Zero Mode and Symmetry Breaking on the Light Front

We discuss the spontaneous symmetry breaking (SSB) on the light front (LF) in view of the zero mode. We first 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 singularity. We then adopt the Discretized Light-Cone Quantization (DLCQ) which was first introduced by Maskawa and Yamawaki to establish the trivial LF vacuum and was re-discovered by Pauli and Brodsky in a different 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 mπ is introduced as an infrared regulator. The NG-boson zero mode integrated over the LF must exhibit singular behavior ∼ 1/mπ in the symmetric limit mπ → 0 in such a way that the LF charge is not conserved even in the symmetric limit; Q̇ 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 satisfies Q|0〉 = 0 and Q̇ 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 satisfies Q|0〉 6= 0 and Q̇ = 0. We demonstrate such a peculiarity in a concrete model, the linear sigma model, where the role of zero-mode constraint is clarified. To appear in Proc. of International Workshop “New Nonperturbative Methods and Quantization on the Light Cone”, Les Houches, France, Feb. 24 -March 7, 1997. E-mail address: [email protected]