[Paper Review] Charmonium current-current correlators with Mobius domain-wall fermion

Katsumasa Nakayama, Brendan Fahy+1 more|arXiv (Cornell University)|Nov 30, 2015

Quantum Chromodynamics and Particle Interactions4 references3 citations

TL;DR

This study computes charmonium current-current correlators using $n_f=2+1$ Möbius domain-wall fermions on lattices with spacings $a = 0.083$, $0.055$, and $0.044$ fm to extract the $ar{\mathrm{MS}}$ charm quark mass and strong coupling constant $\alpha_s$ at $\mu = 3$ GeV. By matching lattice moments to continuum perturbative QCD predictions up to $O(\alpha_s^3)$ and including non-perturbative gluon condensate corrections, the authors determine $m_c(3\,\text{GeV}) = 0.9948(26)(16)(64)\,\text{GeV}$ and $\alpha_{\overline{\mathrm{MS}}}(3\,\text{GeV}) = 0.2514(74)(11)(58)$, with uncertainties from perturbation truncation, statistics, and systematics.

ABSTRACT

We calculate the charmonium correlators on the lattice with $n_f = 2+ 1$ Moebius domain wall fermion, and extract the charm quark mass and the strong coupling constant. Time moments are defined by current-current correlators, which have been calculated in the continuum theory by perturbation theory. We extract the charm quark mass by matching the lattice results with the corresponding perturbative QCD calculations, using the recently generated ensembles by the JLQCD collaboration at lattice spacings $a = 0.083, 0.055$, and $0.044$ fm.

Motivation & Objective

• To precisely determine the charm quark mass and strong coupling constant $\alpha_s$ in the $\overline{\mathrm{MS}}$ scheme using lattice QCD with $n_f=2+1$ Möbius domain-wall fermions.
• To reduce systematic errors by employing reduced moments $R_n$ that suppress discretization effects and improve matching between lattice and continuum perturbation theory.
• To account for non-perturbative power corrections via the gluon condensate $\langle (\alpha_s/\pi)G^{\mu\nu}G_{\mu\nu} \rangle$ by treating it as a free parameter in a simultaneous fit to multiple moments.
• To achieve sub-1% precision in $m_c$ and $\alpha_s$ by combining multiple moment ratios and assessing all major systematic uncertainties, including finite volume, $O(a^4)$, and electromagnetic/hyperfine effects.

Proposed method

• Compute the pseudoscalar current-current correlator $G(t)$ on the lattice using Möbius domain-wall fermions at three lattice spacings, then define time moments $G_n = \sum_t (t/a)^n G(t)$ for even $n > 4$.
• Construct reduced moments $R_n = \frac{a m_{\eta_c}^{\mathrm{lat}}}{2a m_{\mathrm{bare},c}^{\mathrm{lat}}} \left( \frac{G_n}{G_n^{(0)}} \right)^{1/(n-4)}$ to minimize discretization effects.
• Match lattice $R_n$ to continuum perturbative predictions $r_n = (g_{2n}/g_{2n}^{(0)})^{1/(2n-4)}$ derived from the vacuum polarization function $\Pi(q^2)$ expanded in $\alpha_s$ up to $O(\alpha_s^3)$.
• Include non-perturbative corrections via the gluon condensate by modifying the perturbative coefficients $C_{k-1} \to C_{k-1} + \frac{16\pi^2}{3Q_q^2} \frac{\langle (\alpha_s/\pi)G^{\mu\nu}G_{\mu\nu} \rangle}{(2m_c)^4} A_k$, treating the condensate as a fit parameter.
• Simultaneously fit $R_6/R_8$, $R_8$, and $R_{10}$ to extract $m_c(\mu)$, $\alpha_s(\mu)$, and the normalized gluon condensate $\langle (\alpha_s/\pi)G^2 \rangle / m_c^4$ at $\mu = 3$ GeV.
• Estimate systematic errors from perturbation truncation, statistical uncertainties, lattice spacing dependence, finite volume, $O(a^4)$ effects, disconnected contractions, electromagnetic shifts, and hyperfine splitting.

Experimental results

Research questions

• RQ1 What is the precise value of the $\overline{\mathrm{MS}}$ charm quark mass at $\mu = 3$ GeV, determined from lattice QCD with Möbius domain-wall fermions and matched to continuum perturbation theory?
• RQ2 How accurately can the strong coupling constant $\alpha_s$ be extracted at $\mu = 3$ GeV using lattice moments of charmonium current-current correlators?
• RQ3 To what extent do non-perturbative gluon condensate corrections improve the consistency between lattice data and perturbative predictions in the moment method?
• RQ4 How do various systematic uncertainties—such as finite volume, $O(a^4)$ effects, and electromagnetic/hyperfine shifts—affect the final determination of $m_c$ and $\alpha_s$?
• RQ5 Can the ratio of reduced moments $R_n/R_{n+2}$ provide a more robust determination of $m_c$ and $\alpha_s$ by reducing truncation errors in perturbation theory?

Key findings

• The charm quark mass at $\mu = 3$ GeV is determined as $m_c(3\,\text{GeV}) = 0.9948(26)(16)(64)\,\text{GeV}$, with statistical, perturbative truncation, and systematic errors reported separately.
• The strong coupling constant at $\mu = 3$ GeV is found to be $\alpha_{\overline{\mathrm{MS}}}(3\,\text{GeV}) = 0.2514(74)(11)(58)$, with uncertainties from perturbation theory, statistics, and systematics.
• The normalized gluon condensate is extracted as $\langle (\alpha_s/\pi)G^{\mu\nu}G_{\mu\nu} \rangle / m_c^4 = 0.0007(14)(0)(1)$, with the first error from perturbation truncation and the second from statistics.
• The final result for $m_c$ at the charm scale is $m_c(\mu = m_c) = 1.2769(91)\,\text{GeV}$, obtained via renormalization group evolution.
• The value of $\alpha_s$ at the $Z$-boson mass scale is $\alpha_{\overline{\mathrm{MS}}}(M_Z) = 0.1174(20)$, consistent with global fits to the Standard Model.
• Systematic error analysis shows that the dominant uncertainties arise from $O(a^4)$ effects (0.5% for $m_c$, 1.9% for $\alpha_s$) and perturbation theory truncation (0.3% and 2.9%, respectively).