Trace-anomaly-subtracted $\sigma$-mass for Heavy Quarks and Matching from Lattice QCD

Technical Summary: Trace-Anomaly-Subtracted $\sigma$-Mass for Heavy Quarks and Matching from Lattice QCD

Problem Statement
The precise determination of quark masses is critical for testing the Standard Model, yet the inherent color confinement of Quantum Chromodynamics (QCD) prevents the direct experimental observation of isolated quarks. Consequently, there is no unique theoretical definition of a "physical" quark mass. While the perturbative pole mass is a standard definition, it is plagued by a leading infrared (IR) renormalon ambiguity. This divergence prevents the pole mass from being meaningful to all orders in perturbation theory, introducing an intrinsic uncertainty that limits precision. Conversely, the $\overline{\text{MS}}$ mass is free from IR renormalons but is scheme- and scale-dependent, making it less intuitive as a "physical" mass parameter. The authors address the need for a mass definition that is both scheme/scale-invariant and free from the leading IR-renormalon ambiguity, while also being amenable to non-perturbative extraction from Lattice QCD.

Methodology
The authors investigate the contribution of the trace anomaly of the energy-momentum tensor (EMT) to the perturbative pole mass. Their methodology proceeds through several rigorous steps:

1. Identification of the IR-Renormalon Source: The authors demonstrate that the leading IR-renormalon divergence in the perturbative pole mass of a massive quark resides entirely within the contribution from the trace anomaly of the EMT. They utilize an exact identity between the on-shell matrix element of the EMT trace-anomaly operator and the derivative of the bare self-energy function with respect to the auxiliary dimensional regularization scale ($\hat{\mu}$).
2. Derivation of the $\sigma$-Mass Formula: By isolating the trace-anomaly contribution, they define the "trace-anomaly-subtracted $\sigma$-mass" ($m_\sigma$) as the remainder of the pole mass after subtracting this specific contribution. They derive an explicit, all-order conversion formula connecting $m_\sigma$ to the perturbative pole mass ($m_{\text{os}}$) and the $\overline{\text{MS}}$ mass ($m$). This formula relies solely on the QCD $\beta$-function, the quark-mass anomalous dimension $\gamma_m$, and the pole-to-$\overline{\text{MS}}$ mass conversion factor ($C_m$).
3. High-Order Perturbative Expansion: Using the derived master formula and existing literature results for the pole-to-$\overline{\text{MS}}$ conversion factor (up to four loops) and the $\beta$-function and $\gamma_m$ (up to five loops), the authors compute the perturbative relationship between $m_\sigma$ and $m_{\text{os}}$ up to five loops, and between $m_\sigma$ and $m$ up to four loops.
4. Lattice QCD Matching Strategy: To facilitate non-perturbative determination, the authors outline a strategy to extract $m_\sigma$ directly from Lattice QCD. This involves an intermediate, regularization-independent momentum-subtraction (RI/MOM) scheme. They calculate the perturbative matching factor ($C_{\text{RI}\to m_\sigma}$) connecting the RI/MOM mass to $m_\sigma$ up to three loops (with four-loop anomalous dimensions) and analyze the convergence properties of this series.

Key Contributions and Results

• Proof of IR-Renormalon Cancellation: The paper provides a rigorous proof that the leading IR-renormalon singularity in the perturbative pole mass is fully captured by the trace-anomaly contribution. Consequently, the subtracted $\sigma$-mass is proven to be free from this leading ambiguity.
• Compact All-Order Conversion Formulas:
  • The authors derive the ratio $Z_\sigma = m_\sigma / m_{\text{os}}$ (Eq. 17) and the ratio $m_\sigma / m$ (Eq. 18).
  • A notable feature of these formulas is that the perturbative result for $Z_\sigma$ at order $O(\alpha_s^N)$ depends on the pole-to-$\overline{\text{MS}}$ conversion factor $C_m$ only up to $O(\alpha_s^{N-1})$. This allows for higher-order predictions without requiring new, complex diagrammatic calculations.
• Explicit Five-Loop Results:
  • The authors present the explicit five-loop expansion for the ratio $m_\sigma / m_{\text{os}}$ (Eq. 20) and the four-loop expansion for $m_\sigma / m$ (Eq. 21).
  • They provide updated numerical values for the $\sigma$-mass of the top quark ($m_t^\sigma = 158.67 \pm 0.29$ GeV) and the bottom quark ($m_b^\sigma = 3.97^{+0.04}_{-0.03}$ GeV), derived from current PDG averages.
• Lattice Matching Coefficients:
  • The paper derives the perturbative matching factor $C_{\text{RI}\to m_\sigma}$ up to three loops (Eq. 23) and its logarithmic derivative up to four loops (Eq. 24).
  • Numerical evaluations at a benchmark scale ($\mu_s = 2 m_{\text{RI}}$) show that the perturbative series for $C_{\text{RI}\to m_\sigma}$ exhibits better apparent convergence than its $\overline{\text{MS}}$ counterpart.
  • The authors propose a method to determine $C_{\text{RI}\to m_\sigma}$ at scales far from the initial perturbative domain by solving the renormalization group equation (Eq. 24), thereby minimizing reliance on the perturbative $\alpha_s$ in non-perturbative regions.

Significance and Claims
The paper claims that the trace-anomaly-subtracted $\sigma$-mass definition offers a unique solution to the challenges of quark mass definitions. Its significance rests on three pillars:

1. Theoretical Robustness: It is unique for each quark flavor, scheme- and scale-invariant, and free from the leading IR-renormalon ambiguity that plagues the pole mass.
2. High-Precision Perturbative Control: The availability of explicit five-loop conversion relations allows for high-precision comparisons between different mass schemes.
3. Lattice Feasibility: The proposed matching strategy via the RI/MOM scheme enables the direct extraction of $m_\sigma$ from Lattice QCD without the need to pass through the scale-dependent $\overline{\text{MS}}$ mass as an intermediate step.

The authors conclude that due to these theoretical merits and the feasibility of direct extraction, the $\sigma$-mass is a promising candidate to serve as a unique common reference scheme for comparing or combining quark masses of the same flavor obtained under different definitions. They explicitly note that while the treatment of internal massive quark loops is approximated (treating them as massless in the effective Lagrangian), the resulting mass effects are small (below the percent level for $b$ and $c$ quarks) and comparable to residual uncertainties in current higher-order calculations.