Single spin asymmetry in $e+p\to e'+B^\uparrow+X$

This paper investigates a novel single spin asymmetry in unpolarized electron-proton scattering where the outgoing electron's momentum is asymmetric relative to the transverse spin of a leading baryon, proposing two theoretical explanations: twist-three fracture functions and the spin-dependent odderon.

The Gist

The Cosmic "Left-Right" Mystery: A Simple Guide to the Hatta & Teryaev Paper

Imagine you are playing a game of billiards. You hit a white cue ball (the electron) toward a cluster of colored balls (the proton). Usually, in physics, if you don't know anything about the spin or "rotation" of the balls, you’d expect the results to be totally random. If you hit the cluster, the pieces should fly off in every direction with no predictable pattern.

But this paper explores a strange, "exotic" phenomenon: What if the pieces fly off with a consistent tilt?

Specifically, the authors are looking at a scenario where the electron hits a proton, and one of the "shrapnel" pieces (a baryon, like a neutron or a Lambda particle) comes flying out spinning in a certain direction. Even though the electron itself wasn't spinning, the way that piece flies out might show a "left-right" preference based on its spin.

Here is how they break down this mystery using two different "detective frameworks."

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1. The "Tangled Threads" Approach (Twist-Three Fracture Functions)

The Concept: High-energy, small-scale physics.

Imagine the proton isn't just a solid ball, but a massive, swirling ball of yarn made of tiny threads (quarks and gluons). When the electron hits the proton, it doesn't just bounce off; it rips through these threads.

In "ordinary" physics, we usually look at how the threads are arranged before the hit. But this paper looks at the "Fracture Function." Think of this as studying the specific way the yarn tangles and snaps during the explosion.

The authors argue that because these threads are constantly interacting and "tugging" on each other as they break, they create a tiny, invisible "twist" in the debris. This twist causes the outgoing particle to prefer one side over the other. It’s like if you snapped a spinning top—the way the pieces fly apart tells you exactly how the top was wobbling at the moment of impact.

2. The "Ghostly Shadow" Approach (The Odderon)

The Concept: High-energy, large-scale physics (The Regge Limit).

Now, imagine a different scenario. Instead of a violent collision, imagine the electron is like a flashlight beam passing near a dark, swirling cloud (the proton).

In this world, there are two types of "ghostly" forces that govern how things move:

• The Pomeron: Think of this as a friendly, predictable wind that moves everything in a standard way.
• The Odderon: Think of this as a strange, "antisocial" shadow-wind. It’s much harder to detect and behaves differently than the Pomeron.

The authors suggest that the "left-right" asymmetry happens because of an interference between these two. It’s like two different waves in a pool hitting each other. Where the "friendly wind" and the "shadow wind" overlap, they create a specific pattern—a ripple that pushes the outgoing particles to the left or the right.

By measuring this tilt, scientists can finally "see" the Odderon, a particle-like force that has been theorized for decades but is incredibly difficult to pin down.

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Why does this matter? (The "Recoil Polarimetry" Problem)

The big "catch" in this whole theory is that to see this effect, you have to be able to measure the spin of the piece flying away. This is like trying to determine if a single splinter flying out of a broken window is spinning clockwise or counter-clockwise while it's mid-air.

It is incredibly difficult, but the authors point out that new technology (like the upcoming Electron-Ion Collider) is making this possible.

The Bottom Line

This paper is a roadmap for a new way to "interrogate" the proton. Instead of just smashing it to see what's inside, we can look at the spin and direction of the debris to understand the invisible, "T-odd" forces that hold the universe together. It’s moving from just seeing the explosion to understanding the subtle, rhythmic "dance" of the particles within it.

Technical Summary

Technical Summary: Single Spin Asymmetry in $e + p \to e' + B^\uparrow + X$

Authors: Yoshitaka Hatta and Oleg V. Teryaev

1. The Problem

The paper investigates a novel class of Transverse Single Spin Asymmetries (SSAs) in unpolarized Deep Inelastic Scattering (DIS). Traditionally, SSAs are studied by colliding unpolarized projectiles with transversely polarized targets. However, the authors propose a "reciprocal" effect: even in unpolarized collisions, an asymmetry can arise if one measures the transverse polarization of a leading baryon ($B$) in the target fragmentation region (e.g., a proton, neutron, or $\Lambda$ particle).

