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I asked a question here a few days ago and got some fantastic answers so I'm going to continue.

Let me preface this by saying I know quarks do not actually have 'colour', but colour is some sort of label to help us understand and calculate what decays are possible and what bound states can exist.

Ok, so in the first part of a lecture I had this morning my lecturer said;

"Gluons can interact, so let's assign them arbitrary colours."

And then later said;

"Because gluons have colour, they can self-interact, unlike chargeless photons."

This argument seems $very$ circular to me, so what's actually going on? I appreciate 'why' questions tend to be harder to answer than 'how' questions, but having said that, why are gluons like this?

Is it that the actual answer is unknown, or that it's simply a question for a fourth-year course rather than a third-year one? Or maybe it's a question whose answer is 'just because it explains what we observe'?

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    $\begingroup$ An $\operatorname{SU}(3)$ gauge symmetry contains a self-interacting term, unlike a $\operatorname{U}(1)$ one. This is due to the fact that the generators aren't commutative. $\endgroup$ – Slereah Feb 13 '19 at 16:26
  • $\begingroup$ Ah yes, it was said that there are 9 colour combinations of gluon/anti-gluon pairs, and only 8 orthogonal ones are coloured, is the 9th colourless one the self-interacting term you mention? $\endgroup$ – T. Smith Feb 13 '19 at 16:29
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    $\begingroup$ See Wikipedia for why color was historically theorized: Shortly after the existence of quarks was first proposed in 1964, Oscar W. Greenberg introduced the notion of color charge to explain how quarks could coexist inside some hadrons in otherwise identical quantum states without violating the Pauli exclusion principle. $\endgroup$ – Qmechanic Feb 13 '19 at 16:42
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    $\begingroup$ Is your question why quarks have color, or why gluons have color? $\endgroup$ – G. Smith Feb 13 '19 at 17:09
  • $\begingroup$ QCD is much more complicated than QED because gluons don't just transmit the color charge they also feel it (whereas photons don't feel EM). Thus it's not just quarks exchanging gluons, gluons also exchange gluons, so things get rather fractal. OTOH, color confinement prevents the direct observation of color charge, free quarks and gluons are impossible. IOW, quarks and gluons are only ever virtual particles. $\endgroup$ – PM 2Ring Feb 25 '19 at 0:00
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High energy particle collisions involve the production of a plethora of different particles. One example is $\Delta^{++}$, which consists of three up-quarks in the same spin state. Since quarks are fermions, Pauli's exclusion principle seems to be violated, unless there is another quantum number (we call it colour) in which the three up-quarks are different from each other (so there have to be at least three dictinct "values" for colour).

Looking now at the annihilation of an electron and a positron, two processes can be compared: $e^+e^- \to q\bar q$ (resulting in hadron production) and $e^+e^- \to \mu^- \mu^+$. Quantum field theory tells us that the cross sections of these two processes are proportional to the squared charges of the products, summed over all possibile quark species, i.e.

\begin{equation} \frac{\sigma(e^+ e^- \to \text{hadrons})}{\sigma(e^+ e^- \to \mu^+\mu^-)} = n_c \sum_{\substack{\text{quark}\\\text{flavours}}} e_q^2, \end{equation}

where $e_q$ is the electric charge of the respective quarks and $n_c$ is the number of distinct colour charges. Experimental evidence strongly points to $n_c = 3$, such that there are three distinct colour charges (called red, green and blue).

In Theoretical Physics, particularly in Quantum Field Theory, the governing equations of a theory are usually constructed in a way that they are invariant with respect to certain symmetry groups, corresponding to symmetries and conserved quantities in the properties of the described particles. Having figured out that there are three distinct colour charges, one can try to find the symmetry group for the theory of strongly interacting quarks.

It turns out that using the Lie group $SU(3)$ is consistent with all the considered observations (including the non-existence of free coloured particles etc.). Constructing a suitable Lagrangian that is invariant under $SU(3)$ results in interaction terms including objects that can be interpreted as gluons, with all the properties you are probably dealing with in your lecture. The theory developed this way is called Quantum Chromodynamics (QCD).

The reason why there are only 8 gluons even though naively one might expect 9 is of group theoretical nature. It has to do with representations/generators of $SU(3)$ and is a topic by itself.

To learn more about this you can take a course on Quantum Field Theory as soon as you have all the preliminaries or check out literature on gauge symmetries in particle physics (e.g. Chris Quigg, Gauge Theories of the Strong, Weak, and Electromagnetic Interactions).

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Your title and your text are asking different questions, the first about quarks and the second about gluons. Furthermore what you are really asking is not "Why do .. Have colour" but "Why do we say .. Have colour", as the answer to the first is something like "If they didn't have colour they wouldn't be quarks, they'd be leptons". But having got all that out of the way let's answer the question I think you're asking.

We say quarks have colour, as @Qmechanic points out, because there are some baryons which are composed of 3 identical quarks (in the same spin state) like the $\Delta^{++}$ which is made of 3 $u$ quarks, so the Pauli principle requires another quantum numbers with (at least) 3 values. At the risk of confusing people this is called colour with 3 charges, redness and blueness and greenness.

The directions chosen for the $r$, $g$ and $b$ axes are arbitrary. We use this in the gauge symmetry argument (like for charges and the photon) and require the Lagrangian (or the Dirac equation) not to change if the axes are rotated locally. To make this work we find we need to introduce a new term which on inspection turns out to describe a massless particle, i.e. we predict the gluon.

However the gluon terms introduced are more complicated than the photon. As well as the space-time index they have to have two indices corresponding to colour. So we write $A_\mu$ but $g_\mu^{ab}$ where $a$ and $b$ run over $rgb$. (One is actually an anti colour.) And there are gluon interaction terms, hence gluons are coloured and self-interact.

Self interaction is nothing to do with the colourlessness combination. Just as two spin half particles can combine to give a spin 1 triplet and a spin 0 singlet, so a colour and an anti colour combine to give a coloured octet or a colourless singlet. $3 \times \overline 3= 8+1$

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The other responses seem to be answering questions like "Why do we say quarks have color?" or "Why did we hypothesize that quarks have color?"

However, if the question is literally, why do quarks and gluons have color... this is just a basic postulate of the standard model. You can say it more technically, that there's a local SU(3) gauge symmetry and that quarks and antiquarks transform in the 3 and 3bar representations of that symmetry, and then you can say how that leads to "having color".

But in the end, it's just a postulate (or a hypothesis, or an axiom if you are thinking deductively)... an assumption that is apparently correct, one of the assumptions that defines the best quantum field theory we have for describing particle physics. One of the starting points, in terms of which everything else about hadrons is explained. In your words: "just because it explains what we observe".

To find a genuine explanation of why quarks and gluons are like that, you need something more fundamental than the standard model. In a grand unified theory, you might say that quarks have three colors because the scalar potential breaking the unified symmetry group has certain properties, such that it leaves an unbroken SU(3). In a string theory model, you might say that quarks have three colors because they are actually open strings, one end of which terminates on a stack of three branes. A cosmological theory which explains why string theory ends up in a particular vacuum, or an anthropic argument as to why we would be likely to find ourselves in a particular kind of vacuum, would be other kinds of explanation.

So there are definitely unproven theories which offer possible explanations. But for now, again in your words, the actual answer is indeed unknown.

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