Advances in High Energy Physics

Advances in High Energy Physics / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 709492 | 34 pages |

Nearly Supersymmetric Dark Atoms

Academic Editor: George Siopsis
Received04 Mar 2011
Accepted20 Apr 2011
Published22 Aug 2011


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.

1. Introduction

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 [14] and cosmic ray signatures [511] has motivated reexamining the standard assumptions about the identity of dark matter. Most models of dark matter assume that dark matter is an elementary particle with no relevant or long-range interactions. If supersymmetry is present in these models, the supersymmetric mass splittings are so large that the supersymmetric structure of dark matter is unimportant. This article provides a framework to illustrate the exact opposite case; dark matter is composite with long-range interactions, and supersymmetry breaking effects are small.

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, [1225], require several scales to reconcile the anomalies with the null results of other direct detection experiments. A hierarchy of scales is also employed in the exciting dark matter scenario [26] to explain the 511 keV signal from INTEGRAL/SPI. Additional structure in the dark sector is also motivated by positron excesses in cosmic ray data, which might be a result of cascade decays in the dark sector. Examining the standard model, one finds a variety of different bound state systems: mesons and baryons, nuclei, atoms, and molecules. Given the prevalence of bound states in standard model systems, it is natural to explore the possibility [2734] that dark matter is composed of bound states in a separate sector.

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 [3542] and experimental signatures [4351].

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 𝒯-matrix elements and then diagonalizing the Hamiltonian. At each step, the calculation is not supersymmetric, although the final answer is supersymmetric. Ultimately, the states have organized themselves into supersymmetric multiplets, and the admixtures of different supersymmetric particles that each composite state consists of is known. For instance, a spin zero fermion-fermion bound state will mix with a spin zero scalar-scalar bound state. Since phenomenological applications depend on these admixtures, it would be convenient to understand their structure and how they generalize to other bound state systems. Similarly, phenomenological studies would be made easier by understanding how bound state interactions are constrained by supersymmetry. This article develops a simple formalism to do this using off shell superfields.

The organization of the paper is as follows. Section 2 reviews nonrelativistic supersymmetric bound states, focusing on how supersymmetry organizes the spectrum and superspin wavefunctions of the states. The free effective action is also introduced, which will form the basis for computing the supersymmetric interactions of the bound states. Section 3 incorporates the effects of supersymmetry breaking into the spectrum for the case where the dominant source of supersymmetry breaking is the soft masses of the scalar constituents. Section 4 computes the interactions of the bound states when interacting with weakly coupled external gauge interactions. Section 5 constructs a realistic model of nearly supersymmetric atomic dark matter. Section 6 discusses possible directions of future research for models along these lines, including recombination and the formation of supersymmetric molecules. Section 7 makes some concluding remarks.

2. Nonrelativistic Supersymmetric Bound States

This section studies how nonrelativistic supersymmetric bound states organize themselves into supermultiplets. Section 2.1 outlines a general procedure for determining the composition of nonrelativistic bound states formed from massive superfields. When applicable, this procedure has the advantage of sidestepping a detailed perturbative calculation in favor of some superfield algebra. This procedure is illustrated in the particular case of bound states formed from two chiral multiplets. Section 2.2 continues the study of this particular example by introducing an effective field theory description of the ground state. This will provide the basis for Section 4, in which bound state interactions are discussed.

2.1. Wavefunctions from Superspace

Nonrelativistic bound states have a structure that is simple to understand because they benefit from a good expansion parameter: the velocity 𝑣. This is especially the case for two-body systems, where an expansion in powers of 𝑣 not only helps to organize calculations but also determines the relevant scales of the problem. The gross structure of the spectrum can be organized into principle excitations split by energies of order 𝑚prin𝜇𝑣2,(2.1) where 𝜇 is the reduced mass. Fine structure effects are the next order correction in the nonrelativistic expansion, appearing as 𝑚FS𝜇𝑣4.(2.2) Recent papers [52, 53] have computed the fine structure of supersymmetric hydrogen through explicit calculation. This section rederives these results by considering how supersymmetry acts on the bound states in the nonrelativistic limit. The organization of the spectrum into supermultiplets does not depend on the details of the binding dynamics except for specific quantities, such as energy splittings. Most notably, the superspin wavefunctions are completely determined by supersymmetry alone if there are no accidental degeneracies in the spectrum. This method of using supersymmetry to fix the superspin wavefunctions is applicable to a wider class of nonrelativistic bound states than Coulombic bound states and more cleanly delineates which quantities depend upon dynamics versus the structure of supersymmetry.

For simplicity, assume that the bound state is supported by a central potential that is spin independent at 𝒪(𝑣2). This is true for a wide range of composite states, including those bound together by the exchange of light vector or chiral multiplets. The ground state then has a non-degenerate radial wavefunction with 𝑙=0 [54] and factorizes as ||||||Ψ=𝜓(𝑟)𝒮(2.3) to leading order, where |𝒮 is the superspin part of the wavefunction. At leading order in 𝑣, the supercharges act only on |𝒮, leaving |𝜓(𝑟) intact, because gradients of nonrelativistic wavefunctions are suppressed, 𝜕𝑖𝜓𝒪(𝑣). Since 𝜓(𝑟) has trivial angular dependence, decomposing |𝒮 into irreducible representations decomposes |Ψ into irreducible supersymmetry representations, Ω𝑗, where 𝑗 refers to the spin of the Clifford vacuum (e.g., Ω0 is the chiral multiplet). At 𝒪(𝑣4), the Hamiltonian is typically spin dependent, and any degeneracy among the Ω𝑗’s will generically be lifted in the absence of any special symmetries. For Ω𝑗’s, that are accidentally degenerate at 𝒪(𝑣2), there can be large mixing that depends on the details of the dynamics, though in many cases the appropriate mass eigenstates are determined by the action of additional symmetries on the supermultiplets.

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 (𝐸, 𝐸𝑐, 𝑃, 𝑃𝑐) with Dirac masses 𝑚𝑒 and 𝑚𝑝 satisfying 𝑚𝑒𝑚𝑝. The binding dynamics respect parity, under which the coordinates and superfields transform as 𝑥𝜇(1)𝜇𝑥𝜇,𝜃𝛼𝜃̇𝛼,𝑃𝑃𝑐,𝐸𝐸𝑐.(2.4) The dynamics also respect a 𝑈(1)𝑅-symmetry and a 𝑈(1)𝑒×𝑈(1)𝑝 flavor symmetry. The charges of the component fields are taken to be 𝑝̃𝑝𝛼𝑝𝑐̇𝛼̃𝑝𝑐̃e𝑒𝛼𝑒𝑐̇𝛼̃𝑒𝑐𝑈(1)𝑅101101𝑈(1)𝑒+𝑝111111.(2.5) Significantly, the 𝑈(1)𝑅 symmetry and 2 parity do not commute and combine into an 𝑂(2)𝑅 symmetry. This can be seen by considering the “selectrons” ̃𝑒 and ̃𝑒𝑐. Parity, 𝒫, acts upon the selectrons as 𝒫̃𝑒=̃𝑒𝑐,(2.6) while under a 𝑈(1)𝑅 transformation, 𝑅(𝛼), the selectrons transform as 𝑅(𝛼)̃𝑒=𝑒𝑖𝛼̃𝑒,𝑅(𝛼)̃𝑒𝑐=𝑒𝑖𝛼̃𝑒𝑐,(2.7) so that [𝒫,𝑅(𝛼)]̃𝑒0. Thus 𝑈(1)𝑅 and 2 are not a direct product and instead combine as the semidirect product 𝑈(1)𝑅2𝑂(2)𝑅. This is important because 𝑂(2)𝑅 has two-dimensional irreducible representations that are realized in the bound state spectrum. In particular, any state that transforms nontrivially under 𝑈(1)𝑅 must sit in an 𝑂(2)𝑅 doublet.

For this system, the superspin wavefunction |𝒮 in (2.3) decomposes as two chiral multiplets and one vector multiplet (𝖁), as can be verified by counting degrees of freedom. As a consequence of the 𝑂(2)𝑅 symmetry, however, the two chiral multiplets combine into a hypermultiplet 𝕳 so that the decomposition of |𝒮 reads ||𝒮=2Ω0Ω1/2=𝕳𝖁.(2.8) Both 𝕳 and 𝖁 are charged under the global 𝑈(1)𝑒+𝑝 flavor symmetry of the theory. The superspin wavefunctions of the ground state are fixed by (super)symmetry at leading order because 𝕳 and 𝖁 are irreducible under the full symmetry group and, therefore, insensitive to mixing.

