Estimate of the Collins fragmentation function in a chiral invariant approach

TL;DR

The paper estimates the Collins fragmentation function $H_1^{\perp}$ for pions at a soft scale using the Manohar-Georgi chiral invariant model. It computes the unpolarized fragmentation function $D_1$ at tree level and demonstrates that $H_1^{\perp}$ arises from one-loop corrections, with explicit expressions based on Cutkosky-imaginary parts. The results show $D_1$ is reasonably reproduced and $H_1^{\perp}$ grows with $z$, yielding a rising $H_1^{\perp}/D_1$ ratio and predicting SIDIS single-spin asymmetries around 10% and $e^+e^-$ azimuthal asymmetries around 5%, offering a pathway to extract transversity and the Collins function at low scales. These findings provide low-scale inputs for evolution and phenomenology, highlighting the role of chiral dynamics and loop effects in T-odd fragmentation and emphasizing the need for experimental data to constrain the transversity distribution and $H_1^{\perp}$.

Abstract

We predict the features of the Collins function, which describes the fragmentation of a transversely polarized quark into an unpolarized hadron, by modeling the fragmentation process at a low energy scale. We use the chiral invariant approach of Manohar and Georgi, where constituent quarks and Goldstone bosons are considered as effective degrees of freedom in the non-perturbative regime of QCD. To test the approach we calculate the unpolarized fragmentation function and the transverse momentum distribution of a produced hadron, both of which are described reasonably well. In the case of semi-inclusive deep-inelastic scattering, our estimate of the Collins function in connection with the transversity distribution gives rise to a transverse single spin asymmetry of the order of 10%, supporting the idea of measuring the transversity distribution of the nucleon in this way. In the case of e+ e- annihilation into two hadrons, our model predicts a Collins azimuthal asymmetry of about 5%.

Figures (15)

• Figure 1: Lowest-order unitarity diagram describing the fragmentation of a quark into a pion.
• Figure 2: One-loop corrections to the fragmentation of a quark into a pion contributing to the Collins function. The hermitian conjugate diagrams (h.c.) are not shown explicitly.
• Figure 3: One-loop self-energy, and vertex corrections.
• Figure 4: Model result for the unpolarized quark fragmentation function $D_1^{u \rightarrow \pi^+}$ (solid line) and comparison with a parametrization of Ref. Kretzer:2000yf (grey line).
• Figure 5: Model result for the average hadron transverse momentum for different choices of the cutoff: $\mu^2=0.5$ GeV$^2$ (dotted line), $\mu^2=1$ GeV$^2$ (solid line), $\mu^2=1.5$ GeV$^2$ (dashed line) and comparison with a fit to experimental results from DELPHI Abreu:1996na (grey line).