Sommerfeld enhancement from unstable final-state particles in dark matter annihilation

Abstract

We study the Sommerfeld enhancement of the annihilation cross section of dark matter into heavier unstable particles. In this process, the annihilation products become non-relativistic near the kinematical threshold. If they experience long-range interactions with each other, their wave function is distorted from a plane wave, and the annihilation cross section can be significantly enhanced. When evaluating the Sommerfeld enhancement from the long-range interactions between the annihilation products, the decay of the products needs to be taken into account. We treat this issue by including the decay width in the Schrödinger equations of the two-body wave function of the annihilation products. We find that bound states of the annihilation products with a narrow decay width enhance the annihilation cross section through a resonant effect. At the same time, this formulation automatically includes the annihilation process with off-shell final state particles, which is relevant for a wide decay width. We show that the resonant effect significantly affects the prediction of the dark matter relic abundance.

Figures (9)

• Figure 1: Top: Annihilation cross sections of $\chi_1\bar{\chi}_1$ as a function of $E_2$ under the attractive Coulomb potential, $V(r)=-\alpha/r$. We take $m_1=1~{\rm TeV}$, $m_2/m_1=1.01$, $\alpha=0.2$, $a=10^{-7}~{\rm GeV^{-2}}$, and $\Gamma/m_2 = 10^{-4}$. The vertical line at $E_2=0$ indicates the threshold energy of $\chi_1\bar{\chi}_1\to\chi_2\bar{\chi}_2$. The black dotted line shows the cross section including the final-state SE obtained by the cutoff method (\ref{['eq:luoSE']}) with $v_{\rm cut}=10^{-2}$. The red dashed line (free) shows the cross section without SE with $\Gamma/m_2=10^{-4}$. The red solid line (full result) shows the cross section including the final-state SE defined in eq. (\ref{['eq:ourSE']}). Bottom: Annihilation cross sections of $\chi_1\bar{\chi}_1$ in the vicinity of $E_2=0$. Parameters and lines are the same as in the top panel.
• Figure 2: Thermally averaged annihilation cross section as a function of $x=m_1/T$. The parameters are the same as in figure \ref{['fig:integrand']}. The red solid line indicates the full result obtained from eq. (\ref{['eq:sigvave']}) with $\Gamma/m_2 = 10^{-4}$. The red dashed line indicates the free result obtained from eq. (\ref{['eq:sigvave']}) with $S_f(E_2,\Gamma)=1$ and $\Gamma/m_2 = 10^{-4}$. The black dotted line indicates the cutoff method result obtained from eq. (\ref{['eq:sigvavecut']}) with $v_{\rm cut}=10^{-2}$.
• Figure 3: Top: Thermally averaged annihilation cross sections as a function of $m_2/m_1$ with $x=30$. We take the same parameters as in figure \ref{['fig:sigmav_ave']}. The vertical lines indicate the mass ratios shown in the bottom panel. Bottom: $\sigma v_{\rm rel}$ as a function of $\sqrt{{\cal E}_1/E_2}$. We focus on the region where $E_2<0$ in this panel. The solid cyan, dashed magenta, and dotted black lines indicate the results with $m_2/m_1=1.01, 1.0050$, and $1.0012$, respectively. The vertical lines indicate the minimum mass ratio for which the $1s$ and $2s$ bound states contribute to the annihilation cross section.
• Figure 4: Left: $\sigma v_{\rm rel}$ as a function of $\sqrt{{\cal E}_1/E_2}$ for $\Gamma/m_2=10^{-4}$ (cyan solid line), $\Gamma/m_2=10^{-3}$ (magenta dashed line), and $\Gamma/m_2=10^{-2}$ (black dotted line). Right: $\sigma v_{\rm rel}$ as a function of $\sqrt{{\cal E}_1/E_2}$ for $\alpha=0.2$ (cyan solid line), $\alpha=0.1$ (magenta dashed line), and $\alpha=0.05$ (black dotted line).
• Figure 5: Thermally averaged annihilation cross section as a function of $\Gamma/m_2$. In both panels, we set $m_1=1~{\rm TeV}$, $m_2/m_1=1.01$, $x=30$, and $a=10^{-7}~{\rm GeV^{-2}}$. In the left panel, we set $\alpha=0.2$, while we set $\alpha=0.05$ in the right panel.