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.)