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) × UA(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 UA(1) current is a problem (the so-called UA(1) problem): it can be seen that the spontaneous breakdown of the SU(N) × SU(N) implies the same for the UA(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.

Gravity exists, so if there is any truth to supersymmetry then any realistic supersymmetry theory must eventually be enlarged to a supersymmetric theory of matter and gravitation, known as supergravity. Supersymmetry without supergravity is not an option, though it may be a good approximation at energies far below the Planck scale.

There are two leading approaches to the construction of the theory of supergravity. First, supergravity can be presented as a theory of curved superspace. This approach is analogous to the development of supersymmetric gauge theories in Sections 27.1-27.3; the gravitational field appears as a component of a superfield with unphysical as well as physical components, like the unphysical C, M, N, and ω components of the gauge superfield V. The task of deriving the full non-linear supergravity theory in this way is forbiddingly complicated, and so far has not been freed of steps that are apparently arbitrary. At one point or another in the derivation, it has been necessary simply to state that some set of constraints on the graviton superfield are the proper ones to adopt.

Here we will follow a second approach that is less elegant but more transparent. In our discussion here, we begin in Sections 31.1-31.5 with the case where the gravitational field is weak, analyzing supergravity by the same flat-space superfield methods that we used in Chapters 26 and 27 to study ordinary supersymmetry theories.