Three-nucleon force in relativistic three-nucleon Faddeev calculations

Three-nucleon force in relativistic three-nucleon Faddeev calculations

H. Witała M. Smoluchowski Institute of Physics, Jagiellonian University, PL-30059 Kraków, Poland    J. Golak M. Smoluchowski Institute of Physics, Jagiellonian University, PL-30059 Kraków, Poland    R. Skibiński M. Smoluchowski Institute of Physics, Jagiellonian University, PL-30059 Kraków, Poland    W. Glöckle Institut für theoretische Physik II, Ruhr-Universität Bochum, D-44780 Bochum, Germany    H. Kamada Department of Physics, Faculty of Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan    W. N. Polyzou Department of Physics and Astronomy, The University of Iowa, Iowa City, IA 52242
July 11, 2019
Abstract

We extend our formulation of relativistic three-nucleon Faddeev equations to include both pairwise interactions and a three-nucleon force. Exact Poincaré invariance is realized by adding interactions to the mass Casimir operator (rest Hamiltonian) of the non-interacting system without changing the spin Casimir operator. This is achieved by using interactions defined by rotationally invariant kernels that are functions of internal momentum variables and single-particle spins that undergo identical Wigner rotations. To solve the resulting equations one needs matrix elements of the three-nucleon force with these properties in a momentum-space partial-wave basis. We present two methods to calculate matrix elements of three-nucleon forces with these properties. For a number of examples we show that at higher energies, where effects of relativity and of three-nucleon forces are non-negligible, a consistent treatment of both is required to properly analyze the data.

pacs:
21.45.-v, 21.45.Ff, 25.10.+s, 24.10.Jv

I Introduction

High precision nucleon-nucleon potentials such as AV18 AV18 (), CDBonn CDBOnucleon-nucleon (), Nijm I, II and 93 NIJMI () provide a very good description of the nucleon-nucleon data set up to about 350 MeV. When these forces are used to predict binding energies of three-nucleon systems they underestimate the experimental bindings of and by about 0.5-1 MeV Friar1993 (); Nogga1997 (). This missing binding energy can be restored by introducing a three-nucleon force into the nuclear Hamiltonian Nogga1997 ().

Also the study of elastic nucleon-deuteron scattering and nucleon induced deuteron breakup revealed a number of cases where the nonrelativistic description using only pairwise forces is insufficient to explain the data. Generally, the studied discrepancies between a theory using only nucleon-nucleon potentials and experiment become larger with increasing energy of the three-nucleon system. Adding a three-nucleon force to the pairwise interactions leads in some cases to a better description of the data. The elastic nucleon-deuteron angular distribution in the region of its minimum and at backward angles is the best studied example wit98 (); sek02 (). The clear discrepancy in these angular regions at energies up to  MeV nucleon lab energy between a theory using only nucleon-nucleon potentials and the cross section data can be removed by adding a modern three-nucleon force to the nuclear Hamiltonian. Such a three-nucleon force must be adjusted with each nucleon-nucleon potential separately to the experimental binding of and  wit98 (); wit01 (); sek02 (). At energies higher than  MeV current three-nucleon forces only partially improve the description of cross section data and the remaining discrepancies, which increase with energy, indicate the possibility of relativistic effects. The need for a relativistic description of three-nucleon scattering was also raised when precise measurements of the total cross section for neutron-deuteron scattering abf98 () were analyzed within the framework of nonrelativistic Faddeev calculations wit99 (). Nucleon-nucleon forces alone were insufficient to describe the data above  MeV. The effects due to relativistic kinematics considered in wit99 () were comparable at higher energies to the effects due to three-nucleon forces. These results showed the importance of a study taking relativistic effects in the three nucleon continuum into account.

In witrel1 (); witrel2 () the first results on relativistic effects in the three-nucleon continuum have been presented. The dynamics was defined by a three-nucleon center of momentum Hamiltonian or mass operator including only pairwise interactions. The mass operator was used to calculate three-nucleon scattering observables. The input to that approach is a “Lorentz boosted” nucleon-nucleon potential, which generates the nucleon-nucleon -matrix in a moving frame by solving a standard Lippmann-Schwinger equation. To get the nucleon-nucleon potential in an arbitrary moving frame one needs the interaction in the two-nucleon center of momentum system, which appears in the relativistic nucleon-nucleon Schrödinger or Lippmann-Schwinger equation. The relativistic Schrödinger equation in the two-nucleon center of momentum system differs from the nonrelativistic Schrödinger equation just by the relativistic form for the kinetic energy. Current realistic nucleon-nucleon potentials are defined and fit by comparing the solution of the nonrelativistic Schrödinger equation to experimental data. Up to now nucleon-nucleon potentials refitted with the same accuracy in the framework of the relativistic nucleon-nucleon Schrödinger equation do not exist. Such refitting can be, however, avoided by solving a quadratic integral equation whose solution is a relativistic potential which is phase-equivalent to a given input high-precision nonrelativistic nucleon-nucleon potential kamada1 (). An alternative equivalent approach towards a relativistic nucleon-nucleon t-matrix in another frame is provided in CSP ().

