Nuclear Many-Body and Medium Effects in Nuclear Interactions and Reactions: Proceedings of the Kyudai-Rcnp International Symposium

Proceedings of the Kyudai-RCNP International Symposium Nuclear Many-Body and Medium Effects in Nuclear Interactions an...

Author: K. Hatanaka | T. Noro | K. Sagara | H. Sakaguchi | H. Sakai

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Proceedings of the Kyudai-RCNP International Symposium

Nuclear Many-Body and Medium Effects in Nuclear Interactions and Reactions

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Proceedings of the Kyudai-RCNP International Symposium

Nuclear ManymBody and Medium Effects in Nuclear Interactions and Reactions 25-26 October 2002

Fukuoka, Japan

Edited by K. Hatanaka T. Nor0 K. Sagara

H. Sakaguchi H. Sakai

World Scientific Singapore New Jersey. London Hong Kong

Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224

USA once: Suite 202,1060 Main Street, River Edge, NJ 07661 UK once: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book i s available from the British Library.

NUCLEAR MANY-BODY AND MEDIUM EFFECTS IN NUCLEAR INTERACTIONS AND REACTIONS Proceedings of the Kyudai-RCNPInternational Symposium Copyright 0 2003 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical. including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-362-X

Printed in Singapore by World Scientific Printers (S) Pte Ltd

V

PREFACE

The Kyudai-RCNP International Symposium on Nuclear Many-Body and Medium Effects in Nuclear Interactions and Reactions (MEDIUM02) was held at Kyushu University (Kyudai) from October 25 to 26, 2002. The symposium is hosted by Department of Physics, Kyushu University and jointly sponsored by RCNP, Osaka University and Faculty of Sciences, Kyushu Univerisity as well as the University of Tokyo and Kyoto Univerisity. This is the third meeting of this series following the workshop held in Kyoto in early 1997 and the symposium held in Osaka in late 1998. These symposium are motivated by experimental studies on nuclear medium effects by using protons and light ions at intermediate energies and intended t o give opportunities for lively discussions among experimental and theoretical scientists on relatively limited subjects in nuclear physics. During the symposium, the following topics have been discussed: Nuclear few-body reactions and three-body forces, Nuclear interactions in medium, Medium effects in nuclear reactions, Properties of nuclear medium, Related topics. About 70 people including 10 foreign people from 7 countries participated in the symposium and the number of scientific talks given was 33. The organizers would like t o express their thanks to all the speakers and participants for their excellent talks and fruitful discussions. An acknowledgment is also expressed to the students of Kyushu University and other universities for their help which contributes to the success of this symposium. K. Hatanaka (RCNP, Osaka Univ.) T. Nor0 (Dept. of Phys., Kyushu Univ.) K. Sagara (Dept. of Phys., Kyushu Univ.) H. Sakaguchi (Dept. of Phys., Kyoto Univ.) H. Sakai (Dept. of Phys., Univ. of Tokyo)

vi i

Contents Preface T . Nor0

I. Nuclear Matter Quenching of Weak Interactions in Nucleon Matter S. Cowell and V. R. Pandharipande Pseudoscalar Mesons in Nuclear Medium T. Tatsumi

3 18

II. Pion Condensation, Compressibility, Pionic States Pion Condensation Based on Relativistic Formulation M. Nakano, T. Tatsumi, L. G. Liu, H. Matsuura, T . Nagasawa, K. Makino, N. Noda, H. Kouno and A. Hasegawa

33

Compressional-Mode Giant Resonances in Deformed Nuclei 41 M. Itoh, M. Fujimura, M. Fujiwara, K. Hara, H. P. Yoshida, H. Sakaguchi, M. Uchida, T . Ishikawa, T. Kawabata, T. Murakami, H. Takeda, T . Taki, S. Terashima, N. Tsukahara, Y. Yasuda, M. Yosoi, U. Garg, M. Hedden, B. Kharraja, M. KOSS,B. K. Nayak, S. Zhu, H. Akimune, M. N. Harakeh and M. Volkerts Isoscalar Giant Dipole Resonance and Nuclear Incompressibility 48 M. Uchida, H. Sakaguchi, M. Itoh, M. Yosoi, T . Kawabata, H. Takeda, Y. Yasuda, T . Murakami, T . Ishikawa, T . Taki, N. Tsukahara, S. Terashima, U. Garg, M. Hedden, B. Kharraja, M. KOSS, B. K. Nayak, S. Zhu, M. Fujiwara, H. Fujimura, K. Hara, E. Obayashi, H. P. Yoshida, H. Akimune, M. N. Harakeh and M. Volkerts High Resolution Study of Pionic 0- States in l 6 0 T. Wakasa, G. P. A. Berg, H. Fujimura, K. Fujita, K. Hatanaka, M. Itoh, J. Kamiya, Y. Kitamura, E. Obayashi, N. Sakamoto, Y. Sakemi, Y. Shimizu and H. P. Yoshida, H. Sakaguchi, H. Takeda, M. Uchida, Y. Yasuda, M. Yosoi and T . Kawabata

53

viii

IU. Relativistic Effects, Dibaryon, NN Interactions Nuclear Medium Effects on the Coulomb Sum Values

63

T. Suzuki and H. Kurasawa Y-Scaling Analysis of the Deuteron within the Light-Front Dynamics Method

71

M. K. Gaidarov, M. V. Ivanov and A. N. Antonov

Search for Super-Narrow Dibaryon Resonances by the pd -+ p d X and pd -+p p X Reactions

78

A. Tamii, M. Hatano, H. Kato, Y. Maeda, T. Saito, H. Sakai, S. Sakoda, N. Uchigashima, K. Hatanaka, D. Hirooka, J. Kamiya, T. Wakasa, K. Sekiguchi, T. Uesaka and K. Yak0 A Quark-Model NN Interaction and its Application to the Three-Nucleon and Nuclear-Matter Problems

84

Y. Fujiwara, M. Kohno, Y. Suzuki, C . Nakamoto, K. Miyagawa and H. Nemura

N. Few Body System Quark Mass Dependence of the Nuclear Forces

99

E. Epelbaum, W. Glockle and U. G. Meissner Different Types of Discrepancies in 3N Systems

107

K. Sagara

Polarization Transfer Measurement for Deutron-Proton Scattering and Three Nucleon Force Effects

115

K. Sekiguchi, H. Sakai, H. Okamura, A. Tamii, T . Uesaka, K. Suda, N. Sakamoto, T. Wakasa, Y. Satou, T. Ohnishi, K. Yako, S. Sakoda, H. Kato, Y. Maeda, M. Hatano, J. Nishikawa, T . Saito, N. Uchigashima, N. Kalantar-Nayestanaki and K. Ermisch

Search for Three Nucleon Force Effects in pd Elastic Scattering at 250MeV Y. Shimizu, K. Hatanaka, Y. Sakemi, T . Wakasa, H. P. Yoshida, J. Kamiya, T. Saito, H. Sakai, A. Tamii, K. Sekiguchi, K. Yako, Y. Maeda, T. Noro, K. Sagara and V. P. Ladygin

120

ix

Study of Three-Nucleon-Force via Neutron-Deuteron Elastic Scattering at 250MeV 125

Y. Maeda, H. Sakai, A. Tamii, S. Sakoda, H. Kato, M. Hatano, T . Saito, N. Uchigashima, H. Kuboki, K. Hatanaka, Y. Sakemi, T . Wakasa, J. Kamiya, D. Hirooka, Y. Shimizu, Y. Kitamura, K. Fujita, N. Sakamoto, H. Okamura, K. Suda, T. Ikeda, K. Itoh, K. Yako, K. Sekiguchi, M. B. Greenfield and H. Kamada Search for Anomaly around Space Star Configuration in pd Reaction

130

T . Ishida, T. Yagita, S. Ochi, S. Nozoe, K. Tsuruta, F. Nakamura and K. Sagara

V. Four Body and 3He Scattering Backward Elastic P - ~ H Scattring ~ at Intermediate Energies

137

Yu. N. Uzikov Study of the pf3He Backward Elastic Scattering

149

K. Hatanaka, Y. Sakemi, T . Wakasa, H. P. Yoshida, J. Kamiya, Y.Shimizu, H. Okamura, T . Uesaka, K. Suda, H. Ueno, K. Sagara, T. Ishida, S. Ishikawa, M. Tanifuji, A. P. Kobushkin and E. A. Strokovsky Polarization Observables in the 4N Scattering with the 3N Calculations

157

H. Kamada Faddeev-Yakubovsky Calculation above 4N Break-up Threshold

164

E. Uzu, H. Kamada and Y. Koike Study of the Spin Structure of 3He(3H)via dd 3Hen(3Hp)Reactionat Intermediate Energies

T . Saito, M. Hatano, H. Kato, Y. Maeda, H. Sakai, S. Sakoda, A. Tamii, N. Uchigashima, V. P. Ladygin, A. Yu. Isupov, N. B. Ladygina, A. I. Malakhov, S. G. Reznikov, T . Uesaka, K. Yako, T . Ohnishi, N. Sakamoto, K. Sekiguchi, H. Kumasaka, J. Nishikawa, H. Okamura, K. Suda and R. Suzuki

169

X

Study of the Spin Dependent 3He-Nuc€eusInteraction at 450MeV J. Kamiya, K. Hatanaka, Y. Sakemi, T. Wakasa, H. P. Yoshida, E. Obayashi, K. Hara, K. Kitamura, Y. Shirnizu, K. Fujita, N. Sakamoto, Y. Shimbara, T. Adachi, H. Sakaguchi, M. Yosoi, M. Uchida, Y. Yasuda, T. Kawabata and T. Nor0

174

VI. Nuclear Correlations Properties of Nucleons and their Interaction in the Nuclear Medium W. H. Dickoff

181

Determination of the Gamow-Teller Quenching Factor via the ''Zr(n,p) Reaction at 293 MeV 193 K. Yako, H. Sakai, M. Hatano, H. Kato, Y. Maeda, T. Saito, K. Sekiguchi, A. Tamii, N. Uchigashima, K. Hatanaka, J. Kamiya, Y. Kitamura, Y. Sakemi, Y. Shimizu, T. Wakasa, H. Okamura, K. Suda, M. B. Greenfield, C. L. Morris and J. Rapaport Gamov-Teller Sum Rule with the A Isobar M. Ichimura

198

Two-step Effects in Analysis of Nuclear Responses Y. Nakaoka, T. Wakasa and M. Ichimura

203

VII. Quasi-free scattering Relativistic Predictions of Spin Observables for Exclusive Proton Knockout Reactions 213 G. C. Hillhouse, J. Mano, R. Neveling, A. A. Cowley, S. M. Wyngaardt, K. Hatanaka, T. Nor0 and B. I. S.Van Der Ventel Study of In-Medium NN Interactions by using (p,2p) Reactions T. Noro, H. Sakaguchi, 0. V. Miklukho, S. L. Belostotski, K. Hatanaka, J. Kamiya, A. U. Kisselef, H. Takeda, T. Wakasa, Y. Yasuda and H. P. Yoshida Dependence of the Complete Set of Spin Transfer Coefficients on Effective Interaction in Nuclear Medium K. Ogata, M. Kawai, Y. Watanabe, Sun Weili and M. Kohno

223

231

xi

Vm.Nucleon-Nucleon Interactions and Medium Effects Nucleus-Hydrogen Scattering - A Probe of Neutron Matter K. Amos

241

( p , p ’ ) Reactions below 350MeV: A Tool to Study the Effective Interaction in the Nuclear Medium E. J. Stephenson

253

Probing Medium Effects on the Nucleon-Nucleon Interaction in Nuclear Matter and Nuclei F. Sammarruca

261

Extraction of Neutron Density Distributions from Proton Elastic Scattering at Intermediate Energies H. Takeda, H. Sakaguchi, S. Terashima, T. Taki, M. Yosoi, M. Itoh, T. Kawabata, T. Ishikawa, M. Uchida, N. Tsukahara, Y. Yasuda, T. Noro, M. Yoshimura, H. Fujimura, H. P. Yoshida, E. Obayashi, A. Tamii and H. Akimune

269

Program

275

List of Participants

279


I. NUCLEAR MATTER


3

QUENCHING OF WEAK INTERACTIONS IN NUCLEON MATTER *

S. COWELL AND V. R. PANDHARIPANDE Department of Physics, University of Illinois at Urbana-Champaign, 1110 W . Green St., Urbana, IL 61801, U.S.A.

1. Introduction

Weak interactions in nucleon matter occur during the beta-decay of nuclei, electron and muon capture reactions, neutrino-nucleus scattering and in various astrophysical environments, such as evolving stars, neutron stars and supernovae Recently there has been much interest in weak interactions in the sun 'i2, those of and l6O due to their use in neutrino detectors searching for neutrino oscillations and in interactions of neutrinos with dense matter in neutron stars and supernovae '. The work presented here is within the context of neutron stars and supernovae. Most of the gravitational energy released in core-collapse supernovae is carried away by neutrinos. Therefore, accurate calculations of neutrino interactions with matter are essential for supernovae simulations and neutron star cooling models. Current calculations of neutrino interactions utilize effective interactions between nucleons and bare weak interaction operators within the framework of the shell model, random phase approximation (RPA), etc. Typically the calculated rate of weak interactions is larger than observed; for example, a factor of 0.6 brings the calculated pf-shell GT transition rates in agreement with experiment Similarly, continuum RPA calculations lo of neutrino interactions with I2C require an overall scaling of 0.7 to bring them into agreement with experimental results. In contrast, using quantum Monte Carlo or Faddeev methods to calculate nuclear wave functions from realistic models of bare nuclear forces, and bare weak operators, the beta-decay matrix elements have been calculated for light nuclei with A 5 7 l1>l2.The calculated values for 3H, 6He and 39495,6,

-

899.

-

'This work has been partially supported by the US NSF via grant PHY 00-98353.

4

7Be are within 5 % of the observed, and better agreement is obtained after including weak pair currents. The weak muon capture by 3He has also been calculated l 3 with realistic wave functions with similar success. However, quantum Monte Carlo calculations are possible only in A 5 12 nuclei. Alternative methods need to be developed for calculating weak transition rates in larger nuclei, neutron stars and supernovae from realistic interactions. Some of the current methods incorrectly use bare weak operators with effective nuclear interactions. When one uses a strong Hamiltonian H e f f containing effective interactions, the basis states of the calculation, [ * I ) have been transformed to model states I @ I ) such that

One must then use an effective weak operator defined such that:

The effective strong and weak interactions depend upon the chosen transformation and must be calculated consistently. There are several ways to obtain consistent sets of effective operators and interactions starting from the bare. For example, one can introduce a model space and employ the Lee-Suzuki similarity transformation l4 as in the no core shell model approach 15. In the present work we use the correlated basis (CB) approach l 6 , l 7 ,evolved out of variational theories of quantum liquids 18. A brief description of CB theory is presented in Sections 2 and 3. Matrix elements of operators between CB states are generally calculated using cluster expansions 19. We begin with the simplest, lowest order two-nucleon cluster approximation to study the general properties of the weak onebody effective operators and of the two-body strong interactions in CB for nucleon matter at densities p = 0.08, 0.16 and 0.24 fm-3 and for proton fraction x p = p p / p = 0.2, 0.3, 0.4 and 0.5. We study the density, proton fraction and momentum dependence of the operators and the interactions. The effective one-body Fermi (F), Gamow-Teller (GT) and neutral current operators have been calculated 2o and their results are discussed in Sects. 4 through 6. The results for the CB two-nucleon interaction are presented in Sect. 7, and we conclude in Sect. 8.

5 2. Correlated Basis Theory The correlated states are defined as: i<j

where l a x ) are uncorrelated Fermi gas or shell model states and Fij are pair correlation operators. Here we concentrate on weak interactions in nucleon matter and therefore use Fermi gas states, I@x).The SII denotes a symmetrized product necessary because the Fij do not commute. The states I X ) are generally not orthonormal; they have to be orthonormalized, I X ) + orthonormal I X ) , through a series of Lowdin and Schmidt transformations l7 preserving the diagonal matrix elements of the Hamiltonian. (XI H IX) = (XlHlX)/(XlX) = (@XI -&vq p g ? ( i j )

c

+ i ~

= atk

t X ~ % x 1~'1)

F N The .

,

(13)

neutral vector FGME is:

1 + -(1 - 2sin2 Ow)(-r,Z)= -0.2314 2

f 0.2686

(14)

for proton and neutron particle-hole pairs respectively. The above two terms nearly cancel for uncorrelated protons. The correlations influence each operator differently and the final CB result for proton NV current matrix element depends sensitively on ki,kf,p and z p . Fortunately the CBME is also small 20, and the proton NV current is not likely to have a significant contribution to the v-nucleus interaction.

11 Neutron

0.8 -

0.7

p = 0.5 Po

-

1

1

1

1

1

,

1

1

,

1

1

Neutron ~ N Vas a function of q and proton fraction zp for ki = ~The. solid, dotted, dashed and dash-dot lines show results for z p = 0.5, 0.4, 0.3, and 0.2.

Figure 5. kj =

k

~

Figure 5 shows the density and xp dependence of q N V for neutron NV current. It has a significant xp dependence absent in the q~ and ~ G T however, it is relatively insensitive to the magnitudes of ki and k f . 6. Neutral-Axial-Vector Effective Weak Operator

The neutral-axial-vector quenching factor, ~ N , Afor neutron and proton particle-hole pairs are plotted in Figure 6 for p = po and the considered proton fraction values. In these matrix elements Ici = Cf = k F N . The charge-changing and neutral axial vector operators (OGT and O N A ) ,appropriately scaled, can be interpreted as the three components of an isospin vector operator. In symmetric nuclear matter the expectation values of these three components are equal as one can not quantify the isospin axis. The stars in Fig. 6 are results obtained for ~ G in T symmetric nuclear matter. Unlike for the GT CBME, there is a noticeable dependence of V N A on the proton fraction xp originating from the 7; terms in the NA effective weak operator.

7. Correlated Basis Interaction The two-body effective interaction of Eq. (4) is given by: 1 v$y(ij) = Fij [vijFij - -(V’Fij) - (V‘. ( F V ’ ) F F ( V F ) * m m

+

V)] (15)

,

12

I

p= 0.16 fm.3

'r

Neutron and Proton 7 ) N A as a function of q and proton fraction x p for k F N . The solid, dotted, dashed and dash-dot lines show results for x p = 0.5, 0.4, 0.3, and 0.2. The stars are results for ~ ) G Tat x p = 0.5. Figure 6.

ki = kf =

in the two-body cluster approximation. The V' operate to the left while V to the right. The energy expectation values of correlated states I X ) are obtained by using this v$y(ij) in 1st order with FG wave functions, 9 ~ , as in the Hartree-Fock approximation. The vGB has a momentum dependence via the ( V F i j ). V terms which contribute to the matter energy via exchanges. This contribution is much desmaller than that of the momentum independent, static terms in ":v fined as:

In the present work we have considered only the static part of Fij. We therefore keep only the dominant, static part of the Argonne v; interaction containing terms with the six static operators, Or3=1'6,and neglect the spinorbit terms. In this approximation the is a static operator having six terms with 0 P = l i 6 :

IJF;~

p=1,6

The Landau-Migdal effective interactions used in studies of weak interactions in nuclei and nucleon matter are obtained from the spin-isospin susceptibilities of nucleon matter. We have therefore studied these susceptibilities with the and The energy of nucleon matter with

''

~ $ 7 ~$7'.

13

densities

P N and ~ PNJ

can be expressed as:

E ( p , z , y , z )= Eo(P) + ET(Pb2 + E d d y 2 + E U T ( P ) Z 2

1

(18)

+ Pnl - PPt - P P J ) / P Y = (Pnf - PnJ + P P t - PPJ)/P >

(19) (20)

.

(21) and Eo(p) is

2

= (Pnf

7

z = (Pnt - Pnl - P P t +.PPJ)IP

The T , D and DT susceptibilities are proportional to E;:,,,, the energy of symmetric nuclear matter with z = y = z = 0. We have calculated the E,,u,u,(p) using the obtained from the Fij at p = f , 1 and PO. Results using the us? are given by full lines in 7, while those with the simpler us?” by dashed lines. The momentum dependent part of us?, responsible for the difference between us? and is much smaller than the static part. Both us? and us?” have a density dependence due to that of Fij. However, it has very little effect on E,, E, and E,, (see Fig. 7). The stars in Fig. 7 show the values of E,(p) extracted from recent variational calculations 24 of symmetric nuclear matter (SNM) and pure neutron matter (PNM) with the Argonne v18 and Urbana IX interactions, assuming that Eq. (18) is valid up to 2 = 1 for y = z = 0. The two-body v!: seems to provide a fair approximation to the E, so obtained. We also consider the spin susceptibility of PNM given by the inverse of E f N M ( p )defined as:

~$7

~$7’

+ (22) The results obtained with the ~ $ 7and ~$7”are shown in the lower part E P N M ( p , y )= E f N M ( p ) E r N M ( p ) y 2.

of Fig. 7 along with those of quantum Monte Carlo calculations 28 with the static part of Argonne v18 and Urbana-IX interactions. The two-body effective interactions obtained from the Fij of SNM give fairly accurate values of E f N M . Fig. 8 shows Eo(p)and E T N M ( p calculated ) from the us: at the three values of p. The stars in this figure give results of the recent variational calculations 24 with the full Argonne v18 and Urbana IX interactions. At low densities the two-body is not a bad approximation; however, the Eo(p) obtained from it does not show a minimum at PO. The 3-body interaction and cluster contributions are repulsive and are essential to obtain the minimum. The 2-body is more accurate in predicting the susceptibilities than the equation of state, E o ( p ) . This is partly because the contributions of TFGand to the E,,,,,,(p) add. The contribution of TFGto the E f N M

~$7

~$7

~$7

14

Figure 7. E u , ( p ) (upper set), E r ( p ) (middle set) and & ( p ) (lower set) of SNM are shown in the upper half, and E F N M in the lower half. In each set, results are shown using Fij for p = (upper), 1 (middle), and ;po (lowest). Solid lines show results obtained with v:; and the dashed lines for Stars denote values of E,(p) from variational calculations and values of E r N M ( p )from quantum Monte Carlo calculations. The dash-dot line in the lower half is the Fermi-gas E u ( p ) .

3

vFF.

Figure 8. Eo(p) for SNM (lower set of curves) and PNM (upper set of curves). In each set, the upper most curves are results using Fij for p = $, the middle and for p = 1, and the lowest for p = i p o . Solid lines show the results for the dashed lines v:.' Stars and crosses denote values obtained for E o ( p ) from variational calculations.

~$7

15

is shown by the dotted line in Fig. 7, it is about half of the total. For this reason even relatively simple estimates 29 of E f N M are not too different from the current state of the art 2 8 . In contrast, the contributions of and TFGto the Eo(p) have opposite signs, and large cancellations occur. Therefore the many-body clusters and interactions are relatively more important in the calculation of Eo(p). When the momentum dependent term of Eq. (15) is negligible, the offdiagonal matrix elements of are just the Fourier transform of The is weaker than the bare u , particularly at large values of q, as shown in Figs. 9-11. Perturbative corrections typically involve a loop integration over the momentum transfer q with a q2 phase-space factor. Hence in these figures we compare q2vgEs(q)with q2v,(q).

wzy

....

p=orp,,

- Pp == P.I

--

0

~$7".

~$7

~$7"

5

,*'

p,,

-------

..-

,'

..'.

-'-.,central

,I'

Ban;

2

r

'*.

4

6

'.

8 0 .0

Figure 9. The Fourier transforms, multiplied by q 2 , of the central and ui . uj components of u : r calculated using F;j at p = k , l , are shown by dotted, solid, and dash-dot lines respectively. The dashed lines show q2 times the Fourier transforms of the bare interaction.

8 . Conclusions

We have calculated the effective weak operators and nuclear interactions consistently using the same correlated basis states. The calculated twobody effective interactions explain the spin-isospin susceptibilities of SNM and PNM, however, higher order terms are necessary to obtain the correct compressibility. The effective one-body weak operators are quenched

16 4Wr

I

I

I

I

2

4

6

8

Figure 10. The Fourier transforms, multiplied by q 2 , of the ~i components of .?v: Notation as in Fig. 9.

2

4

6

. ~ and j

ui 'u3T ; " T ~

8

Figure 11. The Fourier transforms, multiplied by q 2 , of the Sij and ~i . ~ j S i j components of ve";;". Notation as in Fig. 9.

with respect to the bare by 20 to 25 % and also have a dependence on the momentum transfer. In the future we intend to calculate the interactions of neutrinos with nucleon matter using these effective operators and interactions. On the other hand we also need to calculate the higher order effective interactions such as the three-nucleon V$?(ijk) and the weak effective pair currents to obtain reliable results. Nevertheless, it appears that

17

-

interactions of neutrinos with nucleon matter are overestimated by 25 % when bare operators are used along with effective nuclear interactions. References 1. Q.R. Ahmad, et al (SNO collaboration), Phys. Rev. Lett. 87, 071301 (2001). 2. L.E. Marcucci, R. Schiavilla, M. Viviani, A. Kievsky, S. Rosati and J.F. Beacom, Phys. Rev. C 63,015801 (2001). 3. J. Kleinfeller et al, in Neutrino ‘96, edited by K. Enquist, H. Huitu, and J. Maalampi (World Scientific, Singapore, 1997). 4. L.B. Auerbach, et al, Phys. Rev. C 64,065501 (2001). 5. L.B. Auerbach, et all nucl-ex/0203011. 6. Proceedings of the First International Workshop on Neutrino-Nucleus Interactions (NuIntOl), edited by M. Sakuda, to be published by Nucl. Phys. B Proceedings Supplement. 7. M. Prakash, J.M. Lattimer, R.F. Sawyer and R.R. Volkas, Ann. Rev.Nuc1. Part. Sci. 51,295 (2001). 8. K. Langanke, D.J. Dean, P.B. Radha, Y. Alhassid and S.E. Koonin Phys. Rev. C 52,718 (1995). 9. G. Martinez-Pinedo, A. Poves, E. Caurier and A.P. Zuker, Phys. Rev. C 53, R2602 (1996). 10. E. Kolbe, K. Langanke and P. Vogel, Nucl. Phys. A 652,91 (1999). 11. R. Schiavilla, et al, Phys. Rev. C 58, 1263 (1998). 12. R.B. Wiringa and R. Schiavilla, Phys. Rev. C 65,054302 (2002). 13. J.G. Congleton and E. Truhlik, Phys. Rev. C 53,956 (1996). 14. K. Suzuki and S. Y. Lee, Prog. Theor. Phys. 64,2091 (1980). 15. P. Navratil, J.P. Vary and B.R. Barrett, Phys. Rev. C 62,054311 (2000). 16. S. Fantoni and V.R. Pandharipande, Nucl. Phys. A A473, 234 (1987). 17. S. Fantoni and V.R. Pandharipande, Phys. Rev. C 37, 1697 (1988). 18. J.W. Clark, Prog. in Part. and Nucl. Phys. 2,89 (1979). 19. V.R. Pandharipande and R.B. Wiringa, Rev. Mod. Phys. 51,821 (1979). 20. S. Cowell and V. R. Pandharipande, preprint(2002). 21. S. Fantoni, B.L. Friman and V.R. Pandharipande, Nucl. Phys. A A399, 51 (1983). 22. A. Fabrocini and S. Fantoni, Nucl. Phys. A A503, 375 (1989). 23. A. Akmal and V.R. Pandharipande, Phys. Rev. C 56,2261 (1997). 24. J. Morales, V.R. Pandharipande and G. Ravenhall, preprint (2002). 25. R.B. Wiringa, V.G.J. Stoks, and R. Schiavilla, Phys. Rev. C 51,38 (1995). 26. B.S. Pudliner, V.R. Pandharipande, J. Carlson, S.C. Pieper, and R.B. Wiringa, Phys. Rev. C 56,1720 (1997). 27. A. Arima, K. Shimizu, W. Bentz and H. Hyuga, Advances in Nuclear Physics, 18,l(1987). 28. S. Fantoni, A. Sarsa and K.E. Schmidt, Phys Rev Lett 87,181101 (2001). 29. V.R. Pandharipande, V.K. Garde and J.K. Srivastava Phys Lett B, 38B,485 (1972).

18

PSEUDOSCALAR MESONS IN NUCLEAR MEDIUM *

T. TATSUMI Department of Physics, Kyoto University Kyoto, 606-8502, Japan E-mail: [email protected]. ac.jp

The behavior of pseudoscalar mesons in nuclear medium is reviewed with an emphasis on the possibility of their Bose-Einstein condensation in dense matter. In particular pion condensation is reexamined in detail, stimulated by recent theoretical and observational developments.

1. Introduction Recently much attention has been paid for high density QCD: hadron matter at relatively low density and quark matter a t high density are typical subjects there. As a key concept which goes through hadron and quark worlds chiral symmetry is realized in both matter, but in a different way. Parity-even and odd quantities interplay in this context. Since the vacuum is good parity state, there is no expectation value of parity-odd operators. However, they may have finite values in matter (parity violation); Bose-Einstein condensation of pseudoscalar mesons in hadronic matter or nonvanishing of the parity-odd mean-field, (q-y5raq),in quark matter. We consider the particle-hole operator here and discuss how the pseudoscalar quantity becomes nonvanishing in hadronic matter and what are its implications, by studying the behavior of pseudoscalar mesons in nuclear medium. Pseudoscalar mesons (T,K ) have some salient features; they are the lightest hadrons without and with strangeness and considered as the Nambu- Goldstone bosons as results of spontaneous breaking down of chiral symmetry. Since they are bosons, they may lead t o the Bose-Einstein condensation in some situations. To understand the behavior of these mesons in nuclear medium, we also take into account the effects of resonances. A(1232) strongly couples with 'This work is supported by the grant-in-aid for scientific research fund of the ministry of education, culture, sports, science and technology (11640272, 13640282).

19

nucleon by the p wave 7rN interaction and A(1405) gives rise to a peculiar feature in s- wave K N scattering near threshold. The mass difference between these resonances and nucleons is small (O(mT)),so that they should play important roles even for low energy phenomena (energy-momentum ) are interested in. Thus chiral symmetry and resscale 0 ( ( 2 - 3 ) m T ) we onances may characterize the behavior of these mesons in nuclear medium. N

log t

(years)

Figure 1. Cooing curves for stars with medium FP EOS, taken from ref.2. Solid curve shows standard cooling of a 1.2Ma neutron star and dashed curve shows pion cooling of a 1.4Mo star. The data of c,2,4, 5 indicate cooler stars: (c)3C58, (2) Vela, (4) Geminga and (5) RXJ 1856-3754.

There have been performed and planed many nuclear experiments to reveal the behavior of these mesons in nuclear medium. On the other hand, observations of compact stars have also provided information about it. Very recently appeared an interesting data about the surface temperature of young pulsar '. They reported that the pulsar inside the historical supernova 3C58 shows too low surface temperature to be explained by the standard cooling scenario, which essentially assumes usual neutron matter inside it (see Fig 1). The importance of this observation is in the age (- 103yr) of this pulsar. As we shall see later if this fast cooling is attributed t o the presence of pion condensation, the star cools very rapidly in the early neutrino-emitting phase and the difference of the surface temperature from the standard cooling scenario becomes remarkable there. So we can hope t o see the evidence of pion condensation more clearly for young

20

pulsars. In Fig. 1 we present our theoretical cooling curve with observational data. We can see that pion cooling can explain cooler stars including 3C58 '. 2. Kaon Condensation

The low-energy K N interaction is specified by three kinds of the s- wave interaction, the K N cr term, the energy-dependent Tomozawa-Weinberg term and the resonant term with A(1405). Empirically the scattering amplitude shows an interesting behavior in the I = 0 channel due to the existence of the resonance A( 1405) below the threshold. Recently some people propose a possibility of the deeply bound kaonic nuclei in relation to the property of this resonance '. On the other hand, the interaction becomes relatively weak in the I = 1 channel, and it is relevant when we consider kaon ( K - ) ) condensation in neutron stars. Since we know the non-resonant s- wave terms can be well described in terms of SU(3) x SU(3) chiral symmetry, kaon condensation has been discussed on the basis of chiral Lagrangians 4,5. These s- wave terms cooperatively work to give a large decrease of the effective energy of kaons in nuclear medium. When the energy reaches the electron chemical potential as density increases, kaons begin to condense through the reaction, nn -+ npK-.

(1)

This is very similar to the Bose-Einstein condensation of alkali atoms '. The most important conseqiience of kaon condensation is the large softening of the equation of state, which leads to an interesting phenomenon, delayed collapse of protoneutron stars t o produce the low-mass black holes 6,7,8,9,10.

3. Pions in Nuclear Medium

Since pions couple with particle -hole and A- hole states with the same quantum number, we can study the properties of these states as well as pions themselves by considering the pion propagation in nuclear medium ll. In Fig. 2 we show the longitudinal spin-isospin excitation (pionic) modes in the energy (w)-momentum (k) plane.

3.1. Pionic Excitations within the F e m i Liquid Theory Consider the pion propagator in nuclear medium at density p ~ : D,'(w, k;p ~ =)w2 - TTI:

-

k2 - II(k,W; p ~ ) ,

21

4

3.5

$

3

2.5 2 1.5 1

0.5 0

0 0.5 1

1.5

2 2.5 3 3.5 4 k(md

Figure 2. Schematic view of the spin-isospin modes in symmetric (N=Z) nuclear matter. N N - ’ and AN-’ denote the continuum spectrum of the particle-hole and A-hole excitations, respectively. “2s” means the zero-sound mode, which corresponds to the Gamow-Teller state in asymmetric matter.

due to the self-energy II. For the p wave T N interaction given by the particle-hole polarization function U(O),

a

, it is simply

in the lowest order, where we have taken into account the nucleon particle

- hole (ph) and A- hole (Ah) states . The polarization functions UAo’ are further given in terms of the Lindhard functions L,

--

13,

+

where we introduced the form factor I? = (A2 - &)/(A2 k 2 ) with the cut-off A O(lGeV), and the T N N and T N A coupling constants are fx” 1 and f T N A 2(Chew - Low value), respectively. the Lindhard functions La are explicitly evaluated to be, e.g.

-

aThe s- wave interaction is negligible in symmetric ( N = 2)nuclear matter. bWe, hereafter, use the nonrelativistic approximation for nucleons. See ref. 12 for a relativistic treatment .

22 with

a =w

-

k2/2m>, bi = kvk,

(6)

for T - propagation. Here we introduced the effective mass my, for nucleons.

Figure 3. Examples of the particle-hole and A- hole interactions in the spin-isospin channel. They are written as sum of the one-pion exchange interaction and the phenomenological zero-range interaction with the Landau- Migdal parameters.

It is well known that the lowest order calculation is not sufficient to discuss the behavior of the pion in nuclear medium: we must take into account the correlations between ph and Ah states. They can be easily incorporated in the spirit of Landau Fermi-liquid theory. Since we are interested in the region of w,k (2 - 3)m,, full ph-ph, ph-Ah or Ah-Ah interaction should be separated into two terms, depending on their length scales: we explicitly treat the long-range (0(mG1)) interactions by way of pion, ph and Ah propagation, while the short-range (O(m,' 0.2fm)) interactions are replaced by the momentum-independent local interactions parametrized by the Landau-Migdal parameters; e.g.

-

N

where the strength in the spin-isospin channel is measured in the pion unit. It can be extended to include the isobar degrees of freedom:

23

with the transition spin and isospin operators, S and T . Then we have the p wave self-energy of the pion by considering the one- line irreducible diagrams and the Dyson equations;

where

IIN = -k2UN

=

- k 2 UN (0) [I

+ (&A

n~ = -k 2 UA = -k2U!, 0)[1f (&N

- & A ) u A (0)1 / -

0 7

&a)U$)]/D

(10)

with

D

=1

+ &Nuj$)'+ g b a u A(0) + (&N&A

r2

(0) (0)

- g N A ) U N VA .

(11)

We can study the spin-isospin modes in another way, starting from the ph and Ah propagation within RPA. Both ways are equivalent with each other for the longitudinal modes. Considering the correlation function between the generalized spin-isospin density operator,

we have the same excitation spectra of the spin-isospin modes as before l4vt5. Indeed the nuclear response function in the longitudinal spin-isospin channel is defined as follows;

with the free pion propagator &(w, k) = (w2 - rn;

-

k2)-l.

3.2. Pion Condensations

3.2.1. Neutral Pion Condensation First consider neutral pion condensation in symmetric (N=Z) nuclear matter, which require the following condition: the softening of the longitudinal spin-isospin mode

N O , kc)

4

00,

Imnp(O, kc;p c ) 0: wO(w) --t 0,

(14)

24 or equivalently W O , kc; P c )

0,

-+

(15)

in terms of the pion propagator. It is t o be noted that pion condensation by no means implies a naive Bose-Einstein condensation of pions, but the softening of the longitudinal spin-isospin mode with the critical momentum k,. On the other hand, such mode is unstable and the Lindhard function has an imaginary part before the critical density pc. We can see a peculiar enhancement of the strength function a t small energy near the critical density. The pion condensed phase can be represented in terms of chiral transformation as follows:

Inc)= U(Bv(kc. r),6A(kc . r))lnormal)

with (normall7rInormal) = 0. Then (7r)

= 7rc # 0.

(17)

As an example, 7rc =

(O,O, Acoskcz),

(TO

condensation).

(18)

It would be worth mentioning that 7ro condensation gives rise to a magnetic ordering of nuclear matter: it exhibits a liquid-crystalline nature with one-dimensional anti-ferromagnetic ordering, called Alternating-Layer-Spin [ALS] structure (see Fig. 4 ) 17. Note that since U(a)lnormal) = . Inormal), U E S U ( 2 ) , in the isospin space, the isospin rotated condensate, iic = R7rc = (17rcl sinBcos4, J7rcIsinBcos4, 17rcI COSB), R E O ( 3 ) is also possible with the same energy, which means 7r*, 7ro condensation. Indeed all the propagators for 7ro,7r* become identical in symmetric nuclear matter. Also note that g h N should be replaced by (&N g N N ) / 2 < f&N in neutron ( 2 = 0) matter. Finally, since w = 0 in this case, a potential description is possible instead of the explicit introduction of pion field. This aspect has been emphasized in the study of the Alternating-Layer-Spin structure in the condensed phase 17.

+

3.2.2. Charged Pion Condensation Next consider the charged pion condensation in neutron matter, which should have a direct relevance with neutron star phenomena. Consider,

25

I&

7

.........................

J I J I

I/

I / I .........................

&I

.......................

I/

\ L \ L 0

0 0

Figure 4. Alternating-Layer-Spin [ALS] structure associated with 7ro condensate (cx Acos k c z ) in symmetric nuclear matter. Bold arrows denote proton spins and thin arrows neutron spins.

e.g., the

T+

propagator: W

D ; : ( w , k ; p ~ )= u 2 - m t - k 2 - T p - I l , ( w , k ; p s ) . 2fT

(19)

Note that there appears the isovector s- wave coupling term cx ( p n - pp) besides the p wave term. Poles of D,S' include the energies of T* mesons and the pn-' collective mode with the same quantum number of K + , called IT$, besides single ph and Ah excitations. The threshold condition for the charged pion condensation is W*t

+ w,-

= 0,

(20)

which implies the n $ ~ -pair condensation. In terms of the propagator we have

which are called the double-pole condition. The condensed phase can be represented as

Inc)= exp(i

1

V3k, . rd3z)exp(iQ:O)Inormal),

and we see (T)= nC=

(sin O cos k, . r, sin O sin k, . r, 0).

26

= O . 8 m ~(bottom) to m ) = 0 . 6 m ~ (top) by O . l m ~step. Similarly, the solid lines in the right panel show those in the three Q cases from Q = 0.85 (top) to Q = 0.95 (bottom) by 0.05 step. The dashed curve denoted by “U” means those by the universality ansatz.

28 Recently sophisticated variational calculations have been done for symmetric nuclear matter and pure neutron matter, using modern potentials They also found that there are phase transitions to pion condensation a t low densities, 2po and 1 . 3 ~ for 0 symmetric nuclear matter and pure neutron matter, respectively. It would be interesting to compare these values with our results. The critical density for charged pion condensation is presented in Fig. 6. The behavior is almost the same as that for 7ro condensation and the critical density is low; po < pc < 2p0 for gLA < 1 and m& = 0 . 8 m ~ It . would be interesting to refer the works by Tsuruta et al. 2,25 in this context: they set charged pion condensation at pc = 2 . 5 ~ in 0 their calculation of neutron star cooling. 23924.

5 4

1 0 0

0.2 0.4 0.6

0.8

1

g’M Figure 6. Critical density for charged pion condensation in pure neutron matter. rn;. = 0 . 8 m ~and Q = 0.9 are used here. The symbol “U” means that by the universality ansatz.

4. Summary and Concluding Remarks

We have seen three recent results, which may support possible existence of pion condensation a t low densities. The new experiment on the GamowTeller resonance tells us the universality ansatz about the Landau-Migdal parameters by no means hold; &A should be much less than g&N or gLA. The critical densities of pion condensations are pc 1.5po(N = Z, r n L / r n N = 0.8, &A = 1) and p c 2.4po(Z = 0, m&/mN = 0.8, ghA = N

N

29

1) for neutral pion condensation, while pc N 2.5po(Z = O , m > / r n N = 0.8, gha = 1) for charged pion condensation. As another theoretical work, a new calculation of nuclear matter with a modern potential has also suggested the phase transition at pc 2po(N = 2 ) and pc 1.3po(Z = 0), which corresponds to neutral pion condensation

-

N

23

Besides these theoretical developments, the current observation about the surface temperature of a neutron star inside 3C58 suggests we need exotic cooling mechanisms beyond the standard cooling scenario. We have seen that a consistent calculation about pion cooling have been done by taking into account nucleon superfluidity in a proper way, and it can explain the data =. Unfortunately these are indirect evidences for pion condensation, and we hope for a direct evidence by heavy-ion collision experiments in near future. Finally I would like give a comment about another theoretical aspect of pion condensation. It means a spontaneous violation of parity in nuclear matter and the condensed phase can be described as a chirally rotated state. We may also consider its analog in quark matter: nonvanishing of the parity-odd mean-filed ( p y 5 P q ) . It would be interesting in this context to refer to recent studies about ferromagnetism in quark matter, where a magnetic ordering is realized under the axial-vector mean-field 26.

Acknowledgments The author thanks T. Suzuki, H. Sakai, M. Nakano, S. Tsuruta, T. Takatsuka, T. Muto and R. Tamagaki for their collaboration.

References 1. P. Slane, D.J. Helfand and S.S. Murry, ApJ 571, L45 (2002). 2. S. Tsuruta, M.A. Teter, T. Takatsuka, T. Tatsurni and R. Tarnagaki, A p J 571, L571 (2002). 3. T. Kishimoto, Phys. Rev. Lett. 83, 4701 (1999). Y. Akaishi and T. Yarnazaki, Phys. Rev.C65, 044005 (2002); Phys. Lett. B535, 70 (2002). 4. T. Muto and T. Tatsurni, Pjys. Lett. B283, 165 (1992). 5. T. Tatsurni and M. Yasuhira, Phys. Lett. B441, 9 (1998); Nucl. Phys. A653, 133 (1999); Nucl. Phys. A670 218c (2000). 6. G. E. Brown and H. A. Bethe, ApJ423, 659 (1994). 7. T. W. Baurngarte, S. L. Shapiro and S. Teukolsky, ApJ 443, 717 (1995); 458, 680 (1996). 8. M. Yasuhira and T. Tatsumi, Nucl. Phys. A663&664, 881c (2000); Nucl. Phys. A680, 102c (2001); Nucl. Phys. A690, 769 (2001).

30 T. Tatsumi and M. Yasuhira, Proc. of INPC 2001, American Inst. of Phys, 476 (2002). 9. J. A. Pons, S. Reddy, P. J. Ellis, M. Prakash and J. M. Lattimer, Phys. Rev. C62, 035803 (2000). 10. J. A. Pons, J. A. Miralles, M. Prakash and J. M. Lattimer, A p J 553, 382 (2001). 11. A.B. Migdal et al., Phys. Rep. 192, 179(1990). 12. M. Nakano et al.,Int.J.Mod.Phys. E10,459(2001). M. Nakano, in this proceedings. 13. A.L. Fetter and J.D. Walecka, Quantum Theory of Many Particle Systems, McGraw-Hill Inc., 1971. 14. E. Shiino et al., Phys. Rev. C34, 1004 (1986). 15. M. Ichimura and Nakaoka, in this proceedings. 16. W. M. Alberico, M. Ericson and A. Molinari, Nucl. Phys. A379, 429 (1982). 17. R. Tamagaki et al., Prog. Theor. Phys. Suppl. 112, (1993). 18. J. Meyer-ter-Vehn, Phys. Rep. 74, 323 (1981). 19. F. Osterfeld, Rev. Mod. Phys. 64, 491 (1992), and references therein. 20. T. Wakasa et al, Phys. Rev. C55, 2909 (1997). T. Wakasa et al, in this proceedings. 21. T . Suzuki and H. Sakai, Phys. Lett. B455, 25 (1999). 22. T. Suzuki, H. Sakai and T. Tatsumi, Proc. RCNP Int. Sympo. on Nuclear Responces and Medium Effects, Univ. Academy Press, 77 (1999). 23. A. Akmal and V.R. Pandharipande, Phys. Rev. C56, 2261 (1997). 24. H. Heiselberg and V.R. Pandharipande, Ann. Rev. Nucl. Part. Sci. 50, 481 (2000). 25. H.Umeda, K . Nomoto, S. Tsuruta, T. Muto and T. Tatsumi, ApJ431, 309 ( 1994). 26. T. Tatsumi, Phys. Lett. B489, 280 (2000).

II. PION CONDENSATION, COMPRESSIBILITY, PIONIC STATES


33

PION CONDENSATION BASED ON RELATIVISTIC FORMULATION

MASAHIRO NAKANO University of Occupational and Environ.m.enta1Health E-mail: [email protected] p TOSHITAKA TATSUMI Department of Physics, Kyoto University, Kyoto, Japan LIANG-GANG LIU Department of Physics, Zhongshan University, Guangzhou, China HIROYUKI MATSUURA GRIPS, Tokyo 162-0056, Japan TAISUKE NAGASAWA Department of Physics, Kyushu University KEN-ICHI MAKIN0,NOBUO NODA,HIROAKI KOUNO AND AKIRA HASEGAWA Department of Physics, Saga University, Saga 840-8502, Japan

The critical density of neutral pion condensation is investigated based on the relativistic framework and compared with nonrelativistic results. The particlehole and delta-hole polarizations of the pion selfenergy are calculated in the relativistic way by using a new set of Landau-Migdal parameters derived from a recent experimental data. It is concluded that the use of relativistic particle-holeand delta-holeexcitations for the pion selfenergy increases the critical density compared with the non-relativistic result, however still leads to condensation for densities from 2 t o 4 times normal nuclear matter density.

*This work is supported by national science foundation of china.

34 1. Introduction

Strong attractive force of pion is believed to bring about pion condensation at a high density. It is one of important problems to clarify the critical density of the pion condensation since it may affect high density systems such as heavy-ion collisions and the interior of neutron stars. There are many studies on the pion c ~ n d e n s a t i o n . 'These ~ ~ ~ ~calculations ~~ are, however, mainly based on non-relativistic formalism. Therefore the main purpose of this paper is to report results based on the relativistic framework. Recently new experimental information on the Landau-Migdal parameters is obtained from the analysis of Gamow-Teller(GT) giant rese nance by Wakasa et al.576 Using this new set, it is interesting to investigate the critical phenomena in the relativistic framework.

2. Formalism

Pion condensation is a kind of phase transitions from a normal ground state to the condensed state of a pionic spin-isospin mode. Due to the strong p-wave interaction, the collective excitation with the same quantum number as the pion becomes soft through the particle-hole-pion interaction and the A-hole-pion interaction. The effects of the particle-hole or A-hole excitations in nuclear matter are taken into account in the pion selfenergy I I ( k , k o ) . The pion selfenergy is given by a sum of the nucleon particlehole excitation and the A-hole excitation. The pion selfenergy is generally complex, however, the imaginary part is exactly zero when the excitation energy is zero. Once we obtain the pion selfenergy, we can calculate the critical density of the pion condensation. The critical density is given by the following condition for neutral pion.

and

This condition means the excitation energy of the pionic mode becomes zero at a certain momentum. Therefore it is important t o calculate the pion selfenergy.

35

We start with the following Lagrangian. The interaction Lagrangian is given for nucleon and delta-isobar.

where the coupling strength f r N A = 2fr" and f r N N = 0.988 is taken in this calculation. Using the explicit form of the nucleon and delta-isobar interactions, we obtain following pion selfenergy within random phase approximation

(RPA) .

where T and T are isospin operators. The G and Gpu are nucleon and delta propagators respectively, which are given in particle-hole-antiparticle (PHA) representation:.'

G N ( d = GP(q)+ G h ( d + G"(q)

(6)

G,4(q) = G A ( d + GA(q)

(7)

where GP, Gh and G" represent the propagation of particle, hole and antiparticle in nuclear matter, respectively, and given by

+

M* 1 - nq GP(q) = - -Zy,qp Eq - 40 - ZE 2Eq M* 729 Gh(q)= - - i r p q p Eq - qo ZE 2Eq -iypq, M* 1 Ga(q)= Eq qo - ZE' 2Eq Note that Zypqp = iyiqi - y0Eq for the positive energy, and iypq, = iyiqi y0Eq for the negative energy in our notation,* and Eq is on-shell energy of nucleon with momentum of q: E , =

+ +

+

+

+

d m .

36

Similarly the propagators of positive energy delta-isobar and antideltaisobar are given by

d m .

where E t is on shell energy of delta-isobar, The projection operator of delta-isobar onto the positive (or negative) energy state is given,

Inserting these propagators, we obtain following expression of the selfenergy for a neutral pion in symmetric (N=Z) nuclei.

