The effect of the scalar unparticle on the production of Higgs - radion at high energy colliders

An attempt is made to present the influence of the scalar unparticle on some scattering processes in the Randall - Sundum model. The contribution of the scalar unparticle on the production of Higgs - radion at high energy colliders is studied in detail. We evaluate the production cross-sections in the electron-positron ($e^{+}e^{-}$), photon-photon ($\gamma\gamma$) and gluon-gluon ($gg$) collisions, which depend strongly on the collision energy $\sqrt{s}$, the scaling dimension $d_{U}$ of the unparticle operator $\mathcal{O}_{U}$ and the energy scale $\Lambda_{U}$. Numerical evaluation shows that the cross - sections for the pair production of scalar particles are much larger than that of the associated production of the scalar particle with unparticle in the same condition


I Introduction
The Standard model (SM) is the successful model in describing the elementary particle physics. Recently, the 125 GeV Higgs is discovered by the ATLAS and CMS collaborations [1,2], which has completed the particle spectrum of the SM. Although the SM has been considered to be successful model, the model suffers from many theoretical drawbacks. In 1999, Lisa Randall and Raman Sundrum suggested the Randall-Sundrum (RS) model to extend the SM and solve the hierarchy problem naturally [3,4]. The RS setup involves two three-branes bounding a slice of 5D compact anti-de Sitter space. Gravity is localized at the UV brane, while the SM fields are supposed to be localized at the IR brane. The separation between the two 3-branes leads directly to the existence of an additional scalar called the radion (φ ), corresponding to the quantum fluctuations of the distance between the two 3-branes [5][6][7].
In the Lagrangian of the Standard model, the scale invariance is broken at or above the electroweak scale [8,9]. The scale invariant sector has been considered as an effective theory at TeV scale and that if it exists, it is made of unparticle suggested by Geogri [10,11]. Based on the Banks-Zaks theory [12], unparticle stuff with nontrivial scaling dimension is considered to exist in our world. The invariant Banks-Zaks field can be connected to the SM particles [13]. Recently, the possibility of the unparticle has been studied with CMS detector at the LHC [14,15].
The effects of unparticle on properties of high energy colliders have been intensively studied in Refs. [16][17][18][19][20][21][22][23][24][25][26]. However, the influence of scalar unparticle on the production of particles at the high energy colliders have not yet been concerned in the RS model. In this work, the contribution of the scalar unparticle on the production of Higgs -radion at the e + e − , γγ and gg colliders are studied in detail. The layout of this paper is as follows. In Section II, we give a review of the RS model and the mixing of Higgs-radion. The contribution of the scalar unparticle on the production of Higgsradion at high energy colliders are calculated in Section III. Finally, we summarize our results and make conclusions in Section IV.

II A review of Randall-Sundrum model and the mixing of Higgsradion
The RS model is based on a 5D spacetime with non -factorizable geometry. The single extra dimension is compactified on an S 1 /Z 2 orbifold of which two fixed points accommodate two three-branes (4D hyper-surfaces), the UV brane and the IR brane. The four dimensional effective action is obtained by integrating out the extra dimension. The classical action describing the above set-up is given by [3] where M is the five dimensional Planck scale, G = detG M N , Λ is a bulk cosmological constant, R is the 5D Ricci scalar. 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 kr c π 35 (r cthe compactification radius and k -the bulk curvature). Thus the weak and the gravity scales can be naturally generated. Consequently, the hierarchy problem is addressed. The gravity-scalar mixing is described by the following action [5] where ξ is the mixing parameter, R(g vis ) is the Ricci scalar for the metric g µν vis = Ω 2 b (x)(η µν + εh µν ) induced on the visible brane, Ω b (x) = e −krcπ (1+ φ 0 Λ φ ) is the warp factor, φ 0 is the canonically normalized massless radion field,Ĥ is the Higgs field in the 5D context before rescaling to canonical normalization on the brane. With ξ = 0, there is neither a pure Higgs boson nor pure radion mass eigenstate. This ξ term mixes the h 0 and φ 0 into the mass eigenstates h and φ as given by where Z 2 = 1+6γ 2 ξ (1 − 6ξ) = β −36ξ 2 γ 2 is the coefficient of the radion kinetic term after undoing the where m h 0 and m φ 0 are the Higgs and radion masses before mixing. The new physical fields h and φ in (4) are Higgs-dominated state and radion, respectively Feynman rules for the couplings of Higgs and radion are showed as follows where b 3 = 7, b 2 = 19/6, b Y = −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 with m i is the mass of the internal loop particle (including quarks, leptons and W boson), m s is the mass of the scalar state (h or φ).
Here, τ f = 2m f ms 2 , τ W = 2m W ms 2 denote the squares of fermion and W gauge boson mass ratios, respectively. There are four independent parameters Λ φ , m h , m φ , ξ that must be specified to fix the state mixing parameters. We consider the case of Λ φ = 5 TeV and m 0 M P = 0.1, which makes the radion stabilization model most natural [6,7].

