Study of light kaonic nuclei with a Chiral SU(3) - based KbarN interaction

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by  Akinobu Dote

EPJ manuscript No.
(will be inserted by the editor)
Study of light kaonic nuclei with a
¯
KN
Chiral SU(3) - based interaction
Akinobu Dot´e1 and Wolfram Weise2
7
0 1 IPNS/KEK,1-1 Ooho, Tsukuba, Ibaraki, Japan, 305-0801
0 2 Physik-Department,Technische Universit¨at Mu¨nchen, 85747 Garching, Germany
2

n Received: date/ Revised version: date
a
J Abstract. Wehaveinvestigatedtheprototypekaonicnucleus,ppK−,usingthemethodofAntisymmetrized
8
Molecular Dynamics(AMD).InthepresentstudyweusearealisticNN potentialwith stronglyrepulsive
1 core, and a Chiral SU(3)-based K¯N interaction which is energy-dependent and includes both s- and p-
wave interactions. We find that no self-consistent solutions exist when the range parameter of the K¯N
1
interaction is less than 0.67 fm. Due to the strong repulsion of the NN interaction at short distance, the
v
two nucleons in ppK− keep a distance of about 1.2 fm and the total binding energy of ppK− does not
0
exceed about 50 MeV in thescope of our current analysis.
5
0
1 PACS. 13.75.Jz Kaon-baryoninteractions–21.30.Fe Forcesinhadronicsystemsandeffectiveinteractions
0 – 21.45.+v Few-body systems
7
0
/ Kaon-nuclear systems, with a K− tightly bound to a Weinvestigatelightkaonicnucleiinafullymicroscopic
h
t nuclear core, have recently received considerable interest. way employing the variational Antisymmetrized Molecu-
-
Previousstudies[1,2,3]suggestedlotsofexoticproperties lar Dynamics (AMD) method. In the AMD framework,
l
c oflightkaonic nuclei,asa consequenceofthe stronglyat- singlenucleonwavefunctions ϕ andthekaonwavefunc-
u tractive I = 0 K¯N interaction. Several experiments have tion ϕ are constructed as f|oliliows;
n | Ki
been performed by various groups [4,5,6,7]. Recently the
:
v FINUDAgroupreportedmeasurements[5]whichtheyin- Nn Zi 2
Xi btearspicreptraostostigynpaelssofofrktahoenifcornmuactleioi.nWofhpilpeKth−iscliunstteerrpsr,etthae- |ϕii =α=1Cαi exp"−ν(cid:18)r− √να(cid:19) # |σii|ταii, (1)
r X
a tionisnotwithoutcontroversy,thequestfordeeplybound NK ZK 2
kaon-nuclearstateshasattractedagreatamountofactiv- ϕ = CKexp ν r α τK . (2)
ity in the field. | Ki α − − √ν!  | α i
α=1
X
 
The trial wave function of each nucleon (or the kaon) is
Ourinvestigationsareaimedatadetailedunderstand-
described as a superposition of N (N ) Gaussian wave
ing of the conditions under which kaonic nuclei can exist. n K
− packets whose centers Zi ( ZK ) are different from
As a first step we explore the ppK system, focusing on { α} { α}
each other; ν is a width parameter of the Gaussian wave
the following two points: the role of the repulsive core of
packets; σ represents a spin wave function, or ;
tShUe(3N)-NbasinedteKr¯aNctiionntearnacdtitohne.IifmtphleekmaeonntiactniouncleoifaareCdheinrsael |ταii and||τiαiKi are isospin wave functions and| ↑hiave|t↓hie
following form:
andcompactassuggestedbypreviouscalculations[1,2,3],
the strong repulsion of the NN interaction at short dis-
1 1
tance is expected to be very important in acting against τi = +γi p + γi n , (3)
suchcompressionintoverydenseconfigurations.TheK¯N | αi (cid:18)2 α(cid:19)| i (cid:18)2 − α(cid:19)| i
interactionisakeyingredientinthestudyofkaonicnuclei. τK = 1 +γK K¯0 + 1 γK K− (4)
In the present calculations we adopt a theoretically moti- | α i 2 α | i 2 − α | i
(cid:18) (cid:19) (cid:18) (cid:19)
vated interaction instead of the previous phenomenologi-
cal one [1]. The s-wave part of this interaction is derived where γi and γK are variational parameters.
