How Reliable is the Mean-Field Nulear Matter
Desription for Supporting Chiral Eetive Lagrangians?
A. Delno 1
, M.Malheiro 1
, and T. Frederio 2
1
Institutode Fsia,UniversidadeFederalFluminense,
24210-340,Niteroi,R.J.,Brazil
2
InstitutoTenologiodaAeronautia
12331S~aoJose dosCampos,S~aoPaulo,Brazil
Reeivedon20May,2001
Thelinkbetweennon-linearhiraleetiveLagrangiansandtheWalekamodeldesriptionofbulk
nulearmatteris questioned. Thisfat is by itselfduetothe MeanField Approximation (MFA)
whihinnulearmatermakesthepitureofanuleon-nuleoninterationbasedonsalar(vetor)
mesonexhangeequivalenttothedesriptionofanulearmatterbasedonattrativeandrepulsive
ontatinterations. WepresentalinearhiralmodelwherethislinkbetweentheWalekamodel
andanunderlyingtohiralsymmetryrealizationstillholds,duetoMFA.
Reently, a hiral eetive Lagrangian desription
of nulearmatter has been suggested[1℄. Suh a
pro-posal wasbased on adding afour-fermion Lagrangian
term to a general Lagrangian proposed by Weinberg
[2℄,desribingtheinterationsofpionsandnuleonsin
whih thespontaneously broken SU(2) xSU(2) hiral
symmetry is non-linearly realized. In short, the
on-strutednon-linearhiralLagrangianis
L
NLC =L
Weinberg + L
4 N
; (1)
where
L
4 N =
1
2 G
2
s (
) 2
1
2 G
2
v (
)
2
; (2)
It turned out from that study that, at the mean-eld
level,theequationsofstateforthis modelarenothing
but those obtainedfrom the standardWaleka model
[3℄for innitenulearmatter. Therefore,onegetsthe
pitureofaonnetionbetweenanon-linearhiral
La-grangianand abaryoniphenomenologialmodel. At
the level ofmean-eld this is orret. Inorder to
un-derstandhowfarweanproeedbeyondthispitureit
isimportanttoseewhereallthisomesfrom.
Tostudythepropertiesofhadronimatter,Waleka
[3℄ proposed a simple renormalizable model based on
eld theory, whih is often referred to as Quantum
Hadrodynamis (QHD). In this model nuleons
inter-at through the exhange of and ! mesons, with
simulating medium rangeattration and ! simulating
short-range repulsion. The usual approah to solve
whih the meson elds are replaed by their
expeta-tionvalues. A uriousaspetof thismodel isthat the
salar (vetor) m
(m
!
) masses and g
(g
!
) oupling
onstantsin the equations of statefor innitenulear
matter are eliminated in favour of C 2
= g
2
M
2
=m 2
and C 2
! =g
2
! M
2
=m 2
!
[4℄. Sine C 2
andC
2
!
are tted
toreproduethenulearmatterbulkproperties,values
of m
and m
!
beome irrelevant. It meansthat this
model annot distinguish betweenarbitraryvaluesfor
themesonimasses.Suharbitrarinessanbeextended
to inlude innitemesoni masses. If this is the ase
, oneis led to thesituation where the interation
be-tween the nuleons is zero-range. By using attrative
and repulsive ontat interations it is easy to show
that the above onjeture is orret and therefore, a
ZeroRangeModel(ZRM)isequivalenttotheWaleka
model forinnitenulear matterat theMFA level. It
meansthat,ifwestartwith
L
ZRM =
i
M + 1
2 G
2
s (
) 2
1
2 G
2
v (
)
2
;
(3)
where 's are baryoni elds, we arrive at the same
equationsofstateoftheWalekamodelifonejust
mul-tipliesbyM 2
theouplingonstants. Thisstrange
pi-ture is therefore onstruted from MFA itself applied
for innitenulear matter, andhasnothingstritly to
do with hiral invariane. The Lagrangian given by
Eq.(1) andtheLagrangiangivenbyEq.(3) leadto the
sameequationsofstateobtainedfromtheWaleka
La-grangianatMFAlevelforinnitenulearmatter. They
obtained also for the self-oupling nonlinear !
model [5,6℄, if one adds terms suh as a( ) 3 and b( ) 4
to theaboveLagrangianwhihin thismodel
beomepurethree-nuleon-andfour-nuleon-fores
re-spetively. A modiation of the Waleka model,
in-ludinghigherordersofmany-bodyfores,assuggested
byZimanyiandMoszkowski[7-9℄alsobeomes
equiva-lentto ZRMfor innitenulearmatter inMFA ifone
replaes G 2
s
inEq. (3)byG 2
s
=(1+(
)=M).
