A framework and implementation for data-driven trigger efficiency estimation at LHCb

This paper introduces a data-driven framework for estimating trigger efficiencies at LHCb and presents the design, implementation, and performance of the centralised software package TriggerCalib, which facilitates these calculations and uncertainty estimations for physics analyses.

The Gist

Imagine you are running a massive, high-speed train station (the LHCb experiment) where millions of trains (particle collisions) pass through every second. Your job is to find a few specific, rare trains carrying precious cargo (interesting physics events) while ignoring the millions of empty or boring ones.

To do this, you have a team of security guards (the Trigger System) who stand at the gates. They have to make split-second decisions: "Let this train through" or "Stop that one."

The Problem: How Good Are the Guards?

In physics, you need to know exactly how often your guards let the right trains through. If they miss 10% of the precious cargo, your final count of "treasure found" will be wrong.

Usually, you'd try to test this by watching the guards in a simulation (a video game version of the station). But real life is messy, and the guards' decisions are so complex that the video game can't perfectly mimic them.

If you tried to test them by recording every single train that passes, you'd need a camera system so huge it would cost more than the universe. So, you can't just count everything.

The Solution: The "TISTOS" Method

The paper introduces a clever trick called TISTOS (Trigger Independent of Signal / Trigger On Signal). Think of it as a "Tag-and-Probe" game.

Imagine you are looking for a specific type of passenger (the Signal) on the train.

1. The Tag (TOS): You find a passenger who definitely triggered the guard's attention because they were the one the guard was looking for. (e.g., "This person has a red hat, and the guard only stops people with red hats.")
2. The Probe (TIS): You look at the same train and ask: "Did the guard stop this train even if the red-hat person wasn't there?" Maybe the guard stopped the train because of a passenger with a blue hat elsewhere on the train.

By comparing these two groups, you can mathematically figure out the guard's efficiency without needing to see every single train that was stopped. It's like deducing how good a bouncer is by seeing how often he lets people in when the VIP is there versus when the VIP isn't there.

The New Tool: "TriggerCalib"

Before this paper, every scientist had to build their own custom calculator to do this math. It was like every chef in a restaurant having to invent their own knife. It was slow, prone to errors, and took forever.

The authors built TriggerCalib, a "universal kitchen knife" (a software package).

• What it does: It automates the TISTOS math.
• Why it's great: Instead of taking days to set up a calculation, a scientist can now do it in minutes. It's a central, reliable tool that everyone can use, ensuring everyone is cutting their data the same way.

Dealing with "Noise" (Background)

In a train station, there's always noise—people loitering, fake tickets, or random crowds (called Combinatorial Background). This noise can trick your efficiency calculation.

The paper shows three ways to filter out this noise using TriggerCalib:

1. Sideband Subtraction: Imagine looking at the crowd just outside the VIP area. If you see 100 random people there, you assume there are about 100 random people mixed in with the VIPs, and you subtract them.
2. Fit-and-Count: You use a mathematical curve to draw a line between "VIPs" and "Randoms" and count how many fall on the VIP side.
3. sPlot: A fancy statistical trick that assigns a "weight" to every person. If a person looks 90% like a VIP, they get a high weight; if they look 10% like a VIP, they get a low weight. You sum up the weights to get the true VIP count.

The Result

The authors tested this new tool using a "practice run" with simulated data (a fake train station). They found that all three noise-filtering methods gave the same answer, proving the tool works.

They also explained how to calculate the uncertainty (how much you can trust the result). Just like a weather forecast saying "70% chance of rain," they now have a precise way to say, "The guard lets 97.3% of the right trains through, give or take 0.1%."

In a Nutshell

This paper is about building a standardized, easy-to-use toolbox that helps particle physicists accurately measure how well their "security guards" (triggers) are working. By using clever math (TISTOS) and a new software package (TriggerCalib), they can stop guessing and start knowing exactly how much "treasure" their experiment is finding, even when the data is messy and noisy.

Technical Summary

1. Problem Statement

In modern particle physics experiments like LHCb, multi-stage trigger systems are essential for filtering collision data. Accurately estimating the efficiency of these triggers is critical for physics analyses but presents significant challenges:

• Simulation Limitations: Simulated samples cannot perfectly model the complex, non-linear nature of trigger selection algorithms.
• Data Scarcity: Evaluating efficiency as a simple fraction of accepted events in recorded data is impossible because unfiltered data (needed for the denominator) is prohibitively large to store.
• Redundancy Issues: While recorded data contains triggered events, the redundancy between different trigger selections is insufficient to calculate efficiency as a simple ratio without introducing biases.
• Implementation Fragmentation: Historically, the "TISTOS" method (the standard data-driven approach) was re-implemented from scratch in nearly every physics analysis. This required significant time, effort, and specialized trigger expertise, leading to potential inconsistencies and validation errors.

