Perturbative computations of neutron-proton scattering observables using renormalization-group invariant χ EFT up to N 3 LO

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I. INTRODUCTION
Nuclear potentials used in ab initio [1] computations of atomic nuclei [2] are almost exclusively derived using chiral effective field theory (χEFT) [3][4][5] based on Weinberg power counting (WPC) [6,7].Such potentials [8][9][10][11][12][13][14], now derived up to the fifth chiral order [15][16][17], have furnished a wide range of structure and reaction predictions across the nuclear chart [18,19], but at the same time they grapple with the renormalization challenge inherent to chiral nuclear forces [20].Indeed, numerical studies [21] of the nucleon-nucleon scattering amplitude have shown that the contact operators, accounting for unresolved short-range physics, already at leading order (LO) in WPC are not sufficient to renormalize the singular nature [22] of the one pion-exchange potential.Consequently, LO predictions based on WPC exhibit an unphysical dependence on the cutoff Λ that regularizes the amount of high-momentum (or short-range) physics that is resolved.
Several PCs leading to renormalization-group (RG) invariant nucleon-nucleon amplitudes have been proposed in the past two decades [23][24][25][26][27][28][29][30][31][32][33][34][35].They can collectively be referred to as modified Weinberg power countings (MW-PCs).However, we typically know very little about their predictive power for nuclei beyond the lightest-mass systems [36].The one exception is the recent study by Yang et al. [37] that presented the first ab initio predictions of binding energies in 4 He, 6 Li, and 16 O using χEFT potentials up to next-to-leading order (NLO) in several different MWPCs.The calculations in that work revealed an α-decay instability in the ground states in 6 Li and 16 O.Subsequent analyses brought forward probable causes for this instability as originating in (i) overfitting of the low-energy constants (LECs) that parameterize the short-range interactions [38] and (ii) underestimating the importance of few-nucleon forces [39] at LO in MWPC.* oliver.thim@chalmers.se The notable absence of MWPC-based predictions for heavier-mass nuclei is likely due to a variety of factors.Firstly, potentials based on WPC are easier to implement in present ab initio computer codes as one straightforwardly sum leading and sub-leading corrections to the potential before solving the Schrödinger equation, whereas in MWPC sub-leading corrections should be added in perturbation theory [40].Secondly, there exists several widely available computer codes for evaluating matrix elements of chiral nucleon-nucleon and threenucleon potentials, as well as currents, to very high orders in WPC.Finally, it is currently prohibitively costly to converge ab initio predictions of nuclear properties at the large values of the cutoff required for analyzing RGinvariance in MWPC.
In light of these facts we certainly see the utility of WPC, which might provide a consistent EFT framework provided that renormalization is interpreted in a fashion where the cutoff never exceeds the order of the breakdown scale [41][42][43].However, the existence of MWPCs, where renormalization does allow for the cutoff to be taken far beyond the breakdown scale, calls for a continued effort.We note that it was shown in Ref. [44] that one can encounter so-called exceptional points in the cutoff domain beyond the breakdown scale for which the cutoff independence of the phase shifts significantly deteriorates.Given the fundamental importance of RG-invariance it should be seriously explored whether MWPC approaches can furnish a realistic and predictive framework for ab initio nuclear physics.
In this paper, we contribute to the meager list of quantitative predictions grounded in RG-invariant formulations of χEFT.To the best of our knowledge, and somewhat surprisingly, nucleon-nucleon scattering observables have not been computed in MWPC beyond LO [41].Here, we present predictions for integrated and differential cross-sections, as well as polarization observables, for elastic neutron-proton (np) scattering up to next-tonext-to-next-to-leading order (N 3 LO) in the MWPC of Long and Yang [30,32,45], where higher-order correc-tions to the potential are treated perturbatively [21,40].This work serves as an important step in the development and uncertainty quantification of any model of the nuclear interaction [46][47][48][49][50].
In Section II we review how to construct potentials in the PC of Long and Yang, describe how to numerically compute the scattering amplitude in distorted-wave perturbation theory, and explain how we calibrated LEC values.In Section III we present results for scattering observables up to N 3 LO, and we summarize and conclude in Section IV.

