Hypernuclear production by ($gamma, K^+$) reaction within a relativistic model

a r X i v :0710.4888v 2 [n u c l -t h ] 19 A p r 2008Hypernuclear production by (γ,K +)reaction within a relativistic

model

R.Shyam 1,2,H.Lenske 2,and U.Mosel 2

1

Saha Institute of Nuclear Physics,Kolkata,India 2Institute f¨u r Theoretische Physik,Universit¨a t Giessen,D-35292Giessen,Germany

(Dated:April 19,2008)

Abstract Within a fully covariant model based on an e?ective Lagrangian picture,we investigate the hypernuclear production in photon-nucleus interaction on 16O target.The explicit kaon produc-tion vertex is described via creation,propagation and decay into relevant channel of N ?(1650),N ?(1710),and N ?(1720)intermediate baryonic resonance states in the initial interaction of the incident photon with one of the target protons.Bound state nucleon and hyperon wave functions are obtained by solving the Dirac https://www.wendangku.net/doc/ba5694651.html,ing vertex parameters determined in the previous studies,contributions of the N ?(1710)baryonic resonance dominate the total production cross sections which are found to peak at photon energies below 1GeV.The results show that photo-production is the most appropriate means for studying the unnatural parity hypernuclear states,thus accessing the spin dependence of the hyperon-nucleon interaction.PACS numbers:21.80.+a,13.60.-r,13.75.Jz

Lambda hypernuclei are the most extensively studied hypernuclear systems,both exper-imentally(see,e.g.,a recent review[1])as well as theoretically[2,3,4].They have,tra-ditionally,been produced by stopped as well as in-?ight(K?,π?)and(π+,K+)reactions. Alternatively,Λ-hypernuclei can also be produced with proton as well as electromagnetic probes.The feasibility of hypernuclear production with proton beams has been investigated in Refs.[5,6].Recently,discrete hypernuclear states have been produced for the?rst time in electron induced reactions on light nuclear targets at the Jlab[7,8,9].First measurements of the(γ,K+)reaction on a nuclear target(12C)have been reported long ago[10,11].In-terest in this?eld has been revived as a number of experiments are planned for this reaction at accelerators MAMI-C in Mainz,ELSA in BONN and also at Jlab(see,e.g.,Ref.[12]).In this paper we report a theoretical study of the(γ,K+)reaction on a16O target.

In contrast to the hadronic reactions[(K?,π?),and(π+,K+)]which are con?ned mostly to the nuclear surface due to strong absorption of both K?andπ±,the(γ,K+)reaction occurs deep in the nuclear interior due to weaker interactions of both photon and K+with the nucleus.This property makes this reaction an ideal tool for studying the deeply bound hypernuclear states provided the corresponding production mechanism is reasonably well understood.Unlike the hadronic reactions which excite predominantly the natural parity hypernuclear states,both unnatural and natural parity states are excited with comparable strength in the(γ,K+)reaction.This is because sizable spin-?ip amplitudes are present in the elementary p(γ,K+)Λreaction due to the fact that the photon has spin1and small angles dominate this reaction.This feature persists in the hypernuclear photoproduction. Furthermore,Since,a proton in the target nucleus is converted into a hyperon,this reaction leads to the production of neutron rich hypernuclei(see,e.g.,Ref.[13])which may carry exotic features such as a halo structure.

Several theoretical investigations of the(γ,K+)reaction on nuclear targets have been reported in the literature[14,15,16,17,18,19,20].In these studies,the kaon photopro-duction amplitudes on nuclei are calculated within an impulse approximation by determining expectation values of the operator for the elementary p(γ,K+)Λproduction process between initial and?nal states of the reaction.This operator is constructed either by using the Feyn-man diagrammatic approach where graphs corresponding to Born terms and resonance terms in s and u channels,are included[15,16,20,21],or phenomenologically by parameterizing the experimental cross sections for the elementary process[18,19].Although in Ref.[17],

A γΛB

K +FIG.1:Representation of the type of Feynman diagrams included in our calculations.The elliptic shaded area represents the optical model interactions in the outgoing channel.

Dirac spinors have been used for bound state wave functions in the initial and ?nal channels,a full covariant calculation of this reaction is still missing.

In this paper,we study the A (γ,K +)ΛB reaction within a fully covariant model by retaining the ?eld theoretical structure of the interaction vertices and by treating the baryons as Dirac particles moving in a static nuclear mean-?eld.This type of approach has previously been used in Ref.[6]to describe the hypernuclear production in proton-nucleus collisions.In our model,the initial state interaction of the incoming photon with a bound proton leads to excitations of N ?(1650)[12+],and N ?

