Energy Correlators Within Jets in Transversely Polarized Proton-Proton Collisions at $\sqrt{s} = 200$ GeV

The STAR collaboration at RHIC reports the first measurement of one- and two-point energy correlators within jets in transversely polarized proton-proton collisions at $\sqrt{s}=200$ GeV, observing significant spin-dependent asymmetries that provide a novel, precise probe of nucleon transversity and nonperturbative dynamics for future three-dimensional nucleon tomography.

The Gist

Imagine you are trying to understand the inner workings of a proton (a tiny particle inside an atom that makes up the core of every atom in your body). For a long time, physicists have known that protons aren't just solid marbles; they are chaotic, swirling storms of even smaller particles called quarks and gluons.

The big mystery is: How do these invisible, high-speed particles decide to stick together to form the visible matter we see? And how does the proton's "spin" (its internal rotation) influence this process?

This paper from the STAR Collaboration at the Relativistic Heavy Ion Collider (RHIC) is like a new, ultra-high-speed camera that finally lets us take a snapshot of this chaotic process in 3D.

Here is the breakdown of what they did, using simple analogies:

1. The Experiment: The "High-Speed Smash"

Think of the proton as a spinning top. The scientists at RHIC took two of these spinning tops and smashed them together at nearly the speed of light (200 GeV of energy).

• The Twist: They didn't just smash them randomly; they made sure the tops were spinning sideways (transversely polarized).
• The Result: When they collide, they don't just bounce off; they explode into a shower of new particles, forming a "jet" (a cone of debris shooting out).

2. The New Tool: "Energy Correlators"

In the past, physicists tried to study these jets by looking at individual particles one by one, like trying to understand a forest by counting every single leaf. It's messy and hard to see the big picture.

This paper introduces a new tool called Energy Correlators.

• The Analogy: Imagine you are in a dark room with a bunch of people running around shouting. Instead of trying to hear every single word, you measure how the sound energy is distributed.
  • One-Point Correlator: You measure how much "noise" (energy) is coming from a specific angle relative to the center of the jet.
  • Two-Point Correlator: You measure how the noise from two different people relates to each other. Are they shouting in sync? Are they far apart or close together?

This method is special because it ignores the "noise" of the messy details and focuses purely on the flow of energy. It's like measuring the wind patterns in a hurricane rather than counting every raindrop.

3. The Discovery: The "Spin" Effect

The most exciting part of this paper is what they found when they looked at the spin of the proton.

• The Expectation: Standard physics (called "perturbative QCD") says that if you smash protons together, the spin shouldn't really matter much in the final explosion. It's like throwing two spinning tops together; the spin shouldn't change the direction of the debris much.
• The Reality: The scientists saw huge differences based on the spin!
  • When the proton spun one way, the debris (specifically pions, which are like the "bricks" of matter) flew out in a specific pattern.
  • When the proton spun the other way, the pattern flipped.

The Metaphor: Imagine a sprinkler head spinning. If you spin it clockwise, the water sprays slightly to the left. If you spin it counter-clockwise, it sprays to the right. The scientists found that the proton's spin acts like that sprinkler, directing the flow of energy in a way that standard theories didn't fully predict.

4. The "Transition Zone"

The paper also discovered a specific "sweet spot" or scale where the rules change.

• Large Angles: Far away from the center of the jet, the particles behave like free, independent runners (this is the "perturbative" world).
• Small Angles: Close to the center, the particles start sticking together, forming the final bricks of matter (this is the "non-perturbative" world where the magic happens).

The scientists found that the spin effect (the sprinkler direction) is strongest right at the boundary where the free runners start sticking together. This tells us exactly where and when the proton's internal spin starts to dictate how matter is built.

5. Why Does This Matter?

This is a "first" measurement. It's the first time anyone has used this specific "energy flow" camera on spinning protons.

• The Map: It provides a new, clearer map of the proton's internal structure.
• The Future: This sets the stage for the Electron-Ion Collider (EIC), a future super-microscope. The techniques used here will allow scientists to create a full 3D "tomography" (like a CT scan) of the proton, showing exactly where the quarks and gluons are and how they spin.

Summary

In short, the STAR team smashed spinning protons together and used a new way of measuring energy flow to discover that the proton's spin acts like a steering wheel, directing how the debris flies out. This helps us understand the fundamental rules of how the universe builds matter from pure energy, bridging the gap between the invisible quantum world and the visible world we live in.

Technical Summary

1. Problem Statement and Motivation

The fundamental challenge in Quantum Chromodynamics (QCD) is understanding the multidimensional structure of the nucleon, specifically how quarks and gluons evolve into observable hadrons (hadronization) and how spin-momentum correlations manifest in the nonperturbative regime.

• The Gap: While Energy-Energy Correlators (EECs) have been successfully used in unpolarized collisions to study the transition from perturbative parton showers to nonperturbative hadronization, their application to transversely polarized collisions was previously unexplored.
• The Goal: To extend EEC measurements to polarized proton-proton ($p^\uparrow p$) collisions to probe spin-dependent fragmentation dynamics. This aims to provide sensitivity to the nucleon's transversity (a chiral-odd parton distribution function) and map the angular scale where spin-dependent nonperturbative dynamics emerge.
• Theoretical Context: Perturbative QCD predicts spin-dependent effects from hard scattering to be negligible ($\sim 10^{-4}$). Therefore, any sizable azimuthal asymmetry observed in energy correlators is a direct signature of nonperturbative dynamics, such as the Collins effect or interference fragmentation functions.

2. Methodology

The analysis utilizes data collected by the STAR detector at the Relativistic Heavy Ion Collider (RHIC).

