Where the Matter Potential Comes From: Deriving Wolfenstein

The standard textbook derivation of solar neutrino oscillation invokes the Wolfenstein potential $V_{\mathrm{CC}}=\sqrt{2}{G}_F{n}_e$ that electron neutrinos pick up while travelling through matter. The MSW effect then follows: at the resonance density where the matter potential matches the vacuum oscillation phase, the mass eigenstates of the in-matter Hamiltonian rotate maximally, and a flavor neutrino can be converted with high efficiency from one mass eigenstate to another. This is the centerpiece of the solar neutrino problem’s resolution and continues to govern matter effects in atmospheric, accelerator, and supernova neutrinos.

The potential itself is usually presented as a fact to be accepted: it is what makes the math work for the Sun. But the derivation is not difficult, and seeing where the factor of $\sqrt{2}$, the Fermi constant, and the electron number density actually come from is a useful exercise. The answer is coherent forward scattering — a phenomenon familiar from the index of refraction for light in a transparent medium, applied to neutrinos and the weak interaction. This post is about how to extract the matter potential from the underlying Standard-Model weak Lagrangian, why only the electron flavor sees the charged-current term, and what the result implies.

The setup

Consider an electron neutrino propagating through a medium at constant electron density $n_e$. The neutrino interacts weakly with the medium electrons, but at energies relevant to oscillation physics — keV to GeV — the interaction is dominated by forward scattering, in which the final-state momenta of both neutrino and electron are exactly the same as in the initial state. Forward scattering does not produce any observable change in the system: no energy transfer, no recoil. But the amplitude for forward scattering picks up a phase, and that phase shows up in the effective Hamiltonian governing neutrino propagation.

The analogy with light in matter is direct. A photon propagating through a transparent medium does not scatter incoherently very often — the medium is, after all, transparent. But the forward-scattering amplitude builds up coherently and gives the photon an effective index of refraction $n\ne 1$. The neutrino in a weak-interaction medium picks up an analogous “weak index of refraction” — a potential term that shifts its effective energy.

The Fermi interaction at low energies

At the energies typical of solar neutrinos (sub-MeV to tens of MeV), the W boson cannot be produced on-shell. The four-fermion Fermi effective interaction captures the relevant physics: when an electron neutrino and an electron meet, the W-boson exchange is contracted to a point and the interaction Lagrangian becomes

${\mathcal{L}}_{\mathrm{CC}}=-\frac{G_F}{\sqrt{2}}\left[{\bar{\nu }}_e{\gamma }^{\mu }(1-{\gamma }^5)e\right]\left[\bar{e}{\gamma }_{\mu }(1-{\gamma }^5){\nu }_e\right]$

with $G_F$ the Fermi constant and the $(1-{\gamma }^5)$ factors enforcing the V-A structure of the weak interaction. This is the charged-current piece. There is also a neutral-current term proportional to $G_F$ that involves a $Z$-boson exchange and couples all three neutrino flavors equally to all the electrons, protons, and neutrons in the medium.

For oscillation physics what matters is the difference in the effective Hamiltonian between different flavors. The neutral-current term affects all flavors equally and drops out of the flavor-difference physics — it just adds a common phase to all eigenstates and shifts the oscillation phase by a known irrelevant offset. The charged-current term, however, couples specifically to the same-flavor charged lepton in the medium. In ordinary matter the medium contains electrons but no muons or taus, so only the electron-neutrino flavor picks up the charged-current term.

Forward scattering and the index of refraction

The forward-scattering amplitude $f(0)$ for an electron neutrino scattering off a single medium electron at zero momentum transfer can be computed from the Fermi Lagrangian. In the non-relativistic limit for the electron and at zero momentum transfer, the calculation yields

$f(0)=-\frac{G_F}{\sqrt{2}\pi }\cdot E_{\nu }$

with $E_{\nu }$ the neutrino energy. The negative sign is fixed by the convention that an attractive potential has positive $f(0)$.

In a medium of $n_e$ electrons per unit volume, the neutrino accumulates forward-scattering amplitude over its path at a rate proportional to $n_{e}f(0)$. The standard result from coherent scattering theory is that this gives an effective potential

$V_{\mathrm{CC}}=-\frac{2\pi n_e}{E_{\nu }}f(0)=\sqrt{2}{G}_F{n}_e$

This is the Wolfenstein matter potential. The factor of $\sqrt{2}$ is the same one that appears in the Fermi Lagrangian; the factors of $E_{\nu }$ cancel between the forward-scattering amplitude and the index-of-refraction kinematics. The result is independent of the neutrino energy — a curious and important feature.

