Renormalization in tensor field theory and the melonic fixed point

This thesis focuses on renormalization of tensor field theories. Its first part considers a quartic tensor model with $O(N)^3$ symmetry and long-range propagator. The existence of a non-perturbative fixed point in any $d$ at large $N$ is established. We found four lines of fixed points parametrized by the so-called tetrahedral coupling. One of them is infrared attractive, strongly interacting and gives rise to a new kind of CFT, called melonic CFTs which are then studied in more details. We first compute dimensions of bilinears and OPE coefficients at the fixed point which are consistent with a unitary CFT at large $N$. We then compute $1/N$ corrections. At next-to-leading order, the line of fixed points collapses to one fixed point. However, the corrections are complex and unitarity is broken at NLO. Finally, we show that this model respects the $F$-theorem. The next part of the thesis investigates sextic tensor field theories in rank $3$ and $5$. In rank $3$, we found two IR stable real fixed points in short range and a line of IR stable real fixed points in long range. Surprisingly, the only fixed point in rank $5$ is the Gaussian one. For the rank $3$ model, in the short-range case, we still find two IR stable fixed points at NLO. However, in the long-range case, the corrections to the fixed points are non-perturbative and hence unreliable: we found no precursor of the large $N$ fixed point. The last part of the thesis investigates the class of model exhibiting a melonic large $N$ limit. We prove that models with tensors in an irreducible representation of $O(N)$ or $Sp(N)$ in rank $5$ indeed admit a large $N$ limit. This generalization relies on recursive bounds derived from a detailed combinatorial analysis of Feynman graphs involved in the perturbative expansion of our model.