The transition operator for elastic scattering is given by a straightforward one-body integral equation, which of course requires the knowledge of PUP, which is a many-body operator. For a brief review we follow the spectator expansion of PUP as introduced in Ref. [54] in contrast to Ref. [6] where the expansion of T is considered. Following those references, we assume the presence of two-body forces only for the present discussion. The extension to many-body forces is not precluded by the formulation. With this assumption the operator U can be expanded as U = ∑ i = 1 A U i , (5) where U _i is given by U i = v 0 i + v 0 i G 0 E Q ∑ j = 1 A U j , (6) provided that V = ∑ i = 1 A v 0 i , where the two-body potential v _0i acts between the projectile and the ith target nucleon. Through the introduction of an operator τ _i which satisfies τ i = v 0 i + v 0 i G 0 E Q τ i , (7) Eq. 6 can be rearranged as U i = τ i + τ i G 0 E Q ∑ j ≠ i U j . (8) This rearrangement process can be continued for all A target particles, so that the operator for the optical potential can be expanded in a series of A terms of the form U = ∑ i = 1 A τ i + ∑ i , j ≠ i A τ i j + ∑ i , j ≠ i , k ≠ i , j A τ ijk + ⋯ . (9) This is the Spectator Expansion for U, where each term is treated in turn. The separation of the interactions according to the number of interacting nucleons has a certain latitude, due to the many-body nature of G ₀(E), which needs to be considered separately. In the following we will concentrate on the leading-order term, which is still a many-body operator due the presence of G ₀(E). The next-to-leading order term in this spectator expansion for U has been formally derived and connected to standard three-body equations in Ref. [54].

When using the leading-order term of the spectator expansion as given in Eq. 7, for elastic scattering only Pτ _i P, or equivalently ⟨Φ₀|τ _i |Φ₀⟩ needs be considered. With this in mind, Eq. 7 can be re-expressed as τ i = v 0 i + v 0 i G 0 E τ i − v 0 i G 0 E P τ i = τ i ̂ − τ i ̂ G 0 E P τ i , (10) or 〈 Φ 0 | τ i | Φ 0 〉 = 〈 Φ 0 | τ i ̂ | Φ 0 〉 − 〈 Φ 0 | τ i ̂ | Φ 0 〉 1 E − E A − h 0 + i ε 〈 Φ 0 | τ i | Φ 0 〉 , (11) where τ i ̂ is defined as the solution of τ i ̂ = v 0 i + v 0 i G 0 E τ i ̂ . (12)

The combination of Eqs 10, 2 corresponds to the leading-order Watson optical potential [3, 4]. In ab initio structure calculations the one-body densities or ground state wave functions for protons and neutrons are calculated separately, so that Eq. 11 allows to combine e.g. for proton scattering of a nucleus the proton-neutron interaction ( τ ̂ i = p n ) with the neutron one-body density and the proton-proton interaction with the proton one-body density. The sum over i then adds both to obtain the driving term ⟨ Φ 0 | τ i ̂ | Φ 0 ⟩ the integral equation, Eq. 11.

If the projectile-target-nucleon interaction is assumed to be the same for all target nucleons and if iso-spin effects are neglected then the KMT approximation ( A − 1 A ⟨ Φ 0 | τ i ̂ | Φ 0 ⟩ ) can be derived from the leading-order Watson potential [5]. When working with momentum space integral equations, the numerical implementation of Eq. 11 is straightforward [40, 41, 45, 55]. Working in coordinate space with differential equations does not allow an equally straightforward implementation, and thus the KMT prescription is the most favorable alternative. A comparison between leading-order Watson potential and the KMT prescription is shown in Figure 1 for elastic proton scattering from ⁸He at 71 MeV laboratory kinetic energy. Despite the relatively large difference between the proton and neutron densities for this nucleus the KMT prescription agrees with the exact Watson description very well up to momentum transfers of about 2 fm⁻¹.

Since Eq. 11 is a one-body integral equation, the principal problem is to find a solution of Eq. 12, which due to many-body character of G ₀(E) is still a many-body integral equation, and in fact no more easily solved than the starting point of Eq. 1.

For most practical calculations the so-called closure approximation to G ₀(E) is implemented [56] turning Eq. 12 into a one-body integral equation. This approximation replaces H _A by a constant that is interpreted as an average excitation energy, and is justified when the projectile energy is large compared to typical excitation energies of the nucleus. The closure approximation is very successfully applied for elastic scattering around 80 MeV and higher.

