Perturbation Theory for Path Integrals of Stiff Polymers

TL;DR

This paper develops a systematic perturbation theory for the path integrals of stiff polymers modeled as wormlike chains, addressing boundary condition issues and enabling calculation of distribution functions and correlations.

Contribution

It introduces a modified theoretical framework with regularization procedures for boundary conditions, allowing large-stiffness expansions for physical quantities of stiff polymers.

Findings

Derived normalized end-to-end distribution functions in arbitrary dimensions

Calculated moments and correlation functions of the polymer model

Established a method for handling divergent Feynman integrals with boundary conditions

Abstract

The wormlike chain model of stiff polymers is a nonlinear $\sigma$-model in one spacetime dimension in which the ends are fluctuating freely. This causes important differences with respect to the presently available theory which exists only for periodic and Dirichlet boundary conditions. We modify this theory appropriately and show how to perform a systematic large-stiffness expansions for all physically interesting quantities in powers of $L/\xi$, where $L$ is the length and $\xi$ the persistence length of the polymer. This requires special procedures for regularizing highly divergent Feynman integrals which we have developed in previous work. We show that by adding to the unperturbed action a correction term ${\cal A}^{\rm corr}$, we can calculate all Feynman diagrams with Green functions satisfying Neumann boundary conditions. Our expansions yield, order by order, properly normalized end-to-end distribution function in arbitrary dimensions $d$, its even and odd moments, and the two-point correlation function.