Hadrons in Dense Resonance-Matter: A Chiral SU(3) Approach

28 Pages·0.23 MB·English

by  D. Zschiesche

Hadrons in Dense Resonance-Matter: A Chiral SU(3) Approach
D. Zschiesche1, P. Papazoglou1, S. Schramm1, J. Schaffner-Bielich2, H. St¨ocker1 and
W. Greiner1
1Institut fu¨r Theoretische Physik, Postfach 11 19 32, D-60054 Frankfurt am Main, Germany
2Riken BNL Research Center, Brookhaven National Lab, Upton, New York 11973
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Abstract

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A nonlinear chiral SU(3) approach includingthespin 3 decuplet is developed
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to describedense matter. Thecoupling constants of the baryon resonances to

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thescalarmesonsaredeterminedfromthedecupletvacuummassesandSU(3)
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0 symmetry relations. Different methods of mass generation show significant
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0 differencesinthepropertiesofthespin-3 particles andinthenuclearequation
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of state.
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I. INTRODUCTION
The investigation of the equation of state of strongly interacting matter is one of the
most challenging problems in nuclear and heavy ion physics. Dense nuclear matter exists
in the interior of neutron stars, and its behaviour plays a crucial role for the structure
and properties of these stellar objects. The behaviour of hadronic matter at high densities
and temperatures strongly influences the observables in relativistic heavy ion collisions (e.g.
flow, particle production,...). The latter depend on the bulk and nonequilibrium properties
of the produced matter (e.g. pressure, density, temperature, viscosity,...) and the properties
of the constituents (effective masses, decay widths, dispersion relations,...). So far it is
not possible to determine the equation of state of hadronic matter at high densities (and
temperatures) from first principles. QCD is not solvable in the regime of low momentum
transfersandfinitebaryondensities. Thereforeonehastopursuealternativewaystodescribe
the hadrons in dense matter. Effective models, where only the relevant degrees of freedom
for the problem are considered are solvable and can contain the essential characteristics of
the full theory. For the case of strongly interacting matter this means that one considers
hadrons rather than quarks and gluons as the relevant degrees of freedom. Several such
models like the RMF model(QHD) and its extensions (QHD II, nonlinear Walecka model)
successfully describe nuclear matter and finite nuclei [1–5]. Although these models are
effective relativistic quantum field theories of baryons and mesons, they do not consider
essential features of QCD,namely broken scale invariance andapproximate chiral symmetry.
Including SU(2) chiral symmetry in these models by adding repulsive vector mesons to
the SU(2)-linear σ-model does neither lead to a reasonable description of nuclear matter
ground state properties nor of finite nuclei [6]. Either one must use a nonlinear realization
of chiral symmetry [7,8] or include a dilaton field and a logarithmic potential motivated
by broken scale invariance [9,10] in order to obtain a satisfactory description of nuclear
matter. Extending these approaches to the strangeness sector leads to a number of new,
undetermined coupling constants due to the additional strange hadrons. Both to overcome
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this problem and to put restrictions on the coupling constants in the non-strange sector
the inclusion of SU(3) [11] and chiral SU(3) [12,13] has been investigated in the last years.
Recently [13] it was shown that an extended SU(3) SU(3) chiral σ ω model can describe
× −
nuclear matter ground state properties, vacuum properties and finite nuclei simultaneously.
This model includes the lowest lying SU(3) multiplets of the baryons (octet), the spin-0
and the spin-1 mesons (nonets) as physical degrees of freedom. The present paper will
discuss the predictions of this model for high density nuclear matter, including the spin
3 baryon resonances (decuplet). This is necessary, because the increasing nucleonic fermi
2
levels make the production of resonances energetically favorable at high densities. The
paper is structured as follows: Section II summarizes the nonlinear chiral SU(3) SU(3)-
×
model. Section III gives the baryon meson interaction, with main focus on the baryon
meson-decuplet interaction and the constraints on the additional coupling constants. In
section IV the resulting equations of motions and thermodynamic observables in the mean
field approximation are discussed. Section V contains the results for dense hadronic matter,
followed by the conclusions.
