Photon splitting in a laser field
Photon splitting in a laser field
A. Di Piazza,1, ∗ A. I. Milstein,1, 2, † and C. H. Keitel1, ‡
1Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany
2Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
(Dated: November 3, 2018)
Abstract
Photon splitting due to vacuum polarization in a laser field is considered. Using an operator
technique, we derive the amplitudes for arbitrary strength, spectral content and polarization of
the laser field. The case of a monochromatic circularly polarized laser field is studied in detail and
the amplitudes are obtained as three-fold integrals. The asymptotic behavior of the amplitudes for
various limits of interest are investigated also in the case of a linearly polarized laser field. Using
the obtained results, the possibility of experimental observation of the process is discussed.
PACS numbers: 12.20.Ds (QED Specific Calculations), 42.62.-b (Laser Applications), 42.50.Xa (Optical
Tests in Quantum Electrodynamics)
∗Electronic address: dipiazza@mpi-hd.mpg.de
†Electronic address: milstein@inp.nsk.su
‡Electronic address: keitel@mpi-hd.mpg.de
http://arxiv.org/abs/0704.0695v2
mailto:dipiazza@mpi-hd.mpg.de
mailto:milstein@inp.nsk.su
mailto:keitel@mpi-hd.mpg.de
I. INTRODUCTION
Vacuum polarization is known to result in such nonlinear Quantum Electrodynamics
(QED) effects as coherent photon scattering and photon splitting in an external electromag-
netic field. Photon scattering in the electric field of atoms (Delbrück scattering) has been
investigated in detail both theoretically and experimentally [1, 2, 3, 4]. Recently, an essen-
tial progress in the understanding of the photon splitting process in an atomic electric field
was achieved [5, 6] and the first successful observation of photon splitting was performed
[7]. It turned out that higher orders of the perturbation theory with respect to Zα play an
important role and drastically modify the cross sections of Delbrück scattering and photon
splitting at high photon energy (here, Z is the nuclear charge number, α = e2 = 1/137 is the
fine-structure constant with e being the electron charge, the system of units ~ = c = 1 is
used). Therefore, the theoretical and experimental investigation of photon splitting gives the
possibility to verify QED in the presence of strong external field when the nonperturbative
effects in the field strength (contributions of higher-order terms) are very important.
Photon splitting in a constant and uniform electromagnetic field for arbitrary values of
the field strength was considered in numerous papers because of the potential astrophysical
implications of the process (see, e.g., Refs. [8, 9, 10, 11, 12]). The possibility of observing
photon splitting in the electric fields of single crystals at high photon energies was investi-
gated in [13]. Photon splitting due to vacuum polarization in a strongly magnetized plasma
was discussed in [14].
Photon splitting in a laser field with arbitrary strength and frequency content has never
been considered before for an arbitrary energy of the incoming photon. The kinematics of
photon splitting depends essentially on the structure of the external field. In the Coulomb
field, the wave vector k1 of the initial photon and the wave vectors k2 and k3 of the final ones
are, in general, not coplanar. In a constant and uniform electromagnetic field, the initial and
final photons are almost collinear (up to nonzero photon mass due to vacuum polarization
in the external field). In a classical electromagnetic plane wave, due to energy-momentum
conservation, the initial and final photons are coplanar in the frame where the laser wave
propagates in the direction anti-parallel to k1. Throughout the paper we refer to this frame
as the laboratory frame. The structure of the photon splitting amplitude in a laser field
depends on the value of two Lorentz-invariant parameters. In the laboratory frame they are
given by
, χ =
, (1)
where ω1 is the energy of the initial photon, ω0 and E are the characteristic frequency
and electric field strength of the laser pulse, Ec = m
2/|e| = 1.3 × 1016V/cm is the critical
electric field and m is the electron mass. For a monochromatic laser field, ω0 and E in Eq.
(1) are the laser frequency and the root mean square of the laser electric field. For η ≪ 1
and χ ≪ 1, photon splitting in a monochromatic laser wave was considered in [15] in the
leading approximation with the help of the Euler-Heisenberg Lagrangian. The contribution
to the amplitudes of photon splitting linear in χ but at arbitrary η is nothing but the usual
photon-photon scattering which was studied in detail theoretically (see, e.g., [16]) but never
observed experimentally [17]. The possibility to observe “laser-assisted” photon-photon
scattering by using the next generation of petawatt lasers has been investigated in [18, 19].
Other processes related to photon-photon scattering and initiated by vacuum polarization
in the presence of the superposition of two laser fields, or a laser field and a Coulomb field
have been considered in [20, 21, 22, 23].
For the strongest optical (ω0 ≈ 1 eV) laser beam available today the laser intensity is
I ≈ 1022W/cm2 and the corresponding electric field amplitude is E ≈ 10−4Ec [24]. The
Extreme Light Infrastructure (ELI) project aims at reaching laser intensity values higher
than 1025W/cm
corresponding to E ≈ 10−2Ec. The X-ray Free-Electron Laser (X-FEL)
facilities presently being developed at DESY (Hamburg, Germany) and at SLAC (Stanford,
USA) are intended to provide spatially coherent and highly brilliant beams of synchrotron
radiation with single-photon energies up to 8÷12 keV at a maximum intensity ≈ 1015W/cm2
corresponding to a field strength of E ≈ 10−7Ec [26]. The above mentioned projects have ini-
tiated intensive investigations of various QED processes that can be experimentally observed
when using these future facilities [27, 28, 29]. On the other hand, sources of high-energy
photons are nowadays available in a wide range of photon energies spanning from tens of
MeV up to hundreds of GeV. It will then be possible to reach experimentally unexplored
regions for the values of the parameters η and χ where vacuum polarization effects in a laser
field, and photon splitting in particular, are essentially nonperturbative.
In the present paper, we derive the amplitudes of photon splitting in a plane wave
described by an arbitrary four-vector potential Aµ(φ) = (0,A(φ)), where φ = κx with
κ = (1,κκκ), κ2 = 0, and κκκ · A(φ) = 0. At a first stage, we do not assume that the plane
wave is monochromatic or even periodic. It is convenient, without loss of generality, to
consider the process in the laboratory frame, i.e. in the frame where k1 is anti-parallel to
the vector κκκ. The calculations are performed by using the operator technique developed
in Refs. [30, 31] for the calculation of the polarization and mass operators in a laser field.
Alternative forms of these operators were derived independently in Refs. [32, 33] by means
of a different method. The results of Refs. [30, 31] and Refs. [32, 33] are in agreement
with each other. The operator technique allows one to find the amplitudes without using
the explicit form of the Green’s function for the Dirac equation in the external field. This
circumstance considerably simplifies the calculations.
The rest of the paper is organized as follows: in Section II we describe the general prop-
erties of the photon splitting amplitudes. Then, starting from the operator representation of
the amplitudes, in Sections III and IV we calculate the explicit expressions of the amplitudes
for a general vector potential of the form A(φ). In Section V we consider a monochromatic
laser field with particular emphasis on the case of a circularly polarized laser field. In Section
VI we analyze various asymptotic forms of the amplitudes both for a circular and for a lin-
early polarized field. The possibility of the experimental observation of photon splitting in a
laser field is discussed in detail in Section VII. Finally, in Section VIII the main conclusions
of the paper are presented.
