# Dynamical chiral symmetry breaking

I don't know if it is the correct sub-forum, if I choose wrong then feel free to move the thread.

I was listening to a talk today using DCSB. I think I could get a glimpse on some other parts of the talk and found some ideas intriguing. I would like to understand them better, but I cannot really figure out what exactly is meant by DCSB. Some collegues explained to me it is nothing else than Spontaneous Symmetry breaking but I would disagree.

If it is SSB, why does one speak of DCSB, just because of the momentum dependence of the masses?

I would be very grateful if someone could explain the differences between the two models to me.

Thank you very much :)

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dynamical symmetry breaking is a form of spontaneous symmetry breaking, but the converse does not hold. The Higgs mechanism is also a form of SSB, but it is not dynamical.

In gauge theory of the standard model you deal with some gauge group, like SU(2)xU(1) (the electroweak interaction). If the gauge group is unbroken then all particles involved must be massless. Since particles have mass, the gauge group is broken and you would like to have some mechanism which causes this breaking.

To break the gauge symmetry what you need is a coupling of the gauge fields to a condensate. The condensate arises due to some form of SSB. When you have a condensate you essentially define a non-trivial vacuum of the theory. This vacuum / condensate breaks the gauge symmetry explicitly (either completely or to some subgroup), et voila: mission accomplished.

One way to accomplish this is the Higgs mechanism: the gauge fields are coupled to some scalar potential which, for example, is the Mexican hat potential. When the energy scale sits in the classical minimum of the scalar potential the Higgs fields condenses and acquires a vacuum expecation value (i.e. SSB).

But you can get away with other condensates as well. This is where DSB comes in.

DSB essentially introduces a new non-Abelian gauge coupling (technicolor) between the existing fermions (quark, fermions, neutrino's for electroweak breaking). The basic idea is that at 'low' energies (where you want the original gauge theory to be broken) this new gauge coupling condenses bilinear combinations of the fermions. So you form a bilinear of two fermions, i.e. $q_L \bar{q}_R$ where L,R denotes the chirality and the bar denotes the antiparticle, and let this combination acquire a non-trivial expecation value. Depending on the nature of the new gauge coupling you can condense multiple and types of bilinears.

The result is now "precisely" the same: a condensate has formed which breaks the original gauge symmetry. It's SSB without the Higgs.

One thing I should add for clarity: In the case of the Higgs mechanism the scalar potential has a specific shape which has some free parameter in it. This parameter fixes the energy scale of the Higgs mechanism (it's in some sense the Higgs mass). So to generate the standard model you need to fine-tune this free parameter, which is problematic (the standard model suffers from hierarchy problems: very large differences between the different energy scales. The free parameter needs to be fine tuned very carefully to generate the SM, but why would nature choose that path? Supersymmetry allows for mechanisms to get rid of this fine tuning, but that's another story).

In DSB you do not have this freedom: once the non-Abelian gauge group becomes strongly coupled, confinement kicks in and the energy scale of the symmetry breakings is generated dynamically. Hierarchy of the different energy scales comes in naturally, due to the way non-Abelian gauge couplings work.

OK, that was quite a rant.

Thank you :) I have read everything, so far my knowledge was solely on SSB in the elctroweak sector of the SM.

But the gauge symmetry is preserved even with Higgs field and it is only the ground state that is broken, that is why we use SSB, isn't it? And gauge symmetry is also preserved on Lagrangian level in DSB and broken in the ground state, right?

Is it necessarily technicolor that is used for DSB, probably there also exist other theories that use DSB?

Is it correct that in DSB on contrary to SSB, where the symmetry is already broken on tree level, the symmetry is only broken when higher order terms are took into account?

And then, why does one talk about non-perturbative DSB, if DSB only comes into play for higher orders?

I guess there will be more questions tomorrow. Thanks again very much for your extensive answer :)

I have to warn you that my knowledge on this topic is limited, but I'll try my best. Perhaps someone else can fill in the gaps (or errors) ;)

Thank you :) I have read everything, so far my knowledge was solely on SSB in the elctroweak sector of the SM.

But the gauge symmetry is preserved even with Higgs field and it is only the ground state that is broken, that is why we use SSB, isn't it? And gauge symmetry is also preserved on Lagrangian level in DSB and broken in the ground state, right?
Yes, gauge symmetry is preserved in the action and broken only by the non-trivial vacuum / ground state.

Is it necessarily technicolor that is used for DSB, probably there also exist other theories that use DSB?
Technicolor models refers specifically to the introduction of an extra, non-Abelian gauge coupling which causes confinement at low energies (very similar to QCD) with the additional property that another gauge coupling is broken.

Superconductivity is another example of DSB: the electrons form cooper pairs, which condense and therefore define a non-trivial vacuum. The gauge group which is broken due to coupling with this non-trivial vacuum is the electromagnetic force. The physical effect due to this broken symmetry is the Meissner effect: the superconductor does not allow magnetic fields to enter the superconductor. The idea is: the photon becomes effectively massive within the superconductor and therefore have a finite penetration depth -- much like the weak force. The underlying coupling which causes the formation of cooper pairs is a combined effect of screening due to the lattice and phonon-electron interaction -- this force is not a non-Abelian gauge coupling. So this is not an example of technicolor.

But as far as the standard model is concerned, it seems we only deal with gauge couplings, although I do not know why something like e.g. a Yukawa coupling does not accomplish the same thing (it probably does though!). I do know that a non-Abelian gauge coupling is appealing, since it naturally incorporates the hierarchy problem (if you know what that is).

Is it correct that in DSB on contrary to SSB, where the symmetry is already broken on tree level, the symmetry is only broken when higher order terms are took into account?
I don't know. I've read these statements, but haven't gone so deep to check them myself.

But again, you do not want to make a distinction between DSB and SSB. DSB is a form of SSB, but there are other mechanisms which cause SSB as well which are not dynamical (like the Higgs mechanism). SSB 'simply' means you deal with a ground state configuration which does not possess the symmetry of the original theory. When the theory is cooled down it breaks the symmetry, but there is some ambiguity associated with this. But since it has to make a choice, we call the resulting effect spontaneous. The scale at which this symmetry breaking can occur can be dynamical or 'put in by hand'.

And then, why does one talk about non-perturbative DSB, if DSB only comes into play for higher orders?
I'm not sure I understand this question.. What I do know is: the DSB only appears when the new non-Abelian gauge coupling becomes strongly coupled (sidenote: non-Abelian gauge theories are asymptotically free, so they always become very weakly coupled at high energies. Moving the energy scale down increases the coupling strength). But performing perturbative calculations of a strongly coupled theory beyond tree-level is very hard, if not impossible. So if DSB only comes into play beyond tree level, you got a serious problem if you want to say anything quantitively about it using perturbation theory. Another way to say something meaningful about DSB at strong coupling is to use non-perturbative methods.

The most popular non-perturbative method is to put the theory on a lattice, which is probably the reason why many people doing lattice QCD also look at technicolor.

Come to think of it (and this is just a personal brainfart): undoubtly someone is looking at trying to apply ADS/CFT to technicolor models -- which is a non-perturbative method originating from string theory.

I guess there will be more questions tomorrow. Thanks again very much for your extensive answer :)
No problem.