Decoherence in weak localization. II. Bethe-Salpeter calculation of the cooperon

This is the second in a series of two papers (Papers I and II) on the problem of decoherence in weak localization. In Paper I, we discussed how the Pauli principle could be incorporated into an influence functional approach for calculating the cooperon propagator and the magnetoconductivity. In the present paper, we check and confirm the results so obtained by diagrammatically setting up a Bethe-Salpeter equation for the cooperon, which includes self-energy and vertex terms on an equal footing and is free from both infrared and ultraviolet divergences. We then approximately solve this Bethe-Salpeter equation by the ansatz C̃(t)=C̃⁰(t)e^−F(t), where the decay function F(t) determines the decoherence rate. We show that in order to obtain a divergence-free expression for the decay function F(t), it is sufficient to calculate C̃¹(t), the cooperon in the position-time representation to first order in the interaction. Paper II is independent of Paper I and can be read without detailed knowledge of the latter.