Effective chemical potential in spontaneous baryogenesis

Models of spontaneous baryogenesis have an interaction term $\partial_\mu\theta j^\mu_B$ in the Lagrangian, where $j^\mu_B$ is the baryonic current and $\theta$ can be a pseudo-Nambu-Goldstone boson. Since the time component of this term, $\dot{\theta} j^0_B$, equals $\dot{\theta} n_B$ for a spatially homogeneous current, it is usually argued that this term implies a splitting in the energy of baryons and antibaryons thereby providing an effective chemical potential for baryon number. In thermal equilibrium, one {then obtains} $n_B \sim \dot{\theta} T^2$. We however argue that a term of this form in the Lagrangian does not contribute to the single particle energies of baryons and antibaryons. We show this for both fermionic and scalar baryons. But, similar to some recent work, we find that despite the above result the baryon number density obtained from a Boltzmann equation analysis can be proportional to $\dot{\theta} T^2$. Our arguments are very different from that in the standard literature on spontaneous baryogenesis.