# Counterterms in Truncated Conformal Perturbation Theory

## TL;DR

This paper explores the Hamiltonian approach to renormalization in deformed conformal field theories, matching conformal perturbation theory results and analyzing counterterms in the truncated conformal space approach, with applications to $^4$ theories.

## Contribution

It provides an explicit Hamiltonian framework for perturbative renormalization, matching conformal perturbation theory up to third order and analyzing counterterms in the truncated conformal space approach.

## Key findings

- Explicit match between Hamiltonian and conformal perturbation theory up to third order.
- Identification of non-covariant and non-local counterterms in the truncated conformal space approach.
- Proposal of a smooth cutoff method to handle subleading oscillations.

## Abstract

We investigate the perturbative renormalisation of deformed conformal field theories from the Hamiltonian perspective. We discuss the relation with conformal perturbation theory, to which we provide an explicit match up to third order in the coupling, and show how second-order anomalous dimensions in the Wilson-Fisher fixed points are straightforwardly computed in the Hamiltonian framework. The second part of the paper focuses on the cutoff employed in the truncated conformal space approach of Yurov and Zamolodchikov. We discuss the appearance of non-covariant and non-local counterterms to second order in the cutoff, which we concretise in the $\phi^4$ theories, and find a smooth cutoff to deal with subleading oscillations.