The Cosmology of Composite Inelastic Dark Matter
Composite dark matter is a natural setting for implementing inelastic dark matter — the mass splitting arises from spin-spin interactions of constituent fermions. In models where the constituents are charged under an axial gauge symmetry that also couples to the Standard Model quarks, dark matter scatters inelastically off Standard Model nuclei and can explain the DAMA/LIBRA annual modulation signal. This article describes the early Universe cosmology of a minimal implementation of a composite inelastic dark matter model where the dark matter is a meson composed of a light and a heavy quark. The synthesis of the constituent quarks into dark hadrons results in several qualitatively different configurations of the resulting dark matter composition depending on the relative mass scales in the system.
DAMA/LIBRA has an on-going 8.9 annual modulation signal of their single hit rate [1, 2, 3]. If this signal is caused by dark matter scattering off their NaI crystals, then DAMA, when combined with other null direct detection searches, suggests that dark matter is scattering inelastically to an excited state split in energy by [6, 7, 8, 9]. Inelastic dark matter (iDM) is a testable scenario with XENON10 and ZEPLIN3 performing dedicated reanalyses of their data [4, 5]. IDM will be decisively confirmed or refuted in the XENON100, LUX, or CRESST science run this year [10, 11].
Composite inelastic Dark Matter (CiDM) is a model of iDM, where the scale is dynamically generated by hyperfine interactions of a composite particle . (See [13, 14, 15, 16, 17, 18, 19, 20] for other examples of composite dark matter.) In the original CiDM model, the dark matter consists of a spin-0 meson, , that has a single heavy constituent quark. Adjacent in mass to is the spin-1 dark meson, , and with the mass scales chosen in , the mass splitting between and is . The dark-matter origin in CiDM is non-thermal and requires a primordial asymmetry between the number densities of heavy quarks to heavy antiquarks. This article addresses the early Universe cosmology of this minimal CiDM model, calculating the abundance of bound states relative to states, dark baryon states or other exotic configurations of the dark quarks, as well as the direct detection properties of the various components.
The results of this article show that a wide range of final abundances of dark hadrons is possible, depending on the spectroscopy of the dark sector states. Some spectra have dark matter dominantly in the form of , while other spectra have dark baryons dominating the abundance. In the latter case, where the dark baryons are the predominant dark matter constituent, the residual component interacts sufficiently strongly to account for DAMA’s signal. In all cases, exotic dark matter components arise in CiDM with novel elastic scattering properties – nuclear recoil events are suppressed at low-energy by a dark matter form factor. The relative abundance of to is typically . Existing searches for down-scattering () in direct detection experiments are not constraining, but may be feasible in the near future.
The organization of this article is as follows. Sec. 1.1 briefly reviews a specific implementation of CiDM presented in  that will be used throughout this analysis. The spectroscopy of this model is discussed in Sec. 2 and the synthesis of dark meson and dark baryon states in the early Universe in Sec. 3. Sec. 3.4 is the primary result of this article and the qualitatively different results of the synthesis calculation are classified here. Sec. 4 addresses the upscattered fraction of dark pions and its implications for direct detection. Sec. 5 summarizes constraints on the dark baryon fraction that arise from direct dark matter searches such as CDMS and comments on the novel properties of the elastic scattering processes. Sec. 6 concludes with the outlook and further possibilities.
1.1 Review of Axial CiDM Model
A wide variety of hidden sectors weakly coupled to the Standard Model have been considered in the literature, and the possibility that dark matter is charged under hidden sector gauge forces has received considerable recent attention [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. In CiDM, dark matter is a bound state with constituents charged under a hidden sector gauge group of the form , where the gauge boson kinetically mixes with Standard Model hypercharge gauge boson, and the group condenses at the GeV scale. Theories in which dark matter is charged under a new GeV-scale gauge group, as in CiDM, predict a variety of multi-lepton signals in -factories and -factories [33, 34, 35, 36, 37], low-energy upcoming fixed-target experiments [38, 39, 40, 41], and distinctive astrophysical signatures [42, 43, 44].
