Looking for the WIMP Next Door
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 sufficient 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 defined, 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. We determine this thermalization floor as a function of equilibration temperature for the first 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.
The existence of some form of dark matter (DM) constituting 26 of the present-day energy budget of our universe is well-established through its gravitational imprint on baryonic matter Ade:2015xua (). No evidence to date indicates that DM must interact in any way beyond gravitationally. The cosmological history of DM, however, will typically require DM to have some non-gravitational interaction(s) responsible for establishing its observed relic abundance, and these interactions can leave potentially observable footprints. For instance, the cosmic coincidence of the “weakly interacting massive particle (WIMP) miracle” implies that new stable weak-scale particles with weak interactions that freeze out of the thermal Standard Model (SM) plasma in the early universe can provide a good DM candidate. The cosmic abundance of WIMPs is directly determined by their coupling to the SM, 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) Escudero:2016gzx (), 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 Abercrombie:2015wmb (); Alexander:2016aln ().
This class of thermal relics, however, represents only a fraction of possible identities for dark matter. Hidden sector freezeout (HSFO Pospelov:2007mp (); Feng:2008ya (); Feng:2008mu (); ArkaniHamed:2008qn ()), where the DM relic abundance is chiefly 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 floor. The parameter space for WIMPs next door is bounded: the DM mass must lie between (to preserve the successful predictions of Big Bang Nucleosynthesis (BBN) and (from perturbative unitary), while the coupling between the SM and the HS must be sufficiently 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,
We will use two simple reference models in this work,
HSFO-VP: fermionic DM , annihilating to vector mediators, , that couple to the SM through the vector portal; and
HSFO-HP: fermionic DM , annihilating to scalar mediators that couple to the SM through the Higgs portal.
These models of HSFO can be probed via complementary methods across different 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 significant portions of the parameter space. Indirect detection remains a powerful probe, provided the DM has an appreciable -wave annihilation cross-section, as in our minimal vector portal model. Our minimal Higgs portal model, on the other hand, freezes out through -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 Sec. 2, where we establish the physical parameter space of our models. In Sec. 3 we discuss different experimental avenues to test this parameter space. In Secs. 4 and 5 we show the consequences for vector and Higgs portal models respectively, and in Sec. 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 Pospelov:2007mp (); Feng:2008ya (); Feng:2008mu (); ArkaniHamed:2008qn (). 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 sufficiently rapidly to avoid disrupting the successful predictions of BBN, thus generally requiring 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 define as a WIMP next door. We will focus here on the well-motivated cases 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 sufficiently large to ensure that the SM and the dark sector were in thermal equilibrium for some temperature above the DM freezeout temperature, , the existence of the SM thermal bath is then sufficient 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 Hodges:1993yb (); Berezhiani:1995am (); Adshead:2016xxj (), 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 , and become more important in comparison with as the temperature drops. Thus , the minimum value of consistent with thermalization at a temperature , does not depend on the unknown reheating temperature of the universe (provided ) or other unknown UV particle content. This cosmic origin for DM also significantly sharpens predictivity by limiting the degree to which the temperature of the dark sector can differ from the temperature of the SM.
In order to determine the thermalization floor , we have to distinguish between two cases: first, when the hidden sector contains (at least) one relativistic species at , and second, when all species in the hidden sector are already nonrelativistic at thermalization, for all masses. The value of is the point where a bosonic species contribution to drops by a factor of 2, and is our definition for when a species transitions from relativistic to non-relativistic. In the first 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 efficient 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 , 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 first case where the HS has a radiation bath at . In this cosmology the lower bound of the thermalization floor 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 . 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 .