In our previous studies with only nucleon-nucleon interactions we found that when the non-relativistic form of the kinetic energy is replaced by the relativistic one and a proper treatment of the relativistic dynamics is included, the elastic scattering cross section is only slightly influenced by relativity. Only at backward angles and higher energies are the elastic cross sections increased by relativity witrel1 (). It is exactly the region of angles and energies where the effects of three-nucleon forces are also significant wit01 (). Also, for nucleon-deuteron breakup reactions regions of phase space were found at higher energies of the incoming neutron where relativity significantly changes the breakup cross sections witrel3 (); skibbr (). For some spin observables large effects due to relativity and three-nucleon forces have been reported in nucleon-deuteron breakup for an incoming deuteron energy of  MeV, some of which seem to be supported by proton-deuteron data brkim (). These observations call for three-nucleon continuum relativistic Faddeev calculations which include three-nucleon forces. Only such consistent calculations should be used to analyze the data in cases when both relativity and three-nucleon force effects are large.

The paper is organized as follows. Sec. II provides the conceptual basis for the choice of the momentum-space representation and the definition of spin in the relativistic context. In Sec. III we summarize the formalism underlying relativistic three-nucleon Faddeev calculations with only nucleon-nucleon interactions, presented in detail in witrel1 (); witrel2 (). In Sec. IV we focus on the three-nucleon Faddeev equation with an included three-nucleon force and discuss two methods to compute matrix elements of the three-nucleon force in the partial wave basis used in our relativistic calculations. In Sec. V we apply our formulation to elastic nucleon-deuteron scattering and breakup and show and discuss the results. Sec. VI contains our conclusions and summary. Appendixes A and B formulate three-nucleon forces in the momentum space representation adapted to Poincaré invariance.

Ii Relativistic dynamics

Relativistic invariance of a quantum theory means that the Poincaré group (inhomogeneous Lorentz group) is a symmetry group of the theory. This requires the existence of a unitary representation of the Poincaré group Wigner:1939cj (). The Poincaré group has ten generators, six Lorentz generators , and four spacetime translation generators, . The dynamics of the system is given by the Hamiltonian, . The Lie algebra has two polynomial invariants,

(1)

where is the Pauli-Lubanski vector paulilubanski ()

(2)

It satisfies

(3)

Equation (1) implies that the Hamiltonian can be expressed in terms of the mass operator, , where . Thus, given a representation for , the dynamics is defined by the mass operator , which plays the same role in Poincaré invariant quantum mechanics as the center of mass Hamiltonian does in Galilean invariant quantum mechanics.

In the absence of interactions the mass operator becomes the invariant mass operator of three non-interacting relativistic particles. The full interaction is defined by

(4)

For a system of three particles interacting with short-range interactions, two-body interactions are defined by

(5)

where is obtained from by turning off all interactions in that involve particle . The difference

(6)

defines a three-body interaction. With these definitions the mass operator has the form

(7)

This has the same form as the non-relativistic three-body center of mass Hamiltonian with two and three-body forces, except the non-relativistic kinetic energy is replaced by the relativistic invariant mass of the non-interacting system. As in the non-relativistic case, bound and scattering eigenstates of this mass operator can be computed using the Faddeev equations with two and three-body interactions. For identical nucleons the coupled relativistic Faddeev equations can be replaced by a single equation. Details are discussed in the next section.

In addition to the constraints imposed by discrete symmetries, translational invariance, and particle exchange symmetry, there are non-trivial constraints on the interactions due to both the Poincaré symmetry and cluster properties. The constraints on the interaction due to Poincaré invariance come from the commutator

(8)

which means that interactions appearing in must be generated by the operators in the commutator. One way to satisfy the constraints due to Poincaré invariance was suggested by Bakamjian and Thomas Bakamjian:1953kh (). Their construction adds interactions to the mass Casimir operator that commute with the spin Casimir operator

(9)

The required interactions commute with and are independent of the total momentum and commute with the non-interacting three-body canonical spin operator.

Spin is associated with rotational degrees of freedom that appear in the rest frame. Because the Lorentz boost generators, , do not form a closed sub-algebra, a sequence of Lorentz boosts that map the rest frame to the rest frame can generate a rotation. Thus in order to obtain a well-defined relativistic spin it is necessary to define a standard procedure for measuring the spin. This normally requires the specification of a special frame where spins can be compared (usually the rest frame) and a standard set of Lorentz transformations , parameterized by momentum, that transform arbitrary frames to the special frame. The three-body canonical spin is defined in terms of the Pauli-Lubanski vector by kei91 ()

(10)

where is the rotationless Lorentz transformation-valued function of the four momentum ,

(11)

This Lorentz transformation (11) satisfies

(12)

Equations (3) and (12) can be used to show that the components of satisfy commutation relations. The spin is not a four vector because is a matrix of operators, rather than a constant Lorentz transformation. Under Lorentz transformation the canonical spin Wigner rotates

(13)

where . The spin Casimir operator is independent of the choice of boost (11) used to define the spin. The non-interacting (kinematic) canonical spin, , is obtained from (10) by replacing , in (10) and in (11). Thus, Poincaré invariance can be satisfied provided the interactions and commute with .

