DIRAC PHENOMENOLOGY

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DIRAC PHENOMENOLOGY

J. Tjon

To cite this version:

J. Tjon. DIRAC PHENOMENOLOGY. Journal de Physique Colloques, 1990, 51 (C6), pp.C6-111-

C6-123. �10.1051/jphyscol:1990609�. �jpa-00230872�

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Colloque C 6 , supplgment au n022, T o m e 51, 15 novembre 1990

DIRAC PHENOMENOLOGY

J.A. TJON

Institute for Theoretical Physics, University ^of Utrecht. ^NZ-3508 T A

Utrecht, The Netherlands

Resume

-

L'approche relativiste de la diffusion de proton basee sur l'interaction B deux nucleons est revue. En utilisant les models d'echange d'un boson, le potential optique de Dirac est construit dans l'approximation d'impulsion relativiste. Ce potential presente une douce dependence en energie et contient beaucoup moins de potentiels scalaire et vector du champ fort que ceux obtenus par la phenom6nologie de Dirac. D'autres effets tel que le blocking de Pauli ainsi que les contributions de la polarization du vide sont necessaires B la bonne description des observables de diffusion elastique de proton sur llenti&re region d'energie intermediare. Des resultats recents d'effets hors-couche sont discutCs.

Abstract

-

A relativistic approach to proton scattering based on the two-nucleon interaction is reviewed. Using relativistic one boson exchange models, the Dirac optical potential is constructed in the relativistic impulse approximation. It exhibits a smooth energy dependence and has considerably less strong scalar and vector potentials than obtained from Dirac phenomenology. Additional effects like vacuum polarization contributions and Pauli blocking effects are needed to provide a good description of the elastic proton observables over the entire medium energy region. Recent results on off-shell effects are discussed.

In the past decade considerable attention has been paid to the role of relativity in nuclear systems. Both in nuclear structure and reactions it has been suggested that relativistic effects like the contributions from the so-called 2-graph are significant. Interesting results have been obtained within a relativistic mean field approach of a renormalizable field theory of hadrons and mesons 111. Furthermore, using the Dirac equation to describe elastic proton nucleus scattering remarkably good agreement is found with the experimental polariza- tion data, employing phenomenological Dirac optical potentials of a scalar and vector type.

Recent examples can be found in ref. 121. Here I would like to review some of the work which has been carried out with the Maryland group to see whether there is a possible microscopic description of the Dirac analysis in terms of an underlying meson theory. Due to the complexity of the many-body problem of the nucleus, we have to rely on approximations to determine the elastic scattering process of a nucleon on a nucleus.

A basic assumption which can be made is that the proton is described as a relativistic spin 112 particle and that its motion is governed by a Dirac equation

D i rac

[ - E 7 0 + P 7 + m + U l f = O ,

opt

where E is the energy of the incoming proton and p its three momentum. To have a microscopic description we may attempt to determine the Dirac optical potential in terms of the basic NN interaction. According to the nonrelativistic multiple scattering theory of Kerman, McManus and Thaler ( K M T ) 1 3 1 , the impulse approximation, shown schematically in fig. 1, can be used to calculate the optical potential for intermediate energies. Assuming that this also holds for the relativistic case, we have for the case of a spin zero nucleus 141

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990609

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where q=p-p' is the momentum transfer and r is a known kinematical factor.

Fig. 1

-

The diagrammatic representation of the first-order optical potential.

Furthermore, ~ N N is the off shell energy-dependent NN scattering amplitude and

3,

^{are the}

Dirac orbitals, describing the occupied single-particle states in the nucleus. In eq. (2) a Dirac spin trace is taken over the struck nucleon. In practical calculations it is usually assumed that the NN amplitude varies slowly as compared to the nuclear wave function, so that the dependence of ~ N N on the integration variable k can be neglected. This leads to the so- called tp approximation for the optical potential

w i t h

corresponding to the Fourier transform of the density distribution function of the nucleus.

