Benchmark calculation of n-3H and p-3He scattering
M. Viviani, A. Deltuva, R. Lazauskas, J. Carbonell, A.C. Fonseca, A. Kievsky, L.E. Marcucci, S. Rosati
aa r X i v : . [ nu c l - t h ] S e p Benchmark calculation of n - H and p - He scattering M. Viviani a , A. Deltuva b , R. Lazauskas c , J. Carbonell d ,A. C. Fonseca b , A. Kievsky a , L.E. Marcucci e , a , and S. Rosati e , a a INFN-Pisa, 56127 Pisa, Italy b Centro de F´ısica Nuclear da Universidade de Lisboa,P-1649-003 Lisboa, Portugal c IPHC, IN2P3-CNRS/Universit´e Louis Pasteur BP 28,F-67037 Strasbourg Cedex 2, France d CEA-Saclay, IRFU/SPhN,F-91191 Gif-sur-Yvette, France e Department of Physics,University of Pisa, 56127 Pisa, Italy (Received November 1, 2018)The n - H and p - He elastic phase-shifts below the trinucleon disintegration thresholds are cal-culated by solving the 4-nucleon problem with three different realistic nucleon-nucleon interactions(the I-N3LO model by Entem and Machleidt, the Argonne v potential model, and a low- k modelderived from the CD-Bonn potential). Three different methods – Alt, Grassberger and Sandhas, Hy-perspherical Harmonics, and Faddeev-Yakubovsky – have been used and their respective results arecompared. For both n - H and p - He we observe a rather good agreement between the three differenttheoretical methods. We also compare the theoretical predictions with the available experimentaldata, confirming the large underprediction of the p - He analyzing power.
PACS numbers: 21.45.+v, 21.30.-x, 24.70.+s, 25.10.+s
I. INTRODUCTION
The four–nucleon (4N) system has been object of in-tense studies in recent years. In first place, this systemis particularly interesting as a “theoretical laboratory”to test the accuracy of our present knowledge of thenucleon–nucleon (NN) and three nucleon (3N) interac-tions. In particular, the effects of the NN P-waves and ofthe 3N force are believed to be larger than in the A = 2 or3 systems. Moreover, it is the simplest system where the3N interaction in channels of total isospin T = 3 / d + d → He + γ or p + He → He + ν e + e + (the hep process) play important roles in solar models and in thetheory of big-bang nucleosynthesis.Nowadays, the 4N bound-state problem can be nu-merically solved with good accuracy. For example, inRef. [1] the binding energies and other properties of the α -particle were studied using the AV8 ′ [2] NN interaction;several different techniques produced results in very closeagreement with each other (at the level of less than 1%).More recently, the same agreement has also been ob-tained considering different realistic NN+3N force mod-els [3–6].In recent years, there has also been a rapid advance insolving the 4N scattering problem with realistic Hamil-tonians. Accurate calculations of four-body scatteringobservables have been achieved in the framework of theAlt-Grassberger-Sandhas (AGS) equations [7–11], solvedin momentum space, where the long-range Coulomb in- teraction is treated using the screening and renormal-ization method [12, 13]. Also solutions of the Faddeev-Yakubovsky (FY) equations in configuration space [14–18] and the application of the Hyperspherical Harmonics(HH) expansion method [19] to the solution of this prob-lem have been reported [20, 21].In addition to these methods, the solution of the 4Nscattering problem has been obtained also by using theresonating group model (RGM) method [22–25]. Calcu-lations of scattering observables using the Green’s func-tion Monte Carlo method are also underway [26].The 4N studies performed so far have evidenced severaldiscrepancies between theoretical predictions and exper-imental data. Let us consider first n - H elastic scatter-ing. Calculations based on NN interaction models dis-agree [5, 7, 20] rather sizeably with the measured to-tal cross section [27], both at zero energy and in the“peak” region ( E n ≈ . p - He elastic scattering is more in-teresting since there exist several accurate measurementsof both the unpolarized cross section [29–31] and theproton analyzing power A y [31–33]. The calculationsperformed so far (with a variety of NN and NN+3N in-teractions) have shown a large discrepancy between the-ory and experiment for A y [8, 17, 31, 33, 34]. In ad-dition, at the Triangle Universities Nuclear Laboratory(TUNL), there has been recently a new set of accuratemeasurements of other p - He observables (the He ana-lyzing power A y and some spin correlation observablesas A yy , A xx , A xz , A zx , and A zz ) at E p = 1 .
60, 2 .
25, 4 . .
