Dvali states in his paper on Three-Form Gauging of axion Symmetries and Gravity that
“As usual, in the Higgs phase the electric field is completely screened in the vacuum,”
and in another occasion in the same paper that
“There is a well known way to get rid of any constant electric field in the vacuum (i.e. to screen it), which is by putting the gauge theory in the Higgs phase.”
It may sure be well-known among others, but I for one, am not aware of it. Can someone explain to me with an illustrative example (QED) how this works?