General derivation of complete monotonicity in the azimuthal angle

Establish a general derivation of the alternating-sign derivative hierarchy for the vacuum cusp anomalous dimension and the boundary/interface infrared logarithmic coefficient with respect to the azimuthal angle on the reflection-symmetric locus, namely prove that $(-1)^{n+1}\partial_\psi^n\Gamma(\chi)\geq0$ and $(-1)^{n+1}\partial_\psi^n A_{\rm IR}\geq0$ for all $n\geq1$.

Background

The paper derives monotonicity and concavity in the azimuthal angle from reflection positivity of two-point functions of displacement operators. Numerical studies in free scalar and Maxwell theories, weakly coupled planar N=4\mathcal N=4 super-Yang–Mills theory, and the holographic D3–D5 defect CFT indicate a stronger property: the signs of all successive azimuthal derivatives alternate. This hierarchy is not obtained by the two-point reflection-positivity argument; higher-moment positivity instead produces nonlinear inequalities involving connected cumulants and does not directly imply sign constraints on individual higher derivatives of the logarithmic coefficients.

The unresolved problem is to find a general principle or proof establishing this complete-monotonicity hierarchy beyond the examples examined in the paper, for both the ordinary vacuum cusp anomalous dimension and the boundary/interface-sensitive infrared coefficient where applicable.

References

Beyond these inequalities, our explicit examples mentioned below exhibit evidence for a stronger infinite hierarchy in the azimuthal angle. On the reflection-symmetric locus, we find that the negatives of the logarithmic coefficients with nontrivial $\psi$-dependence obey an alternating-sign derivative hierarchy, \begin{equation} (-1){n+1}\partial_\psin \Gamma(\chi)\geq 0, \qquad (-1){n+1}\partial_\psin A_{\rm IR}\geq 0\qquad n\geq 1. \end{equation} We do not presently have a general derivation of this property and therefore regard it as a conjecture.

— Positivity constraints on the dynamics of cusped impurities in CFTs  (2608.28531 - Chandra, 28 Aug 2026) in Section 1, Introduction and summary of results

We assume that Rindler positivity extends to the defect-ray operators considered here; provides a justification for defects admitting suitable endpoint operators, while a general proof for arbitrary defect lines is not presently available.

— Positivity constraints on the dynamics of cusped impurities in CFTs  (2608.28531 - Chandra, 28 Aug 2026) in Section 2, subsection “Bounds on Lorentzian continuations of logarithmic coefficients,” footnote accompanying Eq. (\ref{eq:refpos})