Aharonov–Bohm effect for a fermion field in a planar black hole “spacetime”
Eur. Phys. J. C
AharonovBohm effect for a fermion field in a planar black hole ?spacetime?
M. A. Anacleto 2
F. A. Brito 1 2
A. Mohammadi 2
E. Passos 0 2
0 Instituto de Fi?sica, Universidade Federal do Rio de Janeiro , Caixa Postal 21945, Rio de Janeiro , Brazil
1 Departamento de Fi?sica, Universidade Federal da Parai?ba , Caixa Postal 5008, Joa?o Pessoa, Parai?ba 58051970 , Brazil
2 Departamento de Fi?sica, Universidade Federal de Campina Grande , Caixa Postal 10071, Campina Grande, Parai?ba 58429900 , Brazil
In this paper we consider the dynamics of a massive spinor field in the background of the acoustic black hole spacetime. Although this effective metric is acoustic and describes the propagation of sound waves, it can be considered as a toy model for the gravitational black hole. In this manner, we study the properties of the dynamics of the fermion field in this ?gravitational? rotating black hole as well as the vortex background. We compute the differential cross section through the use of the partial wave approach and show that an effect similar to the gravitational AharonovBohm effect occurs for the massive fermion field moving in this effective metric. We discuss the limiting cases and compare the results with the massless scalar field case.

Unruh [1,2] in 1981 suggested a theoretical method based on
the fact that, considering the sound wave motion, a change
of a subsonic runoff for supersonic flow forms an event
horizon analogous to an event horizon of a gravitational black
hole. Since then, the study of analog models of gravity [3?
19] has become an important field where one investigates
the Hawking radiation as well as to improve the theoretical
understanding of quantum gravity. For such analog models
there are many examples, so we highlight gravity waves [20],
water [21], slow light [22?24], optical fibers [25] and
electromagnetic waveguides [26]. Specially in fluid systems, the
propagation of perturbations of the fluid has been analyzed
in many analog models of acoustic black holes, such as the
models of superfluid helium II [27], atomic Bose?Einstein
a email:
b email:
c email:
d email:
condensates [28,29] and onedimensional Fermi degenerate
noninteracting gas [30] that were elaborated to create a sonic
black hole geometry in the laboratory. More specifically, in
a quantum liquid the quasiparticles are bosons (phonons)
in 4He and bosons and fermions in 3He which move in
the background of effective gauge and/or gravity simulated
by the dynamics of collective modes. These quasiparticles
are analogs of elementary particles of lowenergy effective
quantum field theory. In particular, phonons propagating in
the inhomogeneous liquid can be described by the
effective lagrangian mimicking the dynamics of an scalar field
in the curved spacetime given by the effective acoustic
metric where the free quasiparticles move along geodesics. In
superfluid 3HeA, the effective quantum field theory contains
chiral fermion quasiparticles where the collective bosonic
modes interact with these elementary particles as gauge fields
and gravity. These advances are important for a better insight
into the understanding of quantum gravity. In addition, the
study of a relativistic version of acoustic black holes was
presented in [31?34]. Furthermore, the acoustic black hole
metrics obtained from a relativistic fluid in a
noncommutative spacetime [35] and Lorentzviolating Abelian Higgs
model [36,37] have been considered. The thermodynamics
of acoustic black holes in two dimensions was studied in [38].
The authors in [39?41] studied acoustic black holes in the
framework of neoNewtonian hydrodynamics and in [42]
the effect of neoNewtonian hydrodynamics on the
superresonance phenomenon was analyzed.
In 1959, Aharonov and Bohm showed that when the wave
function of a charged particle passing around a region with
the magnetic flux, despite the magnetic field being negligible
in the region through which the particle passes, experiences a
phase shift as a result of the enclosed magnetic field [43]. The
Aharonov?Bohm (AB) effect, which is essentially the
scattering of charged particles, has been used to address several
issues in planar physics and it was experimentally confirmed
by Tonomura [44] ? for a review see [45]. The effect can
also be simulated in quantum field theory as for example
by using a nonrelativistic field theory for bosonic particles
which interact with a Chern?Simons field [46]. More
specifically, in [47?50] was addressed the AB effect considering
the noncommutative spacetime and in [51] the effect was
obtained due to violation of Lorentz symmetry in quantum
field theory.
