Nuclear physics in colourful worlds

Nuclear Physics A462 (1987) 701-726 North-Holland, Amsterdam

NUCLEAR PHYSICS IN COLOURFUL WORLDS* Quantumchromodynamics and nuclear binding H. MIJTHER 1, C.A. ENGELBRECHT 2 and G.E. BROWN

Department of Physics, State University of New York at Stony Brook, Stony Brook, N Y 11794, USA Received 14 January 1985 (Revised 20 June 1986)

Abstract: When quantumchromodynamics (QCD) is generalized from SU(3) to an SU(Nc) gauge theory, where /V~ is the number of colours, it depends on only two parameters: N,. and the bare quark mass mq. A more general understanding of nuclear physics can be achieved by considering what it would be like in worlds with the number of colours different from 3, and bare quark masses different from the "empirical" ones. Such an investigation can be carried out within a framework of meson-exchange interactions. The empirical binding energy of nuclear matter results from a very near cancellation between attractive and repulsive terms which are two orders of magnitude larger and may be expected to depend sensitively on the parameters of QCD. It is indeed found that our world is wedged into a small comer of the two-dimensional manifold of mq versus N c. If the number of colours were decreased by one, or the bare quark masses raised by more than 20%, nuclear matter would become unbound. By tracing the origin of this state of affairs, one obtains a clearer picture of the relative importance of various effects on the behaviour of the bulk nuclear matter. In particular, correlations like those embodied in the Coester band of saturation points appear to have a broader degree of validity than is implied by fits to the actual physical world only.

1. Introduction Deep wisdom

in t h e p r o b l e m

of nucleon-nucleon

i n t e r a c t i o n s is a f f o r d e d b y

t h i n k i n g a b o u t q u a n t u m c h r o m o d y n a m i c s ( Q C D ) in t h e w a y p r o p o s e d b y 't H o o f t 1). H e c o n s i d e r e d Q C D g e n e r a l i z e d f r o m S U ( 3 ) to a n S U ( N c ) g a u g e g r o u p , i.e. h e i m a g i n e d t h e r e c o u l d b e N~, n o t o n l y 3, c o l o u r s . I n this g e n e r a l i z a t i o n , gc = 1~No is t h e e x p a n s i o n p a r a m e t e r a n d f u n c t i o n s as t h e c o u p l i n g c o n s t a n t . I f o n e a s s u m e s c o n f i n e m e n t , t h e n Q C D is e q u i v a l e n t to a l o c a l field t h e o r y o f m e s o n s ( a n d g l u e b a l l s ) in t h e l a r g e Arc limit. T h u s Q C D is e q u i v a l e n t , in a w e l l d e f i n e d w a y , to a m e s o n theory. W h e r e d o t h e b a r y o n s c o m e f r o m ? W i t t e n 2) h a s g i v e n s t r o n g a r g u m e n t s t h a t t h e y m u s t a r i s e as s o l i t o n s o l u t i o n s o f t h e ( n o n l i n e a r ) m e s o n field e q u a t i o n s . V e r y r o u g h l y , t h e b a r y o n is c o m p o s e d o f Nc q u a r k s . A s N c - ~ oo, so m u s t t h e n u c l e o n m a s s g o to * Research supported in part by the US Department of Energy under contract DE-AC02 76ER/3001. i Permanent address: Institut fiir Theoretische Physik, Universit/it Tiibingen, Auf der Morgenstelle 14, 7400 T~ibingen, W.-Germany. 2 Permanent address: Institute of Theoretical Nuclear Physics, University of Stellenbosch, 7600 Stellenbosch, South Africa.

0375-9474/87/$03.50 ~) Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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H. Miither et al. / QCD and nuclear binding

infinity linearly in the n u m b e r of colours No. This means that

mNoc N~ = -

1

g~

.

(1)

Such a proportionality to the inverse of the coupling constant is indicative of soliton solutions. The nucleon-nucleon ( N N ) interaction, when described in terms of mesonexchange, is also of order No. We give Witten's argument in a slightly different form. The original coupling constant gq between quarks and gluons must be rescaled as gq/x/Nc so that a smooth transition to Arc ~ oo can be made for the one-loop gluon vacuum polarization. The coupling of the nucleon to a meson can then be read off from fig. 1. The two quark-gluon vertices bring in a factor (gq/x/N~) 2. The sum over colours c in the nucleon gives a factor No, as does the sum over c' in the meson. A factor 1/~/Nc is contributed by the meson normalization so that the net dependence on Arc is given by

g~

(2)

1 = g~,/N~.

I f we now do second-order perturbation theory to get the N N interaction, two factors ~/Nc come from the emission and absorption of a meson. The difference between intermediate and initial state energies, which comes in the denominator, involves only a meson energy, which is of order one in No. Thus, the net interaction is ocN~. It is true that different components of the N N interaction yield matrix elements which are of the same general size as the nucleon mass. The attraction from scalar meson exchange (o- or e) in boson-exchange models is g2 e - m r V~ =

47r

(3)

r

C

C~

9'

C

C Fig. 1. Coupling of a meson, denoted by coloured quarks c'~', to a nucleon, containing coloured quark c. The wavy line represents a gluon.

H. Miither et al. / QCD and nuclear binding

with g:/&r

- 4-7 and

m,

-

500 MeV. The short-range

703

repulsion

from w-exchange

v =& eernwr o 4rr r with a value typical

for a one-boson-exchange

(4)

potential

is thus of the order GeV at a range determined

of gt/4n

- 10-14, and this

by the mass of the IX The net binding

energy of nuclear matter, E/A = -16 MeV, is then only -l-2% sizes of the components making up the NN interaction. Viewed easy to see how a change

‘) of only l-2%

is

in

m,

can change

of the “natural” in this way, it is

the nuclear

binding

energy by a factor 2. The generalized QCD discussed above has only two parameters which are relevant for the description of nuclear physics: l/N, and m4, the bare (“current”) mass value of the light quarks (u and d). Witten has emphasized “) that nuclear physics, as we know it, occurs in a very special place in the two-dimensional manifold spanned by the two variables mq and N,. The problem is that we do not know precisely where this place is, because the caricature theory of weakly interacting bosons holds for large N,, whereas N, = 3 is not large. In order to connect N, = 3 and N, = co at this time,

we must introduce

a model.

