狄拉克粒子落入史瓦西-修正引力（MOG）黑洞的双分支探测器响应
Two-branch detector response for Dirac infall into a Schwarzschild--MOG black hole
AI总结：
该研究分析自旋-1/2探测器落入史瓦西-MOG黑洞的响应，通过双分支处理量子化标量场，分解细致平衡比并明确局域与全局散射贡献，得出特定条件下的出射比及修正项规律。
AI中文摘要：
我们研究一个局域自旋-1/2探测器落入史瓦西-修正引力（MOG）黑洞的过程。该探测器通过电荷守恒的两能级跃迁与中性无质量标量场相互作用，其平动波包满足带MOG荷的狄拉克方程。在外视界附近，分离后的径向系统简化为具有指数Θ_D = E_H/(2ℏκ_α)的平方反比方程。对于所有未来指向的穿越轨迹，规范不变的视界能量满足E_H = m_ψ U_H > 0。我们在全局归一化的Boulware散射基中量子化标量场，并保留在无穷远处出射的模式的两个径向通量分支。该处理未将局域出射近似与完整模式等同。有限径向门χ_p(x) = (x/L_χ)^p e^(-x/L_χ)给出闭式形式的激发和吸收概率密度，包含入射贡献和散射相位干涉项。在受控的近视界近似下，完整细致平衡比可分解为分支分辨的出射比和双分支因子。当近视界、绝热、高能隙和分支隔离条件全部满足时，仅出射比趋近于exp(-2πν/κ_α)。主导的开关修正由(ν/κ_α)/(S_± L_χ)控制，而非仅由(S_± L_χ)⁻¹控制。在弱MOG展开中，局域修正可分离为表面引力、轨迹 prefactor 和有限门项。完整响应还包含来自全局散射的贡献。该分离明确了哪些项源于局域视界几何，哪些依赖于探测器协议和标量在视界区域外的传播。
英文摘要：
We study a localized spin-$1/2$ detector falling into a Schwarzschild--MOG black hole. The detector interacts with a neutral massless scalar field through a charge-preserving two-level transition, while its translational wave packet obeys the MOG-charged Dirac equation. Near the outer horizon, the separated radial system reduces to an inverse-square equation with index $Θ_{\rm D}=E_{\rm H}/(2\hbarκ_α)$. The gauge-invariant horizon energy satisfies $E_{\rm H}=m_ψU_{\rm H}>0$ for every future-directed crossing trajectory. We quantize the scalar field in a globally normalized Boulware scattering basis and retain both radial-flux branches of a mode that is outgoing at infinity. This treatment does not identify a local outgoing ansatz with a complete mode. A finite radial gate $χ_p(x)=(x/L_χ)^p e^{-x/L_χ}$ gives closed-form excitation and absorption probability densities that include the ingoing contribution and scattering-phase interference. Within the controlled near-horizon approximation, the full detailed-balance ratio factorizes into a branch-resolved outgoing ratio and a two-branch factor. Only the outgoing ratio approaches $\exp(-2πν/κ_α)$ when the near-horizon, adiabatic, high-gap, and branch-isolation conditions all hold. The leading switching correction is controlled by $(ν/κ_α)/(S_\pm L_χ)$, not by $(S_\pm L_χ)^{-1}$ alone. In a weak-MOG expansion, the local correction separates into surface-gravity, trajectory-prefactor, and finite-gate terms. The full response also contains a contribution from global scattering. This separation shows which terms follow from the local horizon geometry and which depend on the detector protocol and scalar propagation outside the horizon region.