Furthermore, several other analogs of the AB effect
were found in gravitation [52?56], fluid dynamics [57,58],
optics [61] and Bose?Einstein condensates [62,63]. In [57]
it was shown that the background flow velocity v plays the
role of the electromagnetic potential A, and the integrated
vorticity in the core = (? ? v) ? dS plays the role of
the magnetic flux = B ? dS. Thus, surface waves on
water crossing an irrotational (bathtub) vortex experience an
analog of the AB effect. Also, the gravitational analog of
the electromagnetic AB effect which is purely classical is
related to the particles constrained to move in a region where
the Riemann curvature tensor vanishes. However, a
gravitational effect arises from a region of nonzero curvature from
which the particles are excluded.
An interesting system was investigated in [64], where it
was shown that planar waves scattered by a draining bathtub
vortex develops a modified AB effect that has a dependence
on two dimensionless parameters, related to the circulation
and draining rates [65]. It has been found an inherent
asymmetry even in the lowfrequency regime which leads to novel
interference patterns. More recently in [66?68], the analysis
made in [64] was extended to a Lorentzviolating and
noncommutative background [69?75], which allows the system
to have persistence of phase shifts even if circulation and
draining vanish.
One of the theoretical methods of investigating the
underlying structure of the spacetime is studying the solution of
field equations for the fermion fields, besides the bosonic
ones, in a curved geometry. The behavior of matter fields in
the vicinity of black holes results in the better understanding
of their properties. Fermion fields were analyzed in a Kerr
black hole, in the near horizon limit, as well as in the case
of Reissner?Nordstr?m black hole [76,77]. In [78] nonzero
Dirac fermion modes in the spacetime of a black hole cosmic
string system was considered. The authors studied the
nearhorizon behavior of fermion fields which results in
superconductivity in the case of extremal charged dilaton black
hole.
Although the effective sonic black hole spacetime is not
the one fermion fields would observe, but it can be used as
a toy model and a mathematical tool to study and as a result
understand better the dynamics of the massive and
massless fermion fields in a gravitational rotating black hole as
well as a vortex background and shed light on the
underlying physics. In the present study we consider the dynamics
of a massive Dirac spinor field in a curved spacetime and
apply the acoustic black hole metric to obtain the differential
cross section for scattered planar waves which leads to an
analog AB effect. Besides that, we compare the results with
the massless scalar field case.
The paper is organized as follows. In Sect. 2 we briefly
introduce the acoustic black hole. In Sect. 3 we compute the
differential cross section due to the scattering of planar waves
that leads to an analog AB effect. Finally in Sect. 4 we present
our final considerations.
2 Acoustic black hole The acoustic line element in polar coordinates is governed by
ds2 = (c2 ? v2)dt 2 + 2(vr dr + v? d?)dt ? dr 2 ? r 2d?2,
where c = ?dh/d? is the velocity of sound in the fluid and
v is the flow/fluid velocity. We consider the flow with the
velocity potential ? (r, ?) = D ln r + C ? whose flow/fluid
velocity is given by
Thus, Eq. (1) can be rewritten as follows:
ds2 =
c2 ?
where C and D are the constants of the circulation and
draining rates of the fluid flow. The radius of the ergosphere given
by g00(re) = 0, and the horizon, the coordinate singularity,
given by grr (rh ) = 0, are
re =
?C 2
, rh =
We set c = 1, and choose the following change of variables
[64]:
D C D
dt? = dt ? r f (r ) dr, d?? = d? ? r 3 f (r ) dr,
where f (r ) = 1 ? D2/r 2, and which results in the following
line element:
ds2 =
f (r ) ? r 2
Therefore, the metric can be written in the form
= ?
0
? f (r )
0
We know that this effective metric is acoustic which describes
the propagation of sound waves. It is trivial that the Dirac
field does not obey the acoustic metric. In the case it exists in
the considered media, it would have its own metric with its
own ?speed of light?. We are interested in the dynamics of the
fermion field in this background as a toy model to investigate
the physical properties in contrast with the scalar field case.
In the following, we study the scattering of monochromatic
planar waves of frequency ? for a massive fermion by the
draining vortex, a process governed by two key quantities;
circulation and draining.
