In this paper, we study a proposal put forward recently by Bodendorfer, Mele
and M\”unch and Garc\’\i{}a-Quismondo and Marug\’an, in which the two
polymerization parameters of spherically symmetric black hole spacetimes are
the Dirac observables of the four-dimensional Ashtekar’s variables. In this
model, black and white hole horizons in general exist and naturally divide the
spacetime into the external and internal regions. In the external region, the
spacetime can be made asymptotically flat by properly choosing the dependence
of the two polymerization parameters on the Ashtekar variables. Then, we find
that the asymptotical behavior of the spacetime is universal, and, to the
leading order, the curvature invariants are independent of the mass parameter
$m$. For example, the Kretschmann scalar approaches zero as $K \simeq
A_0r^{-4}$ asymptotically, where $A_0$ is generally a non-zero constant and
independent of $m$, and $r$ the geometric radius of the two-spheres. In the
internal region, all the physical quantities are finite, and the Schwarzschild
black hole singularity is replaced by a transition surface whose radius is
always finite and non-zero. The quantum gravitational effects are negligible
near the black hole horizon for very massive black holes. However, the behavior
of the spacetime across the transition surface is significantly different from
all loop quantum black holes studied so far. In particular, the location of the
maximum amplitude of the curvature scalars is displaced from the transition
surface and depends on $m$, so does the maximum amplitude. In addition, the
radius of the white hole is much smaller than that of the black hole, and its
exact value sensitively depends on $m$, too.
