Critical amplitudes of Edwards' continuous polymer chain model

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Abstract
We investigate the ε-expansion of the critical amplitudes associated with Edwards' two-parameter model describing interacting continuous polymer chains in d dimensions ( d = 4 - ε). For any dimensionless quantity H (z,ε), which is a function of the Zimm et al.-Yamakawa parameter z and of ε only, and which scales for z → ∞, like H ( z, ε) = A ( ε) Zσ ( ε) , where σ (ε) is a critical exponent, we give a simple method for calculating the critical amplitude A as a function of ε. We show that in general A (ε )= [h(ε)]-σ(s), where h (ε) has an ε-series expansion, which we calculate to second order in ε. This holds true for the general case where σ (ε) starts like a non vanishing constant, i.e. σ(ε) = O (1) . If σ (ε) =O (εn), then the critical amplitude A (ε) reads A (ε) . = [h[n] (ε)]-σ(ε)/εn , where, again, h [n] (ε) has a well-defined ε-expansion, which we calculate to second order. We apply these results to calculate the critical amplitude of the general connected partition function Z N of N chains, N ≽ 1. The critical amplitudes of the end-to-end and of the gyration swellings of a single polymer chain are also calculated. We compare the asymptotic end-to-end swelling with former numerical results for continuous polymer chains. The asymptotic expansion of the entropy of a long continuous polymer chain is given