Chiral symmetry and Fermion mass non-divergences

07Mar08

Scalar quadratic divergence

One of the main motivators for new physics at the TeV scale is the quadratic divergence in the Higgs mass from radiative corrections:

m_H^2 \sim \int d^4k \frac{1}{k^2} \sim \Lambda^2

This means that for a natural UV completion of the Standard Model we would expect new physics entering not too far above the Higgs mass. This is a general feature of scalar particles which have renormalisable (i.e. “good at low energies”) four-point interactions.

Fermion linear divergence…

But what about fermions? Naively, doing the same analysis as above, we have the following diagrams:

The fermion needn’t interact with a vector (e.g. it could interact with a scalar), but Lorentz invariance forces the interaction to be with a boson and hence the propagators lead to the following radiative corrections to the fermion mass:

m_f \sim \int d^4k \frac{1}{k^3} \sim \Lambda + m_f^{(0)} \log \Lambda

So we should expect a linear divergence in the fermion masses that would point to new physics, shouldn’t we? (This, of course, would be a bit problematic since the fermions themselves span a range of masses.) Why don’t we ever talk about the fermion linear divergence?

… or not!

Because it happens to not be there. We know that in the limit of zero fermion mass, chiral symmetry protects fermion masses. Namely, \psi \rightarrow \exp(i\alpha \gamma_5)\psi prevents mass terms from being generated at loop level. (The mass term violates the symmetry, and there’s no way to produce symmetry-violating terms out of symmetry-abiding Feynman rules.)

But what good is this if we know that there are explicit mass terms in the Standard Model? I.e. chiral symmetry is broken! Here’s something that might be surprising: even though the symmetry is broken, it’s still useful! (This is an example of a “life lesson from physics.”)
Chiral symmetry is broken in such a way that it is restored in the limit where all the masses go to zero. Compare this to the divergence structure above. In the limit m_f^(0) \rightarrow 0, we expect the one-loop mass to be zero. The logarithmic divergence indeed vanishes, but the linear divergence seems to stick around! What does this apparent inconsistency mean? The linear divergence must not be there!!

You can attach more words to this and say something about being continuously connected to the chiral theory, but the point is that consistency in the unbroken limit requires the divergence structure to only be logarithmic. Don’t get me wrong, the theory we’re talking about has broken chiral symmetry — we’re just able to extract useful information because chiral symmetry is broken in a specific way (by explicit mass terms only).

Further discussion: so what?

I should say a few further words about this and why it’s kind of neat. First of all, this is an example of what Professor Lykken calls “symmetry naturalness” (see SSI04). The fermion mass parameters may `unnaturally light’ compared to the scale of the SM cutoff, but the [particularly broken] chiral symmetry gives a reason why radiative corrections don’t push the masses to this scale.

Secondly, we should note that the above discussion only holds for Dirac mass terms, m_D\bar\psi\psi. Majorana mass terms, m_M\psi^\dag\psi, are chirally invariant! This means that they are not protected against radiative corrections. We would hence expect Majorana masses to naturally live at the appropriate cutoff scale. Update (9 Mar 08): This is incorrect! Majorana masses are protected by a chiral symmetry. See my next post.

This has particular relevance in the neutrino seesaw mechanism. In the seesaw one balances a light mass against a very heavy mass. For neutrinos, the light masses come from the chirally protected Dirac mass terms from electroweak symmetry breaking. Observations of neutrino oscillations (and hence estimates of neutrino masses) suggest that the high scale must be very heavy. This is okay since this high scale comes from a Majorana mass term (allowed since the right-handed neutrino is a gauge-singlet), which is unprotected and pushed up to the cutoff scale. Viewed this way, neutrino masses are an indication of an interesting high scale where we expect new physics. Update (9 Mar 08): The reason why the Majorana mass term is large isn’t because it’s not chirally protected, but rather that it is generated by physics at a different scale.

