Revisiting (logarithmic) scaling relations using renormalization group

Abstract

We explicitly compute the critical exponents associated with logarithmic corrections (the so-called hatted exponents) starting from the renormalization group equations and the mean field behavior for a wide class of models at the upper critical behavior (for short and long range $\phi^n$-theories) and below it. This allows us to check the scaling relations among these critical exponents obtained by analysing the complex singularities (Lee-Yang and Fisher zeroes) of these models. Moreover, we have obtained an explicit method to compute the $\hat{\coppa}$ exponent [defined by $\xi\sim L (\log L)^{\hat{\coppa}}$] and, finally, we have found a new derivation of the scaling law associated with it.