Threshold electron-positron pair production by a polarized electron in a strong magnetic field
Resonant e⁺e⁻–-pair production by an electron in a magnetic field near the process threshold is analytically studied. Using the Nikishov's theorem we estimate the number of events in the magnetic field equivalent to laser wave in the SLAC experiment. The obtained estimate is in reasonable agree...
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| Цитувати: | Threshold electron-positron pair production by a polarized electron in a strong magnetic field / O.P. Novak, R.I. Kholodov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 102-104. — Бібліогр.: 14 назв. — англ. |
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irk-123456789-1070052016-10-11T03:02:30Z Threshold electron-positron pair production by a polarized electron in a strong magnetic field Novak, O.P. Kholodov, R.I. Section B. QED Processes at High Energies Resonant e⁺e⁻–-pair production by an electron in a magnetic field near the process threshold is analytically studied. Using the Nikishov's theorem we estimate the number of events in the magnetic field equivalent to laser wave in the SLAC experiment. The obtained estimate is in reasonable agreement with the experimental data. Аналитически исследовано образование e⁺e⁻–-пары электроном в магнитном поле вблизи порога процесса. Произведена оценка числа событий в эксперименте SLAC с использованием теоремы Никишова. Полученное число событий согласуется с экспериментальными данными. Аналітично досліджено утворення e⁺e⁻–-пари електроном в магнітному полі поблизу порогу процесу. Одержана оцінка числа подій в експерименті SLAC з використанням теореми Нікішова. Знайдене число подій узгоджується з експериментальними даними. 2012 Article Threshold electron-positron pair production by a polarized electron in a strong magnetic field / O.P. Novak, R.I. Kholodov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 102-104. — Бібліогр.: 14 назв. — англ. 1562-6016 PACS: 12.20.-m, 13.88.+e http://dspace.nbuv.gov.ua/handle/123456789/107005 en Вопросы атомной науки и техники Institute of Applied Physics NAS of Ukraine |
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Section B. QED Processes at High Energies Section B. QED Processes at High Energies |
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Section B. QED Processes at High Energies Section B. QED Processes at High Energies Novak, O.P. Kholodov, R.I. Threshold electron-positron pair production by a polarized electron in a strong magnetic field Вопросы атомной науки и техники |
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Resonant e⁺e⁻–-pair production by an electron in a magnetic field near the process threshold is analytically studied. Using the Nikishov's theorem we estimate the number of events in the magnetic field equivalent to laser wave in the SLAC experiment. The obtained estimate is in reasonable agreement with the experimental data. |
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Article |
| author |
Novak, O.P. Kholodov, R.I. |
| author_facet |
Novak, O.P. Kholodov, R.I. |
| author_sort |
Novak, O.P. |
| title |
Threshold electron-positron pair production by a polarized electron in a strong magnetic field |
| title_short |
Threshold electron-positron pair production by a polarized electron in a strong magnetic field |
| title_full |
Threshold electron-positron pair production by a polarized electron in a strong magnetic field |
| title_fullStr |
Threshold electron-positron pair production by a polarized electron in a strong magnetic field |
| title_full_unstemmed |
Threshold electron-positron pair production by a polarized electron in a strong magnetic field |
| title_sort |
threshold electron-positron pair production by a polarized electron in a strong magnetic field |
| publisher |
Institute of Applied Physics NAS of Ukraine |
| publishDate |
2012 |
| topic_facet |
Section B. QED Processes at High Energies |
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http://dspace.nbuv.gov.ua/handle/123456789/107005 |
| citation_txt |
Threshold electron-positron pair production by a polarized electron in a strong magnetic field / O.P. Novak, R.I. Kholodov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 102-104. — Бібліогр.: 14 назв. — англ. |
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Вопросы атомной науки и техники |
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AT novakop thresholdelectronpositronpairproductionbyapolarizedelectroninastrongmagneticfield AT kholodovri thresholdelectronpositronpairproductionbyapolarizedelectroninastrongmagneticfield |
| first_indexed |
2025-07-07T19:21:37Z |
| last_indexed |
2025-07-07T19:21:37Z |
| _version_ |
1837017174459285504 |
| fulltext |
THRESHOLD ELECTRON-POSITRON PAIR PRODUCTION
BY A POLARIZED ELECTRON IN A STRONG
MAGNETIC FIELD
O.P. Novak ∗and R.I. Kholodov
Institute of Applied Physics NAS of Ukraine, Sumy, Ukraine
(Received October 31, 2011)
Resonant e+e−-pair production by an electron in a magnetic field near the process threshold is analytically studied.
