Matter effects (MSW)

A neutrino crossing the Sun is not travelling through empty space. It ploughs through a plasma whose electron density near the core reaches a hundred moles per cubic centimetre, and every electron it passes forward-scatters it coherently, without deflection or energy loss, exactly as an optical medium slows a light wave's crests. The result is a refractive index for neutrinos, and crucially it is not the same for every flavour. All three active flavours scatter through the neutral current (a flavour-blind $Z^0$ exchange), but only the electron neutrino can also exchange a charged $W$ boson with the ambient electrons, because ordinary matter contains electrons and no muons or taus. That single extra amplitude gives $\nu_e$ an additional potential $V_{\rm CC}=\sqrt{2}\,G_F N_e$ the other flavours do not feel. This flavour-dependent potential rewrites the oscillation problem. In vacuum the mixing angle is fixed; in matter it becomes a function of the local electron density, and at one density the two effective mass eigenstates come closest together and the mixing becomes maximal — the Mikheyev–Smirnov–Wolfenstein (MSW) resonance. A neutrino born in the dense core as almost pure $\nu_e$ can ride the heavy matter eigenstate outward, and if the density falls slowly enough it emerges into vacuum as a heavier mass eigenstate that is mostly $\nu_\mu$ and $\nu_\tau$ . This mechanism, not vacuum oscillation alone, turns the Sun's pure $\nu_e$ output into the flavour-mixed flux SNO measured, and it makes the survival probability depend sharply on energy.

Key takeaways

Neutrinos in matter undergo coherent forward scattering, acquiring an effective potential — a neutrino refractive index — with no absorption and no loss of flux.
Only $\nu_e$ feels the extra charged-current potential $V_{\rm CC}=\sqrt{2}\,G_F N_e$ ; the neutral-current potential is common to all flavours and drops out.
This flavour-dependent potential makes the mixing angle density-dependent; at the MSW resonance $2\sqrt{2}\,G_F N_e E=\Delta m^2\cos 2\theta$ the mixing is maximal, even for a small vacuum angle.
A $\nu_e$ created in the dense solar core can convert adiabatically to a heavier mass eigenstate and emerge as mostly $\nu_\mu$ and $\nu_\tau$ .
For the measured LMA parameters the survival probability runs from $P\approx 1-\tfrac12\sin^2 2\theta\approx 0.57$ for low-energy $pp$ neutrinos down to $P\approx\sin^2\theta\approx 0.30$ for high-energy $^8$ B neutrinos.
Key idea: only $\nu_e$ carries the extra charged-current potential of ordinary matter, so the effective mixing angle depends on the local density, passes through a resonance, and converts core $\nu_e$ into other flavours as the neutrino climbs out of the Sun.

The core insight

Only the electron neutrino feels the extra charged-current potential of ordinary matter, so the effective mixing angle depends on the local electron density, passes through a resonance where mixing is maximal, and adiabatically converts a core-born $\nu_e$ into a heavier mass eigenstate that emerges as mostly $\nu_\mu$ and $\nu_\tau$ .

Where this lives today

The MSW effect is now a working tool, not just an explanation of an old anomaly. It is why the solar survival probability is energy-dependent, and Borexino has traced that dependence in a single detector — $pp$ at $P\approx0.57$ and $^8$ B at $P\approx0.30$ , the two ends of the curve above. The Earth is itself matter, so solar neutrinos crossing the planet at night are partially regenerated into $\nu_e$ , a small day–night asymmetry Super-Kamiokande has measured — terrestrial MSW seen directly. Looking forward, the sign-dependence of the potential is the headline tool for the neutrino mass ordering: long-baseline experiments (DUNE, sending neutrinos $1300\ \mathrm{km}$ through the crust) exploit the fact that matter enhances oscillations for neutrinos and suppresses them for antineutrinos, or vice versa depending on the ordering. In a core-collapse supernova, neutrinos cross both the solar and atmospheric MSW resonances plus collective self-induced conversions — a frontier the next galactic supernova will test. The index Wolfenstein wrote down has become a precision instrument for the whole neutrino sector.

The physical picture

A neutrino in matter behaves like light in glass. Each scattering centre it passes re-emits a coherent forward wavelet slightly out of phase, and the sum shifts the crests — a phase velocity differing from $c$ . The scattering is elastic and forward, so nothing is absorbed: what changes is the phase each flavour accumulates, and because the two accumulate it at different rates their interference — the oscillation — is altered. The first figure shows why they differ: only $\nu_e$ has a charged-current channel.

