A step towards exploring the features of Gravidilaton sector in Randall-Sundrum scenario via lightest Kaluza-Klein graviton mass

In this paper we study the role of the 5D Gauss-Bonnet corrections and two loop higher genus contribution to the gravity action in type IIB string theory inspired low energy supergravity theory in the light of gravidilatonic interactions on the lightest Kaluza-Klein graviton mass spectrum. From the latest constraints on the lightest Kaluza-Klein graviton mass as obtained from the ATLAS dilepton search in 7 TeV proton-proton collision, we have shown that due to the presence of Gauss-Bonnet and string loop corrections, the warping solution in an ${\bf AdS_{5}}$ bulk is quite distinct from Randall-Sundrum scenario. We discuss the constraints on the model parameters to fit with present ATLAS data.

• Additionally while determining the value of the model parameters from the proposed setup we also require that the bulk curvature is less than the five dimensional Planck scale M 5 so that the classical solution can be trusted [21][22][23].
In the present article first we compute the warping solution in presence of 5D Gauss-Bonnet as well as gravidilaton coupling and the two loop higher genus string loop correction. Further using the solution we have discussed the detailed phenomenological features of lightest Kaluza-Klein graviton mass in the light of the constraint obtained from the ATLAS dilepton search. We further compare our results with the results obtained from the well known Randall-Sundrum model and comment on the present status of both of them in the light of present collider constraints. In this analysis we use the combined phenomenological bounds on Gauss-Bonnet coupling α 5 obtained from Higgs diphoton and dilepton decay channels [24] and from Higgs mass from the ATLAS [25] and CMS [26] data within 5σ C.L.. This bound lies below the upper bound of viscosity-entropy ratio [27] and satisfies the unitarity bound [27][28][29][30][31][32][33] on the GB coupling. We have also discussed the explicit dependence and the phenomenological feature of the lightest Kaluza-Klein graviton mass on the 5D Gauss-Bonnet coupling, gravidilaton coupling and the two loop higher genus string loop correction by scanning our analysis throughout the allowed parameter space in the perturbative regime of the proposed setup. We start our discussion with the following 5D action of the two brane warped geometry model given by [6]: (1) with A, B, C, D = 0, 1, 2, 3, 4(extra dimension) and a conformal two-loop string coupling constant A 1 . Here i signifies the brane index, i = 1(hidden), 2(visible) and L f ield i is the Lagrangian for the fields on the ith brane where ith the brane tension V i and φ(y) represent the dilaton field which is dynamical in the bulk with respect to the extra dimensional coordinate 'y'. The background metric describing slice of the AdS 5 is given by, where r c is the dimensionless quantity in the Planckian unit representing the compactification radius of extra dimension. Here the orbifold points are y i = [0, π] and periodic boundary condition is imposed in the closed interval −π ≤ y ≤ π.
Varying the action stated in equation (1) and neglecting the back reaction of all the other brane/bulk fields except gravity and dilaton, the five dimensional Bulk Einstein's equation turns out to be, where the five dimensional Einstein's tensor and the Gauss-Bonnet tensor is given by: Similarly varying equation(1) with respect to the dilaton field the gravidilaton equation of motion turns out to be: where the five dimensional D'Alembertian operator is defined as: To solve equation (3) and equation (6) we assume that the dilaton is weakly coupled to gravity (weak coupling θ 1 ) and the bulk cosmological constant (weak coupling θ 2 ) since the Gauss-Bonnet coupling is an outcome of perturbative correction to gravity at the quadratic order. Now including the well known Z 2 orbifolding symmetry at the orbifold points, y i = [0, π], for perturbative regime of solution due to the presence of very weak couplings θ 1 , θ 2 and α (5) we can neglect the contribution from first two terms in the right hand side of Eq (6) in the bulk. The contribution from the left hand side in Eq (6) autometically vanishes within bulk. Finally we are left with only the last term in the right hand side of Eq (6) from which we get the following solution of the dilaton degrees of freedom within the bulk [6]: where c 1 and c 2 are the integration constants to be determined from the value of φ(y) at the boundaries. We write the dimensionless exponent of the dilaton factors by substituting Eq (8) at the orbifold point y i = π: where we redefine the exponents by using the symbols χ 1 and χ 2 . In the present context we have chosen that the two different dilaton couplings are connected through, θ 1 = −θ 2 for which we have: For numerical estimations we take the dilation couplings θ 1 , θ 2 to be small to keep it within the perturbative regime of solution and we set the arbitrary integration constants c 1 , c 2 to a desired value for which the dimensionless exponents of the dilaton factors are fixed at: Such a value of the dimensionless exponent of the dilaton factor produce a large enhancement even for small value of dilaton coupling parameter θ 1 , θ 2 and moderate values for and φ(0) and φ(π). In our subsequent calculation this enhancement factor will play a significant role. As we will see later such a choice is inspired from the requirement of Planck to TeV scale warping as well as to keep the mass of the first excited state of the Kaluzu-Klein mode graviton above the bound set by LHC which is 1.01 TeV as can be seen from the Table (I). Thus this choice sets a bound on the dilaton coupling consistent with LHC constraint.
