Measurement of event shape variables using charged particles inside jets in proton-proton collisions at $\sqrt{s}$ = 13 TeV

Using 138 fb$^{-1}$ of proton-proton collision data at $\sqrt{s}$ = 13 TeV collected by the CMS detector, this paper presents a measurement of five event shape variables derived from charged particles inside jets, showing general agreement between the corrected distributions and various theoretical predictions for multijet production.

The Gist

The Big Picture: A Cosmic Smackdown

Imagine the Large Hadron Collider (LHC) at CERN as the world's most powerful slingshot. Scientists fire two tiny particles (protons) at each other at nearly the speed of light. When they collide, it's like smashing two Swiss watches together at full speed.

The result isn't just a broken watch; it's an explosion of tiny fragments (quarks and gluons) that fly out in all directions. These fragments quickly clump together to form "jets" of new particles, like debris from a car crash forming piles of metal.

This paper is about measuring the shape of that debris field.

The Problem: Too Much Noise

In the real world, when you try to study a car crash, you have to deal with a lot of background noise: other cars passing by, wind, and rain. In the LHC, this "noise" is called pileup. Because the slingshot fires so fast, dozens of other collisions happen at the exact same time, creating a messy cloud of extra particles that muddies the data.

The Solution: The scientists decided to ignore the "dust" (neutral particles) and only look at the "shrapnel" that has an electric charge. Why? Because charged particles leave a clear trail in the detector, allowing scientists to trace them back to the exact moment of the main crash, ignoring the background noise. It's like looking only at the license plates of the cars involved in the crash, ignoring the wind and the rain.

The Five "Shapes" They Measured

The researchers looked at five specific ways to describe the shape of the particle explosion. Think of these as different ways to describe the aftermath of a party:

1. Transverse Thrust ($\tau_\perp$): The "Back-to-Back" Test

   • The Analogy: Imagine two people throwing snowballs at each other. If they are perfectly aligned, the snowballs fly straight back and forth. If they throw wildly in all directions, the snowballs scatter everywhere.
   • What they found: They measured how "straight" the jets were. If the jets are perfectly back-to-back, the score is low. If they are scattered like a spherical explosion, the score is high.

2. Third-Jet Resolution ($Y_{23}$): The "Third Wheel" Detector

   • The Analogy: Imagine a dance floor with two main couples dancing perfectly. Suddenly, a third person crashes the dance. This variable measures how "awkward" that third person is. Are they just a tiny, shy dancer on the edge, or are they a massive, loud intruder?
   • What they found: This tells them how often a third jet (a third "couple" of particles) appears in the collision.

3. Jet Broadening ($B_{tot}$): The "Spread Out" Meter

   • The Analogy: Imagine throwing a handful of confetti. If you throw it tight, it falls in a small pile. If you throw it wide, it covers the whole room. This measures how "spread out" the energy is in the collision.
   • What they found: They checked if the energy was concentrated in a tight beam or if it was scattered loosely.

4. Total Jet Mass ($\rho_{tot}$): The "Heaviness" Check

   • The Analogy: If you have a pile of feathers and a pile of bricks that take up the same amount of space, the bricks are "heavier" (more massive). This measures how much "stuff" is packed into the jets.
   • What they found: They looked at the mass of the particle clusters to see if the models predicted the right amount of "stuff."

5. Total Transverse Jet Mass ($\rho^T_{tot}$): The "Sideways Weight"

   • The Analogy: Similar to the previous one, but this only cares about the weight of the particles moving sideways, ignoring how fast they are moving forward or backward. It's like weighing a suitcase while it's lying on its side.

The Great Debate: Theory vs. Reality

The scientists took their measurements (the "Reality") and compared them to three different computer simulations (the "Theories"):

• PYTHIA 8: A very popular, well-tuned simulation.
• HERWIG 7: Another famous simulation that thinks about particle interactions slightly differently.
• MADGRAPH5: A simulation that tries to calculate the very first split-second of the crash with extreme precision.

