The $K^{+}K^{-}$ final state interaction was investigated based on both the
$K^{+}K^{-}$ invariant mass distributions measured at excess energies of Q = 10
and 28 MeV and the near threshold excitation function for the $pp \to
ppK^{+}K^{-}$ reaction. The $K^+K^-$ final state enhancement factor was
parametrized using the effective range expansion. The effective range of the
$K^+K^-$ interaction was estimated to be: $\mathrm{Re}(b_{K^{+}K^{-}}) = -0.1
\pm 0.4_{stat} \pm 0.3_{sys} \mathrm{fm}$ and $\mathrm{Im}(b_{K^{+}K^{-}}) =
1.2^{+0.1_{stat}+0.2_{sys}}_{-0.2_{stat}-0.0_{sys}} \mathrm{fm}$, and the
determined real and imaginary parts of the $K^+K^-$ scattering length amount
to: $|\mathrm{Re}(a_{K^{+}K^{-}})| = 8.0^{+6.0_{stat}}_{-4.0_{stat}}
\mathrm{fm}$ and $\mathrm{Im}(a_{K^{+}K^{-}}) =
0.0^{+20.0_{stat}}_{-5.0_{stat}} \mathrm{fm}$.Comment: 5 pages, 3 figures, article accepted for publication in Phys. Rev. 

Moskal, P.

Silarski, M.

Physical Review C

English

arXiv

Silarski, Michał

Moskal, Paweł

Jagiellonian Univeristy Repository

Combined analysis of the $K^{+}K^{−}$ interaction using near-threshold $pp \rightarrow ppK^{+}K^{−}$ data

The K+K− final state interaction was investigated based on both the K+K− invariant mass distributions measured at excess energies of Q=10 and 28 MeV and the near threshold excitation function for the pp→ppK+K− reaction. The K+K− final state enhancement factor was parametrized using the effective range expansion. The effective range of the K+K− interaction was estimated to be Re(bK+K−)=−0.1±0.4stat±0.3sysfm and Im(bK+K−)=1.2+0.1stat+0.2sys−0.2stat−0.0sysfm, and the determined real and imaginary parts of the K+K− scattering length amount to |Re(aK+K−)|=8.0+6.0stat−4.0statfm and Im(aK+K−)=0.0+20.0stat−5.0statfm

