Radion stabilization in higher curvature warped spacetime
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Abstract
We consider a five dimensional AdS spacetime in presence of higher curvature term like \(F(R) = R + \alpha R^2\) in the bulk. In this model, we examine the possibility of modulus stabilization from the scalar degrees of freedom of higher curvature gravity free of ghosts. Our result reveals that the model stabilizes itself and the mechanism of modulus stabilization can be argued from a geometric point of view. We determine the region of the parametric space for which the modulus (or radion) can to be stabilized. We also show how the mass and coupling parameters of radion field are modified due to higher curvature term leading to modifications of its phenomenological implications on the visible 3brane.
1 Introduction
Till date, Standard Model (SM) of particle Physics is a widely accepted theory to describe the interactions of fundamental particles. Despite its enormous successes, the model is plagued with divergence of the Higgs mass due to radiative corrections which may run up to Planck scale. An unnatural fine tuning is needed to confine the Higgs mass within TeV scale.
Many attempts have been made to address this problem by considering the theories beyond SM of particle Physics. Few such candidates are – supersymmetry, technicolor and extra dimensions. Among many such attempts [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], RandallSundrum (RS) model [6] of warped extra dimension draws special attention since it resolves the gauge hierarchy problem without choosing any intermediate scale in the theory.
In RS model, two 3branes are embedded in a five dimensional anti deSitter spacetime compactified on a \(M_4 \times S_1/Z_2\) orbifold. The distance between these two branes is assumed to be \(\sim \) of Planck length so that the required hierarchy between the two branes is generated. One of the crucial aspects of this braneworld scenario is to stabilize the distance between the branes (known as modulus or radion). For this purpose, it is necessary to generate an appropriate radion potential with a stable minimum consistent with the value proposed in RS model in order to solve the hierarchy problem. Goldberger and Wise (GW) proposed a mechanism [16] to create such a radion potential by introducing a bulk scalar field with appropriate boundary values. Subsequently the phenomenology of radion field has also been studied extensively. This radion phenomenology [17, 18, 19, 20] along with the study of RS graviton [21, 22, 23, 24, 25] and RS black holes [26, 27, 28] are considered to be the testing ground of warped extra dimensional models in collider experiments [29, 30]. As the present experimental lower bound of the first graviton KKmode mass climbs above 3 TeV, the RStype resolution of the naturalness problem is undoubtedly under pressure. The question that how the Higgs is so much lighter than the 5dimensional Planck scale needs to be settled properly. However, in any higher dimensional model with gravity in the bulk, the modulus must be stabilized to appropriate value to extract a meaningful low energy effective theory on the brane. The present work aims to address this issue specially in the context of a higher dimensional model where the fundamental curvature scale is of the order of Planck scale.
It is well known that Einstein–Hilbert action can be generalized by adding higher order curvature terms which naturally arise from diffeomorphism property of the action. Such terms also have their origin in String theory due to quantum corrections. F(R) [31, 32, 33], Gauss–Bonnet (GB) [34, 35, 36] or more generally LanczosLovelock gravity are some of the candidates in higher curvature gravitational theory.
Higher curvature terms become extremely relevant at the regime of large curvature. Thus for RS bulk geometry where the curvature is of the order of Planck scale, the higher curvature terms should play a crucial role. In general inclusion of higher curvature terms in the action leads to the appearance of ghost from higher derivative terms resulting into Ostragradsky instability. The Gauss–Bonnet model (a special case of Lanczos–Lovelock model) is however free of this instability due to appropriate choice of various quadratic combinations of Riemann tensor, Ricci tensor and curvature scalar . Some important modified solutions of the Randall–Sundrum model in presence of Gauss–Bonnet terms have been obtained by Kim et al. [37, 38] in the context of both static and inflationary scenario. A GB modified warped solution and it’s phenomenological implications was also discussed in [39].

Is the RS braneworld modified by F(R) gravity, stabilized even without introducing an external stabilizing field?

If the modulus can be stabilized in the dual scalar–tensor model, does it mean that it is also stabilized in the original F(R) model?

Does the scalar kinetic and potential terms for the purpose of modulus stabilization correspond to a F(R) model which is free of ghostlike instability?

