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

Abstract

We present a detailed version of our recent work on the RG approach to multicritical scalar theories with higher derivative kinetic term φ(−) k φ 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. 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. 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 2 theory in d c = 6, also some OPE coefficients. * = 0 pattern of mixing, which includes non-derivative couplings as well, in this case is given by the following table

Multi-Critical Multi-Field Models: A CFT Approach to the Leading Order

“…In this case additional information is also given by some special structure constants (OPE coefficients) derived at LO in the CFT+SDE framework which can be also obtained analyzing the expansion of the beta functional for v in the second order in the deformations [22]. We also note that for non unitary theories, with standard kinetic terms but odd potential interactions or higher derivative theories which can have also derivative interactions, results in the -expansion obtained assuming conformal symmetry have been found to be in full agreement with renormalization group analysis [20,24].…”

“…The results are the same as those obtained with perturbative RG methods, which if treated at functional level give rise to a very compact and effective computational framework. Actually for certain models (unitary multi-critical) one can easily reconstruct the functional perturbative RG (FPRG) equations [21][22][23][24][25] starting from the obtained CFT relations. Before discussing which relations are implied by assuming the CFTs and the lagrangian description in the general multi-field case, let us illustrate how to deal with the simpler theories with a single scalar field [13].…”

Leading order CFT analysis of multi-scalar theories in $$d>2$$ d > 2

“…Again interesting constraints on the symmetries should be investigated both in this framework and in the functional perturbative RG to get access to their universal data. Among these possible extensions we would like to include the study of nonunitary higherderivative theories, which have recently been investigated in the single-field case in [24,25] and which are important to extend critical theories to higher dimensions.…”

“…This method has also another important advantage which is relevant for this paper: it can be used with a lot of generality in that very little of the system under investigation must be specified a priori. For example, it can be used for theories with rather arbitrary interacting potential (as we will do in this paper), as well as for families of scalar theories with both unitary and nonunitary interactions [22,23], and even for higher derivative theories in which the derivative interactions are present at criticality [24,25]. Since it may also happen that internal symmetries emerge at critical points [26][27][28][29][30], this method can be taken as a starting point to investigate theories not constrained a priori by any symmetry, even including supersymmetry [31,32].…”

The Search for the Universality Class of Metric Quantum Gravity

“…This candidate can be found by following the same logic of our guiding principles, but applying it to the gauge degrees of freedom, rather than to the metric. An interesting aspect of this idea is that a higher multicriticality should emerge from the competition of derivative interactions of various orders, which is known to lead to critical field theories in the scalar case [78,79]. Some of us plan to come back to this topic soon.…”