# Symmetry Groups

## Main Question or Discussion Point

I have a question regarding symmetry groups. I've often heard that the Standard Model is a SU(3) x SU(2) x U(1) theory. From what I understand these groups contain the symmetries under which the Lagrangian function is invariant. If so, what does every one of the 3 groups above contain (what symmetry do they correspond to) and does specifying the symmetry groups give a complete description of the Lagrangian ?

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I have a question regarding symmetry groups. I've often heard that the Standard Model is a SU(3) x SU(2) x U(1) theory. From what I understand these groups contain the symmetries under which the Lagrangian function is invariant.
That's not an exhaustive list of the symmetries. It includes the local gauge symmetries which are the more important symmetries because they govern the dynamics of the theory.
If so, what does every one of the 3 groups above contain (what symmetry do they correspond to)
These symmetries generate the conserved dynamic charges (through Noether theorem) of the theory which are the color charge [SU(3)], the isospin charge [SU(2)], and the hypercharge [U(1)]. The isospin and hypercharge are spontaneously broken but one U(1) sub-symmetry remains unbroken and generates the electric charge conservation.
and does specifying the symmetry groups give a complete description of the Lagrangian?
It goes a long way, but it is not complete. You still have to specify the field content beyond the gauge fields, what representation of the gauge symmetry do they belong to, and what other non-gauge interactions (such as a Yukawa interaction) are present. Most of the standard model parameters go into completely describing the Yukawa interaction.

ChrisVer
Gold Member
Isn't the Yukawa term just one more possible term due to the SM gauge symmetry group?

As dauto points out this is just the gauge symmetry group, and there are other symmetries (for example everything in the SM are also representations of the Poincaré group). There are various matter fields with handedness.

SU(3) is the strong interaction - dimSU(3) = 8 gauge bosons

SU(2)xU(1) is, in a manner of speaking, the electroweak interaction: dim = 3+1 gauge bosons. This breaks to the weak and electromagnetic interactions where 3 representations gain mass (w gauge bosons) and one (photon) remains massless due to the Higgs. Excuse my abuse of terminology.

ChrisVer
Gold Member
in complete (I think), SU(2)xU(1) contains 4 gauge bosons (massless). This is spontaneously broken via the Higgs Mechanism, under which the 3 out of the 4 gauge bosons gain mass (Ws and Z), into one U(1) symmetry as a remain, which gives you the massless boson (photon).

But I guess knowing the symmetries of your problem allows you to write down the lagrangian terms (the allowed ones are those who are invariant under the symmetry transformations- and Yukawa term is such)