Electron-positron pair photo-production with radiation of a photon in magnetic field at nonresonant regime
Theoretical investigations of quantum electrodynamic processes in strong magnetic field are carried out. Such the processes may occur between colliding heavy ions. Magnetic fields of the nuclei are added and electric fields of nuclei mutually compensate one another in that region. The electron-posit...
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| Cite this: | Electron-positron pair photo-production with radiation of a photon in magnetic field at nonresonant regime / P.I. Fomin, R.I. Kholodov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 111-114. — Бібліогр.: 10 назв. — англ. |
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| citation_txt | Electron-positron pair photo-production with radiation of a photon in magnetic field at nonresonant regime / P.I. Fomin, R.I. Kholodov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 111-114. — Бібліогр.: 10 назв. — англ. |
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| description | Theoretical investigations of quantum electrodynamic processes in strong magnetic field are carried out. Such the processes may occur between colliding heavy ions. Magnetic fields of the nuclei are added and electric fields of nuclei mutually compensate one another in that region. The electron-positron pair production by a photon in the case when one additional photon is emitted in external magnetic field under nonresonant condition has been investigated. Kinematics of the process and the resonance conditions in approximation of strong magnetic field and weakly excited electron (positron) states (ultra-quantum approximation) have been studied. The resonant conditions have the place, when the photon energy is close to the splitting between Landau levels. The differential probability of nonresonant process has been obtained. The probability of the process is three order of magnitude less the resonant case.
Проводятся теоретические исследования квантово-электродинамических процессов в сильном магнитном поле. Такие процессы могут происходить между сталкивающимися тяжелыми ионами. В этой области магнитные поля ядер складываются, а электрические поля взаимно компенсируются. Исследуется процесс рождения электрон-позитронной пары фотоном в случае, когда излучается один дополнительный фотон во внешнем магнитном поле при нерезонансных условиях. Изучаются кинематика процесса и условия резонанса в приближении сильного магнитного поля и слабо возбужденных состояний электронов (позитронов). Резонансные условия имеют место, когда энергия фотона близка к расстоянию между уровнями Ландау. Получена дифференциальная вероятность нерезонансного процесса в единицу времени. Вероятность такого процесса на три порядка меньше резонансного случая.
Проводяться теоретичні дослідження квантово-електродинамічних процесів в сильному магнітному полі. Такі процеси можуть відбуватися при зіткненні важких іонів. В області між ними магнітні поля ядер складаються, а електричні поля взаємно компенсуються. Досліджується процес народження електрон-позитронної пари фотоном у випадку, коли випромінюється один додатковий фотон в зовнішньому магнітному полі при нерезонансних умовах. Вивчаються кінематика процесу і умови резонансу в наближенні сильного магнітного поля і слабо збуджених станів електронів (позитронів). Резонансні умови мають місце, коли енергія фотона близька до відстані між рівнями Ландау. Знайдено диференційну ймовірність нерезонансного процесу в одиницю часу. Ймовірність такого процесу на три порядки менша за резонансний випадок.
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ELECTRON-POSITRON PAIR PHOTO-PRODUCTION WITH
RADIATION OF A PHOTON IN MAGNETIC FIELD AT
NONRESONANT REGIME
P.I. Fomin 1,2 and R.I. Kholodov 1∗
1Institute of Applied Physics NAS of Ukraine, Sumy, Ukraine
2Bogolyubov Institute for Theoretical Physics NAS of Ukraine, Kiev, Ukraine
(Received November 1, 2011)
Theoretical investigations of quantum electrodynamic processes in strong magnetic field are carried out. Such the
processes may occur between colliding heavy ions. Magnetic fields of the nuclei are added and electric fields of nuclei
mutually compensate one another in that region. The electron-positron pair production by a photon in the case
when one additional photon is emitted in external magnetic field under nonresonant condition has been investigated.
Kinematics of the process and the resonance conditions in approximation of strong magnetic field and weakly excited
electron (positron) states (ultra-quantum approximation) have been studied. The resonant conditions have the place,
when the photon energy is close to the splitting between Landau levels. The differential probability of nonresonant
process has been obtained. The probability of the process is three order of magnitude less the resonant case.
PACS: 12.20.-m, 13.88.+e
1. INTRODUCTION
FAIR (Facility for antiproton and ion research
project) is one of the largest research project today.
It will be erected at GSI (Darmstadt) in the next
few years. Quantum-electrodynamic test in strong
electromagnetic fields, for example under heavy ions
collisions, is one of the important applied task of this
project.
