Toy Long-Baseline Oscillation Analysis

• The toy analysis encapsulates neutrino oscillation mechanisms via a modified Hamiltonian that integrates mass mixing and velocity-induced terms.
• It employs Hamiltonian diagonalization to derive effective mixing angles and oscillation probabilities for accurate event spectrum simulations.
• The framework facilitates systematic evaluation of standard and nonstandard oscillation effects, offering actionable insights for long-baseline experiments.

A toy long-baseline oscillation analysis refers to a self-contained, algorithmic, and numerically tractable framework for modeling neutrino flavor transitions in accelerator or reactor beams over distances ($L$) of hundreds to thousands of kilometers, typically within the Earth's crust, and spanning the multi-GeV regime ($E$). The primary aim is precise computation of oscillation probabilities ($P_{\mu e}$, $P_{\mu\mu}$, etc.) and simulated event spectra for detector and phenomenological studies, often exploring extensions such as modified propagation (e.g., velocity-induced effects) alongside canonical three-flavor oscillation in matter. Such analyses extract the essential features of neutrino phenomenology using physics-motivated parameterizations, closed-form formulas, and numerically efficient routines, as demonstrated in (Feldman et al., 2012, Asano et al., 2011, Banik et al., 2014), and (Denton et al., 3 May 2024).

1. Formulation of the Oscillation Hamiltonian

The evolution of neutrino flavor states in long-baseline experiments is governed by a Schrödinger-like equation: $$i\,\frac{d}{dt}\,|\nu_\alpha\rangle = H|\nu_\alpha\rangle$$ where, for ultra-relativistic neutrinos, $t \simeq L$. The most general effective Hamiltonian in the flavor basis incorporates both the mass-mixing and matter effects: $$H = \frac{1}{2E}\left[U M^2 U^\dagger + 2E U_v V U_v^\dagger + A\right]$$ with $M^2 = \text{diag}(m_1^2, m_2^2, m_3^2)$ (masses), $V = \text{diag}(v_1, v_2, v_3)$ (maximum attainable velocities for each flavor), $U$ (standard PMNS mixing matrix parameterized by $\theta_{12}$, $\theta_{13}$, $\theta_{23}$), $U_v$ (velocity mixing), and $A = 2E V_{cc}$ (matter potential, $V_{cc} = \sqrt{2} G_F N_e$ acts exclusively on $\nu_e$) (Banik et al., 2014, Denton et al., 3 May 2024, Feldman et al., 2012).

By absorbing velocity effects into an effective mass term, $M'^2 \equiv M^2 + 2E^2 V$, the Hamiltonian becomes: $H = \frac{1}{2E} \left[U M'^2 U^\dagger + A\right]$ This encapsulates standard three-flavor oscillations, velocity-induced modifications, and matter effects in a unified formalism relevant for both canonical and exotic oscillation scenarios. In practical toy analyses, the solar-scale splitting ($\Delta m_{21}^2$) is often neglected for analytic tractability in the dominant 1–3 sector (Banik et al., 2014).

2. Diagonalization and Effective Oscillation Parameters

Toy analyses proceed by diagonalizing the Hamiltonian to obtain effective mass eigenvalues $\lambda_i$ and mixing angles in matter. For the two-flavor-like dominant sector:

• The leading vacuum-plus-velocity splitting is

$$\Delta M^2 = (m_3^2 - m_1^2) + 2 E^2 (v_3 - v_1) \equiv \Delta m_{31}^2 + 2 E^2 \Delta v_{31}$$

• The effective mixing angle in matter,

$$\sin 2\theta_{13}^m = \frac{\Delta M^2 \sin 2\theta_{13}}{\sqrt{[\Delta M^2 \cos 2\theta_{13} - A]^2 + [\Delta M^2 \sin 2\theta_{13}]^2}}$$

• The two relevant eigenvalues,

$$\lambda_\pm = \frac{A + \Delta M^2 \cos 2\theta_{13} \pm \sqrt{[\Delta M^2 \cos 2\theta_{13} - A]^2 + [\Delta M^2 \sin 2\theta_{13}]^2}}{4E}$$

The diagonalizing matrix $\widetilde{U}$ can be constructed from rotations in the 1–3 and (optionally) 1–2 sectors (Banik et al., 2014, Denton et al., 3 May 2024). For full three-flavor systems and/or arbitrary constant-density matter, closed-form diagonalization using the characteristic equation $$X(\lambda) = \lambda^3 - A_1 \lambda^2 + B \lambda - C = 0$$ and efficient eigenvector–eigenvalue identities, as provided in NuFast, yield all quantities required to build the oscillation amplitudes (Denton et al., 3 May 2024).

