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INTEGRAL CHARACTERISTICS OF BREMSSTRAHLUNG
AND PAIR PHOTOPRODUCTION IN A MEDIUM
V. N. BAIER AND V. M. KATKOV
Budker Institute of Nuclear Physics, Novosibirsk, 630090, Russia
E-mail: baier@inp.nsk.su
The bremsstrahlung of an electron and e−e+-pair creation by a photon in a medium
is considered in high-energy region, where influence of the multiple scattering on the
processes (the Landau-Pomeranchuk-Migdal (LPM) effect) becomes essential. The
integral characteristics: the radiation length and the total probability of radiation
and pair photoproduction are analyzed under influence of the LPM effect.
1 Introduction
When a charged particle is moving in a medium it scatters on atoms. With
probability ∼ α this scattering is accompanied by a radiation. At high en-
ergy the radiation process occurs over a rather long distance, known as the
formation length lc:
lc =
l0
1 + γ2ϑ2c
, l0 =
2εε′
m2ω , (1)
where ω is the energy of emitted photon, ε(m) is the energy (the mass) of a
particle, γ = ε/m is the Lorenz factor, ε′ = ε − ω, ϑc is the characteristic
angle of photon emission, the system h¯ = c = 1 is used.
Landau and Pomeranchuk were the first who showed that if the formation
length of bremsstrahlung becomes comparable to the distance over which the
multiple scattering becomes important (when the mean angle of multiple scat-
tering is of the order of the characteristic angle of photon emission ∼ 1/γ),
the bremsstrahlung will be suppressed 1. Migdal 2 developed the quantitative
theory of this phenomenon.
New activity with the theory of the LPM effect (see 3, 4, 5) is connected
with a very successful series of experiments performed at SLAC recently (see
6, 7). In these experiments the cross section of the bremsstrahlung of soft
photons with energy from 200 keV to 500 MeV from electrons with energy
8 GeV and 25 GeV is measured with an accuracy of the order of a few percent.
Both LPM and dielectric suppression are observed and investigated. These
experiments were the challenge for the theory since in all the mentioned pa-
pers calculations are performed to logarithmic accuracy which is not enough
for description of the new experiment. The contribution of the Coulomb cor-
lpm2ws: submitted to World Scientific on February 1, 2008 1

rections (at least for heavy elements) is larger than experimental errors and
these corrections should be taken into account.
We developed the new approach to the theory of the Landau-
Pomeranchuk-Migdal (LPM) effect 8 basing on the quasiclassical operator
approach 9. In this paper the cross section of the bremsstrahlung process
in the photon energies region where the influence of the LPM is very strong
was calculated with a term ∝ 1/L , where L is characteristic logarithm of
the problem, and with the Coulomb corrections taken into account. In the
photon energy region, where the LPM effect is ”turned off”, the obtained
cross section gives the exact Bethe-Maximon cross section (within power ac-
curacy) with the Coulomb corrections. This important feature was absent in
the previous calculations. Some important features of the LPM effect were
considered also in 10, 11, 12, 13.
The crossing process for the bremsstrahlung is the pair creation by a
photon. The created particles undergo here the multiple scattering. It should
be emphasized that for the bremsstrahlung the formation length (1) increases
strongly if ω ≪ ε. Just because of this the LPM effect was investigated at
SLAC at a relatively low energy. For the pair creation by a photon with energy
ω the formation length lp =
2ε(ω − ε)
m2ω attains maximum at ε = ω/2 and this
maximum is lp,max = (ω/2m)λc. Because of this even for heavy elements
the effect of multiple scattering becomes noticeable at photon energies ω ≥
10 TeV. Starting from these energies one has to take into account the influence
of a medium on the pair creation and on the bremsstrahlung hard part of
the spectrum in electromagnetic showers being created by the cosmic ray
particles of the ultrahigh energies. These effects can be quite significant in
the electromagnetic calorimeters operating in the detectors on the colliders in
TeV range.
In the present paper the radiation length is calculated under influence of
the LPM effect. The total probability of photon radiation and the integral
probability of the pair creation are considered also.
2 Influence of the multiple scattering on the bremsstrahlung
2.1 Bremsstrahlung spectrum at high energy
The spectral radiation intensity obtained in 8 (see Eq.(2.39)) has the form
dI = ωdW = αm
2xdx
2π(1− x) Im
[
Φ(ν)− 1
2Lc
F (ν)
]
, x = ωε , (2)
lpm2ws: submitted to World Scientific on February 1, 2008 2

