Why is it not possible to measure the individual neutrino mass from $\beta-$decay? Why do we have to rely only on neutrino oscillations, to measure the mass squared differences of neutrinos? Why is it not possible to measure the neutrino masses directly, say, from $\beta-$decay?


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*When Pauli hypothesized the existence of neutrinos to save the conservation of energy and angular momentum in $\beta-$decay, why did he assumed the neutrinos to be massless? 

*How did experimentalists at that time concluded that neutrinos were massless?

*What about the measurement of energy in the 3-body decay $n\rightarrow p+e^-+\bar{\nu}_e$, in the sophisticated experiments today, and determination of the individual neutrino masses from that? 
One has $$E_n=E_p+E_e+E_\nu\Rightarrow m_nc^2+T_n=m_pc^2+T_p+m_ec^2+T_e+m_\nu c^2+T_\nu.$$ $m_p,m_n,m_e$ are known. Therefore, by measuring the kinetic energies $T_n,T_p,T_e$ and $T_\nu$ one can determine the mass of $\nu_e$.
Why is this not possible experimentally? 
Is this because $T_\nu$ is not measurable? Or is this because $\nu_e$ being a flavour state, does not have a definite mass?
 A: There cannot be enough accuracy in the momentum and energy measurements of the end products of beta decay, only limits can be determined because of measurement errors.
See this link  for a recent review.

The paper reviews recent experiments on tritium β-spectroscopy searching for the absolute value of the electron neutrino mass m(ν_e). By use of dedicated electrostatic filters with high acceptance and resolution, the uncertainty on the observable m^2(ν_e) has been pushed down to about 3 eV^2. The new upper limit of the mass is m(ν_e) < 2 eV at 95% C.L. In view of erroneous and unphysical mass results obtained by some earlier experiments in β-decay, particular attention is paid to systematic effects. The mass limit is discussed in the context of current neutrino research in particle- and astrophysics. A preview is given of the next generation of β-spectroscopy experiments currently under development and construction; they aim at lowering the m^2(ν_e)-uncertainty by another factor of 
  100, reaching a sensitivity limit m(ν_e) < 0.2 eV. 

A: 
Why do we have to rely only on neutrino oscillations, to measure the mass squared differences of neutrinos? 

I don't think we do. It's just a hypothesis. 

Why is it not possible to measure the neutrino masses directly, say, from $\beta-$decay?

Because rest mass is a measure of energy-content for a body at rest, and nobody has every seen a neutrino at rest.  

When Pauli hypothesized the existence of neutrinos to save the conservation of energy and angular momentum in $\beta-$decay, why did he assumed the neutrinos to be massless? 

He didn't. See his 1930 letter to Lise Meitner and others:
"Dear Radioactive Ladies and Gentlemen,
As the bearer of these lines, to whom I graciously ask you to listen, will explain to you in more detail, because of the "wrong" statistics of the N- and Li-6 nuclei and the continuous beta spectrum, I have hit upon a desperate remedy to save the "exchange theorem" of statistics and the law of conservation of energy. Namely, the possibility that in the nuclei there could exist electrically neutral particles, which I will call neutrons, that have spin 1/2 and obey the exclusion principle and that further differ from light quanta in that they do not travel with the velocity of light. The mass of the neutrons should be of the same order of magnitude as the electron mass and in any event not larger than 0.01 proton mass. The continuous beta spectrum would then make sense with the assumption that in beta decay, in addition to the electron, a neutron is emitted such that the sum of the energies of neutron and electron is constant…"


  
*How did experimentalists at that time concluded that neutrinos were massless?
  

They looked at the curve on the beta decay spectrum. See section VII of Fermi's paper here. The measured curves were in line with the μ=0 plot. See this lecture re tritium beta decay. Paul Dauncey of the HEP group at Imperial. He says the small energy release makes it easier to observe the effects of a non-zero mass.


  
*What about the measurement of energy in the 3-body decay $n\rightarrow p+e^-+\bar{\nu}_e$, in the sophisticated experiments today, and determination of the individual neutrino masses from that? 
  

Measuring energy isn't measuring mass. A photon can have considerable energy but no mass. If the photon travels at less than c it exhibits an "effective" mass. If you trap it in a mirror-box all of its energy-momentum is effective as mass, and the box is harder to move as a result. If the neutrino travels at c it has no mass, full stop.    

One has $$E_n=E_p+E_e+E_\nu\Rightarrow m_nc^2+T_n=m_pc^2+T_p+m_ec^2+T_e+m_\nu c^2+T_\nu.$$ $m_p,m_n,m_e$ are known. Therefore, by measuring the kinetic energies $T_n,T_p,T_e$ and $T_\nu$ one can determine the mass of $\nu_e$.
  Why is this not possible experimentally? 

Because when the neutrino travels at c,  $T_\nu = E_\nu$ and that's all you know. 

Is this because $T_\nu$ is not measurable? Or is this because $\nu_e$ being a flavour state, does not have a definite mass?

We're happy with the neutrino energy, that's how they were predicted in the first place. But we don't have any real evidence of what the neutrino is doing as it propagates. I recently asked if we have any evidence of slower-than-light neutrinos. The answer is no. All we know is that we aren't detecting as many neutrinos as we expect. The rest is hypothesis. 
