Search for the 𝑛𝑛Λ state
via the 3H(e, e’K+)X reaction at JLabKazuki Suzuki

1
Abstract
In 2013, the existence of the 𝑛𝑛Λ bound state with a binding energy of about
0.5 MeV was claimed by the HypHI collaboration at GSI. Considering that the
ground-state binding energy of hypertriton that is the isosinglet state (I = 0) is as
small as 130 keV, the 𝑛𝑛Λ (I = 1) state is hard to be bound. In contrast, several
theoretical calculations suggest the existence of a resonant state near the threshold.
Therefore, it is desirable to conduct 𝑛𝑛Λ search experiments that can explore a wide
range from the bound region to the resonance region. In the electro-productionof hypernuclei, the 𝑛𝑛Λ state can be investigated by using a tritium target (𝑝𝑛𝑛)by changing a proton to Λ. We performed a search experiment for the 𝑛𝑛Λ statesusing the 3 H(𝑒 , 𝑒 0 𝐾 + )X reaction at the experimental Hall A in Jeﬀerson Laboratory.
The experiment was performed using a 4.32-GeV electron beam. The momenta of
the scattered electron and 𝐾 + were measured by the existing spectrometers, HRS-L
and HRS-R, respectively. The central momenta of the scattered electron and 𝐾 +were set to cover the expected kinematics of 𝑛𝑛Λ production, which were 2.22 and1.82 GeV/c, respectively. To estimate the various eﬃciency and correction factors
required for cross-section analysis, a Monte Carlo simulation was newly developed
to reproduce the experimental data in the framework of Geant4.
There seemed some enhancement around the energy threshold of the 𝑛𝑛Λ boundstate in the cross-section spectrum for the 3 H(𝛾∗ , 𝐾 + )X reaction. However, noclear peak structures were observed. Therefore, spectral ﬁts with assumed peak
position (𝐵Λ ) and decay width (Γ) were performed to quantitatively discuss theexcess above the expected background distribution. Typical theoretical predictions
that show the small and large decay widths were used for the ﬁts, which are(−𝐵Λ , Γ) = (0.25, 0.8) MeV and (0.55, 4.7) MeV. As a result of the ﬁtting, no peakstructure with a statistical signiﬁcance exceeding 3𝜎 was observed. The upper
limit of the diﬀerential cross section at the 90%-conﬁdence level was obtained tobe 21 and 31 nb/sr, respectively, when (−𝐵Λ , Γ) = (0.25, 0.8) and (0.55, 4.7) MeV
were assumed. In addition, the cross-section upper limit with various 𝐵Λ and Γother than the above ﬁxed assumptions were obtained to provide data that are
independent on the theoretical predictions.
The upper limit of the 𝑛𝑛Λ production cross section was obtained for the ﬁrst
time. In the theoretical calculations, the existence of the 𝑛𝑛Λ resonant state strongly
depend on the models of the Λ𝑛 interaction and calculation methods. The present
result would constrain the interaction models and calculation methods. Theoretical
calculations of the 𝑛𝑛Λ production cross section to be compared with the present
result are highly desirable.

2

Contents

Chapter 1

Introduction

9

1.1

Baryon-Baryon Interactions . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.2

Few body hypernulcear system . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3𝑛𝑛Λ experiment by HypHI collaboration . . . . . . . . . . . . . . . . . . . 11

1.4

Theoretical calculations for 𝑛𝑛Λ state . . . . . . . . . . . . . . . . . . . . . 12
1.4.1

Bound state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4.2

Resonant state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5

Current status of study on 𝑛𝑛Λ state . . . . . . . . . . . . . . . . . . . . . 16

1.6

Our experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.7

Purpose of this paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 2

2.1

Experiment(𝑒 , 𝑒 0 𝐾 + ) reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1

2.2

19

Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Electron beam provided by Jeﬀerson Laboratory . . . . . . . . . . . . . . 23
2.2.1

CEBAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.2

Beam measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3

Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4

Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5

Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6
Chapter 3

2.5.1

Tracking detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5.2

Cherenkov counters . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.3

Scintillation counters . . . . . . . . . . . . . . . . . . . . . . . . . . 33

DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Analysis35

3.1

Optics calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2

Number of target nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1

Vertex cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Contents

3.3

3.4

3.5

Chapter 43
3.2.2

Density change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.3

3 H decay and contamination. . . . . . . . . . . . . . . . . . . . . 44

Particle identiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.1

Coincidence time method . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.2

Cherenkov cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Beam electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.1

Beam charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.2

Raster correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5.1

Tracking eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5.2

Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.5.3

DAQ eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Monte Carlo Simulation

58

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2

Event generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3

4.4

Chapter 5

4.2.1

Beam electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.2

Scattering electron and virtual photon . . . . . . . . . . . . . . . . 60

4.2.3

Radiation eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.4

Fermi motion and oﬀ-shell eﬀect . . . . . . . . . . . . . . . . . . . 63

4.2.5

Hyperon production . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Geant4 setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.1

Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.2

Magnetic ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.3

Missing mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Evaluation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4.1

𝐾 + decay and absorption . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4.2

Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.4.3

Energy loss correction . . . . . . . . . . . . . . . . . . . . . . . . . 73

Results and Discussion

78

5.1

Derivation of the diﬀerential cross section . . . . . . . . . . . . . . . . . . 78

5.2

Missing mass resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3

5.2.1

Angular resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2.2

Momentum resolution . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.3

Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Contents

5.4

4
5.3.1

Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3.2

Mass accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Upper limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4.1

Fitting function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4.2

Spectral ﬁt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4.3

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Chapter 6

Summary

96

Appendix A

Summary of theoretical study

100

A.1

Bound state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

A.2

Resonant state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Appendix B
References

Determination of the magnetic setting of the HRS Geant4

105
111

5

List of Tables

1.1

Theoretical calculation of complex resonance energies. . . . . . . . . . . . . . . 14

1.2

Theoretical calculations for the bound state. . . . . . . . . . . . . . . . . . . . . 15

1.3

Theoretical calculations of the resonant state. . . . . . . . . . . . . . . . . . . . 15

2.1

Major reaction parameters of the present experiment. . . . . . . . . . . . . . . 24

2.2

Design value of HRS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3

Basic information of the Cherenkov detectors. . . . . . . . . . . . . . . . . . . . 32

2.4

Trigger condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1

The total beam charge irradiated to each target obtained by accumulating the
BCM information on each run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2

VDC plane eﬃciency and tracking eﬃciency in a typical run. . . . . . . . . . . 53

4.1

Material list of the particle trajectory in HRS. . . . . . . . . . . . . . . . . . . . 69

4.2

The measurement values of cell thickness. . . . . . . . . . . . . . . . . . . . . . 76

5.1

Eﬃciencies and correction factors used for the cross-section calculation in
Eq. (5.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2

Decomposition of momentum resolution. . . . . . . . . . . . . . . . . . . . . . 84

5.3

Relative errors for the eﬃciencies and correction factors that contributed tothe uncertainty on the cross-section calculation. . . . . . . . . . . . . . . . . . . 87

6

List of Figures

1.1

Level schemes of the A=4 hypernuclear system. . . . . . . . . . . . . . . . . . . 10

1.2

Level schemes of the A=3 hypernuclear system. . . . . . . . . . . . . . . . . . . 11

1.3

Invariant mass distributions of 𝑑 + 𝜋− and 𝑡 + 𝜋− at GSI. . . . . . . . . . . . . . 12

1.4

The hypercentral potential for the 𝑛𝑛Λ system. . . . . . . . . . . . . . . . . . . 14

1.5

The reaction process of Λ production. . . . . . . . . . . . . . . . . . . . . . . . . 16

1.6

Momentum transfer and hypernuclear cross section for each Λ production
process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1

A diagram of (𝑒 , 𝑒 0 𝐾 + ) reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2

The angle dependence of the cross section of Λ photo-production. . . . . . . . 21

2.3

The energy dependence of the cross section of 𝑝(𝛾, 𝐾 + )Λ reaction. . . . . . . . 22

2.4

Momentum correlations of 𝑒 0 and 𝐾 + for the reactions in this experiment. . . . 23

2.5

The schematic view of CEBAF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6

The schematic view of arc method. . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7

The schematic view of BCM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.8High Resolution Spectrometers in Hall A. . . . . . . . . . . . . . . . . . . . . . 27

2.9

The magnetic conﬁguration of HRS. . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.10

The target assembly used in this experiment. . . . . . . . . . . . . . . . . . . . 29

2.11

A schematic diagram of HRS detectors. . . . . . . . . . . . . . . . . . . . . . . . 30

2.12

Schematic diagram of the vertical drift chamber. . . . . . . . . . . . . . . . . . 31

2.13

Momentum dependence of the number of photo-electrons in the Cherenkov
detectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.14

Schematic diagram of S2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.15

Trigger timing with T5 condition. . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1

The carbon foil proﬁle reconstructed with Eq. (3.5). . . . . . . . . . . . . . . . . 37

3.2

A drawing of the sieve slit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3

Picture of sieve slits set in front of the Q1 magnet. . . . . . . . . . . . . . . . . . 39

List of Figures
3.4

7

The sieve slit patterns reconstructed with the initial angle parameters of
Eqs. (3.2) and (3.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5

The sieve slit patterns reconstructed with optimized parameters of Eqs. (3.2)
and (3.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6

H(𝑒, 𝑒 0 𝐾 + )Λ/Σ0 missing mass spectrum reconstructed by pre and post optimized parameters of Eq. (3.4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.7

Correlation of subtraction (𝑧 𝐿 − 𝑧 𝑅 )and average (𝑧 𝐿 + 𝑧 𝑅 )/2 of the vertex proﬁle. 41

3.8

𝑧diﬀ dependence of the survival ratio of the tritium target. . . . . . . . . . . . . 42

3.9

Distribution of 𝑧 𝐿 and 𝑧 𝑅 when 𝑧diﬀ cut was applied. . . . . . . . . . . . . . . . 43

3.10

Survival ratio of the target with the 𝑧mean cut. . . . . . . . . . . . . . . . . . . . 43

3.11

The change of the tritium gas density as a function of beam current. . . . . . . 44

3.12

Coincidence time distribution without Cherenkov cut. . . . . . . . . . . . . . . 46

3.13

Correlation between the number of photo-electrons and the coincidence timeof AC1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.14

Correlation between the number of photo-electrons and the coincidence timeof AC2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.15

Missing-mass distribution of 𝑝(𝑒 , 𝑒 0 𝐾 + ) reaction without Cherenkov cut andthat of 𝑝(𝑒 , 𝑒 0𝜋+ ) reaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.16

Coincidence time distribution before and after Cherenkov cut. . . . . . . . . . 50

3.17

Beam image of 𝑥 vs. 𝑦 at the target, which was reconstructed by using the
BPM information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.18

The estimation of the wire eﬃciency. . . . . . . . . . . . . . . . . . . . . . . . . 52

3.19

VDC hit information estimated from experimental data. . . . . . . . . . . . . . 52

3.20

Strategy of MC simulation for plane eﬃciency estimation. . . . . . . . . . . . . 53

3.21

Run dependence of tracking eﬃciency. . . . . . . . . . . . . . . . . . . . . . . . 54

3.22

TDC timing information when a multiple hit event occurs. . . . . . . . . . . . 55

3.23

Coincidence time distribution in the multiplicity event. . . . . . . . . . . . . . 56

3.24

Run dependence of DAQ dead time. . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1

Strategy for missing mass reconstruction with Geant4. . . . . . . . . . . . . . . 59

4.2

The schematic view of the 3 H(𝑒, 𝑒 0 𝐾 + )𝑋 reaction. . . . . . . . . . . . . . . . . . 60

4.3

Schematic view of internal and external radiation. . . . . . . . . . . . . . . . . 61

4.4

Feynman diagrams contributed to the internal radiation. . . . . . . . . . . . . . 61

4.5

The eﬀect of the internal radiation. . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.6

Spectral function of the process 3 H(e, e0p)2n. . . . . . . . . . . . . . . . . . . . 64

4.7

Diﬀerential cross sections for 𝛾 + 𝑝 → 𝐾 + + Λ. . . . . . . . . . . . . . . . . . . . 66

4.8

Diﬀerential cross sections for 𝛾 + 𝑝 → 𝐾 + + Σ0 . . . . . . . . . . . . . . . . . . . 67

List of Figures8

4.9

Visualization of the HRS Geant4 simulator. . . . . . . . . . . . . . . . . . . . . 68

4.10

Residuals between the experimental data and the simulation. . . . . . . . . . . 70

4.11

Comparison with the Geant4 simulation and data in the case of hydrogen target. 71

4.12

Comparison with Geant4 simulation and data in the case of tritium target. . . 72

4.13

𝐾 + momentum dependence of 𝐾 + survival ratio in HRS-R path. . . . . . . . . 72

4.14

Acceptance estimated by the HRS Geant4 and SIMC. . . . . . . . . . . . . . . . 74

4.15

The name scheme of cell thickness. . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.16

An illustration of the energy-loss correction for the the target cell. . . . . . . . 76

4.17

𝑦𝑇0 dependence of energy loss of electron in HRS-L. . . . . . . . . . . . . . . . . 77

5.1

Diﬀerential cross section for the 3 H(𝑒 , 𝑒 0 𝐾)𝑛𝑛Λ reaction. . . . . . . . . . . . . . 79

5.2

Strategy of the mixed event analysis. . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3

Strategy of the simple Monte Carlo simulation for angle resolution estimation. 82

5.4

Result of Monte Carlo simulation for the sieve slit. . . . . . . . . . . . . . . . . 83

5.5

Comparison between simulation and data of sieve-slit pattern. . . . . . . . . . 83

5.6

Momentum resolution estimated by the HRS Geant4. . . . . . . . . . . . . . . 85

5.7

Momentum distribution before and after energy correction. . . . . . . . . . . . 86

5.8

Missing mass distribution of 3 H(𝑒 , 𝑒 0 𝐾 + )𝑋 reaction assuming hydrogen target. 88

5.9Missing mass distributions for 𝑝(𝑒, 𝑒 0 𝐾 + )Λ and 3 H(𝑒, 𝑒 0+ )𝑛𝑛Λ reactions estimated with the HRS Geant4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.10

𝑛𝑛Λ signal estimated with the HRS Geant4. . . . . . . . . . . . . . . . . . . . . 90

5.11

Template functions for the 𝑛𝑛Λ signal. . . . . . . . . . . . . . . . . . . . . . . . 91

5.12

The correction for the radiative tale. . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.13

The change in histograms for 3 H(𝑒, 𝑒 0 𝐾 + )𝑛𝑛Λ reaction due to binning. . . . . . 93

5.14

The results of the spectral ﬁts assuming the 𝑛𝑛Λ signal by theoretical calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.15

The 90% C.L. of the upper limit of the 𝑛𝑛Λ cross section when the spectral ﬁt
was performed by changing 𝐵Λ and natural width. . . . . . . . . . . . . . . . . 95

A.1

The pole trajectory of the 𝑛𝑛Λ state. . . . . . . . . . . . . . . . . . . . . . . . . . 103

B.1

The momentum acceptance of HRS-L and HRS-R. . . . . . . . . . . . . . . . . 106

B.2

The 𝑦 0 distribution at VDC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

B.3

The comparison between experimental result and simulation. . . . . . . . . . . 108

B.4

The simulation results of momentum acceptance. . . . . . . . . . . . . . . . . . 109

B.5

The simulation results of momentum resolution. . . . . . . . . . . . . . . . . . 110

9Chapter 1

Introduction

1.1

Baryon-Baryon Interactions

A nucleus is composed of protons and neutrons; they are called necleons. Nucleon–
nucleon interactions are well understood through scattering experiments, and in the 1990s,
several groups succeeded in constructing a realistic nucleon–nucleon interaction model that
reproduced the scattering data in 𝜒2 ∼ 1 [1–3]. There exists only one two-body bound system,
deuteron, which consists of a proton and a neutron. There are no bound systems in 𝑝–𝑝 and𝑛–𝑛 systems. This property makes our present world as what it is today. If the nucleon forceswere diﬀerent from those in this universe, our world would be completely diﬀerent, and there
is a high possibility that stars and life would not be created. Is this exquisite ﬁne tuning reallya coincidence?
To answer this question, it is important to extend the world of SU(2), which consists of
u-quarks and d-quarks, to the world of SU(3), which consists of u-quarks, d-quarks, ands-quarks. By adding one more degree of freedom, we can see nucleon interactions from ahigher perspective view. The Λ particle is the lightest baryon that includes an s-quark, and it
is important to understand the Λ-nucleon interaction. The Λ particle decays in only 263 ps,
which makes scattering experiments diﬃcult. Therefore, the interaction between Λ-particles
and nucleons has been mainly obtained by creating Λ hypernuclei, which are bound states ofΛ-particles and nucleons. Since the discovery of hypernuclei in 1953, numerous hypernuclei

have been investigated.

1.2

Few body hypernulcear systemtransition energy and firmly es
experimental methods, have been awaited. Recently, the π −of 4Λ He, as shown in Fig. 1 (onmomentum in the 4Λ H → 4 He þ π − weak decay was preThe J-PARC E13 experimentcisely measured at MAMI-C [5], and the obtained value ofbeam line in the J-PARC Had
BΛ (4Λ Hð0þ Þ) ¼ 2.12  0.01ðstatÞ  0.09ðsystÞ MeV
is[10]. The 4 HeðK − ; π − Þ reactio
consistent with the emulsion value.
4 Heð1þ Þ, which was populated
The BΛ difference for the excited 1þ states provides
Λ
Chapter 1 Introduction
additional important information on the spin dependentof the K − þ10n → Λ þ π − proce
CSB effect from which the origin of CSB can be studied.
1.5 GeV=c was chosen consideþ
The BΛ values for the 1 state are obtained via thesection of the spin-flip Λ pro
beam intensity. A 2.8 g=cm2 -th
1þ → 0þ γ-ray transition energies. The 4Λ H γ ray wasmeasured three times, and the 4Λ Hð1þ ; 0þ Þ energy spacing
irradiated with a total of 2.3 ×
Many-body calculations based on the baryon-baryon
interactions are available forfew-body
was determined to be 1.09  0.02 MeV as the weighted(K − =π − ¼ 2 ∼ 3) was delivered
systems due to theaverage
recent improvements
super computerof these three ofmeasurements(1.09power.
 0.03 The
MeVΛN interaction
intensity ofcan
3 × 105 over a 2.1 s[6], 1.04
 0.04 MeV[7],and
1.114  hypernuclear
0.030 MeV [8]),asoccurring
every 6 s. Incident K
be precisely extracted
by comparing
withthefew-body
structures.
While these
shown
in
Fig.
1
(on
the
left).
On
the
other
hand,
observation
wereparticlestudies have yielded various
results, new puzzles have arisen that cannot be explained by theidentified and m
4of the Λ He γ ray was reported only once by an experimentbeam line spectrometer and th
simple ΛN interaction
with alone.
stopped K − absorption on a 7 Li target, which claimed
Spectrometer (SKS) [11], respeþþ
3
4 to be 1.15
4  0.04 MeV [7].
were
the
(1
,
0
)
energyspacing
The established s-shell hypernuclei are Λ H, Λ H, and Λ He. In the four-body (A= 4)detected
system, by a Ge detecto
rounding
This result4 suggests
significantlylarge
CSB andeffecttwo
alsostatesin withbeen
found,
spinstheof 0target. Through a
H and 4ΛaHe—have
two types of hypernuclei—Λ
between these spectrometer sy
the 1þ statewith ΔBΛ ð1þ Þ ¼ 0.29  0.06 MeV. However,and 1 have been measured,in eachsystem.
The existenceof chargeandsymmetrybreaking
rays fromhypernuclei were mea
this 4Λ He γ-ray
spectrumis statisticallyinsufficient,
4(CSB) has been reported
in recently
high-precisionthroughmeasurements
Figure 1.2
surrounding
the target is shown
identificationof developedthe Λ He hyperfragment
high [4,5].
4
4
0
The
detector
energyγ
rays
attributed
to
the
He
→
He
þ
π
weak
Λ
shows level schemes of the A = 4 hypernuclei. Although
there is no CSB in the excited setting in SKS
spectroscopic experiments vi
decay seems to be ambiguous.
state with 𝑆 = 1, there
is
a
CSB
of
Δ𝐵
=
350
keV
in
the
groundstate.
Consideringthat SksMinus hadΛIn order to clarify this situation, we performed a γ-ray(SksMinus).
Δ𝐵(3 He −3 H) = 70spectroscopic
keV in ordinally
nuclei, the
CSB is [9],veryinlarge
in hypernuclei.
experimentfor 4Λeﬀect(∼100 msr)Any
for detecting the oHe atofJ-PARC
which
þ
4
ratoryscattering
angle range
the1
excited
state
of
He
was
directly
producedvia
thetheoretical explanation for this eﬀect hasΛnot yet been found, and further theoretical work is4− −−HeðK ; π Þ reaction with a 1.5 GeV=c K beam, and γ
needed.
rays were measured using germanium (Ge) detectors with

