Direct Detection is testing Freeze-in

Dark Matter (DM) may belong to a hidden sector that is only feebly interacting with the Standard Model (SM) and may have never been in thermal equilibrium in the Early Universe. In this case, the observed abundance of dark matter particles could have built up through a process known as Freeze-in. We show that, for the first time, direct detection experiments are testing this DM production mechanism. This applies to scenarios where the SM and hidden sectors communicate through a light mediator particle of mass less than a few MeV. Through the exchange of such light mediator, the very same FIMP candidates can have self-interactions that are in the range required to address the small scale structure issues of collisionless cold dark matter.


I. INTRODUCTION
There is overwhelming evidence for dark matter (DM) in the Universe, but its precise nature still eludes us. A key question is to explain the observed abundance of DM and several mechanisms have been proposed. A particularly attractive possibility is that DM is a weakly interacting massive particle (WIMP) that was in thermal equilibrium in the Early Universe, down to a temperature at which its number density became so Boltzmann suppressed that it chemically decoupled from the thermal bath, a process known as Freeze-out (FO). The observed relic density is given by the population that was left at the time when DM freezed-out, leading to a DM abundance inversely proportional to the cross section responsible for chemical equilibrium where n DM is the DM number density and s the entropy density of the Universe. This cross section may be the DM annihilation cross section, or it may be effective, the relevant processes being co-annihilation of the DM with other exotic particles [1]. Regardless, agreement with cosmological observations requires that σv ≈ 3 × 10 −26 cm 3 /s, equivalent to a picobarn, characteristic of weak interactions. This has triggered a vast experimental program that aims at looking for a WIMP. In particular, direct detection experiments search for the recoil of nuclei from collisions with WIMPs from the local halo of DM [2, 3]. Recently, the XENON1T collaboration has released its latest data [4]. Their strongest constraints on DMnucleon spin-independent (SI) cross section, following a 1 tonne×year exposure, have reached σ SI ≈ 7 × 10 −47 cm 2 for a 30 GeVDM particle mass, setting a new landmark * thambye@ulb.ac.be † mtytgat@ulb.ac.be ‡ jvdecast@ulb.ac.be § lavdheyd@ulb.ac.be in the quest for a WIMP. In this short note, we put forward the fact that, for the first time, direct detection experiments, in particular XENON1T, have also reached the sensitivity required to test another mechanism for the DM abundance: Freeze-in (FI). The FI mechanism rests on the possibility that a DM particle may be only feebly interacting with the known SM, hence a feebly interacting massive particle (FIMP). Unlike a WIMP, a FIMP may have never been in thermal equilibrium in the Early Universe [5][6][7]. Instead, its abundance could have slowly built up through SM particles collisions, for example SM + SM → DM + DM pair creation. In this case As it is well known, the abundance of DM is typically a very small number, Y DM ≈ 4×10 −10 (GeV/m DM ). In the case of FO, this is explained by Boltzmann suppression of the DM abundance at the time of thermal decoupling.
In the case of FI mechanism, this is simply provided by a very small cross section for DM production. Indeed, focusing on the case of renormalizable interactions, the required coupling must be tiny, typically of order 10 −10 . As simple and natural as the freeze-in scenario may be, given the smallness of such coupling, one may wonder if we will ever be able to test it experimentally? We emphasize here the fact that FI is already being tested by XENON1T for the important class of FIMPs for which the SM and DM are coupled through a light mediator particle. The exchange of such a light mediator between DM particles can also induce large self-interaction, a possibility that is of much interest because it may alleviate the possible shortcomings of collisionless dark matter on small scales (core/cusp [8] and too-big-to-fail [9] problems, see [10] for a review). We show that indeed several FIMP candidates that can meet these additional issues are also being tested by XENON1T.

