PHYSICAL REVIEW D 96, 115021 (2017) Light dark matter: Models and constraints

Simon Knapen,1,2 Tongyan Lin,1,2,3 and Kathryn M. Zurek1,2 1Theory Group, Lawrence Berkeley National Laboratory, Berkeley, California 94709 USA 2Berkeley Center for Theoretical Physics, University of California, Berkeley, California 94709 USA 3Department of Physics, University of California, San Diego, California 92093, USA (Received 30 October 2017; published 26 December 2017) We study the direct detection prospects for a representative set of simplified models of sub-GeV dark matter (DM), accounting for existing terrestrial, astrophysical and cosmological constraints. We focus on dark matter lighter than an MeV, where these constraints are most stringent, and find three scenarios with accessible direct detection cross sections: (i) DM interacting via an ultralight kinetically mixed dark photon, (ii) a DM subcomponent interacting with nucleons or electrons through a light scalar or vector mediator, and (iii) DM coupled with nucleons via a mediator heavier than ∼100 keV.

DOI: 10.1103/PhysRevD.96.115021

I. INTRODUCTION with semiconductors [40,41], atoms [42], graphene [43], and scintillators [44]. In recent years, new ideas to search for dark matter (DM) When the DM mass is below a MeV, new targets and have changed the direct detection landscape. As highly detection techniques sensitive to meV energy depositions sensitive searches for the weakly interacting massive – particle (such as LUX [1], PandaX-II [2], XENON1T must be sought. Superconductors [45 47] and Dirac [3] and SuperCDMS [4]) have not yet turned up a signal, materials [48] have been proposed to detect sub-MeV the urgency to develop techniques to search for DM outside DM with coupling to electrons, while superfluid helium of the ∼10 GeV–10 TeV mass range has increased. [49,50], though multiphonon and multi-roton excitation, Outside of this window, there are theoretically well- can have sensitivity to such light DM with coupling to motivated candidates that are consistent with the observed nucleons. The same experiments are sensitive to small history of our Universe. DM may reside in a hidden sector energy depositions can also detect bosonic DM (produced – communicating via a mediator coupled to both sectors (see nonthermally) via absorption [47,51 54]. (See Ref. [55] for for example [5–10]). The relic abundance can be fixed in a a recent summary of dark matter detection proposals.) variety of ways, including via particle-antiparticle asym- Direct detection experiments do not, however, exist in metry as in asymmetric DM [11–13], strong dynamics isolation from other types of probes. If the DM particle can [14,15], freeze-in [16,17], or other thermal histories for be probed via electron or nucleon couplings in direct example [18–24]. Cosmology (via structure formation detection experiments, this necessarily implies the presence bounds on warm DM) indicates that such DM candidates, of other constraints from the early Universe, structure populated through thermal contact with the standard model formation in the late Universe, as well as laboratory probes. (SM) at some epoch, must be heavier than ∼few keV to tens Most of these constraints are model dependent in one way of keV [25,26]. Thus the keV–10 GeV mass window is a or another, and to get a sense of their relation to the size of natural place to consider searching for light DM. the couplings probed in direct detection experiments, one Because of the existence of well-motivated candidates must commit to a set of benchmark models. Our goal in this (such as asymmetric DM) pointing to the GeV scale, a work is to present a unified and complete picture of these number of direct detection experiments targeting WIMPs constraints on a variety of simplified models for sub-GeV through nuclear recoils have also been actively working to DM, for interactions with both electrons and nucleons. increase their sensitivity to smaller energy depositions For DM having mass between an MeV and a GeV, the and lighter DM candidates. Such experiments include importance of cosmological history and terrestrial con- CRESST-II [27], DAMIC [28], NEWS-G [29], PICO straints was highlighted by for instance in Refs. [56–59]. [30], SENSEI [31], and SuperCDMS [32–35]. In addition, When the DM becomes lighter than an MeV, a tapestry of there are a number of recent proposals to detect nuclear DM self-interaction, stellar emission, Neff and terrestrial recoils of sub-GeV dark matter, such as liquid helium constraints comes to the fore. When the DM has an electron detectors [36], bond breaking in molecules [37], and defect coupling, some of these constraints were discussed in creation in crystal lattices [38,39]. Furthermore, for DM Ref. [45] and considered more extensively in Ref. [46,60], masses in the MeV-GeV range, the largest energy depo- while scalar-mediated nucleon couplings have also been sitions are typically achieved not via nucleon interactions, discussed in Ref. [60]. but via scattering off electrons. New detection methods In all our benchmark models, the DM interacts with the sensitive to electron interactions have thus been proposed SM via the exchange of a mediator (whether scalar or

2470-0010=2017=96(11)=115021(26) 115021-1 © 2017 American Physical Society KNAPEN, LIN, and ZUREK PHYSICAL REVIEW D 96, 115021 (2017) vector) in the t-channel. The mediator will be constrained Massless mediator: When the mediator mass is below by virtue of the fact that it couples to SM particles. The me, stellar constraints generally put a very strong bound on scattering cross section in direct detection experiments is αT, such that only mϕ ≪ 1 MeV gives a realistic direct typically parametrized by detection cross section. For example, when mϕ < 100 keV, −26 the constraints require αn ≲ 8 × 10 for scalar nucleon 4πα αχ σ¯ ∼ T μ2 ; ð Þ couplings. Stellar constraints in the massless regime are, DD ð 2 þ q2Þ2 Tχ 1 mϕ 0 however, strongly model dependent—a kinetically mixed dark photon, for example, has a much smaller production in μ χ where Tχ is the reduced mass of the DM with the target the star than a scalar. For a light mediator, DM self- (whether electron or nucleon), and q0 is a reference value interactions also become important; for example, the for the momentum transfer. (Precise equations and con- constraint on the coupling αχ is ventions will be established for each benchmark model in 3=2 the corresponding section.) The scattering is defined by the −10 mχ αχ ≲ 6 × 10 × ð4Þ light or heavy mediator regimes, when the momentum 1 MeV transfer q is much smaller or much larger than the mediator mass mϕ. In these regimes, the scattering cross section is ≈ 10 ≈ 10−3 for mχvDM=mϕ and vDM . However, this con- χ 2 4 straint is much weaker if we consider a relic which is only α αχ μ χ keV σmassless ≃ 1 × 10−39 cm2 T T a subcomponent of all the DM. DD 10−30 m q e Here, we study three broad classes of simplified models: 2 4 α αχ μ χ 5 MeV (i) Real scalar dark matter coupled to nucleons σmassive ≃ 2 10−40 2 T T DD × cm −16 : through a hadrophilic scalar. A scalar mediator 10 me mϕ interacting with nucleons can be generated by the ð2Þ scalar coupling to top quarks or to heavy colored vector-like fermions. We show the corresponding For light mediators, even with small couplings the rate is constraints in Fig. 1. Models of this type also potentially observable in a direct detection experiment generate mediator-pion interactions, which are not sensitive to low momentum-transfer scattering. Clearly, relevant for direct detection but do matter for the to understand the parameter space for direct detection, thermal history of the universe. We find, generally, we must understand the nature of the constraints on the that there are two parameter regimes where sub- α couplings of the mediator to the DM, χ, and of the MeV DM may be detectable in a low threshold α mediator to the target, T. The relevant constraints depend experiment (e.g. a superfluid helium target [49,50]) of course on the mass and spin of the mediator, and whether with a kg-year exposure. the couplings are predominantly to nucleons, electrons or In the first case, the DM scatters via a very light both. The interplay between the various bounds is best mediator (typically keV mass or lighter) having understood in terms of the mediator mass regime in which small enough couplings such that it does not cool they dominate. stars. For such small couplings the mediator also Massive mediator: In the massive mediator regime decouples from the SM thermal bath before the QCD (mϕ ≳ 1 MeV), stellar constraints are absent or substan- phase transition, leading to a contribution of the Δ tially reduced such that the couplings of the mediator to the mediator to Neff of at most: α ∼ 10−9 target can be as large as T . The remaining bounds on these couplings primarily come from rare meson 4 ð Þ 4=3 Δ ≈ gSM Tνdec ≈ 0 06 ð Þ processes (such as B → Kϕ) and beam dump experiments. Neff 7 ð Þ . 5 gSM TQCD DM self-interaction bounds place a mild limit on αχ, with g ðTν Þ and g ðT Þ the number of 1 2 2 SM dec SM QCD 1 keV = mϕ degrees of freedom in the SM thermal bath at αχ ≲ 0.02 : ð3Þ mχ 1 MeV neutrino decoupling and before the QCD phase transition, respectively. Note that this value of α α Δ For realistic direct detection cross sections, χ and T are Neff can be tested by CMB S4 experiments however large enough that the mediator and the DM are [61]. In order to have large enough direct detection generally in thermal equilibrium with the SM in the early cross sections, we consider a subcomponent of the universe. This can lead to a contribution to the number of DM so as to evade constraints from DM self- relativistic degrees of freedom (parameterized in terms of interactions, as can be seen in Fig. 5. number of effective neutrino species Neff), with constraints The second case is where the mediator is fairly from big bang nucleosynthesis (BBN) and the cosmic massive (typically in the 100 keV to 1 MeV mass microwave background (CMB). range). Here, for detectable cross sections, the

115021-2 LIGHT DARK MATTER: MODELS AND CONSTRAINTS PHYSICAL REVIEW D 96, 115021 (2017) 3 10 when the mediator is heavier. Bounds are shown in Fig. 7. (As such, this model shares many qualitative features with the Higgs-portal dark matter model 10 6 B K Xe 1 n K considered in Ref. [56] for mDM > MeV.) Similar

5th force to the nucleon case, for sub-MeV DM scattering via

n 9 a light mediator we find sizable direct detection

y 10 SN1987a cross sections only for a subcomponent of the total HB stars DM; this is illustrated in Fig. 9. For the massive RG stars 10 12 mediator case, BBN constraints from thermalization of the mediator via its couplings to the electron are tt particularly strong due to improved deuterium mea- 10 15 surements [63]. As shown in Fig. 10, this thermal- eV keV MeV GeV ization consideration implies that a low-threshold m experiment (such as a superconducting detector with 10 3 B K kg-year exposure [45,46]) will not have sensitivity to

K sub-MeV DM scattering via a massive mediator. (iii) Dirac fermion dark matter, coupled to a kinetically 10 6 − n Xe mixed dark photon or a B L gauge boson.

