Limit Cycles and Conformal Invariance

Abstract

There is a widely held belief that conformal field theories (CFTs) require zero beta functions. Nevertheless, the work of Jack and Osborn implies that the beta functions are not actually the quantites that decide conformality, but until recently no such behavior had been exhibited. Our recent work has led to the discovery of CFTs with nonzero beta functions, more precisely CFTs that live on recurrent trajectories, e.g., limit cycles, of the beta-function vector field. To demonstrate this we study the S function of Jack and Osborn. We use Weyl consistency conditions to show that it vanishes at fixed points and agrees with the generator Q of limit cycles on them. Moreover, we compute S to third order in perturbation theory, and explicitly verify that it agrees with our previous determinations of Q. A byproduct of our analysis is that, in perturbation theory, unitarity and scale invariance imply conformal invariance in four-dimensional quantum field theories. Finally, we study some properties of these new, "cyclic" CFTs, and point out that the a-theorem still governs the asymptotic behavior of renormalization-group flows.

31 pages, 4 figures. Expanded introduction to make clear that cycles discussed in this work are not associated with unitary theories that are scale but not conformally invariant

References in corpus (3)

• Comments on Unparticles
• Limit Cycles in Four Dimensions
• On Limit Cycles in Supersymmetric Theories

Cited by in corpus (90)

• Convexity and Liberation at Large Spin
• Deformations of Superconformal Theories
• Effective Theory of a Light Dilaton
• Anomalies, Renormalization Group Flows, and the a-Theorem in Six-Dimensional (1,0) Theories
• Conformal Invariance in the Long-Range Ising Model
• On Scale and Conformal Invariance in Four Dimensions
• Constraints on RG Flow for Four Dimensional Quantum Field Theories
• Higgs mass naturalness and scale invariance in the UV
• Exotic RG Flows from Holography
• Standard Model Vacuum Stability and Weyl Consistency Conditions
• The local Callan-Symanzik equation: structure and applications
• Spontaneously Broken Conformal Symmetry: Dealing with the Trace Anomaly
• Higher-Order Scheme-Independent Series Expansions of ${\gamma }_{\bar{\psi }\psi ,IR}$ and ${\beta }_{IR}^{\prime }$ in Conformal Field Theories
• Constraints on 3- and 4-loop $\beta$-functions in a general four-dimensional Quantum Field Theory
• General Properties of Multiscalar RG Flows in $d=4-\epsilon$
• The a theorem for Gauge-Yukawa theories beyond Banks-Zaks
• Scheme-Independent Series Expansions at an Infrared Zero of the Beta Function in Asymptotically Free Gauge Theories
• Structures on the Conformal Manifold in Six Dimensional Theories
• On the particle spectrum and the conformal window
• Higher-Loop Structural Properties of the $\beta$ Function in Asymptotically Free Vectorial Gauge Theories
• Consequences of Weyl Consistency Conditions
• Defect $a$-Theorem and $a$-Maximization
• Symmetric Superfluids
• Limit Cycles in Four Dimensions
• Four-Loop Gauge and Three-Loop Yukawa Beta-Functions in a General Renormalizable Theory
• Unimodular gravity with external sources
• Consistency of local renormalization group in d=3
• Weyl versus Conformal Invariance in Quantum Field Theory
• The a-function for gauge theories
• New Methods for Conformal Correlation Functions
• Infrared Fixed Point Physics in $\mathrm{SO}(N_c)$ and $\mathrm{Sp}(N_c)$ Gauge Theories
• Scale invariance vs conformal invariance
• Supercurrent, Supervirial and Superimprovement
• Physics of the Non-Abelian Coulomb Phase: Insights from Padé Approximants
• Scale-invariant breaking of conformal symmetry
• Dilaton: Saving Conformal Symmetry
• Scheme-Independent Calculations of Anomalous Dimensions of Baryon Operators in Conformal Field Theories
• Perturbative c-theorem in d-dimensions
• RG Limit Cycles and Unconventional Fixed Points in Perturbative QFT
• Dualities from higher-spin supergravity
• Higher-Order Scheme-Independent Calculations of Physical Quantities in the Conformal Phase of a Gauge Theory
• Scheme-Independent Calculations of Physical Quantities in an $\mathcal{N}=1$ Supersymmetric Gauge Theory
• Minimal Distances Between SCFTs
• Is boundary conformal in CFT?
• On Ambiguities and Divergences in Perturbative Renormalization Group Functions
• Scheme-Independent Calculations of Properties at a Conformal Infrared Fixed Point in Gauge Theories with Multiple Fermion Representations
• Weyl Consistency Conditions in Non-Relativistic Quantum Field Theory
• Two-loop renormalization of multiflavor ${\varphi }^3$ theory in six dimensions and the trace anomaly
• Conformal scaling and the size of $m$-hadrons
• On Limit Cycles in Supersymmetric Theories
• Constraints on Perturbative RG Flows in Six Dimensions
• CFT data and spontaneously broken conformal invariance
• On Newton-Cartan local renormalization group and anomalies
• The a-theorem and entanglement entropy
• Comments on scale and conformal invariance in four dimensions
• A functional RG equation for the c-function
• Conformal symmetry of QCD in $d$-dimensions
• Scale without conformal invariance in membrane theory
• The a-function in six dimensions
• Hidden global conformal symmetry without Virasoro extension in theory of elasticity
• Scale Invariance vs. Conformal Invariance: Holographic Two-Point Functions in Horndeski Gravity
• Conformal invariance in three dimensional percolation
• Aspects of Ultra-Relativistic Field Theories via Flat-space Holography
• Scale Invariance, Conformality, and Generalized Free Fields
• The local RG equation and chiral anomalies
• Scheme-Independent Series for Anomalous Dimensions of Higher-Spin Operators at an Infrared Fixed Point in a Gauge Theory
• Holographic RG flows and boundary conditions in a 3D gauged supergravity
• (Delta a) curiosities in some 4d susy RG flows
• Discrete scale invariance in holography revisited
• The dilaton Wess-Zumino action in higher dimensions
• Gravity duals of cyclic RG flows, with strings attached
• Renormalization and redundancy in 2d quantum field theories
• Interacting scale but non-conformal field theories
• Conformal invariance from scale invariance in non-linear sigma models
• ${\beta }_{IR}^{\prime }$ at an Infrared Fixed Point in Chiral Gauge Theories
• Euclidean M-theory background dual to three-dimensional scale invariant field theory without conformal invariance
• Large-$N_c$ and Large-$N_F$ Limits of SU($N_c$) Gauge Theories with Fermions in Different Representations
• Holographic R-symmetric flows and the τ_U conjecture
• Anomalous Dimensions at an Infrared Fixed Point in an SU($N_c$) Gauge Theory with Fermions in the Fundamental and Antisymmetric Tensor Representations
• Fractal dimension and the counting rule of the Goldstone modes
• Superspace formulation of the local RG equation
• The local renormalization of super-Yang-Mills theories
• Weyl Consistency Conditions from a local Wilsonian Cutoff
• Tuning the Cosmological Constant, Broken Scale Invariance, Unitarity
• RG Flows and Dynamical Systems
• Scale without Conformal Invariance in bottom-up Holography
• The Weyl anomaly in interacting quantum field theory on curved spacetimes
• Divergences in anomalous dimension matrices of quarks at three loops: Explanation and simple solution
• General Gauge-Yukawa-Quartic $\beta$-Functions at 4-3-2--Loop Order
• Gradient flows for $\beta$ functions via multi-scale renormalization group equations