To illustrate still further the risks of assoiating
hiral symmetry to the Waleka model we show
be-low that within MFA one annot even distinguish if
this symmetry is realized in anon-linear [1℄ or linear
form. We start with the well-known
Nambu-Jona-Lasinio(NJL)model[10℄,whihhasalinearrealization
of hiral symmetry. We inlude a vetor interation
in a urrent-urrentform and theLagrangianremains
invariant under a linear hiral transformation of the
baryoneld L LC = i + 1 2 G 2 s (( ) 2 +( i 5 ~ ) 2 ) 1 2 G 2 v ( ) 2 ; (4)
For stati, innite nulear matter, the three-vetor
momentum and spindependentinteration averageto
zero due to rotational symmetry and the Lagrangian
redues to[11℄
L LC = i + 1 2 G 2 s ( ) 2 1 2 G 2 v ( ) 2 ; (5)
whih is idential to that of Eq. (3) , exept for the
baryonimassterm. IntheMFA, welinearize the
in-teration in Eq. (3) by losing the Fermi loop [12℄.
It meansreplaing ( ) 2 by 2 h i. In
nulear matter we have only
= 1 and
o . Here
h
iisthevauum(groundstate)expetationvalue
of the operators. Fora Lorentz-invariant and
parity-onserving vauum, the only non-vanishing term is
h
i.
So,in thesamespiritofthedynamialquarkmass
generation mehanism of the NJL model [13℄, we an
assoiatethenuleonrestmass M to
M = G
2 s h i va = G 2 s M (2) 3 Z 0 d 3 k E(k) (6)
therefore obtaining a mean eld Lagrangianidential
to theZRMmodelinMFA
L= i (M G 2 s h i) G 2 v ( o )h o i ; (7)
where now h
i means the expetation value of
these operatorsinthenulearmattergroundstate(all
stands for the degeneray fator equalto four. By
lookingat theproblem in thisperspetivewehave
in-trodued aonstraint, asin the standardNJL model.
Thus, in order to havea non trivialnuleon mass
so-lutionfor theEq. (6), theoupling onstant G 2
s must
be greaterthan aritialvalue G
rit = (4 2 )=( 2 ).
Hereistheut-owhihxesthebarenuleonmass
Mgivenbythegapequation
1 C 2 s 2 2 Z =M 0 x 2 dx p
(1+x 2
)
=0 (8)
where we have identied G 2
s
to the Waleka oupling
onstant C 2
s =M
2
and x = k=M is a dimensionless
variable. Byxing M=938.27MeV and thevalue of
C 2
s
whih gives the orret nulear matter saturation
properties [14,15℄ we have obtained = 328:5MeV.
Then, Eq.(7)will givealsothe sameequation ofstate
of the Waleka model, but now the hiral symmetry
is 'realized' in a linear way. We are aware that our
drawbak to this hiral approah is the existene of
a zero-frequeny mode, the Nambu-Goldstone boson,
whih in this aseis apseudosalar-isovetor
nuleon-antinuleonmodethatweannotidentifyto thepion.
In summary, all the arguments we have presented
stress the diÆulties of extrating , from a nulear
matterdesriptionat the MFA level, ajustiation of
Walekamodel omingfrom reenthiraleetive
La-grangians[16℄. Tobemorespei,ifoneaddsto the
WeinbergLagrangian[2℄thefour-nuleonterm, atthe
MFAlevelonearrivesataequivalentWalekamodelas
presentedinref. [1℄. However,ifweaddanyotherterm
involving eld derivatives whih expliitly breaks the
hiralsymmetrywearrivealsoatanequivalentWaleka
model. Wehave shown that the sameproedure may
beextendedtoamodelwherethehiralsymmetryan
berealizedlinearly. Toonlude,webelievethatwhat
isinfat behindallofthisisthefatthat theMFA of
Walekamodelforinnitenulearmattergivessimply
thesameresultsofZRM.Welaimthatismore
appro-priate and onsistent to think just about a distorted
piturefurnished byMFAthan tolaimanyhiral
re-alizationofhadroniphenomenologialmodel. In this
approximation, the pion whih realizesthe non-linear
hiralsymmetryannotoupletothenuleons. So,the
onlyway to onludesomethingabouthiral eetive
Lagrangiansdesribingbulknulear matterproperties
istogobeyondthisapproximation,wherethe
identi-ationwiththeWalekamodeldesriptionwillbelost.
This work hasbeenpartially supported by CNPq,
Brazil.
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