II. FORMALISM
In χEFT, scattering amplitudes are expanded in a dimensionless ratio (Q/Λ b ) ν .Here, ν indicates the chiral order, Λ b is the underlying high-momentum scale of χEFT, and Q denotes the relevant low-energy scale.For nucleon-nucleon scattering, we assume Q ≈ max(p, m π ), where p is the relative momentum in the center of mass (c.m.) frame of the interacting nucleons, and the pion mass m π is the relevant low-energy mass scale.In this work we adopt a nomenclature where LO scales as (Q/Λ b ) 0 while sub-leading orders are denoted by their relative scaling to LO.As such, NLO scales as 2 and so on.In what follows, we summarize relevant details regarding the MWPC that we use in this work, define the potential terms V (ν) entering at each chiral order, and explain how we performed the perturbative calculations of scattering amplitudes.

A. The nucleon-nucleon interaction potential in the Long and Yang power counting
We employ the MWPC of Long and Yang [30,32,40,51], which adheres to the following overarching principles: • The chiral order of a pion-exchange diagram, along with the necessary counterterms for renormalizing pion loops, is determined by the naive dimensional analysis (NDA) of its non-analytic part.This follows the same principle as in Weinberg Power Counting (WPC).
• Counterterms are promoted to lower chiral order only when needed to fulfill the requirement of RGinvariance.
• All corrections to the potential beyond LO are included perturbatively to obtain RG-invariant amplitudes.
One-pion exchange (OPE) enters at LO in χEFT and must be treated non-perturbatively, at least in the low partial waves where it is sufficiently strong.The singular nature of OPE is increasingly alleviated by the centrifugal barrier.Thus, at some point in the partial-wave expansion there is sufficient angular momentum ℓ to furnish a perturbative treatment of OPE [29,52,53] and consider it sub-leading.
At LO in the MWPC by Long and Yang, the OPE potential V (0) 1π is considered non-perturbative in the 1 S 0 , 3 P 0 , 1 P 1 , 3 P 1 , 3 S 1 − 3 D 1 and 3 P 2 − 3 F 2 channels.OPE is attractive in 3 P 0 and 3 P 2 .Renormalization requires promotion of counterterms to the corresponding channels of the LO contact potential V (0) ct [21], thereby extending it beyond the canonical non-derivative 1 S 0 and 3 S 1 counterterms.At sub-leading orders (ν > 0), two pion-exchange, V (ν) 2π , as well as higher-order contact potentials, V (ν) ct , enter perturbatively according to the principles presented in the beginning of this subsection.The contributions to the potential up to N 3 LO in the 1 S 0 , 3 P 0 , 1 P 1 , 3 P 1 , 3 S 1 − 3 D 1 and 3 P 2 − 3 F 2 channels are listed in the third column of Table I labeled "non-perturbative (at LO) channels".
See Appendix A for detailed expressions of the potentials appearing in Table I.Following Long and Yang, we do not consider any higher-order corrections to OPE and employ potential expressions where pion loops are treated in dimensional regularization.For the sub-leading twopion exchange potential V (3) 2π we use pion-nucleon LECs c 1 , c 3 , c 4 with central values from the Roy-Steiner analysis in Ref. [54].
Let us now turn to the channels with ℓ > 1 (and without any coupling to ℓ ≤ 1).For these channels we consider OPE to be perturbative and consequently set it to zero at LO.We follow Ref. [52] and suppress two-pion exchanges by the same chiral power as OPE.Up to N 3 LO, there are no contact potentials in the perturbative channels, and the contributions are listed in the last column of Table I.Other suggestions for the PC in perturbative channels are discussed by, e.g., Pavón Valderrama et al. [27].

