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Faddeev calculation of a K−pp quasi-bound state N.V. Shevchenko∗,1 A. Gal,2 and J. Mareˇs1 1Nuclear Physics Institute, 25068 Rˇeˇz, Czech Republic 2Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel (Dated: February 9, 2008) We report on the first genuinely three-body K¯NN −πΣN coupled-channel Faddeev calculation − − insearchforquasi-boundstatesintheK ppsystem. ThemainabsorptivityintheK psubsystem − is accounted for by fitting to K p data near threshold. Our calculation yields one such quasi- bound state, with I = 1/2, Jπ = 0−, bound in the range B ∼ 55−70 MeV, with a width of Γ∼90−110MeV.These resultsdiffersubstantially from previousestimates, andare atoddswith theK−pp→Λp signal observed bythe FINUDAcollaboration. 7 PACSnumbers: 11.80.Jy, 13.75.Jz,21.45.+v 0 0 2 The issue of K¯ nuclear quasi-bound states has at- motivated at present. n tracted considerable interest recently, motivated by ear- In this Letter we report on the first K¯NN πΣN a liersuggestionsfor(anti)kaoncondensationindensemat- coupled-channel Faddeev calculation which is ge−nuinely J − ter[1]andbyextrapolationsofK opticalpotentialsfit- three-body calculation, searching for quasi-bound states 3 − − − tedtoK atomdata[2,3]. TheseK atomstudiessug- that are experimentally accessible through a K pp fi- 2 gested K¯ nuclear potential depths about 150 200 MeV nal state. Coupled-channel three-body Faddeev calcula- at nuclear-matter density ρ = 0.16 fm−3, al−though po- tions were reported for K−d, with an emphasis on other 2 0 − v tentialsevaluatedbyfittingtoK plow-energydatagive entities than on quasi-bound states [15]. We note that − − 2 substantially lower values, about 100 MeV [4] or even as the K d system is not as favorable as the K pp sys- 2 lowas50MeV[5]dependingonhowtheK¯N in-mediumt tem for strong binding, since the relative weight of the 0 matrixisconstructed. ItwaspointedoutthatK¯ nuclear I = 0 K¯N interaction with respect to the weakly at- 0 states, if bound by over 100 MeV where the K¯N πΣ tractive I =1 K¯N interaction is 1:3 for K−d and 3:1 1 → − main strong-decay channel is closed, might become suf- for(K pp) . Bydoingcoupledchannelcalculations, 6 I=1/2 ficiently narrow to be observed [6, 7, 8]. Yamazaki and with two-body input fitted to available low-energy data, 0 / Akaishi [9], in particular, discussed few-body K¯ nuclear we wish to determine the scale of binding energy and h configurationsinwhichthestronglyattractiveI =0K¯N width expected for few-body K¯ nuclear systems. t - interactionismaximized. ItistheI =0coupled-channel In the present work we solve non-relativistic three- l c s-wave interaction that generates a resonance in the πΣ body Faddeev equations in momentum space, using the u coupled channel about 27 MeV below the K−p thresh- Alt-Grassberger-Sandhas (AGS) form [16]. The AGS n old, the quasi-bound Λ(1405) [10]. The lightest K¯ nu- equations for three particles are: v: clear configuration maximizing the I = 0 K¯N interac- i tionistheI =1/2K¯(NN) statewithS =L=0and U11 = T2G0U21+T3G0U31 X Jπ =0− [11]. The significaIn=c1e of identifying this poten- U = G−1+T G U +T G U (1) 21 0 1 0 11 3 0 31 ar tiallylow-lyingquasi-boundstateintheK−ppmassspec- U = G−1+T G U +T G U , 31 0 1 0 11 2 0 21 trum of suitably chosen production reactions has been recentlyemphasized[12]. However,becausethe