In this lecture on neutrino physics, Prof. Feruglio defines the Jarlskog invariant as $$J=\text{Im}(U_{\alpha i}^{*} U_{\beta i}^{\,} U_{\alpha j}^{\,} U_{\beta j}^{*})\tag{1}$$ where $U$ is the neutrino mixing matrix with elements $U_{\alpha i}$. Here, $\alpha$ labels neutrino flavours ($e,\mu$ or $\tau$) and $i$ labels neutrino mass eigenstates such that
$$|\nu_\alpha\rangle=\sum_{i=1}^{3}U^*_{\alpha i}|\nu_i\rangle.$$ On the other hand, this well-cited paper defines $$J=\text{Im}(U_{e2}^{\,}U_{e3}^{*} U_{\mu 2}^{*}U_{\mu 3}^{\,}).\tag{2}$$
- Clearly, these two definitions are different because, in general, none of the entries of $U$ is zero. Which one of these definitions is correct and why?
- Moreover, what does the expression (1) mean? But it implies a sum over $\alpha,\beta, i$ and $j$? The expansion of this term would be different upon whether there are these sums in the definition or not.