II. GENERAL DISCUSSION
The amplitude M of the photon splitting process of a photon with four-momentum k1 =
(ω1,k1) into two photons with four-momenta k2 = (ω2,k2) and k3 = (ω3,k3) with k
1 = k
k23 = 0 is represented by the Feynman diagram shown in Fig. 1. The corresponding formal
expression for M in the Furry representation reads
M = ie3
d4xTr〈x|e−ik1x ê1
P̂ −m+ i0
eik2x ê∗2
P̂ −m+ i0
eik3x ê∗3
P̂ −m+ i0
+(k2 ↔ k3 , e2 ↔ e3) , (2)
where e1µ is the initial photon polarization four-vector, e2µ and e3µ are the polarization
four-vectors of the final photons, P̂ = Pµγµ with Pµ = i∂µ− eAµ(x) and γµ being the Dirac
matrices. The factor (4π)3/2, contained in the external photon wave functions, has been
included into the formula for the probability of the process. The calculation of the amplitude
M is essentially simplified by using the Green’s function D(x2, x1) of the “squared” Dirac
equation:
D(x2, x1) = 〈x2|
P̂2 −m2 + i0
|x1〉 . (3)
As shown in [5], the expression ofM can be written as a sum of two contributions containing
either three or two Green’s functions D(x2, x1):
M ≡ Mµνρe1µe∗2νe∗3ρ =M (2) +M (3) . (4)
The term M (2) has the form
M (2) =M
3 (5)
1 = −ie3(e∗2e∗3)
d4xTr〈x|e−ik1x(2e1P − ê1k̂1)
P̂2 −m2 + i0
ei(k2+k3)x
P̂2 −m2 + i0
|x〉 ,
2 = −ie3(e1e∗3)
d4xTr〈x|eik2x(2e∗2P + ê∗2k̂2)
P̂2 −m2 + i0
ei(k3−k1)x
P̂2 −m2 + i0
|x〉 ,
3 = −ie3(e1e∗2)
d4xTr〈x|eik3x(2e∗3P + ê∗3k̂3)
P̂2 −m2 + i0
ei(k2−k1)x
P̂2 −m2 + i0
|x〉 .
The term M (3) is given by
M (3) =
dxTr〈x|e−ik1x(−ê1k̂1 + 2e1P)
P̂2 −m2 + i0
×eik2x(ê∗2k̂2 + 2e∗2P)
P̂2 −m2 + i0
eik3x(ê∗3k̂3 + 2e
P̂2 −m2 + i0
+(k2 ↔ k3 , e2 ↔ e3) . (6)
For each photon with the four-momentum k
i (i = 1, 2, 3) it is convenient to introduce
two vectors ε
, (λi = 1, 2)
− kiaλi
µ , εi,λiεj,λj = −δλiλj
a0λ = 0 , (aλ)
2 = 1 , a1 · a2 = 0 , aλ · κκκ = 0. (7)
Due to gauge invariance, it is possible to write the tensor Mµνρ introduced in Eq. (4) for
real photons in the form (see Ref. [30]):
Mµνρ = −
λ1,λ2λ3=1,2
Rλ1λ2λ3ε
εν2,λ2ε
, (8)
For the coefficients Rλ1λ2λ3 we have
Rλ1λ2λ3 = Mµνρε
εν2,λ2ε
. (9)
So, the coefficients Rλ1λ2λ3 can be calculated from Eqs. (5) and (6) by the substitution
i −→ ε
. Using the coefficients Rλ1λ2λ3 , we write the amplitude M of photon splitting in
the laser field as
M = −
λ1,λ2λ3=1,2
Rλ1λ2λ3(ε1,λ1e1)(ε2,λ2e
2)(ε3,λ3e
3) . (10)
This expression of the amplitude M is very convenient because it automatically provides
the fulfillment of the gauge invariance principle. As a consequence, it is not necessary to
perform the subtraction in the calculation of the lowest-order photon splitting amplitude,
described by the photon-photon scattering diagrams.
We perform the calculations in terms of the helicity amplitudes Mσ1σ2σ3 , with σj = ±1.
In this case, the polarization vector eσ of each external photon with the wave vector k
and the energy ω satisfies the relations eσ · k = 0 and eσ × k = iσωeσ. Due to momentum
conservation in the laser field, the final transverse momenta k2⊥ and k3⊥ (with respect to the
laser propagation direction) are equal in modulus and opposite in direction: k2⊥ = −k3⊥ ≡ q
and q · κκκ = q · k1 = 0. We direct the vector a1 along the vector q, so that a2 = κκκ × a1,
e1 ,σ1 = −
(a2 + iσ1a1) ,
e2 ,σ2 =
(a2 − iσ2a2 × k2/ω2) ,
e3 ,σ3 =
(a2 − iσ3a2 × k3/ω3) . (11)
The corresponding polarization four-vectors ei,σi = (0, ei ,σi) have the following products
with the four-vectors εi,λi:
ε1,λ1e1,σ1 =
(iσ1δλ1,1 + δλ1,2) ,
ε2,λ2e2,σ2 =
(iσ2δλ2,1 − δλ2,2) ,
ε3,λ3e3,σ3 =
(iσ3δλ3,1 − δλ3,2) . (12)
These useful relations allow us to state that the amplitude Mσ1σ2σ3 can be obtained from
the coefficient Rλ1λ2λ3 by the substitution in the latter:
aλ1 → −
(iσ1a1 + a2) , aλ2 →
(−iσ2a1 + a2) , aλ3 →
(−iσ3a1 + a2) . (13)
The energies ω2,3 and the parallel components k2,3 ‖ = κκκ ·k2,3 of the momenta of the final
photons are functions of the transverse momentum q and of the energy Ω absorbed from
the laser field, Ω = ω2 + ω3 − ω1. It is convenient to introduce the quantities
ν2,3 =
κk2,3
1∓ cos θ0
, cos θ0 =
. (14)
The angle θ0 coincides with the angle between the vectors k2 and κκκ in the frame where
k2 = −k3, so that q2 ≤ ω1Ω. Then
ω2 = ω1ν2 + Ων3 , k2‖ = −ω1ν2 + Ων3
ω3 = ω1ν3 + Ων2 , k3‖ = −ω1ν3 + Ων2 . (15)
It follows from Eqs. (5) and (6) that, like the amplitude M , the coefficients Rλ1λ2λ3 can
also be presented as the sum Rλ1λ2λ3 = R
λ1λ2λ3
λ1λ2λ3
. In order to explain the application
of the operator technique to the calculation of the photon splitting amplitude, we present
the derivation of the contribution R
λ1λ2λ3
. The calculation of R
λ1λ2λ3
can be performed in a
similar way and only the final result is given for this contribution.
III. CALCULATION OF R
λ1λ2λ3
We direct the vector κκκ along the z-axis and pass from the variables t and z to the variables
φ = κx = t− z and T = (t+ z)/2. In this way
p0 = i∂t = −pφ − pT/2 , pz = −i∂z = −pφ + pT/2 , pφ = −i∂φ , pT = −i∂T .
