Accessing baryon-antibaryon generalized distribution amplitudes in $e^{\pm} γ\to e^{\pm} B \bar{B}$

This paper investigates the feasibility of extracting baryon-antibaryon generalized distribution amplitudes from the $e^\pm \gamma \to e^\pm B \bar{B}$ process using QCD factorization and numerical estimates, demonstrating that such a measurement is achievable at the Belle II experiment.

The Gist

Imagine the universe is built out of tiny, invisible Lego bricks called quarks. When these bricks snap together to form larger structures like protons or neutrons (which we call baryons), they create a complex, 3D puzzle. Scientists want to see the "blueprint" of how these bricks are arranged inside.

This paper is about a new, clever way to take a picture of that blueprint, specifically for protons and their antimatter twins, antiprotons.

The Problem: Invisible Blueprints

Usually, to see inside a proton, scientists smash things together. But protons are tricky; they are often unstable or hard to isolate. It's like trying to study the inside of a fragile, spinning top by throwing it against a wall—you might break it before you see the gears.

The paper proposes a different approach: instead of smashing, let's gently "scan" the proton using light.

The Experiment: A Cosmic Dance

The authors describe a process called $e^\pm \gamma \to e^\pm B \bar{B}$. Let's break that down into a story:

1. The Setup: Imagine an electron (a tiny particle of electricity) and a photon (a particle of light) colliding.
2. The Magic Trick: When they collide, they don't just bounce off. Instead, they briefly transform into a pair of new particles: a baryon (like a proton) and an antibaryon (its antimatter opposite).
3. The Goal: The scientists want to measure exactly how this transformation happens. By studying the angles and speeds of the new particles, they can reverse-engineer the "Generalized Distribution Amplitudes" (GDAs).

What are GDAs?
Think of GDAs as a 3D map of the proton's internal traffic. They tell us how the quarks are moving and sharing energy inside the proton when it's being created from pure energy. The paper focuses on "chiral-even" GDAs, which is a fancy way of saying they are looking at the specific type of traffic flow that doesn't flip the "handedness" of the particles.

The Two Paths (The Analogy)

The paper explains that this collision can happen in two different ways, like two different routes to the same destination:

• Route A (The QCD Path): The electron and photon fuse directly into a quark-antiquark pair, which then instantly snaps together to form the proton-antiproton pair. This path is governed by the strong nuclear force (QCD) and contains the "GDAs" the scientists want to measure.
• Route B (The Bremsstrahlung Path): The electron emits a photon first (like a car braking and flashing its lights), and that photon then creates the proton-antiproton pair. This path is well-understood and acts as a known "background noise."

The Solution: Tuning the Radio

Here is the tricky part: Route A (the one with the new information) and Route B (the known background) happen at the same time. They interfere with each other, like two radio stations playing on the same frequency.

The authors realized that if you compare what happens when you use a negative electron versus a positive electron (positron), the "noise" from Route B stays the same, but the "signal" from Route A flips. By subtracting the two results, the background noise cancels out, leaving only the pure signal of the GDAs.

They also looked at polarization. Imagine the proton isn't just a ball, but a spinning top. By measuring which way the proton spins after the collision, they can get even more details about the internal map, specifically the "imaginary" parts of the blueprint that are usually hidden.

The Results: Is it Possible?

The authors did some math and created computer models to see if this could actually work in a real experiment. They focused on the Belle II facility in Japan, a massive particle accelerator.

• The Good News: Their calculations show that there is a specific "sweet spot" in the energy levels where the signal (the GDAs) becomes strong enough to be seen clearly above the background noise.
• The Prediction: They estimate that with the current capabilities of Belle II, scientists could successfully extract these GDAs for the first time.

The Bottom Line

This paper is a "feasibility study." It doesn't claim to have measured the GDAs yet. Instead, it provides the instruction manual and the map for how to do it.

It tells experimentalists: "If you set your machine to these specific energy settings and look for these specific spin patterns, you will be able to see the internal structure of the proton in a way we haven't been able to before."

In short, they have designed a new camera lens that might finally let us take a clear photo of the invisible gears inside the building blocks of our universe.

Technical Summary

Technical Summary: Accessing Baryon-Antibaryon Generalized Distribution Amplitudes in $e^\pm\gamma \to e^\pm B \bar{B}$

Problem and Motivation
Generalized Distribution Amplitudes (GDAs) and Generalized Parton Distributions (GPDs) are fundamental hadronic matrix elements of light-cone bilocal operators used for hadron tomography. While GPDs are accessed via Deeply Virtual Compton Scattering (DVCS), GDAs are the $s$-$t$ crossed counterparts, accessible through hadron pair production. While meson GDAs (e.g., $\pi\pi$) have been studied, the extraction of baryon-antibaryon GDAs remains an open challenge. Unlike stable hadrons, many baryons are unstable, making their GPDs difficult to probe; however, GDAs can be studied via pair production ($\gamma^*\gamma \to B\bar{B}$ or $\gamma^* \to B\bar{B}\gamma$).

The specific problem addressed in this work is the theoretical formulation and numerical estimation of the process $e^\pm\gamma \to e^\pm B \bar{B}$ (where $B$ is a spin-1/2 baryon like the nucleon, $\Lambda$, or $\Sigma$). This process involves two competing mechanisms: a QCD-driven subprocess ($\gamma^*\gamma \to B\bar{B}$) sensitive to baryon GDAs, and a QED-driven bremsstrahlung subprocess ($e^\pm \to e^\pm \gamma^*$ followed by $\gamma^* \to B\bar{B}$) described by timelike electromagnetic form factors (EM FFs). The goal is to determine if the GDAs can be isolated from the interference between these amplitudes and whether polarization observables can provide additional constraints.

