K − d→πΣn reactions and structure of the Λ(1405)
Shota Ohnishi, Yoichi Ikeda, Tetsuo Hyodo, Emiko Hiyama, Wolfram Weise
aa r X i v : . [ nu c l - t h ] A ug K − d → π Σ n reactions and structure of the Λ(1405)
S Ohnishi , , Y Ikeda , T Hyodo , E Hiyama , W Weise , Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan RIKEN Nishina Center, Wako, Saitama 351-0198, Japan Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan ECT*, Villa Tambosi, I-38123 Villazzano (Trento), Italy Physik Department, Technische Universit¨at M¨unchen, D-85747 Garching, GermanyE-mail: s [email protected]
Abstract.
We report on the first results of a full three-body calculation of the ¯
KNN - πY N amplitude for the K − d → π Σ n reaction, and examine how the Λ(1405) resonance manifests itselfin the neutron energy distributions of K − d → π Σ n reactions. The amplitudes are computedusing the ¯ KNN - πY N coupled-channels Alt-Grassberger-Sandhas (AGS) equations. Two typesof models are considered for the two-body meson-baryon interactions: an energy-independentinteraction and an energy-dependent one, both derived from the leading order chiral SU(3)Lagrangian. These two models have different off-shell properties that cause correspondinglydifferent behaviors in the three-body system. As a remarkable result of this investigation, it isfound that the neutron energy spectrum, reflecting the Λ(1405) mass distribution and width,depends quite sensitively on the (energy-dependent or energy-independent) model used. Henceaccurate measurements of the π Σ mass distribution have the potential to discriminate betweenpossible mechanisms at work in the formation of the Λ(1405).
1. Introduction
Understanding the structure of the Λ(1405) with spin-parity J π = 1 / − and strangeness S = − KN threshold energy. The Λ(1405) can be considered as a ¯ KN quasi-bound state embeddedin the π Σ continuum [1, 2]. Guided by this picture, ¯ KN interactions which reproduce themass of Λ(1405) and two-body scattering data have been constructed phenomenologically [3, 4].On the other hand, ¯ KN interactions have been studied for a long time based on chiralSU(3) dynamics [5, 6, 7]. Between the phenomenological and chiral SU(3) ¯ KN interactions,subthreshold ¯ KN amplitudes are quite different [8]. The phenomenological model describesΛ(1405) as a single pole of the scattering amplitude around 1405 MeV. The ¯ KN amplitudefrom the interaction based on chiral SU(3) dynamics has two poles, one of which located notat 1405 MeV but around 1420 MeV [9, 10]. The differences in the pole structure come fromthe different off-shell behavior, especially as a consequence of the energy-dependence of the¯ KN interaction. The ¯ KN interaction based on chiral SU(3) dynamics is energy-dependent,and its attraction becomes weaker as one moves below the ¯ KN threshold energy. Hencethe (upper) pole of the ¯ KN amplitude shows up around 1420 MeV. On the other hand, thephenomenological ¯ KN interaction is energy-independent and strongly attractive so that thepole shows up around 1405 MeV. These differences are enhanced in the so-called few-bodykaonic nuclei, such as the strange dibaryon resonance under discussion in the ¯ KN N - πY N oupled system [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. How a possible signature of thisstrange dibaryon resonance shows up in the resonance production reaction is also of interest asit reflects the two-body dynamics of the Λ(1405) [22].One of the possible kaon-induced processes forming the Λ(1405) is K − d → Λ(1405) n . Thesignature of the Λ(1405) was observed in an old bubble-chamber experiment that measuredthe π Σ invariant mass distribution in the K − d → π + Σ − n reaction [23]. A new experiment isplanned at J-PARC [24]. Theoretical investigations of the K − d → π Σ n reaction have previouslybeen performed in simplified models assuming a two-step process [25, 26, 27, 28].In this contribution we examine how the Λ(1405) resonance shows up in the K − d → π Σ n reaction by making use of the approach based on the coupled-channels Alt-Grassberger-Sandhas (AGS) equations developed in Refs. [19, 20, 21, 22]. This is the first calculation ofthis process which incorporates the full three-body dynamics.
