View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector Eur. Phys. J. C (2016) 76:612 DOI 10.1140/epjc/s10052-016-4464-3 Regular Article - Theoretical Physics Rapidly rotating pulsar radiation in vacuum nonlinear electrodynamics V. I. Denisov1,I.P.Denisova2,A.B.Pimenov1,V.A.Sokolov1,a 1 Physics Department, Moscow State University, Moscow 119991, Russia 2 Moscow Aviation Institute (National Research University), Volokolamskoe Highway 4, Moscow 125993, Russia Received: 5 September 2016 / Accepted: 23 October 2016 / Published online: 8 November 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com 13 Abstract In this paper we investigate the corrections of high-field (surface fields more than Bp > 10 G) radio loud vacuum nonlinear electrodynamics on rapidly rotating pulsar pulsars can be explained by pair-production suppression due radiation and spin-down in the perturbative QED approach to photon splitting [19]. (post-Maxwellian approximation). An analytical expression In this paper we calculate vacuum nonlinear electrody- for the pulsar’s radiation intensity has been obtained and ana- namics corrections to electromagnetic radiation of rapidly lyzed. rotating pulsar and analyze pulsar spin-down under these cor- rections. This paper is organized as follows. In Sect. 2, we present 1 Introduction vacuum nonlinear electrodynamics models and discuss their main physical properties and predictions. In Sect. 3 pulsar Vacuum nonlinear electrodynamics effects is an object that radiation in post-Maxwellian approximation is calculated. piques a great interest in contemporary physics [1–4]. First of Section 4 is devoted to an analysis of the pulsar’s spin-down all it is related to the emerging opportunities of experimental vacuum nonlinear electrodynamics influence. In the last sec- research in terrestrial conditions using extreme laser facili- tion we summarize our results. ties like extreme light infrastructure (ELI) [5–7], Helmholtz International Beamline for extreme fields (HIBEF) [8]. It opens up new possibilities in fundamental physics tests [9– 2 Vacuum nonlinear electrodynamics theoretical models 11] with an extremal electromagnetic field intensities and particle accelerations that have never been obtained before. Modern theoretical models of nonlinear vacuum electrody- At the same time the investigation of the effects of vac- namics suppose that the electromagnetic field Lagrange func- uum nonlinear electrodynamics in astrophysics gives us an tion density L = L(I(2), I(4)) depends on both independent ki kl mi additional opportunity to carry out versatile research using invariants I(2) = FikF and I(4) = FikF Flm F of the the natural extreme regimes of strong electromagnetic and electromagnetic field tensor Fik. The specific relationship gravitational fields with intensities unavailable yet in lab- between the Lagrange function and the invariants depends oratory conditions. Compact astrophysical objects with a on the choice of the theoretical model. Nowadays the most strong field, such as pulsars and magnetars, are best suited promising models are Born–Infeld and Heisenberg–Euler for vacuum nonlinear electrodynamics research. Nowadays, electrodynamics. there are many effects of vacuum nonlinear electrodynam- Born–Infeld electrodynamics is a phenomenological the- ics predicted in the pulsar’s neighborhood. For example vac- ory originating from the requirement of self-energy finite- uum electron–positron pair production [12] and photon split- ness for a point-like electrical charge [20]. In subsequent ting [13], photon frequency doubling [14], light by light scat- studies, the attempts of quantization were performed [21,22] tering, and vacuum birefringence [15], transient radiation ray and also it was revealed that Born–Infeld theory describes bending [16,17] and normal waves delay [18]. Some of the the dynamics of electromagnetic fields on D-branes in string predicted effects are indirectly confirmed by astrophysical theory [23–25]. As the main features of Born–Infeld electro- observations. For instance the evidence of the absence of the dynamics one can note the absence of birefringence (how- ever, there are modifications of the Born–Infeld theory [26] a e-mail: [email protected] with the vacuum birefringence predictions) and dichroism for 123 612 Page 2 of 8 Eur. Phys. J. C (2016) 76 :612 electromagnetic waves propagating in external electromag- predictions were experimentally proved in Delbrück light netic field [27]. Furthermore, this theory has a distinctive by light scattering [30], nonlinear Compton scattering [31], feature – the value of electric field depends on the direction Schwinger pair production in multiphoton scattering [1]. At of approach to the point-like charge. This property was noted the same time the recent astrophysical observations [32,33] by the authors