Constraints on nuclear matter parameters of an Effective Chiral Model

0.21 MB·English

by  T. K. Jha

Constraints on nuclear matter parameters of an Effective Chiral Model
T. K. Jha1∗ and H. Mishra1,2,3†
1 Theoretical Physics Division, Physical Research Laboratory, Navrangpura, Ahmedabad, India - 380 009
2 Institut fu¨r Theoretische Physik, J. W. Goethe Universita¨t,
Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany
3 School of Physical Sciences, Jawaharlal Nehru University, New Delhi, India 110067
(Dated: January 2, 2009)
Withinaneffectivenon-linearchiralmodel,weevaluatenuclearmatterparametersexploitingthe
uncertainties in the nuclear saturation properties. The model is sternly constrained with minimal
free parameters, which display the interlink between nuclear incompressibility (K), the nucleon
effectivemass(m⋆),thepiondecayconstant(fπ)andtheσ−mesonmass(mσ). Thebestfitamong
the various parameter set is then extracted and employed to study the resulting Equation of state
(EOS). Further, we also discuss the consequences of imposing constraints on nuclear EOS from
9
Heavy-Ioncollision and other phenomenological model predictions.
0
0
PACSnumbers: 21.65.-f,13.75.Cs,97.60.Jd,21.30.Fe
2

n
a I. INTRODUCTION with the repulsive vector forces provides the saturation
J mechanismfornuclearmatter[2]. Theestimatefromthe

2 Particle Data Group quotes the mass of this scalar me-
The framework of Quantum Hadrodynamics [1, 2] as son‘f0(600)’orσ mesoninthe range(400 1200)MeV
] an elegant and consistent theoretical treatment of finite [13]. A recent est−imate however, for sigma−meson mass
h
t nucleiaswellasinfinitenuclearmatterlaiddownthepil- is found to be 513 32 MeV [14].
- larsofrelativistictheorieswhichseemtoprovidesolution ±
l Phenomenologically, parallel to the well known σ ω
c to the so called “the Coester band” problem [3, 4]. How- −
u model,preferablyknownastheWaleckamodel[1,2,15],
ever,ourpresentknowledgeofnuclearmatterisconfined
n chiralmodels [16, 17,18, 19, 20, 21,22] havebeen devel-
[ around nuclear saturation density (ρ0 ≈3×1014gcm−3) oped and were applied to nuclear matter studies. Chiral
and therefore, in order to have some meaningful correla-
symmetry is a symmetry of strong interactions in the
2 tions while extrapolating to higher densities, the nuclear
v limit of vanishing quark masses and is desirable in any
equation of state (EOS) must satisfy certain minimum
3 relativistic theory. However, because the current quark
criteriaquantifiedasthe“nuclear saturation properties”,
3 masses are small but finite this symmetry can be con-
which are the physical constants of nature. Basically it
2 sidered as an approximate symmetry. This symmetry is
4 is understood that the inherited uncertainty at ρ0 gets spontaneouslybrokeninthegroundstate. Inthecontext
1. more pronounced at higher densities (3 −10 ρ0), rele- of σ models, the σ field (which carries the quantum
vant to astrophysical context such as the modeling of − −
1 numbers of the vacuum) attains a finite vacuum expec-
8 neutron stars. In this context, the two most important tation value σ = σ0 = fπ. Equivalently, the potential
0 quantities which play vital role and are known to have h i
for the σ field attains a minimum at f [23, 24]. The
: substantial impact on the EOS are the nucleon effec- − π
v value of f reflects the strength of the symmetry break-
π
tive mass and the nuclear incompressibility [5, 6]. Ironi-
i ing and experimentally it is found to be f 131 MeV
X cally, these two properties are not very well determined π ≈
[13].
