[Paper Review] Towards precise relativistic b quarks on the lattice

Craig McNeile, C. T. H. Davies+4 more|arXiv (Cornell University)|Oct 15, 2009

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper presents a method to calculate bottom quark properties using highly improved staggered quarks (HISQ) on the lattice, enabling precise, relativistic calculations without relying on effective field theories like NRQCD. By extrapolating from charm-mass to bottom-mass regimes across multiple lattice spacings, the authors achieve consistent results for bottomonium and B-meson spectroscopy and decay constants, with a projected 2–3% uncertainty on the bottom quark mass.

ABSTRACT

We discuss the status of our ongoing efforts to improve on our calculation of the $D_s$ decay constant. We show preliminary results on the ratio of the charm to the strange quark mass. We also present preliminary results for spectroscopy, decay constants and bottom quark mass obtained by performing calculations with highly improved staggered quarks at masses above the c mass and close to the b mass.

Motivation & Objective

• To extend the use of the highly improved staggered quark (HISQ) action to bottom quarks, enabling relativistic lattice QCD calculations with high precision.
• To overcome discretization errors in heavy quark calculations by using a relativistic formulation that avoids renormalization issues via PCAC relations.
• To perform a joint continuum and heavy quark mass extrapolation from charm to bottom quark masses using fine lattice ensembles.
• To determine the bottom quark mass with high accuracy using moments of pseudoscalar current correlators and continuum perturbation theory.
• To test the consistency of the method by comparing extrapolated results for B-meson masses, decay constants, and splittings with experimental data and previous NRQCD calculations.

Proposed method

• Utilizes the HISQ action, a relativistic discretization that removes tree-level $a^2$ errors and reduces taste-symmetry breaking via gauge field smearing.
• Employs PCAC (partially conserved axial current) to calculate decay constants without renormalization, reducing systematic errors.
• Performs joint $a \to 0$ and $M_h \to M_b$ extrapolations using ensembles with lattice spacings from 0.045 fm to 0.15 fm and heavy quark masses above charm.
• Calculates reduced moments of the pseudoscalar current-current correlator to extract $aM_h$, combining lattice data with high-order continuum perturbation theory.
• Uses the $g_n(\alpha_{\overline{MS}}, \mu/M_h)$ functions from Chetyrkin et al. to relate lattice moments to the bottom quark mass in the $\overline{\text{MS}}$ scheme.
• Applies the same method used for charm quark mass determination to obtain the bottom quark mass with controlled systematic errors.

Experimental results

Research questions

• RQ1 Can the HISQ action be used reliably for bottom quarks with controlled discretization errors?
• RQ2 Is the joint continuum and heavy quark mass extrapolation from charm to bottom quark masses feasible and accurate?
• RQ3 Can the bottom quark mass be determined with 2–3% uncertainty using relativistic lattice QCD and moments of current correlators?
• RQ4 Are the extrapolated values for $f_{B_s}$, $M_{B_s}$, and $\Delta M_{bs}$ consistent with experimental data and previous NRQCD results?
• RQ5 Can the relativistic HISQ approach achieve precision comparable to NRQCD for bottomonium and B-meson properties?

Key findings

• The method achieves a projected total uncertainty of 2–3% on the bottom quark mass using reduced moments of the pseudoscalar current correlator.
• Preliminary results for $\Delta M_{bs} = M_{B_s} - M_{B_b}/2$ and $f_{B_s}$ show smooth dependence on lattice spacing and heavy quark mass, with extrapolated values consistent with experiment and previous NRQCD calculations.
• The $f_{D_s}$ decay constant is calculated with 2% accuracy, now only about 2σ away from experiment after updated experimental values.
• The $m_c/m_s$ ratio is determined with high precision, and when combined with the 1% accurate $m_c$, yields a $\approx 1.5\%$ determination of $m_s$.
• The HISQ action enables consistent, unrenormalized calculation of decay constants via PCAC, eliminating a major source of systematic error.
• The approach is viable for bottom quark physics, and finer lattices (e.g., 0.03 fm) could further reduce extrapolation uncertainties.