Particle asymmetries from quantum statistics

We consider a class of baryogenesis models where the Lagrangian in the visible sector is charge-parity ($CP$) invariant and a baryon asymmetry is produced only when quantum statistics is taken into account. The $CP$ symmetry is broken by matter effects, namely the assumption that the primordial plasma contains another asymmetric species, such as dark matter. Out-of-equilibrium baryon number violating decays can then generate an asymmetry through Bose enhancement and/or Pauli blocking of certain decay channels.

Figure 1

Decay widths as a function of $M_{\phi }/T$. The left plot compares numerical computations (solid lines) to the analytic approximations (dashed lines) of the decay asymmetry discussed in the text for a fixed value of $m_{{\varphi }_D}/T$. Contributions from Bose enhancement due to ${\varphi }_D$ and Pauli blocking due to ${\psi }_D$ are isolated by choosing ${\mu }_{{\varphi }_D}\ne 0$ and ${\mu }_{{\psi }_D}\ne 0$ one at a time. The chemical potentials are assumed to be small, such that the resulting $\mathrm{\Delta }\mathrm{\Gamma }$ is linear in ${\mu }_i/T$. The right plot compares the various physical rates to Hubble for fixed masses $m_{{\varphi }_D}=m_{{\psi }_D}=m_{{\psi }_B}=300\mathrm{GeV}$, $M_{\phi }=100\mathrm{TeV}$, and $\mathrm{\Delta }{n}_{{\varphi }_D}/s={10}^{-5}$. As before, the solid lines are numerical calculations while the dashed lines correspond to analytic approximations. The dips in $\mathrm{\Delta }\mathrm{\Gamma }$ show where these approximations break down, namely where $M_{\phi }/T\sim 1$ and $m_{{\varphi }_D}/T\sim 1$.

Figure 2

Evolution of the baryon and dark matter asymmetries during $\phi$ decay for radiation-dominated (left plot) and matter-dominated (right plot) initial conditions. We chose ${\rho }_{\phi ,i}/{\rho }_{R,i}={10}^{-3}$ in the former case, and ${\rho }_{\phi ,i}/{\rho }_{R,i}=10^3$ in the latter. The solid lines show the numerical solution of the Boltzmann equations, while the dashed lines show the analytic approximations for the final abundances discussed in the text. Note that the bulk of the visible asymmetry $Y_{{\psi }_B}$ is produced before $\phi$ decays. We used $M_{\phi }=100\mathrm{TeV}$, $T_{\mathrm{RH}}=2.5\mathrm{TeV}$, $Y_{{\varphi }_D,i}={10}^{-5}$, and $T_i=3T_{\mathrm{RH}}$ in these examples.

Figure 3

Ratio of baryon to DM asymmetry yield at late times as a function of the initial $\phi$ density. The left (right) dashed lines show the approximate analytic solutions in the radiation-dominated (matter-dominated) universe as discussed in the text. We see that these analytic solutions very quickly become good approximations in their respective regions of validity.

Figure 4

Numerical solutions to the Boltzmann equations describing asymmetry production through Pauli-blocked decays of $\phi$. The left panel shows the evolution of the various number densities as a function of the scale factor. The dotted line is the analytic approximation of Eq. (36). The dashed line shows that the net baryon number, $Y_B=Y_{{\psi }_B}-Y_{{\mathrm{\Phi }}_{DB}}$, is produced at early times, before $\phi$ decays. The right panel shows the dependence of the final baryon yield on the decay temperature of $\phi$, $T_{\mathrm{RH}}$. The red dashed line is the analytic solution for $Y_{{\psi }_B}$, while the purple dashed line is the initial DM asymmetry. This initial asymmetry is diluted by the $\phi$ entropy injection at low $T_{\mathrm{RH}}$. The initial conditions and parameter values used are described in the text.

Figure 5

Thermally averaged cross section for ${\psi }_B{\psi }_B\leftrightarrow {\overline{\psi }}_B{\overline{\psi }}_B$ as a function of $x=m_{{\psi }_B}/T$ for $a=0$, $b=1$, and $m_{{\psi }_B}/M_{\phi }=0.1$. The solid blue line is the standard thermal average computed from the numerical integral in Eq. (a10), which includes the resonant enhancement for $s\approx M_{\phi }^2$. The solid orange line shows the same cross section with the real intermediate state contribution subtracted to avoid double counting in solving the Boltzmann equations. The dashed line is the analytical large $x$ approximation from Eq. (a9).