Although

quark

degrees

of

freedom have been integrated out once we go over to the boson model, we shall assume that couplings of bosons to nucleons scale with N, as they do in quark models. Witten’s ‘) original large-NC limits, e.g., for the nucleon mass, were taken from constituent quark model descriptions. Later, Ioffe and Shifman ‘) showed that this counting is correct for quarks with (small) current quark masses. There are indications that using the quark model to do the scaling with N, is sensible from the work of Karl and Paton “) and of Jackson et al. '). They show that the ratios of couplings found by Adkins, Nappi and Witten “) for the skyrmion supposed to result in the N, = 00 limit - is the same as those which would be obtained from the naive quark model as N, + 00. For example, the ratio -g?rNA ( goes

g,NN

to the usual nonrelativistic

*

>

=- 9 (N,+5)(N,-1) 2 (X+2)(X+2)

quark model value of g for N, = 3, and to the value

3 in the limit N, + 00. The latter value is that found

by Adkins,

Nappi

Given the quark model for scaling with N,, we wish to investigate of bulk properties

of nuclei

on

mq

and N,. How do changes

and Witten ‘). the dependence

in these parameters

change the binding energy and saturation density of nuclear matter? We believe that an investigation of nuclear matter in this more general context can give us a better perspective in the understanding of nuclear matter problems, even though the latter involve particular values of m, and N, only. The nonlinear lagrangians, such as that of Skyrme, which are presumed to result in the large N, limit, are very different from the efictive lagrangians used in nuclear physics 9-11) which give linear meson field equations. Presumably, the latter deal

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H. Miither et al. / QCD and nuclear binding

with the same mesons, etc., as the former, but the connection between these lagrangians has not been made. By way of example of this connection, however, ref. 3) shows that the large nonlinearities in the field equations of the linear model can be essentially removed by loop corrections. We assume that something like this happens in general, and that one will end up with the essentially linear field equations of the boson exchange models 9-11). We shall use the latter, which give results in good agreement with phenomenology, in our calculations, After this introduction we will discuss in sect. 2 of this paper the construction of one-boson-exchange models for the N N interaction assuming different values for mq and /V~. Special attention will be given to the description of the o-exchange, which is of dominant importance for the binding energy of nuclei. The approximation for the m a n y - b o d y theory which is used for the calculation of the nuclear matter properties is briefly described in sect. 3. The results of our investigations are presented in sect. 4 and some conclusions are given in sect. 5.

2. One-boson-exchange for different Nc and mq 2.1. ONE-BOSON-EXCHANGE AND NUCLEAR BINDING In the spirit of the large-No limit of QCD, the interaction between two nucleons should be described in terms of the exchange of physical mesons. This idea is strongly supported by the success of the one-boson-exchange potentials (OBEP) in describing low-energy nuclear physics 9,~0). Using such an OBEP for the nucleonnucleon ( N N ) interaction one can obtain an excellent fit to the two-nucleon data, i.e. the scattering phase shifts and the properties of the deuteron. A list of the mesons which are typically taken into account is given in table 1. One can try to adjust the parameters of such an OBEP, the masses of the mesons, the coupling constants and the form factor for the m e s o n - n u c l e o n vertices in order to reproduce the empirical N N data. However, the masses of the mesons and also most of the coupling constants are very well determined from other experimental data, and therefore the freedom to vary these variables is highly restricted. Only the scalar-isoscalar g-meson, which is responsible for the medium range attractive part of the N N interaction, may be interpreted as a phenomenological ingredient of the OBE model, for which mass and coupling constant can be freely adjusted. But even the o--exchange can be determined in a microscopic way. One way is to use dispersion relations and determine the effects of the g-exchange from experimental zrN and ~-zr scattering data u.~2). An alternative way is to calculate directly the irreducible 27r-exchange terms (cross-box terms and terms involving intermediate A excitations), which are simulated by a o--exchange in OBEP ~o,~3). Keeping all these restrictions for a variation of the OBEP parameters in mind, one must consider it as a real success of the OBE model of the N N interaction that it is possible to fit the N N data using an OBEP with the same or better accuracy

H. Miither et al. / QCD and nuclear binding

705

as that obtained with phenomenological models of the N N interaction, which have up to 50 or so free parameters 14). The OBEP potential which we consider in this p a p e r for the description of the "real" world (number of colours Nc = 3 , the current masses of the light quarks mu+ rna = 11 MeV, see sect. 2.2), can be regarded as a modification of the OBEP determined by Holinde et al. lS). They derived the OBEP from covariant perturbation theory. The matrix elements are calculated in m o m e n t u m space using the helicity state representation. The parameters were adjusted to fit the deuteron data and N N phase shifts up to an energy of 330 MeV. I f this potential is used in a calculation of the binding energy of nuclear matter, using the B r u e c k n e r - H a r t r e e - F o c k approximation (see sect. 3 below), a saturation point is obtained at a binding energy of -12.4 MeV per nucleon and a density corresponding to a Fermi m o m e n t u m kF of 1.55 fm -1. We have slightly readjusted the parameters of the OBEP of ref. is). The masses of the mesons were updated to the most recent values provided by the Particle Data G r o u p 16) and the pion coupling constant was increased from the particularly low value of g2/47r = 13.0 to a more realistic value of 14.5. Furthermore we have slightly modified the coupling constants for o-- and &exchange in order to improve the description of the binding energy of nuclear matter. Using the parameters displayed in table 1 a nuclear matter binding energy of -15.4 MeV per nucleon is obtained at a saturation density given by kv = 1.52 fm -1. This saturation density could be moved down to the empirical value corresponding to kF = 1.36 fm -1 by reducing the sigma coupling strength in table 1 from 4.5 to 4.2 (in which case the binding energy at TABLE 1 M e s o n p a r a m e t e r s u s e d for O B E P Meson

jr

T

m [MeV]

g2/4r;

n" o" r/ p to 8

00÷ 0110+ 1-

1 0 0 1 0 1 0

138.0 500.0 548.8 769.0 782.6 983.0 1020.0

14.5 4.5 6.0 1.5 14.0 4.5 7.0

Listed are the spin a n d p a r i t y of the m e s o n s ( J ~ ) , the i s o s p i n (T), the m a s s e s ( m ) a n d the c o u p l i n g constants. F o r the p - m e s o n a t e n s o r c o u p l i n g with f / g = 3.5 was i n c l u d e d in a d d i t i o n to the v e c t o r c o u p l i n g . For the form factors the s a m e p a r a m e t r i z a t i o n was u s e d as in ref. ~5). A l t h o u g h the ~b-meson s h o u l d not be c o u p l e d to n u c l e o n s if it has the q u a r k s t r u c t u r e sg, it was k e p t in the O B E P used here in o r d e r to c o n f o r m as closely as p o s s i b l e to the p o t e n t i a l o f ref. ~5).

706

H. Miither et al. / QCD and nuclear binding

saturation changes to about - 1 0 MeV), or by including the three-body interaction of Jackson et al. 3). 2.2. THE INFLUENCE OF THE QUARK MASSES The pion can be understood as a collective excitation of the non-perturbative physical vacuum. I f the u and d quarks were massless, the Q C D lagrangian would have chiral symmetry and the pion would correspond to a massless Goldstone boson. The small current quark masses mu and md break this symmetry and provide the pion with mass. This already indicates that the pion mass will depend in a very special and quite sensitive way on the masses of the light quarks. By using sum rules derived from current algebra, the masses of the pseudoscalar mesons of the flavour SU(3) octet may be written in terms of commutators of appropriate axial charges m 2 =f~r2(0][Q~, [Hm, Q~,]][0)

(5)

with the symmetry breaking hamiltonian 5,x7-~9) H ~ = ~, rnsFlyqs.

(6)

f I f one assumes (t~u)0 = (dd)o = (qq)o, this gives for the pion m ~ = - f 2 2 ( q q ) o ( m u + m,O .

(7)

The various mass relations can be satisfied reasonably well with 5) mu = 4 MeV and ma = 7 MeV (the sum mu + ma will henceforth be denoted by mq with a value of 11 MeV in the physical world). With f,r = 93 MeV, the real pion mass is then obtained assuming (t~q)o = -(246.5 MeV) 3. If electromagnetic mass splittings are ignored, the masses of the pseudoscalar mesons can be written as m~r = Cox/m--qq,

(8)

mr: = C o - - q ,

(9)

m , = Cox/(2 - 4 a 2) ms + 2a2mq,

(10)

where the wave function of the ~/ is assumed to be [7) = a l a u ) + a [ d d ) - 4 1 - 2 , ~ 2 [~s).

