Trace of Λ(1405) resonance in low energy K^{-}+\, ^{3}\mathrm{He}\rightarrow (π^{0}Σ^{0})+d reaction
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Trace of Λ (1405) resonance in low energy K − + He → ( π Σ ) + d reaction J. Esmaili , S. Marri , M. Raeisi and A. Naderi Beni Department of Physics, Faculty of Basic Sciences, Shahrekord University, Shahrekord, 115, Iran Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran Department of Physics, Payame Noor University, P. O. Box 19395-3697, Tehran, IranReceived: date / Revised version: date
Abstract.
In present work, we investigated K − + He reaction at low energies. The coupled-channel Faddeev AGSequations were solved for ¯ KNd − πΣd three-body system in momentum representation to extract the scattering am-plitudes. To trace the signature of the Λ (1405) resonance in the πΣ invariant mass, the deuteron energy spectrum for K − + He → πΣd reaction was obtained. Different types of ¯ KN − πΣ potentials based on phenomenological andchiral SU(3) approaches were used. As a remarkable result of this investigation, it was found that the deuteron en-ergy spectrum, reflecting the Λ (1405) mass distribution and width, depends quite sensitively on the ¯ KN − πΣ modelof interaction. Hence accurate measurements of the πΣ mass distribution have the potential to discriminate betweenpossible mechanisms at work in the formation of the Λ (1405). PACS.
An important issue in various aspects in strangeness nuclearphysics is the structure of Λ (1405) resonance, which has beenfound to be a highly controversies topic in studying the antikaon-nucleon interaction. The Λ (1405) resonance is a bound state of ¯ KN , which exclusively decays into the πΣ ( I = 0) channelvia the strong interaction. The ¯ KN − πΣ interaction, which isa fundamental ingredient of the antikaonic nuclear clusters [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] is also strongly dom-inated by Λ (1405) resonance. The existence of Λ (1405) reso-nance was first predicted by Dalitz and Tuan [17,18] in 1959showing that the unitarity in coupled-channel ¯ KN − πΣ systemleads to the existence of Λ (1405) . As early as in 1961 an exper-imental evidence of this resonance was reported in the invariantmass spectrum of the πΣ resulting from K − p → πππΣ reac-tion [19] at 1.15 GeV.The ¯ KN interaction models which reproduce the mass of Λ (1405) resonance and two-body scattering data can be di-vided into two classes: those constructed phenomenologically [5,6,20] and those derived based on chiral SU(3) dynamics [7,8,9,10,11]. Even though the phenomenological and the chiralSU(3) ¯ KN interaction models produce comparable results atand above ¯ KN threshold, they differ significantly in their ex-trapolations to sub-threshold energies [21]. The phenomeno-logical ¯ KN potentials are constructed to describe the Λ (1405) as a single pole of the scattering amplitude around 1405 MeV,corresponding to a quasi-bound state of the ¯ KN system witha binding energy of about 30 MeV. On the other hand, the ¯ KN − πΣ coupled-channels amplitude resulting from chiralSU(3) dynamics has two poles. The two poles are commonly characterized as following: the first pole in the complex en-ergy plane is located quite close to the ¯ KN threshold with asmall imaginary part, around 10-30 MeV, and a strong cou-pling to ¯ KN . In turn, the second one is wider, with a rela-tively large imaginary part around 50-200 MeV, coupling morestrongly to the πΣ channel and its pole position shows moredependence on the specific theoretical model [22,23,24,25,26,27]. This different pole structure comes from different off-shellproperties of the ¯ KN interactions. The ¯ KN interactions basedon the chiral SU(3) dynamics are energy-dependent, and thatin the sub-threshold become less attractive than those proposedby the energy-independent phenomenological potentials [21].The πΣ mass spectrum is a suitable tools to study the ¯ KN reaction below the ¯ KN threshold. As it is impossible to per-form the scattering experiment in the πΣ channel directly, theresonance properties can be extracted by analyzing the invari-ant mass distribution of the πΣ final state in reactions that pro-duce Λ (1405) resonance. During the past decades, many ex-perimental and theoretical searches were carried out to investi-gate the possible observation of Λ (1405) resonance. Braun etal., studied the K − d reaction and reported a resonance energyaround 1420 MeV [28]. In Ref. [29], the pion induced reaction ( π − p → K + πΣ ) was investigated and the mass of the reso-nance was found to be consistent with 1405 MeV. Using photo-production