r and they posses large uncertainty. The nuclear incom-
a Time and again, the aforesaid facts and figures em-
pressibility derived from nuclear measurements and as-
phasize the need to address the importance of imposing
trophysical observations exhibit a broad range of values
constraints to the EOS to narrow down the uncertain-
K = (180 800) MeV [7]. Further the non-relativistic
− ties both experimentally and theoretically. Arguably, to
and the relativistic models fails to agree to a commom
address these issues, one needs a model that has the de-
consensus. The non-relativistic calculations predict the
sired attributes of the relativistic framework and which
compression moduli in the range K = (210 240)MeV
− canbesuccessfullyappliedtovariousnuclearforceprob-
[8, 9, 10], whereas, relativistic calculations predicts it in
the range (200 300) MeV [11, 12]. Apart from that lembothinthevicinityofρ0aswellasathigherdensities
− with the same set of parameters. With this motivation,
we are inevitably marred by the uncertainty in the de-
we choose a model [25] which embodies chiral symme-
termination of mass of the scalar meson (σ-meson). The
try and has minimum number of free parameters (total
attractiveforceresultingfromthescalarsectorisrespon-
five)toreproducethesaturationproperties. Thesponta-
sible for the intermediate range attraction which, along
neousbreakingofchiralsymmetryrelatesthemassofthe
hadrons to the vacuum expectation value of the scalar
field and thus naturally restricts the parameters of the
∗email: [email protected] model. Therefore, the present study, apart from test-
†email: [email protected] ing the reliability of the model, puts valuable constraint
2
onthe EOSbasedonthepiondecayconstantandbrings tiontothedynamicallygeneratedmassofthevectorme-
outcorrelationsbetweenbetweenthepiondecayconstant son. Withouthigherorderinscalarfieldinteractions,the
(f ), the σ meson mass (m ), the nuclear incompress- model was first employed to study high density matter
π σ
ibility (K)−and the nucleon effective mass (m⋆). [28]. To bring down the resulting high incompressibil-
In section 2, we briefly describe basic ingredients of ity,non-linearinteractioninthescalarfieldwasincluded
thehadronicmodelandtheenergyandpressureofmany in later work [29] and subsequently applied to study nu-
baryonic system is computed following the mean-field clear matter at finite temperature [30]. The success of
ansatz. Subsequent section (Section 3) describes the the model then motivated us to generalize it to include
methodology to evaluate the model parameters. In Sec- the octet of baryons and to study hyperon rich matter
tion4,weextractthebestfitamongthevariousparame- and properties of neutron star [25, 31]. However, in ear-
ter ofthe modelandapply it to study the resultingEOS lier works, the parameter sets that were employed were
ofsymmetricnuclearmatter. Intheresultanddiscussion not studied and analyzed in detail with respect to the
section, we discuss the consequences of imposing various inherent vacuum properties of chiral symmetry. More-
constraintsonthe modelparametersandfinally,we con- over, rather than a phenomenological fit the parameters
clude with some important findings of this work. must be constrained meaningfully, so that the resulting
EOSismorerealisticandpurposeful. Motivatedbythis,
wepresentlytrytoexploretheconsequencesofimposing
II. THE EFFECTIVE CHIRAL MODEL stringent constraint on the model parameters not only
with properties known at saturation density but also on
Using the chiral sigma model with dynamically gen- the resulting EOS with other phenomenological model
erated mass for vector meson, Glendening studied finite predictionsandexperimentaldataathighdensity. Inad-
temperature aspects of nuclear matter and its applica- ditiontothat,the correlationbetweenvariousquantities
tion to neutron stars [26]. However, there the ρ meson with the vacuumvalue ofthe scalarfield naturallyspells
and its isospin symmetry influence was not con−sidered. out definite interlink between them. We now proceed to
Although a nice framework respecting chiral symmetry, describe the salient features of the present model. The
a drawback was its unacceptable high incompressibility effectiveLagrangianofthemodelinteractingthroughthe
and in the subsequent extension of the model [27], the exchangeofthe pseudo-scalarmesonπ, the scalarmeson
mass of the vector meson is not generated dynamically. σ, the vector meson ω and the iso-vector ρ meson is
−
The model that we consider [25] in our present analy- given by:
sis embodies higher orders of the scalar field in addi-
1
= ψ¯B iγµ∂µ gωγµωµ gρρ~µ ~τγµ gσ σ+iγ5~τ ~π ψB
L (cid:20) − − 2 · − · (cid:21)
(cid:0) (cid:1) (cid:0) (cid:1)
+1 ∂ ~π ∂µ~π+∂ σ∂µσ λ x2 x2 2 λb x2 x2 3 λc x2 x2 4
2 µ · µ − 4 − 0 − 6m2 − 0 − 8m4 − 0
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
1 1 1 1
F F + g 2x2ω ωµ R~ R~µν + m2ρ~ ρ~µ . (1)
−4 µν µν 2 ωB µ − 4 µν · 2 ρ µ·
The first line of the above Lagrangian represents the for the vector bosons through spontaneous breaking of
interaction of the nucleon isospin doublet ψ with the the chiral symmetry with scalar field attaining the vac-
B
aforesaid mesons. In the second line we have the kinetic uumexpectationvaluex0. Thenthe massofthe nucleon
and the non-linear terms in the pseudo-scalar-isovector (m), the scalar (m ) and the vector meson mass (m ),
σ ω
pion field ‘~π’, the scalar field ‘σ’, and higher order terms are related to x0 through
ofthescalarfieldintermsoftheinvariantcombinationof
thetwoi.e.,x2 =~π2+σ2. Finallyinthelastline,wehave
the field strength and the mass term for the vector field m=gσx0, mσ =√2λx0, mω =gωx0 . (2)
‘ω’ and the iso-vector field ‘ρ~’ meson. g ,g and g are
σ ω ρ Toobtaintheequationofstate,wereverttothemean-
the usual meson-nucleon coupling strength of the scalar,
field procedure in which, one assumes the mesonic fields
vectorandtheiso-vectorfieldsrespectively. Hereweshall
tobeuniformi.e.,withoutanyquantumfluctuations. We
be concerned only with the normal non-pion condensed
recall here that this approach has been extensively used
state of matter, so we take <~π >=0 and also m =0.
