Inclusive jet cross section in $pp$ collisions at $\sqrt{s} = 200$ and $510$ GeV

This paper presents double-differential inclusive jet cross sections measured by the STAR detector in proton-proton collisions at $\sqrt{s} = 200$ and $510$ GeV, providing new constraints on the poorly known gluon parton distribution function in the proton through comparisons with next-to-next-to-leading order perturbative QCD calculations.

The Gist

Imagine the proton not as a solid marble, but as a bustling, chaotic city. Inside this city, there are tiny citizens called quarks (the "up" and "down" workers) and a massive, invisible cloud of energy called gluons that holds the city together. Gluons are the glue, but they are also the most mysterious part of the city because they are hard to count and map.

This paper is like a report from a team of cosmic detectives (the STAR Collaboration) who went to a giant particle collider (the RHIC) to take a snapshot of this city in action. They smashed two protons together at incredibly high speeds and watched what happened.

Here is the story of their investigation, broken down simply:

1. The Crash and the "Jets"

When two protons crash, it's like two high-speed trains colliding. The tiny particles inside them (the partons) bounce off each other violently. Because these particles can't exist alone for long, they immediately turn into sprays of new particles.

Think of these sprays as fireworks or shrapnel flying out in a tight cone. In physics, we call these cones "Jets." By studying the direction and speed of these jets, the detectives can figure out what kind of particle caused the crash in the first place. In this specific experiment, they were mostly looking for gluons, the "glue" citizens.

2. The Challenge: The "Background Noise"

The problem with watching these fireworks is that the collision isn't clean. There is a lot of "background noise."

• The Underlying Event: Imagine trying to hear a single violin in a stadium full of people cheering, clapping, and shouting. The "cheering" is the Underlying Event (UE)—random debris from the collision that isn't part of the main jet.
• The Solution: The team used a clever trick called the "Off-Axis Cone" method. Instead of just looking at the jet, they looked at the empty space 90 degrees away from it. They measured how much "noise" was in that empty space and subtracted it from the jet's energy. It's like measuring the background noise in a quiet room next door and subtracting that amount from your recording to hear the violin clearly.

3. The Detective Work: Unfolding the Truth

The detectors (the "cameras" of the experiment) aren't perfect. Sometimes they miss a particle, or they think a particle is heavier than it is.

• The Unfolding Process: To fix this, the team used a massive computer simulation (a "digital twin" of the experiment). They compared what the camera actually saw with what should have happened in the perfect simulation. By mathematically reversing the camera's mistakes, they could "unfold" the data to reveal the true, unbiased picture of the jets.

4. The Big Discovery: Mapping the Glue

The main goal was to map the gluon distribution. Think of the proton as a bag of marbles. We know exactly how many red marbles (up quarks) and blue marbles (down quarks) are in there. But the bag is also filled with invisible, shifting sand (gluons).

• The Gap in Knowledge: Previous experiments at giant machines (like the LHC in Europe) mapped the sand when the proton was moving very fast, but they missed the "middle" speed range.
• The STAR Contribution: This experiment ran at specific speeds (200 and 510 GeV) that act like a "Goldilocks zone." They found a lot of gluons in a range that other machines had missed. It's like finding a hidden neighborhood in a city that no one had a map for yet.

5. Did the Theories Match?

The detectives compared their real-world data with the "Theoretical Blueprints" (mathematical models called QCD and computer programs like Pythia).

• The Result: The blueprints were close, but not perfect. The real jets were slightly different than the models predicted.
• Why it matters: This tells the scientists that their blueprints need a little tweaking. Just like a GPS app that gets you to the right city but misses the turn, the models need to be updated to account for the specific behavior of gluons at these speeds.

Why Should You Care?

You might ask, "Why do we need to know about invisible glue in tiny particles?"