reactions, the CLAS [30,31,32] and LEPS [33,34]collaboration investigated the Λ (1405) resonance signal. Sev-eral theoretical studies have been done to analyze the CLASdata using different interaction models for ¯ KN system, whichare based on chiral SU(3) dynamics [35,36,37,38,39] and phe-nomenological approachs [40]. Other interesting experimentswere also performed at GSI by HADES collaboration [41] and a r X i v : . [ nu c l - t h ] F e b J. Esmaili, S. Marri, M. Raeisi and A. Naderi Beni: Trace of Λ (1405) resonance in low energy K − + He → ( π Σ ) + d reaction at J-PARC as an E31 experiment [42] to clarify the nature ofthe Λ (1405) by using pp and K − d reactions, respectively. Inthe E31 experiment, the πΣ mass spectra are measured for allcombinations of charges, i.e., π ± Σ ∓ and π Σ . To establish atheoretical framework for a detailed analysis of K − d reaction,different theoretical studies were performed. The theoretical in-vestigations of K − d → πΣ + n have been performed in [43,44,45,46] using a two-step process. The differential cross sec-tions of K − d reaction were also studied in Refs. [47,48] us-ing three-body Faddeev method and it was demonstrated thatthe K − d reaction can be a useful tool for studying the sub-threshold properties of the ¯ KN interaction.The K − + He reaction was studied in Ref. [49] usingvariational method. The Σπ invariant-mass spectrum in theresonant capture of K − at rest in He were calculated by acoupled-channel potential for ¯ KN − πΣ interaction. The re-sults in Ref. [49] confirmed the Λ (1405) ansatz and the pro-posed predictions by chiral-SU(3) ( M ∼ MeV/ c ) wereexcluded. The authors finally proposed more stringent test byusing a He target. An experimental search for ¯ KN N boundstate was performed at J-PARC by using the in-flight K − + He reaction at 1 GeV/c [50]. In the E15 experiment, the πΣ in-variant mass spectrum resulting from K − + He reaction wasmeasured for two combinations of charges, i.e., π ± Σ ∓ pn . Itwas shown that the production cross section of the Λ (1405) p is ∼ times larger than that of the K − pp bound state observedin Λp invariant mass which is an important information on theproduction mechanism of the K − pp bound state [50].The purpose of the present work is to explore the πΣ in-variant mass spectra resulting from the K − + He → πΣd reaction. The problem can be solved using methods developedwithin three-body theories. To reduce the four-body K − + He system to a three-body system, we considered a p − d clusterstructure for He nuclei (Fig. 1). To study this reaction, the Fad-deev amplitudes for ¯ KN d − πΣd system were calculated at realscattering energies. One of the aims is to study the role of dif-ferent off-shell properties of the underlying interactions as theyare realized in chiral SU(3) dynamics versus phenomenologicalpotential models. With this method, we investigated how wellthe Λ (1405) resonance manifests itself in the three-body ob-servable. To study the dependence of the K − + He reactionon the fundamental ¯ KN − πΣ interaction, different interac-tion models derived from chiral SU(3) and phenomenologicalapproaches, were included in our calculations.The paper is organized as follows: in Sect. 2, we will ex-plain the Faddeev formalism used for the three-body ¯ KN d system and give a brief description of scattering amplitude for K − + He → πΣd reaction. The two-body inputs of the cal-culations and the extracted results for πΣ mass spectra are pre-sented in Sect. 3 and in Section 4, we give conclusions. K − + He reaction In the present work, the possible signature of the Λ (1405) reso-nance in πΣ mass spectrum resulting from K − + He → πΣd reaction was studied. We used the three-body Faddeev AGSequations [51]. As there are three different particles in the sys-tem under consideration, we will have the following partitions of the ¯ KN d three-body system, defining the interacting pairsand their allowed spin and isospin quantum numbers (1) : ¯ K + ( N d ) s = ; I = , (2) : N + ( ¯ Kd ) s =1; I = , (3) : d + ( ¯ KN ) s = ; I =0 . (1)The quantum numbers of the ¯ KN d are I = 0 and s = , inactual calculations, when we include isospin and spin indicesthe number of configurations is equal to three, correspondingto different possible two-quasi-particle partitions.The key point of the present calculations is the separablerepresentation of the scattering amplitudes in the two-body sub-systems. The separable potentials for two-body subsystems aregiven by V I i i ( k, k (cid:48) ) = g I i i ( k ) λ I i i g I i i ( k (cid:48) ) , (2)where g I i i ( k ) is used to define the form