Exploring twist-3 chiral even generalized parton distributions of light sea quarks in the proton using the light front model

It is an extremely challenging task to understand the internal three-dimensional structure of a proton in terms of its constituent partons (quarks, gluons) [1, 2, 3, 4, 5, 6]. With the help of parton distribution functions (PDFs), the one-dimensional function of the longitudinal momentum fraction and extensive study of the internal structures are carried out. PDFs are forward, diagonal matrix elements that provide a probabilistic interpretation for the operator chosen. Unlike PDFs, GPDs also depend on the squared momentum transfer ($t$) and the longitudinal momentum transfer (or skewness $\xi$). GPDs are off-forward matrix elements of non-local operators and can be accessed experimentally through processes like deeply virtual Compton scattering (DVCS) [7, 8, 9, 10] or deeply virtual meson production (DVMP) [11, 12, 13, 14]. Experimental data for these processes have been collected by collaborations like H1 [15, 16], ZEUS [15, 17], and fixed-target experiments at HERMES [18, 19], COMPASS [20], and JLab [21, 22]. Considerable progress on the theoretical side has also been observed. Because of the perturbative nature of GPDs, they can’t be directly calculated from the first principle of QCD. Although ongoing lattice calculations will provide more insides. Therefore, model calculations are carried out, such as the MIT bag model [23], the light-front constituent quark model [2, 24, 25, 26], the NJL model [27], the color glass condensate model [28], the chiral quark-soliton model [29, 30], the Bethe-Salpeter approach [31, 32] and the meson cloud model [33, 34].

Parton distribution functions (PDFs) are categorized by their twist, which refers to their order in a $1/Q$ expansion, where $Q$ denotes the hard scale associated with the underlying physical process. While most research on generalized parton distributions (GPDs) has focused on leading-twist contributions, higher-twist GPDs, particularly those of twist-3, remain comparatively under-explored. Nevertheless, twist-3 GPDs provide valuable insights into the partonic structure of hadrons. Notably, there is a connection between quark orbital angular momentum within the nucleon and twist-3 GPDs [6, 35, 36]. The amplitude of deeply virtual Compton scattering (DVCS) at twist-3 can be expressed in terms of twist-3 GPDs through the Compton form factor [37]. Some studies suggest that twist-3 GPDs can also offer insights into the average transverse color Lorentz force acting on quarks [38],[39]. However, experimentally determining higher-twist distributions is challenging, as they are difficult to disentangle from leading-twist contributions. Moreover, higher-twist contributions are typically smaller, owing to suppression factors. The twist-3 GPDs have been investigated in the quark target model [40, 41, 42] and scalar diquark model [42] for the nucleon. The twist-4 proton GPDs were studied within the light-front quark-diquark model (LFQDM) recently [43], and the chiral-even twist-3 GPDs of the proton have also been investigated by the Lattice QCD [44]. The calculations of the twist-3 GPDs chiral-even GPDs of the protons have been carried out in the basis light-front quantization (BLFQ collaboration) [45] with the overlap representations of light-font wave functions.

In this work we have developed a light-front spectator model for proton to study the light sea quarks ($\bar{u},\bar{d}$), which are considered as spin-$1/2$, and the remaining spin-$1$ particle is considered as a spectator. We have used the momentum wave function predicted by the soft-wall Ads/QCD. The model parameters are determined by fitted unpolarized sea quarks PDF from CTEQ global analysis at scale $Q_{0}=2\ GeV$. We have investigated the behaviors of twist-3 GPDs at different energy scales and x-values. The higher-order Mellin moments of GPDs are also investigated, and we have given the values of Mellin moments in table [2-3], which are calculated from the model under study. In the forward limit and chiral even case, we have only one PDF, $g_{T}$, and we have investigated the PDF at different values of ‘$-t$’.

In Section I, we present an introduction to the subject matter. Section II provides a detailed calculation of the overlap representation for the twist-3 generalized parton distributions (GPDs). In Section III, we discuss the light-front model used in this study. This is followed by a fit of the unpolarized parton distribution functions (PDFs), calculated from the model, to the CT18NNLO data in order to evaluate the model parameters. In Part III.2 of Section III, we derive the expression for the twist-3 GPDs. Finally, the results and discussion are presented in Section IV.

