Gravitational form factors and transverse spin sum rule in a light front quark-diquark model in AdS/QCD

The gravitational form factors are related to the matrix elements of the energy-momentum tensor $T^{\mu \nu }$. Using the light front wave functions of the scalar quark-diquark model for a nucleon predicted by the soft-wall AdS/QCD, we calculate the flavor-dependent $A(Q^2)$, $B(Q^2)$ and $\overline{C}(Q^2)$ form factors. We also present all the matrix elements of the energy-momentum tensor in a transversely polarized state. Further, we evaluate the matrix element of the Pauli-Lubanski operator in this model and show that the intrinsic spin sum rule involves the form factor $\overline{C}$. The longitudinal momentum densities in the transverse impact parameter space are also discussed for both unpolarized and transversely polarized nucleons.

Figure 1

Plots of nucleon gravitational form factor (a) $A(Q^2)$ and (b) $B(Q^2)$ for both $u$ and $d$ struck quarks. The quark contributions are shown in (b) and (c) and the diquark contributions are shown in (e) and (f).

Figure 2

Plots of the nucleon gravitational form factor $\overline{C}(Q^2)$. The contributions of the quark and diquark are shown in (b) and (c).

Figure 3

The longitudinal momentum densities for the active $u$ quark in the transverse plane, upper panel for an unpolarized nucleon, lower panel for a nucleon polarized along the $x$ direction. (b), (d) are the top views of (a), (c) respectively.

Figure 4

The longitudinal momentum densities for the active $d$ quark in the transverse plane, upper panel for an unpolarized nucleon, lower panel for a nucleon polarized along the $x$ direction. (b), (d) are the top views of (a), (c) respectively.

Figure 5

The longitudinal momentum densities of a nucleon in the transverse plane for the active quark $u$, upper panel for an unpolarized nucleon, lower panel for a nucleon polarized along the $x$ direction. (b), (d) are the top views of (a), (c) respectively.

Figure 6

The longitudinal momentum densities of a nucleon in the transverse plane for the active quark $d$, upper panel for an unpolarized nucleon, lower panel for a nucleon polarized along the $x$ direction. (b), (d) are the top views of (a), (c) respectively.

Figure 7

The momentum density asymmetry $({\rho }_T(b)-\rho (b))$ in the transverse plane for a nucleon polarized in the $x$ direction, (a) for a $u$ quark, (b) for a $d$ quark, (c) for a nucleon when the active quark is $u$ and (d) for a nucleon when the active quark is $d$.