factor of the interactingtwo-body subsystem with relative momentum k and isospin I and λ I i i defines the strength of the interaction. The two-bodyinteractions are also labeled by the i values to define simulta-neously the spectator particle and interacting pair. Using sepa-rable potentials, we can define the two-body t-matrices in thefollowing form T I i i ( k, k (cid:48) ; z ) = g I i i ( k ) τ I i i ( z − p i η i ) g I i i ( k (cid:48) ) , (3)where the τ I i i ( z ) -functions are the two-body propagators em-bedded in three-body system and p i is the spectator particlemomentum. The reduced mass η i is also given by η i = m i ( m j + m k ) / ( m i + m j + m k ) . (4)The whole dynamics of ¯ KN d three-body system is de-scribed in terms of the transition amplitudes K I i I j i,m ; j,n , whichconnect the quasi-two-body channels characterized by Eq. 1. InFig. 2, the three different rearrangement channels of the ¯ KN d are represented. The Faddeev AGS equations for ¯ KN d systemscan be expressed by K I ¯ K I ¯ K ¯ K,m ; ¯
K,n = (cid:88) rr (cid:48) M I ¯ K I N ¯ K,m ; N,r τ I N N ( rr (cid:48) ) K I N I ¯ K N,r (cid:48) ; ¯
K,n + (cid:88) rr (cid:48) M I ¯ K I d ¯ K,m ; d,r τ I d d ( rr (cid:48) ) K I d I ¯ K d,r (cid:48) ; ¯ K,n K I N I ¯ K N,m ; ¯
K,n = M I N I ¯ K N,m ; ¯
K,n + (cid:88) rr (cid:48) M I N I ¯ K N,m ; ¯
K,r τ I ¯ K ¯ K ( rr (cid:48) ) K I ¯ K I ¯ K ¯ K,r (cid:48) ; ¯
K,n + (cid:88) rr (cid:48) M I N I d N,m ; d,r τ I d d ( rr (cid:48) ) K I d I ¯ K d,r (cid:48) ; ¯ K,n K I d I ¯ K d,m ; ¯ K,n = M I d I ¯ K d,m ; ¯ K,n + (cid:88) rr (cid:48) M I d I ¯ K d,m ; ¯ K,r τ I ¯ K ¯ K ( rr (cid:48) ) K I ¯ K I ¯ K ¯ K,r (cid:48) ; ¯
K,n + (cid:88) rr (cid:48) M I d I N d,m ; N,r τ I N N ( rr (cid:48) ) K I N I ¯ K N,r (cid:48) ; ¯
K,n . (5)Here, the operators K I i I j i,m ; j,n are the three-body transitionamplitudes, which describe the dynamics of the three-body ¯ KN d . Esmaili, S. Marri, M. Raeisi and A. Naderi Beni: Trace of Λ (1405) resonance in low energy K − + He → ( π Σ ) + d reaction 3 pK d d π Σ He Fig. 1. (Color on line) Diagrammatic representation for K − + He → ( πΣ ) + d reaction using a p − d cluster structure for He nuclei. system. To define the spectator particles or interacting parti-cles in each subsystem, we used the i, j and k indices andthe isospin of the interacting particles is defined by I i . Sincesome potentials are rank- , the indices n, l, m are used to de-fine, which term of the sub-amplitudes is used. The operators M I i I j i,m ; j,n are Born terms, which describe the effective particle-exchange potential realized by the exchanged particle betweenthe quasi-particles in channels i and j . The Born terms are de-fined by M I i I j i,m ; j,n ( p i , p j ; z ) = Ω I i I j × (cid:90) +1 − dx g I i i,m ( q i ) g I j j,n ( q j ) z − p i m i − p j m j − ( (cid:126)p i + (cid:126)p j ) m k , (6)where the parameters Ω I i I j are the spin and isospin couplingcoefficients. The momenta (cid:126)q i ( (cid:126)p i , (cid:126)p j ) and (cid:126)q j ( (cid:126)p i , (cid:126)p j ) are given interms of (cid:126)p i and (cid:126)p j . We use the relations (cid:126)q i = − (cid:126)p j − m j m j + m k (cid:126)p i ,(cid:126)q j = (cid:126)p i + m i m i + m k (cid:126)p j , (7)where m k is exchanged particle or quasi-particle mass and x isdefined by x = ˆ p i · ˆ p j .For the K − + He reaction, the initial state in the labora-tory frame contains an incoming kaon, the projectile, and one He , the target, at rest. There are three possibilities for the fi-nal state. In one, the He survives scattering, i.e. the final statecontains a kaon and a He . This is called elastic scattering. Theother is where the system goes to ( ¯ KN ) + d and ( ¯ Kd ) + N channel. Before we proceed to discus about the obtained results, weshall begin with a survey on the two-body interactions, whichare the central input to our present three-body calculations. Forall two-body interactions, the angular momentum is taken to bezero and all potentials have the separable form in momentumrepresentation. Different phenomenological and chiral based potentials wereused to describe the ¯ KN − πΣ interaction, which is the mostimportant input in the ¯ KN d − πΣd three-body system. Thephenomenological potentials SIDD and SIDD from Ref. [53]are constructed to reproduce the SIDDHARTA [54] experimentresults. SIDD and SIDD potentials reproduce the one- andtwo-pole structure of Λ (1405) , respectively. The potentials havethe following form V Iαβ ( k, k (cid:48) ) = g Iα ( k ) λ Iαβ g Iβ ( k (cid:48) ) . (8)Here, the strength parameters and the form factors of thetwo-body