The other non-trivial constraint on the interactions is imposed by cluster properties. The problem arises due to the non-linear relation between the two-body interaction in the two-body problem and the corresponding two-body interaction, , in the three-body problem. Cluster properties relate to the Poincaré generators for the interacting pair and spectator . Unfortunately each mass operator constructed by requiring cluster properties commutes with a different spin Casimir operator, which means that linear combinations of these interactions will break Poincaré invariance. Coester relform1 () observed that these interactions could be replaced by phase-equivalent interactions that commute with . These interactions are designed to satisfy cluster properties in the three-body rest frame. Using the Bakamjian-Thomas construction linear combinations of the phase equivalent ’s can be added in a manner that preserves the overall Poincaré invariance. While these interactions do not lead to generators that satisfy cluster properties, cluster properties in the three-body rest frame and Poincaré invariance of the matrix ensures that the three-body -matrix retains cluster properties in all frames.

To construct two-body interactions, the two-body interactions in the two-body problem that commute with the two-body canonical spin are replaced by phase equivalent two-body interactions in the three-body problem that commute with the three-body canonical spin. The phase equivalent interactions are identified in the rest frame of the three-body system. They are determined in all other frames by the requirement that the three-body spin remains kinematic (in the Bakamjian-Thomas construction this choice fixes the representation of the boost generators).

To construct interactions that commute with the three-body kinematic canonical spin it is useful to introduce momenta and spin variables that have the same Wigner rotation properties as the three-body kinematic canonical spin. This is because the kinematic canonical spin can be constructed out of these degrees of freedom using conventional methods for adding angular momenta.

The desired momentum operators are the relativistic analog of Jacobi momenta. In the non-relativistic case Jacobi momenta can be defined using Galilean boosts to the two and three-body rest frames. In the relativistic case the Galilean boosts are replaced by the rotationless boost (11) and the relevant Jacobi momenta are relform1 ()

(14)
(15)

In terms of these variables

(16)

where the two-body invariant mass is

(17)

The vector variables satisfy

(18)

The relevant property of these momentum vectors is that they experience the same Wigner rotations as the three-body kinematic canonical spin (13),

(19)

Similarly,

(20)

where the last line follows from the property of the rotationless boosts (11) that the Wigner rotation of a rotation is the rotation kei91 ()

(21)

for any . Thus the and all undergo the same Wigner rotations as the three-body kinematic canonical spin.

Next we introduce single-particle spins with the same property. Single-particle canonical spins can be constructed from single-particle Poincaré generators using

(22)

where the operators on the right side of (22) are constructed by replacing all of the three-body generators in (2), (10) and (11) by the corresponding one-body generators.

Under kinematic Lorentz transformations the single-particle canonical spins experience Wigner rotations, , that depend on the single-particle momenta. These rotations differ from the Wigner rotations experienced by , and . This can be changed by introducing new single-particle spin operators that replace the rotationless boost in (22) by a two step boost,

(23)

These two boosts agree when . Note that both of these boosts transform , so they differ by momentum dependent rotations. We call these spins three-body constituent spins to distinguish them from single-particle canonical spins. The constituent spin operators are defined by relform1 ()

(24)

When , which means that single-particle canonical spins and three-body constituent spins agree in the three-body rest frame. For a three-body system the total spin is identified with total angular momentum in the three-body rest frame, which is the sum of the single-particle angular momenta. The angular momentum of a single particle in the three-body rest frame is the sum of the single-particle constituent spin and a contribution from the single particle orbital angular momenta.

A calculation, using the property (21), shows that under Lorentz transformations

(25)

Wigner rotates with the same rotation as the vectors and and the three-body kinematic canonical spin. The three-body kinematic canonical spin is the sum of the orbital angular momenta associated with and and the single-particle three-body constituent spins. The requirement that an interaction commutes with the kinematic three-body canonical spin is equivalent to the requirement that the interaction have a rotationally invariant kernel when expressed in terms of these variables. Thus the required interactions in the Bakamjian-Thomas construction are given by kernels of the form

(26)

where the reduced kernel is a rotationally-invariant function of and the three-body constituent spins.

Two-body interactions in the two-body problem have a similar form

(27)

where

(28)

is the two-body relative momentum and the magnetic quantum numbers are associated with the two-body constituent spins

(29)

When these interactions are embedded in the three-body Hilbert space the kernels (27) are replaced by kernels that are rotationally invariant functions of the three-body Jacobi momenta and the three-body constituent spins. In order to satisfy cluster properties given by (28) is replaced by the given by (15), the are replaced by the corresponding , and the two-body constituent spins (29) are replaced by

(30)

These operators represent two-body constituent spins in the three-body rest frame. They agree with the two-body constituent spins (29) that they replace in the three-body rest frame, but are defined so they remain unchanged by canonical boosts out of the three-body rest frame. This ensures that they undergo the same Wigner rotations as the kinematic three-body canonical spin under kinematic Lorentz transformations. Thus, the kernels (27) are related by

(31)

Here the unbarred magnetic quantum numbers are three-body constituent spins while the barred magnetic quantum numbers are the two-body constituent spins in the three-body rest frame.

