Collins asymmetries for pion-in-jet production in polarized $\ell p$ collisions at the EIC

This paper investigates Collins azimuthal asymmetries in pion-in-jet production within polarized lepton-proton collisions at the Electron-Ion Collider using a simplified TMD approach, demonstrating that these processes offer a theoretically cleaner and more direct probe of the transversity distribution and its sea-quark components compared to polarized proton-proton scattering, thereby providing a crucial test for the universality of the Collins function and TMD factorization.

The Gist

Imagine the proton not as a solid marble, but as a bustling, chaotic city made of tiny, speeding particles called quarks and gluons. Physicists have long wanted to map this city in 3D, understanding not just where the particles are, but how they spin and move. This paper is a blueprint for a new way to take a "snapshot" of that city using a future machine called the Electron-Ion Collider (EIC).

Here is the story of the paper, broken down into simple concepts:

1. The Goal: Mapping the Spin

Think of the quarks inside a proton like dancers. Some spin one way, some the other. A specific property called "transversity" describes how these dancers spin sideways relative to their direction of travel. It's a very tricky property to measure because it's hidden inside the chaos of the proton.

To see it, scientists use a trick: they smash particles together and watch what flies out. If they can spot a specific pattern in how the debris flies, they can deduce how the original dancers were spinning. This pattern is called the Collins asymmetry.

2. The Old Way vs. The New Way

• The Old Way (pp collisions): In the past, scientists smashed two protons together (like two busy cities crashing into each other). It was messy. The "debris" (particles flying out) came from many different sources, including heavy, invisible "gluons" that acted like fog, making it hard to see the specific spin of the quarks. It was like trying to hear a single violin in a full orchestra where the drums were playing too loud.
• The New Way (ℓp collisions): This paper proposes a cleaner experiment. Instead of smashing two protons, they smash a lepton (a lightweight particle, like an electron) into a proton.
  • The Analogy: Imagine throwing a ping-pong ball (the lepton) at a bowling ball (the proton). Because the ping-pong ball is so light and clean, it mostly hits the individual dancers (quarks) inside the bowling ball without getting tangled up in the "fog" (gluons). This makes the signal much clearer.

3. The "Jet" and the "Pion"

When the collision happens, a quark gets knocked out and zooms away. It doesn't travel alone; it drags a swarm of new particles with it, forming a cone-shaped spray called a jet.

• Inside this jet, the scientists look for a specific particle called a pion (a type of light meson).
• They are looking at how the pion wobbles or rotates as it flies out of the jet. If the pion wobbles in a specific direction relative to the proton's spin, it proves the quark had a specific sideways spin.

4. The "Ghost" Contribution (Quasireal Photons)

The authors realized that in this specific setup, there's a sneaky extra player. Sometimes, the incoming electron acts like a flashlight, shooting out a "quasireal photon" (a burst of light that acts like a particle) which then hits the proton.

• The Paper's Finding: They calculated that this "flashlight" effect is actually quite strong—it adds a lot of extra data. However, the good news is that it doesn't ruin the clarity. Even with this extra light, the "quark" signal remains the star of the show, and the "gluon" noise stays quiet.

5. Why This Matters (The "Sea" of Quarks

Inside the proton, there are "valence" quarks (the main residents) and a "sea" of temporary quarks that pop in and out.

• The Discovery: Because this new method (lepton-proton collision) is so clean, it allows scientists to see the "sea" quarks much better than before. In the old messy proton-proton crashes, the sea quarks were drowned out. Here, the authors predict we can finally get a good look at the spin of these fleeting, sea-quark residents.

6. The Bottom Line

The authors ran the numbers for the future Electron-Ion Collider (EIC). They found that:

• The "clean" method works beautifully.
• The extra "flashlight" effect (quasireal photons) is important to include but doesn't mess up the results.
• This process offers a much clearer window into the transversity (sideways spin) of quarks, especially the elusive ones in the "sea."

In summary: This paper is a proposal to use a cleaner, more precise "camera" (lepton-proton collisions) to take a high-definition photo of the spinning quarks inside a proton. It promises to clear up the fog that has obscured our view for years, allowing us to finally see the "sea" of quarks and test if our theories about how these particles behave are correct.

Technical Summary

Technical Summary: Collins Asymmetries for Pion-in-Jet Production in Polarized $\ell p$ Collisions at the EIC

Problem and Motivation
The study of the three-dimensional structure of hadrons relies heavily on Transverse Momentum Dependent (TMD) factorization and the extraction of parton distribution and fragmentation functions. While significant progress has been made in Semi-Inclusive Deep Inelastic Scattering (SIDIS), $e^+e^-$ annihilation, and polarized proton-proton ($pp$) collisions, the universality of the Collins fragmentation function (FF) and the validity of TMD factorization in different processes remain critical open questions. Previous analyses of hadron-in-jet production in $pp$ collisions (Refs. [44–49]) have provided constraints on the Collins FF and transversity distribution but are complicated by the significant role of gluon-initiated channels in the denominator of the asymmetry, which obscures the direct extraction of quark transversity.

