# Why Electron & Neutrino Form Doublet in Electroweak Model

• ericthebee
For massless particles, this means a Lagrangian with a global SU(2) symmetry. However, for this symmetry to act on particles with the same spacetime properties, the only option is to combine the left electron and neutrino spinors into a doublet. This is supported by the fact that singlets are uncharged and do not fit into the SU(2) representation. Therefore, the left electron and neutrino spinors are combined into a doublet in the Weinberg-Salam model. In summary, the left electron and neutrino spinors are combined into a doublet in the Weinberg-Salam model in order to maintain both Lorentz and gauge invariance in the Lagrangian.

#### ericthebee

Hello, I've unfortunately a trivial question, but one I couldn't answer by reading my references.
In the Weinberg-Salam model, why do we take the left electron and neutrino spinors to form a doublet?
I'd like a "physical" answer, if possible, and a more formal one, which I think is based on the SU(2) representations.
My teacher's notes start with writing down a Lagrangian for massless electron and neutrino, and then he literally says "...we must now determine the internal symmetries of the Lagrangian. It's obvious that an internal symmetry must act on particles which have the same space-time properties. The only possibility is then to put together the electron and the neutrino (left parts) in a doublet". Perhaps I'm just too stupid, but it's unintelligible to me.

Thanks everyone :).

You have two particles, so your choices are one doublet or two singlets. Since singlets are uncharged, that eliminates two singlets as a possibility.

Technically, you could have had them in singlets (anomaly cancelation aside, having them in singlets would mean also putting the quarks in singlets and then there would be no fermions charged under SU(2)). However, this is not what we observe in Nature.

The general idea is to build a Lagrangian which is both Lorentz and gauge invariant.