m_b^{\overline{\rm MS}}(m_b) = 4.0$–$4.4$ GeV. This provides a robust bottom-mass input for supersymmetric analyses with large $\tan\beta$ and potential Yukawa unification constraints.

Abstract

The value of the bottom quark mass at M_Z in the DR-bar scheme is an important input for the analysis of supersymmetric models with a large value of tan(beta). Conventionally, however, the running bottom quark mass extracted from experimental data is quoted in the MS-bar scheme at the bottom mass scale. We describe a two loop procedure for the conversion of the bottom quark mass from MS-bar to DR-bar scheme. The Particle Data Group value m_b(m_b) = 4.2 \pm 0.2 GeV in the MS-bar scheme corresponds to a range of 2.65-3.03 GeV for m_b(M_Z) in the DR-bar scheme.

Figures (1)

• Figure 1: $m_b(M_Z)$ as a function of $m_b^{\overline{\rm MS}} (m_b)$. The solid lines show the results for $m_b^{\overline{\rm DR}}(M_Z)$ using the first procedure described in the text, where we first run $m_b^{\overline{\rm MS}}$ to $M_Z$, and then convert to the $\overline{\rm DR}$ mass at the same scale. The lines correspond from top to bottom to values of $\alpha_s(M_Z)_{\overline{\rm MS}}=0.1152,0.1172,0.1192$. The dashed lines show the corresponding results when we first convert to the $\overline{DR}$ scheme, and then evolve to $Q=M_Z$. Finally, the dotted line shows the value of $m_b^{\overline{\rm MS}} (M_Z)$ for $\alpha_s(M_Z)_{\overline{\rm MS}}=0.1172$.