On-shell Matrix Elements of the EMT Trace in Gauge Theories and Heavy Quark Masses

TL;DR

The paper proves a diagrammatic identity: the forward matrix element of the EMT trace on a single on-shell state equals the perturbative pole mass to all loop orders in gauge theories, without relying on predefined operator renormalization or Ward identities. A fermion case is established exactly, complemented by a general insertion theorem for $\mathcal{O}_{F}[\xi]$ in vacuum diagrams and a gauge-boson analysis showing on-shell vanishing of the matrix element, clarifying the trace-anomaly’s role. Explicit three-loop calculations in perturbative QED and QCD reveal nonzero trace-anomaly contributions to on-shell masses and show that the trace anomaly captures all leading IR renormalon effects at this order, motivating a trace-anomaly-subtracted $\sigma$-mass, $m_{\sigma}=Z_{\sigma} m_{\mathrm{os}}$, which is gauge-invariant and scheme-/scale-independent. The work provides analytic three-loop relations between $m_{\sigma}$, $m_{\mathrm{os}}$, and $\overline{\mathrm{MS}}$ masses, and offers numerical insights for electrons and heavy quarks, with potential implications for heavy-quark phenomenology and the nonperturbative understanding of mass generation in QFT.

Abstract

We present a novel diagrammatic proof of the identity between the forward matrix element of the energy-momentum-tensor (EMT) trace operator over a single particle's on-shell state and its perturbative pole mass to any loop orders in perturbative gauge theories (with gauge bosons and fermions), without appealing to any pre-laid operator renormalization conditions or Ward identities. The proof is based on the equation of mass-dimensional analysis in dimensional regularization, the topological properties of contributing Feynman diagrams and the on-shell renormalization condition. Considering for definiteness perturbative QED and QCD with at most one fermion kept massive, we have verified the aforementioned identity, up to three loops, for all elementary particles through direct computation of dimensionally-regularized matrix elements of the relevant bare operators. We observe interestingly that the trace-anomaly contribution seemingly contains all leading-renormalon effects (explicitly verified up to three loops), and we propose accordingly a new scheme- and scale-independent trace-anomaly-subtracted mass definition for heavy $t$-,$\,b$-,$\,c$-quarks and electrons. A list of amusing numbers is subsequently presented for the composition of their perturbative pole masses.

Figures (3)

• Figure 1: The first row of sample diagrams is for the self-energy function of a massive fermion at three loops; the second row represents the counterpart in the contribution to the matrix element of the fermion-mass operator $m \bar{\psi} \psi$ (with the black dot denoting the vertex factor $i\, m$); the last row of example diagrams illustrates some of the three-loop contribution with an insertion of $\mathcal{O}_{F}[\xi]$, indicated by the circled cross, as defined in eq. \ref{['eq:EMTtrace_GF']}.
• Figure 2: The Feynman rules for the local composite operator $\mathcal{O}_{F}[\xi]$ with zero-momentum insertion, represented by the circled cross (all momenta are outgoing).
• Figure 3: The reduction of the insertion of a degree-2 vertex onto an internal gauge boson propagator inside a generic diagram (note the relative minus sign in the r.h.s.).