Charge asymmetry in the differential cross section of high-energy bremsstrahlung in the field of a heavy atom

0.26 MB·English

by  P. A. Krachkov

Charge asymmetry in the differential cross section of high-energy
bremsstrahlung in the field of a heavy atom
P.A. Krachkov1,2,∗ and A. I. Milstein1,†
1Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
2Novosibirsk State University, 630090 Novosibirsk, Russia
5
1 (Dated: January 19, 2015)
0
2 Abstract

n
a The distinction between the charged particle and antiparticle differential cross sections of high-
J

6 energy bremsstrahlung in the electric field of a heavy atom is investigated. The consideration is
1
based on the quasiclassical approximation to the wave functions in the external field. The charge
]
h
p asymmetry (the ratio of the antisymmetric and symmetric parts of the differential cross section)
-
p arises due to the account for the first quasiclassical correction to the differential cross section. All
e
h
evaluations are performed with the exact account of the atomic field. We consider in detail the
[

1 charge asymmetry for electrons and muons. For electrons, the nuclear size effect is not important
v
7 while for muons this effect should be taken into account. For the longitudinal polarization of the
9
8
initial charged particle, the account for the first quasiclassical correction to the differential cross
3
0
section leads totheasymmetryinthecross section with respecttothereplacement ϕ ϕ, where
1. → −
0 ϕ is the azimuth angle between the photon momentum and the momentum of the final charged
5
1
particle.
:
v
i
X
PACS numbers: 12.20.Ds, 32.80.-t
r
a
Keywords:
∗Electronic address: peter˙[email protected]
†Electronic address: [email protected]
1
I. INTRODUCTION
The theoretical investigation of high-energy bremsstrahlung and high-energy particle-
antiparticle photoproduction in the electric field of a heavy nucleus or atom has a long
history because of importance of these processes for various applications; for the latter
process see reviews in Refs. [1, 2]. These processes should be taken into account when
considering electromagnetic showers in detectors, they also give the significant part of the
radiative corrections in many cases. Therefore, it is necessary to know the cross sections of
these processes with high accuracy. In the Born approximation, the cross sections of both
processes have been obtained forarbitraryenergies ofparticles andforarbitraryatomic form
factors [3, 4] (see also Ref. [5]). The Coulomb corrections to the cross section, which are
the difference between the exact in the parameter η = Zα cross section and the Born cross
section, are very important (here Z is the atomic charge number, α is the fine-structure
constant, ~ = c = 1). There are formal expressions for the Coulomb corrections to the cross
sections exact in η and energies of particles [6]. However, the numerical computations based
on these expressions become more and more difficult when energies are increasing, and, for
instance, the numerical results for e+e− photoproduction have been obtained so far only for
the photon energy ω < 12.5 MeV [7].
At high energies of initial particles, the final particle momenta usually have small angles
with respect to the incident direction. In this case typical angular momenta, which provide
the main contribution to the cross section, are large (l E/∆ 1, where E is energy and
∼ ≫
∆ is the momentum transfer). This is why the quasiclassical approximation, based on the
account of largeangular momenta contributions, becomes applicable. In this approximation,
the wave functions and the Green’s functions of the Dirac equation in the external field have
very simple forms which drastically simplify their use in specific calculations. The wave
functions, obtainedintheleadingquasiclassical approximationfortheCoulomb field, arethe
famous Furry-Sommerfeld-Maue wave functions [8, 9] (see also Ref. [5]). The quasiclassical
Green’s function have been derived in Ref. [10] for the case of a pure Coulomb field, in
Ref. [11] for an arbitrary spherically symmetric field, in Ref. [12] for a localized field which
generally possesses no spherical symmetry, and in Ref. [13] for combined strong laser and
atomic fields.
