Dark matter and dark radiation

We explore the feasibility and astrophysical consequences of a new long-range $U(1)$ gauge field (“dark electromagnetism”) that couples only to dark matter, not to the standard model. The dark matter consists of an equal number of positive and negative charges under the new force, but annihilations are suppressed if the dark-matter mass is sufficiently high and the dark fine-structure constant $\widehat{\alpha }$ is sufficiently small. The correct relic abundance can be obtained if the dark matter also couples to the conventional weak interactions, and we verify that this is consistent with particle-physics constraints. The primary limit on $\widehat{\alpha }$ comes from the demand that the dark matter be effectively collisionless in galactic dynamics, which implies $\widehat{\alpha }\lesssim {10}^{-3}$ for TeV-scale dark matter. These values are easily compatible with constraints from structure formation and primordial nucleosynthesis. We raise the prospect of interesting new plasma effects in dark-matter dynamics, which remain to be explored.

Figure 1

The allowed values of dark $g_{\mathrm{light}}$ (those degrees of freedom relativistic at $T_{\mathrm{BBN}}$) and $g_{\mathrm{heavy}}$ (the remaining dark degrees of freedom) arising from BBN constraints Eqs. (8, 9). The allowed regions correspond to 95% confidence levels for $\xi (T_{\mathrm{RH}})=1$ and a visible sector $g_{*\mathrm{vis}}=106.75$ (middle in red), $\xi (T_{\mathrm{RH}})=1$ and $g_{*\mathrm{vis}}=228.75$ (corresponding to MSSM particle content, upper in blue), and $\xi (T_{\mathrm{RH}})=1.4(1.7)$ and $g_{*\mathrm{vis}}=106.75(228.75)$ (lowest in yellow). The minimal dark sector model of this paper is noted by a black star at $g_{\mathrm{light}}=2$ and $g_{\mathrm{heavy}}=3.5$.

Figure 2

Pair annihilation/creation of dark matter $\chi$ into dark photons $\widehat{\gamma }$ via $t$ and $u$-channel exchange diagrams. These processes keep the dark sector in thermal equilibrium until the $\chi$ particles become nonrelativistic.

Figure 3

The allowed regions of $\widehat{\alpha }$ vs $m_{\chi }$ parameter space. The relic abundance allowed region applies to models in which $U(1{)}_D$ is the only force coupled to the dark matter; in models where the DM is also weakly interacting, this provides only an upper limit on $\widehat{\alpha }$. The thin yellow line within the relic abundance allowed region represents the allowed region from correct relic abundance assuming ${\Omega }_{\mathrm{DM}}{h}^2=0.106\pm 0.08$, $\xi (T_{\mathrm{RH}})=1$, $g_{*\mathrm{vis}}\approx 100$, and $g_{\mathrm{heavy}}+g_{\mathrm{light}}=5.5$ while the surrounding blue region is $g_{*\mathrm{vis}}=228.75(60)$, $\xi (T_{\mathrm{RH}})=1(0.1)$, and $g_{\mathrm{heavy}}+g_{\mathrm{light}}=100(5.5)$ at the lower (upper) edge. The diagonal green line is the upper limit on $\widehat{\alpha }$ from effects of hard scattering on galactic dynamics; in the lower red region, even soft scatterings do not appreciably affect the DM dynamics. We consider this to be the allowed region of parameter space.

Figure 4

Feynman diagrams leading to $\gamma /\widehat{\gamma }$ mixing. The vertex in (a) can be expanded into that shown in (b), because the only particle to which the $\widehat{\gamma }$ couples is $\chi /\overline{\chi }$. Since the mass and $SU(2{)}_L$ charge of these two particles are the same, yet they possess opposite $U(1{)}_D$ charge, the sum of the $\chi$ and $\overline{\chi }$ diagrams in (b) is zero, and the overall mixing vanishes.

Figure 5

Feynman diagram leading to $\widehat{\gamma }$ interactions with SM fermions $f$. The vertex in (a) can be expanded into that shown in (b), because the only particle with an interaction with $\widehat{\gamma }$ is the $\chi /\overline{\chi }$. Since the mass and $SU(2{)}_L$ charge of these two particles are the same, yet the $U(1{)}_D$ charges are opposite, the sum of the $\chi$ and $\overline{\chi }$ diagrams in (b) is zero, and the overall coupling of $f$ to $\widehat{\gamma }$ is therefore zero as well.

Figure 6

The leading order interaction of the dark sector with SM fermions. The dark photons $\widehat{\gamma }$ couple to a loop of $\chi$ particles, which couple through two $SU(2{)}_L$ gauge bosons to SM fermions. Coupling through a single $SU(2{)}_L$ boson is zero due to the tracelessness of ${\tau }^a$.