Looking for the WIMP next door

Journal of High Energy Physics, Feb 2018

Jared A. Evans, Stefania Gori, Jessie Shelton

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Looking for the WIMP next door

HJE Looking for the WIMP next door Jared A. Evans 0 1 2 3 Stefania Gori 0 1 2 Jessie Shelton 0 1 3 0 345 Clifton Court , Cincinnati, OH 45221 , U.S.A 1 1110 West Green Street, Urbana, IL 61801 , U.S.A 2 Physics Department, University of Cincinnati 3 Department of Physics, University of Illinois at Urbana-Champaign , USA We comprehensively study experimental constraints and prospects for a class of minimal hidden sector dark matter (DM) models, highlighting how the cosmological history of these models informs the experimental signals. We study simple `secluded' models, where the DM freezes out into unstable dark mediator states, and consider the minimal cosmic history of this dark sector, where coupling of the dark mediator to the SM was su cient to keep the two sectors in thermal equilibrium at early times. In the well-motivated case where the dark mediators couple to the Standard Model (SM) via renormalizable interactions, the requirement of thermal equilibrium provides a minimal, UV-insensitive, and predictive cosmology for hidden sector dark matter. We call DM that freezes out of a dark radiation bath in thermal equilibrium with the SM a WIMP next door, and demonstrate that the parameter space for such WIMPs next door is sharply de ned, bounded, and in large part potentially accessible. This parameter space, and the corresponding signals, depend on the leading interaction between the SM and the dark mediator; we establish it for both Higgs and vector portal interactions. In particular, there is a cosmological lower bound on the portal coupling strength necessary to thermalize the two sectors in the early universe. Beyond Standard Model; Cosmology of Theories beyond the SM; Thermal - We determine this thermalization oor as a function of equilibration temperature for the rst time. We demonstrate that direct detection experiments are currently probing this cosmological lower bound in some regions of parameter space, while indirect detection signals and terrestrial searches for the mediator cut further into the viable parameter space. We present regions of interest for both direct detection and dark mediator searches, including motivated parameter space for the direct detection of sub-GeV DM. Parameter space for minimal hidden sector freezeout and the WIMP 1 Introduction 2 3 4 5 Higgs portal 4.1 4.2 4.3 4.4 5.1 5.2 5.3 5.4 Direct, indirect, and accelerator constraints Vector portal Thermal freezeout and indirect detection Direct detection Accelerator and other mediator constraints Beyond the minimal model of dark vector interactions Thermal freezeout and indirect detection Direct detection Accelerator and other mediator constraints Beyond the minimal model of dark scalar interactions 6 Summary and conclusions A Thermal (de)coupling A.1 Vector decoupling B Sommerfeld enhancement C Bounds from dwarf galaxies matter [1]. No evidence to date indicates that DM must interact in any way beyond and thus this class of models makes sharp predictions for signals accessible to a variety of experiments. While the parameter space for thermal WIMPs is now acutely limited by the interplay of null results at direct and indirect detection experiments and at the Large Hadron Collider (LHC) [2], thermal DM that freezes out directly to SM particles via new beyond-the-SM (BSM) mediator(s) similarly has a cosmological abundance directly set by the strength of its interactions with the SM, and has thus driven the terrestrial DM discovery program in recent years. These models, too, are becoming increasingly challenged by the lack of signals to date [3, 4]. This class of thermal relics, however, represents only a fraction of possible identities for dark matter. Hidden sector freezeout (HSFO [5{8]), where the DM relic abundance is chie y determined by interactions internal to a thermal dark sector with little to no involvement of the SM, provides a much broader class of models. In this paper, we survey the current constraints and future discovery prospects in the simplest exemplars of hidden sector freezeout. In these simple and minimal models, DM is a thermal relic that annihilates not to SM states, but to pairs of dark mediators that subsequently decay via small couplings into the SM. We take these small couplings to be the leading interaction between the HS and the SM, and consider the well-motivated and generic case where this interaction is renormalizable. Any theory where DM arises from an internally thermalized dark sector must also address the question: how was this dark sector populated in the early universe? The most minimal cosmological history for a dark sector is for it to interact strongly enough with the SM that the two sectors were in thermal equilibrium at early times. In this case, the existence of a thermal SM plasma in the early universe guarantees the population of the dark sector. We call DM that freezes out from a thermal dark radiation bath in thermal equilibrium with the SM a WIMP next door. Mandating this cosmological history for the dark sector imposes a lower bound on the interactions between the dark sector and the SM today, the thermalization oor. The parameter space for WIMPs next door is bounded: the DM mass must lie between 1 MeV (to preserve the successful predictions of Big Bang Nucleosynthesis (BBN) and few TeV (from perturbative unitary), while the coupling between the SM and the HS must be su ciently strong to thermalize the dark sector with the SM prior to DM freezeout. The aim of this paper is to establish this bounded parameter space for two minimal models of HS freezeout and systematically map out how this parameter space can be tested in indirect detection, accelerator, and direct detection through a variety of experiments spanning the cosmic, energy, and intensity frontiers. The characteristic signatures of hidden sector freezeout are largely dictated by the Lorentz quantum numbers of the DM and the mediator, together with the choice of portal operator. We focus here on dark sectors which have a leading renormalizable coupling with the SM, through either the vector portal interaction, or the Higgs portal interaction, 2 cos ZD B ; 2 S2jHj2: { 2 { (1.1) (1.2) We will use two simple reference models in this work, HSFO-VP: fermionic DM , annihilating to vector mediators, ZD, that couple to the SM through the vector portal; and HSFO-HP: fermionic DM , annihilating to scalar mediators s that couple to the SM through the Higgs portal. These models of HSFO can be probed via complementary methods across di erent experimental frontiers. Direct searches for the dark mediator are the most sensitive test at accelerator-based experiments, far outpacing more traditional collider searches for DM that rely on a missing energy signature. Direct detection experiments can access the cosmological lower bound on the portal coupling in signi cant portions of the parameter space. Indirect detection remains a powerful probe, provided the DM has an appreciable s-wave annihilation cross-section, as in our minimal vector portal model. Our minimal Higgs portal model, on the other hand, freezes out through p-wave interactions, placing traditional cosmic ray signals largely out of reach. The constraints on our simple reference models provide a reasonably general guide to the physics of more complicated hidden sectors, as we discuss below. We begin with a discussion of WIMPs next door in section 2, where we establish the physical parameter space of our models. In section 3 we discuss di erent experimental avenues to test this parameter space. In sections 4 and 5 we show the consequences for vector and Higgs portal models respectively, and in section 6 we summarize our results. Three appendices describe details of our calculations of thermal scattering rates, Sommerfeld enhancements, and bounds from dwarf galaxies. 2 Parameter space for minimal hidden sector freezeout and the WIMP next door In hidden sector freezeout, DM is part of a larger dark sector that is thermally populated in the early universe. As the universe expands and cools, the relic abundance of DM is determined by the freezeout of its annihilations to a dark mediator state, ! with little to no involvement of SM particles [5{8]. In the simplest realizations of hidden sector freezeout, these dark mediators, , are cosmologically unstable, decaying into the SM through a small coupling. These decays must occur su ciently rapidly to avoid disrupting the successful predictions of BBN, thus generally requiring . 1 s, and providing a cosmological lower bound on the strength of the coupling of the mediators to the SM. When the interaction that allows to decay to the SM is the leading interaction between the two sectors, it will additionally control the thermalization of the dark sector and the SM in the early universe. Requiring the SM and the dark sector to be in thermal equilibrium prior to DM freezeout is the simplest and most minimal cosmology for the origin of the dark sector. DM freezing out from a thermal dark radiation bath in equilibrium with the SM radiation bath is what we de ne as a WIMP next door. We will focus here on the well-motivated cases { 3 { where the vector or scalar portal operators (1.1){(1.2) mediate the leading interactions between sectors, and establish the observable consequences. If the portal coupling is su ciently large to ensure that the SM and the dark sector were in thermal equilibrium for some temperature above the DM freezeout temperature, Teq > Tf , the existence of the SM thermal bath is then su cient to guarantee the population of the dark sector. If, on the other hand, the portal interactions cannot thermalize the two sectors prior to DM freezeout, then some other mechanism, such as asymmetric reheating [9{11], must be invoked to populate the dark thermal bath in the early universe. When the leading interaction between sectors is renormalizable, this minimal cosmology is additionally UV-insensitive: the scattering rates controlling thermalization obey / T , and become more important in comparison with H / T 2=MP l as the temperature drops. Thus min(Teq), the minimum value of consistent with thermalization at a temperature Teq, does not depend on the unknown reheating temperature of the universe (provided TRH > Teq) or other unknown UV particle content. This cosmic origin for DM also signi cantly sharpens predictivity by limiting the degree to which