B. A perturbative treatment of nucleon-nucleon scattering amplitudes
The perturbative computation of nucleon-nucleon scattering amplitudes proceeds in two steps.First, we solve the Lippmann-Schwinger (LS) equation for the LO amplitude in the 1 S 0 , 3 P 0 , 1 P 1 , 3 P 1 , 3 S 1 − 3 D 1 and 3 P 2 − 3 F 2 channels.Note that the LO potential is identically zero in all other channels.Second, we perturbatively include higher-order potential corrections to the amplitude, accounting for the distortion due to the non-perturbative LO solution where necessary.In the following, we explain this procedure in detail, see also Refs.[30,32,53].
The neutron-proton Hamiltonian in the center-of-mass (c.m.) frame can be written where p denotes the c.m. momentum and m N = 2m n m p /(m n + m p ) the nucleon mass.The projectile energy in the laboratory frame will be denoted T lab .Furthermore, V I denotes the LO potential, and V II denotes the sum of all sub-leading potentials, which formally can be infinitely many.The PC helps us identify important and less important contributions to the scattering amplitude T and therefore facilitates a meaningful truncation of V II .With the notation for the chiral potentials V (ν) introduced in Section II A, V I and V II read The LO amplitude, T (0) , is obtained (nonperturbatively) by solving the LS-equation where the free resolvent is given by and H 0 = p 2 /m N .We use a notation where we suppress the explicit dependence on the c.m. scattering energy, E, for the resolvents and amplitudes.
In WPC, higher-order corrections are accounted for non-perturbatively by solving the LS-equation for the sum V I +V II .In MWPC, however, potentials beyond LO, i.e., the corrections (V II ), enter in perturbation theory to obtain RG invariant results [40].Indeed, higher-order corrections should be amenable to a perturbative treatment.If not, they are non-perturbative in nature and belongs at LO. Distorted-wave perturbation theory has been applied to compute scattering amplitudes in several previous studies, see, e.g., Refs.[28,30,32,51,53,55].The perturbation series for the scattering amplitude can be derived and expressed in various ways.The one that we find most instructive follows Refs.[56,57].First, using the two-potential trick, the T -operator for the Hamiltonian in Eq. ( 1) is written in the form where the Møller wave operators are defined as and the full LO resolvent reads Inserting Eq. ( 3) in Eq. ( 6) gives for the full T -operator (10) Expanding both sums and organizing terms according to their chiral orders ν yields the expressions for the first-, second-, and third-order corrections to the LO amplitude as A diagrammatic representation of amplitudes up to NLO is presented in Fig. 1.Note that the full amplitude at, e.g., third order (N 3 LO) is given by the sum T (0) +T (1) + T (2) + T (3) .Clearly, the distorted-wave corrections in Eqs. ( 11) to ( 13) simplify dramatically when applied to the channels where OPE is perturbative such that T (0) = 0, Ω + = 1, and Ω † − = 1.In these channels we therefore recover ordinary perturbation theory.
The distorted-wave corrections to the amplitudes T (ν>0) can alternatively be obtained as solutions to a set of modified LS-type equations, discussed in more detail in Refs.[58,59], which read We use this formulation to verify our numerical implementation of Eqs. ( 11) to (13).We note that the alternative approach of modified LS-equations requires a matrix inversion at each order, whereas the distorted-wave approach requires matrix multiplications only.However, the number of matrix multiplications increases rapidly as the chiral order is increased.For example, at ν = 10, Eqs. ( 11) to ( 13) require an order of magnitude more matrix multiplications than the modified LS equations in Eq. (14).In this study we only go to ν = 3 for which the number of matrix multiplications of the two formulations are similar.