II. FRAMEWORK
We consider the slow, out-of-equilibrium production of DM from the thermal bath of SM particles during the arXiv:1807.05022v2 [hep-ph] 13 Aug 2018 radiation dominated era. Typical processes include DM pair production from one (SM → DM + DM or 1 → 2) or two SM particles (SM + SM → DM + DM or 2 → 2). The general FI scenario we refer to here is based on the assumption that these processes are mediated by particles lighter than the SM and DM particles involved (see [6,11] for non-thermal production through heavy mediators). In this case, the DM is essentially produced at a temperature T approximately equal to the mass of the heaviest particle involved, T FI ≡ max[m DM , m SM ] [5,7,12,13]. Hence, such FI is an infrared dominated production mechanism, which rests only on dynamics around T ∼ T FI . The fact that DM is dominantly produced around T FI stems from the T dependence of the rate for DM production (per unit volume), γ 1→2 and/or γ 2→2 , relative to the Hubble expansion rate H (times the entropy density). In particular, at T T F I the production rate decreases rapidly because of Boltzmann suppression: if m DM > m SM this is due to exponential suppression of the number of SM particles with energy large enough to produce DM particles; if instead m DM < m SM , it is the abundance of SM particles that is Boltzmann suppressed. The number of DM particles produced around T ∼ T F I is thus essentially determined by the rate of production at T FI times the age of the Universe at T FI , t FI ≈ 1/H(T FI ). For instance, for the case of annihilation, where n SM,eq is the abundance of the relevant SM particles at equilibrium. Observations require Y DM ≈ 4× 10 −10 (GeV/m DM ), a very small number that is easily achieved if the underlying interactions are feeble. For instance, if the cross section in (2.1) is quadratic in a coupling parameter, say, κ, then typically one needs κ = O(10 −10 ) as we will see in an explicit model below. Let us emphasize here that FI and FO are complementary mechanisms, corresponding to opposite corners of the parameter space of a DM model. Starting from FI, with a tiny effective coupling κ driving a 2 → 2 cross section, one can reach the FO regime by continuously increasing κ. The DM abundance first increases with κ, eventually overshooting the observed relic abundance, and does so as long as the 2 → 2 does not reach equilibrium. When κ is large enough, DM is in thermal equilibrium and one enters the FO regime, in which the relic density decreases with κ [5,13].
In the sequel, we focus on a class of FI scenarios with a rather light mediator, with mass below O(MeV). The reason that such FI scenarios can already be tested by current direct detection experiments stems from the fact, already emphasized in [13,14], that DM elastic collisions with nuclei proceed through the t-channel, which for a light enough mediator results in a boost of the cross section that can largely compensate the smallness of the coupling required for FI. To illustrate this, we consider the particularly simple model of millicharged DM. In its minimal version, millicharged DM consists of a single massive particle (χ in the sequel) charged under a new U (1) gauge symmetry, with fine structure constant α = e 2 /4π. Just like the electron in the SM, the stability of χ is guaranteed by gauge symmetry (see e.g. [15]). The χ field naturally couples to the SM through mixing of its gauge field strength tensor with the one of U (1) Y hypercharge gauge field a mechanism known as kinetic mixing [16]. Because of kinetic mixing, the χ particle (and its antiparticle) couples to the photon with a millicharge where tan θ = tan θ W / √ 1 − 2 and θ W is the Weinberg angle; κ is the parameter that controls the coupling between the DM and electrically charged SM particles, hence both FI production cross sections and the direct detection elastic collisions. This simple model is one instance of a generic scenario in which DM belongs to a hidden sector (HS) that is coupled with the SM through one of its so-called portals [17]. Clearly, the HS may be more complex, containing more than one massive particle (including a full mirror copy of the SM [18,19]). Also, the U (1) may or may not be spontaneously broken. In the latter case, the SM and HS particles interact feebly with each other through exchange of massive gauge bosons, one being the ordinary Z gauge boson, the other one being a new γ state, also called dark photon (see e.g. [13]).

III. RECASTING DIRECT DETECTION LIMITS
Direct detection experiments report their results in the plane m DM − σ DM,n where n stands for a nucleon (typically the neutron). Specifically, σ DM,n refers to a total elastic collision cross section, which is considered to be either spin-dependent or spin-independent. We focus on the latter because it provides the strongest constraints and it is the one relevant for the FIMP scenario described at the end of section II.
