ELSEVIER Nuclear Physics A73 1 (2004) 114-120

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The Gamow-Teller states and sum rule in relativistic models

Haruki Kurasawa”, Toshio Suzukib and Nguyen Van Giai”

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

bDepartment of Applied Physics, Fukui University, Fukui 910-8507, Japan and RIKEN, 2-l Hirosawa, Wako-shi, Saitama 351-0198, Japan

‘Institut de Physique NuclCaire, CNRSIN2P3, 91406 Orsay Cedex, France

The giant Gamow-Teller(GT) states and the GT sum rule are investigated in relativistic models. It is shown that the Ikeda-Fujii-Fujita sum rule value is quenched by 6% owing to relativistic effects. The quenched amount is taken by the nucleon-antinucleon states. This fact, together with the recent experiment which has observed 90% of the sum rule value, implies that the contribution of the A-hole states to the quenching is strongly reduced. As a result, the Landau-Migdal parameter gha which dominates the critical density of the pion condensation becomes much smaller than what was believed before.

1. INTRODUCTION

The quenching of the Gamow-Teller (GT) strength is a long standing problem experi- mentally and theoretically in nuclear physics. The summary of the experimental data up to 1983 by Gaarde[l] h s ows that the experimentally observed strength was less than 60% of the Ikeda-Fujii-Fujita(IFF) sum rule value. Those data were obtained from experiments which analysed the giant GT resonance region, assuming that there was no GT states in the higher energy region.

There were two conjectures to explain theoretically the quenching of the GT strength. The first one is to relate the quenching to the coupling of the GT state with the A-hole states[2]. Since the coupling is caused by the repulsive force, a part of the GT strength of the nucleon sector is absorbed by the A-hole states in the time-like region which is not observed in usual experiments. If all of the quenching about 40% is assumed to be due to this coupling, the value of the Landau-Migdal(LM) parameter gha is estimated to be about 0.75[3]. Th’ is result was crucial in particular for the study of the critical density of the pion condensation, since the critical density strongly depends on the value of gka. For example, in symmetric nuclear matter, the critical density ranges from 7 times the normal nuclear density up to infinity for the values of ghn from 0.75 to 0.8[4,5]. Thus, if the total quenching is due to the A-hole states the pion condensation may not be expected even in neutron stars.

The other possibility for quenching is to distribute the GT strength over a higher excitation energy region which was not analysed experimentally[6]. In particular it was shown that the tensor forces are responsible for the distribution of the strength. From

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H. Kurasawa et al. /Nuclear Physics A731 (2004) 114-120 115

the analyses of nuclear magnetic moments also, a small value of gha was expected[‘l]. Recently, the Tokyo group[8] h as p f er ormed new experiments on the CT states of “Nb

and 27Si by the (p, n) reaction, They analysed the data not only for the giant GT resonance region, but also for the higher excitation energy region up to 50 MeV by employing the multipole decomposition method. Finally they have found 88% of the IFF sum rule value in “‘Nb and 84% in 27Si. Most of the missing strength in 1983 has been observed in higher excitation energy region, as expected in the second conjecture mentioned above.

Thus, the quenching of the GT strength is shown experimentally to be of the order of 10%. This fact implies that the value of the LM parameter gha is very small compared with that believed in 1980’s. If we estimate the value in random phase approximation (RPA) by assuming that all of the 10% quenching is due to the A-hole states, we obtain gha = 0.21 in the quark model and 0.15 in Chew-Low model[9]. In fact, this estimation depends on the unknown value of the LM parameter ghl\ for scattering of A-hole states themselves. Fortunately, however, the dependence is very weak. In the present estimation we assumed that gh,=O.6.

Because of the small value of gka, the critical density of the pion condensation becomes lower. For the above small values the critical density is less than 2 times the normal nuclear density, even if we assume the value of gha to be more than 1.0[5]. The finding of the weak coupling between the particle-hole states and A-hole states may force us to re-investigate the pion condensation in more detail and also other spin-dependent response functions, as in quasielastic (p, n) reaction[lO].

