[Paper Review] Microscopic Stern-Gerlach Effect and Thomas Spin Precession as an Origin of the SSA

V. Abramov|ArXiv.org|Oct 7, 2009

Molecular spectroscopy and chirality3 citations

TL;DR

This paper proposes a semi-classical mechanism for single-spin asymmetries (SSA) in high-energy hadronic collisions, attributing them to the microscopic Stern-Gerlach effect in chromomagnetic fields and Thomas spin precession in chromoelectric fields generated by spectator quarks. The model explains the observed $A_N(x_F)$ and $P_N(p_T)$ dependencies in $p^ ightarrow p$ and A+A collisions, including oscillatory behavior due to quark spin precession in effective color fields (ECF), with predictions matching data from E704, BRAHMS, and STAR experiments.

ABSTRACT

The single-spin asymmetry and hadron polarization data are analyzed in the framework of a phenomenological effective-color-field model. Global analysis of the single-spin effects in hadron production is performed for h+h, h+A, A+A and lepton+N interactions. The model explains the dependence of the data on $x_{F}$, $p_{T}$, collision energy $\sqrt{s}$ and atomic weights $A_{1}$ and $A_{2}$ of colliding nuclei. The predictions are given for not yet explored kinematical regions.

Motivation & Objective

• To explain the origin of single-spin asymmetries (SSA) in inclusive hadron production across $h+h$, $h+A$, $A+A$, and lepton+N reactions.
• To account for the observed dependence of SSA on $x_F$, $p_T$, $ oot s o$, and atomic weights $A_1$, $A_2$ in hadronic collisions.
• To provide a phenomenological model based on effective color fields (ECF) from spectator quarks and antiquarks, consistent with quark counting rules.
• To predict oscillatory behavior in $A_N$ and $P_N$ as functions of kinematic variables, testable at existing accelerators.
• To unify the microscopic origin of SSA through chromomagnetic Stern-Gerlach forces and chromoelectric Thomas precession in a semi-classical framework.

Proposed method

• Model the effective color field (ECF) as a superposition of QCD string fields from spectator quarks and antiquarks, with spatial dependence $E^{(3)}_Z o -2ar{α}_s u_A / ho^2 ext{exp}(-r^2/ ho^2)$ and $B^{(2)}_φ o -2ar{α}_s u_A r / ho^3 ext{exp}(-r^2/ ho^2)$.
• Treat the detected hadron's valence quark as a 'probe' experiencing Stern-Gerlach forces $f_x, f_y$ via its chromomagnetic moment $\mu^a_Q = s g^a_Q g_S / 2M_Q$.
• Model spin precession using Bargman-Michel-Telegdi equations (3)–(4), with precession frequency $\omega_A = g_S \alpha_s \nu_A S_0 (g^a_Q - 2 + 2M_Q/E_Q) / (M_Q \rho^2 c)$.
• Include Thomas precession via an effective Hamiltonian term $U = \mathbf{s} \cdot \boldsymbol{\omega}_T$, with $\boldsymbol{\omega}_T \approx [\mathbf{F} \mathbf{v}] / M_Q$.
• Apply quark counting rules to determine color charge weights: $\lambda \approx -1/8$ for quark pairs, $1$ for antiquarks, and $-\tau\lambda$ for target spectators with $\tau \approx -0.0562$.
• Derive analyzing power $A_N = -D \delta p_x$ with $\delta p_x$ from spin-dependent momentum shift due to precession and field gradients, including oscillatory terms in $x_A$.

Experimental results

Research questions

• RQ1 How do microscopic Stern-Gerlach forces in chromomagnetic fields and Thomas precession in chromoelectric fields contribute to single-spin asymmetries in hadron production?
• RQ2 What is the origin of the observed $A_N(x_F)$ and $P_N(p_T)$ dependencies in $p^\uparrow + p \to \pi^+ + X$ and $\Lambda$ production in A+A collisions?
• RQ3 Why does the model predict oscillatory behavior in $A_N$ and $P_N$ as functions of $x_A$, $p_T$, and $\sqrt{s}$, and how does this arise from quark spin precession?
• RQ4 How do the effective color field (ECF) parameters—such as $\nu_A$, $\rho$, $S_0$, and $g^a_Q$—determine the magnitude and sign of the asymmetry?
• RQ5 To what extent do the model predictions for $A_N$ and $P_N$ at high energies ($\sqrt{s} = 500$ GeV) and in nuclear collisions (Au+Au, Cu+Cu) match existing experimental data?

Key findings

• The model predicts a negative $A_N$ for $p^\uparrow + p \to \pi^+ + X$ at $\sqrt{s} = 200$ GeV and $x_F \approx 0.6$, and for $\sqrt{s} = 130$ GeV in the range $x_F \in [0.1, 0.6]$, due to $u$-quark spin precession in strong ECF.
• For $\sqrt{s} = 500$ GeV, the model predicts strongly oscillating $A_N(x_F)$ behavior, as shown in Fig. 2b, due to enhanced precession frequency and field gradients.
• The model reproduces the STAR data on $P_N(p_T)$ for $\Lambda$ hyperons in Au+Au collisions at $\sqrt{s} = 62$ and $200$ GeV, with oscillations arising from $s$-quark spin precession in strong color fields.
• The effective color field strength increases dramatically at $\sqrt{s} > 70$ GeV, leading to enhanced spin precession and oscillatory asymmetries in nuclear collisions.
• The global fit yields $\lambda = -0.1321 \pm 0.0012$, consistent with the expected value, providing strong support for the ECF model.
• The model predicts oscillating $P_N(\eta)$ behavior in S+S, Cu+Cu, and Au+Au collisions at $\sqrt{s} = 7$ and $9$ GeV, with pseudorapidity dependence due to precession in the fragmentation region.