  Decay amplitude of $H \to \gamma Z$ process via one $W$ loop in the unitary
gauge is presented. The divergent integrals including those of high divergence
orders typical of unitary gauge are arranged to cancel to get the
electromagnetic $U(1)$ gauge invariant finite result, hence no contribution to
the renormalization constant of $Z\gamma$ mixing in this 1-loop subprocess. For
the calculation of the Feynman diagrams employing the Feynman rules, all the
integrations of the propagator momenta and all the delta-functions representing
the 4-momentum conservation of every vertex are retained in the beginning.
Therefore, the ambiguity of setting independent loop momentum for divergences
worse than logarithmic does not exist, and shift of integrated variable in such
divergent integrals is eschewed. The calculation are done in 4-dimension
Minkowski momentum space without the aid of any regularization. The correct
treatment on the surface terms for the quadratic and logarithmic tensor
integral is one of the key points.
  This part I is devoted to the calculation details and the indications from
the key surface terms. Comparing with other gauge(s) and complete results for
$H \to \gamma Z$ are left for part II.
