Fermi four point vs Higgs condensate: differences?

[arivero]

The fermi four point interaction (consider its modern version, quark up to down plus electron plus antineutrino) is not renormalisable

Then we introduce an intermediate boson W in the middle.

Then we give mass to this W using the higgs mechanism.

Then we produce the higgs via a fermion condensate.

eh? But then the fermion higgs vertex is really a four fermion vertex, isn't it? Was not the problem we were solving in first place?

 

[Bill_K]

Then we produce the higgs via a fermion condensate.

The Higgs field is a boson condensate.

 

[tom.stoer]

The Higgs is not a condensate of a more fundamental or different field

 

[fzero]

As far as I know, all 4-fermion vertices in composite models are generated by integrating out heavy gauge bosons of some extended high energy gauge group. The low-energy theory (with the 4-fermion interaction) should therefore be viewed in the context of effective field theory, in which case there is no problem with renormalizability. As you say, it's precisely the same problem as the V-A model of the weak interactions and it is solved in an analogous way.

 

[arivero]

Hmm yep, surely the point is renormalizability and the infrared.

 

[fzero]

Hmm yep, surely the point is renormalizability and the infrared.

Is there a particular IR divergence that you believe is a problem? The usual nonrenormalizability of the 4-fermion interaction is a UV problem. In the EFT approach this goes away, because above the cutoff we have a renormalizable gauge theory.

 

[arivero]

Yep, that was the point. The nonrenormalizability of the 4 Fermion is UV, and the condensation of 2 fermions, whose condensate then becomes a Higgs (and then the mass term is a 4 fermion) is a IR issue.

Consider the example where there is no Higgs, then the W and Z get QCD scale masses via the IR mechanism, so the electroweak interaction still is short distance, but in the UV the problem dissappears. Interesting.

 

[fzero]

I'm still not sure where the question is. In the SM with a fundamental Higgs, at scales $E\ll\Lambda_\mathrm{EW}$ we have an effective field theory where integrating out the massive gauge bosons leaves a 4-fermion interaction.

In technicolor models we have some technifermions which get a mass at a scale $\Lambda_\mathrm{tc}$. In the minimal models, there is some gauge group $G_\mathrm{tc}$ that is confining at some scale $\Lambda_\mathrm{tc} \gtrsim \Lambda_\mathrm{EW}$. The technifermions are charged under $G_\mathrm{tc}$, but none of the SM fields are. If the technifermions also have weak charges, they can induce masses for the weak gauge bosons. From the EFT perspective, these mass terms appear from loop diagrams with technifermions running in the loop. SM fermions do not get a mass from a minimal technicolor model. They don't couple directly to the technifermions and, since the weak interaction is chiral, loops of massive gauge bosons don't generate fermion masses.

The latter point is the same reason why we need explicit Yukawa couplings to the Higgs in the SM to generate fermion masses. In your QCD toy model, for this reason, the leptons remain massless.

In order to generate masses for the SM fermions, we need to go to extended technicolor, where we have some high energy gauge group $G_\mathrm{etc}$ that contains the SM gauge group. The charges must be such that SM fermions can become technifermions by emitting an ETC gauge boson. After condensation at some scale $\Lambda_\mathrm{etc}$, fermion mass terms are obtained from loop diagrams where a SM fermion emits a massive ETC gauge boson to become a massive technifermion (which can change chirality), which then reabsorbs the ETC gauge boson to become a SM fermion (possibly different from the original one) again.

There are no 4-fermi operators in the UV theories and the models that are considered are renormalizable. The 4-fermi operators in the EFT below $\Lambda_{etc}$ aren't even the ones that give rise to SM fermion masses. The condensation necessary to generate EW scale masses is IR in a certain sense in the ETC theories, but it is definitely UV physics from our perspective.