Gravity stabilizes itself
aa r X i v : . [ g r- q c ] A ug Gravity stabilizes itself
Sumanta Chakraborty ∗ and Soumitra SenGupta † Department of Theoretical Physics, Indian Association for the Cultivation of Science, Kolkata 700032, India
June 25, 2018
Abstract
We show that a possible resolution to the stabilization of an extra spatial dimension (radion) canbe obtained solely in the context of gravitational dynamics itself without the necessity of introducingany external stabilizing field. In this scenario the stabilized value of the radion field gets determinedin terms of the parameters appearing in the higher curvature gravitational action. Furthermore, themass of the radion field and its coupling to the standard model fields are found to be in the weak scaleimplying possible signatures in the TeV scale colliders. Some resulting implications are also discussed.
Gravity has become the stumbling block in our search for a unified theory, which probably will lead toan understanding of the origin of our universe to the late time cosmic acceleration. On the other hand,even though the standard model of strong and electroweak interactions can explain a vast landscape ofexperimental results, it continues to have some longstanding unresolved issues, which strongly suggests tolook for physics beyond the standard model. One of the major drawback of the standard model is thenecessity of fine tuning, which originates from the large hierarchy between the electroweak and the Planckscale, known as the gauge hierarchy problem. It is remarkable that gravity provides a very novel solutionto this fine tuning problem through the existence of extra dimensions. Such a gravity based resolutionof the gauge hierarchy problem was elegantly described by Randall and Sundrum, where a single extradimension with manifold structure S /Z was assumed, resulting in two branes (hypersurfaces of (3 + 1)dimensions) located at orbifold fixed points with positive and negative tensions. Subsequently, startingfrom the Einstein’s equations in the bulk (higher dimensional spacetime) with a negative cosmologicalconstant, they could show that physical mass of a field confined on the negative tension brane is in theweak scale, due to an exponential suppression, whose origin traces back to gravity. There have beennumerous works later on to clarify some of the disadvantages of this model, as well as in generalizationsto more complex settings (for a representative class of works see [1–13]). One of the key feature of theRandall-Sundrum model is the appearance of an additional four dimensional massless scalar field havingno dynamics. This is an undesirable feature, since without a stabilization mechanism one cannot arrive atthe desired exponential suppression.Unfortunately, gravity sector alone could not cure this problem. One had to introduce an additionalscalar field in the bulk, whose action when integrated over the extra spatial dimension, provided the ∗ [email protected] † [email protected] only gravitational interactions! Thisis achieved by introducing higher curvature corrections to the Einstein-Hilbert action, which is expected,since the bulk spacetime is governed by Planck scale physics. We further delineate on the phenomenologyof the associated radion field, whose potential is being supplied by the higher curvature corrections anddemonstrate the significance for collider physics. The phenomenological study enables one to probe theproperties of gravitational physics, in particular that of higher curvature gravity, using colliders, leadingto new avenues of exploration.We have organized the paper as follows: We start with a brief introduction to higher curvature gravityand the particular model we will be interested in. Proceeding further we demonstrate how one can haveboth the exponential warping and radion stabilization in this scenario, the main theme of this work. Finallywe discuss the radion phenomenology and comment on possible collider signatures of our model beforepointing out future directions of exploration. It is generally believed that at high energies (or, small length scales) the Einstein-Hilbert action must besupplemented with higher curvature corrections respecting the diffeomorphism invariance of the action.There are several possibilities for the same, two such candidates being f ( R ) theories of gravity and Lanczos-Lovelock models of gravity. The Lanczos-Lovelock models are more complicated, but is free of ghosts dueto its quasi-linear structure [24–29]. On the other hand the f ( R ) models need special care and must satisfyfew conditions in order to ensure its ghost free behavior. The success of f ( R ) models lie in its excellentmatch with observations as far as the cosmological arena is considered. Further, f ( R ) models with acertain constraint on its parameters can also evade the solar system tests as well [30–44]. In this work, itwill be sufficient for our purpose to focus on the f ( R ) theories of gravity, satisfying a couple of constraintsensuring its ghost free behavior.We will work with a five dimensional spacetime consisting of a single extra spacelike coordinate y . Theextra dimension will assumed to be compact with a S /Z orbifold structure. Alike the Randall-Sundrumscenario, two branes are located at the orbifold fixed points y = 0 , π respectively, with y and − y identified.The bulk gravitational action is assumed to be of the following form A = Z d xdy √− g (cid:20) κ { f ( R ) } − Λ (cid:21) = Z d xdy √− g (cid:20) κ (cid:8) R + αR − | β | R (cid:9) − Λ (cid:21) , (1)where, κ is the five dimensional gravitational constant with mass dimension − /
2, Λ being the negativebulk cosmological constant with mass dimension 5, α and β are the higher curvature couplings havingmass dimensions of − − f ′ ( R ) > f ′ ( R ) < f ′′ ( R ) > α > α > | β | . Note that the model must satisfy these criterion at all curvature scales. An exact warpedgeometric solution to the above gravitational action has been derived recently in [45], which reads ds = f ( y ) h e − A ( y ) η µν dx µ dx ν + r c dy i ; A ( y ) = kr c y + κ v
12 exp (cid:18) − b κ r c y (cid:19) . (2)Here v is a constant of mass dimension 3 /
2, we have also defined b = (9 κ p | β | / α ) and k = p − Λ κ / f ( y ) appearing in Eq. (2),having the following structure f ( y ) = " √ κ v (cid:18) − b κ r c y (cid:19) − √ | β | κ v α exp (cid:18) − b κ r c y (cid:19) − / . (3)Note that the solution derived above has the desired exponential warping, through the term kr c y in thewarp factor. Such that on the visible brane at y = π , the physical mass of any field will be suppressed byexp( − kr c π ), leading to weak scale behavior. This is alike the Randall-Sundrum scenario, where introducinggravity alone leads to the desired weak scale phenomenology with a choice of kr c ≃
12. We would liketo emphasize the fact that the effect of higher curvature gravity is through some combinations of α and β both, keeping only α or β is not sufficient to get the desired warping. The next hurdle is to provide astabilization mechanism for the radion, which will drive the radion field to its value r c , compatible withthe exponential suppression. This is what we will elaborate on in the next section. In the original Randall-Sundrum scenario, one needs to introduce a bulk scalar field in order to achievethe stabilization of the radion field. The reason being, the Lagrangian for the Randall-Sundrum scenario,which is the Ricci scalar, has no potential term for the radion field. The bulk field introduces such apotential, and thereby satisfies both the suppression of the Planck scale and the stabilization of the radionfield. We will show that, since our action in Eq. (1) has higher curvature terms, if one evaluates the samefor the metric given in Eq. (2) and Eq. (3), it will naturally lead to a potential for the radion field andhence one can stabilize the same without ever introducing a bulk scalar field. The additional degree offreedom originating from the higher curvature terms actually plays the role of a stabilizing field. As aconsequence, the stabilized value would depend upon the parameters α and β appearing as the couplingsof the higher curvature terms. It will turn out that the condition kr c ≃
