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An effective lagrangian with broken scale and chiral symmetry applied to nuclear matter and finite nuclei
NUCLEAR PHYSICS A
Nuclear Physics A571 (1994) 7 13-732
North-Holland
An effective lagrangian with broken scale and chiral symmetry applied to nuclear matter and finite nuclei Erik K. Heide, Serge Rudaz, Paul J. Ellis School of Physics and Astronomy, University ofMinnesota, Minneapolis, MN 55455, USA
Received 3 August 1993 (Revised 13 October 1993) Abstract We study nuclear matter and finite nuclei in the mean field approximation with a chiral lagrangian which generalizes the linear u model and also accounts for the QCD trace anomaly by means of terms which involve the u and I fields as well as the glueball field 4. The scale invariant term which leads to an omega meson mass, after symmetry breaking, is strongly favored to be of the form o,~&$~ by the bulk properties of nuclei; they also rather strongly constrain the other parameters. A reasonable description of the closed shell nuclei oxygen, calcium and lead can be achieved and this can be improved by including a term (w&wp)2 in the lagrangian. This leads to a level of agreement with the data that is comparable to that of the nonlinear Walecka model.
1. Introduction We are still far from being able to transform the lagrangian of quantum chromodynamits (QCD) into an effective lagrangian involving mesons and baryons which could be used at the mean field level for nuclear matter or finite nuclei. Nevertheless it is possible to incorporate the broken global chiral and scale symmetries of QCD into an effective lagrangian. In particular the notion of (broken) scale invariance has been modeled by a scalar glueball potential [ 1). An early attempt to introduce this into an approp~ately modified Walecka-type lagrangian was made in ref. [ 21. An alternative approach is to include the glueball potential in the lagrangian of the linear sigma model, supplemented by a contribution from the vector-isoscalar omega field [ 3,4] (for related work in the quark sector see ref. [ 51). The sigma model is particularly attractive since it incorporates (spontaneously broken ) chiral symmetry, another feature suggested by QCD. A common problem associated with these models is that the compression modulus for equilibrium nuclear matter is at least a factor of two larger than current estimates in the range 200300 MeV. Of course one can arbitrarily correct this by adding terms to the lagrangian, for instance cr3 and 04, and choosing the parameters to produce the desired result [6]. We have recently suggested [ 71 a more satisfactory approach, however, where the form of the potential which breaks scale invariance is modified in a reasonable way to include 0375-9474/94/$07.00
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714
a contribution
E.K. Heide et al. / Eflective lagrangian
from the cr field as well as the glueball field, 4. This was found to yield
K M 390 MeV, which, while still on the large side, is decidedly more reasonable. There still remains uncertainty
concerning
the form of the omega mass term, fm&+wP,
where the explicitly dimensionful
parameter
rni must be replaced by the square of a scalar
field (multiplied by a dimensionless coupling constant) in order to satisfy the dictates of scale invariance. The question arises as to whether the coupling be to a2 or 4’ or some linear combination of the two. In refs. [ 3,7] the first choice was made. One of the purposes of the present paper is to examine this choice in nuclear matter and finite nuclei; we shall find that the properties of finite nuclei are sensitive to this question. A more basic purpose of this investigation is to determine whether our effective lagrangian, which saturates nuclear matter in the mean field approximation, is able to give a sensible account of finite nuclei. This is particularly pressing since Fumstahl and Serot [ 8 ] have recently suggested that chiral models are inherently unable to give a satisfactory description of nuclei; this work was based on the approach of Boguta [ 91 and did not include the glueball field, so there was no attempt to incorporate broken scale invariance. The effective lagrangian that we employ, together with the necessary formalism for finite nuclei and nuclear matter is given in sect. 2. Our results are displayed in sect. 3, first for nuclear matter and then, in subsect. 3.2, for the closed shell nuclei 160, “‘%a and *‘*Pb. In subsect. 3.3 we discuss a refinement of the basic model which improves agreement with the data. Our conclusions are given in sect. 4. It is of interest to compare the low density expansion to that of the standard Walecka model [lo], this we discuss briefly in the appendix. This suggests a simple approximation in which the glueball field is frozen at its vacuum value and the formalism is outlined in the appendix. Examples are given in the text which show that the frozen glueball model reproduces our complete results quite accurately.
2. Theory We write the effective lagrangian
in the form
&T = LO -
v, ,
where in this schematic equation we have separated out the scale invariant part, Lo, from the potential Vi which induces the breaking of scale and chiral invariance. (The effects of explicit chiral symmetry breaking arising from the nonvanishing of current quark masses in the QCD lagrangian are neglected.) We write Lo in terms of the chiralinvariant combination of sigma and pi fields, tr and n, the glueball field q!~and the field of the omega vector meson wg. Specifically
(2)
715
E.K. Heide et al. / Eflective lagrangian
Here the field strength tensor is defined in the usual way, Fpy = d,w, - &w,. As we have remarked in the introduction, the form of the scale invariant o mass term is not a priori obvious, exotic choices In order to divergence of
so we have chosen a linear combination cr2 and 4’ fields (of course more are possible) and we shall examine the effect of varying the coefficients. obtain the scale breaking term & in the lagrangian, we recall that the the scale current in QCD is given by the trace of the “improved” energy-
momentum tensor and so the scalar potential Noether’s theorem, the effective trace anomaly
VG(4, CJ,n) is chosen to reproduce, [ 1,111
via
where @i runs over the scalar fields (4, g, n} and EVat is the vacuum energy. The proportionality 0; 0: $4 is suggested by the form of the QCD trace anomaly [ 121, e;(x)
= y
F;v(x)F”““(x) >
where F& (x ) is the gluon field strength tensor and /3 (g ) is the usual QCD beta function. Eq. (3) involves a purely gluonic, color-singlet, scalar, dimension-four operator, and also allows for the recovery of the low energy theorems which follow from broken scale invariance [ 13 1. Then in ref. [ 7 ] we suggested the form
where c = &,/a~ and the subscript 0 indicates the vacuum value. Here the logarithmic terms contribute to the trace anomaly and are such that Eq. (3) is satisfied with eVac= -$B4:(1 - 6). Thi s requirement uniquely specifies the second term on the right in Eq. (5). The third term is needed to ensure that in the vacuum 4 = 40, g = a0 and n = 0. To retain the physically necessary feature that EyI?c< 0, given that B > 0, we require that + 00 as is physically sensible. For further 6< l.ProvidedO<6< l,Vo+~fordor4 insight regarding the parameter 6, we recall that the QCD beta function and nf flavors is given at the one loop level by /3(g) =
with NC colors
-$$ (1- z) +W&?) 3
where the first number in parentheses arises from the (antiscreening) self-interaction of the gluons and the second, proportional to nr, is the (screening) contribution of quark pairs. Recalling Eq. (4), a value of 6 = 4/33 is suggested for the present case with nr = 2 and NC = 3. Since one cannot rely on the one-loop estimate for j?(g), we shall examine the effect of modifying the value of S.
E.K. Heide et al. / Effective lagrangian
716
The careful reader will have noticed that L:ee, given by Eqs. (2) and (5), does not contain the standard potential term of the linear sigma model which we have previously written [3,7] in scale invariant form as f1(02 + n2 - 42/[2)2. We found that very small values of ;i were favored by the predicted compression modulus in nuclear matter. We have also found that departures from small values yield binding energies of nuclei which are much too low. Therefore in this presentation we make the welcome simplification of setting I = 0, thus eliminating this term. Note that while the term is present in the standard linear sigma model, it is not necessary for the breaking of chiral invariance. This can be carried out just as well by Vo in Eq. ( 5 ). Since we are interested in finite nuclei it is necessary to include the vector-isovector rho field B, and the Maxwell field A,, in our total lagrangian. Thus we take &,a = I& + .C’, where ,C’ =
f p”- ;B,, . B#” + ;Gpqi2b,. b”
- +
-Ny’[ig,b,.r+
ie(l
+q)A,]N.
(7)
Here the field strength tensors are fpy = +A, - &A, and B,, = a,b, - &b,. Since the effect of the p meson is relatively small, particularly in light nuclei, we have taken the simplest form in L’ which is not chirally invariant. Note, however, that we have made the p meson mass term scale invariant by coupling to 42, we shall comment further upon this later. It is economical to write the sigma and glueball fields in terms of their ratios to the vacuum values, viz. x(r)
=
For r -+ 00 we obtain the vacuum
4(r)
-5
v(r)
40
= $.
(8)
values x = v = 1. Now, in the vacuum,
we require
that the rho and the omega masses take their physical values so that we can write the mass terms in the form ~rn~x2b, - b”
;rnZ, [Rov2 + (1 - &)x2]
and
wPwP.