The Lagrangian of the axial CiDM model in  is given by
and the gauge charges are
The fermion sector has an chiral flavor symmetry, broken down to a vector-like by Yukawa interactions with a dark Higgs boson, . The vacuum expectation value of , , causes the Abelian gauge field to acquire a mass and the fermions to pair into a light and a heavy Dirac fermion,
with masses and , respectively. In this minimal model, is near the electroweak scale while the other quark has a mass at or beneath the confinement scale, .
In analogy to the Standard Model without weak interactions, the flavor symmetry renders the lightest mesons and baryons charged under this symmetry stable. The lightest stable bound state is a meson containing a single and a , which will be denoted as . This dark meson is the dark matter candidate in . Because the constituents are fermions, is paired with a vector state, . The spin-spin interactions of the constituents generate a hyperfine splitting
where is the ’t Hooft coupling, and the parameters are chosen such that the splitting is at the scale suggested by DAMA/LIBRA. The hyperfine structure is described in more detail in Sec. 2.3.
Another key feature of this model is that the dark gauge boson, , couples axially to the dark quarks. The coupling between the and the dark mesons are constrained by parity and all leading order scattering channels are forbidden but the transition . mixes with the Standard Model photon and mediates dark meson/baryon scattering off SM nuclei. In particular, up-scattering can explain the DAMA/LIBRA annual modulation signal. CiDM direct detection phenomenology is discussed in detail in [12, 45, 46].
The mass of the dark Higgs, , is radiatively unstable and introduces a second gauge hierarchy problem to the dark matter - Standard Model theory. The solution to the Standard Model’s gauge hierarchy problem may also solve this new hierarchy problem. Supersymmetric extensions of these models may solve both hierarchy problems at once, and may introduce new phenomena into the theory. For instance, if the only communication of supersymmetry breaking to the dark sector occurs through kinetic mixing, then the dark sector may be nearly supersymmetric, resulting in nearly supersymmetric bound states [47, 48]. These susy bound states may have different scattering channels and the phenomenology may be different than minimal CiDM .
2 Dark Matter Synthesis Prelude: Hadron Spectroscopy
The origin of the dark matter abundance in CiDM is non-thermal, possibly a baryogenesis-like process that generates a non-zero number, see [66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77] for examples. Most often, particularly at large , no net dark baryon number is generated, resulting in a zero number. This article addresses how the residual and asymmetry is configured at late times, i.e. whether the states are in dark mesons or in dark baryons.
At temperatures above , there is a thermal bath of gluons and free quarks and anti-quarks. When the temperature drops below , the heavy dark quarks rapidly annihilate, and only the asymmetric abundance of remains. The non-relativistic heavy quarks are much heavier than the confining scale and can form Coulombic bound states through antisymmetric color channels. Light antiquarks have binding energies comparable to the confinement scale and therefore never bind at temperatures . At confinement, the heavy quarks are dominantly screened by light antiquarks to form singlets.
The evolution of the and asymmetry after confinement is dominantly determined by the spectroscopy of the bound state systems. After the heavy quark mass, the largest energy scale in the problem is the Coulombic binding energy between the heavy quarks. The energetics of these heavy quark configurations guide the synthesis of baryon states through hadron binding reactions. The analysis of Sec. 3 illustrates the qualitative behavior of the synthesis of the and asymmetry in the early Universe. The fraction of dark mesons synthesizing into dark baryons depends sensitively on the masses of the light hadrons and the results are parameterized by the mass of the lightest hadron.
2.1 Coulombic Binding Energies
The Coulombic binding energies of collections of heavy quarks is relevant for dark matter synthesis. Collections of heavy quarks can form quasi-Coulombic bound states that are deeply bound relative to the confinement scale when . The deepest bound states of heavy quarks are antisymmetric color configurations. This section computes the binding energy of multiple bound states using a variational approach.
where is the ordinary gauge coupling and the running ’t Hooft coupling is defined as
and is the following combination of Casimirs for -rank antisymmetric tensors of
The Casimirs for rank antisymmetric tensors are
where in the above expression.
Notice that the potential in Eq. 6 for combining fundamentals into antisymmetric tensors is attractive but suppressed for fixed ’t Hooft coupling in contrast to a quark-antiquark potential.
The confinement scale, , is determined by the ’t Hooft coupling evaluated at a scale
for the minimal model with a single light flavor.
