Experiments of Strangeness Nuclear Physics at J-PARCTomofumi NAGAE (Kyoto University)

Progress on J-PARC Hadron Physics,

Nov.30-Dec.2, 2014

Contents-Introduction of J-PARC

E19 : Search for penta-quark Θ+

E10 : Search for Neutron-rich hypernucleus 6ΛH

E15 & E27 : Search for ”K-pp” →T. Hashimoto & Y. Ichikawa

Future program

(E13→H. Tamura)

(E07→H. Ekawa and others)

E05 : Spectroscopy of S=-2 systems

Summary

3Photo in July of 2009

J-PARC Facility(KEK/JAEA）

South to North

Neutrino Beams　(to Kamioka)

JFY2009 Beams

Hadron Exp. Facility

Materials and Life Experimental

Facility50 GeV Synchrotron

JFY2008 Beams

3 GeV Synchrotron

CY2007 Beams

Linac

Hadron Experimental Hall

K1.8

KL

K1.1BR

High p (in construction)SKS

K1.8BRK1.1

First beam in Feb. 2009World highest intensity Kaon beams !

30~50 GeV Primary Beam

Productiontarget (T1)

60m x 56m

Successful data taking of E19 in Oct. - Nov. 2010

~40 people from KEK, Kyoto, Tohoku, Tokyo, Nara WU, Osaka, JAEA, UNM, INFN, Torino, Seoul, ITEP, JINR

2010/10/5

~40peopleFromKEK,KyotoU.,TohokoU.,U.Tokyo,NaraWU,OsakaU.,JAEA, UNM(USA),INFN(Italy),Seol N.U.,ITEP,JINR

2009/10/23 2010/11/4

16

~272 hoursusing pion beam

World Facilities in the 21st Century

J-PARCJLab

DA NE

GSI/FAIRMainz

(e,e’K+)

(e,e’K+)

(K-,K+), (K-,π-)

(K-,π-)

HI, π, anti-p

For Strangeness Nuclear Physics

SNP Program History2010: Oct.-Nov.

E19: Penta-quark search in π-p→K-X at 1.92 GeV/c

First physics data taking in Hadron Hall

2012: Feb. , after the Earthquake

E19: π-p→K-X at 2 GeV/c

2012: June

E27: d(π+,K+) for K-pp , a pilot run 5 kW / 270 kW

SNP Program History2012: Dec. 10 kW

E10: (π-,K+)6ΛH

2013: March - May 20 kW

E15: 3He(K-,n) for K-pp

Radiation Accident

E13: Hypernuclear γ-ray spectroscopy; 4ΛHe, 19ΛF

Beams at K1.8: π± → K-

0

3.5

7

10.5

14

2010.10 2012.2 2012.6 2012.12 2013.1 2015.2 2015.5 2017.4

π x10^6 K- x10^6

3 3.3 5 10 15 20 20

50

100SX Power (kW)

E19E27

E10

E19 : Search for penta-quark Θ+

J-­‐PARC E19 Collabora0on

Our Approach ( J-­‐PARC E19 )1. Pion induced reac0on

Complementary to photo-­‐produc0on (LEPS/CLAS). Expect sizable produc0on cross sec0on. => High sta0s0cs

π− + p → K− + Θ+

2. High resolu0on missing mass spectroscopy – K1.8 beam line & SKS : ΔM = 2 MeV (FWHM)

Conclusive result by higher sensi0vity !!

2.6σ

KEK#PS&E522

ΔM&~&13.4&MeV&&

(FWHM)&

Previous&experiment

E19 Experimental setup

J-­‐PARC K1.8

π− beam

1.92/2.01 GeV/c

Dedicated to the (π,K) reaction spectroscopy

Results

• Missing mass of (π,K) @ sca\ering angle: 2—15 deg (Lab) • No peak structure was observed. • 2nd run has wider acceptance than 1st run.

π− + p → K− + X

E19-2nd

pπ = 2.01 GeV/c

E19-1st

Shirotori et al., PRL 109, 132002 (2012).

pπ = 1.92 GeV/c

Moritsu et al., PRC 90 (2014) 035205.