When , scatterings are the dominant process responsible for equilibrating the two sectors. When , scatterings become dominant. This temperature scaling is evident in Fig. 1, where we show and scattering rates in each of our models as a function of the temperature. In the absence of mass thresholds, at high temperatures, while . 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 floor that we obtain is an initial estimate, computed up to a factor of . The resulting new cosmological lower bound on portal couplings is shown in Fig. 2 as a function of , in the regime where . The thermalization floor is insensitive to the mediator mass as long as 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 between DM and the mediator. Simplified model approaches can be effective at highlighting the key physical features of classes of DM theories Cheung:2013dua (); deSimone:2014pda (); Berlin:2015wwa (); Abdallah:2015ter (); Abercrombie:2015wmb (), 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 defined and bounded parameter space. The dark matter-dark mediator coupling, , is fixed by the dark matter relic abundance, while the coupling of the dark sector to the SM is bounded from below by the thermalization floor. Previous estimates of these thermalization floors (e.g. Feng:2010zp (); Krnjaic:2015mbs ()) have considered a subset of processes and/or studied equilibration at a fixed 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 depends on the structure of symmetry-breaking in the dark sector Cline:2014dwa (). Perturbative unitarity constraints in specific models can further tighten the upper bounds on the DM mass (e.g., Hedri:2014mua ()). 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, . This occurs for TeV in both simplified 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 effect 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 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 MeV are restricted by BBN, which mandates that Cyburt:2015mya (). When the dark sector is in thermal contact with the SM at temperatures MeV, we must then have both DM and the mediator be nonrelativistic by MeV. We here impose the simple requirement MeV. A more careful treatment of the regions shown in green in Fig. 2 where the dark sector has departed from equilibrium with the SM prior to DM freezeout would relax these bounds slightly. A detailed treatment of this region is interesting, but beyond the scope of this paper.
3 Direct, Indirect, and Accelerator Constraints
WIMPs next door give rise to signals in many different kinds of experiments. In this section, we briefly discuss the relevant experimental results and their application to our simple models, highlighting how signatures can differ from traditional WIMP models.
Both our vector portal and Higgs portal
models have a leading spin-independent 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 Aprile:2017iyp (), LUX
Akerib:2016vxi (); Akerib:2015rjg () and PandaX-II Cui:2017nnn ()
at higher masses, while CDMSlite Agnese:2015nto () and CRESST-II
Angloher:2015ewa () set the strongest limits at lower
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 Ackermann:2015zua (); Fermi-LAT:2016uux ()
and Planck constraints on DM annihilations near recombination
Ade:2015xua (); Slatyer:2015jla (). 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 flux Aguilar:2015ooa () can potentially give
more powerful constraints on hadronic annihilation channels than
searches with gamma rays Giesen:2015ufa (), the difficulty in
accurately determining propagation parameters remains a serious
hurdle. We follow Elor:2015bho () in considering AMS-02 positron
results Aguilar:2013qda (), 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
Ackermann:2015zua (); Fermi-LAT:2016uux () have good sensitivity.
Meanwhile, CMB limits are mainly sensitive to the net energy deposited
in the - plasma by DM annihilations near recombination
Padmanabhan:2005es (), and are thus robust and nearly
model-independent. The HAWC experiment can place constraints on very
high dark matter masses Albert:2017vtb () 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
Laevens:2015una (); Kirby:2015bxa (). In principle, H.E.S.S. should have sensitivity to our DM models when TeV, but they do not provide enough information to allow their results to be reliably reinterpreted.
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 Curtin:2013fra (); Martin:2014sxa (). Precision tests of 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 Krnjaic:2015mbs (); Chang:2016ntp (), 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 Flacke:2016szy ().
4 Vector portal
We first consider a simple vector portal model, containing a fermionic DM, , and a dark photon, . This type of model has been studied extensively in the literature, especially to address cosmic ray anomalies (HEAT, PAMELA, and ATIC first Cholis:2008vb (); ArkaniHamed:2008qn (), and more recently the Galactic Center excess Hooper:2012cw (); Abdullah:2014lla (); Martin:2014sxa (); Cline:2014dwa ()).
In the following, we define our model and establish notation. We introduce a massive dark photon , the gauge boson for a new dark symmetry, that interacts with the SM through kinetic mixing with SM hypercharge Galison:1983pa (); Holdom:1985ag (). The dark photon mass could arise from the Stueckelberg mechanism Stueckelberg:1938zz (); Feldman:2007wj () 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 , which could become the leading process in the regime Cline:2014dwa (); we discuss this possibility further in Sec. 4.4. The dark vector Lagrangian is thus given by
where is the Weinberg angle and is the dimensionless kinetic mixing parameter. Additionally, we introduce a Dirac fermion with unit charge under and with mass to serve as DM. Making the standard field redefinition to diagonalize the hypercharge and boson kinetic terms rescales the dark coupling , and results in the following mass matrix for the neutral gauge bosons after electroweak symmetry breaking,
in the basis . Here and , with the mass of the SM boson before mixing. The resulting massive eigenstates are
with mixing angle
The massive eigenvalues are
We can now compute the couplings of the SM fermions and the DM with the gauge boson:
where , are the hypercharge and isospin of the (Weyl) fermion . The physical boson acquires a (vector-like) coupling to :
Our simple model can be described by four independent free parameters, which we take to be and .
