EPJ manuscript No. (will be inserted by the editor) FZJ–IKP(TH)–2007–03 + ΛN scattering length from the reaction γd K Λn → A. Gasparyan1,2, J. Haidenbauer3, C. Hanhart3, and K. Miyagawa4 7 1 Instituteof Theoretical and Experimental Physics, 117259, B. Cheremushkinskaya25, Moscow, Russia 0 2 Gesellschaft fu¨r Schwerionenforschung(GSI), Planck Str.1, D-64291 Darmstadt, Germany 0 2 3 Institut fu¨r Kernphysik(Theorie), Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany 4 Simulation Science Center, OkayamaUniversity of Science, 1-1 Ridai-cho, Okayama700-0005, Japan n a J Received: date/ Revised version: date 0 3 Abstract. The perspects of utilizing the strangeness-production reaction γd K+Λn for the determina- → tion of the Λn low-energy scattering parameters are investigated. The spin observables that need to be 1 measuredinordertoisolatetheΛnsinglet(1S0)andtriplet(3S1)statesareidentified.Possiblekinematical v regions where the extraction of theΛn scattering lengths might befeasible are discussed. 0 9 PACS. 11.55.Fv Dispersion relations – 13.75.-n Hadron-induced low- and intermediate-energy reactions 0 – 13.75.Ev Hyperon-nucleoninteractions – 25.40.-h Nucleon-inducedreactions 1 0 7 1 Introduction Inourpreviouspapers[2,3]wedevelopedamethodfor 0 a quantitative study of the final-state interactions in pro- / h The experimental information on the ΛN interaction at duction reactions with large momentum transfer such as, -t lowenergiesisratherpoorand,moreover,ofratherlimited e.g.,pp K+Λporpp K0Σ+p.Ingeneral,themethod cl accuracy [1]. Specifically, the available data do not allow canbea→ppliedwhenthe→reisastronginteractioninoneof u a reliable determination of the ΛN low-energy (1S , 3S ) the producedtwo-bodysubsystems,andinadditionthere 0 1 n scattering parameters. Therefore, it has been suggested are no other channels with near-by thresholds that cou- v: in the past to consider inelastic processes where the ΛN ple strongly to that system. Also, the interaction in the i system is produced in the final state and to exploit the other final two-body subsystems should be weak. Then it X occuring final-state interactionfor the extractionof those is possible to reconstruct the elastic two-body (ΛN, say) r scattering parameters. With this aim in mind, we [2,3], amplitude (or at least its threshold value – the scatter- a but also other groups [4,5], have recently looked at the ing length) via the invariant mass dependence of the pro- reaction pp K+Λp which can be studied experimen- duction amplitude in the region where the relative ΛN tally at the →COSY facility in Ju¨lich and where concrete momentum is small. The idea is to separate the different experiments have been already performed [6,7,8,9,10]. momentum scales appearingin the problem. In fact there In the present paper we want to investigate the per- are three scales one has to deal with: by assumption—we spects of utilizing another strangeness-production reac- lookonly atverystrong finalstate interactionsleading to tion, namely γd K+Λn [11,12,13,14,15,16,17,18,19, large scattering lengths—a very small scale given by the 20],forthedeterm→inationoftheΛN low-energyscattering inverse of the scattering length, 1/a, of the relevant final- parameters. In this case pertinent experiments have been state interaction, the inverse range of forces in the case announcedalready long time ago at CEBAF[21], but are of the elastic scattering, which is usually larger then the also possible at ELSA in Bonn [22], at the present JLAB former scale, and – the largest scale – the inverse range facility [23], and the future MAMI-C project in Mainz of the production operator.From the point of view of an- [24]. In this paper we want to discuss the differences and, alytical properties of the amplitude the latter two scales in particular, the merits but also possible disadvantages are roughly given by the corresponding closest left hand ofconsideringthephoton-inducedstrangenessproduction. singularities. It