S. Akar, E. Cogneras, S. Monteil, S. Ordonez-Soto*  

 

B2KShh' \(\mu\)-group Meeting

 

April 14th, 2025

 

Time-integrated Amplitude Analysis of the decay \(B_{s}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\) with Run I and Run II data

Outline

Sebastian Ordoñez-Soto

April 14th, 2025

  • Introduction
  • Analysis strategy
  • Towards the nominal DP model
    • Mass fit
    • Efficiency maps
    • Background model
  • Preliminary Dalitz plot fit
  • Conclusion and outlook

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Introduction

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Sebastian Ordoñez-Soto

April 14th, 2025

Introduction

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

April 14th, 2025

  • The Amplitude Analysis of the \(B_{s}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\) provides access to the following observables:
    • The direct \(CP\) asymmetries of the quasi-two-body decays with an untagged time-integrated (TI) Dalitz analysis:
      • Flavour specific: \(\bar{B}_{s}^{0}\rightarrow K^{*+}_{0}(1430)\pi^{-}\), \(\bar{B}_{s}^{0}\rightarrow K^{*+}(892)\pi^{-}\)
      • \(CP\)-eigenstates: \(B_{s}^{0}\rightarrow \rho^{0}K_{S}^{0}\)
    • Fit fractions of the different amplitudes contributing to the decay.

Physics motivation

Data sample

  • The data used in this analysis corresponds to: B2pipiKS Secondary Optimization
    • ​The cut ProbNNp < 0.9 has been included.
    • Let's recall the charm and charmonia vetoes applied:
      • Charm meson vetoes (30 MeV): \(D^{+}_{(s)}\rightarrow K_{S}^{0}\pi^{+}\), \(D^{0}\rightarrow \pi^{+}\pi^{-}\)
      • Charmonia vetoes (48 MeV): \(J/\psi\rightarrow \pi^{+}\pi^{-}\), \(\chi_{c0}\rightarrow \pi^{+}\pi^{-}\)

Introduction

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

April 14th, 2025

The general PDf describing the dynamics of the decay \(B_{s}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\) reads:

Dalitz signal PDF

\mathcal{S}(s_{+},s_{-},t,r_{\text{tag}}) = \frac{\text{d} \Gamma_{B^0_{S},\bar{B}^{0}_{S}}(t)}{\text{d}t} = \frac{ 1}{2}e^{-\Gamma t} \left( |\mathcal{A}(s_{+},s_{-})|^2 + |\bar{\mathcal{A}}(s_{+},s_{-})|^2 \right)\left[ \cosh(\Gamma yt) - K \sinh(\Gamma yt) + r_{\text{tag}}\left(C\cos(\Gamma x t) - S \sin(\Gamma x t)\right)\right]
  • \(\mathcal{A}_{f} = \langle f|H_{\Delta F=1}|B_{s}^{0}\rangle\) and \(\bar{\mathcal{A}}_{f} = \langle f|H_{\Delta F=1}|\bar{B}_{s}^{0}\rangle\)
  • \(S\), \(C\) and \(K\) are the time-dependent \(CP\) asymmetries
  • \(r_{t}\) tagging parameter
  • \(\Gamma\) and \(\Delta m\) are the decay rate and mass difference, respectively. 
  • After integration and with \(r_{\text{tag}} = 0\) simplifies as:
\mathcal{S}(s_{+},s_{-}) \propto |\mathcal{A}(s_{+},s_{-})|^2 + |\bar{\mathcal{A}}(s_{+},s_{-})|^2
  • \(\mathcal{A}\) is parametereized by the isobar model:
|\mathcal{A}(s_+, s_-) |^2 = \left| \sum_j a_j e^{i\phi_j} F_j(s_+, s_-) \right|^2
  • \(a_{j}\) and \(\phi_{j}\) describe the relative magnitude and phase.

Analysis Strategy

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Sebastian Ordoñez-Soto

April 14th, 2025

Analysis Strategy

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

April 14th, 2025

 This analysis follows a similar strategy to that of the DP analysis of the decay \(B_{d}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\) with Run I data.

