Quark-Model Baryon-Baryon Interaction Applied to the Neutron-Deuteron Scattering (III)

Quark-Model Baryon-Baryon Interaction Applied to the Neutron-Deuteron Scattering (III)

Yoshikazu Fujiwara and Kenji Fukukawa

1 Introduction

The three-nucleon () system is a good place to study the underlying nucleon-nucleon () interaction, since many techniques to solve the system exactly are well developed nowadays.[1, 2] Ample experimental data are already accumulated especially for the low-energy neutron-deuteron () and proton-deuteron () scattering and extensive studies to detect the force have been carried out based on the modern meson-exchange potentials,[3, 4] and more recently, on the chiral effective field theory.[5, 6] Most of the researches to such a direction are concerned with higher energies than 100 MeV for the nucleon incident energy in the laboratory (lab) system, since the force effect is expected to be revealed more prominently than in the low energies.[3] On the other hand, the discrepancies of various observables between the theory and experiment in the MeV region, are not resolved even by the recent accurate treatment of the Coulomb force.[7, 8, 9] This is particularly true for the nucleon-induced deuteron breakup processes. It is therefore worth while reexamining the interaction itself if the present-day realistic force is the most appropriate one to start with.

In previous papers,[10, 11] referred to as I and II hereafter, we have applied the quark-model (QM) baryon-baryon interaction fss2 to the neutron-deuteron () elastic scattering. This interaction model, fss2, describes available data in a comparable accuracy with the modern meson-exchange potentials.[12] By eliminating the inherent energy dependence of the resonating-group kernel, fss2 was found to yield a nearly correct triton binding energy, the -wave scattering length, and the low-energy eigenphase shifts without reinforcing it with the three-body force.[13, 14, 15] The predicted elastic differential cross sections have sufficiently large cross section minima at MeV and , in contrast to the predictions by the standard meson-exchange potentials.[10] The so-called puzzle at low-energies MeV is largely improved in this model.[11] In this paper, we continue these studies by examining the breakup processes with various decaying kinematics for the energy range MeV. The main motivation is to find if the quite different off-shell properties, originating from the strong nonlocality of the QM baryon-baryon interaction, give some influence to the breakup differential cross sections. In contrast to the elastic scattering amplitude, the breakup amplitude covers a wide momentum region of the three-body phase space. It will be found unfortunately that the fss2 gives predictions similar to the meson-exchange potentials and does not improve much the discrepancies between the theory and the experiment.

The organization of this paper is as follows. In , the formulation of the breakup differential cross sections is given in terms of the direct breakup amplitude. Various kinematical configurations for the three-body decay are introduced in . A minimal description of the three-nucleon breakup kinematics is given in Appendix A. The isospin factors for the breakup amplitudes are derived in Appendix B. The comparison with the experimental data is presented in for energies and 65 MeV. The difference from the predictions by meson-exchange potentials are discussed in detail. We close this paper with a summary of this series of investigations in .

2 Formulation

2.1 Breakup differential cross sections

Following the notation of Refs. 1, I and II, the three-body breakup amplitude is given by

(2.1)

In order to derive the breakup differential cross sections, we start from the Fermi’s golden rule

(2.2)

and divide it by the incident flux . Here, , is the nucleon mass, and is the incident momentum related to the energy, , in the center-of-mass (cm) system. We obtain in the cm system

(2.3)

where Eq. I(2.88)*)*)*)In the following, we cite equations of the previous paper I (or II), with adding I (or II) in front of the equation number. is used to perform the -integral. In Eq. (2.3), is the spin-isospin quantum numbers in the -coupling scheme and the subscript 0 in the matrix element implies the on-shell condition with . Here, is related to the deuteron binding energy through . In this paper, we use the notation and to specify the quantum numbers in the -coupling scheme: i.e.,

(2.4)

with , and and being the three-particle spin and isospin wave functions, respectively. In Eq. (2.4), are the angular functions. For the initial state, we use channel-spin representation as for the elastic scattering. We take the sum of Eq. (2.3) over all the spin and isospin quantum numbers and divide by the initial spin multiplicity 6. The selection of the detected particles in the final state is controlled by the isospin projection operator , the explicit form of which will be specified later. The breakup differential cross sections of the scattering are therefore calculated from

