Systematic study of bottomonium production in proton-proton collisions at LHC energies

This paper presents a comprehensive leading-order NRQCD study of $\Upsilon(nS)$ production in proton-proton collisions at LHC energies, demonstrating that theoretical predictions for cross-sections and ratios, including feed-down contributions, align well with experimental data from ALICE, ATLAS, CMS, and LHCb while exhibiting saturation behavior at high transverse momenta.

The Gist

Imagine the Large Hadron Collider (LHC) as the world's most powerful "particle blender." Scientists smash protons together at nearly the speed of light to see what tiny pieces fly out. One of the most interesting things they look for is a specific type of particle called bottomonium (specifically the $\Upsilon$ or "Upsilon" family).

Think of bottomonium as a "heavy-duty atom" made of two super-heavy particles (bottom quarks) holding hands. Because they are so heavy, they are like the "elephants" of the particle world, making them easier to study than their lighter cousins.

Here is a simple breakdown of what this paper does, using some everyday analogies:

1. The Goal: Predicting the "Family Reunion"

The author, Biswarup Paul, is trying to predict exactly how many of these "Upsilon elephants" are created when protons smash together. But there's a catch: these elephants don't always appear directly.

• Direct Production: Imagine a baby elephant being born right in the middle of the collision.
• Feed-down (The "Grandparent" Effect): Sometimes, a bigger elephant (a heavier version of the Upsilon) is born first. This bigger elephant is unstable and quickly sheds some weight (decays) to become a smaller, more stable Upsilon.

The paper calculates both the "babies born directly" and the "babies born from the transformation of bigger relatives." The author found that if you ignore the "grandparents" (the heavier states decaying into lighter ones), your prediction is wrong. You have to count the whole family tree to get the right numbers.

2. The Tool: The "Rulebook" (NRQCD)

To make these predictions, the author uses a theoretical framework called NRQCD (Non-Relativistic Quantum Chromodynamics).

• The Analogy: Think of the laws of physics as a massive, complex rulebook for how particles interact. This rulebook has two parts:
  1. The Hard Part (Short Distance): This is like calculating the initial crash of two cars. We know the rules of the crash very well because it happens fast and cleanly.
  2. The Soft Part (Long Distance): This is like the messy aftermath—how the metal crumples, how the airbags deploy, and how the wreckage settles. This part is fuzzy and hard to calculate directly.

The author uses a method called "Factorization" to separate these two. They calculate the crash perfectly, and then use a "fudge factor" (called a Matrix Element) to estimate the messy aftermath based on real-world data. It's like knowing exactly how hard the car hit, and then using past crash data to guess how much the car will bounce.

3. The Experiment: Checking the Scoreboard

The author ran these calculations for collisions at the LHC (the world's biggest particle collider) at different energy levels (7 and 13 TeV). They then compared their "theoretical scoreboard" against the actual "real-world scoreboard" kept by four giant teams: ALICE, ATLAS, CMS, and LHCb.

• The Result: The author's predictions matched the real data very well, especially when the particles were moving fast (high momentum).
• The "Saturation" Surprise: The most interesting finding is what happens at very high speeds. As the particles get faster and faster, the ratio of different types of Upsilon particles stops changing. It hits a "ceiling" or a "flatline."
  • The Metaphor: Imagine pouring water into a bucket with a hole. At first, the water level rises quickly. But eventually, the water flows out as fast as it comes in, and the level stays constant. The paper suggests that at very high speeds, the production of these particles hits a similar "steady state" where the rules of the game become universal, regardless of which specific type of Upsilon you are looking at.

4. Why Does This Matter?

You might ask, "Why do we care about these heavy elephants?"

• Testing the Rules: By seeing if the predictions match the data, scientists are testing the fundamental laws of the universe (Quantum Chromodynamics). If the math works, it means we understand how the "glue" holding the universe together works.
• The "Puzzle": While this paper solved the "how many" puzzle (the cross-sections), the author notes that the "how they spin" puzzle (polarization) is still tricky. It's like knowing exactly how many cars crashed, but still not fully understanding why they spun in the specific directions they did.
• A Clean Lab: Because bottom quarks are so heavy, they are less "noisy" than lighter particles. This makes them a cleaner, clearer laboratory to study the fundamental forces of nature.

In a Nutshell

This paper is a detailed report card for a specific type of heavy particle. The author used a sophisticated mathematical model to predict how often these particles are made and how they transform into each other. The model passed the test with flying colors, matching real-world data from the world's most powerful particle accelerators. Most importantly, it discovered that at very high speeds, the production of these particles follows a simple, predictable pattern, suggesting a deep, underlying order to the chaotic world of particle collisions.

Technical Summary

1. Problem Statement

The production of heavy quarkonia (specifically bottomonium, $\Upsilon(nS)$) in high-energy hadronic collisions is a critical testing ground for Quantum Chromodynamics (QCD). While the Non-Relativistic QCD (NRQCD) factorization formalism is the standard framework for describing these processes, several challenges persist:

• Discrepancies in Polarization: Theoretical predictions for quarkonium polarization often fail to match experimental data, even at Next-to-Leading Order (NLO).
• Universality of LDMEs: Different global fits yield inconsistent Color-Octet (CO) Long-Distance Matrix Elements (LDMEs), questioning their universality.
• Feed-down Complexity: Experimental measurements of "prompt" production include contributions from direct production and decays of higher excited states (feed-down). A systematic, transparent calculation isolating these contributions at Leading Order (LO) is necessary to establish a robust baseline before tackling more complex NLO corrections or polarization puzzles.
• High-$p_T$ Behavior: The behavior of cross-section ratios at very high transverse momentum ($p_T$) requires further investigation to understand the transition to fragmentation-dominated regimes.