In the leading quasiclassical approximation, the cross sections for pair photoproduction
2
and bremsstrahlung have been obtained in [14–18]. The first quasiclassical corrections to the
spectra of both processes, as well as to the total cross section of pair photoproduction, have
been obtained inRefs. [19–22]. Recently, the first quasiclassical correction to the fullydiffer-
ential cross section was obtained in Ref. [23] for e+e− pair photoproduction and in Ref. [24]
for µ+µ− pair photoproduction. As a result, the charge asymmetry in these processes (the
asymmetry of the cross section with respect to permutation of particle and antiparticle mo-
menta) was predicted. This asymmetry is absent in the cross section calculated in the Born
approximation and also in the cross section exact in the parameter η but calculated in the
leading quasiclassical approximation. Thus, the charge asymmetry appears solely due to the
quasiclassical corrections to the Coulomb corrections. The difference between the atomic
field and the Coulomb field of a nucleus results in the modification of the cross sections
(effect os screening). The influence of screening on the Coulomb corrections to e+e− pair
photoproduction is small for the differential cross section and for the total cross section
[15]. However, screening is important for the Born term. The quantitative investigation of
the effect of screening on the Coulomb corrections to the photoproduction cross section is
performed in Ref. [19].
The influence of screening on the bremsstrahlung cross section in an atomic field is more
complicated. It is shown in Refs. [16, 20] that the Coulomb corrections to the differential
cross section are very susceptible to screening. However, the Coulomb corrections to the
cross section integrated over the momentum of final charged particle (electron or muon)
are independent of screening in the leading approximation over a small parameter 1/m r ,
e scr
where r Z−1/3(m α)−1 is a screening radius and m is the electron mass. The quanti-
scr e e
∼
tative investigation of the effect of screening on the Coulomb corrections to the spectrum of
bremsstrahlung is performed in Ref. [20]. The differential cross section of bremsstrahlung,
calculated in the leading quasiclassical approximation, is the same for e+ and e− (for µ+ and
µ−). Therefore, topredict thecharge asymmetry (thedifference between thebremsstrahlung
differential cross section for particles and antiparticles in the atomic field), one should per-
form calculations in the next-to-leading quasiclassical approximation. This is the main goal
of our paper. The result is obtained exactly in the parameter η. Besides, for the case of
muons the nuclear size effect is taken into account.
The bremsstrahlung differential cross section from high-energy charged particle in an
3
atomic field, dσ(p,q,k,η), can be written as
dσ(p,q,k,η) = dσ (p,q,k,η)+dσ (p,q,k,η),
s a
dσ(p,q,k,η)+dσ(p,q,k, η)
dσ (p,q,k,η) = − ,
s
2
dσ(p,q,k,η) dσ(p,q,k, η)
dσ (p,q,k,η) = − − , (1)
a
2
wherekisthephotonmomentum, pandq aretheinitialandfinalchargedparticlemomenta,
respectively. Evidently, the bremsstrahlung differential cross section from high-energy an-
tiparticle can be obtained from dσ(p,q,k,η) by the replacement η η, so that it is equal
→ −
to dσ (p,q,k,η) dσ (p,q,k,η). We show that the antisymmetric part of the differential
s a
−
cross section, dσ (p,q,k,η), is independent of screening in the kinematical region which
a
provides the main contribution to the antisymmetric part of the spectrum.
The paper is organized as follows. In Sec. II we derive the general expression for the
quasiclassical matrix element of the process. In Sec. III we find in the quasiclassical approx-
imation all structures of the Green’s function of the squared Dirac equation for a charged
particle in arbitrary localized potential and the corresponding structures of the wave func-
tions. We obtain the leading terms and the first quasiclassical corrections as well. Using
these wave functions, we derive in Sec. IV the matrix element of the process and the corre-
sponding differential cross section for arbitrary localized potential and in the particular case
of the pure Coulomb field. In Sec. V we investigate in detail the charge asymmetry in high-
energy bremsstrahlung from electrons. In this case the nuclear size effect is not important.
In Sec. VI we investigate the charge asymmetry in high-energy bremsstrahlung from muons
which is sensitive to the deviation at small distances of the nuclear atomic field from the
pure Coulomb field. Finally, in Sec. VII the main conclusions of the paper are presented.
