Geometry of polymer chains near the theta-point and dimensional regularization

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Abstract
We consider the ab initio continuum theory of polymer chains in a theta solvent, interacting via two- and three-body forces, with a short range cutoff s0 and in continuous space dimension d(2<d≤3). We calculate the full expression in terms of d and s0 of the squared end-to-end distance R2 of a single chain, and of its squared radius of gyration R2G, to first order in the interactions. We show explicitly how the cutoff is removed in any d, up to d=3, to yield an effective three-parameter theory. We show how this procedure is exactly equivalent to dimensional regularization. This completely disproves recent claims that for polymers in a θ solvent, cut-off theories and dimensional regularization would not be equivalent near three dimensions, or in three dimensions. We locate numerical errors in calculations by others, which explain the discrepancies. We also give to all orders the general relation existing between cut-off and dimensional regularizations for the three-parameter model, for 2<d<3.

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