Dark matter in the Randall-Sundrum model with non-universal coupling

We consider simplified dark matter models (DM) interacting gravitationally with the standard model (SM) particles in a Randall-Sundrum (RS) framework. In this framework, the DM particles interact through the exchange of spin-2 Kaluza-Klein (KK) gravitons in the $s$-channel with the SM particles. The parameter space of the RS model with universal couplings to SM particles is known to be strongly constrained from the LHC data. We are thus led to consider models with non-universal couplings. The first model we consider in this study is a top-philic graviton model in which only the right-handed top quarks are taken to interact strongly with the gravitons. In the second, the lepto-philic model, we assume that only the right-handed charged leptons interact strongly with the gravitons. We extend the study to include not only the scalar, vector and spin-1/2 fermions but also spin-3/2 fermionic dark matter. We find that there is a large parameter space in these benchmark models where it is possible to achieve the observed relic density consistent with the direct and indirect searches and yet not to be constrained from the LHC data.


Introduction
Dark matter (DM) existence has been inferred from several cosmological and astrophysical observations at different scales. At the galactic scale, we have the observation of the flattening of rotation curves, weak lensing measurements at the scale of galactic clusters and the CMB and large scale structure observations at the cosmic scale point towards the existence of DM in the Universe. The dark matter constitutes roughly 75% of the entire matter existing in the Universe. The Planck collaboration has measured the DM density to great precision and has given the value of relic density Ω DM h 2 = 0.1198 ± 0.0012 [1]. (Ω DM is the DM mass density in units of critical density and h ∼ 0.7 is today's Hubble constant in the units of 100 km/s/Mpc.) The nature of the DM particles, however, remains elusive. One of the ways to unravel the nature of DM particles is to search for its non-gravitational interaction with the Standard Model (SM) particles, but none has been manifested so far. Direct searches of DM particles from the local galactic halo aim to measure recoil of nucleons through scattering in underground targets. In direct detection, the important quantity is the spin-independent (SI) and spin-dependent (SD) scattering cross-sections in a non-relativistic (NR) regime. These measurements have reached a sensitivity level where σ SI > 8 × 10 −47 cm 2 for DM masses ∼ 30 GeV has been ruled out in Dark Side 50 (2016) [2], LUX [3], XENON [4], and PANDA [5]. Indirect detection aims at unveiling excess cosmic rays produced by DM annihilation or decay in the galaxy or beyond. The indirect detection of DM is made through the emission of monochromatic gamma rays by the satellite-based γ-ray observatory Fermi-LAT [6,7] and ground-based Cherenkov telescope H.E.S.S [8]. Collider searches by the ATLAS and CMS collaborations [9,10] at the Large Hadron Collider (LHC) aim at identifying signatures of production of DM particle involving missing energy ( / E T ) accompanied by a single (monojet + / E T ) or two jets (dijet + / E T ) events. No experimental observation so far has made any confirmed detection and as a result, a large DM parameter space has been excluded. Most of the theoretical effort has been invested as the hypothesis that DM is a weakly interacting massive particle (WIMP) with mass lying between several GeV to a few TeV. WIMPS emerge in attempts to address the hierarchy problem underlying the staggering difference between the Planck Scale O(10 16 GeV) on the one hand and the electro-weak scale O(100 GeV) on the other. WIMPS provide the simplest production mechanism for massive relics from the early Universe. Freeze-out from the thermal plasma for particles having weak scale interactions and mass lying between GeV to TeV range can naturally account for the observed relic density. The fact that recent detection experiments notably by XENON1T have not found a signal puts a severe strain on the parameter space of one of the most attractive theories to address the hierarchy problem namely, the supersymmetric (SUSY) theories and, most of the parameter space of natural simple SUSY is ruled out [11]. Null results are constraining more and more of the parameter space of current DM theories. Randall-Sundrum (RS) formalism in the 5D warped extra-dimensions [12] with its Kaluza-Klein (KK) gravitons interacting purely gravitationally with the SM particles provide another possible solution to the hierarchy problem. The RS model with universal coupling to all SM particles is, however, very seriously constrained in its parameter space by the collider (LHC) data [13] and one is thus led to consider the bulk RS model with non-universal couplings. In one of these models, KK graviton interacts strongly with only the top-quark and this making the top-quark loops important for its production and decay at the colliders [14][15][16][17][18]. Recently, a top-philic KK graviton model with non-universal couplings to SM fields [19] in which only the right-handed top-quarks interact strongly with the KK graviton has been considered as a possible solution to the gauge hierarchy problem. In this framework, the 5D warped space-time has two boundaries corresponding to UV and IR branes. The SM fields in this framework are 5D objects which either propagate in the bulk or are located on the branes. The interaction between the particles is given by the overlap of particle wave functions thereby making the interactions hierarchical. The first KK graviton is assumed to have a maximum near the IR brane so that it would couple strongly with the fields near the IR brane. It's coupling to SM particles would decrease exponentially depending on the variations of the SM wave function from the IR to the UV brane. Since the search for DM particles interacting non-gravitationally with SM particles has come under severe constraints from current observational data, one has been led to explore the interesting possibility of DM interacting with the SM particles by purely gravitational interaction in the RS framework [13,[20][21][22][23].
It was observed that in most of the benchmark models considered in the literature, the desired values of the relic density could only be attained for the case of vector DM with mass near the s-channel KK graviton resonance [23] or for very low Graviton and DM mass [22] in a very narrow window in the parameter space. We consider here bulk RS model in which (i) only the right-handed top-quarks (top-philic) interact strongly with KK graviton [22] or (ii) only the right-handed leptons (lepto-philic) interact strongly with the KK gravitons. We also extend the study to consider spin-3/2 DM particle, along with the scalar, vector and spin-1/2 fermionic DM particles.
In Section 2, we describe our benchmark models in the RS framework. In Section 3, we calculate the relic abundance through KK graviton exchange with DM particles. The constraints arising from the direct and indirect searches are obtained. Section 4 is devoted to the results and conclusions. The decay width and DM annihilation cross-section expressions are given in the Appendix A and Appendix B respectively.

