Modelling quark distribution and fragmentation functions

TL;DR

The paper combines a non-local operator description of quark distributions and fragmentations with a covariant spectator (diquark) model to estimate functions for nucleons and pions, ensuring correct support and crossing properties. By evaluating multiple Dirac projections, it yields both leading and subleading transverse-momentum dependent functions and uncovers relations implied by Lorentz invariance and discrete symmetries. The results show qualitative agreement with low-scale parton distributions and fragmentation data, while clarifying the model’s limitations, notably the absence of sea quarks and QCD evolution. This framework provides a tractable, covariant means to explore the full set of distribution and fragmentation functions, including higher-twist and T-odd structures in fragmentation.

Abstract

The representation of quark distribution and fragmentation functions in terms of non-local operators is combined with a simple spectator model. This allows us to estimate these functions for the nucleon and the pion ensuring correct crossing and support properties. We give estimates for the unpolarized functions as well as for the polarized ones and for subleading (higher twist) functions. Furthermore we can study several relations that are consequences of Lorentz invariance and of C, P, and T invariance of the strong interactions.

Figures (12)

• Figure 1: Diagrammatic representation of the correlation functions $\Phi$ and $\Delta$.
• Figure 2: The $\delta$-function constraint in the $\sigma$-$\tau$ plane (using quark momentum $k$ and hadron momentum $P$) coming from fixing $x$ and $\bm k_T^2$ in the expression for the distribution functions $F(x,\bm k_T^2)$ (and similarly for the fragmentation functions $D(z,z^2 \bm k_T^2)$) and the full integration regions for the $\bm k_T$ integrated functions $F(x)$ (and similarly for $D(z)$). The latter region is determined by $\bm k_T^2 \ge 0$ and $M_R^2 \ge 0$.
• Figure 3: The constraint in the $\sigma-\tau$ plane coming from fixing the spectator mass $M_R$ (compare with Fig. \ref{['support']}).
• Figure 4: Twist two distributions for the nucleon. The plots on the top represent $f_1(x)$, the ones on the middle show $g_{1}(x) / a_R$ and at the bottom we have $h_{1}(x) /a_R$. The plots on the left correspond to $\Lambda = 0.4$ GeV and the ones on the right to $\Lambda = 0.6$ GeV. The full line corresponds to $M_R = 0.6$ GeV and the dashed line to $M_R = 0.8$ GeV.
• Figure 5: Twist two distributions for the nucleon. The plot at the top shows $x f_1^s(x)$ (full line) and $x f_1^a(x)$ (dashed line) for $M_s=0.6$ GeV, $M_a = 0.8$ GeV and $\Lambda=0.5$ GeV. The plot on the middle shows $x f_1^u(x)$ (full line) and $x f_1^d(x)$ (dashed line) for the same values of the parameters. The third plot shows the low scale ($\mu^2 = 0.23$ GeV$^2$) valence distributions of Glück, Reya and Vogt grv95.