54 MeV, which has allowed a phase-shift analysis(PSA) [35]. A preliminary comparison with these datahas been reported in Ref. [21].In order to have definite answer about the ability of thedifferent interaction models to reproduce the experimen-tal data it is certainly of interest to establish the accuracyreached by the theoretical methods in the solution of the A = 4 scattering problem. In a previous benchmark, theresults obtained by different groups working with differ-ent techniques were found to be at variance with eachother [17]. Clearly, this situation should be clarified be-fore questioning the ability of present NN+3N force mod-els to describe the experimental data beyond the bindingenergy of He. This is the purpose of the present paper,in which we present low energy n - H and p - He scatter-ing results obtained by three different groups, using in-dependent methods to solve the four-body problem, i.e.,the AGS equations, the variational HH expansion, andthe FY equations.The potentials used in this paper are the I-N3LO modelby Entem and Machleidt [36], with cutoff Λ = 500 MeV,the Argonne v (AV18) potential model [37], and alow- k model derived from the CD-Bonn potential [38].The I-N3LO potential has been derived using an effec-tive field theory approach and the chiral perturbationtheory up to next-to-next-to-next-to-leading order. TheAV18 potential is a phenomenological potential having arather strong repulsion at short interparticle distances.The low − k potentials have been obtained separating theHilbert space into low and high momentum regions andusing the renormalization group method [38] to integrateout the high-momentum components above a cutoff Λ.The low − k potential adopted in this work is obtainedstarting from the realistic CD-Bonn potential [39] andusing a smooth cutoff Λ = 2 . − . The cut of thehigh-momentum part is reflected in configuration spacein an almost total absence of the repulsion at short in-terparticle distances. Note that the first and third modelare non-local, while AV18 is local in configuration space.The three potentials reproduce equally well the np and pp data, and are a representative set of the large varietyof modern NN potential models. We note finally that I-N3LO and AV18 interactions, without the inclusion of asuitable 3N interaction model, largely underestimate the He binding energy B ( He). On the contrary, with theadopted low- k potential model we have B ( He) = 29 . .
30 MeV.This paper is organized as follows. In Section II, abrief description of the methods used for this benchmarkis reported. In Section III, a comparison between theresults obtained within the different schemes is shown.In Section IV, the theoretical calculations are comparedwith the available experimental data. The conclusions will be given in Section V.
II. METHODS
In order to solve the 4N scattering problem we em-ploy the AGS equations, the HH method, and the FYequations. The various procedures are briefly describedbelow.The total kinetic energy, T c.m. , in the center of mass(c.m.) and the nucleon kinetic energy, E N ( N = p , n ),in the laboratory reference frame are given by T c.m. = q µ , E N = 43 T c.m. , (1)where µ = (3 / M N is the reduced mass of the 1 + 3system, M N is the nucleon mass, and q the magnitude ofthe relative momentum between the two clusters. A. AGS Equations
The AGS equations [40] are integral equations for thefour-body transition operators. They are well-definedonly with short-range potentials. Nevertheless, togetherwith the screening and renormalization method [8, 41],they can be applied also to the reactions involvingcharged particles. In the 4N system we use the isospinformalism and solve the symmetrized form of the AGSequations [7]. In this case there are only two distinctfour-particle partitions, one of the 3 + 1 type and one ofthe 2 + 2 type. We choose those partitions to be (12,3)4and (12)(34) and denote them in the following by α = 1and 2, respectively. The corresponding transition op-erators U βα for the initial states of the 3 + 1 type, asappropriate for the n - H and p - He scattering, obey theintegral equations U = − ( G T G ) − P − P U G T G U + U G T G U , (2) U = ( G T G ) − (1 − P )+(1 − P ) U G T G U . (3)Here G = ( E + iǫ − H ) − is the free resolvent, E beingthe energy of the 4N system and H the free Hamiltonian,and P ij is the permutation operator of particles i and j . The (short-range) two-nucleon potential V s enters theAGS equations via the two-nucleon transition matrix T = V s + V s G T and the 3+1 and 2+2 subsystem transitionoperators U α = P α G − + P α T G U α , (4)where P = P P + P P and P = P P . The 3+1elastic scattering amplitudes are given by h p f |T | p i i =3 