Four-dimensional gravity on supersymmetric dilatonic domain walls

We investigate the localization of four-dimensional metastable gravity in supersymmetric dilatonic domain walls through massive modes by considering several scenarios in the model. We compute corrections to the Newtonian potential for small and long distances compared with a crossover scale given in terms of the dilatonic coupling. 4D gravity behavior is developed on the brane for distance very much below the crossover scale, while for distance much larger, the 5D gravity is recovered. Whereas in the former regime gravity is always attractive, in the latter regime due to non-normalizable unstable massive graviton modes present on the spectrum, in some special cases, gravity appears to be repulsive and signalizes a gravitational confining phase which is able to produce an inflationary phase of the Universe.

Furthermore, have also been considered in the literature thick brane realizations of metastable localization of gravity in the literature [7][8][9][10][11][12][13][14][15]. In most of these cases one has considered five-dimensional theories of scalar fields coupled to gravity in the realm of 'fake supergravity' where several issues were discussed. One should, though, also consider such realizations in supergravity domain wall solutions [16].
In the present study, we shall consider the supersymmetric dilatonic solution that can be found from a higher dimensional supergravity theory that after specific compactifications, such as toroidal compactification, turns to a simpler five-dimensional theory of a scalar field coupled to gravity [17][18][19][20] in the same fashion of those conceived in [2]. We shall investigate the possibility of localizing four-dimensional metastable gravity in supersymmetric dilatonic domain walls through massive gravitational modes by considering several scenarios, including those in which nonnormalizable unstable massive graviton modes determine dominant repulsive gravity.
The paper is organized as follows. In Sec. II we make a brief review on the mechanism of gravity localization on braneworlds. In Sec. III we present four scenarios by properly adjusting one of the free parameters of the theory. In all of these cases is recovered the four dimensional attractive gravity in the regime r ≪ r c . The first case gives the expected result since the localized gravity is attractive everywhere. However, due to non-normalizable unstable massive graviton modes present on the spectrum, the following three cases assume a new behavior at large distance, i.e., at r ≫ r c . In this regime they exhibit a repulsive gravity with exponentially increasing potential which signals a gravitational confining phase that is able to produce an inflationary phase of the Universe [24]. This effect has also been addressed in earlier similar studies on time-like extra-dimensions [25,26]. Finally in Sec. IV we make our final comments.

II. GRAVITY ON DILATONIC DOMAIN WALLS
Let us briefly introduce the mechanism behind gravity localization in braneworlds. We consider a general dilatonic (D − 2)−brane solution in a spacetime in D dimensions discussed in Refs. [17][18][19][20]. The spacetime is assumed to be the direct product M D−1 × K of the (D − 1)−dimensional spacetime M D−1 and some noncompact internal space K. We shall focus our attention to obtain solutions describing braneworlds as the worldvolume of domain walls. The Einstein-frame action for the dilatonic domain wall is where κ D is the D−dimensional gravitational constant, and the cosmological constant term is multiplied by the dilaton factor which depends on an arbitrary dilaton coupling parameter a in any D-dimensional spacetime. The dilatonic domain wall solution given in terms of a conformally flat form is where For future discussions it is useful to define Λ in terms of the space curvature 1/L. In the limit of sufficiently small curvature (L ≫ ℓ s ) being ℓ s the string length, the supergravity approximation takes place. In order to study the spectrum of the graviton in the dilatonic domain wall background, we consider the metric describing the small fluctuation h µν (x ρ , z) of the (D − 1)−dimensional Minkowski spacetime embedded into the conformally flat D−dimensional spacetimeḡμν where |h µν | ≪ 1. By using the linearized Einstein equations in the transverse traceless gauge, i.e., ∂ µ h µν = 0, and the metric fluctuation in the form h µν (x ρ , z) =h µν , we obtain the Schroedingerlike equation with the potential where the prime denotes derivatives with respect to z. In this paper, we shall discuss the properties of the KK modes for different values of ∆ taking Q > 0. We explore the scenarios generated by the Schroedinger-like potentials that arises from four choices for the range of ∆ , and we study the asymptotic behavior of the corrections for the Newtonian potential, which are derived from the wave functions that emerge from the corresponding Schroedinger-like equations. This means to investigate how easy four-dimensional massive gravity can be localized on the brane. Notice that the structure of this potential allows to write the Schroedinger-like equation in a quadratic form [2] where A(z) = 1 2 ln C(z). The zero mode obeying H ψ 0 = 0 is given by It is easy to conclude from the equation (9) that for Q > 0 there exists normalizable zero mode only in the range ∆ ≤ −2, [19,20]. In the following, we consider four cases which do not fall into this range, and do not exhibit zero mode. This is because we aim to do a complementary study of gravity localization on supersymmetric dilatonic domain walls by extending to massive modes the previously analysis mainly based on zero modes [17][18][19][20]. This produces localization of four-dimensional gravity for distances very much smaller than a crossover scale r c .