π to obtain field-theoretical EoS for high density matter
The interaction of the scalar and the pseudoscalar [27], and gets increasingly valid when the source terms
mesonswiththevectorbosongeneratesadynamicalmass are large [2]. The details of the present model and its
3
attributes such as the derivation of the equation of mo- the isovector field (ρ) is respectively given by
tionofthe mesonfieldsandits equationofstate(ε &P)
canbefoundinourprecedingwork[25,31]. Forthesake
ρ
B
of completeness however, we write down the meson field ω0 = g x2, (3)
equations in the mean-field ansatz. The vector field (ω), XB ω
the scalar field (σ) (in terms of Y =x/x0 =m⋆/m) and
2
b c 2c c ρ 2c ρ
(1 Y2) (1 Y2)2+ (1 Y2)3+ σ ω B σ S =0 (4)
− − m2c − m4c2 − m2Y4 − mY
ω ω
Intheabove,‘k ’isthefermimomentaofthebaryonand
F
γ = 4 (symmetric matter) is the spin degeneracy factor.
g
ρ
ρ03 = m2I3 ρB. (5) For symmetric nuclear matter (N = Z), we neglect the
XB ρ contribution from the ρ meson. The nucleon effective
massisthenm⋆ Yma−ndc g2/m2 arec g2/m2
The quantity ρB and ρS are the vector and the scalar arethescalarand≡vectorcoupσli≡ngpσaramσ etersωth≡ateωnterωs
density defined as,
in our calculations.
γ kF
3
ρ = d k, (6)
B (2π)3 Z
o
γ kF m⋆d3k The total energy density ‘ε’ and pressure ‘P’ of sym-
ρ = . (7) metric nuclear matter for a given baryon density is:
S (2π)3 Z √k2+m⋆2
o
γ kF m2(1 Y2)2 b c c ρ2
ε = k2dk k2+m⋆2+ − (1 Y2)3+ (1 Y2)4+ ω B (8)
2π2 Z 8c − 12c c − 16m2c2c − 2Y2
o p σ ω σ ω σ
γ kF k4dk m2(1 Y2)2 b c c ρ2
P = − + (1 Y2)3 (1 Y2)4+ ω B (9)
6π2 Zo √k2+m⋆2 − 8cσ 12cωcσ − − 16m2c2ωcσ − 2Y2
The meson field equations for ω (eqn 3) and σ-meson properties of nuclear matter that a EOS has to satisfy,
(eqn. 4) are solved self-consistently at a fixed baryon they arethe Binding energy per nucleon( 16.3)MeV,
density to obtain the respective field strengths and the the saturation density (ρ0 0.153 fm−3≈),−the nuclear
≈
corresponding energy density and pressure is calculated. incompressibility (167 380)MeV, the nucleon effective
mass (m⋆/m = 0.75 −0.90) and the asymmetry energy
−
coefficient (J = 32 4)MeV, all defined at ρ0, the nu-
±
clear saturation density. However, it can be seen that
III. EVALUATION OF MODEL PARAMETERS
the uncertainty in their values enables us to extract and
study the parameters within the specified range or with
Having calculated the the thermodynamic quantities
thevariationthereof,inordertoanalyzetheireffectona
suchasthe energydensityandthe pressure,ourprimary
particularEOS.Thefiveparametersofthepresentmodel
aim is to evaluate the set of parametersfor the EoSthat
thataretobeevaluatedarethethreemeson-nucleoncou-
satisfies the nuclear matter properties defined at normal
pling constants (C ,C ,C ) and the two higher order
σ ω ρ
nuclear matter density (ρ0) at zero temperature. As dis- scalar field constants (b & c).
cussedearlier,adesirableandvalidEoSmustsatisfythe
The individual contributions to the energy density for
saturation properties of symmetric nuclear matter and
symmetric nuclear matter (eqn. (8)) can be abbreviated
theparametersofthemodelcanbe adjustedtofitthose.
as,
Similar procedure has been adopted in Ref. [32, 33] to
evaluate the parameters of the mean-field models.