The XENON1T collaboration has published their so far strongest constraints on DM-nucleon SI cross sections following a 1 tonne×year exposure, pushing it down to 7 × 10 −47 cm 2 for a 30 GeV DM mass. The XENON1T limits have been derived assuming the scattering as being due to a short range contact interaction, mediated by a heavy mediator. Here we consider the possibility that the interaction between a DM particle and a nucleus N is mediated by a light γ , so as to lead to a long-range interaction, with t-channel propagator where m N is the nucleus mass (e.g. a Xenon nucleus) and E R is the recoil energy probed in direct detection experiments, which is typically 5 keV. If the DM-N differential cross section scales as 1/E 2 R , which can lead to a large enhancement of the collision rate at small E R . Because of this, the XENON1T results do not directly apply to the case of a light mediator. Nevertheless, it is possible to get rather conservative limits by recasting their constraints based on an E R independent cross section. To this end, we begin with the differential rate of collisions (the number of events per second per unit of recoil energy) with N T the number of targets, n DM the local number density of DM and f ⊕ ( v) its velocity distribution in the Earth frame, which we take to be Maxwellian with r.m.s velocity σ v = 270 km/s in the Galactic reference frame. In (3.3), dσ/dE R is the DM-N differential cross section for a given recoil energy E R . The integration is made on all velocities that are allowed for a given E R , that is to say satisfying v ≥ v min = m N E R /2µ 2 χN with µ χN the DM-N reduced mass. It is convenient to write dσ/dE R as [20] dσ where µ χp is the DM-proton reduced mass and Z the nucleus atomic number. The F (qr A ) is the nucleus form factor, which for concreteness we take from [20,21]. For a SI contact interaction and no isospin violation, we can simply set σ χp ≡ σ χn = σ DM,n and replacing Z by the mass number A. Instead, in the millicharged scenario and dark photon with mass m γ where α is the QED fine structure constant.
If (3.2) is satisfied, it is most natural to express the direct detection constraints in the plane m χ − κ. To do so, we can exploit two facts. First, while the differential rates through a light and massive mediator have distinct shapes for DM candidates of same mass, they can be similar for distinct DM masses, at least within a given recoil energy E R range. Second, due to detection efficiency, nuclear form factor and velocity distribution, the differential rates fall rapidly at low and high recoil energies and this regardless of the type of interactions, so only a finite range of E R is relevant for each candidates. These allow to map (m DM , σ DM,n ) onto (m χ , κ). In practice, Blue solid: best fit for a DM candidate with long range interactions with (mχ, κ) = 68 GeV, 3 × 10 −11 . The error is ∆ ≈ 20%. The dashed curves are for the same candidates, but not taking into account XENON1T efficiency.
we take the upper limit on σ DM,n from XENON1T corresponding to a given DM candidate, to get a proxy for the observable differential rate, where (E R ) is the detector efficiency from Fig. 1 in [4] and with dR/dE R given by (3.3) and (3.4) with a constant cross section σ DM,n and Z → A. For this candidate (m DM , σ DM,n ), we then determine the couple (m χ , κ) that would have a similar observable differential rate (dR χ /dE R ) exp . Concretely, we consider the pair (m χ , κ) that minimizes the quadratic distance between the two rates where R exp is the total measurable rate from (3.6) and dR χ /dE R is the differential rate corresponding to a candidate with a light mediator.
This procedure is illustrated by Fig. 1. The dashed red line represents the differential rate obtained for a constant cross section (σ DM,n = 10 −46 cm 2 ) and a DM mass (m DM = 15 GeV) taken from the XENON1T upper bound. The dashed blue line is the differential rate obtained from the cross section (3.5) that minimizes ∆, here corresponding to the candidate (m χ , κ) = 68 GeV, 3 × 10 −11 . The solid lines correspond to the observable differential rates, taking into account efficiency. We emphasize that the matching is dominated by low recoil energies E R < 10 keV corresponding to higher expected event numbers. In this example, ∆ ≈ 20%. Over all the DM mass range we consider, ∆ never exceeds O(30%)

IV. FREEZE-IN VS DIRECT DETECTION
Scanning over the XENON1T exclusion limits, the solid black line in Fig. 2 gives the upper limits on the coupling κ as a function of the DM mass m χ . We emphasize that these limits only use the XENON1T constraints in the range 1 GeV m DM 50 GeV. This stems from the ∝ 1/E 2 R behavior of the cross section, leading to events in the low E R region, which for a heavy mediator corresponds to relatively lighter DM candidates. In the same Fig. 2, the solid green line gives the κ corresponding to the observed DM abundance, along the FI scenario depicted in section II (see [13]). In the millicharged model, FI is set by annihilation of SM particles into DM pairs, ff /W + W − → χχ, and by Z decay, Z → χχ [13,22]. The dip at m χ m Z /2 corresponds to Z decay threshold, m χ ≤ m Z /2. Fig. 2 reveals that XENON1T is testing for the first time a FI scenario, excluding millicharged FIMP candidates within the 45 GeV < m DM < 100 GeV range. We also show the limits from the 2017 PANDAXII results [23], following the same procedure we used for recasting XENON1T data. PANDAXII limits almost reach the FI parameter range. Finally, we show the prospects for XENON1T for 4 years of exposure and for the future LZ experiment [24] (for 1000 days). XENON1T should probe the millicharged FI scenario for m χ from 45 GeV up to ∼ 400 GeV, whereas LZ could test it all the way from m DM ∼ 15 GeV to ∼ 4 TeV.
