# Neutrino flavor and mass

Neutrinos with specific mass don't have a unique flavor and neutrinos with specific flavor don't have unique mass.

Let's call the neutrinos with specific mass $$\nu_1, \nu_2, \nu_3$$ and the neutrinos with specific flavor $$\nu_e, \nu_\mu, \nu_\tau$$.

According to the subatomic stories of Fermilab

\begin{align} \nu_1 &= \{ \nu_e \} \\ \nu_2 &= \{ 0.5 \nu_\tau, 0.5\nu_\mu \} \\ \nu_3 &= \{ 0.3 \nu_e, 0.3\nu_\mu, 0.3\nu_\tau \} \end{align}

and

\begin{align} \nu_e &= \{ \text{mix of } \nu_1, \nu_2, \nu_3 \} \\ \nu_\mu &= \{ \text{mix of } \nu_1, \nu_2, \nu_3 \} \\ \nu_\tau &= \{ \text{mix of } \nu_1, \nu_2, \nu_3 \} \\ \end{align}

Since $$\nu_1$$ has always the flavor of an $$\nu_e$$, can we say that the mass of $$\nu_\mu$$ and $$\nu_\tau$$ depend on the mass of the $$\nu_e$$?

• "according to ?????" What is your source for this obviously wrong non-fact? – Cosmas Zachos Jun 30 at 14:43
• youtube.com/… (understanding neutrino oscillations like the pros) – user268802 Jun 30 at 14:54
• iI is the channel of fermilab on youtube where they explain some particle physics stuff. – user268802 Jun 30 at 14:56
• Don messed up, glibly. He should have provided ellipsis ... after $\nu_e$ to indicate the less than 50% components of the other flavors. He says "mostly" in words, to indicate this, where green is electron neutrino... He oversimplifies and ends up confusing you. Avoid physics videos... – Cosmas Zachos Jun 30 at 15:11
• Right. Popular science expositions not infrequently confuse more than explain... – Cosmas Zachos Jun 30 at 15:18

The {vectors} in your question are the rows and columns of the neutrino mass mixing matrix, which is unitary. That means your statement (which you assign to a source) that

$$\nu_1 = \{ \nu_e \} \qquad\text{and}\qquad \nu_3 = \{0.3 \nu_e, \cdots \}$$

can not be correct. That's too much neutrino.

The middles of the allowed ranges (which aren't the same as the best-fit values, for reasons that require about an hour to explain to a new grad student) are

\begin{align} \nu_1 &= \{ 0.67 \nu_e,\ 0.13 \nu_\mu,\ 0.16\nu_\tau \} \\ \nu_2 &= \{ 0.30 \nu_e,\ 0.33 \nu_\mu,\ 0.35\nu_\tau \} \\ \nu_3 &= \{ 0.02 \nu_e,\ 0.50 \nu_\mu,\ 0.47\nu_\tau \} \end{align}

If tables of numbers make you a little cross-eyed, like they do me, a commenter linked to this explanation which includes the following graphic:

(with $$\color{green}{\text{electron}},\ \color{yellow}{\text{muon}},\ \color{cyan}{\text{tau}}$$ neutrinos represented by the three colors.)

These are consistent with the values in your question, with two changes. First, my $$\nu_2, \nu_3$$ are swapped relative to yours --- which may be a difference between "normal hierarchy" versus "inverted hierarchy," or may just an arbitrary labeling. Second, your question has $$\nu_1 = \{\nu_e\}$$, where the reality is $$\nu_1=\{\text{mostly } \nu_e\}$$.

None of the flavor neutrinos $$\nu_e, \nu_\mu, \nu_\tau$$ has a well-defined mass, and the masses of the $$\nu_1, \nu_2, \nu_3$$ are independent parameters.