Sebastian Ordoñez-Soto

LHCb@LPCA group meeting

April 2nd, 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 2nd, 2025

Analysis strategy
Mass fit
Background model
Outlook

Analysis Strategy

Sebastian Ordoñez-Soto

Selection of the signal

Efficiency evaluation over the DP

Background model

Dalitz plot Fit

Mass fit on data

The analysis consists of mainly five stages, from the signal selection to the final Dalitz plot fit.

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

April 2nd, 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

March 24th, 2025

Sebastian Ordoñez-Soto

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

Analysis Strategy

Done?

Latest "official" version is in progress

March 24th, 2025

Homemade mass fit

Sebastian Ordoñez-Soto

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

The fraction of each contribution in the $B_{s}$ signal region are crucial for the AmAn.

The ultimate goal is to employ the mass fit of the BF analysis.

Features of the temporary homemade mass fit

The signal MC is fitted using a Symmetric Crystal Ball PDF.
The partial bkg MC is fitted using: Argus $\circledast$ Symmetric Crystal Ball PDF

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 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

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

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

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

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.
Background model done.

Done!

March 24th, 2025

Sebastian Ordoñez-Soto

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

Efficiency maps from BF analysis

March 24th, 2025

2018-DD

2018-LL

Sebastian Ordoñez-Soto

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

Efficiency maps from BF analysis

March 24th, 2025

2017-DD

2017-LL

Sebastian Ordoñez-Soto

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

Efficiency maps from BF analysis

March 24th, 2025

2016-DD

2016-LL

Sebastian Ordoñez-Soto

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

Efficiency maps from BF analysis

March 24th, 2025

2015-DD

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

March 24th, 2025

Sebastian Ordoñez-Soto

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

$B^{0}_{d}$ background model

Model from $B_{d}^{0}\rightarrow K_{S}^{0}\pi^{-}\pi^{+}$ Run I analysis

Sebastian Ordoñez-Soto

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

$B^{0}_{d}$ background model

Solely physics $\Rightarrow$ No efficiency

March 24th, 2025

Sebastian Ordoñez-Soto

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

$B^{0}_{d}$ background model

Solely physics $\Rightarrow$ No efficiency

March 24th, 2025

Sebastian Ordoñez-Soto

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

$B^{0}_{d}$ background model

February 3rd, 2025

Sebastian Ordoñez-Soto

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

$B^{0}_{d}$ background model

Physics + efficiency + vetoes

March 24th, 2025

2018-DD

Sebastian Ordoñez-Soto

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

$B^{0}_{d}$ background model

Physics + efficiency + vetoes

March 24th, 2025

2018-DD

Sebastian Ordoñez-Soto

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

Background model

March 24th, 2025

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

Combinatorial bkg. model

$B_{d}^{0}\rightarrow K_{s}\pi\pi$ bkg. model

Using Truth match MC for signal

Using No Truth Match MC for signal

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 (baseline) model.

Done!

April 2nd, 2025

Sebastian Ordoñez-Soto

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

Dalitz plot Fit

Signal Model based on SM expectations

The preliminary results presented here include all the Run 2 data.

\overline{B}_s \to \rho^0 K^0_S; \quad \overline{B}_s \to K^{*+}(892) \pi^-; \quad \overline{B}_s \to K_0^{*+}(1430); \quad \overline{B}_s \to K^0_S \pi^+ \pi^- \text{ (NR)}

April 2nd, 2025

Sebastian Ordoñez-Soto

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

Dalitz plot Fit

Amplitudes and phases

Raw asymmetries

April 2nd, 2025

The preliminary results presented here include all the Run 2 data.

Sebastian Ordoñez-Soto

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

Dalitz plot Fit

April 2nd, 2025

The preliminary results presented here include all the Run 2 data.

Sebastian Ordoñez-Soto

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

Dalitz plot Fit

April 2nd, 2025

The preliminary results presented here include all the Run 2 data.

Sebastian Ordoñez-Soto

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

Dalitz plot Fit

April 2nd, 2025

The preliminary results presented here include all the Run 2 data.

