Photon Splitting in Strongly Magnetized Medium with Taking Into Account Positronium Influence

EPJ Web of Conferences 191, 08011 (2018) https://doi.org/10.1051/epjconf/201819108011 QUARKS-2018

Photon splitting in strongly magnetized medium with taking into account positronium inﬂuence

1, 2, 2, 1, Anikin R.A. ∗, Chistyakov M.V. ∗∗, Rumyantsev D.A. ∗∗∗, and Shlenev D.M. ∗∗∗∗ 1Division of Physics, Yaroslavl Higher Military School of Air Defense, Moscow Prospect 28, 150001 Yaroslavl, Russian Federation 2Division of Theoretical Physics, Department of Physics, Yaroslavl State P.G. Demidov University, Sovietskaya 14, 150000 Yaroslavl, Russian Federation

Abstract. The process of the photon splitting, γ γγ, is investigated in → strongly magnetized vacuum with taking into account positronium inﬂuence. The dispersion properties of photons and the new polarization selection rules are obtained. The absorption rate of the leading photon splitting channels are calculated with taking account of the photon dispersion and wave function renormalization.

1 Introduction

The process of photon splitting into two photons, which is forbidden in vacuum by charge conjugation symmetry of QED known as Furry’s theorem, becomes allowed in the presence of external magnetic field. This process was investigated in numerous papers because of it’s possible applications in astrophysics (for detailed review see pioneer work [1] and recent paper [2]). It may play an important role in explanation of peculiarities in the gamma-ray spectra of some radio pulsars, the radio quiescence of soft gamma repeaters (SGRs) and anomalous X-ray pulsars (AXPs) and mechanisms of the SGR burst [3]. Note, that the scale of the magnetic field of the above objects may reach the critical value B = m2/e 4.41 cr × 1013 G 1 or even exceed it [4]. Positronium influence on dispertion properties of photon was investigated in [5] on ex- ample of process ν νγ in strong magnetic field. Analysis shows that positronium effects → may significally change kinemaitics and probability of this process by altering dispertion properties of photon. Therefore, it’s interesting to calculate the probability of photon splitting process in strong magnetic field with taking into account variations of dispertion properties and kinematics due to positronium influence.

∗e-mail: [email protected] ∗∗e-mail: [email protected] ∗∗∗e-mail: [email protected] ∗∗∗∗e-mail: [email protected] 1We use the natural units c = = k = 1, m is the electron mass, e > 0 is the elementary charge.

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 191, 08011 (2018) https://doi.org/10.1051/epjconf/201819108011 QUARKS-2018

2 Photon dispersion with taking into account positronium inﬂuence We begin to consider the process γ γγ with investigation of the photon dispersion prop- → erties. The propagation of an electromagnetic radiation in any active medium is conveniently described in terms of normal modes (eigenmodes). In turn, the polarization and dispersion properties of the normal modes are deﬁned by the eigenvectors and eigenvalues of the polarization operator , respectively [6]. Pαβ Hereafter we use the following notations: the four-vectors with the indices and belong ⊥ to the Euclidean {1, 2} subspace and the Minkowski {0, 3} subspace correspondingly. Then for arbitrary 4-vectors Aµ, Bµ one has

µ µ A = (0, A1, A2, 0), A = (A0, 0, 0, A3), ⊥ (AB) = (AΛB) = A1B1 + A2B2, ⊥ (AB) = (AΛB) = A0B0 A3B3, − where the matrices Λµν = (ϕϕ)µν, Λµν = (˜ϕϕ˜)µν are constructed with the dimensionless 1 ρσ tensor of the external magnetic ﬁeld, ϕµν = Fµν/B, and the dual tensor,ϕ ˜µν = 2 εµνρσϕ . The matrices Λ and Λ are connected by the relation Λµν Λµν = gµν = diag(1, 1, 1, 1), µν µν − − − − and play the roles of the metric tensors in the perpendicular ( ) and the parallel ( ) subspaces ⊥ respectively. It had been shown in [5] that positronium inﬂuence leaves physical polarisation eigenvectors 2 just the same as in magnetized vacuum without positronium contribution.

(1) (qϕ)α (2) (qϕ˜)α εα (q) = ,εα (q) = . (2) q2 2 ⊥ q Eigenvalue for photon of mode 1 also remains unchanged from the case of magnetized vacuum without positronium contribution. This leads to almost vacuum dispertion law for photon of the first mode. α κ(1) = q2 (3) −3π ⊥ Taking into account of positronium influence modifies the eigenvalue for photon of mode 2 with respect to the case of magnetized vacuum without positronium contribution.