Then, we use in Eq. (5) the representation of the operator Green’s function derived in [31]
P̂2 −m2 + i0
ds exp[is(P̂2 −m2)]
ds exp(−ism2)
[Â(φ+ 2spT )− Â(φ)]
× exp
ds′ (p⊥ − eA(φ+ 2s′pT ))2
exp(2ipφpT ) . (16)
As before, the index ⊥ indicates transverse quantities with respect to the laser propagation
direction κκκ. Now, we pass in Eq. (5) from the integration over the variables T and x⊥ to
the corresponding momenta pT and p⊥ in accordance with
dT 〈T |...|T 〉 →
〈pT |...|pT 〉 ,
dx⊥〈x⊥|...|x⊥〉 →
(2π)2
〈p⊥|...|p⊥〉 .
We also exploit the relations
exp(2ispφpT )g(φ) exp(−2ispφpT ) = g(φ+ 2spT ) , exp(2ispφpT )|φ〉 = |φ− 2spT 〉 , (17)
valid for an arbitrary function g(φ) of the variable φ. Besides, the matrix elements 〈pT |...|pT 〉
and 〈p⊥|...|p⊥〉 become
〈pT | exp[i2ω1T (ν2 + ν3 − 1)|pT 〉 = 2πδ
2ω1(ν2 + ν3 − 1)
〈p⊥| exp[−ix⊥ · (k2⊥ + k3⊥)]|p⊥〉 = (2π)2δ(k2⊥ + k3⊥) . (18)
The factors (18) are common to all amplitudes. They will be omitted below in the expres-
sions of the amplitudes and will be taken into account in the usual way in the formula for
the probability of the process. The matrix element 〈φ|...|φ〉 also becomes the δ-function con-
taining the variable pT in its argument. Therefore, the integration over pT becomes trivial
as well as the integration over p⊥. Following the procedure pointed out above and using the
useful relations
2 = Ων3κ
µ + ν2k
1 + q
µ , k
3 = Ων2κ
µ + ν3k
1 − qµ ,
k2k3 = 2Ωω1 , k2k1 = 2Ωω1ν3 , k3k1 = 2Ωω1ν2 , q
2 = 4Ωω1ν2ν3 , (19)
with qµ = (0,q), we arrive at the following result for R
λ1λ2λ3
λ1λ2λ3
= − me
(2π)2
j=1,2,3
ds1 ds2
(s1 + s2)2
exp[−i(s1 + s2)]
dφ exp
iΩφ + i
ϕj − i(s1 + s2)
dy∆2(yϕj)−
dy∆(yϕj)
−2aλj ·
dy∆(yϕj) +
(s1 + s2)
2s1s2
aλj ·∆(ϕj)
. (20)
Here we defined the quantities
δ(1) = δλ2λ3 , δ(2) = δλ1λ3 , δ(3) = δλ1λ2 ,
(ϕ1, ϕ2, ϕ3) =
s1 + s2
(1, −ν2, −ν3) , ∆(u) =
[A(φ+ 4uω1/m
2)−A(φ)] . (21)
In the parametrization of the Green’s function we made the substitution s → s/m2. As it
has to be according to the Furry theorem, the contribution (20) contains only odd powers
of the external field. The explicit result of the integration over φ depends on the form of
the function A(φ). For a monochromatic wave, the integral over φ can be easily performed
(see Section V).
IV. CALCULATION OF R
λ1λ2λ3
The calculation of the contribution R
λ1λ2λ3
is similar to that of R
λ1λ2λ3
, but considerably
more cumbersome. In order to bring the answer into a relatively short form, we introduce
the convenient notation
S = s1 + s2 + s3 , τ1 =
s2ν2 − s3ν3
, τ2 = ν2 − τ1 , τ3 = ν3 + τ1 ,
D1(y) = ∆(s2τ2y + s3τ3(1− y)) , D2(y) = ∆(s2τ2y) , D3(y) = ∆(s3τ3y) ,
dy[s1D1(y) + s2D2(y) + s3D3(y)] ,
F0 = SF
dy[s1D
1(y) + s2D
2(y) + s3D
3(y)] ,
dy [s2(D1(y)−D2(y)) + s3(D1(y)−D3(y))] , (22)
and use the relation s1τ1 = s2τ2 − s3τ3. Before taking the trace over the γ-matrices, we
obtain the following compact representation for R
λ1λ2λ3
λ1λ2λ3
ds1 ds2 ds3
dφ exp
4ω1Ωs2s3
+ F0 − S + ρ ·G
−2aλ3 ·N3 +
ε̂3,λ3 k̂3
mκ̂d̂3
−2aλ1 ·N1 −
ε̂1,λ1 k̂1
mκ̂d̂2
−2aλ2 ·N2 +
ε̂2,λ2k̂2
mκ̂d̂1
δ(1)aλ1 · (N1 − V ) + δ(2)aλ2 · (N2 + ν2V ) + δ(3)aλ3 · (N3 + ν3V )
+(2 ↔ 3 , ρ → −ρ) , (23)
where
, N1 = F −
, N2 = F −D2(1) +
N3 = F −D3(1) +
, V =
(d1 + d2 + d3) ,
D2(1)−D3(1)
, d2 =
D2(1)
, d3 =
D3(1)
. (24)
The substitution (2 ↔ 3 , ρ → −ρ) in Eq. (23) results in the replacement N1(ρ) →
N1(−ρ), N2(ρ) → N3(−ρ), and N3(ρ) → N2(−ρ). In fact, the role of the term with the
substitution (2 ↔ 3 , ρ → −ρ) is to provide the fulfillment of the Furry theorem (R(3)λ1λ2λ3
contains only odd powers of the external field). Therefore, the sum of the two terms is twice
the odd part of the first term with respect to the external vector potential A(φ). Finally,
we take the trace and obtain:
λ1λ2λ3
ds1 ds2 ds3
dφ exp
4ω1Ωs2s3
+ F0 − S + ρ ·G
− 4(aλ1 ·N1)(aλ2 ·N2)(aλ3 ·N3) + 4(aλ2 ·N2)(aλ3 ·N3)(aλ1 · V )
−4ν3(aλ1 ·N1)(aλ2 ·N2)(aλ3 · V )− 4ν2(aλ1 ·N1)(aλ3 ·N3)(aλ2 · V )
+(aλ1 ·N1)
[ρ · ρ]λ2λ3
2ν2ν3
+ [ρ · (d1 − d2 − d3)]λ2λ3 − 2ν2ν3[d1 · (d2 + d3)]λ2λ3
−(aλ2 ·N2)
[ρ · ρ]λ1λ3
+ [ρ · (d1 + d2 − d3)]λ1λ3 − 2ν3[d3 · (d1 + d2)]λ1λ3
−(aλ3 ·N3)
[ρ · ρ]λ1λ2
+ [ρ · (d1 − d2 + d3)]λ1λ2 − 2ν2[d2 · (d1 + d3)]λ1λ2
+(aλ1 · ρ)[ρ · d1]λ2λ3 −
(aλ2 · ρ)[ρ · d3]λ1λ3 −
(aλ3 · ρ)[ρ · d2]λ1λ2
[aλ1 · d1]λ2λ3 −
[aλ2 · d3]λ1λ3 −
[aλ3 · d2]λ1λ2
(aλ1 · ρ)(aλ2 · ρ)(aλ3 · ρ)−
4ν2ν3
[δ(1)(aλ1 · ρ) + δ(2)(aλ2 · ρ) + δ(3)(aλ3 · ρ)]
δ(1)aλ1 · (N1 − V ) + δ(2)aλ2 · (N2 + ν2V ) + δ(3)aλ3 · (N3 + ν3V )
+U1 − ν2 U2 − ν3 U3 + 2ν2ν3 U4
+ (2 ↔ 3 , ρ → −ρ) . (25)