is clear then that in the case of elastic Wealsoinvestigatethespindependenceoftheproduction scatteringthelocationofthosesingularitiesisdetermined amplitude and identify those spin-dependent observables bythemassofthe exchangedmeson,whereasforthe pro- thatneedtobemeasuredtoenableaseparationofthe1S duction reaction it is fixed in most cases by the value of 0 and3S partialwaves.Finally,wepresentquantitativere- the required momentum transfer q [25]. The production 1 sultswithinamodelcalculationforoneoftheobservables amplitude itself is free of the left hand singularities of inquestioninorderto demonstratethe kindofsignalone the elastic amplitude, but has the same right hand cut. could expect in a concrete experiment. Dispersion theory enables to factorize the left hand sin- 2 A.Gasparyan et al.: ΛN scattering length. gularities from the elastic amplitude and to represent the to produce the KΛN system at threshold is around 600 production amplitude as a product (see Refs. [26,27,28]) MeV/c.However,incontrasttotheNN inducedreaction, here a new small scale might enter the reaction depend- 1 ∞ δ(m′2) ing on the kinematics: for forward going kaons at suffi- A(s,t,m2)=exp dm′2 "π Zm20 m′2−m2−i0 # coinelnytlbyyhtihghe semnearlglybitnhdeinignteenrmeregdyiaotfetnhuecdleeounteirsonoffb–esfhoerlel Φ(s,t,m2), (1) the photoncouples.Thenquasifree productiondominates × thereactionandthedispersionintegralmethodcannotbe where the exponent contains the full information on the appliedanymore.Therefore,one hasto impose additional righthandsingularitiesoftheelasticamplitudeandthere- kinematicalconditionstoensurethatquasifreeproduction mainingfactorΦpossessesonlylefthandsingularitiesand, is not allowed or at least strongly suppressed. therefore,incaseoflargemomentum-transferreactions,is InSec.2weconsiderthespinstructureofthereaction only weakly dependent on m2, the invariant mass of the amplitude for γd K+Λn and we derive those spin ob- consideredtwo-bodysubsystem,e.g.ofΛN.InEq.(1)δis → servablesthatneedtobemeasuredinordertoseparatethe theelasticΛN phaseshiftandm =m +m .sandtare 0 Λ N ΛN spin-singlet and spin-triplet states. In Sec. 3 we esti- thetotalc.m.energysquaredandthe4-momentumtrans- mate uncertainties of the extracted ΛN scattering length fer (from one of the initial particles to the kaon) squared, that could arise from the interaction in the other final respectively. In a more realistic situation when inelastic states (KΛ, KN). Concrete results for the spin observ- channels are present at higher energy (as is the case for able that projects on the spin-triplet state are presented ΛN due to the openingoftheΣN channel)one canwrite in Sec. 4, based on a model calculation by Yamamura et down a similar expression where the integration involves al. [15]. Furthermore, as a test we apply the dispersion only the range where the final state interaction is strong integral method described above to those specific model [2] predictions for extracting the Λn 3S scattering length. 1 1 m2max δ(m′2) We also discuss issues concerning the kinematical regions A(m2)=exp dm′2 where experiments should be preferably performed in or- "π Zm20 m′2−m2−i0 # der to ensure a reliable determination of the scattering Φ˜(m2,m2 ) . (2) lengths. Specifically, we identify the kinematical condi- × max tions, where the quasifree production is not allowed or Here Φ˜(m2,m2 ) is also a slowlyvarying function of m2 strongly suppressed and where then the dispersion inte- max providedthatδissufficientlysmallinthevicinityofm . gralmethodcanbereliablyapplied.Thepapercloseswith max NeglectingthemassdependenceofΦ˜(m2,m2 )thescat- a brief Summary. max tering length a in a specific partial wave S can be then S expressed in terms of the differential partial production cross section σ : 2 Spin observables S 1 m +m Λ N aS = lim An important issue for the extraction of the low-energy m2→m20 2π (cid:18) √mΛmN (cid:19) scattering parameters is the separation of the different m2max m2 m2 