About the Processes with two Fermions and two
Photons in an Intensive Laser wave (Nonlinear and
Spin Effects)
Sergey M. Sikach
Institute of Physics National Academy of Sciences of Belarus, e-mail: [email protected]
The pattern of calculation of amplitudes of a series of processes in the field of
an intensive laser wave, in which two fermions p; p and two real photons k1 ; k2 participate, is considered. In relation to one-photon processes, these processes are of
the second order on α if the wave intensity ξ
1 (i.e., actually absorption from the
wave only one quantum). Otherwise, they are competing and essentially nonlinear. Onephoton processes have a number of the important physical applications. For example,
γ e and γγ colliders work on their basis. Some of these processes calculated in [1]
in Diagonal Spin Basis (see [2] in this issue). Two-photons back Compton-effect in
intensive laser wave occurs together with one-photon process and thus influences the
formation of spectr and polarisation of photon beam. The amplitudes of two-photon
annihilation of a fermion pair in an intensive laser wave are applicable for calculating of
induced decay of orthopositronium into two photons. And in this approach there is no
necessity to assume the relative momentum equal to zero.
In DSB the calculation is conducted at the level of reaction amplitudes. It essentially
simplifies both the calculation and the form of obtained results; those combinations of
amplitudes which describe the spin effects are easy to calculate. And these effects are
especially essential in nonlinear processes. The calculations are conducted in covariant
form. Besides compactness, this provides independence of the frames of reference and
energy modes. The masses of fermions are taken into account precisely and are not
assumed equal each other (heavy leptons, modes of various decays).
The process under consideration is described by 2 5 amplitudes. Because of parity
conservation, we need to calculate 16 amplitudes (in our approach, they are described
by 8 formulas). Therefore, here we can present only the pattern of calculation. The
necessary formulas are cited in accordance with monograph [3] with the indication
paragraph number,where §40 “An electron in the field of the plane electromagnetic
wave”; §89 “An annihilation of a positronium”; §101 “Radiation of a photon by an
electron in the field of an intensive electromagnetic wave”. Unlike [3], we assume
γ5 iγ 0 γ 1 γ 2 γ 3 (opposite sign) and elementary charge e 0.
Process Laser e e γ1 γ2 described by amplitude
M ie
2ω M2ωd 2qx 2q‘ e 4
1
2
i k1 k2 qq‘iα1 sin φ α2 cos φ 0
0
CP675, Spin 2002: 15th Int'l. Spin Physics Symposium and Workshop on Polarized Electron
Sources and Polarimeters, edited by Y. I. Makdisi, A. U. Luccio, and W. W. MacKay
© 2003 American Institute of Physics 0-7354-0136-5/03/$20.00
666
(1)
M v̄δ p 1 ¼
Q êλ
q̂
2
e
Âk̂Q1
2pk
k̂ m ê
2qk
1
1
λ1 êλ1
2pke k̂Â ū p q̂ k̂ m
2qk ê 2
(2)
(3)
λ2
2
The expressions in square brackets are obtained from the Volkov solutions for an
electron (positron) in the wave field.
A al1 cos ϕ µ l2 sin ϕ is the 4-potential of the wave (101.2), µ is the polarization,
ξ 2 m2 µ
k is the quasimomentum,
ϕ kx, ξ eam is the wave intensity, q µ pµ
2kp
l p l p
l p l p
m2 m2 1 ξ 2 1014 α1 ea 1 1 α2 ea 2 2 1016
kp kp
kp kp
If in tetrad nA (4), (6) from [2] for fermion line we make the replacement p
k1 ,
p
k2 , then we obtain tetrad hA common for both photons. In this case, the vector
1
eλ h
iλ h2 simultaneously describes both the emitted photons: k 1 with helicity
2 1
λ1 λ and k2 with λ2 λ . Thus, if λ1 λ2 0, then only one operator êλ enters into
the amplitude, if λ1 λ2 2, then êλ and êλ enter into the amplitude. This fact also
extremely simplifies both the structure and the calculation of the reaction amplitudes.
