  The Loop Vertex Expansion (LVE) is a constructive technique using canonical
combinatorial tools. It works well for quantum field theories without
renormalization, which is the case of the field theory studied in this paper.
Tensorial Group Field Theories (TGFT) are a new class of field theories
proposed to quantize gravity. This paper is devoted to a very simple TGFT for
rank three tensors with U(1)group and quartic interactions, hence nicknamed
$U(1)$-$T^4_3$. It has no ultraviolet divergence, and we show, with the LVE,
that it is Borel summable in its coupling constant.
