Theoretical uncertainties of the elastic nucleon-deuteron scattering observables
aa r X i v : . [ nu c l - t h ] M a r Theoretical uncertainties of the elastic nucleon-deuteronscattering observables
R. Skibi´nski, Yu. Volkotrub, J. Golak, K. Topolnicki, and H. Wita la
M. Smoluchowski Institute of Physics,Jagiellonian University, PL-30348 Krak´ow, Poland (Dated: October 18, 2018)
Abstract
Theoretical uncertainties of various types are discussed for the nucleon-deuteron elastic scat-tering observables at the incoming nucleon laboratory energies up to 200 MeV. We are especiallyinterested in the statistical errors arising from uncertainties of parameters of a nucleon-nucleoninteraction. The obtained uncertainties of the differential cross section and numerous scatteringobservables are in general small, grow with the reaction energy and amount up to a few percentat 200 MeV. We compare these uncertainties with the other types of theoretical errors like trun-cation errors, numerical uncertainties and uncertainties arising from using the various models ofnuclear interaction. We find the latter ones to be dominant source of uncertainties of modern pre-dictions for the three-nucleon scattering observables. To perform above mentioned studies we usethe One-Pion-Exchange Gaussian potential derived by the Granada group, for which the covari-ance matrix of its parameters is known, and solve the Faddeev equation for the nucleon-deuteronelastic scattering. Thus beside studying theoretical uncertainties we also show a description of thenucleon-deuteron elastic scattering data by the One-Pion-Exchange Gaussian model and compareit with results obtained with other nucleon-nucleon potentials, including chiral N LO forces fromthe Bochum-Bonn and Moscow(Idaho)-Salamanca groups. In this way we confirm the usefulnessand high quality of the One-Pion-Exchange Gaussian force.
PACS numbers: 21.45.-v, 13.75.Cs, 25.40.Cm . INTRODUCTION One of the main goals of nuclear physics is to establish properties of the nuclear in-teractions. After many years of investigations we are now in position to study details ofthe nuclear forces both from the theoretical as well as the experimental sides. It has beenfound that the three-nucleon (3 N ) system, which allows to probe also the off-energy-shellproperties of the nuclear potential, is especially important for such studies. Moreover, toobtain a precise description of the 3 N data one has to supplement the two-nucleon (2 N )interaction by a 3 N force acting in this system. Currently the structure of 3 N force is stillunclear and many efforts are directed to fix 3 N force properties. However, in order to obtaintrustable and precise information from a comparison of 3 N data with predictions based ontheoretical models it is necessary to take into account, or at least to estimate, in additionto the uncertainties of data also the errors of theoretical predictions.The precision of the experimental data has significantly increased and achieved in recentmeasurements a high level, see e.g. Refs. [1–5] for examples of state-of-the-art experimentalstudies in the three-nucleon sector. Precision of these and other experiments has become sohigh that the question about the uncertainties of the theoretical predictions is very timely [6].In the past the theoretical uncertainties for observables in three-nucleon reactions were es-timated by comparing predictions based on various models of nuclear interactions [7] or byperforming benchmark calculations using the same interaction but various theoretical ap-proaches [8–12]. Such a strategy was dictated by a) a common belief that a poor knowledgeabout the nuclear forces, reflected by the existence of very different models of nuclear in-teraction, is a dominant source of the theoretical uncertainty, b) lack of knowledge aboutthe correlations between nucleon-nucleon ( N N ) potential parameters, c) using inconsistentmodels of 2 N and 3 N forces, and last but not least d) a magnitude of uncertainties of ex-perimental data available at the time. Nowadays these arguments, at least partially, are nolonger valid due to the above mentioned progress in experimental techniques, progress inthe derivation of consistent 2 N and 3 N interactions, e.g. within the Chiral Effective FieldTheory ( χ EFT) [13–17] and due to availability of new models of nuclear forces, where freeparameters are fixing by performing a careful statistical analysis [18, 19]. As a consequence,the estimation of theoretical uncertainties has become again an important issue in theoreticalstudies.An extensive introduction to an error estimation for theoretical models was given inRef. [20], followed by a special issue of J. Phys. G: Nucl. Part. Phys. [21]. In the latterreference many applications of the error estimation to nuclear systems and processes arediscussed. However, omitting the few-nucleon reactions, the authors focus mainly on modelsused in direct fitting to data or on models used in nuclear structure studies. Among theother papers focused on the estimation of theoretical uncertainties of
N N interaction werefer the reader to works by A.Ekstr¨om et al. [22], R.Navarro P´erez et al. [19, 23, 24]and to a recent work by P.Reinert et al. [25]. Simultaneously, the Bayesian approach toestimate uncertainties in the 2 N system was derived in Ref. [26] with some applicationsshown again in Ref. [21]. Beyond the 2 N system, the uncertainty of theoretical modelshas been recently studied in the context of nuclear structure calculations for which such anevaluation is important also from a practical point of view. Namely, predictions for many-nucleon systems require not only a huge amount of advanced computations but also rely,e.g. in the case of the No-Core shell model [27], on extrapolations to large model spaces.A knowledge of precision of the theoretical models is important for efficient use of available2omputer resources.Studies of theoretical uncertainties in few-nucleon reactions are less advanced. Beside theabove mentioned attempts to estimate their magnitudes by means of benchmark calcula-tions most efforts in the field were orientated to estimate uncertainties present in the χ EFTapproach [28]. In this case three sources of theoretical uncertainties have been investigated:the truncation of the chiral expansion at a finite order (what results in the so-called trunca-tion errors), the introduction of regulator functions (what results in a cut-off dependence),and the procedure of fixing values of low-energy constants. A simple prescription how toestimate the truncation errors was proposed by E.Epelbaum and collaborators for the 2 N system [29] and adopted also for 3 N systems, for the case where predictions were based ona two-body interaction [30] only. It was found that both for pure nuclear systems [30], aswell as for electroweak processes [31] the magnitude of truncation errors strongly decreaseswith the order of chiral expansion and at the fifth order (N LO) it becomes relatively small.The prescription of Ref. [29] is in agreement with the Bayesian approach [26], see also therecent work [32] for a discussion of the Bayesian truncation errors for the
N N observables.The dependence of the chiral predictions on used regulator functions and their parametershas been studied since the first applications of chiral potentials to the 2 N and 3 N sys-tems [33–35]. The regulator dependence of chiral forces was broadly discussed in the past,see e.g. [36] and various regulator functions were proposed. The non-local regularizationin the momentum space was initially used and estimations of the theoretical uncertaintiesof the 2 N and many-body observables related to regulators were made by comparing pre-dictions obtained with various values of regularization parameters. It was found that thenon-local regularization leads to an unwanted dependence of observables on the parametersused. This dependence was especially strong for predictions for the nucleon-deuteron ( N d )elastic scattering based on 2 N and 3 N forces at the next-to-next-to-next-to leading order(N LO) of chiral expansion [37] and for the electromagnetic processes in the 3 N systemswhen also the leading meson-exchange currents were taken into account [38, 39]. Theseresults were one of the reasons for introducing another, the so-called ”semi-local” method ofregularization of chiral forces. Such an improved method was presented and applied to the N N system in Refs. [29, 40], leading to weak cut-off dependence of predictions in two-bodysystem at chiral orders above the leading order. Similar picture of weak dependence of pre-dictions based on the chiral forces of Refs. [29, 40] was found for
N d elastic scattering [30]and for various electroweak processes [31]. Also the nuclear structure calculations confirmedthis observation [41, 42].The estimation of the theoretical uncertainties arising from an uncertainty of the potentialparameters (which we will call in the following also a statistical error) has not been studiedyet, to the best of our knowledge, in
N d scattering. Within this paper we investigatehow such statistical uncertainties propagate from the
N N potential parameters to the
N d scattering observables. We also compare them with the remaining theoretical uncertaintiesfor the same observables. To this end we use, for the first time in
N d scattering, the One-Pion-Exchange (OPE) Gaussian
N N interaction derived recently by the Granada group [19].The knowledge of the covariance matrix of the OPE-Gaussian potential parameters is adistinguishing feature of this interaction. This is also crucial for our investigations as we usea statistical approach to estimate theoretical uncertainties. Namely, given the covariancematrix for the potential parameters, we sample 50 sets of the potential parameters and, aftercalculating for each set the 3 N observables, we study statistical properties of the obtainedpredictions. The OPE-Gaussian interaction is described briefly in Sec. II and our method3o obtain statistical errors is discussed step by step in Sec. III. The OPE-Gaussian force hasbeen already used, within the same method, to estimate the statistical uncertainty of the H binding energy [43] which was found to be around 15 keV ( ≈ . N d elastic scattering observables obtained with the OPE-Gaussian force and compare themwith predictions based on the AV18
N N potential [44]. We also discuss various estimatorsof uncertainties in hand for the 3 N scattering observables. In Sec. IV we compare, for a fewchosen observables, the theoretical uncertainties arising from various sources, including thetruncation errors and the regulator dependence. Here, beside the OPE-Gaussian potentialand other semi-phenomenological N N forces, we also use the chiral interaction of Ref. [29, 40]and, for the first time in the
N d scattering, the chiral N LO interaction recently derived bythe Moscow(Idaho)-Salamanca group [17]. Finally, we summarize in Sec. V.