This paper addresses the need for a complementary process to test TMD factorization and Collins function universality. The authors propose studying pion-in-jet production in polarized lepton-proton ($\ell p^\uparrow$) collisions, specifically $\ell p^\uparrow \to \text{jet } \pi X$, at the Electron-Ion Collider (EIC). This process is theoretically simpler than the $pp$ case because the leading-order (LO) contribution involves standard lepton-quark scattering, minimizing gluon contamination in the asymmetry's denominator and offering a clearer access to the transversity distribution, including its sea-quark component.

Methodology
The authors employ a simplified TMD approach where transverse momentum effects are restricted to the fragmentation mechanism, assuming a collinear initial state. The calculation is performed within a leading-order factorization framework but is extended beyond pure LO by including the contribution from quasireal photon exchange via the Weizsäcker-Williams (WW) approximation (Equivalent Photon Approximation).

• Formalism: The azimuthal asymmetry $A_{UT}^{\sin(\phi_S - \phi_H^h)}$ is computed as the ratio of the polarized cross-section difference to the unpolarized sum.
  • LO Contribution: Involves the $q\ell \to q'\ell'$ subprocess. The numerator convolutes the collinear quark transversity distribution ($h_1^q$) with the Collins TMD-FF ($\Delta N D_{h/q\uparrow}$).
  • WW Contribution: Accounts for the process where the incoming lepton acts as a source of real photons ($\gamma N \to \text{jet } h X$). This introduces partonic channels such as $q\gamma \to qg$, $q\gamma \to gq$, and $g\gamma \to q\bar{q}$. Crucially, the numerator of the asymmetry in the WW sector is dominated by the $q\gamma \to qg$ channel, which still involves the transversity distribution, while the denominator includes gluon-initiated channels.
• Inputs: The calculations utilize transversity and Collins FFs extracted from global fits to SIDIS and $e^+e^-$ data (Refs. [30, 32, 33, 35, 42, 49]). Unpolarized parton distribution functions (PDFs) are taken from the MSHT20nlo set, and unpolarized fragmentation functions from the DEHSS set.
• Kinematics: Predictions are generated for EIC kinematics at three center-of-mass energies ($\sqrt{s} = 45, 105, 141$ GeV), covering forward and backward jet rapidity regions ($\eta_j$) and various transverse momentum ranges ($p_{jT}$).

Key Results

1. Impact of WW Contribution: The inclusion of the quasireal photon exchange (LO+WW) significantly enhances the unpolarized cross-section, particularly in the backward rapidity region where the ratio of (LO+WW) to LO exceeds 2. However, the quark-initiated channels remain dominant (80–90%) in the total cross-section, even with the inclusion of gluon-initiated WW channels.
2. Access to Transversity: Unlike $pp$ collisions, where gluon contributions dominate the denominator of the Collins asymmetry, the $\ell p$ process maintains a dominance of quark-initiated channels. This preserves a "clear access" to the transversity distribution, minimizing the dilution effects seen in hadronic collisions.
3. Asymmetry Predictions:
   • $\pi^+$ Production: Asymmetries are generally flat and small (around 1%) across rapidities.
   • $\pi^-$ Production: Asymmetries show a more pronounced dependence, growing to 2–8% in the forward rapidity region. This behavior is driven by the interplay between the up-quark transversity distribution and the unfavored Collins FF.
   • $z$ Dependence: A distinct feature is observed in the dependence on the pion's momentum fraction $z$. For $\pi^+$, asymmetries increase with $z$ (reaching ~8% in the forward region) due to the dominance of the favored Collins FF coupled with up-quark transversity. Conversely, $\pi^-$ asymmetries remain relatively flat. At low $z$, $\pi^+$ asymmetries are suppressed due to the vanishing favored Collins FF.
4. Sea-Quark Sensitivity: The backward rapidity region is identified as a unique kinematic window to probe the sea-quark component of the transversity distribution, which is poorly constrained by current data.

Significance and Claims
The paper claims that the study of Collins azimuthal asymmetries in $\ell p$ collisions represents a "complementary and theoretically simpler process" compared to $pp$ scattering. Its primary significance lies in:

• Testing Universality: Providing a stringent test of the universality of the Collins function across different hard scattering processes (SIDIS, $e^+e^-$, $pp$, and now $\ell p$).
• Validating TMD Factorization: Offering a cleaner environment to verify TMD factorization hypotheses in processes involving a third energy scale (the virtuality of the exchanged boson).
• Flavor Separation: Enabling a more direct extraction of the transversity distribution, particularly its sea-quark component, which is difficult to isolate in other processes due to gluon dominance.
• Future Measurements: The authors argue that future EIC measurements of these asymmetries could represent a "further step" in testing the TMD framework and refining the parametrization of Collins functions, particularly regarding their flavor and $z$ dependencies.

The authors remain modest regarding theoretical uncertainties, noting that their results rely on a Gaussian parametrization of intrinsic transverse momentum and a LO treatment of jet dynamics. They acknowledge that a more flexible parametrization and a higher-order treatment of jet fragmentation (TMD jet FFs) are necessary for a more robust picture, which they intend to address in future work.