Analysis of virtual meson production in a ($1+1$)-dimensional scalar field model

Light-front time-ordered amplitudes are investigated in the virtual scalar meson production process in ($1+1$) dimensions using the scalar field model extended from the conventional Wick-Cutkosky model. There is only one Compton form factor (CFF) in the ($1+1$)-dimensional computation of the virtual meson production process, and we compute both the real and imaginary parts of the CFF for the entire kinematic regions of $Q^2>0$ and $t<0$. We then analyze the contribution of each and every light-front time-ordered amplitude to the CFF as a function of $Q^2$ and $t$. In particular, we discuss the significance of the “cat’s ears” contributions for gauge invariance and the validity of the “handbag dominance” in the formulation of the generalized parton distribution (GPD) function used typically in the analysis of deeply virtual meson production processes. We explicitly derive the GPD from the “handbag” light-front time-ordered amplitudes in the $-t/Q^2\ll 1$ limit and verify that the integrations of the GPD over the light-front longitudinal momentum fraction for the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi and Efremov-Radyushkin-Brodsky-Lepage regions correspond to the valence and nonvalence contributions of the electromagnetic form factor that we have recently reported [Phys. Rev. D 103, 076002 (2021)]. We also discuss the correspondence of the GPD to the parton distribution function for the analysis of the deep inelastic lepton-hadron scattering process and the utility of the new light-front longitudinal spatial variable $\tilde{z}$.

Figure 1

Relevant Feynman diagrams for the reaction of ${\gamma }^{*}(q)+\mathcal{M}(p)\to S(q^{\prime })+\mathcal{M}(p^{\prime })$. (a) The $s$-box diagram, (b) the $u$-box diagram, and (c) the cat’s ears diagram.

Figure 2

The LF time-ordered diagrams for the scattering amplitudes (${\mathcal{M}}_s^{\mu },{\mathcal{M}}_u^{\mu },{\mathcal{M}}_c^{\mu }$) corresponding to the $(s,u,c)$-channels.

Figure 3

The LF time-ordered effective tree diagrams in the $s$- and $u$-channels for a charged target case. (a),(b) The valence and nonvalence contributions to ${\mathcal{M}}_{s,\mathrm{ET}}^{\mu }$, respectively, and (c),(d) the valence and nonvalence contributions to ${\mathcal{M}}_{u,\mathrm{ET}}^{\mu }$, respectively.

Figure 4

Diagrams for GPDs in different kinematic regions for the $\zeta >0$ case. The sum of ${\mathcal{M}}_{s,\mathrm{hand}}^{+\mathrm{DVMP}}$ and ${\mathcal{M}}_{u,\mathrm{hand}}^{+\mathrm{DVMP}}$ corresponds to the valence diagram (a) defined in the DGLAP ($\zeta \le x\le 1$) region and the sum of ${\mathcal{M}}_{s,\mathrm{twist}}^{+\mathrm{DVMP}}$ and ${\mathcal{M}}_{u,\mathrm{stret}}^{+\mathrm{DVMP}}$ corresponds to the nonvalence diagram (b) defined in the ERBL ($0\le x\le \zeta$) region. The small white blob in the figure represents the nonlocality of the constituent–gauge-boson vertex.

Figure 5

Role of the $c$-channel contributions in the gauge invariance condition of $\mathrm{Re}[q·\mathcal{M}]$ and $\mathrm{Im}[q·\mathcal{M}]$ for a charged target. The solid line is for the full amplitude while the dashed and dotted lines are for the amplitudes without the $c$-channel contributions.

Figure 6

The Compton form factor ${\mathcal{F}}_c^{\mathrm{VMP}}(Q^2,t)$. (a) Its real part, (b) its imaginary part, and (c) its modulus for three constituent masses, $m_{Q_1}=m_{Q_2}=1.9$, 2.0, and 2.1 GeV at the momentum transfer squared $t=t_{\mathrm{th}}\simeq -0.7593{\mathrm{GeV}}^2$.

Figure 7

(Left) Three-dimensional and (right) contour plots for the real part, imaginary part, and modulus of the Compton form factor ${\mathcal{F}}_c^{\mathrm{VMP}}$.

Figure 8

The real and imaginary parts of the Compton form factor determined from ${\mathcal{M}}_{s+u+c+ET}^{+}$ and ${\mathcal{M}}^{+DVMP}$. (a) $\mathrm{Re}[\mathcal{F}(Q^2)]$ for $t=t_{\text{th}}\simeq -0.7593{\mathrm{GeV}}^2$, (b) $\mathrm{Re}[\mathcal{F}(Q^2)]$ for $t=-2{\mathrm{GeV}}^2$, (c) $\mathrm{Im}[\mathcal{F}(Q^2)]$ for $t=t_{\text{th}}\simeq -0.7593{\mathrm{GeV}}^2$, and (d) $\mathrm{Im}[\mathcal{F}(Q^2)]$ for $t=-2{\mathrm{GeV}}^2$.

Figure 9

Three-dimensional plots of $H(\zeta ,x,t)$ for two parameter sets. (a) $(m_{Q_1},m_{Q_2})=(2,2)\mathrm{GeV}$ and (b) $(m_{Q_1},m_{Q_2})=(1,3)\mathrm{GeV}$, in the region of $0\le x\le 1$ and $-4{\mathrm{GeV}}^2\le t\le 0$.

Figure 10

The ordinary valence PDF $q_v(x)$ of the scalar target multiplied by $x$ for $m_{Q_1}=m_{Q_2}=1.9$, 2.0, and 2.1 GeV.

Figure 11

Longitudinal probability density $\varrho (\tilde{z})$ for the scalar target with $m_{Q_1}=m_{Q_2}=1.9$, 2.0, and 2.1 GeV in the LF coordinate space $\tilde{z}$. (a) $\varrho (\tilde{z})$ in linear scale for the range of $0\le \tilde{z}\le 50$, and (b) $\varrho (\tilde{z})$ in logarithmic scale in the range of $0\le \tilde{z}\le 200$.

Figure 12

The first three moments ${\overline{F}}_n(\zeta ,t)\equiv {\overline{F}}_n(\zeta )$ ($n=1$, 2, 3) of $H(\zeta ,x,t)$ given by Eq. (43) for (a) the weakly bound helium target with $m_{Q_{1(2)}}=2\mathrm{GeV}$ and $M_T=3.7\mathrm{GeV}$ and (b) the strongly bound pion target with $m_{Q_{1(2)}}=0.25\mathrm{GeV}$ and $M_T=0.14\mathrm{GeV}$, where $0\le \zeta \le 1$ corresponds to $0\le -t\le \infty$.

Figure 13

Electromagnetic form factor $F_{\mathcal{M}}(t)$ of the helium target as the first moment of $H(\zeta ,x,t)$ obtained for $m_{Q_1(2)}=2\mathrm{GeV}$. The results are obtained by using (a) the plus current and (b) the minus current.