One-Loop Beta Functions in QFT

• One-loop beta functions are defined as the leading-order logarithmic scale dependence of couplings in quantum field theory, derived from first-order quantum fluctuations.
• They are computed using perturbative techniques such as Feynman diagrammatics, heat kernel methods, and cohomological approaches, applicable across gauge, tensor, and gravity models.
• The analysis of one-loop beta functions provides key insights into UV/IR behaviors, fixed point structures, asymptotic freedom or safety, and implications for anomalies and topological sectors.

A one-loop beta function, denoted typically as $\beta(g)$, is a fundamental object in quantum field theory (QFT) and quantum statistical mechanics, encoding the leading-order logarithmic scale dependence of couplings under renormalization group (RG) flow. At one-loop order, the beta function captures contributions from first-order quantum fluctuations—typically those associated with elementary Feynman diagrams involving a single loop—serving as a baseline for understanding ultraviolet (UV) and infrared (IR) properties, critical phenomena, and universality classes across a broad spectrum of physical models. One-loop beta functions provide both explicit quantitative predictions for RG evolution and nontrivial constraints on fixed points, anomalies, and topological sectors.

1. Formal Definitions and Renormalization Conditions

The one-loop beta function is defined as the response of the renormalized coupling $\lambda^{\text{ren}}$ (or $g$) to changes in the regularization scale %%%%3%%%% or $\mu$ (as appropriate), holding bare couplings fixed. Schematically, in the momentum-space subtraction scheme for a rank-3 $U(1)^3$ tensor model (Geloun et al., 2011), beta and anomalous dimension functions are introduced as: $$\beta_\epsilon(\lambda) = \Lambda \frac{\partial \lambda_\epsilon^{\text{ren}}}{\partial \Lambda}\Big|_{\text{bare fixed}}, \qquad \gamma_\epsilon(\lambda) = \Lambda \frac{\partial \ln Z_\epsilon}{\partial \Lambda}\Big|_{\text{bare fixed}},$$ with wavefunction renormalizations $Z_\epsilon$ determined from two-point self-energies and renormalized couplings fixed by four-point functions at vanishing external momenta.

In the Batalin-Vilkovisky formalism (Elliott et al., 2017), the beta function acquires a cohomological meaning as the first-order deformation class in the local obstruction-deformation complex: $$\beta^{(1)} = [\,\mathcal{O}_\beta^{(1)}\,] \in H^0(\mathcal{O}_{\text{loc}}(\mathcal{E}),d_{\text{cl}}),$$ where $\mathcal{O}_\beta^{(1)}$ is the infinitesimal generator of dilations within the quantum effective interaction family.

In lattice gauge theory, such as the Schrödinger functional scheme for Möbius domain wall fermions (Murakami et al., 2017), the beta function is extracted from the logarithmic divergence coefficient in the one-loop variation of the effective action with respect to a background field parameter.

2. Computational Techniques and Diagrammatics

The computation of one-loop beta functions proceeds via perturbation theory, typically leveraging either Feynman diagrammatic expansions, background field methods, heat kernel techniques, or algebraic cohomology.

Diagrammatic approach: One-loop diagrams contributing to self-energy and vertex functions provide the $1/\epsilon$-pole structure or logarithmic divergences dictating renormalization constants $Z$ and their scale dependence. For instance, in noncommutative scalar QED$_4$ (Ghasemkhani et al., 2016), the beta coefficient combines contributions from gauge loops, matter loops, and ghost loops, with noncommutative effects introducing non-Abelian-like terms.

Heat kernel and proper-time methods: The Barvinsky-Vilkovisky generalized Schwinger-DeWitt technique (Jack, 2020), and theta-function cutoff schemes in gravity theories (Percacci et al., 2013, Percacci et al., 2010), give one-loop RG evolution by evaluating the trace of the second heat-kernel coefficient $a_2(x)$ for the effective operator on the background.

Cohomological/Index-theoretic methods: In certain gauge theories, the beta function is constructed as a scale anomaly computed via index theorems on twistor space (Bittleston et al., 30 Oct 2025), relating the scale dependence of determinant bundles to topological invariants.

3. Explicit One-Loop Beta Functions in Representative Theories

Yang-Mills and Gauge Theories

The canonical one-loop beta function for Yang-Mills in $SU(N_c)$, with $n_f$ flavors in representation $R$ (Bittleston et al., 30 Oct 2025), is: $$\beta(g) = -\frac{g^3}{16\pi^2}\frac{11 C_A - 4 T_R n_f}{3},$$ with $C_A$ the adjoint quadratic Casimir and $T_R$ the Dynkin index.

Tensor Field Theories

In rank-3 $U(1)^3$ tensor field theories (Geloun et al., 2011): $$\beta(\lambda) = -\lambda^2 \kappa + O(\lambda^3), \qquad \gamma(\lambda) = 0 + O(\lambda^2),$$ with $\kappa = \partial_{\ln\Lambda} S > 0$ and $S$ a strand-wise log-divergent sum. The model is perturbatively asymptotically free in the UV.

Infinite-Parameter Gauge Theories

An infinite-tower of couplings $g_n$ for generalized functionals $f(x)$ yields (Krasnov, 2015): $$\mu \frac{\partial}{\partial\mu} f(x) = \frac{1}{(4\pi)^2} \frac{1}{6} \, \mathrm{Tr}_{\text{adj}} \left\{ 6 f' (f'')^{-1} f' - 3 f' (f'')^{-1} f' (f'')^{-1} f' \right\}.$$ Expanding $f(x)$ as a series, individual beta functions $\beta_n$ are defined for each coupling.

