The Angular Factor Between the Scalar and Vector Pairing Channels of a Single Contact Interaction

A single attractive contact interaction for a massless Dirac fermion pairs in both a scalar (0+) and a vector (1-) channel, and its Fierz decomposition assigns the two channels the same coefficient. It is therefore tempting to write the vector channel's gap equation by transplanting the scalar one, with the coupling relabelled and nothing else changed. This note computes the quantity the transplant assumes away: the ratio F of the two channels' pair susceptibilities at a common Fermi surface. Three values suggest themselves a priori — F = 1 (the transplant), F = 1/3 (the longitudinal angular average), and the value the computation gives. The vector pair wavefunction is everywhere transverse to the relative momentum, its total weight per pair is twice the scalar's, and that weight is distributed equally over the channel's three components. The susceptibility per component is therefore 4/3 against the scalar's 2, and F = 2/3 exactly, so the vector channel's effective coupling is (2/3) of the scalar channel's coupling. The scalar channel, computed by the same machinery as a control, returns 1 identically, and a Monte Carlo evaluation of the susceptibility tensor reproduces F = 0.666667. The factor is a property of the pair states, not of the interaction vertex, which is why no Fierz analysis of the vertex can contain it. Files: the note (PDF and LaTeX source), a Python script that reruns the overlap checks and the Monte Carlo evaluation reported in the note's verification table, and a README with the summary, verification steps, and citation.