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 ${\Lambda }_p{\Lambda }_L=m$, where ${\Lambda }_p$ and ${\Lambda }_L$ 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 (${\Lambda }_p=0\mathrm{}$) in $6-\epsilon$ dimensions with $\nu =\frac{1}{2}+0.092\mathrm{}\epsilon$ and $\eta =-0.073\mathrm{}\epsilon$ is located. An old result of Zimm and Stockmayer for the radius of gyration of a Gaussion branched polymer is rederived.