Let us first consider the spin-isospin sum by neglecting the initial spin quantum numbers for the time being. The effect of the permutation in is defined by

(2.6)

if the function does not contain the spin-isospin degree of freedom. In fact, we should use

(2.7)

where is the permutation operator in the spin-isospin space and the bra-ket notation is used for the spin-isospin degree of freedom. Using these notations and the completeness relationship in the spin-isospin space, , we find

(2.8)

Here we separate the , sum into the diagonal part () and the off-diagonal part (). In the off-diagonal part, we specify and by the cyclic permutations of (123) (=(123)-cyclic). For these terms, the - term and the - term are complex conjugate to each other. Thus we obtain

(2.9)

where implies the sum over the three cyclic permutations of .

The extension to , incorporating the isospin projection operator , is rather easy. Here, is specified as

(2.10)

depending on the species of particles 1 and 2 detected. We use and the notation . Then, by defining

(2.11)

we obtain

(2.12)

The spin-isospin factors in Eq. (2.12) are calculated by separating the spin-isospin state to the spin and isospin parts, . We find

(2.13)

where and , and Eq. (B.1) is used for the spin part. We also extend the definition in Eq. (B.1) for the spin part to the isospin part as in Eq. (B.2). Using the definition of in Eq. (B.2), we can write the matrix elements in Eq. (2.13) as

(2.14)

for a cyclic permutation of (123). Thus we find

(2.15)

where the generalized Pauli principle is used for the two-nucleon part of . The isospin factors are explicitly given in Appendix B.

We assign the direct breakup amplitude to in Eq. (2.15) through

(2.16)

with . The partial wave decomposition is given by

(2.17)

where the prime on the sum implies that we take all the orbital angular momentum sum for the coupling scheme of ; i.e., the sum over only with . (Note the extra factor for the scattering amplitude.) It is convenient to define

(2.18)

by the solutions, , of the basic AGS equation in Eq. I(2.59), and the elastic scattering amplitude in Eq. I(2.92). The partial-wave amplitude for the direct breakup, ,†)†)†)This corresponds to the direct term of in Eq. I(2.92). is expressed as

(2.19)

if we use the the spline interpolation for a particular value of . For the practical calculations, it is convenient to adopt a particular coordinate system with in Eq. (2.17). Then the basic direct breakup amplitude in the spin-isospin space is calculated from

(2.20)

with . The differential cross sections in the cm system are given by

(2.21)

The breakup differential cross sections in the lab system, , are specified by the two directions , , and the energy measured along the locus of the - energy plane. They are obtained from Eq. (2.21) by a simple change of the phase space factor [1]

(2.22)

The details of the three-body kinematics are summarized in Appendix A.

2.2 Three-nucleon breakup kinematics

Assuming that we detect two outgoing particles 1 and 2, the breakup differential cross sections are specified by two polar angles , , and a difference of azimuthal angles , in addition to the energy determined from the kinematical curve (-curve) in the and plane. The starting value of the arc length is quite arbitrary and we follow the convention by the experimental setup. In Appendix A, we have parametrized the locus in the - plane with an angle , and the starting point is uniquely determined by specifying . We also assume that the beam direction of the incoming particle is the axis and set , [16] which determines the -axis.

It is customary to classify the three-body breakup kinematics into the following six categories based on the classical (or geometrical) argument:[1, 4]


  1. The quasi-free scattering (QFS): one of the nucleons in the final state is at rest in the lab system.

  2. The final-state interaction (FSI): the relative momentum of the two outgoing nucleons is equal to zero.

  3. The collinear configuration (COLL): one of the outgoing nucleons is at rest in the cm system, and the other two have momenta back to back.

  4. The symmetric space star configuration (SST): the three nucleons emerge from the reaction point in the cm system, keeping equal momenta with relative to each other and perpendicular to the beam direction (on the - plane in the cm system).

  5. The coplanar star configuration (CST): the same with the symmetric space star configuration, but with the three momenta lying on the reaction plane.

