Chirality in $(\vec{p},2p)$ reactions induced by proton helicity

This paper demonstrates that longitudinally polarized protons can induce chirality in the final states of intermediate-energy $(\vec{p},2p)$ reactions involving non-coplanar momenta, a phenomenon quantified by the analyzing power $A_z$ and explained through the coupling of incident proton helicity to the orbital chirality of single-particle wave functions.

The Gist

The Big Idea: Finding the "Handedness" of the Atomic Nucleus

Imagine you are looking at a pair of gloves. A left-handed glove and a right-handed glove look almost identical, but you can never stack them perfectly on top of each other. They are chiral (handed). In the world of physics, scientists have long wondered: Do atomic nuclei have a "handedness" too?

Usually, the answer is "no." If you look at a nucleus in a mirror, it looks exactly the same as the real thing. Nature is generally fair; it doesn't prefer left over right. However, this paper proposes a clever trick to force a nucleus to show its "handedness" during a collision.

The Setup: The High-Speed Pinball Game

To do this, the scientists propose a specific experiment called a $(\vec{p}, 2p)$ reaction. Here is how it works:

1. The Cue Ball (The Beam): They fire a stream of protons (tiny particles) at a target nucleus (like Oxygen-16). But there's a catch: these protons are longitudinally polarized.
   • Analogy: Imagine spinning a top. Usually, a top spins around its vertical axis. "Longitudinal polarization" means the protons are spinning like a bullet, with their spin pointing exactly in the direction they are flying.
2. The Collision: One of these spinning protons hits a proton inside the target nucleus.
3. The Ejection: The hit knocks the target proton out. Now, you have two protons flying away from the crash site (the original beam proton and the knocked-out proton), plus the leftover nucleus.

The Secret Sauce: The "Non-Coplanar" Dance

If these three objects (the two protons and the leftover nucleus) flew off in a flat, 2D sheet (like a sheet of paper), the experiment wouldn't work. The paper argues that for "handedness" to appear, the particles must fly off in a 3D, non-flat shape.

• Analogy: Imagine three people running away from a central point. If they all run in a straight line or a flat circle, there is no "twist." But if one runs forward, one runs up, and one runs sideways, they form a twisted, spiral shape. That twist is the chirality.

How They Measure It: The "Left vs. Right" Score

The scientists define a score called $A_z$ (Longitudinal Analyzing Power).

• The Mirror Test: They look at the pattern of the three flying particles. Then, they imagine a mirror image of that pattern.
• The Comparison: They ask: "Did the reaction happen more often in the 'real' pattern or the 'mirror' pattern?"
• The Result: If the numbers are different, the nucleus has shown a preference for one "handedness" over the other. The paper predicts this difference will be huge.

Why Does This Happen? (The Intuitive Explanation)

The paper offers a simple reason why this works, based on three factors:

1. The Spin Match: When the spinning "bullet" proton hits the proton inside the nucleus, they prefer to be spinning in the same direction (like two dancers spinning the same way). This forces the proton inside the nucleus to align its spin with the beam.
2. The Orbit: Protons inside a nucleus don't just sit still; they orbit the center. Because the spin is now locked to the beam direction, the proton's orbit gets "tilted" relative to the beam.
3. The Absorption (The Wall): This is the most important part. As the two protons fly out, they have to pass through the rest of the nucleus.
   • Analogy: Imagine two runners leaving a stadium. One takes a short path out the front door. The other has to run through a crowded hallway to get out.
   • Because of the "tilted" orbit caused by the spin, one of the protons has to travel through more "crowd" (nuclear matter) than the other. The nucleus "absorbs" or blocks the one taking the long path.
   • This blocking effect is different for the "real" pattern versus the "mirror" pattern. This difference creates the measurable "handedness."

The Results: A Clear Signal

The authors ran computer simulations using Oxygen-16. They found:

• When the particles fly in a flat line (coplanar), the "handedness" score is zero.
• When they fly in a twisted, 3D shape (non-coplanar), the score jumps up.
• Interestingly, the sign of the score flips depending on which "orbit" the proton came from (like $p_{1/2}$ vs. $p_{3/2}$). It's like saying the left-handed glove looks different from the right-handed glove depending on which finger you are looking at.