Going beyond the closure approximation in the spirit of the spectator expansion we want to single out one target nucleon i and write G ₀(E) as G 0 E = E − h 0 − H A + i ε − 1 = E − h 0 − h i − ∑ j ≠ i v i j − H i + i ε − 1 , (13) where the target Hamiltonian is expanded as H _A = h _i + ∑ _j≠i v _ij + H ⁱ with v _ij being the interaction between target nucleons i and j, and H ⁱ being an (A-1)-body operator containing all higher order effects. Realizing that ∑ _j≠i v _ij ≡ W _i and thus H ⁱ = H _A −h _i −W _i does not have an explicit dependence on the ith particle, then H ⁱ may be replaced by an average energy E ⁱ which is akin to the effective binding energy between the ith nucleon and the A−1 spectator. This is not an approximation since G ₀(E) may be regarded as G 0 E = E − E i − h 0 − h i − W i − H i − E i + i ε − 1 (14) and (H ⁱ −E ⁱ ) should be set aside to be treated in the next order of the expansion of the propagator G ₀(E). In this order of the expansion G ₀(E) becomes G i E = E − E i − h 0 − h i − W i + i ε − 1 , (15) and Eq. 12 reads τ i ̂ = v 0 i + v 0 i G i E τ i ̂ . (16) In order to connect the above expression with the free NN amplitude t 0 i = v 0 i + v 0 i g i t 0 i (17) with g i = E − E i − h 0 − h i + i ε − 1 . (18) algebraic relations between the resolvents lead to τ i ̂ = t 0 i + t 0 i G i W i g i E τ i ̂ . (19) Defining G i W i = g i T i with T i = W i + W i g i T i leads to τ i ̂ = t 0 i + t 0 i g i T i g i τ i ̂ . (20) The three-body character of the above expression becomes more evident if one defines it as a set of coupled equations as τ i ̂ = t 0 i + t 0 i g i X i X i = T i g i τ i ̂ . (21)

Though the spectator expansion of the operator U in terms of active particles is defined in Eq. 9, we see that this expansion is performed in terms of quantities which contain many-body propagators. Each of the ingredients τ _i , τ _ij , etc. may themselves be expanded in a spectator expansion, i.e. expanding the many-body propagator also according to the number of active participants. The corrections to the propagator in the leading-order term of U contributions that arise from the Q space, whereas the terms arising from the propagator remain in the P space at first order level. Thus their contribution may be more relevant for elastic scattering.

In an explicit treatment of G _i (E) it is necessary to consider the explicit form of ∑ _j≠i v _ij = W _i , which is a priori a two-body operator. In the framework of ab initio nuclear structure calculations this will involve two-body densities. In earlier work [54, 57, 58] the quantity W _i was treated as one-body operator, specifically a mean-field potential. This was a physically reasonable choice, though being outside the strict demands of the spectator expansion. However, those studies revealed that the next order in the propagator expansion has little effect on elastic scattering observables at energies larger than 100 MeV, while the description of differential cross section and spin-observables for elastic scattering from ⁴⁰Ca at 48 MeV showed considerable improvement with respect to experiment [57]. Obviously this type of calculation will need to be explored within an ab initio approach. In Ref. [57] the energy E ⁱ of Eq. 18 was set to zero.

As illustrated in this section, deriving a multiple scattering expansion for elastic NA scattering means projecting on the ground state of the target in order to obtain a Lippman-Schwinger type equation for the transition amplitude and obtaining an operator U for the effective interaction, which is defined in the space Q = 1 − P. In this spirit, the spectator expansion contains therefore two pieces, namely the expansion of the operator U in terms of active particles in the scattering process as well as the expansion of target Hamiltonian H _A in the propagator G ₀(E) in a similar fashion. Thus it is very difficult to define a single expansion parameter which governs the convergence of the expansion.

