A Quark-Meson Coupling Model for Nuclear and Neutron Matter

nucl-th/9403015 18 Mar 1994 Adelaide February y A Corresp ondence to email athomasphysicsadelaideeduau density dep endence internal structure of the nucleon and QHDclear Finally understanding we use of the the mo del relationship b to etween study this the mo del which incorp orates the to nucleon quark meson investigate An University University explicit quark bags Department Physics b ound the ADPT Dr K Saito email ksaitonuclphystohokuacjp prop erties of of the quark condensate inmedium by Division mo del based Adelaide the of neutron coupling selfconsistent exchange Physics Sendai of March A b oth Tohoku South W on K Abstract nuclear and a mean eld and Saito Thomas matter Australia mo del College Mathematical Japan and neutron y description of of of for ! Pharmacy matter and nuclear Physics Australia nonoverlapping mesons is used We establish and a

Recently there has b een considerable interest in relativistic calculations of innite

nuclear matter as well as dense neutron matter A relativistic treatment is of course

essential if one aims to deal with the prop erties of dense matter including the equation of

state EOS The simplest relativistic mo del for hadronic matter is the Walecka mo del

often called Quantum Hadro dynamics ie QHDI which consists of structureless

nucleons interacting through the exchange of the meson and the time comp onent of the

meson in the meaneld approximation MFA Later Serot and Walecka extended the

mo del to incorp orate the isovector mesons and QHDI I and used it to discuss

systems like neutron stars with N Z QHD has proven to b e a very p owerful to ol for

the treatment of a wide range of nuclear phenomena However it should b e noted

that a strong excitation of nucleonantinucleon pairs due to the meson elds is somewhat

unlikely from the p oint of view of the quark mo del b ecause the meson elds create only

one quarkantiquark pair at a time An extension of such a mo del to include the structure

of the nucleon may provide some new insight into the prop erties of hadronic matter

A quarkmeson coupling QMC mechanism for the saturation of nuclear matter was

initially prop osed by Guichon In his simple mo del nuclear matter consists of non

overlapping nucleon bags b ound by the selfconsistent exchange of scalar and vector

mesons in the MFA This mo del has b een generalised to include nucleon Fermi motion

as well as the centerofmass cm correction to the bag energy Using this mo del it

has proven p ossible to investigate b oth the change of nucleon prop erties in nuclear matter

and nuclear structure functions An alternative version of this mo del based on the

color dielectric soliton mo del has b een developed by Naar and Birse who also applied

it to deepinelastic scattering To treat asymmetric nuclear matter ie N Z the

mo del requires some extension In this work we show that it is p ossible to include the

contribution of the isovector vector meson in the study of such systems We shall see

that it is very interesting to compare this new mo del with QHDI I and to investigate the

prop erties of hadronic matter in terms of quark degrees of freedom

Let the meaneld values for the the time comp onent and the time comp onent

in the third direction of isospin elds in uniformly distributed nuclear matter with

N Z b e and b resp ectively The nucleon is describ ed by the static spherical MIT

bag in which quarks interact with those mean elds The Dirac equation for a quark

eld in a bag is then given by

0 z

i m V V V

q q

q q q q q

where V g V g and V g b with the quarkmeson coupling constants g g

q z

and g The bare quark mass is denoted by m and is the third comp onent of the Pauli

matrix Here we deal with u and d quarks qu or d only The normalized ground

state for a quark in the nucleon is then given by

j xr R

q 0

i tR

r t N e

i r j xr R

q 1

where

quark V for RV

q q

2 3 2 2

N R j x Rm x

q q

0 q

2 2

with x Rm and the quark spinor The eective quark mass m is

q q

dened by

m V m g m

q q

The b oundary condition at the surface provides the equation for the eigenvalue x

j x j x

0 q 1

Using the SU spinavor nucleon wave function the nucleon energy is given by

p p

E V E V

bag N

n n

proton

V for E V

bag

neutron

The sign of the eld value in this pap er is opp osite to that in Ref in order to simplify the

comparison with QHD

Here the bag energy is

q 0

E B R

bag

with B the bag constant and z the usual parameter which accounts for zerop oint motion

In order to correct for spurious cm motion in the bag the mass of the nucleon

at rest is taken to b e

M E hp i

bag cm

2 2 2

hp i and hp i is the exp ectation value of the quark momentum squared where hp i

q q cm

xR Inside nuclear matter the eective nucleon mass M is also given by minimizing

eq with resp ect to R To test the sensitivity of our results to the radius of the free

nucleon R we vary B and z to x R f m and the free nucleon mass

0 0 0

M MeV The values of B and z for several choices of radius for the free nucleon

N 0

Table B and z for some bag radii m MeV

0 q

R f m

B MeV

are summarized in table

The Pauli principle induces Fermi motion of the nucleon Since the energy of a moving

