Relativistic Chiral Theory of Nuclear Matter and QCD Constraints

0.09 MB·English

by  G. Chanfray

Relativistic Chiral Theory of Nuclear Matter and QCD
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Constraints
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n G. Chanfraya M. Ericsona,b
a aIPN Lyon, IN2P3/CNRS, Universit Lyon1, France
J b Theory Division, CERN, Geneva, Switzerland
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h Abstract
t
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l Wepresentarelativisticchiraltheoryofnuclearmatterwhichincludestheeffectofconfinement.
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u Nuclear binding is obtained with a chiral invariant scalar background field associated with the
n radial fluctuations of the chiral condensate Nuclear matter stability is ensured once the scalar
[ responseofthenucleondependingonthequarkconfinementmechanismisproperlyincorporated.

Alltheparametersarefixedorconstrained byhadronphenomenology andlatticedata.Agood
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description of nuclear saturation is reached, which includes the effect of in-medium pion loops.
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Asymmetrypropertiesofnuclearmatterarealsowelldescribedoncethefullrhomesonexchange
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and Fock terms are included.
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Key words: Chiralsymmetry,confinement, nuclearmatter
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1 PACS: 24.85.+p11.30.Rd12.40.Yx13.75.Cs21.30.-x
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General context. The aim of this talk is to discuss the possible relation between funda-
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v mental properties of low energy QCD, namely chiral symmetry and confinement, with
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X the rich structure of the nuclear many-body problem. More precisely with the introduc-
tion of the concept of nucleon substructure adjustement to the nuclear environment [1]
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a we will bring some constraints from lattice QCD data into the modelling of many-body
forces. In the relativistic mean-field approachesinitiated by Walecka, the nucleons move
in an attractive scalar and in a repulsive vector background fields. This provides a very
economical saturation mechanism and a spectacular well known success is the correct
magnitude of the spin-orbit potential since the large vector and scalar fields contribute
to it in an an additive way. Now the question of the very nature of these background
fields and their relationshipwith the QCD condensates has to be elucidated. To address
this problem we formulate the effective theory in terms of the fields associated with the
fluctuations of the chiral condensates in a matrix form (W = σ+i~τ ·~π) by going from
cartesian to polar coordinates i.e., going from a linear to a non linear representation :
PreprintsubmittedtoElsevier 16January2009
W = σ + i~τ ·~π = SU = (f + s)exp(i~τ ·ϕ~ /f ). In ref [2] we made the physical as-
π π π
sumptiontoidentifythechiralinvariantscalarfields=S−f ,associatedwithradial(in
π
order to respect chiral constraints) fluctuations of the condensate, with the background
attractive scalar field.
In this approach the Hartree energy density of nuclear matter, including ω exchange,
writes in terms of the order parameter s¯= hsi : E0/V = ε0 = (4d3p/(2π)3)Θ(pF −
p)Ep∗(s¯) + V(s¯) + gω2/2mω2ρ2,whereEp∗(s¯)= p2 + MN∗2(s¯)isRthe energyofaneffec-
tivenucleonwiththeeffectiveDiracmassM∗(s¯)p=M +g s¯.g =M /f isthescalar
N N S S N π
couplingconstantofthe sigmamodel.Heretwoseriousproblemsappear.Thefirstoneis
the fact that the chiral effective potential, V(s), contains an attractive tadpole diagram
which generates an attractive three-body force destroying matter stability. The second
one isrelatedtothe nucleonsubstructure.The nucleonmasscanbe expandedaccording
to MN(m2π) = a0 + a2m2π + a4m4π + Σπ(mπ,Λ)+..., where the pionic self-energy is
explicitely separatedout. The a2 parameter is related to the non pionic piece of the πN
sigmatermanda4tothenucleonQCDscalarsusceptibility.Accordingtothelatticedata
−3
analysis a4 is found to be negative, (a4)latt ≃−0.5GeV [3], but much smaller than in
our chiral effective model, (a4)Chiral =−3fπgS/2m4σ ≃−3.5GeV−3, where the nucleon
is seen as a juxtaposition of three constituent quarks getting their mass from the chiral
condensate. The common origin of these two failures can be attributed to the absence
of confinement. In reality the composite nucleon should respond to the nuclear environ-
