Sommerfeld Enhancement from Quantum Forces for Dark Matter

Abstract

Quantum forces are long-range interactions that arise only at the loop level. In this work, we study the Sommerfeld enhancement of dark matter (DM) annihilation cross sections caused by quantum forces. One notable feature of quantum forces is that they are subject to coherent enhancement in the presence of a background of mediator particles, which occurs in many situations in cosmology. We show that this effect has important implications for the Sommerfeld enhancement and DM physics. For the first time, we calculate the Sommerfeld factor induced by quantum forces for both bosonic and fermionic mediators, including the background corrections. We observe several novel features of the Sommerfeld factor that do not exist in the case of the Yukawa potential, such as temperature-induced resonance peaks for massless mediators, and having both enhancement and suppression effects in the same model with different DM masses. As direct applications, we discuss the DM phenomenology affected by the Sommerfeld enhancement from quantum forces, including thermal freeze-out, CMB spectral distortion from DM annihilation, and DM indirect detection. We highlight one particularly interesting effect relevant to indirect detection caused by the Sommerfeld enhancement in a non-thermal background of bosonic mediators in the galaxy, in which case the DM mass is shifted due to the background correction and the effective cross section for DM annihilation can be either enhanced or suppressed. This may be important for DM searches in the Milky Way or its satellite galaxies.

Figures (30)

• Figure 1: A schematic diagram of the Sommerfeld enhancement induced by quantum forces. Left: Quantum forces in vacuum. Right: Quantum forces in the presence of a background of mediator particles. The small blue dots on the internal lines represent the background particles, which is equivalent to putting one of the propagators on-shell. When there exists a background, the total effective potential that contributes to the Sommerfeld enhancement is given by the sum of the vacuum potential $V_0$ and the background potential $V_{\rm bkg}$.
• Figure 2: The vacuum potentials from the three effective operators $\mathcal{O}_{S}, \mathcal{O}_{F}$, and $\mathcal{O}_{\widetilde{F}}$. The solid lines correspond to the full expressions in Eqs. (\ref{['eq:VS0']}), (\ref{['eq:VF0']}) and (\ref{['eq:VF0tilde']}). The dashed lines represent the regularization of the potential at distances below $r=1/\Lambda$ using Eq. (\ref{['eq:Vreg']}). For definiteness, we have fixed the mediator mass to be $10^{-3} \Lambda$. As a result, all three vacuum potentials begin to exponentially decrease at $r\gtrsim 10^3/\Lambda$.
• Figure 3: Sommerfeld enhancement from the two-scalar potential $V_0^S$ in Eq. (\ref{['eq:VS0']}), where the regularization procedure in Eq. (\ref{['eq:Vreg']}) is used to get a physical result. We plot the Sommerfeld factor $S$ as a function of the normalized reduced DM mass $\eta_\chi\equiv M/\Lambda$. Left: We fix the mediator mass to be $\eta_m\equiv m_\phi/\Lambda=10^{-3}$ and change the velocity. Right: We fix the velocity to be $v=10^{-3}$ and change the mediator mass . The dots with arrows on each curve represent the masses for which the momentum transfer exceeds the cutoff scale, i.e., $|\mathbf{q}|\sim Mv \geq \Lambda$; the exact value of $S$ to the right of these dots depends on the UV completion of the effective operator. To the left of these dots, the value of $S$ is determined by a single parameter controlling the UV physics; see the text around Eq. (\ref{['eq:Vreg']}) for details.
• Figure 4: Same conventions as Fig. \ref{['fig:scalarvacuum']}, but with the attractive two-fermion potential $V_0^F$ in Eq. (\ref{['eq:VF0']}).
• Figure 5: Sommerfeld suppression from the repulsive two-fermion potential $V_{0}^{\widetilde{F}}$ in Eq. (\ref{['eq:VF0tilde']}), where the regularization procedure in Eq. (\ref{['eq:Vreg']}) is used. Conventions are the same as Fig. \ref{['fig:scalarvacuum']}.