(11)

G o o d agreement with the K and 77 masses is then obtained with ms = 136 MeV and a value of 0.189 for a 2. (The value for the pure octet state would be a2=0.167 whereas a mixing angle of 11 ° with the singlet would increase this value to 0.26.) We have assumed m~ and m, to depend on the sum of the light quark masses mq according to (8) and (10). For the other nonstrange mesons (except the or, which following the discussion given in the previous section, is considered separately in sect. 2.4) a bag model description was assumed, leading to mass values of the form

mq2

(12)

H. Miither et al. / QCD and nuclear binding

707

For the mesons i = p, to, 8 and ~b the constants Ci were adjusted to reproduce the masses of the physical mesons. Since we are interested only in the dependence of the meson masses on the quark masses mq it is justified to ignore the explicit dependence of mi on the residual interaction between the quarks. The same picture was used to represent the dependence of baryon masses on mq. For baryons containing N~ quarks this gives mB = Ncx/ C ~ + (3mq)2 .

(13)

As will be seen in the following section, this expression was used for an average mass of the nucleon and the A isobar. Their individual masses are obtained if the mass splitting term (15) is added to (13).

2.3. DEPENDENCE OF THE PARAMETERS ON Arc As Witten 2) has shown, the masses of all the physical mesons should be independent of the number of colours Arc. At first sight the expression (7) for the pion mass seems to be in disagreement with this result, since (t~q)o is obviously proportional to No. However, this dependence is counterbalanced by the fact that also the coupling constant for pion d e c a y f 2 is proportional to No. According to Ioffe and Shifman 2o), the baryon masses also depend on the quark condensate 3

2

-

mB ~ N c ( q q ) o ,

(14)

which is consistent with the colour dependence already given in eq. (13). Eqs. (8), (10) and (12) are therefore the final expressions which we used for the masses of the physical mesons. They do not change with Arc. It is also clear that the number of species of different mesons would not change with Arc if the number of flavours is kept constant. The situation is different in the case of the baryons, which must contain Arc quarks to build a colour singlet of SU(Nc) To preserve the fermion character of the baryons, the number of colours must be odd: Nc = 2 n ~ + 1 . Even if quarks of only two flavours (u and d) are considered, the number of possible baryon states increases rapidly with N~. For N~ = 1 the entire baryon spectrum collapses to the four states of the nucleon which, with S = T = ½, constitutes the fundamental quadruplet of SU(4). For larger numbers of colours, the wave function of the N~ quarks may be decomposed into its colour (singlet), spin, isospin and orbital components. For Nc = 3 the 64 states of 4 ® 4 ® 4 combine into four multiplets of SU(4) containing nine different (S, T) multiplets. If all the quarks are in the same lowest orbital state, however, the only spin-isospin multiplets allowed are the nucleon N(~, 1 ~) 1 and the A (~, 23-)of the fully symmetric 20-plet of SU(4). When Nc = 5, the 1024 states of 4 ® 4 ® 4 ® 4 ® 4 in the same way combine into 25 multiplets of

708

H. Miither et al. / QCD and nuclear binding

SU(4) containing a total of 100 different (S, T) multiplets. Again symmetry of the orbital part of the wave function reduces this multitude of baryons to a nucleon 1 1 3 3 N(~, ~), a delta (~, ~) and a superdelta (5,_~) of the 56-plet of SU(4). This pattern is repeated, the totally symmetric multiplet always consisting of nc + 1 baryons: N~I, A33, • • • • With the isospin of the pion being T = 1, the emission or absorption of a pion thus continues to connect the nucleon NH only with N , itself or with a single A33, unless the orbital wave functions of the quarks are disturbed. The masses of these nc + 1 baryon members of the symmetric multiplet would be split by the exchange of gluons. This splitting may be represented 2~) by a Fermi-Breit interaction AEs = A ~ Y~tr,. tri i#j

=4AS(S + I)-3NcA,

(15)

where the summations go over all the quarks and we have used ~i ~i = 2S for the spin of the baryon. To produce the splitting of 294 MeV between the nucleon and delta masses, the value A = 24.5 MeV must be used for N¢ = 3. Since A will go 22) like N7 ~, we therefore choose A = (73.5/Arc) MeV. The difference between the masses of a33 and N , thus decreases with increasing Arc. If Nc is treated as a continuous variable, this difference already becomes smaller than the pion mass when Arc/> 6.4. With this value for A, the constant CB in eq. (13) was chosen so that (13) plus (15) gives the nucleon mass of 938.9 MeV. As has already been discussed in the introduction, Witten also showed 2) that the coupling strengths (g~) of the physical mesons to the nucleon will in general be proportional to N~. An apparent exception to this rule is encountered in the case of the pion. According to the Goldberger-Treiman relation, the pseudoscalar pionnucleon coupling constant satisfies the constraint mN -gAg w N N -- - - f~,

(16)

Now we already know that the nucleon mass my varies as N~ while the fundamental coupling f= used in PCAC theory varies as x/No. In a simple quark model, the value of the axial vector coupling constant is proportional to the expectation value of ~'i O'3"/'3,i;summed over all Nc quarks, in the state of the proton with spin up. This yields 7) gA=½(N~ + 2 ) .

(17)

This N~ dependence of gA has been disputed by Ioffe and Shifman 2o), who base their conclusion on the equation

g~+If°° dV[cr,,-(v)-cr~,+(v)]=~,

f~

~

....

v

(18)

H. Miither et al. / Q C D and nuclear binding

709

obtained from the Adler-Weisberger sum rule. They argue that it is natural to expect each of the two terms on the left-hand side of eq. (18) to have the same N, dependence, namely N~-~, as the right-hand side, from which they conclude that gA goes like a constant to leading order in No. However, we believe this conclusion to be incorrect. The left- and right-hand side of eq. (18) may be traced back 23) to a commutation rule v

leijkQk ,

(19)

which links the SU(2) axial vector and vector charges. When matrix elements of (19) are taken between physical nucleon states, each of the two terms on the left-hand side goes like N~ due to the tr¢ matrix element used in (17). On the other hand, the right-hand side contains a matrix element of ~" only, which goes like N °. This is at the root of the remarkable cancellation of the two leading orders of N,. on the left-hand side of eq. (18). We have concluded therefore, that the pion-nucleon coupling strength, unlike the other mesons, varies like 2

g ~ N N Nac

47r

(20)

to leading order in No. This may seem to contradict Witten's arguments for coupling strengths g i2- No. One has to keep in mind, however, that Witten's argument is purely based on the colour dependence and does not account for the spin-isospin structure of the mesons and baryons involved. Furthermore, if the N N interaction due to the exchange of a pion is calculated assuming pseudoscalar coupling, the coupling always enters in the form (g~NN/mN), SO that effectively two powers of Arc are removed from eq. (20) by the Arc dependence of the nucleon mass. In fact, the pseudovector coupling provides a more consistent description of low energy phenomena 24). The coupling constant for pseudovector coupling f~NN is related to g w N N by 2mN gcrNN --

mcr

'fTr NN •

(21)

