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

Abstract

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.

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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.

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.

Currently, the best theoretical description of fundamental matter and its gravitational interaction is given by the Standard Model (SM) of particle physics and Einstein's theory of General Relativity (GR). These theories contain a number of seemingly unrelated scales. While Newton's gravitational constant and the mass of the Higgs boson are parameters in the classical action, the masses of other elementary particles are due to the electroweak symmetry breaking. Yet other scales, like ΛQCD associated to the strong interaction, only appear after the quantization of the theory. We reevaluate the idea that the fundamental theory of nature may contain no fixed scales and that all observed scales could have a common origin in the spontaneous break-down of exact scale invariance. To this end, we consider a few minimal scale-invariant extensions of GR and the SM, focusing especially on their cosmological phenomenology. In the simplest considered model, scale invariance is achieved through the introduction of a dilaton field. We find that for a large class of potentials, scale invariance is spontaneously broken, leading to induced scales at the classical level. The dilaton is exactly massless and practically decouples from all SM fields. The dynamical break-down of scale invariance automatically provides a mechanism for inflation. Despite exact scale invariance, the theory generally contains a cosmological constant, or, put in other words, flat spacetime need not be a solution. We next replace standard gravity by Unimodular Gravity (UG). This results in the appearance of an arbitrary integration constant in the equations of motion, inducing a run-away potential for the dilaton. As a consequence, the dilaton can play the role of a dynamical dark-energy component. The cosmological phenomenology of the model combining scale invariance and unimodular gravity is studied in detail. We find that the equation of state of the dilaton condensate has to be very close to the one of a cosmological constant. If the spacetime symmetry group of the gravitational action is reduced from the group of all diffeomorphisms (Diff) to the subgroup of transverse diffeomorphisms (TDiff), the metric in general contains a propagating scalar degree of freedom. We show that the replacement of Diff by TDiff makes it possible to construct a scale-invariant theory of gravity and particle physics in which the dilaton appears as a part of the metric. We find the conditions under which such a theory is a viable description of particle physics and in particular reproduces the SM phenomenology. The minimal theory with scale invariance and UG is found to be a particular case of a theory with scale and TDiff invariance. Moreover, cosmological solutions in models based on scale and TDiff invariance turn out to generically be similar to the solutions of the model with UG. In usual quantum field theories, scale invariance is anomalous. This might suggest that results based on classical scale invariance are necessarily spoiled by quantum corrections. We show that this conclusion is not true. Namely, we propose a new renormalization scheme which allows to construct a class of quantum field theories that are scale-invariant to all orders of perturbation theory and where the scale symmetry is spontaneously broken. In this type of theory, all scales, including those related to dimensional transmutation, like ΛQCD, appear as a consequence of the spontaneous break-down of the scale symmetry. The proposed theories are not renormalizable. Nonetheless, they are valid effective theories below a field-dependent cut-off scale. If the scale-invariant renormalization scheme is applied to the presented minimal scale-invariant extensions of GR and the SM, the goal of having a common origin of all scales, spontaneous breaking of scale invariance, is achieved.