Abstract. Properties of axion production in γµ− collision are considered in
detail using the the Feynman diagram method. The differential cross-sections are
presented and numerical evaluations are given. Based on results obtained it was
found that the cross sections depend strongly on the polarization factors of µ−
beams and are much larger than in axion production in γe− collision. In addition,
some estimates for experimental conditions are derived from our results.

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JOURNAL OF SCIENCE OF HNUE
Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 132-137
This paper is available online at
PROPERTIES OF AXION PRODUCTION IN
COLLISION
Dao Thi Le Thuy1 and Le Nhu Thuc2
1Faculty of Physics, Hanoi National University of Education
2Department of Post Graduate, Hanoi National University of Education
Abstract. Properties of axion production in γµ− collision are considered in
detail using the the Feynman diagram method. The differential cross-sections are
presented and numerical evaluations are given. Based on results obtained it was
found that the cross sections depend strongly on the polarization factors of µ−
beams and are much larger than in axion production in γe− collision. In addition,
some estimates for experimental conditions are derived from our results.
Keywords: Standard Model, axion, γµ−, strong-CP.
1. Introduction
The Standard Model (SM) expectations always resulted in very good agreement
with the huge number of experimental observations made in the past 40 years and more.
The recent discovery of a Higgs boson at the LHC (Large Hadron Collider) seemingly
completes the discovery program for all matter states predicted by the SM. Nevertheless,
some aspects of the theory remain unexplained and it is possible to make the assumption
that SM is not a definitive theory but an approximation at low energy (that is, the energy
currently accessible for particle physics experiments) of a more fundamental theory. The
strong CP problem is big mystery that is unexplained in the SM. Among various candidate
solutions proposed so far, the Peccei-Quinn mechanism is the most attractive candidate
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 [8].
At present, the axion mass is constrained by laboratory [5] and by 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 the order 10−5 eV, it is a good candidate to be dark
matter of the universe. Besides, an axino (the fermionic partner of the axion) naturally
appears in SUSY (supersymmetry) models which acquires a mass from three-loop
Received September 25, 2013. Accepted October 30, 2013.
Contact Le Nhu Thuc, e-mail address: thucln@hnue.edu.vn
132
Properties of axion production in γµ− collision
Feynman diagrams in a typical range of between a few eV to a maximum of 1 keV [4].
The candidates for dark matter can appear in a different model, in the 3-3-1 models [7]
or in the supersymmetric and superstring theories [2]. The light particles with a two
photon interaction can transform into photons in external electric or magnetic fields 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 of such experiments have been published [3]. The experiment CAST (CERN
Axion Solar Telescope) [1] at CERN (European Organization for Nuclear Research)
searches 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, the result shows
that, the cross-section of radions may give observable values at the moderately high
energies [11]. In a previous paper [14], we considered axion production in unpolarized
and polarized γe− collision using the Feynman diagram 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. In this
paper, we consider the axion production in unpolarized and polarized γµ− collision using
the Feynman diagram method in the PQWW (Peccei- Quinn - Weinberg - Weilcezek)
axion model. In this model, we have an axion - lepton interaction (we focus on the Model
I) as [5]:
La−ℓY = i
a
v
(χmee¯γ5e+ χmµµ¯γ5µ+ χmτ τ¯ γ5τ), (1.1)
From this interaction, we obtain the matrix elements in the next section.
2. Production of axion in γµ− collision
We get the following expression for the matrix elements for the production of axions
in γµ− collision, when the beam of µ− is unpolarized or polarized.