III
The contribution of the scalar unparticle on the production of Higgs -radion at high energy colliders The effects of unparticle on properties of high energy colliders have been intensively studied in Refs. [16][17][18][19][20][21][22][23][24][25][26]. In the rest of this work, we restrict ourselves by considering only scalar unparticle. The scalar unparticle propagator is given by [9,11] where The effective interactions for the scalar unparticle operators are given by where G αβ denotes the gauge field strength and f stands for a standard model fermion. Feynman rules for the couplings of the scalar unparticle in the RS model are showed as follows Using the above formulas, we will study the effect of the scalar unparticle on some high energy scatterings in the RS model. We note here that in our previours works [27][28][29] we have shown that the detection of scalar particles in the RS model at high energy colliders would provide a clear evidence of new physics beyond the SM. Now we will investigate the contribution of the scalar unparticle on the production of Higgs -radion in the RS model at high energy colliders, such as e + e − , γγ and gg collisions in which Feynman diagrams are considered in detail in Appendix A.
1. The e + e − → hh/φφ collisions Now we consider the collision process in which the initial state contains electron and positron, the final state contains the couple of the scalar particles (Higgs or radion). We note here that the contribution of the scalar unparticle is by the propagator in the s -channel where X is Higgs or radion. The transition amplitude is given by Here, is the square of the collision energy. From the expressions of the differential cross-section [30] dσ d(cosψ) = 1 32πs where is the scattering angle. The model parameters are chosen as: λ f f = λ hh = λ φφ = λ 0 = 1, Λ U = 1000 GeV, 1 < d U < 2 in case of the scalar unparticle [22], m h = 125 GeV, m φ = 10 GeV [27,28]. We give estimates for the cross-sections which depend on the collision energy √ s, the scaling dimension d U of the unparticle operator O U and the energy scale Λ U as follows i) In Fig.1 we plot the total cross-sections as the function of d U . The collision energy is chosen as √ s = 500 GeV and 1.1 ≤ d U ≤ 1.9. From the Fig.1 we can see that in case of the additional scalar unparticle propagator, the cross sections decrease rapidly as d U increases and they are flat when d U > 1.6. ii) In Fig.2 we evaluate the dependence of the total cross-sections on the collision energy √ s. The collision energy is chosen in the range of 500 GeV≤ √ s ≤ 1000 GeV (ILC), the various d U is chosen as 1.1, 1.3, 1.5, 1.7, respectively. The figure shows that the total cross-sections decrease when the collision energy √ s increases. It is worth noting that with the contribution of the scalar unparticle propagator, the cross-sections for pair production of scalar particles are much enhanced. iii) In Fig.3 we evaluate the dependence of the total cross-sections on the Λ U at the fixed collision energy, √ s = 500 GeV. In case of the additional scalar unparticle propagator, the cross-sections increase rapidly in the region of 2 TeV ≤ Λ U ≤ 5 TeV. Note that here we only plot the maximum cross-sections based on Fig.1 In this section, we investigate the associated production of the scalar particle with unparticle at high energy e + e − colliders in which the scalar unparticle contribution on the scattering process is in the final state The transition amplitude can be given as follows With the parameters chosen as above, we give some estimates for the cross-sections with the contribution of scalar unparticle as follows i) In Fig.4 we plot the total cross-sections as the function of d U . We can see from the figure that the curve of the cross-sections is similar to Fig.1. That is, the cross-sections decrease rapidly as d U increases.
ii) In Fig.5 we evaluate the dependence of the total cross-sections on the collision energy √ s with the various d U . The result shows that the cross-sections decrease as the collision energy √ s increases. Note that the curve of the cross-sections is flat at very high energies.
iii) The dependence of the total cross-sections on the Λ U at the fixed collision energy, √ s = 500 GeV is shown Fig.6. The figures show that the total cross-section for the associated production in the e + e − → U h collision is about 10 3 times larger than that in the e + e − → U φ collision. Numerical values for the production cross section with d U = 1.1 are given in detail in Table 1. We can see from Table  1 that the cross-sections for the pair production of scalar particles are much larger than that of the associated production of scalar particles with unparticle under the same conditions. It is worthing to note that, when the collision energy increases, the total cross-section in the e + e − → φφ collision is insignificantly larger than that in e + e − → hh collision. 3. The γγ → hh/φφ collisions In this section, we consider the collision process in which the initial state contains the couple of photons, the final state contains the couple of scalar particles. The Feynman diagram is given by We obtain the results in the s, u, t -channels Now we estimate the production cross-sections with the contribution of the scalar unparticle propagator as follows i) In Fig.7 we plot the total cross-sections in the γγ → hh/φφ collisions as the function of d U . The collision energy is chosen as √ s = 3000 GeV (CLIC) and 1.1 ≤ d U ≤ 1.9. We can see from the figure that, the curve goes through the minimum value at d U = 1.65 and then increases rapidly with d U . ii) In Fig.8 we plot the total cross-sections as a function of the collision energy √ s. The collision energy region is 1T eV ≤ √ s ≤ 5T eV . The total cross-sections decrease gradually as √ s increases with the fixed d U . iii) In Fig.9 we plot the dependence of the total cross-sections on the energy scale Λ U with the parameters chosen as above. The figure shows that the cross-sections decrease gradually as the Λ U increases.