An example of ficng result

Upper limit for Θ+ produc0on cross sec0onSignal: Gaussian with fixed experimental width. B.G.: 2nd order polynomialFicng results of each mass

Upper Limit (90%C.L.)

p Upper limit for differential cross section averaged from 2 to 15 deg: < 0.28 µb/sr @ 1.50 – 1.56 GeV/c2

This limit is an order of magnitude smaller than that of KEK-­‐E522.

Summary of E19J-­‐PARC E19 is a pentaquark Θ+ search experiment with high sta0s0cs and high resolu0on.

π− p → K− Θ+ reac0on J-­‐PARC K1.8 B.S. and SKS

Result of E19-­‐1st and 2nd run was presented. No peak structure was observed in MM spectrum. Upper limit for Θ+ produc0on cross sec0on was obtained to be 0.28 µb/sr @ 1.50 – 1.56 GeV/c2

Upper limit on Θ+ decay width was also discussed. Comparing with theore0cal calcula0on, at most decay width should be less than 0.36 and 1.9 MeV for JP = ½+ and ½-­‐, respec0vely. This is extremely small as width of hadron resonance.

E10 : Neutron-Rich Hypernuclei

J-PARC E10 collaborationM. Agnello, J.K. Ahn, S. Ajimura, Y. Akazawa, N. Amano, K. Aoki, H.C. Bhang,

N. Chiga, M. Endo, P. Evtoukhovitch, A. Feliciello, H. Fujioka, T. Fukuda, S. Hasegawa, S. Hayakawa, R. Honda, K. Hosomi, S.H. Hwang, Y. Ichikawa,

Y. Igarashi, K. Imai, N. Ishibashi, R. Iwasaki, C.W. Joo, R. Kiuchi, J.K. Lee, J.Y. Lee, K. Matsuda, Y. Matsumoto, K. Matsuoka, K. Miwa, Y. Mizoi, M. Moritsu,

T. Nagae, S. Nagamiya, M. Nakagawa, M. Naruki, H. Noumi, R. Ota, B.J. Roy, P.K. Saha, A. Sakaguchi, H. Sako, C. Samanta, V. Samoilov, Y. Sasaki, S. Sato,

M. Sekimoto, Y. Shimizu, T. Shiozaki, K. Shirotori, T. Soyama, H. Sugimura, T. Takahashi, T.N. Takahashi, H. Tamura, K. Tanabe, T. Tanaka, K. Tanida, A.O. Tokiyasu, Z. Tsamalaidze, M. Ukai, T.O. Yamamoto, Y. Yamamoto,

S.B. Yang and K. YoshidaPolitecnico di Torino, Pusan National University, RCNP, Tohoku University,

KEK, Seoul National University, Osaka University, JINR, INFN, Kyoto University, Osaka Electro-Communication University, JAEA, BARC,

Virginia Military Institute, RIKEN

2014.3.12 2

Glue-like role of Λ hypernuclei

Large area un-explored!

Expand hypernuclear chart by using new spectroscopic tools

Production reactionsMissing-mass spectroscopy

Non-CX reactions (NCX) (K-,π-), (π+,K+)

Single CX reactions (SCX) (e,e’K+)

thin target available (K-,π0), (π-,KS)

new spectrometer Double CX reactions (DCX)

(π-,K+), (K-,π+) access neutron-rich hypernuclei

L. Majling NP A585 (1995) 211c possible with existing setup very low cross section !

roughly 1/1000 of NCX

SCXDCX

J-PARC E10

Mixing and n-rich hypernucleus 6ΛHPossible observation of mixing effect in 6ΛH structure

Y. Akaishi and T. Yamazaki, Frascati Phys. Ser. XVI (1999) 59

Normal ΛN interaction (“glue” effect) BΛ ~ 4.4 MeV

Coherent ΛN-ΣN mixing BΛ ~ 4.4 + 1.4 MeV

Prediction of Akaishi

Structure of 6ΛH should be investigated experimentally

Prediction of Gal and MillenerCoherent ΛN-ΣN mixing

ΔBΛN-ΣN ~ 0.1 MeV

A. Gal and D.J. Millener, Phys. Lett. B 725 (2013) 445

Studies of Neutron-rich Hypernuclei (2)DAΦNE (FINUDA)