Here, we introduced the operator
[X · Y ]λiλj ≡ (X · aλi)(Y · aλj ) + (X · aλj )(Y · aλi)− (X · Y )(aλi · aλj ) (26)
for any two vectors X and Y and we used the abbreviations
U1 = −(aλ1 · ρ)[d2 · d3]λ2λ3 + (aλ2 · ρ)[d2 · d3]λ1λ3 + (aλ3 · ρ)[d2 · d3]λ1λ2
+2(aλ1 · ρ)(aλ2 · d2)(aλ3 · d3)− (d2 · ρ)[d3 · aλ2 ]λ1λ3 − (d3 · ρ)[d2 · aλ3 ]λ1λ2 ,
U2 = (aλ1 · ρ)[d1 · d2]λ2λ3 − (aλ2 · ρ)[d1 · d2]λ1λ3 + (aλ3 · ρ)[d1 · d2]λ1λ2
+2(aλ2 · ρ)(aλ1 · d2)(aλ3 · d1)− (d1 · ρ)[d2 · aλ3 ]λ1λ2 − (d2 · ρ)[d1 · aλ1 ]λ2λ3 ,
U3 = (aλ1 · ρ)[d1 · d3]λ2λ3 + (aλ2 · ρ)[d1 · d3]λ1λ3 − (aλ3 · ρ)[d1 · d3]λ1λ2
+2(aλ3 · ρ)(aλ1 · d3)(aλ2 · d1)− (d1 · ρ)[d3 · aλ2 ]λ1λ3 − (d3 · ρ)[d1 · aλ1 ]λ2λ3 ,
U4 = (aλ1 · d2)[d1 · d3]λ2λ3 + (aλ1 · d3)[d1 · d2]λ2λ3 − (aλ1 · d1)[d2 · d3]λ2λ3
+2(aλ1 · d1)(aλ2 · d2)(aλ3 · d3) + (d1 · d2)[d3 · (δ(2)aλ2 − δ(3)aλ3)]
−(d1 · d3)[d2 · (δ(2)aλ2 − δ(3)aλ3)]− (d2 · d3)[d1 · (δ(2)aλ2 + δ(3)aλ3)] . (27)
Equations (20) and (25) are valid for an arbitrary vector potential A(φ) and they are our
starting point for further consideration. The simplest case A(φ) = −E0φ, where E0 is
a constant vector perpendicular to κκκ, corresponds to the so-called crossed field with the
electric field E = E0 and the magnetic field H = κκκ × E0 , |H| = |E|. In this case there
is no energy and momentum transfer from the field so that Ω = 0 and q = 0 (the momenta
k2 and k3 of the final photons are parallel to the momentum k1 of the initial photon). Note
that in this case the expansion of the amplitudes over E0/Ec starts with the terms of order
of (E0/Ec)
3 while the linear terms in E0/Ec vanish. We have checked that for a crossed field
our amplitudes agree with those found in [11].
V. PHOTON SPLITTING IN A MONOCHROMATIC PLANE WAVE
Below, we consider the external field to be a monochromatic plane wave with the fre-
quency ω0. In this case, the helicity amplitude Mσ1σ2σ3 has the form
Mσ1σ2σ3 =
2πδ(Ω− nω0)Mn, σ1σ2σ3 (28)
after the integration over the variable φ. In the above equation the prime means the sum-
mation over odd numbers n and where Mn, σ1σ2σ3 denotes the invariant amplitude of the
photon splitting process with absorption of n laser photons. Using the usual Fermi golden
rule, we obtain for the photon splitting rate dẆn
dẆn, σ1σ2σ3 =
|Mn, σ1σ2σ3 |2
ω1Ωn cos θ0
|Mn, σ1σ2σ3 |2 dν2 dφq , (29)
where Ωn = nω0, cos θ0 is defined in Eq. (14), and φq is the azimuth angle of the vector
q in the plane perpendicular to κκκ. We remind the reader of the fact that q is a two-
dimensional vector perpendicular to κκκ and k1. Besides, dq/(ω1Ωn cos θ0) coincides with the
differential of the solid angle of the vector k2 in the frame where ω1 = Ωn. It is also useful
to express the rate dẆn, σ1σ2σ3 in terms of the polar angle θ2 = ∠(k2,κκκ) (or of the polar
angle θ3 = ∠(k3,κκκ)). Using the expressions (15) valid for any Ω we obtain the relations
ν2,3 =
cot2(θ2,3/2)
, cot(θ2/2) cot(θ3/2) =
. (30)
As a result, we find from Eq. (29) that
dẆn, σ1σ2σ3 =
|Mn, σ1σ2σ3 |2
sin θ2 dθ2 dφq[
sin2(θ2/2) +
cos2(θ2/2)
]2 . (31)
In the general case of an elliptically polarized and monochromatic laser wave, the vector
potential reads
A(φ) = A1 cos(ω0φ) +A2 sin(ω0φ) , (32)
where A1 · A2 = A1 · κκκ = A2 · κκκ = 0. Generally speaking, the vectors A1 and A2 are
not parallel to the unit vectors a1 and a2 introduced in Eq. (7), respectively, as there is an
azimuth asymmetry of the photon splitting amplitude for an elliptically polarized laser wave
(we point out that a1 has been chosen to be parallel to q). However, in a circularly polarized
laser field it is |A1| = |A2|, then the photon splitting rate is symmetric with respect to the
azimuth angle φq of the vector q, and the vector A1 can be directed along a1.
The photon splitting rate can be converted into a cross section according to the relation
, Φ =
1 + A
(ξ21 + ξ
2) , (33)
with Φ being the laser photon flux and ξ1,2 = eA1,2/m. Note that this cross section has
a restricted meaning because in the presence of a strong field it still depends on the laser
flux through the parameters ξ1,2. In terms of the parameters ξ1,2 and η, the parameter χ
introduced in Eq. (1) has the form χ = η
(ξ21 + ξ
2)/2.
The integral over φ in Eqs. (20) and (25) can be taken analytically for an elliptically
polarized laser wave. However, the result is very cumbersome even in the case of linear laser
polarization. This is why we limit ourselves to the case of a circularly polarized laser field.