II. FORMALISM
The formalism of the momentum space Faddeev equation is one of the standard techniquesto investigate 3 N reactions and has been described in detail many times, see e.g. [45, 46].Thus we only briefly remind the reader of its key elements.For a given N N interaction V we solve the Lippmann-Schwinger equation t = V + V ˜ G t to obtain matrix elements of the 2 N t operator, with ˜ G being the 2 N free propagator.These matrix elements enter the 3 N Faddeev scattering equation which, neglecting the 3 N force, takes the following form T | φ i = tP | φ i + tP G T | φ i . (2.1)The initial state | φ i is composed of a deuteron and a momentum eigenstate of the projectilenucleon, G is the free 3 N propagator and P is a permutation operator.The transition amplitude for the elastic N d scattering process h φ ′ | U | φ i contains the finalchannel state | φ ′ i and is obtained as h φ ′ | U | φ i = h φ ′ | P G − | φ i + h φ ′ | P T | φ i , (2.2)from which observables can be obtained in the standard way [45].Equation (2.1) is solved in the partial wave basis comprising all 3 N states with thetwo-body subsystem total angular momentum j ≤ N angular momentum J ≤ .Since we obtained the bulk of our results with the OPE-Gaussian interaction [19], webriefly remind now the reader of a structure of this potential. A basic concept at the heartof this force is analogous to the one stated behind the well-known AV18 interaction [44]. TheOPE-Gaussian potential V ( ~r ) is composed of the long-range V long ( ~r ) and the short-range V short ( ~r ) parts V ( ~r ) = V short ( ~r ) θ ( r c − r ) + V long ( ~r ) θ ( r − r c ) , (2.3)where r c =3 fm and the V long ( ~r ) part contains the OPE force and the electromagnetic correc-tions. The V short ( ~r ) component is built from 18 operators ˆ O n , among which 16 are the same4s in the AV18 model. Each of them is multiplied by a linear combination of the Gaussianfunctions F k ( r ) = exp ( − r / (2 a k )), with a k = a k , and the strength coefficients V k,n : V short ( ~r ) = X n =1 ˆ O n " X k =1 V k,n F k ( r ) . (2.4)The free parameter a present in the F k ( r ) functions together with the parameters V k,n have been fixed from the data. It is worth noting that to this end the ”3 σ self-consistentdatabase” [18] was used. It incorporates 6713 proton-proton and neutron-proton data,gathered within the years 1950 to 2013, in the laboratory energy range E lab up to 350 MeV.The careful statistical revision of data and the fitting procedure allowed the authors ofRef. [19] to confirm good statistical properties of their χ fit, e.g. by checking the normalityof residuals. The χ /data for the OPE-Gaussian force is 1.06 as fitted to data enumeratedin Ref. [18]. We have been equipped by the authors of Ref. [19] with 50 sets of parameters { V k,n , a } obtained by a correlated sampling from the multivariate normal distribution witha known covariance matrix (see [47] for details). The OPE-Gaussian model, as having asimilar structure to the AV18 force, but being fitted to the newer data can be regarded as arefreshed version of the standard AV18 model. In the N N sector these two potentials leadto a slightly different description of phase shifts, especially at energies above 150 MeV inthe F and D partial waves [19]. Thus it seems to be interesting to compare predictionsfor N d scattering given by both potentials.Beside the OPE-Gaussian and the AV18 models we show in Sec. IV predictions based ontwo chiral forces at N LO, derived by R.Machleidt and collaborators [17] and by E.Epelbaumand collaborators [29, 40]. In the case of the first of these forces the non-local regularization,applied directly in momentum space, has been used. The regulator function is taken as f ( p ′ , p ) = exp ( − ( p ′ Λ ) − ( pΛ ) ), where n depends on regarded operators (e.g. n = 4 forthe one-pion exchange potential). Three values of the cutoff parameter Λ (450, 500 and550 MeV) were suggested for this potential and are also used in this paper. In the caseof the N LO potential and Λ = 500 MeV the χ / data = 1 .
15 for the combined neutron-proton and proton-proton data in the energy range 0-290 MeV [17]. In this paper we showfor the first time the predictions of this new chiral potential at N LO for the
N d elasticscattering observables. As mentioned above, in the approach of Refs. [29, 40] the semi-localregularization of nuclear forces is performed in coordinate space with the regulator function f ( r ) = [1 − exp ( − ( rR ) )] , where r is the distance between nucleons and R is the regulatorparameter. The authors of Ref. [29] suggested five values of the regulator R =0.8, 0.9, 1.0,1.1, and 1.2 fm. The best description of the N N observables is achieved with R =0.9 fmand R =1.0 fm, and leads to the χ / data ≈ .
14 at R =0.9 fm for the N LO force [25] whenusing the “3 σ -self-consistent database” from Ref. [18]. This value is comparable with theones obtained for the semi-phenomenological potentials. III. THE OPE-GAUSSIAN PREDICTIONS FOR ND SCATTERING AND THEIRSTATISTICAL ERRORSA. Determination of statistical uncertainty in 3N system
To determine the theoretical uncertainty arising from the 2 N potential parameters wetook the following steps: 5. We prepared various sets of the potential parameters.Actually, this step had been already taken by the Granada group as a part of theirstudy of the statistical uncertainty of the H binding energy. They provided us withfifty sets ( S i with i = 1 , . . . ,
50) of 42 potential parameters (drawn from the mul-tivariate normal distribution with known expectation values and covariance matrix)and one set of expectation values of potential parameters ( S ). Such a relatively bigsample of fifty-one sets allows us to obtain statistically meaningful conclusions.2. For each set S i ( i = 0 , , . . . ,
50) we calculated the deuteron wave function and the t matrix, solved, at each considered energy, the Faddeev equation (2.1), calculatedthe scattering amplitude (Eq. (2.2)) and finally computed observables. As a resultthe angular dependence of various scattering observables is known for each set ofparameters S i .The predictions obtained in such a way allow us to study:a) for a given energy E , an observable O , and a scattering angle θ , the empirical probabilitydensity function of the observable O ( E, θ ) resulting when various sets S i , ( i = 1 , . . . , O , both the angular and energy dependencies of results based onvarious sets S i .Based on these studies we can conclude on the measure of statistical uncertainties andquality of elastic N d scattering data description. This is a content of the next two subsec-tions.
