Even in weakly coupled QFTs, perturbation theory breaks down when one considers amplitudes with a large number $n$ of legs. The series cleverly organizes as a double expansion in $g^2$ and $g^2n$. I show how the series in $g^2n$ can be fully captured by a semiclassical expansion around a non-trivial solution. Focussing on $U(1)$ symmetric $|\phi|^4$ theory in $4$ and $4-\epsilon$ dimension, I derive explicit and consistent all order results for the anomalous dimension of the complex operator $\phi^n$. When restricting to the Wilson-Fisher fixed point and working on the cylinder, the dominant trajectory is seen to correspond to a superfluid phase for the conserved U(1). This creates a correspondence between, on one side, the spectrum of operators and fusion coefficients and and on the other the spectrum of hydrodynamics modes and their interactions. The results also nicely match Monte Carlo simulations in 3D, compatibly with the stunt of taking $\epsilon=1$. [please email alejandro.cabo_bizet@kcl.ac.uk for the Zoom link]

By generalizing the recent proof of the a-theorem, I derive constraints on the possible UV and IR asymptotics of 4D Lorentz invariant unitary quantum field theory. Within perturbation theory the only possible RG flow asymptotic is given by conformal field theory. I 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 definite technical sense.

The hierarchy problem can be represented as a tension between the need for a large cut-off scale suggested, for instance, by flavor physics and the need for a low cut-off scale suggested by naturalness in electroweak symmetry breaking. I will illustrate how this tension could be largely alleviated if the Standard Model flowed to an approximate CFT above the weak scale with a specific relation among the scaling dimensions of the Higgs sector fields. To investigate the viability of that scenario one is led to ask the following simple question: in an arbitrary CFT, given a scalar operator phi, and the operator S=phi phi defined as the lowest dimension scalar S which appears in the OPE phi phi, what is the bound (that is d(S) is smaller than f(d(phi))) on the scaling dimensions of the two operators? I will present a derivation of the bound based on general considerations of OPE, conformal block decomposition, and crossing symmetry. The function f(d(phi)) is computed numerically. When d(phi) goes to 1, one has f(d(phi))=2+O(sqrt(d(phi)-1)), which shows that the free theory limit is approached continuously. An analogous bound can be derived in 2D where some non-trivial consistency check can be made. I will discuss the relevance of the result for the hierarchy problem and illustrate the directions of future investigation.