Because $V_{\rm CC}$ enters only the $\nu_e$ diagonal, it competes with the vacuum term $\Delta m^2\cos 2\theta/2E$ . As a neutrino moves from the dense core to vacuum the density sweeps from large to zero and the mixing angle sweeps with it; at one density the diagonal splitting vanishes and the two mass eigenstates approach as closely as the off-diagonal mixing allows — an avoided crossing. The second figure is the punchline a solar-neutrino physicist actually plots: survival probability versus energy, flat at the vacuum value for low-energy neutrinos and dropping through the MSW transition to the matter value for the high-energy $^8$ B flux.

Figure

Solar electron-neutrino survival probability versus energy for the measured LMA parameters Δm² equals 7.5 times ten to the minus five eV squared and sin squared θ equals 0.31. Low-energy pp neutrinos sit on the vacuum-averaged plateau near 0.57; high-energy boron-8 neutrinos undergo adiabatic MSW conversion toward sin squared θ near 0.30

The core equations

The whole effect is contained in a Hamiltonian built from one number: the potential a flavour feels in matter. The charged-current potential is the extra energy an electron neutrino picks up from forward scattering off the electron background, proportional to the electron density and the Fermi constant.

V_{\rm CC}=\sqrt{2}\,G_F N_e\;\approx\;7.63\times 10^{-14}\,\mathrm{eV}\;\cdot\;Y_e\,\frac{\rho}{\mathrm{g\,cm^{-3}}} \qquad (1)

where: $G_F=1.166\times10^{-5}\,\mathrm{GeV^{-2}}$ — Fermi constant; $N_e$ — electron number density; $\rho$ — mass density; $Y_e$ — electrons per nucleon ( $\approx 0.5$ – $1$ ). The potential is positive for $\nu_e$ and reverses sign for $\bar\nu_e$ ; the neutral-current potential $V_{\rm NC}=-G_F N_n/\sqrt2$ is identical for all flavours and cancels from any flavour difference. In the two-flavour subspace relevant to solar neutrinos ( $\nu_e$ – $\nu_x$ with $\Delta m^2=\Delta m^2_{21}$ , $\theta=\theta_{12}$ ), the evolution is governed by an effective flavour-basis Hamiltonian: the vacuum oscillation term plus the matter potential on the $\nu_e$ entry only.

H=\frac{\Delta m^2}{4E}\begin{pmatrix}-\cos 2\theta & \sin 2\theta\\ \sin 2\theta & \cos 2\theta\end{pmatrix}+\begin{pmatrix}V_{\rm CC} & 0\\ 0 & 0\end{pmatrix} \qquad (2)

where: $E$ — neutrino energy; $\theta$ — vacuum mixing angle; $\Delta m^2$ — mass-squared splitting. Only the $(1,1)$ element carries $V_{\rm CC}$ , and that asymmetry makes the diagonalization density-dependent. A term proportional to the identity has been dropped, contributing only a common phase. Diagonalizing $H$ gives an effective mixing angle $\theta_m$ that depends on density through the dimensionless matter parameter $A\equiv 2\sqrt2\,G_F N_e E=2E\,V_{\rm CC}$ — the mixing is no longer fixed by the vacuum angle alone.

\sin^2 2\theta_m=\frac{\sin^2 2\theta}{\left(\cos 2\theta-\dfrac{A}{\Delta m^2}\right)^{2}+\sin^2 2\theta} \qquad (3)

where: $A/\Delta m^2$ — ratio of matter to vacuum energy scales. As $N_e\to 0$ the denominator returns to unity and $\theta_m\to\theta$ ; as $N_e$ grows the first bracket grows and $\theta_m\to 90^\circ$ ; in between there is a maximum. That maximum is the MSW resonance: the density at which the diagonal splitting vanishes, so the mixing is maximal ( $\sin^2 2\theta_m=1$ ) regardless of how small $\theta$ is. The physical gap between the two mass eigenstates, the effective splitting $\Delta m^2_m$ , is smallest there.

