Evidence for an Inverted Neutrino Mass Hierarchy from the T2K $\theta_{1 3}$ Result and $\nu$-Process Nucleosynthesis

The synthesis of $^{11}$B and $^{7}$Li via neutrino-induced nucleon emission (the $\nu$-process) is sensitive to the neutrino mass hierarchy if the $\theta_{1 3}$ mixing angle is large enough. This arises because, when there is significant 13 mixing, the average electron neutrino energy for the charged-current neutrino reactions is larger for a normal mass hierarchy than for an inverted hierarchy. This mixing occurs in the carbon shell and hence affects the nucleosynthesis of $^{11}$B and $^{7}$Li in the helium shell of core-collapse supernovae. Recent constraints on $\theta_{1 3}$ from the T2K collaboration indicates that indeed $\theta_{1 3}$ is large enough to induce substantial mixing. Moreover, there is also recent recent evidence from SiC $X$ grains in meteorites for the existence of $\nu$-process $^{11}$B and $^{7}$Li encapsulated in some grains. We show here that these two new results hint at a marginal ($1 \sigma$) preference for an inverted neutrino mass hierarchy. The analysis of more $X$ grains enriched in Li and B could substantially improve this limit.