  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.
