# What really enforces technical naturalness of electron mass?

Technical or 't Hooft naturalness A parameter $$\theta$$ in the Lagrangian of a field theory is said to be natural, if in the limit of vanishing $$\theta$$, the theory has some enhanced symmetry. If this happens, the smallness of the parameter $$\theta$$ is said to be natural.

An example Let us consider the theory of QED with massless electrons. It can be shown that the electron mass does not receive quantum corrections and remains zero. If we repeat the same calculation with massive QED, we find that the bare electron mass $$m_0$$ receives a correction which itself is proportional to $$m_0$$ i.e. $$m_0\to m=m_0+\frac{3\alpha}{4\pi}m_0\ln\Big(\frac{\Lambda^2}{m_0^2}\Big)\tag{1}$$ where $$\Lambda$$ is the cut-off. Thus, massive QED reproduces the result of massless QED in the limit $$m_0\to 0$$. This clearly shows that if $$m_0$$ is zero in the classical action to start with it will remain zero; if $$m_0$$ is nonzero but small to start with, it will remain small (or is it?). In this sense, the smallness of electron mass it technically natural.

Question But I am still uncomfortable with the role of symmetry here and cannot fully digest the idea of technical naturalness. Because if the symmetry is anomalous in the limit $$m_0\to 0$$, what is it that stabilizes the electron mass against large quantum corrections? Since the symmetry is anomalous, we cannot say for sure that it is the symmetry that protects $$m_0$$ from receiving large correction. What is really going on at the heart of the matter?

To forbid this mass term, we do not need a full chiral U(1) symmetry, but only a $$\mathbb{Z}_2$$ symmetry under which, say, the left-handed electron picks up a sign. This $$\mathbb{Z}_2$$ is anomaly free and can thus forbid the mass term also at the quantum level.

• Instanton contributions can break the Z2 and U(1) symmetry, see journals.aps.org/prl/abstract/10.1103/PhysRevLett.61.794 (although not in QED, since the gauge group is abelian). – thedoctar Apr 17 at 12:33
• @thedoctar I am aware that discrete symmetries can have anomalies, but as you agree this one doesn't. – user178876 Apr 17 at 13:51
• So how do I see that the $Z_2$ axial symmetry is anomaly free? – Thomas Apr 18 at 18:18
• @Thomas AFAIK the first paper on this is by Krauss & Wilczek, then there were papers by Ibañez and Ross, by Banks and Dine and subsequently several others who derived the anomaly constraints for discrete symmetries. The arguably cleanest derivation is based on the path integral method, but this has been done much later. Here it is sort of pointless since the U(1)$_\mathrm{em}$ does not have instantons. – user178876 Apr 18 at 20:36

In a general chiral gauge theory in D-dimensions, the anomaly only appears in the one-loop diagram with (D-1) external gauge bosons (arXiv:0802.0634).

The chiral anomaly you are talking about only appears in the triangle diagram (in 4 dimensions), i.e. 3 external photons with an electron loop. Adler, Bell and Jackiw proved this perturbatively, but a more modern treatment can be found in arXiv:0802.0634.

So the loop corrections to the electron mass don't break the chiral symmetry. Only the triangle diagram does.

The non-conservation of left- and right-handed fermions only happens with non-zero electric/magnetic fields, since the divergence of the chiral current is proportional to the field-strength tensor.

See chapter 19.2 of Peskin & Schroeder, as well as problem 19.1.

I'd also recommend following Fukikawa's analysis, where the anomaly comes from the Jacobian of the field measure in the path-integral. Also done in P&S.

EDIT: I've neglected non-perturbative effects in this answer (which is okay for the photon, since abelian gauge groups have no topological configurations). See the comments section.

It's not just the chiral anomaly that can muddle the argument that chiral symmetry protects masslessness. If we start with massless quarks the strong interactions would still cause chiral symmetry breaking and give the quarks a "constituent" mass. Indeed in the real world the "current" masses of the the $$u$$ and $$d$$ quarks are rather small and most of the masses of the hadrons containing them comes from chiral symmetry breaking and so are proportional to $$\Lambda_{\rm QCD}$$ rather than $$m_u$$ or $$m_d$$.
• @thedoctor Are you thinking of the $\eta'$ mass? -- or is there a machanism that gives a mass directly to the fermions? – mike stone Apr 17 at 12:40