Question: flavor transformations probabilities

Hi, I'm looking into flavor transformations, trying to implement #242
I have a probably silly question about the definition of probabilities.
In many cases we have prob_ee defined, and other probabilities derived from it:
prob_ex:

snewpy/python/snewpy/flavor_transformation.py

Line 203 in 06a3f7b

return 1. - self.prob_ee(t,E)

prob_xx:

snewpy/python/snewpy/flavor_transformation.py

Line 220 in 06a3f7b

return (1. + self.prob_ee(t,E)) / 2.

prob_xe:

snewpy/python/snewpy/flavor_transformation.py

Line 237 in 06a3f7b

return (1. - self.prob_ee(t,E)) / 2.

so it's defined as

$$ \begin{bmatrix} P_{ee}& 1-P_{ee} \\ (1- P_{ee})/2& (1+ P_{ee})/2 \end{bmatrix} $$

My questions:

Shouldn't our matrix rows and columns add up to to 1, to conserve total number of neutrinos❓
Shouldn't our matrix be symmetrical❓

Or maybe I'm mistaken in the definitions of the flux?

Example

In case of ThreeFlavorDecoherence we set $P_{ee}==\frac{1}{3}$, so we have the matrix

$$ \begin{bmatrix} 1/3& 2/3 \\ 1/3& 2/3 \end{bmatrix} $$

let's say I have initial flux $F^0_e = 1$, $F^0_x = 0$, then we have final flux

$$ \begin{aligned} F_e =& F^0_e P_{ee} + F^0_x P_{ex} = 1/3\\ F_x =& F^0_e P_{xe} + F^0_x P_{xx} = 1/3 \end{aligned} $$

so our total flux $\sum F^0 = 1$ becomes $\sum F = 2/3$ 😞

What you’re missing in the example is that our convention is that $F_x$ is the flux of either $\nu_\mu$ or $\nu_\tau$; it’s not the sum of both fluxes. So the total flux is $\Sigma F = F_e + 2 \times F_x = 1$.

For the matrix, it’s a bit trickier to visualise the effect; but that would effectively give you a factor of 2 in the second row; and then the first row/column would sum up to 1 (as expected; corresponding to the initial and final $F_e$) and the second row/column would sum up to 2 (corresponding to the initial/final sum of $F_\mu = 1$ and $F_\tau = 1$).

Thank you for clarification! I get the matrix now.
I just hope we have the same definition of $F_x$ in all our models. I will check the preSN ones.

Still this looks rather counter-intuitive to me, when it's displayed as a matrix, and might be misleading for the users as well (probably someone else is expecting like me to have prob_ex == prob_xe).

If we would have $F_x = F_\nu+F_\tau$, we have a standard two-flavor conversion matrix, and chaining several transformations becomes just a matter of matrix multiplication. but for $F_x = F_\nu = F_\tau$ it's not that trivial.