Stable bound states of asymmetric dark matter

The simplest renormalizable effective field theories with asymmetric dark matter bound states contain two additional gauge singlet fields, one being the dark matter and the other a mediator particle that the dark matter annihilates into. We examine the physics of one such model with a Dirac fermion as the dark matter and a real scalar mediator. For a range of parameters the Yukawa coupling of the dark matter to the mediator gives rise to stable asymmetric dark matter bound states. We derive properties of the bound states including nuggets formed from $N\gg 1$ dark matter particles. We also consider the formation of bound states in the early Universe and direct detection of dark matter bound states. Many of our results also hold for symmetric dark matter.

Figure 1

Ground state binding energy (in blue curves) of the two-body DM bound state in the ${\alpha }_{\chi }-m_{\varphi }$ space, for DM mass equal to 1 TeV (left), 100 GeV (middle), and 10 GeV (right), respectively. No two-body bound state exists in the dark blue region, and only one ($1s$) bound state exists in the lighter blue region. The yellow region is not consistent with BBN and LUX constraints in the minimal model described by Eq. (1). The black dashed curves are the lower bounds on ${\alpha }_{\chi }$ from having a large enough annihilation rate for the ADM, in the cases ${\alpha }_I=0$ (upper) and ${\alpha }_I={\alpha }_{\chi }$ (lower).

Figure 2

Upper bound on the $\varphi -h$ mixing parameter ${\mu }_{\varphi h}v/m_h^2$ from dark matter direct detection (LUX), assuming all DM are free particles today. The solid and dashed curves correspond to ${\alpha }_{\chi }=0.1$ and 0.01, respectively. We fix $m_{\varphi }=500\mathrm{MeV}$.

Figure 3

Feynman diagrams for two-body ADM bound state formation (dissociation) with an on-shell $\varphi$ emission (absorption).

Figure 4

Temperature dependence of the two-body ADM bound state formation (red labelled by ${\mathrm{\Gamma }}_{\chi \chi \to B\varphi }$), dissociation (blue labelled by ${\mathrm{\Gamma }}_{\varphi B\to \chi \chi }$) rates in together with the Hubble expansion rate (black). In both cases, we set ${\mu }_{\varphi h}v/m_h^2={10}^{-7}$ such that both DM direct detection and BBN bounds can be satisfied.

Figure 5

Same as Fig. 4, but the parameters are chosen such that, at the temperature when ${\mathrm{\Gamma }}_{\text{diss}}=H$, the formation rate satisfies $P={\mathrm{\Gamma }}_{\text{form}}/H=1$ (solid) or $P={\mathrm{\Gamma }}_{\text{form}}/H=5\%$ (dashed).

Figure 6

Same parameter space as Fig. 1. The thick red curves represent where the critical condition like Fig. 5 is reached; i.e., at the temperature when ${\mathrm{\Gamma }}_{\text{diss}}=H$, the formation rate satisfies $P={\mathrm{\Gamma }}_{\text{form}}/H=1$ (solid) or $P={\mathrm{\Gamma }}_{\text{form}}/H=5\%$ (dashed). The dot-dashed lines are where $m_{\varphi }=B{E}_0$ is satisfied.

Figure 7

Feynman diagram for direct detection in the $SU(2)$ model.