Neutrino Oscillations, Coherence, and Matter Effects

Neutrino oscillations are interference among mass eigenstate amplitudes produced and detected coherently through flavor-tagging weak interactions. In vacuum the relative phase is $\Delta m_{ij}^2L/(2E)$; in matter, coherent forward scattering adds a flavor-dependent potential to the Hamiltonian. The textbook probability is valid only when production and detection cannot resolve the mass eigenstates, their wave packets still overlap, and experimental averaging and density variation are treated consistently.

Required background. Lepton Mixing, PMNS Parameters, and Majorana Phases fixes $U_{\alpha i}$, mass labels, and phase conventions. Linear ODEs, Evolution Operators, and Wronskians supplies ordered evolution for a position-dependent Hermitian Hamiltonian.

Helpful background. Multiparticle States, Statistics, and Fock Organization supplies the production and detection state language behind the wave-packet conditions.

Coherent flavor amplitudes in vacuum

Use $\Delta m_{ij}^2=m_i^2-m_j^2$, choose $\Delta m_{21}^2>0$, and define the produced state

$$|\nu_\alpha\rangle=\sum_iU_{\alpha i}^*|\nu_i\rangle.$$

For an ultrarelativistic component centered at energy $E$, $E_i\simeq E+m_i^2/(2E)$ up to a common term. Propagation over $L\simeq t$ gives

$$\mathcal A_{\alpha\beta}(L,E) =\sum_iU_{\beta i} \exp\!\left(-i\frac{m_i^2L}{2E}\right) U_{\alpha i}^*.$$

This phase does not require different masses to have exactly equal energy or exactly equal momentum. A localized production–propagation–detection amplitude gives the same leading phase when the mass alternatives are coherent; imposing either mnemonic beyond its accuracy can create a factor-of-two error. A field-theoretic wave-packet derivation and its limiting assumptions are given by Beuthe 2003, §§ 3–6, pp. 123–188.

Let

$$X_{\alpha\beta}^{ij} =U_{\alpha i}^*U_{\beta i} U_{\alpha j}U_{\beta j}^*, \qquad \Delta_{ij}=\frac{\Delta m_{ij}^2L}{4E}.$$

Unitarity then yields

$$\begin{aligned} P_{\alpha\beta}={}&\delta_{\alpha\beta} -4\sum_{i>j}\operatorname{Re}X_{\alpha\beta}^{ij} \sin^2\Delta_{ij}\\ &+2\sum_{i>j}\operatorname{Im}X_{\alpha\beta}^{ij} \sin(2\Delta_{ij}). \end{aligned}$$

For antineutrinos in vacuum, $U\to U^*$, so the CP-odd line changes sign. Majorana column phases cancel from $X$. The conversion useful for laboratory baselines is

$$\Delta_{ij}=1.2669\ldots\, \frac{\Delta m_{ij}^2}{\mathrm{eV}^2} \frac{L}{\mathrm{km}} \frac{\mathrm{GeV}}{E}.$$

Every numerical evaluation must state whether its code expects this dimensionless conversion or natural-unit inputs.

In a two-flavor subspace,

$$P_{e\to x}^{\rm vac} =\sin^2(2\theta) \sin^2\!\left(\frac{\Delta m^2L}{4E}\right), \qquad P_{e\to e}=1-P_{e\to x}.$$

At the exact synthetic point $\theta=\pi/6$ and $\Delta m^2L/(4E)=\pi/2$, the probabilities are $3/4$ and $1/4$.

Constant-density matter

Coherent forward charged-current scattering of electron neutrinos adds

$$V_{\rm CC}=\sqrt2G_FN_e$$

for neutrinos in ordinary unpolarized matter. Neutral-current terms common to the active flavors are proportional to the identity and do not affect active–active oscillations. In the flavor basis,

$$H_f=\frac{1}{2E}U \operatorname{diag}(m_1^2,m_2^2,m_3^2)U^\dagger +\operatorname{diag}(V_{\rm CC},0,0),$$

up to any multiple of the identity. For antineutrinos, use $U\to U^*$ and $V_{\rm CC}\to-V_{\rm CC}$. Matter can therefore produce a neutrino–antineutrino asymmetry even when the intrinsic phase $\delta$ is CP conserving; comparing the two is not by itself a clean CP test. The forward-scattering potential was derived by Wolfenstein 1978, §§ II–III.

For two flavors $(\nu_e,\nu_x)$ define $\Delta=\Delta m^2/(2E)$ and remove the trace:

$$H_2=\frac{\Delta}{2} \begin{pmatrix} -\cos2\theta&\sin2\theta\\ \sin2\theta&\cos2\theta \end{pmatrix} +\frac{V}{2} \begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$

Its eigenvalue splitting and effective angle are

$$\begin{aligned} \Delta_m&=\sqrt{(\Delta\cos2\theta-V)^2 +(\Delta\sin2\theta)^2},\\ \sin2\theta_m&=\frac{\Delta\sin2\theta}{\Delta_m}, \qquad \cos2\theta_m=\frac{\Delta\cos2\theta-V}{\Delta_m}. \end{aligned}$$

Thus at constant density,

$$P_{e\to x}^{\rm mat} =\sin^2(2\theta_m)\sin^2\!\left(\frac{\Delta_mL}{2}\right).$$

Maximal effective mixing occurs when $V=\Delta\cos2\theta$, but only for the compatible signs of $V$, $\Delta m^2$, and $\cos2\theta$. Calling this a resonance without recording those signs loses the ordering information.

