# Unclear approximation in demonstration regarding neutrino oscillations

 P: 20 I'm stucked in a passage of Particle Physics (Martin B., Shaw G.) in page 41 regarding neutrino oscillations. Having defined $E_i$ and $E_j$ as the energies of the eigenstates $\nu_i$ and $\nu_j$, we have: $E_i - E_j = \sqrt{m^2_i - p^2} - \sqrt{m^2_j - p^2} \approx \frac{m^2_i - m^2_j}{2p}$ It can be useful to know that here natural units are used ($c=1$) and that the masses of the neutrino are considered much smaller than their momenta ($m << p$) Still, I can't understand where the $\frac{m^2_i - m^2_j}{2p}$ comes from. Does anyone have any idea?
 PF Gold P: 370 $$E_i - E_j = \sqrt{m^2_i + p^2} - \sqrt{m^2_j + p^2} \approx \frac{m^2_i - m^2_j}{2p}$$ since up to the second order: $$\sqrt{m^2 + p^2} \approx p + \frac{m^2}{2 p}$$ I just don't understand why the formula involves only one momentum. Why is it not: $$E_i - E_j = \sqrt{m^2_i + p_i^2} - \sqrt{m^2_j + p_j^2} \approx p_i - p_j + \frac{m^2_i - m^2_j}{2p}$$ Any idea?
 P: 876 because you consider that the only difference in energies comes from the mass differences - or in other words you consider $p_{i}=p_{j}$ (momentum conservation).
 P: 876 I don't think it's a symmetry... I think it has to do with the fact that the momentum is described by the flavor and not by the mass eigenstates... in other words, when you expand a flavor eigenstate: $v_{f}$ it has to have some momentum $p$ then the expanded ones should keep the same momentum...and all the differences are supposed to come from the masses
 P: 20 I'm sorry, but something is missing for me. If we expand, we get: $\sqrt{m^2 + p^2} + m \frac{2m}{2\sqrt{m^2 + p^2}}$ and considering $p>>m$: $p + \frac{m^2}{p}$ So, I'm missing the factor 2 next to $p$.
 Mentor P: 6,243 Use the first two terms of a binomial expansion for the last line of \begin{align} \sqrt{m^2 + p^2} &= p \sqrt{1 + \frac{m^2}{p^2}} \\ &= p \left(1 + \frac{m^2}{p^2}\right)^{\frac{1}{2}} \end{align}