Complete description of polarization effects in the nonlinear Compton scattering II. Linearly polarized laser photons arxiv:hep-ph/0311v 10 Dec 003 D.Yu. Ivanov 1, G.L. Kotkin, V.G. Serbo 1 Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia Novosibisk State University, Novosibirsk, 630090, Russia December 10, 003 Abstract We consider emission of a photon by an electron in the field of a strong laser wave. Polarization effects in this process are important for a number of physical problems. We discuss a probability of this process for linearly polarized laser photons and for arbitrary polarization of all other particles. We obtain the complete set of functions which describe such a probability in a compact form. 1 Introduction Let us consider emission of a photon by an electron in the field of a strong laser wave. The complete description of polarization effects in the case of the circularly polarized laser wave was considered in our paper [1]. The present paper is a continuation of Ref. [1] and we use the same notations 1. Thus, we deal with the process of nonlinear Compton scattering eq + n γ L k eq + γk, 1 when the electromagnetic laser field is described by 4-potential A µ x = A µ cos kx, where A µ is the amplitude of this field. Such a process with absorbtion of n = 1,, 3, 4 linearly polarized laser photons was observed in recent experiment at SLAC []. The method of calculation for such process was developed by Nikishov and Ritus [3]. It is based on the exact solution of the Dirac equation in the external electromagnetic plane 1 Below we shall quote formulas from paper [1] by a double numbering, for example, Eq. 1.1 means Eq. 1 from Ref. [1]. 1
wave. Some particular polarization properties of this process for the linearly polarized laser photons were considered in [3, 4]. In the present paper we give the complete description of the nonlinear Compton scattering for the case of linearly polarized laser photons and arbitrary polarization of all other particles. We follow the method of Nikishov and Ritus in the form presented in [5] 101. In the next section we briefly describe the kinematics. The cross section, including polarization of all particles, is obtained in Sect. 3. In Sect. 4 the polarization of final particles, averaged over azimuthal angle, is obtained. In Sect. 5 we summarize our results and compare them with those known in the literature. In Appendix we give a comparison of the obtained cross section in the limit of weak laser field with the known results for the linear Compton scattering. Kinematics All kinematical relations derived in Sect. of Ref. [1] are valid for the considered case as well. In particular, the parameter of nonlinearity is defined via the mean value of squared 4-potential in the same form ξ = e A mc µ xa µ x = e 1 mc A µa µ, 3 where e and m is the electron charge and mass, c is the velocity of light. Therefore, the amplitude of 4-potential can be presented in the form mc A µ = ξ e µ L e, e Le L = 1, 4 where e µ L is the unit 4-vector describing the polarization of the laser photons. In the frame of reference, in which the electron momentum p is anti-parallel to the initial photon momentum k we call it as a collider system, we direct the z-axis along the initial electron momentum p. For the discussed problem it is convenient to choose the x-axis along the direction