B. Measure of statistical uncertainty
Our first task is to choose an estimator of the theoretical uncertainties in question. Due toa big complexity of calculations required to obtain the 3 N scattering observables we are notable a priori to determine analytically the probability distribution function of the resulting3 N predictions and consequently to choose the best estimator to describe the dispersion ofresults. In Figs. 1 and 2 we show the empirical distributions (histograms) of the cross section dσ/d Ω and the nucleon analyzing power A y at the nucleon laboratory energy E =13 MeV andat four c.m. scattering angles: θ c . m . = 30 ◦ , 75 ◦ , 120 ◦ and 165 ◦ . The same observables at thesame θ c . m . angles but at E =200 MeV are shown in Figs. 3 and 4, respectively. It is clear thatthe distribution of the predictions cannot be regarded as the normal distribution. To obtainquantitative information on the distribution we have performed the Shapiro-Wilk test [48],which belongs to the strongest statistical tests of normality. As is seen from the obtainedP-values (the smaller P-value the more unlikely the predictions are normally distributed)given in Figs. 1-4, in many cases the resulting distributions of the cross section and thenucleon analyzing power cannot be regarded with high confidence as normal distributions.This restricts a choice of the dispersion estimators - neither the commonly used confidenceinterval nor the usual estimators for the standard deviation can be used directly as theyare tailored to the normal distribution. Thus we considered the following estimators for thestatistical error of the observable O ( E, θ ) (at a given energy and a scattering angle):6. ∆ ≡ ( max i ( O i ) − min i ( O i ) ), where the minimum and maximum are takenover all predictions based on different sets of the N N potential parameters S i , i =1 , , . . . , ∆ ≡ ( max i ( O i ) − min i ( O i ) ), where the minimum and maximum are taken over34 (68% of 50) predictions based on different sets of the N N potential parameters; Theset of 34 observables is constructed by disposing of the 8 smallest and the 8 biggestpredictions for the observable O ( E, θ );3. IQR – the half of standard estimator of the interquartile range being the differencebetween the third and the first quartile
IQR = Q − Q . For the sample of size 50this corresponds to taking the half of difference between the predictions on 37th and13th position in a sample sorted in the ascending order. The flexibility in applyingthis measure to the non-normal distribution is a great asset to the IQR;4. σ ( O ) – the sample standard deviation σ ( O ) = q n − P ni =1 ( x i − ¯ x ) , where ¯ x is theusual mean value. The disadvantage of this estimator is that on formal grounds itcannot be applied to samples from an arbitrary probability distribution.The ∆ and the σ ( O ) are sensitive to the possible outliers in the sample and thustaking them as estimators of dispersion can lead to overestimation of the statistical error.On the other hand the IQR is calculated using only half of the elements in the sample andthus can lead to underestimation of the theoretical uncertainty. Thus we decided to adaptthe ∆ as an optimal measure of predictions’ dispersion and consequently as an estimatorof the theoretical uncertainty in question. The same choice has been made in a study ofthe statistical error of the H binding energy in Ref. [47]. The similarity to the standarddeviation is one more advantage of ∆ since the comparison of the theoretical errorswith the experimental (statistical) uncertainties, delivered usually in the form of standarddeviations, is finally unavoidable.However, in Tab. I we compare values of the above mentioned estimators for the N d elasticscattering differential cross section at three energies of the incoming nucleon and at four c.m.scattering angles. By definition IQR ≤ ∆ ≤ ∆ and indeed this is observed inTab. I. The magnitudes of the ∆ is very close to the measure based on the samplestandard deviation σ ( dσ/d Ω) and in practice it does not matter which of these estimators isused. The relative uncertainty (exemplified in the Tab. I for the sample standard deviation)remains below 1% for all scattering angles at E = 13 MeV and E = 65 MeV, and onlyslightly exceeds it at E = 200 MeV. In Tab. I we also show values of the differential crosssection obtained with the central values of the OPE-Gaussian potential parameters andmean values of predictions calculated separately for the 50 ( M ) or 34 ( M ) sets ofparameters S i . Also here in most of the cases dσ/d Ω( S ) ≈ M ≈ M , what shows,that the predictions based on sets S i for i = 0 cluster around dσ/d Ω( S ) evenly. The otherobservables behave in a similar way. C. Nucleon-deuteron elastic scattering observables from the OPE-Gaussian model
In the following we present predictions obtained with the OPE-Gaussian
N N interactionfor various observables in the elastic neutron-deuteron scattering process at incoming nucleonlaboratory energies E = 13 MeV, 65 MeV, and 200 MeV. We will focus on the elastic7 [MeV] θ c . m . [deg] d σ /dΩ(S ) ∆ ∆
68% 12
IQR σ ( dσ/d Ω) M M
30 134.9970 0.1780 0.1025 0.0635 0.0954 (0.132%) 135.0040 135.010013.0 75 51.3274 0.0315 0.0153 0.0110 0.0149 (0.061%) 51.3283 51.3295120 9.7437 0.0347 0.0181 0.0118 0.0179 (0.356%) 9.7421 9.7420165 103.1210 0.1085 0.0420 0.0230 0.0462 (0.105%) 103.1190 103.119030 23.7000 0.1785 0.0812 0.0569 0.0824 (0.753%) 23.7137 23.709265.0 75 2.3630 0.0134 0.0060 0.0040 0.0057 (0.568%) 2.3630 2.3630120 0.7787 0.0035 0.0015 0.0011 0.0016 (0.451%) 0.7786 0.7785165 4.7537 0.0174 0.0076 0.0060 0.0075 (0.366%) 4.7532 4.753530 3.7626 0.0351 0.0164 0.0097 0.0162 (0.325%) 3.7634 3.7625200.0 75 0.2088 0.0018 0.0008 0.0005 0.0008 (0.839%) 0.2087 0.2087120 0.0585 0.0006 0.0004 0.0003 0.0003 (1.069%) 0.0589 0.0589165 0.1645 0.0022 0.0009 0.0007 0.0009 (1.356%) 0.1647 0.1647
TABLE I: The differential cross section d σ /dΩ obtained with the expectation values of theOPE-Gaussian potential parameters (set S ), various estimators of its dispersion (see text)and mean values taken from 50 ( M ) or 34 ( M ) predictions. In case of the samplestandard deviation σ ( dσ/d Ω) also the relative magnitude σ ( dσ/d Ω) / ( dσ/d Ω( S )) ∗ − ]. σ /d Ω ( Θ c.m. =30°) [mb sr -1 ]246810 51.285 51.3 51.315 51.33 51.345 51.36d σ /d Ω ( Θ c.m. =75°) [mb sr -1 ]02468109.708 9.72 9.732 9.744 9.756 9.768 9.78d σ /d Ω (Θ c.m. =120°) [mb sr -1 ]0246810 102.99 103.02 103.05 103.08 103.11 103.14 103.17 103.2 103.23d σ /d Ω ( Θ c.m. =165°) [mb sr -1 ]02468101214 a)P = 0.13 b)P = 0.24c)P = 0.49 d)P = 0.60 FIG. 1: The histograms and the P-values for the Shapiro-Wilk test for the elastic
N d scattering differential cross section d σ /dΩ [mb sr − ] at the incoming nucleon laboratoryenergy E =13 MeV and the scattering angle: a) θ c . m . = 30 ◦ , b) θ c . m . = 75 ◦ , c) θ c . m . = 120 ◦ ,and d) θ c . m . = 165 ◦ , obtained with 50 sets of the OPE-Gaussian potential parameters.scattering cross section d σ /dΩ, the nucleon vector analyzing power A y , the nucleon tonucleon spin transfer coefficients K y ′ y , and the spin correlation coefficients C y , y . However, wewill also give examples for other observables.The N d cross section is shown in Fig. 5. Apart from the solid line which representspredictions based on the OPE-Gaussian force when the expectation values of its parameters8 .0247 0.02475 0.0248 0.02485 0.0249 0.02495 0.025A y ( Θ c.m. =30°)024681012 0.03645 0.0366 0.03675 0.0369 0.03705 0.0372A y ( Θ c.m. =75°)02468100.1288 0.1292 0.1296 0.13 0.1304 0.1308 0.1312 0.1316A y ( Θ c.m. =120°)024681012 0.0161 0.0162 0.0163 0.0164 0.0165A y ( Θ c.m. =165°)0246810 a)P = 0.93 b)P = 0.90c)P = 0.99 d)P = 0.41 FIG. 2: The histograms and the P-values for the Shapiro-Wilk test for the nucleonanalyzing power A y in N d elastic scattering at the incoming nucleon laboratory energy E =13 MeV and the scattering angle: a) θ c . m . = 30 ◦ , b) θ c . m . = 75 ◦ , c) θ c . m . = 120 ◦ , and d) θ c . m . = 165 ◦ , obtained with 50 sets of the OPE-Gaussian potential parameters. σ /d Ω ( Θ c.m. =30°) [mb sr -1 ]0246810121416 0.2065 0.207 0.2075 0.208 0.2085 0.209 0.2095 0.21 0.2105d σ /d Ω ( Θ c.m. =75°) [mb sr -1 ]024681012140.0582 0.0585 0.0588 0.0591 0.0594 0.0597d σ /d Ω (Θ c.m. =120°) [mb sr -1 ]0246810 0.1624 0.1632 0.164 0.1648 0.1656 0.1664d σ /d Ω ( Θ c.m. =165°) [mb sr -1 ]02468101214 a)P = 0.13 b)P = 0.21c)P = 0.5 d)P = 0.6 FIG. 3: The same as in Fig. 1 but at E =200 MeV.(set S ) are used, we also show the red band representing the range of predictions obtainedwith the same 34 sets S i as used to calculate ∆ , and the blue dashed curve showingresults obtained with the AV18 interaction. The nucleon-deuteron data (at the same ornearby energies) are also added for the sake of comparison. The predictions based on theOPE-Gaussian force are in agreement with the predictions based on the AV18 potential.Only small, ( ≈ .
