Radial sensitivity of kaonic atoms and strongly bound $\bar K$ states

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by  N. Barnea

Radial sensitivity of kaonic atoms and strongly bound K¯ states
N. Barnea1,∗ and E. Friedman1,†
1Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
(Dated: February 9, 2008)
ThestrengthofthelowenergyK−-nucleusrealpotentialhasrecentlyreceivedrenewedattention
in view of experimental evidence for the possible existence of strongly bound K− states. Previous
fitstokaonic atom dataled toeither ‘shallow’ orto ‘deep’ potentials, where only theformer are in
agreement with chiral approaches but only the latter can produce strongly bound states. Here we
explore the uncertainties of the K−-nucleus optical potentials, obtained from fits to kaonic atom
data,usingthefunctionalderivativesofthebest-fitχ2 valueswithrespecttothepotential. Wefind
thatonlythedeeptypeofpotentialprovidesinformation whichisapplicabletotheK− interaction
in the nuclearinterior.
PACSnumbers: 13.75.Gx, 21.10.Gv,25.80.Dj
7
0
0 I. MOTIVATION AND BACKGROUND 0
2

n
Sincetheearlydaysofkaonicatomexperimentsitwas
a
J known [1, 2] that due to the strength of the K− nu- −50
9 clear absorption the real part of the K−-nucleus poten- F (84)
tρ (130)
tial played a secondary role compared to the imaginary

2 part of the potential and consequently it could not be V)
v determined uniquely from fits to the data for a single Me −100
0 target nucleus. More recent ‘global’ fits to large sets V (R
2
of data encompassing the whole of the periodic table
0 FB (84)
showed [3] that although traditional ‘tρ’ potentials yield
1
−150
1 reasonably good fits to the data, the use of phenomeno-
6 logicaldensity-dependentt(ρ)amplitudesleadstosignif-
0 icantly better fits. When extrapolated into the interior DD (103)
h/ of nuclei the real part of the potentials is typically 180 −200
t MeV deep for the density-dependent variety whereas for 0 2 4 6
- r (fm)
l the ‘tρ’ potentials it is typically less than 100 MeV. We
c note that chiral-motivated K−-nucleus potentials [4, 5]
u
are shallower than the phenomenological ‘tρ’ potentials. FIG. 1: Real part of the K−-Ni optical potential for various
n modelsandthevaluesofχ2 for65datapointsinparentheses,
: Figure 1 shows, as an example, the real part of the
v see text.
K− optical potential for Ni obtained from global fits to
i
X kaonic atoms data using several phenomenological mod-
ar ethlsefroeratlhaenidntimeraagcitnioanr.yTpahretssiomfptlheesteff‘teρc’taivpeptr-omacahtrwixhaerree reports on candidates for K¯-nuclear bound states in the
determinedfromfitstothedatayieldsaχ2 of130forthe rangeofbindingenergyBK¯ ∼100−200MeV[7,8,9,10].
However, very recently a possible explanation of the ex-
65datapointsusedinthefit. Addinganadjustablenon-
perimental results of FINUDA [10] was given in terms
linear term leads to the deep potential ‘DD’ of Ref. [3]
of Λp final state interaction [11] and the observed width
with a χ2 of 103 and a greatly increased depth. Also
wasshowntobeatoddswithrecentFaddeevcalculations
shown in the figure is another potential using a geomet-
[12].
rical approach to the density-dependence of t (the ‘F’
potential of Ref.[6]), leading to χ2 of 84. The similarity Obviously, the ‘tρ’-type of potential cannot gener-
ate strongly bound nuclear states in the energy range
of the two deep potentials and the great difference com-
paredto the shallow one areclearly observed. The other BK¯ ∼ 100 − 200 MeV, whereas the deeper potentials
(‘DD’ and‘F’) mightdo. Inthe presentworkweaddress
curve ‘FB’ with its error band, also of χ2 = 84, is dis-
thequestionofhowwellistherealpartoftheK−-nucleus
cussedbelow. Aconsequenceofthedepthofthepotential
potentialdetermined,withitsabilitytosupportstrongly
is the ability to support a strongly bound state, a ques-
bound states as the topic of interest.
