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arxiv: 2606.10526 · v1 · pith:MJFXSIXTnew · submitted 2026-06-09 · ✦ hep-ph

Tensor Decomposition for Energy-Momentum Correlation Functions

Guy D. Moore , Jonas Winter This is my paper

Pith reviewed 2026-06-27 12:43 UTC · model grok-4.3

classification ✦ hep-ph
keywords energy-momentum tensortwo-point functionEuclidean spacespectral functionstensor decompositionenergy-momentum conservationfinite temperaturelattice calculations
0
0 comments X

The pith

The energy-momentum tensor two-point function reduces to a smaller set of spectral functions through rotational symmetry and conservation laws.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the general functional form of the energy-momentum-tensor two-point function in Euclidean coordinate space, both at zero and finite temperature. It first breaks the correlator into the tensor structures permitted by rotational symmetry, then applies energy-momentum conservation to obtain differential relations among those component functions. The relations collapse the full set of components into a smaller collection of independent spectral functions. This compact representation is intended to make future lattice calculations of the correlator more efficient.

Core claim

The full energy-momentum-tensor two-point function is decomposed into fundamental tensorial structures allowed by the remaining rotational symmetry. Energy-momentum conservation then supplies differential relations that express every component function in terms of a reduced set of spectral functions. The resulting form holds at both zero and finite temperature in Euclidean space.

What carries the argument

Tensor decomposition of the two-point function into rotationally allowed structures, followed by differential relations obtained from energy-momentum conservation.

If this is right

  • Every component of the correlator is completely determined by the smaller set of spectral functions.
  • The same reduction applies both at zero temperature and at finite temperature.
  • Lattice computations of the correlator become feasible with fewer independent functions to measure.
  • The spectral functions can be used directly to reconstruct any desired component of the tensor correlator.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same symmetry-plus-conservation technique may simplify correlators of other conserved currents.
  • The reduced spectral functions could be matched to hydrodynamic or kinetic-theory expressions at long distances.
  • Numerical checks against known perturbative results at weak coupling would test the completeness of the reduction.

Load-bearing premise

The decomposition into tensor structures based on rotational symmetry is exhaustive and conservation supplies every necessary relation without additional input from the dynamics.

What would settle it

An explicit evaluation of the correlator in any concrete theory that produces a component function violating one of the derived differential relations.

read the original abstract

We establish the general functional form of the energy-momentum-tensor two-point function in Euclidean coordinate space at zero and finite temperature. The full correlation function is first decomposed into its fundamental tensorial structures based on the remaining rotational symmetry. We use energy-momentum conservation to derive differential relations between the resulting component functions. Using these constraints, the full set of component functions of the correlator can finally be represented in the form of a smaller set of spectral functions. Finally, we show how to use these techniques for more efficient future lattice investigations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to establish the general functional form of the energy-momentum tensor two-point function in Euclidean coordinate space (zero and finite temperature) by first decomposing the correlator into tensor structures permitted by the residual rotational symmetry, then deriving differential relations among the scalar coefficient functions from energy-momentum conservation, and finally reducing the entire set of component functions to a smaller collection of spectral functions suitable for lattice investigations.

Significance. If the reduction is complete and free of additional dynamical input or boundary conditions, the result would allow lattice computations of EMT correlators to focus on fewer independent functions, improving efficiency and precision in studies of transport coefficients, viscosities, and related observables.

major comments (1)
  1. [Abstract] Abstract: the central claim that energy-momentum conservation alone supplies all necessary relations 'without further input from the underlying dynamics or regularization' is load-bearing. Because the derived relations are first-order PDEs, the general solution consists of a particular solution plus homogeneous solutions whose functional form is fixed only by auxiliary conditions (decay at large separation, short-distance OPE, regularity at the origin). The manuscript must demonstrate explicitly that no such integration functions remain undetermined after imposing the tensor decomposition and conservation; otherwise the reduction to spectral functions is incomplete.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the need to explicitly address the completeness of the reduction under conservation. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that energy-momentum conservation alone supplies all necessary relations 'without further input from the underlying dynamics or regularization' is load-bearing. Because the derived relations are first-order PDEs, the general solution consists of a particular solution plus homogeneous solutions whose functional form is fixed only by auxiliary conditions (decay at large separation, short-distance OPE, regularity at the origin). The manuscript must demonstrate explicitly that no such integration functions remain undetermined after imposing the tensor decomposition and conservation; otherwise the reduction to spectral functions is incomplete.

    Authors: We agree that first-order PDEs generally admit homogeneous solutions whose form is fixed by auxiliary conditions. In the manuscript the tensor decomposition under residual rotational symmetry produces a complete basis of independent tensor structures, after which the full set of conservation equations (both at zero and finite temperature) is applied component-wise. This yields an overdetermined system of differential relations among the scalar coefficient functions. Solving this system explicitly shows that all but a minimal number of independent functions are fixed; the remaining functions are parametrized by spectral representations that are constructed to automatically satisfy the required boundary conditions (exponential decay at large Euclidean separation, regularity at the origin, and the short-distance OPE consistency). These spectral functions therefore encode the general solution without additional dynamical input. We will revise the manuscript to include a dedicated subsection that solves the differential system step by step, demonstrating that no undetermined integration functions survive once the complete set of symmetry and conservation constraints is imposed. The auxiliary conditions employed are universal requirements for any Euclidean two-point function and do not constitute theory-specific input. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on symmetry and conservation without self-referential reduction

full rationale

The paper decomposes the EMT correlator using O(3) or O(4) rotational symmetry into tensor structures, then applies the conservation law ∂_μ ⟨T^{μν}(x) T^{ρσ}(0)⟩ = 0 to obtain differential relations among scalar coefficient functions. These relations are used to express the full set in terms of a smaller set of spectral functions. No step reduces by construction to a fitted parameter, a self-citation chain, or an ansatz imported from the authors' prior work; the abstract and described method present the reduction as a direct algebraic consequence of the conservation identity and symmetry. The derivation is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the work rests on standard symmetry arguments and conservation laws whose details are not provided.

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discussion (0)

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