h Ψ ( p f ) |U | Ψ ( p i ) i where the factor 3 results from thesymmetrization and | Ψ α ( p j ) i are properly normalizedinitial/final channel state Faddeev components.In order to include the Coulomb interaction V C be-tween the protons in the p - He scattering we use thescreening and renormalization approach [8, 41]. We addto the nuclear pp potential the screened Coulomb one V R ( r ) = V C ( r ) exp ( − ( r/R ) n ). Thus, the AGS equa-tions with V s + V R are well-defined but all transitionoperators and the resulting amplitudes depend on thescreening radius R . The renormalization procedure inthe R → ∞ limit yields the full p - He transition ampli-tude h p f |T ( C ) | p i i = h p f | t c.m.C | p i i + lim R →∞ h p f | [ T ( R ) − t c.m.R ] | p i i Z − R , (5)where h p f | t c.m.C | p i i and h p f | t c.m.R | p i i are the proper andscreened Coulomb amplitudes between the c.m. of twocharged clusters, respectively; the former is known an-alytically. The renormalization factor Z R is defined inRef. [8]. Thus, the long- and Coulomb-distorted short-range parts in the scattering amplitudes are isolated andtheir infinite R limit is calculated separately. The long-range part of the amplitude h p f | t c.m.R | p i i is of two-bodynature and its R → ∞ limit after renormalization is h p f | t c.m.C | p i i . The Coulomb-distorted short-range part[ T ( R ) − t c.m.R ] is calculated by solving the AGS equationsfor V s + V R numerically at a finite R that is sufficientlylarge to get R -independent results after the renormal-ization. In other words, the R → ∞ limit is reachedwith sufficient accuracy at finite R . However, R mustbe considerably larger than the range of the nuclear in-teraction thereby leading to a slower partial-wave con-vergence. The right choice of the screening, i.e., theexponent n , is essential in dealing with this difficulty.For a fast convergence with R we have to ensure that V R ( r ) approximates well V C ( r ) for r < R and simulta-neously vanishes smoothly but rapidly for r > R , provid-ing a comparatively fast convergence of the partial-waveexpansion. Using the optimal value n = 4 we obtainreasonably converged results with R ranging from 10 to15 fm and including two-proton partial waves with or-bital angular momentum up to 10. The R -convergence isslower at lower energies; the worst cases are the S wavesat E p = 2 .
25 MeV where we estimate the accuracy ofour phase-shift results to be around 1%. In contrast, the n - H results are converged very well, considerably betterthan 0.2%, as demonstrated in Ref. [7] where also thedetails on the included partial waves can be found.
B. HH Expansion
The wave function describing a n - H or p - He scatter-ing state with total angular momentum quantum num-bers
J, J z , incoming relative orbital angular momentum L , and channel spin S ( S = 0 ,
1) can be written asΨ
LS,JJ z = Ψ LS,JJ z C + Ψ LS,JJ z A , (6) where the part Ψ LS,JJ z C describes the system in the re-gion where the particles are close to each other and theirmutual interactions are strong. Hence, Ψ LS,JJ z C vanishesin the limit of large inter-cluster distances. This partof the wave function is written as a linear expansion P µ c LSJµ Y µ , where Y µ is a set of basis functions con-structed in terms of the HH functions (for more details,see, for example, Ref. [19]).The other part Ψ LS,JJ z A describes the relative motion ofthe two clusters in the asymptotic regions, where the 1+3interaction is negligible (except eventually for the long-range Coulomb interaction). In the asymptotic regionthe wave functions Ψ LS,JJ z reduces to Ψ LS,JJ z A , whichmust therefore be the appropriate asymptotic solution ofthe Schr¨odinger equation. Let us consider, for example,the p - He case. Then, Ψ
LS,JJ z A can be decomposed as alinear combination of the following functionsΩ ± LS,JJ z = X l =1 h Y L ( ˆ y l ) ⊗ [ φ ( ijk ) ⊗ s l ] S i JJ z × (cid:18) f L ( y l ) G L ( η, qy l ) qy l ± i F L ( η, qy l ) qy l (cid:19) , (7)where y l is the distance between the proton (particle l )and He (particles ijk ), q is the magnitude of the relativemomentum between the two clusters, s l the spin state ofparticle l , and φ is the He wave function. Moreover, F L and G L are the regular and irregular Coulomb function,respectively, with η = 2 µe /q . The function f L ( y ) =[1 − exp( − βy )] L +1 in Eq. (7) has been introduced toregularize G L at small y , and f L ( y ) → y is large,thus not affecting the asymptotic behavior of Ψ LS,JJ z .Note that for large values of qy l , f L ( y l ) G L ( η, qy l ) ± i F L ( η, qy l ) → exp h ± i (cid:0) qy l − Lπ/ − η ln(2 qy l ) + σ L (cid:1)i , (8)where σ L is the Coulomb phase-shift. Therefore, Ω + LS,JJ z (Ω − LS,JJ z ) describe the asymptotic outgoing (ingoing) p - He relative motion. Finally,Ψ
