Neutrino Oscillations in Matter: the MSW Effect

L. Wolfenstein observed in 1978 [Wol78] that ``the effect of coherent forward scattering must be taken into account when considering the oscillations of neutrinos travelling through matter''. In fact, he discovered that in general if there exists an interaction through which neutrinos can change flavor (this could be neutrino oscillations, but also a flavor changing neutral current component for example), this flavor change can be enhanced or even only possible if the neutrinos travel through matter. The latter is clear in the case of massless neutrinos but with off-diagonal neutral current interactions (connecting different neutrino flavors): there are no ``oscillations'' in vacuum but in matter neutrinos can change flavor.

In 1984, S.P. Mikheev and A. Yu. Smirnov [Mik85] noticed that for specific oscillation and matter density parameters, this enhancement could develop a resonance behavior.

We will treat here the two-neutrino case for clarity. T.K. Kuo and J. Pantaleone have shown [Kuo87] that in most cases the three-neutrino formalism can be reduced to a two-neutrino-like situation with two resonances instead of one.

Neutrino propagation through matter differentiates neutrino flavors because matter is nearly exclusively composed of first generation leptons and quarks. This singles out the electron neutrino, which can propagate while having charged-current interactions with electrons in addition to the neutral-current interactions. Let us consider the potential energy the electron neutrino acquires due to these charged-current interactions. For the energies we are considering, and supposing the sun is unpolarized and at rest, it can be written as

$$V_{\nu_{e}} = \sqrt{2} G_{F} N_{e}$$ (3.31)

with $N_{e}$ the electron density. In the two-neutrino mixing case^3.1

$$\left(\begin{array}{c} \nu_{e}\\ \nu_{\alpha} \end{array}\right)... ...{array}{cc} cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array}\right)$$ (3.32)

( $\alpha = \mu$ or $\tau$), the evolution of flavor content as a function of propagation can be described by [Gel95]

$$i \frac{d}{dx} \left(\begin{array}{c} \nu_{e}\\ \nu_{\alpha} \en... ...E} {\bf M}^{2}\left(\begin{array}{c} \nu_{e}\\ \nu_{\alpha} \end{array}\right)$$ (3.33)

The matrix ${\bf M}^{2}$ can be written

$${\bf M}^{2} = \frac{1}{2} \left[ R_{\theta} \left( \begin{array}{... ...t{2} G_{F} N_{e} & 0 \\ 0 & - \sqrt{2} G_{F} N_{e} \end{array} \right) \right]$$ (3.34)

where the first term on the right-hand side is the usual vacuum oscillations term (see equation (2.44)) while the second term is the one due to $\nu_{e}- e$ charged-current scattering. Note that we have subtracted a piece proportional to the identity matrix so that ${\bf M}^{2}$ appears in a more symmetric form.

We are now ready to define the matter eigenstates

$$\left(\begin{array}{c} \nu_{1m} \\ \nu_{2m} \end{array}\right) =... ...heta_{m}}^{T} \left(\begin{array}{c} \nu_{e}\\ \nu_{\alpha} \end{array}\right)$$ (3.35)

which we obtain by diagonalizing ${\bf M}^{2}$:

$$R_{\theta_{m}}^{T} {\bf M}^{2} R_{\theta_{m}} = \frac{1}{2} \left( \begin{array}{cc} -\Delta_{m} & 0 \\ 0 & \Delta_{m} \end{array} \right)$$ (3.36)

In this equation $\Delta_{m} = \Delta m^{2} \sqrt{(a-cos2\theta)^{2} + sin^{2}2\theta}$ with $a = 2 \sqrt {2} E G_{F} N_{e}/\Delta m^{2}$. The newly defined matter mixing angle $\theta_{m}$ appearing in $R_{\theta_{m}}$ is such that

$$sin^{2}2\theta_{m} = \frac{sin^{2}2\theta}{(cos2\theta - a)^{2} + sin^{2}2\theta}$$ (3.37)

and we see immediately that even if the vacuum mixing angle is very small, we will have maximal mixing in matter if the condition $a = cos2\theta$ is satisfied.

Figure 3.7: $sin^{2}2\theta _{m}$ as a function of $a = 2 \sqrt {2} E G_{F} N_{e}/\Delta m^{2}$ for three different values of the vacuum mixing angle: $sin^{2}2\theta = 4 \ 10^{-2}$ (solid line), $sin^{2}2\theta = 1 \ 10^{-2}$ (dashed line) and $sin^{2}2\theta = 4 \ 10^{-3}$ (dotted line).

Figure 3.7 shows the value of $sin^{2}2\theta _{m}$ as a function of $a$ for three different vacuum mixing angles. The resonance is the so-called MSW effect.

Combining the solar neutrino data with one of the solar models (for the predicted neutrino flux and the evolution of the electron density in the sun), oscillation parameters can be computed taking into account the MSW effect^3.2. Such a calculation has been made for the most recent results and the latest Bahcall-Pinsonneault solar model [Bah95] by N. Hata and P. Langacker [Hat97]. The allowed regions are shown in figure 3.8.

Figure: Oscillation [Hat97] parameter space (shaded areas) allowed by the solar neutrino observations at 95 % C.L. assuming the Bahcall-Pinsonneault 95 Standard Solar Model [Bah95] and taking into account matter effects (MSW). The allowed regions are also shown for the individual experiments: Homestake (dot-dashed line), Kamiokande and Super-Kamiokande combined (solid line), SAGE and GALLEX combined (dashed line), as well as regions excluded by the Kamiokande $^{8}B$ spectrum and day-night observations (dotted lines).