Massless RG Flows in Quantum Field Theory

Updated 30 December 2025

Massless RG flows are continuous trajectories in theory space that connect gapless fixed points while preserving local operator spectra.
They are characterized by strict anomaly matching, fusion ring homomorphisms, and a gradient descent in the C-function across various dimensions.
These flows underpin the classification of conformal interfaces and emergent gapless phases in high-energy, condensed matter, and mathematical physics.

A massless Renormalization Group (RG) flow is a continuous interpolation between quantum field theory (QFT) fixed points—typically conformal field theories (CFTs)—under changes of energy scale, with a spectrum that remains gapless at all intermediate energy scales. Such flows are central objects of study across high-energy and condensed matter theory, and are characterized by a host of structural, algebraic, and anomaly-theoretic constraints, as well as their realization in diverse field theoretic, algebraic, and holographic contexts.

1. Mathematical and Physical Definition

A massless RG flow is a one-parameter trajectory in theory space—a space of couplings or Hamiltonians—connecting a UV fixed point (usually a CFT) to an IR fixed point that is likewise either a CFT or a (scale-invariant but not necessarily conformal) field theory. These flows are classified by the property that no gap opens in the spectrum: local operators or excitations with arbitrarily small energies persist throughout the flow. The physical content of massless RG flows is most transparent in terms of their nontrivial infrared critical (gapless) behavior and the mapping of UV operator spectra into IR critical data (Gukov, 2015, Delmastro et al., 2022).

Morse-theoretically, massless RG flows correspond to soliton-like trajectories flowing between two critical points (fixed points) of a Morse-Bott functional—the C-function—on the theory space. If the RG β-functions are of gradient form with respect to this functional, the flow is literally the gradient descent trajectory for the C-function, which monotonically decreases along the RG (Gukov, 2015).

2. Structures and Criteria Across Dimensions

Six Dimensions: (1,0) SCFTs

In 6D (1,0) supersymmetric CFTs, massless RG flows are realized by spontaneous symmetry breaking on moduli spaces of vacua—specifically, the tensor or Higgs branches (Cordova et al., 2015). On the tensor branch, real scalar VEVs in tensor multiplets break conformal invariance but leave the R-symmetry unbroken; in the deep IR, the effective theory consists of a decoupled IR CFT plus free tensors. The flow is “massless” in that there is no gap: the dilaton is a Nambu–Goldstone boson associated with broken dilatations. Matching of the ’t Hooft anomalies and the Green–Schwarz mechanism enforce the consistency of the new infrared effective theory and ensure the anomaly mismatch is always a perfect square—guaranteeing a monotonic “a-theorem” even for flows ending in scale- but not conformally invariant theories (SFTs). On the Higgs branch, hypermultiplet VEVs break both conformal and R-symmetries, again yielding additional massless Goldstone modes. A unifying linear relation links the Weyl a-anomaly to the coefficients in the total anomaly polynomial for both branches.

3. Algebraic and Fusion-Ring Interpretation

The structure of massless RG flows between rational CFTs is elegantly encoded in the language of fusion rings and ring homomorphisms (Fukusumi et al., 29 Jun 2025). Each rational CFT possesses a fusion ring $R$ of primary fields or anyons, with fusion rules specified by nonnegative integers $N_{ab}^c$ . A massless RG flow from a UV CFT to an IR CFT is mathematically modeled by a ring homomorphism $\phi: R \to R'$ , preserving the identity and all fusion products. The kernel $I = \ker\phi$ encodes those anyons that “condense” (become trivial in the IR), and the image (or, by the first isomorphism theorem, $R/I$ ) specifies the fusion structure of the IR theory. This formalism naturally captures anyon condensation, generalized global symmetry breaking, and emergence phenomena, and connects to domain walls and boundary conditions in effectively topologically ordered phases.

Explicit computations in key RG flows—such as Tricritical Ising → Ising, $(\mathrm{SU}(2)_1)^n \to \mathrm{SU}(2)_n$ , and more exotic fractional-coefficient flows—demonstrate rich and sometimes unexpected algebraic structures, supporting classification schemes for CFT interfaces and gapped phases arising via RG (Fukusumi et al., 29 Jun 2025).