Even though the spins in (30) transform the same way as the three-body constituent spins, they differ from the three-body constituent spins (24) by the Wigner rotation

(32)

When the two-body interactions are embedded in the three-body system the spins are identified with the two-body constituent spins in the three-body rest frame, as would be expected by cluster properties, but in other frames they are defined to remain unchanged with respect to canonical boosts out of the three-body rest frame. The Wigner rotations (48) and (64) arise because the two-body subsystem is moving in the three-body rest frame; however because the Wigner rotations in (32) are functions of the rather than the , both spins in (32) undergo the same Wigner rotations under kinematic Lorentz transformation. Because of this it is also possible to construct the three-body canonical spin using partial wave methods directly in a mixed representation involving the barred spins in the interacting pair and the unbarred spin for the spectator. In the mixed representation the two-body interaction in the three-body Hilbert space has the simple form

(33)

For the two-body problem in the three-body Hilbert space it is advantageous to use (33) because spins (30) do not require Wigner rotations. However, with this choice each interacting pair of particle must be treated using a permuted basis which requires Wigner rotations in the permutation operators. The three-body forces are naturally expressed by a rotationally invariant kernel in the three-body constituent spins. When they are transformed to a mixed basis that involves the spin (30) for one pair, then it is necessary to transform two of the three-body constituent spins with the Winger rotations in (32). The calculations performed in this work use a partial wave projection of the mixed basis (33), although the Wigner rotations in the three body-interaction are not yet included.

Iii Relativistic three-nucleon Faddeev equations with nucleon-nucleon forces

The nucleon-deuteron scattering with neutron and protons interacting through only a nucleon-nucleon interaction is described in terms of a breakup operator satisfying the Faddeev-type integral equation wit88 (); glo96 ()

(34)

The two-nucleon -matrix is the solution of the Lippmann-Schwinger equation with the interaction . The permutation operator is given in terms of the transposition operators, , which interchanges nucleons i and j. The incoming state describes the free nucleon-deuteron motion with relative momentum and the deuteron state vector . Finally is resolvent of the three-body center of mass kinetic energy. Transition operators for the elastic nd scattering, , and breakup, , are given in terms of by wit88 (); glo96 ()

(35)
(36)

This is our standard nonrelativistic formulation, which is equivalent to the nonrelativistic three-nucleon Schrödinger equation plus boundary conditions. The formal structure of these equations in the relativistic case remains the same but the ingredients change. As explained in  relform () the relativistic three-nucleon rest Hamiltonian (mass operator) has the same form as the nonrelativistic one, only the momentum dependence of the kinetic energy and the relation of the pair interactions in the three-body problem to the pair interactions in the two-body problem change. Consequently all the formal steps leading to (34) and (36) remain the same.

The free relativistic invariant mass of three identical nucleons of mass has the form  witrel2 () (see Eq.(16))

(37)

with spectator momentum and the free two-body mass operator expressed in terms of the relative momentum in the center of momentum frame by (see Eq.(17))

(38)

As introduced in  relform1 () and in Eq.(5) the pair forces in the relativistic three-nucleon mass operator are related to the two-body forces in the two-body problem, , by

(39)

where reduces to the interaction for , which acts in the two-body center of momentum frame. The momentum dependence ensures that the resulting three-nucleon scattering matrix satisfies space-like cluster properties in all frames relform1 ().

The transition matrix that appears in the kernel of the Faddeev equation (34) is obtained by solving the relativistic Lippmann-Schwinger equation as a function of

(40)

The input two-body interaction is computed by solving the nonlinear equation kamada1 ()

(41)

where is a nonrelativistic nucleon-nucleon potential fitted to the nucleon-nucleon data basis and where anticommutator . In case of that equation reduces to a nonlinear equation for the relativistic two-body interaction . Therefore the problem of refitting all two-nucleon data when changing from a nonrelativistic to a relativistic Lippmann-Schwinger equation is avoided. The nonlinear equation (41) can be solved by iteration kamada1 (). An alternative approach to determine is described in CSP ().

The new relativistic ingredients in (34) and (36) will therefore be the -operator (40) (expressed in partial waves) and the resolvent of the three-nucleon invariant mass

(42)

with given by (37). is the total three-nucleon invariant mass expressed in terms of the initial neutron momentum relative to the deuteron by

(43)

with the deuteron rest mass. Related to the choice of the permutation operator P the pair is chosen as .

Currently the Faddeev equation (34) in its nonrelativistic form is numerically solved for any nucleon-nucleon interaction using a momentum space partial-wave decomposition. Details are presented in wit88 (). Projecting (34) on such a basis turns it into a coupled set of two-dimensional integral equations. As shown in witrel1 (); witrel2 (), in the relativistic case we can keep the same formal structure, though the permutation operators are replaced by the corresponding Racah coefficients for the Poincaré group which include both Jacobians and Wigner rotations that do not appear in the nonrelativistic permutation operators glo96 (); book ().

In the nonrelativistic case the partial-wave projected momentum-space basis is

(44)

where p and q are the magnitudes of standard Jacobi momenta (see glo96 (); book ()), obtained by transforming single particle momenta to the rest frame of a two- or three-body system using a Galilean boost, and are two-body quantum numbers with obvious meaning, refer to the third, spectator nucleon, taken as the nucleon and described by the momentum q, is the total three-nucleon angular momentum and the rest are isospin quantum numbers. In the relativistic case this basis is replaced by the Poincaré irreducible states defined as witrel2 ()

(45)
(46)
(47)
(48)

where is given by (65) in Appendix A and by (133) in Appendix B. These states are labeled by the same quantum numbers as the corresponding non-relativistic basis states.

