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Bottom Quark Mass from Upsilon Mesons

A. H. Hoang

Abstract

The bottom quark pole mass $M_b$ is determined using a sum rule which relates the masses and the electronic decay widths of the $Υ$ mesons to large $n$ moments of the vacuum polarization function calculated from nonrelativistic quantum chromodynamics. The complete set of next-to-next-to-leading order (i.e. ${\cal{O}}(α_s^2, α_s v, v^2)$ where $v$ is the bottom quark c.m. velocity) corrections is calculated and leads to a considerable reduction of theoretical uncertainties compared to a pure next-to-leading order analysis. However, the theoretical uncertainties remain much larger than the experimental ones. For a two parameter fit for $M_b$, and the strong $\bar{MS}$ coupling $α_s$, and using the scanning method to estimate theoretical uncertainties, the next-to-next-to-leading order analysis yields 4.74 GeV $\le M_b\le 4.87$ GeV and $0.096 \le α_s(M_z) \le 0.124$ if experimental uncertainties are included at the 95% confidence level and if two-loop running for $α_s$ is employed. $M_b$ and $α_s$ have a sizeable positive correlation. For the running $\bar{MS}$ bottom quark mass this leads to 4.09 GeV $\le m_b(M_{Υ(1S)}/2)\le 4.32$ GeV. If $α_s$ is taken as an input, the result for the bottom quark pole mass reads 4.78 GeV $\le M_b\le 4.98$ GeV (4.08 GeV $\le m_b(M_{Υ(1S)}/2)\le 4.28$ GeV) for $0.114\lsim α_s(M_z)\le 0.122$. The discrepancies between the results of three previous analyses on the same subject by Voloshin, Jamin and Pich, and Kühn et al. are clarified. A comprehensive review on the calculation of the heavy quark-antiquark pair production cross section through a vector current at next-to-next-to leading order in the nonrelativistic expansion is presented.

Bottom Quark Mass from Upsilon Mesons

Abstract

The bottom quark pole mass is determined using a sum rule which relates the masses and the electronic decay widths of the mesons to large moments of the vacuum polarization function calculated from nonrelativistic quantum chromodynamics. The complete set of next-to-next-to-leading order (i.e. where is the bottom quark c.m. velocity) corrections is calculated and leads to a considerable reduction of theoretical uncertainties compared to a pure next-to-leading order analysis. However, the theoretical uncertainties remain much larger than the experimental ones. For a two parameter fit for , and the strong coupling , and using the scanning method to estimate theoretical uncertainties, the next-to-next-to-leading order analysis yields 4.74 GeV GeV and if experimental uncertainties are included at the 95% confidence level and if two-loop running for is employed. and have a sizeable positive correlation. For the running bottom quark mass this leads to 4.09 GeV GeV. If is taken as an input, the result for the bottom quark pole mass reads 4.78 GeV GeV (4.08 GeV GeV) for . The discrepancies between the results of three previous analyses on the same subject by Voloshin, Jamin and Pich, and Kühn et al. are clarified. A comprehensive review on the calculation of the heavy quark-antiquark pair production cross section through a vector current at next-to-next-to leading order in the nonrelativistic expansion is presented.

Paper Structure

This paper contains 16 sections, 102 equations, 13 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The Basic Ideas and Notation
  3. Calculation of the Moments
  4. Calculation of the Nonrelativistic Correlators
  5. Determination of the Short-distance Coefficients
  6. The Dispersion Integration
  7. Some Comments to the Moments
  8. Experimental Moments the Fitting Procedure
  9. Numerical Results and Discussion
  10. Determination of $M_b$ and $\alpha_s$ without Constraints
  11. Determination of $M_b$ with Constraints on $\alpha_s$
  12. Comments on Previous Analyses
  13. Conclusions and Outlook
  14. NNLO Corrections from $\delta H_{kin}$, $V_{\hbox{\tiny BF}}$ and $V_{\hbox{\tiny NA}}$
  15. Inverse Laplace Transforms
  16. ...and 1 more sections

Figures (13)

  • Figure 1: Graphical representation of the longitudinal and the transverse gluon exchange including the corresponding Feynman rules for the momentum exchange $q=(q^0,\vec{q})$. The exchange of a longitudinal gluon is instantaneous in time because its does not have an energy dependence. As a consequence the longitudinal exchange can be described by an instantaneous potential. The exchange of a transverse gluon, on the other hand, is retarded in time and, in general, cannot be described in terms of an instantaneous potential.
  • Figure 2: Graphical representation of the resummation of Coulomb ladder diagrams to all orders. The quark-antiquark propagation contains the nonrelativistic kinetic energy. The resummation is carried out explicitly by calculating the Green function the nonrelativistic Schrödinger equation with the Coulomb potential at the Born level, see Eq. (\ref{['CoulombGreenfunctioncomplete']}).
  • Figure 3: Typical diagrams describing the exchange of a transverse gluons (in Coulomb gauge) in the background of the Coulomb exchange of longitudinal gluons. Longitudinal lines with a $\sum$ sign represent the summation of Coulomb ladder diagrams to all orders, see Fig. \ref{['figladder']}.
  • Figure 4: Vertex diagram in Coulomb gauge responsible for the potential non-Abelian potential $V_{\hbox{\tiny NA}}$.
  • Figure 5: Symbols describing the interactions potentials $V_c^{(0)}$, $V_c^{(1)}$, $V_c^{(2)}$, $V_{\hbox{\tiny BF}}$ and $V_{\hbox{\tiny NA}}$ and the kinetic energy correction $\delta H_{\hbox{\tiny kin}} = -\vec{\nabla}^4/4 M_b^3$.
  • ...and 8 more figures