potential are labeled by particle indices α and β totake into account the coupling between ¯ KN and πΣ systems.To study the dependence of the results to the ¯ KN − πΣ modelof interaction, we also used the potential given by Akaishi andYamazaki [1] and the new potential given in Ref. [55] which thefirst one is an extremely deep potential. We referred to these po-tentials as AY and Rev-A potentials, respectively. The Rev-Amodel is a chiral based but energy-independent potential whichis a rather new and probably not well known. Five different in-teraction models (A-E) are presented in Ref. [55]. As the Amodel reproduce the lowest value of χ , the A version was cho-sen to be used in present calculations. Most of chiral potentialsin the literatures are not suitable for Faddeev calculation or atleast will make the calculations difficult. The last ¯ KN poten-tial that we used in our calculations is an energy-dependentchiral potential (Chiral-IKS). The parameters of the energy-dependent chiral based potential are presented in Ref. [11]. InTable 1, the pole position(s) of the ¯ KN system for all modelsof interaction are presented.Two models of interaction were used to describe the pd in-teraction. The first one is a two terms potential, which includesthe short range repulsive part of the interaction V NdA ( k, k (cid:48) ) = (cid:88) m =1 g NdA ; m ( k ) λ NdA ; m g NdA ; m ( k (cid:48) ) , (9)where the functions g Ndi ( k ) are the form factors of pd interac-tion and are parametrized by Yamaguchi form [56]: g NdA ; m ( k ) = 1 k + ( Λ NdA ; m ) . (10)We refer to the two-term potential as V NdA . The parametersof the V NdA potential are adjusted to reproduce the p − d phase-shifts [57]. The physical values for data fitting were obtained J. Esmaili, S. Marri, M. Raeisi and A. Naderi Beni: Trace of Λ (1405) resonance in low energy K − + He → ( π Σ ) + d reaction K K Kd d dN N N
Fig. 2. (Color on line) Diagrammatic representation for different partitions of the ¯ KNd system. Defining the interacting particles, we will havethree partitions, namely ¯ K + ( Nd ) , ( ¯ Kd ) + N and ( ¯ KN ) + d . The anti-kaon is defined by green circle, the nucleon by red circle and thedeuteron by brown circle. Table 1.
The pole position(s) (in MeV) of the ¯ KN system for different phenomenological and chiral based models of the ¯ KN − πΣ interaction. X and X stands for a one- and a two-pole version of the SIDD potentials. The quantities p and p represent the first and second pole of ¯ KN − πΣ system for each potential. SIDD SIDD Chiral − IKS AY Rev − A p . − i . . − i . . − i . . − i . . − i . − . − i . . − i . − − by solving the Lippmann-Schwinger equations without inclu-sion of the Coulomb interaction into the p − d system. We usedalso a one-term potential for pd interaction and its parameterare adjusted to reproduce the He binding energy and pd scat-tering length. We refer to the rank one potential as V NdB . Theparameters of the the V NdA and V NdB potentials are presented inTable 2
Table 2.
The parameters of V NdA and V NdB potentials to describe the pd interaction. The range parameters are in MeV and the strength pa-rameters are in fm − . V NdA potential: Λ NdA ;1 Λ NdA ;2 λ NdA ;1 λ NdA ;2
115 152 -0.0404 0.2967 V NdB potential: Λ NdB λ NdB
We need also a potential model to describe the interactionbetween antikaon and deuteron. A one-channel complex poten-tial with rank- were used to describe K − d interaction V ¯ Kd ( k, k (cid:48) ) = (cid:88) m =1 g ¯ Kdm ( k ) λ ¯ Kdm ; Complex g ¯ Kdm ( k (cid:48) ) . (11)The parameters of this potential are given in Ref. [58]. Thecomplex strength parameters and range parameters of the po-tential are adjusted to reproduce the K − d scattering length a K − d and also the effective range r K − d [58]. ¯ KN d − πΣd To take the coupling between ¯ KN and πΣ channels into ac-count, the formalism of Faddeev equations should be extendedto include the πΣd channel. In the present subsection, the πΣd channel of the ¯ KN d system has not been included directly andone-channel Faddeev AGS equations are solved for three-body ¯ KN d system and we approximated the full coupled-channelinteraction by using the so-called exact optical ¯ KN ( − πΣ ) po-tential [52]. Therefore, the decaying to the πΣd channel istaken into account through the imaginary part of the optical ¯ KN ( − πΣ ) potential. To study the possible signature of the Λ (1405) resonance in the π Σ mass spectra in the K − +( N d ) → πΣ + d reaction, first we should define break-upamplitude. As we do not include the lower lying channels di-rectly into the calculations, the only Faddeev amplitude, whichcontribute in the scattering amplitude is K d, ¯ K . Therefore, thebreak-up amplitude can be expressed as T ( πΣ )+ d ← ¯ K +( Nd ) ( (cid:126)k d , (cid:126)p d , ¯ p ¯ K ; z ) = (cid:88) n =1 g I =0 πΣ ( (cid:126)k d ) × τ I =0( πΣ ) ← ( ¯ KN ) ( z − p d η d ) K d,
