How strong are the experimental constraints on the neutrino helicity really? It is commonly accepted that all neutrinos have left-handed helicity and all anti-neutrinos are right-handed helicity.$^1$  The experimental base comes from Goldhaber's famous experiment performed in the 1950's. Its results are consistent with only left-handed neutrinos, but the uncertainties are actually rather large. I have been searching for more recent, improved measurements of the neutrino helicity, but everything I found online were only references to Goldhaber's original experiment.
It looks to me like there were no serious attempts to reproduce or improve Goldhaber's more than 60-year old result, which I find somewhat intriguing, given that the result had a major impact on the development of the Standard Model of particle physics. Yet, it is often made to appear like the question was unambiguously settled experimentally by the Goldhaber experiment.
So my questions are: have there been any experiments which were able to improve the uncertainty (say, to the 1%-level)? Or is a better measurement simply too difficult, even with modern hardware and techniques? If there really were no lab experiments for a better value of the amount of left-handed neutrinos, are there other constraints, e.g. from cosmology? If not, then I would say that in principle right-handed neutrinos are still a possibility which should be more thoroughly tested experimentally, since it would have a major impact on particle physics.
${^1}_\textrm{For the purpose of this question I'll not consider reference frames in which the observer is faster than the (massive) neutrino.}$
 A: It is true that the Goldhaber's experiment has not been recently repeated directly in the present (I think, as I couldn't find any recent paper trying to repeat the experiment with higher precision). However, neutrino helicity and the inexistence of the right handed neutrino is a direct consequence of Dirac's equation for massless particles. 50+ years latter there is no experimental or theoretical evidence that this result could not be true. However even if not measured directly, the neutrino helicity is a very important mechanism in recent discoveries such as the Higgs boson and its interactions, meaning that if the neutrino helicity theory was incorrect, other recent experiments would have shown very different results.
In addition, recently it has been proved that the neutrino must have mass, meaning that the existence of right-handed neutrinos is actually true, but to measure this is very hard, taking in account how difficult neutrino detection his and how low is the probability of producing one of those. Here you have a link with more information about the topic:
http://ctp.berkeley.edu/neutrino/neutrino5.html
