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Yes, you would have to introduce another gauge field. For example in the Standard Model there is gauge invariance under $SU(3)\times SU(2) \times U(1)$, and so there are three gauge fields: the gluons, the $W^\pm, Z$ weak gauge bosons and the photon. In general terms, it is simpler to argue like this: if you have gauge invariance under a Lie group $G$, the ...
Indeed, the two effects are very much related! I don't know how your background is, so let me start by defining the four-vector $x^\mu=(t,x,y,z)=(t,\vec{x})$ such that $x^0=t$ and $x_i=x,y,z$ for $i=1,2,3$. (Note that it is convention that greek indices run from $0$ to $3$ (space-time) while latin indices run from $1$ to $3$ (space only). Summation over ...