the initial and nal photons. These selection rules are based on the collinear inematics of photon splitting in the case of the subcritical magnetic el

Transcription

1 Photon Splitting in a Strong Magnetic Field M.V. Chistyaov, A.V. Kuznetsov and N.V. Miheev Division of Theoretical Physics, Department of Physics, Yaroslavl State University, Sovietsaya 14, Yaroslavl, Russian Federation Abstract The process of photon splitting! in a strong magnetic eld is investigated both below and above the pair creation threshold.the partial amplitudes and the splitting probabilities are calculated taing account of the photon dispersion and large radiative corrections near the resonance. The non-linear electromagnetic process of the photon splitting in external eld is one of the most exiting eect of the quantum electrodynamics. It is interesting to note that possible manifestations of this process belong to the two completely oposite elds of our physical world. On one hand this eect could manifest itself in the microcosm, in the electric eld of a heavy atoms. On the other hand, possible manistations of this process in macrocosm are discussed during a long time, namely, in the astrophysical objects, where strong magnetic elds could exist. In this tal we consider the procees of the photon splitting in the strong magnetic eld. Theoretical study of this process has a rather long history [1,, 3]. Adler was the rst who considers the photon splitting as a possible mechanism of the production linearly polarized photons in a pulsar magnetic eld. In his wor the comprehensive investigation of this process was carried out for the magnetic eld below the critical value, B e = m =e ' 4: G (m is the electron mass, e > 0 is the elementary charge). There were found the selection rules connected whith the defenite polarization states of 1

2 the initial and nal photons. These selection rules are based on the collinear inematics of photon splitting in the case of the subcritical magnetic eld. The recent progress in astrophysics has drawn attention again to the photon splitting induced by a magnetic eld. It has found application in the study of the spectral formation of gamma ray burst (GRB) from neutron stars and has also been used in models of soft gamma ray repeaters (SGR), where it soften the photon spectrum. It is supposed for these objects to have very strong magnetic eld which can exceed essentialy the critical value. The good review on this topic could be found in [4]. Therefore the study of the photon splitting has been continued. In some recent papers [5, 6] this process was studied in the magnetic eld much greater than the critical value. In our opinion there are some facts which are not taen into account in this wors. The limit of the collinear inematics is not aplicable in the strong magnetic eld (B B e ). Therefore ones cannot use Adler's selection rules. The radiative corrections can give a large contribution near the pair creation threshold. In this tal, photon splitting in a strong magnetic eld is investigated both below and above the pair-creation threshold, with taing account of the noncollinearity of the inematics and large radiative corrections. 1 The inematic of photon splitting. The inematic of the photon splitting in magnetic eld is dened by the vacuum polarization. The dispersion relations for the photons of dierent polarizations could be found from the following equations! () ; ;P =0 (1) where P are the eigenvalues of the photon polarization operator. Only two of these equations correspond to the real photons with the polarization vectors " () = (') q " (?)? = ( ~') q ()

3 These are the so-called parallel and perpendicular modes in Adler's notation. Here ' = F =B ~' = 1" ' are the dimensionless tensor of the external magnetic eld and the dual tensor, =! ; 3? = 1 +. Magnetic eld is directed along the third axes, B =(0 0 B). The corresponding eigenvalues P are singular at the points of the socalled cyclotron resonances. In the limit of the strong magnetic eld (B B e )thelowest of them is only essential and corresponds to the pair creation threshold, = 4m p. Therefore we shall further analyse the inematical region < (m + m +eb). In this region P tae the following oneloop approximate form P '; 3 P? ';! 4m eb H (3) where H(z) = H(z) =; 1 z 1 p arctan p ; 1 z>1 z ; 1 z ; 1 z p ln 1+p! 1 ; z 1 ; z 1 ; p 1 ; z ++i z p z<1: 1 ; z One can consider the eigenvalues of the polarization operator as a photon eective masses squared, m eff. There is the eective mass of photon which denes the inematics of the photon splitting. This process is inematically allowed only when one of the nal photons of the?-mode is created in the inematic region < 4m. Realy, if all photons are in the region > 4m the limit of the collinear inematic is realized because of the relatively small (jm effj! ) eective massesof the photons in this region. The photon of the?-mode aquares positive and the photon of the -mode negative eective mass squared. It means, that only?-mode could split. On the other hand it is nown, see [], that only transition!??is allowed in the collinear inematic. Therefore the photon splitting is forbidden when > 4m for all photons in the procces. If the nal photon of the?-mode is created in the region < 4m the inematic of photon splitting could be far from collinearity. This is due to the fact that the photon of this mode aquares signicant eective mass, m eff < 0 jm effj!, in this region. Moreover, the large value of the polarization operator near the resonance 3