Before doing such detailed investigations, however, we should ask whether or not all the quenching comes from the coupling with A-hole states. Since the quenching amount is observed to be small, we should explore other small contributions to the quenching which may reduce further the A-hole contribution. From this point of view, we notice relativistic effects on the IFF sum rule itself[ll]. S ince the sum rule is nothing but the commutation relation between spin-isospin operators, the total sum rule value itself should not be changed. However, there is a possibility that the GT strength is redistributed in the time-like region which is not observed such as in (p, n) reaction, The purpose of the present paper is to investigate such possibilities. We will show that about 6% of the sum rule value is taken by the nucleon-antinucleon states in the time-like region[l2]. As a results, the contribution from the A-hole states is expected to be much smaller.

2. THE SUM RULE IN NON-RELATIVISTIC MODELS

In non-relativistic models the GT states are excited by the operators:

Since these operators satisfy the commutation relation:

1 -J-+fly, r-gy] = 27,,

we obtain the IFF sum rule for p- and ,D+ transitions:

(1)

(I&+&-I )-(IQ-Q+I)=2(N-G (2)

116 H. Kuvasawa et al. /Nuclear Physics A731 (2004) 114-120

Notice that our definition (1) of the GT operators differs from the usual one and conse- quently, the difference of our IFF sum rule value with the familiar 3(N - 2) result is just a matter of definition. If we assume that

&+I ) = 0,

then we simply obtain

( I&+&-I ) =‘4N-4. (3)

3. THE GT STRENGTH IN RELATIVISTIC MODELS

First we discuss the GT strength in nuclear matter in order to see relativistic effects in an analytic way [ 111, and next we will estimate it in finite nuclei. When we assume that the nuclear mean field is provided by Lorentz scalar and vector potentials[l3], the nucleon field operator is described as

Nz) = /g&g ua P exdip . z) 4P) + 4P) exd-ip. x) am), ( 1 (4

where we have used the notations: u,(p) = uo(p) 17) ( a = g,~ ), etc. with the positive and negative spinors as

In the above equation, E denotes the Pauli spinor. In nuclear matter, the Lorentz vector potential is not relevant for the present discussions, while the Lorentz scalar potential is included in the nucleon and antinucleon effective mass M* as Ep = dm. For nuclear excitations with no momentum transfer, the excitation operator is given by

For the p- and /3+ excitations in N > 2 nuclei, it is written explicitly as, respectively,

F-I ) = v’+i3p c (%(P)&+(P) &(P)w~,(P) WV

(6)

(7)

where r z ~~7~. The calculation of the matrix elements is straightforward and we obtain

(lF+F-I) = 4& 1% [e$“) (1 - 8;)) (M*2 +p$) + (1 - Qt’) (P” - I$)] , (8)

( IF-F+1 ) = 4& /g (1 - 0:)) ($2 -P$) (9)

H. Kurasawa et al. /Nuclear Physics A731 (2004) 114-120 117

Table 1 The GT strength of the single-particle transitions in 48S~ The values in the parentheses following the single-particle quantum numbers show the binding energy in MeV. T is the value of the GT strength.

r&j dey T lP3/2(-39.41) %3,2(-1.09) 0.000 lp+(-39.41) +(-1.16) 0.002 b&-36.23) %a/+1.09) 0.005 lf~/z(-10.00) lf5,2(-1.16) 8.629 lf7,2(-10.00) lf7,2(-9.59) 6.361

Total 14.997

Here we have defined the nuclear volume V = A (37?/2k$), and the step function 8# = B(lci - ipi) for i =p or n with the Fermi momenta k, (protons) and k, (neutrons). The above equations show that each of the above matrix element is divergent, but that their difference gives the IFF sum rule value,

( 1 F+F- 1 ) - ( 1 F-F+ 1 ) = 2(N - 2). (10)

Thus, the anti-nucleon space is necessary for the sum rule in relativistic models. In other words, we need the complete set for the excited states. The nucleon-antinucleon states, however, are in the time-like region, so that their strengths do not contribute to the GT transitions induced, for example, by (p, ) n, reactions. As a result the total GT strength to be observed is quenched.

