By solving the Bethe-Goldstone equation, we have obtained the $\bar{K}$
optical potential from the $\bar{K}N$ effective interaction in nuclear matter
at T=0. We have extended the model by incorporating finite temperature effects
in order to adapt our calculations to the experimental conditions in heavy-ion
collisions. In the rank of densities ($0-2\rho_0$), the finite temperature
$\bar{K}$ optical potential shows a smooth behaviour if we compare it to the
T=0 outcome. Our model has also been applied to the study of the ratio between
$K^+$ and $K^-$ produced at GSI with $T$ around 70 MeV. Our results point at
the necessity of introducing an attractive $\bar{K}$ optical potential.Comment: 7 pages, 4 figures. Contribution to the proceedings of Mesons & Light
  Nuclei '01 (2-6th July, Prague

A. Polls

A. Ramos

I. The Model At T

Laura Tolós

English

arXiv

L. Tolós

Crossref

Finite temperature effects on the K̄ optical potential

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Finite temperature effects on the K̄ optical potential
Laura Tolós, A.Polls, A.Ramos
Departament d’Estructura i Constituents de la Matèria, Universitat de Barcelona,
Diagonal 647, 08028 Barcelona, Spain
(November 2, 2018)
Abstract
By solving the Bethe-Goldstone equation, we have obtained the K̄ optical
potential from the K̄N effective interaction in nuclear matter at T = 0.
We have extended the model by incorporating finite temperature effects in
order to adapt our calculations to the experimental conditions in heavy-ion
collisions. In the rank of densities (0− 2ρ0), the finite temperature K̄ optical
potential shows a smooth behaviour if we compare it to the T = 0 outcome.
Our model has also been applied to the study of the ratio between K+ andK−
produced at GSI with T around 70 MeV. Our results point at the necessity
of introducing an attractive K̄ optical potential.
PACS: 12.38.Lg, 13.75.Jz
Keywords: K̄N interaction, Kaon-nucleus potential, finite temperature, heavy-ion collisions
I. THE MODEL AT T = 0
The K̄ optical potential in nuclear matter is computed using a K̄N effective interaction,
G-matrix, derived microscopically from a meson-exchange potential [1]. The G-matrix is
given by the solution of the Bethe-Goldstone equation
G(w) = V + V
Q
w −H0 + iη
G(w), (1)
which takes into account the different channels (K̄N , πΣ, πΛ), which are coupled by the
strong interaction.
At lowest order in the Brueckner-Hartree-Fock (BHF) theory, the single-particle potential
for antikaons UK̄ at T = 0 is given by
UK̄(k, E
qp
K̄
) =
∑
N≤F
〈K̄N | GK̄N→K̄N(w = E
qp
N + E
qp
K̄
) | K̄N〉, (2)
in which Eqp
K̄
is self-consistently determined.
In order to introduce in-medium effects on the intermediate states of the G-matrix, we
have employed mean field single-particle propagators for the baryons. The single-particle
potential for nucleons has been derived from a σ − ω model, while for Λ and Σ, we have
1
followed the parameterization of Ref. [2]. Finally, the pion is dressed with the momentum
and energy dependent self-energy of Ref. [3], that incorporates the coupling to particle-hole,
∆-hole and two particle-two hole excitations.
Fig.1 shows the real and imaginary part of UK̄ as a function of kK̄ at T = 0 for ρ = ρ0
including or not the dressing of pions. The long-dashed line has been calculated determining
the complex K̄ single particle energy self-consistently. However, the pion self-energy in the
intermediate states is not included. This procedure is similar to the one presented in Ref.
[4] with the inclusion here of a σ − ω model for the nucleon dressing. The presence of
the Λ(1405) resonance, dynamically generated in our model, introduces a strong energy
dependence on the K̄N amplitude which governs, in turn, the behaviour of UK̄ . The Pauli
blocking on the intermediate nucleons has a repulsive effect on the resonance, produced by