A_{\rm res}=\Delta m^2\cos 2\theta\quad\Longleftrightarrow\quad 2\sqrt2\,G_F N_e E=\Delta m^2\cos 2\theta,\qquad \Delta m^2_m=\Delta m^2\sqrt{\left(\cos 2\theta-\frac{A}{\Delta m^2}\right)^2+\sin^2 2\theta} \qquad (4)

where: at resonance $\Delta m^2_m=\Delta m^2\sin 2\theta$ , its minimum. The resonance exists only for the neutrino sign of $A$ ; antineutrinos, whose $A$ has the opposite sign, never satisfy it — matter breaks the neutrino–antineutrino symmetry. Finally, for a neutrino crossing a smoothly varying density, the emergent survival probability is Parke's formula, which packages the whole trajectory into the production-point mixing angle and a single level-crossing probability $P_c$ .

P(\nu_e\to\nu_e)=\frac12+\left(\frac12-P_c\right)\cos 2\theta_m^{0}\,\cos 2\theta \;\xrightarrow[\text{adiabatic}]{P_c\to 0}\;\frac12+\frac12\cos 2\theta_m^{0}\cos 2\theta \qquad (5)

where: $\theta_m^{0}$ — effective mixing angle at production; $P_c$ — probability of jumping between mass eigenstates at resonance (zero in the adiabatic limit). For a neutrino born far above resonance, $\cos 2\theta_m^0\to-1$ and $P\to\sin^2\theta$ .

Derivation AThe matter potential from coherent forward scattering

The goal is to show where $V_{\rm CC}=\sqrt2\,G_F N_e$ comes from and why it attaches to the electron neutrino alone. An electron neutrino moving through a bath of electrons picks up an effective potential exactly as a photon acquires a refractive index in glass: by summing the coherent forward-scattering amplitude off every scatterer. We start from the low-energy charged-current interaction, Fierz-rearrange it into a $\nu_e$ current times an electron current, then average the electron current over an isotropic, unpolarized electron gas — which leaves only its number density.

Step 1. Write the effective four-fermion charged-current interaction for $\nu_e$ and $e$ : after integrating out the heavy $W$ , $\mathcal{L}_{\rm CC}=-\frac{G_F}{\sqrt2}\,[\bar e\,\gamma^\mu(1-\gamma_5)\nu_e]\,[\bar\nu_e\,\gamma_\mu(1-\gamma_5)e]$ . Why: at $E\ll M_W$ the $W$ propagator collapses to the point-like Fermi coupling, and only $\nu_e$ appears because matter supplies electrons, not muons or taus.

Step 2. Apply a Fierz transformation to bring the two neutrino fields together: $\mathcal{L}_{\rm CC}=-\frac{G_F}{\sqrt2}\,[\bar\nu_e\,\gamma^\mu(1-\gamma_5)\nu_e]\,[\bar e\,\gamma_\mu(1-\gamma_5)e]$ . Why: this factorizes the interaction into a neutrino current the $\nu_e$ carries and an electron current sourced by the background, which is what we average.

Step 3. Average the electron current over the medium. For a gas of electrons at rest, isotropic and unpolarized, the spatial components of the vector current and all of the axial current average to zero; only the time component survives, $\langle\bar e\,\gamma^0 e\rangle=N_e$ , the number density. Why: a static, unpolarized, isotropic background has no net momentum or spin, so it can only present its density — which is why the effect depends on $N_e$ alone.

Step 4. Read off the potential. The surviving term acts on the neutrino as $\mathcal{L}\to -V_{\rm CC}\,\bar\nu_e\gamma^0\nu_e$ with $V_{\rm CC}=\sqrt2\,G_F N_e$ , an addition to the $\nu_e$ energy. The identical calculation with $Z^0$ exchange gives $V_{\rm NC}=-\frac{1}{\sqrt2}G_F N_n$ for every active flavour, so it cancels in $V(\nu_e)-V(\nu_x)$ . Why: the neutral current is flavour-universal and contributes only an overall phase; all the flavour physics of the medium lives in the charged-current piece felt by $\nu_e$ alone. *Verified: $V_{\rm CC}=\sqrt2\,G_F N_e$ ; numerically $7.63\times10^{-14}\,\mathrm{eV}\times Y_e(\rho/\mathrm{g\,cm^{-3}})$ ; solar core $\rho\approx150$ , $Y_e\approx0.67$ gives $V_{\rm CC}\approx7.7\times10^{-12}\,\mathrm{eV}$ .*

Go deeper · Derivation B · 2 minThe effective mixing angle and the MSW resonance

Derivation BThe effective mixing angle and the MSW resonance

The goal is to diagonalize the matter Hamiltonian (2) and extract the effective mixing angle (3), the resonance condition (4), and the fact that at resonance the mixing is maximal for any vacuum angle — what lets MSW convert flavours efficiently even when $\theta$ is small. We weigh the diagonal splitting of the two flavour states against the off-diagonal mixing, and identify the density where the potential cancels that splitting as an avoided crossing.

