N-d Elastic Scattering Using the Hyperspherical Harmonics Approach with Realistic Local and Non-Local Interactions
L.E. Marcucci, A. Kievsky, L. Girlanda, S. Rosati, M. Viviani
aa r X i v : . [ nu c l - t h ] M a y N − d Elastic Scattering Using the Hyperspherical HarmonicsApproach with Realistic Local and Non-Local Interactions
L.E. Marcucci , , A. Kievsky , L. Girlanda , S. Rosati , , and M. Viviani Department of Physics “E. Fermi”,University of Pisa,Largo B. Pontecorvo 3,I-56127, Pisa, Italy Istituto Nazionale di Fisica Nucleare,Sezione di Pisa, Largo B. Pontecorvo 3,I-56127, Pisa, Italy
Abstract
The application of the hyperspherical harmonic approach to the case of the N − d scatteringproblem below deuteron breakup threshold is described. The nuclear Hamiltonian includes two-and three-nucleon interactions, in particular the Argonne v , the N3LO-Idaho, and the V low − k two-nucleon, and the Urbana IX and N2LO three-nucleon interactions. Some of these modelsare local, some are non-local. Also electromagnetic effects are included. Accurate calculationsfor many scattering observables at various center-of-mass energies are performed and the resultsare compared with the available experimental data. Furthermore, a χ analysis of some of theHamiltonian models has been performed to compare their capability to describe the scatteringprocess. . INTRODUCTION One of the main inputs for any study on nuclear systems within a non-relativistic frame-work is the model used to describe the nuclear interaction, i.e. the nuclear Hamiltonian.Nowadays, it is common practice to use, at least for few-nucleon systems, Hamiltonian mod-els composed of a two-nucleon plus, for A ≥
3, a three-nucleon interaction (TNI). The mod-ern two-nucleon interaction models have a large number of parameters and can reproducethe deuteron properties and the nucleon-nucleon scattering data up to the pion thresholdwith a χ /datum ≃
1. Among them, the Argonne v (AV18) [1] and the charge-dependentBonn (CDBonn) [2] explicitly include charge-symmetry-breaking terms in the nuclear in-teraction, in order to reproduce equally well the np and pp data. Recently, a number oftwo-nucleon interaction models have been derived by many authors within an effective fieldtheory (EFT) approach, up to next-to-next-to-next-to leading order (N3LO) [3, 4]. In par-ticular, the N3LO model of Ref. [4] (N3LO-Idaho) reaches the same level of accuracy of theCDBonn model.The available models for the TNI contain, on the contrary to the two-nucleon interactionmodels, a small number of parameters, usually fixed to reproduce the H and/or He bindingenergies and, in some cases, the nuclear matter equilibrium density. Among the differentexisting models, we quote only those ones of the Urbana and Tucson-Melbourne families.Although constructed within different frameworks, these two families of potentials haveshown to give similar results, when used in conjunction with a given two-nucleon interactionmodel. Therefore, we have considered the Urbana IX [5] (UIX) TNI in conjunction with boththe AV18 and N3LO-Idaho two-nucleon interaction models. Finally, it should be noticedthat within the EFT approach mentioned above, also TNIs appear at the next-to-next-toleading order (N2LO) [6]. In particular, we will consider the local version of this N2LO TNI,as given in Ref. [7].More recently, a new class of two-nucleon interactions has been obtained ( V low − k po-tentials). With the purpose of eliminating from the semi-phenomenological high precisiontwo-nucleon potentials the high-momentum parts, the two-nucleon Hilbert space has beenseparated into low- and high-momentum regions and the renormalization group method hasbeen used to integrate out the high-momentum components above a cutoff Λ [8]. The valuefor Λ is typically fixed in A > A ≥ A ≥ A = 3 scattering problem, which has been the object of a large number ofinvestigations [9]. Traditionally, the A = 3 scattering problem with realistic Hamiltonians issolved using the Faddeev equations. On the other hand, we have developed in recent yearsa variational approach, based on the expansion of the wave functions on the hypersphericalharmonics (HH) basis (for a recent review, see Ref. [10]). This method has proven tobe very efficient in the description of bound and scattering states in few-nucleon systems.In Ref. [11] the HH expansion with correlations factors (the correlated HH – CHH – andpair-correlated HH – PHH – expansions) has been used to describe A = 3 bound states,whereas the extension to scattering states has been discussed in Ref. [12]. The inclusion ofcorrelation factors was motivated by the short range repulsion of the two-nucleon potentialwhich induces particular configurations in the wave function difficult to describe using thebare expansion. In fact, in Ref. [13] the HH expansion without correlation factors has beenused to describe the A = 3 bound state, with the AV18 interaction. The conclusion was thata much higher number of states are necessary when the bare expansion is used. The sameobservation has been done in the A = 4 system [14] and is a direct consequence of using localinteractions, which result to have a strong repulsion at short distances. The implementationof the HH method in momentum-space has been done in Ref. [15] for the A = 3 , N − d scattering states using either local or non-local potentials. We will showthat we can apply the method in both configuration and momentum spaces. Second, wewill present a detailed comparison between the predictions of the different models, local andnon-local, at low center-of-mass energies, for n − d as well as p − d scattering. Moreover,we will consider the Coulomb potential plus the magnetic moment (MM) interaction thatwas shown to give sizable contributions [16]. To our knowledge, this is the first time thatnon-local two- plus three-nucleon potentials are used to describe p − d scattering at very lowenergies.The paper is organized as follows: in Sec. II, the HH method for the low-energy scattering3roblem is described, putting more emphasis on those new developments of the methodnecessary in order to use non-local interaction models. In Sec. III, the results for the zero-energy scattering lengths and low-energy elastic scattering observables are presented anddiscussed. Some concluding remarks are given in Sec. IV. II. FORMALISM
In this section we present the HH method for scattering states. The method for boundstates has been most recently reviewed in Ref. [10], and its main characteristics are brieflysummarized in the following subsection.