II. LAGRANGIAN OF THE NONLINEAR CHIRAL SU(3) MODEL
We use a relativistic field theoretical model of baryons and mesons based on chiral sym-
metry and scale invariance to describe strongly interacting nuclear matter. In earlier work
the Lagrangian including the baryon octet, the spin-0 and spin-1 mesons has been developed
[13]. Here the additional inclusion of the spin-3 baryon decuplet for infinite nuclear matter
2
will be discussed. The general form of the Lagrangian then looks as follows:
= + + + + + . (1)
kin BW VP vec 0 SB
L L L L L L L
W=X,Y,V, ,u
X A
is the kinetic energy term, includes the interaction terms of the different baryons
kin BW
L L
with the various spin-0 and spin-1 mesons. contains the interaction terms of vector
VP
L
mesons with pseudoscalar mesons. generates the masses of the spin-1 mesons through
vec
L
3
interactions with spin-0 mesons, and gives the meson-meson interaction terms which
0
L
induce the spontaneous breaking of chiral symmetry. It also includes the scale breaking
logarithmic potential. Finally, introduces an explicit symmetry breaking of the U(1)
SB A
L
symmetry, theSU(3) symmetry, andthechiralsymmetry. These termshave beendiscussed
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in detail in [13] and this shall not be repeated here. We will concentrate on the new terms
in , which are due to adding the baryon resonances.
BW
L
III. BARYON MESON INTERACTION
consists of the interaction terms of the included baryons (octet and decuplet) and
BW
L
the mesons (spin-0 and spin-1). For the spin-1 baryons the SU(3) structure of the couplings
2
to all mesons are the same, except for the difference in Lorentz space. For a general meson
field W they read
1
= √2gW α [B BW] +(1 α )[B BW] gW Tr(B B)TrW , (2)
LOW − O8 OW O F − OW O D − O1√3 O
(cid:16) (cid:17)
with [B BW] := Tr(B WB B BW) and [B BW] := Tr(B WB + B BW)
F D
O O − O O O O −
2Tr(B B)TrW. The different terms to be considered are those for the interaction of spin-1
3 O 2
baryons (B), with scalar mesons (W = X, = 1), with vector mesons (W = V , = γ ),
µ µ
O O
with axial vector mesons (W = , = γ γ ) and with pseudoscalar mesons (W = u , =
µ µ 5 µ
A O O
γ γ ), respectively. For the spin-3 baryons (Dµ) one can construct a coupling term similar
µ 5 2
to (2)
= √2gW [Dµ D W] gW [Dµ D ]TrW , (3)
LDW − D8 O µ − D1 O µ
where [Dµ D W]and[Dµ D ]areobtainedfromcoupling [1¯0] [10] [8] = [1]+[8]+[27]+
µ µ
O O × ×
[64] and [1¯0] [10] [1] to an SU(3) singlet, respectively. In the following we focus on the
× ×
couplings of the baryons to thescalar mesons which dynamically generate the hadron masses
and vector mesons which effectively describe the short-range repulsion. For the pseudoscalar
mesons only a pseudovector coupling is possible, since in the nonlinear realization of chiral
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symmetry [13] they only appear in derivative terms. Pseudoscalar and axial mesons have a
vanishing expectation value at the mean field level, so that their coupling terms will not be
discussed in detail here.
Scalar Mesons
The baryons and the scalar mesons transform equally in the left and right subspace.
