Manifestation of an excited electron in e⁺e⁻ → γγ reaction
The differential cross section and some polarization observables have been calculated for the e⁺e⁻ → γγ reaction taking into account the contribution of the excited electron. The spin correlation coefficients were calculated for the case when both beams are polarized. We consider two approaches for...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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nasplib_isofts_kiev_ua-123456789-1949662025-02-23T18:06:39Z Manifestation of an excited electron in e⁺e⁻ → γγ reaction Прояв збудженого електрона в реакції e⁺e⁻ → γγ Проявление возбуждённого электрона в реакции e⁺e⁻ → γγ Gakh, G.I. Konchatnij, M.I. Merenkov, N.P. Gakh, A.G. Nuclear physics and elementary particles The differential cross section and some polarization observables have been calculated for the e⁺e⁻ → γγ reaction taking into account the contribution of the excited electron. The spin correlation coefficients were calculated for the case when both beams are polarized. We consider two approaches for the excited electron contribution: the eeγγ contact interaction and the exchange of the excited electron in t- and u-channels. Numerical estimations are given for the excited electron contribution to the differential cross section and spin correlation coefficients for various values of the electron beam energy and excited electron mass. Диференціальний переріз та деякі поляризаційні спостережувані обчислені для реакції e⁺e⁻ → γγ з врахуванням внеску збудженого електрона. Коефіцієнти спінової кореляції були отримані для випадку, коли обидва пучки поляризовані. Було розглянуто два підходи для врахування внеску збудженого електрона: eeγγ контактна взаємодія та обмін збудженим електроном у t- і u-каналах. Числові оцінки для внеску збудженого електрона у величину диференціального перерізу та коефіцієнтів спінової кореляції приведені для деяких величин енергії електронного пучка та маси збудженого електрона. Дифференциальное сечение и некоторые поляризационные наблюдаемые вычислены для реакции e⁺e⁻ → γγ с учетом вклада возбужденного электрона. Коэффициенты спиновой корреляции были получены для случая, когда оба пучка поляризованы. Было рассмотрено два подхода для учета вклада возбуждённого электрона: eeγγ контактное взаимодействие и обмен возбужденным электроном в t- и u-каналах. Численные оценки для вклада возбужденного электрона в величину дифференциального сечения и коэффициентов спиновой корреляции приведены для некоторых величин энергии электронного пучка и массы возбужденного электрона. 2021 Article Manifestation of an excited electron in e⁺e⁻ → γγ reaction / G.I. Gakh, M.I. Konchatnij, N.P. Merenkov, A.G. Gakh // Problems of Atomic Science and Technology. — 2021. — № 3. — С. 14-18. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 12.20.-m, 13.40.-f, 13.60.-Hb, 13.66.-a, 13.88.+e DOI: https://doi.org/10.46813/2021-133-014 https://nasplib.isofts.kiev.ua/handle/123456789/194966 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Nuclear physics and elementary particles Nuclear physics and elementary particles |
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Nuclear physics and elementary particles Nuclear physics and elementary particles Gakh, G.I. Konchatnij, M.I. Merenkov, N.P. Gakh, A.G. Manifestation of an excited electron in e⁺e⁻ → γγ reaction Вопросы атомной науки и техники |
| description |
The differential cross section and some polarization observables have been calculated for the e⁺e⁻ → γγ reaction taking into account the contribution of the excited electron. The spin correlation coefficients were calculated for the case when both beams are polarized. We consider two approaches for the excited electron contribution: the eeγγ contact interaction and the exchange of the excited electron in t- and u-channels. Numerical estimations are given for the excited electron contribution to the differential cross section and spin correlation coefficients for various values of the electron beam energy and excited electron mass. |
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Article |
| author |
Gakh, G.I. Konchatnij, M.I. Merenkov, N.P. Gakh, A.G. |
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Gakh, G.I. Konchatnij, M.I. Merenkov, N.P. Gakh, A.G. |
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Gakh, G.I. |
| title |
Manifestation of an excited electron in e⁺e⁻ → γγ reaction |
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Manifestation of an excited electron in e⁺e⁻ → γγ reaction |
