  In this article, we study a class of closed connected orientable PL
$4$-manifolds admitting a semi-simple crystallization and which have an
infinite cyclic fundamental group. We show that the manifold in the class
admits a handle decomposition in which the number of $2$-handles depends upon
its second Betti number and other $h$-handles ($h \leq 4$) are at most $2$.
More precisely, our main result is the following. For a closed connected
orientable PL $4$-manifold having a semi-simple crystallization with the
fundamental group as $\mathbb{{Z}}$, we have constructed a handle decomposition
for $M$ as one of the following types:
  $(1)$ one $0$-handle, two $1$-handles, $1+\beta_2(M)$ $2$-handles, one
$3$-handle and one $4$-handle,
  $(2)$ one $0$-handle, one $1$-handle, $\beta_2(M)$ $2$-handles, one
$3$-handle and one $4$-handle,
  where $\beta_2(M)$ denotes the second Betti number of manifold $M$ with
$\mathbb{Z}$ coefficients.
