CP Hypothesis Fit and Asymmetry Analysis in Periodic Distributions

Overview

A repository for plotting in the CPinHToTauTau analysis. This script performs a fit for three CP hypotheses :

• 🔴 CP-even
• 🔵 CP-odd
• 🟢 CP-maximal mixing

and computes the asymmetries between one another. The approach can be applied to a general case where geometric distributions are fitted OUTPUT/version_name/<process_name>/* and their asymmetries are analysed OUTPUT/Asymmetry_*.pdf.

Features

• Fit for CP Hypotheses : The script provides fits to the data for the three CP hypotheses : CP-even, CP-odd, and maximal mixing.
• Asymmetry Computation : Asymmetries between these hypotheses for each output plot are computed.
• Spin Analysing Power Visualization : The results are plotted separately to analyse the spin-analyzing power of the underlying reconstruction method.

Uses

Output (.pickle) from Columnflow

How to prepare the files ?

1. Copy the needed pickle files Use scripts/phi_cp_merges_hists.sh to:

   • Select files by CF production version, era, and process.
   • Is performed from the within cf.MergeHistograms
   • Copies them to your CERNBox.

2. Organise the files locally Place the following structure inside the INPUT directory:

   version_name/
   ├── ggF
   │   ├── sm
   │   ├── mm
   │   └── cpo
   └── VBF
       ├── sm
       ├── mm
       └── cpo
   

3. Rename the files Use scripts/copy_merge_and_rename.sh to:

   • Remove process and CP hypothesis from filenames.
   • => Ready to use for the PhiCPfit_BarPlot_DESYTau_cf_0p3.ipynb plotting script.

Produces

• Cosine fit for Φ_CP distributions in the H → ττ → τ_lτ_h channel for multiple hypothesis
• Plots the Asymmetry between different hypotheses

Formulas for the Fitting and their Errors

1. Fit Formula

   The model function for the fit analysis is :

   $fit = f(x) = a \cdot \cos(x + c) + b$

   Where $a$ is the amplitude, $b$ is the offset, and $c$ is the phase shift parameter.

2. Fit Parameter Errors

   The uncertainties on the fit parameters $a$, $b$, and $c$ are determined using the Hesse matrix from the Minuit minimisation :

   $\sigma_a, \sigma_b, \sigma_c = \text{Minuit.Hesse}(\text{fit})$

   These errors are propagated into the asymmetry calculation.

3. Asymmetry Formula

   The asymmetry between two distributions $\text{fit}_{1}$ and $\text{fit}_{2}$ is computed by :

   $A_{1,2} = \frac{1}{N} \sum_{i=1}^{N} \left| \frac{\text{fit}_{1}^i - \text{fit}_{2}^i}{\text{fit}_{1}^i + \text{fit}_{2}^i} \right|$

   Where $N$ is the number of elements in the arrays.

4. Asymmetry Error Formula

   The error of the asymmetry is calculated using error propagation for the two distributions $\text{fit}_{1}$ and $\text{fit}_{2}$ :

   $\sigma_A = \frac{1}{N} \sqrt{\sum_{i=1}^{N} \left( \left( \frac{2 \cdot \text{fit}_{2}^i \cdot \text{err}_1^i}{(\text{fit}_{1}^i + \text{fit}_{2}^i)^2} \right)^2 + \left( \frac{2 \cdot \text{fit}_{1}^i \cdot \text{err}_2^i}{(\text{fit}_{1}^i + \text{fit}_{2}^i)^2} \right)^2 \right)}$

Thanks

Thanks to Gourab Saha for providing the original Φ_CP fit code for the HToTauTau Analysis :
Gourab's Repo