THE ROLE OF 3-BODY FORCES IN THE PHOTODISINTEGRATION OF 3-BODY SYSTEMS

Edward L. Tomusiak¹, Victor D. Efros², Winfried Leidemann³ and Giuseppina Orlandini³

1) Department of Physics and Engineering Physics and Saskatchewan Accelerator Laboratory, University of Saskatchewan, Saskatoon, Canada S7N 0W0
2) Russian Research Centre "Kurchatov Institute", Kurchatov Square 1, 123182 Moscow, Russia
3) Dipartimento di Fisica, Università di Trento, I-38050 Povo, Italy and Istituto Nazionale di Fisica Nucleare, Gruppo collegato di Trento

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abstract:

$\textstyle \parbox{15truecm}{\footnotesize \parindent=0pt We describe the Loren... ...ects of 3N forces are illustrated for the case of the UrbanaVIII model. \par }$

1. Introduction

Experimental data on the photodisintegration of the trinucleons has been available for a rather long time. The seventeen year period from 1964 to 1981 saw at least thirteen different publications giving measured cross sections, mainly for ³He, in the low excitation range $\omega$ < 50 MeV. The experimental techniques included bremsstrahlung beams, fixed or variable energy photon beams, and p- $\gamma$ capture. The most recent published data giving total cross sections for ³H and 3-body breakup cross sections for ³He are in the 1981 paper of Faul, Berman, Meyer, and Olson [1]. New data [2] for two-body breakup in both ³He and ³H are presently being analyzed. This data results from a simultaneous measurement of both ³He and ³H using the photon tagger at the Saskatchewan Accelerator Laboratory.

The reason for this effort is to hopefully learn more about NN and, in particular, 3N forces. Clearly the trinucleons are natural laboratories for studying 3N forces just as the deuteron was ( and still is ) for the NN case. A major difference, however, is that whereas in the deuteron case the non-relativistic quantum mechanics is relatively simple, the computational difficulties in the 3N continuum are non trivial. This explains the lag in theoretical progress in this field. Until recently the only quantum mechanically rigorous calculations of trinucleon photodisintegration were those of Barbour and Phillips [3] and of Gibson and Lehman [4] in the 1970's. Although these authors solved the bound and continuum Faddeev equations their dynamical models included only S-state forces through separable interactions. It hardly needs saying that before anything can be inferred about the role of 3N forces we must first be able to perform accurate calculations using realistic modern NN interactions. Such calculations applied to the two-body breakup channels of ³H and ⁴He have been reported by Sandhas el al [5]. That work employed separable representations of various modern potential models in order to solve the Faddeev-type equations. Here, however, we are interested in the total absorption cross sections which require the three-body break-up cross sections as well. On the surface it would appear that such calculations would naturally be more difficult than those just for two-body breakup. However a theoretical breakthrough in this regard was made my Efros, Leidemann,and Orlandini [6], henceforth referred to as ELO , who demonstrated the feasibility and accuracy of the Lorentz integral transform technique for the calculation of inclusive cross sections. In this talk I will outline their method and show results for what are in fact the first theoretical results for trinucleon photodisintegration using modern realistic potentials including a three-body force.

2. Method

The response $\cal {R}$ ( $\omega$ ) of a system to an external probe $\cal {O}$ is given by

$\displaystyle \cal {R}$ ( $\displaystyle \omega$ )	=	< $\displaystyle \Psi_{0}^{}$ \| $\displaystyle \cal {O}$ $\displaystyle \delta$ (H - E₀ - $\displaystyle \omega$ ) $\displaystyle \cal {O}$ \| $\displaystyle \Psi_{0}^{}$ >
	=	$\displaystyle \int$ dE \| < E\| $\displaystyle \cal {O}$ \| $\displaystyle \Psi_{0}^{}$ > \|² $\displaystyle \delta$ (E - E₀ - $\displaystyle \omega$ )
	+	$\displaystyle \sum_{b}^{}$ \| < b\| $\displaystyle \cal {O}$ \| $\displaystyle \Psi_{0}^{}$ > \|² $\displaystyle \delta$ (E_b - E₀ - $\displaystyle \omega$ ) .	(1)

This would appear to require a knowledge of all continuum and bound state wavefunctions. For simplicity in presentation we assume that the only bound state is the ground state | $\Psi_{0}^{}$ > and that the matrix element < $\Psi_{0}^{}$ | $\cal {O}$ | $\Psi_{0}^{}$ > vanishes. To sidestep the calculation of continuum wavefunctions one requires a method which will allow closure to be invoked. This is accomplished by taking a transform of $\cal {R}$ ( $\omega$ ) with some kernel $\cal {K}$ ( $\sigma$ , $\omega$ ) i.e.