Using the Nikishov’s theorem we estimate the number of events in the magnetic field equivalent to laser wave in the
SLAC experiment. The obtained estimate is in reasonable agreement with the experimental data.
PACS: 12.20.-m, 13.88.+e
1. INTRODUCTION
Quantum electrodynamic processes in strong mag-
netic field keep their urgency for both theoretical and
experimental study. Strong magnetic field modifies
physical processes and allows new ones to occur, like
e+e− pair production by a moving electron.
The field strength is measured with respect to the
so called critical Schwinger field Bc = m2c3/e� ≈
4.4 · 1013 G. When magnetic field strength B is com-
parable with the critical one, the quantum electrody-
namical treatment of QED processes is necessary.
Critical and subcritical magnetic fields are not
feasible in laboratory conditions at present time. The
strongest field was created using explosive genera-
tors [1] and had the strength of about 30 MG, which
is still much less than Bc. Nevertheless, neutron stars
are believed to have surface magnetic fields within the
range from 1012 G for radiopulsars to 1015 G for mag-
netars. Thus, QED processes in magnetic field are of
great importance in astrophysics.
It is necessary to mention the possibility to study
QED processes in subcritical magnetic field in exper-
iments on heavy ion collisions [2]. If the impact para-
meter has order of magnitude ∼ 10−10 cm, then mag-
netic field of moving ions can approach magnitude of
∼ 1012 G in the region between the ions. These ex-
periments could be carried out at the facilities like
LHC or FAIR, which is under construction at the
present time in GSI (Darmstadt, Germany).
It is necessary to notice that reaction of e+e− pair
production by an electron in intense laser field was
experimentally studied at SLAC [3]. Up to 106 ± 14
events were reported to be observed in collisions of
∼ 50 GeV electron beam with terawatt laser pulses.
In Ref. [4] this process was studied numerically.
In the present work the second-order process of
e+e− pair production by an electron in strong mag-
netic field (see Figure) is analytically studied near
the process threshold. Analytical expressions for the
total process rate were obtained for subcritical mag-
netic field strength B � Bc. It is shown, that the
main contribution to the rate is determined by the
resonant case in agreement with Refs. [3, 4].
2p
1p
p
k
1p
2p
+p
p
k
+p
Feynman diagrams of e+e−-pair production by
an electron in the magnetic field. Double lines
represent solutions of Dirac equation for the electron
in the magnetic field
2. PROCESS RATE
The process of e+e− pair production of electron in
magnetic field can be described by two exchange
Feynman graphs as on the Figure, where double lines
represent solutions of Dirac equation in magnetic
field. The calculations have been made in frame of
Furry picture, thus the following condition should be
satisfied:
b =
B
Bc
� 1. (1)
The process is considered near the threshold,
when the final particles occupy ground Landau levels
and the initial electron energy is (hereinafter rela-
tivistic units with � = c = 1 are used):
E = 3m. (2)
∗Corresponding author E-mail address: novak-o-p@ukr.net
102 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 102-104.
Thus, final spin states are determined.
Lorentz transformation to the reference frame
moving along magnetic field does not change the field,
therefore without loss of generality one can set longi-
tudinal momentum of the initial electron to zero
pz = 0.
The process probability amplitude reads:
Sfi = iα
∫∫
d4xd4x′×
× [
(Ψ̄2γ
μΨ)Dμν(Ψ̄′
1γ
νΨ′
+)−
−(Ψ̄1γ
μΨ)Dμν(Ψ̄′
2γ
νΨ′
+)
]
, (3)
where Dμν is the photon propagator,
Dμν =
gμν
(2π)4
∫
d4k e−ik(x−x′) 4π
kiki
. (4)
When process kinematics allows to be k2 = 0,
then a resonant divergence arises. In this case the in-
termediate photon becomes real and the interaction
range goes to infinity [5]. The divergence was elimi-
nated introducing small imaginary part into the prop-
agator denominator using Breit-Wigner prescription:
ω → ω − i
Δ
2
, (5)
where Δ is the corresponding state width. Nonres-
onant contribution to the process is negligible com-
pared to the resonant part.
The general expressions for the process rate near
the threshold with account of spin projections is ob-
tained in Ref. [6]. Taking into account, that interfer-
ence of the diagrams can be neglected, the rate takes
on the following form:
W+ ≈ α2m
3π2
√
3l!