Dark matter in the Randall-Sundrum framework
In the Randall-Sundrum framework, the particle interaction with the massive spin-2 gravitons Y µν is purely gravitational and is through the energy-momentum tensor being given by where T µν i is the energy-momentum tensor of the i th particle, c i is the corresponding coupling and Λ is the scale of KK graviton interaction. We assume the DM fields to be either a real scalar, a real vector, a vector-like spin-1/2 Dirac fermion or a spin-3/2 fermion. The DM fields are further assumed to be SM singlets which do not carry any SM charge. It is also assumed that the DM particles are odd under a discrete Z 2 symmetry so that there is no mixing between the DM and SM fields. It is assumed that the DM particles live on the IR brane. The mass of the DM particle is taken to be less than the mass of the gravitons for simplicity so that there is no DM annihilation into KK graviton states. We consider two benchmark models depending upon the relative placement of SM particles on or near the branes Model A In the first benchmark model (Top-philic KK graviton) the right-handed topquarks alone are assumed to be located on the IR brane, SU (3) C and U (1) Y gauge bosons live in the bulk and the rest of the SM fields including the SU (2) L gauge bosons live on the UV brane or close to it.

Model B
In the second benchmark model (lepto-philic KK graviton), the right-handed leptons are assumed to live on the IR brane and only the U (1) Y gauge bosons are assumed to live in the bulk with the rest of the SM particles including the gauge bosons live close to or on the UV brane.
In the present framework, the graviton alone is expected to interact strongly with particles on the IR brane while the coupling with W ± , SM Higgs boson and other fermion is weak. The coupling of graviton with the gauge bosons living in the bulk is suppressed by the volume factor of the order of 1/ ln(M P l /M IR ) ∼ 0.03 with the IR brane scale taken to be at O(TeV) scale. This suppression factor is of the order of one-loop suppression factor α/4π for the EW or colour gauge bosons, α being the corresponding fine structure constant (α = α S for colour gluons and α em for electro-weak gauge bosons). The interaction Lagrangian relevant for the two benchmark models is this given by where D µ = ∂ µ + i(2/3) g 1 B µ + ig s t a G a µ is the covariant derivative for the right-handed top quark field, G µ and B a µ are the gluon and U (1) Y gauge boson fields respectively. The coupling c tt are scaled by appropriate loop suppression and c tt > c gg α S /4π ∼ c 1 α/4π other couplings.
where D µ = ∂ µ + i 2 3 g 1 B µ is the covariant derivative for the right-handed leptons.
The DM particles scalars (S), vectors (V ), spin-1/2 fermions (χ) and spin-3/2 fermions (Ψ) are taken to be present on the IR brane and interact with KK gravitons with a coupling strength of order one, through the energy-momentum tensor. The interaction Lagrangian is given as In above energy-momentum tensor, V αβ is the vector field tensor. The spin-3/2 field Ψ α satisfies the Euler-Lagrange equation with ∂ ν Ψ ν = 0 and γ ν Ψ ν = 0.
The spin-2 KK graviton polarisation sum is given by p µ Π 2 µν,αβ = 0 and Π 2 µν,αβ η µν = 0 for on-shell Y µν . The polarisation sum of spin-3/2 fermions After electroweak symmetry breaking, the coupling between Y µν and U (1) Y gauge bosons is written in terms of coupling of Y µν with photons and Z bosons as 14) The couplings of c γγ , c γZ and c ZZ can be obtained from the coupling c 1 and are given as Since the gravitons couple strongly with the right-handed top quarks in benchmark model A and with right-handed charged leptons in model B, the top quark and lepton triangle loop contribution to the graviton-gluon and graviton-U (1) Y gauge bosons can typically be of the same order as the corresponding tree-level couplings. The resulting effective couplings are evaluated in Refs. [19,24] and are given Appendix A where KK graviton decay expressions are also listed.
