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Publication# Semiclassical methods in conformal field theories scrutinized by the epsilon-expansion

Résumé

Conformal Field Theories (CFTs) are crucial for our understanding of Quantum Field Theory (QFT). Because of their powerful symmetry properties, they play the role of signposts in the space of QFTs. Any method that gives us information about their structure, and lets us compute their observables, is therefore of great interest. In this thesis we explore the large quantum number sector of CFTs, by describing a semiclassical expansion approach. The idea is to describe the theory in terms of fluctuations around a classical background, which corresponds to a superfluid state of finite charge density. We detail the implementation of the method in the case of U (1)-invariant lagrangian CFTs defined in the epsilon-expansion. After introducing the method for generic correlators, we illustrate it by performing the computation of several observables.First, we compute the scaling dimension of the lowest operator having a given large charge n under the U (1) symmetry. We demonstrate how the semiclassical result in this case bridges the gap between the naive diagrammatic computation (which fails at too large n) and the general large-charge expansion of CFTs (which is only valid for n large enough).Second, we apply the method to the computation of 3- and 4-point functions involving the same operator. This lets us derive some of the OPE (Operator Product Expansion) coefficients.Finally, we consider the rest of the spectrum of charge-n operators, and propose a way to classify them by studying their free-theory equivalent. In the free theory, we construct the complete set of primary operators with number of derivatives bounded by the charge.We also find a mapping between the excited states of the superfluid and the vacuum states of standard quantization, which is valid when the spin of said states is bounded by the square root of the charge.

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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.

The thesis is dedicated to two groups of questions arising in modern particle physics and cosmology. The first group concerns with the problem of stability of the electroweak (EW) vacuum in different environments. Due to its phenomenological significance, the problem attracts high attention in recent research. We contribute to this research in two directions.
First, we study decay rate of the EW vacuum at the inflationary stage of the universe. While in a low density, low temperature environment characteristic of the present-day universe the Standard Model EW vacuum is safely long-lived, the situation may be different during inflation. We estimate tunneling transition via Coleman-De Luccia instanton in this case and confirm that it is exponentially suppressed, contrary to the claims made in the literature.
Second, we compute the lifetime of the EW vacuum in a scale-invariant extension of the Standard Model and gravity, known as the Higgs-Dilaton theory. The theory passes phenomenological tests and provides us with a plausible cosmological scenario. To confirm its viability, it is necessary to check if the EW vacuum in this theory is sufficiently safe. We perform this check and find that features of the Higgs-Dilaton theory yield additional stabilization of the low-energy vacuum, compared to the Standard Model case.
Another group of questions addressed in the thesis is related to the hierarchy problem. Combining quantum scale invariance with the absence of new degrees of freedom above the EW scale leads to stability of the latter against perturbative quantum corrections. Nevertheless, the hierarchy between the weak and the Planck scales remains unexplained. We suggest that this hierarchy can be a manifestation of a non-perturbative effect relating low-energy and strong-gravity domains of the theory. To support this suggestion, we construct instanton configurations and investigate their contribution to the vacuum expectation value of the Higgs field.
The effect we find relies on properties of the theory in the ultraviolet regime. Non-minimal coupling of the Higgs field to the Ricci scalar and an approximate Weyl invariance of the theory in this regime are important ingredients of the mechanism. Dynamical gravity plays a crucial role in the effect as it leads to existence of instanton solutions suitable for generating the EW scale.

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.