Let me address your questions one by one.

*Why is it said that lepton number is conserved in Standard Model (SM)? How do I know that lepton number is an Abelian charge?*

The SM Lagrangian is invariant under the fermion transformations,
$$
\psi \to e^{iL\theta}\psi
$$
where $L$ is assigned such that $e^-$, $\mu^-$ and $\tau^-$ leptons and lepton-neutrinos have $L=1$, whilst their antiparticles have $L=-1$, and everything else has $L=0$. This global $U(1)$ symmetry corresponds to lepton number conservation - $L$ is what we call lepton number. Lepton number is by construction the Abelian charge corresponding to that $U(1)$ global symmetry.

*Why is this conservation not as sacred as electric charge conservation?*

The lepton $U(1)$ global symmetry is *accidental*. We simply cannot write a gauge invariant, renormalizable operator in our Lagrangian that breaks conservation of lepton number. There is no reason to expect that physics beyond the SM respects lepton number conservation.

The electric charge $U(1)_{em}$ local symmetry was a principle on which the SM was built. The SM would not be renormalizable if $U(1)_{em}$ was explicitly or anomalously broken. It would be catastrophic if $U(1)_{em}$ were broken.

*How does one mathematically distinguish between lepton number and electric charge?*

They are the conserved charges associated with different $U(1)$ symmetries. In general, when you have multiple $U(1)$ symmetries, charge assignment is somewhat arbitrary, since one can pick different linear combinations of the original $U(1)$ generators as the symmetries. In this case, however, the $U(1)_{em}$ is local, so there is no mixing with the global lepton number $U(1)$.

This post imported from StackExchange Physics at 2014-04-13 14:44 (UCT), posted by SE-user innisfree