Eur. Phys. J. C (2018) 78:12 https://doi.org/10.1140/epjc/s10052-017-5498-x Regular Article - Theoretical Physics Effects of the fermionic vacuum polarization in QED M. F. X. P. Medeiros1,a, F. E. Barone2,b, F. A. Barone1,c 1 IFQ-Universidade Federal de Itajubá, Av. BPS 1303, Caixa Postal 50, Itajubá, MG CEP 37500-903, Brazil 2 Niterói, RJ, Brazil Received: 3 December 2017 / Accepted: 22 December 2017 / Published online: 9 January 2018 © The Author(s) 2018. This article is an open access publication Abstract Some effects of vacuum polarization in QED due radiative corrections to the Aharonov–Bohm scattering [10], to the presence of field sources are investigated. We focus the interaction between two solenoids [11], vacuum currents on effects with no counter-part in Maxwell electrodynamics. produced around a Dirac string [12,13], Bremsstrahlung and The the Uehling interaction energy between two stationary pair production in the Aharonov–Bohm potential [14] and so point-like charges is calculated exactly in terms of Meijer-G on. functions. Effects induced on a hydrogen atom by the vacuum The vacuum polarization can be modified by the presence polarization in the vicinity of a Dirac string are considered. of an external field. The most common situations studied in We also calculate the interaction between two parallel Dirac this context are the effects produced by the presence of an strings and corrections to the energy levels of a quantum par- external magnetic field in the Coulomb interaction and in the ticle constrained to move on a ring circumventing a solenoid. hydrogen-like atoms [15–20]. Modifications in the nuclear Coulomb field induced by a strong laser field were also con- sidered in the literature [21]. 1 Introduction The vacuum polarization was also studied in coordinate space in reference [22], where the Green’s function was cal- Since the establishment of the QED, the effects regarding culated in higher orders beyond the Uehling term. the vacuum polarization had been drawing attention mainly In this paper we study some effects produced by the vac- in situations with no counterpart in classical electrodynam- uum polarization of the fermionic field. We focus on sit- ics. In this context, we can mention the Uehling potential uations with no counterpart in classical electrodynamics, [1], which is the electromagnetic potential associated with a studying setups where sources for the electromagnetic field single point-like stationary charge corrected in lowest order can interact via the vacuum polarization. In Sect. 2 we start in the fine structure constant. The calculation of this poten- by studying the standard interaction between two point-like tial is usually found in the literature perturbatively in the steady charges corrected by the vacuum polarization, in low- momentum space for a point-like source [2–5] and was also est order in the fine structure constant. We calculate exactly generalized for a charge distribution with finite radius [6]. the Uehling interaction between them. Our result has two Recently, the Uehling potential was calculated exactly in main advantages: it is an exact result (in lowest order in the terms of Bessel Integral functions [7]. In the context of non fine structure constant) valid for any distance between the relativistic quantum mechanics, the Uehling potential can charges and the result is given by a simple expression writ- lead, for instance, to effects on hydrogen-like atoms [8]. ten in terms of a K-Bessel functions and Meijer-G functions, Atomic effects of QED not accounted by Uehling potential what makes it easier to be plotted, once those functions are were also studied in the literature [9]. well known in the literature. Another interesting scenario created by the vacuum polar- In Sect. 3 we find out the field produced outside a Dirac ization, with no counterpart in classical electrodynamics, is string due to the vacuum polarization, in lowest order in the related to the effects which emerge around solenoids and fine structure constant. We show that we have a magnetic field Dirac strings. In this context we can mention, for instance, the outside the string anti-parallel to the internal magnetic flux and we calculate the corrections in order α (the fine struc- ture constant) in the energy levels of a