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Publication# Numerical investigations of 3D CFTs using conformal bootstrap methods

Abstract

Conformal field theories (CFTs) play a very significant role in modern physics, appearing in such diverse fields as particle physics, condensed matter and statistical physics and in quantum gravity both as the string worldsheet theory and through the AdS/CFT correspondence. In recent years major breakthroughs have been made in solving these CFTs through a method called numerical conformal bootstrap. This method uses consistency conditions on the CFT data in order to find and constrain conformal field theories and obtain precise measurements of physical observables. In this thesis we apply the conformal bootstrap to study among others the O(2)- and the ARP^3- models in 3D.

In the first chapter we extend the conventional scalar numerical conformal bootstrap to a mixed system of correlators involving a scalar field charged under a global U(1) symmetry and the associated conserved spin-1 current J. The inclusion of a conserved spinning operator is an important advance in the numerical bootstrap program. Using numerical bootstrap techniques we obtain bounds on new observables not accessible in the usual scalar bootstrap. Concentrating on the O(2) model we extract rigorous bounds on the three-point function coefficient of two currents and the unique relevant scalar singlet, as well as those of two currents and the stress tensor. Using these results, and comparing with a quantum Monte Carlo simulation of the O(2) model conductivity, we give estimates of the thermal one-point function of the relevant singlet and the stress tensor. We also obtain new bounds on operators in various sectors.

In the second chapter we investigate the existence of a second-order phase transition in the ARP^3 model. This model has a global O(4) symmetry and a discrete Z_2 gauge symmetry. It was shown by a perturbative renormalization group analysis that its Landau-Ginzburg-Wilson effective description does not have any stable fixed point, thus disallowing a second-order phase transition. However, it was also shown that lattice simulations contradict this, finding strong evidence for the existence of a second-order phase transition. In this chapter we apply conformal bootstrap methods to the correlator of four scalars t transforming in the traceless symmetric representation of O(4) in order to investigate the existence of this second order phase transition. We find various features that stand out in the region predicted by the lattice data. Moreover, under reasonable assumptions a candidate island can be isolated. We also apply a mixed t-s bootstrap setup in which this island persists. In addition we study the kink-landscape for all representations appearing in the t times t OPE for general N. Among others, we find a new family of kinks in the upper-bound on the dimension of the first scalar operator in the "Box" and "Hook" representations.

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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.

Our knowledge of our universe is deeply related to our understanding of physics at the sub- atomic scale. Indeed, at its early stage, our universe was so hot and so dense that the only relevant interactions were those between fundamental particles. Therefore, to improve our comprehension of the phenomena that take place over very large scales, we should improve our understanding of the physics at extremely small ones. The so-called Standard Model of particle physics (SM) provides us with a description of these fundamental particles and their interactions. Despite its enormous success, this model fails to explain several experimental evidences. In this thesis, we focus on the following three questions that are not answered by the SM. What is the so-called dark matter that is present in every galaxy? Why is our present universe made of matter with almost no trace of anti-matter? Finally, why can a neutrino of one family transform into a neutrino of another family? In this thesis, we show that the simple addition of three new neutrinos to the Standard Model allows us to answer the three questions above. The model that realizes this scenario is known as Neutrino Minimal Standard Model (νMSM). In this work, we give a comprehensive summary of all known constraints in the νMSM. We present the first complete quantitative study of the parameter space of the model where no physics beyond the νMSM is needed to simultaneously explain the three phenomena cited above. Moreover, these new particles can be looked for using current day experimental techniques and our results provide a guideline for future experimental searches.