Ξ-nucleus potential for Ξ^- quasifree production in the ^9Be(K^-, K^+) reaction
aa r X i v : . [ nu c l - t h ] J a n J-PARC-TH-230
Ξ-nucleus potential for Ξ − quasifree production in the Be( K − , K + ) reaction Toru Harada
1, 2, ∗ and Yoshiharu Hirabayashi Research Center for Physics and Mathematics,Osaka Electro-Communication University,Neyagawa, Osaka, 572-8530, Japan J-PARC Branch, KEK Theory Center,Institute of Particle and Nuclear Studies,High Energy Accelerator Research Organization (KEK),203-1, Shirakata, Tokai, Ibaraki, 319-1106, Japan Information Initiative Center, Hokkaido University, Sapporo, 060-0811, Japan (Dated: January 8, 2021)
Abstract
We study phenomenologically a Ξ − production spectrum of the Be( K − , K + ) reaction at 1.8GeV/ c within the distorted-wave impulse approximation using the optimal Fermi-averaged K − p → K + Ξ − amplitude. We attempt to clarify properties of a Ξ-nucleus potential for Ξ − - Li, comparingthe calculated spectrum with the data of the BNL-E906 experiment. The results show a weakattraction in the Ξ-nucleus potential for Ξ − - Li, which can sufficiently explain the data in the Ξ − quasifree region. The strength of V Ξ0 = − ± W Ξ0 = − − p → Ξ n , ΛΛ transitions innuclear medium. It is difficult to determine the value of W Ξ0 from the data due to the insufficientresolution of 14.7 MeV FWHM. The energy dependence of the Fermi-averaged K − p → K + Ξ − amplitude is also confirmed by this analysis, and its importance in the nuclear ( K − , K + ) reactionis emphasized. PACS numbers: 21.80.+a, 24.10.Eq, 25.80.Hp, 27.20.+n ∗ Electronic address: [email protected] . INTRODUCTION Recently, Nakazawa et al. [1] reported the first evidence of a bound state of the Ξ − - N system which was identified by the “KISO” event in the KEK-E373 experiment. Thisresult supports that the Ξ-nucleus potential has a weak attraction of V Ξ ≃ −
14 MeV in theWood-Saxon (WS) potential, as suggested by previous analyses [2–4]. However, there stillremains an uncertainty about the nature of the S = − N interactionand Ξ N -ΛΛ coupling in nuclei due to the limit to the available data. More experimentalinformation is needed for the understanding of Ξ hypernuclei. Recently, Nagae et al. [5]have performed an accurate observation of the Ξ − production spectrum in double-chargeexchange reactions ( K − , K + ) on C targets at 1.8 GeV/ c in the J-PARC E05 experiment,and their analysis is now ongoing. The double-charge exchange reactions such as ( K − , K + )on nuclear targets provide to produce neutron-rich Ξ hypernuclei, e. g., the neutron excess of( N − Z ) / ( N + Z ) = 0.25 for a Ξ − - Li system, which is populated on Be. The behavior of theΞ − in the neutron-excess environment is strongly connected with the nature of neutron stars[6] in which the baryon fraction is found to depend on properties of hypernuclear potentials[7].Kohno [8] examined theoretically Ξ − production spectra for the quasifree (QF) interactionregion in the ( K − , K + ) reactions on Be and C targets in the semiclassical distorted wavemethod, using the Ξ-nucleus potential derived from the next-to-leading order (NLO) in chiraleffective field theory. However, it has shown that the calculated Ξ − QF spectrum on Beseems to be insufficient to reproduce the experimental data, so that quantitative informationon the Ξ-nucleus potential for Ξ − - Li ( − He) may be inreliable.In this paper, we investigate phenomenologically the Ξ − QF spectrum produced via the Be( K − , K + ) reaction at 1.8 GeV/ c in order to extract valuable information on the Ξ-nucleus(optical) potential for the Ξ − - Li system from the data of the BNL-E906 experiment [8, 9].We attempt to clarify properties of the Ξ-nucleus potential for Ξ − - Li and to understand amechanism of the Ξ − QF spectrum in comparison with the data [9]. Thus we demonstrate thecalculated Ξ − QF spectrum in the Be( K − , K + ) reaction within the distorted-wave impulseapproximation (DWIA), taking into account the energy dependence of the K − p → K + Ξ − amplitude in the optimal Fermi-averaging procedure [10, 11].2 I. CALCULATIONSA. Distorted-wave impulse approximation