A quick numerical check: at the centre of the Sun the electron density is approximately $n_e\approx 6\times 10^{25}$ cm${}^{-3}$, and using $G_F\approx 1.166\times 10^{-5}$ GeV${}^{-2}$ gives

$V_{\mathrm{CC}}^{\mathrm{core}}\approx 7.6\times 10^{-12}\text{eV}$

That looks tiny, but the relevant comparison is to the vacuum oscillation Hamiltonian $\Delta m^2/2E$. For solar mass-squared difference $\Delta m_{21}^2\approx 7.5\times 10^{-5}$ eV² and a ${}^8$B neutrino at $E=10$ MeV, the vacuum scale is

$\frac{\Delta m_{21}^2}{2E}\approx 3.7\times 10^{-12}\text{eV}$

The matter potential at the solar core is about twice the vacuum oscillation scale — meaning matter effects are not small in the Sun’s interior, and the MSW resonance condition $V_{\mathrm{CC}}=\Delta m^2\cos 2\theta /2E$ is reached at a sub-core density. This is the central reason MSW works.

Why coherence matters

The Wolfenstein potential’s existence depends on the coherence of forward scattering: amplitudes add over electrons, so the effect grows linearly with $n_e$. Incoherent neutrino-electron scattering, by contrast, would give an effect proportional to $n_e\times \sigma$ where $\sigma$ is the (tiny) total cross-section. The latter is many orders of magnitude smaller for solar neutrinos and does not produce a useful potential.

Coherence is preserved by two conditions: the medium must be transparent (no significant inelastic scattering at the relevant energies), and the neutrino’s de Broglie wavelength must be large compared to the inter-electron spacing. Solar neutrinos easily satisfy both — the medium is essentially perfectly transparent to weak interactions over centimeter scales, and the neutrino wavelength at MeV energies is around $10^{-13}$ m while the inter-electron spacing in the solar interior is about $10^{-10}$ m. The coherent forward scattering builds up over the propagation, and the matter potential is well-defined.

Antineutrinos and the sign

For antineutrinos, the charged-current matter potential flips sign: $V_{\mathrm{CC}}({\bar{\nu }}_e)=-\sqrt{2}{G}_F{n}_e$. The sign change has a clean physical reason: antineutrinos couple to the opposite W chirality, and the forward-scattering diagram involves the CP-conjugated process with an opposite-sign amplitude. The consequence for oscillation is that matter enhances or suppresses neutrino oscillations differently from antineutrino oscillations, breaking the CP symmetry of vacuum mixing. This is what allows long-baseline accelerator experiments to separate the matter-effect contribution from the genuine CP-violating contribution by comparing neutrino and antineutrino running.

Higher-order effects

The leading Wolfenstein potential is dominant for solar, atmospheric, and accelerator-baseline physics, but several higher-order effects are relevant in particular settings.

Neutral-current scattering on the neutron component of matter contributes an additional flavor-blind term proportional to $G_F{n}_n/\sqrt{2}$. In neutron-rich environments such as the proto-neutron star of a core-collapse supernova this can matter for the detailed dynamics, although it does not affect flavor differences directly.

Neutrino-neutrino refraction — the term proportional to $V_{\nu \nu }\sim G_F{n}_{\nu }$ that drives collective oscillations in supernovae — adds a nonlinear self-interaction term to the effective Hamiltonian. The collective-oscillations post discussed this in detail.

Loop corrections introduce small $\alpha$-suppressed energy- and density-dependent corrections to the leading potential. These are too small to matter for solar or accelerator oscillation analyses but begin to be relevant for proposed precision measurements at $10^{-4}$ accuracy.

For most purposes, the leading Wolfenstein potential $V_{\mathrm{CC}}=\sqrt{2}{G}_F{n}_e$ is the only one that needs to be remembered.

Summary

The Wolfenstein matter potential arises from coherent forward scattering of electron neutrinos off the electrons in a medium, mediated by W-boson exchange at low energy. The amplitudes from each electron add coherently across the medium, producing an effective potential $V_{\mathrm{CC}}=\sqrt{2}{G}_F{n}_e$ that is flavor-discriminating because only the electron flavor has a same-flavor charged lepton available in ordinary matter. Numerically the potential at the solar core is about twice the vacuum oscillation scale for ${}^8$B neutrinos, making matter effects dominant inside the Sun and driving the MSW resonance that resolves the solar neutrino problem. For antineutrinos the potential flips sign, providing the lever that separates matter-induced asymmetries from CP-violating asymmetries in long-baseline experiments. The derivation, while elementary, makes clear why the seemingly tiny weak forward-scattering amplitude turns into a dominant effect inside dense matter — and why it discriminates flavor at all.