(9)
Thus, for the omega, taking R. to be 1 gives a pure u coupling, whereas the value 0 yields a pure x coupling. The effective mass of the nucleon, in the usual way, is M’ (r ) = Mv (r). In the mean field approximation n = 0 and the field equations are obtained from Lagrange’s equations. We specialise to the time independent, spherically symmetric case. The Dirac equation for the nucleon and the equation for the photon field are of the form given by Horowitz and Serot [ 141 and do not need to be repeated here. As usual for the vector fields only the time-like components survive and also just the isovector z-component for the p field. The equations for the 4, u, w and p can be written 4;Dx
- 2Bo(2 - 8)x = 4B0[x3(lnx
- mi(l
+ 2BoSu
-6lnv)
- Rw)w&
-x
+ 6~1 + BoS[x(v2--x2)
- rnzb,&
,
+ 2(x -v)]
717
E.K. Heide et al. / Effective iagrangian
2Bo8v + 2BoSX = Mps - BoS Doo - rniwo Dbo-
=
-gdB
+
mibo = -gpp3 -
x; - X2Y -I 2(V -X)
m~[R,(v2mi(l -x’)bo,
- miRow&, >
1) + (1 -&)(x2-
1)100,
(10)
where the densities ps = (NN), pB = (iVyoN) and p3 = f (~y”73N) can be expressed in terms of the components of the nucleon Dirac spinors in the usual way [ 141. In Eq. (10) we have made the definitions D = d2/dr2 + (2/r)d/dr and Bo = Bq$. The terms linear in the fields, i.e., the kinetic energy and mass terms, have been separated out on the left of these equations. The equations of ref. [ 71 for nuclear matter are regained by setting the derivatives to zero so that the results are independent of 40 and a,-,, however these quantities are needed for finite nuclei. They are also needed to obtain the vacuum scalar masses and we note that the mass matrix is not diagonal. In order to solve Eqs. ( 10) by an iterative Green’s function technique [8,14] it is necessary to go to a representation which is diagonal in the limit that r + CCLWe solve in terms of the quantities ( 1 - x ) and ( 1 - v ) which go to zero in the same limit. The energy-momentum tensor can be used to obtain the total energy of the system in the standard way [ lo]. Subtracting constants so that the energy is measured relative to the vacuum, we obtain 00 E
=
5,(2j,
+ 1)-2x I
a +
drr2{Mvps
+
gowOpB
+
gpbOp3
0
2Bo[x4(lnx
- 6lnv
+ $) - $1 + iBoJ[x2(2v2-
- mi[R,v' + (1 - Rw)X2]wi
- m~X2b~}.
3x2) + l] (11)
In the first term on the right the ea are the Dirac single particle energies and jm is the total angular momentum of the single particle state. In nuclear matter this term becomes 4 c, (&oo + Jk* + M*2). Making this replacement, and using Eq. (lo), the expression for the energy density of nuclear matter given in ref. [ 71 can be obtained.
3. Results 3.1. NUCLEAR MATTER In nuclear matter we have four parameters to consider. We will choose Bo and Ci = g;M2/mi to fit the saturation properties. The binding energy/nucleon we take to be 16 MeV. In refs. [ 3,7] we took a saturation density of 0.16 fmm3, but this yields central densities in lead which are too high. We have therefore used 0.148 fmm3 (kF = 1.3 fm-’ ), which is the value favored by Serot and coworkers in mean field calculations [ 14,8].
718
E.K. Heide et al. / Efective lagrangian TABLE 1 Values of the parameters and the
derived quantities for nuclear matter
Set
R,
336
I IF II III IIIF IV V VI VII
0.0 0.0 0.5 1.0 1.0 0.0 0.0 0.0 0.0
4 4 4 4 4 1 2 8 16
235 231 255 269 265 336 281 194 160
156 152 82.6 51.3 49.2 153 154 158 156
116 123 127 132 135 122 119 107 88
0.71 0.72 0.80 0.84 0.85 0.72 0.72 0.69 0.65
383 341 356 377 356 350 360 442 666
6.5 5.6 5.9 6.1 5.7 5.8 6.0 7.5 10.6
VIIIF
0.0
4
214
227
114
0.65
267
4.4
Having
WK
fixed these two parameters,
we then have the parameter
of the o mass term, R,,
and the coefficient of the quark contribution to the trace anomaly, 6, which can be varied. We show in Table 1 several parameter sets that we have considered. The designation F here indicates the frozen glueball model where x is kept equal to 1; see discussion in the appendix. (Parameter set VIIIF is for later use in subsect. 3.3 and will be excluded from the present discussion.) One might hope that the parameters would show a reasonable correspondence with the independently determined quantities, although precise agreement would not be expected for an effective theory such as this. In this sense the values for C’i in Table 1 are in acceptable agreement with the value 103 f 36 deduced from data on the o coupling constant [ 151 and the known mass. Turning to ]evac],QCD sum rule studies suggest [ 16 ] a value of (240 MeV)4 which would not favor the extreme variations of 6 (sets IV and VII), although bag model estimates [ 171 range as low as (146 MeV)4. The values of Cj, given in Table 1 for later use, have been obtained by fitting to a symmetry energy of 35 MeV. Since exchange effects are known to make a significant contribution to the symmetry energy we shall not make a direct comparison to experiment. Turning to the derived quantities, nuclei generally favor effective masses below 0.7 [ 81, which is the case for sets VI and VII. However, here the compression moduli, K, are in excess of 400 MeV. Pearson [ 181 has made the point that the leptodermous expansion, which is one method used to fit the data, permits values of K up to about 400 MeV with little difference in the x2 fits. Defining the third derivative of the binding energy per particle as S = kz (d3/dk:) (E/A) (evaluated at equilibrium), the data indicate a linear relation between S/K and K (see ref. [ 191). We have listed values of S/K in Table 1. Cases VI and VII are welI off the allowed error band, but the remaining cases are quite close to, although slightly off, the band. The parameter sets designated with an F in Table 1 correspond to the frozen glueball model (x = 1) and we see that they give a close correspondence with the complete
E.K. Heide et al. / E#.?ctive la~angi~~
719
300
____ _____._.._... se+
250
-_-
..-..-..-..-..-. - .-
200
%c z
,F
Set ill set
,,,F
150
I I q W
loo
50
0
0
1
2
3
4
5
6
P/P, Fig. 1. The binding energy/particle as a function of density for nuclear matter; the density p is given as the ratio to the equilibrium value p. = 0.148 fmm3. The parameter sets used are indicated and the designation F indicates the frozen glueball model.
results. The point is further made in Figs. 1 and 2 where the agreement is seen to be particularly good at low density, as we argue in the appendix. The lower panel of Fig. 2 shows that in the complete model x remains very close to unity over the whole region of densities considered. We remark that the curves for v will increase linearly with kF for sufficiently large density (a discussion of the high density behavior was given in ref. [ 3) ), although it is debatable
whether this theory is applicable
in that region.
3.2. FINITE NUCLEI When we turn to finite nuclei two extra parameters are involved, namely the vacuum values of the scalar fields, $0 and 00. Thus we have a total of four free parameters. Let us first eliminate two of these. We shall discuss changes relative to parameter set I of Table 1 with a0 = 110 MeV, which will turn out to be our preferred parameter set in this subsection. First consider reducing 6 by using sets IV and V of Table 1 (and reasonable values for the parameters not specified). We find little sensitivity in the predicted properties of nuclei. However, if we increase 6 using sets VI and VII, for which K is large, we find a significant reduction in the binding energy, particularly for set VII (40Ca is less bound by 2: MeV, for example). We conclude that a value of 6 in the nei~borho~ of 4/33 is
E.K. Hetie et al. I E$%vtive~a~ran~i~
720 1.0 0.8 A
0.6 0.4
_________ ___.__._Set 0.2
IF
--Set Ill t ___ .._.._..-.I_..-.. - Set l#F
1.2
1.1 x 1.0
0.9
0.8
Fig. 2. The fields v and x corresponding to Fig. 1.
reasonable and fix on this value, although nuclei are not sensitive to the precise figure. Next consider the ratio of the vacuum scalar fields, ( = #o/~~ Using set I we have varied C in the range 0.7-2.1, which corresponds to varying the higher scalar mass, ma, between 3 and 1 GeV. Very little change is observed in the nuclear properties. The salient point is that the mixing between the glueball and the sigma is small, d 3%, and it is the sigma which directly couples to nucleons. Henceforth we choose (, such that nz, = 1.5 GeV because this seems reasonable in view of QCD sum rule estimates for a scalar glueball mass in the range 1-2 GeV 1201. The values of C are listed in Table 2. We are then left to consider the variation of R. and a~: a few comments on the expected magnitude of o. are in order here. In the original Gell-Mann-L&y linear sigma model [ 2 11, a calculation of the matrix element of the axial vector current responsible for pion decay led to the identification CO = .fX = 93 MeV. In a more general chiral effective lagrangian in~~mting vector mesons (see ref. [22] for an early review) this identification no longer follows due to the necessity of including the al axial vector meson, together with the p. Briefly, cross-terms of the form (I: - Qc appear in the full lagrangian which are eliminated by the replacement a: --e a: + &Pa, with an appropriate parameter t;. This, in turn, leads to a change in the coeffkient of the pion kinetic term, 0,~ * Pit, and a resealing of the pion field is necessary to bring this to canonical form. One defines a ~nor~liz~d field SR = z I/2=, Now a calculation of the full axial current yields
721
E.K. Heide et al. /Effective lagrangian TABLE 2
Bulk properties of nuclei for various parameter sets Set
m<
0
c
(MeV) Experiment I/110 IF/l 10 I/93 I/120 II/l 10 III/l 10
506 508 598 464 598 664
1.4 1.4 1.4 1.6 1.8
VIIIF/ 100
481
-
Ca
Pb
BEIA (MeV)
ret, (fm)
BE/A (MeV)
rch (fm)
BE/A (MeV)
ret, (fm)
1.98
2.73
8.55
3.48
7.86
5.50
7.35 7.86 9.41 6.25 10.08 10.98
2.64 2.62 2.52 2.72 2.49 2.44
1.96 8.35 9.38 7.16 9.85 10.45
3.41 3.40 3.32 3.41 3.29 3.26
1.33 7.54 8.04 6.92 8.22 8.48
5.49 5.49 5.45 5.52 5.45 5.44
7.18
2.69
7.91
3.45
1.44
5.53
a0 = Z:‘*fn. The precise form of Z, is model dependent. In the simplest model, which does not include the physics of broken scale invariance one finds [22], using the KSRF = fi. We do not expect either this precise functional relation, that Zi/* = m,,/m, form or value to be maintained in more realistic models, but we do anticipate Z,“* > 1. Accordingly, as is kept as a free parameter, which we can presume to be larger than 93 MeV. Two further remarks are in order. First, the additional terms involving the al will not contribute in the mean-field approximation. Second, from Eq. (2) after symmetry breaking, the nucleon mass is given by A4 = gas. The physical pion-nucleon coupling is defined in terms of the renormalized pion field to be gnNN = Z,“*g. So, in fact, one still has M = go0 = g*NNfn as the approximate form of the Goldberger-Treiman relation. With this in mind, we first consider variation of tre keeping R, = 0. The results are given in Table 2, where the notation I/93, for example, implies use of parameter set I from Table 1 with as = 93 MeV. (Again we remark that parameter set VIIIF is for later use in subsect. 3.3 and is excluded from the present discussion.) The theoretical values of the binding energy/particle include a correction for the c.m. kinetic energy [ 231. The charge densities, and therefore the radii quoted, are corrected for the finite size of the proton [ 141 and for c.m. effects [24]. We see that the choice 00 = 93 MeV (fourth row in Table 2) yields poor binding energies, radii which are too small and, as shown in Fig. 3 for %a, an oscillatory charge density. This figure indicates that increasing a0 to 110 MeV improves matters greatly. One can view this as reducing the magnitude of the derivative of v in Eq. ( 10) or as reducing m< to - 500 MeV (Furnstahl and Serot [ 81 stress that values in this neighborhood are to be preferred). In fact as regards the charge radii and densities the value 120 MeV gives even better results, however we see from Table 2 that in this case the binding energies/particle are l- 1i MeV too low. Therefore we prefer the parameter set I/ 110. Now we consider variation of R, with a9 = 110 MeV. As R, is increased from zero
722
E.K Heide et al. / Efiective lagrangian
0.08
0.02
0.00 0
1
2
3
4
5
6
7
f- k-9 Fig. 3. Comparison of the experimental charge density 1251 for %a for the parameter sets indicated.
