# What was the motivation for thinking the weak interaction could be described by a Yang-Mills theory?

In some sense, describing the strong force using an $$SU(3)$$ Yang-Mills theory makes perfect sense: Yang-Mills theories describe massless bosons, of which the gluon is clearly a member, while the two most common hadrons were observed (in deep inelastic scattering) to consist of three components, motivating an $$SU(3)$$ symmetry for the associated interaction. What confuses me is why the weak force would be thought to be described by such a theory, given that it's governed by massive bosons. Clearly, if one wants to unify electromagnetism and the weak interaction, a gauge theory of some sort is needed, and we can of course describe how a massive boson could arise due to spontaneous symmetry breaking from such a theory. But unless one was specifically trying to unify these two interactions, is there a specific reason a Yang-Mills theory would be attempted? (And if the answer is that the Yang-Mills formulation arose from an attempted unification—what was the motivation for suspecting the weak and electromagnetic forces could be unified, rather than any other two?)

Put more simply, my question is this: Electromagnetism and the strong interaction are both mediated by massless particles, which require a gauge-invariant interaction term (and therefore require the fermions to have a symmetry under some Lie group). Massive particles don't require gauge-invariant interaction terms (since they don't undergo gauge transformations) and therefore don't need to couple to fermions with such a symmetry. So is there any good reason for thinking that massive bosons also are governed by a gauge theory, despite not having a gauge in the traditional sense?

• There isn't any requirement about a Yang-Mills theory needing to describe a massless boson. – Triatticus Jul 14 '20 at 2:44
• This is a question for HSM, if there ever was one. Your imagined logical path based on unification is deeply unsound, beyond being orthogonal to the actual development of the theory, well described by numerous accounts. – Cosmas Zachos Jul 14 '20 at 2:57
• Right (post-spontaneous symmetry breaking, the weak interaction is a massive Yang-Mills theory), but I don’t know why one would postulate a Lie group symmetry to start with. For QED/QCD, it makes sense, since you need the symmetry to offset the gauge transformation. – laaksonenp Jul 14 '20 at 2:58
• @cosmas zachos Thanks for the link. I realize the approach is ahistorical (I’m mainly thinking about how one would motivate it on the far side of the theory), but which parts are incorrect? I haven’t studied non-Abelian gauge theory in much detail, as you can probably tell. – laaksonenp Jul 14 '20 at 3:01
• A long story. You might start by finding out how the theory actually developed. The book by Pais is superb. – Cosmas Zachos Jul 14 '20 at 3:04

What you are asking is "why the discovery by experiments of particle interactions happened the way they happened", a history of interactions.

The electromagnetic interaction had a complete formulation with Maxwell's equation, but it broke down not explaining: spectrums of atoms, black body radiation and the photoelectric effect. Quantum mechanics was invented to fit this data, to start with.

Then quantum field theory for electromagnetic interactions emerged and could numerically predict scattering and decay data due to electromagnetic processes.

Then the cosmic ray, to start with, and then accelerator data found the zoo of resonances at present found in the pdg.

To start with, the success of QED and the Feynman diagrams used for specific calculations led to describe the data with Regge theory , (a string model that is having a revival of interest now with string theories). It worked well for the strong interaction resonances appearing in accelerator experiments. Next, came the four fermi interaction, the vector dominance model, theorists were working hard to fit the data.

And then came the quark model in the data displaying spectacular symmetries , SU(3) symmetries , the eightfold way..

The discovery of the omega minus was the triumph of the weak SU(3) model

Note this SU(3) is not fundamental it is an emergent one from the basic inetractons of quarks.

The eightfold way may be understood in modern terms as a consequence of flavor symmetries between various kinds of quarks. Since the strong nuclear force affects quarks the same way regardless of their flavor, replacing one flavor of quark with another in a hadron should not alter its mass very much, provided the respective quark masses are smaller than the strong interaction scale—which holds for the three light quarks. Mathematically, this replacement may be described by elements of the SU(3) group. The octets and other hadron arrangements are representations of this group.

Presently the fundamental symmetry for weak interactions is SU(2)

This focusing on group structures made theorists start looking for fundamental group theories to unify weak and electromagnetic, and led to the SU(3) for strong interactions.

So it is the serendipity in time evolution of data and theoretical research that lead to the existing SU(3)xSU(2)xU(1) present theory, not sitting down and thinking "how to use gauge groups".

• Your last sentence says it all: gauge theories entered the picture at the very last moment, and in the face of serious cultural resistance. People were talking about the "heavy intermediate vector boson" of the weak interactions and its peculiar chiral couplings long before it could be deniably shoehorned into gauge theories. The mental picture of the question seems to have jumped out of a GUT project proposal... – Cosmas Zachos Jul 14 '20 at 12:26