Therefore, in contrast to the linear realization of chiral symmetry, an f-type coupling is
allowed for the baryon-octet-meson interaction. In addition, it is possible to construct mass
terms for baryons and to couple them to chiral singlets. Since the current quark masses in
QCDaresmallcomparedtothehadronmasses, wewillusebaryonicmasstermsonlyassmall
corrections to the dynamically generated masses. Furthermore a coupling of the baryons to
the dilaton field χ is also possible, but this will be discussed in a later publication. After
insertion of the vacuum matrix X , (Eq.A4), one obtains the baryon masses as generated
h i
by the vacuum expectation value (VEV) of the two meson fields:
1
m = m gS (4α 1)(√2ζ σ) (4)
N 0 − 3 O8 OS − −
2
m = m gS (α 1)(√2ζ σ)
Λ 0 − 3 O8 OS − −
2
m = m + gS (α 1)(√2ζ σ)
Σ 0 3 O8 OS − −
1
m = m + gS (2α +1)(√2ζ σ)
Ξ 0 3 O8 OS −
with m = gS (√2σ + ζ)/√3. The parameters gS , gS and α can be used to fit the
0 O1 O1 O8 OS
baryon-octet masses to their experimental values. Besides the current quark mass terms
discussed in [13], no additional explicit symmetry breaking term is needed. Note that the
nucleon mass depends on the strange condensate ζ! For ζ = σ/√2 (i.e. f = f ), the
π K
masses are degenerate, and the vacuum is SU(3) -invariant. For the spin-3 baryons the
V 2
procedure is similar. If the vacuum matrix for the scalar condensates is inserted one obtains
the dynamically generated vacuum masses of the baryon decuplet
m = gS (3 α )σ +α √2ζ (5)
∆ D − DS DS
h i
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mΣ∗ = gDS 2σ +√2ζ
h i
mΞ∗ = gDS (1+αDS)σ +(2−αDS)√2ζ
h i
m = gS 2α σ +(3 α )√2ζ
Ω D DS − DS
h i
The new parameters are connected to the parameters in (3) by gW = √120(1 α )gS
D8 − − DS D
and gW = √90gS. gS and α can now be fixed to reproduce the masses of the baryon
D1 D D DS
decuplet. As in the case of the nucleon, the coupling of the ∆ to the strange condensate is
nonzero.
It is desirable to have an alternative way of baryon mass generation, where the nucleon
and the ∆ mass depend only on σ. For the nucleon this can be accomplished for example
by taking the limit α = 1 and gS = √6gS . Then, the coupling constants between the
OS O1 O8
baryon octet and the two scalar condensates are related to the additive quark model. This
leaves only one coupling constant to adjust for the correct nucleon mass. For a fine-tuning
of the remaining masses, it is necessary to introduce an explicit symmetry breaking term,
that breaks the SU(3)-symmetry along the hypercharge direction. A possible term already
discussed in [12,14], which respects the Gell-Mann-Okubo mass relation, is
= m Tr(BB BBS) m Tr(BSB), (6)
∆m 1 2
L − − −
whereSa = 1[√3(λ )a δa]. Asinthefirstcase, onlythreecouplingconstants, g 3gS ,
b −3 8 b− b Nσ ≡ O8
m and m , are sufficient to reproduce the experimentally known baryon masses. Explicitly,
1 2
the baryon masses have the values
m = g σ (7)
N Nσ
−
1 2
m = g σ g √2ζ +m +m
Ξ Nσ Nσ 1 2
−3 − 3
2 1 m +2m
m = g σ g √2ζ + 1 2
Λ Nσ Nσ
−3 − 3 3
2 1
m = g σ g √2ζ +m ,
Σ Nσ Nσ 1
−3 − 3
For the baryon decuplet the choice α = 0 yields coupling constants related to the additive
DS
quark model. We introduce an explicit symmetry breaking proportional to the number of
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strange quarks for a given baryon species. Here we need only one additional parameter m
Ds
to obtain the masses of the baryon decuplet:
m = g [3σ] (8)
∆ ∆σ
mΣ∗ = g∆σ 2σ +√2ζ +mDs
h i
mΞ∗ = g∆σ 1σ +2√2ζ +2mDs
h i
m = g 0σ +3√2ζ +3m
Ω ∆σ Ds
h i
For both versions of the baryon-meson interaction the parameters are fixed to yield the
baryon masses of the octet and the decuplet. The corresponding parameter set C , has been
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discussed in detail in [13].
