The transverse momentum dependent distribution functions in the bag model

Abstract

Leading and subleading twist transverse momentum dependent parton distribution functions (TMDs) are studied in a quark model framework provided by the bag model. A complete set of relations among different TMDs is derived, and the question is discussed how model-(in)dependent such relations are. A connection of the pretzelosity distribution and quark orbital angular momentum is derived. Numerical results are presented, and applications for phenomenology discussed. In particular, it is shown that in the valence-x region the bag model supports a Gaussian Ansatz for the transverse momentum dependence of TMDs.

Figures (7)

• Figure 1: (a) The unpolarized functions $f^{\perp u}(x)$, $f_1^u(x)$, $e^u(x)$ vs. $x$ from the bag model at the low scale. The $d$-quark distributions are factor two smaller compared to the unpolarized $u$-quark distributions according to the SU(6)-flavour factors in Eqs. (\ref{['Eq:wafe-function-SU(6)-pol']}, \ref{['Eq:fperp']}). (b) The polarized functions $g_T^{\perp u}(x)=-h_{1T}^{\perp u}(x)$, $g_{1T}^{\perp u}(x)=-h_{1L}^{\perp u}(x)$, $h_1^u(x)$, $g_1^u(x)$ vs. $x$. The $d$-quark distributions are factor four smaller and have opposite sign compared to the $u$-quark distributions according to the SU(6)-flavour factors in Eqs. (\ref{['Eq:wafe-function-SU(6)-pol']}, \ref{['Eq:fperp']}). (c) The polarized functions $h_T^{\perp u}(x)$, $g_L^{\perp u}(x)=-h_T^u(x)$, $g_T^u(x)$, $h_L^u(x)$ vs. $x$. The $d$-quark functions are as in (b).
• Figure 2: For the unpolarized TMD $f_1^q(x,k_\perp)$ (a) the (1/2)-moment defined in Eq. (\ref{['Eq:def-avpT(x)']}), (b) the derivative of the (1)-moment and the regularized (1)-moment as discussed in the text, and (c) $\langle p_T(x)\rangle$ in comparison to $(\pi\langle p_T^2(x)\rangle/4)^{1/2}$. In the Gauss-model the two quantities would be equal. (The dotted marks the value $\langle p_T(x)\rangle=0.25\,{\rm GeV}$.)
• Figure 3: (a) $\langle p_T^2\rangle$ of $f_1^q$ as function of $x$. Solid line: computed using the exact definition in Eq. (\ref{['Eq:def-avpT2(x)']}). Dashed line: using the Gauss model relation, Eq. (\ref{['Eq:Gauss-width']}). (b) and (c) $f_1^q(x,p_T)$ vs. $p_T$ for selected values of $x$. The thin dotted lines are the respective Gauss model approximations obtained from the Gaussian widths from Fig. \ref{['Fig03:f1-vs-Gauss']}a.
• Figure 4: The Gaussian widths as defined in Eq. (\ref{['Eq:Gauss-width']}) vs. $p_T$ for various TMDs.
• Figure 5: The $p_T$-dependence of various TMDs for selected values of $x$. The thin dotted lines are the respective Gauss model approximations obtained from the Gaussian widths from Fig. \ref{['Fig04:avpT2-vs-Gauss']}.