5. Uncertainties#
5.1. Model loading#
Of the 18 models, there are 9 with a unique expression tree.
Show number of mathematical operations per model
5.2. Statistical uncertainties#
5.2.1. Parameter bootstrapping#
n_bootstraps = 100
default_functions = jax_functions[default_model_title]
bootstrap = ParameterBootstrap(model_file, default_model.decay, default_model_title)
bootstrap_parameters = bootstrap.create_distribution(n_bootstraps, seed=0)
bootstrap_parameters_t = {k: v[None].T for k, v in bootstrap_parameters.items()}
n_events = 100_000
transformer = create_data_transformer(default_model)
phsp_sample = generate_phasespace_sample(default_model.decay, n_events, seed=0)
phsp_sample = transformer(phsp_sample)
5.2.2. Mean and standard deviations#
assert stat_intensities.shape == (n_bootstraps, n_events)
assert stat_polarimetry.shape == (3, n_bootstraps, n_events)
assert stat_polarimetry_norm.shape == (n_bootstraps, n_events)
n_bootstraps, n_events
(100, 100000)
5.2.3. Distributions#
5.2.4. Comparison with default values#
5.3. Systematic uncertainties#
5.3.1. Mean and standard deviations#
n_models = len(models)
assert syst_intensities.shape == (n_models, n_events)
assert syst_polarimetry.shape == (3, n_models, n_events)
assert syst_polarimetry_norm.shape == (n_models, n_events)
n_models, n_events
(18, 100000)
syst_polarimetry_times_I = [
syst_polarimetry_norm * syst_intensities,
*(syst_polarimetry * syst_intensities),
]
syst_polarimetry_times_I = jnp.array(syst_polarimetry_times_I)
syst_alpha_times_I_mean = syst_polarimetry_times_I.mean(axis=1)
syst_alpha_times_I_diff = (
syst_polarimetry_times_I - syst_polarimetry_times_I[:, None, 0]
)
assert syst_alpha_times_I_diff.shape == (4, n_models, n_events)
assert jnp.nanmax(syst_alpha_times_I_diff[:, 0]) == 0.0
syst_alpha_times_I_extrema = jnp.abs(syst_alpha_times_I_diff).max(axis=1)
intensity_diff_with_model_0 = syst_intensities - syst_intensities[0]
intensity_extrema = jnp.nanmax(intensity_diff_with_model_0, axis=0)
5.3.2. Distributions#
5.4. Uncertainty on polarimetry#
For each bootstrap or alternative model \(i\), we compute the angle between each aligned polarimeter vector \(\vec\alpha_i\) and the one from the default model, \(\vec\alpha_0\):
The solid angle can then be computed as:
The statistical uncertainty is given by taking the standard deviation on the \(\delta\Omega\) distribution and the systematic uncertainty is given by taking finding \(\theta_\mathrm{max} = \max\theta_i\) and computing \(\delta\Omega_\mathrm{max}\) from that.
5.5. Decay rates#
0: Default amplitude model
1: Alternative amplitude model with K(892) with free mass and width
2: Alternative amplitude model with L(1670) with free mass and width
3: Alternative amplitude model with L(1690) with free mass and width
4: Alternative amplitude model with D(1232) with free mass and width
5: Alternative amplitude model with L(1600), D(1600), D(1700) with free mass and width
6: Alternative amplitude model with free L(1405) Flatt’e widths, indicated as G1 (pK channel) and G2 (Sigmapi)
7: Alternative amplitude model with L(1800) contribution added with free mass and width
8: Alternative amplitude model with L(1810) contribution added with free mass and width
9: Alternative amplitude model with D(1620) contribution added with free mass and width
10: Alternative amplitude model in which a Relativistic Breit-Wigner is used for the K(700) contribution
11: Alternative amplitude model with K(700) with free mass and width
12: Alternative amplitude model with K(1410) contribution added with mass and width from PDG2020
13: Alternative amplitude model in which a Relativistic Breit-Wigner is used for the K(1430) contribution
14: Alternative amplitude model with K(1430) with free width
15: Alternative amplitude model with an additional overall exponential form factor exp(-alpha q^2) multiplying Bugg lineshapes. The exponential parameter is indicated as alpha
16: Alternative amplitude model with free radial parameter d for the Lc resonance, indicated as dLc
17: Alternative amplitude model obtained using LS couplings
5.6. Average polarimetry values#
The components of the averaged polarimeter vector \(\overline{\alpha}\) are defined as:
The averages of the norm of \(\vec\alpha\) are computed as follows:
\(\left|\overline{\alpha}\right| = \sqrt{\overline{\alpha_x}^2+\overline{\alpha_y}^2+\overline{\alpha_z}^2}\), with the statistical uncertainties added in quadrature and the systematic uncertainties by taking the same formula on the extrema values of each \(\overline{\alpha_j}\)
\(\overline{\left|\alpha\right|} = \sqrt{\int I_0\left(\tau\right) \left|\vec\alpha\left(\tau\right)\right|^2 \mathrm{d}^n \tau \; \big / \int I_0\left(\tau\right)\,\mathrm{d}^n \tau}\)
Cartesian coordinates:
Polar coordinates:
Averaged polarimeter values for each model (and the difference with the default model):
Tip
These values can be downloaded in serialized JSON format under Exported distributions.
Tip
A potential explanation for the \(xz\)-correlation may be found in Section XZ-correlations.
5.7. Exported distributions#
Exported 100x100 JSON grids for each bootstrap (statistics & systematics)
Exported 100x100 JSON grids for each model
[download] Default amplitude model
[download] Alternative amplitude model with K(892) with free mass and width
[download] Alternative amplitude model with L(1670) with free mass and width
[download] Alternative amplitude model with L(1690) with free mass and width
[download] Alternative amplitude model with D(1232) with free mass and width
[download] Alternative amplitude model with L(1600), D(1600), D(1700) with free mass and width
[download] Alternative amplitude model with free L(1405) Flatt’e widths, indicated as G1 (pK channel) and G2 (Sigmapi)
[download] Alternative amplitude model with L(1800) contribution added with free mass and width
[download] Alternative amplitude model with L(1810) contribution added with free mass and width
[download] Alternative amplitude model with D(1620) contribution added with free mass and width
[download] Alternative amplitude model in which a Relativistic Breit-Wigner is used for the K(700) contribution
[download] Alternative amplitude model with K(700) with free mass and width
[download] Alternative amplitude model with K(1410) contribution added with mass and width from PDG2020
[download] Alternative amplitude model in which a Relativistic Breit-Wigner is used for the K(1430) contribution
[download] Alternative amplitude model with K(1430) with free width
[download] Alternative amplitude model with an additional overall exponential form factor exp(-alpha q^2) multiplying Bugg lineshapes. The exponential parameter is indicated as alpha
[download] Alternative amplitude model with free radial parameter d for the Lc resonance, indicated as dLc
[download] Alternative amplitude model obtained using LS couplings
averaged-polarimeter-vectors.json (34.0 kB)
Tip
See Import and interpolate for how to use these grids in an an analysis and see Determination of polarization for how to use these fields to determine the polarization from a measured distribution.