Gluon photon mixing -- why not? In the electroweak theory there is a mixing between forces which results in the photon being part weak and mostly electromagnetic and the Z0 being mostly weak and partly electromagnetic.  Why doesn't the neutral gluon and photon (and for that matter the Z0) have a mixing angle?  If the forces are all united in a GUT, how does the GUT not combine the stong force with the other forces with a mixing angle for the neutral particles?
 A: In the standard model the Higgs field only carries hypercharge and isospin, but no color. As a result the gluon does not mix with the photon or $Z$. 
Remember that the original gauge fields, $U(1)$ hypercharge $B_\mu$ and $SU(2)$ isospin $W_\mu^a$, both couple to the Higgs. The photon $A_\mu$ and the $Z_\mu$ emerge as diagonal eigenstates. 
The requirement on a GUT is simply to reproduce the hypercharge and color assignments in the SM.
If there was a colored Higgs, or a composite scalar field that carries color, then mixing would occur. It has been suggested that this does happen in QCD at very high baryon density. Quark form Cooper pairs, and the pair condensate is a composite colored Higgs field, see Goldstone bosons, quark and glon masses counting in color-flavor locking QCD
A: "...Why doesn't the neutral gluon and photon (and for that matter the Z0) have a mixing angle?.."
The gluon belongs to the representation of the QCD color group $SU_{c}(3)$ while the photon belongs to the representation of the electroweak $SU_{W}(2)\times U_{Y}(1)$. Moreover, the Higgs doublet it uncharged under the QCD color group and there is no possibility to obtain the mixing term through the Higgs mechanism.
"...If the forces are all united in a GUT, how does the GUT not combine the stong force with the other forces with a mixing angle for the neutral particles?.."
The answer is very simple. Absense of the mixing between the photon and the gluon is the fact which must be incorporated by any of the realistic GUTs.
A: That has to do with gauge invariance, and the fact that the electroweak and strong groups are distinct. Let's recall the two types of interactions that occur in gauge theories. The most important one is between matter and gauge bosons: say if $\psi$ is a fermion of charge $e$ with respect to a gauge group that has gauge boson $A_\mu$, then there is an interaction
$$e \times A_\mu \bar{\psi} \gamma^\mu \psi$$
(suppressing indices etc.). Clearly, this type of interaction doesn't mix two different gauge bosons. Second, there can be self-interactions, in the case of non-abelian gauge groups (cubic and quartic interactions between gluons in Yang-Mills). Again, different gauge bosons don't mix there.
A: The photon is a superposition of B and W3. Both belong to the electroweak symmetry group and for this reason can mix. In contrast gluons belong to the SU(3) symmetry group that is defined as independent from SU(2). Therefore photons cannot interact or "mix" with gluons directly. Higher order processes may be possible, but have not been observed due to their extremely low probability and high energies required. 
