Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Question: flavor transformations probabilities #249

Closed
Sheshuk opened this issue May 26, 2023 · 2 comments
Closed

Question: flavor transformations probabilities #249

Sheshuk opened this issue May 26, 2023 · 2 comments
Labels
question Further information is requested

Comments

@Sheshuk
Copy link
Contributor

Sheshuk commented May 26, 2023

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:

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

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

prob_xe:
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:

  1. Shouldn't our matrix rows and columns add up to to 1, to conserve total number of neutrinos❓
  2. 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 $\sum F = 2/3$ 😞

@Sheshuk Sheshuk added the question Further information is requested label May 26, 2023
@JostMigenda
Copy link
Member

JostMigenda commented May 26, 2023

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 $F_\tau = 1$).

@Sheshuk
Copy link
Contributor Author

Sheshuk commented May 26, 2023

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.

@Sheshuk Sheshuk closed this as completed May 26, 2023
Sign up for free to join this conversation on GitHub. Already have an account? Sign in to comment
Labels
question Further information is requested
Projects
None yet
Development

No branches or pull requests

2 participants