Two-nucleon systems in three dimensions

The Two-Nucleon System in Three Dimensions 1 J. Golak1 , W. Glöckle2 , R. Skibiński1 , H. Witala1 , D. Rozpedzik , ι 1 3 4 5 K. Topolnicki , I. Fachruddin , Ch. Elster , and A. Nogga 1 M. Smoluchowski Institute of Physics, Jagiellonian University, PL-30059 Kraków, Poland arXiv:1001.1264v1 [nucl-th] 8 Jan 2010 2 Institut für Theoretische Physik II, Ruhr-Universität Bochum, D-44780 Bochum, Germany 3 Departemen Fisika, Universitas Indonesia, Depok 16424, Indonesia 4 Institute of Nuclear and Particle Physics, Department of Physics and Astronomy, Ohio University, Athens, OH 45701, USA and 5 Forschungszentrum Jülich, Institut für Kernphysik (Theorie), Institute for Advanced Simulation and Jülich Center for Hadron Physics, D-52425 Jülich, Germany (Dated: January 8, 2010) Abstract A recently developed formulation for treating two- and three-nucleon bound states in a threedimensional formulation based on spin-momentum operators is extended to nucleon-nucleon scattering. Here the nucleon-nucleon t-matrix is represented by six spin-momentum operators accompanied by six scalar functions of momentum vectors. We present the formulation and provide numerical examples for the deuteron and nucleon-nucleon scattering observables. A comparison to results from a standard partial wave decomposition establishes the reliability of this new formulation. PACS numbers: 21.45.-v, 21.30.-x, 21.45.Bc 1 I. INTRODUCTION A standard way to obtain scattering observables for nucleon-nucleon (NN) scattering is to solve the Schrödinger equation either in momentum or coordinate space by taking advantage of rotational invariance and introduce a partial wave basis. This is a well established procedure and has at low energies (below the pion production threshold) a clear physical meaning. At higher energies the number of partial waves needed to obtain converged results increases, and approaches based on a direct evaluation of the scattering equation in terms of vector variables become more appealing. Especially the experience in three- and four-nucleon calculations [1, 2] shows that the standard treatment based on a partial wave projected momentum space basis is quite successful at lower energies, but becomes increasingly more tedious with increasing energy, since each building block requires extended algebra and intricate numerical realizations. On the other hand for a system of three bosons interacting via scalar forces the relative ease with which a three-body bound state [3] as well as three-body scattering [4] can be calculated in the Faddeev scheme when avoiding an angular momentum decomposition altogether has been successfully demonstrated. Thus it is only natural to strive for solving the three nucleon (3N) Faddeev equations in a similar fashion. Recently we proposed a three-dimensional (3D) formulation of the Faddeev equations for 3N bound states [10] and 3N scattering [11] in which the spin-momentum operators are evaluated analytically, leaving the Faddeev equations as a finite set of coupled equations for scalar functions depending only on vector momenta. One of the basic foundations of this formulation rests on the fact that the most general form of the NN interaction can only depend on six linearly independent spin-momentum operators, which in turn dictate the form of the NN bound and scattering state. Here we extend the formulation of the NN bound state given in [10] to NN scattering and provide a numerical realization. There have been several approaches of formulating NN scattering without employing a partial wave decomposition. A helicity formulation related to the total NN spin was proposed in [5], which was extended to 3N bound state calculations in [6]. The spectator equation for relativistic NN scattering has been successfully solved in [7] using a helicity formulation. Aside from NN scattering, 3D formulations for the scattering of pions off nucleons [8] and protons off light nuclei [9] have recently been successfully carried out. In Section II we introduce the formal structure of our approach starting from the most general form of the NN potential. We derive the resulting Lippmann-Schwinger equation and show how to extract Wolfenstein parameters and NN scattering observables. Numerical realizations of our approach that employ a recent chiral next-to-next-leading order (NNLO) NN force [12–14] as well as the standard one-boson-exchange potential Bonn B [15] are presented in Section III. The scalar functions, which result from the evaluation of the spinmomentum operators and have to be calculated only once are given in Appendices A and B. Finally we conclude in Section IV. The more technical information necessary to perform calculations with the chiral potential is given in Appendix C. In Appendix D the Bonn B potential is presented in the form required by our formulation. II. THE FORMAL STRUCTURE We start by projecting the NN potential on the NN isospin states | tmt i, with t = 0, mt = 0 being the singlet and t = 1, mt = −1, 0, 1 the triplet. We assume that isospin is conserved, 2 but allow for charge independence and charge symmetry breaking, and thus for a dependence on mt , ht′ m′t | V | tmt i = δtt′ δmt m′t V tmt (2.1) Furthermore, the most general rotational, parity and time reversal invariant form of the off-shell NN force can be expanded into six scalar spin-momentum operators [17], which we choose as w1 (σ 1 , σ 2 , p′ , p) w2 (σ 1 , σ 2 , p′ , p) w3 (σ 1 , σ 2 , p′ , p) w4 (σ 1 , σ 2 , p′ , p) w5 (σ 1 , σ 2 , p′ , p) w6 (σ 1 , σ 2 , p′ , p) = = = = = = 1 σ1 · σ2 i (σ 1 + σ 2 ) · (p × p′ ) σ 1 · (p × p′ ) σ 2 · (p × p′ ) σ 1 · (p′ + p) σ 2 · (p′ + p) σ 1 · (p′ − p) σ 2 · (p′ − p) (2.2) Each of these operators is multiplied with scalar functions which depend only on the momenta p and p′ , leading to the most general expansion for any NN potential tmt V ≡ 6 X vjtmt (p′ , p) wj (σ 1 , σ 2 , p′ , p) (2.3) j=1 The property of Eq. (2.1) carries over to the NN t-operator, which fulfills the LippmannSchwinger (LS) equation ttmt = V tmt + V tmt G0 ttmt , (2.4) with G0 (z) = (z − H0 )−1 being the free resolvent. The t-matrix element has an expansion analogous to the potential, tmt t ≡ 6 X ′ ′ t ttm j (p , p) wj (σ 1 , σ 2 , p , p) (2.5) j=1 Inserting Eqs. (2.3) and (2.5) into the LS equation (2.4), operating with wk (σ 1 , σ 2 , p′ , p) from the left and performing the trace in the NN spin space leads to X X ′ t Akj (p′ , p)vjtmt (p′ , p) Akj (p′ , p)ttm j (p , p) = j j Z X ′ ′′ ′′ t + d3 p′′ (2.6) vjtmt (p′ , p′′ )G0 (p′′ ) ttm j ′ (p , p) Bkjj ′ (p , p , p). jj ′ The scalar coefficients Akj and Bkjj ′ are defined as   Akj (p′ , p) ≡ Tr wk (σ 1 , σ 2 , p′ , p) wj (σ 1 , σ 2 , p′ , p)   Bkjj ′ (p′ , p′′ , p) ≡ Tr wk (σ 1 , σ 2 , p′ , p) wj (σ 1 , σ 2 , p′ , p′′ ) wj ′ (σ 1 , σ 2 , p′′ , p) (2.7) (2.8) Here all spin dependencies are analytically evaluated, and the coefficients only depend on the vectors p, p′ , and p′′ . The explicit expressions for the coefficients are given in Appendix A. 3 ′ t Thus we end up with a set of six coupled equations for the scalar functions ttm