1; ¯
K,n ( p d , ¯ p ¯ K ; z ) , (12)where (cid:126)k i is the relative momentum between the interacting pair( jk ) and ¯ p ¯ K is the initial momentum of ¯ K in ¯ KN d center ofmass. The quantity K I i I j i ( m ) ,j ( n ) is the Faddeev amplitude, whichis derived from Faddeev equation (5). . Esmaili, S. Marri, M. Raeisi and A. Naderi Beni: Trace of Λ (1405) resonance in low energy K − + He → ( π Σ ) + d reaction 5 πΣ (MeV/c )0246 d σ / d E d ( - M e V - ) SIDD SIDD Chiral-IKSAYRev-A p K - = 50 MeV/c (a) πΣ (MeV/c )0123 d σ / d E d ( - M e V - ) SIDD SIDD Chiral-IKSAYRev-A p K - = 100 MeV/c (b) πΣ (MeV/c )012 d σ / d E d ( - M e V - ) SIDD SIDD Chiral-IKSAYRev-A p K - = 150 MeV/c (c) πΣ (MeV/c )048 d σ / d E d ( - M e V - ) SIDD SIDD Chiral-IKSAYRev-A p K - = 250 MeV/c(d) Fig. 3. (Color on line) The πΣ mass spectra for K − + He → π Σ + d reaction. Different types of ¯ KN − πΣ potentials were used.The kaon incident momentum is around p cm ¯ K = 50 − MeV/c. In panels (a), (b), (c) and (d), the values of ¯ P K are 50, 100, 150 and 250MeV/c, respectively. The blue dashed and red dash-dotted curves show the mass spectrum with SIDD and SIDD potential, respectively. Theextracted results for Chiral-IKS, AY and Rev-A potentials are also depicted by black solid, brown dash-dot-dotted and green dash-dash-dottedlines, respectively. Using Eq.(12), we define the break-up cross section of ¯ K +( N d ) → d + ( π Σ ) as follows: dσdE d = ω ( Nd ) ω ¯ K z ¯ p ¯ K m π m Σ m d m π + m Σ + m d (cid:90) dΩ p d dΩ k d p d k d × (cid:88) if | T ( πΣ )+ d ← ¯ K +( Nd ) ( (cid:126)k d , (cid:126)p d , ¯ p ¯ K ; z ) | , (13)where E d is the deuteron energy in the center-of-mass frame of πΣ , which is defined by E d = m d + p d η d , (14)and the energies ω ( Nd ) and ω ¯ K are the kinetic energy of ¯ K and N d in the initial state.Since the input energy of the AGS equations is above the ¯ KN d threshold, the moving singularities will appear in the three-body amplitudes. To remove these standard singularities,we have followed the same procedure implemented in Refs. [59,60]. Using the so called “point-method”, we computed the crosssection of K − + ( N d ) → πΣ + d reaction.Starting from Faddeev AGS equations 5, the π Σ invari-ant mass for K − +( N d ) → πΣ + d reaction was calculated. Inour calculation, we studied the dependence of the mass spec-trum on the fundamental ¯ KN − πΣ interaction by using fivedifferent interaction models reproducing various pole structurefor Λ (1405) resonance. With this method, we extracted the πΣ mass spectrum for different incident antikaon momentum p cm ¯ K = 50 − /c . The extracted mass spectrum for mo-menta p cm ¯ K = 50 − /c is strongly affected by thresholdeffects, but for the momentum p cm ¯ K = 250MeV /c the signatureof the resonance is clearly visible. Furthermore, it was foundthat the πΣ mass spectrum, reflecting the Λ (1405) mass distri-bution and width, depends quite sensitively on the ¯ KN − πΣ model of interaction. J. Esmaili, S. Marri, M. Raeisi and A. Naderi Beni: Trace of Λ (1405) resonance in low energy K − + He → ( π Σ ) + d reaction A definitive study of the three-body ¯ KN d system could bealso performed using the standard energy-dependent ¯ KN inputpotential, too [11]. The energy-dependent potentials providea weaker ¯ KN attraction for lower energies than the energy-independent potentials. Therefore, the quasi-bound state in ¯ KN system resulting from the energy-dependent potential happensto be shallower. In Fig. 3, a comparison is made between theresults obtained for the chiral-IKS ¯ KN − πΣ and the calcu-lated mass spectra for other potentials. For energy-dependentpotential the peak structure is not located at the position of thefirst pole given in table 1. These results are in agreement withthe statement that the Λ (1405) spectrum is the superposition oftwo independent states and one can not see two different polestructure in the Λ (1405) spectrum [22,26].The key point in the present calculations is the detection ofdeuteron which is a loosely bound system. Although, for largemomenta the signal of Λ (1405) overcomes the kinetic effects,at these energies, the probability for the deuteron (as a looselybound system) to survive the reaction will decrease and onewould expect a rather low reaction rate for K − + He → πΣd reaction. Therefore, an accurate measurements of the πΣ massdistribution at an optimized value of momentum in a possiblefuture experiment could give a good reaction rate and less kine-matical effects.To study the dependence of the πΣ invariant mass on the N d model of interaction, in Fig. 4, we calculated the πΣ massspectra for two different N d interaction, including a one-term( V NdB ) and a two-term ( V NdA ) potential. Comparing the ex-tracted invariant mass spectra for one-term potential and thecorresponding mass spectra for the two-term potential, we cansee that the
