Bayesian analysis of muon capture on the deuteron in chiral effective field theory

We compute the muon capture on deuteron in the doublet hyperfine state for a variety of nuclear interactions and consistent nuclear currents. Our analysis includes a detailed examination of the theoretical uncertainties coming from different sources: the single-nucleon axial form factor, the truncation of the interaction and current chiral expansion, and the model dependence. Moreover, we study the impact of the use of different power counting scheme for the electroweak currents on the truncation error. To estimate the truncation error of the chiral expansion of interactions and currents we use the most modern techniques based on Bayesian analysis. This method enables us to give a clear statistical interpretation of the computed theoretical uncertainties. Finally, we provide the differential capture rate as function of the kinetic energy of the outgoing neutron which may be measured in future experiments. Our recommended theoretical value for the total doublet capture rate is ${\mathrm{\Gamma }}_{\mathrm{th}}=395\pm 10{\mathrm{s}}^{-1}$ ($68\%$ confidence level). We calculated also the capture rate in the quartet hyperfine state, which turns out to be in the range [13.3–13.8] ${\mathrm{s}}^{-1}$ depending on the adopted nuclear interaction.

Figure 1

The Gaussian process modeling of the current chiral expansion coefficients and its diagnostics for the EMN550 interaction using the Bochum power counting. (a) The simulators (solid lines, i.e., our calculation) along with the corresponding Gaussian process emulators (dashed lines) and their $2\sigma$ intervals (bands). The data used for training are denoted by dots. (b) The Mahalanobis distances compared to the mean (interior line), $50\%$ (box), and $95\%$ (whiskers) credible intervals of the reference distribution. (c) The pivoted Cholesky diagnostics versus the index along with $95\%$ credible intervals (gray lines). (d) The credible interval diagnostics for the truncation error bands. The $1\left(2\right)\sigma$ is represented with the dark (light) gray band. The steepness of the orange line indicates that the N2LO and the N3LO are very close to each other (see text for more details).

Figure 2

Same as Fig. 1 for the EMN550 interaction using the JLab-Pisa power counting.

Figure 3

The differential capture rate as function of the neutron energy $E_1^{\prime }$ computed with the EMN550 interaction fixed at N3LO for various order of the current N2LO (orange) and N3LO (green) using the Bochum power counting. The bands represent the $2\sigma$ truncation errors at each order.

Figure 4

The differential capture rate as function of the neutron energy $E_1^{\prime }$ computed with the EMN550 interaction fixed at N3LO for various order of the current NLO(orange), N2LO (green), and N3LO (blue) using the JLab-Pisa power counting. The bands represent the $2\sigma$ truncation errors at each order.

Figure 5

Same as Fig. 1 but for the truncation errors of the interaction chiral expansion. The results reported in the figure are obtained using the EMN550 interaction fixing the order of the chiral current at N2LO for the interaction at NLO, and at N3LO in the remaining cases.

Figure 6

The differential capture rate as function of the neutron energy $E_1^{\prime }$ computed with the EMN550 interaction at NLO (orange), N2LO (green), and N3LO (blue). The bands represent the $2\sigma$ truncation errors at each order. Note that the chiral order of the current used in this analysis is N2LO when the interaction is at NLO and N3LO in the rest of the cases.

Figure 7

The recommended differential capture rate as function of the kinetic neutron energy $E_1^{\prime }$. The green, blue, and red bands represent respectively the $99\%, 95\%$, and $68\%$ CL. Note that the difference among the bands can be appreciated only at the peak of the spectra.

Figure 8

Comparison of the theoretical results obtained in our work using the Bochum (BPC) and JLab-Pisa (JPPC) power counting with the $\chi \mathrm{EFT}$ results of Refs. [10, 11, 15]. The errors in the previous works have been assumed to be the limit of a uniform distribution and then divided by $\sqrt{3}$ to obtain the $68\%$ CL. Note that only in Ref. [15] and in the present work various sources of uncertainty besides the model dependence were considered. The red dashed line represent the arithmetic mean of all the theoretical results.

Figure 9

Capture rate as function of the value of $c_D$ for the NVIa interaction. The green band represent the error on ${\mathrm{\Gamma }}^{3/2}$ due to the truncation error on the chiral expansion of the current and the interaction. The red dashed lines represent the present error bar on $c_D$. The blue horizontal line is the error band corresponding to an hypothetical result of MuSun ${\mathrm{\Gamma }}^{3/2}=393.5{\mathrm{s}}^{-1}$ with a $1.5\%$ total error.

Figure 10

Capture rate as function of the value of $r_A^2$ for the NVIa interaction. The green band represent the error on $\mathrm{\Gamma }$ due to the truncation error on the chiral expansion of the current and the interaction. The red dashed lines represent the $68\%$ CL on $\mathrm{\Gamma }$ considering the error on $r_A^2$ as reported in Ref. [14]. The blue dashed lines represent the predicted $68\%$ CL on $\mathrm{\Gamma }$ with an error of $10\%$ on $r_A^2$.