A reproducible calculation uses $\Delta=1$, $\theta=\pi/6$, and neutrino-sign $V=1/2$. Then

$$H_2=\begin{pmatrix}0&\sqrt3/4\\\sqrt3/4&0\end{pmatrix},$$

so a state initially $\nu_e$ converts completely at $L=2\pi/\sqrt3$. A Hermitian matrix exponential must reproduce transition probability one to the declared tolerance, while $V\to-V$ supplies the antineutrino check.

Varying density and adiabatic evolution

For a profile $N_e(x)$, solve

$$i\frac{d}{dx}\nu_f(x)=H_f(x)\nu_f(x), \qquad S(x_f,x_i)=\mathcal P \exp\!\left[-i\int_{x_i}^{x_f}H_f(x)\,dx\right].$$

Matrices at different positions generally do not commute, so replacing the profile by its mean is not an identity. In an instantaneous two-flavor matter basis, adiabatic following requires

$$\left|\frac{d\theta_m}{dx}\right|\ll\Delta_m$$

through the relevant region. A small gap makes this condition hardest near the avoided crossing. If it fails, compute nonadiabatic transitions or integrate the ordered equation; do not splice constant-density probabilities incoherently. Standard vacuum and matter evolution, including the MSW limit, are summarized in Workman et al. 2022 with 2023 update, §§ 14.3–14.4, pp. 8–15 (PDF).

Coherence, localization, and averaging

The plane-wave result describes an interference term only if two distinct conditions hold:

1. Production and detection coherence. Their energy–momentum uncertainties must be too large to determine which mass $m_i$ was emitted or absorbed. If the alternatives are distinguishable, probabilities—not amplitudes—are summed.
2. Propagation coherence. Wave packets associated with different group velocities must still overlap at detection.

For relativistic Gaussian packets with effective spatial width $\sigma_x$,

$$|v_i-v_j|\simeq\frac{|\Delta m_{ij}^2|}{2E^2}, \qquad L_{\rm coh}^{ij}\sim \frac{4\sqrt2E^2\sigma_x}{|\Delta m_{ij}^2|},$$

where the numerical factor follows a common standard-deviation convention while the scaling is robust. Localization also requires the production–detection region to be small enough compared with the oscillation length $L_{\rm osc}^{ij}=4\pi E/|\Delta m_{ij}^2|$; otherwise the phase is averaged at the source or detector. The dependence on the production and detection widths is treated carefully by Giunti 2004, §§ 3–5, pp. 108–120.

Loss of wave-packet overlap and finite experimental resolution can lead to the same limiting probability but are physically different. When all relative phases are averaged,

$$\overline P_{\alpha\beta} =\sum_i|U_{\alpha i}|^2|U_{\beta i}|^2.$$

Energy-bin averaging, baseline distributions, source size, detector response, and environmental decoherence therefore need separate nuisance or dynamical models. Labeling an averaged spectrum “coherent propagation” is incorrect even if the displayed mean equals the decoherent limit.

Independent checks and limits

• Normalization: $S^\dagger S=I$ for a Hermitian closed-system Hamiltonian, so $\sum_\beta P_{\alpha\beta}=1$.
• Zero baseline: $S(0)=I$ gives $P_{\alpha\beta}(0)=\delta_{\alpha\beta}$.
• Degenerate mass: if every $m_i^2$ is equal and $V=0$, all relative phases vanish and flavor cannot oscillate.
• Identity shift: $H\to H+c(x)I$ changes only a common phase and no probability.
• CP and matter: in vacuum antineutrinos complex-conjugate $U$; in matter they also reverse $V$. Omitting either operation is a sign error.
• Resolution: an explicit convolution over flux, cross section, baseline, and detector response should approach the coherent and fully averaged limits continuously.
• Stop rule: the closed Hermitian evolution above does not cover absorption, stochastic media, nonstandard production/detection, decay, or a general open system without additional operators.

Current fitted contours, realistic Earth or solar profiles, flux and cross-section models, and detector response belong to dated specialist records. Leptonic CP Observables and Combined Inference gives the durable likelihood and degeneracy contract without freezing those results into propagation theory.

Common pitfalls

Using $L/E$ without units. The natural-unit phase is dimensionless; kilometers, GeV, and electronvolt-squared require the conversion factor shown above. A missing factor of four or an inconsistent unit can mimic a changed splitting.

Calling every suppression decoherence. Wave-packet separation, localization failure, energy smearing, baseline averaging, absorption, and environmental decoherence have different physical and statistical models. Identify which interference term is lost and why.

Reading a neutrino–antineutrino difference as intrinsic CP violation. Ordinary matter is not CP symmetric and $V$ reverses sign. A combined analysis must retain the density, ordering, cross-section, and detector nuisances before attributing a difference to $\delta$.

References

• Beuthe, Mikael. “Oscillations of Neutrinos and Mesons in Quantum Field Theory.” Physics Reports 375 (2003): 105–218. DOI.
• Giunti, Carlo. “Coherence and Wave Packets in Neutrino Oscillations.” Foundations of Physics Letters 17 (2004): 103–124. DOI.
• Wolfenstein, Lincoln. “Neutrino Oscillations in Matter.” Physical Review D 17 (1978): 2369–2374. DOI.
• Workman, R. L., et al. (Particle Data Group). “Review of Particle Physics.” Progress of Theoretical and Experimental Physics 2022 (2022): 083C01, with 2023 update, review 14, “Neutrino Masses, Mixing, and Oscillations.” DOI.