The Generalized Parton Distributions (GPDs) are defined as non-forward matrix elements of the quark-quark correlator function. In light cone coordinate, the quark-quark correlator [46] is commonly written as,

	$\displaystyle F^{\Gamma}_{[\Lambda^{\prime},\Lambda]}(x,\xi,t)=\frac{1}{2}\int\frac{dz^{-}}{2\pi}e^{ix\bar{P}^{+}z^{-}}\langle p^{\prime},\Lambda^{\prime}|\bar{\psi}\bigl{(}-\frac{z}{2}\bigl{)}\Gamma$
	$\displaystyle\mathcal{W}\bigg{(}\frac{-z}{2},\frac{z}{2}\bigg{)}{\psi}\bigl{(}+\frac{z}{2}\bigl{)}|p,\Lambda\rangle{\biggl{|}}_{z^{+}=0,\bm{z}_{T}=0}$		(1)

Where $\Gamma$ is the Dirac bilinear and $\Lambda,\Lambda^{\prime}$ are the target initial and final state helicities. $\mathcal{W}$ is the Wilson line, which plays a critical role in ensuring gauge invariance of the nonlocal operator product. It is a path-ordered exponential that connects the quark fields along the light cone, compensating for gauge transformations along the path and maintaining the physical observability of the GPDs. The general definition of $\mathcal{W}$ is,

$$\mathcal{W}\bigg{(}\frac{-z}{2},\frac{z}{2}\bigg{)}=P\ exp\bigg{(}ig\int_{a}^{b}dx^{-}A^{+}(x^{-}n_{-})\bigg{)}$$

(2)

Here $P$ denotes the path ordering from $a$ to $b$. We have to consider the light-cone gauge where $A^{+}=0$. So the value of $\mathcal{W}$ is $1$, which also makes it $SU(3)$ color gauge invariant.

The key Dirac bilinears relevant to twist-3 chiral-even generalized parton distributions are given by $\Gamma=\gamma^{j},\gamma^{j}\gamma_{5}$. Consequently, the parameterization of Equation 1, as provided in [46, 47], for these bilinears is expressed as follows:

$$F^{\gamma^{j}}_{\Lambda\Lambda^{\prime}}=\left[\frac{\Delta^{j}}{2P^{+}}(E_{2T}+2\tilde{H}_{2T})+\frac{i\Lambda\epsilon^{jk}\Delta^{k}}{2P^{+}}(\tilde{E}_{2T}-\xi E_{2T})\right]\delta_{\Lambda\Lambda^{\prime}}+\left[\frac{-M(\Lambda\delta_{j1}+i\delta_{j2})}{P^{+}}H_{2T}-\frac{(\Lambda\Delta^{1}+i\Delta^{2})\Delta^{j}}{2MP^{+}}\tilde{H}_{2T}\right]\delta_{\Lambda-\Lambda^{\prime}}$$

(3)

$$F^{\gamma^{j}\gamma_{5}}_{\Lambda\Lambda^{\prime}}=\left[\frac{i\epsilon^{jk}\Delta^{k}}{2P^{+}}(E^{{}^{\prime}}_{2T}+2\tilde{H}^{{}^{\prime}}_{2T})-\frac{\Lambda\Delta^{j}}{2P^{+}}(\tilde{E}^{{}^{\prime}}_{2T}-\xi E^{{}^{\prime}}_{2T})\right]\delta_{\Lambda\Lambda^{\prime}}+\left[\frac{M(\delta_{j1}+i\Lambda\delta_{j2})}{P^{+}}H^{{}^{\prime}}_{2T}-\frac{i\epsilon^{jk}(\Lambda\Delta^{1}+i\Delta^{2})\Delta^{k}}{2MP^{+}}\tilde{H}^{{}^{\prime}}_{2T}\right]\delta_{\Lambda-\Lambda^{\prime}}$$

(4)

