Full node-resolved impurity vertex for inter-node scattering

Determine the full valley-space impurity-ladder eigenvalues for inter-node scattering in a multi-Weyl superconductor, including the effects of the anisotropic Fermi-surface measure and the resulting protection factors for the conventional and monopole pairing channels.

Background

The analysis in the paper is restricted to valley-diagonal, intra-node, momentum-independent scalar disorder. In that projected kernel, the monopole-channel protection factor is obtained exactly as ηm=1/(J+2)\eta_m=1/(J+2), while the conventional channel is treated under an assumed Anderson-protection condition, ηs=1\eta_s=1.

For generic short-range disorder, inter-node processes are not negligible. The anomalous impurity kernel then contains matrix elements from both paired nodes and need not coincide with the normal-state kernel. A complete node-resolved calculation could therefore change both protection factors and determine whether the disorder-driven crossing persists under generic intervalley scattering.

References

We do not compute the inter-node channel eigenvalues: that requires diagonalizing the full valley-space impurity ladder with the anisotropic measure, including the possibility that inter-node scattering also pair-breaks the conventional channel (\eta_s<1). The reason is structural: for zero-center-of-mass inter-node pairing the anomalous ladder contains one impurity matrix element from each paired node, K{\rm an}(k,k')\sim\overline{U_{+}(k,k')U_{-}{*}(-k,-k')}, which is not generically equal to \overline{|U_{+}(k,k')|2}. Replacing it by the normal kernel |O|2 requires the sewing identity of App.~\ref{app:band_sewing_gap} and is not a consequence of the disorder being scalar; a full node-resolved treatment could therefore shift both \eta_s away from unity and \eta_m away from 1/(J+2). This is the central technical problem left open here, and we leave it for future work.

— Disorder-tuned crossing of monopole and conventional pairing instabilities in multi-Weyl semimetals  (2608.24587 - Muñoz et al., 25 Aug 2026) in Section 5, “Beyond intra-node scalar disorder: scope” (Sec. \ref{sec:dissym}); also identified as the principal open problem in Section 9, “Status of the results” (Sec. \ref{sec:status})