5th force Because of strong constraints from BBN, we con-

n 9 sider only the case of a light mediator with suffi-

y 10 SN1987a ciently small couplings that it does not thermalize HB stars with the SM. As has been noted elsewhere [41,48], RG stars 10 12 and shown explicitly in Fig. 11, scattering via a kinetically mixed dark photon is consistent with all G G current bounds in the 10 keV—1 GeV mass range. 15 10 For mχ ≲ 1 MeV, such DM can be probed with eV keV MeV GeV Dirac materials and superconductors, although in the m latter case the reach is substantially reduced due to 1 FIG. 1. Constraints on a sub-GeV scalar mediator, given in in-medium effects. For mχ > MeV various other targets have sensitivity, in particular semiconduc- terms of the effective scalar-nucleon coupling yn. In the top panel, we assume that the nucleon interaction is generated by a ϕ-top tors. For the B − L gauge boson, there are strong coupling and in the bottom panel, we assume it is generated by a fifth force and stellar constraints, as can be seen in ϕ-gluon coupling (for instance from a heavy colored fermion). Fig. 13. Scattering of sub-MeV DM through such a We show limits from fifth force [64] and neutron scattering mediator would be detectable by either a superfluid searches [65] (orange), rare meson decays (green), and stellar helium or a Dirac material target, but only for cooling limits from HB stars [66] (red), RG stars [66] (purple) subcomponent DM that evades self-interaction con- and SN1987A (blue). straints, shown in Fig. 13. This paper is organized to consider each of these models in turn: scalar DM coupling to a hadrophilic scalar mediator in coupling is large enough that the mediator tends to Sec. II, to a leptophilic scalar mediator in Sec. III, and DM thermalize with the pions in the early universe, scattering via a vector mediator in Sec. IV. We summarize giving rise to ΔN ≈ 4=7 for a sub-MeV mediator. eff the results in Sec. V. We emphasize that current prospects This value is in tension at the 2σ level with the N eff are often dictated by the capabilities of particular target derived from recently improved measurements of materials for detection of sub-MeV dark matter—in the the deuterium abundance [62,63]. The scenario case of nucleon couplings, superfluid helium and in the where the mediator and the dark matter both case of electron couplings, superconductors and Dirac thermalize with the SM is moreover firmly excluded, materials. The general considerations studied in this paper and we must place a strong limit on αχ to avoid motivate the search for materials with even stronger thermalization of the DM with the mediator. The sensitivity to light dark matter. resulting parameter space is shown in Fig. 6. (ii) Real scalar dark matter coupled to an leptophilic II. HADROPHILIC SCALAR MEDIATOR scalar mediator. Similar to the nucleon case, a mediator coupling to the electrons is constrained In our first model we assume a hadrophilic scalar by fifth force experiments and stellar cooling argu- mediator ϕ has couplings exclusively to the SM hadrons, ments when the mediator is light, and predominantly specifically pions and nucleons. The coupling to nucleons by beam dump and other accelerator experiments is the most consequential for direct detection and the

115021-3 KNAPEN, LIN, and ZUREK PHYSICAL REVIEW D 96, 115021 (2017) majority of the constraints, and we therefore parametrize decays, which are primarily sensitive to the coupling of the model in terms of the low energy effective Lagrangian ϕ with top quarks. (1) First consider a coupling to top quarks of the form 1 1 1 L ⊃− 2χ2 − 2 ϕ2 − ϕχ2 − ϕ¯ ð Þ mχ mϕ yχmχ yn nn; 6 ϵ m 2 2 2 L ⊃ pffiffiffi t ϕtt¯ ð8Þ 2 v where χ is a real scalar that composes all or part of the DM. On its own, this potential has runaway directions, but with v ¼ 246 GeV. This in turn induces the gluon these can be stabilized by adding in quartic couplings for coupling the scalars without affecting the dark matter phenome- α 1 ϵ s a aμν nology. One may further verify that with the normali- L ⊃ ϕGμνG with ¼ ; ð9Þ 4Λ Λ 3πv zation in Eq. (6), the perturbativity condition is yχ ≲ 4π for mϕ ≪ mχ, although we will conservatively require yχ < 1. which at low energies maps to a nucleon coupling This will be relevant for the light mediator regime, where 2 we will consider subcomponent dark matter such that the mn −4 L ⊃ ynϕnn¯ with yn ¼ −ϵ ≈−2.6 × 10 ϵ; self-interaction constraints are relaxed. We further elaborate 3bv on unitarity and vacuum stability in Appendix A. ð10Þ In order to account for both the heavy and light mediator limits, we will parametrize the direct detection cross section where b ¼ 29=3 is the first coefficient of the QCD on nucleons as β-function. Here we assumed that mϕ is below the strange quark mass, and neglected the small light 2 2 2 y yχ μχ σ ≡ n n ð Þ quark contributions to the nucleon mass [70]. While n 4π ð 2 þ 2 2Þ2 ; 7 we do not explicitly consider couplings to the lighter mϕ vDMmχ quarks, they would not substantially change the μ ϕ where χn is the DM-nucleon reduced mass, and mχvDM is constraints, provided that does not mediate a ¼ 10−3 a reference momentum transfer with vDM . The dark large flavor-changing interaction. For example, a matter scattering “form factors” (as defined, for example, model where ϕ couples to the SM quarks through in Ref. [40]) in the heavy and light mediator limits are minimal flavor violation (MFV) would be consistent 2ð 2Þ¼1 2ð 2Þ¼ð Þ4 4 F q and F q vDMmχ =q , respectively. with our setup, though we here assume that the With the normalization of the Yukawa coupling in coupling to leptons can be neglected. Eq. (6), the direct detection cross section also has the (2) Alternatively, if the mediator couples to a colored same scaling as for fermionic χ. The main difference in our vectorlike generation, we also generate the gluon results comes in when we consider the cosmology and the interaction in Eq. (9), where Λ is now a function of the mass (m ) and coupling of the heavy generation. contributions of the dark sector to Neff. As we will see, both Q the scalar and the fermionic DM case are in tension with We assume for simplicity that these additional BBN measurements if they are in equilibrium with the SM particles are outside the reach of the LHC, with below the QCD phase transition. If the dark sector mQ ≈ 5 TeV and Λ a free parameter. The map to the decouples before the QCD phase transition, the resulting low energy theory is given by N depends on the number of degrees of freedom of χ and eff 2π m m could be observable with CMB Stage IV. y ¼ − n ≈−0.65 n : ð11Þ n b Λ Λ Below we outline the primary constraints on yn and yχ arising from meson decays, fifth-force experiments, stellar Although there is no direct coupling to the top emission, and DM self-interactions. We then turn to con- quark at tree-level, Eq. (8) is still generated radia- straints from cosmology, which are sensitive to therma- tively and can lead to rare meson decays. The lization of the mediator and/or the DM. For mχ between leading contribution is 100 MeV and 1 GeV, this model can be probed in direct detection experiments such as CRESST [27], SuperCDMS pffiffiffi α2 m2 ϵ ≈ 2 s Q v ð Þ [34] and NEWS [29,67]. In addition, for mχ ≪ 1 GeV, this log 2 : 12 π m Λ model could be accessible to proposed low-threshold experi- t ments with superfluid helium [36,49,50,68], crystal defect Similar couplings to the lighter quarks are generated techniques [37,39,69] and magnetic bubble chambers [53]. as well, but are less relevant to the meson constraints In this work, we consider two possible origins of the we consider here. This induced coupling to the top nucleon interaction in Eq. (6): one with the mediator quark can also be written as ϵ ≈−17yn. coupling to the top, and one with the mediator coupling As we will see, the meson constraints depend on ϵ rather to a vectorlike generation of heavy, colored particles. This than yn. For a fixed value of yn, they are therefore weaker in distinction is only important for bounds from meson the model with a vectorlike generation. The compiled

115021-4 LIGHT DARK MATTER: MODELS AND CONSTRAINTS PHYSICAL REVIEW D 96, 115021 (2017) constraints on ϕ for these two scenarios are summarized in The width for K → πϕ follows the same expression, with Fig. 1, and are described in detail below. the appropriate substitution of the masses and Cabibbo- Kobayashi-Maskawa (CKM) matrix elements. The form ð Þ ≈ 1 A. Terrestrial constraints factor for this process is well approximated by f0 q in the low q limit [75]. For each scenario, the resulting bounds ϵ ≪ 1. Meson constraints on can then be converted to a bound on yn.Formϕ mπ, the strongest bounds come from the K → πϕ process: A light scalar with a coupling to hadrons appears → ϕ → πϕ −8 in exotic meson decays, like B K and K . yn ≲ 4.2 × 10 ðtop couplingÞð17Þ Generally, the dominant contribution to these decay rates −6 comes from a top-W loop (see Fig. 2), which explains the yn ≲ 9.3 × 10 ðvectorlike generationÞ: ð18Þ need to specify the origin of the nucleon coupling in Eq. (6). The constraints are shown in green on Fig. 1. In both models outlined above, the (indirect) coupling to the top quark opens up the radiative decay ϕ → γγ, with 2. 5th force constraints width of For mediator masses below ∼100 eV, various 5th force experiments become relevant (see e.g. [64,76] for recent 4 2α2 m3 Γ ¼ qt Nc ϕ ϵ2 ð Þ reviews.). This is important in particular for the cosmo- ϕ→γγ 144π3 2 13 v logical fate of ϕ: at least for values of yn that can give rise to detectable values of the direct detection cross section, 5th ¼ 2 3 ¼ 3 with qt = and Nc . In the absence of a competing force constraints prevent ϕ from being light enough to ϕ decay mode, this results in a lifetime for of behave as dark radiation at late times [45,60]. We will comment more on the cosmology in Sec. II C. 3 1 τ ≈ MeV 108 ð Þ At low energies, the presence of the massive, scalar c × 2 × cm: 14 mϕ ϵ mediator introduces an attractive Yukawa potential between two macroscopic objects, of the form We will therefore always treat ϕ as missing energy in these y2 1 decays, regardless of whether it decays to photons or to an ð Þ¼− n −mϕr ð Þ V r 4π e 19 invisible state (e.g. the DM). The relevant flavor measure- r ments to compare with are per nucleon pair. 5th force constraints are conventionally parametrized in terms of a modification to the gravitational BrðB → Kνν¯Þ < 1.6 × 10−5 ½71 potential þ1 15 −10 m1m2 BrðK → πνν¯Þ¼1.73 . × 10 ½72: ð15Þ VðrÞ¼−G ð1 þ αe−mϕrÞð20Þ −1.05 N r ’ The partial width for B → Kϕ is given by (e.g. [73]), with m1;2 the test masses in the potential, GN Newton s constant and α parametrizing the strength of the additional ≫ ϕ j j2 ð Þ2 2 − 2 2 force. Since mn me, and since couples to all nucleons, Csb f0 mϕ mB mK Γ → ϕ ¼ ξðm ;m ;mϕÞ we make the identification B K 16π 3 − B K mB mb ms 2 2 2 yn Mpl 3mbmt VtsVtb α ¼ ð Þ C ¼ 4π 2 : 21 sb 16π2 3 mn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv ϵξða; b; cÞ¼ ða2 − b2 − c2Þ2 − 4b2c2 In particular, for the mediator masses most relevant for us, the strongest constraints arise from Casimir force experiments 2 2 f0ðqÞ¼0.33ð1 − q =38 GeV Þ: ð16Þ [64], and are shown as the orange region in Fig. 1.Formϕ ≈ −12 1 eV and yn ≲ 10 the bound crosses over to the stellar where f0ðqÞ parametrizes the hadronic form factor [74]. constraints discussed below.

FIG. 2. Diagrams generating B → Kϕ (left) and K → πϕ (right).