When the beam µ− is not polarized, we have:
- For s-channel:
Ms =
iemµ
λa(q2s +m
2
µ)
v¯(k1)γ5qˆsγµu(p1)ϵ
µ(p2), (2.1)
- For u-channel:
Mu =
iemµ
λa(q2u +m
2
µ)
v¯(k1)γµqˆuγ5u(p1)ϵ
µ(p2), (2.2)
133
Dao Thi Le Thuy and Le Nhu Thuc
- For t-channel:
Mt =
ieαgaγ
q2t πfa
qtµp2ρgνβgασε
µνρσv¯(k1)γ
βu(p1)ϵ
µ(p2), (2.3)
For interfering between s-channel and u-channel:
MsM
+
u =
−e2m2µ
λa(q2s +m
2
µ)(q
2
u +m
2
µ)
[16(qsk1)(qup1) + 8m
2
µ(qsqu)], (2.4)
For interfering between s-channel and t-channel:
MsM
+
t =
8ie2αgaγm
2
µ
λaπfa(q2s +m
2
µ)q
2
t
×
{−[(qtqs)(p1p2)− (qtp1)(qsp2)] + [(qtqs)(p2k1)− (qtk1)(qsp2)]} (2.5)
For interfering between u-channel and t-channel:
MuM
+
t =
8ie2αgaγm
2
µ
λaπfa(q2u +m
2
µ)q
2
t
×
{−[(qtqu)(p1p2)− (qtp1)(qup2)] + [(qtqu)(p2k1)− (qtk1)(qup2)]} (2.6)
When the beam µ− is polarized, we have:
- For s-channel: when the beam of µ− in the initial state is left polarized and the
beam of µ− in the finial state is right polarized, we have:
MsRL =
−iemµ
2λa(q2s +m
2
µ)
v¯(k1)(1− γ5)qˆsγµu(p1)ϵµ(p2), (2.7)
when the beam of µ− in the initial state is right polarized and the beam of µ− in the finial
state is left polarized, we have:
MsLR =
iemµ
2λa(q2s +m
2
µ)
v¯(k1)(1 + γ5)qˆsγµu(p1)ϵ
µ(p2), (2.8)
- For u-channel: when the beam of µ− in the initial state is left polarized and the
beam of µ− in the finial state is right polarized, we have:
MuRL =
−iemµ
2λa(q2u +m
2
µ)
v¯(k1)(1− γ5)γµqˆuu(p1)ϵµ(p2), (2.9)
when the beam of µ− in the initial state is right polarized and the beam of µ− in the finial
state is left polarized, we have:
134
Properties of axion production in γµ− collision
MuLR =
iemµ
2λa(q2u +m
2
µ)
v¯(k1)(1 + γ5)γµqˆuu(p1)ϵ
µ(p2), (2.10)
- For t-channel: when the beam of µ− in the initial state and the finial state are right
polarized, we have:
MtRR =
ieαgaγ
2πfaq2t
qtµp2ρgνβgασε
µνρσv¯(k1)(1− γ5)γβu(p1)ϵα(p2), (2.11)
when the beam of µ− in the initial state and the finial state are left polarized, we have:
MtLL =
ieαgaγ
2πfaq2t
qtµp2ρgνβgασε
µνρσv¯(k1)(1 + γ5)γ
βu(p1)ϵ
α(p2), (2.12)
- For interfering between s-channel and u-channel when the beam of µ− is
polarized, we have:
MsRLM
+
uRL =MsLRM
+
uLR =
−e2m2µ
λ2a(q
2
s +m
2
µ)(q
2
u +m
2
µ)
[8(qsk1)(qup1)], (2.13)
MsRLM
+
uLR =MsLRM
+
uRL =
−e2m2µ
λ2a(q
2
s +m
2
µ)(q
2
u +m
2
µ)
[4m2µ(qsqu)]. (2.14)
3. Discussion
From the matrix elements for the production axion above, we calculate the
differential cross section (DCS) and the total cross section (TCS) in the center-of-mass
frame and to discuss the following issues: Note that, in our calculation, we choose:
mµ = 0, 581GeV , λa = 247GeV , α = 1137 , ma = 6.10
−10GeV , fa = 1010GeV ,
gaγ = 0, 36 (DFSZ model).
* The evaluation of the cross section depends on the polarization factor, the
evaluation of the DCS depends on cos θ (Figure 1) and TCS depends on the center of
mass energy (Figure 2).
From Figure 1, we can see that, the polarization of the beam µ− is very clear
influence on the scattering process and particularly the DCS depends strongly on cos θ.
For P1 = P2 = 0 (line 1) which is a case of the beam of µ− unpolarization, for P1 = −1,
P2 = 1 or P1 = 1, P2 = −1 (line 2) then the DCS is bigger than in the case of
unpolarization, for P1 = P2 = 1 or P1 = P2 = −1 (line 4). In this case, the contribution of
t-channel scattering is mainly in the collision process. However, this contribution is very
small (about 1012 times) compared with the contribution of the s-channel and u-channel
when P1 = 1, P2 = −1 or P1 = −1, P2 = 1.
