The local Callan-Symanzik equation: structure and applications

The local Callan-Symanzik equation describes the response of a quantum field theory to local scale transformations in the presence of background sources. The consistency conditions associated with this anomalous equation imply non-trivial relations among the beta-function, the anomalous dimensions of composite operators and the short distance singularities of correlators. In this paper we discuss various aspects of the local Callan-Symanzik equation and present new results regarding the structure of its anomaly. We then use the equation to systematically write the n-point correlators involving the trace of the energy-momentum tensor. We use the latter result to give a fully detailed proof that the UV and IR asymptotics in a neighbourhood of a 4D CFT must also correspond to CFTs. We also clarify the relation between the matrix entering the gradient flow formula for the beta-function and a manifestly positive metric in coupling space associated with matrix elements of the trace of the energy momentum tensor.

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

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-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

Electrical conductivity of hot Abelian plasma with scalar charge carriers

Oleksandr Sobol

We study the electrical conductivity of hot Abelian plasma containing scalar charge carriers in the leading logarithmic order in coupling constant alpha using the Boltzmann kinetic equation. The leading contribution to the collision integral is due to the Moller and Bhabha scattering of scalar particles with a singular cross section in the region of small momentum transfer. Regularizing this singularity by taking into account the hard thermal loop corrections to the propagators of intermediate particles, we derive the second order differential equation which determines the kinetic function. We solve this equation numerically and also use a variational approach in order to find a simple analytical formula for the conductivity. It has the standard parametric dependence on the coupling constant sigma approximate to 2.38T/(alpha log alpha(-1)) with the prefactor taking a somewhat lower value compared to the fermionic case. Finally, we consider the general case of hot Abelian plasma with an arbitrary number of scalar and fermionic particle species and derive the simple analytical formula for its conductivity.

2019

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

Lorenzo Vitale

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EPFL2016