(1720)[3

contributions become relatively large at photon energies in excess of2.1GeV[20].Since,in this paper we have concentrated only on investigating the role of baryonic resonances in the (γ,K+)reaction and have ignored the nucleon intermediate states,we have omitted these terms to keep our model as simple as possible.Furthermore,in this exploratory study,to reduce the further computational complications we have used plane waves(PW)to describe the relative motion of the outgoing kaon which is justi?ed by the relatively weaker mutual interaction in this channel.

The e?ective Lagrangians for the electromagnetic couplings of spin-1

)ˉΨN?ΓμνΨN Fμν+h.c.,(1)

4m N

where the operatorΓμνisγ5σμν(σμν)for odd(even)parity resonances.Fμνrepresents the electromagnetic?eld tensor:Fμν=?νAμ??μAν.We have used the notations of Ref.[25] through out in this paper.For the spin-3

)ˉΨαN?Θαμ(z1)γνΓΨN Fμν

2m N

eg2N?Nγ

?(

particle.This involves the o?shell projector

2

Θαμ(z)=gαμ?1

?elds.The choice of this parameter is arbitrary and in earlier

2

studies it has been treated as a free parameter to be determined by?tting to the data(see, e.g.[22]).For a more detailed discussion we refer to[22,26,27].The electromagnetic coupling constants g1,g2are related to the helicity couplings[A1/2,3/2](see,e.g.Ref[24]) which are taken from Ref.[22].There these are determined in a coupled channels K-matrix method by?tting simultaneously to all the available data for transitions fromγN to?ve meson-baryon?nal states,πN,ππN,ηN,KΛ,and KΣfor center of mass energies ranging from threshold to2GeV including all the baryonic resonances up to spin≤3

TABLE I:Coupling constants for the N?ΛK vertices and the helicity amplitudes used in the calculations[22,28].

vertex g A1/2A3/2

(10?3GeV?1/2)(10?3GeV?1/2)

L N?

3/2ΛK+

=

g N?

3/2

ΛK+

2

and spin-3

widths in denominators of resonance propagators to account for the fact that they have ?nite lifetime for decay into various channels.

Spinors ψ(p )are solutions of the Dirac equation in momentum space for a bound state problem in the presence of an external potential ?eld [6,30]

p /ψ(p )=m N ψ(p )+F (p ),

(5)where

F (p )=δ(p 0?E )

d 3p ′V s (?p ′)ψ(p +p ′)?γ0 d 3p ′V 0v (?p ′)ψ(p +p ′) .(6)

In Eq.(6),the real scalar and timelike vector potentials V s and V 0v represent,respectively,the momentum space local Lorentz covariant interaction of single nucleon or Λwith the remaining (A ?1)nucleons.We denote a four momentum by p =(p 0,p ).The magnitude of the three momentum p is represented by k ,and its directions by ?p .p 0is the time like component of p .Spinors ψ(p )and F (p )are written as

ψ(p )=δ(p 0?E )

f (k )Y m j ?1/2j (?p )?i

g (k )Y m j ?′1/2j (?p ) ,F (p )=δ(p 0?E ) ζ(k )Y m j

?1/2j (?

p )?iζ′(k )Y m j ?′1/2j (?p )

,(7)where f (k )[ζ(k )]is the radial part of the upper component of the spinor ψ(p )[F (p )].Similarly g (k )[ζ′(k )]are the same of their lower component.f (k )and g (k )represent Fourier transforms of radial parts of the corresponding coordinate space spinors.ζ(k )are related to f ,g and the scalar and vector potentials (see Ref.[30]for more details).We have de?ned ?′=2j ??with ?and j being the orbital and total angular momenta,and

Y m j

?1/2j (?p )=

m ?μ

Y ?m ?(?p )χ1/2μ,

where Y represents the spherical harmonics and χ1/2μthe spin space wave function of a spin-1d ?=1

(E γ+E A )2p K

where Eγand E A are the total energies of incident photon and the target nucleus,respec-tively while m A and m B are the masses of the target and residual nuclei,respectively. R represents summation over all the resonances.M i and M f are initial and?nal spin states, respectively,and?is the photon polarization.