j (p , p), ′ which depend for fixed |p| on two other variables, |p | and the cosine of the relative angle between the vectors p′ and p, given by p̂′ · p̂. Since Eqs. (2.3) and (2.5) are completely general, any arbitrary NN force can be cast into this form and serve as input. Finalizing the formulation, we only need to antisymmetrized in the initial state by applying (1 − P12 )|pi|m1 m2 i|tmt i, and consider the on-shell t-matrix element for given tmt : m ′ t (2π)2 ttm m′1 m′2 ,m1 m2 (p , p) 2    tmt ′ on−shell t tmt ′  m 2 ′ ′ s |m1 m2 i . (2.9) = − (2π) hm1 m2 | t (p , p) + (−) t (p , −p)P12 2 tmt ≡ − Mm ′ m′ ,m m 1 2 1 2 s Here P12 interchanges the spin magnetic quantum numbers for the initial particles, m represents the nucleon mass. For the on-shell condition, characterized by |p′ | = |p| the vectors p − p′ and p + p′ are orthogonal. Under this condition, the operator σ 1 · σ 2 can be represented as a linear combination of the operators wj , j = 4 − 6 [18], i.e. 1 σ 1 · (p × p′ ) σ 2 · (p × p′ ) ′ 2 (p × p ) 1 + σ 1 · (p + p′ ) σ 2 · (p + p′ ) ′ 2 (p + p ) 1 σ 1 · (p − p′ ) σ 2 · (p − p′ ) + ′ 2 (p − p ) σ1 · σ2 = (2.10) We can use the relation of Eq. (2.10) for internal consistency checks of the calculations. However, in order to keep the most general off-shell structure of Eq. (2.5), we need to keep all six terms. We will come back to the numerical implications of this fact below. From Eq. (2.9) we read off that the scattering matrix is given by tmt Mm ′ m′ ,m m 1 2 1 2 6  X m 2 ′ ′ ′ ′ t = − (2π) ttm j (p , p) hm1 m2 |wj (σ 1 , σ 2 , p , p)|m1 m2 i 2 j=1  t tmt ′ ′ ′ ′ + (−) tj (p , −p) hm1 m2 |wj (σ 1 , σ 2 , p , −p)|m2 m1 i (2.11) On the other hand the standard form of the on-shell t-matrix for given quantum numbers tmt [18] reads in the Wolfenstein representation tmt = atmt hm′1 m′2 |m1 m2 i Mm ′ m′ ,m m 1 2 1 2 ctmt − i hm′1 m′2 |w3 (σ 1 , σ 2 , p′ , p)|m1 m2 i ′ |p × p | mtmt + hm′ m′ |w4 (σ 1 , σ 2 , p′ , p)|m1 m2 i |p × p′ |2 1 2 (g + h)tmt + hm′1 m′2 |w5 (σ 1 , σ2 , p′ , p)|m1 m2 i (p + p′ )2 (g − h)tmt hm′1 m′2 |w6 (σ 1 , σ 2 , p′ , p)|m1 m2 i + ′ 2 (p − p ) 4 (2.12) s Due to the action of P12 in Eq. (2.9), which interchanges m1 with m2 , the two parts of Eq. (2.11) yield different results. Again, standard relations [18, 19] must be applied to extract the Wolfenstein parameters: atmt = ctmt = mtmt = (g + h)tmt = (g − h)tmt = 1 Tr (M) 4   1 w3 (σ 1 , σ 2 , p′ , p) −i Tr M 8  |p × p′ |  w4 (σ 1 , σ 2 , p′ , p) 1 Tr M 4 |p × p′ |2   1 w5 (σ 1 , σ 2 , p′ , p) Tr M 4 (p + p′ )2   w6 (σ 1 , σ 2 , p′ , p) 1 Tr M 4 (p − p′ )2 (2.13) It is straightforward to work out those relations starting from Eq. (2.11). In order to simplify tmt ′ ′ ′ t the notation we write tj ≡ ttm j (p , p), t̃j ≡ tj (p , −p), and x ≡ p̂ · p̂ and obtain h1 i 3 1 atmt = t1 + (−)t t̃1 + t̃2 + p4 (1 − x2 )t̃4 + p2 (1 − x)t̃5 + p2 (1 + x)t̃6 2 2 2
√ ctmt = ip2 1 − x2 t3 − (−)t t̃3 h1 1 1 mtmt = t2 + p4 (1 − x2 )t4 + (−)t t̃1 − t̃2 + p4 (1 − x2 )t̃4 2i 2 2 2 2 −p (1 − x)t̃5 − p (1 + x)t̃6 h i 1 1 4 tmt 2 2 t 1 2 g = t2 + p (1 + x)t5 + p (1 − x)t6 + (−) t̃1 − t̃2 − p (1 − x )t̃4 2 2 2 h i htmt = p2 (1 + x)t5 − p2 (1 − x)t6 + (−)t − p2 (1 − x)t̃5 + p2 (1 + x)t̃6 (2.14) It remains to consider the particle representation. For the proton-proton or neutron-neutron system the isospin is t = 1. Thus the above given Wolfenstein parameters are already the physical ones and enter the calculation of observables. In the case of the neutron-proton system both isospins contribute and the physical amplitudes are given by 21 (a00 + a10 ), 1 00 (c + c10 ), etc. 2 Once the Wolfenstein parameters are known, all NN observables can readily be calculated taking well defined bilinear products thereof [18]. For example, the spin averaged differential cross section I0 is given as 41 Tr MM † . For completeness, we also give the derivation of the deuteron which carries isospin t = 0 and total spin s = 1. We employ the operator form from Ref. [5],     1 2 hp|Ψmd i = φ1 (p) + σ 1 · p σ 2 · p − p φ2 (p) |1md i 3 2 X ≡ φk (p) bk (σ 1 , σ 2 , p)|1md i, (2.15) k=1 where |1md i describes the state in which the two spin- 21 states are coupled to the total spin-1 and the magnetic quantum number md . The definition of the operators bk can be easily read off the first line of Eq. (2.15). The two scalar functions φ1 (p) and φ2 (p) are related in a 5 simple way to the standard s- and d-wave components of the deuteron wave function, ψ0 (p) and ψ2 (p) by [5] ψ0 (p) = φ1 (p), 4p2 ψ2 (p) = √ φ2 (p). 3 2 (2.16) Next we use the Schrödinger equation in integral form projected on isospin states, Ψmd = G0 V 00 Ψmd . (2.17) Inserting the explicit expression of Eq. (2.15) we obtain   i h 1 2 φ1 (p) + σ 1 · p σ 2 · p − p φ2 (p) |1md i = 3 Z 6 X 1 vj00 (p, p′ ) wj (σ 1 , σ2 , p, p′ ) d 3 p′ p2 Ed − m j=1   i h 1 ′2 ′ ′ ′ φ2 (p′ ) |1md i, × φ1 (p ) + σ 1 · p σ 2 · p − p 3 (2.18) where Ed is the deuteron binding energy. We remove the spin dependence by projecting from the left with h1md |bk (σ 1 , σ2 , p) and summing over md . This leads to 1 X h1md |bk (σ 1 , σ 2 , p) md =−1 Z 1 X 1 p2 m md =−1 2 X φk′ (p)bk′ (σ 1 , σ 2 , p)|1md i = k ′ =1 6 X 3 ′ dp vj00 (p, p′ )wj (σ 1 , σ 2 , p, p′ ) Ed − j=1 2 X φk′′ (p′ ) bk′′ (σ 1 , σ 2 , p′ )|1md i . × (2.19) k ′′ =1 Defining the scalar functions Adkk′ (p) ≡ 1 X h1md |bk (σ 1 , σ2 , p)bk′ (σ 1 , σ2 , p)|1md i (2.20) md =−1 and d ′ Bkjk ′′ (p, p ) ≡ 1 X h1md |bk (σ 1 , σ 2 , p)wj (σ 1 , σ 2 , p, p′ )bk′′ (σ 1 , σ2 , p′ )|1md i, (2.21) md =−1 we obtain for Eq. (2.19) 2 X k ′ =1 Adkk′ (p)φk′ (p) = 1 Ed − p2 m Z d 3 p′ 6 X j=1 6 vj00 (p, p′ ) 2 X k ′′ =1 ′ d ′ Bkjk ′′ (p, p )φk ′′ (p ) . (2.22) d Note that Adkk′ and Bkjk ′′ are both independent of the interaction. Therefore, these coefficients can be prepared beforehand for all calculations of the deuteron bound state, which consists of two coupled equations for the functions φ1 (p) and φ2 (p). The summation over ′ d md guarantees the scalar nature of the functions Adkk′ (p) and Bkjk ′′ (p, p ), which are given in Appendix B. The azimuthal angle can be trivially integrated out, leading to the final form of the deuteron equation 2 X Adkk′ (p)φk′ (p) = k ′ =1 2 Z X 2π Ed − p2 m k ′′ =1 ∞ ′ ′2 Z ′ dp p φk′′ (p ) 0 1 dx −1 6 X d ′ vj00 (p, p′, x)Bkjk ′′ (p, p , x), (2.23) j=1 where x ≡ p̂′ · p̂. III. NUMERICAL REALIZATION A. The deuteron For a numerical treatment of Eq. (2.23), it is convenient to first define ′ Zk,k′ (p, p ) ≡ Z 1 dx −1 6 X d ′ vj00 (p, p′ , x)Bkjk ′ (p, p , x) (3.1) j=1 and then assume that the integral over p′ will be carried out with some choice of Gaussian points and weights (pj , gj ) with j = 1, 2, . . . , N. This leads to  2 2 X N  X X p2j d 1 d 2 Akk′ (pi ) φk′ (pj ) = Ed Akk′ (pi ) φk′ (pi ). (3.2) gj pj Zkk′ (pi , pj ) + δij 2mπ 2π ′ ′ j=1 k =1 k =1 Eq. (3.2) can be written as a so-called generalized eigenvalue problem Rξ = Ed Y ξ, (3.3) or 2N X Rll′ ξl′ = Ed 2N X Yll′ ξl′ , (3.4) l′ =1 l′ =1 where l = i + (k − 1)N, ξl′ = φk′ (pj ), l′ = j + (k ′ − 1)N, p2j d Rll′ = gj p2j Zkk′ (pi , pj ) + δij A ′ (pi ) 2mπ kk 1 Yll′ = δij Adkk′ (pi ). 2π 7 (3.5) TABLE I: The parameters of the chiral potential of Ref. [13] in order NNLO. The LEC’s are given for the cutoff combination Λ= 600 MeV and Λ̃= 700 MeV. The pion decay constant Fπ and masses are given in MeV. The constants ci are given in GeV−1 , CS and CT in GeV−2 and the other Ci in GeV−4 gA Fπ mπ 0 mπ ± m c1 c3 c4 1.29 92.4 134.977 139.570 938.919 -0.81 -3.40 3.40 CS CT C1 C2 C3 C4 C5 C6 C7 -112.932 2.60161 385.633 1343.49 -121.543 -614.322 1269.04 -26.4880 -1385.12 TABLE II: Meson parameters for the Bonn B potential [15]. The σ parameters shown in the table are for NN total isospin 0. For NN total isospin 1 they should be replaced by mσ = 550 MeV, 2 gα 4π = 8.9437, Λα = 1.9 GeV and n = 1. 2 gα 4π meson mα [MeV] π η δ σ ρ ω 138.03 548.8 983 720 769 782.6 fα gα 14.4 3 2.488 18.3773 0.9 6.1 24.5 0 Λα [GeV] n 1.7 1.5 2 2 1.85 1.85 1 1 1 1 2 2 Since Ad11 6= 0, Ad12 = Ad21 = 0 and Ad22 6= 0, the matrix Y is diagonal and can be easily inverted, we encounter an eigenvalue problem