spin components in the ΛN system. In Ref. [2] we have P dm′2 max− shown that by measuring specific spin observables in the × Zm20 sm2max−m′2 reaction NN NKΛ one can project on the production → 1 1 d2σ of spin-singlet or spin-triplet states. Let us now discuss S log . × m′2 m2 (m′2 m2) p′ dm′2dt what observables can be used to disentangle the different − 0 − (cid:26) (cid:18) (cid:19)(cid:27) spin states for the reaction γd K+Λn. (3) → p We startfromthegeneralformforthe matrixelement of the process γd K+Λn : A detailed analysis of the uncertainties of Eq. (3) has → shown that the theoretical error of the extracted value M = A(ǫ ǫ )+B (ǫ ǫ ) forthescatteringlengthisoftheorderof0.3fm[2].Note, d· γ · d× γ however, that a possible influence of meson-baryon inter- 2 +C(ij) ǫ iǫ j +ǫ jǫ i δij(ǫ ǫ ) , (4) actions in the other two-body subsystems has not been d γ d γ − 3 d· γ (cid:18) (cid:19) explicitly included into this estimate so far—we will do this below. whereǫ andǫ arethepolarizationvectorsofthephoton γ d Inthispaperwepresentresultsforanotherstrangeness- and deuteron, respectively. If we assume the ΛN system production process that is a possible candidate for the to be in an S–wave, then we have only the (normalized) extraction of the ΛN scattering length, namely γd initialmomentumpˆandtheoutgoingkaonmomentumq′ K+Λn. This reaction satisfies formally the main cond→i- available to construct the structures for the coefficients. tion needed for the dispersion integral method to be ap- If the final ΛN system is in a spin triplet state we have plied: The momentum transfer in this reaction is large in addition S′, the spin vector of the final state, that has comparedto the typicalrangeofthe finalstateΛN inter- to appear linearly in the coefficients A, B and C. Parity action.Therequiredc.m.momentumoftheinitialphoton conservation demands that both pˆ and q′ appear either A.Gasparyan et al.: ΛN scattering length. 3 in an odd number or in an even number. Thus, we have – ǫ p. In this case C(ij)t vanishes, and the spin- d for the spin singlet case: sing⊥letamplitudes (proportionaltoA andC(ij)s)are s symmetric with respect to an interchange of ǫ and As =as, d ǫ ,whereasthespin-tripletamplitude(proportionalto Bs =bs(q′ pˆ), Bγt) is antisymmetric. This allows to construct combi- × C(ij)s =cspˆipˆj +c q′iq′j +c q′ipˆj . (5) nationsofspinobservablescontainingonlyspin-singlet 1 2 3 orspin-tripletcontributions(twocombinationsforeach On the other hand, for the spin triplet final state we get spin), namely At = atS′ (q′ pˆ), dσ Bt = bt1S′·+(S×′·pˆ) bt2pˆ+bt3q′ +bt4q′(S′·q′) (2+√2T200 −√3(T2l2+T2l−2))dmΛn0dΩq C(ij)t = ++Sc(′St4·S(′q×′i′(×qpˆ)′ipˆ×)c(cid:0)pt7ˆcq)t1j(cid:0)pˆ′j+ipˆ+j(S+ct8′pcˆ×jt2qq′.i(cid:1)′q)′ij(cid:0)+ct5cqt3′qj′i+pˆjc(cid:1)t6pˆj(cid:1)(6) =∼−−√a3s(+√232T(c1c0s1++(cTs3(2lp2ˆq+)T+2l−cs22q))2d)m2dΛσn0d(Ω10q) Notethatthecoefficien(cid:0)tsas,bs,etc.(cid:1)arefunctionsof(q′)2 (cid:12) (cid:12) (cid:12) dσ (cid:12) andq′ pˆ.Asignificantsimplificationallowingonetosep- (2+√2T0 +(cid:12)√3(Tl +Tl )) 0 (cid:12) arated·ifferentspinstatescanbeachievedifweassumeq′ 20 (cid:12) 22 2−2 dmΛndΩq (cid:12) tobealongthebeamdirection(inparticularthenBs and =√3( √2Tc +(Tl +Tl )) dσ0 At vanish). This means that one considers the situation − 10 22 2−2 dm dΩ Λn q wherethekaonisemittedeitherinforwardorinbackward bt +bt +bt(pˆq)+btq2 2, (11) direction. Then we can look at two different cases: ∼| 1 2 3 4 | – ǫ pˆ. As real photons are transverse (λ = 1), As with d γ || ± aannddlCon(igji)tsuddionanlottarcgoenttpriobluartiez.aTtihonus(λfor=re0a)l,opnhloytothnes T2l±2 = Ω2±2;λ1λ3Ωλl2λ4Mλ1λ2Mλ∗3λ4 d spin-triplet state contributes through bt and C(ij)t. λ1,λX2,λ3,λ4 1 ∗ Hence this is the case where we can study the spin- / M M λ1λ2 λ1λ2 triplet final-state. The observable that provides access λX1,λ2 to the longitudinal target polarization is Tc = Ω Ωc M M∗ 10 10;λ1λ3 λ2λ4 λ1λ2 λ3λ4 dσ (1 √2T0) 0 bt +ct +(ct +ct)(pˆq) λ1,λX2,λ3,λ4 − 20 dmΛndΩq ∼| 1 8 6 