Prior to our detailed study of truncation uncertainties of a chiral NN interaction in elastic NA scattering observables we need to choose a specific chiral NN interaction. Here we want to focus on the EKM chiral NN interaction [18, 19] with a semi-local coordinate space regulator of R = 1 fm, which has a breakdown scale of Λ _b = 600 MeV. This interaction gives a slightly better description of the ground state energies in the upper p-shell than a similar, more recent interaction with a semi-local momentum space regulator. For consistency with the leading-order optical we only use the NN potentials, omitting three-nucleon forces, which appear at N2LO in the chiral expansion, both in the structure and the scattering part of the calculations. Including three-nucleon forces consistently in both, the structure and scattering calculations requires going beyond the leading-order optical potential, and is beyond the scope of this work. Though initial attempts of incorporating three-nucleon forces as an effective density-dependent NN force in the scattering part have been presented [46], they can not yet be considered as systematic consideration of three-nucleon forces in NA scattering. For similar reasons, we restrict most of our results to N2LO since three-nucleon force contributions at N3LO and N4LO are significant [71].

Next, the translationally-invariant one-body density needed for the scattering calculation can be obtained using the NCSM approach, in which the nuclear wavefunction is expanded in Slater determinants of harmonic oscillator basis functions [30]. Ideally, one uses a sufficiently large basis to ensure convergence of this expansion, but in practice observables depend on both the many-body basis truncation, N _max (defined as the total number of harmonic oscillator quanta in the many-body system above the minimal configuration), and on the harmonic oscillator scale ℏΩ. In Table 1 we give the ground state binding energies and point-proton radii of ⁴He, ¹²C, and ¹⁶O obtained with the EKM chiral NN potential [18, 19] with a semi-local coordinate space regulator of R = 1 fm (note that at N2LO we did not include any three-nucleon forces).

For ⁴He we can obtain nearly converged results for both the binding energy and the proton radius, and these results agree, to within their estimated numerical uncertainties (the first set of uncertainties in Table 1), with Yakubovsky calculations using the same NN potential [71]. However, for larger nuclei such as ¹²C and ¹⁶O we are more limited in the N _max values that can be reached on current computational resources. ¹

To assess the relative size of chiral truncation uncertainties compared to other known uncertainties, e.g. the harmonic oscillator parameters N _max and ℏΩ, we employ a pointwise truncation procedure and study reaction observables that are not functional quantities, e.g. reaction cross sections at a specified laboratory energy. This pointwise approach was previously implemented in Refs. [36, 43] and it starts by assuming the expansion parameter Q and reference scale y _ref are known. From there, we can apply Eq. 27 to calculate the coefficients c _n , which are treated as independent draws from the same underlying distribution. The properties of this distribution can be learned from Bayesian techniques and the posterior distribution for the prediction can be readily calculated with its associated credible intervals. For more details, see Ref. [36].

In order to estimate the chiral truncation uncertainties of the obtained ground state binding energies and radii, we apply the pointwise approach with Q ≈ .3 as the effective expansion parameter, following Ref. [31]. These uncertainties are listed as the second set of uncertainties in Table 1, starting from NLO. Here we see that for the energies, the chiral uncertainties are at least of the same order as the estimated numerical uncertainties; however, the uncertainties of the radii of ¹²C and ¹⁶O are clearly dominated by their systematic dependence on the basis parameter ℏΩ.

To illustrate the pointwise approach for scattering observables, Figure 3 shows the reaction cross sections for proton scattering from ⁴He at 65 MeV and ¹⁶O at 100 MeV. For each case, the result is shown as a function of N _max, and variations with respect to ℏΩ are indicated. While more obvious for the smaller nucleus where the NCSM can better converge, in both cases the uncertainty resulting from the chiral truncation remains larger than the uncertainty arising from the many-body method. To better illustrate this point, we present the reaction cross section for ⁴He with a scale starting from 115 mb and with a range of only 45 mb, while using the full range of 600 mb for ¹⁶O. While larger model spaces will better converge the NCSM results, smaller truncation uncertainties will only be achieved by higher chiral orders, despite the noticeable dependence of the radii on the harmonic oscillator parameter ℏΩ, in particular for the heavier nuclei, in the current calculations. Note however that even at N3LO we anticipate the chiral truncation uncertainties will be larger than the indicated variations with respect to the harmonic oscillator parameter ℏΩ due to the rather large value of the expansion parameter Q in the scattering calculation.

