Axial Anomaly and Fermion Mass

[scientist7]

It is often said that fermions are protected from large mass corrections by chiral symmetry. My question is does the axial anomaly generate corrections to fermion masses, and if so, doesn't this ruin the protection afforded by chiral symmetry to some extent?

Thanks,
Ben

 

[blechman]

The anomaly certainly cannot generate fermion masses within the context of perturbation theory.

You can, in principle, get fermion masses from instantons. Check out, for example, John Terning's TASI 2002 lecture notes (or his textbook which is based on those notes) to see how this is sometimes used to generate "gaugino" masses in supersymmetric theories.

However, for SM fermions even these effects are under control - they're exponentially suppressed, first of all. Also you are still breaking the gauge symmetry, which means that you must have a Higgs vev insertion, so that the WORST these contributions can do is shift the Yukawas. In particular, you still don't have a correction that brings the fermion masses up to the Planck scale (like the Higgs boson and the hierarchy problem) and therefore there is still no problem.

I haven't thought too hard about this question, but there might not even be an instanton effect for fermions in the SM - I'm not sure. But I would claim that even if there is, then it's irrelevant.

 

[scientist7]

Thank you for the response, blechman.

The suppression factor, e^{-8 \pi^{2}/g^{2}}), is usually small, but not so small for SU(3) instantons since the coupling constant is large at low energy.

So if instanton effects are big enough to solve the axial U(1) problem (and the effects are really visible since the symmetry would otherwise be there yet one finds that nature doesn't respect this symmetry at all!), it seems like they might be able to have a significant effect on fermions. Well maybe not Planck scale sized, but I wonder what the size of corrections would be?

I actually own the Terning book, though admittedly I don't have a mastery of it at all. I remember him getting gaugino masses through holomorphy arguments, but these cannot apply to just the non-SUSY Standard Model. So I have no idea how one would calculate, if they exist, instanton contributions to SM fermion masses.

 

[blechman]

I'm not convinced there is any contribution to the quark masses from QCD instantons:

Instanton bubbles can generate a 't Hooft operator det {Qbar Q} where the det is over the flavor structure. See Terning, Ch 7. You would then need pairs of fermions to "annhiliate" into a Higgs, which then goes to its vev, leaving one pair left over. so you would have something like:

Qbar Q * (yv)^p * I

where y=yukawa coupling, v=higgs vev, p=number of pairs that are annihilated, and I is the Instanton amplitude (see, for example, Coleman's "Aspects of Symmetry", ch 7).

As you can see, even in the QCD case, I is still quite small, and so the correction is TINY relative to the tree-level Yukawa coupling. This is why I say the correction is irrelevant for SM fermions.

That being said: these effects might be there, and be important, in other contexts, such as the gaugino masses I mentioned earlier, as well as various aspects of technicolor models where there is strong coupling and chains of broken gauge groups (so there are other sources of breaking besides the Higgs). So these effects might be there in that case. But they should be irrelevant to the SM fermions themselves, as things are.

ADDED:
BTW: it is these operators that allow for violation of B and L in the early universe (both of which are anomalous, while B-L is not), and may potentially play a role in baryogenesis, for example. I just thought I'd mention that while I'm here...

 

[Haelfix]

Its precisely in this regime where there may be a hitch, b/c while those operators are tiny in vacuum, they might not be in a thermal bath.

 

[blechman]

Its precisely in this regime where there may be a hitch, b/c while those operators are tiny in vacuum, they might not be in a thermal bath.

Yes, I agree that the story is more complicated in this regime. However, from the point of view of asking if fermions have a "hierarchy problem" like the Higgs due to these instanton effects, I think the answer is simply, "No."