The antisymmetric tensor is defined by $\epsilon^{ij}=1=-\epsilon^{ji}$ and $P=(p+p^{\prime})/2$ is the average proton momentum, $\Delta=p-p^{\prime}$ is the momentum transfer with $t=\Delta^{2}=-\Delta_{T}^{2}$ and $\xi=-\Delta^{+}/2P^{+}$ is the skewness parameter. We denote ‘$\pm$’ as the proton helicity. Since $j=1,2$, the parameterization given in equations 3 and 4 can be reduced according to possible helicity combinations. These combinations are listed in Appendix A from equations 36-43. Using these equations, we can derive the explicit formula of the GPDs in overlap representations, which are given below,

	$\displaystyle-\frac{2i\Delta^{1}\Delta^{2}}{MP^{+}}\tilde{H}_{2T}$	$\displaystyle=\big{(}F_{[+-]}^{\gamma^{1}}+F_{[-+]}^{\gamma^{1}}\big{)}$
		$\displaystyle\hskip 56.9055pt+i\big{(}F_{[+-]}^{\gamma^{2}}-F_{[-+]}^{\gamma^{2}}\big{)}$		(5)

	$\displaystyle-\frac{4M}{P^{+}}H_{2T}-\frac{\Delta^{2}_{T}}{MP^{+}}\tilde{H}_{2T}$	$\displaystyle=\big{(}F_{[+-]}^{\gamma^{1}}-F_{[-+]}^{\gamma^{1}}\big{)}$
		$\displaystyle-i\big{(}F_{[+-]}^{\gamma^{2}}+F_{[-+]}^{\gamma^{2}}\big{)}$		(6)

	$\displaystyle\frac{\Delta^{1}+i\Delta^{2}}{P^{+}}(\tilde{E}_{2T}-\xi E_{2T})$	$\displaystyle=\big{(}F_{[++]}^{\gamma^{1}}-F_{[--]}^{\gamma^{1}}\big{)}$
		$\displaystyle+i\big{(}F_{[++]}^{\gamma^{2}}-F_{[--]}^{\gamma^{2}}\big{)}$		(7)

	$\displaystyle\frac{\Delta^{1}+i\Delta^{2}}{P^{+}}(E_{2T}+2\tilde{H}_{2T})$	$\displaystyle=\big{(}F_{[++]}^{\gamma^{1}}+F_{[--]}^{\gamma^{1}}\big{)}$
		$\displaystyle+i\big{(}F_{[++]}^{\gamma^{2}}+F_{[--]}^{\gamma^{2}}\big{)}$		(8)

	$\displaystyle\frac{2i\Delta^{1}\Delta^{2}}{MP^{+}}\tilde{H}{}_{2T}^{\prime}$	$\displaystyle=\big{(}F^{\gamma^{1}\gamma_{5}}_{[+-]}-F^{\gamma^{1}\gamma_{5}}_{[-+]}\big{)}$
		$\displaystyle\hskip 56.9055pt-i\big{(}F^{\gamma^{2}\gamma_{5}}_{[+-]}+F^{\gamma^{2}\gamma_{5}}_{[-+]}\big{)}$		(9)

	$\displaystyle\frac{\Delta^{1}+i\Delta^{2}}{P^{+}}(E_{2T}^{\prime}+2\tilde{H}_{2T}^{\prime})$	$\displaystyle=\big{(}F^{\gamma^{1}\gamma_{5}}_{[++]}+F^{\gamma^{1}\gamma_{5}}_{[--]}\big{)}$
		$\displaystyle+i\big{(}F^{\gamma^{2}\gamma_{5}}_{[++]}+F^{\gamma^{2}\gamma_{5}}_{[--]}\big{)}$		(10)

	$\displaystyle\frac{4M}{P^{+}}H_{2T}^{\prime}+\frac{\Delta^{1}+i\Delta^{2}}{MP^{+}}\tilde{H}_{2T}^{\prime}$	$\displaystyle=\big{(}F^{\gamma^{1}\gamma_{5}}_{[+-]}+F^{\gamma^{1}\gamma_{5}}_{[-+]}\big{)}$
		$\displaystyle+i\big{(}F^{\gamma^{2}\gamma_{5}}_{[+-]}-F^{\gamma^{2}\gamma_{5}}_{[-+]}\big{)}$		(11)