FIG. 2 (color online). A schema
setup around the liquid 4 He target
FIG.
1(coloronline).
Levelschemesofthemirrorhypernuclei,
Figure 1.1 Level schemes of the A=4 hypernuclear system [5].
4
4
4
þ
4
þ
conducting dipole magnet (2.5 T);
H
and
He.
Λ
bindingenergies
(B
)
of
Hð0
Þ
and
Heð0
Þ
Λ
Λ
Λ
Λ
Λ
counter hodoscope; BAC1,2 andare taken from past emulsion experiments [2]. BΛ (4Λ Heð1þ Þ) andcounters with n ¼ 1.03; SDC1,2 a
BΛ (4Λ Hð1þ Þ) are obtained using the present data and past3 γ-ray3
H is the lightest
A currently established hypernucleus in the A = 3 system
is H.
data [6–8], respectively. Recently, BΛ (4Λ Hð0þ Þ) ¼Λ2.12Λ electromagnetic shower counter tohypernucleus consistingof𝑝,0.09ðsystÞ
𝑛, and ΛMeV
withwasspin
1/2 andisospin
0. Its bindingwas
π 0 energy
0.01ðstatÞ
obtainedwithan independent
decay. Hyperball-J
consists oftechnique[5].
roundedby
PWO
counters
for bacmeasured to be 𝐵 = 130 ± 50 keV by emulsion experiments [6]. Considering that the bindingΛenergy of the deuteron composed of 𝑝 and 𝑛 is 2.22 MeV, we assume that 3Λ H is a system in222501-2
which the deuteron and Λ are weakly bound. In contrast, the recent STAR
experiment using
relativistic heavy ion collisions has claimed that 𝐵Λ = 0.41 ± 0.12 ± 0.11 MeV [7], and further
experimental conﬁrmation is desirable. Given the shallow bound state of 3Λ H, as shown

Chapter 1 Introduction

11

in Fig. 1.2, the existence of other three-body hyper-nuclei (𝑁 𝑁Λ) is unlikely. However, as
mentioned above, there are still many puzzles left to be answered in few-body hypernuclei.
Λ𝑁 𝑁 three-body forces or Λ𝑁-Σ𝑁 conversion, which are not yet well understood, may be
the key to solve these problems. Such eﬀects may lead to the emergence of new boundand resonant states of 𝑁 𝑁Λ. In particular, the 𝑁 𝑁Λ state with I=1 and S=1/2 has been
an important topic in hyper-nuclear physics since the 2010s, starting from the experimental
results by HypHI collaboration described in the next section.

-BΛ

3
ΛHe

3
ΛH

p
Λ

?

p

p

No data

3
Λn

Λ

?

n

n
Λ

No data?

n

T=1This study0pΛ

0.13 MeV

n

T=0

Bound

Figure 1.2 Level schemes of the A=3 hypernuclear system.

1.3

𝑛𝑛Λ experiment by HypHI collaboration

The HypHI collaboration at GSI attempted to produce hypernuclei using a relativistic ion
beam. A 2A GeV beam of 6 Li was irradiated onto a 12 C target. The initial goal of theexperiment was to generate Λ, 3Λ H, and 4Λ H to demonstrate the feasibility of the production
method. The results have been reported in Refs. [8–10]. In the same dataset, the 𝑛Λ and 𝑛𝑛Λ
bound states were searched by measuring 𝑑 + 𝜋− , 𝑡 + 𝜋− invariant masses [11]. The results areshown in Figure 1.3. The upper ﬁgures show the invariant mass distributions of 𝑑 + 𝜋− , andthe lower ﬁgures show the invariant mass distributions of 𝑡 + 𝜋− . On the left, the decay vertex
was selected at −10 < Z < 30 cm, and on the right, at −2 < Z < 30 cm. Here, the target was
set at −6 < Z < −2 cm. In this experiment, the existence of both bound states of 𝑛Λ and 𝑛𝑛Λ
is reported. In particular, the 𝑛𝑛Λ bound state was observed with a signiﬁcance of 5.0𝜎 in thewide Z region (b1) and 5.2𝜎 in the narrow Z region (b2). The width at (b2) was 5.4 ± 1.4 MeV,

cally the nn stateregion, cross-checked with the rapidity regions of detected3]; however, recent
helium and lithium isotopes.
tates [24] indicate
After the identification of the particles and fragments ofrations from Dalitz
interest, the invariant mass distributions of d + π − and t + π −nels of such a state,
were studied with the identical rules for the secondary vertex
π − and the first
selections applied in the case of , 3 H, and 4 H [28]. Figure 1
Chapter 1 Introduction
evised calculationshows the resultant invariant mass distributions of d + π − inn that the existencepanels (a1) and (a2) and t + π − in panels (b1) and (b2). Theenergydecay
was −𝐵± 1.1 ±(Z)2.2 wasMeV.requested to be set
Λ = −0.5
tion by Dalitz andand the binding
longitudinalvertexposition
hypernuclei [27],iscarded; however,they discussed the200(a1) d+π −(a2) d+π −ms of scintillatings for the secondary
doscope walls were
g and time-of-flighta large acceptancertexing was placed
he expected decay
net, two separated
ated in such a way
negatively charged

150
100
50

−2 < Z < 30 cm

100
Counts / (2.8 MeV)Counts / (2.8 MeV)

−10 < Z < 30 cm

80
60
40
20
0

0
2.04

2.06
2.08
Mass (GeV)

2.1

2.04

2.06
2.08
Mass (GeV)(b1) t+π −
80
60

2.1

(b2) t+π −−10 < Z < 30 cm−2 < Z < 30 cm

60
Counts / (2.8 MeV)Counts / (2.8 MeV)

12

50

40
cles and fragments
30
ification based on
40
easurements of the20
20
ith the hodoscope10
he invariant mass00d, and a lifetime
2.98
3
3.02
3.04
2.98
3
3.02
3.04
Mass (GeV)
Mass (GeV)
erved decay vertex
ental method was−
FIG.Invariant
1. (Color
online)Invariantdistributionsd The+ πupperFigure 1.3
mass
distributionsof 𝑑 +mass
𝜋− and𝑡 + 𝜋− at GSIof[11].
ﬁguresH, and 4 H, whose−−−
final-statecandidateinpanels
(a1)
and
(a2)
and
of
t
+
π
in
panelsshow 𝑑 + 𝜋 and the bottom ﬁgures show 𝑡 + 𝜋 invariant mass. The diﬀerence between
8]. In this Rapid(b1)andandright(b2).
Panels(a1) and(b1)cut.
are for −10 cm < Z < 30 cm,
the leftis theproduction
vertexes and discussions
and panels (a2) and (b2) are for −2 cm < Z < 30 cm. Observedt + π − that might
distributions are represented by the filled-in circles. The hashedorange (gray) region represents one standard deviation of the fitted

on procedures, themodel centered at the solid blue (gray) line of the total best fit.
ged fragments was 1.4 TheTheoretical
black and colored
dotted lines respectively
calculations
for 𝑛𝑛Λ show
statethe separate
een the measured
contributions of the signal and the background. The open triangle
um from the trackdata corresponding
invarianttomass
distributionof
Variousrepresents
theoreticalthecalculations
have been to
performed
investigate
the existenceof the 𝑛𝑛Λ
ope separation three-bodywasthesystem.
mixed event
analysis.
If its binding energy is deﬁned as −𝐵 = 𝑀− 2𝑀 − 𝑀 , the observed
Λ

𝑛𝑛Λ

𝑛

Λ

state can be a bound state with −𝐵Λ < 0 or a resonant state with −𝐵Λ > 0. In the following

041001-2

sections, these two possibilities will be described for each of the theoretical studies so far. The
details of the theoretical studies are summarized in Appendix A.

1.4.1

Bound stateChapter 1 Introduction

13

Many calculations on the bound state of 𝑛𝑛Λ have been carried out by using various interaction models and techniques of few-body calculation. The currently established hypernuclei,
in particular 3Λ H, 4Λ H, and 4Λ He, were used as a reference to see whether the bound state of𝑛𝑛Λ can exist or not.
The ﬁrst theoretical calculation for a three-body system including hyperons was reported
by Downs and Dalitz [12] in 1958, only six years after the discovery of the hypernuclei. The
calculations were carried out with a variational method. They concluded that it is diﬃcult
to bound the 𝑁 𝑁Λ system with I = 1. After that, the ﬁrst calculation incorporating Λ-Σconversion was performed by Miyagawa et al. [13]. Faddeev method was used to calculate the
𝑁 𝑁Λ three-body system. The interaction due to Λ-Σ conversion was found to be attractive
and played an important role in the binding of hypertriton. In the case of (𝐼, 𝐽) = (1, 1/2),
which is the same quantum number as the 𝑛𝑛Λ state, it was also found to be attractive, but
still not enough to bind it.
After the report by HypHI collaboration, theoretical calculations were carried out by
Hiyama [14], Gal [15] and Garcilazo [16, 17] to explain the result with novel interaction
models. They used diﬀerent interactions and calculation methods. For instance, Hiyama
et al. performed variational calculations with NSC97f as YN interaction which incorporate
Λ𝑁-Σ𝑁 coupling. Garcilazo et al. used the chiral constituent quark model to derive the interactions [16, 17]. In this model, a baryon is represented as a three-body cluster of constituent
quarks, and its mass is generated by spontaneous breaking of chiral symmetry. The Faddeev
equation was solved to calculate the 𝑛𝑛Λ bound state using this interaction. However, allcalculations show that the existence of the bound state is diﬃcult to explain within our current
understanding of hypernuclei.
On the other hand, Ando et al. considered the possibility of 𝑛𝑛Λ bound states from thenew perspective of Eﬃmov states. The 𝑛𝑛Λ state was calculated by solving the coupledintegral equation using pion-less eﬀective ﬁeld theory (Z𝜋 EFT) of the leading order [18]. As aconsequence of the calculation, the possibility of the existence of Eﬃmov states was suggested.
However, since the experimental data to determine the parameters of three-body forces arevery scarce, no quantitative discussion has been presented.
As mentioned above, almost all theoretical calculations have shown that the existence ofbound states is diﬃcult. Theoretical studies on bound states are summarized in Tab. 1.2.

1.4.2

Resonant stateChapter 1 Introduction

14

Table 1.1 Theoretical calculation of complex resonance energies taken from Refs. [19].

Λ𝑛 potential
E0 (MeV)A

B

C

0.551- 2𝑖 4.698

0.456- 2𝑖 4.885-0.149- 2𝑖 5.783

A calculation of resonant states is more diﬃcult than the bound state, so the calculation
results are limited. The resonant states in 𝑛𝑛Λ three-body systems were studied by Belyaev
et al. [19] for the ﬁrst time. The resonant state was searched for by calculating the zeros of thethree-body Jost function. Three Λ𝑛 potentials were proposed by Nemura [20] et al., denoted
A, B, and C. Figure 1.4 shows the potential of the S-state under the assumption of thesepotentials. In all of them, there is a pocket of weak attraction. The results of the calculation
with these potentials are shown in Tab. 1.1. It can be seen that there is a broad resonance in
A and B, while there is no resonance in C.
218

V.B. Belyaev et al. / Nuclear Physics A 803 (2008) 210–226

1.4 The hypercentral
potential
forforthesystem[19].
Λ𝑛Fig.Figure
3. The hypercentral
potential definedby Eq. (30)the𝑛𝑛Λ
systemΛnn with
theThethreediﬀerent
choices (A,B,potentials
and C) of the Λninteraction.
are shown as A, B, and C.

called the hypercentral approximation, [L] = [Lmin ]. We assume that the two-body subsystems
Immediately
afterstates
the HypHI
three calculations
means that on resonant states were reported by
are in the S-wave(1 = report,
2 = 0), which
Afnan et al. [21], Filikhin
et al. [22] and Kamada et al. [23]. They came to diﬀerent conclusions.
3λ = λmin = .
2Afnan et al. performed
a Faddeev calculation using a separate potential. The calculation
So,
in
the
minimalapproximation,
the infinite
system(14), However,
we remain by
withonly one
result was 𝐸 = −0.107 − 𝑖0.622 MeV,insteadand noofresonant
state
existed.
multiplying
equation,
the strength
 of 1 𝑆0 , 3 𝑆1 components
 by 1.05, a resonant state appeared. On the other hand,
λmin (λmin + 1)
2
2
∂r + k −u(E,r) = 2MUu(E, r),(29)
Filikhin reported
a resonant
state ofratherwide widthof 𝐸 = 0.2 − 𝑖1 MeV with NSC97f
r2
potential. Kamada also reported a resonant state of 𝐸 = 0.25 − 𝑖0.40 MeV with YN Nĳmegenwhere all unnecessary subscripts are dropped, and the brackets on the right-hand side mean thefollowing integration

jj ∗
jj
U (r) = Φ[Lzmin ] (ω)(U12 + U13 + U23 )Φ[Lzmin ] (ω) dω.
(30)
From the mathematical point of view, Eq. (29) looks exactly like the two-body radial Schrödinger
equation. The only difference is that the angular momentum is not an integer number.
The explicit expression for the integral (30) is given in Appendix B. The hypercentral po-

Chapter 1 Introduction

15

potential (Nĳmegen89).
Theoretical calculations using the pionless eﬀective ﬁeld theory (Z𝜋EFT) were recently reported by Schäfer et al. [24, 25]. No 𝑛𝑛Λ bound state was found, but there existed a resonant
state depending on the interaction. The interaction models NSC97f and Chiral EFT (NLO)
showed the existence of resonant states, while Alexander B and Chiral EFT (LO) showed noresonant states. The pole position of the resonant state diﬀers depending on the model, but it
exists in the energy range of 𝐸 ≤ 0.3 MeV and the width range of 1.16 ≤ Γ ≤ 2.00 MeV. These
calculations on resonant states are summarized in Tab. 1.3.
Table 1.2 Theoretical calculations for the bound state.

AuthorMethodYN interaction

Bound

Regards

Downs [12]variational method

exponential

×-

Miyagawa [13]Faddeev

Nĳmegen89

×-Hiyama [14]variational method

NSC97f×-

Gal [15]

Faddeev

NSC97e,f

×-

Garcilazo [16, 17]

Faddeev

CCQM

×

-

𝜋EFT
Z

4

Eﬁmov state

Minesota×V ↑ 50% boundAndo [18]Belyaev [19]coupledintegral equation
hypersphericalharmonics

Afnan [21]Faddeev

Yamaguchi etc.

×Λ𝑛 ↑ 25% bound

Filikhin [22]Faddeev

NSC97f×-

Kamada [23]Faddeev

Nĳmegen89

×

YN ↑ 20% bound

Table 1.3 Theoretical calculations of the resonant state.

AuthorMethodBelyaev [19]hypersphericalharmonics

YN interaction

Resonance

Minesota

Regards-×Λ𝑛 ↑ 5% resonanceAfnan [21]Faddeev

Yamaguchi etc.

Filikhin [22]Faddeev

NSC97f-Kamada [23]Faddeev

Nĳmegen89-

Schäfer [24, 25]

IACCC, CSM

NSC97f etc.

-

1.5

Current status of study on 𝑛𝑛Λ stateChapter 1 Introduction

16

The present experimental data other than 𝑛𝑛Λ are consistent and does not accept theexistence of 𝑛𝑛Λ bound state. As mentioned in the previous section, almost all theoretical
calculations have shown that the existence of 𝑛𝑛Λ bound states is diﬃcult. In contrast, many
papers suggest the existence of 𝑛𝑛Λ resonant states, but there are large uncertainties in theirbinding energies and widths. Therefore, an experiment for investing 𝑛𝑛Λ state is desired.
There are many experimental diﬃculties in the 𝑛𝑛Λ production. It is diﬃcult to detect
neutron-rich hypernuclei such as 𝑛𝑛Λ in an emulsion experiment that captures trajectories.
In the conventional counter experiments using (𝜋+ , 𝐾 + ) or (𝐾 − , 𝜋− ) reactions, the unstabletarget (𝑛𝑛𝑛) was required because the neutron is converted to Λ as shown in Fig. 1.5. Since theinvariant mass methods such as the GSI experiment or emulsion methods detect weak decayed
particles, only bound states can be searched for. Furthermore, since ion-beam reactions arerather new reactions to be used for hyper-nuclear productions, the feasibility of the reaction
should be investigated.

(𝐾 − , 𝜋 − ) 𝐾 − 𝑢ത
𝑠

𝑢ത
−
𝑑 𝜋

𝑑
𝑑
𝑢

𝑠
𝑑 Λ
𝑢

(𝜋 + , 𝐾 + ) 𝜋 + u ҧ
𝑑
𝑑
𝑛 𝑑
𝑢

𝑢 𝐾+
𝑠ҧ

𝑛

𝑠
𝑑 Λ
𝑢

𝑒

(𝑒, 𝑒 ′ 𝐾 + )

𝑒′
𝛾∗

𝑝

𝑢
𝑑
𝑢

𝑢 𝐾+
𝑠ҧ
𝑠
𝑑 Λ
𝑢

Figure 1.5 The reaction process of Λ production.

1.6

Our experimentChapter 1 Introduction

17

In this study, the electro-production of 𝑛𝑛Λ in the 3 H(𝑒 , 𝑒 0 𝐾 + )X reaction is presented.
Since the (𝑒, 𝑒 0 𝐾 + ) reaction converts a proton to Λ, the experiment can be performed using
the 3 H target. The use of the 3 H target is very restricted due to its radioactivity. We can,however, use it at the Jeﬀerson Laboratory (JLab) thanks to thorough safety control. JLab canprovide electron beams with high accuracy. The advantages of hypernuclear production inthe (𝑒, 𝑒 0 𝐾 + ) reaction are as follows:• Excitation of spin-ﬂipped states,• Absolute energy measurement using the 𝑝(𝑒, 𝑒 0 𝐾 + )Λ/Σ0 reactions,• High mass resolution.
Figure 1.6 shows the momentum transfer of the Λ particle by (𝐾 − , 𝜋− ), (𝐾 + , 𝜋+ ), and (𝑒, 𝑒 0 𝐾 + )reactions. In the (𝐾 − , 𝜋− ) reaction, where the momentum transfer is small, Λ is generated
on the same orbit as the neutron causing the reaction. In contrast, in the (𝑒, 𝑒 0 𝐾 + ) reaction,
the momentum transfer of Λ is large. It is relatively easy to produce a deeply bound stateof Λ or spin-ﬂipped states. In addition, the Λ (1115.683 ± 0.006 MeV) and Σ0 (1192.642 ±0.024 MeV) produced by the 𝑝(𝑒 , 𝑒 0 𝐾 + )Λ/Σ0 reactions can be used to accurately calibrate themass. Since there is no appropriate target in the (𝐾 − , 𝜋− ) and (𝜋+ , 𝐾 + ) reactions, the binding
energy of 12
C measured in the emulsion is used as a reference. The electron beam used inΛthe (𝑒, 𝑒 0 𝐾 + ) reaction is a primary beam, which allows measurements with higher energy
resolution than hadronic beams, which are secondary beams. Therefore, sub-MeV resolution
has been achieved in previous experiments, and the best resolution can be achieved in counter
experiments.
In contrast, the disadvantages of the (𝑒, 𝑒 0 𝐾 + ) reaction are as follows:• Low production cross section,• High background event due to electromagnetic reaction.
The (𝑒, 𝑒 0 𝐾 + ) reaction has a much lower cross section than the hadronic reaction due to theproperty of the electromagnetic reactions. This is compensated by using a high intensity
electron beam at JLab. In addition, it is important to remove background events such as
M𝜙ller scattering and bremsstrahlung, which occur at a very high rate. Even a very low
fraction of background can be a serious problem since a very high intensity beam is requiredin electro-hypernuclear production.
Hypernuclear studies using electron beams started in 2000. Four experiments (E89-009 [26],E91-016 [27, 28], E01-011 [29], E05-115 [30–34]) were performed in Hall C and one experiment(E94-107 [35]) was performed in Hall A to establish the hypernuclear production by theChapter 1 Introduction

18(𝑒, 𝑒 0 𝐾 + ) reaction. In particular, in the most recent E05-115 experiment, various hypernuclei
in the light- to medium-heavy nuclear regions were successfully produced, and a resolutionof 0.5 MeV (FWHM) was achieved [30].