M. Agnello et al. PRL 108 (2012) 042501. 6Li(stopped K-,π+) reaction

measure also weak decay cut on T(π+)+T(π-)

3 events of candidates 2.9±2.0×10-6/(stopped K-)

HeH

HLiK66

66

+→

+→+−

Λ

Λ+−

π

π

(π-,K+) Reaction MechanismTwo-step reaction

One-step reaction

Λ+→++→+

+→+Λ+→++−

+−

KpnpnKpKKp

00

00

,,πππ

π

)()(, npKp Λ→ΣΣ+→+ −−+−π

pion beam momentum 1.05 GeV/c

1.2 GeV/c

Theory Harada et al. ~1.6 nb/sr ~1.2 nb/sr

pion beam momentum 1.05 GeV/c

1.2 GeV/cTheory Harada et al. 2.4 nb/sr 5.4 nb/sr

It is important to study ΛN-ΣN mixing for DCX reaction

IA≠0 IA≠0Λ

I=0Σ

I=1

E10 Experimental Setup•K1.8 Beam Analyzer -1.2GeV/c π- Beam Δp/p ~3.3×10-4

•SKS Spectrometer 0.9GeV/c scattered K+

Δp/p~1.0×10-3 dΩ = 120 msr

•Target 6Li (95.54% enriched) C and (CH2)n

Results : Missing-mass spectrum• Select scattering angle from

2 deg to 14 deg <- No acceptance ambiguity

• In the missing mass region around 4ΛH+2n , no peak structure was observed.

• If the production crosection is ~10nb/sr, the events should be measured more than 60 events.

Data does not favor FINUDA’s data.

Discussion• 6Li(π-,K+) reaction populates 1+ state of 6ΛH ← 6Li(1+) at

forward angle • It needs spin-flip to generate 0+ state

• In the case of Scenario-1,2, width of 1+ state is narrow (Gal and Millener).

• → We can not find any narrow peak structures.

• In the case of scenario-3, wave function is broad. • If states existed, it would not be detected (Hiyama).

Summary of E10Successfully Performed the measurement of 6Li(π-,K+) reaction at pπ- = 1.2 GeV/c.

In the missing-mass spectrum of 6Li(π-,K+) reaction, no significant peak was observed.

An upper limit of production cross section was estimated to be 1.2 nb/sr.

It indicates that 0+ and 1+ states are unbound if cross section is about 10 nb/sr.

Need reconsideration of 6ΛH structure

E15 & E27

J-PARC E15 Collaboration

T. Hashimoto@SOTANCP3, May 29, 2014

The J-PARC E15 collaboration

2

J-PARC E27 Collaboration

Prog. Theor. Exp. Phys. 2013, 00000 (8 pages)DOI: 10.1093/ptep/0000000000

Inclusive spectrum of the d(π+,K+) reactionat 1.69 GeV/c

Yudai Ichikawa1,2, Tomofumi Nagae1, Hyoungchan Bhang3, Stefania Bufalino4,Hiroyuki Ekawa1,2, Petr Evtoukhovitch5, Alessandro Feliciello4, Hiroyuki Fujioka1,Shoichi Hasegawa2, Shuhei Hayakawa6, Ryotaro Honda7, Kenji Hosomi2,Kenichi Imai2, Shigeru Ishimoto8, Changwoo Joo3, Shunsuke Kanatsuki1,Ryuta Kiuchi2, Takeshi Koike7, Harphool Kumawat9, Yuki Matsumoto7,Koji Miwa7, Manabu Moritsu10, Megumi Naruki1, Masayuki Niiyama1,Yuki Nozawa1, Ryota Ota6, Atsushi Sakaguchi6, Hiroyuki Sako2, Valentin Samoilov5,Susumu Sato2, Kotaro Shirotori10, Hitoshi Sugimura2, Shoji Suzuki8,Toshiyuki Takahashi8, Tomonori Takahashi11, Hirokazu Tamura7,Toshiyuki Tanaka6, Kiyoshi Tanida3, Atsushi Tokiyasu10, Zviadi Tsamalaidze5,Bidyut Roy9, Mifuyu Ukai7, Takeshi Yamamoto7 and Seongbae Yang3