3 Fermion field in the acoustic black hole background
Now let us consider a spinor field in the background of the
sonic black hole. The dynamics of a massive spinor field in
curved spacetime is described by the Dirac equation
where D? = ?? + ?, ? ? are the Dirac matrices in curved
spacetime and ? are the spin connections. Let us choose the
following representation:
? (0) = ? 3, ? (1) = i ? 2, ? (2) = ?i ? 1.
For the geometry at hand, using the relation ? ? = e(?a)? (a),
the gamma matrices take the form
= ?1f ? 3, ? 1 = i f sin ?? 1 + cos ?? 2 ,
= r 2C? f ? 3 + ri sin ?? 2 ? cos ?? 1 .
The only nonzero Christoffel symbols are
2 C(C2+D2) ,
? = 41 ? (a)? (b)e(?a)??e(b)? ,
one can find the spin connection components for the system
as follows:
In the Dirac equation there appears
? i
(1 + ? f )
cos ? ? 1 ? sin ? ? 2 ,
Now, we substitute the following spinor field in the Dirac
equation:
where j = ?1/2, ?3/2, . . .. Replacing the corresponding
parameters in the Dirac equation, we obtain
f ?r +
f ?r ?
n? 2 = n2 ? C (?? + M f ? 2n?) ? D2(?2 + ?2),
m? 2 = m2 ? C (? + M f ? 2m?) ? D2(?2 + ?2).
?2C m + D2(3? + 2M f )
2(? + M f )r
2C ? 2C m + D2(3? ? 2M f )
2(? ? M f )r
Choosing ? = C ?, ? = C M f and ? 2 = D2(?2 + ?2), we
have
n? 2 = n2 + 2?n + (? ? ?) ? ? 2,
m? 2 = m2 + 2?m ? (? + ?) ? ? 2.
Therefore for large r , ignoring the terms O[ ?14 ] and O[ ?14 ],
one can obtain analytic solutions 1 2
It is not difficult to show numerically that these solutions
besides satisfying the second order Eqs. (20) and (21), satisfy
the first order Eqs. (18) and (19) in the limit r ? ?.
Comparing this result with the one in the Minkowski space, we
obtain the following approximate expressions for the phase
shift for the upper and lower components of the spinor fields:
m=??
where n = m ? 1. For n, m ? + ?2 + ? + ? 2, one
can expand the above expressions. Thus, we obtain
The first term in both expressions, the dominant contribution,
is exactly the same as the bosonic case considered in [64].
Now we can calculate the differential scattering cross
section, which is given by
For the ?m when m = 0 one needs to use Eqs. (32)?
(33). Therefore, the differential scattering coss section at
long wavelengths, ? + ?2 + ? + ? 2 1, for a massive
fermion in the acoustic black hole background is given as
follows:
?
2
?
2
Obviously, in the case ? = ? + ? 2 the two expressions for
? > ?+? 2 and ? < ?+? 2 are the same. In the absence of the
circulation, the differential scattering cross section reduces
to
which is ? independent. This is an expected result, since
in the absence of the circulation the only term that
contributes in the long wavelength regime to the
differential cross section is m = 0, polar symmetric term. The
same happens for the bosonic case which is equal to
? D2?. In the absence of the circulation, the result for the
fermionic case is exactly the same as the bosonic one in
the zero mass limit. The above expression vanishes when
the draining is also zero. This is also an expected result,
because without circulation and draining there is no source
for the scattering and we should recover the Minkowski
result.
Expanding the expression for the differential scattering
cross section in Eq. (38) for small angles results in
Therefore, for ? ? 0 and small ? the dominant term in the
differential scattering cross section is the first term in the
above expression. It means that for small angles the
circulation plays more important role compared to the draining.
Using ? = C ? and ? = (?2 ? M 2f )1/2, the first term in the
above expression can be expanded as
for small M f /?. The first term which is the result for
massless fermion, matches exactly the one for the massless boson
case.
Now one can compare the results for the fermionic and
bosonic case considering a massless fermion, M f = 0. To
compare these results, we need the same arbitrary energy
scale for both cases. We call this energy scale M .
Figures 1, 2 show the differential cross section for the
massless boson and fermion for two cases ? = ?/5 and
? = ?/100 with C = 0.01M ?1, choosing D = 0.01M ?1
and also the special case where there is no draining, in the
absence of the black hole. As can be seen, the result for the
fermionic case tends to the bosonic one for small angles.
