Flavor, transverse momentum, and azimuthal dependence of charged pion multiplicities in SIDIS with 10.6 GeV electrons

This paper reports high-precision measurements of charged pion multiplicities and their azimuthal modulations in semi-inclusive deep inelastic scattering off proton and deuteron targets using a 10.6 GeV electron beam at Jefferson Lab, revealing consistent transverse momentum dependencies and significant $\pi^-$ azimuthal asymmetries that will enable improved determinations of quark transverse momentum distributions.

The Gist

Imagine the inside of a proton (the tiny particle at the center of every atom) not as a solid marble, but as a bustling, high-speed highway of invisible traffic. This paper is like a traffic report from a very specific, high-energy experiment where scientists tried to understand how this traffic behaves when hit by a fast-moving electron.

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

The Experiment: A High-Speed Collision Course

Think of the Jefferson Lab as a massive, high-tech racetrack. The scientists fired a beam of electrons (like tiny, super-fast bullets) at two different targets: a tank of liquid hydrogen (pure protons) and a tank of liquid deuterium (protons mixed with neutrons).

When these electron "bullets" hit the protons, they didn't just bounce off; they shattered the proton's internal structure, creating a spray of new particles. The scientists were specifically interested in catching two types of "debris" from this crash:

1. Kaons: A specific type of particle (like a specific model of car in the traffic jam).
2. Protons: The original heavy particles that got knocked loose.

They used giant, precise "cameras" (spectrometers) to track where these particles went, how fast they were moving, and what angle they took.

The Goal: Mapping the "Traffic Rules"

Physicists have two main theories about how this traffic works:

1. The "Hard" Theory (TMD): This predicts that if you smash things hard enough, the particles fly out in very specific, predictable patterns based on strict mathematical rules. It's like a perfectly choreographed dance.
2. The "Soft" Theory: This suggests that in the middle of the chaos, things are messy, fuzzy, and don't follow the strict dance steps. It's more like a crowded mosh pit where people bump into each other randomly.

The scientists wanted to see which theory matched reality for Kaons and Protons.

What They Found: The Kaon Story

The Good News: When they looked at positively charged Kaons (K+), the data matched the "Hard" theory predictions quite well. It was as if the traffic followed the choreographed dance steps perfectly.
The Bad News: When they looked at negatively charged Kaons (K-), the reality was very different. There were far fewer of them than the theory predicted. It's like the theory said there should be 100 red cars, but the camera only saw 10.
The Angle: They also checked if the particles were spinning or wobbling in a specific direction (azimuthal modulation). For Kaons, the answer was essentially "no." They weren't wobbling; they were just flying straight out.

What They Found: The Proton Story

This is where things got really interesting. The scientists looked at the protons that got knocked loose.
The Surprise: The "Hard" theory predicted that protons would be rare in this specific type of crash. But the cameras saw way more protons than expected—sometimes 10 times more!
The Explanation: The scientists realized that the experiment was happening in the "Soft" central region (the mosh pit). The strict "Hard" rules don't apply here. Instead, the data matched a computer simulation called "Lund Monte Carlo," which is designed to model messy, chaotic particle creation. It's like realizing you can't predict the movement of a crowd in a mosh pit using a ballet manual; you need a model that accounts for the chaos.

The Takeaway

• For Kaons: The universe is a bit of a mix. Sometimes it follows the strict rules (K+), and sometimes it breaks them completely (K-).
• For Protons: The universe is messy. In the conditions of this experiment, protons behave like a chaotic crowd, not a choreographed dance. The old, strict rules don't work here; we need a model that understands the "soft" chaos.

In short: The scientists fired electrons at protons to see how the debris flies. They found that while some particles (positive Kaons) follow the rules, others (negative Kaons and all protons) do things that the old rulebooks didn't predict. This tells us that in the messy middle of a particle collision, the "soft" chaos is just as important as the "hard" rules.

Technical Summary

Technical Summary: Flavor, Transverse Momentum, and Azimuthal Dependence of Charged Kaon and Proton Multiplicities in SIDIS

Problem and Motivation
Semi-inclusive deep-inelastic scattering (SIDIS) serves as a primary tool for probing the three-dimensional internal structure of the nucleon. While the process is well-described by the convolution of parton distribution functions (PDFs) and fragmentation functions (FFs) in specific kinematic regimes, the validity of transverse-momentum-dependent (TMD) factorization is kinematically constrained. Previous studies have extensively mapped pion production, but data for identified kaons and protons in the Jefferson Lab kinematic range ($3 < Q^2 < 6$ GeV$^2$, $0.3 < x < 0.6$) remain scarce. Specifically, there is a lack of existing publications reporting cross sections or multiplicities of identified protons in SIDIS kinematics. Furthermore, the applicability of TMD factorization depends on the hadron rapidity ($y_h$); while pions often fall within the current fragmentation region ($y_h < -1$), protons and kaons at these energies may reside in the "soft" central region or target fragmentation region, where factorization theorems are not yet established. This paper addresses the need for high-precision multiplicity data for charged kaons and protons to test TMD predictions and phenomenological models in this intermediate kinematic regime.