	$\displaystyle-\biggl{(}\frac{\Delta^{1}+i\Delta^{2}}{P^{+}}\biggl{)}(\tilde{E}_{2T}^{\prime}-\xi E_{2T}^{\prime})$	$\displaystyle=\big{(}F^{\gamma^{1}\gamma_{5}}_{[++]}-F^{\gamma^{1}\gamma_{5}}_{[--]}\big{)}$
		$\displaystyle+i\big{(}F^{\gamma^{2}\gamma_{5}}_{[++]}-F^{\gamma^{2}\gamma_{5}}_{[--]}\big{)}$		(12)

In this section to investigate the sea quarks ($\bar{u},\bar{d}$) chiral even generalized parton distribution in proton, we have formulated a light cone spectator model following the steps described in [48]. To generate the sea quark degree of freedom, we have considered the Fock state $|qqq\bar{q}q\rangle$. The proton is then reduced to a two-particle system. Sea quark is considered an active parton having spin-1/2, and all the other valance quarks and gluons are considered spin-1 spectators.

Throughout the calculations, we have followed the convention $x^{\pm}=x^{0}\pm x^{3}$. We have considered the frame of reference where the transverse momentum of a proton is zero such that $P=(P^{+},\frac{M^{2}}{P^{+}},0_{T})$. The two particle Fock state is $|\lambda_{q},\lambda_{X},xP^{+},\bm{k}_{T}\rangle$, with $\lambda_{q}$ and $\lambda_{X}$ being the helicities of the sea quark and the spectator. We have used the notation ‘$\pm$’ for quark and ‘$\uparrow(\downarrow)$’ for spectator.

The two particle Fock-state expansion can be written as [48] with proton helicity $J_{Z}=\pm\frac{1}{2}$ as

	$\displaystyle\Psi^{\pm}_{\text{Two particle}}$	$\displaystyle=\int\frac{dxd^{2}{\mathbf{k}}_{T}}{16\pi^{3}\sqrt{x(1-x)}}\times\Biggr{[}\psi^{\pm}_{+}(x,\vec{k}_{T},\uparrow)|+,\uparrow;xP^{+},\vec{k}_{T}\rangle++\psi^{\pm}_{+}(x,\vec{k}_{T},\downarrow)|+,\downarrow;xP^{+},\vec{k}_{T}\rangle$
		$\displaystyle\hskip 142.26378pt+\psi^{\pm}_{-}(x,\vec{k}_{T},\uparrow)|-,\uparrow;xP^{+},\vec{k}_{T}\rangle+\psi^{\pm}_{-}(x,\vec{k}_{T},\downarrow)|-,\downarrow;xP^{+},\vec{k}_{T}\rangle\Biggr{]}$		(13)

The two particle LFWFs of proton with $J_{Z}=+1/2$ is,

	$\displaystyle\psi_{+}^{+}(x,\bm{k}_{T},\uparrow)$	$\displaystyle=-\sqrt{2}\frac{-k_{1}+ik_{2}}{x(1-x)}\varphi(x,\bm{k}_{T})$
	$\displaystyle\psi_{+}^{+}(x,\bm{k}_{T},\downarrow)$	$\displaystyle=-\sqrt{2}\frac{k_{1}+ik_{2}}{1-x}\varphi(x,\bm{k}_{T})$
	$\displaystyle\psi^{+}_{-}(x,\bm{k}_{T},\uparrow)$	$\displaystyle=-\sqrt{2}\biggl{(}M-\frac{M_{X}}{1-x}\biggl{)}\varphi(x,\bm{k}_{T})$
	$\displaystyle\psi^{+}_{-}(x,\bm{k}_{T},\downarrow)$	$\displaystyle=0$		(14)