1Department of Physics, Kyoto University, Kyoto 606-8502, Japan2ASRC, Japan Atomic Energy Agency, Ibaraki 319-1195, Japan3Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea4INFN, Istituto Nazionale di Fisica Nucleare, Sez. di Torino, I-10125 Torino, Italy5Joint Institute for Nuclear Research, Dubna, Moscow Region 141980, Russia6Department of Physics, Osaka University, Toyonaka 560-0043, Japan7Department of Physics, Tohoku University, Sendai 980-8578, Japan8High Energy Accelerator Research Organization (KEK), Tsukuba, 305-0801, Japan9Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai, India10Research Center for Nuclear Physics, Osaka 567-0047, Japan11RIKEN, Saitama 351-0198, Japan∗E-mail: yudai@scphys.kyoto-u.ac.jp

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .We have measured an inclusive missing-mass spectrum of the d(π+,K+) reaction at thepion incident momentum of 1.69 GeV/c in the laboratory scattering angles between 2

and 16 with the missing-mass resolution of 2.7 MeV/c2(FWHM). In this letter, we firsttry to understand the spectrum as a simple quasi-free picture based on several knownelementary cross sections considering the neutron/proton Fermi motion in deuteron.While major spectrum structures are well understood in this picture, we have observedtwo distinct deviations; one peculiar enhancement at 2.13 GeV/c2 is due to a thresh-old cusp of the ΛN → ΣN conversion process, and the other notable thing is a shiftof a broad bump structure by about 22 ± 0.4 MeV/c2 toward low mass side mainlycontributed from hyperon resonance productions of Λ(1405) and Σ(1385)+/0.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index Kaonic nuclei, Λ(1405), Strangeness physics

1. Introduction

The measurement was carried out at the K1.8 beam line [1] of the J-PARC hadron exper-

imental hall by using a π+ beam at 1.69 GeV/c with a typical beam intensity of 3× 106

per 6-seconds spill cycle with a spill length of about 2 seconds. A liquid deuterium target

c⃝ The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License

(http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

New type of Strange matterStrange Mesons (K, K-) in nuclei

S =0 NucleiS =-1 NucleiS =-2 Nuclei

0

100

200

300

400

500

Excitation Energy(MeV)

A [Z] A [Z+1] A [Z+2]

π-⊗A [Z+1]Λ ⊗ [Z]A-1

Σ ⊗ [Z+1]A-1-Σ ⊗ [Z]A-10Σ ⊗ [Z-1]A-1+

K ⊗ [Z+1]A-K ⊗ [Z]A0_

Ξ ⊗ [Z+1]A-1-Ξ ⊗ [Z]A-10

ΛΛ⊗ [Z]A-2

K-,K+( )K-, π+( ), π-,K+( )K-, π-( ), π+,K+( )K-,N( )

Kaonic Nuclei

s u

K- Meson

K-pp Searches at J-PARCE15 : 3He(K-,n/p)”K-pp”, “K-pp”→Λp, Σ0p at 1 GeV/c

K-“n”→n+”K-“, ”K-“+”pp”→K-pp

Exclusive measurement

K-pp→Λp, Σ0p

Isospin dependence

E27 : d(π+,K+) with proton(s) coin. at 1.69 GeV/c

Λ(1405) as a doorway; π+”n”→K+Λ*(1405), Λ*p→K-pp

Semi-exclusive

K-pp→p+Y, p+p+π+(γ, π)

E05 : Spectroscopy of S=-2 systems

S=-2 World

Spectroscopic Study of Ξ-Hypernucleus, 12ΞBe, via the 12C(K-,K+) Reaction

Discovery of Ξ-hypernuclei as a peak

Measurement of Ξ-nucleus potential depth and width of 12ΞBe

J-PARC E05 T. Nagae et al.

