Dark matter may be coupled to dark radiation: light degrees of freedom that mediate forces between dark sector particles. Cosmological constraints favor dark radiation that is colder than Standard Model radiation. In models with fixed couplings between dark matter and the Standard Model, these constraints can be difficult to satisfy if thermal equilibrium is assumed in the early universe. We construct a model of asymmetric reheating of the visible and dark sectors from late decays of a long-lived particle (for instance, a modulus). We show, as a proof of principle, that such a model can populate a sufficiently cold dark sector while also generating baryon and dark matter asymmetries through the out-of-equilibrium decay. We frame much of our discussion in terms of the scenario of dissipative dark matter, as in the Double-Disk Dark Matter scenario. However, our results may also be of interest for other scenarios like the Twin Higgs model that are in danger of overproducing dark radiation due to nonnegligible dark-visible couplings.
Article funded by SCOAP3
g_{*s}(t)$, the only way to achieve such a low $\xi$ is to increase $g_{*s,\text{vis}}^{\text{dec}}$. We assume that the dark sector at low temperatures has only the fermions $\chi$ and ${\overline \chi}$ (which may be thought of as a ``dark electron'' and ``dark positron'') and the U(1) dark photon $\phi$, and that the dark fermions froze out from the dark radiation some time since the two sectors decoupled, ie. $g_{*s,D}^{\text{dec}} = 2 + \frac{7}{8} \times 4 = 5.5$, $g_{*s,D}(t) = 2$. Then we must have \begin{equation} \frac{5.5}{2} \frac{g_{*s,\text{vis}}(t)}{g_{*s,\text{vis}}^{\text{dec}}} \lesssim \frac{1}{1000} \end{equation} or, given 2 present day visible degrees of freedom from the photon, $g_{*s,\text{vis}}^{\text{dec}} \gtrsim 5500$. This is not very likely. However, if we allow $\xi \lesssim 0.25$ instead, which may just barely evade the bounds on dark acoustic oscillations~\cite{Cyr-Racine-ml-2013fsa}, we ``only'' need roughly 352 degrees of freedom in the visible sector at decoupling. This may, very optimistically, be possible due to beyond the Standard Model physics at temperatures slightly above the current LHC bounds, say $T_{\text{dec}} = 10 \text{ TeV}$. Now we come to the basic origin of the tension: decreasing $\xi$ requires that many Standard Model degrees of freedom fall out of thermal equilibrium with the Standard Model \emph{after} the dark radiation does. However, the tighter the coupling between dark sector particles and the Standard Model, the longer they will stay in equilibrium with one another. Thus the CMB constraints prefer small couplings and an early decoupling, whereas a detectable signal in direct detection experiments prefers late decoupling. ]]>
= \frac{\zeta(3)}{\pi^3} \frac{T^5}{\Lambda^4} \approx 0.039 \frac{T^5}{\Lambda^4},
\eeq
where $\zeta(3) \approx 1.2$ is the Riemann zeta function of 3. We compare this to the Hubble expansion rate
\beq
H = \sqrt{\frac{\pi^2 g_*}{90}} \frac{T^2}{M_p}.
\eeq
Decoupling happened when $H \approx \Gamma$, i.e.~at the temperature
\beq
T_{\rm dec} \approx \left(\sqrt{\frac{\pi^2 g_*}{90}} \frac{\Lambda^4}{M_p} \frac{\pi^3}{\zeta(3)}\right)^{1/3} \approx 19~{\rm GeV} \left(\frac{\Lambda}{100~{\rm TeV}}\right)^{4/3} \left(\frac{g_*}{352}\right)^{1/6}.
\label{eq:decouplinglight}
\eeq
or, conversely, that to obtain a desired decoupling temperature we require the coupling to be suppressed by
\begin{equation}
\label{2:lambdadef}
\Lambda \approx 1.1 \times 10^7 \text{ GeV} \left(\frac{T_{\text{dec}}}{10 \text{ TeV}} \right)^{3/4} \left(\frac{352}{g_*}\right)^{1/8}.
\end{equation}
Again, this calculation is approximately correct if the abundance of the $\chi$ particles is not Boltzmann-suppressed at the time of decoupling: $m_\chi \lesssim T_{\rm dec}$. For the dark electron masses considered in refs.~\cite{Fan-ml-2013yva,Fan-ml-2013tia}, this is a reasonable assumption. For the heavier dark proton, it may not be. If we consider a model in which the dark electron has negligible scattering cross section but the dark proton scatters, we find that the dark sector decouples at a temperature of
\begin{equation}
T_{\rm dec} \approx \frac{m_\chi}{29 - 4 \log(\Lambda/{\rm TeV})}.
