Abstract
We employ perturbative RG and $\epsilon$-expansion to study multi-critical single-scalar field theories with higher derivative kinetic terms of the form $\phi(-\Box)^k\phi$. We focus on those with a $\mathbb{Z}_2$-symmetric critical point which are characterized by an upper critical dimension $d_c=2 n k/(n-1)$ 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.
Highlights
Universal large distance behavior of many physical systems are well described within the theoretical framework of quantum/statistical field theory at criticality
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 marginal derivative operators at criticality
We concentrate on those theories with upper critical dimension dc 1⁄4 2nk/ðn − 1Þ, which can be fixed by the requirement that φ2n be a marginal operator
Summary
Universal large distance behavior of many physical systems are well described within the theoretical framework of quantum/statistical field theory at criticality. In particular we find that theories with k > 1 admit the simple LG description of generalized Wilson-Fisher type with φ2n critical interaction only when n − 1 and k are coprime numbers In this case we confirm the anomalous dimensions, pushing the computation to order ε2 for the relevant operators, and calculate an infinite family of operator product expansion (OPE) coefficients, a finite number of whom which have so far been accessible with other methods coincide with these earlier results. In this work we are interested in using perturbation theory to investigate critical theories which are smooth deformations of the generalized free CFT, LF We concentrate on those theories with upper critical dimension dc 1⁄4 2nk/ðn − 1Þ, which can be fixed by the requirement that φ2n be a marginal operator. We refer to theories with coprime k and n − 1 as “first type” and otherwise as “second type.” Based on this classification, in the following we consider these two types of theories in turn and perform an ε-expansion by moving away from the critical dimension to d 1⁄4 dc − ε and employ perturbative RG in the functional form
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