Neutrino Oscillations in Vacuum and Matter
Fall School on Interdisciplinary Neutrino Science · Zhuhai, 12 November 2026
Surprise: \(\nu_e\), \(\nu_\mu\), \(\nu_\tau\) have no mass
- In QFT a fundamental particle is a quantum wave: an excitation of a quantum field with a mass, \(E_i=\sqrt{p^2+m_i^2}\).
- There are no \(\nu_e\), \(\nu_\mu\), \(\nu_\tau\) fields with a mass. The massive fields are \(\nu_1\), \(\nu_2\), \(\nu_3\).
- \(\nu_e\), \(\nu_\mu\), \(\nu_\tau\) are the combinations that the weak interaction creates and absorbs. For two of them:
\[ \boxed{ \begin{aligned} |\nu_e\rangle&=\cos\theta\,|\nu_1\rangle+\sin\theta\,|\nu_2\rangle\\ |\nu_\mu\rangle&=-\sin\theta\,|\nu_1\rangle+\cos\theta\,|\nu_2\rangle \end{aligned}} \]
- One state space, two pairs of axes, turned by the mixing angle \(\theta\). For \(\nu_e\) and the slow clock below: \(\theta_{12}\approx34^\circ\).
Here \(\nu_x=\nu_\mu\).
Geoneutrino Example
One source at distance \(L\): \(P_{ee}(L)\). All sources of the Earth closer than \(L\): \(\bar P_{ee}(L)=\int_0^L w(x)\,P_{ee}(x)\,dx\Big/\!\int_0^L w(x)\,dx\).
A volume \(dV\) emitting \(n\) antineutrinos per cm³ per s gives the flux \(n\,dV/4\pi x^2\); around the detector \(dV=x^2dx\,d\Omega\), so \(d\Phi=\frac{n}{4\pi}\,dx\,d\Omega\). Only the solid angle inside a shell \(a<r<b\) is left; \(\beta\) is the angle from the vertical, \(r^2=R^2+x^2-2Rx\cos\beta\):
\[ \frac{d\Phi}{dx}=\frac{n}{2}\,\bigl[\cos\beta(a)-\cos\beta(b)\bigr], \qquad \cos\beta(r)=\frac{R^2+x^2-r^2}{2Rx}\ \ \text{clipped to}\ [-1,1]. \]
- Crust: \(d\Phi/dx\approx n/2\) for \(x<H=35\) km, \(\approx nH/2x\) beyond. A uniform ball: \(\Phi=\frac{nR}{2}\).
- Each shell emits \(n=\rho\,(\ell_{\rm U}c_{\rm U}+\ell_{\rm Th}c_{\rm Th})\), \(\ell_{\rm U}=7.46\times10^7\), \(\ell_{\rm Th}=1.62\times10^7\) ν̄ per kg per s. Crust: 2.7 g/cm³, U 1.3 ppm, Th 5.6 ppm; mantle: U 12 ppb, Th 48 ppb; core: none.
The detector does not know where a neutrino came from; it counts all of them:
\[ S(L)=\sum_{\rm shells}\int_0^L \frac{d\Phi}{dx}\,P_{ee}(x)\,dx, \qquad w(x)=\frac{1}{S_0}\sum_{\rm shells}\frac{d\Phi}{dx}, \qquad S_0:\ \text{the whole Earth without oscillations.} \]
Continental crust 35 km (Rudnick & Gao 2003); mantle: bulk silicate Earth (McDonough & Sun 1995) minus the crust; TNU after Fiorentini et al. 2007. Spectrum: U and Th chains (S. Enomoto) times the IBD cross section (Vogel & Beacom 1999), U : Th = 30.3 : 9.0 TNU. Oscillations: NuFIT 6.1, normal ordering. Earth picture: radii to scale, interior colors schematic; surface: NASA Blue Marble.
Atmospheric Neutrino Example
A \(\nu_\mu\) born in the atmosphere at height \(h\approx15\) km; the zenith angle fixes the path: \(L=15\) km from above, \(437\) km from the horizon, \(12\,757\) km from below.
The path from the birth point at height \(h\) to a detector on the surface, \(R=6371\) km:
\[ L(\theta_z)=\sqrt{(R+h)^2-R^2\sin^2\theta_z}-R\cos\theta_z , \qquad h\approx15\ \text{km}. \]
Three flavors in vacuum, one energy and one direction:
\[ P(\nu_\mu\to\nu_\beta)=\Bigl|\sum_{i=1}^{3}U_{\beta i}\,U^*_{\mu i}\,e^{-i m_i^2L/2E}\Bigr|^2 . \]
- No averaging over \(L\) or \(E\): atmospheric detectors reconstruct the energy and the direction of each event.
