Renormalization in TGFTs

• Renormalization Group of Tensorial Group Field Theories is a framework combining multi-scale analysis and tensor invariance to control divergences in quantum gravity models.
• It redefines locality and connectedness using combinatorial methods and gauge-invariant subgraph classifications such as melopoles and melordering.
• The approach demonstrates renormalizability in both Abelian (U(1)) and non-Abelian (SU(2)) models, paving the way for exploring continuum spacetime emergence.

Tensorial Group Field Theories (TGFTs) are a broad class of quantum field theories in which fields are defined on products of group manifolds, typically incorporating combinatorial non-locality and tensor invariance. The renormalization group (RG) theory of TGFTs underpins their perturbative consistency, determines their continuum limit, and fundamentally shapes their role as candidate pre-geometric frameworks for quantum gravity. TGFTs generalize many traditional quantum field theory concepts—locality, connectedness, subtraction of divergences—by integrating tensor model methods, group-theoretic backgrounds, and gauge invariance constraints. This article provides a comprehensive synthesis of the renormalization group in TGFTs, focusing on foundational Abelian U(1) and non-Abelian SU(2) models, with an emphasis on the rigorous multi-scale approach, power counting, Wick ordering, combinatorial structures, and implications for quantum gravity.

1. Multi-Scale Analysis and Propagator Decomposition

The core advancement in TGFT renormalization is the introduction of a multi-scale analysis analogous to Wilsonian renormalization but adapted to fields living on group manifolds. Rather than relying on spacetime momentum, scales are defined through the spectrum of the kinetic operator (often a Laplacian on the group). A prototypical propagator on the group $G^d$ takes a heat-kernel Schwinger representation: $$C(g_1,\ldots,g_d;g_1',\ldots,g_d') = \int_0^\infty d\alpha\,e^{-\alpha m^2}\int dh\prod_{\ell=1}^d K_\alpha(g_\ell h g_\ell'{}^{-1})$$ where $K_\alpha$ is the heat kernel on $G$, and integration over $h$ enforces the closure constraint (gauge invariance) (Carrozza et al., 2012, Carrozza et al., 2013). The Schwinger parameter $\alpha$ is sliced logarithmically: $$C^\rho = \sum_{i=0}^\rho C_i\,,\qquad C_i = \int_{M^{-2i}}^{M^{-2(i-1)}} d\alpha\,e^{-\alpha m^2}\ldots$$ with $M > 1$ a fixed base. This decomposition enables tracking divergent contributions across scales, leading to uniform exponential bounds on amplitudes: $$|A_\mu| \leq K^{L(\mathcal{G})} \prod_i \prod_{k=1}^{k(i)} M^{\omega(\mathcal{G}_i^{(k)})}$$ where $\omega(\mathcal{G}_i^{(k)})$ is the divergence degree for connected components at scale $i$ (Carrozza et al., 2012).

This multi-scale approach, akin to a constructive QFT strategy, is critical for establishing power counting, controlling overlapping divergences, and building a precise RG framework on combinatorially non-local, group-theoretic spaces.

2. Generalization of Locality and Connectedness

TGFTs are fundamentally non-local: their interactions are not pointwise but organized by tensor invariance corresponding to convolutions along colored or stranded graphs. Accordingly, standard notions of locality and connectedness must be reformulated.

• Connectedness is defined via the factorization properties of the incidence matrix $\epsilon_{e f}$, where $e$ are lines (propagators) and $f$ are closed faces arising from the stranded structure. A subgraph is connected when its incidence matrix cannot be decomposed into blocks (Carrozza et al., 2012).
• Quasi-locality (traciality) generalizes locality: a subgraph is tracial if, after contraction by colored $k$-dipoles (propagator lines with accompanying colored lines between the same vertices), the interaction contracts to a connected tensor invariant. Dipole contractions "shrink" pairs of vertices while maintaining index ordering.
• Contraction of high subgraphs is performed by recursive dipole moves, identifying tracial (quasi-local) subgraphs suitable for renormalization. This framework enables factorization of divergent subgraphs into structures absorbable by tensor invariant counterterms, preserving symmetry and tensorial invariance (Carrozza et al., 2012).

These refinements are nontrivial and unique to TGFTs, where Feynman diagrams encode not spacetime diagrams but combinatorial gluings of fundamental quantum geometric building blocks.

3. Wick Ordering, Melonic Tadpoles, and Melordering

The renormalization of TGFTs requires a generalization of Wick ordering, adapted to the leading divergent structures in these theories: melonic tadpoles ("melopoles").

• Melopoles are maximal, tracial, one-vertex reducible subgraphs dominating the large-$N$ expansion and responsible for the leading divergences.
• Melordering is a subtraction scheme whereby interactions (bubbles) are decomposed into sums of "meloforests" (collections of melopoles extracted via dipole contraction), and for each melopole $m$ a subtraction operator $\tau_m$ is defined. The renormalized (melordered) interaction is: $$\Omega_\rho(I_b) = \sum_{\mathcal{S}\in\mathcal{S}(b)} \prod_{m\in\mathcal{S}} (-\tau_m) I_{b,\rho}$$ where $\mathcal{S}(b)$ ranges over meloforests of the bubble $b$ (Carrozza et al., 2012).
• The effect of melordering is to systematically subtract the divergences associated with all maximal melonic tadpoles, ensuring renormalization introduces only quasi-local, tensor-invariant counterterms.

This prescription is vital, as the combinatorial non-locality in TGFTs requires counterterms to respect the underlying tensor and gauge invariance structures. Melordering guarantees that the renormalized perturbative expansion remains finite and coherent with the model's invariance principles.