  1. Selection of \(B_{s}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\) signal candidates
  2. Fit the mass spectrum \(K_{S}^{0}\pi^{+}\pi^{-}\) and define a signal window around \(B_{s}^{0}\) signal peak.
    • Determine the fraction of signal and bkg. (Combinatorial and \(B_{d}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\))
  3. Obtain the spline histogram of the efficiency variation across the DP from MC.
  4. Determine a model for the different background components.
  5. Fit simultaneously the different samples an educate the final model.

Precedents

Stages

Analysis Strategy

Sebastian Ordoñez-Soto

Selection of the signal
Efficiency evaluation over the DP
Background model
Simultaneous Dalitz plot Fit
Mass fit on data 

Schematic of the analysis workflow

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Analysis Strategy

Done!

Inherited from the BF Analysis

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Analysis Strategy

Done?

Pending latest version from the simultaneous fit.

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Analysis Strategy

Efficiency maps available, also inherited from BF analysis.

Done!

March 24th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Analysis Strategy

Work in progress!

First fit with a simple model.

April 14th, 2025

Towards the nominal DP model

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Sebastian Ordoñez-Soto

April 14th, 2025

 \(K_{S}^{0}\pi^{+}\pi^{-}\) Invariant Mass fit

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

A preliminary fit to the invariant mass, not simultaneous, has been done to determine in the signal mass window (\([\mu_{B_{s}^{0}}-1.4\cdot\sigma_{B_{s}^{0}}, \mu_{B_{s}^{0}}+3.0\cdot\sigma_{B_{s}^{0}}]\)):

  • \(f_{\text{sig}}\): Fraction of \(B_{s}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\) signal 
  • \(f_{\text{comb bkg.}}\): Fraction of combinatorial bkg. 
  • \(f_{B_{d}^{0} \text{ bkg.}}\): Fraction of \(B_{d}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\)  bkg. 

This preliminary fit follows the same procedure than the one of the BF:

  1. The signal MC, \(B_{s}^{0}\) and \(B_{d}^{0}\), is fitted using a Symmetric Crystal Ball PDF.
  2. The partial bkg. MC is fitted using: Argus \(\circledast\) Symmetric Crystal Ball PDF.
  3. The combinatorial bkg. is fitted using a first order polynomial, with the slope floated. 
  4. The shape parameters (\(\alpha_{R,L},n_{R,L}\), etc) are fixed in the data fit.

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

 \(K_{S}^{0}\pi^{+}\pi^{-}\) Invariant Mass fit

April 14th, 2025

Summary of fractions for each component which are inputs for the Dalitz plot fit.

  In the signal window:

  • Run 2 DD: \(N_{\text{Sig}}^{B_{s}^{0}} \approx 775 \)
  • Run 2 LL: \(N_{\text{Sig}}^{B_{s}^{0}} \approx 345\)

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Efficiency across DP

2018-DD

2018-LL

April 14th, 2025

  • The efficiency maps in the square Dalitz plot (sDP) have been obtained in the BF analysis.
  • For this analysis the spline technique will be the baseline for the efficiency modeling.
  • The maps are smoothed out by a 2D cubic spline and are symmetrized.

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Background model

  • The combinatorial background under the signal is modeled using candidates from the right-hand sideband
    • \([5450,5750] \text{MeV}\)
  • Due to the limited statistics, a combined sample including all Run 2 data (DD + LL) is used.
  • The resulting background sDP distribution is smoothed and used as input for the combinatorial background model.

April 14th, 2025

Combinatorial background

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Background model

\(B_{d}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\) background

  • The \(B_{d}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\) background is estimated using the model for this decay from Run I:
    • Paper: Amplitude analysis of the decay \(B_{d}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}\) and first observation of the CP asymmetry in \(B_{d}^{0}\rightarrow K^{*}(892)\pi^{+}\).
    • Large  toy  MC generated using CRAFT.
    • The toy MC is weighted by the corresponding efficiency map.