| title_full |
Manifestation of an excited electron in e⁺e⁻ → γγ reaction |
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Manifestation of an excited electron in e⁺e⁻ → γγ reaction |
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Manifestation of an excited electron in e⁺e⁻ → γγ reaction |
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manifestation of an excited electron in e⁺e⁻ → γγ reaction |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2021 |
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Nuclear physics and elementary particles |
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https://nasplib.isofts.kiev.ua/handle/123456789/194966 |
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Manifestation of an excited electron in e⁺e⁻ → γγ reaction / G.I. Gakh, M.I. Konchatnij, N.P. Merenkov, A.G. Gakh // Problems of Atomic Science and Technology. — 2021. — № 3. — С. 14-18. — Бібліогр.: 10 назв. — англ. |
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ISSN 1562-6016. ВАНТ. 2021. № 3(133) 14
https://doi.org/10.46813/2021-133-014
MANIFESTATION OF AN EXCITED ELECTRON
IN e e REACTION
G.I. Gakh1, M.I. Konchatnij1*, N.P. Merenkov1, A.G. Gakh2
1National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine;
2V.N. Karazin Kharkiv National University, Kharkiv, Ukraine
The differential cross section and some polarization observables have been calculated for the e e reac-
tion taking into account the contribution of the excited electron. The spin correlation coefficients were calculated for
the case when both beams are polarized. We consider two approaches for the excited electron contribution: the
ee contact interaction and the exchange of the excited electron in t- and u-channels. Numerical estimations are
given for the excited electron contribution to the differential cross section and spin correlation coefficients for vari-
ous values of the electron beam energy and excited electron mass.
PACS: 12.20.-m, 13.40.-f, 13.60.-Hb, 13.66.-a, 13.88.+e
INTRODUCTION
The Standard Model (SM) passed successfully test
in precision experiments. In spite of this huge success,
there are other issues like the replication of the fermion
families, dark matter, baryogenesis etc. that are still not
understood within the framework of the SM and ad-
dressing them needs physics beyond the SM. Some pos-
sible candidate is the compositeness for quarks and lep-
tons [1]. The existence of the excited states is the natural
consequence of the composite models of quarks and lep-
tons. The increase of the number of quarks and leptons is
often considered as the hint that these particles have sub-
structure. The most convincing proof that quarks and
leptons have substructure would be the discovery of the
excited states of ordinary quarks and leptons.
At present, there is no completely predictable model
describing the substructure of the quarks and leptons.
Therefore, the best thing, that can be done for the search
of the substructure effects, is to perform the necessary
phenomenological analysis. The review of possible ef-
fects of the substructure of the quarks and leptons,
which can be displayed in various reactions, is given in
Ref. [2].
Even though there is no evidence for the excited lep-
tons in the experimental studies performed in HERA,
Tevatron, ATLAS and center of mass system (CMS)
(see the references in [3]), the colliders with higher cen-
ter-of-mass energy and luminosity, planned to be in-
stalled in the future, will continue to search for their
discovery. A possible discovery of the any excited fer-
mion will be a direct proof of the lepton and quark
compositeness. The most recent experimental results on
the excited electron mass are provided by the OPAL and
the ATLAS collaborations [4]. Under some assumption,
the mass exclusion limits of the excited electrons are
103 2
e
m GeV for pair production ( e e e e )
and 3000 e
m GeV for single production
( ep e X e X ).
In this paper we investigate the influence of the ee
contact interaction on the angular distribution in the reac-
tion of the two-photon annihilation of e e -pair:
2 1 1 2( ) ( ) ( ) ( ) e p e p k k (1)
It was taken into account not only the contribution of
the interference of this contact interaction with standard
QED mechanism but also the contribution of the contact
interaction itself. The influence of the contact interac-
tion on the polarization observables in this reaction has
been investigated for the case of both polarized beams.
