Energy-Momentum Tensor Matrix Elements

• Energy-momentum tensor matrix elements represent the conserved densities and fluxes in QFT, decomposed into Lorentz-invariant gravitational form factors.
• They provide a systematic framework to link observable quantities such as momentum fraction, angular momentum, and the D-term, clarifying internal force distributions.
• Experimental and lattice QCD studies leverage these form factors to map hadronic mechanical properties, advancing our understanding of nucleon structure.

The energy-momentum tensor (EMT), $T^{\mu\nu}(x)$, is a conserved local operator encoding the densities and fluxes of energy and momentum in quantum field theories (QFT) and classical field systems. Its matrix elements between on-shell states and the associated Lorentz-decomposition are central to the investigation of internal properties of hadrons, nuclei, atoms, electromagnetic and gravitational fields, and even the analytic structure of conformal field theories. Matrix elements of $T^{\mu\nu}$ not only define conserved charges via Noether’s theorem but also generate fundamental gravitational form factors (GFFs), serving as “mechanical tomography” for composite systems.

1. Lorentz Structure and Canonical Decomposition

In relativistic QFT, the most general Lorentz-covariant decomposition of the symmetric, conserved EMT between massive spin-$s$ on-shell states $\ket{p,s}$, $\ket{p',s'}$ is determined by Poincaré symmetry: $$\langle p',s'|\,T^{\mu\nu}(0)\,|p,s\rangle = \bar u_{s'}(p')\left[ \frac{2P^\mu 2P^\nu}{8M}A(q^2) + \frac{i(2P)^{\{\mu}\sigma^{\nu\}\rho}q_\rho}{8M}B(q^2) + M(q^\mu q^\nu - g^{\mu\nu}q^2)C(q^2) \right] u_s(p)$$ where $P=(p+p')/2$, $q=p'-p$, and $A$, $B$, and $C$ are Lorentz-invariant scalar form factors (${\{\mu\nu\}}$ denotes symmetrization). For spin-½ hadrons (e.g., nucleons), this basis underlies the classic “Ji decomposition” and can be recast in terms of the $\gamma^\mu$, $\sigma^{\mu\nu}$, and $(q^\mu q^\nu - g^{\mu\nu}q^2)$ Dirac structures. The form factors map onto physical observables: momentum fraction, total angular momentum, and the so-called $D$-term associated with internal mechanical forces (Lowdon et al., 2017, Hudson et al., 2016).

Extension to transitions involving external spin-1 and spin-3/2 states (notably $N \to \Delta$) introduces additional tensor structures, resulting in up to nine independent form factors (five conserved, four non-conserved), as first systematically enumerated in (Kim, 2022). The reduction to three familiar invariants occurs under equal-mass, parity-conserving circumstances.

For massless particles in unitary, local, Lorentz-invariant theories, the EMT matrix elements are further constrained by the Weinberg–Witten theorem. For helicity $|h|\le 1$, the decomposition admits up to four independent form factors, only two for scalar ($h=0$), three for spinor ($|h|=1/2$), and all four for vector bosons, with explicit covariant multipole expansion as shown in (Lorcé et al., 2020).

2. Form Factor Constraints and Sum Rules

The matrix element parametrization admits rigorous global constraints derived from fundamental symmetry and conservation requirements. The distributional-matching approach shows:

• $A(0) = 1$ (momentum sum rule): all of the hadron’s momentum is attributed to its constituents,
• $B(0) = 0$ (no anomalous gravitomagnetic moment): the net gravitomagnetic dipole vanishes,
• $C(0)$ (the $D$-term) remains undetermined, reflecting internal mechanical structure (Lowdon et al., 2017).

These identities arise independently from the transformation properties of the Poincaré algebra and the on-shell nature of states, not from conservation laws for individual currents. In composite systems, the momentum and angular momentum sum rules are recast as

$J_{q,g} = \frac12 [A_{q,g}(0) + B_{q,g}(0)]$

where $A_{q,g}(0)$ and $B_{q,g}(0)$ are quark and gluon contributions. For the nucleon, $A(0)=1$, $B(0)=0$ imply total spin $1/2$ (Hudson et al., 2016). Higher-spin and transition matrix elements obey analogous reduction formulas, with conservation enforcing linear relations among form factors (Kim, 2022).

Sum rules involving the EMT also emerge in light-front quantization, underpinning the Burkardt sum rule and higher-twist analogs for transverse-momentum-dependent distributions (TMDs) (Lorcé, 2015, Lorcé, 2015).

3. Gravitational Form Factors and Physical Densities

EMT matrix elements in the Breit frame, upon Fourier transformation in $\vec{q}$, yield spatial distributions of mass, pressure, and shear within the target:

• The mass (energy) density $\epsilon(r)$ is governed by $A(t)$ and the $D$-term,
• The pressure $p(r)$ and shear force $s(r)$ distributions depend only on the $D$-term,
• The mechanical (mass/pressure/shear) radius follows from slopes and values of $A$, $C$ at $t=0$ (Meziani, 8 May 2025, Hudson et al., 2016).

For the proton, experimental determinations of gluonic GFFs (notably $A_g(t)$, $D_g(t) = 4C_g(t)$) via J/$\psi$ photoproduction have established

$$\sqrt{\langle r_m^2\rangle_g} = 0.755 \pm 0.067\,\mathrm{fm},\quad D_g(0) = -1.80 \pm 0.53$$

with consistent suppression of $B_g(0)$. These densities represent the “internal pressure map” of the nucleon (Meziani, 8 May 2025).