In presence of dilaton the solution of the five dimensional bulk Einstein Gauss Bonnet equation of motion at leading order in GB coupling (α 5 ) turns out to be [6]: Also the localized brane tension can be computed as: where the brane tension V 1 and V 2 are localized at the position of orbifold fixed points, y i = [0, π], where the hidden and visible branes are placed respectively. However, it is clearly observed from Eq (12) and Eq (13) that within the bulk both the warp factor and the brane tension varies with the extra dimensional coordinate 'y' due to the presence of the dynamical dilaton degrees of freedom within the bulk. Here we have discarded the other branch of solution of k + (+ve branch) which diverges in the limit α 5 → 0, bringing in ghost fields [11,[34][35][36][37][38]. So that we are concentrating only on the -ve branch of the solution which we call further as, k − := k M .  (21)) (using Eq (22)  Here to study the outcome from our proposed setup we fix the model parameters as, Gauss-Bonnet coupling α5 = 5 × 10 −7 (which is consistent with the solar system constraint [41], combined constraint from the Higgs mass and favoured decay channels H → (γγ, ττ ) [24] using ATLAS [25] and CMS [26] data) and string two-loop correction A1 = 0.05 with M P l ≈ 10 19 GeV and Higgs mass mH = 125 GeV (within the 5σ statistical C.L. of ATLAS and CMS). Throughout the analysis additionally we have maintained another constraint between the gravidilaton coupling and the dilaton coupling with the 5D cosmological constant in AdS space-time, θ2 = −θ1. This implies at the leading order approximation, θ2φ(π) = −θ1φ(π) = 11 at the visible brane.
In the limit α 5 → 0, A 1 → 0, θ 1 → 0 and θ 2 → 0 we retrieve asymptotically the same result as in the case of RS model with [16,17]: and the barne tension is given by, with Λ 5 < 0. Now expanding Eq (12) in the perturbation series order by order around α 5 → 0, A 1 → 0, θ 1 → 0 and θ 2 → 0 we can write: (16) For the graviton, the Kaluza Klein mass spectra for n-th excited state in presence of gavidilatonic and Gauss-Bonnet coupling by applying Neumann (-) and Dirichlet (+) boundary conditions at the orbifold point y i = π where the visible brane is placed, can be written as [6,39]: in presence Gauss-Bonnet coupling and string loop correction. For the numerical estimations we use the +ve Dirichlet branch throughout the article. Furthermore the modified 4D effective Planck mass in presence of Gauss-Bonnet coupling can be expressed in terms of 5D mass scale as [6]: within the range 0.01 < ǫRS < 0.10. We have also shown the present status of the lower limit of the lightest Kaluza-Klein graviton mass for LHC ATLAS dilepton search by the green curve as depicted in table (I). Here for this plot we fix the model parameters as, α5 = 5 × 10 −7 , A1 = 0.05, θ2 = −θ1, θ2φ(π) ∼ 11 and θ1φ(π) ∼ −11. Here the allowed region is in the upper half of the green curve. The rest of the region (below the geen curve) is ruled out. Now using Eq (16) on Eq (18) the 5D quantum garvity scale at the position of visible brane y = π turns out to be: (19) where we use the fact that, e −2kMrcπ << 1 approximation holds good in Eq (18). Here additionally we use the fact that, k RS = Z T M P L , where Z T is a dimensionless tuning parameter. Now for the sake of clarity one can write Z T as: where we introduce an additional parameter, ǫ RS = kRS M5 with the restriction on the parameter, 0.01 < ǫ RS < 0.1, as used in [21,40]. This requirement emerges from the fact that the bulk curvature must be smaller than the Planck scale so that the classical solutions for the bulk metric given by the proposed model can be trusted. On the other hand string theory also supports this favoured range within the background of Klebanov Strassler throat geometry motivated D3 − D3 brane-antibrane setup [23]. It is important to mention here that only for RS model Z T ≈ ǫ RS , as M 5 ∼ M P l approximation holds good in RS setup. Further substituting Eq (20) in Eq (19) we