UV Fixed Points in Large N_f Theories

• UV fixed points in large-Nf theories are defined by a resummed beta function that suggests an interacting conformal field theory in the ultraviolet regime.
• Dominant fermion bubble corrections induce a universal singularity at K* = 3, emphasizing the need for full resummation to control scheme-dependent higher-order effects.
• Observable operator dimensions, such as those of fermion masses and glueball states, provide critical tests for unitarity and the physical viability of asymptotic safety.

Ultraviolet (UV) fixed points in large-$N_f$ field theories represent scenarios in which a quantum field theory with many fermion flavors flows to a non-trivial interacting conformal field theory (CFT) in the ultraviolet regime, rather than becoming trivial or encountering a Landau pole. Such fixed points underlie the notion of "asymptotic safety" beyond the classic paradigm of asymptotic freedom and are of broad interest for quantum field theory, critical phenomena, the conformal window, and extensions of the Standard Model. Large-$N_f$ techniques provide the key analytic leverage for identifying, classifying, and understanding these fixed points across a range of gauge, Yukawa, and Gross–Neveu-type models, in both integer and continuous dimensions.

1. Structure of UV Fixed Points in Large $N_f$ Gauge Theories

The organizing principle for large-$N_f$ analyses is the dominance of fermion-induced corrections in the renormalization group (RG) flow equations. The resummed beta function for the gauge coupling $K$ (a rescaled ’t Hooft-like parameter) takes the schematic form: $$\beta(K) = \frac{2K^2}{3} \left[ 1 + \frac{F_1(K)}{N_f} + \frac{F_2(K)}{N_f^2} + \cdots \right].$$ At leading order in $1/N_f$, $F_1(K)$ encodes all fermion-bubble corrections. Its non-abelian component develops a logarithmic singularity (pole) at $K^* = 3$, universally associated with the emergence of a nontrivial UV fixed point in non-Abelian theories (Antipin et al., 2017, Antipin et al., 2018, Cacciapaglia et al., 2020). Explicitly, the condition for the fixed point is

$1 + \frac{F_1(K^*)}{N_f} = 0,$

which for large $N_f$ gives $K^* = 3 - \delta$, with $\delta \sim \exp[-c N_f]$ for an appropriate $c > 0$.

This structure is robust when generalizing to multiple fermion representations: the pole remains dictated by the non-Abelian gauge structure and is insensitive to the detailed matter content, provided the effective multiplicity $N = \sum_i n_i T(R_i)$ is large (Cacciapaglia et al., 2020).

2. Scheme Dependence, Higher-Order Contributions, and Truncation Reliability

Higher-order $1/N_f$ corrections—$F_2$, $F_3$, …—are strongly scheme dependent and introduce additional singularities. Under a general renormalization scheme transformation,

$$K = \widetilde{K} \left(1 + t_1 \frac{\widetilde{K}}{N_f} + t_2 \frac{\widetilde{K}^2}{N_f^2} + \cdots \right),$$

the higher-order terms mix with derivatives of $F_1(K)$, yielding increasingly singular contributions of the form $F_1^{(n)}(K)\sim (K-3)^{-n}$. The result is that, unless all $t_n$ vanish (i.e., only one privileged scheme exists), higher-order corrections will diverge more strongly than the leading term as $K\rightarrow 3$ (Pinoy et al., 22 Jul 2025). This undermines the reliability of any finite-order truncation, and only full resummation or non-perturbative control can render the fixed point prediction meaningful.

In particular, even if a finite set of terms suggests a UV fixed point at $K^*\simeq3$, in all but one scheme the expansion becomes unreliable in the neighborhood of the fixed point, as higher orders dominate due to their enhanced singularity.

3. Existence and Physical Interpretation of the UV Fixed Point

The existence of a solution to $1 + F_1(K^*)/N_f=0$ at $K^*\simeq 3$ signals a putative UV fixed point and thus the possibility of asymptotic safety. This “Safe QCD” scenario has been extensively developed and analyzed in the context of the conformal window and beyond (Antipin et al., 2017, Antipin et al., 2018, 1011.5917).

However, the physical interpretation is tightly constrained by the need for scheme-independence and the possible dominance of higher-order singular corrections. The positive result for UV safety, as originally conjectured, is robust only if the resummed (all-order) beta function preserves the dominance and sign of the leading singularity and no pathological higher-order effects invalidate the expansion (Pinoy et al., 22 Jul 2025).

In resummed toy models, factorially growing singular contributions may nevertheless sum to an analytic function with a fixed zero at $K^*=3$. An explicit example is

$$\beta_{\mathrm{try}}(K) = \frac{2 K^2}{3} \exp\left[\frac{1}{N (K-3)}\right],$$

whose fixed point at $K=3$ survives all singular terms only after resummation. This illustrates that a useful fixed point can only be strictly established by resumming the full tower of singularities.

4. Observable Properties at the Fixed Point: Operator Dimensions

At the candidate UV fixed point, operator scaling dimensions (in particular, anomalous dimensions) play a crucial role in assessing the consistency of the theory.