+k

-, -k].

where

(14) +

(4

(q

+

+ k ) p k p = (f+ k) . k +

Ef+k . ko

(15)

= (f+ k) . f- Et+k . Eq

(16)

-

+

k)p!?p

and the following form factor is introduced for the vertex,

rn; r,, = A2 A2 + k; -

with the same value A =1000MeV for both pion selfenergies.' Here we comment the PHA representation we used. There are two merits in PHA representation.

37 The one is that PHA representation does not include the unphysical components. On the other hand, the density part in traditional densityFeynmann (DF) representation has unphysical components as shown in Ref.7. Moreover, DF representation cannot describe exactly meson propagator of delta-hole excitations. The second merit is that PHA representation has a similar expression to the nonrelativistic one. The only different point is the spin trace part in the numerator. Actually an approximation of Eqs.(16),(17) brings the same expression of the pion selfenergies used in a nonrelativistic calculation. In the non-relativistic limit, the momentum is negligible ( k l 4 .To simplify the calculations, the collective transition densities were used. More details of the MD analysis were described in Ref. 15. The results for the L=O and L=2 strength distributions calculated using the EWSR are shown in Fig. 2. Both the GMR and GQR strengths have well-defined peaks but extend to high excitation energies. The cross section for each L component is well separated by a MD analysis. However, the strength distributions obtained from the comparison with the DWBA cross sections depend on the transition densities used in the analysis. Thus, a possible reason of these excesses in the EWSR fractions is that the collective

45

Figure 3. The peak energies for the HE (closed circles) and LE (closed squares) components of the GMR are plotted as a function of the nuclear deformation 6. The LE components of the GQR are also plotted with the closed triangles. The open squares are the peak energies for the GMR predicted the fluid-dynamical (FD) model, and the open triangles are those of the cranking model (see the text). The lines are drawn to guide the eye.

transition densities of the GMR and the GQR used in this analysis are not valid in the high excitation energy region. Therefore, further analysis was carried out for the energy region, 9 MeV to 18 MeV for the GMR and 9 MeV to 16 MeV for the GQR. At first, the GMR and GQR strengths for 144Smwere fitted with a BreitWigner function, respectively, as shown in Fig. 2 . Next, using the each width for 144Sm,the GMR and GQR strengths for the other targets were fitted with two Breit-Wigner functions. Figure 3 compares the centroid energies of each component with those from the two theoretical

models viz. the adiabatic cranking model and the fluid-dynamical model '. Both the models reproduce the peak energies of the high-excitationenergy (HE) components of the GMR. However, the low-excitation-energy (LE) components are higher in energy than the theoretically predicted values. A similar behavior has been observed for 154Smby Youngblood et al. 3. This difference was caused by taking the GQR energy in the spherical nucleus as 63A-lI3 MeV phenomenologically. In this experiment, that was about 65A-1/3 MeV in '44Sm. The isoscalar odd-parity giant resonances, ISGDR and HEOR, are also expected to couple, resulting in shifting of the strengths to lower excitation energies and broadening of the width of HEOR and ISGDR because of the K-splitting 8. Figure 4 shows the results of the MD analysis for the ISGDR and HEOR in 144Smand 154Sm. In the spherical nucleus 144Sm,the ISGDR strength has two distinct components as reported in Refs. 16, and the HEOR strength has a bump peaking at E, -22 MeV. In the deformed nucleus 154Sm,on the other hand, the LE component of the ISGDR strength appears to split into two components (K=O and K=l) and to be enhanced, whereas the HE component does not show any significant change. For the HEOR, the strength distribution is rather broad, without a discernible

46

ISGDR

Figure 4. Comparison of the ISGDR and HEOR strength distributions in 144Sm and 154Sm. The solid lines are drawn to guide the eye. T h e HEOR strength in '54Sm is enhanced at Ex= 12 - 17 MeV where the LE component of the ISGDR exists.

peak, and the strength is shifted toward low excitation energies. This broadening of the HEOR in deformed nuclei and the strength shift t o lower excitation energies were reported also by Morsch et al. 17. The coupling between the ISGDR and the HEOR is expected for the K=O and K = l components. The enhancement of both the ISGDR and HEOR strengths near E, -12.5 MeV in 154Sm, in comparison with 144Sm, is evidence for a coupling between the two modes. A direct comparison of the observed ISGDR strength in 154Sm with theoretical predictions is, however, complicated by the nature of the LE component of the ISGDR in spherical nuclei.

According t o recent theoretical work on the ISGDR this LE component is of "non-bulk" origin - only the HE component of the ISGDR strength corresponds to a compressional-mode. On the other hand, considering the effects of deformation on the ISGDR and HEOR, Ref. takes into account only the coupling between the HEOR and the compressional-mode ISGDR. Further theoretical work to investigate the effect of deformation on the "non-bulk" LE component of the ISGDR strength is clearly most urgently warranted. 18,19120921722,

*

4. Summary

Clean inelastic scattering spectra, free from instrumental background, have been measured for the 144-154Smnuclei. The spectra have been decomposed into contributions of various multipoles by a multipole decomposition analysis using DWBA angular distributions obtained in the framework of the density-dependent single-folding model. The strength distributions for the GMR, ISGDR, GQR and HEOR have been determined for the spherical nucleus 144Sm and the deformed nucleus 154Sm. We have obtained the systematic behaviors for the splitting of the GMR strength due to the coupling between the GMR and GQR, the broadening of the GQR width

47

due to K-splitting in deformed nuclei. The strength distributions of the GMR and GQR in 154Srnare in good agreement with the calculations by Abgrall et al and by Nishizaki and And6 '. For the ISGDR, the effects of deformation are different for the low- and high-excitation-energy components in 154Sm. The coupling between the ISGDR and HEOR has been evidenced by enhancement and splitting of the low-excitation-energy component of the ISGDR, the broadening of the HEOR, and the shift of the HEOR strength towards lower excitation energies.

Acknowledgments We gratefully acknowledge the RCNP cyclotron staff for providing halo-free beams. This work supported in part by the US-Japan Cooperative Science Program of the JSPS and the US National Science Foundation (grants Nos. INT-9910015 and PHY-9901133), and the University of Notre Dame.

References 1. U. Garg et al., Phys. Rev. Lett. 45 (1980) 1670. 2. T. Kishirnoto et al., Phys. Rev. Lett. 35 (1975) 552. 3. D.H. Youngblood, Y.-W. Lui, and H.L. Clark, Phys. Rev. C 60 (1999) 067302. 4. Y. Abgrall et al., Nucl. Phys. A346 (1980) 431. 5. T. Suzuki and D.J. %we, Nucl. Phys. A289 (1977) 461. 6. N. Auerbach and A. Yeverechyahu, Phys. Lett. 62B (1976) 143. 7. D. Zawischa, J. Speth, and D. Pal, Nucl. Phys. A311 (1978) 445. 8. S. Nishizaki and K. Ando, Prog. of Theor. Phys. 73 (1985) 889. 9. M. Fujiwara et al., Nucl. Instr. Meth. in Phys. Res. A422 (1999) 484. 10. M. Itoh et al., RCNP Ann. Rep. (1999) pp.7. 11. M. Itoh et al., Nucl. Phys. A687 (2001) 52c. 12. B. Bonin et al., Nucl. Phys. A430 (1984) 349. 13. G.R. Satchler and D.T. Khoa, Phys. Rev. C 55 (1997) 285. 14. A. Kolomiets, 0. Pochivalov, and S. Shlorno, Phys. Rev. C 61 (2000) 034312. 15. M. Itoh et al., Phys. Lett. B 549 (2002) 58, and references therein. 16. H.L. Clark, Y.-W. Lui, and D.H. Youngblood, Phys. Rev. C 63 (2001) 031301(R) . 17. H.P. Morsch et al., Phys. Lett. 119B (1982) 311. 18. I. Hamarnoto, H. Sagawa, and X.Z. Zhang, Phys. Rev. C 57 (1998) R1064. 19. G. Colo et al., Phys. Lett. B 485 (2000) 362. 20. J. Pikarewicz, Phys. Rev. C 62 (2000) 051304(R). 21. D. Vretenar, A. Wandelt, and P. Ring, Phys. Lett. B 487 (2000) 334. 22. S. Shlorno and A.I. Sanzhur, Phys. Rev. C 65 (2002) 047308.

48

ISOSCALAR GIANT DIPOLE RESONANCE AND NUCLEAR INCOMPRESSIBILITY

M. UCHIDA, H. SAKAGUCHI, M. ITOH, M. YOSOI, T. KAWABATA, H. TAKEDA Y. YASUDA, T. MURAKAMI, T. ISHIKAWA, T. TAKI, N. TSUKAHARA, S. TERASHIMA Department of Physics, Kyoto Universiry, Kyoto 606-8502, JAPAN

U. GARG, M. HEDDEN, B. KHARRAJA, M. KOSS, B. K. NAYAK, S. ZHU Physics Department, University of Notre Dame, Notre Dame, IN 46556, USA M. FUJIWARA, H. FUJIMURA, K. HARA, E. OBAYASHI, H. P. YOSHIDA Research Centerfor Nuclear Physics, Osaka Universiv, Osaka 567-0047, Japan

H. AKIMUNE Department of Physics, Konan University, Kobe, 658-8501, Japan

M. N. HARAKEH, M. VOLKERTS Kernfisisch Versneller Instituut, 9747 AA Groningen, The Netherlands

The isoscalar giant dipole resonance(1SGDR) in 208Pb has been studied via inelastic ascattering of 400MeV at extremely forward angles, including . ' 0 Energy spectra virtually free from instrumental background have been obtained with a ray-tracing technique, and the ISGDR strength distribution has been extracted using a multipole-decomposition analysis. These results lead to a value for nuclear incompressibility that is consistent for both the isoscalar dipole and monopole modes.

Incompressibility of nuclear matter has been studied experimentally and theoretically due to its fundamental importance in defining the equation of state for nuclear matter, describing various phenomena from collective excitationsof nuclei

49 to supernova explosions in the cosmos. The only direct way to experimentally determine the nuclear incompressibility is to measure the compressional-mode giant resonances, the isoscalar giant monopole resonance (ISGMR) and the isoscalar giant dipole resonance (ISGDR), which are characterized with A T = A S = AL = 0, and AT = A S = 0, AL = 1, respectively. Although the ISGMR has been well investigated since its discovery about 25 years ago, the location of ISGDR is still ambiguous due to its small amplitude of the excitation. One major concern with the ISGDR data so far has been that the value of the nuclear incompressibility extracted from the centroid of the ISGDR strength distribution was significantly different from that obtained from the known ISGMR energies. It is now well established that the appropriate way to extract the value of the incompressibility of nuclear matter (Knm)from the ISGMR energies is to compare the experimental centroids of the ISGMR with those obtained from RPA calculations with various suitable effective interactions with different Knm values. The problem arose when one compared the available ISGDR data' with the theoretical centroids of the ISGDR, calculated using the same interactions that appear to reproduce the available ISGMR data well 233,4953697.Therefore, we performed new measurements on the ISGDR where this apparent discrepancy between the theory and experimental data has been resolved, thanks to high quality data that allowed the extraction of the ISGDR strength up to an excitation energy of 30MeV. The present experiments have been performed at the Research Center for Nuclear Physics (RCNP), Osaka University, using inelastic a-scattering at 400 MeV and extremely forward angles, including 0". Detailes of the expermental procedure and the data reduction are described in the relevant article of Itoh et d. in this proceedings. Figure 1 (a) shows the 0" spectrum of 208Pb(cu,a')reaction after subtraction of the instrumental background as described above. A prominent bump corresponding to ISGMR and the isoscalar giant guadrupole resonance(1SGQR) in 208Pbis observed at Ex 10 - 15MeV and another bump(ISGDR+HEOR) is visible at E, 22MeV. The energy spectra were sliced in 1 MeV bins and reconstructed in terms of measured angles, thus obtaining the angular distributions for each bin. Figure 1 (b) and (c) show the angular distributions for two energy bins near the peaks of ISGMR and ISGDR. The ISGDR strength distribution was obtained by using a multipoledecomposition analysis (MDA). In this approach, the experimental angular distribution for each energy bin was fitted with the superposition of each multipolarity as follows:

-

-

50

AT&, AL=l

2

I

12.5

10

Figure 1 . (a) Energy spectrum of the 208Pb(a,a') reaction at gC.,,,. = 1.0'2' after subtraction of background. (b) Angular distributions of Ez = 14.5MeV for the 208Pb(a,a')reaction at 400 MeV. (c) Results for Ex = 23.5 MeV.

IS 11.5 20 ?35 25 27.5 M

Figure 2. Strength distributions of the ISGDR in 208Pb.The fits to the ISGDR strength distribution for 10 < E x < 28MeV are superimposed. The solidcurves are the results with two BreitWigner functions.

where (&a JdRdE)",""".are the calculated DWBA cross sections corresponding to 100% energy-weighted sum rule(EWSR) for each multipole at that energy, using the code ECIS95 g. Fitting parameters are the fractions of EWSR's, UL(E,), which are related to the strength values, SL(E,), as follows:

where ml is defined as: ml = C E,SL(E,). Standard transition densities and deformation parameters were used in the calculations lo. Transition potentials were constructed by folding the transition densities with an effective a-N interaction 11: V(l.'-

?I,po(r')) =

213

I

-l+$12/av

- V ( 1 + Pvp0 ( r ) ) e - i W ( l +pwpo 213 ( rt ) ) e - l ~ - ~ 1 2 / ~ w ,

(3)

where po(r') is the ground-state density. The parameters of the a-N interaction used in the calculations were obtained from fits to the elastic-scatteringcross sections measured in a separate experiment and are: V = 26.7 MeV, W = 15.5 MeV, 2 QV = aw = 4.4 fm and pv = PW = -1.9fm2. The results of the fits are shown in Fig. 1 (b) and (c) for two 1 MeV energy bins. In Fig2, The ISGDR strength clearly has two distinct components. The low-energy component of the ISGDR, near the energy of the ISGMR and IVGDR, has now been observed in

51

several nuclei l . To determine the centroid energy of the ISGDR, which is derived according to the definition: E, = ml/mo [mo = SL(E~)], the experimental strength distribution was fitted with two Breit-Wigner functions. As mentioned earlier, the well-accepted procedure for extracting K,, from energies of the compressional modes involves comparison of the experimental energies with those calculated using suitable effective interactions that have different nuclear incompressibilities. Table 1 lists the present experimental results compared with some of previously published results, and with theoretical calculations. The centroid of the experimental HE-ISGDR strength determined for '08Pb is now close to the theoretical predictions. All these calculations employ the interactions that give K,, in the range of 215 - 225MeV. It may be concluded, therefore, that a value of K,, 220 MeV is consistent with the observed properties of both the compressional modes in 208Pb.

-

Table 1. Centroid energies of ISGDR and ISGMR for 208Pb. EHE-GDR (MeV)

ELE-GDR (MeV)

23.0 f 0 . 3

12.7 f 0.2

13.5 f 0.2

19.9 f0.8

12.2 f 0.6

14.17 f 0.28

22.4 f 0.5

-

-

Morsch et ul. l 3

21.3 f 0.8

-

13.8

Djalali et al. l 4

21.5 f 0.2

-

13.9

22.6 f 0.2

-

-

14

14.1

217

10.9

14.I

215

This work Clark et ul.

'

Davis et al. l 2

Adams er al. l 5 Hamamoto et al. CoIb et a[.

23.4 23.9 (22.9 t )

N

EGMR (MeV)

Knm(MeV)

Piekarewicz

24.4

-8

13.1

224

Vretenar er ai.5

26.01

10.4

14.1

271

15

14.48

230

11.1

14.3

-

Shlomo and Sanzhur Gorelik and Urin

-

25.0 22.7

N

t Including effects of continuum and 2p - 2h coupling 16.

52 In summary, we have performed 208Pb(a,a') measurements at 4 0 0 M e V to study the ISGDR. The strength distribution of the ISGDR was obtained up to E, 3 0 M e V by using MDA. T h e ISGDR strength distribution has two components. With our results, both the ISGMR and the ISGDR provide a consistent value of the incompressibility of infinite nuclear matter.

-

References 1. H.L. Clark, Y.-W. Lui, and D.H. Youngblood, Phys. Rev. C 63 (2001) 031301(R). 2. I. Hamamoto, H. Sagawa, and X.Z. Zhang, Phys. Rev. C 57 (1998) R1064. 3. G. Co16 et nl., Phys. Lett. B 485 (2000) 362. 4. J. Pikarewicz, Phys. Rev. C 62 (2000) 051304(R); Phys. Rev. C 64 (2001) 024307. 5. D. Vretenar et al., Phys. Rev. C 65 (2002) 021301(R); Phys. Lett. B 487 (2000) 334, 6. S. Shlomo and A.I. Sanzhur, Phys. Rev. C 65 (2002) 044310. 7. M.L. Gorelik and M.H. Urin, Phys. Rev. C 64 (2001) 047301. 8. B. Bonin et al., Nucl. Phys. A 430 (1984) 349. 9. J. Raynal, Program ECIS 95 NEA 0850/14(1996). 10. M.N. Harakeh and A. van der Woude, Giant Resonances:Fundamental HighFrequency Modes of Nucler Excitation (Oxford University Press, Oxford, 2001). 11. A. Kolomiets, 0. Pochivalov, and S . Shlomo, Phys. Rev. C 61 (2000) 034312. 12. B.F. Davis et al., Phys. Rev. Lett. 79 (1997) 609. 13. H. P. Morsch et al., Phys. Rev. Lett. 45 (1980) 337; Phys. Rev. C 28 (1983) 1947. 14. C. Djalali et al., Nucl. Phys. A 380 (1982) 42. 15. G. S. Adams et af.,Phys. Rev. C 33 (1986) 2054. 16. G. Co16 et al., RIKEN Review 23 (1999) 39; and private communication.

53

HIGH RESOLUTION STUDY OF PIONIC 0- STATE IN "0

T. WAKASA, G. P. A. BERG, H. FUJIMURA, K. FUJITA, K. HATANAKA, M. ITOH, J . KAMIYA, Y. KITAMURA, E. OBAYASHI, N. SAKAMOTO, Y. SAKEMI, Y. SHIMIZU AND H. P. YOSHIDA Research Center f o r Nuclear Physics (RCNP), Ibaraki, Osaka 567-004 7, Japan E-mail: [email protected]

H. SAKAGUCHI, H. TAKEDA, M. UCHIDA, Y. YASUDA AND M. YOSOI Department of Physics, Kyoto University, Kyoto 606-8502, Japan

T. KAWABATA Center f or Nuclear Study (CNS), University of Tokyo, Tokyo 113-0033, Japan The cross sections and analyzingpowers of the ' 6 0 ( p , p ' ) 1 6 0 ( O - ,T = 1) scattering were measured at a bombarding energy of 295 MeV and an angular range of 14 O 5 5 30'. The isovector 0 - state a t E , = 12.80 MeV is clearly separated from the neighboring states with an energy resolution of A E N 30 keV. The data have been compared with distorted wave impulse approximation (DWIA) calculations. The analyzing powers are sensitive to the effective nucleon-nucleon ( N N ) interaction used in DWIA calculations, and our data support the medium modification of the N N interaction in nuclei. The DWIA calculation employing a random phase approximation (RPA) response function predicts an enhancement of the cross sections around a momentum transfer of p N 1.7 fm-', and it gives a reasonable agreement with the data.

1. Introduction Isovector J" = O-, O* -+ OF excitations are of particular interest since they carry the simplest pion-like quantum number. At low momentum transfers, they have been investigated in beta decay and muon capture e~perirnents'.~!~. Axial-vector and pseudoscalar currents are responsible for these first-forbidden transitions in nuclear weak processes. Gagliardi e t al.' reported an enhancement of the decay rate by more than a factor of 3 for the first-forbidden beta decay of the 120 keV, 0- state in "N. This enhancement can be explained by considering meson-exchange effects4. The ( p ,n) and (p,p') reactions are suited to study these transitions for

54

a wide range of momentum transfer5. Orihara et aL6 measured the angular distribution for the l60(p,n)16N(O-, 0.12 MeV) reaction at Tp = 35 MeV. They reported discrepancies between distorted wave Born approximation (DWBA) calculations and their data in the large momentum transfer region of q = 1.4-2.0 fm-' that might be due to an enhancement of the pion probability in the nucleus However, in the proton inelastic scattering t o the 0-, T = 1 state in l60at Tp = 65 MeV, such an enhancement was not observed12. The differences between ( p , n ) and ( p ,p') results might indicate contributions from complicated reaction mechanisms as these low incident energies. At intermediate energies of Tp > 100 MeV, where reaction mechanisms are expected to be simple, there are data only for the 0-, T=O transition at Tp = 13513>14,18014, 200 MeV15, 318 MeV16, and 400 MeV17. Most of these measurements were not performed with sufficient energy resolution to separate the OW, T = 0 state at Ex = 10.96 MeV from its strong neighboring doublet (3+ and 4+) which is only about 140 keV away. It should be noted that there are no published experimental data for the 0-, T = 1 state at Ex = 12.80 MeV in this energy region. In this article, we present the measurement of cross sections and analyzing powers for the excitation of the 0-, T=l (12.80 MeV) unnatural-parity state in l60using 295 MeV inelastic proton scattering. The results will be compared with distorted wave impulse approximation (DWIA) calculations with shell-model wave functions. This provides information on tensor and spiri-spin components of effective nucleon-nucleon ( N N ) interactions. Furthermore, the data will be compared with DWIA calculations employing random phase approximation (RPA) response functions in order to assess the pionic enhancement in a large momentum-transfer region.

2. Experimental Methods

The measurement was carried out by using the West-South (WS) beam line and Grand Raiden (GR) spectrometer at the Research Center for Nuclear Physics (RCNP), Osaka University. The WS beam line and GR spectrometer are described in detail in Refs. [19,20]. Here we only present a brief description of the experimental apparatus and discuss details relevant t o the present experiment. The high resolution WS beam line20 has been designed and constructed to accomplish complete matching including both lateral and angular dispersion and focus matching with the high-resolution Grand Raiden spec-

55 1.0

103

0.5

y* 0 0

-0 5

-Global

10-2

10

20

%.m,

OM

30

40

-1.0

10

(deg)

20

%.m.

30

40

(deg)

Figure 1. Differential cross sections u c . m . and analyzing powers A Statistical errors are smaller than the d a t a points.

of elastic scattering.

trometer at RCNP. The WS beam line consists of six dipole magnets with a total bending angle of 270". This beam line is divided into five sections. The beam is focused horizontally and vertically at the end of each section. Beam line polarimeter systems positioned at the ends of the first and second sections allow the measurement of all polarization components of the beam. They are separated by a bending angle of 115" for the determination of horizontal components of the beam polarization. In dispersive mode, lateral and angular dispersions of the WS beam line are b16 = 37.1 m and b26 = -20.0 rad, necessary to satisfy dispersion matching conditions for Grand Raiden. The magnifications of the beam line are ( M z ,My) = (-0.98,0.89) and (-1.00, -0.99) for dispersive and achromatic modes, respectively. A windowless and self-supporting ice target21 was used as an oxygen target. The thin ice target with a thickness of 14.1 mg/cm2 was mounted on a thin aluminum frame attached t o a copper frame that was cooled down t o 77K using liquid nitrogen. Scattered particles were momentum-analyzed by the GR spectrometer. The spectrometer consists of two dipole (D1 and D2) magnets, two quadrupoles (Q1 and Q2), a sextupole (SX), and a multipole (MP). The spectrometer is characterized by a high resolving power of R = 37,000.

3. Data Reduction The elastic scattering data on l60are shown in Fig. 1. Differential cross sections were normalized to the known p p cross section a t flab = 14"

+

56

12.0

12.5

13.0

E x c i t a t i o n e n e r g y of l6O (MeV) Figure 2. A typical energy spectrum of the l 6 0 ( p , p ' ) scattering at T p = 295 MeV and = 30'. Results of Hyper-Gaussianpeak-fittingare also shown.

by utilizing the hydrogen present in the ice target. The beam energy was determined to be 295 f1MeV, based on the kinematic energy shift between elastic scattering from 'H and l60.The beam polarization was continuously monitored with the hydrogen polarimeter in the WS beam line. Its typical value was 0.70 f 0.01. The hydrogen in the ice target limited the useful scattering angles for inelastic scattering on l60to larger than 14". At smaller angles, the p + p events overlap the l60excited states of interest in this measurement. The elastic scattering data were analyzed using optical model potentials generated phenomenologically. The solid curves in Fig. 1 are the results with the global optical potential optimized for l6022. The gray bands represent the results by using several optical potentials parameterized for nuclei from 12C to 208Pbwith a smooth mass number dependence22. The global optical potential for l60shown by the solid curves reproduces the experimental data fairly well not only for cross sections but also for analyzing powers. Thus, in the following,we will use this optical potential in DWIA calculations for inelastic scattering.

57 10-1

10-2 h

h

56

10-3

v

d lo-'

-0.1-

10

20

4.m.

30

(deg)

10

PO

4.m.

30

(deg)

Figure 3. Angular distribution for the isovector 0 - state via l 6 0 ( p , p ' ) scattering at T p = 295 MeV. The solid and dashed curves are the results of DWIA calculations. See text for details.

4. Results and Discussion

Figure 2 shows the excitation energy spectrum of the l 6 0 ( p , p ' ) scattering at Tp= 295 MeV and el& = 30". The isovector 0- state at E, = 12.80 MeV is clearly separated from the neighboring states with an energy resolution of A E = 29-34 keV depending on the reaction angle. We have performed DWIA calculations by using the computer code DWBA9823 in which the knock-on exchange amplitude is treated exactly. The one-body density matrix elements (OBDME) for the isovector 0- transitions of the l 6 0 ( p , p ' )scattering were obtained from Ref. [24]. This shellmodel calculation was performed in the 0s-OplsOd-Oflp configurations by using phenomenological effective interactions. The single particle radial wave functions were generated by using a Woods-Saxon potential, the depth of which was adjusted t o reproduce the binding energy. The effective NN interaction was taken from the t-matrix parameterization of the free NN interaction by Franey and Love25 at 325 MeV. Figure 3 compares the preliminary result of the angular distribution for the isovector 0- state with the DWIA Calculation. The calculation reproduces the cross sections around the 2nd maximum at 14' without a normalization factor, while it underestimates and slightly misses the 3rd maximum. Furthermore the analyzing power data are not reproduced by this calculation completely by giving the opposite sign. We have also performed a DWIA calculation by using the density- and energy-dependent

58

Figure 4.

The results of DWIA+RPA and DWIA+free calculations.

in-medium t-matrix evaluated from the Gmatrices26. The Gmatrix calculations were performed by using the Paris NN potential. The results are shown in Fig. 3 as the dashed curves. The calculated cross sections give a similar angular distribution compared with those with the Franey and Love free t-matrix, but they are larger by a factor, of 2. On the contrary, the results of DWIA calculations for the analyzing powers depend on the choice of the t-matrix. The calculated analyzing powers with the in-medium t-matrix give the correct sign, but they are smaller compared with the experimental data. Finally we compared our experimental data with the DWIA+RPA calculation. The RPA calculations are performed without the commonly used universality ansatz (ghN = &A = & A ) , namely all of the g’s are treated i n d e ~ e n d e n t l y ~The ~ . nonlocality of the mean field is treated by an effective mass m*. These parameters in the present RPA calculation are (dNN, &A, & A ) = (0.6, 0.4, 0.5) and m*(0)=0.7m~. The formalism of DWIA calculations is described in Ref. [28]. The results of DWIASRPA and DWIA+free response calculations are shown in Fig. 4 as solid and dashes curves, respectively. The DWIA+RPA calculation predicts an enhancement of the 3rd maximum of the cross sections compared with the 2nd maximum, and it reproduces the 2nd and 3rd maxima simultaneously with a normalization factor of 0.5. Thus the experimental data supports the enhancement of the pionic 0- mode in nuclei as

59

is predicted in the RPA calculation. Acknowledgments

We are grateful to M. Ichimura and H. Sakai for their helpful correspondence. This work is supported in part by the Grants-in-Aid for Scientific Research Nos. 12740151, and 14702005 of the Ministry of Education, Science, Sports and Culture of Japan. References 1. 2. 3. 4. 5.

G. A. Gagliardi et al., Phys. Rev. Lett. 48,914 (1982). P. Guichon et al., Phys. Rev. C19,987 (1979). E. G. Adelberger et al,, Phys. Rev. Lett. 46,695 (1981). K. Kubodera et al., Phys. Rev. Lett. 40,755 (1978). W. G. Love et al., in Proceedings of International Conference on Spin Excitations, Telluride, Colorado, 1982, edited by F. Petrovich et al., (1982). 6. H. Orihara et al., Phys. Rev. Lett. 49,1318 (1982). 7. C. H. Llewellyn Smith, Phys. Lett. 128B,107 (1983). 8. M. Ericson and A. W. Thomas, Phys. Lett. 128B,112 (1983). 9. B. L. Friman, V. R. Pandharipande, and R. B. Wiringa, Phys. Rev. Lett. 51, 763 (1983). 10. E. L. Berger, F. Coester, and R. B. Wiringa, Phys. Rev. D29,398 (1984). 11. D. Stump, G. F. Bertsch, and J. Pumlin, A I P Conf. Proc. 110,339 (1984). 12. K. Hosono et al., Phys. Rev. C30,746 (1984). 13. J . J. Kelly, et al., Phys. Rev. C39,1222 (1989). 14. J. J. Kelly et al., Phys. Rev. C41,2504 (1991). 15. R. Sawafta et al. IUCF Scientific and Technical Report, May 1988-April 1989, p.19. 16. J. J. Kelly et al., Phys. Rev. C43,1272 (1991). 17. J . D. King et al., Phys. Rev. C44,1077 (1991). 18. W. M. Alberico, M. Ericson, and A. Molinari, Nucl. Phys. A379,429 (1982). 19. M. Fujiwara et al., Nucl. Instrum. Methods Phys. Res. A422,484 (1999). 20. T. Wakasa et al., Nucl. Instrum. Methods Phys. Res. A482,79 (2002). 21. T. Kawabata et al., Nucl. Instrum. Methods Phys. Res. A459,171 (2001). 22. S. Hama et al., Phys. Rev. C41,2737 (1990). 23. J. Raynal, Program DWBA98, NEA 1209/05,1999. 24. T. Kawabata et al., Phys. Rev. C65,064316 (2002), and references therein. 25. M. A. Franey and W. G. Love, Phys. Rev. C31,488 (1985). 26. H. V. von Geramb, A I P Conf. Proc. 97,44 (1982). 27. K. Nishida and M. Ichimura, Phys. Rev. C51,269 (1995). 28. K. Kawahigashi, K. Nishida, A. Itabashi, and M. Ichimura, Phys. Rev. C63, 044609 (2001).


III. RELATIVISTIC EFFECTS, DIBARYON, N N INTERACTIONS


63

NUCLEAR MEDIUM EFFECTS ON THE COULOMB SUM VALUES

HARUKI KURASAWA Department of Physics, Faculty of Science, Chiba University, Chiba 263-8522, Japan

TOSHIO SUZUKI Department of Physics, Fukui University, Fukui 910-8507, Japan and RIKEN, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan

Nuclear medium effects on the Coulomb sum values are investigated in a relativistic model. The Coulomb sum values of non-relativistic models are strongly quenched by the antinucleon degrees of freedom with effective mass due to the Lorentz scalar potential. The quenching is consistent with the recent analysis of the experimental data. Within the same framework, the quenching is understood as a result of modification of the nucleon size or of the omega-meson mass. Effects of the antinucleon degrees of freedom on giant resonance states etc. are also discussed.

1. Introduction

In non-relativistic models, there is Coulomb sum rule on the longitudinal response function for electron scattering off nuclei. The Coulomb sum is defined by

where q and qo denote the four-momentum transfer and its time-component from the electron to the nucleus, respectively, and 2 stands for the number of the protons. The longitudinal response function is given by

64

with the time-component of the nuclear four-current in non-relativistic models, Z

exp{iq

JO(Q) =

'

rk}GE,(Q).

(3)

k= 1

The Sachs form factor G E , ( Qtakes ) into account the size of the proton. In non-relativistic models, thus the nucleon form factor is factorized, so that one can define the response function which depends on nuclear structure only,

Then, the closure property gives

This provides the Coulomb sum rule:

The most of non-relativistic calculations satisfies the sum rule in the region IqI > 400 MeV which is about 2pF, p~ being the Fermi momentum. Experimentally the Coulomb sum rule has been studied for a long time, since it is a fundamental sum rule of non-relativistic models. Although there were disagreements between experiment, recently Morgenstern et al.' have been reanalysed the world data, and found that the sum rules are broken. In the region of 141 > 400, the sum rule value is quenched by 20 t o 30 % in medium heavy and heavy nuclei, while they are exhausted in light nuclei. The purpose of the present paper is to explore the quenching in terms of the medium effects in a relativistic model. In the following section we will show our results which reproduce very well the experimental data. The quenching is due to the antinucleon degrees of freedom with medium effects. In section 3, we will explain the reason why the antinucleon degrees of freedom yield the quenching of the sum rule values, and in section 4, will show that the antinucleon degrees of freedom are also required in the description of the giant resonance states, etc. in the same way as in the Coulomb sum values. The final section will be devoted t o a brief summary.

65 2. The Coulomb Sum in a Relativistic Model

The relativistic nuclear four-current is given by

where Fl(q2)and F2(q2) represent the Dirac and Pauli form factors, respectively. First, it must be explored whether or not the non-relativistic reduction of the nuclear four-current is appropriate for the present discussions. This was already studied in our previous paper2 and we showed that the relativistic current only did not provide us with the quenching of the Coulomb sum values. I t should be noted, however, that the results do not mean relativistic effects to be negligible. Various relativistic effects are cancelled each other, and therefore we need a self-consistent relativistic model t o estimate the Coulomb sum value. Moreover, we should remember that the Sachs form factor used in the non-relativistic current Eq.(3) includes parts of the relativistic effects. Next we explored effects of the RPA correlations in the nucleon space with use of the relativistic u-w mode13y4, since the pmeson exchanges contribute a little to the Coulomb sum values4. Our calculations have been performed for nuclear matter for simplicity, expecting that the the Coulomb sum values are almost A-independent. This fact was confirmed in the relativistic Hartree approximation2, and in RPA by comparing the Coulomb sum values of nuclear matter with those of the local density approximation5. In Fig.1 are shown the results by the thin solid curve together with the world data provided by Morgenstern et a1.ly6 The thin dashed curve is obtained in the Hartree approximation, which is similar to the non-relativistic one2. In the RPA calculation the baryon-meson form factors are taken into account, which reduce a little the particle-hole correlations. T h results show that we obtain the quenching of the Coulomb sum values due to the RPA correlations, but it is not enough to explain the experimental data. When a sum rule is broken, there is a possibility which requires new degrees of freedom. For examples, the enhancement of the Thomas-ReicheKuhn sum rule for photo-absorption indicates the existence of the mesonexchange current. The quenching of the Gamow-Teller sum rule is explained by introducing the A-degrees of freedom in nuclei. Those contribute also to the transverse response function, but are not relevant to the present problem of the longitudinal response. This fact is one of the reasons why the breaking of the Coulomb sum rule is serious for non-relativistic models. The one of the possibilities t o break the Coulomb sum rule is effects

66

0.01 I

0.0

I

I

0.2

I

I

0.4

I

I

I

0.6

I

0.8

I

I

1.0

I

1.2

141 (GeV/c)

Figure 1. The Coulomb sum values as a function of the momentum transfer 191. The thin and thick dashed curves are calculated in the mean field and the Hartree approximation, respectively. In the Hartree approximation the antinucleon degrees of freedom are taken into account. The thin and thick solid curves are obtained in RPA based on the mean field and the renormalized Hartree approximation, respectively. The experimental data are taken from ref.'-6

of the antinucleon degrees of freedom. These effects are able to be taken into account consistently in the relativistic model. Since the antinucleons are excited in the time-like region of the four-momentum space, they are not excited directly through electron scattering. It is, however, possible to contribute to the longitudinal response through virtual excitations in RPA correlations. Finally we have performed the relativistic RPA calculations including the antinucleon space7i8. The RPA correlations are calculated using the basis obtained with the renormalised Hartree approximation. The baryonmeson(a, u)coupling constants are determined so as to reproduce the density and binding energy of nuclear matter. The results are shown in Fig.1 by the thick solid curveg. The thick dashed curve shows the results without the RPA correlation, but using the renormalised Hartree basis. It is seen that the antinucleon degrees of freedom well reproduce the experimental data. In particular, the present calculations show that the quenching in the region IqI > 700 MeV is mostly due to the antinucleon degrees of

67 freedom.

3. Medium Effects on the Nucleon size and the w-Meson

Mass In this section we discuss the reason why we obtain the quenching due to the antinucleon degrees of freedom. We present two ways to understand the reason of the quenching.

Figure 2. The Dirac form factor F1(q2).It contains a contribution from nucleonantinucleon excitations through the w-meson exchange.

The first one is given in terms of the Dirac form factor of the nucleon. The Dirac form factor includes a contribution from N - excitations as shown in Fig.2. Those excitations are included in the present RPA calculations, taking into account the Pauli principle. In the free space, N and N have the free nucleon mass M , but in nuclear medium, the effective mass

M*.

M*

= M - Us,

(8)

Us being the Lorentz scalar potential. This fact modifies the Dirac form factor and the nucleon size which is given approximately by

(P)M

(9)

We have shown that the modification of the proton size is expressed in the present model as”,

which is estimated to be, for our parameters,

((r;)*/(r;))’”

= 1.146

( M > M*

= 0.731M).

(11)

The above swelling of the proton size means the quenching of the proton form factor F;(q2)< Fl(q2). As a result, we have obtained the quenching of the Coulomb sum values.

68

The second way to interpret the quenching is according to the vector meson dominance model. The present RPA calculation yields the self-energy of the w-meson which includes the N - N excitations with the effective mass. This means that the w-meson mass is modified in the nuclear medium from m, to the effective mass m:. In the present model, it is obtained to be m: = 0.696m,. In fact the swelling of the nucleon size in Eq.(lO) is expressed in terms of the effective mass",

The above expression is understood according to the Brown-Rho's arguments12. The Dirac form factor is approximately given by

which provides us the proton size (T;)

= ($)s

+ ($)

= (0.811fm)2,

(14)

V

forA = 840 MeV. Now when we notice the fact that

A = 840 MeV

M

m, = 783 MeV,

(15)

the isoscalar part of the form factor is approximately written as

='(--)A 2h2- +'(-)(

A2

2

2

A2-q2

q2 ).(la) m: - q 2

The above replacement gives the expression of the proton size:

This result shows that if the w-meson mass is modified from m, to the effective mass m:, then we have the same equation as Eq.(12) for the amount of the modification of the proton size. Thus the quenching of the Coulomb sum values is understood by the antinucleon degrees of freedom with the effective mass which is smaller than the free mass. Those yield the swelling of the nucleon size, that is, the quenching of the nucleon form factor. The relationship between the effective mass and the nucleon size may be interpreted more naively according to the uncertainty principle as to the energy and time.

69 4. Antinucleon Effects on Low-Energy Phenomena

In the previous sections we have shown that N-N excitations play an important role in the RPA correlations. As an effect of the N-N states, we could explain the quenching of the Coulomb sum values. In this section, we will show that these N-N excitations are necessary not only for the Coulomb sum values, but also for low-energy phenomena in the relativistic model. The effect of N-N states on the Coulomb sum values is order of 1qI2 as was shown in Eq.(9). These effects appear also at IqI -+ 0, where the RPA correlations are responsible for nuclear collective states. The existence of the giant monopole states are well established, of which excitation energy is about wo = 79/A1l3 MeV. In nuclear matter, it is given in terms of the Landau-Migdal parameters Fo and F1 as13

where E F denotes the Fermi energy. The above expression is the same as the one of non-relativistic models, if we replace E F by the nucleon mass. There are two ways to obtain the above Landau-Migdal parameters. The first way is from the second derivative of total energy density with respect to the occupation number of the quasiparticle. The second way is, after calculating the RPA response function directly in the configuration space of the u - w model, by comparing it with that of the Landau theory. In the second way we can prove that we should include N-N states in the configuration space of RPA14. Namely, when we take into account the NN states in the RPA, we obtain the correct forms of the Landau-Migdal parameters as

where we have defied the following notations, introducing mo as the bare mass of the w-meson in the renormalization procedure,

70

On the other hand, if we neglect the N-N states, we cannot obtain the denominators in Eq.(19), and as a result we have the value of Fo < -1, which implies the collapse of the nucleus. The N-N states are also required in the description of the centre of mass motion and the giant quadrupole states in RPA14. The nuclear current and magnetic moment are also described correctly, when we take into account the N-N excitations in RPA14. Thus, the antinucleon degrees of freedom are necessary in the description of both low and high energy phenomena. In low energy phenomena, however, those effects are hidden in excitation energies or the magnetic moment, but in high energy phenomena they appear as the quenching of the Coulomb sum values. 5 . Conclusion

The quenching phenomena of the Coulomb sum values, observed in the world data, is well reproduced in the relativistic RPA where the (T and wmeson exchanges are assumed. The quenching is understood as a result of the modification of the nucleon size or the w-meson mass. These modification is caused by the antinucleon degrees of freedom with the effective mass in RPA. The antinucleon degrees of freedom is also necessary for the description of low energy phenomena in the relativistic RPA.

References 1. J. Morgenstern et al., Phys. Lett. B515 (2001) 269. 2. S. Nishizaki, T. Maruyama, H. Kurasawa and T. Suzuki, Nucl. Phys. A485 (1988) 515. 3. H. Kurasawa and T. Suzuki, Nucl. Phys. A445 (1985) 685. 4. H. Kurasawa and T. Suzuki, Phys. Lett. B173 (1986) 377. 5. H. Kurasawa and T. Suzuki, to be published. 6. J. Morgenstern, private communication. 7. H. Kurasawa and T. Suzuki, Nucl. Phys. A490 (1988) 571. 8. H. Kurasawa and T. Suzuki, Prog. Theor. Phys. 86 (1991) 773. 9. H. Kurasawa and T. Suzuki, to be published. 10. H. Kurasawa and T. Suzuki, Phys. Lett. B208 (1988) 160; 211B (1988) 500. 11. H. Kurasawa and T. Suzuki, Prog. Theor. Phys. 84 (1990) 1030. 12. G. E. Brown and M. Rho, Phys. Lett. B222 (1989) 324. 13. S. Nishizaki, H. Kurasawa and T. Suzuki, Nucl. Phys. A462 (1987)687. 14. H.Kurasawa and T. Suzuki, Phys. Lett. R474 (2000) 262.

71

Y-SCALING ANALYSIS OF THE DEUTERON WITHIN THE LIGHT-FRONT DYNAMICS METHOD

M.K. GAIDAROV, M.V. IVANOV AND A.N. ANTONOV Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria The concept of relativistic scaling is applied to describe the most recent data from inclusive electron-deuteron scattering a t large momentum transfer. We calculate the asymptotic scaling function f(y) of the deuteron using its relationship with the nucleon momentum distribution. The latter is obtained in the framework of the relativistic light-front dynamics (LFD) method, in which the deuteron is described by six invariant functions fi (z=l,...,6) instead of two (S and D waves) in the nonrelativistic case. Comparison of the LFD asymptotic scaling function with other calculations using S and D waves corresponding to various nucleon-nucleon potentials, as well as with the Bethe-Salpeter result is made. It is shown that for IyI > 400 MeV/c the differences between the LFD and the nonrelativistic scaling functions become larger.

1. Introduction

High-energy electron scattering from nuclei can provide important information on the wave function of nucleons in the nucleus. In particular, using simple assumptions about the reaction mechanism, scaling functions can be deduced that, if shown to scale (i.e., if they are independent of the momentum transfer), can provide information about the momentum and energy distribution of the nucleons. Several theoretical studies have indicated that such measurements may provide direct access to studies of short-range nucleon-nucleon (NN) correlation effects. Since West’s pioneer work 4 , there has been a growth of interest in yscaling analysis, both in its experimental and theoretical aspects. This is motivated by the importance of extracting nucleon momentum distributions from the experimental data. West showed that in the impulse approximation, if quasielastic scattering from a nucleon in the nucleus is a dominant reaction mechanism, a scaling function F ( y ) could be extracted from the measured cross section which is related to the momentum distribution of the nucleons. In the simplest approximation the corresponding scaling variable ‘7’1~

72

y is the minimum momentum of the struck nucleon along the direction of the virtual photon. In principle, the scaling function F ( y , Q 2 ) depends on both y and momentum transfer Q2 (-Q2is the square of the four-momentum transfer), but a t sufficiently high Q2 values the dependence on Q2 should vanish yielding scaling. Recently inclusive electron scattering has been studied a t the Thomas Jefferson National Accelerator Facility (TJNAF) with 4.045 GeV incident beam energy from C, Fe and Au targets to Q2 M 7 (GeV/c)’. Data were also taken using liquid targets of hydrogen and deuterium ‘. The data presented in represent a significant increase of the Q2 range compared to previous SLAC measurement 7, in which an approach to the scaling limit for heavy nuclei is suggested but for low values of IyI < 0.3 GeV at momentum transfers up to 3 (GeV/c)2, and, moreover, a scaling behaviour is observed for the first time at very large negative y (y= -0.5 GeV/c). From theoretical point of view the extended region of y measured at TJNAF is of significant importance since this is a regime where the nucleon momentum distribution is expected to be dominated by short-range NN correlations. The aim of our work is using the nucleon momentum distribution n ( k ) obtained with the LFD method t o calculate the deuteron scaling function. The result for the asymptotic function is compared with the recent TJNAF data measured at six values of Q2. In particular, the scaling behavior observed for very large negative y providing momenta higher than those corresponding to existing experimental data for n ( k ) may allow to distinguish the properties of the covariant LFD method from the POtential approaches. The comparison with the BetheSalpeter (BS) result for the scaling function l1 serves as a test for the consistency of both covariant approaches treating the deuteron relativistically in the case of y-scaling. 516

2. Basic relations in the y-scaling method The scaling function is defined as the ratio of the measured cross section to the off-shell electron-nucleon cross section multiplied by a kinematic factor:

where Z and N are the number of protons and neutrons in the target nucleus, respectively, up and u, are the off-shell cross sections, M is the (v is the energy loss) is the mass of the proton, and 141 = three-momentum transfer . In analysing quasielastic scattering in terms of the y-scaling a new variable y = y ( q , v ) is introduced. Then the nuclear

d

m

73 structure function which is determined using the spectral function P ( k ,E ) as

s

E , , L(,q , v )

F ( q , v ) = 27r

Em

kdkP(k,E ) ,

(2)

ri

can be expressed in terms of q and y rather than q and v (see Eq. (1)).In Eq. (2) E = Em,, E i - , is the nucleon removal energy with E i - l being the excitation energy of the final A - 1 nucleon system. The most commonly used scaling variable y is obtained l 2 starting from relativistic energy conservation, setting k = y, = 1 and the excitation kq. energy E>-,=O, and is defined through the equation

+

v

+ MA = ( M 2+ q2 + y 2 + 2yq)lI2 + ( M ; - l +

2 112

1~ )

,

(3)

where M A is the mass of the target nucleus and MA-^ is the mass of the A - 1 nucleus. Therefore, y represents the longitudinal momentum of a nucleon having the minimum removal energy ( E = E,,,, i.e. Ei_,=O). At high values of q a pure scaling regime is achieved, where k,, M Iy - ( E - Ern,,)[ and Eq. (2) becomes F(%Y)

+

f(Y) = 2

q m

dEJm

kdkP( k , E ) .

l~-(E-E~,,,,t)l

E,,,,,

(4)

In Eq. (4)the particular behavior of P ( k ,E ) at large k and E is used in order to extend the upper limits of integration to infinity ' . In the deuteron one always has Ei-,=O, so that the spectral function is entirely determined by the nucleon momentum distribution n ( k ) , i.e. P ( k , E ) = n(k)G(E- Em,,), and, consequently, k,,, = IyI for any value of q. The scaling function (4) reduces to the longitudinal momentum distribution 00

f ( y ) = 2 7 r s k dk n ( k ) . IYI

(5)

3. Results for the nucleon momentum distribution and the asymptotic scaling function of deuteron The LFD calculations have shown (for more details, see Ref. ') that, as expected, the most important contributions to the total n ( k ) give terms related to the f l , f 2 and fs functions

n(k)-n1(k)

+ n2(k) +

725(k).

(6)

The contributions of 1 2 1 , 722, 7212 = n1 +n2 and 725 are compared in Figure 1. It can be seen that, while the functions f l and f 2 give a good description

74

of the y-scaling data of n ( k ) for k < 2 fm-’ (like the S- and D-wave functions in the nonrelativistic case), it is impossible to explain the highmomentum components of n ( k ) a t k > 2 fm-’ without the contribution of the function f s . We note that the deviation of the total n ( k ) from the sum 7x12 = n1 n2 starts at k around 1.8 fm-’. All this shows the important role of NN interactions which incorporate exchange of relativistic mesons in the case of the deuteron.

+

0

1

2

3

4

5

k [fm”]

Figure 1. The nucleon momentum distribution in deuteron. The contributions of n1, n z , niz = ni +nz and 125 are presented. The y-scaling data are from z . The normalization is: J n(/c)&L= 1.