If the braneworld scenario is stabilized consistently then how the radion mass and coupling parameters will change from that of RS scenario due to the presence of higher order curvature terms?
The paper is organized as follows: Following two sections are devoted to brief reviews of RS scenario and conformal relationship between F(R) and scalar–tensor (ST) theory. In Sect. 4, we extend our analysis of Sect. 3 for the specific F(R) model considered in this work. Section 5 extensively describes the modulus stabilization, radion mass and coupling for the dual ST model while Sect. 6 addresses these for the original F(R) model. After discussing the equivalence, the paper ends with some conclusive remarks in Sect. 7.
2 Brief description of RS scenario and its stabilization via GW mechanism
3 Transformation of a F(R) theory to scalar–tensor theory
4 RS like spacetime in F(R) model and corresponding scalar–tensor theory
5 Modulus stabilization, radion mass and coupling in scalar–tensor (ST) theory
5.1 Modulus stabilization
At this stage, we mention that the value of the stabilized modulus should be \(kr_c \simeq 12\), in order to solve the gauge hierarchy problem. Equation (22) clearly indicates that such magnitude of \(kr_c\) can be achieved without any fine tuning of the parameters [16]. For example, \(v_h/v_v = 1.5\) and \(m_{\Phi }/k = 0.2\) yields \(kr_c \simeq 12\) [16].
Furthermore to derive the stabilization condition in scalar–tensor theory, the backreaction of the scalar field on spacetime geometry is neglected. It can be shown from [16], that this is valid as long as the stress energy tensor for the scalar field is less than the bulk cosmological constant which in turn implies that \(v_h^2/M^3\) and \(v_v^2/M^3\) are less than unity, where \(v_h\) and \(v_v\) are the boundary values of the scalar field. Now using Eqs. (24) and (25), we can determine the conditions of negligible backreactions in terms of the parameters appearing in the original F(R) theory.
The effect of backreaction, though small, shall also incorporated in Sect. 6.3. We will show that the backreaction modifies all the quantities described via Eqs. (18, 19, 20, 22), though the modification is small in the limit \(\kappa v_h< 1\).
5.2 Radion potential
5.3 Coupling between radion and Standard Model fields
Being a gravitational degree of freedom, radion field interacts with brane energy–momentum tensor and the couplings of interaction are constrained by four dimensional general covariance. From the five dimensional metric ansatz (see Eq. (26)), it is clear that the induced metric on visible brane is \((\frac{{\tilde{\Psi }}}{f})^2\eta _{\mu \nu }\) (where \(f=\sqrt{24M^3/k}\)) and consequently \({\tilde{\Psi }}(x)\) couples directly with SM fields.
Now we turn our focus on modulus stabilization as well as on radion mass and coupling for the original F(R) model (Eq. (8)) by using the stabilization condition of the corresponding scalar–tensor theory.
6 Modulus stabilization, radion mass and coupling in F(R) model
6.1 Modulus stabilization
6.2 Radion potential
6.3 Radion potential with backreaction
6.4 Coupling between radion and Standard Model fields
s The radion field arises as a scalar degree of freedom on the TeV brane and has interactions with the Standard Model (SM) fields. From the five dimensional metric (Eq. (41)), it is clear that the induced metric on visible brane is \((\frac{\Psi }{f})^2e^{\kappa v_v\frac{1}{\sqrt{3}}}\eta _{\mu \nu }\) (where \(f = \sqrt{\frac{24M^3}{k}} [1  \frac{20}{\sqrt{3}}\alpha k^2\kappa v_h]\)) and consequently \(\Psi (x)\) couples directly with SM fields.
The form of \(v_h\) and \(v_v\) in terms of five dimensional Ricci scalar can be extracted from Eqs. (24) and (25). It is evident that the coupling \(\lambda _L\), \(\lambda _R\) is modified by the factor \((k\pm 2m)d\), in comparison to the coupling given in Eq. (64) for the fermion fields confined on the TeV brane.
From above analysis we note that the coupling between radion and SM fields is suppressed due to the presence of higher curvature parameter \(\alpha \) which in turn modifies the phenomenology on visible 3brane.
Before concluding, we mention about a recent work [46] , where a higher curvature gravity model with \(R^4\) terms in the action is considered. The corresponding conformally transformed scalar action includes a quartic term which resembles closely to the scalar action considered in reference [11, 13]. It has been shown that with such a specific choice of the scalar action one can estimate the exact modification of the warp factor due to the effects of the backreaction of the scalar field on the background geometry and thus it enables us to address the role of backreaction on the stabilization issue and also on various parameters of the lowenergy effective action. However in such a model, the backreacted geometry can be exactly determined if the scalar mass and the quartic term in the potential are interrelated and in the limit of the quartic term going to zero, the mass term also goes to zero. Therefore there is no smooth limit which takes this model to that considered by the GW where only quadratic mass term was present. The work reported in this article however has a different goal from that of [46]. Here we show that in the leading order quadratic curvature correction to Einstein action in the bulk we find a dual scalar tensor theory which under certain approximation is similar to the original GW scalar action which has a quadratic mass term only. We therefore explore and reexamine the originally proposed GoldbergerWise modulus stabilization condition in the light of higher curvature gravity models where such a stabilizing field appears naturally from higher curvature degrees of freedom with a minimal curvature extension.
7 Conclusion

The model comes as a self stabilizing system due to the presence of higher curvature term \(\alpha R^2\). This is in sharp contrast to a model with only Einstein term in the bulk where the modulus can not be stabilized without incorporating any external degrees of freedom such as a scalar field. However for the higher curvature gravity model, this additional degree of freedom originates naturally from the higher curvature term and plays the role of a stabilizing field. It may also be noted that for \(\alpha \rightarrow 0\), the stabilization condition (Eq. (39)) leads to zero brane separation.

We scan the parametric space of \(\alpha \) for which the modulus is going to be stabilized. Our result reveals that for \(\alpha > 0\), the interbrane separation becomes negative which is an unphysical situation. Thus the braneworld we have considered is stabilized only when \(\alpha < 0\). This puts constraints on the F(R) model itself. Moreover the distance between the branes increases with the value of the parameter \(\alpha \) which is evident from Fig. 1. Thus the results obtained in this work clearly bring out the correlation between a geometrically stable warped solution resulting from negative bulk curvature and the stability of the higher curvature F(R) model free from ghosts.

Quadratic term in curvature also generates the radion potential with a stable minimum. We find the radion mass as well as radion coupling with SM fields. The expressions of mass (Eq. (47)) and coupling (Eq. (62) and Eq. (63)) clearly indicate that the radion mass is enhanced while the coupling is suppressed in comparison to the scenario where only Einstein gravity resides in the bulk [17]. Thus the cross section between radion and SM fields is overall suppressed due to the presence of higher order curvature terms in five dimensional gravitational action leading to a possible explanation of the invisibility of the radion field in the present experimental resolution.

The effect of backreaction on the radion potential and its minima are studied. It is shown that the corrections due to the backreaction is indeed small in the limit \(\kappa v_h< 1\). The possible correction terms for the backreaction are determined.
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