O. Hashimoto, H. Tamura / Progress in Particle and Nuclear Physics 57 (2006) 564–653

571

Hypernuclear production
for typical crossreactionsversus momentum
transfer.
Figure Fig.
1.6 4. Momentum
transfer crossand section
hypernuclear
sectionfor each Λ
production
process. The ﬁgure is taken from Ref. [36].
Table 1Comparison of Λ hypernuclear production reactions
Z = 0
Neutron to ΛZ = −1
Proton to Λ

(π + , K +Purpose)(π − , K 0 )1.7of this paper
In-flight (K − , π − )In-flight (K − , π 0 )Comment
Stretched, high spin
Substitutional−), π 𝑛𝑛Λ
Stopped(K −collaboration,, π 0)
Stopped(K −ofBoundstateswere reported
by GSIbut the theoretical calculations(e, e K 0 ) by negative results. However,(e, e K + ) there are many positive theoretical calculations
are dominatedSpin-flip, unnatural parity(γ , K 0 )(γ , K + )on the existence of the resonant state, and experimental results obtained using other methods
are strongly desired. The counter experiment at JLab using the (𝑒, 𝑒 0 𝐾 + ) reaction is the
only methodthat can𝑛𝑛Λ inspectroscopy.
a missing-mass
spectroscopy.
This method
has beenthat are used
for produceΛ hypernuclear
Typically,larger momentum
transfers
correspondto smaller hypernuclear
cross
sections.
Characteristicsof thesereactionsforestablishedby experimentsin the past 20
years,
and boundand resonant
statescan be searchedmeson and electromagnetic beams are also compared in Table 1. Each reaction has its
for with very high sensitivity.
own advantages and plays its role in a complete program of hypernuclear spectroscopy.
+ ,theIn thisstudy,only
we thereportsectionmeasurementsof the 3andH(𝑒,the𝑒 0 𝐾 + )X
, π −results) and (πofK +cross) reactions
havebeen used extensively,
However,(K −the+ ) reaction has only beenfeasibilityof the (e,e Kproperties
as describedin arereaction.
In Chapter
2, theof the (𝑒 , 𝑒 0 𝐾 + )demonstrated
reaction andrecently,
its experimental
setup
Section 4.
described. After that, an analysis of the obtained data is performed in Chapter 3. In Chapter
The momentum transferred to the recoiling hypernucleus is shown in Fig. 5 as a function
4, Monte
Carlobeam
Simulations,necessary
analysis
this experiment,
explained.
of incident
momentum.
The (K − ,for
π −the) reaction
has ofa “magic
momentum” are
where
the
recoilmomentumzero.discussion,
It thus preferentially
populatesissubstitutional
Chapter
5 contains
thebecomes
results andand the summary
given at thestates,end. inwhich a nucleon is converted to a Λ hyperon in the same orbit with no orbital angularmomentum transfer (L = 0). On the other hand, the (π + , K + ) reaction and also the19Chapter 2

Experiment

2.1(𝑒, 𝑒 0 𝐾 + ) reaction

In the experiment, the Λ hypernucleus was produced by the (𝑒, 𝑒 0 𝐾 + ) reaction using thehigh-energy electron beam. A schematic diagram of this reaction is shown in Fig. 2.1. The
electron beam is scattered from the target and virtual photons are emitted at a certain probability. Hyperons are produced by the reaction of a nucleon in the nucleus with the virtualphoton. In this process, 𝐾 + s are produced to preserve strangeness. We measured the incident
electron, scattered electron (𝑒 0), and 𝐾 + in this experiment. By measuring the momentum
and energy of these three particles, the mass of the hypernucleus 𝑀𝑋 is reconstructed withthe missing mass method, as follows:𝑀𝑋 =√(𝐸 𝑒 + 𝑀𝑡 − 𝐸 𝑒 0 − 𝐸𝐾 )2 − (®
𝑝 𝑒 − 𝑝® 𝑒 0 − 𝑝®𝑌 )2 ,(2.1)

where 𝑝® and 𝐸 represent the momentum vector and the energy, and the subscripts show theparticle. 𝑀𝑡 is the mass of the target.
The hyperon was produced with the following elementary reactions,𝛾∗ + 𝑝 → 𝐾 + + Λ,
∗

+

0

∗

+

−

𝛾 +𝑝 → 𝐾 +Σ ,𝛾 +𝑛 → 𝐾 +Σ .

(2.2)
(2.3)
(2.4)

For the Λ hypernuclear production, Eq. (2.2) was used. In this reaction, a proton in the
nucleus was converted to Λ. Other reaction methods such as (𝜋− , K− ), (K+ , 𝜋+ ) convert aneutron to Λ. These various reactions can be used to perform complementary hypernuclear
studies because there are only a limited number of stable nuclei that can be used as targets.
Furthermore, a major advantage of this reaction is that Λ and Σ0 particles can be measured
using a proton target, as shown in Eqs. (2.2) and (2.3). Since the masses of these particles have

Chapter 2 Experiment

20

been measured very accurately, we can use these masses to achieve spectrometer calibration
with high accuracy.

K+
e’

𝜙𝐾

𝜃𝑒𝑒 ′

p

𝜃𝛾𝐾

γ*
e

Λ,Σ0

Figure 2.1 A diagram of (𝑒, 𝑒 0 𝐾 + ) reaction. The diagram assumes a one-photon approximation.

2.1.1

Kinematics

The triple diﬀerential cross section of (𝑒, 𝑒 0 𝐾 + ) reaction is deﬁned as

(

𝑑3 𝜎
𝑑𝜎𝑈
𝑑𝜎𝑃 √
𝑑𝜎𝐼
𝑑𝜎𝐿
=Γ
+ 𝜖𝐿
+𝜖
+ 𝜖 𝐿 (1 + 𝜖)𝑑𝐸 𝑒 0 𝑑Ω𝑒 0 𝑑Ω𝐾
𝑑Ω𝐾
𝑑Ω𝐾
𝑑Ω𝐾
𝑑Ω𝐾

(

)
(2.5)

)

𝑑𝜎
=Γ
,
𝑑Ω𝐾

(2.6)

where 𝜎𝑈 , 𝜎𝐿 , 𝜎𝑃 , and 𝜎𝐼 are the unpolarized transverse, longitudinal, polarized transverse,
and interference cross section, respectively [37–39]. Ω𝑒 and Ω𝐾 are the solid angle of thespectrometer for each particle. 𝑑𝜎/𝑑Ω𝐾 is the diﬀerential cross section for (𝛾(∗) , 𝐾 + ) reaction.
Here, Γ is a parameter introduced to convert the cross section of electron-production to thatof photo-production,
Γ=𝐸𝛾 𝐸𝑒0𝛼,222𝜋 𝑄 1 − 𝜖 𝐸 𝑒(2.7)

where 𝛼 is the ﬁne structure constant, 𝐸 𝛾 is the eﬀective photon energy (𝐸 𝛾 = 𝜔 + 𝑞 2 /2𝑚 𝑝 ),and 𝑄 2 is deﬁned as 𝑄 2 = −𝑞 2 . The transverse polarization 𝜖 and longitudinal polarization

Chapter 2 Experiment21

𝜖 𝐿 are deﬁned as follows:(2|®𝑞 | 2
𝜃0
𝜖 = 1+
tan2 𝑒
2
2
𝑄
𝜖𝐿 =

) −1
,

(2.8)

𝑄2
𝜖,
𝜔2

(2.9)

where 𝜔 and 𝑞 are the energy and momentum of the virtual photon, which are deﬁned as
𝜔 = 𝐸𝑒 − 𝐸𝑒0 ,𝑞® = 𝑝® 𝑒 − 𝑝® 𝑒 0 .

22(2.10)
第1章

序論(2.11)SAPHIR EPJA19(2004) 251.
(W=2.120 GeV)
CLAS PRC73(2006) 035202.
(W=2.131 GeV)
CLAS PRC81(2010) 025201.
(W=2.125 GeV)LEPS PRC73(2006) 035214(W=2.163 GeV)SLA PRC58(1998) 75.
(W=2.110 GeV)
KM
PRC61(1999) 012201.
(W=2.110 GeV)
RPR3 PRC73(2006) 045207.
(W=2.130 GeV)
RPR2011 PRL108(2012) 182002. (W=2.130 GeV)

0.5

dσ/dΩCMK [µb/sr]

0.4

0.3

0.2

0.1

0.0

0

50

100

θCM
[degree]γK150

図 1.6 p(γ, K + )Λ 反応微分断面積の理論計算 [16–19] と実験データ [22–25] (角度依存性)

dσ/dΩCMK [µb/sr]

Figure 2.2 The angle dependence of the cross section of Λ photo-production [40].
SAPHIR EPJA19(2004) 251. (W=2.120 GeV)
0.5
CLAS PRC69(2004) 042201R. (W=2.131 GeV)
CLAS PRC73(2006) 035202. (W=2.131 GeV)+The 𝐾 momentum in this experiment were determinedmaximizethe yield of Λ. The
CLAS PRC82(2010)to025202.
(W=2.125 GeV)
0.4
+
LEPS
PRC73(2006) 035214(W=2.163 GeV)
angular dependence of the 𝑝(𝛾, 𝐾 )Λ reaction [40] is shown in Fig. 2.2. The cross section
KM PRC61(1999) 012201.
(W=2.110 GeV)SLA PRC58(1998)
75. are few(W=2.110
GeV)
increases in the forward region (𝜃 CM ∼ 0◦ ). Although
thereexperimental
data and𝛾𝐾

0.3

the theoretical calculations depend on the calculation methods, most of them support anCM
increase of the cross section in the very forward region. Therefore, we adopted 𝜃𝛾𝐾
∼ 0◦
0.2

as the central angle. The center-of-mass energy dependence of the 𝑝(𝛾, 𝐾 + )Λ reaction crossCM= 0.9) is shown in Fig. 2.3. The cross-section reaches
section [41] at the forward angle (cos 𝜃𝛾𝐾
0.1√
the maximum around 𝑊(= 𝑠) = 1.8 GeV and then gradually decreases in the higher energy
0

50

100

150

region. To reduce the 𝐾 + decay in a spectrometer, we adopted 𝑊 = 2.1 GeV, which is a sightly
0.0

0
150
larger energy region. The
momentum50of 𝐾 +CM
under100
these conditions
is 1.82 GeV/c.
θγ K [degree]
The virtual photon ﬂux increases as it becomes more forward as shown in Eq. (2.7). At the
図 1.7p(γ, K + )Σ0 反応微分断面積の理論計算 [16, 17] と実験データ [22, 23, 25–27] (角度依存性)

same time, background events from bremsstrahlung and M𝜙llor scattering also increase at theChapter 2 Experiment22√

2.3 The
dependenceof theangles.
cross sectionof 𝑝(𝛾,𝐾 + )Λ reaction[41]. W(=are𝑠) shown
butions for γ + p →Figure
K+ +forenergy
selected
c.m. kaon
CLAS
results(blue circles)
represents
thecenter-of-massenergies.
Data from SAPHIR (open stars [18] and triangles [8]) and older experiments [41] (black squares) areagrangian calculations computed by KAON-MAID [5] (solid red) and Ireland et al. [12] (dashed black)dal et al. [20,21] (dot-dashed blue).

forward region. Owing to the mechanical constraints of spectrometers, they can only movein a range over 12.5◦ relative to the beam. These background events are suﬃciently small atCMthis angle.035202-17
Therefore, to set the 𝜃𝛾𝐾∼ 0◦ , the angle of the spectrometer was determined to

be 13.2◦ for both the electron and 𝐾 + measurements.

Under the kinematic range mentioned above, the momenta of 𝑒 0 and 𝐾 + pair are plottedin Fig. 2.4. Each point is the pair of electron and 𝐾 + momentum obtained by a simulation.
The black points represent 3 H(𝑒, 𝑒 0 𝐾 + )𝑛𝑛Λ events, the green points represent 𝑝(𝑒 , 𝑒 0 𝐾 + )Λ,and the red points represent 𝑝(𝑒 , 𝑒 0 𝐾 + )Σ0 events. First, the central momentum of 𝑒 0 was set
to cover the expected kinematics of 𝑛𝑛Λ production, which was 2.22 GeV/c. The kinematics
setting is called the Physics-Mode (𝑀phys. ), and the momentum acceptance is shown with the
red line in Fig. 2.4. We also used the 𝑝(𝑒, 𝑒 0 𝐾 + )Λ and 𝑝(𝑒, 𝑒 0 𝐾 + )Σ0 events as calibration. Since
the 𝑝(𝑒, 𝑒 0 𝐾 + )Σ0 event is almost out of acceptance in the Physics-Mode, we took data in the
Calibration-Mode (𝑀calib. ). In this mode, the momentum of 𝑒 0 is reduced to 2.10 GeV/c. The
momentum acceptance in the calibration mode is shown with the blue line in Fig. 2.4. Here,the momentum of 𝑒 0 instead of 𝐾 + was lowered in the calibration-mode to avoid increasing
the decay fraction of 𝐾 + . The parameters in these settings are listed in Tab. 2.1.

2.2

Electron beam provided by Jeﬀerson Laboratory

pK (GeV/c)

Chapter 2 Experiment

23

Physics-Mode

Λ

Σ0

nnΛ

Calibration-Modepe’ (GeV/c)
Figure 2.4 Momentum correlations of 𝑒 0 and 𝐾 + for the reactions in this experiment.
The 𝑝(𝑒, 𝑒 0 𝐾 + )Λ, 𝑝(𝑒, 𝑒 0 𝐾 + )Σ0 and 3 H(𝑒, 𝑒 0 𝐾 + )𝑛𝑛Λ reactions are shown in green, red andblack dots, respectively. Blue dashed line and red solid line show the acceptances of thecalibration (𝑀calib. ) and the physics modes (𝑀phys. ), respectively.

The experiment was carried out at the Thomas Jeﬀerson National Accelerator Facility (JLab)in Virginia, USA. The continuous electron beam accelerator (CEBAF) at JLab was used to
extract high-energy electrons. The electrons accelerated by the CEBAF accelerator are injected
into one of four experimental Halls A, B, C, or D. Figure 2.5 shows the drawing of the CEBAF
accelerator and four experimental halls.

2.2.1

CEBAF

The CEBAF can provide a continuous beam of 100% duty with an intensity of up to 200 𝜇A.
A schematic diagram of the CEBAF is shown in Fig. 2.5. First, the electrons are accelerated to
123 MeV by the injector. The electrons are then injected into a ring with a circumference ofapproximately 1400 m. The ring consists of two parts: a linear accelerator for acceleration anda circular ring for circumference. Each time the electrons passe through the linear accelerator,
they are accelerated by 1.1 GeV and are ejected into each experimental hall. The orbiting
ring consists of ﬁve diﬀerent paths, and electrons take diﬀerent paths depending on theirChapter 2 Experiment

24

Table 2.1 Major reaction parameters of the present experiment. The 𝑝(𝑒, 𝑒 0 𝐾 + )Λ reaction√
was assumed for calculations of 𝑠 and the momentum transfer 𝑞.

Calibration Mode

Physics Mode

(Mcalib. )(Mphys. )Reaction(GeV/c)𝑝 cent.
𝑒0𝑝(𝑒, 𝑒 0 𝐾 + )Λ/Σ0𝑝(𝑒, 𝑒 0 𝐾 + )Λ3 H(𝑒 , 𝑒 0 𝐾 + )𝑛𝑛Λ

2.100

2.218

𝑝 𝐾cent. (GeV/c)
𝑄 2 (GeV/c)2

0.479

0.505

𝜃𝑒𝛾 (deg)

11.9

13.2

𝑞 (GeV/c)√𝑠(= 𝑊) (GeV)

0.497

0.389

2.13

2.07

𝜖

0.769

0.794

𝜖𝐿

0.075

0.092

1.823

energy. The maximum energy was originally 6 GeV, but it was upgraded to 12 GeV. Electronsof 4.32 GeV were used in this experiment. The beam current was limited to 22.5 𝜇A for thesafety of the tritium target.

2.2.2

Beam measurement

In this section, the methods of measuring the position, direction, energy, and charge ofthe incident electron beam are described. The beam position was measured using the beam
position monitor (BPM). Two BPMs were installed, located at 7.524 m and 1.286 m upstreamof the target, respectively. The BPM consisted of four antennas, which were pointed in the
direction of the beam. Each time the beam passed by, the BPM measured its voltage. This
allows the BPM to achieve a position accuracy of 100 𝜇m. The directions of the beam can also
be determined using information from the two BPMs. Although the BPM is a nondestructive
measurement device, the harp must ﬁrst be used to calibrate the BPM, which is a destructive
measurement. The harp was scanned in the direction perpendicular to the beam axis, andthe absolute position was determined by reading the signal of the wire.
In this experiment, we adopted a raster system in which the electrons moved periodicallyin the x- and y-axes to avoid thermal eﬀects on the target cell. The raster is caused by four
dipole magnets placed 2 m upstream of the target. Two of each are used for the horizontal
and vertical directions. The period is 25 kHz, and the electrons are shaken in the range of2 × 2 mm2 . The BPM was used to compensate for this eﬀect as shown in Sec. 3.4.2.

Chapter 2 Experiment25

tical to use the SuperHarps sin
trusive measurement. Instead, b
are used. A BPM consists of aoriented parallel to the electron
signal from each antenna is con
is proportional to the distancetenna. From these four signals,
determined [11]. The BPMs pro
the position and need regular h
to be calibrated and properly us

Year
1999
2001
2002
2014
2018
2021
Figure 1: CEBAF schematic, the injector sends the beam to two linear accelerators (linacs),
whichbeam to theHallsof
A,CEBAF
B, C or D.
Figure2.5deliver
Thetheschematic
view
[42].

Surveyed A
Angle (◦
34.30234.29534.39934.25934.25734.259

Table 2: Bending angle of the ARC meGroup throughout the years.

2. ARC Measurement Method
The ARC measurement is b
The electron energy was measured using the arc method [42,43]. Dipole magnets were usedtron moves in a circular trajecto
ARC
used
the energy frommeasurement
the harps,
to bend the electronsThe
when
they
arefortransported
CEBAF comesto Hallfrom
A. Four
two each
momentumof the electrons desouth linac and goes to Hall A, as is layout in Figure 1. It is afield and its bending aat the front and rear of the magnet, were installed. By measuring the position ofmagnetic
the particle
40 m beam section with 9 dipole magnets, as shown in Figure
Rwith the harp, the2.bendingof theareparticle
be obtained.
Furthermore,
the energy
Eight of angle
the magnets
used tocan
deflect
the beam, thebend
◦ingbeangle
is of byapproximately
and it has 𝐵Δ𝑙been ofsurveyed
E = k of the electrons canderivedmeasuring 34.3
the integrated
the magnet. Since theseveral times during the past several years, Table 2 summarizes
magnetic ﬁeld of the magnets in the arc cannot be measured directly, another magnet (Ninththe results of the surveys.
wherek = 0.299792
GeV rad/TDipole in Fig. 2.6), which is connected to the magnets in series with the power
supply,isis the field integral of the dipol
prepared externally. By constantly measuring this magnetic ﬁeld, the energy of bending
the electrons
angle of the beam in
measured using the ninth dipol
can be calculated.
beam
line.
The beam charge was measured using a beam current monitor (BCM). The BCM
comprises
The energy measurement canan Unser Monitor located in the center with RF cavities located upstream and downstream.
is in dispersive or achromatic m
The amount of charge can be measured as an oscillation frequency. The advantage
ofthe
Unser
quiresquadrupoles to be off
willfollow
Equation 1. Howeve
is that it is self-calibrated, but it cannot be used for a long period of time because the oﬀset
precise alignment and focus of t
due to drift current changes in a few minutes. In contrast, RF is not self-calibrated,but 1it have to be made.
to Equationis very stable and can maintain an accuracy of 0.5% for several months. The BCM
achieves
ductionruns the quadroples are
and
in
some
high accuracy by combining these. First, the RF is calibrated using the results of the Unseroccasions, the quadpersive mode for Energy measu
Figure 2: ARC beam line section. Eight magnets are used to bend the beam
calibration. This RF
used
to ismeasure
the amountofa charge.
allows the amount34.3is◦ , then
the ninthmagnetused as reference
to perform
direct field Thismeasurement.
of charge to be determinedwithin an accuracy of 1.0 %.