We have so far neglected the mass of the dark photon, an approximation which is valid as long as m γ √ 2m N E R ∼ 40 MeV, taking E R to be around 5 keV as typical recoil energy, see Fig. 1. Thus for a MeV dark photon, our results still apply. As soon as m γ 40 MeV, the collinear enhancement is lost, which results in less stringent constraints on κ and thus no direct detection test of FI. 1 Indeed, the FI abundance itself is insensitive to the mass of the dark photon, at least provided m γ ≤ 2 m χ , values that lie outside the parameter range we consider here.

V. SELF-INTERACTING DARK MATTER
A massless dark photon leads to infinite range forces, both between DM particles and between DM and ordinary matter. Such interactions could be in conflict  with observations of the dynamics of galaxies [25,26]. Also, a millicharged DM would be deflected by the magnetic field in galaxies [27][28][29] or clusters [30]. In particular, a millicharged particle from the DM halo could be repelled by the coherent magnetic field in our Galaxy, which could deplete the local abundance of DM, thus decreasing the reach of direct detection experiments [27]. A massive dark photon in the range O(MeV) circumvents all these potential complications [13]. This may also be a blessing as a ∼ MeV mediator can induce a large self-interaction cross section (noted here σ T ). Provided 0.1cm 2 /g < σ T /m χ < 10cm 2 /g, this could alleviate the small scale structure issues of collisionless cold dark matter [10,31]. Fig. 3 shows the candidates that satisfy this requirement for α = 5 × 10 −5 and setting the velocity of DM particles to 10 km/s, relevant for the core/cusp problem in dwarf spheroidal galaxies [10]. This value of α is chosen so that over all the mass we consider, the candidates are in the freeze-in regime (see Ref. [13]). The black solid line encloses FIMP candidates that XENON1T is currently testing and the dashed lines are for XENON1T with 4 years of exposure (black) and LZ for 10 3 days (blue). Here, we show both the attractive (top) and repulsive (bottom) channels. Clearly, there is a large overlap between the regions of m χ − m γ satisfying the selfinteractions constraint, satisfying the relic density constraint through FI, and that can be tested by current and future direct detection experiments. We emphasize that self-interacting DM scenarios based on FO and a light mediator are severely constrained by CMB data [14,32]. This is unlike the FI considered here, which is basically showing DM candidates with attractive (top) and repulsive (bottom) self-interaction that can alleviate the smallscale structure issues. Red: 1 cm 2 /g ≤ σT /mχ ≤ 10 cm 2 /g; Blue: 0.1 cm 2 /g ≤ σT /mχ ≤ 1 cm 2 /g. The lines delimitate the FIMP region probed by the current XENON1T (black solid), XENON1T after 4 years of exposure (black dashed) and the future LZ (blue dashed).
unconstrained, as it relies on much smaller couplings. For such small coupling α , one may check (see Ref. [13]) that the hidden sector never thermalizes and that the only way to reach the cosmic abundance is through the Freeze-in mechanism.