A. The HH Method for Bound States
The nuclear wave function for the three-body system with total angular momentum
J, J z can be written as | Ψ JJ z i = X µ c µ | Ψ JJ z µ i , (2.1)where | Ψ JJ z µ i is a suitable complete set of states, and µ is an index denoting the set ofquantum numbers necessary to completely specify the basis elements.The coefficients of the expansion can be calculated using the Rayleigh-Ritz variationalprinciple, which states that h δ c Ψ JJ z | H − E | Ψ JJ z i = 0 , (2.2)where δ c Ψ JJ z indicates the variation of Ψ JJ z for arbitrary infinitesimal changes of the linearcoefficients c µ . The problem of determining c µ and the energy E is then reduced to ageneralized eigenvalue problem, X µ ′ h Ψ JJ z µ | H − E | Ψ JJ z µ ′ i c µ ′ = 0 . (2.3)The main difficulty of the method is to compute the matrix elements of the Hamiltonian H with respect to the basis states | Ψ JJ z µ i . Usually H is given as a sum of terms (kinetic energy,two-body potential, etc.). The calculation of the matrix elements of some parts of H canbe more conveniently performed in coordinate space, while for other parts it could be easier4o work in momentum space. Therefore, it is important that the basis states | Ψ JJ z µ i havesimple expressions in both spaces. The HH functions indeed have such a property.Let us first consider the expression of the HH functions in coordinate space. The internaldynamics of a system of three nucleons of identical mass m is conveniently described interms of the Jacobi vectors x p , x p , constructed from a given particle permutation denotedwith p , which specifies the particle order i, j, k , and given by x p = 1 √ r j − r i ) , x p = r
23 ( r k −
12 ( r i + r j )) . (2.4)Here p = 1 corresponds to the order 1,2,3. It is convenient to replace the moduli of x p and x p with the so-called hyperradius and hyperangle, defined as [17] ρ = q x p + x p , (2.5)tan φ p = x p x p . (2.6)Note that ρ does not depend on the particle permutation p . The complete set of hyper-spherical coordinates is then given by { ρ, Ω ( ρ ) p } , withΩ ( ρ ) p = [ ˆ x p , ˆ x p ; φ p ] , (2.7)and the suffix ( ρ ) recalls the use of the coordinate space.The expansion states | Ψ JJ z µ i of Eq. (2.1) are then given by | Ψ JJ z ( ρ ) µ i = f l ( ρ ) Y { G } (Ω ( ρ ) ) , (2.8)where f l ( ρ ) for l = 1 , . . . M is a complete set of hyperradial functions, chosen of the form f l ( ρ ) = γ s l !( l + 5)! L (5) l ( γρ ) e − γ ρ . (2.9)Here L (5) l ( γρ ) are Laguerre polynomials, and the non-linear parameter γ is variationallyoptimized. As an example, for the N3LO-Idaho potential, it can be chosen in the interval6–8 fm − .The functions Y { G } (Ω ( ρ ) ) are written as Y { G } (Ω ( ρ ) ) = X p =1 h Y LL z [ G ] (Ω ( ρ ) p ) ⊗ [ S ⊗
12 ] SS z i JJ z [ T ⊗
12 ]
T T z , (2.10)5here the sum is performed over the three even permutations. The spins (isospins) ofparticle i and j are coupled to S ( T ), which is itself coupled to the spin (isospin) of thethird particle to give the state with total spin S (isospin T, T z ). The total orbital angularmomentum L and the total spin S are coupled to the total angular momentum J, J z . Thefunctions Y LL z [ G ] (Ω ( ρ ) p ), having a definite value of L, L z , are the HH functions, and are writtenas [13]: Y LL z [ G ] (Ω ( ρ ) p ) = h Y ℓ ( ˆ x p ) ⊗ Y ℓ ( ˆ x p ) i LL z N [ G ] (cos φ p ) ℓ (sin φ p ) ℓ P ℓ + ,ℓ + n (cos 2 φ p ) . (2.11)Here Y ℓ ( ˆ x p ) and Y ℓ ( ˆ x p ) are spherical harmonics, N [ G ] is a normalization factor and P ℓ + ,ℓ + n (cos 2 φ p ) is a Jacobi polynomial, n being the degree of the polynomial. The grandangular quantum number G is defined as G = 2 n + ℓ + ℓ . The notations [ G ] and { G } ofEqs. (2.11) and (2.10) stand for [ ℓ , ℓ ; n ] and { ℓ , ℓ , L, S , T , S, T ; n } , respectively, and µ of Eq. (2.8) is µ = { G } , l . Note that each set of quantum numbers { ℓ , ℓ , L, S , T , S, T } is called “channel”, and the antisymmetrization of Y { G } (Ω ( ρ ) ) requires ℓ + S + T to beodd. In addition, ℓ + ℓ must be even (odd) for positive (negative) parity. To be no-ticed that after the sum on the permutation in Eq. (2.10), some states inside the sub-spacespanned by G are linearly dependent. These states have been identified and removed fromthe expansion [10, 13].In this work, we have considered modern two-body potential models which act on specificspin and angular momentum states of the two-body system. Due to the presence of the sumover the permutations in the expression for Y { G } , a given particle pair is not in a definiteangular and spin state. However, the HH functions with the grand angular quantum number G constructed in terms of a given set of Jacobi vectors x p , x p , defined starting from theparticle order i, j, k , can be always expressed in terms of the HH functions constructed, forinstance, in terms of x p =1) , x p =1) . In fact, the following relation holds Y LL z [ ℓ ,ℓ ; n ] (Ω ( ρ ) p ) = X ℓ ′ ,ℓ ′ ,n ′ a ( p ) ,Lℓ ,ℓ ,n ; ℓ ′ ,ℓ ′ ,n ′ Y LL z [ ℓ ′ ,ℓ ′ ; n ′ ] (Ω ( ρ )( p =1) ) , (2.12)where the sum is restricted to the values ℓ ′ , ℓ ′ , and n ′ such that ℓ ′ + ℓ ′ + 2 n ′ = G . Thecoefficients a ( p ) ,Lℓ ,ℓ ,n ; ℓ ′ ,ℓ ′ ,n ′ relating the two sets of HH functions are known as the Raynal-Revaicoefficients [18], and could be computed rather easily using the orthonormality property ofthe HH functions, namely a ( p ) ,Lℓ ,ℓ ,n ; ℓ ′ ,ℓ ′ ,n ′ = Z d Ω ( ρ )( p =1) (cid:16) Y LL z [ ℓ ,ℓ ; n ] (Ω ( ρ )( p =1) ) (cid:17) ∗ Y LL z [ ℓ ′ ,ℓ ′ ; n ′ ] (Ω ( ρ ) p ) . (2.13)6lso the spin-isospin states can be recoupled to obtain states where the spin and isospinquantum numbers are coupled in a given order of the particles. The result is that theantisymmetric functions Y { G } can be expressed as a superposition of functions constructedin terms of a given order of particles i, j, k , each one having the pair i , j in a definite spinand angular momentum state. When the two-body potential acts on the pair of particles i , j , the effect of the projection is easily taken into account.The expansion states of Eq. (2.1) in momentum space can be obtained as follows. Let ~ k p , ~ k p be the conjugate Jacobi momenta of the Jacobi vectors, given by ~ k p = 1 √ p j − p i ) , ~ k p = r
23 ( p k −
12 ( p i + p j )) , (2.14) p i being the momentum of the i -th particle. We then define a hypermomentum Q and a setof angular-hyperangular variables as Q = q k p + k p , Ω ( Q ) p = [ˆ k p , ˆ k p ; ϕ p ] , (2.15)where tan ϕ p = k p k p . (2.16)Then, the momentum-space version of the wave function given in Eq. (2.8) is | Ψ JJ z ( Q ) µ i = g G,l ( Q ) Y { G } (Ω ( Q ) ) , (2.17)where Y { G } (Ω ( Q ) ) is the same as Y { G } (Ω ( ρ ) ) of Eq. (2.10) with x ip → k ip , and g G,l ( Q ) = ( − i ) G Z ∞ dρ ρ Q J G +2 ( Qρ ) f l ( ρ ) . (2.18)With the adopted form of f l ( ρ ) given in Eq. (2.9), the corresponding functions g G,l ( Q ) canbe easily calculated, and they are explicitly given in Ref. [15]. B. The HH Method for Scattering States Below Deuteron Breakup Threshold