The measurement of the bending angle is done by wire scanners called SuperHarps. In practice, the wires of the SuperHarps are moved across the beam path, and when the beamhits the wire, the signal is collected by a photomultiplier tube
(PMT). Based on the signal, the exact position of the beam isreconstructed. There are two SuperHarps in the ARC; one at

2.1. Determining ARC Field In

The field integral is determi
dipole magnet located in a bu
shown in Fig. 3. This magnet
other dipoles, and is used as ameasure the field integral of theis fully occupied by the vacuum

tron moves in a circular trajecto
momentum of the electrons demagnetic field and its bending aR
E = k 26

The ARC used for the energy measurement comes from thesouth linac and goes to Hall A, as is layout in Figure 1. It is a
40 m beam section with 9 dipole magnets, as shown in Figure
2. Eight of the magnets are used to deflect the beam, the bending angle is of approximately 34.3◦ and it has been surveyed
Chapter 2 Experiment
several times during the past several years, Table 2 summarizes
the results of the surveys.

where k = 0.299792 GeV rad/Tis the field integral of the dipol
bending angle of the beam in
measured using the ninth dipol
beam line.
The energy measurement canis in dispersive or achromatic mquires the quadrupoles to be off
will follow Equation 1. Howeve
precise alignment and focus of t
to Equation 1 have to be made.
duction runs the quadroples are
and in some occasions, the quadpersive mode for Energy measu

Figure 2: ARC beam line section. Eight magnets are used to bend the beam
Figure 2.6 The schematic view of arc method [42].
34.3◦ , the ninth magnet is used as reference to perform a direct field measurement.

2.1. Determining ARC Field In

ThermalIsolation
The measurement of the bending
angle
is done by wire scanners called SuperHarps. In practice, the wires of the SuperThermally
Regulated
Harps are movedAluminum
across theEnclosure
beam path,and when
the beam
Downstreamtube
hits the wire, theUpstream
signal is collected by a photomultiplier
Cavity(PMT). Based on thesignal, the exact position ofCavity
the beam isreconstructed. There are two SuperHarps in the ARC; one atUnser
the entrance and one at the exit of
the ARC. Both are used toToroids
reconstruct the bend angle.
Calibration
Wire
During the actual operationof the experiments, it is not prac-

The field integral is determi
dipole magnet located in a bu
shown in Fig. 3. This magnet
other dipoles, and is used as ameasure the field integral of theis fully occupied by the vacuum
The fringe effects are accoun
the use of two translating coils [in series and precisely spaced fro
2

Beam
Direction
Beam
Magnetic Shielding

Figure 3-3:
2.7 Theschematic viewof BCM.
The ﬁgurefrom Monitor
Ref. [44] (BCM).
FigureSchematicof theHall-A
Beamis taken
Current

immediately after, and subsequently monitoring the current using the RF cavities.

2.3calibration
Spectrometer
The
process for these devices is described in section 3.3.1.
TheA,charge
deliveredby theincoming electron
beam is Spectrometers
determined byintegrating
In Hallthere aretwo existing
spectrometers,
High Resolution(HRSs).
These
two spectrometers
the measured
currenthave
overalmost
time.the same conﬁguration; the one on the left is called
HRS-L, and the one on the right is called HRS-R. A schematic diagram of the HRS installed inHall A is shown in Fig. 2.8. We measured the scattered electron in HRS-L and 𝐾 + in HRS-R.

3.1.3
Tritiumtargetassembly
The HRS was
designed asa multipurpose
spectrometer to achieve high momentum resolution (Δ𝑝/𝑝 = 1.0 × 10−4 , FWHM) over a wide range of momentum (0.3–4.0 GeV/c). The

The Hall-Aoftarget
systemconsistsis ofa “ladder”
verticallyalign a given
conﬁguration
the HRS
spectrometershown
in Tab.that
2.2. moves
The magnets
used toin HRS-L
target with the incoming electron beam (see Fig. 3-4). The ladder is equipped withfive gas cells followed by a series of solid targets. The gas cells include tritium,
deuterium, hydrogen, helium-3, and an empty cell for background studies. The solid
targets are a 25-cm dummy and optics targets, and the carbon hole, raster, aluminum,Chapter 2 Experiment

27

HRS-L

HRS-R

Figure 2.8 High Resolution Spectrometers in Hall A. The electron beam is directed fromthe left to right.

and HRS-R are identical. Both have a QQDQ conﬁguration with three quadrupole magnets
and one dipole magnet. Originally, the QQQDQQ conﬁguration was required to achieve ahigh performance. Instead, by using an index dipole containing a Q component as D, suﬃcient performance with the QQDQ conﬁguration could be achieved. Q1, Q2, D, and Q3 aresuperconducting magnets and are always cooled with liquid helium.
One of the most important features of this spectrometer is that it is vertically bent. A majoradvantage of vertical bending is that there is little deterioration in resolution when a longtarget is used in the beam axis direction. In horizontal bending, the momentum resolution
deteriorates because the shift in the z-axis direction appears in the dispersive direction. In
contrast, in vertical bending, the deviation in the beam axis direction appears as the deviationin the non-dispersive direction, which reduces the deterioration of resolution. Therefore, highresolution can be achieved even with a gas target such as the one used in this experiment. The
experimental resolution is discussed in Sec. 5.2. A layout of the magnets is shown in Fig. 2.9.

Chapter 2 Experiment

28

Q3

targetQ1 Q2

D5m

Figure 2.9 The magnetic conﬁguration of HRS.
Table 2.2 Design value of HRS.

2.4

Bending angle

45◦

Optical length23.4 mMomentum range

0.3–4.0 GeV/c

Momentum acceptance|𝛿𝑝/𝑝| < 4.5%

Momentum resolutionΔ𝑝/𝑝 = 1.0 × 10−4 (FWHM)

Solid angle

6.0 msrTarget

The targets were tritium and hydrogen. In addition, 2 H, 3 He, empty and carbon targets
were also installed. Since the tritium and hydrogen target are gaseous, a cigar-type cell with alength of 25 cm in the beam direction was used. Tritium was subject to very strict regulations
due to its radiation. Therefore, a sealed-oﬀ type cell was used, which was designed to preventgas leakage. The cell was cooled to 40 K to avoid cell breakdown due to heat, and the pressure
at this condition was 0.3 MPa. The thickness of the gas was 84.8 ± 0.8 mg/cm2 for tritium
and 70.8 ± 0.4 mg/cm2 for hydrogen. The heat generated in the target by the electron beam

Chapter 2 Experiment29

irradiation causes convection in the gas, which changes the local density of the target [45].
This will be discussed in Sec. 3.2.2. A picture of the target is shown in Fig. 2.10. Each target
was set in the direction of gravity, and the electron beam was irradiated to the target bymoving it in this direction.

25 cm
3H
2H
1H

electron

3He

empty
C

Figure 2.10 The target assembly used in this experiment. This ﬁgure is taken from [45].

2.5

Detectors

The charged particles entering the spectrometer were detected by a set of detectors locatedat the far end. Figure 2.5 shows an overview of the HRS-L and HRS-R detector array. Each
uses mostly the same detectors, but the detectors for particle identiﬁcation (PID) are diﬀerent
in the HRS-L and HRS-R.
Two identical vertical drift chambers (VDCs) are installed in front of the HRS-L detector
array to reconstruct the momentum and angle of the particles by measuring their positions
and angles. Scintillation counters, S0 and S2, are installed behind the VDCs. These are used
to make the trigger and measure the timing of particles. Between S0 and S2, a gas Cherenkov
detector is installed to remove 𝜋− , which are background events. In HRS-R, VDCs, S0, and S2
are installed in the same conﬁguration as in HRS-L. Instead of the gas Cherenkov detectors,
two aerogel detectors (AC1 and AC2) are installed to remove 𝜋+ and 𝑝 background events.

Chapter 2 Experiment

30

S2

HRS-L

S2

HRS-R

S0

S0

AC2
AC1

GC

VDC2
VDC1

VDC2
VDC1
e-

1m

K+

1m

Figure 2.11 A schematic diagram of HRS detectors. Left and right ﬁgures show HRS-L
and HRS-R detectors, respectively.

2.5.1

Tracking detectors

VDCs are used for particle tracking [46]. They are installed on the focal plane of the
HRS optics. Each spectrometer has two VDCs—upper VDC and lower VDC—as shown in
Fig. 2.12, which are placed horizontally at a distance of 33.5 cm. The central trajectory particle
is incident on the VDC in the 45◦ direction since the particle is bent by the dipole magnet.
The VDC consists of two layers, u- and v-layers 2.6 cm apart, and they are orthogonal to eachother. 368 wires are attached at intervals of 4.24 mm, and the eﬀective area is 211.8 cm in thedispersive direction and 28.8 cm in the non-dispersive direction. The VDCs have an applied
voltage of −4 kV, and the gas composition is Ar (62%) and ethane (38%). The amount ofmaterial per VDC is 7.8 × 10−4 𝑋0 (radiation length). The position and angle resolution are
𝜎𝑥(𝑦) ∼ 100 𝜇m and 𝜎𝑥0 (𝑦0 ) ∼ 0.5 mrad, respectively. Here, 𝑥(, 𝑦), 𝑥 0(, 𝑦 0) are the position andangle of (non-)dispersive direction.
A special method called the cluster method was employed for particle tracking. VDCs aretilted in the direction of 45◦ relative to the direction of the particle trajectory, so that multiple
wires coherently emit signals for a single particle. The number of wires that emit signals is anaverage of 3 ∼ 7 wires for each particle. Since these wires emit signals in the order in which
the particles pass, the time information can be used to determine the position and angle ofthe incident particles. Since this can be done in the u- and v-layers, the position and angleof the particle can be determined completely using a single VDC. The second VDC can alsomeasure the position and angle of the particles and was used for a consistency check.

Chapter 2 Experiment
110

31

K.G. Fissum et al. / Nuclear Instruments and Methods in Physics Research A 474 (2001) 108–131

Fig. 1. Schematic layout of the VDCs (not to scale). The rectangular area of each wire frame aperture is 2.118 m  0.288 m (see Section
3.2.1). The U and VFigure
sense wires
are orthogonal
to eachdiagram
other and lie
the horizontalof chamber
the laboratory.
They are inclined at an
2.12
Schematic
ofintheverticalplane
drift
[46].
angle of 451 with respect to both the dispersive and non-dispersive directions. The lower VDC coincides (essentially) with thespectrometer focal plane. The vertical oﬀset between like wire planes in 0.335 m.

All outside surfaces of the Stesalit frames areplated witha 70 mm thickcounterslayer of copper to
2.5.2
Cherenkov
suppress RF pickup and static charge buildup.

Further, after assembly, the entire chassis iscovered with an outer sheathing of 125 mm thickcopper. The individual copper plates forming thisThe measured particles were electrons in HRS-L and 𝐾 + in HRS-R. However, a very large
number of background events were also present. Therefore, Cherenkov detectors suitable foreach spectrometer were installed.
The main background particles in HRS-L are 𝜋− . Therefore, a gas Cherenkov detector(atmospheric pressure, 𝑛 = 1.00041) ﬁlled with CO2 was installed to reject 𝜋− [47]. Since only
electrons emit photons in the momentum region of this experiment (2.1 ∼ 2.3 GeV), the 𝜋− wasrejected by selecting photon emission events. Basic information regarding the gas Cherenkov

Chapter 2 Experiment32

detector is shown in Tab. 2.3. The average of the number of photo-electrons at high energy
electrons is approximately 7.3. The calculation results of the number of photo-electrons in
the GC are shown on the left of Fig. 2.13. The momentum region of this experiment is shownby the gray shaded area.
The background particles for HRS-R are 𝑝 and 𝜋+ . In general, the production cross sectionof 𝐾 + is considerably low compared to the background particles, so a more powerful PID
is required in HRS-R. Therefore, two aerogel Cherenkov detectors—AC1 and AC2—were
installed [48]. The basic information of ACs is shown in Tab. 2.3. The average number ofphoto-electrons for high energy electrons is typically 8 for AC1 and 30 for AC2. The calculation
results of the number of photo-electrons in AC1 and AC2 with the respective particles areshown in Fig. 2.13. The momentum region of this experiment (1.7 ∼ 1.9 GeV) is shown by thegray shaded area. AC1 emits photons for 𝜋+ and 𝐾 + and AC2 for 𝜋+ .

refractive index

radiator length (cm)

PMT

𝑁PMT

GC

1.00041120

ET 9390 KB

10

AC1

1.015

9

Burle RCA 8854

24

AC2

1.055

5

Photonis XP 4572B

26

n. p. e (a. u.)n. p. e (a. u.)

Table 2.3 Basic information of the Cherenkov detectors.

e- (GC)

π+ (AC1)
π+ (AC2)
K+ (AC1)
K+ (AC2)
p (AC1)π- (GC)

pe (GeV/c)p (AC2)

pK (GeV/c)

Figure 2.13 Momentum dependence of the number of photo-electrons in the Cherenkov
detectors. Left and right show the number of photo-electrons of the gas Cherenkov (GC)detector and aerogel Cherenkov (AC) detectors, respectively. The gray shaded areas show
the momentum region of this experiment.

Chapter 2 Experiment

2.5.3

33

Scintillation counters

There are two scintillation counters, S0 and S2. These were installed for particle timing
information and triggering. S0 is one large plate made from plastic scintillator (BICRON 408)
and is placed just after the VDC. The signal is read from PMTs (3-inch XP2312B) in bothsides. The active area is 190 × 40 cm2 and the thickness is 10 cm. S2 is located at the far
end of the spectrometer. The timing information of S2 is used to calculate the coincidence
time (Sec. 3.3.1). S2 is composed of 16 scintillators made from plastic scintillator (EJ-230).
The schematic diagram is shown in Fig. 2.14. The signal is read from PMTs (2-inch Photonis
2282B) in both sides. The active area is 220 × 54 cm2 and the thickness is 2 inch.

2.6

DAQ

Data from the detectors were taken using the FASTBUS and VME modules with ADC and
TDC. The DAQ trigger was formed with NIM modules. These DAQ trigger events were
processed with CODA (CEBAF Online Data Acquisition). The beam information was stored
using EPICS
CHAPTER
12.software.
TRIGGER SCINTILLATOR COUNTERS

Discriminator

93

Splitters

17 in

etc.

F = 60 lbs

88 in
Figure 2.14
Schematic[49].
Figure 12.2: The layout
on theframe diagramof the of
S2S2paddles
and electronics.

from each PMT are sent to a passive 90/10% splitter, with the greater and lesser portions
sent to the on-frame discriminator and Fastbus ADCs,respectively. The discriminator isa Phillips-Scientific model 706 with the threshold set at 10mV. Both NIM outputs areused on each channel, with one line as input for the trigger-logic and the other goingto a TDC after passing through a NIM-ECL converter and a delay of some 880ns. The

Chapter 2 Experiment34

The scintillators, S0 and S2, were used for triggering. The trigger conditions are shown in
Tab. 2.4. T1 and T4 were used for data acquisition in a single arm. T1 is for HRS-L and T4 for
HRS-R, and the condition for data acquisition is S0⊗S2 in each spectrometer. In contrast, T5
is the trigger for the coincidence condition, and the condition for data acquisition is that S0
and S2 of the left and right spectrometers emit the signal simultaneously. The main data wereacquired by T5. T1 and T4 were acquired to measure the properties of the single spectrometer.
These were acquired simultaneously with the T5 data. The rates were typically 1 kHz, 2 kHz,and 60 Hz for T1, T4, and T5, respectively. An overview of the trigger timing is shown in
Fig. 2.6. The signal widths for S0 and S2 were set at 50 ns for HRS-L and 150 ns for HRS-R,
respectively, and the timing was adjusted so that the HRS-L signal came later.
Table 2.4 Trigger condition.

T1(S0⊗𝑆2)LT4(S0⊗𝑆2)R

T5(S0⊗𝑆2)L ⊗ (𝑆0 ⊗ 𝑆2)R

HRS-L

50 ns

HRS-R
150 nstriggertimingFigure 2.15 Trigger timing with T5 condition.

35

Chapter 3

Analysis
In this chapter, the detail of a data analysis to calculate the cross section of the3 H(𝑒, 𝑒 0 𝐾 + )𝑛𝑛Λ reaction is described.

First, the optics calibration to obtain the missing

mass spectrum is explained. After that, event selections to improve the signal to noise ratio(S/N) are described. The eﬃciencies that originate from the various event selections andhardware speciﬁcations etc., and other correction factors are described. There are correction
factors which need to be evaluated by using a Monte Carlo simulation. Details about the MC
simulation, and the correction factors that we evaluated by the MC simulation are describedin the next Chapter (Chap. 4).

3.1

Optics calibration

The missing mass was reconstructed with the following equation,
𝑀𝑋 =√(𝐸 𝑒 + 𝑀𝑡 − 𝐸𝐾 − 𝐸 𝑒 0 )2 − ( 𝑝®𝑒 − 𝑝®𝑘 − 𝑝®𝑒 0 )2 .