Methodology
The authors employ the framework of collinear QCD factorization, valid in the generalized Bjorken regime ($Q^2 \gg \hat{s}, \Lambda_{QCD}^2$), where the leading-twist amplitude is a convolution of GDAs and a perturbatively calculable coefficient function.

1. Theoretical Framework:

   • Kinematics: The process is analyzed in the center-of-mass frame of the baryon-antibaryon pair. Variables include the invariant mass squared of the pair ($\hat{s}$), the virtual photon momentum squared ($Q^2$), and the baryon polar angle ($\theta$).
   • Amplitudes:
     • Subprocess (a): The C-even channel $\gamma^*\gamma \to B\bar{B}$ is described by twist-2 chiral-even GDAs ($\Phi^q_V, \Phi^q_S, \Phi^q_A, \Phi^q_P$). The hadron tensor is expressed in terms of four timelike Compton Form Factors (FFs) ($F_V, F_S, F_A, F_P$).
     • Subprocess (b): The C-odd channel involves the bremsstrahlung of a virtual photon from the lepton, coupling to the baryon pair via timelike EM FFs ($G_E, G_M$).
   • Interference: The total cross-section includes the pure QCD term, the pure QED (bremsstrahlung) term, and the interference term between them. The authors derive differential cross-sections for unpolarized cases and single-spin correlations (dependent on the baryon spin vector $S_1$).

2. Observables for Extraction:

   • Lepton Charge Asymmetry: By comparing $e^-\gamma \to e^- B\bar{B}$ and $e^+\gamma \to e^+ B\bar{B}$, the pure QCD and QED terms cancel, leaving only the interference term, which contains the GDAs.
   • Angular Asymmetries: The authors define a forward-backward asymmetry ($\bar{A}_{FB}$) and a specific asymmetry $\bar{A}(\theta, \phi)$ based on the exchange $(\theta, \phi) \to (\pi-\theta, \pi+\phi)$. These isolations rely on the different charge conjugation properties of the two subprocesses.
   • Single-Spin Correlation: The paper calculates cross-sections dependent on the baryon spin vector. While unpolarized cross-sections probe the real parts of FF products, spin-dependent terms access the imaginary parts, offering complementary information.

3. Numerical Estimates:

   • The authors model the proton EM FFs using a two-component description fitted to experimental data.
   • For the proton-antiproton GDAs, they utilize a model derived from $\pi\pi$ GDAs and evolution equations, constrained by sum rules related to quark momentum fractions ($R_q$).
   • Axial-vector GDAs are assumed to be proportional to the axial charge due to a lack of specific data.
   • Calculations are performed for kinematics relevant to the Belle II experiment ($e^-\gamma \to e^- p\bar{p}$), varying $Q^2$ (15 and 30 GeV$^2$) and $\hat{s}$ (40 and 50 GeV$^2$).

Key Results

• Cross-Section Behavior: The bremsstrahlung contribution (pure QED) is dominant at high $Q^2$ but decreases significantly as $Q^2$ lowers. Conversely, the pure QCD (GDA) contribution is suppressed at high $Q^2$ but increases by approximately an order of magnitude when $Q^2$ drops from 30 to 15 GeV$^2$.
• Kinematic Windows: The authors identify a kinematic region (lower $Q^2$) where the GDA contribution becomes comparable to, or potentially larger than, the bremsstrahlung contribution, facilitating its extraction.
• Interference Term: The interference term, accessible via charge asymmetry or angular asymmetries, is found to be comparable in magnitude to the pure QCD contribution at $Q^2 = 30$ GeV$^2$.
• Polarization Effects: The single-spin correlation ratio $R(S_{1\uparrow})$ is estimated to be sizable. The analysis shows that the spin vector of the produced baryon can be perpendicular to the hadron plane (y-axis) or have components in the x-z plane depending on the specific FF interference terms.
• Feasibility: Numerical estimates suggest that with the luminosity of Belle II, a first extraction of baryon-antibaryon GDAs is feasible.

Significance and Claims
The paper claims to provide the first comprehensive theoretical study of baryon-antibaryon GDAs in the $e^\pm\gamma \to e^\pm B \bar{B}$ channel. Its primary significance lies in:

1. Extending GDA Studies: Moving beyond meson pairs to spin-1/2 baryons, including unstable hyperons ($\Lambda, \Sigma$), whose spin vectors can be determined via decay ($\Lambda \to N\pi$).
2. Methodological Proposal: Demonstrating that lepton charge asymmetry and specific angular asymmetries can isolate the interference term, thereby allowing the extraction of GDAs despite the presence of large QED backgrounds.
3. Experimental Guidance: Providing numerical estimates and identifying specific kinematic regions (lower $Q^2$) where GDA effects are enhanced, offering a roadmap for future measurements at Belle II, the proposed Super Tau-Charm Facility (STCF), and electron-ion colliders.

The authors maintain a modest stance regarding the precision of their predictions, noting that the current estimates are order-of-magnitude due to the poor knowledge of proton-antiproton GDAs and the unknown phases of the form factors. They emphasize that further theoretical development of GDAs and precise experimental analysis are necessary for definitive extractions.