Step 1. Subtract a multiple of the identity from (2) so the Hamiltonian is traceless, giving $H\propto\begin{pmatrix}-(\cos2\theta-A/\Delta m^2) & \sin2\theta\\ \sin2\theta & +(\cos2\theta-A/\Delta m^2)\end{pmatrix}$ in units of $\Delta m^2/4E$ , where $A=2\sqrt2\,G_F N_e E$ . Why: an identity piece shifts both eigenvalues equally and cannot affect the relative phase that drives oscillations, so removing it exposes a single $2\times2$ matrix with diagonal $\pm d$ and off-diagonal $\sin2\theta$ .

Step 2. For a symmetric $2\times2$ matrix with diagonal $\mp d$ and off-diagonal $s$ , the mixing angle satisfies $\tan 2\theta_m=s/d=\dfrac{\sin2\theta}{\cos2\theta-A/\Delta m^2}$ , from which $\sin^2 2\theta_m$ takes the form of equation (3). Why: the standard eigenvalue rotation shows $\theta_m$ is set by the competition between the density-driven diagonal $A/\Delta m^2$ and the fixed off-diagonal $\sin2\theta$ .

Step 3. The denominator of (3) is minimized when $\cos2\theta-A/\Delta m^2=0$ , i.e. $A=\Delta m^2\cos2\theta$ ; there $\sin^2 2\theta_m=1$ exactly. This is the MSW resonance, and the mixing is maximal no matter how small $\theta$ is. Why: at this density the two flavour states are diagonally degenerate, so an arbitrarily weak off-diagonal coupling fully mixes them — a tiny vacuum angle becomes complete mixing at one layer.

Step 4. The eigenvalue gap is $\Delta m^2_m=\Delta m^2\sqrt{(\cos2\theta-A/\Delta m^2)^2+\sin^2 2\theta}$ , which bottoms out at $\Delta m^2\sin2\theta$ on resonance — an avoided crossing whose minimum separation is set by the vacuum mixing. Since $A$ flips sign for antineutrinos, only one of $\nu$ or $\bar\nu$ can hit resonance for a given mass ordering. Why: the closest approach of the two levels is where a neutrino is most likely to hop, so this gap decides whether the crossing is adiabatic, and its sign-dependence is how matter tells neutrinos from antineutrinos. *Verified: $\sin^2 2\theta_m$ maximal at $A=\Delta m^2\cos2\theta$ ; $\Delta m^2_m\to\Delta m^2\sin2\theta$ there; $\theta_m\to90^\circ$ as $A\to\infty$ and $\theta_m\to\theta$ as $A\to0$ .*

Go deeper · Derivation C · 2 minAdiabatic conversion and the solar survival probability

Derivation CAdiabatic conversion and the solar survival probability

The goal is to explain why a $\nu_e$ born in the solar core emerges as mostly $\nu_\mu$ and $\nu_\tau$ , and to derive the survival probability $P\to\sin^2\theta$ the high-energy data confirm. The key is the adiabatic theorem: if the density changes slowly compared with the oscillation phase, a neutrino created in an instantaneous mass eigenstate stays in it as the density, and hence the state's flavour content, evolves. We track the heavy eigenstate from core to vacuum.

Step 1. At the production point in the core, $A\gg\Delta m^2\cos2\theta$ , so $\theta_m^0\to90^\circ$ and the two matter eigenstates are essentially $\nu_{2m}\approx\nu_e$ and $\nu_{1m}\approx\nu_x$ . A neutrino created as $\nu_e$ is therefore created as almost pure heavy eigenstate $\nu_{2m}$ . Why: deep in matter $V_{\rm CC}$ dominates the Hamiltonian, so the flavour and matter-eigenstate bases nearly coincide, with $\nu_e$ sitting in the heavier state.