In this case the answer is essentially simpler (but not simple). Thus we set |A1| = |A2| = A,
ξ1 = ξ2 = ξ = eA/m and direct A1 along a1 and A2 along a2 (for positive helicity (p. h.))
or A2 along −a2 (for negative helicity (n. h.)). Only the case of positive helicity of the
laser field will be considered below and the amplitudes for negative helicity of the laser
field polarization can be obtained by means of the relation Mσ1σ2σ3(n. h.) = −Mσ̄1σ̄2σ̄3(p. h.),
where σ̄i denotes the helicity opposite to σi.
Now, by using the substitution rule of Eqs. (13) and by carrying out the change of
variables s1 = us and s2 = (1− u)s in Eq. (20), we obtain for the nonzero contributions of
n,σ1σ2σ3 the following result
n,++− = G(1) +G(−ν3) , M
n,+−+ = G(1) +G(−ν2) , M
n,−−− = G(−ν2) +G(−ν3) ,
G(ν) = δn1
du exp
1 + ξ2
1− sin
eiϑ − sinϑ
sin ϑ
u(1− u)
; ϑ = 2u(1− u)sην . (34)
In order to write M
n,σ1σ2σ3 in a compact form, we introduce the following definitions
fj = sin(2ηsjτj)e
i2ηsjτj , ζ =
, φ0 = arg ζ ,
4η2Sτ1τ2τ3
|f1|2 +
|f2|2 −
|f3|2
− ξ2S ,
Z = ξ|ζ |ρns2s3
, ρn = 2
nην2ν3 , S = s1 + s2 + s3 . (35)
In terms of these functions, the contribution M
n,σ1σ2σ3 has the form
M (3)n,σ1σ2σ3 =
ds1 ds2 ds3
+ F0 − S
e−i(n+j)φ0 Jn+j(Z)Bj,σ1σ2σ3 , (36)
where Jl(Z) are ordinary Bessel functions. The coefficients Bj,σ1σ2σ3 are expressed via the
functions fj introduced in Eq. (35) and they are presented in the Appendix. The expressions
(34) and (36) are valid for any value of the parameters η and ξ = χ/η. As it is clear from
Eq. (36), the photon splitting amplitudes are not equal to zero for any number of absorbed
laser photons. This is because the angle between the two final photons is not zero (ρn 6= 0)
and the conservation of the projection Jz of the total angular momentum does not imply any
selection rule. Note that the expansion of the amplitude Mn,σ1σ2σ3 = M
n,σ1σ2σ3 +M
n,σ1σ2σ3
for a fixed n (see Eqs. (34) and (36)), contains in general all odd powers k of the parameter
χ (or, equivalently, ξ) starting from k = n. The terms with k > n in Mn,σ1σ2σ3 correspond
to rescattering processes of absorption and emission of laser photons with a net absorption
of n laser photons.
In the next Section we consider the asymptotics of the amplitudesM
n,σ1σ2σ3 andM
n,σ1σ2σ3
in some limiting cases.
VI. ASYMPTOTICS OF THE AMPLITUDES
Various limiting cases of the photon splitting amplitudes for a monochromatic plane wave
are studied in this Section.
A. Amplitudes for small η and small χ
We first consider the case of both small η and χ, η ≪ 1 and χ ≪ 1. Then, the leading
contribution to the amplitudes is determined by the ratio η/χ2 [15]. For η/χ2 ≫ 1, the
nonzero amplitudes in the leading approximation are given by the linear terms in χ in
Eqs.(34) and (36) and correspond to n = 1. We obtain
M1,+++ = 11N , M1,+−− = −2(1− ν2ν3)N , M1,−−+ = 11ν22N ,
M1,−+− = 11ν
3N , N = −
ime3ηχ
. (37)
These amplitudes are nothing but the usual photon-photon scattering amplitudes for small
Mandelstam variables s = 4ηm2, t = −4ην2m2 and u = −4ην3m2 (see, e.g., [16]). For
η/χ2 ≪ 1, the leading contributions are proportional to χ3 and correspond to both n = 1
and n = 3. The nonzero amplitudes in the leading approximation are given by
M1,+−+ =M1,++− =M1,−−− = −i
M3,+−− = iÑ , Ñ =
2me3χ3ν2ν3
. (38)
These results are in agreement with those obtained in [15].
B. Amplitudes for small η and fixed χ
We proceed now to the region of parameters η ≪ 1 and χ fixed. In this case the leading
terms of the amplitudes in a circularly polarized laser field contain all powers of χ but only
for n = 1 and n = 3 they still are not equal to zero. The reason is the following: at η ≪ 1,
in a frame where ω1 ≫ ω0, the vectors k2 and k3 are almost parallel to k1 and anti-parallel
to κκκ. Since the clockwise polarized laser field contains photons with projection of the total
angular momentum Jz = +1 then, due to the conservation of Jz, only the values n = 1
and n = 3 are allowed and only the amplitudes M1,++−, M1,+−+, M1,−−− and M3,+−− are
different zero. For these amplitudes we found in the leading approximation with respect to
M1,++− =
ime3χ3√
ds1 ds2 ds3 e
−i(S+ψ0)
− i(z1 + z3)
z1z2z3 − ν2ν3z1 + ν2z3 + ν3z2t23 − ν2ν3(1 + t23)
+ G(1) + G(−ν3) ;
M1,−−− =
ime3χ3√
ds1 ds2 ds3 e
−i(S+ψ0)
− i(z2 + z3)
z1z2z3 − ν2ν3z1t21 + ν2z3 + ν3z2 − ν2ν3(1 + t21)
+ G(−ν2) + G(−ν3) ;
M3,+−− =
ime3χ3√
ds1 ds2 ds3 e
−i(S+ψ0)
z1z2z3 − ν2ν3z1t21 + ν2z3t22 + ν3z2t23 + 2ν2ν3t1t2t3
. (39)
Here we introduced the following quantities:
G(ν) = ν me
du [1− 2u(1− u)]e−i(s+ψ1) ;
z1 = 2x1(x2ν2 + x3ν3) + 2x2x3 − 1 , z2 = 2x1(−x2ν2 + x3ν3)− 2x2x3 + ν2 ,
z3 = 2x1(x2ν2 − x3ν3)− 2x2x3 + ν3 , xj =
, tj = 1− 2xj ,
χ2S3[x22x
3 + 2x1x2x3(ν2x2 + ν3x3) + x
1(ν2x2 − ν3x3)2] ,
ν2χ2u2(1− u)2s3 . (40)
The amplitudeM1,+−+ is obtained fromM1,++− by the replacement ν2 ↔ ν3. Equations (39)
are independent of the parameter η and are valid for arbitrary values of χ. The linear terms
of the expansion of these amplitudes with respect to χ vanish. Then, the expansion starts
with the terms proportional to χ3 that coincide with Eqs. (38). As it is known, both the
electromagnetic invariants FµνF
µν and ǫµνρσF
µνF ρσ are equal to zero for a plane wave, F µν
being the tensor of the electromagnetic field. The Euler-Heisenberg Lagrangian depends on
these two invariants but not on χ which in our case has the form χ =
−(eFµνkν1)2/(2m3).
