Vacuum loops in light-front field theory

We demonstrate that vacuum diagrams in the genuine light front (LF) field theory are non-zero, in spite of simple kinematical counter-arguments (positivity and conservation of the LF momentum p+, absence of Fourier zero mode). Using the light-front Hamiltonian (time-ordered) perturbation theory, the vacuum amplitudes in self-interacting scalar λϕ3(1+1) and λϕ4(1+1) models are obtained as p=0 limit of the associated self-energy diagrams, where p is the external momentum. They behave as Cλ2μ−2 in D=2, with μ being the scalar-field mass, or diverge in D=4, in agreement with the usual “equal-time” form of field theory, and with the same value of the constant C. The simplest vacuum diagram with two internal lines is analyzed in detail displaying the subtle role of the small k+ region and its connection to the p=0 limit. However, the vacuum bubbles in the genuine light-front field theory are nonvanishing not due to the Fourier mode carrying LF momentum k+=0 (as is the case in the LF evaluation of the covariant Feynman diagrams), in full accord with the observation that the LF perturbation theory formula breaks down in the exact zero-mode case. This is made explicit using the DLCQ method - the discretized (finite-volume) version of the theory, where the light-front zero modes are manifestly absent, but the vacuum amplitudes still converge to their continuum-theory values with the increasing “harmonic resolution” K.