Supersymmetry organizes nonrelativistic pairs of free particles into supermultiplets, determining the bound state wavefunctions at leading order in 𝑣 in terms of the constituent particles. The organization of pairs of free particles into supermultiplets is found by putting 𝐸, 𝐸𝑐, 𝑃, and 𝑃𝑐 on shell and constructing all possible superfield bilinears. The resulting bilinears will have spins ranging from 0 to 1. For example, the superfields 𝑃 and 𝐸𝑐 yield the bilinears 𝑃𝐸𝑐, 𝒟𝛼𝑃𝐸𝑐, 𝑃𝒟̇𝛼𝐸𝑐, and 𝒟𝛼𝑃𝒟̇𝛼𝐸𝑐. These bilinears can then be decomposed into irreducible supersymmetry representations with the help of projection operators, which in the case of spin zero superfields are given by 𝒫1=𝒟2𝒟216,𝒫2=𝒟2𝒟216,𝒫𝑇𝒟=𝒟2𝒟8,(2.9) where 𝒫1+𝒫2+𝒫𝑇=1 [55].

The decomposition is simplified by noting that the same state can appear in many different bilinears. In fact, the bilinears 𝒫2𝑃𝐸=𝑃𝐸,𝒫1𝑃𝑐𝐸𝑐=𝑃𝑐𝐸𝑐,𝒫𝑇𝑃𝐸𝑐(2.10) contain all the states as can be verified by counting degrees of freedom. Expanding the first two bilinears in (2.10) using the nonrelativistic fields. (The superscript 𝐷 indicates that the spinor is in the Dirac basis, where 𝛾0 is diagonal.) 𝑒̃𝑝=𝑖𝑚𝑝𝑡2𝑚𝑝𝜙𝑝,Ψ𝐷𝑝=𝑒𝑖𝑚𝑝𝑡𝜓𝑝𝑖𝜎2𝑚𝑝𝜓𝑝(2.11) gives the superfields (cf. [56]) 𝑃𝐸𝜙𝑝𝜙𝑒+2Θ𝑎𝑐𝜃𝜓𝑎𝑝𝜙𝑒+𝑠𝜃𝜙𝑝𝜓𝑎𝑒Θ2𝑠2𝜃𝜙𝑝𝜙𝑒𝑐+𝑐2𝜃𝜙𝑝𝑐𝜙𝑒𝑠2𝜃𝜓𝑝𝜓𝑒0,𝑃𝑐𝐸𝑐𝜙𝑝𝑐𝜙𝑒𝑐+2Θ𝑎𝑐𝜃𝜓𝑎𝑝𝜙𝑒𝑐+𝑠𝜃𝜙𝑝𝑐𝜓𝑎𝑒Θ2𝑐2𝜃𝜙𝑝𝜙𝑒𝑐+𝑠2𝜃𝜙𝑝𝑐𝜙𝑒+𝑠2𝜃𝜓𝑝𝜓𝑒0,(2.12) where the dimensionless Θ𝛼=𝑚𝑝+𝑚𝑒𝜃𝛼 has been introduced, and the mixing angle 𝜃 is defined by tan2𝑚𝜃=𝑒𝑚𝑝.(2.13) These two superfields have 𝑈(1)𝑅-charges of ±2 and transform into each other under parity; they correspond to the two Ω0’s in 𝕳. The 𝖁 wavefunctions are found by decomposing the bilinear 𝒫𝑇𝑃𝐸𝑐, which gives a complex vector (curl) superfield with components 𝐷𝑐2𝜃𝜓𝑝𝜓𝑒0+𝑠2𝜃𝜙𝑝𝑐𝜙𝑒𝜙𝑝𝜙𝑒𝑐2,𝜆1𝑠𝜃𝜓𝑝𝜙𝑒𝑐𝑐𝜃𝜙𝑝𝑐𝜓𝑒,𝜆2𝑠𝜃𝜓𝑝𝜙𝑒𝑐𝜃𝜙𝑝𝜓𝑒,𝑣𝜇𝜓𝑝𝜎𝜓𝑒.(2.14) Going to the parity eigenbasis and introducing notation for the various states gives 𝑣𝖁=𝜇,||𝑣=||𝜓𝑝𝜓𝑒1,𝜒+,𝜒𝑐+,||𝜓𝜒+=𝑐𝜃||𝜙𝑝+𝜓𝑒𝑠𝜃||𝜓𝑝𝜙𝑒+,𝜒,𝜒𝑐,||𝜓𝜒=𝑐𝜃||𝜙𝑝𝜓𝑒𝑠𝜃||𝜓𝑝𝜙𝑒,𝜍,||𝜍=𝑐2𝜃||𝜓𝑝𝜓𝑒0+𝑠2𝜃2||𝜙𝑝+𝜙𝑒||𝜙𝑝𝜙𝑒+,𝜔𝕳=+,||𝜔+=12||𝜙𝑝+𝜙𝑒+||𝜙𝑝𝜙𝑒,𝜔,||𝜔𝑐=2𝜃2||𝜙𝑝𝜙𝑒+||𝜙𝑝+𝜙𝑒+𝑠2𝜃||𝜓𝑝𝜓𝑒0,𝜉+,𝜉𝑐+,||𝜓𝜉+=𝑐𝜃||𝜓𝑝𝜙𝑒++𝑠𝜃||𝜙𝑝+𝜓𝑒,𝜉,𝜉𝑐,||𝜓𝜉=𝑐𝜃||𝜓𝑝𝜙𝑒+𝑠𝜃||𝜙𝑝𝜓𝑒,𝜛+,||𝜛+=12||𝜙𝑝+𝜙𝑒++||𝜙𝑝𝜙𝑒,𝜛,||𝜛1=2||𝜙𝑝+𝜙𝑒+||𝜙𝑝𝜙𝑒+,(2.15) where (𝜓𝑝𝜓𝑒)0=(1/2)𝜖𝑎𝑏𝜓𝑎𝑝𝜓𝑏𝑒, 𝜙𝑝/𝑒±=(1/2)(𝜙𝑝/𝑒±𝜙𝑝𝑐/𝑒𝑐), and 𝑐𝜃,𝑠𝜃 are cos𝜃 and sin𝜃, respectively. These are the same wavefunctions found in [52, 53] by means of a detailed perturbative calculation in the particular case of supersymmetric hydrogen.

Although the 𝖁 and 𝕳 wavefunctions have been determined here without specifying the binding dynamics, the mass splitting between 𝖁 and 𝕳 can only be determined by doing a dynamical calculation. In the absence of any special symmetries, however, it is expected that 𝑚FS𝑚𝖁𝑚𝕳 will be at the fine structure scale, 𝑚FS𝒪(𝑣4𝜇), and in the case of supersymmetric hydrogen one finds 𝑚FS=12𝛼4V𝜇.(2.16)

The states in (2.15) are organized according to their 𝑂(2)𝑅 representations, with simple transformation properties under parity, because the breaking of 𝑂(2)𝑅 plays an important role in lifting degeneracies in the spectrum once supersymmetry is broken. The states 𝜒±, 𝜉± and 𝜛± transform in two-dimensional representations of 𝑂(2)𝑅 with 𝑈(1)𝑅 charges of 1, 1, and 2, respectively. For example, the doublet 𝜛𝜛=+𝑖𝜛(2.17) transforms irreducibly as 𝒫𝜛𝜎3𝜛,𝑅(𝛼)𝜛𝑒2𝑖𝛼𝜎2𝜛.(2.18) The states 𝑣𝜇, 𝜍 and 𝜔± are invariant under 𝑅(𝛼) and, thus, transform as 𝑂(2)𝑅 singlets.

To illustrate the action of supersymmetry on the ground states, consider the heavy proton limit, 𝜃0. In this limit, supersymmetry clocks the states of the heavier constituent, leaving the valence particle intact. In particular, the 𝖁 states consist of a light electron orbiting a heavy proton multiplet, and the 𝕳 states consist of a light selectron orbiting a heavy proton multiplet.

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 𝑆𝑈(3) baryons can be found by studying the decomposition of superfield trilinears. In this case, acting with the projection operators in (2.9) on spin zero trilinears does not give all of the wavefunctions, and a spin 1/2 trilinear is necessary. Similarly, the study of the bound states of a massive chiral and a massive vector superfield requires higher spin projections.