What we have in our hand is the set of five saturation ε=ε +ε +ε , (10)
k σ ω
4
where, Consequently, the respective energy contributions can
be expressed in terms of these specified values at the
ε = γ kF k2dk k2+m⋆2 , (11) saturation density. Using eqn. (14) and eqn. (15), they
k 2π2 Z are given as,
o p
m2(1 Y2)2 b 1 1
εσ = − (1 Y2)3 εσ = ρ0(m a1) (2εk+m⋆ρs) (16)
8c − 12c c − 2 (cid:20) − − 3 (cid:21)
σ ω σ
c
2 4
+ (1 Y ) , (12)
16m2c2c − and
ω σ
and
1 1
c ρ2 εω = 2(cid:20)ρ0(m−a1)− 3(4εk−m⋆ρs)(cid:21), (17)
ε = ω B, (13)
ω 2Y2
where ρ is the scalar density defined in eqn. (7), an-
s
where ρ = ρ +ρ is the total baryon density which
B n p alytically which is given by,
is the sum of the neutron density ‘ρ ’ and the proton
n
density ‘ρ ’. The relative neutron excess is then given
p
bthyeδnu=cl(eρanr −maρtpt)e/rρsBa.tuArattitohnedsetannsidtyaradnsdtaδte=ρB0.=Coρn0-, ρ = 1 m⋆ k E ln kF +EF m⋆2 . (18)
sequently, the standard state is then specified by the ar- s π2 (cid:20) F F − (cid:16) m⋆ (cid:17) (cid:21)
In the above equations, m⋆ = Ym is the effective nu-
gument (ρ0,0), and the energy per particle is e(ρ0,0) = cleon mass and E = k2 +m⋆2 is the effective energy
ε/ρ0-m=a1=-16.3MeVforsymmetricnuclearmatter. F F
ThenuclearmatterEOSderivedearliercanbeexpressed of the nucleon caryingpmomenta kF.
Fromeqn. (13),the vectorcoupling (C ) canbe read-
in terms of the nuclear energy density ε as, ω
ily evaluated using the relation
ε=εk+εσ+εω =ρ0(m−a1). (14) 2Y2
C = ε , (19)
From the the equilibrium condition P(ρ0,0) = 0, we ω ρ20 ω
have,
with ε given by eqn. (17), for a specified value of Y =
ω
∂ε m⋆/m defined at ρ0.
P = ε + ρ
B Similarly, using the equation of motion for the scalar
− ∂ρ
B
field (eqn. (4)), the scalar coupling can be calculated
1 1
= ε m⋆ρ ε + ε = 0. (15) using the relation
k S σ ω
3 − 3 −
mY b c 2c c ρ2
C = (1 Y2) (1 Y2)2+ (1 Y2)3+ σ ω B . (20)
σ 2ρ (cid:20) − − m2c − m4c2 − m2Y4 (cid:21)
S ω ω
In the above expression, the higher order scalar field
couplings constants ‘b’ and ‘c’ are unknown, but they
canbesolvedsimultaneouslytoobtaintherespectivepa-
2c c2ρ2 c(1 Y2) m2c
rameters. To compute the constants of the higher order b = σ ω B + − + ω
Y4(1 Y2)2 m2c (1 Y2)
scalar field, we use the equation of motion of the scalar ω
− −
field (eqn. (4)) and eqn. (12). From eqn. (12), we get 2cσcωmρS
. (22)
− Y(1 Y2)2
−
c(1 Y2) 16ε c c 2m2c 4b
σ ω σ ω
− = + . (21)
m2c (1 Y2)3 − (1 Y2) 3 Substituting eqn. (21) in eqn. (22) leads us to the ex-
ω
− −
pressiontocalculatethehigherorderscalarfieldconstant
Fromthe equationofthe motionofscalarfield, we get ‘b’, which is,
5
IV. RESULTS AND DISCUSSIONS
6c c mρ 6c c2ρ2 48ε c c
b = Y(1σ ωY2S)2 + Y4(1σ ωYB2)2 − (1 σYσ2)ω3 The spontaneous breaking of chiral symmetry lends
− − − mass to the Hadrons and relates them to the vacuum
3m3c
+ ω . (23) expectation value (VEV) of the scalar field (x0), which
(1 Y2) is what is shown in eqn. (2). Immediately, what fol-
−
lows from the third term in eqn. (2) is that, the VEV of
Similarly, the higher order constant ‘c’ in the scalar
the scalar field which has a minimum potential at f is
field can be computed from the relation, π
related to the vector coupling constant C through the
ω
relation x0 = fπ = mω/gω = 1/√Cω. Thus the vec-
8c c2m3ρ 8c c3m2ρ2 48ε c c2 tor coupling constant is explicitly constrained from the
c = σ ω s σ ω B σ σ ω
Y(1 Y2)3 − Y4(1 Y2)3 − (1 Y2)4 vacuum value of the pion decay constant. Similarly the
2c2−m4 − − scalar meson mass can be given by mσ =m √Cω/√Cσ.
+ ω . (24) The model is then sternly constrained and exude the re-
(1 Y2)2
− lationship between various quantities with the VEV of
The calculation of C is straight forward, but eqn. the scalar field.