We consider here the extension of the HH technique to describe N − d scattering statesbelow deuteron breakup threshold, when both local and non-local interaction models areconsidered. 7ollowing Ref. [12], the wave function Ψ LSJJ z N − d describing the N − d scattering state withincoming orbital angular momentum L and channel spin S , parity π = ( − ) L , and totalangular momentum J, J z , can be written asΨ LSJJ z N − d = Ψ LSJJ z C + Ψ LSJJ z A , (2.19)where Ψ LSJJ z C describes the system in the region where the particles are close to each otherand their mutual interactions are strong, while Ψ LSJJ z A describes the relative motion betweenthe nucleon N and the deuteron in the asymptotic region, where the N − d nuclear interactionis negligible. The function Ψ LSJJ z C , which has to vanish in the limit of large interclusterseparations, can be expanded on the HH basis as it has been done in the case of boundstates. Therefore, applying Eq. (2.1), the function Ψ LSJJ z C can be casted in the form | Ψ LSJJ z C i = X µ c µ | Ψ JJ z µ i , (2.20)where | Ψ JJ z µ i is defined in Eqs. (2.8) and (2.17) in coordinate- and momentum-space, re-spectively.The function Ψ LSJJ z A is the appropriate asymptotic solution of the relative N − d Schr¨odinger equation. It is written as a linear combination of the following functions,Ω λLSJJ z = X p =1 Ω λLSJJ z ( p ) , (2.21)where the sum over p has to be done over the three even permutations necessary to anti-symmetrize the functions Ω λLSJJ z , andΩ λLSJJ z ( p ) = X l =0 , w l ( x p ) R λL ( y p ) nh [ Y l ( ˆ x p ) ⊗ S ] ⊗ i S ⊗ Y L ( ˆ y p ) o JJ z × [ T ⊗
12 ]
T T z . (2.22)Here the spin and isospin quantum numbers of particles i and j have been coupled to S and T , with S = 1, T = 0 for the deuteron, w l ( x p ) is the deuteron wave function componentin the waves l = 0 , y p is the distance between N and the center of mass of the deuteron,i.e. y p = q x p , Y l ( ˆ x p ) and Y L ( ˆ y p ) are the standard spherical harmonic functions, andthe functions R λL ( y p ) are the regular ( λ ≡ R ) and irregular ( λ ≡ I ) radial solutions of therelative two-body N − d Schr¨odinger equation without the nuclear interaction. These regular8nd irregular functions, denoted as F L ( y p ) and G L ( y p ) respectively, have the form R RL ( y p ) ≡ F L ( y p ) = 1(2 L + 1)!! q L C L ( η ) F L ( η, ξ p ) ξ p ,R IL ( y p ) ≡ G L ( y p ) = (2 L + 1)!! q L +1 C L ( η ) f R ( y p ) G L ( η, ξ p ) ξ p , (2.23)where q is the modulus of the N − d relative momentum (related to the total kinetic energyin the center of mass system by T cm = q µ , µ being the N − d reduced mass), η = 2 µe /q and ξ p = qy p are the usual Coulomb parameters, and the regular (irregular) Coulomb function F L ( η, ξ p ) ( G L ( η, ξ p )) and the factor C L ( η ) are defined in the standard way [19]. The factor(2 L + 1)!! q L C L ( η ) has been introduced so that F and G have a finite limit for q →
0. Thefunction f R ( y p ) = [1 − exp( − by p )] L +1 has been introduced to regularize G L at small valuesof y p . The trial parameter b is determined by requiring that f R ( y p ) → y p , thus not modifying the asymptotic behaviour of the scattering wave function. A value of b = 0 .
25 fm − has been found appropriate. The non-Coulomb case of Eq. (2.23) is obtainedin the limit e →
0. In this case, F L ( η, ξ p ) /ξ p and G L ( η, ξ p ) /ξ p reduce to the regular andirregular Riccati-Bessel functions and the factor (2 L + 1)!! C L ( η ) → η → LSJJ z A can be written in the formΨ LSJJ z A = X L ′ S ′ h δ LL ′ δ SS ′ Ω RL ′ S ′ JJ z + R JLS,L ′ S ′ ( q )Ω IL ′ S ′ JJ z i , (2.24)where the parameters R JLS,L ′ S ′ ( q ) give the relative weight between the regular and irregularcomponents of the wave function. They are closely related to the reactance matrix ( K -matrix) elements, which can be written as K JLS,L ′ S ′ ( q ) = (2 L + 1)!!(2 L ′ + 1)!! q L + L ′ +1 C L ( η ) C L ′ ( η ) R JLS,L ′ S ′ ( q ) . (2.25)By definition of the K -matrix, its eigenvalues are tan δ LSJ , δ LSJ being the phase shifts. Thesum over L ′ and S ′ in Eq. (2.24) is over all values compatible with a given J and parity π . Inparticular, the sum over L ′ is limited to include either even or odd values since ( − L ′ = π .The matrix elements R JLS,L ′ S ′ ( q ) and the linear coefficients c µ occurring in the expansionof Ψ LSJJ z C of Eq. (2.20) are determined applying the Kohn variational principle [20], whichstates that the functional[ R JLS,L ′ S ′ ( q )] = R JLS,L ′ S ′ ( q ) − D Ψ L ′ S ′ JJ z N − d |L| Ψ LSJJ z N − d E , L = m √ ~ ( H − E ) , (2.26)9as to be stationary with respect to variations of the trial parameters in Ψ LSJJ z N − d . Here E isthe total energy of the system, m is the nucleon mass, and L is chosen so that h Ω RLSJJ z |L| Ω ILSJJ z i − h Ω ILSJJ z |L| Ω RLSJJ z i = 1 . (2.27)As described in Ref. [21], using Eqs. (2.20) and (2.24), the variation of the diagonal func-tionals of Eq. (2.26) with respect to the linear parameters c µ leads to the following systemof linear inhomogeneous equations: X µ ′ h Ψ JJ z µ |L| Ψ JJ z µ ′ i c µ ′ = − D λLSJJ z ( µ ) . (2.28)Two different terms D λ corresponding to λ ≡ R, I are introduced and are defined as D λLSJJ z ( µ ) = h Ψ JJ z µ |L| Ω λLSJJ z i . (2.29)The matrix elements R JLS,L ′ S ′ ( q ) are obtained varying the diagonal functionals of Eq. (2.26)with respect to them. This leads to the following set of algebraic equations X L ′′ S ′′ R JLS,L ′′ S ′′ ( q ) X L ′ S ′ ,L ′′ S ′′ = Y LS,L ′ S ′ , (2.30)with the coefficients X and Y defined as X LS,L ′ S ′ = h Ω ILSJJ z + Ψ LSJJ z ,IC |L| Ω IL ′ S ′ JJ z i ,Y LS,L ′ S ′ = −h Ω RLSJJ z + Ψ LSJJ z ,RC |L| Ω IL ′ S ′ JJ z i . (2.31)Here Ψ LSJJ z ,λC is the solution of the set of Eq. (2.28) with the corresponding D λ term. Asecond order estimate of R JLS,L ′ S ′ ( q ) is given by the quantities [ R JLS,L ′ S ′ ( q )], obtained bysubstituting in Eq. (2.26) the first order results. Such second-order calculation provides asymmetric reactance matrix. This condition is not a priori imposed, and therefore it is auseful test of the numerical accuracy.In the particular case of q = 0 (zero-energy scattering), the scattering can occur only inthe channel L = 0 and the observables of interest are the scattering lengths. Within thepresent approach, they can be easily obtained from the relation (2 J +1) a Nd = − lim q → R J J, J ( q ) . (2.32)10n alternative way to solve the scattering problem, used when q = 0, is to apply thecomplex Kohn variational principle to the S -matrix, as in Ref. [21]. In this way, the Kohnvariational principle of Eq. (2.26) becomes[ S JLS,L ′ S ′ ( q )] = S JLS,L ′ S ′ ( q ) + i h Ψ + ,L ′ S ′ JJ z N − d |S| Ψ + ,LSJJ z N − d i . (2.33)Here Ψ + ,LSJJ z N − d = Ψ LSJJ z C + Ψ + ,LSJJ z A , (2.34)with Ψ LSJJ z C given in Eq. (2.20) andΨ + ,LSJJ z A = X p =1 Ω + LSJJ z ( p )Ω + LSJJ z ( p ) = ( i ˜Ω RLSJJ z ( p ) − ˜Ω ILSJJ z ( p ) )+ X L ′ S ′ S JLS,L ′ S ′ ( q )( i ˜Ω RL ′ S ′ JJ z ( p ) + ˜Ω IL ′ S ′ JJ z ( p ) ) . (2.35)The functions ˜Ω λLSJJ z ( p ) are the same as in Eq. (2.22), with R RL ( y p ) = F L ( η, ξ p ) /ξ p and R IL ( y p ) = f R ( y p ) G L ( η, ξ p ) /ξ p . Note that, with the above definition, the reactance K -matrixelements can be related to the S -matrix elements as K JLS,L ′ S ′ ( q ) = ( − i )[ S JLS,L ′ S ′ ( q ) − δ LL ′ δ SS ′ ] [ S JLS,L ′ S ′ ( q ) + δ LL ′ δ SS ′ ] − . (2.36)The calculation involving Ψ LSJJ z C has been performed with the HH expansion incoordinate- or in momentum-space, depending on what is more convenient, as it has beenexplained for the bound state in the previous subsection. Some difficulties arise for the cal-culation of the potential energy matrix elements which involve Ω λLSJJ z , i.e. h Ψ JJ z µ | V | Ω λLSJJ z i present in Eq. (2.29), and h Ω λ ′ L ′ S ′ JJ z + Ψ L ′ S ′ JJ Z ,λ ′ C | V | Ω λLSJJ z i of Eq. (2.31), with λ, λ ′ = R , I .In the present work, we consider both two- and three-nucleon interactions, and therefore V = X i In this section we present our results for n − d and p − d scattering observables atcenter-of-mass energies below deuteron breakup threshold. The interaction models whichhave been used are the AV18 and the N3LO-Idaho two-nucleon, and the AV18/UIX, N3LO-Idaho/UIXp and the N3LO-Idaho/N2LO two- and three-nucleon interactions. Note thatthe AV18 and AV18/UIX results are the same as those ones first obtained in Ref. [24], usingthe PHH expansion. We have considered also the V low − k model, obtained from the AV18two-nucleon interaction with a cutoff parameter Λ equal to 2.2 fm − . The cutoff param-13ter has been chosen so that the triton binding energy is 8.477 MeV, when the completeelectromagnetic interaction is used, including neutron charge distribution and MM interac-tion effects. On the other hand, when no electromagnetic effects are considered, the tritonbinding energy has been found to be 8.519 MeV. In the scattering problem, only the pointCoulomb interaction has been considered, except when differently indicated.Before presenting the results for the considered low-energy N − d observables, we discussthe pattern of convergence for some representative quantities, i.e. the n − d doublet zero-energy scattering length a nd and the p − d J π = 1 / + , / − phase shifts and mixing anglesat center-of-mass energy E cm = 2 . J π = 1 / + and 1 / − are given in Tables I and II, respectively. The notation is the same as in Eq. (2.11).To be noticed that the scattering channels in the case of J π = 1 / − are ordered for increasingvalues of ℓ + ℓ . This is true also for all the channels here considered, except those for J π = 1 / + (see Table I), where the ordering respects an “historical choice”, first done in thecase of the three-nucleon bound state in Ref. [25].In Table III we present the results for a nd and p − d J π = 1 / + , / − phase shifts andmixing angles ( δ LSJ , ǫ ) at E cm = 2 . M in the hyperradial functions (see Eqs. (2.9) and (2.18)). All the 23 (25) angularmomentum-spin-isospin channels of Table I (II) are considered for J π = 1 / + (1 / − ), andHH functions up to grand angular momentum G = 20 (21) for all the channels have beenincluded. From inspection of the table, we can conclude that the use of M = 28 is enoughto reach an accuracy of at least 0.002 fm for the scattering length and four significant digitsfor the phase shifts and mixing angles. In fact, for other p − d scattering channels at someof the considered values of E cm , even M = 24 and M = 20 has been found enough to reachthe same degree of accuracy.To study the convergence on the HH expansion, as it has been done in Ref. [14], wehave separated the HH functions into classes having particular properties and we have takeninto account the fact that the convergence rates of the different classes are rather different.For instance, we expect that the contribution of the HH functions with lower values of ℓ ,α = ℓ ,α + ℓ ,α to be the most important. Therefore, for all the J π scattering states,except J π = 1 / + , the different classes are classified with increasing value of ℓ ,α , up to ℓ ,α ≤ 6, and among those ones with the same ℓ ,α , we have included first the contributions14 ABLE I: Angular momentum, spin and isospin quantum numbers for the first 23 channels con-sidered in the expansion of the J π = 1 / + core wave function. α ℓ α ℓ α L α S α T α S α T α ABLE II: Same as Table I, but for the first 25 channels considered in the expansion of the J π = 1 / − core wave function. α ℓ α ℓ α L α S α T α S α T α ABLE III: n − d doublet scattering length a nd in fm and p − d J π = 1 / + , / − phase shifts δ LSJ and mixing angles ǫ at E cm = 2 . M . All the channels ofTables I and II are included with grand angular momentum for all the channels set equal to 20 for J π = 1 / + and 21 for J π = 1 / − . M = 4 M = 8 M = 12 M = 16 M = 20 M = 24 M = 28 a nd δ , , -3.611 -3.583 -3.572 -3.570 -3.570 -3.569 -3.569 δ , , -43.28 -34.69 -32.41 -31.96 -31.82 -31.78 -31.77 ǫ 12 + δ , , -8.270 -7.756 -7.608 -7.581 -7.576 -7.575 -7.575 δ , , ǫ − of the HH functions with lower ℓ ,α . Finally, the T α = 3 / J π = 1 / − case, the channels have been classified in 6 classes, includingchannels 1–3, 4–6, 7–10, 11–14, 15–18, and 19–25 of Table II, respectively. In the case of J π = 1 / + , the classification follows the footsteps of Ref. [25], and therefore the channelshave been classified in 5 classes, including channels 1–3, 4–8, 9–12, 13–18, and 19–23 ofTable I, respectively. We have then called G i , for each class i , a number such that each stateof class i has the grand angular momentum G ≤ G i , and we have increased G i until we havereached convergence. Then, keeping G i fixed at this value, we have included the states of thefollowing class, and increased G i +1 again until we have reached convergence. The results forthe zero-energy scattering length and the low-energy phase shifts and mixing angles obtainedwith this procedure are given in Table IV for J π = 1 / + and V for J π = 1 / − . Here, M = 28Laguerre polynomials in