prospect of photoproduction of medium-heavy …

January 29, 2004 14:12 WSPC/Trim Size: 9in x 6in for Proceedings motoba˙ws

PROSPECT OF PHOTOPRODUCTION OFMEDIUM-HEAVY HYPERNUCLEI∗

T. MOTOBA

Lab. of Physics, Osaka Electro-Communication University,Neyagawa 572-8530, Japan†

andPhysics Department, Brookhaven National Laboratory,

Upton, New York 11973-5000, U.S.A.

P. BYDZOVSKY AND M. SOTONA

Nuclear Physics Institute, 250 68 Rez near Prague, Czech Republic

K. ITONAGA

Miyazaki Medical College, Miyazaki 889-1692 Japan

K. OGAWA

Department of Physics, Chiba University, Chiba 263-8522, Japan

O. HASHIMOTO

Department of Physics, Tohoku University, Sendai 980-8578, Japan

Basic characters of the elementary hyperon photoproduction process are clarified.Novel aspects of photoproduction of hypernuclei are demonstrated by choosing typ-ical medium-heavy nuclear targets, which should provide an important opportunityof studying dynamical coupling between a hyperon and nuclear core excitation.

1. Introduction

In the spectroscopic study of hypernuclei, the γ-ray measurements in recentyears have achieved a great success in providing us with remarkably high

∗Work done in part of the Japan-Europe Joint Research Projects supported by JapanSociety for the Promotion of Science (2000-2002).†Permanent address; E-mail:[email protected]

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resolution data on their energy levels 1. However, it is not always easy toproduce hypernuclei abundantly enough to make such measurements pos-sible, in view of the practical kinematical conditions. As for the reactionspectroscopy, there are many challenges to be done for investigating hyper-nuclear structures. Recently the photo-/electro-production of hyperons andhypernuclei has attracted much attention in strangeness nuclear physics.

First, the high-energy electron beams became available at the Jeffer-son National Laboratory (JLab) and its beam intensity is so high that canovercome practically the small hyperon production cross sections. In ad-dition to the elementary process experiments, the first experiment on thenuclear target, 12C(e, e′K+)12

Λ B, was successfully carried out 2, providing apromising result which is consistent with the theoretical calculation 3.

Secondly, the photoproduction of hypernuclei has a characteristic novelmerit of exciting unnatural parity high-spin states which cannot be reachedthrough the (K−, π−) and (π+, K+) reactions. Another merit to be addedis to convert a proton into Λ so that proton-defficient mirror hypernucleiare produced.

Next, an emphasis should be put on the energy resolution of the electro-production which is expected to be several times better than the secondarymesonic beams: the former typical value is ∆E 0.5 MeV 4, whereas thelatter best value is 1.45 MeV in the KEK-SKS achievement 5,6.

In this report, first we discuss basic properties of hyperon photopro-duction process and, secondly, theoretical predictions are demonstrated bychoosing medium-heavy nuclear targets such as 28Si and 40Ca. Determi-nation of energy position of typical high-spin states with unnatural parityhelps us test various models of Y-N potential models 7.

2. Kinematics and Nature of the γp → ΛK+ Reaction

Characteristics of hypernuclear production reaction depend basically onthe magnitude of the momentum transfer and the nature of the elementaryamplitudes describing the hyperon photoproduction process.

First we compare the n(K−, π−)Λ, n(π+, K+)Λ, and p(γ, K+)Λ reac-tions. In Fig. 1 the hyperon recoil momenta qΛ in these reactions areplotted as a function of the projectile momentum. It is well known thatthe small values of q

(K,π)Λ = 50 − 150 MeV/c involved in the (K−, π−) re-

action at pK 0.8 GeV/c favor the hypernuclear excitation with recoillessor very small angular momentum transfer (∆L ≤ 1, ∆S = 0). As for the(π+, K+) reaction, the sizable momentum transfer q

(π,K)Λ 400 MeV/c


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π

π

γ

° θ

°

°

θ

°

Figure 1. Hyperon recoil momentum qΛ as a function of projectile lab momentum. Twocurves for each reaction correspond to the meson lab scattering angles: θlab= 0 and 10

deg.

together with the spin-nonflip dominance leads to the selective excitationof hypernuclear natural parity states with maximum aligned angular mo-mentum (J = Lmax = ln + lΛ of [(nlj)−1

n (nlj)Λ]J in case of Jtarget = 0+).It is interesting to see that the momentum transfer q

(γ,K)Λ is quite similar

to q(π,K)Λ , suggesting its high-spin selective excitation as in the (π+, K+)

reaction. The Λ recoil amounts to 353 MeV/c (θlab=0 deg) − 425 MeV/c(10 deg) at pγ = 1.3 GeV/c which is chosen in the actual experimentalkinematics.