2. Methodology: The TISTOS Framework

The paper revisits and formalizes the TISTOS (Trigger Independent of Signal / Trigger On Signal) method, a fully data-driven approach to estimate trigger efficiencies.

Core Concepts

• Tag-and-Probe Approach: The method identifies a subsample of "tag" candidates (selected by a specific trigger) and evaluates the fraction that are also "probe" candidates (selected by the trigger of interest).
• Classification Categories: Trigger decisions are classified relative to a physics signal candidate based on shared detector hits:
  • TOS (Trigger On Signal): The signal candidate itself is sufficient to fire the trigger.
  • TIS (Trigger Independent of Signal): Another object in the event (unrelated to the signal) is sufficient to fire the trigger.
  • TISTOS: The intersection where both conditions are met.
  • TOB (Triggered On Both): The signal and the rest of the event are individually insufficient, but their combination triggers the system.
• Efficiency Formula: The trigger efficiency ($\epsilon_{Trig}$) is estimated using the relationship between these categories:
  $$\epsilon_{Trig} \approx \frac{N_{Trig} \times N_{TISTOS}}{N_{TIS} \times N_{TOS}}$$
  Where $N$ represents the yield of candidates in each category. This formula allows efficiency calculation using only triggered data.

Handling Correlations

A major source of bias is the kinematic correlation between the "tag" (TOS) and "probe" (TIS) samples. To mitigate this, the method requires evaluating efficiency in narrow phase-space bins (e.g., transverse momentum $p_T$, pseudorapidity $\eta$). As binning becomes finer, the bias from correlations diminishes, and the estimate approaches the truth-level simulation value.

Background Mitigation

To isolate the true signal yield from background noise within these bins, the paper implements three distinct statistical methods:

1. Sideband Subtraction: Defines signal and sideband windows in a discriminating variable (e.g., invariant mass) to estimate and subtract background density.
2. Fit-and-Count: Uses extended probability density functions (PDFs) to fit signal and background components simultaneously in each bin.
3. sPlot: Performs a global likelihood fit to calculate per-event weights (sWeights) for signal and background, allowing the extraction of signal yields in specific phase-space regions without re-fitting every bin.

3. Key Contributions: The TriggerCalib Package

The primary contribution of this work is TriggerCalib, a centralized, Python-based software package that implements the TISTOS framework.

• Centralization: It replaces the need for analysts to re-implement complex TISTOS logic in every analysis, reducing implementation time from days to minutes.
• Flexibility: It supports arbitrary phase-space binning schemes and allows analysts to choose any discriminating variable.
• Background Handling: It natively supports the three mitigation methods (Sideband, Fit-and-Count, sPlot) and includes tools to validate the independence of variables (using Likelihood Ratio and Kendall's $\tau$ tests) to ensure sPlot validity.
• Uncertainty Propagation: It includes a novel implementation for calculating statistical uncertainties that correctly account for the correlations between TIS, TOS, and TISTOS categories (using a generalized Wilson interval).
• Simulation Corrections: It generates correction weights ($w_i = \epsilon_{data} / \epsilon_{sim}$) to apply to simulated samples, enabling the correction of trigger responses in rare decay channels where direct data-driven calculation is impossible due to low statistics.

4. Results and Validation

The package was validated using a simulated sample of $B^+ \to J/\psi (\mu^+\mu^-) K^+$ decays, mimicking LHCb Run 3 conditions (2024).

• Consistency: All three background mitigation methods (Sideband, Fit-and-Count, sPlot) produced consistent trigger efficiency results ($\approx 97.3\%$) with small statistical uncertainties.
• Bias Testing: The paper demonstrates that while kinematic correlations (e.g., between $p_T$ and invariant mass) exist and can bias sWeights, the impact on the final efficiency ratio is often negligible if the bias affects TIS and TOS categories equally.
• Performance: The tool successfully handles the complexity of the LHCb trigger lines (HLT1 and HLT2) and provides a robust workflow for both direct efficiency estimation and simulation correction.

5. Significance

This paper represents a critical step forward in LHCb data analysis infrastructure:

1. Standardization: By providing a single, validated implementation of the TISTOS method, it ensures consistency across different physics analyses and reduces the risk of implementation errors.
2. Efficiency: It drastically lowers the barrier to entry for trigger efficiency studies, allowing physicists to focus on physics results rather than statistical machinery.
3. Precision: The rigorous treatment of statistical uncertainties and systematic biases (via phase-space binning and background modeling) enhances the precision of LHCb measurements, particularly for rare processes where trigger corrections are vital.
4. Future-Proofing: The framework is designed to handle the high-luminosity data of LHCb Run 3 and beyond, supporting the fully software-based trigger system.

In summary, the paper bridges the gap between theoretical trigger efficiency frameworks and practical application, offering a robust, open-source tool that standardizes and accelerates precision measurements in heavy-flavour physics.