the temperature of the dark sector can di er from the temperature of the SM. In order to determine the thermalization oor min(Teq), we have to distinguish between two cases: rst, when the hidden sector contains (at least) one relativistic species at Teq, and second, when all species in the hidden sector are already nonrelativistic at thermalization, Teq < m=2:46 for all masses. The value of T = m=2:46 is the point where a bosonic species contribution to g drops by a factor of 2, and is our de nition for when a species transitions from relativistic to non-relativistic. In the rst case, the energy in the hidden sector radiation bath is the same per degree of freedom as in the SM, and thermalization requires that inter-sector reactions are e cient enough to transfer a sizable amount of energy per SM degree of freedom. In the second case, all hidden sector species have exponentially suppressed number densities at Teq, and the energy that must be transferred from the SM to thermally populate the hidden sector is thus exponentially reduced. The resulting bounds on minimal coupling strengths are correspondingly much weaker. We focus here on the rst case where the HS has a radiation bath at Teq. In this cosmology the lower bound of the thermalization oor is typically far more stringent than the lower bound from requiring mediator decays to occur prior to BBN. We require that the two sectors thermalize at least at Teq = Tf . For simplicity, we consider minimal models that consist only of a dark matter species and a dark mediator . In order to have a dark radiation bath at DM freezeout, we thus require the mediator to have m < 2:46 Tf . When T m , 2 $ 2 scatterings (SM) $ (SM)(SM) are the dominant process responsible for equilibrating the two sectors. When T m , 1 $ 2 scatterings $ (SM)(SM) become dominant. This temperature scaling is evident in gure 1, where we show 1 $ 2 and 2 $ 2 scattering rates in each of our models as a function of the temperature. In the absence of mass thresholds, 2$2 / T at high temperatures, while 1$2 / m2 =T . The SM has many mass thresholds, which makes the temperature dependence of the net scattering rates less transparent. Full details of the calculation of these scattering rates are presented in appendix A; as discussed there, the thermalization oor that we obtain is an initial estimate, computed up to a factor of 2. The resulting new { 4 { Γ2→2 Zdt→gt Zdg→tt Zde→γe Zdb→gb Zdu→gu Zdππ→ππ 1 Γ2→2 st→gt sg→tt ss→h→bb sb→gb sc→gc sππ→ππ V) 1 e G ( Γ 10-3 10-6 on the mediator mass) as a function of the temperature. Dotted lines correspond to the 2 $ 1 rates after the mediator becomes non-relativistic, T < m=2:46, with m the mediator mass. The gray region at low temperatures corresponds to the uncertain regions near QCD 300 MeV, above which s becomes large, and the leading order calculation is unreliable, and below which 3 $ 2 pion processes (included in the 2 $ 2 rate, shown in light blue) dominate the equilibration. The gray region at high temperature is near the electroweak phase transition Tc = 160 GeV; see appendix A for more details. Left: HSFO-VP. In this model the 2 $ 2 scattering rate is nearly linear with temperature above the chiral phase transition. After pion processes become ine ective and QED processes dominate, the scaling is nearly linear again. Right: HSFO-SP. In this model the 2 $ 2 rate is more sensitive to mass thresholds. It drops sharply after the chiral phase transition (kaon processes have been neglected). cosmological lower bound on portal couplings is shown in gure 2 as a function of Tf , in the regime where m . 0:1 m . The thermalization oor is insensitive to the mediator mass as long as 2 $ 2 rates dominate the scattering, a condition that holds generically (but not always) when the mediator is relativistic at the time of freezeout. Both our minimal models can be described by four independent parameters, namely the DM mass, the mediator mass, the portal coupling , and the coupling D between DM and the mediator. Simpli ed model approaches can be e ective at highlighting the key physical features of classes of DM theories [3, 12{15], and, in that spirit, our simple HSFO models can be taken as useful guides to the physics of a general WIMP next door, as we discuss further below. We emphasize, however, that our minimal HSFO models are, themselves, UV-complete and self-consistent. WIMPs next door have a sharply de ned and bounded parameter space. The dark matter-dark mediator coupling, D, is xed by the dark matter relic abundance, while the coupling of the dark sector to the SM is bounded from below by the thermalization oor. Previous estimates of these thermalization oors (e.g. [16, 17]) have considered a subset of processes and/or studied equilibration at a xed temperature. As for standard WIMPs, the upper limit on the mass of DM is TeV-scale, arising when the interaction governing freezeout becomes non-perturbative. The precise value of this upper bound will depend in detail on the particle content of the dark sector. For instance, for DM freezing out via annihilations to massive dark photons, the upper bound { 5 { Never in Thermal Equilibrium 1 10-3 10-6 10-9 10-3 θ10-4 n i S10-5 10-6 10-8 temperature, Tf . The orange region is where the dark sector is in thermal equilibrium with the SM at freezeout, while the blue region has the two sectors never in thermal equilibrium. In the green region, the SM and hidden sector were in equilibrium at some higher temperature (here near the QCD phase transition), but fell out of equilibrium by Tf , so that the temperatures of the two sectors may drift apart (A.11). The hatched regions are near either the chiral or electroweak phase transitions, where our calculation is less reliable. Right: thermal coupling regions for the scalar model as a function of the freezeout temperature, Tf . Colors and hatching are as in the vector model. The kink in the blue region near 4 GeV is when the e ectiveness of the equilibration from the ss ! f f process near 30 GeV exceeds that from top processes near 200 GeV. For more details, see appendix A. depends on the structure of U( 1 ) symmetry-breaking in the dark sector [18]. Perturbative unitarity constraints in speci c models can further tighten the upper bounds on the DM mass (e.g., [19]). We will indicate in our parameter spaces where obtaining the correct relic abundance in our simple models requires the dark matter-dark mediator coupling to become non-perturbative, D 1. This occurs for m 10{150 TeV in both simpli ed models, where the lower end of the mass range is for small DM-dark mediator mass splitting, and the upper end is for large splitting. The Sommerfeld enhancement (discussed in appendix B) included in our freezeout calculation heavily sculpts this range. When the Sommerfeld e ect becomes very large, our numerical freezeout calculation becomes less reliable, and we will further indicate these regions in presenting our parameter space. However, as the phenomenology does not undergo qualitative changes in this mDM TeV region of parameter space, we will not discuss it in detail. Meanwhile, the number of relativistic degrees of freedom that can be present at temperatures T . 2me MeV are restricted by BBN, which mandates that 2:3 < Ne < 3:4 [20]. We here impose the simple requirement mDM; mmed > MeV. A more careful treatment, accounting for the energy deposited into the SM plasma by mediator decays as well as the temperature drift in regions shown in green in gure 2 where the dark sector has departed { 6 { from equilibrium with the SM prior to DM freezeout would modify these bounds slightly. A detailed treatment of this region is interesting, but beyond the scope of this paper.1 3 Direct, indirect, and accelerator constraints WIMPs next door give rise to signals in many di erent kinds of experiments. In this section, we brie y discuss the relevant experimental results and their application to our simple models, highlighting how signatures can di er from traditional WIMP models. Direct detection. Both our vector portal and Higgs portal models have a leading spinindependent scattering cross-section with nuclei. Unlike for traditional WIMPs, the size of this cross-section is not directly related to the dark matter annihilation cross-section: it is proportional to the square of the portal coupling and can be parametrically small. We will demonstrate that both current and proposed direct detection experiments have the sensitivity to test cosmologically interesting values of the portal coupling. Currently, the best constraints on spin-independent DM-nucleus scattering come from XENON1T [23], LUX [24, 25] and PandaX-II [26] at higher masses, while CDMSlite [27] and CRESSTII [28] set the strongest limits at lower masses.2 We show the current limits, along with projections for several future experiments [31{36], in gure 3. In the gure, we also present the neutrino oor for both xenon and calcium tungstate (CaWO4) [37, 38]. Indirect detection. In contrast to direct detection, results from indirect detection searches are insensitive to the (small) portal coupling, and test the dark matter annihilation cross-section directly. There are multiple sensitive probes of dark matter annihilation in the universe. The most important for our models are the Fermi-LAT limits on dark matter annihilation in dwarf galaxies [40, 41] and Planck constraints on DM annihilations near recombination [1, 42]. Charged cosmic rays are another important source of information about galactic DM annihilation, but are subject to much larger systematic uncertainties arising from their propagation within the galaxy. While AMS-02 measurements of the cosmic antiproton ux [43] can potentially give more powerful constraints on hadronic annihilation channels than searches with gamma rays [44], the di culty in accurately determining propagation parameters remains a serious hurdle. We follow [45] in considering AMS-02 positron results [46], which can place bounds on leptonic channels where searches in photons have little reach, but neglecting antiproton searches, as they constrain channels for which the far less uncertain gamma-ray searches of [40, 41] have good sensitivity. Meanwhile, CMB limits are mainly sensitive to the net energy deposited in the e - plasma by DM annihilations near recombination [47], and are thus robust and 1Our main interest is in the regime where the mediator is light compared to the DM, and thus we will show results in the parameter space where m =m < 1. However, there is a narrow window of viability with m < m < 2m , commonly known