Theories of dark matter that support bound states are an intriguing possibility for the identity of the missing mass of the Universe. This article proposes a class of models of supersymmetric composite dark matter where the interactions with the Standard Model communicate supersymmetry breaking to the dark sector. In these models, supersymmetry breaking can be treated as a perturbation on the spectrum of bound states. Using a general formalism, the spectrum with leading supersymmetry effects is computed without specifying the details of the binding dynamics. The interactions of the composite states with the Standard Model are computed, and several benchmark models are described. General features of nonrelativistic supersymmetric bound states are emphasized.
The nature of dark matter is unknown, and its relation to the Standard Model (SM) is an open question. The recent spate of anomalies in direct detection experiments [
Recent anomalies have several common features that motivate considering dark sectors that support bound states. Bound states naturally enjoy a hierarchy of different scales. Inelastic dark matter explanations of DAMA, for example, [
Fermions with gauge interactions are a ubiquitous ingredient in theories beyond the Standard Model. It is plausible that there are additional gauge sectors that SM fermions are not charged under. If there are no SM particles directly charged under the new gauge interaction, then experimental limits on decoupled gauge sectors are extremely weak. If supersymmetry breaking is only weakly mediated to the dark sector, perhaps through dark matter’s interactions with the Standard Model, then the magnitude of supersymmetry breaking effects can be extremely small. This allows for the possibility that dark matter is nearly supersymmetric. If there are any bound states in the dark sector, the spectrum will exhibit near Bose-Fermi degeneracy. Such weakly coupled hidden sectors also naturally sit near the GeV scale, which makes for interesting dark matter phenomenology [
Investigating nearly supersymmetric bound states arising from perturbative Coulombic interactions is a relatively intricate process, and the standard techniques from quantum mechanics involve computing first and second order
The organization of the paper is as follows. Section
This section studies how nonrelativistic supersymmetric bound states organize themselves into supermultiplets. Section
Nonrelativistic bound states have a structure that is simple to understand because they benefit from a good expansion parameter: the velocity
For simplicity, assume that the bound state is supported by a central potential that is spin independent at
As an example that illustrates this decomposition, consider the model bound state system that will form the main subject of this article. It consists of four massive chiral superfields (
For this system, the superspin wavefunction
Supersymmetry organizes nonrelativistic pairs of
The decomposition is simplified by noting that the same state can appear in many different bilinears. In fact, the bilinears
Although the
The states in (
To illustrate the action of supersymmetry on the ground states, consider the heavy proton limit,
This method of calculating superspin wavefunctions through decomposing products of superfields is general and can be applied to a wide class of nonrelativistic supersymmetric bound state problems. For example, the superspin wavefunctions of nonrelativistic
This prescription for finding the superspin wavefunctions can also be applied to the excited states. For a given spatial wavefunction
Thus, provided that a given
Once the ground state spectrum is known, it is important to determine how the various states interact with one another as well as with the SM. There are a variety of interactions, many of which are related through supersymmetric Ward identities. Superfields, thus, offer a convenient method for packaging all these interactions into manifestly supersymmetric forms. This section uses the standard off-shell superfield formalism to formulate an effective free action for the ground state, postponing until Section
The previous section calculated the composition of nonrelativistic supersymmetric bound states using supersymmetric group theory, focusing on the particular example of bound states formed from two chiral superfields. This section builds on Section
The leading supersymmetry breaking effects can be calculated by folding in the perturbed rest energies of the constituents with the ground state superspin wavefunctions calculated in Section
The bound state spectrum has two effective mass scales for supersymmetry breaking effects. The first scale is set by the
The soft supersymmetry breaking Lagrangian for the chiral-chiral bound state system introduced in Section
The leading supersymmetry breaking perturbation on the ground state spectrum is encapsulated in the perturbing Hamiltonian
Supersymmetry breaking effects begin to grow in complexity beyond the rest mass perturbation. The next most important term in the nonrelativistic expansion is the kinetic energy perturbation
In this section, the ground state spectrum with weakly broken supersymmetry is presented by diagonalizing the perturbation
The ground state spectrum for the case
In the absence of supersymmetry, breaking the hypermultiplet contains the degenerate pair of positive parity scalar bound states
In the supersymmetric limit, the ground state contains three parity odd scalars, one of which,
In the limit,
In the absence of supersymmetry, breaking the vector multiplet (hypermultiplet) contains the degenerate pair of
The vector state
Composite systems have a wide range of interactions that are controlled by selection rules and form factors that result in these systems having a much richer phenomenology than elementary particles. This section uses the effective field theory of Section
Section
Section
In models such as that of Section
The interactions of neutral bound states with an external vector superfield,
Before considering the supersymmetric case, it is instructive to review the leading interactions of the photon with the spin-singlet ground state of regular hydrogen. The leading elastic interaction comes from the charge radius operator
The next step is to find the set of operators necessary to satisfy the supersymmetric Ward identities.
