The (3+1)-dimensional scalar field model analysis of beam spin asymmetry in the electroproduction of a scalar meson off a scalar target

TL;DR

The paper develops a (3+1)-dimensional scalar-field framework to study exclusive scalar meson electroproduction off a scalar target, enabling independent extraction of two Compton form factors ${\cal F}_{1}$ and ${\cal F}_{2}$ and the beam spin asymmetry (BSA). By computing the exact four-point function with light-front time-ordered diagrams, it demonstrates that the BSA is nonzero due to the relative phase between ${\cal F}_{1}$ and ${\cal F}_{2}$, and analyzes both the real and imaginary parts across kinematics. In the deeply virtual meson production (DVMP) limit, the amplitude reduces to a single CFF tied to the leading-twist GPD $H(x,\zeta,t)$, which is decomposed into DGLAP and ERBL regions and obeys polynomiality of Mellin moments; the forward limit recovers the PDF $q(x)$ and the EM form factor via standard sum rules. Numerically, the study confirms a nonzero BSA for realistic kinematics, maps the DVMP-breaking effects, and discusses the implications for extracting leading-twist GPDs at facilities like JLab and the EIC, while outlining avenues to extend the framework to gravitational form factors, chiral-odd GPDs, and TMDs.

Abstract

We explore exclusive scalar meson electroproduction off a scalar target in the (3+1)-dimensional scalar field model. This model analysis is a straightforward extension of the previous (1+1)-dimensional model analysis presented in Phys. Rev. D \textbf{105}, 096014 (2022). In contrast to the (1+1)-dimensional model, the (3+1)-dimensional model allows us to compute the beam spin asymmetry (BSA), which is proportional to the imaginary part of the product of the two Compton form factors (CFFs) that appear in the hadronic current of the present scalar meson electroproduction process. We compute both real and imaginary parts of the CFFs and note that the BSA is detectable for $-t/Q^2 \gtrsim 0.1$ although it gets quite small in the kinematic region $-t/Q^2 \ll 0.1$ where the factorization of the generalized parton distribution (GPD) is attainable. We find the analytic forms of the leading twist GPD for the DGLAP and ERBL regions in the (3+1)-dimensional scalar field model, confirming its uniqueness independent of the hadronic current component. While we verify that the GPD sum rule for the total result of summing the DGLAP and ERBL regions holds for all components of the hadronic current, we note that the respective correspondence of the DGLAP and ERBL regions to the valence and non-valence parts of the electromagnetic form factor holds only for the light-front plus component of the hadronic current but not for any other components of the hadronic current. We discuss the polynomiality of the GPD up to the second moments and remark on accessible ranges of kinematics to measure the BSA and CFFs with respect to the future experimental efforts of extracting the leading-twist GPDs.

Figures (17)

• Figure 1: Momentum assignments for exclusive meson electroproduction. Here, $p$ and $p'$ denote the four-momenta of the initial and final scalar target states, while $q$ and $q'$ represent the momenta of the incoming virtual photon and the produced meson, respectively.
• Figure 2: Schematic of the Target Rest Frame, where $\vec{\mathbf{q}} = -\left|\vec{\mathbf{q}}\right| \hat{z}$ so that we have a spacelike virtual photon with $q^+ = q^0+q^3 < 0$. Note that the red arrows point in the $\hat{x}$, $-\hat{y}$, and $-\hat{z}$ directions.
• Figure 3: Scaling variable $x_A$ as a function of $-t$ for ${\bm\Delta}_\perp=0$ and various $Q^2$ values, using $(M_T, M_S) = (3.7, 0.98)$ GeV to model $f_0(980)$ production off a $^4{\rm He}$ target.
• Figure 4: Relevant covariant diagrams for the reaction $\gamma^{*}(q)+\mathbf{h}(P) \rightarrow \mathbf{m}(q') + \mathbf{h}'(P')$: (a) S-Channel, (b) U-Channel, and (c) "cat-ears" diagram (denoted C-Channel).
• Figure 5: The LF time-ordered diagrams for the scattering amplitudues $(J^+_S, J^+_U, J^+_C)$ corresponding to (S, U, C)-channels.