Why does the Standard Model not unify $SU(3)$ and $SU(2)\times U(1)$?

Question

I am struggling with the definition of a unification. I read this question and i am wondering why the standard model is not unifying strong and weak force according to the given definition of a unification, namely that "'Unification' refers to explaining two sets of phenomena (theories) which were previously unrelated, and combining them into a single cohesive description."

Isn't the Standard Model exactly doing this with all three forces? It gives a frame (even in one lagrangian) that explains all three forces.

Regarding to this question I am wondering what we mean when we say the lagrangian of the SM is $SU(3)\times SU(2)\times U(1)$ invariant. In my understanding there is a part which is $SU(3)$ invariant and a part that is $SU(2)\times U(1)$ invariant.

So my questions are:

• What is the precise definition of a unification in QFT?
• Why is the standard model not a unified theory for the strong and electroweak forces (and why is it for the electroweak theory?)
• In which sense is the SM-lagrangian $SU(3)\times SU(2)\times U(1)$ invariant? Or alternatively in which sense is it invariant if we define the charge of each field under each specific group?

Answer 1

1) I'm not aware of a precise definition of unification in QFT. In gauge theories, practitioners tend to mean that the gauge bosons transform in the adjoint representation of a single simple Lie group, for example SO($N$) or SU($N$) or $E_8$. Transforming under a single simple group means they must all have the same coupling strength at sufficiently high energies where the group remains unbroken. Recall that for each simple Lie group, one can have a separate $\frac{1}{2g^2} \operatorname{Tr} F^2$ type kinetic terms with a different gauge coupling $g$.

2) In this sense, the standard model is not unified. There are three kinds of gauge bosons: gluons that transform under the adjoint of SU(3) and the photon, W, and Z bosons which transform under SU(2)$\times$U(1), all a priori with different couplings. The smallest simple Lie group which contains SU(3)$\times$SU(2)$\times$U(1) as a subgroup is SU(5). As I understand it, proton lifetime measurements have pretty much ruled out SU(5) unification. This Wikipedia article has a more in depth discussion.

In my choice of definition, I would say that the electroweak theory is not unified in the standard model. I would say that it allows for successful employment of the Higgs mechanism, which leads to nontrivial mixing between the SU(2) and U(1) factors at low energy.

3) The standard model is SU(3)$\times$SU(2)$\times$U(1) invariant in the usual sense. If one transforms the gauge bosons under the adjoint representations of their respective groups and if one transforms the fermions under the appropriate fundamental and singlet representations, the Lagrangian shifts at most by a total derivative. The SU(3) and SU(2)$ \times $U(1) groups do not split nicely in their action on the Lagrangian. For example, the quarks transform in the fundamental of SU(3) and are also charged under the E&M U(1) subgroup of SU(2) $\times$ U(1).