The quantum-electrodynamic (QED) processes in
the presence of strong magnetic field close to the crit-
ical value of about 1013 G may accompany fast heavy
nuclei collisions. The magnetic field is generated by
two colliding nuclei, that play role of two current.
The magnetic field produced by colliding nuclei in
the region between them at the moment of the clos-
est approach has order of magnitude about 1012 G in
the case, when that region has the size of Compton
wavelength of electron. The electric fields of nuclei
mutually compensate one another in that region.
We consider, that the series of quasi-equidistant
narrow peaks in the electron-positron distribution of
total energy, observed more then ten years ago in
heavy ions collision at GSI, Darmstadt [1,2], is a re-
sult of movement of an electron-positron pair in such
magnetic field in that region. Narrow lines are the
resonant pair production on the Landau levels [3].
The first theoretical works for study of the process
of electron-positron pair photoproduction in mag-
netic field were performed in the middle of the last
century yet [4]. There are monographs that are de-
voted to the first order QED processes in magnetic
field [5,6]. Spin polarization effects are considered
in [7]. However, it should be noted, that similar
quantum-electrodynamic processes can be accompa-
nied by emitting of additional photon. Such a process
in resonant condition has been considered in previous
work [8].
This work is devoted to the study of such the
process at nonresonant regime. In this work we use
the relativism system of units: � = 1, c = 1.
2. PROBABILITY AMPLITUDE AND
RESONANT CONDITION
Probability amplitude of the process is described
by Feynman diagrams that are shown in Fig. 1. Wave
lines in the diagrams correspond to wave functions of
photons. External and internal solid lines are wave
functions and Green’s functions correspondingly of
electrons (positrons) in a homogenies magnetic field.
So amplitude of probability is
Sfi = − ie2(2π)4
4SV
δ3(k − k′ − p+ − p−)√
ωω′m+m−ε+ε−
×
×
⎡
⎣ ∞∑
ng=0
eiΦg
∑16
1 Bgi
g2
0 − ε2
g
+
∞∑
nf=0
eiΦf
∑16
1 Bfi
f2
0 − ε2
f
⎤
⎦ .
(1)
Here,
ε± =
√
(m±)2 + (p±)2, (2)
m± = m
√
1 + 2l±h, (3)
εg,f =
√
m2 + 2ng,fhm2 + p2
g,f , (4)
∗E-mail: kholodovroman@yahoo.com
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 111-114.
111
g2
0 = (ω − ε+)2,
f2
0 = (ω − ε−)2, (5)
h = H/Hc, (6)
where Hc = 4.41 · 1013 G is the critical magnetic
field strength, l− and l+ are Landau levels of elec-
tron (positron), Φg and Φf are phases of direct and
exchange processes, Bgi and Bfi are the factors, that
take into account spin-polarization properties of par-
ticles.
The process is studied in the lowest Landau lev-
els approximation (LLL or ultraquantum approxima-
tion) and the following conditions are true [9]:
h � 1,
l ∼ 1.
(7)
For a charged particle moving in a uniform mag-
netic field, the laws of conservation of energy ε and
the longitudinal external field component of the mo-
mentum p (for definiteness, the magnetic field is di-
rected along the z), so the following relations have
place for this process:
ω = ω′ + ε− + ε+,
kz = k′
z + p− + p+,
(8)
where ω and kz are frequency and the longitudinal
momentum of the initial photon, the primed vari-
ables correspond to the final photon, ε− and p− are
energy and longitudinal momentum of the electron,
for positron the same variables are ε+, p+. Analysis
of these expression gives kinematics of the process.
In order to avoid resonance it is necessary to in-
vestigate resonant conditions. Such situation has the
place at the poles of Green’s function of electron,
when intermediate state goes to mass shell. For elec-
tron in magnetic field these conditions have the form
ω′ = mh(ng − l−),
ω′ = mh(nf − l+). (9)
It means that the frequency of the final photon is
equal to the distance between Landau levels. So non-
resonant case is located between two neighboring res-
onances.
Evidently, the process has a threshold defined by
the next expressions:
ω = 2m,
l+ = l− = 0,
μ+ = −μ− = 1,
(10)
where μ− (μ+) is the sign of spin projection of elec-
tron (positron).
We consider the process near the threshold. Let
the frequency of the initial photon be
ω = 2m + amh2, (11)
where a ∼ 1. In this case, the frequency of the final
photon takes on the form
ω′ = κmh2,
0 < κ < a.