$\displaystyle \Phi$ ( $\displaystyle \sigma$ )/TD>	=	$\displaystyle \int$ $\displaystyle \cal {K}$ ( $\displaystyle \sigma$ , $\displaystyle \omega$ ) $\displaystyle \cal {R}$ ( $\displaystyle \omega$ ) d $\displaystyle \omega$
	=	< $\displaystyle \Psi_{0}^{}$ \| $\displaystyle \cal {O}$ $\displaystyle \cal {K}$ ( $\displaystyle \sigma$ , H - E₀) $\displaystyle \cal {O}$ \| $\displaystyle \Psi_{0}^{}$ > .	(2)

For example the choice $\cal {K}$ = ( $\omega$ + $\sigma$ )^-1 where $\sigma$ > 0 is known as a Stieltjes transform. It would allow $\Phi$ ( $\sigma$ ) to be calculated as

$\displaystyle \Phi$ ( $\displaystyle \sigma$ ) = < $\displaystyle \Psi_{0}^{}$ | $\displaystyle \cal {O}$ | $\displaystyle \tilde{\Psi}$ ( $\displaystyle \sigma$ ) >

(3)

where

( H - E₀ + $\displaystyle \sigma$ )| $\displaystyle \tilde{\Psi}$ ( $\displaystyle \sigma$ ) > = $\displaystyle \cal {O}$ | $\displaystyle \Psi_{0}^{}$ > .

(4)

It has been shown [7] however that from a practical point of view this transform is unstable with respect to the inversion required to extract $\cal {R}$ ( $\omega$ ). However in a later publication [8] ELO showed that stability problems are reduced substantially through the use of the Lorentz transform with kernel

$\displaystyle \cal {K}$ ( $\displaystyle \sigma$ , $\displaystyle \omega$ ) = [( $\displaystyle \omega$ - $\displaystyle \sigma_{R}^{}$ )² + $\displaystyle \sigma_{I}^{2}$ ]^-1 .

(5)

With this kernel the transform is given by

$\displaystyle \Phi$ ( $\displaystyle \sigma$ ) = < $\displaystyle \tilde{\Psi}$ ( $\displaystyle \sigma$ )| $\displaystyle \tilde{\Psi}$ ( $\displaystyle \sigma$ ) >

(6)

where

( H - E₀ - $\displaystyle \sigma_{R}^{}$ + i $\displaystyle \sigma_{I}^{}$ )| $\displaystyle \tilde{\Psi}$ ( $\displaystyle \sigma$ ) > = $\displaystyle \cal {O}$ | $\displaystyle \Psi_{0}^{}$ > .

(7)

The crucial point here from a computational point of view is that $\tilde{\Psi}$ ( $\sigma$ ) is square integrable so that numerical solutions to Eq.(7) can be obtained using modified bound state techniques. One computes $\tilde{\Psi}$ ( $\sigma$ ) for a series of values of $\sigma_{R}^{}$ and then inverts to obtain $\cal {R}$ ( $\omega$ ). Details of the inversion technique can be found in [8].

The technique we have adopted for solving the 3N bound state problem is to expand the bound state in a set of correlated hyperspherical harmonics (CHH) according to

$\displaystyle \Psi$ = $\displaystyle \tilde{\omega}$ $\displaystyle \sum$ c_i $\displaystyle \phi_{i}^{}$

(8)

where the $\phi_{i}^{}$ are a totally antisymmetric basis set constructed from a spatial part $\chi_{i,\mu}^{}$ and a spin-isospin part $\theta_{\mu}^{}$

$\displaystyle \phi_{i}^{}$ = $\displaystyle \sum_{\mu}^{}$ $\displaystyle \chi_{i,\mu}^{}$ $\displaystyle \theta_{\mu}^{}$ .

(9)

The correlation operator $\tilde{\omega}$ is introduced to improve convergence. For simple potential models without short-range repulsion is was sufficient to take $\tilde{\omega}$ in the state independent form [9]

$\displaystyle \tilde{\omega}$ = $\displaystyle \prod_{i<j}^{}$ f (r_ij) .