The excited electron can also contribute to the reaction
(1) by its exchange in t- and u-channels. The effect of
the excited electron contribution can be seen as distor-
tion in the angular distribution beyond the region of the
forward or backward scattering. We investigate also the
influence of the excited electron on the polarization ob-
servables for the case of both polarized beams.
1. MECHANISM OF EXCITED ELECTRON
EXCHANGE
The standard QED mechanism of the reaction (1) is
described by two Feynman diagrams. The production of
the excited electron in the intermediate state (in t- and u-
channels) in this reaction is described by two additional
Feynman diagrams.
The differential cross section of the reaction (1) can
be written as follows (the average over the polarizations
of the initial beams was done)
1
2
2
2
2 [ ( 4 )]
128
d M s s m
d W
(2)
where is the photon energy in the CMS of the reac-
tion (1), W is the total energy of the initial beams, m
is the electron mass, 2 s W The matrix element of the
reaction (1) is the sum of two contributions, namely:
M (this part corresponds to the pure QED mechanism)
and exM which corresponds to the excited electron con-
tribution.
Then the differential cross section of the reaction (1)
can be written, in this approach, as
int exd d dd
d d d d
(3)
where the first term is the cross section of this reaction
corresponding to the QED mechanism, the second one –
contribution of the interference of two mechanisms
(QED and the excited electron) and the last term – the
contribution of the excited electron itself.
ISSN 1562-6016. ВАНТ. 2021. № 3(133) 15
The matrix element M can be written as
2 2 11 1
2 2
1 21 2
12
ˆ ˆˆˆ( )
( )[
( )
ˆ ˆˆˆ( )
] ( ),
( )
A m Ap kM e u p
t m
A m Ap k u p
u m
(4)
where e is the electron charge; 1 2( )A A is the polariza-
tion 4-vector of the first (second) photon;
1 2 1 2( ) ( )p p k k is the 4-momentum of the electron
(positron), first (second) photon, respectively;
2 2
1 1 1 2( ) ( ) t p k u p k
At large energies (where at last time the experi-
ments, investigating this reaction, were done) it is pos-
sible to neglect by the electron mass m (where it is pos-
sible) and then the differential cross section of the reac-
tion (1), caused by the standard QED mechanism (3),
has the form
2 2
2
1
1
d x
d s x
(5)
where x cos is the angle between the electron and
photon momenta. The coordinate frame in CMS of the
reaction (1) is chosen as: z axis is directed along the
initial electron momentum and photon momentum lies
in the xz plane (the reaction plane). The expression (5)
is valid for all angles except forward and backward scat-
tering (i.e., = 0, 180), where it is necessary to take
into account the electron mass in the denominator.
We assume that the spin of the excited electron is
1/2 and its interaction with the electromagnetic field is
described by following effective Lagrangian [5]
( )
2
e e
eL ee u F h cuM
(6)
where ( ) 2 , M is the excited electron
mass, is the dimensionless coupling constant and F
is the photon field-strength tensor. Then the part of the
matrix element, describing the contribution of the excit-
ed electron to the reaction (1), can be written as
2 2 2 1 11
2 2
1 1 2 21
12
ˆ ˆˆ ˆˆ( )
( ) ( )[
( )
ˆ ˆˆ ˆˆ( )
] ( ).
( )
ex
A k M A kpeM u p
M t M
A k M k Ap u p
u M
(7)
As it was already mentioned above, at high energies
one can neglect the electron mass (we assume also that
M m and experimental data suggest this assump-
tion). Using expression (7) as matrix element exM and
expression (4) for the matrix element M one can ob-
tain the following formula for the interference contribu-
tion (in this approximation)
2 2
2 2 2 2 1
2 [1 (1 )][(1 ) ]
intd
x y x y x
d M
(8)
where 22 y M s In the same approximation the
term, caused by the contribution of the excited electron
itself, has the form
2 4 2
3 2 2 2
6
2 2 2 3 2 2 2
[4 (1 )(9 )
8
6 (1 ) (1 ) ][(1 ) ]
exd ys y y x x
d M
y x x y x
(9)
The following ratio will be used for the estimation of
the excited electron contribution to the differential cross
section
( )
int ex
ex
dd d
R
d d d
(10)
Let us consider the influence of the excited electron
on the polarization observables of the reaction (1) for
the case when both beams have arbitrary polarization.
