Emergence of dynamical tensor fields in composite models of gravity (2512.08435v1)

Abstract: We study the composite models of gravity and investigate how dynamical tensor fields emerge by utilizing the functional Renormalization Group. In this paper, we consider two models: One is the fermionic theory. Another is the scalar theory. In both cases, we introduce an auxiliary tensor field corresponding to a composite field of the energy-momentum tensor by means of the Hubbard-Stratonovich transformation. We derive the flow equations for the field renormalization factors of the auxiliary tensor field and show the tensor field becomes dynamical in the infrared regime. The resulting kinetic terms are interpreted as gauge-fixed terms.

• The paper demonstrates that quantum fluctuations in matter fields generate kinetic terms for auxiliary tensor fields in composite gravity frameworks.
• The authors employ the functional renormalization group with optimized Litim regulators to derive flow equations and compute precise kinetic term coefficients.
• Results indicate that the emergent tensor field dynamics follow gauge-fixed kinetic structures, challenging the restoration of full diffeomorphism invariance.

Emergence of Dynamical Tensor Fields in Composite Models of Gravity

Background and Motivation

The pursuit of a quantum theory of gravity remains an open challenge, particularly due to the nonrenormalizability of the Einstein-Hilbert action within standard perturbative formulations. Composite models of gravity propose a compelling alternative wherein the graviton is recast not as a fundamental degree of freedom, but rather as an emergent composite excitation of more elementary fields. This paradigm draws motivation from analogous phenomena observed in other areas of quantum field theory, such as the composite nature of pions in QCD via the Nambu–Jona-Lasinio mechanism.

The paper "Emergence of dynamical tensor fields in composite models of gravity" (2512.08435) analyzes how dynamical tensor fields associated with gravity emerge in such composite frameworks, focusing on the generation of kinetic terms through quantum fluctuations of underlying matter fields. The theoretical approach employs the functional Renormalization Group (fRG) to systematically derive the flow equations governing these emergent gravitational degrees of freedom.

Model Construction and Auxiliary Field Formalism

Two archetypal models are considered:

• Fermionic Model: A $U(N_f)$ theory in which the elementary fields are Dirac spinors $\psi$, with strong four-fermion interactions constructed from the energy-momentum tensor $T^{(f)}_{\mu\nu}$.
• Scalar Model: A massless $O(N)$ theory where interactions are built from the scalar energy-momentum tensor $T^{(s)}_{\mu\nu}$.

In both cases, composite interactions quadratic in the energy-momentum tensor are decoupled via the Hubbard–Stratonovich transformation, introducing auxiliary symmetric tensor fields ($H_{\mu\nu}$ for the fermionic model, $C_{\mu\nu}$ for the scalar model) that serve as proxies for the emergent metric (graviton-like) degrees of freedom. Notably, these fields are initially non-dynamical, lacking kinetic terms at the level of the bare action.

Renormalization Group Flows and Dynamicalization

The central technical achievement is the evaluation of quantum-induced kinetic terms for these auxiliary tensor fields. Utilizing the Wetterich equation, the authors derive the flow equations for mass parameters and field renormalization factors ($Z_{Hi}$, $Z_{Ci}$) in both the fermionic and scalar cases. The loop corrections, computed within the fRG framework and employing optimized Litim regulators, engender momentum-dependent two-point functions whose structure supports the emergence of dynamic (propagating) tensor field modes in the infrared.

The precise RG flow equations feature explicit coefficients, with strong numerical results for the ratios of kinetic term contributions (e.g., $A_i$ and $B_i$ for the fermionic and scalar models, respectively). These coefficients are fixed and independent of the UV cutoff or the number of field flavors, being determined purely by the algebraic structure of the underlying interaction and RG scheme.

Kinetic Term Structure and Gauge Fixing

Upon solving the RG flow equations and imposing IR masslessness, the kinetic terms of the emergent tensor fields are explicitly computed. After field renormalization, the resulting kinetic Lagrangians for $H_{\mu\nu}$ and $C_{\mu\nu}$ take the generic form: $$\mathcal{L} = \frac{1}{4} (\partial_\lambda h_{\mu\nu})^2 + a_1 (\partial_\lambda h_{\mu\nu}) (\partial_\nu h_{\mu\lambda}) + a_2 (\partial_\mu h_{\mu\nu}) (\partial_\nu h) + a_3 (\partial_\mu h)^2$$ where the coefficients $a_1$, $a_2$, $a_3$ are determined by the ratios of $Z_{Hi}$ or $Z_{Ci}$. Crucially, these structures differ from the diffeomorphism-invariant kinetic term of linearized general relativity. The specific ratios $(a_1,a_2,a_3)$ match those of non-covariant gauge-fixed kinetic terms, demonstrating a strong claim that the emergent composite models naturally induce gauge-fixed forms rather than fully invariant general relativistic kinetic structures.

The implication is that, while composite gravity models dynamically generate massless, propagating tensor fields analogous to gravitons, the precise form of their dynamics is intrinsically intertwined with gauge-fixing-like contributions arising from quantum loops (within the truncation considered), rather than emergent diffeomorphism invariance.

Figure 1: Single graviton exchange process in the fermionic (left) and scalar (right) theories.

Gravitational Coupling Extraction

Gravitational interactions are encoded in the coupling between the dynamical tensor fields and the energy-momentum tensor of matter. The effective gravitational constant $G_N$ is identified from the single-graviton exchange amplitude, with the relationship $G_N = \kappa^2/(4\pi Z_0)$, where $\kappa$ is the coupling and $Z_0$ is the field renormalization in the IR limit. The emergence of $G_N$ in this form demonstrates the compositeness-induced suppression expected from the large-$N$ or strong-coupling dynamics.

Implications, Limitations, and Future Directions

The explicit emergence of dynamical tensor fields in both fermion and scalar composite models validates the fRG framework as an effective tool for investigating pregeometric scenarios. The strong numerical results for kinetic term coefficients and the absence of diffeomorphism-invariant structures at leading order raise important theoretical questions. In particular, achieving full general relativistic invariance may require extended truncations, inclusion of additional composite operators, or the evaluation of higher-order nonperturbative vertices.

The approach is limited to quadratic truncations and specific regularization schemes. Evaluating multi-graviton vertices, graviton self-interactions, and matter-graviton couplings beyond the quadratic order is necessary for a deeper understanding and to potentially restore the expected invariances. Methods such as dynamical bosonization in the fRG context are highlighted as promising for future advances, with precedents in gauge and strong-interaction theories [Gies:2001nw, Pawlowski:2005xe].

Conclusion

The paper establishes, within composite gravity frameworks, that quantum fluctuations of fundamental matter fields can induce finite kinetic terms for auxiliary tensor fields that model emergent gravitons (2512.08435). These kinetic terms correspond to gauge-fixed rather than diffeomorphism-invariant structures, both in fermion and scalar models, substantiated by strong, parameter-independent numerical results for all contributing coefficients. The theoretical implications suggest that generating true general relativistic kinetics in composite models may require more complex RG treatments, advancing both the practical definition of composite gravity theories and their foundational understanding within quantum field theory.

Future research directions involve probing multi-graviton vertices, dynamical bosonization within RG flows, and the pursuit of minimal truncations capable of restoring full symmetry, thereby aligning emergent gravity more closely with observable gravitational phenomena.