Measurement of Solar and other neutrino Mixing angles

Question

According to the neutrino flavour oscillation formula $$ P_{\alpha\beta}=4\sum\limits_{i<j}U_{\alpha i}U_{\beta i}^*U_{\alpha j}U_{\beta j}^* \sin^2\frac{(m_i^2-m_j^2)L}{4E}.$$ Hence, there can be three probabilities on the RHS $P_{e\mu},P_{e,\tau},P_{\mu\tau}$. Let us consider $$P_{e\mu}=4\sum\limits_{i<j}U_{e i}U_{\mu i}^*U_{e j}U_{\mu j}^* \sin^2\frac{(m_i^2-m_j^2)L}{4E}.\tag{1}$$ Expanding (1), we get, $$P_{e\mu}=4U_{e 1}U_{\mu 1}^*U_{e 2}U_{\mu 2}^* \sin^2\frac{(m_1^2-m_2^2)L}{4E}+4U_{e 1}U_{\mu 1}^*U_{e 3}U_{\mu 3}^* \sin^2\frac{(m_1^2-m_3^2)L}{4E}+4U_{e 2}U_{\mu 2}^*U_{e 3}U_{\mu 3}^* \sin^2\frac{(m_2^2-m_3^2)L}{4E}.$$ Similar expressions hold for $P_{\mu\tau}, P_{e\tau}$.

1. From the Solar neutrino experiment, can/does one measure all probabilities $P_{e\mu},P_{\mu\tau},P_{e\tau}$?

2. It appears to me that there are various unknowns such as 3 mass squared differences, all 3 mixing angles $\theta_{12},\theta_{23},\theta_{13}$, CP phase. Therefore, how does one measure the solar neutrino mixing angle $\theta_{12}$? Is it possible to eliminate all this unknowns except $\theta_{12}$ in favour of various oscillation probabilities $P_{\alpha\beta}$, $L$, and $E$?

Historically, solar mixing angle was measured first. Am I correct? Without any information of other mixing angles. Therefore, one must have eliminated other mixing angles and phases in favour of $P_{\alpha\beta}$, $L$, and $E$. Is that right?

I'm not interested in experimental subtleties (such as neutrinos from Sun can't be strictly monochromatic and there must be an energy spread etc).

Answer 1

The neutrinos of interest to SNO were all quite low energy (a few MeV at most). This has implication to how they are allowed to interact.

In general the interaction allowed to solar neutrinos in a heavy water detector come in three types \begin{align*} \nu_e + d &\longrightarrow 2p + e \tag{Charged-current} \\ \nu_x + d &\longrightarrow p + n + \nu_x \tag{Neutral-current} \\ \nu_x + e &\longrightarrow \nu_x + e \tag{Elastic electron scattering} \;. \end{align*} Crucially the charged-current reaction only occurs for electron neutrinos while the other reactions are possible with all flavors.

SNO was not sensitive to the difference between fluxuations to mu-type or tau-type neutrinos. Just how many are not detected as electron type, because there is insufficient energy for for reactions like $$\nu_l + d \to 2p + l \;,$$ for $l$ a heavy lepton ($\mu$ or $\tau$).

This means SNO measured both the flux of electron neutrinos and the total flux of solar neutrinos at one time. This gives us $P_{e\mu} + P_{e\tau}$ or equivalently $P_{ee}$.

The data has been analyzed for mixing parameters, but SNOs core accomplishmnet was unambiguously resolving the solar neutrino deficit issue.