coupling where G is the free three-body Green’s function and 0 ofthetwo-bodyK−pchanneltotheabsorptiveπY chan- T , i=1,2,3,aretwo-bodyTmatricesinthethree-body i nels was substituted by an energy-independent complex space for the pair excluding particle i. These equations K¯N potential, the results for binding energy and width definethree unknowntransitionoperatorsU describing ij oftheK−ppsystem[9]provideatbestonlyaroughesti- the elastic and re-arrangementprocesses: mate. Recently, the FINUDA collaboration at DAΦNE, − U : 1+(23) 1+(23) Frascati,presented evidence in K stopped reactions on 11 → several nuclear targets for the process K−pp Λp, in- U : 1+(23) 2+(31) (2) 21 −→ → terpreting the observed signal as due to a K pp deeply U : 1+(23) 3+(12) , 31 bound state [13]. However, this interpretation has been → challenged in Refs. [3, 14]. Given this unsettled exper- with Faddeev indices i,j = 1,2,3 denoting simultane- imental search for a quasi-bound K−pp state, precise ously a given particle and its complementary interacting three-bodycalculationsfortheK−ppsystemappearwell pair. SincetheK¯N two-bodysubsystemisstronglycou- pledtootherchannels,particularlyviatheΛ(1405)resonance to the I =0 πΣ channel, it is necessary to extend theAGSformalisminordertoincludethesechannelsex- ∗Correspondingauthor: [email protected] plicitly. Thus, all operators entering the AGS equations 2 become 3 3 matrices: G Gαβ = δ Gα which is diagonal in×the channel spa0ce→, and0T Tααββ 0where α,β TABLEI:StrengthparametersλαI=β0,1 (inunitsfm−2)forthe i → i K¯N−πΣpotentials(7)with rangeparameter β=3.5 fm−1, are channel indices as follows: corresponding to aK−p =(−0.70+i 0.60) fm. TNN 0 0 T1 = 10 T1ΣN 0  (3) λKI¯=N0,K¯N λKI¯=N0,πΣ λIπ=Σ0,πΣ λKI¯=N1,K¯N λKI¯=N1,πΣ λIπ=Σ1,πΣ  0 0 T1ΣN  -1.370 1.414 -0.176 0.007 1.734 -0.340 TKK 0 TKπ TKK TKπ 0  2 2   3 3  respectively. The calculation of the kernels Z involves T = 0 TπN 0 T = TπK Tππ 0 . 2 2 3 3 3 transformationfrom one set of Jacobicoordinatesto an-  TπK 0 Tππ   0 0 TπN  2 2 3 other one and isospin recoupling as well. The position We assign particle labels (1,2,3) to (K¯,N,N) in chan- of the three-body pole was searched as a zero of the de- terminant of the kernel of the system of integral equa- nel 1, to (π,Σ,N) in channel 2 and to (π,N,Σ) in channel 3. Here TNN, TπN and TΣN are one-channel T- tions on the corresponding unphysical sheet. More de- matrices, whereas TKK, Tππ, TπK and TKπ are the tails on the extended AGS equations and the numeri- elements of the two-channel TKN−πΣ matrix, account- cal procedure are relegated to an expanded version of ing for K¯N K¯N and πΣ πΣ elastic processes, this paper. Here it suffices to mention that by assum- and for K¯N →πΣ and πΣ K→¯N inelastic transitions, ing charge independence, s-wave pairwise interactions, → → and antisymmetrizing over the two nucleons, we end up respectively. We neglect the I = 1 inelastic transition K¯N πΛ since experimentally it is outweighed by the in a system of nine coupled integral equations. This is K¯N → πΣ transition, and also since the I = 1 K¯N theminimaldimensionalityofanyFaddeevcalculationin → the I = 1/2 Jπ = 0− sector which attempts to account configuration plays a minor role in the structure of the − explicitly for the strong absorptivity of the K¯N interac- I =1/2K ppsystemunderdiscussion. Uponthisexten- tionsnearthreshold. Withinthisscheme,theinteraction