If ( f 2 N N ) is assumed to be the basic coupling constant, depending like the other meson coupling constants linearly on Arc and to be independent of mq, eq. (21) induces the correct Arc dependence, as well as a dependence on mq, in (20). Therefore we actually used the expression 2 g~rNN

47r

m~No

= 0.1044 - - - 5 - m~

(22)

to determine the Arc and mq dependence of grrNN. Eq. (17) gives the correction to the next order in (1/No) to gA, and therefore to g~NN- One may argue that it would be better to incorporate this correction in (22). The only effect would be to cause

710

H. Miither et aL/ QCD and nuclear binding

g , SN to increase somewhat more slowly with No. We decided, therefore, rather to keep the leading Nc dependence only everywhere, except in the baryon mass splitting (15). The coupling of the pseudoscalar meson ~/, which is actually not of significant importance for our present investigations, is also divided by the nucleon mass in calculating the nucleon-nucleon interaction. One might ask if it should, by the same argument as for the pion, also be treated differently. The answer is no. In the case of the pion the additional Nc dependence in eq, (17) came from the expectation value of the operator ~i cry"of the axial isovector coupling. The special cry"correlations of the nucleon induced an additional N~ dependence. In the case of the ~/, being an isosclar particle, the corresponding operator would be Zi or. The expectation value of this operator is of the order N o for the nucleon and therefore does not give an additional N~ dependence for gvNN-

2.4. P R O P E R T I E S

OF THE SCALAR MESON

The "scalar meson" cr used in our OBEP is not a real meson but a phenomenological way of describing the exchange of two correlated pions in a state with J = 0 and T = 0 in the crossed channel (t-channel). In other words, the cr-exchange describes loop corrections. A natural way of treating this effect is by relating it via dispersion relations to ~rN and ~Tr scattering data 11,12,24-26). The results of such investigations can be expressed in terms of an effective tr-exchange, which is represented by a weighting function p(t). The kinematical invariant t corresponds to the square of the four-momentum transfer in the N N channel and would be given by the square of the total energy in the c.m. system of the Ngl channel. The weighting function p(t) indicates then the strength of the exchange of an effective o--meson with mass x/t. This weighting function can be related to the N + N ~ 7r + ~ helicity amplitude

f°(t) q Ifo(t)l 2 p(t) = 8,~./(4~ ~

(23)

where 2 q2 = ¼t - m,~,

(24)

K2 = m 2N - ~ t1,

and p(t) is defined in the interval 4m~<~ t<~4m~. We begin by studying the contributions to f°(t) from the diagrams shown in fig. 2, which will be denoted fN(t) and fa(t), suppressing the other labels. The nucleon contribution f ~ was calculated in refs. 12,24) and is given by g~NN

fN(t) = - ~

m~hNtan -1

-

t

,

(25)

H. Miither et al. / QCD and nuclear binding

711

Fig. 2. Contributions to two-pion amplitude arising from intermediate nucleon and delta poles. where

hN = (it -- rn2 ) / 2qK.

(26)

The ingredients for the calculation o f f a ( t ) are given in refs. 12,27). To obtain the quantity corresponding to the square brackets in eq. (25), one should use the amplitude

A+(t, z ) -

e~o(t) + alS + a2s 2 m2a - s '

(27)

given in eq. (5) of ref. 12) and project out the s-wave part. Here the kinematical invariant s is given by

s = K 2 - q2+2iqKz,

(28)

while ao(t), al and a2 are given by 2

ao(t)=½(ma + m N ) ( t - 2 m 2 )

-

_ 2x--

mN

2

2

2

1

2

(mE--m,~)-r6m-----~(mN--m~) --~mN(ma +mN) ,

2

oq = g(m,,a + raN) q- m,~ 3ma

mN 3m~ ( m ~ - r n ~ )

mN

a2=o

armZ"

(29)

The coupling constant to be used can be read off from table 2 of ref. result is

~ ( ~ ) 2 [ f~(t)=

27).

The final

a°(t)+m2al+m4aa2 (-~a) m~2 2 1 h a t a n -1 -al mN--m~+~t

2 2 __, )/ -- a2( m2a + mN + rn,~ zt ,

J

(30)

where

ha = (½t + m ] - m 2 - m~ )/2qK .

(31)

712

H. Miither et al. / QCD and nuclear binding

The calculated curves in ref. 12) are reproduced if one uses for the coupling constant f~Na the value 2.0, which would give the correct decay width of the isobar 28). The form of the coupling in eq. (30) suggests that it is proportional to 2

g~,Na-

4mNma ~c2 ~ J~N~

(32)

rrl~r

and this is the way we assumed the coupling to scale when Nc and mq were varied. In addition to the nucleon and A pole term contribution to f(t) one also has to consider the ~r~" rescattering effects. Instead of calculating the ~rzr scattering effects directly within a certain model, one may also deduce the amplitude by analytical continuation of the experimental results on ~rN scattering from its physical regime ( t < 0 ) to the t-region of interest here. This has been done by Nielsen and Oades, as reported in refs. 12,24). We took these effects into account by simply multiplying our result for IfN(t)+fa (t)l 2 by a factor to get the Nielsen-Oades values. This factor of course depends on t. For the calculation of p(t) for different values of Nc and mq, this correction factor was assumed to be a function of t/m 2~ only, since the ,r~" scattering is of order (N~) °. This is still not the complete story, since the 2zr contribution to the N N interaction should only contain terms which are irreducible with respect to intermediate N N terms to avoid overcounting in solving the N N scattering equation. Therefore one has to subtract from the weighting function p(t) the contributions from the iterated one-pion exchange. This was done as described in ref. 26). An idea of the relative importance of the various effects can be provided by giving the results for one particular case, e.g. for ~/t = 500 MeV in the case of the normal values of Nc and mq. In this case, the ratio of our calculated fN and fa is about 3 : 2, the value of the correction factor to I f s + f a l 2 to account for pion rescattering is 1.37, while the subtraction of the repeated one-pion contribution reduces p(t) by 14%. Having set up the machinery for calculating p(t) for the physical values of Nc and mq, one can now easily perform the same calculations for other values of Nc and mq by scaling g , NN, m~, mN and ma according to the rules developed in sects. 2.2. and 2.3. Typical results of such calculations are shown in fig. 3. The ratio of mq to its physical value is denoted by Rq and the dimensionless plots show mEp(t) against t/m~2 for selected choices of N~ and Rq. Apart from the physical values N o = 3 , Rq= 1, we show one example where Rq is changed (Rq= 1.8) and one example where Nc is changed (No = 2.3). The first important result obtained was the stability of the shapes of the curves when No, Rq or both were changed not too far from their physical values. All the curves indicate the "or-meson" really has a broad mass distribution. The success of the phenomenological o--meson in OBE potentials, however, shows that the effect of the distributed or may be represented by the exchange of a single meson of mass - 5 0 0 MeV (corresponding to t = 13.1m2). Since the mass distributions change only very slowly as a function of Nc and mq, this should also hold for other than the

H. Miither et aL / QCD and nuclear binding I

I

I

1

l

I

I

I

I

I

/

1

713 I

1

/

\

NC=3 RQ=I.8

/,,/"~~-~ 0.03

~-

NC=3 /" ~ Rq=' ~ / ~ x ~ 1 . . . . - - - - - ' x \

-

~ "X ~\\\

/

0.02

I

\

xNx.

