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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?

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

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Particles that carry color participate in strong interactions. Gluons are "bicolored", carrying one positive unit of color and one negative unit. They hence interact with each other via the strong force.

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