N d interaction can affect the mass spectra espe-cially, for energies above the ¯ K + N d mass threshold. How-ever, the mass spectrum in energy region around the ¯ KN poleposition did not change seriously by changing the N d modelof interaction. πΣ (MeV/c )048 d σ / d E d ( - M e V - ) SIDD (A)SIDD (B)p K - = 250 MeV/c Fig. 4. (Color online) The invariant mass spectra for K − + He reac-tion. We used the one-term (solid curves) and two-term (square sym-bols) potential for Nd interaction to study the dependence of the πΣ mass spectra to the Nd model of interaction. The symbols A and B are corresponding to V NdA and V NdB , respectively. In our calculations,the one-pole version of SIDD potential was used to describe the ¯ KN interaction. ¯ KN d − πΣd In subsection 3.1, we solved the one-channel AGS equationsfor ¯ KN d system and the decaying to the lower lying chan-nels is included by using the so-called exact optical potentialfor ¯ KN interaction. In one-channel Faddeev calculations theeffect of the τ πΣ → πΣ amplitude was excluded. Based on chi-ral unitary approach, the first and second pole of Λ (1405) haveclearly different coupling nature to the meson-baryon channels;the higher energy pole dominantly couples to the ¯ KN channel,while the lower energy pole strongly couples to the πΣ chan-nel. Due to the different coupling nature of these resonances,the shape of the Λ (1405) spectrum can be different dependingon the initial and final channels. In the ¯ KN → πΣ amplitude,the initial ¯ KN channel gets more contribution from the higherpole with a larger weight. Consequently, the spectrum shapehas a peak around 1420 MeV coming from the higher pole [22,26]. This is obviously different from the πΣ → πΣ spectrumwhich is largely affected by the lower pole. Therefore, the ex-tracted mass spectra in subsection 3.1 can not reproduce ex-actly the possible experimental mass spectra. To calculate theexact πΣ mass spectra for ¯ K + ( N d ) reaction, we solved theFaddeev equations for coupled-channel ¯ KN d − πΣd system.In addition to the above mentioned reaction, we need an inter-action model for Σd and πd subsystems. In present calcula-tions, the effect of πd interaction is neglected. To describe the Σd interaction, we used a one term complex potential in a formgiven in Eq. 11. To define the parameters of the Σd interaction,we used the Σd scattering length, which can be extracted fromthe Faddeev equations of the three-body Σ ( N N ) s =1 ,I =0 sys-tem. The antisymmetric Faddeev equations for Σd system canbe given by K sI,s (cid:48) I (cid:48) Σ,Σ = (cid:88) s (cid:48)(cid:48) I (cid:48)(cid:48) M sI,s (cid:48)(cid:48) I (cid:48)(cid:48) Σ,N τ s (cid:48)(cid:48) I (cid:48)(cid:48) N ( K s (cid:48)(cid:48) I (cid:48)(cid:48) ,s (cid:48) I (cid:48) N ,Σ − K s (cid:48)(cid:48) I (cid:48)(cid:48) ,s (cid:48) I (cid:48) N ,Σ ) K sI,s (cid:48) I (cid:48) N ,Σ − K sI,s (cid:48) I (cid:48) N ,Σ = 2 M sI,s (cid:48) I (cid:48) N ,Σ + (cid:88) s (cid:48)(cid:48) I (cid:48)(cid:48) M sI,s (cid:48)(cid:48) I (cid:48)(cid:48) N ,Σ τ s (cid:48)(cid:48) I (cid:48)(cid:48) Σ K s (cid:48)(cid:48) I (cid:48)(cid:48) ,s (cid:48) I (cid:48) Σ,Σ − (cid:88) s (cid:48)(cid:48) I (cid:48)(cid:48) M sI,s (cid:48)(cid:48) I (cid:48)(cid:48) N ,N τ s (cid:48)(cid:48) I (cid:48)(cid:48) N ( K s (cid:48)(cid:48) I (cid:48)(cid:48) ,s (cid:48) I (cid:48) N ,Σ − K s (cid:48)(cid:48) I (cid:48)(cid:48) ,s (cid:48) I (cid:48) N ,Σ ) (15)where the Σd scattering length is given by a Σd = − π µ Σd K , Σ,Σ ( p → , p (cid:48) → z = − E d ) (16)where µ Σd is the reduced mass of Σd and E d is the binding en-ergy of deuteron. To solve Eq. 15, we need as input a potentialmodel for ΣN − ΛN and N N interactions. In our three-bodystudy, to describe the singlet and triplet ΣN interaction, weused the potentials given in Ref. [52] and for triplet N N in-teraction, we choose a potential of PEST type [61], which is aseparablization of the Paris potential. The Σd scattering lengthvalue obtained with the two above mentioned ΣN − ΛN and N N potentials is a Σd = − .
