# 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?
• – Cosmas Zachos Jan 7 '18 at 15:37

• 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? – Martin C. Jan 8 '18 at 18:56