Gravitational Multipole Form Factors (GMFFs)

• GMFFs are Lorentz-invariant coefficients that encode hadronic energy-momentum distributions, including momentum fractions, angular momentum, pressure, and shear forces.
• They enable a multipolar decomposition in the Breit frame, yielding clear interpretations such as energy monopoles/quadrupoles and angular-momentum dipoles/octupoles.
• Interpretative challenges arise from the D-term’s sensitivity to twist-three dynamics and methodological differences in quark-gluon operator analysis.

Searching arXiv for the cited GMFF-related papers and recent context. arXiv_search query: "all:gravitational multipole form factors hadron energy-momentum tensor" arXiv_search query: "id:(Tanaka, 2019) OR id:(Yao et al., 2024) OR id:(Ji et al., 2021) OR id:(Dehghan et al., 20 Jul 2025)" Gravitational multipole form factors (GMFFs) are the Lorentz-invariant coefficients that parameterize matrix elements of the energy-momentum tensor (EMT) and thereby encode how composite systems couple to a spin-$2$ probe, how energy and momentum are distributed, and how internal stresses are organized. In hadron structure, the standard setting is the matrix element of the symmetric, gauge-invariant QCD EMT, whose form factors govern momentum fractions, angular momentum, pressure, shear forces, and trace-related structure; for higher-spin targets these invariant form factors can be reorganized into explicitly multipolar combinations in the Breit frame, such as energy monopoles and quadrupoles, angular-momentum dipoles and octupoles, and mechanical multipoles. A broader usage also exists in classical gravity, where renormalized source multipoles or Newtonian inner multipole moments play an analogous role as gravitational shape descriptors, but the narrow contemporary GMFF literature is centered on EMT matrix elements of hadrons (Tanaka, 2019, Dehghan et al., 20 Jul 2025, Almeida et al., 2021).

1. Conceptual scope and terminology

In the hadronic literature, GMFFs arise from matrix elements of the Belinfante-improved, gauge-invariant QCD EMT,

$T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$

with separate quark and gluon pieces that are gauge invariant but not separately conserved. This distinction is essential: the familiar form factors $A$, $B$, $D$, and $\bar C$ are normally defined from the symmetric EMT, and the angular-momentum relation then takes the Ji form. Within this usage, GMFFs are not independent of the EMT formalism; they are its invariant content, interpreted through multipole language once static limits or Breit-frame densities are considered (Tanaka, 2019).

The term is nevertheless used with broader scope in some adjacent literatures. In the effective field theory of compact objects, electric-type multipoles $I^{iji_1\cdots i_r}$ and magnetic-type multipoles $J^{iji_1\cdots i_r}$ act as scale-dependent source couplings in the radiative sector, with tail-of-tail effects inducing renormalization and a classical renormalization-group flow (Almeida et al., 2021). In Newtonian precision-gravity work, inner multipoles $q_{lm}$ and outer multipoles $Q_{lm}$ describe extended mass distributions through

$T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$0

so that the multipoles function as closed-form shape descriptors of solids such as cylinders, prisms, polygonal bodies, and cones (Stirling et al., 2017). A narrow definition therefore identifies GMFFs with off-forward EMT form factors, whereas a broad definition encompasses classical multipolar gravitational couplings as well.

2. EMT decompositions and multipole bases

For the nucleon, the standard quark/gluon EMT decomposition in the symmetric basis is

$T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$1

with $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$2, $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$3, and $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$4. In this basis, $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$5 controls momentum and energy flow, $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$6 carries gravitomagnetic or spin-flip structure, $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$7 controls the traceless stress sector, and $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$8 is the non-conserved $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$9 term present for the separate quark and gluon sectors. A frequently used alternative basis replaces $A$0 by $A$1, with

$A$2

so that the three independent nucleon GFFs are written directly as $A$3, $A$4, and $A$5 (Tanaka, 2019, Yao et al., 2024).

For spin-0 hadrons, the decomposition simplifies because there is no spin-flip structure. One standard form is

$A$6

where $A$7 is the momentum or mass form factor and $A$8 is the D-term–type mechanical form factor. In another widely used spin-0 convention,

$A$9

so that the complete symmetric spin-0 gravitational structure is exhausted by $B$0 and $B$1 (Tanaka, 2019, Sain et al., 18 May 2026).

The explicit multipole nomenclature becomes richer for spin-$B$2 systems. In the $B$3 and $B$4, the full EMT initially contains ten invariant form factors, of which seven survive for the conserved total EMT: $B$5 In the Breit frame these are reorganized into GMFFs with direct static meaning: $B$6 The resulting interpretation is explicitly multipolar: $B$7 is an energy monopole, $B$8 an energy quadrupole, $B$9 an angular-momentum dipole, $D$0 an angular-momentum octupole, and $D$1 mechanical multipoles associated with pressure and shear (Dehghan et al., 2023, Dehghan et al., 20 Jul 2025).

