# Lepton Number Conservation

What is the global symmetry of the electroweak Lagrangian that gives rise to lepton number conservation?

As I understand it, electric charge is some linear combination of the conserved quantities corresponding to the $SU(2)$ symmetry and the $U(1)$ symmetry of the electroweak theory. If something similar is true for lepton number, then perhaps it would not be easiest to give the global symmetry directly. This is perfectly fine of course.

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## 1 Answer

The symmetry is not a mix-up of the gauge symmetries, it's a global symmetry. The symmetry is a phase rotation of the lepton fields in one generation, so lepton number, muon number, and tau number are each conserved, ignoring neutrino oscillations. You multiply the electron-neutrino doublet and electron-singlet field by an opposite phase. Since you choose the fields to diagonalize the Higgs couplings, this doesn't affect either their electroweak or (standard dimension 4) Higgs interaction, because each spinor action term multiplies two pieces $\bar{\psi}$ and $\psi$ that rotate in opposite sense.

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