# Form of Coulomb’s law before the electroweak phase transition

I’ve been thinking about the electroweak phase transition in the early universe, and I have some questions about the form of Coulomb’s law during that epoch.

As I understand it, before the electroweak phase transition, the electromagnetic and weak forces were still united as one force. The Higgs had not yet obtained a non-zero VEV, and the three $W$ particles that correspond to the generators of $SU(2)$, and the $B$ that corresponds to the generator of $U(1)$, were still massless. Here my understanding becomes confused: is it the case that the electroweak force was mediated by all of these particles, i.e. that it was a long range force mediated by four massless gauge bosons? If so, can we derive a classical result that corresponds to something like Coulomb’s law for the electroweak interaction (for e.g. two electrons)? As a very naive guess I could imagine that it looks like Coulomb’s law, but with the electric charge replaced by some combination of the weak isospin and/or hypercharge.

So my questions are:

1. Is the above picture essentially correct? I would appreciate it if anyone could correct any misconceptions I may have.
2. What was/were the gauge boson(s) associated with this united electroweak force? The three massless $W’s$, and the $B$?
3. Can the form of the electroweak interaction during that epoch be derived (‘Coulomb’s law’ at that time) at tree level? If not, can we make an educated guess?
• Jan 7 '18 at 15:37

The answer here by Emilio shows how to derive Coulomb's law from Maxwell's equation. There are two questions answered here and here how Coulomb's law comes out of quantum electrodynamics.

So the question is reduced to the before symmetry breaking case, which has all the SU(3)xSU(2)xU(1) symmetries , but where the bosons exchanged have zero mass, and also all the particles in the particle table. Charges though are still there, electrons and quarks etc, it is only the masses that are affected. As masses do not enter in the derivations, linked above, there should be no difference to the attraction or repulsion for the appropriate charges and the corresponding potential, imo.

• Thank you for the answer! I don’t find the argument completely satisfactory, but I’m not sure I know why :). Maybe the question I intended to ask is more something like ‘phenomenologically, what does it mean to say that electromagnetism and the weak force were combined as the electroweak force in the early universe?’ There were no photons, only B bosons which later combined with a component of the W field to form photons. There must surely be some observational difference in the interaction of the elementary leptons! I could post that as a separate question? Jan 8 '18 at 18:56
• The wikipedia article en.wikipedia.org/wiki/Electroweak_interaction displays two very different Lagrangians for the electroweak force before and after symmetry breaking: do these two different Lagrangians really give the same expression at tree level for two electrons scattering off each other? Jan 8 '18 at 18:57
• Quantum numbers are not lost in the transition,the electrons will interact with the zero mass u(1) part and the weak interaction with the weak interaction part conserving lepton numbers etc. The group algebra will be the same. It is just the masses that appear after symmetry breaking, and I did not see masses taking a part in deriving from field theory the coulomb force ( I may be wrong as I am not a theorist, and may be overlooking something). Jan 8 '18 at 19:04