[Paper Review] Energy-momentum tensor in QCD: nucleon mass decomposition and mechanical equilibrium

Cédric Lorcé, Andreas Metz+2 more|arXiv (Cornell University)|Sep 24, 2021

Quantum Chromodynamics and Particle InteractionsPhysics and Astronomy171 references58 citations

TL;DR

This paper re-examines the nucleon mass decomposition in QCD by analyzing the energy-momentum tensor (EMT) and its role in mechanical equilibrium and the virial theorem within quantum field theory. It demonstrates that the so-called 'quantum anomalous energy' is not a genuine contribution to the nucleon mass due to the constraints of translation symmetry, resolving a long-standing debate about the physical interpretation of the trace anomaly in mass sum rules.

ABSTRACT

We review and examine in detail recent developments regarding the question of the nucleon mass decomposition. We discuss in particular the virial theorem in quantum field theory and its implications for the nucleon mass decomposition and mechanical equilibrium. We reconsider the renormalization of the QCD energy-momentum tensor in minimal-subtraction-type schemes and the physical interpretation of its components, as well as the role played by the trace anomaly and Poincar\'e symmetry. We also study the concept of "quantum anomalous energy" proposed in some works as a new contribution to the nucleon mass. Examining the various arguments, we conclude that the quantum anomalous energy is not a genuine contribution to the mass sum rule, as a consequence of translation symmetry.

Motivation & Objective

• To clarify the physical interpretation of the nucleon mass decomposition in QCD, particularly the role of the trace anomaly and proposed 'quantum anomalous energy'.
• To resolve the controversy over whether the trace part of the energy-momentum tensor contributes independently to the nucleon mass.
• To establish a scheme-independent, Poincaré-covariant framework for separating mass contributions from mechanical equilibrium constraints.
• To critically assess the validity of the four-term energy decomposition based on traceless and trace parts of the EMT.
• To demonstrate that the concept of 'quantum anomalous energy' as a distinct mass contribution is inconsistent with translation symmetry.

Proposed method

• Derives the quantum field theory version of the virial theorem using spacetime dilatation currents and the energy-momentum tensor (EMT).
• Analyzes the expectation value of the EMT under infinitesimal dilatations to derive the virial theorem as a condition for mechanical equilibrium.
• Applies the virial theorem to the nucleon state, showing it corresponds to the constraint of mechanical equilibrium in bound states.
• Examines the renormalization of the EMT in dimensional regularization with minimal subtraction (MS) schemes, focusing on operator mixing and scale dependence.
• Distinguishes between the traceless and trace parts of the EMT, showing they do not mix under Lorentz transformations and are separately conserved in certain limits.
• Uses the Heisenberg equation of motion and commutator algebra to derive the virial condition ⟨T^ii⟩ + ⟨F·x⟩ = 0 in stationary states, linking it to mechanical equilibrium.

Experimental results

Research questions

• RQ1 Is the 'quantum anomalous energy' a genuine contribution to the nucleon mass, as claimed in some recent works?
• RQ2 How does the virial theorem in quantum field theory relate to mechanical equilibrium in hadronic bound states?
• RQ3 What is the correct physical interpretation of the trace part of the energy-momentum tensor in nucleon mass decomposition?
• RQ4 Can the nucleon mass be decomposed into quark and gluon contributions without mixing with mechanical equilibrium constraints?
• RQ5 Why do different decompositions of the EMT (two-term, three-term, four-term) lead to conflicting physical pictures, and which is scheme-independent and physically meaningful?

Key findings

• The virial theorem in QFT, derived from spacetime dilatation symmetry, is equivalent to the condition of mechanical equilibrium in a bound state.
• The 'quantum anomalous energy' proposed as a new contribution to the nucleon mass is not physically valid because it violates translation symmetry.
• The trace part of the EMT contributes one-quarter of the nucleon mass, while the traceless part contributes three-quarters, consistent with a virial-like balance.
• The four-term energy decomposition (traceless and trace parts) is mathematically valid but mixes mass information with mechanical equilibrium constraints.
• In the MS scheme, the renormalized EMT components are well-defined, and the trace anomaly contributes to the mass sum rule, but not as a new independent energy term.
• The physical interpretation of EMT components must respect Poincaré symmetry and translation invariance; breaking these leads to unphysical conclusions like the existence of 'quantum anomalous energy'.