The alpha-theorem and the asymptotics of 4D quantum field theory

We study the possible IR and UV asymptotics of 4D Lorentz invariant unitary quantum field theory. Our main tool is a generalization of the Komargodski-Schwimmer proof for the alpha-theorem. We use this to rule out a large class of renormalization group flows that do not asymptote to conformal field theories in the UV and IR. We show that the only possible UV and IR asymptotics described by perturbation theory have a vanishing trace of the stress-energy tensor, and are therefore conformal. Our arguments hold even for theories with gravitational anomalies. We also give a non-perturbative argument that excludes theories with scale but not conformal invariance. This argument holds for theories in which the stress-energy tensor is sufficiently nontrivial in a technical sense that we make precise.

Perturbative and non-perturbative aspects of RG flows in quantum field theory

Lorenzo Vitale

This thesis explores two aspects of the renormalization group (RG) in quantum field theory (QFT). In the first part we study the structure of RG flows in general Poincaré-invariant, unitary QFTs, and in particular the irreversibility properties and the relation between scale and conformal invariance. Within the formalism of the local Callan--Symanzik equation, we derive a series of results in four and six-dimensional QFTs. Specifically, in the four dimensional case we revisit and complete existing proofs of the

a

-theorem and of the equivalence between scale and conformal invariance in perturbation theory. We then present an original derivation of similar results in six-dimensional QFTs. In the second part we present the Hamiltonian Truncation method and study its applicability to the numerical solution of non-perturbative RG flows. We test the method in the Phi^4 model in two dimensions and show how it can be used to make quantitative predictions for the low-energy observables. In particular, we calculate the numerical spectrum and estimate the critical coupling at which the theory becomes conformal. We also compare our results to previous estimates. The main original ingredient of our analysis is an analytic renormalization procedure used to improve the numerical convergence. We then adapt the method in order to treat the strongly-coupled regime of the model where the Z2 symmetry is spontaneously broken. We reproduce perturbative and non-perturbative observables and compare our results with analytical predictions.

EPFL2016

Constraints on perturbative RG flows in six dimensions

Lorenzo Vitale

When conformal field theories (CFTs) are perturbed by marginally relevant deformations, renormalization group (RG) flows ensue that can be studied with perturbative methods, at least as long as they remain close to the original CFT. In this work we study such RG flows in the vicinity of six-dimensional unitary CFTs. Neglecting effects of scalar operators of dimension two and four, we use Weyl consistency conditions to prove the alpha-theorem in perturbation theory, and establish that scale implies conformal invariance. We identify a quantity that monotonically decreases in the flow to the infrared due to unitarity, showing that it does not agree with the one studied recently in the literature on the six-dimensional O-3 theory.

Springer2016

Studies in strongly coupled quantum field theories and renormalization group flows

Bernardo Zan

This thesis presents studies in strongly coupled Renormalization Group (RG) flows. In the first part, we analyze the subject of non-local Conformal Field Theories (CFTs), arising as continuous phase transitions of statistical models with long-range interactions. Specifically, we study the critical long-range Ising model in a general number of dimension: first we show that it is conformally invariant, and then we study in depth the different regimes of the theory. We find an example of an infrared duality, to our knowledge the first non-local example of such phenomenon. The second part of the thesis deals with walking theories and weakly first order phase transi- tions, meaning Quantum Field Theories that show approximate scale invariance over a range of energies, in a general number of dimensions. We discuss several example in the high energy as well as the statistical mechanics literature, and show that these theories can be understood as an RG flow passing between two complex CFTs, i.e. non-unitary theories living at complex values of the couplings. Combining the conformal data of these complex CFTs and conformal perturbation theory, we describe observables of the walking theory. Finally, we give the explicit example of the two dimensional Potts model with more than four states.

EPFL2019

Perturbative and non-perturbative aspects of RG flows in quantum field theory

Lorenzo Vitale

a

EPFL2016