as the forbidden region [21, 22]. In this region, annihilation after freezeout is generally kinematically inaccessible, which has the important consequence of removing indirect detection constraints on DM. In this region the cosmological constraints on the portal coupling are qualitatively distinct, and we will not discuss it further here. 2While this work was being completed, the CRESST collaboration published limits on dark matter in the 140 MeV{500 MeV mass range [29, 30]. These constraints are not treated in this work. { 7 { XENON1T (2017) ν Floor (Xe) XENON1T [23], LUX [24, 25], PandaX-II [26] CDMSlite [27], and CRESST-II [28]. Also shown in dashed lines are the projected limits from argon-based DEAP-3600 as well as its proposed 50 ton-year upgrade [31]; the xenon-based experiments XENON1T [32], LUX-ZEPLIN (LZ) [33], and DARWIN [34]; the CaWO4-based CRESST-III and its Phase 2 upgrades [35] and the projected limits from SuperCDMS [36] for both silicon and germanium for both the interleaved Z-sensitive Ionization and Phonon (iZIP) detectors (thin, dotted) and those run in high voltage (HV) mode (dashed). The coherent neutrino scattering oor is shown for both CaWO4 and xenon [37, 38]. The oor for silicon, germanium, and argon is very similar to xenon, while more complex materials, like the CaWO4 in CRESST and the proposed EURECA [39] experiment, can have a substantially di erent oor. nearly model-independent. The HAWC experiment can place constraints on very high dark matter masses [48] in the highly Sommerfeld-enhanced regime; these constraints are currently exceeded by the CMB constraints everywhere, but may become more important as HAWC collects more data, or our understanding of the Triangulum II dwarf galaxy, which dominates HAWC's sensitivity, improves [49, 50]. In principle, H.E.S.S. should have sensitivity to our DM models when m TeV, but they do not provide enough information to allow their results to be reliably reinterpreted.3 3The results of ref. [51], which appeared while this work was being completed, indicate HESS is likely to be slighty more sensitive than dwarfs in this regime. { 8 { Accelerator. On the collider front, there are several potential discovery avenues for hidden sector dark matter. The direct production of DM (or of an invisible mediator) in events with large missing energy is no longer the leading signal, as we will demonstrate below. Rather, the leading accelerator signal is the direct production of the dark mediator, followed by its decay back to visible SM states. Mediators can be produced through rare Kaon and B-meson decays, directly through their interaction with electrons and quarks at LEP and LHC, at lower energy colliders such as Babar, and at beam dump and other intensity frontier experiments such as NA62. They can also be produced in exotic Higgs decays [52, 53]. Precision tests of Z and Higgs couplings can also constrain the mixing between dark and visible states. Astrophysical and cosmological constraints on dark mediators. Beyond the standard suite of DM search strategies, models with long-lived dark mediator states face several additional constraints from astrophysical and cosmological observations. As the requirement that the dark sector be thermalized with the SM places lower bounds on the coupling of the dark mediator, these constraints will largely be important for the HSFO-HP model in the sub-GeV regime where small Yukawa couplings help increase the mediator lifetime. Most constraining here are cooling in Supernova 1987A [17, 54], and early universe limits on the dark scalar lifetime coming from potential disruptions of isotope abundances produced during BBN or dilutions of neutrino and/or baryon abundances [55]. 4 Vector portal We rst consider a simple vector portal model, containing a fermionic DM, , and a dark photon, ZD. This type of model has been studied extensively in the literature, especially to address cosmic ray anomalies (HEAT, PAMELA, and ATIC rst [8, 56], and more recently the Galactic Center excess [18, 53, 57, 58]). In the following, we de ne our model and establish notation. We introduce a massive dark photon Z^D, the gauge boson for a new dark U( 1 )ZD symmetry, that interacts with the SM through kinetic mixing with SM hypercharge [59, 60]. The dark photon mass could arise from the Stueckelberg mechanism [61, 62] or from a dark Higgs mechanism. For the sake of minimality, we will assume a Stueckelberg origin, so that the only dark sector particles in our model are the dark vector and the dark matter. Including a dark Higgs boson could open up additional annihilation channels, such as could become the leading process in the regime mDM mZD ; mhD [18]; we discuss this possibility further in section 4.4. The dark vector Lagrangian is thus given by ! ZDhD, which LZD = 1 ^ B 4 ^ B 1 ^ ZD 4 ^ ZD + 2 cos W + 1 2 ^ ZD ^ B m2ZD0 Z^D ZD ; ^ (4.1) where W is the Weinberg angle and is the dimensionless kinetic mixing parameter. Additionally, we introduce a Dirac fermion with unit charge under U( 1 )ZD and with mass m to serve as DM. Making the standard eld rede nition to diagonalize the hypercharge and ZD boson kinetic terms rescales the dark coupling gD = g^D=p1 2= cos2 W , and { 9 { We can now compute the couplings of the SM fermions and the DM with the ZD gauge boson: gZDf = gZD = gD cos ; g cos W sin (T 3 cos2 W Y sin2 W ) + cos sin W Y ; where Y , t3 are the hypercharge and isospin of the (Weyl) fermion f . The physical Z boson acquires a (vector-like) coupling to : gZ = gD sin : Note that this coupling is -suppressed, contrary to the corresponding coupling of the ZD. In fact, if we expand the couplings in (4.6) and (4.7) to leading order in we nd results in the following mass matrix for the neutral gauge bosons after electroweak symmetry breaking, m2ZD (T3 cos2 W Y sin2 W ) + Y tan W ! Our simple model can be described by four independent free parameters, which we take to be D; ; m and mZD . 4.1 When DM is heavier than the dark photon, it can annihilate via the only annihilation channel in the small limit, and in this limit the thermally averaged ! ZDZD. This is cross-section for this annihilation, be important during freezeout, h vi = h vi0 hS0(v)i, which we implement via a Hulthen potential as described in appendix B. Requiring that this reaction yields the observed relic abundance as measured by Planck, DMh of m ; mZD . The same annihilation cross-section governs indirect detection signals. In order to assess the constraints from indirect detection, we utilize the measurement of dwarf galaxies from the Fermi-LAT and DES collaborations [40]. The energy ux of photons from DM annihilations in an astrophysical source can be expressed as where we have specialized to Dirac DM. Here dN=dE is the number distribution of photons from a single DM annihilation, and J is the astrophysical J -factor, describing the line-ofsight density of dark matter in the direction of the source [64]. We use the 41 dwarf galaxies within the nominal sample of [40] to obtain limits on the DM annihilation cross-section using the procedure outlined in appendix C. The corresponding upper bounds on the cross section as a function of the dark matter mass mX and of the mass ratio mZD =mX are shown in gure 4; the red regions show where the thermal relic abundance is excluded. These regions appear in two distinct places: the region in the lower right is where Sommerfeld enhancements are important, while in the upper left they are not (see appendix B for details). The ux of positrons observed in the AMS-02 experiment [45, 46] can constrain photonpoor annihilation channels. In order to set constraints, we use the limit for one-step e+e channels from [45] and compare this to h v i Br(ZD ! e+e ). In principle, considering all ZD decay modes would slightly improve this result, but achieving this mild improvement is beyond the scope of this work. The resulting exclusion is shown in orange in the right panel of gure 4. Dark matter annihilation during the era of recombination can broaden the surface of last scattering and distort the CMB anisotropies through the injection of electrons and photons into the plasma. For WIMPs annihilating with a velocity-independent cross-section, the e ect of this energy injection can be accurately encapsulated by a redshift-independent e ciency parameter fe (mDM), which depends on the DM mass and the species of particles produced by DM annihilations [65]. Planck results together with results from ACT, SPT, and WMAP limit fe (m ) h vi =m < 14 pb c/TeV [42], allowing for robust bounds to be placed on dark matter models. The fe values for DM annihilation to pairs of SM (4.9) (4.10) mZD 10-1 10-3 mZD mZD =mX . 4.2 Direct detection surements of dwarf galaxies [63] in the HSFO-VP model. The red regions indicate where Fermi dwarf measurements exclude the thermal relic abundance. The brown region on the right of the plot illustrates where the freezeout coupling as determined with and without Sommerfeld enhancement deviates by more than a factor of two so that the determined value of gD becomes inaccurate (B.7). The green region in the lower left is where mZD < 2me. Right: indirect detection bounds on the HSFO-VP parameter space. The thermal relic abundance is excluded by Fermi dwarf s (red), positrons at AMS-02 (orange) [45], and CMB spectral distortions from Planck, SPT, ACT, and WMAP (purple). The constraints are sensitive to the precise locations of the Sommerfeld resonances in the lower right region, which are here only approximately determined. Brown and green regions as in the left gure. particles have been computed in [42, 66]. Due to the rather soft dependence on m for all branching ratios except for photons and leptons, we use the fe values in [42] evaluated at m =2 for non-leptonic channels, together with fe f V V !4`(m ) from [42]. We derive a net fenet e from [66]. For leptonic channels we use fenet(m ; mZD ) = X Br(ZD ! ``)feV V !4`(m ) + Br(ZD ! XX)feXX m 2 (4.11) X X6=` where mZD governs the branching ratios. The resulting CMB bound is shown in purple in the right panel of gure 4, as a function of the dark matter mass mX and of the mass ratio Direct detection experiments are an excellent test of this minimal model, and over much of the parameter space place the most stringent constraints on the portal coupling . The spinaveraged, non-relativistic amplitude-squared for DM to scatter o of a nucleus is mediated by the exchange of both dark vector and Z bosons, and is given by M NR(ER) 2 = M 4m mN 2 = gD2 2A2F 2(ER) fn(ZD) m2ZD + 2mN ER + m2Z fn(Z)sW m2ZD 2 ; (4.12) nucleus, and