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Correlator with tensor currents and two masses at two loops

Terry Generet, Nico Gubernari, Eetu Loisa

TL;DR

The paper addresses the problem of computing the vacuum-to-vacuum correlator of two flavour-non-diagonal tensor currents with two massive quarks at next-to-leading order, preserving full momentum dependence in the variable $q^2$. The authors implement a complete analytical two-loop calculation using IBP reduction to four master integrals, renormalise masses and currents in the $\overline{\text{MS}}$ scheme (with OS checks), and provide results and imaginary parts in machine-readable form, including the moments $\\chi_T(Q^2;k)$ used in dispersive analyses. They report numerical results for $\\chi_T(Q^2;k)$ and $\\chi_{AT}(Q^2;k)$ with four-scale uncertainty estimates, and they quote precise values such as $\\chi_T(0;3)=(4.92\pm0.22)\times10^{-4}\,\text{GeV}^{-2}$ and $\\chi_{AT}(0;3)=(2.39\pm0.14)\times10^{-4}\,\text{GeV}^{-2}$ for representative masses, finding good agreement with lattice QCD where available. The work resolves inconsistencies in previous tensor-current analyses, provides essential input for unitarity bounds and QCD sum rules, and demonstrates perturbative stability with NLO corrections reducing scale uncertainties, while indicating the need for NNLO to further tighten predictions for dispersive applications.

Abstract

We calculate the vacuum-to-vacuum correlator of two quark tensor currents with two massive quarks, retaining full momentum dependence. For the first time, we include perturbative corrections up to next-to-leading order. Our fully analytical expressions are provided in machine-readable form. Furthermore, we present numerical results for various input parameters, including an estimate of the scale uncertainties. Our results are essential input for applications of dispersive methods, including unitarity bounds and QCD sum rules.

Correlator with tensor currents and two masses at two loops

TL;DR

The paper addresses the problem of computing the vacuum-to-vacuum correlator of two flavour-non-diagonal tensor currents with two massive quarks at next-to-leading order, preserving full momentum dependence in the variable . The authors implement a complete analytical two-loop calculation using IBP reduction to four master integrals, renormalise masses and currents in the scheme (with OS checks), and provide results and imaginary parts in machine-readable form, including the moments used in dispersive analyses. They report numerical results for and with four-scale uncertainty estimates, and they quote precise values such as and for representative masses, finding good agreement with lattice QCD where available. The work resolves inconsistencies in previous tensor-current analyses, provides essential input for unitarity bounds and QCD sum rules, and demonstrates perturbative stability with NLO corrections reducing scale uncertainties, while indicating the need for NNLO to further tighten predictions for dispersive applications.

Abstract

We calculate the vacuum-to-vacuum correlator of two quark tensor currents with two massive quarks, retaining full momentum dependence. For the first time, we include perturbative corrections up to next-to-leading order. Our fully analytical expressions are provided in machine-readable form. Furthermore, we present numerical results for various input parameters, including an estimate of the scale uncertainties. Our results are essential input for applications of dispersive methods, including unitarity bounds and QCD sum rules.

Paper Structure

This paper contains 7 sections, 20 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Analytical calculation
  3. Theoretical framework and one-loop result
  4. Two-loop result
  5. Comparison with previous results
  6. Numerical results
  7. Summary and conclusion

Figures (3)

  • Figure 1: Leading-power contribution to the OPE at LO (a) and NLO (b)-(d). The crossed circles denote the insertion points of the tensor current.
  • Figure 2: Comparison of our LO (green) and NLO (blue) results with lattice QCD dataHarrison:2024iad (magenta) for different values of $m_1$. The central values are obtained with all scales fixed to $\mu_b \equiv \overline{m}_b(\overline{m}_b) = 4.183\,\text{GeV}$, whereas the procedure used to determine the associated uncertainties is described in the text.
  • Figure 3: $\chi_T(0;3)$ (left) and $\chi_{AT}(0;3)$ (right) as functions of $\overline{m}_1(\overline{m}_b)$. The LO results are shown in green, while the NLO results are shown in blue. The central values are obtained with all scales fixed to $\mu_b \equiv \overline{m}_b(\overline{m}_b) = 4.183\,\text{GeV}$, whereas the procedure used to determine the associated uncertainties is described in the text.