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Why don't we have a theory solely of weak interaction, like QED or QCD? I.e $SU(2)$ gauge theory describing neutrinos and massive $W^{+-}$ bosons.

But instead we have a unified electroweak theory under $SU(2)\times U(1)$ gauge symmetry involving symmetry breaking by Higgs field. What is the reason for this? Is it something related to problem of giving a mass to chiral fermions?

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    $\begingroup$ I don't understand why you think we should "have" this theory. There is no part of the Standard Model where there's a standalone $\mathrm{SU}(2)$ gauge group - before EW symmetry breaking you have the unified electroweak $\mathrm{U}(1)\times\mathrm{SU}(2)$, after symmetry breaking the W/Z are massive and so not gauge - the only "gauge left" is the $\mathrm{U}(1)$ part, that's the whole point of symmetry breaking. In what regime do you think this "pure weak theory" is supposed to live? $\endgroup$
    – ACuriousMind
    Sep 13, 2022 at 21:02
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    $\begingroup$ ...but if you give them masses, then there's no gauge symmetry (no matter whether the mass comes from the Higgs or not)! $\endgroup$
    – ACuriousMind
    Sep 13, 2022 at 21:08
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    $\begingroup$ You seem to have misunderstood my initial comment: There is no problem in principle with a $\mathrm{SU}(2)$ gauge theory, it's just that this theory doesn't occur in any of the phases of the Standard Model - but your question sounds as if you think it should be there, that's what is unclear to me. $\endgroup$
    – ACuriousMind
    Sep 13, 2022 at 21:13
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    $\begingroup$ Group theory. When you look at the commutator of the weak charges, known long before the SM, you must have learned in your course it is manifestly not orthogonal to the (vector) electric charge, so the charged weak interactions cannot be separated from electromagnetism. This is basically why Glashow got his Nobel prize, no? $\endgroup$ Sep 13, 2022 at 21:24
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    $\begingroup$ Should be a basic part of your SM course: the above commutator must be a linear combination of the Electric charge and the neutral current/charge. This is how it is found, and its crazy asymmetry is understood. Review the PDG review. I recommend to students to learn it by heart, so they won't have to ask such questions. $\endgroup$ Sep 13, 2022 at 21:52

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You are asking three questions, really:

  1. Why can't you isolate the weak interactions from electromagnetism, group theoretically?
  2. How do fermions get a mass consistently with the known global chiral symmetries of the WI?
  3. Why do you have to gauge these symmetries?

The fitting of these three points is the quasi-magic of the solution to the SM puzzle.

  1. The first was solved by Glashow in 1961. The Feynman—Gell-Mann theory of the charged weak interactions relied on the two left-chiral charges $$ 2T_+=\int d^3x ~~\nu_e^\dagger (1-\gamma^5) e, \qquad T_-= T_+^\dagger \\ \leadsto ~~~ [T_+,T_-]= \int d^3x ~~~\left ( \nu_e^\dagger (1-\gamma^5) \nu_e -e^\dagger (1-\gamma^5) e \right )/2. $$ But this is not orthogonal to the vector EM charge $Q=\int d^3x ~~ e^\dagger (1-\gamma^5) e $ : it contains it, linearly combined together with an improbable ("cockeyed") neutral-current charge, of spectacularly counterintuitive chirality; unknown back then, and only discovered after a decade. So the weak interactions are inseparable from electromagnetism.

  2. The mass term for the electron, $m_3 \bar e ~ e$, is not invariant under the action of the chiral $T_{\pm}$. Introducing the Higgs and SSB of the global symmetry magnificently solves that problem.

  3. The 4-Fermi interaction suggests massive intermediate vector bosons. Renormalizability/computability all but dictates that the vector bosons be gauge bosons made massive by the Higgs mechanism. Note this is the iffiest part of the puzzle set by Weinberg and solved by 't Hooft and Veltman...

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Without the Higgs mechanism we can't have a gauge theory with massive bosons, the reason is very simple, let's take the example of why we can't have a massive electromagnetic photon:

Electromagnetism is described by the group $U(1)$, this means that under a gauge transformation our vector boson $A_{\mu}$ transforms as:

$$A_{\mu} \rightarrow A_{\mu} + \partial_{\mu}f$$

Mass terms for bosons are of the form : $m A_{\mu} A^{\mu}$, but we have that under a gauge transformation this transforms as:

$m A_{\mu} A^{\mu}\rightarrow m A_{\mu} A^{\mu} + 2m A_{\mu}\partial^{\mu}f+ m \partial_{\mu}f\partial^{\mu}f$, i.e. it's not gauge invariant.

For more complex groups, like $SU(2)$, the reason is exactly the same: the mass term is not gauge invariant.

In conclusion it's not possible to have a gauge theory where the vector bosons are massive, because it would break the gauge invariance, for this reason it was not possible to formulate a quantum theory of the weak interaction without the Higgs Mechanism, because since this is a "contact" interaction, we already knew that it would have been mediated by massive boson.

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    $\begingroup$ "Because it would break the gauge invariance" is not a sufficiently strong condition for discounting the theory. For instance, massive U(1) theory is perfectly well-defined provided the current is conserved: see here. The real reason why the same procedure fails for a non-abelian gauge theory is given in the answer here. $\endgroup$ Sep 14, 2022 at 2:53

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