LS,JJ z A = X L ′ S ′ (cid:20) δ LL ′ δ SS ′ Ω − LS,JJ z − S JπLS,L ′ S ′ Ω + L ′ S ′ ,JJ z (cid:21) , (9)where the parameters S JπLS,L ′ S ′ are the S -matrix elementswhich determine phase-shifts and (for coupled channels)mixing parameters at the energy T c.m. . Of course, thesum over L ′ and S ′ is over all values compatible withthe given J and parity π . In particular, the sum over L ′ is limited to include either even or odd values such that( − L ′ = ( − ) L = π .The S -matrix elements S JπLS,L ′ S ′ and coefficients c LSJµ occurring in the HH expansion of Ψ
LS,JJ z C are determinedby making the functional[ S JπLS,L ′ S ′ ] = S JπLS,L ′ S ′ − D Ψ L ′ S ′ ,JJ z | H − E | Ψ LS,JJ z E (10)stationary with respect to variations in the S JπLS,L ′ S ′ and c LSJµ (Kohn variational principle). In the above equation, E = T c.m. − B ( He) is the energy of the system, B ( He)being the He binding energy. By applying this principle,a linear set of equations is obtained for S JπLS,L ′ S ′ and c LSJµ .This linear system is solved using the Lanczos algorithm.This method can be applied in either coordinate ormomentum space, and using either local or non-local po-tentials [19] (see also Ref. [42] for an application to the A = 3 scattering problem). The first step is a partialwave decomposition of the asymptotic functions Ω ± LS,JJ z ,a task which can be rather time consuming, in particularfor the J π = 2 − state. After this decomposition, the cal-culation of the matrix element in Eq. (10) is fast. Then,the problem reduces to the solution of the linear system,which is performed using an iterative method (however,this solution has to be repeated several times due to thenecessity to extrapolate the results, see below).The expansion of the scattering wave function in termsof the HH basis is in principle infinite, therefore a trun-cation scheme is necessary. The HH functions are es-sentially characterized by the orbital angular momentumquantum numbers ℓ i , i = 1 ,
3, associated with the threeJacobi vectors, and the grand angular quantum number K (each HH function is a polynomial of degree K ). Thebasis is truncated to include states with ℓ + ℓ + ℓ ≤ ℓ max (with all possible re-coupling between angular and spinstates appropriate to the given J ). Between these states,we retain only the HH functions with K ≤ K max . Inthe calculation we have included only states with totalisospin T = 1.The numerical uncertainty comes from the numericalintegrations needed to compute the matrix elements ofthe Hamiltonian and the truncation of the basis. It hasbeen checked that the numerical uncertainty of the cal-culated phase-shifts related to the numerical integrationis small (around 0 . j ≤ j max = 8 (at the considered energies, greatervalues of j max are completely unnecessary). The largestuncertainty is thus related to the use of a finite basis.The convergence with ℓ max is rather fast and the value ℓ max = 6 have been found to be sufficient. The mainproblem is related to the slow convergence of the re-sults with K max . This problem can be partly overcomeby performing calculations for increasing values of K max and then using some extrapolation rule (see for exampleRef. [31]) to get the “ K max → ∞ ” result. This procedurehas an uncertainty which can be estimated. A detailedstudy of this problem will be published elsewhere [43].The convergence of the quantities of interest in term of K max is slower when NN potentials with a strong repul-sion at short interparticle distance are used such as forthe AV18 potential. In this case we have estimated theuncertainty to be of the order of 0 . k models. In these case, the uncertainty hasbeen estimated to be at most 0 . C. FY Equations in Configuration Space
In late sixties, Yakubovsky [44] has managed to gener-alize the three-body equations derived by Faddeev [45] toan arbitrary number of particles. These equations wereprimarily derived for a system of particles submitted toshort range pair-wise potential V s . Nevertheless it be-comes possible to include also repulsive Coulomb inter-action if these, from now on called Faddeev-Yakubovskyequations, are formulated in configuration space. To thisaim, we split the Coulomb potential V C into two parts(short and long range), V C = V s.C + V l.C . The splittingprocedure is quite arbitrary, one should only take carethat the long range part V l.C of the Coulomb potentialapproaches sufficiently fast the full Coulomb interaction V C when any of interparticle distances becomes large.The simplest application of FY equations is the prob-lem of four identical particles. They result into a set oftwo differential equations coupling the two so called FYcomponents, namely K , and H , and have the form: E − H − V s − X i 3, in the so-called j − j coupling scheme [5, 15]. Bystudying the convergence of the calculated phase-shiftswith respect to the size of the partial wave basis, wehave concluded that this truncation scheme should pro-vide results accurate at 1% level.The numerical implementation of these equations isdescribed in detail in Ref. [15]. III. RESULTS In this section we present the phase-shifts for the mostrelevant waves calculated using the three different meth-ods described above. The selected energies for n - H are E n = 1, 2, 3 . p - He are E p = 2 . . 