4 leads to the neccessity of taing acount a large radiative corrections which reduce to renormalization of photon wave-function. " () q! " () Z Z ;1 =? The Amplitude of the Process! in the Strong Magnetic Field. The process of the photon splitting in the magnetic eld is depicted by two Feinman diagrams, see the Fig.1. The amplitude of this process can be written in the following form M = e 3 Z d 4 Xd 4 Y Spf^"() ^S(Y )^"( 00 ) ^S(;X ; Y )^"( 0 ) ^S(X)g e ;ie (XFY )= e i(0 X; 00 Y ) +("( 0 ) 0 $ "( 00 ) 00 ) (5) X = z ; x Y = x ; y where = (! ) is the 4-vector of the momentum of initial photon with the polarization vector ", 0 and 00 are the 4-momenta of nal photons, S(X) is the electron propagator in the magnetic eld [7]. In the case of the strong eld it is advantegeous to use the asymptotic expression of the electron propagator which could be presented in the following form ^S(X)' S a (X)= ieb exp(;ebx? 4 d p = p 0 p 3 ; = 1 (1 ; i 1 ): Z ) d p () (p) + m p ; m ;e ;i(px) (6) Substituting this in Eq. (5) and integrating, one would expect to obtain the amplitude which depends linearly on the eld strength, namely, as B 3 =B, where B in the denominator arises from the integration over d X? d Y?. However, two parts of the amplitude (5) cancel each other exactly. Thus, the asymptotic form of the electron propagator (6) only shows that the linearon-eld part of the amplitude is zero and provides no way of extracting the next term of expansion over the eld strength. As the analysis shows, that could be done by the insertion of two asymptotic (6) and one exact propagator ^S(x) into the amplitude (5), with all 4

5 interchanges. It is worthwhile now to turn from the general amplitude (5) to the partial amplitudes corresponding to denite photon modes, and?, which are just the stationary photon states with denite dispersion relations in a magnetic eld. There are 6 independent amplitudes and only two of them are of physical interest. We have obtained the following expressions, to the terms of order 1=B 3= ( M 0 ' 00 )( 0 ~' 00 ) 4m!? = i4 [( 0 ) ( 00 H )??] 1= ( 0 ) 3= ( M 0!?? = i4 e 00 ( ) [( 0 ) ( 00 ( 00 )H )?] 1= + ( 0 )H 4m ( 00 )!) As for remaining amplitudes, we note that M!! (7) 4m ( 0 ) (8)! is equal to zero in this approximation. On the other hand, the photon of the? mode due to its dispersion can split into two photons only in the inematic region > 4m where the tree-channel?! e + e ; [8] strongly dominates. 3 The Probability of the Photon Splitting. Although the process involves three particles, its amplitude is not a constant, because it contains the external eld tensor in addition to the photon 4- momenta. The general expression for the splitting probability can be written in the form W! 0 00 = g Z jm! 3! 00j 0 Z Z 0Z 00 (! () ;! 0( 0 ) ;! 00( ; 0 )) d3 0 (9)! 0! 00 where the factor g =1; is inserted to account for possible identity of the nal photons. The factors Z account for the large radiative corrections which reduce to the wave-function renormalization of a real photon with definite dispersion! =! (). The integration over phase space of two nal photons in Eq. (9) has to be performed using the photon energy dependence 5