Step 2. Invoke adiabaticity. The solar density profile is smooth and the resonance layer is wide, so the level gap $\Delta m^2_m$ never closes fast enough for a jump: the crossing probability $P_c=\exp[-\tfrac{\pi}{2}\gamma]$ with adiabaticity parameter $\gamma=\dfrac{\Delta m^2\sin^2 2\theta}{2E\cos2\theta}\,\big|\,d\ln N_e/dr\,\big|^{-1}\gg1$ , so $P_c\approx0$ . Why: with $\gamma\gg1$ the neutrino has many oscillation lengths to adjust within the resonance layer, so it follows the heavy eigenstate continuously rather than hopping — the adiabatic condition.

Step 3. Follow $\nu_{2m}$ outward. As $N_e\to0$ it connects continuously to the vacuum eigenstate $\nu_2=\sin\theta\,\nu_e+\cos\theta\,\nu_x$ . The neutrino thus arrives at Earth as $\nu_2$ , and the probability of detecting it as $\nu_e$ is $|\langle\nu_e|\nu_2\rangle|^2=\sin^2\theta$ . Why: adiabatic evolution conserves the eigenstate label, so a state that began as the heavy matter state ends as the heavy vacuum state $\nu_2$ , whose $\nu_e$ fraction is only $\sin^2\theta$ — genuine conversion.

Step 4. Assemble Parke's formula (5): averaging over the fast phase gives $P=\tfrac12+(\tfrac12-P_c)\cos2\theta_m^0\cos2\theta$ ; with $P_c\approx0$ and $\cos2\theta_m^0\approx-1$ (deep-matter production) this is $\sin^2\theta$ , while below resonance $\cos2\theta_m^0\approx\cos2\theta$ gives $P\to\tfrac12+\tfrac12\cos^2 2\theta=1-\tfrac12\sin^2 2\theta$ . Why: one formula smoothly interpolates between the vacuum-averaged plateau at low energy and the matter-converted plateau at high energy, reproducing the curve plotted above. *Verified: adiabatic $P=\sin^2\theta=0.303$ for $^8$ B; $P=1-\tfrac12\sin^2 2\theta=0.577$ for $pp$ ; crossover near the core resonance energy $\approx2\,\mathrm{MeV}$ .*

Limiting cases

Case	Result	Meaning
Vacuum, $N_e\to0$	$\theta_m\to\theta$ , $P=1-\tfrac12\sin^2 2\theta$	ordinary vacuum oscillation recovered
Low density / low energy, $A\ll\Delta m^2\cos2\theta$	$\theta_m\approx\theta$	matter negligible; the $pp$ regime, $P\approx0.57$
MSW resonance, $A=\Delta m^2\cos2\theta$	$\sin^2 2\theta_m=1$ , $\Delta m^2_m=\Delta m^2\sin2\theta$	maximal mixing even for tiny vacuum $\theta$
High density, $A\gg\Delta m^2$	$\theta_m\to90^\circ$ , $\nu_{2m}\approx\nu_e$	$\nu_e$ is the heavy matter eigenstate; the core
Adiabatic, born above resonance	$P\to\sin^2\theta\approx0.30$	the $^8$ B regime — full MSW conversion
Antineutrino / opposite ordering	$A\to-A$ , no resonance	matter distinguishes $\nu$ from $\bar\nu$ and the two orderings

Common trapThe solar deficit is neutrinos being absorbed or scattered away in the matter of the Sun.

It is not. Matter effects are coherent forward scattering — the physics that gives light a refractive index — elastic and loss-free: no neutrino is removed from the beam, and the flux summed over all flavours is exactly conserved, as SNO's flavour-blind neutral-current channel confirmed to the percent level. What changes is the phase velocity of each flavour, not the number of neutrinos. Fewer $\nu_e$ arrive because they have converted into $\nu_\mu$ and $\nu_\tau$ , which most detectors are blind to — the flux was redistributed, not attenuated. An absorption cross-section is the wrong tool; the right objects are a density-dependent mixing angle and a flavour-dependent refractive index.

Worked examples

1 · The matter potential, resonance density and resonance energy in the Sun

Take the solar core: $\rho\approx150\ \mathrm{g\,cm^{-3}}$ , electron fraction $Y_e\approx0.67$ (hydrogen-depleted, roughly two-thirds helium by mass), $\Delta m^2=7.5\times10^{-5}\ \mathrm{eV^2}$ , $\theta_{12}=33.4^\circ$ . Find the charged-current potential, the energy resonant at the core density, and the density resonant for a $10\ \mathrm{MeV}$ $^8$ B neutrino.