It is an operator in Dirac space, characterized by three density functions

In most calculations, the wave functions are taken to be the Hartree-Dirac orbitals as

determined by Horatitz and Serot / S / . The calculated vector nuclear form factor p, is well in accordance with the electron scattering data. In fig. 2 the three form factors are shown for the case of 40~a. Although is of the same magnitude as the scalar and vector form factors ps and p, its contribution is small in view of the prefactor q/m.

Fig. 2

-

The nuclear form factors for 40~a. Solid, dashed and dottea curves are the vector, scalar and tensor form factors respectively.

From eq. (3) we see that in order to determine the optical potential we need the NN amplitude in the full Dirac space. A simple and therefore attractive way to determine this operator was suggested by McNeil. Shepard and Wallace / 6 , 7 / . It is well known, that the physical NN scattering process can be characterized by five invariant amplitudes due to P, C and T invariance. As a consequence, one may express the corresponding amplitude through the set of five invariant functions as used in the analysis of the p-decay experiments

w i t h

and where the amplitudes F, are Lorentz scalars. They can, in principle, be determined from NN scattering data or alternatively from the phase parameters obtained in a phase shift analysis. Assuming that the representation holds for the full Dirac space, one can use this to construct the optical potential. The above version of the relativistic impulse

approximation has been called IA1. Its predictions of the analyzing powers of elastic p - 4 0 ~ a scattering at around 500 MeV is well in accordance with the experimental data.

A technical but important ingredient in their analysis is that they explicitly make use of the covariant structure of the NN amplitudes to arrive at the optical potential in the laboratory system. Besides the boost transformations occurring in the arguments of the invariant

amplitudes also. Wigner rotations are therefore present in the cesulting NN matrix elements in the laboratory system of the nucleus. In the calculation, the NN amplitudes are furthermore

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assumed to be only dependent on the momentum transfer q and the laboratory energy Elab. i.e.

the initial and final momentum of the proton is assumed to be on the energy shell (Ipl=lp'l and ~ ~ ~ ~ = ( ~ ~ + m ~ ) ~ 1 ~ ) . Because of the above approximation, only the on-shell value of the NN t-matrix is needed to calculate the optical potential. When the momentum dependent pieces of the spinors are neglected this results in a local optical potential, so that the Dirac equation for the proton can be solved in coordinate space. The Dirac optical potential constructed in this way has large scalar and vector components similar to that found in the Dirac phenomenological analysis. At 5 0 0 MeV laboratory energy they are of the order of 5 0 0

MeV in strength, to be compared with typically 5 0 MeV for the nuclear interaction. Further- more, they exhibit strong energy dependence as s h a m in fig. 3 for the potentials at r=O.

Fig. 3

-

Real part of the scalar and vector potentials for 4 0 ~ a at r=O. Solid line corresponds to the IAl optical potential, while crosses are the IA2 results.

Although the representation (6) is obviously a rather natural one there is no theoretical ground for its validity. In particular, the operators ( 7 ) do not form a complete set and as a result the /?-decay representation leads to ambiguities in the full Dirac space l8.91. An analysis of the complete representation for the N N amplitude has been carried out using parity, time reversal and charge conjugation invariance of the strong interaction 1101. The result is that 56 operators are needed, considerably more than the 5 Fermi covariants in IAl.

Clearly, apart from the typical N N process, more is needed to uniquely determine this large set of invariant amplitudes. This can be done by having an explicit relativistic dynamical model for the N N interaction, including the negative energy spinor states.

Such a model has been studied by us to descr5be the NN interaction at medium energies 1111.

