{'fraction_non_alphanumeric': 0.03416549460853258, 'fraction_numerical': 0.016350210970464137, 'mean_word_length': 4.311235605353253, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
3N Scattering in a Three-Dimensional Operator Formulation

A recently developed formulation for a direct treatment of the equations for two- and three-nucleon bound states as set of coupled equations of scalar functions depending only on vector momenta is extended to three-nucleon scattering. Starting from the spin-momentum dependence occurring as scalar products in two- and three-nucleon forces together with other scalar functions, we present the Faddeev multiple scattering series in which order by order the spin-degrees can be treated analytically leading to 3D integrations over scalar functions depending on momentum vectors only. Such formulation is especially important in view of awaiting extension of 3N Faddeev calculations to projectile energies above the pion production threshold and applications of chiral perturbation theory 3N forces, which are to be most efficiently treated directly in such three-dimensional formulation without having to expand these forces into a partial wave basis.


I. INTRODUCTION
The three-nucleon (3N) system is the first nontrivial case to learn about the action of nucleonnucleon (NN) and three-nucleon (3N) forces in bound states and scattering observables. Below about 200 MeV laboratory projectile energy 3N scattering can be well treated in a momentum space representation based on the Faddeev equations in a partial wave representation [1] to calculate elastic nucleon-deuteron (Nd) scattering as well as breakup processes. A different approach using a coordinate space representation and based on hyperspherical harmonics is equally precise for elastic Nd scattering [2].
Over many years a rich set of Nd data has been accumulated [1,3,4,5], which is not only a very valuable source of information about the spin-and momentum-dependence of nuclear forces but also about the reaction mechanism of multiple rescattering processes. The nuclear forces under consideration today are on the one hand the so-called (semi)-phenomenological high precision forces [6,7,8], describing the NN data up to the pion production threshold perfectly well. The three-nucleon forces on a semi-phenomenological level are much less developed [9] and are not constructed in a consistent manner with respect to corresponding NN forces. Nevertheless those NN and 3N forces describe 3N data (elastic as well as breakup cross sections and numerous spin observables) often spectacularly well. However, there are exceptions where one finds serious discrepancies between that theoretical prediction and the data, especially for some spin observables at the higher region of that energy range [3,4,5].
In recent years effective chiral perturbation theory links nuclear forces to the symmetry of QCD and diagrammatically builds up the nuclear forces in a systematic expansion [10]. In this fashion NN, 3N and even 4N forces are at present consistently generated up to next-to-next-to-next leading order (N 3 LO). Applications of these forces in the few-nucleon sector [11] and for light nuclei [12] are quite successful. However, the effective theory is limited in the energy regime it can be applied to, which is related to the smallness of the parameter underlying that expansion. A typical upper limit for the applicability of those chiral forces is 100-150 MeV nucleon laboratory projectile energy.
Considering the nucleon projectile energies above the pion production threshold and even going into the GeV region, no similar systematic approach to nuclear forces is yet available. However, this energy region imposes challenging questions not only about the nuclear force, but also about the underlying reaction mechanism. One should expect an increased importance of 3N forces , the influence of baryon resonances, meson production, to name a few. Another challenging question is an investigation of the limit, where hadronic degrees of freedom apply and where subnuclear degrees of freedom must be explicitly considered. In order to enter this energy region in the 3N sector various challenges have to be overcome. On the technical side the standard partial-wave decomposition (PWD) has to be given up due to the strongly increasing number of partial wave states that need to be summed for a converged result. To face this challenge the direct use of momentum vectors, i.e. a 3-dimensional (3D) formulation of the problem turned out to be a promising path. This has been documented in various studies for three-boson scattering carried out in a Faddeev scheme in momentum space [13]. In addition, the high energy region requires that Galilean invariance has to be replaced by Poincaré invariance. For the case of three-boson scattering this has successfully been achieved [14,15], where the Poincaré invariant Faddeev equations have been solved for projectile energies in the GeV regime. As already said the enormous challenge will be to develop the underlying dynamics.
It turns out that Poincaré invariant formulation of the 3N system is already important at quite low laboratory energy. Relativistic effects are discernable in some regions of the breakup phasespace starting at the energy of the incoming nucleon about 65 MeV [16]. For Nd elastic scattering vector analyzing power they contribute to the famous analyzing power puzzle at energies around 10 MeV [17].
In this paper we shall focus on the question how to incorporate spin and isospin degrees of freedom into the bosonic three-body calculations so that the successful approach of Ref. [13] is applicable to nucleons. The aim is to reduce the formulation with spin/isospin degrees of freedom to scalar, spin independent functions of vector momenta in the same spirit as already presented for the 2N and 3N bound states in Ref. [18]. The idea is to use the original structure of the NN forces consisting of scalar operators in spin-and momentum-space and scalar functions which only depend on momenta. This carries over to the NN t-operator which is a central building block in the Faddeev scheme. In addition, 3N forces appear naturally in this formulation.
