Abstract. Axion production in unpolarized and polarized γe− collision are considered in detail using the Feynman diagram method. The cross-sections are presented and numerical evaluations are given. The results show that the axion can be dark matter of the universe. Some estimates for experimental conditions are given from our results.

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JOURNAL OF SCIENCE OF HNUE
Interdisciplinary Science, 2013, Vol. 58, No. 5, pp. 11-16
This paper is available online at
AXION PRODUCTION IN UNPOLARIZED
AND POLARIZED γe− COLLISION
Dao Thi Le Thuy and Le Nhu Thuc
Faculty of Physics, Hanoi National University of Education
Abstract. Axion production in unpolarized and polarized γe− collision are
considered in detail using the Feynman diagram method. The cross-sections are
presented and numerical evaluations are given. The results show that the axion
can be dark matter of the universe. Some estimates for experimental conditions are
given from our results.
Keywords: Axion, axino, DCS, TCS.
1. Introduction
The strong CP problem is a big, unexplained mistery in the Standard Model of
particle physics. Among the various candidate solutions that have been proposed thus far,
the Peccei-Quinn mechanism is the most attractive candidate as a 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 [8].
At present, axion mass is constrained by laboratory [5], astrophysical and
cosmological considerations [12, 13] to 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 the dark matter of the
universe. In addition, an axino (the fermionic partner of the axion) naturally appears in
SUSY models [4] which acquires a mass from three-loop Feynman diagrams in a typical
range of between a few eV to a maximum of 1 keV [14]. Candidates for dark matter
can appear in different models, such as the 3-3-1 models [7] or in supersymmetric and
superstring theories [2]. Light particles with a two photon interaction can be transformed
into photons in an external electric or magnetic field by an effect first discussed by
Primakoff [9]. This effect is the basis of Sikivie’s methods for the detection of axions
in a resonant cavity [10]. Various terrestrial experiments to detect invisible axions by
making use of their coupling to photons have been proposed [6] and results from such
experiments have appeared recently [3]. The experiment CAST [1] at CERN searches
Received January 15, 2013. Accepted May 24, 2013.
Contact Le Nhu Thuc, e-mail address: thucln@hnue.edu.vn
11
Dang Thi Le Thuy and Le Nhu Thuc
for axions from the sun or other sources in the universe. Recently, several authors have
analyzed the potential of CLIC (Compact Linear Collider) based on the γe− collisions
to search for radion in the Randall-Sundrum (RS) model and the result shows that the
cross-section of radions may give observable values at moderately high energies [11]. In
this paper, we consider axion production in polarized and unpolarized γe− collision 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 for experimental
analyses. Beam polarization is also an indispensable tool used to identify and study new
particles and their interactions.
2. Axion production γe− collision
The Feynman diagrams for the collision process γe− through the s, t, u - channel
are drawn as Figure 1. From that, we get the following expression for the matrix element
for the production axion in a γe− collision when the beam of e− causes either polarization
and unpolarization.
Figure 1. Feynman diagram of γe− collision
When the beam of e− is not polarized, we can use the Feynman rules to make
calculations and obtain the square of the scattering amplitude as follows:
∣∣Ms∣∣2 = 2e2m2e
λ2as
(s−m2a)(1 + cos θ), (2.1)
for s-channel. ∣∣Mu∣∣2 = e2m2e
λ2a
8s
(s−m2a)(1 + cos θ)
, (2.2)
for u-channel.
∣∣Mt∣∣2 = 8e2
λ2γ(s−m2a)(1− cos θ)
{4s2 + (s−m2a)2(1 + cos θ)2}, (2.3)
for t-channel, with λγ =
4πfa
αgaγ
.
MsM
+
u =
4e2m2e
λ2as(1 + cos θ)
[(s−m2a)(1− cos θ)− 2s], (2.4)
12
Axion production in unpolarized and polarized γe− collision
for interfering between s-channel and u-channel; for interfering between s-channel and
t-channel; interfering between u-channel and t-channel, not give us the results.
