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

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