The mechanism, caused by the exchange of the excited
electron, does not lead to a non-zero polarization effects
in the case when only one beam is polarized (at least, in
the lowest order of the perturbation theory) since the
reaction of the excited electron production conserve the
space parity (it is seen from the expression for the La-
grangian (6)).
In the case when the initial beams have arbitrary po-
larizations, the differential cross section of the reaction
(1) can be written as
0
1 2 1 2 1 2
1 2 1 2
(1
)
zz z z xx x x yy y y
zx z x xz x z
dd C C C
d d
C C
(11)
where ijC are the spin correlation coefficients, 1 2( )
is
the unit vector along the electron (positron) polarization
in its rest system. Thus, iz describes the longitudinal
polarization of the beams, and ( )ix iy the transverse
polarization of the beams and polarization vector lies in
the reaction plane (orthogonal to the plane). Let us note
that xz zxC C coefficients are proportional to the electron
mass and, therefore, they are zero in the high energy
limit. The rest coefficients in this limit have the form
2 2
2 2 2 1
0 2
4
2 2 2 2 2
2
{1 (1 )(1 )[(1 ) ]
(1 ) [(1 ) ] }
xx
sC y x y x
s M
x y x
y
yy xxC C (12)
2
2 2 1
0
2
2 2 2 2 1
{(1 )(1 )
2 [1 (1 )][(1 ) ]
zzC x x
s
x y x y x
y
4
3 2 2 2 2 2
2
2 3 2 2 2
[ 4 (1 )( 7) 2 (1 )
(1 ) ][(1 ) ] }
y y x x y x
y
x y x
where 0 0 d d is the differential cross section of
the reaction (1) for the case when all particles are unpo-
larized.
Let us do numerical estimations for the contribution
of the excited electron to the observables of the reaction
(1). We investigate the dependence of the effect, caused
by the contribution of the excited electron, on the excit-
ed electron mass M (we use M=150 and 300 GeV), on
the total energy of the beams (W=200 and 500 GeV)
and on the coupling constant value ( =1 and 0.1). The
influence of these parameters on the ratio exR is given
in Fig. 1.
ISSN 1562-6016. ВАНТ. 2021. № 3(133) 16
Fig. 1. The ratio exR for various values of the excited
electron mass M, the total energy of the beams W and
the coupling constant =1 (up) and =0.1 (down)
We see that at fixed beam energy the most sensitivi-
ty to the contribution of the excited electron takes place
at the angles far from the region of the forward scatter-
ing since the cross section, caused by the pure QED
mechanism, has sharp peak at forward scattering. Note
that in this angular region, where the sensitivity to the
contribution of the excited electron is maximal, the
cross section decreases appreciably and it requires more
time to collect comparable statistics. It turns out that at
fixed beam energy the excited electron contribution to
the ratio exR decreases strongly with the increase of the
mass M. The increase of the beam energy at the fixed
mass M leads to the appreciable increase of the excited
electron contribution. The sensitivity of the excited elec-
tron contribution to the coupling constant (at fixed
values of M and W) is very strong. For example, if we
reduce the to one-tenth (from 1 to 0.1) the ratio exR
reduces by a factor of 100. Thus, the investigation of the
reaction (1), at future high energy linear electron-
positron colliders (CLIC or ILC with W 500 GeV),
can give more strict constraints on the excited electron
parameters (mass and coupling constant).
We did the numerical estimations of the spin corre-
lation coefficients xxC and zzC . They are given in
Fig. 2. The analysis of their behaviour depending on the
excited electron mass, the total energy of the beams was
done in the same way as for the ratio exR . One can see
from Fig. 2 that the spin correlation coefficients have
appreciable values over a wide range of the scattering
angles (especially for the zzC quantity).