7 / M M∗ . (12) + ctq2 2, (7) λ1λ2 λ1λ2 5 | λX1,λ2 where T200 is defined by Here the upper index (c or l) refers to circularly or linearly polarized photons. Therefore, the only possi- T0 = Ω M M∗ 20 20;λ1λ3 λ1λ2 λ3λ2 bility toobtainapurespin-singletΛN final-stateis to λ1X,λ2,λ3 perform a double polarization experiment. ∗ / M M (8) λ1λ2 λ1λ2 λX1,λ2 3 Influence of the meson-baryon interaction withλ ,λ beingthedeuteronspinprojectionontothe 1 3 photonmomentumandλ2,λ4 thecircularpolarization AsmentionedintheIntroduction,inthederivationofEq. of the photon. The operators Ωij are defined by (3) we assumed that the interactions in the other two- body subsystems in the final state are small. This con- 1 0 0 1 0 0 3 1 cerns the KΛ and the KN systems. The reason was that Ω = 0 0 0 , Ω = 0 2 0 , 10 2  20 2 −  for excess energies in the order of 100 to 200 MeV, the r 0 0 1 r 0 0 1 − kinetic energy in those subsystems is large and does not     vary strongly with the relativeΛN momentum, when the 0 0 1 0 0 0 latter system is considered near its threshold for the ex- Ω22 =√3 0 0 0 ,Ω2−2 =√3 0 0 0 , tractionoftheΛN scatteringlength,withtherelativeΛN     0 0 0 1 0 0 momentum, when the latter system is considerednear its     threshold for the extraction of the ΛN scattering length, and therefore the energy dependence of the production 1 0 0 1 Ωc = , Ωl = − , (9) amplitude should not change significantly. It was noted, 0 1 1 0 (cid:18) − (cid:19) (cid:18)− (cid:19) however, in [29] that one should still be cautious because see, e.g., Ref. [30]. A complete description of the po- of possible effects due to the presence of N∗ resonances larizationobservablesforsuchkindofreactionscanbe in the KΛ system. Therefore, in the following we are go- found in Ref. [31]. ing to derive some qualitative estimates as to how large 4 A.Gasparyan et al.: ΛN scattering length. the effect of such resonancescan be for the extracted ΛN scattering length. Note that this issue is relevantfor both 0,2 reactions γd KYN and pp KYN, although the de- → → tails might differ. In particular the relative importance of contributions from resonancesand of the backgroundwill ]0,1 m depend on the specific reaction mechanisms. We assume f [ herethatthe energydependence oftheproductionampli- a δ 0 tude is modified by a factor 1 -0,1 Φ= , (13) M2 M2 +iΓ M ΛK − R R R -0,2 averaged over the ΛN c.m. angle, i.e. we consider only 0 200 400 600 800 1000 ε [MeV] theresonancecontributionbutneglectabackground.Here M and Γ are the Breit-Wigner mass and width of the R R resonance,respectively.Ingeneralanontrivialinterference Fig.1.Errorintheextractedscatteringlengthduetothepres- of the resonance amplitude with the backgroundcan pro- enceof aresonancestructureintheKΛsubsystemdepending duce a stronger mass dependence of the production am- ontheavailableexcessenergyǫ.Thesolidlineshowstheresult plitude. On the other hand the sum of all partial waves ofthedispersion integralwhile thedashedline correspondsto intheΛK systemwillhavethe oppositeeffect.Therefore, an approximation, cf. discussion in Sec. 3. we believe that the above approximation is reasonable in order to estimate the uncertainty in the extracted ΛN over x and removing a constant prefactor one obtains scattering length induced by the presence of resonances in the ΛK system. For simplicity we consider only an S- Cp2 4k2p2 wave resonance. Expanding Φ in terms of the ΛN c.m. <Φ> 1+ + . ∼ ∆M2+iΓ M 3(∆M2+iΓ M )2 momentum p one gets R R R R (16) 1 Φ Fortheproductionamplitudesquaredonegetsthefollow- ≈ ∆M2+iΓ M Cp2 2kpx R R− − ing mass dependence 1 ≈ ∆M2+iΓ M 2Cp2∆M2 R R A2 1 + Cp2+2kpx 4k2p2x2 | | ∼ (∆M2)2+Γ2M2 1+ + , R R × (cid:18) ∆M2+iΓRMR (∆M2+iΓRMR)2(cid:19) + 8k2p2((∆M2)2−ΓR2MR2). (17) (14) 3((∆M2)2+Γ2M2)2 R R where C = 2+ mΛ−mn s−m2K−(mΛ+mn)2 +1 , k is the ThecorrespondingcontributiontotheΛN scatteringlength mn 2mΛ(mΛ+mn) is (see Ref. [2]) kaon momentum in the c(cid:16).m. system of ΛN (at(cid:17)p = 0), x is the cosine of the angle