EPJ manuscript No. (will be inserted by the editor) Analysis of (π±,K+) and (K−,K+) hypernuclear production spectra in distorted wave impulse approximation Hideki Maekawa, Kohsuke Tsubakihara and Akira Ohnishi 7 0 Department of Physics, Faculty of Science, Hokkaido UniversitySapporo 060-0810, Japan 0 2 Received: date/ Revised version: date n a Abstract. We study the hyperon-nucleus potential with distorted wave impulse wave approximation J (DWIA) using Green’s function method. In order to include the nucleon and hyperon potential effects 4 in Fermi averaging, we introduce the local optimal momentum approximation of target nucleons. We can 2 describethequasifreeΛ,Σ andΞ productionspectrainabetterwaythaninthestandardFermiaveraged t-matrix treatments. 1 v PACS. 21.80.+a Hypernuclei– 24.50.+g Direct reactions 6 6 0 1 Introduction One of the methods to evaluate the hyperon-nucleus 1 potential is to analyze the quasi free (QF) spectrum in 0 the continuum region [5]. Recent observation of inclusive 7 0 Studyofhyperon-nucleon(YN)interactionhasanadvan- (π−,K+) spectra on heavy nuclear targets performed at tage that the contributions of meson and quark exchange / KEK[6]hasmadeourunderstandingofΣ-nucleuspoten- h are different from those in NN interaction, then it may tialastepforward.Inthedistortedwaveimpulseapproxi- t give an opportunity to separate or distinguish them. For - mation(DWIA)analyses,itissuggestedthattherepulsive l example, Λ does not couple with pions directly then the c real potential of 90 MeV or more would be necessary to u strength of the middle range central attraction would be reproducetheexperimentalspectra[6].Sincethisveryre- n differentinmesonandquarkexchangepictures.Thesitua- pulsive Σ potential in nuclei cannot be supported by any : tionwouldbeclearerforΣ hyperons.Duetotheisovector v theoretical models, it is necessary to verify the validity natureofthe diquarkpairinΣ,the Pauliblockingeffects i of approximations and prescriptions in the reaction the- X betweenquarksappearinamoredirectmannerinΣN in- ory currently used for the analysis. Recently, Harada and r teraction.TheΣ potentialinnuclearmatteratsaturation Hirabayashi pointed out that on-shell condition in Fermi a density is predicted to be around+30MeV (repulsion)in averaging (optimal Fermi averaging) for t-matrix of ele- a quark cluster model YN potential [1], while the poten- mentary process is important to understand the shape of tial is less repulsive or attractive in many of the hadronic theQFspectrum[8],andtheiranalysissuggeststhatΣ−- YN potential models. nucleuspotentialhastherepulsivefeatureinthecenterof Hyperon potential in nuclear matter is also important nuclei[9].ASemiClassicalDistortedWave(SCDW)anal- to understandcompactastrophysicalobjects suchasneu- ysisbyKohnoetal.[10]alsosuggeststherepulsivenature. tron stars. The Λ hyperon-nucleus potential has been in- In these works,while the former is basedon a fully quan- vestigated in the bound region extensively, and its depth tum treatment, the nucleon and hyperon potential effects has been known to be about 30 MeV [2]. For Σ hyperon, are included in the latter. If the on-shell condition is im- the bound state spectroscopy is difficult, because of the portantandthedifferenceoftheinitial(nucleon)andfinal strong Λ conversion, ΣN → ΛN. In 4ΣHe, which is the (hyperon)potentialsislarge,itwouldbenecessarytotake only case of observed Σ (quasi) bound state [3], the cou- account of the effects of the kinematics modification due pling effects is strong and the repulsive contribution in tothepotentialenergyintheon-shellconditionoftheele- the T =3/2, 3S1 channel is suppressed, then it does not mentaryprocessinnuclearenvironmentwithinaquantum