  Chiral effective field theory ($\chi$EFT), as originally proposed by
Weinberg, promises a theoretical connection between low-energy nuclear
interactions and quantum chromodynamics (QCD). However, the important property
of renormalization-group (RG) invariance is not fulfilled in current
implementations and its consequences for predicting atomic nuclei beyond two-
and three-nucleon systems has remained unknown. In this work we present a first
and systematic study of recent RG-invariant formulations of $\chi$EFT and their
predictions for the binding energies and other observables of selected nuclear
systems with mass-numbers up to $A =16$. Specifically, we have carried out ab
initio no-core shell-model and coupled cluster calculations of the ground-state
energy of $^3$H, $^{3,4}$He, $^{6}$Li, and $^{16}$O using several recent
power-counting (PC) schemes at leading order (LO) and next-to-leading order
(NLO), where the subleading interactions are treated in perturbation theory.
Our calculations indicate that RG-invariant and realistic predictions can be
obtained for nuclei with mass number $A \leq 4$. We find, however, that
$^{16}$O is either unbound with respect to the four $\alpha$-particle
threshold, or deformed, or both. Similarly, we find that the $^{6}$Li
ground-state resides above the $\alpha$-deuteron separation threshold. These
results are in stark contrast with experimental data and point to either
necessary fine-tuning of all relevant counterterms, or that current
state-of-the-art RG-invariant PC schemes at LO in $\chi$EFT lack necessary
diagrams -- such as three-nucleon forces -- to realistically describe nuclei
with mass number $A>4$.