S=-1 S=-2 (Multi-Strangeness System)

S-2S(K-,K+) Spectroscopy ＠J-PARC

K-+p→K++Ξ-　＠~1.8 GeV/c S-2S: (2010-2015)

Acceptance~60 msr ∆p/p<5x10-4(FWHM) ∆E=1.5 MeV

Acceptance∆Ω (msr)

Energy Resolution ∆E (MeV)

BNL 19 14

SKS+ 25 3

S-2S 60 1.5 1m

TOF

Q1: vertical focus

Q2: horizontal focus

D1: 70 deg. bend

Tracking Detector 1,2

Tracking Detector 3,4

Aerogel Cherenkov

Water Cherenkov

S-2S Spectrometer

S-2S Construction

1.2 ~ 1.4 GeV/c ~9m

∆p/p=5x10-4

60msr 37 t, 8.7 T/m12 t, 5.0 T/m86 t, 1.5T

Q2D1:

Q1

Coupling between Ξ hypernuclei and double-Λ hypernuclei

T.Harada, Y.Hirabayashi, A.Umeya, PLB 690 (2010) 363.

Theoretical Calculation for 16ΛΛC,

via Ξ- doorways in the 16O(K-,K+) reaction at 1.8 GeV/c

two-step process

one-step process

16O(K-,K+) 7 ~12 nb/sr

p excited states of double-Λ hyp. p sensitive to ΞΝ-ΛΛ coupling strength.

T. Harada et al. / Physics Letters B 690 (2010) 363–368 367

Fig. 3. Partial-wave decomposition of the calculated inclusive spectrum by the one-step mechanism near the 14C + Λ + Λ threshold in the 16O(K −, K +) reactionat 1.8 GeV/c (0). VΞ = −14 MeV and v0

Ξ N,ΛΛ = 500 MeV were used. The la-bels 0+(s2

Λ), 1−(sΛ pΛ) and 2+(p2Λ) denote the Jπ ΛΛ nuclear states of (0sΛ)2,

(0sΛ)(0pΛ) and (0pΛ)2 coupled with 14C(0+), respectively. The labels 2+(s2Λ),

1−(2+ ⊗ sΛ pΛ) and 2+(2+ ⊗ p2Λ) denote the states of (0sΛ)2, (0sΛ)(0pΛ) and

(0pΛ)2 coupled with 14C(2+), respectively.

such ΛΛ excited states below the 14C + Λ + Λ threshold will bemeasured experimentally at the J-PARC facilities [3].

On the other hand, it is extremely difficult to populate the0+ ground state with 14C(0+) ⊗ s2

Λ at ω ≃ 352.3 MeV (BΛΛ ≃24.9 MeV) and also the 2+ excited state with 14C(2+) ⊗ s2

Λ atω ≃ 359.6 MeV (BΛΛ ≃ 17.5 MeV) in the one-step mechanismvia Ξ− doorways in the (K −, K +) reactions. The high momen-tum transfer of qΞ ≃ 400 MeV/c necessarily leads to the non-observability with %L = 0. Thus the integrated cross section ofthe 0+ state is found to be about 0.02 nb/sr, of which the q de-pendence is approximately governed by a factor of exp(− 1

2 (bqΞ )2)

where a size parameter b = 1.84 fm. There is no production in the2+ state with 14C(2+) ⊗ s2

Λ under the angular-momentum conser-vation in the 16O(K −, K +) reactions by the one-step mechanism.The contribution of these states to the ΛΛ spectrum in the one-step mechanism is completely different from that in the two-stepmechanism as obtained in Refs. [7,8].

In the (K −, K +) reaction, ΛΛ hypernuclear states can be alsopopulated by the two-step mechanism, K −p → π0Λ followed byπ0 p → K +Λ [7–9], as shown in Fig. 1(a). Following the procedureby Dover [7,9], a crude estimate can be obtained for the contribu-tion of this two-step processes in the eikonal approximation usinga harmonic oscillator model. The cross section at 0 for quasielasticΛΛ production at pK − = 1.8 GeV/c in the two-step mechanism,which is summed over all final state, is given [9] as

!

f

"dσ (2)f

dΩL

#

0≈ 2πξ

p2π

$1r2

%"α

dσ

dΩL

#K − p→π0Λ

0

×"α

dσ

dΩL

#π0 p→K +Λ

0N pp

eff , (11)

where ξ = 0.022–0.019 mb−1 is a constant nature of the angulardistributions of the two elementary processes, pπ ≃ 1.68 GeV/cis the intermediate pion momentum, and ⟨1/r2⟩ ≃ 0.028 mb−1

is the mean inverse-square radial separation of the proton pair.N pp

eff ≃ 1 is the effective number of proton pairs including the nu-

clear distortion effects [7]. The elementary laboratory cross section(αdσ /dΩL)0 is estimated to be 1.57–1.26 mb/sr for K − p → π0Λand 0.070–0.067 mb/sr for π0 p → K +Λ depending on the nuclearmedium corrections. This yields

