# Contraction of Lorentz indices in gluon propagator of QCD

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?

• Do you know what the mathematical expressions for 3- and 4-gluon vertices are? Feb 18, 2023 at 19:55

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.
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.