The scaling function for deuteron calculated within the LFD method is shown in Figure 2. It is compared with the TJNAF experimental data for all measured angles. The Q2 values are given for Bjorken z = Q 2 / 2 M v = 1 and correspond t o elastic scattering from a free nucleon. It is seen from Figure 2 that the relativistic LFD scaling function is in good agreement with the data in the whole region of negative y available. As known, the scaling breaks down for values of y > 0 due t o the dominance of other inelastic processes beyond the quasielastic scattering. Our LFD deuteron scaling function is also compared in Figure 2 with the scaling function obtained within the BS formalism ll. A small difference between the two results is observed for y < -400 MeV/c but, at the same time, the theoretical LFD

75 scaling function is closer to the experimental data in the same region of y. The fact that both LFD and BS functions reveal similar behavior is a strong indication in favor of the consistency of the two relativistic covariant approaches in case of the y-scaling.

10' a

-

lo"

Q2=0.97 Q2=2.78 Q2=4.24

4

Q2=1.94 Q2=3.53 Q2=4.92

2 lo' z * 2 lo" h

10"

10.' -loo0 -900 -800

-700

-600

-5M)

-400 -300 -200

-100

0

Y [MeV]

Figure 2. The scaling function of deuteron. The experimental data for different Q2 values are from '. The solid and dashed curves represent the LFD calculations of this work and BS result of Ref. ll.

In Figure 3 the asymptotic relativistic LFD and BS scaling functions f ( y ) are compared with the nonrelativistic ones, calculated with various NN potentials, such as the Argonne v18 13, the Nijmegen-I1 l4 and Paris 1980 15. It is shown that for IyI > 400 MeV/c both LFD and BS curves start to deviate from the nonrelativistic scaling functions. The result for f(y) calculated using the Nijmegen-I1 NN potential is in better agreement with the experimental data than those using other potentials and is in accordance with the result for n ( k ) shown in '. For instance, by a thorough comparison between the relativistic Bethe-Salpeter and the nonrelativistic scaling functions of deuteron it has been found in l 1 that the two functions start to sensibly differ also at IyI > 400 MeV/c. Thus, the necessity to treat realistically the relativistic dynamics inside the deuteron in a way different from the potential approaches becomes apparent. In this sense, the results

76

calculated for both momentum distribution and asymptotic scaling function confirm the abilities of the LFD method to describe with a good accuracy the experimental data measured at high momentum transfers.

1

-1000 -900

.

1

.

1

.

-800 -700

1

.

1

.

400 -500

1

.

-400

1

.

1

.

1

-300 -200

.

1

-100

.

0

Figure 3. The asymptotic scaling function of deuteron. Line convention refering to calculations using LFD and potential approaches, as well as BS result l 1 is given (see also the text). The experimental data l6 are given by the full circles.

4. Conclusions

In the present paper inclusive electron-deuteron scattering data have been analyzed in terms of the y-scaling function within the light-front dynamics method. For this purpose, the nucleon momentum distribution in deuteron has been used in order to calculate the asymptotic scaling function. For the trivial case of deuteron, for which the structure function (Eq. (4)) coincides with the longitudinal momentum distribution (Eq. (5)) we have found a good agreement of the calculated scaling function with the experimental data. Thus, the concept of relativistic y-scaling can be introduced in the LFD relativistic description of inclusive quasielastic eD scattering, in the same way as it is done in the conventional nonrelativistic approach, i.e. by introducing a scaling function (which, in the scaling regime, is nothing but the nucleon longitudinal momentum distribution), and in terms of the same

77 variable y. It has been pointed out that for IyI > 400 MeV/c the differences between the LFD and the nonrelativistic scaling functions are very large. Exploring the light-front dynamics, we continue in this paper our analysis of important deuteron characteristics. The effective inclusion of the relativistic nucleon dynamics and of short-range NN correlations can be better seen when analyzing electron scattering at high momentum transfer from complex nuclei, for which a proper theoretical y-scaling analysis is still lacking. Although scaling violation effects due to final-state interactions (FSI) sharply decrease with increasing momentum transfer, a consistent treatment of both FSI and nucleon binding must be made in order t o perform a precise comparison with the new TJNAF data. Such an investigation is in progress.

Acknowledgments This work was partly supported by the Bulgarian National Science Foundation under the Contracts Nrs.@-809 and @-905.

References 1. Xiangdong Ji and J. Engel, Phys. Rev. C40, R497 (1989). 2. C. Ciofi degli Atti, E. Pace and G. SalmB, Phys. Rev. C43, 1155 (1991). 3. M.K. Gaidarov, A.N. Antonov, S.S. Dimitrova and M.V. Stoitsov, Int. J. Mod. Phys. E4, 801 (1995). 4. G.B. West, Phys. Rep. 18, 263 (1975). 5. J. Arrington et al., Phys. Rev. Lett. 82, 2056 (1999). 6. J. Arrington, PhD Thesis (1998). 7. D.B. Day et al., Phys. Rev. Lett. 5 9 , 427 (1987). 8. A.N. Antonov, M.K. Gaidarov, M.V. Ivanov, D.N. Kadrev, G.Z. Krumova, P.E. Hodgson and H.V. von Geramb, Phys. Rev. C65, 024306 (2002). 9. J. Carbonell and V.A. Karmanov, Nucl. Phys A 5 8 1 , 625 (1995). 10. J. Carbonell, B. Desplanques, V.A. Karmanov and J.-F. Mathiot, Phys. Rep. 300, 215 (1998) (and references therein). 11. C. Ciofi degli Atti, D. Faralli, A.Yu. Umnikov and L.P. Kaptari, Phys. Rev. C60, 034003 (1999). 12. E. Pace and G. SalmB, Phys. Lett. B110, 411 (1982). 13. R.B. Wiringa, V.G.J. Stoks and R. Schiavilla, Phys. Rev. C51, 38 (1995). 14. V.G.J. Stoks, R.A.M. Klomp, C.P.F. Terheggen and J.J. de Swart, Phys. Rev. C49, 2950 (1994). 15. M. Lacombe, B. Loiseau, J.M. Richard, R. Vinh Mau, J . CbtC, P. PirBs and R. de Tourreil, Phys. Rev. C21, 861 (1980). 16. D. Day e t al., Annu. Rev. Nucl. Part. Sci. 40, 357 (1990).

78

SEARCH FOR SUPER-NARROW DIBARYON RESONANCES BY THE pd + p d X AND p d + p p X REACTIONS

A. TAMII, M. HATANO, H. KATO, Y . MAEDA, T. SAITO, H. SAKAI, S. SAKODA AND N. UCHIGASHIMA Department of Physics, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan

K. HATANAKA, D. HIROOKA, J. KAMIYA AND T. WAKASA Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-004 7, Japan

K. SEKIGUCHI The Institute of Physical and Chemical Research, Wako, Saitama 351-0198, Japan

T. UESAKA AND K. YAK0 Center for Nuclear Study, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan Dibaryons which have very narrow decay widths, called super-narrow dibaryons, are searched for by the p+d scattering at E p = 295 MeV. Th e experiment was carried out at the Research Center for Nuclear Physics employing a two-arm magnetic spectrometer. A good mass resolution of 0.95 MeV and background free condition have been achieved. No resonance has been observed in the mass region of 1896-1914 MeV in contradiction with the experimental result at the Institute for Nuclear Research. T h e upper limits of the dibaryon production cross section have been determined.

1. Introduction

One of the interesting predictions of QCD is the possibility of the existence of six-quark states, i.e. dibaryons. Much experimental work has been devoted to the search for the dibaryons as well as theoretical predictions of their masses and quantum numbers.l Among many possible dibaryons, those which have very narrow decay widths, called super-narrow dibaryons (SND), are of particular interest. The SNDs are supposed to satisfy the

79

following two assumptions.

(1) The SNDs have a symmetric wave function in terms of nucleons. (2) The mass of SNDs is less than that of two nucleons and a pion. Due to the first assumption their decay into two nucleons is forbidden by the Pauli principle, and due to the second assumption they cannot decay into two nucleons and a pion. Consequently the SNDs must decay by emitting a gamma ray into two nucleons. The weakness of the electromagnetic force comparing with the strong force makes their decay width as small as 1 keV. A few SND candidates have been reported. Khrykin et al. measured pp + yyX reaction at Ep=216 MeV at the Joint Institute for Nuclear Research (JINR). They observed a candidate at 1923 MeV.2 But the resonance was not observed in the proceeding pp 3 yypp measurement by the WASA/CELSIUS collaboration at U p p ~ a l a Khrykin .~ et al. have made an additional measurement and have claimed the existence of a resonance at about 1956 MeV.4 Recently at Institute for Nuclear Research (INR) a group have reported two possible SND resonances below the pion emission threshold at 1905 and 1924 MeV, and later another one at 1942 MeV.' They measured the pd + p X reaction at Ep=305 MeV in coincidence with a proton or a deuteron from the decay of X . They employed a two-arm mass spectrometer which consisted of BGO and plastic scintillators Since the observed widths of the resonances were equivalent to the experimental mass resolution of 3 MeV, the resonances are considered as SND candidates. The production cross section of the resonance at 1905 MeV was estimated to be 8 f 4 pb/sr on the assumption that the dibaryon mainly decays into the p+n+y channel. The observed resonances were, however, placed on a large number of unknown background events such that the presence of the resonances was marginal. Our experiment employing a two-arm magnetic spectrometer is designed to search for the resonance at 1905 MeV with the same kinematical conditions but with much better mass resolution and lower background. 2. Experiment

The experiment was performed at the Research Center for Nuclear Physics, Osaka University. Figure 1 illustrates the experimental setup. We employed a proton beam with an energy of 295 MeV, which was accelerated sequentially by the AVF and ring cyclotrons. The beam intensity was 15-20 nA. A deuterated polyethylene (CD2) target7 with a thickness

80 of 44 mg/cm2 was placed in a scattering chamber. The CD2 target was tilted by 20" in order to reduce multiple scattering of outgoing protons. Beam Dump

Grand Raiden

LAS

70.0"

p

300MeV

Figure 1. Schematic view showing the kinematic coordinates used for the dibaryon search along with the magnetic spectrometers, GR for the detection of scattered protons and LAS for decay charge particles (protons or deuterons) from the dibaryons.

The momenta of scattered protons were analyzed by using the high resolution magnetic spectrometer Grand Raiden (GR)8. The GR was placed at 70". The protons were detected at the focal plane by a pair of multiwire drift chambers (MWDC) of vertical drift type. The energy range of the detected protons was 108-118 MeV, which corresponds to a missing mass range of 1895-1915 MeV. The produced dibaryons are considered to decay immediately into d+y or p+n+y. The large acceptance spectrometer (LAS)' was placed at 34.9" to detect either a proton or a deuteron from the dibaryon decay. The decay particles were detected at the focal plane of the LAS by a pair of MWDCs and two planes of plastic scintillation counters. The energy ranges of detected protons and deuterons were 74-130 and 112198 MeV, respectively. Data were also taken with a carbon target with a thickness of 3 mg/cm2 for subtracting background events from carbon. Absolute value and resolution of the missing mass have been calibrated by measuring the deuteron mass by the p+d elastic scattering at 0lab = 70". As shown in Fig. 2, the resolution was 0.95 MeV which was better than the one at INR by a factor of three. The main ingredient (0.9 MeV) of the resolution was the uncertainty of the scattering angle arising from the angular spread of the beam (0.2") together with multiple scattering in the target (0.14"). The phase space of the decay products covered by the LAS has been

81 I

'

~

~

'

I

"

'

~

"

'

~

"

'

'

"

I

~

~

"

l

'

'

~

'

p + d elastic scattering s

-

0 2000

-

4

v

-

1872

1873

1874

1875

1876

1877

1878

1879

Missing Mass (MeV)

Figure 2. Missing mass of the p+d elastic scattering for checking the experimental mass resolution. The obtained resolution was 0.95 MeV.

simulated by a Monte Carlo method for each of two decay channels. In the simulation of the p+n+y decay channel, uniform three-body decay of the dibaryon was assumed. 3. Results and Discussions The missing mass spectra of the pd + p X reaction are shown in Fig. 3 for the (a) d+y and (b) p+n+y channels. The experimental data are plotted by the solid circles with statistical error bars. No significant peak has been observed over the measured mass range. The results are consistent with zero with reduced chi-squares of 1.0 and 1.1 for the d+y and p+n+y channels, respectively. The solid curve represents the estimated yields on the assumptions of a 8 pb/sr cross section5 at 1905 MeV and a small decay width comparing with the experimental resolution. We conclude that a resonance with a cross section of 8pb/sr has not been observed. The upper limits of the dibaryon production cross section have been determinedlo as shown in Fig. 4. The upper limits are less than 2 pb/sr a t the 90% confidence level in the mass region of 1897-1911 MeV. An improved experiment is planned in the spring of 2003. The experiment will cover the mass region of all the three candidates and provide an order higher statistical sensitivity. We will be able t o draw decisive conclusion on the existence of all the candidates observed at the INR.

4. Acknowledgments We are indebted to the RCNP cyclotron crew for the excellent proton beam. We wish t o thank I. Daito for valulable information on the production of

82

1900 1905 1910 Missing Mass (MeV)

Figure 3. Dibaryon production cross section multiplied by the branching ratio as a function of missing mass of the p d + p X reaction for the (a) d+y and (b) p + n + y decay channels. The experimental results are shown by solid circles with statistical error bars. The solid curve in (b) represents the expected data on assumptions of a differential cross section of 8 pb/sr at 1905 MeV.

thin CD2 targets.

References 1. For example, see a review in B. Tatischeff et al., Phys. Rev. C 59 (1999) 1878. 2. A.S. Khrykin et al., in Proc. VII Int. Conf. on Meson-Nucleon Physics and the Structure of the Nucleon, TRIUMF, Vancouver, 1997. 3. H. CalCn et al., Phys. Lett. B 427 (1998) 248. 4. A.S. Khrykin et al., Phys. Rev. C 64 (2001) 034002. 5. L.V. Fil'kov et al., Phys. Rev. C 61 (2000) 044004. 6. L.V. Fil'kov et al., preprint nucl-th/0101021. 7. Y. Maeda et al., Nucl. Instrum. Methods Phys. Res. A 490 (2002) 518. 8. M. h j i w a r a e t al., Nucl. Instr. and Meth. in Phys. Res. A 422 (1999) 484. 9. N. Matsuoka e t al., RCNP Annual Report 1991. 10. A. Tamii e t al., Phys. Rev. C. 65 (2002) 047001.

83

Figure 4. Upper limits of the dibaryon production cross section multiplied by the branching ratio at the 90% confidence level as a function of dibaryon mass for the d+y (dashed curve) and p+n+7 (solid curve) channels. The result of Fil'kov et al. is plotted by the open square.

84

A QUARK-MODEL N N INTERACTION AND ITS APPLICATION TO THE THREE-NUCLEON AND NUCLEAR-MATTER PROBLEMS

Y. FUJIWARA,l M. KOHN0,2 Y. SUZUKI,3 C. NAKAMOT0,4 K . MIYAGAWA~AND H. NEMURA~ 'Department of Physics, Kyoto University, Kyoto 606-8502, Japan E-mail: [email protected] Physics Division, Kyushu Dental College, Kitakyushu 803-8580, Japan Department of Physics, Niigata University, Niigata 950-2181, Japan 4Suzuka National College of Technology, Suzuka 510-0294, Japan Department of Applied Physics, Okayama Science University, Okayama 700-0005, Japan Institute of Particle and Nuclear Studies, K E K , Tsukuba 305-0801, Japan Baryon-baryon interactions of the complete baryon octet ( B s ) are investigated in a unified framework of the ( 3 q ) - ( 3 q )resonating-group method, which employs the spin-flavor quark-model wave functions. Model parameters are determined to reproduce properties of the nucleon-nucleon system and the low-energy cross section data for the hyperon-nucleon interaction. Some characteristic features of the B8B8 interactions predicted by the most recent model fss2 are discussed. The nucleon-nucleon sector of this quark-model potential is used to calculate Gmatrices for nuclear and neutron matter, and to solve the Faddeev equation of the three-nucleon bound state. The calculated binding energy for 3H is the closest to the experiment among many results obtained by detailed Faddeev calculations employing modern realistic nucleon-nucleon potentials.

1. Introduction

One of the purposes of studying baryon-baryon interactions in the quark model (QM) is to obtain the most accurate understanding of the fundamental strong interaction in a natural picture, in which the short-range part of the interaction is relevantly described by the quark-gluon degree of freedom and the medium- and long-range parts by the dominated meson-exchange processes. In the spin-flavor su6 QM, the baryon-baryon interactions for all the octet baryons (Bs = N , A, C and E) are treated entirely equivalently with the well-known nucleon-nucleon ( N N ) interaction. We have recently proposed comprehensive QM potentials for general baryon-octet

85

baryon-octet (B8Bs) interactions, which are formulated in the (3q)-(3q) RGM using the spin-flavor SU, QM wave functions, the one-gluon exchange Fermi-Breit interaction and effective meson-exchange potentials (EMEP's) acting between quarks. The early version, the model FSS,1-3 includes only the scalar (S) and pseudoscalar (PS) meson-exchange potentials as the EMEP's, while the renovated one fss2 4,5 also introduces the vector (V) meson-exchange potentials and the momentum-dependent Bryan-Scott terms for the S and V mesons. Owing t o these improvements, the model fss2 in the nucleon-nucleon ( N N ) sector has attained the accuracy comparable to that of one-boson exchange potentials (OBEP's). These QM potentials can be used for various types of many-body calculations, which include the G-matrix calculations of baryonic matter and the Faddeev calculations of few-baryon systems. This project, however, involves a non-trivial problem of determining how to extract the effective two-baryon interaction from the microscopic quark-exchange kernel. The basic baryon baryon interaction, formulated in the RGM, is non-local and energy dependent. Furthermore, the RGM equation sometimes involves redundant components, due to the effect of the antisymmetrization, which is related to the existence of the Pauli-forbidden states at the quark level. In such a case, the full off-shell T-matrix is not well defined in the standard procedure. These problems, however, are all resolved by recent investigations for formulating three-cluster equations using two-cluster RGM kernel. As the first step t o apply our QM potentials to baryonic many-body problems, we here discuss two applications; one is the G-matrix calculation of the nuclear and neutron matter and the other is the Faddeev calculation for the three-nucleon ( 3 N ) bound state." 819

2. Formulation

The present model is a low-energy effective model which introduces some of the essential features of QCD characteristics. The color degree of freedom of quarks is explicitly incorporated into the non-relativistic spin-flavor SU, quark model, and the full antisymmetrization of quarks is carried out in the RGM formalism. The gluon exchange effect is represented in the form of the quark-quark interaction. The confinement potential is a phenomenological ?-type potential, which has a favorable feature that it does not contribute to the baryon-baryon interactions. We use a color analogue of the Fermi-Breit (FB) interaction] motivated from the dominant one-gluon exchange process in conjunction with the asymptotic freedom of QCD. We

86 ,

90

90

80

80-

70

70

.

.

.

.

,

np

- .

.

f552 FSS ...... SP99

~

60

-

50

-g 3400 -

-

20 10 0

- 10

-10 -

- 20

-20

-30

0

. .

-401 0

(b)

i o lbo

150 260 250 360 350

-..

"

50

-300

"

'

100 150 200 250 300 3QO

TW (MeV)

TM (MeV)

Figure 1. Comparison of np phase shifts for J _< 2 with the phase-shift analysis SP99 by Arndt et a1.12 The solid curves are the results by fss2 and the dotted curves by FSS.

postulate that the short-range part of the baryon-baryon interaction is well described by the quark degree of freedom. This includes the short-range repulsion and the spin-orbit force, both of which are successfully described by the FB interaction. On the other hand, the medium range attraction and the long-range tensor force, especially afforded by the pions, are extremely non-perturbative from the viewpoint of QCD. These are therefore most relevantly described by the effective meson exchange potentials (EMEP). In the previous model called FSS,1-3 only the scalar and pseudoscalar mesons are introduced. The model fss2 is the most advanced model which also includes the vector-meson exchange EMEP. The full QM Hamiltonian H consists of the non-relativistic kinetic-energy term, the phenomenological confinement potential U;', the colored version of the full FB interaction with explicit quark-mass dependence, and the EMEP U s generated from the scalar (fl=S), pseudoscalar (PS) and vector (V) meson exchange potentials acting between quarks: *v5

Up

87

-

0

CD

s Lo

-5

50

I

100 150 200 250 300 350

TW (MeV) 10

Figure 2.

The same as Fig. 1 but for higher partial waves with J

5 4.

The RGM equation for the relative-motion wave function x(r) reads

(4(3q)K%)lE- HIA { 4 ( 3 q ) 4 ( 3 q ) x ( r ) l ) = 0.

(2)

We solve this RGM equation in the momentum representation." If we rewrite the RGM equation in the form of the Schrodinger-type equation as [E - Ho - VRGM(&)]x(T) = 0, the potential term, VRGM(E) = VD + G + E K , becomes nonlocal and energy dependent. Here VD represents the direct potential of EMEP's, G includes all the exchange kernels for the interaction and kinetic-energy terms, and K is the exchange normalization kernel. and set up with We calculate the plane-wave matrix elements of VRGM(E), the Lippmann-Schwinger equation of the RGM T-matrix. This approach is convenient to proceed to the G-matrix calculation^.^^^ Faddeev calculations using these exchange kernels are also possible with some special considerations of the Pauli forbidden states.*y9

88 3. N N Properties and the G-matrix Calculations of Nuclear Matter Figures 1 and 2 display some important low-partial wave N N phase-shift parameters predicted by the model fss2, in comparison with the phase shift analysis SP99 1 2 . The previous result by FSS is also shown with the dotted curves. Due t o the inclusion of V mesons, the N N phase shifts of the fss2 at the non-relativistic energies up t o = 350 MeV are greatly improved, and now have attained the accuracy almost comparable to that of oneboson exchange potentials. The good reproduction of the N N phase-shift parameters in fss2 continues up t o fiab 600 MeV,4 where the inelasticity of the S-matrix becomes appreciable. For the correct evaluation of the triton binding energy, it is well known that the proper reproduction of the deuteron D-state probability and the 'So effective range parameters is essential. Table 1 shows the deuteron properties predicted by FSS and fss2, in comparison with the experiment. The predictions by the Bonn-B potential,13 which has a smaller deuteron D-state probability, is also shown for comparison. Figure 3 shows saturation curves of symmetric nuclear matter, obtained by fss2 and FSS.' They depend on the prescription how one deals with the energy spectrum of the intermediate single-particle (s.P.) states. In both cases of the QTQ and the continuous choices, they are very similar t o the predictions by meson exchange potentials. In the continuous choice, the saturation curves of fss2 and FSS are very similar to that of the BonnB potential. Unfortunately, our results share the common unsatisfactory feature of any non-relativistic models, that the saturation point does not deviate from the Caester line. The s.p. potentials of the nucleon and hyperons in symmetric nuclear matter are shown in Figs. 5 ( k =~ 1.35 fm-l) and 6 ( k =~ 1.07 fm-l) for

-

Table 1. The deuteron properties by fss2 and FSS in the isospin basis. The results by the Bonn-B potential l3 are also shown for comparison. A small difference in FSS from Table IV of Ref. 2 is due to the numerical inaccuracy in the previous calculation. The effect of the meson exchange current is not included in the calculation of Qd and pd. Ed

(MeV)

PD 1) = A

2.256

D / A ~ 0.0267 rms (fm) 1.963 Qd (fm2) 0.283 112 ( U N ) 0.8464

2.225

2.2246

2.224644 f 0.000046

0.0253 1.960 0.270 0.8485

0.0264 1.968 0.278 0.8514

0.0256 f0.0004 1.9635 fO.0046 0.2860 f0.0015 0.857406 f 0.000001

89 -5 /

-continuous choice

zr 3

fss2 -50

Figure 3. Nuclear matter saturation curves obtained by fss2 and FSS, together with the results of the Paris potential" and the Bonn model-B (BonnB) potential13. The choice of the intermediate spectra is specified by "QTQ" and "cont." The result for the Bonn-B potential in the continuous choice is taken from the non-relativistic calculation in Ref. 19.

0

"

1

2

"

3

"

4 5 P 1 Po

6

' 7

8

Figure 4. Same as Fig. 3 but for the neutron matter saturation curve predicted by fss2. The normal density po corresponds to the Fermi momentum k; = 1.35 fm-'. The results of the Nijmegen soft-core potential (NSC97) 2o and the Bonn-B potential l3 are also shown for comparison.

the model fss2. For the standard Fermi momentum k~ = 1.35 fm-', which corresponds t o the normal density PO, these are fairly deep potentials. For the comparison with the depth of the s.p. potentials in the finite nuclei, one has to take a smaller value for k~ because of the surface effects. If we assume k~ = 1.07 fm-', the s.p. potentials become much shallower, and the depth becomes almost comparable with the empirical values, -50 MeV for N , -30 MeV for A, and -10 -14 MeV for Z. The sign of the C s.p. potential is still controversial. The saturation curve for neutron matter, predicted by fss2, is also shown in Fig 4. The results by the Nijmegen soft-core potential (NSC97) 2o and the Bonn-B potential l 3 are also shown for comparison. Since the strongly attractive 3S1 3D1 channel does not exist in this case, the total energy per neutron in neutron matter becomes repulsive in any models. We find that our quark model potential gives very similar results to the standard

-

+

90

symmetric nuclear matter I+=I .35fm-'

r

-

-100;

'

'

'

'

2

thick line: continuous thin line: QTQ . ' ' ' 4 k [fm-'1

N

I

Figure 5 . The s.p. potentials V , ( k ) in symmetric nuclear matter, predicted by fss2, for the normal density po with k~ = 1.35 fm-l.

-100;

'

'

.

' 2

thick line: continuous thin line: QTQ I ' ' ' ' 4 k [fm-'1

Figure 6. The same as Fig. 5 but for half of the normal density ( p = 0 . 5 ~ 0 with ) k F = 1.07 fm-'.

meson-exchange potentials] as long as the N N interaction is concerned. 4. Characteristics of the B8Bs Interactions by fss2

Since our quark-model parameters are fixed by using the experimental data in the N N and Y N sectors, the B8B8 interactions beyond the strangeness S = -1 are all model prediction^.^ For the systematic understanding of these interactions] it is convenient to discuss them by using the SU3 representation basis for the two-baryon systems. This is because the quarkmodel Hamiltonian is approximately SU3 scalar, and the interactions with the same SU3 label (Xp) should have very similar characteristics] as long as the flavor symmetry breaking is negligible. Table 2 illustrates how the two baryon systems in the isospin basis are classified as the superposition of the SU3 basis. It has entirely different structure between the flavor symmetric and antisymmetric cases. In the 'SOstate, for example, there appear many states having the dominant (22) components. The && interactions for these states should be very similar to that for the N N 'So state. The (ll)s component is completely Pauli forbidden and is characterized by the strong repulsion originating from the quark Pauli principle. The (00) component in the H-particle channel is attractive from the color-magnetic interaction. On the other hand, 3S1state in the flavor antisymmetric case is converted t o the (30) state in the larger side of the strangeness. This (30) state is almost

91

Pauli forbidden, and the interaction is also strongly repulsive. Therefore, the EE interaction is not so attractive as N N , since they are combinations of (22) and (30). The other SU3 state (ll), turns out t o have very weak interaction. After all, the strangeness S = -2 case is most difficult, since it is a turning point of the strangeness. It is also interesting to see that EX channel with I = 312 should be fairly attractive, since the same ( 2 2 ) and (03) SU3 states as the N N system appear in this single isospin channel. Figure 7 shows fss2 predictions of the ' S Ophase shifts for various B8& interactions having the pure ( 2 2 ) configuration. Although CC interaction with the isospin I = 2 is very similar to the N N interaction, the interactions generally get weaker as the strangeness involved becomes larger. This is a combined effect of the flavor symmetry breaking in the quark and mesonexchange contributions. In particular, the EE interaction has the lowest rise of the phase shift, which is about 30". Accordingly, the the ZZ total cross sections become much smaller than the other systems. We find that the N N interaction is the strongest and has the largest cross sections among any combinations of the octet baryons. Some of the following characteristics of fss2 for the && interactions Table 2. The relationship between the isospin basis and the flavor-SU3 basis for the B8Bs systems. The flavor-SU3 symmetry is given by the Elliott notation ( X p ) . P denotes the flavor exchange symmetry, S the strangeness, and I the isospin.

S 0

-1

-2

-3 _ .

-4

-

P = $1 (symmetric) 'E

or

"0

P = -1 (antisymmetric) 'E

or

'0

92 90

-d

60

0

30

,----_

----__.

o

Q

1

-30

0

'

-30 0

I 200

400

600

800

1000

p I& ( M e W

Figure 7. 'SO phase shifts for various B8Bs interactions with the pure (22) state. The model is fss2.

s=-2

so

-60 0

200

600 P lab (MeV/c)

400

800

I

1000

Figure 8. fss2 'So phase shifts in the AAZN-CC coupled-channel system with the isospin Z = 0.

are very much different from the Nijmegen predictions given by Stoks and Rijken 21. There is no bound state in the BsBs system, except for the deuteron. =z total cross sections are not so large as the N N total cross sections. Z N interaction has a strong isospin dependence like the C N interaction. E-C- (namely, EE(I = 3/2)) interaction is fairly attractive.

--

A brief comment on the AA interaction predicted by the model fss2 follows. Figure 8 shows the phase shift curves for the full coupled channel calculation in the H-particle channel. The maximum peak of the A h phase shift is at most 20°, which is much smaller than the previous predictions by various models. This result is in good agreement with the recent experimental data for the double A hypernucleus :AHe. The finding of this event, called the Nagara event,22 is the most important contribution, since the assignment of the decaying scheme is very definite. The ABAAvalue extracted from this event is ABAA= 1.01f0.20 MeV, which implies a weak attraction. Our G-matrix calculation of this system yields an almost right answer, ABAA= 1.12 1.24 MeV, by taking into account the important contribution from the a-particle rearrangement energy.23

-

5. Faddeev Calculation of the Three-Nucleon System Since our QM BsBs interaction describes the short-range repulsion very differently from the meson-exchange potentials, it is interesting to examine

93

the three-nucleon system predicted by fss2 and FSS. Here we solve the Faddeev equation for 3H, by directly using the QM RGM kernel.’’ Our Faddeev calculation is the full 50-channel calculation up to the maximum angular momentum J = 6, and the values are almost completely converged as seen in Table 3. The fss2 prediction, -8.52 MeV, seems to be overbound, compared with the experimental value Eexp(3H) = -8.48 MeV. In fact, this is not the case, since all these calculations use the np interaction, which is more attractive than the nn interaction. The effect of the charge dependence is estimated to be -0.19 MeV.13 If we take this into account, our result is still 150 keV less bound. The charge root-mean-square radii of 3H and 3He are correctly reproduced. An important point here is that we can reproduce enough binding energy of the triton, without reducing the deuteron D-state probability. The self-consistent energy of the 2-cluster RGM kernel, E , has a clear physical meaning related to the decomposition of the total triton energy E into the kinetic-energy and potential-energy contributions through E = E / 3 + (Ho)/6. Table 4 shows this decomposition, together with the results of CD-Bonn and AV18 potential^.^^ We find that our QM potentials give a moderate amount of the kinetic-energy contribution just between these potentials, which have very different strengths of the tensor force.

Table 3. The three-nucleon bound state properties predicted by the Faddeev calculation with fss2 and FSS. The np interaction is used in the isospin basis. The column “channels” implies the number of two-nucleon channels included, and nmaxis the dimension of the diagonalization for the Faddeev equation. E(3H) is the ground state energy, and v / K and are the charge rms radii for 3H and 3He, respectively, including the proton and neutron size corrections. The Coulomb force and the relativistic corrections are neglected. & ( 2 N )is the 2 N expectation value determined self-consistently.

JG

I model

channels

2 ch 5 ch f ~ ~ 2 10 ch 18 ch 34 ch 50 ch 2 ch 5 ch FSS 10 ch 18 ch 34 ch 50 ch

nmax

2,100 5,250 10,500 18,900 35,700 112,500 2,100 5,250 10,500 18,900 35,700 112,500

&(2N) (MeV) 2.361 4.341 4.249 4.460 4.488 4.492 2.038 3.999 3.934 4.160 4.175 4.177

E(3H) (MeV) -7.807 -8.189 -8.017 -8.439 -8.514 -8.519 -7.675 -8.034 -7.909 -8.342 -8.390 -8.394

J ( T ~ ) ~ H

(fm) 1.80 1.75 1.76 1.72 1.72 1.72 1.83 1.78 1.78 1.74 1.74 1.74

J(r2)sHe

(fm) 1.96 1.92 1.94 1.90 1.90 1.90 1.99 1.95 1.97 1.93 1.92 1.92

94

6. Summary We have extended the (3q)-(3q)RGM study of the the N N and Y N interactions 1-4 to the strangeness S = -2, - 3 and -4 sectors without introducing any extra parameters, and have clarified some characteristic features of the B8B8 interaction^.^ The results seem to be reasonable, if we consider i) the spin-flavor SUe symmetry, ii) the weak pion effect in the strangeness sector, and iii) the effect of the flavor symmetry breaking. The N N sector of this QM potential is applied to the nuclear matter calculation in the G-matrix formalism and to the Faddeev calculation of the three-nucleon bound state lo. In these calculations, the necessary G-matrices and the off-shell T-matrices are directly derived from the RGM exchange kernel. For the saturation curves of nuclear and neutron matter] our QM potential gives very similar results to the standard mesonexchange potentials. In the Faddeev calculation, a large binding energy of the triton is obtained for our QM potentials, fss2 and FSS, without reducing the deuteron D-state probability. The charge root-mean-square radii of 3H and 3He are also correctly reproduced. In view of the fact that fss2 reproduces the N N phase shifts better than FSS, the results of fss2 are more meaningful. These results are the closest to the experiment among many results obtained by Faddeev calculations employing modern realistic N N interaction models. Since both models fss2 and FSS have a common feature in describing the short range correlation by the quark exchange kernel, it is important to clarify the mechanism in which the QM potentials give larger 3H binding energy than the meson-exchange potentials. The off-shell behavior of the RGM T-matrix is closely connected to this alternative description of the short-range correlations. More detailed study on this point is now under way.

Table 4. Decomposition of the total triton energy E into the kinetic-energy and potential-energy contributions: E = ( H o ) (V). The unit is in MeV. In the present model, this is given by the expectation value E of the two-cluster Hamiltonian with respect to the Faddeev solution, which is determined self-consistently.

+

model f~s2 FSS CD-Bonn AV18

E

E

(Ho)

4.492 4.177 3.566 5.247

-8.519 -8.394 -8.012 -7.623

43.99 41.85 37.42 46.73

(V) -52.51 -50.25 -45.43 -54.35

Ref.

24

I

95

Acknowledgments This research is supported by Japan Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture (12640265, 14540249).

References 1. Y. Fujiwara, C. Nakamoto and Y. Suzuki, Phys. Rev. Lett. 76, 2242 (1996). 2. Y. Fujiwara, C. Nakamoto and Y. Suzuki, Phys. Rev. C 54, 2180 (1996). 3. T. Fujita, Y. Fujiwara, C. Nakamoto and Y. Suzuki, Prog. Theor. Phys. 100, 931 (1998). 4. Y. Fujiwara, T. Fujita, M. Kohno, C. Nakamoto and Y. Suzuki, Phys. Rev. C 65, 014002 (2002). 5. Y. Fujiwara, M. Kohno, C. Nakamoto and Y. Suzuki, Phys. Rev. C 64,054001 (2001). 6. M. Kohno, Y. Fujiwara, T. Fujita, C. Nakamoto and Y. Suzuki, Nucl. Phys. A674, 229 (2000). 7. Y. Fujiwara, M. Kohno, C. Nakamoto and Y. Suzuki, Prog. Theor. Phys. 104, 1025 (2000). 8. Y. Fujiwara, H. Nemura, Y. Suzuki, K. Miyagawa and M. Kohno, Prog. Theor. Phys. 107, 745 (2002). 9. Y. Fujiwara, Y. Suzuki, K. Miyagawa, M. Kohno and H. Nemura, Prog. Theor. Phys. 107, 993 (2002). 10. Y. Fujiwara, K. Miyagawa, M. Kohno, Y. Suzuki and H. Nemura, Phys. Rev. C 66, 021001(R) (2002). 11. Y. Fujiwara, M. Kohno, C. Nakamoto and Y. Suzuki, Prog. Theor. Phys. 103, 755 (2000). 12. Scattering Analysis Interactive Dial-up (SAID), Virginia Polytechnic Institute, Blacksburg, Virginia R. A. Arndt: Private Communication. 13. R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989). 14. 0. Dumbrajs et al., Nucl. Phys. B216, 277 (1983). 15. N. L. Rodning and L. D. Knutson, Phys. Rev. C 41, 898 (1990). 16. David M. Bishop and Lap M. Cheung, Phys. Rev. A 20, 381 (1979). 17. I. Lindgren, in Alpha-, Beta-, Gamma-Spectroscopy, edited by K. Siegbahn, Vol. 11, (North-Holland, Amsterdam, 1965), p. 1623. 18. M. Lacombe, B. Loiseau, J. Richard, R.Vinh Mau, J. C6tC, P. Pirits and R. de Tourreil, Phys. Rev. C 21, 861 (1980). 19. R. Brockmann and R. Machleidt, Phys. Rev. C 42, 1965 (1990). 20. Th. A. Rijken, V. G. J. Stoks and Y. Yamamoto, Phys. Rev. C 59, 21 (1999). 21. V. G. J. Stoks and Th. A. Rijken, Phys. Rev. C 59, 3009 (1999). 22. H. Takahashi et al., Phys. Rev. Lett. 87, 212502 (2001). 23. M. Kohno, Y. Fujiwara and Y. Akaishi, submitted to Phys. Rev. C. 24. A. Nogga, H. Kamada and W. Glockle, Phys. Rev. Lett. 85, 944 (2000).


IV.FEW BODY SYSTEM


99

QUARK MASS DEPENDENCE OF THE NUCLEAR FORCES

E. EPELBAUM* AND w. GLOCKLE

Ruhr-Universitat Bochum, Institut fur Theoretische Physik 11, 0-44870 Bochum, Germany

U.-G. MEISSNER Uniuersitat Bonn, Helmholtz-Institut fur Strahlen- und Kernphysik (Th) 0-53115 Bonn, Germany

The behaviour of the nuclear forces as a function of light quark masses (or, equivalently, pion masses) is investigated in the framework of chiral effective field theory at next-to-leading order. We discuss the explicit and implicit quark mass dependence of the one- and two-pion exchange contributions and local short-distance operators. The unknown m,-dependent short-range contribution is estimated by means of dimensional analysis. We calculate various observables for different values of mq and discuss in detail the deuteron properties in the chiral limit.

1. Introduction

Chiral Perturbation Theory (CHPT) is a well established modelindependent and systematic tool for calculating low-energy properties of hadronic systems based upon the approximate and spontaneously broken chiral symmetry of QCD. Starting from the most general chiral invariant effective Lagrangian for Goldstone bosons (pions in the two flavor case of light up and down quarks) and matter fields (nucleons, A-xcitations, . . .) low-nergy S-matrix elements are calculated via a simultaneous expansion in small external momenta and quark masses (or, equivalently, the pion massa). For two and more nucleons the interaction becomes too strong to be treated perturbatively and an additional nonperturbative resummation of the amplitude is necessary. In practice, this is achieved by solving the 'This work is partially supported by the DFG under contract no. GL-87/34-1. aIn the isospin limit employed in this work the pion mass M , is given by: M: = 27?zB(1 O(7iz/Ax)), where B is a constant and mq is the average light quark mass.

+

100

corresponding integral equation for the scattering amplitude with the kernel derived within the low-energy expansion1. Since the absolute values of the running up and down quark masses at the scale 1 GeV, mu N 5 MeV, m d N 9 MeV, are rather small 2 , one expects that hadronic properties at low energies do not change strongly in the chiral limit (CL) of M , + 0. This feature is crucial for the chiral expansion to make sense and is certainly true for the 7r and TN systems, where the interaction becomes arbitrarily weak in the CL and for vanishing external momenta. The purpose of this work is to look at the NN system in the CL (and, in general, for values of the pion mass different from the physical one), which is much more complicated due to the nonperturbative aspect. We stress that the M,-dependence of the nuclear force is not only of academic interest, but also of practical use for interpolating results from lattice gauge theory, see for more discussion. For example, the S-wave scattering lengths have been calculated recently on the lattice using the quenched approximation 4 . Another interesting application is related to imposing bounds on the time-dependence of fundamental couplings from the two-nucleon sector, as discussed in '. 2. M,-dependence of the nuclear force

The energy-independent and hermitean NN potential can be obtained using the method of unitary transformation and following the lines of ref. '. At leading order (LO) the potential is given by the (static) one-pion exchange (OPE) and two non-derivative contact interactions. At next-to-leading order (NLO) the following corrections have to be accounted for: contact terms with two derivatives or one M,-insertion; renormalization of the OPE; renormalization of the contact terms; two-pion exchange (TPE). For a detailed discussion of the NLO corrections the reader is referred to 8 . The OPE and T P E at NLO are given by

101

where {is the nucleon momentum transfer, a' (7) refer to spin (isospin) Pauli matrices, g.4 = 1.26 and F, = 92.4 MeV are the nucleon axial vector coupling and pion decay constant, respectively, and the function L ( q ) is given by:

Here and in what follows we denote the value of the pion mass by & inl , order to distinguish it from the physical one denoted by M,. The implicit M,-dependence of the OPE is given by A, which represents the relative shift in the ratio gA/F, compared t o its physical value and is found to be, see ref.

':

Here i d , 218 and 216 are low-energy constants (LECs) related to pion and pion-nucleon interactions. We use the following values for these LECs: 14 = 4.3 ', 216 = -1.23+0.32 -0.53 GeV-2 lo and 218 = -0.97 GeV-2. The constant z 1 8 is fixed from the value of the Goldberger-Treiman discrepancy. Note further that for the LEC 2 1 6 we use an average of three values given in l o , which result from different fits. The shown uncertainty is defined such as t o cover the whole range of values from lo. The remaining &f,-dependence of the nuclear force at NLO is given by the T P E and by the short range terms of the form

'

vcont= M: M,

[Ds +

where the constants

PS,T

DT(z1 . $2)

"I

- (Ps + ~ T ( a ' 1. 8 2 ) ) In - , M,

(5)

are given by':

Here CT is the LEC of the leading contact term proportional to a'l .&. All other contact terms do not depend on the pion mass and the values for the corresponding LECs c s , ~C, I , . . . can , ~ be adopted from the analysis of l 1 performed for the physical pion mass. The essential difficulty in extrapolating the nuclear forces in the pion mass is due to the fact that one cannot disentangle the NLO short-range contribution given in eq. (5) from the LO one using the NN data.b Thus the LECs D s and DT are unknown. In bThe LECs D)s,T can however be fixed in processes including pions such as e.g. piondeuteron scattering at higher orders. Such an analysis is not yet available.

102

order to proceed further we assume natural values for these constants, i.e.: -

where

DS,T = -

F,A2,

QS,T

-

1,

and A, N 1 GeV. In l 2 we have shown that all values of the dimensionless coefficients a related to the contact terms lie at NLO in the range -2.1 . . .3.2 for all cut-offs employed. In the following we will make a conservative estimate for ( Y S , T :

-3.0

< (YS,T < 3.0.

(7)

Certainly, the lack of information about the values of DS,T is the main source of uncertainty of our analysis. We have adopted the same procedure to regularize the LS equation as in i.e. the potential V(@’,p’) is mul(\$I), fFpon whose precise tiplied by the regulator functions fEpon form is given in l l . In the next section we will show how various observables behave with MT. 3. Results for NN observables

3.1. Phase shifts

Having specified the NN interaction at arbitrary value of MT we are now in the position to calculate observables. Let us begin with the behavior of the NN phase shifts in the CL. First of all we stress that the OPE in eq. (1) leads t o a significant scattering even in higher partial waves due to the Coulomb-like pion propagator in the CL. Further, no effective range expansion of the form k21+l cot S l ( k )

1

= -a6

+ T1-k22 + v3c4 + . . . ,

exists due to the vanishing pion mass.‘ In eq. (8) k is the c.m. momentum and 1 the angular momentum. It is easy to derive the low-momentum behavior of Sl(k) at least for large 1, where the potential becomes weak and one can use the Born approximation for the T-matrix. It is then sufficient to look at on-the-energy shell matrix elements of the OPE in the CL V$bE(k),which strongly dominantes the nuclear interaction a t low momenta. Since V&(k) = 0 in all spin-singlet (s = 0) and V$k,(k) = y in the spin-triplet (s = 1) channels, where the constant y depends on =The maximal radius of convergence of the effective range expansion is proportional to M: and goes to zero in the CL.

103

the partial wave, we expect in the CL a strong reduction of &(k) in the s = O-channels and linear growth in Ic of &(k) in the s = l-channels at low momenta. The numerical analysis performed in confirms these expectations. We also stress that we have not found new bound states in the CL in agreement with the previous work by Bulgac et al. 1 3 , although a strong enchancement of &(k) is observed in many casesd Last but not least, we found smaller (in magnitude) and more natural values for the two S-wave scattering lengths in the chiral limit a c ~ ( ~ S= o )-4.1 f 1.6'::; fm and u ~ ~ (= ~1.5f0.4+,0:,2 S ~ ) fm, where the the first indicated error refers to the uncertainty in the value of D3s1 and (t16 being set to the average value d16 = -1.23 GeV-2 while the second indicated error shows the additional uncertainty due to the uncertainty in the determination of d 1 6 as described before. 3.2. Deuteron properties

We have also calculated the deuteron binding energy (BE) as a function of M,. Our results for the cut-off A = 560 MeV are depicted in Fig. 1. According to our complete NLO analysis, the deuteron is stronger bound in the chiral limit with the BE BgL = 9.6 f 1.9'::: MeV. It is worthwhile to take a closer look at the deuteron wave function in the CL. It is easy to see that the asymptotic form of the S- and D-state wave functions u ( r ) and w ( r ) deviates from the one in the physical case,

due to the long-range nature of the one-, two-, . . ., pion exchanges in the (rn denotes the nucleon mass), As and q are the CL. Here a asymptotic S-state normalization and D/S ratio, respectively. The OPE in the CL generates a tensor potential, which in coordinate space reads 1 v p ( r ) = a+?7 , (10)

%&).

+

where a? = -3& (1 2A Here and in what follows the corresponding isospin factors are projected onto the deuteron channel. The TPE in the CL generates both central and tensor parts

VZ"(r) = agf

,

V$,(r) = a?-$,

dFor example 61(k) in the 3Po partial wave reaches pared with N 11' in the physical case.

N

32' in the maximum to be com-

104

Figure 1. Deuteron BE versus Mr.The shaded areas show allowed values. The light shaded band corresponds t o our main result with the uncertainty due t o the unknown LECs Ds,T. T h e dark shaded band gives the additional uncertainty due to the uncertainty of d16. The heavy dot shows the BE for the physical case &fT = MT.

-a4.

where ( y 2GK = 3 ( 2 264r3F: d-w:-1) , a? = Notice that we omit here all zero-range contributions. At large distances the deuteron wave functions

-

0.6

Q -

E 0.4 3 h

v

= 0.2 n

"0

2

4

6

8

r [fml Figure 2. Deuteron wave functions in the CL compared with the ones in the physical case. The upper and lower bands (solid and dashed lines) refer to the S- and D-wave functions u ( r ) and w(r) in the CL (in the physical case), respectively. For remaining notations see Fig. 1.

105

obey the coupled equations

+

where V c ( r ) E V S T ( r ) VT(T) , E V $ T ( ~ )V $ n ( ~ )One . can easily read off the asymptotic form of the deuteron wave functions in the chiral limit from these equations:

The coefficients Pi and 7i can be calculated from eqs. ( 1 2 ) . For example, P 2 , 3 , 4 , 7 2 , 3 , 4 are given by: P2

= P3 =

7 4 = a

2

arnqa?

Jz

1

P 4 = a m

- 377) m - ( a p ) 2 m (4

(a?))"m ( 2 - J z q )

+ a (a? + h a p 17) 8

+ a ((Y?q + 2ap (Jz- 11))

811 We note that higher-order terms in the low-nergy expansion of the NN interaction in the CL change the asymptotics of the deuteron wave functions. Using dimensional arguments, it is easy to see that Qn-correctionse affect only coefficients Pi, -yi with i 2 2n + 2 . Consequently, N-pion exchange modifies coefficients Pi, 7i with i 2 2N. Let us now discuss how various deuteron properties change in the chiral limit. In table 1 we compare the NLO results in the physical case and in the CL. We do not consider the asymptotic S-state normalization A s and D/S ratio 71 since the wave functions have different asymptotic form in the CL. Notice that the smaller deuteron size in the CL has to be expected due to a stronger binding. A more short-range nature of the deuteron in the CL is also clearly visible in the deuteron wave function presented in Fig. 2 . It is also interesting that the probability for the deuteron to be in the D-state is enhanced in the CL, which is mainly due to the stronger tensor force resulting from the OPE. The quadrupole moment takes a smaller value in the CL. Its relatively large uncertainty due to the unknown LECs DS,T indicates significant sensitivity of QD to short-range physics. eLO terms are counted as Qo.