The Results:

• The Good News: For the "Back-to-Back" and "Sideways Weight" measurements, the computer models (especially PYTHIA 8) matched the real data almost perfectly. The theories are doing a great job here.
• The Bad News: For the "Spread Out" and "Heaviness" measurements, the models started to drift apart from reality.
  • PYTHIA 8 tended to think the collisions were "heavier" and more "spread out" than they actually were.
  • MADGRAPH5 tended to think there were fewer "third wheels" (extra jets) than there actually were.

The Takeaway: We Need Better Recipes

Think of the computer models as recipes for predicting how the universe behaves.

• For simple, clean collisions, the recipes are perfect.
• For messy, complex collisions (where lots of particles are created), the recipes are slightly off. They are either adding too much salt (mass) or not mixing the ingredients (hadronization) quite right.

Why does this matter?
The scientists aren't just trying to be pedantic. If the recipes are wrong, it means our understanding of the "glue" that holds the universe together (Quantum Chromodynamics) is incomplete. By finding exactly where the recipes fail, they can tweak the ingredients to make the models better. This is crucial because if we can't predict the "background noise" of the universe perfectly, we might miss a signal of something completely new and exciting (like dark matter or new physics) hiding in the data.

In short: The CMS team took a very messy, high-speed crash, filtered out the noise, measured the shape of the debris in five different ways, and told the computer models: "You're doing great, but you need to fix your recipe for the messy parts."

Technical Summary

1. Problem and Motivation

Quantum Chromodynamics (QCD) describes the strong interaction, but while perturbative QCD accurately models high-energy parton production, the transition from partons to stable hadrons (hadronization) and parton showers (PS) relies on phenomenological models. Validating these models is crucial for understanding the Standard Model and searching for new physics.

Previous studies of Event Shape Variables (ESVs) at the LHC primarily used reconstructed jets as inputs. However, jet-based measurements introduce theoretical uncertainties related to jet clustering algorithms, collinear/soft emissions, and reduced sensitivity to the hadronization process. Furthermore, the high pileup conditions (average 34 additional interactions per crossing) in Run 2 complicate jet reconstruction.

The core problem addressed: How to measure ESVs with higher precision and reduced sensitivity to detector effects and pileup, thereby providing a more stringent test of hadronization models and QCD predictions.

The proposed solution: This paper measures five ESVs using charged particles inside central jets rather than the jets themselves. Charged particles are associated with specific collision vertices, making their momentum measurement significantly less affected by pileup than neutral particles or full jet energy.

2. Methodology

Data and Simulation

• Dataset: Proton-proton collision data collected by the CMS detector at $\sqrt{s} = 13$ TeV during Run 2 (2016–2018), corresponding to an integrated luminosity of 138 fb$^{-1}$.
• Event Selection: Multijet events were selected using High-Level Trigger (HLT) dijet triggers based on the average transverse momentum ($H_{T,2}$) of the two leading jets. Events required $\geq 2$ jets with $p_T > 30$ GeV within $|\eta| \leq 2.1$.
• Inputs: Only charged particles within the selected jets (and within the tracker acceptance $|\eta| < 2.5$) were used to calculate the variables.
• Monte Carlo (MC) Generators: Predictions were compared against three leading-order (LO) generators:
  • PYTHIA 8 (CP5 tune): Uses $p_T$-ordered parton showers and the Lund string model for hadronization.
  • HERWIG 7 (CH3 tune): Uses angular-ordered parton showers and the cluster model for hadronization.
  • MADGRAPH5_aMC@NLO + PYTHIA 8: Generates tree-level matrix elements (2$\to$2, 2$\to$3, 2$\to$4) matched to PYTHIA 8 showers.

Event Shape Variables (ESVs)

Five infrared- and collinear-safe variables were defined using the four-momenta of charged particles:

1. Complement of Transverse Thrust ($\tau_\perp$): Measures the alignment of particles in the transverse plane.
2. Third Jet Resolution Parameter ($Y_{23}$): Sensitive to jet clustering and substructure; determines the scale at which a 3-jet event becomes a 2-jet event.
3. Total Jet Broadening ($B_{tot}$): Measures the spread of momentum perpendicular to the thrust axis.
4. Total Jet Mass ($\rho_{tot}$): Normalized squared invariant mass of particles in hemispheres.
5. Total Transverse Jet Mass ($\rho^T_{tot}$): The transverse plane equivalent of total jet mass.

