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A Yang-Mills theory can be constructed for any Lie group that is compact and semisimple. The motivation behind this is discussed in this question. Is there any physical reason we choose $SU(3)$ or $U(1)$ for QCD and QED respectively? I am still learning about Yang-Mills theory, but so far I get the impression that these groups are chosen either because they're relatively simple compared to other groups and because the math just works out. Is there any physical reason for choosing these groups over others?

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  • $\begingroup$ Behaviour of $\beta function$: Asymptotic freedom of QCD (nonabelian gauge group $\rm SU(3)_c$) vs asymptotic slavery of QED (abelian gauge group $\rm U(1)$). You find this in all good text books on particle physics. $\endgroup$
    – Hyperon
    Dec 30, 2023 at 8:55
  • $\begingroup$ @Hyperon Isn't the $\beta$ function behavior a consequence of the choice of Lie group and not the other way around? $\endgroup$
    – CBBAM
    Dec 30, 2023 at 9:25
  • $\begingroup$ The choice of a Lie group is related to local gauge invariance. For QED, it was already known in classical electrodynamics than gauge invariance was isomorphic to $U(1)$. For QCD I'll let others anwer, but one possible clue is the number of generators: $SU(N)$ has $N^2-1$ generators, which helps select an appropriate Lie group. $\endgroup$
    – Miyase
    Dec 30, 2023 at 9:52
  • $\begingroup$ @CBBAM See my answer explaing the history of the physical motivation for using a nonabelian gauge theory for the description of strong interactions. $\endgroup$
    – Hyperon
    Dec 30, 2023 at 10:43

2 Answers 2

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Well let me say first that the "physical reason" is that the predictions of QED and QCD match hundreds and hundreds of detailed experiments fantastically well. But of course just a link to the body of scientific output of the LHC, or SLAC, or the measurement of $(g-2)_e$ is not a particularly insightful answer. I assume what you are looking for is a simple empirical reason distinguishing QFTs based on these Lie groups from others.

For QED perhaps the most basic answer is that the photon does not self-interact, and so the Lie group must be Abelian. Then $U(1)$ is the only compact choice.

Of course historically $U(1)$ was 'chosen' first in classical electrodynamics. Then maybe your physical answer would be that electric charge was discovered to be one-dimensional, given by some integer multiple of $e$ at long distances.

Along these lines, perhaps the most basic data to pin down QCD from the point of view of Lie groups is the dimensions of the representations. For the quarks, I think historically this information could first be gleaned from measurements of the 'R-ratio', $$ R \equiv \frac{\sigma(e^+ e^- \rightarrow \text{hadrons})}{\sigma(e^+ e^- \rightarrow \mu^+ \mu^-)}, $$ the ratio of the cross-sections for electron-positron annihilation to produce hadrons vs. a muon-antimuon pair. Of course we can't see the quarks directly, only the colorless bound states they combine into (the hadrons). Physically, this is taking place through an $s$-channel photon exchange, a QED process. Then the color quantum number of the quarks is just a bystander here, and one must sum over all the colors of quarks. Finding that the data fit a factor of $N_c=3$ then tells us the dimension of the representation of the quarks. Here's a 1974 review from Burton Richter, but the experimental situation at that point was still confusing, because we have multiple light quarks and hadronic physics is messy.

For gluons, there must be a similar statement in terms of the production cross-section for three-jet events, in which gluons were originally experimentally discovered as recalled in this nice historical manuscript by Stella and Meyer. Certainly there you are also summing over the 8 possible internal orientations of the gluons, though to my knowledge there isn't quite as clean a quantity to point to in this case as the R-ratio.

So given these empirical determinations of the dimension of the adjoint (8) and of the matter fields (3), this may be enough information to point to $G=SU(3)$ Yang-Mills, at least if you assume $G$ should be a simple group.

Of course the real, historical answer is not Lie-group-theoretic, and it's not just these dimensions that particle physicists collectively measured in the 1970s but tons of differential cross-sections and decay rates and energy splittings etc. etc. that are all consistent with the quantum field theoretic predictions of QCD.

But I agree that it's an interesting question and point of view to take, and I regret not having a sharper answer for you, or knowing where one exists to point you to. It would be great to read a review from one of the early QCD theorists if they were thinking seriously about, for example, $Sp$ or $SO$ groups; if I find one I'll update this answer.

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    $\begingroup$ Thank you, this was exactly the kind of answer I was looking for! $\endgroup$
    – CBBAM
    Dec 31, 2023 at 8:56
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    $\begingroup$ Of course any number of copies of $U(1)$, i.e. $U(1)\times \dots \times U(1)$, is abelian and compact $\endgroup$ Dec 31, 2023 at 9:41
  • $\begingroup$ You're right, the fact that we only have one photon is also needed there. $\endgroup$
    – SethK
    Dec 31, 2023 at 11:19
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Experiments on deep inelastic scattering performed at the end of the 1960's and early 1970's suggested that quarks confined in hadrons were behaving essentially like free particles at small distances (i.e. large energies). This was a big challenge for a possible quantum field theoretic description of the strong interaction, as at that time all known QFTs had turned out having positive $\beta$ functions, meaning that the corresponding interaction becomes strong at small distances. In 1973, Frank Wilczek, David Gross and David Politzer computed the $\beta$ function of a nonabelian gauge theory finding that it was negative. This came as big surprise at that time, although earlier remarks by Gerard 't Hooft in 1972 and a paper by Iosif Khriplovich in 1969 had already pointed in this direction. This surprising result was (among others) one of the main arguments for the introduction of QCD based on the nonabelian gauge group $\rm SU(3)_c$ as the theory of strong interactions and at the same time the "rehabilitaion" of quantum field theory.

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  • $\begingroup$ Thanks, this partly answers my question but I'm still not clear on why SU$(3)$ was seen as a potential solution to the asymptotic freedom problem before any attempts to compute its $\beta$ function. Was it just a guess that turned out to be right or was there a deeper reason that led Wilczek, Gross, and Politzer to compute its $\beta$ function? $\endgroup$
    – CBBAM
    Dec 30, 2023 at 23:00
  • $\begingroup$ @CBBAM The task to compute of the $\beta$ function of a (pure) Yang Mills theory was given to Frank Wilczek as a thesis work by his supervisor David Gross and, at the same time, to David Politzer by Sidney Coleman. As the work of Khriplovich and 't Hoofts remark was appeaently unknown to them, this problem was supposed to complete the list of beta functions of all types of quantum field theories. The expectation was $\beta >0$, as it had been in all previous cases and the result $\beta <0$ come out as a big surprise. $\endgroup$
    – Hyperon
    Dec 31, 2023 at 9:44
  • $\begingroup$ @CBBAM This in turn resuscitated the attempt to find a quantum field theoretic description of the strong interaction that had already been given up for some time with (nowadays completly forgotten) alternate approaches advocated by prominent physicists of that time. The new result for the Yang-Mills beta function paved the way for the formulation of QCD based on the gauge group SU(3) as a common effort by Heinrich Leutwyler, Harald Fritzsch and Murray Gell-Mann. Details can be found e.g. in the highly recommendes book "Particles, Fields, Quantum" by Gerhad Ecker (Springer Nature 2019). $\endgroup$
    – Hyperon
    Dec 31, 2023 at 9:54

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