Greybody Factors, Absorption Cross Sections and Hawking Radiation of Holonomy-Corrected Schwarzschild Black Holes

Abstract: We study greybody factors, absorption cross sections and Hawking energy-emission rates for minimally coupled massless scalar, electromagnetic and massless Dirac test fields on the loop-quantum-gravity-inspired holonomy-corrected Schwarzschild black hole. The geometry is controlled by a dimensionless holonomy parameter, and the radial wave equations are solved by direct numerical integration with first- and sixth-order WKB estimates as complementary checks. The scalar, electromagnetic and Dirac channels respond differently: the dominant scalar mode becomes more transparent, the electromagnetic threshold shifts slightly upward, and the dominant Dirac mode is only mildly modified. The scalar absorption cross section retains the universal low-frequency limit, the electromagnetic cross section changes mainly in the infrared, and the Dirac cross section develops a strongly suppressed low-frequency tail. Since the Hawking temperature falls monotonically, thermal suppression dominates the radiative output. Thus the holonomy correction enhances low-lying scalar transmission but suppresses Hawking radiation overall, with the electromagnetic sector most strongly quenched and the fermionic sector dominant once $α$ is appreciable.

• The paper uses direct numerical integration, validated by sixth-order WKB calculations, to show that holonomy corrections enhance scalar transparency, mildly suppress electromagnetic transmission, and leave Dirac transmission nearly unchanged.
• The study finds that at α = 0.9, scalar absorption peaks rise from approximately 26.4 to 45.4 in units of r_h², while electromagnetic and Dirac cross sections approach similar high-frequency behavior with distinct low-frequency changes.
• Although scalar transmission increases, the falling Hawking temperature suppresses total emission sharply at α = 0.9, reducing scalar, electromagnetic, and Dirac powers to 1.17 × 10⁻², 5.64 × 10⁻⁶, and 1.75 × 10⁻⁴ of their Schwarzschild values, respectively.

Overview

This paper computes greybody factors, absorption cross sections, and Hawking energy-emission rates for minimally coupled massless scalar, electromagnetic, and massless Dirac test fields propagating on the loop-quantum-gravity-inspired holonomy-corrected Schwarzschild geometry of Alonso-Bardaji, Brizuela, and Vera. The exterior metric is

$$ds^2 = -F(r)\,dt^2 + \frac{dr^2}{G(r)} + r^2 d\Omega_2^2,\quad F(r)=1-\frac{r_h}{r},\quad G(r)=\left(1-\frac{\alpha r_h}{r}\right)\left(1-\frac{r_h}{r}\right),$$

where $\alpha = r_0/r_h = \lambda^2/(1+\lambda^2)$ is the dimensionless holonomy parameter, with $r_0$ the minimal areal radius replacing the classical singularity and $r_h=2M$ the horizon radius held fixed in all comparisons. The limit $\alpha \to 0$ recovers Schwarzschild; the family terminates at $\alpha \to 1^-$. The Hawking temperature is $T_H = \sqrt{1-\alpha}/(4\pi r_h)$, so the correction cools the hole monotonically — a fact that ultimately dominates all radiative observables.

The key methodological choice is that all quantitative results derive from direct numerical integration of the Schrödinger-like master equations in the tortoise coordinate, with first- and sixth-order WKB estimates used only as barrier-top cross-checks. The flux-imbalance diagnostic $|\gamma_\ell + |\mathcal{R}_\ell|^2 - 1|$ remained below $7.3\times10^{-5}$ throughout.

Geometry and field dynamics

The effective model arises from polymerizing the angular extrinsic-curvature variable via $$b_\lambda(K_\varphi)=\sin(\lambda K_\varphi)/\lambda$$, combined with an anomaly-free recombination of the Hamiltonian and diffeomorphism constraints so that the constraint algebra remains first class. This closure is what licenses interpreting the solution as a spacetime (a black bounce connecting two asymptotically flat regions) rather than a gauge-fixed ansatz.

All three test fields reduce to one-dimensional scattering problems, $\alpha = r_0/r_h = \lambda^2/(1+\lambda^2)$0, but the holonomy parameter enters each sector differently:

• Scalar: the potential contains both the centrifugal term and a derivative term $\alpha = r_0/r_h = \lambda^2/(1+\lambda^2)$1; increasing $\alpha = r_0/r_h = \lambda^2/(1+\lambda^2)$2 lowers the dominant $\alpha = r_0/r_h = \lambda^2/(1+\lambda^2)$3-wave peak from $\alpha = r_0/r_h = \lambda^2/(1+\lambda^2)$4 to $\alpha = r_0/r_h = \lambda^2/(1+\lambda^2)$5 at $\alpha = r_0/r_h = \lambda^2/(1+\lambda^2)$6.
• Electromagnetic: the potential retains its Schwarzschild form in areal radius; $\alpha = r_0/r_h = \lambda^2/(1+\lambda^2)$7 enters only through the tortoise map, which stretches and broadens the barrier.
• Dirac: supersymmetric partner potentials built from $\alpha = r_0/r_h = \lambda^2/(1+\lambda^2)$8 deform only mildly, since the correction acts through the tortoise factor and the superpotential derivative.