The basis states (48) are used for the evaluation of the partial wave representation of the permutation operator P with Wigner rotations of spin states for nucleons and included. In the relativistic case we adopt the following short-hand notation for the Poincaré irreducible three-body states, which also includes isospin quantum numbers coupled in the same order:

(49)

Equipped with that, projecting (34) onto the basis states one encounters, using the nonrelativistic notation of Ref. book ()

(50)

This is evaluated by inserting the complete basis of single-particle states and using (48). It can be expressed in a form which resembles closely the corresponding non-relativistic expression  book (); glo96 ()

(52)

where all ingredients are defined in Appendix 64 of Ref. witrel2 (). The rotational invariance of the nucleon-nucleon interaction in this basis ensures that all three nucleon-nucleon interactions commute with the spin Casimir operator of the non-interacting three-nucleon system. This allows the interactions to be added in a manner that preserves the underlying Poincaré symmetry.

Due to the short-range nature of the nucleon-nucleon interaction it can be considered negligible beyond a certain value of the total angular momentum in the two-nucleon subsystem. Generally with increasing energy will also increase. For we set the t-matrix to zero, which yields a finite number of coupled channels for each total angular momentum J and total parity of the three-nucleon system. To achieve converged results at incoming nucleon laboratory energies below  MeV all partial wave states with total angular momenta of the two-nucleon subsystem up to and all total angular momenta of the three-nucleon system up to must be taken into account. This leads to a system of up to 143 coupled integral equations in two continuous variables for a given and parity. For the details of the numerical performance we refer to glo96 (); book (); witrel1 (). The solution of these equations can be used to construct an exactly Poincaré invariant scattering operator.

Iv Relativistic three nucleon Faddeev equations with a three-nucleon force included

In the standard nonrelativistic formulation when in addition to pairwise interactions between three nucleons also a three-nucleon force is included, a new term appears in a potential energy of the three-nucleon system

(53)

Each is symmetric under exchange of the nucleons and ( and ). In the -exchange three-nucleon force is a contribution to the three-nucleon potential from (off-shell) rescattering of a pion on nucleon 1.

When a three-nucleon force is acting then on top of rescatterings among three nucleons induced by pairwise forces only, which are summed up in integral equation (34), additional rescatterings induced by three-nucleon force and nucleon-nucleon force appear.

Therefore Faddeev equation (34) changes to

(54)

with one new contribution in the leading term and in the kernel glo96 (); hub97 (). While the breakup transition operator preserves its form (36), in the elastic scattering operator two new contributions appear glo96 (); hub97 ()

(55)

The second term is due to a single interaction of three-nucleons via a three-nucleon force and the fourth results from rescattering among three nucleons induced by two- and three-nucleon forces with a three-nucleon force as the final interaction.

After projecting on a partial-wave momentum-space basis equation (54) becomes a system of 2-dimensional coupled integral equations which can be solved numerically exactly for any nuclear force. Since the three-nucleon force is short-ranged its inclusion needs to be carried through only for all total angular momenta of the three nucleon system up to . As mentioned in section III, the longer ranged two-nucleon interactions require states up to . For details of the formalism and numerical performance in case of the nonrelativistic formulation when three-nucleon force is included we refer to Refs. glo96 (); wit88 (); hub93 ().

For relativistic calculations without a three-nucleon force, briefly described in previous section, the details of the numerical treatment are given in witrel1 (); witrel2 (). When a three-nucleon force is added two new terms in (54) contain the free three-nucleon propagator . Since in the basis (see Appendix B) the three-nucleon invariant mass is diagonal, is given by

(56)

That means that performing integrations over momenta and in the intermediate states during the calculation of matrix elements for these new terms, the simple pole singularity occurs for momenta at , where is given by the total three-nucleon center of momentum energy through .

For a given -value the momentum is the solution of . The treatment of that pole, as well as of the deuteron bound state pole in , which occurs at for channels containing the deuteron quantum numbers, was done using subtraction method wit88 (); glo96 ().

The nonrelativistic treatment of (54) requires matrix elements of calculated in a partial-wave basis with standard Jacobi momenta: . In the relativistic calculations, however, one needs them in the new, relativistic basis . The generation of three-nucleon force partial-wave matrix elements is the most time consuming part of three-nucleon continuum Faddeev calculations. One way to reduce the computer time is to perform a transformation of the existing, standard Jacobi momenta matrix elements to the relativistic basis. In Appendix A we give the expression (132) for such transformation which is valid in the general case, when in addition to boost also Wigner spin rotations are taken into account. The complex structure of that transformation, where in addition to the summation over numerous intermediate states with geometrical coefficients, also involved are two integrations and two interpolations over the momenta and , prevents, due to the large amount of computing time and computer resources required, the application of that transformation in fully converged calculations.

It seems thus unavoidable that in order to get matrix elements one must start from a commonly given expression for a three-nucleon force in terms of individual nucleons momenta and their spin and isospin operators and to apply to that expression the recently proposed automatized partial wave decomposition automa1 (); automa2 (). To that aim we derived in Appendix B relation (149) which allows to express matrix element of a three-nucleon force in a 3-dimensional relativistic basis by its matrix element in the individual nucleons momenta basis. In this basis and undergo identical Wigner rotations under kinematic boost of the three-nucleon system. The nucleon spins are defined to be the three-body constituent spins (canonical spins measured by using a rotationless boost to the three-body center of momentum frame). To use them in the basis the spins for the pair must be Wigner rotated before they are coupled. The alternative is to use the representation where three spins are three-body constituent spins; in this case all three of the two-body interactions will have Wigner rotations that convert the two-body constituent spins in the three body rest frame to three-body constituent spins. The three-nucleon force will have no Wigner rotations. In this representation all of the spins can be coupled using standard partial wave methods. For our calculations we work in the basis, but do not account for the Wigner rotations in the three-nucleon interaction for the reasons discussed in the previous paragraph. This allows us to treat the spins in the three-nucleon force using conventional methods. This assumes the Wigner rotations in the three-nucleon force can be neglected. Neglecting these Wigner rotations has no effect on the relativistic invariance or -matrix cluster properties.