Vector mesons
For the spin-1 baryons two independent interaction terms with spin-1 mesons can be
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constructed, in analogy to the interaction of the baryon octet with the scalar mesons. They
correspond to the antisymmetric (f-type) and symmetric (d-type) couplings, respectively.
Fromtheuniversalityprinciple[15]andthevectormesondominancemodelonemayconclude
that the d-type coupling should be small. Here α = 1, i.e. pure f-type coupling, is used.
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It was shown in [13], that a small admixture of d-type coupling allows for some fine-tuning
of the single-particle energy levels of nucleons in nuclei. As in the case of scalar mesons,
for gV = √6gV , the strange vector field φ sγ s does not couple to the nucleon. The
O1 O8 µ ∼ µ
remaining couplings to the strange baryons are then determined by symmetry relations:
g = (4α 1)gV
Nω V − O8
√2
2 g = (2α +1)gV
gΛω = 3(5αV −2)gOV8 Λφ − 3 V O8
(9)
g = √2(2α 1)gV
gΣω = 2αVgOV8 Σφ − V − O8
g = 2√2α gV .
gΞω = (2αV −1)gOV8 Ξφ − V O8
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In the limit α = 1, the relative values of the coupling constants are related to the additive
V
quark model via:
2 g √2
g = g = 2g = g = 2gV g = g = Ξφ = g . (10)
Λω Σω Ξω 3 Nω O8 Λφ Σφ 2 3 Nω
Note that all coupling constants are fixed once e.g. g is specified. For the coupling of
Nω
the baryon resonances to the vector mesons we obtain the same Clebsch-Gordan coefficients
as for the coupling to the scalar mesons. This leads to the following relations between the
coupling constants:
g = (3 α )g g = √2α g
∆ω DV DV ∆φ DV DV
−
gΣ∗ω = 2gDV gΣ∗φ = √2gDV
(11)
gΞ∗ω = (1+αDV)gDV gΞ∗φ = √2(2 αDV)gDV
−
gΩω = αDVgDV gΩφ = √2(3 αDV)gDV .
−
In analogy to the octet case we set α = 0, so that the strange vector meson φ does
DV
not couple to the ∆-baryon. The resulting coupling constants again obey the additive quark
model constraints:
3
g∆ω = gΣ∗ω = 3gΞ∗ω = 3gDV gΩω = 0 (12)
2
3
gΩφ = gΞ∗φ = 3gΣ∗φ = √2g∆ω g∆φ = 0
2
Hence all coupling constants of the baryon decuplet are again fixed if one overall coupling
g is specified. Since there is no vacuum restriction on the ∆-ω coupling, like in the case
DV
of the scalar mesons, we have to consider different constraints. This will be discussed in
section V.
IV. MEAN-FIELD APPROXIMATION
The terms discussed so far involve the full quantum field operators. They cannot be
treated exactly. Hence, to investigate hadronic matter properties at finite baryon density we
adopt the mean-field approximation. This nonperturbative relativistic method is applied to
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solve approximately the nuclear many body problem by replacing the quantum field opera-
tors by their classical expectation values (for a recent review see [16]), i.e. the fluctuations
around the vacuum expectation values of the field operators are neglected:
σ(x) = σ +δσ σ σ; ζ(x) = ζ +δζ ζ ζ (13)
h i → h i ≡ h i → h i ≡
ω (x) = ω δ +δω ω ω; φ (x) = φ δ +δφ φ φ.
µ 0µ µ 0 µ 0µ µ 0
h i → h i ≡ h i → h i ≡
The fermions are treated as quantum mechanical single-particle operators. The derivative
terms can be neglected and only the time-like component of the vector mesons ω ω
0
≡ h i
and φ φ survive if we assume homogeneous and isotropic infinite baryonic matter.