Greybody factors and absorption

The three sectors respond distinctly to the deformation:

Sector	Half-transmission shift ($\alpha = r_0/r_h = \lambda^2/(1+\lambda^2)$9)	Behavior
Scalar $r_0$0	$r_0$1: $r_0$2	Enhanced transparency
Electromagnetic $r_0$3	$r_0$4: $r_0$5	Mildly suppressed
Dirac $r_0$6	$r_0$7: $r_0$8	Essentially invariant

The scalar absorption cross section preserves the universal low-frequency limit $r_0$9 [Das-Gibbons-Mathur], while its first oscillation peak grows from $r_h=2M$0 at $r_h=2M$1 (Schwarzschild) to $r_h=2M$2 at $r_h=2M$3 for $r_h=2M$4. The electromagnetic cross section changes mainly in the infrared before converging to a common geometric-optics envelope peaking near $r_h=2M$5. The Dirac cross section develops a strongly suppressed low-frequency tail — its first plotted point drops from $r_h=2M$6 to $r_h=2M$7 across the same range — while its broad maximum grows only mildly ($r_h=2M$8) at nearly fixed frequency $r_h=2M$9.

The WKB validation is instructive: at benchmark points near half-transmission, first-order WKB errors range from $\alpha \to 0$0 to $\alpha \to 0$1, whereas sixth-order continued-WKB errors fall to $\alpha \to 0$2–$\alpha \to 0$3. The authors note that Padé resummation cannot be applied to greybody factors, so higher WKB order does not guarantee monotonic improvement — reinforcing their reliance on direct integration.

Hawking radiation: thermodynamics wins

Despite enhanced scalar transmission, integrated emission powers collapse because $\alpha \to 0$4:

$\alpha \to 0$5	$\alpha \to 0$6	$\alpha \to 0$7	$\alpha \to 0$8	$\alpha \to 0$9	$\alpha \to 1^-$0
0.0	0.0796	$\alpha \to 1^-$1	$\alpha \to 1^-$2	$\alpha \to 1^-$3	$\alpha \to 1^-$4
0.6	0.0503	$\alpha \to 1^-$5	$\alpha \to 1^-$6	$\alpha \to 1^-$7	$\alpha \to 1^-$8
0.9	0.0252	$\alpha \to 1^-$9	$T_H = \sqrt{1-\alpha}/(4\pi r_h)$0	$T_H = \sqrt{1-\alpha}/(4\pi r_h)$1	$T_H = \sqrt{1-\alpha}/(4\pi r_h)$2

Relative suppressions at $T_H = \sqrt{1-\alpha}/(4\pi r_h)$3 are $T_H = \sqrt{1-\alpha}/(4\pi r_h)$4 (scalar), $T_H = \sqrt{1-\alpha}/(4\pi r_h)$5 (electromagnetic), and $T_H = \sqrt{1-\alpha}/(4\pi r_h)$6 (Dirac). The electromagnetic channel is quenched most aggressively because transmission and temperature act in the same direction there; the Page-style aggregate $T_H = \sqrt{1-\alpha}/(4\pi r_h)$7 becomes fermion-dominated once $T_H = \sqrt{1-\alpha}/(4\pi r_h)$8 is appreciable. At $T_H = \sqrt{1-\alpha}/(4\pi r_h)$9, converting to mass units via $|\gamma_\ell + |\mathcal{R}_\ell|^2 - 1|$0 reproduces Page's Schwarzschild benchmarks, providing a normalization check on the entire pipeline.

Limitations and open questions

The analysis is deliberately restricted to test fields on a fixed background, so backreaction of the radiation on the geometry is neglected — justified semiclassically only when individual quanta carry negligible energy relative to $|\gamma_\ell + |\mathcal{R}_\ell|^2 - 1|$1. The multipole sums are truncated ($|\gamma_\ell + |\mathcal{R}_\ell|^2 - 1|$2–$|\gamma_\ell + |\mathcal{R}_\ell|^2 - 1|$3), adequate for the thermal peaks but not for high-frequency geometric-optics precision. The Page-style aggregate omits gravitons and is explicitly not a complete particle inventory. Massive fields, nonminimal couplings, sparsity diagnostics of the evaporation cascade, and robustness under localized static deformations of the exterior remain unaddressed, as does a combined transmission–ringing treatment linking these greybody factors to the known quasinormal spectra of this geometry.

Conclusion

The paper establishes that holonomy corrections split scalar, electromagnetic, and fermionic transmission already at the level of the effective one-dimensional scattering problem — enhancing the scalar $|\gamma_\ell + |\mathcal{R}_\ell|^2 - 1|$4-wave, mildly suppressing the electromagnetic channel, and leaving the dominant Dirac threshold pinned — while the monotonically falling Hawking temperature suppresses all emission channels overall, most dramatically the electromagnetic one. The central result is quantitative: even at maximal deformation, the black hole is locally more transparent to low-lying scalar modes yet globally dimmer by up to five orders of magnitude in its radiative output.