2.1.1. Excited State Wavefunctions

This prescription for finding the superspin wavefunctions can also be applied to the excited states. For a given spatial wavefunction |𝑛𝑙, the various excited states can be built by acting with supersymmetry on the Clifford vacua defined by the particle content, ||||Ω𝑛𝑙𝑠||𝑎,𝑛𝑙||Ω𝑠,||𝑎𝑛𝑙2||Ω𝑠,(2.19) where |Ω𝑠, 𝑎|Ω𝑠, and 𝑎2|Ω𝑠 are the superspin wavefunctions derived in the previous section. For example, in the case considered above, |Ω𝑠 is either |Ω0 or |Ω1/2, and the raising operators fill out the various states in 𝖁 and 𝕳. Decomposing (2.19) into supermultiplets is equivalent to switching to the basis ||||Ω𝑛𝑙𝑠,𝑎||||Ω𝑛𝑙𝑠,𝑎2||||Ω𝑛𝑙𝑠,(2.20) since irreducible representations of supersymmetry are obtained by acting with the raising operator on Clifford vacua that are irreducible representations of the rotation group. This basis switch is just a matter of Clebsch-Gordon algebra and results in the decomposition 𝑙Ω𝑠=Ω|𝑙𝑠|Ω|𝑙+𝑠|.(2.21) For example, in the case considered above, where the bound state is formed from two chiral multiplets, the decomposition gives Ω𝑙0Ω0Ω=𝑙0Ω0Ω1/2=Ω𝑙1/2Ω𝑙Ω𝑙Ω𝑙+1/2(2.22) with the two Ω𝑙 related to one another by parity.

Thus, provided that a given Ω𝑗 does not undergo large mixing, the excited state angular/superspin wavefunctions can be found just as for the ground state. One does a (single) superfield calculation as before to determine Ω𝑠 and then transforms from the basis of (2.19) to that in (2.20) using Clebsch-Gordan coefficients.

2.2. Effective Action for the Ground State

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 4 a discussion of ground state interactions.

𝕳 is described by two chiral superfields that satisfy the following relations on shell 1𝑃𝐸,2𝑃𝑐𝐸𝑐.(2.23) A second set of chiral superfields, 1𝑐 and 𝑐2, is introduced to give the 𝐹-terms of 1 and 2 dynamics. The free Lagrangian for 𝕳 is given by 𝕳=𝑑4𝜃𝛿𝑖𝑗𝑖𝑗+𝑖𝑐𝑐𝑗+𝑑2𝜃𝛿𝑖𝑗𝑚𝑖𝑐𝑗+h.c.(2.24) The equations of motion which follow from (2.24) then result in the identification 1𝑐𝒫1𝑃𝑐𝐸,𝑐2𝒫2𝑃𝑐𝐸.(2.25) With 𝑖 and 𝑐𝑖 identified as above, the appropriate 𝑈(1)𝑅 and 𝑈(1)𝑒+𝑝 charges are given by 121𝑐𝑐2𝑈(1)𝑅2200𝑈(1)𝑒+𝑝2222(2.26) so that the Lagrangian is properly invariant under 𝑈(1)𝑅 and 𝑈(1)𝑒+𝑝. Parity acts on the composite superfields as 12𝑐12𝑐.(2.27) So that the Lagrangian is also invariant under parity.

𝖁 is described by an off-shell field, 𝒱, and an action consistent with the on-shell constraint 𝒱𝒫𝑇𝑃𝐸𝑐. 𝒱 is a charged vector superfield—a general superfield with no Lorentz index 𝒱𝑥,𝜃,𝜃𝒱𝑥,𝜃,𝜃.(2.28) The action is written with the help of the supersymmetric field strengths 𝒲1𝛼1=4𝒟2𝒟𝛼𝒱,𝒲2𝛼1=4𝒟2𝒟𝛼𝒱,(2.29) which have 𝑈(1)𝑅 charges of +1. Under parity 𝒱 and 𝒲𝑖 transform as 𝒱𝒱,𝒲𝛼1𝒲2̇𝛼.(2.30) The free Lagrangian, which is properly invariant under 𝑈(1)𝑅 and parity, is given by 𝖁=𝑑4𝜃2𝑚2𝒱𝒱𝑑𝒱+2𝜃12𝒲𝛼1𝒲2𝛼+h.c.(2.31) Varying the action yields the equation of motion 𝒟𝛼𝒲𝛼1=2𝑚2V𝒱, implying that 𝒫𝑇𝒱=𝒱 on shell.

3. Supersymmetry Breaking in the Ground State

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 2 by incorporating the effects of weak supersymmetry breaking on the ground state spectrum. The exact changes to the spectrum resulting from supersymmetry breaking depend on the details of the binding dynamics. In many theories, however, supersymmetry breaking level splittings induced by the binding dynamics are accompanied by powers of the velocity, 𝑣. Consequently in the nonrelativistic limit, supersymmetry breaking in the bound state spectrum will be dominated by the differences in the rest energies of the constituent fermions and bosons. For such theories, the resulting spectrum is insensitive to the details of the binding dynamics.

3.1. Constituent Mass Effects

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 2. This leading order effect is straightforward to compute if the effective scale of supersymmetry breaking in the bound states spectrum, 𝑚soft, is smaller than the scale of principle excitations 𝑚soft𝑚prin𝒪𝜇𝑣2.(3.1) In this case, mixing with excited states is unimportant, and the incorporation of supersymmetry breaking into the bound state spectrum reduces to a finite dimensional quantum mechanical perturbation theory problem.

The bound state spectrum has two effective mass scales for supersymmetry breaking effects. The first scale is set by the 𝑈(1)𝑅-preserving soft masses, 𝑚𝑅-pres, while the second is set by the 𝑈(1)𝑅-violating 𝐵-term masses, 𝑚𝑅-viol. The breaking of the 𝑈(1)𝑅 symmetry induces splittings between states that are doublets under the 𝑂(2)𝑅 symmetry. In many implementations of dark sector supersymmetry breaking, 𝑈(1)𝑅-violating soft terms will be suppressed relative to the 𝑈(1)𝑅-preserving soft terms, and for simplicity the relative ordering of the scales is taken to be 𝑚𝑅-pres,𝑚FS𝑚𝑅-viol(3.2) throughout, where 𝑚FS=𝒪(𝜇𝑣4).

The soft supersymmetry breaking Lagrangian for the chiral-chiral bound state system introduced in Section 2 contains a 𝑈(1)𝑅-preserving piece, 𝑅-presΔ2̃𝑒||||̃𝑒2+||̃𝑒𝑐||2+Δ2̃𝑝||||̃𝑝2+||̃𝑝𝑐||2(3.3) and additional supersymmetry breaking terms that break the 𝑈(1)𝑅 symmetry: 𝑅-viol𝐵𝑒𝑚𝑒̃𝑒̃𝑒𝑐+𝐵𝑝𝑚𝑝̃𝑝̃𝑝𝑐+h.c.(3.4) For simplicity, the soft parameters are assumed to obey the relations Δ2̃𝑒Δ2̃𝑝,𝐵𝑝𝐵𝑒𝐵.(3.5) In the presence of Δ2̃𝑒 and 𝐵𝑒, the selectron mass eigenstates become ̃𝑒±=12̃𝑒±̃𝑒𝑐(3.6) with masses 𝑚̃𝑒±=𝑚𝑒+𝛿𝑚̃𝑒±𝑚𝑒+12Δ2̃𝑒𝑚𝑒±12𝐵.(3.7) Analogous expressions hold for the mass eigenstates ̃𝑝±. See Section 5.3 for details on a particular implementation of supersymmetry breaking in the dark sector that satisfies the above assumptions.