Let us consider production of Ξ hypernuclear states in the nuclear ( K − , K + ) reaction.According to the Green’s function method [12] in the DWIA, an inclusive K + double-differential laboratory cross section of the Ξ − production on a nuclear target with a spin J A (its z -component M A ) [13–15] is given by d σd Ω dE = 1[ J A ] X M A S ( E ) (1)with [ J A ] = 2 J A + 1. The strength function S ( E ) is written as S ( E ) = − π Im X αα ′ Z d r d r ′ F α † Ξ ( r ) G αα ′ Ξ ( E ; r , r ′ ) × F α ′ Ξ ( r ′ ) , (2)where G αα ′ Ξ is a complete Green’s function for a Ξ hypernuclear system, F α Ξ is a Ξ productionamplitude defined by F α Ξ = β f K − p → K + Ξ − χ ( − ) ∗ p K + χ (+) p K − h α | ˆ ψ p | Ψ A i , (3)and α ( α ′ ) denotes the complete set of eigenstates for the system. The kinematical factor β denotes the translation from a two-body K − - p laboratory system to a K − -nucleus laboratorysystem. f K − p → K + Ξ − is a Fermi-averaged amplitude for the K − p → K + Ξ − reaction in nuclearmedium [11, 14, 15]. h α | ˆ ψ p | Ψ A i is a hole-state wave function for a struck proton in the target. χ ( − ) p K + and χ (+) p K − are distorted waves for outgoing K + and incoming K − mesons, respectively.The laboratory energy and momentum transfers are ω = E K − − E K + and q = p K − − p K + ,respectively; E K + and p K + ( E K − and p K − ) denote an energy and a momentum of theoutgoing K + (incoming K − ), respectively.Due to a high momentum transfer q ≃ c in the nuclear ( K − , K + ) reactionfor K + forward-direction angles of θ lab = 1.5 ◦ –8.5 ◦ at p K − = 1.8 GeV/ c , we simplify thecomputational procedure for χ ( − ) p K + and χ (+) p K − , using the eikonal approximation [15]. Toreduce ambiguities in the distorted-waves, we adopt the same parameters used in calculationsfor the Λ and Σ − QF spectra in nuclear ( π ± , K + ) and ( K − , π ± ) reactions [11, 16, 17]. Herewe used the total cross sections of σ K − = 28.9 mb for the K − N scattering and σ K + = 19.43b for the K + N scattering, and α K − = α K + = 0, as the distortion parameters. We alsotook into account the recoil effects, which are very important to estimate the hypernuclearproduction cross section for a light nuclear system [18], leading to an effective momentumtransfer having q eff ≃ (1 − /A ) q ≃ . q for A = 9.Recently, the authors [10] have found the strong energy dependence of the K − p → K + Ξ − reaction in the nuclear medium, together with the angular dependence for θ lab . Therefore,we emphasize that such behavior of f K − p → K + Ξ − plays a significant role in explaining theshape of the spectrum in the nuclear ( K − , K + ) reaction [10] as well as those in the nuclear( π ± , K + ) reactions [11, 16, 17]. Because f K − p → K + Ξ − provides to modify the spectral shapeincluding the Ξ − QF region widely, thus one must extract carefully information concerningthe Ξ-nucleus potential from the data.
B. Wave functions
For the Be target, the single-particle (s. p.) description of protons is assumed for sim-plicity. We simulate the calculated results of the s. p. energies of the nucleons and theroot-mean-square (rms) radius of h r N i / for their wave functions in the liner combinationof atomic orbits (LCAO) models [19] which well describe the ground state of Be(3 / − g . s . ; T = 1/2) as α + α + n clusters. Thus we compute the s. p. wave functions for the protonsin 0 p and 0 s , using the WS potential with R = r A / and a = 0.67 fm and omitting thespin-orbit potential; the strength parameter of the potential is adjusted to be V N = − R = 1 . A / = 3.33 fm which may be rather largedue to the structure of α + α + n . Here we obtain the s. p. energies of −
22 MeV for 0 p / and −
35 MeV for 0 s / , which are consistent with the data of the proton separation energiesin Be( p , 2 p ) reactions indicating widths of 8 MeV for 0 p and 13 MeV for 0 s [20, 21]. Thecharge radius of Be(3/2 − g . s . ) is estimated to be 2.53 fm, which is in good agreement with thedata of 2 . ± .