One form of the Faddeev equations for 3N scattering is based on the multiple scattering series for the breakup process which can be summed into a Faddeev integral equation for a transition operator T |Φ in our standard notation [19]. Here |Φ is the initial product state of a deuteron and a momentum eigenstate of the projectile nucleon. Analogous to the NN t-operator there will be an operator form for three-nucleon transition operator T |Φ . However the number of scalar spin-momentum operators will be enormously high, which makes it not advisable to rewrite the Faddeev equation for T |Φ into a coupled set of equations for the accompanying scalar momentum dependent functions. The experience with high energy three-boson scattering [13] suggests that it is promising to generate instead the multiple scattering series, which will automatically generate the operator expansion order by order. We want to use this insight as starting point for our study.
In Section II we introduce the necessary formal ingredients. The lowest order in the multiple scattering series is worked out in Section III. The next term, which is second order in the NN t-operator, is constructed in Section IV. In this order, the free 3N propagator appears for the first time, leading to the notorious moving logarithmic singularities. This can be avoided totally as shown in [20,21]. We apply this new method in which the logarithmic singularities are replaced by single poles, which then can be handled in a similar fashion as the poles in the 2N Lippmann-Schwinger equation. Having the second order under control it is obvious to go to the next order.
However, we will not give this obvious continuation explicitly. The inclusion of 3NF's, which is most interesting from a physical point of view will be skipped in this first formal attempt. However, we do not expect principal difficulty according to the experience for the 3N bound state, where the inclusion of 3NF's has been worked out explicitly [18]. The calculation of observables based on the 3-dimensional form is described in Section V. Various Appendices provide further information.
Finally we conclude in Section VI.

II. THE FORMAL INGREDIENTS
Our standard form of the Faddeev equation for 3N scattering is given by [1,19] T |φ = tP |φ + (1 + tG 0 )V (1) where t is the t-operator for the NN pair 23, G 0 the free 3N propagator, P the sum of a cyclical and an anticyclical permutation and V Knowing T |φ amplitudes for Nd elastic and breakup scattering are given by the matrix elements [1,22] The state describing three free nucleons is given by |Φ 0 .
The iteration of Eq. (1) generates the Faddeev multiple scattering series. When neglecting 3NFs, the iteration leads to As we learned from 3-boson scattering driven by a 2-body force of Malfliet-Tjion type [13] which incorporates typical properties of the NN force, namely an intermediate range attraction and a short range repulsion, the convergence of the series from Eq. (4) improves with increasing energy [23].
First, we introduce the three possible 3N isospin states |γ a = |(t a 1 2 )T a M T : in which the 2N isospin t is coupled with the isospin 1 2 of the third particle to the total isospin T = 1 2 or 3 2 . As is well known [24] in isospin space the 2N t-operator has the form We assume conservation of t a but allow for charge independence and charge symmetry breaking which leads to the coupling of T = 1 2 and 3 2 states: The linear combination of np t-operators in t = 0 and t = 1 states with pp (nn) t-operators for proton-deuteron (pd) (neutron-deuteron (nd)) scattering is given in [25]. Furthermore, the permutation operator P in the 3N isospin space reads [24] where F taTaT b is essentially a 6j-symbol [24] and P sm ij are transpositions of the nucleons ij acting only in spin and momentum spaces. The 3N momentum space is spanned by the standard Jacobi momenta p and q [22]. Combining spin/isospin space with the momentum space leads to the permutation operator with π( q, q ′ ) = 1 2 q + q ′ , Next we use the operator form expansion of the off-shell NN t-operator [18,26] in the 3N spin and momentum spaces where E q = E − 3 4m q 2 and E the total c.m. energy. Due to parity and time reversal invariance exactly 6 terms of scalar spin-momentum operators w j are possible [27]. They are Using this operator representation of the NN force, the functions t (11) are scalar functions and depend only on three variables, the magnitudes of the vectors p and p ′ , and the angle between them,p ·p ′ .
Finally, we use the operator form of the initial state as given in Ref. [28] p q| γ 0 |Φ ≡ p q|φ Here φ 1 (p) and φ 2 (p) are proportional to the standard s-and d-wave components of the deuteron wave function. The state |1m d is the pure spin 1 two-nucleon state with spin magnetic quantum number m d , and m 10 is the initial state spin magnetic quantum number of the projectile nucleon.
The vector q 0 is the initial relative momentum in the Nd system. All these are the necessary ingredients to work out the terms in the multiple scattering series of Eq. (4).
Here we used the isospin property of the initial state: t 0 = 0, T 0 = 1/2. Then p q| γ a |tP |Φ Note that we used the fact that the two operators O k from Eq. (13) depend quadratically on the momenta. Then we define and The scalar expressions a jk and b jk have to be worked out such that each σ(i) occurs only once.
The explicit expressions are given in Appendix A.