ω
(20),(23)and(24)canbesolvedsimultaneouslynumeri- Inthepresentcalculation,wetakethevalueofthesat-
callyforagiveninitialvaluesofC ,bandc,thesolution uration density to be ρ0 = 0.153fm−3 [5], which agrees
σ
ofwhichwouldthus returnthe setofvaluesforadesired with the observed charge and mass distribution of finite
value of Y at ρ0. nuclei. Saturationdensityimpliesthatthepressureofthe
Finally, for studying asymmetric matter, we need to system is zero and the system will remain in this state
incorporatetheeffectofiso-vectorρ mesonandthecou- if left undisturbed. The binding energy per nucleon is
pling for the ρ meson has to be ob−tained by fixing the fixed at an empirical value B/A m = 16.3 MeV [5].
asymmetryene−rgycoefficientJ 32 4MeV [34]atρ0. Withthe uncertaintyinthe nucle−oneffec−tive massatρ0,
Accordingly, the ρ meson cou≈pling±constant (C ) can we calculate the parameters of the present model in the
ρ
be fixed using the r−elation, range Y =m⋆/m=(0.75 0.90). In order to assure the
−
existenceofalowerboundfortheenergy,wedemandthat
thecoefficient‘c’inthequarticscalarfieldterm,remains
c k3 k2
J = ρ F + F , (25) positive. The corresponding related quantities, such as
12π2 6 (k2 +m⋆2) the pion decay constant, the scalar meson mass and the
F
p nuclear incompressibility are also calculated. The asym-
whTerheuscρth≡emgρ2o/dmel2ρpaanrdamkFete=rs(a6rπe2eρvBa/luγa)1t/e3d.solvingequa- mobettariyneednepragryamcoeteeffirsciaerneteinslisfitxeeddinatTaJbl≈eI,3w2hMereeVt.heTrhee-
tions (19), (20), (23), (24) and (25) self-consistently, for lationship between the vector coupling constant and the
the specified or desired values of the properties of sym- piondecayconstantcanbeeasilyvisualized. Strongerthe
metric nuclear matter at saturation point. Further it is vector coupling (repulsion), lower is the value of the chi-
also required that the EOS so obtained has a reasonable ral condensate and vice-versa. From the tabulated data,
nuclear incompressibility which is defined as the curva- wefindthatthecalculatedsigmamesonmassispredicted
ture of the energy curve at the saturation point and is within (340 700) MeV for m⋆/m = (0.75 0.90). Al-
given as, though the v−alues obtained from the analysi−s of neutron
scattering off lead nuclei [5, 35] is consistent with the
∂2(ε/ρ ) range m⋆/m = (0.80 0.90), a lower nucleon effective
2 B −
K =9 ρ0 . (26) mass is is known to reproduce the finite nuclei proper-
∂ρB (cid:12)0 ties,suchasthe spin-orbiteffects splitting correctly[36].
(cid:12)
Incompressibility is a poorly know(cid:12)n quantity experi- Also, we find that as we move to higher effective mass
mentally, for the fact that some sort of theoretical mod- region, the incompressibility of the matter starts to fall
eling comes in these calculations. Apart from that, the or the EOSgets softer. However,within the incompress-
other quantity with large uncertainty is the nucleon ef- ibility range of K = (200 - 300) MeV, the present model
fective mass. The wide range of values determined from predicts higher nucleon effective mass.
experiments of these two quantities motivates us to an- Nuclear matter saturation is a consequence of the in-
alyze and study the EOS with these variations. Further terplay between the attractive (scalar) and the repulsive
wealsoneedtolookintoissuesrelatedtosomeindispens- (vector) forces and hence the variation in the coupling
able elements, such as the σ meson mass and the pion strength effects other related properties as well. Fig.
−
decay constant ‘f ’, while we want to achieve a proper 1(A)reflectsthesame,wherewehaveplottedthenuclear
π
frameworkforstudyingnuclearmatteraspects. Werecall incompressibility for the evaluated parameter sets of the
that in the present work,our aim is to describe and cor- presentmodelasafunctionofthenucleoneffectivemass.
relate these physical quantities in a coherent and unified For better correlation between them, the corresponding
approach. ratio of the scalar and vector coupling is also indicated.
6
TABLE I: Parameter sets of the effective chiral model that satisfies the nuclear matter saturation properties such as binding
energy per nucleon B/A−m = −16.3 MeV, nucleon effective mass Y = m⋆/m = (0.75−0.90) and the asymmetry energy
coefficientisJ ≈32MeVatsaturationdensityρ0=0.153fm−3. Thenucleon,thevectormesonandtheisovectorvectormeson
masses are taken to be 939 MeV, 783 MeV and 770 MeV respectively and cσ =(gσ/mσ)2, cω =(gω/mω)2 and cρ =(gρ/mρ)2
are the corresponding coupling constants. B =b/m2 and C = c/m4 are the higher order constants in the scalar field. Other
derived quantities such as the scalar meson mass ‘mσ’, the pion decay constant ‘fπ’ and the nuclear matter incompressibility
(K) at ρ0 are also given.