Y −1 R ξ = Ed ξ, (3.6) which is of the same type and dimension as is being solved for the deuteron wave function in a standard partial wave representation, where one calculates the s- and d-wave components, ψ0 (p) and ψ2 (p). The connection between the two solutions, (φ1 (p), φ2 (p)) and (ψ0 (p), ψ2 (p)), given by Eqs. (2.16) provides a direct check of the numerical accuracy. As a first example we use a chiral NNLO potential [13], which for the convenience of the reader is briefly described in Appendix C. For the specific calculation performed here we take the neutron-proton version of this potential and employ the parameters listed in Table I. We consistently use these potential parameters in the 3D and the PW calculations. In the first case we solve Eq. (3.6) for φ1 (p) and φ2 (p) and then use Eqs. (2.16) to obtain ψ0 (p) and ψ2 (p). In the second case we employ the standard partial wave representation of the potential and solve the Schrödinger equation directly for ψ0 (p) and ψ2 (p). Both methods give the same value for the deuteron binding energy, namely Ed =-2.19993 MeV and s-state probability Ps =95.291 %. The wave functions are identical as can be seen in Fig. 1. As second NN force we choose the Bonn B potential [15], which has a more intricate structure due to the different meson-exchanges and the Dirac spinors. The operator form of this potential, corresponding to the basis of Eq. (2.2) is derived in Appendix D and the parameters are given in Table II. In this case the nucleon mass is set to m= 939.039 MeV. 8 Again we have an excellent agreement between the 3D and the partial wave based calculation for the deuteron binding energy, Ed =-2.2242 MeV, the s-state probability (Ps = 95.014 %) and the wave functions, which are displayed in Fig. 2. In summary, we confirm that the 3D approach gives numerically stable results, which are in perfect agreement with the calculations based on standard partial wave methods. 14 0 -0.05 3/2 ] 10 ψ2 (p) [fm ψ0 (p) [fm 3/2 ] 12 8 6 4 -0.1 -0.15 -0.2 2 0 -0.25 0 0.5 1 1.5 2 0 1 -1 2 3 4 -1 p [fm ] p [fm ] FIG. 1: The s-wave (left) and d-wave (right) component of the deuteron wave function as a function of the relative momentum p for the chiral NNLO potential specified in the text. Crosses represent results obtained with the operator approach and solid lines are from the standard partial wave decomposition. B. NN scattering observables t can be solved for a The inhomogeneous LS equation (2.6) for the six components ttm j ′ fixed value of p. For the vectors p̂ and p̂ we choose the explicit representation p̂ = (0, 0, 1) p p̂′ = ( 1 − x′ 2 , 0, x′ ) p p p̂′′ = ( 1 − x′′ 2 cos ϕ′′ , 1 − x′′ 2 sin ϕ′′ , x′′ ) (3.7) so that the scalar products become p̂′ · p̂ = x′ p̂′′ · p̂ = x′′ p p p̂′ · p̂′′ = x′ x′′ + 1 − x′ 2 1 − x′′ 2 cos ϕ′′ ≡ y. (3.8) Let us now calculate the integral term on the right-hand-side of Eq. (2.6) for a positive p2 energy of the NN system, Ec.m. ≡ m0 : ′ ′ Sk (p , p, x ) ≡ Zp̄ 0 dp′′ p′′ 2 p20 1 fk (p′′ ; p′ , p, x′ ), 2 ′′ − p + iǫ 9 (3.9) 14 0 -0.05 3/2 ] 10 ψ2 (p) [fm ψ0 (p) [fm 3/2 ] 12 8 6 4 -0.1 -0.15 -0.2 2 0 -0.25 0 0.5 1 1.5 2 0 1 -1 2 3 4 -1 p [fm ] p [fm ] FIG. 2: The same as in Fig. 1 but for the Bonn B potential [15]. where m Z1 6 X dx′′ j,j ′=1 −1 Z2π fk (p′′ ; p′ , p, x′ ) ≡ fk (p′′ ) ≡ dϕ′′ Bkjj ′ (p′ , p′′ , p, x′ , x′′ , ϕ′′ ) vj (p′ , p′′ , y) tj ′ (p′′ , p, x′′ ) . (3.10) 0 Here the index tmt for the t-matrix element is omitted for simplicity. For the momentum integration in Eq. (3.9) an upper bound p̄ is introduced, since the contributions to the integral for larger momenta are insignificant, the potential and the t-matrix are essentially zero. Then the integral of Eq. (3.9) can be treated in a standard fashion and one obtains ′ ′ Sk (p , p, x ) = Zp̄ 0 p′′ 2 fk (p′′ ) − p20 fk (p0 ) 1 dp + p0 fk (p0 ) 2 2 ′′ 2 p0 − p ′′   p̄ + p0 − iπ . ln p̄ − p0 (3.11) It is tempting to solve Eq. (2.6) by iteration and then sum the resulting Neumann series with a Padé scheme. The determinant of the 6 × 6 matrix A(p′ , p, x′ ), which appears on both sides of (2.6), can be easily calculated with the result 4 2   2 2 8 1 − x′ det(A) = −65536 p8 p′ p2 − p′ . (3.12) In particular, this determinant is zero for p′ = p and x′ = ±1. However, by a careful choice of the p, p′ and x′ points, it is possible to work with non-zero values of det(A), so that the matrix A can be inverted. In this case Eq. (2.6) can be written as t(p′ , p, x′ ) = v(p′ , p, x′ ) + A−1 (p′ , p, x′ ) S(p′ , p, x′ ), (3.13) where t(p′ , p, x′ ), v(p′ , p, x′ ) and S(p′ , p, x′ ) denote now six-dimensional vectors with components tj , vj and Sj . Note that S(p′ , p, x′ ) contains the unknown vector t(p′ , p, x′ ). We arrive at the following iteration scheme: t(1) (p′ , p, x′ ) = v(p′ , p, x′ ) t(n) (p′ , p, x′ ) = v(p′ , p, x′ ) + A−1 (p′ , p, x′ )S (n−1) (p′ , p, x′ ), 10 for n > 1, (3.14) where S (n−1) (p′ , p, x′ ) is calculated using the vector t(p′ , p, x′ ) from the previous iteration, i.e. t(n−1) (p′ , p, x′ ). However, our experience with this iteration scheme is discouraging. Numerically det(A) can be very close to zero, and in such cases the rank of matrix A can vary from 2 to 5. As a consequence, it is very difficult to maintain numerical stability for this iterative method. Another drawback of using the inverse of A is that it is impossible to obtain the on-shell matrix element t(p0 , p0 , x′ ) directly. One would have to rely on numerical interpolations for calculating on-shell matrix elements. For this reason we decided to solve Eq. (2.6) directly as a system of inhomogeneous coupled algebraic equations. To this aim we first perform a discretization with respect to the different variables in the problem. As typical grid sizes we take nx = 36 Gaussian points for the x′′ integration, and use the same grid for the x′ points. Furthermore, we use np = 36 Gaussian points for the p′ and p′′ grids, which are defined the interval (0, p̄ = 40 fm−1 ). These points are distributed in such a way that p0 is avoided and the same number of points is put symmetrically into two narrow intervals on each side of p0 [16]. Such a choice proved advantageous in the treatment of the 1 S0 channel for the PWD calculations and is kept here. In addition, p0 is added to the set of p′ points. Finally, we choose nϕ′′ = 60 Gaussian points for the ϕ′′ integration. Thus, we arrive at a system of 6 × (np + 1) × nx linear equations of the form Hξ = b, (3.15) where the vector ξ represents all unknown values of tj (p′ , p, x′ ) for fixed p. If we choose from the very beginning p = p0 , then the solution of Eq. (3.15) contains the on-shell t-matrix in the operator form, namely tj (p0 , p0 , x′ ). It is clear that for the on-shell t-matrix the solution cannot be unique, since the six operators become linearly dependent on each other (see Eq. (2.10)). In principle, one therefore expects that Eq. (3.15) is non-invertible and that tools like a singular value decomposition are required for the solution. However, we found that this is not required since the standard LU decomposition of Numerical Recipes [20] worked safely for both interactions, all the considered laboratory energies and different choices of the mesh points. Interestingly, the actual solution for the on-shell t-matrix is not unique as expected and depends even on the optimization level of the compiler. However, the observables turn out to be stable and unique. Of course, setting p = p0 is not necessary. For p 6= p0 the system of equations (3.15) has a unique and smooth solution and afterwards the interpolation to the on shell case can be safely performed. The path to NN observables is straightforward. From Eq. (2.11) we evaluate first the scattering matrix M for all possible spin projections m′1 , m′2 , m1 , and m2 , noting that on-shell tmt ′ ′ t ttm j (p , p) = tj (p0 , p0 , x ) (3.16) tmt ′ ′ t ttm j (p , −p) = tj (p0 , p0 , −x ). (3.17) and Since we use a set of x′ points which is symmetric with respect to x′ = 0, no interpolation is required and M is easily obtained. Before we can make use of Eq. (2.13), we calculate 11 matrix elements of the modified operators wj appearing in (2.13), in the same representation as for the matrix M: D E w3 (σ 1 , σ2 , p′ , p) m′1 m′2 m m 1 2 |p × p′ | D E w4 (σ 1 , σ2 , p′ , p) m′1 m′2 m m 1 2 |p × p′ |2 E D w5 (σ 1 , σ2 , p′ , p) m m m′1 m′2 1 2 (p + p′ )2 E D w6 (σ 1 , σ 2 , p′ , p) m m (3.18) m′1 m′2 1 2 . (p − p′ )2 For this calculation symbolic software like Mathematica c [21] proves very useful. In the next step, the Wolfenstein parameters are calculated as sums over m′1 , m′2 , m1 and m2 . For example 1 X X δm′ m δm′ m , M tm′ t ′ atmt = 4 ′ ′ m ,m m1 m2 ,m1 m2 1 1 2 2 m1 ,m2 ctmt 1 2 D 1 X X w3 (σ 1 , σ 2 , p′ , p) ′ ′ E tmt m1 m2 . Mm = −i m m ′ m′ ,m m 1 2 1 2 1 2 8 ′ ′ m ,m |p × p′ | m1 ,m2 1 (3.19) 2 Finally, the NN observables result from the Wolfenstein parameters as simple bilinear expressions [18]. In Figs. 3–6 we compare a selected set of observables calculated with the new 3D method to results obtained by using a standard partial wave decomposition, employing the same potentials we used for the deuteron calculations. For the chiral potential we chose two laboratory kinetic energies 13 and 150 MeV, whereas for the Bonn B potential the higher energy is chosen to be 300 MeV. We made sure that in all cases a sufficient number of partial waves is included to obtain converged results in the standard PWD approach. For all the energies considered our converged PWD results agree perfectly with predictions obtained from the new 3D approach. In Figs. 7–8 we demonstrate the convergence with respect to different maximum total angular momenta jmax towards the results calculated using our new 3D method for the differential cross section and the asymmetry A. Here we employ the Bonn B potential and show the calculations for the neutron-neutron and neutron-proton cases separately. As one can see, quite a sizeable number of partial waves is required for a converged calculation at 300 MeV. Finally, in Fig. 9 we display the Wolfenstein amplitudes for neutron-proton scattering at 300 MeV laboratory kinetic energy. Again we compare partial wave based calculations for different maximum total angular momenta jmax to the 3D calculation. We observe that the maximum number of partial waves needed for obtaining a converged result is quite different for the different amplitudes. IV. SUMMARY AND CONCLUSIONS Two nucleon scattering at intermediate energies of a few hundred MeV requires quite a few angular momentum states in order to achieve convergence of e.g. scattering observables. We formulated and numerically illustrated an approach to treating the NN system 12 47.8 63 47.6 62 σ [mb/sr] 47.4 61 47.2 47 60 46.8 59 46.6 58 46.4 46.2 57 R 0 20 40 60 80 100 120 140 160 180 0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 -0.1 20 40 60 80 100 120 140 160 180 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 20 40 60 80 100 120 140 160 180 0.14 0 0.12 -0.1 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 -0.2 0.1 -0.3 0.08 A -0.4 0.06 -0.5 0.04 -0.6 0.02 -0.7 0 -0.8 0 20 40 60 80 100 120 140 160 180 0.03 0.96 0.02 0.94 0.01 0.92 0 D 0 0.9 -0.01 0.88 -0.02 -0.03 0.86 -0.04 0.84 -0.05 0.82 0 20 40 60 80 100 120 140 160 180 Θc.m. [deg] 0 20 40 60 80 100 120 140 160 180 Θc.m. [deg] FIG. 3: Selected observables for the neutron-neutron (left panel) and neutron-proton (right panel) system at the projectile laboratory kinetic energy 13 MeV as a function of the center of mass angle θ for the chiral NNLO potential [13]. Crosses represent results obtained with the operator approach and solid lines represent fully converged results from the standard PWD. For the definition of the R, A and D observables see e.g. [18]. 13 48.2 64 48 63 σ [mb/sr] 47.8 62 47.6 47.4 61 47.2 60 47 59 46.8 58 46.6 46.4 57 R 0 0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 -0.1 20 40 60 80 100 120 140 160 180 0.16 0 0.14 -0.1 0.12 -0.2 0.1 -0.3 0.08 -0.4 0.06 -0.5 0.04 -0.6 0.02 -0.7 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 -0.8 0 D 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 A 20 40 60 80 100 120 140 160 180 20 40 60 80 100 120 140 160 180 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0 20 40 60 80 100 120 140 160 180 Θc.m. [deg] 0 20 40 60 80 100 120 140 160 180 Θc.m. [deg] FIG. 4: The same as in Fig. 3 for the Bonn B potential [15]. 