having moved the threshold of intermediate states to higher energies. The self-consistent
incorporation of the K̄ properties on the K̄N G-matrix moves the resonance back in energy,
but it dilutes as the density increases. If we now include the dressing of pions (solid line),
we obtain a less attractive real part, going from −84 MeV to −73 MeV at k = 0 MeV, and
a smoother behaviour for the imaginary part. This can be understood by looking again to
the K̄N amplitude, specially for L = 0. Due to the additional dressing of the pions, the
resonance dissolves even faster loosing structure, although it shows a tendency to move to
higher energies.
II. TEMPERATURE EFFECTS
The introduction of temperature in the calculation affects the Pauli blocking of the in-
termediate nucleon states in the G-matrix. In addition, the K̄ optical potential is calculated
according to
UK̄(k, E
qp
K̄
) =
∫
d3k n(k, T ) 〈K̄N | GK̄N→K̄N (w, T ) | K̄N〉 (3)
where n(k, T ) is the nucleon momentum distribution at finite temperature. By looking
at Fig.2, one observes that, as the temperature increases, UK̄ is less attractive and shows a
smoother behaviour in momentum. At finite T , the sum over momenta in Eq.(3) is extended
over the corresponding Fermi distribution, and higher nucleon momentum translates into
a weaker interaction. Moreover, the effective interaction has also changed. However, this
last effect is less important. At T = 70 MeV, the momentum dependence is very smooth,
reducing in a factor of three the difference between low and high momentum. However, UK̄
is still attractive. This fact can be used to understand the enhancement of the observed
ratio K−/K+ in heavy-ion collisions at GSI energies.
III. RATIO K−/K+
Heavy-ion collisions provide a unique possibility to create a dense and hot nuclear system
to study the in-medium properties of hadrons, like in-medium effects on the K̄. The mea-
sured ratio of particles K−/K+ gives us information about the necessity of an in-medium
attractive potential for the K−, already corroborated in the analysis of kaonic atoms.
2
In direct nucleon-nucleon collisions, the production of K+ and K− is governed by
quite distinct thresholds (for NN → K+ΛN , the threshold is at 1.58 GeV and for
NN → K+K−NN is 2.5 GeV), and the K+ multiplicity exceeds the K− one by 1-2 orders
at a given energy above thresholds. However, this large difference disappears for nucleus-
nucleus collisions (C + C, Ni + Ni [5,6]), where the data nearly fall on the same curve.
Furthermore, we observe that this ratio is approximately constant for C + C, Ni+Ni and
Au+Au [5–7], although absorption ofK− via K−N → Y π is supposed to be higher for heav-
ier nuclei [8]. It turns out that both effects can be interpreted by assuming an in-medium
attractive UK̄ [9,10]
Analysis of K+/K− in Ni + Ni at GSI energies [5,6] gives a ratio around 30. In Ref.
[11], it was shown that this ratio can be explained in terms of a thermal model, i.e., the final
state can be described as a hadronic gas in chemical equilibrium at a given temperature.
Imposing exact strangeness conservation, one obtains,
K+
K−
=
gK+V
∫ d3p
(2π)3
e
−E
K+
T (gK−V
∫ d3p
(2π)3
e
−E
K−
T + gΛV
∫ d3p
(2π)3
e
−EΛ
T
+
µB
T + gΣV
∫ d3p
(2π)3
e
−EΣ
T
+
µB
T )
gK−V
∫ d3p
(2π)3
e
−E
K−
T (gK+V
∫ d3p
(2π)3
e
−E
K+
T )
, (4)
where gα are the spin-isospin degeneracies and V is the interaction volume. The above
expression shows that the presence of K+ must be compensated by Λ, Σ and K− to conserve
strangeness equal to zero, while K− can only be compensated by K+.
It is found that the data [12] can be explained using a T = 70 ± 10 MeV and µB =
720± 20 MeV. G.Brown [13] introduced the notion of ”broad-band equilibration” in heavy-
ion processes at GSI energies. In this interpretation, due to the compensation between
the increase in µB as the density increases and the density dependence of UK̄ , the ratio is
