To study the dependence of our predictions on the regulator value Λ, we choose the five-fold differential cross sections ( d 5 σ d Ω 1 d Ω 2 d S ) 400 and ( d 5 σ d Ω 1 d Ω 2 d S ) 550 obtained with Λ = 400 MeV and Λ = 550 MeV, respectively. Having them at our disposal, we construct (ϕ ₁ = 0°) δ 400 − 550 θ 1 , θ 2 , ϕ 2 , S ≡ d 5 σ d Ω 1 d Ω 2 d S 400 − d 5 σ d Ω 1 d Ω 2 d S 550 1 2 d 5 σ d Ω 1 d Ω 2 d S 400 + d 5 σ d Ω 1 d Ω 2 d S 550 (5) and then, for given θ ₁ and θ ₂, we find its maximum over the remaining variables Δ 400 − 550 ≡ Δ 400 − 550 θ 1 , θ 2 ≡ max ϕ 2 , S δ 400 − 550 θ 1 , θ 2 , ϕ 2 , S . (6) Here and in the following, the calculations are performed with the NN interaction at the highest available order, that is, at N⁴LO⁺, supplemented by the 3NF at N²LO.

The resulting Δ^400−550 at E = 200 MeV is shown in Figure 1A. The white area shows the kinematically forbidden region. The symmetry with respect to the diagonal, as shown in Figure 1, reflects the fact that two nucleons (1 and 2) are indistinguishable. Some small deviations from symmetry seen in the figure are due to the finite grid of points on the S-curves used in the calculations. When interpreting the results obtained for Δ^400−550, it is important to keep in mind that the differences have already been maximized with respect to the not explicitly shown kinematic variables ϕ ₂ and S. This implies that the actual residual cutoff dependence of the calculated cross sections, encoded in the quantity δ ^400−550, is on average much smaller than the deviations shown in Figure 1 for Δ^400−550. The same comment applies to the results shown for the dependence on the chiral order and the effects of the 3NF.

We observe that Δ^400−550 is spread over the range (−0.440, 0.260), so the two regulator values can yield predictions that diverge by more than 30%. In the majority of the available area in the (θ ₁ and θ ₂) subspace Δ^400−550 ∈ (−30% − 5%), the maximal Δ^400−550 values are concentrated at small θ ₁ and θ ₂ angles. It is interesting to note that the lowest Δ^400−550 values also occur at relatively small polar angles. The particular configurations with extreme Δ^400−550 in these regions are related to the FSI.

Among the configurations leading to Δ^400−550 shown in Figure 1A, there are very likely some with very small cross sections, which can result in large Δ^400−550. Such configurations are not of interest when planning feasible measurements. Thus, in Figure 1B, we show the same Δ^400−550 but after imposing additional conditions on the cross sections: ( d 5 σ d Ω 1 d Ω 2 d S ) 400 > 0.01 [mb sr⁻² MeV⁻¹] and ( d 5 σ d Ω 1 d Ω 2 d S ) 550 > 0.01 [mb sr⁻² MeV⁻¹] and on the energies: E ₁ > 10 MeV and E ₂ > 10 MeV. Notably, these conditions remove many configurations, producing more white space in the graph, and Δ^400−550 decreases on average. While the spread of the Δ^400−550 variation remains nearly unchanged, the distribution of the Δ^400−550 values changes, resulting in, on average, smaller in magnitude values of Δ^400−550. In particular, the configurations with |Δ^400−550| ≥ 0.289 occupy less than 1% of the θ ₁–θ ₂ phase space shown.

To display the full information on the specific configurations leading to the Δ^400−550 values shown in Figure 1B, Figure 2 shows the corresponding energy E ₁ and the relative azimuthal angle ϕ ₁₂, which by our choice of ϕ ₁ = 0° is equivalent to ϕ ₂, as functions of the polar angles θ ₁ and θ ₂. The analogous figure not shown for E ₂ is a mirror image of that for E ₁ with respect to the diagonal θ ₁ = θ ₂. In most cases, E ₁ takes on small (below approximately 30 MeV) or large (above approximately 160 MeV) values, and only in about a quarter of configurations, do the largest Δ^400−550 occur at intermediate E ₁ energies. The ϕ ₁₂ angles corresponding to the most pronounced Δ^400−550 are usually above approximately 130°; however, for some combinations (θ ₁ and θ ₂), the smallest ϕ ₁₂ is preferred.