For functional quantities y(x) we employ a correlated approach that includes information at nearby values of x. This approach is better for observables such as a differential cross section, which we know does not vary wildly from values at nearby angles. It also starts from Eq. 27 and treats the coefficients c _n (x) as independent draws from an underlying Gaussian process. This Gaussian process encodes information about the correlation length ℓ, and the qualities of the underlying distribution can be learned from the order-by-order results. This training is followed up by testing procedures which seek to confirm the Gaussian process has been appropriately fit to the available results, and if not, to diagnose potential issues. From a well-fit Gaussian process we can then extract truncation uncertainties for the functional quantities. For more details and applications, see Refs. [36, 43].

In the following examples, we examine proton scattering for ⁴He, ¹²C, and ¹⁶O at various projectile energies and compare to the available experimental data. In each case, we show the convergence with respect to chiral order and the resulting decrease in the size of the chiral truncation uncertainties, as well as discuss any associated physics insights. To avoid concerns about the expansion parameter increasing at larger angles, we mostly restrict our analysis to forward angles where we expect the expansion parameter to be independent of the scattering angle.

For proton scattering on ⁴He, we see good agreement with experiment for the differential cross sections (Figure 4) at lower projectile energies. Below 100 MeV, most data points fall within the 2σ uncertainty band, and at 100 MeV a majority of the data points are within the 1σ band. At the highest energy of 200 MeV, the chosen interaction seems unable to reproduce the experimental data, though this is not uncommon for scattering from ⁴He.

The analyzing powers for proton scattering on ⁴He (Figure 5) is more complicated. For the lower energies of 65 and 71 MeV, the experimental data shows a near zero value, regardless of scattering angle. In the scattering of a spin-1/2 particle from a spin-0 nucleus, this indicates that there is no spin-orbit force at play. This behavior is only reproduced by the LO result, for which the chiral NN interaction only contains the one-pion exchange and contact terms, which do not produce a spin-orbit force. At NLO the two-pion exchange diagrams are responsible for reproducing the NN p-waves and thus provide a spin-orbit force that leads to a non-zero value for the analyzing power in NA scattering. At N2LO there are no new terms in the two-nucleon sector, and thus A _y does not change its shape at that chiral order. Therefore, one needs to conclude that in this case other physics which goes beyond the leading order NA effective interaction may be needed to describe the analyzing power.

For the higher energy of 200 MeV, all of the experimental data points are within the 2σ uncertainty band, though there is a slight offset in the shape. In all cases, the analyzing power is more difficult to reproduce using this interaction, though other interactions have done better [39, 41].

For proton scattering from ¹²C, the differential cross sections (Figure 6) are reliably reproduced by the central value of the N2LO calculations up to 100 MeV laboratory kinetic energy, and systematically over-predict at higher energies. As the projectile energy increases, the expansion parameter increases and as a result uncertainty bands become larger. This is most noticeable at 160 MeV: the experimental data is within the 1σ band, but the size of that band, as well as the 2σ band, are so large that they are not practically useful. The gray bars in the cross section panels for N2LO indicate the momentum transfer up to where we expect the expansion parameter to be dominated by the c. m. momentum p _NA . Once the momentum transfer exceeds the value given by the bar, the uncertainty is dominated by the momentum transfer q, and is thus underrepresented by the method we use. Note that the vertical bar is at the same scattering angle θ _c.m., but different momentum transfer q, as function of the projectile energy since p _NA is a function of the projectile energy as given in Eq. 29. Looking at the lower energies, the increasing agreement with experiment in the first peak and minimum as higher orders in the chiral NN interaction are included gives the correct trend. Minima in the differential cross section correlate with the size of the target nucleus. It is well well known [31], and also evident from Table 1, that the nuclear binding energy calculated with the LO of the chiral NN interaction is way too large and correspondingly the radius much too small. Only when going to NLO and N2LO the binding energy as well as the radius move into the vicinity of their experimental values. This finding from structure calculations is corroborated by the calculations in Figure 6, where with increasing chiral order the calculated first diffraction minimum moves towards smaller momentum transfers indicating a larger nuclear size.

The analyzing powers for proton scattering on ¹²C are at 65 MeV also almost zero for small momentum transfers and rise at q = 1.2 fm⁻¹ to its maximum value of +1. This is captured by the NLO calculation where spin-contributions occur in the NN interaction (Figure 7). For 65 MeV, the experimental data is mostly within the 2σ band until approximately θ _c.m. = 60°, where we expect the expansion parameter to being increasing and the uncertainty bands to thus be underestimates. For 122 MeV, the very forward direction is inside the 1σ band, but the overall shape of the experimental data is not well captured by this interaction.