The charge radius operator in (
A variety of processes cause the decay of the excited states to the ground state. For example, supersymmetric hydrogen inherits the (fast) electric dipole and magnetic dipole transitions of regular hydrogen. Decays from
The states in
Only (
The operator in (
The various decay channels induced by the interactions in (
Supersymmetry restricts the form of possible interactions significantly, and these restrictions are particularly severe for interactions connecting two chiral superfields. For instance, the only allowed single-photon operator, up to possible additional factors of
The restriction to operators of the form in (
In the supersymmetric limit, the hypermultiplet is exactly stable. Once supersymmetry is broken and decays within the hypermultiplet become kinematically allowed, it is interesting to ask what decay channels determine the relaxation timescale. This question is complicated by the fact that supersymmetry breaking enters the physics of decays in a number of ways. On the one hand, supersymmetry breaking perturbs eigenvalues and eigenstates; this opens up phase space, changes the equations of motion, and induces decay channels through mixing. On the other hand, supersymmetry breaking perturbs the effective interactions of the nonrelativistic constituents. The rest of this section considers these possibilities in more detail, with the conclusion that eigenstate mixing in the magnetic spin-flip operator, (
In the presence of soft masses, the supersymmetric operators in (
Next consider how the magnetic spin-flip operator in (
This section outlines the dominant interactions between dark atoms and an axial
The leading
In models like that of Section
Higher dimension operators also contribute to the interactions with the standard model. For example, the axial spin flip operator
Two body decays mediated by
This section constructs a minimal model for a nearly supersymmetric dark sector that supports Coulombic bound states. Section
Abelian field strengths are gauge invariant, and, therefore, no symmetry principle forbids mixed field strength terms [
Consider a minimal example where a dark
In addition to driving
This section adds light, charged matter to the dark sector. The charged matter consists of four chiral superfields
The interactions of the
The Higgs trilinear coupling,
Although the hidden sector is supersymmetric at tree level, at the loop level small supersymmetry breaking effects are induced through the kinetic mixing portal to the SSM. This section discusses the strength with which the constituent particles’ masses feel supersymmetry breaking. These soft masses determine the leading supersymmetry breaking effects in the ground state spectrum, as discussed in Section
The soft parameters to be calculated (see (
Section
This section constructs three benchmark models to illustrate the scales that emerge in the hidden sector. Although doing detailed direct detection phenomenology is outside the scope of this paper, in both cases, we aim to construct spectra compatible with iDM phenomenology. In particular, we require that the bound states have a mass
It is possible to meet these criteria; however, some tension exists between meeting all three criteria simultaneously. In models consistent with these requirements,
Trying to match the CoGeNT/DAMA anomaly [
Following the above logic, we choose the MSSM parameters shown in Table
The parameters of the dark sector for the three benchmark points are chosen to be as shown in Table
0.005 | 1.5 | 0.004 | 0.15 | 1.2 | ||
0.005 | 1.1 | 0.004 | 0.25 | 1.4 | ||
0.005 | 0.5 | 0.004 | 0.35 | 1.0 |
The first two choices for
0.24 | |||||||
0.29 | |||||||
0.40 |
Supersymmetry breaking effects are encapsulated in the soft parameters as shown in Table
The gauge-mediated contribution to
The resulting spectra are shown in Figure
The ground state spectrum of the three benchmark models. In all cases, the small, unlabeled splittings are of order
From the results in Section
The benchmark models discussed in Section
In a realistic model where the dark matter sector is nearly supersymmetric, the hidden
If dark atoms are relevant for cosmology, then a significant fraction of the supersymmetric dark electrons and protons must recombine into supersymmetric atoms. The recombination of nonsupersymmetric hydrogen-like dark matter atoms is studied in [
Even if gaugino emission is subleading as expected, there are new electric transitions between the different, nondegenerate superspin levels with
If supersymmetric atoms can form, it is possible that these atoms may further aggregate into supersymmetric molecules. This section briefly explores this possibility by examining the role that Bose/Fermi statistics plays in atoms and molecules.
The ground state of regular diatomic hydrogen is a
In the Standard Model, further aggregation into molecules larger than H2 is prevented by the Pauli exclusion principle, which forbids more than two electrons from being in the same orbital. In supersymmetric atoms, this aggregation is not forbidden by Pauli because electrons can convert into their scalar superpartners, selectrons. Supersymmetric bound states will share orbitals more effectively and, hence, are bound more strongly. For nonrelativistic molecules composed of
In order for dark atoms to be a sizeable fraction of the universe’s dark matter, there needs to be a chemical potential for
Composite dark matter offers a rich phenomenology that has only barely been explored in comparison to models in which dark matter is an elementary particle. Nonrelativistic bound states offer one general class of composite dark matter models, and these typically involve a new mass scale that is incorporated either by hand or through a Higgs mechanism. In the later case, the Higgs mass is radiatively unstable, and supersymmetry is a natural way to stabilize its mass. Supersymmetry breaking can be weakly communicated to this sector, particularly if the interactions of the dark sector with the Standard Model are the dominant link to the supersymmetry breaking sector. If this is the case, then the composite dark matter will form nearly supersymmetric multiplets, and the phenomenology of these states can be radically different from the nonsupersymmetric case.
Atomic inelastic dark matter [
This article has also constructed tools to help in the study of quasiperturbative supersymmetric bound states and in incorporating supersymmetry breaking into the bound states. These tools illustrate that the form of the weakly broken spectrum and the composition of the various states is in many cases dictated entirely by supersymmetry at leading order. Supersymmetry also imposes strict restrictions on the allowed interactions, which simplifies the matching of effective interaction operators. More generally supersymmetric bound states offer a rich laboratory for studying supersymmetric dynamics and interactions.
J. G. Wacker thanks E. Katz for useful conversations during the course of this work. M. Jankowiak and T. Rube would like to acknowledge helpful conversations with Michael Peskin as well as collaboration with Daniele Alves in early stages of this work. T. Rube is a William R. and Sara Hart Kimball Stanford Graduate Fellow. M. Jankowiak, S. R. Behbahani, and J. G. Wacker are supported by the US DOE under contract number DE-AC02-76SF00515. M. Jankowiak, S. R. Behbahani, T. Rube, and J. G. Wacker receive partial support from the Stanford Institute for Theoretical Physics. J. G. Wacker is partially supported by the US DOE’s Outstanding Junior Investigator Award and the Sloan Foundation. J. G. Wacker thanks the Galileo Galilei Institute for their hospitality during the early stages of this work.