Musings

LazyWeb Chiral Symmetry Breaking

This past week, in the Geometry and String Theory Seminar, as part of our series on differential cohomology, Dan gave a talk about his derivation of the Wess-Zumino term in the Chiral Lagrangian. It’s a pretty little application of generalized differential cohomology theories and it has several advantages over, say, Witten’s derivation

1. For a general 4-manifold, $X$, the signature, $\sigma(X)$ is an obstruction to finding a 5-manifold, $B$, such that $X=\partial B$. Witten, of course, worked on $X=S^4$, which has vanishing signature, but this construction clearly doesn’t make sense for general $X$.
2. To define integration in the generalized differential cohomology theory in question (which Dan, unimaginatively, dubs differential “E-theory”) requires a choice of spin structure on $X$. In particular, the requirement that $X$ be a spin manifold, which was clearly present in QCD, but not apparent in previous derivations of the Wess-Zumino term from low-energy considerations, re-emerges here.
3. When quantizing on $X=M_{(3)}\times \mathbb{R}$, the class in differential $E$-theory, whose integral over $X$ is the Wess-Zumino term, gives a natural $\mathbb{Z}/2$ grading to Hilbert space (for $N_c$ odd). This is what we call baryon number (compare the Skyrme model).

Anyway, that wasn’t what I really wanted to talk about.

In the course of a subsequent discussion, the following point arose. The 't Hooft Anomaly Matching Conditions ensure that there must be some light degrees of freedom in the low energy theory on which the chiral symmetries are realized. If we assume that the theory confines, then there are really only two possibilities¹:

1. There are massless composite fermions which transform in appropriate representations to saturate the 't Hooft Anomaly Matching Conditions. 2. The chiral symmetry is spontaneously broken, and is realized nonlinearly on some Goldstone boson fields, whose Wess-Zumino term provides the anomalous variation satisfying the 't Hooft Anomaly Matching constraints.

Clearly, possibility 1 is impossible for $N_c$ even. But, for $N_c$ odd, there are all sorts of oddball special cases to check. John Terning found some oddball solutions. Surely there are others.

There’s a broad class of theories, however, where one can essentially prove that chiral symmetry breaking must take place (as it’s the only way to satisfy the Anomaly Matching constraints). Does anyone know of a systematic statement of what’s known?