Besides that, the left graphs in Figs. 1 and 2 show that for the
limit ? ? 0 the differential cross section for the spinor field
goes to a nonzero constant in contrast with the boson field.
To see this more closely let us expand Eq. (38) for small ?
where ? > ? + ? 2, which is the case for the graphs in Figs.
1 and 2, as
C 3/2? cot(?/2) ?? + O[?].
As can be seen, in the limit ? ? 0, the scattering cross
section is equal to C ?/2 for the fermion case which is only
dependent on the circulation. We think this happens due to the
fact that the fermionic field has an intrinsic angular
momentum which interacts with the angular momentum of the black
hole originating from the circulation.
4 Conclusion
In this paper, we have studied the differential cross section of
a massive spinor field in the background of the acoustic black
hole spacetime, as a toy model for the gravitational rotating
black hole, using the partial wave approach. We have
investigated the scattering of planar waves in a fermionic system
by a background vortex as an analog for a rotating black
hole. Because of the form of the spinor field which has upper
and lower components, we have calculated the phase shifts
for these components separately and then averaged them. We
have worked with three dimensionless parameters ?, ? and ?
related to the circulation, fermion mass and draining,
respectively. We have obtained the differential cross section at long
wavelengths and discussed the limiting cases including small
angles and also in the absence of the circulation. One could
see that the dominant contribution in the small angle limit
for the fermionic case is exactly the same as the bosonic one
when the mass of the fermion is zero. This dominant
contribution comes from the circulation term. Therefore, for small
angles the contribution of the circulation is more important
than the draining one. Furthermore, considering the cross
section in the absence of the circulation shows that the result
is the same for the bosonic and fermionic cases when the
fermion is massless. In this limit, the result is independent of
the angle ?. The reason for that is in the absence of the
circulation the only term that contributes in the long wavelength
regime to the differential cross section is a polar
symmetric term. Finally, we have shown that in contrast with the
bosonic case in the limit ? ? 0, considering ? > ? + ? 2,
the scattering cross section goes to a nonzero constant equal
to C ?/2 for the fermion case which is only dependent on the
circulation.
The similar scattering behavior for both boson and
fermion for small angles seems to be in accord with both
being scattered between the horizon and the ergosphere
radius. At this regime both enjoys the Penrose effect of
gaining energy after the scattering (the superresonance effect).
As Figs. 1, 2 (right panel) show, this indeed agrees with
the increasing of the differential cross section as the
frequency increases. This precisely happens as long as the
frequency belongs the interval 0 < ? < m H , where m is the
azimuthal mode number and H is the angular velocity of the
black hole [36, 37, 66?68]. On the other hand, for large angles,
both particles tend to be scattered outside the ergoregion and
in turn they almost keep the scattering constant as Figs. 1
and 2 (left panel) show. This is particularly more accurate for
fermions. A detailed analysis of the superresonance effect for
fermion fields should be addressed elsewhere. Besides that,
in a future work, we plan to study a more realistic model
considering the dynamics of the fermionic fields interacting
with a black hole, gravitational or as an analog in a condensed
matter system.
