Flavored Quantum Boltzmann Equations

TL;DR

This work derives quantum Boltzmann equations for flavored mixing in a time-dependent CP-violating background using the Closed Time Path formalism, applying a two-flavor scalar toy model coupled to a thermal bath. It demonstrates that CP asymmetries can arise from coherent flavor oscillations when the oscillation time is comparable to the wall time, and shows how collisions decohere these oscillations and drive the system toward thermal equilibrium. The results illuminate the interplay between oscillations, CP violation, and collisions, revealing a resonance-like enhancement when $\tau_{\text{osc}} \sim \tau_{\text{w}}$ and a suppression of asymmetries in fast or highly collisional regimes. The framework provides a first-principles, density-matrix treatment of flavored transport relevant to weak-scale baryogenesis and related leptogenesis contexts, and outlines clear directions for extension to spacetime-dependent masses, higher-order effects, and fermionic/MSSM realizations.

Abstract

We derive from first principles, using non-equilibrium field theory, the quantum Boltzmann equations that describe the dynamics of flavor oscillations, collisions, and a time-dependent mass matrix in the early universe. Working to leading non-trivial order in ratios of relevant time scales, we study in detail a toy model for weak scale baryogenesis: two scalar species that mix through a slowly varying time-dependent and CP-violating mass matrix, and interact with a thermal bath. This model clearly illustrates how the CP asymmetry arises through coherent flavor oscillations in a non-trivial background. We solve the Boltzmann equations numerically for the density matrices, investigating the impact of collisions in various regimes.

Figures (7)

• Figure 1: Leading-order self-energy graphs that induce the collision terms in the Boltzmann equations, corresponding to (a) coherent forward scattering, and (b) non-forward scattering ($\phi A \leftrightarrow \phi A$) and annihilation ($\phi \phi^\dagger \leftrightarrow A A$).
• Figure 2: Left panel: mass eigenvalues $m_{1,2}(t)$ as a function of time. Right panel: components of $\vec{B}_\Sigma$ as a function of time: $(\vec{B}_\Sigma)_x$ (red), $(\vec{B}_\Sigma)_y$ (green), $(\vec{B}_\Sigma)_z$ (blue). Input parameters are as in Table \ref{['tab:baseline']}.
• Figure 3: Evolution of flavor polarization and CP asymmetry in the absence of collisions, starting from a $CP$-invariant, pure $L$-handed initial state. Left column: time dependence of the particle polarization vector components $p_x$ (red), $p_y$ (green), and $p_z$ (blue), for different values of $\tau_{\text{w}}=40/T,20/T,10/T,5/T$ (from top to bottom). Middle column: time dependence of the anti-particle polarization vector components $\tilde{p}_x$ (red), $\tilde{p}_y$ (green), and $\tilde{p}_z$ (blue), for different values of $\tau_{\text{w}}=40/T,20/T,10/T,5/T$ (from top to bottom). Right column: time dependence of the flavor diagonal $CP$ asymmetry $n_L(k,t)$ for different values of $\tau_{\text{w}}=40/T,20/T,10/T,5/T$ (from top to bottom). In all cases $k=3T$ and all other input parameters (except $\tau_{\text{w}}$) are as in Table \ref{['tab:baseline']}, corresponding to $\tau_{\rm osc} \simeq 35/T$ (at $t=0$).
• Figure 4: Maximum value of $CP$ asymmetry $|n_L(k,t)|$ for $k=3 T$ as a function of $\tau_{\rm osc}(t=0)/\tau_{\text{w}}$. Except for $\tau_{\text{w}}$ that is varied, all other input parameters are as in Table \ref{['tab:baseline']}.
• Figure 5: Evolution of flavor polarizations with no collisions, flavor-blind collisions, and flavor-sensitive collisions. We plot the components $p_x(k,t)$ (red), $p_y(k,t)$ (green), and $p_ z (k,t)$ (blue) [left panels] and $n_L(k,t)$ [right panels], for $k=3 T$, as a function of time in several regimes: no collisions (1st row); flavor-blind collisions, $y_L = y_R=1$, $g_{\textrm{eff}} = 200$ (2nd row); flavor-sensitive collisions with $y_L = 1$, $y_R=0.8$, $g_{\textrm{eff}} = 200$ (3rd row); flavor-sensitive collisions with $y_L = 1$, $y_R=0.5$, $g_{\textrm{eff}} = 200$ (4th row). In all cases we use equilibrium initial conditions at $t_{\rm in}= - 15/T$. All other input parameters are as in Table \ref{['tab:baseline']}. The dotted line in the 2nd row represents the nonzero density $n_L$ surviving at late times corresponding to a non-vanishing chemical potential $\mu_L$. In the 3rd and 4th rows, the late-time density goes to zero since $y_L\not=y_R$. See text for additional details.