Abstract. The properties of axions are reviewed and the production of axion
in e+e− and ae− collision is considered in supersymmetric theories using the
Feynman diagram method. Based on our results we comment on the general
implications of our study for the production of axion in process annihilation of
ordinary matter and properties of axion in process collision of its with ordinary
matter, and also for signature of axion in the dark matter of universe.

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2014, Vol. 59, No. 7, pp. 101-111
This paper is available online at
PRODUCTION OF AXION IN e+ e− AND ae− COLLISION
Dao Thi Le Thuy1 and Le Nhu Thuc2
1Faculty of Physics, Hanoi National University of Education
2Department of Postgraduate Studies, Hanoi National University of Education
Abstract. The properties of axions are reviewed and the production of axion
in e+e− and ae− collision is considered in supersymmetric theories using the
Feynman diagram method. Based on our results we comment on the general
implications of our study for the production of axion in process annihilation of
ordinary matter and properties of axion in process collision of its with ordinary
matter, and also for signature of axion in the dark matter of universe.
Keywords: Axion, ALPs, pNGB.
1. Introduction
The standard model in particle physics has succeeded in describing physics below
the electronweak scale. It is not, however, a complete theory because of theoretical
problems. One is the strong CP problem. Among various candidate solutions proposed
so far, the Peccei-Quinn mechanism is the most attractive candidates for the solution of
the strong CP problem, where, the CP-violating phase θ (θ 6 10−9) is explained by the
existence of a new pseudo-scalar field, called the axion [10].
At present, the axion mass is constrained by laboratory [7] astrophysical and
cosmological considerations [13] placing it between 10−6 eV and 10−3 eV. If the axion
has a mass near the low limit of order 10−5 eV, it is a good candidate for dark matter of the
universe. In addition, an axino (the fermionic partner of the axion) naturally appears in
SUSY models [5] which acquires a mass from three-loop Feynman diagrams in a typical
range of between a few eV up to a maximum of 1 keV [14]. The candidates for dark
matter can appear in different models, in the 3-3-1 models [8] or in the supersymmetric
and superstring theories [3]. Light particles with a two photon interaction can transform
into photons in external electric or magnetic fields by an effect first discussed by
Primakoff [11]. This effect is the basis of Sikivie’s methods for the detection of axions
in a resonant cavity [12]. Various terrestrial experiments to detect invisible axions by
Received September 9, 2014. Accepted October 25, 2014.
Contact Le Nhu Thuc, e-mail address: thucln@hnue.edu.vn
101
Dao Thi Le Thuy and Le Nhu Thuc
making use of their coupling to photons have been proposed [6] and the results of such
experiments have appeared recently. The CERN Axion Solar Telescope (CAST) [1] is
a helioscope experiment which aims to detect axion and axion-like particles (ALPs)
emitted from the Sun. The detection principle is based on the axion coupling to two
photons, which triggers their conversion into photons of the same energy as they propagate
through a transverse magnetic field [15]. CAST has tracked the Sun in three different
campaigns (2003-2004 [16]), 2005-2006 [1] and 2008 [2]) with a 9.26 m long, 9 Tesla
strong, decommissioned LHC dipole test magnet while measuring the flux of X-rays at
the exits of both bores with four different low-background detectors [4]. In this paper, we
consider the axion production in unpolarized and polarized e+e− collision and the axion
annihilation in unpolarized and polarized ae− processes using the Feynman method. The
polarization of electron and positron beams at the colliders gives a very effective means to
control the effect of the MS processes on the experimental analyses. Beam polarization is
also an indispensable tool in identifying and studying new particles and their interactions.