It consists of meson exchanges to describe the nuclear force within the framework of the field theoretical Bethe Salpeter equation. Furthermore, the two nucleon channels are assumed to be coupled to NA and AA states. Figure 4 shows the diagrammatic representation of the dynamical equations for the model. The standard set of r , U , W , p , )1 and 6 mesons is used for the direct interaction in the N N channel, while the transition interaction to NA and

AA

states is taken to be given by the one pion and p exchange contribution. Within the framework of these dynamical equations, including also the negative energy spinor states in the N N channel, no reasonable description could be found using a pure pseudoscalar r N interaction, due to the too strong coupling between the positive and negative energy states. Using on the other hand a pseudo vector coupling for the pion, the NN phases could reasonably well be reproduced. In

Fig. 4

-

Diagrammatic representation of the relativistic one-boson-exchange model with isobars.

all these calculations a strong meson nucleon form factor was used of a monopole type with a cutoff mass of typically 1.3 nucleon mass. In the actual construction of the Dirac optical potential, the relativistic quasipotential version of the model was used. The resulting IA2 Dirac optical potential 141 calculated with the complete NN representation as derived from the dynamical model is more complicated and has six components, the maximal number allowed by time reversal invariance. It has the form

where K = (p+p8)/2. Although the most important components are the scalar and vector ones, the other terms in eq. (8) are also important to get a proper description of the elastic proton scattering observables. As can be seen in fig.. 3 the strengths of S and V are considerably less than found in the IA1. Furthermore, they exhibit a much smoother energy behavior. The strong energy dependence in the IA1 potential could be traced back to the treatment of the pion using the Fermi covariants 1121. Extending the NN amplitude to the full Dirac space using the representation ( 6 ) tends to favor 75 coupling for the pion in view of the pseudoscalar operator

~ = 7 ~ ~ 7 ~ ~ .

With the optical potential (8) the angular momentum reduced Dirac equation can be solved in coordinate space including the Coulomb interaction of the proton with the nucleus. As an example, the resulting elastic cross section and polarization observables are shown in fig. 5 for the case of 4 O ~ a at 200 MeV laboratory energy. As can be seen, the agreement of the IA2 is substantially better than the IAl prediction. Comparing the IA1 and IA2 predictions at higher energies, the experimental data are closer to the IA1. In particular, the first sharp spike in the analyzing power, at some time quoted as the Dirac signature 1131, is not well reproduced and rounded off in the IA2.

One interesting question which can be raised is whether the IA2 results depends on the NN model used. For that purpose, a study was undertaken by Maung et al. 1141, using OBE models with a different quasipotential prescription 1151. In their case, one of the nucleons is put on mass shell. In the model no isobars are present. As effective Lagrangian for the rN interaction a mixed type is assumed of the form

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Fig, 5

-

IA1 (dashed line) and IA2 predictions of the elastic cross section U(@), analyzing power Ay and spin rotation function Q for 4 0 ~ a at 200 MeV as a function of the scattering angle 8 .

In one of the models four mesons are used while A=0.25. In the second one six mesons are taken, together with a pure pseudovector rN coupling (X=O). A X squared fit to experimental data was carried out up to 300 MeV. Although the models are quite different, only small differences are found in the elastic proton scattering observables. In fig. 6 the various model predictions are shown for 4 0 ~ a at 300 MeV. Somewhat larger effects are found at lower energies.

Since the elastic proton scattering observables are quite sensitive to the difference between ps and p,, we may expect an important effect from vacuum polarization contributions. In the quantum hadrodynamics model of Serot and Walecka /l/, filled negative energy states of the Dirac sea are shifted in energy due to the scalar and vector potentials. The energy

correction can be determined in a mean field approximation, and is expressed in terms of the effective nucleon mass M*. From this we may calculate for the finite nucleus the correction to the single particle density function in the local density approximation (LDA)

Fig. 6

-

IA2 predictions for 4 0 ~ a at 300 MeV. The solid and dashed curves correspond to the two NN models with X=0.25 and X=O respectively, while the dashed curve is the model with isobars included.

Another effect which is expected to play an important role at lower energies is the Pauli blocking effect. Following Murdock and Horowitz 1161 we may include this contribution in the LDA by modifying the NN t-matrix by a density dependent factor as calculated for nuclear matter. Both vacuum polarization and Pauli blocking effects have been studied in the IA2.