The specific observable is the azimuthal asymmetry $\sin(\phi_{\ell'} - \phi_{s_B})$, which describes a left-right correlation between the momentum of the outgoing scattered electron ($\ell'$) and the transverse spin vector of the produced baryon ($s_B$). This effect is "naively T-odd," meaning it is consistent with QCD symmetries but requires the presence of imaginary phases (absorptive parts) in the scattering amplitudes, typically generated by final-state interactions (FSI).

2. Methodology

The authors analyze this phenomenon using two distinct theoretical frameworks corresponding to different kinematic regimes:

• High-$Q^2$ Limit (Collinear Factorization): The authors employ the framework of twist-three fracture functions. Unlike standard distribution or fragmentation functions, "fracture functions" describe the production of a hadron in the target fragmentation region. They introduce the T-odd Fracture Function (TOFF), which accounts for the correlation between the target remnant and the final-state baryon's spin. The calculation involves decomposing the hadronic tensor $W^{\mu\nu}$ and identifying the interference between transversely (T) and longitudinally (L) polarized virtual photons.
• High-Energy/Regge Limit (Dipole Picture): In the limit where $p \cdot q \gg Q^2$ (diffractive DIS), the process is viewed as a color dipole (a $q\bar{q}$ pair from the virtual photon) scattering off the proton. The authors model the asymmetry as an interference effect between the Pomeron (a C-even exchange) and the Odderon (a C-odd exchange).

3. Key Contributions

• Generalization of the Christ-Lee Proposal: The paper revisits a 1966 proposal by Christ and Lee regarding T-violation and modernizes it into a study of QCD spin physics, specifically focusing on how FSI generate spontaneous polarization in the target fragmentation region.
• Identification of TOFF: They provide a rigorous definition of the T-odd fracture function $F_{odd}(x_B, x_L, t)$ and show how it relates to the collinear twist-three functions $u_T^R$ (a "recoil" version of the $u_T$ function used in initial-state SSAs).
• Pomeron-Odderon Interference Mechanism: They demonstrate that the SSA in the Regge limit is a direct probe of the spin-dependent Odderon. They show that while the pure Odderon contribution to the cross-section is proportional to the square of the Odderon amplitude, the SSA is proportional to the interference term, making it linearly sensitive to the Odderon and potentially easier to detect.

4. Results

• Twist-Three Result: In the high-$Q^2$ regime, the asymmetry is driven by the function $u_T^R(x_B, x_L, t)$. The authors derive the differential cross-section, showing that the asymmetry is a leading-order effect in $\alpha_s$ (zeroth order) because it does not require high-$p_T$ hadron production.
• Odderon Estimate: Using Gaussian models for the Pomeron ($T$) and Odderon ($O$) amplitudes, the authors provide a crude numerical estimate for the asymmetry $A_N$. They suggest that for the asymmetry to be experimentally measurable (e.g., $|A_N| \gg 0.1\%$), the dimensionless parameter $C$ (representing the strength of the Odderon) must be at least $0.01$.
• C-Parity Requirement: They note that for the Pomeron-Odderon interference to survive integration over the final state $X$, one must select final states without definite C-parity (e.g., by measuring charged hadrons like $\pi^+$), as the interference term is C-parity odd.

5. Significance

This work provides a new theoretical roadmap for exploring the "fine details" of the strong interaction. Its significance is three-fold:

1. New Observables: It introduces a new class of SSA observables that can be studied even without polarized beams, provided that recoil polarimetry (measuring the spin of the outgoing baryon) is available.
2. Experimental Relevance: The proposed measurements are highly relevant for future facilities like the Electron-Ion Collider (EIC) and current experiments at Jefferson Lab, where recoil proton/baryon polarization measurements are being developed.
3. Probing the Odderon: It offers a unique, potentially higher-rate method to study the elusive spin-dependent Odderon through interference rather than direct production.