05 and 5 . 54 MeV, corresponding to cases where exper-iments have been carried out.The states considered are those with J π = 0 ± , 1 ± ,and 2 − . The scattering in other J π states is dominatedby the centrifugal barrier and therefore the phase-shiftsare smaller and not very sensitive to the interaction andthe method used to calculate them. Note that, for the J π = 2 − state, we have chosen to report only the P phase-shift, since the F phase-shift and the relativemixing parameter are in any case very small. Never-theless the coupling between the P and F waves hasbeen included in the calculations, since the presence ofthe F component in the asymptotic part of the wavefunction has a sizable effect on the P phase-shift.Let us remember that the S -matrix for elastic n - Hand p - He scattering has dimension 1 for J π = 0 ± statesand dimension 2 for J > 0. In the first case, the S -matrix is parametrized as usual as S JπLS,LS = exp(2 iδ JπLS ).For J > 0, since the S -matrix is unitary and symmetric,we can write it as S = O T S D O , (14)with S D a diagonal matrix written as( S D ) LS,L ′ S ′ = δ LL ′ δ SS ′ e iδ JπLS , (15)where δ JπLS is the phase-shift (in the Blatt-Biederharn rep-resentation) of the wave LS . Due to the unitarity prop-erties, δ JπLS is a real number. The matrix O in Eq. (14)is parametrized as O = (cid:20) cos ǫ Jπ sin ǫ Jπ − sin ǫ Jπ cos ǫ Jπ (cid:21) , (16)where ǫ Jπ is the so called mixing parameter of the given J π state. Clearly the values of the phase-shifts and mix-ing parameters may depend on the (arbitrary) choice onthe coupling scheme between the spin of the two clus-ters and the spherical harmonic function Y L ( y ) in the asymptotic functions Ω ± LS (see, for example, Eq. (7)). Itcan be shown that the phase-shifts defined as discussedabove are independent on such choices, while the mixingparameter, on the contrary, depends on them. Neverthe-less, it is easy to establish the linear relation to trans-form the mixing parameter from one coupling scheme toanother. In the following, we chose to report the mixingparameters defined in the LS coupling scheme by Eq. (7).Moreover, in the following tables, the values reported ina column labeled as S +1 L J (using the “spectroscopicnotation”) are relative to the phase-shift δ JπLS .In Table I we present the phase-shifts, mixing parame-ters, and total cross sections for n - H scattering obtainedusing the I-N3LO potential at the selected energies. Byinspecting the table, we can notice the good agreementbetween the three different techniques. The maximal de-viation of the results is less than 1%, fully in line with theestimated errors. Furthermore, the agreement betweenthe results of AGS and HH techniques is even better,only in a few cases the HH and AGS results differ by morethan 0.5%. The strongest deviation, of the order of 0.4deg, is observed with the FY results at the largest stud-ied energy. This slightly larger deviation might be due tothe necessity by the FY method to perform the transfor-mation of the aforementioned potential to configurationspace. In this respect, we note that the AGS calcula-tion is fully performed in momentum space, while in theHH calculation, part of the needed matrix elements arecalculated in momentum space (those involving Ψ LS,JJ z C ,which have to be calculated with more accuracy), andpart in configuration space (those involving Ψ LS,JJ z A ).In Table II we present the same n - H results obtainedusing the AV18 potential. In this case, the convergenceof the HH expansion is more problematic, in particulardue to the necessity to extrapolate the HH results. Weobserve that we still have a very good agreement for the S , P , P , and 1 + phase-shifts, while the differencesin the 1 − phase-shifts appear to be more enhanced.In Table III we have reported the phase-shifts obtainedwith the low − k potential derived from CD-Bonn. In thiscase, the calculations has been performed using the AGSand HH methods, only. We again observe an overall goodagreement between the results obtained by the two