14 8 14 7.5 12 σ [mb/sr] 7 10 6.5 8 6 6 5.5 4 5 4.5 2 R 0 20 40 60 80 100 120 140 160 180 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4 -0.45 0 20 40 60 80 100 120 140 160 180 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 0 20 40 60 80 100 120 140 160 180 0.3 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 Θc.m. [deg] 0.6 0.2 0.4 0.1 0.2 A 0 -0.1 0 -0.2 -0.2 -0.3 -0.4 -0.4 -0.5 -0.6 D 0 20 40 60 80 100 120 140 160 180 0.5 1 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0 0 -0.1 -0.2 -0.2 -0.4 0 20 40 60 80 100 120 140 160 180 Θc.m. [deg] FIG. 5: The same as in Fig. 3 for the projectile laboratory kinetic energy being 150 MeV. 15 3.9 11 10 9 8 7 6 5 4 3 2 1 3.8 σ [mb/sr] 3.7 3.6 3.5 3.4 3.3 3.2 0 20 40 60 80 100 120 140 160 180 0.5 20 40 60 80 100 120 140 160 180 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 0.4 0.3 0.2 R 0 0.1 0 -0.1 -0.2 -0.3 0 20 40 60 80 100 120 140 160 180 0.2 0 20 40 60 80 100 120 140 160 180 1 0.8 0.1 0.6 A 0 0.4 -0.1 0.2 0 -0.2 -0.2 -0.3 -0.4 -0.4 -0.6 D 0 20 40 60 80 100 120 140 160 180 0.7 1 0.6 0.8 0.5 0.6 0.4 0.4 0.3 0.2 0.2 0 0.1 -0.2 0 -0.4 -0.1 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 Θc.m. [deg] -0.6 0 20 40 60 80 100 120 140 160 180 Θc.m. [deg] FIG. 6: The same as in Fig. 4 for the projectile laboratory kinetic energy being 300 MeV. 16 4.2 4 σ [mb/sr] 3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 0 20 40 60 80 100 120 140 160 180 0.3 0.2 0.1 A 0 jmax= 3 jmax= 6 jmax= 9 jmax=12 jmax=15 jmax=18 -0.1 -0.2 -0.3 -0.4 0 20 40 60 80 100 120 140 160 180 Θc.m. [deg] FIG. 7: The convergence of the PWD results for the differential cross section and the depolarization coefficient A [18] for neutron-neutron scattering based on different numbers of partial waves determined by the maximal total angular momentum jmax of the NN system (lines) with respect to the result of the three-dimensional calculation (crosses) for projectile laboratory kinetic energy 300 MeV and the Bonn B potential [15]. working directly with momentum vectors and using spin-momentum operators multiplied by scalar functions, which only depend on the momentum vectors. This approach is quite natural, since any general NN force being invariant under time-reversal, parity and Galilei (or Lorentz) transformations can only depend on six linear independent spin-momentum operators. The representation of the NN potential using spin-momentum operators leads to a 17 σ [mb/sr] 11 10 9 8 7 6 5 4 3 2 1 jmax= 3 jmax= 6 jmax= 9 jmax=12 jmax=15 jmax=18 0 20 40 60 80 100 120 140 160 180 1 0.8 0.6 A 0.4 0.2 0 -0.2 -0.4 -0.6 0 20 40 60 80 100 120 140 160 180 Θc.m. [deg] FIG. 8: The same as in Fig. 7 for neutron-proton scattering. system of six coupled equations of scalar functions (depending on momentum vectors) for the NN t-matrix, once the spin-momentum operators are analytically calculated by performing suitable trace operations. We calculated deuteron properties and NN scattering observables using two different NN potentials, one derived from chiral effective field theory and one from meson exchange. For all cases we found perfect agreement between the calculations based on our new method and conventional calculations using a partial wave basis. This work is intended to serve as starting point towards treating three-nucleon systems without partial waves. The theoretical formulation has already been given for the 3N bound state (including 3N forces) in [10] and for 3N scattering in [11]. For a much simpler case when 18 0.4 jmax = 9 jmax = 6 0.2 jmax = 3 0.0 -0.2 0.05 0.2 0.00 0.1 -0.05 0.0 Re (c) [fm] Im (a) [fm] jmax = 15 Im (c) [fm] Re (a) [fm] 0.8 0.6 0.4 0.2 0.0 -0.2 Im (m) [fm] 0.10 0.0 0.05 -0.2 0.00 -0.4 -0.05 0.2 0.10 0.0 0.05 -0.2 0.00 -0.4 -0.05 0.08 0.0 0.04 -0.2 0.00 -0.4 0 20 40 60 80 100 120 140 160 Im (g) [fm] 0.2 Im (h) [fm] Re (h) [fm] Re (g) [fm] Re (m) [fm] 0.4 0 20 40 60 80 100 120 140 160 180 Θc.m.[deg] Θc.m.[deg] FIG. 9: The Wolfenstein parameters for neutron-proton scattering for projectile laboratory kinetic energy 300 MeV calculated with the Bonn B potential [15]. Results of the 3D calculation are given by the crosses. The convergence of the PWD results for increasing values of maximum angular momentum jmax is shown by the different curves labeled in the figure. The left panels show the real parts of the amplitudes, whereas the imaginary parts are displayed in the right panels. 19 spin- and isospin-degrees are neglected the feasibility of three-body scattering calculation in the GeV regime has already been demonstrated [4], even including Poincaré symmetry [22]. Since our approach leads to coupled equations of scalar functions of momentum vectors, the generalization to include spin-degrees of freedom appears feasible. In the 3N system not only the number of partial waves increases rapidly, but also 3N forces appear as new dynamical input. In particular in the chiral approach the number of 3NF contributions proliferates with order of expansion of the theory. In higher orders many complicated terms contribute to the 3N force [13]. In this case a traditional partial wave decomposition of the 3NF poses a serious problem which has a chance to be alleviated by a direct three-dimensional treatment. Acknowledgments We thank Dr. Evgeny Epelbaum for providing us with a code for the operator form of the chiral NNLO potential. This work was supported by the Polish Ministry of Science and Higher Education under Grants No. N N202 104536 and No. N N202 077435 and in part under the auspices of the U. S. Department of Energy, Office of Nuclear Physics under contract No. DEFG02-93ER40756 with Ohio University. It was also partially supported by the Helmholtz Association through funds provided to the virtual institute “Spin and strong QCD”(VHVI-231). The numerical calculations were partly performed on the supercomputer cluster of the JSC, Jülich, Germany. Appendix A: Coefficients for NN scattering In this appendix we present the expressions A and B given in Eqs. (2.7) and (2.8) for NN scattering. The coefficients Aij (p′ , p) can be obtained in terms of the following four functions F A F A3 (p′ , p) F A4 (p′ , p) F A5 (p′ , p) F A6 (p′ , p) = = = = 20 4(p × p′ )2 4(p′ + p)2 4(p′ − p)2 4(p′2 − p2 )2 (A1) (A2) (A3) (A4) The non-zero coefficients Aij (p′ , p) are: A11 (p′ , p) A22 (p′ , p) A24 (p′ , p) A25 (p′ , p) A26 (p′ , p) A33 (p′ , p) = = = = = = A44 (p′ , p) = A55 (p′ , p) = A56 (p′ , p) = A66 (p′ , p) = 4 12 A42 (p′ , p) = F A3 (p′ , p) A52 (p′ , p) = F A4 (p′ , p) A62 (p′ , p) = F A5 (p′ , p) −2F A3 (p′ , p) 1 2 ′ A (p , p) 4 24 1 2 ′ A (p , p) 4 25 A65 (p′ , p) = F A6 (p′ , p) 1 2 ′ A (p , p) 4 26 (A5) (A6) (A7) (A8) (A9) (A10) (A11) (A12) (A13) (A14) All other Aij (p′ , p) = 0. The non-vanishing coefficients Bikj (p′ , p′′ , p) can be expressed by means of the following 25 functions F B: F B3a (p′ , p′′ , p) = 4(p × p′′ )2 F B3b (p′ , p′′ , p) = 4(p′′ × p′ )2 F B3c (p′ , p′′ , p) = 4(p × p′ )2 (A15) (A16) (A17) F B4a (p′ , p′′ , p) = 4(p′′ + p)2 F B4b (p′ , p′′ , p) = 4(p′ + p′′ )2 F B4c (p′ , p′′ , p) = 4(p′ + p)2 (A18) (A19) (A20) F B5a (p′ , p′′ , p) = 4(p′′ − p)2 F B5b (p′ , p′′ , p) = 4(p′ − p′′ )2 F B5c (p′ , p′′ , p) = 4(p′ − p)2 (A21) (A22) (A23) F B6a (p′ , p′′ , p) = −8(p′′ × p′ ) · (p × p′′ ) F B6b (p′ , p′′ , p) = −8(p × p′ ) · (p′′ × p′ ) F B6c (p′ , p′′ , p) = −8(p × p′ ) · (p × p′′ ) (A24) (A25) (A26) F B7 (p′ , p′′ , p) = 4{(p × p′ ) · p′′ }2 (A27) F B8a (p′ , p′′ , p) = 2{(p′ + p′′ ) · (p′′ + p)} F B8b (p′ , p′′ , p) = 2{(p′ + p) · (p′ + p′′ )} F B8c (p′ , p′′ , p) = 2{(p′ + p) · (p′′ + p)} (A28) (A29) (A30) 21 F B9a (p′ , p′′ , p) = 2{(p′ − p′′ ) · (p′′ − p)} F B9b (p′ , p′′ , p) = 2{(p′ − p) · (p′ − p′′ )} F B9c (p′ , p′′ , p) = 2{(p′ − p) · (p′′ − p)} (A31) (A32) (A33) F B10a (p′ , p′′ , p) = 2{(p′ + p′′ ) · (p′′ − p)} F B10b (p′ , p′′ , p) = 2{(p′ + p) · (p′ − p′′ )} F B10c (p′ , p′′ , p) = 2{(p′ − p) · (p′′ + p)} (A34) (A35) (A36) F B11a (p′ , p′′ , p) = 2{(p′ − p′′ ) · (p′′ + p)} F B11b (p′ , p′′ , p) = 2{(p′ − p) · (p′ + p′′ )} F B11c (p′ , p′′ , p) = 2{(p′ + p) · (p′′ − p)} (A37) (A38) (A39) The non-zero Bikj (p′ , p′′ , p): B122 (p′ , p′′ , p) B124 (p′ , p′′ , p) B125 (p′ , p′′ , p) B126 (p′ , p′′ , p) B133 (p′ , p′′ , p) B142 (p′ , p′′ , p) B212 (p′ , p′′ , p) = B221 (p′ , p′′ , p) = 12 B214 (p′ , p′′ , p) = F B3a (p′ , p′′ , p) B215 (p′ , p′′ , p) = F B4a (p′ , p′′ , p) B216 (p′ , p′′ , p) = F B5a (p′ , p′′ , p) B233 (p′ , p′′ , p) = F B6a (p′ , p′′ , p) B241 (p′ , p′′ , p) = F B3b (p′ , p′′ , p) 1 B144 (p′ , p′′ , p) = F B6a (p′ , p′′ , p)2 16 B145 (p′ , p′′ , p) = = = = = = = = = = B146 (p′ , p′′ , p) = B154 (p′ , p′′ , p) = B164 (p′ , p′′ , p) B415 (p′ , p′′ , p) = B416 (p′ , p′′ , p) = B514 (p′ , p′′ , p) B614 (p′ , p′′ , p) = B451 (p′ , p′′ , p) = B461 (p′ , p′′ , p) B541 (p′ , p′′ , p) = B641 (p′ , p′′ , p) = F B7 (p′ , p′′ , p) B152 (p′ , p′′ , p) B155 (p′ , p′′ , p) B156 (p′ , p′′ , p) B162 (p′ , p′′ , p) B165 (p′ , p′′ , p) B166 (p′ , p′′ , p) B313 (p′ , p′′ , p) (A40) (A41) (A42) (A43) (A44) (A45) (A46) (A47) B251 (p′ , p′′ , p) = F B4b (p′ , p′′ , p) F B8a (p′ , p′′ , p)2 F B10a (p′ , p′′ , p)2 B261 (p′ , p′′ , p) = F B5b (p′ , p′′ , p) F B11a (p′ , p′′ , p)2 F B9a (p′ , p′′ , p)2 B323 (p′ , p′′ , p) = F B6c (p′ , p′′ , p) (A48) (A49) (A50) (A51) (A52) (A53) (A54) B412 (p′ , p′′ , p) = B421 (p′ , p′′ , p) = F B3c (p′ , p′′ , p) 1 B414 (p′ , p′′ , p) = F B6c (p′ , p′′ , p)2 16 B512 (p′ , p′′ , p) = B521 (p′ , p′′ , p) = F B4c (p′ , p′′ , p) B515 (p′ , p′′ , p) = F B8c (p′ , p′′ , p)2 B516 (p′ , p′′ , p) = F B11c (p′ , p′′ , p)2 B612 (p′ , p′′ , p) = B621 (p′ , p′′ , p) = F B5c (p′ , p′′ , p) (A55) = = = = = = = 22 (A56) (A57) (A58) (A59) (A60) B615 (p′ , p′′ , p) = F B10c (p′ , p′′ , p)2 B616 (p′ , p′′ , p) = F B9c (p′ , p′′ , p)2 B331 (p′ , p′′ , p) = B332 (p′ , p′′ , p) = F B6b (p′ , p′′ , p) 1 B441 (p′ , p′′ , p) = F B6b (p′ , p′′ , p)2 16 B551 (p′ , p′′ , p) = F B8b (p′ , p′′ , p)2 B561 (p′ , p′′ , p) = F B10b (p′ , p′′ , p)2 B651 (p′ , p′′ , p) = F B11b (p′ , p′′ , p)2 B661 (p′ , p′′ , p) = F B9b (p′ , p′′ , p)2 B111 (p′ , p′′ , p) = 4 B244 (p′ , p′′ , p) = −p′′2 F B7 (p′ , p′′ , p) 1 B245 (p′ , p′′ , p) = − F B3b (p′ , p′′ , p)F B4a (p′ , p′′ , p) 4 +F B7 (p′ , p′′ , p) 1 B246 (p′ , p′′ , p) = − F B3b (p′ , p′′ , p)F B5a (p′ , p′′ , p) 4 +F B7 (p′ , p′′ , p) 1 B254 (p′ , p′′ , p) = − F B3a (p′ , p′′ , p)F B4b (p′ , p′′ , p) 4 +F B7 (p′ , p′′ , p) 1 B255 (p′ , p′′ , p) = − F B4b (p′ , p′′ , p)F B4a (p′ , p′′ , p) 4 +F B8a (p′ , p′′ , p)2 1 B256 (p′ , p′′ , p) = − F B4b (p′ , p′′ , p)F B5a (p′ , p′′ , p) 4 +F B10a (p′ , p′′ , p)2 1 B264 (p′ , p′′ , p) = − F B3a (p′ , p′′ , p)F B5b (p′ , p′′ , p) 4 +F B7 (p′ , p′′ , p) 1 B265 (p′ , p′′ , p) = − F B5b (p′ , p′′ , p)F B4a (p′ , p′′ , p) 4 +F B11a (p′ , p′′ , p)2 1 B266 (p′ , p′′ , p) = − F B5b (p′ , p′′ , p)F B5a (p′ , p′′ , p) 4 +F B9a (p′ , p′′ , p)2 23 (A61) (A62) (A63) (A64) (A65) (A66) (A67) (A68) (A69) (A70) (A71) (A72) (A73) (A74) (A75) (A76) (A77) (A78) B424 (p′ , p′′ , p) = −p2 F B7 (p′ , p′′ , p) 1 B425 (p′ , p′′ , p) = − F B3c (p′ , p′′ , p)F B4a (p′ , p′′ , p) 4 +F B7 (p′ , p′′ , p) 1 B426 (p′ , p′′ , p) = − F B3c (p′ , p′′ , p)F B5a (p′ , p′′ , p) 4 +F B7 (p′ , p′′ , p) 1 B524 (p′ , p′′ , p) = − F B3a (p′ , p′′ , p)F B4c (p′ , p′′ , p) 4 +F B7 (p′ , p′′ , p) 1 B525 (p′ , p′′ , p) = − F B4a (p′ , p′′ , p)F B4c (p′ , p′′ , p) 4 +F B8c (p′ , p′′ , p)2 1 B526 (p′ , p′′ , p) = − F B5a (p′ , p′′ , p)F B4c (p′ , p′′ , p) 4 +F B11c (p′ , p′′ , p)2 1 B624 (p′ , p′′ , p) = − F B3a (p′ , p′′ , p)F B5c (p′ , p′′ , p) 4 +F B7 (p′ , p′′ , p) 1 B625 (p′ , p′′ , p) = − F B4a (p′ , p′′ , p)F B5c (p′ , p′′ , p) 4 +F B10c (p′ , p′′ , p)2 1 B626 (p′ , p′′ , p) = − F B5a (p′ , p′′ , p)F B5c (p′ , p′′ , p) 4 +F B9c (p′ , p′′ , p)2 B442 (p′ , p′′ , p) = −p′2 F B7 (p′ , p′′ , p) 1 B452 (p′ , p′′ , p) = − F B3c (p′ , p′′ , p)F B4b (p′ , p′′ , p) 4 +F B7 (p′ , p′′ , p) 1 B462 (p′ , p′′ , p) = − F B3c (p′ , p′′ , p)F B5b (p′ , p′′ , p) 4 +F B7 (p′ , p′′ , p) 1 B542 (p′ , p′′ , p) = − F B3b (p′ , p′′ , p)F B4c (p′ , p′′ , p) 4 +F B7 (p′ , p′′ , p) 1 B552 (p′ , p′′ , p) = − F B4c (p′ , p′′ , p)F B4b (p′ , p′′ , p) 4 +F B8b (p′ , p′′ , p)2 1 B562 (p′ , p′′ , p) = − F B4c (p′ , p′′ , p)F B5b (p′ , p′′ , p) 4 +F B10b (p′ , p′′ , p)2 24 (A79) (A80) (A81) (A82) (A83) (A84) (A85) (A86) (A87) (A88) (A89) (A90) (A91) (A92) (A93) 1 B642 (p′ , p′′ , p) = − F B3b (p′ , p′′ , p)F B5c (p′ , p′′ , p) 4 +F B7 (p′ , p′′ , p) 1 B652 (p′ , p′′ , p) = − F B5c (p′ , p′′ , p)F B4b (p′ , p′′ , p) 4 +F B11b (p′ , p′′ , p)2 1 B662 (p′ , p′′ , p) = − F B5c (p′ , p′′ , p)F B5b (p′ , p′′ , p) 4 +F B9b (p′ , p′′ , p)2 (A94) (A95) (A96) B224 (p′ , p′′ , p) B225 (p′ , p′′ , p) B226 (p′ , p′′ , p) B242 (p′ , p′′ , p) B252 (p′ , p′′ , p) = = = = = −2F B3a (p′ , p′′ , p) −2F B4a (p′ , p′′ , p) −2F B5a (p′ , p′′ , p) −2F B3b (p′ , p′′ , p) −2F B4b (p′ , p′′ , p) (A97) (A98) (A99) (A100) (A101) B262 (p′ , p′′ , p) B422 (p′ , p′′ , p) B522 (p′ , p′′ , p) B622 (p′ , p′′ , p) B222 (p′ , p′′ , p) = = = = = −2F B5b (p′ , p′′ , p) −2F B3c (p′ , p′′ , p) −2F B4c (p′ , p′′ , p) −2F B5c (p′ , p′′ , p) −24 (A102) (A103) (A104) (A105) (A106) 1 F B6a (p′ , p′′ , p)F B7 (p′ , p′′ , p) 4 B345 (p′ , p′′ , p) = B435 (p′ , p′′ , p) = −2{(p′′ + p) · p′ }F B7 (p′ , p′′ , p) B346 (p′ , p′′ , p) = B436 (p′ , p′′ , p) = −2{(p′′ − p) · p′ }F B7 (p′ , p′′ , p) B344 (p′ , p′′ , p) = B355 (p′ , p′′ , p) = 1 {2F B3c (p′ , p′′ , p) − F B6b (p′ , p′′ , p) 2 −F B6c (p′ , p′′ , p)} F B8a (p′ , p′′ , p) 1 B356 (p′ , p′′ , p) = − {2F B3c (p′ , p′′ , p) + F B6b (p′ , p′′ , p) 2 −F B6c (p′ , p′′ , p)} F B10a (p′ , p′′ , p) 1 B365 (p′ , p′′ , p) = {2F B3c (p′ , p′′ , p) − F B6b (p′ , p′′ , p) 2 +F B6c (p′ , p′′ , p)} F B11a (p′ , p′′ , p) 1 B366 (p′ , p′′ , p) = − {2F B3c (p′ , p′′ , p) + F B6b (p′ , p′′ , p) 2 +F B6c (p′ , p′′ , p)} F B9a (p′ , p′′ , p) 1 B434 (p′ , p′′ , p) = F B6c (p′ , p′′ , p)F B7 (p′ , p′′ , p) 4 25 (A107) (A108) (A109) (A110) (A111) (A112) (A113) (A114) B534 (p′ , p′′ , p) = B543 (p′ , p′′ , p) = 2{(p′ + p) · p′′ }F B7 (p′ , p′′ , p) B634 (p′ , p′′ , p) = −B643 (p′ , p′′ , p) = 2{(p′ − p) · p′′ }F B7 (p′ , p′′ , p) 1 B535 (p′ , p′′ , p) = − {2F B3b (p′ , p′′ , p) + F B6a (p′ , p′′ , p) 2 −F B6b (p′ , p′′ , p)} F B8c (p′ , p′′ , p) 1 B536 (p′ , p′′ , p) = − {2F B3b (p′ , p′′ , p) + F B6a (p′ , p′′ , p) 2 +F B6b (p′ , p′′ , p)} F B11c (p′ , p′′ , p) 1 B635 (p′ , p′′ , p) = − {2F B3b (p′ , p′′ , p) − F B6a (p′ , p′′ , p) 2 −F B6b (p′ , p′′ , p)} F B10c (p′ , p′′ , p) 1 B636 (p′ , p′′ , p) = − {2F B3b (p′ , p′′ , p) − F B6a (p′ , p′′ , p) 2 +F B6b (p′ , p′′ , p)} F B9c (p′ , p′′ , p) 1 F B6b (p′ , p′′ , p)F B7 (p′ , p′′ , p) 4 B453 (p′ , p′′ , p) = −B354 (p′ , p′′ , p) = 2{(p′ + p′′ ) · p}F B7 (p′ , p′′ , p) B463 (p′ , p′′ , p) = B364 (p′ , p′′ , p) = −2{(p′ − p′′ ) · p}F B7 (p′ , p′′ , p) 1 B553 (p′ , p′′ , p) = − {2F B3a (p′ , p′′ , p) − F B6c (p′ , p′′ , p) 2 +F B6a (p′ , p′′ , p)} F B8b (p′ , p′′ , p) B443 (p′ , p′′ , p) = 1 {2F B3a (p′ , p′′ , p) + F B6c (p′ , p′′ , p) 2 +F B6a (p′ , p′′ , p)} F