Greybody factors of charged black holes with axion hair

1. Problem Statement

The paper investigates the greybody factors (transmission probabilities) of test particles (spin-0 scalars and spin-1 photons) propagating in the background of charged black holes (BHs) with axion hair.

• Context: Standard Reissner-Nordström (RN) black holes possess an electric-magnetic duality; their quasinormal modes and greybody factors depend only on the total charge ($q_T = \sqrt{q_E^2 + q_M^2}$) and mass, regardless of the ratio between electric ($q_E$) and magnetic ($q_M$) charges.
• The Issue: In theories where an axion field ($\phi$) couples to the electromagnetic field via a Chern-Simons term ($-\frac{\lambda}{4}\phi F_{\mu\nu}\tilde{F}^{\mu\nu}$), this duality is broken. The resulting "hairy" black hole solution depends explicitly on the mixing angle $\alpha$ (where $q_M = q_T \sin \alpha$) and the coupling constant $\lambda$.
• Goal: To compute the greybody factors for these hairy BHs without relying on approximations (like WKB) and to determine if the deviation from the standard RN solution is observable, potentially serving as a probe for axions and magnetic monopoles.

2. Methodology

A. Theoretical Framework

The authors utilize Einstein-Maxwell-Axion (EMA) theory with the action:
$$S = \int d^4x\sqrt{-g} \left[ \frac{M_{Pl}^2}{2}R - \frac{1}{2}\partial_\mu\phi\partial^\mu\phi - \frac{1}{2}m_\phi^2\phi^2 - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} - \frac{\lambda}{4}\phi F_{\mu\nu}\tilde{F}^{\mu\nu} \right]$$

• Assumptions: The axion is ultralight ($m_\phi \ll r_h^{-1}$), allowing the mass term to be neglected in the near-horizon regime. The background is static and spherically symmetric.
• Background Solution: They solve the coupled differential equations for the metric functions $f(r), h(r)$, the electric potential $A_0(r)$, and the axion profile $\phi(r)$ numerically using a shooting method, fixing the ADM mass and total charge.

B. Effective Potentials

The authors derive the effective potentials ($V(r)$) governing the motion of test particles on this background:

• Spin-0 (Scalar): The radial equation reduces to a Schrödinger-like form with potential:
  $$V_{s=0}(r) = \frac{l(l+1)f}{r^2} + \frac{\sqrt{fh}}{r}(\sqrt{fh})'$$
• Spin-1 (Photon): The potential is:
  $$V_{s=1}(r) = \frac{l(l+1)f}{r^2}$$
  Note: The spin-1 potential lacks the second term present in the spin-0 case, which is crucial for the $l=0$ mode behavior.

C. Calculation Techniques

The paper employs two distinct methods to evaluate the greybody factors:

1. Analytic Lower Bound (Sec. IV): Using the rigorous bound method (Visser et al.), they derive a lower bound for the transmission coefficient $T$:
   $$T \geq \text{sech}^2 \left[ \int_{-\infty}^{\infty} \Theta(x) dx \right]$$
   This provides a qualitative check and confirms that $T$ depends on the magnetic charge ratio.
2. Direct Numerical Integration (Sec. V & VI): To obtain precise values, they solve the wave equations directly.
   • Boundary Conditions: Purely ingoing waves at the horizon ($r \to r_h$) and a superposition of ingoing and outgoing waves at spatial infinity ($r \to \infty$).
   • Matching: They expand the wave functions and background fields in power series near the horizon and at infinity (up to 9th order). They integrate inward from infinity and outward from the horizon to an intermediate point ($r_m \approx 3M$), matching the wave function and its derivative to solve for the transmission ($T$) and reflection ($R$) coefficients.

3. Key Contributions

1. Breaking of Electric-Magnetic Duality: The study explicitly demonstrates that for hairy BHs, the greybody factor is not solely determined by the total charge $q_T$. It depends on the specific ratio of electric to magnetic charges ($\alpha$) and the axion coupling ($\lambda$).
2. High-Precision Numerical Method: The authors move beyond approximate WKB methods to provide exact numerical solutions for the transmission coefficients of spin-0 and spin-1 particles in this specific hairy BH background.
3. Multipole Dependence: They identify that the deviation from the RN solution is most pronounced for higher multipole moments ($l \geq 1$).
4. Symmetry Verification: The paper proves (Appendix B) that the transmission coefficient is symmetric (invariant under direction of incidence), validating the use of the "infalling" boundary condition to model Hawking radiation emission.

4. Key Results

• Effective Potentials:
  • For Spin-0 ($l=0$): The presence of axion hair and magnetic charge increases the peak of the potential barrier but lowers the tail. This leads to a complex frequency dependence: $T < T_{RN}$ at high frequencies and $T > T_{RN}$ at low frequencies.
  • For Spin-0 ($l \geq 1$) and Spin-1: The effective potential is suppressed (lowered) compared to the RN case when $\lambda \neq 0$ and $\alpha \neq 0$. Consequently, the transmission coefficient $T$ is higher than in the RN case ($T > T_{RN}$) across most frequencies.
• Transmission Coefficients ($T$):
  • The deviation $T - T_{RN}$ increases with the multipole index $l$.
  • For $l=1$ and $l=2$, the maximum deviation reaches the order of $10^{-2}$ (approx. 0.5% to 0.75%).
  • The deviation is most significant in the intermediate frequency regime where $T \approx 0.5$.
• Lower Bounds: The analytic lower bound analysis confirms that $T_{min}$ increases as the magnetic charge fraction ($q_M$) and coupling magnitude ($|\lambda|$) increase, consistent with the suppression of the potential barrier.

5. Significance and Implications

• Probing Axions and Monopoles: The results suggest that precise measurements of greybody factors (potentially via gravitational wave ringdown amplitudes or future electromagnetic observations of BH shadows) could distinguish between standard RN black holes and those with axion hair.
• Magnetic Monopoles: Since the hairy solution requires non-zero magnetic charge, detecting these deviations would provide indirect evidence for the existence of magnetic monopoles (which could have been absorbed by BHs in the early universe).
• Astrophysical Constraints: The study provides a theoretical framework to constrain the axion-photon coupling constant $\lambda$ and the mass of axions using BH observations, complementing existing bounds from CMB polarization and SN1987A.
• Methodological Robustness: The numerical technique developed here is applicable to other "hairy" black hole solutions, offering a tool for future investigations into non-trivial BH physics beyond the no-hair theorem.

In summary, the paper establishes that axion hair breaks the electric-magnetic degeneracy of charged black holes, leading to measurable differences in greybody factors that scale with the multipole moment, offering a new observational window into axion physics and magnetic monopoles.