The total Hamiltonian for a system of heavy quarks is
In the ground state configuration, the color indices of the quarks are antisymmetric with all other quantum numbers being symmetrized. In particular, all the quarks will have the same spatial wave function in the ground state. The ansatz for the bound state system will be
where is a hydrogen-like ground state wave function of the form
where is the variational parameter. Evaluating the expectation value of gives an estimate of the binding energy
Minimizing with respect to gives the inverse Bohr radius for the heavy quark system:
and binding energies of
where the running of the ’t Hooft coupling has been neglected in the differentiation. For the case of the binding energy reduces to
precisely the analogue of the Rydberg constant for the potential in (6).
The estimates of (16) are crucial for two dominant rearrangement processes that occur
Ignoring the effects of running on the ’t Hooft coupling, the energy released in these reactions is
The energy release from the first few reactions are tabulated in Table 1. The first reaction releases the least energy and is a potential bottleneck in the chain reaction that takes .
Eqs. 16 and 2.1 are the main results of this section. They will completely determine whether heavy quark synthesis into bound states takes place before confinement. The role of light states (in particular light quarks) during the unconfined phase is irrelevant due to the large gluon entropy at such temperatures. After confinement, when the states are color-neutralized by light quarks, the estimates of (16) and (2.1) are still expected to hold, since they physically correspond to deeply bound states of heavy quarks and are typically an order of magnitude larger than and the binding energies of light quarks. The energetics of the rearrangement reactions (18) after confinement will be completely determined by the results of (16) and the parametric dependence on the mass of the lightest state in the spectrum, that is typically present among the final states (more about that on Sec. 3.3).
2.2 Hadronization of Bound States
There are two ways of constructing color singlets from heavy quarks in an antisymmetric color configuration: mesons and baryons. For mesons, the color is neutralized by light antiquarks in an antisymmetric color configuration, while for baryons the color is neutralized by light quarks in a color antisymmetric state. In QCD, the analogues to these multi-heavy quark systems would be di-heavy baryon and tetraquark states of the form
In the dark sector of CiDM, the corresponding states will be denoted by for meson states111 refers to all spin configurations of and dark quarks. Specifically, refers to both and . with heavy quarks and by for baryon states with heavy quarks. For convenience, and .
Due to the difference in the number of light quarks necessary to hadronize the color of and , these states will have different masses. Using a constituent picture of the light quarks, the masses of mesons and baryons are given by
In Eq. 2.2, parametrizes the constituent mass of the light quarks confined in a bag of size
In principle this leading order effective mass could be -dependent, due to changes in the light quark wavefunctions. For simplicity, this article will take .
The term in Eq. 2.2 arises from the spin-spin interaction amongst the light quarks and a constituent model of this interaction gives
Evaluating the expectation values above for a totally spin-symmetric state, one finds
Using the system in QCD one infers: . This article will use .
Both the constituent masses and spin-spin interactions cause the mesons to be lighter than the baryons when and the baryons to be lighter than the mesons in the complimentary case.
The most relevant reactions in the early Universe that synthesize stable and states have rates that are exponentially sensitive to total binding energy differences. The estimates of the binding energies have an uncertainty of that has been absorbed into the unknown constants in Eq. 2.2. In practice, the binding energy differences are larger than by roughly an order of magnitude so that the Coulombic spectroscopy dominantly determines the synthesis of into other species of dark hadrons.
2.3 Hyperfine Structure Spectroscopy
The stable dark hadrons have a large degeneracy of their ground states from the suppressed spin-spin interactions of the heavy quarks with the light quarks. Both in dark mesons and dark baryons, same-flavor quarks have antisymmetrized colors and symmetrized spins. Therefore, heavy quarks are in a spin configuration while light quarks are in a spin configuration for mesons and spin configuration for baryons. The resulting range of spins for dark mesons and baryons is, respectively,
Hence, the degeneracy of a bound state containing heavy quarks (either mesonic or baryonic) is equal to . Spin-spin interactions break this degeneracy and introduce hyperfine splittings, resulting in the lowest spin configuration being the lowest energy configuration. Such splittings can be very suppressed with respect to the dark matter mass and offer a natural mechanism to generate the scale suggested by DAMA.