of the theory and also eliminated by them in point on the absence of the vacuum birefringence effect the subsequent model development [28]. which favors the Born–Infeld theory prediction. The mea- Lagrangian function in Born–Infeld electrodynamics has surements performed for the speed of light in vacuum show the following form: that it does not depend on wave polarization with the accuracy δc/c < 10−28. So clarification of the vacuum nonlinear elec- ⎧ ⎫ ⎨ ⎬ trodynamics status requires the expansion of the experimen- 1 a2 a4 a4 L =− 1 − I( ) − I( ) + I 2 − 1 , tal test list both in terrestrial and astrophysical conditions. π 2 ⎩ 2 4 (2) ⎭ 4 a 2 4 8 The main hopes as regards this way to proceed are assigned (1) to the experiments with ultra-high intensity laser facilities [4] and astrophysical experiments with X-ray polarimetry [34] where a is a characteristic constant of theory, the inverse in pulsars and magnetars neighborhood. value of which has a meaning of maximum electric field for As follows from the Lagrangians (1)–(2) vacuum non- the point-like charge. For this constant only the following linear electrodynamics’ influence becomes valuable only − − , ∼ / estimation is known: a2 < 1.2 × 10 32 G 2. in strong electromagnetic fields, comparable to E B 1 a , ∼ The other nonlinear vacuum electrodynamics – the Hei- for Born–Infeld theory and E B Bc for Heisenberg– senberg–Euler model [15,29] – was derived in quantum field Euler electrodynamics [35]. In the case of relatively weak , << theory and describes one-loop radiative corrections caused by fields (E B Bc) the exact expressions (1) and (2) can be vacuum polarization in a strong electromagnetic field. Unlike decomposed and written [36] in the form of a unified para- the Born–Infeld electrodynamics, this theoretical model pos- metric post-Maxwellian Lagrangian: sesses vacuum birefringent properties in a strong field. 1 2 The effective Lagrangian function for Heisenberg–Euler L = I( ) + ξ (η − η )I + η I( ) , π 2 2 1 2 2 (2) 4 2 4 (5) theory has the following form: 32 ∞ ξ = / 2 = . × −27 −2 α 2 −σ σ where 1 Bc 0 5 10 G , and the post- I(2) Bc e d 2 L = − xyσ ctg(xσ)cth(yσ) Maxwellian parameters η1 and η2 depend on the choice of 16π 8π 2 σ 3 0 the theoretical model. In the case of Heisenberg–Euler elec- η η σ 2 trodynamics the post-Maxwellian parameters 1 and 2 are + (x2 − y2) − 1 dσ, (2) coupled to the fine structure constant α [37]: 3 2 3 13 where B = m c /eh¯ = 4.41 × 10 G is the value of the α − 7α − c η = = 5.1 × 10 5,η= = 9.0 × 10 5. (6) characteristic field in quantum electrodynamics, e and m are 1 45π 2 180π the electron charge and mass, α = e2/hc¯ is the fine structure constant, and for brevity we use the notations For Born–Infeld electrodynamics these parameters are equal to each other and they can be expressed through the i 1 field induction 1/a typical of this theory [37]: x =−√ (B2 − E2) + i(BE) 2Bc 2 2 2 η = η = a Bc < . × −6. 1 1 2 4 9 10 (7) − (B2 − E2) − i(BE) , (3) 4 2 The electromagnetic field equations for the post-Maxwellian vacuum electrodynamics with the Lagrangian (5) are equiv- 1 1 y = √ (B2 − E2) + i(BE) alent [35] to the equations of Maxwell electrodynamics of 2Bc 2 continuous media, 1 + (B2 − E2) − i(BE) . (4) ∂ F + ∂ F + ∂ F = 0, (8) 2 m ik i km k mi Many attempts to find the experimental status for each ∂ Qki 4π of these theories were made a long time ago, but nowadays =− jk, (9) it still remains ambiguous. There is experimental evidence ∂xi c in favor of each of them. Heisenberg–Euler electrodynamics with the specific nonlinear constitutive relations [18] 123 Eur. Phys. J. C (2016) 76 :612 Page 3 of 8 612 ki = ki + ξ (η − η ) ki + η ki , where Fki is the electromagnetic field tensor of the rotating Q F 1 2 2 I(2) F 4 2 F(3) (10) (0) magnetic dipole m in Maxwell electrodynamics and f ik is ki kn mi the vacuum nonlinear correction. Substituting (13)into(10) where F( ) = F Fnm F is the third power of the elec- 3 and retaining only the terms linear in a small value f ik it can tromagnetic field tensor. The tensor Qik can be separated be found that into two terms Qki = Fki + Mki, one of which, Mki, will have a meaning similar to the matter polarization tensor in Qik f ik + Fik + Mik , (14) electrodynamics of continuous media. (0) (0) Also it should be noted that in a post-Maxwellian approx- ik ik = ik( nj ) imation the stress-energy tensor T and Poynting vector S where M(0) M F(0) is the polarization tensor calcu- have the form lated in the approximation of the Maxwell electrodynamics nj ik field F(0). The electromagnetic field equations (8)–(9) with ik = 1 ( + ξη ) ik − g T 1 1 I(2) F(2) 2I(2) account of (13)–(14) then will take the form 4π 8 (0) (0) (0) + ξ(η + η ) 2 − η ξ , ∂m F + ∂i F + ∂k F + ∂m fik + ∂i fkm + ∂k fmi = 0, 1 2 2 I(2) 4 2 I(4) (11) ik km mi ki ki ∂ f ki ∂ F( ) ∂ M( ) 4π + 0 + 0 =− jk.