6 on the momenta,! =! (), which can be found from the dispersion equations (1). A calculation of the splitting probability (9) is rather complicated in the general case. In the limit m! sin eb, where is an angle between the initial photon momentum and the magnetic eld direction, we derive the following analytical expression for the probability of the channel!? : W!? ' 3! sin (1 ; x)[1 ; x +x + (1 ; x)(1 + x) ln(1 + x) ; 16 ; x ; x 1 ; x ln 1 m ] x = x! sin 1: (10) Within the same approximation we obtain the spectrum of nal photons in the frame where the initial photon momentum is orthogonal to the eld direction: dw!? d! 0 ' 3 q! 0 +! ; m! <!0 <!; m (! ;! 0 ) ; 4m (11) q(! 0 ;!) ; 4m where!! 0 are the energies of the initial and nal photons of the mode. We have made numerical calculations of the process probabilities for both channels, which are valid in the limit! sin eb. Our results are represented in Figs.,3. The photon splitting probabilities below and near the pair-creation threshold are depicted in Fig.. In this region the channel!?? (allowed in the collinear limit) is seen to dominate the channel!? (forbidden in this limit). For comparison we show here the probability obtained without considering the noncollinearity of the inematics and radiative corrections (the dotted line 3 )which is seen to be inadequate. For example, this probability becomes innite just above the threshold. As is seen from Fig. 3, both channels give essential contributions to the probability at high photon energies, with the \forbidden" channel dominating. It should be stressed that taing account of the photon polarization leads to the essential dependence of the splitting probabilities on the magnetic eld, while the amplitudes (7), (8) do not depend on the eld strength value. 6

7 4 Conclusions The collinear limit is inadequate approximation for the photon splitting! in a strong magnetic eld (B B e ), because of the signicant deviation of the photon dispersion in the strong eld from the vacuum dispersion. The \allowed" channel!?? is not comprehensive for the splitting in the strong eld. The \forbiden" channel!? is also essential, moreover, it dominates at high energies of the initial photon. The partial amplitudes and the splitting probabilities are calculated in the strong eld limit for both channels. The amplitudes do not depend on the eld strength value, while the probabilities do depend essentially, due to the photon polarization in the strong magnetic eld. This wor was supported in part by the INTAS Grant N and by the Russian Foundation for Basic Research Grant N References [1] S.L. Adler, J.N. Bahcall, C.G. Callan and M.N. Rosenbluth, Phys. Rev. Lett. 5 (1970) 1061 Z. Bialynica-Birula and I. Bialynica-Birula, Phys. Rev. D (1970) 341. [] S.L. Adler, Ann. Phys. N.Y. 67 (1971) 599. [3] V.O. Papanian and V.I. Ritus, Sov. Phys. JETP 34 (197) 1195, 38 (1974) 879 R.J. Stoneham, J. Phys. A1 (1979) 187 V.N. Baier, A.I. Milstein and R.Zh. Shaisultanov, Sov. Phys. JETP 63 (1986) 66. [4] A.C. Harding, M.G. Baring and P.L. Gonthier, Astrophys.J. 476 (1997) 46. [5] V.N. Baier, A.I. Milstein and R.Zh. Shaisultanov, Phys. Rev. Lett. 77 (1996) [6] C. Wile and G. Wunner, Phys. Rev. D55 (1997)

8 [7] J. Schwinger, Phys.Rev V.8. P.664. [8] N.P. Klepiov, Zh. Esp. Teor. Fiz. 6 (1954) 19. () ; ; ; ; ; x ' & u $ ;;;;; % u Q Q z y ( 0) ( 00) + ( 0 $ 00) Figure 1: The Feynman diagram for photon splitting in a magnetic eld. The double line corresponds to the exact propagator of an electron in an external eld. 8

9 0. W W 0 sin 3 a 0.15 b 0.1 1a b ! sin m Figure : The dependence of the probability of photon splitting! on energy, below and near the pair-creation threshold: 1a 1b { for the \forbidden" channel!? and for the magnetic eld strength B =10 B e and 10 3 B e, correspondingly a b { for the \allowed" channel!?? for the eld strength B = 10 B e and 10 3 B e 3 { for the channel!?? in the collinear limit without taing account of large radiative corrections. Here W 0 =(=) 3 m. 9

10 W W 0 sin 0 1a 15 1b 10 5 b a ! sin m Figure 3: The probability of photon splitting above the pair-creation threshold: 1a 1b {for the \forbidden" channel!? and for the magnetic eld strength B =10 B e and 10 3 B e, correspondingly a b { for the \allowed" channel!?? for the eld strength B =10 B e and 10 3 B e. 10