Back up

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 $T$ 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 $f$

\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

Outlook

The installation of the (pre) Dalitz plot Fit is complete and ready for the first fits.
- First results expected in the two coming weeks.
The Paris colleagues are working again in the BF mass fit $\Rightarrow$ New results
- New inputs for the AmAn.
The plan after the Bs2KsPiPi AmAn is to continue with the update of the Bd2KsPiPi AmAn.
- This will be the starter point for the TD AmAn.

February 19th, 2025

"Mass window optimization"

Mass window optimization

Analysis

Sebastian Ordoñez-Soto

January 30th, 2025

Fractions during the installation

Fractions after optimisation

Mass window optimization

Sebastian Ordoñez-Soto

Analysis

January 30th, 2025

Mass window optimization

Sebastian Ordoñez-Soto

January 30th, 2025

Analysis

Home made massfit

Homemade mass fit

Sebastian Ordoñez-Soto

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

February 3rd, 2025

About the fits below:

Include an unnecessary veto, it turns out that our tuples already included them.
- I saw them in CRAFT in the scripts used during the installation of the analysis. Unfortunately, there was a wrong number in the D+ range.
The combinatorial background slope was fixed.

Homemade mass fit

Sebastian Ordoñez-Soto

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

February 3rd, 2025

Homemade mass fit

Sebastian Ordoñez-Soto

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

February 3rd, 2025

Homemade mass fit

Sebastian Ordoñez-Soto

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

February 3rd, 2025

Homemade mass fit

Sebastian Ordoñez-Soto

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

February 3rd, 2025

Combinatorial bkg.

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

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

Motivation

S. Ordonez-Soto

December 16th, 2024

Introduction

The TI AmAn of $B_{s}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}$ aims to: Search for CP violation in decay in quasi-two-body states such as $B_{s}\rightarrow K^{*-}\pi^{+}$ or $B_{s}\rightarrow K_{S}^{0}\rho^{0}$
- CP asymmetry parameter:

\boxed{\mathcal{A}_{CP} = \frac{|\bar{\mathcal{A}}_f|^2 - |\mathcal{A}_f|^2}{|\bar{\mathcal{A}}_f|^2 + |\mathcal{A}_f|^2}}

\boxed{\text{The group has already performed a time-integrated analysis of $B^{0}\rightarrow K_{S}\pi^{+}\pi^{-}$ with the LHCb Run I data}}

$B_{(s)}^{0}\rightarrow K_{S}^{0}hh'$ decays mostly proceed through quasi-two-body intermediate states
- Resonant states can be either flavour-specific states or CP eigenstates
- Excellent laboratories to study CP violation in decay and in interference between decay and mixing

LHCB-PAPER-2017-033

Dalitz plot technique

S. Ordonez-Soto

December 16th, 2024

Dalitz plot analysis of $B_{s}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}$

\boxed{\mathrm{d}\Gamma = \frac{1}{(2\pi)^3} \frac{|\mathcal{A}|^2}{32 m_{B_{s}^0}} \mathrm{d}s_{+} \mathrm{d}s_{-} }

\boxed{m_{K_S^0\pi^+}^2 \equiv s_+ = (p_+ + p_0)^2, \quad m_{K_S^0\pi^-}^2 \equiv s_- = (p_- + p_0)^2}

$B_{s}\rightarrow K_{S}^{0}\pi^{-}\pi^{+}$ phase space

Using the Dalitz plot variables to describe the $B0→K0π+π−B^0 \to K^0 \pi^+ \pi^-$ Bs2KSpipi decay, the partial decay rate is given by:

To describe the kinematic and dynamics of 3-body decays, the analysis employs the Dalitz plot formalism.

S. Ordonez-Soto

December 16th, 2024

Dalitz plot analysis of $B_{s}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}$

Dalitz plot technique

m' = \frac{1}{\pi} \arccos\left(2\frac{m_{\pi^+\pi^-} - m_{\pi^+\pi^-}^{\text{min}}}{m_{\pi^+\pi^-}^{\text{max}} - m_{\pi^+\pi^-}^{\text{min}}} - 1\right)

\theta' = \frac{(\theta_H)_{\pi^+\pi^-}}{\pi}

Square Dalitz plot

The boundaries of the $B0→K0π+π−B^0 \to K^0 \pi^+ \pi^-$ DP are crucial, as most interference effects between amplitudes involving intermediate resonances occur in these regions.

