$B$-jet fragmentation with $B^{\pm} \to J/\psi K^{\pm}$ decays in $\sqrt{s} = 13$ TeV $pp$ collisions at LHCb

Using 5.4 fb$^{-1}$ of 13 TeV proton-proton collision data collected by the LHCb detector, this study measures the jet fragmentation functions and radial profiles of $B^{\pm}$ mesons reconstructed via $J/\psi K^{\pm}$ decays, revealing an increasing contribution from gluon fragmentation as jet transverse momentum rises.

The Gist

Imagine you are a detective trying to understand how a massive, chaotic explosion happens inside a tiny, invisible box. In the world of particle physics, that "explosion" is a collision between two protons (the building blocks of atoms) traveling at nearly the speed of light.

This paper from the LHCb collaboration at CERN is essentially a forensic report on what happens after that explosion, specifically focusing on a very heavy, rare type of particle called a B-meson (pronounced "B-meson").

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Big Picture: The "Shrapnel" Problem

When two protons smash together, they don't just bounce off each other. They shatter into a shower of smaller particles. Think of it like two watermelons smashing into each other at high speed. You get a spray of seeds, rind, and juice flying everywhere.

In physics, these "seeds" are called quarks and gluons. They are the fundamental ingredients of matter. But here's the catch: you can never see a single quark or gluon on its own. Nature has a rule (called confinement) that says they must immediately stick together to form a stable "ball" called a hadron (like a proton or a B-meson).

The big mystery this paper tackles is: How do these invisible ingredients decide to stick together? Do they form a tight, neat ball, or do they scatter widely? This process is called fragmentation.

2. The Detective Work: Tracking the "B-Meson"

The scientists used a giant, high-tech camera called the LHCb detector to watch these collisions. They were looking for a specific type of "shrapnel": the B-meson.

• The Clue: B-mesons are heavy and unstable. They don't last long. They quickly decay (fall apart) into other particles.
• The Fingerprint: The scientists looked for a very specific decay pattern: A B-meson turning into a J/ψ (which looks like a pair of muons, heavy cousins of electrons) and a K-meson (a type of kaon).
• The Data: They analyzed data from 2016–2018, which is like looking at a photo album of 5.4 billion of these collisions.

3. The Three Questions They Asked

Once they found a B-meson inside a "jet" (the spray of particles from the collision), they asked three specific questions to understand the fragmentation process. They used three variables, which we can think of as a 3D map:

A. The "Speed Ratio" ($z$): How much of the jet did it take?

Imagine a jet is a delivery truck full of packages. The B-meson is one specific package.

• Question: Did the B-meson take up most of the truck's cargo space (high momentum), or was it just a small package at the back?
• The Finding: As the jets got faster (more energy), the B-mesons started taking up less of the total momentum. This suggests that at high speeds, the "truck" is being driven by gluons (the glue holding quarks together) rather than just the heavy quark itself. It's like the truck is now carrying a lot of extra "glue" that dilutes the B-meson's share.

B. The "Sideways Wiggle" ($j_T$): How far did it drift?

Imagine the jet is a straight arrow flying through the air.

• Question: Did the B-meson fly straight with the arrow, or did it wobble to the side?
• The Finding: As the jets got faster, the B-mesons started wiggling more to the side. This is a sign that the "glue" (gluons) is doing more of the work. When a heavy quark splits into a pair of heavy quarks (a process called gluon splitting), they don't always fly perfectly straight; they kick sideways.

C. The "Spread" ($r$): How wide is the spray?

• Question: Is the B-meson hugging the center of the jet, or is it drifting toward the edge?
• The Finding: Similar to the sideways wiggle, the B-mesons were found further away from the center of the jet as the energy increased.

4. The "Dead Cone" and the "Ghost"

The paper mentions a concept called the "Dead Cone Effect."

• The Analogy: Imagine a heavy person running through a crowd. Because they are heavy, they can't turn quickly. They leave a "cone" of empty space behind them where no one can follow closely.
• The Physics: Heavy quarks (like the one inside the B-meson) are too heavy to emit radiation (gluons) at very small angles. They leave a "dead zone" around them.
• The Twist: The paper finds that while this effect exists, at very high energies, the gluons (which are massless and fast) start taking over the fragmentation process. The "ghost" of the gluon is becoming more important than the heavy quark itself.

5. The Computer Simulation vs. Reality

The scientists compared their real-world data to a computer simulation called Pythia.