set cσ cω cρ B C mσ Y fπ K
(fm2) (fm2) (fm2) (fm2) (fm4) (MeV) (MeV) (MeV)
1 5.916 3.207 5.060 1.411 1.328 691.379 0.75 110.185 1098
2 6.047 3.126 5.087 0.822 0.022 675.166 0.76 111.601 916
3 6.086 3.031 5.107 0.485 0.174 662.642 0.77 113.346 809
4 6.005 2.933 5.131 0.582 2.650 656.183 0.78 115.238 737
5 6.172 2.825 5.155 -0.261 0.606 635.287 0.79 117.403 638
6 6.223 2.709 5.178 -0.711 0.748 619.585 0.80 119.890 560
7 6.325 2.585 5.200 -1.381 0.089 600.270 0.81 122.740 491
8 6.405 2.451 5.222 -1.990 0.030 580.876 0.82 126.039 440
9 6.474 2.323 5.242 -2.533 0.300 562.500 0.83 129.465 391
10 6.598 2.159 5.265 -3.340 0.445 536.838 0.84 134.378 344
11 6.772 1.995 5.285 -4.274 0.292 509.644 0.85 139.710 303
12 7.022 1.823 5.305 -5.414 0.039 478.498 0.86 146.131 265
13 7.325 1.642 5.324 -6.586 0.571 444.614 0.87 153.984 231
14 7.865 1.451 5.343 -8.315 0.502 403.303 0.88 163.824 199
15 8.792 1.249 5.362 -10.766 0.354 353.960 0.89 176.552 168
16 7.942 1.041 5.388 -6.908 15.197 339.910 0.90 193.437 163
Oncomparisonwiththeincompressibilityboundinferred can be seen that both these physical quantities are in-
from heavy ion collision experiment (HIC)[37], we find versely proportional to each other. A lower value of f
π
that the EOS with lower nucleon effective mass is ruled leads to a higher value of m . However, the experimen-
σ
out. ThepresentmodelfavorsEOSforwhichthenucleon talboundofthepiondecayconstantseemstoagreewith
effectivemassm⋆/m>0.82,i.e.,themassofthenucleon slightly higher value of m , which agree with the upper
σ
in the nuclear medium drops to less than 20% of its bound of the experimental bound on m [14]. Precisely,
σ
≈
mass at ρ0. Equivalently, the agreement with the exper- theconstraintoffπ seemstoagreewithmσ (560 22)
≈ ±
imental flow data in the density range 2 < ρ/ρ0 < 4.6 MeV. Figure 2(B) shows the nucleon effective mass Y =
seem to favor repulsion (higher effective mass) in mat- m⋆/masafunctionofthepiondecayconstant. Thecon-
ter at high density. From the plot, it can be seen that straintoff agreewithEOSwithY =(0.82 0.84)(Set
π
−
the EOS becomes much softer with increasing ratio of 8, 9 & 10; Table I). However the corresponding incom-
C /C . pressibilityfallsintherangeK (344 440)MeV,which
σ ω
≈ −
is on the higher side of presently acceptable bounds ([8]
Figure 1(B) shows the variation of incompressibility
- [12]). With combined constraints such as those on nu-
as a function of scalar meson mass obtained for vari-
clear incompressibility inferred from HIC data [37], the
ous parameter sets. Recent experimental estimate for
limits on sigma meson mass [14], the pion decay con-
scalar meson mass m =513 32 MeV [14] is compared
σ ± stant[13]andthenucleoneffectivemass[35],thebestfit
with the present calculation. From the figure, we find
from the parameters can be extracted. Thus we choose
that the EOS with Y = (0.84 0.86) (Set 10, 11 & 12;
− parameter set 9, 11 and 13 of Table I to study the cor-
Table I) seems to agree with the combined constraint
responding EOS and compare it with the experimental
fromtheHICflowdataandtheexperimentalmesonmass
data[37]aswellasothersuccessfulrelativisticmeanfield
range. Further, it is worth noticing that, lowerthe value
models such as NL3 parameterization [11] and the non-
of m lower is the value of incompressibility for matter
σ relativistic DBHF [38] calculations.
and vice-versa. The heavy ion collision estimate seems
to agree with σ meson mass within (350 550) MeV.
− −
We know that the nuclear incompressibility and the σ-
meson mass both are poorly determined quantities and V. EQUATION OF STATE AT T =0
therefore, some sort of correlation between the two will
help to minimize the uncertainties around them.
The selected parameters for further study is high-
Figure 2(A) shows the obtained sigma mass as a func- lightedin bold fonts in Table I.The resulting energyper
tion of the vacuum value of the pion decay constant. It nucleon for symmetric nuclear matter is calculated for
7
1200
1.845 2.297 3.394 7.630
700
C σ /Cω (A) (A)
900
600
)
) V
MeV 600 Me 500 m σ = 513 + 32
(
K ( m σ
f π =
400
300 (130 + 5)
300
0
0.75 0.8 0.85 0.9 100 120 140 160 180 200
Y = m*/m f π (MeV)
1200
(B)
0.9
900
m =
σ
(513 + 32)
) m0.85
MeV 600 m*/
( =
K Y 0.8
f π =
(130 + 5)
300
HIC Estimate 0.75
(B)
0300 400 500 600 700 100 120 140 160 180 200
m σ (MeV) f π (MeV)
FIG. 1: (Color online)(A)- Nuclear matter incompressibility FIG. 2: (Color online)(A) - The scalar meson mass ‘mσ’ as
asafunctionofthenucleoneffectivemassfortheparameters a function of the vacuum value of the scalar field ‘fπ’. We
of thepresent model. Also plotted is thecorresponding ratio takefπ =130±5MeV[13],which isshown withredvertical
of scalar to vector coupling on the opposite x-axis. (B)- In- lines. The blue lines denotes the range of the mass of mσ
compressibility as a function of obtained sigma meson mass from experiment [14]. (B) - Ratio of nucleon effective mass
for the parameter sets enlisted in Table I. The upper and to baremass is plotted as a function of fπ.