2018-DD

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Preliminary signal model

April 14th, 2025

  • A preliminary signal model is proposed based on Standard Model expectations. 
    • \(\overline{B}_s \to \rho^0 K^0_S\)
    • \(\overline{B}_s \to K_0^{*+}(1430)\pi^{-}\) 
    • \(\overline{B}_s \to K^0_S \pi^+ \pi^- \text{ (NR)}\)
    • \(\overline{B}_s \to K^{*+}(892) \pi^-\)
  • The model parameterizes the signal amplitude as the isobar sum of three resonances (and conjugates) plus a non-resonant (NR) contribution.

*Chosen as reference \(\Rightarrow\) \(x=2\), \(y=0\) and \(\bar{y} = 0\)

*

Dalitz Plot fitting

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Sebastian Ordoñez-Soto

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dalitz plot Fit

  • A simultaneous fit to DD+LL 2018 data has been done using the simple signal model.

April 14th, 2025

2018-LL \(\Rightarrow\)

2018-DD \(\Rightarrow\)

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dalitz plot Fit

April 14th, 2025

2018-LL \(\Rightarrow\)

2018-DD \(\Rightarrow\)

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dalitz plot Fit

April 14th, 2025

Fit results

  • The simple model presents a reasonable description of the Dalitz plot data. 

Amplitudes

Phases

Dalitz plot PDF

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dalitz plot Fit

  • A simultaneous fit to DD+LL 2018 data has been done using the simple model + \(f_{2}(1270)\).

April 14th, 2025

2018-LL \(\Rightarrow\)

2018-DD \(\Rightarrow\)

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dalitz plot Fit

April 14th, 2025

2018-LL \(\Rightarrow\)

2018-DD \(\Rightarrow\)

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dalitz plot Fit

April 14th, 2025

Amplitudes

Phases

Dalitz plot PDF

Fit results

  • A simultaneous fit to DD+LL 2018 data has been done using the simple model + \(f_{2}(1270)\).

Conclusion and Outlook

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Sebastian Ordoñez-Soto

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Conclusion and outlook

April 14th, 2025

  • Most elements for the time integrated Dalitz plot are in place.
  • Preliminary simultaneous (only 2018) fit with a simple model  gives a reasonable description of the Dalitz for both DD and LL.
  • There are indications of the potential contribution of \(f_{2}(1270)\) and \(f_{0}(1500)\). 
  • Include, when available, the new inputs (\(f_{\text{sig}}, f_{\text{comb. bkg}}, f_{B_{d}^{0} \text{ bkg}}\)) from the BF analysis.  
  • Run simultaneous fits including all Run 1+2 samples (DD and LL).
  • Incorporate the goodness-of-fit to quantitatively estimate the performance of the fits
  • Educate the final model by adding new resonances and define the nominal model.

Conclusion

Outlook

Thank you!

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Sebastian Ordoñez-Soto

April 14th, 2025

Back up

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Using Truth match MC for signal

 \(K_{S}^{0}\pi^{+}\pi^{-}\) Invariant Mass fit

April 14th, 2025

Homemade mass fit

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

March 24th, 2025

Homemade mass fit

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

March 24th, 2025

Homemade mass fit

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

March 24th, 2025

Homemade mass fit

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Current results from the BF analysis:

Results with the homemade fit:

March 24th, 2025

Homemade mass fit

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

March 24th, 2025

Using No Truth Match MC for signal

Homemade mass fit

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

March 24th, 2025

Using No Truth Match MC for signal

Homemade mass fit

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

March 24th, 2025

Using No Truth Match MC for signal

Homemade mass fit

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

March 24th, 2025

Using No Truth Match MC for signal

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Efficiency across DP

2018-DD

2018-LL

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Efficiency maps from BF analysis

2017-DD

2017-LL

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Efficiency maps from BF analysis

2016-DD

2016-LL

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Efficiency maps from BF analysis

2015-DD

2015-LL

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery without asym

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery without asym

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery without asym

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery without asym

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery without asym

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery without asym

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery without asym

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery without asym

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery without asym

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery without asym

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery with asym

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

\(B^{0}_{d}\) background model: Art gallery

April 14th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Goodness-of-fit test

In short, this is what we do at this stage: 

  • We do an unbinned maximum likelihood fit of a PDF to the data.
  • This fitted PDF is then used to extract the value of some observables from the data.

