RG Limit Cycles and Unconventional Fixed Points in Perturbative QFT

Abstract

We study quantum field theories with sextic interactions in $3-\epsilon$ dimensions, where the scalar fields ${\varphi }^{ab}$ form irreducible representations under the $O(N{)}^2$ or $O(N)$ global symmetry group. We calculate the beta functions up to four-loop order and find the Renormalization Group fixed points. In an example of large $N$ equivalence, the parent $O(N{)}^2$ theory and its anti-symmetric projection exhibit identical large $N$ beta functions which possess real fixed points. However, for projection to the symmetric traceless representation of $O(N)$, the large $N$ equivalence is violated by the appearance of an additional double-trace operator not inherited from the parent theory. Among the large $N$ fixed points of this daughter theory we find complex CFTs. The symmetric traceless $O(N)$ model also exhibits very interesting phenomena when it is analytically continued to small non-integer values of $N$. Here we find unconventional fixed points, which we call "spooky." They are located at real values of the coupling constants $g^i$, but two eigenvalues of the Jacobian matrix $\partial {\beta }^i/\partial g^j$ are complex. When these complex conjugate eigenvalues cross the imaginary axis, a Hopf bifurcation occurs, giving rise to RG limit cycles. This crossing occurs for $N_{\mathrm{crit}}\approx 4.475$, and for a small range of $N$ above this value we find RG flows which lead to limit cycles.

arXiv commentv4: typo in eq. (41) corrected and discussion near it improved

References in corpus (37)

• On Renormalization Group Flows in Four Dimensions
• D-branes, Quivers, and ALE Instantons
• 4d Conformal Field Theories and Strings on Orbifolds
• Towards the F-Theorem: N=2 Field Theories on the Three-Sphere
• On Conformal Theories in Four Dimensions
• On the RG running of the entanglement entropy of a circle
• Seeing a c-theorem with holography
• Conformality Lost
• The $a$-theorem and the Asymptotics of 4D Quantum Field Theory
• Walking, Weak first-order transitions, and Complex CFTs
• F-Theorem without Supersymmetry
• Large N limit of orbifold field theories
• Exact Results in Non-Supersymmetric Large N Orientifold Field Theories
• String Expansion as Large N Expansion of Gauge Theories
• Limit Cycles and Conformal Invariance
• Walking, Weak first-order transitions, and Complex CFTs II. Two-dimensional Potts model at $Q>4$
• Unitarity violation at the Wilson-Fisher fixed point in 4-epsilon dimensions
• Bosonic Tensor Models at Large $N$ and Small $\epsilon$
• Limit cycles in quantum theories
• RG Flows and Bifurcations
• (In)validity of large N orientifold equivalence
• Perturbative Search for Fixed Lines in Large N Gauge Theories
• Large N Field Theory and AdS Tachyons
• Perturbative Gauge Theory and Closed String Tachyons
• Necessary and sufficient conditions for non-perturbative equivalences of large N orbifold gauge theories
• Seeking Fixed Points in Multiple Coupling Scalar Theories in the $\epsilon$ Expansion
• Effective potential in non-supersymmetric SU(N) x SU(N) gauge theory and interactions of type 0 D3-branes
• The Renormalization Group Limit Cycle for the 1/r^2 Potential
• Generalized $F$-Theorem and the $\epsilon$ Expansion
• Prismatic Large $N$ Models for Bosonic Tensors
• Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense of $O(N)$ Symmetry with Non-integer $N$
• Can Renormalization Group Flow End in a Big Mess?
• Limit Cycles in Four Dimensions
• RG flows, cycles, and c-theorem folklore
• Atomic collapse in graphene and cyclic RG flow
• The a-function in three dimensions: beyond leading order
• Renormalization group procedure for potential $-g/r^2$

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• Emergent Supersymmetry at Large $N$
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