2α πνx κ(2) eB H(x) + (4a) −π 1 ν2 x − − q2 q2 1 x x = ,y= ⊥ , H(x) = arctan 1 , 4m2 4m2 √x(1 x) 1 x − − − α eB 2 ν = [ln(4.5u 2.44ln(ln(0.15u))] , u = 32 . 2 − αq2 ⊥ Dispertion law for photon of the mode 2 signiﬁcally diﬀers from the case of magnetized vacuum case without positronium contribution. Single branch splits in two, upper converges to dispertion of photon with unbound electron-positron pair in loop, lower - to positronium spectral line.

2Symbols 1 and 2 correspond to the and polarizations in a pure magnetic ﬁeld in Adler’s notation [1]. ⊥

2 EPJ Web of Conferences 191, 08011 (2018) https://doi.org/10.1051/epjconf/201819108011 QUARKS-2018

In Fig. 1, the dispersion law of the mode 2 photon is depicted as the solution of the equation

q2 κ(2) = 0 (5) − It follows from Eq. (4a) that the eigenvalue of the polarization operator κ(2) becomes large near the electron-positron pair production threshold (see Fig. 1). This means that the renormalization of the wave function for a photon of this polarization should be taken into account: (2) (2) (2) 1 ∂κ (q) ε (q) ε (q) Z , Z− = 1 . (6) α → α 2 2 − ∂ω2 Modifications of the dispertion properties due to positronium influence define new selection rules of photon splitting process. The existence of kinematic area with q2 > 0 for the upper branch of photon of the mode 2 opens new photon splitting channel γ γ γ , which 2 → 1 1 was considered forbidden in the magnetic field, while the splitting channels γ γ γ and 1 → 2 2 γ γ γ allowed in case of magnetized vacuum without positronium contribution, are for- 1 → 1 2 bidden in this area. In the region q2 < 0, polarization selection rules are the same ones as in the magnetized vacuum [1].

3 The probability of the photon splitting

Photon absorption rate due to the photon splitting can be deﬁned in the following way:

g 3 λλ 2 d k Wλ λ λ = λ λ λ ZλZλ Zλ δ(ωλ(k) ωλ (k k ) ωλ (k )) , (7) → 32π2ω M → − − − ω ω λ λ λ where λ - polarisation mode of initial photon, λ and λ - polarisation modes of ﬁnal photons, the factor g = 1 (1/2) δ is inserted to account for the possible identity of the ﬁnal λλ − λλ photons. Here qα = (ωλ, k) - four-momentum of initial photon, qα = (ωλ , k) and qα =

(ωλ , k) - four-momenta of ﬁnal photons. Amplitudes of the photon splitting process in magnetized vacuum without positronium

contribution λ λ λ where obtained earlier in [7]. They can be presented as follows: M →

/ 2 α 3 2 (qϕq)(qϕ˜q) q 1 12 = i 4π H , (8) 2 2 2 1/2 2 M → − π [q q q ] 4m ⊥ ⊥      

/ 2 2 α 3 2 (qq) q q 1 22 = i 4π (qq) H + (qq) H , (9) 2 2 2 1/2 ⊥ 2 ⊥ 2 M → − π [q q q ] 4m 4m ⊥               

2 11 = 1 12(q q) (10) M → M → ↔ Absorption rates of the channels γ γ γ and γ γ γ were calculated numerically 1 → 1 2 1 → 2 2 for the magnetic ﬁeld strenght B = 200Bcr. In Fig. 2 and Fig. 3 they are depicted as functions of the initial photon energy. Analytical expression for the absorption rate of the channel γ γ γ could be obtained 2 → 1 1 from [2] and has the following form:

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3 2 2 2 α q q q 2 W2 11 = Z2 H ⊥ Θ(q ) , → 8π2 4m2 ω F  q2    ⊥ (11)             2 (z) = 2 ln z 1 + z− F −

4 Conclusion

The obtained results show that positronium influence modifies the polarization selection rules in comparison with a case of magnetized vacuum without taking into account of positronium influence. In particular, the splitting channel γ γ γ , which was considered forbidden, is 2 → 1 1 allowed. In the case of the channels γ γ γ and γ γ γ positronium has an ambiguous 1 → 1 2 1 → 2 2 influence on absorption rate. It splits one dispertion branch for photon of the 2 mode in two, but also decreases phase space for these processes. Existing area of q2 < 0 implies additional constraints on probability of process γ γ γ . As a result, absorption rate due to channels 1 → 2 2 γ γ γ and γ γ γ becomes suppressed in comparison with a case of magnetic field 1 → 1 2 1 → 2 2 without positronium contribution.