It is crucial to determine the level of agreement between the fit PDF and the data from a statistical argument (null-hypothesis significance test) \(\Rightarrow\) a goodness-of-fit (g.o.f).

The unbinned Point-to-Point Dissimilarity Method will be used \(\Rightarrow\) event by event 

  • Notation:
    • \(\vec{s} = (s_{-},s_{+}) = (s_{K_{S}\pi^{-}},s_{K_{S}\pi^{+}}) \)​​ 
    • \(f(\vec{s})\) and \(f_{0}(\vec{s})\): parent/true PDF of the data and test PDF, respectively.
    • \(T\): test statistic (TS) \(\Rightarrow\) Larger values correspond to a worse level of agreement.
    • \(H_{0}\): Null hypothesis \(\Rightarrow\) "The two samples are drawn from the same PDF".
    • \(p\): \(p\)-value \(\Rightarrow\) Significance of any discrepancy between the data and the test PDF
p = \int_T^{\infty} g_{f_0}(T')\, dT' = P(T_{perm} \ge T | H_{0})

How good are your fits? Unbinned multivariate goodness-of-fit tests in high energy physics: http://arxiv.org/abs/1006.3019

April 2nd, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Goodness-of-fit test

February 19th, 2025

In short, this is what we do at this stage: 

  • We do an unbinned maximum likelihood fit of a PDF to the data.
  • This fitted PDF is then used to extract the value of some observables from the data.

It is crucial to determine the level of agreement between the fit PDF and the data from a statistical argument (null-hypothesis significance test) \(\Rightarrow\) a goodness-of-fit (g.o.f).

We will use the unbinned Point-to-Point Dissimilarity Method. 

  • Notation:
    • \(\vec{s} = (s_{-},s_{+}) = (s_{K_{S}\pi^{-}},s_{K_{S}\pi^{+}}) \)​​ 
    • \(f(\vec{s})\) and \(f_{0}(\vec{s})\): parent PDF of the data and test PDF, respectively.
    • \(T\): test statistic (TS) \(\Rightarrow\) Larger values correspond to a worse level of agreement.
    • \(p\): \(p\)-value \(\Rightarrow\) Significance of any discrepancy between the data and the test PDF
p = \int_T^{\infty} g_{f_0}(T')\, dT' = P(T_{perm} \ge T | H_{0})

You want to know whether two sets of points in phase space come from the same underlying distribution

The p-value is the probability of getting a test statistic TTT as large or larger than the observed one, under the null hypothesis.

“If the samples really come from the same distribution, how often would I get a dissimilarity as large as the one I observed, just due to chance?”

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Point-to-Point Dissimilarity Method

Ideally, the difference between \(f\) and \(f_{0}\) could be estimated from the T statistic:

T = \frac{1}{2} \int \left( f(\vec{s}) - f_0(\vec{s}) \right)^2 \, d\vec{s}

It is plausible to postulate a weighting function (WF) \(\psi(|\vec{s}-\vec{s}'|)\) which correlates the difference between the PDF's at different points, such that:

T = \frac{1}{2} \int \int \left( f(\vec{s}) - f_0(\vec{s}) \right) \left( f(\vec{s}') - f_0(\vec{s}') \right) \psi(|\vec{s} - \vec{s}'|) \, d\vec{s} \, d\vec{s}'.
T = \frac{1}{2} \int \int \left[ f(\vec{s}) f(\vec{s}') + f_0(\vec{s}) f_0(\vec{s}') - 2 f(\vec{s}) f_0(\vec{s}') \right] \psi(|\vec{s} - \vec{s}'|) \, d\vec{s} \, d\vec{s}'.
T = \frac{1}{n_d(n_d - 1)} \sum_{\substack{i,j>i}}^{n_d} \psi(|\vec{s}_i^d - \vec{s}_j^d|) + \frac{1}{n_{\mathrm{mc}}(n_{\mathrm{mc}} - 1)} \sum_{\substack{i,j>i}}^{n_{\mathrm{mc}}} \psi(|\vec{s}_i^{\mathrm{mc}} - \vec{s}_j^{\mathrm{mc}}|) - \frac{1}{n_d n_{\mathrm{mc}}} \sum_{i,j}^{n_d, n_{\mathrm{mc}}} \psi(|\vec{s}_i^d - \vec{s}_j^{\mathrm{mc}}|).