(10)

However modern NN interactions contain strong state-dependent repulsion at small NN separations. For these cases we use a state dependent correlation operator of the form

$\displaystyle \tilde{\omega}$ = $\displaystyle \cal {S}$ $\displaystyle \prod_{i<j}^{}$ $\displaystyle \sum_{S,T}^{}$ f_ST(r_ij) $\displaystyle \tilde{P}_{ST}^{}$ (ij)

(11)

where P_ST(ij) are projection operators onto nucleon pairs (ij) with spin S and isospin T and where $\cal {S}$ is a particle symmetrization operator. This form of correlation operator is easily incorporated into the calculations by first constructing the set of coefficients < $\theta_{\mu^\prime}^{}$ | $\tilde{\omega}$ | $\theta_{\mu}^{}$ > and then writing

$\displaystyle \Psi$ = $\displaystyle \sum_{i}^{}$ $\displaystyle \tilde{\chi}_{i,\mu^\prime}^{}$ $\displaystyle \theta_{\mu^\prime}^{}$

(12)

where

$\displaystyle \tilde{\chi}_{i,\mu^\prime}^{}$ = $\displaystyle \sum_{\mu}^{}$ < $\displaystyle \theta_{\mu^\prime}^{}$ | $\displaystyle \tilde{\omega}$ | $\displaystyle \theta_{\mu}^{}$ > $\displaystyle \chi_{i,\mu}^{}$ .

(13)

With this form of correlation it follows that if only the (ij) pair are within the healing distance then

$\displaystyle \Psi$ $\displaystyle \approx$ [ $\displaystyle \sum_{ST}^{}$ f_ST(r_ij) $\displaystyle \tilde{P}_{ST}^{}$ (ij)] $\displaystyle \sum_{i}^{}$ c_i $\displaystyle \phi_{i}^{}$

(14)

as required. The correlation functions f_ST(r) are chosen as follows. For r < r₀ , the healing distance, f_ST(r) is chosen to be the zero energy pair wave function in the corresponding ST state. Healing is insured by imposing the conditions f_ST(r)=1 for r > r₀ and f_ST(r₀)=0. The ST= 13 and 31 cases are determined from the ¹S₀ and ³S₁ partial waves of the NN potential. However for the ST=11 and 33 cases the ¹P₁ and ³P₁ potentials are not sufficiently attractive to obtain a healing distance. Therefore we introduce an additional intermediate range central interaction such that a healing distance of 10 fm is obtained.

**Figure 1:** State-dependent correlation functions g_ST(r) for ST=00,01,10, and 11.
$\begin{figure} \begin{center}\leavevmode \epsfxsize 12.cm \epsffile{edfig1.eps} \end{center}\end{figure}$

The presence of other nucleons in a nucleus alters the free correlations of a nucleon pair. We try to incorporate such effects by the introduction of a scaling factor $\alpha_{ST}^{}$ for the argument of the correlation function, i.e. instead of f_ST(r) we take as new correlation function g_ST(r) = f_ST( $\alpha_{ST}^{}$ r). The scaling factors are fitted in order to obtain the highest possible binding energy for the ground state taking an expansion of the ground-state wave function up to K = 10. The fits lead to modified healing distances of r₀/ $\alpha$ . For ST = 13, 31 we find $\alpha$ -values rather close to 1, while there are larger effects for the other two channels ( $\alpha_{33}^{}$ $\simeq$ 1.35, $\alpha_{11}^{}$ $\simeq$ 0.5). These correlations functions are illustrated in Fig. 1.

3. Results and Discussion

We calculate the total photoabsorption cross section for energies below pion threshold by using the unretarded dipole operator

$\displaystyle \vec{D}\,$ = $\displaystyle \sum_{i=1}^{Z}$ ( $\displaystyle \vec{r}_{i}^{}$ - $\displaystyle \vec{R}_{cm}^{}$ ) .

(15)

The total photoabsorption cross section is given in terms of the dipole response function $\cal {R}$ by

$\displaystyle \sigma_{tot}^{}$ = 4 $\displaystyle \pi^{2}_{}$ (e²/ $\displaystyle \hbar$ c)E $\displaystyle \cal {R}$ (E)

(16)

First we review some results obtained earlier with the use of semi-realistic potential models. The Trento (TN) [8] and Malfliet-Tjon (MT) I+III [10] potentials are central even interactions which give a reasonable fit to the ¹S₀ and ³S₁ phase shifts up to meson threshold. Both models give the same binding energies of 8.7 MeV and 8.0 MeV for ³H and ³He respectively. Inclusive photodisintegration cross sections were computed for these potential models by ELO [11].