Inserting then Eqs. (16) and (17) into Eq. (15) yields p q| γ a |tP |Φ We see the expected structure, a sum over the product of scalar operators (a jk , b jk ) multiplied by scalar functions. Note that It remains to display the singularity structure for the calculation of the physical amplitudes and the treatment of the second order.
The NN t-matrix in the pd isospin space has the general structure where and α TaT b , β TaT b are numerical values related to Clebsch Gordon coefficients.
The np t-matrix t 00 np for t = 0 has a deuteron pole Therefore, Eq. (18) can be decomposed further, which retains the structure of scalar spin-momentum dependent operators and momentum dependent scalar functions.
As will be shown for some examples in Appendix B the coefficients C jlk from Eq. (28) together with the other coefficients from Eqs. (29) through (31) will depend on the following types of scalars.
It is sufficient to consider the first integral in Eq. (27) going along with the coefficients C jlk .
The remaining integrals will be treated as exemplified for X 2 . The integral has the form If Y does not depend on q 2 then X 2 = 0. Otherwise, due to the scalar nature of Y , one must have the following structure and therefore The scalars α, β, γ are then determined by multiplying Eq. (44) by p s , q s , and q 0s , respectively.
This leads to three equations for α, β, γ. The three integrals have to be determined numerically.
Correspondingly, despite being more involved, X 3 can be determined One has three external momenta in the function Y , namely p, q , and q 0 . Thus, there are at most 9 possibilities: where the Q Ls Q ′ Lt belong to the set Q Ls Q ′ Lt = [p s p t , p s q t , p s q 0t , q s p t , q s q t , q s q 0t , q 0s p t , q 0s q t , q 0s q 0t ] .
By appropriate multiplications from the left one will have nine equations for the coefficients A L , generated by 9 corresponding integrals. Thus X 3 takes the form Furthermore, Now we have 27 possibilities and consequently 27 equations need to be solved with the result where Q Ls Q ′ Lt Q ′′ Lu are out of the set The appropriate 27 integrals need to be determined numerically. Finally, X 5 can be handled like X 2 , and X 6 like X 3 . All the integrals occurring in Eq. (34) and the ones following from that are of the type where f consist of scalar products of the momenta p, q, q 0 , and q 2 .
We use the decomposition of the NN t-matrix from Eq. (20) in the pd isospin space and the deuteron pole structure from Eq. (23). Then we obtain In numerical implementation one can either follow the standard path [1] dealing with the free propagator singularity and leading to moving logarithmic singularities, whose treatment is well controlled and documented in [1,13] or one applies the new way [20,21] which avoids that complication totally. We suggest the second path, which for the convenience of the reader is detailed again in the Appendix D.
Summarizing, the second order amplitude of Eq. (32) using all information given above will have the structure where O f are spin-momentum dependent scalar operators and S f only momentum dependent scalar functions, all depending just on the external momenta p, q, and q 0 . Further P f is either 1, P s 12 P s 23 or P s 13 P s 23 . This form will be the input for the next order in Eq. (4), namely tP G 0 (tP G 0 tP |Φ ) where the kernel tP G 0 is exactly treated as above and the first order term tP |Φ is replaced by that second order form. Obviously this will go on like that and the total amplitude p q| γ a |T |Φ will appear in the form of the right hand side of Eq. (58).
It remains to present the calculation of the observables, which is described for the example of the breakup cross section in the next section.

V. THE ND BREAKUP CROSS SECTION
The full breakup amplitude according to Eq. (3), projected on spin states and using Eq. (8) is given by The simplest approach is to apply the permutations to the left, using Therefore, Using now the general form of the right hand side of Eq. (58) one obtains (2) , q (2) , q 0 )O f ( σ(1), σ(2), σ(3), p (2) , q (2) , q 0 )P f |1m d |m 10 As shown in Ref. [1], the density matrix for the final state is given by where is the expression of Eq. (63), where the set of spin magnetic quantum numbers is given by Furthermore, ρ i is the density matrix for the initial state, with [S ν ] being the complete set of 2N spin matrices [1]. For an unpolarized initial state one has With this, the unpolarized breakup cross section is given up to a phase-space factor as Since three nucleon forces appear naturally in the form of scalar operators in spin-and momentum-vectors and scalar functions depending only on momenta, the present formulation can be extended to include also 3N forces in a straightforward manner. This feature is especially important in view of the chiral perturbation theory approach, where the multitude of 3N forces at N 3 LO appears to be most efficiently treated directly in this three-dimensional formulation without having to expand these forces into a partial wave basis.
Remember x =q 2 ·q, thus it appears natural to chooseq =ẑ and then the x-integration can be carried out trivially.
The two Θ-functions restrict the integrations in q 2 and p ′ into an area whose size depends on the magnitudes of the spectator momentum q . It results The important point is that the free propagator appears now as a simple pole.
We rewrite .
In the second integral we integrated first over q 2 and then over p ′ . In both cases we have just a simple pole either in p ′ or in q 2 .