II. GENERAL DISCUSSION
The differential cross section of bremsstrahlung in the electric field of a heavy atom
reads [5]
αωqε
dσ = q dΩ dΩ dω M 2, (2)
k q
(2π)4 | |
where dΩ and dΩ are the solid angles corresponding to the photon momentum k and the
k q
final charged particle momentum q, ω = ε ε is the photon energy, ε = p2 +m2,
p q p
−
p
4
ε = q2 +m2, and m is the particle mass. Below we assume that ε m and ε m.
q p q
≫ ≫
The mpatrix element M reads
M = dru¯(−)(r)γ e∗u(+)(r)exp( ik r) , (3)
q · p − ·
Z
where γµ are the Dirac matrices, u(+)(r) and u(−)(r) are the solutions of the Dirac equation
p q
in the external field, e is the photon polarization vector. The superscripts ( ) and (+)
−
remind us that the asymptotic forms of u(−)(r) and u(+)(r) at large r contain, in addition
q p
to the plane wave, the spherical convergent and divergent waves, respectively. The wave
functions u(+)(r) and u(−)(r) have the form [24]
p q
u¯(−)(r) = u¯ [f (r,q) α f (r,q) Σ f (r,q)],
q q 0 1 2
− · − ·
u(+)(r) = [g (r,p) α g (r,p) Σ g (r,p)]u ,
p 0 − · 1 − · 2 p
φ χ
ε +m ε +m
p q
u = σ p , u = σ q , (4)
p q
s 2εp  · φ s 2εq  · χ
ε +m ε +m
p q
   
where φ and χ are spinors, α = γ0γ, Σ = γ0γ5γ, and σ are the Pauli matrices. The
following relations hold
g (r,q) = f (r, q), g (r,q) = f (r, q), g (r,q) = f (r, q). (5)
0 0 1 1 2 2
− − − −
The wave functions in the atomic field can be found from the Green’s function D(r , r ε)
2 1
|
of the “squared” Dirac equation in this field using the relations [24]
expipr
1u(+)(r ) = lim D(r ,r ε )u , p = pn ,
4πr1 p 2 −r1→∞ 2 1| p p − 1
expiqr
2u¯(−)(r ) = lim u¯ D(r ,r ε ), q = qn ,
4πr2 q 1 −r2→∞ q 2 1| q 2
(6)
where n = r /r , n = r /r , and
1 1 1 2 2 2
1
D(r , r ε) = r r
2 1| h 2| ˆ2 m2 +i0| 1i
P −
= r (ε V(r))2 +∇2 m2 +iα ∇V(r)+i0 −1 r . (7)
2 1
h | − − · | i
(cid:2) (cid:3)
Here ˆ = γµ , = (ε V(r),i∇), and V(r) is the atomic potential. It follows from
µ µ
P P P −
Eq. (7) that the Green’s function D(r , r ε) can be written as
2 1
|
D(r , r ε) = d (r ,r )+α d (r ,r )+Σ d (r ,r ). (8)
2 1 0 2 1 1 2 1 2 2 1
| · ·
5
It is convenient to calculate the matrix element for definite helicities of the particles. Let
µ , µ , and λ be the signs of the helicities of initial charged particle, final charged particle,
p q
and photon, respectively. We fix the coordinate system so that ν = k/ω is directed along
z-axis and q lies in the xz plane with q > 0. Denoting helicities by the subscripts, we have
x
1+µ σ n 1+µ 1 θ2 1+µ
φ = p · p p 1+ p (1+µ σ n ) p ,