sionintochannelspace,theunknownoperatorsU assume the most general matrix form: U Uαβ. Substituting oftherelativelyenergeticpionwiththeslowbaryonswas ij → ij neglected,partlybecauseitsp-wavenaturewouldrequire these new 3 3 operators into the AGS system of equa- × an extension of the present s-wave calculation. tions we obtain the system to be solved. The input separable potentials for the T-matrices (4) Assuming charge independence, three-body quasi- are given in momentum space by bound states are labelled by isospin. The isospin ba- sis is used throughout our calculation within a coupling Vαβ(k ,k′)=λαβgα(k )gβ(k′), (7) schemethatensuresthatwearesearchingforanI =1/2 I α β I I α I β quasi-bound state. Assuming pairwise s-wave meson- ′ where k ,k are two-particle relative momenta in the baryoninteractions,asappropriateto theK¯N πΣsys- α β temneartheK¯NN threshold,ands-wavebaryo−n-baryon two-body respective channels, and λαIβ are strength- parameterconstants. Fortheα=β =(NN) channel, interactionslimitedtothe1S configurationasappropri- I=1 0 we have used a separable approximationof the Parispo- ate to pp, the total spin and total orbital angular mo- tential [17], corresponding to the one-rank potential (7) mentumofthethree-bodysystemareS =L=0. Tensor with λNN = 1 and a form factor: forces are not operative for this situation, which also re- I − inforces the neglect of coupling to πΛN since the strong 1 6 cNN ΣN ΛN transition is dominated by the tensor force gNN(k)= i,I=1 . (8) in th→e 3S YN configuration. Hence, the K¯NN πΣN I=1 2√π Xk2+(βNN )2 1 − i=1 i,I=1 systemexploredinthisFaddeevcalculationhasquantum numbers I =1/2, L=0, S =0, Jπ =0−. The constants cNN and βNN are listed in Ref. [17]. i,I=1 i,I=1 In order to reduce the dimension of the integral equa- For the S = 1 interactions, the form factors gα(k ) tions,aseparableapproximationforthetwo-bodyT main Eq. (7) were−parameterized by a Yamaguchi formI α trices is used: 1 Tiα,Iβi =|giα,Iiiτiα,Iβihgiβ,Ii| , (4) gIα(kα)= (kα)2+(βIα)2 . (9) where I is the conserved isospin of the interacting pair. i For the I = 3/2 ΣN interaction we made two different [for α = β our generalized T-matrices coincide with the choicesofλΣN andβΣN . Thefirstchoice,labelled(i) usualones.] Forseparabletwo-bodyT-matrices,theAGS I=3/2 I=3/2 below, reproduces the scattering length a = 3.8 fm equations may be rewritten using a new kernel and un- I=3/2 and effective range r = 4.0 fm of the Nijmegen known functions: I=3/2 Model F [18]. The second choice, labelled (ii) below, re- Zαβ δ gα Gα gβ (5) producesthe mostrecentNijmegen YN phaseshifts [19] ij,IiIj ≡ αβh i,Ii| 0| j,Iji using a scattering length a = 4.15 fm and effective I=3/2 Xαβ gα GαUαβ Gβ gβ , (6) range r = 2.4 fm. For the I = 1/2 ΣN interaction ij,IiIj ≡ h i,Ii| 0 ij,IiIj 0| j,Iji I=3/2 3 FIG. 1: Calculated K−p → K−p cross sections, for three FIG. 2: Calculated K−p → π+Σ− cross sections, for three different sets of K¯N −πΣ parameters, in comparison with different sets of K¯N −πΣ parameters, in comparison with themeasured cross sections (see text). themeasured cross sections (see text). we reproduced the value quoted by Dalitz [20] for the Nogami[11]. WethenperformedfullK¯NN πΣN three- scattering length a = 0.5 fm. bodycalculationsforthethreesetsofK¯N −πΣparame- For the I =0,1 KI¯=N1/2 πΣ−coupled-channel potentials, tersandforthetwosets(i)and(ii)of(ΣN)−I=3/2parame- theparametersλαβ an−dβα inEqs.