Direct Detection with Dark Mediators

We introduce dark mediator Dark matter (dmDM) where the dark and visible sectors are connected by at least one light mediator $\phi$ carrying the same dark charge that stabilizes DM. $\phi$ is coupled to the Standard Model via an operator $\bar q q \phi \phi^*/\Lambda$, and to dark matter via a Yukawa coupling $y_\chi \overline{\chi^c}\chi \phi$. Direct detection is realized as the $2\rightarrow3$ process $\chi N \rightarrow \bar \chi N \phi$ at tree-level for $m_\phi \lesssim 10 \ \mathrm{keV}$ and small Yukawa coupling, or alternatively as a loop-induced $2\rightarrow2$ process $\chi N \rightarrow \chi N$. We explore the direct-detection consequences of this scenario and find that a heavy $\mathcal{O}(100 \ \mathrm{GeV})$ dmDM candidate fakes different $\mathcal{O}(10 \ \mathrm{GeV})$ standard WIMPs in different experiments. Large portions of the dmDM parameter space are detectable above the irreducible neutrino background and not yet excluded by any bounds. Interestingly, for the $m_\phi$ range leading to novel direct detection phenomenology, dmDM is also a form of Self-Interacting Dark Matter (SIDM), which resolves inconsistencies between dwarf galaxy observations and numerical simulations.


Introduction
In this letter, we present Dark Mediator Dark Matter (dmDM) to address two important gaps in the DM literature: exploring mediators with dark charge, and non-standard interaction topologies for scattering off nuclei. Additional details and constraints will be explored in a companion paper [1].
The existence of dark matter is firmly established by many astrophysical and cosmological observations [2], but its mass and coupling to the Standard Model (SM) particles are still unknown. Weakly Interacting Massive Particles (WIMPs) are the most popular DM candidates since they arise in many theories beyond the SM, including supersymmetry, and may naturally give the correct relic abundance [3]. However, improved experimental constraints -from collider searches, indirect detection and direct detection [4,5,6,7,8,9,10,11] -begin to set tight limits (with some conflicting signal hints) on the standard WIMP scenario with a contact interaction to quarks. This makes it necessary to look for a more complete set of DM models which are theoretically motivated while giving unique experimental signatures. Left: the 2 → 3 process at tree-level. Right: the loop-induced 2 → 2 process. The arrows indicate flow of dark charge.

Dark Mediator Dark Matter
Given its apparently long lifetime, most models of DM include some symmetry under which the DM candidate is charged to make it stable. An interesting possibility is that not only the DM candidate, but also the mediator connecting it to the visible sector is charged under this dark symmetry. Such a 'dark mediator' φ could only couple to the SM fields in pairs.
As a simple example, consider real or complex SM singlet scalars φ i coupled to quarks, along with Yukawa couplings to a Dirac fermion DM candidate χ. The terms in the effective Lagrangian relevant for direct detection are where . . . stands for φ, χ mass terms, as well as the rest of the dark sector, which may be more complicated than this minimal setup. This interaction structure can be enforced by a Z 4 symmetry. To emphasize the new features of this model for direct detection, we focus on the minimal case with a single mediator n φ = 1 (omitting the i-index). However, the actual number of dark mediators is important for interpreting indirect constraints [1].
The leading order process for DM-nucleus scattering is χN →χN φ if m φ O(10 keV). However, an elastic scattering χN → χN is always present at loop-level since it satisfies all possible symmetries, see Fig. 1.
Which of the two possibilities dominates direct detection depends on the size of the Yukawa couplings y φi χ as well as the dark mediator masses.
Previous modifications to WIMP-nucleon scattering kinematics include the introduction of a mass splitting [12,13,14]; considering matrix elements |M| 2 with additional velocity-or momentum transfer suppressions (for a complete list see e.g. [15]), especially at low DM masses close to a GeV [16]; light scalar or 'dark photon' mediators (see e.g. [17] which give large enhancements at low nuclear recoil); various forms of composite dark matter [18,19,20,21,22] which may introduce additional form factors; DM-nucleus scattering with intermediate bound states [23] which enhances scattering in a narrow range of DM velocities; and induced nucleon decay in Asymmetric Dark Matter models [24]. Notably missing from this list are  alternative process topologies for DM-nucleus scattering. This omission is remedied by the dmDM scenario.
dmDM is uniquely favored to produce a detectable 2 → 3 scattering signal at direct detection experiments. This is because it contains two important ingredients: (1) a light mediator with non-derivative couplings to enhance the cross section, compensating for the large suppression of emitting a relativistic particle in a non-relativistic scattering process, and (2) a scalar as opposed to a vector mediator, allowing it to carry dark charge (without a derivative coupling). This imposes selection rules which make the 2 → 2 process subleading in y χ . These ingredients are difficult to consistently implement in other model constructions without violating constraints on light force carriers.
The effect of strong differences between proton and neutron coupling to DM have been explored by [25].
To concentrate on the kinematics we shall therefore assume the operatorqqφφ * /Λ is flavor-blind in the quark mass basis. Above the electroweak symmetry breaking scale this operator is realized asQ L Hq R φφ * /M 2 .
It can be generated by integrating out heavy vector-like quarks which couple to the SM and φ [1], giving This UV completion allows for large direct detection cross sections without being in conflict with collider bounds, but may be still probed at the 14 TeV LHC.