012 fm in electron elastic scatterings on Be [22]. Note that we must tunein the energies of the s. p. states for the protons as well as the matter rms radius of h r N i / for their wave functions, leading to the fact that the shape of the calculated QF spectrumin the ( K − , K + ) reaction sufficiently explain the data.To calculate the Ξ − QF spectrum in the nuclear ( K − , K + ) spectrum within the DWIA,we use the Green’s function method [12], which is one of the most powerful treatments in the4alculation of a spectrum describing not only bound states but also continuum states withan absorptive potential for spreading components. Because non-spin-flip processes seem todominate in the K − p → K + Ξ − reaction at 1.8 GeV/ c [23], hypernuclear configurations of[ J πC ⊗ j π Ξ ] J πB with J πB = 3/2 + , 5/2 + , 1/2 − , 3/2 − , 5/2 − , 7/2 − , 3/2 − , 5/2 − , · · · , are populatedin − He with T B =3/2; we take the Li core nucleus states with J πC = 2 + , 1 + , 2 − , and 1 − that are given in (3 / − ⊗ p − / , / ) + , + and (3 / − ⊗ s − / ) − , − configurations formed by aproton-hole state on Be(3 / − g . s . ), and the Ξ − with j π Ξ = ℓ Ξ ⊗ / + , 3/2 − , 1/2 − , · · · that are given in ℓ Ξ ≤
15 being enough to converge in calculations for the Ξ − spectrum.Here the components of Ξ n and ΛΛ channels are not considered explicitly because theΞ − p → Ξ n , ΛΛ transition processes may be described as a spreading imaginary potentialin Ξ bound and continuum regions. III. Ξ -NUCLEUS POTENTIAL The Ξ-nucleus final states are obtained by solving the Schr¨odinger equation (cid:20) − ~ µ ∇ + U Ξ ( r ) + U Coul ( r ) (cid:21) Ψ Ξ = E Ψ Ξ , (4)where µ is the Ξ-nucleus reduced mass, U Ξ is the Ξ-nucleus potential, and U Coul is theCoulomb potential. The Ξ-nucleus potential for Ξ − - Li is given by U Ξ ( r ) = V Ξ ( r ) + iW Ξ ( E, r )= [ V Ξ0 + iW Ξ0 g ( E )] f ( r ) (5)with the assumption of the WS form f ( r ) = [1 + exp { ( r − R ) /a } ] − , (6)where R = r A / and a denote a radius and a diffuseness of the potential, respectively. V Ξ0 is a strength parameter for the real part of the potential; W Ξ0 is a strength parameter for theimaginary part of the potential, which denotes the Ξ − absorption processes including theΞ − p → Ξ n , ΛΛ reactions. g ( E ) is an energy-dependent function which increases linearlyfrom 0.0 at E = E th (Λ) to 1.0 at E = 20 MeV with respect to Ξ − threshold, as often usedin nuclear optical models [24], where E th (Λ) = − . IG. 1: Real and imaginary parts of the Ξ-nucleus potential U Ξ for Ξ − - Li at the energy E =20 MeV, as a function of the distance between the Ξ − and the Li nucleus. Solid and dashedcurves denote the calculated values for V Ξ0 = −
17 MeV and for W Ξ0 = − R = r A / = 1.57 fm where r = 0.783 fm and a = 0.722 fm. Li(2 +g . s . ) has a bound state at the neutron binding energy of B n =2.03 MeV with respect to the n + Li g . s . threshold [20]; the matter rms radius of h r i / =2.39 ± r , a ) inEq. (6) must be used, as we shall mention below.To determine the parameters of ( r , a ) for the nuclear core in the WS form, we adopt afolding-model potential obtained by convoluting the nuclear one-body density for Li with atwo-body Ξ − N force. We assume the s.p. density of the spherical shell model for simplicity;the modified harmonic oscillator (MHO) model is used in a systematic description of asize and a density distribution for Li isotopes with A = 6–9 [25]. For Li(2 +g . s . ), we choosecarefully the MHO size parameters of b s = 1.42 fm and b p = 1.95 fm with center-of-massand nucleon-size corrections, adjusting the matter rms radius of h r i / = 2.39 fm [25].Following to the procedure in Ref. [17], we use the WS form with the parameters of ( r , a )adjusted to give a best least-squares fit to the radial shape of the form factor obtained byfolding a gaussian range of a Ξ N = 1.2 fm into the matter MHO density distribution [17].The parameters of the resultant WS form in Eq. (6) are r = 0.783 fm, a = 0.722 fm, and R = r A / = 1.57 fm, which reproduce the radial shape of the form factor very well; therms radius of the potential denotes h r i / V = (cid:20)Z r V Ξ ( r ) d r . Z V Ξ ( r ) d r (cid:21) / = 2 .