12 is essentially a condition onthese higher curvature couplings and the bulk cosmological constant.As we have already laid out the principles involved, we will proceed directly to the computation andwill work exclusively with the higher curvature action presented in Eq. (1). Given the metric ansatz inEq. (2) one can evaluate the Ricci scalar in a straightforward manner. Then one has to substitute theRicci scalar in the gravitational part of the action and obtain the corresponding f ( R ) Lagrangian, whichto leading orders in the coupling parameters reads (for details see Appendix A), f ( R ) ≃ − k − √ k ( κ v ) e − b r c y/κ + k (cid:0) κ v (cid:1) p | β | α e − b r c y/κ . (4)3iven this form of the f ( R ) Lagrangian, one can substitute the same in the bulk action, i.e., Eq. (1)and then integrate out the extra dimensional coordinate y over the interval [0 , π ], thanks to the orbifoldsymmetry. In this integration it will turn out that the contribution coming from the lower limit y = 0 isindependent of the radion field r c and hence adds a constant contribution to the radion potential, while thecontribution from y = π does have dependence on the radion field and shall serve as the radion potential.Introducing a new field φ ( r c ) = Φ exp[ − kr c π ], where Φ = p /kκ , we finally obtain the potential for theradion field (or, equivalently for the new field φ ) to yield, V ( φ ) = Z dy (cid:20) f ( R )2 κ − Λ (cid:21) ≃ k κ (cid:18) φ Φ (cid:19) " − √ κ v (cid:18) φ Φ (cid:19) δ + 9 κ v p | β | α (cid:18) φ Φ (cid:19) δ + constant (5)where, δ = (2 b /kκ ) = (9 p | β | / kα ), is a dimensionless constant. The usefulness of this quantity φ follows from the fact that one can upgrade it immediately to the status of a four dimensional scalar fieldwith r c → r ( x ), spacetime dependent brane separation. Such that the vacuum expectation value of thesame is given by r c and thus φ ( r c ) will denote the vacuum expectation value of φ [14, 15]. Note that thefield φ is merely a constant depending on the radion field. If the field is being upgraded to depend onthe spacetime coordinates, then the potential structure will remain identical, however it will inherit thecanonical kinetic term in the action as well (see, for example [15]).Note that the above potential is very much similar to the one obtained in [14,15] using bulk scalar field,with δ identified as ( m / k ), where m is the mass of the bulk scalar. Thus the higher curvature termsact as a source for the radion mass as we will explicitly illustrate later. Further in the above scheme thecondition δ < ∂V /∂φ = 0, leading to the following expression for r c (see Eq. (20) in Appendix A), kr c = 16 α π p | β | r − Λ κ √ κ v √ | β | kα q √ κ v √ | β | kα − (6)Thus with κ v ∼
40 and p | β | /kα ≃ /
20, the logarithmic term becomes of order unity and thenone readily obtains kr c ≃
12, the value desired for exponential warping. Hence starting from a puregravitational action, with higher curvature corrections, one can produce an exponential warping as wellas a proper stabilized value for the brane separation without ever introducing any additional structure.Further the desired warping to address the hierarchy problem leads to a relation between the highercurvature couplings and the bulk cosmological constant. This completes what we set out to prove, i.e.,derivation of the exponential warping leading to weak scale physics and a proper stabilization mechanismfor the brane separation, both from the gravitational dynamics alone.For the sake of completeness we would like to discuss on the choice of f ( R ) gravity and its role in radionstabilization. Since we are working in the higher curvature regime, where bulk effects are important, itmakes sense to add terms like R n to the Einstein-Hilbert action, with n positive. The first such choicecorresponds to adding a R term, which would lead to the original Goldberger-Wise scenario in thescalar tensor representation. The next leading order term in a bulk spacetime with negative cosmological4onstant, free of ghosts correspond to the one presented in Eq. (1). Interestingly for this situation onecan solve for the full scalar coupled Einstein’s equations as depicted in [45] and the situation will differconsiderably from the Goldberger-Wise scenario. In principle one can add more higher order terms tothe Lagrangian, however in those scenarios one would not be able to solve the full back-reacted problemin scalar-tensor representation. Moreover, these terms will be further suppressed and will contributeinsignificant corrections over and above Eq. (6). Thus the scenario presented in this work captures all theessential features and is simple enough to be solved in an exact manner. This motivates the choice presentedin Eq. (1). Given the above, it will be worthwhile to spend some time discussing the corresponding scenarioin the scalar-tensor representation [45], which will bring out the difference of our approach with the existingones. This is what we elaborate in the next section. It is well known that any f ( R ) gravity model is mathematically equivalent to a dual scalar-tensor repre-sentation [21, 36–38, 46–52]. The mathematical equivalence follows from the transformation of the Jordanframe action to the Einstein frame aka conformal transformation. Surprisingly, this equivalence holds inlower dimensions as well, viz., if one starts from a higher dimensional action and projects on to a lowerdimensional hypersurface the field equations are still connected by conformal transformation, provided oneexercise caution about the boundary contributions. Despite the mathematical equivalence, there are situ-ations where the two scenarios are not physically equivalent, e.g., in cosmological scenarios the f ( R ) framemay lead to late time acceleration, while the Einstein frame advocates late time deceleration [48, 53, 54].The issue of physical inequivalence becomes important if the spacetime inherits a singularity or is undergo-ing a quick evolution phase. In particular, using reconstruction technique [55–58] it is possible to generate f ( R ) theories explaining the early inflationary phase to late time accelerating phase of the universe, allof which ultimately results into finite time future singularity. The existence of future singularity oftenbreaks the equivalence with scalar-tensor representation [48, 59]. However in absence of singularity [60]equivalence of f ( R ) theories with scalar tensor theories does exist. The situation discussed in this workhas no such singularity in the spacetime structure, as evident from Eq. (2) and Eq. (3) respectively. Thusone may safely use the equivalence between f ( R ) and scalar tensor theories.Given this input, it will be worthwhile to explore the corresponding situation in this context, namelyhow the radion stabilization is affected as one considers the dual picture in the Einstein frame and contrastthe same with the stabilization already discussed in Section 3. For the f ( R ) action under our consideration,the corresponding action in the Einstein frame becomes, A = Z d x √− g (cid:20) R κ − g ab ∇ a ψ ∇ b ψ − V ( ψ ) − Λ (cid:21) (7)where ψ is the scalar field in the dual picture defined as, κ ψ = (2 / √