Unfolding and Corrections

• Detector Effects: Raw distributions were corrected for detector resolution, acceptance, and efficiency using a 2D unfolding procedure (simultaneously unfolding the ESV and $H_{T,2}$) via the TUnfold framework.
• Response Matrices: Constructed using PYTHIA 8 (CP5) simulation passed through a full GEANT4 detector simulation.
• Systematic Uncertainties: Major sources included Jet Energy Scale (JES), Jet Energy Resolution (JER), pileup subtraction, track reconstruction efficiency, and theoretical model dependence (varying between PYTHIA, HERWIG, and MG5). Total relative uncertainties ranged from 2% to 6%.

3. Key Contributions

• First CMS Measurement with Charged Particles in Jets: This analysis pioneers the use of charged particles inside jets for ESV measurements at the LHC, demonstrating a method to mitigate pileup effects and isolate hadronization dynamics more effectively than jet-only approaches.
• Comprehensive Unfolding: The application of 2D unfolding allows for a direct comparison with particle-level theoretical predictions across a wide phase space of $H_{T,2}$ (from ~83 GeV to >573 GeV).
• Model Discrimination: The study provides a detailed breakdown of how different QCD models (specifically hadronization models) perform against data for variables sensitive to different physical processes (hard scattering vs. non-perturbative evolution).

4. Results

The unfolded distributions were compared with the MC predictions across eight $H_{T,2}$ ranges:

• $\tau_\perp$ and $\rho^T_{tot}$ (Transverse Momentum Sensitive):

  • PYTHIA 8 and HERWIG 7 show good overall agreement with data for both back-to-back 2-jet events and spherical multijet events.
  • Since these variables rely heavily on transverse momentum, they are less sensitive to the specific details of hadronization modeling in the longitudinal direction.

• $\rho_{tot}$, $B_{tot}$, and $Y_{23}$ (Full Momentum/Structure Sensitive):

  • PYTHIA 8 tends to overestimate the fraction of multijet, spherical events (predicting broader distributions than observed).
  • HERWIG 7 shows similar behavior to PYTHIA 8 for these variables.
  • MADGRAPH5_aMC@NLO + PYTHIA 8 performs well at lower $H_{T,2}$ but overestimates the 2-jet fraction (underestimating multijet events) at higher $H_{T,2}$. This suggests issues in the matching of Matrix Elements (ME) with Parton Showers (PS) at high energies.

• Energy Dependence:

  • Agreement between data and PYTHIA 8 improves as $H_{T,2}$ increases.
  • Agreement with HERWIG 7 remains relatively constant.
  • Agreement with MADGRAPH5_aMC@NLO + PYTHIA 8 worsens at higher energies.

5. Significance and Conclusion

• Hadronization Modeling: The discrepancy in variables sensitive to full momentum ($\rho_{tot}, B_{tot}$) versus those sensitive only to transverse momentum ($\tau_\perp, \rho^T_{tot}$) suggests that current models may have shortcomings in describing fragmentation and hadronization processes, particularly in how energy is distributed longitudinally within jets.
• Matrix Element Matching: The failure of the MG5+PYTHIA 8 combination at high $H_{T,2}$ highlights the challenges in correctly combining fixed-order calculations with parton showers for high-multiplicity final states.
• Future Impact: These results provide critical constraints for tuning future MC event generators. By isolating charged particles, the analysis reduces pileup-induced systematic uncertainties, offering a cleaner probe of QCD dynamics. The authors conclude that while general agreement exists, the understanding of energy flow in events across different energy scales requires further refinement, particularly in non-perturbative QCD modeling.

Data Availability: Tabulated results are available in the HEPData record associated with this analysis.