C. Numerical implementation
We project potentials and amplitudes to a partial-wave basis of states |p, ℓ, s, j⟩ following the prescription in Ref.
[60]1 .Here, p = |p|, while s, ℓ, j denote the quantum numbers of the two-nucleon spin, orbital angular momentum, and total angular momentum, respectively.Partialwave matrix elements are denoted by where the conserved quantum numbers s and j are given as superscripts.
In the LS-equation, as well as in Eqs. ( 11) to ( 13), infinite momentum integrals appear and all potentials are regulated according to where we choose a regulator function at all orders up to N 3 LO.In the calibration of the LECs, we use the cutoff values Λ = 500 MeV and Λ = 2500 MeV.Using Eqs. ( 7) to ( 9), the terms in Eqs. ( 11) to ( 13) can be expanded to sums of products of the form A 1 G + 0 A 2 , of varying length.The A i 's are either T (0) or V (ν) with ν = 1, 2, 3.For example, the NLO correction in Eq. ( 11) reads Clearly, the fundamental matrix elements that need to be evaluated at sub-leading orders are always of the form where we omit the s and j quantum numbers that are identical for the ket and the bra.In Appendix B we show how to evaluate Eq. ( 19) using ordinary matrix products and Gauss-Legendre quadrature.Longer products, e.g., of the form A 1 G + 0 A 2 G + 0 A 3 , are straightforwardly reduced to the form in Eq. ( 19) by the associativity of matrix products.Knowing this, and the distributive property with respect to addition, we can also reduce the computational complexity of evaluating the perturbation series for T by computing and storing the composite operators Ω † − , Ω + , and G + 1 .For separable potentials of Yamaguchi type [61], both the distorted-wave series and the LS equation can be solved analytically.We exploit this to verify our numerical implementation and to inspect the stability of the perturbative expansion.Numerical and analytical results for semi-realistic and separable Yamaguchi potentials in the 1 S 0 and 3 S 1 − 3 D 1 channels agree to at least single precision.

D. Calibrating the low-energy constants
Our focus in this work is to predict and analyze the description of np scattering observables in MWPC and specifically the PC of Long and Yang.To enable quantitative calculations, we calibrate the values of the unknown LECs using the same approach as Long and Yang, i.e., by tuning the contact LECs to achieve a good reproduction of the low-energy Nijmegen phase shifts [62] at selected scattering energies.
Before discussing the details of the calibration, it is important to remember that the order-by-order amplitudes are computed perturbatively and their sum is unitary only up to perturbative corrections.We therefore consider it natural to compute phase shifts perturbatively as well and proceed in this work by expanding the np S-matrix and match to chiral orders, see Appendix C for details.In doing so the phase shifts are real by construction.If one instead solves for the partial-wave Smatrix non-perturbatively from the order-by-order sum of T (ν) amplitudes, the corresponding phase shifts will have a non-zero imaginary part that increases with scattering energy.Indeed, Fig. 2 shows phase shifts computed perturbatively and non-perturbatively in the two channels 1 D 2 and 3 D 2 .Note that there are no LECs that need to be calibrated in these channels at the orders considered in this work.The imaginary part of the nonperturbative phase shift increases with scattering energy.