V Results

To study the importance of a consistent treatment of both relativity and a three-nucleon force we numerically solved the three-nucleon Faddeev equations for neutron-deuteron scattering at the neutron laboratory energies , , and  MeV. As dynamical input we took the nonrelativistic nucleon-nucleon potential CD Bonn CDBOnucleon-nucleon () and TM99 three-nucleon force TM (); TMgl (); TM99 (). The cut-off parameter of that three-nucleon force was adjusted to in units of the pion mass, , to give, together with the CD Bonn potential, the experimental binding energy of H. At each energy we generated solutions of nonrelativistic and relativistic three-nucleon Faddeev equation, without and with TM99 three-nucleon force included. For relativistic case we produced, starting from the CD Bonn potential and solving nonlinear equation (41) at the required spectator nucleon momenta , the relativistic, on-shell equivalent interaction with boost effects incorporated exactly. That interaction served as dynamical input to calculate, using the relativistic Lippmann-Schwinger equation (40) the relativistic off-shell t-matrix that appears in Faddeev equations.

Since in witrel2 () it was found that effects of Wigner spin rotations are practically negligible in the studied energy range, we neglected them in the present study. When performing relativistic calculations with three-nucleon force included one requires matrix elements of the TM99 three-nucleon force in a relativistic momentum space basis, where the relative momentum of two nucleons in their c.m. system, , replaces standard Jacobi momentum . That momentum together with spectator nucleon momentum , equal in magnitude and opposite to the total momentum of the free pair in three-nucleon center of momentum system, unambiguously define the configuration of three nucleons. Since it is the region of small and not large momenta which is most important when solving Faddeev equations it seems reasonable to assume that the momenta and do not differ substantially. Therefore, in order to avoid calculations of the TM99 three-nucleon force matrix elements in a relativistic basis we assumed, that the matrix elements in a relativistic and nonrelativistic bases are equal:

(57)

That assumption allowed us to use the existing matrix elements of the TM99 three-nucleon force.

To check quality of the approximation (57) we compared the matrix element of the TM99 3NF in the relativistic basis, , calculated according to (149) and using automatized partial wave expansion of Ref. automa2 () (what corresponds to neglection of Wigner spin rotations in (158)), with the corresponding matrix element in the standard, nonrelativistic basis, , at a number of the spectator momentum values. In Figs. 1 and 2 we exemplify the typical behavior showing at a number of q’ values and at a fixed , taking two different values of , the dependence of these matrix elements for a particular channel . As expected, clear differences between these matrix elements occur only at very large values of the spectator momentum q, where magnitudes of these matrix elements are small. This justifies application of the approximation (57) in the present study.

The approximation (57) can be investigated also directly for the three-dimensional matrix elements, comparing and . The connection between these matrix elements is given by (149) in Appendix B. They depend on momentum vectors and spin-isospin quantum numbers in the initial and final state. In Fig. 3 we show a particularly simple case, where , all four momenta are parallel to the unit vector and all spin magnetic quantum numbers are equal . We display and for several values as a function of . We see how the difference develops gradually with increasing , resembling the picture seen for partial wave decomposed matrix elements.

Transition amplitudes for elastic neutron-deuteron scattering and breakup based on that set of four solutions of three-nucleon Faddeev equations, are used to predict numerous observables for both reactions. By comparing these observables conclusions on how strongly three-nucleon force effects depend on relativity were drawn. In the following subsections we show and discuss results for the cross section and numerous spin observables, separately for elastic scattering and breakup reactions.

v.1 Elastic scattering

At higher energies of the incoming nucleon three-nucleon forces play significant role in determining the angular distribution of the elastic neutron deuteron scattering. The clear evidence of three-body force effects start to develop at  MeV for scattering angles close to a minimum of the cross section, which at  MeV occurs at wit98 (); wit01 (). With increasing energy of the three-nucleon system not only the magnitude of predicted three-nucleon force effect increases but it also influences the cross section in a wider range of angles, which at  MeV covers wit98 (); wit01 (). The standard -exchange three-nucleon forces, such as TM99 TM99 () or Urbana IX uIX (), are able to account for existing discrepancies between theoretical cross sections obtained with realistic nucleon-nucleon potentials and data only up to  MeV. Data at larger energies in a region of angles ranging from the cross section minimum up to are drastically underestimated even when -exchange three-nucleon forces are included in the calculations. This is exemplified on Fig. 4, where solid (red) lines are nonrelativistic predictions based on the CD Bonn potential alone and dotted (blue) lines are results obtained when the CD Bonn potential was combined with the TM99 three-nucleon force.

