Global proof of monotonicity for the Abrikosov–Gor'kov scaling function

Prove that \(h'(x)=\psi'(x)+(x-\tfrac12)\psi''(x)>0\) for every \(x>1\), thereby establishing analytically the global monotonicity of the Abrikosov–Gor'kov scaling function and rigorously validating the necessary-and-sufficient crossing criterion.

Background

The paper derives the crossing criterion for the two pairing channels using monotonicity of the Abrikosov–Gor'kov scaling function. Positivity of the relevant derivative is established analytically for 12<x≤1\tfrac12<x\le1 and asymptotically for large xx, while the intermediate range is checked numerically.

A closed-form proof covering all x>1x>1 is not supplied. Thus, although the criterion is presented as necessary and sufficient conditional on the monotonicity verification, its fully analytic global justification remains unresolved.

References

For large x the Stirling expansion gives h'(x)=1/(6x{3})+O(x{-4})>0. In the intermediate range we verify Eq.~eq:hprime numerically (h'=0.443,\, 3.88\times10{-2},\, 1.77\times10{-3},\,1.37\times10{-6} at x=1,2,5,50; positive on a 2\times10{5}-point scan of x\in[\tfrac12+10{-4},500], with h'\sim1/(6x3)>0 controlling x>500), but we have not found a closed-form proof of positivity for all x>1. Accordingly we state Eq.~eq:crosscrit as necessary and sufficient given the monotonicity of the AG scaling function established analytically at both ends of the interval and verified numerically in between---not as a fully proved ``if and only if.''

— Disorder-tuned crossing of monopole and conventional pairing instabilities in multi-Weyl semimetals  (2608.24587 - Muñoz et al., 25 Aug 2026) in Appendix, “Status of the criterion and its derivation” (App. \ref{app:criterion}), following Eq. \(\ref{eq:hprime}\)