115021-5 KNAPEN, LIN, and ZUREK PHYSICAL REVIEW D 96, 115021 (2017) In addition, bounds on new forces with masses as heavy discussed above—light scalars can be reabsorbed on nuclei as an MeV can also be obtained from low-energy neutron with a mean free path smaller than the core. If ϕ decays to experiments [77–79]. We show limits from neutron-Xe dark matter, then the decay length may be much shorter −7 ϕ scattering, derived in Ref. [79], giving a limit yn < 10 for than the reabsorption mean free path. Then the question mϕ ≪ MeV. The dark matter itself can mediate a force of whether the energy is lost depends on the mean free path through virtual effects, which implies that there is still a of the dark matter. As we will see in the next section, for 2 χ (weaker) constraint for mχ ≪ mϕ, even if the mediator mϕ > mχ scenario we must typically require that does itself is too short ranged [80]. not thermalize with ϕ in the early universe to evade BBN bounds. This puts a sufficiently stringent upper bound on yχ such that the ϕ → χχ decay is not relevant for SN1987A. B. Astrophysical constraints Reference [82] did not provide a calculation of trapping via scalar reabsorption. We estimate that trapping is 1. Stellar emission −7 relevant for yn ≳ 10 simply by taking the results for Light bosons with small couplings to electrons or axions [82–84]. Our justification is the following: despite nucleons can be emitted in stars, giving rise to rapid the different parametrics of scalar and axion production, cooling. New energy loss processes are constrained in a the weak-coupling constraints on axions and on scalars are number of stellar systems, giving strong limits on the −11 quite similar, with yn ≲ 5 × 10 for axions. Since the coupling of the light boson (see [81] for a review). Here we production rate and absorption rates are related by detailed consider limits from horizontal branch (HB) stars, red Γ ðωÞ¼ −ω=T Γ ðωÞ balance, prod e abs , we find to leading order giants (RG) and supernova 1987A (SN1987A). l−1 ∝ ϵρ 4 that the absorption mean free path is given by abs =T Horizontal branch (HB) and red giant (RG) stars have ϵ temperatures close to T ≈ 10 keV, required for helium with the energy loss rate [81]. Hence, we expect the ratio burning in the core. Bosons of mass up to ∼10–100 keV of the yn at the trapping boundary to yn at the weak- can thus be emitted in the core, escaping the star and coupling limit to be similar for both axions and scalars. leading to a new form of energy loss. The lifetime of There are a number of caveats in the bounds above, aside horizontal branch stars is measured by the ratio of the from the estimate of the trapping regime we have used. First, the weak-coupling result in Ref. [82] was obtained abundance of red giant to that of horizontal branch stars, with a simplified model of the SN core, and the result can and would be shortened depending on the energy loss rate vary by up to an order of magnitude depending on the core ϵ. Existing constraints on bosons with small couplings to temperature and radius. For instance, see Ref. [85] for a nucleons primarily utilize a condition ϵ ≲ 10 erg=g=s. discussion of systematic uncertainties for SN1987A This approximate condition applies both for HB and RG bounds on dark photons. In addition, the dominant pro- stars; in the latter case, the constraint is due to the fact duction mode in this case is nucleon-nucleon scattering that additional energy loss can delay the onset of helium with ϕ emission. Existing results have been calculated with ignition. For a detailed discussion, see Ref. [81]. the approximation of one-pion exchange. As discussed in More massive mediators can be constrained by the subsequent work [86–88], one-pion exchange is not an luminosity of SN1987A, where the core temperature was accurate description of nucleon-nucleon scattering data; around T ≈ 30 MeV. Here the requirement is that any new ϵ ≲ 1019 instead, they used a soft theorem description along with energy loss satisfies erg=g=s [81]. In addition, nucleon-nucleon scattering data to calculate production ρ ∼ 1015 3 due to the large core density ( g=cm rather than rates, leading to differences of up to an order of magnitude ρ ∼ 104 3 g=cm in HB stars), light bosons emitted in the in some models. Finally, the result of Ref. [82] did not core may be reabsorbed before escaping. This leads to a account for production due to mixing with the longitudinal trapping regime, where the coupling of the bosons is large component of the photon, an effect discussed in Ref. [66]. enough that they do not efficiently escape the core. In this Bounds from HB and RG stars. Constraints on scalars regime, the new particles can still modify energy transport coupling to baryons from stellar emission were given in −11 within the star and may be constrained, but this requires Refs. [89,90], with yn ≲ 4.3 × 10 from HB stars [81]. detailed modeling beyond the scope of this work. This result was derived assuming the Compton process Bounds from SN1987A. For a scalar with coupling γ þ He → He þ ϕ for mϕ ≲ 10 keV. Recently, constraints ϕ¯ yn nn, constraints from SN1987A were derived in the on light scalars were updated in Ref. [66], which included weak-coupling limit in Ref. [82]. Following these results, the effects of in-medium mixing on the production of we require that the energy loss per unit mass be scalars. To summarize, the scalar can mix with longitudinal 19 ϵ < 10 erg=g=s; taking a fiducial set of parameters photon polarization modes in a star, leading to an additional 14 3 T ¼ 30 MeV and ρ ¼ 3 × 10 g=cm , this gives a limit contribution to the rate. This production mechanism is ≲ 10−10 of yn .(Formϕ close to T, we simply assume a possible as long as mϕ < ωp, where the plasma frequency −mϕ=T Boltzmann suppression of e in the rate.) This bound ωp is also the oscillation frequency of the longitudinal does not apply to large yn due to the trapping effect mode. For horizontal branch stars ωp ∼ 2 keV and for red

115021-6 LIGHT DARK MATTER: MODELS AND CONSTRAINTS PHYSICAL REVIEW D 96, 115021 (2017) giants ω ∼ 20 keV. We use the constraints given in α2π p σborn ≈ χ ð 2 − 1Þ ð Þ V 2 4 log R : 26 Ref. [66], which come from production in HB stars mχv via brem off ions (“continuum”) and mixing effects (“resonant”), and from production in RG cores via mixing For instance, taking v ¼ 10−3 and R ¼ 10, we have effects. It is also possible that for sufficiently large 3=2 couplings the scalars may be trapped in RG and HB stars −10 mχ αχ ≲ 6 × 10 × : ð27Þ as in the case of SN1987A, though this requires detailed 1 MeV modeling of the energy transport in the star [91,92].ForRG and HB stars, the couplings at which trapping would likely Furthermore, assuming dark matter self-interactions are be relevant are also in a regime where the terrestrial responsible for the deviations of observed halo shapes from constraints become important, so we do not consider this ΛCDM, it is possible to fit the interaction cross section to possibility further. the shapes of dwarf galaxies, elliptical galaxies, and clusters; this was carried out in Ref. [95]. Remarkably, – 2. Dark matter self-interactions they found that a mediator mass on the 1 10 MeV scale χ (depending on the dark matter mass) was favored by the When composes all of the dark matter, there are data. significant constraints on dark matter self-interactions from The self-interaction constraints are substantially relaxed the shapes of halos (see Ref. [93] for a review). Because of when χ is only a fraction of the dark matter. In addition, the the low momentum transfer involved in the scattering, the effects of a strong self-interaction may enter a new regime self-interaction constraints on the dark matter coupling, – 2 where the dark matter behaves as a fluid [96 98]. This αχ ¼ yχ=4π, are particularly strong in the limit of a light −3 would form an additional isothermal component of the mediator with mϕ ≲ mχv and v ∼ 10 . Bullet-cluster and Milky Way’s dark matter halo. For the parameter space we halo shape observations tell us that DM self-interactions consider with light (but not massless) mediators, we expect should satisfy that dissipative effects are kinematically suppressed by σ finite mϕ. Thus, as a representative case, we will take ≲ 1–10 2 ð Þ Ω Ω ≈ 0 05 cm =g; 22 χ= DM . in relaxing the SIDM constraints [99]. mχ Interestingly, partially interacting dark matter may also where σ is the self-interaction cross section. have some connections with some discrepancies in large For scattering of distinguishable particles, the relevant scale structure measurements [100,101]. cross section is the transfer cross section σ , which is T C. Cosmology the scattering weighted by momentum transfer (see Appendix B). However, for the particular model at hand The detailed thermal history of the universe must be with identical particles, we instead use the viscosity cross addressed in scenarios where the dark matter and/or section, defined as mediator are relativistic during BBN and recombination, Z since the light particles may contribute to the effective dσ σ ¼ Ω 2θ ð Þ number of relativistic degrees of freedom Neff. In the V d sin ; 23 ≈ 3 046 dΩ standard model, Neff . . The deviation from the standard model value can be written as in order to regulate the forward and backward scattering X 4 divergences [94]. For our benchmark model, the non- 4 T ΔN ¼ g i ð28Þ relativistic Born cross section is eff 7 i i Tν 2 4 2 αχπ R þ 2R þ 2 σborn ≈ ½1 þ 2 − 1 ð Þ with gi and Ti the effective degrees of freedom and the V 2 4 2 2 log R 24 mχv R ðR þ 2Þ temperature of the various relativistic species in the dark sector, and Tν the neutrino temperature after electron with v the relative velocity and R ¼ mχv=mϕ. In the heavy decoupling in the standard cosmology. mediator limit with R ≪ 1 and cross section bound of In general, the strongest bounds come from CMB 1 cm2=g, the corresponding constraint on the coupling constraints on light degrees of freedom present at late constant is times, but they can vary significantly depending on the assumed cosmology and data sets included in the fit. The 1 2 2 1 = mϕ α ≲ 0 025 keV ð Þ most stringent constraint from CMB (namely Planck) and χ . 1 : 25 CMB ¼ 3 04 0 18ð1σÞ mχ MeV large scale structure gives Neff . . [102]. However, this fit is for the minimal extension of the Meanwhile, for light mediators with R ≫ 1, standard cosmology, and is modified in the presence of

115021-7 KNAPEN, LIN, and ZUREK PHYSICAL REVIEW D 96, 115021 (2017) other physics. For instance, if an additional eV-scale instrumental in setting the relic abundance of χ.1 We sterile neutrino is included, the 95% CL constraints therefore extend our simplified model with a real scalar CMB 3 7 eff 0 38 a with couplings of the form weaken to Neff < . , mν;sterile < . eV, while if the mass of active neutrinos is included, the 95% CL 1 2 1 2 1 2 2 constraint from CMBP plus large scale structure is L ⊃− maa − yamaϕa − λχ a : ð31Þ CMB ¼ 3 2 0 5 0 32 2 2 4 Neff . . , mν < . eV. Another modifica- tion to the standard picture is if neutrinos have a somewhat We take ma ≪ eV such that a is a small contribution to the large self interaction—instead of free-streaming, they may energy density at late times, with the main constraints behave as a fluid at late times. For instance, Ref. [103] coming from CMB Neff bounds. CMB ¼ 3 0 0 3ð1σÞ found that Neff . . in this scenario which In order to determine Neff, we must determine the ratio of again increases the uncertainty. Finally, as noted in the temperature of the dark sector relative to the neutrino Ref. [63], the bounds presented by the Planck collabora- temperature. This quantity depends on the cosmological tion generally assume a particular relationship between history, specifically on the temperature at which the dark the He fraction and the baryon asymmetry. sector decoupled from the standard model bath, as well as In this paper, we choose to be as agnostic as possible the number of dark degrees of freedom that were in about the detailed cosmological history, such that the most equilibrium when this decoupling occurred. In particular, robust bounds on our model come from BBN only. In for sufficiently small ya, it is possible that a does not come particular, a recent combined fit of He and D abundances into equilibrium until after the dark sector decouples from from Ref. [63], driven by improved errors in the measured the SM. At that point, a can then be responsible for the BBN ¼ 2 89 0 28ð1σÞ relic abundance of ϕ and χ. For instance, as long as D abundance, gives Neff . . . Using the η ≲ 10−1ð Þ ϕ 2D likelihood for Neff and the baryon-to-photon ratio ya eV=ma , then a is not in equilibrium with 2σ until after ϕ decouples from the SM bath (which occurs at shown in Ref. [63], the bound on Neff is T ≈ mπ, as we discuss below). On the other hand, the decay ϕ → ϕ ΔNBBN ≲ 0.5; ð29Þ aa is in equilibrium through the mass threshold for eff 3=2 −5 mϕ eV which implies roughly 2σ tension with a single real scalar y ≳ 10 ; ð32Þ a 200 keV m with a temperature similar to that of the neutrinos a Δ BBN ≈ 0 57 ( Neff . ). In similar spirit, Ref. [63] finds that the allowing for efficient depletion of the ϕ abundance. CMB constraint is Δ BBN Therefore, in this scenario our estimates of Neff depend only on whether ϕ and χ have equilibrated with the SM. ΔNCMB ≲ 0.6 ð30Þ ≈ 300 eff Above the QCD phase transition TQCD MeV, from Eqs. (9)–(10) we can write the coupling of ϕ with at recombination, where this bound uses only CMB data gluons in terms of yn, and no assumptions about the He fraction from BBN are α made. The next stage of CMB experiments can signifi- sbyn a a ϕGμνGμν; ð33Þ cantly improve on these results, with a projected sensitivity 8πmn σð CMBÞ ≈ 0 04 of Neff . from CMB Stage IV [61]. Both low mass dark matter and mediators may contribute where we assume the ϕ coupling to light quarks is Δ negligible. Thermal scatterings such as gg → ϕg can bring to Neff. In this section, we are considering the most optimistic case where both the dark matter and the mediator the mediator into equilibrium. Since the coupling in (33) is are real scalars, such that gϕ ¼ gχ ¼ 1. If the dark sector given by an irrelevant operator, the mediator drops out of was ever in thermal contact with the SM, it is also necessary to introduce a mechanism to avoid ϕ and/or χ having too 1There are a number of other ways one could reduce any excess ≳ 10 density of non-relativistic ϕ and/or χ. First, due to its (indirect) large an abundance: for mϕ eV, the relic abundance of ϕ ϕ coupling with the top quark, has a radiative decay to photons, becomes a non-negligible component of the dark matter given in Eq. (13). While this process can lead to the decay and for much larger masses it exceeds the observed density ϕ ≳ 10−7 pof ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibefore it becomes nonrelativistic,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi one needs yn × ≈ 1 −6 ¯ of DM. (For mϕ eV, close to the boundary of the fifth 1 MeV=mϕ (yn ≳ 4 × 10 × 1 MeV=mϕ) for the ϕtt force constraints, ϕ behaves as hot DM, but is less than 1% (ϕGG) coupling model. This is only satisfied in a very small of the dark matter.) part of the parameter space allowed by the meson constraints. It is ϕ ϕνν¯ In the case where ϕ becomes thermalized with the SM, also a priori possible to have decay to neutrinos via a portal. This is viable for mϕ ≳ 10 MeV, and excluded for mϕ ≲ 10 MeV the simplest solution for its relic abundance is to introduce if ϕ is in equilibrium with the neutrinos during BBN [57]. For an additional light degree of freedom, as also considered mϕ ≪ 1 MeV it is also possible that the mediator enters equi- in [60]. This additional degree of freedom can also be librium only after BBN but before ϕ becomes nonrelativistic.