(3.1)

Here, 𝑝® and 𝐸 represent the momentum vector and the energy, and the subscripts show theparticle. 𝑀𝑡 is the mass of the target. The momentum and the angle at the production point
were reconstructed with the backward transfer matrix (BTM) for which the inputs are theparticles position and angle at VDC. The BTM is deﬁned as follows:
𝑥𝑇0 =
𝑦𝑇0 =
𝑝𝑇 =

𝑛1
∑
𝑎+𝑏+𝑐+𝑑+𝑒=0
𝑛2∑
𝑎+𝑏+𝑐+𝑑+𝑒=0
𝑛3∑
𝑎+𝑏+𝑐+𝑑+𝑒=0

𝐶 𝑥0 (𝑎, 𝑏, 𝑐, 𝑑, 𝑒)𝑥 𝑎 𝑥 0𝑏 𝑦 𝑐 𝑦 0𝑑 𝑧𝑇𝑒(3.2)

𝐶 𝑦0 (𝑎, 𝑏, 𝑐, 𝑑, 𝑒)𝑥 𝑎 𝑥 0𝑏 𝑦 𝑐 𝑦 0𝑑 𝑧𝑇𝑒(3.3)

𝐶 𝑝 (𝑎, 𝑏, 𝑐, 𝑑, 𝑒)𝑥 𝑎 𝑥 0𝑏 𝑦 𝑐 𝑦 0𝑑 𝑧𝑇𝑒 ,(3.4)

Chapter 3 Analysis

36

where 𝑥(, 𝑦), 𝑥 0(, 𝑦 0), 𝑝, 𝑧𝑇 are the position and angle of (non-)dispersive direction, themomentum and the vertex positon along the beam direction. The angles 𝑥 0 and 𝑦 0 are deﬁned
as 𝑝 𝑥 /𝑝 𝑧 and 𝑝 𝑦 /𝑝 𝑧 , respectively. The variables with and without the subscript T are the onesat the production point and those at VDC, respectively. The 𝑛 stand for the powers of eachpolynomial function. 𝐶 𝑥0 ,𝑦0 ,𝑝 are the parameters to be optimized in the calibration.
At ﬁrst, the production point 𝑧𝑇 is required to calculate Eqs. (3.2), (3.3), (3.4). 𝑧𝑇 is reconstructed with the following equation,
𝑧𝑇 =

𝑛4
∑

𝐶 𝑧 (𝑎, 𝑏, 𝑐, 𝑑)𝑥 𝑎 𝑥 0𝑏 𝑦 𝑐 𝑦 0𝑑

(3.5)

𝑎+𝑏+𝑐+𝑑=0

The parameters of Eq. (3.5) were optimized with carbon multi foil data. Ten thin foils(45 mg/cm2 ) were placed at 2.5 cm intervals, and the reconstructed data are shown in Fig. 3.1.
These optimizations were performed so that the events make peaks at the ideal foil positions.
The MINUIT package in ROOT framework [50, 51] was used to minimize 𝜒2 of the result of
Eq. (3.5). In this analysis, the 3rd-order parameters (𝑛4 = 3) were used.
The angle parameters of Eqs. (3.2) and (3.3) were calibrated with sieve slits, which havemulti holes. A drawing of the sieve slit is shown in Fig. 3.2. The slit has two types of holes—
large holes with a radius of 3.0 mm and small holes with a radius of 2.0 mm. There are twolarge holes, which are convenient for checking the direction. These slits were placed in frontof the Q1 magnet and are shown in Fig. 3.3. Sieve-slit patterns of HRS-L and R with the initial
parameters of Eqs. (3.2) and (3.3) are shown in Fig. 3.4. The black points show the positionof the small hole, and red points show the position of the large hole. The reconstructed hole
images at the sieve-slit for both HRS-L and HRS-R were not at the right position with theinitial parameters. The parameter optimization needed to be performed taking into accountthe ﬁnite size of the beam and the ﬁnite thickness of target. The particle-position images atthe sieve slits after the optimization are shown in Fig. 3.5 in which the right and left panels
are for HRS-L and HRS-R, respectively. The hole image is much clear in HRS-L because theelectrons have larger energy loss due to the radiation and more probability to be stopped
in the slit plate compared to the hadron particles in HRS-R. The hadron particles punchedthrough the sieve slit, and they caused large backgrounds in the HRS-R image. The 4th-orderpolynomials (𝑛1,2 = 4) were used to optimize the angle parameters. Here, the parameters
related to 𝑧𝑇 were limited up to the second order; the parameters tuned to be zero if the powerof 𝑧𝑇 became more than 2.
Finally, missing mass distribution of the elementary process from the hydrogen target wasused to optimize the momentum parameters of Eq. (3.4). Two spectrometer settings wereused in the experiment: 𝑀phys. and 𝑀calib. as shown in Tab. 2.1. The 𝑛𝑛Λ production data
were taken with 𝑀phys. . However, only 𝑝(𝑒 , 𝑒 0 𝐾 + )Λ production which is the energy calibration

37

Counts (/0.3cm)

Chapter 3 AnalysiszT (cm)Figure 3.1 The carbon foil proﬁle reconstructed with Eq. (3.5). Ten carbon foils were used
as targets. The targets were placed at 2.5 cm intervals. No foil was placed at 𝑧 = 7.5 cm.

source was in the acceptance. For purpose of better calibration, data with the lowered central
momentum for 𝑒 0 detection (𝑀calib. ) were taken to simultaneously measure the Λ and Σ0
productions as shown in Fig. 2.4. The parameters were tuned so that the events make Λ andΣ0 peaks are at PDG values [52]. The missing mass distribution obtained by the optimization
are shown in Fig. 3.6. The 5th order of polynomials (𝑛3 = 5) were used for the momentum.
The parameters with the power of more than 2 for 𝑧𝑇 were set to zero as were for the angle
polynomial functions. It is noted that the parameters for the momentum function (Eq. (3.4)) in
𝑀phys. and 𝑀calib. are the same, but, for a conversion, a scale factor was applied by 2.218/2.210
which is the ratio of the central momentum settings. The missing-mass resolution of the peakof 𝑝(𝑒, 𝑒 0 𝐾 + )Λ reaction after the all above calibrations was 𝜎 = 1.3 ± 0.1 MeV/c2 when a ﬁt
was performed with a Gaussian function over a range of |MX − MΛ | < 2 MeV/c2 .

3.2

Number of target nuclei

A gas target was enclosed in an aluminum cell with a length of 25 cm along the beam line.
When the density is 𝜌 mg/cm2 , the Avogadro number is 𝑁𝐴 , and the atomic weight is A, thenumber of targets 𝑁targ with the unit of cm−2 is expressed by the following equation:
𝑁targ =

𝜌𝑁𝐴
A

(3.6)

Chapter 3 Analysis

38

Figure 3.2 A drawing of the sieve slit.

The density of tritium gas was 84.8 ± 0.8 mg/cm2 and A = 3.016 [53], so the number of target
nuclei is (1.69 ± 0.02) × 1022 cm−2 . The number of target nuclei used for the analysis werereduced from this value due to the vertex cut, eﬀects of a density reduction and tritium decay
as shown in the following sections (Sec. 3.2.1–3.2.3).

3.2.1

Vertex cut

The vertex 𝑧𝑇 can be reconstructed for each spectrometer using Eq. (3.5). The vertex 𝑧𝑇
reconstructed by using the tracking information of HRS-L is denoted as 𝑧 𝐿 , and the vertex 𝑧𝑇
reconstructed by HRS-R as 𝑧 𝑅 . The vertex resolution can be improved by taking the averageof both,𝑧mean =

𝑧𝐿 + 𝑧𝑅.
2(3.7)

Figure 3.7 shows the correlation between 𝑧mean and 𝑧diﬀ = 𝑧 𝐿 − 𝑧 𝑅 . The region in the red box
corresponds to the events originating from the target nuclei, whereas the region in the blackbox corresponds to the events originating from the aluminum (Al) cell. The other events are
considered to be accidental coincidence events originating from the cell and the target. The
Al cell has atoms with the larger proton number than the gas target, which leads to the larger
amount of electromagnetic background. In the region where 𝑧diﬀ is larger than the vertex

Chapter 3 Analysis

39

HRS-R
HRS-L

SS

SS

X (cm)X (cm)

Figure 3.3 Picture of sieve slits set in front of the Q1 magnet. The sieve slits were removed
in the production data taking.

Y (cm)Y (cm)

Figure 3.4 The sieve slit patterns reconstructed with the initial angle parameters of
Eqs. (3.2) and (3.3). The left and right panels show the 𝑥 vs. 𝑦 images for HRS-L and
HRS-R, respectively. The markers show the expected position of the hole. The black andred markers represent the positions for the small and large holes, respectively.

resolution, it consists of the following three elements of accidental coincidence:
Alleft &Alright ,(3.8)

Alleft &Target,(3.9)

Alright &Target.

(3.10)

X (cm)40

X (cm)

Chapter 3 Analysis

Y (cm)Y (cm)Counts ( /2 MeV)

Figure 3.5 The sieve slit patterns reconstructed with the optimized parameters of Eqs. (3.2)
and (3.3). Left and right panels show the 𝑥 vs. 𝑦 image for HRS-L and HRS-R, respectively.
The markers show the expected position of the holes. The black and red markers represent
the positions for small and large holes, respectively.

After tuning
Before tuning

MM - MΛ (MeV)
Figure 3.6 H(𝑒 , 𝑒 0 𝐾 + )Λ/Σ0 missing mass spectrum reconstructed by pre and post optimized parameters of Eq. (3.4).

Therefore, a cut condition of 𝑧diﬀ was required to remove these background events. The
survival ratio due to the 𝑧 diﬀ cut was estimated by using Λ production events from thehydrogen gas target. The |𝑧diﬀ | cut dependence of the survival ratio is shown in Fig. 3.8. In
this analysis, the events in the range of |𝑧 diﬀ | < 2.0 cm were selected. This resulted in a target
survival ratio of 84.3 ± 6.6%.
In addition, a 𝑧 mean cut was required to exclude the aluminum events from the entrance and41

zL-zR (m)Chapter 3 Analysis

Selected

Al cell

Al cell(zL + zR)/2 (m)Figure 3.7 Correlation of subtraction (𝑧 𝐿 − 𝑧 𝑅 )and average (𝑧 𝐿 + 𝑧 𝑅 )/2 of the vertex proﬁlein the log scale. The selected area and Al cell region are shown with red and black boxes,
respectively.

exit of the Al cell. Figure 3.9 shows the 𝑧L and 𝑧R distributions when the above 𝑧diﬀ cut wasapplied. The peaks around ±12.5 cm are events from the Al cell, and those in between are
events from the gas target. A spectral ﬁt was performed to evaluate the survival ratio of thegas component after the Al-cell events are cut out with the 𝑧𝑇 selection. A Gaussian function
was assumed for events from the Al cell. For the gas target, a convolution of a quadratic
function and a Gaussian function were used as shown in the following equation.

∫ ∞N(z) =−∞

𝑑𝑧 𝑓 (𝑧 0)𝐺(𝑧 − 𝑧 0 , 𝜎)(3.11)

where 𝑓 (𝑧) is the quadratic function and 𝐺(𝑧 − 𝑧 0 , 𝜎) denotes the Gaussian. We assumed thatthe width parameter of 𝐺(𝑧 − 𝑧 0 , 𝜎) is the same as that of the Gaussian for the Al cell. Fit results
for 𝑧 L and 𝑧R are shown in Fig. 3.9. The thickness of the Al cell is thin enough (∼ 400 𝜇m)
that the width of the Gaussian represents the vertex resolution. Obtained vertex resolution
were 𝜎𝑧 𝐿 = 0.53 ± 0.02 cm and 𝜎𝑧 𝑅 = 0.50 ± 0.02 cm. Similarly, the vertex resolution for 𝜎𝑧mean
was found to be 0.38 ± 0.02 cm. The survival ratio of the gas component as a function ofX𝑧mean (; |𝑧mean | < X𝑧mean ) is shown in the right panel of Fig. 3.10. The survival ratio was foundto be 84.7 ± 0.8% with X𝑧mean = 10 cm which is the condition used in the present analysis.
The contamination of the Al-cell events was evaluated to be less than 0.1% relative to theremaining events in the case of X𝑧mean = 10 cm.

42

Survival Ratio

Chapter 3 Analysis

zdiff (m)Figure 3.8 𝑧diﬀ dependence of the survival ratio of the tritium target.

The excess of events can be seen on the left and right sides of the Al peaks. This is the area
where the Gaussian function cannot represent the Al cell. Considering the possibility that
this event also occurs inside the Al cell, we took this event as a systematic error. The diﬀerence
between the data and the ﬁtting function in the outer region of the cell was taken and inverted
around the position of the cell, and the contamination rate was calculated assuming thatthe same amount of contribution from aluminum exists in the inner region of the cell. The
systematic error of the survival ratio in this test was less than 0.1%. Combining the above
cuts for 𝑧 diﬀ and 𝑧mean , the eﬃciency for vertex cuts was estimated to be 71.5 ± 5.6% and the
Al contamination was less than 0.1% relative to the remaining events with these cuts.

3.2.2

Density change

The gas target was irradiated by a large number of electron beams. In this process, energy
is lost at the cell entrance, cell exit and target, and heat of 15 W is produced. In contrast,
the heat from tritium decay is negligible at 50 mW. The heat on the path of beam caused adensity reduction of gas [45]. This eﬀect depends on the beam current and target. The local
density reaches equilibrium in a few seconds after the electron beam irradiates the cell, andif the current is constant, the local density remains constant.
This eﬀect has already been estimated by measuring the electron scattering using HRS-

Sum

43

zleft

counts (/1.5 mm)counts (/1.5 mm)

Chapter 3 Analysis

zrightSum

TargetTargetAlAlvertex z (m)vertex z (m)

Sum

zmean

Survival Ratio

counts (/1.5 mm)

Figure 3.9 Distribution of 𝑧 𝐿 and 𝑧 𝑅 when 𝑧diﬀ cut was applied. The ﬁtting results arealso shown. See the text for the functions.

TargetTargetAlAlvertex z (m)X𝑧mean (m)Figure 3.10 Survival ratio of the target with the 𝑧mean cut. (Left) Distribution of 𝑧mean(Right) Survival ratio of target and Al. These were calculated with the ﬁtting result of theleft ﬁgure.

L [45]. The density change of the gas was evaluated from a change of counting rate in the
spectrometer. The results are shown in Fig. 3.11. The ﬁgure shows the density correction 𝑓when the beam current 𝐼beam was set to one at 0 𝜇A. The blue points are the data points, andthe solid line is the value of the density correction 𝑓 ﬁtted by a quadratic function.
2𝑓 (𝐼beam ) = 𝑎𝐼beam+ 𝑏𝐼beam + 𝑐(3.12)For the tritium target, 𝑎 = (1.06 ± 0.36) × 10−4 , 𝑏 = (−6.8 ± 0.89) × 10−3 , and 𝑐 = 1. ± 0.003.
The 95% conﬁdence interval due to statistical error is shown in the blue band, and the one
including the systematic error is shown in the gray hatched area. This error is mainly due
to the uncertainty in the current measurement of the BCM. The density is evaluated to beChapter 3 Analysis

44

reduced to 0.901 ± 0.03 at the beam current of 22.5 𝜇A.

Figure 3.11 The change of the tritium gas density as a function of beam current. The
local heat along the beam path caused the density reduction. The experiment for the 𝑛𝑛Λmeasurement was performed at the beam current of 22.5 𝜇A. The ﬁgure is taken from
Ref. [45].

3.2.3

3 H decay and contamination

Tritium nuclei decay in the weak decay mode as follows,3

H → 3 He + e− + 𝜈 e .

(3.13)

Therefore, the loss of the 3 H nuclei and the contamination of the 3 He nuclei need to beconsidered as𝑛3 H (𝑡) = 𝑛30H 𝑒 −𝑡/𝜏3 H(3.14)

𝑛3 He (𝑡) = 𝑛30He + 𝑛30H (1 − 𝑒 −𝑡/𝜏3 H ),(3.15)

where 𝜏3 H = (12.32 ± 0.02 years)/ln(2) is the lifetime of 3 H and 𝑛 is the number of nuclei.
The initial contamination of 3 He when the cell was ﬁlled with tritium gas was reported inRef. [54];𝜖0 =𝑛30He (𝑡)𝑛30He (𝑡) + 𝑛30H (𝑡)= 0.059%.

(3.16)

Chapter 3 Analysis

45

Our experiment was conducted approximately 90 days after the gas was ﬁlled. From Eq. (3.14),
we estimated that 1.4 ± 0.3% of the tritium had already decayed at the experiment.
3 He nuclei also produces a background event because a Λ is produced with the 𝑝(𝑒, 𝑒 0 𝐾 + )Λreaction. The 3 He contamination in the tritium cell was estimated to be 1.4 ± 0.3% from
Eq. (3.15). The number of 3 He shown above is the number of nuclei and must be multipliedby the cross-section ratio. For this, we refer to Ref. [55]. This reports the Fermi momentum
dependence of the proton in R(p) = 𝜎3 He (𝑝)/𝜎3 H (𝑝). We calculated the weighted average of R
according to the Fermi momentum of tritium nucleus [56] and estimated as Rave = 2.12 ± 0.03.
The number of 3 He events in the 3 H(𝑒, 𝑒 0 𝐾 + )𝑋 reaction was estimated to be 3.0 ± 0.7%.
The tritium cell also contained a small amount of hydrogen nuclei. The Λ production eventsfrom the hydrogen target produce a broad peak at −𝐵Λ = 40 ∼ 50 MeV on the 3 H(𝑒, 𝑒 0 𝐾 + )X
spectrum and does not aﬀect the region of interest (−𝐵Λ = −20 ∼ 20 MeV). It is noted that theevents can be used for energy calibration as shown in Sec. 5.3.2.

3.3

Particle identiﬁcation

Various particles other than electron and 𝐾 + enter the HRS-L and HRS-R. In the HRS-L, 𝜋−
is major background, and in the HRS-R, 𝜋+ and 𝑝 are major background. Therefore, particle
identiﬁcation (PID) must be performed to remove them. PID was performed by analysisof time measurement (coincidence time analysis shown in Sec. 3.3.1) and light yields in the
Cherenkov counters (Sec. 3.3.2).

3.3.1

Coincidence time method

We deﬁne 𝑡0 as the timing when the hyperon production event occurs. Since the electrons
and 𝐾 + pass through the spectrometer (∼ 23 m), the timings for the detector to measure
the particles diﬀer from 𝑡0 . These timings are denoted as 𝑡 𝑒,det and 𝑡 𝐾,det . The velocity canbe calculated by using the mass of the particles and the momentum reconstructed by BTM.
Finally, the timing at the hyperon production can be calculated in each spectrometer (𝑡 𝑒,0 , 𝑡 𝐾,0 )by using the velocity and the trajectory of the particle. If the particle assumptions are correct,
the estimated timing should be the same (𝑡 𝑒 ,0 = 𝑡 𝐾,0 ). Therefore, if we deﬁne the coincidence
time as follows:𝑡coin = 𝑡 𝐾,0 − 𝑡 𝑒,0 ,(3.17)

the coincidence events for electron and 𝐾 + can be selected by choosing those with coincidence
times near 0 ns. Figure 3.12 shows the calculation results of coincidence time when the electron

Chapter 3 Analysis

46

and 𝐾 + mass were assumed. The peak near 0 ns is the coincidence event between the electron
and 𝐾 + . In contrast, the peak on the left (𝑡coin ∼ −3 ns) is the coincidence events between theelectron and 𝜋+ , and the peak on the right (𝑡coin ∼ 8 ns) is the coincidence events between theelectron and 𝑝. There is also a periodic structure every 2 ns. This is accidental coincidence
events due to the fact that the incident electron has a bunch structure every 2 ns. Since thisstructure also exists in the region of electron and 𝐾 + coincidence events, we estimated thenumber of accidental coincidence events in the region using side-band data. Here, we selectedfour bunches in the region of 𝑡coin = [3–5], [13–15], [15–17], [17–19] ns as side-band data. The
black line in Fig. 3.12 shows the bunch structure estimated from the side-band data. The time
resolution was estimated to be 0.3 ns by ﬁtting the (𝑒 0 , 𝐾 + ) coincidence peak. The events of|𝑡coin | < 0.7 ns were selected to avoid 𝜋+ contamination. The eﬃciency of the coincidence

Counts (/70 ps)time cut is describedin the next section.
w.o. cut𝝅+ − 𝒆 −selected𝒑 − 𝒆−

𝑲+ − 𝒆 −tcoin (ns)Figure 3.12 Coincidence time distribution without Cherenkov cut. The blue line showsthe experimental data. The black line shows the accidental background evaluated bycollecting some accidental bunches.

3.3.2

Cherenkov cut

The gas Cherenkov detector (GC) at HRS-L identiﬁed the electrons, and the aerogel detectors(AC1 and AC2) at HRS-R identiﬁed 𝐾 + . Figures 3.13 and 3.14 show the correlation between
the number of photo-electrons and the coincidence time in AC1 and AC2. Events in the blue

Chapter 3 Analysis

47boxes are considered to be 𝜋+ and 𝑝. 𝐾 + events are seen in the red box around 𝑡coin = 0 ns.
The following conditions were applied in the analysis to improve the purity of 𝐾 + :n.p.e. (AC1) < 3.0,
1 < n.p.e. (AC2) < 23.0.
The 𝐾 + eﬃciency with the Cherenkov cuts was obtained using events from the𝑝(𝑒, 𝑒 0 𝐾 + )Λ/Σ0 reactions. First, a missing mass distribution for (𝑒 0 , 𝐾 + ) coincidence events
was obtained without the Cherenkov cuts. The blue line in Fig. 3.15 shows the distribution.
The background from (𝑒 0 , 𝜋+ ) coincidence events exist in this distribution. Therefore, amissing mass distribution was reconstructed with (𝑒 0 , 𝜋+ ) coincidence events to remove 𝜋+contamination. This distribution was scaled to 𝜋+ contamination ratio as shown in Fig. 3.15
and was subtracted from the distribution for the (𝑒 0 , 𝐾 + ) coincidence events. This distribution
was compared to the distribution with the Cherenkov cuts. The scale with the smallest
residuals of these distributions were searched for. This scale indicate the 𝐾 + eﬃciency. Whilechanging the 𝜋+ contamination ratio, the scale where residuals were the smallest was set as
the 𝐾 + eﬃciency. The 𝐾 + eﬃciency with the Cherenkov cuts was estimated to be 91.4 ± 6.3%.
The eﬃciency of coincidence time cut and 𝜋+ contamination ratio were evaluated by ﬁtting
the coincidence time distribution. Here, the 𝜋+ contamination ratio was deﬁned as 𝑁𝜋 /(𝑁𝜋 +𝑁𝐾 ). The ﬁtting result with the double-Gaussian functions is shown on the left of Fig. 3.16.
The accidental coincidence components have already been subtracted. The result shows that
there are residual events to the right of 𝐾 + peak. The 𝜋+ peak may be distorted from theGaussian function because the coincidence time was reconstructed with the assumption of
𝐾 + mass for particles in HRS-R as shown in Sec. 3.3.1. A exponential function was added torepresent this event, since the events on the right side of 𝐾 + peak were assumed to be the taildue to 𝜋+ events. The ﬁtting result with the function with the exponential tail for 𝜋+ is shownon the right of Fig. 3.16. We took the average of the cases with and without the exponentialfunction. From these ﬁttings, the contamination of 𝜋+ was estimated to be 2.4 ± 1.8%. In
contrast, the contamination from protons was negligibly small.
In addition, we selected the events with |𝑡coin | < 0.7 ns to improve 𝐾 + purity. The eﬃciencyof the event selection was estimated to be 98.4 ± 0.5% from the ﬁtting of Fig. 3.16. In addition,
the data of the timing information was partly broken. Some bunch of events could be foundwhere the coincidence time was shifted on the order of 𝜇s. When the number of events in abunch is large enough, we could increase the statistics by correcting the coincidence time forproper timing. Some bunches had so few events that it was diﬃcult to correct the coincidence
time. These events were cut and taken into account as an eﬃciency, which was estimated to
be 97.7 ± 0.2%. A total eﬃciency due to the coincidence-time event selections was estimatedto be 96.1 ± 0.5%.

n.p.e. of AC1

Chapter 3 Analysis

48

π+
p

K+
selectedtcoin (ns)Figure 3.13 Correlation between the number of photo-electrons and the coincidence timeof AC1. The typical regions of 𝜋+ , 𝐾 + and 𝑝 are shown in solid boxes.