  6. The non-standard configuration (NS): the other non-specific configurations.

These are mathematically distinguished by particular values of the lab momentum , the relative momenta and , etc., and provide a rough guidance to which portion of the two-nucleon -matrix is responsible at the final stage of the reaction, according to the structure of the direct breakup amplitudes in Eq. (2.19). For example, Ref. 1 argues that the first Born term of the QFS with is approximately a product of an on-shell two-nucleon -matrix and the deuteron state at zero momentum. It is known that the force effect is rather small for the QF condition.[4] On the other hand, the collinear configurations with are expected to be sensitive to the force intuitively, and the experimental study by Correll et al.[17] was carried out to study the effect of the force intensively in the reaction around these configurations at the deuteron incident energy MeV. Furthermore, the FSI is characterized by , for which the half off-shell -matrix in Eq. (2.19) generates a large peak corresponding to the positive-energy bound state near the zero-energy threshold. Although the height of the peak is influenced by the background amplitude , the FSI peak is usually nicely reproduced. The disagreement with the data is reported at the early stage for the SST configuration, which is still an unsolved problem called space star anomaly.[18] It should be noted, however, that the disagreement between the theory and experiment is also seen in some other coplanar star and non-standard configurations, for which off-shell properties of the two-nucleon -matrix is expected to play a role in different ways. We will examine these case by case in the next section.

3 Results and discussion

3.1 reaction at MeV

It is important to take enough number of discretization points and partial waves to get well converged results, especially for the breakup differential cross sections. In this paper, we take --=6-6-5 in the notation introduced in of I, unless otherwise specified. This means that the three intervals, and , are discretized by the six-point Gauss Legendre quadrature for each, and the total number of the discretization points for is 35 (). The three-body model space is truncated by the two-nucleon angular momentum , which depends on the incident energy of the neutron. We find that is large enough for MeV. The Coulomb force is entirely neglected in the present calculations.



Fig. 1: Breakup differential cross sections for the reaction with MeV. The experimental data are the deuteron incident reaction, , with the energy MeV.[17]

We first investigate the Correll et al.’s experiment;[17] i.e., reaction with the deuteron incident energy MeV. This corresponds to the nucleon-induced breakup reaction of the nucleon incident energy of 8 MeV. We generate the direct breakup amplitude for the deuteron incident reaction by adding an extra phase factor to each term of Eq. (2.20), corresponding to the change from to in Eq. (2.17).‡)‡)‡)We appreciate Professor H. Witała for informing us about this phase change. The decay kinematics for the deuteron incident reaction is discussed in Appendix A. The breakup differential cross sections for the reaction with MeV are compared with the Correll et al.’s data in Fig. 1, with respect to two collinear (COLL1, COLL2) and two non-standard (NS1, NS2) configurations. The starting point is chosen as the collinear points or the nearest point, as is discussed in Appendix A. The dashed curve, the solid curve, and the bold solid curve correspond to the S+D (i.e., and only), , and cases, respectively. The solid curves almost overlap with bold curves and is actually good enough at this energy. We also see that even the restriction to the S+D model space is not too bad. If we compare our results with meson-exchange predictions in Ref. 1, we find that they are very similar to each other. The calculated values are somewhat too small especially in COLL1, COLL2 and NS1, although to less extent for the meson-exchange predictions. There is no exact collinear point in the case of NS2, and the best agreement with experiment is obtained in this case.

3.2 reaction at MeV

The breakup differential cross sections for the reaction at MeV are displayed in Figs. 2 and 3, together with the experimental data by Gebhardt et al.[19] The figure number, fig. 5 etc., in each panel corresponds to the original one in Ref. 19. The large two peaks seen in fig.5 - fig.13 are the final state interaction peaks with on the lower side and those with on the higher side. On the whole, the comparison with experiment gives fair agreement, but some discrepancies found in Ref. 19 still persist. In Ref. 19, the experimental data are compared with the solutions of the AGS equations in the -method, using a charge-dependent modification of the Paris potential. Their results and ours are strikingly similar to each other, sharing the same problems for the detailed fit to the experiment. The peak heights for the final state interaction peaks are not precisely reproduced in fig. 5, fig. 7, fig. 8 and fig. 10, probably because we did not take into account the charge dependence of the two-nucleon interaction. In fig. 9, our result is worse than the theoretical calculation in Ref. 19. The collinear point is realized at MeV in fig. 11, at MeV in fig. 13, at MeV in fig. 14, and at MeV in fig. 15. In fig.11, we have obtained a smooth curve around the collinear point, just as the theoretical calculation in Ref. 19. The breakup differential cross sections around the collinear points are well reproduced. In fig. 16, the flat structure between - 7 MeV is just the same as the theoretical calculation in Ref. 19. The experimental data of Ref. 19 for the symmetric space star configuration is plotted in the first panel of Fig. 6. Here again, we have obtained very similar result with the theoretical calculation in Ref. 19.