The calculation of the GT strength in the nucleon degrees of freedom only provides us with

( 1 F+F- / ) M 2 (1 - iv;) (N - 2) , (11)

where Z+ stands for the Fermi velocity, WF = kF/EF ( Ep = dm ). If we take the value of the nucleon effective mass to be M* = 0.6M and the Fermi momentum kF = 1.36fm-1 as in usual relativistic models, the Fermi velocity becomes 21~ = 0.43, which produces a quenching of the GT strength by 12%. If we use the free nucleon mass, the quenching is about 5%. The quenched amount is taken by the nucleon-antinucleon states, as mentioned before. We note that Eq.(ll) has been derived by expanding the exact equation in terms of (N - 2)/A, but not in terms of vF[I1,12].

The analytic formula obtained above makes it clear why the GT strength is quenched by relativistic effects. Since the quenching amount depends on the Fermi velocity, however, we must estimate the GT strength in finite nuclei[l2]. For this purpose, we use the mean field approximation with the NL-SH Lagrangian[l4]. The results are shown in Table 1 for 48Sc, where the GT strengths between the bound single-particle states are listed. The strength to the continuum states are expected to be small. The quantum numbers of the initial(fina1) state are indicated by n!j(n’!‘j’), while the numbers in the parenthesis denote the single-particle energies. The values of the column T shows the GT strength. It

118 H. Kurasawa et al. /Nuclear Physics A731 (2004) 114-120

is seen that the total GT strength is 14.997 which is 93.7% of the sum rule value 16 in the present definition. The quenching amount, 6.3%, is a little smaller than that in nuclear matter, as expected from the consideration on the nucleon effective mass. In “Nb also, we obtain a similar amount of the quenching(6.4%). If a half of the observed quenching amount is thus due to the relativistic effects, the room for the coupling with the A-hole states is very limited. In this case, the value of the LM parameter gha becomes much smaller, 0.12 in the quark model and 0.079 in Chew-Low model.

4. THE PAUL1 BLOCKING EFFECTS

In relativistic models which assume the nucleus to be composed of Dirac particles and various mesons, it is known that the Pauli blocking terms should be taken into account for some physical quantities if one adopts an RPA approach. This means that all the density- dependent terms in the RPA correlation function should be kept in the calculations. For example, in the description of the centre of mass motion and the giant monopole resonance in RPA, the Pauli blocking terms are not negligible[l5]. Since in the present case, the quenched part of the GT strength is taken by the nucleon-antinucleon states, we should also estimate the effects of the Pauli blocking terms on the GT strength. For this purpose, the RPA correlation function is calculated by introducing the LM parameter g’ as a contact term in the nuclear Lagrangian as

(12)

In nuclear matter the RPA correlation function can be calculated in an analytic way. The result of the excitation energy WGT and strength S GT of the GT state is given by[11,12]

WGT = 8k; N-Z

g5% 2A ’ SGT =

1 - 2$/3 (1+ &?543a2))2 2 (N - 4 > (13)

where the term depending on K M -2kg wg/5 stems from the Pauli blocking terms. When we take the value of the LM parameter to be g’ = 0.6 determined from the analysis of the excitation energy of the GT state, we find that the effects of the Pauli blocking terms on the GT excitation energy and strength are negligible:

2g5/c/(37r2) M -0.005. (14

When WF = n = 0, the above equations of WGT and So, are reduced to their non-relativistic expressions[ 161.

5. CONCLUSION

The Landau-Migdal(LM) parameters gkN and gka dominate the spin- and isospin- dependent structure of nuclei. The best way to determine those values phenomenologically is to use experimental data on giant Gamow-Teller(GT) states.