S. Ordonez-Soto

December 16th, 2024

\mathcal{S}(s_{+},s_{-},r_{\text{tag}}) \propto \left( |\mathcal{A}(s_{+},s_{-})|^2 + |\bar{\mathcal{A}}(s_{+},s_{-})|^2 \right)\left[ \frac{1-K}{1-y^2}+r_{\text{tag}}\left(\frac{C+x_{d}\cdot S}{1+x_{d}^{2}}\right)\right]

After integration over time:

Finally, with $r_{\text{tag}}=0$ the Dalitz signal PDF becomes simply:

\boxed{\mathcal{S}(s_{+},s_{-}) \propto |\mathcal{A}(s_{+},s_{-})|^2 + |\bar{\mathcal{A}}(s_{+},s_{-})|^2}

Dalitz plot signal PDF

Dalitz plot analysis of $B_{s}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}$

\boxed{\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]}

The time and DP dependent signal PDF accounting for the transitions $\mathcal{A}_{f} = \langle f|H_{\Delta F=1}|B_{s}^{0}\rangle$ and $\bar{\mathcal{A}}_{f} = \langle f|H_{\Delta F=1}|\bar{B}_{s}^{0}\rangle$ reads:

Isobar model

S. Ordonez-Soto

December 16th, 2024

\begin{align*} |\mathcal{A}(s_+, s_-) |^2 &= \left| \sum_j c_j F_j(s_+, s_-) \right|^2 = \left| \sum_j a_j e^{i\phi_j} F_j(s_+, s_-) \right|^2, \\ |\overline{\mathcal{A}}(s_+, s_-) |^2 &= \left| \sum_j \overline{c}_j \overline{F}_j(s_+, s_-) \right|^2 = \left| \sum_j \overline{a}_j e^{i\overline{\phi}_j} \overline{F}_j(s_+, s_-) \right|^2. \end{align*}

F_j(s_+, s_-, L) = \mathcal{R}_j(s) \times X^L(|\vec{p}'|r) \times X^L(|\vec{q}'|r) \times T_j(L, \vec{p}, \vec{q}),

The total amplitude is approximated as a coherent sum of terms:

The spin-dependent dynamical function is rewritten as:

Dalitz plot analysis of $B_{s}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}$

RBW

Flatté

LASS

Total Dalitz plot PDF

S. Ordonez-Soto

December 16th, 2024

\boxed{\mathcal{P}_{\text{tot}} = f_{\text{sig}} \frac{\epsilon(s_{+}, s_{-}) \mathcal{S}(s_{+}, s_{-})}{\iint_{\text{DP}} \epsilon(s_{+}, s_{-}) \mathcal{S}(s_{+}, s_{-}) \text{d}s_{+} \text{d}s_{-}} + \sum_{i} f_{\text{bkg}_{i}} \frac{\mathcal{B}_{i}(s_{+}, s_{-})}{\iint_{\text{DP}} \mathcal{B}_{i}(s_{+}, s_{-}) \text{d}s_{+} \text{d}s_{-}},}

\mathcal{S}(s_+, s_-) = \sum_{j=1}^{N} \left( \left| \frac{a_j}{a_r} e^{i(\phi_j - \phi_r)} F_j(s_+, s_-) \right|^2 + \left| \frac{\bar{a}_j}{a_r} e^{i(\bar{\phi}_j - \bar{\phi}_r)} \bar{F}_j(s_+, s_-) \right|^2 \right).

The signal Dalitz plot PDF in this framework is then given by:

The total Dalitz plot PDF is written as:

with $f_{\text{sig}} + \sum_{i} f_{\text{bkg}_{i}} = 1$. Three main components of the PDF are clearly distinguished:

Signal PDF ($\mathcal{S}(s_{+},s_{-})$):
Background models ($\mathcal{B}_{i}(s_{+},s_{-})$)
Efficiency maps ($\epsilon(s_{+},s_{-})$)

Dalitz plot analysis of $B_{s}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}$

Physical observables from DP fit

S. Ordonez-Soto

December 16th, 2024

c_j = x_j + i y_j \, , \\ \bar{c}_j = \bar{x}_j + i \bar{y}_j \, .