Fig. 2: Breakup differential cross sections for the reaction with MeV. The experimental data are taken from Ref. 19 with the same figure number, fig. 5 etc., in each panel.




Fig. 3: The same as Fig. 2, but for other kinematical configurations.

3.3 Quasi-free scattering

We show in Fig. 4 the breakup differential cross sections for the quasi-free scattering at energies - 65 MeV. We find some deviation from the experimental data at the peak position for all the energies. Detailed investigation of the Coulomb effect in Ref. 8 has revealed that this overestimation at the peak position is reduced to some extent. However, the reduction is probably not large enough except for the case of MeV. Figure 8 of Ref. 8 implies that this reduction is energy dependent. The large overestimation at MeV may not be resolved only by the Coulomb effect. The direct incorporation of the Coulomb force is necessary for our QM interaction. In Fig. 4, we can see that the roles of higher partial waves are important for higher energies. The partial waves up to is clearly necessary for MeV. For the energies, and 65 MeV, we need more partial waves up to .





Fig. 4: Breakup differential cross sections for the quasi-free scattering (QFS). The experimental data for the reaction are taken from Refs. 20 for 10.5 MeV, 21 for 13 MeV, 22 for 19 MeV, 23 for 22.7 MeV, and 24 for 65 MeV. For the reaction with MeV, the experimental data shown by bars are taken from Ref. 20.

3.4 Final state interaction

Four examples of the breakup differential cross sections for the final state interaction are displayed in Fig. 5. Almost all the data are for the reaction. In the 10.5 MeV and 13 MeV cases, the data are shown with open circles, while the data with bars, some dots and diamonds. The lower peaks are the final state interaction peaks with , while the upper with . Here we find that the higher peaks are slightly too small. The Coulomb correction increases the peak height a little, [8] and improves the fit to the experiment to some extent. We probably need more careful treatment of the charge dependence of the interaction, just as in the previous 10.3 MeV case. We also see that the minimum point at - 12 MeV for the MeV reaction is too low. The three-nucleon force might be necessary to increase the cross sections and get a good fit to the experiment.[28]



Fig. 5: Breakup differential cross sections for the final state interaction (FSI). The experimental data are taken from Ref. 20 (: open circles, : filled circles with bars) for 10.5 MeV, Refs. 26, 27, 25 (: bars, filled circles, filled diamonds) and 21 (: open circles) for 13 MeV, 28 () for 16 MeV, and 22 () for 19 MeV.

3.5 Symmetric space star configurations

As is well known, a large discrepancy appears in the breakup differential cross sections in the symmetric space star configurations.[18] This is seen in Fig. 6, where our results in the various model spaces are compared with the and data. A strange thing is that any theoretical calculations of the reaction at MeV deviate largely from the old and new data [26, 27, 18], although the deviation is not much for the 10.3 MeV, 19 MeV and 65 MeV data. The data at 13 MeV in Ref. 21 are more than smaller than the data. A theoretical study of the Coulomb effect for the symmetric space star configuration in Ref. 8 shows that it is generally very small irrespective of the energy (see Fig. 6 of Ref. 8). Our results at MeV, and are located just between the lower data and the higher data, which is very similar to other predictions by the meson-exchange potentials. In the other geometrical configurations at 13 MeV, the cross sections in the space star 2 case () are about half of the experimental values and those in the space star 3 case () are almost smaller than the experiment. The same situation happens in the Faddeev calculations by the Paris potential.[26] (See Figs. 29 and 30 of Ref. 27.) Note that we need enough partial waves for the convergence of the symmetric space star configurations in particular, which was already pointed out in Ref. 1.





Fig. 6: Breakup differential cross sections for the symmetric space star (SST) configurations. The experimental data are taken from Refs. 29, 19 () and 20 (: [10.5 MeV]) for 10.3 MeV, 26, 27, 18 () and 21 () for 13 MeV, 22 () for 19 MeV, and 30 () for 65 MeV.