0
≡ h i
Additionally, duetoparityconservationwehave π = 0. Thebaryonresonancesaretreated
i
h i
as spin-1 particles with spin-3 degeneracy. After these approximations the Lagrangian (1)
2 2
reads
0 0
+ = ψ [g γ ω +g γ φ +m ]ψ
LBM LBV − i iω 0 iφ 0 ∗i i
i
X
1 χ2 1 χ2
2 2 2 2 4 4 4
= m ω + m φ +g (ω +2φ )
Lvec 2 ωχ2 2 φχ2 4
0 0
1 σ4
2 2 2 2 2 2 4 2
= k χ (σ +ζ ) k (σ +ζ ) k ( +ζ ) k χσ ζ
0 0 1 2 3
V 2 − − 2 −
1 χ4 δ σ2ζ
4 4
+ k χ + χ ln ln
4 4 χ4 − 3 σ2ζ
0 0 0
2
χ 1
= m2f σ +(√2m2 f m2f )ζ ,
VSB χ0! " π π K K − √2 π π #
with the effective mass m of the baryon i, which is defined according to section III for
∗i
i = N,Λ,Σ,Ξ,∆,Σ ,Ξ ,Ω.
∗ ∗
Now it is straightforward to write down the expression for the thermodynamical potential
of the grand canonical ensemble, Ω, per volume V at a given chemical potential µ and at
zero temperature:
Ω γ
i 3
= d k[E (k) µ ] (14)
V −Lvec −L0 −LSB −Vvac − (2π)3 i∗ − ∗i
i Z
X
The vacuum energy (the potential at ρ = 0) has been subtracted in order to get a
vac
V
vanishing vacuum energy. The γ denote the fermionic spin-isospin degeneracy factors. The
i
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single particle energies are E (k) = k2 +m 2 and the effective chemical potentials read
i∗ i ∗i
q
µ = µ g ω g φ.
∗i i − ωi − φi
The mesonic fields are determined by extremizing Ω(µ,T = 0):
V
∂(Ω/V) χ χ δ σ2ζ
2 2 2 2 2 3
= ω m +k χ(σ +ζ ) k σ ζ + 4k +1+4ln 4 ln χ + (15)
∂χ − ωχ20 0 − 3 4 χ0 − 3 σ02ζ0!
χ 1
+ 2 m2f σ +(√2m2 f m2f )ζ = 0
χ20 " π π K K − √2 π π #
∂(Ω/V) δχ4
2 2 2 3
= k χ σ 4k (σ +ζ )σ 2k σ 2k χσζ 2 + (16)
0 1 2 3
∂σ − − − − 3σ
2
χ ∂m
+ m2f + ∗i ρs = 0
χ0! π π i ∂σ i
X
∂(Ω/V) δχ4
2 2 2 3 2
= k χ ζ 4k (σ +ζ )ζ 4k ζ k χσ + (17)
0 1 2 3
∂ζ − − − − 3ζ
2
χ 1 ∂m
+ √2m2 f m2f + ∗i ρs = 0
χ0! " K K − √2 π π# i ∂ζ i
X
∂(Ω/V) χ g
2 4 3 iω
= m ω 4g ω + = 0 (18)
∂ω − χ0! ω − 4 i ρi
X
∂(Ω/V) χ g
2 4 3 iφ
= m φ 8g φ + = 0 (19)
∂φ − χ0! φ − 4 i ρi
X
The scalar densities ρs and the vector densities ρ can be calculated analytically for the case
i i
T = 0, yielding
d3k m γ m k +E
ρs = γ ∗i = i ∗i k E m 2ln Fi F∗i (20)
i iZ (2π)3 Ei∗ 4π2 " Fi F∗i − ∗i m∗i !#
kFi d3k γ k3
ρ = γ = i Fi . (21)
i i 0 (2π)3 6π2
Z
The energy density and the pressure follow from the Gibbs–Duhem relation, ǫ = Ω/V +
µ ρi and p = Ω/V. The Hugenholtz–van Hove theorem [17] yields the Fermi surfaces
i i
−
aPs E (k ) = k2 +m 2 = µ .
∗ Fi Fi ∗i ∗i
q
V. RESULTS FOR DENSE NUCLEAR MATTER
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