Neutrino masses and sterile neutrinos

[ChrisVer]

I am wondering... And I may be wrong but please correct me...
In general we have some constraints on the masses of the 3 flavored neutrinos $m_{\nu_e}, m_{\nu_\mu} , m_{\nu_\tau}$ and so there must be some constrain on the values of the 3 neutrino masses $m_1, m_2, m_3$. Am I right?
Also the probability of flavor change is oscillating with distance with a frequency that's proportional to $\Delta m_{ij}^2$ (i,j in [1,3] ).
My question is what happens if there is a 4th sterile neutrino, of the acceptable mass of $m_\nu \sim 50~\text{GeV}$ ?
I'd think then that the additional $m_4$ that we would have to introduce would have to be very large as well (so that its combination with $m_i$ is large)...and finally $\frac{\Delta m _{4i}^2}{E} \approx \frac{m_4^2}{E} = \frac{2500 ~\text{GeV}^2}{5 \cdot 10^{-3}~\text{GeV}} = 5 \cdot 10^5~\text{GeV}$
Is this right so long? I'd guess then that a reasonable distance for oscillation to occur would be at approximately $L \sim 10^{-21} m$?
http://pdg.lbl.gov/2011/reviews/rpp2011-rev-neutrino-mixing.pdf
(used 13.15 together with 13.12)

 

[Orodruin]

In general we have some constraints on the masses of the 3 flavored neutrinos mνe,mνμ,mντmνe,mνμ,mντ m_{\nu_e}, m_{\nu_\mu} , m_{\nu_\tau} and so there must be some constrain on the values of the 3 neutrino masses m1,m2,m3m1,m2,m3m_1, m_2, m_3. Am I right?

No, the flavour states do not have definite masses, the definite masses are properties of the mass eigenstates. If you want to talk about effective masses of flavour eigenstates there are several different combination of the mass eigenstate masses which are relevant in different applications.

What is constrained better is the mass squared differences, the absolute scale is still quite unknown.

My question is what happens if there is a 4th sterile neutrino, of the acceptable mass of mν∼50 GeVmν∼50 GeVm_\nu \sim 50~\text{GeV} ?

A neutrino of that mass would be kinematically distinguishable from the light neutrinos. It would therefore quickly decohere and there would be no oscillations.

 

[arivero]

No, the flavour states do not have definite masses, the definite masses are properties of the mass eigenstates. If you want to talk about effective masses of flavour eigenstates there are several different combination of the mass eigenstate masses which are relevant in different applications.

Hmm is there some invariant independent of the different combinations? Trace of the mass matrix, for instance?

 

[Orodruin]

Hmm is there some invariant independent of the different combinations? Trace of the mass matrix, for instance?

The trace is just the sum of the masses. You would need to find a setting where this is relevant to make it an observable. The most interesting invariant with regards to mixing is probably the Jarlskog invariant which measures the degree of CP violation.