Partial wave analysis of the Dirac fermions scattered from Reissner - Nordstr\"om charged black holes

The asymptotic form of Dirac spinors in the field of the Reissner-Nordstrom black hole are derived for the scattering states (with $E>mc^2$) obtaining the phase shifts of the partial wave analysis of Dirac fermions scattered from charged black holes. The elastic scattering and absorption are studied giving analytic formulas for the partial amplitudes and cross sections. A graphical study is performed for analysing the differential cross section (forward/backward scattering) and the polarization degree as functions of scattering angle.


Introduction
The problem of the quantum fermions scattered from Schwarzschild black holes was studied either in particular cases [1,2] or by using combined analytical and numerical methods [3]- [6]. Recently we performed an analytic study of this process proposing a version of partial wave analysis that allowed us to write down closed formulas for the scattering amplitudes and cross sections [7]. The analytic approach improves our understanding of the quantum mechanisms that governs the fermion scattering by black holes.
In the present letter we would like to extend this analytic study to the problem of the Dirac fermions scattered from Reissner-Nordström charged black holes since it seems that this problem was neglected so far. The studies performed in the existing literature have been concentrating manly on scalar field [11,8,9,10] and electromagnetic scattering [12,13,14,15] on charged black holes . For these reasons we concentrate in studding the problem of fermion scattering on a Reissner-Nordström charged black hole by using analytically and graphically methods. This phenomenon could be interesting since along side with the gravitational interaction we can study the effect of the interaction between the charges of the black hole and the fermion charge upon the scattering process. In addition the results related to the fermion absorbtion by the Schwarzschild black hole will be modified depending on the electric atraction/repulsion between the black hole charges and the fermion charge.
We deduce the asymptotic form of the Dirac spinors in the Reissner-Nordström geometry deriving the phase shifts and the partial amplitudes of elastic scattering as well as the absorption cross section. We present the principal analytic results without extended examples or comments that exceed the space of this short paper. We use the methods and notations of Ref. [7] and Planck's natural units with G = c = = 1.
Let us consider the Dirac equation, iγαDαψ − mψ = 0 of a free spinor field ψ of mass m, written with our previous notations [7] in the frame {x; e} defined by the Cartesian gauge [16,17], sin θ sin φ dr + r cos θ sin φ dθ +r sin θ cos φ dφ , in the central gravitational and Coulomb field of a charged black hole of mass M and charge Q > 0 with the Reissner-Nordström line element defined on the radial domain D r = (r + , ∞) where and r ± = M ± M 2 − Q 2 provided Q < M. The corresponding Coulomb potential gives the potential energy Q r → V (r) = eQ r , of the fermion carrying the elementary electric charge e = ± √ α 1 . In what follows we study the scattering solutions of the Dirac equation in the asymptotic domain where r ≫ r + .
We have shown [16] that in the gauge we consider here the spherical variables of the Dirac equation can be separated just as in the case of the central problems in Minkowski spacetime [18]. Consequently, the particle-like energy eigenspinors of energy E, are expressed in terms of radial wave functions, f ± E,κ , and usual four-component angular spinors Φ ± m j ,κ [18]. These spinors are orthogonal to each other being labeled by the angular 1 In this system α ≃ 1 137 is the fine structure constant while the electron mass is m e = √ α G ≃ 4.178 10 −23 .
quantum numbers m j and which encapsulates the information about the quantum numbers l and j = l ± 1 2 as defined in Refs. [18,19] (while in Ref. [6] κ is of opposite sign). We note that the antiparticle-like energy eigenspinors can be obtained directly using the charge conjugation as in the flat case [20]. Thus the problem of the angular motion is completely solved remaining with a pair of radial wave functions, f ± , (denoted from now on without indices) which satisfy two radial equations that can be written in compact form as the eigenvalue problem H r F = EF of the radial Hamiltonian [16], in the space of two-component vectors, The resulting radial problem cannot be solved analytically as it stays such that we are forced to resort to the same method of approximation as in Ref. [7] by using a convenient Novikov's dimensionless coordinate [21,22]. In the present case we chose the Novikov coordinate corresponding to the event horizon of radius r + defined as Then, by changing the variable, multiplying with x −1 (1 + x 2 ) and introducing the notations we rewrite the exact radial problem in the form EF = 0 where the new matrix operator (14) is suitable for further approximations.
For very large values of x, we can use the Taylor expansion of these equations with respect to 1 x neglecting the terms of the order O(1/x 2 ). We obtain thus the asymptotic radial problem [23,7] which can be rewritten as E ′ F = 0 where the new matrix operator takes the form after reversing between themselves its lines and introducing the notations As in the Dirac-Coulomb case [19] it is useful to put in diagonal form the terms proportional to x by using the matrix for transforming the radial doublet as Then we obtain the new system of radial equations where ν = ε 2 − µ 2 . These equations can be solved analytically for any values of ε but here we restrict ourselves to the scattering modes corresponding to the continuous spectrum ε ∈ [µ, ∞). These solutions can be expressed in terms of Whittaker functions as [23,7] f where we denote The integration constants must satisfy [23] C We observe that the functions M ρ ± ,s ( [24]. These solutions will help us to find the scattering amplitudes of the Dirac particles by charged black holes, after fixing the integration constants.