µp 4cos(θ /2)   ≈ 4 8 p · p  
p 1 µp (cid:18) (cid:19) 1 µp
− −
   
1+µ σ n 1+µ 1 θ2 1+µ
χ = q · q q 1+ q (1+µ σ n ) q ,
µq 4cos(θ /2)   ≈ 4 8 q · q  
q 1 µq (cid:18) (cid:19) 1 µq
− −
1    
e = (e +iλe ), (9)
λ x y
√2
where θ and θ are the polar angles of the vectors p and q, respectively. The unit vectors e
p q x
and e are directed along q and k q, respectively, where the notation X = X (X ν)ν
y ⊥ ⊥
× − ·
for any vector X is used. We also introduce the vectors θ = p /p, θ = q /q, and the
p ⊥ q ⊥
matrix = u u¯ , which can be written as
F pµp qµq
1
= (a +Σ b )[γ0(1+PQ)+γ0γ5(P +Q)+(1 PQ) γ5(P Q)],
F 8 µpµq · µpµq − − −
µ p µ q
p q
P = , Q = , (10)
ε +m ε +m
p q
where a and b are defined from
µpµq µpµq
1
φ χ† = (a +σ b ). (11)
µp µq 2 µpµq · µpµq
We obtain from Eq.(9)
θ2 iµ
a = 1 pq ν [θ θ ],
µµ p q
− 8 − 4 · ×
µ
a = e θ ,
µµ¯ µ pq
√2 ·
1 i
b = µ 1 (θ +θ )2 + ν [θ θ ] ν
µµ p q p q
− 8 4 · ×
(cid:26) (cid:20) (cid:21) (cid:27)
µ i
+ (θ +θ )+ [θ ν],
p q pq
2 2 ×
1
b = √2e (e ,θ +θ )ν, (12)
µµ¯ µ µ p q
− √2
where θ = θ θ and µ¯ = µ. The matrix element M, Eq. (3), can be written as follows
pq p q
− −
M = dre−ik·rSp (f α f Σ f )γ e∗(g α g Σ g ) . (13)
{ 0 − · 1 − · 2 · λ 0 − · 1 − · 2 F}
Z
6
Note that only the terms with (P+Q) and (1+PQ) in , Eq. (10), contribute to the matrix
F
element (13) because it contains the odd number of the gamma-matrices.
In the quasiclassical approximation the relative magnitude of the functions f , f , g ,
0 1,2 0
and g is different, so that
1,2
f l f l2f , g l g l2g , d l d l2d , (14)
0 ∼ c 1 ∼ c 2 0 ∼ c 1 ∼ c 2 0 ∼ c 1 ∼ c 2
where l ε/∆ 1 is the characteristic value of the angular momentum in the process,
c
∼ ≫
∆ = q+k pisthemomentumtransfer. Tofindthedistinctionbetweenthedifferentialcross
−
sectionofbremsstrahlung fromparticlesandantiparticles, itisnecessary totakeintoaccount
the first quasiclassical corrections to the functions f , g , f , and g , while the functions
0 0 1 1
f and g can be taken in the leading quasiclassical approximation. Let us introduce the
2 2
quantities
(A , A , A , A , A ) = dr exp( ik r)(f g , f g , f g , f g ,f g ). (15)
00 01 10 02 20 0 0 0 1 1 0 0 2 2 0
− ·
Z
In terms of these quantities, the matrix element M has the form
M = δ δ (e∗, θ A 2A +2µ A )
µpµq λµp λ − q 00 − 10 p 20
h mµ (p q)
+δ (e∗, θ A +2A +2µ A ) p − δ δ A . (16)
λµ¯p λ − p 00 01 p 02 − √2pq µqµ¯p λµp 00
i
Below we calculate all quantities in (16) for arbitrary atomic potential V(r) which includes
the effect of screening and the nuclear size effect as well.