tionthatwashighlightedagainrecentlyby experimental
Estimating the uncertainties of hadron-nucleus poten-
tials as function of position is not a simple task. For
example, in the ‘tρ’ approach the shape of the potential
∗Electronicaddress: [email protected] isdeterminedbythenucleardensitydistributionandthe
†Electronicaddress: [email protected] uncertainty in the strength parameter, as obtained from
2
χ2 fits to the data, implies a fixed relative uncertainty tostartthediscussionofvariationsbyassumingthatthe
at all radii, which is, of course, baseless. Details vary globalopticalpotentialisdefinedonadiscretesetofgrid
when more elaborate forms such as ‘DD’ or ‘F’ are used, points η ,i = 1...N , so that χ2 depends on a set of
i
but one is left essentially with analytical continuation N para{meters V = V(}η ). The variation of χ2 due to a
i i
intothe nuclearinteriorofpotentials thatmightbe well- small change in these parameters is simply
determined only close to the nuclear surface. ‘Model-
independent’methodshavebeenusedinanalysesofelas- N ∂χ2
dχ2 = dV . (1)
tic scattering data for various particles [13] to alleviate ∂V i
X i
this problem. However, applying e.g. the Fourier-Bessel i=1
(FB) method in globalanalysesof kaonic atom data end
The equivalent expression for the continuous function
upintoofew terms inthe series,thus makingthe uncer-
V(η)canbeobtainedbytakingthelimitforaverydense
taintiesunrealisticintheirdependence onposition. This
grid, leading to [15]
is illustrated in Fig. 1 by the ‘FB’ curve, obtained by
adding a Fourier-Bessel series to a ‘tρ’ potential. Only δχ2
three termsinthe seriesareneededto achieveaχ2 of84 dχ2 = dη δV(η), (2)
Z δV(η)
and the potential becomes deep, in agreement with the
othertwo‘deep’solutions. Theerrorbandobtainedfrom where
the FB method [13] is, nevertheless, unrealistic because
only three FB terms are used. However, an increase in δχ2[V(η)] χ2[V(η)+ǫδσ(η η′)] χ2[V(η)]
= lim lim − −
the number of terms is found to be unjustified numeri- δV(η′) σ→0ǫ→0 ǫ
cally in this case. (3)
In the present work we adopt a functional-derivative isthefunctionalderivatives(FD)ofχ2[V]. Thenotation
approach in order to get more realistic position- δ (η η′) stands for an approximated δ-function. From
σ
dependence of the uncertainties of K−-nucleus poten- Eq. (−2) it is seen that the FD determines the effect of a
tials. The method is applied to the two types of po- local change in the optical potential on χ2. Conversely
tentials (‘shallow’ and ‘deep’) mentioned above and it is itcanbe saidthattheopticalpotentialsensitivitytothe
shownthat deep potentials within nuclei are reliably ob- experimentaldataisdeterminedbythemagnitudeofthe
tainedfromfitstoexperimentalresultsforkaonicatoms. FD.InpracticethecalculationoftheFDwascarriedout
by multiplying the best fit potential by a factor
f =1+ǫδ (η η′) (4)
II. METHOD σ
−
using a normalized Gaussian with a range parameter σ
The radial sensitivity of exotic atom data was ad- for the smeared δ-function,
dressed before [3] with the help of a ‘notch test’, intro-
ducingalocalperturbationintothepotentialandstudy- δ (η η′)= 1 e−(η−η′)2/2σ2. (5)
ing the changes in the fit to the data as function of po- σ − √2πσ
sition of the perturbation. The results gave at least a
semi-quantitative information on what are the radial re- For finite values of ǫ and σ the FD can be approximated
gions which are being probed by the various types of by
exotic atoms. In fact, the difference in that respect be-
tween deep and shallow kaonic atom potentials could be δχ2[V(η)] 1 χ2[V(η)(1+ǫδσ(η−η′))]−χ2[V(η)] .