the expansion of the hyperradial function are included, and againthe N3LO-Idaho two-nucleon potential is used.From the cases presented in the Tables, and as well as for all cases taken in consideration,we can observe that (i) the last classes of channels, corresponding to the T α = 3 / p − d phase shifts and mixing angles, but negligible ones17o the n − d ones. (ii) The T α = 1 / ℓ ,α (fourth classfor J π = 1 / + and fifth one for J π = 1 / − ) give negligible contributions. This implies that ℓ ,α ≤ J π = 1 / + , when the non-local potential N3LO-Idahois used, values of G up to 80 have been found to be necessary (see Table IV). On theother hand, in the case of the local AV18, we have verified that within the HH expansion(i.e. without the correlation) G = 160 is needed to reach the same degree of accuracy.This is related to the fact that the AV18 potential is more repulsive at short interparticledistances, and therefore the corresponding wave functions in that region are more difficult tobe constructed. In fact, when the calculation is performed using the V low − k potential model,which is very soft at short interparticle distances, it is sufficient to set G = 40. A completelyidentical pattern of convergence is found for all other J π waves. (iv) The convergence of theother classes is usually faster than for the first class, as it is evident for the cases reportedin Tables IV and V. For the J π = 1 / + case, we obtain convergence with just G , , , = 20.For J π = 1 / − , we have to consider fairly large values of G only for the fourth class (upto G = 51), since the channels belonging to this class (the channels 11-14 as reportedin Table II) are needed to describe pairs in orbital angular momentum ℓ = 2. Namely,together with the channels of the first class, they are needed to have a good descriptionsof the pairs in the deuteron waves. We have also found that the convergence rate of theseclasses does not depend much on the non-local interaction model. For example, with the V low − k potential, convergence is achieved with G , , , = 20 for J π = 1 / + and G , = 31, G , , = 21 for J π = 1 / − . However, note that for the AV18 potential model, we need toset G = 90, G , , = 40 for J π = 1 / + and G = 61, G , = 41, G = 91, G = 21 for J π = 1 / − . A similar pattern of convergence has been found for all the calculated quantities.From now on, all the results which will be presented have been obtained at convergence inthe basis expansion.The results for the n − d and p − d doublet and quartet scattering lengths are given inTable VI and are compared with the available experimental data [26, 27]. The results for theAV18 and AV18/UIX have been taken from Ref. [10]. Comparing the theoretical and exper-imental results for a nd and a nd , we can conclude that a nd is very little model-dependent(as well as a pd ), and there is a satisfactory agreement between theory and experiment. On18 ABLE IV: Convergence of the n − d doublet scattering length a nd in fm and p − d J π = 1 / + phase shifts δ LSJ and mixing angles ǫ at E cm = 2 . M = 28. G G G G G a nd δ , , δ , , ǫ 12 + 50 1.245 -3.577 -32.11 1.24860 1.243 -3.577 -32.09 1.24870 1.242 -3.577 -32.08 1.24880 1.242 -3.577 -32.08 1.24880 16 1.112 -3.572 -31.18 1.24080 20 1.112 -3.572 -31.17 1.23880 20 16 1.100 -3.569 -31.09 1.23980 20 20 1.100 -3.569 -31.09 1.23980 20 20 16 1.099 -3.569 -31.09 1.23980 20 20 20 1.099 -3.569 -31.09 1.23980 20 20 20 16 1.099 -3.569 -31.04 1.24180 20 20 20 20 1.099 -3.569 -31.04 1.241 the contrary, a nd is strongly model-dependent, and only the inclusion of the TNI brings thetheoretical value close to the experimental one. However, some disagreement still remains,and the recent measurement of Ref. [27] is not well described by any of the potential modelsconsidered. Though, the N3LO-Idaho/UIXp and N3LO-Idaho/N2LO models give slightlybetter results. Note that the AV18/UIX results obtained including also MM interactioneffects are a nd = 0 . 590 fm and a nd = 6 . 343 fm. Finally, the V low − k results are in remark-able disagreement with the experimental data, and a sizable difference from the AV18/UIXresults is also observed. Therefore, even when the cutoff parameter of the V low − k interactionmodel is fixed to reproduce the triton binding energy, the doublet scattering length is notwell reproduced. This observation seems to suggest that the S -wave sensitive scatteringobservables, like the scattering lengths, are not properly described by simply increasing theattraction, but a right balance between attraction and repulsion of the nuclear force has to19 ABLE V: Same as Table IV but for p − d J π = 1 / − phase shifts δ LSJ and mixing angles ǫ at E cm = 2 . G G G G G G δ , , δ , , ǫ − 61 -7.416 21.24 5.54471 -7.413 21.25 5.54581 -7.412 21.25 5.54591 -7.411 21.25 5.54591 11 -7.382 21.53 5.61991 21 -7.380 21.55 5.62291 31 -7.379 21.55 5.62291 31 15 -7.367 21.77 5.70491 31 21 -7.367 21.77 5.70591 31 21 31 -7.372 22.02 5.79991 31 21 41 -7.370 22.03 5.79891 31 21 51 -7.369 22.04 5.79891 31 21 51 15 -7.369 22.04 5.79891 31 21 51 21 -7.369 22.04 5.79891 31 21 51 21 15 -7.342 22.05 5.81891 31 21 51 21 21 -7.340 22.05 5.819 be reached. Such a balance cannot be achieved with just one parameter, as the cutoff Λ ofthe V low − k interaction. Further analysis of these aspects is currently underway [28].The p − d elastic scattering observables have been studied at different values of center-of-mass energy E cm . Since we have considered several interaction models, we first focusour attention on the two-nucleon only models, i.e. the AV18 and the N3LO-Idaho. Thedifferential cross section dσ/d Ω, the proton vector analyzing power A y , the deuteron vectorand tensor analyzing powers iT , T , T and T , as function of the center-of-mass angle θ cm , are given in Figs. 1, 2, 3, 4, 5 and 6, respectively. The data are taken from Refs. [29,30, 31, 32, 33, 34], as indicated in detail in the figure captions. By inspection of the figures,we can observe that: (i) theory and experiment are in disagreement for the A y and iT ABLE VI: n − d and p − d doublet and quartet scattering lengths in fm calculated with the HHtechnique using different Hamiltonian models.Interaction a nd a nd a pd a pd AV18 1.275 6.325 1.185 13.588AV18/UIX 0.610 6.323 -0.035 13.588N3LO-Idaho 1.099 6.342 0.876 13.646N3LO-Idaho/UIXp 0.623 6.343 -0.007 13.647N3LO-Idaho/N2LO 0.675 6.342 0.072 13.647 V low − k ± ± ± ± observables (the well-known “ A y -puzzle” [24, 35]); (ii) no differences between the AV18and the N3LO-Idaho curves can be seen for the differential cross sections; (iii) the N3LO-Idaho curves are systematically closer to the data than the AV18 ones for the