Fig. 2. The spin correlation coefficients for various
values of the excited electron mass M,
the total energy of the beams W
Note that the coefficient 1zzC for the pure QED
mechanism. Such behaviour of the spin correlation coef-
ficients is important since at small angles the cross sec-
tion is large and this circumstance permits to collect more
data. The spin correlation coefficients do not change
strongly with increasing of the total energy W. The de-
pendence of these coefficients on the excited electron
mass reduces with the increase of the energy W.
Note that the proposed future linear colliders ILC
and CLIC cover, at first run, the region W 500 GeV
(with longitudinal beam polarization). Both colliders
have posible upgrades to 1 (ILC) and 3 TeV (CLIC) [6].
2. CONTACT ee INTERACTION
Earlier, the contact ee interaction was investigated
in a number papers (see, for example [7, 8]) where the
contribution of this contact interaction was taken into
account to the reaction (1). Besides, in the paper [8] it
was considered the manifestation of this contact interac-
tion in the reactions 3 e e F F ll
(where F designates arbitrary fermion). In these papers
the contribution of contact interaction was taken into
account on the level of its interference with standard
QED mechanism. The influence of the initial particle
polarizations on the observable characteristics of these
processes was not investigated in these papers.
The experimental search for various types of the
contact interaction are being done at present at the lep-
ton, lepton-hadron and pure hadron colliders. The refer-
ences on the experimental search for these contact inter-
ISSN 1562-6016. ВАНТ. 2021. № 3(133) 17
actions see in the paper [9]. Using the results of various
experiments it was obtained the lower limits for the cor-
responding energy scales
The matrix element of the reaction (1) is the sum of
M (the pure QED mechanism) and ciM which de-
scribes the contribution of the contact ee interaction.
Then the differential cross section of the reaction can be
written, in this approach, as a sum of three contribu-
tions, namely, by Eq. (3), where instead of ex index it
is necessary to put ci index (the contact interaction
contribution). Then, in this case, the second term de-
scribes the contribution of the interference of two mech-
anisms (QED and the contact interaction), and the last
one the contribution to the cross section of the contact
interaction itself.
The effective Lagrangian of any contact interaction
is constructed using the fields of particles known at pre-
sent and is proportional to the lowest possible power of
1 which depends on the dimensionality of the fields
entering the Lagrangian. When constructing this La-
grangian, we demand that fermion currents correspond-
ing this Lagrangian conserve the helicity. This assump-
tion is necessary, for example, for various types of the
composite models. This condition ensures that masses
of known particles are much less than the energy scale
The contact interaction for two fermions and two
bosons was considered, in general case, in the paper
[10]. For the case of the ee , the Lagrangian of the
contact interaction can be written as
2
4( ) 2
ieL ee F F h c (13)
where is the derivative, is the electron wave
function, F is the strength of the electromagnetic
field. The dimensions of the fields participating in the
effective Lagrangian (12) lead to the fact that it is pro-
portional to 4 The dimensionless coefficient de-
termines the strength of the interaction. The transition
amplitude corresponding to the contact interaction is
real value and can be both positive and negative (and
each sign of the parameter is associated with differ-
ent value of the energy scale ).
The matrix element ciM , corresponding to the effec-
tive Lagrangian (12), can be written in the following
general form (in the impulse representation)
2
2 142 ( ) ( )
CI
eM V v p u p (14)
where
1 2 1 2 1 2 1 2 1
1 1 2 1 2 2 1 1 2
[( )
( )] 1 2.
V k k p A p k k A A
k p k A A k A p A
(15)
For the interference contribution (between QED and
the contact interaction mechanisms) to the differential
cross section of the reaction (1) one can obtain the fol-
lowing simple expression valid at high energies
2
2
4 (1 )
2
intd s x
d
(16)
In the same approximation ( 0m ), the term corre-
sponding to the contact interaction mechanism itself has
the form
2 3
2 4
8 (1 )
16
CId s x
d
(17)
Let us consider the influence of the contact interac-
tion on the polarization observables in the reaction (1)
for the case when initial particles are polarized. Since in
our case the contact interaction does not violate the
space parity then non-zero observables arise in the case
when only both beams are polarized. The pure QED
mechanism of this reaction (at least without taking into
account the radiative corrections) does not lead to the
polarization effects when only one beam is polarized.