tech-niques, except for the lowest energy where the differencesare sizeable.The total cross sections σ t are found to be in agreementwithin 0.05 b at all considered energies. By comparingthe values obtained using the different potentials, we canobserve the following well-known characteristics: (i) atlow energies the I-N3LO and AV18 models overpredictsthe experimental cross section. For example, at E n = 1MeV, σ expt t ≈ . σ I − N3LO t ≈ σ AV18 t ≈ . k potential the cal-culated σ t is quite close to the experimental one. Thisbehavior is related to the strict relation between the to-tal cross section at low energy and the triton bindingenergy [14, 20, 23, 28]. (ii) at the peak (around E n = 3 . σ expt t ≈ . 45 b [27]. In this case we note that theAV18 and low- k potential models underpredict sizeablythe experimental value, while σ I − N3LO t is quite close toit.Let us now consider p - He scattering. The phase-shiftsand mixing parameters obtained within the three meth-ods have been reported in Tables IV, V, and VI, corre-sponding respectively to the I-N3LO, AV18, and the low- k NN potential models. Here the differences between thevarious techniques are larger than in the n - H case, espe-cially at low energy and for the J π = 0 ± states. For theAV18 potential, we note that the HH results are morelessintermediate between the AGS and FY results.In Table VI we have also reported the phase-shifts andmixing parameters obtained by the recent PSA [35]. Notethat the low- k potential used in this work is the onlypotential which does not underestimate the He bind-ing energy. The PSA estimates have rather large errors.However, it is possible to draw some conclusions aboutthe capability of this low- k potential model to describethe experimental data. As can be seen, the PSA S-wavephase-shifts seem to be well reproduced (except for the S phase-shift at 5 . 54 MeV) by the calculations. Alsothe P and P agree well, but for these cases the exper-imental errors are large. On the other hand, we note asizeable underestimation of the large P and P phase-shifts.Let us now see how the fairly good agreement found forthe phase-shifts and mixing parameters calculated withthe three different methods reflects on the observables.We have considered the differential cross section and theneutron (proton) analyzing power A y for n - H ( p - He)elastic scattering at the considered energies, as functionsof the c.m. scattering angle. Furthermore, we have alsoconsidered the triton ( He) analyzing power A y . Thisobservable is in fact rather sensitive to small variationsof the phase-shifts in the kinematical regime consideredin this paper.In Figs. 1 and 2 we have reported the results obtainedusing the AGS equation (solid lines), the HH expansionmethod (dashed lines), and the FY equations (dottedlines) using the I-N3LO potential. As can be seen byinspecting the two figures, the three curves almost al-ways perfectly coincide and cannot be distinguished. Wehave also reported the experimental data for the n - Hdifferential cross section [46] and the three p - He observ-ables [29–33, 35]. We note that the differences betweenthe three calculations, where they can be appreciated, arein any case always smaller than the experimental errors.The agreement between the three calculations whenthe AV18 potential is adopted is again rather satisfac-tory, as can be seen in Figs. 3 and 4. A small disagree-ment can be observed only for the A y observable (seethe panels in the last row of Fig.4). This observableis also rather sensitive to the small D-wave and F-wavephase-shifts not reported in Tables II and V. We alreadyknow that the AV18 model contains a stronger repulsionat short interparticle distance than the I-N3LO. As dis- cussed above, the convergence of the HH method for thiscase is more problematic and consequently the calcula-tions have a larger uncertainty. In spite of these difficul-ties, the agreement in the considered observables is stillquite good.Let us consider now the low- k potential, which has norepulsion at short interparticle distance. Consequently,in this case, we expect a good agreement between theresults of the different techniques. For this potential, thecalculations have been performed using the AGS (solidcurves) and HH (dashed curves) methods, only, and thecorresponding results are reported in Figs. 5 and 6. Thetwo curves are practically indistinguishable, confirmingthat for soft potentials the convergence of the calculationsis excellent.Finally, in the literature for p - He scattering, there ex-ist measurements of other spin correlation observables( A yy , A xx , A zz , A xz , and A zx ). Also for these observ-ables we have found a good agreement between the pre-dictions