If the photon were massive how do the electric and magnetic field change? According to electrodynamics the photon is massless. That is due to gauge invariance. But if the photon were massive  what would be the change in the electric and magnetic field? Is the massive nature of the photon connected with magnetic monopole?
 A: 
Allowing photons to bear mass, the electric and magnetic fields of a steadily moving charge
  are no longer perpendicular to each other, as anticipated from Biot-Savart law.  The electric and
  magnetic fields of such a particle depend on the gauge potentials,
  φ
  and
  ~
  A
  . The orthogonality relations
  of the particle fields and the direction of motion depend on the mass of the photon. The non-relativistic
  correction to the particle fields was found to be related to the Lorenz gauge condition. It is shown
  that the existence of magnetic monopoles inside matter is inevitable when magnetic field is applied in a
  conductor. Their existence is a manifestation of the massive nature of the photon inside matter. Neither
  electric nor magnetic current is separately conserved for photons, but their sum is. Massive photons
  are found to produce electric and magnetic fields. A force proportional to the square of the current is
  found to act along the wire,
  F=
  1
  2
  μ
  0
  I
  2
  , where
  μ
  0
  is vacuum permeability.

Citation from:
http://onlinewww.jpier.org/PIERM/pierm34/18.13111603.pdf