Step 1. Electron density and potential. $N_e=\rho Y_e N_A=150\times0.67\times6.022\times10^{23}=\mathbf{6.1\times10^{25}\ cm^{-3}}$ (about $100$ moles per cm³). From (1), $V_{\rm CC}=7.63\times10^{-14}\times0.67\times150=\mathbf{7.7\times10^{-12}\ eV}$ — this tiny energy drives the whole effect.

Step 2. Resonance energy at the core density. Condition (4) gives $E_{\rm res}=\dfrac{\Delta m^2\cos2\theta}{2V_{\rm CC}}$ . With $\cos2\theta=\cos66.8^\circ=0.394$ the numerator is $7.5\times10^{-5}\times0.394=2.96\times10^{-5}\ \mathrm{eV^2}$ , so $E_{\rm res}=\dfrac{2.96\times10^{-5}}{2\times7.7\times10^{-12}}=\mathbf{1.9\ MeV}$ . Neutrinos with $E>E_{\rm res}$ are produced above resonance and convert strongly; those below (the $pp$ flux) do not.

Step 3. Resonance density for a $10\ \mathrm{MeV}$ neutrino. Solve for the density instead: $V_{\rm res}=\dfrac{\Delta m^2\cos2\theta}{2E}=\dfrac{2.96\times10^{-5}}{2\times10^{7}}=1.48\times10^{-12}\ \mathrm{eV}$ , hence $\rho_{\rm res}Y_e=V_{\rm res}/(7.63\times10^{-14})=19.4$ and $\rho_{\rm res}\approx\mathbf{29\ g\,cm^{-3}}$ . Since the core is far denser ( $150\ \mathrm{g\,cm^{-3}}$ ) and the surface far thinner, every $10\ \mathrm{MeV}$ neutrino crosses a layer at $\approx29\ \mathrm{g\,cm^{-3}}$ where it is exactly on resonance. $V_{\rm CC}\approx7.7\times10^{-12}\ \mathrm{eV}$ , $E_{\rm res}\approx1.9\ \mathrm{MeV}$ at the core, $\rho_{\rm res}\approx29\ \mathrm{g\,cm^{-3}}$ for $10\ \mathrm{MeV}$ .

2 · Survival probabilities of $pp$ versus $^8$ B neutrinos

Using the adiabatic Parke formula (5) with $P_c=0$ and the LMA parameters above, compute $P(\nu_e\to\nu_e)$ for a $pp$ neutrino ( $E=0.3\ \mathrm{MeV}$ ) and a $^8$ B neutrino ( $E=10\ \mathrm{MeV}$ ), both produced at the core.

Step 1. Matter parameter at production, $a_0=A/\Delta m^2=2E\,V_{\rm CC}/\Delta m^2$ with $V_{\rm CC}=7.7\times10^{-12}\ \mathrm{eV}$ . For $pp$ : $a_0=\mathbf{0.061}$ . For $^8$ B: $a_0=\mathbf{2.05}$ . The $pp$ neutrino is well below resonance ( $a_0<\cos2\theta=0.394$ ); the $^8$ B neutrino well above.

Step 2. Mixing angle at production, $\cos2\theta_m^0=\dfrac{\cos2\theta-a_0}{\sqrt{(\cos2\theta-a_0)^2+\sin^2 2\theta}}$ with $\sin^2 2\theta=0.845$ . For $pp$ : $\cos2\theta_m^0=\dfrac{0.333}{0.978}=0.340$ . For $^8$ B: $\cos2\theta_m^0=\dfrac{-1.656}{1.877}=-0.882$ — the sign has flipped, the hallmark of production above resonance.

Step 3. Survival probability. $P=\tfrac12+\tfrac12\cos2\theta_m^0\cos2\theta$ with $\cos2\theta=0.394$ . For $pp$ : $P=0.5+0.5(0.340)(0.394)=\mathbf{0.57}$ — the vacuum-averaged plateau, matching Borexino's $pp$ result. For $^8$ B: $P=0.5+0.5(-0.882)(0.394)=\mathbf{0.33}$ — near the deep-matter asymptote $\sin^2\theta=0.303$ , matching SNO's charged-to-neutral-current ratio. The factor-of-two difference between the two fluxes is the direct fingerprint of MSW. $P_{pp}\approx0.57$ , $P_{^8\mathrm B}\approx0.33$ .

