Radion tunneling in modified theories of gravity
Abstract
We consider a five dimensional warped spacetime where the bulk geometry is governed by higher curvature F(R) gravity. In this model, we determine the modulus potential originating from the scalar degree of freedom of higher curvature gravity. In the presence of this potential, we investigate the possibility of modulus (radion) tunneling leading to an instability in the brane configuration. Our results reveal that the parametric regions where the tunneling probability is highly suppressed, corresponds to the parametric values required to resolve the gauge hierarchy problem.
1 Introduction
Over the last two decades models with extra spatial dimensions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] have been increasingly playing a central role in search for physics beyond standard model of elementary particle [14, 15] and Cosmology [16, 17]. Such higher dimensional scenarios occur naturally in string theory and also are viable candidates to resolve the well known gauge hierarchy problem. Depending on different possible compactification schemes for the extra dimensions, a large number of models have been constructed. In all these models, our visible universe is identified as a 3brane embedded in a higher dimensional spacetime and is described through a low energy effective theory on the brane carrying the signatures of extra dimensions [18, 19, 20].
Among various extra dimensional models proposed over last several years, warped extra dimensional model pioneered by Randall and Sundrum (RS) [6] earned a special attention since it resolves the gauge hierarchy problem without introducing any intermediate scale (between Planck and TeV) in the theory. Subsequently different variants of warped geometry model were extensively studied in [17, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. A generic feature of many of these models is that the bulk spacetime is endowed with high curvature scale \(\sim \) 4 dimensional 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 from quantum corrections. In this context F(R) [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43], GaussBonnet (GB) [44, 45, 46] or more generally Lanczos–Lovelock gravity are some of the candidates in higher curvature gravitational theory.
In general the higher curvature terms are suppressed with respect to Einstein–Hilbert term by Planck scale. Hence in low curvature regime, their contributions are negligible. However higher curvature terms become extremely relevant in a region with large curvature. Thus for bulk geometry where the curvature is of the order of Planck scale, the higher curvature terms should play a crucial role. Motivated by this idea, in the present work, we consider a generalized warped geometry model by replacing Einstein–Hilbert bulk gravity action with a higher curvature F(R) gravitational theory [41, 42, 47, 48, 49, 50, 51, 52, 53].
One of the crucial aspects of higher dimensional two brane models is to stabilize the interbrane separation (also known as modulus or radion). For this purpose, one needs to generate a suitable radion potential with a stable minimum [21, 22, 23]. The presence of such minimum guarantees the stability of the modulus field. In Goldberger–Wise stabilization mechanism [21, 22], an external bulk scalar field was invoked to create such a stable radion potential. However, when the bulk is endowed with higher curvature F(R) gravity, then apart from the metric there is an additional scalar degree of freedom originating from higher derivative terms of the metric. It has been shown that such a scalar degree of freedom can play the role of the stabilizing field for appropriate choices of the underlying F(R) model [26, 27].
It is important to analyze the exact nature of the resulting radion potential to explore whether there exists a metastable minimum for the radion from which it can tunnel and leads to an instability of the braneworld [54, 55, 56, 57, 58, 59]. In this paper, we aim to determine the radion tunneling in the presence of higher curvature gravity in the bulk.
Our paper is organized as follows: following section is devoted to brief review of the conformal relationship between F(R) and scalar–tensor (ST) theory. In Sect. 3, we extend our analysis of Sect. 2 for the specific F(R) model considered in this work. Section 4 extensively describes the tunneling probability for the dual ST model while Sect. 5 addresses these for the original F(R) model. After discussing the equivalence, the paper ends with some concluding remarks in Sect. 6.
2 Transformation of a F(R) theory to scalar–tensor theory
3 Warped spacetime in F(R) model and corresponding scalar–tensor theory
4 Radion potential and tunneling probability in scalar–tensor (ST) theory
Figure 2 clearly depicts that the tunneling probability increases with the parameter b and asymptotically reaches to unity at large b. It is expected, because with increasing value of b, the height as well as the width \(\big (\) both are \(\varpropto \frac{1}{b^2} \big )\) of the potential barrier decreases and as a consequence, \(P_{ST}\) increases. Moreover, \(P_{ST}\) becomes zero as b tends to zero, because for \(b \rightarrow 0\), the height of the potential barrier goes to infinite and as a result, \(P_{ST} = 0\). This character of global minimum (as b tends to zero) actually overlaps with the Goldberger–Wise result [21, 22].
However, resolution of the gauge hierarchy problem requires \(k\pi <r_c>_+ = 36\), which in turn makes \(b \simeq \frac{1}{3\sqrt{3}}\) (for \(a=1\)). With these values of a and b, \(P_{ST}\) becomes drastically suppressed and comes as \(\sim 10^{32}\). This small value of tunneling probability guarantees the stability of the interbrane separation (and hence of the radion field) at \(<r_c>_+\). Thus it can be argued that the smallness of tunneling probability is intimately connected with the requirement of resolving the gauge hierarchy problem. Further it may be mentioned that these values of a and b are consistent with the condition \(e^{b\kappa v_h}<\frac{\kappa k\sqrt{\Lambda }}{ab}\), necessary for neglecting the backreaction of the scalar field \(\Psi \) in the background spacetime (as mentioned earlier).
Now we turn our focus on radion potential as well as on radion tunneling probability for the original F(R) model (Eq. (5)).
5 Radion potential and tunneling probability in F(R) model
Figure 3 clearly depicts that the radion potential goes to zero at \(T(x)=0\) and reaches a constant value asymptotically at large value of T(x). Comparing Figs. 1 and 3, it is clear that the nature of radion potential does not change in comparison to that in ST theory. However, due to the conformal factor, the extremas of the potential are shifted in F(R) model, which is clear from Eq. (33).
Figure 4 reveals that just as in ST theory, \(P_{F(R)}\) (tunneling probability in F(R) model) increases with increasing value of b and acquires the maximum value (\(= 1\)) asymptotically at large b. For \(b\rightarrow \infty \), the higher curvature parameter (\(\alpha \)) goes to zero (see Eq. (9)) and the action reduces to Einstein–Hilbert action. This in turn makes the brane configuration unstable [22] and as a consequence the tunneling probability becomes unity. As the parameter b decreases, the effect of higher curvature term starts to contribute and as a result, the modulus is stabilized at a certain separation and hence the probability for tunneling becomes less than one. Furthermore for \(b\rightarrow 0\), higher curvature parameter \(\alpha \rightarrow \infty \), which in turn makes the height of the radion potential barrier infinity (height \(\varpropto \frac{1}{b^2}\)) and thus the potential acquires a global minimum. As a consequence, the tunneling probability tends to zero, which is shown in Fig. 4). The character of global minimum actually mimics the result of Goldberger and Wise [21]. It is expected because for \(b\rightarrow 0\), the bulk scalar potential in the present context (\(U(\Psi )\)) becomes quadratic (all the other terms are proportional to higher power of b and can be neglected) as same as the potential considered in [21].
Finally we examine whether the solution of gauge hierarchy problem in F(R) model leads to a small value of the tunneling probability or not. We find that the resolution of gauge hierarchy problem requires \(k\pi <d>_+ = 36\), which in turn makes \(b = \frac{\sqrt{2}}{3}\). For this value of \(b, P_{F(R)}\) is highly suppressed and takes the value of \(\sim 10^{32}\). Therefore, in original F(R) theory, the requirement for solving the gauge hierarchy problem is correlated with the smallness of radion tunneling probability (a similar analysis is also obtained in ST theory as discussed in Sect. 4).
6 Conclusion