Methodology
The experiment was conducted in 2018–2019 at Jefferson Lab's Hall C using 10.2 and 10.6 GeV electron beams incident on liquid hydrogen and deuterium targets. Data were collected using the High Momentum Spectrometer (HMS) for scattered electrons and the Super High Momentum Spectrometer (SHMS) for detected hadrons.

• Particle Identification (PID):
  • Kaons: Identified using a combination of aerogel Cherenkov detectors, Heavy Gas Cherenkov (HGC) counters, time-of-flight (TOF) measurements, and calorimeter energy deposition. Cuts were optimized to maintain efficiencies of ~94–97% while suppressing pion and proton backgrounds to below 10%.
  • Protons: Identified via aerogel and HGC signals, TOF, and calorimeter cuts. Due to the significantly longer flight time of protons compared to lighter particles, contamination from pions, kaons, and positrons was reduced to less than 1%.
• Analysis Framework:
  • Yields were normalized by beam charge and corrected for spectrometer efficiencies, trigger dead time, and radiative effects.
  • Multiplicities were extracted in bins of hadron longitudinal momentum fraction ($z$), transverse momentum ($P_t$), and azimuthal angle ($\phi^*$).
  • Azimuthal distributions were fitted to the functional form $M_0[1 + A \cos(\phi^*) + B \cos(2\phi^*)]$ to extract the multiplicity $M_0$ and modulation parameters $A$ and $B$.
  • Radiative corrections utilized fragmentation function models (DSS for kaons; an empirical fit for protons) and Monte Carlo simulations (SIMC) that accounted for energy loss, multiple scattering, and particle decays.

Key Contributions
This work provides the first comprehensive set of SIDIS multiplicities for identified protons and an extensive dataset for charged kaons in the Jefferson Lab kinematic range. The paper maps these observables across a grid of $0.3 < z < 0.7$ and $P_t < 0.6$ GeV, covering $0.3 < x < 0.6$ and $3 < Q^2 < 6$ GeV$^2$. It explicitly evaluates the kinematic affinity to TMD factorization using the Boglione et al. rapidity framework, distinguishing between the current fragmentation region, the soft central region, and the target fragmentation region.

Results

• Kaons:

  • Multiplicities ($M_0$): The $K^+$ multiplicities for the deuteron target agree well with predictions using CTEQ5 PDFs and DSS FFs. However, $K^+$ multiplicities on the proton target are larger than predicted. $K^-$ multiplicities are significantly lower than $K^+$ values and fall well below DSS predictions across most of the kinematic range, except at the highest invariant mass squared ($W^2$) values where TMD affinity is highest.
  • Azimuthal Modulations: Both parameters $A$ and $B$ are consistent with zero within uncertainties, showing no significant dependence on $W^2$.
  • Ratios: The $K^+/ \pi^+$ ratios increase with $z$ and $W^2$, aligning with Lund Monte Carlo (LEPTO/JETSET) and CTEQ/DSS trends. Conversely, $K^-/\pi^-$ ratios are very small and remain below both Lund and DSS predictions, though a trend of increasing with $W^2$ is observed.

• Protons:

  • Multiplicities ($M_0$): Proton multiplicities are observed to be an order of magnitude larger than TMD predictions at low $W^2$, decreasing to a factor of two at the highest $W^2$. This discrepancy is attributed to the data residing in the "soft" central rapidity region ($-1 < y_h < 1$), where TMD factorization is not applicable.
  • Target Dependence: No significant difference is observed between proton and deuteron targets.
  • Azimuthal Modulations: Parameter $B$ is consistent with zero. Parameter $A$ is consistently positive with an average value of approximately 0.01, showing no clear dependence on $W^2$.
  • Model Comparison: The Lund Monte Carlo (LEPTO/JETSET) provides a qualitative description of the proton-to-pion ratio trends, particularly the strong decrease with increasing $W^2$, whereas DSS-based predictions fail to capture the magnitude of the effect.

Significance and Claims
The paper claims that the kinematic range of this experiment spans the transition between the "soft" central region and the region of good affinity for TMD factorization for kaons, while sitting squarely in the soft central region for protons. This positioning explains the observed discrepancies with high-energy TMD-based predictions: the proton data, being in the soft region, exhibits multiplicities far exceeding TMD expectations.

The authors conclude that while TMD factorization is not the appropriate framework for the bulk of the proton data, the HERMES-tuned LEPTO/JETSET event generator offers a qualitative description of the results for both kaons and protons. They suggest that with further tuning, such generators could serve as practical tools for understanding non-perturbative contributions to SIDIS in kinematic regimes where rigorous factorization theorems are not yet available. The paper provides the raw numerical data in full 3D grids to facilitate future global fits and model development.