- No matter yet: the Earth’s electrons change these curves at a few GeV (slide “Matter turns the axes”, tab Picture).
\(h\approx15\) km: the most probable production height of vertical neutrinos from pion decay (Gaisser & Stanev, Phys. Rev. D 57, 1977 (1998)). NuFIT 6.1 best fit (IC24 + SK-atm), normal ordering: \(\sin^2\theta_{12}=0.3088\), \(\sin^2\theta_{13}=0.02248\), \(\sin^2\theta_{23}=0.470\), \(\delta=212^\circ\), \(\Delta m^2_{21}=7.537\times10^{-5}\) eV², \(\Delta m^2_{31}=2.511\times10^{-3}\) eV². Earth picture: radii to scale (PREM), interior colors schematic; surface: NASA Blue Marble.
Vacuum = Birefringent Crystal
Calcite on graph paper: every line is seen twice. Photo: APN MJM, Wikimedia Commons, CC BY-SA 3.0.
Only \(\nu_e\) feels the electrons
\[ \boxed{\,n_{\nu_e}-1=-\frac{V}{E}<0\,}, \qquad V=\sqrt2\,G_F\,n_e . \]
- Only \(\nu_e\) exchanges a \(W\) with the electrons; the \(Z\) part is the same for all flavors.
- \(V>0\) is a repulsion: \(p=E-V\). The wavelength grows, the phase velocity exceeds \(c\), as for X-rays on free electrons.
- The group velocity stays below \(c\): \(E=\sqrt{p^2+m^2}+V\) gives \(v_g=dE/dp=p/\sqrt{p^2+m^2}<1\).
- \(\bar\nu_e\): \(V<0\), \(n>1\).
\[ n-1=-\frac{V}{E}\begin{pmatrix}1&0\\0&0\end{pmatrix}: \ \text{the axes } \nu_e,\nu_\mu \text{ stay.} \]
The same scattering, written for \(\nu_1,\nu_2\) (\(U_{e1}=\cos\theta\), \(U_{e2}=\sin\theta\)):
\[ n-1=-\frac{V}{E}\begin{pmatrix}\cos^2\theta&\sin\theta\cos\theta\\\sin\theta\cos\theta&\sin^2\theta\end{pmatrix}. \]
- One scattering turns \(\nu_1\) into \(\nu_2\): the amplitude is \(\propto U_{e1}U_{e2}^*=\sin\theta\cos\theta\neq0\).
- Add the vacuum, \(\mathrm{diag}(n_1,n_2)\): the sum is not diagonal.
The crystal turns its axes, \(\nu_{1,2}\to\nu_{1m,2m}\), and its indices change, \(n_{1,2}\to n_{1m,2m}\).
A plane wave \(e^{ikz}\) crosses a slab \(dz\) with \(N\) scatterers per volume, each scattering \(f(\theta)\,e^{ikr}/r\). The first Fresnel zones add up to
\[ e^{ikz}\Bigl[1+\frac{2\pi i}{k}\,N f(0)\,dz\Bigr]=e^{inkz} \quad\Rightarrow\quad n-1=\frac{2\pi N f(0)}{k^2}. \]
Only the forward amplitude \(f(0)\) enters.
\(W\) exchange at \(q^2\ll M_W^2\), after the Fierz rearrangement (coefficient \(+1\) for anticommuting fields):
\[ \mathcal L=-\frac{G_F}{\sqrt2}\,[\bar\nu_e\gamma^\mu(1-\gamma_5)\nu_e]\,[\bar e\gamma_\mu(1-\gamma_5)e]. \]
Forward, left-handed \(\nu_e\), unpolarized \(e^-\) at rest: \(\bar u_\nu\gamma^\mu(1-\gamma_5)u_\nu=4p^\mu\), \(\langle\bar u_e\gamma_\mu(1-\gamma_5)u_e\rangle=2k_\mu\):
\[ \mathcal M_{\rm fwd}=-4\sqrt2\,G_F\,E\,m_e, \qquad f(0)=\frac{\mathcal M_{\rm fwd}}{8\pi m_e}=-\frac{G_F E}{\sqrt2\,\pi}<0 . \]
Born: \(f=-\dfrac{E}{2\pi}\displaystyle\int U\,d^3r\), so \(U=\sqrt2\,G_F\,\delta^3(\mathbf r)\). A vector exchange between like charges: repulsion; \(\bar\nu_e\): \(V\to-V\).
\[ V=N_e\!\int U\,d^3r=-\frac{2\pi N_e f(0)}{E}=\sqrt2\,G_F N_e>0, \qquad n-1=-\frac{\sqrt2\,G_F N_e\,E}{p^2}\simeq-\frac{V}{E}<0 . \]
The role of \(\omega_0\). Light on an oscillator: \(f(0)=r_e\,\omega^2/(\omega_0^2-\omega^2)\); a free electron, \(\omega_0=0\): \(f(0)=-r_e\), \(n<1\) (X-rays). Here the pole is the \(W\): for \(\nu_e e^-\) it sits in the \(u\) channel and is never reached, \(n<1\) at all energies; for \(\bar\nu_e e^-\) it is in the \(s\) channel, at \(E=M_W^2/2m_e\approx6.3\) PeV (Glashow): below it \(n>1\).
