The e−e+ production via γγ collision in the randall-sundrum model at the ILC

HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0066 Natural Sciences, 2018, Volume 63, Issue 11, pp. 28-33 This paper is available online at THE e−e+ PRODUCTION VIA γγ COLLISION IN THE RANDALL-SUNDRUM MODEL AT THE ILC Bui Thi Ha Giang Faculty of Physics, Hanoi National University of Education Abstract. Taking into account the 125 GeV Higgs boson and radion propagators, we investigate the e−e+ production via γγ collision in the Randall-Sundrum model at International Linear Collider (ILC). The observable cross-sections are calculated for the diphoton collision. The cross-sections are evaluated with dependence on the collision energy √ s, the polarization of electron and positron beams. Keywords: ILC, diphoton collision, cross-section. 1. Introduction The gauge hierarchy between the eletroweak scale and the Plank scale was solved naturally by the Randall-Sundrum (RS) model in 1999 [1]. The RS model involves two three-branes bounding a slice of 5D compact anti-de Sitter space taken to be on an S1/Z2 orbifold extra dimension. Gravity is localized at the UV brane, while the Standard model fields are supposed to be localized at the IR brane. The additional scalar called radion (φ) corresponds to the fluctuations of the size of the extra dimension [2-5]. Radion can be the lightest particle in the RS model. In the effective four dimensional theory, the Higgs boson and the radion can be mixed since radion and Higgs fields have the same quantum numbers [6, 7]. In 2012, the 125 GeV Higgs signal is discovered by the ATLAS and CMS collaborations [8, 9], which has completed the particle spectrum of the Standard model. The international linear collider (ILC) [10], which is an e−e+ collider with high energy and luminosity, will supply the chance to detect new physics. The potential of the ILC can be enhanced by considering γγ and γe− collisions with the photon beam generated by the backward Compton scattering of electron and laser beams [11-14]. In this work, we investigate the e+e− production via γγ collision taking into account the Higgs and radion propagators. The layout of this paper is as follows. In Section II, we review the Higgs-radion mixing in the RS model. The e−e+ production of γγ collision is calculated in Section III. Finally, we summarize our results and make conclusions in Section IV. 2. Content 2.1. The Higgs-radion mixing in the Randall-Sundrum model In the RS model, the values of the bare parameters are determined by the Planck scale and the applicable value for size of the extra dimension is assessed by krcpi ≃ 35 (rc - the Received September 4, 2018. Revised: November 9, 2018. Accepted November 16, 2018 Contact Bui Thi Ha Giang, e-mail address: giangbth@hnue.edu.vn 28 The e−e+ production via γγ collision in the randall-sundrum model at the ILC compactification radius and k - the bulk curvature). Thus the weak and the gravity scales can be naturally generated. The Higgs boson and radion can be mixed with the gravity-scalar mixing described by the following action[6] Sξ = ξ ∫ d4x √ gvisR(gvis)Hˆ +Hˆ, (2.1) where ξ is the mixing parameter, R(gvis) is the Ricci scalar for the metric gµνvis = Ω2b(x)(ηµν + εhµν) induced on the visible brane, Ωb(x) = e−krcpi(1 + φ0Λφ ) is the warp factor, φ0 is the canonically normalized massless radion field, Hˆ is the Higgs field in the 5D context before rescaling to canonical normalization on the brane. With ξ 6= 0, there is neither a pure Higgs boson nor pure radion mass eigenstate. This ξ term mixes the h0 and φ0 into the mass eigenstates h and φ as given by h0 = dh+ cφ, φ0 = bh+ aφ, (2.2) where a = −cosθ Z , b = sinθ Z , c = sinθ + 6ξγ Z cosθ, d = cosθ − 6ξγ Z sinθ, Z2 = 1 + 6γ2ξ (1− 6ξ) is the coefficient of the radion kinetic term after undoing the kinetic mixing, γ = υ/Λφ, υ = 246 GeV. The mixing angle θ is tan 2θ = 12γξZ m2h0 m2φ0 −m2h0 (Z2 − 36ξ2γ2) , (2.3) where