constant over a large range of densities, not only for ρ ∼ 1
4
ρ0, but also up to 2ρ0.
In Fig.3, we show the importance of dressing K−. Plotting the ratio K+/K− as function
of density, it can be seen that the effect of including UK̄ reduces the ratio that, otherwise,
would increase steadily (short-dashed line). It is also seen that dressing the pions induces
appreciable differences in the calculations (long-dashed line).
Once the full dressing is included, Fig.4 shows the sensitivity of the ratio to the chemical
potential, and hence to density, as well as the importance of introducing the Σ in the
balance equation. The ”broad-band equilibration” is hardly reproduced when the Σ is taken
into account, and one obtains large differences depending on the character (attractive or
repulsive) of the Σ optical potential.
IV. CONCLUSIONS
Performing a microscopic calculation of the Brueckner-Hartree-Fock type , we conclude
that dressing pions in the calculation of UK̄ introduces significant differences, around 13%
at zero momentum. Moreover, the introduction of temperature gives rise to a smoother
behaviour in momentum, being attractive even at momentum as large as 500 MeV, relevant
in heavy-ion collisions. Heavy-ion collisions are the perfect scenario to test in-medium K̄
properties, specially in understanding the ratio K−/K+. The attempts to reproduce this
ratio require an attractive UK̄ . The ratio shows a measurable density dependence. The
3
”broad-band equilibration” is far from being reproduced if hyperons like Σ are taken into
account, showing large differences with regard to only including Λ hyperons.
ACKNOWLEDGMENTS
We are very grateful to Dr. Jürgen Schaffner-Bielich for the useful discussions that have
made possible this work and L.T. wishes to acknowledge Brookhaven National Laboratory
for kind hospitality during her stay.
4
REFERENCES
[1] A. Müller-Groeling, K. Holinde and J. Speth, Nucl. Phys. A513 (1990) 557.
[2] S.Balberg and A.Gal Nucl. Phys. A625 (1997) 435.
[3] E. Oset and A. Ramos, Nucl. Phys. A671 (2000) 481.
[4] L.Tolós, A. Ramos, A.Polls and T.T.S.Kuo Nucl. Phys. A690 (2001) 547.
[5] R. Barth et al., Phys. Rev. Lett. 78 (1997) 4007; F. Laue et al., Phys. Rev. Lett. 82
(1999) 1640.
[6] M.Menzel et al.Phys. Lett. B (2000) 26.
[7] A.Förster, Ph.D Thesis, TU Darmstad.
[8] P.Senger , Nucl. Phys. A685 (2001) 312.
[9] W. Cassing, E.L. Bratkovskaya, U. Mosel, S. Teis and A. Sibirtsev, Nucl. Phys. A614
(1997) 415; E.L. Bratkovskaya, W. Cassing and U. Mosel, Nucl. Phys. A622 (1997)
593.
[10] G.Q. Li, C.-H. Lee and G.Brown, Nucl. Phys. A625 (1997) 372; ibid, Phys. Rev. Lett.
79 (1997) 5214.
[11] J.Cleymans, H.Oeschler and K.Redlich, Phys. Rev.C59 (1999) 1663
[12] J.Cleymans, D.Elliot, A.Keränen and E.Suhonen, Phys. Rev.C57 (1998) 3319
[13] G.Brown, M.Rho and C.Song, Nucl. Phys.A690 (2001) 184
5
FIGURES
0 50 100 150 200 250 300 350 400 450 500
kK(MeV/c)
−90
−80
−70
−60
−50
−40
−30
Im
 U
K
(M
eV
)
−90
−80
−70
−60
−50
−40
−30
−20
R
e 
U
K
(M
eV
)
not dressing pions
dressing pions
FIG. 1. UK̄ as a function of kK̄ for T = 0, including or not the dressing of pions at ρ = 0.17fm
−3
0 50 100 150 200 250 300 350 400 450 500
kK(MeV/c)
−80
−70
−60
−50
Im
 U
K
(M
eV
)
−80
−70
−60
−50
−40
R
e 
U
K
(M
eV
)
T=0
T=20 MeV
T=40MeV
T=70 MeV
FIG. 2. UK̄ as a function of kK̄ for different T , at ρ = 0.17fm
−3
6
0 0.1 0.2 0.3 0.4 0.5 0.6
ρΒ (fm
−3
)
0
50
100
150
200
K
+
/K
−
full dressing
not dressing pions
not dressing K
−
FIG. 3. K+/K− as a function of density at T = 70MeV , including or not UK̄
0 0.1 0.2 0.3 0.4 0.5 0.6
ρΒ (fm
−3
)
0
10
20
30
40
50
60
70
80
K
+
/K
−
including only Λ
including Λ+Σ(attractive at ρ0)
including Λ+Σ(repulsive at ρ0)
FIG. 4. K+/K− as a function of density at T = 70MeV , included UK̄ , including or not Σ
7


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