,7/

--_-

0.01

4

8

12

16

20

24

28

l/m~ Fig. 3. Plot of the spectral function (eq. (23)) for 27r exchange in the T = 0, J 0+ channel as function of t. Results are shown for the cases (Nc = 3, Rq = 1.8), (Arc= 3, Rq = 1) and (N~ = 2.3, Rq = 1). In each case the broken line indicates the results without and the solid line the results with the repeated one-pion contribution subtracted. =

physical values o f Arc a n d mq. The stability o f the curves as a function o f t / m ~ in fig. 3 implies that the mass o f the or-meson in our O B E P should scale directly with the pion mass

m¢(Ne, m q ) = Cm~(Nc, mq).

(33)

The coupling constant (g2) which determines the magnitude o f the N N interaction due to o--exchange, s h o u l d be p r o p o r t i o n a l to the area u n d e r the curves in fig. 3. In fixing the w a y g~ should scale, we have in fact simply used the value o f rn~p(t) at t/rn~ = 13.1. W h e n the d e p e n d e n c e o f g~ on Rq is investigated, the results clearly indicate a proportionality g~ oc x/Rq oc m=

(34)

for a range o f Rq f r o m 0.2 to 5. W h e n Arc is changed, problems were encountered for larger values o f Arc ( > 4 ) with the subtraction o f the iterated or-exchange: the iterated terms b e c a m e larger than the total contribution o f f N to p(t). This m a y be due to the fact that p s e u d o s c a l a r coupling was used for the calculation o f the iterated ~r terms 26). Therefore we assumed that the coupling constant o f the o--meson

714

H. Miither et al. / QCD and nuclear binding

depends on Nc in the same way as the other mesons, or, incorporating eq. (34): g2

47r

(35)

CNcm~.

This linear dependence on No, which is suggested by the large-N~ limit of QCD 2), is supported by our results for p(t) without subtracting the iterated 7r-exchange.

3. Calculation of the nuclear binding energy

After the parameters of the OBE potential are determined as a function of Arc a n d mq it is straightforward to evaluate matrix elements for the N N interaction and

to perform a calculation of the nuclear matter binding energy within a given approximation of a many-body theory. The matrix elements of the N N interaction are calculated in momentum space using the techniques which are described e.g. in ref. 9). For the many-body calculation we have used the Brueckner-Hartree-Fock (BHF) approach 27) which we will briefly review in the following. A realistic N N interaction V contains very strong, repulsive components of short range, which in the OBE model are essentially described by to-exchange. Therefore one must consider a many-body theory which takes into account the two-body correlations induced by these strong, short-range components. In the framework of the Brueckner-Bethe theory this is done by solving the Bethe-Goldstone equation G ( W ) = V+ V

Q W - Ho

G( W ) .

(36)

With this equation, which is very similar to the Lippmann-Schwinger equation, one tries to solve the two-nucleon problem in the nuclear medium. The Pauli operator Q ensures that the interacting nucleons obey the Pauli principle and are not scattered into states which are occupied by other nucleons. It is defined for two-nucleon states Ik, k') in a momentum state representation by

QIk, k,) = ~lk, k'),

if k > kF and k ' > kF otherwise,

(37)

where the Fermi momentum k F for the non-interacting nucleons is related to the density p of the nuclear matter under consideration by p

2k 3 3rr2.

(38)

For the energies of the intermediate particle states, defined by Ho in eq. (36), we have made the so-called "conventional choice" and replaced Ho by the kinetic energy. The starting energy W is defined self-consistently in terms of the BHF

H. Miither et al. / QCD and nuclear binding

715

single-particle energies k2

fkF

ek = ~m + Jo d3k'(kk']G( W = ek + ek,)lkk')

(39)

and the total energy E can be calculated from E =

d3kT--+½ zm

d3k d3k ' (kk'lG(ek + ek,)lkk').

(40)

The self-consistent solution of (39) can be obtained by a simple iterative method. We perform this iteration by parametrizing the single-particle spectrum of the occupied states (hole-states) k2 ek = 2m---~+ UB

(41)

in terms of an effective nucleon mass m* and a constant, attractive potential UB. For a given choice of m* and UB the BHF single-particle energies ek can be calculated for some values of k according to eq. (39). The calculated values for ek can then be used to determine a new set of optimal parameters ms* and UB following (41). This procedure is repeated until convergence is obtained. The energy of nuclear matter for a given density p or Fermi momentum kF can then be calculated using (40). These calculations have to be performed for different densities and one then identifies the saturation point by determining the minimum of the resulting energy versus density curve. The energy per nucleon and the density at the saturation point can be compared with the empirical values: E / A ~ - 1 6 MeV, p =0.17 fm -3 corresponding to kF ~ 1.36 fm -~. The B H F approach is the lowest order approximation of the Brueckner-Bethe hole-line expansion. In this approximation the effects of two-body correlations are taken into account, while the effects of real three-body correlations and those involving more nucleons are in general ignored. This approximation is obviously justified for systems of low density, for which the average distance of two neighbouring particles is large compared to the range of the strong components of the interaction. Numerical calculations 30) indicate that this is the case for normal nuclear matter up to kF-- 1.5 fm-'. For larger densities the effects of three-body correlations may become more and more important and change the calculated saturation properties. In the investigations discussed in this paper we change the range of the underlying N N interaction and explore nuclear matter systems at densities which are in some cases considerably larger than the empirical value. One has to expect that higher order terms of the many-body theory, which are not taken into account in the B H F approximation, will change the results for the energy especially in cases where the density or the range of the interaction are large. Nevertheless, we think that the BHF approximation is justified for an investigation

H. Miither et al. / Q C D and nuclear binding

716

of the general dependence of the nuclear binding energy on quark masses and number of colours. Until now we have only discussed the binding energy of the nuclear system. However, we also would like to investigate the features of the nuclear excitation spectra as a function of Nc or mq. In particular, we would like to study whether there exist combinations of Nc and mq for which the normal nuclear matter system becomes unstable against an excitation mode. To attack these questions, we consider nuclear matter as a Fermi liquid applying the methods of the Landau theory 31). Within this framework the relevant parameters for the description of collective modes with m o m e n t u m transfer q = 0 are the effective mass rn*, parametrizing the quasiparticle spectrum at the Fermi m o m e n t u m , and the Landau parameters which define the residual interaction. We want to determine these parameters by starting from the Brueckner G-matrix calculated for a starting energy W which corresponds to the interaction of two nucleons with m o m e n t u m kF. The residual interaction between particle-hole states with m o m e n t u m transfer q = 0 can then be parametrized

G(k, k') =

.t7-2

2m*LkF

{ft-f'"g

I • ~'2-+- g i l t I • 0"2-1- g " r 1 • T 2 0 " l • 0 " 2 } .

(42)

The functions f f ' , g and g' depend only on one angle variable 0L and are made dimensionless by dividing the right-hand side of eq. (42) by the density of states at the Fermi surface 2rn*kF/rr 2. Each of the functions is expanded in a Legendre series like e.g.

f = Y. F,P1(cos 0L)-

(43)

I

The coefficients F~ and corresponding coefficients FI, G~, GI for the other functions are the Landau parameters. The parameters for l = 0 characterize the sign and the strength of the residual interaction. A Landau parameter for l = 0 which is less than - 1 indicates that the system is unstable against a corresponding excitation. The Landau parameter F1 defines the effective mass by m* = m{1 + i F 1 }

.

