I am an undergraduate student working on a project on neutrino oscillations. I am currently trying to code the probability of neutrino oscillations in matter (MSW effect). I am using the following formulae: $$ P (\nu_{\alpha} \rightarrow \nu_{\beta}) = \sin^2 ( 2\theta_M ) \, \sin^2 \bigg(\frac{\Delta m^2_M \, x}{4E} \bigg), $$ where $$ \sin^2 (2\theta_M) = \frac{\sin^2(2\theta)}{\sin^2(2\theta) + ( \cos(2\theta) - x)^2} $$ and $$ \Delta m^2_M = \Delta m^2 \sqrt{\sin^2 (2\theta) + (\cos(2\theta) - x )^2} $$ with the vacuum mixing angle $\theta$ and $ x = \displaystyle{\frac{2\sqrt{2} \, G_F \, N_e \, E}{\Delta m^2}} $ in terms of the electron density $N_e$, neutrino energy $E$ and the mass difference squared in vacuum $\Delta m^2$. $G_F$ is the Fermi constant. The math is taken from here on page 16.

My questions are the following:

1) To my understanding, $x$ is dimensionless, but the argument of the second sine squared in the first equation I have written down (the transition probability) needs to be dimensionless as well. How is this possible? In the formula for the transition probability of the vacuum oscillations the $x$ is replaced by the distance $L$ which has units of meters (or km). This makes sense and in that case the argument of the sine squared is dimensionless. Shouldn't the matter oscillations also depend on the distance? This would give the right units.

2) I have tried to find some data on the electronic density of the Sun (not only the core, but also the other layers) and I couldn't find much. Is there any source you could recommend to look up this information? As I am considering solar neutrinos, I would ideally also need data on the Earth's atmosphere and various layers.

Any help would be very much appreciated!


1 Answer 1


Welcome to Physics.StackExchange!

Unfortunately, the PDF you linked from NobelPrize.org has a mistake. $x$ is not dimensionless in the first formula, since it is the length travelled by the in-flight neutrino, and should be denoted by $L$ or similar. In the second and third formulas the $x$ has the correct dimensionless form.

For the density profile, Earth density is usually modelled by PREM model, and the external link of the page provides some data. Raw data of density profile of Sun has been posted in this Astronomy.StackExchange question.

  • $\begingroup$ Thanks for your reply! So the first formula needs to have in the argument of the second sine squared the mass difference squared times x times L divided by 4E? Or should I take remove x completely? Moreover, the link you posted is about the density of the Sun, but I would need the electronic density. Is there data on that or is there some way to obtain it from the density? $\endgroup$
    – Lodovico
    Commented Feb 22, 2020 at 7:19
  • $\begingroup$ In the first formula just replace the $x$ by $L$, then it is correct. For the electron density in Sun, have you looked at this StackExchange question? $\endgroup$
    – Zeick
    Commented Feb 24, 2020 at 15:08
  • $\begingroup$ I'm not sure if the StackExchange question you mentioned is helpful to me, however the following article helped me regarding the electronic density of the Sun's core: iopscience.iop.org/article/10.1088/0004-637X/765/1/14/meta. $\endgroup$
    – Lodovico
    Commented Feb 26, 2020 at 22:37

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