Therefore only the terms (38) can be obtained by means of this effective Lagrangian but
not the total answer (39). Thus, the applicability of the Euler-Heisenberg Lagrangian to
the calculation of the photon splitting amplitudes is restricted by the conditions η ≪ 1 and
χ≪ 1. It is worth noting that the region of applicability of the small-χ asymptotics is very
narrow. For instance, at χ = 0.1 the difference between the leading terms (38) and the
exact result (39) is already around 20%. We illustrate the importance of high-order terms
in χ in Fig. 2 where we show the total photon splitting rate Ẇ at intermediate values of
the parameter χ. The solid line is obtained by using the amplitudes (39), the dashed line
by employing Eq. (38).
At ω1 ≫ m it is very difficult to measure the energy of the final photons with the accuracy
necessary to distinguish between the process of photon splitting with one-photon absorption
and three-photon absorption. However, there is another possibility to distinguish these two
processes which is based on the use of the angular distribution of the final photons. At
ω1 ≫ ω0, the final photons are almost parallel to the initial one so that ϑ2,3 ≪ 1 where
ϑ2 = ∠(k2,k1) and ϑ3 = ∠(k3,k1). It follows from Eq. (31) that
dẆn, σ1σ2σ3 =
|Mn, σ1σ2σ3 |2
ϑ2 dϑ2 dφq(
)2 . (41)
Using the expressions (39), we have tabulated the ratio
dẆ1, σ1σ2σ3
(dẆ1, σ1σ2σ3 + dẆ3, σ1σ2σ3)
. (42)
The result of this tabulation at χ = 1 is shown in Fig. 3. It can be seen that R noticeably
differs from unity at angles ϑ2 such that ϑ2 ≫
ω0/ω1.
It is interesting to compare the amplitudes of photon splitting in a circularly polarized
laser field with those in a linearly polarized field. In the latter case for η ≪ 1 and fixed χ,
the leading terms of the amplitudes are not zero for all odd n. This is due to the fact that
a linearly polarized laser beam contains photons with both Jz = +1 and Jz = −1. In this
sense the situation reminds us of the properties of the polarization operator in the laser field
[30]. In the case of a linearly polarized laser field it is convenient to perform the calculations
of the amplitudes in terms of the linear polarization of photons. Since the vectors k2 and
k3 are almost parallel to k1, the azimuth asymmetry disappears. Therefore, we can choose
A(φ) = a1A1 cos(ω0φ) and we have χ = ηξ1/
2 with ξ1 = eA1/m. The polarization of each
photon is parallel either to a1 or to a2 so that our amplitudes coincide with the coefficients
Rλ1λ2λ3 in Eq. (9). Note that a2 = κκκ × a1 and thus a2 is a pseudo-vector. Therefore, the
nonzero amplitudes in the collinear approximation are those which either do not contain
or contain twice the indices λ = 2, namely, Mn,111, Mn,122, Mn,212, and Mn,221. Putting
η equal to zero in Eqs. (20) and (25) and performing the integral over φ, we obtain
the leading contributions of the amplitudes at η ≪ 1 for a linearly polarized laser field
Mn,111 =
2me3χ3
ds1 ds2 ds3 e
−i(S+ψ0)
− i(z1 + z2 + z3)D
n (ψ0)
2S2χ2
+SD(2)n (ψ0)
2z1z2z3 − ν2ν3(1 + t21)z1 + ν2(1 + t22)z3 + ν3(1 + t23)z2
−4ν2ν3 + ν2ν3(1 + t1)(1 + t2)(1 + t3)
+ G̃n(1) + G̃n(−ν2) + G̃n(−ν3) ;
Mn,122 =
2me3χ3
ds1 ds2 ds3 e
−i(S+ψ0)
− iz1D
n (ψ0)
2S2χ2
−SD(2)n (ψ0)ν2ν3(1− t1)
(1 + t1)z1 + 1− t2t3
+ G̃n(1) ;
Mn,212 =
2me3χ3
ds1 ds2 ds3 e
−i(S+ψ0)
− iz2D
n (ψ0)
2S2χ2
+SD(2)n (ψ0)ν3(1− t3)
(1 + t3)z2 − ν2(1− t1t2)
+ G̃n(−ν2) , (43)
where
D(1)n (y) = Jn−1
(y)− iJn+1
(y) ,
D(2)n (y) =
1 + i
(y)− i
1− in+ 1
(y) ,
G̃n(ν) = ν
2 me3χ
du [1− 2u(1− u)]e−i(s+ψ1)D(1)n (ψ1) ,
and the other notations are presented in Eq. (40). The amplitudes Mn,221 are obtained
from Mn,212 by the replacement ν2 ↔ ν3. In Fig. 4 we show the photon splitting rate Ẇ
in a linearly polarized laser field as a function of the number n of laser photons absorbed.
The rate has been calculated from the above amplitudes (43) with χ = 1. The figure shows
that, as the number n increases, the photon splitting rate decreases quite rapidly and in a
non-perturbative way.
The leading contributions to the amplitudes at χ≪ 1 have the form [see also Eq. (38)]
M1,111 =
Ñ , M3,111 = −
M1,122 =M1,212 =
Ñ M3,122 =M3,212 = −
Ñ . (45)
Note that at η ≪ 1 there is a simple relation between the amplitudes of the photon
splitting in a crossed field and those calculated in a linearly polarized laser field [11]. The
amplitudes in a crossed field depend only on the parameter χ while those in a laser field
depend only on χ in the leading approximation for η ≪ 1. In this case, it can easily be
shown that
(lin)
n,λ1λ2λ3
(χ) =
einφM
(cross)
λ1λ2λ3
(χ sinφ) . (46)
Therefore, due to Parseval’s identity, we have
|M (lin)n,λ1λ2λ3(χ)|
|M (cross)λ1λ2λ3(χ sinφ)|
2 . (47)
For the same reason, in the case of a circularly polarized laser field the following relation
holds
n, σi
|M (circ)n,σ1σ2σ3(χ)|
|M (cross)σ1σ2σ3(χ)|
2 , (48)
where the indices σi denote the helicities of the external photons. We emphasize that Eq.
(48) is not valid without the summation over σi. This is because the rotational symmetry
around the vector κκκ is absent in a crossed field (see discussion below Eq. (27)), as is the
conservation of the projection of the total angular momentum Jz. Therefore, it is impossible
to obtain the amplitudes M
(circ)
n,σ1σ2σ3(χ) in a circularly polarized field from those in a crossed
field by using any Fourier transformation as it was done in Eq. (46).
Now, at χ≫ 1 the asymptotics (39) obtained for the circularly polarized field read
M1,++− = Nc
ψ̃4/3
4(z1 + z3)ψ̃ + z1z2z3 − ν2ν3z1
+ν2z3 + ν3z2t
3 − ν2ν3(1 + t23)
8Γ2(1/3)
5Γ(2/3)
(1− ν1/33 )
M1,−−− = Nc
ψ̃4/3
4(z2 + z3)ψ̃ + z1z2z3 − ν2ν3z1t21
+ν2z3 + ν3z2 − ν2ν3(1 + t21)
2(1/3)
5Γ(2/3)
2 + ν
M3,+−− = Nc
ψ̃4/3
z1z2z3 − ν2ν3z1t21 + ν2z3t22 + ν3z2t23 + 2ν2ν3t1t2t3
,(49)
where
me3(6χ)1/3e−iπ/6Γ(1/3)
ψ̃ = x22x
3 + 2x1x2x3(ν2x2 + ν3x3) + x
1(ν2x2 − ν3x3)2 ,
x1 = 1− u , x2 = uw , x3 = u(1− w) , (50)
and zj and tj are expressed via xj as in Eq. (40). Analogously, at χ ≫ 1 we find from Eq.