Juelich Shared Electronic Resources

PHYSICAL REVIEW C 88, 025205 (2013)Combined analysis of the K+ K− interaction using near-threshold pp → ppK+ K− dataM. Silarski1,* and P. Moskal1,21Institute of Physics, Jagiellonian University, PL-30-059 Cracow, Poland2Nuclear Physics Institute, Research Center Ju¨lich, D-52425 Ju¨lich, Germany(Received 28 December 2012; revised manuscript received 29 July 2013; published 19 August 2013)The K+K− final state interaction was investigated based on both the K+K− invariant mass distributionsmeasured at excess energies of Q = 10 and 28 MeV and the near threshold excitation function for thepp → ppK+K− reaction. The K+K− final state enhancement factor was parametrized using the effectiverange expansion. The effective range of the K+K− interaction was estimated to be Re(bK+K− ) = −0.1 ±0.4stat ± 0.3sys fm and Im(bK+K− ) = 1.2+0.1stat+0.2sys−0.2stat−0.0sys fm, and the determined real and imaginary parts of theK+K− scattering length amount to |Re(aK+K− )| = 8.0+6.0stat−4.0stat fm and Im(aK+K− ) = 0.0+20.0stat−5.0stat fm.DOI: 10.1103/PhysRevC.88.025205 PACS number(s): 13.75.Lb, 14.40.BeI. INTRODUCTIONThe strength of the K+K− interaction is a crucial quantityregarding the formation of a hypothetical kaon–antikaonbound state. Existence of such a state could explain the natureof the a0(980) and f0(980) scalar mesons [1,2], whose massesare very close to the sum of the K+ and K− masses.1 Amongmany theoretical investigations [7–11] the K+K− interactionwas studied also experimentally in the pp → ppK+K−reaction with COSY-11 and ANKE detectors operating atthe COSY synchrotron in Ju¨lich, Germany [12–18]. Theexperimental data collected systematically below [12–16]and above [18] the φ meson threshold reveal a significantenhancement in the shape of the excitation function near thekinematical threshold, which may be due to the final stateinteraction (FSI) in theppK+K− system. The indication of theinfluence of the pK− final state interaction was found in bothCOSY-11 and ANKE data in the ratios of the differential crosssections as a function of the pK and the ppK invariant masses,RpK = dσ/dMpK−dσ/dMpK+, RppK = dσ/dMppK−dσ/dMppK+,where a significant enhancement in the region of both the lowpK− invariant mass MpK− and the low ppK− invariant massMppK− is observed [15,19]. The phenomenological modelbased on the factorization of the final state interaction intointeractions in the pp and pK− subsystems, neglecting theK+K− potential, does not describe the whole experimentalexcitation function for the pp → ppK+K− reaction,underestimating the data very close to the kinematicalthreshold [15,20]. This indicates that in the low-energyregion the influence of the K+K− final state interactionmay be significant [15,19,20]. Motivated by this observationthe COSY-11 Collaboration has recently estimated thescattering length of the K+K− interaction based on thepp → ppK+K− reaction measured at excess energies ofQ = 10 and 28 MeV [19]. As a result of the analysis theK+K−*michal.silarski@uj.edu.pl1Besides the standard interpretation as qq¯ mesons [3], theseresonances were also proposed to be qqq¯q¯ states [4], hybridqq¯/meson–meson systems [5], or even quarkless gluonic hadrons [6].scattering length was determined based on the low-energyproton-proton (Mpp) and K+K− (MKK ) invariant massdistributions (so-called Goldhaber plot) shown in Fig. 1 [19].In this article we combine the Goldhaber plot distributionestablished by the COSY-11 group with the experimentalexcitation function [12–15,18] near threshold and determinethe K+K− scattering length with better precision compared tothe previous results. We have also extracted the effective rangeof the K+K− interaction.II. DESCRIPTION OF THE FINAL STATE INTERACTIONIN THE ppK+ K− SYSTEMAs in the previous analysis [19] we use the factorizationansatz proposed by the ANKE group with an additional termdescribing the interaction in the K+K− system. We assumethat the overall enhancement factor originating from final stateinteraction can be factorized into enhancements in the proton-proton, the two pK−, and the K+K− subsystems:2FFSI =Fpp(k1) ×Fp1K− (k2) ×Fp2K− (k3) ×FK+K− (k4), (1)where kj stands for the relative momentum of particles inthe corresponding subsystem [19]. The proton-proton scat-tering amplitude was taken into account using the followingparametrization:Fpp = eiδpp(1S0) sin δpp(1S0)Ck1,where C stands for the square root of the Coulomb penetrationfactor [21]. The parameter δpp(1S0) denotes the phase shiftcalculated according to the modified Cini-Fubini-Stanghelliniformula with the Wong-Noyes Coulomb correction [22–24].Factors describing the enhancement originating from theinteraction in the pK− subsystems are parametrized usingthe scattering length approximationFpK− = 11 − ikapK− .2In this model we neglect the pK+ interaction since it is repulsiveand weak [15].025205-10556-2813/2013/88(2)/025205(4) ©2013 American Physical SocietyM. SILARSKI AND P. MOSKAL PHYSICAL REVIEW C 88, 025205 (2013)FIG. 1. Experimental Goldhaber plots for the pp → ppK+K−reaction. The solid lines of the triangles show the kinematicallyallowed boundaries. Individual events are shown in (a) and (b)as black points. The superimposed squares represent the samedistributions but binned into intervals of M = 2.5 MeV/c2 (M =7 MeV/c2) widths for an excess energy of Q = 10 (28) MeV,respectively. The area of the square is proportional to the numberof entries in a given interval. The figure was adapted from [19].The pK− scattering length apK− was estimated both the-oretically [25–30] and experimentally based mainly on thekaonic hydrogen atom measurements [31,32]. As shown inRef. [33] different approaches result in a slightly differentapK− values. Therefore, in our analysis we have assumed thepK− scattering length to be equal to the mean of all thevalues from elaborations [25–32] summarized in Ref. [33]:apK− = (−0.65 + 0.78i) fm.The K+K− FSI was parametrized using the effective rangeexpansion,FK+K− = 11aK+K−+ bK+K− k242 − ik4,where aK+K− and bK+K− are the scattering length and theeffective range of theK+K− interaction, respectively. We haveperformed a fit to the experimental data treating aK+K− andbK+K− as free parameters. Moreover, we have repeated theanalysis for every quoted apK− to check how their differentvalues change the result. This allowed us also to estimate thesystematic error due to the pK− scattering length used in theestimation of aK+K− and bK+K− .It is worth mentioning that there is a similar phenomenolog-ical model of the K+K− final state interaction which takes intoaccount the elastic and charge-exchange interaction allowingfor the K0K0 ⇀↽ K+K− transitions. This FSI should generatea significant cusp effect in the K+K− invariant mass spectrumnear the K0K0 threshold (details can be found in [34]).Another contribution to this effect may be also generated bythe kaons rescattering to scalars, eg.: KK → f0(980) → KKand KK → a0(980) → KK . However, the ANKE data canbe described well without introducing the cusp effect [34],thus we neglect it in this analysis. We also cannot distinguishbetween the isospin I = 0 and I = 1 states of the K+K−system. However, as pointed out in [34], the productionwith I = 0 is dominant in the pp → ppK+K− reactionindependent of the exact values of the scattering lengths.In the fit we do not take into account influence of thef0(980) and a0(980) production. There exist only very roughexperimental estimates of upper limits for production of theseresonances in the N -N collisions. In fact, up to now in thesereactions there has not been found any signal of these particles.The theoretical estimations result in negligible cross sectionsfor the pp → f0pp → K+K−pp resonant contribution withrespect to the nonresonant one [35] (the upper limit ofthe cross section for this reaction is estimated to be about1 × 10−4 nb/MeV at Q = 5 MeV and 4 × 10−2 nb/MeV at50 MeV [35]). Also the branching ratios off0(980) and a0(980)are very poorly known. However, according to the Particle DataGroup (PDG) a0(980) dominantly decays to ηπ0 and the ππchannel is dominant for the f0(980) meson [36]. Thus, the f0resonance contribution to the near-threshold pp → ppK+K−reaction is expected to be negligible. Moreover, regardingthe a0(980) resonance, following Ref. [34] the K+K− pairsare produced in proton-proton collisions mainly with isospinI = 0. Thus, a0(980) would have to decay to K+K− throughisospin violation, which is an additional suppressing factor.According to Ref. [35] for energies up to Q = 115 MeV(DISTO measurement [18]) the production of resonant K+K−pairs should not produce any significant enhancement in theK+K− invariant mass.III. DETERMINATION OF THE K+ K− SCATTERINGLENGTH AND EFFECTIVE RANGEIn order to estimate the strength of the K+K− interactionthe experimental Goldhaber plots, determined at excess en-ergies of Q = 10 and 28 MeV, together with the total crosssections measured near the threshold were compared to theresults of the Monte Carlo simulations treating the K+K−scattering length aK+K− and effective range bK+K− as unknowncomplex parameters. To determine aK+K− and bK+K− we haveconstructed the following χ2 statistics:χ2 (aK+K− , bK+K− , α)=8∑i=1(σexpti − ασmi)2(σexpti)2+ 22∑j=110∑k=1[βjNsjk − Nejk + Nejkln(NejkβjNsjk)], (2)where the first term was defined following the Neyman’s χ2statistics, and accounts for the excitation function near the025205-2COMBINED ANALYSIS OF THE K+K− INTERACTION . . . PHYSICAL REVIEW C 88, 025205 (2013)threshold for the pp → ppK+K− reaction. σ expti denotes theith experimental total cross section measured with uncertaintyσexpti and σmi stands for the calculated total cross sectionnormalized with a factor α which is treated as an additionalparameter of the fit. σmi was calculated for each excess energyQ as a phase space integral over five independent invariantmasses [37]:σm =∫π2 |M|28s√−B dM2ppdM2K+K−dM2pK−dM2ppK−dM2ppK+ .Here s denotes the square of the total energy of the systemdetermining the value of the excess energy, and B is a functionof the invariant masses with the exact form to be found inNyborg’s work [37]. The amplitude for the process |M|2contains the FSI enhancement factor defined in Eq. (1) and itdepends on the parameters aK+K− and bK+K− . The second termof Eq. (2) corresponds to the Poisson likelihood chi-squarevalue [38] describing