Three-nucleon forces and their importance in three-nucleon sys- tems and heavier nuclei

In the past two decades, several laboratories have produced a large amount of data for cross sections, analyzing powers, and other spin observables from various reactions in the three-nucleon system. The experimental results are moderately described by only using the two-nucleon potentials in Faddeev-type calculations. The remaining discrepancies should, in principle, and aside from Coulomb and relativistic e ffe ts, be removed once the e ffects of three-nucleon forces are implemented. High precision data on elastic and break-up reactions show, however, that even after the inclusion of these e ffects, the picture is not complete yet and some ingredients are still missing in the calculations. With the advent of new frameworks within which two and three-nucleon forces can be properly implemented in the calculation of observables in heavy nuclei, it is essential that these forces are better understood.

In order to have a comprehensive picture of the three-body forces, one should not only study masses and spin-averaged cross section but also observables which involve spin such as analyzing powers, spin-transfer coefficients, and spin-correlation coefficients for a large energy and angle range. Only with such a reservoir of data, one can draw a coherent picture of the interplay between the two and three-body forces. The understanding of the nature of forces is, of course, strongly coupled to the theoretical efforts which attempt to generate nuclear forces [18][19][20] in a consistent manner such as those in the effective field theory approach [21,22].
On the experimental front, there have been many experiments in the past decades. More recently, high-precision measurements have been performed at various laboratories around the world. Some results have been presented in this conference and can be found elsewhere in these proceedings.
At KVI, we set up a program with our Polish collaborators to study many of the observables for various reactions of interest. These include elastic proton-deuteron scattering [7][8][9][10][13][14][15][16], protondeuteron break-up reaction [23][24][25] and proton-deuteron capture reaction [26,27]. In this presentation, only a fraction of the data obtained in the past few years will be discussed. For a comprehensive investigation of three-nucleon systems at intermediate energies refer to Ref. [28].

