Initial Baryon Stopping and Angular Momentum in Heavy-Ion Collisions

Noncentral heavy-ion collisions create fireballs with large initial orbital angular momentum that is expected to induce strong vorticity in the hot bulk fluid and generate global spin polarization of the produced particles. As the collision beam energy $\sqrt{s_{\rm NN}}$ decreases to approach the two-nucleon-mass threshold, this initial angular momentum approaches zero. One may thus expect that the observed global spin polarization should reach a maximum and then drop to zero as increased stopping competes with decreased initial momentum. Recent experimental measurements, however, appear to show a continual rise of hyperon polarization even down to $\sqrt{s_{\rm NN}} =$ 2.42 GeV, suggesting a peak very near threshold which is difficult to interpret and calls for a better understanding of angular momentum initial conditions, especially at low energy. Here, we develop a new Glauber-based initial state model (“Glauber+”) to investigate the initial distribution of angular momentum with respect to rapidity as well as the dependence of this distribution on initial baryon stopping across a wide range of collisional beam energy. We estimate that the angular momentum per produced final charged particle at mid-rapidity peaks around 5 GeV, which may present a potential challenge to an interpretation of the spin polarization measurements near threshold as being a consequence of the initial angular momentum of the colliding system.

💡 Research Summary

The paper addresses a puzzling observation in non‑central heavy‑ion collisions at low beam energies: global hyperon polarization continues to rise even as the center‑of‑mass energy √sₙₙ approaches the two‑nucleon mass threshold (≈ 1.88 GeV). Conventional wisdom predicts that the initial orbital angular momentum (OAM) of the system, which drives vorticity and spin polarization, should vanish at threshold, leading to a non‑monotonic dependence of polarization on √sₙₙ with a peak at intermediate energies. To resolve this tension, the authors develop an extended Glauber‑type initial‑state model, dubbed “Glauber+”, that incorporates the rapidity loss (baryon stopping) and the increase of the effective nucleon mass immediately after the first binary collisions.

Key ingredients of the model are:

1. Standard Glauber geometry – Woods‑Saxon nuclear densities generate participant thickness functions T_A,B(x,y) and participant densities n_A,B(x,y). The impact parameter b defines the transverse offset of the two nuclei.
2. Elasticity (coefficient of restitution) e – Defined as the ratio of post‑collision rapidity gap to the initial rapidity gap (e = (Y′_B – Y′_A)/(Y_B – Y_A)). e = 1 corresponds to no interaction, e = 0 to full stopping, and e = –1 to a perfectly elastic bounce. This single parameter controls how much longitudinal momentum is lost.
3. Effective mass λ – After the first encounter each nucleon becomes a “wounded nucleon” with an increased mass λ m_N, where λ ≥ 1. λ is derived from energy–momentum conservation equations (6)–(9) together with the chosen e and the local participant density ratio n_A/n_B.
4. Rapidities after collision – Using the conservation equations the post‑collision rapidities Y′_A and Y′_B are expressed analytically (Eq. 14, 15) as functions of (x,y), e, λ, and the beam rapidity Y_beam.
5. Baryon and angular‑momentum densities – The net‑baryon rapidity distribution is built from delta‑functions at Y′_A and Y′_B weighted by n_A and n_B (Eq. 20). The local OAM density is d²J_y/dxdy = –x λ m_N