{ α} { α}
from chiral SU(3) theory [8] and includes the strong en- The total wave function is constructed by antisym-
ergy dependence resulting from coupled-channel dynam- metrizing a set of nucleon wave functions, Eq. (1), and
ics. An energy dependent p-wave interaction dominated combining it with the kaon wave function, Eq. (2): Φ =
| i
by the Σ(1385) is also incorporated. [ϕ (a) ] ϕ .Thiswavefunctionisprojectedontoan
i K
A| i ⊗| i
2 Akinobu Dot´e, Wolfram Weise: Studyof light kaonic nuclei with a Chiral SU(3) - based K¯N interaction
3500 2.5
3000 AvA1K8 2.0 RIme[[ FFKK--pp ]]
2500 Av18-like Re[ FK-n ]
1.5
Im[ FK-n ]
2000
m] 1.0
MeV] 1500 [fm
[ Fc 0.5
1000
0.0
500
-0.5
0 Λ(1405)
-1.0
-500
300 350 400 450 500
0.0 0.2 0.4 0.6 0.8 1.0 1.2
[fm] Kaon energy ω [MeV]
Fig. 1. Comparison of NN potentials in 1S0 channel. “AK”
0.25
istheNN potentialused inthepreviousstudy.“Av18”isthe
Argonne v18 potential. “Av18-like” is the NN potential used
in thepresent study. 0.20 Re[ CK-p ]
Im[ CK-p ]
0.15
eigenstate of angular momentum (J), isospin (T), charge 3m]
(mCix)eadndstaptaer,itEyq(sP.)(:3|)PJanPdTP(C4)P,PanΦdi.tGheivecnhatrhgee cphraorjgece-- [fp,cm 0.10 Σ(1385)
tion (PC), we can adequately treat the K−p/K¯0n mixing C 0.05
whichiscausedbytheK¯N interaction.Allvariationalpa-
rameters {Cαi,Ziα,γαi;CαK,ZKα, γαK}, which are complex 0.00
numbers, are determined by the frictional cooling equa-
tion. Details of the AMD method are explained in Ref. -0.05
[3]. 300 350 400 450 500
The NN interaction used here is guided by the Ar- Kaon energy ω [MeV]
gonnev18(Av18)potential[9].Fig.1showsthe1S chan-
0
nel of the potential used in our study (“Av18-like”), the Fig. 2. Thes-waveamplitudeFKN(ω)(upperpanel) and the
original Argonne v18 potential (“Av18”), and the poten- p-wavescattering “volume” CKN(ω)(lower panel).
tialusedinthepreviousstudies(“AK”)[1,2,3].Evidently,
thepresentrealisticpotentialhasastronglyrepulsivecore
ory[8].Itsenergydependencereflectsprimarilythestrong
incontrasttothepreviouslyusedpotential.Sincethispre-
K¯N πΣ coupled-channels dynamics which drives the
vious potential has been constructed using the G-matrix
↔
Λ(1405)resonance.The C (ω) is an updated versionof
method [1], the repulsive-core part is smoothed out. KN
The s-wave and p-wave K¯N interactions have the fol- the parameterization introduced long ago in Ref. [10]. It
features the dominance of the Σ(1385) in p-wave I = 1
lowing forms:
channel for which CK−n ≃ 2CK−p holds. Fig. 2 shows
v (r r ,ω)= 2π ω+MN F (ω) FKN(ω) and CKN(ω).
KN,S N K KN The procedureofthe presentvariationalcalculationis
− − ωM
(cid:18) N (cid:19) as follows. In a first step we treat the p-wave K¯N inter-
1
exp[ (r r )2/a2], (5) actionperturbativelyandomitthe imaginarypartsofthe
×π3/2a3s − N − K s s-andp-waveK¯N interactions.The totalHamiltonianHˆ
v (r r ,ω)= 2π ω+MN C (ω) is
KN,P N K KN
− − ωM
(cid:18) N (cid:19) Hˆ =Hˆ +Re[Vˆ (ω)] (7)
1 0 KN,P
×π3/2a3∇exp[−(rN −rK)2/a2p]∇, (6) Hˆ0 Tˆ+VˆNN +Re[VˆKN,S(ω)]+VˆCoul. TˆCM. (8)
p ≡ −
where MN is the nucleon mass and ω is the kaon energy. Here, Tˆ, VˆNN and VˆCoul. correspond to the total kinetic
The present K¯N potentials have a strong energy depen- energy, NN central potential and Coulomb potential, re-
dence.Thes-andp-wavepotentialshaveGaussianshapes spectively. We use all channels of the Av18-like potential
with range parameters as and ap. FKN(ω) and CKN(ω) as our VˆNN, although channels other than 1S0 turn out
are the s-wave K¯N scattering amplitude and the energy nottobeimportantinppK−.Vˆ (ω)andVˆ (ω)are
KN,S KN,P
dependent p-wave K¯N scattering “volume”, respectively. the s- and p-wave K¯N potentials of Eqs. (5) and (6), re-
The amplitude F (ω)is derivedfromChiralSU(3) the- spectively.Thecenter-of-massmotionenergyofthewhole
KN
Akinobu Dot´e, Wolfram Weise: Studyof light kaonic nuclei with a Chiral SU(3) - based K¯N interaction 3
500 Table 1. Self consistent solution of ppK− with various range
Ref parameters. “a” is the range parameter of K¯N interaction
a=0.90 in fm. T, V(NN), V(KN,S), V(KN,P) and V(Coul.) are
400 a=0.80 the total kinetic energy, the NN potential energy, the s-wave
V] a=0.70 K¯N potentialenergy,thep-waveK¯N potentialenergyandthe
e
M a=0.67 Coulombenergy,respectively.TotalE.isthetotalenergyhHˆi.