Since the vacuum polarization in the above phenomenological way may have large uncertainties, the sensitivity of bpvac has been explored by including 0, 50 and 100% of bpvac to ps. The results are shown in fig. 7 for the case of 4 0 ~ a at various energies 1171. From this we see that the agreement with the experimental data improves significantly,, especially at higher energies. For comparison, also shown is the result NRIA1, which is obtained by keeping only the positive energy spinor components in the optical potential. Apart from the relativistic kinematic effects, this corresponds effectively to using the nonrelativistic Schrbdinger equation. This result shows the effect of ne lecting the 2-graph contribution. Calculations with the 162 potential for other nuclei like 160 and 208~b yield predictions well in

accordance with the experiments as can be seen for example in fig. 8 for the case of 500 MeV.

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Fig. 7

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IA2 predictions of elastic p-40~a observables at various energies. Pauli blocking and vacuum polarization contributions to the optical potential are included. Solid, dashed and dotted lines correspond to 0, 50 and 100% of bpvac respectively.

In the studies discussed up to now, the implicit assumption is made that only the on-shell NN amplitudes can be used to determine the Dirac optical potential. Clearly, in principle however, the off-shell NN amplitudes are needed. Recently some.studies were undertaken to see how reliable such an on-shell approximation is. For such a calculation, the momentum space Dirac equation has to be solved. Non-local contributions can also readily be included in such a framework. In fig. 9 are shown the results of an IA2 calculation 1181 where in the tp approximation the full off-shell NN t-matrix has been included together with the IA2 with on- shell amplitudes. We see that the off-shell effects are small. Modifying only the

++

^states

with the off-shell NN amplitude yields also a result very close to the on-shell one. This should be contrasted with the predictions of Elster and Tandy 1101 (see fig. 10), which are based on a KMT type of optical potential analysis.. In the Elster and Tandy calculation some relativity is built in by including the so-called Holler factor 1201.

To understand the apparent discrepancy we have carried out a detailed analysis of the

difference between the two approaches 1211. First of all we may keep only the positive states in the IA2, yielding effectively the KMT optical potential. In this approximation some relativistic effects have been kept which are essentially of a kinematic nature. They follow from the transformation properties due to special relativity. This requires that in addition to the MUller factor also, the Wigner rotation and boost effects in the Mandelstam arguments of the invariant functions have to be included. The effect of turning the Wigner rotation off in our calculations is shown in fig. 11 for the polarization obsenrables. Considering the first dip in the analyzing power, we see that the effect is quite sizable, giving rise to substantially more off-shell sensitivity. The effect from the boost in the arguments of the invariant NN amplitudes is found to be small in this energy region.

1 0 ' - 1 I , (a , 1 I , I

0 500 MeV

103

- -

k 10'

-

V)

I; G

^1°'

-

Y

-

^loo

^-

2

b lV1

-

29 (deg)

10'

PB 500 MeV 10'

Fig. 8

-

^IA2 predictions at 500 MeV for 160 and 208~b. The meaning of the curves are the same as in fig. 7.

Another difference is that the Coulomb interaction is included in an approximate way by Elster and tandy, without essentially worrying about the Coulomb singularity. Ours consists of a modified Vincent and Phatak 1221 method. Instead of a sharp cutoff in coordinate space of the Coulomb interaction, a &shell charge is introduced well outside the strong interaction region, to get a well behaved potential at large distances. The effect of the &shell charge is then eliminated in coordinate space and then subsequently properly matched to the pure Dirac Coulomb wave function in coordinate space to extract the Coulomb modified phases. using the Elster-Tandy way of the Coulomb treatment an additional off-shell sensitivity is found.