1. Understanding the Universe: Protons make up almost all the visible matter in the universe. If we don't understand how they are built, we don't fully understand reality.
2. The "Primordial Soup": This data serves as a "control group" for studying Quark-Gluon Plasma. When scientists smash heavy gold atoms together, they create a super-hot soup that existed just after the Big Bang. To know if the soup is behaving strangely, they first need to know exactly how protons behave when they don't make soup. This paper provides that baseline.
3. Better Simulations: The data helps tune the computer games (simulations) physicists use to predict future experiments, making our future discoveries more accurate.

In a nutshell: The STAR team took a high-speed photo of colliding protons, cleaned up the blurry background noise, and used math to reveal a hidden map of the "glue" holding our universe together. They found that our current maps were slightly off, giving scientists a new puzzle to solve and a better foundation for understanding the cosmos.

Technical Summary

1. Problem Statement and Motivation

The internal structure of the proton is described by Parton Distribution Functions (PDFs), which quantify the momentum fraction ($x$) carried by quarks and gluons. While Deep Inelastic Scattering (DIS) has provided extensive constraints on PDFs, the gluon distribution at intermediate-to-high momentum fractions ($x \gtrsim 0.1$) remains less constrained, particularly compared to the precision available at TeV-scale colliders (Tevatron, LHC).

• The Gap: Existing inclusive jet measurements at the TeV scale are sensitive to low-$x$ gluons. Measurements at lower energies ($\sqrt{s} = 200$ and $510$ GeV) at the Relativistic Heavy Ion Collider (RHIC) are uniquely sensitive to gluons with $x \gtrsim 0.1$.
• The Challenge: Previous measurements at RHIC were limited in kinematic coverage or pseudorapidity ($\eta$). Furthermore, separating the "underlying event" (UE) contributions (beam remnants and multiple parton interactions) from the hard scattering jet signal is difficult. At RHIC energies, UE contributions differ significantly from TeV-scale colliders, requiring specific correction methods rather than generic extrapolations from higher-energy tunes.
• Goal: To report the first double-differential inclusive jet cross sections at $\sqrt{s} = 510$ GeV and an extended measurement at $\sqrt{s} = 200$ GeV, providing high-precision data to constrain the gluon PDF, tune Monte Carlo (MC) generators, and serve as a baseline for quark-gluon plasma studies.

2. Methodology

Experimental Setup

• Detector: Data were collected using the STAR detector at RHIC in 2012. Key components included the Time Projection Chamber (TPC) for charged particle tracking ($|\eta| < 1.3$) and the Barrel/Endcap Electromagnetic Calorimeters (BEMC/EEMC) for energy measurement.
• Luminosity: Integrated luminosities of $17 \text{ pb}^{-1}$ ($\sqrt{s}=200$ GeV) and $42 \text{ pb}^{-1}$ ($\sqrt{s}=510$ GeV) were achieved. Luminosity was determined via Van der Meer scans using Zero-Degree Calorimeters (ZDC), with total systematic uncertainties of 5.6% and 5.2%, respectively.
• Triggers: Events were selected using Jet Patch (JP) triggers in the BEMC/EEMC, with thresholds adjusted for each energy (e.g., 3.5–7.3 GeV at 200 GeV; 5.4–14.4 GeV at 510 GeV).

Jet Reconstruction and Correction

• Algorithm: Jets were reconstructed using the anti-$k_T$ algorithm (FastJet) with a resolution parameter $R=0.5$. Inputs included TPC tracks (charged) and calorimeter towers (neutral).
• Underlying Event (UE) Correction: A novel off-axis cone method was applied. Instead of treating UE as a global non-perturbative correction, the UE transverse momentum density ($\rho$) was measured in two cones located $\pm 90^\circ$ in azimuth ($\phi$) from the jet axis but at the same pseudorapidity ($\eta$). The shift $\delta p_T = A_{jet}(\rho_1 + \rho_2)/2$ was subtracted from the raw jet $p_T$. This allowed for a data-driven, $\eta$-dependent correction, separating UE effects from hadronization.
• Unfolding: To correct for detector resolution and acceptance, an unfolding process was employed.
  • Embedding: Pythia 6 events (Perugia 2012 tune) were embedded into real zero-bias data to simulate detector response.
  • Matrix Inversion: An unfolding matrix $A$ was constructed from matched detector-particle jet pairs. The particle-level spectrum was recovered via $x = A^{-1}b$.
  • Efficiency: Corrections were applied for tracking efficiency and trigger biases.