• The Prediction: The computer thought the B-mesons would stay very close to the center of the jet and keep a high share of the momentum.
• The Reality: The real data showed the B-mesons were more spread out and had less momentum share than the computer predicted.
• The Conclusion: The computer models are a bit too optimistic about how "isolated" heavy particles are. In reality, the messy process of gluon fragmentation is more significant than the models thought.

Summary: Why Does This Matter?

Think of the universe as a giant Lego set. We know how the big bricks (protons) are made, but we don't fully understand the instructions for how the tiny, invisible pieces (quarks and gluons) snap together to form the final structures.

This paper provides a new, high-resolution instruction manual for how heavy particles form.

1. It confirms that gluons play a bigger role in making heavy particles than we thought, especially at high speeds.
2. It gives physicists new data to fix their computer models (like Pythia) so they can predict future experiments better.
3. It helps us understand the fundamental rules of the "strong force," the glue that holds our entire universe together.

In short: The scientists took a snapshot of a high-speed crash, found a heavy particle, and realized it wasn't behaving exactly how the textbooks said it should. This means the textbooks need a little update!

Technical Summary

1. Problem and Motivation

The hadronization process—where color-charged quarks and gluons form color-neutral bound states (hadrons)—remains one of the least understood aspects of Quantum Chromodynamics (QCD). While collinear fragmentation functions (FFs) are well-constrained for light quarks, heavy-flavor hadrons (specifically beauty/b-quarks) are poorly understood due to the additional non-negligible mass scale ($m_Q$) they introduce.

Previous measurements of $B$-hadron fragmentation were primarily conducted in $e^+e^-$ collisions (e.g., by ALEPH, OPAL, DELPHI). However, these environments lack direct access to gluon-initiated jets at leading order, making it difficult to constrain gluon fragmentation functions ($g \to B$). Proton-proton ($pp$) collisions offer a complementary environment where gluons are accessible at leading order. This paper aims to fill this gap by measuring jet fragmentation functions (JFFs) for $B^\pm$ mesons in $pp$ collisions at $\sqrt{s} = 13$ TeV, specifically focusing on:

• Collinear JFFs: Probability of a $B^\pm$ carrying a fraction $z$ of the jet momentum.
• Transverse-Momentum-Dependent (TMD) JFFs: Distribution of $B^\pm$ transverse momentum ($j_T$) relative to the jet axis.
• Radial Profile: The angular distance ($r$) of the $B^\pm$ from the jet axis.
• Joint Distributions: Correlations between these variables (e.g., $z$ vs. $j_T$).

2. Methodology

Data and Event Selection

• Dataset: Proton-proton collision data collected by the LHCb detector during 2016–2018, corresponding to an integrated luminosity of 5.4 fb$^{-1}$ at $\sqrt{s} = 13$ TeV.
• Decay Channel: $B^\pm \to J/\psi K^\pm$, with $J/\psi \to \mu^+\mu^-$.
• Kinematic Selection:
  • $B^\pm$ candidates reconstructed with invariant mass in $5.15 < m_{\mu\mu K} < 5.55$ GeV/$c^2$.
  • Jets reconstructed using the anti-$k_t$ algorithm with resolution parameter $R=0.5$ (AK5).
  • Jet kinematics restricted to $10 < p_{T,\text{jet}} < 100$ GeV/$c$ and $2.5 < y_{\text{jet}} < 4.0$ (forward rapidity region).
  • Only events with exactly one reconstructed primary vertex (PV) are used.

Signal Extraction

• Sideband Subtraction: To isolate the signal from combinatorial background, the analysis uses the invariant mass distribution of the $B^\pm$ candidate.
  • Signal Region: Defined dynamically based on $p_{T,B^\pm}$ (ranging from $2\sigma$ to $4\sigma$ around the peak).
  • Background Modeling: A second-order Chebyshev polynomial models the combinatorial background, while the signal is modeled by the sum of two Double-Sided Crystal Ball (DSCB) functions.
  • Procedure: Distributions of observables are measured in the signal region and subtracted by a scaled version of the sideband distribution to remove fake contributions.