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10 6 10 6 B

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y 10 y 10 BBN BBN Neff 0.06 Neff 0.06 2 2 45 cm 45 cm 12 10 12 10 10 , N 10 N A 1 MeV A 1 MeV, m m tt GG 10 15 10 15 eV keV MeV GeV eV keV MeV GeV m m

FIG. 3. Thermalization history in relation to the mϕ vs yn plane, where the gray shaded regions are the constraints from Fig. 1. Below the dashed blue line the dark sector decouples from the SM before the QCD phase transition. Approximate values of ΔNeff are shown for both regions (see text). A representative cross section contour for fixed mχ, and where αχ is chosen to saturate self-interaction bounds, is indicated by the dotted gray line. equilibrium with the SM as the universe cools. Estimating −7 mχ 3 2 λ ≈ 3 × 10 : ð35Þ αs byn the cross section as σ ∝ 2 2 , we find this process is out of MeV 64π mn equilibrium by T ¼ 300 MeV for y ≲ 10−9. (For more n Self-interaction bounds can still be satisfied, since χ − χ detailed estimates of these rates, see Appendix C.) scattering is only generated through a loop of a particles This qualitative boundary is shown by the dashed blue λ λ4 line in Fig. 3, in relation to the terrestrial and astrophysical and thus is higher order in , scaling as while the λ2 constraints discussed in the previous sections. In the annihilation cross section scales as . Since this way of supernova trapping window, we see that the mediator setting the relic density via thermal freeze-out in the dark always remains in equilibrium (region B). This is only sector is essentially independent of the direct detection and relevant for the ϕGG model, where the meson constraints other phenomenology, we choose to be agnostic about the are somewhat weaker. The dashed gray line is intended to specific mechanism whenever possible. ≳ 10−9 ϕ give some intuition on the possible direct detection cross If yn (region B in Fig. 3), remains in equilib- sections, which will be discussed in more detail in the next rium throughout the QCD phase transition and even after- section. wards due its coupling with pions/nucleons. While the Then, taking y ≪ 10−9 (region A in Fig. 3) such that the scattering on nucleons is suppressed by the baryon-to- n η ≈ 6 10−10 mediator decoupled before the QCD phase transition, the photon ratio × , the mediator still scatters with pions. In particular, the gluon coupling in (33) induces a contribution to Neff from the dark sector is at most pion coupling [70], X 4 3 X 4 g ðTν Þ = ΔNBBN ≈ g SM dec ≈ 0.06 g ð34Þ byn ϕ∂π†∂π ð Þ eff 7 i g ðT Þ i : 36 i SM QCD i 18mn ≳ 10−9 ϕ where we took g ðT Þ ≈ 61.75 and g the degrees of For yn , this keeps in equilibrium until around SM QCD i ≈ freedom of the species in the dark sector which are in T mπ when pions decouple. In this case, since pion and equilibrium before decoupling. This case is unconstrained muon decoupling occurs more or less simultaneously, it is with current CMB or BBN data but may be probed by CMB conservative to assume that the dark sector has the same temperature as the neutrinos. This implies Stage IV [61], depending on the value of gSM at the temperature where the dark sector decouples. For values 4 X of y consistent with the stellar constraints, the dark sector ΔNBBN ≈ g ; ð Þ n eff 7 i 37 decouples from the SM at T ≈ 100 TeV, such that there is i plenty of room for a suitable mechanism to set the dark sector relic density. In particular, if χ is assumed to be all of summing over the dark degrees of freedom gi just below the dark matter, then annihilation of χχ → ϕϕ is not pion decoupling. If χ and ϕ are both in equilibrium at this Δ BBN ≈ 1 14 ϕ sufficient to obtain the correct relic abundance—this is time, Neff . , which is firmly excluded. If only is Δ BBN ≈ 0 57 because of the bounds on yχ from dark matter self- in thermal contact with the SM we have Neff . and Δ CMB ≈ 0 72 2σ interactions. However, the annihilation χχ → aa can set Neff . , both of which are in roughly tension Δ CMB the correct abundance if with current data. The Neff number was obtained by

115021-9 KNAPEN, LIN, and ZUREK PHYSICAL REVIEW D 96, 115021 (2017) transferring the energy density of ϕ to the light scalar a, where we assumed that a and ϕ were not in equilibrium ≈ with one another until after T mπ. In what follows, we 10 3 will consider the 2σ tension associated with ϕ-SM equi- librium as permissible, but we will insist that the dark matter χ does not thermalize with the mediator. Its relic y 10 6 density must therefore have a different origin, such as from m 10 3m the interaction with the scalar a (as discussed above), or SIDM from interactions with the neutrinos [20]. m 500 keV 10 9 SIDM Before turning to the direct detection prospects, we first Thermalization discuss the implications of forbidding ϕ-χ equilibrium. 10 3 10 2 10 1 100 101 102 103 Possible χ thermalization mechanisms are annihilation of ϕϕ → χχ or ϕ → χχ decay, with the latter possible only if m MeV mϕ > 2mχ. If the decay is open, it dominates the thermal- FIG. 4. In the yχ vs mχ plane, we compare limits from self- ization process, since it is lower order in yχ. The thermal- interactions (assuming Ωχ =ΩDM ¼ 1) to our bound on thermal- ization conditions at a particular temperature T are ization of χ with ϕ, using the conditions given in Eq. (38). In the −3 light mediator case (mϕ < 10 mχ), we do not place a bound mϕ Γ ∼ ð Þ 2 ϕ→χχ H T if mϕ > mχ on χ thermalization since the values of Neff satisfy current T bounds even with 2 degrees of freedom. For the massive mediator ð Þh σ i ∼ ð Þ 2 ð Þ −9 nϕ T v ϕϕ→χχ H T if mϕ < mχ; 38 benchmark (mϕ ¼ 500 keV), and for couplings above yn ≈ 10 , we require that at most ϕ is in equilibrium with the SM to avoid where the factor mϕ=T in the decay rate accounts for the Neff bounds; this gives the thermalization bound shown (dashed Lorentz boost of ϕ at high temperatures. We evaluate these blue line). Above mχ ¼ Tπ, we assume that the dark matter can conditions at T ≈ Max½mϕ;mχ or T ≈ mπ, whichever is annihilate away efficiently and deposit entropy back into the pions. 3 lower. As long as ϕ is relativistic, we have nϕðTÞ ≈ 0.38T . In all expressions, we neglected thermal corrections to the 2 ≪ potential, which is justified as long as yχT mϕ. The ϕ-χ thermalization constraint for mχ ≳ mπ, since in this decay rate is given by case the energy density of χ can still be deposited into the 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SM sector rather than in dark radiation. As can be seen in yχ mχ Γ ¼ 1 − 4 2 2 ð Þ Fig. 4, for a massive mediator in the supernova trapping ϕ→χχ 32π mχ=mϕ; 39 mϕ window (region B in Fig. 3), the thermalization constraint pffiffiffi is almost always dominant over the self-interaction con- and, in the limit where mχ ϕ ≪ s, the cross section is ; straint. Instead, for the light mediator limit (region A in 4 2 Fig. 3), self-interaction bounds are important. yχ mχ σϕϕ→χχ ≈ : ð40Þ Finally, we note that ΔN may be additionally sup- 16π s2 eff pressed in more elaborate models, as has been studied in With s ∼ T2, it is clear that both decay and scattering some detail for light sterile neutrinos. Possible examples are become more important compared to Hubble as the temper- late time entropy production [105,106], non-conventional ature drops, so that χ will enter equilibrium as the universe cosmological evolution of the mass parameters [107,108] or cools. For our numerical results, we use thep fullffiffiffi expression a late dark sector phase transition [109].Thismayremove for σϕϕ→χχ (without expanding in mχ;ϕ ≪ s) and numeri- the need for demanding that the DM does not equilibrate, cally evaluate the thermally averaged cross section [104] and could open more parameter space. 1 hσ i¼ ϕϕ→χχv 4 2 D. Results 8mϕTðK2ðmϕ=TÞÞ Z As suggested by the results in Figs. 1 and 3, terrestrial ∞ pffiffiffi pffiffiffi 2 × dsσðs − 4mϕÞ sK1ð s=TÞ; ð41Þ and astrophysical constraints indicate two possible regimes 4 2 mϕ where direct detection of sub-MeV dark matter is conceiv- able in this simplified model: where K1 2 are modified Bessel functions of the second −12 ; (i) mϕ ≪ mχ and yn ≲ 10 (Region A in Fig. 3): The kind. The resulting constraint on yχ, as derived from dark sector decouples from the SM before the QCD Eq. (38), as well as the self-interaction constraint of phase transition, which cools the dark sector relative Δ Sec. II B 2, are shown in Fig. 4 for two benchmark points. to the SM sector enough such that Neff satisfies The feature around mϕ ¼ 2mχ clearly indicates where current bounds [Eq. (34)]. SIDM constraints, shown ϕ → χχ decay becomes relevant. We do not impose a in Fig. 4, provide the strongest constraints on yχ

115021-10 LIGHT DARK MATTER: MODELS AND CONSTRAINTS PHYSICAL REVIEW D 96, 115021 (2017) m 10 3 m m 10 3 m 36 36 10 10 color center color center

He 2 5th force SN1987a He 2 phonon 39 39 10 phonon 10 HB stars yr 2 2 1 kg RG stars He NR cm yr cm 10 42 1 kg 10 42

n n yr kg He NR 100 yr kg 100 10 45 10 45 1 0.05 DM DM Asymmetric DM 10 3 10 2 10 1 100 101 102 103 10 3 10 2 10 1 100 101 102 103 m MeV m MeV

−3 FIG. 5. Direct detection cross section as function of the dark matter mass, where the mediator mass is mϕ ¼ 10 mχ . We show the case that χ is all the dark matter (left) and where χ composes 5% of the dark matter (right). In the former case yχ is fixed by saturating the self- interaction constraint, while in the latter case we take yχ ¼ 1 and assume χ is a complex scalar with an asymmetric relic abundance. The blue lines indicate the projected reach with superfluid helium in the multi-phonon and nuclear recoil modes [50], assuming that the nuclear recoil mode includes energies from 3 meV up to 100 eV. We also show projected reach for color centers [39], where in this case we show their sensitivity for the massless mediator limit.

m 500 keV mm 36 36 10 NEWS CRESST 10 NEWS CRESST

color center superCDMS color center superCDMS

K He phonon 2 B G He phonon 2 G 10 39 10 39 fifth force BK

2 K 2

SN1987a K cm 42 cm 42

n 10 yr n 10 yr 1 kg HB stars 1 kg NR He RG He NR yr stars SN1987a yr kg kg 100 1000 10 45 10 45 1 1 DM DM

10 3 10 2 10 1 100 101 102 103 10 3 10 2 10 1 100 101 102 103 m MeV m MeV

FIG. 6. Constraints and detection prospects in the direct detection cross section vs mχ plane, for the heavy mediator regime. We scan over yn and fix yχ by demanding that the dark matter does not thermalize with ϕ for mχ mπ, the self-interaction constraint is used instead. We show the projected reach for NEWS-G and SuperCDMS [55], as well as proposed experiments with superfluid helium [50] or color centers [39]. The orange shaded region is excluded by CRESST [27]. For all of the accessible direct 4 detection for mχ < 100 MeV, we note that ϕ is in equilibrium with the SM until after the QCD phase transition and thus ΔNeff ≈ 7, which is in ≈2σ tension with current BBN and CMB bounds.

