--- title: Gravitational Multipole Form Factors (GMFFs) url: https://www.emergentmind.com/topics/gravitational-multipole-form-factors-gmffs type: topic --- # Gravitational Multipole Form Factors (GMFFs) 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:1905.01934 OR id:2409.15547 OR id:2110.14781 OR id:2507.14840" 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 [1905.01934] [2507.14840] [2107.02634]. ## 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 [1905.01934]. 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 [2107.02634]. In Newtonian precision-gravity work, inner multipoles \(q_{lm}\) and outer multipoles \(Q_{lm}\) describe extended mass distributions through \[ V = -4\pi G \sum_{l=0}^\infty\sum_{m=-l}^l \frac{1}{2l+1} q_{lm}Q_{lm}, \] so that the multipoles function as closed-form shape descriptors of solids such as cylinders, prisms, polygonal bodies, and cones [1707.01577]. 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 \[ \langle p'|T_{q,g}^{\mu\nu}|p\rangle = \bar{u}(p')\Bigl[A_{q,g} (t)\gamma^{(\mu}\bar P^{\nu)} +B_{q,g}(t)\frac{\bar P^{(\mu}i\sigma^{\nu)\alpha}\Delta_\alpha}{2m_N} + D_{q,g}(t)\frac{\Delta^\mu\Delta^\nu -g^{\mu\nu}t}{4m_N} + \bar{C}_{q,g}(t)m_Ng^{\mu\nu}\Bigr] u(p), \] with \(\Delta=p'-p\), \(\bar P=(p+p')/2\), and \(t=\Delta^2\). In this basis, \(A\) controls momentum and energy flow, \(B\) carries gravitomagnetic or spin-flip structure, \(D\) controls the traceless stress sector, and \(\bar C\) is the non-conserved \(g^{\mu\nu}\) term present for the separate quark and gluon sectors. A frequently used alternative basis replaces \((A,B)\) by \((A,J)\), with \[ J(t)=\frac12\bigl(A(t)+B(t)\bigr), \] so that the three independent nucleon GFFs are written directly as \(A(Q^2)\), \(J(Q^2)\), and \(D(Q^2)\) [1905.01934] [2409.15547]. For spin-0 hadrons, the decomposition simplifies because there is no spin-flip structure. One standard form is \[ \langle h(p')|T_{q}^{\mu \nu}|h(p)\rangle = \frac{1}{2}{\Theta}_{1q}(t)\left(tg^{\mu\nu}- \Delta^\mu \Delta^\nu\right)+ \frac{1}{2}{\Theta}_{2q}(t) \bar{P}^\mu\bar{P}^\nu+ \Lambda^2\bar{C}^h_q(t)g^{\mu \nu}, \] where \(\Theta_2\) is the momentum or mass form factor and \(\Theta_1\) is the D-term–type mechanical form factor. In another widely used spin-0 convention, \[ \langle \Psi(P^+,\vec{P}{\,'}^{\perp})|T^{\mu\nu}(0)|\Psi(P^+,\vec{P}^{\perp})\rangle = 2\bar{P}^{\mu}\bar{P}^{\nu} A(Q^2) +\frac{1}{2}\left(q^\mu q^\nu-q^2 g^{\mu\nu}\right)D(Q^2), \] so that the complete symmetric spin-0 gravitational structure is exhausted by \(A\) and \(D\) [1905.01934] [2605.18350]. The explicit multipole nomenclature becomes richer for spin-\(3/2\) systems. In the \(\Delta\) and \(\Omega^-\), the full EMT initially contains ten invariant form factors, of which seven survive for the conserved total EMT: \[ F_{1,0},\;F_{1,1},\;F_{2,0},\;F_{2,1},\;F_{4,0},\;F_{4,1},\;F_{5,0}. \] In the Breit frame these are reorganized into GMFFs with direct static meaning: \[ \varepsilon_0,\ \varepsilon_2,\ \mathcal J_1,\ \mathcal