As that happens, the real part of the phase shift and the (real-valued) perturbative phase shift differ progressively.This is consistent with observations in Ref. [63] and the perturbative and non-perturbative approaches of defining the phase shifts are consistent within the theoretical uncertainty due to omitted and higher-order amplitudes.
An alternative approach would therefore be to work with the real part of the perturbative amplitude as done in Refs.[64,65].
In the calibration of LECs, we do not account for uncertainties stemming from the Nijmegen phase shifts or the truncation of the χEFT expansion.While we are aware of the potential risk of overfitting in doing so, we opted for a simple approach to establish a first quantitative potential and a baseline understanding.The application of Bayesian inference methods [47][48][49] to quantify the posterior probability distributions for the values of the LECs in MWPC [38], though more robust, requires considerably more efforts.In this work, we focus on studying the effectiveness of MWPC for realistic description of elastic np scattering.
The T lab values of the Nijmegen phase shifts used as calibration data are listed in Table II for each channel and order.The calibrated LECs up to N 3 LO are compiled in Table III in Appendix A. We use a naming-convention where capital letters C, D, E, . . .denote LECs with dimension MeV −2 , MeV −4 , MeV −6 , . .., respectively.
Each LEC receives perturbative corrections at subsequent orders from where it was first introduced.As an example, the LO LEC C1 S0 is expanded into contributions where the superscript enumerates the perturbative correction and not the chiral order.In the following we will exemplify the calibration procedure by discussing in detail how we calibrated the LECs in the 1 S 0 channel.
At LO we calibrate the LEC C (0) 1 S0 such that the LO 1 S 0 phase shift, δ (0) , reproduces the Nijmegen phase shift at T lab = 5 MeV.Two LECs are present in the 1 S 0 channel of the NLO potential: 1 S0 .The latter is a perturbative correction to the LO LEC.These two LECs are calibrated such that the LO phase shift plus the perturbative NLO correction, i.e., δ (0) + δ (1) , reproduce the Nijmegen phase shifts at T lab = 5 and 25 MeV.The role of C (1) 1 S0 is to ensure that the NLO correction vanishes for T lab = 5 MeV.At N 2 LO we have the LECs C (2) 1 S0 calibrated to phase shifts at energies T lab = 5, 25 and 50 MeV.Finally, at N 3 LO the LECs C (3) 1 S0 are calibrated to reproduce the phase shifts at T lab = 5, 25, 50 and 75 MeV.An analogous scheme is employed for the remaining partial waves and LECs.We calibrate all LECs for two different momentum cutoffs: Λ = 500 and 2500 MeV.
For the channels where OPE is perturbative there are no LECs present that need to be calibrated.As a consistency check we compute and reproduce the scattering phase shifts of Ref. [52]. Figure 3 shows our fit of the phase shifts in the channels where OPE is non-perturbative.The bands indicate the variation due to the two different cutoff values.There is an overall order-by-order convergence in all channels up to around T lab = 100 MeV and we can reproduce the known results of [30,32,51].The degree of cutoff sensitivity varies notably among different channels.For instance, channels like 1 P 1 and 3 F 2 show minimal sensitivity to the cutoff value, while 3 P 2 and ϵ 1 demonstrate a more pronounced dependency.The calibration in the 3 P 0 channel was particularly challenging at the higher chiral orders and the calibration energies needed to be shifted to relatively high values at N 3 LO, as seen in Table II.