Representative results
In this section, some demonstrative results are presented in the elastic proton-deuteron scattering as well as the break-up channel. In Fig. 2, results are shown for the cross sections, dσ/dΩ, the vector, A y , and tensor, A yy , analyzing powers, the induced polarizations, P y ′ , and the vector, K y ′ y , and tensor, K y ′ yy spin-transfer coefficients in the elastic deuteron-proton scattering [14]. This experiment was performed with a 180 MeV deuteron beam on a solid CH 2 or a liquid hydrogen target [29]. P y ′ is the same as the analyzing power in the inverse reaction (with a negative sign). We used the information from our earlier measurements on the analyzing power to fix the overall small asymmetry of the detection system. It should, therefore, be noted that for P y , only the shape of the observable should be compared with the results of the calculations.
The error bars, which are included for each data point, are for some data points smaller than the symbol size of the point. This error accounts for the statistical uncertainties and a very small point-topoint (PTP) uncertainty. The statistical uncertainties come from the spin-dependent cross sections and the statistical uncertainty in determining the incoming-beam polarization with the in-beam polarimeter [30]. The PTP error accounts for a very small instability of the experimental apparatus over long periods of time and background subtraction. In addition to the statistical and the PTP errors, there are other types of systematic errors. These errors originate from the target thickness measurement, the estimation of the angular opening of the detector, the total collected charge, and the systematic error of the incoming beam polarization. The resultant systematic error is typically 5% for the cross sections and ≤ 3% for all other spin-dependent observables.
The calculations used in the figures of this article and in almost all the studies done in the past few years come from two groups: Bochum-Cracow group [4,31] which, for these calculations has used CD-Bonn potential [1] with TM' three-nucleon force [19], and Hanover-Lisbon group [32] which has used a slightly modified CD-Bonn potential and includes the effect of the three-body forces through the interaction with ∆ [33]. The Bochum-Cracow group also uses the AV18 nucleon-nucleon potential [2] in combination with UIX three-body force [20].
For the cross sections, it is clear that the inclusion of the 3NF (three-nucleon forces) is needed and the magnitude of the correction due to 3NF is generally in agreement with the trend seen in Fig. 2 for the intermediate energies. For the analyzing powers (vector and tensor), calculations including 3NF seem to describe the data better than those based only on NN, with the exception of CDB+∆ [33] which comes close to the results of the calculations based only on two nucleon forces. This picture is completely opposite for the spin-transfer coefficients where the calculations based on 2NF agree with those from CDB+∆ and are also in better agreement with the experimental data. It is obvious that the full spin structure of 3NF is not understood at this energy either and needs further attention. The shape of the induced polarization is in good agreement with the results of all calculations.
For the proton-deuteron break-up reaction, two measurements have been performed using 130 MeV deuteron beam with SALAD and the liquid target developed for this purpose [23][24][25]34] and three measurements have been performed at beam energies of 100 (deuteron), 135 (proton) and 190 (proton) MeV with BINA [35]. Experiments were also performed in Jülich with 130 MeV deuterons [36]. Due to the rich kinematics of the three-body final state, many kinematical configurations have been measured and analyzed. Here, only a very small sample of the analyzing-power results are shown for the experiment done at an incident proton beam energy of 190 MeV. The results shown in Fig. 3 are for configurations where the two protons scatter to forward angles of (25 • , 25 • ) (left panels) and (25 • , 20 • ) (right panels). The azimuthal opening angles between the two protons vary between 180 • (top panels) to 20 • (bottom panels). The results of various calculations are also shown. As can be seen from the top panels, various models agree with each other and also with experimental data for co-planar geometries. However, as one decreases the co-planarity angle from 180 • to 20 • , serious disagreements between various calculations set in and more importantly, they all disagree with the experimental data. Adding a 3NF to two-nucleon potential increases the disagreement even further. Even the effect of relativity [37] goes in the wrong direction. This problem needs to be further investigated.
Finally, in Fig. 4, a global comparison is shown in which all the available data for the cross sections and analyzing powers in the proton-deuteron elastic scattering in the range of 50 to 250 MeV/nucleon have been compiled and compared with theoretical calculations. For the cross sections, comparisons are made with the results of two different groups as mentioned before: Bochum-Cracow group (BC) and Hanover-Lisbon group (HL). In the latter calculations, the Coulomb effect is also included but the effects on the elastic cross sections are rather small for almost all the data points shown here. In addition to cross sections, vector analyzing powers for both polarized proton and deuteron beams are displayed. For this observable, only the results of the calculations from the Hanover-Lisbon group are used as they systematically have also taken the Coulomb force into account. Also for this observable, the effect of Coulomb is negligible for a very large part of the phase space. This figure is taken from Ref. [28] where many comparisons of this sort are made for various observables and different reactions. All the comparisons made (of which a number are shown in the figure) give a clear message, namely that the calculations based only on two-nucleon forces are not sufficient to describe the data. However, the addition of three-nucleon forces available in the literature, only sometimes helps the situation. The discrepancies generally grow with increasing incident beam energies but also when going to more backward angles. This is clearly seen in Fig. 4. For these particular observables, the addition of 3NF improves the disagreement to some extent but clearly not enough. For some other observables, the addition of 3NF even worsens the agreement. This global analysis is meant to point to observables and regions of phase space where theoretical efforts should be spent in the coming years. The spin structure of nuclear forces remain as illusive as their central part.

Conclusions
In order to understand the properties of the three-nucleon forces, a large data-base including spectra of light nuclei and scattering data in three-body systems with all possible observables measured is mandatory. The measurements presented in this contribution are only part of this effort. The results show unambiguously the fact that two-nucleon potentials are not enough to describe the bulk of data. However, none of the three-nucleon potentials available in the literature is at a stage of describing all the data presented here and elsewhere consistently. For the cross sections, the minimum of the cross section is filled up by adding any of the three-nucleon potentials but the improvement is certainly not sufficient. The self-consistent treatment of the ∆ in the Hanover approach [33] which does a relatively good job in describing the analyzing powers in the elastic proton-deuteron scattering at energies between 100 and 200 MeV, performs less satisfactorily in some regions of phase space of the same observables (vector and tensor analyzing powers) when a deuteron beam is used. The advantage of this calculation is the inclusion of Coulomb force for the first time which is shown to be rather small in the elastic channel except for small scattering angles, but rather sizable for some configurations in the break-up channel. The illusive nature of three-body forces might be exposed completely when the results of calculations based on effective field theory become available for these intermediate energies.
The error band from these calculations at intermediate energies above 50 MeV is so large that a sensible comparison presently makes no sense. On the experimental front, efforts are continuing to investigate the nuclear forces and their manifestation in four-nucleon systems. First high-precision measurements with a deuteron beam of 130 MeV impinging on a liquid deuterium target have been performed and results are emerging [38]. For the four-body systems, exact calculations beyond the threshold are still not available. However, from the four-body scattering, regions have been selected where the quasi-free scattering is dominant and comparisons have been made with the three-body system [39].