Many theoretical attempts have been made to describe the elementaryhyperon photoproduction processes. In the hypernuclear calculations, weadopt four isobaric models denoted hereafter as Adelseck-Wright (AW2) 11,Adelseck-Saghai (AS1) 12, Williams-Ji-Cotanch (C4) 13 and Saclay-Lyon A(SLA) 14. In Fig. 2 we show how well these theoretical amplitudes repro-duce the experimental differential cross sections at a fixed kaon scatteringangle. Predictions of all models are in good agreement with the data forenergies up to 1.4 GeV, but for the higher energies the 3 models AW2,AS1, and C4 deviate remarkably from the new SAPHIR data 10 (trian-gle data points) which were not included in the fitting procedure. In thepresent hypernuclear calculations the cross section estimates are performedat Elab

γ ≈ 1.3 GeV where all the four models provide acceptable descriptionof the elementary process. In more detail the SLA model gives 15 − 20%overestimates with respect to the experimental differential cross sectionsat θCM

K < 15 deg(θlabK < 6.5 deg) This suggests the amount of theoretical


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1

1.5

2

2.5

3dσ

/dΩ

lab (

µb/s

r)

AW2AS1C4SLA

1 1.2 1.4 1.6 1.8 2

Eγ (GeV)

0

0.2

0.4

dσ/d

Ωc.

m. (

µb/s

r)

AW2AS1C4SLA

θK

lab = 3 deg

θK

c.m. = 90 deg

γ + p ⎯→ Λ + K+

Figure 2. Experimental and theoretical differential cross sections for the γp → ΛK+

reaction are plotted as a function of the photon lab energy at a fixed kaon scattering

angle. See text for the models denoted as AW2, AS1, C4, and SLA. The data are from

Refs. 8 (solid circles), 9 (empty circles), and 10 (triangles).

uncertainty in predicting the hypernuclear production rate with this model.Next we emphasize that the unique character of hypernuclear photo-

production comes from the spin-flip dominance. The elementary transitionmatrix for γp → ΛK+ can be expressed in terms of four complex amplitudes(f0, g0, g1, and g−1) defined in the lab frame of Fig. 3 as:

M ≡< k − p,p|t|k, 0 >Lab= ε0(f0 + g0σ0) + εx(g1σ1 + g−1σ−1) (1)

Here εx and ε0 denote the unit vectors describing the photon polarizationand σm is the Pauli spin operator for the baryon. The amplitudes arenormalized as follows:(

dσ

dΩ

)lab

=(2π)4p2

KEKEΛ

pK(EΛ + EK) − EγEK cos θK(|f0|2+|g0|2+|g1|2+|g−1|2) (2)

where the energies, momenta, and angle are all in the laboratory frame.In order to demonstrate the spin-flip dominance, we list in Table 1 the

squared magnitudes of the four complex amplitudes. One sees that, at smallangles concerned here, the spin-dependent amplitudes (g0, g1, g+1) clearly


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Figure 3. Hyperon photoproduction kinematics in the laboratory frame

dominate over the the spin-nonflip one (f0). Although all of the modelswere obtained in fitting to the available scattering data, one also noticesthat these amplitudes are considerably different from each other. Espe-cially these models predict very different polarization values, respectively,for which the experimental data are very few.

We remark that the important two factors in the elementary process— its large momentum transfer and spin-flip dominance — give rise to theselective excitation of hypernuclear high-spin states with unnatural parity.

Table 1. Magnitudes of the spin-independent (f0) and spin-dependent ampli-tudes (g’s) in units of nb/sr/GeV2. The evaluation is made at Elab

γ = 1.3 GeV.

The laboratory cross sections are in µb/sr.

θKlab Model |f0|2 |g0|2 |g1|2 |g−1|2 dσ/dΩL Pol(Λ)

AW2 11 0.0001 0.369 0.188 0.192 1.78 −0.0053 deg AS1 12 0.0071 0.398 0.191 0.206 1.91 −0.055

C4 13 0.0002 0.609 0.290 0.299 2.85 −0.019

SLA 14 0.0024 0.451 0.224 0.222 2.14 −0.016

AW2 11 0.0011 0.296 0.209 0.218 1.65 −0.007

10 deg AS1 12 0.0548 0.375 0.175 0.205 1.84 −0.142C4 13 0.0011 0.569 0.190 0.217 2.23 −0.066

SLA 14 0.0255 0.392 0.186 0.178 1.78 −0.053

3. Excitation Spectra Predicted for 28Si(γ, K+)28Λ Al

In order to demonstrate the characteristics of photoproduction of hypernu-clei, here we choose 28Si as a typical nuclear target. The excitation spectrahave been evaluated at Eγ = 1.3 GeV and θK

L = 3 deg which correspondto the kinematical condition in the experimental proposal. In the first sub-section, for demonstration the 28Si target ground state is assumed to havethe lowest proton-closed shells [s4p12(0d5/2)12

pn]. In the second subsection,we extend the calculation to employ realistic wave funtions solved in the


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[s4p12(sd)12pn] full shell model space.