with theoretical predictions
to 0.5 or 1.0 we see that the binding energies become too large and shows the wrong A-dependence and also the radii become too small. Table 2 also shows that the mass of the lighter scalar meson, m,: , increases sharply. Further, as we see for “sPb in Fig. 4 the charge density begins to develop oscillations which are not present in the data. Similar problems were noted by Furnstahl and Serot [ 81 who used R, values of 0.76 or 1, although their model differs significantly from ours. As pointed out by Price, Shepard and McNeil f26] this is likely to signal the onset of a situation where the ground state of nuclear matter is no longer uniform, but exhibits periodic density fluctuations. We conclude that nuclei strongly favor Rw = 0. We have also compared the case I/ 110 with the frozen glueball approximation, case IF/l 10, in Table 2. There is a close correspondence between the results, as there is for the predicted charge densities which we illustrate for I60 in Fig. 5. Finally we turn to the single particle energies. The levels near the Fermi surface in lead are shown in Figs. 6 and 7 for neutrons and protons respectively. The relative positions of the neutron and proton levels are sensitive to the treatment of the p meson [ 141. We have used a scale-invariant 4’ coupling in the mass term, but the results differ negligibly if a standard mass term is used. However there is a difference if a t-r2coupling is used because this reduces Ci and leads to N 2 MeV less (more) binding for protons (neutrons). Since
723
E.K. Heide et al. / Eflective lagrangian
0.08
0.07
0.06
s
0.05
x .z ii
0.04
n 8-l
“‘Pb
0.03
b 4
Experiment .._._ _._. Set 1/110 --~ - - -..- - - - Set 111/110
0.02
0.01
0.00 0
1
2
3
4
(fm)5
6
7
8
9
r
Fig. 4. As for Fig. 3, but for 20*Pb.
Figs. 6 and 7 clearly show that this is undesirable we have not considered this possibility further. The results with the sets II/l 10 (R, = 0.5) and I/93 show immediate problems since the major shell closure is incorrectly given due to the position of the proton 1h9,2 and the neutron 1il 1,2 levels. This problem disappears with the preferred parameter set, I/ 110, and the single particle spectrum is much more reasonable, although a larger proton 1h9,2-3s1,2 splitting would be desirable. This may reflect the spin-orbit splittings which are, on average, 65% of the experimental values here and also in the other nuclei we have considered. They could be improved by reducing the effective mass in Table 1 below 0.7. This can be achieved by taking a small negative value for Rw, although we find that this leads to a larger compression
modulus
and lower binding
energies.
3.3. FINE TUNING OF THE MODEL While the results obtained for finite nuclei in the preceding subsection are reasonable, it is natural to ask whether significant improvement can be achieved by including additional terms consistent with the scale and chiral properties of our effective lagrangian. We have not made an exhaustive study, but we have investigated the effect of including an additional term of the form (o+~#)~. Such a term is automatically scale invariant and has previously been studied in connection with the Walecka model [ 281. For present
E.K. Heide et al. / Eflecticrive lagrangian
724
0.10
0.08
m5i “r; 25
0.06
x 5
5
cl
6,
0.04
P 0 if 0.02
0.00 2
3
4
5
t- b-d
Fig. 5. As for Fig. 3, but for lag. The data are from ref. 1271.
purposes it is sufficient to employ the frozen glueball model and it is straightforward to extend the equations to include the additional term in the lagrangian which we write as [ (G~)2~rop]2; note that this has the same sign as the w mass term. Bearing in mind the previous results, we fix R, = 0 and 6 = 4/33 and refit nuclear matter. We then have Cd/g” as a free parameter and we judge that the best results for nuclear matter and nuclei (here 00 = 100 MeV) are obtained with Ga/g, = 0.21. In fact two sets of parameters can be obtained with this value of G4f go, both of which yield the~odyn~ica~y stable solutions. The parameters we select are Iisted in Table 1 as set VIIIF; the second solution we reject on the grounds that it corresponds to a 50% larger value of C& a 25O41larger compression modulus and a rather low value of M& = 0.504. With our chosen parameters we find that relative to the standard omega mass term, the (G~wo )4 contribution to the equations is a 2OW effect at equilibrium nuclear matter density; the effect is smaller at lower density. Table 1 shows that, in comparison to set I, the present value of Ci shows a substantial increase (it is about a factor of two larger than the experimental value quoted earlier). The effective mass M& is slightly smaller which will improve the spin-orbit splittings. Also the compression modulus K has been significantly reduced and lies within the often-quoted range of 200-300 MeV. This reflects the softening of the equation of state. In fact the point defined by K and S/K lies just within the allowed error band referred to in subsect. 3.1.
125
E.K. Heide et al. / Efective lagrangian 2 0
-~
__=
.--------_
2gs,2
__--
-2
/
,*,’
-4 % 3
h/2
-’
--._. *-
*. ‘\
-6
$ ‘P l5
-6 -10
3 r CY
-12
a, P
-14
_A::*" 3py2: 2f5/2+-_:;::::---
*.
-
..--;_.-____ ,- -;,*-
__-3P3/2 ‘~::_--_-::.r__ _---__ lhJ/2 _--2f,,z ----_____.**-+ lh9/2
-
___+-I_-
_-’
____----...;
.lily2 3PVZ 3~~2 2f5/2 ,2f7/2
lb/z
\\
iz -16
-16
-20 EXPERIMENT
set l/l10
set II/110
set l/93
-22
208 Pb Neutron Energy Levels Fig. 6. Occupied and unoccupied neutron levels near the Fermi energy in *OsPb.The experimental data are compared with predictions for the parameter sets indicated.
Turning to finite nuclei, our aim is to compare our results with the best that can be obtained in mean field calculations with the Walecka model extended to include nonlinear o3 and o4 terms. As a representative of the latter we choose parameter set C of Fumstahl, Price and Walker [29] (designated set B in ref. [ 81) which we refer to as FPW here. The comparison with our results using parameter set VIIIF/ 100 for the charge densities of I60 40Ca and *08Pb are shown in Figs. S-10. It is clear that there is close agreement between the two calculations, both of which yield a good account of the data. In Fig. 11 we make the corresponding comparison for the single particle levels of *“Pb. Again the two calculations agree quite closely and yield a fairly reasonable account of the experimental spectra. It is noteworthy that the separation between the occupied and unoccupied levels is much better with set VIIIF/ 100 than was the case with set I/l 10 in the previous subsection. The spin-orbit splittings are also better - on average they are 82Oh of the
E.K. Heide et
726
al. / Efective iagrangian
2
,-
0
f a 8 ‘p 9 a, 9 tB
u
ol .E v,
-2
-
lh3/2
-*_,*
2$/z
-‘_
-4
_
lb/z:
-
‘.
..
-.
.I
-8
-10
.
h/z
_
2ck/2
‘.
_’
\\ ‘. ,_._:~~___-----
\\
\\
I-* .,. ___-_ ---
_- _:=; -._J*.
I__-____.---
t&2
--.c --__
___-’ __.-
,/-
-12
-
*\
c3sy2
,h9/2
___---
2&/2
_____
, 2dy2
------_-197/2
2f7/2
*-
-6 3%/Z ‘, 2d3/2 -_:::::_
_---
I’ ~‘-‘-l______-r,’ .. I’ ,I .\ .___....---I’ ,I
lhJ@
%, ‘*. ‘,
-14
<\ -
‘\ lgs/2
-16
-._ -.
-.
‘\
\.
-.
lg,/2
~.___Lr----
*.
‘----._ --__ ---._ *-. ---
EXPERIMENT -22
set l/110
set II/110
lgs/2
set l/93
t 208
Pb Proton Energy Levels
Fig. 7. As for Fig. 6, but for protons.
experimental values versus 65% with set I/ 110 and 88% for FPW. The binding energies and radii obtained with our parameters are given in the last row of Table 2. As would be expected from the charge distributions we obtain radii that are in reasonable agreement with the data and with the predictions of FPW (not shown). Our binding energies are low by - $ MeV. This is a modest discrepancy, however FPW do better in this respect with deviations ;5 f MeV.