4.1 Thermal Freezeout and Indirect Detection
When DM is heavier than the dark photon, it can annihilate via . This is the only annihilation channel in the small limit, and in this limit the thermally averaged cross-section for this annihilation,
is independent of . For sufficiently heavy DM (), Sommerfeld enhancement can be important during freezeout, , which we implement via a Hulthén potential as described in Appendix B. Requiring that this reaction yields the observed relic abundance as measured by Planck, Ade:2015xua (), fixes for each choice of . 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 Ackermann:2015zua (). The energy flux of photons from DM annihilations in an astrophysical source can be expressed as
where we have specialized to Dirac DM. Here is the number distribution of photons from a single DM annihilation, and is the astrophysical -factor, describing the line-of-sight density of dark matter in the direction of the source Geringer-Sameth:2014yza ().
We use the 41 dwarf galaxies within the nominal sample of Ackermann:2015zua () 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 and of the mass ratio are shown in Fig. 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 flux of positrons observed in the AMS-02 experiment Aguilar:2013qda (); Elor:2015bho () can constrain photon-poor annihilation channels. In order to set constraints, we use the limit for one-step channels from Elor:2015bho () and compare this to . In principle, considering all 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 Fig. 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 effect of this energy injection can be accurately encapsulated by a redshift-independent efficiency parameter , which depends on the DM mass and the species of particles produced by DM annihilations Finkbeiner:2011dx (). Planck results together with results from ACT, SPT, and WMAP limit pb c / TeV Slatyer:2015jla (), allowing for robust bounds to be placed on dark matter models. The values for DM annihilation to pairs of SM particles have been computed in Slatyer:2015jla (); Madhavacheril:2013cna (). Due to the rather soft dependence on for all branching ratios except for photons and leptons, we use the values in Slatyer:2015jla () evaluated at for non-leptonic channels, together with from Madhavacheril:2013cna (). For leptonic channels we use from Slatyer:2015jla (). We derive a net
where governs the branching ratios. The resulting CMB bound is shown in purple in the right panel of Fig. 4, as a function of the dark matter mass and of the mass ratio .
4.2 Direct Detection
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 spin-averaged, non-relativistic amplitude-squared for DM to scatter off of a nucleus is mediated by the exchange of both dark vector and bosons, and is given by
with the atomic number, and and the couplings of the boson with up and down quarks in (4.10). Here we have retained the momentum dependence in the propagator from exchange, as it is needed to accurately describe the scattering when , where the DM-nucleus reduced mass is .
where is the local DM density, is an experiment-specific selection efficiency, and is the mean inverse speed Freese:2012xd () defined by
for which, following the experiments, we use the expression in Ref. Lewin:1995rx ().
where we have defined the nucleon-DM reduced mass .
However, for the HSFO-VP model, is sensitive to the recoil energy once . In order to correctly account for this important effect, we will determine the excluded cross-section via
where the function is determined separately for each experiment. Given masses for the DM and dark vector, the relic density constraint fixes . The latest XENON1T Aprile:2017iyp (), LUX Akerib:2016vxi (); Akerib:2015rjg (), PandaX-II Cui:2017nnn (), Super-CDMS Agnese:2014aze (), CDMSlite Agnese:2013jaa () and CRESST-II Angloher:2015eza () searches then determine the maximum allowed value of the portal coupling . We show these upper bounds in the left panel of Fig. 5. Sensitivity is greatest at small values of , thanks to the behavior of the nuclear matrix element. However, the sensitivity saturates when (the threshold energy of the experiment), and the propagator in the matrix element is dominated by the momentum. Over a sizable region where and GeV, current direct detection limits are sensitive enough to exclude values of the portal coupling at and below the thermalization floor. This region is shown in blue in the left panel of Fig. 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 Fig. 5, we show the direct detection parameter space consistent with our HSFO-VP WIMP next door (in tan). As Fig. 5 shows, this cosmological history for DM can predict spin-independent cross-sections well below the neutrino floor.
