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In our Standard Model course we wrote down the term for the quark-gluon interaction of the left handed $SU(2)_L$ quark doublet $q_L$ as: $\mathscr{L} = … + \bar{q}_L \gamma^\mu iG^a_\mu(x) \lambda^a q_L + …$, where $\lambda^a$ are the eight Gell-Mann matrices and $G^a_\mu(x)$ describes the Gluon field. This interaction term is proportional to a seperate up and down quark interaction with the gluon: $\mathscr{L} = … + \bar{u}_L \gamma^\mu iG^a_\mu(x) \lambda^a u_L + \bar{d}_L \gamma^\mu iG^a_\mu(x) \lambda^a d_L + …$ So there is no color mixing term between the up and down quark, since otherwise the Lagrangian would contain a term proportional to: $\bar{u}_L \gamma^\mu G^a_\mu(x) \lambda^a d_L +$ hermitian conjugate.

Now my problem is that after an extensive search on google I‘m convinced that terms of this form should exist: an up and a down quark should be able to exchange their color charge via a gluon.

Is the mistake in the Lagrangian, which should respect terms of this form? How should it look like correctly? Or are there simply no such interactions? enter image description here

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In the Lagrangian, the possible vertices of the theory are given. The piece of Lagrangian given in your question contains two such vertices: one describing the interaction of an up quark with the gluon field and one describing the interaction of a down quark with the gluon field.

What the Lagrangian doesn't contain is a vertex which describes an up quark emitting a gluon and turning into a down quark (or the other way around). That would not be consistent with conservation of electric charge, for instance. The suggested vertex $\bar{u}_{L}\gamma^{\mu}i G_{\mu}^{a} \lambda^{a}d_{L}$ would describe such a process.

The diagram you drew is not described by such a vertex, but by the two correct vertices from the original Lagrangian, connected by a propagator. Look closely: in the diagram, there is a down quark emitting a gluon (corresponding to $\bar{d}_{L}\gamma^{\mu}i G_{\mu}^{a} \lambda^{a}d_{L}$) and an up quark absorbing one ($\bar{u}_{L}\gamma^{\mu}i G_{\mu}^{a} \lambda^{a}u_{L}$). The diagram contains two vertices, not one.

In short: it is possible for quarks of different types to exchange gluons and change color, but this does not happen within one vertex.

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    $\begingroup$ First of all thank you very much! I have one more question concerning your last sentence: Do you mean, that a color exchange between quarks of different flavor is possible, but such a process does not occur in nature? If so, why? $\endgroup$ – user205636 Sep 4 '18 at 10:08
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    $\begingroup$ It does occur, according to the diagram you drew, but a process of the form "d -> u + gluon" (which would be the process described by your suggested Lagrangian) does not, for reasons including conservation of electric charge. Note that the exchange of a gluon between two quarks is something else than a quark turning into another one while emitting a gluon. $\endgroup$ – Stijn B. Sep 4 '18 at 10:11
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The strong interaction is not flavour changing so there is no local vertex that will do what you want. While the suggested interaction vertex is not permissible because it spoils the $\mathrm{U}(1)$ charge invariance, if you extend to the weak sector, you can have however a vertex of the form you wrote $$\sim \frac{g_W}{\sqrt{2}} \bar{u}_L \gamma^{\mu} d_L W_{\mu}^+ + \text{h.c}$$

but note this is a flavour changing vertex not a colour one.

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