fn(ZD) and fn(Z) are given by where A is the mass number of the target nucleus, F 2(ER) is the Helm nuclear form factor [67, 68] as a function of the recoil energy ER, sW sin , mN is the mass of the f (X) = n 1 A Z(2gu;X + gd;X ) + (A Z)(gu;X + 2gd;X ) ; (4.13) with Z the atomic number, and gX;u and gX;d the couplings of the boson X with up and down quarks in (4.8). Here we have retained the momentum dependence in the propagator from ZD exchange, as it is needed to accurately describe the scattering when mZD . N v , where the DM-nucleus reduced mass is N = m mN =(m + mN ). When the scattering amplitude does not depend on the DM velocity, then the event detection rate per unit detector mass in the experiment can be expressed as [69, 70] R M NR(ER) = 2 m dER M NR(ER) 2 (ER) (ER): where is the local DM density, (ER) is an experiment-speci c selection e ciency, and (ER) is the mean inverse speed [70] de ned by Z 1 0 Z (ER) = v>vmin(ER) v f (v) 3 d v for which, following the experiments, we use the expression in ref. [68].4 If the amplitude is independent of the recoil energy to leading order, it is reasonable to approximate M NR(ER) ! M NR(0) in (4.14), allowing for the particle physics contributions to the rate to be entirely factorized from the experimental and astrophysical inputs. Experimental results are typically expressed in terms of a cross-section that has been factorized in this manner and further simpli ed by de ning an e ective per-nucleon cross-section, facilitating comparison between di erent experiments. For the HSFO-VP model, this DM-nucleon cross-section is 0 n = 2 n M NR(0) 2 A2 = where we have de ned the nucleon-DM reduced mass n = m mn=(m + mn). However, for the HSFO-VP model, M NR(ER) is sensitive to the recoil energy once m2ZD . 2mN ER 2 N v2 . In order to correctly account for this important e ect, we will determine the excluded cross-section via 4While this is experimental usage, a more accurate expression for (ER) can be found in ref. [70]. n = (4.14) (4.15) (4.16) (4.17) mZD 10-3 -4 -3 -2 -1 0 -8 -9 -5 -6 -7 1 10 102 103 104 10-9 HSDM Vector Allowed Within the shaded blue region, the two sectors were not in thermal equilibrium at freezeout. The combined region excluded by indirect detection constraints from gure 4 is shown by the black region. Above the dashed pink line is where ZD is non-relativistic at the freezeout temperature (mZD =Tf > 2:46). The brown region on the right of the plot illustrates where the freezeout coupling as determined with and without Sommerfeld enhancement deviates by more than a factor of two (B.7). Right: the tan region shows the direct detection parameter space for WIMPs next door in the HSFO-VP model. At low dark matter masses, Planck excludes all dark vector masses and couplings. Shown in green are three speci c lower bounds on the direct detection cross section for the mass ratios mZD =m = 10 1; 10 2, and 10 3 . We also show the neutrino oor for xenon and CaWO4 (used in CRESST) with dashed purple and blue lines, respectively. where the function R is determined separately for each experiment. Given masses for the DM and dark vector, the relic density constraint xes D. The latest XENON1T [23], LUX [24, 25], PandaX-II [26], Super-CDMS [71], CDMSlite [72] and CRESST-II [73] searches then determine the maximum allowed value of the portal coupling . We show these upper bounds in the left panel of gure 5. Sensitivity is greatest at small values of mZD =m , thanks to the 1=m4ZD behavior of the nuclear matrix element. However, the sensitivity saturates when m2ZD . 2mN Emin (the threshold energy of the experiment), and the propagator in the matrix element is dominated by the momentum. Over a sizable region where mZD =m . 0:01 and m 10 GeV, current direct detection limits are sensitive enough to exclude values of the portal coupling at and below the thermalization oor. This region is shown in blue in the left panel of gure 5; see appendix A.1 for details of its determination. Future direct detection experiments will be able to test this cosmological origin for DM over a broader range of DM and mediator masses. In the right panel of gure 5, we show the direct detection parameter space consistent with our HSFO-VP WIMP next door (in tan). As gure 5 shows, this cosmological history for DM can predict spin-independent cross-sections well below the neutrino oor. 10-1 10-2 10-3 10-4 10-6 10-7 10-8 10-9 HSDM Vector Allowed 10-3 10-2 10-1 10 100 1000 1 mZD (GeV) vector mass and portal coupling. In the red (blue) shaded regions, indirect (direct) detection alone excludes that point in f ; mZD g parameter space for all values of m > mZD . In the white region, > mZD are excluded by a combination of direct and indirect detection. The future direct detection sensitivity is determined by assuming CRESST-III Phase 2 and DARWIN ( gure 3) are placing their nominal limits. 4.3 Accelerator and other mediator constraints Direct searches for the ZD are the leading terrestrial signal of this model. As summarized in [74{77] and in gure 6, there are many constraints on massive vector bosons kinetically mixed with SM hypercharge. Most of these results come from searches for rare meson decays, beam dump experiments, precision electroweak tests, direct production at BaBar or the LHC, and Supernova 1987A. Figure 6 shows the parameter space for a vector portal WIMP next door as a function of mZD and . As shown there, Supernova 1987A uniquely probes the thermalization oor in a limited range of dark photon masses at around few 10 2 GeV. Furthermore, especially at low masses, terrestrial searches for dark photons bound more tightly than the direct detection constraints do alone. Figure 6 highlights the unique capability of direct detection experiments to probe otherwise challenging regions of dark photon parameter space. Within the blue region, direct detection excludes dark photons for any choice of m > mZD . Indirect detection from CMB experiments cuts o the entire region of dark photon parameter space below HJEP02(18) 400 MeV (red region), while a combination of both direct and indirect detect results exclude > mZD in a region up to 4 GeV for a range of portal couplings. Future direct detection experiments, DARWIN and CRESST-III Phase 2 [34, 35], will greatly cut into this range (green line), even excluding down to the thermalization oor near 500 MeV. However, while the net impact of these direct mediator searches is generally subdominant to the combined constraints from direct and indirect detection for the minimal HSFO-VP model, it is important to emphasize that they provide complementary information. In particular, gure 6 shows that any dark photon discovered in meson decays or at high-energy colliders is su ciently strongly coupled to the SM to populate a dark radiation bath in the early universe, regardless of the identity of dark matter. Further, as we will discuss below, simple extensions to the minimal model can suppress the direct detection cross-section, thus leaving dark photon searches as the leading terrestrial test of vector portal WIMPs next door. A summary of all constraints is shown in gure 7 as a function of m and mZD =m . As before, we show the union of -independent constraints from Fermi dwarfs, AMS-02 positrons, and the CMB with the black shaded region. The shaded green region denotes where the most important bound on comes from collider, beam dump, and supernova searches. Above the pink dashed line, the mediator was non-relativistic at dark matter freezeout, and thus the oor from thermalization can be much lower. As in gure 4, the solid green in the lower left corner has mZD < 2me, causing issues with BBN, and the brown region on the right of the plot illustrates where the freezeout coupling as determined with and without Sommerfeld enhancement deviate by a factor of 2. We do not show constraints from mono-X searches at the LHC, since they are generically weaker than the constraints coming from direct detection experiments. In particular, the most stringent mono-X constraint arises from the ATLAS and CMS mono-jet searches performed with Run II data [78, 79]. Values of as small as 0:1 are only probed in a small region of parameter space at around m 100 GeV and m (0:8 1)mZD . 4.4 Beyond the minimal model of dark vector interactions While we have worked with a minimal two-species model consisting only of fermionic DM and the vector mediator, the salient features of this model are representative of the behavior of a broad class of dark sectors with a vector mediator. In this section, we brie y discuss the modi cations of the dark sector phenomenology obtained by introducing new dark degrees of freedom (or altering the assumed quantum numbers of DM), and argue that our minimal two-species HSFO-VP model provides a reasonable general guide to the characteristic sizes and locations of signals for vector portal WIMPs next door. To begin with, any additional relativistic species in the thermal plasma at freezeout will to 2 contribute to the Hubble parameter, thereby requiring a mild increase of the value of D needed to obtain the thermal relic abundance. The DM relic abundance is proportional g S=pg at freezeout. Neglecting the logarithmic sensitivity of the freezeout temperature to g S, we can thus simply estimate the e ect of adding additional equilibrated dark species by rescaling D to absorb the shift in g S=pg . This is a minor quantitative e ect, particularly at relatively high DM masses where g SM (Tf ) 10-1 10-2 10-3 -3 -2 -1 0 -10 -4 -5 -6 -7 -8 -9 1 10 102 103 104 mχ (GeV) region shows the combined indirect detection constraints (see gure 4) and excludes all values of for the indicated masses given TSM = THS at the time of freezeout. Regions where direct detection constraints are superseded by constraints on the mediator (collider, beam dump, supernova, etc.) are shown in green. Labelled contours show the maximum value of permitted by either direct detection or mediator constraints, whichever bound is stronger. The blue region indicates where thermalization cannot occur prior to DM freezeout, the brown contour illustrates where the freezeout calculation is less reliable, and above the pink dashed line the mediator is non-relativistic during freezeout. The solid green region in the lower left is where mZD < 2me. An excellent motivation for introducing additional dark species is to provide a dark Higgs mechanism to generate mZD [8, 18, 80]. Our model assumed a Stueckelberg mechanism for simplicity. Using a dark Higgs to generate mZD is a generic alternative scenario, but with a dark Higgs