  The $\alpha$-R\'enyi entropy in the gapless models have been obtained by the
conformal field theory, which is exact in the thermodynamic limit. However, the
calculation of its finite size effect (FSE) is challenging. So far only the FSE
in a single interval in the XX model has been understood and the FSE in the
other models and in the other conditions are totally unknown. Here we report
the FSE of this entropy in disjointed intervals $A = \cup_i A_i$ under a
uniform dilation $\lambda A$ in the XY model, showing of a universal scaling
law as
  \begin{equation*}
  \Delta_{\lambda A}^\alpha = \Delta_A^\alpha \lambda^{-\eta} \mathcal{B}(A,
\lambda),
  \end{equation*}
  where $|\mathcal{B}(A, \lambda)| \le 1$ is a bounded function and $\eta =
\text{min}(2, 2/\alpha)$ when $\alpha < 10$. We verify this relation in the
phase boundaries of the XY model, in which the different central charges
correspond to the physics of free Fermion and free Boson models. We find that
in the disjointed intervals, two FSEs, termed as extrinsic FSE and intrinsic
FSE, are required to fully account for the FSE of the entropy. Physically, we
find that only the edge modes of the correlation matrix localized at the open
ends $\partial A$ have contribution to the total entropy and its FSE. Our
results provide some incisive insight into the entanglement entropy in the
many-body systems.