59 + i .
71 fm − (17) . Esmaili, S. Marri, M. Raeisi and A. Naderi Beni: Trace of Λ (1405) resonance in low energy K − + He → ( π Σ ) + d reaction 7 The whole dynamics of coupled-channel ¯ KN d − πΣd three-body system is described in terms of the transition amplitudes K αβ ; I i I j i,m ; j,n . The superscripts α, β = 1 , are included to take intoaccount the coupling between ¯ KN d and πΣd systems. TheFaddeev AGS equations for ¯ KN d − πΣd system can be ex-pressed by K I ¯ K I ¯ K ¯ K,m ; ¯
K,n = (cid:88) rr (cid:48) ( M ,I ¯ K I N ¯ K,m ; N,r τ I N N ( rr (cid:48) ) K I N I ¯ K N,r (cid:48) ; ¯
K,n + M ,I ¯ K I d ¯ K,m ; d,r τ ,I d d ( rr (cid:48) ) K ,I d I ¯ K d,r (cid:48) ; ¯ K,n + M ,I ¯ K I d ¯ K,m ; d,r τ ,I d d ( rr (cid:48) ) K ,I d I ¯ K d,r (cid:48) ; ¯ K,n ) K I N I ¯ K N,m ; ¯
K,n = M I N I ¯ K N,m ; ¯
K,n + (cid:88) rr (cid:48) ( M I N I ¯ K N,m ; ¯
K,r τ ,I ¯ K ¯ K ( rr (cid:48) ) K ,I ¯ K I ¯ K ¯ K,r (cid:48) ; ¯
K,n + M ,I N I d N,m ; d,r τ ,I d d ( rr (cid:48) ) K ,I d I ¯ K d,r (cid:48) ; ¯ K,n + M ,I N I d N,m ; d,r τ ,I d d ( rr (cid:48) ) K I d I ¯ K d,r (cid:48) ; ¯ K,n ) K I d I ¯ K d,m ; ¯ K,n = M I d I ¯ K d,m ; ¯ K,n + (cid:88) rr (cid:48) ( M I d I ¯ K d,m ; ¯ K,r τ I ¯ K ¯ K ( rr (cid:48) ) K I ¯ K I ¯ K ¯ K,r (cid:48) ; ¯
K,n + M I d I N d,m ; N,r τ I N N ( rr (cid:48) ) K I N I ¯ K N,r (cid:48) ; ¯
K,n ) K I d I ¯ K d,m ; ¯ K,n = (cid:88) rr (cid:48) M I d I π d,m ; π,r τ I π π ( rr (cid:48) ) K I π I ¯ K π,r (cid:48) ; ¯ K,n K I d I ¯ K π,m ; ¯ K,n = (cid:88) rr (cid:48) ( M I π I d π,m ; d,r τ I d d ( rr (cid:48) ) K I d I ¯ K d,r (cid:48) ; ¯ K,n + M I π I d π,m ; d,r τ I d d ( rr (cid:48) ) K I d I ¯ K d,r (cid:48) ; ¯ K,n ) . (18)The break-up amplitude for ¯ K +( N d ) → ( πΣ )+ d reactionin terms of the Faddeev transition amplitudes can be given by T ( πΣ )+ d ← ( Nd )+ ¯ K ( (cid:126)k d , (cid:126)p d , ¯ P ¯ K ; z )= (cid:88) n g I =0 πΣ ( (cid:126)k d ) τ I =0 πΣ ← ¯ KN ( z − E d ( (cid:126)p d )) K d,
1; ¯
K,n ( p d , ¯ P ¯ K ; z )+ (cid:88) n g I =0 πΣ ( (cid:126)k d ) τ I =0 πΣ ← πΣ ( z − E N ( (cid:126)p d )) K d,
1; ¯
K,n ( p d , ¯ P ¯ K ; z )+ (cid:88) n (cid:104) [ π ⊗ Σ ] I =0 ⊗ d | π ⊗ [ Σ ⊗ d ] I =1 (cid:105) g I =1 Σd ( (cid:126)k π ) × τ I =1 Σd ← Σd ( z − E π ( (cid:126)p π )) K π,
1; ¯
K,n ( p π , ¯ P ¯ K ; z ) , (19)where the momenta (cid:126)p π and (cid:126)k π are given by (cid:126)p π = (cid:126)k d − m π m π + m Σ (cid:126)p d (cid:126)k π = − m d m Σ + m d (cid:126)k d − m Σ ( m π + m Σ + m d )( m π + m Σ )( m Σ + m d ) (cid:126)p d . (20)As one see from Eq. 19, in coupled-channel calculationsplus the K d,
1; ¯
K,n amplitude, the effect of the K d,
1; ¯
K,n and K π,
1; ¯
K,n are also included which accordingly, produces a moreprecise mass spectrum for πΣ . Inserting the new break-up am-plitude (Eq. 19) in Eq. 13, we can calculate the πΣ mass spec-trum for ¯ K + ( N d ) → ( πΣ ) + d reaction. In Fig. 5, we cal-culated the πΣ mass spectrum using different potential mod-els for ¯ KN − πΣ interaction. As one can see from Fig. 5, in the fully coupled-channel calculations the resonance part ofthe mass spectrum is stronger and a more clear peak structurecan be seen in πΣ invariant mass. However, the observed peakstructure of each model of ¯ KN − πΣ interaction is locatedat lower energies than those presented in Table 1, due to themomentum distribution in p − d subsystem. By comparison ofresults using one-channel and coupled-channel Faddeev equa-tions, it may be possible to study the effect of τ πΣ → πΣ ampli-tude on πΣ invariant mass. As can be seen in Fig. 5, the ex-tracted mass spectra in coupled-channel calculations are ratherdifferent from those by one-channel