J_3,\ D_0,\ D_2,\ D_3. \] The resulting interpretation is explicitly multipolar: \(\varepsilon_0\) is an energy monopole, \(\varepsilon_2\) an energy quadrupole, \(\mathcal J_1\) an angular-momentum dipole, \(\mathcal J_3\) an angular-momentum octupole, and \(D_0,D_2,D_3\) mechanical multipoles associated with pressure and shear [2307.14880] [2507.14840]. ## 3. Symmetry constraints, operator relations, and QCD structure Conservation and Poincaré symmetry impose the first layer of constraints. Because the total EMT is conserved, \[ \partial_\mu T^{\mu\nu}(x)=0, \] the non-conserved pieces cancel in the total matrix element, \[ \bar C_q(t)+\bar C_g(t)=0. \] In the forward limit, \[ A_q(0)+A_g(0)=1, \] and the Ji relation gives \[ J_{q,g}=\frac12\left[A_{q,g}(0)+B_{q,g}(0)\right]. \] These statements fix the normalization of the momentum and total angular-momentum sectors but do not determine the D-term, which remains dynamical [1905.01934]. Tanaka’s operator analysis makes the non-conserved and mechanical sectors more precise. The separate quark and gluon EMTs satisfy \[ \partial _\nu T_q^{\mu \nu } = - \bar \psi g{F^{\mu \nu }{\gamma _\nu }\psi,\qquad \partial _\nu T_g^{\mu \nu } = - F_a^{\mu \nu }D_{ab}^\rho F_{\rho \nu }^b, \] so \(\bar C\) 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 \[ \int_{-1}^{1} dxx\, H^{q} (x, \eta,t) =A_q (t)+ \eta^2D_q(t), \qquad \int_{-1}^{1} dxx\,E^{q} (x, \eta,t) =B_q(t)- \eta^2D_q(t). \] The standard twist-2 polynomiality relation is therefore compatible with, but does not exhaust, the higher-twist operator content of the D-term [1905.01934]. The trace sector introduces a second structural layer. In the forward limit, \[ g_{\mu \nu }\langle p|T_{q,g}^{\mu \nu }|p\rangle = 2m_N^2\left(A_{q,g}(0)+4\bar C_{q,g}(0)\right), \] and renormalization invalidates naive classical trace identities. The one-loop expressions quoted for \(g_{\mu\nu}T_q^{\mu\nu}\) and \(g_{\mu\nu}T_g^{\mu\nu}\) show that \(\bar C_{q,g}(0)\) is constrained by the QCD trace anomaly and acquires nontrivial scale dependence [1905.01934]. 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, \[ T_a^{\mu\nu}=\bar T_a^{\mu\nu}+\hat T_a^{\mu\nu}, \] 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 [2503.23008]. At large momentum transfer, perturbative QCD and counting arguments imply additional structure. For the nucleon, the large-\(|t|\) analyses summarized in the literature give \[ A_{q,g}(t)\sim \frac{1}{(-t)^2},\qquad B_{q,g}(t)\sim \frac{\ln^2(-t/\Lambda_c^2)}{(-t)^3},\qquad C_{q,g}(t)\sim \frac{\ln^2(-t/\Lambda_c^2)}{(-t)^3}, \] with \(\bar C_q=-\bar C_g\) and \(\bar C=0\) for the total EMT [2203.13493]. For the pion and proton gluon sectors, the perturbative large-\(|t|\) results are \[ A_g^\pi(t)=C_g^\pi(t)\sim \frac{1}{-t},\qquad A_g^p(t)\sim \frac{1}{(-t)^2},\qquad C_g^p(t)\sim \frac{\ln^2(-t/\Lambda^2)}{(-t)^3}, \] so the mechanical form factor is more suppressed in the proton because it requires helicity flip and orbital angular momentum [2101.02395]. ## 