(12)
Fig. 1. Feynman diagrams of the process of
e+e−-pair production by a photon with photon
emission in a magnetic field. Solid lines repre-
sent solutions of Dirac equation for an electron
in a magnetic field
3. PROBABILITY PER UNIT OF TIME
Standard rules of quantum electrodynamics give
expression for process probability in time unit in the
form
dW =
1
T
|Sfi|2 V S2
(2π)7
d3kd2pd2p+, (13)
where V , S, T are normalizing constants. Taking into
account the condition (7) the process probability can
be written as follows:
dW =
α2πh2e−2/h
√
a − κ
κ|ez|2|Y |2dω′dΩ, (14)
where
|Y |2 =
a − κ
κ2
K +
Δ
√
a − κ
κ
L + M, (15)
K =
1
2
(1 + ξ′3)(1 − u2), (16)
L =
1
2
(1 + ξ′3) sin(2θ) (cos(Δφ) − cos(Δφ − Λ))+
+ξ′2 sin(θ′) (cos(Δφ) − cos(Δφ − Λ))−
−ξ1 (sin(Δφ) − sin(Δφ − Λ)) ,
(17)
M = (1 + u2) − ξ′3(1 − u2)+
+
(
(1 − u2) − ξ′3(1 + u2)
)
cos(2Δφ − Λ)+
+2ξ′2u sin(2Δφ − Λ),
(18)
u = cos(θ′),
Δφ = φ′ − φ,
Λ = 2κh sin(θ′)sin(Δφ).
(19)
Here ξ′1, ξ′2, ξ′3 are the Stokes parameters of the fi-
nal photon, θ′ and φ′ are the final photon polar and
azimuthal angles
The angular dependence of the quantity |Y |2, that
is the differential process probability in relative units
is shown in Fig. 2 for linear polarized final photon
when Stokes parameter ξ′3 = 1 .
112
Fig. 2. The angular dependence of the differen-
tial process probability in relative units
After integrating over angles the differential prob-
ability has the form
dW
dω′ = α2 2π2
3
h2e−2/h(1 + ξ3)Z, (20)
where ξ3 is the Stokes parameter of the initial photon
and
Z =
√
a − κ
κ
(1 + ξ′3) +
2κ√
a − κ
(2 − ξ′3). (21)
Figure 3 shows the spectral distribution of the
process probability.
Fig. 3. Spectral distribution of the probability
for various values of Stokes parameter ξ′3
In order to estimate obtained probability it is nec-
essary to consider unpolarized final photons.
dW
dκ
= α2 4π2
3
mh4e−2/h
(√
a − κ
κ
− 4κ√
a − κ
)
.
(22)
The total probability
W =
a∫
0
dW
dκ
dκ (23)
diverges logarithmically at the lower limit of the in-
tegral. The reason of the infrared divergence is radi-
ation of ultrasoft photons [10]. After elimination of
this divergence we have for the total probability of
the process the next form
W = α2 4π2
3
mh4e−2/h
√
a
(
ln
a
κmin
+
16a
3
)
. (24)
The numeric value of total probability in time unit is
W ≈ 106 s−1, (25)
when a = 1 and h = 0.1, what means H = 4.41 ·
1012 G and ω − 2mc2 = 0.01 mc2.
4. CONCLUSIONS
Nonresonant probability of photon emission in the
process of photon pair production with photon emis-
sion strong depend on the polarization of the final
photon and on its motion direction also; the total
probability diverges logarithmically by reason of in-
frared divergence in the process of radiation of ultra-
soft photons. Total probability for the nonresonant
process of photon emission 106 s−1 is three order of
magnitude less, than resonant one.
We thank V. Yu. Storizhko, S. P. Roshchupkin and
O.P. Novak for useful discussions.
References
1. I. Koenig, E. Berderman, F. Bosch, et al. // Z.
Phys. A. 1993, v. 346, p. 153.
2. R. Bar, A. Balanda, J. Baumann, et al. // Nu-
clear Phys. A. 1995, v. 583, p. 237.
3. P.I. Fomin, R.I. Kholodov // Dopovidi of NAS
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4. N.P. Klepikov // Zh. Eksp. Teor. Fiz. 1954,
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5. A.A. Sokolov and I.M. Ternov. Synchrotron Ra-
diation from Relativistic Electrons. New York:
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6. A.I. Nikishov // Tr. Fiz. Inst. im. P.N. Lebe-
deva, Akad. Nauk SSSR. 1979, v. 111, p. 152.
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9. O.P. Novak, R.I. Kholodov // Phys. Rev. D.
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10. A.I. Akhiezer and V.B. Berestetskiy. Quantum
Electrodynamics / 4th ed. Moscow: “Nauka”,
1981; New York: “Wiley”, 1965.