quantum particle con- a e-mail: [email protected] strained to move on a ring (2-D quantum rigid rotor). In Sect. b e-mail: [email protected] 4 we show that a hydrogen atom, in its ground state, interact c e-mail: [email protected] 123 12 Page 2 of 9 Eur. Phys. J. C (2018) 78 :12 with the string via a kind of Zeeman effect. This interaction Substituting the Fourier integrals for the field configura- falls down very quickly when the distance between the atom tion and for the external source, and the string increases. This force is attractive when the total 4 μ d p μ angular momentum of the electron is parallel to the internal A (x) = A˜ (p)e−ipx, M (2π)4 M magnetic flux of the string, and repulsive in the opposite case. 4 μ d p μ − In Sect. 5 we show that it emerges an interaction between J (y) = J˜ (p )e ip y, (7) two Dirac strings due to the vacuum polarization of the (2π)4 fermionic field. We calculate this interacting force exactly, and the integral (4)inEq.(6) and using the fact that α up to order , for any distance between the strings when d4 yei(p−p )y = (2π)4δ4(p − p), we can show that they are parallel to each other. Section 6 is devoted to some comments and final remarks. ˜μ ( ) = ˜ ( ) ˜μ( ). AM p DM p J p (8) With the aid of Eqs. (8) and (5) we can write the Fourier 2 Uehling interaction transform of the external source as a function of the Fourier transform of the gauge field obtained from the Maxwell elec- In this section we calculate exactly the Uehling interaction trodynamics, between two point-like steady charges. The gauge sector of the classical electrodynamics can be 2 μ p μ described by the Lagrangian J˜ (p) =− A˜ (p). (9) 4π M 1 μν μ 1 μ 2 L =− Fμν F − Jμ A − (∂μ A ) (1) μ( ) = μ( ) 16π 8π For a steady external source, J x J x , one can show that the energy stored in the electromagnetic field is where the last term is a gauge fixing one, Aμ is the electro- given by μν μ ν ν μ magnetic field, F = ∂ A − ∂ A is the field strength μ and J is the external source. 3 4 1 μ EM = d xd y Jμ(x)DM (x − y)J (y). (10) From the Lagrangian (1) one obtain the dynamical equa- 2 tion It is well known in the literature that the net QED effects ∂μ∂ ν = π ν μ AM 4 J (2) of the fermionic vacuum bubbles can be taken into account by a correction in the gauge field propagator [2,5], as follows for which the corresponding propagator DM (x − y) satisfies the differential equation 4π D˜ (p) = D˜ (p) 1 + (p) =− 1 + (p) . (11) M p2 μ 4 ∂μ∂ DM (x − y) = 4πδ (x − y). (3) where The sub-index M in (3) means that we have the quantities 1 calculated for the Maxwell theory. α v2 1 − 1 v2 ( ) =− v 3 , The solution for (3) is given by the Fourier integral in the p d 2 (12) μ π v2 + 4m − 1 four momentum p , 0 p2 4 α d p ˜ −ip(x−y) with standing for the fine structure constant and m,the DM (x − y) = DM (p)e , (4) (2π)4 mass of the electron. The corrected propagator for the electromagnetic field is where given by the Fourier integral 4π ˜ ( ) =− 4 DM p (5) d p − ( − ) p2 D(x − y) = D˜ (p)e ip x y . (13) (2π)4 is the Fourier transform of the propagator DM (x − y) The solution for the field equation (2) is given by the inte- In the presence of an external source, the corrected field gral configuration is μ ( ) = 4 ( − ) μ( ). μ( ) = 4 ( − ) μ( ). AM x d xDM x y J y (6) A x d xD x y J y (14) 123 Eur. Phys. J. C (2018) 78 :12 Page 3 of 9 12 Substituting the second Eqs. (7) and (13)in(14), we can The right and side of Eq. (20) is the same as the one found show that in the calculations of the Uehling potential [2]. We shall cal- 4 culate it exactly in this section. μ d p ˜ ˜μ −ipx A (x) = D(p)J (p)e The first term on the right hand side of Eq. (20)is (2π)4 the coulombian interaction between the charges. This well 4 d p μ − = D˜ (p) 1 + (p) J˜ (p)e ipx. (15) known result [3,23–27] is obtained from EM (the interac- ( π)4 M 2 tion energy given by the Maxwell electrodynamics) and the where, in the second line, we used Eq. (11). propagator defined in (5) and (4). The integral in the second With the aid of Eq. (8), we can rewrite Eq. (15)intheform term can√ be calculated by changing the integration variables q = p 1 − v2, as follows 4 μ d p μ − ( ) = ˜ ( ) + ( ) ipx · A x AM p 1 p e 3 ip a (2π)4 d p e = 1 / 4 (2π)3 p2(1 − v2) + 4m2 (1 − v2)3 2 μ d p μ − = A (x) + A˜ (p)(p)e ipx.