In Figs. 1 and 2 we have plotted the KK-graviton-gauge boson effective couplings, decay width and branching ratios in the final states relevant to the benchmark models A and B respectively. For the purpose of illustration we have taken the couplings c gg , c tt , c ll and c 1 defined in Eqs. (2.2) and (2.3) to be equal to 1. The KK-graviton interaction scale factor Λ is fixed at 1 TeV and the dark matter mass is taken to be equal to 300 GeV. We see from Figs. 1 and 2 that after the onset of KK-graviton decay into DM particles   2) are taken to be equal to 1. The KK-graviton interaction scale factor Λ is fixed at 1 TeV and the dark matter mass at 300 GeV. Panels (1c -1f) show the branching ratios separately for the scalar, spin-1/2, vector and spin-3/2 dark matter.  3) are taken to be equal to 1. The KK-graviton interaction scale factor Λ is fixed at 1 TeV and the dark matter mass at 300 GeV. Panels (2c -2f) show the branching ratios separately for the scalar, spin-1/2, vector and spin-3/2 dark matter.
at 600 GeV, the decay width for the case of spin-3/2 DM particle rises rapidly with the increase in graviton mass m Y in comparison to the case of scalar, spin-1/2 and vector DM particles. This is a general feature of spin-3/2 particles [25,26] and can be seen from the decay width expression (A. 19) of Y into ΨΨ. In Figs. 1c to 1f we have shown the branching ratios separately for the scalar, spin-1/2, vector and spin-3/2 dark matter in the benchmark model A and likewise for the model B in Figs. 2c to 2f. We note that the graviton branching ratio into spin-3/2 DM pairs approaches one with the increase in graviton mass.

Thermal relic density
In the early Universe, DM is in thermal equilibrium with the hot dense plasma and as the Universe expands and cools, it freezes out. The evolution of the number density of DM matter n DM is governed by the Boltzmann equation n eq DM is the equilibrium density of the DM at temperature T and is given by where g is the number of degrees of freedom (g = 1, 2, 3) and 4 respectively for scalar, spin-1/2, vector and spin-3/2 fermionic DM. σ|v is the thermal averaged cross-section of DM annihilation channels into the SM states. The thermally averaged cross-section can be written as a one-dimensional integral over the centre of mass energy square s as where K 1 and K 2 are the modified Bessel functions of the second kind, the annihilation cross-section σ ann depends only on the masses, s and couplings of the DM and SM particles involved. The Boltzmann equation can be solved to give the thermal relic density Here h is the Hubble parameter today, g * (x F ) is total number of dynamic degrees of freedom near the freeze-out temperature T F and x F = m DM /T F is obtained by solving and c is of order 1. Scattering cross-sections as a function of s and DM mass m DM and couplings are given in the Appendix B. Since the mediator KK graviton state carries spin-2, the annihilation rates into final SM states in the benchmark models considered here, are velocity suppressed. The suppression depends on the spin of the initial DM and final SM particles. The maximum suppression occurs for the scalar and spin-1/2 DM's and there is no suppression for the vector and spin-3/2 DM particles in the dominant annihilation channels. We first examine the benchmark model A (top-philic) in which only the right-handed top quarks and the DM particles live on the IR-brane.
We have computed the relic density in the two benchmark models considered here numerically. We have generated the input model files using LanHEP [27] which calculates all the required couplings and Feynman rules by using the Lagrangian given in Section 2. particles in the benchmark models A and B respectively. The panels (3a -3d) and (4a -4d) correspond to the KK-graviton interaction scale Λ = 500 GeV, 1 TeV, 2 TeV and 3 TeV respectively. It can be seen from these figures that the distinction with respect to the contribution to the relic density arising from the scalar, spin-1/2, vector or spin-3/2 dark matter particles tends to diminish with the increase in the interaction scale Λ.