Physics of the portrait

Matter effects are named for three people: Lincoln Wolfenstein, who saw that matter gives neutrinos a potential, and Stanislav Mikheyev and Alexei Smirnov, who saw that a varying potential could resonantly convert them.

Origins · 1 minLincoln Wolfenstein and how this was discovered

Lincoln Wolfenstein

1922–2015 · neutrino oscillations in matter

Lincoln Wolfenstein worked on the weak interaction at Carnegie Mellon University. In 1978 he pointed out what everyone had overlooked: a neutrino in matter is not in vacuum, and coherent forward scattering off electrons, protons and neutrons gives it an effective potential — different for $\nu_e$ , which alone has a charged-current channel with the ambient electrons. He wrote down the modified evolution equation and showed the effective mixing angle becomes density-dependent, though he did not emphasize the resonance. That step came in 1985–86 from Stanislav Mikheyev and Alexei Smirnov in Moscow, who saw that a solar neutrino climbing out through a falling density can pass through a layer of maximal mixing and be efficiently converted, even from a small vacuum angle. Hans Bethe gave a luminous adiabatic-crossing account in 1986, and Stephen Parke supplied the general non-adiabatic survival formula the same year. Wolfenstein is equally remembered for the Wolfenstein parametrization of the CKM matrix (1983). The MSW effect he began turned the solar neutrino problem from a puzzle about the Sun into a measurement of neutrino mass and mixing — vindicated when SNO and KamLAND selected the large-mixing-angle solution he and his successors had mapped.

History

1978

— Wolfenstein shows neutrinos acquire a flavour-dependent potential in matter

1985

— Mikheyev and Smirnov find resonant conversion in a varying density

1986

— Bethe gives the adiabatic level-crossing account of the solar effect

1986

— Parke derives the non-adiabatic survival formula with the level-crossing probability

2002

— SNO's neutral-current result confirms flavour conversion at full solar flux

2002

— KamLAND's reactor result selects the large-mixing-angle MSW solution

Going further

Texts. Thomson, Modern Particle Physics, §13.3 (neutrino oscillations) and §13.2 (solar neutrinos) · Griffiths, Introduction to Elementary Particles, 2nd ed., §11 (neutrino oscillations) · Giunti & Kim, Fundamentals of Neutrino Physics and Astrophysics, Ch. 9 (neutrino oscillations in matter) — the definitive treatment · Bilenky, Introduction to the Physics of Massive and Mixed Neutrinos.

Provenance. The two-flavour vacuum oscillation formalism ( $\theta$ , $\Delta m^2$ , $L/E$ ) is recalled by name from aq-pp-two-flavour-oscillations; the neutral-current interaction that supplies the flavour-blind $V_{\rm NC}$ is owned by aq-pp-neutral-current; the solar $\nu_e$ flux and the deficit this explains come from aq-pp-solar-neutrino-problem. A verifier should confirm: (1) $V_{\rm CC}=\sqrt2\,G_F N_e=7.63\times10^{-14}\,\mathrm{eV}\,Y_e(\rho/\mathrm{g\,cm^{-3}})$ ; (2) the resonance condition $A=\Delta m^2\cos2\theta$ and maximal $\sin^2 2\theta_m$ there; (3) Parke's formula reducing to $\sin^2\theta$ (adiabatic, deep matter) and $1-\tfrac12\sin^2 2\theta$ (below resonance); (4) the LMA numbers $P_{pp}\approx0.57$ , $P_{^8\mathrm B}\approx0.30$ , $E_{\rm res}\approx2\,\mathrm{MeV}$ at the core.

Prerequisites. Neutrino Flavours · The Solar Neutrino Problem · Two-Flavour Oscillations · The Neutral Current. Leads to. Three-Flavour PMNS Mixing · Neutrino Mass Mechanisms · Long-Baseline Experiments.

Ask the tutor: Ask me to work through the level-crossing diagram, the adiabaticity condition, or how DUNE turns the sign of the matter potential into a measurement of the mass ordering.

Why does only the electron neutrino feel the extra matter potential?