(43) for a linearly polarized laser field
Mn,111 = Nn
ψ̃4/3
4(z1 + z2 + z3)ψ̃ + 2z1z2z3 − ν2ν3(1 + t21)z1 + ν2(1 + t22)z3
+ν3(1 + t
3)z2 − 4ν2ν3 + ν2ν3(1 + t1)(1 + t2)(1 + t3)
8Γ2(1/3)
5Γ(2/3)
(1− ν1/32 − ν
Mn,122 = Nn
ψ̃4/3
4z1ψ̃ − ν2ν3(1− t1)
(1 + t1)z1 + 1− t2t3
8Γ2(1/3)
5Γ(2/3)
Mn,212 = Nn
ψ̃4/3
4z2ψ̃ + ν3(1− t3)
(1 + t3)z2 − ν2(1− t1t2)
2(1/3)
5Γ(2/3)
, Nn =
me3(6χ)1/3eiπ/3Γ(1/6)Γ(n/2− 1/6)
25/672π5/2Γ(n/2 + 7/6)
. (51)
The dependence on n in these expressions is contained in the coefficients Nn which at large
n rapidly decrease as n−4/3. Instead, the dependence of the amplitudes on the parameter χ,
both for a linearly and a circularly polarized laser field, is weak and proportional to χ1/3.
C. Amplitudes for large η and fixed ξ = χ/η
The structure of the photon splitting amplitudes at large η and fixed ξ = χ/η is rather
interesting because the function G(ν) introduced in Eq. (34) and which enters the contri-
butions M
n,σ1σ2σ3 has the form
G(ν) = δn1
imξe3
4η(−ν − i0)
1 + ξ2
sinϑ eiλϑ
1 + ξ2
1 + ξ2(1− sin2 ϑ/ϑ2)
with λ = sign(ν), and therefore contains different powers of log η as well as some functions of
ξ2. However, when we add the corresponding asymptotics of M
1,σ1σ2σ3
which is the leading
term in the limiting case under discussion, any dependence on η and ξ is cancelled out
and we obtain the high-energy asymptotics of photon-photon scattering in vacuum [34].
Therefore, analogously to the case of photon-photon scattering in vacuum, the high-energy
asymptotics of the amplitudes of photon splitting in a laser field is independent of the energy
of the initial photon as well as of the parameter ξ. More precisely, these statements are valid
for any value of ξ which satisfies the inequality ξ2 ≪ η.
VII. POSSIBILITY OF EXPERIMENTAL OBSERVATION OF PHOTON SPLIT-
TING IN A LASER FIELD
We now discuss the possibility of experimental observation of photon splitting in a laser
field. It is natural to consider the principal scheme of the experiment which was used in the
first successful observation of photon splitting in a Coulomb field [7]. In this experiment
a tagged photon beam was used and the energy of each final photon was measured by
a calorimeter. It was necessary to confirm two-photon events, and the distance d between
photons in the calorimeter was about one centimeter. Assuming a distance l ≈ 25 m between
the calorimeter and the interaction region between tagged photons and laser field, then the
angles ϑ2 = ∠(k2,k1) and ϑ3 = ∠(k3,k1) have to be larger than ϑ0 = 4 × 10−4 rad. At
ω1 ≫ ω0 the final photons are almost parallel to the initial one so that ϑ2,3 ≪ 1. It follows
from Eq. (30) that in this limit
ϑ2 = 2
, ϑ3 = 2
. (53)
If, for instance, ν2 > ν3, then for n = 1
4ν3ω0
= 25× 106 ν3ω0
. (54)
For the planned photon energy of the X-FEL at DESY [26], with ω0 = 1 keV, the latter
condition becomes ω1 < 25ν3/ν2 GeV and is not restrictive. Then, we can essentially reduce
l or ω1, or increase ϑ0. The planned intensity of the X-FEL is about 2 × 1015 W/cm2
corresponding to E/Ec ≈ 10−7 and ξ ≈ 5 × 10−5. In this case, the amplitudes of photon
splitting coincide with those of photon-photon scattering. Therefore, it is convenient to
choose ω1 = m
2/ω0 = 250 MeV for which η = 1. This value of η corresponds to the
threshold of electron-positron pair production and therefore the background initiated by
this process will be strongly suppressed. Besides, at η = 1 the cross section of photon-
photon scattering is almost maximal and is given by σ ∼ 10−30 cm2 (see e. g. [16] on pg.
572). Now, from Eq. (33) we find Ẇ ≈ 60 sec−1. The X-FEL pulses will have a duration of
about 100 fs and the interval between pulses will be about 93 ns. If we take, for example, a
flux of 108 tagged photons per second [35], we obtain about two photon splitting events per
hour. However, the very small laser beam-size makes this estimation essentially smaller. In
consequence, the possibility of experimental observation of photon splitting in a laser field
becomes problematic when using the X-FEL and the scheme of Ref. [7].
We discuss now the possibility of using a very strong optical laser as that described
in [25]. In that case ω0 = 1 eV, and the laser intensity will be of order of 10
25 W/cm
corresponding to E/Ec ≈ 4.6 × 10−3. For a value of the angle ϑ0 = 10−4 we obtain, e. g.
for ν3/ν2 = 1/4, approximately 100 MeV as the upper limit for ω1. For these parameters we
find η = 4× 10−4 and χ = 0.9. Therefore, we have to use Eq. (39) but not the asymptotics
(38) because high-order corrections in χ are important. Using the numerical results shown
in Fig. 2 we obtain Ẇ = 4 × 108 sec−1 at χ = 0.9. Though the value of Ẇ is very large,
the duration of each pulse is 10 fs and the laser repetition rate is only 1 Hz [25]. For a total
flux of 108 photons per second [35] and for an electron bunch revolution frequency equal to
1 MHz in the accelerator, we obtain an average number of tagged photons per pulse equal
to 100. If all the photons pass through the laser pulse, we obtain again approximately two
events per hour. However, the transverse beam size of a focused optical laser is even smaller
than that of an X-FEL and the reduction due to the tagged and laser beams overlapping is
larger.
Our negative conclusion about the possibility of observing photon splitting in a laser
field when using the scheme discussed is mainly due to the short duration and small size
of the laser beam as well as to the small flux of tagged photons. On the one hand, short
and focused laser beams are closely coupled with large intensity of the laser pulses. On the
other hand, the use of tagged photons, which essentially restricts the statistics, is needed
to suppress the large background due to many other processes (double Compton scattering
with the experimental equipment like collimators, bremsstrahlung accompanied by e+ − e−
pair production and so on). Of course, we cannot exclude the existence of other schemes of
the experiment that avoid the difficulties pointed out in this Section.