observed. However, the extent of the perturbation was δV(η′) ≈ V(η′) ǫ
somewhat arbitrary and in the present work we report (6)
on extending that approach to a mathematically well- The parameter ǫ was used for a fractional change in the
defined limit. potential and the limit ǫ 0 was obtained numerically
→
In order to study the radial sensitivity of global fits for several values of σ and then extrapolated to σ =
to kaonic atom data, it is instructive to define the ra- 0. Good numerical stability and good convergence were
dial position parameter in a ‘natural’ way using, e.g., obtained in all cases. Here η and σ are dimensionless
the knownchargedistributionforeachnuclearspeciesin variables.
the data base. The radial position r is then defined as
r =R +ηa ,whereR anda aretheradiusanddiffuse-
c c c c
ness parameters, respectively, of a two-parameter Fermi III. RESULTS AND CONCLUSIONS
(2pF)chargedistribution[14]. Inthatwayηbecomesthe
relevant radial parameter when handling together data The K−-nucleus potentials used in the present work
for several nuclear species along the periodic table. The are taken from recent global fits [6] to kaonic atom data
valueofχ2canberegardednowasafunctionalofaglobal from6LitoU,atotalof65datapoints. Atwo-parameter
optical potential V(η), i.e. χ2 = χ2[V(η)]. The param- fit with a tρ potential yields a total χ2 value of 130
eter η is a continuous variable, however it is instructive whereas a four-parameter fit with t(ρ) amplitudes yields
3
40

F potl. 0 0
C
F tρ
−50 −50
20 R V)
e
M −100 −100
V) R tρ potl. V (R −150 −150
V/
δ 0
2δΧ/( C −2000 2 4 6 −2000 2 4 6
40 20
F tρ
-20 −1m) 20 10
V) (fR 0 0
V/R
δ
-40-6 -4 -2 0 2 4 6 8 2Χ/( −20 −10
η δ
−40 −20
FIG. 2: Functional derivatives of χ2 with respect to the full 0 2 4 6 0 2 4 6
r (fm) r (fm)
complex (C) and real (R) potential as function of η, where
r = Rc + ηac, with Rc and ac the radius and diffuseness
parameters, respectively, of the charge distribution. Results FIG. 3: Real potentials (top) and FD (bottom) for the ‘F’
are shown for the tρ potential and for the t(ρ) ‘F’ potential potential(left)andthetρpotential(right)forK−interaction
withNi. Theregionsbetweentheverticaldottedlinesindicate
of Ref. [6].
where thepotentials are determined reliably, see text.
χ2=84 for the 65 data points. In calculating FD for
0 0
global fits radial positions were defined in terms of units
of diffuseness relative to the charge radius, as described F tρ
−50 −50
above. Forseveralofthelightestnucleiharmonicoscilla-
tor densities are more appropriate and had indeed been eV)
M −100 −100
used in the fits of Ref. [6]. These nuclei have been ex- V (R
cluded from the calculations of FD for the global fits so
−150 −150
as to have full consistency in the use of 2pF density dis-
tributions. That left 50 data points in the calculations
−200 −200
of the FD. 0 2 4 0 2 4
60 20
Figure 2 shows the FD for relative variations in the 40 F 10 tρ
risnhegaolwtphnoe)treFenDvteiaawllisathanddrdeiisntpivtehictetytfouoflltthhceoemFimpDal,egxhinepnaocrtyeentphtoieatedl.nifftIinearslepn(enccoetst- −1V) (fm)R 200 −100
betweentheFDforthecomplexpotentialandforthereal V/R −20 −20
δ
potentialinthe figurearethe correspondingFDwithre- 2Χ/( −40 −30
spect to the imaginary potential. We note that the FD δ
−60 −40
forthedeep‘F’potentialisdominatedbytherealpartof 0 2 4 0 2 4
the potential whereasfor the ‘tρ’ potentialthe two parts r (fm) r (fm)
makesimilarcontributionstotheFD.Theappearanceof
regionswithnegativeFD neednot be surprisingbecause FIG. 4: Sameas fig. 3 but for 12C.