135
Dao Thi Le Thuy and Le Nhu Thuc
Figure 1. The DCS as a function of cosθ
Figure 2. The TCS as a function of
√
s
From Figure 2, for P1 = P2 = 1 (line 4) the TCS does not change. In other cases,
the TCS decrease, when the center of mass energy increases. This means that we only need
to consider the collision process γµ− in the low energy region and it is also favorable to
the receivers of the axion in experiments.
Figure 3. The TCS as a function of the polarization factor of µ− beam
* When the evaluation of the total cross section depends on the polarization factor,
we note that, P1 is the polarization factor of the beam of µ− in the initial state and P2
136
Properties of axion production in γµ− collision
is the polarization factor of the beam of µ− in the finial state. From figure (3), we can
see that, the TCS gets the maximum value when the beams of µ− are fully polarized
(P1 = −1, P2 = 1 or P1 = 1, P2 = −1). This result is larger than the result of the
collision process γe− by about 105 times. In addition, the TCS is equal to zero when
P1 = −1, P2 = −1 or P1 = 1, P2 = 1.
4. Conclusion
In this paper, the properties of axion production in unpolarized and polarized γµ−
collision are calculated in detail. The results show that, the cross sections depend strongly
on the polarization factors of µ− beams (P1, P2) and scattering angle (θ) and they get a
maximum value when the beams of µ− are fully polarized. This value is larger than the
production of axion in γe− collision by about 105 times. However, the cross sections for
axion production at high energy are very small, so that the direct production of dark matter
particles is in general not expected to lead to easily observable signals in γµ− collision.
REFERENCES
[1] S. Aune, et al., 2009. New Micromegas detectors in the CAST experiment. Nucl. Instrum.
Meth. A604, 15.
[2] K. S. Babu, I.Gogoladze and K.Wang, 2003. Stabilizing the axion by discrete gause
symmetries. Phys. Lett. B560, 214.
[3] C. Hagmann, D. Kinion, W. Stoeffl, K. van Bibber (LLNL, Livermore), E. Daw, H. Peng,
L.J. Rosenberg (MIT, LNS), J. LaVeigne, P. Sikivie, N.S. Sullivan (Florida U.) et al., 1998.
Results from a high sensitivity search for cosmic axions. Phys. Rev. Lett. 80, pp. 2043-2046.
[4] J .E. Kim, 1984. A Common Scale for the Invisible Axion, Local SUSY GUTs and Saxino
Decay. Phys. Lett. B136 378.
[5] J. E. Kim, 1987. Light Pseudoscalars, Particle Physics and Cosmology. Phys. Rept., 150, 1.
[6] H. N. Long, D.V.Soa and T.A.Tuan, 1995. Electromagnetic detection of axions. Phys. Lett.
B357, 469.
[7] H. N. Long and N.Q.Lan, 2003. Selfinteracting dark matter and Higgs bosons in the SU(3)(C)
x SU(3)(L) x U(1)(N) model with right-handed neutrinos. Europhys. Lett. 64, 571.
[8] R. D. Peccei and H.R.Quinn, 1977. CP Conservation in the Presence of Instantons. Phys.
Rev. Lett. 38, 1440.
[9] H. Primakoff, 1951. Photoproduction of neutral mesons in nuclear electric fields and the
mean life of the neutral meson. hys. Rev. 81, 899.
[10] P. Sikivie, 1985. Detection Rates for ’Invisible’ Axion Searches. Phys. Rev. D32, 2988.
[11] D. V. Soa, D. T. L. Thuy, N. H. Thao and T. D. Tham, 2012. Radion production in
gamma-electron collisions. Mod. Phys. Lett. A27, 1250126.
[12] G. G. Raffelt, 1990. Astrophysical methods to constrain axions and other novel particle
phenomena. Phys. Rept. 198, 1.
[13] M. S. Turner, 1990. Windows on the Axion. Phys. Rept. 197, 67.
[14] D. T. L.Thuy and L. N. Thuc, 2013. Axion production in unpolarized and polarized γe−
collision. Journal of Science of HNUE, Vol. 58, No. 5, p. 11.
137