For proton scattering from ¹⁶O, the differential cross sections (Figure 8) are similar to the ¹²C case. Namely, the lower energies do reasonably well at describing the data within the 2σ bands, but as the projectile energy increases the uncertainty bands increase to unhelpful sizes. At the lowest energy of 65 MeV, we see a better and better reproduction of the first minimum in the differential cross section as the chiral order increases. Again, this first minimum is known to be related to the size of the nucleus, so this is an important feature to reproduce from both a structure, see Table 1, and reaction perspective.

The analyzing powers for proton scattering on ¹⁶O (Figure 9) are again similar to the ¹²C case. At lower energies (65 and 100 MeV), we again see a good reproduction to within 1σ or 2σ of the forward direction data, but beyond θ _c.m. = 60°, the experimental data is outside the uncertainty bands. At the higher energy of 135 MeV, many of the experimental data are within the uncertainty bands but for a nucleus of this size, the expansion parameter has already increased such that the resulting uncertainty bands are unhelpfully large.

As stated toward the beginning of the section we omit three-nucleon forces for consistency with the leading-order optical potential which only treats two active nucleons. Those three-nucleon forces already appear at N2LO in the chiral expansion, however, including them consistently in the structure as well as reaction calculation requires going beyond the leading-order optical potential and is beyond the scope of this work. For the sake of investigating truncation errors in the chiral NN force, one may carry out inconsistent calculation in the sense that the structure part of the calculation is kept fixed at N2LO, and in the reaction part higher orders in the NN force are used. Proceeding in this fashion is sensible, since the scattering calculation is more sensitive to the NN force compared to the structure calculation, provided this structure calculation gives a reasonable description of the ground state one-body density. To show how the chiral truncation error develops when higher chiral orders in the NN interaction are introduced, we show in Figure 10 proton scattering from ¹⁶O at 100 MeV projectile energy, where the higher chiral orders are only employed in the scattering part through the corresponding Wolfenstein amplitudes. In both, the differential cross section as well as the analyzing power the two most right panels depicting the inconsistent calculation show that the uncertainty bands become smaller when higher chiral orders in the NN interaction are included. However, these uncertainty bands are not necessarily realistic due to missing higher-body effects, which include higher orders in the chiral force as well as higher orders in the multiple scattering expansion. Therefore, we can not draw firm conclusions from the fact that data are outside the uncertainty estimates. Nevertheless, it is obvious that the decrease in the uncertainties in the chiral truncation is rather slow due to the large expansion parameter. Furthermore, the medians of the calculations shown in Figures 8, 9 do not change when higher chiral orders are considered in Figure 10, which further indicates that the smaller error bands of the higher order chiral truncations may be artificial.

Even while restricting our analysis to a region where we expect the expansion parameter to be constant, we can still observe effects on the uncertainty bands if the expansion parameter is large, as noted in many of the results at larger projectile energies. In fact, this behavior will place limits on the size of nucleus that can be considered with this approach, since p _NA as defined by Eq. 29 will continue to increase as A increases, yielding Q > 1 eventually. While this situation is not ideal, we non-etheless find support for it in our analysis after examining the posteriors for Q, in accordance with Refs. [36, 43].

In Figure 11, we calculated posteriors for the differential cross sections in proton scattering from ⁴He, ¹²C, and ¹⁶O at the energies previously discussed. From these, we can extract a single best guess for the value of Q based on the order-by-order calculations and compare that to the expectation for Q based on Eq. 30. For ¹⁶O, the largest nucleus considered, we see generally good agreement between the expected value of Q and the best guess value from the posteriors (Figure 11C). However, as the nucleus decreases in size and as the laboratory energy decreases, some differences begin to emerge between the two values. In Figure 11B for ¹²C, the comparisons are roughly similar to the ¹⁶O case, but for the ⁴He analysis (Figure 11A), the differences are more pronounced, especially for the lower laboratory energies. A similar analysis of neutron scattering on ¹²C did not show any significant differences between the two values [43], which implies ⁴He may be the outlier in this approach. This analysis may imply scattering from ⁴He with projectiles at lower energies could be analyzed with a smaller expansion parameter Q, though the higher energy results still favor the larger expansion parameter. As the smallest nucleus considered here, it may also point to the few-body character of ⁴He, which has not historically been well captured in an optical potential approach.