3) ln(1 + f ′ ) and the correspondingpotential becomes V ( ψ ) = 332 α ψ − κ k δ ψ (8)The expression for the potential brings out the key difference between the Goldberger-Wise stabilizationmechanism and the one advocated here — the potential for the scalar field in Goldberger-Wise mechanism5acks the quartic term present in our analysis. Incidentally, the presence of this quartic term helps to solvethe full back-reacted problem, while the original stabilization proposal was without incorporating the back-reaction of the scalar field. Hence the stabilized value of the radion field derived above incorporates theeffect of the quartic potential, as well as the back-reaction of the scalar field on the spacetime geometryand differs from the standard scenarios. The above structure of the potential also shows the reason forneglecting further higher curvature terms (e.g., R ) in the action . The scalar tensor representation withsuch higher curvature terms will involve more complicated potentials and hence cannot be solved in fullgenerality by incorporating the back-reaction as well. Further the R term in the action leads to theleading order departure from the Goldberger-Wise action, which we have studied in this paper. Additionalhigher curvature terms would lead to further sub-leading corrections and thus can be neglected.Given the action in the Einstein frame, one can invoke the corresponding solution (derived in Appendix A)and integrate out the extra dimensional part present in the action. This will result in a potential for theradion field r c , alike the Goldberger-Wise mechanism, whose minima would yield the following stabilizedvalue of the radion field, kr c ≃ α π p | β | r − Λ κ p | β | κ ψ kα ! (9)where ψ is the value of the dual scalar field in the y = 0 brane. Comparison with Eq. (6) reveals thatthe leading order behavior (i.e., the term outside logarithm) of the stabilized value of the radion field isidentical in both Jordan and Einstein frame. This observation explicitly demonstrates that, in both theseframes the radion is stabilized to the desired value necessary for exponential suppression of the Planckscale. Thus unlike various scenarios with either singularity or a quick evolution (where the two framesare physically inequivalent) in this particular situation the physical equivalence between the two frames ismanifest.At this stage, we may point out another alternative possibility of stabilizing the radion field, by incor-porating quantum effects of the bulk scalar field at nonzero temperature [61, 62]. In particular, one canthink of the resulting thermal fluctuations to generate a modulus potential which may inherit a minimum,thereby stabilizing the brane separation. The brane separation necessary to solve the hierarchy probleminvolves considering a low temperature limit of the free energy associated with the bulk quantum field,which may have connections with the AdS/CFT correspondence [61, 62]. However, in this context as well,one generally neglects the effect of back-reaction and treats the bulk quantum field to be siting on thefixed Anti-de Sitter background, unlike the scenario we have depicted. Thus after explicitly establishingthe differences between our approach and the existing ones, we now concentrate on the phenomenology ofthe radion field, viz. mass of the radion field and its interaction with the standard model fields in the nextsection. In this section we will briefly discuss two possible applications of the stabilized radion field in two diversephysical contexts. The first application will discuss the implications of this stabilized radion from theperspective of particle physics, while the other will address the effect of the radion field on the inflationaryparadigm. In both these contexts we will demonstrate that the stabilization of the radion field (which has It is evident that, the higher curvature terms are more and more suppressed, and we choose to work with the first twoleading order corrections, compatible with ghost free criterion, to the Einstein-Hilbert action.
Given the potential V ( φ ), one can immediately obtain the mass of the φ excitation by computing ∂ V /∂φ and then expressing the same using r c , given in Eq. (6). Performing the same, one arrives at the followingexpression for radion mass, m φ ≡ ∂ V∂φ ( r c ) = 5 k κ v (cid:18) φ ( r c )Φ (cid:19) δ δ e − kr c π , (10)where, φ ( r c ) can be obtained from Eq. (6). It is evident from the expression for m φ , that the there is anexponential suppression of the radion mass, which leads to a weak scale value from a Planck scale quantity.Note that this expression is very much similar to the result obtained in [14, 15], with the identificationof δ with m /k . But with one difference, which is caused by the ( φ ( r c ) / Φ) δ term in Eq. (10). Thisresults in a decrease in the radion mass as compared to the Goldberger-Wise scenario described in [15].However at the same time the choice of κ v also becomes important. To see the difference in a quantitativemanner, consider the following situation: κ v ∼ , δ ∼ /
32, using which one obtains kr c ∼
12, leadingto, m φ ∼ . k e − kr c π . While for the standard Goldberger-Wise scenario one would have obtained m φ ∼ . k e − kr c π . This explicitly shows that the radion mass in our approach is lighter (in this casetwo times) compared to the one obtained in [15], depicting quantitatively the difference between thesetwo approaches. Further note that the results in [14, 15] were derived in the context of a bulk scalar field,here we derive the same but from a purely gravitational standpoint. Similar exponential suppression willdrive the masses of the low lying Kaluza-Klein excitations to TeV scale [63–76]. In the context of radionmass however there is one further suppressing factor, namely δ and thus radion mass will be a bit smallercompared to the low lying Kaluza-Klein excitations of the bulk field. Since the radion mass, as presentedin this work is completely of higher curvature origin, this suggests that detection of the radion field φ maybe a hint not only of higher dimensions but also of higher curvature gravity.In order to find out coupling of the radion field to standard model particles, note that in this scenarioradion appears as a gravitational degree of freedom and since gravity couples to all kinds of matter, so willthe radion. Since the standard model fields are confined on the brane hypersurfaces, it is clear that theywill couple to the induced metric on the brane. On the brane located at y = 0, the warp factor becomesunity and the induced metric is proportional to η µν . Thus fields confined to y = 0 hypersurface does notcouple to the radion field. On the other hand, on the visible brane (the hypersurface y = π ) the inducedmetric is ( φ/ Φ) η µν and thus radion field will couple to the standard model fields. Here, φ = φ ( r c ) + δφ ,where δφ is the fluctuations around the minimum value φ ( r c ). In the case of a scalar field, say the standardmodel Higgs h ( x ), one can reabsorb the factors of Φ by redefining h → (Φ /φ ( r c )) h ( x ), such that physicalHiggs mass becomes m e − kr c π . For a Planck scale bare mass m and kr c ≃
12 one immediately obtainsthe physical mass in the weak scale. Further the corresponding interaction term of a standard model fieldwith the radion will involve L int = 1 φ ( r c ) δφT µµ ≡ φ δφT µµ , (11)where T µµ stands for the trace of the energy momentum tensor of the standard model field and Λ φ definesthe coupling of the radion to the standard model fields. (Since radion is a gravitational degree of freedom7t has to couple to some combination of the matter energy momentum tensor). Note that for large Λ Φ ,the coupling becomes small and as a consequence radion will couple weakly to the standard model fields.In particular recent bounds on both radion mass and radion coupling strength shows that they are notindependent, if radion mass is smaller the coupling will be larger and vice versa. To get a numericalestimate, one must provide an estimate for the value of k in Planck units. For k ∼ .
