do we have any idea why the gauge groups are what they are? why would a new particle accelerator help in that?
>>16361678I work in string theory so this isn't my expertise but here goes:While we don't know why the specific interaction values are what they are or where the choice of strong groups come from, the choice of electroweak and higgs gauge groups are pretty basic consequences of 3+1D geometric symmetries whether you allow for extra dimensions or not. The best intro papers without string theory are Naka and Umezawa's original work, but IIRC Karananas wrote a survey article about the state of our understanding.