Higgs Gravitational Form Factors

• Gravitational form factors of the Higgs boson are defined via the energy–momentum tensor and characterize internal energy, pressure, and shear force distributions.
• One-loop electroweak corrections using dimensional regularization reveal key features such as the Higgs energy-radius and necessitate the nonminimal Higgs–gravity coupling.
• These form factors affect scattering amplitudes and unitarity bounds, linking effective field theory predictions to experimental collider and cosmological implications.

The gravitational form factors of the Higgs boson encode the matrix elements of the energy–momentum tensor between Higgs one-particle states, providing a quantitative framework for analyzing the spatial and dynamical structure of the Higgs in gravitational interactions. These quantities, which generalize the concept of electromagnetic form factors to the energy–momentum tensor, become relevant in both quantum field theory and effective field theory analyses. Conceptually, gravitational form factors such as θ₁(q²) and θ₂(q²) represent internal distributions of energy, pressure, and shear forces within the Higgs boson. Recent advances (Beißner et al., 27 Aug 2025, Tanaka, 2018) provide comprehensive one-loop corrections from the Standard Model (SM), and connections to higher-dimensional operators and nonminimal couplings, notably ξH†HR, that mediate Higgs–gravity interactions.

1. Mathematical Framework for Higgs Gravitational Form Factors

The gravitational form factors for a scalar (spin-0) particle such as the Higgs are defined via the energy–momentum tensor matrix element:

$$\langle p' | T_{\mu\nu} | p \rangle = \frac{1}{2}[(g_{\mu\nu}q^2 - q_\mu q_\nu)\,\theta_1(q^2) + P_\mu P_\nu\,\theta_2(q^2)]$$

where $q = p' - p$, $P = p' + p$, and $\theta_1(q^2)$, $\theta_2(q^2)$ are the gravitational form factors (GFFs) of the Higgs boson (Beißner et al., 27 Aug 2025). The normalization conditions are typically set via θ₂(0) = 1, reflecting energy conservation; the D-term, D = –θ₁(0), quantifies pressure/shear. For composite states, such as spin-0 hadrons, additional higher-twist terms enter (Tanaka, 2018), but for the elementary Higgs, only the leading terms are present.

2. One-Loop Electroweak Corrections and Renormalization

At the quantum level, θ₁(q²) and θ₂(q²) receive one-loop corrections from SM particles, including W/Z bosons, fermions, and Higgs self-interactions. The computation proceeds via dimensional regularization and on-shell renormalization, combining propagator self-energy corrections and vertex corrections with an EMT insertion. The result is:

• θ₂(q²): Ultraviolet (UV) finite; its slope at q² = 0 gives the Higgs energy-radius.
• θ₁(q²): Contains a momentum-independent UV divergence:

$$\theta_1^\text{div} = \frac{e^2 M_Z^2 (3M_H^2 + 2m_n^2 - 6M_W^2 - 3M_Z^2)}{24\pi^2 (D - 4) M_W^2 (M_Z^2 - M_W^2)}$$

(D = spacetime dimension, e = electric charge, m_n = fermion mass) (Beißner et al., 27 Aug 2025).

This divergence cannot be cancelled by SM renormalization alone, but is naturally compensated by a counterterm from the nonminimal operator $\xi H^\dag H R$.

3. Role of the Nonminimal Higgs–Gravity Operator

The operator $\xi H^\dag H R$ constitutes the unique gauge-invariant, dimension-4 coupling of the Higgs doublet to the Ricci scalar (Xianyu et al., 2013, Ren et al., 2014). Transformations between Jordan and Einstein frames via a Weyl rescaling reallocate the effects of ξ into gauge-invariant effective interactions or higher-dimensional operators. These alter the Higgs kinetic and gauge couplings, and result in energy-dependent corrections to scattering amplitudes:

• Modified kinetic term:

$$\mathcal{L}_\text{kin} \supset \frac{1}{2}(1+6\xi^2 v^2/M_\text{Pl}^2) (\partial_\mu h)^2 \implies h \rightarrow \zeta h$$

where $\zeta = [1 + 6\xi^2 v^2/M_\text{Pl}^2]^{-1/2}$ (Xianyu et al., 2013, Ren et al., 2014).

• Production and decay rates, as well as Higgs–gauge boson interactions, are universally rescaled.

The nonminimal coupling is both generated radiatively and required to renormalize certain gravitational form factors (notably θ₁).

4. Physical Interpretation: Internal Structure and Spatial Distributions

The gravitational form factor θ₂(q²) facilitates the definition of a mean-square energy–radius via:

$$r^2 = 4\,\left[\frac{d\theta_2(q^2)}{dq^2}\right]_{q^2 = 0}$$

A representative calculation yields $r^2 \sim 1.44 \times 10^{-6}~\text{GeV}^{-2}$ (Beißner et al., 27 Aug 2025), significantly smaller than the Higgs Compton wavelength squared, indicating a compact internal energy distribution. This is a manifestation of quantum corrections: even an elementary scalar presents spatial structure when probed through the EMT.

The D-term, related to θ₁(0), quantifies the net pressure forces within the Higgs and is regularized only in the presence of the nonminimal coupling, again underscoring the necessity of ξ for a consistent theory.

5. Phenomenological Signals: Unitarity Bounds and Scattering Amplitudes

The induced gravitational form factors modify weak boson scattering amplitudes, most notably causing them to scale with energy as ξ²E²/M_Pl². Coupled-channel analyses yield an explicit unitarity bound (Xianyu et al., 2013):

$|ξ| \lesssim \frac{\sqrt{8π}M_\text{Pl}}{3E}$

For an effective cutoff Λ ∼ 10 TeV, ξ can reach $O(10^{14}–10^{15})$ without violating unitarity. These corrections generate distinctive, non-resonant enhancements in cross sections for WW scattering processes, a robust target for present and next-generation collider experiments (Xianyu et al., 2013, Ren et al., 2014).

6. Analogies to Hadronic and Composite Form Factors

Formally, the Higgs gravitational form factors mirror the structures found in composite hadronic systems (Tanaka, 2018), but without higher-twist quark–gluon correlations. For spin-0 hadrons, twist-3 GPD moments determine D-terms linked to internal shear forces; for the elementary Higgs, these terms are absent, leaving a simpler structure governed by the EMT and its one-loop corrections.

In models of Higgs compositeness (Gounaris et al., 2015), the q²-dependence of form factors (especially the triple-Higgs form factor) can reveal substructure or new physics. Deviations from SM Sudakov logarithmic behavior, e.g., resonance peaks or altered slopes, are diagnostic for compositeness or strong-sector dynamics.

7. Experimental and Theoretical Implications

While direct measurement of Higgs gravitational form factors is currently infeasible, they provide a unifying language for effective field theory and quantum field theory investigations. They clarify the renormalization requirements for gravity–matter couplings and specify energy-dependent corrections that can be tested in high-energy weak boson scattering. Cosmological scenarios, such as Higgs inflation, are also controlled by ξ’s value, with unitarity bounds relaxed for large background fields (Ren et al., 2014).

In summary, gravitational form factors of the Higgs boson encapsulate the quantum-induced energy and pressure distributions as probed by gravity, require nonminimal couplings for consistency, and generate distinctive signals in scattering processes at high energies. Their analysis integrates techniques from quantum field theory, effective field theory, and collider phenomenology, providing an essential theoretical bridge between particle physics and gravitational dynamics.