1, the couplingsatisfies the following stringent bound Λ φ > . k ∼
1, in Planck units, the coupling can becomeΛ φ ∼ ∼ W ± and Z with a coupling Λ − φ , which is (TeV) − . These interaction terms will producethree point as well as four point interactions between radion and the gauge bosons due to gauge fixingterms as well as higher loop effects in the effective Lagrangian. Fixing the mass of the standard modelHiggs boson at 125 GeV [77], affects the electroweak parameters such that they become insensitive to theradion mass, since the radion mass is further suppressed by its own vacuum expectation value [82]. Despiteabove, the fact that the mass of the radion field as well as its coupling with standard model fields are inthe TeV scale makes the phenomenology of the radion field a nice testbed for higher dimensional as wellas higher curvature physics in the next generation colliders. Having already discussed the imprints of the stabilization mechanism of the radion field on the phenomeno-logical side, we will presently address the corresponding situation in a cosmological setting. In a moregeneral context, following [65] one should have made the radion field dynamical by considering arbitraryfluctuations around the stabilized value and hence study the dynamics in an arbitrary background. How-ever this will be a complicated exercise, due to presence of higher curvature terms in the gravitationalLagrangian. Thus we concentrate on a specific situation with the radion field depending on time alone,leading to the stabilized value in the cosmological context. Note that this situation has already beenanalyzed in the context of general relativity in [83] and has been elaborated further in the context ofradion stabilization in [23]. Thus one may try to find out the cosmology on the brane in presence of highercurvature terms, whose detailed analysis for various cosmological epochs will be presented elsewhere. Inthis section we will try to answer this question in the context of inflationary paradigm alone. However forcompleteness, we will also provide some general computations. The metric ansatz suitable for our purposecorresponds to, ds = e − A ( y ) r ( t ) (cid:8) − dt + a ( t ) (cid:0) dx + dy + dz (cid:1)(cid:9) + r ( t ) dy (12)where, both the branes are located at y = 0 and π respectively and they are assumed to be expanding withthe scale factor a ( t ). Further, the radion field is assumed to be dynamical, such that, r ( t ) = r c + δr ( t ).Here r c corresponds to the stabilized value of the radion field derived in the earlier sections and δr ( t )corresponds to the fluctuations around the stabilized value. Further A ( y ) should take care of all the extradimensional dependent quantities, which will behave as ky to the leading order. The Ricci scalar derived8or the above ansatz, on a y = constant hypersurface becomes, R = − k + 6 e kry (cid:18) ¨ aa + ˙ a a (cid:19) − aa ˙ rr (6 kry − e kry + e kry (cid:26) ¨ rr (2 − kry ) + ˙ r r kyr (6 kry − (cid:27) (13)Having obtained the Ricci scalar one can derive the Lagrangian with ease which is given in Eq. (1).Derivation of the Lagrangian enables one to obtain the corresponding field equations for a and r , byvarying the scale factor and the radion field respectively. The gravity theory being f ( R ) in nature, thefield equations will definitely inherit higher than second derivatives of the scale factor and the radion field,which reads,3 a re − kry f ( R ) − ra f ′ ( R ) ¨ aa + d dt (cid:8) ra e − kry f ′ ( R ) (cid:9) − ra e − kry f ′ ( R ) (cid:18) ˙ aa (cid:19) + 3 e − kry ˙ rr ˙ aa (6 kry − ra f ′ ( R ) + 3 ddt (cid:18) e − kry ˙ rr (6 kry − ra f ′ ( R ) (cid:19) = p (14)In general it looks sufficiently complicated, however in the case of an exponential expansion (with a ( t ) = e Ht ), i.e., for inflationary scenario the above equation simplifies a lot. In particular it is possible toapproximately solve for the time dependence of the radion field explicitly, which turns out to be decreasingwith time, similar with the corresponding situation in general relativity [23]. Thus as the universe expandsexponentially, the radion field decreases with time, finally attaining the stabilized value as the inflationends. Hence the scenario depicted above can also explain a dynamical procedure for stabilization of theradion field, modulo inflationary paradigm. This provides yet another application of the radion field akahigher curvature gravity in the present context. Gauge hierarchy problem is a very serious fine tuning problem in standard model physics. One avatar ofthe extra dimensional physics as depicted by Randall and Sundrum has the capability of addressing thegauge hierarchy problem by suppressing the Planck scale to weak scale gravitationally . Unfortunately, tostabilize the above scenario one needs to introduce an additional field. In this work, we have shown thatthe introduction of such a stabilizing field is unnecessary and one can achieve both the suppression to weakscale as well as a stabilization mechanism starting from a higher curvature gravitational action alone . Inthis sense gravity stabilizes itself!Starting from a higher curvature gravitational action we have derived an exact warped geometricsolution with the extra spatial dimension having S /Z orbifold symmetry and two 3-branes located atorbifold fixed points y = 0 , π respectively. We would like to emphasize that the above solution is exact witheffects from the higher curvature terms duly accounted for, unlike the original Goldberger-Wise solutionwhere back reaction of the stabilizing field was neglected. In this respect our work is more in tune with [63],where also solutions have been derived with inclusion of back-reaction as well. The warp factor again hasexponential suppression and thus the Planck scale physics will be reduced to weak scale phenomenon onthe visible brane located at y = π . But, surprisingly the higher curvature terms provide a potential forthe radion field as well, whose minima leads to the stabilized value of the radion field gravitationally .Further, the stabilized value depends on the gravitational couplings present in the higher curvature actionas well as the bulk cosmological constant, such that a particular choice of these parameters results to kr c ≃
12, leading to the desired warping. Added to the excitement is the result that the radion mass has9imilar suppression leading to TeV scale physics, which is smaller compared to the low-lying Kaluza-Kleinspectrum of the bulk scalar field. Being a gravitational degree of freedom, radion field couples to thestandard model fields through the trace of the energy momentum tensor with a coupling having strengthTeV − . Once again, we reiterate on the fact that the above analysis has been performed completely in agravitational physics framework.The above results open up a broad spectrum of further avenues to explore. From the observationalpoint of view, the above result brings down the dynamics of the higher curvature gravity to a TeV scalephenomenon, which may become accessible in near future collider. In particular the radion mass whichdepends exclusively on the couplings present in the higher curvature theory may provide the first hinttowards observational signatures of higher curvature gravity besides that of higher dimensions. A morecareful analysis in this direction can be performed following [67–71], where similar analysis for an additionalstabilizing scalar field has been carried out. From the theoretical hindsight, it will be worthwhile tounderstand the phenomenon of generating a potential for the radion field and its subsequent stabilizationin the context of Lanczos-Lovelock (or, Einstein-Gauss-Bonnet for simplicity) gravity. It will also be ofinterest to explore the consequences of making the radion field dynamical, in which case the kinetic term willalso contribute to the gravitational field equations resulting in distinctive cosmological consequences of thishigher curvature brane world scenario, e.g., imprints on the Cosmic Microwave Background, inflationaryscenario and so on. Acknowledgement
Research of S.C. is supported by the SERB-NPDF grant (PDF/2016/001589) from Government of India.
