Morris, Tim R.
(2021)
The continuum limit of the conformal sector at second order in perturbation theory.
*Physical Review D*.
(In Press)

## Abstract

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

**The continuum limit of the conformal sector at second order in perturbation theory - Accepted Manuscript**

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