Nuclear Recoil Spectrum
We start by examining the novel 2 → 3 regime of dmDM. The DM-nucleus collision is inelastic, not by introducing a new mass scale like a splitting, but by virtue of the process topology. The nuclear recoil spectrum is different compared to previously explored scenarios. This is illustrated in Fig. 2, where we compare nuclear recoil spectra of standard WIMPs to dmDM for fixed velocity and different nucleus mass, before convolving with various form factors and the ambient DM speed distribution. The observable dmDM differential cross section is independent of m φ for m φ keV and can be well described by the function where same as the WIMP case for a given DM velocity. (We emphasize that this is a phenomenological description, the actual spectra were produced in MadGraph, see Section 5.) The first factor comes from the light mediator propagator (2m N E r ) −2 as well the integrated phase space of the escaping φ. The cross section suppression (second factor) is more pronounced as the DM becomes lighter or slower, and as the nucleus becomes heavier, both of which reduces E max r . This is because massless φ emission carries away a more significant fraction of the total collision energy if the heavy particle momenta are smaller. The maximum kinematically allowed nuclear recoil is then less likely.
When n φ = 1, the 2 → 2 process will dominate direct detection for Yukawa coupling y χ above some threshold, or if m φ 10 keV. For the purpose of calculating the matrix element, the loop diagram in Fig. 1 (right) is equivalent to the operator y 2 where q = √ 2 m N E r is the momentum transfer in the scattering. Effectively, this is identical to a standard WIMP with aχχN N contact operator, but with an additional 1/E r suppression in the cross section. This gives a similar phenomenology as a light mediator being exchanged at tree-level with derivative coupling.
Note that the relative importance of these two scattering processes is highly model dependent. For example, if n φ = 2 the dominant scalar-DM coupling could beqqφ 1 φ * 2 /Λ 12 . In that case, the 2 → 2 operator above is ∝ y φ1 χ y φ2 χ and can be suppressed without reducing the 2 → 3 rate by taking y φ2 χ y φ1 χ . The scattering behavior of both the 2 → 3 and 2 → 2 regimes necessitates a re-interpretation of all DM direct detection bounds. We will do this below.

Indirect Constraints
Direct detection experiments probe the ratio y χ /Λ and y 2 χ /Λ for 2 → 3 and 2 → 2 scattering respectively. However, indirect constraints on dmDM from cosmology, stellar astrophysics and collider experiments are sensitive to the Yukawa coupling and Λ separately. In [1] we conduct an extensive study of these bounds, including the first systematic exploration of constraints on theqqφφ * /Λ operator with light scalars φ. Since these constraints (in particular, Eqns. 4 and 5 below) provide important context for our results on direct detection, we summarize the two most important results here. For details we refer the reader to [1].
The scalar mediator(s) of dmDM are most stringently constrained from stellar astrophysics and cosmology: • Avoiding overclosure requires m φ eV [28], so we take the heaviest stable φ to be essentially massless, making it a very subdominant dark matter component. This also satisfies structure formation, computed for light sterile neutrinos in [29]. Measurements by the Planck satellite [2] restrict the number of light degrees of freedom during Big Bang Nucleosynthesis, enforcing the bound n φ ≤ 2 for real scalars.
• The coupling of φ to the SM is most constrained from stellar astrophysics. For n φ = 1, observational data on neutron star cooling essentially rules out any directly detectable dmDM model [1]. However, this bound is easily relaxed for n φ = 2 if m φ1 eV, m φ2 ∼ MeV, with a cosmologically unstable The dominant interaction to the SM is assumed to beqqφ 1 φ * 2 /Λ. In that case, φ 2 emission in the neutron star is Boltzmann suppressed due to its core temperature of T 100 keV, and φ 1 emission proceeds via a loop process. The bound on Λ is then weakened to Λ 10 TeV.
• In Supernovae, emission of light invisible particles can truncate the neutrino burst [30]. However, if these particles interact with the stellar medium more strongly than neutrinos they are trapped and do not leak away energy from the explosion. The temperature of supernovae T ∼ 10 MeV is large enough to produce φ 1 , φ 2 at tree-level in the above n φ = 2 scenario, and the scattering cross section with nuclei is much larger than for neutrinos if Λ 10 6 TeV. Therefore this setup is compatible with supernovae constraints. The dark matter yukawa coupling is constrained from observations on large scale structure and (under certain assumptions) from cosmology: • Dark matter self-interaction bounds from bullet cluster observations constrain the DM Yukawa coupling to be y χ 0.13(m χ / GeV) 3/4 [32].
• A thermal relic χ with Ω χ = Ω CDM requires if there is no significant φ 3 term. This also satisfies the above self-interaction bounds.