assuming that χ is the dominant component of the are compared with the reach for superfluid helium [50] and dark matter. color centers [39]. We saturate self-interaction constraints −5 −7 χ (ii) mϕ ≳ 500 keV and 10 ≳ yn ≳ 10 in the ϕGG and include stellar bounds and fifth force constraints. If model (Region B in Fig. 3): The mediator ϕ is in composes all of the dark matter in the light mediator equilibrium with the SM until the pion threshold, regime, direct detection with near future experiments Δ ≈ 4 2σ ≲ 1 leading to Neff 7. This is roughly in tension appears to be challenging for mχ MeV. On the other with BBN and CMB constraints. To avoid firm hand, if χ is a subcomponent of the dark matter with Ω Ω ≲ 0 05 exclusion from BBN/CMB, we further require that χ= DM . , the self-interaction constraints can be χ cannot thermalize with ϕ, which restricts yχ as relaxed as discussed in Sec. II B 2. In this case we impose shown in Fig. 4. the conservative perturbativity bound yχ < 1 for the right In Fig. 5, we present direct detection prospects for panel of Fig. 5. There is a subtlety associated with this mϕ ≪ mχ, fixing the mϕ=mχ ratio. The existing constraints regime: since mϕ

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e e MET e e χ annihilates extremely efficiently to ϕ, reducing its relic 10 3 Fifth force density to negligible levels. A straightforward way around g 2 e this is to consider a complex scalar/Dirac fermion χ with an 10 6 asymmetric relic abundance. In the regime of interest for BBN beam dump SN1987a subcomponent dark matter, the dark sector drops out of 10 9 e equilibrium well above the QCD phase transition, and the y HB stars additional degrees of freedom required for asymmetric dark 10 12 matter are allowed by the BBN bounds.

Both superfluid He and color centers could probe several 10 15 RG stars orders of magnitude of new parameter space in this scenario. For the superfluid He projections shown in this 10 18 paper, we have included the finite mediator mass and eV keV MeV GeV integrated over energies of 3 meV up to 100 eV in the m nuclear recoil mode, leading to a slightly different reach compared to Ref. [50]. FIG. 7. Constraints on a sub-GeV scalar mediator, given in y For mϕ ≳ 500 keV, we require that the dark matter χ terms of the effective scalar-electron coupling e. We show does not thermalize with ϕ, consistent with BBN bounds, terrestrial limits from fifth force searches [64] (orange), accel- ðg − 2Þ and as explained in the previous section. We present the erator [59,113,114] and e constraints [114,115] (green), and stellar cooling limits from HB stars [66] (red), RG stars [66] direct detection prospects in Fig. 6 for two benchmarks, ϕ χ (purple) and SN1987A (blue). Note that the beam dump con- where we fix yχ by saturating the - thermalization straints, derived in Ref. [114], assume only ϕ → eþe− and and SIDM constraint, whichever is strongest. The various ϕ → γγ decay modes are present, with negligible branching of contours indicate the terrestrial and astrophysical con- ϕ to dark matter. Thermalization of ϕ before electron decoupling −9 straints on yn discussed above. Since the self-interaction can occur for ye ≳ 10 , indicated by the dashed yellow line. The constraints are less stringent than the ϕ-χ thermalization shaded yellow region is excluded by BBN constraints, as condition in this scenario, we do not find substantially discussed in the text. different results for subcomponent dark matter. The turn-on 2 2 2 ϕ → χχ ¼ 2 yχy μχ of the decay mode at mχ mϕ= is clearly visible, σ¯ ≡ e e ð Þ e 4π 2 2 2 2 43 and in practice we find that accessible cross sections are ðmϕ þ α meÞ excluded whenever this decay is open. Then, if one allows Δ ≈ 0 57 with μχ the DM-electron reduced mass. For electron for a somewhat large Neff . , we find that there is e available parameter space for mχ > 100 keV with y scattering, the momentum transfer scale is set by the typical n ∼ α between the meson constraints and the supernova trapping in-medium electron momentum q me. As a result, ≪ α window; these cross sections could be probed by experi- here we define the light mediator limit by mϕ me. ments such as the nuclear recoil mode (mχ > 1 MeV) in The leptophilic scalar model can be probed by super- He, color centers, SuperCDMS and NEWS. conductors [45,46], Dirac materials [48], liquid xenon [42,110], graphene [43], scintillators [44], and semicon- III. LEPTOPHILIC SCALAR MEDIATOR ductor detectors [40,41,111,112], among others [55]. The constraints on the scalar mediator with electron Analogous to the model in the previous section, here we coupling are shown in Fig. 7, and described in more detail take real scalar dark matter χ interacting with leptons via a in the remainder of this section. Note that these constraints scalar mediator. As long as the couplings of ϕ do not induce have significant overlap with Higgs mixing models, which large lepton flavor-violation (e.g., with MFV couplings), were considered recently in the context of thermal dark the direct detection cross section and the bulk of the matter with mχ > 1 MeV [56]. The similarities are mostly constraints depend only on the coupling to the electron. in the very stringent stellar constraints on electron cou- The effective Lagrangian is written as plings of light scalars, which in Higgs mixing models 1 1 1 dominate over nucleon couplings despite the Yukawa L ⊃− 2χ2 − 2 ϕ2 − ϕχ2 − ϕ¯ ð Þ 2 mχ 2 mϕ 2 yχmχ ye ee: 42 coupling suppression. Important differences arise in the context of terrestrial constraints, where the absence of a The discussion relating to vacuum stability of the scalar coupling to hadrons lifts some of the accelerator and potential and perturbativity is identical to that in Sec. II and meson decay bounds. Cosmologically, the most important Appendix A, and we take yχ < 1 everywhere. Again, we difference compared to the hadrophilic scalar is that here ϕ consider real scalar dark matter for simplicity, but our can enter into equilibrium as the universe cools, and results also hold for fermion DM modulo important furthermore can remain in equilibrium through T

115021-12 LIGHT DARK MATTER: MODELS AND CONSTRAINTS PHYSICAL REVIEW D 96, 115021 (2017) −16 A. Terrestrial constraints correspond to ge ≲ 7 × 10 in the massless limit. With only a coupling to electrons, the dominant Reference [66] also included the effects of trapping for ≈ 10−6 terrestrial constraints on the mediator are derived from couplings as large ge , which weakens the bounds ð − 2Þ ϕ somewhat in these stars. We show the HB and RG bounds precision measurements of g e [114,115],andfrom direct production in high intensity eþe− colliders or beam in Fig. 7, where we have extrapolated their results to even dump experiments. For mϕ < 2me, ϕ must decay to dark larger couplings (as this region is separately excluded by matter or photons, where the latter decay is suppressed by BBN bounds). α2=16π2.Ifϕ either decays invisibly or outside the Complete constraints from SN1987A have not yet been detector, we can apply the BABAR mono-photon limits derived for this model. We estimate these bounds in the from Ref. [59]. While this constraint is one of the more limit that only production via mixing with the longitudinal robust limits in Fig. 7, it is not competitive with the BBN component of the photon is included, and neglecting direct and stellar constraints, which we discuss below. When production via Compton scattering or electrion-ion inter- þ − actions. As shown in Ref. [66], this is a “resonant” mϕ ≳ 20 MeV the BABAR dark photon search for e e → γðϕ → l−lþÞ production because the energy of the emitted scalars is [113] can be used to set a constraint, ω ¼ ω ω provided that ϕ couples to muons and electrons with L, with L the frequency where the scalar and mass-hierarchical couplings. In particular, while the dark longitudinal photon dispersions cross. From Ref. [66] photon model considered in Ref. [113] has democratic (see Appendix A.4), the energy loss rate from resonant branching ratios between muons and electrons, most production is given by of its sensitivity comes from the muon channel. For ω ω 2 1 ½ϕ → μμ ≫ ½ϕ → ϕ ≃ L L ΠϕL ð Þ Br Br ee , a limit on can be approxi- Qres ω ; 44 4π m L=T − 1 mated using the limit on the dark photon model consid- ϕ e ered by BABAR. We rescale the limit to account for the ϕ where we have used the fully relativistic result. Π L is the hadronic branching fraction in the dark photon model mixing of the scalar with the longitudinal component of the [116], which is absent in the model we consider here. If ϕ photon in the medium, which can be written as has a small or zero branching ratio to muons, the limit is Z eff weaker by an order one factor. A recasting of Ref. [113] is y em mϕ ∞ ΠϕL ≃ e e 2ð ð Þþ ð ÞÞ needed in this case, which we do not attempt here. 2 dpv fe Ep fe¯ Ep π k 0 In the mass range mϕ ≳ 100 keV, electron beam dump ω ω þ 2m2 experiments provide stringent constraints [117–119],as L L vk − ϕ ð Þ ϕ¯ × log ω − ω2 − 2 2 ; 45 derived for the ee coupling in Ref. [114]. The beam dump vk L vk L k v constraints may be relaxed if the visible decays of ϕ are suppressed by a competing decay mode to invisible states. where fe and fe¯ are the phase space distributions for the 2 This is especially likely to occur for mϕ < me, where the electrons, v ¼ p=Ep is the electron momentum, and k ¼ dominant visible mode is the radiative decay to photons. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2 − 2 ϕ Finally, fifth force constraints become important for L mϕ is the 3-momentum of the mediator . Note that mediator masses below an eV; they are, however, weaker the result is proportional to the in-medium mass of the eff by a factor me=mn compared to the hadrophilic scalar, and electron, me ≈ 12 MeV in the core of the supernova. Using so are not competitive with stellar and BBN bounds in the the Raffelt condition on the energy loss per unit mass ϵ ¼ mass range we consider. There are also bounds on light 19 Q=ρ ≲ 10 erg=g=swithT ≈ 30 MeV, ωL ≈ 82 MeV, and scalars from measurements of splittings in positronium ρ ≈ 3 × 1014 g=cm3, we obtain a limit for the weak coupling [120], although these are currently weaker than the bound y ≲ 10−9 ð − 2Þ regime of e for massless scalars. Given that we have from g e. only included resonant production, we expect that the true bounds due to thermal production may be even stronger. B. Astrophysical constraints To derive the trapping regime for SN1987a, we again use The self-interaction constraints on this model are iden- detailed balance to relate the production rate to the absorption λ−1 ¼ Γ ðωÞ ∼ ω4 tical to those for the hadrophilic scalar, and we refer the rate. For resonant production, mfp abs Qres= L. reader to Sec. II B 2. The stellar constraints on the other Requiring that the scalar is reabsorbed within R ≈ 10 km 3 10−7 hand differ quantitatively, as the rate for producing a light leads to a trapping limit of ye > × .Wealsoaccount þ − mediator off an electron is enhanced compared to the rate for trapping due to the decay of ϕ → e e , where we require þ − off a comparatively heavier nucleon. The strongest bounds that the decay length of ϕ → e e is within R ≈ 10 km. The are however still obtained from horizontal branch (HB) and decay of ϕ determines the bound in the trapping regime for ≤ ≤ 30 red giant (RG) stars for mϕ ≲ 100 keV and from SN1987A masses MeV mϕ MeV. (Note that in computing for heavier mediators. We take the HB and RG limits from kinematically allowed decays to eþe−, we use the vacuum eff Ref. [66], which account for plasma mixing effects, and mass me as opposed to the effective mass me .Thisis