(44)

Other Landau parameters can be related to observables like the compressibility or the symmetry energy of nuclear systems.

4. Results and discussion

Before we discuss the saturation properties of nuclear matter as a function of the number of colours Nc and the masses of the light quarks mq, we consider the different contributions to the energy of nuclear matter for the physical values of these parameters (Arc = 3 , R q = 1, if we define as before Rq as the ratio of the light quark masses actually used to their physical values). The contribution of the kinetic

H. Miither et al. / QCD and nuclear binding

717

energy to the energy per nucleon for nuclear matter with Fermi momentum kF can be calculated as T = 3k2/10rnN.

(45)

For the saturation density, which we obtain at kF = 1.52 fm -~ for the OBEP defined in sect. 2, this yields ~29 MeV. As has already been stated before the main attractive contribution to the energy is due to o--exchange. To estimate the contribution from this meson and also the density dependence of this contribution, we use the following procedure: First we determine the self-consistent BHF single-particle spectrum (see eq. (39)) and calculate the energy as a function of kF for the complete OBEP. Keeping the parametrization of the BHF single-particle energies we then determine the energy of nuclear matter for OBEP leaving out the o--exchange. The difference between these two energies is a measure of the cr contribution zaE~. It turns out that AE~ is given reasonably accurately by an expression of the form AE~(kF) = C~k 32 .

(46)

If kF is measured in fm -~, the constant C ~ 2 8 . 5 MeV when N~=3 and Rq---1, which yields zl E~ = - 109 MeV at the saturation density. If this contribution would be assumed to be proportional to the volume integral of the N N interaction, i.e. g 2 / m 2 , the constant C~ should scale like Nc/x/Rq if the dependences given by (33), (35) and (7) are inserted. Actually, one finds that C~r oc Nlc'4/ Rq

(47)

provides a better approximation to the scaling properties. The large attractive contributions to the energy arising from tr-exchange are partly counterbalanced by repulsive short-range components. These repulsive terms may be attributed to to-exchange. Since, however, the to-contributions are very strong, an analysis of the form just outlined for the tr does not make sense for the to-meson. The Born term does not provide a reasonable approximation and the main effect of the to is to introduce short-range correlations in the N N wave functions. For large values of kF, the repulsive terms increase more rapidly with kF than the attractive ~r terms, which contributes to producing a minimum in the energy as a function of kF. In our example for normal nuclear matter (kF = 1.52 fm -~) the repulsive short-range contribution is ~76 MeV, which yields for the sum of kinetic energy plus short-range repulsion plus zl E~ about - 4 MeV. The rest of the energy (about -11 MeV) is mainly due to ~r-exchange. The contribution of the 7r to the energy can be split into two parts: the repulsive one-pion and the attractive two-pion terms. Due to the spin and isospin of the pion the direct matrix elements of VNN(Cr) do not contribute to the energy of nuclear matter. The exchange terms are relatively small and repulsive. Especially because of its tensor components the iterated two-pion exchange terms (second order of VNN in the G-matrix) are large and attractive. For the physical values of Nc and Rq they are larger than the exchange terms of the one-pion piece. This can be seen if we calculate

718

H. Miither et al. / QCD and nuclear binding

a AE~ in the same manner as outlined above for the tr. For smaller values of kF (kF < 1.5 fm -1) AE,~ becomes slightly more attractive with increasing kv- For higher densities the Pauli operator in the Bethe-Goldstone equation (36) suppresses the iterative 2~r terms to some extent and weakens AE,~. Although the dependence of AE~r on k F is relatively weak as compared e.g. to eq. (46) the delicate balance of repulsive to and attractive tr terms causes AE,~ to be an important term in determining the saturation properties of nuclear matter. Keeping these features in mind it is quite easy to understand the way in which the nuclear saturation properties depend on Nc and Rq. Let us first keep Nc = 3 fixed and change the masses of the light quarks. If we consider a decrease of the quark masses (Rq < 1) the repulsive short-range contributions should not change since the to coupling constant is not changed at all and the effect on too, is negligible. The change of the or contribution with Rq was already given in eqs. (46) and (47). At first sight the dependence of AE~ seems to be rather weak. However, one must take into account that already a small enhancement of the attractive tr term by decreasing Rq will shift the point of balance between the attractive and repulsive terms to a considerably larger value for the saturation density since the to-contribution is hardly changed. At the higher density the value of dE~ gets enhanced due to its strong density dependence. A decrease of Rq also leads to a smaller pion mass, a ~r-exchange of longer range and therefore a larger attractive 2~" contribution. These combined effects yield the strong dependence of the binding energy of nuclear matter on Rq displayed in fig. 4. Already for a value of Rq = 0.75 a binding energy per nucleon of -97.6 MeV is obtained (kF ~- 2.28 fm-~), which is about six times larger than for the physical value of Rq ~ 1. Increasing the value of Rq above the physical value, the binding energy and also the saturation density of nuclear matter gets reduced. According to our model calculation nuclei would be unbound if the current quark masses of the light quarks would be 20% larger than their physical values. As a next step we consider the quark masses to be fixed (Rq-~ 1) and change the number of colours N~. In this case the nucleon mass gets larger with increasing N~ and therefore the kinetic energy is reduced. Since the N~ dependence of ~E~ is stronger than the Rq dependence, one might expect that the energy of nuclear matter depends even more strongly on N~ than on Rq. The results displayed in fig. 5, however, show that the dependence is of similar degree. An enhancement of Nc by a factor ~ (N~ = 4) yields an energy (E = -96.0, kF = 2.29 fro-l), which is very close to the one obtained when Rq is only 3 of its normal value (see above). On the case of variable Nc also the repulsive short-range terms get changed. Also the to coupling constant g2NN is scaled like N~. This acts as some kind of balance against the attractive terms discussed above. Going to larger values of Nc the attractive pion contribution AE~ becomes more important than in the case of decreasing Rq. TO understand this fact one should recall that AE~ has a repulsive term which scales like N~ and an attractive term (iterated one-pion exchange) which scales like N 2.

H. Miither et al. / QCD and nuclear binding I

I

719

I

I

~

-2o

- 4 0 ~-

.~

-

60

a3 -80

I

I

I

I

0.8

0.9

1.0

I.I

Ratio

of

quark

masses

Rq

Fig. 4. Binding energy of nuclear matter in MeV per nucleon as function of quark mass ratio fixed number of colours Nc = 3.

Rq for

-20

0)

-40

-60 ._~

-80

I

2.6

I

I

I

:3.0 Number

I

I

3.4 of

Colours

I

3.8 Nc

Fig. 5. Binding energy of nuclear matter in MeV per nucleon as function of number of colours N c for fixed quark mass ratio Rq.