• Data Set:
  • Collision System: Transversely polarized proton-proton collisions at $\sqrt{s} = 200$ GeV.
  • Year: 2015.
  • Integrated Luminosity: $52 \text{ pb}^{-1}$.
  • Beam Polarization: Average $\langle P \rangle = 57\%$ (scale uncertainty $\Delta P/\langle P\rangle = 3.0\%$).
• Detector Components: Time Projection Chamber (TPC), Barrel/Endcap Electromagnetic Calorimeters (EMCal), and Time-of-Flight (TOF) detector.
• Event Selection:
  • Triggers: Jet Patch (JP) triggers requiring transverse electromagnetic energy deposits.
  • Jet Reconstruction: Anti-$k_T$ algorithm ($R=0.6$) using charged tracks (TPC) and neutral energy (EMCal).
  • Kinematic Cuts: Jets with $0 < \eta_{jet} < 0.9$ and $p_{T,jet} > 6.0$ (or $8.4$) GeV/c. Tracks with $p_T > 20$ GeV/c were excluded to minimize resolution issues.
  • Hadron Selection: Momentum fraction $0.1 < z_h < 0.8$.
• Observables Defined:
  1. One-Point Energy Correlator ($\Sigma_{EC}$): Measures the energy-weighted angular distribution of single hadrons relative to the jet axis ($\theta_h, \phi_h$).
  2. Two-Point Energy-Energy Correlator ($\Sigma_{EEC}$): Measures the energy-weighted distribution of hadron pairs ($\pi^+\pi^-$) as a function of their opening angle ($\theta_{hh}$) and relative azimuth ($\phi_{hh}$).
• Analysis Techniques:
  • Particle Identification (PID): Charged pions identified via $dE/dx$ in the TPC using $n_\sigma(\pi)$ cuts to suppress kaon/proton contamination.
  • Asymmetry Extraction: The cross-ratio method was used to extract Transverse Single-Spin Asymmetries ($A_{UT}$), canceling first-order acceptance and luminosity effects.
  • Unfolding: A purity correction matrix was applied to deconvolve the measured asymmetries from particle misidentification (mixing $\pi$, $K$, $p$).
  • Corrections: Simulations (Pythia 6 + Geant 3) were used to correct for kinematic shifts, detector resolution smearing (dilution factors), and underlying event (UE) contamination.

3. Key Contributions

This work represents the first measurement of one- and two-point energy correlators within jets in transversely polarized collisions.

• Novel Probe: It establishes energy correlators as a theoretically clean tool for studying nucleon spin structure, minimizing uncertainties from nonperturbative fragmentation functions by projecting dynamics onto Mellin moments.
• Angular Mapping: It provides the first spatial map of the onset of spin-dependent hadronization, locating specific angular scales where dynamics transition from perturbative to nonperturbative regimes.
• Theoretical Benchmark: The results offer quantitative constraints for global QCD analyses (e.g., JAM3D) and Transverse Momentum Dependent (TMD) evolution formalisms.

4. Results

The study reports significant, non-zero transverse single-spin asymmetries ($A_{UT}$) for $\pi^+$, $\pi^-$, and $\pi^+\pi^-$ pairs.

• Magnitude and Sign:
  • One-Point: $\pi^+$ and $\pi^-$ exhibit opposite signs. Asymmetries increase with jet transverse momentum ($p_{T,jet}$).
  • Two-Point: The $\pi^+\pi^-$ asymmetry grows more rapidly with $p_{T,jet}$ than the one-point $\pi^+$, reaching up to $\sim 6\%$ in the highest $p_{T,jet}$ bin (approx. twice the one-point magnitude).
• Angular Dependence:
  • Low $p_{T,jet}$ ($\le 11.1$ GeV/c): Asymmetries are small ($\sim 1\%$) and vanish for angles $\theta > 0.3$.
  • High $p_{T,jet}$ ($\ge 13.1$ GeV/c): A distinct peak in asymmetry is observed, shifting with $p_{T,jet}$.
  • Two-Point Behavior: The region of nonzero asymmetry broadens at higher $p_{T,jet}$. Notably, sizable asymmetries persist up to $\theta_{hh} \sim 0.3$, suggesting spin correlations remain relevant within the angular region typically associated with the perturbative parton shower.
• Comparison with Theory:
  • Data qualitatively align better with the TMD Evolution formalism than the JAM3D framework (which omits TMD evolution) at low $p_{T,jet}$ and small angles.
  • However, the observed difference between $\pi^+$ and $\pi^-$ asymmetries is smaller than predicted, indicating a need for refinement in the flavor dependence of theoretical models at these energy scales.
  • The transition scale between perturbative and nonperturbative regimes was identified, with asymmetry maxima coinciding with the transition threshold for $p_{T,jet} > 11.1$ GeV/c.

5. Significance

• Validation of EECs in Polarized Collisions: The results demonstrate that energy correlators are a robust, theoretically clean channel for nucleon spin structure studies, complementary to traditional hadron-in-jet observables.
• Insight into Nonperturbative Dynamics: The observation of sizable asymmetries at angular scales overlapping with the perturbative regime suggests that the onset of nonperturbative spin dynamics may occur at larger angular scales than previously thought for unpolarized bulk energy flow.
• Future Impact: These measurements establish a critical baseline for the Electron-Ion Collider (EIC). The methodology and precision achieved here pave the way for high-precision, three-dimensional tomography of the nucleon, specifically targeting the poorly constrained transversity distribution.
• Precision: The experimental precision in several kinematic regions exceeds current theoretical uncertainties, providing new constraints for phenomenological descriptions of transverse-spin dynamics.