The QCD energy-momentum tensor on the lattice: non-perturbative renormalization with $N_f=3$

Technical Summary: The QCD Energy-Momentum Tensor on the Lattice: Non-Perturbative Renormalization with $N_f = 3$

Problem Statement
The energy-momentum tensor (EMT), $T_{\mu\nu}$, is central to Quantum Chromodynamics (QCD), encoding the theory's response to space-time transformations and external gravitational fields. It governs the realization of scale transformations, provides conserved currents for Poincaré invariance, and determines physical observables ranging from hadron gravitational form factors and the origin of hadron masses to the thermodynamic properties and transport coefficients (viscosities) of the QCD plasma.

While the definition and conservation of $T_{\mu\nu}$ follow directly from space-time symmetries in the continuum, its non-perturbative definition on the lattice faces significant challenges. The discretization of space-time explicitly breaks the $SO(4)$ invariance down to the hypercubic group. Consequently, the irreducible nonet representation of the traceless part of $T_{\mu\nu}$ splits into a triplet and a sextet irreducible representations of the hypercubic group. Furthermore, at finite lattice spacing, these multiplets are neither exactly conserved nor do they satisfy exact Ward identities. Crucially, the gluonic and fermionic components of the EMT mix under renormalization. A precise non-perturbative definition is required to satisfy the appropriate Ward identities in the continuum limit, particularly for applications demanding percent-level accuracy, such as the study of the QCD Equation of State and transport properties.

Methodology
The authors generalize a strategy previously applied to pure $SU(3)$ Yang–Mills theory to full QCD with $N_f = 3$ massless flavors. The methodology relies on formulating thermal quantum field theory in a moving reference frame by imposing shifted boundary conditions in the presence of an imaginary chemical potential.

1. Lattice Setup: The theory is discretized using the Wilson-plaquette action for gluons and the $O(a)$-improved Wilson action for fermions. The bare gluonic components of the EMT are defined via the clover discretization of the field strength tensor, while fermionic components are constructed from symmetric covariant derivatives.
2. Shifted Boundary Conditions: The system is defined on a lattice with temporal size $L_0$ and spatial size $L$. Shifted boundary conditions are imposed along the temporal direction:
   • Gauge fields: $A_\mu(x_0 + L_0, \mathbf{x}) = A_\mu(x_0, \mathbf{x} - L_0 \boldsymbol{\xi})$.
   • Fermion fields: $\psi(x_0 + L_0, \mathbf{x}) = -e^{i\theta_0} \psi(x_0, \mathbf{x} - L_0 \boldsymbol{\xi})$, where $\theta_0$ acts as an imaginary chemical potential.
3. Ward Identities: In this framework, exact Ward identities for one-point functions can be derived. Specifically, the thermal expectation value of the space-time components of the EMT is related to the derivative of the free-energy density with respect to the shift vector $\boldsymbol{\xi}$. Additional identities link diagonal and off-diagonal matrix elements.
4. Renormalization Conditions: The traceless EMT decomposes into sextet ($i=6$) and triplet ($i=3$) representations. For each, the renormalized operators are linear combinations of bare gluonic ($T^G$) and fermionic ($T^F$) components:
   $$T^{R,\{i\}}_{\mu\nu} = Z^{\{i\}}_G(g_0^2) T^{G,\{i\}}_{\mu\nu} + Z^{\{i\}}_F(g_0^2) T^{F,\{i\}}_{\mu\nu}$$
   The renormalization constants $Z^{\{i\}}_G$ and $Z^{\{i\}}_F$ are fixed non-perturbatively by imposing discretized versions of the continuum Ward identities. By utilizing two distinct fermionic twist phases ($\theta_0^A = 0$ and $\theta_0^B = 3\pi/10$), the authors construct a system of linear equations that disentangles the gluonic and fermionic contributions.
5. Numerical Implementation: Monte Carlo simulations were performed for various bare couplings ($0 \le g_0^2 \le 0.96$) and temporal lattice sizes ($L_0/a = 4, 6, 8$). Finite-volume effects were minimized by using large spatial volumes ($18 \lesssim L/L_0 \lesssim 70$). The study was restricted to the trivial topological sector, justified by the high temperatures ($T > 5$ GeV) where topological susceptibility is negligible.
6. Continuum Extrapolation: The renormalization constants were extrapolated to the thermodynamic limit ($1/L_0 \to 0$) at fixed $g_0$. Perturbative improvement was applied to subtract discretization effects up to one-loop order, and the remaining $O(a^2)$ effects were handled via global fits to polynomials in $g_0^2$.

Key Contributions and Results
The paper provides a fully non-perturbative determination of the renormalization constants for the traceless components of the QCD energy-momentum tensor with $N_f=3$.

• Renormalization Constants: The authors determine $Z^{\{6\}}_G, Z^{\{6\}}_F$ (sextet) and $Z^{\{3\}}_G, Z^{\{3\}}_F$ (triplet) as functions of the bare coupling squared $g_0^2$ in the range $0 \le g_0^2 \le 0.96$.
• Precision: The results are determined with a final accuracy of a few percent. The relative errors are approximately 3% for the gluonic sextet, 2% for the fermionic sextet, 4% for the gluonic triplet, and 3% for the fermionic triplet at $L_0/a=6$.
• Parametrization: The constants are parametrized by polynomials in $g_0^2$ (e.g., $Z(g_0^2) = 1 + Z^{(1)}g_0^2 + c_2 g_0^4 + c_3 g_0^6$).
  • Fermionic Constants: $Z^{\{6\}}_F$ and $Z^{\{3\}}_F$ are found to be very close to unity and are compatible with one-loop perturbation theory across the entire simulated range.
  • Gluonic Constants: $Z^{\{6\}}_G$ and $Z^{\{3\}}_G$ deviate significantly from one-loop perturbation theory. At $g_0^2 = 0.4$, a deviation of roughly 3% is observed, increasing to approximately 20% at $g_0^2 = 0.96$.
• Systematic Control: The study demonstrates that systematic effects from finite volume, topological sectors, and numerical integration (Gaussian quadratures) are negligible compared to statistical uncertainties. The extrapolation to the $1/L_0 \to 0$ limit is shown to be stable across different fit strategies.

Significance
The authors claim that this work establishes a fully non-perturbative definition of the traceless components of the QCD energy-momentum tensor whose correlation functions satisfy the appropriate Ward identities in the continuum limit. The primary significance lies in the necessity of non-perturbative renormalization for phenomenological applications requiring high precision (errors of a few percent or less), such as the calculation of the QCD Equation of State and transport properties of the QCD plasma.

The results demonstrate that relying on perturbative renormalization is insufficient for the range of couplings relevant to lattice QCD simulations, particularly for the gluonic components where deviations from one-loop results are substantial. Furthermore, these findings pave the way for a non-perturbative matching of the lattice EMT with analogous quantities defined at non-zero flow time (gradient flow), offering a robust theoretical foundation for future studies of strongly interacting matter.