III. NEUTRON-PROTON SCATTERING OBSERVABLES
Here we predict selected np scattering observables up to T lab ≈ 100 MeV using the potentials that were defined and calibrated in Section II.We compute scattering observables from the partial-wave amplitudes by first constructing the spin-scattering matrix, M , by [56,66,67] The angles θ cm ∈ [0, π] and ϕ ∈ [0, 2π] are the polar and azimuthal scattering angles, respectively where the latter is set to zero by cylindrical symmetry.The on-shell scattering momentum, p 0 , is given from the laboratory scattering energy T lab using Eq.(B2) in Appendix B. We compute S (ν)js ℓ ′ ℓ (p 0 , p 0 ), i.e., the S−matrix for a potential up to some chiral order ν, by summing the perturbatively computed T -matrix amplitudes to order ν.Using the conventions applied in this work, the partial-wave relation between the on-shell S-and T -matrix elements is thus given by S (ν)js Note that we do not need to compute phase shifts as a middle step, but rather use the amplitudes directly.This means that the perturbative phase shift computations only indirectly influence the observables through the LECs.We focus our discussion on the differential np scattering cross section and two selected polarizations, and calculate these from the spin-scattering matrix as where σ in ≡ σ i • n for nucleon i, σ i is the Pauli spin matrices, and n is normal to the scattering plane.Figure 4 shows our prediction for these scattering observables in the energy range T lab = 10 to 100 MeV for the two cutoffs Λ = 500 MeV and Λ = 2500 MeV.For the lower scattering energies (T lab ≲ 60 MeV) we observe an order-by-order improvement for all considered observables.Interestingly, the N 3 LO predictions do not always perform better, but in general performs at least as well as N 2 LO.Indeed, for T lab ≈ 100 MeV (rightmost panels of Fig. 4), it appears that the order-by-order improvement in the predictions of the differential cross section and P b polarization deteriorates and N 2 LO can perform better than N 3 LO.This effect is visible also at the level of phase shifts shown in Fig. 3.It is not clear at the moment if this is due to overfitting and (or) an underlying issue with the MWPC that we employ.Our N 3 LO predictions are certainly influenced by the adopted values of sub-leading πN LECs [54].Calculations of other scattering cross observables show that the order-by-order convergence demonstrated in Fig. 4 is representative for all elastic np scattering observables in the PC by Long and Yang.Two-pion exchange is clearly important for achieving a realistic description of scattering observables with T lab ≲ 100 MeV.
The total cross section can be straightforwardly computed from the differential cross section as and predictions for scattering energies up to T lab = 150 MeV are shown in Fig. 5. Also for this obvservable, the agreement with experimental data typically improves order-by-order, at least up to N 2 LO.The improvement 3. Phase shifts in the channels where OPE is non-perturbative and the amplitudes are computed using full distorted-wave perturbation theory.The bands indicate the envelope of the variation due to the two different cutoff values; 500 MeV (dashed line) and 2500 MeV (solid line).Note that LO and NLO results coincide for all channels except 1 S0, which is why the blue NLO band appears to be missing in several panels.The black solid lines show phase shifts from the Nijmegen analysis [62] and the diamond markers indicate the calibration data at T lab values from Table II.
of N 3 LO over N 2 LO is not obvious.At very low energies, the higher-order predictions for the total cross section are much better than the lower-order predictions.This result is somewhat peculiar for a low-energy EFT and likely due to overfitting at the phase shift level.For T lab ≳ 100 MeV, roughly corresponding to 220 MeV relative momentum, the agreement with data even deteriorates at N 3 LO.This is analogous to what was found for the angular-differential observables shown in Fig. 4 and consistent with the observation in Fig. 3 that the phase shifts at N 3 LO might suffer from overfitting at the higher energies.Alternatively, the observed decline in predictive power might indicate the presence of an additional mass scale at 200-300 MeV.Thus, it will be very interesting to study the effects of accounting for the ∆(1232)-isobar in two-pion exchange in this MWPC.
Next, we analyze how the perturbative breaking of uni-tarity in χEFT affects the predictions of total cross sections.Indeed, the computation of S-matrix elements using Eq. ( 23), where the order-by-order contributions of the scattering amplitudes are summed directly to the Smatrix, leads to a perturbative breaking of unitarity.In contrast, amplitudes computed non-perturbatively, i.e., when the potential terms are summed before solving for the scattering amplitude (as is done in WPC), are unitary by construction.In this case, the probability flux in the scattering process is also conserved exactly and the optical theorem can be safely used to compute the total cross section as, e.g., where a(θ cm ) and b(θ cm ) are Saclay-amplitudes computed from the M -matrix [70].We use the difference between total cross sections calculated using Eq. ( 27) and Eq. ( 28) to measure the effects of unitarity breaking.In Fig. 6 we show the relative difference between the cross sections computed using exact integration and the optical theorem as a function of scattering energy.The figure demonstrates how unitarity is restored perturbatively as we go to higher chiral orders.Indeed, the relative difference between the two cross section calculations is limited to 10% for scattering energies up to 40 MeV at NLO, 70 MeV at N 2 LO, and 120 MeV at N 3 LO, respectively.The bands in the figure reflect differences coming from using two cutoff values 500 MeV and 2500 MeV.The bands for NLO and N 2 LO increase smoothly with the scattering energy.The band at N 3 LO shows an artifact from the two different calculations for Λ = 2500 MeV intersecting at some energies leading to very small relative errors.We also note that the cutoff dependencies for the N 2 LO and N 3 LO calculations do not vanish as the scattering energy approaches zero.
We can also discuss this result in terms of the EFT truncation error.For a given chiral order, we argue that the results from the two different cross section calcula-tions should not become significantly different until we reach an energy where the next (omitted) order in the chiral low-energy expansion becomes relevant.This should correspond to the scattering energy for which the truncation error is significant.Breaking unitarity implies that the norm of the partial-wave S-matrix in Eq. ( 23) deviates from unity as , where we also expect C to be of natural size.This scaling of unitarity breaking should be revisited when probability distributions of the LEC values and the hyperparameters of the EFT truncation error have been inferred using a Bayesian approach.