of the laser linear polarization, i.e. along the vector e L. Azimuthal angles ϕ, β and β of vectors k and electron polarizations ζ and ζ are defined with respect to this x-axis. After that, the unit 4-vector e L can be presented in the form e L = e 1 sin ϕ e cosϕ, 5 where the unit 4-vectors e 1 and e are given in 1.19. This equation determines the quantities sin ϕ and cos ϕ in an invariant form, suitable for any frame of reference. As in paper [1], we use the Stokes parameters ξ i and ξ i to describe the polarization of the initial photon and the detector polarization of the final photon, respectively. They are defined 1/. Note, that our definition of this parameter differs from that used in Refs. [3] by additional factor
with respect to the x y z -axes which are fixed to the scattering plane. The x -axis is the same for both photons and perpendicular to the scattering plane: the y -axes are in that plane, in particular, for the initial photon and x k k ; 6 y k k k = ω k 7 y k k k 8 for the final photon. The azimuthal angle of the linear polarization of the laser photon in the collider system equals zero with respect to the xyz-axes, and it is ϕ π/ with respect to the x y z -axes. Therefore, in the considered case of 100 % linearly polarized laser beam one has ξ 1 = sin ϕ, ξ = 0, ξ 3 = cos ϕ. 9 At small emission angles of the final photon θ γ 1, the final photon moves almost along the direction of the z-axis and k k azimuth is approximately equal to ϕ π/. Let ˇξ be the detected Stokes parameters for the final photons, fixed to the xyz-axes. They are connected with ξ by the relations ξ 1 ˇξ 1 cos ϕ + ˇξ 3 sin ϕ, ξ = ˇξ, ξ 3 ˇξ 3 cos ϕ ˇξ 1 sin ϕ. 10 The polarization properties of the initial and final electrons are described in the same form as in paper [1]. 3 The effective cross section As in paper [1], we present the effective differential cross section in the form 3 dσξ i, ζ = r e 4x n F n dγ n, dγ n = δq + n k q k d3 k ω d 3 q q 0, 11 where r e = α/m is the classical electron radius, and F n = F n 0 + 3 =1 We have calculated functions F n F n, G n ξ + G n and i ζ + 3 i,=1 i ζ i ξ. 1 using the standard technic presented in [5] 101. All the necessary traces have been calculated using the package MATEMA- TIKA. In the considered case almost all dependence on the nonlinearity parameter ξ 3 Below we use the system of units in which c = 1, h = 1. 3
accumulates in three functions: f n = 4 [A 1 n, a, b] 4A 0 n, a, ba n, a, b, g n = 4n z n [A 0 n, a, b], 13 h n = 4n a A 0n, a, b A 1 n, a, b, where functions A k n, a, b were introduced in [3] as follows A k n, a, b = π π cos k ψ exp [ i nψ a sin ψ + b sin ψ] dψ π. 14 The arguments of these functions are a = e Ap kp Ap, b = 1 1 kp 8 e A kp 1 kp 15 or in more convenient variables they are a = ξ m el p kp e Lp, b = kp y x ξ. 16 In the collider system one has where a = z n cos ϕ, 17 z n = ξ 1 + ξ n s n was defined in 1.41. Among the functions A 0 n, a, b, A 1 n, a, b and A n, a, b there is a useful relation n b A 0 n, a, b a A 1 n, a, b + 4bA n, a, b = 0. 18 To find the photon spectrum, one needs also functions 13 averaged over the azimuthal angle ϕ: f π dϕ π n = f n 0 π, g dϕ n = g n 0 π. 19 For small values of ξ 0 or y 0, we have a y ξ, b y ξ, 0 A 0, n, a, b y ξ n/, A1 n, a, b y ξ n 1/, f n, g n, h n y ξ n 1, 4
