# Measurement of Solar and other neutrino Mixing angles

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).

• Few of comments. First, Super-K got the first clear evidence of oscillation on atmospheric neutrinos. SNO came later. Second, most early experiments were (at least initially) analyzed in a 2-flavor framework (in effect assuming that one term would dominate the sum). – dmckee --- ex-moderator kitten Apr 14 '17 at 12:21

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 $P_{ee}$.
• SNO cannot distinguish between $\nu_\mu$ and $\nu_\tau$ but can measure the combined flux of $\nu_\mu$ and $\nu_\tau$. Do I get that right? Also, what does the subscript $x$ in $\nu_x$ denote? I guess, it stands for either of $\mu$ and $\tau$. Right? – SRS Jul 14 '18 at 16:49
• Here $x$ stands for any flavor (including electron-type). – dmckee --- ex-moderator kitten Jul 14 '18 at 16:57