The leading supersymmetry breaking perturbation on the ground state spectrum is encapsulated in the perturbing Hamiltonian 𝐻soft=𝛿𝑚̃𝑝±||𝜙𝑝±𝜙𝑝±||+𝛿𝑚̃𝑒±||𝜙𝑒±𝜙𝑒±||.(3.8) The 𝑈(1)𝑅-preserving contributions of 𝐻soft will appear in the combination 𝑚soft12𝛿𝑚̃𝑝++𝛿𝑚̃𝑝+𝛿𝑚̃𝑒++𝛿𝑚̃𝑒.(3.9) The rest energy perturbations can now be read off directly from the supersymmetric wavefunctions in (2.15). For example, consider the state ||𝜍=𝑐2𝜃||𝜓𝑝𝜓𝑒0+𝑠2𝜃||𝜙𝑝+𝜙𝑒||𝜙𝑝𝜙𝑒+2.(3.10) The fermion-fermion component is insensitive to 𝐻soft, but the scalar-scalar component results in a perturbation Δ𝑚𝜍𝜍||𝐻soft||𝜍=12𝑠22𝜃𝛿𝑚̃𝑝++𝛿𝑚̃𝑒+𝛿𝑚̃𝑝+𝛿𝑚̃𝑒+=𝑠22𝜃𝑚soft,(3.11) which is the leading supersymmetry breaking contribution to the mass of 𝜍 in the limit that 𝑚FS𝑚soft. For many physical applications, such as decays or scattering off of SM nuclei, knowing only the leading breaking is sufficient. Using the superspace approach for finding the wavefunctions, as in Section 2.1, the leading supersymmetry breaking can, thus, be found for a broad range of perturbative bound states.

3.2. Subdominant Effects

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 𝐻𝑣2𝑝22𝜇𝛿𝜇𝜇.(3.12) This changes the principle structure of the bound state and leads to a 𝒪(𝑣2) perturbing Hamiltonian 𝐻𝑣2𝑝=22𝜇2cos4𝜃𝛿𝑚̃𝑒±||𝜙𝑒±𝜙𝑒±||+sin4𝜃𝛿𝑚̃𝑝±||𝜙𝑝±𝜙𝑝±||.(3.13) At the level of fine structure, many new effects arise. These include additional kinematic effects from 𝒪(𝑝4) terms and, in the case of supersymmetric hydrogen, gaugino mass effects and 𝐷-term contributions. Incorporating all these effects requires using the 𝒯 matrix and computing all tree-level Feynman diagrams contributing to 𝑒𝑝𝑒𝑝 matrix elements. The 𝒯 matrix is proportional to an effective nonrelativistic Hamiltonian that can be used to do perturbation theory, as in the calculation of the fine structure of supersymmetric hydrogen [52, 53, 57].

3.3. Eigenstates

In this section, the ground state spectrum with weakly broken supersymmetry is presented by diagonalizing the perturbation 𝐻soft. The qualitative features of the spectrum are shown in Figure 1.

3.3.1. Scalars

In the absence of supersymmetry, breaking the hypermultiplet contains the degenerate pair of positive parity scalar bound states 𝜛+ and 𝜔+. In the presence of 𝐻soft, these states mix maximally: 𝜛+𝜔+𝑚soft𝐵𝐵𝑚soft𝜛+𝜔+(3.14)𝐵 characterizes the size of 𝑂(2)𝑅 breaking and mixes states of different 𝑅-charge. The mass eigenstates are 𝜔1+𝜙𝑝+𝜙𝑒+=12𝜛++𝜔+,𝑚𝜔1+=𝛿𝑚̃𝑒++𝛿𝑚̃𝑝+=𝑚soft𝜔+𝐵,2+𝜙𝑝𝜙𝑒=12𝜛+𝜔+,𝑚𝜔2+=𝛿𝑚̃𝑒+𝛿𝑚̃𝑝=𝑚soft𝐵.(3.15)

In the supersymmetric limit, the ground state contains three parity odd scalars, one of which, 𝜍, is in the vector multiplet, and two of which, 𝜔 and 𝜛, are in the hypermultiplet. In the presence of 𝐻soft and 𝐻𝑣2, all three states mix: 𝜍𝜔𝜛𝑚FS+𝑠22𝜃𝑚soft12𝑠4𝜃𝑚soft12𝑠4𝜃𝐵12𝑠4𝜃𝑚soft𝑐22𝜃𝑚soft𝑐22𝜃𝐵12𝑠4𝜃𝐵𝑐22𝜃𝐵𝑚soft𝜍𝜔𝜛.(3.16) Here 𝐵 characterizes the 𝑈(1)𝑅-breaking in this sector and comes about through 𝐻𝑣2 in (3.13) or from the difference in the 𝐵-term masses between the ̃𝑒 and ̃𝑝 and is of the order 𝐵𝒪𝐵𝑣2𝐵,𝒪𝑒𝐵𝑝.(3.17) By specializing to the regime where 𝐵𝑚FS,tan22𝜃𝑚soft,(3.18) so that mixing between the three states occurs primarily between 𝜍 and 𝜔, one obtains simple formulae for the approximate energy levels 𝑚𝜍=𝑚FS+𝑚soft2+𝑚FS𝑐4𝜃𝑚soft2𝑠1+24𝜃𝑚2soft𝑚FS𝑐4𝜃𝑚soft2𝑚,(3.19)𝜔=𝑚FS+𝑚soft2𝑚FS𝑐4𝜃𝑚soft2𝑠1+24𝜃𝑚2soft𝑚FS𝑐4𝜃𝑚soft2𝑚,(3.20)𝜛=𝑚soft.(3.21) Nonzero 𝐵-terms split the 𝑂(2)𝑅 doublet containing 𝜛+ and 𝜛.

In the limit, 𝑚soft/𝑚FS with 0𝜃<𝜋/8 (resp., 𝜋/8<𝜃𝜋/4), the state 𝜍 (resp., 𝜔) becomes the (𝜓𝑒𝜓𝑝)0 bound state. Naively, for 𝜃2𝑚𝑒/𝑚𝑝1/1836 and 𝑚FS=(1/2)𝛼4V𝜇, the splitting between this state and the vector (𝜓𝑒𝜓𝑝)1 should give the hyperfine splitting in regular hydrogen; however, (3.19) yields instead 𝑚(𝜓𝑒𝜓𝑝)1𝑚(𝜓𝑒𝜓𝑝)0=𝑚𝑣𝜇𝑚𝜍𝑠22𝜃𝑚FS2𝛼4V𝑚2𝑒𝑚𝑝.(3.22) This is not the correct hyperfine splitting of regular hydrogen which is 𝑚HFS=83𝛼4𝑚2𝑒𝑚𝑝(3.23) for a point-like proton. This difference arises because 𝑚soft/𝑚FS is not the full decoupling limit. In particular, the ground state of supersymmetric hydrogen contains admixtures of higher principle excitations arising from gaugino exchange at second order in perturbation theory. These effects contribute to the hyperfine splitting in (3.22) but disappear in the full decoupling limit where the gaugino mass goes to infinity, 𝑚V.

3.3.2. Fermions

In the absence of supersymmetry, breaking the vector multiplet (hypermultiplet) contains the degenerate pair of 𝑗=1/2 bound states 𝜉± (𝜒±). In the presence of 𝐻soft, the states of equal parity mix with one another 𝜒±𝜉±𝑇𝑚FS+12𝑠22𝜃𝑚soft±121𝐵4𝑠4𝜃𝑚soft14𝑠4𝜃𝑚soft𝑠4𝜃+𝑐4𝜃𝑚soft±12𝐵𝜒±𝜉±.(3.24) The spectrum is given by 𝑚𝜒±=𝑚FS+𝑚soft2+𝑚FS𝑐22𝜃𝑚soft21+(1/4)𝑠24𝜃𝑚2soft𝑚FS𝑐22𝜃𝑚soft2±12𝑚𝐵,𝜉±=𝑚FS+𝑚soft2𝑚FS𝑐22𝜃𝑚soft21+(1/4)𝑠24𝜃𝑚2soft𝑚FS𝑐22𝜃𝑚soft2±12𝐵.(3.25) For nonzero 𝐵, the 𝑂(2)𝑅 symmetry that ensured the degeneracy of the pair of states 𝜒± as well as the pair of states 𝜉± is broken, and the fermionic spectrum splits completely.

3.3.3. Vector

The vector state |𝑣=|(𝜓𝑝𝜓𝑒)1 is insensitive to 𝐻soft and, as a consequence, does not feel supersymmetry breaking at leading order.

4. Interactions

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 2.2 to study the interactions that arise when composite states inherit gauge interactions from their constituents (cf. [58]).