Acknowledgements We would like to thank CNPq and CAPES for
partial financial support. A. M. thanks PNPD/CAPES for the financial
support.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
1. W. Unruh , Phys. Rev. Lett . 46 , 1351 ( 1981 )
2. W. Unruh , Phys. Rev. D 51 , 2827 ( 1995 )
3. M. Visser , Class. Quantum Gravity 15 , 1767 ( 1998 )
4. G. Volovik , The Universe in a Helium Droplet (Oxford University Press, Oxford, 2003 )
5. L.C.B. Crispino , A. Higuchi , G.E.A. Matsas , Rev. Mod. Phys . 80 , 787 ( 2008 )
6. M. Cadoni , S. Mignemi , Phys. Rev. D 72 , 084012 ( 2005 )
7. M. Cadoni , Class. Quantum Gravity 22 , 409 ( 2005 )
8. L.C. Garcia de Andrade, Phys. Rev. D 70 , 64004 ( 2004 )
9. T.K. Das , Transonic black hole accretion as analogue system . arXiv:grqc/0411006
10. G. Chapline , P.O. Mazur , Superfluid picture for rotating spacetimes . arXiv:grqc/0407033
11. O.K. Pashaev , J.H. Lee , Theor. Math. Phys. 127 , 779 ( 2001 )
12. S. E. Perez Bergliaffa , K. Hibberd , M. Stone , M. Visser , Phys . D 191 , 121 ( 2004 )
13. S.W. Kim , W.T. Kim , J.J. Oh , Phys. Lett . B 608 , 10 ( 2005 )
14. X. H. Ge , S.F. Wu , Y. Wang , G.H. Yang , Y.G. Shen , Int. J. Mod. Phys. D 21 , 1250038 ( 2012 )
15. C. Barcelo , S. Liberati , M. Visser , Living Rev . Rel. 8 , 12 ( 2005 )
16. E. Berti , V. Cardoso , J.P.S. Lemos , Phys. Rev. D 70 , 124006 ( 2004 )
17. V. Cardoso , J.P.S. Lemos , S. Yoshida , Phys. Rev. D 70 , 124032 ( 2004 )
18. P.A. Horvathy , M.S. Plyushchay , Phys. Lett . B 595 , 547 ( 2004 )
19. M.A. Cuyubamba , Class. Quantum Gravity 30 , 195005 ( 2013 )
20. R. Sch?tzhold , W.G. Unruh , Phys. Rev. D 66 , 044019 ( 2002 )
21. G. Rousseaux , C. Mathis , P. Ma?ssa , T.G. Philbin , U. Leonhardt , New J. Phys. 10 , 053015 ( 2008 )
22. U. Leonhardt , P. Piwnicki , Phys. Rev. Lett . 84 , 822 ( 2000 )
23. U. Leonhardt, Nature 415 , 406 ( 2002 )
24. W.G. Unruh , R. Sch?tzhold , Phys. Rev. D 68 , 024008 ( 2003 )
25. T.G. Philbin , C. Kuklewicz , S. Robertson , S. Hill , F. K?nig , U. Leonhardt, Science 319 , 1367 ( 2008 )
26. R. Sch?tzhold , W.G. Unruh , Phys. Rev. Lett . 95 , 031301 ( 2005 )
27. M. Novello , M. Visser , G. Volovik (eds.), Artificial Black Holes (World Scientific, Singapore , 2002 )
28. L.J. Garay , J.R. Anglin , J.I. Cirac , P. Zoller , Phys. Rev. Lett . 85 , 4643 ( 2000 )
29. O. Lahav , A. Itah , A. Blumkin , C. Gordon , J. Steinhauer, arXiv:0906.1337
30. S. Giovanazzi , Phys. Rev. Lett . 94 , 061302 ( 2005 )
31. X. H. Ge , S.J. Sin , JHEP 1006 , 087 ( 2010 )
32. N. Bilic , Class. Quantum Gravity 16 , 3953 ( 1999 )
33. S. Fagnocchi , S. Finazzi , S. Liberati , M. Kormos , A. Trombettoni , New J. Phys. 12 , 095012 ( 2010 )
34. M. Visser , C. MolinaParis , New J. Phys. 12 , 095014 ( 2010 )
35. M.A. Anacleto , F.A. Brito , E. Passos , Phys. Rev. D 85 , 025013 ( 2012 )
36. M.A. Anacleto , F.A. Brito , E. Passos , Phys. Lett . B 703 , 609 ( 2011 )
37. M.A. Anacleto , F.A. Brito , E. Passos , Phys. Lett . B 694 , 149 ( 2010 )