3.4

Beam electron

The number of electrons incident on the target was counted for the cross-section analysis
as shown in the following Section 3.4.1. The electron beam was periodically deﬂected in 𝑥
and 𝑦 directions (raster) to make the beam spot large enough to avoid damage on the target
cell due to the heat (we called it raster beam). The beam spot size with the raster 2 × 2 mm2although the original beam size at Hall A is about rms = 100 𝜇m which is negligibly small forbeam optics analysis of spectrometers. The beam size due to the raster needed to be corrected
event by event in the analysis as shown in Sec. 3.4.2.

3.4.1

Beam charge

The beam charge on the target was measured by BCM (Sec. 2.2.2). The amount of charge irradiated to each target was obtained by integrating the BCM data. The results are summarizedin Tab. 3.1. The error was mainly due to the accuracy of the BCM.

3.4.2

Raster correction

Chapter 3 Analysis

49

n.p.e. of AC2

p

K+

π+

selectedtcoin (ns)Figure 3.14 Correlation between the number of photo-electrons and the coincidence timeof AC2. The typical regions of 𝜋+ , 𝐾 + , and 𝑝 are shown in solid circles.
Table 3.1 The total beam charge irradiated to each target obtained by accumulating the
BCM information on each run.

TargetCharge (C)H2

4.71±0.05

T2

16.95±0.17

The incident electrons were bent by the magnetic ﬁeld and irradiated to the target witha distribution of 2 × 2 mm2 in the 𝑥𝑦-plane to avoid heat concentration from the electron
beam. Here, 𝑥 and 𝑦 directions are dispersive and non-dispersive directions, respectively. The
distribution is shown in Fig. 3.17. Experimental analysis was performed under the assumptionthat events were produced at (𝑥, 𝑦) = (0, 0) mm. The vertex position was corrected with the𝑦 position as follows,
𝑧𝑇0 = 𝑧𝑇 +𝑦𝑦0

(3.18)

Here, 𝑧 and 𝑧 0 are the vertex position before and after the correction and 𝑦 0 is the angle (𝑝 𝑦 /𝑝 𝑧 )of the particle at the target.

50Counts/ (1 MeV/c2)

Chapter 3 Analysis

𝒑 𝒆, 𝒆′ 𝑲+𝒑(𝒆, 𝒆′ 𝝅+ )MX－mΛ (MeV/c2)

w/o Cherencov
w/ Cherencov

𝜒 2 = 47.63/43

Coincidence time (ns)Counts (/70 ps)

Counts (/70 ps)

Figure 3.15 Missing-mass distribution of 𝑝(𝑒, 𝑒 0 𝐾 + ) reaction without Cherenkov cut andthat of 𝑝(𝑒, 𝑒 0 𝜋+ ) reaction.

w/o Cherencov 𝜒 2 = 48.42/45
w/ Cherencov

Coincidence time (ns)Figure 3.16 Coincidence time before and after Cherenkov cut. The left shows the ﬁtting
result assuming the Gaussian function as the 𝜋+ peak. The right shows the ﬁtting result
assuming the Gaussian function and exponential tail as the 𝜋+ peak.

3.5

OthersIn this section, we discuss the eﬃciency evaluation, which has not been discussed above.
In Sec. 3.5.1, tracking eﬃciency of VDCs will be discussed. In Sec. 3.5.2, the treatment ofmultiplicity data is described. A cut was applied to data with multiplicity of 2 or morebecause the multiplicity was small in this experiment. We will discuss the eﬃciency of the

51

X (mm)

Chapter 3 Analysis

Y (mm)
Figure 3.17 Beam image of 𝑥 vs. 𝑦 at the target, which was reconstructed by using the
BPM information.

multiplicity cut. In Sec. 3.5.3, we will discuss DAQ eﬃciency.

3.5.1

Tracking eﬃciency

The VDCs were used for tracking, as described in Sec. 2.5.1. Tracking eﬃciency wasestimated from the experimental data. First, as shown in the left panel of Fig 3.18, a wireeﬃciency was determined based on whether the center wire had a hit when neighbored wires
at both sides of the wire had hits. The wire eﬃciency of a typical run is shown in the rightpanel of Fig. 3.18. The eﬃciency near the center is almost 1, and the eﬃciency decreases
toward the edge. The hit distribution of the particles and the number of wires per hit werealso estimated from the experimental data. The hit distribution of the particles is shown in
Fig. 3.19.
The eﬃciency per layer (Plane eﬃciency) of each VDC was calculated using a MC simulationby using the wire eﬃciency. Here, we assumed that the wire eﬃciencies were independent
with each other. The wire hit information is given to the wire in MC simulation according
to the particle distribution as shown in Fig. 3.20. Then, the wire is set to be ineﬃcient witha certain probability based on the wire eﬃciency. If the cluster is broken by this process,Chapter 3 Analysis

52

the eﬃciency decreases. By repeating this procedure, a plane eﬃciency of each VDC layer
was obtained. The tracking eﬃciency per spectrometer was obtained as the product of theplane eﬃciency of each VDC layer. The eﬃciency for a typical run is summarized in Tab. 3.2.
The tracking eﬃciency for each run was calculated and the result is shown in Fig. 3.21. The

Efficiency

average result was 98.1% with an error of less than 0.1%.

Efficient
wire

Inefficient
wire

Wire number

CountsCountsFigure 3.18 The estimation of the wire eﬃciency. (Left) Example of a wire hit pattern.
(Right) Example of the wire eﬃciency of a layer.

Wire number

Cluster size

Figure 3.19 VDC hit information estimated from experimental data. (Left) Wire hit position in a layer. (Right) Cluster size of one particle.

3.5.2

Multiplicity

Chapter 3 Analysis

53

MC

Set
inefficient wire

Figure 3.20 Strategy of MC simulation for plane eﬃciency estimation.
Table 3.2 VDC plane eﬃciency and tracking eﬃciency in a typical run.

Layer

LHRS Eﬃciency (%)

RHRS Eﬃciency (%)

VDC1 u

99.72

99.78

VDC1 v

99.65

99.64

VDC2 u

99.80

99.78

VDC2 v

99.73

99.80

Tracking eﬃciency

98.90+0.12−0.18

99.18+0.09−0.19

In this experiment, multiple particles may come into each spectrometer for one trigger.
Since we are using multi-hit TDC, these events were recorded. The multiplicities per eventof the left and right spectrometers are denoted as 𝑀𝐿 and 𝑀𝑅 , respectively. The multiplicity
was low in this experiment, so data analysis was performed only in the case of 𝑀𝐿 = 1 and
𝑀𝑅 = 1. The loss of signals which is due to the cut of multiple hits needed to be taken
into account for the cross-section analysis. The ratio of remaining events to that of the total
number of signals after the multiple-hit cuts is deﬁned as 𝜖multi .

54Tracking Efficiency

Chapter 3 Analysis

Run Number
Figure 3.21 Run dependence of tracking eﬃciency.

First of all, we consider (𝑀𝐿 , 𝑀𝑅 ) = (2, 1), (1, 2) as the excluded event. For example, in thecase of (𝑀𝐿 , 𝑀𝑅 ) = (2, 1), two tracks (𝑝 𝐿1 , 𝑝 𝐿2 ) were recorded in HRS-L and one track (𝑝 𝑅1 ) in
HRS-R. The possible events at this time are 𝑝 𝐿1 × 𝑝 𝑅1 and 𝑝 𝐿2 × 𝑝 𝑅1 .
First, the case (𝑀𝐿 , 𝑀𝑅 ) = (1, 2) is considered. The timing information is shown in theupper part of Fig. 3.22. The coincidence time is calculated by considering the time diﬀerence
between the trigger timing and 𝐾 + . Since the trigger timing is determined by the electrons,
the coincidence time can be correctly calculated in both cases. The left panel of Fig. 3.23
shows the coincidence time when the event (𝑀𝐿 , 𝑀𝑅 ) = (1, 2) is selected. It shows the sum of
the distributions of 𝑝 𝐿1 × 𝑝 𝑅1 and 𝑝 𝐿1 × 𝑝 𝑅2 . The distribution of coincidence time is diﬀerent
from (𝑀𝐿 , 𝑀𝑅 ) = (1, 1), and no clear peak structure of 𝐾 + could be observed in this condition.
This may be because multiple events are mostly secondary particles that have been createdby collisions with detectors or magnets, and the physical quantities such as momentum havechanged. Therefore, the region of 𝐾 + was ﬁtted with an exponential function representing
BG and a Gaussian function representing the 𝐾 + events. The upper limit of the integral of thesignal was estimated to be 0.86% and set as a systematic error.
Next, the case (𝑀𝐿 , 𝑀𝑅 ) = (2, 1) is considered. The timing information in this case isChapter 3 Analysis

55

shown in the lower part of Fig. 3.22. In the ﬁrst event 𝑝 𝐿1 × 𝑝 𝑅1 , the coincidence time wascalculated correctly, but in the second event 𝑝 𝐿1 × 𝑝 𝑅2 , the coincidence time could not becalculated correctly because the time diﬀerence between the electron and 𝐾 + could not be
used. However, the probability that a true event is included is the same for both 𝑝 𝐿1 × 𝑝 𝑅1 and𝑝 𝐿1 × 𝑝 𝑅2 . Therefore, the upper limit was determined in 𝑝 𝐿1 × 𝑝 𝑅1 , and the upper limit was
doubled. The calculation was performed in the same manner as in (𝑀𝑅 , 𝑀𝑅 ) = (1, 2) and thesystematic error was estimated to be 0.42%.
The number of events with multiplicity higher than the above is several times less, so thediﬀerence in eﬃciency due to cutting these events is negligible. By taking the sum of both
(𝑀𝐿 , 𝑀𝑅 ) = (1, 2) and (𝑀𝐿 , 𝑀𝑅 ) = (2, 1), the error due to the multiplicity was estimated to be
1.3%.

(ML, MR) = (1, 2)

e1
K1
K2

(ML, MR) = (2, 1)

e1
e2
K1

Figure 3.22 TDC timing information when a multiple hit event occurs.

3.5.3

DAQ eﬃciency

The run dependence of the DAQ dead time is shown in Fig. 3.24. The DAQ dead time
diﬀers between the initial runs and the subsequent runs. In the data acquisition, not only
T5 (coincidence trigger) but also T1 and T4 (single trigger) were acquired to check the statusof the single arm. In the beginning, the DAQ eﬃciency was poor because a lot of data wereacquired by a single trigger. In the meantime, the ratio of the single trigger data acquisition

56(ML,MR) = (1,2)

Coincidence time (ns)Counts (/0.1 ns)

Counts (/0.1 ns)

Chapter 3 Analysis(ML,MR) = (2,1)

Coincidence time (ns)Figure 3.23 Coincidence time distribution in the multiplicity event. Left shows the distribution for (𝑀𝐿 , 𝑀𝑅 ) = (1, 2) and right shows for (𝑀𝐿 , 𝑀𝑅 ) = (2, 1), respectively.

was reduced by changing the pre-scale factor, so the DAQ eﬃciency improved. By averaging
the DAQ dead time for each run with the weight of the number of events, the eﬃciency due
to DAQ dead time was estimated to be 96.5%. The error was less than 0.1%.
In addition, the eﬃciency of S0 and S2, which are the trigger counters, was considered. Thiswas estimated using experimental data and was found to be 98.1 ± 0.1% [44]. Combining theabove, the eﬃciency associated with DAQ was estimated to be 94.6 ± 0.1%.

57

DAQ dead time (%)

Chapter 3 Analysis

Before tuning

Run No.
Figure 3.24 Run dependence of DAQ dead time.

58

Chapter 4Monte Carlo Simulation
In this chapter, the Monte Carlo (MC) simulation used for data analysis is described. Some
physical quantities, such as acceptances and resolution, could not be obtained from only theexperimental data and they were evaluated by the MC simulation. The MC simulation was
built based on Geant4. In this chapter, the outline of the Geant4 simulator is described ﬁrst.
Then, the event generator, which was used as inputs in the MC simulation, is explained. Next,
a setup model of the MC simulation and its validity test by comparing with the experimental
data are shown. Finally, various quantities evaluated by the optimized MC simulation are
given.

4.1

Introduction

The MC simulation based on Geant4 was built to evaluate various quantities such as
spectrometer acceptance, 𝐾 + decay factor, and so on. There is an existing simulation SIMC [57].
SIMC uses the backward transfer matrix (BTM) which was obtained by using an optics
calculation software. Therefore, the optics calculation needs to be redone to generate a new
matrix as an input of SIMC when one studies diﬀerent beam optics. SIMC runs much quicker
than Geant4 because it calculates particle distribution based on the transfer matrix. However,
we wanted to study the optics more detail by changing the magnetic ﬁeld to quantitatively
evaluate systematic errors on the cross-section results.
A new simulator was constructed by the Geant4 [58–60] to enable more detailed simulations
for this analysis. This was built in C++ and provides a toolkit for Monte Carlo simulations.
The Geant4 can change the magnetic ﬁeld in the simulation, so we can easily modify thespectrometer optics. Since the optics of both HRS-L and HRS-R are identical, a simulator
reproducing HRS-L (HRS Geant4) was constructed, and HRS-R was used as a reversal of themagnetic polarity of HRS-L.

Chapter 4 Monte Carlo Simulation

59

The procedure for obtaining the missing mass distribution by the HRS Geant4 is shownin Fig. 4.1. First, the kinematics parameters (𝑝, 𝑥𝑇 , 𝑥𝑇0 , 𝑦𝑇 , 𝑦𝑇0 ) of scattered electrons and 𝐾 +are obtained using the event generator described in Sec. 4.2. Using this information as input,
the particle position and angular distribution on the VDC (𝑥, 𝑦, 𝑥 0 , 𝑦 0) can be obtained bysimulating the particle trajectory using the HRS Geant4. The BTM can be obtained using theEqs. (3.2), (3.3), (3.4), and (3.5). Furthermore, by randomly varying these quantities with theposition and angle resolutions of VDC, the VDC distribution that represents the experimentcan be produced. The physical quantities at the target (𝑥𝑇0 , 𝑦𝑇0 , 𝑝𝑇 , 𝑧), which include the eﬀects
of resolution and bremsstrahlung, can be reconstructed using the BTM. The missing mass
distribution can be calculated by applying this calculation to the scattered electron and 𝐾 + .
Moreover, the resolution of each physical quantity can be estimated by comparing the quantitywith the true value. The details of the resolution will be described in Sec. 5.2.

e’

HRS-L
Geant4

Transfer
MatrixEvent
Generator

K+

Missing Mass
HRS-R
Geant4

Transfer
MatrixFigure 4.1 Strategy for missing mass reconstruction with Geant4.

4.2

Event generator

The MC simulation was used for the particle transportation, and it needed the particles’
information of momentum vectors at the production point. We made a code to calculate themomentum vectors of 𝑒 0 and 𝐾 + at the production point assuming the (𝑒, 𝑒 0 𝐾 + ) reaction. The
schematic view of the event generation is shown in Fig. 4.2. First, the energy of the incident
electron is determined, from which a virtual photon 𝛾∗ is emitted (Sec. 4.2.1–4.2.2). It thenreacts with the target to produce 𝐾 + and hyperon (Sec. 4.2.5). The calculation method is
diﬀerent for the hydrogen target and the tritium target. In the following sections, details of
the generator are given.

4.2.1

Beam electron

Chapter 4 Monte Carlo Simulation

60

K+

γ*

e’

p
Λ,Σ0

e

T
Figure 4.2 The schematic view of the 3 H(𝑒 , 𝑒 0 𝐾 + )𝑋 reaction.

The energy of the incident electrons was determined based on the distribution of theexperimental data. The central momentum of the incident electrons was 4.31 GeV/c, and theenergy resolution was Δ𝐸/𝐸 ∼ 1.0 × 10−4 (FWHM). In addition, the energy was lost in thetarget cell. The energy loss was taken into account as shown in Sec. 4.4.3.

4.2.2

Scattering electron and virtual photon

The generator uses one-photon approximation, which assumes that the incident electron
only emits one virtual photon. The number of virtual photons (𝑁𝛾∗ ) is expressed as∬
𝑁𝛾∗ = 𝑁𝑒Γ𝑑Ω𝑒 0 𝑑𝐸 𝑒 0 .

(4.1)

Here, 𝑁𝑒 is the number of the beam electrons. Γ is expressed asΓ=𝐸𝛾 𝐸𝑒0𝛼2𝜋2 𝑄 2 1 − 𝜖 𝐸 𝑒(4.2)

Each parameter is explained in Sec. 2.1.1. The four momentum of the scattered electron 𝑝 𝑒 0
are determined by using Eq. (4.2). The momentum of the virtual photon can be calculated asfollows
𝑝 𝛾∗ = 𝑝 𝑒 − 𝑝 𝑒 0 .

4.2.3

Radiation eﬀect(4.3)

Chapter 4 Monte Carlo Simulation

61

Material

Material
Target nuclei

External
External

Internal
Figure 4.3 Schematic view of internal and external radiation.

In electron scattering experiments, the radiation eﬀect due to the small mass of the electron
cannot be ignored. The missing mass distribution has a large amount of radiative tails in thehigh mass side as shown in Fig. 3.6. These radiative tails can be categorized into two types:
internal and external radiation. Internal radiation is caused by the scattering of electrons by
the electric ﬁeld of the target nucleus. In contrast, external radiation is caused by scatteringof electrons by materials other than the target nucleus. The term "radiation" is usually used
to refer to the external radiation. Since external radiation is included in the physics process ofGeant4, the only radiation that needs to be included in the generator is the internal radiation.

Figure 4.4 Feynman diagrams contributed to the internal radiation. Panels a, b, and c
represent virtual internal radiation. Panel d represents the real internal radiation.

The internal radiation was calculated in the Ref. [61], and this calculation was incorporated
in the generator. The correction of the cross-section by internal radiation can be expressed as

Chapter 4 Monte Carlo Simulation
follows:

(

62

(

)

)

𝑑𝜎
𝑑𝜎
=
𝑑Ω𝑒 0 exp𝑑Ω𝑒 0𝑒 𝛿𝑣𝑒𝑟𝑡𝑒 𝑥 +𝛿 𝑅.
2
𝐵𝑂𝑅𝑁 (1 − 𝛿 𝑣𝑎𝑐 /2)(4.4)

Here, 𝛿 𝑣𝑒𝑟𝑡𝑒𝑥 , 𝛿 𝑣𝑎𝑐 and 𝛿 𝑅 represent the vertex correction, the vacuum polarization correction
and the real radiative correction, respectively. 𝛿 𝑣𝑒𝑟𝑡𝑒 𝑥 and 𝛿 𝑣𝑎𝑐 come from the Feynman
diagrams in Fig. 4.4(a), (b), and (c), respectively. Since they denote the virtual internalradiation, they do not contribute to the experimental observables. 𝛿 𝑅 comes from the Feynman
diagram shown in Fig. 4.4(d), which contributes to the experiment as internal radiation. The
discussion will therefore be limited to the 𝑒 𝛿 𝑅 part. 𝛿 𝑅 is expressed as follows under 𝑄 2  𝑚 𝑒2 .

