I am not able to get around the following question (if it make sense):

Suppose I can derive the scattering matrix S for any particle scattering process.

Suppose that the standard model is actually correct and fully describe the physics of particles.

Would I be able to derive the Gauge Group of the Standard Model from the symmetries of the $S$-matrix defined as all matrices that commutes with $S$?


I think your question is a well defined one, but the answer is probably not so easy (and I will try to give a partial one).

First of all, you have to realize that $G=\text {SU} (2)_L \times \text{U} (1)_Y$ is not a symmetry of the real-world $S$-matrix, or more generally of the time evolution operator. If this was the case, then there would exist a unitary representation $U(g)$ of $G$, commuting with Poincarè transformation and acting trivially on the vacuum state $\vert 0\rangle$. But we know that this is not the case, for this is inconsistent with the Higgs field having a vacuum expectation value, electrons and quarks having a mass, and so on. In other words: when a symmetry group $G$ is spontaneously broken, the states of the theory do not fall into representations of $G$, and a fortiori there is no notion of the $S$-matrix being $G$-invariant.

My second point, which is somewhat more technical, is that the exact $S$-matrix should connect, strictly speaking, only asymptotic states of stable particles: electrons, photons and neutrinos (which for the sake of discussion we can assume to be massless left-handed fermions). If you wanted to extract exact information from the $S$-matrix, you would also need an exact theory of unstable states: how to express the hadronic resonances you deal with in real world experiments in terms of electrons, photons and neutrinos multiparticle states. I believe this is, at present, an hard task even in simple toy models.

Let's assume that the second difficulty can be bypassed, at least for all practical purposes. Concerning my first point, one would expect that at high enough energy (where the order parameter of electroweak breaking can be neglected) the consequences of electroweak invariance should be recovered. For example, you can directly verify that the tree-level amplitude of the processes $$e^{-}_L e^{+}_R \to e^{-}_L e^{+}_R$$ and $$\nu _L \overline {\nu} _R\to \nu _L \overline {\nu} _R$$ become equal in the asymptotic limit $E\to \infty$.

However, going from here to infering the existence of a symmetry group appears to me a very long way, and I'm not sure there exists any well defined procedure for this kind of inverse problem. For sure, there will be some relations such as the above, but you need a good ansatz on how to represent physical states (which as I said do not come automatically in representations of $G$) in terms of "asymptotic" $G$-multiplets.

To wrap up:

  1. Electroweak symmetry is not an $S$-matrix symmetry.
  2. The exact $S$-matrix would connect only absolutely stable states, which are a small subset of those appearing in real world experiments.
  3. Electroweak symmetry will have some consequence at high enough energy, but in order to infer the existence of a symmetry group you would need to formulate an ansatz about the representation to which scattering states belong to.

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