Sin light of electron in matter Alexander Grigoriev a,b, Sergey Shinkevich a, Alexander Studenikin a,b, Alexei Ternov c, Ilya Trofimov a a Deartment of Theoretical Physics, arxiv:he-h/0611103v1 8 Nov 006 Moscow State University, 11999 Moscow, Russia b Skobeltsyn Institute of Nuclear Physics, Moscow State University, 11999 Moscow, Russia c Deartment of Theoretical Physics, Moscow Institute for Physics and Technology, 141700 Dolgorudny, Russia Abstract We further generalize the owerful method, which we have recently develoed for descrition of the background matter influence on neutrinos, for the case of an electron moving in matter. On the basis of the modified Dirac equation for the electron, accounting for the standard model interaction with articles of the background, we redict and investigate in some detail a new mechanism of the electromagnetic radiation that is emitted by moving in matter electron due to its magnetic moment. We have termed this radiation the sin light of electron in matter and redicted that this radiation can have consequences accessible for exerimental observations in astrohysical and cosmological settings. 1 Introduction In a series of our aers [1 4] we have develoed a rather owerful method of investigation of different henomena that can aear when neutrinos and electrons move in the background matter. The method discussed is based on the use of the modified Dirac equations for the articles wave functions, in which the corresondent effective otentials, that account for the matter influence on articles, are included. It is similar to the Furry reresentation [5] in quantulectrodynamics, widely used for descrition of articles interactions in the resence of external electromagnetic fields. In [1 4] we aly the discussed method for elaboration of the quantum theory of the sin light of neutrino in matter. The sin light of neutrino in matter ax.grigoriev@mail.ru shinkevich@gmail.com studenik@srd.sin.msu.ru a ternov@mail.ru 1
(SLν), one of the four new henomena studied in our recent aers [6], is a new tye of electromagnetic radiation that can be emitted by a massive neutrino (due to its magnetic moment) when the article moves in the background matter. Within quasi-classical treatment the existence of this radiation was roosed and studied in [7], while the quantum theory of this henomenon was develoed in [1 4,8]. It has been shown [1] how the aroach, develoed at first for descrition of a neutrino motion in the background matter, can be sread for the case of an electron roagating in matter. The modified Dirac equation for an electron in matter has been derived [1] and on this basis we have considered the electromagnetic radiation that can be emitted by the electron (due to its magnetic moment) in the background matter. We have termed this radiation as the sin light of electron in matter. It should be noted here that the term sin light was introduced in [9] for designation of the articular sin-deendent contribution to the electron synchrotron radiation ower. Modified Dirac equation for electron in matter Let us consider an electron having the standard model interactions with articles of electrically neutral matter comosed of neutrons, electrons and rotons. Indeed, we account below only for the neutron comonent that can be used as an abrut model for modelling a real situation existed when electrons move in nuclear matter of a neutron star (see, for instance, [10]). We suose that there is a macroscoic amount of the background articles in the scale of an electron de Broglie wave length. Then the addition to the electron effective interaction Lagrangian is eff = f µ( 1 4sin θ W +γ 5 ) ēγ µ e, (1) L (e) where the exlicit form of f µ deends on the background articles densities, seeds and olarizations. The modified Dirac equation for the electron wave function in matter is [1] {iγ µ µ 1 γ µ(1 4sin θ W +γ 5 ) f µ } Ψ e (x) = 0, () where for the case of electron moving in the background of neutrons f µ = G F (n n,n n v). (3) Here n n is the neutrons number density and v is the seed of the reference frame in which the mean momentum of the neutrons is zero. The solution of equation () can be found in analogy with the case of neutrino (for details see [,3]). The comlete set of the electron wave functions in the matter
is: Ψ ε,,s (r,t) = e i(eεt r) L 3/ s m 1+ e 1+ sε 1 ε 1 1+s 3 E ε cα n 1 s 3 e iδ E ε cα n E ε cα n 1+s 3 E ε cα n 1 s 3 e iδ, (4) where L is the normalization length, δ = arctan( / 1 ) and the so called matter density arameter α n is defined as n n α n = G F. (5) The electron energy in matter ( E ε (e) = ε m ) e 1 sα n +me +cα n (6) deends on the electron helicity s = ±1 and the quantity ε = ±1 which slits the solutions into ositive- and negative-frequency branches. 