H. Miither et al. / QCD and nuclear binding

720

For increasing Nc the attractive term becomes more and more dominant. Therefore the AE~ contribution at N o = 4 (Rq= 1) is about twice as large ( A E ~ = - 9 1 MeV) as in the case of Rq = 0.75 (No = 3, AE~ = - 4 6 MeV) although the total energies and the saturation densities are essentially equal. Decreasing Nc reduces the energy so that at Nc = 2.5 nuclear matter gets unbound. Also in this limit the changes of AE~, which becomes repulsive, are very important. As a result of this discussion we conclude that the binding energy of nuclear matter gets reduced by increasing the mass of the quarks a n d / o r by decreasing the number of colours. In the one case (changing Rq) the change of AE~ is more important, whereas in the other case this is rather due to the change of zlE,~. The result, however, is the same in both cases: nuclear matter becomes unbound when one reaches a certain boundary in this Arc versus Rq manifold. The position of this boundary is displayed in fig. 6. The cross corresponds to the values of the binding energy per nucleon and the saturation density in the real, physical world. The closeness of the physical values of Nc and Rq to this borderline of vanishing energy is just another way of expressing the fact that compared to the energy scale set by the mass of the nucleons, nuclear matter or finite nuclei are very loosely bound objects. The fact that this borderline of vanishing energy is almost a straight l'ae in the Rq versus N~ diagram, supports the observation discussed already above that the very similar energies are obtained for nuclear matter as long as the ratio N~/Rq is kept constant. 1

I

I

I

I

I

I 2

I

1 5

t

I

1.8

c~

1.4

O

I0

~= o.6

c~

0.2

Number

of

L

Colours

1 4

i 5

Nc

Fig. 6. Curve of vanishing nuclear binding in the Q C D p a r a m e t e r space of light quark mass ratio and n u m b e r of colours N c.

Rq

H. Miither et al. / QCD and nuclear binding

721

Another interesting feature resulting from our investigations is displayed in fig. 7. Each point in this plot stands for the energy and the saturation density obtained in the nuclear matter calculation for an OBEP which results from a certain choice of Nc and Rq. The strong correlation between the resulting energy and density is remarkable. This behaviour seems to be similar to the so-called "Coester band" of nuclear matter calculations 32). The Coester band is obtained when one displays results for nuclear matter using different models for the N N interation, which are all fitted to the experimental N N phase shifts. In the present paper, however, we are studying N N interactions which are quite different and which will also give quite different phase shifts. That we obtain a very similar correlation between energy and density (but spread over a much wider range of energies and densities) may indicate that this is a general feature of the many-body descriptions of nuclear systems in general and is not so much related to the N N interaction. After having discussed the binding energy of nuclear matter, we also would like to estimate the change of the different excitation modes of nuclear systems as a function of N~ and Rq. As discussed in sect. 3 a rough estimate can be obtained by analysing the effective mass and the Landau parameters of the residual particlehole interaction. Results for these quantities assuming different values of Arc (with Rq = 1) are displayed in table 2. The corresponding results as a function of Rq (with Nc = 3) are listed in table 3. In each case the calculation has been performed at a

0

O~ 0 0 0 0

-25

0 0

-50 g:

~5 -75

o

-I00 o

I

I

I

I

2

5

Fermi

momentum (fm -I)

Fig. 7. Dependence of binding energy o f Fermi m o m e n t u m at the saturation density for nuclear matter obtained with the OBEP used in this work. The OBEP parameters are expressed as a function of the n u m b e r o f quark colours N~ and the ratio of light quark masses to their physical values Rq in sect. 2. Each point on this figure corresponds to a different choice of N~ and Rq.

H. Miither et al. / QCD and nuclear binding

722

density close to the saturation density for this (No, Rq) combination. These densities and the corresponding energies are also given in the tables. With increasing Nc the effective mass decreases drastically. This is true for mE*, which is derived from the Landau parameter F1 (see eq. (44)) and characterizes the single particle energy at the Fermi surface, as well as for m*, the effective mass which was used to parametrize the BHF single-particle spectrum for all hole states (see eq. (41)). Actually it turns out that the values of m* and m* are very close to each other. Of course, comparing the ratios m*/m in table 2 one must take into account that the nucleon mass m increases linearly with No. The decreasing value of m* or m* with increasing binding energy can also be observed in table 3. The density of states at the Fermi surface, which is essentially the product of kF and mE*, is almost constant as a function of Rq and increases linearly with Arc. If one now divides the Landau parameters by the density of states, in order to compare the interaction of two nucleons at the Fermi surface in the different spin-isospin channels (see eq. (43)), one finds that this interaction hardly changes with N~ or Rq. Only the interaction in the isoscalar spin-spin interaction (Go), which is weak at the physical point becomes attractive for large Arc or small Rq. It is the increase of the density of states due to the increase of the bare nucleon mass which is responsible for the large Landau parameters for large N~. The Landau parameters calculated from the bare G-matrix discussed so far can be used to analyse the G-matrix at the Fermi surface. However, they provide a very TABLE 2 Landau parameter as a function of N~ 2.7

3.0

3.5

4.0

5.0

E [MeV] k F [fm -1] rna*[ raN] m~ [mN]

-5.0 1.2 0.70 0.71

-15.2 1.6 0.60 0.60

-45.1 2.0 0.50 0.51

-94.9 2.4 0.43 0.40

-218.0 3.0 0.43 0.39

Fo F~ {bare} G o { G-matrix} G~

- 1.45 0.65 0.16 0.78

- 1.45 0.65 0.05 0.90

- 2.04 0.91 -0.14 1.11

-2.10 0.85 -0.41 1.20

-3.09 0.34 -0.93 1.43

Fo F~ {plus} G O{induced} G~

-0.35 0.35 -0.05 0.91

-0.20 0.34 -0.27 0.99

-0.23 0.69 -0.45 1.11

0.02 1.03 -0.60 0.97

0.33 1.90 -0.77 0.53

Landau parameters calculated for the bare G-matrix and with inclusion of the induced interaction terms are given for different values of Arc (with Rq = 1). For each N c a calculation was performed for a density, given by the Fermi m o m e n t u m kF, which was close to the saturation point for this No. The energy obtained at this density (E) and also the mass parameter defining the B H F spectrum (m*) are listed. The BHF effective mass as well as the effective mass determined from the Landau parameter (mE*) are given in terms of the nucleon mass (mN) which is valid for the actual No.

723

H. Miither et al. / QCD and nuclear binding

TABLE 3 Landau parameters as a function of Rq Rq

1.2

1.0

0.85

0.75

E [MeV] kv [fm-1] m* [mN] mL*[mNl

-0.4 0.8 0.80 0.81

-15.2 1.6 0.60 0.60

-50.3 2.0 0.49 0.50

-97.1 2.2 0.43 0.44

Fo F~ {bare} GO{G-matrix} G~

- 1.46

- 1.45

- 1.65

-1.86

0.74 0.18 0.69

0.65 0.05 0.90

0.65 -0.06 0.96

0.68 -0.16 0.95

Fo F~ {plus} Go {induced} G~

-0.42 0.51 0.11 0.81

-0.20 0.34 -0.27 0.99

-0.22 0.43 -0.39 1.01

-0.33 0.54 -0.42 0.99

The Landau parameters are givenfor different value of Rq (keeping N~ = 3). For further details see table 2. poor approximation for the quasiparticle interaction which is responsible for the excitation modes of nuclear systems. This is especially due to the very attractive value of Fo. The value of Fo being smaller than - 1 indicates that in the G-matrix approximation nuclear matter would be unstable against scalar-isoscalar excitations. The collectivity in this channel, however, makes it necessary to extend the calculation of the residual interaction and take into account the effects of the so-called induced interaction 33). Neglecting any momentum dependence, i.e. evaluating the induced interaction terms from the Landau parameter of lowest order only, we obtain the results in the lower parts of tables 2 and 3. The resulting parameters show a weak dependence only. As a function of No, Fo as well as F~ become more repulsive. This means that not only the compression modulus k2 K = 6~ (1 + Fo) (48) 2mL but also the coefficient for the symmetry energy

1 k~ /3 = 3 2m~ (1 + F~)

(49)

increase even beyond the strong kF dependence. It is interesting to note that the residual interaction in the spin-spin channel (Go) becomes very attractive, so that spin-spin correlations might be expected for systems with large Nc or small Rq. 5.