(7,9)werefitted tersdescribedabove. Thesensitivityoftheresultstothe I=0,1 I=0,1 ΣN interaction was also studied by setting TΣN =0 for to reproduce (i) E = 1406.5 i 25 MeV [10], the Λ(1405) − both I =1/2 and I =3/2. The calculated binding ener- position and width of Λ(1405) which is assumed to be a gies(B = E )andwidths(Γ)arepresentedinTableII quasi-bound state in the K¯N channel and a resonance − B − where the energies are given with respect to the K pp in the πΣ channel, (ii) the branching ratio at rest [21] threshold. ItisseenthattheΣN interaction,dominantly γ = Γ(K−p π+Σ−)/Γ(K−p π−Σ+) = 2.36, and − → → in the I = 3/2 channel, adds only about 3 MeV to the (iii) the K p scattering length aK−p for which we used binding energy (less than 6%) affecting the width by up as a guideline the KEK measured value [22]: to2MeV(lessthan2%). Thisisnegligibleonthescaleof bindingenergiesandwidths displayedinthe tableandis aK−p =(−0.78±0.15±0.03)+i(0.49±0.25±0.12)fm. consistent with the negligible effect (less than 2%) that (10) the YN and πN final-state interactions were found to In order to check the sensitivity of our results to this in- − have in the latest K d Faddeev calculation of Ref. [15]. put, within the quoted errors, we fitted three different In contrast, the calculated binding energies and widths values of aK−p using a range parameter β = 3.5 fm−1. show sensitivity to the fitted K¯N πΣ coupled-channel All three sets of our K¯N πΣ parameters, which also − two-body interactions, giving rise in our calculations to − reproduce the energy and width of Λ(1405) and the up to about 25% variation in B and up to about 15% − − branching-ratio γ, yield low-energy K p K p and variation in Γ. It is worth noting that B increases with K−p π+Σ− cross-sections which are in a→good agree- → Im aK−p, whereas Γ is correlated more with Re aK−p; ment with experimental data, as shownin Figs. 1 and 2. this feature is typical to strong-absorption phenomena Wenotethatthedatapointsinthesefiguresareprecisely where the width gets saturated beyond a critical value those compiled and cited in Ref. [24]. The strength pa- of absorptivity [25]. We have also studied the depen- rameters λαI=β0,1 for the K¯N −πΣ coupled-channel sepa- dence of the calculated binding energy and width on the rable potentials fitted to aK−p =(−0.70+i 0.60)fm are range parameter β within acceptable fits, keeping aK−p given for illustration in Table I. constant, say aK−p = (−0.78+i 0.49) fm. The binding In a test calculation we first switched off the coupling energy changes very little, by about 3 MeV, whereas the of the K¯NN channels to the πΣN channels. This re- width changesappreciably,decreasingfrom115MeV for ducesthenumberofcoupledintegralequationsfromnine β =3 fm−1 to 89 MeV for β =4 fm−1. to three within the three-body K¯NN space. We as- Our calculations confirm the existence of an I = 1/2, sumed the Λ(1405) to be a genuine bound state of the Jπ = 0− three-body quasi-bound state, with apprecia- (K¯N) subsystem, reproducing the real part of the ble width, in the K¯(NN) channel. The width of this I=0 I=1 − K p scattering length of Eq. (10). We found a zero- quasi-boundstateisameasureofitscouplingtotheπΣN wthiedtK¯hNboNuntdhrsetsahtoelda.tTenheirsgbyinEdK¯inNgNen=er−gy43is.7coMnesVidebrealbolwy