81 fm . (7)On the other hand, the spreading imaginary parts of W Ξ0 may represent complicatedcontinuum states of He ∗ , − He ∗ , and He ∗ . Considering the states of He(3 / − g . s . ) locatedat E ex = 0.45 MeV above the n + He threshold [20], we have the Λ emitted thresholdcorresponding to the Λ + He threshold for the Ξ − p → ΛΛ transition. The threshold-energy difference between Ξ − - Li and Λ- He channels accounts for ∆ M = M ( Li) + m Ξ − − M ( He) − m Λ = 23.3 MeV, where M ( Li) = 7471.4 MeV and M ( He) = 7654.1 MeV. Forthe Ξ − p → Ξ n transition, the Ξ emitted threshold for Ξ - He is located at E = 4.3 MeVabove the Ξ − - Li threshold. The spin-orbit potential for Ξ − is also considered to denotea term of V Ξso (1 /r )[ df ( r ) /dr ] σ · L , where V Ξso ≃ V N so ≃ U Coul , we use theattractive Coulomb potential with the uniform distribution of a charged sphere where Z = 3for Ξ − - Li.We attempt to determine the strength parameters of V Ξ0 and W Ξ0 in Eq. (5) phenomeno-logically in comparison with the data of the Be( K − , K + ) reaction. Figure 1 shows the real7 Dc = 9.21 Dc = . Dc = 2.30c min c FIG. 2: Contour plots of the χ -value distribution in the { V Ξ0 , W Ξ0 } plane from fitting to theaverage cross section of ¯ σ . ◦ - . ◦ in the Be( K − , K + ) reaction at p K − = 1.8 GeV/ c . A solid circledenotes the minimum position of χ = 15.2 at ( V Ξ0 , W Ξ0 ) = ( −
17 MeV, − f s = 0.940.The solid lines labeled by ∆ χ = 2.30, 4.61, and 9.21 correspond to 68%, 90%, and 99% confidencelevels for 2 parameters, respectively. and imaginary parts of the Ξ-nucleus potential for Ξ − - Li, choosing the reasonable strengthsof V Ξ0 = −
17 MeV and − ABLE I: The χ -fitting for various strength parameters, V Ξ0 and W Ξ0 , in the WS potential with r = 0.738 fm and a = 0.722 fm for Ξ − - Li. The value of χ /N and the renormalization factor f s are obtained by comparing the calculated spectrum with the N = 17 data points of the averagecross sections of ¯ σ . ◦ - . ◦ for p K + = 1.07–1.39 GeV/ c . The data were taken from Ref. [9]. V Ξ0 W Ξ0 N = 17 data points(MeV) (MeV) χ /N f s +12 0 69.8/17 0.9880 0 37.6/17 0.964 − −
12 0 18.9/17 0.939 −
18 0 15.6/17 0.927 −
24 0 16.8/17 0.914 −
30 0 22.8/17 0.902+12 − − − − − − − − − − − − − − −
10 49.0/17 1.0100 −
10 25.7/17 0.985 − −
10 18.9/17 0.973 − −
10 15.6/17 0.961 − −
10 16.3/17 0.948 − −
10 21.1/17 0.936 − −
10 30.4/17 0.923 V. RESULTSA. χ fitting Tamagawa et al. (BNL-E906 collaboration) reported the experimental data of the Ξ − QFspectra for the K + forward-direction angles of θ lab = 1.5 ◦ –8.5 ◦ in the Be( K − , K + ) reactionsat the incident K − momentum of p K − = 1.8 GeV/ c [9]. The average cross section ¯ σ . ◦ - . ◦ in the laboratory frame was obtained by¯ σ . ◦ - . ◦ ≡ Z θ lab =8 . ◦ θ lab =1 . ◦ (cid:18) d σdp K + d Ω K + (cid:19) d Ω (cid:30) Z θ lab =8 . ◦ θ lab =1 . ◦ d Ω (8)with the detector resolution of 14.7 MeV FWHM [9]. The strength parameters of V Ξ0 and W Ξ0 in Eq. (5) should be adjusted appropriately to reproduce the data of ¯ σ . ◦ - . ◦ .We consider the Ξ − QF spectrum for Ξ − - Li hypernuclear states with J πB , T B = 3 /