between the kaon and Λ in the C∆M2 4k2((∆M2)2 Γ2M2) δa p + − R R , same system and ∼ max (∆M2)2+Γ2M2 3((∆M2)2+Γ2M2)2 (cid:20) R R R R (cid:21) (18) ∆M2 =M2 (p=0) M2 =m2 +m2 + ΛK − R K Λ (s−m2K −(mΛ+mN)2)mΛ M2 . (15) where pmax ≈ 200 MeV/c reflects the limit of the dis- m +m − R persion integral. It is easy to see that the result depends Λ N on two important scales: the resonance width (typically Note that the factor C is of the order of 2 for the consid- 150 200MeV)and∆M2 whichisdeterminedby theex- − ered excess energies up to several hundred MeV because cess energy. In order to obtain a rough idea for the order the mass difference between the Λ and nucleon is small. of magnitude of the corrections to the scattering length In order to estimate the effect that resonances may have let us put ∆M2 = 0. Then δa 4k2p /(3Γ2M2). If ∼ − max R R on the extraction of the scattering length we evaluated wetakeas atypicalexample the massofthe resonanceto the dispersion integral Eq. (3) for the amplitude given in be M = 1700 MeV and its width to be Γ = 150 MeV, R R Eq. (14). Corresponding results are shown in Fig. 1 as then k 400 MeV/c, which yields δa 0.1 fm. The a function of the excess energy. Obviously, the resulting absolute∼value of δa becomes smaller as∼∆−M2 increases. scattering length should be identical to zero if there is InFig.1theresultforδacalculatedbymeansofEq.(18) completely no influence. We see that the deviations due iscomparedtothe valueobtainedfromthe fulldispersion tosuchresonancesaresomewhatdependentontheexcess integral Eq. (3). The two curves turn out to be almost energy but amount to 0.2 fm at most. identical and, therefore, justify the use of our approxima- ± For a qualitativeunderstandingofthe roleofthe vari- tions made in Eqs. (14)-(18). We conclude that for such ousscalesitisinstructivetoproceedasfollows.Averaging excessenergieswheretheavailablephasespacefortheKΛ A.Gasparyan et al.: ΛN scattering length. 5 + system covers the resonance region one has to expect an K additional uncertainty of the extracted scattering length in the order of 0.2 fm—this has to be added to the one γ estimated previously leading to a total theoretical uncer- tainty of 0.5 fm. A more concrete quantitative statement can be made only by analyzing the actual experimental Λ Dalitzplots,whereoneshouldclearlyseewhetherthereis a strong dependence of the production amplitude on the p KΛ invariant mass or not. n d 4 Results and discussion As already said in the Introduction, the reaction γd K+Λn satisfies formally the main condition needed f→or Fig. 2. Diagramcorrespondingtothequasi-freekaonproduc- tion on theproton. the dispersion integral method to be applied: The mo- mentum transfer in this reactionis large comparedto the typical range of the final state YN interaction. The re- quired c.m. momentum of the initial photon to produce new,verysmallscaleintheproductionoperatorcausedby the YNK system at threshold is equal to 584 MeV/c. In thesmalldeuteronbindingenergy.Itisclearthatthispar- ticular production mechanism is dominant when the rel- order to be able to resolve the structure induced by the ative momentum of the two nucleons inside the deuteron ΛN interaction one needs at least data covering ΛN in- variantmassesintherangeof40MeVfromthethreshold, is not large. Thus, it influences primarily forward kaons cf.Ref.[2].Inthatworkwearguedalsothatthepertinent where then the ΛN system is moving in direction of the deuteron momentum in the c.m. system. Note that the experiments shouldbe performedpreferably atsomewhat higher total energies. Then there will be no distortion of peakisshiftedsomewhatawayfromverylowrelativeYN the signal within that 40 MeV range by the (upper) limit momenta because the photon cannot produce a Λ at rest on a proton at rest. of the available phase space. Moreover, effects from pos- sible interactions in the other final states (KN, KΛ) are In Fig. 3 we demonstrate the situation for a concrete kinematically better separated and should not influence modelcalculationwhereresultsfor1−√2T002atEγ =1300 the results for ΛN too much. MeV(ǫ=349MeV)areshownforkaonproductioninfor- An important kinematical constraint for the reaction ward direction (ΘK = 0o). Details of the model calcula- γd K+Λn is the limitation of the kaon angle to very tioncanbefoundinRef.