13 - Chiral anomalies

Summary

Triangle diagram and different renormalization conditions

Introduction

Anomalies have already been mentioned in this book on several occasions. In this chapter we systematically discuss the fermion anomaly in (3+1)-dimensional quantum field theory (Adler 1969, Bell & Jackiw 1969). Its existence can be traced back to the short-distance singularity structure of products of local operators.

To be specific let us consider QCD. As discussed in Chapter 9, apart from being invariant under gauge transformations in the colour space its lagrangian is also invariant under the global SU(N) × SU(N) × U(1) × U_A(1) chiral group of transformations acting in the flavour space.† Fermions belong to a vector-like, i.e. real, representation of the gauge group and for N > 2 to a complex representation of the chiral group (see Appendix E). We know from Chapters 9 and 10 that the Noether currents corresponding to the global flavour symmetry, although external with respect to the strong interaction gauge group, acquire important dynamical sense. The axial non-abelian currents couple to Goldstone bosons (pseudoscalar mesons) and the left-handed chiral currents couple to the intermediate vector bosons, i.e. they are gauge currents of the weak interaction gauge group. Moreover, the conservation of the U(1) current corresponds to baryon number conservation whereas the conservation of the U_A(1) current is a problem (the so-called U_A(1) problem): it can be seen that the spontaneous breakdown of the SU(N) × SU(N) implies the same for the U_A(1) but there is no good candidate for the corresponding Goldstone boson in the particle spectrum.

It is clear from the preceding discussion that Green's functions involving chiral or axial currents are of direct physical interest.