III. FOUR-DIMENSIONAL GRAVITY ON DILATONIC DOMAIN WALLS
In the present section, we focus on four scenarios of 4D gravity on dilatonic domain walls, by considering four ranges of ∆, for Q > 0.
In this case, the potential (7) can be written as whose massive modes solution of the Schroedinger-like eq.(6) is given by [21] ψ m (z) = K 1 (m) 1 in terms of the Bessel functions of first and second kind, J β1 and N β1 respectively, where , and K 1 and F 1 are the normalization factors -See Fig. 1. This problem is similar to that for a particle trapped into an infinite box with a delta function at the origin, which we use to impose the boundary conditions. The jump condition at z = 0, is given by In order to obtain the correction term to the four-dimensional Newton's law between two unit masses on the 3-brane (D = 5), one needs to obtain the probability density of massive modes on the brane. The asymptotic behaviors of |ψ m (0) | 2 are obtained from the normalization factors given by where to find the asymptotic forms of Bessel function as J n (x) ∼ 2 xπ cos (x − 1/2 n π + 1/4 π) and N n (x) ∼ 2 xπ sin (x − 1/2 n π + 1/4 π) for x ≫ 1, J n (x) ∼ (1/2 x) n and N n (x) ∼ −1/π (1/2 x) −n for x ≪ 1. Thus, the asymptotic probability density assumes the simplified forms where we define the crossover scale r c = 1/ǫ 1 . And, the boundary conditions of the potential applied to the wavefunction (11) allows us to determine how the parameter ǫ 1 is related to the graviton masses, i.e.
For |α 1 | ≪ 1 (or in terms of the bulk cosmological constant |Λ| ≫ 1), in the asymptotic regime the solution of the Eq. (17) is approximately given by In such regime, it is reasonable to approximate the potential generated by discrete massive graviton states as a summation of Yukawa-like potentials, which makes the total effective potential to have the form [2, 8,9] V (r) = ψ 2 0 (0) where the first term is contribution of zero mode and the second term corresponds to the correction which is generated by the exchange of KK-modes. However, the set of discrete states may be replaced by a continuous treatment for |α 1 | ≪ 1, such that n → . In this case, we have no zero mode. Then, the contribution for the potential cames from the massive modes (the second term of Eq. (19)), which can be expressed by an integration in two distinct regions, as follows In the limit where the crossover scale is very small, i.e., r c → 0 (∆ → −4 + 2 √ 2, 1 √ |Λ| → 0), the first integral in Eq. (20) is dominant, and we find the familiar behavior of Randall-Sundrum scenario [1] As we previously stated that r c ∼ H −1 , this regime is particularly achieved in the early Universe, where only very small distances can be probed.
In order to analyze the behavior for large and small distance compared with the crossover scale r c , we use the asymptotic forms: ci(x) ∼ γ + ln(x) and si(x) ∼ x − π/2 for x ≪ 1, ci(x) ∼ sin(x)/x and si(x) ∼ −cos(x)/x for x ≫ 1. For small distance, i.e., r/r c ≪ 1 we obtain the following form where γ ≈ 0.577 is the Euler-Masceroni constant. Notice that, at this limit the potential has the correct 4D Newton's law with 1/r scaling. Finally, for large distance, i.e., r/r c ≫ 1, the potential in Eq. (22) gives that recovers the laws of 5D gravity [5,6].
