Derive the transmission-phase matching prefactor

Derive the overall transmission-phase prefactor multiplying the analytically known static factor \(\zeta(3)-9/8\) for low-frequency axial gravitational-wave scattering by a Schwarzschild black hole in dynamical Chern–Simons gravity, using a first-principles dispersive near-zone/far-zone matching calculation.

Background

The paper derives the static dispersive integral governing the transmission-phase asymmetry and obtains the exact factor D2=ζ(3)−9/8D_2=\zeta(3)-9/8. However, the coefficient relating this static factor to the physical linear-in-frequency transmission-phase asymmetry is only identified numerically as approximately 2π22\pi^2. A first-principles matching calculation between the near and far zones is therefore still required to establish this prefactor analytically.

References

Only the overall prefactor $2\pi2$ remains a numerical identification: $c_1/[\zeta(3)-9/8]\to 2\pi2$ to four digits in the two-channel integration, while a first-principles derivation of $2\pi2$---which requires the dispersive near/far-zone matching---is left to future work.

— Analytic parity-violating greybody factors of a dynamical Chern-Simons black hole  (2610.10280 - Rout, 7 Oct 2026) in Section 5.3, “Phase sector: a rotation null result”; Appendix A, Section “The transmission-phase integral”

A closed-form family $d_\ell,\kappa_\ell$ for general $\ell$---and the associated spectral sums entering the total absorbed flux and scalar luminosity---would require the general-$\ell$ axial static solutions and is left to future work.

— Analytic parity-violating greybody factors of a dynamical Chern-Simons black hole  (2610.10280 - Rout, 7 Oct 2026) in Section 6.4, “Multipole dependence”; Appendix B, Section “General-\(\ell\) structure”