[Paper Review] On the interpretation of $Λ$ spin polarization measurements

Wojciech Florkowski, Radosław Ryblewski|arXiv (Cornell University)|Feb 4, 2021

High-Energy Particle Collisions Research4 citations

TL;DR

This paper proposes a corrected method for interpreting $Λ$ hyperon spin polarization measurements in relativistic heavy-ion collisions by properly accounting for Lorentz boosts between the center-of-mass frame and the $Λ$ rest frame. It shows that projecting polarization along the direction of total angular momentum—transformed to the $Λ$ rest frame before averaging—yields a 10% correction for high-transverse-momentum $Λ$'s, improving accuracy for relativistic hyperons.

ABSTRACT

The physics interpretation of the recent measurements of the spin polarization of $Λ$ hyperons produced in relativistic heavy-ion collisions is discussed. We suggest that the polarization measured in the $Λ$ rest frame should be projected along the direction of the total angular momentum that is first transformed to the same frame, and only then averaged over $Λ$'s with different momenta in the center-of-mass frame. While this procedure does not affect the current measurements done in a broad transverse-momentum range, it may become important (represent a correction of about 10\%) for the most energetic hyperons under study (with transverse momenta reaching 4-5 GeV/c). The proposed treatment is generally more appropriate for relativistic $Λ$'s. Throughout the paper, we deliver explicit expressions for various boosts, rotations, and transformations of angular distributions, which may help to compare model predictions with the experimental results.

Motivation & Objective

• To address inconsistencies in the interpretation of $Λ$ spin polarization measurements due to improper frame transformations.
• To resolve the mismatch between the $Λ$ rest frame and center-of-mass (COM) frame directions, especially for relativistic $Λ$'s.
• To provide a frame-invariant method for relating $Λ$ spin polarization to the system's total angular momentum.
• To deliver explicit transformation formulas for angular distributions and polarization vectors across frames, enabling better model-experiment comparison.

Proposed method

• Applies Lorentz transformations to map the total angular momentum vector from the COM frame to the $Λ$ rest frame before polarization projection.
• Uses explicit boost and rotation transformations to relate spatial directions in the COM frame to those in the $Λ$ rest frame.
• Derives frame-invariant polarization components via four-vector formalism, ensuring consistency across reference frames.
• Constructs the projected momentum distribution of decay protons along an arbitrary direction in the $Λ$ rest frame using angular distribution integrals with Dirac delta functions.
• Derives the Jacobian correction for the delta function in angular integrals to handle non-trivial directional mappings.
• Validates the transformation by numerically confirming consistency with the inverse of the weak decay angular distribution.

Figure 1: The center-of-mass (COM) frame for non-central heavy-ion collisions. In this case both the reaction and production planes can be defined.

Experimental results

Research questions

• RQ1 How does the direction of the total angular momentum in the COM frame relate to the direction in the $Λ$ rest frame after Lorentz transformation?
• RQ2 What is the impact of neglecting frame-dependent direction transformations on the measured $Λ$ spin polarization?
• RQ3 How can polarization be consistently projected along the total angular momentum direction when the $Λ$ is relativistic?
• RQ4 What corrections arise from proper frame transformation for high-transverse-momentum $Λ$'s?
• RQ5 Can a frame-invariant formalism be constructed to relate experimental polarization measurements to global angular momentum in heavy-ion collisions?

Key findings

• The standard interpretation of $Λ$ spin polarization neglects the non-trivial boost-induced rotation between the COM frame and the $Λ$ rest frame.
• Projecting polarization along the total angular momentum direction after transforming it to the $Λ$ rest frame yields a more accurate and physically consistent result.
• For $Λ$'s with transverse momenta of 4–5 GeV/c, the correction to polarization magnitude is approximately 10%, making it significant for high-$p_T$ measurements.
• The proposed method ensures frame-invariance of polarization direction by using a four-vector formalism for the direction vector.
• Explicit analytical expressions are derived for the transformation of angular distributions and polarization components across frames, enabling direct comparison with model predictions.
• Numerical checks confirm the consistency of the derived transformation with the inverse of the weak decay angular distribution.

Figure 2: The center-of-mass (COM) frame for $p$ + $p$ collisions.