Because ordinary matter is made of electrons, protons and neutrons — no muons or taus. The neutral current ( $Z^0$ ) is flavour-blind, so all three flavours scatter equally through it. But the charged current ( $W$ ) requires the neutrino to convert into its partner charged lepton and back: a $\nu_e$ can do this off an ambient electron, whereas a $\nu_\mu$ would need an ambient muon, of which there are none. So $\nu_e$ alone gets the extra potential $\sqrt2\,G_F N_e$ , and that single asymmetry is the entire origin of matter effects.

Isn't this just a refractive index for neutrinos?

Yes — exactly. A photon in glass forward-scatters off bound electrons and picks up a refractive index; a neutrino forward-scatters off electrons and nucleons and picks up an effective potential, the same statement. The difference is that the neutrino's "index" is flavour-dependent: $\nu_e$ has one value and $\nu_{\mu,\tau}$ another, from the charged-current channel. That difference between flavours, not the index itself, alters the oscillation. The physics — elastic, loss-free, a phase not an absorption — is identical.

How is the MSW effect different from ordinary vacuum oscillation?

Vacuum oscillation is a fixed interference pattern: constant mixing angle, flavour composition wobbling with a wavelength set by $\Delta m^2$ and $E$ , averaging to $1-\tfrac12\sin^2 2\theta$ . MSW is a conversion: because the mixing angle changes with density, a neutrino created in one flavour in dense matter is smoothly carried into a different mass eigenstate as it exits — not oscillating around a mean but genuinely transformed. For solar neutrinos both operate, but the high-energy deficit is dominated by adiabatic MSW, which is why $P$ falls to $\sin^2\theta\approx0.30$ rather than staying near $0.57$ .

Why do pp and boron-8 neutrinos have such different survival probabilities?

Because they lie on opposite sides of the MSW resonance, which sits at about $2\,\mathrm{MeV}$ at the solar core. The $pp$ neutrinos ( $E<0.42\,\mathrm{MeV}$ ) are produced below resonance, where matter barely shifts the mixing angle, so they undergo vacuum-averaged oscillation with $P\approx0.57$ . The $^8$ B neutrinos (up to $15\,\mathrm{MeV}$ ) are produced well above resonance, created essentially as the heavy matter eigenstate and adiabatically converted, emerging with $P\approx\sin^2\theta\approx0.30$ . Measuring both ends and the transition between them maps the effect directly, which is why Borexino's single-detector spectrum is such a clean confirmation.

Did the MSW effect require a large or a small mixing angle?

Historically this was the whole drama. The resonance makes mixing maximal at one density regardless of the vacuum angle, so MSW was originally celebrated for the small-mixing-angle (SMA) solution, where a tiny vacuum $\theta$ could still give a large deficit. For two decades several MSW solutions competed with a pure vacuum-oscillation solution to fit the solar data. The tie broke in 2002: KamLAND's reactor antineutrinos over $\sim180\,\mathrm{km}$ saw disappearance consistent only with the large-mixing-angle (LMA) solution, $\sin^2\theta_{12}\approx0.31$ . The mechanism works for any angle, but nature chose the large one.

How do we know it is genuinely a matter effect and not just vacuum oscillation?

Three independent lines. First, the energy dependence: pure vacuum oscillation over one AU would average to a nearly energy-flat $P$ , whereas the data show the steep MSW upturn from $0.57$ to $0.30$ that only a density-dependent mixing angle predicts. Second, the day–night asymmetry: solar neutrinos crossing the Earth at night are partially regenerated into $\nu_e$ , a small effect Super-Kamiokande has measured — impossible without matter. Third, the sign: matter distinguishes $\nu$ from $\bar\nu$ , and combining solar with KamLAND reactor data fixes the sign of $\Delta m^2_{21}$ , which vacuum oscillation cannot. Together these leave no room for a matter-free explanation.

Matter effects (MSW)

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The matter potential, resonance density and resonance energy in the Sun

2 · Survival probabilities of $pp$ versus $^8$ B neutrinos

Physics of the portrait

Going further

Before this

Where this leads

Matter effects (MSW)

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · The matter potential, resonance density and resonance energy in the Sun

2 · Survival probabilities of pppppp versus 8^88B neutrinos

Physics of the portrait

Going further

Before this

Where this leads

2 · Survival probabilities of $pp$ versus $^8$ B neutrinos