2, usingthe Green’s function method [12], in order to be compared with the data of the Be( K − , K + )reaction at the BNL-E906 experiment [9]. Calculating the spectra for θ lab = 1.5 ◦ –8.5 ◦ , weestimate the average cross section for the corresponding ¯ σ . ◦ - . ◦ in Eq. (8). To make a fitto the spectral shape of the data, we will introduce a renormalization factor of f s into theabsolute value of the calculated spectrum because the eikonal distortion and the amplitudeof f K − p → K + Ξ − would have some ambiguities [10, 15]. The detector resolution of 14.7 MeVFWHM is also taken into account. We obtain the values of χ for fits to the data points of N = 17 in p K + = 1.07–1.39 GeV/ c , varying the strengths of ( V Ξ0 , W Ξ0 ) and f s ; we assumedthe value of 0.018 µ b/sr/MeV c − as a constant background. Thus we estimate the averagecross section in Eq. (8), calculating the spectra for θ lab = 1.5 ◦ –8.5 ◦ in the parameter regionof V Ξ0 = ( − W Ξ0 = ( − χ point.Figure 2 displays the contour plots of χ -value distribution for ¯ σ . ◦ - . ◦ . The minimumvalue of χ is found to be χ = 15.2 at V Ξ0 = −
17 MeV, W Ξ0 = − f s = 0.940,leading to belt-like regions of ∆ χ = 2.30, 4.61, and 9.21 which correspond to 68%, 90%,and 99% confidence levels for 2 parameters, respectively, where ∆ χ ≡ χ − χ . We findthat the value of χ is almost insensitive to W Ξ0 . This fact implies that the parameter of W Ξ0 cannot be determined from the BNL-E906 data due to the insufficient resolution of 14.7MeV FWHM. Nevertheless, we recognize that the calculated spectrum for V Ξ0 ≃ −
17 MeVseems to be in good agreement with the data when W Ξ0 ≃ − IG. 3: Calculated spectrum for ¯ σ . ◦ - . ◦ in the WS potential with V Ξ0 = −
17 MeV, W Ξ0 = − r = 0.738 fm, and a = 0.722 fm, together with the data of the Be( K − , K + ) reaction at p π − = 1.8 GeV/ c [9]. The calculated spectrum is normalized by f s = 0.940 for fits to the data. Solid,dashed, and dot-dashed curves denote the contributions of total, s -hole, and p -hole configurations,respectively. The calculated values are folded with a detector resolution of 14.7 MeV FWHM. value of χ /N = 15.2/17= 0.89, and the standard deviation of σ ≃ χ values of χ /N in calculations when V Ξ0 = − − − − −
6, 0, and+12 MeV, and W Ξ0 = − −
5, and 0 MeV, comparing the calculated spectra with the data.Note that the absolute values of the calculated cross section can explain the magnitude ofthe data, as seen by f s ≃ σ . ◦ - . ◦ in the best-fitcalculation, comparing them with the data of the BNL-E906 experiment at p K + = 1.07–1.39GeV/ c . We recognize that an attraction in the Ξ − - Li potential is needed to reproduce thedata. The contribution of p -hole configurations is larger than that of s -hole configurationsin the Ξ − QF region of p K + = 1.2–1.4 GeV/ c , whereas the former is similar to the latter11 ABLE II: Binding energies B Ξ − and widths Γ Ξ − of the Ξ − -nucleus ( nℓ ) bound states for Ξ − – Li( − He). The strengths of V Ξ0 = −