A Appendix: Calculational Details
In this section, we provide all the calculational details to supplement the results presented in the maintext. In order to make the calculation simple, we will introduce the following definitions: k = r − Λ κ A = κ v
12 ; A = 2 b κ = 9 p | β | α ; A = √ κ v A = − √ | β | κ v α . (15)We further divide the appendix in two parts, the first done depicts the situation with f ( R ) gravity, whilethe other illustrates the dual scalar-tensor description. A.1 Stabilization in f(R) gravity
The Ricci scalar computed for the metric presented in Eq. (2) turns out to be R = 1 r c f ( y ) (cid:2) f ( y ) (cid:0) A ′′ − A ′ (cid:1) + 4 f (4 A ′ f ′ − f ′′ ) + f ′ (cid:3) , (16)10here ‘prime’ denotes differentiation with respect to the extra spatial coordinate y . Evaluation of the Ricciscalar, given the functions A ( y ) and f ( y ) can be performed to leading orders of exp( − A r c y ) yielding R = − k + e − A r c y (cid:26) A A + 323 kA A − k A (cid:27) + e − A r c y n kA A − kA A + 32 A A − A A + 209 k A + 23 A (cid:18) A A + 323 kA A − k A (cid:19) o ≡ − k + Ee − A r c y + F e − A r c y . (17)Let us now write down the higher curvature Lagrangian of Eq. (1) using the above expression for the Ricciscalar, L = (cid:0) − k + Ee − A r c y + F e − A r c y (cid:1) + α (cid:0) − k + Ee − A r c y + F e − A r c y (cid:1) − | β | (cid:0) − k + Ee − A r c y + F e − A r c y (cid:1) = n − k + α (cid:0) − k (cid:1) − | β | (cid:0) − k (cid:1) o + e − A r c y n E − αk E − | β | (cid:0) − k (cid:1) E o + e − A r c y n F + αE − αk F − | β | E (cid:0) − k (cid:1) − | β | F (cid:0) − k (cid:1) o = P + Qe − A r c y + Re − A r c y . (18)Thus the bulk gravitational action reads,2 κ A = Z d x Z dyr c f ( y ) / e − A ( y ) L = Z d x Z dyr c (cid:0) A e − A r c y + A e − A r c y (cid:1) − / e − kr c y (cid:0) P + Qe − A r c y + Re − A r c y (cid:1) = Z d x Z dyr c e − kr c y (cid:20) P + (cid:26) Q − P A (cid:27) e − A r c y + (cid:26) R − QA + 209 P A (cid:27) e − A r c y (cid:21) . (19)Integrating out the extra spatial coordinate, leads to a potential term for the radion field, which contributesonly at y = π and becomes,2 κ V ( r c ) = e − kr c π (cid:18) P k (cid:19) + e − kr c π e − A r c π (cid:26) Q − P A k + A (cid:27) + e − kr c π e − A r c π (cid:26) R − QA + P A k + 2 A (cid:27) + terms independent of r c . (20)Such that the minima of the potential, leading to stabilized value of r c can be obtained by equating ∂V /∂r c = 0 resulting in kr c = 16 α π p | β | r − Λ κ (cid:0) R − QA + P A (cid:1)(cid:0) P A − Q (cid:1) + q(cid:0) Q − P A (cid:1) − P (cid:0) R − QA + P A (cid:1) . (21)11he above expression can be enormously simplified by keeping terms to the leading order. In particular,one will have P = − k , Q = − (40 / k A and R = 80 kA A leading to Eq. (5), where we have assumed A > k > A and A >
1. It turns out that in order to have the stabilized value of kr c ≃
12, the aboveconditions are identically satisfied. Introduce the field φ , such that the above potential becomes,2 κ V ( φ ) = (cid:18) P k (cid:19) (cid:18) φ Φ (cid:19) + (cid:18) φ Φ (cid:19) A k (cid:26) Q − P A k + A (cid:27) + (cid:18) φ Φ (cid:19) A k (cid:26) R − QA + P A k + 2 A (cid:27) + terms independent of r c . (22)This is the expression, which along with the previous identifications has been used to arrive at Eq. (6).Given the potential V ( φ ), one can compute,2 κ ∂V∂φ = 1Φ (cid:18) P k (cid:19) φ Φ + 1Φ (cid:18) φ Φ (cid:19) A k (cid:18) A k (cid:19) (cid:26) Q − P A k + A (cid:27) + 1Φ (cid:18) φ Φ (cid:19) A k (cid:18) A k (cid:19) (cid:26) R − QA + P A k + 2 A (cid:27) , (23)as well as, 2 kκ ∂ V∂φ = 1Φ P φ Φ + 1Φ (cid:18) φ Φ (cid:19) A k (cid:18) A k (cid:19) (cid:26) Q − P A (cid:27) + 1Φ (cid:18) φ Φ (cid:19) A k (cid:18) A k (cid:19) (cid:26) R − QA + 209 P A (cid:27) . (24)Thus the minimum of the potential corresponds to ∂V /∂φ = 0, P + (cid:18) φ Φ (cid:19) A k (cid:26) Q − P A (cid:27) + (cid:18) φ Φ (cid:19) A k (cid:26) R − QA + 209 P A (cid:27) = 0 . (25)Thus the radion mass becomes, m φ = ∂ V∂φ ( r c ); ∂V∂φ ( r c ) = 0= 12 kκ φ Φ " − (cid:18) φ Φ (cid:19) A k (cid:26) Q − P A (cid:27) − (cid:18) φ Φ (cid:19) A k (cid:26) R − QA + 209 P A (cid:27) + (cid:18) φ Φ (cid:19) A k (cid:18) A k (cid:19) (cid:26) Q − P A (cid:27) + (cid:18) φ Φ (cid:19) A k (cid:18) A k (cid:19) (cid:26) R − QA + 209 P A (cid:27) . (26)This helps to obtain the radion mass as in Eq. (10).12 .2 Stabilization in scalar-tensor representation Since any f ( R ) theory of gravity has a dual description it would be interesting to understand the situationpresented above in the dual scalar-tensor theory as well. This will provide a similar setup to the originalGoldberger-Wise scheme but with back-reaction included. In this case the gravitational action is given byEq. (7), with the following solution, ds = e − A ( y ) η µν dx µ dx ν + r c dy ; A ( y ) = A + kr c y + κ ψ e − A r c y ; ψ ( y ) = ψ e − A r c y (27)Note that the correspondence between Jordan and Einstein frame relates the parameter A appearing herewith that earlier. The Ricci scalar for the above metric can be evaluated leading to, R = 1 r c (cid:0) A ′′ − A ′ (cid:1) = − a + (cid:18) b κ + 409 a b (cid:19) ψ − b ψ (28)where, a is a constant to be identified with 3 k , while, b = κ A /
