nLab counterterm

Idea

In perturbative quantum field theory one way to construct an S-matrix $\mathcal{S}$ via ("re"-)normalization of time-ordered products/Feynman amplitudes is to consider a sequence of UV cutoffs $\Lambda$ with corresponding effective S-matrices $\mathcal{S}_\Lambda$ and then form the limit as $\Lambda \to \infty$, which exists if in the course one applies suitable interaction vertex redefinitions $\mathcal{Z}_\Lambda$

$$\mathcal{S}(g S_{int} + j A) = \underset{\Lambda \to \infty}{\lim} (\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda)(g S_{int} + j A)$$

(See at effective QFT this prop.).

This may be read as saying that as one “removes the UV-cutoff” the original interaction action functional $g S_{int} + j A$ is to be corrected by “counterterm-interactions”

$$S_{counter, \Lambda} \;\coloneqq\: \left( \mathcal{Z}_\Lambda - id \right) \left( g S_{int} + j A \right) \,.$$

Details

See at effective action around this remark.

References

The rigorous formulation of ("re"-)normalization via UV cutoff and counterterms in causal perturbation theory/perturbative QFT is due to

• Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen, section 5.2 of Perturbative Algebraic Quantum Field Theory and the Renormalization Groups, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (arXiv:0901.2038)

• Michael Dütsch, Connection between the renormalization groups of Stückelberg-Petermann and Wilson, Confluentes Mathematici, Vol. 4, No. 1 (2012) 12400014 (arXiv:1012.5604)

• Michael Dütsch, Klaus Fredenhagen, Kai Keller, Katarzyna Rejzner, appendix A of Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization, J. Math. Phy. 55(12), 122303 (2014) (arXiv:1311.5424)

reviewed in

• Michael Dütsch, section 3.8 of From classical field theory to perturbative quantum field theory, 2018

See also

• Kevin Costello, section 8 Renormalisation and the Batalin-Vilkovisky formalism (arXiv:0706.1533)