When the beam of e− is polarized, we have the square of the scattering amplitude
as follows:
- When the beam of e− in the initial state is left polarized and the beam of e− in the
finial state is right polarized, we have,
∣∣MsRL∣∣2 = e2m2e
λ2as
(s−m2a)(1 + cos θ), (2.5)
for s-channel.
- When the beam of e− in the initial state is right polarized and the beam of e− in
the finial state is left polarized, we have,
∣∣MsLR∣∣2 = e2m2e
λ2as
(s−m2a)(1 + cos θ), (2.6)
for s-channel.
- When the beam of e− in the initial state is left polarized and the beam of e− in the
finial state is right polarized, we have,
∣∣MuRL∣∣2 = e2m2e
λ2a
4s
(s−m2a)(1 + cos θ)
, (2.7)
for u-channel.
- When the beam of e− in the initial state is right polarized and the beam of e− in
the finial state is left polarized, we have,
∣∣MuLR∣∣2 = e2m2e
λ2a
4s
(s−m2a)(1 + cos θ)
, (2.8)
for u-channel.
- When the beam of e− in the initial state, the finial state is right polarized, we have,
∣∣MtRR∣∣2 = 4e2
λ2γ(s−m2a)(1− cos θ)
{4s2 + (s−m2a)2(1 + cos θ)2}, (2.9)
for t-channel.
- When the beam of e− in the initial state, the finial state is left polarized, we have,
∣∣MtLL∣∣2 = 4e2
λ2γ(s−m2a)(1− cos θ)
{4s2 + (s−m2a)2(1 + cos θ)2}, (2.10)
13
Dang Thi Le Thuy and Le Nhu Thuc
for t-channel.
- For interference between the s-channel and u-channel when the beam of e− is
polarized, we have:
MsLRM
+
uLR = MsRLM
+
uRL =
2e2m2e
λ2as(1 + cos θ)
[(s−m2a)(1− cos θ)− 2s] (2.11)
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: me = 5, 1.10
−4GeV , λa = 247GeV , α =
1
137
, ma = 6.10
−10GeV ,
fa = 10
10GeV , gaγ = 0, 36 (DFSZ model), we have:
- When the evaluation of the cross section depends on the polarization factor, we
evaluate the DCS depending on cos θ (Figure 2) where TCS depends on the center of mass
energy (Figure 3).
Figure 2. The DCS as a function of cosθ
Figure 2 shows that the DCS depends strongly on cos θ and the DCS’s reduce with
an increasing value of cos θ, the DCS quickly reduse when the value of cos θ is in the
range of -1 to -0.9 and the DCS slowly reduse when the value of cos θ is in the range of
-0.9 to 1. For P1 = P2 = 0 (line 1), it is the case of the beam of e
− becoming unpolarized.
For P1 = −1, P2 = 1 or P1 = 1, P2 = −1 (line 2), the DCS is bigger than in the
case of unpolarization. For P1 = P2 = 1 or P1 = P2 = −1 (line 4), the contribution of
t-channel scattering is mainly in the collision process. However, this contribution is very
small (about 10−9 times) compared to the contribution of the s-channel and u-channel
when P1 = 1, P2 = −1 or P1 = −1, P2 = 1.
From Figure 3, for P1 = P2 = 1 (line 4), the TCS does not change. In other cases,
when the center of mass energy increases, then the TCS decreases.
14
Axion production in unpolarized and polarized γe− collision
Figure 3. The TCS as a function of
√
s
Figure 4. The TCS as a function of the polarization factor of e−
- An evaluation of the total cross section depends on the polarization factor, and
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 4, we can see that the TCS
reaches a maximum value when P1 = −1, P2 = 1 or P1 = 1, P2 = −1. In this state, the
beams of are fully polarized. In addition, the TCS is equal to zero when P1 = −1, P2 =
−1 or P1 = 1, P2 = 1.