obtained by the three different methods, for allthe potential models considered here. The comparison ofthe theoretical predictions and the experimental data forthese observables will be discussed in the next section. IV. COMPARISON WITH EXPERIMENTALDATA In this section we discuss the comparison between thetheoretical calculations and the experimental data. Weconsider here only p - He scattering since for this processthe experimental data are more abundant and precise.The figures presented in this section can be consideredas an update of previous comparisons [7, 8, 17, 21, 31].For the three observables considered so far ( dσ/d Ω, A y ,and A y ), the comparison between theory and experi-ment can be inferred already from Figs. 1–6. However,in order to better appreciate the differences in the predic-tions obtained by the three potential models as comparedto the experimental data, we summarize again in Fig. 7the results for dσ/d Ω, A y , and A y . In order to take intoaccount the (slight) different predictions obtained usingthe three different theoretical methods, we have decidedto present the calculated observables for each potential asbands. Each band contains the results obtained by usingthe three different methods. As can be seen from Fig. 7,the differential unpolarized cross sections obtained us-ing the I-N3LO potential (red bands) agree well with theexperimental data. With the other two potentials we ob-serve some disagreement, in particular around θ c . m . ≈ A y are found to depend on the potentialmodel. Here, we observe the well known underpredictionof the experimental data by the theoretical calculations.Interestingly, the results obtained with the low- k poten-tial are in a better agreement with the experimental A y .A similar situation is found also for A y , as can be seenin the three lower panels of Fig. 7. It is worthy to notethat the effect of supplementing the AV18 potential withthe Urbana 3N force model [47] has been found to be al-most negligible for this observable [31]. The inclusion ofthe new chiral 3N potential derived in Ref. [48] is understudy [43] (see Ref. [20] for a preliminary report).In Fig. 8, we report the results found for the A yy and A xx spin correlations at the three different proton en-ergies. As can be seen, for these two observables thepredictions obtained with the three potentials are almostidentical. We observe that the calculated A yy is slightlyat variance with respect to the experimental data, whilethe A xx observable is reasonably well reproduced by thecalculations.Finally, in Fig. 9 we compare the results obtained forthe A xz , A zx , and A zz spin correlation observables. Inthis case, only the E p = 5 . 54 proton laboratory energy isconsidered, since only for this energy experimental dataexist. Also in this case, the sensitivity to the differentpotential models is not significant. Moreover, the calcu-lations reproduce well the (few) experimental data. V. CONCLUSIONS In this work, we have studied several low energy n - H and p - He elastic observables by using three dif-ferent approaches, the HH, AGS and FY techniques.Around four years ago, some of the authors of thepresent paper presented very accurate solutions of the 4-nucleon scattering problem using the AGS technique [7–9]. They were able to take into account the long-rangeCoulomb interaction using the screening-renormalizationmethod [12, 13]. In recent years, also the accuracy of thecalculations performed using the HH and FY techniquesincreased [18, 20, 21]. Therefore, it becomes appropriateto compare the results obtained by the different methodsin order to test their capability to solve the 4N scatteringproblem. This is the primary aim of the present paper.Another important motivation is to provide a set of solidconverged results in the literature, which could representuseful benchmarks for future applications in A = 4 scat-tering.In the present paper we have shown that for I-N3LOand the selected low- k potential model, which have a“soft” repulsion at short interparticle distances (the low- k model has no repulsion at all), the results obtainedby the different techniques are in very good agreement.With the AV18 potential, the agreement is not so perfect,although the (slight) differences can be appreciated onlyfor some small polarization observables. We can concludetherefore, that the A = 4 scattering problem is nowadayssolved with a very good accuracy, better than 1%.Concerning the comparison with the experimentaldata, we have confirmed the large underprediction of the p - He A y observable, a problem already put in evidencesome time ago [14, 33, 34], and certainly related to the N − d “ A y puzzle”. For this observable we have