115021-13 KNAPEN, LIN, and ZUREK PHYSICAL REVIEW D 96, 115021 (2017) because the thermal corrections to the electron mass drop Analogous to the discussion in Sec. II C, we say the rapidly and become smaller than the bare me for R beyond a mediator thermalizes if the thermally averaged rate is few km, depending on the model assumed. Hence, while we greater than the Hubble expansion HðTÞ at T≈ eff ½1 use the in-medium me for production in the core, we simply max MeV;mϕ . This yields the dashed yellow line in use me ¼ 511 keV to calculate decay.) Fig. 7; below this line, the mediator does not come in thermal contact with the standard model while electrons and C. Cosmology the mediator are both still relativistic. For mϕ ≪ 1 MeV, ≲ 5 10−10 For large enough couplings, the mediator ϕ will be in this value is ye × and independent of mϕ.Above ∼ thermal contact with the standard model through annihi- the electron threshold, the mediator decouplespffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi at T mϕ þ − − − ∼ lation (e e → γϕ) and Compton scattering (e γ → e ϕ). and the bound on ye therefore scales as mϕ=Mpl. þ − ≳ 5 10−10 ϕ The decays ϕ → e e (if mϕ > 2me)andϕ → γγ (if For ye × , can enter equilibrium with electrons ∼ mϕ < 2me) also contribute to equilibrating ϕ with the before T MeV, potentially running afoul of BBN. In standard model. Both the scattering and decay processes particular, any light degrees of freedom in thermal equilib- are IR dominated, as compared to the Hubble expansion, rium with γ=e will decrease the deuterium abundance [57]. þ − such that the mediator enters thermal equilibrium as the Here, the presence of ϕ dilutes the entropy release from e e universe cools. This is qualitatively different from the annihilation at the electron mass threshold, which has a nucleon coupling model, where the coupling with bigger effect than increasing Neff alone. This is because the the standard model was provided by the UV dominant, photon temperature Tγ=Tν is reduced during BBN, leading to dimension-five ϕGG operator. In practice we find that a increased baryon-to-photon ratio η and thus a more efficient decays are always subdominant to the Compton and conversion of deuterium into He. We compare the results of annihilation processes, regardless of mϕ. In the limit Ref. [57] with new measurements of the helium fraction Yp 2 2 s ≫ mϕ;me, the cross sections are and of D/H [62,63] in Fig. 8. It follows that mϕ below me is in   tension with current measurements, regardless of whether ϕ α 2 5 σ ≈ ye s þ ð Þ is in equilibrium with other dark degrees of freedom. This is eγ→eϕ log 2 2 2 46 s me þ mϕ the meaning of the yellow shaded region in Fig. 7. ϕ χ 2α 2 The bounds further strengthen if and are both in σ ≈ ye s ð Þ equilibrium with the SM. Similar to the discussion above, ee→γϕ log 4 2 : 47 s me Fig. 8 indicates that two real scalars with mass below ≈5 MeV are in tension with the new deuterium measure- The thermally averaged cross section for annihilation is ments. For this statement, we used the complex scalar obtained by replacing mϕ → m in Eq. (41), while the e benchmark from [57], implicitly assuming mχ ≈ mϕ,as corresponding formula for Compton scattering is well as the absence of other dark degrees of freedom in the ϕ χ 1 thermal bath. (The case for a nondegenerate and would hσ i¼ eγ→eϕv 2 3 require a dedicated study, which we do not attempt here.) 16m T K2ðm =TÞ Z e e We note that for mϕ ≳ 10 MeV, ϕ could in principle ∞ pffiffiffi pffiffiffi 2 transfer its entropy back to the SM thermal bath before × dsσðs − meÞ sK1ð s=TÞ: ð48Þ 2 me neutrino decoupling, for instance by freezing out against

0.265 2 Real scalar 2.6 1 Complex 1 0.260 2 2.4 5

0.255 10 2 p

Y 2.2 0.250 1 H D

0.245 2.0 Real scalar 1 Complex 0.240 1.8 10 2 10 1 100 101 10 2 10 1 100 101 m MeV m MeV

FIG. 8. For a real (complex) scalar degree of freedom in equilibrium with electrons/photons, we compare the results of Ref. [57] with the updated measurements of the BBN abundances of helium (Yp) and deuterium (D=H) from Ref. [62,63]. We show the central values of the measurements (solid blue lines) along with 1σ and 2σ bands.

115021-14 LIGHT DARK MATTER: MODELS AND CONSTRAINTS PHYSICAL REVIEW D 96, 115021 (2017) 3 3 m 10 e m 10 e 10 36 10 36 SENSEI 39 39 10 fifth force SENSEI 10 DAMIC DAMIC SuperCDMS G2 G2 10 42 SuperCDMS 10 42 SN1987a Al SC 2 2 45 Dirac materials Al SC 45 Dirac materials cm 10 cm 10 e e HB stars stars 10 48 10 48 RG stars HB stars 51 RG 51 10 1 10 0.05 DM Asymmetric DM DM 10 54 10 54 10 3 10 2 10 1 100 101 102 103 10 3 10 2 10 1 100 101 102 103 m MeV m MeV

FIG. 9. Direct detection cross section as function of the dark matter mass for the case that χ is all the dark matter (left) and when χ composes 5% of the dark matter (right). In the former case yχ is fixed by saturating the self-interaction constraint, while in the latter case −3 we take yχ ¼ 1 and assume χ is a complex scalar with an asymmetric abundance. We take mϕ ¼ 10 μχe, such that we are in the light mediator limit. The various lines indicate the reach for SuperCDMS-G2þ, SENSEI-100g, DAMIC-1K [55], an aluminum superconducting (Al SC) target [46], or a Dirac material [48], assuming kg-year exposure in all cases. See Ref. [55] for other proposals that could probe this parameter space. Current bounds from Xenon10 and Xenon100 are present only for cross sections above the range shown.

the electrons. In this case the mχ ≳ me bound should be SIDM constraints provide the strongest bounds on sufficient, even if χ remains in equilibrium with the yχ (see Fig. 4), if χ is the dominant component of the electrons through off-shell ϕ exchange. This is an impor- dark matter. The available parameter space in this tant difference with the hadrophilic scalar in Sec. II: once scenario is shown in the left-hand panel of Fig. 9, the temperature drops below mπ in the nucleon coupling where we included the expected DAMIC and model, ϕ and/or χ have no more means to effectively SuperCDMS G2þ reach as representative examples communicate with the SM, and all their entropy must be for semiconductor targets, as well as the reach for dumped into other dark sector degrees of freedom. This Dirac materials and superconductors. We find that Δ ≫ 1 increases Neff,evenifmχ;ϕ MeV. this scenario is not accessible with existing propos- For mχ ≲ 1 MeV, other dark sector states must be als, in agreement with earlier findings [46]. χ present in order to set the χ relic abundance, similar to We also show the case where is 5% of the total the interactions in Eq. (31). Clearly, the precise constraints dark matter density, in the right-hand panel of Fig. 9, from BBN depend on the details of how the relic abundance assuming that SIDM constraints are lifted and ¼ 1 of χ is set and a dedicated study is necessary to map out the setting yχ . Similar to the hadrophilic model, full allowed parameter space. In what follows, we will here we take χ to be a complex scalar with an restrict ourselves instead to two conservative benchmark asymmetric relic abundance. The symmetric com- χ points as far as the heavy mediator regime is concerned: ponent of the abundance is then rapidly depleted χχ¯ → ϕϕ ϕ¯ (i) Fix mϕ ¼ 10 MeV, and allow χ to thermalize with ϕ through annihilation. The ee coupling remains challenging even in this case, and only as long as mχ >me. superconductors are expected to have sensitivity. It (ii) Fix mϕ ¼ mχ > 5 MeV, and allow χ to thermalize with ϕ, such that the bound for a complex scalar in is however conceivable that the stellar constraints Fig. 8 is satisfied. could be lifted or weakened in a more sophisticated model, and that reach for Dirac materials or semi- conductor targets could be recovered. D. Results (ii) mϕ ≳ 10 MeV and mχ ≳ me, and mϕ ¼ mχ > 5, −10 Motivated by the combination of cosmological, astro- both with ye ≳ 5 × 10 : For these couplings, the physical, and terrestrial constraints discussed above, we mediator and the dark matter are in equilibrium with consider two regimes (similar to the hadrophilic scalar) the SM in the early universe, but disappear from the which are demarcated according to whether the mediator is thermal bath sufficiently early to satisfy BBN con- very light or massive: straints (in particular from the deuterium abun- −15 (i) mϕ ≪ mχ and ye ≲ 10 , where the bound on ye dance). Figure 10 shows the resulting parameter arises from stellar constraints. The dark sector is space in this massive mediator scenario. In the left never in equilibrium with the standard model and panel, we take mϕ ¼ 10 MeV, motivated by the

115021-15 KNAPEN, LIN, and ZUREK PHYSICAL REVIEW D 96, 115021 (2017) m 10 MeV m m 36 36 10 Dirac materials 10 Dirac materials XENON XENON

beam e 2 e g g dump e 2 10 39 10 39 e Al SC Al SC 2 SENSEI 2 SENSEI cm cm DAMIC DAMIC e e 10 42 10 42 SuperCDMSbeam G2 BBN SuperCDMS G2 dump

BBN SN1987a

RG stars

1 HB stars 1 DM DM SN1987a 10 45 10 45 10 3 10 2 10 1 100 101 102 103 10 3 10 2 10 1 100 101 102 103 m MeV m MeV

FIG. 10. Direct detection cross section as function of the dark matter mass for χ being all the dark matter, for two different massive mediator benchmarks. yχ is fixed by saturating the self-interaction constraint. The various lines indicate the reach for SuperCDMS-G2þ, SENSEI-100g, DAMIC-1K [55], an aluminum superconducting (Al SC) target [46], or a Dirac material [48], assuming kg-year exposure in all cases. See Ref. [55] for other proposals that could probe this parameter space. Current bounds from Xenon10 and Xenon100 are also shown [42,110]. Note that the constraint from beam dumps in the right panel (for masses of MeV to 100 MeV) can be lifted if ϕ decays to invisible states.

ð − 2Þ window between the beam dump and g e rough boundary between the two. In the light mediator constraints in Fig. 7. There is detectable and poten- regime, the constraints are driven by stellar cooling and tially unconstrained parameter space when the DM fifth force bounds. In the massive mediator regime, cross is heavier than the electron mass, although further sections were instead limited due to terrestrial bounds, and studies of the BBN predictions in this case are Neff bounds on thermalization of the dark sector (with Neff required. In this part of parameter space one finds primarily important for sub-MeV dark matter). ye ≪ yχ, such that the beam dump constraints do not The vector mediator cases differ notably from the scalar apply if the ϕ → χχ decay mode is kinematically mediated models in that (i) stellar constraints on the accessible (mχ < 5 MeV in the left-hand panel of mediator decouple in the massless limit and (ii) the Fig. 10). BBN bounds on the massive mediator scenario are even In the right panel of Fig. 10, we instead fix more stringent than those of Sec. III C, given the larger mϕ ¼ mχ and exclude the mϕ ¼ mχ < 5 MeV from number of degrees of freedom. Due to the BBN bounds, BBN considerations (yellow region), as discussed in realistic cross sections for mχ ≲ MeV are difficult to obtain the previous section. The terrestrial bounds allow with a massive mediator. Moreover, for mχ ≳ MeV massive fairly large σ¯ e for mχ > 10 MeV. Note also that we vector mediator models have been discussed extensively in have conservatively applied the beam dump con- the literature already [121–126], especially for the case of a straints, which excludes a range of cross sections in kinetically mixed dark photon. For these reasons, we will the mass range from 1 MeV to 100 MeV—however, focus exclusively on the available parameter space in the these could be lifted if ϕ can decay to invisible states light mediator regime. (either by taking mϕ > 2mχ or by introducing other dark sector states). A. Kinetically mixed dark photon The interactions for this model are given by IV. VECTOR MEDIATORS

1 2 0 0μ 1 0μν 0 ϵ μν 0 0 μ We now discuss vector mediators, concentrating on the L ⊃¼ − m 0 AμA − F Fμν − F Fμν − yχAμχγ¯ χ; simplest two anomaly-free extensions to the SM: a kineti- 2 A 4 2 ð1Þ ð Þ cally mixed dark photon and a U B−L gauge boson. For 49 these benchmark models we assume Dirac dark matter, and as such the self-interaction constraints are slightly different where now we consider Dirac fermion dark matter. from the real scalar dark matter in the previous sections. We For simplicity, we assume that the mass of the dark vector review this in Appendix B. In the scalar mediator scenarios was generated by the Stueckelberg mechanism. The kinetic discussed in Sec. II and III, constraints were broadly mixing, parametrized by ϵ, gives rise to a coupling of the characterized by a “light mediator” regime and a “massive dark photon with electrons and protons. From this, we mediator” regime, with mϕ ∼ a few hundred keV as the define the reference electron-scattering cross section as