bag with momentum k is

proton

k M k V V for

neutron

the total energy p er nucleon at the nuclear density is given by

q Z Z

k k

F F

n p

2 2 2 2 2 2

k V M dk m m m b E

tot

B B B

Here is the dierence b etween the proton and neutron densities and k and

3 p n F

k are the Fermi momenta for protons and neutrons resp ectively

3 3

and k k

n B p n p

F F

p n

3 3

The eld created by uniformly distributed nucleons is determined by baryon number

conservation to b e

V g g

B B

2 2

g m m

where we dene a new coupling constant g g while the and meanelds are

given by the thermo dynamic conditions

E E

tot tot

and

R R

B B

Since the eld value is expressed by

where we take g g the total energy p er nucleon can b e rewritten as

Z Z q

2 2

k k

F F

p n

m g

2 2

dk k M E

B 3 tot

3 2 2

m m

B B B

Therefore using eq the value of the eld is given by

Z Z

k k

F F

p n

M M

N N

2 3

M k

We should note that the expression for the total energy eq is identical to that of

QHDI I The eect of the internal quark structure enters entirely through the eective

nucleon mass eqs and and the selfconsistency condition SCC for the eld

Let us consider the SCC further The eective quark mass m is given by eq If

the nucleon were simply made of three massive constituent quarks the nucleon mass in

vacuum and that in matter would satisfy

M m and M m

N q

and hence using eq the eective nucleon mass could b e

M M g

where we dened g g Since one nds

from eq the SCC gives

g g

Z Z

k k

F F

p n

g M

2 3

M k

Together with eq we then nd the SCC in the heavy quark mass limit

M g M

Z Z

k k

F F

p n

M g

dk M

2 3

k M

This is exactly the SCC for QHDI I

However in the more realistic case of light quarks the internal structure of the nucleon

signicantly alters the SCC In particular the u and d quarks in the bag mo del like current

quarks have very small masses and must therefore b e treated in a fully relativistic manner

Using eqs and one nds

E V M

q bag

g S

M E R E

bag bag

g C

where S is the scalar density of the nucleon bag in matter

q q

S dr

q q

Hence we nd the SCC for the eld in the case where the nucleon has the quark

structure

Z Z

k k

F F

n p

M g

dk C g

2 3

M k

Clearly the eect of the internal quark structure is completely absorb ed into the scalar

density factor C Thus we see that it is p ossible to unify the present QMC mo del and

QHD in the MFA the energy p er nucleon and the SCC for the eld are resp ectively

given by eq and eq with

M QHD

for g

QMC C

The scalar density factor is shown in g as a function of the scalar eld strength for

0.45

0.4

0.35

0.3 C 0.25

0.2

0.15

0.1 0 0.1 0.2 0.3 0.4 0.5

RVσ

Figure Scalar density factors for various bag radii as a function of RV m

MeV The solid dotted and dashed curves show the results for R and f m

resp ectively

three bag radii We see that it is much smaller than unity and that the dep endence on

the bag radius is not strong

Now we are in a p osition to present numerical results First we take m MeV and

2 2

determine the coupling constants g and g so as to t the binding energy MeV

and the saturation density f m for equilibrium nuclear matter Furthermore

we choose the meson coupling constant g so as to repro duce the bulk symmetry

energy MeV The coupling constants and some calculated prop erties of nuclear

Table Coupling constants and calculated prop erties of equilibrium nuclear matter The

eective nucleon mass M the nuclear compressibility K and the symmetry energy a

are quoted in MeV The b ottom row is for QHDI I We take m MeV m

MeV and m MeV

x R

2 2 2

K a M g g R g

4 0

R x

0 0

QHDI I

matter at saturation density are listed in table Because the nucleon is extended these

coupling constants are related to the coupling constants at the meson p oles g im

g im by

or or

ik r

g im g j dre

q q

jk j=im

ik r y

dre j g im g

jk j=im

Using eqs and the value of the coupling constant at each meson p ole is ob

2 2 2

tained for example g im g im and g im for

R f m It should b e p ointed out that the strength of the eld is considerably

less than that required in QHD b ecause the cm correction to the bag provides a new

source of repulsion Furthermore the present mo del provides values for the nuclear

compressibility K well within the exp erimental range K MeV In

the last two columns of table we show the relative mo dications with resp ect to their

values at zero density of the bag radius R and the lowest eigenvalue x at the saturation

density The changes are small see also Ref

(MeV) 20 N

10 - M tot

E 0

-10

-20 0 0.5 1 1.5 2 2.5 3 ρ / ρ

Β 0

Figure Energy p er nucleon for symmetric nuclear matter and for neutron matter The

dotdashed curve is the saturation curve for nuclear matter with R f m The solid

2 2

curve for g and the dotted curve for g show the results for neutron

matter with the same bag radius

The total energy p er nucleon for b oth symmetric nuclear matter and for neutron matter

with R f m are presented in g We see that ignoring the meson coupling yields

a shallow b ound state of neutron matter around but it b ecomes unbound when

B 0

the meson contribution is introduced In g we show the equation of state EOS

for neutron matter for b oth R and f m The meson contribution is again

0 36

35 p = ε )

