Simplified approach to extracting nucleon transversity in collinear factorization using near-side energy-energy correlators

This paper proposes a novel strategy using near-side energy-energy correlators within the dihadron fragmentation framework to extract the nucleon's transversity parton distribution function in collinear factorization, thereby eliminating the need to model intrinsic transverse momentum or resonances while yielding analytical results analogous to standard unpolarized extractions.

The Gist

Imagine the nucleus of an atom (the nucleon) as a bustling, crowded city. Inside this city, tiny particles called quarks zoom around. Some of these quarks have a special property called "transversity," which is like a unique spin or orientation they carry, distinct from their usual forward motion. Physicists have been trying to map out exactly how these quarks are oriented, but it's been a messy job.

The Old Way: Navigating a Foggy City
Previously, scientists tried to figure out this orientation by looking at how quarks break apart into pairs of particles (like a pi-meson pair). However, this method was like trying to navigate the city while wearing foggy glasses.

• The Problem: To make sense of the data, they had to guess how the particles were moving sideways (intrinsic transverse momentum) and account for a chaotic mix of temporary particle "resonances" (like traffic jams that appear and disappear). This required building complex, messy models with hundreds of adjustable knobs to fit the data. It was like trying to solve a puzzle while the pieces kept changing shape.

The New Approach: A Clear, High-Definition Map
This paper proposes a brand-new, simplified strategy. The authors suggest using a tool called an "Energy-Energy Correlator" (EEC), specifically looking at particles that fly out in nearly the same direction (the "near-side").

Think of the EEC as a high-definition camera that doesn't just take a picture of the particles, but measures how their energy is distributed relative to each other.

• The Magic Trick: By focusing on this specific angle, the new method acts like a filter that removes the "fog." It completely bypasses the need to model the messy sideways movements or the confusing resonance traffic jams.
• The Result: The math becomes incredibly simple. Instead of dealing with complicated, multi-variable functions, the equations look just like the standard, clean formulas physicists use to measure other, simpler properties of the atom. It's like switching from a tangled ball of yarn to a straight, smooth thread.

What They Did and Found
The team didn't just write down the theory; they ran the numbers to see if it would actually work in the real world.

• The Simulation: They used existing data from major particle physics experiments (like BELLE, COMPASS, and HERMES) and simulated what future experiments (like the Electron-Ion Collider) would see.
• The Prediction: They found that the "signal" (the asymmetry caused by the quark's orientation) is strong enough to be measured. In some scenarios, the effect could be as large as 20%, which is huge in the world of particle physics.
• The Promise: They showed that by re-analyzing old data or taking new measurements with this specific "near-side" angle, scientists can extract the transversity map of the nucleon without the headache of the old, complicated models.

Why It Matters
Understanding this "transversity" is crucial because it helps calculate the "tensor charges" of the proton and neutron. Think of these charges as the fundamental ID cards of the atom's building blocks. Knowing them helps scientists test the Standard Model of physics and look for new, unknown forces (Beyond the Standard Model) that might explain mysteries like why the universe is made of matter instead of antimatter.

In short, this paper offers a "simplified approach" that turns a difficult, model-dependent guessing game into a clean, direct measurement, making it much easier to understand the hidden spin of the building blocks of our universe.

Technical Summary

1. Problem Statement

The nucleon transversity parton distribution function (PDF), denoted as $h_1(x)$, is a fundamental quantity in QCD describing the transverse polarization of quarks inside a transversely polarized nucleon. It is crucial for determining the nucleon's tensor charges ($\delta u, \delta d$), which are vital inputs for testing Beyond the Standard Model (BSM) physics, such as free neutron beta decay and nucleon electric dipole moments.

Currently, extracting $h_1(x)$ faces significant theoretical and phenomenological challenges:

• TMD Factorization: The standard approach uses Transverse Momentum Dependent (TMD) factorization with single-hadron final states. This requires modeling the intrinsic transverse momentum ($k_T$) of partons and the Collins fragmentation function, often involving complex functional forms and non-perturbative parameters.
• Collinear Factorization with Dihadrons: An alternative uses dihadron final states in collinear factorization. However, this relies on Dihadron Fragmentation Functions (DiFFs), specifically $H_1^{\sphericalangle}(z, M_h)$ and $D_1(z, M_h)$, which depend on the invariant mass of the dihadron pair ($M_h$). The $M_h$ dependence is complicated by resonance structures (e.g., $\rho$ mesons), requiring the fitting of $\mathcal{O}(100)$ parameters to model the DiFFs accurately.

The authors aim to develop a strategy to extract $h_1(x)$ within collinear factorization that avoids the complexities of modeling intrinsic transverse momentum and the resonant $M_h$ dependence of standard DiFFs.

2. Methodology

The authors propose a novel framework utilizing Near-Side Energy-Energy Correlators (EECs) in the context of Semi-Inclusive Deep-Inelastic Scattering (SIDIS) and electron-positron ($e^+e^-$) annihilation.