Due to the presence of higher curvature gravity in the bulk, a potential term for the radion field is generated, as shown in Eq. (39). This is in sharp contrast to a model with only Einstein term in the bulk where the modulus potential can not be generated 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. It may also be noted that the radion potential goes to zero as the higher curvature parameter \(\alpha \rightarrow 0\).

The radion potential (\(V_{F(R)}\)) has a minimum (\(<d>_+\)) and a maximum (\(<d>_\)) respectively where the height between minimum and maximum of the potential depends on both the parameters \(\alpha \) and n. Moreover, \(V_{F(R)}\) becomes zero at \(T(x) = 0\) (T(x) is the radion field) and reaches a constant value asymptotically at large T(x), as depicted in Fig. 3.

According to GW mechanism, the modulus is stabilized at \(<d>_+\). But due to quantum mechanical effect, there exists a possibility of tunneling for the radion field from \(d = <d>_+\) to \(d = 0\), which in turn makes the aforementioned brane configuration unstable. We calculate this tunneling probability (\(P_{F(R)}\)) which depends on the parameters a and b (a and b can be written in terms of \(\alpha \) and n, see Eq. (9)). For a certain choice of \(a, P_{F(R)}\) increases with increasing value of b, as demonstrated in Fig. 4. It may be observed that this behaviour of \(P_{F(R)}\) with the parameter b is expected, because the height of the potential barrier decreases as b increases and as a result, \(P_{F(R)}\) increases. Finally we find that the solution of gauge hierarchy problem requires \(k\pi <d>_+ = 36\), which in turn highly suppresses the tunneling probability and as a consequence, \(P_{F(R)}\) comes as \(\sim 10^{32}\). This small value of the tunneling probability guarantees the stability of interbrane separation at \(<d>_+\). Therefore, it can be argued that the smallness of tunneling probability is interrelated with the requirement of solving the gauge hierarchy problem.
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