**Figure 2:** Inclusive ³He photodisintegration cross sections computed from the TN, MT potential models. Also shown as points are results taken from the Gibson-Lehman calculation. Here $\omega$ is the photon energy above threshold.
$\begin{figure} \begin{center}%\leavevmode\epsfxsize 8.5cm \epsffile{edfig2.eps} \end{center}\end{figure}$

Fig. 2 shows the helium cross section for these cases as well as for the earlier calculations of Gibson and Lehman [4]. The Gibson-Lehman results are plotted as points to make the distinction clearer. The differences are probably attributable to the better fit to the phase shifts obtained with the local TN and MT models. Prior to obtaining the above results ELO [12] used the MT and TN potential models to calculate the total photoabsorption cross section for ⁴He. There is a scarcity of data for this process but one can piece together available ( $\gamma$ ,p) and ( $\gamma$ ,n) data to present "data", some of which misses mainly the three- and four-body breakup channels.

**Figure 3:** Total ⁴He photoabsorption cross section with MT and TN potentials. Experimental cross sections: shaded area ref. [18] (photon scattering); sum of partial cross section for total neutron breakup ref. [19] and two-body proton breakup ref. [20] (dotted curve with typical size of the experimental error). Also shown doubled experimental cross sections for two-body proton breakup ref. [21] (open circles) and for two-body neutron breakup ref. [22] (triangles) and ref. [23] (full circles).
$\begin{figure} \begin{center} \epsfig{file=edfig3.eps,width=9.5cm}\end{center}\end{figure}$

Fig. 3 shows their results, which appear to track the data well until the three-body breakup threshold is reached. The implication of the disagreement is unclear but does invite further theoretical and experimental work.

We have moved beyond the central semi-realistic potential models to include more realistic models containing both short range repulsion and non-central terms. At present these are the TRSB [13] soft-core potential model and the hard-core Argonne AV14 model [14]. Whereas both these potential models give a good account of two-nucleon properties, they underbind the triton by 1.0 MeV in the case of the TRSB and 0.8 MeV in the case of the AV14. A 3N force, the UrbanaVIII model [15] has been constructed to provide this extra binding when taken together with the AV14 model. The results quoted below for the TRSB potential have been published by ELO [8]. Ground state properties computed with these three potential models are shown in Table 1.

**Table 1:** ³H Ground state properties for TRSB, AV14, and AV14 + UrbVIII potential models.
Potential	E_B (MeV)	$\sqrt{<r^2>}$ (fm)	P_D	TRK (MeV-mb)
TRSB	7.50	1.65	8.65	60.2
AV14	7.70	1.65	8.84	66.9
AV14+UrbVIII	8.51	1.59	9.38	70.1

One notes from this table the increase of the TRK sum rule for those potentials with stronger short-ranged NN repulsion. As well, the 3N force increases the TRK sum. We shall soon see that this arises from the increased strength at higher energies which results from addition of the 3N force to the NN potential.

**Figure 4:** ³He total photoabsorption cross section with AV14 potential plus URBVIII 3N-force (full curve); points with error bars are experimental data from ref. [15]; the two light grey lines delineate the data of ref. [1].
$\begin{figure} \begin{center}%\leavevmode\epsfxsize 12.cm \epsffile{edfig4.eps} \end{center}\end{figure}$

We display first the inclusive ³He cross section in Fig. 4 for the complete AV14+UrbVIII model. Although the error bars are large it would appear that our calculation tracks more closely with the data of Fetisov el al [16]. The potential model dependence of this calculation is illustrated

**Figure 5:** Peak and tail regions of ³He total photoabsorption cross section for various potential models. Points with error bars are experimental data from ref. [15]; the two light grey lines delineate the data of ref. [1].
$\begin{figure} \begin{center}%\leavevmode\epsfysize 12.cm \epsffile{edfig5.eps} \end{center}\end{figure}$

in Fig. 5 where both the peak and tail regions are illustrated separately. One notes that the effect of the 3N force in the peak region is to lower the peak in the direction of the data. At the higher energy end above 60 MeV the NN potential with the strongest short range repulsion, the AV14, has the most strength. Addition of the 3N force increases this strength by another $\approx$ 15% in the tail region from 60 MeV to meson threshold. This accounts for the increase in the TRK sum rule when the 3N force is added to the AV14 potential. We have yet to complete tests using other NN and 3N potential models. Only then, and when more data is available, will we be able to say whether or not total photoabsorption cross sections require 3N forces for their description. It may be that we will require accurate calculations of exclusive channels , specifically, the three-body breakup channel. For example O'Connell [17] has proposed a set of three-body breakup kinematics which should be sensitive to 3N forces.