!

f

"dσ (2)f

dΩL

#

0≃ 0.06–0.04 µb/sr, (12)

which is half smaller than ∼ 0.14 µb/sr at 1.1 GeV/c. Consider-ing a high momentum transfer q ≃ 400 MeV/c in the (K −, K +)reactions by comparison with the (π+, K +) reaction [39], we ex-pect that the production probability for the ΛΛ bound states doesnot exceed 1% in the quasielastic ΛΛ production, so that an es-timate of the ΛΛ hypernucleus in the two-step mechanism maybe on the order of 0.1–1 nb/sr. This cross section is smaller thanthe cross section for the ΛΛ 1− states we mentioned above in theone-step mechanism. Consequently, we believe that the one-stepmechanism acts in a dominant process in the (K −, K +) reactionat 1.8 GeV/c (0) when v0

Ξ N,ΛΛ = 400–600 MeV. This implies thatthe (K −, K +) spectrum provides valuable information concerningΞ N–ΛΛ dynamics in the S = −2 systems such as ΛΛ and Ξ hy-pernuclei, which are often discussed in a full coupling scheme [40].

4. Summary and conclusion

We have examined theoretically production of doubly strangehypernuclei in the DCX 16O(K −, K +) reaction at 1.8 GeV/c withinDWIA calculations using coupled-channel Green’s functions. Wehave shown that the Ξ− admixture in the ΛΛ hypernuclei playsan essential role in producing the ΛΛ states in the (K −, K +) reac-tion.

In conclusion, the calculated spectrum for the 16Ξ− C and 16

ΛΛChypernuclei in the one-step mechanism K − p → K +Ξ− via Ξ−

doorways predicts promising peaks of the ΛΛ bound and excitedstates in the 16O(K −, K +) reactions at 1.8 GeV/c (0). It has beenshown that the integrated cross sections for the significant 1− ex-cited states in 16

ΛΛC are on the order of 7–12 nb/sr dependingon the Ξ N–ΛΛ coupling strength and also the attraction in theΞ–nucleus potential. The Ξ− admixture probabilities are on theorder of 5–9%. The sensitivity to the potential parameters indicatesthat the nuclear (K −, K +) reactions have a high ability for thetheoretical analysis of precise wave functions in the ΛΛ and Ξhypernuclei. New information on ΛΛ–Ξ dynamics in nuclei fromthe (K −, K +) data at J-PARC facilities [3] will bring the S = −2world development in nuclear physics.

Acknowledgements

The authors are obliged to T. Fukuda, Y. Akaishi, D.E. Lanskoy,T. Motoba and T. Nagae for many discussions. This work was sup-ported by Grants-in-Aid for Scientific Research on Priority Areas(Nos. 17070002 and 20028010) and for Scientific Research (C)(No. 22540294).

References

[1] For example in: A. Gal, R.S. Hayano (Eds.), Special Issue on Recent Advances inStrangeness Nuclear Physics, Nucl. Phys. A 804 (2008) 1.

[2] C.B. Dover, A. Gal, Ann. Phys. 146 (1983) 309.[3] T. Nagae, et al., J-PARC proposal E05, http://j-parc.jp/NuclPart/Proposal_e.html.[4] T. Fukuda, et al., Phys. Rev. 58 (1998) 1306.[5] P. Khaustov, et al., Phys. Rev. C 61 (2000) 054603.[6] S. Tadokoro, H. Kobayashi, Y. Akaishi, Phys. Rev. C 51 (1995) 2656;

H. Maekawa, K. Tsubakihara, A. Ohnishi, Eur. Phys. J. A 33 (2007) 269;S. Hashimoto, M. Kohno, K. Ogata, M. Kawai, Prog. Theor. Phys. 119 (2008)1005.

Summary

π era → K era

E19, E10, E27 E15, E13, E07, E05, …

PRL, PRC, PLB, PTEP(2)