BLED WORKSHOPS
IN PHYSICS
VOL. 1, NO. 1
Proceedins of the Mini-Workshop
Few-Quark Problems (p. 49)
Bled, Slovenia, July 8-15, 2000
Exact treatment of the Pauli operator in nuclear
matter?
Michio Kohno1??, K. Suzuki2, R. Okamoto2 and S. Nagata31Physics Division, Kyushu Dental College, Kitakyushu 803-8580, Japan2Department of Physics, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan3Department of Applied Physics, Miyazaki University, Miyazaki 889-2192, Japan
Abstract. Exact formulae are derived for the matrix element of the Pauli operatorQ in the
Bethe-Goldstone equation and the binding energy per particle in nuclear matter. Numer-
ical calculations are carried out, using the Bonn B potential and the quark model Kyoto-
Niigata potential fss2. The exact treatment of the operator Q brings about non-negligible
attractive contribution to the binding energy compared with the standard angle average
approximation. However the difference is rather small, which quantitatively demonstrates
the good quality of the angle average prescription in nuclear matter calculations.
The Pauli principle plays an essential role in the nucleon-nucleon scattering in
nuclear medium. It constrains single particle momenta of intermediate two par-
ticles to be above the Fermi momentum kF. The Pauli operator Q is defined asQ = 12X jihj (k - kF)(k - kF) :
The operator Q depends not only on the magnitude of total and relative mo-
menta of scattering two nucleons but also on their relative angles. Properties of
this angular dependence, owing to which partial waves with different angular
momenta are coupled, were investigated in the early stage of the development
of the Brueckner theory. Werner presented explicit coupled equations in 1959 [1].
However, since the numerical calculations are rather involved, the standard angle
average approximation has been introduced to avoid the difficulty.
Thematrix element of the operatorQ between angular-momentum-coupling
states is given ashKk(`1S1)J1M1T1Tz1jQjK 0k 0(`2S2)J2M2T2Tz2i= Æ(K-K 0)Æ(k - k 0)k2 ÆS1S2ÆT1T2ÆTz1Tz2Q(`1J1M1; `2J2M2 : S1T1kKKK) :
Wederived [2] useful analytic expressions for theQ(`1J1M1; `2J2M2 : S1T1kKKK)
asQ(`1J1M1; `2J2M2 : STkKKK)? Talk delivered by Michio Kohno?? E-mail: kohno@kyu-dent.ac.jp
50 M. Kohno= f`1ST f`2ST f Æ`1`2ÆJ1J2ÆM1M2 x0 + XL>0;L=even(-1)S+M1p4 ^̀1 ^̀2 Ĵ1 Ĵ2L̂3h`10`20jL0ihJ1 -M1J2M2jLMiYLM(K; K)W(`1J1`2J2; SL)[PL+1(x0) - PL-1(x0)℄g ;
where K and K are the polar angles of the c.m. momentum K, ^̀ p2` + 1,f`ST  1 - (-1)`+S+T2 and x0 = 8>><>>: 0 for k <qk2F - K2=4;K2=4+k2-k2FKk for qk2F - K2=4 < k < kF + K=2;1 otherwise.
We also presented practical expressions for the nucleon single particle potential,
based on which numerical calculations of the ground state energy were carried
out.
1.2 1.4 1.6 1.8
–14
–12
–10
E/A  [MeV]
kF [fm
–1]
Bonn–B potential
average
exact
1.2 1.4 1.6 1.8
–14
–12
–10
E/A  [MeV]
kF [fm
–1]
quark model fss2
average
exact
(a) (b)
Fig. 1. Energies per nucleon in symmetric nuclear matter as a function of the Fermi mo-
mentum kF: (a) Bonn-B potential [3] and (b) quark model potential fss2 [4].
Calculated saturation curves with the Bonn B potential [3] and the new ver-
sion of the quark model Kyoto-Niigata potential fss2 [4] are shown in Fig. 1,
where results with the exact Pauli operator and the angle averaged one are com-
pared. The exact treatment of the Pauli operator brings about attractive contri-
butions to the binding energy per nucleon at any nuclear densities. However the
difference is rather small, although the results somewhat depend on the nucleon-
nucleon interaction employed. This quantitatively confirms the good quality of
the angle average approximation. The same conclusion was obtained by Schiller,
Müther and Czerski [5]. It is suggested that the angle average treatment of the
Pauli operator in considering more than three-body correlations is reliable.
Exact treatment of the Pauli operator ... 51
References
1. E. Werner, Nucl. Phys. 10 (1959), 688-697.
2. K. Suzuki, R. Okamoto, M. Kohno and S. Nagata, Nucl. Phys. A665 (2000) 92-104.
3. R. Machleidt, Adv. Nucl. Phys. 19 (1989) 189.
4. Y. Fujiwara, talk given in this workshop.
5. E. Schiller, H. Müther and P. Czerski, Phys. Rev. C59 (1999) 2934; Phys. Rev. C60 (1999)
059901.