Figure 3 shows the differential cross section d 5 σ d Ω 1 d Ω 2 d S as a function of the arc-length S for three configurations that maximize Δ^400−550 at one S point. The left panel shows a configuration where both θ _i are small (θ ₁ = 12.5° and θ ₂ = 7.5°), while the central panel shows the case where θ ₁ = θ ₂ = 27.5°. For the first configuration, the maximal δ ^400−550(θ ₁, θ ₂, ϕ ₂, S) occurs in the maximum of the FSI peak at S ≈ 134 MeV. In this case, the softer interaction yields a larger cross section. On the contrary, in the case of the configuration shown in Figure 3B, predictions based on the SMS potential with Λ = 400 MeV are smaller around S = 111 MeV than those with the cutoff Λ = 550 MeV. Finally, in Figure 3C, we give an example of the configuration from a different position in the θ ₁ − θ ₂ plane, namely, for large θ ₁ and small θ ₂. Here, again, the cross section resulting from the interaction at Λ = 400 MeV exceeds the one obtained at Λ = 550 MeV, leading to δ ^400−550(172.5°, 2.5°, 177.5°, 92 MeV) = 0.128.

Similar to the previously defined Δ^400−550, we also studied Δ N2LO−N4LO+ ≡ Δ N2LO−N4LO+ θ 1 , θ 2 ≡ max ϕ 2 , S δ N2LO−N4LO+ θ 1 , θ 2 , ϕ 2 , S (7) with δ N2LO−N4LO+ θ 1 , θ 2 , ϕ 2 , S ≡ d 5 σ d Ω 1 d Ω 2 d S N2LO − d 5 σ d Ω 1 d Ω 2 d S N4LO+ 1 2 d 5 σ d Ω 1 d Ω 2 d S N2LO + d 5 σ d Ω 1 d Ω 2 d S N4LO+ (8) and where ( d 5 σ d Ω 1 d Ω 2 d S ) N2LO ( d 5 σ d Ω 1 d Ω 2 d S ) N4LO+ are the differential cross sections obtained with the NN force at N²LO (N⁴LO⁺) supplemented in both cases by the 3NF at N²LO. The regulator value Λ = 450 MeV is used. The free parameters of the 3NF, c _D and c _E , were determined separately when the 3NF was combined with the N²LO or N⁴LO⁺ NN force, in such a way that the ³H binding energy [47] and the differential cross section for the neutron-deuteron elastic scattering data [48] are properly described. The definition (8) shows that both δ ^N2LO−N4LO+ (θ ₁, θ ₂, ϕ ₂, E ₁) and subsequently Δ ^N2LO−N4LO+ carry partial information about the convergence of the results to the chiral order.

Figure 4A shows that there are big differences between the predictions based on both combinations of two- and three-nucleon forces in the whole available kinematic area when no additional constraints are imposed on the cross sections and nucleon energies. Δ ^N2LO−N4LO+ varies in the range (−200%, 200%). The picture is not as symmetric as in Figure 1A. This shows that in many cases, the magnitude of Δ ^N2LO−N4LO+ depends on a very precise position on the S-curve, which is understandable when we deal with small values of the cross sections. In fact, once the additional conditions on the cross sections and energies are applied, the symmetry is restored, as shown in Figure 4B. The values of Δ ^N2LO−N4LO+ are now substantially smaller and range approximately from −30%, 30%. The inclusion of the higher chiral order contributions in the NN interaction decreases the breakup cross section, which leads to positive Δ ^N2LO−N4LO+ , in configurations where one of the θ _i is small (below 20°) and the second θ _j takes intermediate values in the range θ _i ∈ (60°, 120°). In the other parts of the allowed θ ₁ and θ ₂ space, the negative Δ ^N2LO−N4LO+ values prevail. Specifically, for one of the θ _i in the range (30°, 60°) and another one in the range (30°, 90°), Δ ^N2LO−N4LO+ takes the smallest values corresponding to ( d 5 σ d Ω 1 d Ω 2 d S ) N4LO+ > ( d 5 σ d Ω 1 d Ω 2 d S ) N2LO .

Figure 5 shows E ₁ and ϕ ₁₂ for configurations maximizing Δ ^N2LO−N4LO+ for given θ ₁ and θ ₂. The picture is similar to that of Figure 2—again, the energy of the first nucleon takes the largest possible values, while the energy of the second nucleon remains small. ϕ ₁₂ is above 150° for most of the configurations; however, there are also configurations, clustered around the diagonal or around a straight line intersecting the diagonal at θ ₁ = θ ₂ = 60° with ϕ ₁₂ below 20°.