2. Content
2.1. Properties of axions in axion - electron coupling
We have the lagrangian density [4]
$ =
1
2
(∂µa)(∂
µa)− 1
2
m2aa
2 − gaγ
4
Fµν eF µνa− gae ∂µa
2me
ψeγ5γ
µψe, (2.1)
where a is the axion field,ma is its mass, Fµν and eF µν are the electromagnetic field tensor
and its dual,me is the electron mass, and ψe is the electron field. The coupling constant gaγ
has units of energy−1 while gea is a dimensionless Yukawa coupling. Particles featuring
this type of lagrangians are often called axion-like particles (ALPs) and they appear as
pseudo-Nambu-Goldstone bosons (pNGB), associated with a global shift symmetry a→
a+ const [4]. The last term in eq. (1) is the interaction between electron and axion. This
interaction is of derivative nature, and it is linear in a with no higher-order terms.
In 1977 Peccei and Quinn proposed one such symmetry to solve the strong CP
problem [10] with the additional condition that it should be color anomalous. The
resulting pNGB was called axion [10]. The axion receives its mass from chiral symmetry
breaking after mixing with the pseudoscalar mesons through the color anomaly, its
magnitude being
ma =
√
z
1 + z
mπfπ
fa
w 6meV 10
9GeV
fa
, (2.2)
wheremπ is the neutral pion mass and fπ is the pion decay constant. For the ration z = mumd
of up to down quark masses we use the canonical value z s 0.56 although the allowed
range is z = 0.35− 0.60 [9].
The axion has a model-independent contribution to its tow-photon coupling coming
from the above-mentioned mixing with mesons and can also have a model-dependent part
102
Production of axion in e+e− and ae− collision
if the PQ symmetry has electromagnetic anomaly. The two contributions sum to
gaγ =
α
2πfa
(E
N
− 2
3
(4 + z)
(1 + z)
)
w α
2πfa
(E
N
− 1.29
)
, (2.3)
where α is the fine-structure constant and E/N the ration of the electromagnetic and color
anomalies of the PQ symmetry.
The coupling to electrons has a model-dependent contribution proportional to
an O(1) coefficient Xe arising only in non-hadronic axion models and a very small
model-independent contribution induced at one-loop via the photon coupling,
gae = Xe
me
fa
+
3α2
4π
me
fa
(E
N
log
fa
me
− 1.29log ∧
me
)
, (2.4)
where ∧ is an energy scale close to the QCD confinement scale. Based on a new
calculation, the axion-electron Yukawa coupling gae and axion-photon interaction strength
gaγ using the CAST phase-I data (vacuum phase) are limited. For ma 6 10 meV / c2 it
was found gaγgae < 8.1× 10−23GeV −1 at 95% CL [4].
2.2. Axion production in e+e annihilation
The Feynman diagrams for the annihilation process e+e− through s, t, u - channel
are presented in Figure 1. From that, we get the following expression for the matrix
element for the production axion when the beam of e− unpolarization and polarization.
Figure 1. Feynman diagram of e+e− collision
When the beam of e+ and e− are not polarized, using the Feynman rules to
calculate, we obtain the scattering amplitudes as follows:
Ms =
4ieαgaγ
4πfa
qsµk1ρgσαgνβε
µνρσv(p2)γ
βu(p1)ε
α(k1), (2.5)
for the s-channel.
Mu =
iemeχ
v(q2u −m2e)
v(p2)γµ(bqu +me)γ5u(p1)εµ(k1), (2.6)
for the u-channel.
Mt =
iemeχ
V (q2t −m2e)
v(p2)γ5(bqt +me)γµu(p1)εµ(k1), (2.7)
103
Dao Thi Le Thuy and Le Nhu Thuc
for the t-channel.
