Dark Matter in a Three-Brane Randall-Sundrum Scenario out of the Evanescent Limit

Abstract

The Nature of Dark Matter (DM), that constitutes approximately 25\% of the energy density in the Universe, is still eluding us. An intriguing possibility is that DM does indeed interacts with SM particles only gravitationally (the only mean by which we have detected it so far), albeit in an extra-dimensional scenario yet it has not been possible to detect it by some non-gravitational means. In a three-brane Randall-Sundrum setup, with DM located on a Deep Infra-Red GeV-TeV brane, and the SM on an Infra-Red TeV-PeV one, it was shown to be possible to recover the observed DM relic abundance and somewhat relax the hierarchy problem, whilst avoiding LHC stringent bounds on DM and KK graviton masses that constrain severely similar two-brane setups. The phenomenological results, however, have been obtained under the assumption that the bulk curvatures on the left ($k_1$) and the right ($k_2$) of the intermediate IR-brane are identical, $k_2 \to k_1$. Since the brane tension $σ_{\rm IR}$ of the intermediate brane is proportional to $k_2 - k_1$ and, therefore, vanishes, it is clear that this limit (for which the IR-brane becomes {\em evanescent}) is unphysical. We could say that this is {\em no brane}: if the brane tension of a brane vanishes, there is no brane in the bulk (pun intended). In this paper, therefore, we study in detail the theoretical framework needed to explore this interesting phenomenological possibility, {\em out of the evanescent brane limit}. We show that most of the formulæ$\,$used in the {\em evanescent limit} are still valid for ${\cal O}(1)$ differences between $k_1$ and $k_2$ (thus, introducing no new unjustified hierarchy in the bulk). Once the relevant couplings of radions and KK gravitons are computed, we study the (enlarged) parameter space of the model looking for the region in which the relic DM abundance is recovered.

Figures (8)

• Figure 1: Schematic representation of the three-brane setup used in this work in conformal coordinates.
• Figure 2: Three-brane model KK graviton wave-functions as a function of the conformal coordinate $z$. Left panel: $\hat{\chi}^{(n)}(z)$, for KK number $n=1$ (solid), $n=2$ (dashed) and $n=3$ (dot-dashed) for $\bar{M}_{\rm eff}$ = 10 TeV. Right panel: $\hat{\chi}^{(1)}(z)$ for three different values of the intermediate brane position corresponding to $\bar{M}_{\rm eff} = 10$ TeV (solid), $20$ TeV (dashed) and $100$ TeV (dot-dashed). The blue vertical lines represent $z(L_1)$ in the three cases.
• Figure 3: The potential $V(\omega, \xi)$ for several choices of ($\bar{\omega}, \bar{\xi}$) in the plane $(R_1, R_2)$. Left panel: $\bar{\omega} = 1 \times 10^{-3}, \bar{\xi} = 1 \times 10^{-1}$; Middle panel: $\bar{\omega} = 1 \times 10^{-3}, \bar{\xi} = 9 \times 10^{-1}$; Right panel: $\bar{\omega} = 9 \times 10^{-1}, \bar{\xi} = 9 \times 10^{-1}$. The red dot represents the numerical minimum of the potential, whereas the black dot is the analytical result. In the white region, the potential is not real for the considered choice of $\bar{\omega}$ and $\bar{\xi}$.
• Figure 4: The ratio of the masses of the scalar modes $r_1$ and $r_2$. Left: as a function of the curvature splitting parameter $\delta k$, for several different values of $\epsilon_1 = 0.1$ (green), $0.05$ (blue) and $0.01$ (red), with $\epsilon_2 = 0.01$ in all cases. Right: as a function of $\epsilon_2/\epsilon_1$ for several different values of $\delta k = 0.01$ (red), $0.1$ (blue) and $1$ (green), with $\epsilon_1 = 0.1$ in all cases. In both panels, the region where the difference between $k_1$ and $k_2$ is ${\cal O}(1)$ ( i.e., the non-evanescent regime) is depicted as a grey area.
• Figure 5: Left panel: The total decay width of the first KK graviton $G_1$ as a function of $\delta k$. The DIR and IR scales are fixed to $\Lambda_{\rm DIR} = 8$ TeV and $\Lambda_{\rm IR} = 30$ TeV, respectively. The first KK graviton mass is $m_1 = 1$ TeV, whereas the radion mass is $m_{r_2} = 100$ GeV. The blue (dashed) line stands for $\Gamma \left ( G_1 \to {\rm SM \, SM}\right )$. The red (dashed) line stands for $\left ( G_1 \to r_2 \, r_2 \right )$. Eventually, the red (solid) line is the total decay width. Right panel: Branching ratios of the first KK graviton $G_1$ (scale on the left vertical axis) and its lifetime $\tau$ (scale on the right vertical axis), as a function of the KK graviton mass $m_1$. The solid lines correspond to $\tau$. Dashed and dotted lines represent the BR's ${\rm BR}(G_1 \to r_2 \, r_2)$ (dotted) and ${\rm BR} (G_1 \to {\rm SM \, SM})$ (dashed). The color code works as follows: red stands for $\delta k = 0.1$, green for $\delta k = 1$ and blue for $\delta k = 10$.