4. Conclusions Our primary purpose has been to see whether an effective lagrangian which includes the breaking of scale and chiral invariance can make sensible predictions for finite nuclei. Finite nuclei are expected, and found, to be a much more severe test than nuclear matter. In this endeavour we discarded the conventional quartic potential term of the linear
E.K. Heide et al. / Efective lagrangian
727
x
T&
c
n
& b
0.04
6
-
160
, 0.02
o.oo
v I
0
1
2
3
4
5
r k-9
Fig. 8. The charge density of I60 obtained with parameter set VIIIF/lOO, which includes an (~+ofl)~ contribution, is compared with that of the nonlinear Walecka model denoted by FF’W [ 291. The experimental data are also shown.
sigma model. Instead we used the potential Vo of Eq. (5) to spontaneously break chiral invariance as well as to explicitly break scale invariance in accord with the trace anomaly. The form of the scale invariant mass term for the w meson was found to be critical and a pure coupling to the glueball field, dew,&‘, was required, although a small effect from the sigma field cannot be ruled out. A similar coupling for the p mass term was also favored. It is interesting that the properties of nuclei can be used to provide such strong constraints
on the form of the coupling of the scalar glueball to mesonic fields appearing
in the chiral effective lagrangian. In the minimal version of the model we found it necessary to increase the value of the vacuum value of the sigma field, a~, from the naive expectation of 93 MeV to somewhere in the neighborhood of 110 MeV. With these caveats the bulk properties of 0, Ca and Pb were quite well accounted for and, while the spin-orbit splittings were about 65% of the desired values, the basic structure of the single particle spectra was reasonable. As a by-product we were able to define a simplified, frozen glueball model which was able to reproduce the results of the complete model quite accurately, but, by itself, would seem highly arbitary. By introducing a term (o,o’)* into the lagrangian and adjusting the strength, im-
728
E.K. Heide et al. / Effective lagrangian 0.10
0.00
0.02
---
0.00
0
1
2
3
4
5
6
7
r b-9
Fig. 9. As for Fig. 8, but for 40Ca.
proved results for nuclear matter and finite nuclei were obtained with 09 = 100 MeV. We mention in particular the compression modulus of 267 MeV and the major shell separation at the Fermi surface in Pb. In fact our results are quite comparable to those obtained with the standard Walecka model extended to include nonlinear c3 and o4 terms [ 29,8]. It is remarkable that the present effective chiral lagrangian, whose form is motivated by QCD, can give an acceptable description of both nuclear matter and finite nuclei in the mean field approximation.
We thank G.F. Bertsch and A. Vischer for useful comments and C.J. Horowitz for giving us a copy of his computer code for finite nuclei which was suitably modified for present purposes. One of us (PJE) is grateful for the hospitality provided by the Institute for Theoretical Physics of the University of California at Santa Barbara where a portion of this work was carried out. We acknowledge partial support from the Department of Energy under contracts No. DE-FG02-87ER40328 and DE-AC02-83ER40105. A grant for computing time from the Minnesota Supercomputer Institute is gratefully acknowledged.
E. K. Heide et al. / Eflective lagrangian
729
0.06
‘t &
0.05
.f=
E
0.04
2 th
0.03
5 6
0.02
0.01
0
1
2
4
3
6
7
8
9
Fig. 10. As for Fig. 8, but for 208Pb.
Note added After this work was completed
we received a preprint
from Fumstahl
and Serot [ 301
who have studied the same problem, restricting themselves to an o mass term of the form a20,,wJ’. In agreement with our conclusions, they point out that the results for nuclei are poor. As we have stressed, nuclei require that the coupling be to the glueball field rather than the sigma field.
Appendix LOW DENSITY EXPANSION
AND FROZEN GLUEBALL MODEL
It is of interest to study the low density expansion of Eqs. ( 10) and ( 11) in the case of nuclear matter. Examination of Eq. ( 10) indicates that we can expand v
= l+apBU+bk:)+O(k~),
x
= 1+
cpB(1
+
dk;)
+
o&j,
(A.11
where pB is the baryon density and kF the Fermi momentum. Further to this order the coupling between WC,and v and x can be neglected. Then the equation of motion for x
f_i .-
f? %
z t
--.
-
2f7/2
IhS&
‘. *. *. \. ‘. .___“,c-----
------__
Set VlllF/tOO
‘.
--
FPW
.’ _'
208 Pb Neutron Energy Levels
‘,
--. --._
--
--
_-_“____L’,
-1..-----__ ---‘=‘-----,____-_-_-,---~-----_ ---_-_-____:-
--_ --_--
_----_--___-_ -- - __.--',' --__ ___-------
EXPERIMENT
lii3/2
-
----__ *,_--
\,
----
-
\-I"-
31)3/Z
2fy
3~~2
299/z
Cl/Z
1115/Z
lhg/;l
2f7/2
w2
Qv2
*.
‘\
\*
‘.
*.
‘.
____-------
** ------...._*-_
‘r
-..c::::,:=
set VlllF/100
** *I
*-. ‘.
-IS._
_-c-c- -----
---___
~~~~~~~~~_~-
,P----llll”.e-
,’
d
-Y
208 Pb Proton Energy Levels
EXPERIMENT
-
-
8.
,,____“--._-----------~--
---_ --.,
,*
,*’
.’ ,’ *’ +’ .” .--
,’
,I’
-----Ic__” 2d3/2 lhql/2 ~\_-__---“------1”.“..____ 2&/z
3$..
--.._
3PVJ 3P3/2 I lh3/2 2f5/2
I
-
lhw2
lill/2
-’
1tr3/2 ?f7/2
%9/2
II
kJw
‘v/2
=/2
lb2
#’ 2%2
3s1/2
lh9/2
2f7y2
lil3/2
_
Fig. 11wOccupied and unoccupied, neutron and proton levels near the Fermi energy in 208Pb obtained with parameter set VIIIF/lOO, which includes an (o~w~)~ contribution, are compared with those of the nonlinear Walecka model denoted by F’PW [ 291. The experimental data are -I-- 3.-_
-20
-18
-16
-14
-12
-10
-8
-6
a .s
P ii W
-4
%c
-2
0
2
E.K. Heide et al. /Effective lagrangian indicates
731
that d=b,
The equation
“=A. a
of motion for u yields M(2 - 6) a = -4&&l -S)
Substituting
(A.21
b=-&.
’
64.3)
into Eq. ( 1 1 ), we find the energy/particle
64.4) The first three terms are the energy of a Fermi gas and the succeeding terms are the o and scalar meson contributions. This is of exactly the same form as the standard Walecka model [lo], except that the scalar meson mass is replaced by 4&S(l--6) a,2(2-6)
m=
1/Z = m m <
>
>
40
[2Bo(2 - S)]‘/2 LXm<.
(4.5)
where m< (m,) are the lower (upper) eigenvalues of the vacuum mass matrix (see ref. [7] for an explicit discussion). The mass, m, is approximately m< , provided that 40/oo is not too large so that the mixing of the sigma and glueball states is small. Now the fact that d = b in Eq. (13), means that, to the order we are considering, we can obtain the correct result for v with x = 1, provided that BO is replaced by B’ = B 2(1-d) 0 O 2-6 In this approximation,
we see that the scalar potential, I&(4,(T,n
= 0) -&c
= $&8(V2
In Eq. (lo), with x = 1, the glueball equation
(‘4.6)
. I& takes the simple form .
- 1 -lnv*)
(‘4.7)
of motion can be dropped and we are left
with the single scalar equation a;Dv - 2B06u = Mps - Boa (’ i-V2) v Taking care that the energy is consistently occ E = 5(2j, a +
obtained cc
+ 1)-2x
-m~R,w&.
(A.81
we find
drr*{Myp,
+
gd’-)OpB
+
gpbop3
s 0
2BoS In u - m~R,v*w~}
.
64.9)
Using the low density expansion for nuclear matter and replacing BO by Bh in Eqs. ( 19) and (20) this expression gives the same result as the complete result of Eq. (11). In practice to saturate nuclear matter correctly a small adjustment of C’i is needed and BA is only approximately given by ( 17).