where
Φ(ν) =
∫ ∞
0
dze−it
[
r1
(
1
sinh z −
1
z
)
− iνr2
(
1
sinh2 z
− 1z2
)]
= r1
(
ln p− ψ
(
p+ 1
2
))
+ r2
(
ψ(p)− ln p+ 1
2p
)
,
F (ν) =
∫ ∞
0
dze−it
sinh2 z
[r1f1(z)− 2ir2f2(z)] ,
f1(z) =
(
ln ̺2c + ln
ν
i − ln sinh z − C
)
g(z)− 2 cosh zG(z),
f2(z) =
ν
sinh z
(
f1(z)−
g(z)
2
)
, g(z) = z cosh z − sinh z,
G(z) =
∫ z
0
(1 − y coth y)dy
= z − z
2
2
− π
2
12
− z ln
(
1− e−2z
)
+
1
2
Li2
(
e−2z
)
,
t = zν , r1 = x
2, r2 = 1+ (1 − x)2, t = t1 + t2, z = νt. (3)
here α = 1/137, z = νt, p = i/(2ν), ψ(x) is the logarithmic derivative of
the gamma function, Li2 (x) is the Euler dilogarithm. Use of found form
of Φ and the last representation of function G(z) simplifies the numerical
calculation. The term with Φ(ν) in (2) describes the intensity in logarithmic
approximation, the term with F (ν) is the first correction. The parameters in
these formulas are
ν2 = iν20 , ν20 = |ν|2 ≃ ν21
(
1 +
ln ν1
L1
ϑ(ν1 − 1)
)
, ν21 =
ε
εe
1− x
x ,
εe = m
(
8πZ2α2naλ3cL1
)−1 , Lc ≃ L1
(
1 +
ln ν1
L1
ϑ(ν1 − 1)
)
, L1 = ln
a2s2
λ2c
,
as2
λc
= 183Z−1/3e−f , f = f(Zα) = (Zα)2
∞
∑
k=1
1
k(k2 + (Zα)2) , (4)
here Z is the charge of the nucleus, na is the number density of atoms in
the medium, λc = 1/m is the electron Compton wavelength. The LPM effect
manifests itself when
ν1(xc) = 1, xc =
ε
εe + ε
. (5)
In the case ε ≪ εe in the hard part of spectrum (1 ≥ x ≫ xc) the
parameter ν21 ≃ xc/x≪ 1 and the contribution into the integral (3) give the
lpm2ws: submitted to World Scientific on February 1, 2008 3

region z ≪ 1.
Im Φ(ν) ≃ r1
ν21
6
+ r2
ν21
3
, −Im F (ν) = −1
9
(r2 − r1)ν21 (1 +O(ν41 )). (6)
Substituting into (2) we have
dI
dx =
2Z2α3naε
3m2
[
r1
(
L1 −
1
3
)
+ 2r2
(
L1 +
1
6
)
]
(7)
This is the Bethe-Maximon intensity spectrum (with the Coulomb correc-
tions) in case of complete screening (if one neglects the contribution of atomic
electrons) written down within power accuracy (omitted terms are of the or-
der of powers of 1/γ), see e.g. Eq.(18.30) in 14. So, to obtain it in the limit
considered one has to take into account the both terms in brackets in (2).
At very strong multiple scattering ν0 ≫ 1 or ε≫ εe one can omit e−it in
the integrand of function F(ν) (3). Integrating over z we obtain
−Im F (ν) = π
4
(r1 − r2) +
ν0√
2
(
ln 2− C + π
4
)
r2, (8)
where we take into account the next terms of the decomposition in the term
∝ r2. Under the same conditions (ν0 ≫ 1) the function Im Φ(ν) is
Im Φ(ν) = π
4
(r1 − r2) +
ν0√
2
r2. (9)
Thus, at ν0 ≫ 1 the relative contribution of the first correction
dW 1
dω is defined
by
r = dW
1
dW c =
1
2Lc
(
ln 2− C + π
4
)
≃ 0.451Lc
. (10)
In the case ε ≥ εe the intensity spectrum differs from the Bethe-Maximon
one at x ∼ 1 also. When ε ≫ εe we find in the interval not very close to the
end of the spectrum (x = 1):
dI
dx ≃
2
√
2Z2α3naε
m2
√ εex
ε(1− x)
(
1 +
1
4L1
ln
ε(1− x)
εex
)
[
x2
+2(1− x)
(
1− π
2
√
2
√ εex
ε(1− x)
)
]
, ε(1− x) ≫ εex. (11)
lpm2ws: submitted to World Scientific on February 1, 2008 4