9

106

Table 1. Deuteron properties. physical case 2.17

chiral limit 9.6 f 1.9':::

0.860

:::T 1.266f 0.085 ::it 0.820 f0.002 :::T

3.6

10.5 f 0.3

0.274 1.975

0.247 f 0.030

2::;

4. Summary

To conclude, we did not find dramatic changes in the properties of the NN systems in the CL, such as the appearance of new bound states. Various observables like the deuteron binding energy and the S-wave scattering lengths are shown to be more natural in the CL. 5. Acknowledgements

I am grateful to organizers of the workshop and especially to Kenshi Sagara for inviting me. I would also like t o thank Hiroyuki Kamada for many interesting discussions and for his hospitality dyring my stay in Japan. References 1. S. Weinberg, Nucl. Phys. B363, (1991) 3. 2. J. Gasser, and H. Leutwyler, Phys. Rep. (287,(1982) 77. 3. S.R. Beane, et al., Nucl. Phys. A700,(2002) 377. 4. M. Fukugita, et al., Phys. Rev. D52, (1995) 3003. 5. S.R. Beane and M.J. Savage, hep-ph/0206113. 6. S. Okubo, Progr. Theor. Phys., Japan, 12, (1954) 603. 7. E. Epelbaoum, W. Glockle and Ulf-G. Meiher, Nucl. Phys. A637, (1998) 107. 8. E. Epelbaum, Ulf-G. MeiOner, and W. Glockle, nucl-th/0207089, Nucl. Phys. A, in print. 9. J. Gasser, and H. Leutwyler, Ann. Phys. 158,(1984) 142. 10. N. Fettes, doctoral thesis, published in Jul-Report 3814,(2000). 11. E. Epelbaum, W. Glockle and Ulf-G. MeiOner, Nucl. Phys. A671, (2000) 295. 12. E. Epelbaum, Ulf-G. MeiSner, W. Glockle, and Ch. Elster, Phys. Rev. C65, (2002) 044001. 13. A. Bulgac, G.A. Miller, and M. Strikman, Phys. Rev. C56, (1997) 3307.

107

DIFFERENT TYPES OF DISCREPANClES IN 3N S Y m M S Kenshi SAGARA

Department ofPhysics, Kyushu University, Hakozaki, Higashi-ku, Fukuoka, 812-8.581 Japan E-mail: [email protected] Discrepancies between exact 3-nucleon (3N) calculations based on realistic 2-nucleon (2N) potentials and accurate experiments on 3N systems are described and discussed. Some of the scalar type discrepancies have been understood as the effects of a 2~-exchange 3-nucleon force (2n3NF), however, vector- and tensor-type discrepancies can not be explained by 2x3NF and are still open problems.

1. Introduction To understand three-nucleon force (3NF) is one of the basic subjects in nuclear physics. 3N systems are the best field to study 3NF because Faddeev equations for 3N systems can be exactly solved with 2NF by the aid of recent computers. If 3N calculations for an observable based on realistic 2NF remarkably disagree with experimental results, we call the disagreement as discrepancy. Origin of the discrepancy may be the existence of 3NF or wrong prediction of the off-energy-shell part of 2NF. Systematic study on energy and angular dependence of the discrepancy is important to investigate the origin of the discrepancy. So far 3N experiments have been made on 3N bound states, Nd scattering, Nd breakup reactions, Nd capture reactions, photo-disintegration of 3N bound states, electron elastic and inelastic scattering on 3N bound states. In this paper, discrepancies in 3N binding energy, cross section and spin observables of Nd scattering, Nd breakup and Nd capture are discussed. Historically, the discrepancy in 3N binding energy was first found in late 1 9 7 0 ’ ~and ~ discrepancy in Ay of Nd scattering which is now well-known as Ay puzzle was pointed out in 1986. Since then several discrepancies have been reported, however, they have not been confirmed in a wide energy and angular range yet except a systematic discrepancy in the cross section

108

minimum of Nd scattering which was found at 2-18 MeV in 1994 (Ref.l), and was confirmed at 135 MeV in 1996 and named as Sagara discrepancy (Ref.2). In 1998 Witala et al. found that the discrepancies in 3N binding energy and in Nd scattering cross section minimum in wide energy range are excellently explained by introducing the same 2x3NF (Ref.3). Existence of 2x3NF has become convincing. Since 1998 many 3N experiments have been made in a wide energy range searching for 3NF effects and several kinds of discrepancies have been found. The discrepancy in Nd scattering cross section minimum is always eliminated when 2z3NF is introduced, however, other discrepancies are not always reproduced by 2NF+2x3NF indicating the existence of origins other than 2z3NF. 2. Scalar type discrepancies 2.1. Binding energies and Nd scattering cross section As described above, both the discrepancies in 3N binding energy and in Nd scattering cross section minimum are well explained by the same 2x3NF. The binding energy and the cross section are scalar quantities. We expect therefore the simple picture that all the scalar quantities in 3N systems are well reproduced by introducing 2x3NF.

2.2. Nd breakup cross section (SS anomaly) A famous exception to our expectation is the space star (SS) anomaly in nd breakup reaction. At the SS configuration, momenta of three outgoing nucleons form a regular triangle in a plane perpendicular to the beam axis in the c.m. system. The nd breakup cross section at SS configuration was reported to exceed calculated value by 20-30 % at En = 13 MeV and by about 10 % at 10.5 MeV. Above 16 MeV the nd breakup SS cross section becomes close to calculations and SS anomaly disappears. Predicted effects of 2z3NF are less than 1% in the energy range concerned and can not explain SS anomaly. Curiously no SS anomaly has been found in the pd breakup cross section at the same energy region. Recently we have made a systematic measurement of pd breakup cross section at 23 different angular configurations around SS at 13 MeV (Ref.4, See References in Ref.4 for previous study on SS anomaly). The measured pd cross sections are 7-15 % smaller than the calculated nd cross sections in the angular range where the measurement was made. From pd calculations by E.O. Alt et al. with a simple S-wave NN interaction, Coulomb force is

109

expected to reduce the cross section by about 10 % at SS configuration at 13 MeV. This fact means no SS anomaly may exists in pd breakup at 13 MeV, contrary to the case of nd breakup. To get a definite conclusion pd calculations with realistic NN potentials are necessary, however, such calculations have not been made yet due to difficulty in treating Coulomb force in 3N calculations. If it is confirmed that SS anomaly appears only in nd breakup, we must investigate the charge symmetry breaking in either 2NF or in 3NF. On the other hand, if there is no SS anomaly in both nd and pd breakup, then we could have expectation that all the scalar observables in 3N system are reproduced well with 2NF+2.n3NF. At 250 MeV, an incomplete measurement of pd breakup cross section is in progress to see 2n3NF effects and relativistic effects in pd breakup.

2 3 . pd capture cmss section Recent measurement of pd radiative capture at 100 MeV ( E d = 200 MeV) revealed that noticeable discrepancy exists in pd capture cross section and the discrepancy is well explained by 2.n3NF (See section 4.4. below).

3. Vector type discrepancies 3.1. Vectoranalyzing powers in Nd scattering Well-known Ay puzzle in Nd scattering appears in the energy region below EN = 30 MeV, where 2n3NF has small effects on Nd scattering and can not solve the puzzle. Nd scattering Ay arises mainly from P-wave parts of 2N interactions, while Nd scattering cross section comes up from S-wave parts. While Sagara discrepancy in Nd scattering cross section becomes large monotonically with energy and is well explained by 2n3NF up to 250 MeV, Ay puzzle fades out above 30 MeV. This curious energy dependence of Ay puzzle is hard to understand and therefore Ay puzzle has been an open problem for over 15 years. Below 30 MeV, iTll in pd scattering has a discrepancy similar to Aypuzzle. At 70-135 MeV, pd scattering iTll is not reproduced by 2NF alone and is well described by the prediction based on 2NF+2n3NF (Ref. 5). At 150-190 MeV, pd scattering Ay takes the values between the predictions with 2NF alone and with 2NF+2n3NF (Ref. 6). At 250 MeV, pd scattering Ay show angular dependence different from the both the predictions with 2NF and with 2NF+2.n3NF (Ref. 7).

110

The small discrepancies in A,, and iTll above 70 MeV have features different from that of A, puzzle. It is apparent that Ay puzzle has an origin which is different from 2n3NF. The discrepancy at higher energy has at least two origins, one of which is 2n3NF. To find the origin(s) is an interesting work.

3.2. Vectoranalyzing powers in Nd breakup So far no discrepancy has been reported in vector analyzing powers of Nd breakup. This may be partly due to low counting rate in kinematically complete experiments of Nd breakup. Incomplete experiment can be made with high statistics. Such an experiment is in progress at 250 MeV.

3.2. Vectoranalyzing powers in pd capture Recent measurement of pd capture at 100 MeV (Ed = 200 MeV) has revealed that noticeable discrepancy exists in pd capture iTl1 and the discrepancy is well explained by 2n3NF similarly to the case of pd scattering iTll at 70-135 MeV (See section 4.4.).

4. Tensor type discrepancies

4.1. Tensor analyzing powers in pd scattering Tensor analyzing powers of pd scattering have also been measured in 70-135 MeV range. Experimental results are different from both the predictions with 2NF and with 2NF+2n3NF. There are visible effects of 2n3NF on the tensor analyzing powers, and the effects reduce the discrepancies in some angular region and increase them in other region. At the energy range below about 30 MeV, such discrepancies have not been reported.

4.2. Polarization transfer coeficknh in pd scattering Several polarization transfer coefficients of pd scattering have also been measured in 70-135 MeV range. Similarly to the case of analyzing powers, finite discrepancies of complicated feature are seen also in the polarization transfer coefficients, and the discrepancies can not be removed by the introduction of 2n3NF.

4.3. Tensor analyzing powers in pd breakup Measurement of tensor analyzing powers in pd breakup reaction is suffered from low counting rate. Therefore if finite discrepancies exist, they are hard to detect. Remarkable discrepancies in A, and A,, of H(d,pp)n reaction at

111

26 MeV (Ed = 52 MeV) was reported at a certain kinematical configuration (Ref.8). However, because counting rate is very low at the configuration and the experimental results are easily influenced by backgrounds, a new confirming measurement is now in progress.

( calc(2NF)

- exp ) /exp

4 Discrepancy in Scalar obs

Sagara discrepancy

Discrepancy in Vector obs.

Fig. 1. Schematic view of normalized discrepancies in 3N observables between calculations based on 2NF and experiments. Energy ranges of accelerators available for 3N experiments in Japan are shown in the bottom with the range of chiral perturbation theory.

112

4.4. Tensoradyzing powers in pd capture Recently we have measured analyzing powers of pd capture at 100 MeV (Ed= 200 MeV) and revealed that a very large discrepancy exists in pd capture A,, which is an order of magnitude larger than effects of 2n3NF and also a small discrepancy in Ayy,as seen in Fig. 2 (Ref. 9). Since pd capture has small cross section, we used a liquid hydrogen target and detected recoil 3He simultaneously in a wide angular range. The measurement of A, was made in a vertical reaction plane with the beam polarization axis in the vertical direction. This was the first measurement of pd capture A,, with beams from cyclotrons. Previous measurements of were made below Ed = 17.5 MeV with beams from tandem accelerators. Contrary to the tensor analyzing powers, cross section and Ayd (proportional to iTll) are well reproduced by a calculation with 2NF+2n3NF.

.. ... ... .

.. ..... ..

I

.

.

.. ..... .:.

I

.

I

I

..

..... :..

.

I

..

..... :..

.

I

..

..... :.

.

I

..

..... :.

.

..

:.

0.2

.

I

.....

.

0.15

.

0.1

0.05 Ph 2-

0

-0.05 -0.1 -0.15

.

.

... ... .... .... ... ... . .. ............................................ ... : . ..

.

... .. ...

..

...

...

.... .. :...

I

I

I

I

-

-0.2

.. I

.

.. ,

.

.. I

.. I

~

.

...

..

..

...

..

.

20 40 60 80 100 120140160180

0

0.2

!

0.1

i......... 1..

.

!

.

I

1

'

'

2NF

'-

2NF+3NF

I

-1

0

20 40 60 80 100120140160180

0

20 40 60 80 100 120 140 160 180

0CM

= 200 MeV. Fig. 2. Cross section, A,", Ayyand A, of pd radiative capture at Theoretical curves are calculations with and without 2z3NF by Kamada (Ref.10).

113

Fig. 3 shows the energy dependence of pd capture A,, and A, at ,8 = 90". Experimental values of A, and Ayyare nearly equal at 17.5 MeV and at 200 MeV, and we expect that they are nearly equal in all the energy range below 200 MeV. Calculated A,, and An are nearly equal at low energy and become different above 50 MeV. Hence, discrepancy in A,, is expected to appear above 50 MeV. The A,, discrepancy is an open problem. 0.1

0

17

----_

......... :.**.:.. ............... w--.................. i.................. i...................

..........

2

---::;:? -

6

-0.1

x f k

r;

-0.2

3

-0.3 -0.4

...............

....

i.................. ;................. 4................

-

-0.5 I 0

A, JOU&UI et al. (1986) A,Anklinetal.(1998) A, Ttts et al. (1988)

I

1

.....,................

A n DATA

-

I

I

I

I

I

I

50

100

150

200

250

300

Ed

[MeV]

Fig.3. Energy dependence of pd capture A,, and An at 0, = 90". Calculations are made by Kamada (Ref. 10).

5. Prospects

There are various kinds of discrepancies between 3N experiments and 3N calculations based on 2NF. Scalar discrepancies in 3N binding energies and Nd scattering cross section are excellently explained by introducing 2n3NF. However, well-known Ay puzzle in Nd scattering below 30 MeV is not explained by 2n3NF and has been an open problem for over 15 years. Recently, remarkable anomaly in pd capture A, has been found. Besides the above outstanding discrepancies, there have been reported small finite

114

discrepancies in the energy region above about 70 MeV. These discrepancies indicate the existence of new origins other than 2x3NF. Several theoretical proposals for the new origins have been made, especially for the origin of Ay puzzle, by modifying 2NF or by introducing new 3NF’s. However, satisfactory solution has not been obtained yet. Relativistic effect may also be a candidate and should be taken into accounts. There are scalar, vector and tensor type discrepancies. It is natural to consider that there are three kinds of origins for three kinds of discrepancies and the origins have a common root. In this sense, we pay attention to the chiral perturbation theory (xPT) for QCD Lagrangian of nuclear field, because various kinds of 3NF’ (including 27c3NF) and also 4NF’s come out from the chiral expansion of QCD Lagrangian. To determine the expansion coefficients, we need many precise experimental data of various kinds of polarizations. Since the expansion converges energy below about 100 MeV, precise data sets below 100 MeV are necessary. In Fig. 1, the energy range of xPT is also marked. Kyushu University will move suburbs in 10 years from 2005, and we are requiring a new accelerator instead of the present tandem accelerator. The desirable energy range of the new accelerator is also shown in Fig. 1. The energy range necessary for xPT is covered by accelerators in Japan including the new accelerator. The author wishes to thank all the collaborators in our 3N experiments and H. Kamada and theorists in Bochum group for their calculations and valuable discussions.

Refemnces 1) K. Sagara et al., Phys. Rev. C50 576 (1994). 2) Y. Koike and S. Ishikawa, Nucl. Phys. A631 683c (1998). 3) H. Witala et a]., Phys. Rev. Lett. 81 1183 (1998). 4) T. Ishida et al., presented in this symposium. 5) K. Sekiguchi et al., Phys. Rev. C65 034003 (2002). 6) K. Ermisch et al., Phys. Rev. Lett. 86 5864 (2001). 7)H. Hatanaka et al., Phys. Rev. C66 044002 (2002). 8) L. M. Qin et al., Nucl. Phys. A587 252 (1995). 9) T. Yagita et al., Proc. of APFB02 (Shanghai, Aug. 2002). 10) H. Kamada, private communication.

115

POLARIZATION TRANSFER MEASUREMENT FOR DEUTERON-PROTON SCATTERING AND THREE NUCLEON FORCE EFFECTS

K. SEKIGUCHI*, H. SAKAI *>t>t,H. OKAMURA §, A. TAMII t , T. UESAKA $, K. SUDA 5, N. SAKAMOTO *, T. WAKASA q, Y . SATOU 11, T. OHNISHI *, K. Y A K 0 i , S. SAKODA t , H. KATO t , Y. MAEDA t , M. HATANO t , J. NISHIKAWA 5 , T. S A I T O ~N. , UCHIGASHIMA t , N. KALANTAR-NAYESTANAKI ** , AND K. ERMISCH ** * R I K E N , Saitama 351-0198, Japan

t Department of Physics, University of Tokyo, 113-0033, Japan Center for Nuclear Science, University of Tokyo, 113-0033, Japan

5 Department of Physics, Saitama University, Saitama 338-8570,Japan RCNP, Osaka University, Osaka 567-0047, Japan

11 Department of Physics, Tokyo Institute of Technology, Tokyo152-8551, Japan ** Kernfysisch Versneller Institvut, NL-9747 A A Groningen, The Netherlands Precise measurements of the deuteron to proton polarization transfer coefficients for the d-p elastic scattering has been made at 135 MeV/u at RIKEN Accelerator Research Facility. The obtained results are compared with the Faddeev calculations based on modern nucleon-nucleon forces together with Tucson-Melbourne, a modification thereof closer to chiral symmerty and Urbana-Argonne type of three nucleon forces.

1. Introduction One current interest has been focused on the threenucleon force (3NF) effects in the nucleon-deuteron ( N d ) elastic scattering at intermediate energies ( E / A 60 MeV). This is partly because recent advance in computational resources has made it possible to obtain rigorous numerical Faddeev calculations for the threenucleon scattering processes by using twonucleon(2N) and threenucleon forces (3NF). Therefore it has allowed us to study of 3NF properties by direct comparison between such theoretical predictions and precisely measured data. In Refs. [1,2] we have reported the precise measurement of the cross section and the deuteron analyzing powers for d-p elastic scattering a t

>

116

incoming deuteron energies of 70, 100, and 135 MeV/u. The data have been compared with the Faddeev calculations with or w/o 3NFs. For the cross section, the large discrepancy between the data and the calculations w/o 3NFs has been found in the cross section minimum and it is essentially removed by taking into account 3NFs. The vector analyzing power A t is also explained by the predictions incorporating 3NFs. However the tensor analyzing power data are not reproduced by any theoretical prediction and these results indicate that the present day 3NF models have deficiencies in the spin parts. In order to assess further the spin parts of the 3NF, we have measured the deuteron-to-proton polarization transfer coefficients for d-p elastic scattering, which are expected theoretically to have strong sensitivities to the spin dependent parts of 3NF. 2. Experiment

The experiment was performed a t the RIKEN Accelerator Research Facility using tensor and vector polarized deuteron beams of 135 MeV/u 3 . A liquid hydrogen (19.8 mg/cm2) or CH2 (93.4 mg/cm2) target was bombarded and scattered protons were momentum analyzed by the magnetic spectrograph SMART '. The polarization of the scattered protons were measured with the focal-plane polarimeter DPOL 5. The measured observables were the deuteron to proton polarization transfer coefficients ( K $ , K!: - K&, and K!:) in the angular range of & m , = 90" - 180". The relation between the polarizations and the observables is expressed as Py'

(g)

=

with

where x, y, and z are the coordinates of the incident deuterons; x', y', and z' are those of the emitted protons; and denotes the cross section with unpolarized beams. In order to get a particular polarization state on the target the deuteron polarization axis was rotated by a spin rotation system Wien Filter downstream of the polarized ion source. It was directed normally to the scattering plane for the measurement of K:' and K$,. In case of KZ: the rotation was performed into the scattering plane so

(%)

117

that polarization axis pointed sideways, perpendicular to the beam direction. For the K i i measurement, the spin symmetry axis was additionally rotated in the reaction plane and was aligned at an angle 0 to the beam direction. Then we measured KZL and K:; separately to obtain KZL - K:; . The beam polarizations was monitored by using the d-p elastic scattering and the actual magnitudes were 60 - 80% of the theoretical maximum values throughout the measurement. Note this measurement also yielded an induced polarization ( P I ) of the outgoing protons.

3. Results and Discussions

0

120

60

4.m.

180

[degl

Figure 1. First 3 normalized frequencies versus release location for clamped simply supported beam with internal slide release.

Figure 1 shows a part of the experimental data KZj - K:; with open squares. The statistical errors are only shown. The statistical errors are smaller than 0.03 for all the polarization transfer coefficients, and 0.01 for the induced polarization PY'. The systematic uncertainties for the polar-

118

ization transfer coefficients are estimated to be 3% at most. In Fig. 1, four theoretical predictions in terms of Faddeev theory are shown together with the experimental results. The dark (light) shaded band in the figure is the Faddeev calculations with (w/o) Tucson-Melbourne (TM) 3NF based on the modern nucleon-nucleon(") potentials, namely CDBonn 8 , AV18 ', Nijimegen I, I1 and 93 lo. The solid line is the calculation with including Urbana IX 3NF l 1 based on AV18 potential. The dotted line is the predictions in which TM' 3NF is taken into account and CDBonn potential is considered as the NN potential. The TM' 3NF is a modified version of the TM 3NF closer to chiral symmetry. Comparing the theoretical predictions with the observed values, for K& - K&, the clear discrepancies exist between the data and the 2N force predictions and these deviations are explained well by inclusion of 3NFs. All 3NF potentials considered here (TM, TM', Urbana IX ) provide almost the same 3NF effects (magnitude and direction). However for the other polarization transfer coefficients K$ and KZh which are not shown here, large differences between the data and the 2N force predictions are not reproduced by including the 3NF models. The results of the comparison for the polarization transfer coefficients reveal that the present 3NF models have deficiencies in its spin parts and that these observables are useful to clarify the spin dependence of 3NF effects. 4. Summary

In order to study of the properties of the three nucleon forces, we have measured the deuteron to proton polarization transfer coefficients for d-p elastic scattering at 135 MeV/u which cover the angular range of &.m. = 90" - 180". Highly accurate data have been obtained. These results are compared with the Faddeev calculations with and without the TucsonMelbourne 3NF, a modification thereof closer to chiral symmetry TM', and the Urbana IX 3NF. The large difference are obtained between the data and the 2N force predictions. However not all spin observables are reproduced by incorporating the present three nucleon force models and the results clearly show the deficiency of these models in spin parts.

Acknowledgement We would like to thank H. Witala, W. Glockle and H. Kamada for their strong theoretical support. We would also like to thank S. Nemoto and P.

119

U. Sauer for their useful comments on theoretical issues. We would also like to express our appreciation to the continuous help of the staff of RIKEN Accelerator Research Facility. References H. Sakai et al., Phy. Rev. Lett. 84 (2000) 5288. K . Sekiguchi et al., Phy. Rev. C 65 (2002) 034003. H. Okamura et al., AIP Conf. Proc. 293,84 (1994). T. Ichihara et al., Nucl. Phys. A569, 287c (1994). 5. S. Ishida et al., AIP Conf. Proc. 343,182 (1995). 6. H. Okamura et al., AIP Conf. Proc. 343,123 (1995). 7. S. A. Coon, and M. T. Peiia, Phys. Rev. C 48,2559 (1993). 1. 2. 3. 4.

8. R. Machleidt, Phys. Rev. C 63,024001 (2001). 9. R. B. Wiringa, et al., Phys. Rev. C 51,38 (1995). 10. V. G. J. Stoks, et al., Phys. Rev. C 49,2950 (1994). 11. B. S. Pudliner, et al., Phys. Rev. C 56,1720 (1997).

120

SEARCH FOR THREE NUCLEON FORCE EFFECTS IN p'd ELASTIC SCATTERING AT 250 MEV

Y. SHIMIZU, K. HATANAKA, Y. SAKEMI, T. WAKASA, H.P. YOSHIDA, J. KAMIYA, T. S A I T O ~ H. , S A K A I ~ A. , T A M I I ~ ,K. SEKIGUCHI~, K. YAKO', Y. MAEDA', T . NOR02, K. SAGARA2, AND V.P. LADYGIN3 Research Center for Nuclear Physics (RCNP), Ibaraki, Osakil 567-0047, Japan 'Department of Physics, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan Department of Physics, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan Joint Institute for Nuclear Research (JINR), 141980 Dubna, Russia

The angular distributions of the cross section, the proton analyzing power and all of proton polarization transfer coefficients of p'd elastic scattering were measured at 250 MeV. The present data are compared with theoretical predictions based on exact solutions of the three-nucleon Faddeev equations and modern realistic nucleon-nucleon potentials combined with three-nucleon forces. These results call for a better understanding of the spin structure of the three-nucleon force and very likely for a full relativistic treatment of the three-nucleon continuum.

1. Introduction

One of the fundamental interests in nuclear physics is to establish nuclear forces and understand nuclear phenomena based on the fundamental Hamiltonian. Studies of few-nucleon systems offer a good opportunity to investigate these forces. Owing to intensive theoretical and experimental efforts, an often called new generation of realistic nucleon-nucleon (NN) potentials has been obtain using meson-exchange or other more phenomenological approaches, namely AV18',CD Bonn2, Nijm I, 11, and 933. They describe a rich set of experimental NN data up to 350 MeV. The accuracy of these theoretical predictions is remarkable and gives a x2 per degree of freedom very close t o one. These realistic two nucleon forces (2NF), however, fail to reproduce experimental binding energies for light nuclei where exact solutions of the Schrodinger equation are available, clearly showing underbinding. One can achieve correct three nucleon (3N) and four nucleon (4N) binding energies by including the Tucson-Melbourne (TM)4 or Urbana IX

121

(UIX)5 three-nucleon forces (3NF) which are refined versions of the FujitaMiyazawa force6 based on, a 2r-exchange between three nucleons with an intermediate A excitation. In recent years it became possible to perform rigorous numerical Faddeev-type calculations for the 3N scattering processes by the tremendous advance in computational c a p a b i l i t i e ~ ~ > In ~ ~ addition '. to the first signal on 3NF effects resulting from discrete states, strong 3NF effects were observed in a study of the minima of the N d elastic scattering cross sections at incoming nucleon energies higher than about 60 MeV1'. This discrepancy between the data and predictions exclusively based on NN forces could be removed by including the 2~-exchangeTM 3NF properly adjusted to reproduce the 3H binding energy in the 3 N Hamiltonian". In the present study, we have measured angular distributions of the differential cross section, the proton analyzing power A,, and all of proton polarization transfer (PT) coefficients K:' ,K $ IT:, K:' and K $ for p'd elastic scattering at 250 MeVl1. This energy is slightly above the pion threshold at 215 MeV. Realistic NN energy is slightly above the pion threshold at 215 MeV. Realistic NN potentials have been obtained by analyzing the existing NN data base up to 350 MeV1i2>3.The corresponding proton energy in pd system is 259 MeV to give the same center-of-mass (c.m.) energy. Most of the effects caused by the pion production are expected to be taken into account in the realistic NN potentials. The cross sections of elastic pd scattering shows a smooth energy dependence in the 200 - 300 MeV range indicating a small effect of the pion production but a possibly larger relativistic effect in this energy region12. ]

]

2. Experiment

The measurements were performed at the Research Center for Nuclear Physics (RCNP), Osaka University using the high resolution spectrometer Grand Raiden13, the focal plane polarimeter (FPP)I4, and the Large Acceptance Spectrometer (LAS)15. Polarized protons were produced in an atomic beam polarized ion source", injected into and accelerated by the K = 140 MeV AVF (Azimuthally Varying Field) cyclotron up to 46.7 MeV. Subsequently the beam was injected into the K = 4 0 0 MeV Ring Cyclotron17 and accelerated to the final energy of 250 MeV. The extracted beam from the Ring Cyclotron was transported to the West Experimental Hall via the WS beam line18. The proton polarization was continuously measured with two beam-line polarimeters which are separated by a total

122

bending angle of 115". Both the horizontal and vertical components of the polarization vector were determined. During the measurements typical values of polarization and beam current were 70% and 200 nA, respectively. Differential cross sections, analyzing powers, and a complete set of PTcoefficients were measured for p'd elastic scattering using self-supporting 99% isotopically enriched deuterated polyethylene foils (CD2)19with total thicknesses of 21 and 44 mg/cm2. A 15 mg/cm2 thick, natural carbon target was used to subtract contributions due to scattering on carbon. Precise absolute normalization of the cross sections was achieved by independent measurements with a gaseous target. The uncertainty in the overall normalizations was estimated to be 3% by comparing p p scattering measurements with calculations by the phase-shift analysis program code SAID20. The relative uncertainty in the normalization for the analyzing power of the ppscattering is 1%at Olab = 17" with a CH2 target".

3. Results and Discussion The experimental results for the differential cross section (da/dO), the vector analyzing power (Ay), and the PT-coefficients (K:', K f , K i , K ; ' , and K ; ' ) are shown in Figs.1 and 2. The quoted errors are statistical ones only. For the PT-coefficients, axes i^ and are defined in the laboratory scattering frame and K!' are plotted as function of the c.m. angles.

3'

-

10

2;

101

\

05

P

2, c

-$

100

a

2

00

-0 5

10-1

'0

-1 0

0

60

180

180

e,,

0

60

120

180

(deg)

Figure 1. The differential cross section da/dR (left) and proton analyzing powers (right) of elastic p'd scattering at E p = 250 MeV. The light shaded bands contain N N force predictions (AV18, CD-Bonn, Nijm I, 11, and 93), the dark shaded bands N N TM 3NF predictions. The solid and dashed lines are the AV18 + UIX and CD-Bonn TM' preditions, respectively.

+ +

In the left panel of Fig.1, the differential cross section is compared with theoretical predictions. Discrepancies at intermediate angles can be removed by including 3NFs used in the present study, the TM 3NF, its mod-

123

ified version TM', and the UIX 3NF. At backward angles, the inclusion of the 3NFs significantly reduces the discrepancies but is not sufficient to explain the data completely. While one can probably neglect ppCoulomb force effects a t the present energy, it is very likely that relativistic effects play a role. In the right panel of Fig.1, we compare the analyzing power A , with different nuclear force predictions. These predictions are in good agreement with the experimental data a t forward angles, but deviate dramatical at backward angles larger than 60". In the angular range 60" - 120", 2NF predictions are clearly larger in absolute value than experimental data. By including the TM 3NF the agreement with the data becomes better in the minimum around 60" - 120" but the discrepancies at more backward angles remain. Calculation with the TM' nor UIX 3NF improve the agreements with the data. 10

K:'

10

K

05 00

00

-0 5

-0 5

-1 0

-1 0

10

10

K :'

10

:' 0 5

K '; 06

0 6

04

0

KZ,'

05

05

00

0 0

-0 5

-0 5

-1 0 0

60

120

-1 0 180

0

60

120

180

8 c m (deg)

Figure 2. Polarization transfer coefficients ( K $ , K:', K z ' , K Z ' , and K $ ) of elastic Fd scattering at E p = 250 MeV. For the description of bands and lines see legend of Fig. 1

Present PT data are shown in Fig.2 together with theoretical predictions. For the PT-coefficients in the horizontal plane ( K $ ,K $ , K i , and K f ) calculations with the TM' and UIX 3NF provide similar predictions to those with 2NF only and a better description of the data. On the other hand, predictions of the PT-coefficients in the vertical plane ( K & ' are ) improved by including 3NFs, where UIX comes closest to the data. Overall, these results clearly indicate the spin structure of 3NFs is not properly described by today's models.

124 4. Summary

At intermediate energies our data are the first complete set of PTcoefficients for p’d elastic scattering covering a wide angular range and serve as a good testing ground of the investigation of the spin structure of 3NFs and the effects of relativity. In order to offer further valuable sources of information, a rich spectrum of spin observables will be measured not only for elastic scattering but also for Nd breakup process. For both of them, large 3NF effects have been predicted at intermediate energies. Acknowledgments We thank the RCNP staff for their supports during the experiment. We also wish to thank Professor H. Toki for his encouragement throughout the work. This experiment was performed under the Program No. El46 and R38 at the RCNP. References R.B. Wiringa, V.G.J. Stoks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995). R. Machleidt, F. Sammarruca, and Y. Song, Phys. Rev. C 53, R1483 (1996). V.G.J. Stoks et al., Phys. Rev. C 49, 2950 (1994). S.A. Coon and J.L. Friar, Phys. Rev. C 34, 1060 (1996). B.S. Pudliner et al., Phys. Rev. C 56, 1720 (1997). 6. J. Fujita and H. Miyazawa, Prog. Theor. Phys. 17, 360 (1957). 7. W. Glokle et al., Phys. Rep. 274 107 (1990). 8. A. Kievsky, M. Viviani, and S. Rosati, Phys. Rev. C 64, 024002 (2001). 9. J.L. Friar et al., Phys. Rev. C 42, 1838 (1998). 10. H. Witala et al., Phys, Rev. C 64, 014001 (2001). 11. K. Hatanaka et al., Phys, Rev. C 66, 044002 (2002). 12. H. RohdjeP et al., Phys. Rev. C 57, 2111 (1998). 13. M. Fujiwara et al., Nucl. Instrum. Methods Phys. Res. A 422, 484 (1999). 14. M. Yosaoi et al., AIP Conf. Proc. No. 343, edited by K.J. Heller and S.L. Smith (AIP , New York, 1995), p.157. 15. N. Matsuoka et al., RCNP report, 1992. 16. K. Hatanaka et al., Nucl. Instrum. Methods Phys, Res. A 384, 575 (1997). 17. I. Miura et al., in Proceedings of the 13th International Conference on Cyclotrons and theier Applications, Vancouver, 1992, edited by G. Dutto and M.K. Craddock, World Scientific, Singapore, 1993, p.3. 18. T. Wakasa et al., Nucl. Instrum. Methods Phys. Res. A 482, 79 (2002). 19. Y. Maeda et al., Nucl. Instrum. Methods Phys. Res. A 490, 518 (2002). 20. R.A. Arndt and L.D. Roper, Scattering Analysis Interactive Dial-In Program (SAID), phase-shift solution SP98, Virginia Polytechnic Institute and State University (unpublished); see also Phys. Rev. C 56, 3005 (1997), and references therein. 1. 2. 3. 4. 5.

125

STUDY OF THREE-NUCLEON-FORCE VIA NEUTRON DEUTERON ELASTIC SCATTERING AT 250 MEV

-

Y. MAEDA:H. SAKAI, A. TAMII, S. SAKODA, H. KATO, M. HATANO, T. SAITO, N. UCHIGASHIMA, H. KUBOKI Department of Physics, University of Tokyo 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan K . HATANAKA, Y. SAKEMI, T. WAKASA, J. KAMIYA, D . HIROOKA, Y. SHIMIZU, Y.KITAMURA, K. FUJITA, N. SAKAMOTO Research Center for Nuclear Physics (RCNP) 10-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan H. OKAMURA, K. SUDA, T. IKEDA, K. ITOH Department of Physics, Saitama University 255 Shimo-okubo, Urawa, Saitama, 338-8570, Japan K. Y A K 0 Center for Nuclear Study (CNS), University of Tokyo 2-1 Hirosawa, Wako, Saitama 351-0198, Japan K. SEKIGUCHI The Institute of Physical and Chemical Research (RIKEN) 2-1 Hirosawa, Wako, Saitama 351-0198, Japan M.B. GREENFIELD Division of natural Science, International Christian University Mitaka, Tokyo 181-8585, Japan

H. KAMADA Department of Physics, Kyushu Institute of Technology 1-1 Sensuicho, Tobata, Kitakyushu 804-8550, Japan

* yukieQnuc1.phys.s.u-tokyo.ac.jp

126 To study the three-nucleon force (3NF) without ambiguities from the Coulomb effect, the differential cross sections and vector analyzing powers for the iid elastic scattering at En = 245 MeV have been measured at O,, = 85' - 180'. The data are compared to the results of Faddeev calculations using modern nucleon-nucleon (NN) forces with and without 3NF. Additionaly we have a plan to extend our measurement t o the forward angular region where the Coulomb effect is expected to be large. We carried out the test experiment and measured the differential cross sections atecm = 21'.

1. Introduction

1.1. Motivation

In the intermediate energy region (E/A > 100 MeV), precise data of d p elastic scattering were compared with Faddeev calculations which included modern nucleon-nucleon (NN)forces The calculations without the threenucleon-force (3NF) underestimate the differential cross sections in the backward angular region. These discrepancies are removed by adding the Tucson-Melbourne (TM) 3NF3 into the calculations. This result is considered as a clear signature of the 3NF effects. However the inclusion of the 3NF does not always improve the description of the data of spin observables. In addition to that, since the inclusion of the Coulomb interaction into the calculation is difficult, we compared the d p data and the d n Faddeev calculations in which Coulomb interaction is neglected. To study 3 N F effects in a Coulomb-free system, we have performed the n'd elastic scattering measurements at En = 250 MeV. 1.2. Experiment of the backward scattering We measured the differential cross sections and the vector analyzing powers for the iid elastic scattering at En = 250 MeV in the angular range of 9,, = 85" - 180°4. This measurement was carried out at the ( n , p ) facility5 at the Research Center for Nuclear Physics (RCNP) by detecting recoiled deuterons. The polarized neutron beam was produced by the 7Li(p',G) reaction at E, = 250 MeV. We used the deuterated polyethylene (CD2)6 as deuteron targets. Recoiled deuterons were momentum analyzed by the Large Acceptance Spectrometer (LAS) and were detected at the focal plane. Details are described in references. The results of the differential cross sections and the vector analyzing powers are shown in Figure 1 by solid circles with the Faddeev calculations7. In this analysis, we normalized the cross section data by comparing the results of iip measurements and the calculations with the program code

127

SAID8. Concerning about the differential cross sections, it can be seen that the calculations including 3NF fill the gap seen in the data but still underestimate largely the data. These discrepancies may be an indication of the relativistic effect^^^^^ which are not taken into account in the present calculations. The data of the p'd elastic scattering at same energy1' are also shown in Fig. 1. Both data of the n'd and the p'd are consistent within the systematic error of the iid data.

n

&

5

+

d En=245MeV 1.0

10.00 5.00

- CDBONN CDBONN+TM3NF

--

1.00 0.50

AVlB+UrbanaIX3NF

0.5 h

q 0.0

C -t3

2

0.10

-t3

0.05

-0.5

0

-1.0 50

100

150

0

0c.m. [degl

50

100

150

8c.m. [degl

Figure 1. Differential cross sections and vector analyzing powers for the iid elastic scattering at En = 250 MeV. The solid circles are the results of the previous experiment. The solid square is the result of the test experiment using the NTOF method. The statistical errors are shown in the figures. The open circles are the results of the p'd elastic scattering at E, = 250 MeV". The calculations with CD-BONN potential (thin solid curves), AV18 potential only (thin dashed curves), CD-BONN potential including TM-3NF (thick solid curves) and AV18 potential including Urbana 3NF (thick dashed curves) are also shown.

2. Experiment of the forward scattering We are planing to extend our measurement to the forward angular region, where the Coulomb effect is expected to be large.

128 Liquid scintillator

Veto scintillator

TOF tunnel NPOLZ

Bearn'durnp

Figure 2. A schematic layout of the nd measurement of the forward angular region at the NTOF facility at RCNP.

2.1. Experimental procedure

In the forward angle measurement, we can not apply the same technique used in the backward measurement since the recoiled deuteron energy becomes too low to detect by LAS. Thus we plan to measure the n d elastic scattering by detecting scattered neutrons at the NTOF facility at RCNP. The experimental layout is schematically illustrated in Figure 2. The neutron beam is produced by the 7Li(p,n)7Be*(0")reaction at Ep = 250 MeV in the vacuum chamber of the beam swinger magnet. The neutron beam pass through the neutron exit window of the vacuum chamber and bombard the deuteron target. The distance between the 7Li target and the deuteron target is 2m. The scattering angle of the 2H(n,n)reaction can be varied from Blab = 0" - 40" by moving the 7Li target along the beam trajectory and moving the deuteron target simultaneously. The scattered neutrons run through the 70 meter time-of-flight (TOF) tunnel and be detected by NPOL212i13714.For the purpose of the check of the experimental system, we carried out the test measurement at Blab = 13", namely I ~ C M= 21O. In this experiment, we used the deuterated liquid scintillator BC537 as the deuteron target and performed the coincidence measurement. We also used the liquid scintillator NE213 as the proton target and measured n p elastic reactions. These target were contained in a cylinder with a diameter of 16 cm and a length of 8 cm. To reduce the accidental coincidence events, we applied the n - y discrimination method to these target.

129

2.2. Results

The very preliminary data of the differential cross section is shown in Figure 1 by solid square. It has been shown that the absolute cross section is properly measured by our method. Since this experiment was performed only for the test of the experimental system, this result contains large systematic error and it is not be able t o extract the Coulomb effect from the comparison between this result and Ref. 11. However we can improve the errors and the angular region in the next experiment.

Acknowledgement This project was supported in part by the Ministry of Education, Culture, Sports, Science and Technology of Japan with the Grant-in-Aid for Science Research No. 10304018 and the Research Fellowships of the Japan Society for the Promotion of Science (JSPS) for Young Scientists.

References 1. H. Sakai et al., Phys. Rev. Lett. 84,5288 (2000). 2. K. Sekiguchi et al., Phys. Rev. C65,034003 (2002). 3. S.A. Coon et al., Nucl. Phys. A317,242 (1979) 4. Y. Maeda et al., in Spin 2000, eds. K. Hatanaka et al., AIP Conf. Proc. No. 570 (AIP, 2001), p. 719. 5. K. Yak0 et al., Nucl. Phys. A684,563c (2001) 6. Y. Maeda et al., Nucl. Inst. Meth. A518,490 (2002) 7. H. Kamada, private communication. 8. CNS DAC online Services, http://gwdac.phys.gwu.edu/ 9. H. Rohdjeil et al., Phys. Rev. C57,2111 (1998) 10. H. Witala et al., Phys. Rev. C57,2111 (1998) 11. K. Hatanaka et al., Phys. Rev. A66,044002 (2002) 12. H. Sakai et al., Nucl. Inst. Meth. A320,479 (1992) 13. H. Sakai et al., Nucl. Inst. Meth. A369, 120 (1996) 14. T. Wakasa et al., Nucl. Inst. Meth. A404,355 (1998)

130

SEARCH FOR ANOMALY AROUND SPACE STAR CONFIGURATION IN pd REACTION

T. ISHIDA, T. YAGITA, S. OCHI, S. NOZOE, K. TSURUTA, F. NAKAMURA AND K. SAGARA Department of Physics, Kyushu University, 6-10-1 Hakozaki Fukuoka, 812-8581 Japan H. G. P. G. SCHIECK Institute for Nuclear Physics, University of Cologne, 0-50937 Cologne, Germany Systematic measurement of D(p,pp)n cross section has been made at an incident energy of 13 MeV around the space star configuration. In spite of the anomaly found in nd breakup cross section, no anomaly has been found in pd breakup.

1. Scientific Motivation

A 2r-exchange 3NF ( ~ T ~ N Fwhich ), is adjusted so as to reproduce the 3N binding energies, has been proved to excellently reproduce the cross section of N d scattering at energies below 200 MeV. On the other hand, the analyzing powers and polarization transfer coefficients of N d scattering are not satisfactorily reproduced, even though agreement is seen at restricted energy region. From the above facts, we expect that scalar observables are well reproduced by calculations with 2r3NF. An exception is the “space star (SS) anomaly” in n d breakup cross section at 13 MeV. Calculations underestimate the experimental cross section by an amount of about 30 % and 27r3NF contributes very little (Ref. 18~2).The anomaly has also been reported at 10 MeV (Ref. l),however, no anomaly has been reported above 16 MeV. Also no anomaly has been found in pd breakup in the same incident energy region. Though cross section of pd breakup at SS configuration is below the n d breakup calculation by about 10 % at 13 MeV, the 10 % difference is well predicted by a pd calculation by E. 0. Alt and M. Rauh

131

using a simple S-wave NN interaction (Ref. 5). If SS anomaly exists in nd breakup, the same anomaly should exist also in pd breakup. It is possible in pd breakup that Coulomb force may shift the kinematical configuration where SS anomaly appears. We therefore search for SS anomaly in pd breakup at 13 MeV in a wide angular range around SS configuration. 2. Experimental Setup The pd breakup experiment has been made at Kyushu University Tandem Laboratory (KUTL). The SS configuration is satisfied at &=02=50.5O and +12=120" in the laboratory system, where 8i is the angle between the beam axis and an outgoing i-th proton's momentum and 4 1 2 is the azimuthal angle between two outgoing protons. We made systematic measurements at 23 different pairs of (61,132, 412=120") around the space star configuration, as shown in Figure 1.

63.0 59.5

.

53.5 52.5 50.5 -

56.0

a9

k

k

e

47.5 50.5 53.556.0 59.5 63.0

Figure 1. The angle pairs where the measurements were made.

Figure 2 shows the detector setup. A deuterated polyethylene foil was used as a target. We employed six Si SSD's to make simultaneous measurement of D(p,pp)n reaction a t 9 pairs of (&, &). A monitor SSD was used t o normalize the cross section. Recoil deuterons from p+d scattering pro-

132

duced a lot of accidental coincident events in the energy range of protons from D(p,pp)n. Therefore, we detected the elastically scattered protons by veto SSD's and rejected the accidental coincident events induced by d recoils.

veto SSDO

.-. *.

13MeV p beam

#

oo

0 0 0 0

0-

monitor

SSD Figure 2.

SSD

Experimental setup. (view facing to beam)

We measured the energies of two coincident particles and the time difference between them. The time difference was used t o identify the true (+background) events from the background events. With those techniques we succeeded in drastically reducing the accidental backgrounds.

3. Results and Discussion Out of 23 geometrical conditions, typical results of measured cross section at 81=82=50.5" are shown in Figure 3 and the location of the SS configuration is pointed by an arrow. The solid line and the dashed line represent D(n,nn)p cross sections calculated with 2NF and with 2NF+27r3NF, respectively, by H. Kamada (Ref. 4). The difference between these two kinds of calculation are very small. It indicates that there are little effects of 27r3NF around the SS condition. Our data are consistent with previous data of Cologne group a t the same energy (Ref. 3). The pd breakup data lie under the D(n,nn)p calculation, not only a t the SS point but also along the locus around SS. From D(p,pp)n calculation with a simple S-wave NN interaction (Ref. 5), the Coulomb force is expected to decrease the cross section by about 10%. Therefore the most part of the disagreement between the D(n,nn)p calculation and

133 1.4 '

1.2

ouidata wlo3NF

-

1

0.8 0.6 0.4 0.2 0 -

0

2

4

6 8 S [MeV]

10

12

14

Figure 3. Measured cross section at 81=82=50.5O which includes the space star configuration. The solid circles represent the present data. The solid and the dashed line represent theoretical calculation with 2NF and with 2NF+3NF respectively (Ref. 4). The difference is too small to distinguish. The space star configuration is pointed by an asrow.

the D(p,pp)n experiment in Figure 3 might be attributed to the Coulomb effects. Disagreement between D(n,nn)p calculation and the D(p,pp)n experiment are also seen in the other 22 angle pairs. In order to see how this disagreement varies with the angle, the cross section was averaged around El=E2 point, and is plotted against angle sum of el+&, as in Figure 4. The averaged cross section varies smoothly against &+& and no signature of anomaly is seen. The disagreement gradually increases with &+&. This might be because relative energy between the two outgoing protons decreases and Coulomb repulsion increases as 01 +& increases. This qualitative discussion, however, should be examined quantitatively by a D(p,pp)n calculation with the realistic potentials. Because of difficulty in making correct 3N calculation with Coulomb force, pd breakup calculation with realistic potentials is not available yet above the deuteron breakup threshold. We hope that such a calculation is performed in the near future.

134 1.4 1.2

1 0.8

0.6

0.4 0.2

0' 90

95

100

105

110

115

120

125

130

01 + e2 [degl

Figure 4. The averaged cross section as a function of 81+&.

4. Summary

The cross section of D(p,pp)n breakup reaction at an incident beam energy 13 MeV has been measured systematically at 23 different (01, 6%) pairs around the space star configuration. The pd breakup cross section varies smoothly against the angle sum of &+02 and no anomaly is found, in contrast with the SS anomaly in n d breakup at the same incident energy. For quantitative discussion of the smooth disagreement between the experiment and calculation, calculations with Coulomb force are highly desired.

References 1. J. Strate et al., Nucl. Phys. A501, 51 (1989). 2. H. R. Setze et al., Phys. Lett. B388, 229 (1996). 3. G. Rauprich et al., Nucl. Phys. A535, 313 (1991). 4. H. Kamada, Private Communication. 5. E. 0. Alt and M. Rauh, Phys. Rev. C49, R2285 (1994).

V. FOUR BODY AND 3He SCATTERING


137

BACKWARD ELASTIC P 3 H E SCATTERING AT ITERMEDIATE ENERGIES

YU. N. UZIKOV * Joint Institute for Nuclear Research, L N P , 141980 Dubna, Moscow Region, Russia E-mail: [email protected]

Recent results of study of mechanisms of p3He backward elastic scattering at beam energies 0.1-2 GeV are reviewed. The main attention is paid to a role of intermediate pions analyzed on the basis of the triangle diagrams with the subprocess pd +3Heao and pd* + 3 H e ~ 0 ,where d' denotes a pn pair in the spin-singlet ' S o state. It as shown that these diagrams dominate in the cross section at 0.3-0.8 GeV and their contribution is comparable with that for a two-nucleon transfer at 1-1.5 GeV. It makes much less transparent relationship between the measured cross section at Tp > 0.9 GeV and high-momentum components of the 3 H e wave function, established previously for the two-nucleon transfer mechanism. Spin-spin correlation parameter C,,, is calculated for the two-nucleon transfer.

1. Introduction

The structure of the lightest nuclei at short distances in the nucleon overlap region ?'" < 0.5 fm, zt ed. at high relative momenta between nucleons q N N 1/?'" > 0.4 GeV/c, is a fundamental problem of physics of strong interaction. This structure is probed mainly by electromagnetic processes at high transferred momenta Q However, unknown strength of the meson exchange currents and rather small value of the electromagnetic cross sections restrict possibilities of these processes. An important independent information can be obtained from the hadron-nucleus interactions. Since dynamics of hadron-nucleus interactions is complicated by the excitation/de-excitation of nucleon resonances in the intermediate states, a special type of processes has to be chosen. So, to minimize the screening effects, arising in the backward elastic scattering pd + d p due to excitation of the A(1232) isobar, more preferable is to study a deuteron breakup

-

'.