3.1. A Demonstrative Model with (0d5/2)6p

As the proton shells are closed up to (0d5/2)6p, the final hypernuclear statesare described, respectively, with 1p-1h configurations [(nlj)−1

p (nlj)Λ]J . Forthe single-particle wave functions, we employ the DDHF solutions so as tobe as realistic as possible. The calculated cross sections are summarized inTable 2.

Table 2. Differential cross sections (in µb/sr) for the 28Si(γ, K+)28Λ Al reaction

calculated at Eγ = 1.3 GeV and θLK = 3 deg with the SLA amplitude 14. The

final hypernuclear states are expressed by [(lj)−1N (lj)Λ]J .

0sΛ1/2

0pΛ3/2

0pΛ1/2

0dΛ5/2

0dΛ3/2

1sΛ1/2

(EΛ) (−16.92) (−8.60) (−8.00) (−0.29) (1.29) (0.32)

(p-hole) J=0 − − − 0.0 − −J=1 − 6.9 − 36.7 9.8 −

(0d−15/2

) J=2 43.4 4.1 20.9 0.0 44.6 2.1

J=3 87.8 5.8 115.2 27.2 41.4 4.2J=4 − 189.1 − 0.0 164.4 −J=5 − − − 221.7 − −J=0 0.4 − 0.0 − − 0.1

(0p−11/2

) J=1 51.0 0.9 37.1 − 1.5 26.2

J=2 − 99.3 − 11.4 51.5 −J=3 − − − 115.7 − −J=0 − 0.0 − − 0.0 −J=1 23.3 15.0 0.8 3.6 6.4 12.8

(0p−13/2

) J=2 70.6 0.0 90.9 2.5 21.0 38.8

J=3 − 148.1 − 6.1 144.2 −J=4 − − − 194.8 − −J=0 0.0 − 0.2 − − 0.0

(0s−11/2

) J=1 24.0 19.5 39.0 − 19.8 75.9

J=2 − 58.9 − 39.0 58.4 −J=3 − − − 78.6 − −

The charcteristic result to be emphasized first is the selective excitationof the highest-spin state within each 1h-1p multiplet. In fact one sees inTable 2 that the [d−1

5/2sΛ1/2]3+ , [d−1

5/2pΛ3/2]4− , [d−1

5/2pΛ1/2]3− , [d−1

5/2dΛ5/2]5+ , and

[d−15/2d

Λ3/2]4+ states are very strongly excited and that the cross sections to

the lower-spin states are very smaller. The preferential excitation of thehigh-spin states are attributed to the large momentum transfer (about 300MeV/c) as similarly as in the case of the (π+, K+) reaction.

Secondly, such a novel fact is revealed that the selectively excited statein each combination [j−1

> jΛ>]J has an unnatural parity with the maximum


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spin valiue of J = Jmax = j> + jΛ> = lN + lΛ + 1 = Lmax + 1. In Table 2,

one may refer to the [d−15/2s

Λ1/2]3+ , [d−1

5/2pΛ3/2]4− , and [d−1

5/2dΛ5/2]5+ cases for

reconfirmation. The [p−13/2j

Λ>]J=2−,3+,4− states in the lowest block of Ta-

ble 2 have the similar nature. This kind of selectivity is not seen in theother hypernuclear production processes such as (π+, K+) and (K−, π−)reactions. This is attributed to the spin-flip transition dominance in theelementary hyperon photoproduction reaction. It is also noted that, inthe other combinations such as [j−1

> jΛ<]J or [j−1

< jΛ>]J , the highest spin is

limitted to J ′max = lN + lΛ and accordingly the natural parity (−)lN+lΛ .

The numerical results of Table 2 are schematically shown in Fig. 4 whererelative strengths in each J-multiplet are easily understood.

Λ

ΛΛΛ

Λ

Λ Λ

Λ

Figure 4. Divided contributions to the particle-hole state [j−1p jΛ]J as calculated for the

28Si(γ, K+)28Λ Al reaction at Eγ = 1.3 GeV and θLab

K = 3 deg.