4.3 Accelerator and other mediator constraints
Direct searches for the are the leading terrestrial signal of this model. As summarized in Hoenig:2014dsa (); Curtin:2014cca (); Ilten:2016tkc (); Aaij:2017rft () and in Fig. 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. Fig. 6 shows the parameter space for a vector portal WIMP next door as a function of and . As shown there, Supernova 1987A uniquely probes the thermalization floor in a limited range of dark photon masses at around few GeV. Furthermore, especially at low masses, terrestrial searches for dark photons bound more tightly than the direct detection constraints do alone.
Fig. 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 . Indirect detection from CMB experiments cuts off the entire region of dark photon parameter space below 400 MeV (red region), while a combination of both direct and indirect detect results exclude all values of in a region up to 4 GeV for a range of portal couplings. Future direct detection experiments, DARWIN and CRESST-III Phase 2 Schumann:2015cpa (); Strauss:2016sxp (), will greatly cut into this range (green line), even excluding down to the thermalization floor 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, Fig. 6 shows that any dark photon discovered in meson decays or at high-energy colliders is sufficiently 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 Fig. 7 as a function of and . 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 floor from thermalization can be much lower. As in Fig. 4, the solid green in the lower left corner has , 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- searches at the LHC, since they are generically weaker than the constraints coming from direct detection experiments. In particular, the most stringent mono- constraint arises from the ATLAS and CMS mono-jet searches performed with Run II data CMS:2016pod (); Aaboud:2016tnv (). Values of as small as are only probed in a small region of parameter space at around GeV and .
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 briefly discuss the modifications 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 contribute to the Hubble parameter, thereby requiring a mild increase of the value of needed to obtain the thermal relic abundance. The DM relic abundance is proportional to at freezeout. Neglecting the logarithmic sensitivity of the freezeout temperature to , we can thus simply estimate the effect of adding additional equilibrated dark species by rescaling to absorb the shift in . This is a minor quantitative effect, particularly at relatively high DM masses where .
An excellent motivation for introducing additional dark species is to provide a dark Higgs mechanism to generate ArkaniHamed:2008qn (); Cline:2014dwa (); Bell:2016fqf (). Our model assumed a Stueckelberg mechanism for simplicity. Using a dark Higgs to generate is a generic alternative scenario, but with a dark Higgs comes additional model dependence, especially through the choice of the DM’s U(1) quantum numbers. When the dark Higgs is light, it can furnish additional annihilation modes: depending on the spectrum, both the -wave process and the -wave process can contribute to freezeout Bell:2016fqf (). (We continue to assume that the vector portal coupling dominates the dark sector’s interactions with the SM.) The additional annihilation modes change the specific value of needed to obtain the thermal relic abundance, generically by no more than an amount. These additional annihilation modes also alter the detailed cosmic ray spectrum for indirect detection. When -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 . 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 TuckerSmith:2001hy (). 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 significantly more challenging to observe at direct detection experiments, but some signals are still possible Bramante:2016rdh ().
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 . At relatively high dark vector masses, the axial components of the SM– couplings are sizable, giving rise to spin-dependent cross-sections. The vector SM– couplings yield spin-independent cross-sections suppressed by DM velocities or nuclear recoil momenta, giving small but still potentially interesting signal rates DEramo:2016gos (). Elastic spin-independent cross-sections are also induced at one loop Cirelli:2005uq (); Essig:2007az (); Kopp:2009et (); Freytsis:2010ne (); Haisch:2013uaa (). The size of this contribution is thus sensitive to the UV field 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 Sec. 5.2 below.
