Zero Mode and Symmetry Breaking on the Light Front

TL;DR

This paper explores the challenges of spontaneous symmetry breaking on the light front, showing that zero modes prevent a trivial vacuum and that NG bosons exhibit singular behavior, leading to a modified understanding of NG phenomena.

Contribution

It demonstrates the impossibility of removing zero modes in continuum light-front theory and establishes how SSB can occur only with explicit symmetry-breaking and singular zero-mode behavior in DLCQ.

Findings

Zero modes prevent trivial vacuum in continuum LF theory.

SSB on the LF requires explicit symmetry-breaking mass.

NG boson zero mode exhibits 1/m_{pi}^2 singularity as m_{pi} approaches zero.

Abstract

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 m_pi is introduced as an infrared regulator. The NG-boson zero mode integrated over the LF must exhibit singular behavior sim 1/m_{pi}^2 in the symmetric limit m_{pi} rightarrow 0 in such a way that the LF charge is not conserved even in the symmetric limit; dot{Q} ne 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 dot{Q} ne 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 > ne 0 and dot{Q} = 0. We demonstrate such a peculiarity in a concrete model, the linear sigma model, where the role of zero-mode constraint is clarified.