Consistent, high-quality two-nucleon potentials up to fifth order of the chiral expansion

We present $NN$ potentials through five orders of chiral effective field theory ranging from leading order (LO) to next-to-next-to-next-to-next-to-leading order (N$^4$LO). The construction is consistent in the sense that the same power counting scheme as well as the same cutoff procedures are applied in all orders. Moreover, the long-range parts of these potentials are fixed by the very accurate $\pi N$ LECs as determined in the Roy-Steiner equations analysis by Hoferichter, Ruiz de Elvira and coworkers. In fact, the uncertainties of these LECs are so small that a variation within the errors leads to effects that are essentially negligible, reducing the error budget of predictions considerably. The $NN$ potentials are fit to the world $NN$ data below pion-production threshold of the year of 2016. The potential of the highest order (N$^4$LO) reproduces the world $NN$ data with the outstanding $\chi^2$/datum of 1.15, which is the highest precision ever accomplished for any chiral $NN$ potential to date. The $NN$ potentials presented may serve as a solid basis for systematic {\it ab initio} calculations of nuclear structure and reactions that allow for a comprehensive error analysis.