This can be approximated by:

The average kernel value between pairs of points if both are drawn from distribution fff

\psi_{\text{exp}}(|\vec{s}_i - \vec{s}_j|) = e^{-|\vec{s}_i - \vec{s}_j|^2 / 2\sigma(\vec{s}_i)\sigma(\vec{s}_j)} \\ \psi_{\text{Log}}(|\vec{s}_{i} - \vec{s}_{j}|) = \ln(|\vec{s}_{i} - \vec{s}_{j}| + \epsilon)

April 2nd, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Point-to-Point Dissimilarity Method

The distribution of \(T\) for the case \(f = f_{0}\) is not known... How do we estimate a \(p\)-value?    

  • Permutation sampling 
  • Combine all data points from both samples into a single pool \(n_{d}+n_{mc}\)
  • Randomly reassign points into two new "data" and "MC" samples with sizes \(n_{d}\) and \(n_{mc}\)
  • Compute the test statistic \(T_{perm}\) for each permuted pair.
  • Repeat this process \(n_{perm}\) times to obtain \(\{T_{perm}^{1},...,T_{perm}^{n_{perm}}\}\)
    • The \(p\)-value is the fraction of permutations where \(T_{perm} \geq T \)
  • If \(p\)-value is small:
    • The observed dissimilarity is unlikely under \(H_{0}\).
  • If \(p\)-value is large:
    • The observed dissimilarity could happen by chance.

April 2nd, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Point-to-Point Dissimilarity Method

  • First test of the method using the Run 1 B2KSpipi PDF
    • ~1000 toys
    • \(n_{d}=n_{mc} = 100\) (low statistics)
    • \(n_{perm}\) = 100
    • The logarithmic function is chosen as \(\psi\)

The \(p\)-value is a function of the data and MC, and is therefore itself a random variable

April 2nd, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Towards the nominal DP model

Run 1 procedure for B2KSpipi 

April 2nd, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model

May 2025

\boxed{p_{\text{global}} = 0.38}

Fisher's method

P-values per sub-sample baseline

Global p-value (Fisher) baseline: 0.38

Global p-value (Stouffer) baseline: 0.16

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model+f2

May 2025

\boxed{p_{\text{global}} = 0.5}

Fisher's method

P-values per sub-sample baseline+f2

Global p-value (Fisher) baseline+f2: 0.5

Global p-value (Stouffer) baseline+f2: 0.21

P-values per sub-sample

baseline

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model+f2

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model+f2

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model+f2

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model+f2

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model+f2

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model+f2

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

Baseline model+f2

May 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Dissimilarity test results

May 2025

p-values per sub-sample baseline+f2

Global p-value (Fisher) baseline+f2: 0.12

Global p-value (Stouffer) baseline+f2: 0.1

P-values per sub-sample

baseline

Comparison using gaussian correlation function

Global p-value (Fisher) baseline: 0.12

Global p-value (Stouffer) baseline: 0.11

p-value full Run 2

baseline+f2

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

Run 2 DD merged

February 17th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

February 3rd, 2025

2018 DD

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

February 3rd, 2025

2017 DD

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

February 3rd, 2025

2016 DD

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

February 3rd, 2025

2015 DD

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

Run 2 LL merged

February 17th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

February 3rd, 2025

2018 LL

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

February 3rd, 2025

2017 LL

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

February 3rd, 2025

2016 LL

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

February 3rd, 2025

2015 LL

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

Run 2 DD + LL categories merged

February 19th, 2025

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

February 3rd, 2025

2018 DD + LL categories merged

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

February 3rd, 2025

2017 DD + LL categories merged

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

February 3rd, 2025

2016 DD + LL categories merged

Sebastian Ordoñez-Soto

AmAn of the \(B_{s}^{0}\rightarrow K_{S}^{0} \pi^{+}\pi^{-}\) decay

Combinatorial background model

February 3rd, 2025

2015 DD + LL categories merged