What is the Mathematical description of Weak Interaction at low energies?

Question

Introduction

When I started to study gauge theory the mathematical road map seemed to be quite "simple". After all the concepts and notions about principal the differential geometry of fibre bundles the phrase "give me a symmetry group and then I have a lagrangian density of the theory" have a precise meaning. For instance, the symmetry group $U(1)_{\mathrm{em}}$ and it`s Lie algebra $\mathfrak{u}(1)_{\mathrm{em}}$, are the building blocks to construct the covariant derivative, the field strength and the lagrangian. Furthermore, electromagnetism is a abelian theory, and after $[1]$ we realize the deeps of gauge theory mathematical description.

Now,if you give me just the group $U(1)_{\mathrm{em}}$ it seems that is possible to describe the electromagnetic interaction just following the math. But, if you give me the group $SU(2)$, it seems that this cannot describe, solely, the weak interaction. It seems that I need all the machinery of Spontaneous Symmetry Breaking and the group $SU(2)_{\mathrm{L}} \otimes U(1)_{\mathrm{Y}}$ to descend from a high energy picture, and then study the weak interaction and electromagnetism in a covariant and guage invariant consistent description.

My Question

With the group $SU(3)$, solely, we have the strong interaction mathematical description; with the gauge group $U(1)_{\mathrm{em}}$, solely, we have the electromagnetic interaction mathematical description: separated, low energy theories. Now, is it possible to describe a low energy theory of weak interaction with just a $SU(2)$ group? In other words, how can I study the weak interaction separated from strong interaction and eletroweak unification?

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$[1]$ C. N. Yang and R. L. Mills. Conservation of Isotopic Spin and Isotopic Gauge Invariance

Answer 1

... how can I study the weak interaction separated from strong interaction and eletroweak unification?

You cannot. It is impossible, for a good reason. At very low energies, the weak interactions disappear, and all that's left is electromagnetism. But at no point do EM and weak interactions "separate" from their high energy blending.

The answer to your question is detailed in most good introductory SM outlines, e.g. T D Lee's classic book, or TP Cheng's & LF Li's vademecum, eqns (11.160 - 11.22). It is the heart of S Glashow's (1961) contribution to the SM:

• The weak interactions by themselves cannot be separated from electromagnetism, as they are joined at the hip, so to speak, group theoretically.

Very schematically, the effective piece in the weak Fermi lagrangian density describing, e.g., μ decay (low energy) is $$ -\frac{G_F}{\sqrt{2}} J^\dagger_\lambda(x) J^\lambda ,\\ J_\lambda = \bar \nu_e \gamma_\lambda (1-\gamma_5) e + \bar \nu_\mu \gamma_\lambda (1-\gamma_5) \mu . $$ As a result, the weak charge $$ T_+= \frac{1}{2} \int\!\!d^3x~~ J_0(x)= \frac{1}{2} \int\!\!d^3x~~(\nu_e^\dagger (1-\gamma_5) e + (e\to \mu))\\ T_-=T_+^\dagger. $$

But, at equal times, $$ [T_+, T_-]= \frac{1}{2} \int\!\!d^3x~~(\nu_e^\dagger (1-\gamma_5) \nu_e - e^\dagger (1-\gamma_5) e + (e\to \mu)), $$ a $T_3$ closing with the above $T_\mp$ into an su(2) Lie algebra. However, this charge had not been observed in physics before the advent of the SM, even though it had some overlap with the electromagnetic charge.

Still, this $T_3$ is definitely and visibly not the electric charge acting on leptons,
$$ Q = - \int\!\!d^3x~~(e^\dagger e + \mu^\dagger \mu). $$

The simplest (minimal) construction incorporating both this $T_3$ and the piece of Q trace-orthogonal to it (nowadays called weak hypercharge, $Q-T_3$) was $su(2)\oplus u(1)$, which Glashow speculated about in 1961, before the Weinberg-Salam model, and, despite its "artificiality" according to the brightest and the best of the time, turned out to be absolutely right, to some surprise in the community. Go figure...

With the advent of higher-energy experiments, people discovered neutral current weak interactions involving $T_3$, and confirmed the above mixing structure. In lower energy physics (classical), the weak interactions completely extinguish themselves out of the picture.