III. GREEN’S FUNCTIONS AND WAVE FUNCTIONS
Let us consider the case of arbitrary central localized potential V(r). We expand the
Green’s function D(r , r ε), Eq. (7), up to the second order with respect to the correction
2 1
|
α ∇V(r):
·
1 1 1 1 1 1
D(r , r ε) = r iα ∇V(r) + iα ∇V(r) iα ∇V(r) r ,
2 1 2 1
| h | − · · · | i
H H H H H H
V2(r)
= ε2 m2 2εϕ(r)+∇2 +i0, ϕ(r) = V(r) . (17)
H − − − 2ε
The function D(0)(r , r ε) = r −1 r is the Green’s function of the Klein-Gordon equa-
2 1 2 1
| h |H | i
tion. This function was found in the quasiclassical approximation with the first correction
7
taken into account [12] :
ieiκr 1
D(0)(r ,r ε) = dQexp iQ2 ir dxV(R )
2 1| 4π2r − x
Z (cid:20) Z0 (cid:21)
1 x
ir3
1+ dx dy(x y)∇ V(R ) ∇ V(R ) ,
⊥ x ⊥ y
× 2κ − · 
 Z0 Z0 
2r r
r =r r , R = r +xr +Q 1 2 ,  (18)
2 1 x 1
− κr
r
where Q is a two-dimensional vector perpendicular to r and ∇ is the component of the
⊥
gradient perpendicular to r. Within the same accuracy, D(0)(r , r ε) coincides with the
2 1
|
contribution d(r , r ) to the Green’s function D(r , r ε), Eq. (8).
2 1 2 1
|
Using this formula and Eqs. (4), (6), and (8), we obtain the function f (r,q),
0
i ∞
f (r,q) = e−iq·r dQexp iQ2 i dxV(r )
0 x
−π −
Z (cid:20) Z0 (cid:21)
∞ x
i
1+ dx dy(x y)∇ V(r ) ∇ V(r ) ,
⊥ x ⊥ y
× 2ε − · 
q
 Z0 Z0 
2r
 
r = r +xn +Q , Q n = 0, (19)
x q q
ε ·
s q
where ∇ is the component of the gradient perpendicular to n = q/q. Then we use the
⊥ q
relation
1 i
i∇V(r) = [p, ]+ ∇V2(r), (20)
2ε H 2ε
and write the linear in ∇V(r) term in Eq. (17) as α d (r ,r ), where
1 2 1
·
i
d (r ,r ) = (∇ +∇ )D(0)(r ,r ε)+δd (r ,r ),
1 2 1 1 2 2 1 1 2 1
−2ε |
1 i 1
δd (r ,r ) = r ∇V2(r) r . (21)
1 2 1 2 1
−h | 2ε | i
H H
If we replace V(r) by V(r)+δV(r) in the operator , where δV(r) = iα ∇V2(r)/(2ε)2,
H − ·
then we obtain from Eq. (18)
1
ieiκr 1
δd (r ,r ) = dQexp iQ2 ir dxV(R ) dx∇V2(R ), (22)
1 2 1 −16π2ε2 − x x
Z (cid:20) Z0 (cid:21)Z
0
where R is given in (18). Using Eqs. (4), (6), and (8), we find the function f (r,q),
x 1
1
f (r,q) = (i∇ q)f (r,q)+δf (r,q),
1 0 1
2ε −
8
∞
i ∞
δf (r,q) = e−iq·r dQexp iQ2 i dxV(r ) dx∇V2(r ). (23)
1 −4πε2 − x x
Z (cid:20) Z0 (cid:21)Z
0
where r is given in Eq. (19).