4. Further one can obtain, g µν ∂ µ ψ∂ ν ψ = 1 r c (cid:18) ∂ψ∂y (cid:19) = A ψ e − A r c y (29)Such that the action becomes, A = Z d x Z dyr c e − A " κ (cid:26) − a + (cid:18) κ A κ a A (cid:19) ψ e − A r c y − κ A ψ e − A r c y (cid:27) − A ψ e − A r c y − (cid:26)(cid:18) − Λ − a κ (cid:19) + (cid:18) A
32 + a A (cid:19) ψ − (cid:18) κ A (cid:19) ψ (cid:27) − Λ (30)Hence the potential becomes (except for terms independent of r c ) by introducing R = Φ e − kr c π , V ( R ) = R Φ " (cid:18) − a κ (cid:19) k + (cid:18) A + 2 a A (cid:19) ψ ( R/ Φ) A /k k + A + (cid:18) − κ A (cid:19) ψ ( R/ Φ) A /k k + 2 A (31)In this case as well the minima of the potential originates from the following algebraic equation, (cid:18) − a κ (cid:19) + (cid:18) A + 2 a A (cid:19) ψ ( R/ Φ) A /k + (cid:18) − κ A (cid:19) ψ ( R/ Φ) A /k = 0 (32)Leading to a solution for the stabilized potential, which only differs within the logarithm and hence haveidentical leading order behavior. The mass as well as the coupling with standard model fields work out inan identical fashion. Thus the two frames lead to very much similar expressions for the stabilized value ofthe radion field as well as its mass and couplings with standard model fields. Hence at least in thiscontext the two frames have nearly identical physical behavior.13 eferences [1] L. Randall and R. Sundrum, “A Large mass hierarchy from a small extra dimension,” Phys.Rev.Lett. (1999) 3370–3373, arXiv:hep-ph/9905221 [hep-ph] .[2] H. L. Verlinde, “Holography and compactification,” Nucl.Phys.
B580 (2000) 264–274, arXiv:hep-th/9906182 [hep-th] .[3] A. Djouadi, “The Anatomy of electro-weak symmetry breaking. I: The Higgs boson in the standardmodel,”
Phys.Rept. (2008) 1–216, arXiv:hep-ph/0503172 [hep-ph] .[4] J. Garriga and T. Tanaka, “Gravity in the brane world,”
Phys.Rev.Lett. (2000) 2778–2781, arXiv:hep-th/9911055 [hep-th] .[5] N. Dadhich, R. Maartens, P. Papadopoulos, and V. Rezania, “Black holes on the brane,” Phys.Lett.
B487 (2000) 1–6, arXiv:hep-th/0003061 [hep-th] .[6] C. Csaki, M. Graesser, C. F. Kolda, and J. Terning, “Cosmology of one extra dimension withlocalized gravity,”
Phys.Lett.
B462 (1999) 34–40, arXiv:hep-ph/9906513 [hep-ph] .[7] Z. Chacko and A. E. Nelson, “A Solution to the hierarchy problem with an infinitely large extradimension and moduli stabilization,”
Phys.Rev.
D62 (2000) 085006, arXiv:hep-th/9912186 [hep-th] .[8] A. G. Cohen and D. B. Kaplan, “Solving the hierarchy problem with noncompact extradimensions,”
Phys.Lett.
B470 (1999) 52–58, arXiv:hep-th/9910132 [hep-th] .[9] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, “New dimensions at a millimeter toa Fermi and superstrings at a TeV,”
Phys.Lett.
B436 (1998) 257–263, arXiv:hep-ph/9804398 [hep-ph] .[10] V. Rubakov and M. Shaposhnikov, “Do We Live Inside a Domain Wall?,”
Phys.Lett.
B125 (1983) 136–138.[11] D. J. H. Chung and K. Freese, “Cosmological challenges in theories with extra dimensions andremarks on the horizon problem,”
Phys. Rev.
D61 (2000) 023511, arXiv:hep-ph/9906542 [hep-ph] .[12] J. M. Cline, C. Grojean, and G. Servant, “Cosmological expansion in the presence of extradimensions,”
Phys. Rev. Lett. (1999) 4245, arXiv:hep-ph/9906523 [hep-ph] .[13] T. Nihei, “Inflation in the five-dimensional universe with an orbifold extra dimension,” Phys. Lett.
B465 (1999) 81–85, arXiv:hep-ph/9905487 [hep-ph] .[14] W. D. Goldberger and M. B. Wise, “Modulus stabilization with bulk fields,”
Phys.Rev.Lett. (1999) 4922–4925, arXiv:hep-ph/9907447 [hep-ph] .[15] W. D. Goldberger and M. B. Wise, “Phenomenology of a stabilized modulus,” Phys. Lett.
B475 (2000) 275–279, arXiv:hep-ph/9911457 [hep-ph] .[16] A. V. Lipatov and M. A. Malyshev, “On possible small-x effects in the Kaluza-Klein graviton andradion production at high energies,” arXiv:1612.03811 [hep-ph] .1417] A. Das, T. Paul, and S. SenGupta, “Modulus stabilisation in a backreacted warped geometry modelvia Goldberger-Wise mechanism,” arXiv:1609.07787 [hep-ph] .[18] M. T. Arun and D. Choudhury, “Stabilization of moduli in spacetime with nested warping,” arXiv:1606.00642 [hep-th] .[19] D. Bazeia, M. A. Marques, R. Menezes, and D. C. Moreira, “New braneworld models in the presenceof auxiliary fields,”
Annals Phys. (2015) 574–584, arXiv:1412.0135 [hep-th] .[20] P. Cox, A. D. Medina, T. S. Ray, and A. Spray, “Radion/Dilaton-Higgs Mixing Phenomenology inLight of the LHC,”
JHEP (2014) 032, arXiv:1311.3663 [hep-ph] .[21] S. Anand, D. Choudhury, A. A. Sen, and S. SenGupta, “A Geometric Approach to ModulusStabilization,” Phys. Rev.
D92 no. 2, (2015) 026008, arXiv:1411.5120 [hep-th] .[22] S. Chakraborty and S. SenGupta, “Kinematics of Radion field: A possible source of dark matter,”
Eur. Phys. J.
C76 no. 12, (2016) 648, arXiv:1511.00646 [gr-qc] .[23] S. Chakraborty and S. Sengupta, “Radion cosmology and stabilization,”
Eur.Phys.J.