If there are additional light dark sector species, , then can have open decay modes within the hidden sector, and when these are active one generally expects . Letting decay can eliminate (if is stable) or modify (if is unstable Elor:2015bho ()) cosmic ray and CMB constraints on DM annihilation, and, often for either case, subject the mediator to the generally weaker terrestrial searches for invisible Lees:2017lec (); Banerjee:2017hhz (); Aguilar-Arevalo:2017mqx (). 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 , 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 Fig. 2, a vector portal hidden sector equilibrated with the SM cannot decouple from the SM at sufficiently early times to permit significant departures of the HS temperature from . Interestingly, a model with a portal coupling of 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 Abazajian:2016yjj (). 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 -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 symmetry are important during freezeout. Introducing more states in different representations and allowing coannihilation to be important in determining the relic abundance can significantly alter the phenomenology and open up different 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 amounts, e.g., through affecting Hubble. Indirect detection signals are especially robust, as adding additional annihilation channels, etc., generically changes cosmic ray signals quantitatively but does not suppress them significantly below expectations for an -wave thermal relic. For vector portal models there is very little scope to eliminate indirect detection signals via 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 define a simple reference model for a Higgs portal WIMP next door, HSFO-HP. We consider a Majorana fermion dark matter, , with a scalar mediator, , that interacts with SM states through a (small) Higgs portal coupling. A useful simple model is Shelton:2015aqa () (see also Pospelov:2007mp (); Baek:2011aa (); Dupuis:2016fda ())
where we use the usual conventions for the Higgs potential,
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 (5.20–5.21), we have imposed a discrete symmetry taking , , thus forbidding cubic and linear terms in as well as a Majorana mass for . Imposing this discrete symmetry allows us to expose the essential physics of this theory with the minimum number of parameters.
In order for the fermions to be massive, both and must acquire nonzero vacuum expectation values (vevs), and , where is the vev of ( GeV). Minimizing the potential gives analytic expressions for the vevs,
The dark matter gets a mass of , while the scalars have a simple mass matrix,
yielding the mass eigenvalues
and a mixing angle defined by
where the latter equality holds only for and therefore the mass of the scalar quite different from 125 GeV. In this regime, for small , and , so either or can be viewed as a measure of the strength of coupling between the SM and dark sectors.
We can express the Lagrangian in terms of GeV, GeV and the four free parameters, , , , and . In terms of these parameters, the most important couplings for our discussion, to leading order in and for , are:
It will sometimes be useful to refer to a fine-structure-like constant, .
5.1 Thermal Freezeout and Indirect Detection
Assuming freezeout of the Majorana dark matter is governed by interactions entirely within the hidden sector (i.e., effects proportional to can be neglected), three diagrams dominate the process : - and -channel exchange of , and -channel annihilation through an off-shell . The spin-averaged amplitude, integrated over final state phase space, can be written as
where , and the factor of one-half for identical final state particles is explicit. This quantity, , is related to the cross-section by, , where the relative velocity is conventionally defined as .
Defining , the thermally averaged cross-section, to leading order in , can be written Shelton:2015aqa ()
which is -wave suppressed. This cross-section is then corrected by the Sommerfeld enhancement (B.53) to give .
Thanks to the -wave annihilation cross-section, this model does not yield observable signals from DM annihilations in halos, nor do late annihilations deposit noticeable amounts of energy into the CMB. Thus standard indirect detection strategies do not constrain the minimal HSFO-HP model. Dark matter density spikes surrounding super-massive black holes could potentially boost -wave DM annihilation to observable rates, yielding a point-like gamma-ray signal Shelton:2015aqa (). Additionally, there are regions of parameter space at high where and DM annihilations can proceed through bound state formation and decay. These processes are -wave, and thus although they are unimportant during thermal freezeout, they can leave an imprint on the CMB An:2016kie (). Assuming that the velocity of DM at recombination satisfies and bound states are accessible, an analytic solution for the bound state formation rate (via monopole transitions into -wave bound states) can be written as An:2016kie ()
where , and a factor of appears due to only a single available bound state with spin-0 due to Majorana dark matter An:2016kie (). In practice, the Gamma functions in (5.29) yield roughly a cosecant function with maxima at regularized at the singular points by a tiny imaginary contribution. This bound state decay can be used to bound -wave annihilating dark matter models with , using the condition
