Leading CFT constraints on multi-critical models in d > 2

Abstract

Abstract:We consider the family of renormalizable scalar QFTs with self-interacting potentials of highest monomial φ m below their upper critical dimensions d c = 2m m−2 , and study them using a combination of CFT constraints, Schwinger-Dyson equation and the free theory behavior at the upper critical dimension. For even integers m ≥ 4 these theories coincide with the Landau-Ginzburg description of multi-critical phenomena and interpolate with the unitary minimal models in d = 2, while for odd m the theories are non-unitary and start at m = 3 with the Lee-Yang universality class. For all the even potentials and for the Lee-Yang universality class, we show how the assumption of conformal invariance is enough to compute the scaling dimensions of the local operators φ k and of some families of structure constants in either the coupling's or the -expansion. For all other odd potentials we express some scaling dimensions and structure constants in the coupling's expansion.

Functional perturbative RG and CFT data in the $$\epsilon $$ ϵ -expansion

“…We note that the OPE coefficientC 1 14 is in perfect agreement with that found in [10] using CFT methods, while we did not find any result in the literature for the other two coefficients to compare to. An explanation of why this agreement is expected will be given in Sect.…”

“…1 In our approach we can compute only the subset of OPE coefficients which are "massless" at the upper critical dimension (the others being projected away by the dimensionally regularized scheme) and therefore less sensitive to -corrections induced by a change in the RG scheme. We show that our next-to-leadingorder (NLO) computation gives these OPE coefficients at order O( ) and reproduces the structure constants previously obtained in a CFT framework [10][11][12]. This fact, even if plausible, is in general not obvious because of the possible scheme dependence, and we find it to be supported by the functional approach, which indeed constrains to some extent the possible choices of coupling redefinitions that otherwise would be completely arbitrary.…”

“…3.1. It is also important to stress that we ensured the agreement by choosing the same normalization of [10], that is, by fixing the coefficients of the two point functions.…”

“…All other "massive" C k i j strongly depend on the computational scheme and vanish in dimensional regularization. We observe that some OPE coefficients, including "massive" ones which would thus require a separate investigation, can be or have already been obtained for several universality classes in the -expansion in a CFT framework, with either the CFT/Schwinger-Dyson Bootstrap [10,[13][14][15][16][17], the perturbative conformal block techniques [11,12] or Mellin space methods [18,19]. Also large spin expansion techniques could be useful [20,21].…”

Scaling dimensions in QED3 from the ϵ-expansion

“…[63] proposed an approach based on multiplet recombination, further applied and developed in refs. [64][65][66][67][68][69][70][71][72][73][74][75][76][77][78][79]. Another approach is the analytic bootstrap, either together with the largespin expansion [80], or in its Mellin-space version [81][82][83][84].…”