Corrections and Unfolding

To correct for detector effects (resolution, acceptance, efficiency), a multi-step procedure is applied:

1. Purity Correction: Accounts for fake $B$-jets entering the sample.
2. Unfolding: An iterative Bayesian unfolding procedure (using the RooUnfold package) corrects for detector smearing.
   • Response matrices (RM) are constructed in 4D (for 1D observables) and 6D (for joint distributions) spanning $(z, j_T, r, p_{T,\text{jet}})$.
   • The prior distribution is based on Pythia 8 simulations but is reweighted using data to minimize model bias.
3. Efficiency Correction: Corrects for missed $B$-jets due to tracking, trigger, and Particle Identification (PID) inefficiencies. These are derived using data-driven methods (e.g., tag-and-probe) and applied event-by-event.

Systematic Uncertainties

Major sources include:

• Jet Energy Scale/Resolution (JES/JER): Determined using $Z \to \mu^+\mu^-$ + jet events. This is a dominant uncertainty, particularly at low $p_{T,\text{jet}}$.
• Unfolding Procedure: Variations in the number of iterations and the prior distribution shape.
• Signal Extraction: Variations in the fit model (e.g., using Student's $t$-distribution) and sideband definitions.
• Efficiency: Statistical uncertainties in calibration samples and hadronic interaction effects.

3. Key Contributions

This paper presents the first measurement of the following for $B^\pm$ mesons in jets:

1. TMD Jet Fragmentation Function ($F(z, j_T)$): The joint distribution of collinear momentum fraction and transverse momentum.
2. Joint Distributions: Correlations between $(z, r)$ and $(j_T, r)$.
3. Extended Kinematic Reach: Measurements extend down to $p_{T,\text{jet}} = 10$ GeV/$c$ (compared to 50 GeV/$c$ in previous ATLAS measurements) and cover the forward rapidity region ($2.5 < y < 4.0$), which is sensitive to gluon splitting.

4. Results

Collinear Fragmentation ($z$)

• The measured distribution $F(z)$ shows an increase in the JFF at low $z$ as $p_{T,\text{jet}}$ increases.
• Interpretation: This trend suggests a growing contribution from gluon fragmentation ($g \to B^\pm$) via an intermediate $g \to b\bar{b}$ splitting. In this process, the $b$-quark carries only a fraction of the gluon's momentum, leading to a lower $z$ for the resulting $B$-meson.
• Comparison with Simulation: Pythia 8.186 tends to overpredict the production of isolated $B^\pm$ mesons (high $z$) and underpredict the low-$z$ component, indicating that the simulation may underestimate the contribution of gluon splitting to heavy-flavor production in the parton shower.

Transverse Momentum ($j_T$) and Radial Profile ($r$)

• $j_T$ Broadening: The $j_T$ distribution broadens as $p_{T,\text{jet}}$ increases. This is attributed to both kinematic reach and the increasing contribution of gluon fragmentation, where the $g \to b\bar{b}$ splitting imparts transverse momentum to the $b$ and $\bar{b}$ relative to the gluon axis.
• Radial Profile ($r$): $B^\pm$ mesons are typically found close to the jet axis (small $r$), consistent with high collinear momentum fractions. However, the profile also broadens with increasing $p_{T,\text{jet}}$, further supporting the gluon fragmentation hypothesis.

Correlations

• $(z, j_T)$: A strong anticorrelation is observed: $B^\pm$ mesons with high $z$ (carrying most of the jet momentum) tend to have low $j_T$, while those with low $z$ have higher $j_T$.
• $(z, r)$: Similarly, an anticorrelation exists between $z$ and radial distance $r$.
• $(j_T, r)$: A strong correlation is observed; higher transverse momentum relative to the jet axis corresponds to a larger angular separation from the jet axis.

5. Significance

• QCD Factorization Tests: These measurements provide stringent tests of QCD factorization theorems in the heavy-flavor sector, where the heavy quark mass introduces a new scale.
• Gluon Fragmentation Constraints: The forward rapidity coverage of LHCb and the low $p_{T,\text{jet}}$ reach are crucial for constraining the gluon-to-beauty fragmentation function, a parameter poorly constrained by $e^+e^-$ data.
• Model Validation: The discrepancy between data and Pythia 8 (overprediction of isolated production) highlights the need for improved modeling of heavy-flavor production in parton showers, particularly regarding in-shower $g \to b\bar{b}$ splittings.
• Global Analyses: The results, including the full covariance matrices provided in HEPData, will be integrated into global analyses of fragmentation functions, improving the precision of heavy-flavor hadronization models used in future high-energy physics simulations.

In summary, this work provides a comprehensive, multi-dimensional view of how $B^\pm$ mesons fragment within jets, offering new insights into the interplay between heavy quark mass, gluon dynamics, and the hadronization process in high-energy proton-proton collisions.