The renormalization group flow equation obtained by means of a proper time regulator is used to calculate the two loop $\beta$ function and anomalous dimension $\eta$ of the field for the $O\left(N\right)\mathrm{}$ symmetric scalar theory. The standard perturbative analysis of the flow equation does not yield the correct results for both $\beta$ and $\eta .$ We also show that it is still possible to extract the correct $\beta$ and $\eta$ from the flow equation in a particular limit of the infrared scale. A modification of the derivation of the exact renormalization group flow, which involves a more general class of regulators, to recover the proper time renormalization group flow is analyzed.

• Received 25 February 2002

DOI:https://doi.org/10.1103/PhysRevD.66.105020