K) [ 300 a=0.65 B(K) is the separate binding energy of kaon. All energies are
d B( a=0.60 given in MeV. Rpp (in fm) is the mean distance between the
ne 200 two protons.
ai
bt
O
100 a=0.67 a=0.70 a=0.80 a=0.90
T 469.8 382.2 270.4 188.8
0 V(NN) 15.3 16.1 14.1 12.8
0 100 200 300 400 500 V(KN,S) −404.8 −340.9 −255.0 −191.1
Assumed B(K) [MeV] V(KN,P) −130.6 −88.8 −49.3 −27.1
V(Coul.) −2.2 −2.0 −1.7 −1.4
Fig. 3. Self consistency of the kaon’s binding energy B(K).
Total E. −52.5 −33.4 −21.5 −18.0
The horizontal axis B(K)assumed isthekaon’s bindingenergy
assumed as input in order to determined the strength of K¯N B(K) 195.4 153.7 110.4 84.5
iinntgereancetrigoyn.cTahlceulvaetretdicawlitahxissuBch(Ka)oKb¯taNineidntiesrathcteiokna.on“’Rsebfi”ndis- Rpp 1.04 1.11 1.25 1.41
the line B(K)assumed =B(K)obtained. The curves correspond
to results calculated with different range parameters a.
Here, Tˆ and Tˆ are the kinetic energy and the
N CM(N)
center-of-mass motion energy of the nucleons only.
system, Tˆ , is subtracted. We determine an AMD wave
CM −
WenowshowanddiscussresultsfortheppK system.
function Φ by the energyvariationfor the Hamiltonian
0 −
| i In the present study, the quantum numbers of ppK are
Hˆ0. Then we calculate the total energy Etot which is the assumedtobeJπ =0− andT =1/2,wheretheparity(π)
expectation value of the total Hamiltonian Hˆ with this
includes the intrinsic parity of the kaon. Single nucleons
wave function Φ . Secondly, we must take into account
| 0i and the kaon are represented by two and five Gaussian
the self-consistency of the kaon’s energy since the present
wave packets, respectively. Namely, we set N =2 in Eq.
K¯N interaction is strongly energy dependent. We obtain n
(1) and N = 5 in Eq. (2). We use a common range
K
the self-consistent solution by the following steps. 1. We parameter for both s-wave and p-wave K¯N interactions,
assume a kaon binding energy B(K) as trial in-
assumed a =a a, and investigate the systematics with various
s p
put. Once the kaon’s energy is determined as ω = m ≡
B(K) , the strength of the K¯N interactionis fiKxe−d range parameters a.
assumed
Fig.3showstherealizationoftheself-consistencycon-
and the Hamiltonian is also determined. 2. We perform
theenergyvariationandtreatthep-waveK¯N interaction dition for the kaon’s binding energy B(K), with varying
range parameters. The self-consistent solution should ap-
perturbativelysoastocalculatethetotalenergy.Wecom-
pear on the line “Ref” which indicates B(K) =
pute separately the kaon binding energy, B(K) . 3. assumed
obtained
B(K) . As can be seen, the curves for a 0.67
We compare B(K)obtained with B(K)assumed. If they dif- obtained ≥
fm cross the line “Ref”. This means there exists a self-
fer, we return to step 1 and change the input value of
consistent solution for these cases. However, for a < 0.67
B(K) .Werepeatthecycle1to3untilB(K)
assumed obtained
fm, no self-consistent solution is found.
coincides with B(K) . When B(K) =
assumed obtained
B(K) , a self-consistent solution is found. The self-consistent solutions so obtained are summa-
assumed
rized in Table 1. As the range parameter decreases, the
WecommentbrieflyonthekaonbindingenergyB(K).