Acknowledgements We are grateful to A. V. Kuznetsov for useful discussions and valuable remarks. D. S. expresses his deep gratitude to the organizers of the Seminar “Quarks-2018” for warm hos- pitality.

The paper is partially supported by the YSU Initiative Scientiﬁc Research Activity (Project No. AAAA-A16-116070610023-3).

q2 ┴ 4 m2 1.0000

0.9995

0.9990

0.9985

0.9980

0.9975

q2 ║ 0 20 40 60 80 100 4 m2

Figure 1. The photon dispersion of photon of the 2 mode in a strong magnetic ﬁeld (B/Bcr = 200).

4 EPJ Web of Conferences 191, 08011 (2018) https://doi.org/10.1051/epjconf/201819108011 QUARKS-2018

3 2 2 2 α q q q 2 5 = ⊥ Θ , W2 11 2 Z2 H 2 2 (q ) → 8π 4m ω F  q  (11)    ⊥          4     2 (z) = 2 ln z 1 + z− F −

W 3 W 4 Conclusion 0

2 The obtained results show that positronium influence modifies the polarization selection rules in comparison with a case of magnetized vacuum without taking into account of positronium influence. In particular, the splitting channel γ γ γ , which was considered forbidden, is 1 2 → 1 1 allowed.

In the case of the channels γ1 γ1γ2 and γ1 γ2γ2 positronium has an ambiguous 0 → → 0.0 0.5 1.0 1.5 2.0 2.5 3.0 influence on absorption rate. It splits one dispertion branch for photon of the 2 mode in two, ω 2 m but also decreases phase space for these processes. Existing area of q2 < 0 implies additional constraints on probability of process γ γ γ . As a result, absorption rate due to channels 1 2 2 Figure 2. The photon absorption rate due to the channel γ1 γ1γ2 in a strong magnetic field γ γ γ γ γ γ → → 1 1 2 and 1 2 2 becomes suppressed in comparison with a case of magnetic field (B/Be = 200). Dashed line corresponds to the probability in magnetized vacuum without positron- → → 3 2 1 without positronium contribution. ium influence [7]. Here W = (α/π) m 3.25 10 cm− . 0 ·

Acknowledgements 5 We are grateful to A. V. Kuznetsov for useful discussions and valuable remarks. D. S. expresses his deep gratitude to the organizers of the Seminar “Quarks-2018” for warm hos- pitality. 4

The paper is partially supported by the YSU Initiative Scientiﬁc Research Activity 3 (Project No. AAAA-A16-116070610023-3). W W 0 q2 2 ┴ 4 m2 1.0000

0.9995

0 0.9990 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ω 2 m

0.9985 Figure 3. The photon absorption rate due to the channel γ1 γ2γ2 in a strong magnetic ﬁeld → (B/Be = 200). Dashed line corresponds to the probability in magnetized vacuum without positron- 0.9980 3 2 1 ium inﬂuence [7]. Here W = (α/π) m 3.25 10 cm− . 0 ·

0.9975 q2 ║ References 0 20 40 60 80 100 4 m2 [1] S. L. Adler, Ann. Phys. (N.Y.) 67, 599 (1971).

Figure 1. The photon dispersion of photon of the 2 mode in a strong magnetic ﬁeld (B/Bcr = 200). [2] D. A. Rumyantsev, M. V. Chistyakov, N. S. Stus’, Phys. Rev. D 86, 043007 (2012). [3] D. Lai, Rev. Mod. Phys. 73, 629 (2001). [4] R. C. Duncan, C. Thompson, Astrophys. J. 392, L9 (1992). [5] R. A. Anikin, N. V. Mikheev, Journ. of Exp. and Theor. Phys. 115, 411 (2012).

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[6] A. E. Shabad, Tr. Fiz. Inst. Akad. Nauk SSSR 192, 5 (1988). [7] M. V. Chistyakov, A. V. Kuznetsov, N. V. Mikheev, Phys.Lett. B434, 67 (1998); A. V. Kuznetsov, N. V. Mikheev, M. V. Chistyakov, Yad. Fiz. 62, 1638 (1999) [Phys. At. Nucl. 62, 1535 (1999)].