in particular, at ξ = 0 or at y = 0 f 1 = f 1 = g 1 = h 1 = 1, g 1 = 1 + cos ϕ. 1 The results of our calculations are the following. First, we define the auxiliary functions where we use the notation X n = f n 1 + c n [ 1 r n g n h n cos ϕ ], Y n = 1 + c n g n h n cos ϕ, V n = f n cos ϕ + 1 + c n [ 1 r n g n h n cos ϕ ] sin ϕ, = ξ 1 + ξ, c n and r n is defined in 1.10 and 1.11, respectively. The item F n 0, related to the total cross section 1.35, reads F n 0 = y + y f n s n 1 + ξ g n. 3 The polarization of the final photons ξ f is given by Eq. 1.36 where F n 1 = X n sin ϕ, F n 3 = V n + s n 1 + ξ g n, 4 F n = 1 + ξ The polarization of the final electrons ζ f G n 1 = G n = [ hn ζ 1 sin ϕ Y n ζ ] + y f n s n 1 + ξ g n y ζ 3, is given by Eqs. 1.37, 1.38 with f n s n 1 + ξ g n ζ 1 + hn ζ 1 + ξ 3 sin ϕ, f n s n 1 + ξ g n ζ 1 + ξ Y nζ 3, 5 G n 3 = 1 + ξ y + y + hn ζ 1 sin ϕ Y n ζ f n + y s n 1 + ξ g n ζ 3. At last, the correlation of the final particles polarizations are 11 = 1 = y + y y X n ζ 1 sin ϕ y V n s n 1 + ξ g n ζ Y 1 + ξ nζ 3, [ ] y yv n s n 1 + ξ g y + y n ζ 1 + X n ζ sin ϕ 5
31 = 1 = 3 = 13 = 3 = + 1 + ξ h n ζ 3 sin ϕ, Yn ζ 1 + ξ 1 h n ζ sin ϕ + X n ζ 3 sin ϕ, hn sin ϕ, 1 + ξ = 1 + ξ Y n, [ ] y y f n s n 1 + ξ g n, 6 y + y V n s n 1 + ξ g y y n ζ 1 X nζ sin ϕ + h 1 + ξ n ζ 3 sin ϕ, y y y + y X nζ 1 sin ϕ V n + y + Y 1 + ξ nζ 3, s n 1 + ξ g n ζ 33 = hn ζ 1 + ξ 1 sin ϕ + Y n ζ V n s n 1 + ξ g n ζ 3. 4 Averaged polarization of the final particles In many application it is important to know the averaged over azimuthal angle ϕ polarization of the final photon and electron in the collider system. To find it, we can use the same method as for the linear Compton scattering see Refs. [9], [7] and [8]. We substitute ξ, ζ, ξ and ζ from Eqs. 9, 1.51, 10 and 1.56, respectively, into Eq. 1 and obtain after integration over azimuthal angle ϕ: where dσ n ξ i, ζ dy +G n πr e x ζ ζ + Gn F n 0 = n F 0 + Φ n 1 ˇξ 1 n + F ˇξ + Φn 3 ˇξ 3 + 7 3 ζ 3ζ 3 + ζ i ξ, i,=1 i 1 + f n s n 1 + ξ g n, Φ n 1 = F n 1 cos ϕ F n 3 sin ϕ = 0, F n y = f n s n 1 + ξ g n y ζ 3, Φ n 3 = F n 1 sin ϕ F n 3 cos ϕ, G n = f n s n 1 + ξ g n, 6
G n = y + y f n + y s n 1 + ξ g n, As a result, the averaged Stokes parameters of the final photon are n ˇξ f f F f 1n 0, ˇξ n, ˇξ F n 3n 0 Φn 3 F n 0, 8 and the averaged polarization of the final electron is ζ f n Gn F n 0 ζ, ζ f 3n Gn F n 0 ζ 3. 9 Note that averaged Stokes parameters of the final photon do not depend on ζ and that averaged polarization vector of the final electron is not equal zero only if ζ 0. These properties are similar to those in the linear Compton scattering. 5 Summary and comparison with other papers Our main result is given by Eqs. 3 6 which present 16 functions F 0, F, G and H i with i, = 1 3. They describe completely all polarization properties of the nonlinear Compton scattering in a rather compact form. In the literature we found 4 functions which can be compared with ours F 0, F 1, F, F 3. They enter the total cross section 1.35, 1.54, the differential cross sections 1.53, 1.54 and the Stokes parameters of the final photons 1.36. The function F 0 was calculated in [3], the functions F were obtained in [4]. Our results 3, 4 coincide with the above mentioned ones. The polarization of the final electrons is described by functions G 5. They enter the polarization vector ζ f given by exact 1.38, 1.61 and approximate 1.67 equations. The correlation of the final particles polarizations are described by functions H i given in Eqs. 6. We did not find in the literature any on these functions. At small ξ all harmonics with n > 1 disappear due to properties 0 and 1, dσ n ξ i, ζ ξn 1 at ξ 0. 