732
E.K. Heide et al. / Effective lagrangian References
[ 11 J. Schechter, Phys. Rev. D21 (1980) 3393; A.A. Migdal and M.A. Shifman, Phys. Lett. B114 (1982) 445 [2] J. Ellis, J.I. Kapusta and K.A. Olive, Nucl. Phys. B348 (1991) 345 [3] P.J. Ellis, E.K. Heide and S. Rudaz, Phys. Lett. B282 (1992) 271; B287 (1992) 413 (E) [4] E.K. Heide, Ph.D. Thesis, unpublished (1991); I. Mishustin, J. Bondorf and M. Rho, Nucl. Phys. A555 (1993) 215 [5] G. Ripka and M. Jaminon, Ann. Phys. (NY) 218 (1992) 51; Nucl. Phys. A564 (1993) 505 [6] R.G. Rodriguez and J.I. Kapusta, Phys. Rev. C44 (1991) 870 [7] E.K. Heide, S. Rudaz and P.J. Ellis, Phys. Lett. B293 (1992) 259 [S] R.J. Fumstahl and B.D. Serot, Phys. Rev. C47 (1993) 2338 [9] J. Boguta, Phys. Lett. B120 (1983) 34 [lo] B.D. Serot and J.D. Walecka, Advances in nuclear physics, vol. 16, eds. J.W. Negele and E. Vogt (Plenum, New York, 1986); B.D. Serot, Rep. Prog. Phys. 55 (1992) 1855 [ 111 H. Gomm and J. Schechter, Phys. Lctt. B158 (1985) 449 [ 121 R.J. Crewther, Phys. Rev. Lett. 28 (1972) 1421; M.S. Chanowitz and J. Ellis, Phys. Lett. B40 (1972) 397; Phys. Rev. D7 (1973) 2490; J.C. Collins, A. Duncan and S.D. Joglekar, Phys. Rev. D16 (1977) 438 [ 131 A.I. Vainshtein, V.I. Zakharov, V.A. Novikov and M.A. Shifman, Sov. J. Part. Nucl. 13 (1982) 224 [ 141 C.J. Horowitz and B.D. Serot, Nucl. Phys. A368 (1981) 503 [ 15 ] 0. Dumbrajs et al., Nucl. Phys. B216 ( 1983) 277; J. Hamilton and G.C. Oades, Nucl. Phys. A424 (1984) 447 [ 161 M.A. Shifman, AI. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 448 [ 171 A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn and V.F. Weisskopf, Phys. Rev. D9 (1974) 3471; T. DeGrand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060 [ 181 J.M. Pearson, Phys. Lett. B271 (1991) 12 [ 191 S. Rudaz, P.J. Ellis, E.K. Heide and M. Prakash, Phys. Lett. B285 (1992) 183 [20] M.A. Shifman, Z. Phys. C9 ( 1981) 347; P. Pascual and R. Tarrach, Phys. Lett. B113 (1982) 495 [21] M. Gell-Mann and M. Levy, Nuovo Cimento 16 (1960) 705 [22] S. Gasiorowicz and D.A. Geffen, Rev. Mod. Phys. 41 (1969) 531 [23] J.W. Negele, Phys. Rev. Cl (1970) 1260 1241 L.J. Tassie and F.C. Barker, Phys. Rev. 111 (1958) 940 [25] H. de Vries, C.W. de Jager and C. de Vries, At. Data and Nucl. Data Tables 36 (1987) 495 [26] C.E. Price, J.R. Shepard and J.A. McNeil, Phys. Rev. C41 (1990) 1234; C42 (1990) 247 [27] I. Sick and J.S. McCarthy, Nucl. Phys. A150 (1970) 631 [28] A.R. Bodmer, Nucl. Phys. A526 (1991) 703; S. Gmuca, Z. Phys. A342 ( 1992) 387; Nucl. Phys. A547 (1992) 447 [29] R.J. Fumstahl, C.E. Price and G.E. Walker, Phys. Rev. C36 (1987) 2590 [30] R.J. Fumstahl and B.D. Serot, Phys. Lctt. B316 (1993) 12
Nuclear Physics A571 (1994) 7 13-732
North-Holland
An effective lagrangian with broken scale and chiral symmetry applied to nuclear matter and finite nuclei Erik K. Heide, Serge Rudaz, Paul J. Ellis School of Physics and Astronomy, University ofMinnesota, Minneapolis, MN 55455, USA
Received 3 August 1993 (Revised 13 October 1993) Abstract We study nuclear matter and finite nuclei in the mean field approximation with a chiral lagrangian which generalizes the linear u model and also accounts for the QCD trace anomaly by means of terms which involve the u and I fields as well as the glueball field 4. The scale invariant term which leads to an omega meson mass, after symmetry breaking, is strongly favored to be of the form o,~&$~ by the bulk properties of nuclei; they also rather strongly constrain the other parameters. A reasonable description of the closed shell nuclei oxygen, calcium and lead can be achieved and this can be improved by including a term (w&wp)2 in the lagrangian. This leads to a level of agreement with the data that is comparable to that of the nonlinear Walecka model.
1. Introduction We are still far from being able to transform the lagrangian of quantum chromodynamits (QCD) into an effective lagrangian involving mesons and baryons which could be used at the mean field level for nuclear matter or finite nuclei. Nevertheless it is possible to incorporate the broken global chiral and scale symmetries of QCD into an effective lagrangian. In particular the notion of (broken) scale invariance has been modeled by a scalar glueball potential [ 1). An early attempt to introduce this into an approp~ately modified Walecka-type lagrangian was made in ref. [ 21. An alternative approach is to include the glueball potential in the lagrangian of the linear sigma model, supplemented by a contribution from the vector-isoscalar omega field [ 3,4] (for related work in the quark sector see ref. [ 51). The sigma model is particularly attractive since it incorporates (spontaneously broken ) chiral symmetry, another feature suggested by QCD. A common problem associated with these models is that the compression modulus for equilibrium nuclear matter is at least a factor of two larger than current estimates in the range 200300 MeV. Of course one can arbitrarily correct this by adding terms to the lagrangian, for instance cr3 and 04, and choosing the parameters to produce the desired result [6]. We have recently suggested [ 71 a more satisfactory approach, however, where the form of the potential which breaks scale invariance is modified in a reasonable way to include 0375-9474/94/$07.00
@ 1994 - Elsevier Science B.V. All rights reserved
SSDI0375-9474(93)E0636-M
714
a contribution
E.K. Heide et al. / Eflective lagrangian
from the cr field as well as the glueball field, 4. This was found to yield
K M 390 MeV, which, while still on the large side, is decidedly more reasonable. There still remains uncertainty
concerning
the form of the omega mass term, fm&+wP,
where the explicitly dimensionful
parameter
rni must be replaced by the square of a scalar
field (multiplied by a dimensionless coupling constant) in order to satisfy the dictates of scale invariance. The question arises as to whether the coupling be to a2 or 4’ or some linear combination of the two. In refs. [ 3,7] the first choice was made. One of the purposes of the present paper is to examine this choice in nuclear matter and finite nuclei; we shall find that the properties of finite nuclei are sensitive to this question. A more basic purpose of this investigation is to determine whether our effective lagrangian, which saturates nuclear matter in the mean field approximation, is able to give a sensible account of finite nuclei. This is particularly pressing since Fumstahl and Serot [ 8 ] have recently suggested that chiral models are inherently unable to give a satisfactory description of nuclei; this work was based on the approach of Boguta [ 91 and did not include the glueball field, so there was no attempt to incorporate broken scale invariance. The effective lagrangian that we employ, together with the necessary formalism for finite nuclei and nuclear matter is given in sect. 2. Our results are displayed in sect. 3, first for nuclear matter and then, in subsect. 3.2, for the closed shell nuclei 160, “‘%a and *‘*Pb. In subsect. 3.3 we discuss a refinement of the basic model which improves agreement with the data. Our conclusions are given in sect. 4. It is of interest to compare the low density expansion to that of the standard Walecka model [lo], this we discuss briefly in the appendix. This suggests a simple approximation in which the glueball field is frozen at its vacuum value and the formalism is outlined in the appendix. Examples are given in the text which show that the frozen glueball model reproduces our complete results quite accurately.
2. Theory We write the effective lagrangian
in the form
&T = LO -
v, ,
where in this schematic equation we have separated out the scale invariant part, Lo, from the potential Vi which induces the breaking of scale and chiral invariance. (The effects of explicit chiral symmetry breaking arising from the nonvanishing of current quark masses in the QCD lagrangian are neglected.) We write Lo in terms of the chiralinvariant combination of sigma and pi fields, tr and n, the glueball field q!~and the field of the omega vector meson wg. Specifically
(2)
715
E.K. Heide et al. / Eflective lagrangian
Here the field strength tensor is defined in the usual way, Fpy = d,w, - &w,. As we have remarked in the introduction, the form of the scale invariant o mass term is not a priori obvious, exotic choices In order to divergence of
so we have chosen a linear combination cr2 and 4’ fields (of course more are possible) and we shall examine the effect of varying the coefficients. obtain the scale breaking term & in the lagrangian, we recall that the the scale current in QCD is given by the trace of the “improved” energy-
momentum tensor and so the scalar potential Noether’s theorem, the effective trace anomaly
VG(4, CJ,n) is chosen to reproduce, [ 1,111
via
where @i runs over the scalar fields (4, g, n} and EVat is the vacuum energy. The proportionality 0; 0: $4 is suggested by the form of the QCD trace anomaly [ 121, e;(x)
= y
F;v(x)F”““(x) >
where F& (x ) is the gluon field strength tensor and /3 (g ) is the usual QCD beta function. Eq. (3) involves a purely gluonic, color-singlet, scalar, dimension-four operator, and also allows for the recovery of the low energy theorems which follow from broken scale invariance [ 13 1. Then in ref. [ 7 ] we suggested the form
where c = &,/a~ and the subscript 0 indicates the vacuum value. Here the logarithmic terms contribute to the trace anomaly and are such that Eq. (3) is satisfied with eVac= -$B4:(1 - 6). Thi s requirement uniquely specifies the second term on the right in Eq. (5). The third term is needed to ensure that in the vacuum 4 = 40, g = a0 and n = 0. To retain the physically necessary feature that EyI?c< 0, given that B > 0, we require that + 00 as is physically sensible. For further 6< l.ProvidedO<6< l,Vo+~fordor4 insight regarding the parameter 6, we recall that the QCD beta function and nf flavors is given at the one loop level by /3(g) =
with NC colors
-$$ (1- z) +W&?) 3
where the first number in parentheses arises from the (antiscreening) self-interaction of the gluons and the second, proportional to nr, is the (screening) contribution of quark pairs. Recalling Eq. (4), a value of 6 = 4/33 is suggested for the present case with nr = 2 and NC = 3. Since one cannot rely on the one-loop estimate for j?(g), we shall examine the effect of modifying the value of S.
E.K. Heide et al. / Effective lagrangian
716
The careful reader will have noticed that L:ee, given by Eqs. (2) and (5), does not contain the standard potential term of the linear sigma model which we have previously written [3,7] in scale invariant form as f1(02 + n2 - 42/[2)2. We found that very small values of ;i were favored by the predicted compression modulus in nuclear matter. We have also found that departures from small values yield binding energies of nuclei which are much too low. Therefore in this presentation we make the welcome simplification of setting I = 0, thus eliminating this term. Note that while the term is present in the standard linear sigma model, it is not necessary for the breaking of chiral invariance. This can be carried out just as well by Vo in Eq. ( 5 ). Since we are interested in finite nuclei it is necessary to include the vector-isovector rho field B, and the Maxwell field A,, in our total lagrangian. Thus we take &,a = I& + .C’, where ,C’ =
f p”- ;B,, . B#” + ;Gpqi2b,. b”
- +
-Ny’[ig,b,.r+
ie(l
+q)A,]N.