Why Should We Care?

This isn't just about proving nuclei are twisty. It's a new tool.

• A New Lens: Previously, scientists mostly used beams spinning sideways (transverse polarization) to study nuclei. This paper shows that using beams spinning forward (longitudinal) opens a new window.
• Unstable Nuclei: This method is perfect for studying "exotic" or unstable nuclei (like those found in stars or created in labs) because the signal is strong and clear.
• Fundamental Forces: It might help us understand the "glue" that holds the nucleus together (the nuclear force) and how it interacts with the spin of particles.

Summary

In short, the authors propose using a spinning bullet to knock a proton out of a nucleus. If the debris flies off in a twisted 3D shape, the nucleus will reveal a left-right preference (chirality) that was previously hidden. This preference is caused by the spinning proton getting "stuck" in the nucleus's "crowded hallway" on its way out. It's a clever way to make the invisible "handedness" of the atomic world visible.

Technical Summary

1. Problem Statement

The paper addresses the challenge of inducing and measuring chirality (handedness) in nuclear reactions within the laboratory frame.

• Context: While chirality is observed in weak interactions (e.g., parity violation in $\beta$-decay) and specific nuclear structures (triaxial nuclei), the total Hamiltonian of an isolated nucleus is invariant under mirror reflection. Consequently, mirror-symmetric phenomena occur with equal probability unless an external stimulus breaks this symmetry.
• Gap: Previous studies of nucleon knockout reactions, specifically $(p, pN)$, have primarily utilized transversely polarized proton beams to measure the vector analyzing power ($A_y$). This measures geometric asymmetry in coplanar scattering.
• Objective: The authors propose using a longitudinally spin-polarized (helicity) proton beam to induce chirality in the final states of intermediate-energy $(p, pN)$ reactions. They aim to define a new observable, the longitudinal analyzing power ($A_z$), to quantify the asymmetry between non-coplanar final-state momentum configurations and their mirror images.

2. Methodology

The study employs a combination of intuitive physical reasoning and rigorous theoretical formalism based on the Distorted Wave Impulse Approximation (DWIA).

A. Theoretical Framework

• Reaction Model: The $(p, pN)$ reaction is treated as a quasi-free nucleon-nucleon (NN) scattering process where an incident proton (particle 0) strikes a nucleon inside the target nucleus (particle N), ejecting it (particle 2) while the incident proton becomes particle 1.
• Key Assumption: The NN scattering amplitude has a strong spin dependence ($C_{zz} \approx 1$), meaning scattering is highly favored for spin-parallel pairs. This couples the helicity of the incident proton ($\mu_0$) to the spin of the struck nucleon ($\mu_N$), such that $\mu_0 = \mu_N$.
• Definition of $A_z$:
  The longitudinal analyzing power is defined as the asymmetry between the cross-section of a non-coplanar final state configuration ($\mathbf{K}$) and its mirror reflection ($\tilde{\mathbf{K}}$):
  $$A_z = \frac{\sigma_+(\mathbf{K}) - \sigma_+(\tilde{\mathbf{K}})}{\sigma_+(\mathbf{K}) + \sigma_+(\tilde{\mathbf{K}})}$$
  Alternatively, it can be viewed as the asymmetry between positive and negative beam helicities for a fixed configuration.
• Mechanism of Chirality:
  1. Spin-Orbit Coupling: The incident proton's helicity aligns with the struck nucleon's spin. Due to the orbital angular momentum ($l$) of the bound nucleon, the spin and orbital motion are coupled (LS coupling).
  2. Non-Coplanarity: The orbital motion tilts the NN scattering plane relative to the beam axis, creating a non-coplanar momentum configuration ($\mathbf{K} \neq \tilde{\mathbf{K}}$).
  3. Absorption Effect (Maris Effect): The asymmetry arises from nuclear absorption. In one configuration ($\mathbf{K}$), the lower-momentum ejected nucleon travels a short path through the nucleus and survives. In the mirror configuration ($\tilde{\mathbf{K}}$), the same nucleon must traverse a longer path through the nuclear density, leading to higher absorption and a lower cross-section.