IV. SUMMARY AND OUTLOOK
This work presents a comprehensive analysis of np scattering observables (cross sections and polarizations) utilizing an RG-invariant formulation of χEFT by Long and Yang.We calibrated the LECs by reproducing Nijmegen phase shifts at specific scattering energies, and carried out calculation up to N 3 LO for two values of the rel.diff FIG. 6.The relative difference between total np cross sections (σ) computed by integrating of the differential cross section (27) and the optical theorem (28).The bands indicate cutoff variation as in Fig. 3.The color coding for the orders is the same as Fig. 3.The horizontal dashed line marks a 10% difference.
momentum-space cutoffs, 500 MeV and 2500 MeV.The PC that we employed is fairly representative of a broad class of MWPCs in which corrections beyond LO, based on one-pion exchange, are included perturbatively and the short-range contact potential incorporates countert-erms promoted to renormalize the long-range pion contributions to the scattering amplitudes.A key result of this paper was a quantitative demonstration that RGinvariant χEFT exhibits a steady order-by-order convergence in the description of scattering observables, starting already at LO.A second key result was the realistic reproduction of experimental scattering data in an energy range up to T lab = 100 MeV at N 2 LO.We also found that N 3 LO predictions do not always improve over N 2 LO.A perturbative approach exposes the deficiencies of any PC, not only the possible lack of RG-independence.In fact, using a perturbative approach we found that the accuracy of our N 3 LO predictions for the total np cross section declines as one approaches T lab ≳ 100 MeV.This corresponds to a relative scattering momentum of 220 MeV and might suggest the presence of an additional mass scale at 200-300 MeV.This finding is in accordance with the known mass splitting between the nucleon and the ∆(1232) resonance, but is markedly lower than conventional estimates of the breakdown scale of χEFT residing in the vicinity of the ρ-meson mass.The latter estimate has also been corroborated in a Bayesian study of non-perturbative WPC predictions of nucleon-nucleon scattering observables [71].
Based on our comparison of perturbative and nonperturbative calculations of phase shifts, we speculated that the magnitudes of the imaginary component of the non-perturbative phase shift and the χEFT truncation error are linked.We also investigated the breaking of unitarity at the level of total np cross sections.The connection between perturbative unitarity breaking and the truncation error deserves further attention.
Future work will focus on quantifying posterior probability distributions for the LECs and the EFT truncation error, making predictions beyond the two-nucleon system, and the effects of including the ∆(1232) resonance in the two-pion exchange potential.Fast and accurate emulators [72], adapted to perturbative computations, will likely be essential for rigorous testing of RG-invariant χEFT against nuclear data and to address critical questions regarding, e.g., the construction of LO, the importance of promoting higher-order pion exchanges and many-nucleon forces as one increases the mass num-ber, and the level of fine-tuning in χEFT.
For the πN LECs c 1 , c 3 , c 4 , appearing in V The potential contributions at each order in the channels where OPE is treated non-perturbatively are listed in Table III.We denote counterterms in coupled channels by a 2 × 2 matrix representing ℓ ′ = j ∓ 1 (rows) and ℓ = j ∓ 1 (columns).Table III expands upon Table I in Ref. [30] to also explicitly show the perturbative corrections to LECs present at each order.Table IV summarizes the number of LECs present at each order, excluding the three πN LECs at N 3 LO from the total number.
ct : ct : This appendix gives some more details regarding the implementation of the equations for higher-order corrections to the scattering amplitude in Eqs.(11) to (13).Since all operator products reduce to the form in Eq. ( 19), the implementation can be done in complete analogy with the solution of the partial-wave Lippmann-Schwinger equation using Gauss-Legendre quadrature [75,76].a Sub-leading πN LECs: c 1 , c 3 , c 4 excluded from the total in the last column.
In this appendix we suppress the conserved quantum numbers s and j, and write the resolution of identity in the partial wave basis as Furthermore, for a stationary proton (mass m p ) and an incoming neutron (mass m n ) with kinetic energy T lab in the laboratory frame of reference, the modulus of the c.m.momentum, p 0 , is given by By inserting the resolution of identity in Eq. ( 19) and discretizing the integral using Gauss-Legendre quadrature with momentum points and weights, {k i , w i } N i=1 , we obtain Here, p 0 denotes the on-shell momentum for a given scattering energy T lab given by Eq. (B2).Doing some manipulations and converting the +iϵ prescription to a principal value we obtain [67,76] ⟨p ′ , ℓ All potentials are regulated using Eq. ( 16) and at sufficiently high momentum, Λ, all potential matrix elements are essentially zero.This means that the integral in Eq. (B4) is well represented by the discretized sum where the momentum points and weights {k i , w i } N i=1 are chosen using Gauss-Legendre quadrature in the interval [0, Λ].The last term in the bracket in Eq. (B5) implements the principal-value integral on the interval [ Λ, ∞] analytically since the grid is just doing the integration on [0, Λ] [77].It is possible to have a grid that extends to numerical infinity, but this generally leads to slower convergence with N .For the calculations in this study, we employ Λ = Λ + 1500 MeV, for both Λ = 500 MeV and Λ = 2500 MeV, which we find sufficient for numerical convergence.

