Phenomenology of a two-component dark matter model

We study a two-component dark matter model consisting of a Dirac fermion and a complex scalar charged under new U(1) gauge group in the hidden sector. The dark fermion plays the dominant component of dark matter which explains the measured DM relic density of the Universe. It has no direct coupling to ordinary standard model particles, thus evading strong constraints from the direct DM detection experiments. The dark fermion is self-interacting through the light dark gauge boson and it would be possible to address that this model can be a resolution to the small scale structure problem of the Universe. The light dark gauge boson, which interacts with the standard model sector, is also stable and composes the subdominant DM component. We investigate the model parameter space allowed by current experimental constraints and phenomenological bounds. We also discuss the sensitivity of future experiments such as SHiP, DUNE and ILC, for the obtained allowed parameter space.


Introduction
We have compelling evidence for the existence of non-baryonic dark matter (DM) in the Universe. But, its physical properties remain one of the main mysteries in particle physics today. Weakly Interacting Massive Particles (WIMPs) have been leading DM candidates, since they can provide a right amount of DM relic density with the weak scale masses and couplings in a natural way [1][2][3]. However, several experimental attempts have failed to find clear evidence for WIMPs so far. The sensitivity of the DM direct detection experiments [4][5][6] is now approaching, without positive DM signals, neutrino f loor level which represents an almost irreducible background from coherent neutrino-nucleus scattering [7][8][9][10][11].
On the other hand, the collisionless cold DM (CDM) paradigm describes the large-scale structure of the Universe remarkably well. However, it suffers from the long-standing smallscale structure problems [12]. For instance, while the mass density profile for CDM halos in simulations turns out to steeply increase toward galactic center at small radii [13][14][15], the rotation curves of many observed galaxies show flat central DM density profiles [16][17][18][19]. In order to solve the problems such as the core-cusp problem, DM self-scattering cross section 1 typically has to be of order σ/m DM ∼ 1 cm 2 g −1 [20]. One way to have such a large cross section is to introduce a light mediator in a weakly coupled theory. In this case, the selfscattering cross section becomes large enough at small DM velocities through Sommerfeld enhancement, so that the small scale problems can be solved. However, if the light mediator is unstable and decays to the standard model (SM) particles, a very strong constraint comes from the measurements of the Cosmic Microwave Background (CMB) radiation [21] and it basically rules out [22] such models for solving the small scale problems. Therefore, the strong constraints from the CMB data require that the light mediator should be stable.
In this paper we study a two-component DM model [23], where two species of DM are introduced. One species (a dark Dirac fermion ψ) constitutes a dominant DM component. It has no direct coupling to the SM particles so that the direct DM searches are insensitive to the species, thus avoiding the strong constraints from the direct DM search experiments. The self-interaction of the dark fermion DM is mediated by a dark gauge boson A X , which is so light that the DM self-scattering cross section at present is large enough for addressing the small scale problems. The dark gauge boson A X itself is stable and constitutes a subdominant DM in the Universe. Therefore, the annihilation process ψψ → A X A X at the recombination epoch does not affect the CMB data. We will investigate the model parameter space, allowed by various experimental and phenomenological constraints, and discuss the prospects for finding new physics signals from future experiments. This paper is organized as follows. In section 2, we introduce the two component DM model. In section 3, we consider various experimental and phenomenological constraints on the parameter space of the DM model. We also discuss the sensitivity of the ILC, SHiP and DUNE experiments for the allowed parameter space. We summarize our conclusions in section 4.

A two-component DM model
In this section, we introduce a two component dark matter model [23]. The dark sector is composed of a complex scalar field S and a Dirac fermion field ψ, both of which are charged under a new dark U (1) X gauge group and singlet under the SM gauge group. The Lagrangian for the dark sector is given by the following renormalizable interactions, The covariant derivative is where X is the hidden U (1) X charge operator, and A µ X the corresponding gauge field. Due to the imposition of dark charge conjugation symmetry, there is no kinetic mixing term between A X and the U (1) Y gauge boson of the SM.
The Higgs portal interaction connects the dark sector and the SM sector, and the full scalar potential of the model is given as The Higgs doublet H is written in the unitary gauge after the electroweak symmetry breaking as follows: with v h 246 GeV. The singlet scalar field also develops a nonzero vacuum expectation value, v s , and the singlet scalar field is written as The mass parameters m 2 h and m 2 s can be expressed in terms of other parameters by using the minimization condition of the full scalar potential V , i.e., The mass terms of the scalar fields are where A non-vanishing value of µ 2 hs induces mixing between the SM Higgs field configuration h and the singlet scalar field s as where the mixing angle θ s is given by tan θ s = y with y ≡ 2µ 2 hs /(µ 2 h − µ 2 s ). Then, the physical Higgs boson masses are where h 1 is the SM-like Higgs boson with m h 1 = 125 GeV and h 2 is the singlet-like scalar boson. Scalar cubic interaction terms relevant for our study are given by with The dark gauge boson mass and interactions with the scalar particles, assuming U (1) X charge X = 1 for the dark scalar, are obtained from which gives the dark gauge boson mass m A X = g X v s .