B10b (p′ , p′′ , p) 1 B653 (p′ , p′′ , p) = {2F B3a (p′ , p′′ , p) − F B6c (p′ , p′′ , p) 2 −F B6a (p′ , p′′ , p)} F B11b (p′ , p′′ , p) 1 B663 (p′ , p′′ , p) = − {2F B3a (p′ , p′′ , p) + F B6c (p′ , p′′ , p) 2 −F B6a (p′ , p′′ , p)} F B9b (p′ , p′′ , p) (A115) (A116) (A117) (A118) (A119) (A120) (A121) (A122) (A123) (A124) B563 (p′ , p′′ , p) = 1 B334 (p′ , p′′ , p) = − F B6a (p′ , p′′ , p)F B6c (p′ , p′′ , p) 8 ′ ′′ B335 (p , p , p) = B353 (p′ , p′′ , p) = B633 (p′ , p′′ , p) = B333 (p′ , p′′ , p) = 2F B7 (p′ , p′′ , p) B336 (p′ , p′′ , p) = B363 (p′ , p′′ , p) = B533 (p′ , p′′ , p) = −2F B7 (p′ , p′′ , p) 1 B343 (p′ , p′′ , p) = − F B6b (p′ , p′′ , p)F B6a (p′ , p′′ , p) 8 1 B433 (p′ , p′′ , p) = − F B6c (p′ , p′′ , p)F B6b (p′ , p′′ , p) 8 26 (A125) (A126) (A127) (A128) (A129) (A130) (A131) (A132) 1 {2F B3c (p′ , p′′ , p) − F B6b (p′ , p′′ , p) 16 2 −F B6c (p′ , p′′ , p)} 1 B456 (p′ , p′′ , p) = − {2F B3c (p′ , p′′ , p) + F B6b (p′ , p′′ , p) 16 2 −F B6c (p′ , p′′ , p)} 1 B465 (p′ , p′′ , p) = − {2F B3c (p′ , p′′ , p) − F B6b (p′ , p′′ , p) 16 2 +F B6c (p′ , p′′ , p)} 1 B466 (p′ , p′′ , p) = − {2F B3c (p′ , p′′ , p) + F B6b (p′ , p′′ , p) 16 2 +F B6c (p′ , p′′ , p)} B455 (p′ , p′′ , p) = − 1 {2F B3b (p′ , p′′ , p) + F B6a (p′ , p′′ , p) 16 2 −F B6b (p′ , p′′ , p)} 1 B546 (p′ , p′′ , p) = − {2F B3b (p′ , p′′ , p) + F B6a (p′ , p′′ , p) 16 2 +F B6b (p′ , p′′ , p)} 1 B645 (p′ , p′′ , p) = − {2F B3b (p′ , p′′ , p) − F B6a (p′ , p′′ , p) 16 2 −F B6b (p′ , p′′ , p)} 1 B646 (p′ , p′′ , p) = − {2F B3b (p′ , p′′ , p) − F B6a (p′ , p′′ , p) 16 2 +F B6b (p′ , p′′ , p)} (A133) (A134) (A135) (A136) B545 (p′ , p′′ , p) = − 1 {2F B3a (p′ , p′′ , p) − F B6c (p′ , p′′ , p) 16 2 +F B6a (p′ , p′′ , p)} 1 B564 (p′ , p′′ , p) = − {2F B3a (p′ , p′′ , p) + F B6c (p′ , p′′ , p) 16 2 +F B6a (p′ , p′′ , p)} 1 B654 (p′ , p′′ , p) = − {2F B3a (p′ , p′′ , p) − F B6c (p′ , p′′ , p) 16 2 −F B6a (p′ , p′′ , p)} 1 B664 (p′ , p′′ , p) = − {2F B3a (p′ , p′′ , p) + F B6c (p′ , p′′ , p) 16 2 −F B6a (p′ , p′′ , p)} (A137) (A138) (A139) (A140) B554 (p′ , p′′ , p) = − B445 (p′ , p′′ , p) B446 (p′ , p′′ , p) B454 (p′ , p′′ , p) B464 (p′ , p′′ , p) = = = = −{(p′′ + p) · p′ }2 F B7 (p′ , p′′ , p) −{(p′′ − p) · p′ }2 F B7 (p′ , p′′ , p) −{(p′ + p′′ ) · p}2 F B7 (p′ , p′′ , p) −{(p′ − p′′ ) · p}2 F B7 (p′ , p′′ , p) 27 (A141) (A142) (A143) (A144) (A145) (A146) (A147) (A148) B544 (p′ , p′′ , p) = −{(p′ + p) · p′′ }2 F B7 (p′ , p′′ , p) B644 (p′ , p′′ , p) = −{(p′ − p) · p′′ }2 F B7 (p′ , p′′ , p) 1 B444 (p′ , p′′ , p) = − F B72 (p′ , p′′ , p) 4 B566 (p′ , p′′ , p) = B656 (p′ , p′′ , p) = B665 (p′ , p′′ , p) = B555 (p′ , p′′ , p) = −4F B7 (p′ , p′′ , p) (A149) (A150) (A151) (A152) Appendix B: Coefficients for the deuteron In this appendix we present the expressions Ad and B d given in Eqs. (2.12) for the deuteron. Ad11 (p) = 3 Ad12 (p) = Ad21 (p) = 0 8 Ad22 (p) = p4 3 (B1) ′ d The coefficients Bkjj ′ (p, p ) are explicitly calculated as d B111 (p, p′ ) d B131 (p, p′ ) d B141 (p, p′ ) d B151 (p, p′ ) d B161 (p, p′ ) = = = = = d B121 (p, p′ ) = 3 0 (p × p′ )2 (p′ + p)2 (p′ − p)2 (B2) 4 d d B212 (p, p′ ) = B222 (p, p′ ) = 4(p · p′ )2 − p2 p′2 3 d B232 (p, p′ ) = −8 p · p′ (p × p′ )2 20 56 d B242 (p, p′ ) = − p4 p′4 − 4(p · p′ )4 + p2 p′2 (p · p′ )2 9 9 16 2 ′2 4 4 2 ′2 2 ′2 d ′ B252 (p, p ) = p p (p + p ) − p p (p · p′ ) − (p2 + p′2 )(p · p′ )2 9 9 3 4 2 ′2 2 16 2 ′2 4 2 d ′ ′2 ′ B262 (p, p ) = p p (p + p ) + p p (p · p ) − (p + p′2 )(p · p′ )2 9 9 3 d d d B211 (p, p′ ) = B221 (p, p′ ) = B231 (p, p′ ) = 0 4 d B241 (p, p′ ) = − p2 (p′ × p)2 3 8 4 16 2 d ′ B251 (p, p ) = p + p (p · p′ ) + 4(p · p′ )2 − 3 3 8 4 16 2 d ′ B261 (p, p ) = p − p (p · p′ ) + 4(p · p′ )2 − 3 3 4 2 ′2 pp 3 4 2 ′2 pp 3 (B3) (B4) d d The expressions for the functions B1k2 , k = 1, 6, can be obtained from the functions B2k1 by ′ replacing p ↔ p . 28 Appendix C: Example of a chiral potential For this particular example we will use the next-to-next-to-leading order (NNLO) chiral potential from Ref. [12]. The leading-order (LO) NN potential in the two-nucleon center-of-mass system (CMS) reads [13]: VLO = − 1 gA2 σ 1 · q σ2 · q CS CT τ1 · τ2 + + σ1 · σ2 , 3 2 2 2 3 (2π) 4Fπ q + Mπ (2π) (2π)3 (C1) where q = p ′ − p, and mπ , Fπ and gA denote the pion mass, the pion decay constant and the nucleon axial coupling constants. At next-to-leading order (NLO) a renormalization of the low energy constants (LECs) is required and the contribution from the GoldbergerTreiman discrepancy leads to a modified value of gA . The remaining contributions to the NN potential at this order are 1 τ 1 · τ 2 Λ̃ h 2 48gA4 m4π i 4 2 2 4 2 (q) 4m (5g − 4g − 1) + q (23g − 10g − 1) + L π A A A A (2π)3 384π 2 Fπ4 4m2π + q 2   3gA4 1 LΛ̃ (q) σ 1 · q σ2 · q − σ 1 · σ 2 q 2 − (2π)3 64π 2Fπ4 C2 2 C3 C4 2 C1 q2 + k +( q2 + k ) σ1 · σ 2 + 3 3 3 (2π) (2π) (2π) (2π)3 C5 i C6 C7 + (σ + σ ) · q × k + q · σ q · σ + k · σ1 k · σ2 , (C2) 1 2 1 2 (2π)3 2 (2π)3 (2π)3 VN LO = − where q ≡ |q | and k = 21 (p ′ + p ). The loop function LΛ̃ (q) is defined in the spectral function regularization (SFR) as [12] LΛ̃ (q) = θ(Λ̃ − 2mπ ) ω Λ̃2 ω 2 + q 2 s2 + 2Λ̃qωs ln , 2q 4m2π (Λ̃2 + q 2 ) (C3) q + and s = Λ̃2 − 4m2π . Here, Λ̃ denotes with the following abbreviations: ω = the ultraviolet cutoff in the mass spectrum of the two-pion-exchange potential. The contributions at NNLO again lead to the renormalization and/or redefinition of the LECs CS , CT , C1 , . . . C7 . The only new momentum dependence is due to the following terms: p 4m2π q2
1 3gA2 2m2π (2c1 − c3 ) − c3 q 2 (2m2π + q 2 )AΛ̃ (q) 3 4 (2π) 16πFπ
gA2 c4 1 τ 1 · τ 2 (4m2π + q 2 )AΛ̃ (q) σ 1 · q σ 2 · q − q 2 σ 1 · σ 2 , − 3 4 (2π) 32πFπ VN N LO = − (C4) where c1 , c3 , c4 are new πN LECs and the loop function AΛ̃ (q) is given by AΛ̃ (q) = θ(Λ̃ − 2mπ ) q(Λ̃ − 2mπ ) 1 arctan . 2q q 2 + 2Λ̃mπ 29 (C5) The expressions of the potential given in Eqs. (C1), (C2), and (C4) show that this potential can be readily expressed in the operators wj , j = 1, 6 of Eq. (2.2). The chiral potential we consider in this example is the sum of V ≡ VLO + VN LO + VN N LO . (C6) It requires regularization when inserted into the Lippmann-Schwinger equation, which is achieved by introducing a regulated potential of the form Vreg (p ′ , p ) ≡ e−(p ′ 4 /Λ4 ) V (p ′ , p ) e−(p4 /Λ4 ) , (C7) with the cut-off parameter Λ. Appendix D: Scalar functions for the Bonn B potential For the convenience of the reader we give the expressions for the Bonn B potential from Ref. [15] in a form which is more suited for our three-dimensional calculations. The expressions for the exchange of pseudo-scalar (ps), scalar (s), and vector (v) mesons are given by r r 2 2 gps [(p′ − p)2 ] Ops m m Fps ′ Vps (p , p) = (2π)3 4m2 E ′ E (p′ − p)2 + m2ps W ′ W r r gs2 m m Fs2 [(p′ − p)2 ] Os ′ Vs (p , p) = (2π)3 4m2 r E ′ r E (p′ − p)2 + m2s W ′ W 1 m m Fv2 [(p′ − p)2 ] (gv2 Ovv + 2gv fv Ovt + fv2 Ott ) Vv (p′ , p) = , (D1) (2π)3 4m2 E ′ E (p′ − p)2 + m2v W ′W p where mα are the masses of the exchanged mesons, m the nucleon mass, E = m2 + p2 and W = m + E. The crucial quantities are the operators Ops , Os , Ovv , Ovt and Ott , which