polarizationobservables, especially for A y and iT . The reason of this behaviour is well known [16] andis related to the MM interaction. In fact, the AV18 potential model has been constructedkeeping the electromagnetic interaction separated from the nuclear one. The electromagneticinteraction includes the MM one, as well as higher-order corrections to the pp Coulombpotential as two-photon exchange, Darwin-Foldy and vacuum polarization terms. The MMinteraction effects are known to be sizable in N − d elastic scattering [16]. On the contrary, theN3LO-Idaho potential model keeps as electromagnetic interaction only the point Coulombpotential and MM effects are indirectly included in the nuclear part of the interaction bythe fitting procedure. From this observation, we can guess that the results obtained withthe two-nucleon potentials AV18 and N3LO-Idaho should be comparable when the AV18calculation includes also the MM effects. To verify this hypothesis, we have calculated the p − d elastic scattering observables at two values of E cm , 1.33 and 2.0 MeV, using the AV18,AV18+MM, and N3LO-Idaho potential models. The results are given in Figs. 7 and 8,respectively. From inspection of the figures, we can notice that the AV18+MM results forthe A y and iT vector polarization observables are larger than the AV18 alone ones in21 d σ / d Ω [ m b / s r] θ cm [deg] d σ / d Ω [ m b / s r] θ cm [deg] θ cm [deg] E cm =266 keV E cm = 431 keV E cm = 666 keVE cm = 1.33 MeV E cm = 1.66 MeV E cm = 2.0 MeV FIG. 1: p − d differential cross section for E cm = 0 . , . , . , . , . 66 and 2.0 MeV calcu-lated with the AV18 (dashed lines) and the N3LO-Idaho (solid lines) two-nucleon potential models.Data are from Ref. [29] at E cm = 0 . 266 MeV, from Ref. [30] at E cm = 0 . 431 MeV, from Refs. [29](solid circles), [31] (empty circles), and [32] (empty squares) at E cm = 0 . 666 MeV, from Refs. [32](empty squares – E p = 1 . 993 MeV), [33] (solid circles), and [34] (empty circles – E p = 2 . 08 MeV)at E cm = 1 . 33 MeV, from Refs. [33] (solid circles) and [34] (empty circles – E p = 2 . 53 MeV) at E cm = 1 . 66 MeV, from Refs. [33] (solid circles), [32] (empty squares – E p = 2 . 995 MeV), and [34](empty circles) at E cm = 2 . the maximum region, and that the AV18+MM and N3LO-Idaho curves are quite close toeach other for all the observables considered. Although this analysis should be performedsystematically at any value of E cm and for any observable, given the conclusions of Ref. [16],it can be expected that a similar behaviour still holds. Therefore, we can conclude that thenon-local N3LO-Idaho and the local AV18 two-nucleon interactions give similar results once22 A y θ cm [deg] A y θ cm [deg] θ cm [deg] E cm =266 keV E cm = 431 keV E cm = 666 keVE cm = 1.33 MeV E cm = 1.66 MeV E cm = 2.0 MeV FIG. 2: Same as Fig. 1, but for the proton vector analyzing power A y . Data are from Ref. [30] at E cm = 0 . 431 MeV, from Ref. [31] at E cm = 0 . 666 MeV, from Ref. [33] at E cm = 1 . , . 66 and 2.0MeV. the MM effects are included in the AV18 calculation. For this reason, we have chosen to usethe N3LO-Idaho two-nucleon interaction model in the continuation of our study.In order to have a meaningful comparison with the data, the TNI cannot be neglected inthe calculation. Therefore, we present in Figs. 9, 10, 11, 12, 13 and 14 the results for the dif-ferent observables, obtained with the N3LO-Idaho two-nucleon, and the N3LO-Idaho/UIXpand N3LO-Idaho/N2LO two- and three-nucleon interaction models. From inspection of thefigures, we can observe that the TNI effects are sizable, especially for the polarization ob-servables, and the N3LO-Idaho/N2LO potential model gives a slightly better descriptionof the data than the N3LO-Idaho/UIXp one. In particular, it is interesting to notice thatthe A y and iT observables are better described at every value of E cm , except for iT at E cm = 1 . 66 MeV, although even in this case all the curves are very close to each other.23 i T θ cm [deg] i T θ cm [deg] θ cm [deg] E cm =266 keV E cm = 431 keV E cm = 666 keVE cm = 1.33 MeV E cm = 1.66 MeV E cm = 2.0 MeV FIG. 3: Same as Fig. 1, but for the deuteron vector analyzing power iT . Data are from Ref. [30]at E cm = 0 . 431 and 1.33 MeV, from Ref. [31] at E cm = 0 . 666 MeV, from Ref. [33] at E cm = 1 . For a better comparison between the different potential models and the data, a χ analysishas been carried only for those observables, except the differential cross section, for whichthe number of data N is N ≥ 7. In particular, following Ref. [36], χ / datum = 1 N X i ( f expi − f thi ) (∆ f i ) , (3.1)where f expi is the i th datum at center-of-mass angle θ i , ∆ f i is its experimental error, and f thi is the theoretical value at the same angle. The results are given in Table VII for E cm = 0 . , . , . 66 and 2.0 MeV. The N3LO-Idaho, N3LO-Idaho/UIXp and N3LO-Idaho/N2LO interaction models have been considered. From inspection of the table we cannotice that all the values for χ /datum are comparable, although the ones obtained with theN3LO-Idaho two-nucleon interaction are usually higher than the ones obtained with two-24 T -0.03-0.02-0.0100.010.02 -0.04-0.0200.020.040 30 60 90 120 150 180 θ cm [deg] -0.04-0.0200.02 T θ cm [deg] -0.04-0.0200.02 0 30 60 90 120 150 180 θ cm [deg] -0.04-0.0200.020.04 E cm =266 keV E cm = 431 keV E cm = 666 keVE cm = 1.33 MeV E cm = 1.66 MeV E cm = 2.0 MeV FIG. 4: Same as Fig. 1, but for the deuteron tensor analyzing power T . Data are from Ref. [30]at E cm = 0 . 431 MeV, from Ref. [31] at E cm = 0 . 666 MeV, from Refs. [33] at E cm = 1 . 66 and 2.0MeV. and three-nucleon interactions, except for the tensor analyzing power T and T . This isa well-known and still unclear issue, i.e. T and T are better described, as the energyincreases, by two-nucleon only interaction models, even at 30.0 MeV [37]. Among the two-plus three-nucleon interaction models, the N3LO-Idaho/N2LO performs slightly better thanthe N3LO-Idaho/UIXp.The p − d elastic scattering observables at E cm = 0 . 