VIII. CONCLUSIONS
In conclusion, we have derived compact expressions for the amplitudes of the photon
splitting process in a plane wave of arbitrary form, intensity and polarization (see Eqs.
(20) and (25)). In the simpler case of a monochromatic circularly polarized laser field,
the amplitudes have the form of three-fold integrals (see Eqs. (34) and (36)). Moreover, we
analyzed in detail various asymptotics of the amplitudes in the case of a monochromatic wave
with linear or circular polarization. We have demonstrated that the angular distribution
of the probabilities corresponding to different numbers of absorbed photons is noticeably
different; this permits us the distinction between these processes for the case where the laser
photon energy is less than the accuracy of the final photon energy measurement.
The photon splitting amplitudes depend on the two Lorentz-invariant parameters η and
χ. For arbitrary values of these parameters, the amplitudes both in the case of a circularly
and a linearly polarized laser field are not equal to zero for any (odd) number n of absorbed
laser photons. This is because the angle between the two final photons is not zero and the
conservation of the projection Jz of the total angular momentum (in the case of a circularly
polarized laser field) does not imply any restriction. However, we have shown that in the
limit of small η the leading contribution to the amplitudes for a circularly polarized laser
comes from the terms with n = 1 and n = 3, while for a linearly polarized laser field the
leading contributions contain the terms with any odd n.
Finally, we have demonstrated that the scheme previously employed in the experiment of
photon splitting in an atomic field together with new generation X-FEL sources and strong
optical lasers is not suitable for the corresponding experiment in a laser field. Therefore, it
is necessary to find another more appropriate experimental configuration to observe photon
splitting in a laser field.
Acknowledgments
We would like to thank K. Z. Hatsagortsyan for valuable discussions. A. I. M. gratefully
acknowledges the great hospitality and the financial support he has received during his visit
at Max-Planck-Institute for Nuclear Physics. The work was supported in part by the RFBR
under grants 0502-16079 and 06-02-04018.
APPENDIX: COEFFICIENTS FOR THE HELICITY AMPLITUDES
Here we present the coefficients Bj,σ1σ2σ3 used in Eq. (36). We introduce the abbreviations
, d1 =
(τ1 − τ2)f2
2τ1τ2
(τ1 + τ3)f3
2τ1τ3
, d2 =
(τ2 − τ1)f2
2τ1τ2
(τ1 − τ3)f3
2τ1τ3
(τ2 + τ1)f2
2τ1τ2
− (τ1 + τ3)f3
2τ1τ3
, h = d1 + d2 + d3 =
(τ2 + τ1)f2
2τ1τ2
(τ1 − τ3)f3
2τ1τ3
− 1− ih , g2 =
− 1− 2if2 + iν2h , g3 =
− 1− 2if3 + iν3h . (A.1)
Then, the nonzero coefficients for the contribution M
n,+++ are
B2,+++ = 4ρn ξ
g2g3 + ν2ν3d
B1,+++ = 4ξ
g∗1g2g3 + ν2ν3g
1 + (ν3g2 + ν2g3)|h|2 − iν2ν3h∗(h2 + d21)
−4ρ2nξx1
+ id1
− 4iξ
(g2 + g3) ,
B0,+++ = −4ρnξ2
+ ig∗1d1 + h
∗d1 +
ρ3nx1(4x2x3 − 1)
B−1,+++ =
[(4x2x3 − 1)g∗1 + ih∗] , (A.2)
The nonzero coefficients for the contribution M
n,++− are
B1,++− =
[(4x1x2 − 1)g2 + iν2h] ,
B0,++− = 4ρnξ
− x1g∗3
− ig2d∗3 − ν2hd∗3 + ν2
+ x1ν3
nx3(4x1x2 − 1)
B−1,++− = 4ξ
−g∗1g2g∗3 + ν2(ν3g∗1 − g∗3)|h|2 + ν3g2d∗23 − iν2ν3h(h∗2 + d∗23 )
4ρ2nξx3
g∗1 + id
(g∗1 + g
B−2,++− =
3 − ν3d∗23
. (A.3)
Finally, the nonzero coefficients for the contribution M
n,+−− are
B0,+−− =
4x1x2x3ρ
B−1,+−− = ρ
4x2x3 − 1
g∗1 −
4x1x2 − 1
g∗2 −
4x1x3 − 1
−2i(1− 2x1)d∗1 +
2i(1− 2x3)
d∗3 +
2i(1− 2x2)
B−2,+−− = 4ρnξ
3 − g∗1
g∗2 +
+ ig∗1d
1 + ig
3 + ig
+x1ν2ν3d
d∗22 +
d∗23 + ν2d
2 + ν3d
3 − d∗2d∗3
B−3,+−− = 4ξ
3(g∗1g
3 + ν2ν3g
1 − ν3g∗2d∗23 − ν2g∗3d∗22 + 2iν2ν3d∗1d∗2d∗3) . (A.4)
The coefficients for the amplitude M
n,+−+ can be obtained fromM
n,++− by the replacement
2 ↔ 3. The coefficients for the amplitudes M (3)n,−−−, M
n,−+− and M
n,−++ can be obtained
with the help of the substitutions
Bj,−++ = −B∗−j,+−−(S → −S) , Bj,−+− = −B∗−j,+−+(S → −S) ,
Bj,−−− = −B∗−j,+++(S → −S) , (A.5)
where the replacement S → −S means that after complex conjugation, it is also necessary
to change the sign of the terms containing S in Eqs. (A.2), (A.3) and (A.4).
FIG. 1: Feynman diagram for the photon splitting amplitude given by Eq. (2). This diagram
corresponds to the Furry representation and the thick line denotes the electron propagator (Green’s
function) in the laser field. The wavy lines symbolize the external photons. The diagram with the
permutation k2 ↔ k3 has to be added.
FIG. 2: The photon splitting rate Ẇ in a circularly polarized laser field in units (π2m2α3/ω1)×10−4
as a function of χ at η ≪ 1. The solid line is obtained by using the amplitudes (39), the dashed
line is based on the small-χ asymptotics (38). The rate is averaged over the polarization of the
initial photon, summed up over the polarizations of the final ones and integrated over ν2 and φq.
FIG. 3: The ratio R defined in Eq. (42) calculated at χ = 1 with the help of Eq. (39) as a function
of the angle ϑ2, measured in units of
ω0/ω1.
FIG. 4: The photon splitting rate Ẇ in a linearly polarized laser field in units (π2m2α3/ω1)×10−4
as a function of the number n of laser photons absorbed. The rate is calculated from the amplitudes
in Eqs. (43) with χ = 1, is averaged over the polarization of the initial photon, summed up over
the polarizations of the final ones and integrated over ν2 and φq.
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	Introduction
	General discussion
	Calculation of R123(2)
	Calculation of R123(3)
	Photon splitting in a monochromatic plane wave
	Asymptotics of the amplitudes
	Amplitudes for small  and small 
	Amplitudes for small  and fixed 
	Amplitudes for large  and fixed =/
	Possibility of experimental observation of photon splitting in a laser field
	Conclusions
	Acknowledgments
	Coefficients for the helicity amplitudes
	References