comes additional model dependence, especially through the choice of the DM's U( 1 )D quantum numbers. When the dark Higgs is light, it can furnish additional annihilation modes: depending on the spectrum, both the s-wave ! ZDhD process and the p-wave ! hDhD process can contribute to freezeout [80]. (We continue to assume that the vector portal coupling dominates the dark sector's interactions with the SM.) The additional annihilation modes change the speci c value of D needed to obtain the thermal relic abundance, generically by no more than an O( 1 ) amount. These additional annihilation modes also alter the detailed cosmic ray spectrum for indirect detection. When p-wave contributions are important, the expected annihilation cross section for CMB and galactic signals can be decreased, generically by a factor of no more than O( 1 ). Thus introducing these additional annihilation modes generally changes indirect detection signals quantitatively but not qualitatively. On the other hand, a dark Higgs can drastically impact direct detection. A Stueckelberg mass for the dark photon requires Dirac dark matter, and thus yields unsuppressed spin-independent direct detection cross-sections. However, a dark Higgs mechanism allows the Dirac spinor to split into two Majorana mass eigenstates, the lighter of which is dark matter [81]. If the mass splitting is small so that the Majorana states are nearly degenerate (pseudo-Dirac), then the leading spin-independent cross-section is now inelastic. Inelastic scattering is signi cantly more challenging to observe at direct detection experiments, but some signals are still possible [82]. However, as the mass splitting increases, the dominant direct detection signals come from elastic processes. These processes can arise at tree level, from the now axial-vector coupling of the DM to ZD. At relatively high dark vector masses, the axial components of the SM{ZD couplings are sizable, giving rise to spin-dependent cross-sections. The vector SM{ZD couplings yield spin-independent cross-sections suppressed by DM velocities or nuclear recoil momenta, giving small but still potentially interesting signal rates [83]. Elastic spin-independent cross-sections are also induced at one loop [84{88]. The size of this contribution is thus sensitive to the UV eld content of the dark sector. Finally, while the coupling of the dark Higgs to the SM is sub-leading for thermalization, the exchange of the dark and SM Higgses gives a spin-independent cross-section, and, depending on the size of the dark Higgs-SM couplings, could provide the leading direct detection signal; for further discussion of Higgs-portal direct detection, see section 5.2 below. If there are additional light dark sector species, , then ZD can have open decay modes within the hidden sector, and when these are active one generally expects Br(ZD ! ) ' 1. Letting ZD decay can eliminate (if is stable) or modify (if is unstable [45]) cosmic ray and CMB constraints on DM annihilation, and, often for either case, subject the mediator to the generally weaker terrestrial searches for ZD ! invisible [89{91]. As the vector portal coupling is the leading interaction between sectors, these new light states will have longer lifetimes than the dark vector, which can potentially lead to stringent cosmological constraints. In particular, as BBN does not allow for an additional radiation species with the SM temperature TSM, a model that maintains equilibrium with the SM through the vector portal has limited prospects for including stable dark radiation. As illustrated by the small green region in the left panel of gure 2, a vector portal hidden sector equilibrated with the SM cannot decouple from the SM at su ciently early times to permit signi cant departures of the HS temperature from TSM. Interestingly, a model with a portal coupling of 2 10 9 and DM that freezes out after the chiral phase transition may provide a WIMP next door that permits additional dark radiation. As this model could eliminate indirect detection signals, it would open up interesting parameter space for GeV-scale DM, together with dark radiation signals that could be observable at CMB-S4 [92]. Verifying the existence of this dark radiation window, via a more detailed calculation of the thermal production rate of dark photons from the hadronic plasma near the chiral phase transition, is an interesting topic for future studies. Our reference model assumes fermionic DM. If DM is instead a complex scalar, the story is qualitatively unchanged: the leading annihilation cross-section is s-wave, while the leading direct detection cross-section is spin-independent and unsuppressed. Once again, the introduction of a dark Higgs would make only minor changes to the indirect detection signals, while potentially introducing sizeable and model-dependent changes to the direct detection signals. Finally, in our HSFO-VP model and the variants above, annihilations of only one representation of the dark U( 1 ) symmetry are important during freezeout. Introducing more states in di erent representations and allowing coannihilation to be important in determining the relic abundance can signi cantly alter the phenomenology and open up di erent areas of parameter space, but represents a much greater departure from the minimal model discussed here. To summarize, our minimal model provides a good guide to the essential physics of vector portal WIMPs next door. Many possible additions to the dark sector would change signals qualitatively, by O( 1 ) amounts, e.g., through a ecting Hubble. Indirect detection signals are especially robust, as adding additional annihilation channels, etc., generically changes cosmic ray signals quantitatively but does not suppress them signi cantly below expectations for an s-wave thermal relic. For vector portal models there is very little scope to eliminate indirect detection signals via ZD decays to dark radiation. On the other hand, direct detection signals are especially sensitive to the origin of dark symmetry breaking. The direct detection signals for the minimal model we present are maximally predictive; in a dark sector with a dark Higgs mechanism, direct detection signals can be suppressed by model-dependent amounts. 5 Higgs portal Here we de ne a simple reference model for a Higgs portal WIMP next door, HSFO-HP. We consider a Majorana fermion dark matter, , with a scalar mediator, S, that interacts with SM states through a (small) Higgs portal coupling. A useful simple model is [93] (see also [5, 94, 95]) 1 2 L = Lkin (yS) ( + H.c.) + 2 s S2 2 4! s 4 S 2 S2jHj2 V (jHj); where we use the usual conventions for the Higgs potential, V (H) = 2jHj2 + jHj4: We should also add to this Lagrangian the interaction of the Higgs with the quarks, leptons and gauge bosons of the SM. These interactions will be inherited by the dark scalar through its mixing with the SM Higgs. In the Lagrangian in (5.1){(5.2), we have imposed a discrete symmetry taking S ! S, ! i , thus forbidding cubic and linear terms in S as well as (5.1) (5.2) this assumption, e.g., by considering a model with a less stringent symmetry structure, allows the DM to have a bare mass m0 independent of the scalar VEV. Unless there is some theoretical reason to expect m0, to be at the same scale as yvs, one would generally expect either one or the other to dominate. The case m the results shown above. However even when m m0 yvs m0 is well-described by yvs, the value of the Yukawa coupling y determined in our freezeout calculation will not change much: freezeout is dominated by the t- and u-channel processes in (5.8), which are independent of vs. Thus the thermal relic result for y is largely determined by m , regardless of the origin of this mass. Therefore, at a xed m , ms, and sin , introducing an independent bare DM mass results in a smaller vs, and thus a larger value for (5.6). As discussed in appendix A, most of the scattering rates important for thermalization are dependent on sin , but the process ss ! f f depends instead directly on . Decoupling the DM mass from vs thus makes the ss ! f f process more important relative to the other processes. As can be seen from gure 1 right, the ss ! f f processes peak near 30 GeV and die o sharply afterwards, so that adding a bare DM mass means that for T 30 GeV, thermalization becomes more e cient for the same value of sin . Viewed in terms of , the case treated in detail here where m = yvs results in the largest parameter space above the thermalization oor, i.e., it allows thermalization for smaller . The case where m yvs has smaller sin for a xed and thus typically requires larger to thermalize. Insofar as direct detection cross-sections depend on sin , and for most temperatures thermalization is controlled by sin , the ability of direct detection experiments to probe the thermalization oor is largely una ected by the introduction of a bare Majorana DM mass. The exception is in regions T 30 GeV where thermalization is dominated by the ss ! f f process so that the thermalization oor is located at smaller values of sin . It is worth bearing in mind that our reference model, strictly speaking, predicts the scale of its own symmetry breaking phase transition. As discussed in appendix A, this phase transition happens long before DM freezeout in our reference model; however, this is an important caveat to keep in mind when considering dark Higgs sectors with a di erent symmetry structure. Unlike in the vector case, making the dark matter Dirac instead of Majorana only provides quantitative shifts and no qualitative changes, as this change does not alter the leading spin-independent matrix element for direct detection. A much more substantial change arises when one considers scalar dark matter: in this case the leading annihilation channel ! ss is now s-wave, and indirect detection signals become important and constraining [53, 58]. (An unsuppressed s-wave annihilation cross-section could also arise for fermionic DM in the presence of CP-violation [146], or if the dark sector contains a light pseudo-scalar a in addition to the light scalar s, such that important annihilation channel [147].) Broadly speaking, the indirect detection signals and constraints in the presence of an s-wave annihilation cross-section are generally similar to the results found for the HSFO-VP model. In particular, constraints from the CMB are nearly identical, while limits from cosmic ray searches are qualitatively similar, achieving sensitivity to annihilation cross sections at the same order of magnitude. ! as can be an Again, adding additional light states can allow for invisible mediator decays (and, in the presence of a leading s-wave annihilation cross section, mediator decays into stable dark states would allow indirect detection signals to be re-suppressed). Also as before, these additional dark sector states can face strong cosmological constraints; these can become especially acute for low-mass unstable dark sector states, which can become very long-lived when the Higgs portal is the leading interaction between sectors. Importantly, Higgs portal WIMPs next door are amenable to larger temperature drifts between the SM and hidden sector temperatures (see right panel of gure 2), which allows more scope for relativistic species at BBN. Stable dark radiation species would give visible signals in CMB-S4. To summarize, one of the major di erences between vector portal and Higgs portal WIMPs next door is that the Higgs portal models o er more opportunities to include dark radiation. There is slightly more model-dependence in the detailed location of the thermalization oor, as di erent choices in constructing the dark Higgs sector can alter the relationship between sin and . Straightforward extensions and variations of our minimal HSFO-HP model allow for the introduction of an s-wave annihilation cross-section and thus reintroduce indirect detection signals and constraints, while leaving direct detection signals qualitatively undisturbed. 