When the beams of e+ and e− are polarized, we have:
- For the s-channel, when the beams of e+ and e− are same left polarized or same
right polarized then scattering amplitude do not zero and in other cases the scattering
amplitude equal zero, so we have
MsLL =
ieαgaγ
2πfa
qsµk1ρgσαgνβε
µνρσv(p2)γ
β(1− γ5)u(p1)εα(k1), (2.8)
and
MsRR =
ieαgaγ
2πfa
qsµk1ρgσαgνβε
µνρσv(p2)γ
β(1 + γ5)u(p1)ε
α(k1), (2.9)
- For the u-channel, we obtain scattering amplitudes as follows:
+ When the beam of e+ and e− are same left polarized, we have
MuLL =
−iem2eχ
v(q2u −m2e)
v(p2)
(1 + γ5)
2
γµu(p1)ε
µ(k1), (2.10)
+ When the beam of e+ and e− are same right polarized, we have
MuRR =
iem2eχ
v(q2u −m2e)
v(p2)
(1− γ5)
2
γµu(p1)ε
µ(k1), (2.11)
+ When the beam of e+ is left polarized and the beam of e− is right polarized,
we have
MuLR =
iem2eχ
v(q2u −m2e)
v(p2)
(1 + γ5)
2
γµbquu(p1)εµ(k1), (2.12)
+ When the beam of e+ is right polarized and the beam of e− is left polarized,
we have
MuRL =
iem2eχ
v(q2u −m2e)
v(p2)
(γ5 − 1)
2
γµbquu(p1)εµ(k1), (2.13)
- For the t-channel, similar to the above, we have
MtLL =
iem2eχ
v(q2t −m2e)
v(p2)
(1 + γ5)
2
γµu(p1)ε
µ(k1), (2.14)
+ When the beam of e+ and e− are same right polarized, we have
MtRR =
iem2eχ
v(q2t −m2e)
v(p2)
(γ5 − 1)
2
γµu(p1)ε
µ(k1), (2.15)
104
Production of axion in e+e− and ae− collision
+ When the beam of e+ is left polarized and the beam of e− is right polarized,
we have
MtLR =
iem2eχ
v(q2t −m2e)
v(p2)
(1 + γ5)
2
γµbqtu(p1)εµ(k1), (2.16)
+ When the beam of e+ is right polarized and the beam of e− is left polarized,
we have
MtRL =
iem2eχ
v(q2t −m2e)
v(p2)
(γ5 − 1)
2
γµbqtu(p1)εµ(k1), (2.17)
A straightforward calculation yields the following differential cross section (DCS)
and total cross section (TCS) in the center-of-mass frame, in our calculation, we choose:
χ = 1,me = 5, 1.10−4GeV , v = 247GeV , α = 1137 ,ma = 6.10
−10GeV , fa = 1010GeV ,
gaγ = 0, 36 (DFSZ model) and
√
s = 3TeV and thus we have
- For the evaluation of the total cross section which depends on the polarization
factors, we note that, P1 is the polarization factor of the beam in the initial state and P2 is
the polarization factor of the beam in the finial state. From Figure 2, we can see that, the
TCS attains maximum value when P1 = −1, P2 = 1 or P1 = 1, P2 = −1, this means that,
the beams of e+ and e− are fully polarized. In addition, the TCS is equal to zero when
P1 = −1, P2 = −1 or P1 = 1, P2 = 1.
- From the evaluation of the cross section which depends on the polarization factors,
we see that the DCS depends on cos θ (Figure 3) when the beam of e+ and e− are not
polarized (dσ0) and (Figure 4) when the beams of e+ and e− are polarized (dσ). We can
see that, the DCS depends strongly on cos θ, in particular, the DCS gets a minimum value
when cos θ = 0 and a maximum value when cos θ = ±1, and we have ration dσ
dσ0
= 2.
Figure 2. The TCS as a function of the polarization factor of e−
105
Dao Thi Le Thuy and Le Nhu Thuc
Figure 3. The DCS as a function of cosθ
when the beams of e+ and e− are not polarized
Figure 4. The DCS as a function of cosθ
when the beams of e+ and e− are polarized
- For the evaluation of the total cross section (TCS) which depends on the center of
mass energy (
√
s), we determine that the TCS depends on
√
s (Figure 5) when the beam
of e+ and e− are not polarized and (Figure 6, Figure 7) when the beam of e+ and e− are
polarized. We can see that, the TCS quickly decrease, when the center of mass energy
increases. This means that we only need to consider the collision process e+e− in the low
energy region and it is also favorable to the receivers of the axion in experiments.