R. D. Peccei, hep-ph/9906509, Eqs. (91)–(92); E. Kh. Akhmedov, hep-ph/0001264, Eqs. (84)–(88); L. Wolfenstein, Phys. Rev. D 17, 2369 (1978). \(Z\) exchange is common to all flavors and drops out.
Matter turns the axes of the crystal
After a length \(L\) each state collects the phase \(E\,(1-n)\,L\). In the basis \(\nu_1,\nu_2\):
\[ E\,(1-n)\,L=\underbrace{\varphi_0\,\mathbf{1}}_{\text{same for all}} +\underbrace{\frac{L}{2E}\begin{pmatrix}0&0\\0&\Delta m^2\end{pmatrix}}_{\text{vacuum}} +\underbrace{V L\begin{pmatrix}\cos^2\theta&\sin\theta\cos\theta\\\sin\theta\cos\theta&\sin^2\theta\end{pmatrix}}_{\text{matter}} \]
\(\varphi_0\) is not seen. Take out \(\dfrac{\Delta m^2L}{2E}=2\pi\dfrac{L}{L_{\rm osc}}\):
\[ E\,(1-n)\,L=\varphi_0\,\mathbf{1}+2\pi\,\frac{L}{L_{\rm osc}}\, \boxed{\begin{pmatrix}0&0\\0&1\end{pmatrix}+A\begin{pmatrix}\cos^2\theta&\sin\theta\cos\theta\\\sin\theta\cos\theta&\sin^2\theta\end{pmatrix}}, \qquad A=\frac{2EV}{\Delta m^2}. \]
The box is a matrix of numbers: the crystal.
In its own axes \(\nu_{1m},\nu_{2m}\) the crystal is diagonal: each axis keeps its own phase, nothing mixes.
\[ \psi_{km}(L)=e^{-2\pi i\,\lambda_k L/L_{\rm osc}}\,\psi_{km}(0), \qquad \lambda_{1,2}=\tfrac12\bigl(1+A\mp R\bigr), \qquad R=\sqrt{(\cos2\theta-A)^2+\sin^22\theta}. \]
\[ \nu_e=\cos\theta_m\,\nu_{1m}+\sin\theta_m\,\nu_{2m}, \qquad \sin2\theta_m=\frac{\sin2\theta}{R}, \qquad \cos2\theta_m=\frac{\cos2\theta-A}{R}. \]
The phase between the axes grows as \(2\pi R\,L/L_{\rm osc}\). In vacuum, \(A=0\): \(R=1\), \(\theta_m=\theta\).
\[ \boxed{P(\nu_e\to\nu_\mu)=\sin^22\theta_m\,\sin^2\!\left(\frac{\Delta m^2_m\,L}{4E}\right)} \]
\[ \sin^22\theta_m=\frac{\sin^22\theta}{(\cos2\theta-A)^2+\sin^22\theta}, \qquad \Delta m^2_m=\Delta m^2\sqrt{(\cos2\theta-A)^2+\sin^22\theta}. \]
The vacuum formula with \(\theta\to\theta_m\) and \(\Delta m^2\to\Delta m^2_m\).
- \(A=0\): the vacuum.
- Resonance at \(A=\cos2\theta\): \(\sin^22\theta_m=1\) for any vacuum angle (Mikheyev, Smirnov, Wolfenstein).
- \(A\gg1\): \(\theta_m\to90^\circ\), \(\nu_e\) is itself an axis of the crystal, the flavor freezes.
- \(A<0\) (antineutrinos for \(\Delta m^2>0\)): no resonance. Matter tells \(\nu\) from \(\bar\nu\), and the sign of \(\Delta m^2\).
Solar Neutrinos
Curve: three flavors, adiabatic MSW, NuFIT 6.1, BS05(OP) density where \(^8\)B neutrinos are made. Points: Borexino with high-metallicity solar fluxes, Nature 562, 505 (2018); values as quoted in arXiv:2105.13858.
The Textbook
Quantum Field Theory for Experimentalists and Beyond
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