17 MeV and W Ξ0 = − . − .
5) MeV are used in the WSpotential for the Ξ − bound region. These values are estimated in combination with the Ξ-nucleuspotential U Ξ = V Ξ + iW Ξ and the Coulomb potential U Coul . V Ξ + U Coul + iW Ξ V Ξ + U Coul V Ξ U Coul ( nℓ ) − B Ξ − Γ Ξ − rms − B Ξ − rms − B Ξ − rms − B Ξ − rms(MeV) (MeV) (fm) (MeV) (fm) (MeV) (fm) (MeV) (fm) W Ξ0 = − . S − − − − S − × − − − P − × − − − W Ξ0 = − . S − S − × − P − × − in the region of p K + < c where the recoil momentum grows into q >
540 MeV/ c .Consequently, we confirm that the Ξ potential for Ξ − - Li has a weak attraction in the realpart of the WS potential with r = 0.738 fm and a = 0.722 fm; V Ξ0 = − ± W Ξ0 = − . (9)This potential provides the ability to explain the Be( K − , K + ) data at the BNL-E906experiment. Several authors [2, 3] attempted to determine the values of V Ξ0 for fits to theshape and magnitude of the Ξ − QF spectra from the data of the C( K − , K + ) reaction [3].They suggested that the Ξ-nucleus potential has a weak attraction of V Ξ0 ≃ −
14 MeV inthe WS potential. It is shown that the results of Eq. (9) in our analysis are considerablyconsistent with the results of the previous studies [2, 3].12 . Ξ − -nucleus bound states In Table II, we show the numerical results of binding energies and widths of the Ξ − -nucleus ( nℓ ) bound states for Ξ − - Li, where ( nℓ ) denote the principal and angular momentumquantum numbers for the relative motion between Ξ − and Li. By solving the Schr¨odingerequation of Eq. (4) with the WS potential U Ξ and the finite Coulomb potential U Coul , weobtain a complex eigenvalue as a Gamow state, E nℓ = − B Ξ − − i Γ Ξ − , (10)where B Ξ − and Γ Ξ − denote a binding energy and a width of the bound state, respectively.When we use V Ξ0 = −
17 MeV in the WS potential, we confirm that there exists a veryshallow Ξ − (1 S ) bound state due to the weak attraction in the Ξ-nucleus potential even ifthe Coulomb potential is switched off; the binding energy accounts for B Ξ − (1 S ) = 0 . h r i / = 5.56 fm. When the Coulomb potential is switched on,the binding energy is significantly shifted downward in comparison with the correspondingCoulomb eigenstate, as seen in Table II. Thus this state is often regarded as a “Coulomb-assisted” Ξ − -nucleus bound state; B Ξ − (1 S ) = 1 .
90 MeV and h r i / = 3.96 fm.A Ξ − hyperon bound in nuclei must be absorbed by strong interaction via the Ξ − p → ΛΛconversion process. To estimate the width of the Ξ − bound state, we assume the value of W Ξ0 = − . W Ξ ( E ) at the Ξ − threshold ( E =0.0 MeV). Thus we obtain the width of Γ Ξ − (1 S ) = 1.12 MeV, together with B Ξ − (1 S ) = 1 . W Ξ0 = − . − p → ΛΛ conversion in the Ξ N NLO potential [28, 29], we obtain Γ Ξ − (1 S ) = 0.669 MeV and B Ξ − (1 S ) = 1 .