2 34

32 p(dyn/cm

10 31 log 30

12 12.5 13 13.5 14 14.5 15 log ε (g/cm3)

Figure The equation of state for neutron matter The dashed curve for R f m

and the dotted curve for R f m show the results when the rho meson contribution

is ignored The solid and dotdashed curves are the full calculations for R and

f m resp ectively

signicant in the EOS while the dep endence on the radius of the free nucleon is weak

Although the shap e of the EOS for neutron matter in the present mo del is qualitatively

similar to that in QHDI I it is much softer than in QHDI I b ecause of the lower nuclear

compressibility resulting from the internal quark structure of the nucleon

Having shown that the QMC mo del provides an acceptable description of the satu

ration prop erties of nuclear matter we now apply it to study the b ehaviour of the quark

condensate in matter Recent work has linked the quark condensate to a wide range of

nuclear phenomena and it is an imp ortant parameter in any nite density calculation

using QCD sumrules From this p oint we shall deal only with symmetric nuclear mat

ter and introduce a tiny current quark mass for b oth u and d quarks for example m

MeV Accordingly we must readjust the bag parameters and the coupling constants to

t the free nucleon mass and the saturation energy and density We nd that B and z

b ecome slightly reduced and enhanced resp ectively from those values shown in table

2 2

and that the coupling constants are g and g

for R f m resp ectively

The quark condensate at low density can b e related to the nucleon term which

is obtained by using the HellmannFeynman theorem Since we nd

from eqs and the term for a nucleon in free space is

m C

N q

where C is the scalar density factor for a massive quark We note that it do es not

change much from that in the massless quark case and that the bag radius dep endence

is not strong unlike g If we use C and m MeV we obtain

q N

exp

MeV which is much smaller than the empirical value MeV However it

should b e noted that there are many other p otential contributions to such as the

meson cloud of the nucleon and its strange quark content

Following Cohen et al the dierence b etween the quark condensates in matter

and vacuum is given by

Q Q

E M E m E m

M m m m m m

q q q

C E E E

m C M m m

where E is the total energy density E Q is the inmedium quark condensate

B tot B

h jq q j i and Q that in vacuum We have used eqs and and some relations

B B

m m m

N N

suggested by Cohen et al and with

m m m m M

q q q q

to derive the last equation Using eqs and and the GellMann and

2 2

OakesRenner relation m Q m f we nd

2 2

Q m g m

B N 0 0 B

g g

2 2 2 3 2

Q m g g m f C

0 0 0

where m is the pion mass MeV and f the pion decay constant MeV Note

that the ratio of the quark condensates Q Q is a simple function of the mean

eld g and In g the eld values are plotted at low density We see that

B 0

300

250

200 − σ σ

g 150

100

0 0.2 0.4 0.6 0.8 1 ρ / ρ

Β 0

Figure Mean eld value of the meson at low densities with m MeV The

solid dotted and dashed curves are for R and f m resp ectively

the dep endence on the radius of the bag is very weak at low density and that the mean

eld value is well approximated by a linear form for

B 0

g V MeV

2 2

and C we nd a g Hence if we assume MeV and take g

simple formula for the ratio at low densities

B B B

0 0

which is consistent with the mo delindep endent prediction for the ratio of quark conden

sates prop osed by Cohen et al At the inmedium quark condensate is

B 0

reduced by roughly from its value in vacuum

Furthermore since the eective nucleon mass can b e expanded at low densities as

M m V m M

N q

q N

m =m

M C g

we obtain a relation b etween the quark and nucleon masses

M M dM

N N

m m dm

q q

m =m

where we have used eq This relation is satised within a few p ercent up to

B 0

If we now solve eq for M M we nd

C M

M M

N N

and using eq and C we nd

N 0

Comparing this with eq we see that Q Q is essentially M M at low

B N

density This result is intermediate b etween the cubic dep endence found by Brown and

Rho and the linear dep endence found by Cohen et al If on the other hand

we were to take the limit of heavy quarks considered earlier it is easy to show that one

recovers the linear dep endence

Finally we would like to add some caveats concerning the present mo del The basic

idea is that the mesons are lo cally coupled to the quarks This is certainly common for

pions in mo dels like the chiral or cloudy bag but may b e less justied for vector mesons

which are not collective states The mo del is b est suited to low and mo derate density

regions certainly no higher than b ecause shortrange qq correlations are

B 0

not taken into account At high densities these would b e exp ected to dominate The

pionic cloud of the nucleon should b e considered explicitly in any truly quantitative

study of the prop erties of the nucleon inmedium

In summary we have presented the quarkmeson coupling mo del based on a mean eld

description of nucleon bags interacting through the exchange of and mesons and

shown that it can provide an excellent description of the prop erties of b oth symmetric and

asymmetric nuclear matter The relationship to QHD and particularly the role of the

internal structure of the nucleon was claried Finally the mo del was shown to provide

a reasonable description of the quark condensate inmedium The present mo del seems

to b e the simplest p ossible extension of QHD to incorp orate explicit quark degrees of

freedom

This work was supp orted by the Australian Research Council

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