3. Symmetry constraints, operator relations, and QCD structure

Conservation and Poincaré symmetry impose the first layer of constraints. Because the total EMT is conserved,

$D$2

the non-conserved pieces cancel in the total matrix element,

$D$3

In the forward limit,

$D$4

and the Ji relation gives

$D$5

These statements fix the normalization of the momentum and total angular-momentum sectors but do not determine the D-term, which remains dynamical (Tanaka, 2019).

Tanaka’s operator analysis makes the non-conserved and mechanical sectors more precise. The separate quark and gluon EMTs satisfy

$D$6

so $D$7 is directly tied to interaction-dependent quark-gluon operators rather than to an autonomous static density. In light-cone gauge, the same analysis gives approximate relations that connect the nucleon D-term to a twist-three quark-gluon correlator, while the second Mellin moments of quark GPDs determine

$D$8

The standard twist-2 polynomiality relation is therefore compatible with, but does not exhaust, the higher-twist operator content of the D-term (Tanaka, 2019).

The trace sector introduces a second structural layer. In the forward limit,

$D$9

and renormalization invalidates naive classical trace identities. The one-loop expressions quoted for $\bar C$0 and $\bar C$1 show that $\bar C$2 is constrained by the QCD trace anomaly and acquires nontrivial scale dependence (Tanaka, 2019). A related warning appears in the twist analysis of the chiral quark-soliton model: the full EMT splits into a traceless twist-2 part and a trace twist-4 part,

$\bar C$3

so leading-twist GPD moments do not determine the complete EMT structure; in that framework, the twist-2 pressure and shear-force distributions are both repulsive, and twist-4 contributions are therefore required to secure stability (Kim et al., 29 Mar 2025).

At large momentum transfer, perturbative QCD and counting arguments imply additional structure. For the nucleon, the large-$\bar C$4 analyses summarized in the literature give

$\bar C$5

with $\bar C$6 and $\bar C$7 for the total EMT (Tong et al., 2022). For the pion and proton gluon sectors, the perturbative large-$\bar C$8 results are

$\bar C$9

so the mechanical form factor is more suppressed in the proton because it requires helicity flip and orbital angular momentum (Tong et al., 2021).

4. Multipole interpretation, static densities, and radii

The mechanical interpretation of GMFFs is made explicit once Breit-frame Fourier transforms are introduced. In the nucleon analysis of Roberts and collaborators, the energy density, pressure, and shear-force distributions are

$I^{iji_1\cdots i_r}$0

$I^{iji_1\cdots i_r}$1

and the longitudinal normal-force distribution is

$I^{iji_1\cdots i_r}$2

In this representation the D-term alone governs pressure, shear, and the mechanical radius, whereas the mass radius depends on $I^{iji_1\cdots i_r}$3 and $I^{iji_1\cdots i_r}$4, and in the quoted density formula also on $I^{iji_1\cdots i_r}$5 (Yao et al., 2024).

The corresponding radii are defined by

$I^{iji_1\cdots i_r}$6

with the practical forms

$I^{iji_1\cdots i_r}$7

For the proton in that framework,

$I^{iji_1\cdots i_r}$8

and the predicted ordering is

$I^{iji_1\cdots i_r}$9

with $J^{iji_1\cdots i_r}$0 and $J^{iji_1\cdots i_r}$1 for $J^{iji_1\cdots i_r}$2 (Yao et al., 2024).

Mesonic systems display a different pattern. In the continuum-QCD analysis of pion and kaon GFFs, the reported radii satisfy

$J^{iji_1\cdots i_r}$3

so the mechanical radius exceeds the charge radius and the charge radius exceeds the mass radius in both mesons (Xu et al., 2023). In BLFQ, the same qualitative hierarchy is found for the 2D impact-parameter radii,

$J^{iji_1\cdots i_r}$4

with

$J^{iji_1\cdots i_r}$5

for the pion, and

$J^{iji_1\cdots i_r}$6

for the kaon, although the same work emphasizes that the low-$J^{iji_1\cdots i_r}$7 D-term is especially sensitive to small-$J^{iji_1\cdots i_r}$8 and possible zero-mode effects in its truncated light-front extraction (Sain et al., 18 May 2026).

For higher spin, the static interpretation itself becomes multipolar. In the $J^{iji_1\cdots i_r}$9, the Breit-frame decomposition produces an energy density

$q_{lm}$0

an angular-momentum density driven by $q_{lm}$1 and $q_{lm}$2, and a stress tensor with monopole and quadrupole pieces governed by $q_{lm}$3. The monopole components dominate numerically, but the quadrupole pieces are genuine higher-order deformations rather than kinematic artifacts (Dehghan et al., 20 Jul 2025).

5. Representative systems, methods, and reported values

The contemporary GMFF literature is methodologically heterogeneous. Continuum Schwinger-function methods, light-cone sum rules, three-point QCD sum rules, top-down holography, BLFQ, perturbative QCD, and pion mean-field approaches all compute EMT form factors, but they do so in different operator sectors and with different degrees of control over quark-gluon separation, higher twist, and large-$q_{lm}$4 behavior.