• Definition of EEC-DiFFs:
  Instead of standard DiFFs dependent on $M_h$, the authors define new non-perturbative functions, $D_{h_1h_2/i}(z_\chi, Q^2)$ and $H_{h_1h_2/i}(z_\chi, Q^2)$, where $z_\chi = \frac{1}{2}(1 - \cos \chi)$ and $\chi$ is the opening angle between the two hadrons.
  These functions are constructed by integrating standard DiFFs over the relative transverse momentum and invariant mass, weighted by kinematic factors, effectively projecting the dihadron system onto the near-side collinear region ($\chi \approx 0$).
  $$D_{h_1h_2/i}(z_\chi, Q^2) \propto \int d\xi_1 d\xi_2 d^2\vec{R}_T \, \delta\left(z_\chi - \frac{R_T^2}{Q^2}\frac{\xi^2}{\xi_1^2\xi_2^2}\right) \xi_1\xi_2 D_{1}^{h_1h_2/i}(\dots)$$

• Theoretical Framework:
  The authors derive Leading-Order (LO) analytical expressions for transverse-spin dependent observables in SIDIS and $e^+e^-$ annihilation.

  • SIDIS: The cross-section involves a convolution of the transversity PDF $h_1(x)$ and the unpolarized PDF $f_1(x)$ with the new EEC-DiFFs.
  • $e^+e^-$: The cross-section involves convolutions of EEC-DiFFs from the quark and antiquark jets.
  • Key Simplification: The resulting expressions (Eqs. 7 and 8) have the exact same structure as standard collinear factorization for single-hadron observables (e.g., $f_1 \otimes D_1$). Crucially, the new EEC-DiFFs depend only on the single variable $z_\chi Q^2$ (or $Z$), eliminating the dependence on the invariant mass $M_h$.

• Modeling Strategy:
  To generate numerical predictions, the authors relate the new EEC-DiFFs to existing DiFFs extracted from experimental data (JAMDiFF analysis). They expand the dependence on the momentum fraction difference ($\zeta$) using Legendre polynomials up to the second order, constrained by positivity bounds. This allows them to map existing $M_h$-dependent data to the new $z_\chi$-dependent functions without introducing new free parameters for the resonance structure.

3. Key Contributions

1. Novel Factorization Scheme: Established a rigorous framework for extracting $h_1(x)$ using near-side EECs within collinear factorization, bypassing the need for TMD factorization or complex $M_h$ modeling.
2. Simplified Functional Form: Demonstrated that the new EEC-DiFFs ($D(Z)$ and $H(Z)$) are single-variable functions, making the extraction of $h_1(x)$ mathematically analogous to extracting unpolarized PDFs ($f_1$) or helicity PDFs ($g_1$).
3. Analytical Derivations: Provided complete LO analytical results for transverse-spin asymmetries in both SIDIS and $e^+e^-$ annihilation, including the specific azimuthal angle dependencies required to isolate the transversity contribution.
4. Numerical Predictions: Generated robust predictions for future facilities (EIC, BESIII) and re-analyses of existing data (BELLE, COMPASS, HERMES), quantifying uncertainties from both non-perturbative inputs and the modeling of the $\zeta$-dependence.

4. Results

The paper presents numerical predictions for three key observables:

• SIDIS Asymmetry ($A_{UT, EEC}^{\sin(\phi+\phi_S)}$):

  • Predicted for the EIC kinematics ($x=0.01, 0.1, 0.3$).
  • The asymmetry increases with $x$ and the hard scale $Q$.
  • At $x=0.3$ and $Q=10$ GeV, the asymmetry reaches up to 20%, making it highly measurable.
  • Predictions for HERMES and COMPASS kinematics fall within the expected ranges, suggesting feasibility with existing data.

• $e^+e^-$ Asymmetry ($A_{EEC}^{\cos(\phi+\bar{\phi})}$):

  • Predicted for BELLE ($Q \approx 10.6$ GeV) and BESIII ($Q \approx 3.65$ GeV).
  • At BELLE energies, the asymmetry reaches approximately 8% for large opening angles ($\bar{\chi} = 30^\circ$).
  • At BESIII, the asymmetry is smaller (<1%) but still non-zero.

• Unpolarized EEC ($EEC_{\pi^+\pi^-}^{e^+e^-}$):

  • The shape of the unpolarized EEC distribution matches the "standard" EEC behavior observed at higher energies, peaking near the transition scale from quarks/gluons to hadrons ($\sim 2.6$ GeV).
  • Crucially, the distribution shows no resonance structures (unlike the $d\sigma/dzdM_h$ spectrum), confirming that the near-side EEC approach successfully smooths out the complex resonance dynamics that plague traditional DiFF analyses.

• Uncertainty Analysis:

  • The uncertainty in the asymmetries is dominated by the statistical uncertainty of the non-perturbative JAMDiFF inputs, not the modeling of the $(a,b)$ parameters for the Legendre expansion.
  • The uncertainty in the unpolarized EEC is dominated by the $(a,b)$ parameter scan.

5. Significance

This work represents a paradigm shift in the phenomenology of nucleon transversity:

• Simplification: It reduces the extraction of $h_1(x)$ from a complex, multi-parameter problem involving resonances and TMDs to a standard collinear factorization problem. This lowers the barrier for precision extraction.
• Feasibility: The large predicted asymmetries (up to 20%) indicate that this method is experimentally viable using data from existing facilities (BELLE, COMPASS, HERMES) and future high-luminosity colliders (EIC, BESIII).
• Theoretical Cleanliness: By removing the $M_h$ dependence, the approach avoids the "modeling trap" of resonance parameters, leading to more robust and model-independent determinations of the nucleon tensor charges.
• Future Outlook: The authors note that this framework can be extended to proton-proton collisions (e.g., STAR measurements of EECs in jets), opening new avenues for transversity studies in hadronic collisions.

In summary, the paper provides a theoretically sound and experimentally promising pathway to precisely measure the nucleon transversity PDF, potentially resolving long-standing uncertainties in the tensor charges of the nucleon.