For the fermion mass operator: $\gamma_m^* = \lim_{A\to 3} \frac{C_R}{2T_R N_f}$ in fundamental representations, yielding typically $\gamma_m^* \sim 1/20$ at large $N_f$—comfortably within unitarity and bootstrap bounds (Antipin et al., 2017).

However, the anomalous dimension of the glueball operator $\mathrm{Tr}~F^2$ is directly tied to the derivative of the beta function at the fixed point: $$\gamma_{F^2} = -\beta'(\alpha^*) + \frac{2 \beta(\alpha^*)}{\alpha^*}.$$ At the would-be fixed point $K^*\to 3$, this anomalous dimension can exhibit exponential growth with $N_f$ and violate the unitarity bound by orders of magnitude (Ryttov et al., 2019). This signals either the decoupling of the operator from the spectrum or that the fixed point is unphysical.

Baryon operator dimensions, by contrast, remain small and within unitarity for all considered $N_f$.

5. Beyond Gauge Theories: Yukawa, Scalar, and Gross–Neveu Theories

Large-$N_f$ methods naturally extend to gauge–Yukawa systems and purely fermionic models (e.g., Gross–Neveu theories):

• In gauge–Yukawa systems, a controlled $1/N_f$ expansion for the beta functions of all couplings is possible, leading to similar UV fixed point structure as in pure gauge theories, provided the gauge group is non-Abelian and no Abelian factors induce additional singularities (Antipin et al., 2018).
• In three-dimensional Gross–Neveu theories with sextic interactions, a line of exactly marginal UV fixed points (i.e., a conformal manifold) emerges at large $N_f$. Universal scaling dimensions and critical phase diagrams can be extracted analytically (Cresswell-Hogg et al., 2022).

6. Dimensionality, Abelian Limit, and Phase Structure

The large-$N_f$ fixed point structure is sensitive to the spacetime dimension and the abelian or non-Abelian nature of the gauge group:

• For non-Abelian theories, the essential UV fixed point mechanism persists from $d=4$ into the $d=4+2\epsilon$ expansion and can even support genuinely interacting fixed points in $d=5$ (e.g., for $SU(2)$ with $n_f\leq 4$) as shown using ε-expansions resummed with Padé–Borel techniques (Cesare et al., 2021, Cesare et al., 2022).
• In Abelian QED, the corresponding pole in the beta function (at $A=15/2$ rather than $A=3$) leads to exponentially large fermion mass anomalous dimensions at the purported fixed point, violating unitarity bounds and casting doubt on the physicality of asymptotic safety in Abelian gauge theory (Antipin et al., 2017).

7. Outstanding Issues and Requirements for Asymptotic Safety

The existence of nontrivial UV fixed points in large-$N_f$ non-Abelian gauge theories is supported at leading order but undermined by the proliferation of scheme-dependent, increasingly singular corrections at higher orders (Pinoy et al., 22 Jul 2025). The only way to give these fixed points a physical interpretation is to employ complete resummation or develop non-perturbative frameworks (such as conformal bootstrap or lattice simulations) that can control or circumvent these singularities.

Furthermore, for matter representations, semi-simple gauge groups, and their applications (including chiral and Grand Unified models), the structure of the UV fixed point is preserved as long as the effective non-Abelian coupling dominates and the corresponding singularity remains at $K^*=3$ (Cacciapaglia et al., 2020).

Absent resummation or additional control, any assertion of asymptotic safety in large-$N_f$ theories based on finite-order results must be viewed as provisional.

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Summary Table: Core Structural Features of Large-$N_f$ UV Fixed Points

Feature	Non-Abelian Gauge Theories	Abelian (QED)	Gross–Neveu, Yukawa, Scalar Extensions
Singular Point of $\beta(K)$	$K^*=3$	$K^*=15/2$	Model-dependent (typically controlled by marginal sextic or quartic interactions)
Scheme Dependence	Absent at LO, severe at subleading orders	Absent at LO, but anomalous dimension explosion at $K^*$	Controlled via $1/N_f$ critical point analysis
Glueball $\gamma$	Exponentially divergent as $N_f\uparrow$	Irrelevant; unitarity violation for mass operator	Universality class determined by marginal interactions
Baryon $\gamma$	Small, within unitarity	N/A	N/A
Physical UV FP	Tentative, requires resummation	Not physical	Marginal lines/fixed point manifolds exist

In conclusion, the paper of UV fixed points in large-$N_f$ theories yields a rich structure governed by the resummed behavior of the gauge beta function, with the non-Abelian case demonstrating a universally located interacting fixed point at $K^*=3$ at leading order. However, the breakdown of the $1/N_f$ expansion at subleading orders due to increasingly singular, scheme-dependent corrections severely limits the reliability of this result, unless the expansion can be fully resummed or handled non-perturbatively. These realities impose strict constraints on claims of asymptotic safety in such models and outline the challenges for future analysis.