In this case, for the potential in Eq. (7) the general solution of the Schroedinger-like equation (6) for massive modes is given by [21] ψ m (z) = K 2 (m) Another time, using the Lommel's formula, J n+1 (x) N n (x) − N n+1 (x) J n (x) = 2/π x and the asymptotic form of the Bessel functions, the probability density reduces to where we define the crossover scale r c = 1/ǫ 2 . Here, to compute the correction to the four-dimensional Newtonian potential generated by the massive modes, we use the formula [1] where the first term is the contribution of the zero mode, and the second term corresponds to the corrections generated by the exchange of massive Kaluza-Klein modes As in the earlier case, there are no zero modes and we divide this integral into two regions limited by the crossover scale as Thus, the Newton's law correction for massive modes at a distance r is given as follows. In the limit where the crossover scale is very small, i.e., r c → 0 (∆ ≪ Q), the first integral in (32) is dominant and gives the familiar behavior of Randall-Sundrum scenario [1], for the critical values of the dilaton coupling a 2 ∼ 4 (D−1) (D−2) 2 in this limit, so that Now, for the crossover scale being very large, i.e., r c → ∞ (∆ ≫ Q), the second integral in (32) is dominant and we can get δV (r) approximately by [5,12] δV (r) = e − r Once more, we use the asymptotic forms of ci(x) and si(x), to examine the large and small distance behavior. For small distance, i.e., r/r c ≪ 1 we obtain the following form Interestingly, at short distance the computed potential has the correct 4D Newton's law with 1/r scaling. The next leading correction is given by the logarithmic repulsion term in (35). On the other hand, for large distance, i.e., r/r c ≫ 1, the potential in Eq. (34) gives which describes the laws of 5D gravity [5,6].
The following analysis is pretty similar to the previous one. In the present case, for the potential (7) written as the Schroedinger-like equation gives the following solution for massive modes [21], in terms of the Bessel functions of first and second kind, J β2 and N β2 , respectively, where , and the normalization factors K 3 and F 3 -See Fig. 3. The jump condition at z = 0 is The asymptotic behavior of |ψ m (0) | 2 comes from the normalization factors given by Anew, using the Lommel's formula, J n+1 (x) N n (x) − N n+1 (x) J n (x) = 2/π x and the asymptotic form of the Bessel functions, the probability density become where we define the crossover scale r c = 1/ǫ 3 . Again, we have no zero mode, consequently, all the contribution to the Newtonian potential comes from the massive modes given by Eq. (31), that we shall integrate in the two regions as (2 α 2 − ǫ 3 ) When the crossover scale is very small, i.e., r c → 0 (Q → ∞), the first integral in (44) is dominant and we naturally find the familiar behavior of the Randall-Sundrum scenario [1] For the crossover scale being very large, i.e., r c → ∞ (Q → 0) the second integral in (44) is dominant, and using the relation E 1 (i x) = −ci(x) + i si(x) we have [5,12] δV (r) ∼ 1 4 Now, from the asymptotic form of ci(x) and si(x), we examine the large and small distance behavior comparing with the crossover scale. For small distance, i.e., r/r c ≪ 1, we obtain Notice that at short distances the potential has the correct 4D Newtonian 1/r scaling, with a subsequently correction by the logarithmic repulsion term in (47). Finally, for large distance, i.e., r/r c ≫ 1, the potential (46) reduces to which is in accordance with the laws of 5D gravity [5,6].
In the previous potentials all the scenarios with r c → 0 implies large space curvature (L ≪ ℓ s ), because Λ becomes large and then the validity of supergravity approximation breaks down.