Since effects of relativity for predictions based on two-nucleon forces only are restricted to very backward angles witrel1 () (see also Fig. 4 where dashed (blue) lines are relativistic predictions based on the CD Bonn potential), the drastic discrepancy between data and theory seen at  MeV would indicate that at such large energies shorter-ranged three-nucleon force components, not taken into account in these calculations, start to play significant role. The possibility, that including such three-nucleon force contributions would indeed help to improve description of the cross section data is further supported by an interesting pattern revealed when the TM99 three-nucleon force is included into relativistic calculations. Namely, when a consistent treatment of relativity and a three-nucleon force as described in the present study is made, then the resulting changes of the cross section are not a simple incoherent sum of effects due to relativity, seen when two-nucleon forces alone are acting, and three-nucleon force effects found in nonrelativistic calculations. The relativity modulates effects exerted by the TM99 three-nucleon force on the cross section found in nonrelativistic calculations and the magnitude of this modulation depends from the scattering angle. While at backward angles the nonrelativistic cross section with a three-nucleon force included is further enhanced by relativity, in a region of center of momentum angles near the cross section minimum the magnitude of three-nucleon force effects seen in nonrelativistic calculations is strongly reduced by relativity (dashed-dotted (brown) lines in Fig. 4).

Also elastic scattering polarization observables reveal such incoherent and angle-dependent modulation of three-nucleon force effects by relativity. The details, however, depend on the particular spin observable under study and every conceivable scenario can be found.

For elastic scattering spin observables effects of relativity, when only two nucleon forces are acting, were found to be small witrel1 (). It is exemplified by nearly overlapping solid (red) and dashed (blue) lines in Figs. 5-15. Adding three-nucleon force in nonrelativistic calculations leads to substantial effects for some polarization observables, especially at higher energies sek02 (); wit01 (). The resulting picture, however, is quite complex. Some of those three-nucleon force effects are supported by the data. For some observables they deteriorate the data description.

For tensor analyzing powers , and relativistic effects are non-negligible even at  MeV (see Fig. 5) and clearly increase with increasing energy as seen in Figs. 6, 7 and 9. When three-nucleon force is added in the relativistic calculations the resulting effect depends on the observable and the energy.

For large three-nucleon force effects remain. At  MeV and  MeV they are practically identical in magnitude to three-nucleon force effects found in nonrelativistic calculations and nonrelativistic and relativistic predictions for at these energies are practically overlapping (see dotted (blue) and dashed-dotted (brown) lines in Figs. 5, 6 and 9). At  MeV, however, adding three-nucleon force in relativistic calculations leads to angle dependent modulations of the magnitude of three-nucleon force effects, similar to that found for the cross section (see Fig. 9).

For a drastically different scenario occurs. Large effects of the TM99 three-nucleon force are seen for that observable in nonrelativistic calculations at  MeV and  MeV in wide range of angles and they practically vanish when relativity is included. As a result the dashed-dotted (brown) line practically overlaps with pure two-nucleon relativistic and nonrelativistic predictions (see Fig. 5 and 6).

For (Fig. 5 and 6) the large effects of the three-nucleon force seen in nonrelativistic calculations are simply reduced by relativity. For the tensor analyzing power , for which data exist only at and  MeV, the influence of relativity induces both modulation and reduction of nonrelativistic three-nucleon force effects (Fig. 7).

The TM99 three-nucleon force acts differently on the nucleon, , and deuteron, , vector analyzing powers. While three-nucleon force effects for are rather small even at  MeV (Fig. 8 and 15), for they are significant (Fig. 5, 6 and 8). For and , but more clearly displayed due to larger effects for the deuteron vector analyzing power, both reduction and modulation of nonrelativistic three-nucleon force effects by relativity was found. That reduction and modulation depend on angle and energy.

A similar picture was found for numerous spin correlation coefficients, as exemplified by different theoretical predictions shown in Figs. 9-13. Again all scenarios are available: total reduction by relativity of large three-nucleon force effects seen in nonrelativistic calculations (e.g. at and  MeV for in Fig. 10, at and  MeV at in Fig. 11), practically the same three-nucleon force effects in nonrelativistic and relativistic calculations ( at  MeV in Fig. 9, at and  MeV in Fig. 12), angle dependent modulation of nonrelativistic three-nucleon force effects by relativity ( at  MeV in Fig. 9, at and  MeV in Fig. 10, and at and  MeV in Fig. 13).

The polarization transfer coefficients are not exceptions; also for them a similar complex influence of relativity on nonrelativistic three-nucleon force effects have been found as shown in Figs. 14 and 15.

The comparison of nonrelativistic predictions based on -exchange three-nucleon force’s revealed for spin observables a complex, angle and energy dependent pattern of discrepancies between data and theory wit01 (); sek02 (); sek_eltransfer (); maedand (); hat02 (). The nontrivial interplay between the -exchange three-nucleon forces and relativity suggests that the inclusion of further three-nucleon force mechanisms, like forces of shorter range, is needed to improve the description of elastic scattering polarisation data.

v.2 Breakup

Theoretical study of exclusive breakup reaction performed at different incoming nucleon energies revealed regions of breakup phase-space where large three-nucleon force effects have been found zoln1 (). The effects, similarly to elastic scattering, generally increase with energy. With increasing energy also the effects of relativity increase witrel3 (); skibbr (), revealing for exclusive breakup cross section a characteristic pattern when viewed as a function of the angles of detected nucleons. Largest effects where found when two of three outgoing nucleons are detected coplanarly on both sides of the beam. Keeping one of the detectors at a constant position and changing the polar angle of the second, regions of phase space were found in which nonrelativistic breakup cross section was increased or decreased by relativity skibbr (). In these specific configurations effects of three-nucleon force’s on breakup cross section, both in norelativistic as well as in relativistic calculations, are practically negligible (see Fig. 16).

