Uncovering novel phase structures in $$\Box ^k$$ □k scalar theories with the renormalization group

Abstract

Abstract We present a detailed version of our recent work on the RG approach to multicritical scalar theories with higher derivative kinetic term $$\phi (-\Box )^k\phi $$ ϕ(-□)kϕ and upper critical dimension $$d_c = 2nk/(n-1)$$ dc=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 $$n-1$$ 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 $$\epsilon $$ ϵ . 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 $$n-1$$ n-1 have a common divisor we unveil a novel interacting structure at criticality. $$\Box ^2$$ □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 $$\Box ^2$$ □2 theory in $$d_c=6$$ dc=6 , also some OPE coefficients.