Direct detection
The tree-level nucleon-DM scattering through the massive spin-2 KK graviton propagator, after integrating out the massive graviton field tends to the effective Lagrangian whereT µν and T are traceless and trace part of the energy-momentum tensor viz.
For the DM-nucleon scattering, the relevant SM energy-momentum tensor is the one that involves light quarks and gluons. In the top-philic RS model A considered here, The trace and traceless part of T a µν are given by respectively. The matrix element of the trace part of the energy-momentum tensor T g µν between nucleon and state can be evaluated easily [28] and are given by where f (N ) The nucleon-matrix element of the traceless part of the energy-momentum tensor is given by the second moment of the gluon distribution function of the factorisation scale.
where g (2 : µ) = 1 0 x g 2 (x, µ) dx and is ∼ 0.464 and the matrix element is suppressed by α S /4π in comparison to the matrix element of the trace part of the energy-momentum tensor between nucleon states. The contribution of the traceless part of the energy-momentum tensor to the nucleon-DM scattering turns out to be ∼ 10% compared to the trace contribution. Thus in the N-R limit, neglecting small momentum dependent terms, the DM-nucleon scattering in the lab frame can be calculated from the trace part of the energy-momentum tensor. In this approximation, the spin-independent scattering cross-section does not depend on the spin of the DM particle. The spin-independent DM-nucleon scattering crosssection is given by where m N the nucleon mass and µ = m DM m N m DM +m N is the reduced mass. In the lepto-philic model B considered here, DM-nucleon scattering arises through KK graviton-photon effective coupling which is suppressed by α/4π. The DM-nucleon scattering cross-section is further suppressed by (α/4π) 2 and does not give any meaningful constraint and is much below the sensitivity level achieved in the current or planned direct detection experiments [29][30][31].
Using the expression Eq. (3.12) we have plotted the spin-independent DM-Nucleon scattering cross-section in the bench-mark model A as a function of DM mass in Fig. 5. The panels (5a -5d) correspond to the KK-graviton interaction scale Λ = 500 GeV, 1 TeV, 2 TeV and 3 TeV respectively. The parameter set used in the computation are consistent with the observed relic density given in Figs. 3 and 4. The upper limits from PANDA 2x-II 2017 [5] and XENON-1T [32,33] are also shown with the forbidden region. (d) Figure 6: Velocity-averaged cross-section σv γγ in the benchmark model A . The current upper bounds from the Fermi-LAT [7] and H.E.S.S. [8,34] data are shown. All points on the contour satisfy the observed relic density. The panels (6a -6d) correspond to the KK-graviton interaction scale Λ = 500 GeV, 1 TeV, 2 TeV and 3 TeV respectively.