found the simplified expression for the 5D quantum gravity scale in terms of ǫ RS which turns out to be: On the other hand to solve the hierarchy problem the brane localized Higgs mass can be written as: where we introduce a new parameter m CUT defined as, physically represents the cut off scale of the theory, above which new physics beyond standard model is expected to appear. A natural choice for this would be Planck or quantum gravity scale beyond which standard model will not be valid. Now using Eq (16) we introduce a new parameter ǫ M defined as: (24) Further using Eq (24) and Eq (22) in the graviton Kaluza Klein mass spectra as stated in Eq (17), the first Kaluza-Klein excitation (n = 1) becomes: 25) where we again use the fact that, e −2kMrcπ << 1 and the lightest graviton mass for Randall Sundrum model is given by: where x 1 = 7π/4 be the root of the Beesel function of the order 1 as obtained from Eq (17). Here in Eq (25) we introduce a new parameter, Θ T given by: which signifies the multiplicative uplifting factor of the lightest Kaluza Klein graviton mass spectra for the proposed model compared to the lightest graviton mass for Randall Sundrum model. The five dimensional action describing the interaction between bulk graviton and visible Standard Model fields dominated by fermionic contribution on the brane is given by where T αβ SM (x) represents the energy momentum or stress energy tensor containing all informations of Standard Model matter fields on the visible brane and h αβ (x, y) be the bulk graviton degrees of freedom. In this context K (5) is the coupling strength describing the tensor fluctuation in the context of graviton phenomenology. After substituting the Kaluza-Klien expansion for graviton degrees of freedom: and rescaling the fields appropriately, the effective four dimensional action turns out to be It is evident from equation (30) that while the zero mode couples to the brane fields with usual gravitational coupling ∼ 1/M P l which we have taken as unity, the coupling of the KK modes are ∼ e kMrcπ /M P l ∼ T eV −1 which is much larger than the coupling of massless graviton. Though such feature is also observed for the graviton KK modes in the usual RS model, here the graviton KK mode coupling depends on the GB coupling α 5 . In the present context the values of k M though increases with α 5 , the enhancement of the graviton KK mode mass causes the overall decrease in the detection cross section. Thus the absence of any signature of graviton KK modes, as reported by ATLAS data in dilepton decay processes, may be explained by GB coupling in warped geometry models.
In table (I) we present a comparative study between the lower limit of the lightest Kaluza-Klein graviton mass for n = 1 mode from the proposed theoretical model, the well known Randall-Sundrum (RS) model and the LHC ATLAS dilepton search in 7 TeV proton-proton collision. Additionally we have shown that the 5D mass scale of the proposed model is lying within the window 1.56M P l < M 5 < 4.95M P l for 0.01 < ǫ RS < 0.1. For RS model, the lower limit of the graviton KK mode mass lying within the window, 0.22 T eV < m G,RS 1 < 1.02 T eV for 0.01 < ǫ RS < 0.1. On the other hand the latest data from ATLAS predicts the graviton KK mode mass lying within 1.01 T eV < m G,ATLAS 1 < 2.22 T eV for 0.01 < ǫ RS < 0.1. This implies a serious conflict between graviton KK modes as predicted in RS model and the result reported by ATLAS Collaboration. But for the proposed model the lightest bound of the KK graviton mass estimated as 1.68 T eV < m G,RS 1 < 16.82 T eV which is above the recent lower bound of the KK graviton mass measured from LHC ATLAS dilepton search and lies within the parameter space for the future probing region of LHC. By taking into consideration of the enhancement of coupling between SM fields and graviton, it may be observed from the table (I) that for ǫ RS = 0.07 or higher, the lower bound of the graviton KK mode exceeds that predicted from ATLAS data.