2.2 Integral characteristics of bremsstrahlung
Now we turn to the integral characteristics of radiation. The total intensity
of radiation in the logarithmic approximation can be presented as (see (2))
I
εL
0
rad = 2
εe
ε Im
[
∫ 1
0
dx
g
√ x
1− x (2(1− x) + x
2)
+
∫ 1
0
x3dx
1− x
(
ψ(p+ 1)− ψ
(
p+ 1
2
))
+ 2
∫ 1
0
xdx (ψ (p+ 1)− ln p)
]
,(12)
where
p = gη
2
, η =
√ x
1− x, g = exp
(
iπ
4
)
√
L1
Lc
εe
ǫ ,
L0rad is the radiation length in the logarithmic approximation. The relative
energy losses of electron per unit time in terms of the Bethe-Maximon radia-
tion length L0rad:
I
εL
0
rad in gold is given in Fig.1 (curve 1), it reduces by 10%
(15% and 25%) at ε ≃ 700 GeV (ε ≃ 1.4 TeV and ε ≃ 3.8 TeV) respectively,
and it cuts in half at ω ≃ 26 TeV. This increase of effective radiation length
can be important in electromagnetic calorimeters operating in detectors on
colliders in TeV range. The contribution of the correction terms r (see (10))
is r ≃ 0.451/Lc.
In Eqs.(7) and (11) we can use the main terms of decomposition only.
The main term in (7) gives after the integration over x the standard ex-
pression for the radiation length Lrad without influence of multiple scattering.
I
ε =
αm2
4πεe
(
1 +
1
9L1
− 4π
15
ε
εe
)
≃ L−1rad
(
1− 4π
15
ε
εe
)
,
1
Lrad
=
2Z2α3naL1
m2
(
1 +
1
9L1
)
=
1
L0rad
(
1 +
1
9L1
)
(13)
The integration over x of the main term in (11) gives (terms ∝
√
εe/ε in
the square brackets are neglected)
I0 ≃
9πZ2α3na
√εεe
4
√
2m2
L1
[
1 +
1
4L1
(
ln
ε
εe
− 46
27
)
+ r0
]
, (14)
where r0 = (ln 2− C + π/4) /2L1. The corrections (without terms ∝ 1/L1) to
(14) are calculated in Appendix B of 13(see Eq.(B.11)). The complete result
is
I
εLrad
≃ 5
2
√
εe
ε
[
1− 2.37
√
εe
ε − 4.57
εe
ε +
1
4L1
(
ln
ε
εe
− 0.3455
)]
(15)
lpm2ws: submitted to World Scientific on February 1, 2008 5

C2 = 2C −
5
8
+ 12
∫ ∞
0
ln z
(
1
z3 −
cosh z
sinh3 z
)
dz ≃ 1.96 (16)
In the case ε ≫ εe we can calculate the integral probability of radia-
tion starting with Eq.(11). Conserving the main term, dividing it by xε and
integrating over x we find
W0 =
11πZ2α3na
2
√
2m2
√
εe
ε L1
[
1 +
1
4L1
(
ln
ε
εe
+
8
11
)
+ r0
]
(17)
The correction terms to Eq.(16) are calculated in Appendix B of 13(see
Eq.(B.13)). Substituting them we have
W = 11πZ
2α3na
2
√
2m2
√
εe
ε L1
[
1− 1.23
√
εe
ε + 1.65
εe
ε +
1
4L1
(
ln
ε
εe
+ 2.53
)]
.
(18)
Ratio of the main terms of Eqs.(15) and (18) gives the mean energy of
radiated photon
ω¯ = 9
22
ε ≃ 0.409ε. (19)
3 Influence of multiple scattering on pair creation process
The probability of the pair creation by a photon can be obtained from the
probability of the bremsstrahlung with help of the substitution law:
ω2dω → ε2dε, ω → −ω, ε→ −ε, (20)
where ω is the initial photon energy, ε is the energy of the created electron.
Making this substitution in Eq.(2) we obtain the spectral distribution of the
pair creation probability (over the energy of the electron)
dW cp
dε =
αm2
2πεε′ Im
[
Φp(ν)−
1
Lc
Fp(ν)
]
,
Φp(ν) = ν
∫ ∞
0
dte−it
[
s1
(
1
sinh z −
1
z
)
− iνs2
(
1
sinh2 z
− 1z2
)]
= s1
(
ln p− ψ
(
p+ 1
2
))
+ s2
(
ψ(p)− ln p+ 1
2p
)
,
Fp(ν) =
∫ ∞
0
dze−it
sinh2 z
[s1f1(z)− 2is2f2(z)] ,
s1 = 1, s2 =
ε2 + ε′2
ω2 , ε
′ = ω − ε. (21)
lpm2ws: submitted to World Scientific on February 1, 2008 7

