Wolfenstein potentials for neutrinos induced by ultra-light mediators

New physics can emerge at low energy scales, involving very light and very weakly interacting new particles. These particles can mediate interactions between neutrinos and usual matter and contribute to the Wolfenstein potential relevant for neutrino oscillations. We compute the Wolfenstein potential in the presence of ultra-light scalar and vector mediators and study the dependence of the potential on the mediator mass $m_A$, taking the finite size of matter distribution (Earth, Sun, supernovae) into consideration. For ultra-light mediators with $m_{A}^{-1}$ comparable to the size of the medium ($R$), the usual $m_{A}^{-2}$ dependence of the potential is modified. In particular, when $m_{A}^{-1}\gg R$, the potential does not depend on $m_{A}$. Taking into account existing bounds on light mediators, we find that for the scalar case significant effects on neutrino propagation are not possible, while for the vector case large matter effects are allowed for $m_{A} \in [2\times10^{-17}$, $4\times10^{-14}]$ eV and the gauge coupling $g\sim 10^{-25}$.