[15].Letusmentionherethatthe forw→ard or very backward directions because only then calculation is done in the impulse approximation includ- a separation of the spin-singlet and spin-triplet states is ing the YN final-state interaction, utilizing the deuteron possible, as shown in Sec. 2. Unfortunately, there are in- wave function of the Nijmegen93 potential [32] and the dications that the total count rate could be very small in NSC97f YN force [33]. The elementary kaon-production the backward region. For example, the model calculation amplitude on the nucleon (γN KΛ) is derived from a → of Salam and Arenh¨ovel [17] suggests that the cross sec- setoftree-levelFeynmandiagramswhere the free param- tions drop dramatically in that angular range, cf. their eters have been fixed so that all available K+Λ, K+Σ0, Fig.12.This canbe easilyunderstoodwithin the impulse and K0Σ+ photoproduction data in the relevant energy approximation. In this case the spectator nucleon carries regionarereproduced[34]. Additionalproduction mecha- necessarilyalargemomentumforkaonsproducedinback- nisms involving, e.g., KN rescattering or the πN KΛ → ward direction and for such large momenta the deuteron process, considered in Ref. [17], are not included in this wave function is strongly suppressed. Additional produc- model. However, those mechanisms contribute predomi- tion mechanisms that involve two-step processes, consid- nantly for kaon production at backward angles [17] and ered also in Ref. [17], relax the situation somewhat. But are not so important for the forward angles we consider. still it could be difficult to perform measurements for the ThemodelcalculationpresentedinFig.3clearlyshows backward region and one has to wait for concrete exper- the presence of a bump due to quasi-free kaon produc- iments in order to see whether sufficient statistics can be tion. It occurs at fairly small Λn invariant masses and, achieved. therefore, makes a reliable determination of the Λn scat- Therefore, in the following we will concentrate on re- tering length from data impossible. Thus, for extracting sults for forward angles. However, in this case there is the Λn scattering lengthfrom forward-angledata one has a particular singularity of the production amplitude that to consider the reaction γd K+Λn for energies below → imposessomerestrictionsontheapplicationofourmethod. the appearance of this quasi-free peak, i.e. at excess en- It is the so-called quasi-free production mechanism (see ergies 40 50 MeV. First of all one should note that the Fig. 2). When the available excess energy in the ΛnK+ influence−of the ΛK interaction is not necessarily much system is around 90 MeV or more then the production of strongerthenathigherexcessenergies,sincewearewithin the ΛK+ system is possible on a single proton, resting in the resonance region in both cases. Therefore, the uncer- thedeuteronrestframe.Therefore,thiseffectintroducesa tainty of the method remains the same. This issue was 6 A.Gasparyan et al.: ΛN scattering length. addressed already in the previous section. Another prob- mechanism remains dominant even at low energies). In a lem is the limited phase space at low excess energies.The pureS wavesituationtheallowedoperatorstructuregiven phase space is proportional to q′ p′ dm . Since we inEqs.