Operator Product Expansion on a Fractal: The Short Chain Expansion for Polymer Networks

TL;DR

This paper proves a short chain expansion in polymer field theory, demonstrating factorization of partition functions for networks with long and short chains, and applies it to fractal polymers and multifractal measures.

Contribution

It introduces a rigorous short chain expansion for polymer networks, extending field theory methods to fractal polymers and multifractal measures.

Findings

Factorization of partition sum for polymer networks with long and short chains.

Explanation of the scaling of the second virial coefficient in bimodal solutions.

Application of the method to correlations of multifractal measures on polymers.

Abstract

We prove to all orders of renormalized perturbative polymer field theory the existence of a short chain expansion applying to polymer solutions of long and short chains. For a general polymer network with long and short chains we show factorization of its partition sum by a short chain factor and a long chain factor in the short chain limit. This corresponds to an expansion for short distance along the fractal perimeter of the polymer chains connecting the vertices and is related to a large mass expansion of field theory. The scaling of the second virial coefficient for bimodal solutions is explained. Our method also applies to the correlations of the multifractal measure of harmonic diffusion onto an absorbing polymer. We give a result for expanding these correlations for short distance along the fractal carrier of the measure.