Heavy Quark Masses from Sum Rules in Four-Loop Approximation

TL;DR

This work determines the charm and bottom MS-bar quark masses by applying low-n moment sum rules to the e+e- → hadrons cross-section near heavy-quark thresholds, incorporating recent four-loop corrections. By carefully modeling the background and using BES data in the charm region and Υ resonances in the bottom region, the authors extract m_c(3 GeV)=0.986(13) GeV (equivalently m_c(m_c)=1.286(13) GeV) and m_b(10 GeV)=3.609(25) GeV (equivalently m_b(m_b)=4.164(25) GeV), with substantially reduced theoretical uncertainties due to four-loop contributions. The analysis emphasizes stability across moments n=1–3 and provides a precise framework for translating threshold-region data into short-distance MS-bar masses. These results offer improved inputs for precision QCD phenomenology and GUT-related Yukawa unification studies, and underscore the value of four-loop calculations in heavy-quark mass determinations.

Abstract

New data for the total cross section $σ(e^+e^-\to{hadrons})$ in the charm and bottom threshold region are combined with an improved theoretical analysis, which includes recent four-loop calculations, to determine the short distance $\bar{\rm MS}$ charm and bottom quark masses. A detailed discussion of the theoretical and experimental uncertainties is presented. The final result for the $\bar{\rm MS}$-masses, $m_c(3 {GeV})=0.986(13)$ GeV and $m_b(10 {GeV})=3.609(25)$ GeV, can be translated into $m_c(m_c)=1.286(13)$ GeV and $m_b(m_b)=4.164(25)$ GeV. This analysis is consistent with but significantly more precise than a similar previous study.

Figures (10)

• Figure 1: Feynman diagrams contributing to $R(s)$ at order $\alpha_s^2$. A secondary charm quark pair is produced through gluon splitting.
• Figure 2: Sample singlet diagram contributing to $\Pi(q^2)$ which arises at order $\alpha_s^3$ for the first time. The cuts (indicated by the dashed lines; not all possible cuts are shown) represent contributions to $R(s)$.
• Figure 3: $R(s)$ for different energy intervals around the charm threshold region. The solid line corresponds to the theoretical prediction, the uncertainties obtained from the variation of the input parameters and of $\mu$ are indicated by the dashed curves. The inner and outer error bars give the statistical and systematical uncertainty, respectively.
• Figure 4: $\delta R_{\rm uds(c)}^{(2)}$ for $2~\hbox{GeV}\le \sqrt{s}\le 5~\hbox{GeV}$. Both the exact result (solid line) and the approximations including terms of order $(s/M_c^2)^2$ (dashed) and $(M_c^2/s)^2$ (dash-dotted), respectively, are shown and $\mu^2=s$ has been chosen.
• Figure 5: $\delta R_{\rm uds(c)}^{(3)}$ for $2~\hbox{GeV}\le \sqrt{s}\le 5~\hbox{GeV}$. The approximations including terms of order $(s/M_c^2)^2$ (dashed) and $(M_c^2/s)^2$ (dash-dotted), respectively, are shown and $\mu^2=s$ has been chosen.