Higgs boson production in γµ− collision in the randall-sundrum model

1. Introduction The LHC discovery of a 126 GeV Higgs boson [1, 2] confirms the great success of the SM in describing electroweak phenomena, but with SM there are still many unanswered issues [3]. One is the question of hierarchy that arises from the quadratic divergence of the Higgs mass, suggesting the presence of some underlying physics in the gauge symmetry breaking mechanism that is so far unknown. Another issue is that the SM does not specify the Yukawa structures, has no justification for the number of generations, and it does not offer an explanation for the large hierarchy of the fermion masses, which exceed a range of five orders of magnitude in the quark sector and a much wider range when neutrinos are included. The gauge hierarchy problem is one of the driving theoretical reasons for the invention and necessity of new physics at the electroweak scale. Its strength can be quantified as a technical naturalness problem induced by the instability of the electroweak scale under radiative corrections, which gains severity given the hierarchy of 16 orders of magnitude between the electroweak and the Planck scale MP L. Models with compact extra dimensions explain this hierarchy in terms of geometry and at the same time, the hierarchical structures observed in the fermionic masses and in the mixing angles via so-called geometrical sequestering [4]. This can be achieved naturally within the framework of a warped extra dimension, first proposed by Randall and Sundrum (RS) [5]. There one studies the SM on a background consisting of Minkowski space, embedded in a slice of five-dimensional anti de-Sitter geometry (AdS5) with curvature k. The fifth dimension is an S1/Z2 orbifold of size r and has two branes, the UV and the IR brane, located at orbifold fixed points.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0043 Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 137-145 This paper is available online at HIGGS BOSON PRODUCTION IN γµ− COLLISION IN THE RANDALL-SUNDRUMMODEL 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 production of the Higgs boson in the γµ− collision process with the polarization of the muon beam is studied indetail. The value of differential cross-section is greatest when the angle between the direction of the beam Higgs boson and beam µ− approximately 180 degrees. For the total cross-section is greatest in high energy region. Based on this results, we hope that the reaction can give observable cross-section in Larger Hadron Collider (LHC) at the high degree of polarization. Keywords: Higgs boson, RS, cross-section. 1. Introduction The LHC discovery of a 126 GeV Higgs boson [1, 2] confirms the great success of the SM in describing electroweak phenomena, but with SM there are still many unanswered issues [3]. One is the question of hierarchy that arises from the quadratic divergence of the Higgs mass, suggesting the presence of some underlying physics in the gauge symmetry breaking mechanism that is so far unknown. Another issue is that the SM does not specify the Yukawa structures, has no justification for the number of generations, and it does not offer an explanation for the large hierarchy of the fermion masses, which exceed a range of five orders of magnitude in the quark sector and a much wider range when neutrinos are included. The gauge hierarchy problem is one of the driving theoretical reasons for the invention and necessity of new physics at the electroweak scale. Its strength can be quantified as a technical naturalness problem induced by the instability of the electroweak scale under radiative corrections, which gains severity given the hierarchy of 16 orders of magnitude between the electroweak and the Planck scale MPL. Models with compact extra dimensions explain this hierarchy in terms of geometry and at the same time, the hierarchical structures observed in the fermionic masses and in the mixing angles via so-called geometrical sequestering [4]. This can be achieved naturally within the framework of a warped extra dimension, first proposed by Randall and Sundrum (RS) [5]. There one studies the SM on a background consisting of Minkowski space, embedded in a slice of five-dimensional anti de-Sitter geometry (AdS5) with curvature k. The fifth dimension is an S1/Z2 orbifold of size r and has two branes, the UV and the IR brane, located at orbifold fixed points. Received December 1, 2015. Accepted December 20, 2015. Contact Le Nhu Thuc, e-mail address: thucln@hnue.edu.vn 137 Dao Thi Le Thuy and Le Nhu Thuc In the original RS model, there are two new particles beyond the Standard Model. One is a spin-2 graviton (and its Kaluza-Klein excitations) and the other is a scalar-field radion φ which is a metric fluctuation along the extra dimension. The radion acquires the mass of the order of the electroweak scale due to the Goldberger-Wise mechanism and it could be a lightest extra particle in the RS model [6, 7]. The radion, therefore, is expected to be the first signature of warped extra dimension models in direct search experiments such as the LHC. Phenomenology of the radion can be characterized by two parameters, radion mass mφ and scale parameter Λφ. The search experiments of the Higgs boson at the LHC give stringent constraints on the parameters of the radion [8]. Furthermore, there could be a mixing between the radion and the SM Higgs boson through the scalar-curvature mixing term in the four dimensional effective action [9, 10]. The characteristic features of the radion have been studied, including the phenomenological aspects of the radion in various colliders [11-17 ]. In this paper, we study the production of the Higgs boson at a photon collider which has been proposed as the γµ− collider. 2. Content 2.1. A review of RS model The experiments at the LHC have already used the dijet invariant mass to constrain the mass of these new resonances [18, 19]. The RS model is one of a number of new physics models which can solve the large hierarchy problem of the weak and the Plank scale. The RSmodel is based on a 5D space-time with the fifth dimension compacted on an S1/Z2 orbifold which has two fixed points, φ = 0 and φ = π. They correspond to the high energy brane and the brane we live on, respectively. Graviton is the only particle that can propagate through the bulk between these two branes. The 5-dimensional warped matric is given by [20]: ds2 = e−2kr|φ|(ηµν + 2 M 3 2 PL hµν)dx µdxν − r2dφ2, 0 ≤ |φ| ≤ π, (2.1) with ηµν = diag(1,−1,−1,−1) and where φ is the five dimensional coordinate, k is a scale of order of the Plank scale, r is compactification radius of the extra dimensional circle and hµν is the graviton metric. Solving the 5-dimensional Einstein equation and using Eq. (2.1), we can get the relation between the 4-dimensional reduced Plank scale M¯PL and the 5-dimensional Plank scale MPL [5], M¯PL = M3 PL k (1− e−2krπ). (2.2) The physical mass m of a field in a 4-dimension scale is related to the fundamental mass parameter m0 as follows: m = e−krπm0. (2.3) Thus the hierarchy problem can be solved by assuming kr ≈ 12 [20]. The 4-dimensional effective Lagrangian is L = − φ0 Λφ T µµ − 1 ΛˆW T µν(x) ∞∑ n=1 hnµν , (2.4) 138 Higgs boson production in γµ− collision in the randall-sundrum model where Λφ ≡ √ 6MPLΩ0 is the VEV of the radion field and ΛˆW ≡ √ 2MPLΩ0. T µν(x) is the energy-momentum tensor of the TeV brane which is localized on the SM fields. The T µµ is the trace of the energy-momentum tensor which is given at the tree level as T µµ = ∑ fmf f¯f − 2m2WW+µ W−µ −m2ZZµZµ + (2m2h0h20 − ∂µh0∂µh0) + ... (2.5) The gravity-scalar mixing arises at the TeV brane by [9] Sξ = −ξ ∫ d4x √−gvisR(gvis)Hˆ+Hˆ, (2.6) where R(gvis) is the Ricci scalar for the induced metric on the visible brane or TeV brane, g µν vis = Ω2b(x)(η µν + εhµν) . Hˆ is the Higgs filed before re-scaling, i.e., H0 = Ω0Hˆ . The parameter ξ denotes the size of the mixing term. 2.2. The matrix element of Higgs boson production in γµ− collisions In this paper we are interested in the production of the Higgs boson in high energy γµ− colliders when the µ− beams are polarized, µ−(p1) + γ(p2)→ µ−(k1) + h(k2). (2.7) Here pi, ki stand for the momentum of the particle, respectively. There are three Feynman diagrams contributing to reaction (2.7) and representing the s, u, t - channel exchange depicted in Figure 1. Figure 1. Feynman diagrams for µ−γ → µ−h Using Feynman rules, the matrix element for this process can be written as in the following cases: + For the s - channel: Ms = egmµ(d+ γb) 2mW (q2s −m2µ) εµ(p2)u¯(k1)(qˆs +mµ)γ µu(p1). (2.8) MsLR = egmµ(d+ γb) 2mW (q2s −m2µ) εµ(p2)u¯L(k1)(qˆs +mµ)γ µuR(p1). (2.9) MsRL = egmµ(d+ γb) 2mW (q2s −m2µ) εµ(p2)u¯R(k1)(qˆs +mµ)γ µuL(p1). (2.10) 139 Dao Thi Le Thuy and Le Nhu Thuc MuLL = egmµ(d+ γb) 2mW (q2s −m2µ) εµ(p2)u¯L(k1)(qˆs +mµ)γ µuL(p1). (2.11) MsRR = egmµ(d+ γb) 2mW (q2s −m2µ) εµ(p2)u¯R(k1)(qˆs +mµ)γ µuR(p1). (2.12) + For the u - channel: Mu = egmµ(d+ γb) 2mW (q2u −m2µ) εµ(p2)u¯(k1)γ µ(qˆu +mµ)u(p1). (2.13) MuLR = egmµ(d+ γb) 2mW (q2u −m2µ) εµ(p2)u¯L(k1)γ µ(qˆu +mµ)uR(p1). (2.14) MuRL = egmµ(d+ γb) 2mW (q2u −m2µ) εµ(p2)u¯R(k1)γ µ(qˆu +mµ)uL(p1). (2.15) MuLL = egmµ(d+ γb) 2mW (q2u −m2µ) εµ(p2)u¯L(k1)γ µ(qˆu +mµ)uL(p1). (2.16) MuRR = egmµ(d+ γb) 2mW (q2u −m2µ) εµ(p2)u¯R(k1)γ µ(qˆu +mµ)uR(p1). (2.17) + For the t - channel: Mt = ieCγ q2t [(p2qt)η µν − pµ2qνt ]u¯(k1)γµu(p1)εv(p2). (2.18) MtLR = ieCγ q2t [(p2qt)η µν − pµ2qνt ]u¯L(k1)γµuR(p1)εv(p2). (2.19) MtRL = ieCγ q2t [(p2qt)η µν − pµ2qνt ]u¯R(k1)γµuL(p1)εv(p2). (2.20) MtLL = ieCγ q2t [(p2qt)η µν − pµ2qνt ]u¯L(k1)γµuL(p1)εv(p2). (2.21) MtRR = ieCγ q2t [(p2qt)η µν − pµ2qνt ]u¯R(k1)γµuR(p1)εv(p2). (2.22) Here uL(pi) = 1− γ5 2 u(pi), uR(pi) = 1 + γ5 2 u(pi), pi = p1ork1; Cγ = − α 2πv [gfV ∑ i e2iN i cFi(τi)− (b2 + bY )gr][11]. Using these matrix elements, we evaluate the differential and total cross-section for Higgs boson production in the γµ− collider in the next section. 140 Higgs boson production in γµ− collision in the randall-sundrum model 2.3. Numerical results In this section, we present the numerical results for a differential and total cross-section for Higgs boson production in the γµ− collider when the µ− beams are polarized. From the expression of the cross-section: dσd(cos θ) = 1 64π |~k| |~p| |M |2 whereM is the matrix element, we assess the number, make the identification and evaluate the results obtained from the dependence of the differential cross-section by cos θ , and the total cross-section fully follows √ s and polarization factors of the µ− beams (P1, P2).We obtain some estimates for the cross-section as follows: Figure 2. Differential cross-section of the process γµ− → hµ− as a function of cosθ for the s - channel Figure 3. Differential cross-section of the process γµ− → hµ− as a function of cos θ for the u - channel We show in Figure 2, Figure 3, Figure 4 and Figure 5 the behavior of dσ/d cos θ at fixed collision energy √ s = 3000GeV (this energy can be done in LHC, in future can up to 14TeV). We have chosen a relatively low Higgs boson mass, mh = 126GeV , and the polarization coefficients are P1 = P2 = 0, −1, P1 = −1, P2 = 1, respectively. We see that the differential cross-section 141 Dao Thi Le Thuy and Le Nhu Thuc of the t-channel is largest when P1 = P2 = −1 and when cos θ approaches -1. For the case P1 = −1, P2 = 1, we only obtain the differential cross-section by s, u-channels, and for t-channel it is zero. In the other cases, the differential cross-sections were very small. Figure 4. Differential cross-section of the process γµ− → hµ− as a function of cos θ for the t - channel Figure 5. Differential cross-section of the process γµ− → hµ− as a function of cos θ for interference between the s-channel and the u-channel In Figures 6- 9, we plot total cross-section as a function of collision energy √ s for s, u and t - channels. The polarization coefficients are P1 = P2 = 0, −1, P1 = −1, P2 = 1 and the collision energy is in the region 500GeV ≤ √s ≤ 3000GeV . The figure shows that the total cross section has a maximum value at P1 = P2 = −1 for the t - channel. For the s and u - channels, the total cross-section obtained in the low energy region is greater than in the higher energy region, but this is not true for the t-channel. 142 Higgs boson production in γµ− collision in the randall-sundrum model Figure 6. Total cross-section of the process γµ− → hµ− as a function of the collision energy √ s for the s - channel Figure 7. Total cross-section of the process γµ− → hµ− as a function of the collision energy √ s for the u - channel Figure 8. Total cross-section of the process γµ− → hµ− as a function of the collision energy √ s for the t - channel 143 Dao Thi Le Thuy and Le Nhu Thuc Figure 9. Cross-section of the process γµ− → hµ− as a function of collision energy √ s for interference between the s-channel and the u-channel 3. Conclusion In this paper, we have calculated production of the Higgs boson in the γµ− collider when the µ− beams are polarized, with the results showing that contributions from the t-channel are greatest when the initial and final µ− beams are completely polarized and identical, left or right. The value of the differential cross-section is greatest when the angle between the direction of the beam Higgs boson and µ− beam approximately 180 degrees. The total cross-section is greatest in the high energy region so the ability to capture the Higgs boson is better in the high energy region than in the low energy region. 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