Baryogenesis from the Berry phase

The spontaneous baryogenesis scenario explains how a baryon asymmetry can develop while baryon violating interactions are still in thermal equilibrium. However, generation of the chemical potential from the derivative coupling is dubious since the chemical potential may not appear after the Legendre transformation. The geometric phase (Pancharatnam-Berry phase) results from the geometrical properties of the parameter space of the Hamiltonian, which is calculated from the Berry connection. In this paper, using the formalism of the Berry phase, we show that the chemical potential defined by the Berry connection is consistent with the Legendre transformation. The framework of the Berry phase is useful in explaining the mathematical background of the spontaneous baryogenesis and also in calculating the asymmetry of the nonthermal particle production in time-dependent backgrounds. Using the formalism, we show that the mechanism can be extended to more complex situations.

Figure 1

A straight motion with $m_R=\mathrm{\Lambda }+ig\phi$ is shown in the left, and the rotational oscillation with $m_R=|m_R|e^{i\theta (t)}$ is shown in the right.

Figure 2

Particle creation area (nonadiabatic area) for $m_D=i\mathrm{\Lambda }+g\phi (t)$ is shown in the middle. In both sides, matter and antimatter creation for $m_R=i\mathrm{\Lambda }+g_1\phi +i{g}_2\frac{{\phi }^2}{M_{*}}$ is shown. In the left, $s=-1$ creation starts earlier than $s=+1$, while in the right, $s=+1$ creation starts earlier than $s=-1$.

Figure 3

The top figure shows the original Landau-Zener tunneling. Equation (90) for $s=\pm$ is shown in the bottom. There is the shift of the oscillation due to the sign of the helicity. The tunneling occurs when $-s|\mathit{k}|+\frac{{\theta }_0^{s(i)}{m}_{\theta }}{2}\sin m_{\theta }{t}_{\mathit{k}}^{s(i)}=0$. From the picture, we can find the relation $t_{\mathit{k}}^{+(i)}\simeq t_{\mathbf{k}}^{-(i)}+\frac{\pi }{m_{\theta }}$, which is exact when the amplitude does not change with time.