  In this paper, we study the line bundle mean curvature flow defined by Jacob
and Yau. The line bundle mean curvature flow is a kind of parabolic flows to
obtain deformed Hermitian Yang-Mills metrics on a given K\"ahler manifold. The
goal of this paper is to give an $\varepsilon$-regularity theorem for the line
bundle mean curvature flow. To establish the theorem, we provide a scale
invariant monotone quantity. As a critical point of this quantity, we define
self-shrinker solution of the line bundle mean curvature flow. The Liouville
type theorem for self-shrinkers is also given. It plays an important role in
the proof of the $\varepsilon$-regularity theorem.
