Transverse-momentum-dependent quark distribution functions of spin-one targets: Formalism and covariant calculations

We present a covariant formulation and model calculations of the leading-twist time-reversal even transverse-momentum-dependent quark distribution functions (TMDs) for a spin-one target. Emphasis is placed on a description of these three-dimensional distribution functions which is independent of any constraints on the spin quantization axis. We apply our covariant spin description to all nine leading-twist time-reversal even $\rho$ meson TMDs in the framework provided by the Nambu–Jona-Lasinio model, incorporating important aspects of quark confinement via the infrared cutoff in the proper-time regularization scheme. In particular, the behaviors of the three-dimensional TMDs in a tensor polarized spin-one hadron are illustrated. Sum rules and positivity constraints are discussed in detail. Our results do not exhibit the familiar Gaussian behavior in the transverse momentum, and other results of interest include the finding that the tensor polarized TMDs—associated with spin-one hadrons—are very sensitive to quark orbital angular momentum, and that the TMDs associated with the quark operator ${\gamma }^{+}{\mathbf{\gamma }}_T{\gamma }_5$ would vanish were it not for dynamical chiral symmetry breaking. In addition, we find that 44% of the $\rho$ meson's spin is carried by the orbital angular momentum of the quarks, and that the magnitude of the tensor polarized quark distribution function is about 30% of the unpolarized quark distribution. A qualitative comparison between our results for the tensor structure of a quark-antiquark bound state is made to existing experimental and theoretical results for the two-nucleon (deuteron) bound state.

Figure 1

Graphical representation of the correlator of Eq. (11). The dots labeled by $\alpha ,\beta$ indicate the Dirac indices of the quark field operators, the line labeled by the momentum $k$ represents the active quark, and the thick line labeled by the momentum $p$ and the polarizations ${\varepsilon }^{\mu *}$ and ${\varepsilon }^{\nu }$ represents its parent spin-one target. The shaded oval represents the spectator states.

Figure 2

The $\overline{q}q$ Bethe-Salpeter equation in the random phase approximation. The double-line represents the $\overline{q}q$ $t$ matrix, the single line is the dressed quark propagator, and the dark shaded circle is the $\overline{q}q$ interaction kernel.

Figure 3

Feynman diagrams for the $\rho$-meson TMDs in the NJL model. The shaded circles represent the $\rho$-meson Bethe-Salpeter vertex functions of Eq. (47) and the solid lines the dressed quark propagator. The operator insertion has the form $\mathrm{\Gamma }\delta \left(x-\frac{k^{+}}{p^{+}}\right)$, with $\mathrm{\Gamma }$ given in Eq. (23). The left diagram gives the valence $u$ quark TMDs and the right diagram the valence $\overline{d}$ antiquark TMDs in the ${\rho }^{+}$ meson.

Figure 4

Results for the TMDs $f(x,{\mathit{k}}_T^2),g_L(x,{\mathit{k}}_T^2)$, and $h(x,{\mathit{k}}_T^2)$ for the ${\rho }^{+}$ meson. These TMDs are defined in Eqs. (16, 17, 18), and when integrated over ${\mathit{k}}_T$ give the PDFs $f\left(x\right),g\left(x\right)$, and $h\left(x\right)$ which are familiar from the spin-half case. The TMDs are given in units of ${\mathrm{GeV}}^{-2}$ and ${\mathit{k}}_T^2$ in ${\mathrm{GeV}}^2$.

Figure 5

Results for $g_T(x,{\mathit{k}}_T^2)$ and $h_L^{\perp }(x,{\mathit{k}}_T^2)$ of a ${\rho }^{+}$ meson. These TMDs are defined in Eqs. (17) and (18), where the pretzelosity TMD $h_T^{\perp }(x,{\mathit{k}}_T^2)$ vanishes identically in our calculations. The TMDs are given in units of ${\mathrm{GeV}}^{-2}$ and ${\mathit{k}}_T^2$ in ${\mathrm{GeV}}^2$.

Figure 6

Results for the tensor polarized TMDs $f_{LL}(x,{\mathit{k}}_T^2),f_{LT}(x,{\mathit{k}}_T^2)$, and $f_{TT}(x,{\mathit{k}}_T^2)$ for the ${\rho }^{+}$ meson. These TMDs are defined in Eqs. (16), and are given in units of ${\mathrm{GeV}}^{-2}$, with ${\mathit{k}}_T^2$ in ${\mathrm{GeV}}^2$.

Figure 7

The solid line is the TMD $f(x,{\mathit{k}}_T^2)$ for $x=0.5$ and the dotted line is a Gaussian fit, of the form given in Eq. (56), to this result.

Figure 8

We illustrate results for the unpolarized, helicity, transversity, and tensor polarized PDFs of the ${\rho }^{+}$ meson. The left panel gives results at the model scale $Q_0^2=0.16{\mathrm{GeV}}^2$ and the right panel presents the PDF results, multiplied by $x$, at the scale $Q^2=5{\mathrm{GeV}}^2$. The NLO nonsinglet DGLAP evolution was performed using the codes of Refs. [48, 49, 50].

Figure 9

The positivity constraints expressed by Eq. (37) (upper panel), Eq. (38) (middle panel), and Eq. (39) (lower panel). At the endpoints of the PDFs where $x=\{0,1\}$, we find the results $f_{LL}=-\frac{3}{2}f,\left|g\right|=\frac{3}{2}f$, and $\left|h\right|=\frac{M}{m_{\rho }}\frac{3}{2}f$.