\boxed{\mathcal{A}_{CP} = \frac{|\bar{c}_j|^2 - |c_j|^2}{|\bar{c}_j|^2 + |c_j|^2} = \frac{(\bar{x}_j^2 - x_j^2) + (\bar{y}_j^2 - y_j^2)} {(\bar{x}_j^2 + x_j^2) + (\bar{y}_j^2 + y_j^2)}. }

Dalitz plot analysis of $B_{s}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}$

A fit to data allows to measure the isobar parameters, i.e., the relative magnitudes of the isobar amplitude

From there, the direct CP asymmetry for an amplitude $j$ can be derived:

FF_{j} = \frac{\iint_{\text{DP}} |c_j F_j(s_+, s_-)|^2 \, \mathrm{d}s_+ \mathrm{d}s_-}{\iint_{\text{DP}} \left|\sum_j c_j F_j(s_+, s_-)\right|^2 \, \mathrm{d}s_+ \mathrm{d}s_-}

The fit fraction of a given amplitude is obtained also as result from the fit:

Steps of the Amplitude Analysis

S. Ordonez-Soto

December 16th, 2024

The analysis strategy is summarized in the following steps:

Fit the invariant mass spectrum $K_{S}\pi^{+}\pi^{-}$.
- Define a signal window around $B_{s}^{0}$ signal peak.
- Determine the fractions of each component in this signal window: $f_{\text{sig}}$ and $f_{\text{bkg}_{i}}$
Model the distribution of the backgrounds in the Dalitz plot of $B_{s}\rightarrow K_{S}^{0}\pi^{-}\pi^{+}$.
- Obtain a smoothed histogram with the model for each background.
Estimate the efficiency variation across the DP.
- Obtain a map from signal MC after the selection and with all of the corrections incorporated.
Fit simultaneously the different samples (with shared IM parameters for all categories).
- Define a nominal model according to SM predictions.
- Educate the final model by adding potential contributions to the baseline.

Analysis strategy

S. Ordonez-Soto

December 16th, 2024

\boxed{[m_{B_{s}^{0}}-1.4\cdot\sigma_{B_{s}^{0}},m_{B_{s}^{0}}+3\cdot\sigma_{B_{s}^{0}}]}

Invariant mass fit corresponding to $B_{s}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}$ 2018 data

$K_{S}^{0}\pi^{+}\pi^{-}$ Invariant mass fit

Analysis strategy

$B_{s}^{0}\rightarrow K_{S}^{0}\pi^{+}\pi^{-}$ Dalitz plot considering only data within the signal window

$K_{S}^{0}\pi^{+}\pi^{-}$ Invariant mass fit

S. Ordonez-Soto

December 16th, 2024

Analysis strategy

$K_{S}^{0}\pi^{+}\pi^{-}$ Invariant mass fit

S. Ordonez-Soto

December 16th, 2024

Analysis strategy

Distribution of the backgrounds

S. Ordonez-Soto

December 16th, 2024

Analysis strategy

It is clear that the most relevant source of bkg. within the signal window is the combinatorial bkg.

This contribution can be modeled using the data from the right side-band.

\boxed{[5500,5750]\text{MeV}}

One of the challenges in this analysis is the modeling of other sources of background, primarily from the B2KSpipi signal tail, which, although small, is not negligible.

Efficiency variation across the DP

S. Ordonez-Soto

December 16th, 2024

Analysis strategy

The efficiency of signal events is affected by selection cuts, geometrical acceptance, and trigger efficiency, leading to non-uniform event distribution across the Dalitz plane.

\boxed{ \varepsilon^{\text{tot}} = \varepsilon^{\text{geom}} \times \varepsilon^{\text{sel|geom}} \times \varepsilon^{\text{PID|sel\&geom}} }

This efficiency is determined by using MC samples with appropriate corrections, e.g., PIDcorr.