Theoretical Comparison

• Hadronization Correction: A multiplicative factor ($C_{had}$) was derived from Pythia to bridge the gap between particle-level data and parton-level perturbative QCD (pQCD) calculations.
• pQCD Calculations: Comparisons were made against Next-to-Next-to-Leading Order (NNLO) calculations using the NNLOJet framework and FastNLO grids. Various modern PDF sets were tested: CT18, MSHT20, NNPDF4.0, and HERAPDF2.0.

3. Key Contributions

1. First STAR Measurement at 510 GeV: This paper presents the first inclusive jet cross-section results from STAR at $\sqrt{s} = 510$ GeV, extending the kinematic reach significantly.
2. Separation of UE and Hadronization: By employing the off-axis cone method, the analysis explicitly separates the underlying event correction from the hadronization correction. This is a methodological improvement over previous analyses where these were often combined into a single non-perturbative correction.
3. Extended Kinematic Coverage: The data covers $0.07 < x_T < 0.5$ at 200 GeV and $0.03 < x_T < 0.31$ at 510 GeV (where $x_T = 2p_T/\sqrt{s}$), directly probing the poorly constrained high-$x$ gluon region.
4. Systematic Uncertainty Breakdown: The paper provides a detailed breakdown of uncertainties, identifying the Jet Energy Scale (JES) as the dominant systematic source (reaching up to 20-35% at high $p_T$), followed by tracking efficiency and unfolding uncertainties.

4. Results

• Cross Sections: Double-differential cross sections ($d^2\sigma/dp_T d\eta$) were measured in two $\eta$ ranges ($|\eta| < 0.5$ and $0.5 < |\eta| < 0.9$) across a wide $p_T$ range (up to ~80 GeV/c at 510 GeV).
• Comparison with Monte Carlo:
  • Pythia 6 (STAR tune): Describes the shape of the cross-section well but underpredicts the magnitude by ~30%.
  • Pythia 8 (Detroit tune): Overestimates the cross-section by 20–40%, indicating that current tunes based on TeV-scale data do not extrapolate perfectly to RHIC energies.
• Comparison with NNLO pQCD:
  • When compared to NNLO calculations with modern PDFs (CT18, MSHT20, NNPDF4.0), the measured cross sections are larger than predictions at low $p_T$ and smaller (by ~20%) at high $p_T$.
  • HERAPDF2.0 Exception: The data agrees remarkably well with predictions using the HERAPDF2.0 set (fitted primarily from DIS data), suggesting that modern PDF sets incorporating LHC data may over-constrain the gluon distribution at high $x$.
• Scale Breaking: The ratio of invariant cross sections between 200 and 510 GeV shows scale-breaking effects (deviations from unity) similar to those observed at the Tevatron and LHC, providing a test of the factorization scale dependence in pQCD.

5. Significance

• Gluon PDF Constraints: These results provide critical constraints on the gluon PDF in the region $x > 0.1$, a region where TeV-scale data has limited sensitivity. The discrepancy with modern PDF sets suggests a need for re-evaluation of the gluon distribution at high $x$.
• Event Generator Tuning: The data offers a unique baseline for tuning Monte Carlo generators (like Pythia) specifically for RHIC energies, improving the modeling of underlying events and hadronization at lower $\sqrt{s}$.
• Heavy-Ion Baseline: Precise pp cross-sections are essential for calculating nuclear modification factors ($R_{AA}$) in heavy-ion collisions. This data serves as the fundamental reference for studying jet quenching and the properties of the Quark-Gluon Plasma (QGP) at RHIC.
• QCD Validation: The agreement with HERAPDF2.0 and the specific trends in scale-breaking provide a rigorous test of the perturbative QCD framework and the validity of factorization at these energy scales.