30 We checked that in this limit our expression for dσξ i, ζ coincides with the result known for the linear Compton effect, see Appendix. Acknowledgements We are grateful to I. Ginzburg, M. Galynskii, A. Milshtein, S. Polityko and V. Telnov for useful discussions. This work is partly supported by INTAS code 00-00679 and RFBR code 0-0-17884; D.Yu.I. acknowledges the support of Alexander von Humboldt Foundation. 7
Appendix: Limit of the weak laser field At ξ 0, the cross section 11 has the form where dσξ i, ζ = r e 4x F dγ; dγ = δp + k p k d3 k ω d 3 p E, 31 F = F 0 + 3 3 F ξ + G ζ + H i ζ i ξ. 3 =1 To compare it with the cross section for the linear Compton scattering, we should take into account that the Stokes parameters of the initial photon have values 9 and that our invariants c 1, s 1, r 1 and auxiliary functions transforms at ξ = 0 to Our functions i,=1 c 1 c = 1 r, s 1 s = r1 r, r 1 r = X 1 c, Y 1 c1 ξ 3, V 1 ξ 3. y x, 33 1 F 0 = + s 1 ξ 3, F 1 = cξ 1, F 3 = s + 1 + c ξ 3, 34 1 F = yscζ + y + c ζ 3 ys ζ 1 ξ 1 + ys cζ + sζ 3 ξ 3 coincide with those in [7]. Our functions G 1 = 1 + c + s ξ 3 ζ s ξ 1ζ 3, G = 1 + c + s ξ 3 ζ ysc 1 ξ 3 ζ 3, 35 [ ] 1 G 3 = ysξ 1 ζ 1 + ysc1 ξ 3 ζ + 1 + y c + s ξ 3 ζ 3. coincide with functions Φ given by Eqs. 31 in [8]. At last, we check that our functions H 11 = H 1 = y y + y c ζ 1 ξ 1 + y [ ] s + + c ξ 3 ζ1 + [ ] y ξ 3 + s 1 ξ 3 ζ ysc1 ξ 3 ζ 3, y + y cξ 1 ζ ysξ 1 ζ 3, H 31 = ycs 1 ξ 3ζ 1 + ys ξ 1ζ + cξ 1 ζ 3, H 1 = ys H = ycs 1 ξ 3, H 3 = y [ y s 1 ξ 3 ], H 13 = s ζ 1 + 1 + c + y ξ 3 ζ c y ξ 1 ζ ys ξ 1 ζ 3, 8 ξ 1, 36
= y y c ξ 1 ζ 1 + + y s ζ + 1 + c + y c ys H 33 = ξ 1 ζ sc 1 ξ 3 ζ + s ζ 3 + 1 + c ξ 3 ζ 3 H 3 ξ 3 ζ + ysc1 ξ 3 ζ 3, coincide with the corresponding functions from [6]. At such a comparison, one needs to take into account that the set of unit 4-vectors, used in our paper see Eqs. 1.19, 1.3, and one used in paper [6] are different. The relations between our notations and the notations used in [6] are given by Eqs. 1.75. References [1] D.Yu. Ivanov, G.L. Kotkin, V.G. Serbo, Complete description of polarization effects in the nonlinear Compton scattering. I. Circularly polarized laser photons. hep-ph/031035. [] C. Bula, K. McDonald et al., Phys. Rev. Lett. 76, 1996 3116. [3] A.I. Nikishov, V.I. Ritus, Zh.Eksp.Teor.Fiz. 46, 776 1964; Trudy FIAN 111 1979 Proc. Lebedev Institute, in Russian. [4] Ya.T. Grinchishin, M.P. Rekalo, Yad. Fiz., 40 1984 181. [5] V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii, Quantum electrodynamics Nauka, Moscow 1989. [6] A.G. Grozin, Proc. Joint Inter. Workshop on High Energy Physics and Quantum Field Theory Zvenigorod, ed. B.B. Levtchenko, Moscow State Univ. Moscow, 1994 p. 60; Using REDUCE in high energy physics Cambridge, University Press 1997. [7] I.F. Ginzburg, G.L. Kotkin, S.L. Panfil, V.G. Serbo, Yad. Fiz. 38, 101 1983; I.F. Ginzburg, G.L. Kotkin, S.L. Panfil, V.G. Serbo, V.I. Telnov, Nucl. Instr. Meth. 19, 5 1984. [8] G.L. Kotkin, S.I. Polityko, V.G. Serbo, Nucl. Instr. Meth. A 405, 30 1998. [9] I.I. Goldman, V.A. Khoze, ZhETF 57, 918 1969; Phys. Lett. B 9, 46 1969. 9