In QED, the photon propagator has a factor of $g^{\mu \nu}$, and both $\mu$ and $\nu$ contract with the $\gamma$ matrix indices, which come from the fermion antifermion photon vertices on either end of the propagator.

In QCD, the gluon propagator also has a factor of $g^{\mu \nu}$. When the gluon is propagating between two quark-antiquark gluon vertices, we are able to contract the indices of $g^{\mu \nu}$ with the gamma matrix indices that come from these vertices. However if at least one of the vertices at the endpoints of the gluon propagator is a three-gluon vertex (such as in the process $q + \overline{q} \rightarrow g + g$), or a $4$-gluon vertex, how do we contract the indices on the metric?

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    $\begingroup$ Do you know what the mathematical expressions for 3- and 4-gluon vertices are? $\endgroup$
    – Ghoster
    Feb 18, 2023 at 19:55

1 Answer 1


An $n$-point interaction vertex is directly related to the $n$-point function of the theory. More precisely, it is the $n$-point function with external lines removed. In fact that is the operational way by which you can compute vertices: write down $\langle \Phi_1(x_1)\cdots \Phi_n(x_n)\rangle$ for whatever fields you have, remove the external propagators and you are going to end up with the interaction vertex.

For your question we don't need the explicit calculation though, we just need to understand the tensor structure of the vertices. This can be done by looking at the corresponding correlators.

The $3$-gluon vertex follows from $\langle A_\mu^a(x)A_\nu^b(y)A_\sigma^c(z)\rangle$ so that the tensor structure of the vertex, written in momentum space, is $\Gamma^{abc}_{\mu\nu\sigma}(k_1,k_2,k_3)$. On the other hand the $4$-gluon vertex follows from the four-point function$\langle A_\mu^a(x)A_\nu^b(y)A_\sigma^c(z)A_\rho^d(w)\rangle$ and hence it has tensor structure $\Gamma^{abcd}_{\mu\nu\sigma\rho}(k_1,k_2,k_3,k_4)$.

The ghost-gluon vertex follows from $\langle c^a(x)\bar c^b(y)A_\mu^c(z)\rangle$ so that the vertex has tensor structure $\Gamma^{abc}_\mu(k_1,k_2,k_3)$.

The quark-gluon vertex on the other hand follows from $\langle \psi_{i\alpha}(x)\bar \psi_{j\beta}(y)A^{\mu a}(z)\rangle$ where $i,j$ are representation indices and $\alpha,\beta$ are spinor indices. As a result it has the following tensor structure $\Gamma^{a \mu}_{ij;\alpha\beta}(k_1,k_2,k_3)$, or supressing the spinor indices $\Gamma^{a\mu}_{ij}(k_1,k_2,k_3)$ which is now a matrix in spinor space.

As you can see, all these vertices do have four-vector indices which can contract with gluon propagators. In fact, the number of four-vector indices matches the number of gluon propagators to which it can attach.


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