Properties of axion production in γµ-collision

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. 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