(5,6)simplifiessignificantlyand,inparticular,the Λn are interested in the region of sma×ll re×lative momenta p′ reactionamplitudedoesnotdependonthedirectionofthe in the Λn system in any case the suppression enters only kaon momentum anymore. Consequently, all expressions duetothefactorq′.Theconcreteeffectofthesuppression inSec.2arevalidforarbitraryangles.Therefore,onecan depends, of course, on the actual shape of the mass spec- work with observables integrated over the kaon angle in trum, but to get a rough estimate one can compare the the c.m. system which means that a significant enhance- ′ q values for different excess energies at the Λn threshold ment of the experimental statistics can be achieved. In (p′ = 0). For example, for the excess energy 50 MeV this additionthe angularintegrationallowsto getridofinter- value is about 2.5 times smaller than for 300 MeV. This ference terms between the S- and (small) P-waves that means that the suppression is not such a serious problem depend linearly on the kaon momentum so that possible in our case. influences from the energy dependence of the production operator,which is primarily due to terms linear in q′, are minimized. 30 V/Sr] 25 pFlSaIne wave e M nb/ 20 [K Ω d σ)d/dM/TΛ20n1105 b/MeV/Sr] 00,,65 PFlSaIne wave √1-2 5 Ω [nK0,4 ( d /Λn 0 M 0,3 0 10 20 30 40 50 60 d m -m -m [MeV] σ/ Λn Λ n d T)200,2 2 √1- 0,1 ( 0 y units] 1000 pFlSaIne wave 0 10 m2Λ0n-mΛ-mn [M3e0V] 40 50 bitrar 800 ar } [ σT)d/d{Phase space20 246000000 e} [arbitrary units] 340000 PFlSaIne wave √(1-2 00 10 20 30 40 50 60 hase spac 200 mΛn-mΛ-mn [MeV] d{P σd/ 100 T)20 2 Fig. 3. Top: Model results for the spin-dependent observable √1- 1−√2T002 at Eγ = 1300 MeV and ΘK = 0o as a function of ( 00 10 20 30 40 the Λn invariant mass mΛn. The dashed line is the impulse mΛn-mΛ-mn [MeV] approximation while the solid line is the full result including the Λn final-state interaction. Bottom: Same results but the phase-space factor is diveded out. Fig. 4. Top: Model results for the spin-dependent observable 1 √2T002 at Eγ = 850 MeV and ΘK = 0o as a function of − the Λn invariant mass mΛn. The dashed line is the impulse Aninterestingsideaspectatlowexcessenergiesisthat approximation while the solid line is the full result including then also the kaons should be predominantly produced in the Λn final-state interaction. Bottom: Same results but the an S-waverelativeto the Λn system(unless the quasifree phase-space factor is divededout. A.Gasparyan et al.: ΛN scattering length. 7 In Fig. 4 we show predictions ofthe model calculation ΛN scattering lengths with an accuracy similar to the [15] for the spin observable 1 √2T0 for E = 850 MeV reaction pp K+Λp. Thus, we believe that the photon- − 02 γ → (ǫ=41.5MeV)andforwardkaons.Thedashedlineisthe induced reaction is an interesting alternative for extract- result for the impulse approximation while the solid line ingtheΛN scatteringlengthsanditisalsoveryusefulfor corresponds to the full model including the ΛN FSI. It cross-checking results obtained from the purely hadronic is obvious how strongly the ΛN interaction modifies the strangeness production. observableforinvariantmassesclosetotheΛN threshold. When applyingthe dispersionintegralmethodto this ob- servable, cf. Ref. [2] for details, we obtain the scattering Acknowledgments length of -2.06 fm for the 3S partial wave. This has to 1 be compared with a = -1.70 fm of the YN model [33] A.G.thankstheInstitutfu¨rKernphysikattheForschungs- t used for the model calculation. Thus, the extracted scat- zentrum Ju¨lich for its hospitality during the period when tering length differs from the one utilized in the model the present work was carried out. Furthermore, he would calculationby about0.4fm,whichis inline with the un- liketoacknowledgefinancialsupportbythegrantNo.436 certainty that is expected for the method [2]. Specifically, RUS 17/75/04 of the Deutsche Forschungsgemeinschaft onehastokeepinmindthatthepresentmodelcalculation andbytheRussianFundforBasicResearch,grantNo.06- includes alsothe uncertaintiesdiscussedin Sec.3 because 02-04013. it is based on an elementary kaon-production amplitude that involves resonances in the ΛK channel [34]. References 5 Summary 1. G. Alexanderet al., Phys. Rev. 173, 1452 (1968). 2. 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