115021-16 LIGHT DARK MATTER: MODELS AND CONSTRAINTS PHYSICAL REVIEW D 96, 115021 (2017)

2 2 2 mA' eV 4yχαϵ μχ 30 σ¯ ≡ e ð Þ 10 e 2 2 2 2 : 50 ðm 0 þ α m Þ A e 15 eV SN 10 33 10 0 ' Xenon 10 The scattering form factor for a light A is given by m A 2 2 4 4 4 Al SC F ðq Þ¼α m =q . 36 e 10 Freeze BBN in

Unlike constraints on scalar mediators, when a vector 2 eV 39 12 mediator couples to SM particles proportional to electric cm SENSEI 10 10

e ' DAMIC charge, the stellar constraints decouple due to in-medium m A 4 eV 0 → 0 42 SuperCDMS G2 effects as mA . We review the derivation of this effect 10 10 ' in Appendix D. For dark photons, these in-medium effects m A 45 have been accounted for in the Sun, red giants, and 10 WD Dirac materials horizontal branch stars in Refs. [127–129], while the RG SN1987A limits have recently been updated in 10 48 Refs. [66,85]. In all cases, the limits on the kinetic mixing 10 3 10 2 10 1 100 101 102 103 parameter ϵ scale as 1=mA0 . The in-medium suppression of mX MeV the A0 coupling with the electromagnetic current also implies thermalization of A0 is a negligible effect, thus FIG. 11. Dark matter scattering via a kinetically mixed dark 0 ≪ avoiding cosmological bounds on A0. photon, in the limit where mA keV. The shaded regions show These bounds are however for direct emission of the A0, millicharged DM limits (from BBN, SN1987A, and WD/RG) [130] as well as Xenon10 bounds [42]. The dashed lines are but light dark matter can also be emitted via an off-shell upper bounds on the cross section, for several benchmarks photon in the medium. As reviewed in Appendix D, this where we have considered roughly the largest allowable ϵ: process does not decouple in the massless limit. One way to −15 −3 (a) mA0 ¼ 10 eV, ϵ ¼ 10 , where stellar and fifth force 0 → 0 χ −12 −6 see this is that for mA , is effectively millicharged 0 constraints have decoupled (b) mA ¼ 10 eV, ϵ ¼ 10 , with respect to the SM photon in the stellar medium. Hence consistent with the CMB bounds shown in [131], and the stellar and BBN constraints on millicharged particles −4 −8 (c) mA0 ¼ 10 eV, ϵ ¼ 10 , consistent with the stellar bounds can be applied to the DM [130] and are directly sensitive to in [127]. Here the DM-mediator coupling yχ is fixed by saturating ϵ the combination yχ=e, which is the effective millicharge of SIDM constraints. For larger mA0 (up to 100 keV), the bounds the DM in the mA0 → 0 limit. In other words, in the on ϵ are much stronger. Similar to Fig. 10, we also show the reach massless A0 limit, both the direct detection cross section for various direct detection proposals. as well as the stellar and BBN constraints become inde- pendent of m 0 . We can therefore map the millicharge A model is anomaly free, it suffices to consider Dirac constraints directly in the mχ vs σ¯ plane, indicated by the e neutrinos, or to add a set of 3 (heavy) sterile neutrinos. shaded regions in Fig. 11. Furthermore, χ gains its relic In order to avoid complicating the cosmology, here we abundance through production via an off-shell photon, as in follow the latter avenue, which implies that Uð1Þ is the stellar medium. The solid blue line in Fig. 11 labels B−L broken softly by the Majorana neutrino masses at a scale couplings where the correct abundance can be achieved ∼m 0 =g − . For most of the parameter space of interest, the through freeze-in [40,41,122]. A B L sterile neutrinos can be as heavy as a few GeV. The relevant Even though the stellar constraints decouple for constraints for Uð1Þ − are summarized in Fig. 12, while m 0 ≪ eV, astrophysical constraints and tests of deviations B L A the alternative case with an unbroken Uð1Þ is discussed from Coulomb’s law still constrain the mixing parameter ϵ as B−L in Ref. [134]. For the sake of generality, we allow for a function of mA0 . For a summary, see for example 0 ð1Þ Refs. [127,132,133]. (The fifth force constraints we have different values for the coupling of A to the SM U B−L considered previously are for macroscopic neutral systems, current (gB−L) and to the dark matter [yχ, as in Eq. (49)]. We and are generally not relevant for the kinetically mixed dark note that in parts of the parameter space we consider, the photon.) For reference, we have included a number of hierarchy between these couplings may be rather large and, benchmark lines in Fig. 11:foreachm 0 we have selected in the absence of further model building, requires a perhaps A − χ the largest allowed ϵ and we have fixed yχ by saturating unnaturally large B L charge for . SIDM constraints, such that direct detection cross sections above the dashed line are excluded for the corresponding 1. Terrestrial and astrophysical constraints 0 ð1Þ value of mA . We see that for sub-MeV dark matter, ultralight For a U B−L gauge boson, the coupling to electrons mediators are required to satisfy both SIDM constraints and and protons behaves in much the same way as the dark for χ to live on the freeze-in line. photon. However, the situation is somewhat different due to the additional coupling of the vector with neutrons, B. B − L gauge boson which does not decouple in the mA0 → 0 limit (see ð1Þ Next we consider gauging the U B−L symmetry of the Appendix D). For the sun, HB, and RG stars, emission standard model, with gauge coupling gB−L. To ensure the from electrons dominates and the effect of the nucleon

115021-17 KNAPEN, LIN, and ZUREK PHYSICAL REVIEW D 96, 115021 (2017) 0 0 10 3 e e A A → νν decay can keep the A in equilibrium with the

g 2 e neutrinos after the neutrinos decouple from the electron- 6 0 ≲ 10 10 beam dump photon plasma. For mA MeV, this is excluded when BBN comparing the deuterium abundance [62,63] with the 10 9 SN1987a predictions in [57], similar to what was done in Fig. 8 L

B 5th force for the leptophilic scalar. For m 0 ≳ 10 MeV, the neutrinos

g A 10 12 can remain in equilibrium with the electron through off- 0 shell A exchange which would also increase Neff. The Sun RG −ν → −ν 10 15 HB dominant process in this case is e e scattering with a cross section of

10 18 4 eV keV MeV GeV gB−L s σ − − ≈ ð Þ e ν→e ν 6π 4 ; 51 mA mA0 ð1Þ FIG. 12. Constraints on a sub-GeV U B−L mediator. Shown for which we require that the thermal averaged rate at are limits from fifth force searches [64] and neutron scattering T ≈ 1 MeV is smaller than the Hubble expansion. The [65] (orange), from BBN (yellow), and from stellar cooling in HB constraints from the decay and scattering processes are stars [66] (red), the sun [66] (purple), red giants [129] (purple), indicated by the yellow line in Fig. 12. We note that our and SN1987A (light blue). Shown in green are also beam dump estimates do not include plasma corrections, and refer to [135,136], BABAR [113] and ðg − 2Þ [137] constraints. e [134] for a treatment of these corrections in the context of a B − L boson with Dirac neutrinos. In the heavy mediator regime, where both the dark matter coupling is mild, becoming important only for mA0 ≲ −2 and the mediator thermalize with the SM, mχ 0 ≳ 10 MeV 10 eV [66]. For SN1987A, the dominant production is ;A is allowed by BBN for reasonably large coupling. Since nucleon-nucleon scattering. While SN1987A constraints our focus is primarily on sub-MeV DM, we do not further have not been derived for the Uð1Þ case, we can obtain B−L elaborate here, other than noting that the beam dump approximate constraints by combining previous results in constraints in Fig. 12 do not apply if yχ ≫ g − and the literature. In the weak coupling regime, we use limits B L 2 0 on Uð1Þ gauge bosons derived in Ref. [88] and the limits mχ

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10 DAMIC 10 SN1987a Al SC SN1987a SENSEI Al SC 10 39 SENSEI 10 39 DAMIC

force SuperCDMS G2 yr 42 42 fifth 1 kg 10 SuperCDMS G2 10 sun HB stars fifth force kg yr He NR 1 WD yr 2 2 Dirac materials 100 kg He NR 1 kgk yr 45 WD kg yr 45 cm cm Dirac materials 100 phonon 10 1 kg yr 10 RGHe 2 e e phonon 100 kg RGHe 2 yr 100 HB stars 10 48 kg yr 10 48

51 51 10 1 10 0.05 sun DM Asymmetric DM DM 10 54 10 54 10 3 10 2 10 1 100 101 102 103 10 3 10 2 10 1 100 101 102 103 m MeV m MeV

FIG. 13. Direct detection cross section vs mχ for the case that χ composes all the dark matter (left), or that χ is a 5% sub-component of the dark matter (right). We show the projected reach with a kg-year exposure for SuperCDMS G2þ, SENSEI-100g, DAMIC-1K [55], superfluid helium [50], an aluminum superconductor (Al SC) target [46], and Dirac materials [48]. Constraints labeled with “WD” and “RG” refer to stellar constraints on χ production itself, as in Fig. 11, while those from the sun, HB stars and SN1987a are bounds on emission of A0, as in Fig. 12.

ð1Þ Apriorione may expect that the constraints on a U B mediator to be a significant exception to arguments gauge boson would be greatly relaxed, in particular on presented in this section. the cosmology front since the baryon density drops sharply during the QCD phase transition. However, this V. CONCLUSIONS is generically not the case, since the Uð1Þ always B We have considered simplified models for light DM, develops a radiative kinetic mixing term with the standard scattering off nucleons or electrons via scalar or vector model photon from quark loops, which reintroduces mediators in direct detection experiments. These models couplings to electrons. Given the rather large number are tightly constrained by stellar cooling arguments and of flavors in the SM, this term is not particularly small, fifth force experiments (in the light mediator limit, for although a mild cancellation exists between the up and mediator masses below an MeV), or by BBN and CMB down sectors. In particular, the mixing parameter is constraints on thermalization of the dark sector (in the case Z of more massive mediators). The bounds restrict much of X 1 ϵ ≈−gBe Nc ð1 − Þ the parameter space that can be probed by current proposals 2 Qf dxx x π 3 0 f for light DM direct detection, especially for dark matter   with mass below an MeV. We highlight the simplified xð1 − xÞΛ2 þ m2 QCD f ð Þ models of sub-MeV DM that satisfy all constraints: ×log ð1 − ÞΛ2 þ 2 52 x x mf (i) DM interacting with nucleons via a mediator heavier than ∼100 keV, though this is in ∼2σ tension with with f running over all 6 quarks and Qf and mf the quark current BBN bounds on Neff; electric charges and masses, respectively. We cut the (ii) DM interacting with the standard model via a Λ ≈ 1 ultralight kinetically mixed dark photon, and where running off at QCD GeV, since there are no addi- tional states carrying baryon number below this scale. the DM is populated nonthermally through a mecha- The parameter Λ is the high energy threshold at which nism such as freeze-in; ϵ ≈ 0. A natural choice for Λ is the GUT scale, which (iii) A DM subcomponent interacting with nucleons or electrons via a very light scalar or vector mediator. yields ϵ ≈−0.33gB, but in principle a scale as low as a few TeV is possible. If we follow the latter (more Given the importance of the BBN, CMB, and stellar conservative) avenue and fix Λ ≈ 5 TeV, we find cooling constraints in our understanding of sub-GeV dark sectors, further exploration of these bounds and under- ϵ ≈−0.05g . It is important to keep in mind that these B standing their model-dependence is certainly warranted. values are estimates at best, since one expects sizable higher order QCD corrections to this operator. The ð1Þ ACKNOWLEDGMENTS generic point, however, is that the coupling of a U B mediator to leptons is not parametrically suppressed to We thank Claudia Frugiuele, Dan Green, Ed Hardy, Rob the extent that stellar and BBN constraints from electrons Lasenby, Zoltan Ligeti, Sam McDermott, Michele Papucci, ð1Þ can be neglected, and as such we do not expect a U B Surjeet Rajendran, Sean Tulin, Haibo Yu, and Yiming