*Work supported partially by BMBF Heisenberg-Landau program

138

pd + ( p p ) n in the pd + dp kinematics but with the singlet pp('S0)-pair z. An obvious advantage of the 3 H e nucleus in comparison with the deuteron is a more intensive high momentum component of the 3He wave function. Furthermore, due to a larger mass of 3He as compared t o the deuteron, much more high internal momenta can be probed in a nucleus in the p3He backward elastic scattering than in the pd -+dp at the same beam energy. 2. Two-nucleon transfer mechanisms

Over the past view years an analysis of p3He backward elastic scattering was performed on the basis of the DWBA method using a three nucleon bound state wave function obtained in Ref.6 from a solution of the Faddeev equations for the RSC N N potential. In the plane wave approximation, the most general expression for the two-nucleon transfer amplitude of the process 0 { 123) + (023) 1 takes the form 3,475

+

+

where cpij(k,ij) = p(qij,pk) is the Faddeev component of the full wave function Q(i,j,k ) = cplk + 'pki+ 'pij of three nucleon bound state {ijlc), x p ( 0 ) ( x p ~ ( l )is) the spin-isotopic spin state of the initial (final) proton; L23 = E + q"23/m 3p:/4m, m is the nucleon mass, E is the 3 H e binding energy. The subscript i (f) in Eq.(l) refers to the initial (final) nucleus. In the Feynman diagram approach the terms ~py+'p!~, 'py+'pzl+ and 'py+'p:', correspond to the diagrams a, b and c in Fig.1, respectively. The studies 3,4,5 suggest that this process at initial energies Tp = 1 - 2 GeV can give unique information about the high momentum component of the 3He wave function ~ ~ ~ ( 9p1) 2 3 -, specifically for high relative momenta, 423 > 0.6 GeV/c, of the nucleon pair (23) in the ' S Ostate and low momenta of the nucleon "spectator", p l < 0.1 GeV/c. For comparison, available data on ed elastic scattering at Q2 < 1.4 (GeV/c)' essentially probe the internal momenta in the deuteron wave function at q < 4p 0.55 GeV/c ', where ep p is the pion mass. This important feature of the ~ ~ H e + ~ Hprocess follows from a dominance of the mechanism of sequential transfer (ST) of the proton-neutron pair (Fig.lc) in this process over a wide range of initial energies Tp = 0.1 - 2 GeV, except for the region of the ST dip at around 0.3 GeV. Other mechanisms of two-nucleon transfer, like the interacting pair

+

-

139

exchange (Fig. l a ) , non-sequential np-transfer (Fig.lb) 4 , and direct pNscattering l 1 > l 2involve very high internal momenta both in q 2 3 and p l in

Figure 1 . T h e two-nucleon transfer mechanisms of the backward elastic p 3 H e scattering: a - interacting pair exchange (IPE), b - nonsequential transfer (NST), c - sequential transfer (ST). The off-shell "-scattering amplitudes are denoted as black circles.

the 3 H e wave function and, on the whole, give much smaller contributions. The deuteron exchange, discussed in Refs. does not arise here as an independent mechanism being included in the sum of diagrams in Fig.1. 9310112,

3. Role of intermediate pions

However, by analogy with pd backward elastic scattering 1 3 , one should expect a significant contribution of mechanisms related to excitation of nucleon isobars in the intermediate state and followed by production of virtual pions. Such mechanisms were discussed in Refs. l4l5 on the basis of the triangle diagram of the one pion exchange (OPE) with the subprocess p d -+ 3He7r0 (Fig. 2b) and in Refs. l5>l6 with the subprocess 7rd + 7rd. The energy dependence of the cross section ~ ~ H e - + ~ H and e palso its the absolute value were explained to some extent in Refs. 15916. However, a common drawback of the models is a neglect of both (i) the contribution of the singlet deuteron d' in 3He (i.e. where the pn pair is in the spin-singlet ' S Ostate) and (ii) distortions arising due to rescattering in the initial and final states. In paper l7 both these effects (i)-(ii) are considered and it is shown that there is an effective cancellation between them. When calculating the OPE diagrams depicted in Fig.2, we proceed from the formalism of Ref. 5 , which takes into account the two-body d + p configuration of the 3He nucleus. It allows one to express the cross section of p3He scattering through the experimental cross section of the reaction p d +3He 7ro. In order to calculate the contribution of the meson production 14115,16

140

Figure 2. The OPE mechanism of the backward elastic p3He scattering with the intermediate deuteron (a), singlet pn-pair ( b ) , biproton ( p ~ ( )c )~.

on the virtual singlet deuteron d* = (pn),and biproton pair pp in the 'So state in 3He one has to use the d* N configuration of 3He. Furthermore, a concrete mechanism of the reaction pd* +3He7r0 has to be elaborated. As the first step, this mechanism is assumed here to be the same as for the pd +3He7r0 subprocess. According to Ref.18, the two-body mechanism depicted in Fig. 3 explains reasonably well the cross section of the reaction pd +3 H e r o in forward (0, = O"), and backward (8, = 180") direction at Tp 0.3 - 1.0 GeV. Within the similar model, explanation of the tensor analyzing power T20(0, = 180") was reasonably explained in Ref.19 but using the d and d* in the intermediate state of the diagram in Fig. 3 instead of the "-loop. At higher energies, Tp > 1 GeV, this mechanism fails to reproduce the second peak in the excitation function of the pd +3 HeKO reaction. In this region the 3-body mechanisms l8 was found more important, nevertheless it also underestimates considerably the experimantal cross section at 8, = 180".

+

-

3.1. Elements of formalism of the OPE model

Since the mechanism of the reaction pd -+3He7ro is not well established, a complete microscopical description of the reaction ~ ~ H e + ~ Hwithin ep the OPE mechanism cannot be achieved at present. For this reason, when estimating here the contribution of the singlet ( N N ) ppairs, we consider mainly the region of energies Tp = 0.3 - 1 GeV, where the contribution of the spectator diagram in Fig.3 is valuable. Considering the coherent sum of the OPE amplitudes Md Md* + Mpp with the deuteron Md (Fig. 2b), singlet deuteron Md* (Fig. 2c) and biproton Mpp (Fig. 2d) we use the overlap integrals 3He - d and 3He - d" from 2 1 parametrized in Ref. The 3He-d overlap wave function contains the

+

'.

141

S- and D-components. As was shown in Ref. 5, the D-component of the 3He-d overlap integral is negligible in the O P E amplitude. For the singlet deuteron there is only the S-component in the overlap integral 3He-d*. Keeping the S-wave in the 3 H e - d overlap wave function, we find that there is no interference term between the triplet ( M d ) and singlet ( M i . + M p p ) amplitudes in the squared and spin-averaged sum l h f d ibfd* MpPl2.This feature simplifies the theoretical analysis significantly. Thus, the squared spin averaged O P E amplitude of the p3He backward elastic scattering has the form

+

+

here M is the mass of the 3He nucleus ( m and p were defined in Sec. 2);

+ 7

Ept = m2 p,, and p’ are the total energy and momentum of the secondary proton in the laboratory system; fn” is the coupling constant and F,”(k2) is the monopole formfactor in the 7rNN vertex. pj (p-ljt) is the spin projection of the initial (final) particle j ( j = p , h ) , X is the spin projection of the deuteron. The amplitudes of the subprocesses p d -+ 3He7r0, pd* + 3He7ro and ~ ( p p+ ) ~3He7r+ are denoted as T[jA(d), T,$(d*) and

5

T/:(pp), respectively. The isospin factors and 2 in the second term in the curl brackets in Eq. (2) do not depend on the mechanism of the process p ( N N ) + 3He7r. The spectroscopic factors for the triplet deuteron, Sp”, and the singlet one, are taken here as Sid = SP”. = 1.5 according to 2 1 J 9 . The nuclear formfactor for the singlet (s) and triplet ( t ) channels are given as

s$*,

2

Gs3t= I i ~ F f ( f i+) W$(fi,b)( ,

(3)

where

w:~((~T,z) = l m j l ( ~ r ) ~ ; , t ( r ) ( i ~ +1)exp ( - i i ~ ) d r .

(4)

Here U,“(r)( U t ( r ) )is the S-component of the 3He-d (3He-d*) overlap integral, j l is the spherical Bessel function. The kinematical variables IC, 6 and are determined by the proton energy Epl. The formulas can be found in Ref. in the same notations.

142 We introduce the following relation for the squared triplet and singlet amplitudes of the p ( N N ) , , t -+3Hen processes

IT(d*) + 2 T(PP)l2 =

GIT012,

(5)

where CIis the dynamical factor. After that the c.m.s. cross section of the p3He backward elastic scattering can be written as

where sij is the square of the invariant mass of the system j + i, and qij is the relative momentum in this system. The distortion factor D(T,) is given in Ref.' in the Glauber approximation in terms of parameters of forward p N and p3He scattering amplitudes.

n:

Figure 3.

The spectator model of the p ( N N ) , , t

+3

H e n reaction.

3.2. Contribution of the singlet deuteron and biproton Evaluation of the factor Cl for the spectator model (Fig. 3 ) of the p ( N N ) , , t + 3He7r reaction reveals that there is an enhancement for the singlet deuteron and biproton in comparison with the deuteron. We assume here that the p N + d n subprocess dominates in the upper vertex of the diagram in Fig. 3 and that the amplitude p N + ( N N ) , n is negligible. This is true in the A-region] as was shown recently 22. After that we find that the interference of the singlet amplitudes Md* and Mpp is constructive and the factor CI in Eqs. (5,6) equals to independently on the beam energy. Since the singlet and triplet formfactors are related numerically by

TI

143

Figure 4. C.m.s. cross section of elastic p 3 H e scattering at the scattering angle O,, = 180' as a function of the kinetic energy of the proton beam. Experimental data are from Refs. l5 (o), 30 (black square), 31 (open square), 32 (a), and 33 (diamond). Model calculations on the basis of the OPE mechanism and without distortions correspond to the full thin line (for the deuteron in the intermediate state) and the dash-dotted line (for the d+d* + p p ) . OPE calculations including distortions in the initial and final states correspond to the thick dashed line (for the deuteron) and the full thick line (for the d d' p p ) . The result for the nondistorted S T cross section with CD Bonn w.f. is given by the dotted line. Note that the distortion factor for the S T mechanism differs from the one for OPE.

+ +

G" M 1.5 Gt (cf. Ref.

19) in the kinematical region under discussion, the total enhancement factor due to the contribution of the singlet N N pairs to the p3He + 3 Hep cross section is about 12. Thus, the contribution of the singlet N N pairs can be taken into account in this approximation by variation of the effective spectroscopic factor of the deuteron in the 3 H e , Sid + M S$(l+ 1.5Cr). In the numerical calculation, we use the parametrizations from Ref. l9 for the 3 H e - d and 3 H e - d*

144

overlap integrals 21 obtained for the Urbana N N potential. We have found that the final result is almost the same when the RSC parametrization from Ref. is used. The experimental cross section of the reaction pd -+3 H e r o for the backward scattered pions (Ocm = 180') is taken from 23. The cut-off parameter A, in the monopole formfactor of the r N N vertex is ranged in the interval of A, = 0.65 - 1.3 GeV/c. The lower case A, = 0.65 GeV/c corresponds to the result of the analysis of the p p + pnr+ reaction a t 0.8 GeV 24 performed in the r + p exchange model 25. The upper case A, = 1.3 GeV/c corresponds to the value used in the full Bonn N N model 2 6 .

3.3. Numerical results The result of our calculation are shown in Fig. 4. One can see that the OPE model with the deuteron describes reasonable the energy dependence of the cross section for Tp = 0.4 - 1.5 GeV, although it underestimates the magnitude. The calculated cross section is smaller then the experimental data by factor of around 7 for A, = 0.65 GeV/c, and by about 1.5-2 for A, = 1.3 GeV/c. The contributions of the singlet deuteron d' and the p p pair increase the cross section considerably. As a result the cross section for p 3 H e +3 H e p is now overestimated in the region of Tp = 0.4 - 1.5 GeV by factor of 2-3 and 5-10 for A, = 0.65 GeV/c and 1.3 GeV/c, respectively. The distortion factor D(T,) reduces the OPE (d d* + p p ) cross section of the p 3 H e -+ 3 H e p process by one order of magnitude (full thick line) and brings it in qualitative agreement with the data. The discrepancy with the data in the region of the first shoulder, Tp = 0.3 - 0.6 GeV, can be attributed t o others terms in the pd* -+3 H e r o amplitude, like the two-step mechanism 27, which are not considered here.

+

4. The ST mechanism revisited

Turning back to the pure two nucleon transfer mechanism 3,4,5 we should note that the three-nucleon bound state wave function based on the Reid RSC N N potential most likely contains too strong high momentum components as compared to modern N N potentials. In order to corroborate that we show here, in the framework of the S-wave formalism of Ref. 5 , that for the Faddeev 3H wave function 28 based on the CD Bonn N N interaction 29 the ST cross section at Tp > 0.5 GeV/c is by a factor of 30 smaller than for the RSC. Nevertheless the predicted cross section is still comparable with

145

-2

lo

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 5. T h e same as in Fig. 4, but the theoretical curves show the results of calculations for the S T mechanism in the plane wave approximation and with different 3He wave functions: Reid RSC (dashed thin line); CD Bonn (dotted). T h e ST cross section for the C D Bonn wave function with distortions taken into account is shown by full thick line. The deuteron plus singlet deuteron exchange calculated in the plane-wave approximation with the 3He-d(d*) overlap integrals from Ref. l9 is shown by dashed-dotted line.

the experimental data at Tp > 0.9 GeV (Fig.5). a One can see from Figs. 4, 5 that the ST mechanism definitely dominates at Tp < 0.3 GeV and > 1.5 GeV and is still very important in the region 0.9-1.5 GeV. 5. Spin-spin correlation In general case of angular momentum and P-parity conservation, the amplitude of the p3He-+ 3Hep process consists of six independent spin ampliaInclusion of the D-waves will lead t o an additional increase of the calculated cross section (see Ref.4).

146

tudes. For the case of the ST-mechanism in the S-wave approximation for the H e wave function there are only two independent spin amplitudes

F = 5 (Ado l [

1 + MI) + (Mo - -Mi)ui 3

'02

1

,

(7)

where MO ( M I )is the amplitude of the spin-singlet (spin-triplet) nppair transfer, as defined by Eq.(9) in Ref. 5 ; (TI and u2 are the Pauli spin operators acting on the spin states of the proton and 3 H e nucleus, respectively. In case of collinear kinematics the number of amplitudes equals to three and the most general form of the amplitude is the following

F =A

+ B u I . u2 + C(u1 . m)(u2.m),

(8)

where m is the unit vector in a direction of the proton beam. The unpolarized cross section dao and spin-spin correlation parameter C,,, (for definition see 3 4 ) can be written in collinear kinematics as

+ 21BI2 + IB + CI2), R e (BA* - ( B + C ) B * )

duo = K (IA12 cy>y

= IAI2

(9)

+ 21BI2 + IB + CI2'

where K is the phase space factor. At present seems unrealistic to evaluate C,, for the O P E mechanism. The results for the ST+NST and deuteron (d d * ) exchange are shown in Fig.6. For the pure deuteron exchange, only the triplet amplitude MI is nonzero in the S-wave approximation for the 3 H e - d overlap integral. In this case one can find CyYy = - $ independently on the beam energy. Adding the singlet deuteron exchange, i.e. the d d* exchange mechanism, leads to MO # 0. We show numerically (Fig.6), that C,,, for the d+d*- exchange mechanism differs from that for the ST+NST. At low energies, Tp < 0.3 GeV these two mechanisms give approximately the same unpolarized cross section (see Fig.5). A measurement of C,,, planned at RCNP 35 can allow to discriminate these two mechanisms.

+

+

6 . Conclusion

Present analysis shows that the intermediate pions play very important role in the ~ ~ H e + ~ Hprocess e p at Tp > 0.3 GeV. While the OPE diagram with only the subprocess pd -+3 H e r o seems t o be not sufficient to explain the absolute value of the ~ ~ H e - - + ~cross H e psection, mainly due to depressive distortions, after inclusion of the d' and p p pairs the OPE contribution increases drastically. As a result, the O P E model describes reasonably the energy dependence of the differential cross section at Tp = 0.3 - 1.5 GeV.

147 3 0.4

O" 0.2 0 -0.2 -0.4

-0.6

-0.6

2 -1 0.2 0 -0.2 -0.4

-0.6 -0.8

Figure 6. Calculated spin-spin correlation parameter in the p3He+ 3Hep process. a:The d d' exchange in the PWBA with the 3He-d(d*) overlap integrals from Ref.lg (dashed-dotted line), d exchange only (dashed). b: the NST+ST mechanism with the RSC 3 H e wave function in the PWBA (dotted line), the NSTSST with the CD Bonn w.f. 28 in the PWBA (full thin, the same in a) and in the DWBA (full thick).

+

Using the CD Bonn wave function instead of the RSC one for the 3He nucleus diminishes the role of the ST+NST mechanism, but does not change the main conclusion of Ref. '. Namely significant contribution of the ST mechanism for Tp > 1 GeV allows one to probe essentially the high momentum components of the 3He wave function. However, the relation between the observables and the high-momentum structure of the 3He wave function becomes much less transparent due to uncertainties in the contribution of the OPE mechanism. Future progress in analysis of this process requires to clarify the mechanism of the reaction pd +3He7r0, specifically at Tp > 1 GeV, and to take into account off-shellness of the virtual subprocesses P ( N N ) ~+3He7r0 ,~ and the N N continuum as well. Polarization measurements 35 can give an important new information. Acknowledgements. I am thankful1 to K.Hatanaka for warm hospi-

148

tality at RCNP (Osaka University) a n d fruitful discussions.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

F.Gross, Gilman. J.Phys.G: Part.Nuc1. 28 (2002) Yu.N.Uzikov. J.Phys.G: Part.Nuc1. 28 (2002) B13. A.V. Lado, Yu.N. Uzikov, Phys. Lett. B279 (1992) 16. L.D. Blokhintsev, A.V. Lado, Yu.N. Uzikov, Nucl. Phys. A 597 (1996) 487. Yu.N. Uzikov, NucLPhys. A 644 (1998) 321. R.A. Brandenburg, Y. Kim, A. Tubis, Phys. Rev. C 1 2 (1975) 1368. L.D. Blokhintsev, E. Dolinskiy. Yad.Fiz.5 (1967) 797. R. Schiavilla, V.R. Pandharipande. Phys.Rev.C65 (2002) 064009. L. Lesniak, H. Lesniak, Acta Phys. Po1.B 9 (1978) 419. M.S. Abdelmonem, H.S. Sherif, Phys. Rev. C36 (1987) 1900. M.J. Paez, R.H. Landau, Phys. Rev. C 29(1984) 2267. Yu.N. Uzikov, Elem. Part. At. Yadr. 2 9 (1998) 1010 (Part. Nucl. 29 (1998) 417). N.S. Graigie, C. Wilkin, Nucl. Phys. B14 (1969) 477. G.W. Barry, Phys. Rev. D7 (1973) 1441. P. Berthet et al., Phys. Lett. 1 0 6 B (1981) 465. A.P. Kobushkin et al., nucl-th/0112078. Yu.N. Uzikov, J.Haidenbauer, (in preparation). J.M. Laget, J.F. Lecolley, Phys. Lett B194 (1987) 177. J.-F. Germond, C. Wilkin, J. Phys. G 14 (1988) 181. Ch.H. Haiduk, A.M. Green, M.E. Sainio, Nucl. Phys. A337 (1980) 13. R. Schiavilla, V.R. Pandharipande, R.B. Wiringa, Nucl. Phys. A449 (1986) 219. Yu.N. Uzikov, C. Wilkin, Phys. Lett. B545 (2001) 191. P. Bertheth et al., Nucl. Phys. A443 (1985) 589; V.N. Nikulin et al., Phys. Rev. C 54 (1996) 1732. J. Hudomaij-Gabitsch et al., Phys. Rev.Cl8 (1978) 2666. A. Matsuyama, T.-S.H. Lee, Phys. Rev. C 34 (1986) 1089; 0. Imambekov, Yu.N. Uzikov, Sov. J. Nucl. Phys. 47 (1988) 695. R. Machleidt, K. Holinde, Ch. Elster, Phys. Rep. 149 (1987) 1. G. Faldt, C. Wilkin, Nucl. Phys. A 587 (1995) 769; L.A. Kondratuyk, Yu.N. Uzikov, J E T P Lett. 63 (1996) 1. V. Baru, J. Haidenbauer, C. Hanhart, J.A. Niskanen, nucl-th/0207040. R. Machleidt, Phys. Rev. C 6 3 (2001) 024001. H. Langevin-Joliot et al., Nucl. Phys. A158 (1970) 309. V.I. Komarov et al., Yad. Fiz. 11 (1970) 399. R. Frascaria et al., Phys. Lett. 6 6 B (1977) 329. L.G. Votta et al., Phys. Rev. C 10 (1974) 520. G.G. Ohlsen, Rep. Prog. Phys. 35 (1972) 717. K.Hatanaka, In: Proc. of MEDIUM02 (this issue).

149

S T U D Y OF THE

P + 31?E

BACKWARD ELASTIC SCATTERING

K. HATANAKA~, Y. SAKE MI^, T. WAKASA~, H.P. Y O S H I D A ~ , J. K A M I Y A ~ Y. , SHIMIZU~,H. O K A M U R A ~ , T. U E S A K A ~ , K. S U D A ~ , H. UEN03, K. SAGARA4, T. ISHIDA4, S. ISHIKAWA5, M. TANIFUJ15, A.P. KOBUSHKIN1i6, AND E.A. STROKOVSKY 197 'Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan Department of Physics, Saitama University, Urawa, Saitama 338-8570, Japan R I K E N (Institute for Chemical and Physical Research), Wako, Saitama 351-01 98, Japan Department of Physics, Kyushu University, Hakozaki, Fukuoka 81 2-8581, Japan Department of Physics, Hosei University, Fujimi , Tokyo 102-8160, Japan Bogolyubov Institute for Theoretical Physics, 03143 Kiev, Ukraine LPP, Joint Institute for Nuclear Researches, 141980 Dubna, Russia We will measure the differential cross section and the spin-spin correlation C y y of the p 3 H e backward elastic scattering at 200 - 400 MeV and at the Research Center for Nuclear Physics (RCNP), Osaka University. There are several cross section data but mainly at energies higher than 400 MeV. Our calculations show that at the energy region of the RCNP Ring Cyclotron, the cross section and spin-spin correlations are sensitive to the 3He wave function and the reaction mechanisms.

1. Introduction

It is well known that it is possible to use nucleon-nucleus scattering as a probe of the spin structure of nuclei since target-related observables are extremely sensitive to small spin-dependent parts of the target wave function. It is this sensitivity that makes these spin observable data such a severe test of theoretical models. In addition, one can gain information about the nucleon-nucleus reaction mechanism, the spin dependence of the nucleon-nucleon interaction in the nuclear medium, and the off-shell behavior of the N-N amplitudes. The 3He and 4He nuclei are the most attractive in this aspect as well as for searches of effects of 3N forces because these nuclei are the most dense.

150

Proton scattering from 3He is especially interesting, since the large spin effects present when one third of the nucleons in a target has unpaired spin. In the first order, the relative magnitude of spin-dependent effect is expected to behave inversely proportional to the target mass number. 3He provides the best opportunity t o observe the spin dependence in scattering from a dense nucleus, but good understanding of its structure is a prerequisite of such a program. Measurements of analyzing powers in elastic pion scattering’ have shown that the asymmetries observed in the 3He system are much larger than in either the 13C or 15N system. In addition, the availability of Faddeev calculations of the 3He ground state wave function make it attractive target from a theoretical standpoint. Methodologically, the predicted approximate alignment of the spin of the odd neutron with the 3He spin indicates that one may be able to use polarized 3He as a substitute for a polarized neutron target and hence t o measure properties of the fundamental N-N scattering system. These expectations are supported by data already available. + In particular, spin + correlations and analyzing powers in 3He(p’, 2p) and 3He(@,p n ) quasielastic scattering were recently measured as a function of the transfered momentum and missing momentum at IUCF Cooler Ring2. At momentum transfer larger than 500 MeV/c, the 3He(p’,pn) spin observables are in good agreement with free p - n scattering observables, and therefore 3He can indeed serve as a good polarized neutron target. Nowadays, 3He targets are widely used to investigate such neutron’s properties as form factors in a quasielastic or a deep inelastic scattering. It must be emphasized again that satisfactory knowledge of the 3He structure is a necessary pre-requisite for such studies.

-

-

-

2. Theoretical framework For several decades considerable efforts have been done t o investigate structure of the lightest nuclei (the deuteron, 3He, 4He) a t short distances between the constituent nucleons. Significant progress was achieved both in theory and experiment, first of all because high quality data on spindependent observables were obtained with both h a d r ~ n i c and ~ > ~electromagnetic probes6. Large part of these investigations consists of study of elastic backward (in the center of mass system) proton - nucleus scattering (EBS). This process involves large momentum transfer It1 and therefore a belief exists that EBS can provide an access to the high momentum components of the wave function of the lightest nuclei.

151

From those investigations it became obvious that there is no theoretical model at present which quantitatively describes the existing data, even for the simplest reaction, pd EBS (Ref. 5 and references therein). Surprisingly, the wide gap exists between precise and detailed data base collected during decades and rather unprecise (even qualitatively) theoretical understanding of these reactions. The elastic backward P ( ~ HP~) , ~ scattering H ~ is studied in much less detail than the pd EBS. But presently, high intensity beams of polarized protons in combination with polarized 3He targets' give an opportunity to perform detailed studies of p3He EBS with spin dependent observables. This, in turn, demands careful theoretical study of the reaction mechanism. In the present paper the g H e EBS up to Tp 1 GeV is considered. In this particular case a request from experiment is to find an adequate connection of this reaction with the structure of the 3N system and t o get quantitative estimations for sensitivities of its cross section and spin-dependent observables t o the existing 3He wave functions of. General formalisms are found in Ref. 8. The one-deuteron-exchange (ODE) of Fig. l a shows that such connection exists. Still, it is obvious that at T p > 200 MeV this mechanism fails to reproduce detailed structure of existing data in both the nonrelativi~tic~ and relativistic" approaches. Among other important mechanisms the socalled direct mechanism (DIR) of Fig. l b may also play important role in this reaction, as it was pointed in Refs. 11 and 12. Nevertheless ODEfDIR also cannot reproduce the "shoulder" in the measured energy dependence of the elastic differential cross section at O,, = 180" near Tp -500 MeV13. Note that similar situation takes place in the backward pd scatteringa. Some authors try t o connect this shoulder in the pd and p3He scattering with pion e ~ c h a n g e ' ~ ~ ' ~ ~Figure ' ~ ~ l' c~ demonstrates ~ ' ~ ~ ~ ~ . this mechanism for the case of p3He scattering (later on it will be abbreviated as PI). Calculations of the PI amplitude with the pd + r3He subprocess need care, because most of the previous attempts were theoretically inconsistent and sometimes double-counting was not avoided. Indeed the intermediate pion can be created by many ways, the most important of which are illustrated by diagrams of Figs. lcl-c3. For example, it may come from 2 N pair of 3He (see diagram ( c l ) of Fig. 1). It is clear that such pion exchange conN

aIt was shown that one-neutron-exchange with the empirical momentum distribution of the nucleons in the deuteron extracted from A ( d , p ) X breakup describes the general behavior of the pd EBS cross section 14, but there is a small room for some additional contributions.

152

+

+

Figure 1. Reaction mechanisms of the p 3He elastic backward scattering: one deuteron exchange (ODE, (a)), direct mechanism (DIR, (b)) and the "triangle" diagram, (c). The diagrams (cl), (c2) and ( c 3 ) are subprocesses contributing to the triangle diagram.

tributes t o the 3He wave function and this diagram should be contained in (2A91.9, exchange, which may be important at Tp 2 1 GeV20. The diagram (c2) is already contained in DIR. So in our further calculations we take only diagram (c3) for the PI mechanism, where momentum from the incoming proton to the outgoing proton is transfered by the backward pion elastic scattering on the intermediate deuteron. The differential cross section of the .rrd -+ r d subprocess has a sharp resonance structure at energy

153 near 200 MeV21. Similar mechanism for the backward p d scattering was discussed in Refs. 18 and 22. 3. Numerical calculations and comparison between

experimental results and predictions In the present calculations the wave function is projected out from the 3He wave function obtained by the Faddeev c a l ~ u l a t i o nwith ~ ~ Argonne V18 potential24. The standard parameterizationz5 of the form factor FT(qz)= (A2 - p 2 ) / ( A z- q 2 ) with = 1300 MeV and f:"/4" = 0.08 was used. Amplitudes are taken from the partial wave analysis by Virginia group26. It must be emphasized here, that in most of the previous studies (Refs. 12 and 15-19.) people either built special theoretical models for the subprocess or made simplifying approximations to replace the amplitude by experimental data on the corresponding cross sections. It is proven by experience that such procedures are not satisfactory. Agg1

0.0

0.2

0.4

0.6

0.8

1.0

Tp,GeVIc

1.2

1.4

1.6

1.8

Figure 2: Differential elastic cross section at & , = 180'. The bold solid line represents ODE+DIR+PI for the wave function with Argonne V18 potential. The dot-dashed and dashed lines represent the ODE and DIR mechanisms, respectively. The dotted line is for the PI mechanism. The thin solid line represents ODE+DIR+PI for the wave function with Urbana potential. Data are taken from Refs. 13 and 27-29. Dataz7128~z9 were extrapolated to Bc, = 180' by us.

Results of the calculations are compared with experimental data for the differential cross section on Fig. 2. To demonstrate, up to which extent the reaction is sensitive to the potential model, we also provide calculations

154

based on Urbana potential3'. One sees that (qualitatively) Urbana potential gives similar result, but it is systematically larger (with factor 1.5-2.5) in comparison with the result with Argonne potential. I t also overestimates the experimental data. Predictions for the polarization correlation C,, are displayed in Fig. 3. Figs. 2 and 3 demonstrate that at T p z 200 MeV the cross section and the spin-dependent observables should have a sharp structure which comes from the interference of two mechanisms, ODE and

PI.

ODE only, AV18

Figure 3: Predictions of the model for the polarization correlation C,, . The lines are the same as at Fig. 2.

0,O

0,2

0,4

0,6

0,8

1,0

Tp,GeV

4. Experiment

The experiment will be done at the RCNP. We use vertically polarized protons at incident energies of 200, 300 and 400 MeV. Elastically scattered 3He particles are measured by Grand Raiden sitting at 0". Magnetic rigidities of measured 3He are about 72 % of the primary protons in the present energy range. The primary beam should be stopped near the exit of the

155

first dipole magnet ( D l ) of Grand Raiden. We have enough experiences on this technique and know it is possible to eliminate backgrounds from the beam stop installed in the D1 magnet. The Large Acceptance Spectrograph (LAS) is set around 50" and used as a luminosity monitor. From results of E144, clean spectra are expected with a double slit system31. A spin exchange type7 polarized 3He target is used. The target cell, made of Corning7056, has a cigar shape7 of 30 mmq5 x 100 mm. The entrance and the exit windows of the cell are as thin as 100 pm, in order t o reduce background events. The shape of the window is hemispherical so as to sustain a high internal pressure of more than 8 atom. At 8 atom., the target thickness is 2 x loz1 nuclei/cm2 which corresponds to 11 mg/cm'. The 100 pm thick entrance and exit windows contain Si and 0 nuclei of 2.6 x lo2' and 5.2 x lo2' /cm' in total, respectively. A diode laser is employed during measurements and the direction of the 3He polarization will be reversed within a proper interval, in order to reduce the systematic errors of polarization observables. Details of the polarization handling system are described elsewhere7. During the experiment, The 3He polarization is measured with Adiabatic Fast Passage Nuclear Magnetic Resonance (AFP-NMR) method7. However, this gives only relative value of the polarization. The target polarization can be measured independently using the 3J%(pi T + ) ~ H reac~ tion, where polarization is inferred from beam related asymmetries3'. This + + reaction has the spin relation; $ 4 Of 0-, and the spin-spin correlation C,, takes the constant value of 1.0. We will calibrate the target polarization by this reaction at 400 MeV. We are now preparing for a polarized target at the RCNP. The first measurements are expected in the fall of 2003.

+3

+

References 1. B. Larson et al., Phys. Rev. Lett. 67,3356 (1991); Yi-Fen et al., Phys. Rev.

Lett. 66,1959 (1991); R. Tacik et al., Phys. Rev. Lett. 63,1784 (1989). 2. M.A. Miller et al., Phys. Rev. Lett. 74,502 (1995). 3. D.K. Hasell et al., Phys. Rev. C34 236 (1986); E.J. Brash et al., Phys. Rev. C52 807 (1995); 0. HLser et al., Phys. Lett. B343 36 (1995). 4. V.G. Ableev et al., Pis'ma ZhETF 37 (1983) 196 [JETP Lett. 37 (1983) 2331; Nucl. Phys. A393 (1983) 491, A411 541(E); Pis'ma ZhETF 45 (1987) 467 [JETP Lett. 45 (1987) 5961; JINR Rapid Comm. 1[52]-92 (1992) 10; Few Body Systems 8 (1990) 137; C.F. Perdrisat et al., Phys. Rew. Lett 59 (1987) 2840; V. Punjabi et al., Phys. Rev. C39 (1989) 608.

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5. L.S. Azhgirey et al., Yad. Fiz. 61 (1998) 494 (Phys. At. Nucl. 61 (1998) 4321; V. Punjabi et al., Phys. Lett. B350 (1995) 178. 6. I. Sick, Progress in Particle and Nuclear Physics 47 (2001) 245; A.P. Kobushkin and Ya.D. Krivenko, nucl/th 01 12009. 7. T. Uesaka et al., Nucl. Instr. Meth. A402 (1998) 212. 8. A.P. Kobushkin et al., nucl-th/0112078 (2001). 9. H. Leiniak and L. Leiniak, Acta Phys. Pol. B9 (1978) 419. 10. A.P. Kobushkin, In: Proc. Int. Conf. DEUTERON-93, ed. V.K. Lukyanov (Dubna, 14-18 Sept., 1993) p.71. 11. S.A. Gurvitz, J.-P. Dedonder and R.D. Amado, Phys. Rev. C19 (1979) 142;S.A. Gurvitz, Phys. Rev. C20 (1979) 1256. 12. Yu.N. Uzikov, Nucl. Phys. A644 (1998) 321. 13. R. Frascaria et al., Phys. Lett. 66B (1977) 329; P. Berthet et al., Phys. Lett. 106B (1981) 465. 14. A.P. Kobushkin, J. Phys. G 12 (1986) 487. 15. N.S. Cragie and C. Wilkin, Nucl. Phys. B14 (1969) 477. 16. V.M. Kolybasov and N.Ya. Smorodinskaya, Phys. Lett. B37 (1971) 272; Yad. Fiz. 17 (1973) 1211 [Sov. J . Nucl. Phys. 17 (1973) 630). 17. G.W. Barry, Phys. Rev. D7 (1973) 1441. 18. L.A. Kondratyuk and F.M. Lev, Yad. Fiz. 26 (1977) 294 [Sov. J . Nucl.Phys. 26 (1977) 1531. 19. A. Nakamura and L. Satta, NucLPhys. A445 (1985) 706. 20. L.D. Blokhintsev, A.V. Lado and Yu.N. Uzikov, Nucl.Phys. A 597 (1996) 487. 21. H. Garsilazo and T. Mitsutani, R N N Systems (World Scientific, Singapore, 1990). 22. J.C. Anjos, A. Santoro, F.R.A. Simao and D. Levy, Nucl.Phys. A356 (1981) 383. 23. Y. Wu, S. Ishikawa, T. Sasakawa, Few-Body Syst. 15 (1993) 145. 24. R.B. Wiringa, V.G.J Stokes, R. Schiavilla, Phys. Rev. C 51 (1995) 38. 25. R. Machleidt, K. Holinde and Ch. Elster, Phys. Rep. 149 (1981) 1. 26. R.A. Arndt, 1.1. Strakovsky and R.L. Workman, Phys. Rev. C50 (1994) 1796. 27. C.C. Kim et al., Nucl. Phys. 58, (1964) 32. 28. L.G. Vottaet al., Phys. Rev. C10 (1974) 520. 29. H. Langevin-Joliot et al., Nucl. Phys. A158 (1970) 309. 30. R. Schiavilla, V.R. Pandharipande and R.B. Wiringa, Nucl. Phys. A449 (1986) 219. 31. H. Ueno, Proposal of El44 and private communications. 32. G.G. Ohlsen, Rep. Prog. Phys. 35 760 (1972).

157

POLARIZATION OBSERVABLES IN THE 4N SCATTERING WITH THE 3N CALCULATIONS

H. KAMADA Department of Physics, Faculty of Engineering, Kyushu Institute of Technology, 1-1 Sensuicho, Tobata, Kitakyushu 804-8550, Japan E-mail: [email protected]

'H(4

n)3He reaction at Ed = 270 MeV is theoretically investigated to compare with the recent RIKEN data. The method is characterized that using pure Faddeev rigorous 3N scattering amplitude one fixs the equal velocities of the particles forming the outgoing particle. The vector analyzing power A, agrees rather well with the data but the tensor analyzing powers A,, A,, and A,, do not gain satisfactory.

1. Introduction

In the deuteron properties' the D-state probability (PD)is most decisive factor for the binding energies in the few-body systems. The components of the deuteron, the triton and the alpha particle mainly consist of the S-wave, however, their binding energies are easily quantitatively varied by the D-wave. In Fig. 1 the correlation between the PD of the deuteron and the binding energies of the triton is shown. These abbreviated potential names refer therein2. In general, larger PD makes smaller binding energies. There is a Tjon plot3 which describes the strong correlation of the binding energies between the triton and the alpha particle, which is shown in Fig. 2. The calculations were rigorously performed3 by the Faddeev equations and Yakubovsky ones with the modern realistic forces including with the three-nucleon forces (3NF). According to the Tjon plot for the sake of adjusting the strength of the three-body force to the triton binding energy data, the theoretical prediction of the alpha particle binding energy seems to be well decribed. However, it does not mean simply that the gap from data occurs only from the 3NF. There has been being no good argument to obtain the PD precisely both in experiment and in theory. The deuteron has the couple states 3S1-3D1 which are combined by the tensor force in nucleon-nucleon inter-

158

action. Therefore, the phase shift el of nucleon-nulceon elastic scattering is expected to bring much information about the tensor interaction. Nevertheless, obtaining the well X2/Ndata M 1 in the nucleon-nucleon scattering the modern realistic potentials have still been kept the room of uncertainty of the PD.

s

”8.50 X

BonnA x BonnB

6 8.00

2

.d

a

2 7.50 a c

Ni j megenx Paris

x X

AV14 RUHRPOT x RSC

0

Figure 1. The correlation between the D-state probability in the deuteron and the binding energies of the triton

Recently, measurement4 of the 3€6e(4 p)4He reaction for Ed = 270 MeV at =EN was carried out as an investigation of the high momentum component of the deuteron wave function and the PD. According to the one-nulceon exchange model5 (ONE) the polarization correlation coefficient C// for the 31ife(4 p)4He reaction may be a unique probe to the P D 4 . There are some models extended from ONE. The SUT group reported a tiny deviation from the ONE where their calculations are based on a 3He-n-p and d-d-p three cluster model6. The Hosei group analyzed7 the deuteron tensor analyzing power T20 and no with the 3He-

159

n-p cluster model using the analogy between 3He and the proton. They concluded that the ONE describes the global feature of the experimental data. The three-nucleon model (TNM) has proposed directly using the threenucleon scattering amplitude. It is shown that the theoretical predictions of the TNM gives better description of the experimental data than ONE at E d = 140, 200 and 270 MeV. The A , is identically zero when we use only the plane wave impulse approximation like as the ONE. This success in the 31fe(& p)4He reaction encourages us to apply the TNM t o the 'H(& n)3He reaction. The essence of TNM are also applied to investigate the " C ( 6 d ) reaction at Ed = 270 MeV by Satoul0. The agreement between theory and data compares well with that for the ( p , p') reactions at comparable incident energies/nucleon.

30

29 28 b)

27 d

5 25 24 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 E(3H) [MeV]

Figure 2. The Tjon plot3. There is a strong correlation of the binding energies between the triton and the alpha particle.

2. Model and Discussion

Here, the TNM is applied to the 'H(6 r ~ ) ~ H reaction. e The reaction mechanism is shown in Fig. 3. There are intermediate particles, neutron n,

160

proton 5 and the deuteron d' in the process. As well as the 3€fe(cf, p)4He reaction we assume that only the two particles (d' and 5) forming the 3He have equal velocities.

This assumption fixes all kinematics uniquely as a static limit. The subprocess n d-+ n d' in Fig. 3 is described by the amplitude U .

+

+

n

d d'

a

d

-

He

1

/

P Figure 3.

Diagram of the reaction mechanism.

The three-body amplitude U is rigorously calculated by the Faddeev scheme with the AV18 potential. The effective incident energy E i f for U is about 250 MeV vs. the true deuteron incident energy Ed = 270 MeV. The predictions of the full Faddeev solution are given in Figs. 4 - 7. These are compared with recent data''. For the deuteron vector analyzing power A,, and tensor ones A,,, A,, and A,, the qualitative behavior of our model is similar to that of the experimental data. However, quantitatively they are over-estimated except for A,. This situation already took place in the 3€fe(d7 P ) ~ reaction H ~ at the higher energy. Our approach is based on the realistic modern nucleon-nucleon potential. If one has no doubt the model potential the theoretical uncertainty arises only from the subprocess mechanism. As shown in Fig. 3 our fournucleon scattering amplitude is mainly assumed to consist of the threenucleon one U . In fact we did not take into account a important process which in the initial state the polarized deuteron and unpolarized one are interchanged. The over-estimation for the polarization observables may cause from this reason. There are many missing subprocess.

161 0.4

0.3 0.2

2

0.1

0 -0.1

-0.2 140

145

150

155 160 165 170 Scattering angle [deg]

175

180

Figure 4. The deuteron vector analyzing power A, for the pd elastic scattering at E ; j f = 2 5 0 MeV as a function of 8;", corresponding to the 2H(&n)3He reaction for Ed=270 MeV.

x

2

Scattering angle [deg] Figure 5 . The same as in Fig. 4 for the deuteron tensor analyzing power A,,.

3. Four-body equations and outlook

The 2 H ( 4 7 ~ ) ~ H reaction e is the four-body problem. The Yakubovsky equations yield fully the four-body scattering amplitude. They had been solved in low energy region Even in the three-body system there are two kinds of singularities in the integral Faddeev equations, one of them ap-

162 0.8

0.7 0.6

0.5 0.4 x

4

0.3 0.2

0. I 0 -0.I

-0.2 -0.3 .-

140

145

150

155

160 165 170 Scattering angle [deg]

175

180

Figure 6. The same as in Fig. 4 for the deuteron tensor analyzing power A,,

0 -0.2

-0.4 N

2

-0.6

-0.8 -1 -1.2

140

145

150

155

160

165

170

175

180

Scattering angle [deg] Figure 7. The same as in Fig. 4 for the deuteron tensor analyzing power A,, .

pears as a moving logarithmic singularity2. The four-body equations have more complicate structure of these singularities than the three-body ones. Because of the technical difficulties they had not been solved beyond the positive energy threshold yet. These singularities turn out from the free propagator, two-body t-matrix, three-body and 2+2 partition t-matrices, where the new threshold channels are opened. The complex analytical continuation method has been developed in

163

atomic and molecule p h y s i ~ s ' ~ They . investigated the two-body scattering amplitude by sampling ones at negative energies. The scheme is usefully extended15 that, instead of calculating the amplitude at the positive energy E ir where the integral kernel of the Lippmann-Schwinger type equation has a singularity because of infinitesimal 6 , the sampling amplitudes are sampled around the E i& with the finite small E avoiding from the singularity to switch over the solution by the analytical continuation. The new scheme starts to be applied to the Yakubovsky equations. Beyond the four-body breakup threshold the feasibility study is first tried and done16.

+

+

Acknowledgments

Author would like t o thank Prof. H. Sakai, Dr. T. Uesaka and Dr. Y. Satou for fruitful discussion in Institute of Physical and Chemical Research (RIKEN), Japan. References 1. e.g., J. Carlson and R. Schiaviila, Rev. Mod. Phys. 70, 743 (1998). 2. W. Glockle, H. Witala, D. Hiiber, H. Kamada and J. Golak, Phys. Rep. 274, 107 (1996). 3. A. Nogga, H. Kamada, W. Giockie, B. R. Barrett, Phys. Rev. C 6 5 , 054003 (2002). 4. T. Uesaka, H. Sakai, H. Okamura, T. Ohnishi, Y . Satou, S. Ishida, N. Sakamoto, H. Otsu, T. Wakasa, K . Itoh, K . Sekiguchi, T. Wakui, Phys. Lett. B467, 199 (1999). 5. A. P. Kobushkin et al., Phys. Rev. C 5 0 , 2627 (1994). 6. S. Gojuki, H. Kamada, E. Uzu and S . Oryu, Few-Body Systems Suppl. 12, 501 (2000). 7. M. Tanifuji et al., Phys. Rev. C 6 1 , 024602 (2000). 8. H. Kamada et al., Prog. Theor. Phys. 104, 703 (2000). 9. T. Uesaka et al., Nucl. Phys. A 6 8 4 , 606c (2001). 10. Y . Satou et al., Phys. Lett. B549, 307 (2002). 11. T. Saito et al., in this proceeding (2003); T. Saito, Genshikaku Kenkyu 4 6 , 75 (2002). 12. A. C. Fonseca, Nucl. Phys. A631, 675c (1998); B. J. Crowe I11 et al., Phys. Rev. C61, 034006 (2000). 13. E. Uzu et al., Prog. Theor. Phys. 90, 937 (1993); Few-Body Systems Suppl. 99, 97 (1995); Few-Body Systems 22, 65 (1997). 14. L. Schlessinger, Phys. Rev. 167, 1411 (1968). 15. H. Kamada, Y. Koike and W. Giockle, in preparing (2003); LANL, nucl-th/0301018 (2003). 16. E. Uzu et al., in this proceeding (2003).

164

FADDEEV-YAKUBOVSKY CALCULATION ABOVE 4-BODY BREAK-UP THRESHOLD

EIZO UZU Department of Physics, Faculty of Science and Technology, Tokyo University of Science 2641 Yamazaki, Noda, Chiba 278-8510, Japan e-mail: [email protected] HIROYUKI KAMADA Department of Physics, Faculty of Engineering, Kyushu Institute of Technology, 1 - 1 Sensuicho, Tobata, Kitakyushu 804-8550, Japan YASURO KOIKE Department of Physics, Hosea University, 2-1 7-1 Fujimi, Chiyoda-ku, Tokyo 102-8160, Japan This is the first attempt to solve the four-body Faddeev-Yakubovsky equation at an energy above the 4-body break-up threshold. T h e calculation is carried out with minimal inputs employing a separable type of the Paris potential, and numerical convergence is checked. About two digits convergence is obtained below the 4-body break-up threshold, however, one or less above that.

1. Introduction

The Schodinger or Lippmann-Schwinger (LS) equations are employed to solve a problem of the quantum mechanics based on the non-relativistic scheme. The former needs the boundary condition which is complicated in 3- or more-body system, however the latter includes it automatically and one formula is applicable to both of the bound and scattering systems. Therefore we usually employ the 3-body Faddeev and 4-body FaddeevYakubovsky (FY) equations based on the LS equation which the difficulty of the boundary condition appears as singularities of the Green's function. We already know how to treat it in the 3-body system, however, do not in the 4-body system, thus there are no example to solve the FY equation

165

above the 4-body break up threshold, until now. Recently two of the authors (HK and YK) found a new method. It is applied successfully to the 2- and 3-body system. This paper reports the first attempt to apply it to the FY equation above the 4-body break-up threshold of the 4-nucleon system containing two protons and two neutrons. Following section 2 describes what is the difficulty to treat the singularities, new method is proposed in section 3, inputted conditions are shown in section 4, and section 5 shows our numerical results and discussions. 2. Difficulty of Singularities

(P3W

4He (1)

n3He

dd

I I

I I

(11)

(11)

dpn

m n

I

I

(111)

E (IV)

Figure 1. Rough illustration of the threshold energies for 4He. Right side is higher energy than left.

Figure 1 shows a rough illustration of the threshold for 4He. We can calculate in area (I) easily due to no singularities. For instance the bound state is in this area. In area (11), merely 2-body break-up channels are open. The propagators for [3+1] and [2+2] subsystems are expressed in a form similar to the 2-body free Green’s function as (1)

where E is the energy, p is the reduced mass, and p is the momentum in the center of mass (CM) system. When we take the limiting value of E + 0, one fixed pole appears on the real momentum axis which we can calculate with the principal value and residue. In area (111),3-body break-up channel is open. When we consider three particles named 1, 2, and 3, the 2-body propagator is expressed in a form similar t o the 3-body Green’s function as 1

Go =

P? - Pi - PIP2 E+i&- -

166

where pl and p a are the momenta in CM system for particle 1 and 2, respectively, v1 and v~ are the reduced mass for pairs (2, 3) and (3, l), respectively, and m3 is mass for the particle 3. When we take the limiting value E -+ 0, a pole appears on the real p l axis, and it moves depending on p z . Fortunately the pole is in some limited area, then the integration pass is able to be avoided from this area on the complex plane. This is a n outline of the contour deformation method. In area (IV), the 4-body break-up channel is open. When we consider four particles named 1, 2, 3, and 4, the Green’s function is expressed as

where p l , p2, and p3 are the momenta in CM system for particle 1, 2, and 3, respectively, ~ 1 4 ,~ 2 4 and , v34 are the reduced mass for pairs (1, 4), (2, 4), and (3, 4), respectively, and m4 is mass for the particle 4. When we take the limiting value E -+ 0, a pole appears on the real pl axis, and it moves depending on p2 and p3. Nobody has been successful to trace it due to the motion is more complicated than in the 3-body case and there were no examples to calculate the FY equation in this area.