9% at E =13 MeV and ≈ E =200 MeV), differences are seenin the minimum of the cross section. Similarly to the AV18, the OPE-Gaussian modelclearly underestimates the data at two higher energies reflecting the known fact of growingimportance of a 3 N force [49, 50]. The statistical error arising from the uncertainty of the9 .65 0.651 0.652 0.653 0.654 0.655A y ( Θ c.m. =30°)024681012 -0.514 -0.512 -0.51 -0.508 -0.506 -0.504A y ( Θ c.m. =75°)0246810-0.36 -0.355 -0.35 -0.345A y ( Θ c.m. =120°)051015 0.125 0.13 0.135A y ( Θ c.m. =165°)02468101214 d)P = 0.41c)P = 0.9 b)P = 0.9a)P = 0.95 FIG. 4: The same as in Fig. 2 but at E =200 MeV. Θ c.m. [deg]10100 d σ /d Ω [mb sr -1 ] Θ c.m. [deg]110 0 30 60 90 120 150 180 Θ c.m. [deg]0,1110 a) b) c) FIG. 5: The
N d elastic scattering cross section d σ /dΩ [mb sr − ] at the incoming nucleonlaboratory energy a) E =13 MeV, b) E =65 MeV, and c) E =200 MeV as a function of thec.m. scattering angle θ c . m . . The black curve represents predictions obtained with thecentral values of the OPE-Gaussian parameters, the red band reflects statisticaluncertainty discussed in this subsection, and the blue dashed curve represents predictionsbased on the AV18 force. The data are in b) from Ref. [51] ( pd black pluses) and [52] ( nd orange circles) and in c) from Ref. [53] ( pd , E = 198 MeV, violet squares), Ref. [54] ( pd , E = 180 MeV, orange x’s), and Ref. [55] ( pd , E = 198 MeV, black circles). N N force parameters is in all cases very small and red bands are hardly visible in Fig. 5.The OPE-Gaussian force delivers predictions which are very close to the AV18 resultsalso for the most of the polarization observables at the energies studied here. Likewise thedispersion of predictions remains small for most of the polarization observables. Below wediscuss a few of them, choosing mainly ones with the largest statistical uncertainties.Let us start, however, with the nucleon analyzing power A y , shown in Fig. 6. Here theuncertainties remain negligible at all energies and also the differences between predictions10
30 60 90 120 150 180 Θ c.m. [deg]00.050.10.150.20.25 A y Θ c.m. [deg]-0.6-0.4-0.200.2 0 30 60 90 120 150 180 Θ c.m. [deg]-0.4-0.200.20.40.6 a) b) c) FIG. 6: The nucleon analyzing power A y for N d elastic scattering at the same energies asused in Fig. 5 as a function of the c.m. scattering angle θ c . m . . Curves and band as inFig. 5. The data in a) are from Ref. [56] ( nd black pluses), in b) are from Ref. [51] ( pd black pluses) and Ref. [52] ( nd orange circles), and in c) are from Ref. [53] ( pd violetsquares), Ref. [3] ( pd E = 200 MeV orange circles), Ref. [57] ( pd E = 197 MeV blacktriangles up), and Ref. [58] ( pd blue x’s).based on the OPE-Gaussian force and the ones obtained with the AV18 potential are tiny.Thus we see that the OPE-Gaussian model does not deliver any hint on the nature of theA y puzzle at E = 13 MeV.We have chosen the nucleon to nucleon spin transfer coefficient K y ′ y and the spin correla-tion coefficient C y , y to demonstrate, in Figs. 7 and 8, respectively, changes of the statisticalerrors when increasing the reaction energy. For both observables dispersion of the resultsgrows with energy, and while at lowest energy E =13 MeV it is negligible, at E =200 MeVits size is bigger, although it remains small ( ∆ < . y , y compari-son with the data reveals that the spread of the OPE-Gaussian results is still smaller thanuncertainties of experimental results.In Fig. 9 we show two observables for which the difference between the AV18 predictionsand the OPE-Gaussian results is especially big already at the two lower energies. They arethe spin correlation coefficient C xx , y -C yy , y at E =13 MeV and the deuteron to nucleon spintransfer coefficient K x ′ yz at E =65 MeV. The difference between both predictions amountsto ≈
19% at the minimum of C xx , y -C yy , y , while the statistical error of the OPE-Gaussianresults is only ≈ x ′ yz these differences amount to ≈
23% and ≈ N N interaction.The statistical errors grow with the reaction energy. Thus in Fig. 10 we show for E =200 MeV a few observables with the largest uncertainties. Beside the spin transfer coefficientK y ′ y already shown in Fig. 7 they are the deuteron tensor analyzing powers T and T andthe nucleon to deuteron spin transfer coefficient K x ′ x ′ y -K y ′ y ′ y . While the bands representingthe theoretical uncertainties are clearly visible, they still remain small compared to theexperimental errors for both analyzing powers. The differences between predictions basedon the AV18 potential and the OPE-Gaussian force are small. This is true also for the otherNd elastic scattering observables both at E = 200 MeV and at the lower energies, so we11
30 60 90 120 150 180 Θ c.m. [deg]0.40.60.8 K yy’ Θ c.m. [deg]0.20.40.60.8 0 30 60 90 120 150 180 Θ c.m. [deg]0.70.80.9 a) b) c) FIG. 7: The nucleon to nucleon spin transfer coefficient K y ′ y at the incoming nucleonlaboratory energy a) E =13 MeV, b) E =65 MeV and c) E = 200 MeV as a function of thec.m. scattering angle θ c . m . . See Fig.5 for a description of band and curves. Θ c.m. [deg]-0.10.00.10.20.30.4 C y,y Θ c.m. [deg]00.20.40.6 0 30 60 90 120 150 180 Θ c.m. [deg]-0.200.20.40.6 a) b) c) FIG. 8: The spin correlation coefficient C y , y at the incoming nucleon laboratory energy a) E =13 MeV, b) E =65 MeV and c) E =200 MeV as a function of the c.m. scattering angle θ c . m . . See Fig.5 for a description of band and curves. In c) data are from Ref. [57] ( pdE = 197 MeV, orange circles) and Ref. [3] ( pd E = 200 MeV, black pluses).conclude that the OPE-Gaussian force yields a similar description of this process comparedwith the AV18 potential. IV. COMPARISON OF VARIOUS THEORETICAL UNCERTAINTIES IN NDSCATTERING
It is interesting to compare the statistical error ∆ obtained in the previous sectionwith the other uncertainties (like the uncertainty arising from using the various models ofnuclear interaction, the uncertainty introduced by the partial wave decomposition approach,the truncation errors of chiral predictions and the uncertainties originating in the cut-off12
30 60 90 120 150 180 Θ c.m. [deg]-0.08-0.040 C xx,y -C yy,y a) Θ c.m. [deg]-0.12-0.0600.06 K yzx’ b) FIG. 9: The spin correlation coefficient C xx , y -C yy , y at the incoming nucleon laboratoryenergy E =13 MeV (a) and the deuteron to nucleon spin transfer coefficient K x ′ yz at theincoming nucleon laboratory energy E =65 MeV (b) as a function of the c.m. scatteringangle θ c . m . . Curves and band are as in Fig.5. Θ c.m. [deg]-0.5-0.4-0.3-0.2-0.10.0 T Θ c.m. [deg]-0.6-0.4-0.20.0 T a) b) Θ c.m. [deg]-0.6-0.4-0.200.2 K yx’x’ -K yy’y’ c) FIG. 10: The deuteron tensor analyzing powers T (a) and T (b) and the nucleon todeuteron spin transfer coefficient K x ′ x ′ y -K y ′ y ′ y (c) for E = 200 MeV as a function of thecenter of mass scattering angle θ c . m . . See Fig.5 for description of bands and curves. TheT and T data are from Ref. [1] ( pd E = 186 . pdE = 200 MeV black circles)dependence of chiral forces) present in the elastic N d scattering studies and specifically inour approach.The accuracy of predictions arising from the algorithms used in our numerical scheme,which comprises, among others, numerical integrations, interpolations and series summa-tions, is well under control. This has been tested, e.g. by using various grids of mesh points,or more generally by benchmark calculations involving different methods to treat
N d scat-tering [9–12]. The main contribution to theoretical uncertainties comes in our numericalrealization from using a truncated set of partial waves. Typically we restrict ourselves to13artial waves with the two-body total orbital momentum j ≤
