# Spin conservation in $\beta^+$ decay

In $\beta^+$ decay, a proton (consisting of 2 up and 1 down quarks) decays into a neutron (1 up and 2 down quarks), a positron and an electron neutrino.

If I'm not mistaken, all quarks, the positron and the neutrino each have a spin of 1/2. So in the overall process, there is a net gain of spin 1. How is this possible? Where does the extra spin come from?

(If spin conservation didn't matter on the other hand, I wouldn't see any argument for the production of the neutrino.)

• Spin and angular momentum are vector quantities. You can have two spin-1 objects that form a system with a net angular momentum of zero. – dukwon Jul 18 '17 at 10:16

• From a modern perspective, it is an integral part of the Standard Model, which wouldn't survive without the neutrinos, so the question makes very little sense. In this particular reaction the neutrino is required because otherwise you'd have a different reaction, which can range over a wide range of possibilities with multiple constraints, so again the question makes very little sense. You can argue that the neutrino is there to preserve lepton number (or more specifically electron number) or that the only available $e$, $W^+$ vertices involve a neutrino, though. – Emilio Pisanty Jul 18 '17 at 11:58
• From a modern perspective the neutrino is present because lepton number must be conserved at the $W$-$e^+$ vertex. Though historical development of lepton-number as quantity we expect to be conserved was driven in part by the observation of neutrinos in beta decay processes. – dmckee Jul 18 '17 at 15:14