We investigate the electromagnetic form factor $F(q^2)$ of the meson by using the solvable ${\varphi }^3$ scalar field model in ($1+1$) dimensions. As the transverse rotations are absent in ($1+1$) dimensions, the advantage of the light-front dynamics (LFD) with the light-front time $x^{+}=x^0+x^3$ as the evolution parameter is maximized in contrast to the usual instant form dynamics (IFD) with the ordinary time $x^0$ as the evolution parameter. In LFD, the individual $x^{+}$-ordered amplitudes contributing to $F(q^2)$ are invariant under the boost, i.e., frame independent, while the individual $x^0$-ordered amplitudes in IFD are not invariant under the boost but dependent on the reference frame. The LFD allows us to get the analytic result for the one-loop triangle diagram which covers not only the spacelike ($q^2<0$) but also timelike ($q^2>0$) region. Using the analytic results, we verify that the real and imaginary parts of the form factor satisfy the dispersion relations in the entire $q^2$ space. Comparing with the results in ($3+1$) dimensions, we discuss the transverse momentum effects on $F(q^2)$. We also discuss the longitudinal charge density in terms of the boost-invariant variable $\tilde{z}=p^{+}{x}^{-}$ in LFD.

• Received 12 January 2021
• Accepted 2 March 2021

DOI:https://doi.org/10.1103/PhysRevD.103.076002

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Published by the American Physical Society