{

(

𝛼 𝑒𝑚
Δ𝐸 𝑒 Δ𝐸 𝑒 0
𝛿𝑅 =
ln
𝜋
𝐸𝑒 𝐸𝑒0

)[

(

)

]

(

)

(

)

(

˜
𝑄2
1 2 𝐸˜ 𝑒
1 2 𝑄2
𝜋2
2 𝜃𝑒
ln
−
1
−
ln
+
ln
−
+
Sp
cos
2
2
3
2
𝑚 𝑒2
𝑚 𝑒2
𝐸˜ 𝑒 0

)}

.
(4.5)

Here, the tilde symbol indicates the parameters in the center-of-mass frame, and Δ𝐸 𝑒 andΔ𝐸 𝑒 0 are the energy loss of the incident and scattered electrons due to internal radiation,
respectively. Sp denotes Spence’s function, which is deﬁned as follows:
Sp(𝑥) = −

∫ 𝑥

𝑑𝑡

0

ln(1 − 𝑡)
.
𝑡

(4.6)

It is the ﬁrst term in Eq. (4.5) that contains the energy loss term, and the other terms do notcontribute to energy loss. Therefore, the contribution of internal radiation can be describedas follows:

(

𝑒

𝛿𝑅

Δ𝐸 𝑒 Δ𝐸 𝑒 0∼
𝐸𝑒 𝐸𝑒0

(

Δ𝐸 𝑒
=
𝐸𝑒
where

[

)𝑎 (

)𝑎
(4.7)

Δ𝐸 𝑒 0
𝐸𝑒0

(

𝛼 𝑒𝑚
𝑄2
𝑎=
ln
𝜋
𝑚 𝑒2

)𝑎
,

(4.8)

)]
.

(4.9)

When an electron with energy 𝐸 loses energy Δ𝐸, the distribution is expressed as follows:(

𝑎 Δ𝐸
𝐼int (𝐸, Δ𝐸, 𝑎) =
Δ𝐸 𝐸

)𝑎
.

(4.10)

Here, 𝑎/Δ𝐸 is the normalization factor. This equation is the same function as that for externalradiation, where the thickness of material 𝑡 is replaced by 𝑎. In the generator, the energy loss
was calculated for both the incident and scattered electrons. The result is shown in Fig. 4.5.
The external radiation alone does not reproduce the experimental data, but the inclusion ofinternal radiation reproduces the data well.

Counts /(1 MeV/c2)

Chapter 4 Monte Carlo Simulation

63

data
w/o internal radiation

Λ

w/ internal radiation

Σ0

MX – mΛ (MeV/c2)
Figure 4.5 The eﬀect of the internal radiation. The missing mass spectrum of the𝑝(𝑒 , 𝑒 0 𝐾 + )Λ/Σ0 reactions with and without the internal radiation is shown using a green
and red histogram, respectively.

4.2.4

Fermi motion and oﬀ-shell eﬀect

The process of a virtual photon reacting with a proton in a target nucleus is considered. In
this experiment, a hydrogen target and a tritium target were used. In the case of the hydrogentarget, the proton is assumed to be at rest. In contrast, in the case of the tritium target, theprotons are bound in the nucleus and have Fermi momentum.
Fermi motion was incorporated using the spectral function 𝑆(𝐸, 𝑘). The spectral function is
the probability of ﬁnding a nucleon with momentum 𝑘 in a state with removal energy 𝐸. The
removal energy 𝐸 is expressed as follows:∗
𝐸 = |𝐸𝐴 | − |𝐸𝐴−1 | + 𝐸𝐴−1,

(4.11)

where 𝐸𝐴 is the binding energy of the target nucleus, 𝐸𝐴−1 is the binding energy of the residual∗
nucleus, and 𝐸𝐴−1
is the excitation energy of the residual nucleus. The momentum of the
proton 𝑘®𝑝 can be obtained by using the spectral function. In the case of a tritium target, theresidual nucleus has two neutrons. By denoting their momentum as 𝑘®𝑛1 , 𝑘®𝑛2 , the following

Chapter 4 Monte Carlo Simulation

64

equation is obtained:
2
𝑘®𝑟𝑒
𝑙

∗
𝐸𝐴−1
=

( 𝑘®𝑛1 − 𝑘®𝑛2 )2
4𝑀

=

𝑀

(4.12)

Here, 𝑘®𝑟𝑒 𝑙 is the relative momentum of the two neutrons. The angle of 𝑘®𝑟𝑒𝑙 was uniformly
distributed. Based on Eq. (4.12) and conservation of momentum, the magnitude of the
momentum of the two neutrons can be determined as follows:√| 𝑘®𝑛1 | =| 𝑘®𝑛2 | =𝑘®𝑝2

2𝑘 ®𝑟𝑒𝑙+4√2
𝑘 ®𝑟𝑒𝑙++ | 𝑘 ®𝑟𝑒𝑙 || 𝑘®𝑝 | cos 𝜃,(4.13)− | 𝑘 ®𝑟𝑒𝑙 || 𝑘®𝑝 | cos 𝜃.

(4.14)𝑘®𝑝2
4Theoretical calculations by Cioﬁ were used as the spectral function. For the tritium target,
the result published in 1980 [62] is shown in Fig. 4.6. However, since the calculations used are
outdated and there is a lack of experimental information to conﬁrm this, the spectral function
result [63, 64] for 3 He is used in this work. Assuming isospin symmetry, the theoretical
calculations for neutrons and protons in 3 He were used to calculate the momentum of protons
and neutrons in 3 H. The uncertainty of assumption has been reported by one of the JLabtritium experiments [55] and was estimated to be less than 3% based on the Fermi momentum
calculation by Wiringa [56].
R F p I. ISTICUCLEQNNUCI EQNPND YHK~~ ~P(k, Eg (f~ )g k(fm-'I

E(MeV)
2

Spectral function
ion oog the proce» 3H{e, g 'p) 2s . The conti5uous liness re p resent the fu ll calculation,cwhereasreas thedot-dashed lines correspon
res nd too the calcu l at'ionn performed omittingin the triton &= 2 wav
Fg~

Figure 4.6 Spectral function of the process 3 H(e, e0 p)2n. This ﬁgure is taken from Ref. [62].
~

onl

the

I =0 component (dot-'', are presented;
ubstantialansubs
differences
starting from k -1 fm are o serv
w- neutron energy
low values of thee tworgy E . It can becontribution f rom thee @wave is
his fact casts severa~ 1 fm ') of calcu 1a t'ovalidity (at ki'
withh wavee functions containing on y

be discussed ins ectral function antained by Dieperinrink et al.. using thee Faddeafunction of Ref . 1 a) are shown forbe seen that the variationa 1 spe
spectral function
r er content off h'ig h m
momentum components; this featu
ture is even moreore apprec able ahigher values off
It has recen tl y been shown 22
r momentum conten t ofo the varia-

Nucleons in nuclei are oﬀ-mass shells because they have Fermi momentum. However, as
will be discussed incomponent.
a later section,the 𝛾 + 𝑁 → 𝐾 + +E.𝑌 reaction was calculated using thet'
A more importantnt q uestion con
oncerns the extent- od wave functions ea

e ercentages give rise to different
ig. 4 the spectral functionspto the RSC andare compare;d it
i can be seen th a t th'tfer. The probbl em as to whether
this difference can bee exp erimenat ll y d t t d 'll""t"1

in the calculation of various o se(Sec. VII).
Finally, we would like tot men
entioni that the numerical accuracy of the calcu 1a t'on
ion of our spectra 1
function has been
een checked by compuuting,for eachif k the momentum dis
distribution
ri u
in two different ways: (i) by using thee eexpression P(k)Chapter 4 Monte Carlo Simulation

65

cross-section results from the CLAS experiment [41]. The target of the CLAS experimentis hydrogen, so the proton is an on-mass shell. The on-mass-shell approximation [65] wasapplied to solve this issue. The proton mass (𝑀 𝑝∗ ) assumed to be oﬀ-mass shell as follows:

(
2
𝑀 𝑝∗
=

√
𝑀𝑇 −

𝑀𝑛2 + | 𝑘®𝑛1 | 2 −√𝑀𝑛2 + | 𝑘®𝑛2 | 2)2− | 𝑝®𝐹 | 2(4.15)

where 𝑀𝑇 and 𝑀𝑛 are the masses of the tritium nucleus and neutron, respectively, and 𝑝®𝐹 is
the momentum of the proton.

4.2.5

Hyperon production

The reaction between the virtual photon and the target nucleus is considered here. In thecase of a tritium target, the reaction 𝛾∗ +3 H → 𝐾 + + 𝑌 should be considered. However, theimpulse approximation, in which 𝛾 ∗ reacts with a single nucleon, can be used due to the highenergy of the virtual photon. Therefore, the reaction 𝛾∗ + 𝑝 → 𝐾 + + 𝑌 is considered. Since
experimental data and theoretical calculations on hyperon production with virtual photons
are scarce, the reaction is assumed to be with real photons. There are three types of hyperon
production reactions to be considered:𝛾 + 𝑝 → 𝐾 + + Λ,
𝛾 + 𝑝 → 𝐾 + + Σ0 ,+(4.16)−

𝛾+𝑛 → 𝐾 +Σ .
The data from the CLAS experiment [41] were used to take into account the energy andangular dependence of these reaction cross-sections. The CLAS experiment was performed
at Hall B in JLab and measured the hyperon-production cross section on a proton target using
real photons over wide energy and angular ranges. The experimental results are shown in
Fig. 4.7 and 4.8. Since there were no data available for the neutron target, the energy andangle dependence of 𝛾 + 𝑝 → 𝐾 + + Λ reaction were applied assuming isospin symmetry.
These results were used to obtain the four momenta of 𝐾 + and the hyperons.

4.3

Geant4 setup

The momentum vectors of the scattered electrons and 𝐾 + created by the generator are used
as input for the HRS Geant4. The construction of the HRS Geant4 is described below.

Chapter 4 Monte Carlo Simulation

66

DIFFERENTIAL CROSS SECTIONS FOR γ + p → K + + Y . . .

PHYSICAL REVIEW C 73, 035202 (2006)

FIG. 8. (Color online) Differential cross sections for γ + p → K + + . The number in each panel designates W (=are results of the amplitude fits [Eq. (3)] discussed in the text.
+√s). The solid lines

Figure 4.7 Diﬀerential cross sections for 𝛾 + 𝑝 → 𝐾 + Λ [41]. The number in each panel√shows
𝑊(= 𝑠).
The parameters of the fit may be used to gain some insightreal numbers. The range over which each parameter is plotted

into the reaction mechanism, unraveling effects from interference among partial waves. Figure 9 shows the coefficientsfrom the fit using Eq. (3). The ai were taken to be purely

4.3.1

Geometry

depended upon its significance, as estimated by the statistical
F test. Mostly, the higher partial waves are not significant
near threshold, but our angular coverage is also less complete

The geometrical information of the HRS spectrometer was taken from Ref. [66]. The visualized result of the HRS Geant4 is shown in Fig. 4.9. Since the Q1 and D magnets considerably
contribute to acceptance, the design was reproduced by 3D-CAD
andonline)
incorporated
theFIG. 9. (Color
Amplitude fit to intothedifferential cross sections for γ + p → K + +simulator. In contrast, Q2 and Q3, which do not contributesigniﬁcantly
to the. Thecoefficients are definedin Eq.acceptance,(3). The
solid vertical lines mark the well-known N ∗ res-were incorporated in a simple cylindrical shape. The target
cellS11is
also
incorporated(1650),
P11 (1710),
and P13 (1720). in its
onances
The dotted line marks the  0 threshold, and theactual shape because it is important for the calculation of theloss.
Asposition.
for detectors,
dashedenergyline marks theD13 (1895)
the position and size of the VDC and S2 were also incorporated. The target chamber, vacuum
ﬁlms, air, etc. were also incorporated to calculate the energy loss in the spectrometer pathcorrectly. The material list is summarized in Tab. 4.1.
035202-9

4.3.2

Magnetic ﬁeld

The magnetic ﬁeld is one of the most important parameters that determine the yield andresolution of an experiment. For the dipole magnet, the magnetic ﬁeld was calculated using
the TOSCA code. TOSCA is a software which can calculate the 3D magnetic ﬁeld maps usingChapter 4 Monte Carlo Simulation

67

DIFFERENTIAL CROSS SECTIONS FOR γ + p → K + + Y . . .

PHYSICAL REVIEW C 73, 035202 (2006)

FIG. 11. (Color online) Differential cross sections for γ + p → K + +  0 . The number in each panel designates W (=are results of the amplitude fits [Eq. (3)] discussed in the text.
+
0√s). The solid lines

Figure 4.8 Diﬀerential cross sections for 𝛾 + 𝑝 → 𝐾 + Σ [41]. The number in each panel√
shows 𝑊(= 𝑠).

the ﬁnite element method. In contrast, for the quadrupole magnets (Q1, Q2, Q3), an empiricalformula, the Kato formula [67], was used.

[

𝑦 2 + 3𝑥 2 𝑑 2 𝐻(𝑠)𝐵 𝑥 = 𝑔 𝑦 𝐻(𝑠) −
12𝐺 2
𝑑𝑠 2

[

]
]

𝑥 2 + 3𝑦 2 𝑑2 𝐻(𝑠)𝐵 𝑦 = 𝑔𝑥 𝐻(𝑠) −
12𝐺2
𝑑𝑠 2
𝑥𝑦 𝑑𝐻(𝑠)𝐵𝑧 = 𝑔
𝐺 𝑑𝑠
𝑧𝑠=
𝐺
035202-11
𝐻(𝑠) = (1 + exp(𝐶0 + 𝐶1 𝑠 + 𝐶2 𝑠 2 + 𝐶3 𝑠 3 ))−1

FIG. 12. (Color online) Amplitude fit to the differential crosssections for γ + p → K + +  0 .
The coefficients are defined in
Eq. (3). The solid vertical lines
mark the well-known N ∗ resonances S11 (1650), P11 (1710), andP13 (1720). The dotted line marksthe  0 threshold, and the dashedline marks the D13 (1895) position.

𝐶0 = −0.9842, 𝐶1 = 6.3375, 𝐶2 = −3.5134, 𝐶3 = 0.9895(4.17)(4.18)(4.19)(4.20)(4.21)(4.22)The 𝑧 direction is the depth direction of a quadrupole magnet. 𝐺 is the bore radius and 𝑔 is
the magnetic ﬁeld gradient.
As described above, an empirical formula was used for the magnetic ﬁeld, which does notcompletely reproduce the actual magnetic ﬁeld. Therefore, the magnetic ﬁeld was optimized
to reproduce the experimental values such as VDC hit distributions. The optimization processChapter 4 Monte Carlo Simulation68

Detectors

Q3

Dipole

Q2
Q1

25 cm

Target
H2 ,T2

Scattering
Chamber

Figure 4.9 Visualization of the HRS Geant4 simulator.

is described below. Since the optics are equal for both HRS-L and HRS-R arms, the parametersof the magnetic ﬁeld were varied while keeping the ratio of each magnetic ﬁeld strength asfollows:
𝐵𝐷𝐿 : 𝐵𝑄1𝐿 : 𝐵𝑄2𝐿 : 𝐵𝑄3𝐿 = 𝐵𝐷𝑅 : 𝐵𝑄1𝑅 : 𝐵𝑄2𝑅 : 𝐵𝑄3𝑅 ,(4.23)

where 𝐵𝐷 is the magnetic ﬁeld of the dipole magnet, 𝐵𝑄 is that of the quadrupole magnet, Lstands for HRS-L, and R stands for HRS-R. Furthermore, the ratio of 𝐵𝑄2 to 𝐵𝑄3 did not change
since the same magnets were used for Q2 and Q3. Therefore, the parameters to be optimized
were 𝐵𝐷 , 𝐵𝑄1 , and 𝐵𝑄2 . The 𝐵𝐷 was adjusted so that the central ray passed through the central
orbit, and the momentum and mass distributions were consistent with the experimental data.
The 𝐵𝑄1 and 𝐵𝑄2 were determined so that the angle distribution at VDC and target would beconsistent between the experimental data and the simulation. The details of the optimization
are shown in Appendix B. The residuals of the distributions from the experimental data wereChapter 4 Monte Carlo Simulation

69

Table 4.1 Material list of the particle trajectory in HRS. The total amount of thickness isshown in two diﬀerent ways because S2 does not aﬀect to the 𝐾 + absorption.

SectionElement

Density (g/cm3 )Length (cm)Thickness (g/cm2 )Gas Target

T2

3.16×10−3

25.0

0.079

Target Cell

Al70752.81

0.04/sin(13.2)

0.492

Scattering Chamber

Al70752.81

0.02790.078

Vacuum Entrance

Ti

4.54

0.0127

0.058

Vacuum Exit

Ti

4.54

0.0127

0.058

Air

Air

1.29×10−3

139+50

0.248

VDC

Al

2.7

0.0069

0.037

S0

CH2

1.03

1.0

1.03

S2

CH2

1.03

5.08

5.23

AC1

0.10

9.0

0.90

AC2

0.20

5.0

1.00

Total

10.22

Total(w/o S2)

4.99

calculated by changing the 𝐵𝑄1 and 𝐵𝑄2 . The magnetic ﬁeld setting where the residual was
the minimum value was adopted as the best setting. The results are shown in Fig. 4.10. The
axis shows the ratio with respect to the best setting. The experimental settings are shown as
black dots. There is a diﬀerence of up to 3% in the magnetic ﬁeld between the experimental
setting and the best setting. The diﬀerence can be attributed to the accuracy of the HRS
Geant4 setup, such as geometry and magnetic ﬁeld. Therefore, the region within the redline in Fig. 4.10, which includes the experimental settings, was adopted as 𝐵𝑄1 and 𝐵𝑄2 . The
diﬀerence in this line was assumed to be a systematic error.

4.3.3

Missing mass

The missing mass distribution calculated by the HRS Geant4 for a proton target is shownin Fig. 4.11. The purple points show the experimental data, and the colored histogram is theresult of the simulation. Green represents 𝑝(𝑒 , 𝑒 0K+ )Λ events, and red represents 𝑝(𝑒 , 𝑒 0K+ )Σ0
events. The HRS Geant4 can reproduce the mass resolution and the tail of the high mass regionof the peaks since the HRS Geant4 includes matter in the particle path and radiation eﬀects.
Systematic errors resulting from the diﬀerence of the missing mass distribution between theexperimental data and the simulation are discussed in Sec. 5.4.1.
The missing mass distribution for a tritium target is shown in Fig. 4.12. The left ﬁgure shows

70

𝐵𝑄1

Residue (/103)

Chapter 4 Monte Carlo Simulation

Best

Experiment
𝐵𝑄2
Figure 4.10 Residuals between the experimental data and the simulation. The residuals
were calculated for the angular distribution at VDC and target. The 𝐵𝑄1 and 𝐵𝑄2 are theratios with respect to the best setting.

the missing mass distribution calculated assuming a tritium target, and the right ﬁgure shows
the missing mass distribution calculated assuming a hydrogen target. The left ﬁgure showsthat the data and the simulation agree well in the high mass side. In contrast, near thethreshold, there is a discrepancy between the data and the simulation due to the Λ𝑛 ﬁnal
state interaction or on-mass-shell approximation. As can be seen from the right ﬁgure, thepartial contamination of the proton target results in contamination of events generated by the𝑝(𝑒 , 𝑒 0 𝐾 + )Λ reaction. The number of quasi free Σ0 events is negligibly small in this kinematic
setting.

4.4

Evaluation results

The HRS Geant4 simulator was used to calculate the various quantities required for analysis.