Dark matter phenomenology
In this model, we have two DM candidates and one new scalar particle h 2 in addition to the SM particles. The dark fermion ψ is stable due to dark fermion number conservation. Therefore, it is a DM candidate. If m A X < 2 m ψ , the dark gauge boson A X cannot decay to the dark fermion pair ψψ and is also stable thanks to dark charge conjugation symmetry, thus providing another DM candidate. In this work, we assume A X is much lighter than the dark fermion, keeping in mind the small-scale problems. Accordingly, the dark gauge boson also becomes a DM candidate. The dark sector communicates with the SM sector through the Higgs portal interaction. The new scalar h 2 is in thermal equilibrium with the SM particles if the interaction rate is larger than the expansion rate of the universe. The annihilation amplitute of h 2 h 2 → bb is proportional to λ hs m b /m 2 h 1 so that the thermal equilibrium condition reads as [23] At T ∼ m ψ ∼ 100 GeV, it requires that λ hs > 10 −7 for the thermal equilibrium. In turn, it implies that sinθ s > 3.2 × 10 −9 for m h 2 = 10 MeV and v s = 2 GeV. As we will see in the next section, the constraint that h 2 lifetime should be smaller than 1 second requires larger sin θ s value (therefore, larger λ hs value) so that the thermal equilibrium condition is fulfilled. In this work, we require it is the case. Then the dark sector would be in thermal equilibrium with the SM sector at the early universe. The relic density of the dark fermion is determined by the annihilation process ψψ → A X A X at the time of thermal freezeout. There is also another annihilation process ψψ → A X h 2 , but its contribution to the relic density is negligible. The annihilation cross section × relative velocity is determined by the dark fermion mass m ψ and coupling g ψ ≡ g X X ψ . Note that g ψ is independent of g X because the dark U(1) charge X ψ can be chosen as we want. The relevant approximate formula for the process is given by [23] For m ψ = 100 GeV and g ψ = 0.2, it results in Ω ψ h 2 = 0.12. Then, the dark fermion constitutes the dominant DM component with the correct DM relic density which we observe today.
Besides the DM fermion ψ, we consider a case that the dark gauge boson A X constitutes the subdominant DM component whose relic density is much smaller than the dominant one. The annihilation process A X A X → h 2 h 2 at the early universe determines the relic density of the dark gauge boson. In this work, we will fix that m h 2 = 0.8 m A X to guarantee no kinematic suppression for the process, and require that the annihilation cross section × relative velocity is at least 10 times larger than the canonical value for the correct DM relic density in order to make the dark gauge boson A X be the subdominant DM.
With a light vector mediator A X , the DM fermion self-interacting cross section would be enhanced at a small DM velocity by the Sommerfeld enhancement. For each combination of the dark fermion ψ and the vector mediator A X mass, with the g ψ coupling value which is adjusted to give a correct relic density Ω ψ h 2 , the Sommerfeld enhancement factor can be calculated. It was shown that the DM self-interacting cross section could explain astrophysical observables at dwarf galaxy scale with the masses roughly in the following range [22]: We study particle phenomenology and future discovery potential of the new physics model in this mass range of m A X assuming m A X ∼ m h 2 . Since m ψ and g ψ do not affect the particle phenomenology discussed in this work, one can easily choose m ψ and g ψ to satisfy the current relic density observation.

SM-like Higgs decays
With the light new particles (m h 2 m A X m h 1 ), the SM-like Higgs h 1 has two additional decay modes: For Because the scalar mixing angle θ s in the limit of small mixing is given by we get following relations Then, the two decay rates are almost the same to each other, The decay rate for the exotic Higgs decay channels can be written in terms of the new physics parameters as The decay products of the exotic Higgs decays would appear invisible to the detector. Direct searches for invisible Higgs decays have been carried out with the ATLAS and the CMS detectors at the Large Hadron Collider (LHC). The recent results of the upper limit on the branching ratio of the invisible Higgs decays correspond to about 0.1 at 95% confidence level [24]. Here, we will demand that the branching ratio of the exotic Higgs decay, Br(h 1 → exo) is less than 0.1 for searching allowed parameter spaces of the two-component DM model.