666 and 2.0 MeV have been calculatedalso using the two-nucleon only potential model V low − k , obtained from the AV18 with acutoff parameter Λ equal to 2.2 fm − , as already used for the calculation of the scatteringlengths. The results are given in Figs. 15 and 16, respectively. Together with the V low − k results, we have shown also the bare AV18 and the AV18/UIX ones. From inspectionof the figures, we can observe that the V low − k results are very similar to the AV18/UIX25 T θ cm [deg] -0.00500.0050.010.015 T θ cm [deg] -0.01-0.00500.0050.010.0150.02 0 30 60 90 120 150 180 θ cm [deg] -0.01-0.00500.0050.010.0150.02 E cm =266 keV E cm = 431 keV E cm = 666 keVE cm = 1.33 MeV E cm = 1.66 MeV E cm = 2.0 MeV FIG. 5: Same as Fig. 1, but for the deuteron tensor analyzing power T . Data are from the samereferences as in Fig. 4. ones. This can be understood noticing that the considered observables are sensitive to P -and D -wave scattering. The P -wave phase shifts and mixing angles are influenced by theUIX TNI attraction term, which is reproduced, within the V low − k approach, by fitting thecutoff parameter Λ. In fact, the J π = 1 / − phase shifts and mixing angle ( δ 12 12 , δ 32 12 , ǫ − )obtained at E cm = 2 . V low − k potential models are( − . , . , . − . , . , . − . , . , . V low − k results for N − d scattering at low energies, we can conclude thatthe V low − k and AV18/UIX results are close to each other for observables sensitive to P - and D -wave scattering, like vector and tensor analyzing powers. Further work on these aspectsis currently underway.The n − d elastic scattering observables, including differential cross section, neutron vectoranalyzing power A y , deuteron vector and tensor analyzing powers iT , T , T and T , at26 T -0.012-0.01-0.008-0.006-0.004-0.0020 -0.015-0.01-0.00500 30 60 90 120 150 180 θ cm [deg] -0.025-0.02-0.015-0.01-0.005 T θ cm [deg] -0.035-0.03-0.025-0.02-0.015-0.01-0.005 0 30 60 90 120 150 180 θ cm [deg] -0.04-0.03-0.02-0.01 E cm =266 keV E cm = 431 keV E cm = 666 keVE cm = 1.33 MeV E cm = 1.66 MeV E cm = 2 MeV FIG. 6: Same as Fig. 1, but for the deuteron tensor analyzing power T . Data are from the samereferences as in Fig. 4. E cm = 1 . 33 and 2.0 MeV are given in Figs. 17 and 18, respectively. The experimental dataare from Refs. [38, 39, 40] and Refs. [41, 42] at E cm = 1 . 33 MeV and 2.0 MeV, respectively.The different curves are obtained using the N3LO-Idaho, N3LO-Idaho/UIXp and N3LO-Idaho/N2LO potential models. From inspection of the figures, we can observe that all thecurves are very close to each other, especially for the differential cross section dσ/d Ω and thetensor analyzing powers T , T and T , although some small differences are appreciable.Moreover, some differences are present for the A y and iT vector polarization observablesat the peak, even if TNI effects are small. Comparing the calculations with the data, we canobserve that the calculated dσ/d Ω at E cm = 1 . 33 MeV is much lower than the measuredone for large values of the center-of-mass angle θ cm . Such a discrepancy however disappearsat E cm = 2 . p − d case, the n − d vector analyzing powers A y are poorly described27 30 60 90 120 150 180 θ cm [deg]0200400600800 d σ / d Ω [ m b / s r] θ cm [deg] 00.010.020.030.04 A y θ cm [deg]00.0050.010.0150.02iT θ cm [deg] -0.04-0.03-0.02-0.0100.010.02T θ cm [deg]-0.00500.0050.010.0150.02T θ cm [deg] -0.025-0.02-0.015-0.01-0.0050 T FIG. 7: Theoretical results for p − d differential cross section dσ/d Ω, and polarization observables A y , iT , T , T and T , at E cm = 1 . 33 MeV are compared to the experimental data. Thecalculation are done using the AV18 (dashed lines), the AV18+MM (dotted-dashed lines), and theN3LO-Idaho (solid lines) interactions. The data are from Refs. [32] (empty squares – E p = 1 . E p = 2 . 08 MeV) for the differential cross section,and from Refs. [33] and [30] for the A y and iT polarization observables, respectively. The incidentproton (deuteron) is E p = 2 . E d = 4 . 30 60 90 120 150 180 θ cm [deg]0100200300400500 d σ / d Ω [ m b / s r] θ cm [deg] 00.010.020.030.040.050.06 A y θ cm [deg]00.010.020.03iT θ cm [deg] -0.04-0.0200.02T θ cm [deg]-0.0100.010.02T θ cm [deg] -0.03-0.02-0.010 T FIG. 8: Same as Fig. 7 but for E cm = 2 . E p = 2 . 995 MeV), and [34] (empty circles) for the differential cross section, andfrom Ref. [33] for the polarization observables. The incident proton (deuteron) is E p = 3 . E d = 6 . d σ / d Ω [ m b / s r] θ cm [deg] d σ / d Ω [ m b / s r] θ cm [deg] θ cm [deg] E cm =266 keV E cm = 431 keV E cm = 666 keVE cm = 1.33 MeV E cm = 1.66 MeV E cm = 2.0 MeV FIG. 9: p − d differential cross section for E cm = 0 . , . , . , . , . 66 and 2.0 MeV calcu-lated with the N3LO-Idaho (dashed lines), the N3LO-Idaho/UIXp (dotted-dashed lines) and theN3LO-Idaho/N2LO (solid lines) two- and three-nucleon interaction models. Data are from thesame references as in Fig. 1. by the theory in the maximum region, but it should be noticed that the N3LO-Idaho/N2LOgives again a better description of the observables than the N3LO-Idaho/UIXp Hamiltonianmodel. IV. CONCLUSIONS Following our previous studies on the HH method revisited to work in momentum-space [10, 15], we have implemented our technique to study the N − d elastic scatteringproblem at center-of-mass energies below deuteron breakup threshold, using both local andnon-local realistic nuclear interactions. Using this method, it is possible to accurately cal-30 A y θ cm [deg] A y θ cm [deg] θ cm [deg] E cm =266 keV E cm = 431 keV E cm = 666 keVE cm = 1.33 MeV E cm = 1.66 MeV E cm = 2.0 MeV FIG. 10: Same as Fig. 9, but for the proton vector analyzing power A y . Data are from the samereferences as in Fig. 2. culate N − d scattering observables at very low energies, including the contribution fromthe Coulomb potential as well as from higher order electromagnetic terms, such as the MMinteraction. In particular, it is the first time that nuclear model including non-local two-nucleon interactions plus TNIs are used to describe p − d scattering at very low energies.We have studied several observables, as scattering lengths, differential cross section, vectorand tensor analyzing powers, and we have compared our results with the available experi-mental data. Our main conclusions can be summarized as follows: (i) the results obtainedfrom the local AV18 and the non-local N3LO-Idaho two-nucleon interaction are quite dif-ferent from each other, especially for the vector polarization observables A y and iT in themaximum region. (ii) The differences between AV18 and N3LO-Idaho results are stronglyreduced when the MM effects are included in the AV18 calculation. To be noticed thatthe MM effects are indirectly included in the nuclear N3LO-Idaho interaction, since in the31 i T θ cm [deg] i T θ cm [deg] θ cm [deg] E cm =266 keV E cm = 431 keV E cm = 666 keVE cm = 1.33 MeV E cm = 1.66 MeV E cm = 2.0 MeV FIG. 11: Same as Fig. 9, but for the deuteron vector analyzing power iT . Data are from thesame references as in Fig. 3. fitting procedure for this model only the point Coulomb interaction between pp is used.