We have chosen the reaction16O(γ,K+)16ΛN for the?rst numerical application of our model,as this reaction is well known to have a simple structure.The initial state is a doubly closed system.The binding energies(BE)of1s1/2,1p3/2and1p1/2single particle states in

16

Λ

N hypernucleus are taken to be13.20MeV,2.50MeV and2.04MeV,respectively which are the same as those given in Ref.[17].The BE of the1p3/2and1p1/2nucleon hole states in16O are taken as18.40MeV and12.13MeV,respectively[18].Because of the large separation in the binding energies of the nucleon hole states,the hypernuclear spectrum is clearly divided

into4groups corresponding to con?gurations[p?1

1/2,sΛ],[p?1

3/2

,sΛ],[p?1

1/2

,pΛ],and[p?1

3/2

,pΛ].

The con?guration mixing is negligible except may be for the Jπ=1+hypernuclear state. AnyΛN residual interaction that may lead to con?guration mixing as considered in Ref.[32] is neglected in our study.

Spinorsψ(p)for bound hypernuclear and nucleon states are obtained by Fourier trans-formations of the corresponding coordinate space spinors which are determined by solving the Dirac equation with scalar and vector potential?elds(V s and V v,respectively)with a Woods-Saxon radial form with(reduced radius)r s=r v=0.983fm,and(di?useness) a s=0.70fm and a v=0.58fm.For a given state,the depths of these?elds(V0s and V0v) have been searched so as to reproduce the BE of that state.For the single particleΛstates 1s1/2,1p3/2and1p1/2the values of(V0s and V0)were(-204.78MeV and165.89MeV),(-228.49MeV and185.96MeV),and(-251.80and203.96MeV),respectively,while for each of the nucleon hole states1p3/2and1p1/2,they were(-445.0MeV and360MeV).We see that the strengths of the mean?eld self energies for theΛstates are about1/2of those of the nucleon states which is line with the results of the microscopic Dirac-Bruckner calculations as shown in Ref.[4].

It is shown in Ref.[30]that spinors calculated in this way provide a good description of the experimental nucleon momentum distributions for various nucleon orbits.Fig.2shows, e.g.,the momentum space1p1/2Λand1p3/2Λhyperon spinors as functions of the momentum transfer(q),for the16ΛN hypernucleus.In the lower panel of Fig.2we show the momentum distribution of theΛhyperon for these states in16ΛN.We note that only for q<1.5fm?1,is

FIG.2:(Upper panel)(|g(q)|)(dashed line) components of the momentum space spinors for1p3/2Λand1p1/2Λorbits in16ΛN hypernucleus. (Lower panel)hyperon momentum distributions(de?ned asρ(q)=[|f(q)|2+|g(q)|2])for the same states.

the magnitude of the lower component(|g(q)|)substantially smaller than that of the upper component(|f(q)|).In the region of q pertinent to the kaon production,|g(q)|may not be negligible.In fact,it has been shown earlier[17]that the relativistic e?ects resulting from the small component of Dirac bound states are large for the kaon photoproduction reactions on nuclei.

The threshold for the kaon photoproduction on16O is about680MeV.The momentum transfer involve in this reaction at zero degrees is above500MeV/c.In Fig.3,we investigate the contributions of the various resonance intermediate states to the total cross section for

populating the(1p?1

3/2,1sΛ

1/2

)2?state in the16O(γ,K+)16ΛN reaction as a function of photon

energy.We see that the individual contributions of the N?(1710)intermediate state by far dominate the cross sections.In comparison to this,cross sections corresponding to the N?(1650)state are at least one order magnitude smaller and those of the N?(1720)state are smaller by3-4orders of magnitude(these contributions are omitted from this?gure consequently).The nearly negligible contribution of this resonance has also been noted in the study of A(p,K+)ΛB reaction[6].It is worthwhile to note that in Ref.[20]individual contributions of various resonances are not speci?ed although the resonance terms as a whole are shown to dominate both elementary and nuclear photo-kaon production cross sections.

Another noteworthy aspect of Fig.3is that cross sections peak at photon energies around 900MeV,which is about200MeV above the production threshold for this reaction.In-terestingly,the total cross section of the elementary p(γ,K+)Λreaction also peaks about

FIG.3:Individual contributions line)resonance states to the total cross section for the16O(p,K+)NΛN reaction for the indicated con?guration as a function of photon energy.The solid line shows the coherent sum of the contributions of these two and N?(1720)resonance states.The individual contributions of the N?(1720)resonance are 3-4four orders of magnitude smaller and are not shown here separately.