In summary the Lorentz transform technique has allowed the computation of cross sections which would have been very difficult by standard means. In addition to using other realistic NN potentials and other 3N forces we are also proceeding with the calculation of the transverse operators including explicit exchange currents. Finally, it appears possible that a slight modification of these methods will allow the calculation of exclusive cross sections as well.

Bibliography

1

D. D. Faul, B. L. Berman, P. Meyer and D. Olson, Phys. Rev. C 24 (1981) 849, and references therein

2

See a description of SAL Experiment 071 by G. Feldman at http://gwis.circ.gwu.edu/ ndl/LightNuclei.htm

3

I.M. Barbour and A.C. Phillips, Phys. Rev. C 1 (1970) 165

4

B.F. Gibson and D.R. Lehman, Phys. Rev. C 11 (1975) 29 and Phys. Rev. C 13 (1976) 477

5

W. Sandhas, W. Schadow, G. Ellerkmann, L.L. Howell, and S.A. Sofianos, Nucl. Phys. A 631 (1998) 210; also see W. Schadow and W. Sandhas, Nucl. Phys. A 631 (1998) 588

6

V. D. Efros, W. Leidemann and G. Orlandini, Phys. Lett. B 338 (1994) 130

7

V. D. Efros, W. Leidemann and G. Orlandini, Few-Body Syst. 14 (1993) 151

8

V. D. Efros, W. Leidemann, and G. Orlandini, Few-Body Syst., in print

9

V. D. Efros, W. Leidemann and G. Orlandini, Phys. Rev. Lett. 78 (1997) 432

10

R. A. Malfliet and J. Tjon, Nucl. Phys. A 127 (1969) 161

11

V. D. Efros, W. Leidemann and G. Orlandini, Phys. Lett. B 408 (1997) 1

12

V. D. Efros, W. Leidemann and G. Orlandini, Phys. Rev. Lett. 78 (1997) 4015

13

R. De Tourreil, B. Rouben and D. W. L. Sprung, Nucl. Phys. A 242 (1975) 465; J. Côte, R. De Tourreil, B. Rouben and D. W. L. Sprung, Nucl. Phys. A 273 (1976) 269

14

R. B. Wiringa, R. A. Smith, and T.L. Ainsworth, Phys. Rev. C 29 (1984) 1207

15

R. B. Wiringa, Phys. Rev. C 43 (1991) 1585

16

V. N. Fetisov, A. N. Gorbunov and A. T. Varfolomeev, Nucl. Phys. A 71 (1965) 305

17

J. O'Connell, Lecture Notes in Physics 260 (1986) 235

18

D. P. Wells, D. S. Dale, R. A. Eisenstein, F. J. Federspiel, M. A. Lucas, K. E. Mellendorf, A. M. Nathan and A. E. O'Neill, Phys. Rev. C 46 (1992) 449

19

B. L. Berman, D. D. Faul, P. Meyer and D. L. Olson, Phys. Rev. C 22 (1980) 2273

20

G. Feldman, M. J. Balbes, L. H. Kramer, J. Z. Williams, H. R. Weller, and D. R. Tilly, Phys. Rev. C 42 (1990) 1167

21

R. Bernabei et al., Phys. Rev. C 38 (1988) 1990

22

L. Ward, D. R. Tilly, D. M. Skopik, N. R. Robertson and H. R. Weller, Phys. Rev. C 24 (1981) 317

23

J. Asai, G. Feldman, R.E.J. Florizone, E.L. Hallin, D.M. Skopik, J.M. Vogt, R.C. Haight and S.M. Sterbenz, Few-Body Syst. Suppl. 7 (1994) 136

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$\displaystyle \cal {R}$ ( $\displaystyle \omega$ )	=	< $\displaystyle \Psi_{0}^{}$ \| $\displaystyle \cal {O}$ $\displaystyle \delta$ (H - E₀ - $\displaystyle \omega$ ) $\displaystyle \cal {O}$ \| $\displaystyle \Psi_{0}^{}$ >
	=	$\displaystyle \int$ dE \| < E\| $\displaystyle \cal {O}$ \| $\displaystyle \Psi_{0}^{}$ > \|² $\displaystyle \delta$ (E - E₀ - $\displaystyle \omega$ )
	+	$\displaystyle \sum_{b}^{}$ \| < b\| $\displaystyle \cal {O}$ \| $\displaystyle \Psi_{0}^{}$ > \|² $\displaystyle \delta$ (E_b - E₀ - $\displaystyle \omega$ ) .	(1)