For a larger value of the coupling α and for a non too larger DM mass, the HS thermalizes and the relic abundance cannot be obtained through the FI regime. One is instead in the so-called reannihilation regime [12]. This regime corresponds to a situation in which interactions within the hidden sector are fast compared to the expansion rate, while the energy transfer from the visible sector is slow, so that the temperature of the hidden sector is less than that of the visible sector, see [12] for more details. We show in Fig.4 the candidates that alleviate the small scale structure problem for α = 10 −3 . In this case, the current XENON1T limits are only probing reannihilation cases. However, Xenon1T should start probing the FI regime after 4 years of exposure and this in the range between m χ ≈ 400 GeV and m χ ≈ 480 GeV. LZ experiment will test FI from m χ ≈ 400 GeV up to 4 TeV. showing DM candidates with attractive (top) and repulsive (bottom) self-interaction that can alleviate the small-scale structure issues. Red: 1 cm 2 /g ≤ σT /mχ ≤ 10 cm 2 /g; Blue: 0.1 cm 2 /g ≤ σT /mχ ≤ 1 cm 2 /g. The lines delimitate the FIMP region probed by XENON1T after 4 years of exposure (black dashed) and the future LZ (blue dashed). The shaded region corresponds to the reannihilation region which is already probed by XENON1T experiment. We emphasize that another example of scenario with a light mediator we could have considered to show that direct detection is testing FI, with possible connections to self-interactions, is the one of Ref. [14], based on the Higgs rather than the kinetic mixing portal. 2 Other works are dealing with direct detection and selfinteracting DM, see e.g. [33] and/or dark photons, see e.g. [34], but with a distinct emphasis. The latter deals with DM in the form of dark photons. The former rests on the necessity to suppress, through weak decay into SM particles, the cosmic abundance of dark photons to avoid over-closure of the Universe. We stress that our millicharged FI scenario is meant to be illustrative. We could, for instance, have considered a scenario based on Higgs portal, like in [14] (see also [35,36]), instead of kinetic mixing to show that direct detection is testing FI, with possible connections to self-interactions.

VI. CONCLUSIONS
In this work, we have brought together three corners of DM phenonemology: the FI mechanism for DM creation in the Early Universe, direct detection of DM from the galactic halo and self-interacting DM as a way to address the small scale structures of collisionless DM. In particular, we have shown that XENON1T data, based on 1 tonne× year exposure, is testing for the first time the parameter space of a very simple scenario of FI, based on millicharged DM coupled to the SM through a dark photon with mass below ∼ 40 MeV. The very same dark photon may induce very large self-interaction for DM, in a range 0.1 cm 2 /g < σ T /m DM < 10 cm 2 /g. Such FI scenario has several interesting features, and is currently less constrained than the case of FO. Similar results could be anticipated also for other light mediator FI scenarios, such as based on a Higgs portal.
Our main result can be read from Fig. 2 where we confront, in the plane m χ − κ, FIMP candidates with an abundance that matches cosmological observations, to the exclusion limits from the XENON1T and PANDAX II direct detection experiments, including the reach of LZ (for 1000 days) and XENON1T (for 4 years). These limits have been obtained by recasting the constraints from these experiments on the DM-nucleon SI elastic collision cross section based on a contact interaction, or equivalently mediated by a heavy particle. The approach we have used is straightforward, but is only approximate and does not take full account of the data available to the DM direct detection experimentalists. Given the relevance of the FI mechanism and of its possible relation with selfinteracting DM, we believe that it would be important that direction detection collaborations provide in the future their own bounds on scenarios involving a collision cross section proportional to 1/E 2 R .

ACKNOWLEDGMENTS
This work is supported by the FRIA, the "Probing dark matter with neutrinos" ULB-ARC convention, the IISN convention 4.4503.15 and the Excellence of Science (EoS) convention 30820817. Drukier, G. Gelmini and D. N. Spergel, Limits on Cold 2 A difference though between Higgs and kinetic mixing portal based scenarios, concerns the fact that Higgs portal scenarios are more constrained by Big Bang Nucleosynthesis (BBN) than kinetic mixing scenarios. In the former case, a light mediator in the ∼ 1-100 MeV range might be excluded [14] because in this case the decay of the light mediator into an electron and a positron is still active during BBN (as a result of the electron Yukawa coupling suppression of this decay). For the kinetic mixing scenario it can be checked that the γ which are produced through FI have in general already decayed away by the BBN time. Thus to determine if for a scalar portal there exists a region combining FI, self-interactions and direct detection testability (as we found above for the kinetic mixing case) is still to be investigated further.