the same energy above the corresponding production threshold(910MeV).Although the absolute magnitude of our near peak energy cross sections are comparable to those reported in Ref.[17],their drop o?with the beam energy is faster than those of these authors.Such a strong drop of the cross sections with beam energy beyond the peak position was also seen in the calculations of(p,K+)reactions on nuclei within a similar model.It may partly be due to lack of the kaon-nucleus distortion e?ects in our calculations.It should be men-tioned here that calculations performed with only the upper component of the bound state is about6-8%smaller than the total cross sections shown in https://www.wendangku.net/doc/ba5694651.html,rger e?ects of the lower components are seen in the angular distributions at larger angles.However,since cross sections are quite small at these angles in comparison to those at smaller ones,the total cross sections shows less sensitivity to the lower component.More comprehensive investigation of the relativistic e?ects due to the lower component is made in Refs.[17]where it is shown that the cross sections obtained by using fully relativistic small component are signi?cantly di?erent from those where the small component is related to the large component by a free space relation.It is also found that the di?erences between cross sections obtained by using Dirac or Schr¨o dinger solutions for the upper component are non-negligible.

In Fig4,we present the excitation spectrum for four groups of(ΛN?1)hypernuclear states involving bound1s and1pΛorbitals in the hypernucleus16ΛN.The relative excitation strengths for a given J state of each group of[(n?j)?1p,(n?j)Λ]con?guration,are obtained

γE = 900 MeV ?1[(nlj), (nlj)p Λ

]Configuration ,1s (1p ?1

1/2)Λ1/2?11/2)Λ1/2?13/2)Λ1/2(1p ?1 ,1p 3/23/2

)Λ(1p ,1p (1p ,1s γO (, K ) +16

Λ16N 1R e l a t i v e S t r e n g t h 0.20.4

0.60.81.01+0+21??0??3+2+1

+0+FIG.4:Bound state excitation spectrum of hypernucleus 16Λ

N.by dividing the total cross section of that J state by that of the state having the maximum cross section within that group.We ?rst note that within each group the highest J state is most strongly excited which is in line with the results presented in Refs.[17,32].Further-

more,unnatural parity states within each group [except for the ground state (1p ?

11/2,1s Λ1/2)

con?guration],are preferentially excited by this reaction.For example,within the group

(1p ?

13/2,1p Λ3/2),cross section of the 3+state is larger than that of the 2+state by about a fac-tor of 2.5and by more than an order of magnitude than that of the 0+state.The unnatural parity states are excited through the spin ?ip process.Thus,kaon photoproduction on nuclei is an ideal tool to investigate the structure of unnatural parity hypernuclear states.The addition of unnatural parity states to the spectrum of hypernuclei is expected to constrain the spin dependent part of the e?ective Λ?N interaction more tightly.

In summary,we studied the hypernuclear production by (γ,K +)reaction on 16O within a covariant model where in the initial collision of the photon with a target proton,N ?(1710),N ?(1650)and N ?(1720)baryonic resonances are excited which subsequently propagate and decay into a Λhyperon which gets captured in one of the nuclear orbits,and a K +which goes out.Wave functions of nucleon and Λbound states are obtained by solving the Dirac equation with appropriate potential ?elds.The distortion e?ects in the K +channel have not been included in this study.However,as shown in Refs.[17,32],these e?ects are weak for reactions on p -shell nuclei but they may be more signi?cant for heavier systems.Since,we have not included the nucleon intermediate states (Born terms),the absolute magnitudes of our total cross sections may be uncertain to the extent of about 10%.

Using the vertex constants determined in previous studies,the excitation of N ?(1710)resonance dominates the hypernuclear production process.Similar results were also found

in the previous studies of the A(p,K+)ΛB reaction.The total production cross sections peak at photon energies which are above the corresponding production threshold by almost the same amount of energy as is the position of the maximum in the elementary cross section away from its respective threshold.

Our calculations con?rm that the(γ,K+)reaction on nuclei selectively excites the high spin unnatural parity states,which makes it an ideal tool for investigating the spin-?ip transitions which are only weakly excited in reactions induced by hadronic probes.Therefore, electromagnetic hypernuclear production provides a fuller knowledge of hypernuclear spectra and will impose more severe constraints on the models of theΛN interaction,particularly on its the poorly known spin dependent part.To this end it is important to extend our model to include distortion e?ects in the?nal channel so that the mechanism of this reaction can be understood more properly.

This work has been supported by Sonderforschungsbereich/Transregio16,Bonn-Giessen-Bochum of the German Research Foundation.One of the authors(RS)acknowledges the support of Abdus Salam International Centre for Theoretical Physics in form of a senior associateship award.

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