6 Summary and conclusions In this paper, we have comprehensively assessed the current constraints on and discovery prospects for a class of minimal hidden sector freezeout models, where the DM relic abundance is set by the freezeout of a single DM species, , into a dark mediator, . We consider the well-motivated scenario where the leading interaction between the dark sector and the SM is renormalizable, for two simple reference models, HSFO-VP, where the mediator is a dark photon kinetically mixed with SM hypercharge, and HSFO-HP, where the mediator is a dark scalar that mixes with the SM Higgs boson. In both cases, the interaction with the SM renders the mediator cosmologically unstable. Experiments across the cosmic, intensity, and energy frontiers provide complementary information about the nature and cosmic history of hidden sector DM in these reference models, and leading signals can di er substantially from models of more traditional WIMP-like dark matter. We have carefully considered the cosmology of these HSFO models. When the interaction between the SM and dark sectors is su ciently strong to ensure that the two sectors achieve thermal equilibrium prior to dark matter freezeout, the cosmology is highly predictive. When the leading interaction between the dark sector and the SM is renormalizable, this minimal cosmology is additionally UV-insensitive: the scattering that works to equilibrate the two sectors becomes increasingly important, relative to the Hubble rate, as the universe expands. Thus, as we emphasize, our simple models of HSFO de ne a minimal and robust dark cosmology. We de ne the WIMP next door as dark matter that freezes out from a dark radiation bath in thermal equilibrium with the SM. One major consequence for WIMPs next door is the existence of a new cosmological lower bound on the portal coupling, the thermalization oor min(Tf ): the minimum value of that allows the dark sector to reach thermal equilibrium with the SM before DM freezes out at the temperature Tf . We provide an initial computation of this thermalization oor for both Higgs and vector portal couplings. This bound is signi cantly more stringent than the bound from BBN over the vast majority of parameter space, and can be terrestrially interesting. While obtained in the context of our minimal models, these results should generally serve as a good guide for thermally populated DM in more general `next door' hidden sectors. WIMPs next door provide a sharply predictive scenario for hidden sector DM. Requiring that the dark radiation bath attains thermal equilibrium with the SM prior to DM freezeout enforces a relationship between the temperatures of the two sectors, so that the parameter space is bounded and clearly de ned. Both the DM and mediator masses are bounded from below by BBN and from above by perturbativity, while the coupling of the mediator to the SM is bounded from below by min(Tf ). Dark matter direct detection experiments can access a signi cant portion of the parameter space for WIMPs next door, and in some regions can probe all the way down to the thermalization oor min(Tf ). Low-mass (m . 10 GeV) WIMPs next door with unsuppressed s-wave annihilation cross-sections (like in the HSFO-VP model) can be robustly excluded from their impact on the CMB. On the other hand, WIMPs next door with a velocity-suppressed p-wave annihilation cross-section (like in the HSFO-HP model) predict interesting direct detection signals for DM candidates in the low-mass range that o er an attractive target for low-threshold direct detection experiments. Additionally, unlike in the case of standard WIMPs, a notable fraction of the viable WIMP next door parameter space dwells underneath the coherent neutrino scattering oor, providing a target for future direct detection experiments that would need some ability to distinguish these signals from the neutrino background. The leading accelerator signature of WIMPs next door is the production of dark mediators. A variety of experiments currently constrain both kinetically-mixed vectors and Higgs-mixed scalars. Both existing experiments, such as LHCb and NA62, and proposed experiments, like SHiP, MATHUSLA, CODEX-b, and FASER, project sensitivity to signi cant regions of unexplored territory, and will either lead to a revolutionary discovery or greatly improve the constraints on this parameter space. At higher masses, the multipurpose LHC experiments have the best opportunities to discover the mediators in exotic Higgs decays (Higgs portal) or through direct production (vector portal). By contrast the traditional LHC mono-X searches give little hope of nding DM. One avenue for future work is improving on our estimates of the thermalization oor, most critically through the SM's two phase transitions. In particular, a careful treatment of dark photon production through the chiral phase transition could be important for understanding windows for dark radiation and thus low mass (m . 10 GeV) vector portal WIMPs next door. Acknowledgments We are grateful to H. An, G. Kribs, A. Long, S. McDermott, B. di Micco, T. Robens, V. Martinez Outschoorn, M. Williams, Y. Zhang, and J. Zupan for useful discussions. The work of JS and JAE is supported in part by DOE grant DE-SC0015655. JS acknowledges additional support from DOE grant DE-SC0017840. SG acknowledges support from the University of Cincinnati. SG is supported by a National Science Foundation CAREER Grant No. PHY-1654502. JAE and SG thank the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293, where part of this work was performed. SG is grateful to the hospitality of the Kavli Institute for Theoretical Physics in Santa Barbara, CA, supported in part by the National Science Foundation under Grant No. NSF PHY11-25915, where some of the research reported in this work was carried out. A Thermal (de)coupling In this appendix, we describe in detail our estimate of the minimum portal coupling necessary to thermalize the hidden sector with the Standard Model in the early universe, and point out some ways to improve on our treatment. We always assume that the particles of the hidden sector (i.e., the DM, , and the dark mediator, ) rapidly thermalize among themselves and can be characterized by a single temperature. As we focus on the regime where the mediator forms a radiation bath at the time of equilibration, this assumption is well justi ed. We begin with some general comments. First, we are mainly interested in the thermal interaction rates between particles at temperatures T & m, where classical statistics do not apply. Once nal state blocking and/or enhancement factors can no longer be neglected, the evaluation of collision terms becomes signi cantly more technically involved. Fortunately, the SM thermal bath is dominated by fermions: empirically, classical statistical (\MaxwellBoltzmann") treatments of relativistic scattering processes involving fermions provide a reasonable approximation to the full quantum statistical expressions, agreeing within a factor of . 2 (see, e.g., [11]). Thus we employ classical statistics to evaluate the rates for 2 $ 2 scattering processes like f ! (g= )f , f ! hf , and the crossed processes (g= ) ! f f , h ! f f , etc. The other major simplifying approximation we make is to neglect 2 $ 2 scatterings with EW gauge bosons. This is a good approximation thanks in large part to the sheer numerical dominance of quarks in the SM plasma, combined with s > 2 and the larger color factors present in QCD scattering amplitudes. The processes f ! (W; Z)f 0 and their crosses are thus numerically unimportant compared to f ! gf at high temperatures at our level of precision. At temperatures T mW , the W , Z masses render these scatterings irrelevant. Meanwhile all-bose processes such as V ! V V are only important for a small range of temperatures at and below the electroweak crossover Tc 160 GeV [148] before Boltzmann suppression kicks in. A study of dark mediator production from electroweak boson scattering in this regime is interesting, but involves a careful treatment of (evolving, nonperturbative) thermal masses, and is beyond the scope of this paper. Above the electroweak crossover, the leading scattering processes that mediate thermalization have a di erent structure, as discussed further for each model below. We incorporate three-loop running of s above the chiral phase transition (everywhere in this work, we use QCD 300 MeV). However, below the QCD phase transition, 2 $ 3 ! pion processes, e.g., + 0 + 0ZD, dominate. Due to the qualitative similarities, these are lumped into our 2 $ 2 processes in the discussion below. In order to estimate these processes, we expand the chiral Lagrangian to leading order in fp; m g=4 f to compute the relevant cross-sections. As the thermally averaged cross-section receives important contributions from values of s where this expansion is no longer reliable, we introduce a simple regulator that ensures (s) has physical high s behavior ( (s the extremely broad QCD (f0) resonance would be expected to perform the bulk of the unitarization of pion scattering, we de ne our multiplicative regulator