4. Power Counting, Divergence Degrees, and Renormalizability

After multi-scale decomposition and identification of quasi-local subgraphs, one classifies divergences via power counting, which, in TGFTs, depends on lines $L$, faces $F$, and the rank $r$ of the incidence matrix.

For a connected subgraph $\mathcal{G}$ in an Abelian U(1) rank-$d$ model: $$\omega(\mathcal{G}) = -2L(\mathcal{G}) + F(\mathcal{G}) - r(\mathcal{G})$$ (Carrozza et al., 2012, Samary et al., 2012).

Classification for the $d=4$ Abelian TGFT with gauge invariance:

• $\omega(\mathcal{G})=1$: vacuum melopole,
• $\omega(\mathcal{G})=0$: non-vacuum melopole or submelonic vacuum graph,
• $\omega(\mathcal{G})\leq -1$: all other graphs.

For just-renormalizable models (e.g., U(1)$^6$ $\varphi_6^4$, U(1)$^5$ $\varphi_5^6$):

• Only graphs with a degree of divergence $\omega=0$ or greater require renormalization,
• The number and structure of divergent graphs matches exactly the set of allowed interactions,
• Super-renormalizability arises when the divergence degree is so negative that only a finite number of divergences occur (e.g., in U(1)$^5$ $\varphi_5^4$) (Samary et al., 2012).

Power counting is strongly shaped by the rank $d$ and the combinatorial properties of the bubble interactions, fundamentally differing from scalar or gauge QFTs in their engineering dimensions and divergence patterns.

5. Gauge Invariance, Closure Constraint, and Structural Constraints

A defining feature of many TGFTs is the imposition of a gauge invariance or closure constraint, crucial for their geometric interpretation and renormalization properties.

• The closure constraint enforces invariance under simultaneous (diagonal) group action: $$\varphi(hg_1,\ldots,hg_d) = \varphi(g_1,\ldots,g_d),\quad \forall h\in G$$ which in momentum space is realized as a delta function $\delta(\sum_{i=1}^d p_i)$, tightly correlating the strand variables (Carrozza et al., 2012, Samary et al., 2012, Carrozza et al., 2013).
• This constraint fundamentally alters the combinatorics and divergence structure, manifesting in the form of strict power counting formulas and non-trivial face-connectedness of subgraphs.
• In renormalization, this ensures that divergences exhibit the same tensorial and trace invariance as the original interactions, allowing for absorption into a finite set of couplings consistent with the symmetries.

In models with non-Abelian gauge groups (e.g., SU(2)), the closure constraint remains pivotal, with divergences classified through face-connected subgraphs, and counterterms maintaining geometric and group-theoretic content essential for quantum gravity applications (Carrozza et al., 2013).

6. Perturbative Renormalizability, RG Flows, and Fixed Points

Through the application of the above techniques, it is established that several classes of TGFTs are just-renormalizable or super-renormalizable:

• For Abelian U(1)$^d$ models, the classification by rank and interaction degree yields just-renormalizable examples (e.g., $\varphi_6^4$ on U(1)$^6$ and $\varphi_5^6$ on U(1)$^5$) and super-renormalizable truncations (e.g., $\varphi_5^4$ on U(1)$^5$) (Samary et al., 2012).
• For non-Abelian models with geometric closure, as in SU(2) Boulatov-type models in $d=3$, just-renormalizability is proven up to degree-6 tensor invariants (Carrozza et al., 2013).
• Exact renormalization equations (in the Polchinski and Wetterich/FRG sense) have been formulated and analyzed for both Abelian and non-Abelian models, allowing extraction of non-perturbative beta functions, demonstration of asymptotic freedom in select cases, and mapping of fixed point structures (Krajewski et al., 2015).
• In the Abelian U(1)$^4$ super-renormalizable case, the main theorem is that melordered interactions result in finite Schwinger functions at every order, with only melopole counterterms needed and a convergent sum over scale attributions:

$|A^R| \leq K^{L(\mathcal{G})}$

(Carrozza et al., 2012).

The concrete RG flows in these models, as established via multi-scale and combinatorial methods, can exhibit fixed points corresponding to different phases. In highly controlled settings, this suggests the potential for geometrogenesis—a transition from a pre-geometric quantum phase to an emergent continuum spacetime (Benedetti et al., 2014, Benedetti et al., 2015).

7. Implications for Quantum Gravity and Extensions

The rigorous control over the renormalization group in TGFTs has foundational implications for quantum gravity:

• Perturbatively renormalizable TGFTs, under strong symmetry and combinatorial constraints, constitute well-defined QFTs of discrete pre-geometric structures.
• The multi-scale methodology, combinatorial classification, and subtraction schemes (e.g., melordering) equip these models with the necessary apparatus to paper the continuum limit in a background-independent setting.
• The geometric content enforced by gauge invariance and closure constraints bridges TGFTs to spin foam models, loop quantum gravity, and the program to derive continuum gravitational dynamics from quantum discrete pre-geometric degrees of freedom.
• Robustness under RG flows, super-renormalizability, and the presence of well-controlled fixed points open avenues for further paper, particularly regarding non-Abelian models, phase diagrams, and non-perturbative quantum gravity scenarios.
• Constructive techniques, possibly leveraging the loop vertex expansion, present plausible future directions for establishing mathematically complete quantum gravity models built on the TGFT formalism (Carrozza et al., 2012).

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In sum, the renormalization group of tensorial group field theories incorporates rigorous multi-scale analysis, redefined QFT structural notions, and combinatorial tensorial methods to resolve divergences, classify interactions, and underpin the construction of background-independent quantum field theories for discrete quantum geometry. These tools have enabled the proof of renormalizability in a set of Abelian and non-Abelian TGFTs, clarified the role of gauge invariance and closure constraints, and laid a systematic foundation for the emergence of continuum spacetime from quantum gravity models.