## Solar neutrino oscillation experiments - how do they extract the parameters?

In oscillation experiments that use solar neutrinos, it seems that the mixing angle $\theta_{12}$ and the mass squared difference $\Delta m_{12}^2$ can be determined from comparing measurements of the neutrino flux to theoretical models of what the Sun should be emitting.

However, I am struggling to find out how these quantities are actually determined. When I look up specific experiments like Super-kam and SNO, all I keep finding is just the stated value of these quantities from a "combined analysis of all solar neutrino experiments".

Can anyone please explain briefly how they determine the values of these parameters, or link me to something useful?

Or, is it that it is actually impossible to determine these values using the data from just one observatory?

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 Mentor Qualitative description: If you know the parameters and the initial neutrino composition (e.g. electron-antineutrinos from the sun) and some other parameters like the energy of the neutrinos, you can calculate the composition (electron/muon/tau neutrinos) at your detector. Therefore, if you can measure this composition, you can put constraints on the parameters. I think that it is possible to estimate all values with a single detector, but usually these detectors are good at a specific set of parameters only, and worse at the determination of others. Therefore, you combine the measurements of all neutrino detectors to calculate the set of parameters which fits the best to all measurements at the same time. This is the usual way to combine experimental results and is used in many fields, not just neutrino physics.

 Quote by mfb Qualitative description: If you know the parameters and the initial neutrino composition (e.g. electron-antineutrinos from the sun) and some other parameters like the energy of the neutrinos, you can calculate the composition (electron/muon/tau neutrinos) at your detector. Therefore, if you can measure this composition, you can put constraints on the parameters. I think that it is possible to estimate all values with a single detector, but usually these detectors are good at a specific set of parameters only, and worse at the determination of others. Therefore, you combine the measurements of all neutrino detectors to calculate the set of parameters which fits the best to all measurements at the same time. This is the usual way to combine experimental results and is used in many fields, not just neutrino physics.
hmmm. It no doubt becomes easier to constrain things when you've already got some of the pieces of the puzzle in place, but what I'm trying to understand is how things were done in the early experiments when no parameter values were initially known at all.

If I'm not mistaken, solar neutrinos were the first to have parameters measured, at the kamiokande detector. Also I think a 2-flavour approximation was valid rather than requiring a full 3-flavour treatment, just electron neutrinos oscillating to muon neutrinos. So, the probability was

$P(\nu_{e} \rightarrow \nu_{e}) = 1 - sin^{2}2\theta_{12}sin(\frac{1.27*\Delta m_{12}^{2}L}{4E})$

So, experimenters would have known the distance L they travelled from the Sun to the detector, and I'm guessing they could somehow measure the energy E from the rings of Cerenkov light produced when the neutrinos interacted with kamiokande's volume of water. They found the probability from the ratio of the observed flux of electron neutrinos to the predicted flux.
That means they could fill in the variables P, L and E for the oscillation probability equation shown above, but that still leaves 2 unknown variables $\theta_{12}, \Delta m_{12}^{2}$. How could they have determined their values? It seems to me that to calculate one of them, you would have to take some arbitrary guess at the other's value?
How did they get around this?

Mentor

## Solar neutrino oscillation experiments - how do they extract the parameters?

 Quote by Doofy that still leaves 2 unknown variables $\theta_{12}, \Delta m_{12}^{2}$. How could they have determined their values? It seems to me that to calculate one of them, you would have to take some arbitrary guess at the other's value? How did they get around this?
You can't, if you have only a single source and detector, and a fixed energy (or fixed distribution of energy). All you can do is map out an "allowed" region on a plot of $\theta_{12}$ versus $\Delta m_{12}^{2}$. That is, you can say that certain combinations of the two quantities are possible, and the others are not.

Different experiments cover different energy ranges and use different source-to-detector distances. When you combine their results, the "allowed" region shrinks to the intersection of the individual regions.