to be m ) / s 1). As Reg(s) = <8 1 h : m2 2 (s m 2)2+m2 2 i m s > m ; (A.1) is chosen to enforce the desired UV behavior and m ) are set to 500 (600) MeV. While this assumption is grounded in physical expectations, it is a somewhat arbitrary choice and a di erent regulator could modify the results signi cantly. Further, while pion processes are included, kaons and other light QCD resonances, have been neglected. These states have a higher Boltzmann suppression, which should generally make their contribution subleading, but their contribution may still be considerable for scalar production thanks to the larger strange Yukawa coupling. Lastly, near the phase transition the strong self-interactions of the hadronic plasma likely make signi cant thermal corrections to the mediator production rates. For these reasons, our min(Teq) results in this region are much more uncertain than the roughly factor of two precision that we have elsewhere. In gures 1 and 2, we shade temperatures near both the electroweak (T 160 GeV) and QCD (T 300 MeV) phase transitions to highlight the large uncertainties in these areas. At the level of precision we are using, we can check whether a process is in equilibrium simply by comparing it to the Hubble rate, requiring int(T ) > H(T ) = r 4 3g (T ) T 2 45 Mpl : (A.2) Here the e ective number of relativistic degrees of freedom g (T ) includes degrees of freedom from the dark mediator as well as those of the SM. We have checked that this simple equilibration criterion reproduces the results of a full numerical treatment of the energy transfer rate between sectors to within an O( 1 ) factor; see also [149, 150]. The ratio of scattering rates 1$2= 2$2 in both HSFO-VP and HSFO-HP models is shown in gure 12. Regions below the dashed pink line in gure 12 have a radiation bath at freezeout. The gure demonstrates that 2 $ 2 scattering processes dominate thermalization in almost all of this region. A.1 Vector decoupling In our vector model, the 2 $ 2 rate is dominated by the scattering processes ZDf ! g= f , ZDg= ! f f , ZDf ! hf , and ZDh ! f f . For each of these cross-sections, 12!34, the mχ 1 10-1 10-2 1 0 -1 -2 102 1 10-1 10-2 1 0 -1 -2 -3 102 In regions below the pink dashed line, the dark mediator furnishes a radiation bath at the time of DM freezeout. In the region where 1$2 & 2$2, the true value of min(Teq) is lower than the value we quote from 2 $ 2 processes alone. Left: HSFO-VP. For dark vector masses near the Z-pole, the 1 $ 2 grows very large. Right: HSFO-HP. thermal average given by Maxwell-Boltzmann statistics is [151] h 12!34viT ne2q = Z 1 psmin g2 s 2 2s(m21 +m22) (m21 m22)2 K1 p T s 8 2m21K2 T m1 where psmin = Max[m1 + m2; m3 + m4] and Kn is the modi ed Bessel function of the second kind. This expression for the cross section is related to the total reaction rate by 12!34(s)dps; (A.3) ZD int;2!2(T ) = X X;Y;Z h ZDX!Y Z viT neXq: (A.4) 8We use 2Yeq(Tf ) our freezeout calculation. If int;ZD (T ) < H(T ) for all T > Tf , then the two sectors are not in thermal equilibrium at the time of freezeout and may have completely di erent temperatures. The resulting value of min(Tf ) required for the sum of these 2 $ 2 scattering rates to equilibrate the dark sector with the SM is indicated by the blue curve in the left panel of gure 13. Individual contributions to the scattering rate are shown in gure 1. In this model, Tf with the larger splitting for m 10 TeV and the smaller splitting for m m =(30 20), 1 GeV.8 The rate of inverse decay processes can potentially be larger than the 2 $ 2 rate. We continue to neglect nal state blocking and stimulated emission factors in evaluating this rate. Here this approximation is reasonable as the 1 $ 2 rate becomes most important in comparison with the 2 $ 2 rate as T mZD . We include all SM species in evaluating this Y0(Tf ) as the de nition of our freezeout temperature, as determined numerically in rate, including EW gauge bosons. We compute the 1 $ 2 rate as ZD int;1!2 = 1 nZD;eq Z d3p (2 )3 fZD;eq(E) mZD E ZD ; using the Bose-Einstein distribution feq(E) for the ZD. This yields ZD int;1!2(T ) = ZD 8 > > > < >>> 12 3 T 2 mZD K1 K2 mZD T mZD T T mZD T . mZD where ZD / 2 is the zero-temperature width of the dark vector. Here in the top line we have used the Bose-Einstein result for the relativistic nZD;eq, while the bottom line gives the Maxwell-Boltzmann result. At low temperatures, the Bessel function ratio asymptotes and ZD / mZD , this results in a rough parametric scaling of to unity. As the interaction rate for the 2 $ 2 processes scales as int;2$2 / T in the UV ffewg f s or EM g m2ZD ; so decays and inverse decays are typically unimportant for thermalization unless Tf . mZD . A comparison of the minimum allowed value for only the 2 $ 2 processes (which do not depend on the vector mass) and only the 1 $ 2 process for xed values of mZD =Tf is shown in the left panel of gure 13. As the width of the dark photon rapidly increases for xed at mZD mZ , much smaller values of can thermalize the two sectors in this region. Below the QCD con nement scale, 2 $ 3 pion processes brie y dominate, but after they become subdominant near temperature of 50 MeV, 2 $ 2 process becomes proportional to EM , and decays and inverse decays become relatively more important for thermalization. For mZD the thermalization. m Tf , the large width of ZD allows for this to dominate The narrow green region in gure 2 corresponds to a scenario where at higher temperatures near 200 MeV, the hidden sector and standard model were in thermal equilibrium, but have since decoupled. The temperatures of the two sectors are then allowed to drift apart. For the vector model, this is a very narrow region of parameter space, so we defer our detailed discussion of this interesting region to the scalar decoupling section near (A.11). Finally, depending on the origin of the dark vector mass, the dark vector model may implicitly contain a symmetry-breaking phase transition where the mass of the dark vector is generated. In this case, when both SM and dark sectors are in the unbroken phase, the leading scattering process responsible for bringing the two sectors into thermal equilibrium is f f ! [152]. As this process depends on the dark Yukawa coupling, and D S, for the purposes of determining the minimal portal coupling that can yield thermalization, the unbroken UV interaction rate is unimportant compared to the interaction rate after both electroweak symmetry breaking and dark symmetry breaking. (A.5) (A.6) (A.7) 10-6 10-7 ϵ10-8 10-9 1→2 mZD = 1 Tf 2→2 10 1/3 10-1 10-2 10-3 10-4 inθ10-5 S 10-6 10-7 10-8 10-9 1→2 ms Tf = 1 2→2 10 1/3 10-3 10-2 10-1 10 100 1000 10-3 10-2 10-1 10 100 1000 104 processes and only the 1 $ 2 process for xed values of m =Tf . Left: the HSFO-VP model for three choices of mZD =Tf where m =Tf very small masses and for mZD 20. For mZD . Tf , 1 $ 2 processes are unimportant except at mZ . For mZD Tf , 1 $ 2 processes can dominate everywhere. Right: as left gure, but for the HSFO-HP model. Here when ms . Tf , 1 $ 2 processes are unimportant except at very small masses and for very large masses, where the very large decay rate into W and Z bosons takes over. Again, for ms Tf , 1 $ 2 processes are dominant. The dark scalar model predicts the critical temperature, Tc, of its phase transition from the symmetric vacuum (hSi = 0) to the broken vacuum where S develops a VEV. This phase transition occurs comfortably prior to DM freezeout, Tc Tf , as we now demonstrate. This model exhibits a second-order phase transition, so Tc occurs when the second derivative of the thermal potential at the origin changes sign. To estimate the critical temperature and understand its relation to other mass scales in the dark sector, it su ces to consider the one-loop approximation to the thermal e ective potential for S, yielding V 00(0) = 1 2 T 3ms4y2 h 8m2 ms i + y2T 2 2 ms2 Sign y2T 2 1 6 1 6 4m2 ms2 4m2 ms2 ms2 (A.8) Setting (A.8) to zero, we can estimate Tc. If Tc Tf , then DM freezeout occurs during the broken phase. At small scalar masses, the largest contributions to (A.8) are 112 y2T 2 812 y2m2 , so that Tc q 3 m 2 0:4m Tf . For larger scalar masses, V 00(0) V 00(0) 112 + 8mms22 y2T 2 12 ms2, and the critical temperature can be much higher than m . Until the electroweak phase transition, the scalar's only tree-level interactions with the SM are with the Higgs multiplet. In the unbroken phase, above both the electroweak and dark sector phase transitions, there is thus a single process controlling the equilibration of 2 ln cosh( (p0 cosh( (p0 + 2jp~j)=2) 2jp~j)=2) 1 1 1 q : 1 the two sectors, ss ! hh.9 As this process involves only bosons with masses mi accurately determine the interaction rate it is necessary to use Bose-Einstein statistics. Using the techniques of [153], the thermally averaged scattering rate can be expressed as an integral over the total CM energy-squared, s, and p, the magnitude of the threemomentum of the CM frame in the rest frame of the plasma. De ning, as usual, the scattering rate UV as the collision term divided by the (equilibrium) number density of one of the initial state particles, we have UV = 1 Z 1 Z 1 dp ds 256 (3)T (2 )3 s0 0 p0 sinh2( p0=2) ln cosh( (p0 cosh( (p0 + 1jp~j)=2) 1jp~j)=2) 1 1 Here p0 ps + p2, = 1=T , and i 4mi2=s. We evaluate this integral numerically. It diverges as the lower limit on s is taken to zero, s0 ! 0, re ecting the divergence in the Bose-Einstein distribution f (E) as E ! 