The electroweak corrections (at one-loop level) can lead
to additional term in the amplitude of this process which
violate the parity and, therefore, can lead to the non-
zero polarization observables.
Let us consider the case when both beams have arbi-
trary polarization. Note that synchrotron radiation in the
electron-positron colliders leads to the polarization of
these beams. This polarization is transverse and its val-
ue is appreciable.
The pure QED mechanism lead to the following
contribution to the differential cross section of the reac-
tion (1) which depends on the polarization of both
beams (in the high energy limit)
1 2
2
1 2 1 2 1 22
( )
1[1 ( )]
1
z z x x y y
d
d
d x
d x
(18)
The interference (between QED and the contact in-
teraction mechanisms) contribution to the differential
cross section of the reaction (1) caused by the polariza-
tion of both beams has the following form (in the high
energy limit)
2
21 2
1 2 1 24
2
1 2
( ) [(1 )( )
2
(1 ) ]
int
x x y y
z z
d s x
d
x
(19)
The term in the differential cross section of the reac-
tion (1), caused by the contribution of the contact inter-
action itself and depending on the polarization of both
beams, has the following form (in the high energy limit)
2 3
2 2 21 2
1 2 1 28
4
1 2
( )
[(1 ) ( )
16
(1 ) ]
CI
x x y y
z z
d s x
d
x
(20)
CONCLUSIONS
We have analyzed the influence of a particular
mechanism (the excited electron contribution), which is
beyond the SM framework, on the observables of the
reaction e e . Two approaches of taking into
account the excited electron contribution has been con-
sidered: the contact ee interaction mechanism and
the presence of the excited electron in the t- and u-
channels. The influence of these mechanisms on the
angular dependence of the differential cross section and
spin correlation coefficients (when both initial beams
ISSN 1562-6016. ВАНТ. 2021. № 3(133) 18
are arbitrarily polarized) have been investigated for the
reaction e e . These effects turned out to be ap-
preciable and their magnitude increases quickly when
initial beam energy grows. Therefore, the experimental
investigation of this reaction on the future lepton collid-
ers may essentially progress to limit the parameters of
the mechanisms under consideration.
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Article received 22.04.2021
ПРОЯВЛЕНИЕ ВОЗБУЖДЁННОГО ЭЛЕКТРОНА В РЕАКЦИИ e e
Г.И. Гах, М.И. Кончатный, Н.П. Меренков, А.Г. Гах
Дифференциальное сечение и некоторые поляризационные наблюдаемые вычислены для реакции
e e с учетом вклада возбужденного электрона. Коэффициенты спиновой корреляции были получены
для случая, когда оба пучка поляризованы. Было рассмотрено два подхода для учета вклада возбуждённого
электрона: ee контактное взаимодействие и обмен возбужденным электроном в t- и u-каналах. Числен-
ные оценки для вклада возбужденного электрона в величину дифференциального сечения и коэффициентов
спиновой корреляции приведены для некоторых величин энергии электронного пучка и массы возбужден-
ного электрона.
ПРОЯВ ЗБУДЖЕНОГО ЕЛЕКТРОНА В РЕАКЦІЇ e e
Г.І. Гах, М.І. Кончатний, М.П. Меренков, А.Г. Гах
Диференціальний переріз та деякі поляризаційні спостережувані обчислені для реакції e e з вра-
хуванням внеску збудженого електрона. Коефіцієнти спінової кореляції були отримані для випадку, коли
обидва пучки поляризовані. Було розглянуто два підходи для врахування внеску збудженого електрона:
ee контактна взаємодія та обмін збудженим електроном у t- і u-каналах. Числові оцінки для внеску збу-
дженого електрона у величину диференціального перерізу та коефіцієнтів спінової кореляції приведені для
деяких величин енергії електронного пучка та маси збудженого електрона.
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