A few examples of d 5 σ d Ω 1 d Ω 2 d S with large |Δ ^N2LO−N4LO+ | are shown in Figure 6. In all the cases, a clear difference between N²LO + N²LO and N⁴LO⁺+N²LO predictions is observed in one of the maxima of the cross section; however, for the configuration shown in the central panel, the largest (negative) δ ^N2LO−N4LO+ (θ ₁, θ ₂, ϕ ₂, S) occurs at the slope of the cross section around S = 160 MeV. A relatively large spread of cross sections and the angles defining this configuration make it, in our opinion, an encouraging case for experimental efforts. In the future, it would be interesting to investigate the origin of the enhanced sensitivity of these configurations to the details of the nuclear Hamiltonian and to study the effects of the isospin-breaking corrections to the NN force considered in ref. [17].

Finally, for the predictions at Λ = 450 MeV, we define accordingly δ 3NF θ 1 , θ 2 , ϕ 2 , S ≡ d 5 σ d Ω 1 d Ω 2 d S N N − d 5 σ d Ω 1 d Ω 2 d S N N + 3 N F 1 2 d 5 σ d Ω 1 d Ω 2 d S N N + d 5 σ d Ω 1 d Ω 2 d S N N + 3 N F (9) and Δ 3NF ≡ Δ 3NF θ 1 , θ 2 ≡ max ϕ 2 , S δ 3NF θ 1 , θ 2 , ϕ 2 , S . (10) In Eq. 9, the two cross sections are obtained at Λ = 450 MeV, with the N⁴LO⁺ NN force alone or combined with the N²LO 3NF.

Our results for Δ ^3NF are shown in Figures 7, 8. The Δ ^3NF values are in the range (−0.378, 0.386) without constraints on the cross sections and energies and in the range (−0.270, 0.386) when these restrictions are taken into account. Initially, in most of the θ ₁ − θ ₂ regions, Δ ^3NF is negative or close to zero. For almost half of the θ ₁ − θ ₂ combinations, we observe the importance of the 3NF as Δ ^3NF is in the range (−0.378, − 0.150). Only for both polar angles below ≈ 25 ° Δ 3NF becomes positive. If configurations with small cross sections and energies are neglected, we find in the most typical case −0.14 < Δ ^3NF < − 0.07. The largest positive Δ ^3NF value occurs either when both polar angles are small or when one of them is small and the other one is very large. The distribution of nucleon energies in the configurations contributing to Figure 7B reveals that the largest |Δ ^3NF | is observed when one of the observed nucleons absorbs nearly all of the kinetic energy, while the energy of the remaining observed nucleon becomes close to the imposed threshold of 10 MeV. For most of the configurations, ϕ ₁₂ is large, and only for configurations close to this diagonal θ ₁ = θ ₂ or close to the line perpendicular to that diagonal at θ ₁ = 60°small relative azimuthal angles are preferred.

Figure 9 shows the differential cross section at three configurations selected from those that maximize Δ ^3NF . Figure 9A shows a case where 3NF lowers the cross section by about 39% at S = 128 MeV. In the two remaining configurations, 3NF increases the cross section by 22% at S = 71 MeV (Figure 9B) and by 23% at S = 187.5 MeV (Figure 9C).

This concludes our discussion of the cross section at E = 200 MeV, and in the remaining part of Section 3, we present similar maps as above, but for the deuteron breakup reaction induced by nucleons with a lower kinetic energy of E = 135 MeV.

Figure 10 shows Δ^400−550, both before and after implementing the cutoff conditions on the energies E ₁ and E ₂ and the exclusive cross sections, obtained with Λ = 400 MeV or Λ = 550 MeV. As in the case of E = 200 MeV, the N⁴LO⁺ NN interaction is used, complemented by the N²LO 3NF. We also maintain the same thresholds for energies and the same cross sections as were used at E = 200 MeV.

Initially, Δ^400−550 varies between −24%, 17% but most often remains in the (−12%, 4%) intervals. The interesting configurations with Δ^400−550 < −20% are typically those with one of the θ _i below at approximately 30° and another θ _i in the (45°, 100°) range. The maximal positive Δ^400−550 occurs only in the part of the θ ₁ − θ ₂ plane where both θ _i are small. After reducing the number of allowed configurations by applying the threshold conditions, Δ^400−550 is found in the (−19%, 12%) range. Maximal values, around Δ^400−550 = 10%, survive at both θ _i small. On the contrary, regions with large negative Δ^400−550 are significantly shrunk. In most of the phase space, Δ^400−550 belongs to (−10%, 0%). Comparing the resulting picture and numbers with those at E = 200 MeV, there is a significant increase in the magnitude of Δ^400−550 when moving to higher energies, on average by a factor of two. This is, of course, perfectly in line with the expectations based on the fact that the truncation uncertainty of the chiral EFT grows with energy.