88 MeV. (Seealso Sect. V B.)
V. DISCUSSIONA. Effects of the real part of the Ξ -nucleus potential To see effects of the attraction in the Ξ-nucleus potential for Ξ − - Li, we discuss theshapes and magnitudes of the calculated spectra. Figure 4 shows the absolute values of thecalculated spectra for ¯ σ . ◦ - . ◦ in the Ξ − QF region, using various strengths of V Ξ0 . We findthat the shape and magnitude of the calculated spectrum are considerably sensitive to the13 - = - V X FIG. 4: Shapes and magnitudes of the calculated spectra for ¯ σ . ◦ - . ◦ in the Be( K − , K + ) reactionat p K − = 1.80 GeV/ c , depending on the strengths of V Ξ0 = − −
12, 0, and +12 MeV in the WSpotential with W Ξ0 = − value of V Ξ0 . This confirms that the value of χ /N is significantly changed by V Ξ0 . The peakposition of the QF spectrum is scarcely shifted downward for p K + , as V Ξ0 changes from − V Ξ0 = ( − p K + > c , corresponding to the region of lower energies E <
140 MeV.On the other hand, the shapes and magnitudes of the spectra with V Ξ0 = ( − E >
140 MeV ( p K + < c ). 14 . Validity of the imaginary part of the Ξ -nucleus potential In Sect. IV A, we have found that the shapes and magnitudes of the calculated spectra arenot so sensitive to the value of W Ξ0 when we change W Ξ0 = ( − W Ξ0 is inevitable dueto the insufficient resolution of 14.7 MeV FWHM. Thus we recognize that it is difficult todetermine the value of W Ξ0 .According to the procedure by Gal, Toker, and Alexander [27], we examine theoreticallyan appropriate parameter for W Ξ0 from a viewpoint of the first order optical ( tρ ) potential, U (1)Ξ ( r ) = t Ξ − p ρ p ( r ) + t Ξ − n ρ n ( r ) (11)in terms of the effective two-body Ξ N elastic t Ξ N scattering matrices in the laboratoryframe, where ρ p,n ( r ) are the proton and neutron densities of the core nucleus. By the opticaltheorem 4 π Im f Ξ N = k Ξ σ tot and considering collisions of zero energy Ξ with bound nucleons,we obtain the imaginary part W (1)Ξ of the optical potential involving the Ξ − p → Ξ n, ΛΛconversion, which is given by W (1)Ξ ( r ) = − (cid:10) v Ξ − p σ (Ξ − p → Ξ n, ΛΛ) (cid:11) × ρ p ( r ) / , (12)where v is the relative velocity of a Ξ − p pair, and h· · · i indicates nuclear medium correctionsto the free space value of vσ arising from Fermi averaging, binding effects, and Pauli principle,etc. The cross section is well approximated up to 300 MeV/ c in the laboratory system bythe form v Ξ − p σ = ( v Ξ − p σ ) / (1 + αv ) , (13)with the two representative parametrization of ( v Ξ − p σ ) = 25 mb and α = 18 for the Ξ − p → Ξ n , ΛΛ reactions, fitting to v Ξ − p σ which are given by the Ξ N NLO potential [28, 29].Taking into account the closure assumption and nuclear medium corrections [27], we obtain h v Ξ − p σ i = 7.02 mb within the Fermi gas model. Using the relation between h vσ i and Im b ,where b is the effective parameter of a complex scattering length for Ξ − p , we roughly estimateIm b = µ h vσ i / π = 0 .
078 fm , (14)15 ABLE III: Comparison of the standard Fermi-averaged differential cross sections ( dσ/d Ω) avlab forthe K − p → K + Ξ − reaction at p K − = 1.8 GeV/ c with the differential cross sections ( dσ/d Ω) freelab for the K − p → K + Ξ − reaction in free space [10]. The values are in unit of mb/sr. θ lab ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ( dσ/d Ω) avlab dσ/d Ω) freelab of which the value corresponds to W Ξ0 = − . W Ξ0 = − χ min , as shown in Fig. 2. If wereplace the momentum distributions of the Fermi gas model by those of the s. p. shell modelfor the finite nuclei, the results may not change. Therefore, we believe that the Ξ-nucleuspotential with V Ξ0 = −
17 MeV and W Ξ0 = − − QF spectrum in the Be( K − , K + ) reaction at p K − = 1.8 GeV/ c .Considering the same manner for only the Ξ − p → ΛΛ conversion [28, 29], we also obtain( v Ξ − p σ ) = 4.5 mb and α = 20. Thus we estimate Im b = 0.018 fm, which corresponds to W Ξ0 = − . W Ξ0 ≃ − − p → ΛΛ coupling is recently predicted to be rather small [8, 30].