Section 4.1 considers the case where the constituents are charged under an unbroken vectorial gauge symmetry 𝑈(1)V such that the composite state is neutral with the following charge and parity assignments 𝐸𝐸𝑐𝑃𝑃𝑐𝑈(1)V1+1+11,VV.(4.1)V does not need to be responsible for binding the chiral multiplets together; for example, the binding could arise from a Yukawa force. The 𝑈(1)V gauge interactions of the constituents induce a number of effective operators, including charge radius, Rayleigh scattering, and spin flip operators. Specializing to the case where the hypermultiplet is lighter than the vector multiplet 𝒱 and 𝑚FS𝑚soft, decays within the ground state are discussed in detail. It is found that the states of the vector multiplet 𝒱 decay relatively quickly down to , while the decays within are much slower.

Section 4.2 briefly considers the case where the constituents are charged under a broken axial gauge symmetry 𝑈(1)A with charge and parity assignments 𝐸𝐸𝑐𝑃𝑃𝑐𝑈(1)A+1+111,AA.(4.2)

In models such as that of Section 5 where A undergoes kinetic mixing with the SSM, these interactions mediate the dominant coupling of dark atoms to the Standard Model. Section 4.2 discusses the allowed scattering channels and finds the leading supersymmetric axial interactions.

4.1. 𝑈(1)𝑉 Interactions

The interactions of neutral bound states with an external vector superfield, V, are characterized by two scales corresponding to the charge radius, 𝑅𝑒, and magnetic radius, 𝑅𝑚. Physically 𝑅𝑒 corresponds to the size of the bound state, 𝑅𝑒𝑟2. In the case of Coulombic bound states, 𝑅𝑒 is given by the Bohr radius, 𝑅𝑒1=𝛼V𝜇. 𝑅𝑚 is just the Compton wavelength, 𝑅𝑚1=𝜇. For convenience, this section will restrict its discussion to supersymmetric hydrogen, although it is generally applicable to chiral-chiral bound states.

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 𝑔V𝑐2𝜃𝑅2𝑒𝜓𝑝𝜓𝑒0𝜕𝜇𝜓𝑝𝜓𝑒0𝜕𝜈V𝜇𝜈,(4.3) which is fully determined by the charge distribution of the bound state. The leading inelastic interaction comes from the magnetic spin-flip operator which is determined by the fermion content 𝑔V𝑅𝑚𝜕𝜇𝜓𝑝𝜓𝑒1,𝜈𝜓𝑝𝜓𝑒0V𝜇𝜈.(4.4) Finally, there is the Rayleigh scattering operator 𝑔V𝑅3𝑒𝑚𝐻𝜓𝑝𝜓𝑒0𝜓𝑝𝜓𝑒0V𝜇𝜈V𝜇𝜈,(4.5) which makes the sky blue. All other operators are higher order in either 𝑔V or 𝜇1.

The next step is to find the set of operators necessary to satisfy the supersymmetric Ward identities. 𝒱𝒱 interactions can be important for scattering processes if the states of 𝒱 are long lived but have a subdominant effect on the lifetimes of the states in 𝒱. Because the leading 𝒱 decay is relatively fast, 𝒱 tends to be short lived, and, therefore, 𝒱𝒱 interactions are ignored here. The interactions are found in the following, since they determine the relaxation timescale of the supermultiplet.

The charge radius operator in (4.3) only depends on the charge distribution, and, therefore, the scalar-scalar bound states must share identical (diagonal) interactions 𝑔V𝑐2𝜃𝑅2𝑒𝜙𝑝±𝜙𝑒±𝜕𝜇𝜙𝑝±𝜙𝑒±𝜕𝜈V𝜇𝜈.(4.6) Rewriting (4.3) and (4.6) in terms of the wavefunctions in (2.15), the charge radius interactions become 𝑔V𝑐2𝜃𝑅2𝑒𝜔±𝜕𝜇𝜔±+𝜛±𝜕𝜇𝜛±+𝜍𝜕𝜇𝜍𝜕𝜈V𝜇𝜈.(4.7) Similarly, the spin-flip and Rayleigh scattering operators become 𝑔V𝑅𝑚𝜕𝜇𝑣𝜈𝑐2𝜃𝜍+𝑠2𝜃𝜔V𝜇𝜈,𝑔(4.8)V𝑅3𝑒𝑚𝑒+𝑚𝑝𝜔±𝜔±+𝜛±𝜛±+𝜍𝜍V𝜇𝜈V𝜇𝜈,(4.9) respectively. The operators in (4.7) to (4.9) represent the leading single-photon and two-photon interactions for the scalar states in 𝒱 and . Several interactions remain to be found, for example, the leading single-photino interactions as well as the interactions for the fermionic states. The coefficients of the remaining interactions are found by forming operators from the effective fields of Section 2.2. The matching coefficients are determined by expanding the supersymmetric operators in terms of their components and identifying the corresponding interactions from (4.7) to (4.9). This procedure allows for the various supersymmetric interactions to be systematically enumerated by building upon the known interactions of regular hydrogen.

4.1.1. Interactions of the Hypermultiplet with Higher States

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 𝒱 to , however, are not as fast and merit further discussion.

The states in 𝒱 are connected to through two one-photon operators of dimension five 𝒱V=𝑐𝑀𝑠2𝜃𝑔V𝑅𝑚𝑑2𝜃𝑐1𝒲1+𝑐2𝒲2𝒲V+h.c.(4.10)+𝑐𝑀𝑔V𝑔,𝜃V𝑅𝑚𝑑4𝜃𝑐1+𝑐2𝒱𝒟𝒲V+h.c.(4.11) These two operators are the most general forms for 𝒱 interactions mediated by 𝑈(1)V. Higher dimensional operators can be reduced to these two forms with additional factors of 𝜕2 acting on 𝒲V by using the matter field equations of motion.

Only (4.10) contains the magnetic spin-flip interaction, 𝜔𝜕𝜇𝑣𝜈𝐹𝜇𝜈V. The factor of 𝑠2𝜃 is fixed by comparison with (4.4). In supersymmetric hydrogen, some of the component interactions contained in (4.10) arise from 𝒪(𝛼V) mixing between the ground state and higher principle excitations. For example, excited 𝜔+ states (2𝑝, 3𝑝, etc.) mix with 𝑣𝜇, allowing for 𝑣𝜇 to decay to 𝜔+ through electric dipole transitions. This mixing with excited states is the origin of the “electric” interaction 𝜔+𝜕𝜇𝑣𝜈𝐹𝜇𝜈V contained in the operator of (4.10). In this sense, the operator of (4.10) is neither purely magnetic or electric. The role that excited state mixing plays in ensuring this supersymmetric result is familiar from the calculation of the supersymmetric spectrum in [52, 53, 57], where second order perturbation theory is needed to determine the spectrum to 𝒪(𝛼4V).

The operator in (4.11) does not mediate decays in the supersymmetric limit. This can be seen by using the equations of motion to replace 𝒟𝒲V with the current 𝒥V. Decays through this operator are kinematically forbidden because the mass splitting between 𝒱 and is much smaller than the mass of any particle charged under 𝑈(1)V. For this reason, we leave the coefficient of this operator undetermined, noting, however, that it can only come in at higher order than 𝑔V𝑅𝑒, since it contains off-diagonal scalar-scalar transitions, which do not arise from charge radius scattering.

The various decay channels induced by the interactions in (4.10) cause each state in 𝒱 to have the same inclusive decay width to the states of in the supersymmetric limit—otherwise the component propagators of 𝒱 would have different poles. Therefore, the decay width can be calculated by considering the state with the simplest decay modes, in this case 𝜍: 𝒱V𝑐𝑀𝑔V𝑅𝑚𝑚V𝑖𝑠2𝜃22𝑖𝜉+𝛾5+𝜉ΛV𝜍.(4.12) Here ΛV is the four-component Majorana gaugino of 𝑈(1)V, and 𝑚V is its mass. This gives the decay rate Γ𝒱V||𝑐𝑀𝑔V𝑠2𝜃𝑅𝑚𝑚V||2𝑚2FS4𝜋𝑚V=||𝑐𝑀||2𝛼9V𝜇.(4.13) This is a factor of 𝜇/𝑚FS faster than the corresponding spin-flip transition in regular hydrogen, which scales as 𝛼𝑚3FS. This is because the decays are dominated by ΛV emission rather than V𝜇 emission, for which the amplitude carries an additional factor factor of 𝐸1/2, where 𝐸 is the energy of the emitted gauge particle.

