Production of axion in e+ e− and ae− collision

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 REFERENCES [1] Arik E., et al., 2009. [CAST Collaboration], JCAP 0902, 008. [2] Arik E., et al., 2011. [CAST Collaboration], Phys. Rev. Lett. 107, 261302. [3] Babu K. S., Gogoladze I. and Wang K., 2003. Stabilizing the axion by discrete gause symmetries. Phys. Lett. B 560, 214. [4] Barth K., et al., 2013. CAST constraints on the axion-electron coupling. JCAP 05, 010. [5] Kim J. E., 1984. 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