Field Theory for the Statistics of Branched Polymers, Gelation, and Vulcanization

Abstract

This paper presents a field theory for the statistics of branched polymers and for the gelation transition. The gelation transition with ${\ensuremath{\Lambda}}_{p}{\ensuremath{\Lambda}}_{L}=m$, where ${\ensuremath{\Lambda}}_{p}$ and ${\ensuremath{\Lambda}}_{L}$ are the fugacities for polymer number and loop number, is shown to be in the same universality class as the $m$-state Potts model. A new fixed point governing the statistics of branched polymers in the dilute limit (${\ensuremath{\Lambda}}_{p}=0$) in $6\ensuremath{-}\ensuremath{\epsilon}$ dimensions with $\ensuremath{\nu}=\frac{1}{2}+0.092\ensuremath{\epsilon}$ and $\ensuremath{\eta}=\ensuremath{-}0.073\ensuremath{\epsilon}$ is located. An old result of Zimm and Stockmayer for the radius of gyration of a Gaussion branched polymer is rederived.