Ok, so we’ve mentioned the scalars and fermions. What about vector bosons? Gauge symmetry helps us here in the same way chiral symmetry did for fermions. One might be clever and ask: what about vector particles that aren’t gauge bosons? Because the gauge symmetry is `constructed’ rather than `inherent,’ we can make any theory of vector bosons gauge-invariant, see Prof. NAH’s lecture, and hence the construction carries over. (This, by the way, is why you always see vectors as gauge particles.)

More generally, the general lesson is that symmetries that are broken in a particular way can still end up being very useful. This is what is meant by the `accidental symmetries’ of the standard model. An example is the U(1)^5 flavour symmetry that is broken by Yukawa couplings. (See this older post, or this newer post.) Higher-dimensional operators can break such symmetries, but they are suppressed by powers of the cutoff.

I’ll make one last note, since I gave my split-SUSY talk recently. For natural theories, the Higgs mass divergence is a problem. Finely-tuned (a.k.a. “split”) supersymmetry turns this problem on its head by using it as a feature. Split-SUSY pushes the SUSY-breaking scale far above the TeV scale, which drags up the masses of the unprotected SUSY scalars. These scalars, which mediate troublesome processes like loop-level proton decay, then decouple from the low energy theory. On the other hand, the SUSY fermions, which give us nice things like a dark matter candidate, are chirally protected and can continue to hang out at the phenomenologically-interesting TeV scale.

[Special thanks to my office-mates Luis and Tracey for chatting to me about this.]

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7 Responses to “Chiral symmetry and Fermion mass non-divergences”

  1. 1 piscator

    On split susy: aren’t the gauginos in susy all Majorana fermions? So what is it that can protect their mass, and why should they all be light?

  2. Ah! Thanks for catching me on a mistake, Piscator! The gauginos are indeed Majorana fermions. They *are* protected by chiral symmetry, I was just sloppy in my statements about Majorana fermions! I’ll correct this shortly. See: http://fliptomato.wordpress.com/2008/03/09/correction-majorana-fermions-and-chiral-symmetry/

  3. Funny post adventure. A further fast thought I got while reading it: Given that the failure of preservation of axial vector current is proportional to the Dirac mass, we could reverse the interpretation and *define* Dirac masses as a measurement of the failure of this preservation.

  4. Hi Alejandro! Since I’ve had a bad track record with quick responses, I’ll not say anything that commits me to a particular (non)factual statement. Do Majorana mass tersm also fail to preserve the axial current? The Majorana mass term is actuall of the form nu^c nu, where the c refers to charge conjugation. There’s a sigma^2 hidden in the charge conjugation matrix, so I’m not sure. (The answer is probably simple, but that’s what I thought about my previous corrections… and subsequent corrections-to-corrections…)

  5. 5 Alejandro Rivero

    In any case, your series on chirality is better than the wikipedia (check the entries for Spinos, Majorana Fermion, and Majorana Equation. No, better dont check them; it is painful, and not only for extra dimensions).
    About majorana mass, the great thing is that it is a mass in any case, because it appears in the equation of Klein-Gordon derived from Majorana Equation. Ex. 3.4(a) in Peskin-Schroeder, and eq 2.31 ff in Zee. There I see that the major source of notational confusion is that a solution of the Majorana’s Equation is not immediate or forcefully a Majorana spinor; the later applies when one requests the solution to be also a solution of Dirac’s Equation.

  6. 6 Alejandro Rivero

    I took a pause to go to the cinema () and I did not answered the question in #4. Worse, I introduced more confusiong by referring to Majorana “equation”. Kind of ashamed I am; so I will also try a non-commitment by appealing to authority. Alvaro de Rujula in http://ccdb4fs.kek.jp/cgi-bin/img/reduced_gif?198109208+4+16 says that “the axial current… becomes a conserved current in the majorana m –> 0 limit”. This limit rescues the conservation of lepton number.


  1. 1 Correction: Majorana fermions and chiral symmetry « An American Physics Student in England

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