Not the current working HEAD REFERENCE which points in our current branch.
You can directly see
HEAD if you look into the file
.git/HEAD. What you find in there is all there is to it, there is nothing more.
HEAD does not, strictly, point to the current branch but is the commit that is currently checked out. It is, by definition, the first entry that
git log lists.
If you do a
git checkout branchname then, yes, indeed,
.git/HEAD will contain the
ref of the branch you checked out, probably something like
refs/head/branchname. This in turn is literally the file that contains the actual commit hash; you can see it in
.git/refs/head/branchname. But if you do
git checkout commithash then
.git/HEAD will contain said hash, and you will be in a "headless" state (git will give you a longish message explaining what that is). In this case, there is no "current branch".