3.6 Coplanar star configurations

Let us move to the coplanar star configurations in Figs. 7 and 8. The agreement between our calculation and the data is satisfactory in general, but some deviations still exist. For example, in the first panel of 13 MeV, the new data [18] are much closer to the calculation than the old data [27], but some underestimation still exists in the calculation. The underestimation of the cross sections at the minimum point MeV of CST2, and also at the final state interaction peaks around MeV in CST3 and in CST4 are the common feature with the meson-exchange potentials. See Figs. 11, 12 and 13 of Ref. 27. For 16 MeV reactions, we have given a comparison not only for the coplanar star configuration, but also for the intermediate star (IST) configuration, both of which are very similar to the predictions by other models given in Ref. 28. For MeV data, the curves are not plotted as a function of but by for the second particle. For this and MeV reactions, we find a large contribution of higher partial waves up to . In the symmetric backward plane star configuration of 65 MeV, denoted by CST2, the original experimental data are shifted to the larger side of by 3.5 MeV, since the starting position of does not seem to be the same between our calculation and the experiment.





Fig. 7: Breakup differential cross sections for the coplanar star (CST) configurations at the energies and 16 MeV. The result for the intermediate star (IST) configuration is also shown at MeV. The experimental data are taken from Refs. 18, 27 () for 13 MeV, 28 () for 16 MeV.


Fig. 8: The same as Fig. 7, but for the energies and 65 MeV. The experimental data are taken from Refs. 31 () for 22.7 MeV and 30 () for 65 MeV.

3.7 Collinear configurations

The comparison for the collinear configurations are displayed in Figs. 9 and 10. For these configurations, the comparison with the experiment is generally good. The Coulomb force has an appreciable effect to increase the breakup cross sections at the collinear point, especially on the low-energy side, [8] which is much more important than the force effect. In the first panel with MeV (COLL1), we find some kinematical mismatch of the final state interaction peak at MeV. For COLL2 - COLL5 with MeV, the small breakup cross sections around the collinear points (minimum points) move to better direction to fit the experimental data by the expected Coulomb effect. This would also be true for the minimum point for MeV. On the other hand, the breakup cross sections in COLL1 - COLL4 for MeV seem to be slightly overestimated.





Fig. 9: Breakup differential cross sections for the collinear (COLL) configurations. The experimental data are taken from Ref. 20 ( and ) for 10.5 MeV and Refs. 18, 27 () for 13 MeV.




Fig. 10: The same as Fig. 9, but for other geometrical configurations and energies. The experimental data are taken from Refs. 27 () for 13 MeV, 22 () for 19 MeV, and 32 () for 65 MeV.

3.8 Non-standard configurations

The comparison with the experimental data for the non-standard configurations is shown in Figs. 11 and 12. Here we find big deviation from the experimental data in some cases, again a common feature with the meson-exchange predictions. These are the NS2 scattering of 13 MeV, and the scattering of 22.7 MeV and 65 MeV. A large number of figures, NS1 - NS9, for 13 MeV are very similar to the predictions by the Paris potential in Ref. 27. The huge final state interaction peaks in NS4, NS7 and NS8 are very similar to the results by the Malfliet-Tjon potential. In 22.7 MeV and 65 MeV cases, the calculated results are completely off the experimental data.





Fig. 11: Breakup differential cross sections for the non-standard (NS) configurations. The experimental data are taken from Ref. 27 ().




Fig. 12: The same as Fig. 11, but for other geometrical configurations and energies. The experimental data are taken from Refs. 27 () for 13 MeV, 23 () for 22.7 MeV, and 1 () for 65 MeV.

4 Summary

In this and previous papers,[10, 15, 11] we have applied the quark-model (QM) baryon-baryon interaction fss2 to the neutron-deuteron () scattering in the Faddeev formalism for composite particles. The main motivation is to investigate the nonlocal effect of the short-range interaction in a realistic model, reproducing all the two-nucleon properties and yet based on the naive three-quark structure of the nucleons. The calculations are carried out by the 15-point Gaussian nonlocal potential constructed from fss2, which is accurate enough to reproduce the converged triton binding energy of fss2 with the accuracy of 15 keV and the phase shift parameters with the difference of less than .[15, 33] The potential keeps all the nonlocal effects of the original fss2, including the energy-dependent term of the QM resonating-group method (RGM). This energy dependence is eliminated by the standard off-shell transformation utilizing the square root of the normalization kernel for two three-quark clusters. It is extremely important to deal with this energy dependence properly, since an extra nonlocal kernel from this procedure is crucial to reproduce all the elastic scattering observables below MeV.[10, 11]

