Bayesian analysis of nucleon-nucleon scattering data in pionless effective field theory

We perform Bayesian model calibration of two-nucleon $(NN)$ low-energy constants (LECs) appearing in an $NN$ interaction based on pionless effective field theory $(\overline{)\ensuremath{\pi}}\mathrm{EFT})$. The calibration is carried out for potentials constructed using naive dimensional analysis in $NN$ relative momenta $(p)$ up to next-to-leading order $[\mathrm{NLO},O({p}^{2})]$ and next-to-next-to-next-to-leading order $[\mathrm{N}3\mathrm{LO},O({p}^{4})]$. We consider two classes of $\overline{)\ensuremath{\pi}}\mathrm{EFT}$ potential: one that acts in all partial waves and another that is dominated by $s$-wave physics. The two classes produce broadly similar results for calibrations to $NN$ data up to ${E}_{\mathrm{lab}}=5$ MeV. Our analysis accounts for the correlated uncertainties that arise from the truncation of the $\overline{)\ensuremath{\pi}}\mathrm{EFT}$. We simultaneously estimate both the EFT LECs and the parameters that quantify the truncation error. This permits quantitative estimates of the $\overline{)\ensuremath{\pi}}\mathrm{EFT}$ breakdown scale, ${\mathrm{\ensuremath{\Lambda}}}_{b}$: the 95% intervals are ${\mathrm{\ensuremath{\Lambda}}}_{b}\ensuremath{\in}[50.11,63.03]$ MeV at NLO and ${\mathrm{\ensuremath{\Lambda}}}_{b}\ensuremath{\in}[72.27,88.54]$ MeV at N3LO. Invoking naive dimensional analysis for the $NN$ potential therefore does not lead to consistent results across orders in $\overline{)\ensuremath{\pi}}\mathrm{EFT}$. This exemplifies the possible use of Bayesian tools to identify inconsistencies in a proposed EFT power counting.