Functional renormalization group for the $U(1)\text{−}{T}_5^6$ tensorial group field theory with closure constraint

This paper is focused on the functional renormalization group applied to the $T_5^6$ tensor model on the Abelian group $U(1)$ with closure constraint. For the first time, we derive the flow equations for the couplings and mass parameters in a suitable truncation around the marginal interactions with respect to the perturbative power counting. For the second time, we study the behavior around the Gaussian fixed point, and show that the theory is nonasymptotically free. Finally, we discuss the UV completion of the theory. We show the existence of several nontrivial fixed points, study the behavior of the renormalization group flow around them, and point out evidence in favor of an asymptotically safe theory.

Figure 1

An example of bubble of order 4.

Figure 2

Typical contribution to the two-points observable. The dashed line represents the contraction with the propagator ${\partial }_k{R}_k$.

Figure 3

Contributions coming from the six-points interactions to the four-point interaction.

Figure 4

Contribution to the four-point interaction involved two vertices.

Figure 5

Contribution to the flow of ${\lambda }_3$.

Figure 6

Contribution to the flow of ${\lambda }_2$.

Figure 7

Phase portrait in the plans $({\overline{\lambda }}_1,{\overline{\lambda }}_3)$ for ${\overline{\lambda }}_2=0$ (on the left) and in the plan $({\overline{\lambda }}_1,{\overline{\lambda }}_2)$, for ${\overline{\lambda }}_3=0$.

Figure 8

Qualitative behavior of the RG trajectories around an IR fixed point. The critical surface is spanned by the relevant directions in the IR, and the arrows are oriented toward the IR direction. This illustrates the scenario of asymptotically safety.

Figure 9

Graphical representation of the vertices of valence 8.