  We define a new invariant $w$ for hyperelliptic Lefschetz fibrations over
closed oriented surfaces, which counts the number of Dirac braids included
intrinsically in the monodromy, by using chart description introduced by the
second author. As an application, we prove that two hyperelliptic Lefschetz
fibrations of genus $g$ over a given base space are stably isomorphic if and
only if they have the same numbers of singular fibers of each type and they
have the same value of $w$ if $g$ is odd. We also give examples of pair of
hyperelliptic Lefschetz fibrations with the same numbers of singular fibers of
each type which are not stably isomorphic.