4. Massless Flows in Two-Dimensional Theories

RG flows in two dimensions admit detailed characterization via field-theoretic, S-matrix, and bootstrap methods. In 1+1D integrable models with $A_n$ symmetry, complete S-matrix bootstrap equations restrict possible massless flows to three universal types: minimal, diagonal, and saturated cases. These S-matrices uniquely correspond to TBA (thermodynamic Bethe ansatz) systems and Y-systems with specific central charges, predicting the allowed UV and IR CFTs connected by massless flows (Ahn, 25 Sep 2025). Not all previously conjectured flows survive the bootstrap and TBA constraints: only these three classes yield UV-complete massless flows, as seen by explicit analysis of plateau equations and locality.

Perturbed WZW models triggered by adjoint primaries yield massless RG flows when the coupling $\lambda>0$ and certain selection rules are satisfied; for instance, in $\mathrm{SU}(N)_k$ WZNW, the flow to $\mathrm{SU}(N)_1$ is allowed only when $N$ and $k$ are coprime, as dictated by center symmetry and mixed anomaly matching (Lecheminant, 2015, Kikuchi, 2022). Anomaly-matching, “half-integral” conditions on line operators, and modular data determine when a flow remains gapless or ends in a massive phase.

2d QCD with gauge group $G$ and massless quarks flows in the IR to either CFTs or TQFTs depending on the specific matter representation content, with operator dimensions and RG operator maps computable exactly via embedding and coset decompositions. Deformations with relevant operators can drive the system through a cascade of massless flows (“tumbling”), often realized as a Higgs mechanism for the gauge group, providing non-perturbative scenarios for dynamical symmetry breaking (Delmastro et al., 2022).

5. Universal Monotonic Quantities and Theorems

A central feature of massless RG flows is the monotonic decrease of certain quantities—such as the Zamolodchikov $c$ -function in 2D, the $a$ -anomaly in even dimensions, and various generalizations in supersymmetric and holographic settings.

In six-dimensional (1,0) SCFTs, the change $\Delta a = a_{UV} - a_{IR}$ is quantifiably positive for all nontrivial tensor-branch flows, derivable both from explicit anomaly matching and from the dilaton effective action. The precise relation $\Delta a = (24576\pi^3/7) b^2$ connects the a-anomaly difference to the square of the dilaton kinetic coupling, and the value of $a$ for all SCFTs is given by a universal linear formula in terms of their 't Hooft anomaly coefficients (Cordova et al., 2015).

In holographic duals, massless flows correspond to multi-domain-wall spacetimes, and the monotonicity of holographic $a(l)$ - and $c(l)$ -functions is enforced by energy-flux positivity and can be explicitly established for Lovelock and quasi-topological gravities. Criteria derived from the sign, monotonicity, and positivity of these central functions delineate the region of parameter space for which massless flows are admissible (Sotkov et al., 2012).

6. Structural Stability and Non-Perturbative Control

Non-perturbative stability of massless RG flows is established in certain settings, as in the 2D Gross-Neveu model, where the use of the Bauerschmidt–Brydges–Slade framework allows one to prove that for sufficiently small initial coupling, the flows remain uniformly close to their quadratic approximation, and all “irrelevant tails” remain bounded in volume and UV cutoff (Dimock et al., 2023). This provides a mathematically rigorous foundation for the control of massless RG flows in weakly coupled fermionic systems.

The structural stability formalism is widely applicable to other fermionic and potentially bosonic models, provided one can construct appropriate Banach norms and multiscale decompositions.

7. Topological and Moduli Space Structure

The topology of the space of CFTs and their relevant deformations encodes the classification and counting of massless RG flows (Gukov, 2015). Using Bott–Morse theory, each (potentially degenerate) critical manifold (conformal manifold) is assigned an index equal to the count of relevant operators. The moduli space of RG flows between two CFTs is given by the intersection of unstable manifolds (UV) and stable manifolds (IR), with dimension equal to the difference of indices. Topological obstructions to a single, globally smooth C-function—such as the appearance of dangerously irrelevant operators transitioning to relevance along a flow—manifest as phase transitions on the RG trajectory, closely tied to breakdowns in the gradient-flow picture and manifest “branch flips” in the $a$ -function or conformal manifolds.

A practical upshot is that the topology of theory space, encoded in Morse–Bott polynomials, constrains the number of possible distinct massless flows and the existence of RG walls, with applications ranging from minimal models in 2D to Argyres–Douglas cascades and SQCD in 4D (Gukov, 2015).

In sum, massless RG flows are deeply constrained objects, simultaneously governed by anomaly relations, fusion ring algebra, S-matrix bootstrap equations, monotonicity theorems, and topological data of theory space. Their rigorous study interlinks high-energy, condensed matter, and mathematical approaches, yielding universal mechanisms for the classification of gapless phases, conformal interfaces, and emergent symmetry phenomena across dimensions.