• asymptotic safety
• Yang-Mills theory
• lattice
• simulations
• monte carlo methods
• resummation

Perturbatively renormalizable and asymptotic free QFTs have become the basic ingredient for the construction of fundamental theories. Since the QFT of E.H. gravity is perturbatively non-renormalizable, the general understanding is that it can be interpreted only as an EFT.
However Weinberg pointed out that for a QFT to be a complete theory it is sufficient to satisfy the condition of Asymptotic Safety, that is essentially based on the existence of an UV fixed point of the RG-flow in the space of couplings. If gravity satisfies this condition of non-perturbative renormalizability, it can be expected to be a well defined theory at all energy scales, but there is no clear evidences so far.
Given the importance of this topic, we have decided to study the SU(2) Yang-Mills theory in 5 dimensions, that is simpler than gravity, but has elements in common with it.
The aim of this thesis is to investigate if this theory has a non-gaussian UV fixed point in 5 or more dimensions. To do this, we need information about the non-perturbative behavior of the theory. Firstly, we tried to extrapolate this information through the Resummation technique. The idea is to resum the perturbative expansion of the 4 dimensional beta function and analytically continue it in D = 4 + 2ϵ. The results obtained seem to confirm the presence of the fixed point in 5 dimensions, the systematic errors do not allow us to make strong statements.
Therefore, we have decided to face the problem from a non-perturbative point of view through lattice simulations. The phase diagram of the SU(2) lattice gauge theory with fundamental plus adjoint Wilson’s action is characterized by a line of first order transition dividing two phases, but no critical point. We have estimated the position of this line and investigated the possibility of a critical point located at infinity studying the scaling of the Creutz ratios. From our analysis we can exclude the existence of a critical point for the SU(2) lattice gauge with fundamental plus adjoint action, even if it is possible that another action intersects the critical surface for some values of its couplings. However we have seriously questioned the existence of a non-gaussian UV fixed point for the 5 dimensional SU(2) Yang-Mills theory and excluded a lattice action for possible further studies.