3. Oscillation Probability Calculation

Transition probabilities are computed as

$$P_{\alpha\to\beta}(L,E) = \Big| \sum_{i=1}^3 \widetilde{U}_{\alpha i}^* \widetilde{U}_{\beta i} e^{-i \lambda_i L} \Big|^2$$

For the 1–3 dominant approximation: \begin{align*} P_{\mu \to e} &\simeq \sin^2\theta_{23} \sin^2 2\theta_{13}^m \sin^2\left(\frac{\Delta\lambda L}{2}\right) \ P_{\mu \to \mu} &\simeq 1 - \sin^2 2\theta_{23} [\cos^2\theta_{13}^m \sin^2(\lambda_- L) + \sin^2\theta_{13}^m \sin^2(\lambda_+ L)] \end{align*} where $\Delta\lambda = \lambda_+ - \lambda_-$ (Banik et al., 2014).

In full three-flavor toy codes, all transitions are generated using: \begin{align*} P_{\alpha\alpha} &= 1 - 4 \sum_{i<j} |\widetilde{U}{\alpha i}|^2 |\widetilde{U}{\alpha j}|^2 \sin^2(\Delta_{ij}) \ P_{\alpha\beta} &= -4\sum_{i<j} R^{ij}_{\alpha\beta} \sin^2(\Delta_{ij}) - 8\widetilde{J} \sin(\Delta_{21}) \sin(\Delta_{31}) \sin(\Delta_{32}) \end{align*} with $R^{ij}_{\alpha\beta}$ built from the squared matrix elements, and $\Delta_{ij} = (\lambda_i - \lambda_j) L / (4E)$ (Denton et al., 3 May 2024, Feldman et al., 2012).

The inclusion of velocity-induced effects generalizes $$\Delta m_{ij}^2 \to \Delta m_{ij}^2 + 2E^2 \Delta v_{ij}$$ throughout, offering a direct avenue to probe Lorentz-violating new physics in oscillation spectra (Banik et al., 2014).

4. Event Spectrum and Detector Modeling

Toy analyses translate probabilities into expected event counts using simplified models for detector response, flux, and cross section. For a water-Čerenkov detector of mass $M$ and a neutrino beam flux $\phi(E)$, with charged-current cross section $\sigma_{CC}(E)$,

$$N_{e-\text{like}}(\Delta E) \simeq N_{\text{target}} \times \phi(E) \times \sigma_{CC}(E) \times P_{\mu\to e}(E) \times \Delta E$$

$$N_{\mu-\text{like}}(\Delta E) \simeq N_{\text{target}} \times \phi(E) \times \sigma_{CC}(E) \times P_{\mu\to \mu}(E) \times \Delta E$$

Simulations over realistic energy ranges (e.g., $E=1$–$10$ GeV, $L=730$–$1300$ km) yield rates such as $N_e \sim 10^3$ and $N_\mu \sim 10^4$ for standard scenarios. Velocity-induced splittings ($\Delta v_{31}\sim 10^{-24}$) can produce $\mathcal{O}(20\%)$ modifications in appearance event rates and noticeable shifts in spectral shapes (Banik et al., 2014). Effects at the level of $10\%$ in the spectrum would be regarded as a clear signal of nonstandard oscillation physics.

Efficient computation and statistical analysis (e.g., binned $\chi^2$ fits) are implemented via high-level pseudocode, such as that in NuFast (Denton et al., 3 May 2024), and in Python, C++, or Fortran, permitting evaluation of the likelihood space across the full parameter manifold with rapid iteration.

5. Implementation: Algorithmic Workflow and Computational Benchmarks

Toy long-baseline analyses utilize stepwise routines:

1. Setup parameter and energy arrays: Select $\theta_{ij}$, $\Delta m_{ij}^2$, $\delta$, $L$, $E$, $\rho$, $Y_e$.
2. Hamiltonian assembly: Compute $H_\text{vac}$, add matter potential $A=2\sqrt{2} G_F N_e E$, optionally include velocity terms.
3. Numerical or analytic diagonalization: Use routines (e.g., scipy, LAPACK, NuFast) to extract eigenvalues and effective mixing.
4. Compute probabilities: Via closed-form expressions or direct amplitude evolution.
5. Event spectrum simulation: Apply flux, cross-section, detection efficiency models.
6. Statistical analysis: Generate pseudo-experiments, propagate uncertainties, perform fits.