Gravity and Sigma Models

In higher-derivative gravity (Jack, 2020), with $C^2$ and $R^2$ couplings $f_2, f_0$: $$\beta_{f_2^{-2}} = \frac{1}{162}\big\{312 - \ldots\big\}, \qquad \beta_{f_0^{-2}} = \ldots$$ and for nonlinear sigma models (0910.0851), the four-derivative couplings $\lambda$, $f_1$, $f_2$ satisfy: $$\beta_\lambda = -\frac{n-1}{8\pi^2} \lambda^2, \qquad \beta_{f_1}, \, \beta_{f_2} = (\ldots)\lambda,$$ with asymptotic freedom for $\lambda$ and nontrivial fixed points in $f_i$ for suitable $N$.

Enhanced Tensor Field Theories

In quartic enhanced tensor field theories (Geloun et al., 2023), certain couplings (e.g., $\lambda_+$, $\lambda_\times$) are marginal and remain fixed at one loop, with other couplings flowing to zero or growing linearly, depending on model details:

• Model $+$: $\beta_{\lambda_+} = 0$ (fixed line), other couplings are relevant and suppressed in UV.
• Model $\times$: $\beta_{\lambda_\times} = 0$, but certain kinetic terms grow linearly, precluding asymptotic freedom or safety.

4. RG Flow, Fixed Points, and Asymptotic Behavior

One-loop beta functions determine the location and nature of RG fixed points:

• Gaussian fixed point: $\beta(g^*) = 0$ at $g^* = 0$, typically UV attractive if $\beta'(0)<0$ (asymptotic freedom, e.g., tensor $\phi^4$ models (Geloun et al., 2011), Yang-Mills (Elliott et al., 2017)).
• Non-Gaussian fixed points: arise due to nontrivial zeros of $\beta(g)$ or multivariate flows, e.g., in topologically massive gravity (Percacci et al., 2010, Percacci et al., 2013), nonlinear sigma models (0910.0851), enhanced tensor models (Geloun et al., 2023). Asymptotic safety is realized where all relevant operators flow to finite values in the UV, contingent on the critical surface dimensions and coupling structure.

Marginal couplings may remain constant (as in certain tensor or gravitational Chern-Simons sectors), and relevant couplings vanish in the UV for asymptotic freedom, or approach a finite fixed point in the asymptotic safety scenario.

5. Special Features and Topological Sectors

• Topological mass terms and anomalies: Certain couplings may possess vanishing one-loop beta functions due to topological quantization or balance between bosonic and fermionic degrees of freedom. In topologically massive gravity (Percacci et al., 2010, Percacci et al., 2013), the Chern-Simons coupling $\nu$ satisfies $\beta_\nu=0$ to one loop.
• Melonic dominance and tensor locality: In tensor models (Geloun et al., 2011), only melonic diagrams contribute to leading-order divergences, simplifying the RG analysis and excluding anomalous interactions.
• Cohomological/Index-theoretic identification: The beta function may be interpreted as a cohomological obstruction to scale invariance (Elliott et al., 2017) or computed via index theorems on twistor spaces (Bittleston et al., 30 Oct 2025), linking RG flow to topological invariants.

6. Methodological Comparisons and Notable Resolutions

Coordinate-space versus momentum-space techniques may yield discrepancies if derivative ordering and gauge choices are not carefully accounted for, as resolved in renormalisable gravity studies (Jack, 2020). Universal features at one loop, such as scheme independence, are typically tied to logarithmic divergences, as evidenced in proper-time cutoff analyses (Percacci et al., 2013). Lattice implementations, such as domain wall fermion discretizations (Murakami et al., 2017), provide nonperturbative confirmations of continuum RG predictions even in the presence of explicit symmetry-breaking boundary terms.

7. Applications, Generalizations, and Implications

One-loop beta functions underpin:

• Classification of UV/IR properties (asymptotic freedom, safety)
• Verification of renormalizability and universality in generalized models (e.g., infinite-parameter gauge theories (Krasnov, 2015), higher-derivative sigma models (0910.0851))
• Model construction with nontrivial RG behavior in quantum gravity, string theory, and novel field-theoretic structures (DBI, noncommutative QED, enhanced tensors)
• Rigorous formulation of quantum anomalies, topological invariants, and deformation cohomology in mathematical QFT frameworks

A plausible implication is that one-loop beta functions, while simple in computational structure, encode robust and universal constraints on field-theoretic model-building, provide direct links between perturbative RG and deeper topological properties, and serve as a diagnostic for asymptotic regimes and continuum-limit physics.

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Key References:

• 3D Tensor Field Theory: Renormalization and One-loop $β$-functions (Geloun et al., 2011)
• Asymptotic Freedom in the BV Formalism (Elliott et al., 2017)
• The one-loop analysis of the beta-function in the Schroedinger Functional for Moebius Domain Wall Fermions (Murakami et al., 2017)
• One Loop Beta Functions in Topologically Massive Gravity (Percacci et al., 2010)
• Beta Functions of Topologically Massive Supergravity (Percacci et al., 2013)
• One-loop $\mathbfβ$ function of noncommutative scalar $QED_{4}$ (Ghasemkhani et al., 2016)
• One loop beta functions and fixed points in Higher Derivative Sigma Models (0910.0851)
• The One-Loop QCD $β$-Function as an Index (Bittleston et al., 30 Oct 2025)
• One-loop beta-functions for renormalisable gravity (Jack, 2020)
• One-loop beta-function for an infinite-parameter family of gauge theories (Krasnov, 2015)
• One-loop beta-functions of quartic enhanced tensor field theories (Geloun et al., 2023)
• $β$-function for topologically massive gluons (Mukhopadhyay et al., 2014)