To study the various hidden phenomenological features within super-Planckian regime of the UV cut-off scale from our proposed setup the scanned parameter space is given by: This bound on GB coupling α 5 is also consistent with the solar system constraint [41], combined constraint from the Higgs mass and favoured decay channels H → (γγ, ττ ) [24] using ATLAS [25] and CMS [26] data within the 5σ statistical C.L. Additionally, the parameter space mentioned in Eq (31)  the lower bound of the lightest KK graviton mass constrained from LHC ATLAS dilepton search. It is important to mention here that the bound on the 5D Gauss-Bonnet coupling obtained from Eq (31) is lying below the upper cut-off on coupling obtained from the Kubo formula i.e. α 5 < 1/4 [4,5,27] obtained in the context of AdS 5 /CFT 4 correspondence.
In Fig (1) we have shown the variation of the lightest Kaluza-Klein graviton mass from the proposed model (represented by red curve) and Randall Sundrum (RS) model (represented by blue curve) with respect to the phenomenological parameter ǫ RS = k RS M 5 , within the range 0.01 < ǫ RS < 0.10 as stated in Eq (31). We have also shown the present status of the lower limit of the lightest Kaluza-Klein graviton mass for LHC ATLAS dilepton search by the green curve in Fig (1). Here the allowed region is in the upper half of the green curve. The rest of the region below the geen curve phenomenologically is ruled out. We have also explored the phenomenological feature of the lightest Kaluza-Klein graviton mass with respect to the dialton coupling χ 2 = θ 2 φ(π) with the 5D AdS 5 cosmological constant Λ 5 at the wall of the visible brane for the proposed theoretical setup in Fig (2). We have also shown the present status of the allowed region for the lower limit of the lightest Kaluza-Klein graviton mass for LHC ATLAS dilepton search by the yellow shaded region in Fig (2). This will constrain the parameter χ 2 within, χ 2 ∼ O(6 − 12.8). This is also consistent with the present theoretical analysis as the proposed setup predicts χ 2 ∼ 11 as mentioned in Eq (31). For both the branch the lightest Kaluza-Klein graviton mass increases exponentially by increasing the dialton coupling χ 2 = θ 2 φ(π) and fixing the other parameters within the allowed parameter space stated in Eq (31). Next in Fig (3(a)) and Fig (3(b)) we have presented the characteristic feature of the lightest Kaluza-Klein graviton mass with respect to the 5D Gauss-Bonnet coupling (α 5 ) for the proposed theoretical setup for ǫ RS = 0.01 and ǫ RS = 0.10 respectively. For both the cases the lightest Kaluza-Klein graviton mass increases by increasing the 5D Gauss-Bonnet coupling (α 5 ) and fixing the other parameters stated in Eq (31). We have also shown the present status of the lower limit of the lightest Kaluza-Klein graviton mass for LHC ATLAS dilepton search by a point in Fig (3(a)) and Fig (3(b)) both. To uplift/increase the lower bound of the lightest Kaluza-Klein graviton mass estimated from the proposed theoretical setup compared to the LHC dilepton search by proposing the 5D Gauss-Bonnet coupling (α 5 ) within , α 5 ∼ O((4.8 − 5.1) × 10 −7 ), as explicitly mentioned in the table (I). Finally, in Fig (4(a)) and Fig (4(b)) we have explicitly shown the behaviour of the lightest Kaluza-Klein graviton mass with respect to the string two-loop coupling A 1 by fixing the rest of the parameters for the proposed theoretical setup for ǫ RS = 0.01 and ǫ RS = 0.10 respectively. For both the cases the lightest Kaluza-Klein graviton mass decreases by increasing the string two-loop coupling A 1 and fixing the other parameters stated in Eq (31). We have also shown the present status of the lower limit of the lightest Kaluza-Klein graviton mass for LHC ATLAS dilepton search by a point in these figures.
To summarize, we say that the perturbative two loop higher genus correction to Einsteins gravity in presence of stringy type IIB gravidilatonic interaction can also be examined through collider experimental tests by studying the hidden phenomenological features of lightest KK mode from graviton mass spectrum. Using the prescription mentioned in this paper one can directly check the validity and justifiability of a higher order gravity or any modified gravity model in presence of stringy higher genus corrections and also constrain the associated parameter space which involves various couplings with such higher order gravity corrections. Thus, in this work, by applying the requirements from latest data we have also elaborately analyzed the multi parameter space dependence on the lightest Kaluza-Klein graviton mass by studying the flow of the running through the crucial parameters proposed in this article. This analysis therefore determines the allowed parameter space for the proposed model and brings out the phenomenological constraint on the value of the stringy parameters in the context of recent LHC experiment by scanning the multiparameter space within a phenomenologically feasible range.