106
Production of axion in e+e− and ae− collision
Figure 5. The TCS as a function of
√
s when the beams of e+ and e− are not polarized
Figure 6. The TCS as a function of
√
s when the beams of e+ and e− are polarized,
with P1 = −1, P2 = 1
Figure 7. The TCS as a function of
√
s when the beams of e+ and e− are polarized,
with P1 = 1, P2 = 1
107
Dao Thi Le Thuy and Le Nhu Thuc
2.3. Axion production in ae annihition
The Feynman diagrams for collision process ae− through the s-channel is drawn
as Figure 8. From this, we get the following expression for the matrix element for the
production axion when the beams of e− are unpolarized and polarized.
Figure 8. Feynman diagram of ae− annihition
When the beams of e− are not polarized, we have the expression for the matrix
element for the production axion in this process as follows:
M =
iem2eχ
2
v2(q2s −m2e)
u(p1)(bqs −me)u(p1), (2.18)
When the beams of e− in the initial state and the beam of e− in the final state are
same left polarized, we have
MLL =
iem2eχ
2
v2(q2s −m2e)
u(p1)
1 + γ5
2
(bqs −me)u(k1), (2.19)
When the beams of e− in the initial state and the beams of e− in the final state are
same right polarized, we have
MRR =
iem2eχ
2
v2(q2s −m2e)
u(p1)
1− γ5
2
(bqs −me)u(k1), (2.20)
When the beams of e− in the initial state are right polarized and the beams of e− in
the final state are left polarized, we have
MRL =
−iem3eχ2
v2(q2s −m2e)
u(p1)
1 + γ5
2
u(k1), (2.21)
When the beams of e− in the initial state are left polarized and the beams of e− in
the final state are right polarized, we have
MLR =
−iem3eχ2
v2(q2s −m2e)
u(p1)
1− γ5
2
u(k1), (2.22)
108
Production of axion in e+e− and ae− collision
The straightforward calculation yields the following differential cross section
(DCS) and total cross section (TCS) in the center-of-mass frame, in our calculation, we
choose: χ = 1, me = 5, 1.10−4GeV , v = 247GeV , α = 1137 , ma = 6.10
−10GeV ,
fa = 10
10GeV , gaγ = 0, 36 (DFSZ model), we have:
- For the evaluation of the total cross section which depends on the polarization
factors, from Figure (9), we can see that, the TCS attains a maximum value when P1 =
1, P2 = 1 or P1 = −1, P2 = −1. In addition, the TCS is equal to zero when P1 = 1, P2 =
−1 or P1 = −1, P2 = 1.
Figure 9. The TCS as a function of the polarization factor of e−
- For the evaluation of the cross section which depends on the polarization factors,
we determine that the DCS depends on cos θ (Figure 10) when the beams of e− are not
polarized. We can see that, the DCS is at a minimum value when cos θ = −1 and a
maximum value when cos θ = 1.
- For the evaluation of the total cross section (TCS) which depends on the center of
mass energy (
√
s), we determine that the TCS depends on
√
s (Figure 11) when the beams
of e− are not polarized (line 1) and when the beams of e− are polarized (line 2). We can
see that, the TCS quickly decrease, when the center of mass energy increases. This means
that we only need to consider the annihition process ae− in the low energy region and it
is also favorable to the receivers of the axion in experiments.
109
Dao Thi Le Thuy and Le Nhu Thuc
Figure 10. The DCS as a function of cosθ
Figure 11. The TCS as a function of
√
s
3. Conclusion
In this paper, we have calculated the production of axion in unpolarized and
polarized e+e− collision and the in process annihition ae−. The results show that, cross
sections depend strongly on the polarization factors of e beams (P1, P2) and the TCS
quickly decrease, when the center of mass energy increases. This means that we can
receive axion in the low energy region and it is also favorable to the receivers of the
axion in experiments.
110
Production of axion in e+e− and ae− collision
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