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CMSDAS: Measurement of B0s→μ+μ− Long Exercise

Introduction

Overview

Teaching: 15 min
Exercises: 0 min
Questions

  • What is the physics motivation for measuring $B_s^0 \to \mu^+\mu^-$?

  • What is the overall analysis strategy?

Objectives

  • Understand why $B_s^0 \to \mu^+\mu^-$ is a sensitive probe of new physics.

  • Know the key ingredients of the branching fraction measurement.

  • Understand the role of the normalization channel.

Physics motivation

The decay $B_s^0 \to \mu^+\mu^-$ is a Flavour-Changing Neutral Current (FCNC) process. In the Standard Model it is loop- and helicity-suppressed, giving a branching fraction of:

$\text{BF}(B_s^0 \to \mu^+\mu^-) \approx 3.66 \times 10^{-9}$

Many beyond-SM scenarios (SUSY, leptoquarks, extra dimensions) predict significant deviations from this value, making it one of the most sensitive indirect probes of new physics at the LHC.

Analysis strategy

The measurement follows the strategy of the CMS Run-2 paper BPH-21-006:

  1. Select $B_s \to \mu\mu$ candidates and classify them into 8 BDT categories based on signal/background discrimination.
  2. Model the signal PDF using a double Gaussian + Crystal Ball shape fitted to MC.
  3. Model background PDFs: combinatorial (Bernstein), peaking (KDE from MC), semileptonic (KDE from MC).
  4. Fit the normalization channel $B^+ \to J/\psi K^+$ in data to extract the observed yield and efficiency.
  5. Perform a simultaneous unbinned maximum likelihood fit across all 8 categories to extract BF($B_s \to \mu\mu$).

Branching fraction formula

The branching fraction is extracted via:

[\text{BF}(B_s \to \mu\mu) = \frac{N_{B_s}}{N_{B^+}} \cdot \frac{\varepsilon_{B^+}}{\varepsilon_{B_s}} \cdot \frac{f_u}{f_s} \cdot \text{BF}(B^+ \to J/\psi K^+)]

where $f_u/f_s$ is the ratio of $B^+$ to $B_s^0$ production fractions.

Key Points

  • $B_s^0 \to \mu^+\mu^-$ is a FCNC decay heavily suppressed in the SM — new physics can enhance it.

  • The branching fraction is extracted from a simultaneous fit across BDT categories.

  • $B^+ \to J/\psi K^+$ serves as the normalization channel to cancel many systematic uncertainties.

Signal MC Fit

Overview

Teaching: 10 min
Exercises: 40 min
Questions

  • How do we model the $B^+ \to J/\psi K^+$ signal peak?

  • How do we apply MC-derived shape parameters to data?

Objectives

  • Fit a double Gaussian model to $B^+ \to J/\psi K^+$ MC.

  • Understand mean shift and resolution scale corrections.

  • Fit the full signal+background model to data.

Task 2.1 — Double Gaussian fit to MC (category 0)

Fit a double Gaussian model to the $B^+ \to J/\psi K^+$ MC in category 0. The invariant mass range is 5.0–5.8 GeV.

Task 2.1

Open task_2_1.py (or task_2_1.C) and run the double Gaussian fit to MC. Record the fitted parameters — you will use them as starting values in later tasks.

python task_2_1.py

Task 2.2 — Fit data with fixed signal shape

Use the MC-derived signal shape (fixed parameters) and fit the $B^+ \to J/\psi K^+$ data with a signal + combinatorial + $J/\psi^+X$ background model.

Task 2.2

Run task_2_2.py. The signal PDF parameters are hard-coded from the Task 2.1 MC fit result. Observe the fit quality and check the yield.

Task 2.3 — Fit data with mean shift and resolution scale corrections

Introduce two free parameters:

These correct for known data/MC differences.

Task 2.3

Run task_2_3.py. Compare the fitted sig_shift and sig_scale to unity/zero. Are the data/MC corrections significant?

Task 2.4 — Repeat for category 1

Repeat Tasks 2.1–2.3 for BDT category 1: fit the MC first, then fit the data with corrections.

Task 2.4

Run task_2_4.py. Note that the MC parameters are different for category 1.

Task 2.5 — $B_s \to J/\psi\phi$ signal fit

Repeat the MC+data fit for the $B_s \to J/\psi\phi$ channel (mass peak near 5.37 GeV).

Task 2.5

Run task_2_5.py. Note the different mass peak position and the simpler background (no $J/\psi^+X$ tail needed for the $B_s \to J/\psi\phi$ channel).

Key Points

  • The signal shape is fixed from MC, with a floating mean shift and resolution scale fitted in data.

  • The combinatorial background uses an exponential; the $J/\psi^+X$ tail uses an error function.

Normalization Channel

Overview

Teaching: 10 min
Exercises: 30 min
Questions

  • How do we fit the normalization channel $B^+ \to J/\psi K^+$?

  • How do we extract the $B^+ \to J/\psi K^+$ yield and efficiency?

  • How do we compute the fs/fu production fraction ratio?

Objectives

  • Fit the $B^+ \to J/\psi K^+$ data and MC to extract signal yield and shape parameters.

  • Fit the $B_s \to J/\psi\phi$ channel to extract the Bs yield.

  • Compute fs/fu from the ratio of the two channel yields.

Task 3.1 — $B^+ \to J/\psi K^+$ normalization fit

Task 3.1

Run task_3_1.py to fit the $B^+ \to J/\psi K^+$ invariant mass distribution in data. Record the signal yield and efficiency for each category.

python task_3_1.py

Task 3.2 — $B_s \to J/\psi\phi$ yield fit

Task 3.2

Run task_3_2.py to fit the $B_s \to J/\psi\phi$ invariant mass distribution. This gives the Bs yield needed to compute the fs/fu ratio.

Task 3.3 — Computing fs/fu

Task 3.3

Run task_3_3.py. This script does pure arithmetic — no ROOT needed. It uses the yields from Tasks 3.1 and 3.2 to compute fs/fu with propagated uncertainty.

Key Points

  • The normalization channel $B^+ \to J/\psi K^+$ cancels many systematic uncertainties.

  • fs/fu is measured from data using $B_s \to J/\psi\phi$ and $B^+ \to J/\psi K^+$.