4.1.2. Interactions within the Hypermultiplet

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 𝜕2, is 𝑑4𝜃Φ1Φ2𝒟𝒲V+h.c.(4.14) In the case of interactions, the only operators of this form allowed by the 𝑂(2)𝑅 and 𝑈(1)𝑒+𝑝 symmetries of the theory are 𝑑4𝜃1122𝒟𝒲V,𝑑4𝜃𝑐11𝑐𝑐22𝑐𝒟𝒲V.(4.15) These operators contain terms like 𝜛+𝜕𝜇𝜛+𝜕𝜈𝐹𝜇𝜈V and 𝜔+𝜕𝜇𝜔+𝜕𝜈𝐹𝜇𝜈V, respectively, and thus correspond to charge radius interactions. Matching to (4.7) then gives the supersymmetric completion of the charge radius interactions V=𝑐𝐸𝑔V𝑐2𝜃𝑅2𝑒𝑑4𝜃1122𝑐11𝑐+2𝑐𝑐2𝒟𝒲V.(4.16) Replacing 𝒟𝒲𝑉 with the current 𝒥V gives atom-ion scattering. Similarly matching onto the Rayleigh scattering operator in (4.9) yields VV=𝑐𝐸𝑔V𝑅3𝑒𝑑4𝜃1𝑐1+2𝑐2𝒲V𝒲V+h.c.(4.17) Just like the operator in (4.16), this operator will mediate decays within the hypermultiplet once supersymmetry is broken.

The restriction to operators of the form in (4.11) is a supersymmetric analog of the statement that any interaction involving two scalars and a field strength can be written as “(derivatives)×𝜙𝜕𝜇𝜙𝜕𝜈𝐹𝜇𝜈,” which implies that transitions between scalar states cannot proceed via single-photon emission. Thus, for example, direct single-photon/-photino decays from the 2𝑠 hypermultiplet to the ground state hypermultiplet are forbidden. The decay will instead proceed through either two-photon/-photino transitions or a cascade decay via magnetic operators of the form in (4.10).

4.1.3. Hypermultiplet Decays

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, (4.10), induces the largest decay rates.

In the presence of soft masses, the supersymmetric operators in (4.16) and (4.17) can mediate decays within the hypermultiplet. In the case of the three-body decays mediated by the Rayleigh scattering operator in (4.17), these soft masses appear in the eight powers of phase space Γ{𝜔1+,𝜔2+,𝜛}𝜉±ΛVVΓ𝜉±𝜔ΛVV||𝑐𝐸𝑔V𝑅3𝑒||2𝑚𝜉±𝑚𝜔864𝜋3𝑚=||𝑐𝐸||2𝛼27V16𝜋2𝑠2𝜃218𝑚soft𝑚FS8𝜇.(4.18) In the case of the two-body decays mediated by the charge radius operator in (4.16), these soft masses appear in phase space as well as in an overall factor of 𝑚2V. This latter factor arises from the modified equations of motion for ΛV, which imply that 𝒟𝒲𝑉𝜃/𝜕ΛV𝑚V𝜃ΛV. The resulting decay rate is Γ{𝜔1+,𝜔2+,𝜛}𝜉±ΛVΓ𝜉±𝜔ΛV||𝑐𝐸𝑔V𝑐2𝜃𝑅2𝑒||2|||𝑚V𝑚𝑚𝜉±|||2𝑚𝜉±𝑚𝜔24𝜋𝑚||𝑐𝐸||2𝑐22𝜃𝛼21V𝑠162𝜃26𝑚soft𝑚FS4𝑚V𝑚FS2𝜇.(4.19) Here two powers of 𝑚soft arise from cancellations between the terms involving and 𝑐 in (4.16). Higher order operators may not have this cancellation.

Next consider how the magnetic spin-flip operator in (4.10) induces decays in the presence of supersymmetry breaking. Mixing between, for example, the fermionic states 𝜒± and 𝜉± allows all the states in to decay down to 𝜔 through (4.10), which contains interactions of the form 𝑐𝑀𝑔V𝑅𝑚𝑠2𝜃𝑚VΛV1𝛾5𝜒𝜛𝜔.(4.20) Comparison with (3.24) shows that, in (4.20), this fermionic mixing is accounted for by making a replacement of the form 𝑠𝜒𝜒+4𝜃4𝑚soft𝑚FS𝜉(4.21) which leads to the decay rate Γ{𝜔1+,𝜔2+,𝜛}𝜉±𝜆VΓ𝜉±𝜔𝜆V||𝑐𝑀𝑔V𝑠2𝜃𝑅𝑚𝑚V||2𝑠4𝜃4𝑚soft𝑚FS2(1/2)𝑠22𝜃𝑚soft24𝜋𝑚=||𝑐𝑀||2𝛼9V𝑠2𝜃𝑠4𝜃82𝑚soft𝑚FS4𝜇.(4.22) This decay rate also receives contributions from supersymmetry breaking in the effective Yukawa operators of the nonrelativistic theory, since the coefficients carry factors of 𝑚𝑒1/2 and 𝑚𝑝1/2 from the nonrelativistic normalization of the scalar constituents in (2.11). These contributions, however, are parametrically smaller by an amount 𝒪(𝑚2FS/𝑚2𝑒). Hence the decay rate (4.22), which is suppressed by only four powers of the largest supersymmetry breaking spurion, 𝑚soft, characterizes the relaxation timescale of the hypermultiplet.

4.2. 𝑈(1)𝐴 Interactions

This section outlines the dominant interactions between dark atoms and an axial 𝑈(1) with charges given in (4.2) and mediated by a vector superfield A. Axial gauge symmetry forbids mass terms for fermions, and, therefore, the gauge symmetry must be broken if nonrelativistic bound states exist. As in the previous section, there are several allowed supersymmetric operators, and the interactions of the vector boson are sufficient to fix the coefficients of the operators.

The leading A𝜇 interactions are determined by the axial charges of the constituents. The scalars 𝜛± in (2.15) have zero axial charge, but the combinations 12𝜔+𝑐2𝜃𝜔±𝑠2𝜃𝜍(4.23) have charges of ±2, respectively. This leads to the inelastic interactions 2𝑖𝑔A𝑐2𝜃𝜔+𝜕𝜇𝜔𝑠2𝜃𝜔+𝜕𝜇𝜍A𝜇.(4.24) Similarly, the fermion-fermion bound states are charged with interactions given by 𝑔A𝑚V𝑐2𝜃𝜍+𝑠2𝜃𝜔𝑣𝜇A𝜇.(4.25) The interactions of (4.23)–(4.25) can be embedded in the following superspace operators: A𝑔A𝑐2𝜃𝑑4𝜃1𝑐𝑐1+2𝑐𝑐2A,𝒱A𝑔A𝑠2𝜃𝑚V𝑑4𝜃𝑐12𝑐𝒱A+h.c.,𝒱𝒱A𝑔A𝑚2V𝑐2𝜃𝑑4𝜃𝒱𝒱A.(4.26) The interactions of the linear superfield eaten by A can be obtained by going out of unitary gauge 𝜋AA+A+𝜋A2𝑚A.(4.27)

In models like that of Section 5, where A𝜇 undergoes kinetic mixing with Standard Model hypercharge, A𝜇 interactions mediate the dominant coupling of dark atoms to the Standard Model. This setup can be used to realize inelastic dark matter because the elastic interaction of the ground state, 𝜔, with A𝜇 is forbidden due to parity. After supersymmetry breaking, (1/2)(𝜔+±𝜛+) become mass eigenstates. The interactions in (4.24)-(4.25) then allow 𝜔 to upscatter to 𝜔1+ and 𝜔2+, which are heavier by an amount 𝒪(𝑚soft), and to 𝑣𝜇, which is heavier by an amount 𝒪(𝑚FS).

Higher dimension operators also contribute to the interactions with the standard model. For example, the axial spin flip operator 𝐷=5𝒱A𝑔A𝑠2𝜃𝑅𝑚𝑑2𝜃𝑐1𝒲1𝑐2𝒲2𝒲A+h.c.(4.28) leads to scattering which can be important in certain regions of parameter space.

Two body decays mediated by A𝜇 are either kinematically forbidden or severely suppressed, since 𝑚A can only be made smaller than 𝑚FS by choosing 𝑔A𝒪(𝛼4V). Similarly, three body decays mediated by an off-shell A𝜇 are subdominant.