1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5
E n erg y o f e le c tro n in T e V
0
2
4
6
8
10
To
ta
l p
ro
ba
bi
lit
y
o
f r
ad
ia
tio
n
Figure 2. The total probability of photon emission W0 in terms of the Bethe-Maximon
radiation length L0rad in gold vs the initial energy electron .
All entering functions are defined in (3).
The total probability of pair creation in the logarithmic approximation
can be presented as (see (21))
W cp
WBHp0
=
9
14
ωe
ω Im
∫ 1
0
dy
y(1− y)
[
(
ln p− ψ
(
p+ 1
2
))
+
(
1− 2y + 2y2
)
(
ψ (p)− ln p+ 1
2p
)
]
, p = bs
4
, (22)
where
s = 1√
y(1− y)
, b = exp
(
iπ
4
)
√
L1
Lc
ωe
ω , ωe = m
(
2πZ2α2naλ3cL1
)−1 ,
here WBHp0 is the Bethe-Maximon probability of pair photoproduction in the
logarithmic approximation. Note that ωe is four times larger than εe, in gold
ωe = 10.5 TeV. This is just the value of photon energy starting with the
lpm2ws: submitted to World Scientific on February 1, 2008 8

LPM effect becomes essential for the pair creation process in heavy elements.
The total probability of pair creation W cp in gold is given in Fig.1 (curve 2),it
reduced by 10% at ω ≃ 9 TeV and it cuts in half at ω ≃ 130 TeV.
4 Conclusion
In this paper we considered the influence of multiple scattering on the
bremsstrahlung process at any energy including the high-energy region (ε ≥
εe), where all the spectrum of radiation is distorted. In this region the total in-
tensity of radiation diminishes and respectively the radiation length increases.
The cross section of e−e+ pair creation by a photon changes essentially if the
photon energy ω ≥ ωe = 4εe, see Eq.(4).
If we restrict to the main terms of the decomposition Eq.(15) in asymp-
totic region ε ≫ εe, then the intensity of radiation and the corresponding
radiation length can be written as
I ≃ 9
16
√
π
2
Zα2
(
εna ln
(
9πZ2α2εnaa4s2
))1/2 , Lrad =
ε
I(ε) . (23)
The integral cross section of radiation follows from the integral probability of
radiation (18)
σ = Wna
≃ 11
8
√
π
2
Zα2√εna
(
ln
(
100πZ2α2εnaa4s2
))1/2 . (24)
We have from for the total probability of pair creation by a photon at ω ≫ ωe
and the corresponding cross section
Wp ≃
3
4
√
π
2
Zα2
(na
ω ln
(
2πZ2α2ωnaa4s2
)
)1/2
, σp =
Wp
na
(25)
The Eqs.(23)-(25) don’t depend on the electron mass and the cross sections of
bremsstrahlung and pair creation diminish with energy and density na growth.
In this paper we considered the case of an infinitely thick target where
the formation length is much shorter than the thickness of a target. Because
of this we neglected the boundary effects. These effects were considered in
detail in 8,10, they can give quite essential contribution in the soft part of
spectrum depending on the target thickness. We neglected also by effects of
the polarization of a medium. They were considered in detail in 8. The relative
contribution of polarization of a medium into probability of pair creation is
discussed in 15
ω20εε′
ω2m2 ≤
ω20
m2 < 10
−7 ≪ 1, ω20 =
4πe2ne
m , (26)
lpm2ws: submitted to World Scientific on February 1, 2008 9

where ne is the number density of electron in the medium, ω0 is the plasma
frequency. The contribution of polarization of a medium into the total energy
losses in thick target is of the order ω0/m. The polarization of a medium
affects at the soft part of the spectrum only at ω ≤ ωp = γω0 (x ≤ ωp/ε =
ω0/m). Even for heavy elements ω0/m ∼ 2 · 10−4. This contribution was
analyzed in 8.
Acknowledgments
. This work was supported in part by the Russian Fund of Basic Research
under Grant 00-02-18007.
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