3. New Method Now we propose a new method which is in contrast to that mentioned above. First we solve the FY equation with some finite E’S. In this case there are no singularities on the real integration axis. After obtaining several solutions for various E’S, we take the limiting value E -+ 0 with an analytical continuation method numerically. This is applicable not only to the 4-body system, but also to the 2-, 3-, and more than 4-body systems. We have two ways to define E . One is to employ independent E’S for each [3+1], [2+2], and 2-body propagators and 4-body Green’s function, however, it needs to reconstruct programing codes. The other one is to put one E on the free Green’s functions appearing in all process of this calculation. The FY equation is solved with the same way as in area (I),still the singularity on the complex plain has an influence on the calculation, and no one has experience about this effect. We adopt the latter method in this study, and aim to check convergence with the number of integrational mesh points. We fix them between two nucleons as 48, and angular integration in the 3- and 4-body system as 24. And the check is restricted in the [3+1] and [2+2] subsystems, and 4-body system.

167

4. Inputs and Numerical Methods

Due t o this aim, we neglect the Coulomb force and include minimal state channels as follow. In the 2-body subsystem, only 'SO and 3S1-3D1 states are included and the rank-1 PEST potential' is employed for them, which is a separable type of the Paris potential2. In the [3+1] subsystem, we consider merely 1/2+ state of total J and parity and total isospin 1/2 state, and in the [2+2] subsystem, we consider both of the two pairs t o be 'So or 3S1-3D1 states. These subamplitudes are expanded in a separable form using the energy dependent pole expansion3 (EDPE) method and we take 4 ranks. As for the 4-body system we calculate only for O+ of total J and parity and total isospin 0 state. As for a numerical method of the analytical continuation, we employ the point method4.

5. Results and Discussion Obtained scattering amplitudes in area (11) at lMeV below d p n threshold is shown in Table 1 where we adopt 64 mesh points in both of the subsystems and 4-body system. In this case we found two or three digits convergence. Table 2 shows solutions in area (111) at lMeV above d p n threshold where we adopt 48 points in the subsystems and 64 in the 4-body system. We obtain about two digits convergence. Table 3 shows in area (IV) at 12MeV above p p n n threshold, where we adopt 56 points in the subsystems and 72 in the 4-body system. The real parts of all channels converge about one digits, however the imaginary parts of channel [B] and [ C ] don't. We have now two method t o improve this. One is to calculate with principal value and residue as far as it is useful. The other one is as follows. In this study we employ the Legendre-Gauss integration method which the mesh points are defined in area [-1,11 and they are transformed to [0, M). We expect t o improve the solution to contrive the mapping function of this transformation. Further investigation for the latter is in progress.

References 1. J. Haidenbauer and W. Plessas, Phys. Rev. C30, 1822 (1984), Phys. Rev. C32, 1424 (1985). 2. W. N. Cottingham, M. Lacombe, B. Loiseau, J. M. Richard, and R. Vinh Mau, Phys. Rev. D8, 800 (1973); M. Lacombe, B. Loiseau, J. M. Richard, R. Vinh Mau, J. CBtC, P. PirBs, and R. de Tourreil, Phys. Rev. C 2 1 , 861 (1980). 3. S. A. Sofianos, N. J. McGurk, and H. Fiedelday, Nucl. Phys. A 3 1 8 , 2 9 5 (1979). 4. L. Schlessinger, Phys. Rev. 167, 1411 (1968).

168 Table 1. Scattering amplitudes in area (11) at lMeV below d p n threshold. Column [A] shows numerical results for p3H elastic channel, [B] for d d to p3H or reversed reaction channel, and [C] for d d elastic channel. All of the initial and final channels are in S state. The second and third columns for each channels show the real and imaginary parts of on-shell amplitudes, respectively, whose unit is fm-’. The first column indicates the number of solutions into the point method for each E ’ S , and ‘‘exad’ means our calculations with principal value and residue. We set E = 0.5j MeV where j = 1 , 2 , . . ..

PI

I

.~

Im

n 1

Re 0.0026

0.5650

6

0.1501 0.1509 0.1499

0.1362 0.1320 0.1324

IB1

I

I

n

Re-

1

1.515

Im -1.186

0.754

-1.230 -1.223

n 1 2 3 4 5

exact

ICl . _ Re -2.78 -2.12 -1.60 -1.66 -1.63

Im 3.34 3.26 3.30 3.49 3.46

-1.70

3.45

Table 2. Same as for Table 1 but for results in area (111) at lMeV above d p n threshold. We set E = 2.0j MeV where j = 1 , 2 , . . .. n 1 2 3 4 5 6 7

[A1 Re -0.820 -0.628 -0.588 -0.606 -0.583 -0.576 -0.589

Im 1.104 0.837 0.496 0.472 0.480 0.489 0.490

n 1 2 3 4 5 6

PI

Re 0.872 0.408 0.646 0.630 0.635 0.638

Im -0.325 -0.419 -1.076 -1.055 -1.067 -1.043

n 1 2 3 4 5 6

7

PI

Re -1.329 -0.821 -0.442 -0.477 -0.530 -0.392 -0.325

Im 1.228 1.175 1.069 1.164 1.112 1.213 1.044

Table 3. Same as for Table 1 but for results in area (IV) at 12MeV above p p n n threshold. We set E = (3.0 l . 0 j ) MeV where j = 1,2,. . ..

+

14

n 1 2

3 4 5 6

Re -0.363 -0.004 -0.578 -0.509 -0.555 -0.519

Im 0.991 0.693 0.576 0.671 0.681 0.687

n 1 2 3

4 5

[BI Re -0.114 -0.092 -0.566 -0.569 -0.576

Im 0.465 0.207 0.722 1.020 0.648

n 1 2 3

[CI Re -0.476 -0.257 -0.281

Im 0.266 0.568 0.104

169

STUDY OF THE SPIN STRUCTURE OF 3He(3H) VIA & + 3He n (3Hp) REACTION AT INTERMEDIATE ENERGIES

T. SAITO, M. HATANO, H. KATO, Y. MAEDA, H. SAKAI, S. SAKODA, A. TAMII AND N. UCHIGASHIMA Department of Physics, University of Tokyo, 7-3-1 Hongo,Bunkyo, Tokyo 113-0033, Japan

V.P. LADYGIN, A.YU. ISUPOV, N.B. LADYGINA, A.I. MALAKHOV AND S.G. REZNIKOV LHE- JINR, 141980, Dubna, Moscow region, Russia

T. UESAKA AND K. Y A K 0 Center f o r Nuclear Study (CNS), University of Tokyo, 7-3-1Hongo,Bunkyo, Tokyo 113-0033, Japan

T. OHNISHI, N . SAKAMOTO, K. SEKIGUCHI The Institute of Physical and Chemical Research ( R I K E N ) , 2-1 Hirosawa, Wako, Saitama 351-0198

H. KUMASAKA, J . NISHIKAWA, H. OKAMURA, K. SUDA, R. SUZUKI Department of Physics, Saitama University, 255 Shimo-okubo, Saitama, Saitama 338-8570, Japan /

Measurements of the tensor and vector analyzing powers A,, , A,, , A,, , and A , for the i d -+ 3Hen and i d + 3 H p reactions were performed at Ed = 270 and 200 MeV over wide angular range. Tzo at O,, = ' 0 and 180' were also measured at Ed = 270, 200, and 140 MeV. Obtained data were compared with predictions based on one nucleon exchange approximation.

1. Introduction It has been predicted from non-relativistic Faddeev calculations of the three nucleon bound state that the main components of the 3He ground state wave function are a spatially symmetric 5'-state and a small contribution

170

of a D-state [l]. In the last two decades, the structure of a 3He nucleus has been investigated using reactions of quasielastic knockout of the 3He constituent nucleons. The momentum distribution of the constituent nucleons was extracted by plane wave impulse approximation (PWIA) analyses of 3He(e,ep) reaction [2] and 3He(p, 2p)d and 3He(p,pd)p reactions [3]. It was found that the theoretical calculations using modern realistic 3He wave functions did not reproduce the experimentally obtained momentum distribution functions in the region of the internal nucleon momentum q > 250 MeV/c. To investigate the spin structure of 3He, spin correlation for the quasi elastic 3He(p',pN) reactions was measured up to the internal nucleon momentum of q 400 MeV/c, and the distribution function of the nucleon polarization in a 3He nucleus was extracted by a PWIA analysis [4]. The distribution function by Faddeev calculations, however, did not reproduce the experimental data in the region of q > 300 MeV/c. These deviations indicate that the structure of 3He in the highmomentum region has not been clearly understood. Since various kinds of mesons contribute to the nuclear interaction in the high-momentum region, investigation of high-momentum 3He structure may reveal new physics which has not been observed in the low-momentum region. Since the contribution from the D-state component becomes large in the high-momentum region, measurements of polarization observables sensitive to the D-state is necessary to study the high-momentum 3He structure. -#

-

Figure 1.

ONE processes of the i d

2. ONE approximation for the

-+

3Hen(3Hp) reaction.

+ 3He n (3Hp) reaction

In the framework of One Nucleon Exchange (ONE) approximation, tensor analyzing powers for the dd -+ 3Hen and Jd -+ 3 H p reactions at intermediate energies are sensitive to the D-state component of 3He or 3H [ 5 , 61. The ONE processes of these reactions are shown in Figure 1. Let u d ( k d ) and wd(kd) be the S- and D-state radial wave functions of a deuteron in the momentum space, respectively. Similarly, let u h ( k h ) and wh(kh) re-#

171

spectively be the S- and D-state radial functions of a 3He or 3H in the d N cluster configuration. The tensor analyzing powers for the dd --+ 3Hen and dd + 3 H p reactions in the framework of ONE approximation have following characteristics: (i) If only the diagram (A) (see Figure 1) is considered, they are determined by the ratio of the 3He(3H) wave function components w h ( k h ) / u h( k h ) . (ii) Conversely, If only the diagram (B) is considered, they are determined by the ratio of the deuteron wave function c o m p e nents w d ( k d ) / u d ( l C d ) . (iii) If 3He(3H) is scattered at forward angles, the corresponding deuteron internal momentum kd is very large, hence the contribution from the diagram (B) becomes negligible. Consequently, they are determined by the 3He(3H) structure. (iv) Conversely, if 3He(3H) is scattered at backward angles, corresponding internal momentum of 3He(3H) kh is very large, hence the contribution from the diagram (A) becomes negligible. Consequently, they are determined by the deuteron structure. Particularly, the tensor analyzing power T20 at 0,,(3He,3H) = 0" or 180" is simply given by [5, 61

+

T20

=

1 2fiu(k)w(k) - w ( k ) 2 Jz u(k)2+ w(k)2 .

(1)

Here, u ( k ) and w ( k ) are respectively replaced by u h ( k h ) and wh(hh) if O,, = 0", or, by u d ( k d ) and w d ( k d ) if O,, = 180". Thus, the tensor analyzing powers for the dd -+ 3Hen and i d + 3 H p reactions at forward angles are directly related to the D / S ratio of 3He(3H). With a 270 MeV deuteron beam, the 3He(3H) structure can be investigated up to a relative momentum of the d N pair of 600 MeV/c, in principle.

+

-

3. Experiment The experiment was performed at RIKEN Accelerator Research Facility. A polarized deuteron beam extracted from a polarized ion source was accelerated with AVF and Ring Cyclotrons up to the energy of 270, 200, or 140 MeV. The accelerated beam was transported to a spectrometer SMART [7] and were injected onto a target placed in the scattering chamber. Scattered particles (3He,3H, or protons) were momentum analyzed with three quadrupole and two dipole magnets (Q-Q-D-Q-D configuration) and then detected with a multi-wire drift chamber and three plastic scintillators at the focal plane. The direction of the symmetry axis of the beam polarization was controlled with a Wien filter located at the exit of the ion source. The magnitude of the beam polarization was measured with beam-

172

line polarimeters based on the d p elastic scattering. We used a deuterated polyethylene (CD2) sheet [8] as the deuteron target. Measurement with a carbon foil target was also performed to subtract the contribution from the carbon nuclei in the CD2 target. We detected 3He for the 3He n channel. In the case of 3H+p channel, we detected 3N (protons) if 3H were scattered in the forward (backward) angles in the center-of-mass frame.

+

4. Results and Discussion The experimental results of the tensor analyzing power T20 at Ocm = 0" and 180" at Ed = 270, 200, and 140 MeV are presented in Figure 2. The results for the 3He n (3H + p ) channel are presented by filled (open) symbols. The curves are predictions by ONE approximation [ 5 , 61. The upper fives symbols (two filled ones are hidden behind the open ones) and the three curves are T20 a t Ocm = 0". The lower three symbols and a curve are Tzo at O, = 180". The solid, dashed, and dot-dashed curves are respectively calculated using 3He wave functions of Urbana, Paris, and Reid soft core potentials. Paris deuteron wave function was used for these calculations. The ONE predictions reproduced the incident energy dependence and the signs of the experimental data. Since T ~ ato 0" and 180" is directly connected with the D / S ratio of 3He (3H) or deuteron by Eq. (l), the difference in the signs of Tzo at 0" and 180" reflects the difference in the relative sign of u(k) and w ( k ) for 3He (3H) and deuteron. Angular distri-

+

0.0

0.5

1.0

1.5

2.0

2.5

Incident deuteron momentum [GeV/c]

3.0

Figure 2. The experimental results of Tzo at Ocm = 0 ' and 180". The curves are ONE predistions. Explanations are written in the text.

173

butions of the analyzing powers at Ed = 270 and 200 MeV are presented in Figure 3. The meanings of the symbols and curves are same as those in Figure 2. The ONE predictions 161 reproduced the global features of the experimental data at backward angles, where the tensor analyzing powers depend mainly on the deuteron structure. At forward angles, however, significant discrepancies can be found. Since the tensor analyzing powers at forward angles are mainly determined by the 3He or 3H structure, these discrepancies might be naively ascribed to some problems of the wave function of 3He or 3H. However, since the ONE approximation is very crude, calculation with more detail reaction mechanism is needed to extract information of the 3He or 3H structure in the high-momentum region. Further development in theoretical calculations of four body systems is expected. E d = 270 MeV

E d = 200 MeV

8,,

of 3H or 3He [deg]

Figure 3. Angular distributions of the analyzing powers for the zd reactions. See text for the explanations.

+

3Hen(3Hp)

References 1. B. Blankleider and R. M. Woloshyn, Phys. Rev. C29,538 (1984). 2. E. Jans et al., Nucl. Phys. A475,687 (1987). 3. M. B. Epstein et al., Phys. Rev. C32,967 (1985). 4. R. G. Milner et al., Phys. Lett. B379,67 (1996). 5. V. P. Ladygin and N. B. Ladygina, Phys. Atom. Nucl. 59,789 (1996). 6. V. P. Ladygin et al., Part. Nucl. Lett. 3[100]-2000, 74 (2000). 7. T. Ichihara et al., Nucl. Phys. A569,287c (1994). 8. Y. Maeda et al., Nucl. Instrum. Methods. Phys. Res. A490,518 (2002)

174

STUDY OF THE SPIN DEPENDENT 3HE-NUCLEUS INTERACTION AT 450 MEV

J. KAMIYA, K. HATANAKA, Y. SAKEMI, T. WAKASA, H. P. YOSHIDA, E. OBAYASHI, K. HARA, K. KITAMURA, Y. SHIMIZU, K. FUJITA, AND N. SAKAMOTO Research Center for Nuclear Physics (RCNP) Ibaraki, Osaka 567-004 7, Japan Y. SHIMBARA, AND T. ADACHI Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan H. SAKAGUCHI, M. YOSOI, M. UCHIDA, AND Y. YASUDA Department of Physics, Kyoto University, Kyoto 606-8502, Japan T. KAWABATA RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, N Y 11973, USA T. N O R 0 Department of Physics, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan Differential cross sections and induced polarizations of 3He+12C, 58Ni, and gOZr elastic scattering were measured at E3,,=450 MeV. This is the first measurement of the polarization for 3He scattering at intermediate energies. The optical potential parameters including the spin-orbit potential were determined with small uncertainties. The volume integrals per nucleon of the potentials were investigated for 3He and their energy dependence showed the similar behavior to that for protons at intermediate energies. The single folding calculations were compared with the data. The real central and spin-orbit parts of the folded potentials had to be reduced by a few tens of percent in order to reproduce the experimental results.

1. Introduction

Nucleus-nucleus scattering has been well studied in the framework of both non-relativistic and relativistic equations. Although their Hamiltonian can be determined, the strict solution for a many-body system can not be ob-

175 tained unlike the case of a few-body system. As one of approaches to solve the Schrodinger equation, the interaction has been obtained by folding the nucleon-nucleon or nucleon-nucleus interaction by nucleon densities of nucleus. In the folding model approach, different kinds of the effective interactions and their modification in the nuclear matter have been studied To explain the medium modification due to a mutual interaction or Pauli blocking, the density dependent effective interactions have been introduced to the scattering of heavy-ions and light-ions 3 . The spin dependent interaction for nucleus-nucleus scattering is of special interest because it is closely related to the nuclear structure and reaction mechanism. However, its origin has not been studied well, especially for the interaction between complex nuclei. For heavier projectiles, polarization observable measurements are limited to low energies due to the lack of polarized beams. Recently, theoretical investigations have been reported on microscopic folding models including the spin dependent 3He-nucleus interaction '. Calculations showed that the spin-orbit interactions obtained by the microscopic folding models had large effects on cross sections of the elastic scattering and predicted large values of vector analyzing powers even at forward angles at intermediate energies. It is indispensable to measure polarization observables of 3He-nucleus scattering in order t o investigate the microscopic models as well as to determine the optical potential parameters including the spin-orbit potential.

'.

2 . Experiment

The experiment was performed at RCNP, Osaka University. We measured the differential cross sections and induced polarizations, which equals to the analyzing powers in the present condition, of 3He+'2C, 58Ni, and 90Zr elastic scatterings at a bombarding energy of 450 MeV. The polarizations were measured by using the double scattering method. The calorimeter which was dedicated to this experiment was installed after the focal plane polarimeter (FPP) system of the Grand Raiden spectrometer in order to measure the energies of secondary scattered 3He particles. Preceding the polarization measurements, the effective analyzing power of the FPP system was calibrated. At first we measured the absolute value of the polarization p , of 3He+12C elastic scattering at 6lab = 7", where the double folding model predicted the large value of the polarization. Using the measured polarizations p , = 0.547 f 0.018, the effective analyzing power Aiff = 0.232 f 0.010 was obtained with the figure of merit of the polarimeter of

176 I

a

a,

-0.1-

5

10

6c.m.

15

20

26

90

(deg)

6

10

1s

10

25

6c.m. (deg)

so

6c.m. (deg)

Figure 1. Differential cross sections and induced polarizations. Solid curves represent the results of the optical model calculations with spin-orbit potentials which give the minimum x2.

6.563-03.

3. Results and discussions Figure 1 shows the differential cross sections da/dR and induced polarizations Py of 3He elastic scattering off 12C, 58Ni, and "Zr nuclei. The experimental data are shown by the closed circles with statistical uncertainties. The experimental polarizations show the large values in the middle angular range. Solid curves represent the results of the phenomenological optical model calculations with central and spin-orbit potentials. Figure 2 shows the volume integrals per nucleon JR,J/APATas a function of the incident energy divided by mass of the projectile. The upper and lower panels show the volume integrals of real J R / A ~ A and T imaginary J 1 / A p A central ~ potentials] respectively. Closed circles are the present results evaluated from the optical potentials parameters which give the solid curves in Fig. 1. Open circles and squares in the left panels represent JR,J/A,AT for "Fe and 5sNi1respectively. Open circles in the right panels are those for 90Zr. Solid and dotted curves show the volume integrals for protons The dash-dotted curve is the guide for eyes. The present optical model analysis including the spin-orbit term give larger values of the imaginary volume integrals per nucleon JJIAPAT than previous results, which were obtained by analyzing only the cross sections '. Both the real and imaginary volume integrals for 3He show the same energy dependence as that for protons at intermediate energies above 70 MeV/nucleon, where 536.

177

.. . .. .. .

PrDtDIlS

---.-..(, Becchetti-Greenlees

400

5

10

100

200

20 30

50 70 100

E/A,

200

(MeV)

Figure 2. Volume integrals per nucleon for Fe-Ni region (left) and gOZr (right) as a function of incident energy per nucleon. Closed circles were the evaluated from the optical potentials parameters which gave the solid curves in Fig. 1.

the binding energy of 3He is negligible and 3He can be treated as three free nucleons. This argument is closely related to the single folding (SF) model 4 . It is interesting to see the applicability of the SF model a little more in detail. The SF potential was calculated by folding the proton-nucleus optical potential at 150 MeV by the nucleon density distribution of 3He. The spin-orbit component was obtained assuming only the S-state neutron in 3He contributes to the spin-orbit single folding potential. Introducing the renomalization factors ( N R ,N I , Nso) to examine the modification of the strength of the correspondent potentials, the total single folding potential is written as

+ iNIW(&, + NsoF&,

U ( S F ) ( R )= N R v ( > F )( R )

ysF,,

( R ) L. g,

(1)

where WtsF);FtlF,( R ) L. D represent the real central, imaginary central and spin-orbit parts. Figure 3 shows the cross sections and polarizations calculated with the SF potentials. Dashed curves in the upper panels represent the results without renormalization factors, Le., ( N R ,N I ,Nso)=(l,1,l). Solid curves are best-fit calculations allowing the renormalization factors to vary

178

- (1,1,1)

--

(0.85,1.00.0.61) [..:I

-1.0

5

10 *c.m.

15

20

(deg)

25

30

5

(0.81,1.04.0.67)

1

. . . .1 . . . . I . . . . 1 . . . .I . . . . 10

15

0c.m.

20

25

30

(deg)

Figure 3. Differential cross sections and polarizations calculated with the SF potential. The solid and dashed curves represent the calculations with and without renormalization factors.

in order to minimize X2-values. Resulting renormalization factors were ( N ~ , N I , N , , ) = ( ~ .1.00, ~ ~ , 0.61) for 58Ni, and (0.81, 1.04, 0.67) for 90Zr. By using the renormalization factors, experimental data can be reproduced except for the cross sections at backward angles. This results suggest the interaction between point nucleon and target nucleus is modified in the 3He nucleus. One reason why renormalization factors are required is that the SF potential doesn’t include any density dependence of the nucleon-nucleus potentials in the 3He nucleus. The density dependence was important at low energies of E / A a few tens MeV 2,3. The present results suggests the importance of density dependence even in the intermediate energies.

-

References 1. F. Petrovich, et al., Phys. Lett. 71B,259 (1977). 2. W. G. Love, Phys. Lett. 72B,4 (1977). 3. A. M. Kobos, et al., Nucl. Phys. A384, 65 (1982). 4. Y. Sakuragi and M. Katsuma, Nucl. Instrum. Methods Phys. Res. A 402, 347 (1998). 5. F. D. Becchetti, Jr, and G. W. Greenlees, Phys. Rev. 182,1190 (1969). 6. L. G. Arnold, et al., Phys. Rev. C 25, 936 (1982). 7. T. Yamagata, et al., Nucl. Phys. A589,425 (1995).

VI. NUCLEAR CORRELATIONS


181

PROPERTIES OF NUCLEONS AND THEIR INTERACTION IN THE NUCLEAR MEDIUM*

W. H. DICKHOFF Department of Physics, Washington University, St. Louis, Missouri 63130, USA E-mail: [email protected]

The importance of recent experimental results obtained with the (e,e'p) reaction with regard to medium effects on nucleon properties is discussed. An overview is given of theoretical calculations aimed at describing the experimental singleparticle strength distribution. The separate role of short- and long-range correlations on the single-particle strength distribution will be emphasized. The consequences of these new developments for the understanding and interpretation of other data, especially response functions, is also discussed.

1. Introduction

During the last fifteen years considerable progress has been made in elucidating the limits of the nuclear mean-field picture. The primary tool in exhibiting these limits in a quantitative fashion has been provided by the (e,e'p) r e a ~ t i o n ~ The ~ ~ ~physical ~ ~ * . picture associated with the (e,e'p) reaction involves the intimate connection of this reaction with the probability of removing a nucleon with momentum p while keeping the binding energy of this nucleon (or the missing energy) fixed. The resulting cross sections are consequently proportional to the square of the corresponding momentum-space wave functions of these nucleons when distortion and absorption effects of the outgoing proton have been properly accounted for. The results from the NIKHEF facility are shown for four different nuclei in Fig. 13. It is important to realize that the shapes of the wave functions in momentum space correspond closely to the ones expected on the basis of a standard Woods-Saxon potential well (or more involved mean-field wave functions). This is itself an important observation since the (e,e'p) reaction 'This work is supported by the U.S. National Science Foundation under Grant No. PHY-0140316.

182

102

10‘

-t 9-l

2

l*

P. B 10‘

102

-m -lw

0

loo

p,[MeV/c]

m

300

Figure 1. Momentum distributions for various nuclei obtained from the (e,e’p) reaction performed at NIKHEF.

probes the interior of the nucleus, a feat not available with hadronically induced reactions. While the shapes of the valence nucleon wave functions correspond to the basic ingredients expected on the basis of years of nuclear structure physics experience, there is a significant departure with regard to the integral of the square of these wave functions. This quantity is usually referred to as the spectroscopic factor and is shown in Fig. 2 for the data obtained at NIKHEF3. The results shown in Fig. 2 indicate that there is an essentially global reduction of the single-particle strength of about 35 % which needs to be explained by the theoretical calculations. An additional feature obtained in the (e,e’p) reaction is the fragmentation pattern of more deeply bound orbitals in nuclei. This pattern is such that isolated peaks are obtained in the vicinity of the Fermy energy whereas

183

' v

04-

2C

VALENCE PROTONS

0.0

10'

target mass

1 o2

Figure 2. Spectroscopic factors from the (e,e'p) reaction as a function of target mass. Data have been obtained at the NIKHEF facility.

for more deeply bound states a stronger fragmentation of the strength is obtained with larger distance from E F . All these features of the strength need to be explained theoretically. In Sec. 2 some theoretical ingredients of Green's function calculations are presented. In Sec. 3 the results of several calculations employing this many-body technique are discussed. The role of short- and long-range correlations will be addressed and the consequences of the experimental strength distribution for the admixture of high-momentum components in the ground state will be explored. Some more recent data covering a larger missing energy domain will also be discussed. In Sec. 4 the consequences of these experimental and theoretical results for response functions and other relevant physical quantities will be clarified. Finally, conclusions are drawn in Sec. 5.

2. Theoretical Concepts

The relevant theoretical quantity for comparison with the (e'e'p) data is the spectral function associated with the removal of particles from the ground state of the target nucleus. This spectral function is part of the singleparticle propagator describing the properties of a nucleon in the nuclear

184

medium. This quantity is given in its Lehmann representation by

Exact eigenstates and eigenenergies for the A f 1- systems are employed in this definition together with the corresponding quantities for the ground state of the A-particle system using obvious notation. The quantum numbers a and p refer to an appropriately chosen set of single-particle quantum numbers relevant for the problem under study. The first contribution to Eq. (1)involves the propagation of a particle, or, equivalently, is associated with adding a particle to the target nucleus. Information related to this quantity is therefore relevant for issues related to elastic scattering of nucleons from nuclei. The second term of the propagator in Eq. (1) provides information relevant for the (e,e'p) reaction. This can be understood by taking the imaginary part of Eq. (1)for energies w 5 E: - E f l EF

It is clear that this definition of the (hole) spectral function incorporates the simple notion of representing the (energy) probability density for the removal of a particle with quantum numbers a from the ground state of the target nucleus while leaving the system with the hole at an energy w. In the domain of valence hole states in nuclei the corresponding energies refer only to discrete states and the spectral function factors into an energy conserving &function and the removal probability at that energy for an orbital with quantum numbers a. Another useful quantity to gauge the strength of correlations is given by the summed (integrated) strength below the Fermi energy. This occupation number for quantum numbers a is given in various equivalent forms by

n

185

The development of a formal perturbation expansion of the single-particle propagator permits the introduction of the nucleon self-energy C which allows the determination of the propagator from this quantity and the noninteracting (or mean-field) propagator G(') by employing the Dyson equation

G ( a ,P ; w ) = G(')(a,P; w )

+

G(O)(a, y;w ) C ( y ,6 ;w)G(6,P; w ) .

(4)

Y J

Several relevant diagrammatic choices for the nucleon self-energy are available for nuclear systems. Some of these will be discussed in the next section. A clearer understanding of the physical significance of the self-energy and the importance of the Dyson equation can be gleaned by converting the Dyson equation to an equivalent Schrodinger-like equation for the transition amplitudes

in the case of discrete states n in the A - 1 system. Making the choice of the coordinate space representation, one can show5 that these amplitudes obey the following equation

This equation has the form of a Schrodinger equation with a nonlocal potential which is represented by the self-energy. Note that an eigenvalue c; can only be obtained when it coincides with the energy argument of the selfenergy. An important difference with the ordinary Schrodinger equation is related to the normalization of the quasihole wave functions z;;. This result is most conveniently expressed in terms of the single-particle state which corresponds to the quasihole wave function z&. In other words, one can use the eigenstate which diagonalizes Eq.(6), to express the normalization condition. Assigning the notation a q h to this state, one has

The subscript qh refers t o the quasihole nature of this state and the fact that for states very near to the Fermi energy with quantum numbers corresponding to fully occupied mean-field states the normalization yields a number of order 1. In the analysis of an (e,e'p) experiment it is conventional to find a local potential well (mostly of Woods-Saxon type) which will generate a single-particle state at the removal energy for the transition that is studied. This state is further required t o provide the best possible fit t o

186

the experimental momentum dependence of the cross section (with proper inclusion of complications due to electron and proton distortion) The overall factor necessary to bring the resulting calculated cross section into agreement with the experimental data, can then be interpreted as the spectroscopic factor corresponding to the experimental wave function according to Eq.(7). Although the well is local, it provides important information on the self-energy of a nucleon at energies below E F . From the theoretical perspective it is clear that in order to obtain spectroscopic factors in qualitative agreement with the experimental data ( i e . < l ) , one must include an energy-dependent self-energy in Eqs. ( 6 ) and (7). 'l2i3.

3. Theoretical Results In this section the status of the theoretical understanding of the spectroscopic factors that have been deduced from the analysis of this reaction will be reviewed. The qualitative features of the strength distribution can be understood by realizing that a considerable mixing occurs between hole states and two-hole one-particle (2hlp) states. This mixing leads to different strength patterns depending on the location of the orbital under consideration. An orbital in the immediate vicinity of the Fermi energy will still yield a large fragment near its original position since the 2hlp states are quite far in energy. For the same reason, only small components of this orbit appear at 2hlp energies. An orbit which is surrounded by many 2hlp states will yield a strongly fragmented pattern, the width depending on the strength of the mixing interaction. These features are all observed experimentally and can be easily understood. The theoretical description of this mixing requires an energy-dependent self-energy as discussed in the previous section. For medium-heavy nuclei like 48Caand 90Zr a fairly good description of the fragmentation pattern of valence hole states can be obtained by including the coupling to low-lying collective states in the self-energy. This typically requires the additional particle-hole states to be correlated at the level of the Tamm-Dancoff (TDA) or random phase approximation (RPA)'. Diagrammatic contributions t o the self-energy that represent these correlations involve to socalled ring diagrams. For quantitative results one also requires the inclusion of short-range and tensor correlations. These contributions are represented diagrammatically by ladder diagrams which are then inserted into the nucleon self-energy. On the one hand, the inclusion of short-range correlations leads t o a global depletion of mean-field orbitals which ranges from 10% in light nuclei to

187

about 15%in heavy nuclei and nuclear matterl0l5. The combination of these results leads to the typical pattern of occupation numbers that is illustrated in Fig. 3 for protons in ‘%a6. The depletion effect due to short-range 48

Ca

0

Energy Figure 3. Occupation numbers for 48Ca from theoretical calculations including both long- and short-range correlations.

correlations leads to a global depletion of about 10 % for all orbits below the Fermi energy. The coupling to collective surface vibrations reflecting the effect of long-range correlations has a distinctively different pattern. This pattern is characterized by long-range correlations having almost no effect on occupation numbers of orbits very far from the Fermi energy but increasing their effect for orbits as they approach this energy. The mixing of nearby particle states with 2hlp states leads to a small occupation of these orbits of a few percent as illustrated in Fig. 3. Recent experimental data from NIKHEF7i8 for 208Pbconfirm the characteristics of these occupation numbers and thier link to different types of correlations. These data demonstrate that all deeply bound proton orbits are depleted by about the amount expected from nuclear-matter calculations. This depletion effect due to short-range correlations in nuclear

188

matterg is in quantitative agreement with these recent results. This reduction of the single-particle strength of mean-field orbits below the Fermi energy, on the other hand, must be partially compensated by the admixture of high-momentum components in the ground state. These high-momentum nucleons have not yet been unambiguously identified experimentally using the (e,e'p) reaction. The search for these high-momentum components in valence states has not been successfu111,12,as was anticipated by earlier theoretical work13. Indeed, one can understand the absence of high-momentum components near the Fermi energy based on the following argument. In order to admix high-momentum nucleons with 2hlp states in the self-energy one must accomodate the small total momentum of the two holes by an opposite but in magnitude equal momentum for the particle state of the 2hlp component toensure momentum conservation. The energy of this 2hlp state then corresponds roughly to an average two-hole energy with the kinetic energy of the high-momentum nucleon to be subtracted. calculation^^^^^^ confirm that such high-momentum components occur at large binding energies of 100-200 MeV below the Fermi energy. While the results for medium-heavy nuclei are satisfactory, there is a considerable discrepancy for the spectroscopic factors in All theoretical results overestimate the experimental spectroscopic factors by at least 10 % and only those calculations that include long-range correlations come that close16. The collective excitations for this nucleus are of a quite complicated nature and less amenable to a description of the TDA or RPA type. For this reason a new technique was developed to sum collective (microscopically calculated) phonon contributions to all orders in the self-energy using the Faddeev technique17. Results of such calculations generate new theoretical features of the strength in accord with the experimental data and give slight improvements of the theoretical spectroscopic factors18. One may summarize all these theoretical results by identifying the redistribution of strength due to different types of correlations together with their quantative significance. This is done in Fig. 4. The results depicted in Fig. 4 have important consequences for other observables in nuclei. A brief discussion of these effects is given in the next section.

4. Consequences for other physical observables The simplest consequence of the reduction of the quasiparticle (hole) strength in nuclei is observed in inelastic electron scattering to high-spin magnetic states in '08Pb19. These high-spin states of the 12- and 14- kind

189

High-energy strength due to SRC and tensor force

continuum

i

jlrl

100 MeV

LO% ]Coupling to surface i phonons and q giant Resonances -.-c

65% quasihole strength

110%

1

Coupling to surface phonons and Giant Resonances

Location of high-momenti components due to SRC at high missing energy

Spectral strength for a correlated nucleus

Figure 4. The distribution of single-particle strength in nuclei. Several generic diagrams are identified which have unique physical consquences for the redistribution of the singleparticle strength when they are taken into account in the solution of the Dyson equation. The middle column of the figure characterizes the mean-field picture that is used as a starting point of the theoretical description. The right column identifies the location of the single-particle strength of the orbit just below the Fermi energy when correlations are included. The physical mechanisms responsible for the strength distribution are also identified. In the left column the diagram that is responsible for the admixture of highmomentum components in the ground state is depicted. The energy domain of these high-momentum nucleons is at large missing energies.

190 can only be made by simple particle-hole configurations involving very high angular momentum orbits. The mixing to nearby states is quite negligible based on theoretical considerations involving the high-angular momentum involved which yields very small matrix elements for the effective interaction. This feature is further confirmed by experiment since the shapes of the corresponding formfactors are accurately given by the corresponding particle-hole formfactors obtained from mean-field calculations. To get complete agreement with these data it is, however, necessary to reduce the theoretical results by a factor of two. This same factor is expected t o represent the reduction of the strength of both the particle and the hole state involved in this excitation ( M 0.7’). This simple explanation ties different experimental results together and confirms the new theoretical framework for the description of the single-particle strength in nuclei which should be in accord with the results from (e,e’p) reactions. The dressing of the nucleons also has similar consequences for other excited states which display more collective behavior or appear at higher excitation energy. A prime example of modifications of the strength distributions that reflect this dressing is given by the Gamov-Teller (GT) excitations. Conventional RPA calculations always concentrate ail the GT strength at low excitation energy with most of the strength residing in the giant GT resonance. Experimentally, only 50 to 60 % of this strength is observed at low energy. The effect of dressing nucleons on the GT strength distribution is manyfold. By having a correlated Fermi sea and the corresponding distribution of the single-particle strength it is possible to couple particle-hole excitations to more complicated states. The extension of the RPA that takes these effects into account shows that for 48Caa 25 % reduction of the G T strength in the resonance domain is obtained while moving the rest of the strength to higher excitation energyz0 with most of this strength residing below 80 MeV. Dealing directly with dressed particles in determining the response also yields substantial strength for the ( n , p ) reaction on account of the partial occupations depicted in Fig. 3. In the socalled dressed RPA (DRPA) the amount of GT strength in the npchannel is about 5.9 for 48Ca. The corresponding strength in the pn-channel is then further reduced in the resonance region bringing it close to the experimental resultsz1. Even more strength is then moved to the energy domain between 20 and 80 MeV. It is gratifying to note that recent experiments discussed during this workshop confirm the presence of this strength experimentally.

191

5. Conclusions

The aim of this paper has been to paint a picture of the nucleus in which nucleons spend a sizable fraction of their time involved in more complicated excitations. These correlations can be distinguished on the basis of the energy scales that characterize them. Short-range correlations couple low-lying states to high-lying ones in the particle domain through the repulsive core of the nucleon-nucleon interaction. The result of this coupling is a global depletion of the single-particle strength which corresponds to 10 % in light nuclei and 15 % in nuclear matter. Additional correlations are involved in coupling the single-particle motion to low-lying states. These long-range correlations provide an important aditional reduction of the strength for valence hole states. The combined effect of long- and short-range correlations then yields about a 35 % reduction of the single-particle strength for these orbits as observed in the (e,e'p) reaction. The reduction of strength near the Fermi energy must be accompanied by a corresponding admixture of strength corresponding to nearby states which are unoccupied in the meanfield. The results in Fig. 3 indicate that this leads to occupation numbers of these particle states of several percent. Additional strength must be admixed by accomodating high-momentum nucleons in the nuclear ground state. This is graphically illustrated in Fig. 4. This complete picture of the single-particle strength has important consequences for the description of excited states of nuclei. These consequences are reviewed for the magnetic high-spin states in "'Pb and the distribution of the GT-strength for 48Ca. References 1. A. E. L. Dieperink and P. K. A. de Witt Huberts, Ann. Rev. Nucl. Part. Sci. 40,239 (1990). 2. I. Sick and P. K. A. de Witt Huberts, Comm. Nucl. Part. Phys. 20,177 (1991). 3. L. Lapikb, Nucl. Phys. A553,297c (1993). 4. V. R. Pandharipande, I. Sick, and P. K. A de Witt Huberts, Rev. Mod. Phys. 69, 981 (1997). 5. W. H. Dickhoff, in Nuclear Methods and the Nuclear Equation of State, ed. M. Baldo (World Scientific, Singapore, 1999) p. 326. 6. G. A. Rijsdijk, K. Allaart, and W. H. Dickhoff, Nucl. Phys. A550,159 (1992). 7. M.F. van Batenburg, Ph.D. thesis, University of Utrecht (2001). 8. L. Lapikb et al., to be submitted to Phys. Rev. Lett. (2002). 9. B. E. Vonderfecht, W. H. Dickhoff, A. Polls, and A. Ramos, Phys. Rev. C44, R1265 (1991). 10. W. H. Dickhoff, Phys. Rep. 242,119 (1994). 11. I. Bobeldijk et al., Phys. Rev. Lett. 73,2684 (1994). 12. K. I. Blomqvist et al., Phys. Lett. B344,85 (1985).

192

H. Muther and W. H. Dickhoff, Phys. Rev. C 49,R17 (1994). H. Miither, A. Polls, and W. H. Dickhoff, Phys. Rev. C 51, 3040 (1995). M. Leuschner et al., Phys. Rev. C 49,955 (1994). W. J. W. Geurts, K. Allaart, W. H. Dickhoff, and H. Miither, Phys. Rev. C53,2207 (1996). 17. C. Barbieri and W. H. Dickhoff, Phys. Rev. C63,034313 (2001). 18. C. Barbieri and W. H. Dickhoff, Phys. Rev. C65,064313 (2002). 19. J. Lichtenstadt et al., Phys. Rev. C20, 497 (1979). 20. M. G. E. Brand, K. Allaart, and W. H. Dickhoff, Nucl. Phys. A509,1 (1990). 21. G . A. Rijsdijk, W. J. W. Geurts, M. G. E. Brand, K. Allaart, and W. H. Dickhoff, Phys. Rev. C48,1752 (1993). 13. 14. 15. 16.

193

DETERMINATION OF THE GAMOW-TELLER QUENCHING FACTOR VIA THE "Zr(n,p) REACTION AT 293 MEV

K. YAKO, H. SAKAI, M. HATANO, H. KATO, Y. MAEDA, T. SAITO, K. SEKIGUCHI, A. TAM11 AND N. UCHIGASHIMA Department of Physics, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan K. HATANAKA, J. KAMIYA, Y . KITAMURA, Y. SAKEMI, Y. SHIMIZU AND T. WAKASA Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-004 7, Japan H. OKAMURA AND K. SUDA Department of Physics, Saitama University, Saitama, Saitama 338-8570, Japan M.B. GREENFIELD International Christian University, Mitaka, Tokyo 181-8585, Japan C.L. MORRIS Los Alamos National Laboratory, Los Alamos, NM 87545, USA J. RAPAPORT Department of Physics, Ohio University, Athens, Ohio 45701, USA The double differential cross sections a t 0'-12' were measured for the gOZr(n,p) reaction a t 293 MeV in a wide excitation energy region of 0-70 MeV. The experiment was performed by using the ( n , p ) facility at the Research Center for Nuclear Physics. The multipole decomposition (MD) technique was applied to the measured cross sections to extract the GT component in the continuum. After subtracting the contribution of the isovector spin-monopole excitation we obtained the G T strength of Sa+ = 3 . 0 f 0 . 3 f 0 . 8 f 0 . 5 up to 31 MeV excitation. The quenching factor Q was deduced by using the present result and the Sa- value obtained from the MD analysis of the gOZr(p,n ) spectra. The result is Q = 0.83 f0.06 in regards to Ikeda's sum rule value of 3 ( N - 2 ) = 30.

1. Introduction

Gamow-Teller (GT) resonances have been extensively studied since its discovery in 1975.l The GT transition involves the operator LTTand is char-

194

ocal Plane Detectors

Figure 1.

0

1

2 m

A schematic drawing of the RCNP (n,p)facility.

acterized as spin-flip (AS = l), isospin-flip (AT = 1) and no transfer of orbital angular momentum (AL = 0). There exists a model-independent sum rule, So- - Sp+ = 3(N - Z ) , where So- and Sp+ are the GT strength of p- and p+ types, respectively.2 Surprisingly, however, only a half of the GT sum rule value was identified from the (p,n) measurement on targets throughout the periodic table.3 This problem, so-called the quenching of the GT strengths, has been one of the most interesting phenomena in nuclear physics because it is related to non-nucleonic (A-isobar) degrees of freedom in nuclei; the quenching factor sets a strong constraint on the Landau-Migdal parameters, g& and Aa, in the n+p+g' model.4 Recently, Wakasa et al. have measured the angular distribution of the double differential cross sections for the gOZr(p,n)reaction at 295 MeV.5 By performing multipole decomposition (MD) analysis, the GT strengths of Sp- = 28.0 f 1.6 has been obtained in the continuum up to 50 MeV excitation in 90Nb.5Determination of the A-isobar contribution to the GT sum rule, however, requires precise ( n , p ) cross section data at the same energy. For this purpose we have constructed an ( n , p ) facility at Research Center for Nuclear Physics (RCNP) and measured the double differential cross sections for the 90Zr(n,p)90Yreaction at 293 MeV. 2. Experiment

Figure 1 shows a schematic layout of the RCNP (n,p) facility in the WS beam course. A nearly mono-energetic neutron beam was produced by the 7Li(p,n) reaction at 295 MeV. The primary proton beam, after going through the 7Li target, was bent away by 23" by the clearing magnet6 to a

195

beam dump in the floor. The typical intensity of the beam was 450 nA and the thickness of the 7Li target was 320 mg/cm2. About 2 x 106/secneutrons bombarded the target area of 3OW x 20H mm2 downstream by 95 cm from the 7Li target. Three 'OZr targets with thicknesses of 200400 mg/cm2 and a polyethylene (CH2) target with a thickness of 46 mg/cm2 were mounted in a multiwire drift chamber (target MWDC). Wire planes placed between the targets detected outgoing protons and enabled one to determine the target in which a reaction occurred. Charged particles coming from the beam line were rejected by the veto scintillator with a thickness of 1 mm. The 'H(n,p) events from the CH2 target were used for normalization of the neutron beam flux. The position of outgoing protons were detected by six wire planes installed just behind the targets in the target MWDC. Another MWDC, front end MWDC, was installed at the entrance of the Large Acceptance Spectrometer (LAS). The scattering angle of the ( n , p ) reaction was determined by the information from the two MWDCs. The outgoing protons were momentum analyzed by LAS and were detected by the focal plane detectors. Blank target data were also taken for background subtraction. The 'H(n,p) cross sections given by the program SAID7 was used to normalize the "Zr(n,p) cross sections. We have obtained the differential cross sections up to 70 MeV excitation energy over an angular range of 0'-12' with a statistical accuracy of 1.7%/2 MeV-1' at lo-2O. The overall energy resolution expected from the target thicknesses and the energy spread of the beam is 1.5 MeV. The angular resolution is 10 mr which is dominated by the the effect of multi scattering in the 'OZr targets. 3. Analysis

The MD analysis has been performed on the excitation energy spectra to extract the GT strengths. First of all, the cross section data was binned in 2-MeV energy intervals to reduce the statistical fluctuation. For each excitation energy bin from 0 MeV to 70 MeV, the experimentally obtained angular distribution cex*(8cm,E,)has been fitted by means of the least-squares method with the linear combination of calculated distributions ~ ~ ~ ~ ~E,)~ defined ( 8 l ~byb ccalc(8cm,

E,) =

C

calc aAL\Jncpph;Ap(8cm7

(1)

AJ n

where the variables a A J m are the fitting coefficients with positive values. The angular distributions for the following final J K states have been calculated: 1+(AL = 0 ) , 0-, l-, 2-(AL = l), 3+(AL = Z), and 4-(AL = 3).

,

196

90Zr(n,p)90Y at 293 MeV

oo-lo

-

E

2

3r

I

8

0

10

20

30

40

50

60

70

E, (MeV) Figure 2. The result of MD analysis on the double differential cross sections for the gOZr(n,p)gOY reaction at 293 MeV. The upper and lower panels show the result at the angular region close to the maximum of GT and dipole angular distributions, respectively.

The one-body transition densities are calculated from pure lplh configurations. For the transitions with AL 2 1, the active neutron particles are restricted to the 19712, 2d5/2, 2d3/2, 1hlll2, or 3s1l2 shells, while the active proton holes are restricted to the 19912, 2p1l2, 2p3/2, 1f5/2, or 1f7/2 shells by assuming 40Ca to be a core. The GT transitions due to ground state correlation were taken into account by activating the (~ l g ~n1g;t2) / ~ , and (vlggl2,,lg&) configurations. The DWIA calculations are performed by using the computer code DW81.8 The optical potential parameters for the incident proton and for the outgoing neutron are taken from Refs. 9, 10. The effective NN interaction is taken from the t-matrix parameterization of the free NN interaction by Franey and Love at 325 MeV.I1 The minimizing procedure was performed for all the possible 58080 combinations. The combination of the ph configurations at each energy window was chosen so that the x2 value was minimized. Figure 2 shows the result of the MD analysis. The AL = 0 component has a broad ( w 10 MeV in FWHM) bump at E, N 20 MeV mainly due to the isovector spin monopole (IVSM) resonance,12 which is excited through the r2ar operator. The AL = 0 component of the cross section, oAL=o(q,w), is related to the GT strengths through the proportionality relation, a.e. CJAL=o(Q,W) = bGTF(q,w)B(GT), where ~ G is T the GT unit

197

cross section5 and F ( q , w ) is the kinematical correction factor13. The upper limit energy of integration, EFax,is determined so that it corresponds to E,ma" = 50 MeV in the (p,n) work.5 Considering the difference in the and the 'ONb nuclei and the difference in Coulomb energy between the the reaction Q value, we use the Era"value of 31 MeV and obtained a total GT strength of SO+ = 5.4 k 0.3 f 0.9, where the errors are uncertainties of the MD analysis and the GT unit cross section. The contribution of IVSM is estimated by the DWIA calculations in which all the IVSM strengths are assumed to lie below 31 MeV e ~ c i t a t i o n . ~ After subtracting the IVSM contribution of 2.4 f 0.8 GT units, we have obtained a total GT strength of S,+ = 3.Ok0.3(MD)f0.8(IVSM)k0.5(6 up to 31 MeV excitation. By using the S,- value by Wakasa et aL5 the quenching factor Q, which is defined by Q E

s,- -s,+ , has been deduced to be Q = 0.83 f 0.06

3(N - 2 ) in regard to Ikeda's sum rule value of 3(N - 2 ) = 30. Therefore the quenching of the GT strength due to the AN-' admixture into the lplh GT state is significantly smaller than the quenching of 50%, observed in the previous studies3 where the GT strengths in the continuum are not taken into account. Then the Landau-Migdal parameters, &A and ghN, have been determined from the quenching factor. The deduction by Suzuki and sakai4 in Chew-Low model leads to &N x 0.6 and 0.16 < gka < 0.35 for &A = 0.6. Therefore the universality ansatz of the Landau-Migdal parameters, i.e. g k N = &A = &A(= 0.6 0.8),does not hold.

