Energy-momentum tensor from Wilson flow in lattice φ4-theory

Abstract

The energy-momentum tensor (EMT) is the Noether current associated with translations. It is of interest because, first of all, it has physical meaning as it contains the energy density and the momentum density. Moreover, its trace can be related to the beta function so that the scaling behaviour of the theory at hand can be studied. We are particularly interested in the scaling behaviour of strongly coupled theories. To explore the strong coupling regime it is necessary to compute the EMT non-perturbatively, i.e. on the lattice. This complicates matters greatly. On the lattice translation invariance is broken which leads to additional terms in the translation Ward identity from which the EMT is derived. This results in turn in the need to renormalise the EMT on the lattice. In this thesis we extend recent studies on the renormalisation of the EMT in four-dimensional gauge theory to the case of a three-dimensional scalar theory to investigate its divergence structure and the numerical feasibility of the suggested procedure on a more basic level. Furthermore, scalar φ4-theory in three dimensions exhibits an infrared fixed point and can thus serve as a toy model to examine mechanisms for building theories beyond the standard model. Our strategy to renormalise the EMT on the lattice is to identify all possible terms that can mix with both sides of the translation Ward identity. The renormalised EMT is a combination of operators of the same or lower dimension obeying the symmetries of the theory. The mixing is determined by requiring that the renormalised EMT satisfies the correct Ward identities. Using different probes in the translation Ward identity one can compute the coefficients of the EMT by solving a linear system of equations. However, contact terms can arise. One solution is the recently introduced Wilson flow. Its renormalisation properties allow for expectation values free of contact terms. That way the Wilson flow provides for a meaningful theoretical formulation of the EMT on the lattice that can be used in practice. In this thesis we review the renormalisation properties and the phase diagram of scalar φ4-theory in three dimensions, the translation Ward identity and the EMT in the continuum, as well as the gradient flow for scalar theory. A large part is dedicated to the perturbative renormalisation of the EMT on the lattice. Finally, our strategy to compute the renormalisation constants of the EMT in scalar theory non-perturbatively is discussed in detail, and our results for the renormalisation constants are presented.