are given in terms of the Dirac spinors as Ops = 4m2 W ′ W ū(p′ )γ 5 u(p)ū(−p′ )γ 5 u(−p), (D2) Os = −4m2 W ′ W ū(p′ )u(p) ū(−p′ )u(−p), (D3) Ovv = 4m2 W ′ W ū(p′ )γ µ u(p)ū(−p′ )γµ u(−p), n Ovt = mW ′ W 4m ū(p′ )γ µ u(p)ū(−p′ )γµ u(−p)   ′ µ ′ ′ 0 0 ′ − ū(p )γ u(p)ū(−p ) (E − E)(gµ − γµ γ ) + (p2 + p2 )µ u(−p)   o ′ ′ ′ 0µ µ 0 ′ µ − ū(p ) (E − E)(g − γ γ ) + (p1 + p1 ) u(p)ū(−p )γµ u(−p) , n Ott = W ′ W 4m2 ū(p′ )γ µ u(p)ū(−p′ )γµ u(−p)   ′ µ ′ ′ 0 0 ′ − 2m ū(p )γ u(p)ū(−p ) (E − E)(gµ − γµ γ ) + (p2 + p2 )µ u(−p) 30 (D4) (D5)   ′ 0µ µ 0 ′ µ − 2m ū(p ) (E − E)(g − γ γ ) + (p1 + p1 ) u(p)ū(−p′ )γµ u(−p)   ′ ′ 0µ µ 0 ′ µ + ū(p ) (E − E)(g − γ γ ) + (p1 + p1 ) u(p)   o ′ ′ 0 0 ′ × ū(−p ) (E − E)(gµ − γµ γ ) + (p2 + p2 )µ u(−p) ′ (D6) with (p1 + p1 ′ )µ = (E + E ′ , p + p′ ) and (p2 + p′2 )µ = (E + E ′ , −p − p′ ). These operators act in the spin spaces of nucleons 1 and 2: the bilinear forms built with ū(p′ ) . . . u(p) contain σ 1 as acting in the spin space of nucleon 1 and the bilinear forms with ū(−p′ ) . . . u(−p) contain σ 2 and act in the spin space of nucleon 2. The spinors u(q) are normalized according to the definitions given in Ref. [23] and explicitly given as ! r E+m 1 u(q) = (D7) σ ·q . 2m E+m Each vertex is multiplied with a form factor Fα2 [(p′ 2 − p) ] =  Λ2α − m2α Λ2α + (p′ − p)2 2n . (D8) where the values of n and the cutoff parameters Λα are given in Table II. Note that for the three iso-vector mesons (π, δ and ρ) contributing to the Bonn B potential, expressions (D2)–(D6) are additionally multiplied by the isospin factor τ (1) · τ (2). In Ref. [5] this potential was presented in a different operator form. However, in that work one of the six operators was chosen to be σ 1 · (p+ p′ ) σ 2 · (p′ −p) + σ 1 · (p′ −p) σ 2 · (p+ p′ ) , which is an operator that violates time reversal invariance. In practice, this operator is always multiplied with the term (p′2 − p2 ), which also violates time reversal invariance. Therefore, the entire term is invariant as it should be. In principle it is not desirable to work with symmetry violating operators, thus we prefer to use the operators from Eq. (2.2) and rewrite σ 1 · (p + p′ ) σ 2 · (p′ − p) + σ 1 · (p′ − p) σ 2 · (p + p′ ) = −4(p × p′ )2 (p − p′ )2 σ · σ + σ 1 · (p + p′ ) σ 2 · (p + p′ ) 1 2 p′ 2 − p2 p′ 2 − p2 (p + p′ )2 4 + ′2 σ 1 · (p − p′ ) σ 2 · (p − p′ ) + ′ 2 σ 1 · (p × p′ ) σ 2 · (p × p′ ), 2 2 p −p p −p (D9) which is an identity for (p + p′ )(p′ − p) = p′2 − p2 6= 0. Inserting Eq. (D9) into the expressions given in [5] cancels the factor p′2 − p2 and one obtains the following expressions for the operators Oα from Eqs. (D2)-(D6) in terms of the operators wj ≡ wj (σ 1 , σ 2 , p′ , p) from Eq. (2.2):      2m 2m ′ 2 ′2 2 Ops = 1 + ′ (p · p) − p p w2 + 1 + ′ w4 E +E (E +E )   1 2m + −(W ′ − W )2 + 1 + ′ [p′2 + p2 − 2(p′ · p)] w5 4 E +E 31 ( )   2m 1 ′ 2 ′2 2 ′ [p + p + 2(p · p)] w6 + −(W + W ) + 1 + ′ 4 E +E Os = −[W ′ W − (p′ · p)]2 w1 − [W ′ W − (p′ · p)]w3 + w4 Ovv = Ovt Ott = (  ′ ′ 2 ′2 2 2 ′2 ′ ′ [W ( W + (p · p)] + W p + W p + 2W W (p · p) w1

1 + − W ′2 + W 2 p′2 + p2 + 2W ′W (p′ · p) 2 )    ′4  1 2m + 1+ ′ p + p4 − 2(p′ · p)2 w2 2 E +E   2m ′ ′ w4 −[3W W + (p · p)]w3 − 2 + ′ E +E    1 2m ′ 2 ′2 2 ′ − −(W − W ) + 1 + ′ [p + p − 2(p · p)] w5 4 E + E   2m 1 ′ 2 ′2 2 ′ −(W + W ) + 1 + ′ [p + p + 2(p · p)] w6 − 4 E +E 
W′ − W = W ′2 p2 + W 2 p′2 − W ′2 p2 − W 2 p′2 2m   W ′ + W  ′2 2 ′ ′ ′ ′ 2 + 2W W [2W W + (p · p)] − W W − (p · p) w1 m 

1 W ′2 + W 2 p′2 + p2 + 2W ′ W (p′ · p) −   2  ′4  1 2m + 1+ ′ p + p4 − 2(p′ · p)2 2 E +E   ′2 4
′  1 2 ′4 ′2 2 2 − W p + W p − W + W (p · p) w2 ′  2m(E + E) ′ W +W ′ (p · p) w3 − 2W ′ W + m    ′2  W′ + W 1 2 ′ + − W + W − 2m(W + W ) w4 + ′ m 2m(E   + E)  2m 1 ′ 2 −(W − W ) + 1 + ′ [p′2 + p2 − 2(p′ · p)] − 4 E +E 
′ 1 ′2 2 2 ′2 ′2 2 − [W p + W p − W + W (p · p)] w5 ′ m(E  + E)   2m 1 ′ 2 −(W + W ) + 1 + ′ [p′2 + p2 + 2(p′ · p)] − 4 E +E 
′ 1 ′2 2 2 ′2 ′2 2 − [W p + W p + W + W (p · p)] w6 m(E ′ + E)    W ′ + W  ′2 2 W W − (p′ · p)2 [W W + (p · p)] + 2 2 − m ′ ′ 2 32 (D10) (D11) (D12) (D13)   3[W ′ W − m(W ′ + W )] + (p′ · p) 2 [W ′ W − (p′ · p)] + 3+ 2   2m  
(W ′ − W )2 (W ′ − W )2 ′2 2 2 ′2 + 1+ W ′ W (p′ · p) W p +W p +2 1− 2 4m2 4m )
W′ − W ′2 2 2 ′2 w1 W p −W p − m (      (W ′ − W )2 (W ′ − W )2 2m ′ ′ + 2 1− (p′ · p)2 W W (p · p) − 1 + 1+ ′ 4m2 4m2 E +E  2   m + E ′E (W ′ − W )2 + m (W ′p2 + W p′2 ) − 1+ 4m2 E′ + E ) 
 1 W ′2 p4 + W 2 p′4 − W ′2 + W 2 (p′ · p)2 w2 − ′ m(E + E) (     W′ + W (W ′ − W )2 ′ ′ ′ + −W W − (p · p) + 2 2 − (p · p) − 2 1 − W ′W m 4m2 )   3[W ′ W − m(W ′ + W )] + (p′ · p) + 3+ [W ′ W − (p′ · p)] w3 2m2    W′ + W 3[W ′ W − m(W ′ + W )] + (p′ · p) −2 2− − 4+ 2m2 m )     ′ 2
(W − W ) 1 2m + 1+ − 1+ ′ W ′2 + W 2 w4 2 ′ 4m E +E m(E + E) ( 
′ (W ′ − W )2 1 1 ′ ′2 2 2 ′2 ′2 2 1− W W − [W p + W p − W + W (p · p)] − 2 4m2 m(E ′ + E) )     1 (W ′ − W )2 2m + 1+ [p′2 + p2 − 2(p′ · p)] − W ′2 − W 2 w5 1+ ′ 2 4m2 E +E (    ′2 2  (W ′ − W )2 1 1 − 1− W p + W 2 p′2 + (W ′2 + W 2 )(p′ · p) W ′W − − 2 ′ 2 4m m(E + E) )     1 (W ′ − W )2 2m + 1+ [p′2 + p2 + 2(p′ · p)] − W ′2 − W 2 w6 . (D14) 1+ ′ 2 2 4m E +E The values of the parameters are given in Table II. [1] [2] [3] [4] [5] [6] [7] [8] W. Glöckle, H. Witala, D. Hüber, H. Kamada, J. Golak, Phys. Rep. 274, 107 (1996). A. Nogga, H. Kamada and W. Glöckle, Phys. Rev. Lett. 85, 944 (2000). Ch. Elster, W. Schadow, A. Nogga and W. Glöckle, Few Body Syst. 27, 83 (1999). H. Liu, Ch. Elster and W. Glöckle, Phys. Rev. C 72, 054003 (2005). I. Fachruddin, Ch. Elster, W. Glöckle, Phys. Rev. C63, 054003 (2001). S. Bayegan, M. R. Hadizadeh and M. Harzchi, Phys. Rev. C 77, 064005 (2008). G. Ramalho, A. Arriaga and M. T. Pena, Few Body Syst. 39, 123 (2006). G. Caia, V. Pascalutsa and L. E. Wright, Phys. Rev. C 69, 034003 (2004). 33 [9] M. Rodriguez-Gallardo, A. Deltuva, E. Cravo, R. Crespo and A. C. Fonseca, Phys. Rev. C 78, 034602 (2008). [10] W. Glöckle, Ch. Elster, J. Golak, R. Skibiński, H. Witala, H. Kamada, arXiv:0906.0321, Few-Body Syst. DOI 10.1007/s00601-009-0064-1. [11] W. Glöckle, I. Fachruddin, Ch. Elster, J. Golak, R. Skibiński, H. Witala, arXiv:0910.1177, accepted for publication in EPJ A. [12] E. Epelbaum, W. Glöckle, U.-G. Meißner, Nucl. Phys. A 747, 362 (2005). [13] E. Epelbaum, Prog. Part. Nucl. Phys. 57, 654 (2006). [14] E. Epelbaum, H. W. Hammer, and U.-G. Meißner, arXiv:0811.1338. [15] R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989). [16] I. Sloan, Comp. Phys. 3, 332 (1968). [17] L. Wolfenstein, Phys. Rev. 96, 1654 (1954). [18] W. Glöckle, The Quantum Mechanical Few-Body Problem, Springer-Verlag, Berlin-Heidelberg, (1983). [19] M. H. McGregor, M. J. Moravcsik, H. P. Stapp, Annu. Rev. Nucl. Sci. 10, 291 ( 1960). [20] W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes, Cambridge University Press, 1989. [21] Wolfram Research, Inc., Mathematica c , Version 7.0, Champaign, IL (2008). [22] T. Lin, C. Elster, W. N. Polyzou, H. Witala and W. Glöckle, Phys. Rev. C78, 024002 (2008). [23] J.D. Bjorken, S.D. Drell, Relativistic Quantum Mechanics, McGraw-Hill Science/Engineering/Math, 1998. 34 View publication stats