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 ϕ 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 x0 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 x0-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 q2 space. Comparing with the results in ( 3 + 1 ) dimensions, we discuss the transverse momentum effects on F ( q2 ). We also discuss the longitudinal charge density in terms of the boost-invariant variable ˜ z = p + x − in LFD.