Thus,
γ
ê2λ 0 êλ êλ 1 5 k1µ σµν k2ν ;
k1 k2
(4)
2
f k2 ωλ k̂1 f k1 ωλ k̂2 êλ fˆêλ 2eλ f êλ ; êλ fˆêλ k1 k2
1
1 λ γ5 .
2
Let us consider in more detail the case of λ1 λ2 0. Using Volkov’s solution [3] for
the electron in the field of the plane wave (40.7), (101.3) and above relations, we obtain
the spin structure of the amplitude
where f is arbitrary vector, ωλ
M
1
ξ 2 m2 eλ k
1
eλ q
v̄δ p êλ k̂
k1 q k2 q
2kpkp ¼
e
ê k̂Â
λ
2kp
Âk̂êλ
2kp
uδ p (5)
Making series expansion in Bessel functions Jn z (101.7) and integrating in 4-space, we
make sure that amplitude (5) of the reaction sk q q k1 k2 completely coincides, to
an accuracy of the factor preceding the “sandwich”, with the s-channel of single-photon
reaction sk q q k (101.10). sk denotes coherent absorption from the wave of
photons (nonlinearity). The factor itself just reflects that exactly two-photon annihilation
(or a double Compton back- scattering) takes place.
If λ2 λ1 λ eλ eλ eλ in the selected gauge. In this case formula simplifying
1
2
the matrix structure is (B is an arbitrary matrix operator)
êλ Bêλ
1
1
γµ Bγν Tµν
k k γµ Bγν k1µ k2ν k2µ k1ν
λ 1 2
2
2k1 k2
667
k k g
1 2
µν
n iλ ε µνρσ k1 ρ k2 σ (6)
From (6) one gets (4) and more complicate structure in (2) is
2a2
f k2 k1 kωλ f k1 k2 kωλ k̂ Âk̂êλ fˆêλ k̂ k1 k2
For the laser wave we choose
l1 vv kv1kv
kv v
2
(7)
l 2 n2 (8)
ξ m kvkv kvkv
is the Bessel function
vv 2 1k
argument. With such a choice in the DSB we have
Than in (101.6) α2
êλ k̂Â δ
u p 2kp
ξ ê k̂ ∞ isϕ
µδ 2λ
e
2m k s∑
∞
e
Besides,
êλ k̂ωλ
0 and z
α 1 s k
1 µδ 3 Jsµ zωµ
vk
2k2 k
k̂ k̂ωλ
k1 kk1 k2 1
1
êλ k̂ωλ
s k
µδ 3 Jsµ zωµ uδ p vk
2k1 k
k̂ k̂ωλ
k2 kk1 k2 2
A similar expression is obtained for the structure v δ p Âk̂êλ . Than rather trivial calculations by formulas (15),(16) or (17), (18) from [2] remain.
The obtained formulas permit investigating the spin and nonlinear effects of a number
of interesting processes: in the region of conversion (linear colliders); induced decay of
ortopositronium with coherent absorption (radiation) of an odd number of photons.
In the t-channel (when e
e ), this is a double Compton back-scattering (the
Compton back-scattering is a basic process in the operation of γ -colliders [5]); 2γ creation of pairs in the region of conversion (collider testing), etc. Examples of comprehensive studies of single-photon processes are given in [1].
¼
REFERENCES
1.
2.
3.
4.
5.
M.V. Galynskii, S.M. Sikach// Physics of Particles and Nuclei.1998.Vol.29.P.496. [hep-ph/9910284].
S.M. Sikach// In this issue.
L.D. Landau, E.M. Lifshitz// Quantum Electrodinamics. Pergamon, New York.1980.
S.M. Sikach// Proceedings of 10 Annual Seminar Nonlinear Phenomena in Complex Systems,
Minsk.2001.P.297. [hep-ph/0103323].
I.F. Ginzburg, G.L. Kotkin, V.G. Serbo, V.I. Telnov// Nucl.Instr.Meth.1983.Vol.205.P.47.
I.F. Ginzburg, G.L. Kotkin, SL. Panfil, V.G. Serbo, V.I. Telnov// Nucl.Instr.Meth.1984.Vol.219.P.5.
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