ment, i.e., by readjusting its confined quark structure. The resulting polarization of the
nucleon is accounted for by the phenomenological introduction of the scalar nucleon re-
sponse,κ ,inthenucleonmassevolution :M (s)=M +g s+ 1κ s2+.....This
NS N N S 2 NS
constitutes the only change in the expression of the energy density but this has numer-
ousconsequences.Inparticularthea4 parameterismodified:a4 =(a4)Chiral 1 − 23C .
The valueofC ≡(f2/2M )κ whichreproducesthe lattice dataisC ≃1.2(cid:0)5implyin(cid:1)g
π N NS
a strong cancellation effect in a4. Moreover the scalar response of the nucleon induces
an new piece in the lagrangian Ls2NN= − κNSs2N¯N/2 which generates a repulsive
three-body force able to restore saturation.
Applications. The restorationofsaturationpropertieshasbeenconfirmedatthe Hartree
level[4]withavalueofthedimensionlessscalarresponseparameter,C,closetothevalue
estimated from the lattice data. The next step has been to include pion loops on top of
the Hartree mean-field calculation [5]. We use a standard many-body (RPA) approach
′
which includes the effect of short-range correlation (g parameters). Non relativistically,
themainingredientisthefullpolarizationpropagatorΠ (whichalsoincludesthe ∆−h
L
excitations) in the longitudinal spin-isospin channel. To be complete we also add the
transversespin-isospinchannel(Π ) andthe associatedrho mesonexchange.The calcu-
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lation of Eloop = E − E0 is done using the well-known charging formula with residual
′ ′
interaction V =π+g , V =ρ+g :
L T
+∞ 1
E idω dq dλ
loop =3V (V (ω,q)Π (ω,q;λ) + 2V (ω,q)Π (ω,q;λ)) .
V Z 2π) Z (2π)3 Z λ L L T T
−∞ 0
Theparametersassociatedwithspin-isospinphysicsarefixedbynuclearphenomenology
andthe calculationhas no realfree parametersapartfor a fine tuning for C (aroundthe
lattice estimate) and for g (around the VDM value). The result of the calculation is
ω
2
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20 40
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V) 10 V) 30
E / A (Me −055 a (MeS20
−10 10
−15
−20 0
0 0.5 1 1.5 2
−25 ρ/ρ
0 1 ρ / ρ 2 3 0
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Fig. 1. Left panel: Bindingenergy of nuclear matter with gω =8, mσ =850 MeV and C =0.985 with
theFockandcorrelationenergiesontopofσandωexchange.Fullline:fullresult.Dottedline:without
Fock and correlation energies. Dot-dashed line: Fock terms. Decreasing dotted line (always negative):
correlationenergy[5].Rightpanel:AsymmetryenergyversusdensityintheRHFapproach[6].
shownonfig.1.Westresstherelativelymodestvalueofthe correlationenergy(−8MeV
and−7MeV fortheLandTchannels),muchsmallerthatwhatisobtainedfromiterated
pionexchange(planardiagramm)inin-mediumchiralperturbationtheory.Thiseffectis
mainly due to the strong sceening of pion exchange from short-range correlations.
WehavealsoperformedafullrelativisticHartree-Fock(RHF)calculationwiththeno-
tableinclusionoftherhomesonexchangewhichisimportanttoalsoreproducetheasym-
metry properties of nuclear matter [6]. Again in an almost parameter free calculation,
saturationpropertiesofnuclearmattercanbe reproducedwithg =10,m =800MeV
S σ
(lattice),g =2.6(VDM),g =6.4(closetotheVDMvalue3g )andC =1.33(closeto
ρ ω ρ
the lattice value). For the tensor coupling we a first take the VDM value κ =3.7. The
ρ
correspondingasymmetryenergy,a ,isshownonthe rightpaneloffig1.Itisimportant
S
to stress that the rho Hartree contribution (7MeV) is not sufficient when keeping the
VDMvalueforthevectorcouplingconstantg .TheFocktermthroughitstensorcontri-
ρ
butionisnecessarytoreachtherangeofacceptedvalueofa around30MeV.Increasing
S
κ (strong rho scenario) to κ =5 allows a particularly good reproduction of both sym-
ρ ρ
metric and asymmetric nuclear matter. The model also predicts a neutron mass larger
than the proton mass with increasing neutron richness in agreement with ab-initio BHF
calculations [7]. This approachgives very encouragingresults and raisesquestion of how
confinement can generate such a large and positive scalar response of the nucleon, κ .
NS
This is presumably linked to a delicate balance of (partial) chiral symmetry restoration
and confinement mechanism inside the nucleon.
References
[1] P.A.M.Guichon,Phys.Lett. B200(1988) 235.
[2] G.Chanfray,M.EricsonandP.A.M.Guichon,Phys.RevC63(2001)055202.
[3] D.B.Leinweber,A.W.ThomasandR.D.Young,Phys.Rev.Lett92(2004)242002.
[4] G.ChanfrayandM.Ericson,EPJA25(2005)151.
[5] G.ChanfrayandM.Ericson,Phys.RevC75(2007) 015206.
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[6] E.MassotandG.Chanfray,Phys.Rev.C78(2008)015204.
[7] F.Sammarruca,W.BarredoandP.Krastev,Phys.Rev.C71(2005)064306.
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