Are there any non-gauge invariant physical theories that describe anything from the "real world"? I have learnt QFT and the Standard Model and we always had gauge invariance there (in my lectures at least...) I wonder if there are any theories which are not gauge invariant that describe a bit of "reality"?
I have read this post Can we allowe gauge non-invariant terms in a gauge theory? and this one What is the problem with a Lagrangian that isn't gauge invariant? but I could not find a definite answer to my question.
I have also read that the superconducting phase transition breaks $U(1)$ gauge symmetry, but I don't fully understand that bit, it doesn't appear to me a superconducting phase is described by a non gauge invariant Lagrangian, or anything like that.
 A: I published a paper describing a working non-gauge invariant theory of  classical electromagnetism, thereby breaking the spell of the gauge principle. In this approach it is explicit that the Lorentz force only accounts for the change in kinetic momentum of a charge. However, a charge also has potential momentum and only the sum of the two is conserved. It also accounts for spin as a separately conserved contribution to angular momentum, something impossible in gauge theory. The Lorenz condition is interpreted as a consequence of charge conservation. It is still possible to add a "gauge" field to the potential without altering the charge kinetic (Lorentz) force. However, such solutions correspond to nonconserved source distributions that give the same field tensor as the original one.
A: The forces between nucleons can be described in an low-energy effective field theory that gives a good low-energy description of QCD. In this theory, the charged and neutral pions (which are scalar fields) mediate interactions between nucleons (which are fermions). There are no gauge fields.