-

-

References 1. R. Doering, A. Galonsky, D. Patterson, and G. Bertsch, Phys. Rev. Lett., 35, 1691 (1975). 2. K. Ikeda, S. Fujii, and J.I. Fujita, Phys. Rev. Lett., 3, 271 (1963). 3. C. Gaarde et al., Nucl. Phys., A369, 258 (1981). 4. T. Suzuki, and H. Sakai, Phys. Lett., B455, 25 (1999). 5. T. Wakasa, et al., Phys. Rev., C55, 2909 (1997). 6. J. Kamiya et al., RCNP annual report 1998, p. 113. 7. R.A. Arndt and L.D. Roper, Program SAID, (unpublished). 8. M.A. Schaeffer and J. Raynal, Program DW81, (unpublished). 9. S. Quing-biao, F. Da-chun and Z. Yi-zhong, Phys. Rev., C43, 2773 (1991). 10. E.D. Cooper, S. Hama, B.C. Clark and R.L. Mercer, Phys. Rev., C47, 297 (1993). 11. M.A. Franey and W.G. Love, Phys. Rev., C31, 488 (1985). 12. K.J. Raywood et al., Phys. Rev., C41, 2836 (1990). 13. T.N. Taddeucci et al., Nucl. Plays., A469, 125 (1987).

198

GAMOV-TELLER SUM RULE WITH THE A ISOBAR

MUNETAKE ICHIMURA* Faculty of Computer and Information Sciences, Hosei University, 6 7 - 2 Kajino-cho, Koganei-shi, Tokyo 184-8584, JAPAN E-mail: ichimuraak. hosei.ac.jp

The Gamov-Teller sum rule (Ikeda sum rule) S ( f l - ) - S(fl+) = 3 ( N - 2 ) holds model independently when only the nucleon degrees of freedom are considered. We generalize it for the case with the A isobar degrees of freedom. It is descrbed by the isospin z-component carried by the nucleons and that by the A. If the axial week current of the quark model is adopted, the generalized sum rule turned out to be S ( p - ) - S@+) = (;)'33(N - 2 ) .

1. Introduction

Quenching of the celebrated Gamov-Teller(GT) sum rule' is a current great interest of nuclear physics. Ikeda et al.'t3 first predicted concentration of the GT strength, which promoted extensive studies of its distribution. Ikeda3 also showed that the total sum of the P--type strength is proportional to ( N - 2 ) in the simple shell model. Thus this sum rule is usually called Ikeda sum rule. The previous speaker Yako4 reported the latest experimental results of the quenching. As he pointed out, it has been a long-standing question to what extent the A isobar affects the quenching. Since the sum rule is restricted to the nucleon(N) degrees of freedom, we generalize it in the case with the A degrees of freedom as well. The sum rule is derived by Gaarde et al.' as

*Work partially supported by the Grant in-Aid for Scientific Research No. 12640294 of Ministry of Education, Culture, Sport, Science and Technology of Japan.

199

with the sum of the GT-type ,Bftransition strength . , A

denotes a nuclear state and especially where does the parent state. Note that ~ * ( k and ) u ( k ) are the operators on the nucleons and the states @*consist of only the nucleons. Under this condition the sum rule is model independent and valid for any state @O with any complicate configurations. Thus we attached the superscript N on the sums above. 2. Generalization to the N +A system 2.1. G T transition currents

The nuclear G T currents are conventionally expressed as

as appeared in Eq. (2) The corresponding transition currents between the nucleon and the A are usually defined by substituting u and T with the transition operators5 S and T , and St and Tt. Namely

with T* = T, f iTy. We also have t o consider the G T currents between the A’s themselves, which we write

k= 1

Then the week Hamiltonian for the G T processes of the N can be written as

where

.are the leptonic currents.

+ A system

200

2.2. Sum rule

The generalization of the sums corresponding to Eq. (2) should be

1l(QnIjT+AlQo)12

sNfA(0*) =

(7)

n

with

where 9, are the states which consist of N and A and 90can be any parent state with N and A. Then the generalization of the sum rule (1) is expressed by the commutator a s

s~+~(G= T )~ ~ + ~ -( ~p ~- +) ~ ( = p (*ol+ )

C

[j+N;A,j-N;A] po).(9)

a=x , Y > Z

$21

$21

The commutators [ j z , a , and [ j f , a , do not vanish in general, but after summing up over a we can prove that they vanish. Hence the sum becomes

After some manipulation, we obtain the extended sum rule

o) the isospin zHere (Qol EI,t Z ( k ) l Q o )and (901 XI, ~ F ( k ) l ~ represent component carried by the nucleons and by the A's in the parent state Qo, respectively.

20 1

3. Quark model

A way t o specify the coupling ratios

2

and IC,"/cAl2 is t o use the quark model. Corresponding to Eq. (2)' the G T currents of the quark system are written as Ic,""/cAl

and the week Hamiltonian is given by

we obtain the relations

and consequently

Using these relations the sum rule (11) turns out t o be

We may also introduce the quark G T sum rule in unit of I C A as ~~

Q.O&

where is a 3A quark state with the isospin z-component i ( N - 2). In the same way to prove S N ( G T ) ,we obtain

This sum rule in unit of /C,&I2was derived by Osterfeld8. We see that the sum rule for the N+A system exhausts the quark sum rule completely.

202 4. Discussions

It is interesting t o note that the sum rule value is much reduced by the 2 , once the A sector is included. This means that the P--type factor transition is larger in the nucleon sector but the P+-type is larger in the A sector for the system with T, > 0. We must be careful that this sum rule concerns with the difference of the p- and p+ strengths. It is informative t o consider the case where the parent state is a simple nucleon double closed shell of N > 2.Then we get

(g)

SN(p-)= 3 ( N - Z ) ,

SN(p+)= 0

where SNA(p*)are the sum of the transition strength from the ground state to the A-hole states, which are given by

The above results can be read in the appendix of the ref.g There has been many works which concerned with the transitions from N t o A, but they discussed only the case that the parent state consists of only nucleons. Therefore the currents j : were never included. However, they are inevitably needed t o derive the general form of the sum rule, which is valid for any parent state of the N+A system. Then its relation t o the quark G T sum rule is clarified.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

C. Gaarde et al., Nucl. Phys. A334,248 (1981). K. Ikeda, S. Fujii and J.I. Fujita, Phys. Lett. 3,271 (1963). Kiyomi Ikeda, Prog. Theor. Phys. 31,434 (1964). K. Yako, presentation in this symposium. H. Sugawara and F. von Hippel, Phys. Rev. 172,1764 (1968). J,J.J.Kokkedee, The Quark Model (W.A. Benjamin, Inc. New York, 1969) M. Ichimura, H. Hyuga and G.E. Brown, Nucl. Phys. A196, 17 (1972). Franz Osterfeld, Rev. Mod. Phys. 64,491 (1992). C. Gaarde el al., Nucl. Phys. A369,258 (1981).

203

TWO-STEP EFFECTS IN ANALYSIS OF NUCLEAR RESPONSES*

YASUSHI NAKAOKA Department of Physics, University of Tokyo, Bunkyo-ku,Tokyo 11 3-0033, JAPAN E-mail: [email protected] TOMOTSUGU WAKASA~ Research Center for Nuclear Physics, Osaka University, Ibaraki-shi, Osaka 567-0047, JAPAN E-mail: wakasa8rcnp.osalca-u.ac.jp MUNETAKE ICHIMURA~ Faculty of Computer and Information Sciences, Hosea University, Koganei-shi, Tokyo 184-8584, JAPAN E-mail: [email protected]

A charge exchange quasi-elastic l2C(g,z) reaction a t the scattering angles el& = 16', 22', 27' with the incident energy 346 MeV has been measured at RCNP. We analyzed its spin longitudinal and spin transverse cross sections I D , and I D , by evaluating 2-step contributions in addition to the DWIA analysis with RPA correlation. We found that the 2-step contributions play important role to explain the problematic discrepancy seen in I D , between the experimental and the DWIA results. The present analysis supports the appearance of a precursor phenomenon of the pion condensation.

1. Introduction After proposal of the pion condensation in a high density nuclear matter', finding of its precursors in ordinary nuclei has been a long-standing desire. *This paper is presented by M. ICHIMURA t Work partially supported by the Grant in-Aid for Scientific Research Nos. 12740151and 14702005 of Ministry of Education, Culture, Sport, Science and Technology of Japan. *Work partially supported by the Grant in-Aid for Scientific Research No. 12640294 of Ministry of Education, Culture, Sport, Science and Technology of Japan.

204

A promising phenomenon to prove it is the enhancement of the isovector spin longitudinal response function R L ( q ,w ) for the momentum transfer q and the energy transfer w at the quasi-elastic region. This theoretical prediction by Alberico, Ericson and Molinari’ was based on a random phase approximation (RPA) calculation with the one-pion and one-rho-meson exchange potential plus the contact interaction specified by the Landaup g’ interaction). Migdal parameter g”s .( To get information about R L ( q , w ) , measurements of the cross section and the polarization transfer coefficients Dij of the quasi-elastic 12C$0Ca(@7 2)reactions were carried o ~ t ~ 1Note ~ > that ~ . the electron scattering does not work for this purpose. From these data we can construct the spin longitudinal and the spin transverse cross sections6, I D , and ID,, which are suitable to extract R L ( q , w ) and the spin transverse response function RT (q, w ) , because

+ +

ID,

0:

RL, I D ,

C(

RT

in the plane wave impulse approximation (PWIA), where the directions are specified by q (the direction of q ) , n (that of the reaction normal) and p (that perpendicular to q and n). Those data have been analyzed‘ in the distorted wave impulse approximation (DWIA) in cooperation with RPA, which includes the A-isobar degrees and uses the T p gr interaction. The recent analysis’ well reproduced ID,, and seemed to demonstrate the first experimental evidence of the enhancement of R L ( q , w ) due to RPA correlation. It also showed importance of the A mixing. However, I D , is very much underestimated7 so that we should reserve the definite conclusion about the spin responses. The abnormally large observed I D , seems to contradict with the obtained R T ( q , w ) from the ( e , e r ) data, which suggests that we are missing some reaction mechanisms in the analysis of the (p’,,) reactions. So we raised a question whether 2-step processes overcome this discrepancy. A formalism of the 2-step calculation in a plane wave approximation (PWA) and preliminary numerical calculations with a on-energy shell approximation were presented by Nakaoka and Ichimurag. Recently Nakaoka’’ carried out an improved calculation, in which the on-energy shell approximation was taken out. Here we report our analysis of a part of the latest experimental results5 of 12C(@,G), with the 2-step contributions in addition to the DWIA ones. We found that the 2-step contributions are crucial to understand the ob-

+ +

205 served polarized cross sections. The analysis supports the prediction that the precursor phenomena of the pion condensation are seen in ID, of the quasi-elastic (@,6) reactions. 2. Twestep Processes

Here we sketch our method to calculate the 2-step processes in the quasielastic (@, 6) reactions.

2.1. Formalism and Approximations Since full 2-step calculation" is extremely cumbersome, we adopt following approximat ions. (1) Plane Wave Approximation The motion of the scattering nucleon (the incident proton, outgoing neutron and those in the intermediate states) is treated as the plane wave (PWA). Of course we must include the nuclear distortion and absorption. For this purpose we assume that these effects are similar for the 1- and 2-step processes because the scattering nucleon runs on the similar trajectories in the both processes. Based on this idea we evaluate the distortion (absorption) effects from the ratio between the DWIA and the PWIA cross sections. Then we estimate the 2-step contributions by the relation DWIA 2-step with distortion M 2-step in PWA x PWIA' (2) No Correlation In the calculation of the 2-step processes we treat a target to be a simple double closed shell state and do not take into account any further correlations. Note that we took account of the RPA correlations in the DWIA calculation. (3) Never-come-back Approximation We consider only processes, in which the first step creates a particle-hole state (plhl) and the second step again creates a different particle-hole state (p2h2) where pl # p2 and hl # h2. Thus we can neglect the interference between the 1- and 2-step amplitudes. (4) Adiabatic Approximation Following Tamura, Udagawa and Lenske12, we neglected interference among the 2-step amplitudes with the same 2p-2h states created through different sequence. Kawano and Yoshida" called this approximation an adiabatic approximation in contrast with their sudden approximation.

206

Under these conditions the polarized cross sections ID,, I D p are concisely expressed as

with X a (2) ;aza;al

=

c

t:;b;

(d)t:;6;

( d 2 ) t a z b z (‘?2)taibi

(q1)

bzb; bib;

Gw (k)G:, ( k ’ ) R b ; b z

(427 q;; w2)Rb;b1 (41, qi; w1)7

(4)

where K is the geometrical factor, k(k’)are the momenta of the intermediate scattering nucleon, w1(w2)is the transferred energy in the first (second) step (w = w1 wz),and q1(q2)and qi(q5) are the transferred momentum in the first (second) step ( q = q1 q2 = qi qh ), which are related with k(k’)and the incident momentum ki as k = ki - q l , k‘ = ki - 4;. The suffices al,bl specify the spin directions O , q 1 ,n1 ,p l , which denote scalar, q,n,pdirections in the first step, and similarly u2,bz = 07q2,n2,p2.The Green’s function G,, (k)for the intermediate scattering nucleon is given by

+

+

+

with the nucleon mass M , the target mass MT and the total energy E . The nucleon-nucleon t-matrix in the optimal frame13 is written as

t:(k’7 k) = Jrl [Aaoo~lco + Buonalcn + C(aOnak0

+

OOO%J

+ Egoqalcq + FaopWcp] ab

2.2. Numerical method

The formula involves 7-fold multi-integration. The amplitudes t a b ( q ) have 6 different spin structure, thus 64 = 1296 terms appear in the spin sums. Using additional conditions such as the spin saturation we reduce them into 400 termsg. Since the calculation is still extremely cumbersome, we adopted the on-energy shell approximation for the Green’s functions G, (k) in the previous workg.

207

NakaokalO took out this approximation and improved the calculation, in which the Green's functions are calculated for several values of finite E in Eq.(5) and the extrapolation to the limit of E = 0 is taken. A similar method is reported by U d 4 in this symposium. Here we use this method in the present analysis.

3. Numerical Results In the RPA calculations we varied the Landau-Migdal parameters, and looked for a set of the values of gkN,gkA and &A t o reproduce the data as better as possible. In the DWIA and the PWIA calculations, we used the response functions calculated with the radial dependent effective mass r n * ( ~ )of the nucleons in the target, whose value is 0 . 7 r n ~at the center of the nucleus, and with the spreading widths of the particles and the holes phenomenologically7. In the 2-step calculations we used the simplest response functions with m* = m N and without the spreading widths and the RPA correlations. Figure 1compares our calculations with the latest experimental results5 of I D , and ID,, of 12C(@,ii)with the incident energy 346 MeV at three angles of t91ab, = 16",22" and 27", which correspond t o the momentum transfers about q = 1.3, 1.7 and 2.0 fm-l, respectively. The experimental results are represented by the black circles. The dashed, the dash-dotted and the dotted lines denote the results of the DWIA calculations with the RPA correlation (DWIA/RPA), without the correlation (DWIA/Free) and those of the 2-step contribution without the correlation (2-step/Free), respectively. The sums of the DWIA and the 2-step contributions are shown by the full line. Here we present the RPA results with the Landau-Migdal parameters, gkN = 0.7, &A = 0.4,gba = 0.5. For I D , we see that the enhancement due to the RPA correlation plays important role t o explain the experimental data at !!?lab. = 22" and 27". The 2-step contributions make the theoretical results somewhat larger than the experimental data. At !!?lab. = 16", the longitudinal interaction (T 9') is small and thus sensitive t o g"s at this momentum transfer. Therefore the RPA effects are also so. The 2-step effects cause overestimation. I t has been well known that the RPA correlation quenches R T ( q , w ) , which is consistent with the (e, e') data. Reflecting this, calculated results of I D , are also quenched by the RPA correlation. However, the experimental results are much larger than the DWIA/RPA results. We found that the 2-step contributions reduced the discrepancy very much though not enough.

+

208

0

20 40 60 80 100

0

20 40 60 80 100 120

Energy t r a n s f e r qab (MeV) Figure 1. I D , and I D , of 12C($, 3) a t Tp = 346 MeV. The black circles represent the experimental data. The dashed, dash-dotted and dotted lines denote the DWIAIRPA, the DWIA/Fkee, 2-step/Free results, respectively. The full line does the sum of the DWIA/RPA 2-step/Free.

+

4. Summary and Conclusion

We reported the analysis of the latest experimental data of the polarized cross sections IDq and I D , for "C($, 5 ) at various angles. We included the 2-step contributions on top of the DWIA calculations with the RPA correlation. We adjusted the Landau-Migdal parameters.

209

The obtained values were typically ghN = 0.7, gf\ra = 0.4, gLa = 0.5, which are somewhat larger than those previously obtained7j8 without the 2-step contributions. These values are consistent with those obtained from the quenching factor of the Gamov-Teller sum rule15 if we take account of the finite nucleus effects16. We found the 2-step processes play an important role to explain the large discrepancy between the experimental and the DWIA results in ID,. In this symposium Ogata17 also reported similar amount of the 2-step contributions in the semiclassical distorted wave approximation. The present analysis shows that the enhancement of RL is supported by the experiments, which implies the precursor phenomena of the pion condensation. We need more systematic analysis for more definitive conclusion, which will be presented in near future. The authors would like to thank Professors H. Sakai, K. Hatanaka, and H. Okamura for leading the experiments and valuable discussions. The experiments were performed under program numbers E59 and El31 at RCNP. References A.B. Migdal, ZhETF 61 2210 (1971); Sou. Phys.-JETP34 1185 (1972). W.M. Alberico, M. Ericson and A. Molinari, Nucl. Phys. A379 429 (1982). T.N. Taddeucci et al., Phys. Rev. Lett. 73 3516 (1994). T. Wakasa e t al., Phys. Rev, C59 3177 (1999). 5. T. Wakasa et al., RCNP-El31 Collaboration; T. Wakasa, Talk presented in 15th International Spin Physics Symposium (2002). 6. E. Bleszynski, M. Bleszynski and C.A. Whitten Jr., Phys. Rev. C 26 2063 (1982). 7. K. Kawahigashi, K. Nishida, A. Itabashi and M. Ichimura, Phys. Rev. C 63 044609 (2001). 8. M. Ichimura and K. Kawahigashi, ”Challenge of Nuclear Structure”, Proc. of the 7th International Spring Seminar on Nuclear Physics, (ed. by A. Covello, World Scientific, 2002) p. 531 9. Y. Nakaoka and M. Ichimura, Prog. Theor. Phys. 102 599 (1999). 10. Y. N a b k a , Phys. Rev. C 65 064616 (2002). 11. T. Kawano and S. Yoshida, Phys. Rev. C64 024603 (2001). 12. T. Tamura, T. Udagawa and H. Lenske, Phys. Rev. C26 379 (1982). 13. M. Ichimura and K. Kawahigashi, Phys. Rev. C45 1822 (1992) . 14. E. Uzu, Talk in this symposium (2002). 15. T. Suzuki and H. Sakai, Phys. Lett. B 76 25 (1999) . 16. A. Arima et al., Phys. Lett. B 499 104 (2001) . 17. K. Ogata, Talk in this symposium. 1. 2. 3. 4.


W. QUASI-FREE SCATTERING


213

RELATIVISTIC PREDICTIONS OF SPIN OBSERVABLES FOR EXCLUSIVE PROTON KNOCKOUT REACTIONS

G. C. HILLHOUSE1>’, J. M A N 0 3 , R. NEVELINGlTA. A. COWLEY’, S. M. WYNGAARDT4, K. HATANAKA2, T. N O R 0 5 AND B. I. s. VAN DER VENT EL^ ‘Department of Physics, University of Stellenbosch, Private Bag X 1 , Matieland 7602, South Africa Research Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-004 7, Japan Department of Electrical Engineering and Computer Science, Osaka Prefectural College of Technology, Osaka 572-8572,Japan 4Department of Physics, University of the Western Cape, Private Bag Xl7, Bellville 7535, South Africa Department of Physics, Kyushu University, Fukuoka, 812-8581,Japan

We have developed a relativistic distorted wave model, based on the zero-range approximation to the NN interaction, for the description of complete sets of exclusive (P; 2 y ) polarization transfer observables. The predictive power of this model is tested by comparing calculations to recent analyzing power data for the knockout of protons from the 3s112 state in 2osPb at an incident energy of 202 MeV, and for coincident coplanar scattering angles We also study the sensitivity of complete sets of polarization transfer observables, for which no published data exist, to distorting optical potentials and also compare results to nonrelativistic Schrodinger-based calculations.

1. Introduction

One of the most challenging problems in nuclear physics is to understand how the properties of the strong interaction are modified inside nuclear matter. In principle, the exclusive nature of (@,23) reactions can be exploited to knockout protons from deep- t o low-lying single particle states in nuclei, thus yielding information on the density dependence of the nucleon-nucleon (NN) interaction. However, before studying exotic nuclear medium effects, it is important t o test the validity of conventional theoretical models for ‘Present address: Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA.

214

the description of observables which are NOT sensitive to nuclear medium modifications of the NN interaction. Recently, Neveling et al reported that both relativistic (Dirac equation) and nonrelativistic (Schrodinger equation) models, based on the distorted wave impulse approximation (DWIA), completely fail to reproduce exclusive (p’i2p) analyzing power data for the knockout of protons from the 3s1/2 state in 208Pb at an incident laboratory kinetic energy of 202 MeV. For the prediction of energy-sharing cross sections, however, both dynamical models yield spectroscopic factors which are in good agreement with those extracted from ( e ,e’p) studies. Systematic corrections to the nonrelativistic model - such as different kinematic prescriptions for the NN amplitudes, non-local corrections to the scattering wave functions, densitydependent modifications to the free NN scattering amplitudes, as well as the influence of different scattering and boundstate potentials - fail to remedy the analyzing power dilemma, and hence, it is not clear how to improve existing Schrodinger-based analyses. On the other hand, such an exhaustive analysis has not yet been performed within the context of the relativistic DWIA. We have identified a number of possible explanations for the failure of the recent relativistic finite-range DWIA predictions The evaluation of the six-dimensional transition matrix elements associated with the finite-range approximation is subject to large computational time and, hence, numerical errors need t o be closely monitored. The scalar and vector scattering potentials employed are microscopic in the sense that they are generated by folding the NN t-matrix, based on the Horowitz-Love-Franey (HLF) meson-exchange model, with the appropriate Lorentz densities via the t p approximation. However, for the kinematic region of interest, we consider it inappropriate to employ microscopic t p optical potentials, the reason being that HLF parameter sets only exist at 135 MeV and 200 MeV, whereas optical potentials for the outgoing protons are required at energies ranging between 24 and 170 MeV. Thus, employing microscopic potentials involves large, and relatively crude, interpolations and/or extrapolations, leading to inaccurate predictions of the analyzing power, as was reported in Ref. 1. One of the aims of this paper is t o investigate the predictive power of a simple relativistic model which eliminates the above-mentioned uncertainties. Firstly, we consider a zero-range approximation to the relativistic DWIA for which one needs to evaluate less complicated three-dimensional integrals. Secondly, we use appropriate global Dirac optical potentials, as opposed to microscopic t p optical potentials, for obtaining the scattering

21 5

wavefunctions of the Dirac equation. Thirdly, we directly employ the more accurate experimental NN amplitudes rather than adopt the limited HLF parameter sets. In particular, we focus OIL the ability of our zero-range relativistic model to predict the analyzing power for the knockout of protons from the 3s112 state in '08Pb at an incident energy of 202 MeV, and for coincident coplanar scattering angles (28.0",-54.6"). This reaction is mainly localized at the nuclear surface and, hence to first order, one can neglect density-dependent corrections to the NN interaction. Predictions are also compared to corresponding nonrelativistic results. It is well-known that spin observables are better discriminators of subtle physical effects than unpolarized cross sections and, hence these quantities provide the very stringent tests for theoretical models. Unfortunately published data on spin observables, other than the analyzing power, do not exist. However, despite this lack of data, and in anticipation of such measurements in future, we present the first relativistic predictions of complete sets of polarization transfer observables and, in particular, investigate the sensitivity thereof to different global optical potential parameter sets. We also study the influence of distortion effects on spin observables: current qualitative arguments suggest that, since spin observables are ratios of polarized cross sections, distortion effects on the scattering wave functions effectively cancel, and hence simple plane wave models (ignoring nuclear distortion) should be appropriate for studying polarization phenomena This claim, however, has never been studied quantitatively within the context of ($,2@) reactions and, hence we study this issue by comparing the distorted wave results, of complete sets of spin observables, to corresponding plane wave predictions for zero scattering potentials. Our choice of a heavy target nucleus, 'O*Pb, and a relatively low incident energy of 202 MeV, is ideally suited for maximizing the influence of distortion effects, while still maintaining the validity of the impulse approximation, and also avoiding complications associated with the inclusion of recoil corrections in the relativistic Dirac equation In Sec. 2, we briefly discuss the main ingredients of the zero-range approximation to the relativistic DWIA. The spin observables of interest are defined in Sec. 3. Results and conclusions are presented in Secs. 4 and 5, respectively. 't3.

576.

21 6

2. Relativistic distorted wave model

For notational purposes, a general exclusive ( p , 2p) reaction is denoted by A(a,a'b)C, whereby an incident proton, a, knocks out a bound proton, b, from a specific orbital in the target nucleus, A, resulting in three particles in the final state, namely the recoil residual nucleus, C , and two outgoing protons, a' and b, which are detected in coincidence at coplanar laboratory scattering angles, and Ob, respectively. All kinematic quantities are completely determined by specifying the rest masses (mi) of particles, where i = ( a , A, a', b, C ) ,the laboratory kinetic energy (T,) of incident particle a , the laboratory kinetic energy (Tat) of scattered particle a', the laboratory scattering angles $,I and O b , and also the binding energy of the proton that is to be knocked out of the target nucleus. For a ZR approximation to the DWIA, the relativistic distorted wave transition matrix element is given by Oat

where T:gb and 0; represent the effective two-body laboratory kinetic energy and effective center-of-mass scattering angles, respectively, and 8 denotes the kronecker product. The four-component scattering + wave functions, $(F, k i , s i ) , are solutions to the fixed-energy Dirac equation with spherical scalar and time-like vector nuclear optical potentials: $I(+)(?, ;,A, 3,) is the relativistic scattering wave function of the incident particle, a , with outgoing boundary conditions [indicated by the superscript (+)], where k,A is the momentum of particle a in the ( a + A ) center-of-mass + system, and s, is the spin projection of particle a with respect to k a A as the .%quantization axis; $(-I (F, s j ) is the adjoint relativistic scattering wave function for particle j [ j = (u', b)] with incoming boundary conditions [indicated by the superscript (-)I, where zjc is the momentum of particle j in the ( j C) center-of-mass system, and s j is the spin projection of particle j with respect to &C as the ?-quantization axis. The boundstate proton wave function, ( T ) , with single-particle quantum numbers L, J , and M J , is obtained via selfconsistent solution to the Dirac-Hartree field equations of quantum hadrodynamics '. In addition, we adopt the impulse approximation which assumes that the form of the NN scattering matrix in the nuclear medium is the same as that for free NN scattering: the NN scattering matrix, ,"fi is parameterized in terms of five Lorentz invariants (scalar, pseudoscalar, vector, axial-vector, tensor), the so-called IA1

-.

&,

+

$FJMJ

21 7

representation] which are directly related to the nonrelativistic Wolfenstein amplitudes. Note that the periodic nature of the three-dimensional integrand, given by Eq. ( l ) ,ensures numerical stability and rapid convergence.

3. Spin observables The spin observables of interest are denoted by Ditj and are related to the probability that an incident beam of particles] a , with spin-polarization j induces a spin-polarization i‘ for the scattered beam of particles] a‘: the subscript j = ( O 1 e l n 1 ~ is) used t o specify the polarization of the incident beam a along any of the orthogonal directions

and the subscript i’ = (0, e’, n’,s‘) denotes the polarization of the scattered beam a’ along any of the orthogonal directions:

2

= 2’ =

katC

fi’=fi=$

5‘ = i’ - f i x ? .

(3)

The subscript i’ = j = 0 is used to denote an unpolarized beam. With the above coordinate axes in the initial and final channels, the spin observables, Dvj, are defined by

where DnO = P refers to the induced polarization, Do, = A , denotes the analyzing power, and the other polarization transfer observables of interest are D,,, D,!,, Dpe, D,te, and Det,. The denominator of Eq. (4) is related to the unpolarized triple differential cross section, that is, d3a dT,, do,! dRb

o(

Tr(TTt).

(5)

In Eq. (4), the symbols air and aj denote the usual 2 x 2 Pauli spin matrices] namely,

218

ffn

=

0,

=

(p ii)

and the 2 x 2 matrix T is given by

where sa = 5 ; and sat = ki refer t o the spin projections of particles a and a' along the i and i' axes, defined in Eqs. (2) and (3), respectively; the matrix TL";"' is related to the relativistic ( p , 2p) transition matrix element TLJMJ( S a , sat, s b ) , defined in Eq. (I), via

4. Results

First, we study the predictive power of our zero-range relativistic model by comparing predictions to data where previous relativistic and nonrelativistic models fail. For this purpose we focus on the analyzing power, A,, for the knockout of protons from the 3s1/2 state in 208Pb,at an incident energy of 202 MeV, and for coincident coplanar scattering angles (28.0', -54.6'). This reaction is localized in the nuclear surface ', and hence one does not expect the NN interaction to be significantly modified from the corresponding value in free space. We start by studying the sensitivity of the analyzing power to different relativistic global optical potential parameter sets. In particular we consider the following parameter sets from Cooper et al 7 : Fit 1, Fit2, EDAI-208Pb,EDAD-Fitl, EDAD-Fit 2, EDAD-Fit3. The variation of the analyzing power t o these different potentials is represented by the dotted band in Fig. 1. In general, one concludes that the analyzing power is relatively insensitive to different parameter sets, with differences between parameter sets being smaller than the experimental statistical error. Note that, whereas previous relativistic and nonrelativistic distorted wave models fail, our zero-range prediction, based on the impulse approximation, provides a perfect description of the data. Hence, one does not

21 9

d

To, (MeV) Figure 1. Analyzing power plotted as a function of the kinetic energy, T,,, for the knockout of protons from the 3s1/2 state in *08Pb, at an incident energy of 202 MeV, and for coincident coplanar scattering angles ( 2 8 . 0 ° , -54.6'). The dotted band represents the sensitivity of a particular observable to different relativistic global optical potential parameter sets (see text). The dotted line denotes the relativistic plane wave prediction, and the dashed line represents the nonrelativistic distorted wave prediction. The data are from Ref. 1.

need to invoke exotic medium corrections to the NN interaction in order to describe the analyzing power data. Next, we study the influence of relativistic nuclear distortion effects on the analyzing power by comparing our zero-range distorted wave prediction to the corresponding plane wave prediction (with zero scattering potentials): the dotted line in Fig. (1) represents the relativistic plane wave result. We see that the prominent oscillatory structure of the analyzing powers is mostly attributed to distortions of the scattering wave functions. This clearly illustrates the importance of nuclear distortion on the analyzing power, thus refuting, for the first time, qualitative claims that the analyzing power, being a ratio of cross sections, is insensitive to nuclear distortion effects. In Fig. 1, we also compare our relativistic distorted wave (dotted band) calculations to the nonrelativistic (dashed line) prediction reported in Ref. 1. It is clearly seen that our relativistic prediction is superior compared to the corresponding nonrelativistic calculation. This provides evidence for the preference of the analyzing power to relativistic Dirac dynamics. Moreover, this result represents the clearest signature of relativistic dynamics

220 to date. We now turn our attention to the other spin observables of in101

'

I

. , . , . , .

1

0.4 C

d

0.2

0.0 -0.2

90

110uo150nol90

0.9

-

0.3 Ld

'0 0.1

0

-'0

-

-

-0.1

0.5

n

t 3

.... .. .. .. .. ..

..

.. _. . . I

.

. .

0.1

-0.3

-0.7

.. .. 90

110

l30 150

no

0.4

0.2 Ld

d0.0 -0.2

90

110

uo150nol90

TOI(MeV)

-0.5'.

90

' . ' . 110 l30 150

no' . 190'

T, (MeV)

Figure 2. Complete sets of polarization transfer observables, D i ~ jplotted , as a function for the knockout of protons from the 3 ~ 1 1 2state in 20sPb, at of the kinetic energy, Tat, an incident energy of 202 MeV, for coincident coplanar scattering angles (28.0°, -54.6'). The hatched band represents the sensitivity of a particular observable to different relativistic global optical potential parameter sets (see text). The dotted line denotes the relativistic plane wave prediction, and the dashed line represents the nonrelativistic distorted wave prediction.

221 terest, namely the induced polarization ( P ) ,the depolarization parameter (Dnn), and the other polarization transfer observables (Dsrs, D,,e, Dpe and Dpe). Unfortunately published data do not exist for spin observables other than the analyzing power. However, in anticipation of such measurements in future, we present the first relativistic predictions of complete sets of polarization transfer observables. In Fig. 2, we investigate the sensitivity of these spin observables to different global optical potential parameter sets: the hatched band in represents the variation of the spin observables to these different potentials. In general, we see that all spin observables are relatively insensitive to different optical potential parameters. We also compare relativistic distorted wave (hatched band) to corresponding plane wave predictions (dotted line). As is the case for the analyzing power, we see the large influence of the distorting potentials on all spin observables: the oscillatory structure of all spin observables is mainly attributed to distortion effects. Finally, in Fig. 2, we compare relativistic distorted wave (hatched band) to nonrelativistic (dashed line) predictions of spin observables: the nonrelativistic Schrodinger-based calculations are based on the computer code THREEDEE by Chant and Roos 8. For all spin observables, large differences are observed between relativistic and nonrelativistic predictions. Hence, measurements of these quantities should provide further insight into the the role of nonrelativistic Schrodinger versus relativistic Dirac dynamics in intermediate energy nuclear physics.

5. Conclusions

We have developed the first relativistic distorted wave model, based on the zero-range approximation t o the NN interaction, for the description of complete sets of exclusive (9,29) polarization transfer observables. Whereas previous relativistic (finite-range) and nonrelativistic models fail to reproduce analyzing power data for the knockout of protons from the state in 'O*Pb - at an incident energy of 202 MeV, and for coincident coplanar scattering angles (28.0', -54.6') - our model provides a perfect description of the data in question. This successful prediction provides the most convincing evidence to date that the Dirac equation is the most suitable dynamical equation for the description of polarization phenomena in nuclear physics. We also note that, for the particular reaction of interest, the impulse approximation, which assumes that the form of the NN scattering matrix in the nuclear medium is the same as that for free NN scattering, is sufficient. Hence, it is not necessary to invoke density-dependent correc-

222

tions, to the free NN interaction, to describe the data. Encouraged by this initial success, we have extended our model to calculate additional spin observables, namely the induced polarization ( P ), the depolarization parameter (Dnn), and the polarization transfer observables (Dsjs, D,,t, Dpe and Dpt). In particular, we have studied the sensitivity of the latter observables, including the analyzing power, to different relativistic global optical potential parameter sets: generally, we see that all spin observables are relatively insensitive to different optical potential parameter sets. We have also studied the influence of relativistic nuclear distortion effects on spin observables by comparing our zero-range distorted wave prediction to the corresponding plane wave prediction (with zero scattering potentials): we see that the prominent oscillatory structure of the spin observables is mostly attributed to distortions of the scattering wave functions. This clearly illustrates the importance of nuclear distortion on all spin observables, thus refuting, for the first time, qualitative claims that spin observables, being a ratio of cross sections, are insensitive to nuclear distortion effects. In addition, our relativistic distorted wave predictions are compared to nonrelativistic distorted wave calculations: all spin observables are very sensitive to whether the Scrodinger- or Dirac-equation is employed for the underlying equation of motion. To further explore the role of different dynamical models in nuclear physics, and t o provide further constraints on existing models, it is very important to measure complete sets of polarization transfer observables: at present there are no published data for complete sets of exclusive ($,2$) polarization transfer observables. References 1. R. Neveling, A. A. Cowley, G. F. Steyn, S. V. Fortsch, G. C. Hillhouse, J. Mano and S. M. Wyngaardt, Phys. Rev. C 66,034602 (2002). 2. C. J. Horowitz and D. P. Murdock, Phys. Rev. C 37,2032 (1988). 3. C. J. Horowitz and J. Piekarewicz, Phys. Rev. C 50,2540 (1994). 4. C. J. Horowitz and B. D. Serot, Nucl. Phys. A368, 503 (1981). 5. E. D. Cooper, B. K . Jennings and 0. V. Maxwell, Nucl. Phys. A556, 579 (1993). 6. 0. V. Maxwell and E. D. Cooper, Nucl. Phys. A565, 740 (1993). 7. E. D. Cooper, S. Hama, B. C. Clark, and R. L. Mercer, Phys. Rev. C 47, 297 (1993). 8. N. S. Chant and P. G. ROOS,computer program THREEDEE, University of Maryland (unpublished).

223

STUDY OF IN-MEDIUM NN INTERACTIONS BY USING ( p , 2 p ) REACTIONS

T. NORO, H. SAKAGUCHI*, 0. v. M I K L U K H O ~s. , L. BELOSTOTSKI~,K. HATANAKA~,J. KAMIYA~,A. u. KISSELEF~,H. TAKEDA*, T. WAKASA~,Y. YASUDA*, AND H. P. YOSHIDA~ Department of Physics, Kyushu University, Fukuoka 812-8581, Japan E-mail: [email protected] *Department of Physics, Kyoto University, Kyoto 606-8502, Japan Petersburg Nuclear Physics Institute, Gatchina,188350, Russia $Research Center for Nuclear Physics, Osaka University, Ibaraki 567-004 7, Japan Present status of ( p , 2p) studies at incident energies of 392 MeV at RCNP and of 1 GeV at PNPI are presented. Based on several kinds of evidences, it is concluded that the long standing problem, the reduction of analyzing powers and polarizai tions, in this reaction is not caused by a mixture of preequilibrium processes. This fact and the fact that conventional DWIA calculations reasonably reproduce the experimental ( p , 2p) data leading to discrete states, where contribution from the nuclear interior region is not significant, suggest that this reduction is an appearance of some nuclear medium effect or correlation effect. A discussion is also given for a possible finite P - A, caused by a relativistic effect.

1. Introduction

Exclusive measurements of the nucleon knockout reactions] the ( p , 2p) r e actions, give a direct means t o study the nucleon-nucleon (NN) interaction in the nuclear field. In the nuclear field, the NN interaction is expected to be modified because of the Fermi motion and Pauli blocking effects. In addition to these effects, non trivial nuclear medium effects in the hadron level is theoretically predicted. An enhancement of the lower component of a Dirac spinor is predicted in a framework of QHD,' quantum hadrodynamics, and reductions of hadron masses are predicted in a framework of QCD.2i3>4It is one of challenges in current nuclear physics to look for evidences which is caused by these non trivial medium effects. For the proton quasi-free scattering, there is a long standing problem in analyzing powers A,, which are significantly reduced form values predicted

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by using the NN interaction in free space. This reduction in the ( p , 2 p ) reaction is firstly observed in a TRIUMF measurement5 for the reaction leading to the ls1/2 hole state of residual 15N nuclei. The measurement has been extended to several target nuclei at RCNP and found that the reduction monotonically depends on the effective mean density, which is estimated with a distorted wave impulse approximation (DWIA) .6 Since conventional non-relativistic DWIA calculations, even with the g-matrix as a NN interaction in the nuclear field, completely fail to reproduce this feature of the reduction, this phenomenon strongly suggests existing of some medium effect in the NN interaction (see the left panel of Fig. 3). In this article, we report on three kinds of studies related to investigations of the NN interaction in the nuclear medium. Firstly, in the next session, various kinds of experimental and analytic results are examined from the view point of reaction mechanism, a possibility of multi-step contribution to this reaction. Next, we examine the reliability of the DWIA by comparing the experimental data and calculations for knockout of protons from the orbits close to the Fermi surface, where the medium effect is expected t o be small. In section 4,a discussion is given on a possibility of finite P - A,, inequality between the polarization and analyzing power, caused by a relativistic effect. 2. Examination of the reaction mechanism In probing the nuclear interior by using a projectile which interacts with the strong interaction, it is essentially important to examine if or not, or what extent, multi-step processes disturb the probe. Here we present several evidences which show that they are not a major reason of the A , reduction.

Spectra f o r l s l / a - h o l e state and m o m e n t u m dependence In Figure 1, separation-energy spectra corresponding to the nucleon knockout from the l s l p orbit are shown. In the case of the left panel, where the recoil momentum ks is large, considerable yields exist in the region with large separation energies around 60 MeV, where the single-hole response is expected to be small. These yields are caused by pre-equilibrium processes and the spectrum implies that such processes also contribute in the region of the lsl/2-knockout bump. On the other hand, in the case of the right panel, the pre-equilibrium processes seem to give only a minor contribution to the bump region which corresponds to k3=0.7 This situation is more clearly shown in Fig. 2, where the cross sections integrated for all the bump region are plotted as a function of the recoil

225 Figure 1. Separation-energy spectra for the 12C(p,2p)"B reaction. The broad bump in each panel corresponds to the proton knockout from the orbit of the target nuclei. The values of the recoil momentum are given as k3.

i2C(p,2p)iiB

k,= 150 MeV/c

60 40 20 Separation Energy (MeV)

60 40 20 Separation Energy (MeV)

Figure 2. Recoil-momentum dependence of the cross section. The open circles are the integrated cross sections for whole of the 1s112 hole region and the closed circles are the values after subtracting the continuum parts shown as the dashed lines in Fig. 1.

/ o before continuum subtraction 0 after continuum subtraction -100

0

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momentum and a comparison with a DWIA calculation is given. When the pre-equilibrium contributions are subtracted, though with a rough way with dashed lines in Fig. 1, the experimental values are well reproduced by the calculation where single-hole responses are assumed. From these figures, the contribution of the pre-equilibrium processes is estimated to be only ten percent for the whole region of the k3=0 bump and less than a few percent in the peak region of the bump.

Estimation of pre-equilibrium processes with a hybrid model Cowley et al. have measured the 40Ca(p,p'p'') cross section for wide angle and energy regions of the outgoing protons, which include both of the quasifree region and the far outside of the quasi-free region.8 Then they compared the data with a calculation of a hybrid model for the multi-step processes where the first step is estimated with a DWIA and the distortion for one of outgoing protons 'is replaced by an experimental cross section for the ( p , p ' ) or ( n , p ' ) reaction to continuum states, which automatically include multi-step contributions. The result shows a reasonably good agreement for the whole of wide region with no free parameters.

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Since they also estimated the single-step cross sections with usual DWIA, the ratio of the pre-equilibrium processes to the quasi-free scattering is given in their paper. The result is about 25% for the 1s-knockout peak and 10% for the lp-knockout peak. Since this is a result for a mediumheavy target, we can expect much less ratio for a light target as 12C. Comparison of A,, P and D N N At 392 MeV, we have measured the polarizations P and the spin-transfer D", in addition to A,, for the ( p , 2p) reactions. The experimental data and a comparison with DWIA and PWIA calculations are given in Fig. 3 . Even though a little difference between A, and P exists, and the difference will be mentioned again later, both of them show primarily the same amount of reduction from the IA calculations. On the other hand, the measured DNNis reproduced by the calculations well. If the reductions of A, and P are caused by mixture of processes which are less spin dependent than the ( p , 2 p ) reaction, DNNshould also be reduced. These data show that the reduction is not dominantly caused by a mixture of such processes. Polarization data at 1 GeV Recently, we have measured the polarization P of outgoing protons in the ( p , 2p) reactions at 1 GeV by using the synchrocyclotron at PNPI, R ~ s s i a . ~ The result of the target dependence, effectively a density dependence is consistent with the result at 392 MeV. Namely, P is reduced from theoretical predictions as a function of the effective mean density. In addition, P of forward- and backward-outgoing protons give about the same amount of reduction, while those energies are quite different. The fact that consistent reductions are observed for different energies, where contributions of the multi-step processes are expected to be different, shows that the reduction

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is not originated by its reaction mechanism. From those evidences, we now conclude that the reductions are not dominantly caused by a mixture of multi-step processes. 3. Reliability of DWIA calculations

We have measured the cross sections and the analyzing powers for the ( p , 2p) reactions leading to low lying states, which correspond to knockouts from lp312 and 1p112 orbits of the ”C target and ld3/2 and 2 ~ 1 1 2orbits of the 40Ca target. In order to examine the reliability of DWIA calculations, measurements have been performed for various kinds of kinematics. Figure 4 shows a schematic contour map of the recoil momentum, as a function of outgoing angles of two protons, for a ( p , 2 p ) reaction. The kinetic energies of two outgoing protons are fixed. We chose three kinds of lines on this map and performed the measurement along these lines. Along each line, one of the detection angle two-body scattering angle 6” or the two-body relative energy ENNis kept almost constant. Since the outgoing energies are fixed, with minor deviations because of the recoil energies in actual reactions, we can use the same distorted waves for all of data points in DWIA calculations and ambiguities caused by distorted potential are minimized. As the forth kinds of kinematics, measurements are performed at a fixed angle pair with changing the detection energies. The result of the measurement for a 12C target is shown in Fig. 5. The dashed lines and solid lines are DWIA calculations with an empirical global potential and a theoretically calculated potential with a relativistic impulse approximation, respectively. The spectroscopic factors deduced from these comparisons are consistent with those deduced by using (e, e‘p) reactions within 20%, which is comparable with experimental errors in the absolute

Btab,

I

Figure 4. Schematic contour map of the recoil momentum for a (p, 2p) reaction. The 191 and 82 are the angles of two outgoing protons and the k3’ s are values of the recoil momentum. The measurement has been performed along the three lines with fixed proton energies and a t a fixed angle pair with changing detection energies of outgoing protons.

el

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cross sections. The dashed lines are results of the PWIA calculations. The data for 40Ca are not shown because of space, but the result is essentially the same as the 12C case. From these results, we conclude that the DWIA calculations predict the experimental data reasonably well except limited data points and that they are reliable at this energy. 4. Energy off-shell effect caused by a spinor distortion For the NN scattering in free space, the time reversal invariance requires that the polarization P and the analyzing power A, are equal. There is also an equality between two spin transfer coefficients, Dpq= -Dqp, where q and p are the directions of the transfered momentum and an average of incident and final momentum, respectively, in the center of mass system. In the case of quasifree scattering, these equalities are broken in general. Actually, the data in Fig. 3 show a finite difference between P and A,. The data are replotted in Fig. 6 as a function of the ( k - k ’ ) / ( k + k ’ ) , where k and k’ are the relative momenta of two protons in the initial and final channels, respectively. In the same figure, Dpq+Dqpvalues are also plotted and again a finite experimental value is given. Here we discuss about these inequalities caused by a distortion in the Schrodinger equivalent expression of the relativistic DWIA. A trivial distortion effect is given by distorting potentials which is the same as that in a non-relativistic framework. But this effect is negligibly small, at least for the kinematics corresponds to these data, as shown by the dashed lines in the figure. In the relativistic framework, there exists another kind of distortion effect, an enhancement of the lower component of the nucleon spinor in nuclear field. In the Schrodinger equivalent expression, this effect is taken into account as modifications of effective NN amplitudes. In the expression, the t-matrix of the NN scattering in the nonrelativistic framework is equated to the Dirac-spinor matrix elements of Lorentz invariant amplitudes. For the free NN scattering, the t-matrix is expressed with five terms as

under assumptions of parity and time reversal invariances. Here n refers t o a direction normal to the reaction plane, p and q refer to the same directions as mentioned above, and a1 and ~2 are the spin operators for two nucleons. Since the lower component of the Dirac spinor is a function of the nucleon momentum as well as the effective nucleon mass. the difference of

230

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Figure 6. Experimental data and theoretical estimations for the observables which vanish for elastic NN scattering. The dashed lines show a non-relativistic distortion effect and the solid lines represents an effect of relativistic distortion.

the momenta between the initial and final channels causes additional terms in the effective t matrix. Actually it is easily derived that the term D(k2 - k'2)(ul,a2,

+ Ulq02p)

appears if k#k'. This term is equivalent t o the D - t e r n proposed by KMT" and causes finite P-A, and Dqp+Dpqvalues. The value of this amplitude was estimated following a similar procedure t o that proposed by Horowitz and Iqball, by using a five-term approximation for the Lorentz invariant amplitudes. The result is plotted in the figure with solid lines. The plot shows that this off-shell effect causes significant deviations from zero for both of those observables. Even though the calculation gives the opposite sign for D,,+D,, and a refinement of the theoretical estimation will be required, the present result shows that this kind of relativistic effect is not negligible and, conversely, gives new information on reliability of the relativistic treatments of nuclear reactions. References 1. C. J. Horowitz and M. J. Iqbal, Phys. Rev. C33,2059 (1986). 2. G. E. Brown and M. Rho, Phys. Rev. Lett. 66,2720 (1991). 3. R. J. Furnstahl, D. K. Griegel, and T. D. Cohen, Phys. Rev. C46, 1507 (1992).