5. Kinetically Mixed Supersymmetric Hydrogen

This section constructs a minimal model for a nearly supersymmetric dark sector that supports Coulombic bound states. Section 5.1 introduces a minimal Higgs sector and discusses how kinetic mixing of the dark 𝑈(1)A with hypercharge in the supersymmetric Standard Model (SSM) drives gauge symmetry breaking in the hidden sector. Section 5.2 adds matter fields that are charged under a second Abelian gauge symmetry, 𝑈(1)V, that introduces hydrogen-like bound states into the spectrum of the theory. In the low energy limit, this theory reduces to supersymmetric QED with two massive flavors. Section 5.3 discusses how supersymmetry breaking is communicated to the dark sector from the SSM. Finally, Section 5.4 illustrates the scales of the resulting model by calculating three benchmark points.

5.1. Kinetic Mixing

Abelian field strengths are gauge invariant, and, therefore, no symmetry principle forbids mixed field strength terms [59, 60]. Kinetic mixing occurs in extensions of the Standard Model with additional 𝑈(1) gauge factors if there are fields that are charged under both the new 𝑈(1) and hypercharge. In supersymmetric theories, the entire gauge supermultiplet undergoes gauge kinetic mixing, leading to both gaugino kinetic mixing and 𝐷-term mixing [6168]. If there are light fields charged under the new 𝑈(1), then kinetic mixing drives gauge symmetry breaking.

Consider a minimal example where a dark 𝑈(1)A couples to a pair of chiral superfields Φ and Φ𝑐 with charges ±2 (chosen for later convenience). The Lagrangian is given by Hidden=𝑑4𝜃Φ𝑒4𝑔AAΦ+Φ𝑐𝑒4𝑔AAΦ𝑐+𝑑2𝜃14𝒲2A𝜖2𝒲A𝒲𝑌+𝑊0+h.c.,(5.1) where A is the supersymmetric gauge potential of the hidden 𝑈(1)A, 𝒲A is the supersymmetric gauge field strength of A, and 𝒲𝑌 is the supersymmetric gauge field strength of SSM hypercharge. For 𝜖1, the hidden sector is only a small perturbation to the SSM so that all SSM fields will have their normal vacuum expectation values; in particular the SSM Higgs fields will acquire vevs along a non-𝐷-flat direction 𝐷𝑌=𝑔𝑌𝑣24cos2𝛽.(5.2) This SSM vev now acts as a source term for 𝒲A in (5.1) and forces 𝜙, the lowest component of Φ, to acquire a vev, since 𝐷𝑌 acts an effective Fayet-Illiopoulos term for 𝑈(1)A. The resulting effective Lagrangian is 𝐷1=2𝐷2A+𝐷A𝜖𝐷𝑌2𝑔A||𝜙||2||𝜙𝑐||2||𝜙||2=||𝜙𝑐||2+𝜖𝐷𝑌2𝑔A0.(5.3) This 𝐷-term potential has a residual flat direction, which can be lifted by 𝑊0. This section uses a superpotential 𝑊0=𝜆𝑆ΦΦ𝑐𝜇20,(5.4) where 𝑆 is a new singlet chiral superfield. With the addition of 𝑊0, the vevs of all fields are fixed, and there are no massless fermions. It is convenient to let the superfields acquire vevs and to expand around the new field origin Φ=𝑣Acos𝛽A,Φ𝑐=𝑣Asin𝛽A,ΦΦ𝑐=𝜇20,(5.5) where the last expression is enforced by the 𝐹-term for 𝑆. Solving for 𝑣A and tan𝛽A gives 𝑣4A=𝜖𝐷𝑌2𝑔A2+4𝜇40,tan2𝛽A=4𝑔A𝜇20𝜖𝐷𝑌.(5.6) In the limit 𝜇20𝜖𝐷𝑌/𝑔A, tan𝛽A0 and in the opposite limit, tan𝛽A1. Fluctuations around the vacuum in (5.5) can be diagonalized using the field definitions 𝑣Φ=A+𝜋cos𝛽A+sin𝛽A𝑆𝑐,Φ𝑐=𝑣A𝜋sin𝛽A+cos𝛽A𝑆𝑐(5.7) so that the superpotential becomes 𝑊0=𝜆𝑆ΦΦ𝑐𝜇20=𝜆𝑣A𝑆𝑆𝑐+(5.8) clearly showing that 𝑆 picks up a Dirac mass 𝑚𝑆=𝜆𝑣A. In the Kähler term, the super-Higgs mechanism takes place 𝐾=Φ𝑒4𝑔AAΦ+Φ𝑐𝑒4𝑔AAΦ𝑐+𝑆𝑆=𝑆𝑐𝑆𝑐+𝑆𝑆+𝑚2AA+𝜋+𝜋2𝑚A2+(5.9) The superfield 𝜋 is clearly identified as the eaten linear superfield, and the vector field has picked up a mass 𝑚A=22𝑔A𝑣A.(5.10)

In addition to driving 𝑈(1)A gauge symmetry breaking, kinetic mixing also leads to 𝑈(1)𝑅-breaking mass effects in the gaugino sector of the theory. This is because for 𝜖0 gaugino kinetic mixing between 𝑈(1)𝑌 and 𝑈(1)A entangles the 𝑈(1)𝑅-preserving Dirac mass 𝑚A with the 𝑈(1)𝑅-breaking bino mass 𝑀1. The effective Lagrangian for the gauginos is 𝜆=𝜆𝑌𝑖/𝜕𝜆𝑌+𝜆A𝑖/𝜕𝜆A+𝜒𝜋𝑖/𝜕𝜒𝜋𝑀1𝜆𝑌𝜆𝑌+𝜖𝜆A/𝜕𝜆𝑌+𝑚A𝜒𝜋𝜆A+h.c.,(5.11) where 𝜒𝜋 is the fermion component of the linear superfield 𝜋. The three eigenvalues to 𝒪(𝜖2) are 𝑚=𝑚A𝜖12𝑚A𝑀1𝑚A,𝑚A𝜖12𝑚A𝑀1+𝑚A,𝑀1𝜖12𝑀21𝑚21𝑚2A.(5.12) Notice that the two mass eigenvalues at |𝑚|𝑚A are no longer identically the same due to kinetic mixing with 𝜆𝑌, and this introduces 𝑈(1)𝑅 breaking into the hidden sector.

5.2. Charged Matter

This section adds light, charged matter to the dark sector. The charged matter consists of four chiral superfields 𝐸, 𝐸𝑐, 𝑃, and 𝑃𝑐 that have axial charges under 𝑈(1)A. The charge assignments of the dark electron and proton are chosen to be chiral to prevent them from acquiring supersymmetric masses in the absence of gauge symmetry breaking. Once the 𝑈(1)A gauge symmetry is broken, these states acquire masses at a scale set by 𝑚A and tan𝛽A. In addition to new matter fields, the gauge sector is extended by a second gauge group 𝑈(1)V under which 𝐸, 𝐸𝑐, 𝑃, and 𝑃𝑐 have vector-like charges and which will lead to the formation of hydrogen-like bound states in the hidden sector. In summary, the additional matter content has the following charge assignments: 𝐸𝐸𝑐𝑃𝑃𝑐𝑈(1)V1+1+11𝑈(1)A+1+111.(5.13) The superpotential in (5.4) is augmented by Yukawa terms 𝑊=𝑊0+𝑊Yukawa,𝑊Yukawa=𝑦𝑒Φ𝑐𝐸𝐸𝑐+𝑦𝑝Φ𝑃𝑃𝑐+h.c.,(5.14) so that after 𝑈(1)A breaking both 𝐸 and 𝑃 acquire Dirac masses 𝑊Yukawa=𝑚𝑒1+𝜋+cot𝛽A𝑆𝑐𝑣A𝐸𝐸𝑐+𝑚𝑝1+𝜋+tan𝛽A𝑆𝑐𝑣A𝑃𝑃𝑐+h.c.,(5.15) where 𝑚𝑒=𝑦𝑒sin𝛽A𝑣A,𝑚𝑝=𝑦𝑝cos𝛽A𝑣A.(5.16) None of the fields charged under 𝑈(1)V acquires a vev, and, therefore, 𝑈(1)V is a massless gauge multiplet.

The interactions of the 𝑈(1)A vector superfield A with the matter superfields are given by 𝐾=2𝑔AA+𝜋+𝜋2𝑚A𝐸𝐸+𝐸𝑐𝐸𝑐𝑃𝑃𝑃𝑐𝑃𝑐.(5.17) Here the interactions of the 𝜋 have been moved from the superpotential to the Kähler potential with the equations of motion. The 𝜋 fields can have subdominant mixing with the Higgs fields of the SSM and mediate subdominant interactions.