 Quote by jtbell You can't, if you have only a single source and detector, and a fixed energy (or fixed distribution of energy). All you can do is map out an "allowed" region on a plot of $\theta_{12}$ versus $\Delta m_{12}^{2}$. That is, you can say that certain combinations of the two quantities are possible, and the others are not. Different experiments cover different energy ranges and use different source-to-detector distances. When you combine their results, the "allowed" region shrinks to the intersection of the individual regions.
hmm, okay, so say the electron neutrino survival probability was measured at 0.5.
Then the $sin^2(2\theta_{12})sin(\frac{1.27*\Delta m_{12}^{2}}{4E})$ part takes a value of 0.5, and the $sin^2(2\theta_{12})$ and $sin(\frac{1.27*\Delta m_{12}^{2}}{4E})$ parts each must take a value between 0 and 1, so you can find the allowed values of the parameters from that. That makes sense. These must be these plots of closed loops I keep seeing but ignoring, lol.

Do they ever do solar neutrino analysis another way though? What I was thinking was that the sun has a few different neutrino-producing reactions, some with sharply defined neutrino energies, some with broad ranges of neutrino energies - so can they measure these energies and plot a range of L/E from that? Or when people plot a range of L/E is it typically L that gets varied and they just use a source with a well defined energy?

 Mentor Assuming you know the energy distribution of the neutrinos, you can generate (in a computer program) a random sample of theoretical neutrinos with different energies according to that distribution, let them undergo oscillation with specific values of ##\theta_{12}## and ##\Delta m^2_{12}##, and compare the results with what you actually observe. Repeat for different combinations of ##\theta_{12}## and ##\Delta m^2_{12}##, and you can construct a plot of the "allowed region" that I described before. My Ph.D. dissertation (completed almost exactly 30 years ago) used this method to map out upper limits in a plot of ##\theta_{12}## and ##\Delta m^2_{12}## for muon-antineutrino to tau-antineutrino oscillations, using data from an experiment at Fermilab. Obviously I didn't actually find any oscillations, otherwise I'd be at least semi-famous now. The details of the analysis that people do nowadays are surely more sophisticated than what I did, but the general drift is probably similar.

 Quote by jtbell The details of the analysis that people do nowadays are surely more sophisticated than what I did, but the general drift is probably similar.
yeah, there surely can't be that many ways to skin this particular cat. Am I right in thinking that making these plots of allowed regions is not always the aim though?
I think I remember seeing some probability plot where the $sin^2(2\theta)$ part could be read directly from the oscillation amplitude, and by looking at the oscillation wavelength one could get the square mass difference?

 Mentor If the energy resolution (and your model of sun) is good enough, you can measure the oscillation probability for different L/E ratios. In theory, this gives the option to measure both at the same time. Experimental issues might make this impractical, but that is not a theoretical limit. The common approach to determine parameters for experiments like this is to evaluate the likelihood of parameter sets, given the observations. Then you can scan through the whole plane of your physics parameters and get a good idea about their values. This method leads to plots like this or colorful versions like that one.

 Quote by mfb If the energy resolution (and your model of sun) is good enough, you can measure the oscillation probability for different L/E ratios. In theory, this gives the option to measure both at the same time. Experimental issues might make this impractical, but that is not a theoretical limit. The common approach to determine parameters for experiments like this is to evaluate the likelihood of parameter sets, given the observations. Then you can scan through the whole plane of your physics parameters and get a good idea about their values. This method leads to plots like this or colorful versions like that one.
Okay thanks, I'm happy with these allowed region plots in parameter space now, but I also keep seeing "best fit" parameter values being quoted. Is it possible to explain in a nutshell how they get these best fit values, or is it too complex and specific to each experiment?

 Mentor If you have a single measurement, like the radiochemistry experiments, the only way to get a best fit is to combine multiple measurements. The water Cerenkov experiments have some energy resolution, but it is quite poor, so you don't so much measure the DeltaM-sin(theta) plane as exclude points that are inconsistent.