Zero Mode and Symmetry Breaking on the Light Front

TL;DR

This paper investigates the zero mode and spontaneous symmetry breaking on the light front using DLCQ, revealing that NG bosons exhibit singular behavior in the symmetric limit and discussing implications for continuum theories and LF quantum field theory.

Contribution

It clarifies the role of zero modes in symmetry breaking on the light front and compares DLCQ results with continuum theory, highlighting challenges in removing zero modes.

Findings

NG boson zero mode behaves as 1/m_π^2 in the symmetric limit

Zero mode integration leads to violation of current conservation at zero mode

Difficulties in removing zero modes in continuum theory are discussed

Abstract

We study the zero mode and the spontaneous symmetry breaking on the light front (LF). We use the discretized light-cone quantization (DLCQ) of Maskawa-Yamawaki to treat the zero mode in a clean separation from all other modes. It is then shown that the Nambu-Goldstone (NG) phase can be realized on the trivial LF vacuum only when an explicit symmetry-breaking mass of the NG boson $m_{\pi}$ is introduced. The NG-boson zero mode integrated over the LF must exhibit singular behavior $ \sim 1/m_{\pi}^2$ in the symmetric limit $m_{\pi}\to 0$, which implies that current conservation is violated at zero mode, or equivalently the LF charge is not conserved even in the symmetric limit. We demonstrate this peculiarity in a concrete model, the linear sigma model, where the role of zero-mode constraint is clarified. We further compare our result with the continuum theory. It is shown that in the continuum theory it is difficult to remove the zero mode which is not a single mode with measure zero but the accumulating point causing uncontrollable infrared singularity. A possible way out within the continuum theory is also suggested based on the ``$\nu$ theory''. We finally discuss another problem of the zero mode in the continuum theory, i.e., no-go theorem of Nakanishi-Yamawaki on the non-existence of LF quantum field theory within the framework of Wightman axioms, which remains to be a challenge for DLCQ, ``$\nu$ theory'' or any other framework of LF theory.