As can be seen from fig. 10, the two successive approximations yield a prediction close to that of Elster and Tandy. From this we may conclude that small details and approximations may lead to important differences. In this case due to the neglect of a part of relativistic kinematic effects in the NN amplitude and the Coulomb treatment sizable off-shell sensitivity

is found, whereas a complete analysis leads to virtually no dependence. It should be noted that the effects from the various approximations made are in general not additive in nature.

Considering the on-shell analysis, the above differences in treatment do not lead to the large differences as found in the off-shell case.

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CA 200 MeV 1.0

- 8

J \ /

m-.z

1 1 1 1 1 1 1 1 1

5 10 l5 20 25 30 35 40 45 50

Fig. 9

-

^Off-shell IA2 predictions at 200 MeV for 40~a. Solid and dotted lines are the on- shell and off-shell results respectively. The dashed line shows the off-shell effect of only modifying the ++ states.

Fig. 10

-

Off-shell effects in a KMT approach (from ref. 1191). Solid and dashed curves are the off- and on-shell results respectively.

Another important approximation which has been made in the above calculations is the

replacement of the full folding expression of the impulse approximation graph by the optimal factorized form. In a strictly nonrelativistic analysis the full folding calculation of the optical potential has been carried out by Arellano et al. 1231 using the Paris potential. In such a calculation, besides carrying out the k loop integral, the additional complication arises that the energy of the off-shell NN t-matrix also depends on k. They find substantial deviations from the optimal approximation results. Some of their calculations are shown in fig. 12. Near the first dip region of the analyzing power we see that the off-shell tp prediction is close to the on-shell result, although some relativistic kinematics have been included like the M8ller factor in the on-shell tp calculation. Doing a full folding

calculation deepens the first dip considerably, so that a good agreement is obtained with the experimental data. In the relativistic approach, this agreement is achieved by the inclusion of the 2-graph. The large full folding sensitivity is in apparent disagreement with the findings of Elster et al. 1241 and Crespo et al. 1251, who examined the validity of the tp approximation in nuclear matter and finite nuclei, respectively. They conclude that the tp approximation is good. The major difference in these calculations with those of Arellano et al. is that they keep the energy of the NN t-matrix fixed. Since the energy within the loop may vary quite a bit this may be the source of discrepancy. Although good agreement is achieved at lower energies, the predictions without 2-graphs at higher energies fail.

Fig. 11

-

Polarization observables at 200 MeV for 4 0 ~ a using two successive approximations in NRIA. Solid curve is the complete off-shell result. Neglecting the Wigner rotation yields the dashed curve, while further doing the Coulomb treatment as in ref. 1191 and dropping the boost in the argument of the NN amplitudes leads to the dotted curve.

In recent years it has become clear that the original microscopic interpretation in terms of the IA1 analysis turns out to be too simple. More detailed calculations, based on meson theoretical models, show that additional effects have to be considered to get satisfying agreement. In particular, vacuum polarization effects are important, due to the sensitivity of the p-nucleus scattering observables on

PS-&.

Also proper treatment of relativistic effects arising from the Lorentz transformation properties of the NN amplitudes and the Coulomb interaction in the optical potential leads over the entire medium energy region to virtually no sensitivity on off-shell behavior. It is clear that an important objective in

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Fig. 12

-

Full folding result (solid curve) in a strictly nonrelativistic treatment at 200 and 300 MeV for 4 0 ~ a (from ref. 1231). Dashed and dotted curves are the tp on- and off-shell results respectively.

these microscopic studies is to examine in how far the Z-graph contributions are crucial in explaining the data over the whole energy range. One aspect is still missing in the

relativistic analysis done up to now, namely a full folding calculation, before any definite conclusions can be drawn. In conclusion, even if we would succeed to explain the data in terms of a microscopic description based on mesons and hadrons as effective degrees of freedom, the role of the negative energy spinor states within a more basic theory as quantum chromodynamics has to be understood. In particular, since the nucleon is a bound system of quarks, an important problem which has to be addressed in the future is the theoretical understanding within a composite system description of the limits of validity of the Dirac phenomenology.

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