Partial wave analysis
The scattering of Dirac fermions on a charged black hole is described by the energy eigenspinor U whose asymptotic form, for r → ∞ (where the gravitational and Coulomb fields vanish) is given by the plane wave spinor of momentum p and the free spherical spinors of the flat case behaving as since in the asymptotic zone the fermion energy is that of special relativity, E = m 2 + p 2 .
Here we fix the geometry such that p = p e 3 while the direction of the scattered fermion is given by the scattering angles θ and φ which are just the spheric angles of the unit vector n. Then the scattering amplitude depend on two scalar amplitudes, f (θ) and g(θ), that can be studied by using the partial wave analysis.

Phase shifts
The partial wave analysis exploits the asymptotic form of the exact analytic solutions which satisfy suitable boundary conditions that in our case might be fixed at the (exterior) event horizon (where x = 0). Unfortunately, here we have only the asymptotic solutions (20) and (21) whose integration constants cannot be related to those of the solutions near event horizon without resorting to numerical methods [1]- [6]. Therefore, we must chose suitable asymptotic conditions for determining the integration constants. The arguments of Ref. [7] (appendix C) show that in our approach it is necessary to adopt the general asymptotic condition C + 2 = C − 2 = 0 in order to have elastic collisions with a correct Newtonian limit for large angular momenta.
The asymptotic form of the doublet F = TF can be obtained as in Ref. [7] observing that now we must replace νx 2 = p(r − r + ). Thus we obtain the definitive asymptotic form of the radial functions of the scattered fermions by charged black holes, whose point-independent phase shifts δ κ give the quantities Notice that the values of κ and l are related as in Eq. (9), i. e. l = |κ| − 1 2 (1 − sign κ). The remaining point-dependent phase, which does not depend on angular quantum numbers, may be ignored as in the Dirac-Coulomb case [19,6].
We arrived thus at the final result (28) depending on the parameters introduced above that can be expressed in terms of physical quantities m, M, e, , Q... etc.. In addition, assuming that the inequality |β| ≥ |ζ| holds even for p = 0, it is convenient to introduce the new real parameter k obeying that allows us to write simply s = √ κ 2 − k 2 . Then by using Eq. (25b) we obtain our principal new result that holds for massive fermions: Obviously, in the case of the massless neutral fermions (with m = e = 0) we remain with the unique parameter q = r + p since then λ = 0 and k = q. Hereby we see that the parameter s has a special position since this can take either real values or pure imaginary ones regardless the fermion mass.

Elastic scattering
For the real values of s the scattering is elastic since in this case the identity (30) guarantees that the phase shifts of Eq. (28) are real numbers such that |S κ | = 1. Obviously, this happens only when κ (at given p) satisfies the condition wherek = floor(k) is the greater integer less than k. Then, the scalar amplitudes of Eq. (26), which depend on the following partial amplitudes [19,6], give rise to the elastic scattering intensity or differential cross section, and the polarization degree, This last quantity is interesting for the scattering of massive fermions representing the induced polarization for an unpolarized initial beam.

Absorption
The absorption is present in the partial waves for which we have Here we meet a branch point in s = 0 and two solutions s = ±i|s| = ±i √ k 2 − κ 2 , among them we must chose s = −i|s| since only in this manner we select the physical case of |S κ | < 1. More specific, by substituting s = −i|s| in Eq. (28) we obtain the simple closed form |S κ | = |S −κ | = e −2ℑδκ = e −π|s| sinh π(q − |s|) sinh π(q + |s|) showing that 0 < |S κ | < 1 since |s| < q for any (p, κ) obeying the condition (39). Moreover, we can verify that in the limit of the large momentum (or energy) the absorption tends to become maximal since regardless of the fermion mass.
Under such circumstances we can calculate the absorption cross section that reads [6] since for s = −i|s| we have |S −κ | = |S κ | as in Eq. (40). This cross section can be calculated at any time as a finite sum of the partial cross sections whose definitive closed form, is derived according to Eqs. (40) and (42) while the condition (39) introduces the Heaviside step function θ(k − l). Hereby we understand that the absorption arises in the partial wave l for the values of p (or E) satisfying the condition k > l. This means that for any fixed value of l there is a threshold, E l , defined as the positive solution of the equation A similar condition with 44 was obtained in ref. [25] for absorbtion on a dilaton black hole.
This indicates that the fermions with |κ| = l can be absorbed by black hole only if E ≥ E l .
The existence of these thresholds is important since these keep under control the effect of the singularities in p = 0.
Finally, we observe that in the high-energy limit all these absorption cross sections tend sincek(k + 1) ∼ k 2 ∼ r 2 + p 2 .

Graphical discussion of the results
Let us now discuss some physical consequences of our results encapsulated by the formulas presented in the previous sections. For a better understanding of the analytical results we perform a graphical analysis of the differential cross section in terms of scattering angle θ.
All the plots are obtained using the methods described in Ref. [7] where we used a technique proposed some time ago by Yennie et al. [26]. In what follows we focus our analysis on scattering from small or micro black holes (with M ∼ 10 −15 kg) since in this case the wave length of the fermion (λ = 2πh/p) and the Schwarszchild radius (r S = 2M) have the same order of magnitude such that we can observe the presence of glory and orbiting scattering.
Comparing the scattering cross section of a Reissner-Nordström black hole with that of a Schwarzschild black hole (see fig. 1) one can observe several things. First is that in the case of scattering by the Reissner-Nordström black hole the differential cross section will increase if the charge of the black hole and the incoming fermions charge have both the same sign (the point and dash-dotted lines in fig. 1). Thus we can say that the electric repulsion increases the scattering cross section as expected, according to (44), comparatively with the scattering on Schwarzschild black hole (the solid line in fig. 1). In the opposite case, when the black hole charge and the incoming fermions charge have opposite signs then the electric attraction between the two charges will make more fermions to be absorbed by the black hole lowering thus the scattering cross section comparatively with the scattering on Schwarzschild black hole (as can be seen from the doted line in fig.1). Figure 1 also revel us the fact that the width of the glory peak becomes more pronounced when the sign of the total black hole charge is the same with the charge of the fermion, respectively the glory peak's width decreases if the black hole and the fermions have opposite signs. We can conclude that when the charge of the black hole and fermions charge have the same sign, the glory scattering will be a phenomenon which becomes important comparatively with the Schwarzschild case.
In figs. 2-3 we present the differential cross section in terms of scattering angle for the Reissner-Nordström black hole. The effects of electric interaction (between the black hole and fermion charges) on the glory and orbiting scattering can be also observed. We have found that if the black hole has opposite charge than the incoming fermion then, as we increase the charge on the black hole, the glory (i.e. backward scattering at angles close to π) starts to decrees up to a point when it will disappear completely (see fig.2). The same is true for the orbiting scattering (i.e. scattering for θ < π) for which we observe a decrees in oscillations as the black hole charge increases. This should not come as a surprise since   can be seen as resulting from the oscillatory behavior of glory/orbiting scattering as well as from the forward/backward one. The dependence of the absorbtion cross section in terms of energy (E/m), is given in Fig.(5), where the plots were obtained for l = 1, 2, 3, 4. We observe that the modes with small angular momenta have the most important contribution to the absorbtion, because as we increase l the maxima observed in Fig.(5) becomes smaller.
Also we observe that the maxima are shifted to the right as we increase the value of l. Our graphical result for the absorbtion cross section is similar to those obtained in [25], where the absorbtion of scalar particles on dilaton black hole was studied.