the fit to the Goldhaber plots determinedat excess energies Q = 10 MeV (j = 1) and Q = 28 MeV(j = 2) using COSY-11 data [19]. Nejk denotes the numberof events in the kth bin of the j th experimental Goldhaberplot, and Nsjk stands for the content of the same bin in thesimulated distributions. βj is a normalization factor whichis fixed by values of the fit parameters α and aK+K− . It isdefined for the j th excess energy as the ratio of the totalnumber of events expected from the calculated total crosssection σmj (aK+K−) and the total luminosity Lj [14], to the05101520-0.75 -0.5 -0.25 0 0.25(a)Re(bKK) (fm)χ2-χ2    min01020300.9 1 1.1 1.2 1.3(b)Im(bKK) (fm)χ2-χ2    min00.250.50.7510 2.5 5 7.5 10(c)Im(aKK) (fm)χ2-χ2    min02040600 2.5 5 7.5 10(d)|Re(aKK)| (fm)χ2-χ2    minFIG. 2. χ 2 − χ 2min distribution as a function of (a) Re(bK+K− ),(b) Im(bK+K− ), (c) Im(aK+K− ), and (d) |Re(aK+K− )|. χ 2min denotesthe absolute minimum with respect to parameters α, Re(bK+K− ),Im(bK+K− ), |Re(aK+K− )|, and Im(aK+K− ).total number of simulated pp → ppK+K− events Ngenj :βj =LjασmjNgenj.The χ2 distributions (after subtraction of the minimum value)for FK+K− taken in the effective range expansion are presentedas a function of the real and imaginary parts of aK+K−and bK+K− in Fig. 2. The best fit to the experimental datacorresponds toRe(bK+K− ) = −0.1 ± 0.4stat ± 0.3sys fm,Im(bK+K− ) = 1.2+0.1stat+0.2sys−0.2stat−0.0sys fm, (3)|Re(aK+K−)| = 8.0+6.0stat−4.0stat fm,Im(aK+K− ) = 0.0+20.0stat−5.0stat fm,with a χ2 per degree of freedom of χ2/ndof = 1.30. Thestatistical uncertainties in this case were determined at the70% confidence level taking into account that we havevaried five parameters [α, Im(aK+K− ), Re(aK+K− ), Im(bK+K− ),Re(bK+K− )]. Here uncertainties correspond to the range of1101020 50 100Q (MeV)σ (nb)FIG. 3. (Color online) Excitation function for the pp →ppK+K− reaction. Triangle and circles represent the DISTO andANKE measurements, respectively [15,18,40]. The squares areresults of the COSY-11 [12,13,19] measurements. The dashed curverepresents the energy dependence obtained assuming that the phasespace is homogeneously and isotropically populated, and there isno interaction between particles in the final state. Calculations takinginto account the pp and pK− FSIs are presented as the dashed-dottedcurve. The dashed and dashed-dotted curves are normalized tothe DISTO data point at Q = 114 MeV. Solid curve correspondsto the result obtained taking into account pp, pK , and K+K−interactions parametrized with the effective range approximation.These calculations were obtained using the scattering length aK+K−and effective range bK+K− as obtained in this work. The latest datapoint measured by the ANKE group was published recently [40], andthus it was not taken into account in the fit.025205-3M. SILARSKI AND P. MOSKAL PHYSICAL REVIEW C 88, 025205 (2013)values for which the χ2 of the fit is equal to χ2 = χ2min +6.06 [39]. Systematic errors due to the uncertainties in theassumed pK− scattering length were instead estimated as amaximal difference between the obtained result and theK+K−scattering length determined using different apK− values3quoted in Refs. [30,33]. One can see that the fit is in principlesensitive to both the scattering length and effective range;however, with the available low statistics data the sensitivityto the scattering length is very weak.Results of the analysis with inclusion of the interaction inthe K+K− system described in this article are shown as thesolid curve in Fig. 3. One can see that the experimental dataare described quite well over the whole energy range.IV. CONCLUSIONSWe have performed a combined analysis of both total anddifferential cross section distributions for thepp → ppK+K−reaction in view of the K+K− final state interaction. In theanalysis we have used a factorization proposed by the ANKEgroup with an additional term describing interaction in theK+K− system, without distinguishing between the isospinI = 0 and I = 1 states. We have also neglected a possiblecharge exchange interaction leading to a cusp effect in the3Due to the fact that in the case of scattering length the systematicuncertainties are much smaller than the statistical ones we neglectthem in the final result.K+K− invariant mass spectrum, since taking it into accountwould require much more precise data [34]. The K+K−enhancement factor was parametrized using the effective rangeexpansion. Fit to experimental data is very weakly sensitiveto aK+K− , but allows us to estimate the effective range of theK+K−–FSI.All studies of the pp → ppK+K− reaction near threshold[12–16] reveal that in the ppK+K− system the interactionbetween protons and the K− meson is dominant, and aK+K−is relatively small. It seems that this reaction is driven by the(1405) production pp → K+(1405) → ppK+K− ratherthan by the scalar mesons [34], which may, however, contributeto the observed cusp effect by rescattering of kaons. 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Combined analysis of the K + K − interaction using near-threshold p p → p p K + K − data1

M. Silarski

P. Moskal

Crossref

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