Renormalization in tensor field theory and the melonic fixed point

This thesis focuses on the study of the renormalization group flow in 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 \textit{tetrahedral} coupling. One of them is infrared attractive, strongly interacting and gives rise to a new kind of conformal field theories, called melonic CFTs. This melonic CFT is then studied in more details. We first compute dimensions of bilinears and operator product expansion coefficients at the fixed point. The results are consistent with a unitary CFT at large $N$. We then compute $1/N$ corrections to the fixed point. 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 next-to-leading order. Finally, the $F$-theorem is investigated for this model. This theorem states that the free energy of a CFT on the sphere in dimension $3$ decreases along the renormalization group flow. We show that our model respects this theorem. The next part of the thesis investigates sextic tensor field theories in rank $3$ and $5$. In rank $3$, we found two infrared stable real fixed points in short range and a line of infrared 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 infrared stable fixed points at next-to-leading order. 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. Indeed, this limit was lacking for models with ordinary tensor representations of $O(N)$ and $Sp(N)$, such as symmetric traceless or antisymmetric ones. Recently, it was proven that models with tensors in an irreducible representation of $O(N)$ or $Sp(N)$ in rank $3$ indeed admit a large $N$ limit. This proof is here extended in rank $5$. This generalization relies on recursive bounds derived from a detailed combinatorial analysis of Feynman graphs involved in the perturbative expansion of our model