System and framework	GMFF basis	Representative reported results
Proton, symmetry-preserving continuum QCD	$q_{lm}$5	$q_{lm}$6; $q_{lm}$7; $q_{lm}$8
Hyperons, LCSR (quark EMT)	$q_{lm}$9	$Q_{lm}$0; $Q_{lm}$1; $Q_{lm}$2
$Q_{lm}$3, three-point QCD sum rules	$Q_{lm}$4	$Q_{lm}$5; $Q_{lm}$6; $Q_{lm}$7
$Q_{lm}$8, three-point QCD sum rules	same spin-$Q_{lm}$9 GMFF basis	$T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$00; $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$01; $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$02
Pion, top-down holographic QCD	$T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$03	$T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$04; $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$05; glueball dominance

These results illustrate several systematic patterns. First, quark-only calculations naturally give $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$06 and $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$07, while full-EMT calculations recover the expected normalizations (Özdem et al., 2020). Second, the D-term is negative in the standard hadronic examples quoted for the proton, hyperons, $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$08, and $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$09, but its magnitude and even its detailed $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$10-shape are highly method dependent (Yao et al., 2024, Dehghan et al., 2023, Dehghan et al., 20 Jul 2025). Third, higher-spin systems exhibit nonzero energy quadrupoles and angular-momentum octupoles, making the “multipole” label literal rather than merely interpretive (Dehghan et al., 2023, Dehghan et al., 20 Jul 2025).

Meson studies add further structure. In top-down holographic QCD, the pion GFFs are generated through an infinite glueball tower, and the scalar $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$11 sector contributes to $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$12, leading to a faster falloff of $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$13 than $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$14 and yielding the chiral-limit result $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$15 (Fujii et al., 2024). In BLFQ, by contrast, $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$16 is found to be in overall agreement with recent lattice QCD and dispersive results, whereas $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$17 is enhanced in magnitude at low $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$18, a behavior attributed to the use of transverse EMT components and their sensitivity to the small-$T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$19 region and light-front zero-mode effects (Sain et al., 18 May 2026). In the pion mean-field approach, the strange quark is reported to be small in $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$20 and $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$21 but essential in the D-term, and the approximate flavor blindness $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$22 is said to hold only when the strange quark is included (Kim et al., 29 Mar 2025).

6. Classical analogues, interpretive issues, and persistent controversies

A broader GMFF language emerges naturally in classical gravity, but it is not identical to the hadronic EMT usage. In NRGR, composite compact objects are described by a multipolar worldline action with electric moments $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$23 and magnetic moments $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$24, and tail-of-tail processes renormalize these multipoles according to classical RG equations such as

$T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$25

with an analogous equation for $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$26. The radiative multipoles are then the scale-independent observables, while the source multipoles are scale-dependent EFT parameters (Almeida et al., 2021). In Newtonian gravity, the same conceptual role is played by inner moments

$T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$27

which the closed-form literature treats as reusable building blocks for extended solids (Stirling et al., 2017). These classical objects are best viewed as analogues, not substitutes, for hadronic EMT GMFFs.

Several interpretive issues remain nontrivial. One is the status of the D-term sign. Much of the hadronic EMT literature associates a negative D-term with stable internal mechanical structure and with the usual pressure–shear pattern; that interpretation is explicit in the proton, hyperon, $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$28, and $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$29 studies summarized above (Yao et al., 2024, Özdem et al., 2020, Dehghan et al., 2023). By contrast, the momentum-current multipole analysis of hadrons defines a tensor monopole $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$30 and argues, based on the momentum-current distribution in the hydrogen atom, that the sign of the D-term has little to do with mechanical stability (Ji et al., 2021). The disagreement is not a simple contradiction of numerics; it reflects different notions of stability and different emphases on the spatial momentum-current tensor.

A second issue is methodological rather than interpretive. The operator analysis shows that the D-term is sensitive to twist-three quark-gluon correlations and that $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$31 is tied to interaction-dependent nonconservation and trace-anomaly physics, so extracting a D-term from twist-2 GPD moments does not imply a purely twist-2 mechanical interpretation (Tanaka, 2019). Light-front calculations sharpen the same caution in a different language: $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$32 extracted from $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$33 is relatively robust, whereas $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$34 extracted from transverse components such as $T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$35 can be strongly sensitive to small-$T^{\mu\nu}=T_q^{\mu\nu}+T_g^{\mu\nu},$36, nonvalence sectors, and possible zero modes (Sain et al., 18 May 2026). A plausible implication is that GMFF phenomenology is intrinsically more delicate in the mechanical sector than in the momentum sector.

GMFFs therefore occupy a junction between exact symmetry constraints, higher-twist operator structure, spatial stress reconstruction, and, in higher-spin systems, genuine multipolar deformation. Their invariant definitions are straightforward, but their physical interpretation depends sharply on the spin of the target, on whether quark and gluon sectors are separated, on whether one is probing conserved or non-conserved combinations, and on whether the intended meaning is quantum-field-theoretic, classical-radiative, or Newtonian.