Y, (6)
W− ∼ bW+, (7)
where superscript denotes the initial electron spin
projection, α is fine structure constant, l is Landau
level number of the initial electron, b = B/Bc, B is
the magnetic field, Bc is the critical field, and
Y =
∫∫
ds du
∣∣∣e−s2
D
∣∣∣2 , (8)
D =
∫
(s + iq)l
r2 − q2
e−q2−2iuqdq. (9)
Here, the following notations are used:
s = mΩ(x0 − x01),
u = mΩ(x0 − x02),
q = kx/m
√
2b,
r2 = Ω2 − s2, Ω2 = 2/b,
(10)
x0, x01 and x02 are x-coordinates of classical orbit
centers of initial and final electrons respectively, kx
is x-component of intermediate photon momentum.
The analysis shows, that the main contribution
in the integral (9) is made by the summand with sl
when r2 > 0 and the integrand has a singularity.
Taking into account, that the quantity Δ is small,
integration in Eqs. (9)-(8) can be carried out analyti-
cally. After performing corresponding calculations it
could be found that
Y = bπ2
√
π
Ω2le−2Ω2
Δ/m
Γ(l + 1/2)
l!
, (11)
where Γ(l + 1/2) is gamma-function.
Averaging the rate over the initial electron spin
projection, finally we obtain (in CGS units):
W = α2
(
mc2
�
)
b
√
π
6
√
3
Ω2le−2Ω2
Δ/m
Γ(l + 1/2)
(l!)2
. (12)
In the resonant case, the total probability can be
expressed via the product of the rates of the first-
order processes of photon radiation and pair produc-
tion by a single photon. Near the process threshold,
when the condition E ≈ 3m is true and consequently
bl = 4 and l � 1 are also valid, the process rate can
be found in the following form:
W =
√
b
3
√
6
We→γeWγ→ee+
Δ
. (13)
Here, We→γe and Wγ→ee+ are the rates of the corre-
sponding first-order processes [7, 8]:
We→γe = αm
√
π
Ω2le−Ω2
Γ(l + 1/2)l
, (14)
Wγ→ee+ = αm
be−Ω2
√
2 δE/m
, (15)
where δE = E − 3m ∼ mb is the kinematic factor.
The intermediate photon width is determined
mainly by the total radiation rate of the initial elec-
tron. The radiation process was studied in a number
of works, e. g. [8–13].
As an example, let us calculate the rate (12) when
field strength is b = 0.1 (B ≈ 4.4 · 1012 G):
Δ ≈ 7 · 1017s−1, (16)
W = 7 · 103 s−1. (17)
3. DISCUSSION
It should be noted, that according to Lorentz trans-
formation arbitrary electromagnetic field goes to al-
most equal crossed electric and magnetic fields in
the rest frame of a relativistic particle. This means
physical equivalence of any field configuration and
electromagnetic wave, if the field changes slowly in
comparison with characteristic electromagnetic time
(∼ 10−21 s). The latter statement known as Nik-
ishov theorem [14] allows to compare the result (12)
with SLAC experiment on observation of e+e− pair
production by electron in intense laser wave.
103
For this purpose one should calculate the rate (12)
in the equivalent magnetic field Beq and carry out av-
eraging over the wave oscillations [14]:
Wemw =
2
π
π/2∫
0
W (Beq sinφ)dφ, (18)
where Beq = 2Femw and Femw is the strength of an
electromagnetic wave.
However, Eq. (12) is true near the process thresh-
old only, when the condition E ≈ 3m is fulfilled.
Therefore before the comparison with the SLAC ex-
periment it is necessary to pass from the laboratory
frame to the “threshold” one where the electron beam
energy is E ≈ 3m. The amplitude value of equal mag-
netic field in the threshold frame is Beq ≈ 6.1·1012 G,
and b ≈ 0.14 respectively.
When calculating the rate Eq. (12) it is necessary
to take into account limited interaction time as well
as the radiative width (16). Therefore the intermedi-
ate state width is a sum of the radiative width and
the quantity 1/ΔtT , where ΔtT is laser-electron in-
teraction time in the threshold frame.
The number of produced pairs is
Ne+e− = k · Nint(1 − e−WeqΔtT ), (19)
where k = 21 962 is number of collisions of the
electron and laser beams [3], Nint is the number of
electrons in the interaction region. The values of
Nint and ΔtT can be estimated using the data from
Ref. [3]: ΔtL ≈ 0.002 fs, Nint ∼ 1.7 ·108. Finally, the
number of events according to Eq. (19) is
Ne+e− ≈ 80, (20)
which is in reasonable agreement with the experimen-
tal result of 106± 14 indicated in Ref. [3].
We thank V.Yu. Storizhko and S.P. Roshchupkin
for useful discussions.
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