Cutoff effects in Hartree-Fock calculations at leading order of chiral effective field theory

We explore the effects on nuclear bulk properties of using regularization cutoffs larger than the nucleon mass within the chiral effective field theory using a power counting that ensures order-by-order renormalization in the two-nucleon system. To do so we calculate ground-state properties of the $^{16}$O nucleus in the Hartree--Fock approach in a basis made up of plane waves confined in a cube. We find a strong sensitivity to the regularization cutoff through the counter-terms in attractive singular partial waves and to the correction for spurious deeply bound states. This questions the possibility of testing in nuclei the renormalization-group invariance of renormalizable potentials from chiral effective field theory at leading order. A possible way out of this problem is proposed.