Singlet-like Higgs decays
The decay pattern of the singlet-like Higgs h 2 is exactly the same as the SM-like Higgs case of the same mass. But the decay rates are suppressed by a overall factor of sin 2 θ s , compared to the SM-like Higgs case. For a light h 2 of order 10 MeV in mass, its dominant decay channel is to an electron-positron pair and the corresponding decay rate is It would give a rather large lifetime for a light singlet-like Higgs if sinθ s is very small. For instance, when m h 2 = 8 MeV and sinθ s = 2 × 10 −5 , h 2 lifetime is about 1.2 seconds. Such a large lifetime might spoil the successful Big Bang Nucleosynthesis (BBN). For the allowed model parameter space, we will require that h 2 lifetime should be less than 1 second.

Heavy meson decays
We may use rare B decays to search for new physics signals or constrain the new physics model parameters. We focus on B → A X A X decays and B → KA X A X decays, which would appear as B → invisible and B → K + invisible to detector, respectively. The effective Lagrangian relevant for the rare B decays [25] is where h = h 1 cosθ s − h 2 sinθ s , G F is the Fermi constant, and V tq is a relevant CKM matrix element for the process.
The matrix element squared for B 0 meson decays, where the Wilson coefficient For a small m A X m h 2 10 MeV, sinθ s = 0.0002, g X = 0.005, and f B = 190MeV, the branching ratio of the decays B 0 → A X A X is At present, BaBar collaboration has established the most stringent upper limit of 2.4 × 10 −5 at the 90% confidence level for the branching ratio of B 0 → invisible [26]. We require Br(B 0 → A X A X ) should be smaller than 2.4 × 10 −5 for constraining the new physics model parameters.

B → KA X A X , Kh 2 decays and B → K + invisible
For the three-body B decays, B(p B ) → K(p K )A X (k 1 )A X (k 2 ), with q ≡ p B − p K , the matrix element is given by (30) where the hadronic matrix element is related to a hadronic form factor as follows,

< K|s
Then, the decay rate for the process is obtained as where the matrix element squared is . For numerical calculation, we adopt the form factor f K 0 (q 2 ) as follows [27]: For a small m A X = 10 MeV, m h 2 = 0.8 m A X , sinθ s = 0.0002, g X = 0.005, the branching ratio of the decay B → KA X A X is (35) For m A X = 1 GeV, m h 2 = 0.8 m A X , and with the same values for sinθ s = 0.0002, g X = 0.005, the branching ratio is Br(B → KA X A X ) 1.8 × 10 −14 .
The branching ratio of B + → K + νν is predicted to be (4.6 ± 0.5) × 10 −6 in the SM. The search for the decays B + → K + νν was performed at the Belle II experiment. An upper limit on the branching ratio Br(B + → K + νν) of 4.1 × 10 −5 is set at the 90% confidence level [28]. We require that Br(B + → K + + invisible) should be less than 4.1 × 10 −5 .
If h 2 lifetime is large, it would be invisible to detector. In that case, the process B → Kh 2 also contributes to B → K + invisible decays. For B(p B ) → K(p K )h 2 (q) decays, the corresponding matrix element is and the decay rate is where q is the momentum of the final state particles in the B meson rest frame, For m h 2 = 10 MeV, and sinθ s = 0.0002, the branching ratio is about We will consider B → Kh 2 decays as B → K + invisible events if m h 2 < 2m µ .