5. Predictions for observablesconverge with increasing j , as was documented e.g. in [45]. In the following we compare theOPE-Gaussian predictions, shown in the previous section, based on all two-body channelsup to j = 5 with the predictions based on all channels up to j = 4 only to remind the readersome facts about the convergence of our approach. However, since the differences between(not shown here) predictions based on all channels up to j = 6 and those with j max = 5are, based on results with other NN potentials, smaller than this for j max = 5 and j max = 4predictions, the latter difference very likely overestimates the uncertainty arising from ourcomputational scheme. A recent work [59] compares predictions for the elastic N d scatter-ing, based however only on the driving term of Eq. (2.1), obtained within the partial waveformalism with the ones from the ”three-dimensional” approach, i.e. the approach whichtotally avoids the partial wave decomposition and uses momentum vectors. A very goodagreement between the partial waves based results and the ”three-dimensional” ones con-firms that neglecting the higher partial waves in the calculations presented here practicallydoes not affect our predictions.Next, we would like to focus on the truncation errors and the cut-off dependence presentin the chiral calculations and last but not least on the differences between predictions basedon various models of the nuclear two-body interaction.To estimate two types of theoretical uncertainties present when chiral potentials are usedwe calculated the elastic
N d scattering observables using two
N N interactions at the N LO:one delivered by E.Epelbaum et al. [40] (E-K-M force) and the other derived by D.R.Entemet al. [17] (E-M-N force). In the case of the E-K-M model the semi-local regularization withthe cut-off parameter R in the range between 0.8 fm and 1.2 fm is used and the breakdownscale of the χ EFT is 0.4-0.6 GeV [40]. The E-M-N model uses the chiral breaking scale of1 GeV and the cut-off parameter Λ for non-local regularization lies between 450 MeV and550 MeV [17].The truncation errors δ ( O ) ( i ) of an observable O at i -th order of the chiral expansion,with i = 0 , , , . . . , when only two-body interaction is used, can be estimated as [30]: δ ( O ) (0) ≥ max (cid:0) Q | O (0) | , | O ( i ≥ − O ( j ≥ | (cid:1) ,δ ( O ) (2) = max (cid:0) Q | O (0) | , Q | ∆ O (2) | , | O ( i ≥ − O ( j ≥ | (cid:1) ,δ ( O ) ( i ) = max (cid:0) Q i +1 | O (0) | , Q i − | ∆ O (2) | , Q i − | ∆ O (3) | (cid:1) for i ≥ , (4.1)where Q denotes the chiral expansion parameter, ∆ O (2) ≡ O (2) − O (0) and ∆ O ( i ) ≡ O ( i ) − O ( i − for i ≥
3. In addition conditions: δ ( O ) (2) ≥ Qδ ( O ) (0) and δ ( O ) ( i ) ≥ Qδ ( O ) ( i − for i ≥ δ ( O ) ( i ) in case when at higher orders 3N force isnot included into calculations [30].The uncertainty arising from the cut-off dependence can be easily quantified - we justtake the difference between the minimal and the maximal prediction, separately for the E-K-M force and for the E-M-N model. However, one has to be aware that in the case of theE-K-M force the cut-off values between R = 0 . R = 1 . N system. Thus in the following we separately discuss the whole range of regulator values(0 . ≤ R ≤ . . ≤ R ≤ . σ /dΩ,the nucleon analyzing power A y , the deuteron tensor analyzing power T , and the spincorrelation coefficient C y , y is given in Figs. 11-14, respectively. In these figures, in eachsubplot, the predicted value of the observable is given at the bottom horizontal axis andthe vertical lines are used to mark predictions based on different N N forces and length ofthese lines has no meaning. The top horizontal axis shows the percentage relative difference N ( O ) with respect to the OPE-Gaussian prediction and its ticks are calculated as ˜ x =( x − O OPE − Gaussian O OPE − Gaussian ) ∗ ∗ sign ( O OPE − Gaussian ) where x are the ticks values shown at the bottomaxis. In addition, for the sake of figures’ clarity, the ˜ x ’s are rounded to the two digits only.Note, that the magnitude of such a relative difference depends on the magnitude of the OPE-Gaussian prediction and can increase to infinity as the OPE-Gaussian prediction approacheszero. The OPE-Gaussian results (at the central values of the parameters) are represented byvertical red lines, the AV18 ones by the black line, the CD-Bonn predictions by the blue line,the E-K-M N LO R = 0 . LO R = 1 . LO Λ=500 MeV ones by the green line.Horizontal lines represent magnitudes of various theoretical uncertainties and starting fromthe bottom they are: statistical error for the OPE-Gaussian model (the red line), differencebetween OPE-Gaussian predictions based on the j max = 5 and j max = 4 calculations (theorange line), regulator dependence for the E-K-M N LO force in range R =0.8-1.2 fm (thesolid magenta line), the truncation error for the E-K-M N LO R = 0 . LO force in range Λ=450-550 MeV(the solid green line), and truncation error for the E-M-N N LO Λ=500 MeV potential(the dashed green line). Further, subplots in various rows in Figs. 11-14 show predictionsat different incoming nucleon lab. energies, which are E = 13 MeV (top), E = 65 MeV(middle) and E = 200 MeV (bottom). Finally, the various columns show predictions atdifferent scattering angles: 30 ◦ , 75 ◦ , 120 ◦ , and 165 ◦ moving from the left to the right.An analysis of Figs. 11-14 leads to the following conclusions:1. In general, all models investigated here provide similar results, which differ only bya few percent at lower energies but differences between predictions grow with theincreasing energy. There is no single model which gives systematically the smallest orthe biggest value. There are also no two models, whose predictions for all the cases lieclose to each other. Note, the above statements describe general trends but exceptionsfrom this pattern for specific observables and angles are possible.2. At all energies the dominant theoretical uncertainty is the one arising from usingvarious models of the nuclear interaction.3. The statistical errors for the OPE-Gaussian predictions are small (and with no prac-tical importance) for all the considered observables and energies.4. The difference between j max = 5 and j max = 4 predictions, as expected, grows withenergy, however, it remains small, when compared to other uncertainties, even at E = 200 MeV (with the only exceptions of the T at 200 MeV and C y , y at 65MeV). Thus the uncertainty bound with partial wave decomposition and numericalperformance is also negligible. 15. The OPE-Gaussian predictions based on the central values are always inside the rangegiven by the statistical errors. The E-K-M results show monotonic behaviour of thepredicted observables with the regulator value. In the case of the E-M-N force themiddle value of regulator (Λ = 500 MeV) delivers extreme (among the E-M-N ones)predictions in many cases.6. The difference between predictions based on the two chiral N LO models used (E-K-Mand E-M-N) is not smaller than the difference between any other pair of predictionsbased on different
