Conformal window in QCD for large numbers of colours and flavours

Abstract

We conjecture that the phase transitions in QCD at large number of colours N\gg 1 is triggered by the drastic change in the instanton density. As a result of it, all physical observables also experience some sharp modification in the \theta behaviour. This conjecture is motivated by the holographic model of QCD where confinement -deconfinement phase transition indeed happens precisely at temperature T=T_c where $\theta$ dependence of the vacuum energy experiences a sudden change in behaviour: from N^2\cos(\theta/N) at T<T_c to \cos\theta\exp(-N) at T>T_c. This conjecture is also supported by recent lattice studies. We employ this conjecture to study a possible phase transition as a function of \kappa\equiv N_f/N from confinement to conformal phase in the Veneziano limit N_f\sim N when number of flavours and colours are large, but the ratio \kappa is finite. Technically, we consider an operator which gets its expectation value solely from nonperturbative instaton effects. When \kappa exceeds some critical value \kappa> \kappa_c the integral over instanton size is dominated by small-size instatons, making the instanton computations reliable with expected \exp(-N) behaviour. However, when \kappa<\kappa_c, the integral over instaton size is dominated by large-size instantons, and the instanton expansion breaks down. This regime with \kappa<\kappa_c corresponds to the confinement phase. We also compute the variation of the critical \kappa_c(T, \mu) when the temperature and chemical potential T, \mu \ll \Lambda_{QCD} slightly vary. We also discuss the scaling (x_i-x_j)^{-\gamma_{\rm det}} in the conformal phase.