C74 no. 9, (2014) 3045, arXiv:1306.0805 [gr-qc] .[24] S. Chakraborty and N. Dadhich, “Brown-York quasilocal energy in Lanczos-Lovelock gravity andblack hole horizons,”
JHEP (2015) 003, arXiv:1509.02156 [gr-qc] .[25] S. Chakraborty, “Lanczos-Lovelock gravity from a thermodynamic perspective,” JHEP (2015) 029, arXiv:1505.07272 [gr-qc] .[26] S. Chakraborty and T. Padmanabhan, “Geometrical variables with direct thermodynamicsignificance in Lanczos-Lovelock gravity,” Phys.Rev.
D90 no. 8, (2014) 084021, arXiv:1408.4791 [gr-qc] .[27] S. Chakraborty and T. Padmanabhan, “Evolution of Spacetime arises due to the departure fromHolographic Equipartition in all Lanczos-Lovelock Theories of Gravity,”
Phys.Rev.
D90 no. 12, (2014) 124017, arXiv:1408.4679 [gr-qc] .[28] N. Dadhich, “Characterization of the Lovelock gravity by Bianchi derivative,”
Pramana (2010) 875–882, arXiv:0802.3034 [gr-qc] .[29] T. Padmanabhan and D. Kothawala, “Lanczos-Lovelock models of gravity,” Phys.Rept. (2013) 115–171, arXiv:1302.2151 [gr-qc] .[30] L. Pogosian and A. Silvestri, “The pattern of growth in viable f(R) cosmologies,”
Phys. Rev.
D77 (2008) 023503, arXiv:0709.0296 [astro-ph] . [Erratum: Phys.Rev.D81,049901(2010)].[31] S. Capozziello and A. Troisi, “PPN-limit of fourth order gravity inspired by scalar-tensor gravity,”
Phys. Rev.
D72 (2005) 044022, arXiv:astro-ph/0507545 [astro-ph] .[32] S. Capozziello, A. Stabile, and A. Troisi, “The Newtonian Limit of f(R) gravity,”
Phys. Rev.
D76 (2007) 104019, arXiv:0708.0723 [gr-qc] .1533] T. P. Sotiriou, “The Nearly Newtonian regime in non-linear theories of gravity,”
Gen. Rel. Grav. (2006) 1407–1417, arXiv:gr-qc/0507027 [gr-qc] .[34] S. Capozziello, A. Stabile, and A. Troisi, “Fourth-order gravity and experimental constraints onEddington parameters,” Mod. Phys. Lett.
A21 (2006) 2291–2301, arXiv:gr-qc/0603071 [gr-qc] .[35] S. Nojiri, S. D. Odintsov, and S. Ogushi, “Cosmological and black hole brane world universes inhigher derivative gravity,”
Phys. Rev.
D65 (2002) 023521, arXiv:hep-th/0108172 [hep-th] .[36] S. Nojiri and S. D. Odintsov, “Unified cosmic history in modified gravity: from F(R) theory toLorentz non-invariant models,”
Phys.Rept. (2011) 59–144, arXiv:1011.0544 [gr-qc] .[37] T. P. Sotiriou and V. Faraoni, “f(R) Theories Of Gravity,”
Rev.Mod.Phys. (2010) 451–497, arXiv:0805.1726 [gr-qc] .[38] A. De Felice and S. Tsujikawa, “f(R) theories,” Living Rev.Rel. (2010) 3, arXiv:1002.4928 [gr-qc] .[39] S. Chakraborty and S. SenGupta, “Spherically symmetric brane spacetime with bulk f ( R ) gravity,” Eur.Phys.J.
C75 no. 1, (2015) 11, arXiv:1409.4115 [gr-qc] .[40] S. Chakraborty and S. SenGupta, “Effective gravitational field equations on m -brane embedded inn-dimensional bulk of Einstein and f ( R ) gravity,” Eur. Phys. J.
C75 no. 11, (2015) 538, arXiv:1504.07519 [gr-qc] .[41] S. Chakraborty and S. SenGupta, “Spherically symmetric brane in a bulk of f(R) and Gauss-BonnetGravity,”
Class. Quant. Grav. no. 22, (2016) 225001, arXiv:1510.01953 [gr-qc] .[42] S. Nojiri and S. D. Odintsov, “Modified gravity with negative and positive powers of the curvature:Unification of the inflation and of the cosmic acceleration,” Phys. Rev.
D68 (2003) 123512, arXiv:hep-th/0307288 [hep-th] .[43] S. Nojiri and S. D. Odintsov, “Unifying inflation with LambdaCDM epoch in modified f(R) gravityconsistent with Solar System tests,”
Phys. Lett.
B657 (2007) 238–245, arXiv:0707.1941 [hep-th] .[44] W. Hu and I. Sawicki, “Models of f(R) Cosmic Acceleration that Evade Solar-System Tests,”
Phys. Rev.
D76 (2007) 064004, arXiv:0705.1158 [astro-ph] .[45] S. Chakraborty and S. SenGupta, “Solving higher curvature gravity theories,”
Eur. Phys. J.
C76 no. 10, (2016) 552, arXiv:1604.05301 [gr-qc] .[46] J. D. Barrow and S. Cotsakis, “Inflation and the Conformal Structure of Higher Order GravityTheories,”
Phys. Lett.
B214 (1988) 515–518.[47] S. Capozziello, R. de Ritis, and A. A. Marino, “Some aspects of the cosmological conformalequivalence between ’Jordan frame’ and ’Einstein frame’,”
Class. Quant. Grav. (1997) 3243–3258, arXiv:gr-qc/9612053 [gr-qc] .[48] S. Bahamonde, S. D. Odintsov, V. K. Oikonomou, and M. Wright, “Correspondence of F ( R )Gravity Singularities in Jordan and Einstein Frames,” arXiv:1603.05113 [gr-qc] .1649] M. Parry, S. Pichler, and D. Deeg, “Higher-derivative gravity in brane world models,” JCAP (2005) 014, arXiv:hep-ph/0502048 [hep-ph] .[50] R. Catena, M. Pietroni, and L. Scarabello, “Einstein and Jordan reconciled: a frame-invariantapproach to scalar-tensor cosmology,”
Phys. Rev.
D76 (2007) 084039, arXiv:astro-ph/0604492 [astro-ph] .[51] T. Chiba and M. Yamaguchi, “Conformal-Frame (In)dependence of Cosmological Observations inScalar-Tensor Theory,”
JCAP (2013) 040, arXiv:1308.1142 [gr-qc] .[52] S. Bhattacharya, “Rotating Killing horizons in generic F ( R ) gravity theories,” arXiv:1602.04306 [gr-qc] .[53] F. Briscese, E. Elizalde, S. Nojiri, and S. D. Odintsov, “Phantom scalar dark energy as modifiedgravity: Understanding the origin of the Big Rip singularity,” Phys. Lett.