the lower limit for incompressibility inferred from Heavy Ion
Collision data[37]K =(167−380) MeV is shown with hori-
zontalredlines. Recentexperimentalscalarmesonmasslimit
(mσ =513±32) MeV[14]isdepictedwithblueverticallines NloLw3erpdareanmsiteyter(ρs0et=, nu0c.1le4a8rfmmat−t3e)r stahtaunrawtehsatatwselighhatvlye
taken in present calculation, the Binding energy per nu-
cleon almost remains same ( 16.3 MeV). In case of
≈ −
theseparametersandisplottedinFig. 3(A).Forcompar- DBHF, nuclear matter saturates at still higher density.
ison, we plot the same with NL3 parameterization from It is worth noticing that the incompressibility of the pa-
RelativisticMeanField(RMF)calculations[11]andalso rameter chosen in the present model spans within (230
thenon-relativisticrealisticDBHF(Bonn-A)parameter- - 390) MeV, yet the resulting EOS seems to be soft at
ization[38]. Inthe inset,the regionofsaturationdensity higher densities in comparison to that predicted by the
is magnified, where we find nice agreement within rela- NL3 parameter set which has K = 271.6 MeV. Incom-
tivistic mean-field models near ρ0. Although in case of pressibilityofnuclearmatteristhemeasureofthedegree
8
of softness/ stiffness of the EOS. Conventionally, EOS 180 -12
with K < 300 MeV are considered to be soft. But in the
(A)
present case the EOS predicted by the effective model 150 -14
is relatively much softer than the NL3 parameterization
3
although the value for the former is high enough. How- ) 120 -16 0.8 5
evNviLcei3rntistheyiesmocfsasntaotbuceroaumtnipdoanerresdtewonoesldilt,ywifi(twIhnestelhoteopkEloOattS)t.hwTeithEheOKcSu=rinve2th3o1ef m (MeV 900.1 0.15 0.2 Y = Y = 0.8 = 0.87
MeV below saturation density, but the energy predicted − Β 60 Y
is much larger at higher densities. In contrast to that, ρ
the EOS predicted by the effective model gets softer at Ε/
30 DBHF
higher densities. It should be interesting to study the
NL3
consequences of such behavior in the astrophysical con- K = 391
0
text, especially on the globalproperties and structure of K = 303
neutronstarsatphysicallyinterestingdensities(2-5ρ0) K = 231
-30
[39, 40]. 0 0.2 0.4 0.6 0.8
ρ (fm-3)
In Fig. 3(B), the nucleon effective mass in the nu- Β
clear medium is plotted as a function of baryon density
up to 6ρ0. This medium mass modification of nucleon
in nuclear medium is a consequence of the Dirac field
and forms an essential element for the success of the rel- 950
ativistic phenomenology. From the plot, it is interesting
toseethatthenucleonexperiencesrepulsiveforcesinnu- Y = 0.83, K = 391 MeV
900
clear matter at higher densities (ρB > 2ρ0), as a result Y = 0.85, K = 303 MeV
of which the nucleon effective mass increases again for Y = 0.87, K = 231 MeV
the three cases that we study presently. A careful look 850
into Table I reveals the relationship between the cou- V)
plings(bothscalarandvector)andthe resultingnucleon e
M800
effective mass. The model predicts a higher nucleon ef- (
*
fective mass if the ratio of scalar to vector coupling is m
larger but the increase in m⋆ is slower thereafter, which 750
reflects the dominance of attractive force at high densi-
ties. At saturation density, the present model results in
700
much higher nucleon effective mass in comparisonto the (B)
NL3 (m⋆/m=0.60) and DBHF (m⋆/m=0.678).