cchouanpnlienlgs twohtehree πitΣsNhowchsaunpnealss,ainbardoadditiroensotnoapnrcoev.idTinhge larger than the value EK¯NN ≈ −10 MeV estimated by a width which renders the K¯(NN)I=1 bound state into 4 − oration to a K pp bound state. Possible extensions of TABLEII:Calculated energyEK¯NN =EB−iΓ/2(inMeV) the present coupled-channel Faddeev calculation should of the I =1/2, Jπ =0− quasi-bound K¯(NN) state with respect to the K−pp threshold, calculated forId=i1fferent two- includetheI =1πΛchannel,enlargethemodelspaceto body input. E(i) and E(ii) correspond to sets (i) and (ii), re- includep-wavetwo-bodyinteractionsandintroducerela- tivistickinematics. Relyingontheexperienceofcoupled- spectively,of the(ΣN) interaction parameters, whereas I=3/2 − E(0) stands for no ΣN interaction (see text). channel Faddeev calculations of the K d system [15], none of these extensions is expected to change qualita- aK−p (fm) EK(¯i)NN (MeV) EK(¯iiN)N (MeV) EK(¯0N) N (MeV) tively our results and conclusions. −0.78+i 0.49 −55.8−i 49.1 −56.2−i 50.1 −53.4−i 49.2 In conclusion, we performed the first coupled-channel −0.78+i 0.65 −69.4−i 46.8 −70.0−i 47.9 −66.3−i 47.5 three-body Faddeev calculation for the I = 1/2 −0.70+i 0.60 −66.0−i 54.7 −66.5−i 55.8 −63.5−i 54.6 K¯(NN)I=1systeminsearchofaquasi-boundstate. This state canbe reachedin productionreactions aiming at a − final K pp system. It is primarily the large width, here a quasi-bound state, also provides substantial extra at- calculated for a K¯ nuclear state above the πΣ two-body traction through which the binding energy is increased threshold, that poses a major obstacle to observing and from 44 MeV to the range of values shown in the table. identifying K¯ nuclear quasi-bound states. Yet, even for The acceptable parameter sets considered in our calcu- deeper states below the πΣ threshold, in heavier nuclei, lations yield binding in the range B 55 70 MeV, a residual width of order 50 MeV is expected to persist withawidthofΓ 90 110MeV.Alth∼ough−thebinding due to K¯NN YN absorption [3]. ∼ − → energycalculatedhereissimilartothatestimatedbyYa- This work was supported by the GA AVCR grant − mazaki and Akaishi [9] for K pp, our calculated width A100480617 and by the Israel Science Foundation grant is considerably larger than their estimate Γ = 61 MeV 757/05. NVS is grateful to János Révai for many fruit- and is also larger than the width Γ 67 MeV of the ful discussions. AG acknowledges the support of the − ≈ K pp ΛpsignalintheFINUDAexperiment[13]. Our Alexander von Humboldt Foundation and thanks Wol- → rangeofcalculatedbindingenergiesisconsiderablylower framWeise forhis kindhospitalityatTUMuenchenand than B 115 MeV attributed by the FINUDA collab- for stimulating discussions. ≈ [1] G.E. Brown, C.-H. Lee, M. Rho, V. Thorsson, Nucl. 74 (2006) 025206. Phys.A 567 (1994) 937. [15] G. Toker, A. Gal, J.M. Eisenberg, Nucl. Phys. A 362 [2] E.Friedman,A.Gal,C.J.Batty,Phys.Lett.B308(1993) (1981)405;M.Torres,R.H.Dalitz,A.Deloff,Phys.Lett. 6; Nucl. Phys. A 579 (1994) 518; E. Friedman, A. Gal, B 174 (1986) 213; A. Bahaoui, C. Fayard, T. Mizutani, J. Mareˇs, A. Cieply´, Phys. Rev.C 60 (1999) 024314. B. Saghai, Phys.Rev.C 68 (2003) 064001. [3] J.Mareˇs, E.Friedman,A.Gal,Phys.Lett.B606(2005) [16] E.O. Alt, P. Grassberger, W. Sandhas, Nucl. Phys. 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