38. B. Zhang , Thermodynamics of acoustic black holes in two dimensions . arXiv:1606 .00693 [hepth]
39. H. Velten , D.J. Schwarz , J.C. Fabris , W. Zimdahl , Phys. Rev. D 88 , 103522 ( 2013 )
40. A.M. Oliveira , H.E.S. Velten , J.C. Fabris , I.G. Salako , Eur. Phys. J. C 74 , 3170 ( 2014 )
41. J.C. Fabris , O.F. Piattella , I.G. Salako , J. Tossa , H.E.S. Velten , Mod. Phys. Lett A 28 , 1350169 ( 2013 )
42. I.G. Salako , A. Jawad , arXiv:1503.08714
43. Y. Aharonov , D. Bohm , Phys. Rev . 115 , 485 ( 1959 )
44. A. Tonomura , Phys. Rev. Lett . 48 , 1443 ( 1983 )
45. M. Peskin , A. Tonomura , The Quantum Hall Effect (Springer, Berlin, 1989 )
46. O. Bergman , G. Lozano , Ann. Phys. 229 , 416 ( 1994 )
47. M.A. Anacleto , M. Gomes , A.J. da Silva , D. Spehler , Phys. Rev. D 70 , 085005 ( 2004 )
48. M.A. Anacleto , M. Gomes , A.J. da Silva , D. Spehler , Phys. Rev. D 70 , 129905 ( 2004 )
49. M.A. Anacleto , M. Gomes , A.J. da Silva , D. Spehler , Phys. Rev. D 71 , 107701 ( 2005 )
50. M.A. Anacleto , J.R. Nascimento , A.Y . Petrov , Phys. Lett . B 637 , 344 ( 2006 )
51. M.A. Anacleto , Phys. Rev. D 92 , 085035 ( 2015 )
52. L.H. Ford , A. Vilenkin , J. Phys . A 14 , 2353 ( 1981 )
53. V.B. Bezerra , Phys. Rev. D 35 , 2031 ( 1987 )
54. V.B. Bezerra , J. Math. Phys. 30 , 2895 ( 1989 )
55. G.A. Marques , V.B. Bezerra , Mod. Phys. Lett. A 19 , 49 ( 2004 )
56. V.B. Bezerra , G.A. Marques , S.G. Fernandes , J. Math. Phys. 47 , 072504 ( 2006 )
57. M.V. Berry , R.G. Chambers , M.D. Large , C. Upstill , J.C. Walmsley , Eur. J. Phys . 1 , 154 ( 1980 )
58. E. Cerda , F. Lund , Phys. Rev. Lett . 70 , 3896 ( 1993 )
59. C. Coste , F. Lund , M. Umeki , Phys. Rev . E 60 , 4908 ( 1999 )
60. F. Vivanco , F. Melo , C. Coste , F. Lund , Phys. Rev. Lett . 83 , 1966 ( 1999 )
61. D. Neshev , A. Nepomnyashchy , Y.S. Kivshar , Phys. Rev. Lett . 87 , 043901 ( 2001 )
62. U. Leonhardt , P. Ohberg , Phys. Rev. A 67 , 053616 ( 2003 )
63. M. Stone , Phys. Rev . B 61 , 11780 ( 2000 )
64. S.R. Dolan , E.S. Oliveira , L.C.B. Crispino , Phys. Lett . B 701 , 485 ( 2011 )
65. A.L. Fetter , Phys. Rev . 136 , A1488 ( 1964 )
66. M.A. Anacleto , F.A. Brito , E. Passos , Phys. Rev. D 86 , 125015 ( 2012 )
67. M.A. Anacleto , F.A. Brito , E. Passos , Phys. Rev. D 87 , 125015 ( 2013 )
68. M.A. Anacleto , F.A. Brito , E. Passos , Phys. Lett . B 743 , 184 ( 2015 )
69. D. Bazeia , R. Menezes , Phys. Rev. D 73 , 065015 ( 2006 )
70. A. de Souza Dutra , M. Hott , F.A. Barone , Phys. Rev. D 74 , 085030 ( 2006 )
71. D. Bazeia , M.M. Ferreira Jr., A.R. Gomes , R. Menezes , Phys . D 239 , 942 ( 2010 )
72. A. de Souza Dutra , R.A.C. Correa, Phys. Rev. D 83 , 105007 ( 2011 )
73. P.R.S. Carvalho , Phys. Lett . B 726 , 850 ( 2013 )
74. R.A.C. Correa, R. da Rocha , A. de Souza Dutra, Ann. Phys. 359 , 198 ( 2015 )
75. R.A.C. Correa, R. da Rocha , A. de Souza Dutra , Phys. Rev. D 91 , 125021 ( 2015 )
76. I. Sakalli , M. Halilsoy , Phys. Rev. D 69 , 124012 ( 2004 )
77. D. Lohiya , Phys. Rev. D 30 , 1194 ( 1984 )
78. L. Nakonieczny , M. Rogatko , Phys. Rev. D 84 , 044029 ( 2011 )