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In a possible theory like our Standard model but without a Higgs i.e.:

$$ \mathcal{L}=i\bar{\Psi}_f\gamma_\mu D^\mu\Psi_f-\text{Tr}[G^b_{\mu\nu}G^{b\,\mu\nu}] $$

where $b,f$ run over the typical species we have in the standard model (SM), and all fields are in the same representation as in the SM.

In this context it is sometimes stated that, although there is no Higgs, there would be a mass generation mechanism for the gauge bosons of $SU(2)$ because of QCD. This happens via the chiral quark condensate $\langle q_L q_R\rangle\neq 0$. (Or statements like "the gauge bosons eat up the pion")

My question is now, how can I see that this generates a mass for the $SU(2)$-gauge bosons? Usually using methods of spontaneous symmetry breaking, I would put a vacuum expectation value for some field and see that it results in a term that behaves like a mass term. But this won't work here because there is no term involving quarks and bilinear in gauge bosons.

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  • $\begingroup$ I could write this as an answer but it might be not enough to satisfy you. The quark-quark condensate happens because of SU(3) confinement and cannot be study perturbatively. The best way of studying it at a Lagrangean level is to use chiral-perturbation theory where you have explicit "pions" (quark-quark bound states) as the fields. Any good book on QFT and the SM that discusses QCD has a discussion on chiral perturbation theory, I advise you to give it a look! $\endgroup$ – romanovzky Jul 19 '15 at 11:21
  • $\begingroup$ Which books are there for example? Do you know a good one? $\endgroup$ – Nontriviality Jul 22 '15 at 19:22
  • $\begingroup$ I also red about chiral perturbation theory in Schwartz, but many aspects concerning the theory of pions, seem to appear all of a sudden. But how are the two descriptions related? $\endgroup$ – Nontriviality Jul 22 '15 at 19:31
  • $\begingroup$ Unfortunately I am not an expert in chiral PT. I would advise you to read like arxiv.org/abs/hep-ph/0210398. My SM course had some introduction stuff and I can only tell you some general notes. But the idea is very similar to the SM Higgs mechanism, if $\langle q_L q_R \rangle \neq 0$ you are aligning the vev in a preferable $SU(2)$ direction (just like the SM Higgs) as the condensate $q_L q_R$ is not $SU(2)$ singlet. Therefore, a mass will be generated. The details should be clear in a good chiral PT text... sorry for the superficial answer $\endgroup$ – romanovzky Jul 24 '15 at 10:05
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The standard text worth reading is Georgi's Weak Interactions text, which outlines the effective σ models resulting out of chiral symmetry breaking in QCD. This process is quite subtle and elusive and Schwartz's book is no more schematic than standard texts. But, once you've bought the conceit of quark bilinears supplanted/summarized by mesons (and this is pure low energy QCD, nothing to do with the weak interactions; it is a potential separate question I would not be keen to answer!), which you complain about in your comment, it is not hard to see how the pions of the σ-model, this effective low energy theory, also overlap with generators of the EW symmetry, which are thus also SSBroken, in turn, but only by a little.

This, in fact, is the building principle of all technicolor theories: using states resulting out of a technistrong theory chiral symmetry breaking to substitute for the Higgs field in the Higgs mechanism and break EW. So, let me flesh out the point that @romanovzky made schematically.

Take the SM, keep only the lightest quark generation, for simplicity, so the u,d quarks only, and discard the Higgs field. Hence, u and d are now massless. QCD, in the χSB elided here (cf. WP link provided) generates a σ model that summarizes this symmetry breaking into PCAC; actually, here, CAC, conserved axial current, as the pion is massless, that is, the 3 conserved axial currents $$ \vec{A}_\mu= f_\pi \partial_\mu \vec{\pi} +... , $$ where $f_\pi$ is the pion decay constant, of the order of o.1 GeV --this is low energy QCD, after all, and we are interested in features of the peculiar asymmetric vacuum.

Enter the Weak interactions. You couple these axial currents to the axial half of the W, to get schematically and cavalierly, $$ ...+g \vec{A}_\mu \cdot \vec{W}_\mu +... $$ etc... so you are gauging the σ model, in our case, for convenience, the nonlinear one. The EW $SU(2)_L$ overlaps the 3 broken chiral charges of the Axials, so it's broken. (Recall the V-A action on pions, $\delta _{\vec{\theta}_L} \vec{\pi} \sim \vec{\theta}_L\times \vec{\pi} - f_\pi \vec{\theta}_L +...$. You can see the axial variation of the above current-W term may only be cancelled by the variation of the W bilinear in the next paragraph.)

Now the seagull term in the gauge-covariant kinetic term for the pion is, quite schematically, in the leading, pionless term, something like $$ g^2 f_\pi^2 \vec{W}_\mu \cdot \vec{W}^\mu ~, $$ that is, $f_\pi$ has supplanted the standard Higgs EW v.e.v. $v\sim 246$ GeV of the real world; that is, the new notional Ws now have a mass $$ \frac{f_\pi}{246~GeV} M_W\sim 4\cdot 10^{-4} M_W \sim 32 ~MeV, $$ real light... lighter than the strong χSB scale.

You can see how this sort of thing (which happens at some level in real life) is a negligible piece of noise in the big picture of the SM. We have omitted the "second job of the Higgs", namely giving the fermions (e.g. leptons, in case we didn't wish to descend into current quark mass complications) mass in a gauge invariant way, but it can be arranged.

Sophisticated versions of this mechanism undergird Technicolor models, where the interaction is some QCD-inspired strong coupled theory, e.g., Susskind 1979. It's eqn (15) provides the W mass "directly" through the vacuum polarization contribution of the quarks, assuming χSB, but it might come across as more cryptic than the effective Lagrangean sketch outline here.

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