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 \[ \epsilon(r) = m_N \left( \hat A(r) - \frac{1}{4m_N^2}\big[\widehat{(t D)}(r) + \widehat{(t A)}(r) - 2 \widehat{(t J)}(r)\big]\right), \] \[ p(r) = \frac{1}{6m_N}\frac{1}{r^2} \frac{d}{dr} \left[r^2 \frac{d}{dr} \hat D(r)\right], \qquad s(r) = -\frac{1}{4m_N}r \frac{d}{dr} \left[\frac{1}{r} \frac{d}{dr} \hat D(r)\right], \] and the longitudinal normal-force distribution is \[ F^\parallel(r)=p(r)+\frac{2}{3}s(r). \] In this representation the D-term alone governs pressure, shear, and the mechanical radius, whereas the mass radius depends on \(A\) and \(D\), and in the quoted density formula also on \(J\) [2409.15547]. The corresponding radii are defined by \[ \langle r^2 \rangle_{\rm mass} = \frac{\int d^3 r \, r^2 \epsilon(r)}{\int d^3 r \epsilon(r)}, \qquad \langle r^2 \rangle_{\rm mech} = \frac{\int d^3 r \, r^2 F^\parallel(r)}{\int d^3 r F^\parallel(r)}, \] with the practical forms \[ \langle r^2 \rangle_{\rm mass} = \left[\left. -6 \frac{d}{dt} A(t) \right|_{t=0} - 3 \frac{D(0)}{2 m_N^2} \right]\frac{1}{A(0)}, \qquad \langle r^2 \rangle_{\rm mech} = \frac{6 }{\int_0^\infty dt \, [D(t)/D(0)]}. \] For the proton in that framework, \[ D(0)=-3.114(10)_\pm, \] and the predicted ordering is \[ r_{\rm mech} r_\pi^F > r_\pi^{\theta_2},\qquad r_K^{\theta_1} > r_K^{F} > r_K^{\theta_2}, \] so the mechanical radius exceeds the charge radius and the charge radius exceeds the mass radius in both mesons [2311.14832]. In BLFQ, the same qualitative hierarchy is found for the 2D impact-parameter radii, \[ r_{\mathrm{mech}} > r_{\mathrm{mat}}, \] with \[ \sqrt{\langle b_\perp^2 \rangle_{\mathrm{mat}}}=0.35\pm0.04~\mathrm{fm},\qquad \sqrt{\langle b_\perp^2 \rangle_{\mathrm{mech}}}=0.64\pm0.06~\mathrm{fm} \] for the pion, and \[ \sqrt{\langle b_\perp^2 \rangle_{\mathrm{mat}}}=0.31\pm0.03~\mathrm{fm},\qquad \sqrt{\langle b_\perp^2 \rangle_{\mathrm{mech}}}=0.55\pm0.06~\mathrm{fm} \] for the kaon, although the same work emphasizes that the low-\(Q^2\) D-term is especially sensitive to small-\(x\) and possible zero-mode effects in its truncated light-front extraction [2605.18350]. For higher spin, the static interpretation itself becomes multipolar. In the \(\Omega^-\), the Breit-frame decomposition produces an energy density \[ T^{00}(\boldsymbol r,s',s)= \varepsilon_0(r)\delta_{s's} + \varepsilon_2(r)Y_2^{kl}(\Omega_r)\hat Q^{kl}_{s's}, \] an angular-momentum density driven by \(\mathcal J_1\) and \(\mathcal J_3\), and a stress tensor with monopole and quadrupole pieces governed by \(D_0,D_2,D_3\). The monopole components dominate numerically, but the quadrupole pieces are genuine higher-order deformations rather than kinematic artifacts [2507.14840]. ## 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-\(|t|\) behavior. | System and framework | GMFF basis | Representative reported results | |---|---|---| | Proton, symmetry-preserving continuum QCD | \(A,J,D\) | \(D(0)=-3.114(10)_\pm\); \(F^{\mathpzc g}(Q^2)/\sum_{\mathpzc q}F^{\mathpzc q}(Q^2)=0.71(4)\); \(r_{\rm mech}