Zero Mode and Symmetry

We discuss the spontaneous symmetry breaking (SSB) on the light front (LF) in view of the zero mode. We rst demonstrate impossibility to remove the zero mode in the continuum LF theory by two examples: The Lorentz invariance forbids even a free theory on the LF and the trivial LF vacuum is lost in the SSB phase, both due to the zero mode as the accumulating point causing uncontrollable infrared sin-gularity. We then adopt the Discretized Light-Cone Quantization (DLCQ) which was rst introduced by Maskawa and Yamawaki to establish the trivial LF vacuum and was rediscovered by Pauli and Brodsky in a diierent context. It is shown in DLCQ that the SSB phase can be realized on the trivial LF vacuum only when an explicit symmetry-breaking mass of the Nambu-Goldstone (NG) boson m is introduced as an infrared regulator. The NG-boson zero mode integrated over the LF must exhibit singular behavior 1=m 2 in the symmetric limit m ! 0 in such a way that the LF charge is not conserved even in the symmetric limit; _ Q 6 = 0. There exists no NG theorem on the LF. Instead, this singular behavior establishes existence of the massless NG boson coupled to the current whose charge satisses Qj0i = 0 and _ Q 6 = 0, in much the same as the NG theorem in the equal-time quantization which ensures existence of the massless NG boson coupled to the current whose charge satisses Qj0i 6 = 0 and _ Q = 0. We demonstrate such a peculiarity in a concrete model, the linear sigma model, where the role of zero-mode constraint is clariied.