stronglyconstrainthe Σ potentialin nuclearmatter.The mechanicalframeworkinordertounderstandthehyperon analysisofΣ−atomicdatasuggestedaΣ−-nucleuspoten- production spectra in the QF and bound state region. tialhavingashallowattractivepocketaroundthenuclear surface and repulsion inside the nucleus [4], but it is diffi- In this paper, we investigate the hyperon-nucleus po- cult to determine the Σ−-nucleus optical potential in the tential through hyperon production spectra by introduc- innerpartofnucleusfromtheatomicdataunambiguously. ing the local optimal Fermi averagingt-matrix in DWIA, which is expected to possess both of the merits in the Send offprint requests to: [email protected] previous two works. 2 Maekawa et al.,:Analysis of (π±,K+) and (K−,K+) hypernuclearproduction spectra in DWIA 2 Model; Green’s function method and Local InobtainingLOFAt-matrix,wedefinethenucleonand optimal Fermi averaged t-matrix hyperon energy in nuclei and hypernuclei containing the nuclear and hypernuclear potential effects, The Green’s function method has been widely applied to p2 tthhee aandavalynstiasgoefthoyptreerantuctlheearcorenatcintiuounms. aTshiwsemlleatshobdouhnads EB(r)=qp2B +m2B+2mBVB(r)∼mB+2mBB+VB(r), (6) state region on the same footing. In order to include the where B = N or Y. These treatments enable us to in- effects of nucleon Fermi motion and nucleon/hyperonpo- clude the potential effects naturally through the effective tentials into optimal Fermi averaging t-matrix, we intro- mass m∗2 = m2 +2m V (r). Consequently, the LOFA duce the Local Optimal Fermi Averaging t-matrix (LO- B B B t-matrix have the dependence on the collision point r FAt). throughnucleonand hyperonpotentials, V (r). It should B UsingtheFermi’sgoldenrule,thedifferentialcrosssec- be noted that the LOFA t-matrix is equivalent to ordi- tion of (π,K) reaction is written as [11], naryoptimal Fermiaveragingt-matrix when potential ef- fects are switched off. Product of incoming and outgo- d2σ p E = K K |T |2δ(E +E −E −E ), ing distorted meson waves is evaluated in the eikonal ap- dEKdΩK (2π¯h2)2vπ Xf fi π T K H proximation.In(π±,K+)and(K−,K+)reactionsat1.20 GeV/c and 1.65 GeV/c, the isospin averaged cross sec- (1) where the subscripts T and H represent target and pro- tions are assumed to be σ¯Nπ±=34mb, σ¯NK+=18mb and duced hypernucleus, respectively, and v = p /E is the σ¯NK−=40mb, σ¯NK+=30mb, respectively. π π π incident particle velocity. From the angular momentum algebra, we can get the 3 Results partialwavedecompositionof the strengthfunction S(E) in the Green’s function method [12], 3.1 Λ production spectrum d2σ p E K K = S(E) , (2) dEKdΩK (2π¯h2)2vπ First, we calculate the Λ production spectrum using the wellknownparametersfromtheboundstatespectroscopy, S(E)= W[αβα′β′]SαJβMα′β′(E) , (3) i.e. a typical depth of about 30MeV [2], in order to judge XJMXαβ αX′β′ the validity of the present method. 1 Inthecalculation,wehaveassumedtheonebodyWoods- SαJβMα′β′(E)= −πImZ r2drr′2dr′˜jJ∗M(r)φ∗α(r)t¯∗(r) Saxon type hyperon-nucleus optical potential, × GJαMβα′β′(E;r,r′)t¯(r′)˜jJM(r′)φα′(r′) . (4) ¯h2l·s 1df(r) U (r)=(VY +iWY)f(r)+VY +VY(r), Y 0 0 ls (m c2)2 r dr C Heresubscriptsαandβstandforthequantumnumbersof π (7) nucleon and hyperon states, respectively. The coefficient W[αβα′β′] represents the hypernuclear statistical factor. withf(r)=1/(1+er−dR), R=r0(A−1)1/3,whereVlYs The function ˜jJM is called distorted Bessel function [13], and VCY(r) denote the spin-orbit strength and Coulomb potential, respectively. φ (r) is the radial wave function of target nucleon, and α Figure 1 shows the calculated results of Λ produc- J is the total spin of hypernuclei. The Green’s function Gαβα′β′(E;r,r′) contains