2. Three-body Scattering Equations
Throughout this paper, we assume that the three-body processes take place via separable two-body interactions, which have the following form in the two-body center-of-mass (CM) frame, V αβ ( ~q α , ~q β ; E ) = g α ( ~q α ) λ αβ ( E ) g β ( ~q β ) , (1)where ~q α [ g α ( ~q α )] is the relative momentum [form factor] of the two-body channel α ; E is thetotal energy of the two-body system. With this assumption the amplitudes for the quasi-two-body scattering of an “isobar” and a spectator particle, X ij ( ~p i , ~p j ; W ), are then obtained bysolving the AGS equations [29, 30], X ij ( ~p i , ~p j , W ) = (1 − δ ij ) Z ij ( ~p i , ~p j , W )+ X n = i Z d~p n Z in ( ~p i , ~p n , W ) τ n ( W − E n ( ~p n )) X nj ( ~p n , ~p j , W ) . (2)Here the subscripts i, j, n specify the reaction channels; W and ~p i are the total scattering energyand the relative momentum of channel i in the three-body CM frame, respectively; Z ij ( ~p i , ~p j ; W )and τ i ( W − E i ( ~p i )) are the one-particle exchange potential and the two-body propagator.With the quasi-two-body amplitudes, the scattering amplitudes for the break-up process d + ¯ K → π + Σ + N are obtained as T π Σ N - ¯ Kd ( ~q N , ~p N , ~p ¯ K , W ) = g Y π ( ~q N ) τ Y π Y K ( W − E N ( ~p N )) X Y K d ( ~p N , ~p ¯ K , W )+ g Y π ( ~q N ) τ Y π Y π ( W − E N ( ~p N )) X Y π d ( ~p N , ~p ¯ K , W )+ g N ∗ ( ~q Σ ) τ N ∗ N ∗ ( W − E Σ ( ~p Σ )) X N ∗ d ( ~p Σ , ~p ¯ K , W )+ g d y ( ~q π ) τ d y d y ( W − E π ( ~p π )) X d y d ( ~p π , ~p ¯ K , W ) , (3)where X Y K d ( ~p N , ~p ¯ K , W ) is the quasi-two-body amplitude anti-symmetrized for two nucleons; thesubscripts denote the isobars. The notations for the isobars are Y K = ¯ KN , Y π = πY , d = N N , N ∗ = πN and d y = Y N , respectively.In this contribution we employ the first two terms of Eq. (3) as a first step. These termsemerge directly from the Λ(1405) in the final state interaction; they are the dominant parts ofthe full T-matrix. Using this T-matrix, the differential cross section of the break-up process d + ¯ K → π + Σ + N is calculated as: dσdE n = (2 π ) E d E ¯ K W p ¯ K m N m Σ m π m N + m Σ + m π × Z d Ω p N d Ω q N p N q N X ¯ if | < N Σ π | T ( W ) | d ¯ K > | , (4) able 1. Cutoff parameters of ¯ KN - πY interaction.Λ I =0¯ KN (MeV) Λ I =0 π Σ (MeV) Λ I =1¯ KN (MeV) Λ I =1 π Σ (MeV) Λ I =1 π Λ (MeV)E-dep 1000 700 725 725 725E-indep 1000 700 920 960 640where E n is the neutron energy in the center-of-mass frame of π Σ defined by E n = m N + p N η N . (5)
3. Models of Two-body Interaction
We use two-body s -wave meson-baryon interactions obtained from the leading order chiralLagrangian, L W T = i F π T r ( ¯ ψ B γ µ [[ φ, ∂ µ φ ] , ψ B ]) . (6)Here, we examine two interaction models, both of which are derived from the above Lagrangianbut have different off-shell behavior: one is the energy dependent model (E-dep), V αβ ( q ′ , q ; E ) = − λ αβ π F π E − M α − M β √ m α m β (cid:18) Λ α q ′ + Λ α (cid:19) Λ β q + Λ β ! . (7)while the other is the energy independent model (E-indep), V αβ ( q ′ , q ) = − λ αβ π F π m α + m β √ m α m β (cid:18) Λ α q ′ + Λ α (cid:19) Λ β q + Λ β ! , (8)Here, m α ( M α ) is the meson (baryon) mass; F π is the pion decay constant; λ αβ are determinedby the flavor SU(3) structure of the chiral Lagrangian.In the derivation of these potentials we have assumed the so-called “on-shell factorization” [6]for Eq. (7) and q, q ′ ≪ M α for Eq. (8). The cutoff parameters Λ are determined by fittingexperimental data as shown in Table 1.In the E-dep model, the ¯ KN amplitudes have two poles for l = I = 0 in the ¯ KN -physical and π Σ-unphysical sheets, corresponding to those derived from the chiral unitary model [10]. On theother hand, the E-indep model has a single pole that corresponds to Λ(1405). It is interesting toexamine how this difference of the two-body interaction models appears in the neutron energyspectrum of the K − d → π Σ n reaction.