(7)
Here the field strength tensors are fpy = +A, - &A, and B,, = a,b, - &b,. Since the effect of the p meson is relatively small, particularly in light nuclei, we have taken the simplest form in L’ which is not chirally invariant. Note, however, that we have made the p meson mass term scale invariant by coupling to 42, we shall comment further upon this later. It is economical to write the sigma and glueball fields in terms of their ratios to the vacuum values, viz. x(r)
=
For r -+ 00 we obtain the vacuum
4(r)
-5
v(r)
40
= $.
(8)
values x = v = 1. Now, in the vacuum,
we require
that the rho and the omega masses take their physical values so that we can write the mass terms in the form ~rn~x2b, - b”
;rnZ, [Rov2 + (1 - &)x2]
and
wPwP.
(9)
Thus, for the omega, taking R. to be 1 gives a pure u coupling, whereas the value 0 yields a pure x coupling. The effective mass of the nucleon, in the usual way, is M’ (r ) = Mv (r). In the mean field approximation n = 0 and the field equations are obtained from Lagrange’s equations. We specialise to the time independent, spherically symmetric case. The Dirac equation for the nucleon and the equation for the photon field are of the form given by Horowitz and Serot [ 141 and do not need to be repeated here. As usual for the vector fields only the time-like components survive and also just the isovector z-component for the p field. The equations for the 4, u, w and p can be written 4;Dx
- 2Bo(2 - 8)x = 4B0[x3(lnx
- mi(l
+ 2BoSu
-6lnv)
- Rw)w&
-x
+ 6~1 + BoS[x(v2--x2)
- rnzb,&
,
+ 2(x -v)]
717
E.K. Heide et al. / Effective iagrangian
2Bo8v + 2BoSX = Mps - BoS Doo - rniwo Dbo-
=
-gdB
+
mibo = -gpp3 -
x; - X2Y -I 2(V -X)
m~[R,(v2mi(l -x’)bo,
- miRow&, >
1) + (1 -&)(x2-
1)100,
(10)
where the densities ps = (NN), pB = (iVyoN) and p3 = f (~y”73N) can be expressed in terms of the components of the nucleon Dirac spinors in the usual way [ 141. In Eq. (10) we have made the definitions D = d2/dr2 + (2/r)d/dr and Bo = Bq$. The terms linear in the fields, i.e., the kinetic energy and mass terms, have been separated out on the left of these equations. The equations of ref. [ 71 for nuclear matter are regained by setting the derivatives to zero so that the results are independent of 40 and a,-,, however these quantities are needed for finite nuclei. They are also needed to obtain the vacuum scalar masses and we note that the mass matrix is not diagonal. In order to solve Eqs. ( 10) by an iterative Green’s function technique [8,14] it is necessary to go to a representation which is diagonal in the limit that r + CCLWe solve in terms of the quantities ( 1 - x ) and ( 1 - v ) which go to zero in the same limit. The energy-momentum tensor can be used to obtain the total energy of the system in the standard way [ lo]. Subtracting constants so that the energy is measured relative to the vacuum, we obtain 00 E
=
5,(2j,
+ 1)-2x I
a +
drr2{Mvps
+
gowOpB
+
gpbOp3
0
2Bo[x4(lnx
- 6lnv
+ $) - $1 + iBoJ[x2(2v2-
- mi[R,v' + (1 - Rw)X2]wi
- m~X2b~}.
3x2) + l] (11)
In the first term on the right the ea are the Dirac single particle energies and jm is the total angular momentum of the single particle state. In nuclear matter this term becomes 4 c, (&oo + Jk* + M*2). Making this replacement, and using Eq. (lo), the expression for the energy density of nuclear matter given in ref. [ 71 can be obtained.
3. Results 3.1. NUCLEAR MATTER In nuclear matter we have four parameters to consider. We will choose Bo and Ci = g;M2/mi to fit the saturation properties. The binding energy/nucleon we take to be 16 MeV. In refs. [ 3,7] we took a saturation density of 0.16 fmm3, but this yields central densities in lead which are too high. We have therefore used 0.148 fmm3 (kF = 1.3 fm-’ ), which is the value favored by Serot and coworkers in mean field calculations [ 14,8].
718
E.K. Heide et al. / Efective lagrangian TABLE 1 Values of the parameters and the
derived quantities for nuclear matter
Set
R,
336
I IF II III IIIF IV V VI VII
0.0 0.0 0.5 1.0 1.0 0.0 0.0 0.0 0.0
4 4 4 4 4 1 2 8 16
235 231 255 269 265 336 281 194 160
156 152 82.6 51.3 49.2 153 154 158 156
116 123 127 132 135 122 119 107 88
0.71 0.72 0.80 0.84 0.85 0.72 0.72 0.69 0.65
383 341 356 377 356 350 360 442 666
6.5 5.6 5.9 6.1 5.7 5.8 6.0 7.5 10.6
VIIIF
0.0
4
214
227
114
0.65
267
4.4
Having
WK
fixed these two parameters,
we then have the parameter
of the o mass term, R,,
and the coefficient of the quark contribution to the trace anomaly, 6, which can be varied. We show in Table 1 several parameter sets that we have considered. The designation F here indicates the frozen glueball model where x is kept equal to 1; see discussion in the appendix. (Parameter set VIIIF is for later use in subsect. 3.3 and will be excluded from the present discussion.) One might hope that the parameters would show a reasonable correspondence with the independently determined quantities, although precise agreement would not be expected for an effective theory such as this. In this sense the values for C’i in Table 1 are in acceptable agreement with the value 103 f 36 deduced from data on the o coupling constant [ 151 and the known mass. Turning to ]evac],QCD sum rule studies suggest [ 16 ] a value of (240 MeV)4 which would not favor the extreme variations of 6 (sets IV and VII), although bag model estimates [ 171 range as low as (146 MeV)4. The values of Cj, given in Table 1 for later use, have been obtained by fitting to a symmetry energy of 35 MeV. Since exchange effects are known to make a significant contribution to the symmetry energy we shall not make a direct comparison to experiment. Turning to the derived quantities, nuclei generally favor effective masses below 0.7 [ 81, which is the case for sets VI and VII. However, here the compression moduli, K, are in excess of 400 MeV. Pearson [ 181 has made the point that the leptodermous expansion, which is one method used to fit the data, permits values of K up to about 400 MeV with little difference in the x2 fits. Defining the third derivative of the binding energy per particle as S = kz (d3/dk:) (E/A) (evaluated at equilibrium), the data indicate a linear relation between S/K and K (see ref. [ 191). We have listed values of S/K in Table 1. Cases VI and VII are welI off the allowed error band, but the remaining cases are quite close to, although slightly off, the band. The parameter sets designated with an F in Table 1 correspond to the frozen glueball model (x = 1) and we see that they give a close correspondence with the complete
E.K. Heide et al. / E#.?ctive la~angi~~
719
300
____ _____._.._... se+
250
-_-
..-..-..-..-..-. - .-
200
%c z
,F
Set ill set
,,,F
150
I I q W
loo
50
0
0
1
2
3
4
5
6
P/P, Fig. 1. The binding energy/particle as a function of density for nuclear matter; the density p is given as the ratio to the equilibrium value p. = 0.148 fmm3. The parameter sets used are indicated and the designation F indicates the frozen glueball model.
results. The point is further made in Figs. 1 and 2 where the agreement is seen to be particularly good at low density, as we argue in the appendix. The lower panel of Fig. 2 shows that in the complete model x remains very close to unity over the whole region of densities considered. We remark that the curves for v will increase linearly with kF for sufficiently large density (a discussion of the high density behavior was given in ref. [ 3) ), although it is debatable
whether this theory is applicable
in that region.
3.2. FINITE NUCLEI When we turn to finite nuclei two extra parameters are involved, namely the vacuum values of the scalar fields, $0 and 00. Thus we have a total of four free parameters. Let us first eliminate two of these. We shall discuss changes relative to parameter set I of Table 1 with a0 = 110 MeV, which will turn out to be our preferred parameter set in this subsection. First consider reducing 6 by using sets IV and V of Table 1 (and reasonable values for the parameters not specified). We find little sensitivity in the predicted properties of nuclei. However, if we increase 6 using sets VI and VII, for which K is large, we find a significant reduction in the binding energy, particularly for set VII (40Ca is less bound by 2: MeV, for example). We conclude that a value of 6 in the nei~borho~ of 4/33 is
E.K. Hetie et al. I E$%vtive~a~ran~i~
720 1.0 0.8 A
0.6 0.4
_________ ___.__._Set 0.2
IF
--Set Ill t ___ .._.._..-.I_..-.. - Set l#F
1.2
1.1 x 1.0
0.9
0.8
Fig. 2. The fields v and x corresponding to Fig. 1.
