Composite Operators and Mixing

A local operator is not renormalized merely because the fields and couplings in the action have been renormalized. Bringing fields to the same point creates new short-distance singularities; bringing two already-renormalized insertions together creates further contact singularities. The resulting operators generally form a mixing system, and the scale dependence of that system must be compensated by the opposite evolution of any Wilson coefficients multiplying it.

Enter this chapter

Begin with Renormalized Composite-Operator Insertions to understand why local insertions need additional counterterms.

Topics

1. Renormalized Composite-Operator Insertions — Renormalizes a marked local vertex beyond ordinary fields and couplings.
2. Contact Terms and Renormalized Operator Products — Accounts for extra singularities when two insertions coincide.
3. Operator Mixing and Renormalization Matrices — Finds operator sectors closed under ultraviolet subtraction.
4. Operator Anomalous-Dimension Matrices — Derives scale evolution from the operator renormalization matrix.
5. Dual Evolution of Operators and Wilson Coefficients — Makes coefficient evolution compensate operator mixing.
6. Symmetry-Protected Operators, Currents, and Improvement — Separates conserved-charge normalization from allowed current improvements.
7. Nonperturbative Renormalization Schemes and Step Scaling — Defines normalization and finite scale changes without a loop expansion.

How the ideas fit together

Mixing and coefficient evolution are two sides of one scale-independent interaction. If the renormalized operator column obeys $\mu\,dO/d\mu=-\gamma O$, the coefficient column must obey $\mu\,dC/d\mu=\gamma^{\mathsf T}C$ so that $C^{\mathsf T}O$ is unchanged. This cancellation requires a closed sector: off-shell subtraction can need equation-of-motion or BRST-exact operators even when their physical matrix elements vanish. Renormalizing each insertion also leaves a separate problem at coincident points, where contact counterterms are needed. Collins 1984/2023, ch. 6 and § 7.12.

Where to go next

Continue to Renormalization-Group Equations and Running to derive running from independence of the bare theory from the subtraction scale.

References

• Collins, John C. 1984; open-access reissue 2023. Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion. Cambridge Monographs on Mathematical Physics. Cambridge University Press. DOI and Open PDF.