FIG. 2 .
FIG.2.np scattering phase shifts in the 1 D2 (top row) and 3 D2 (bottom row) channels at NLO, N 2 LO, and N 3 LO using a momentum cutoff Λ = 2500 MeV.Phase shifts computed using the perturbative method are shown with black solid lines.The red dashed and dot-dashed lines show the real and imaginary parts, respectively, of the phase shift computed by summing the T -matrix contribution and using the non-perturbative relation between phase shifts and the S-matrix.The black dashed lines show phase shifts from the Nijmegen analysis[62].
FIG.3.Phase shifts in the channels where OPE is non-perturbative and the amplitudes are computed using full distorted-wave perturbation theory.The bands indicate the envelope of the variation due to the two different cutoff values; 500 MeV (dashed line) and 2500 MeV (solid line).Note that LO and NLO results coincide for all channels except 1 S0, which is why the blue NLO band appears to be missing in several panels.The black solid lines show phase shifts from the Nijmegen analysis[62] and the diamond markers indicate the calibration data at T lab values from TableII.

FIG. 4 .
FIG.4.Selection of np scattering observables in the energy interval T lab = 10 to 100 MeV.Experimental data from Refs.[68,69].The bands indicate cutoff variation in the same way as in Fig.3.

FIG. 5 .
FIG.5.Total np cross sections computed by integrating the differential cross sections(27).Panel (a) shows cross sections for a large interval of scattering energies, T lab = 5-150 MeV.Panels (b) and (c) expand results at low-and high-energy intervals, respectively.The bands indicate cutoff variation as in Fig.3.Experimental data from Refs.[68,69].

) 3 P 1 p
′ p Appendix B: Numerical implementation of distorted-wave perturbation theory

TABLE I .
Potential contributions at each in channels where OPE is treated non-perturbatively (column three) and perturbatively (column four).Detailed expressions for the potentials can be found in Appendix A.

TABLE III .
Potential contributions at each chiral order in the channels where OPE is treated non-perturbatively.This table complements the information in TableI.

TABLE IV .
The number of LECs at each order in the Long and Yang PC.