itmeansthatsomelocalvariationsintheshapeofthepo-
tentials may cause further reduction in the values of χ2.
Such variations are impossible in the tρ potential. How- for the real tρ potential is between η = 1.5 and η = 6
ever,suchvariationsareincluded whenthe moreflexible whereas for the F potential it is between−η = 3.5 and
‘F’potentialisintroduced,andtheminimisationprocess η = 4. Recall that η = 2.2 correspond to 90−% of the
essentiallyfollowsimplicitlythevariousFD.Weavoidat central charge density an−d η =2.2 correspond to 10% of
present making a quantitative use of the local values of that density. It therefore becomes clear that within the
the FD, rather we identify with the help of the FD the tρ potential there is no sensitivity to the interior of the
radial regions to which the kaonic atom data are sensi- nucleus whereas with the t(ρ) ‘F’ potential, which yields
tive. greatly improved fit to the data, there is sensitivity to
FromFig. 2itcanbeinferredthatthesensitiveregion regions within the full nuclear density.
4
Figure 3 shows similar results for Ni, this time the at regions of almost the full nuclear matter density and
radial variable is r, the radial position. Here σ is in consequently they may be reliably extrapolated to the
units of fm and therefore the FD is in units of fm−1. On nuclear interior. In contrast the ‘shallow’ type of po-
the left are shown the real potential (top) and the FD tentials, which yield inferior fits to the data, are well
(bottom) for the real ‘F’ potential and on the right are determined only at the nuclear surface and one cannot
shown the corresponding quantities for the tρ potential. infer from these what is the depth of the K¯-nucleus po-
Theregionsbetweenthetwoverticaldottedlinesindicate tential in the nuclear interior. The different sensitivi-
where variation of the potential will affect the fit to the tiesresultfromthe potentialsthemselves: the additional
data, thus suggesting the regions where the potentials attraction provided by the deep potentials enhances the
are determined by experiment. The vertical dashed line atomicwavefunctionswithinthenucleus[3]thuscreating
indicates 50% of the central charge density. It is evident the sensitivity atsmallerradii. We concludethat optical
thatforthe‘F’potentialthesensitivityextendstodepths potentials derived from the observed strong-interaction
of 180 MeV at radii where the density is essentially the effects in kaonic atoms are sufficiently deep to support
full nuclear density. Very similar results are obtained strongly-boundantikaonstates,butitdoesnotnecessar-
also for heavier nuclei, where, e.g. for Pb the distinction ily imply that such states do exist. Moreover, the dis-
betweenthe tρ andthe ‘F’potentialis evengreaterthan crepancy between the very shallow chiral-motivated po-
for Ni, with respect to the sensitivity of the experiment tentials [4, 5] and the deep phenomenological potentials
to depths and densities. remains an open problem.
Finally, in Fig. 4 are shown similar results for 12C,
which is one of the targets studied experimentally [10]
andtheoretically[6]. Againitisseenthatwiththe‘F’po- Acknowledgments
tential the sensitive region is at smaller radii and higher
densities compared to that for the tρ potentials. This work was supported in part by the Israel Science
In summary, it is found that the ‘deep’ K¯-nucleus po- Foundation grant 757/05. We thank A. Gal for many
tentials, which yield excellent fits to all the kaonic atom discussions.
data, are determined reliably to depths of 150-180 MeV
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