2. Methodology

The author employs the Leading Order (LO) NRQCD factorization formalism to calculate bottomonium production in proton-proton ($pp$) collisions at LHC energies ($\sqrt{s} = 7$ and $13$ TeV).

• Theoretical Framework:

  • The cross-section is factorized into a perturbative short-distance part (calculable in pQCD) and a non-perturbative long-distance part (LDMEs).
  • The calculation includes both Color-Singlet (CS) and Color-Octet (CO) intermediate states.
  • The short-distance cross-sections are calculated using CTEQ6L Parton Distribution Functions (PDFs).
  • The strong coupling constant $\alpha_s$ is parametrized according to standard renormalization group equations.

• Feed-down Contributions:

  • The study explicitly distinguishes between direct production and feed-down contributions.
  • $\Upsilon(1S)$: Includes direct production plus feed-down from $\Upsilon(2S)$, $\Upsilon(3S)$, $\chi_{bJ}(1P)$, and $\chi_{bJ}(2P)$.
  • $\Upsilon(2S)$: Includes direct production plus feed-down from $\Upsilon(3S)$ and $\chi_{bJ}(2P)$.
  • $\Upsilon(3S)$: Treated as purely direct/prompt production, as no significant feed-down from higher states (like $\chi_{bJ}(3P)$) is included in this specific LO baseline.

• Parameters and Uncertainties:

  • LDMEs: CS LDMEs are derived from radial wave functions. CO LDMEs are extracted from global fits to experimental data (CDF, ALICE, ATLAS, CMS, LHCb) using a covariance-matrix method to account for correlations.
  • Scale Variation: Theoretical uncertainties are estimated by varying the renormalization ($\mu_R$) and factorization ($\mu_F$) scales independently within $0.5 < \xi_{R,F} < 2$. This scale variation is identified as the dominant source of uncertainty (up to 42%).
  • Other Uncertainties: Variations in the bottom quark mass ($m_b$), branching ratios, and PDF sets are also considered.

3. Key Contributions

• Comprehensive LO Baseline: Provides a transparent, LO-level calculation that isolates individual production channels (CS vs. CO) and feed-down mechanisms, serving as a reference for future NLO studies.
• Systematic Feed-down Analysis: Explicitly quantifies the impact of feed-down from $\chi_{bJ}$ and higher $\Upsilon$ states on the prompt $\Upsilon(1S)$ and $\Upsilon(2S)$ yields, demonstrating their necessity for reproducing experimental shapes.
• High-$p_T$ Saturation Discovery: Identifies and analyzes a clear saturation behavior in the production cross-section ratios ($\Upsilon(2S)/\Upsilon(1S)$, $\Upsilon(3S)/\Upsilon(1S)$, etc.) at $p_T \approx 40$ GeV, extending up to $130$ GeV in CMS data.
• Multi-Experiment Validation: Validates the model against a wide range of experimental data from ATLAS, CMS, LHCb, and ALICE across different rapidity regions (mid, near-forward, and forward) and collision energies.

4. Results

• Cross-Section Agreement:

  • The calculated differential cross-sections ($d\sigma/dp_T$) for $\Upsilon(1S)$, $\Upsilon(2S)$, and $\Upsilon(3S)$ show good agreement with experimental data from all four collaborations.
  • The agreement is particularly strong for $p_T > 4$ GeV. Below this threshold, discrepancies are more pronounced, likely due to non-perturbative effects not fully captured at LO.
  • The inclusion of feed-down contributions is found to be essential to reproduce the correct $p_T$-dependent shape of the prompt $\Upsilon(1S)$ and $\Upsilon(2S)$ cross-sections. Without them, the theoretical curves fail to match the data.

• Cross-Section Ratios:

  • The ratios $\Upsilon(2S)/\Upsilon(1S)$, $\Upsilon(3S)/\Upsilon(1S)$, and $\Upsilon(3S)/\Upsilon(2S)$ are well described by the model across the entire $p_T$ range ($p_T > 0$).
  • Saturation Behavior: A key finding is that these ratios exhibit a clear saturation (flattening) beyond $p_T \approx 40$ GeV. The ratios become nearly constant at high $p_T$, a trend confirmed by CMS data extending up to $p_T \approx 130$ GeV.

• Uncertainty Dominance:

  • The theoretical uncertainty bands are dominated by the choice of renormalization and factorization scales, which can vary the cross-section by up to 42% in the low-to-mid $p_T$ region.

5. Significance and Conclusion

• Validation of NRQCD: The study reaffirms the effectiveness of the NRQCD factorization approach in describing bottomonium production, even at LO, provided that feed-down contributions are rigorously included.
• Physical Interpretation of Saturation: The observed saturation of cross-section ratios at high $p_T$ is interpreted as the transition to a fragmentation-dominated regime. In this regime, short-distance partonic cross-sections become universal across different quarkonium states, and differences are encoded only in constant non-perturbative matrix elements. This confirms the asymptotic scaling behavior predicted by NRQCD.
• Future Directions: While LO provides a solid baseline for cross-sections, the paper notes that polarization observables and precise descriptions at low $p_T$ likely require NLO corrections. The established baseline is crucial for future studies involving proton-nucleus and heavy-ion collisions to disentangle cold and hot nuclear matter effects.

In summary, this paper provides a robust, systematic LO NRQCD analysis that successfully describes LHC bottomonium data, highlights the critical role of feed-down contributions, and identifies a universal scaling behavior in cross-section ratios at high transverse momenta.