B646 (2007) 105–111, arXiv:hep-th/0612220 [hep-th] .[54] S. Capozziello, P. Martin-Moruno, and C. Rubano, “Physical non-equivalence of the Jordan andEinstein frames,”
Phys. Lett.
B689 (2010) 117–121, arXiv:1003.5394 [gr-qc] .[55] S. Nojiri, S. D. Odintsov, and D. Saez-Gomez, “Cosmological reconstruction of realistic modifiedF(R) gravities,”
Phys. Lett.
B681 (2009) 74–80, arXiv:0908.1269 [hep-th] .[56] S. Nojiri and S. D. Odintsov, “Modified gravity and its reconstruction from the universe expansionhistory,”
J. Phys. Conf. Ser. (2007) 012005, arXiv:hep-th/0611071 [hep-th] .[57] S. Nojiri, S. D. Odintsov, A. Toporensky, and P. Tretyakov, “Reconstruction anddeceleration-acceleration transitions in modified gravity,” Gen. Rel. Grav. (2010) 1997–2008, arXiv:0912.2488 [hep-th] .[58] S. Carloni, R. Goswami, and P. K. S. Dunsby, “A new approach to reconstruction methods in f ( R )gravity,” Class. Quant. Grav. (2012) 135012, arXiv:1005.1840 [gr-qc] .[59] K. Bamba, S. Nojiri, and S. D. Odintsov, “The Universe future in modified gravity theories:Approaching the finite-time future singularity,” JCAP (2008) 045, arXiv:0807.2575 [hep-th] .[60] I. Antoniadis, J. Rizos, and K. Tamvakis, “Singularity - free cosmological solutions of thesuperstring effective action,”
Nucl. Phys.
B415 (1994) 497–514, arXiv:hep-th/9305025 [hep-th] .[61] I. H. Brevik, K. A. Milton, S. Nojiri, and S. D. Odintsov, “Quantum (in)stability of a brane worldAdS(5) universe at nonzero temperature,”
Nucl. Phys.
B599 (2001) 305–318, arXiv:hep-th/0010205 [hep-th] .[62] I. H. Brevik, “Can the hierarchy problem be solved by finite temperature massive fermions in theRandall-Sundrum model?,”
Grav. Cosmol. (2002) 31–32, arXiv:hep-th/0106225 [hep-th] .[63] O. DeWolfe, D. Z. Freedman, S. S. Gubser, and A. Karch, “Modeling the fifth-dimension withscalars and gravity,” Phys. Rev.
D62 (2000) 046008, arXiv:hep-th/9909134 [hep-th] .1764] E. A. Mirabelli, M. Perelstein, and M. E. Peskin, “Collider signatures of new large spacedimensions,”
Phys. Rev. Lett. (1999) 2236–2239, arXiv:hep-ph/9811337 [hep-ph] .[65] W. D. Goldberger and M. B. Wise, “Bulk fields in the Randall-Sundrum compactification scenario,” Phys. Rev.
D60 (1999) 107505, arXiv:hep-ph/9907218 [hep-ph] .[66] E. Dudas and M. Quiros, “Five-dimensional massive vector fields and radion stabilization,”
Nucl. Phys.
B721 (2005) 309–324, arXiv:hep-th/0503157 [hep-th] .[67] J. L. Hewett, “Indirect collider signals for extra dimensions,”
Phys. Rev. Lett. (1999) 4765–4768, arXiv:hep-ph/9811356 [hep-ph] .[68] A. Das, R. Hundi, and S. SenGupta, “Bulk Higgs and Gauge fields in a multiply warped braneworldmodel,” Phys.Rev.
D83 (2011) 116003, arXiv:1105.1064 [hep-ph] .[69] H. Davoudiasl, J. Hewett, and T. Rizzo, “Bulk gauge fields in the Randall-Sundrum model,”
Phys.Lett.
B473 (2000) 43–49, arXiv:hep-ph/9911262 [hep-ph] .[70] H. Davoudiasl, J. Hewett, and T. Rizzo, “Experimental probes of localized gravity: On and off thewall,”
Phys.Rev.
D63 (2001) 075004, arXiv:hep-ph/0006041 [hep-ph] .[71] S. Chakraborty and S. SenGupta, “Bulk scalar field in warped extra dimensional models,”
Phys.Rev.
D89 no. 12, (2014) 126001, arXiv:1401.3279 [gr-qc] .[72] S. Chakraborty and S. SenGupta, “Higher curvature gravity at the LHC,”
Phys.Rev.
D90 no. 4, (2014) 047901, arXiv:1403.3164 [gr-qc] .[73] J. L. Hewett and T. G. Rizzo, “750 GeV Diphoton Resonance in Warped Geometries,” arXiv:1603.08250 [hep-ph] .[74] S. B. Giddings and H. Zhang, “Kaluza-Klein graviton phenomenology for warped compactifications,and the 750 GeV diphoton excess,”
Phys. Rev.
D93 no. 11, (2016) 115002, arXiv:1602.02793 [hep-ph] .[75] A. Oliveira, “Gravity particles from Warped Extra Dimensions, predictions for LHC,” arXiv:1404.0102 [hep-ph] .[76] G.-C. Cho, D. Nomura, and Y. Ohno, “Constraints on radion in a warped extra dimension modelfrom Higgs boson searches at the LHC,”
Mod. Phys. Lett.
A28 (2013) 1350148, arXiv:1305.4431 [hep-ph] .[77]
CMS
Collaboration, “Updated measurements of the Higgs boson at 125 GeV in the two photondecay channel,”.[78] H. Davoudiasl, T. McElmurry, and A. Soni, “The Radion as a Harbinger of Deca-TeV Physics,”
Phys. Rev.
D86 (2012) 075026, arXiv:1206.4062 [hep-ph] .[79] H. Kubota and M. Nojiri, “Radion-higgs mixing state at the LHCwith the KK contributions to theproduction and decay,”
Phys. Rev.
D87 (2013) 076011, arXiv:1207.0621 [hep-ph] .[80] M. Frank, B. Korutlu, and M. Toharia, “Radion Phenomenology with 3 and 4 Generations,”
Phys. Rev.
D84 (2011) 115020, arXiv:1110.4434 [hep-ph] .1881] U. Mahanta and A. Datta, “Production of light stabilized radion at high-energy hadron collider,”
Phys. Lett.
B483 (2000) 196–202, arXiv:hep-ph/0002183 [hep-ph] .[82] C. Csaki, M. L. Graesser, and G. D. Kribs, “Radion dynamics and electroweak physics,”
Phys. Rev.
D63 (2001) 065002, arXiv:hep-th/0008151 [hep-th] .[83] C. Csaki, M. Graesser, L. Randall, and J. Terning, “Cosmology of brane models with radionstabilization,”
Phys.Rev.
D62 (2000) 045015, arXiv:hep-ph/9911406 [hep-ph]arXiv:hep-ph/9911406 [hep-ph]