The spin-spin coupling of heavy and light degrees of freedom splits the ground state degeneracy of hadrons. This is studied for Standard Model hadrons in [60, 61, 62]. The quark chromomagnetic moment is suppressed by its constituent mass, , and is given by
where the coupling is evaluated at the effective Bohr radius for the entire, color-neutral bound state. For , , , and is introduced to fix the relationship. The first order correction to the energy levels due to spin-spin coupling of heavy and light degrees of freedom is
where and are the spin operators for the collection of heavy and light quarks, respectively. The energy splittings can be evaluated using
Note that the color expression above applies to both mesons and baryons.
Finally, in Eq. 27 is the ground state wave function (i.e., ) for the light quarks evaluated at the origin and is roughly
For states with multiple light quarks, i.e. all states but and , there could be significant multi-body effects that could change the wave functions. In particular, QCD baryons have a hyperfine splitting smaller than that of mesons by a factor of roughly three. For bottom hadrons
Remarkably similar ratios hold for charm hadrons and even strange and light flavored hadrons. This is evidence for dependence in and can be interpreted as from Eq. 22 being
This dependence in should be universal and applicable to both baryons and mesons.
Combining the results for gives
where for states and for states. The energy splitting between the ground state and the first excited state for dark mesons is
For baryons the hyperfine splitting of the ground state is
Fitting expression (35) to the hyperfine splitting of and in the Standard Model, one finds
for and . Throughout this paper this constant will be taken to be .
QCD indicates , potentially making the smallest hyperfine splitting be in the meson, rather the meson. This opens up an alternate explanation of DAMA: that it is the multi-heavy quark mesons that are responsible for scattering.
2.4 Light States
Just like hybrid mesonic and baryonic bound states, unstable states are in thermal equilibrium with the thermal bath at early times. The unstable spectrum consists of states that carry no net and quantum numbers. There are no Goldstone bosons, since this is a single flavor theory at confinement and the global is anomalous. Therefore the mass of the lightest state is of the order of the confinement scale and could either be a light meson or a glueball state [63, 64, 65]. This section describes the unstable, mass states, estimating their lifetimes and briefly describing their cosmology.
Long lived states are of potential concern to Standard Model BBN. We will show here, however, that a general depletion mechanism causes the abundance of quasi-stable states to be negligible by the time BBN begins. Among the long lived states is the meson, which decays through the operator
to four Standard Model leptons through two off-shell ’s. The resulting decay rate scales as :
giving a typical lifetime of
Scalar glueballs with mix with mesons and inherit the same decay channels. scalar glueballs, on the other hand, mix with and can decay to either on-shell or off-shell ’s or light dark mesons. Their lifetime is expected to be no longer than the lifetime of the states.
Among the short-lived states are vector mesons and glueballs that can decay through mixing with . Their estimated lifetime is :
The existence of short lived states in the spectrum is a leaky bottom mechanism for quasi-stable flavorless hadrons. These long lived states are depleted by scattering into shorter lived ones, e.g., , with cross sections set by :
The residual abundance of quasi-stable particles by the time Standard Model BBN begins is
The most stringent constraints for a late decaying relic with hadronic branching fraction are set by :
The relic abundance (43) is several orders of magnitude below the upper limit (44). This is a conservative bound because the decay products of the dark hadrons are predominantly leptons or photons, which alter the primordial abundances of elements less than hadronic decays. Therefore, light states have a negligible effect on BBN.
3 Dark Matter Synthesis and Evolution
The evolution of the and asymmetry in the early Universe is divided into three stages ordered by the temperature relative to the confinement scale: above, at, or below. Starting at temperatures close to , the thermal abundance of dark -quarks is exponentially suppressed and only the non-thermal component is left to synthesize into composite states. At temperatures above the confinement scale, perturbative techniques are applicable to estimate the cross sections for rearrangement reactions because . For temperatures above confinement, the light quarks can be ignored throughout, and included during confinement to screen the color charge of the heavy quark bound states. Below confinement the light quarks can be treated as spectators. Assuming that approach is justified, the computation is reasonably accurate up to -factors.
3.1 Preconfinement Synthesis
At high temperatures, , the non-Abelian gauge dynamics is unconfined and quasi-perturbative. Light quarks are always relativistic and do not form bound states. Below temperatures , heavy quarks can form Coulombic bound states through antisymmetric color channels (as described in Sec. 2.1) by reactions of the kind
where is the reduced mass of , accounts for Sommerfeld enhancement, and is the ’t Hooft coupling evaluated at the Bohr radius of the bound state .
