Zero mode correction for the critical coupling in (1+1)-dimensional $\phi^4$ theory

Evaluating the effect of the zero momentum mode in the discretized light cone quantization (DLCQ) approach for light front field theory is a long standing problem. Using 1+1 dimension $\phi^4$ theory, we compare the critical coupling calculated in light front with zero mode excluded and included. The critical coupling without zero mode has been obtained by solving the theory in DLCQ, and the critical coupling with zero mode included was recently obtained by solving the theory in light front using a symmetric polynomial basis which was claimed to circumvent the zero mode problem (Burkardt et al. 2016). The critical coupling from these two methods can be compared, and a conclusion can be drawn on whether the zero mode has a significant effect for the DLCQ critical coupling result. We then further compare the zero mode included and excluded cases after their critical couplings are converted to the corresponding values in the equal time scheme. Finally, we discuss the consistency of these converted values with the critical coupling obtained by equal time quantization approaches in literature