5.2 Direct Detection
As in the HSFO-VP model, the HSFO-HP model has a leading spin-independent direct detection cross-section, and direct detection therefore provides a powerful test of this model. The direct detection cross-section is mediated by exchanges of both the dark scalar and the visible Higgs boson. In the regime where , momentum exchange rather than the dark scalar mass dominates the dark scalar propagator. In order to account for this important recoil energy dependence, we follow the procedure discussed in Sec. 4.2. For the HSFO-HP model, the spin-averaged, non-relativistic amplitude-squared for DM-nucleon scattering is
where is the mass of the nucleon and
Here the nucleon matrix elements are
where heavy quark flavors have been removed in the second line Shifman:1978zn (). In principle and could differ substantially, but owing to the dominance of heavy-flavor (i.e., isospin-universal) contributions, they are nearly identical for Yukawa-coupled scalars. We use here and Hill:2014yxa (). If the recoil energy in (5.31) can be ignored, then the DM-nucleon cross-section is
However, it is important to retain the recoil energy dependence in the dark boson propagator to accurately describe scattering rates when ; to handle this, we use the procedure discussed previously in Sec. 4.2. In the left panel of Fig. 8, we show the maximum allowed value of the mixing angle consistent with the current direct detection bounds Aprile:2017iyp (); Akerib:2016vxi (); Akerib:2015rjg (); Cui:2017nnn (); Agnese:2015nto (); Angloher:2015ewa (). The blue region shows where direct detection experiments are probing values of below the thermalization floor. Additionally, the dashed blue-gray region corresponds to where the direct detection bound on implies the dark and SM sectors have decoupled and their formerly equilibrated temperature can drift (A.49). Once again, direct detection experiments can probe all the way down to the cosmological lower bound on the portal coupling in some portions of parameter space. The direct detection parameter space consistent with Higgs portal WIMPs next door is shown in the right panel of Fig. 8, where the tan region is defined by cutting out the region where is within 10% of .
5.3 Accelerator and other mediator constraints
The leading collider and accelerator searches for the HSFO-HP model are again direct searches for the mediator, . Many different low-energy and collider observables are sensitive to Higgs-portal coupled scalars. Additionally, there are astrophysical and cosmological constraints on when it becomes sufficiently light and long-lived. Our results for the current experimental reach for a light mediator, , are presented in Fig. 10. In order to establish these results we need the SM branching ratios of the scalar. These branching ratios are notoriously uncertain for scalar masses in the range between and GeV (see Clarke:2013aya () for more details). We adopt the hadronic branching fraction derived in Ref. Donoghue:1990xh (), supplemented with a simple extrapolation in the very uncertain 1–4 GeV region, as shown in Fig. 9. Features near 1 GeV in Fig. 10 are sensitive to this choice, which is conservative for signals sensitive to muon decays; the overall lifetime is also important for determining projections for SHiP Alekhin:2015byh (), MATHUSLA Chou:2016lxi (), and CODEX-b Gligorov:2017nwh ().
In the rest of this subsection, we will explain in the detail the constraints shown in Fig. 10. While other experiments have constrained this parameter space, such as KTeV AlaviHarati:2000hs () and NA48/2 Batley:2011zz (), these results have been surpassed by the bounds from other experiments and will not be discussed here. For high scalar masses and large portal couplings, additional constraints from perturbativity and electroweak precision tests can be important Lopez-Val:2014jva (); Robens:2016xkb (); however, these constraints are model-dependent and not in our main regime of interest, and we do not discuss them further here.
LHC (ATLAS & CMS): Heavy Higgs Searches — There have been many searches looking for Higgs-like bosons at the LHC. The strongest limits in the region GeV come from a search at ATLAS and two at CMS in the diboson decay channels Khachatryan:2015cwa (); CMS:2016ilx (); ATLAS:2017spa ().
LEP (OPAL, DELPHI, ALEPH & L3): Higgs Searches — The LEP Working Group for Higgs boson searches combined the data from the four experiments at LEP to place very tight constraints on Higgs states that are produced in association with a from 15-115 GeV Barate:2003sz (). For scalar masses below 15 GeV, the tightest constraints come from L3 Acciarri:1996um (), but the ALEPH search Buskulic:1993gi () sets slightly stronger limits below the muon threshold where the scalar is detector stable.
LHCb: — LHCb has made many detailed measurements of the important observables within . Two of these have been specifically interpreted to constrain light Higgs-mixed scalars, in the Aaij:2015tna () and the channels Aaij:2016qsm (). We shift these limits to match our branching ratios.
E949 & E787: invisible — The E949 collaboration at BNL has made the most accurate measurement of Artamonov:2009sz () by measuring the decays of stopped