Multi-critical ${\square }^k$ scalar theories: A perturbative RG approach with $\epsilon$-expansion

Abstract

We employ perturbative RG and $\epsilon$-expansion to study multi-critical single-scalar field theories with higher derivative kinetic terms of the form $\varphi (-\square {)}^k\varphi$. We focus on those with a ${\mathbb{Z}}_2$-symmetric critical point which are characterized by an upper critical dimension $d_c=2nk/(n-1)$ accumulating at even integers. We distinguish two types of theories depending on whether or not the numbers $k$ and $n-1$ are relatively prime. When they are, the theory admits a local potential approximation. In this case we present the beta functional of the potential and use this to calculate some anomalous dimensions and OPE coefficients. These confirm some CFT data obtained using conformal block techniques, while giving new results. In the second case where $k$ and $n-1$ have a common divisor, the theories show a much richer structure induced by the presence of derivative operators. We study the case $k=2$ with odd values of $n$, which fall in the second class, and calculate the functional flows and spectrum. These theories have a phase diagram characterized at leading order in $\epsilon$ by four fixed points which apart from the Gaussian UV fixed point include an IR fixed point with purely derivative interactions.

6 pages, latex, 1 figure; v2: typos corrected, to appear in PRD - Rapid Communications

References in corpus (8)

• Generalized Wilson-Fisher critical points from the conformal OPE
• The analytic structure of conformal blocks and the generalized Wilson-Fisher fixed points
• Leading CFT constraints on multi-critical models in d>2
• Functional perturbative RG and CFT data in the $\epsilon$-expansion
• New universality class in three dimensions: The critical Blume-Capel model
• Inconsistency of Minkowski higher-derivative theories
• Higher dimensional higher derivative ${\varphi }^4$ theory
• Isotropic Lifshitz critical behavior from the functional renormalization group

Cited by in corpus (15)

• Essential Quantum Einstein Gravity
• Functional Renormalization and $\overline{\text{MS}}$
• Leading order CFT analysis of multi-scalar theories in d>2
• Scale and Conformal Invariance in Higher Derivative Shift Symmetric Theories
• Weyl Covariance and the Energy Momentum Tensors of Higher-Derivative Free Conformal Field Theories
• Multi-critical multi-field models: a CFT approach to the leading order
• Indications of isotropic Lifshitz points in four dimensions
• Symmetry and universality of multi-field interactions in $6-\epsilon$ dimensions
• The search for the universality class of metric quantum gravity
• Renormalization of multicritical scalar models in curved space
• Anomalous dimensions of partially-conserved higher-spin currents from conformal field theory: bosonic ${\varphi }^{2n}$ theories
• Symmetry breaking and RG flows with higher dimensional operators
• On RG flows in Generalized Effective Field Theory
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories
• Boundary anomalous dimensions from BCFT: O($N$)-symmetric ${\varphi }^{2n}$ theories with a boundary and higher-derivative generalizations