mh0 and mφ0 are the Higgs and radion masses before mixing. The new physical fields h and φ in (2.2) are Higgs-dominated state and radion, respectively m2h,φ = 1 2Z2 [ m2φ0 + βm 2 h0 ± √ (m2φ0 + βm 2 h0 )2 − 4Z2m2φ0m2h0 ] . (2.4) There are four independent parameters Λφ, mh, mφ, ξ that must be specified to fix the state mixing parameters. We consider the case of Λφ = 5 TeV and m0MP = 0.1, which makes the radion stabilization model most natural [5]. Feynman rules for the couplings of Higgs and radion are showed as follows V (h, f, f) = igffh = −i gmf 2mW (d+ γb) , (2.5) V (φ, f, f) = igffφ = −i gmf 2mW (c+ γa) , (2.6) V (h, γµ(k1), γµ(k2)) =iCγh [(k1k2)η µν − kν1kµ2 ] =− i α 2piυ0 ( (d+ γb) ∑ i e2iN i cFi(τi)− (b2 + bY )γb ) [(k1k2)η µν − kν1kµ2 ] , (2.7) V (φ, γµ(k1), γµ(k2)) =iCγφ [(k1k2)η µν − kν1kµ2 ] =− i α 2piυ0 ( (c+ γa) ∑ i e2iN i cFi(τi)− (b2 + bY )γa ) [(k1k2)η µν − kν1kµ2 ] , (2.8) 29 Bui Thi Ha Giang where b3 = 7, b2 = 19/6, bY = −41/6 are the SU(2)L ⊗ U(1)Y β-function coefficients in the SM. The auxiliary functions of the h and φ are given by F1/2(τ) = −2τ [1 + (1− τ)f(τ)], (2.9) F1(τ) = 2 + 3τ + 3τ(2 − τ)f(τ), (2.10) with f(τ) = ( sin−1 1√ τ )2 (for τ > 1), (2.11) f(τ) = −1 4 ( ln η+ η− − ipi )2 (for τ < 1), (2.12) η± = 1± √ 1− τ , τi = ( 2mi ms )2 . (2.13) mi is the mass of the internal loop particle (including quarks, leptons and W boson), ms is the mass of the scalar state (h or φ). Here, τf = ( 2mf ms )2 , τW = ( 2mW ms )2 denote the squares of fermion and W gauge boson mass ratios, respectively. 2.2. The e−e+ production of γγ collision We consider the collision process in which the initial state contains diphoton, the final state contains electron and positron in the RS model, γ(p1) + γ(p2) → e−(k1) + e+(k2). (2.14) Here, pi, ki (i = 1,2) stand for the momentums. There are three Feynman diagrams contributing to reaction (2.14), representing the s, u, t-channels exchange depicted in Figure1. Figure 1. The Feynman diagrams for γγ → e−e+ The transition amplitude can be given as follows: 30 The e−e+ production via γγ collision in the randall-sundrum model at the ILC Ms = [ −Cγφgeeφ q2s −m2φ − Cγhgeeh q2s −m2h ] εµ(p1) [(p1p2) η µν − pν1pµ2 ] εν(p2)u(k1)v(k2), (2.15) Mu = − ie 2 q2u −m2e u(k1)γ νεν(p2)(/qu +me)v(k2)γ µεµ(p1), (2.16) Mt = − ie 2 q2t −m2e u(k1)γ νεν(p1)(/qt +me)v(k2)γ µεµ(p2). (2.17) Here, qs = p1 + p2 = k1 + k2, qu = p1 − k2 = k1 − p2, qt = p1 − k1 = k2 − p2. The differential cross-section for the subprocess γγ → e−e+ is written as [16] dσ̂(ŝ) dcosψ = 1 32piŝ |−→k 1| |−→p 1| |Mfi| 2, (2.18) where |Mfi|2 = |Ms|2+|Mu|2+|Mt|2+Re (M∗sMu +M∗uMs +M∗sMt +M∗tMs +M∗uMt +M∗tMu), ψ = (−→p 1,−→k 1) is the scattering angle. The effective cross-section σ(s) for the subprocess γγ → e−e+ at the ILC can be calculated as follows σ(s) = ∫ 0.83 4m2e/s dxfγ/e(x) ∫ (cosψ)max (cosψ)min dcosψ dσ̂(ŝ) dcosψ , (2.19) where x = ŝ/s in which √ ŝ is center of mass energy of the subprocess γγ → e−e+,√s is center of mass energy of the ILC. The photon distribution function fγ/e is given by [12] fγ/e = 1 D(ζ) [ (1− x) + 1 1− x − 4x ζ(1− x) + 4x2 ζ2(1− x)2 ] (2.20) where D(ζ) = ( 1− 4 ζ − 8 ζ2 ) ln(1 + ζ) + 1 2 + 8 ζ − 1 2(1 + ζ)2 . (2.21) xmax = ζ 1 + ζ , for ζ = 4.8, xmax = 0.83. With the model parameters chosen as mh = 125 GeV, mφ = 10 GeV, we give some estimates for the cross-sections as follows: (i) In Figure 2, we evaluate the dependence of the differential cross-section on the cosψ in case of Pe− , Pe+ , which are the polarization coefficients of e−, e+ beams, respectively. The collision energy √ s is chosen as 500 GeV. The polarization coefficients are chosen as Pe− = Pe+ = 1; Pe− = 1, Pe+ = −1; Pe− = Pe+ = 0. To make the scattered particles be detected, the scattering angle is chosen as 10o ≤ ψ ≤ 170o [11]. The figure shows that the differential cross-sections dσ/dcosψ decrease first and then increase. (ii) We evaluate the dependence of the total cross-sections on the collision energy √s in Figure 3. The polarization coefficients are chosen as Fig.2. The collision energy is chosen in the range of 500 GeV≤ √s ≤ 1000 GeV (ILC). The figure shows that the total cross-sections decrease rapidly when the collision energy √ s increases. (iii) In Figure 4, the total cross-section is plotted as the function of Pe− , Pe+ . The figure indicates that the total cross-section achieves the minimum value when Pe− = Pe+ = ±1 and the 31 Bui Thi Ha Giang maximum value when Pe− = 1, Pe+ = −1 or Pe− = −1, Pe+ = 1. Numerical values for the production cross-section and the number of events in a year with the contribution of radion and Higgs propagators in the γγ → e−e+ collision at the ILC in case of Pe− = 1, Pe+ = - 1 are given in detail in Table 1. The high integrated luminosity is chosen as L = 10−34cm−1s−1. Figure 2. The differential cross-section as a function of cosψ in case of Pe− , Pe+ , which are the polarization coefficients of e−, e+ beams, respectively Figure 3. The cross-section as a function of the center of mass energy √s of the ILC experiments in case of Pe− , Pe+ , which are the polarization coefficients of e−, e+ beams, respectively Figure 4. The cross-section as a function of the Pe− , Pe+ polarization coefficients of the e−, e+ beams at ILC 32 The e−e+ production via γγ collision in the randall-sundrum model at the ILC Table 1. Numerical values for the production cross-section and the number of events in a year in case of Pe− = 1, Pe+ = −1 at the ILC √ s (GeV) 500 600 700 800 900 1000 σ (fbar) 15667.1 10995.6 8150.21 6287.65 5001.23 4075.05 N ( 106 events) 4.941 3.468 2.570 1.983 1.577 1.285 3. Conclusions In this paper, we have evaluated the production of e−e+ in the RS model via diphoton collision. The result shows that with the high integrated luminosity and the high polarization of electron and positron beams, the signal of Higgs-radion mixing might be detected in future ILC experiments. The differential cross-section is peaked in the backward and forward direction. We show that, in the region 500 GeV≤ √s ≤ 1000 GeV, the total cross-section is in the range of 4075.05 fbar ∼ 15667.1 fbar in case of Pe− = 1, Pe+ = −1. Finally, it is worth noting that the number of events in a year can reach at about 106 events with the high integrated luminosity and at the high degree of polarization. REFERENCES [1] L. Randall and R. Sundrum, 1999. Phys. Rev. Lett. 83 3370. L. Randall and R. Sundrum, 1999. Phys. Rev. Lett. 83 4690. [2] M. Frank, K. Huitu, U. Maitra, M. Patra, 2016. Phys. Rev. D94 055016. [3] Eboos, S.Keizerov, E.Rahmetov, K.Svirina, 2014. Phys. Rev. D90 095026. [4] W.D. Goldberger and M.B. Wise, 1999. Phys. Rev. Lett. 83 4922. [5] C. Csaki, M. L. Graesser and G. D. Kribs, 2001. Phys. Rev. D63 065002. [6] D. Dominici, B. Grzadkowski, J. F. Gunion and M. Toharia, 2003. Nucl.Phys. B671 243. [7] C. Csaki, J. Hubisz and S. J. Lee, 2007. Phys. Rev. D76 125015. [8] G. Aad et al., ATLAS Collaboration, 2012. Phys. Lett. B716 1-29. [9] S. Chatrchyan et al., CMS Collaboration, 2012. Phys. Lett. B716 30-61. [10] T. Behnke et al., arXiv: 1306.6327 [physics.acc-ph]; H. Baer et al., arXiv: 1306.6352[hep-ph]. [11] C.Y. Yue, C. Pang and Y.C. Guo, 2015. J. Phys. G: Nuclear and Particle Physics, Vol. 42, No. 7, 075003. [12] I. F. Ginzburg, G. L. Kotkin, V. G. Serbo and V. I. Telnov, 1983. Nucl. Instr. and Meth. 205 47; I. F. Ginzburg, G. L. Kotkin, S. L. Panfil, V. G. Serbo and V. I. Telnov, 1984. Nucl. Instr. and Meth. 219 5; V. I. Telnov, 2000. Nucl. Phys. Proc. Suppl 82 359. [13] R. Brinkmann, I. Ginzburg, N. Holtkamp, G. Jikia, O. Napoly, E. Saldin, E. Schneidmiller, V. Serbo, G. Silvestrov, V. Telnov, A. Undrus, M.Yurkov, 1998. Nucl.Instrum.Meth. A406, pp.13-49. [14] D. V. Soa et al., 2012. Mod. Phys. Lett. A27 1250126; J. P. Delahaye, 2011. Mod. Phys. Lett. A26 2997. [16] M. E. Peskin and D. V. Schroeder, 1995. An Introduction to Quantum Field Theory, Addision-Wesley Publishing. 33