Conclusions

Quantumchromodynamics generalized to any number of colours Arc has only two parameters: the quark-gluon coupling constant, which should be proportional to

724

H. Miither et al. / QCD and nuclear binding

1/x/Nc in order to obtain finite results for gluon loops in the large Nc limit, and the masses of the light quarks. It was the aim of the present investigation to study the dependence of bulk properties of nuclei on these two fundamental variables. It is our hope that an investigation of exotic systems with unphysical values for Arc and mq may lead to a deeper understanding of real nuclei. Since QCD in the limit of large Arc goes over into a meson theory, we use a one-boson-exchange (OBE) model for the N N interaction and study the dependence of its parameters, which are the masses of the mesons and the meson-nucleon coupling constants, as a function of Nc and mq. Calculating the binding energy of nuclear matter, we find that it depends very strongly on the values considered for Nc and mq. Lowering the mass o f the light quarks yields smaller masses especially for the zr and tr mesons and thereby an increasing range for the corresponding meson exchange potentials. In this case it is in particular the increasing range for the o--exchange which yields a larger binding energy for nuclear matter. Increasing the number of colours Nc yields a larger nucleon mass and larger coupling constants for all mesons. In this case the effects of the enhanced attractive g-exchange are to some extent counterbalanced by more repulsive short-range components in the N N interaction (to-exchange). From the ~r terms the repuslive one-pion exchange term gets enhanced, but the attractive iterated pion-contribution even more so, which turns out to be a major source for the increasing binding energy. Although the mechanisms for the change in energy with Nc and mq are somewhat different, it turns out that the energy of nuclear matter depends to a good approximation on the combination N c / m q only. This is also the reason why the curve of zero energy for nuclear matter in the plane spanned by the parameters N~ and mq (see fig. 6) is almost a straight line (and would, incidentally, pass through the origin when extrapolated). It is interesting to note that zero binding energy is obtained by, for example, reducing the number of colours to 2.5 (keeping mq fixed) or by increasing the current quark masses mq to 1.2 times the physical values (keeping N~ fixed). This shows that already a small change of these parameters leads to unbound nuclei. Studying the saturation densities for nuclear matter calculated for different N~ and mq, we find a strong correlation between the energy and the density of the saturation points calculated for quite different values of N~ and mq (see fig. 7). This correlation is very similar to the so-called Coester band which has been obtained by calculating saturation points of nuclear matter for phase shift equivalent N N potentials. This shows that this strong correlation is not only restricted to potentials which are adjusted to reproduce N N phase shifts. This behaviour might be understood from the fact that the binding energy of nuclear matter results from a delicate balance between large repulsive and attractive contributions which are both strongly density dependent (see e.g. eq. (46)). Changing this balance a little bit can lead to a change of the energy accompanied by a change of the density.

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We also study the excitation modes of nuclear systems by evaluating Landau parameters for the residual interaction at the Fermi surface. The interactions in the different spin-isospin channels in general show a weak dependence on mq and Nc only. The density of states at the Fermi surface (calculated for the saturation points) is almost independent of mq but increases approximately linearly with No. Therefore, collective features show up for larger values of Arc. Especially the spin-spin correlations seem to become important since they are not damped by the terms of the induced interaction. This investigation was prompted by questions which were raised by Ed Witten, and the authors wish to thank him for his comments. They also wish to thank John Durso and Andreas Wirzba for many useful discussions and Tom Kuo for placing some of his computer programmes at their disposal. Two of us (C.A.E. and H.M.) would like to thank the State University of New York at Stony Brook for their support and the nuclear theory group for their hospitality. C.A.E. also wishes to thank the South African Council for Scientific and Industrial Research for financial assistance which made this visit possible.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26)

G. 't Hooft, Nucl. Phys. B72 (1974) 461; B75 (1974) 461 E. Witten, Nucl. Phys. BI60 (1979) 57 A.D. Jackson, M. Rho and E. Krotschek, Nucl. Phys. A407 (1983) 495 E. Witten, private communication M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. 11147 (1979) 385, 448 G. Karl and J.E. Paton, Phys. Rev. D30 (1984) 238 A. Jackson, A.D. Jackson and V. Pasquier, Nucl. Phys. A432 (1985) 567 G.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. 11228 (1983) 552 K. Erkelenz, Phys. Rep. 13C (1974) 191 K. Holinde, Phys. Reports 68C (1981) 121 M. Lacombe, B. Loiseau, J.M. Richard, R. Vinh Mau, J. Cote, P. Pires and R. de Tourreil, Phys. Rev. C21 (1980) 861 J.W. Durso, M. Saarela, G.E. Brown and A.D. Jackson, Nucl. Phys. A278 (1977) 445 K. Holinde, R. Machleidt, A. Faessler, H. Miither and M.R. Anastasio, Phys. Rev. C24 (1981) 1159 R. Reid, Ann. of Phys. 50 (1968) 411 K. Holinde, K. Erkelenz and R. Alzetta, Nucl. Phys. A194 (1972) 161; A198 (1972) 598 Particle Data Group, Rev. Mod. Phys. 56 (1984) 51 M. Gell-Mann, R.J. Oakes and B. Renner, Phys. Rev. 175 (1968) 2195 S. Weinberg, Trans. NY Acad. Sci. 38 (1977) 185 A.I. Vainshtein, V.I. Zakharov, V.A. Novikov and M.A. Shifman, Sov. J. Part. Nucl. 13 (1982) 224 B.L. Ioffe and M.A. Shifman, Nucl. Phys. B202 (1982) 221 A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. DI2 (1975) 147 E. Witten, Workshop on Solitons in nuclear and particle physics, Lewes Center for Physics, June 1984 S.L. Adler, Phys. Rev. Lett. 14 (1965) 1051; W.I. Weisberger, ibid, 1047 G.E. Brown, in Mesons in nuclei, ed. M. Rho and D.H. Wilkinson (North-Holland, 1979) pp. 331-356 D. Amati, E. Leader and B. Vitale, Nuovo Cim. 17 (1960) 68; 18 (1960) 409 M. Chemtob, J.W. Durso and D.O. Riska, Nucl. Phys. B38 (1972) 141

726 27) 28) 29) 30) 31) 32) 33)

H. Miither et al. / QCD and nuclear binding

J.W. Durso, A.D. Jackson and B.J. Verwest, Nucl. Phys. A345 (1980) 471 G.E. Brown and W. Weise, Phys. Reports 22C (1975) 281 B.D. Day, Rev. Mod. Phys. 50 (1978) 495 B.D. Day, Phys. Rev. C24 (1981) 1203 A.B. Migdal, Theory of finite fermi systems (Interscience N.Y., 1967) F. Coester, S. Cohen, B.D. Day and C.M. Vincent, Phys. Rev. CI (1970) 769 S. Babu and G.E. Brown, Ann. of Phys. 78 (1973) 1