Derivation of neutrino oscillation phase factor

Question

As we know, the neutrino $\nu_{\alpha}$ with flavor $\alpha=e,\mu,\tau$ is a linear combination of mass eigenstates: $$ |\nu_{\alpha}\rangle=\sum_iU_{\alpha i}|\nu_i\rangle,\quad i=1,2,3 $$ where the propagation of the mass eigenstates $|\nu_i\rangle$ can be described by plane wave solutions: $$ |\nu_{i}(t)\rangle = e^{ -i ( E_{i} t - \vec{p}_{i} \cdot \vec{x}) } |\nu_{i}(0)\rangle. $$ Suppose that $t=T$ and $|\vec{x}|=L$. Then the phase factor can be given by $$ \begin{split} E_iT-p_iL &= E_i\frac{L}{\sqrt{1-m_i^2/E_i^2}}-\sqrt{E_i^2-m_i^2}L \\ &=E_iL\left(1+\frac{m_i^2}{2E_i^2}\right)-E_iL\left(1-\frac{m_i^2}{2E_i^2}\right) +\mathcal{O}\left(\frac{m_i^4}{E_i^3}\right) \\ &=\frac{m_i^2L}{E_i}+\mathcal{O}\left(\frac{m_i^4}{E_i^3}\right) \end{split} $$ However, this result is not correct. The artilce Neutrino oscillation on Wikipedia says that $$ |\nu_{i}(L)\rangle = e^{ -i m_{i}^2 L/2E }|\nu_{i}(0)\rangle $$ which differs in a factor of $1/2$. I can't figure out what is wrong with my derivation above (Maybe the formula for time interval $T$ is not correct, but I am not sure). Any help is greatly appreciated.

Answer 1

Then the phase factor can be given by [...] $\frac{m_i^2L}{E_i}+\mathcal{O}\left(\frac{m_i^4}{E_i^3}\right)$

The calculation as shown in the question statement above seems correct to me.
But it's not the whole story ...

[...] a factor of $1/2$

The appearance of this factor $\frac{1}{2}$ along with the difference $\Delta~m^2_{jk} := m^2_j - m^2_k$ has been motivated in various ways, according to the (recent) PDG Review article on neutrino mixing (ch. 13).

The quickest way to make it plausible seems to calculate the interference (or beat) of two waves (of equal amplitude):

$$e^{i \phi} + e^{i \theta} = \left( e^{ \frac{1}{2} (i \phi - i \theta) } + e^{ \frac{1}{2}~ (i \theta - i \phi) } \right) \times e^{ \frac{1}{2}~ (i \phi + i \theta) } = 2 ~ \text{Cos}[ \frac{1}{2} (\phi - \theta) ] \times e^{ \frac{1}{2}~ (i \phi + i \theta) }.$$

Substituting "phase factors" as calculated above results in the phase difference

$$\delta \phi_{jk} := \frac{1}{2}~ \left( \frac{m_j^2 L}{E_j} - \frac{m_k^2 L}{E_k} \right)$$

which in "realistic experimental situations" due to $E \gg m ~ c^2$ becomes to good approximation (for a suitable "mean energy" $E$) the more familiar

$$\delta \phi_{jk} \simeq \frac{1}{2}~ \Delta~m^2_{jk}~ \frac{L}{E}.$$

The remaining "high frequency factor" $e^{ \frac{1}{2}~ i~ \left( \frac{m_j^2 L}{E_j} + \frac{m_k^2 L}{E_k} \right) }$ may apparently be considered and treated as a "usual single particle phase factor".

Finally a note on terminology:

the neutrino $\nu_{\alpha}$ with flavor $\alpha = e, \mu, \tau$ is a linear combination of mass eigenstates [...] $

When referring to quarks or to charged leptons, the word "flavor" is synonymous to "mass eigenstate".
I find it at least unfortunate and confusing if this convention isn't followed when referring to neutrinos; and, in correspondence with the terminology used for quarks, I prefer to call $\nu_e, \nu_{\mu},$ and $\nu_{\tau}$ "weak (eigen-)states" of neutrinos (which couple by electro-weak interaction to the charged leptons of the indicated flavor: $e, \mu,$ or $\tau$).