Recent experiments[8] h ave shown that the GT strength is quenched by about lo%, compared with the Ikeda-Fujii-Fujita(IFF) sum rule value. On the one hand, since the

H. Kuvasawa et al. /Nuclear Physics A731 (2004) 114-120 119

quenching is small, we can fix well the value of ghN within nucleon degrees of freedom. It is finally estimated to be about 0.6 in “‘Nb[9]. On the other hand, the small quenching requires detailed investigations to determine the value of gka. In addition to the a-hole contribution, contributions to the quenching of the same order of magnitude should be also taken into account, to fix the value of gha.

We have estimated relativistic effects on the IFF sum rule itself. It has been shown that 6% of the sum rule value is taken by nucleon-antinucleon states in the time-like region which can not be excited, for example, in (p,n) reactions. Consequently, if the only 4% quenching is due to the coupling of the particle-hole states with n-hole states, the value of gha becomes extremely small as 0.12 in the quark model and 0.08 in Chew-Low model.

In fact, in finite nuclei the LM parameters are not enough to describe the GT states, because the momentum is not a good quantum number. The nuclear finite size effects on the value of gha is estimated by assuming the coupling interaction between the particle- hole states and n-hole states to be[17]

v = &a + vNA(T) + VNA(P), (15)

where the last terms denote the transition potentials through the ?r and p-exchange. Since these potentials are attractive, the value of gLA should be increased to reproduce the experimental value of the quenching. The results are as follows,

diA = 0.12 + 0.09 = 0.21 (Quark), (16) 0.08 + 0.08 = 0.16 (Chew - Low), (17)

where the second number in each line shows the corrections to compensate the effects of the finite range forces. Thus, the value of gkA is still much smaller than that believed so far.

In conclusion, since the value of ghA is much smaller than that of ghN, the spin- dependent structure of nuclei should be re-investigated in detail. For doing such in- vestigations, relativistic effects should be taken into account. For the derivation of the quenching factor itself from experiment, a relativistic framework may be required in the analysis.

REFERENCES

1. C. Gaarde, Nucl. Phys. A396 (1983) 127~. 2. M. Rho, Nucl. Phys. A231 (1974) 493;K. Oht a and M. Wakamatsu, Nucl. Phys. A234

(1974) 445. 3. TSuzuki and M. Kohno, Prog. Theor. Phys. 68 (1982) 690. 4. J. Meyer-ter-Vehn, Phys. Rep. 74 (1981) 323. 5. T. Suzuki, H. Sakai and T. Tatsumi, in Proc. of the RCNP Int. Symp. on Nuclear

Responses and Medium Effects, Universal Academic Press,Tokyo, 1999, p.77. 6. G.F. Bertsch and I. Hamamoto, Phys. Rev. C26 (1982) 1323. 7. A. Arima et al., Phys. Lett. B122 (1983) 126. 8. T. Wakasa et al., Phys. Rev. C55 (1997) 2909;Phys. Lett. B426 (1998) 257. 9. T. Suzuki and H. Sakai, Phys. Lett. B455 (1999) 25.

120 H. Kuvasawa et al./Nuclear Physics A731 (2004) 114-120

10. K. Kawahigashi, K. Nishida, A. Itabashi and M. Ichimura, Phys. Rev. C 63 (2001) 044609.

11. H. Kurasawa, T. Suzuki and N. Van Giai, nucl-th/0301074 (Phys. Rev. Lett. 2003, in press).

12. H. Kurasawa, T. Suzuki and N. Van Giai, nucl-th/0306080. 13. B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, ed. E. Vogt and J. Negle (Plenum,

N. Y., 1986). 14. M.M. Sharma, M.A. Nagarajan and P. Ring, Phys. Lett. B312 (1993) 377. 15. J. F. Dawson and R.J. Furnstahl, Phys. Rev. C42 (1990) 2009; H. Kurasawa and T.

Suzuki, Phys. Lett. B474 (2000) 262; Z. Ma, N. Van Giai, A. Wandelt, D. Vretenar and P. Ring, Nucl. Phys. A686 (2001) 173.

16. T. Suzuki, Nucl. Phys. A379 (1982) 110. 17. A. Arima, W. Bentz, T. Suzuki and T. Suzuki, Phys. Lett. B499 (2001) 104.