3 Sin light of electron in matter To the lowest order of the erturbation theory the corresonding quantum rocess is described by the one-hoton emission diagram with the initial ψ i and final ψ f electron states and with the vertex corresonding to the standard electromagnetic interaction with the hoton due to the electron charge e. Thus, the transition amlitude is giving by S fi = ie 4π d 4 x ψ f (x)(γ µ e e ikx µ) ψ i(x), (7) ωl 3 where k µ = (ω,k) and e µ are the hoton momentum and olarization vector, resectively. Choosing the three-dimensional transversal gauge and erforming integration over the time variable, we get π S fi = ie ωl 3πδ(E E ω) d 3 x ψ f (r)(γe )e ikr ψ i (r), (8) The δ-function in the last exression stands for the energy conservation law with E and E being energies of the initial and final neutrino states. Performing further integration over the satial coordinates we get also the momentum conservation law, which together with the energy conservation law, E = E +ω, = +k, (9) leads to the only ossible transition of the electron when its chirality changes from s i = 1 to s f = 1. The corresonding hoton energy then is given by the exression [1]: ω = α n [Ẽ (+α n )cosθ] (Ẽ cosθ) (α n ), (10) 3
where θ is the angle between the directions of initial neutrino and emitted hoton roagation. We also use the following notation Ẽ = E cα n. In the case of relativistic electrons and small values of the matter density arameter α n the hoton energy is 1 ω SLe = 1 β e cosθ ω 0, ω 0 = G F n n β e, (11) here β e is the electron seed in vacuum. From this exressions we conclude that for the relativistic electron the energy range of the SLe may even extend u to energies eculiar to the sectrum of gamma-rays. We also redict the existence of the electron-sin olarization effect in this rocess. 4 Rate and ower of the radiation With the exressions for the amlitude (8) and the emitted hoton energy (10) we arrive to general formulas for the total rate and ower of the radiation: Γ = e π ω (1 0 1+ β β e β e ) ey Ẽ Ẽ (1 ycosθ)sinθdθ, (1) I = e π ω (1 1+ β β e β e ) ey Ẽ Ẽ (1 ycosθ)sinθdθ, (13) 0 where β e = +α n, β e = α n, (14) Ẽ Ẽ are the quantities, describing the initial and final electron grou velocities, and E = E ω, = K e ω, K e = Ẽ cosθ, y = ω cosθ α n. Taking integration in (1) and (13) we obtain: m e [(1+α n ) + ] Γ = e 4 (4α n + ) (+ α n ) +m e [ (4α n + ) ln ( 1+ 4α ] n) 4αn ( +6α n ) (15) (16) and I = e m { ( e (4α n + ) 3 (1+α n)m e + ) ( (4α n + ) 3 ln 1+ 4α ) n 4 [ 3 α n 88α n 4 +3(1+α n )m4 e +30α n(1+α n )m3 e ]} + m ( e 3+88α n (1+α n )) +α n (15+16α n ) 3. (17) 4
Let us estimate the total rate Γ of the radiation, and also the corresonding lifetime T SLe of the electron in resect to the considered rocess using exression (16). Consider the electron with momentum = 1 MeV moving in matter characterized by the number density n n 10 37 cm 3, in this case the matter density arameter is α n = 0.6 10 6. Then, for the rate of the rocess we get Γ 3. 10 10 MeV which corresonds to the characteristic life-time of the electron T SLe 10 s. Finally, we have develoed the aroach to descrition of the matter influence on an electron which is based on the exact solutions of the corresondent modified Dirac equation for the article wave function. The aroach develoed (we have reviously used it for the case of neutrino) is similar to the Furry reresentation in quantum electrodynamics. Note that our focus has been on the standard model interactions of electrons with the background matter. A similar aroach, which imlies the use of the exact solutions of the corresondent modified Dirac equations, can be develoed in the case when electrons interact with different external fields redicted within various extensions of the standard model (see, for instance, [11, 1] and the aer of V.Zhukovsky et al in this book). We have redicted and investigated in some detail a new tye of electromagnetic radiation (the sin light of electron in matter, SLe) that canbeemitted bytheelectron dueits magneticmoment withinthestandardmodelof interaction with the background matter. The obtained SLe energy sectrum shows that for ultra-relativistic electrons it can even extend u to energy range eculiar to the sectrum of gamma-rays. Comaring the rates of the sin light of neutrino and sin light of electron in matter, we have redicted (see also [1]) that the latter is more effective then the former. We have redicted also that the SLe emitted by ultra-relativistic electrons moving in dense astrohysical and cosmological media can have consequences accessible for exerimental observations. References [1] A.Studenikin, J.Phys.A: Math. Gen. 39 (006) 6769. [] A.Studenikin, A.Ternov, Phys.Lett.B 608 (005) 107. [3] A.Grigoriev, A.Studenikin, A.Ternov, Phys.Lett.B 6 (005) 199; Grav.Cosm. 11 (005) 13. [4] A.Grigoriev, A.Studenikin, A.Ternov, Phys.Atom.Nucl. 69 (006) 1940. [5] W.Furry, Phys.Rev. 81 (1951) 115. [6] A.Studenikin, Nucl.Phys.B (Proc.Sul.) 143 (005) 570. [7] A.Lobanov, A.Studenikin, Phys.Lett.B 564 (003) 7; ibid. 601 (004) 171; M.Dvornikov, A.Grigoriev, A.Studenikin, Int.J Mod.Phys.D 14 (005) 309. [8] A.Lobanov, Phys.Lett.B 619 (005) 136; Dokl.Phys. 50 (005) 86. [9] I.M.Ternov, Sov.Phys.Us. 38 (1995) 405; V.A.Bordovitsyn, I.M.Ternov, V.G.Bagrov, Sov.Phys.Us. 38 (1995) 1037. [10] A.Kusenko, M.Postma, Phys.Lett.B 545 (00) 38. [11] D.Colladay, V.A.Kostelecky, Phys.Rev.D 55 (1997) 6760; Phys.Rev.D 58 (1998) 1160. [1] V.Zhukovsky, A.Lobanov, E.Murchikova, Phys.Rev.D 73 (006) 065016. 5