reasonable and fix on this value, although nuclei are not sensitive to the precise figure. Next consider the ratio of the vacuum scalar fields, ( = #o/~~ Using set I we have varied C in the range 0.7-2.1, which corresponds to varying the higher scalar mass, ma, between 3 and 1 GeV. Very little change is observed in the nuclear properties. The salient point is that the mixing between the glueball and the sigma is small, d 3%, and it is the sigma which directly couples to nucleons. Henceforth we choose (, such that nz, = 1.5 GeV because this seems reasonable in view of QCD sum rule estimates for a scalar glueball mass in the range 1-2 GeV 1201. The values of C are listed in Table 2. We are then left to consider the variation of R. and a~: a few comments on the expected magnitude of o. are in order here. In the original Gell-Mann-L&y linear sigma model [ 2 11, a calculation of the matrix element of the axial vector current responsible for pion decay led to the identification CO = .fX = 93 MeV. In a more general chiral effective lagrangian in~~mting vector mesons (see ref. [22] for an early review) this identification no longer follows due to the necessity of including the al axial vector meson, together with the p. Briefly, cross-terms of the form (I: - Qc appear in the full lagrangian which are eliminated by the replacement a: --e a: + &Pa, with an appropriate parameter t;. This, in turn, leads to a change in the coeffkient of the pion kinetic term, 0,~ * Pit, and a resealing of the pion field is necessary to bring this to canonical form. One defines a ~nor~liz~d field SR = z I/2=, Now a calculation of the full axial current yields
721
E.K. Heide et al. /Effective lagrangian TABLE 2
Bulk properties of nuclei for various parameter sets Set
m<
0
c
(MeV) Experiment I/110 IF/l 10 I/93 I/120 II/l 10 III/l 10
506 508 598 464 598 664
1.4 1.4 1.4 1.6 1.8
VIIIF/ 100
481
-
Ca
Pb
BEIA (MeV)
ret, (fm)
BE/A (MeV)
rch (fm)
BE/A (MeV)
ret, (fm)
1.98
2.73
8.55
3.48
7.86
5.50
7.35 7.86 9.41 6.25 10.08 10.98
2.64 2.62 2.52 2.72 2.49 2.44
1.96 8.35 9.38 7.16 9.85 10.45
3.41 3.40 3.32 3.41 3.29 3.26
1.33 7.54 8.04 6.92 8.22 8.48
5.49 5.49 5.45 5.52 5.45 5.44
7.18
2.69
7.91
3.45
1.44
5.53
a0 = Z:‘*fn. The precise form of Z, is model dependent. In the simplest model, which does not include the physics of broken scale invariance one finds [22], using the KSRF = fi. We do not expect either this precise functional relation, that Zi/* = m,,/m, form or value to be maintained in more realistic models, but we do anticipate Z,“* > 1. Accordingly, as is kept as a free parameter, which we can presume to be larger than 93 MeV. Two further remarks are in order. First, the additional terms involving the al will not contribute in the mean-field approximation. Second, from Eq. (2) after symmetry breaking, the nucleon mass is given by A4 = gas. The physical pion-nucleon coupling is defined in terms of the renormalized pion field to be gnNN = Z,“*g. So, in fact, one still has M = go0 = g*NNfn as the approximate form of the Goldberger-Treiman relation. With this in mind, we first consider variation of tre keeping R, = 0. The results are given in Table 2, where the notation I/93, for example, implies use of parameter set I from Table 1 with as = 93 MeV. (Again we remark that parameter set VIIIF is for later use in subsect. 3.3 and is excluded from the present discussion.) The theoretical values of the binding energy/particle include a correction for the c.m. kinetic energy [ 231. The charge densities, and therefore the radii quoted, are corrected for the finite size of the proton [ 141 and for c.m. effects [24]. We see that the choice 00 = 93 MeV (fourth row in Table 2) yields poor binding energies, radii which are too small and, as shown in Fig. 3 for %a, an oscillatory charge density. This figure indicates that increasing a0 to 110 MeV improves matters greatly. One can view this as reducing the magnitude of the derivative of v in Eq. ( 10) or as reducing m< to - 500 MeV (Furnstahl and Serot [ 81 stress that values in this neighborhood are to be preferred). In fact as regards the charge radii and densities the value 120 MeV gives even better results, however we see from Table 2 that in this case the binding energies/particle are l- 1i MeV too low. Therefore we prefer the parameter set I/ 110. Now we consider variation of R, with a9 = 110 MeV. As R, is increased from zero
722
E.K Heide et al. / Efiective lagrangian
0.08
0.02
0.00 0
1
2
3
4
5
6
7
f- k-9 Fig. 3. Comparison of the experimental charge density 1251 for %a for the parameter sets indicated.
with theoretical predictions
to 0.5 or 1.0 we see that the binding energies become too large and shows the wrong A-dependence and also the radii become too small. Table 2 also shows that the mass of the lighter scalar meson, m,: , increases sharply. Further, as we see for “sPb in Fig. 4 the charge density begins to develop oscillations which are not present in the data. Similar problems were noted by Furnstahl and Serot [ 81 who used R, values of 0.76 or 1, although their model differs significantly from ours. As pointed out by Price, Shepard and McNeil f26] this is likely to signal the onset of a situation where the ground state of nuclear matter is no longer uniform, but exhibits periodic density fluctuations. We conclude that nuclei strongly favor Rw = 0. We have also compared the case I/ 110 with the frozen glueball approximation, case IF/l 10, in Table 2. There is a close correspondence between the results, as there is for the predicted charge densities which we illustrate for I60 in Fig. 5. Finally we turn to the single particle energies. The levels near the Fermi surface in lead are shown in Figs. 6 and 7 for neutrons and protons respectively. The relative positions of the neutron and proton levels are sensitive to the treatment of the p meson [ 141. We have used a scale-invariant 4’ coupling in the mass term, but the results differ negligibly if a standard mass term is used. However there is a difference if a t-r2coupling is used because this reduces Ci and leads to N 2 MeV less (more) binding for protons (neutrons). Since
723
E.K. Heide et al. / Eflective lagrangian
0.08
0.07
0.06
s
0.05
x .z ii
0.04
n 8-l
“‘Pb
0.03
b 4
Experiment .._._ _._. Set 1/110 --~ - - -..- - - - Set 111/110
0.02
0.01
0.00 0
1
2
3
4
(fm)5
6
7
8
9
r
Fig. 4. As for Fig. 3, but for 20*Pb.
Figs. 6 and 7 clearly show that this is undesirable we have not considered this possibility further. The results with the sets II/l 10 (R, = 0.5) and I/93 show immediate problems since the major shell closure is incorrectly given due to the position of the proton 1h9,2 and the neutron 1il 1,2 levels. This problem disappears with the preferred parameter set, I/ 110, and the single particle spectrum is much more reasonable, although a larger proton 1h9,2-3s1,2 splitting would be desirable. This may reflect the spin-orbit splittings which are, on average, 65% of the experimental values here and also in the other nuclei we have considered. They could be improved by reducing the effective mass in Table 1 below 0.7. This can be achieved by taking a small negative value for Rw, although we find that this leads to a larger compression
modulus
and lower binding
energies.
3.3. FINE TUNING OF THE MODEL While the results obtained for finite nuclei in the preceding subsection are reasonable, it is natural to ask whether significant improvement can be achieved by including additional terms consistent with the scale and chiral properties of our effective lagrangian. We have not made an exhaustive study, but we have investigated the effect of including an additional term of the form (o+~#)~. Such a term is automatically scale invariant and has previously been studied in connection with the Walecka model [ 281. For present
E.K. Heide et al. / Eflecticrive lagrangian
724
0.10
0.08
m5i “r; 25
0.06
x 5
5
cl
6,
0.04
P 0 if 0.02
0.00 2
3
4
5
t- b-d
Fig. 5. As for Fig. 3, but for lag. The data are from ref. 1271.
purposes it is sufficient to employ the frozen glueball model and it is straightforward to extend the equations to include the additional term in the lagrangian which we write as [ (G~)2~rop]2; note that this has the same sign as the w mass term. Bearing in mind the previous results, we fix R, = 0 and 6 = 4/33 and refit nuclear matter. We then have Cd/g” as a free parameter and we judge that the best results for nuclear matter and nuclei (here 00 = 100 MeV) are obtained with Ga/g, = 0.21. In fact two sets of parameters can be obtained with this value of G4f go, both of which yield the~odyn~ica~y stable solutions. The parameters we select are Iisted in Table 1 as set VIIIF; the second solution we reject on the grounds that it corresponds to a 50% larger value of C& a 25O41larger compression modulus and a rather low value of M& = 0.504. With our chosen parameters we find that relative to the standard omega mass term, the (G~wo )4 contribution to the equations is a 2OW effect at equilibrium nuclear matter density; the effect is smaller at lower density. Table 1 shows that, in comparison to set I, the present value of Ci shows a substantial increase (it is about a factor of two larger than the experimental value quoted earlier). The effective mass M& is slightly smaller which will improve the spin-orbit splittings. Also the compression modulus K has been significantly reduced and lies within the often-quoted range of 200-300 MeV. This reflects the softening of the equation of state. In fact the point defined by K and S/K lies just within the allowed error band referred to in subsect. 3.1.
125
E.K. Heide et al. / Efective lagrangian 2 0
-~
__=
.--------_
2gs,2
__--
-2
/
,*,’
-4 % 3
h/2
-’
--._. *-
*. ‘\
-6
$ ‘P l5
-6 -10
3 r CY
-12
a, P
-14
_A::*" 3py2: 2f5/2+-_:;::::---
*.
-
..--;_.-____ ,- -;,*-
__-3P3/2 ‘~::_--_-::.r__ _---__ lhJ/2 _--2f,,z ----_____.**-+ lh9/2
-
___+-I_-
_-’
____----...;
.lily2 3PVZ 3~~2 2f5/2 ,2f7/2
lb/z
\\
iz -16
-16
-20 EXPERIMENT
set l/l10
set II/110
set l/93
-22
208 Pb Neutron Energy Levels Fig. 6. Occupied and unoccupied neutron levels near the Fermi energy in *OsPb.The experimental data are compared with predictions for the parameter sets indicated.
Turning to finite nuclei, our aim is to compare our results with the best that can be obtained in mean field calculations with the Walecka model extended to include nonlinear o3 and o4 terms. As a representative of the latter we choose parameter set C of Fumstahl, Price and Walker [29] (designated set B in ref. [ 81) which we refer to as FPW here. The comparison with our results using parameter set VIIIF/ 100 for the charge densities of I60 40Ca and *08Pb are shown in Figs. S-10. It is clear that there is close agreement between the two calculations, both of which yield a good account of the data. In Fig. 11 we make the corresponding comparison for the single particle levels of *“Pb. Again the two calculations agree quite closely and yield a fairly reasonable account of the experimental spectra. It is noteworthy that the separation between the occupied and unoccupied levels is much better with set VIIIF/ 100 than was the case with set I/l 10 in the previous subsection. The spin-orbit splittings are also better - on average they are 82Oh of the
E.K. Heide et
726
al. / Efective iagrangian
2
,-
0
f a 8 ‘p 9 a, 9 tB
u
ol .E v,
-2
-
lh3/2
-*_,*
2$/z
-‘_
-4
_
lb/z:
-
‘.