Due to richness of the breakup phase-space also geometrical configurations can be found where both, three-nucleon force and relativistic effects are significant. Exclusive cross sections in some of these configurations are shown as a function of the laboratory energy of one of the outgoing and detected nucleons in Fig. 17 for neutron-deuteron breakup at  MeV. It is seen that including relativity reduces slightly the magnitude of three-nucleon force effects observed in nonrelativistic calculations.

Relativity changes also the magnitude of three-nucleon force effects seen in nonrelativistic calculations for breakup polarization observables. We exemplify that in Fig. 18 at three configurations of exclusive dp breakup at  MeV, for which data have been taken brkim (). Again, influence of relativity on magnitude of three-nucleon force effects change with configuration as shown in Fig. 18 along the S-curve arc length. Especially interesting is the case of polarization-transfer coefficient from the deuteron to the nucleon, , for which inclusion of TM99 three-nucleon force changes completely the S-dependence found in case when only two nucleon-forces were acting. The effect of three-nucleon force is further modified slightly by relativity resulting in a better reproduction of data.

Vi Summary and outlook

We extended our relativistic formulation of three-nucleon Faddeev equations to include also three-nucleon force. The relativistic features are the relativistic form of the free propagator, the change of the nucleon-nucleon potential caused by the boost of the two nucleon subsystem, and the modification of the permutation operators. In present study we neglected Wigner spin rotations induced by these boosts since investigations based on two-nucleon forces only have shown that their effects are negligible. For the momentum-space basis we used the relative momentum of two free nucleons in their c.m. system together with their total momentum in the three nucleon c.m. system, which in this frame is the negative momentum of the spectator nucleon. Such a choice of momenta is adequate for relativistic kinematics and allows to generalize the nonrelativistic approach used to solve the nonrelativistic three nucleon Faddeev equation to the relativistic case in a more or less straightforward manner. That relative momentum in the two-nucleon subsystem is a generalization of the standard nonrelativistic Jacobi momentum . We numerically solved these equations for neutron-deuteron scattering including relativistic features and/or three-nucleon force at the neutron lab energies , , and  MeV. As dynamical input we took the nonrelativistic nucleon-nucleon potential CD Bonn and generated in the two nucleon center of momentum system an exactly on-shell equivalent relativistic interaction. As a three-nucleon force we took the -exchange TM99 force.

By comparing our relativistic calculations without and with the three-nucleon force included we studied influence of relativity on three-nucleon force effects. In studies with two-nucleon forces only it was found that significant relativistic effects for the elastic scattering cross section appear at higher energies and they are restricted only to the very backward angles where relativity increases the nonrelativistic cross section. At other angles the effects are small. Also for spin observables, analyzing powers, spin correlation coefficients and spin transfer coefficients, no significant changes due to relativity have been found when only two-nucleon forces were acting. The similar picture was found for breakup, however, in that case significantly larger effects for the cross section in specific regions of the breakup phase-space have been found.

The results obtained in the present study document that this picture changes dramatically when in addition to the two-nucleon force in a relativistic treatment also a three-nucleon force is acting. For the elastic scattering large changes of the cross section at higher energies, caused by three-nucleon force in large region of angles ranging from around minimum of the cross section up to very backward angles, are further significantly modulated by relativity. Also such modulation in a large, similar to that for the cross section, range of angles have been found for numerous polarization observables. In that case every conceivable scenario of modulations was observed: from wiping out large three-nucleon force effects found in nonrelativistic calculations to their modulations with energy and angle, with strong amplification or reduction of their magnitude. Also for exclusive breakup cross section and polarisation breakup observables in some geometries the relativity influences effects induced by three-nucleon forces. Thus also for that reaction the relativistic treatment when three-nucleon forces are acting is required for proper interpretation of data.

The comparison of our nonrelativistic theory with existing elastic scattering cross section and polarisation data exhibits at the higher energies clear discrepancies. The discrepancies between the theory based on pairwise forces only and data are largest in the region starting from the cross section minimum around up to . At energies up to about  MeV these discrepancies can be removed when current three-nucleon forces, mostly of -exchange character TM (); uIX (), are included in the nuclear Hamiltonian. At the higher energies, however, a significant part of the discrepancy remains and increases further with increasing energy. Especially complex picture exists for spin observables. Here adding -exchange three-nucleon force into nonrelativistic calculations leads to effects which depend on observable. They can be large or negligible, change their magnitude with energy and angle. Similarly to the elastic scattering cross section even after inclusion of three-nucleon force some of the discrepancies remain and increase with increasing energy. This indicates that additional three nucleon forces should be added to the -exchange type forces. Natural candidates in the traditional meson-exchange picture are exchanges like and . This has to be expected since in PT epel2002 () in the order in which nonvanishing three-nucleon force’s appear the first time there are three topologies of forces, the