total binding energy (= “Total E.”) increases rapidly
In kaonic nuclei, B(K) is different from the total energy, −
from 18 MeV to 53 MeV. One observes that the p-wave
E , because the nuclear components rearrange them-
tot K¯N potential contributes significantly to the binding of
selves under the influence of the strong attraction from
thekaon.However,itscontributiondecreasesastherange
the kaon field located at the center of the nucleus. In
parameter increases. Compared to the s-wave potential,
fact, the nuclear part in a kaonic nucleus is in an ex-
the p-wave part is still sufficiently small to justify a per-
cited state relative to the originalnuclear system. We de-
turbative treatment.
fine B(K)= (E E ) as the kaon binding energy,
tot nucl
where E i−s the n−uclear part of the energy calculated Inallcases,the averagedistancebetweenthetwopro-
nucl
as Φ Hˆ Φ / Φ Φ withthecorrespondingpartofthe tons (Rpp) is larger than 1.0 fm. This is attributed to the
0 N 0 0 0
h | | i h | i strongly repulsive core of the NN interaction. If the size
Hamiltonian,
of the core of a single nucleon is assumed to be roughly
0.5 fm, the present result indicates that the cores of the
Hˆ Tˆ +Vˆ Tˆ . (9) two protons remain well separated in space.
N N NN CM(N)
≡ −
4 Akinobu Dot´e, Wolfram Weise: Studyof light kaonic nuclei with a Chiral SU(3) - based K¯N interaction
Table2. Onlys-waveinteractionvsincludingp-waveinterac-
tion. “With p-wave” (“Only s-wave”) is the result calculated
with (without) thep-waveinteraction for a=0.70 fm.
With p-wave Only s-wave
T 382.2 195.3
V(NN) 16.1 18.9
(a) Nucleon (b) Kaon V(KN,S) −340.9 −243.1
V(KN,P) −88.8 0.0
Fig. 4. Density contour of ppK− calculated with the range V(Coul.) −2.0 −1.4
parameter a=0.90 fm. The left (right) panel shows the distri-
butionofthenucleons(thekaon).Thesize scaleis3×3fm2. Total E. −33.4 −30.3
B(K) 153.7 106.2
Rpp 1.11 1.36
Fig. 4 depicts the density distribution of the nucleons
−
and the kaon in ppK calculated with the range param-
eter a = 0.9 fm. The two protons keep a distance and Afurther importantstepwillbe the calculationofthe
the kaon is centered between them. In cases using other decay width. This involves not only the mesonic mode
range parameters,the configuration of nucleons and kaon (K¯N πY) but also the two-nucleon absorption pro-
is essentially the same as that shown in Fig. 4. cess (K¯→NN YN). A reliable estimate of the width of
→
InTable2,weshowaself-consistentsolutionobtained antikaon-nuclear states is crucial in order to assess the
byusingonlythes-waveK¯N interaction(“Onlys-wave”). experimental observability of kaonic nuclei.
Here the range parameter is 0.70 fm. We performed the
same calculation as mentioned before, switching off the We thank Avraham Gal for fruitful and stimulating
p-wave interaction. In this case, the total energy is simi- discussions during his visit in Munich. This work is sup-
lar to that obtained with the p-wave interaction (“With ported in part by BMBF and GSI.
p-wave”).However,thedecompositionofthistotalenergy
into its components is quite different. In the case “With
−
p-wave”, the ppK tends to increase its binding energy
References
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In summary, as a result of the present study of ppK arXiv:nucl-ex/0310018.
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tial and the Chiral SU(3)-based K¯N potential, no self- 6. T.Kishimotoet al.,Prog.Theor. Phys.Suppl.149,(2003)
consistent solutions are found when the range parameter 264; Nucl.Phys. A754, (2005) 383.
oftheK¯N interactionislessthan0.67fm.Themaximum 7. N.Herrmann,Proceedings of “International Conference on
total binding energy of ppK− is 53 MeV. The configura- Exotic Atom and Related Topics (EXA05)” published by
tionoftheppK− clusterissuchthatthetwoprotonskeep Austrian Academy of Science Press, pp.73-81.
8. B. Borasoy, R. Nißler and W. Weise, Eur. Phys. J. A 25,
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the short-range NN interaction. The kaon field occupies
9. R.B.Wiringa,V.G.J.StoksandR.Schiavilla,Phys.Rev.
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The present results should still be considered prelimi-
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tobeinsufficientbecausetheAMDvariationalwavefunc-
tion is based on the independent-particle picture. In the
next forthcoming step a two-nucleon correlation function
will be introduced in the AMD trial wave function. Since
−
thebindingmechanismofppK isdominatedbythestrong
attraction of the K¯N interaction, it is nonetheless ex-
pected that the essential points of the present study will
not be modified substantially once the short-range corre-
lations are adequately dealt with.