Uncovering novel phase structures in ${\square }^k$ scalar theories with the renormalization group

Abstract

We present a detailed version of our recent work on the renormalization group approach to multicritical scalar theories with higher derivative kinetic term of the form $\varphi (-\square {)}^k\varphi$ and upper critical dimension $d_c=2nk/(n-1)$. Depending on whether the numbers $k$ and $n$ have a common divisor two classes of theories have been distinguished which show qualitatively different features. For coprime $k$ and $n-1$ the theory admits a Wilson-Fisher type fixed point with a marginal interaction ${\varphi }^{2n}$. We derive in this case the renormalization group equations of the potential at the functional level and compute the scaling dimensions and some OPE coefficients, mostly at leading order in $\epsilon$. While giving new results, the critical data we provide are compared, when possible, and accord with a recent alternative approach using the analytic structure of conformal blocks. Instead when $k$ and $n-1$ have a common divisor we unveil a novel interacting structure at criticality. In this case the phase diagram is more involved as other operators come into play at the scale invariant point. ${\square }^2$ theories with odd $n$, which fall in this class, are analyzed in detail. Using the RG flows that are derived at quadratic level in the couplings it is shown that a derivative interaction is unavoidable at the critical point. In particular there is an infrared fixed point with a pure derivative interaction at which we compute the scaling dimensions. For the particular example of ${\square }^2$ theory in $d_c=6$ we include some cubic corrections to the flow of the potential which enable us to compute some OPE coefficients as well.

61 pages; v2: typos fixed, section 4.8.1 added, to appear in EPJC

References in corpus (5)

• Asymptotic safety guaranteed
• Generalized Wilson-Fisher critical points from the conformal OPE
• New universality class in three dimensions: The critical Blume-Capel model
• Inconsistency of Minkowski higher-derivative theories
• Isotropic Lifshitz critical behavior from the functional renormalization group

Cited by in corpus (15)

• Essential Quantum Einstein Gravity
• Asymptotic safety of scalar field theories
• 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
• Character Integral Representation of Zeta function in AdS${}_{d+1}$: II. Application to partially-massless higher-spin gravities
• Anomalous dimensions of partially-conserved higher-spin currents from conformal field theory: bosonic ${\varphi }^{2n}$ theories
• Wilson-Fisher fixed points for any dimension
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories