# Scattering of Dirac fermions by spherical massive bodies

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## Abstract

The asymptotic form of Dirac spinors in the field of a Schwarzschild black hole is used for deriving analytically for the first time the phase shifts of the partial wave analysis of Dirac fermions scattered from massive spherical bodies, imagined as black holes surrounded by a surface producing total reflection. A simple model is analyzed by using graphical methods.

## 1 Introduction

In general relativity the study of the scattering of the scalar, electromagnetic or Dirac particles from Schwarzschild [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] black holes remains of actual interest. An effective analyse of the scattering of the massive Dirac fermions from Schwarzschild black holes was performed combining analytical and numerical methods [6, 7, 8, 9, 10]. Recently, starting with a set of asymptotic solutions [11], we propsed an analytic version of partial wave analysis that allowed us to write down closed formulas for the phase shifts giving the scattering amplitudes and cross sections of the fermions scattered from Schwarzschild [12], Reissner-Nordström [13, 14], Bardeen [15] and MOG [16] black holes. Other studies that investigate fermion scattering by different types of spherically symmetric black holes can be found for example in Refs. [17, 18, 19, 20, 21, 22, 23, 24, 25, 26].

In the present paper we would like to extend this analytic study to the problem of the Dirac fermions scattered from a massive spherical body whose exterior radius is larger that the Schwarzschild radius. We consider that this body is electrically neutral generating only a gravitational field with a Schwarzschild metric in its exterior while its surface is able to prevent the natural black hole absorption, reflecting entirely the incident fermion beam.

Recently, in Ref. [27] the authors have studied the elastic scattering of a scalar field in the curved spacetime of a compact object. Moreover, studies of gravitational waves by compact objets also received little attention [28, 29, 30].

Our method is based on the approximative solutions of the Dirac equation in the Schwarzschild geometry [11] that can be used at large distances from the black hole singularity [11, 12]. This is not an impediment when the body is extended enough since then we can use these solutions for fixing different boundary conditions on the exterior surface. In general, the boundary conditions are linear relations between these radial functions. Here we show that there exists a specific boundary condition determining the *total* reflection, preventing absorption in any partial waves where the natural black hole absorption might be possible [12].

Our principal goal here is to study the reflected electron beam by using the partial wave analysis for studying the cross sections and induced polarization. We use an analytical method based on the asymptotic approximation of the radial functions that allows us to derive for the first time closed formulas of the phase shifts and scattering amplitudes in the case of the total reflection on a massive body. Thus we complete our analytical study devoted to the fermion scattering from black holes [12, 13, 14] obtaining new formulas which are in accordance with our previous results.

We must specify that our method based on the asymptotic approximation leads to results which are independent on the exterior radius of the massive body. This creates an *ideal* image of the target which is seen as a point-like particle producing total reflection regardless its physical dimensions. This is somewhat in accordance with the philosophy of the partial wave analysis which exploits mainly the asymptotic zone. Obviously, this ideal image can be corrected at any time by using numerical methods for which the analytical results we present here could be an useful guide. Note that our preliminary numerical investigations show that these corrections are very small remaining thus less relevant.

Under such circumstances, the reflecting surface cannot be characterized *a priori* in terms of scattering parameters such that we must analyze different models by fixing the set the of arbitrary phases of the normalization factors and studying then the resulted scattering intensity and induced polarization for understanding the physical content of the model. Here we consider a simple model that can be compared with the model of the bare black hole we studied earlier [12].

We start in the second section revisiting our approximative scattering solutions of the Dirac equation in Schwarzschild’s geometry [11] which are used for developing the partial wave analysis briefly presented in the next section. In the fourth section the boundary conditions corresponding to the total reflection are proposed calculating the phase shifts for which we obtain simple closed formulas. The next section is devoted to the study of the models we propose here using graphical methods. In the last section we present our concluding remarks. In what follows we use the notations of Refs. [12, 13] and Planck’s natural units with \(G=c=\hbar =1\).

## 2 Scattering Dirac spinors in Schwarzschild’s geometry

*M*,

*g*), having as flat model the Minkowski spacetime \((M_0,\eta )\) with the metric

*m*, may be written with our previous notations [12, 13] in the frame \(\{x;e\}\) defined by the Cartesian gauge,

*M*with the Schwarzschild line element

*l*and \(j=l\pm \frac{1}{2}\) [32, 33]. We note that the antiparticle-like energy eigenspinors can be obtained directly using the charge conjugation as in the flat case [31].

*x*, we can use the Taylor expansion with respect to \(\frac{1}{x}\) of the radial Hamiltonian operator, neglecting the terms of the order \(O(\frac{1}{x^2})\). Thus for \(E>0\) we obtain thus the

*approximative*scattering radial solutions [11, 12]

## 3 Partial wave analysis

*U*whose asymptotic form,

*l*are related as in Eq. (9), i. e.

*s*can take either real values or pure imaginary ones. In the case of the massless fermions (\(m=0\)) we remain with the unique parameter \(k=q=2pM\) since then \(\lambda =0\).

*elastic*while for \(|S_ {\kappa }|<1\) a part of the incident beam is absorbed. Here we study only the elastic scattering with real valued phase shifts for which the scalar amplitudes of Eq. (18),

*partial*amplitudes [10, 33],

Note that all the quantities derived here depend on the fixed *E* (or *p*). When we intend to investigate a domain of energies then we have to speak about functions of *E* (or *p*) as for example \(S_{\kappa }(E), C_{\kappa }(E),\ldots \) etc.

## 4 Boundary conditions and phase shifts

In what follows we consider that the target is a massive spherical body of mass *M* and radius \(R\gg r_0\). In these circumstances we may study the scattering of the Dirac fermions on this target by using exclusively our asymptotic solutions with different boundary conditions on the surface of the radius *R* delimiting the body. These boundary conditions may depend on the properties of the exterior surface which may reflect, absorb or polarize the scattered beam. In our formalism, these properties must be encapsulated in the form of the boundary conditions determining the constant \(C_\kappa \) of the spinor solutions and, implicitly, the scattering amplitudes.

In Refs. [12] and [13] we have shown that in the case of the genuine Schwarzschild and Reissner-Nordström black holes we must take \(C_\kappa =0\). Then for the Schwarzschild black holes we obtain an elastic scattering for \(|\kappa | > \mathrm{floor}(k)\) and absorption for \(1\le |\kappa |\le \mathrm{floor}(k)\). A similar result holds for the Reissner-Nordström black holes.

*r*since the condition \(\partial _r J_{rad}=0\) is fulfilled whenever the functions \(f^{\pm }\) are solutions of the radial equations [12].

*a*and

*b*are complex numbers. The condition of preventing absorption is very selective leaving us with two possibilities

*s*can take either real values when \(|\kappa |>\mathrm{floor} (k)\) or pure imaginary ones \(s=\pm i|s|\) if \(1\le |\kappa |\le \mathrm{floor} (k)\). These three cases must be considered separately denoting from now the quantities \(C_{\kappa }\) and the corresponding \(S_{\kappa }\) given by Eq. (23) as

*s*since in the branch point \(s=0\) we have

*s*.

*s*. Indeed, for \(s=|s|\) the constant (42) can be canceled as,

*s*the boundary condition (37) is no longer satisfied allowing absorption.

Finally, we must specify that in the asymptotic approximation used here we neglect the terms of the order \({\mathscr {O}}(\frac{1}{x^2})\) of the radial functions and implicitly the terms of the order \({\mathscr {O}}(\frac{1}{R})\) in Eq. (37) such that the final result is independent on *R*. This corresponds to the ideal image of the target which is seen as a point-like massive particle. This image can be corrected by resorting to additional numerical methods but, as mentioned, these corrections are very small and less relevant.

## 5 Graphical analysis and discussion

Let us now discuss some physical consequences of our results concerning the fermion elastic scattering from a massive body which reflects totally the incident beam. We use the phase shifts given by Eqs. (43) and (45) where the phases \(\alpha _{\kappa }\) cannot be determined in the ideal approximation considered here. Therefore, these remain free parameters related to the properties of the reflecting surface of the massive body. Bearing in mind that the absence of this surface leads to the phases (48) we understand that we have the opportunity of building different models, postulating the form of the phases \(\alpha _{\kappa }\) and analyzing then *a posteriori* the physical effects supposed to be due to the reflecting surface.

*s*. Therefore, in this model we must take

*l*is given by Eq. (24) for any given \(\kappa \). In what follows we analyze this model focusing on the scattering intensity and induced polarization as functions of the scattering angle \(\theta \).

*m*-th order defined as,

Figure 2 presents plots of the scattering cross section for a fixed value of the fermion’s speed *v* (in units of *c*) and three different values of the parameter *mM*, that can be related to the quantity \(\epsilon =\frac{GME}{\hbar c^3}=\frac{\pi r_S}{v\lambda }\) which gives a convenient dimensionless measure of the gravitational coupling. The relation is \(mM=ME\sqrt{1-v^2}\) and also \(r_S\) stands for the Schwarzschild radius, while \(\lambda =h/p\) is the quantum wavelength.

*M*. We also notice that the scattering intensity takes higher values for non-relativistic fermions compared with relativistic ones. The spiral (orbiting) scattering is not present in the cross section at low values of

*mM*. However, as the value of

*mM*is increased more complex scattering patterns start to appear. Looking at the glory peek we see that it is increasing with

*mM*and at the same time the width it’s narrowing down.

As showed in Refs. [10, 12, 13] an initially unpolarized beam of incident fermions could become partially polarized after the interaction with a black hole. This conclusion remains valid also for the gravitational interaction with a massive spherical body. Figure 3 shows how the polarization varies with the scattering angle for a given value of the speed *v* and of the parameter *mM*. We observe that the massive spherical body generates more complex patterns in the polarization in comparison with the black hole case. The oscillations that appear in the polarization can be correlated with the oscillations present in the scattering cross section, that give rise to glory and spiral scattering.

## 6 Concluding remarks

We presented here a simple model of massive body surrounded by a surface able to reflect totally the incident beam of massive Dirac fermions. We used an asymptotic approximation which is suitable for developing the partial wave analysis in terms of simple closed formulas giving the phase shifts that allowed us to study the principal features of the fermion scattering from the massive bodies reflecting the incident beam. Thus we may compare the scattering from massive bodies with that from black holes finding significant differences in what concerns the profile of the scattering intensity and induced polarization. Thus we may conclude that our analytical method is accurate enough for revealing the principal differences among the massive bodies and bare black holes.

However, our approach can be refined by using numerical methods for improving the boundary conditions on exterior reflecting surfaces where some additional physical effects may be considered. For this reason our further objective is to complete our analytical approach with effective numerical methods for studying more complicated scattering processes.

## Notes

### Acknowledgements

C.A. Sporea was supported by a grant of Ministery of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P1-1.1-PD-2016-0842, within PNCDI III. I. I. Cotăescu was partially supported by a grant of the Romanian Ministry of Research and Innovation, CCCDI-UEFISCDI, project number PN-III-P1-1.2-PCCDI-2017-0371, within PNCDI III.

## References

- 1.J.A.H. Futterman, F.A. Handler, R.A. Matzner,
*Scattering from Black Holes*(Cambridge University Press, Cambridge, England, 1988)CrossRefGoogle Scholar - 2.W.G. Unruh, Phys. Rev. D
**14**, 3251 (1976)ADSCrossRefGoogle Scholar - 3.N.K. Kofiniti, Int. J. Theor. Phys.
**23**, 991 (1984)CrossRefGoogle Scholar - 4.B. Mukhopadhyay, S.K. Chakrabarti, Class. Quantum Grav.
**16**, 3165 (1999)ADSCrossRefGoogle Scholar - 5.A. Al-Badawi, M.Q. Owaidat, Gen. Relativ. Gravit.
**49**, 110 (2017)ADSCrossRefGoogle Scholar - 6.J. Jing, Phys. Rev. D
**70**, 065004 (2004)ADSCrossRefGoogle Scholar - 7.J. Jing, Phys. Rev. D
**71**, 124006 (2005)ADSMathSciNetCrossRefGoogle Scholar - 8.K.H.C. Castello-Branco, R.A. Konoplya, A. Zhidenko, Phys. Rev. D
**71**, 047502 (2005)ADSMathSciNetCrossRefGoogle Scholar - 9.C. Doran, A. Lasenby, S. Dolan, I. Hinder, Phys. Rev. D
**71**, 124020 (2005)ADSCrossRefGoogle Scholar - 10.S. Dolan, C. Doran, A. Lasenby, Phys. Rev. D
**74**, 064005 (2006)ADSCrossRefGoogle Scholar - 11.I.I. Cotăescu, Mod. Phys. Lett. A
**22**, 2493 (2007)ADSCrossRefGoogle Scholar - 12.I.I. Cotăescu, C. Crucean, C.A. Sporea, Eur. Phys. J. C
**76**, 102 (2016)ADSCrossRefGoogle Scholar - 13.I.I. Cotăescu, C. Crucean, C.A. Sporea, Eur. Phys. J. C
**76**, 413 (2016)ADSCrossRefGoogle Scholar - 14.C.A. Sporea, Chin. Phys. C
**41**(12), 123101 (2017)ADSCrossRefGoogle Scholar - 15.C.A. Sporea, Fermion scattering by a class of Bardeen black holes, (under review). arXiv:1806.11462
- 16.C.A. Sporea, MOG black hole scattering. arXiv:1812.09945
- 17.S.R. Das, G. Gibbons, S.D. Mathur, Phys. Rev. Lett.
**78**, 417 (1997)ADSCrossRefGoogle Scholar - 18.W.M. Jin, Class. Quantum Grav.
**15**(10), 3163 (1998)ADSCrossRefGoogle Scholar - 19.C. Doran, A. Lasenby, Phys. Rev. D
**66**, 024006 (2002)ADSMathSciNetCrossRefGoogle Scholar - 20.E. Jung, S.H. Kim, D.K. Park, JHEP
**09**, 005 (2004)ADSCrossRefGoogle Scholar - 21.H. Cho, Y. Lin, Class. Quantum Grav.
**22**(5), 775 (2005)ADSCrossRefGoogle Scholar - 22.A.B. Gaina, G.A. Chizhov, Mosc. Univ. Phys. Bull.
**38N2**, 1–7 (1983)Google Scholar - 23.A.B. Gaina,
*Moscow VINITI*, No. 1970-80 Dep., p. 20 (1980). http://adsabs.harvard.edu/abs/1980MoVIN1970...20G - 24.M. Rogatko, A. Szyplowska, Phys. Rev. D
**79**, 104005 (2009)ADSCrossRefGoogle Scholar - 25.H. Liao, J.-H. Chen, P. Liao, Y.-J. Wang, Commun. Theor. Phys.
**62**, 227–234 (2014)ADSCrossRefGoogle Scholar - 26.A. Ghosh, P. Mitra, Phys. Rev. D
**50**, 7389–7393 (1994)ADSCrossRefGoogle Scholar - 27.S.R. Dolan, T. Stratton, Phys. Rev. D
**95**, 124055 (2017)ADSCrossRefGoogle Scholar - 28.K. Tominaga, M. Saijo, K.-I. Maeda, Phys. Rev. D
**60**, 024004 (1999)ADSCrossRefGoogle Scholar - 29.K. Tominaga, M. Saijo, K.-I. Maeda, Phys. Rev. D
**63**, 124012 (2001)ADSCrossRefGoogle Scholar - 30.S. Bernuzzi, A. Nagar, R. De Pietri, Phys. Rev. D
**77**, 044042 (2008)ADSCrossRefGoogle Scholar - 31.I.I. Cotăescu, Phys. Rev. D
**60**, 124006 (1999)ADSMathSciNetCrossRefGoogle Scholar - 32.B. Thaller,
*The Dirac Equation*(Springer, Berlin, Heidelberg, 1992)CrossRefGoogle Scholar - 33.V.B. Berestetski, E.M. Lifshitz, L.P. Pitaevski,
*Quantum Electrodynamics*(Pergamon Press, Oxford, 1982)Google Scholar - 34.I. D. Novikov,
*doctoral disertation*, Sthernberg Astronomical Institute (1963)Google Scholar - 35.C.W. Misner, K.S. Thorne, J.A. Wheeler,
*Gravitation*(Freeman & Co., San Francisco, 1971)Google Scholar - 36.F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark,
*NIST Handbook of Mathematical Functions*(Cambridge University Press, Cambridge, 2010)zbMATHGoogle Scholar - 37.D.R. Yennie, D.G. Ravenhall, R.N. Wilson, Phys. Rev.
**95**, 500 (1954)ADSCrossRefGoogle Scholar

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