4. T. Hatsuda, Nucl. Phys. A544,27c (1992). 5. C. A. Miller et al., in Proceedings of the 7th International Conference on Polarization Phenomena in Nuclear Physics, (Paris, 1990) C6-595. 6. K. Hatanakaet al., Phys. Rev. Let. 78, 1014 (1997). 7. T. Nor0 et al., Nucl. Phys. A629,324c (1998). 8. A. A. Cowley et al., Phys. Rev. C57,3185 (1998). 9. T. Nor0 et al., in Proceedings of the International Nuclear Physics Conference (Berkeley, 2001) AZP Conf. Proc. 610 p.1034. 10. A. K. Kerman, H. McManus and R. M. Thaler, Ann. Phys. 8, 551 (1959).

23 1

DEPENDENCE OF THE COMPLETE SET OF SPIN TRANSFER COEFFICIENTS ON EFFECTIVE INTERACTION IN NUCLEAR MEDIUM

K. OGATA AND M. KAWAI Department of Physics, Kyushu University, Fukvoka 812-8581, Japan E-mail: kazu2scpOmbox.nc.kyushu-u.ac.jp

Y. WATANABE Department of Advanced Energy Engineering Science, Kyushu University, Kasuga, Fukvoka 816-8580, Japan SUN WEILI Institute of Applied Physics a n d Computational Mathematics, P. 0. Box 8009, Beijing 100088, China

M. KOHNO Physics Division, Kyushu Dental College, Kitakyushu 803-8580, Japan

Multistep direct reactions 40 Ca(p, p ' z ) a t 392 MeV and 40Ca(p,nz) at 346 MeV are analyzed including up to three-step process. The double differential inclusive cross sections and the complete set of spin transfer coefficients Dj, are calculated by the semiclassical distorted wave model and compared with experimental data. We use single particle wave functions in a Woods-Saxon potential incorporating the Wigner transform of a one-body density matrix and also introduce a phenomenological effective mass m* of a nucleon in the target. Analysis of Dj, in terms of an effective interaction in nuclear medium is also done.

1. Introduction

Pre-equilibrium multistep direct (MSD) processes' play important roles in nuclear reactions at intermediate energy. The semiclassical distorted wave (SCDW) model, which was proposed by Luo and Kawai2 in 1991, is one of the quantum mechanical models to describe MSD. In the last decade the SCDW model has been extensively developed to achieve good description of MSD, i.e., inclusion of two- and three-step p r o c e s ~ e sand ~~

232 the incorporation of a single particle wave function of target nuclei by using the Wigner transform (WT) of a one-body density matrix6. As a result, the SCDW model calculation can well reproduce experimental data of double differential inclusive cross sections (DDX) for ( p ,p’z) and ( p , n z ) on several target nuclei in the energy range from about 50 MeV to 200 MeV7. The SCDW model contains no free adjustable parameters, hence the model has a possibility of predicting experimental data of DDX that is unavailable at this stage; this is one of the most important aspects of this study. Besides study on DDX, we made analyses of spin o b s e r v a b l e ~ the ~~~, complete set of spin transfer coefficients Dij in particular, making use of an in-medium NN effective interaction”. Dij contain important information on an effective interaction in nuclear medium, which can be extracted by systematic and detailed studies on Dij. It was shown in Ref.[9], where oneand two-step processes were included and the local Fermi gas model (LFG) was adopted for nuclear wave functions, that the two-step process has quite large contribution to Dij for 40Ca(p,nz)at 346 MeV and 8, = 22O with large energy transfers w ,just the same as in DDX. Quantitative estimation of the contribution of multistep processes to Dij is crucial for detailed analyses of experimental data. The result in Ref.[9] indicates the need to evaluate the contribution from three-step process, which plays significant roles in the description of DDX5i6. Effects of the change of nuclear wave functions from the LFG to a single particle model in WS potential] hereafter referred to the WS model, on Dij are also interesting. In the present paper, we make SCDW model analyses of DDX for 40Ca(p,p’z) at 392 MeV and DDX and Dij for 40Ca(p,n z ) at 346 MeV and 200 MeV, including up to three-step process and with the WS model. In addition to that, we introduce a phenomenological effective mass m* of a nucleon in the target nucleus. The results are compared with experimental data and analyzed in terms of individual steps and effective interactions. In Sec.2, the SCDW model is reviewed very briefly arid inclusion of m* is described. The numerical results are presented and discussed in Sec.3 and a summary and conclusions are given in Sec.4.

2. The semiclassical distorted wave model

We start from the DWBA series expansion of the reaction T matrix. First we make the following assumptions: a) the single particle model for the target nucleus, b) sum of two-body potentials for residual interaction and c) the “never-c~me-back~~ assumption of MSD. Then the final states of the

233

system with different number of steps are different, hence one can treat individual steps separately. In the calculation of one-step DDX and Dii we make the local semiclassical approximation (LSCA) to the distorted waves: X ( r f s ) Z X(r)e f i k ( r ) . s7

(1)

where k ( r ) is the local wave number2 of the leading particle (LP) determined by the flux of X(r). The LSCA, which is the essential approximation to the SCDW model, is not an approximation for the distorted wave itself but for the evaluation of interference between X ( r ) and x(r’). It should be noted that we use a quantum mechanical X(r), hence the effects of diffraction and absorption are precisely included in the SCDW model, in contrast to so-called semiclassical calculations. For nuclear states, we use single particle wave functions in a potential of WS form, incorporating the W T of a one-body density matrix6. As a result of the approximations above, one can obtain formulae for one-step DDX and Dij of simple-closed formg. Extension of the one-step DDX and Dij formulae to the multistep processes has been made with the LSCA and incorporation of the W T as in the one-step process and the additional assumption of the eikonal approximation to the Green functions3. Explicit formulae of the DDX and Dij for the two- and three-step processes are given e l ~ e w h e r e ~ ~ ~ ~ In the SCDW model hitherto, the single particle energy of target nuclei E~ (y = a , p) has been replaced by its kinetic part h2k:/2m. In principle, however, E~ is governed by the following dispersion relation: E~ = h2k:/(2m) UL(r, k:,Ey), where UL is the local potential determined to be equivalent to the essentially nonlocal and energy-dependent one ( U N L ) ;the k-dependence of UL has its origin on the nonlocality of UNL. Expanding UL around k: = K : and E~ = EO and keeping terms up to the first order, one obtains E~ E h2k:/(2m*) UL(r, K : , E O ) , where m* is the effective mass; in this paper we assume a simple WS form for the r-dependence of m’ and disregard its dependence. This can be justified by the fact that we calculate the quasi-elastic scattering where huge number of nuclear states contribute to the whole inclusive processes.

+

+

3. Results and discussion 3.1. Input parameters

The input for the calculation are the same as in Ref.[9] except for the following things. First, we used the half-off-shell G matrix parameterized

234

by the Melbourne group" without the optical model modification" for the Bonn-B force13. Second, as for the single particle potential of the target nucleus, the global WS potential by Bohr and M ~ t t e l s o n 'was ~ adopted. Third, we assumed a WS form of m * ( r ) / mwith the same geometrical parameters as in the Bohr-Mottelson potential; we put m * / m at the center of the nucleus be 0.7 (0.8) for proton (neutron), referring Mahaux and Sartor 5 . 3.2. Results of cross sections and spin transfer coeficients

Figure 1 shows the calculated and measured16 DDX for 40Ca(p,p ' x ) at 392 MeV and the scattering angles ranging 25.5' to 120'. The dashed, dotted and dot-dashed lines represent the one-, two- and three-step cross sections, respectively. The solid line is the sum of them. One sees from the figure that the SCDW model calculation gives very good agreement with the 101)

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Figure 1. T h e energy spectra of DDX for 40Ca(p,p'z) a t 392 MeV for six emission angles, compared with the experimental16 ones. T h e DDX of one-, two- a n d three-step processes are represented by the dashed, dotted and dot-dashed lines, respectively. T h e solid line is the sum of them.

235 experimental data for wide ranges of the emission angles and energy transfers. At 100' and 120' the calculated DDX considerably underestimate the data, which may indicate that contributions of more than three-step processes and/or effects of the short-range correlation between nucleons in the target nucleus are important there. It should be noted that underestimation remains only in regions where the contribution of the three-step process exceeds those of the one- and two-step processes, hence inclusion of four-step process is expected not to spoil the good agreement below 80'. Next we show in Fig.2 the results of Dij for 40Ca(p,112) at 197 MeV, compared with the experimental data17. The solid, dotted and dashed lines represent] respectively, the results including up to three-, two- and one-step processes. Agreement between the calculation and the experimental data is very well for all components of the Dij. One sees that inclusion of the two-step process is essential to reproduce the data, in large w regions in

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H""l*k42. Additionally, the new operator depends on M 131. When converted into NN amplitudes, the new G-matrix elements require new spin operators. To make a DWBA calculation tractable using standard programs, we discarded the J’#J and 1. 1>2 elements, which were anyways small, and retained the M-dependence that has been shown to be potentially important [3]. This new feature of the G-matrix requires that we consider the direction of the quantization axis of the projectile when computing the blocking operator. For the calculation of the G-matrix elements, the axis lies along k,; for the NN effective interaction another choice is to use the direction of P = (k, + k2)/2. We show in Fig. 3 the results for both cases. Large changes are seen at 100 MeV for the natural-parity 2’ state in I6O when the spherical approximation is removed. (These results omit relativistic effects needed for the level of agreement seen in Fig. 2.) There is little dependence on the choice of quantization axis. For the 0’ 0- transition where non-local exchange is essential for obtaining any spin dependence, this quantization axis choice matters. Placing the axis along P helps with the agreement for the cross section. We conclude that the non-spherical treatment of the Pauli operator matters and needs to be considered along with other medium effects.

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Figure 3: Measurements of the cross section and analyzing power for the (p,p’) reaction to the first 2’ state in ‘‘0 at 100 MeV [6] and the T=O, 0- state in ‘‘0 at 200 MeV. The curves represent spherical BHF (short dash) and non-spherical Pauli calculations with the quantization axis along kl (long dash) or P (solid).

3.

Un-natural Parity Transitions

Unnatural-parity transitions offer a special opportunity to check the spindependent parts of the density-dependent effective interaction because of the one-to-one connection between linear combinations of polarization transfer coefficients and the square of the spin-orbit and tensor NN amplitudes [ 10,11]. High-spin stretched states offer the advantage of a simple particle-hole transition density that is well constrained by transverse (e,e’) data, but the large angular momentum transfer often places the transition density at radii where the average nuclear density is less than 20% of the full nuclear matter density. By contrast, low-spin transitions are more sensitive to density-dependent effects but normally have a complicated particle-hole structure that requires careful interpretation. Nevertheless, considering some special cases is helpful. Zero crossings in the predominantly real isovector interaction can be compared against minima in the data. The spin-longitudinal crossing at 0.7 fm-’ appears as a minimum in the cross section for the T=l, O+ 0- transition in I6O while the crossing of the KMT F-amplitude [lo] at 2.2 fm-l appears in the D,

258 combination [2,10,11] of polarization transfer coefficients for a number of &?&2hd%!=l t3ansdm ' * ns kez Fig. 4). the^ is excellent agreement with the low-density values of the zero crossings in the amplitudes.

4 (fm-') Figure 4: Plots of the real part of the spin-longitudinal and KMT F-amplitudes [ 121 at zero (solid) and full (dashed) nuclear density, along with the cross section for the T=l, 0' 0-transition in I6O and the D, combination of polarization transfer coefficients for '% (X), % ( * ),and2%(*). D B H F c a k x h t h r ~ s a z e & c w n ~ € ) r & ~ n(%short dash, I 6 0 solid, and "Si long dash). The size of the spin-dependent isovector terms in the effective interaction a z well ~ mmdudat bw m enturn m w h e axchange dominates, as shown in the left panels of Fig. 5 for the polarized cross sections. The combinations of polarization transfer coefficients associated with each cross section select either the spin-orbit (1s) or one of the three tensor terms (q, n, or p depending on the orientation along which the tensor interaction flips spin) in the effective interaction. Good agreement means that the absolute size of the effective interaction is correct, as formfactors have been adjusted to reproduce (e,e') scattering. However, a decrease in the longitudinal to transverse ratio is clearly seen at momentum transfers of 2-3 fm-', as shown by the right panels where only the linear combinations of polarization transfer coefficients are retained to provide a relative comparison among three different targets. These two combinations rely on the same polarization transfer coefficient combinations that show difficulties for continuum excitations [ 131.

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Figure 5: Data and DBHF calculations for the T=l, I’ state in ‘*C (left panel) [I41 and the T=l stretched states in I%?(x, short dash), I60( ,d i l ) ,d 2 * S i( ,bng da&) [2].

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Figure 6: Measurements and DBHF calculations for the T=O, I’ transition in ‘*C [ 141 and the T=O, 6- transition in %i, both at 200 MeV [2]. The calculations are “hybrid.” In the isoscalar channel where the t-r i n e n firan hnd pxd-m-ge is small, a comparison to the polarized cross sections shows an &estimate in & W L ~ size of& t m s x t e r m , asinbfor I2cat q < I fm-’ and j,6for *‘Si at somewhat higher q. This discrepancy is larger than the problems noted earlier for the isovector channel. It has been suggested by Brown and Rho [15] that the effective masses of the heavier mesons are scaled downward in the nuclear medium. However, in the isovector channel, the loss of tensor attraction in not visible, in part because

260

of the low density of the stretched spin transitions where the signature would be most clean [2]. Any rescaling can be accommodated in the isoscalar channel because natural-parity transitions are sensitive only to the balance between the aU .' ' &I a n md the ~ u k i k h e s o contributions n and not their size. Tests of the tensor spin dependence with the non-spherical Pauli operator showed effects that were much smaller than the differences noted above. The problems described here with the tensor interaction are most likely to be a medium effect because of the quality of the agreement between the free amplitudes and NN data is excellent and no other part of the DWBA integral contains the necessary spin degrees of freedom. Acknowledgments Collaboration with Francesca Sammarruca and Ron C. Johnson on the results reported here is gratefully acknowledged. The author acknowledges financial support under NSF grant PHY-0 100348. References 1. F. Sammarruca, E.J. Stephenson, and K. Jiang, Phys. Rev. C60, 064610 ( 1999). 2. F. Sammarruca, et al., Phys. Rev. C61,014309 (1999). 3. F. Sammarruca, X. Meng, and E.J. Stephenson, Phys. Rev. C62, 014614 (2000). 4. F. Sammarruca and E.J. Stephenson, Phys. Rev. C64, 034006 (2001); F. Sammarruca, contribution to this conference. 5. R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989). 6. H. Seifert, et al., Phys. Rev. C47, 16I5 (1993). 7. R.J. Furnstahl and S.J. Wallace, Phys. Rev. C47, 2812 (1993). 8. J.J. Kelly etal., Phys. Rev. C43, 1272 (1991). 9. W.G. Love and M.A. Franey, Phys. Rev. C24, 1073 (1981). 10. J.M. Moss, Phys. Rev. C26,727 (1982). 11. E. Bleszynski, M. Bleszynski, and C.A. Whitten, Jr., Phys. Rev. C26,2063 (1982). 12. A.K. Kerman, H. McManus, and R.M. Thaler, Ann. Phys. (N. Y.) 8 , 5 5 1 (1959). 13. T.N. Taddeucci et al., Phys. Rev. Lett. 73,335 16 (1994). 14. A.K. Opper ef a].,Phys. Rev. C63,0346 14 (2001). 15. G.E. Brown and M. Rho, Phys. Rev. Lett. 66,2720 (1991).

261

PROBING MEDIUM EFFECTS ON THE NUCLEON-NUCLEON INTERACTION IN NUCLEAR MATTER AND NUCLEI

FRANCESCA SAMMARRUCA* Physics Department, University of Idaho Moscow, ID 83844-0903, USA E-mail: [email protected]

When searching for medium effects, it is important to look broadly. Some approximations/reductions typically applied in many-body theories may not be entirely justified and should be properly examined. Within these scopes, we have explored the role of A isobars in medium modifications of the effective interaction. As a separate but related effort, we have recently developed a microscopic calculation of asymmetric nuclear matter properties. This will allow future applications to asymmetric nuclei.

1. Introduction

This presentation will consists of two main parts: (1) I will review some aspects of our previous work, where we have used (p,p’) reactions as a tool to detect and characterize the effects of the nuclear medium on the nucleon-nucleon (NN) interaction. In particular, I will discuss how the presence of A isobars in the medium generates a stronger density dependence, especially for the central isoscalar interaction, and what we can learn from these observations. (2) Most recently, we have been concerned with the treatment of the nuclear medium when protons and neutrons have different densities. I will present and discuss our results for the asymmetric matter equation of state (EOS). This work will naturally merge into applications to asymmetric nuclei, which will be one of the focal points for nuclear physics in the near future.

*Supported in part by the US Department of Energy under grant No. DE-FG0300ER41148.

262 40. 0.8 h L

v)

\

10.

0.4

-0

E v

G

4.0 0.0

F

1.0

U

0.4

0

A

-0.4 -0.8

0.1

0

20

40

0

20

40

ec.m.(deg) Figure 1 Cross section and analyzing power for (p,p’) scattering at 180 MeV to the 2+ state in 28Si at 1.779 MeV. The curves are as described in the text. The data are from Ref. [4].

2. Exploring Medium Effects Through (p,p’) Reactions

It has been the purpose of our recent work to investigate systematically the signature of the medium as seen through proton induced reactions. We have considered medium effects1*2which originate from Pauli blocking, nuclear binding, and the presence of strong relativistic scalar and vector mean fields included through the Dirac-Brueckner-Hartree-Fock (DBHF) approach to nuclear matter. Elastic proton scattering and natural-parity inelastic transitions are sensitive to these medium effects mostly through the isoscalar central and spin-orbit terms in the effective NN interaction, and thus provide a suitable environment for the evaluation of its density dependence. Within these broad objectives, in a recent work3 we have moved beyond the one-boson-exchange (OBE) picture by including A-isobar degrees of freedom in the baseline NN interaction. We use a coupled-channel model which involves two-meson-exchange diagrams with 7r and p through A intermediate states. In the OBE model, the intermediate range attraction is entirely parametrized through the (T boson. Here, the two-pion exchange diagrams provide about 50% of the intermediate-range attraction, whereas the other 50% is associated with the correlated two-pion contributions and is described by the boson, a scalar boson having approximately one-half the strength of the typical OBE (T. On the other hand, the 7r-p diagrams are (predominantly) repulsive and shorter-ranged. Thus they suppress the

263

two-pion contributions at short distances. Consistent with our previous choices,l we use Thompson-type relativistic two-baryon propagators and apply the pseudo-vector coupling for the pion. The A diagrams are modified in the medium and become strongly density dependent due to the following effects: 1) dispersive effects on the baryon propagators (for both N and A), 2) Pauli blocking on the NA intermediate states, and 3) medium modifications of the Dirac structure for the nucleons involved in those diagrams. We find that the main effect of A states in the medium is t o make the central, isoscalar term in the effective NN interaction more repulsive as a function of increasing density. The consequences of that are apparent in Fig. 1 for 180-MeV proton scattering to the 2+ state in 28Si a t 1.779 MeV, a transition representative of those with significant contributions from the nuclear interior. The interactions used for the different curves are based on the free space (short dash) interaction, DBHF (solid) medium effects, and a Brueckner-Hartree-Fock (BHF) calculation (medium dash). In all cases, the baseline model is the coupled-channel. As we observed previously,lY2 the density dependence of both BHF and DBHF medium effects makes repulsive contributions to the central interaction in the nuclear interior, an enhancement that is particularly large with the coupled-channel model. While observing that the addition of this model component does not improve the description of the natural-parity states, we emphasize again the importance of having a picture that is as complete as possible. Although the role of A isobars for the nuclear force is well established, the theoretical handling of such states within the many-body system may still be incomplete. It has been argued in the literature5 that attractive many-body forces (implied for consistency by the presence of A isobars) would almost entirely cancel the (repulsive) medium effects of those A diagrams. Thus, the latter may need to be examined in a broader context which includes the consideration of many-body forces. Finally, only a complete scenario of medium effects on two-body interactions can facilitate our understanding of other aspects of the many-body problem. Furthermore, it is important to keep in touch with recent advances in the NN sector (e. g. NN potentials based on chiral perturbation theory), since the characterization of medium effects can depend on the nature of the chosen two-body force.6

264

3. The Equation of State for Asymmetric Nuclear Matter The relative simplicity of a homogeneous infinite system makes nuclear matter calculations a convenient starting point for the determination of an effective interaction suitable for finite nuclei. A calculation of symmetric matter is, in fact, the starting point for the effective interactions used in our previous work with symmetric nuclei. Recently, much interest has developed around the study of asymmetric nuclei (in particular, extremely neutron-rich nuclei, and halo nuclei). Rare Isotope Accelerator (RIA) facilities will allow the study of the unique systems which populate the boundaries of the nuclear chart. Thus, it is important and timely to develop microscopic effective interactions which can account for the asymmetry in proton/neutron densities. With that in mind, we have calculated the equation of state for asymmetric matter. We use realistic NN forces and operate within the DBHF framework. In our self-consistent approach, a calculation of nuclear matter properties yields, at the same time, a convenient parametrization of the density dependence in the form of nucleon effective masses. This information can then be used to calculate the scattering matrix at some positive energy, and thus an effective interaction suitable, for example, for proton scattering on asymmetric nuclei. This interaction will be “isospin dependent” , in the sense of being different in the nn, p p , or n p cases. Applications to scattering will be the next step in our pursuit. Asymmetric nuclear matter can be characterized by the neutron density, p n , and the proton density, p p . It is also convenient to define the total density p = pn p p and the asymmetry parameter a = E Pc k . Clearly, a=O corresponds to symmetric matter, and a=l to neutron matter. In terms of cy and the average Fermi momentum, related to the total density in the usual way

+

p=-

2k$ 3iT2

the neutron and proton Fermi momenta can be expressed as

k;4 = k F ( l +

cy)ll3,

and

We use the Thompson relativistic three-dimensional reduction of the Bethe-Salpeter equation. The Thompson equation is applied to nuclear

265 50 I

I

40 c

g

20

aJ

-20

c

0.9

4 1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Figure 2. Energy per nucleon as a function of the Fermi momentum at different levels of asymmetry from symmetric matter to neutron matter.

matter in strict analogy to free-space scattering and reads, in the nuclear matter rest frame,

(4) +

where ij=nn, p p , or n p , and I$ 41, and K are the initial, final, and intermediate relative momenta. The momenta of the two interacting particles in the nuclear matter rest frame have been expressed in terms of their relative + momentum and the center-of-mass momentum, P . The asterix signifies that medium effects are applied to those quantities. The energy of the two-particle system is

qP'E7) = e,r(P,Z)+e;(P,l?)

(5)

and ( E & ) o is the starting energy. The single-particle energy er includes kinetic energy and the nuclear matter potential, to be determined. Q is the

266

Pauli operator for asymmetric matter and prevents scattering to occupied states. The goal is to determine self-consistently the nuclear matter singleparticle potential which, in our case, will be different for neutrons and protons. To facilitate the understanding of the numerical procedure, we will use a schematic notation for the neutron/proton potential. That is, we write un = u n p

+ unn

(6)

=upn

+u p p

(7)

and, for protons u p

where each of the four pieces on the right-hand-side of Eqs.(6-7) depends on the appropriate G-matrix (nn,p p , or n p ) from Eq.(4). Clearly, the two equations above are coupled through the n p part and so they must be solved simultaneously. Furthermore, the G-matrix equation and the two equations above are coupled through the single-particle energy (which includes the single-particle potential). So we have three coupled equations to be solved self-consistently. As done in the symmetric case,7 we parametrize the single particle potential for protons and neutrons in terms of two constants, US,^ and U V , ~(the , scalar and vector potential), through

To facilitate the connection to the usual non-relativistic framework: it is customary to express those in terms of two other constants, defined by

+ US,^

(9)

US,^ + U V , ~

(10)

rn: = mi and U0,i=

The subscript “2’ signifies that these parameters are different for protons and neutrons. Starting from some initial values of m; and U O , ~the , Gmatrix equation can be solved and a first solution for U i ( k i ) is then obtained. These solutions are again parametrized in terms of a new set of constants, and the procedure is repeated until convergence is reached. Finally, the energy per neutron or proton in nuclear matter is calculated as

Ei =<Ti > i- < Ui >

(11)

267

15 0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

kF(frn-l) Figure 3. Nuclear symmetry energy as a function of the Fermi momentum. The solid line is the prediction from the DBHF calculation, whereas the dashed line is obtained with the conventional Brueckner approach.

The EOS, or energy per nucleon as a function of density, is then written as

or e ( k p , c r )=

+

(1 a)en

+ (1 - a)Ep 2

The NN potential used in this work is the relativistic OBEP from Ref. which uses the Thompson equation and the pseudo-vector coupling for the n and 7 mesons. The EOS as obtained from our DBHF calculation is displayed in Fig. 2 as a function of k F and for values of a between 0 and 1. The symmetric matter EOS saturates at kF x 1.4fm-1 with a value of 16.7 MeV, in good agreement with the empirical values. As the neutron density increases (the total density remaining constant), the EOS becomes more repulsive and the minimum shifts towards lower densities. As the system moves towards neutron matter, the “energy well” becomes more and more shallow, until, for (Y larger than 0.8, a bound state no longer exists. We also find that the EOS is linear as a function of a 2 ,that is d ( k ~ , a-) e ( k ~ , O= ) e,a 2 ,

(14)

268

or parabolic versus a. This tendency is shared with the non-relativistic predictions. The nuclear symmetry energy, e,, is shown as a function of density in Fig. 3, where the solid curve is the prediction from the DBHF model and the dashed corresponds to a conventional Brueckner calculation. The curves differ considerably at high density. Empirical constraint on the high-density behaviour of the nuclear symmetry energy (for instance, from studies of energetic reactions with heavy neutron-rich nuclei), would help discriminate between models. In summary, we have developed a microscopic model for the equation of state for nuclear matter when protons and neutrons have different Fermi momenta. The calculation is self-consistent and parameter-free, in the sense that no parameters of the NN force are adjusted in the medium. We hope this effort will complement experimental studies of nuclei with high levels of asymmetry which will take place at RIA facilities in the near future.

Acknowledgments

D. Alonso (UI) is gratefully acknowledged for her contribution to this work. References 1. F. Sammarruca, E.J. Stephenson, and K. Jiang, Phys. Rev. C60, 064610 (1999). 2. F. Sammarruca et al., Phys. Rev. C61, 014309 (2000). 3. F. Sammarruca and E.J. Stephenson, Phys. Rev. C64, 034006 (2001). 4. Q. Chen et al., Phys. Rev. C41, 2514 (1990). 5. H. Miither, Prog. Part. Nucl. Phys. 14, 123 (1984); W.H. Dickoff and H. Miither, Nucl. Phys. A473, 394 (1987). 6. F. Sammarruca, D. Alonso, and E.J. Stephenson, Phys. Rev. C65, 047601 (2002). 7. R. Brockmann and R. Machleidt, Phys. Rev. C42, 1965 (1990). 8. M.I. Haftel and F. Tabakin, Nucl. Phys. A158, 1 (1970). 9. R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989).

269

EXTRACTION OF NEUTRON DENSITY DISTRIBUTIONS FROM PROTON ELASTIC SCATTERING AT INTERMEDIATE ENERGIES

H. TAKEDA, H. SAKAGUCHI, S. TERASHIMA, T. TAKI, M. YOSOI, M. ITOH, T. KAWABATA, T. ISHIKAWA, M. UCHIDA, N. TSUKAHARA, Y. YASUDA Department of Physics, Kyoto University, Kyoto 606-8502, Japan

T. NORO, M. YOSHIMURA, H. FUJIMURA, H.P. YOSHIDA, E. OBAYASHI Research Center for Nuclear Physics, Osaka University, Osaka 567-0047, Japan

A. TAM11 Department of Physics, University of Tokyo, Tokyo 113-0033, Japan

H. AKIMUNE Department of Physics, Konan University, Kobe 658-8501, Japan Cross sections, analyzing powers and spin rotation parameters of proton elastic scattering from 58Ni and 120Sn have been measured at 200-400MeV. Obtained data have been analyzed in the framework of relativistic impulse approximations. In order to explain the 58Nidata, it was necessary to modify N N interactions in the nuclear medium by changing coupling constants and masses of 0 and w mesons. For 120Sn, by assuming the same modification of N N interactions and by using proton densities deduced from charge densities, the neutron density distribution was searched so as t o reproduce 120Sn data a t 300MeV.

1. Introduction

Research fields in nuclear study are remarkably spreading due to the recent developments of radioactive isotope beam facilities all over the world. Nuclei far from the stability line are expected to be different not only quantitatively but also qualitatively. For instance neutron rich unstable nuclei are expected to have anomalous structures such as neutron skin and halo. Neutron distributions in nuclei will provide fundamental information for nuclear structure study. Thus it is indispensable to establish procedures

270

to extract neutron density distributions from experimental information. Protons at intermediate energies are considered to be suitable to extract information inside the nucleus because of the large mean free path in the nuclear medium. Ambiguities due to the target nuclear structure are relatively small in the elastic scattering since the ground state wave functions used for elastic scattering are restricted by charge distributions measured by electron scattering. Although it is hard to obtain neutron distributions from charge distributions, they can be assumed to have same shapes as protons for N N Z nuclei. Thus the proton elastic scattering at intermediate energies has been used to discern various microscopic approaches for nuclear interactions. Applying these models to N # 2 nuclei, neutron density distributions can be extracted from proton elastic scattering data. 2. Experiment

We measured cross sections, analyzing powers and spin rotation parameters of proton elastic scattering from 58Ni and 12'Sn at Ep = 200- 400 MeV. Scattering angles were up to 60' for measurements of cross sections and analyzing powers, and up to 45" for spin rotation parameters. Experiment was performed at Research Center for Nuclear Physics (RCNP). Polarized protons were injected into the pre-accelerator AVF cyclotron, transported to the six sector ring cyclotron, accelerated to final energy of 200-400 MeV and bombarded on the target. Scattered protons were momentum analyzed with a high momentum resolution magnetic spectrometer 'Grand Raiden' and detected by counters at the focal plane. Momenta of scattered protons were measured by detecting their positions with vertical drift chambers. Polarizations of scattered protons were determined using the focal plane polarimeter (FPP), which measured scattering asymmetries in a carbon analyzer block with multi-wire proportional chambers. Scintillators and hodoscopes were used to trigger the data acquisition and for particle identifications. 3. NN Interactions in Medium

Obtained data were analyzed in the framework of relativistic impulse a p proximations (RIA). It has been pointed out' that the RIA model with density dependent coupling constants and masses of exchanged (T and w mesons has been able to explain cross sections and analyzing powers of proton elastic scattering from 58Ni at intermediate energies. In that analysis the neutron distribution has been assumed to be same as protons deduced

27 1

Figure 1. Cross sections, analyzing powers and spin rotation parameters of elastic scattering at 200- 400 MeV.

from the charge distribution except for the normalization factor N / Z . Figure 1 shows the experimental results and some model calculations of cross sections, analyzing powers and spin rotation parameters of 58Ni. Solid circles are our data. Dotted and dashed curves are original RIA calculations developed by Horowitz et aL2 and Wallace et aL3, respectively. Solid curves indicate the medium modified RIA model described above. Modifications of coupling constants and masses are parametrized as

where j refers to the D or w mesons and PO stands for the normal density. Newly measured spin rotation parameters are also well explained by the medium modified RIA model. Figure 2 shows X2-map in (b,, b,) parameter space for 300 MeV. Other six parameters were optimized at each grid point. Very strong correlation between b, and b, is indicated by a narrow valley. Same correlations can be found between a, and a, and in other energies also.

4. Neutron Distribution Search and Discussion For N # Z nuclei such as 12’Sn it can not be expected that the neutron distribution has the same shape as protons. However, the elastic scatter-

272

x2 map (300MeV)

Figure 2. X2-map in (b,,, b,) parameter space for 300 MeV.

Figure 3. in lzoSn.

Deduced neutron distribution

ing is sensitive t o both NN interactions in nuclear medium and density distributions of the target nucleus, the neutron density distribution can be extracted from elastic scattering, assuming the same medium modifications fixed by the 58Ni data. In order t o search the neutron distribution we used a sum of Gaussians (SOG) type distribution;

Normalization condition !pn(r) d3F = N results in the constraint for Qz (CQi= 1). Qi are searched so as t o reproduce lZ0Sn data a t 300MeV, whereas width y and position Ri of each Gaussian are fixed with the values listed in a reference4. All resulting SOG distributions with good reduced

xz (= x 2 / 4 2 xv

2

5 Xvrnin + 1 7

(4)

where v is the number of degrees of freedom and x ; is~the~minimum ~ value of xz, are possible densities and their superposition determines an error band, which is displayed in Fig. 3 as shaded area. Deduced neutron distribution has a bump structure a t the nuclear center. This result is consistent with the wave function of neutrons in 3s1p orbit as expected t o be occupied in lZ0Snnuclei. Solid curves in Fig. 4 are the calculations using the best fit neutron density. Original unmodified RIA calculations with

273

.i

N

x

OcM [degree] Figure 4. Cross sections and analyzing powers of proton elastic scattering from lzoSn at E p = 200 - 400MeV.

relativistic Hartree densities are also displayed in Fig. 4 by dotted curves.

It is notable that our data indicated by solid circles are well explained by the deduced density a t all energies although the density search was performed with 300MeV data only. The difference of the neutron and proton root mean square radii is evaluated as ArnP = 0.116 f 0.O15fm1 which agrees with a result deduced from the SDR sum rule5. In order t o estimate the uncertainty in the deduced distribution due to the ambiguities in our medium modification parameters, we also searched the distributions using various parameter sets indicated as ‘setl’ t o ‘set4’ in Fig. 2. xz of the ‘setl’ and ‘set2’ parameters are about x ; 1, ~ while ~ xz N xzmin+ 2 for the ‘set3’ and ‘set4’ parameters. Left part of Fig. 5 shows the distribution deduced with the ‘setl’ and ‘set2’ parameters. Changes of the distributions are relatively small compared to the error band. However changes become larger if we use the ‘set3’ and ‘set4’ parameters as displayed

+

~

274 0.15

best

e set3

Radius (fm)

Figure 5. Left part shows the distributions deduced with ‘setl’ and ‘set2’ parameters. Deduced distributions with all sets in Fig. 2 are superposed in the right part.

in the right part of Fig.5. In other words, we can obtain the neutron distribution with small errors if the NN interaction in medium are well determined. 5. Acknowledgments We would like t o thank Prof. Hatanaka and the staff members of the RCNP for their support and tuning clear and high intensity beam during the experiment.

References 1. H.Sakaguchi et al., Phys. Rev. C57, 1749 (1998), and references therein. 2. D. P. Murdock and C. J. Horowitz, Phys. Rev. C35, 1442 (1987); C . J. Horowitz et al., Computational Nuclear Physics 1, Springer-Verlag, Berlin, 1991, Chap. 7. 3. J. A. Tjon and S. J. Wallace, Phys. Rev. C32, 1667 (1985); Phys. Rev. C36, 1085 (1987). 4. H.de Vries et al., Atomic Data and Nuclear Data Tables 36, 495 (1987). 5. A.Krasznahorkay et al., Phys. Rev. Lett. 82, 3216 (1999).

275

MEDIUM02 SCIENTIFIC PROGRAM

- October 25 (Friday) Opening

9:OO - 9105

T. Nor0 (Kyushu Univ.) Opening address

Session I (Nuclear matter) Chair: T. Nor0

9~05- 10~25 35 min

V.R. Pandharipande (Univ. Illinois) Quenching of weak interactions in nucleon matter T. Tatsumi (Kyoto Univ.) Pseudo-scalar mesons in medium

25 rnin

Coffee break

lor25 - 0 5 5

Session I1 (Pion condensation, compressibility, pionic states) Chair: W.H. Dickhoff

1055 - 2:20

M. Nakano (Univ. Occupational and Environmental Health) Pion condensation based on relativistic formulation M. Itoh (RCNP) Compressional-modegiant resonances in deformed nuclei M. Uchida (Kyoto Univ.) Isoscalar giant dipole resonance and nuclear incompressibility T. Wakasa (RCNP) High resolution study of mesonic 0 states in I6O

20 min

15 rnin 10 min

15 min

Lunch

12~20- 1350

Session I11 (Relativistic effects, dibaryon, NN interactions) Chair: G.C. Hillhouse

1350 - 15:30

T. Suzuki (Fukui Univ.) Medium effects on the coulomb sum value

20 min

276

M. Gaidarov (INRNE and Kyushu Univ.)

15 rnin

Y-scaling analysis of the deuteron within the light-front dynamics method A. Tamii (Univ. Tokyo) Search for super-narrow dibaryon resonances by the pdO pdXandpdO ppX Y. Fujiwara (Kyoto Univ.) The quark-model NN interaction and its application to three-nucleon and nuclear-matter problems

15 rnin

25 rnin

Coffee break

1 5 ~ 3 -0 16:OO

Session IV (Few body system) Chair: M. Kamimura

1 6 ~ 0 -0 1 8 ~ 0 0

E. Epelbaum (Univ. Paris) Quark (pion) mass dependence of nuclear forces K. Sagara (Kyushu Univ.) Different types of discrepancies in 3N systems K. Sekiguchi (RIKEN) Measurement of d-p elastic scattering at intermediate energies and three-nucleon force Y. Shimizu (RCNP) Three-body effects in pd elastic scattering at 250 MeV Y. Maeda (Univ. Tokyo) Study of three-nucleon-force via pol-n + d elastic scattering at 250 MeV T. Ishida (Kyushu Univ.) Search for anomaly around space star configuration in pd reaction Party

20 min 20 min 10 min 10 min

10 min 10 min

19:OO -

277

- October 26 (Saturday) Session V (Four body and 'He scattering) Chair: K. Amos

9:oo - 11:oo

Yu.N. Uzikov (Dubna) Proton-'He backward elastic scattering at intermediate energies K. Hatanaka (RCNP) Measument of the spin correlation parameters of the backward p3He scattering H. Kamada (Kyushu Inst. Tech.) Polarization observables in the 4N scattering with the 3N calculations E. Uzu (Tokyo Univ. of Science) Faddeev-Yakubovsky calculation above 4N break-up threshold T. Saito (Univ. Tokyo) Measurement of the analyzing powers for the d+d ->'He+n and d+d -> 3H+preactions at intermediate energies J. Kamiya (RCNP) Study of the spin dependent 3He-nucleusinteraction at 450 MeV

25 rnin 15 rnin 15 rnin 10 min 10 min

10 min

Coffee break

11:OO - 11125

Session VI (Nuclear correlations) Chair: M. Yosoi

11:25 - 12:40

W.H. Dickhoff (Washington Univ.) Properties of nucleons and their interaction in the nuclear medium K. Yak0 (CNS) Determination of the Gamow-Teller quenching factor via the gOZr(n,p)reaction at 293 MeV M. Ichimura (Hosei Univ.) Comment on GT sum rule with delta isobar Y. Nakaoka (Univ. Tokyo): presented by M. Ichimura Two-step effects in analysis of nuclear responses

25 min 10 min

5 min 15 min

27 8

Lunch

12140- 14~10

Session VII (Quasi-free scattering) Chair: H. Sakaguchi

14:lO - 15~45

G.C. Hillhouse (Univ. Stellenbosch) Relativistic predictions of complete sets of polarization transfer observables for exclusive (p,2p) reactions T. Noro (Kyushu Univ.) Study of in-medium NN interactions by using (p,2p) reactions K. Ogata (Kyusyu Univ.) Dependence of the complete set of spin transfer coefficients on effective interaction in nuclear medium Coffee break

25 min 20 min 20 min

15:45 - 16~10

Session VIII (Nucleon-Nucleon Interactions and medium effects) 16:lO - 1750 Chair: G.P. Berg K. Amos (Melbourne Univ.) Nucleus-hydrogen scattering - a probe of neutron matter E. Stephenson (IUCF) A Tool to study the effective interaction in the medium F. Sammarruca (Idaho Univ.) Probing medium effects on the nucleon-nucleon interaction in nuclear matter and nuclei H. Takeda (Kyoto Univ.) Medium effects and neutron density distributions via proton elastic scattering Closing H. Sakai (Univ. Tokyo) Closing remarks

25 min 20 min 20 min

10 min

17150-

279

Evgeni EPELBAUM Universite Paris SUD

List of Participants

Ken AMOS School of Physics University of Melbourne Victoria 3010 Australia [email protected]

Shun ASAJI Department of Physics Kyushu University Hakozaki 6- 10-1 Higashiku Fukuoka 812-8581, Japan [email protected]

Georg P. BERG Kernfysisch Versneller Instituut Zernickelaan 25 9747 AA Groningen The Netherlands [email protected]

Willem H. DICKOFF Departmetn of Physics Washington University C.B. 1105, One Brookings Dr. St. Louis MO 63130-4899, USA [email protected]

Institut de Physique Nucleaire Bat. 100 91406 ORSAY Cedex France [email protected]

Kunihiro FUJITA RCNP Osaka University Mihogaoka 10-1 Ibaraki Osaka 567-0047, Japan [email protected]

Yoshikazu FUJIWARA Department of Physics Kyoto University Kitashirakawa Oiwake-cho 1-1 Sakyo-ku Kyoto 606-8502, Japan [email protected]. ac .jp

Mitko K. GAIDAROV Department of Advanced Energy Engineering Science Kyushu University Kasuga Fukuoka 816-8580, Japan [email protected]

Shintarou HASHIMOTO Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected]

280 Kichiji HATANAKA RCNP Osaka University Mihogaoka 10-1 Ibaraki Osaka 567-0047, Japan hat anakaQrcnp.Osaka-u .ac .j p

Greg C. HILLHOUSE Department of Physics University of Stellenbosch Private Bag X1 Matieland 7602 South Africa gchQsun.ac.za

Munetake ICHMURA Faculty of Computer and Information Sciences Hosei University Kajino-cho 3-7-2,Koganei-shi Tokyo 184-8584, Japan ichimura@k. hosei.ac .jp

Takashi ISHIDA Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected]

Masatoshi ITOH RCNP Osaka University Mihogaoka 10-1 Ibaraki Osaka 567-0047, Japan itohQrcnp.osaka-u.ac..jp

Hiroyuki KAMADA DepaTtment of Physics Faculty of Engineering Kyushu Institute of Technology Sensui-cho 1-1, Tobata Kitakyushu 804-8550, Japan kaniadaQmns.kyutech.ac.jp

Masayasu KAMIMURA Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan karni2scpQmbox.nc.kyushu-u.ac..jp

Junichiro KAMIYA RCNP Osaka University Mihogaoka 10-1 Ibaraki Osaka 567-0047, Japan [email protected]. jp

Takahiro KAWABATA CNS University of Tokyo Hirosawa 2-1 Wako Saitama 351-0198, Japan [email protected]

Mitsuji KAWAI Department of Physics Kyushu Univesity Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected]

281

Tatsuji KIMURA Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected]

Hiroo KUMAGAI Fukuoka Institute of Technology Wajiro Higashi 3-30-1 Higashiku Fukuoka 811-0295 Japan kumagai@fit .ac.j p

Satoshi KISHI Department of Physics Kyoto University Kitashirakawa Oiwake-cho 1-1 Sakyo-ku Kyoto 606-8502, Japan [email protected]

Yukie MAEDA Department of Physics University of Tokyo Hongo 7-3-1 Bunkyo Tokyo 113-0033, Japan yukiet3nucl.phys.s.u-tokyo.ac.jp

Yasuro KOIKE Department of Physics Hosei University Fujimi 2-17-1 Chiyoda Tokyo 102-8160, Japan [email protected]

Takuma MATSUMOTO Department of Physics Kyushu University Hakozaki 6- 10-1 Higashiku Fukuoka 812-8581, Japan t [email protected] ushu-u .ac.j p

Hironori KUBOKI Department of Physics University of Tokyo Hongo 7-3-1 Bunkyo Tokyo 113-0033, Japan kubokit3nucl.phys.s.u-tokyo.ac.jp

Shiro MITARAI Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected]

Takashi KUDO Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected]:.jp

Yoshiharu MORI Department of Physics Tohoku University Aoba Atamaki-aza Sendai Miyagi 980-8578, Japan [email protected]

282

Tsuneyasu MORIKAWA Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected]

Shinya NOZOE Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan shinyaO kutl.kyushu-u .ac.jp

Taisuke NAGASAWA Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected]

Kazuyuki OGATA Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected] .ac.jp

Hiroyuki NAKAMURA Kitakyushu National College of Technology Arai 5-20-1 Kokura Minamiku Kitakyushu 802-0985 nakamuraQkct .ac.j p

Ryoji OKAMOTO Department of Physics Faculty of Engineering Kyushu Institute of Technology Sensui-cho 1-1, Tobata Kitakyushu 805-8550, Japan

Masahiro NAKANO University of Occupational and Environmental HeaIth Iseigaoka 1-1 Yahata Nishiku, Kitakyushu Fukuoka 807-8555, Japan [email protected] .ac.jp

Vijay PANDHARIPANDE University of Illinois 337A Loomis Lab of Physics MC-704 1110 West Green Street Urbana IL 61801 U.S.A [email protected]

Tetsuo N O R 0 Department of Physics Kyushu University Hakozaki 6- 10-1 Higashiku Fukuoka 8 12-8581, Japan [email protected]

Kenshi SAGARA Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected]

[email protected]

283 Koichi SAITO Tohoku Pharmaceutical University Aoba Komatsushima 4-4-1 Sendai Miyagi 981-8558 Japan [email protected]

Takaaki SAITO Department of Physics University of Tokyo Hongo 7-3-1 Bunkyo Tokyo 113-0033,Japan [email protected]

Harutaka SAKAGUCHI Department of Physics Kyoto University Kitashirakawa Oiwake-cho 1-1 Sakyo-ku Kyoto 606-8502,Japan [email protected]

Hideyuki SAKAI Department of Physics University of Tokyo Hongo 7-3-1 Bunkyo Tokyo 113-0033,Japan [email protected]

Yasuhiro SAKEMI RCNP Osaka University MihogaokalO-1 Ibaraki Osaka 567-0047,Japan [email protected]

Francesca SAMMARRUCA Department of Physics University of Idaho Moscow Idaho 83844-0903 U.S.A [email protected]

Kimiko SEKIGUCHI Beam Dynamics Division Cyclotron Center (RIKEN) Hirosawa 2-1 Wako Saitama 351-0198,Japan kimi [email protected] p

Yoshifumi SHIMIZU Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581 [email protected]

Youhei SHIMIZU RCNP Osaka University Mihogaoka 10-1 Ibaraki Osaka 567-0047,Japan [email protected] .ac.jp

Shinsuke SHIMOMOTO Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581,Japan shimornoto@kutl. kyushu-u.ac. jp

284 Masato SHIOTA Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected]

Atsushi TAM11 Department of Physics University of Tokyo Hongo 7-3-1 Bunkyo Tokyo 113-0033, Japan [email protected]

Kenji SHIROUZU Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected]

Toshitaka TATSUMI Department of Physics Kyoto University Kitashirakwa Oiwake-cho 1-1 Sakyo-ku Kyoto 606-8502, Japan

Edward J . STEPHENSON IUCF Indiana University 2401 Milo B. Sampson Lane Bloomington IN 47408, U.S.A

Satoru TERASHIMA Department of Physics Kyoto University Kitashirakawa Oiwake-cho 1-1 Sakyo-ku Kyoto 606-8502, Japan

[email protected]

[email protected]

[email protected]

Makoto UCHIDA Toshio SUZUKI Department of Physics Department of Applied Physics Kyoto University Fukui University Kitashirakawa Oiwake-cho 1-1 Bunkyo 3-9-1 Sakyo-ku Fukui Kyoto 606-8502, Japan Fukui 910-8507, Japan uchida@nhscphys .kyoto-u.ac .jp [email protected] p Hiroyuki TAKEDA Department of Physics Kyoto University Kitashirakawa Oiwake-cho 1-1 Sakyo-ku Kyoto 606-8502, Japan [email protected] .ac.j p

Yuriy UZIKOV JINR and RCNP Joint Institute for Nuclear Researches Dubna Moscow Region Russia 141980 [email protected]

285 Eizo UZU Department of Physics Faculty of Science and Technology Tokyo University of Science Yamazaki 2641 Chiba 278-8510, Japan [email protected] .ac.j p

Tomotsugu WAKASA RCNP Osaka University Mihogaoka 10-1 Ibaraki Osaka 567-0047, Japan [email protected]

Yukinobu WATANABE Department of Advanced Energy Engineering, Kyushu University Kasugakoen 6-1 Kasuga Fukuoka 816-8580, Japan [email protected]

Kentaro YAK0 CNS University of Tokyo Hirosawa 2-1 Wako Saitama 351-0198, Japan [email protected] .ac.j p

Naotaka YAMASHITA Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan naot2scpQmbox .nc.kyushu-u .ac.j p

Tsuyoshi YAMADA Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan yamada2scpQmbox .nc.kyushu-u.ac.j p

Masahiro YAMAGUCHI RCNP Osaka University Mihogaoka 10-1 Ibaraki Osaka 567-0047, Japan [email protected]

Yusuke YASUDA Department of Physics Kyoto University Kitashirakawa Oiwake-cho 1-1 Sakyo-ku Kyoto 606-8502, Japan [email protected] .jp

Takahisa YONEMURA Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected]

Hidetomo YOSHIDA RCNP Osaka University Mihogaoka 10-1 Ibaraki Osaka 567-0047, Japan [email protected]

286

Masaru YOSOI Department of Physics Kyoto University Kitashirakawa Oiwake-cho 1-1 Sakyo-ku Kyoto 606-8502, Japan [email protected]

Kouichi ZAIZEN Department of Physics Kyushu University Hakozaki 6-10-1 Higashiku Fukuoka 812-8581, Japan [email protected]


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