650
Fig. 4(A)displaysthepressureasafunctionofbaryon 0 1 2 3 4 5 6
ρ /ρ
density up to nearly 6ρ0 for the selected parameters of B 0
the modelfor symmetric nuclearmatter. The shadedre-
gion corresponds to the experimental HIC data [37] for
symmetric nuclear matter (SNM). Among the three the- FIG. 3: (Color online) (A) - Binding energy per nucleon of
oretical calculations shown, the EOS with Y = 0.85 & symmetricnuclearmatterplottedasafunctionofbaryonden-
0.87 agree very well with the collision data. Precisely, sity up to nearly 5ρ0. For comparison, we also plot thesame
the third set (K = 231 MeV) completely agree with the for NL3 parameter set from the Relativistic Mean Field the-
flow data in the entire density span of 2 < ρB/ρ0 < 4.6. ory[11]aswellasEOSfromDBHF[38]inthenon-relativistic
domain. The inset plot displays the curve in the vicinity of
We now proceed to calculate the EOS of Pure Neutron
nuclear saturation (B) - Variation of Nucleon effective mass
matter (PNM) by taking the spin degeneracy γ = 2 in
in medium as a function of total baryon density for symmet-
eqn. (8) and (9). The inclusion of ρ meson doesn’t
− ric nuclear matter of the selected parameters of the present
seemto affect the EOSsubstantially andsowe refrained
model.
from that. In Fig. 4(B), the case of pure neutron mat-
ter (PNM)is comparedwiththe experimentalflowdata.
Theexperimentalflowdataiscategorizedintermsofstiff
VI. SUMMARY AND CONCLUSIONS
or soft based on whether the density dependence of the
symmetry energy term is strong or week [41]. The EOS
predictedbythepresentmodelseemstoratherlieonthe The effective chiral model provides a natural frame-
softerregime. However,theEOSwithY =0.87,K =231 worktointerlinkthestandardstatepropertiesofnuclear
MeVthoughsatisfythe combinedconstraintratherwell, matter with the inherent fundamental constants such as
isnotconsistentwiththevacuumvalueofthepiondecay the piondecayconstantandtheσ mesonmasswithin a
−
constant. unified approach. With this motivation, the parameters
9
1000 model is that the mass of the vector meson (m ) is gen-
ω
SNM
erated dynamically, as result of which the effective mass
(A)
ofthenucleonacquiresadensitydependenceonboththe
scalarandvectorfields. Theinterplaybetweenthisscalar
and vector forces results in the increase of m⋆ at 3ρ0
100 ≈
) (Fig. 3(B)),asaconsequenceofwhichtheresultingEOS
3
-m is much softer at higher densities.
f
V We also discussed the implication of imposing fun-
e
M damental constraint on the evaluated model parame-
(
P 10 ters. Among various derived quantities, we find that
the pion decay constant is experimentally well known
Y = 0.83; K = 391 MeV quantity in comparison to the nuclear incompressibility
Y = 0.85; K = 303 MeV and σ meson mass, that can put stringent constraint
Y = 0.87; K = 231 MeV on the−model parameters. Employing this constraint
1 (f = 130 5 MeV) rather leave us with few options
π
0 0.2 0.4 0.6 0.8 1 among the w±ide range of parameters enlisted in Table I,
ρΒ(fm-3) while the observed range of σ meson mass do not rule
−
out any. Experimentally determined effective mass from
scatteringofneutronoverPbnuclei[35]seemstogowell
with the presentmodel. Bothofthem favorhighervalue
1000 for nucleon effective mass. In the present calculation we
PNM findthatahighernucleoneffectivemassisendowedwith
(B) reasonable incompressibility too. However, the parame-
terthatagreewell(m⋆ =0.83m;Set9)withthelimitsof
f has incompressibility in the upper bound of the value
π
100 inferred from the flow data. On a comparative analysis
)
-3m of the resulting EOS with that of the HIC data for sym-
f metric nuclear matter as well as pure neutron matter,
V
e parameter set with Y =0.85;K 300 MeV seems to be
M ≈
the idealparameterizationof the presentmodel. The re-
(
P 10 sulting scalarmesonmass mσ 510MeV, is alsoconsis-
≈
tentwiththeexperimentallyobservedmasses[14,42,43].
Stiff EoS (E) Further, a higher value of incompressibility K 300
≈
Soft EoS (E) MeV is known to predict correctly the isoscalar giant
resonance energies in medium and heavy nuclei in the
1 relativistic framework [44]. On account of the aforesaid
0 1 2 3 4 5 6
ρ /ρ argumentsandconstraints,themodelseemstoworkvery
n 0
well within present approach. However, the predictabil-
ity of the model needs to be tested at finite temperature
and high densities. Work is in progress in this direction
FIG.4: (Coloronline)ComparisonoftheHeavyIonCollision
[45]. In this regard, it will also be interesting to study
estimate [37] with the theoretical prediction of the effective
the medium effects on the underlying couplings [46] as
model,(A)forSymmetricnuclearmatter(SNM)caseand(B)
well as the on the meson masses [47] and the pion decay
for Pure Neutron matter (PNM) case.
constant [48].
of the model are evaluated in the mean-field ansatz by
fixing the nuclear matter saturation properties defined
VII. AKNOWLEDGMENT
atρ0 andvaryingthe nucleoneffective massinthe range
Y = m⋆/m = (0.75 0.90). Thus the resulting equa-
−
tion of state not only satisfies the saturation properties OneoftheauthorsHMwouldliketothankInstitutfor
reasonablywellbutalsorelatesthevariousaforesaidfun- Theoretische Physik, University of Frankfurt for warm
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