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

It is shown that longitudinally polarized protons can be used to induce chirality in the final states of the $(\vec{p},pN)$ reaction at intermediate energies, when there exist three final-state particles with non-coplanar momentum vectors. The analyzing power $A_z$ is proposed as a measure of this effect. Theoretical descriptions to obtain $A_z$ based on an intuitive picture as well as a distorted wave impulse approximation are presented, showing that the helicity of incident protons is coupled to the chirality of the orbital motion of a single-particle wave function, resulting in the chirality of the final states and a large $A_z$ value.

💡 Research Summary

The paper proposes a novel way to generate and measure chirality in intermediate‑energy (p, pN) reactions by using a longitudinally polarized proton beam. When the reaction produces three final‑state particles (the incident proton, the knocked‑out nucleon, and the recoil residue) whose momentum vectors are non‑coplanar, the helicity of the incident proton can be transferred to the orbital motion of the struck nucleon, thereby imprinting a handedness on the final state. This handedness is quantified by a longitudinal analyzing power, Az, defined as the asymmetry between cross sections for a given set of momenta K and its mirror‑reflected counterpart ˜K.

The authors first give an intuitive picture: at beam energies of a few hundred MeV the nucleon–nucleon (NN) scattering amplitude is strongly spin‑dependent; spin‑parallel NN pairs have a much larger cross section (Czz≈1). Consequently, the helicity of the incoming proton (µ0) aligns with the spin projection µN of the bound nucleon in the target. For a p1/2 orbital the orbital angular momentum l points opposite to the beam direction, tilting the NN scattering plane out of the beam axis. This tilt makes K and ˜K non‑coplanar. Because the faster outgoing proton (particle 1) can escape the nucleus more easily than the slower particle 2, the absorption of the two configurations differs: the K configuration suffers less attenuation than ˜K, leading to σ(K)>σ(˜K) and a positive Az.

To formalize this, the authors employ the distorted‑wave impulse approximation (DWIA). The transition matrix T is expressed as a product of the NN transition amplitude t̃ and the single‑particle (s.p.) wave function Φnℓj,µN. For ℓ≠0 the azimuthal dependence e^{imφ} of the s.p. wave function introduces a chiral phase that changes sign under mirror reflection (e^{imφ}→e^{-imφ}). The absorption factor DK(R) = χ₁⁎(R)χ₂⁎(R)χ₀(R) encodes the nuclear attenuation of the three distorted waves. Expanding DK in a Fourier series over φ isolates coefficients dm that carry the same m‑dependence as the s.p. wave function. After integrating over φ only the dm term with m = ±1 survives, and the cross sections for spin‑up (σ⁺) and spin‑down (σ⁻) become proportional to |F_{nℓj,µj∓½}(K)|². For a p1/2 orbit σ⁺−σ⁻ is positive, while for a p3/2 orbit it is negative, predicting opposite signs of Az for the two subshells.

Numerical calculations are performed for the 16O(p, 2p)15N reaction at an incident proton energy of 250 MeV. The kinematics are fixed: particle 1 kinetic energy T₁=158 MeV, θ₁=27°, θ₂=56°, and the azimuthal angle φ₂ is varied from 90° to 270°. The DWIA code “pikoe” is used with a Bohr‑Mottelson s.p. potential, the Franey‑Love NN interaction, and the EDAD1 Dirac optical potentials. In the baseline calculation the spin‑orbit term of the optical potential is omitted and Czz=1 is enforced (µ₀=µ_N). The results show that Az vanishes for the coplanar case (ϕ₁₂=180°) and obeys the symmetry Az(ϕ)=−Az(360°−ϕ). For the p1/2 orbit Az reaches +0.4–0.6 around ϕ₁₂≈140°–150° and 210°–220°, while for the p3/2 orbit the same angular region yields Az≈−0.4. Including the spin‑orbit term modifies Az near the edges of the angular range but leaves the central peaks essentially unchanged, confirming that the mechanism is robust.

The paper discusses experimental implications. A sizable Az requires (i) a strong spin‑spin correlation in the NN scattering (Czz≈1), (ii) a non‑zero orbital angular momentum of the struck nucleon, and (iii) a non‑coplanar geometry that differentiates the absorption of the two mirror configurations. Consequently, reactions such as (p, pα) are unlikely to produce large Az, whereas three‑body knockout channels like (p, p ³He) or (p, pt) could be promising. The authors also note that Az provides direct access to the sign of the magnetic quantum number m, offering a new observable to probe single‑particle orbital structure beyond the traditional transverse analyzing power Ay.

In summary, the work establishes that longitudinal proton helicity can be transferred to the orbital motion of a bound nucleon, generating a measurable chirality in the final state of (p, pN) reactions. The longitudinal analyzing power Az serves as a quantitative measure of this effect, with DWIA calculations predicting sizable, orbit‑dependent asymmetries that are experimentally accessible in intermediate‑energy proton‑induced knockout reactions. This opens a novel avenue for studying nuclear single‑particle dynamics and chiral phenomena in hadronic processes.