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|
| id | nasplib_isofts_kiev_ua-123456789-107007 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:44:55Z |
| publishDate | 2012 |
| publisher | Institute of Applied Physics NAS of Ukraine |
| record_format | dspace |
| spelling | Fomin, P.I. Kholodov, R.I. 2016-10-10T20:32:47Z 2016-10-10T20:32:47Z 2012 Electron-positron pair photo-production with radiation of a photon in magnetic field at nonresonant regime / P.I. Fomin, R.I. Kholodov // Вопросы атомной науки и техники. — 2012. — № 1. — С. 111-114. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 12.20.-m, 13.88.+e https://nasplib.isofts.kiev.ua/handle/123456789/107007 Theoretical investigations of quantum electrodynamic processes in strong magnetic field are carried out. Such the processes may occur between colliding heavy ions. Magnetic fields of the nuclei are added and electric fields of nuclei mutually compensate one another in that region. The electron-positron pair production by a photon in the case when one additional photon is emitted in external magnetic field under nonresonant condition has been investigated. Kinematics of the process and the resonance conditions in approximation of strong magnetic field and weakly excited electron (positron) states (ultra-quantum approximation) have been studied. The resonant conditions have the place, when the photon energy is close to the splitting between Landau levels. The differential probability of nonresonant process has been obtained. The probability of the process is three order of magnitude less the resonant case. Проводятся теоретические исследования квантово-электродинамических процессов в сильном магнитном поле. Такие процессы могут происходить между сталкивающимися тяжелыми ионами. В этой области магнитные поля ядер складываются, а электрические поля взаимно компенсируются. Исследуется процесс рождения электрон-позитронной пары фотоном в случае, когда излучается один дополнительный фотон во внешнем магнитном поле при нерезонансных условиях. Изучаются кинематика процесса и условия резонанса в приближении сильного магнитного поля и слабо возбужденных состояний электронов (позитронов). Резонансные условия имеют место, когда энергия фотона близка к расстоянию между уровнями Ландау. Получена дифференциальная вероятность нерезонансного процесса в единицу времени. Вероятность такого процесса на три порядка меньше резонансного случая. Проводяться теоретичні дослідження квантово-електродинамічних процесів в сильному магнітному полі. Такі процеси можуть відбуватися при зіткненні важких іонів. В області між ними магнітні поля ядер складаються, а електричні поля взаємно компенсуються. Досліджується процес народження електрон-позитронної пари фотоном у випадку, коли випромінюється один додатковий фотон в зовнішньому магнітному полі при нерезонансних умовах. Вивчаються кінематика процесу і умови резонансу в наближенні сильного магнітного поля і слабо збуджених станів електронів (позитронів). Резонансні умови мають місце, коли енергія фотона близька до відстані між рівнями Ландау. Знайдено диференційну ймовірність нерезонансного процесу в одиницю часу. Ймовірність такого процесу на три порядки менша за резонансний випадок. We thank V. Yu. Storizhko, S. P. Roshchupkin and
 O. P. Novak for useful discussions. en Institute of Applied Physics NAS of Ukraine Вопросы атомной науки и техники Section B. QED Processes at High Energies Electron-positron pair photo-production with radiation of a photon in magnetic field at nonresonant regime Рождение электрон-позитронной пары фотоном с излучением фотона в магнитном поле в нерезонансном режиме Народження електрон-позитронної пари фотоном з випромінюванням фотона в магнітному полі в нерезонансному режимі Article published earlier |
| spellingShingle | Electron-positron pair photo-production with radiation of a photon in magnetic field at nonresonant regime Fomin, P.I. Kholodov, R.I. Section B. QED Processes at High Energies |
| title | Electron-positron pair photo-production with radiation of a photon in magnetic field at nonresonant regime |
| title_alt | Рождение электрон-позитронной пары фотоном с излучением фотона в магнитном поле в нерезонансном режиме Народження електрон-позитронної пари фотоном з випромінюванням фотона в магнітному полі в нерезонансному режимі |
| title_full | Electron-positron pair photo-production with radiation of a photon in magnetic field at nonresonant regime |
| title_fullStr | Electron-positron pair photo-production with radiation of a photon in magnetic field at nonresonant regime |
| title_full_unstemmed | Electron-positron pair photo-production with radiation of a photon in magnetic field at nonresonant regime |
| title_short | Electron-positron pair photo-production with radiation of a photon in magnetic field at nonresonant regime |
| title_sort | electron-positron pair photo-production with radiation of a photon in magnetic field at nonresonant regime |
| topic | Section B. QED Processes at High Energies |
| topic_facet | Section B. QED Processes at High Energies |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/107007 |
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