Why are reactor experiments measuring theta13, not theta12?

[Aturen]

The question I have came from a talk I saw about the MINOS experiment. They say they can measure theta13 from their beam of mostly muon neutrinos by measuring electron neutrino appearance. Why would this not involve theta12?
Is what is happening that the dominant chain is muon neutrinos going to tau neutrinos before oscillating to electron neutrinos (if so, how does one calculate those rates?), or that the length at which the detector is (735 km) is further than the oscillation length for muon neutrinos directly changing to electron neutrinos (if so, what is the energy to use in calculating the length, L = (4E)/(delta m^2)?).


I have found a few places that have stated that the probability that a muon neutrino oscillates to an electron neutrino is:
$$P(\nu_\mu \to \nu_e) = \sin^2 (2 \theta_{13}) \sin^2 (\theta_{23}) \sin^2 (\Delta m^2_{atm} L/4E)$$

But I don't know how to derive this. I tried using the PMNS 3x3 matrix, but it doesn't simplify well. The probability for neutrino oscillation, assuming two species, is close to this expression, but involves only two sines.

 

[AndyW]

I can understand your confusion about the role of theta12 in the MINOS experiment. Theta12 is indeed an important factor in neutrino oscillations, but it is not directly involved in the measurement of theta13 in this experiment.

To understand why, let's first review the basics of neutrino oscillations. Neutrinos come in three flavors: electron, muon, and tau. However, these flavors are not fixed and can change as the neutrino travels through space. This phenomenon is known as neutrino oscillation, and it is governed by the mixing angles theta12, theta13, and theta23, as well as the difference in masses of the three neutrino species.

In the MINOS experiment, a beam of mostly muon neutrinos is sent from Fermilab to a detector located 735 km away in Minnesota. The goal of the experiment is to measure the appearance of electron neutrinos in the beam, which would indicate that the muon neutrinos have oscillated into electron neutrinos during their journey.

Now, let's look at the equation you provided for the probability of muon neutrino oscillation to electron neutrinos. As you correctly pointed out, it involves theta13, theta23, and the difference in masses of the three neutrino species. But where is theta12?

The key here is to understand that theta12 mainly affects the oscillation between electron and muon neutrinos, not muon and electron neutrinos. In other words, theta12 determines the probability of an electron neutrino turning into a muon neutrino, not the other way around. Since the MINOS experiment is looking for the appearance of electron neutrinos from a beam of muon neutrinos, theta12 is not directly involved.

So why is theta12 not completely ignored in this experiment? This is because it indirectly affects the oscillation between muon and electron neutrinos through its impact on oscillation between electron and muon neutrinos. In other words, theta12 affects the overall probability of neutrino oscillation, but it is not the main factor in determining the appearance of electron neutrinos in the MINOS experiment.

To answer your second question about the calculation of rates, the dominant chain in the MINOS experiment is indeed muon neutrinos oscillating to tau neutrinos before eventually turning into electron neutrinos. The calculation of rates involves using the equation you provided, as well as the energy of the neutrinos and the distance they travel