The γγ → U h/U φ collisions
In this section, we investigate the unparticle contribution on γγ → U h/U φ collisions The transition amplitude can be written as follows We estimate the cross-sections for the associated production as follows i) In Fig.10 we plot the total cross-sections as the function of d U with the parameters chosen as in previous items. The figure shows that the curve of the cross section is similar to Fig.1. We can see that the cross section decreases rapidly as d U increases and it is flat with d U > 1.6. ii) In Fig.11 we evaluate the dependence of the total cross-sections on the collision energy √ s with the fixed d U . The figure shows that when the collision energy √ s increases then the total cross-sections increase gradually. iii) In Fig.12 we plot the dependence of the total cross-sections on the Λ U . The figure shows that in the region 1 TeV ≤ Λ U ≤ 5 TeV the cross-sections decrease gradually as Λ U increases. Some typical values for cross-sections are given in detail in Table 2. The result shows that the crosssections for pair production of scalar particles are much larger than that of the associated production. Moreover, the total cross-section in γγ → U h collision is larger than that in γγ → U φ collision under the same conditions. 5. The gg → hh/φφ collisions Now we consider the gg → hh/φφ process which is similar to the γγ → hh/φφ process. The reaction is given by The transition amplitude for this process can be written as We evaluate the cross-sections as follows i) In Fig.13 we plot the total cross-sections in the gg → hh/φφ collisions as the function of d U . We can see from the figure that the shape of the cross-section is similar to Fig.7. That is, the curve of the cross-sections goes through the minimum value and then increases rapidly with d U .
ii) In Fig.14 we plot the total cross-sections as a function of the collision energy √ s. The figure shows that, the cross-sections decrease as √ s increases. The total cross-section in gg → φφ collision is insignificantly larger than that in gg → hh collision. iii) In Fig.15 we plot the dependence of the total cross-sections on the Λ U . We can see that in the region 1 TeV ≤ Λ U ≤ 5 TeV, the cross-sections decrease as Λ U increases. 6. The gg → U h/U φ collisions Finally, we study the contribution of the scalar unparticle on the associated production in gg → U h/U φ collisions We obtain the transition amplitude in the s, u, t -channels We estimate the cross-sections for associated production as follows i) In Fig.16 we plot the total cross-sections as the function of d U . From the figure we can see that the cross section decreases rapidly as d U increases and it is flat when d U > 1.45. ii) In Fig.17 we evaluate the dependence of the total cross-sections on the collision energy √ s with the fixed d U . The figure shows that when the collision energy √ s increases in the region 1T eV ≤ √ s ≤ 5T eV then the total cross-sections increase. The total cross-section in gg → U h collision is larger than that in gg → U φ collision. iii) In Fig.18 we plot the dependence of the total cross-sections on the Λ U . The figure shows that the cross-sections decrease as Λ U increases. Some numerical values for cross sections in case of d U = 1.1 are given in Table 3.

IV Conclusion
In this paper, we have evaluated the contribution of the scalar unparticle on the production crosssections of Higgs -radion in the Randall-Sundrum model at the (e + e − ), (γγ) and (gg) colliders, which depend strongly on the collision energy √ s, the scaling dimension d U of the unparticle operator O U and the energy scale Λ U . The results indicate that the cross -sections for the pair production of scalar particles are much larger than that of the associated production of scalar particle with the unparticle under the same conditions.
In the e + e − → hh/φφ collisions, the production cross -section decreases as the collision energy √ s increases. With the contribution of the scalar unparticle propagator, the cross-sections for the pair production of scalar particles are much enhanced while the cross-sections for the associated production in e + e − → U h/U φ collisions are very small. Numerical evaluation has shown that the cross-sections for the pair production of scalar particles are about 10 15 times larger than that of the associated production under the same conditions.
In the γγ → hh/φφ collisions, due to the main contribution of scalar unparticle on the propagator in the s-channel, the cross -sections decrease as √ s increases while the cross-sections for the associated production increase as √ s increases. This is because the unparticle couplings in u,t channels give the main contribution on the γγ → U h/U φ scattering process. However, the production cross-sections in γγ → hh/φφ collisions are much larger than that of γγ → U h/U φ collisions (about 10 5 times) under the same conditions.
In the gg → hh/φφ collisions, the cross-sections for the pair production of the scalar particle decrease rapidly first and then increase as √ s increases, while the cross-sections for the associated production increase as √ s increases, which is similar to γγ → U h/U φ process. Numerical evaluation has shown that the cross-sections of the associated production in the gg collisions are much larger than that in the γγ collisions under the same conditions. This is because the scalar couplings with gluon are larger than that with the photon.
Finally, we emphasize that in this work we have considered only on a theoretical basis, other problems concerning the scalar unparticle signals at LHC the readers can see in detail in Ref. [25] .