Figure 5 shows the calculated angular distributions for the pronouncedpeaks. All these differential cross sections decrease quickly as the kaon labscattering angle increases. It is interesting to note that the relative strengthfor the [d−1

5/2dΛ3/2]J=4+ and [d−1

5/2pΛ3/2]J=4− states changes at θL

K 7 deg.

3.2. Use of the (sd)n Full Space Wave Functions

Here we use sophisticated wave functions solved in the(0d5/20d3/21s1/2)11,12

pn full space for 27,28Si. It is remarked that the useof such detailed wave functions should predict several new but minor statesin addition to the pronounced peaks which are predicted in the simplifiedconfiguration.

In order to predict a realistic strength function for the 28Si(γ, K+)28Λ Al

reaction, one has to take the empirical proton-hole widths into account,although they are not always available. Figure 6 shows the result where


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θ

σ

θ

Ω

ΛΛΛ

γ

γ

Λ

Figure 5. Calculated angular distributions of the states excited strongly in28Si(γ, K+)28

Λ Al reaction at Eγ = 1.3 GeV.

the following proton widths are employed tentatively: ΓN (0s−11/2) = 10 MeV,

ΓN (0p−13/2) = 6 MeV, ΓN (0p−1

1/2) = 3 MeV, and ΓN(0d−15/2) = 0 MeV. At the

same time, for the Λ bound states the width ΓΛ(j) = 0.3 MeV is used whichis about half of the energy resolution expected at the Jefferson Lab, whileΓΛ(j) = 1.0 MeV for 0 < EΛ < 2 MeV and ΓΛ(j) = 3 MeV for EΛ > 2MeV are assumed rather arbitrarily. Furthermore the energy splittingsbetween members of the [j−1

N jΛ<]J multiplet are taken from the YNG(ΛN)

h-p interaction 15 derived from the Nijmegen model-D, and it is notablethat the splittings are mostly of the order of 0.1 MeV.

Major 3 doublets (6 peaks) structure obtained with the simlified wavefuctions [d−1

5/2jΛ] (see Table 2) well persist also in the new estimates. It is

quite interesting to note that, for the major peaks, the use of the full spacewave functions result in the reduction of the cross sections by a factor ofabout 0.65 in comparison with the single-j estimate with (0d5/2)6p. It shouldbe also remarked that two pronounced peaks obtained at EΛ ∼ −8.5 MeVcorrespond to the [d−1

5/2pΛ3/2]4− and [d−1

5/2pΛ1/2]3− structure, respectively. As

the hole-particle interactions for high-spin states are generally very small,the energy difference between these two peaks, if separated experimentally,provides us the spin-orbit splitting of the Λ p-state.

The third major doublet obtained at EΛ ∼ 0 MeV includes the unnatu-ral parity highest-spin state [d−1

5/2dΛ5/2]5+ which should get the biggest cross

section. If the dΛ5/2 state is bound, or if it is not bound but the energy


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Figure 6. Theoretical excitation function calculated with the full (sd)n wave functionsfor the 28Si(γ, K+)28

Λ Al reaction at Eγ = 1.3 GeV and θLabK = 3deg.

position is not so high above the threshold, then this peak width mightbe sharp enough to be identified in the experiment with the good energyresolution expected at JLAB. As the partner has the dominant structure of[d−1

5/2dΛ3/2]4+ and the ΛN particle-hole interactions in high-spin states are

very small, the energy difference between these two peaks is almost equal tothe spin-orbit splitting of the d-state Λ. Thus the photo/electro-productionreaction will provide a nice opportunity of looking at such splittings inheavy systems if the energy resolution is good enough.

The present treatment is a direct extension of that for p-shell hypernu-clear photoproduction calculation 3 where always the full p-shell wave func-tions have been easily employed. For the readers’ reference, Fig. 7 showsthe the calculated spectrum for the 12C(γ, K+)12

Λ B at Eγ = 1.3 GeV. Thisis the update of the former prediction 3 by using here the NSC97f ΛN inter-action and the smaller smearing width. This prediction has been confirmedin very good agreement with the recent 12C(e, e′K+)12

Λ B measurement 2.In the case of p-shell hypernuclear production, the involving particles in thelow-lying state are in the s- and p-orbits, so that the ’high-spin’ selectivitymentioned above is realized as the transitions with ∆J = ∆L + ∆S = 2−,2+, and 3+. It is worthwhile to remark that in Fig. 6 there appear sidepeaks at EΛ −16 and −14 MeV in 28

Λ Al as confirmed in 12Λ C (1−2 and

1−3 ). They might be based on the excited states (3/2+ and 7/2+) in 27Alcoupled with an s-state Λ particle.