the hypernuclear Hamiltonian tion spectrum 28Si(π+,K+) at pπ=1.20 GeV/c, θ = 6◦ in comparison with experimental data. The experimen- H then we can get the information of optical potential H tal data are taken from E438 at KEK. Solid line shows U between hyperon and nucleus. Y DWIAresults with LOFAt-matrixwith standardparam- ItwaspointedoutbyHaradaandHirabayashi[8]that onshellkinematicsintheFermiaveragingprocedureroughlyeters V0Λ = −32MeV, VlΛs = 4MeV, r0 = 1.1fm and d = 0.6fm. We find good agreement of the calculated re- decide the shape of the QF spectrum and its prescription sults with data in both of QF and bound state regions. ofthet-matrixisimportant.Wewouldliketoextendtheir ideabyincluding potentialeffects.Here,weintroduceLo- cal Optimal Fermi Averaging t-matrix (LOFAt), 3.2 Σ− production spectrum dp t(s,t)ρ(p )δ4(Pµ(r)−Pµ(r)) t¯(r;ω,q)≡ N N f i , (5) DWIA analysis in the ordinary on-shell Fermi averaging R dpNρ(pN)δ4(Pfµ(r)−Piµ(r)) t-matrix treatmentcanreproduce the (π−,K+) QFspec- R trum shape with the Batty’s density dependent (DD) po- where Pµ (r) denote the total four momenta in the ele- tential and Woods-Saxon potential with 30 MeV repul- i,f mentary initial and final two-body states. We adopt the sion [8], but the absolute values are different in these cal- Fermidistributionfunctionforthetargetnucleonmomen- culations.Itisdesirabletodescribethespectrumshapeas tumdistributionρ(p )andparametersaretakenfrom[11, wellastheyield,andtheLOFAt-matrixwouldbehelpful N 14]. for this purpose. Maekawa et al.,:Analysis of (π±,K+) and (K−,K+) hypernuclearproduction spectra in DWIA 3 20 18 28Si(π+,K+) LOOFFAAtt++DDWWIIAA 2 28Si(π-,K+) 0 eV) 16 pπ=1.20(GeV/c), 6(deg.) eV) pπ=1.20 (GeV/c) -10 r M 14 r M 1.5 θ=6.0 (deg.) b/s 12 b/s µdE ( 1 80 0d5/2-1 µdE ( 1 -50 Batty-DD Ωd 6 0p1/2-1 Ωd -30 +10 2σd/ 24 0p3/2-1 2σd/ 0.5 +90 0 0s1/2-1 0 0 50 100 150 200 0 50 100 150 200 -BΛ (MeV) -BΣ (MeV) 7 Fig. 2. Differential cross section of (π−,K+) reaction on LOFAt+DWIA V) 6 28Si(π+,K+) OFAt+DWIA 2so8SliidtlainrgeesthaotwtshreesinucltidoefnBtamttoym’senDtDumpootfenpπti=al1w.2itGheLVO/cF.ATth+e e M 5 pπ=1.20(GeV/c), 6(deg.) dΛ DWIA,OtherlinearecalculatedresultswithLOFAt+DWIA b/sr 4 0d5/2-1 withpotentialdepthofV0=-50,-30,-10,0,+10,+90MeV(up µ todown),respectively.Imaginary part isfixedtobe-20MeV. E ( 3 pΛ d Ω d 2 2σd/ 1 sΛ 000pps113///222---111 miwmhapitloeedrtt,ahwnehtce[rr1oe7s]ts.hseeccotinotnribauttlioownefrropmK+mureltgiisotnepispruoncdeesrseesstiis- 0-40 -30 -20 -10 0 10 20 In Fig. 4, we show the calculated K+ spectrum in the -BΛ (MeV) boundstateregionof(K−,K+)reactionson27Aland12C Fig. 1. The Λ hypernuclear production spectrum targets with the same potential parameters [(V0Ξ,W0Ξ)= 28Si(π+,K+)intheQFregion(upperpanel)andinthebound (−15MeV,−1MeV)] which explains the QF spectra. We state region (lower panel) at pπ=1.2 GeV/c. Solid line shows have assumed an experimental resolution of 2 MeV. We LOFAt + DWIA results using Λ-nucleus potential depth of findthatboundstatepeaksarepopulatedselectivelyasin 32 MeV. Dotted line shows the Optimal Fermi Averaging t- the case of (π,K) reaction due to high momentum trans- matrix (OFAt) DWIA result. Other lines show hole contri- fer, and these peaks can be identified if the experimental bution with 0s1/2, 0p3/2, 0p1/2 and 0d5/2, respectively, in resolution is improved to be around 2 MeV. LOFAt + DWIA. 4 Summary In Fig.2, we show the Σ− production QF spectrum 28Si(π−,K+) at p =1.2 GeV/c. Calculated results using π We have studied hyperon-nucleus potentials through the Woods-Saxontype optical potentials and Batty’s DD po- QF spectra in (π+,K+) , (π−,K+) and (K−,K+) reac- tential [4] are compared with experimental data [6]. It turns out that experimental data on 28Si target is rea- tionsusingdistortedwaveimpulseapproximation(DWIA) with Local Optimal Fermi Averaging t-matrix (LOFAt) sonably well reproduced in Woods-Saxon type potential treatment. In addition to the on shell kinematics [9], nu- with small repulsion. In the Batty’s DD potential, calcu- cleon and hyperon potential effects are included in the lated result agrees with the experimental data in a wide FermiaveragingprocedureinLOFAt.Wehavefoundthat excitation energy range. We can see the large potential LOFAttreatmentisabettertooltodescribetheQFspec- dependence in the case of LOFAt + DWIA. trumthanstandardFermiaveragingprescriptions.Incom- parison with the Λ production data, we find good agree- ment in both of QF and bound state regions with LOFAt 3.3 Ξ− production spectrum + DWIA. From the comparison with the Σ− production data,LOFAt+DWIAresultpreferslessrepulsiveΣ− po- ThedepthoftheΞ−-nucleuspotentialhasbeensuggested tentialthanthosesuggestedinothertheoreticalmodels[8, to be around 15 MeV from the analysis of the (K−,K+) 10]. This difference may come from the kinematics modi- spectrum in the bound state region[15]. In that analysis, ficationbythe largedifferenceinthe initial(nucleon)and the observed yield in the bound state region is compared final(hyperon)statepotentials.Finally,weinvestigatethe with the calculatedresults,since the experimentalresolu- Ξ− productionspectrum,andcalculatedresultsarefound tion is not enough to distinguish the bound state peaks. to be in good agreement with the experimental QF data − − Figure 3 shows calculated results of Ξ production spec- using the Ξ -nucleus potential depth of 15 MeV. We be- tra in LOFAt + DWIA with potential depth of 15 MeV lievethatthe presentmodificationwouldprovideabetter in comparison with experimental data [16]. Calculated tool for the analysis of spectrum in the QF as well as the curves reproduce the experimental data systematically , bound state region. 4 Maekawa et al.,:Analysis of (π±,K+) and (K−,K+) hypernuclearproduction spectra in DWIA 80 Pb Al 60 2.0 d Ξ 40 U =-15 (MeV) ) Ξ c 20 V/ 1.5 e 0d -1 c) 0 M pΞ 5/2 V/ 60 Ag 0 1.0 5 s e / Ξ 0M 40 b/sr 0.5 0s -1 0p1/2-1 5 20 µ 1/2 0p -1 / ( 3/2 r Ω s 0.0 b/ 600 Cu pd C µ d Ω( σ/ 1.0 d 40 2d pΞ dp 20 0.5 sΞ 0p3/2-1 / 2σ 0 0s1/2-1 d 40 Al 0.0 -20 -15 -10 -5 0 5 10 20 -BΞ- (MeV) 0 20 C Fig. 4. The Ξ−-hypernuclear production spectrum in the bound region at pπ=1.65 GeV/c and 6 (deg.) on Al (upper 0 panel)andC(lower panel)targets with LOFAt+DWIA.Po- 1200 1100 1000 tential parameters are used same as Fig. 3. + K Momentum (MeV/c) Theor. Phys. Suppl. 117 (1994), 227; J. Mares, E. Fried- man,A.GalandB.K.Jenning,Nucl.Phys.A594(1995), Fig. 3. The calculated Ξ−-hypernuclear production spec- 311. tra in the QF region at pπ=1.65 GeV/c and 6(deg.) on C, 5. R. H.Dalitz and A.Gal, Phys. Lett. B 64 (1976), 154. Al, Cu, Ag and Pb targets in comparison with data [16]. 6. H. Noumi et. al., Phys. Rev. Lett. 89 (2002), 072301 [Er- Solid lines show LOFAt + DWIA results with (V0Ξ,W0Ξ) = ratum:Phys.Rev.Lett.90(2003) 049902.]; P.K.Sahaet. (−15MeV,−1MeV), and the experimental resolution is as- al.,Phys. Rev.C 70 (2004), 044613. sumed to be 10 MeV. 7. N. Kaiser, Phys.Rev.C 71 (2005), 068201. 8. T.HaradaandH.Hirabayashi,Nucl.Phys.A744(2004), 323. Acknowledgements 9. T.HaradaandY.Hirabayashi,Nucl.Phys.A767(2006), 206; T. Harada and Y. Hirabayashi, Nucl. Phys. A 759 We would like to thank Prof. A. Gal, Prof. T. Harada (2006), 143. andProf.M.Kohnoforvaluablediscussions.Thisworkis 10. M. Kohno et al.,Prog. Theor. Phys. 112 (2004), 895. 11. E. H. Auerbach, A. J. Balitz, C. B. Dover, A. 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