N N potentials. This suggests that there are substantial differencesin the construction each of these models. Thus it seems mandatory to regard thesemodels independently, as two different models of nuclear forces.7. In numerous cases the two chiral approaches deliver results separated from each otherby more than the estimated uncertainty for their predictions. This again points todifferences between both chiral potentials (and/or to an underestimation of the corre-sponding total theoretical uncertainties).8. In the case of both chiral models, the dominant uncertainty at lower energies arisesfrom the cut-off dependence. This uncertainty is much bigger than the remainingtypes of errors, except for differences between various models. At higher energies thetruncation errors are also important in some specific cases, e.g. the differential crosssection at θ c . m . = 120 ◦ at E = 200 MeV. In the case of A y at E =200 MeV and smallerangles, the truncation errors exceed the regulator dependence for the E-K-M potential.9. In the case of the N LO E-K-M potential, the difference between predictions for R = 0 . R = 1 . N N system) is of the same size as the typical difference between any other pair ofpredictions, what shows strong sensitivity of the observables to the regulator param-eter.10. Comparing the cutoff dependence of both chiral models we can conclude, that thedispersion of their predictions behaves for both models in a correlated way, i.e. abig cutoff dependence for the E-M-N force usually appears together with a big cutoffdependence for the E-K-M potential.11. The truncation errors for the E-M-N force are smaller than these for the E-K-M in-teraction. The reason for this is the bigger value of the chiral breaking scale in theE-M-N approach, which results in different values of Q parameter in Eq. (4.1).Next, it is interesting to compare the size of the theoretical errors presented in Figs. 11-14to experimental errors of available data. In order not to leave the reader with the impressionthat the modern theoretical models of nuclear interactions yield a chaotic description of the N d scattering observables, in Fig. 15 we compare, in a few examples, previously presentedpredictions with the experimental results. This establishes an absolute scale in which onehas to peer at the problem of discrepancies between various theoretical models.Examples given in Fig. 15 show various possible locations of theoretical predictions anddata. The differential cross section at E = 65 MeV and E = 200 MeV at the scatteringangle θ c . m . = 120 ◦ is shown in the upper row and the analyzing power A y for the same angleand energies is displayed below. In the case of the cross section we see that at E = 65 MeV16
30 131 132 133 134 135 136 137 138 139 140d σ /d Ω c.m. ( Θ c.m. =30 o ) [mb sr -1 ]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc-3.70 -2.96 -2.22 -1.48 -0.74 0.00 0.74 1.48 2.22 2.97 3.71N(d σ /d Ω c.m. ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc σ /d Ω c.m. ( Θ c.m. =75 o ) [mb sr -1 ]-1.61 -0.64 0.34 1.31 2.28N(d σ /d Ω c.m. ) [%] 9.5 9.6 9.7 9.8 9.9 10 10.1 10.2d σ /d Ω c.m. ( Θ c.m. =120 o ) [mb sr -1 ]-2.50 -1.48 -0.45 0.58 1.60 2.63 3.66 4.68N(d σ /d Ω c.m. ) [%] 102 102.5 103 103.5 104 104.5d σ /d Ω c.m. ( Θ c.m. =165 o ) [mb sr -1 ]-1.09 -0.60 -0.12 0.37 0.85 1.34N(d σ /d Ω c.m. ) [%]
23 23.2 23.4 23.6 23.8 24d σ /d Ω c.m. ( Θ c.m. =30 o ) [mb sr -1 ]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc-2.95 -2.11 -1.26 -0.42 0.42 1.27N(d σ /d Ω c.m. ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc σ /d Ω c.m. ( Θ c.m. =75 o ) [mb sr -1 ]-2.66 -0.55 1.57 3.68 5.80 7.92N(d σ /d Ω c.m. ) [%] 0.78 0.8 0.82 0.84d σ /d Ω c.m. ( Θ c.m. =120 o ) [mb sr -1 ]0.17 2.74 5.31 7.87N(d σ /d Ω c.m. ) [%] 4.6 4.8 5 5.2 5.4d σ /d Ω c.m. ( Θ c.m. =165 o ) [mb sr -1 ]-3.23 0.98 5.18 9.39 13.60N(d σ /d Ω c.m. ) [%] σ /d Ω c.m. ( Θ c.m. =30 o ) [mb sr -1 ]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc -20.27 -14.95 -9.64 -4.32 0.99 6.31N(d σ /d Ω c.m. ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc σ /d Ω c.m. ( Θ c.m. =75 o ) [mb sr -1 ]-32.95 -23.38 -13.80 -4.22 5.36N(d σ /d Ω c.m. ) [%] 0.05 0.055 0.06 0.065 0.07d σ /d Ω c.m. ( Θ c.m. =120 o ) [mb sr -1 ]-15.18 -6.70 1.78 10.26 18.74N(d σ /d Ω c.m. ) [%] 0.15 0.2 0.25d σ /d Ω c.m. ( Θ c.m. =165 o ) [mb sr -1 ]-9.01 21.32 51.65N(d σ /d Ω c.m. ) [%] FIG. 11: The
N d elastic scattering differential cross section and various theoreticaluncertainties at four scattering angles: θ c . m . = 30 ◦ (1st column), 75 ◦ (2nd column), 120 ◦ (3rd column), and 165 ◦ (4th column) and at three scattering energies: E = 13 MeV (theupper row), E = 65 MeV (the middle row), and E = 200 MeV (the bottom row). Thex-axis at the bottom shows the values of the cross section, the x-axis at the top shows therelative difference of predictions with respect to the OPE-Gaussian results. The verticallines show the position of the cross section obtained with the AV18 force (black line), theOPE-Gaussian force (red line), the E-M-N N LO Λ = 500 MeV force (green line), theCD-Bonn (blue line), the E-K-M N LO R =0.9 fm force (magenta solid line), and theE-K-M N LO R =1.0 fm force (magenta dashed line). The horizontal lines represent (fromthe bottom) the statistical error (red line), the difference between the OPE-Gaussianpredictions with j max = 5 and j max = 4 (orange line) , the regulator dependence for theE-K-M force (magenta solid line), the truncation error for the E-K-M force (magentadashed line), the regulator dependence for the E-M-N force (green solid line), and, at thetop, the truncation error for the E-M-N potential (green dashed line), see text for details.there are discrepancies between various theoretical predictions and the data of differentmeasurements. While the theoretical predictions are close one to each other, the data arescattered. One experimental point overlaps within its statistical error with some of thepredictions, another one would be in agreement with predictions within 3 σ distance and theremaining experimental point is further from the data by more than its 3 σ uncertainty. At E = 200 MeV a clear discrepancy between all predictions, which again are close together,and all data is observed. This discrepancy can be traced back to action of 3 N force athigher energies [49, 50]. The picture is more complex for the analyzing power. Here, at E = 65 MeV the experimental data and predictions differ by more than experimental error17 .023 0.024 0.025 0.026 0.027A y ( Θ c.m. =30 o )OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc-7.40 -3.37 0.66 4.68 8.71N(A y ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc y ( Θ c.m. =75 o )-7.48 -2.04 3.40 8.84N(A y ) [%] 0.128 0.13 0.132 0.134 0.136 0.138A y ( Θ c.m. =120 o )-1.68 -0.14 1.39 2.93 4.47 6.00N(A y ) [%] 0.016 0.0165 0.017 0.0175A y ( Θ c.m. =165 o )-2.01 1.05 4.11 7.18N(A y ) [%] y ( Θ c.m. =30 o )OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc-2.11 0.40 2.91 5.42 7.93N(A y ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc -0.05 -0.04 -0.03A y ( Θ c.m. =75 o )-46.06 -16.85 12.36N(A y ) [%] -0.53 -0.52 -0.51 -0.5A y ( Θ c.m. =120 o )-3.13 -1.18 0.76 2.71N(A y ) [%] 0.078 0.08 0.082 0.084 0.086 0.088A y ( Θ c.m. =165 o )-5.31 -2.88 -0.45 1.98 4.40 6.83N(A y ) [%] y ( Θ c.m. =30 o )OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc-5.09 -3.55 -2.02 -0.49 1.04 2.57 4.10N(A y ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc -0.56 -0.54 -0.52 -0.5 -0.48A y ( Θ c.m. =75 o )-10.07 -6.14 -2.21 1.72 5.65N(A y ) [%] -0.4 -0.3 -0.2 -0.1 0A y ( Θ c.m. =120 o )-14.13 14.40 42.93 71.47 100.00N(A y ) [%] -0.05 0 0.05 0.1 0.15A y ( Θ c.m. =165 o )-138.88 -100.00 -61.12 -22.24 16.64N(A y ) [%] FIG. 12: The same as in Fig. 11 but for the nucleon analyzing power A y . -0.0114 -0.0112 -0.011 -0.0108 -0.0106T ( Θ c.m. =30 o )OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc -5.99 -4.13 -2.27 -0.41 1.45N(T ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc -0.05 -0.049 -0.048 -0.047 -0.046 -0.045T ( Θ c.m. =75 o )-7.80 -5.65 -3.49 -1.33 0.82 2.98N(T ) [%] -0.086 -0.084 -0.082 -0.08 -0.078T ( Θ c.m. =120 o )-9.37 -6.83 -4.28 -1.74 0.80N(T ) [%] -0.0148 -0.0146 -0.0144 -0.0142T ( Θ c.m. =165 o )-4.69 -3.27 -1.86 -0.44N(T ) [%] -0.031 -0.03 -0.029 -0.028 -0.027 -0.026T ( Θ c.m. =30 o )OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc-14.94 -11.23 -7.52 -3.82 -0.11 3.60N(T ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc -0.13 -0.12 -0.11 -0.1T ( Θ c.m. =75 o )-25.40 -15.75 -6.11 3.54N(T ) [%] -0.15 -0.14 -0.13T ( Θ c.m. =120 o )-9.34 -2.05 5.24N(T ) [%] -0.074 -0.073 -0.072 -0.071T ( Θ c.m. =165 o )-4.91 -3.50 -2.08 -0.66N(T ) [%] -0.14 -0.136 -0.132T ( Θ c.m. =30 o )OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc -7.81 -4.72 -1.64N(T ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc -0.38 -0.36 -0.34 -0.32 -0.3T ( Θ c.m. =75 o )-19.46 -13.17 -6.89 -0.60 5.69N(T ) [%] -0.28 -0.24 -0.2 -0.16T ( Θ c.m. =120 o )-20.39 -3.19 14.01 31.20N(T ) [%] -0.18 -0.17 -0.16 -0.15 -0.14T ( Θ c.m. =165 o )-15.12 -8.73 -2.33 4.06 10.46N(T ) [%] FIG. 13: The same as in Fig. 11 but for the deuteron tensor analyzing power T .18 .26 0.265 0.27 0.275C y,y ( Θ c.m. =30 o )OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc-1.70 0.19 2.08 3.97N(C y,y ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc y,y ( Θ c.m. =75 o )-0.51 1.30N(C y,y ) [%] -0.06 -0.05 -0.04 -0.03 -0.02C y,y ( Θ c.m. =120 o )-23.36 -2.80 17.76 38.32 58.88N(C y,y ) [%] 0.322 0.324 0.326 0.328 0.33 0.332C y,y ( Θ c.m. =165 o )-1.11 -0.49 0.12 0.74 1.35 1.97N(C y,y ) [%] y,y ( Θ c.m. =30 o )OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc-4.40 1.98 8.35 14.72N(C y,y ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc y,y ( Θ c.m. =75 o )-1.64 -0.03 1.58 3.19 4.80N(C y,y ) [%] 0.37 0.38 0.39 0.4 0.41C y,y ( Θ c.m. =120 o )-4.54 -1.96 0.62 3.20 5.78N(C y,y ) [%] 0.29 0.295 0.3C y,y ( Θ c.m. =165 o )-1.79 -0.10 1.59N(C y,y ) [%] y,y ( Θ c.m. =30 o )OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc-13.93 -4.36 5.20N(C y,y ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc y,y ( Θ c.m. =75 o )-34.41 -12.55 9.31N(C y,y ) [%] 0.5 0.55 0.6 0.65 0.7 0.75C y,y ( Θ c.m. =120 o )-9.83 -0.81 8.21 17.22 26.24 35.26N(C y,y ) [%] 0.2 0.25 0.3C y,y ( Θ c.m. =165 o )-31.53 -14.41 2.70N(C y,y ) [%] FIG. 14: The same as in Fig. 11 but for the spin correlation coefficient C y , y .but they already agree within the 2 σ range. At E = 200 the experimental statisticalerror is much smaller than the distances between various theoretical predictions and theuncertainties related to the chiral forces. Such a mixed pattern clearly calls for furtherwork on reducing both the theoretical and experimental uncertainties to avoid misleadingconclusions about the properties of the nuclear interactions. The presented here examplesat one scattering angle only show that it is much more reliable to draw conclusions based ona comparison of predictions with data in a wider range of scattering angles and at differentenergies. Especially, these examples do not contradict strong effects of the 3 N force in theminimum of the differential cross section at higher energies [49, 50]. Such conclusions arebased on a systematic comparison of predictions with the data at numerous scattering anglesand energies. V. SUMMARY
We have employed the OPE-Gaussian potential of the Granada group to describe theelastic
N d scattering at energies up to 200 MeV. The OPE-Gaussian potential is one of thefirst models of nuclear forces for which the covariance matrix of its free parameters is known.This gives an excellent opportunity to study the propagation of uncertainties from the 2 N potential parameters to 3 N observables. Therefore, for the same process, we also studiedthe statistical errors of our predictions.The description of data delivered by the OPE-Gaussian force is in quantitative agree-ment with picture obtained using other N N potentials, especially the AV18 model, which19 .7 0.8 0.9 1 1.1d σ /d Ω c.m. [mb sr -1 ] ( Θ c.m. =120 o )OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc-10.10 2.74 15.58 28.42 41.26N(d σ /d Ω c.m. ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc 0.06 0.08 0.1 0.12 0.14d σ /d Ω c.m. [mb sr -1 ] ( Θ c.m. =120 o )OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc 1.78 35.71 69.63 103.56 137.48N(d σ /d Ω c.m. ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc-0.52 -0.5 -0.48 -0.46 -0.44 -0.42 -0.4A y ( Θ c.m. =120 o )OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc -1.18 2.71 6.60 10.49 14.38 18.27 22.17N(A y ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc -0.4 -0.3 -0.2 -0.1 0A y ( Θ c.m. =120 o )OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc-14.13 14.40 42.93 71.47 100.00N(A y ) [%]OPE-G statOPE-G j4-j5EKM regEKM truncEMN regEMN trunc FIG. 15: The same as in Fig. 11 ( dσ/d
Ω, the upper row) and Fig. 12 (A y , the bottom row)at θ c . m . = 120 ◦ for E =65 MeV (left) and E =200 MeV (right) but supplemented by theexperimental points at angles near θ c . m . = 120 ◦ . Vertical and thin horizontal lines are as inFig. 11, and filled rectangles represent experimental data and their statistical errors, as inFigs. 5 and 6.resembles by construction the OPE-Gaussian potential. We found only small discrepan-cies between predictions of these forces, especially at the highest energy investigated here, E = 200 MeV, which can very probably originate from a slightly different behaviour of thephase shifts for the AV18 and the OPE-Gaussian potentials at energies above ≈
150 MeV.It should be noted that the procedure of fixing free parameters for the OPE-Gaussian forcehas been performed with big care for statistical correctness and covers new 2 N data notincluded when fixing the AV18’s parameters.In order to obtain the theoretical uncertainty of our predictions arising from the uncer-tainty of the N N potential parameters, we employed the statistical approach: we computedthe
N d scattering observables using fifty sets of the OPE-Gaussian potential parametersobtained from a suitable multivariate probability distribution. Next, we investigated a dis-tribution of our results and adopted one of estimators of their dispersion, the ∆ , asa measure of the theoretical statistical uncertainty. We also compared such statistical un-certainties for different observables with various types of theoretical errors, including thetruncation errors and a dispersion due to using various models of the nuclear interaction. Acomparison of uncertainties for the N d elastic scattering cross section and a few polarizationobservables for the OPE-Gaussian model with other types of theoretical uncertainties leadsto important conclusions about currently used models of 2 N forces. First, all models of the20 N interaction considered here deliver qualitatively and quantitatively similar predictionsfor the N d elastic scattering observables. None of the interactions yields predictions sys-tematically different from others and also no systematic grouping of predictions is observed.Secondly, we have found that in the case of the chiral forces, at small and medium energies,which are their natural domain of applicability, the dependence of predictions on the valuesof regulators dominates over another types of theoretical errors. At the highest investigatedenergy E = 200 MeV which is at the limit of applicability of chiral forces, the truncationerrors become important. It follows that during a derivation of the chiral models a constantattention should be paid to the regularization methods applied. Current attempts to solvethis problem result in a range of regulator parameters too broad to make the chiral forcessuch a precise tool in studies of nuclear reactions as desired and expected. It would be veryinteresting to check if this conclusion remains valid after taking into account also consistent3 N interaction at the investigated here order (N LO) of chiral expansion.Altogether the presented results clearly show that the modern nuclear experiments andtheoretical approaches for the
N d scattering achieved similar precision. Having in mindthat many investigations are currently focused on studying subtle details of underlyingphenomena, there is a need to further improve precision both in theoretical as well as inexperimental studies. From the theoretical side a continuous progress in deriving consistent
N N and 3 N forces from the χ EFT gives hope that this goal will be achieved.