Each reaction of the type (45) contributes to the Boltzmann equation for the number density of , , as222with appropriate factors of included whenever .
The exponential factor in the Boltzmann equation above accounts for the large gluon entropy that prevents heavy quark bound state formation down to temperatures
where . Plugging in Eq. 17, we see that strong gluon entropy dissociation is effective down to temperatures when confinement takes place for:
For and that corresponds to . Thus for the parameter space favored by DAMA/LIBRA, there is no pre-confinement bound state formation.
3.2 Synthesis at Confinement
It was shown in Sec. 3.1 that dissociation due to relativistic dark gluons inhibits bound state formation down to temperatures . At confinement, this dissociation shuts off since the theory acquires a mass gap. The light quarks and gluons form massive dark hadrons and proceed to decay promptly (Sec. 2.4).
The formation of heavy quark bound states is suppressed since those are dilute at the time of confinement and there is a strong nucleation of light quark-antiquark pairs that screen the color charge of free heavy quarks. So, confinement will preferentially lead to formation of dark mesons over higher- dark hadrons.
It is possible, however, to estimate the abundance of dark hadrons formed at confinement by simple combinatorics. If two heavy quarks are apart by a distance smaller that at confinement, the light-quark screening effect will not operate and the two heavy quarks will bind. Computing the probability that that will happen gives an estimate for the ratio of dark hadrons over dark mesons produced at confinement
These bound states of two heavy quarks will either be screened by two light anti-quarks to form a dark meson, or by light quarks to form a dark baryon. The later is Boltzmann-suppressed over the former for since it has more light constituents.
The formation of dark baryons at confinement depends sensitively on . Using the leading term from Eq. 22
Using QCD as a guide, , leads to
for . The dark baryons can annihilate with into . The residual abundance that does not annihilate will quickly synthesize into .
3.3 Postconfinement Synthesis
After the transition from the unconfined to the confined phase, the dark quarks hadronize dominantly into dark mesons with a suppressed fraction in dark baryons. Heavier dark hadron synthesis occurs through
where is the lightest dark hadron, such as a glueball or light meson, with mass . Once mesons start forming, a chain of reactions can occur that will ultimately lead to formation of dark baryons .
As an illustration consider an theory. The reaction chain down to heavy dark baryon synthesis is:
As a combination of the mass gap in the theory and the fact that the heavy quarks in are not particularly deeply bound, the first reaction (55a) becomes endothermic for a large fraction of the parameter space. On the other hand, the reaction (55b) is always exothermic; therefore, dark baryons formed at confinement process more efficiently than dark mesons. The results listed in Sec. 3.4 assume that all hybrid dark baryons efficiently process into .
Reactions that have hybrid baryons in the final state
are energetically disfavored compared to their mesonic counterparts,
and have negligible contribution to the dark matter synthesis.
Heavy quark binding in the rearrangement reactions (55) requires large momentum transfer , and hence it is expected that the cross sections are controlled by perturbative heavy quark dynamics, as it was during pre-confinement (47). However, there are two important differences between pre- and post-confinement processing:
There is no Sommerfeld enhancement because the lightest degrees of freedom are heavier than the typical momentum transfer in such reactions: . This amounts to a reduction of the post-confinement cross section by a factor of relative to the pre-confinement cross section.
If is heavier than the binding energy, the first reaction (55a) is endothermic, which further suppresses post-confinement synthesis by an additional Boltzmann factor in the thermally averaged cross sections.
Therefore, post-confinement cross sections for heavy hadron processing compares to pre-confinement heavy quark binding (47) as:
For the range of parameters where these reactions are endothermic, processing will be already frozen out by the time of confinement. This demonstrates that no heavy baryons are formed for , or:
However, that is not the only possibility. As one explores other ranges for the confinement scale and the mass of the lightest state , the full numerical solution to the Boltzmann equations reveals that the synthesis of CiDM can allow for much richer range of compositions detailed in Sec. 3.4.
3.4 Synthesis Results
Fig. 1 illustrates the CiDM synthesized spectrum as a function of the parameter space for the case .