In this case, the potential (7) reduces simply to U (z) = −2 Q δ(z) -See Fig. 4. The probability density for the massive modes is given in terms of the scattering states governed by |ψ m (0) | 2 that depends on the magnitude of the transmission T or reflection R coefficients. As usual, the jump condition at z = 0, is obtained from the Schroedinger-like equation by using the properties of the delta function. We now consider the general wave functions for scattered states in the form ψ 1m (z) = e iκ z + R e −iκ z , z < 0 ψ 2m (z) = T e iκ z , z > 0 where C 2 = 1/(2 π M 3 5 ), C 3 = 1/(32 α 2 2 M 3 5 ), and C 4 = 1/M 3 5 , for the cases (ii), (iii), and (iv), respectively. Finally, the resulting potentials are given by the sum of the correction (58) with (35), (47), and (54), for the cases (ii), (iii), and (iv), respectively. This analysis always involves the relationship r/r c ≪ 1, concerning the three cases studied and, interestingly, at short distance the computed potential has the correct attractive 4D Newton's law with 1/r scaling. On the other hand, for large distance, i.e., r/r c ≫ 1, the potential in Eq. (57) gives which implies a scenario of dominant repulsive gravity. The exponential increasing of the potential with r also signals a gravitational confining phase where any particle has infinite energy. As a consequence of such a confinement, no isolated particles can be found such that this phase indeed comprehends a vacuum state in a curved space which is able to produce an inflationary phase of the Universe [24]. This is in accord with the fact that plugging m 2 → −m 2 into Schroedinger-like equation (6) one finds formally the same equation as long as z → −iz, i.e., the extra-dimension becomes a time-like one, and the potential in (7) goes like U (z) → −U (z). This, of course, implies Q → iQ (in the bulk, i.e., z = 0) that from Eq. (4) flips the sign of the bulk cosmological constant Λ. This change of sign leads to a positive cosmological constant and then to a five-dimensional de Sitter space (dS 5 ) which is indeed the aforementioned vacuum developing an inflationary phase of the Universe. This is closely related to earlier studies on time-like extra-dimensions [25,26]. Now, before ending this section, some brief comments about the limit of very small crossover scale are in order. When r c → 0, the additional contribution to the Newtonian potential is dominated by integrals of the type 1 r ∞ 0 m 2n+1 e −mr dm. They impose some restriction on the non-normalizable unstable modes −∞ < m 2 < 0. This can be easily seen by changing the variables m → −im and r → ir to see that the integral get one complex factor i 2n+1 . Recalling that |ψ| 2 ∼ m 2n+1 is always positive, the power of this factor can only assume the values: 2n + 1 = 4, 8, .... This implies attractive gravity depending on each parameter n of the respective cases (ii) and (iii) considered above.

IV. CONCLUSIONS
In this paper, by considering massive graviton modes coming from a dilatonic domain wall solution of fivedimensional supergravity, we have shown that the gravitational potential corresponds to the usual Newton potential which scales with 1/r at short distance and has a five-dimensional behavior scaling with 1/r 2 at large distance compared with the crossover scale r c .We have explored distinct cases which depend on the parameter ∆. In our present analysis, we have chosen to work out with ∆ in four different ranges. We have shown that, in all of these cases is recovered the four dimensional attractive gravity in the regime r ≪ r c . The first case gives the expected result since the localized gravity is attractive everywhere, including the five-dimensional regime at r ≫ r c . However, due to non-normalizable unstable massive graviton modes present on the spectrum, the following three cases assume a new behavior at large distance, i.e., at r ≫ r c . At this regime they exhibit a repulsive gravity with exponentially increasing potential which signals a gravitational confining phase that is able to produce an inflationary phase of the Universe [24]. This has also been addressed in earlier studies on time-like extra-dimensions [25,26]. This study showed that from a five-dimensional supergravity theory with a scalar field describing the dilaton, the emergence of four-dimensional gravity on a 3-brane is possible even if the brane is embedded in an asymptotic five-dimensional flat space, below a crossover scale and the manifestation of extra dimensions does not necessarily occur only at short distances as commonly expected. Localization of gravity in other dilatonic domain walls configurations has also been addressed in [13,22,23].