Does Yang-Mills have free gauge bosons?
Is there any physical problem with a propagating non-Abelian gauge boson? That is, a plane wave mode $A_\alpha^\mu e^{ikx}$, where $A_\alpha^\mu=b_\alpha\epsilon^\mu$, with $b_\alpha$ a constant adjoint-representation gauge vector, and $\epsilon^\mu$ a polarization vector orthogonal to $k^\mu$?
We usually say that gluons are confined due to their strong interactions. But it seems like a single boson wouldn't trigger any interactions at all. The YM Lagrangian is $-\frac{1}{4}F_\alpha^{\mu\nu}F_{\alpha\mu\nu}$, where $F_{\alpha\mu\nu}=\partial_\mu A_{\alpha\nu}-\partial_\nu A_{\alpha\mu}+C_{\alpha\beta\gamma}A_{\beta\mu}A_{\gamma\nu}$. As long as all of the spacetime components of $A_\alpha^\mu$ are parallel to a single adjoint-representation gauge vector, the $C_{\alpha\beta\gamma}A_{\beta\mu}A_{\gamma\nu}$ term vanishes, and the field equation reduces to the wave equation, which is satisfied.
In fact it seems that such a boson would not even emit virtual bosons, so it wouldn't generate a "Coulomb" field. Two wavepackets of the above form could pass near each other without feeling any attraction or repulsion. 298
And if the two of them shared the same $b_\alpha$, then they could even pass through each other with no scattering, just as photons do.
This seems to conflict with the common statement that non-Abelian gauge bosons are "charged". But if we interpret "charge" to mean the timelike components of the conserved currents $J_\alpha^\mu$, then we find that the currents in pure YM are $J_\alpha^\mu=C_{\alpha\beta\gamma}F_\beta^{\mu\nu}A_{\gamma\nu}$. So once again, as long as everything is parallel to a constant gauge vector $b_\alpha$, there is no charge and no current. When we say gauge bosons are "charged", we probably need to clarify that we just mean "transform nontrivially under the gauge group", not that they behave like charged particles.
Once we add in quarks, then all bets are off: quark pair production will happen at a fast rate, leading to more and more interactions. I assume this is why we don't see unconfined gluons in real life. But am I correct that they do exist in YM, or am I making a basic error? 298