Can the Neutrino Oscillation Formula be Simplified Using a Trivial Proof?

[Trifis]

Hi all!

I am not sure how to prove mathematically that the expression for the probability that a neutrino originally of flavor α will later be observed as having flavor β
$P_{α \rightarrow β}=\left|<\nu_{\beta}|\nu_{\alpha}(t)>\right|^2=\left|\sum_{i}U_{β i}^*U_{α i}e^{-iE_it}\right|^2$ (1)
can be equivalently written as
$P_{α \rightarrow β}=δ_{αβ}-4\sum_{i>j}Re(U_{β i}^*U_{α i}U_{β j}U_{α j}^*)\sin^2(\frac{Δm_{ij}^2L}{4E})+2\sum_{i>j}Im(U_{β i}^*U_{α i}U_{β j}U_{α j}^*)\sin(\frac{Δm_{ij}^2L}{4E})$ (2)
Whoever is not familiar with the notation and would still like to contribute "mathematically", all the variables and constants are explained perfectly in the wiki article: http://en.wikipedia.org/wiki/Neutrino_oscillation

The proof has to be trivial, but here I am trying for two hours and still not able to show this for a general case.

 

[ChrisVer]

which part of the formula is confusing you?
how to express $E_{i} t$ in terms of $L, m_{i}$?
This is done by some easy calculations, I think you can find them easy if you search... eg
http://www2.ph.ed.ac.uk/~vjm/Lectures/SHParticlePhysics2012_files/PPNotes5.pdf
eq 16.3
and also by saying $t= \frac{L}{c}$ the time they traveled from distance L up to reaching us.
with c=1, and 16.3 you get your result.

or how to write the sums?
Those sums are in fact two sums, of i and j with the same expressions just change the i's to js and taking the conjugate...
After that you can start breaking the sums in such a way that you'll get something as the final result... for example for i=j you will evenutally get delta because the exponentials will cancel each other out (remind i=j again), and the U matrices are unitary, so you will get the delta Kroenicker by summing them:
$(U^{\dagger} U)_{ab} = \delta_{ab}$ for U unitary.
The rest is more tedious work, but in general that's how you get the Re and I am part (depending of what i,j configurations you sum).

 

[Trifis]

I want to start with (1) and reach (2) step by step.
$P_{α \rightarrow β}=\left|\sum_{i}U_{β i}^*U_{α i}e^{-iE_it}\right|^2=\left|\sum_{i}U_{β i}^*U_{α i}e^{-iE_it}\right|^2=\left|\sum_{i}(Re(U_{β i}^*U_{α i})+iIm(U_{β i}^*U_{α i}))(\cos(E_it)-i\sin(E_it))\right|^2=$
$\left[\sum_{i}Re(U_{β i}^*U_{α i})\cos(E_it)+Im(U_{β i}^*U_{α i})\sin(E_it)\right]^2+\left[\sum_{i}Im(U_{β i}^*U_{α i})\cos(E_it)-Re(U_{β i}^*U_{α i})\sin(E_it)\right]^2$
Is this correct so far? How do we further simplify this expression and get the double sums? Also some trigonometry is involved for sure.

 

[ChrisVer]

why don't you write the square explicitly?

$|\sum_i U_{β i}^* U_{α i} e^{-iE_{i}t}|^{2} = \sum_{i} \sum_{j} U_{β i}^* U_{α i} U_{ α j}^{*} U_{β j} e^{-iE_{i}t} e^{+iE_{j}t}$

$= \sum_{i} \sum_{j} U_{β i}^* U_{α i} U_{α j}^{*} U_{β j} e^{-i (ΔE)_{ij}t}$

that's in general how you proceed...

 

[Trifis]

So you simply write the square of the absolute value as the complex number times its conjugate and you get the Δm differences.
I see how for i=j, the Kronecker delta arises. Sorry but I am too bad at this and I still cannot work out the rest of the terms. Can you take me by hand or show the derivation in detail?

 

[ChrisVer]

because in most of the cases this is an exercises, you should just keep track of your indices and what you get...
I guess could help you if you showed me what you did, where you've reached. In most of cases it's just a to-be-done-carefully calculation, and at one point you have to use trigonometric identities (if I recall well)...
One thing is for sure, after you found the case of delta, you need also to add to this the cases where i is not equal to j... and from that see what you get for that term...

 

[Orodruin]

So you simply write the square of the absolute value as the complex number times its conjugate and you get the Δm differences.
I see how for i=j, the Kronecker delta arises. Sorry but I am too bad at this and I still cannot work out the rest of the terms. Can you take me by hand or show the derivation in detail?

For the remaining terms ($i \neq j$), you have one term for $i > j$ and one term for $j < i$. Consider how they are related by writing out both sums and then changing the summation indices in one of them. After that you can make use of the relations
$$
z + z^* = 2 Re(z), \quad z - z^* = 2 Im(z)
$$
(Also note that $\Delta E_{ij} = - \Delta E_{ji}$ ...)

 

[Trifis]

Thanks for your replies. I have finally worked out the derivation!