\label{eq:decouplingheavy}
\end{equation}
Thus, for $\Lambda \sim {\rm TeV}$, if we would like the dark proton to decouple at temperatures above 1\,TeV we would need its mass to be at least about 30\,TeV. Already when the dark proton mass is 100\,GeV there is limited parameter space for cooling, and decoupling happens near the GeV scale where $g_{*s}$ is known and too small to lead to small $\xi$. Furthermore, for a fixed fraction of the total dark matter abundance, larger dark matter masses imply smaller number densities and hence lower rates of scattering in direct detection experiments. For all of these reasons, the regime in which eq.~\eqref{eq:decouplingheavy} applies is of little interest for us in this~paper.
It could also be the case that dark matter scatters primarily with a heavy Standard Model particle that decouples: for instance, we could have a coupling dominantly to the top quark. In this case, a formula like eq.~\eqref{eq:decouplingheavy} might apply but with the top quark mass appearing in the numerator. However, this leads to small enough $T_{\rm dec}$ that we are stuck with large $\xi$. For the purposes of our argument, it suffices to always take the decoupling temperature to be the larger of equations~\eqref{eq:decouplinglight} and~\eqref{eq:decouplingheavy}, rather than solving the Boltzmann equation for a more precise interpolation between the two.
Now, what does the scale $\Lambda$ mean for dark matter--nucleon scattering at a direct detection experiment? Bounds are usually set on a DM-nucleon scattering cross section
\beq
\label{2:sigmadef}
\sigma = \frac{4 \mu_n^2}{\pi} \left|f_n\right|^2,
\eeq
where $\mu_n = m_\chi m_n/(m_\chi + m_n)$ is the reduced mass of the DM-nucleon system, and $f_n$ is an effective coupling of the DM to the nucleon. This coupling depends on nucleon matrix elements that are used to relate couplings to quarks to couplings to nucleons, as reviewed in~\cite{Belanger-ml-2008sj,Belanger-ml-2008gy}. For the scalar matrix elements of quarks in a nucleon (neutron or proton),
\beq
f^n_{0;q} \equiv \left
\sim 3 \times 10^{-5}~{\rm GeV}^6,
\end{equation}
although there is an order-one uncertainty in existing lattice calculations~\cite{Buchoff-ml-2012bm}. Using this we have the order-of-magnitude estimate:
\begin{equation}
t_{\rm osc} = m_{\rm osc}^{-1} \sim 2.9 \times 10^{8}\,{\rm s}~\left(\frac{10^{-1}}{\lambda}\right)^2 \left(\frac{10^{-1}}{\lambda'}\right)^2 \left(\frac{M_{\rm eff}}{25\,{\rm TeV}}\right)^5, \label{eq:neutronantineutron}
\end{equation}
with $M_{\rm eff}^5 = m_{\tilde d}^4 m_{\tilde b}^4 m_{\tilde N}^4/(\Delta_{db}^2 b_N m_{\rm scalar})$ a combination of the different scalar mass scales appearing in the diagram. This is to be compared with the experimental bound~\cite{kamiokande}
\begin{equation}
t_{\rm osc} \geq 2.7 \times 10^8~{\rm s}.
\end{equation}
From this estimate we see that we are safe from constraints provided that the average mass scale is above about 25\,TeV for couplings of around 0.1. In fact, it is reasonable to take $\Delta_{db} \ll m_{\tilde d, \tilde b}^2$ and also to assume some flavor structure in the $\lambda$ and $\lambda'$ couplings that suppresses them for the first generation quarks, so our model is very safe from nucleon oscillation bounds.
Another potentially important constraint comes from dinucleon decay, $p p \rightarrow K^+ K^+$. Unlike the case of neutron--antineutron oscillation, here the process intrinsically changes flavor. As a result, this process occurs at tree level as illustrated in figure~\ref{dinuc}. The effective Lagrangian is schematically
\begin{equation}
{\cal L}_{\Delta B} \sim \frac{\lambda^2 \lambda'^2 b_{X\overline{X}}^2}{m_N m_{\tilde X}^4 m_{\tilde {\overline X}}^4} \left(u^c d^c s^c\right)^2 + {\rm h.c.},
\end{equation}
which again must be matched to a hadronic matrix element $\left
M_2$ without loss of generality. We focus on a scenario in which the modulus $\tau$ decays to the scalar components of $\phi_1$ and $\phi_2$ which in turn have out-of-equilibrium decays to the fermions in the $\psi$ and $n_d$ multiplet. The decay of the $\phi$s exhibits CP violation from the interference of the tree and 1-loop diagrams in figure~\ref{fig:dddminterference}. We assume that the fermion masses are all relatively small compared to the supersymmetry breaking scale, whereas the scalars in each multiplet acquire soft masses of order $m_{3/2}^2$, roughly of the same order of magnitude as the $\tau$ mass. For simplicity, we can assume that the scalar partners of $\psi$ and $n_d$ are either kinematically out of reach or have suppressed branching ratios, while the modulus frequently decays to the heavy scalars $\phi$. \begin{figure} \centering \includegraphics[width=0.99\textwidth]{fig5.png} ]]>