Lepton number conservation in standard model

Question

1. Why is it said that in standard model lepton number is conserved?

2. How do I know that Lepton number is an abelian charge?

3. Why is this conservation not as sacred as electric charge conservation.

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

Answer 1

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 $U(1)$.

Answer 2

Why is it said that in standard model lepton number is conserved?

Because the standard model is a mathematical model specifically fit to what the data tells us, and lepton number is conserved according to the data.

How do I know that Lepton number is an abelian charge?

It is additive in the number of leptons and antileptons reduce the lepton number in the interaction by their number:

Why is this conservation not as sacred as electric charge conservation.

Lepton number is as sacred as charge conservation as far as the data tell us.

What is not sacred is the partial leptonic family conservation :

In the Standard Model, leptonic family numbers (LF numbers) would be preserved if neutrinos were massless. Since neutrino oscillations have been observed, neutrinos do have a tiny nonzero mass and conservation laws for LF numbers are therefore only approximate. This means the conservation laws are violated, although because of the smallness of the neutrino mass they still hold to a very large degree for interactions containing charged leptons. However, the (total) lepton number conservation law must still hold (under the Standard Model).

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

By the quantum number identification in the group structures of the Standard Model. There exist neutral leptons. An electron neutrino has lepton number one but charge 0.

Please make sure you understand that the SM is a mathematical encapsulation of the data that have been gathered the last fifty and more years. The SM models data up to the energy scales we have experimented with. When new phenomena are observed, ( as with neutrino oscillations) the SM changes to be consistent with the data. It may be that at high energies other models will become necessary to fit observations. At the moment data validate the SM.

Because the lepton number conservation law in fact is violated by chiral anomalies, there are problems applying this symmetry universally over all energy scales. However, the quantum number B − L is much more likely to work and is seen in different models such as the Pati–Salam model.