Light-front dynamic analysis of the longitudinal charge density using the solvable scalar field model in (1+1) dimensions

We investigate the electromagnetic form factor F(q(2)) of the meson by using the solvable phi(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 z = p(+)x(-) in LFD.