0. This divergence is regulated by the thermal masses of the scattering particles, s0 = 4 max(mH (T )2; ms(T )2). With mi(T ) / T , we nd UV / T , as we must. The size of the UV scattering rate thus depends indirectly on couplings internal to the two sectors through their role in determining the thermal masses of S and H. Smaller thermal masses cut o the divergence at a lower value of s0, and hence increase the rate. The relatively large couplings of the SM Higgs to the top quark, electroweak gauge bosons, and (to a lesser extent) itself ensure that mH (T ) determines s0, making UV relatively insensitive to the detailed couplings of s within the hidden sector. We nd that, for a xed value of , the two sectors will thermalize in the unbroken phase only if they could also thermalize within the broken phase as well. In other words, the lower bound on that we nd from requiring UV(T ) = H(T ) at high temperatures is subdominant to the lower bound found by requiring IR(T ) = H(T ) at temperatures below the phase transitions. Thus to understand the process of thermalization through the s-h interaction, it su ces to study the scattering rates in the broken phase. At temperatures below the electroweak phase transition, but before the dark phase transition, the dominant processes are ss ! f f processes via a SM-like Higgs mediator. By far the most important contribution here is ss ! bb, which for temperatures near T 30 GeV bene t from the s-channel enhancement for transit through a nearly on-shell Higgs (regulated with a width set to the SM value neglected).10 As this process depends on rather than sin , h = 4:15 MeV, with thermal e ects (A.9) m vh ym2h + O sin mm2s2 ; sin3 h (A.10) 9The crossed process sh ! sh also assists thermalization by contributing to the energy transfer rate between sectors. While this process does not change the number of dark particles, there exist rapid numberchanging s self-interactions that serve this function. The sh ! sh scattering rate is 10% 20% larger than the ss ! hh scattering rate, but provides a highly subleading contribution to the energy transfer rate once the temperatures of the two sectors are similar. Thus it is su cient to estimate thermalization based on the scattering ss ! hh. 10The treatment of ss ! f f thus also includes the important process where ss ! h; as this process is not considered separately, there is no problem with double counting. decreasing the dark matter mass enhances this rate, and for temperatures T = O(10 GeV) it becomes the dominant factor in determining whether the two sectors have ever been in equilibrium; see gure 1. After both the SM Higgs and the hidden sector scalar have VEVs, many processes can contribute to thermalization. Here the dominant ones are sh ! f f , sf ! hf , sg= sf ! g= f , as well as the ss ! f f processes discussed above, which are una ected at O (sin ) by the dark phase transition and thus remain important in the broken phase. For temperatures T mt, mediators may also be produced through sg $ gg. This rate is logarithmically divergent, and we estimate it by cutting o the log divergence with a nite thermal mass for the gluon. Our estimate indicates sg $ gg is a subdominant contribution. HJEP02(18) Below the QCD phase transition, s processes dominate brie y. Our treatment of these thermal scattering rates follows that outlined in the case of the vector model above. Once again, decays and inverse decays become important when ms & Tf , as can be seen in the right panel of gure 13. Our treatment of the 1 $ 2 scattering rates in the Higgs portal model follows the treatment described for the vector portal above. Again, we include all SM contributions to the scalar width. As the hidden sector scalar couples to SM fermions with strength proportional to the fermion masses, the 2 $ 2 interaction rate drops rapidly after crossing a fermion mass threshold as these massive particles drop out the thermal bath. Thus, it is possible that for some other T > Tf , (A.2) is satis ed, but not at Tf , so that the SM and the hidden sector were at one point in thermal equilibrium, but have since decoupled. When this happens, their temperatures drift apart as T HS = gSSM(T SM)gHSS(TD) 1=3 gSSM(TD)gHSS(T HS) T SM: (A.11) This region is shown in green in the right panel of gure 2. Tracking the detailed temperature evolution of the hidden sector in this region, which can involve cannibal behavior when the scalar is su ciently massive and long-lived [154], is interesting, but beyond the scope of this paper. B Sommerfeld enhancement When DM can interact via the long-range exchange of light mediators, the annihilation rate can exhibit a large enhancement over the tree-level rate, especially at low DM velocities [155{158]. This Sommerfeld enhancement is most pronounced when three basic scenarios are satis ed: the dark ne-structure constant, D, is large; the DM velocity, v, is small; and the mediator is much lighter than the dark matter, R = mDM which can be realized by heavy, thermal dark matter with a light mediator. It is common 1, all of m to de ne the Sommerfeld enhancement through the factorized formula [8, 159, 160], v = S(v) ( v)tree ; (B.1) where v is the full cross-section, ( v)tree is the tree-level cross-section, and S(v) is the velocity-dependent Sommerfeld enhancement. For s-wave DM annihilation, the Sommerfeld enhancement can then be written as [160] using an adaptive fth-order Cash-Karp Runge-Kutta technique. Here h vis = 0 hS0(v)i and h vip = 1vc(x)2 hS1(v)i, where Y use the approximations n =s, x m =T , and vc(x) = . We further q 6 x For all choices of D and R, S0( D; R; v) increases monotonically with decreasing v. For p-wave processes, the Sommerfeld enhancement is [160, 161] 36v2 + 2R 36v2 + ( 2R)2 S1( D; R; v) = S0( D; R; v): (B.4) These analytic results from the Hulthen potential provide a good approximation to scattering from the true Yukawa potential [161] except in the resonant regime where disagreements can become numerically larger. We incorporate the Sommerfeld e ect in two di erent ways. First, the Sommerfeld enhancement can become important during freezeout, especially at large DM masses. In this case, the increased annihilation from the Sommerfeld e ect will reduce the size of the coupling constant necessary to achieve the correct relic abundance. We include Sommerfeld enhancement during freezeout by numerically solving the equation dY dx x2 H(m ) h vi Y 2 Ye2q(x) ; To evaluate the Sommerfeld enhancement to DM annihilations, we make use of the analytic approximation obtained by replacing the Yukawa potential with the Hulthen potential [161], VYukawa = D e m r r S0( D; R; v) = VHulthen = 2 v D cosh 6v R D 1 e r e r ; where 2m sinh 6v R cosh q 36v2 2R2 S0(vc(x)) and hS1(v)i S1(vc(x)): In gures 4, 5, 7, 8, and 11 we will denote (in brown) the region where thermal freezeout with and without the inclusion of the Sommerfeld enhancement gives results for the dark ne structure constant that disagree by more than a factor of 2, Djno SE > 2 DjSE: Outside of this region, the approximation used in (B.6) proves very accurate. Deep within this region, the true coupling is typically smaller than that predicted by the freezeout calculation of (B.5), and in our approximation Sommerfeld resonances will be improperly positioned, by an even larger amount than from the use of the Hulthen potential. We further note that for very large couplings and/or very near resonances, the Sommerfeld enhancement as estimated in (B.3) can violate partial wave unitarity [162], and, again, the condition (B.7) reliably insulates us from this region. (B.2) (B.3) (B.5) (B.6) (B.7) The Sommerfeld enhancement also may greatly a ect the indirect detection of dark matter. Both today and at the era of recombination, dark matter moves very slowly and the Sommerfeld enhancement can substantially increase the annihilation rate. As the Sommerfeld enhancement decreases monotonically with increasing velocities, we will conservatively err on the side of assuming larger velocities. Dark matter in the Milky Way have relative velocities on the order of 10 3, and we use the conservative value of which corresponds to relative velocities of 500 km/s for determining the Sommerfeld enhancement for AMS-02. Dark matter in the smaller dwarf galaxies have characteristic velocities on the order of 10 5{10 4 [163]. In our treatment of dwarf galaxies, we will conservatively use a uniform relative velocity across dwarfs of vGC = 1:7 10 3; vdwarf = 10 4; in computing the Sommerfeld enhancement. Especially in the case of faint dwarf spheroidal galaxies with a small half-light radius, such as Draco II and Segue I recently discovered by the Dark Energy Survey [164, 165], this choice of characteristic velocity could signi cantly underestimate the Sommerfeld enhancement. Lastly, at the time of recombination, the characteristic velocity of dark matter was still dictated by its red-shifted temperature rather than by virialization within a structure. In particular, after the dark matter decouples from the thermal bath in the early universe at TKD, its velocity can be expressed as 100 GeV m 1=2 ; where TCMB 0:27 eV, and we have imposed the bound from Lyman-alpha forest data requiring TKD & 100 eV [166]; the parameter space of interest in this work yields values for TKD well above this lower limit.11 For simplicity, in this work we x vCMB = 10 7; (B.8) (B.9) (B.10) (B.11) which for mass ratios of interest falls well into the regime where the Sommerfeld enhancement does not grow any further with decreasing velocity. C Bounds from dwarf galaxies In this appendix we discuss the procedure we use to set limits on our DM models from Fermi's search for DM annihilations in dwarf galaxies. We consider the 41 dwarf galaxies within the nominal sample of [40]. The Fermi collaboration provides a log-likeihood ratio (LLR) for a signal + background assumption to background only as a function of the injected signal for each of the 41 dwarfs in each of the 24 common energy bins. To use these LLRs in combination to constrain a di erent signal model, it is necessary to account for 11See ref. [167] for discussion of related models in the low TKD regime. pb· 5 0.5 cb· 10 p ( > v<σ 1 HJEP02(18) atic uncertainty shifts in central values compared to those from Fermi-LAT (in blue). Given the value accurate reproduction for 0.5 shift, we use this to approximate a proper treatment of the correlations in systematic uncertainties. Left: bb. Right: + . the correlation of the systematic uncertainty on the J -factors between dwarfs. As this information is not provided, we model it by considering a 0:5 downward shift from the nave central value (using unmeasured = 0:6), as this was determined to replicate constraints on the bb and + annihilation models fairly reliably at lower masses, while slightly underestimating constraints at higher masses (see gure 14). In particular, the region of HSFO-VP parameter space currently excluded by Fermi dwarfs is reliably determined by this choice. After this shift, for each energy bin the 41 dwarf measurements are combined to form a net LLR as a function of the signal injected into that bin. For each point in the m vs mZD parameter space, a dark matter annihilation process, ! ZDZD ! fallg, is generated within Pythia 8 (v8223) [132]12 and the resulting gamma ray spectrum is tabulated into the 24 energy bins. The population of these energy bins scales with h vi. Combining these bins into a 2 with one d.o.f. 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Jared A. Evans, Stefania Gori, Jessie Shelton. Looking for the WIMP next door, Journal of High Energy Physics, 2018, 100, DOI: 10.1007/JHEP02(2018)100