C. Verification of the optimal Fermi-averaged K − p → K + Ξ − amplitude In a previous paper [10], we emphasized the importance of the energy dependence of the K − p → K + Ξ − amplitude of f K − p → K + Ξ − arising from the optimal Fermi-averaging proce-dure [11] in the nuclear ( K − , K + ) reaction. We discuss the calculated Ξ − QF spectra involv-ing the energy dependence of f K − p → K + Ξ − in comparison with the data of the Be( K − , K + )reaction in the BNL-E906 experiment. To see the importance of the energy dependenceof f K − p → K + Ξ − , we also estimate the spectrum in the DWIA using the “standard” Fermi-averaged cross section ( dσ/d Ω) avlab for the K − p → K + Ξ − reaction, which may be given by (cid:16) dσd Ω (cid:17) avlab = Z d p N ρ ( p N ) (cid:16) dσd Ω (cid:17) freelab , (15)where ρ ( p N ) is a proton momentum distribution in the target nucleus, and ( dσ/d Ω) freelab isthe differential cross section for the K − p → K + Ξ − reaction in free space. This spectrum16 IG. 5: Comparison of the calculated spectra for ¯ σ . ◦ - . ◦ with the data of the Be( K − , K + )reaction at p K − = 1.80 GeV/ c [9], using the WS potential with V Ξ0 = −
17 MeV and W Ξ0 = − K − p → K + Ξ − amplitudes for f K − p → K + Ξ − , respectively. A dot-dashed curve denotes thespectrum obtained by β ( dσ/d Ω) avlab = constant. The spectra are folded with a detector resolutionof 14.7 MeV FWHM. is proportional to β ( dσ/d Ω) avlab indicating the energy dependence of β whereas the value of( dσ/d Ω) avlab at each θ lab becomes constant in Eq. (15). In Table III, we show the calculatedvalues of ( dσ/d Ω) avlab and ( dσ/d Ω) freelab [10]. Figure 5 displays the calculated Ξ − QF spec-tra obtained by the optimal and standard Fermi-averaged K − p → K + Ξ − amplitudes inthe Be( K − , K + ) reaction at p K − = 1.8 GeV/ c , together with the spectrum obtained by β ( dσ/d Ω) avlab = constant, omitting the energy dependence of β . We find that the energydependence of f K − p → K + Ξ − acts on the shape and magnitude of the QF spectrum remark-ably, and it makes its width narrower. If we use a constant value for f K − p → K + Ξ − in ourcalculations, the shape and magnitude of the calculated Ξ − QF spectrum cannot explain the17ata qualitatively. We show clearly that the optimal Fermi averaging for the K − p → K + Ξ − reaction provides a good description of the energy dependence of the Ξ − QF spectrum in thenuclear ( K − , K + ) reaction [10]. Therefore, we recognize that the optimal Fermi-averagedamplitudes for f K − p → K + Ξ − is essential to explain the shape and magnitude of the spectrumincluding the Ξ − QF region with a wide energy range. Thus it is required to extract in-formation concerning the Ξ-nucleus potential carefully from the data of the experimentalspectrum.
VI. SUMMARY AND CONCLUSION
We have studied phenomenologically the Ξ − production spectrum of the Be( K − , K + )reaction at 1.8 GeV/ c within the DWIA using the optimal Fermi-averaged K − p → K + Ξ − amplitude. We have attempted to clarify properties of the Ξ-nucleus potential for Ξ − - Li, comparing the calculated spectrum with the data of the BNL-E906 experiment. Wehave performed the χ -fitting to the N = 17 data points for ¯ σ . ◦ - . ◦ , varying the strengthparameters of V Ξ0 and W Ξ0 in the WS potential.In conclusion, we show the weak attraction in the Ξ-nucleus potential for Ξ − - Li,which provides the ability to explain the data for the Ξ − QF region in the Be( K − , K + )reaction at 1.8 GeV/ c , consistent with analyses for previous experiments [1, 3]. Theattraction of V Ξ0 = − ± W Ξ0 = − − p → Ξ n , ΛΛ transitions in nuclearmedium, although it is difficult to determine the value of W Ξ0 from the data due to theinsufficient resolution of 14.7 MeV FWHM. The importance of the energy dependence ofthe Fermi-averaged K − p → K + Ξ − amplitude is confirmed by this analysis. The detailedanalysis is also required for the J-PARC E05 experiment of the C( K − , K + ) reaction at1.8 GeV/ c [5]. This investigation is a subject for future research. Acknowledgments
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