Sommerfeld Enhancements for Thermal Relic Dark Matter

TL;DR

This paper investigates whether Sommerfeld-enhanced annihilation of thermal dark matter can yield large indirect-detection signals without spoiling the correct relic density. By incorporating resonances, the finite mediator lifetime, kinetic decoupling, and self-scattering effects into the freeze-out calculation, it derives maximal present-day boost factors $S_{\rm eff}$ and reveals phenomena like chemical recoupling near resonances. The main finding is that, even under optimistic assumptions, the maximal $S_{\rm eff}$ values (≈7–90 for typical parameters) fall short of explaining PAMELA/Fermi data, and CMB and astrophysical constraints further limit the parameter space. The work also discusses non-minimal models and astrophysical uncertainties, concluding that a fully consistent explanation for the cosmic-ray excesses within this framework is unlikely, though future observations will probe these scenarios more deeply.

Abstract

The annihilation cross section of thermal relic dark matter determines both its relic density and indirect detection signals. We determine how large indirect signals may be in scenarios with Sommerfeld-enhanced annihilation, subject to the constraint that the dark matter has the correct relic density. This work refines our previous analysis through detailed treatments of resonant Sommerfeld enhancement and the effect of Sommerfeld enhancement on freeze out. Sommerfeld enhancements raise many interesting issues in the freeze out calculation, and we find that the cutoff of resonant enhancement, the equilibration of force carriers, the temperature of kinetic decoupling, and the efficiency of self-interactions for preserving thermal velocity distributions all play a role. These effects may have striking consequences; for example, for resonantly-enhanced Sommerfeld annihilation, dark matter freezes out but may then chemically recouple, implying highly suppressed indirect signals, in contrast to naive expectations. In the minimal scenario with standard astrophysical assumptions, and tuning all parameters to maximize the signal, we find that, for force-carrier mass m_phi = 250 MeV and dark matter masses m_X = 0.1, 0.3, and 1 TeV, the maximal Sommerfeld enhancement factors are S_eff = 7, 30, and 90, respectively. Such boosts are too small to explain both the PAMELA and Fermi excesses. Non-minimal models may require smaller boosts, but the bounds on S_eff could also be more stringent, and dedicated freeze out analyses are required. For concreteness, we focus on 4 mu final states, but we also discuss 4 e and other modes, deviations from standard astrophysical assumptions and non-minimal particle physics models, and we outline the steps required to determine if such considerations may lead to a self-consistent explanation of the PAMELA or Fermi excesses.

Figures (12)

• Figure 1: The Sommerfeld enhancement factor $S$ as a function of $\epsilon_{\phi} \equiv m_{\phi}/ (\alpha_X m_X)$ for the constant values of $\epsilon_v \equiv v/\alpha_X$ indicated. The solid red curves are the analytic approximation of Eq. (\ref{['Sapprox']}), and the dashed blue curves are numerical results.
• Figure 2: The self-scattering rate $\Gamma_s$ (solid red) and the Hubble rate $H$ (dashed blue) as functions of $x = m_X/T$ for the values of $m_X$, $m_{\phi}$, $\alpha_X$, and $T_{\text{kd}}$ indicated.
• Figure 3: The relic density $\Omega_X h^2$ (solid red) and $S_{\text{eff}}$ (dotted blue) as a function of the fine-structure constant $\alpha_X$ for the fixed values of $m_X$, $m_{\phi}$, and $T_{\text{kd}}$ indicated and the observed value of $\Omega_X h^2$ (dashed black). In most cases, there is a unique choice of $\alpha_X$ that yields the correct $\Omega_X h^2 = 0.114$ (left). In the presence of strong resonances, however, there are cases where three different choices of $\alpha_X$ all yield the correct $\Omega_X$ (right). In these cases, $S_{\text{eff}}$ varies by about 20--30% between the different solutions.
• Figure 4: The value of $\alpha_X$ required to achieve a relic density of $\Omega_X h^2=0.114$ as a function of the dark matter particle mass $m_X$ (solid red) for $m_{\phi}=250~\text{MeV}$ (left) and $m_{\phi}=1~\text{GeV}$ (right). We also plot the required $\alpha_X$ (dotted blue) if Sommerfeld effects are neglected in the early Universe. The tree level cross section (without Sommerfeld enhancement) is $(\sigma_{\text{an}} v_{\text{rel}})_0=\pi \alpha_X^2/m_X^2$. Because $\alpha_X$ varies over almost two orders of magnitude in this plot, the dips near resonance are not immediately apparent.
• Figure 5: The effective Sommerfeld enhancement factor $S_{\text{eff}}$ (solid red) as a function of $m_X$ for $m_{\phi} = 250~\text{MeV}$ (left) and 1 GeV (right) and the set of $S_{\text{eff}}$-maximizing assumptions listed in Sec. \ref{['sec:maximal']}. Also shown for comparison are $S^0$ (dotted blue), the Sommerfeld factor without resonances and neglecting the Sommerfeld effect on freeze out; $\bar{S}$ (dot-dashed blue), the Sommerfeld factor with resonances but neglecting the Sommerfeld effect on freeze out; and $S_{\text{eff}}^0$ (dashed red), the Sommerfeld factor without resonances, but including the Sommerfeld effect on freeze out.