Allowed parameter space and future prospects
We adopt free model parameters as follows: the singlet-like scalar mass m h 2 , the scalar mixing angle sinθ s , the dark gauge boson mass m A X , and the dark gauge coupling g X , in addition to the dark fermion mass m ψ and the dark U(1) charge X ψ of the dark fermion.
Here, the dark fermion mass m ψ and charge X ψ can be freely adjusted to achieve a right amount of the dark matter relic density as the dominant component of the dark matter. We fix the dark gauge boson mass as m A X = m h 2 /0.8 to have a sufficient annihilation cross section for A X A X → h 2 h 2 , in order to have a small enough relic density as the subdominant component of the dark matter. Then, only 3 free parameters remain: m h 2 , sinθ s , and g X . Now we investigate the allowed model parameter space which satisfy various experimental and phenomenological constraints, and study the possibility for finding new physics signals from future experiments. We show the experimental and phenomenological constraints and the sensitivity of SHiP [29], DUNE [30], and ILC [31] experiments on the parameter plane (m h 2 , sinθ) for the given g X values. Fig. 1(a) shows (m h 2 , sinθ) parameter space for g X = 0.005. On the figure, h 1 → exo line corresponds to the contour line for the branching ratio of the exotic Higgs decays, Br(h 1 → exo) = 10%. The region above the line is excluded by the current upper limit on Br(h 1 → exo) at the LHC. The (yellow-colored) BBN(τ h 2 > 1s) region on the figure corresponds to the parameter space where the lifetime of h 2 exceeding 1 second is predicted. The parameter region of BBN(τ h 2 > 1s) should be avoided for the successful BBN phenomenology. The lines of B → inv and B → K + inv correspond to the contour lines for Br(B → invisible) = 2.4 × 10 −5 and Br(B → K + invisible) = 4.1 × 10 −5 , respectively. The (shaded) regions above the lines are excluded by the current upper limits on the branching ratios at the B-factories.
For annihilation process A X A X → h 2 h 2 , the resulting cross section × relative velocity is proportional to g 4 X /m 2 A X for a fixed mass ratio r = m 2 h 2 /m 2 A X . With r = 0.64, we have density for the subdominant DM A X than the measured DM density of the Universe, in such a way that the relic density and the DM self-interaction cross section of ψ are exclusively determined by m ψ and g ψ . For g X = 0.005, it means that m A X should be smaller than about 0.2 GeV. In turn, it implies that m h 2 should be smaller than about 0.16 GeV, for the fixed r = 0.64. In Fig. 1, the vertical lines of A X A X → h 2 h 2 and m A X = 1 GeV indicate the upper limits on the dark scalar boson mass obtained from Eq. (40) and from Eq. (18) for m h 2 = 0.8m A X , respectively. After imposing current experimental bounds and phenomenological constraints, only small triangular shaped region remains allowed, where 2 MeV m h 2 16 MeV and 2 × 10 −5 sinθ s 3 × 10 −4 . On that figure, we also show the projected sensitivity of the SHiP and DUNE experiments. It indicates that some small part of the allowed parameter region with m h 2 ∼ 10 MeV and sinθ s ∼ 3 × 10 −4 would be probed by the future experiments. We also notice that, in the allowed parameter space, the branching ratio of the exotic Higgs decay is about 0.025 % Br(h 1 → exo) 10%, which might be probed by the future leptonic collider such as ILC whose sensitivity on the branching ratio of the invisible Higgs decay is expected to reach 0.3% level [31]. We also denote the corresponding contour lines for the ILC reach on the figures. Figs. 1(a), 1(b), and 1(c) show the same parameter spaces (m h 2 , sinθ s ) for g X = 0.01, 0.05, 0.1, respectively. By increasing the g X values, the h 1 → exo contour lines get lowered and therefore more parameter spaces are ruled out because the branching ratio of the exotic Higgs decays is proportional to (g X sinθ s /m A X ) 2 . The contour lines for the B → inv and B → K + inv decays get also lowered similarly to the h 1 → exo case. However, the constraints from the exotic Higgs decays are always more stringent than the one from the B decays. For the σ(A X A X → h 2 h 2 ) v rel , it is proportional to g 4 X /m 2 A X . Therefore, the corresponding contour lines of the lower limits on σ(A X A X → h 2 h 2 ) v rel are shifted to larger m A X values when g X increases. Hence, the allowed parameter spaces get broader by increasing g X value. On the other hand, the lifetime of the singlet-like Higgs h 2 does not depend on g X . Therefore, the excluded regions by the upper limit of h 2 lifetime are all the same for different g X .
For g X = 0.005, the SHiP and DUNE sensitivity regions barely touch the allowed parameter space of the model. On the other hand, for larger g X , rather large portion of the allowed parameter space would be explored by the SHiP and DUNE experiments. Also the future ILC experiments would explore a fair amount of the allowed parameter space through the exotic Higgs decays for all g X values.

Conclusions
We studied phenomenology of a two-component DM model, which would provide a solution for the small scale problems while avoiding the strong constraints from the direct DM detection experiments and the CMB observations. A new Dirac fermion introduced as a main component of DM in the Universe, which explains the observed relic density, does not directly interact with the SM particles, thus avoiding strong constraints from the direct DM detection experiments. On the other hand, the main DM component is self-interacting through a light dark gauge boson which plays a subdominant DM and connects with the SM sector via Higgs portal interaction. The self-interacting DM would solve the small scale problem such as the core-cusp problem. We investigated the model parameter space allowed by the current experimental constraints from the Higgs and B decays and the phenomenological bounds from the successful BBN and from the requirement of a large enough annihilation cross section of the dark gauge boson to make it a subdominant DM. We showed the allowed parameter space on the (m h 2 , sinθ s ) plane for various g X values. For g X = 0.005, the region of 2 MeV m h 2 16 MeV and 2 × 10 −5 sinθ s 3 × 10 −4 are allowed by the current experimental constraints and the phenomenological requirements. By increasing g X , both the minimum and the maximum of the allowed m h 2 values are shifted to larger values and the allowed region of sin θ s becomes broader.
We also discussed the sensitivity of the future experiments such as the SHiP, DUNE, and ILC for the obtained allowed parameter space. It turns out that large portion of the allowed parameter space in this model could be explored by the future experiments.