Indirect detection
Observation of diffused gamma rays from the regions of the galaxy such as Galactic centre (GC) and dwarf spheroidal galaxies (dSphs) where DM density appears to be large, imposes bounds on the DM annihilation to the SM particles. Fermi-LAT [7,35] and H.E.S.S [8,34] have investigated DM annihilation as a possible source of incoming photon flux. These experiments are then used to put constraints on the upper limit of velocity averaged scattering cross-sections for various channels which can contribute to the observed photon flux. The γγ photon line search in the photon spectrum in the KK graviton mediated DM annihilation into two photons occur directly as well as through the top-quark triangle loop for the model A and charged lepton loop for model B as discussed in Section 2. In the case of lepto-philic DM model B, the diffused γ-ray flux arises from the annihilation channel into charged leptons. The velocity averaged annihilation cross-section σv + − is, however, several orders of magnitude smaller than the corresponding γγ annihilation cross-section and will not be considered further. The DM is gravitationally bound and is moving typically with velocity roughly lying between 200-500 km/s and the constraints obtained from indirect searches are not sensitive to the specific choice of the velocity. For the case of vector and spin-3/2 DMs, the choice of specific velocity is especially unimportant and v → 0 is an excellent approximation. For our numerical calculation, we have kept the exact expression with the choice of v = 300 km/s. The experimental limits on this mode are given by the Fermi-LAT [7] and H.E.S,S. [8,34] Galactic Centre data sets. The limits depend upon the velocity distribution profile.
In Fig. 6 and Fig. 7 we have plotted the variation of velocity-averaged scattering crosssection σv γγ with the DM mass in bench-mark model A and B respectively. We have also shown the observational constraints for the two photon final state annihilation rates in Fig. 6 and Fig. 7. All parameters are chosen to satisfy the observed relic density. The panels (6a -6d) and (7a -7d) correspond to the graviton interaction scale Λ = 500 GeV, 1 TeV, 2 TeV and 3 TeV respectively for the benchmark models A and B respectively. We find that the annihilation cross-sections for the above processes in the benchmark models A and B are roughly three to four orders of magnitude smaller than the current upper bounds from the Fermi-LAT [7,35] and H.E.S.S [8,34] data.

Results and conclusions
In this article, we have investigated the viability of a scalar, vector, a spin-1/2 or a spin-3/2 fermionic dark matter particle interacting purely gravitationally with the standard model particles mediated by its Kaluza-Klein graviton in the Randall-Sundram framework. Since the RS model with universal coupling to all SM particles is severally constrained in its parameter space by the LHC data [13], we are led to consider the RS model with nonuniversal couplings to SM particles. In the top-philic model A considered here, only the right-handed top quarks are assumed to interact strongly with the gravitons. The colour and U (1) gauge bosons are assumed to live in the bulk. In the second benchmark leptophilic model B only the right-handed charged leptons interact strongly with the gravitons. The dark matter particles of any spin, on the other hand, are assumed to live on the IR brane and thus interact strongly with the gravitons.
We can see from Figs. 3 and 4 that in both these models there exists a large parameter space in which the observed relic density can be obtained for reasonable values of the parameters. In Fig. 5 we have plotted the dark matter-nucleon scattering cross-section as a function of DM mass in model A. Current bounds on spin-independent interactions from experiments like PANDA 2X-II 2017 [5] and XENON1T [32,33] are also shown. Both the models A and B predict dark matter-nucleon scattering cross-sections much below the sensitivity levels achieved in the current or planned direct detection experiments. Another notable feature of the model A is that the leading term in the DM-Nucleon cross-section is independent of the spin of the dark matter particle.
From the velocity-averaged cross-section σv γγ in the benchmark model A (Fig. 6) and model B (Fig. 7), we find that the annihilation cross-sections for the above processes are roughly three to four orders of magnitude smaller than the current upper bounds from the Fermi-LAT [7,35] and H.E.S.S [8,34] data.
In conclusion, the top-philic as well as the lepto-philic models in the Randall-Sundram framework discussed in this work, are capable of explaining the observed relic density for a reasonable set of parameters without transgressing the constraints from the direct and indirect experiments. They are also consistent with the LHC data [13].

A KK graviton-gauge boson effective couplings and decay widths
The effective KK graviton-gauge boson couplings in benchmark models A and B are where Q t = 2/3 and N c = 3.

Model B
KK graviton decay widths relevant for the benchmark models A and B are given as follows The KK graviton decay widths into DM particle pairs are Here, S, χ, V and Ψ are the spin-0, spin-1/2, spin-1 and spin-3/2 DM particles respectively.

B Annihilation cross-sections
The annihilation cross-sections of DM into various final states relevant for the benchmark models A and B are given as follows: B.1 Scalar dark matter

B.2 Vector dark matter
B.3 Spin-1/2 dark matter   The effective coupling constants c eff gg , c eff γγ , c eff γZ and c eff ZZ can be obtained from Eqs. (A.1) and (A.2) for benchmark models A and from Eq. (A.4) for model B respectively replacing the KK graviton mass m Y with the centre of mass energy √ s.