15
Dang Thi Le Thuy and Le Nhu Thuc
3. Conclusion
In this paper, we have calculated the production of axion in unpolarized and
polarized γe− collision. The results show that, cross sections depend strongly on the
polarization factors of e− beams (P1, P2) and have a value that is much smaller than the
production of axion in γµ− collision. However, the cross sections for the axion production
at high energy are very small, much below neutrino production cross section, so that
the direct production of dark matter particles is in general not expected to lead to easily
observable signals in γe− and γµ− collision.
Acknowledgment. The work is supported in part by the Science and Technology
Foundation of the Hanoi National University of Education (HNUE) under grant No.
SPHN-12-216.
REFERENCES
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[3] Hagman.C, Sikivie.P, Sullivan.N.S and Tanner.D.B, 1990. Phys. Rev. D42, p. 1297.
[4] Kim.I. E, 1984. Phys. Lett. B136, p. 387.
[5] Kim.J.E, 1987. Phys. Rep. 150, p. 1.
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[7] Long.H.N and Lan.N.Q, 2003. Europhys. Lett. 64, p. 571.
[8] Peccei.R.D and Quinn. H. R, 1977. Phys. Rev. 38, p. 1440.
[9] Primakoff.H, 1951. Phys. Rev. 81, p. 899.
[10] Sikivie.P, 1985. Phys. Rev. D32, p. 2988.
[11] Soa. D.V, Thuy.D.T.L, Thao.N.H and Tham.T.D, 2012. Mod. Phys. Lett A. 27, No.
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[12] Raffelt.G.G, 1990. Phys. Rep. 198, p. 1.
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JOURNAL OF SCIENCE OF HNUE
Interdisciplinary Science, 2013, Vol. 58, No. 5, pp. 17-21
This paper is available online at
DOPANT EFFECT OF Y ON OPTICAL PROPERTIES OF ZnWO4 CERAMICS
Nguyen Manh An
Faculty of Physics, Hong Duc University
Abstract. We investigate the effect of the rare earth ion, Y on the structure,
absorption and Raman spectroscopy of ZnWO4 ceramics. In the XRD patterns of
Ce doped ZnWO4, some foreign peaks were found and an anormalous change in
cell parameter appeared around x = 0.15. This indicates that the Ce ion has an
effect on the structure of ZnWO4 and suggests a solubility limit of Ce in ZnWO4
ceramics. In addition, we also calculated the Raman active modes by using group
theory and we received 18 modes in Raman active and 16 modes in IR active.
The absorption measurement indicates the band gap of ZnWO4 decreases with
increasing Y content. The reasons for the above changes are discussed in this
presentation.
Keywords: Effect, Y, structure, absorption, Raman spectroscopy, ZnWO4 ceramics.
1. Introduction
Tungstate crystals are perspective scintillating materials for x-ray detectors’ γ-ray
medical tomographs. In particular, the ZnWO4 crystal is a well-known scintillator emitting
light at 480 nm under UV, X-ray and γ-ray excitation. An important parameter of
scintillating material is light emission efficiency and that depends on the transparency
of the host crystal in the visible spectral region. Parasite absorption is usually caused by
various defects of the lattice. Therefore, investigating the nature of defects responsible for
absorption in the ZnWO4 is needed in order to solve material science problems. Doping
of rare-earth ions into ZnWO4 lattice are expected to influence its chemical and physical
properties. However few studies have been reported on doped-ZnWO4 compared to that of
other inorganic compounds [1, 2]. Furthermore, the doping causes a disorder in structure.
The disorder in doped rare earth ZnWO4 ceramics is expected to vary strongly depending
on the doping level and temperature. The dynamic disorder is directly related to electronic
processes and localization in the insulating phase of this material. Therefore, monitoring
the disorder is of significant interest in order to understand the interplay of structural
and optical properties. Raman spectroscopy is an efficient tool for the study of structural
Received April 17, 2013. Accepted June 2, 2013.
Contact Nguyen Manh An, e-mail address: nguyenmanhan@hdu.edu.vn
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