  Emergent symmetry in Dirac system means that the system acquires an
enlargement of two basic symmetries at some special critical point. The
continuous quantum criticality between the two symmetry broken phases can be
described within the framework of Gross-Neveu-Yukawa (GNY) model. Using the
first-order $\epsilon$ expansion in $4-\epsilon$ dimensions, we study the
critical structure and emergent symmetry of the chiral GNY model with $N_f$
flavors of four-component Dirac fermions coupled strongly to an $O(N)$ scalar
field under a small $O(N)$-symmetry breaking perturbation. After determining
the stable fixed point, we calculate the inverse correlation length exponent
and the anomalous dimensions (bosonic and fermionic) for general $N$ and $N_f$.
Further, we discuss the emergent-symmetry and the emergent supersymmetric
critical point for $N\geq4$ on the basis of $O(N)$-GNY model. It turns out that
the chiral emergent-$O(N)$ universality class is physically meaningful if and
only if $N<2N_f+4$. On this premise, the small $O(N)$-symmetry breaking
perturbation is always irrelevant in the chiral emergent-$O(N)$ universality
class. Our studies show that the emergent symmetry in Dirac systems has an
upper boundary $O(2N_f+3)$, depending on the flavor numbers $N_f$. As a result,
the emergent-$O(4)$ and $O(5)$ symmetries are possible to be found in the
systems with fermion flavor $N_f=1$, and the emergent-$O(4)$, $O(5)$, $O(6)$
and $O(7)$ symmetries are expected to be found in the systems with fermion
flavor $N_f=2$. Our result also suggests some rich transitions with
emergent-$Z_2\times O(2)\times O(3)$ symmetry and so on. Interestingly, in the
emergent-$O(4)$ universality class, there is a supersymmetric critical point
which is expected to be found in the systems with fermion flavor $N_f=1$.