observeda moderate dependence on the considered potential mod-els. The low- k potential is found to give a better descrip-tion of the observable when compared with the experi-mental data. However, the same potential does not re-produce well the unpolarized cross section. We have alsofound a small underprediction of the theoretical A y and A yy observables, while other measured observables, suchas A xx , A xz , A zx , and A zz spin correlation coefficients,show less sensitivity to the potential models. They are ingood agreement with the available (sparse) experimentaldata.The discrepancies found, in particular for A y , indicatea serious difficulty of the existing NN force models indescribing the 4N continuum. This difficulty can hardlybe solved by the inclusion of a standard type 3NF, usedto reproduce the few-nucleon binding energies [17, 21,31]. Its origin could rather lie either in the NN forcesthemselves, or in the presence of a 3NF of unknown type.Clearly, an eventual solution of the A = 4 A y problemshould be related in some way to the solution of the N − d “ A y puzzle”.Finally, we conclude noticing that it would be inter-esting to extend the present analysis to p - H, n - He and d - d scattering observables, which have already been cal-culated in the framework of the AGS equations for dif-ferent NN interactions [9, 10]. 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E p S P S D ǫ 1+ 1 P P ǫ − P E p S P S D ǫ 1+ 1 P P ǫ − P ± ± ± ± ± ± 20 16.5 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± k potential. In this Table, only the AGS and HH results are reported. Thephase-shifts and mixing parameters obtained by the recent PSA [35] are also shown. d σ / d Ω [ m b / s r ] A y0 θ c.m. [deg] -0.100.1 A θ c.m. [deg] θ c.m. [deg] θ c.m. [deg] FIG. 1: Differential cross section and neutron and triton analyzing powers A y and A y for n - H elastic scattering at E n = 1, 2,3 . 5, and 6 MeV neutron lab energies as functions of the c.m. scattering angle. Results obtained using the AGS equation (solidlines), the HH expansion method (dashed lines), and the FY equations (dotted lines) using the I-N3LO potential are compared.For most of the cases the three curves coincide and cannot be distinguished. The experimental data are from Ref. [46]. d σ / d Ω [ m b / s r ] Famularo 1954Fisher 2006 A y0 Fisher 2006George 2001 θ c.m. [deg] A Daniels 2010 Mcdonald 1964Fisher 2006 Fisher 2006 θ c.m. [deg] Daniels 2010 Mcdonald 1964 Alley 1993 θ c.m. [deg] Alley 1993Daniels 2010 FIG. 2: Same as Fig. 1, but for p - He elastic scattering at E p = 2 . 25, 4 . 05, and 5 . 54 MeV proton lab energies. The experimentaldata are from Refs. [29–33, 35]. d σ / d Ω [ m b / s r ] A y0 θ c.m. [deg] -0.100.1 A θ c.m. [deg] θ c.m. [deg] θ c.m. [deg] FIG. 3: Same as Fig. 1, but for the AV18 potential. d σ / d Ω [ m b / s r ] Famularo 1954Fisher 2006 A y0 Fisher 2006George 2001 θ c.m. [deg] A Daniels 2010 Mcdonald 1964Fisher 2006 Fisher 2006 θ c.m. [deg] Daniels 2010 Mcdonald 1964 Alley 1993 θ c.m. [deg] Alley 1993Daniels 2010 FIG. 4: Same as Fig. 2, but for the AV18 potential. d σ / d Ω [ m b / s r ] A y0 θ c.m. [deg] -0.100.1 A θ c.m. [deg] θ c.m. [deg] θ c.m. [deg] FIG. 5: Same as Fig. 1, but for the low- k potential. Only the AGS and HH results are reported. d σ / d Ω [ m b / s r ] Famularo 1954Fisher 2006 A y0 Fisher 2006George 2001 θ c.m. [deg] A Daniels 2010 Mcdonald 1964Fisher 2006 Fisher 2006 θ c.m. [deg] Daniels 2010 Mcdonald 1964 Alley 1993 θ c.m. [deg] Alley 1993Daniels 2010 FIG. 6: Same as Fig. 2, but for the low- k potential. Only the AGS and HH results are reported. d σ / d Ω [ m b / s r ] Famularo 1954Fisher 2006I-N3LOAV18low-k A y0 Fisher 2006George 2001 θ c.m. [deg] A Daniels 2010 Mcdonald 1964Fisher 2006 Fisher 2006 θ c.m. [deg] Daniels 2010 Mcdonald 1964 Alley 1993 θ c.m. [deg] Alley 1993Daniels 2010 FIG. 7: (Color online) Differential cross section, proton analyzing power, and He analyzing power for p - He elastic scatteringat E p = 2 . 25, 4 . 05, and 5 . 54 MeV proton lab energies obtained using the I-N3LO (red bands), AV18 (blue bands), and thelow- k (cyan bands) potential models. The experimental data are from Refs. [29–33, 35]. A yy Daniels 2010 Daniels 2010 Daniels 2010Alley 1993 θ c.m. [deg] -0.100.10.2 A xx Daniels 2010 θ c.m. [deg] θ c.m. [deg] Daniels 2010Alley 1993 FIG. 8: (Color online) Same as Fig. 7, but for the spin correlation A yy and A xx observables. The experimental data are fromRefs. [32, 35]. θ [c.m.] [deg] -0.2-0.100.10.20.30.4 Alley 1993 A xz θ [c.m.] [deg] Alley 1993 A zx θ [c.m.] [deg] Alley 1993 A zz FIG. 9: (Color online) Same as Fig. 7, but for the spin correlation A xz , A zx , and A zz observables (at E p = 5 ..