Mixed-Order Parity-Violating Corrections to Rotating Black Holes in Combined Scalar Gauss-Bonnet and Dynamical Chern-Simons Gravity

Abstract

We construct the leading mixed-order $\mathcal{O}(ζ_{\rm GB}ζ_{\rm CS})$ deformation of rotating black holes in a combined scalar Gauss-Bonnet and dynamical Chern-Simons effective theory, the sector in which the parity-even and parity-odd corrections act simultaneously. Using a pseudospectral solution of the coupled mixed scalar and metric equations, we compute horizon and equatorial-geodesic observables--the horizon angular velocity, surface gravity, area, and Wald entropy; the innermost stable circular orbit and photon-ring radii; and photon-ring eikonal estimates of the ringdown--at fixed mass and angular momentum over the spin range $0.1\le a\le0.9$, where the solution is numerically convergent and the perturbative expansion is controlled for sufficiently small dimensionless couplings. We verify the exact Kerr and Schwarzschild limits, reproduce available pure-sector benchmarks, and quantify spectral, residual, and extraction uncertainties. The mixed solution satisfies every component of the field equations, including the one that is not imposed in its construction, and its Wald entropy satisfies the first law to about one part in $10^{5}$. The mixed coefficients exhibit spin dependence qualitatively distinct from the individual sectors within this EFT truncation. All four horizon quantities, and several orbital and eikonal ones, change sign as functions of spin. The ringdown results are leading photon-ring eikonal estimates; a full coupled finite-$\ell$ perturbation calculation is left to future work.