ACKNOWLEDGMENTS
We thank Dr. E. Ruiz Arriola and Dr. R. Navarro P´erez for sending us sets of parametersfor the OPE-Gaussian model and for the valuable discussions. This work is a part of theLENPIC project and was supported by the Polish National Science Center under GrantsNo. 2016/22/M/ST2/00173 and 2016/21/D/ST2/01120. The numerical calculations werepartially performed on the supercomputer cluster of the JSC, J¨ulich, Germany. [1] K. Sekiguchi et al. , Phys. Rev. C , 064001 (2017).[2] G. J. Weisel et al. , Phys. Rev. C , 054001 (2014).[3] B. von Przewoski et al. , Phys. Rev. C , 064003 (2006).[4] St. Kistryn et al. , Phys. Rev. C , 044006 (2005).[5] C. R. Howell et al. , Few-Body Syst. , 127 (1994).[6] The Editors, Phys. Rev. A , 040001 (2011).[7] H. Wita la et al. , Phys. Rev. C , 024007 (2001).[8] A. Kievsky et al. , Phys. Rev. C , 3085 (1998).[9] Th. Cornelius et al. , Phys. Rev. C , 2538 (1990).[10] D. H¨uber et al. , Phys. Rev. C , 1100 (1995).[11] J. L. Friar et al. , Phys. Rev. C , 1838 (1990).[12] J. L. Friar, G. L. Payne, W. Gl¨ockle, D. H¨uber, and H. Wita la, Phys. Rev. C , 2356 (1995).[13] E. Epelbaum, H. W. Hammer, and Ulf-G. Meißner, Rev. Mod. Phys. , 1773 (2009).[14] V. Bernard, E. Epelbaum, H. Krebs, and U. G. Meißner, Phys. Rev. C , 064004 (2008).[15] V. Bernard, E. Epelbaum, H. Krebs, and U. G. Meißner, Phys. Rev. C , 054001 (2011).[16] R. Machleidt and D. R. Entem, Phys. Rep. , 1 (2011).
17] D. R. Entem, R. Machleidt, and Y. Nosyk, Phys. Rev. C , 024004 (2017).[18] R. Navarro P´erez, J. E. Amaro, and E. Ruiz Arriola, Phys. Rev. C , 064002 (2013).[19] R. Navarro P´erez, J. E. Amaro, and E. Ruiz Arriola, Phys. Rev. C , 064006 (2014).[20] J. Dobaczewski et al. , J. Phys. G: Nucl. Part. Phys. , 074001 (2014).[21] Focus issue Focus on Enhancing the Interaction Between Nuclear Experiment and TheoryThrough Information and Statistics of J. Phys. G: Nucl. Part. Phys. , 030301-034033 (2015),Editors: D. G. Ireland and W. Nazarewicz.[22] A. Ekstr¨om et al. , Phys. Rev. Lett. , 192502 (2013).[23] R. Navarro P´erez, J. E. Amaro, and E. Ruiz Arriola, Phys. Rev. C , 054002 (2015).[24] R. Navarro P´erez, J. E. Amaro, and E. Ruiz Arriola, J. Phys. G: Nucl. Part. Phys. , 114001(2016).[25] P. Reinert, H. Krebs, and E. Epelbaum, arXiv:1711.08821 [nucl-th].[26] R. J. Furnstahl, N. Klco, D. R. Phillips, and S. Wesolowski, Phys. Rev. C , 024005 (2015).[27] B. R. Barrett, P. Navratil, and J. P. Vary, Prog. Part. Nucl. Phys. , 131 (2013).[28] E. Epelbaum, H. W. Hammer, and Ulf-G. Meißner, Rev. Mod. Phys. , 1773 (2009).[29] E. Epelbaum, H. Krebs, and Ulf-G. Meißner, Eur. Phys. J. A , 26 (2015).[30] S. Binder et al. , Phys. Rev. C , 044002 (2016).[31] R. Skibi´nski et al. , Phys. Rev. C , 064002 (2016).[32] J. A. Melendez, S. Wesolowski, and R. J. Furnstahl, Phys. Rev. C , 024003 (2017).[33] E. Epelbaum, W. Gl¨ockle, and Ulf-G. Meißner, Nucl. Phys. A , 107 (1998); , 295(2000).[34] E. Epelbaum et al. , Phys. Rev. C , 064001 (2002).[35] D. R. Entem and R. Machleidt, Phys. Rev. C , 041001(R), (2003).[36] E. Marji et al. , Phys. Rev. C , 054002 (2013).[37] H. Wita la, J. Golak, R. Skibi´nski, and K. Topolnicki, J. Phys. G: Nucl. Part. Phys. , 094011(2014).[38] D. Rozp¸edzik et al. , Phys. Rev. C , 064004 (2011).[39] R. Skibi´nski, J. Golak, D. Rozp¸edzik, K. Topolnicki, and H. Wita la, Acta. Phys. Polon. B ,159 (2015).[40] E. Epelbaum, H. Krebs, and Ulf-G. Meißner, Phys. Rev. Lett. , 122301 (2015).[41] P. Maris et al. , EPJ Web of Conf. , 04015 (2016).[42] S. Binder et al. , arXiv:1802.08584 [nucl-th][43] R. Navarro P´erez, A. Nogga, J. E. Amaro and E. Ruiz Arriola, J.Phys.Conf.Ser. , 012001(2016).[44] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C , 38 (1995).[45] W. Gl¨ockle et al. , Phys. Rept. , 107 (1996).[46] W. Gl¨ockle, The Quantum-Mechanical Few-Body Problem (Springer-Verlag, Berlin, 1983).[47] R. Navarro P´erez, E. Garrido, J. E. Amaro, and E. Ruiz Arriola, Phys. Rev. C , 047001(2014).[48] S. S. Shapiro, M. B. Wilk, Biometrika , 591 (1965).[49] H. Wita la, W. Gl¨ockle, D. H¨uber, J. Golak, and H. Kamada, Phys. Rev. Lett. , 1183 (1998).[50] J. Kuro´s- ˙Zo lnierczuk et al. , Phys. Rev. C , 024003 (2002).[51] S. Shimizu et al. , Phys. Rev. C , 1193 (1995).[52] H. R¨uhl et al. , Nucl. Phys. A , 377 (1991).[53] R. E. Adelberg, C. N. Brown, Phys. Rev. D , 2139 (1972).[54] G. Igo et al. , Nucl. Phys. A , 33 (1972).
55] K. Ermisch et al. , Phys. Rev. C , 064004 (2005).[56] J. Cub et al. , Few-Body Syst. , 151 (1989).[57] R. V. Cadman et al. , Phys. Rev. Lett. , 967 (2001).[58] S. P. Wells et al. , Nucl. Instrum. Methods Phys. Res., Sect. A , 205 (1993).[59] K. Topolnicki, J. Golak, R. Skibi´nski, and H. Wita la, Phys. Rev. C , 014611 (2017)., 014611 (2017).