Efficiency maps $\epsilon(s_{+},s_{-})$ used as input for the Dalitz plot fit

Dalitz plot fit

S. Ordonez-Soto

December 16th, 2024

Analysis strategy

First attempt with signal decay model based on SM expectations, then add potential contributions and decide based on s statistical test.

Clermont Root-based Amplitude Fitter Tool

Perform a simultaneous fit of independent samples to determine a common set of parameters by minimizing the sum of all negative log-likelihoods:

-\ln \mathcal{L}_{\text{tot}} = -\sum_{i=1}^{N_{\text{tot}}} \ln \mathcal{P}_{\text{tot}},

Efficiency determination

subWG Kshh

Sebastian Ordoñez-Soto

February 24th, 2025

  if (dkmode == "KK") {
        cut += "h1_PROBNNp" + cut_name + "< 0.5";
        cut += "h2_PROBNNp" + cut_name + "< 0.5";
        if (year == "2018"){
            cut += "Topo_xgb > 0.95";
            cut += "PID_xgb_Bs2KKKS > 0.985";
        }
    } else {
        cut += "h1_PROBNNp" + cut_name + "< 0.9";
        cut += "h2_PROBNNp" + cut_name + "< 0.9";
    }

  if (dkmode == "KK") {
        cut += "h1_PROBNNp" + cut_name + "< 0.9";
        cut += "h2_PROBNNp" + cut_name + "< 0.9";
        if (year == "2018"){
            cut += "Topo_xgb > 0.95";
            cut += "PID_xgb_Bs2KKKS > 0.985";
        }
    } else {
        cut += "h1_PROBNNp" + cut_name + "< 0.9";
        cut += "h2_PROBNNp" + cut_name + "< 0.9";
    }

Previous selection $\epsilon_{0.9}$

Current selection $\epsilon_{0.5}$

Efficiency determination

subWG Kshh

Sebastian Ordoñez-Soto

February 24th, 2025

Efficiency determination

subWG Kshh

Sebastian Ordoñez-Soto

February 24th, 2025

Efficiency determination

subWG Kshh

Sebastian Ordoñez-Soto

February 24th, 2025

Efficiency determination

subWG Kshh

Sebastian Ordoñez-Soto

February 24th, 2025

Efficiency determination

subWG Kshh

Sebastian Ordoñez-Soto

February 24th, 2025

Efficiency determination

subWG Kshh

Sebastian Ordoñez-Soto

February 24th, 2025

Efficiency determination

subWG Kshh

Sebastian Ordoñez-Soto

February 24th, 2025

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

Outline

Analysis Strategy

Analysis Strategy

Analysis Strategy

Homemade mass fit

Homemade mass fit

Homemade mass fit

Homemade mass fit

Homemade mass fit

Homemade mass fit

Homemade mass fit

Homemade mass fit

Homemade mass fit

Homemade mass fit

Analysis Strategy

Efficiency maps from BF analysis

Efficiency maps from BF analysis

Efficiency maps from BF analysis

Efficiency maps from BF analysis

Combinatorial background model

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Background model

Analysis Strategy

Dalitz plot Fit

Dalitz plot Fit

Dalitz plot Fit

Dalitz plot Fit

Dalitz plot Fit

Back up

Goodness-of-fit test

Goodness-of-fit test

Point-to-Point Dissimilarity Method

Point-to-Point Dissimilarity Method

Point-to-Point Dissimilarity Method

Towards the nominal DP model

Outlook

"Mass window optimization"

Mass window optimization

Mass window optimization

Mass window optimization

Home made massfit

Homemade mass fit

Homemade mass fit

Homemade mass fit

Homemade mass fit

Homemade mass fit

Combinatorial bkg.

Combinatorial background model

Combinatorial background model

Combinatorial background model

Combinatorial background model

Combinatorial background model

Combinatorial background model

Combinatorial background model

Combinatorial background model

Combinatorial background model

Combinatorial background model

Combinatorial background model

Combinatorial background model

Combinatorial background model

Combinatorial background model

Combinatorial background model

Motivation

Dalitz plot technique