The pattern of E ₁ for configurations leading to maximal Δ^400−550 at E = 135 MeV is similar to that at E = 200 MeV, as shown in Figure 11. Of course, the lower reaction energy results in lower final state nucleon energies, but again, in some θ ₁ − θ ₂ regions, we obtain combinations of small E ₁ and medium or large E ₂ values, or vice versa, or both energies are about half of the maximal available energy. The pattern for ϕ ₁₂ is even more similar to that for E = 200 MeV—large relative angles dominate the picture, and only for θ ₁ ≈ θ ₂⪅60° ϕ ₁₂ is small. These similarities and the similar patterns shown in Figures 4B, 12B suggest that the considered observables at these two energies may exhibit sizeable correlations.

Next, Figures 12, 13 show the analysis of Δ ^N2LO−N4LO+ and the corresponding kinematic configurations maximizing it. Similar to the case of E = 200 MeV, Δ ^N2LO−N4LO+ is huge: Δ ^N2LO−N4LO+ ∈ (−2.5, 2) for E = 135 MeV if no additional conditions are imposed on the energies and the cross sections. For most of θ ₁ − θ ₂ pairs, we find configurations where Δ ^N2LO−N4LO+ takes values above 80%. Imposing the threshold conditions used here, which limits the possible number of configurations, leads to a much smaller Δ ^N2LO−N4LO+ between −20% and +15%. For a significant part of the phase space, the N⁴LO⁺ NN force increases the cross sections, which results in negative Δ ^N2LO−N4LO+ . Similar to the results at E = 200 MeV, the highest values of Δ ^N2LO−N4LO+ appear at one of the small azimuthal angles, below approximately 20°, and another one in the range of (70°, 180°). Configurations with the largest negative Δ ^N2LO−N4LO+ have at E = 135 MeV a slightly different distribution than observed in Figure 4B—in addition to the previously observed patterns, now also configurations with θ ₁ = θ ₂ ∈ (20°, 30°) contribute. The full range for Δ ^N2LO−N4LO+ at E = 135 MeV is slightly narrower than that at E = 200 MeV.

The distribution of nucleon energies at E = 135 MeV is again reminiscent of that observed at E = 200 MeV, taking into account a different absolute energy scale. More deviations are observed for ϕ ₁₂, as shown in Figure 13B. At E = 135 MeV, the smallest ϕ ₁₂ survives only in one of the two large regions observed for higher energies in Figure 5. Also, for the smallest θ ₁ and θ ₂ medium values of ϕ ₁₂ are now preferred instead of small ϕ ₁₂.

The effects of the 3N interaction at E = 135 MeV are shown in Figure 14. With no additional constraints on energies and cross sections, we find |Δ ^3NF | up to 30%. Actions of the 3NF are possible in both directions —there are configurations where the three-body potential decreases the cross section (positive Δ ^3NF ) or where the cross section is increased (negative Δ ^3NF ). The first configurations are those with θ ₁ and θ ₂ below ≈ 30 ° but above 5°. In the other dominant part of the θ ₁ − θ ₂ plane, Δ ^3NF is negative and in many cases remains below −10%. This does not change much when only cross sections above 0.01 [mb sr⁻² MeV⁻¹] and energies above 10 MeV are considered. Nearly, for all the allowed configurations contributing to Figure 14 Δ ^3NF < − 5%. The largest 3NF effects are clustered around the line parallel to the diagonal in θ ₁ = θ ₂ ≈ 60°. Δ ^3NF above +10% requires both θ _i small. Comparing the results in Figure 14 with those in Figure 7, we observe that the maximal 3NF effects change slightly between E = 135 MeV and E = 200 MeV but are on average larger at higher energies.

The pattern of nucleon energies at which Δ ^3NF is minimized shows that two detected nucleons have intermediate energies in the approximate range of 30 − 70 MeV, as shown in Figure 15. The large ϕ ₁₂ dominates almost all the configurations examined. The only exceptions are some of the configurations on the diagonal θ ₁ = θ ₂ including those yielding the largest positive Δ ^3NF . The medium nucleon’s energies and large relative azimuthal angles again provide good opportunities for measurements.