Chiral odd generalized parton distributions and spin densities in the impact parameter space

In the present work we have studied the chiral odd generalized parton distributions in the impact parameter space by assuming a flexible parametrization in a quark-diquark model. In order to obtain the explicit contributions from the up and down quarks, we have considered both the scalar (spin-0) and the axial-vector (spin-1) configurations for the diquark. We have also studied the spin densities for the up and down quarks in this model for monopole, dipole and quadrupole contributions for unpolarized and polarized quarks in unpolarized and polarized protons.

Figure 1

Vertex tree level diagram defining the quark-diquark model.

Figure 2

${({\mathcal{E}}_T+2{\tilde{\mathcal{H}}}_T)}^q$ ($q=u,d$) as function of $b_{\perp }$ for fixed values of $x$ for the up (left panel) and down (right panel) quarks.

Figure 3

The monopole contribution $\mathcal{H}/2$ for the unpolarized quarks in the unpolarized proton for the up (left panel) and down (right panel) quarks.

Figure 4

The dipole contribution $-\frac{1}{2}{s}_i{b}_j({\mathcal{E}}_T^{\prime }+2{\tilde{\mathcal{H}}}_T^{\prime })/M$ for the transversely polarized quarks in the unpolarized proton for the up (left panel) and down (right panel) quarks.

Figure 5

The sum of monopole $\mathcal{H}/2$ and dipole contribution $-\frac{1}{2}{s}_i{b}_j({{\mathcal{E}}^{\prime }}_T+2{\tilde{\mathcal{H}}}_T^{\prime })$ for the up (left panel) and down (right panel) quarks.

Figure 6

The dipole contribution $-\frac{1}{2}{S}_i{b}_j{\mathcal{E}}^{\prime }$ for the unpolarized quarks in the transversely polarized proton for the up (left panel) and down (right panel) quarks.

Figure 7

The sum of monopole $\mathcal{H}/2$ and dipole contribution $-\frac{1}{2}{S}_i{b}_j{\mathcal{E}}^{\prime }/M$ for the unpolarized quarks in the transversely polarized proton for the up (left panel) and down (right panel) quarks.

Figure 8

The monopole contribution $\frac{1}{2}{s}_i{S}_i({\mathcal{H}}_T-{\mathrm{\Delta }}_b{\tilde{\mathcal{H}}}_T^{\prime }/4M^2)$ for the quarks in the proton polarized in the same direction for the up (left panel) and down (right panel) quarks.

Figure 9

The quadrupole contribution $\frac{1}{2}{s}_i{S}_i(b_i^2-b_j^2){\tilde{\mathcal{H}}}_T^{\prime \prime }/M^2$ for the quarks in the proton polarized in the same direction for the up (left panel) and down (right panel) quarks.

Figure 10

The dipole contribution $\frac{1}{2}{S}_j{b}_i{\mathcal{E}}^{\prime }/M$ for $\widehat{x}$-polarized quarks in a proton polarized in $\widehat{y}$ direction for the up (left panel) and down (right panel) quarks.

Figure 11

The total dipole $\frac{1}{2}[S_j{b}_i{\mathcal{E}}^{\prime }-s_i{b}_j({{\mathcal{E}}^{\prime }}_T+2{\tilde{\mathcal{H}}}_T^{\prime })/M]$ for $\widehat{x}$-polarized quarks in a proton polarized in $\widehat{y}$ direction for the up (left panel) and down (right panel) quarks.

Figure 12

The quadrupole contribution $s_i{S}_j{b}_i{b}_j{\tilde{\mathcal{H}}}_T^{\prime \prime }/M^2$ for $\widehat{x}$-polarized quarks in a proton polarized in $\widehat{y}$ direction for the up (left panel) and down (right panel) quarks.