# Do gluons interact with each other by the strong foce?

I learned that strong force between quarks are mediated by gluons. What does this say about interactions between gluons? Do they interact with each other by the strong force?

• Yes. What does Wikipedia say? – Cosmas Zachos Mar 13 '19 at 15:37
• mutually-interacting gluons are sometimes referred to as "glueballs" – thegreatemu Mar 13 '19 at 20:47

If you take a look at the QCD Lagrangian describing the strong force \begin{align} \mathcal{L}_{QCD}= \overline{q}(i D_\mu \gamma^\mu - m) q - \frac{1}{4} G_{\mu \nu}^a G_a^{\mu \nu}\, , \end{align} you can see in the term $$\frac{1}{4} G_{\mu \nu}^a G_a^{\mu \nu}$$ we have a contraction of the gluon field strength tensor \begin{align} G^a_{\mu \nu} = \partial_\mu A^a_\nu - \partial_\nu A_\mu^a + g f^{abc}A^b_\mu A^c_\nu\, . \end{align} For the non-abelian symmetry of QCD, $$SU(3)_\text{color}$$, the structure constants $$f^{abc}$$ are non-zero, other than in electromagnetism, where the symmetry group is an abelian $$U(1)_\text{em}$$. This means that if we carry out the contraction, we end up with terms containing 3 or even 4 gluon fields. These terms lead to Feynman diagrams where 3 or 4 gluons meet at one vertex, i.e. they interact with each other. So the short answer to this is: yes.