4. Results and Discussion
In Fig.1, we present the differential cross section of K − d → π Σ n [Eq. (4)] computed usingthe E-dep (a) and E-indep (b) models, respectively. We investigate the cross section for initialkaon momentum p labK − = 1000 MeV in accordance with the planned J-PARC experiment [24].Here, we decompose the isospin basis states into charge basis states using ClebschGordancoefficients: the solid curve represents the K − + d → π + + Σ − + n ; the dashed curve refersto the K − + d → π − + Σ + + n ; the dotted curve represents the K − + d → π + Σ + n reaction,respectively. d σ / d E n ( µ b / M e V ) E n -E th (MeV) (a) d σ / d E n ( µ b / M e V ) E n -E th (MeV) (b) Figure 1.
Differential cross section dσ/dE n for K − + d → π +Σ+ n . The initial kaon momentumis set to p labK − = 1000 MeV. Panel (a): the E-dep model; Panel (b) the E-indep model. Solidcurves: π + Σ − n ; dashed curves: π − Σ + n ; dotted curves: π Σ n in the final state, respectively. d σ / d E n ( µ b / M e V ) E n -E th (MeV) (a) d σ / d E n ( µ b / M e V ) E n -E th (MeV) (b) Figure 2.
Contribution of each partial wave component to the differential cross section dσ/dE n for d + K − → π + + Σ − + n . Panel (a): the E-dep model; Panel (b) the E-indep model. The thicksolid curve represents the summation of total orbital angular momentum L = 0 to 14; The thinsolid curve represents L = 0 only; The dashed curve represents L = 1 only; The dotted curverepresents L = 2 only; The dashed-dotted curve represents L = 3 only; The dashed-two-dottedcurve represents L = 4 only, respectively. The initial kaon momentum is set to p labK − = 1000 MeV.We subtract the neutron energy E th at which the amplitudes have the ¯ KN threshold cuspfrom the neutron energy E n , i.e. ¯ KN threshold cusp shows up on the differential cross sectionat E n − E th = 0. Well defined maxima are found at E n ∼ E n ∼ π Σ in the final state. These peak and bump structures appear about 5 MeV higher in energythan the calculated binding energy of the Λ(1405) ( E B ∼
13 MeV for the E-dep model and E B ∼
28 MeV for the E-indep model). The magnitude of the differential cross section for theE-dep model is twice larger than that for the E-indep model, and the interference patterns withbackgrounds are quite different between these two models. This clear difference in the differentialcross section, arising from the model dependence of the two-body interactions, suggests that the K − d → π Σ n reaction can indeed provide useful information on the ¯ KN - πY system.Finally, we examine the contribution of each partial wave component for total orbital angularomentum L to the differential cross section (Fig. 2). We conclude that the s -wave componentis dominant in the low-energy region, but around the ¯ KN threshold higher partial waves suchas the p -wave component become important.In summary, we have calculated the differential cross sections (4) for K − + d → π + Σ + n reactions. We have found peak and bump structures in the neutron energy spectrum, andtherefore it is possible to observe the signal of the Λ(1405) resonance in the physical crosssections. We have also shown that the K − d → π Σ n reactions are useful for judging existingdynamical models of ¯ KN - π Σ coupled systems with Λ(1405). Further improvements of thepresent model to account for the neglected contributions in Eq. (3) and relativistic correctionsare under investigation.
Acknowledgments
The simulation has been performed on a supercomputer (NEC SX8R) at the Research Center forNuclear Physics, Osaka University. This work was partly supported by RIKEN Junior ResearchAssociate Program, by RIKEN iTHES Project and by JSPS KAKENHI Grants Nos. 25800170,24740152 and 23224006.
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