Uncovering novel phase structures in square(k) scalar theories with the renormalization group

We present a detailed version of our recent work on the RG approach to multicritical scalar theories with higher derivative kinetic term phi(-square)(k)phi 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. For coprime k and n - 1 the theory admits a Wilson-Fisher type fixed point. We derive in this case the RG equations of the potential 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. square(2) theories with odd n, which fall in this class, are analyzed in detail. Using the RG flows 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 and, for the particular example of square(2) theory in d(c) = 6, also some OPE coefficients.