..
-.
.I
-8
-10
.
h/z
_
2ck/2
‘.
_’
\\ ‘. ,_._:~~___-----
\\
\\
I-* .,. ___-_ ---
_- _:=; -._J*.
I__-____.---
t&2
--.c --__
___-’ __.-
,/-
-12
-
*\
c3sy2
,h9/2
___---
2&/2
_____
, 2dy2
------_-197/2
2f7/2
*-
-6 3%/Z ‘, 2d3/2 -_:::::_
_---
I’ ~‘-‘-l______-r,’ .. I’ ,I .\ .___....---I’ ,I
lhJ@
%, ‘*. ‘,
-14
<\ -
‘\ lgs/2
-16
-._ -.
-.
‘\
\.
-.
lg,/2
~.___Lr----
*.
‘----._ --__ ---._ *-. ---
EXPERIMENT -22
set l/110
set II/110
lgs/2
set l/93
t 208
Pb Proton Energy Levels
Fig. 7. As for Fig. 6, but for protons.
experimental values versus 65% with set I/ 110 and 88% for FPW. The binding energies and radii obtained with our parameters are given in the last row of Table 2. As would be expected from the charge distributions we obtain radii that are in reasonable agreement with the data and with the predictions of FPW (not shown). Our binding energies are low by - $ MeV. This is a modest discrepancy, however FPW do better in this respect with deviations ;5 f MeV.
4. Conclusions Our primary purpose has been to see whether an effective lagrangian which includes the breaking of scale and chiral invariance can make sensible predictions for finite nuclei. Finite nuclei are expected, and found, to be a much more severe test than nuclear matter. In this endeavour we discarded the conventional quartic potential term of the linear
E.K. Heide et al. / Efective lagrangian
727
x
T&
c
n
& b
0.04
6
-
160
, 0.02
o.oo
v I
0
1
2
3
4
5
r k-9
Fig. 8. The charge density of I60 obtained with parameter set VIIIF/lOO, which includes an (~+ofl)~ contribution, is compared with that of the nonlinear Walecka model denoted by FF’W [ 291. The experimental data are also shown.
sigma model. Instead we used the potential Vo of Eq. (5) to spontaneously break chiral invariance as well as to explicitly break scale invariance in accord with the trace anomaly. The form of the scale invariant mass term for the w meson was found to be critical and a pure coupling to the glueball field, dew,&‘, was required, although a small effect from the sigma field cannot be ruled out. A similar coupling for the p mass term was also favored. It is interesting that the properties of nuclei can be used to provide such strong constraints
on the form of the coupling of the scalar glueball to mesonic fields appearing
in the chiral effective lagrangian. In the minimal version of the model we found it necessary to increase the value of the vacuum value of the sigma field, a~, from the naive expectation of 93 MeV to somewhere in the neighborhood of 110 MeV. With these caveats the bulk properties of 0, Ca and Pb were quite well accounted for and, while the spin-orbit splittings were about 65% of the desired values, the basic structure of the single particle spectra was reasonable. As a by-product we were able to define a simplified, frozen glueball model which was able to reproduce the results of the complete model quite accurately, but, by itself, would seem highly arbitary. By introducing a term (o,o’)* into the lagrangian and adjusting the strength, im-
728
E.K. Heide et al. / Effective lagrangian 0.10
0.00
0.02
---
0.00
0
1
2
3
4
5
6
7
r b-9
Fig. 9. As for Fig. 8, but for 40Ca.
proved results for nuclear matter and finite nuclei were obtained with 09 = 100 MeV. We mention in particular the compression modulus of 267 MeV and the major shell separation at the Fermi surface in Pb. In fact our results are quite comparable to those obtained with the standard Walecka model extended to include nonlinear c3 and o4 terms [ 29,8]. It is remarkable that the present effective chiral lagrangian, whose form is motivated by QCD, can give an acceptable description of both nuclear matter and finite nuclei in the mean field approximation.
We thank G.F. Bertsch and A. Vischer for useful comments and C.J. Horowitz for giving us a copy of his computer code for finite nuclei which was suitably modified for present purposes. One of us (PJE) is grateful for the hospitality provided by the Institute for Theoretical Physics of the University of California at Santa Barbara where a portion of this work was carried out. We acknowledge partial support from the Department of Energy under contracts No. DE-FG02-87ER40328 and DE-AC02-83ER40105. A grant for computing time from the Minnesota Supercomputer Institute is gratefully acknowledged.
E. K. Heide et al. / Eflective lagrangian
729
0.06
‘t &
0.05
.f=
E
0.04
2 th
0.03
5 6
0.02
0.01
0
1
2
4
3
6
7
8
9
Fig. 10. As for Fig. 8, but for 208Pb.
Note added After this work was completed
we received a preprint
from Fumstahl
and Serot [ 301
who have studied the same problem, restricting themselves to an o mass term of the form a20,,wJ’. In agreement with our conclusions, they point out that the results for nuclei are poor. As we have stressed, nuclei require that the coupling be to the glueball field rather than the sigma field.
Appendix LOW DENSITY EXPANSION
AND FROZEN GLUEBALL MODEL
It is of interest to study the low density expansion of Eqs. ( 10) and ( 11) in the case of nuclear matter. Examination of Eq. ( 10) indicates that we can expand v
= l+apBU+bk:)+O(k~),
x
= 1+
cpB(1
+
dk;)
+
o&j,
(A.11
where pB is the baryon density and kF the Fermi momentum. Further to this order the coupling between WC,and v and x can be neglected. Then the equation of motion for x
f_i .-
f? %
z t
--.
-
2f7/2
IhS&
‘. *. *. \. ‘. .___“,c-----
------__
Set VlllF/tOO
‘.
--
FPW
.’ _'
208 Pb Neutron Energy Levels
‘,
--. --._
--
--
_-_“____L’,
-1..-----__ ---‘=‘-----,____-_-_-,---~-----_ ---_-_-____:-
--_ --_--
_----_--___-_ -- - __.--',' --__ ___-------
EXPERIMENT
lii3/2
-
----__ *,_--
\,
----
-
\-I"-
31)3/Z
2fy
3~~2
299/z
Cl/Z
1115/Z
lhg/;l
2f7/2
w2
Qv2
*.
‘\
\*
‘.
*.
‘.
____-------
** ------...._*-_
‘r
-..c::::,:=
set VlllF/100
** *I
*-. ‘.
-IS._
_-c-c- -----
---___
~~~~~~~~~_~-
,P----llll”.e-
,’
d
-Y
208 Pb Proton Energy Levels
EXPERIMENT
-
-
8.
,,____“--._-----------~--
---_ --.,
,*
,*’
.’ ,’ *’ +’ .” .--
,’
,I’
-----Ic__” 2d3/2 lhql/2 ~\_-__---“------1”.“..____ 2&/z
3$..
--.._
3PVJ 3P3/2 I lh3/2 2f5/2
I
-
lhw2
lill/2
-’
1tr3/2 ?f7/2
%9/2
II
kJw
‘v/2
=/2
lb2
#’ 2%2
3s1/2
lh9/2
2f7y2
lil3/2
_
Fig. 11wOccupied and unoccupied, neutron and proton levels near the Fermi energy in 208Pb obtained with parameter set VIIIF/lOO, which includes an (o~w~)~ contribution, are compared with those of the nonlinear Walecka model denoted by F’PW [ 291. The experimental data are -I-- 3.-_
-20
-18
-16
-14
-12
-10
-8
-6
a .s
P ii W
-4
%c
-2
0
2
E.K. Heide et al. /Effective lagrangian indicates
731
that d=b,
The equation
“=A. a
of motion for u yields M(2 - 6) a = -4&&l -S)
Substituting
(A.21
b=-&.
’
64.3)
into Eq. ( 1 1 ), we find the energy/particle
64.4) The first three terms are the energy of a Fermi gas and the succeeding terms are the o and scalar meson contributions. This is of exactly the same form as the standard Walecka model [lo], except that the scalar meson mass is replaced by 4&S(l--6) a,2(2-6)
m=
1/Z = m m <
>
>
40
[2Bo(2 - S)]‘/2 LXm<.
(4.5)
where m< (m,) are the lower (upper) eigenvalues of the vacuum mass matrix (see ref. [7] for an explicit discussion). The mass, m, is approximately m< , provided that 40/oo is not too large so that the mixing of the sigma and glueball states is small. Now the fact that d = b in Eq. (13), means that, to the order we are considering, we can obtain the correct result for v with x = 1, provided that BO is replaced by B’ = B 2(1-d) 0 O 2-6 In this approximation,
we see that the scalar potential, I&(4,(T,n
= 0) -&c
= $&8(V2
In Eq. (lo), with x = 1, the glueball equation
(‘4.6)
. I& takes the simple form .
- 1 -lnv*)
(‘4.7)
of motion can be dropped and we are left
with the single scalar equation a;Dv - 2B06u = Mps - Boa (’ i-V2) v Taking care that the energy is consistently occ E = 5(2j, a +
obtained cc
+ 1)-2x
-m~R,w&.
(A.81
we find
drr*{Myp,
+
gd’-)OpB
+
gpbop3
s 0
2BoS In u - m~R,v*w~}
.
64.9)
Using the low density expansion for nuclear matter and replacing BO by Bh in Eqs. ( 19) and (20) this expression gives the same result as the complete result of Eq. (11). In practice to saturate nuclear matter correctly a small adjustment of C’i is needed and BA is only approximately given by ( 17).
732
E.K. Heide et al. / Effective lagrangian References
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