Wilson-Polchinski exact renormalization group equation for O(N) systems: Leading and next-to-leading orders in the derivative expansion

摘要

With a view to study the convergence properties of the derivative expansionof the exact renormalization group (RG) equation, I explicitly study theleading and next-to-leading orders of this expansion applied to theWilson-Polchinski equation in the case of the $N$-vector model with thesymmetry $\mathrm{O}(N) $. As a test, the critical exponents $% \eta $ and $\nu$ as well as the subcritical exponent $\omega $ (and higher ones) are estimatedin three dimensions for values of $N$ ranging from 1 to 20. I compare theresults with the corresponding estimates obtained in preceding studies ortreatments of other $\mathrm{O}(N) $ exact RG equations at second order. Thepossibility of varying $N$ allows to size up the derivative expansion method.The values obtained from the resummation of high orders of perturbative fieldtheory are used as standards to illustrate the eventual convergence in eachcase. A peculiar attention is drawn on the preservation (or not) of thereparametrisation invariance.