Hyperscaling relations in mass-deformed conformal gauge theories

Abstract

We present a number of analytical results which should guide the interpretation of lattice data in theories with an infrared fixed point (IRFP) deformed by a mass term $\ensuremath{\delta}\mathcal{L}=\ensuremath{-}m\overline{q}q$. From renormalization group (RG) arguments we obtain the leading scaling exponent, $F\ensuremath{\sim}{m}^{{\ensuremath{\eta}}_{F}}$, for all decay constants of the lowest lying states other than the ones affected by the chiral anomaly and the tensor ones. These scaling relations provide a clear cut way to distinguish a theory with an IRFP from a confining theory with heavy fermions. Moreover, we present a derivation relating the scaling of $⟨\overline{q}q⟩\ensuremath{\sim}{m}^{{\ensuremath{\eta}}_{\overline{q}q}}$ to the scaling of the density of eigenvalues of the massless Dirac operator $\ensuremath{\rho}(\ensuremath{\lambda})\ensuremath{\sim}{\ensuremath{\lambda}}^{{\ensuremath{\eta}}_{\overline{q}q}}$. RG arguments yield ${\ensuremath{\eta}}_{\overline{q}q}=(3\ensuremath{-}{\ensuremath{\gamma}}_{*})/(1+{\ensuremath{\gamma}}_{*})$ as a function of the mass anomalous dimension ${\ensuremath{\gamma}}_{*}$ at the IRFP. The arguments can be generalized to other condensates such as $⟨{G}^{2}⟩\ensuremath{\sim}{m}^{4/(1+{\ensuremath{\gamma}}_{*})}$. We describe a heuristic derivation of the result on the condensates, which provides interesting connections between different approaches. Our results are compared with existing data from numerical studies of SU(2) with two adjoint Dirac fermions.