B. Computational Implementation

• Code: Calculations were performed using the pikoe code (DWIA for proton-induced knockout).
• Target System: The specific case studied is $^{16}\text{O}(\vec{p}, 2p)^{15}\text{N}$ at an incident energy of 250 MeV.
• Orbits: Two single-particle orbits were analyzed: $0p_{1/2}$ and $0p_{3/2}$.
• Kinematics:
  • Fixed energies and angles for the two outgoing protons.
  • The azimuthal angle difference ($\phi_{12}$) between the two outgoing protons was varied to introduce non-coplanarity.
  • Coplanar condition corresponds to $\phi_{12} = 180^\circ$.

3. Key Contributions

1. Proposal of $A_z$: Introduction of a new observable specifically designed to detect chirality induced by longitudinal beam polarization in non-coplanar three-body final states.
2. Mechanism Elucidation: A clear theoretical explanation linking the incident proton's helicity to the orbital angular momentum of the target nucleon via strong spin correlations ($C_{zz} \approx 1$), and subsequently to the final state asymmetry via nuclear absorption.
3. Orbital Dependence: Demonstration that the sign of $A_z$ is determined by the total angular momentum $j$ of the orbit relative to the orbital angular momentum $l$ (i.e., $j = l \pm 1/2$).
4. Experimental Feasibility: Identification of specific kinematic regions (non-coplanar angles) where $A_z$ is large and the triple differential cross-section (TDX) is sufficient for measurement.

4. Results

• Zero at Coplanarity: As predicted, $A_z = 0$ when the final state is coplanar ($\phi_{12} = 180^\circ$), confirming that non-coplanarity is a prerequisite for observing this chirality.
• Sign Reversal: A distinct difference was found between the $0p_{1/2}$ and $0p_{3/2}$ orbits:
  • $0p_{1/2}$ ($j < l$): Exhibits a positive $A_z$ in the region $\phi_{12} < 180^\circ$.
  • $0p_{3/2}$ ($j > l$): Exhibits a negative $A_z$ in the same region.
  • This confirms the theoretical prediction that the sign of the asymmetry depends on the LS coupling configuration.
• Magnitude: The calculated values of $A_z$ are significant (reaching up to $\approx \pm 0.6$ in idealized scenarios and remaining robust in realistic calculations including spin-orbit terms and non-parallel spin components).
• Peaks: Realistic calculations show peaks in $A_z$ around $\phi_{12} \approx 140^\circ\text{--}150^\circ$ and $210^\circ\text{--}220^\circ$, where the cross-section is also finite, making these regions optimal for experimental verification.

5. Significance and Future Outlook

• Probing Nuclear Structure: $A_z$ serves as a sensitive probe for the orbital motion of single-particle states in nuclei. It allows researchers to distinguish between spin-orbit partners ($j = l \pm 1/2$) based on the sign of the asymmetry.
• Unstable Nuclei: The method is particularly advantageous for studying unstable nuclei using inverse kinematics (radioactive ion beams). Unlike $A_y$, which requires specific angular coverage, $A_z$ can be measured using $\phi$-symmetric detectors (e.g., solenoid spectrometers), which are more compatible with inverse kinematics setups.
• Fundamental Interactions: The observable may be sensitive to axial-vector couplings in nuclear forces, potentially offering a new avenue to investigate three-nucleon forces (3N) and their rank-1 components.
• Experimental Path: The paper provides a roadmap for future experiments, suggesting that measuring $A_z$ alongside $A_y$ will be crucial for disentangling the origins of observed asymmetries and validating theoretical models of nuclear chirality.

In summary, this work establishes a theoretical foundation for using longitudinally polarized protons to induce and measure chirality in nuclear knockout reactions, offering a powerful new tool for investigating the interplay between spin, orbital angular momentum, and nuclear absorption.