Direct Detection
We compute dmDM nuclear recoil spectra at direct detection experiments by simulating the parton-level process in MadGraph5 [26], and derive the event rates according to where f (v) is the local DM speed distribution (approximate Maxwell-Bolzmann with v 0 ≈ 220 km/s and a v esc ≈ 544 km/s cutoff, boosted into the earth frame v e ≈ 233 [41]), while ρ χ ≈ 0.3 GeV cm −3 is the local DM density [42], and N T is the target number density per kg. dσ N /dE r includes the usual Helm nuclear form factor [43,44], the A 2 coherent scattering enhancement as well as the quark-nucleon form factor for scalar interactions (see [45] for a review). We validated our Monte Carlo pipeline by reproducing analytically known 2 → 2 results. We can make this observation more concrete by mapping dmDM parameters to WIMP parameters. This is possible because both sets of nuclear recoil spectra look roughly like falling exponentials. For each dmDM spectrum with a given mass there is a closely matching WIMP spectrum with some different (lower) mass.
To find the m 2→2 corresponding to each m 2→3 we compare binned WIMP and dmDM distributions and minimize the total relative difference in each bin. The resulting mapping is shown in Fig. 4 1 We have confirmed the validity of this approach with full maximum likelihood fits [47]. are shown in Fig. 5 (left). We include the irreducible neutrino background [46] at the LUX experiment, which serves as an approximate lower border of the observable dmDM parameter space. An identical procedure can be used in the 2 → 2 dominant regime of dmDM. The translation map has similar qualitative features to the previous case since dσ/dE r ∼ E −1 r , except the faked WIMP signal corresponds to somewhat higher mass. The resulting direct detection bounds are shown in Fig. 5 (right).
The probability for any one 2 → 2 nuclear recoil event to lie above experimental detection threshold is much larger than for a 2 → 3 event, due to the less severe recoil suppression. For n φ = 1, this means the former will dominate direct detection unless m φ keV and the Yukawa coupling is very small, y χ 10 −3 < y relic χ . However, as discussed in Section 4, the neutron star cooling constraint requires at least n φ = 2. The 2 → 2 process could then be arbitrarily suppressed, allowing 2 → 3 direct detection with a thermal relic χ.
For the 2 → 3 and 2 → 2 scattering regimes, direct detection probes y χ /Λ and y 2 χ /Λ respectively. The neutron star cooling bound Λ 10 TeV and the bounds on dark matter Yukawa coupling y χ can be combined to be shown in the direct detection planes of Fig. 5. The assumption of a thermal relic then  There are large discoverable regions of dmDM parameter space that are not excluded. Due to the nontrivial dependence of the dmDM recoil spectrum on the target-and dark-matter masses and velocity, signals at several experiments will be needed to differentiate standard WIMPs from our model, but dmDM offers the realistic prospect of TeV-scale heavy quark discoveries pointing the way towards a sensitivity target for direct detection.

Conclusion
Dark Mediator Dark Matter introduces the possibility that dark matter interacts with the standard model via a mediator which also carries dark charge. This "Double-Dark Portal" adds the phenomenon of additional particle emission to the menu of possible interactions with nuclei, serving as an existence proof that this scattering topology can be realized. Direct detection experiments are starting to probe interesting regions of parameter space compatible with a thermal relic and neutron star bounds. For observationally relevant parameters, dmDM also acts as an implementation of SIDM [33,34,35,36,37,38,39,40], which can resolve various inconsistencies between many-body simulations and observations for dwarf galaxies. Even more than many other DM models, dmDM discovery is aided by lowering nuclear recoil thresholds. Further investigation is warranted and includes potential LHC signals, as well as possible leptophilic realizations of the model.