In this paper, we have studied the neutron-induced deuteron breakup differential cross sections for the incident energies MeV, and compared them with available experimental data and the predictions by meson-exchange potentials. We have found that our calculations reproduce almost all the results for the breakup differential cross sections predicted by the meson-exchange potentials, including the disagreement with the experiment. This feature is probably related with the structure of the direct breakup amplitudes in Eqs. (2.18) and (2.19). First, they are constrained by the elastic scattering amplitudes, , in the initial stage. In the final stage of reactions, only the half-off shell two-nucleon -matrix appears owing to the energy conservation for outgoing nucleons. The effect of the completely off-shell -matrix therefore appears only at the stage of solving the basic AGS equations for , for which the present investigations imply that the difference between our QM interaction and the meson-exchange potentials is rather minor. On the whole, the agreement with the experimental data is fair, but there exist some discrepancies in certain particular kinematical configurations, which are commonly found both for our predictions and for meson-exchange predictions. In particular, the space star anomaly of 13 MeV scattering is not improved even in our model. There are severe disagreement of breakup differential cross sections in some of the non-standard configurations. In our model, some overestimations of cross sections are found at the energy MeV. Since these large disagreements can be resolved neither with the Coulomb effect nor by the introduction of the force, systematic studies from more basic viewpoints for the interaction are still needed both experimentally and theoretically.

In spite of the apparent disagreement between the theory and the experiment in some of the breakup differential cross sections, our QM baryon-baryon interaction fss2 was very successful to reproduce almost all other experimental data of the three-nucleon system without reinforcing it with the three-body force. These include: 1) a nearly correct binding energy of the triton,[13] 2) reproduction of the doublet and quartet -wave scattering lengths, and ,[15] 3) not too small differential cross sections of the elastic scattering up to MeV at the diffraction minimum points,[10] 4) the improved maximum height of the nucleon analyzing power in the low-energy region MeV, although not sufficient,[11] and 5) the breakup differential cross sections with many kinematical configurations, discussed in this paper. Many of these improvements are related to the sufficiently attractive interaction in the channel, in which the strong distortion effect of the deuteron is very sensitive to the treatment of the short-range repulsion of the interaction. In our QM interaction, this part is described by the quark exchange kernel of the color-magnetic term of the quark-quark interaction. In the strangeness sector involving the and interactions, the effect of the Pauli repulsion on the quark level appears in some baryon channels. It is therefore interesting to study -deuteron scattering in the present framework, to find the repulsive effect directly related to the quark degree of freedom. Such an experiment is planned at the J-PARC facility.[34]

Acknowledgements

The authors would like to thank Professors K. Miyagawa, H. Witała, H. Kamada and S. Ishikawa for many useful comments. We also thank Professor K. Sagara for useful information on the space-star anomaly, which was obtained through the systematic experiments carried out by the Kyushu university group. This work was supported by the Grant-in-Aid for Scientific Research on Priority Areas (Grant No. 20028003), and by the Grant-in-Aid for the Global COE Program “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. It was also supported by the core-stage backup subsidies of Kyoto University. The numerical calculations were carried out on Altix3700 BX2 at YITP in Kyoto University and on the high performance computing system, Intel Xeon X5680, at RCNP in Osaka University.

A Three-nucleon breakup kinematics

In this appendix, we will discuss the breakup kinematics used in this paper. We choose the standard set of Jacobi coordinate in the momentum space as and set

(A.1)

with being the momentum coordinate of the particle in the lab system. We also choose the -axis as the direction of the incident particle and assume that the magnitude of the incident momentum is in the cm system. This implies that the incident momentum is always and the incident energy is , either the nucleon or the deuteron is the incident particle. Here, is the unit vector of the -axis. In the lab system, the incident momentum and the energy are given by

(A.2)

In the following, all quantities in the initial state are expressed by and , which are determined solely by .

In the experimental setup to detect two particles 1 and 2, the three-particle breakup configurations are uniquely specified with two polar angles , , and a difference of azimuthal angles , in addition to the energy discussed below. These angles are defined through and