x
To transform the third term in (17), we replace i∇V(r) by 1 [p, ]. Then it follows
2ε H
from Eqs. (4), (6), and (8) that the function d (r ,r ) is
2 2 1
i
d (r ,r ) = [∇ ∇ ]D(0)(r ,r ε)+δd (r ,r ),
2 2 1 −(2ε)2 2 × 1 2 1| 2 2 1
1 V′(r) 1
δd (r ,r ) = l r r , (24)
2 2 1 21 2 1
h | 2εr | i
H H
where l = (i/2)(r ∇ r ∇ ) and V′(r) = ∂V(r)/∂r. In (24) we use the
21 2 2 1 1
− × − ×
relation [l, ] = 0. If we replace V(r) by V(r) + δV(r) in the operator , where
H H
δV(r) = r−1V′(r)/(2ε)2, then we obtain from Eq. (18)
1
eiκr 1 V′(R )
δd (r ,r ) = l dQexp iQ2 ir dxV(R ) dx x . (25)
2 2 1 21 16π2ε2 − x R
Z (cid:20) Z0 (cid:21) Z x
0
Substituting this expression in (24), we finally find d (r ,r ),
2 2 1
reiκr 1
d (r ,r ) = dQexp iQ2 ir dxV(R )
2 2 1 −16π2ε2 − x
Z (cid:20) Z0 (cid:21)
1 x
dx dy[∇V(R ) ∇V(R )]. (26)
x y
× ×
Z Z
0 0
The corresponding function f (r,q) is
2
e−iq·r 1
f (r,q) = dQexp iQ2 ir dxV(r )
2 − 4πε2 − x
Z (cid:20) Z0 (cid:21)
∞ x
dx dy[∇V(r ) ∇V(r )]. (27)
x y
× ×
Z Z
0 0
For the Coulomb field V (r) = η/r, we find from (19), (23), and (27)
c
−
f (r,q) = F +(1+n n)F ,
0 A q C
·
f (r,q) = (n +n)ηF ,
1 q B
f (r,q) = iΣ [n n]F , (28)
2 q C
− · ×
where
πη
F (r, q, η) = exp iq r [Γ(1 iη)F(iη,1, iz)
A
2 − · −
(cid:16) (cid:17)
9
πη2eiπ4
+ Γ(1/2 iη)F(1/2+iη,1, iz)],
2√2qr −
i πη
F (r, q, η) = exp iq r [Γ(1 iη)F(1+iη, 2, iz)
B
−2 2 − · −
πη2eiπ4 (cid:16) (cid:17)
+ Γ(1/2 iη)F(3/2+iη, 2, iz)],
2√2qr −
πη πη2eiπ4
F (r, q, η) = exp iq r Γ(1/2 iη)F(3/2+iη,2, iz),
C
− 2 − · 8√2qr −
(cid:16) r (cid:17)
z = (1+n n )qr, n = . (29)
q
· r
Here Γ(x) is the Euler Gamma function and F(α,β,x) is the confluent hypergeometric
function. The results (28) and (29) are in agreement with that obtained in [23].
IV. CALCULATION OF THE MATRIX ELEMENT
The calculation of the quantities A , A , A , A , and A (15) is performed in the
00 01 10 02 20
same way as in Ref.[20]. We present details of this very tricky calculation in Appendix. We
obtain
1
A = drexp[ i∆ r iχ(ρ)] i2ε ε ξ ξ (p +q )
00 ωm4 − · − p q p q ⊥ ⊥
Z ∞ h
+m2(ε ξ ε ξ ) dxx∇ V(r xν) ∇ V(r),
p p q q ⊥ ⊥
− − ·
Z0 i
ε ξ
A = q q drexp[ i∆ r iχ(ρ)]
01 ωm2 − · −
Z
∞
∆ i
i∇ V(r)+ dxx∇ V(r xν) ∇ V(r)+ ∇ V2(r) ,
⊥ ⊥ ⊥ ⊥
× 2ε − · 2ε
p p
Z
h 0 i
ε ξ
A = q q drexp[ i∆ r iχ(ρ)]
02
−2ωε m2 − · −
p
Z
∞
dx[∇V(r xν) ∇V(r)],
× − ×
Z
0
A = A (ε ε , ξ ξ ), A = A (ε ε , ξ ξ ),
10 01 q p q p 20 02 q p q p
− ↔ ↔ − ↔ ↔
∞ m2 m2
χ(ρ) = V(z,ρ)dz, ξ = , ξ = . (30)
p m2 +p2 q m2 +q2
Z−∞ ⊥ ⊥
Substituting Eq. (30) in Eq. (16), we find the matrix element M,
M = δ (ε δ +ε δ )[N (e∗,ξ p ξ q )+N (e∗,ε ξ p ε ξ q )]
− µpµq p λµp q λµ¯p 0 λ p ⊥ − q ⊥ 1 λ p p ⊥ − q q ⊥
10