Project: SK-like Solar Neutrino Fit

Goal

Build one reproducible SK-like solar-neutrino analysis:

\[ {}^8\mathrm{B}\ \text{flux} \to P_{ee}(E_\nu) \to \nu e\ \text{scattering} \to \mu_j \to n_j \to \text{fit}. \]

The final result is a fit of the blinded pseudo-data spectrum for

\[ \sin^2\theta_{12}, \qquad \Delta m^2_{21}. \]

The true values are hidden until the defense.

Data

Download:

data/project/solar_neutrino_project_data.zip

The archive contains:

solar_fluxes.csv
energy_spectra.csv
nu_electron_recoil_cross_sections.csv
oscillation_probabilities_grid.csv
earth_regeneration_grid.csv
earth_regeneration_parameter_scan.csv
blind_pseudo_data_sk.csv
README.md

The oscillation and Earth-regeneration tables were generated with PEANUTS. Students do not need to install PEANUTS.

Homework Milestones

After masterclass 1:

  1. Build the no-oscillation event-rate calculation.
  2. Plot the \({}^8\mathrm{B}\) spectrum and the recoil-electron spectrum.
  3. Check the dependence on the electron kinetic-energy threshold.
  4. Fit the \({}^8\mathrm{B}\) normalization in pseudo-data.
  5. Prepare short theory notes on the Gamow window, the astrophysical \(S\) factor, and why \(\nu_e e\) scattering is larger than \(\nu_{\mu,\tau}e\) scattering.

After masterclass 2:

  1. Add the PEANUTS survival-probability table.
  2. Fit the blinded recoil spectrum in \(\left(\sin^2\theta_{12},\Delta m^2_{21}\right)\).
  3. Compute one-dimensional profiles and two-dimensional contours.
  4. Study the size of Earth regeneration as an extension.
  5. Freeze the result before unblinding.

Required Project Package

Submit one notebook or script directory that reproduces the whole chain:

  1. Read the public tables from the project archive.
  2. Build the event-rate model.
  3. Generate at least one open pseudo-data sample and test the fit machinery.
  4. Fit the blinded spectrum without changing the procedure after looking at it.
  5. Produce plots and numerical tables for the defense.

The result must be reproducible from the public archive and the code shown in the defense.

Part 1: Event Model

Reproduce the first masterclass calculation.

Required outputs:

  1. Plot the normalized \({}^8\mathrm{B}\) spectrum \(f_{^8\mathrm{B}}(E_\nu)\).
  2. Compute the no-oscillation SK-like recoil spectrum.
  3. State the exposure and number of target electrons used.
  4. Check the total number of events above the chosen threshold.
  5. Repeat for at least two thresholds.

Use the practical units printed in the CSV tables. Analytic formulas use \(\hbar=c=1\).

Homework theory from masterclass 1:

  1. Explain the Gamow window for at least three reactions, for example \(p+p\), \({}^7\mathrm{Be}+p\), and \({}^{14}\mathrm{N}+p\).
  2. Define the astrophysical \(S\) factor through \(\sigma(E)=S(E)E^{-1}\exp[-\sqrt{E_G/E}]\).
  3. Find experimental information on one relevant \(S\) factor and give a short critical summary: what was measured, what is extrapolated, and what uncertainty matters for the Sun.
  4. Derive or verify the \(\nu e\) differential cross section used in the table.
  5. Explain why \(\nu_e e\) scattering is larger than \(\nu_{\mu,\tau}e\) scattering.

Part 2: Oscillations

Use

oscillation_probabilities_grid.csv

as a table for

\[ P_{ee}^{day} \left(E_\nu;\sin^2\theta_{12},\Delta m^2_{21}\right). \]

Required outputs:

  1. Plot \(P_{ee}^{day}(E_\nu)\) for several parameter points.
  2. Compute the oscillated recoil spectrum.
  3. Compare it with the no-oscillation spectrum.
  4. Explain why the spectrum is not just rescaled by a constant factor.

Homework theory from masterclass 2:

  1. Derive the two-flavour vacuum probability.
  2. Write the flavour-basis Hamiltonian in matter.
  3. Explain the high-density adiabatic solar limit.
  4. State what is meant by decoherence for solar neutrinos.
  5. Explain what information is lost when only the recoil spectrum is fitted.

Part 3: Fit

For recoil bin \(j\) compute

\[ \mu_j(\theta) = N_eT_{\rm live} \int_{T_j}^{T_{j+1}}dT_e \int dE_\nu\, \Phi_{^8\mathrm{B}}f_{^8\mathrm{B}}(E_\nu) \left[ P_{ee}\frac{d\sigma_{\nu_e e}}{dT_e} + (1-P_{ee})\frac{d\sigma_{\nu_x e}}{dT_e} \right]. \]

Use the Poisson statistic

\[ q(\theta) = 2\sum_j \left[ \mu_j(\theta)-n_j + n_j\ln\frac{n_j}{\mu_j(\theta)} \right], \qquad \Delta q=q-q_{\min}. \]

Required outputs:

  1. Best-fit point.
  2. Two-dimensional contour for \(\Delta q=2.30\) and \(\Delta q=6.18\).
  3. One-dimensional profile in \(\sin^2\theta_{12}\).
  4. One-dimensional profile in \(\Delta m^2_{21}\).
  5. A short statement of what the detector is sensitive to.

Fit checks:

  1. Repeat the fit with \(T_{\min}=3~\mathrm{MeV}\) and \(T_{\min}=6~\mathrm{MeV}\).
  2. Repeat the fit with coarser and finer recoil-energy binning.
  3. Compare the fixed-\(^8\mathrm{B}\)-normalization fit with a fit where \(\alpha_B\) is profiled.
  4. Explain which change affects \(\sin^2\theta_{12}\) most and which affects \(\Delta m^2_{21}\) most.

Part 4: Blinded Spectrum

Use

blind_pseudo_data_sk.csv

as the project data.

Rules:

  1. Decide the fit procedure before unblinding.
  2. Do not tune binning or thresholds after seeing the answer.
  3. Freeze the best fit and contour.
  4. During the defense, compare the frozen result with the unblinded point.

Report:

best sin2theta12 = ...
best dm21 = ... eV^2
68% contour = ...
95% contour = ...

Part 5: Earth Regeneration

This is an optional but recommended extension.

Use

earth_regeneration_parameter_scan.csv

to study

\[ \Delta P_{ee}^{\oplus} = P_{ee}^{night}-P_{ee}^{day}. \]

Answer:

  1. What is the typical size of the Earth effect?
  2. At which energies and nadir angles is it largest?
  3. Is it sensitive to \(\Delta m^2_{21}\)?
  4. Would a day-night observable help the fit?

Theory Questions

Prepare short answers:

  1. What sets the Gamow-window peak and width?
  2. What is the astrophysical \(S\) factor, and why is it introduced?
  3. Why is \(\nu_e e\) scattering larger than \(\nu_{\mu,\tau}e\) scattering?
  4. How does the matter Hamiltonian change the effective mixing angle?
  5. Why does the high-energy solar limit give \(P_{ee}\simeq\sin^2\theta_{12}\)?
  6. Why is \(\Delta m^2_{21}\) harder to measure from the recoil spectrum alone?
  7. What is Earth regeneration?
  8. What is a profile likelihood?
  9. What is a blind analysis, and what bias does it avoid?

Defense

Use this structure:

Question
Input tables
Event-rate calculation
Pseudo-data and blind-data fit
Contours and intervals
Theory interpretation
Limitations
Unblinding result

Keep the presentation short and numerical.

The defense must show enough code or formulas that the result can be reproduced.