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σ

Ω

γ

γ

θ

Λ

Figure 7. Calculated spectrum for the 12C(γ, K+)12Λ B reaction at Eγ = 1.3 GeV and

θLabK = 3 deg.

4. Photoproduction with the 40Ca and 52Cr targets

The next sample target is 40Ca which is doubly LS-closed up to the 0d3/2

shell, so that the situation is different from the 28Si case because 40Ca hasthe uppermost proton orbit of the j<-type. Therefore the highest spin ina [d−1

3/2jΛ>]J multiplet is J ′

max = j< + jΛ> = lN + lΛ = Lmax with a natural

parity. On the other hand, in a [d−13/2j

Λ<]J multiplet the highest spin is

J ′′max = j< + jΛ

< = lN + lΛ − 1 = Lmax − 1, so that this restriction of theangular momentum transfer makes the latter cross sections much smaller.This situation of the j<-closed shell explains why the dominant peak seriesconsists of the natural parity 2+, 3−, and 4+ states accompanied with minorside peaks of the 1+, 2−, and 3+ states, respectively (See Fig. 8). The broadbackground reflects the strengths originating from the proton 0d5/2 shellwhich has the spreading width of several MeV. With this doubly LS-closedtarget, the photoproduction reaction might give a nice example of showingup the Λ single particle energies with good resolution.

In the case of the target 52Cr, which has four protons in the uppermostj> = 0f7/2 shell and the neutron is jj-closed, the dominant peak seriesconsists of the unnatural parity [f−1

7/2jΛ>]J=Jmax states with Jmax = j> +

jΛ> = lN + lΛ + 1 = Lmax + 1 where j> = sΛ

1/2(J = 4−), pΛ3/2(5

+), anddΛ5/2(J = 6−), respectively. On the other hand, the Λ spin-orbit partner

states [f−17/2j

Λ<]J=Lmax gain about 60% production rate of the biggest peak


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Λ

σ

Ω

γ

γ γ=

θ=ο

Λ

Figure 8. Excitation function for the 40Ca(γ, K+)40Λ K reaction calculated in DWIA at

Eγ = 1.3 GeV and θKL = 3 deg.

within each multiplet. The calculated strength function for 52Cr(γ, K+)52Λ V

will be reported elsewhere. See also the report 16.

Acknowledgments

One of the authors (T.M.) would be grateful to Professors L. McLerran,S.H. Kahana, and D.J. Millener for discussion and hospitality with finan-cial support at the Physics Department, Brookhaven National Laboratory(Dec. 2001 − Mar. 2002). His thanks are also due to Professors W. Hax-ton, E. Henley, and G. Bertsch for accepting his 8-month stay with partialfinancial support at the National Institute for Nuclear Theory, Universityof Washington.

This work has been done as a part of the Japan-Europe Joint ResearchProject for “ Electromagnetic Production and Weak Decays of Hypernuclei”(2000−2002) supported by the Japan Society for the Promotion of Science.

References

1. H. Tamura, in these proceedings and references threin.2. T. Miyoshi et al., Phys. Rev. Lett. 90, 232502 (2003).3. T. Motoba, M. Sotona, and K. Itonaga, Prog. Theor. Phys. Suppl. 117, 123

(1994).4. E.V. Hungerford, Prog. Theor. Phys. Suppl.; 117 135 (1994); Strangeness

Nuclear Physics, World Sci., Singapore (2000), p.356; Proposal E89-009; Seealso another proposal E94-107 (F. Galibardi et al.).


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therein.7. T.A. Rijken, Y. Yamamoto, and V.G.J. Stoks, Phys. Rev. C 59, 21 (1999)

and references therein.8. H. Going et al., Nucl. Phys. B26, 121 (1971) and references therein.9. M. Bockhorst et al, Z. Phys. C63, 37 (1994).10. M.Q. Tran et al (SAPHIR Collaboration), Phys. Lett. B445, 20 (1998).11. R.A. Adelseck and L.E. Wright, Phys. Rev. C32, 1681 (1985).12. R.A. Adelseck and B. Saghai, Phys. Rev. C42, 108 (1990).13. R.A. Williams, Ch.R. Ji, and S.R. Cotanch, Phys. Rev. C46, 1617 (1992).14. T. Mizutani, C. Fayard, G.-H. Lamot, and B. Saghai, Phys. Rev. C58, 75

(1998).15. Y. Yamamoto et al.,Prog. Thor. Phys. Suppl. 117 361 (1994).16. M. Sotona et al., in these proceedings.

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