Faddeev calculations and itmay be possible to discriminate between these two approaches.Therefore, the one-channel Faddeev calculations cannot be astrong tool to study the dynamics of Λ (1405) resonance in ¯ K + ( N d ) → ( πΣ ) + d reaction.As one can see in panel (B), an accurate measurements ofthe πΣ mass distribution at p K − = 100MeV / c can differen-tiate the AY and Rev-A potentials from the others and study-ing K − + He reaction at p K − = 250MeV / c , one have achance to discriminate between the other three potentials un-der the consideration. Looking at Fig. 5 one can clearly that forchiral energy-dependent potential, the magnitude of the massspectrum above the ¯ KN threshold is considerably smaller thanthose by other potentials for all kaon incident momenta. There-fore, such a combined study at two different initial energiesshows a big potential to discriminate between possible mecha-nisms of the formation of Λ (1405) resonance. In summary, the Faddeev-type calculations of ¯ KN d systemwith quantum numbers I = 0 and s = were performed. Solv-ing the one-channel and full coupled-channel Faddeev equa-tions for ¯ KN d − πΣd system, we calculated the πΣ massspectrum resulting from K − + He → πΣd reaction by usingthe deuteron mass spectrum. The logarithmic singularities thatappear when solving the AGS equations for the real scatteringenergies have been successfully handled by making use of thepoint method. To investigate the dependence of the resultingmass spectrum on models of ¯ KN − πΣ interaction, differentphenomenological and chiral based potentials having the one-and two-pole structure of Λ (1405) resonance, were used. Wehave examined how well the signature of the Λ (1405) reso-nance manifests itself in the πΣ invariant mass. By comparisonof results using different interaction models, it was found that itmay be possible to discriminate between different approachesdescribing the ¯ KN interaction.The πΣ mass spectrum was calculated for kaon incidentmomentum p cmK − = 50 − MeV/c. However, the kinematicaleffects are important at low momenta and the signal of Λ (1405) is masked, we have found that within our model for momentaabove the 250 MeV/c, a clear bump produced by Λ (1405) res-onance appear in the K − + He → πΣd cross section inthe energy region between the ¯ KN and πΣ thresholds, whichstrongly suggests that the clear signals of Λ (1405) resonanceshould be detected by measuring of πΣ invariant mass distri-butions at the relevant energies.By performing the fully coupled-channel calculations for ¯ KN d − πΣd system, we studied the dependence of the πΣ J. Esmaili, S. Marri, M. Raeisi and A. Naderi Beni: Trace of Λ (1405) resonance in low energy K − + He → ( π Σ ) + d reaction πΣ (MeV/c )024 d σ / d E d ( - M e V - ) SIDD SIDD Chiral-IKSAYRev-A p K - = 50 MeV/c (a) πΣ (MeV/c )01 d σ / d E d ( - M e V - ) SIDD SIDD Chiral-IKSAYRev-A p K - = 100 MeV/c (b) πΣ (MeV/c )00.6 d σ / d E d ( - M e V - ) SIDD SIDD Chiral-IKSAYRev-A p K - = 150 MeV/c (c) πΣ (MeV/c )024 d σ / d E d ( - M e V - ) SIDD SIDD Chiral-IKSAYRev-A p K - = 250 MeV/c(d) Fig. 5. (Color on line) Same as Fig.3, but in the present calculations, the full coupled-channel Faddeev AGS equations for ¯ KNd − πΣd systemare solved. mass spectrum on the τ πΣ → πΣ amplitude. It was shown thatthe full coupled-channel calculations can produce a consider-ably different mass spectrum and the inclusion of the τ πΣ → πΣ amplitude is important for an exact study of the Λ (1405) reso-nance structure.This work has been financially supported by the researchdeputy of Shahrekord University. The grant number was 141/2843. References
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