and for $J_{z}=-\frac{1}{2}$ we have the following LFWFs,

	$\displaystyle\psi_{+}^{-}(x,\bm{k}_{T},\uparrow)$	$\displaystyle=0$
	$\displaystyle\psi^{-}_{+}(x,\bm{k}_{T},\downarrow)$	$\displaystyle=-\sqrt{2}\biggl{(}M-\frac{M_{X}}{1-x}\biggl{)}\varphi(x,\bm{k}_{T})$
	$\displaystyle\psi^{-}_{-}(x,\bm{k}_{T},\uparrow)$	$\displaystyle=-\sqrt{2}\frac{-k_{1}+ik_{2}}{1-x}\varphi(x,\bm{k}_{T})$
	$\displaystyle\psi^{-}_{-}(x,\bm{k}_{T},\downarrow)$	$\displaystyle=-\sqrt{2}\frac{k_{1}+ik_{2}}{x(1-x)}\varphi(x,\bm{k}_{T})$		(15)

$M$ and $M_{X}$ represent the mass of proton and spectator. Here $\varphi(x,\bm{k}_{T})$ is the modified soft-wall Ads/QCD momentum wave function. We have used the wave function that was given in [49] and it has the expression,

	$\displaystyle\varphi(x,\bm{k}_{T})$	$\displaystyle=\sqrt{A}\frac{4\pi}{\kappa}\sqrt{\frac{log[\frac{1}{1-x}]}{x}}x^{\alpha}(1-x)^{3+\beta}$
		$\displaystyle\hskip 56.9055ptExp\biggl{[}-\frac{log[\frac{1}{1-x}]}{2\kappa^{2}x(1-x)}\bm{k}_{T}^{2}\biggl{]}$		(16)

The Ads scale parameter is $\kappa=0.4GeV$ [50], and $A$, $\alpha$, and $\beta$ are model-dependent parameters. They are fixed by fitting unpolarized sea quark PDF with CT18NNLO data set at energy scale $2\ \text{GeV}$. We have considered $M_{X}=0.946\ GeV$. The values of the model parameters for the up- and down-sea quarks are given in 1.

In this section we have presented the analytical results of twist-3 chiral even GPDs of light sea quarks in the proton at zero skewness in the light front spectator framework. We have also calculated the numerical results of Mellin moments as well as twist-3 on the gravitational form factor $B(t)$. The twist-3 PDF ($g_{T}$) is investigated, and we have also examined the Burkhardt-Cottingham sum rule [53] for $\bar{u}$ and $\bar{d}$.

In this paper, we investigate the twist-3 chiral-even generalized parton distributions (GPDs) of light sea quarks, specifically $\bar{u}$ and $\bar{d}$, within the light-front spectator model framework. Utilizing the parameterizations outlined in [21, 47], we express the GPDs through light-front wave functions. Our formalism closely follows the basis light-front quantization (BLFQ) approach presented in [45]. We provide a comprehensive analysis of two-dimensional (2D) and three-dimensional (3D) representations of the GPDs to illustrate their behavior across various values of transverse momentum transfer, $\Delta_{T}^{2}$, and longitudinal momentum fraction, $x$. Additionally, we explore higher-order Mellin moments and, by applying different GPD parameterizations, investigate the gravitational form factor $B(t)$ for $\bar{u}$ and $\bar{d}$ quarks. Lastly, we examine the twist-3 parton distribution function (PDF) $g_{T}$ for the sea quarks.

The investigation of higher-twist generalized parton distributions (GPDs) of sea quarks is less explored compared to the extensive research on leading-twist GPDs. In future studies, we intend to compute higher-twist GPDs for sea quarks at non-zero skewness ($\xi\neq 0$), accounting for contributions from higher Fock states. The study of non-zero skewness is particularly pertinent to the twist-3 cross-section of deeply virtual Compton scattering (DVCS) [37], which will be examined in upcoming experiments at the Electron-Ion Collider (EIC) and the Electron-Ion Collider in China (EicC) [57]. This research will also enable us to investigate the relationships between higher-twist GPDs and orbital angular momentum distributions. Understanding higher-twist effects will yield qualitative insights into the internal structure of the proton and contribute to addressing longstanding questions, such as the proton spin puzzle and the proton mass problem.