Leading twist generalized parton distributions and spin densities in a proton

We evaluate both chirally even and odd generalized parton distributions (GPDs) in the leading twist in a recently proposed quark-diquark model for the proton where the light front wave functions are constructed from the soft-wall AdS/QCD prediction. The GPDs in transverse impact parameter space give the spin densities for different quark and proton polarizations. For longitudinally polarized proton only chiral even GPDs contribute but for transversely polarized proton both chiral even and chiral odd GPDs contribute to the spin densities. We present a detail study of the spin densities in this model.

Figure 1

Plots of chiral-even GPDs in impact parameter space as functions of $b=|b_{\perp }|$ for different values of $x=0.2$, 0.4, 0.6 at ${\mu }^2=10{\mathrm{GeV}}^2$. Left panel for $u$ quark and right panel for $d$ quark. The error bands correspond to $2\sigma$ error in the model parameters. (a) $H_v^u(x,b_{\perp })$, (c) $E_v^u(x,b_{\perp })$, (e) ${\tilde{H}}_v^u(x,b_{\perp })$, (g) ${\tilde{E}}_v^u(x,b_{\perp })$, and (b), (d), (f), (h) are same as (a), (c), (e), (g) respectively but for $d$ quark.

Figure 2

Plots of chiral-odd GPDs in impact parameter space as functions of $b=|b_{\perp }|$ for different values of $x=0.2$, 0.4, 0.6 at ${\mu }^2=10{\mathrm{GeV}}^2$. Left panel for $u$ quark and right panel for $d$ quark. The error bands correspond to $2\sigma$ error in the model parameters. (a) $H_T^u(x,b_{\perp })$, (c) $E_T^u(x,b_{\perp })$, (e) ${\tilde{H}}_T^u(x,b_{\perp })$, and (b), (d), (f) are same as (a), (c), (e) respectively but for $d$ quark

Figure 3

Plots of evolved chiral-even GPDs in impact parameter space as functions of $b=|b_{\perp }|$ for different scales $\mu =1,\sqrt{10},\sqrt{20}\mathrm{GeV}$ and fixed value of $x=0.3$. (a) $H_v^u(x,b_{\perp })$, (c) $E_v^u(x,b_{\perp })$, (e) ${\tilde{H}}_v^u(x,b_{\perp })$, (g) ${\tilde{E}}_v^u(x,b_{\perp })$, and (b), (d), (f), (h) are same as (a), (c), (e), (g) respectively but for $d$ quark

Figure 4

Plots of evolved chiral-odd GPDs in impact parameter space as functions of $b=|b_{\perp }|$ for different scales $\mu =1,\sqrt{10},\sqrt{20}\mathrm{GeV}$ and fixed value of $x=0.3$. (a) $H_T^u(x,b_{\perp })$, (c) $E_T^u(x,b_{\perp })$, (e) ${\tilde{H}}_T^u(x,b_{\perp })$, and (b), (d), (f) are same as (a), (c), (e) respectively but for $d$ quark

Figure 5

In an unpolarized proton: the monopole contribution $\frac{1}{2}H$ (top) for unpolarized quarks, the dipole contribution $-\frac{1}{2}{s}_x{b}_y(E_T^{\prime }+2{\tilde{H}}_T^{\prime })/M$ (middle) for transversely polarized quarks along $\widehat{x}$-direction, and the sum of both (lower). The left (right) panel gives the results for $u$ ($d$) quark.

Figure 6

For unpolarized quarks: the monopole contribution $\frac{1}{2}H$ (top) for an unpolarized proton, the dipole contribution $-\frac{1}{2}{S}_x{b}_y{E}^{\prime }/M$ (middle) for a transversely polarized proton along $\widehat{x}$, and the sum of both (lower). The left (right) panel gives the results for $u$ ($d$) quarks.

Figure 7

The monopole contribution $\frac{1}{2}{s}_x{S}_x(H_T-{\mathrm{\Delta }}_b{\tilde{H}}_T/4M^2)$ (top) and the quadrupole contribution $\frac{1}{2}{s}_x{S}_x(b_x^2-b_y^2){\tilde{H}}_T^{\prime \prime }/M^2$ (lower) for both quarks and the nucleon transversely polarized along $\widehat{x}$. The left (right) panel gives the results for $u$ ($d$) quarks.

Figure 8

The total spin distribution as a sum of monopole, dipole and quadrupole terms, for both quarks and proton polarized along $\widehat{x}$; left (right) panel for $u$ ($d$) quarks.

Figure 9

The dipole contribution $\frac{1}{2}{S}_y{b}_x{E}^{\prime }/M$ (top), the total dipole contribution $\frac{1}{2}[S_y{b}_x{E}^{\prime }-s_x{b}_y(E_T^{\prime }+2{\tilde{H}}_T^{\prime })]/M$ (middle) and the quadrupole contribution $s_x{S}_y{b}_x{b}_y{\tilde{H}}_T^{\prime \prime }/M^2$ (lower) for quarks polarized along $\widehat{x}$ in a nucleon transversely polarized along $\widehat{y}$ direction. Left (right) panel for $u$ ($d$) quarks.

Figure 10

The total spin distribution as a sum of monopole, dipole and quadrupole terms, for quarks polarized along $\widehat{x}$ in a nucleon transversely polarized along $\widehat{y}$ direction; left (right) panel for $u$ ($d$) quarks.

Figure 11

The monopole contribution $\frac{1}{2}H$ (top) and $\frac{1}{2}\tilde{H}$ (middle), and their sum (lower) corresponding to the spin distribution when both the quarks and proton are polarized along longitudinal direction; left (right) panel for $u$ ($d$) quarks.

Figure 12

Plot of $\tilde{H}(x,t)$ GPD vs $-t$ for $d$ quark at scale $\mu =\sqrt{10}\mathrm{GeV}$.