The NuFast algorithm achieves unparalleled speed (e.g., ~45 ns for a $3\times3$ probability call on aggressive compiler flags, a $5$–$10\times$ improvement over prior market routines) and precision ($\Delta P/P$ of $10^{-4}$ without Newton-Raphson, $10^{-9}$ with one iteration) (Denton et al., 3 May 2024). Typical toy codes cover both appearance and disappearance channels over broad energy intervals, matching the requirements for high-statistics Monte Carlo analyses in DUNE, NOvA, and similar experiments.

6. Physical Implications and Extensions

A nonzero velocity splitting introduces an energy-growing phase $\Delta\phi_v \sim E L \Delta v$, with potential to compete against the canonical mass-phase $\Delta m^2 L/(2E)$ at multi-GeV energies. Matter resonance conditions—crucial for MSW effects—are modified to $$A \simeq [\Delta m^2 + 2E^2\Delta v]\cos 2\theta_{13}/(2E)$$, enabling shifted or multiple resonance energies. This suggests novel oscillatory structures that can be probed and constrained by broad-band detectors.

An analysis incorporating both $\Delta m^2$ and $\Delta v$ as fit parameters can set robust bounds on Lorentz-violating effects down to $\Delta v\sim 10^{-25}$, or uncover evidence for new forms of flavor oscillation (Banik et al., 2014). The robustness of toy frameworks enables systematic evaluation of not only standard three-flavor phenomenology but of nonstandard dynamics, CP violation, and mass ordering, under varied experimental conditions (e.g., constant vs. variable density).

7. Representative Parameters, Numerical Examples, and Domain of Validity

Frequently employed oscillation parameters include:

• $\Delta m^2_{21}\sim 7.5\times10^{-5}$ eV$^2$
• $\Delta m^2_{31}\sim 2.4$–$2.6\times10^{-3}$ eV$^2$
• $\theta_{12}\sim 33^\circ$, $\theta_{13}\sim 8.5^\circ$, $\theta_{23}\sim 45^\circ$
• Typical $L=730$–$1300$ km, $\rho=2.8$ g/cm$^3$, $Y_e=0.5$

Numerical evaluation yields,

• For $L=1300$ km and $E=0.5$–$10$ GeV, $P_{\mu e}$ peaks at $E\sim 2$ GeV with amplitude $\sim 0.04$; $P_{\mu\mu}$ dips at similar energies (Denton et al., 3 May 2024).
• Velocity splitting $\Delta v_{31}=10^{-24}$ shifts oscillation peaks and alters amplitudes by $\sim 10$–$20\%$ (Banik et al., 2014).
• Full three-flavor matter corrections, nonstandard effects, and large $\theta_{13}$ perturbative corrections ($s_{13}\sim\sqrt{\epsilon}$, $\epsilon=\Delta m^2_{21}/\Delta m^2_{31}\sim 0.03$) are valid over $E=0.2$–$10$ GeV and $L\lesssim 4000$ km (Asano et al., 2011).

Summary Table: Key Numerical Inputs and Effects

Parameter	Typical Value	Impact on Toy Analysis
Baseline $L$	295–1300 km	Defines $L/E$ oscillation phase
Energy $E$	0.5–10 GeV	Resonance regime, spectral structure
$\Delta m^2_{21}$	$7.5 \times 10^{-5}$ eV$^2$	Solar-sector, subleading in toy codes
$\Delta m^2_{31}$	$2.5 \times 10^{-3}$ eV$^2$	Main atmospheric sector probed
$\theta_{13}$	$8.5^\circ$ (squared $\sim$0.02)	Controls appearance probability, resonance
$\Delta v_{31}$	$0$–$10^{-24}$	Velocity-induced spectral distortions
Matter density $\rho$	$2.8$ g/cm$^3$, $Y_e=0.5$	Sets $A$, resonance shifts

All expressions, code templates, and benchmarks directly reflect the cited arXiv literature (Denton et al., 3 May 2024, Banik et al., 2014, Feldman et al., 2012, Asano et al., 2011). Toy long-baseline oscillation analyses, through compact and flexible modeling, provide a critical interface for algorithmic development, new physics searches, and experimental design across neutrino oscillation research.