(iii) Among the TNIs here considered, the N2LO model performs slightly better than theUIX one. The N3LO-Idaho/N2LO results are in fact generally closer to the experimentaldata than the N3LO-Idaho/UIXp ones. (iv) The V low − k two-nucleon interaction model hasalso been considered, obtained from AV18 with a cutoff parameter Λ = 2 . − , fittedto reproduce the triton binding energy. The V low − k results for those observables sensitive to S -wave scattering, such as the scattering lengths, are in strong disagreement with the exper-imental data and quite different from the corresponding AV18/UIX ones. On the contrary,the results for those observables sensitive to P - and D -wave scattering, such as vector andtensor analyzing powers, are very similar to the corresponding AV18/UIX ones. Furtherstudies on these aspects are currently underway. We expect to extend the present approachto the A = 4 scattering problem below breakup threshold, as already done for zero-energy32 T -0.03-0.02-0.0100.010.02 -0.04-0.0200.020.040 30 60 90 120 150 180 θ cm [deg] -0.04-0.0200.02 T θ cm [deg] -0.04-0.0200.02 0 30 60 90 120 150 180 θ cm [deg] -0.04-0.0200.020.04 E cm =266 keV E cm = 431 keV E cm = 666 keVE cm = 1.33 MeV E cm = 1.66 MeV E cm = 2.0 MeV FIG. 12: Same as Fig. 9, but for the deuteron tensor analyzing power T . Data are from the samereferences as in Fig. 4. scattering in Ref. [10]. [1] R.B. Wiringa, V.G.J. Stoks, and R. Schiavilla, Phys. Rev. C , 38 (1995).[2] R. Machleidt, Phys. Rev. C , 024001 (2001).[3] E. Epelbaum, W. Gl¨ockle, and Ulf-G. Meissner, Nucl. Phys. A , 362 (2005).[4] D.R. Entem and R. Machleidt, Phys. Rev. C , 041001 (2003).[5] B.S. Pudliner et al. , Phys. Rev. Lett. , 4396 (1995).[6] E. Epelbaum et al. , Phys. Rev. C , 064001 (2002).[7] P. Navr´atil, Few-Body Syst. , 117 (2007).[8] S.K. Bogner et al. , Nucl. Phys. A , 79 (2007). T θ cm [deg] -0.00500.0050.010.015 T θ cm [deg] -0.01-0.00500.0050.010.0150.02 0 30 60 90 120 150 180 θ cm [deg] -0.01-0.00500.0050.010.0150.02 E cm =266 keV E cm = 431 keV E cm = 666 keVE cm = 1.33 MeV E cm = 1.66 MeV E cm = 2.0 MeV FIG. 13: Same as Fig. 9, but for the deuteron tensor analyzing power T . Data are from the samereferences as in Fig. 5.[9] See for example W. Gl¨ockle et al. , Phys. Rep. , 107 (1996); J. Carlson and R. Schiavilla,Rev. 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T -0.012-0.01-0.008-0.006-0.004-0.0020 -0.015-0.01-0.00500 30 60 90 120 150 180 θ cm [deg] -0.025-0.02-0.015-0.01-0.005 T θ cm [deg] -0.035-0.03-0.025-0.02-0.015-0.01-0.005 0 30 60 90 120 150 180 θ cm [deg] -0.04-0.03-0.02-0.01 E cm =266 keV E cm = 431 keV E cm = 666 keVE cm = 1.33 MeV E cm = 1.66 MeV E cm = 2.0 MeV FIG. 14: Same as Fig. 9, but for the deuteron tensor analyzing power T . Data are from the samereferences as in Fig. 6.[17] M. Fabre de la Ripelle, Ann. Phys. , 281 (1983).[18] J. Raynal and J. Revai, Il Nuovo Cim. , 612 (1970).[19] C.R. Chen et al. , Phys. Rev. C , 1261 (1989).[20] W. Kohn, Phys. Rev. , 1763 (1948).[21] A. Kievsky, Nucl. Phys. A , 125 (1997).[22] L.E. Marcucci et al. , Few-Body Syst. , 207 (2008).[23] A. Kievsky et al. , Nucl. Phys. A , 402 (1996).[24] A. Kievsky, M. Viviani, and S. Rosati, Phys. Rev. C , R15 (1995).[25] H. Kameyama, M. Kamimura, and Y. Fukushima, Phys. Rev. C , 974 (1989).[26] W. Dilg et al. , Phys. Lett. , 208 (1971).[27] K. Schoen et al. , Phys. Rev. C , 044005 (2003). 30 60 90 120 150 180 θ cm [deg]050010001500200025003000 d σ / d Ω [ m b / s r] θ cm [deg] 00.0050.010.0150.02 A y θ cm [deg]00.0020.0040.0060.008iT θ cm [deg] -0.04-0.03-0.02-0.0100.010.02T θ cm [deg]00.0050.010.0150.020.025T θ cm [deg] -0.015-0.01-0.0050 T FIG. 15: The theoretical values of the p − d differential cross section dσ/d Ω, A y , iT , T , T and T , at E cm = 0 . 666 MeV are compared to the experimental data. The calculation are donewith the AV18 (dashed lines), the AV18/UIX (dotted-dashed lines) and the the V low − k two-nucleoninteraction, obtained from the AV18 with a cutoff parameter Λ = 2 . − (solid lines). The dataare from Refs. [29] (solid circles), [31] (empty circles) and [32] (empty squares) for the differentialcross section, and from Ref. [31] for the polarization observables. The incident proton (deuteron)is E p = 1 . E d = 2 . ABLE VII: χ /datum of the p − d elastic scattering observables at E cm = 0 . , . , . 66 and 2.0MeV, calculated with the N3LO-Idaho two-nucleon only, and the N3LO-Idaho/UIXp and N3LO-Idaho/N2LO two- plus three-nucleon Hamiltonian models. The different number N of experimentaldata is also indicated. The data are from Ref. [31] at E cm = 0 . 666 MeV, and from Ref. [33] at E cm = 1 . , . 66 and 2.0 MeV.0.666 MeV 1.33 MeV 1.66 MeV 2.0 MeV A y iT T T T A y A y iT T T T A y iT T T T N et al. , in preparation (2009).[29] E. Huttel et al. , Nucl. Phys. A , 435 (1983).[30] C.R. Brune et al. , Phys. Rev. C , 044013 (2001).[31] M.H. Wood et al. , Phys. Rev. C , 024005 (2001); ibid. , , 034002 (2002).[32] D.C. Kocher and T.B. Clegg, Nucl. Phys. A , 455 (1969).[33] S. Shimizu et al. , Phys. Rev. C , 1193 (1995).[34] R. Sherr et al. , Phys. Rev. , 662 (1947).[35] H. Witala, D. H¨uber, and W. Gl¨ockle, Phys. Rev. C , R14 (1994).[36] A. Kievsky et al. , Phys. Rev. C , 024005 (2001).[37] A. Kievsky, M. Viviani, and S. Rosati, Phys. Rev. C , 024002 (2001).[38] R.K. Adair et al. , Phys. Rev. , 1165 (1953).[39] J. Weber et al. , Helv. Phys. Acta , 657 (1981).[40] E.M. Neidel et al. , Phys. Lett. B , 29 (2003).[41] P. Schwarz et al. , Nucl. Phys. A , 1 (1983).[42] J.E. McAninch et al. , Phys. Rev. C , 589 (1994). 30 60 90 120 150 180 θ cm [deg]100200300400 d σ / d Ω [ m b / s r] θ cm [deg] 00.010.020.030.040.050.06 A y θ cm [deg]00.010.020.03iT θ cm [deg] -0.04-0.0200.02T θ cm [deg]-0.0100.010.02T θ cm [deg] -0.03-0.02-0.010 T FIG. 16: Same as Fig. 15, but at E cm = 2 . E p = 3 . E d = 6 . 30 60 90 120 150 180 θ cm [deg]02004006008001000 d σ / d Ω [ m b / s r] θ cm [deg] 00.010.020.030.040.05 A y θ cm [deg]00.0050.010.0150.020.025iT θ cm [deg] -0.02-0.0100.010.02T θ cm [deg]-0.00500.0050.010.0150.020.0250.03T θ cm [deg] -0.03-0.02-0.010 T FIG. 17: n − d differential cross section dσ/d Ω, A y , iT , T , T and T , at E cm = 1 . 33 MeVare calculated with the N3LO-Idaho (dashed line), the N3LO-Idaho/UIXp (dotted-dashed line),and the N3LO-Idaho/N2LO (solid line) potential models. The experimental data are of Refs. [38](solid circles) and [39] (empty squares – E n = 2 . 016 MeV) for dσ/d Ω, and Ref. [40] for A y . Theincident neutron (deuteron) is E n = 2 . E d = 4 . 30 60 90 120 150 180 θ cm [deg]0100200300400500 d σ / d Ω [ m b / s r] θ cm [deg] 00.020.040.06 A y θ cm [deg]00.010.020.03iT θ cm [deg] -0.03-0.02-0.0100.010.02 T θ cm [deg]-0.0100.010.020.030.04T θ cm [deg] -0.04-0.03-0.02-0.010 T FIG. 18: Same as Fig. 17, but for E cm = 2 . dσ/d Ω, and Ref. [42] for A y . The incident neutron (deuteron) is E n = 3 . E d = 6 .0 MeV).