For finite systems such as the hydrogen atom, the $D$-term encodes the $r^4$-moment of the pressure distribution. Its sign is not fixed by stability: both $D>0$ and $D<0$ are admissible, controlled by details of the internal force balance rather than the von Laue condition $\int r^2 p(r)\,dr=0$ (Czarnecki et al., 2023). The $D$-term in hydrogen receives a leading Coulomb value $D=\frac12$ with a universal logarithmic $\alpha$ correction.

4. Relation to Generalized Parton Distributions and TMDs

The Mellin moments of generalized parton distributions (GPDs) $H^q(x,\xi,t)$ and $E^q(x,\xi,t)$ provide direct access to EMT form factors: $$\int_{-1}^1 dx \, x \, H^q(x,\xi,t) = A^q(t) + \xi^2 C^q(t), \quad \int_{-1}^1 dx \, x \, E^q(x,\xi,t) = B^q(t) - \xi^2 C^q(t)$$ with $D^q(t) = 4C^q(t)$. These relations underpin the experimental program of “gravitational tomography” via deeply virtual Compton scattering and similar processes (Hudson et al., 2016).

In the light-front framework, a complete parametrization of the nonlocal, gauge-invariant canonical EMT involves 32 amplitudes, with GPD and TMD moments providing partial access. However, the orbital angular momentum (OAM) of partons cannot be fully disentangled at the level of two-parton TMDs alone; only higher-twist or generalized TMDs (GTMDs) probe the gauge-invariant canonical OAM (Lorcé, 2015, Lorcé, 2015).

5. The Energy-Momentum Tensor in Material Media

In macroscopic electromagnetism, the EMT acquires a material dependence. For a simple linear dielectric (index $n$), the electromagnetic EMT is constructed from the energy density

$T^{00} = \frac12 (n^2 \mathbf{E}^2 + \mathbf{B}^2)$

and the Gordon momentum density, $\mathbf{g}_G = n\mathbf{E} \times \mathbf{B}/c$,

$T^{0i} = T^{i0} = n(E \times B)_i$

The stress tensor block $W_{ij}$ ensures symmetry and tracelessness; the full tensor is

$$T^{\alpha\beta} = \begin{pmatrix} \rho_e & c g_G^i \ c g_G^j & W_{ij} \end{pmatrix}$$

Conservation equations, represented via a “material” four-divergence $$\bar\partial_\alpha = \left(\frac{n}{c}\partial_t, \nabla\right)$$, enforce global four-momentum conservation in closed systems, and yield new electromagnetic continuity equations not equivalent to the traditional Abraham–Minkowski constructions (Crenshaw, 2012).

In inhomogeneous dielectrics, the divergence equation acquires a generalized Helmholtz force density source term,

$\bar\partial_\beta T^{\alpha\beta} = F^\alpha$

with $F^\alpha$ capturing exchanges with external mechanical constraints (Crenshaw et al., 2012).

6. Implications for Conformal Field Theory and Analytic Constraints

In conformal field theories (CFTs), the matrix elements of $T^{\mu\nu}$ obey additional analytic constraints from dilation and special conformal invariance. For example, for on-shell massless states, the four form factors $A(q^2)$, $G(q^2)$, $C(q^2)$, $T(q^2)$ satisfy

$$A(0) = G(0) = 1,\quad A(q^2) - 2(\Delta-1)G(q^2) + 12C(q^2) - 4\Delta(\Delta-1)T(q^2) = 0$$

where $\Delta$ is the scaling dimension. For free fields, this fully determines the functional form of the EMT matrix elements, illustrating that all free massless theories are conformally invariant (Lorcé et al., 2020).

7. Experimental Measurements and Phenomenology

Gravitational form factors are now extracted via hard exclusive reactions—deeply virtual Compton scattering, near-threshold quarkonium photoproduction, and lattice QCD calculations. The first experimental determination of the gluonic GFFs for the proton via J/$\psi$ photoproduction at Jefferson Lab Hall C demonstrated consistency with theoretical expectations and lattice, providing gluonic mass radii and $D$-term values (Meziani, 8 May 2025). These approaches are being expanded by the SoLID program at JLab and the EIC, with anticipated reductions in statistical and systematic uncertainties and enhanced sensitivity to the universality properties of the GFFs in different kinematical regimes.

The $D$-term, in particular, is a focus of contemporary hadronic physics—its sign, magnitude, and physical interpretation encapsulating the mechanical stability and spatial pressure profile of hadrons and nuclei (Czarnecki et al., 2023).

Table: Canonical Gravitational Form Factors for Nucleon EMT

Symbol	Physical Meaning	$t=0$ Value/Constraint
$A(t)$	Momentum (mass) fraction	$A(0) = 1$
$B(t)$	Total angular momentum	$B(0) = 0$
$D(t)$	Internal pressure/shear ($D$)	$D(0)$ unconstrained

The matrix elements of the EMT—through their decomposition in terms of invariant form factors—thus serve as a central organizing tool for theoretical and experimental investigation of deep structural properties in QFT, QCD, electrodynamics in media, and CFT, connecting energy, momentum, spin, mechanical stability, and internal force distributions within a mathematically and physically rigorous framework.