115021-19 KNAPEN, LIN, and ZUREK PHYSICAL REVIEW D 96, 115021 (2017) Zhong for useful discussions. We also thank Robert quartics can be neglected in both the thermalization and the Lasenby for discussions leading to an updated SN1987a self-interaction computations. bound for the leptophilic model, and Surjeet Rajendran and For the heavy mediator scenario with electron coupling, Dan Green for discussions leading to a numerical correc- we allowed the mediator to thermalize with the dark matter tion in the BBN constraints for the B-L model. The authors (see Sec. III C). Then yχ is bounded by SIDM constraints are supported by the DOE under Contract No. DE-AC02- and can be as large as ≈0.1–1. The following choice is 05CH11231. T. L. is further supported by NSF Grant enough to stabilize the potential No. PHY-1316783. We thank the Aspen Center for −3 −1 Physics, supported by the NSF Grant No. PHY-1066293, λχ ≈ λϕχ ≈ 10 ; and yχ ≈ 10 ; ðA4Þ for hospitality while parts of this work were completed. 2 again for mχ ≳ mϕ and λϕ ≈ 1. Again since λϕχ; λχ ≪ yχ, APPENDIX A: VACUUM STABILITY FOR the corrections to the χ self-interaction cross section can be SCALAR χ MODEL neglected for our purposes. Finally, the perturbativity constraint requires that the The dark matter interactions in Eq. (6) and Eq. (42) 1 2 one-loop correction to the yχmχϕχ coupling is para- contain a potential for the scalar χ and ϕ, 2 metrically smaller than the tree-level contribution. In the 1 1 1 nonrelativistic limit, the one-loop correction is given by ⊃ 2χ2 þ 2 ϕ2 þ ϕχ2 ð Þ V 2 mχ 2 mϕ 2 yχmχ A1 " 3 yχmχ mϕ mϕ mϕ − qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi which is unbounded from below for finite χ and ϕ → −∞. 2 log tan 16π mχ 4 2 − 2 4 2 − 2 This issue is easily addressed by adding the quartic mχ mϕ mχ mϕ !# couplings 2 2 mϕ − 2mχ − tan−1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA5Þ λϕ λχ λϕχ 4 2 − 2 ⊃ ϕ4 þ χ4 þ ϕ2χ2 ð Þ mϕ mχ mϕ V 4! 4! 4 : A2

yχmχ mϕ ≪ mχ Even though the runaway direction is lifted if all quartics which we require to be smaller than . In the are positive, there may still be dangerous false vacua. There limit, the perturbativity constraint simplifies to is no completely general analytic solution for the positions 2 yχ and energies of the vacua of this potential. However, we can 1 ð Þ 2 < ; A6 check that there exist self-consistent choices for the quartics 16π such that the origin is the only critical point in the potential. ¼ We also require that the quartic couplings do not affect our while for mχ mϕ the result is χ results on self-interactions and thermalization with the 2 yχ mediator, respectively. pffiffiffi < 1: ðA7Þ In the light mediator regime, where mϕ ≪ mχ, the 48 3π potential has a unique minimum at the origin for λϕ ≈ 1, ≲ 4π λχ ≈ 0 and λϕχ ≈ yχ. Since λϕχ only contributes to the self- Both cases are consistent with the requirement of yχ interaction cross section at loop-level, its contribution can from naive dimensional analysis. Throughout this paper we 1 be neglected. λϕχ does contribute to the ϕ-χ thermalization; conservatively impose yχ < . however, in the light mediator case there is no bound on this process since the dark sector is not in thermal contact with the standard model below the QCD phase transition. APPENDIX B: SELF-INTERACTION In the heavy mediator regime for the nucleon coupling CROSS SECTIONS model, where mϕ ≳ mχ, we always require yχ ≪ 1 due to In this appendix, we review scattering of distinguishable the χ − ϕ thermalization constraint. For example, the dark matter particles. Here σT is the transfer cross section, following choice suffices to stabilize the potential at the defined as the scattering cross section weighted by the ≈ λ ≈ 1 origin for mχ mϕ and ϕ : momentum transfer,

−10 −4 Z λχ ≈ λϕχ ≈ 10 ; and yχ ≈ 10 ðA3Þ dσ σ ¼ dΩ ð1 − cos θÞ: ðB1Þ T dΩ where the choice for yχ is representative for the values shown in the blue dashed line in Fig. 4. (For smaller yχ, the In the Born approximation, where αχmχ=mϕ ≪ 1, the vacuum stability condition clearly is easier to satisfy.) Since transfer cross section for DM interacting via a Yukawa 2 λϕχ; λχ ≪ yχ is possible, the presence of these additional potential is [93]

115021-20 LIGHT DARK MATTER: MODELS AND CONSTRAINTS PHYSICAL REVIEW D 96, 115021 (2017) 2 6 2 8παχ 4ζð3Þα T T σborn ≈ ½ ð1 þ 2Þ − 2 ð1 þ 2Þ ð Þ γ ¼ s þ 0 4 ð Þ T 2 4 log R R = R ; B2 a 2 2 × ln 2 . ; C2 mχv π Λ mg where R ≡ mχv=mϕ. When the mediator is heavy, such that resulting in a similar condition on Λ. R ≪ 1, the coupling constant corresponding to a cross Assuming that the results above can also be applied to a 2 section of 1 cm =gis scalar coupling with gluons, we find the following con- dition on the scalar nucleon coupling: 1 2 2 1 keV = mϕ αχ ≲ 0.02 : ðB3Þ mχ 1 MeV −10 yn ≲ few × 10 ðC3Þ For the ultralight mediator limit where R ≳ 1, we instead such that ϕ decouples from the thermal bath before the have QCD phase transition. In the main text we simply use −9 3=2 y ≲ 10 as an approximate condition. −10 mχ n αχ ≲ 10 × ; ðB4Þ 1 MeV

−3 APPENDIX D: IN-MEDIUM COUPLINGS where we took v ∼ 10 . Note that R ≳ 1 also corresponds OF LIGHT VECTORS to the classical regime; in the classical limit, and assuming an attractive potential, analytic formulas for the cross We begin by writing the vacuum Lagrangian in the basis section have been obtained that are valid even for the where the kinetic-mixing term has been rotated away, such nonperturbative regime. However, the Born approximation that our discussion can be applied for the dark photon and is more accurate for αχmχ=mϕ ≪ 1, even in the classical for B − L. Consider the vacuum Lagrangian regime, and so we use the Born result everywhere.2 See for 2 example Refs. [94,142]. Finally, comparing with the results 1 1 m 0 L ¼ − μν − 0 0μν þ A 0μ 0 in Sec. II B 2, we see that the limiting forms of the self- vac 4 FμνF 4 FμνF 2 A A μ α interaction cross section and the resulting bounds on χ are μ 0 μ 0 μ 0 þ J ðeAμ þ gA μÞþgJ 0 A μ þ g J A μ: very similar despite the different models considered. EM SM DM DM ðD1Þ APPENDIX C: THERMALIZATION OF THE MEDIATOR Here we have separated the coupling of the new A0 gauge In Sec. II C, we estimated when a light scalar mediator boson to SM particles into two pieces, the coupling to the EM current and everything else (accounted for in JSM0 ). interacting with gluons would decouple from the standard 0 model thermal bath. A full calculation of the process gg → Additionally, we include the coupling of A with dark ϕ matter, which we write as gDM to account for a possible g requires accounting for thermal gluon masses, which 0 further regulate the t − channel collinear divergence. For large hierarchy between the A couplings with the SM and comparison, here we review some results for light pseu- the dark matter. ≫ doscalar thermalization with gluons that include finite Taking a coupling hierarchy e g, in-medium effects temperature effects. generate mass terms We normalize the pseudoscalar-gluon coupling as α 2 s a ~ a m 4Λ aGμνGμν where a is the pseudoscalar. Reference [143] L ¼ A μ þ ϵ 2 μ 0 ð Þ IM−mass A Aμ mAA A μ; D2 found a pseudoscalar production rate of 2

2 6 2 6 where ϵ ¼ g=e and m is the in-medium mass of the photon αs ζð3ÞT 4αs ζð3ÞT A γ ¼ F3ðm3=TÞ ≃ ð Þ a π3Λ2 ðπÞ3Λ2 C1 due to the charged particle density. Now we rotate to the mass basis by the choice 2 where γa ∼ ngσv is the collision term in the Boltzmann ϵm2 ϵm2 equation and ng is the thermal gluon density. Comparing μ ¼ ~ μ þ A ~ 0μ 0μ ¼ ~ 0μ − A ~ μ A A 2 2 A ;AA 2 2 A ; γ Λ ≲ 3 m 0 − m m 0 − m ϕ=ng withpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the Hubble expansion, this gives × A A A A 109 GeV T=GeV for a production to be out of equilib- ðD3Þ rium at temperature T. Similarly, an earlier result using the hard thermal loop approximation found [144] such that the mass mixing is eliminated up to terms of order Oðϵ2Þ. Then the total in-medium Lagrangian, dropping the 2We thank Haibo Yu for this comment. Oðϵ2Þ terms, becomes

115021-21 KNAPEN, LIN, and ZUREK PHYSICAL REVIEW D 96, 115021 (2017) 2 2 1 1 m 0 J 0 . As shown above, this coupling does not decouple in L ¼ − ~ ~ μν − ~ 0 ~ 0μν þ mA ~ μ ~ þ A ~ 0μ ~ 0 SM IM 4 FμνF 4 FμνF 2 A Aμ 2 A A μ the low mass limit, leading to stellar constraints that are 2 independent of mA0 . For the sun, HB stars, and RG stars, eϵm 0 þ μ ~ 0 þ μ ~ þ A ~ 0 gJSM0 A μ JEM eAμ 2 2 A μ production off of neutrons is a smaller contribution than m 0 − m A A production off of charged particles, so this effect is only 2 ϵ 0 μ ~ 0 mA ~ relevant for mA well below an eV [66].However,ina þ g J A μ − Aμ : ð Þ 0 DM DM 2 − 2 D4 supernova the emission of A in nucleon-nucleon scatter- m 0 mA A ing is important. As described in Sec. IV B,wethus Here we kept the terms of Oðg ϵÞ since it is possible obtain SN1987A bounds on B − L by combining limits DM ð1Þ that g ≫ g. on dark photons from Ref. [85] and limits on U B DM 0 For the piece of the A~ coupling that is proportional to the from Ref. [88], whichever gives the larger effect for a EM current, the mass mixing implies that the effective in- given mA0 ;g. medium coupling will decouple in the limit m 0 ≪ m .In The interactions above also allow production of the A A ~ 0 ~ stars such as the sun, mA can be of order keV, while mA ∼ DM in the star, via an off-shell A or off-shell A. Although 0 20 MeV for SN1987A, leading a significant effect for low the coupling of A~ to SM charged particles is suppressed mass vectors. This is familiar in the case of the dark photon 0 ~ for mA ≪ mA, from Eq. (D4) the induced coupling of A where J 0 ¼ 0 and taking ϵ as the kinetic mixing parameter. ϵ SM with dark matter is given by gDM in this limit. If gDM is Here it is known that the in-medium kinetic mixing parameter relatively large, then production of DM furnishes another 2 2 ≈ 0 is suppressed by mA0 =mA in the small mA limit. Due to a form of stellar constraint, and we can apply limits derived ∼ 0 ≈ ðϵ Þ relative enhancement by E=mA for production of the for the millicharged DM case taking QDM gDM=e . longitudinal modes, the resulting dominant stellar emission This can be directly translated to an upper bound on the ϵ 0 – scales as mA in the low mass limit [127 129]. direct detection cross section (in addition to the constraints − ϵ For B L the same analysis applies, however there is on and gDM from stellar emission and self-interactions, 0 also an interaction of the A~ with neutrons, included in respectively).

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