Abstract
We present a strategy to define non-perturbatively the energy-momentum tensor in Quantum Chromodynamics (QCD) which satisfies the appropriate Ward identities and has the right trace anomaly. The tensor is defined by regularizing the theory on a lattice, and by fixing its renormalization constants non-perturbatively by suitable Ward identities associated to the Poincaré invariance of the continuum theory. The latter are derived in thermal QCD with a non-zero imaginary chemical potential formulated in a moving reference frame. A renormalization group analysis leads to simple renormalization- group-invariant definitions of the gluonic and fermionic contributions to either the singlet or the non-singlet components of the tensor, and therefore of their form factors among physical states. The lattice discussion focuses on the Wilson discretization of quark fields but the strategy is general. Specific to that case, we also carry out the analysis for the on-shell O(a)-improvement of the energy-momentum tensor. The renormalization and improvement programs profit from the fact that, as shown here, the thermal theory enjoys de-facto automatic O(a)-improvement at finite temperature. The validity of the proposal is scrutinized analytically by a study to 1-loop order in lattice perturbation theory with shifted and twisted (for quarks only) boundary conditions. The latter provides also additional useful insight for a precise non-perturbative calculation of the renormalization constants. The strategy proposed here is accessible to Monte Carlo computations, and in this sense it provides a practical way to define non-perturbatively the energy-momentum tensor in QCD.
Highlights
We present a strategy to define non-perturbatively the energy-momentum tensor in Quantum Chromodynamics (QCD) which satisfies the appropriate Ward identities and has the right trace anomaly
The tensor is defined by regularizing the theory on a lattice, and by fixing its renormalization constants non-perturbatively by suitable Ward identities associated to the Poincare invariance of the continuum theory
The latter are derived in thermal QCD with a non-zero imaginary chemical potential formulated in a moving reference frame
Summary
The energy-momentum tensor, Tμν, is a central quantity in a quantum field theory since it groups together the currents associated to the invariance of the theory under spacetime translations, from which those associated to the larger Poincare group and scale invariance can be built. [1, 2] for the Yang-Mills theory and QCD, where it was shown that on the lattice the 10dimensional symmetric tensor Tμν breaks into the sum of a sextet, a triplet and a singlet representation of the hypercubic group Each one of those three parts picks up finite renormalization constants which were computed to 1-loop order in perturbation theory [3,4,5], see [6,7,8]. We investigate the renormalization conditions in perturbation theory, and we compute the renormalization constants and the improvement coefficients of the gluonic and fermionic parts of the energy-momentum tensor to 1-loop order. Conventions, and technical details are reported in several appendices
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