The Higgs trilinear coupling, 𝜆, in (5.4) and the axial gauge coupling, 𝑔A, are taken to be large enough that the masses in the axial/Higgs sector are of order 𝛼V𝑚𝑒 or larger. With this choice of parameters, the axial and Higgs sectors decouple, and the low energy limit of the theory is supersymmetric QED with two massive flavors and weakly broken supersymmetry. Because the axial/Higgs sectors respect the 𝑂(2)𝑅 symmetry, the arguments of Section 2.1 go through, and, in particular, the leading order superspin wavefunctions are as given in (2.15). The dominant residual effect of the axial/Higgs sector is to perturb the mass splitting between the hypermultiplet and vector multiplet. These contributions are suppressed through a combination of coupling constants and/or Yukawa suppression. Although the axial 𝑈(1)A gauge sector plays a subdominant role in the internal dynamics of the hidden sector, it mediates the dominant coupling to the Standard Model. In particular, it mediates supersymmetry breaking, which is the subject of the next section.

5.3. Supersymmetry Breaking

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 3.

The soft parameters to be calculated (see (3.3) and (3.4)) are the 𝑈(1)𝑅-preserving Δ2̃𝑒 and Δ2̃𝑝 and the 𝑈(1)𝑅-breaking 𝐵𝑒 and 𝐵𝑝. The largest soft parameters are the 𝑈(1)𝑅-preserving ones. If supersymmetry breaking is mediated to the SSM through gauge mediation, then these are given by Δ2̃𝑒Δ2̃𝑝𝛼A𝜖2𝑀𝛼2𝐸𝑐,(5.18) where 𝑀𝐸𝑐 is the SSM right handed selectron mass. The next largest soft parameters are the 𝑈(1)𝑅-breaking 𝐵𝜇-type terms, 𝐵𝑒 and 𝐵𝑝. To isolate how 𝑈(1)𝑅-breaking effects are mediated from the SSM, it is useful to integrate out the bino, which generates the operator 𝑂/𝑅=𝜆A𝜖2𝑀1+𝑀21𝜆A+h.c.(5.19)𝐵𝑒 and 𝐵𝑝 are then generated upon the insertion of 𝑂/𝑅 in a loop, with a logarithmically enhanced contribution that is the same for both, 𝐵𝑒𝐵𝑝𝛼𝐵A𝜖2𝑀1𝜋Λlog2UV𝑀21,(5.20) where ΛUV is the messenger scale. The 𝐵-terms feed into the 𝑈(1)V gaugino mass, which is highly suppressed due to the indirect communication of 𝑈(1)𝑅-breaking 𝑚V𝛼V𝐵𝛼4𝜋A𝛼V𝜖2(4𝜋)2𝑀1.(5.21) In this model, 𝜆V is always light and its mass is smaller than the level splittings induced by supersymmetry breaking, which are of order 𝑚soft𝑚V𝑚soft𝛼V𝛼(4𝜋)2𝑀1𝑚𝑒𝑀2𝐸𝑐1.(5.22) This justifies ignoring the contributions from 𝑚V to the ground state energy levels.

Section 3 described how the dominant communication of supersymmetry breaking to the spectrum is through supersymmetry violating perturbations to the rest energies of the constituents. Supersymmetry breaking also introduces several dynamical contributions to bound-state spectroscopy from the exchange of particles from the axial 𝑈(1)A and Higgs sectors. However, these contributions are suppressed for the same reason that the supersymmetric contributions from the axial and Higgs sectors are suppressed.

5.4. Benchmark Models

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 𝑚DM100 GeV and a splitting 𝛿100 keV between the ground state and the next highest state accessible through axial scattering. Furthermore, the (predominantly inelastic) scattering cross section between the ground state and standard model nucleons should be 𝒪(1040) cm2.

It is possible to meet these criteria; however, some tension exists between meeting all three criteria simultaneously. In models consistent with these requirements, 𝜔 or 𝜍 is the lightest state. The three states to which 𝜔/𝜍 can upscatter by exchanging an axial photon with a nucleon are 𝜔1+, 𝜔2+, and 𝑣𝜇. The cross section for 𝜔/𝜎 to upscatter to 𝑣𝜇 is velocity suppressed and can be ignored in the following. The cross section for 𝜔 to upscatter to 𝜔1+ or 𝜔2+ via the operator in (4.24) is given [33, 68] by 𝛼𝜎64𝜋EM𝛼A𝜖2𝑚2N𝑚4A=𝛼EM𝑚2N𝐷2𝑌11+16𝑔2A𝜇40/𝜖2𝐷2𝑌4×1041cm21+16𝑔2A𝜇40/𝜖2𝐷2𝑌(50GeV)4𝐷2𝑌,(5.23) where 𝑚N is the mass of the nucleon. In order to fix 𝜎1040 cm2, one must, therefore, choose 𝜇20𝜖𝐷𝑌/𝑔A. For natural values of the Yukawa couplings, 𝑚DM𝑣A, and so from (5.6) and the requirement that 𝑚DM100 GeV follows the constraint that 𝜖𝑔A(100GeV)2𝐷𝑌.(5.24) Bounds on kinetic mixing [6971] impose further constraints, requiring 𝜖0.005 for 𝑚A1GeV. Finally, the scale of supersymmetry breaking is proportional to 𝜖2𝑔2A, and to get splittings of order 100keV requires that 𝜖𝑔A0.005.

Trying to match the CoGeNT/DAMA anomaly [7274] with light inelastic dark matter is challenging in this specific model because it is difficult to generate an 𝒪(1038cm2) cross section. The primary tension arises from the mediation through the massive axial current. One approach could be to kinetically mix the vector current rather than the axial current. This does not suppress elastic scattering; however, in models of inelastic dark matter where the signal arises from downscattering, rather than upscattering, the elastic rate does not need to be suppressed. This possibility is not pursued further here but illustrates the rich phenomenology possible in composite dark matter models [75, 76].

Following the above logic, we choose the MSSM parameters shown in Table 1 for all three benchmark models.

𝐷 𝑌 𝑀 1 𝑀 𝐸 𝑐 Λ U V

5 0 G e V 1 0 0 G e V 9 0 0 G e V 1 0 0 T e V

The parameters of the dark sector for the three benchmark points are chosen to be as shown in Table 2.

M o d e l 𝜖 𝑔 V 𝑔 A 𝜇 0 𝑦 𝑒 𝑦 𝑝

U n m i x e d 0.005 1.5 0.004 3 0 G e V 0.15 1.2
M i x e d 0.005 1.1 0.004 3 0 G e V 0.25 1.4
H e a v y s c a l a r s 0.005 0.5 0.004 3 0 G e V 0.35 1.0

The first two choices for 𝑔V cause 𝑈(1)V to hit a Landau pole before the GUT scale. The Landau pole can be avoided by embedding 𝑈(1)V into a non-Abelian group, for example, 𝑈(1)V𝑆𝑈(2)V or 𝑈(1)V×𝑈(1)A𝑆𝑂(4). The rather large values of 𝜖 chosen here make for some tension with constraints from BaBar. These constraints may not apply to this model because A𝜇 may cascade decay through the Higgs sector before decaying into Standard Model particles [4351]. These parameters lead to supersymmetric bound state mass scales as shown in Table 3.

M o d e l 𝑣 A 𝑚 A 𝑚 𝑒 𝑚 𝑝 t a n 𝜃 𝑚 P r i n 𝑚 F S

U n m i x e d 4 9 G e V 5 5 0 M e V 3 . 0 G e V 5 3 G e V 0.24 4 6 M e V 1 5 0 0 k e V
M i x e d 4 9 G e V 5 5 0 M e V 5 . 1 G e V 6 2 G e V 0.29 2 2 M e V 2 0 0 k e V
H e a v y s c a l a r s 4 9 G e V 5 5 0 M e V 7 . 1 G e V 4 4 G e V 0.40 1 . 2 M e V 0 . 4 8 k e V

Supersymmetry breaking effects are encapsulated in the soft parameters as shown in Table 4.

M o d e l 𝑚 s o f t 𝐵 𝑚 V

U n m i x e d 2 7 0 k e V 6 . 1 e V 0 . 0 9 e V
M i x e d 1 7 0 k e V 6 . 1 e V 0 . 0 5 e V
H e a v y s c a l a r s