Continuum limit of the conformal sector at second order in perturbation theory

Recently a novel perturbative continuum limit for quantum gravity has been proposed and demonstrated to work at first order. Every interaction monomial $\sigma$ is dressed with coefficient function $f^\sigma_\Lambda(\varphi)$ of the conformal factor field, $\varphi$. Each parametrised by an infinite number underlying couplings, decays large $\varphi$ characteristic amplitude suppression scale which can be chosen common value, $\Lambda_\text{p}$. Although theory in couplings it non-perturbative $\hbar$. At second order perturbation theory, one must sum over all melonic Feynman diagrams obtain particular integral. We show that leads well defined renormalized trajectory thus limit, provided solved starting arbitrary cutoff $\Lambda=\mu$ lies range $0<\mu<a\Lambda_\text{p}$ ($a$ some non-universal number). If $\mu$ above this resulting functions become singular, flow ceases exist, before physical reached. To add well-behaved complementary solution, containing irrelevant determined uniquely first-order interactions, relevant couplings. Even though diverge $\Lambda_\text{p}\to\infty$, domains such diffeomorphism invariance will recovered where disappear replaced effective invariant