4.4.1

𝐾 + decay and absorption

Counts /(1 MeV/c2)

Chapter 4 Monte Carlo Simulation

71

data
𝑝 𝑒, 𝑒 ′ 𝐾 + Λ (sim)Λ

𝑝 𝑒, 𝑒 ′ 𝐾 + Σ0 (sim)Σ0

MX – mΛ (MeV/c2)
Figure 4.11 Comparison with the Geant4 simulation and data in the case of hydrogentarget. The experimental data are shown in purple dots. The simulation results for the𝑝(𝑒 , 𝑒 0 𝐾 + )Λ and 𝑝(𝑒, 𝑒 0 𝐾 + )Σ0 reactions are shown in green and red histograms, respectively.

Most 𝐾 + (𝑐𝜏 = 3.7 m) will decay in the HRS path since the HRS has a long path length of∼ 23 m. This decay rate depends not only on the momentum of 𝐾 + but also on the path lengthof the particles in the spectrometer. This was investigated using the HRS Geant4 simulator.
The results are shown in Fig. 4.13. In the region with large momentum, the survival ratio islarger because the time to reach the spectrometer is shorter. Typically, the survival ratio is
15% for the central momentum (𝑝 𝐾 = 1.82 GeV/c).
The 𝐾 + passes through various materials such as target chambers and vacuum membranes
before being detected by the detector. Since the 𝐾 + N interaction is relatively large, it isnecessary to consider the absorption eﬀect of 𝐾 + in the material to derive the production
cross section. Absorption here refers to the 𝐾 + that were produced but did not reach thedetector. Therefore, it also includes multiple scattering in addition to true nuclear absorption.
The materials that exist on the trajectory of 𝐾 + are shown in Tab. 4.1. The absorption rate 𝑓𝑎𝑏𝑠
in the material is obtained by the HRS Geant4 simulation. For a typical value of the central
momentum 𝑝 𝐾 = 1.82 GeV/c, the result is 8.9%. In the actual analysis, the absorption ratio
depends on momentum, and the data were corrected for each event.

Chapter 4 Monte Carlo Simulation

72

QF Λ (sim)

QF Λ

data
𝑝 𝑒, 𝑒 ′ 𝐾 + Λ (sim)Counts /(2 MeV)Counts /(2 MeV)

data
𝑝 𝑒, 𝑒 ′ 𝐾 + Λ (sim)

QF Λ (sim)

QF Λ

Λ

Λ

MX – MΛ (MeV/c2)-BΛ (MeV)

Survival Ratio

Figure 4.12 Comparison with Geant4 simulation and data in the case of tritium target.
The experimental data are shown as purple dots. The simulation results are shown ascolored histograms. Left and right ﬁgures show missing mass spectra calculated assuming
the tritium target and hydrogen target, respectively.

AcceptancepK (GeV/c)
Figure 4.13 𝐾 + momentum dependence of 𝐾 + survival ratio in HRS-R path. The acceptance region (|𝛿𝑝/𝑝| < 4%) is also shown as a blue line.

4.4.2

AcceptanceAcceptance is the solid angle at which the spectrometer can detect particles. The acceptances of spectrometers are a function of momentum. Therefore, the momentum-dependentacceptances of each spectrometer, ΔΩ𝑒 0 (𝑝 𝑒 0 ) and ΔΩ𝐾 (𝑝 𝐾 ), were obtained and used in theChapter 4 Monte Carlo Simulation

73

analysis. The momentum and angular distributions of the particles were uniformly produced
when estimating the acceptance. When 𝑁gen particles ejected in the range of Ωgen (msr) and𝑁𝑎𝑐𝑐 are detected, the acceptance ΔΩ(𝑝) for a given momentum 𝑝 is expressed as follows:ΔΩ(𝑝) =

𝑁acc (𝑝, cos 𝜃)Ωgen .
𝑁gen (𝑝, cos 𝜃)(4.24)The magnetic ﬁeld was taken from the one described in Sec. 4.3.2. The red point in Fig. 4.10
was used as the central value, and the deviations from the central value of the magnetic ﬁeld
within the red line in Fig. 4.10 were treated as systematic errors. The results of the acceptancefor HRS-L are shown in Fig. 4.14. The blue dots indicate the acceptance of the central value,
and the shaded area indicates the systematic error. The red line is the result of SIMC, i.e., theconventionally used simulator. From this result, the number of virtual photons 𝑁𝛾∗ can beobtained. The number of virtual photons is expressed as follows: 𝑁𝛾∗ = Γint 𝑁𝑒 , where∬Γint =

Γ𝑑Ω𝑒 0 𝑑𝐸 𝑒 0 .

(4.25)

Γ is a quantity calculated using the Eq. (2.7).
The systematic errors were obtained by calculating the acceptance for the magnetic ﬁeld
within the red line in Fig. 4.10, as described above. The errors shown below are averaged
over the momentum acceptance of the experiment. The error of the ΔΓint was estimated to be
8.5%. If we deﬁne ΔΩ𝐾 × 𝜖decay × 𝜖absorp as ΔΩall , the error of ΔΩall was estimated to be 7.6%.
Here, 𝜖decay and 𝜖absorp represent the eﬃciencies of 𝐾 + decay and absorption, respectively.
The diﬀerence of from SIMC in Γint was also 2%, which was within the range of systematic
errors.

4.4.3

Energy loss correction

The particles lose their energy in the material. Therefore, the momentum measured by BCM
and HRS spectrometers are diﬀerent from the momentum at the hyperon production. The
momentum loss should be corrected to calculate the missing mass accurately. The momentum
used for mass reconstruction 𝑝 real is represented as𝑝 real= 𝑝 mea− Δ𝑝 𝑒𝑒𝑒(4.26)mea
𝑝 real+ Δ𝑝 𝑒 0
𝑒0 = 𝑝𝑒0
𝑝 𝐾real = 𝑝 𝐾mea + Δ𝑝 𝐾 ,(4.27)(4.28)

mea
where 𝑝 mea
is the momentum measured with BCM and 𝑝 mea
are the momentum mea𝑒𝑒0 , 𝑝𝐾

sured in HRS-L, HRS-R, respectively. Δ𝑝 is the momentum loss in the material. Especially,

Acceptance (msr)

Chapter 4 Monte Carlo Simulation

SIMC

74

Geant4pe (GeV/c)
Figure 4.14 Acceptance estimated by the HRS Geant4 and SIMC. The result of the HRS
Geant4 is shown as blue points, and the blue shaded area represents the systematic error.
The result of SIMC is shown as a red solid line.

the target cell and air have signiﬁcant eﬀects on the energy loss. The electrons lose theirenergy more than 𝐾 + due to the radiation eﬀect.
As for the target cell, special corrections are required because of the shape. The thickness ofthe target cell is shown in Tab. 4.2 and the name scheme is shown in Fig. 4.15. The thicknessof the target cells for H and 3 H are typically 424.83 𝜇m and 427.50 𝜇m, although it has asystematic error of 25% and 7.6% depending on the position of the cell, respectively. The cell,which is called the cigar-type cell, can be separated into two parts based on the diﬀerence inthe energy loss as shown in Fig. 4.16. The central part is cylindrical and the downstream partis hemispherical. We measured the particles at 13.2◦ to the beam direction; thus, the particles
pass through a distance of 1./sin 13.2◦ (≈ 4.4) times their actual thickness when the particle
crosses the cylindrical part. In this case, the energy loss is ∼ 1 MeV and has a signiﬁcanteﬀect on the energy accuracy and resolution. Moreover, this has 𝑦𝑇0 dependence because
the actual thickness depends on the particle angle. In the case of the hemispherical part,
particles pass through the actual cell thickness and the energy loss is relatively small. We
distinguished whether the particles passed the cylindrical or hemispherical part dependingChapter 4 Monte Carlo Simulation75

on the reaction point 𝑧𝑇 . From the geometric calculations, the division point was determined
to be 𝑧𝑇 = 8.0 cm, which is downstream in the target.
The results of these energy loss estimations are shown in Fig. 4.17. This shows the 𝑦𝑇0 angle
dependence of energy loss. The solid line shows the energy loss of the cylindrical part, andthe dotted line shows that of the hemispherical part. Polynomial functions were used as
the correction function. The momentum of the particles were corrected event by event withthese functions. The systematic error of cell thickness aﬀect the mass accuracy and will be
described in Sec. 5.3.2.
The energy loss of each particle follows a Landau distribution. If the MPVs of two Landau
distributions are 𝑥 MeV and 𝑦 MeV, the MPV of the sum of the two Landau distributions
is not 𝑥 + 𝑦 MeV because the Landau distribution is asymmetric. Therefore, even if theenergy loss is corrected for each particle, there will be a discrepancy when the missing massis reconstructed. This was also estimated by reconstructing the missing mass in the HRS
Geant4.
Beam direction

Entrance
Left

Mid
Left

Exit
Left

Entrance

Exit

Entrance
Right

Mid
Right

Exit
Right

Figure 4.15 The name scheme of cell thickness.

Chapter 4 Monte Carlo Simulation

76

400 μm ×(1/sin 𝜃)

400 μm
𝜃

-12.5

8

0

12.5
z (cm)

Figure 4.16 An illustration of the energy-loss correction for the the target cell.

Table 4.2 The measurement values of cell thickness. The names of location are shown in
Fig. 4.15. These data are taken from [45]. Unit in mm.

Location

Empty cell

3 H Cell

1 H Cell2 H Cell

3 He Cell

Entrance0.254±0.005

0.253±0.004

0.311±0.001

0.215±0.004

0.203±0.007

Exit

0.279±0.005

0.343±0.047

0.330±0.063

0.294±0.056

0.328±0.041

Exit left

0.406±0.005

0.379±0.007

0.240±0.019

0.422±0.003

0.438±0.010

Exit right0.421±0.005

0.406±0.004

0.519±0.009

0.361±0.013

0.385±0.016

Mid left

0.457±0.005

0.435±0.001

0.374±0.004

0.447±0.009

0.487±0.060

Mid right

0.432±0.005

0.447±0.004

0.503±0.005

0.371±0.012

0.478±0.007

Entrance left

0.508±0.005

0.473±0.003

0.456±0.010

0.442± 0.005

0.504±0.003

Entrance right0.424±0.005

0.425±0.003

0.457±0.006

0.332±0.011

0.477±0.011

77

Energy Loss (MeV)

Chapter 4 Monte Carlo Simulation𝑦𝑇′
Figure 4.17 𝑦𝑇0 dependence of energy loss of electron in HRS-L. The solid line shows theenergy loss of the cylindrical part, and the dotted line shows that of the hemispherical part.

78

Chapter 5

Results and Discussion

5.1

Derivation of the diﬀerential cross section

The cross section of the hypernuclear electro-production by the 𝛾 (∗) 𝑝 reaction is deﬁned asfollows:

(

∫

)

𝑑𝜎
=
𝑑Ω𝐾
=

𝑑Ω𝐾
HRS−R

∫

(

𝑑𝜎
𝑑Ω𝐾

𝑑Ω𝐾
HRS−R
𝑁
HYP
∑

1 1
𝑁𝑇 𝑁𝛾∗)

1𝜖 𝑖 ΩHRS−R
𝑖
𝑖=1

,

(5.1)

where 𝑁𝑇 and 𝑁𝛾∗ are the number of target nuclei and virtual photon, 𝜖 𝑖 is the eﬃciency, andΩHRS−R
represents the HRS-R acceptance estimated by the HRS Geant4. The eﬃciencies are𝑖summarized in Tab. 5.1. The cross section of this experiment is shown in Fig. 5.1.
Although the accidental background can be estimated using the side-band data of thecoincidence time, mixed event analysis was used to improve the statistics. First, we divided
the distribution of coincidence time into 2 ns intervals, which are the cycles of the beamelectron. Next, we selected two diﬀerent accidental regions of the distribution. The missing
mass was reconstructed by randomly selecting an event for each region as shown in Fig. 5.2.
By scaling this to the correct number of events, the accidental background with increased
statistics can be obtained. The bunches selected for the side-band data are the same as
described in Sec. 3.3.1. The systematic error was 1.9%, which was determined by the numberof events used in the mixed event analysis.
The missing mass distribution of the 3 H(𝑒 , 𝑒 0 𝐾 + )𝑛𝑛Λ reaction appear to be some excess
around the 𝑛𝑛Λ threshold (−𝐵Λ ∼ 0 MeV). To evaluate this excess quantitatively, the spectral
ﬁt should be performed. The mass resolution and shape of the 𝑛𝑛Λ signal were required toperform the ﬁt and will be estimated in the following section.

Chapter 5 Results and Discussion

79

dσ/dΩ (nb/sr /2 MeV)

3H(e,e’K+)nnΛ

Accidental coincidence−𝐵Λ (MeV)
Figure 5.1 Diﬀerential cross section for the 3 H(𝑒, 𝑒 0 𝐾)𝑛𝑛Λ reaction. The only statisticalerrors are shown in this histogram. The black points show the accidental coincidence background. The distribution was obtained with mixed event analysis using the experimental
data.

5.2

Missing mass resolution

The mass resolution in the 3 H(𝑒, 𝑒 0 𝐾 + )𝑛𝑛Λ reaction needs to be estimated to search for𝑛𝑛Λ state. However, since the signiﬁcant peak has not been observed in the spectrum for thetritium target, it cannot be estimated from the experimental data itself. Therefore, the HRS
Geant4 was used to estimate the mass resolution.
The missing mass can be calculated using the following equation:
𝑀𝑋 =√(𝐸 𝑒 + 𝑀𝑡 − 𝐸𝐾 − 𝐸 𝑒 0 )2 − ( 𝑝®𝑒 − 𝑝®𝑘 − 𝑝®𝑒 0 )2 ,(5.2)

where 𝑀𝑡 is the mass of target. The mass resolution is represented with the error propagation
method as follows:

√
Δ𝑀 =

(

)2 (

)2 (

)2 (

)2 (

𝜕𝑀
𝜕𝑀
𝜕𝑀
𝜕𝑀
𝜕𝑀
Δ𝑝 𝑒 +
Δ𝑝 𝑒 0 +Δ𝑝 𝐾 +
Δ𝜃𝑒𝑒 0 +Δ𝜃𝑒𝐾
𝜕𝑝 𝑒
𝜕𝑝 𝑒 0𝜕𝑝 𝐾𝜕𝜃𝑒𝑒 0
𝜕𝜃𝑒𝐾

)2
(5.3)

Chapter 5 Results and Discussion

80

Table 5.1 Eﬃciencies and correction factors used for the cross-section calculation in Eq. (5.1).

Item

Eﬃciency or correction factor

𝜖 track

0.981

𝜖 decay

0.15 at 𝑝 cent.

𝜖T

0.986

1/𝜖 He

0.97

𝜖 DAQ

0.95

𝜖 ctime0.96

𝜖 absorp

0.91 at 𝑝 cent.

𝜖 density

0.901

𝜖 vertex

0.71

𝜖 PID

0.91

1/𝜖𝜋

0.98

The diﬀerential coeﬃcients are represented as follows:
𝜕𝑀
1
=
𝑀𝐻𝑌𝑃
𝜕𝑝 𝑒

{

1
𝜕𝑀
=−
𝑀𝐻𝑌𝑃
𝜕𝑝 𝑒 0}𝑝𝑒(𝑀𝑡 − 𝐸𝐾 − 𝐸 𝑒 𝑝 ) + 𝑝 𝑒𝑝 cos 𝜃𝑒𝑒 0 + 𝑝 𝐾 cos 𝜃𝑒𝐾 ,
𝐸𝑒

(5.4)

𝑝𝑒0
(𝑀𝑡 + 𝐸 𝑒 − 𝐸𝐾 ) − 𝑝 𝑒 cos 𝜃𝑒𝑒 0 + 𝑝 𝐾 cos 𝜃𝑒 0 𝐾 ,
𝐸𝑒0

(5.5)

{
{

}

}

𝑝𝐾
1
𝜕𝑀
=−
(𝑀𝑡 + 𝐸 𝑒 − 𝐸 𝑒 0 ) − 𝑝 𝑒 cos 𝜃𝑒𝐾 + 𝑝 𝑒 0 cos 𝜃𝑒 0 𝐾 ,𝑀𝐻𝑌𝑃 𝐸𝐾
𝜕𝑝 𝐾
𝜕𝑀
1
=−
𝑝 𝑒 𝑝 𝑒 0 sin 𝜃𝑒𝑒 0 − 𝑝 𝑒 0 𝑝 𝐾 sin(𝜃𝑒𝑒 0 + 𝜃𝑒𝐾 ),𝑀𝐻𝑌𝑃𝜕𝜃𝑒𝑒 01
𝜕𝑀=−𝑝 𝑒 𝑝 𝐾 sin 𝜃𝑒𝐾 − 𝑝 𝑒 0 𝑝 𝐾 sin(𝜃𝑒𝑒 0 + 𝜃𝑒𝐾 ),
𝑀𝐻𝑌𝑃
𝜕𝜃𝑒𝐾

(5.6)
(5.7)
(5.8)

where 𝑀𝐻𝑌𝑃 is a mass of the hypernuclei. Here, 𝜃𝑒 0 𝐾 = 𝜋 − 𝜃𝑒𝑒 0 − 𝜃𝑒𝐾 is assumed because
the reaction plane angle at which 𝐾 + is generated is approximately zero. From Eqs. (5.7) and(5.8), which represent the angular resolution, we can see that the contribution of the angle isinversely proportional to 𝑀𝐻𝑌𝑃 . In contrast, Eqs. (5.4), (5.5), and (5.6), which represent the
momentum resolution, show that the contribution of 𝑀𝐻𝑌𝑃 is small and is less dependent on
the target mass number (A). Although the mass resolution was directly estimated with the
HRS Geant4 as shown in Sec. 4.1, each component needs to be checked for the conﬁrmationof the A dependence of mass resolution.

5.2.1

Angular resolution

Chapter 5 Results and Discussion

81

event0event0event1

event1

event2

event2

event1・・・・・・

eventN

eventNe-K+

Figure 5.2 Strategy of the mixed event analysis. The accidental coincidence event can be
reproduced with randomly mixed events.

The angular resolution was evaluated using the sieve-slit data. It was necessary to subtract
the size eﬀect of the sieve-slit hole because the hole itself has a ﬁnite size. In addition, thesolid angle appears to be smaller toward the edge because the sieve-slit has a thickness of
2.54 cm. Here, Δ𝑥 0 and Δ𝑦 0 were used as angle parameters since it was easier to estimate Δ𝑥 0
and Δ𝑦 0 than Δ𝜃.
A simple Monte Carlo simulation was performed to estimate the angular resolution. A
schematic diagram is shown in Fig. 5.3. We took the sieve-slit data with several carbon foils.
The particle direction was uniformly distributed and the straight line was extended to thesieve slit from the position of carbon foils. When the intersection of the line and the sieve slit
was in the position of holes, the particle was accepted. The thickness of the sieve slit was also
considered. This represents the maximum resolution, i.e., Δ𝑥 0 = 0 rad and Δ𝑦 0 = 0 rad. The
results of the simulation in this state are shown on the left in Fig. 5.4. The straight line was
randomly varied with a Gaussian distribution. An example is shown on the right of Fig. 5.4.
The sigma of the Gaussian was determined to match the simulation results and the sieve-slit
data (Fig. 3.5). The results are shown in the following.
At ﬁrst, only the central hole was simply used although the sieve slit has many holes.

Chapter 5 Results and Discussion

hole

82

Sieve Slit

hole
hole

w/ resolution
Δx’, Δy’w/o resolution

Carbon foils

Figure 5.3 Strategy of the simple Monte Carlo simulation for angle resolution estimation.

The central hole was extracted from the data and projected in the 𝑥 and 𝑦 directions. The
one-dimensional histograms were compared to the simulation. MINUIT method was used
for the comparison, and the resolution as the one with the smallest residue was searched.
The results are shown in Fig. 5.5. In this case, the resolution was Δ𝑥 0 = 2.23 × 10−3 rad andΔ𝑦 0 = 8.95 × 10−4 rad.
To examine the resolution including the edge of the sieve slit, the data of an entire row
and column were also examined. The resolution in this case was Δ𝑥 0 = 2.37 × 10−3 rad andΔ𝑦 0 = 1.52 × 10−3 rad. The Δ𝑦 0 is worse than that estimated from only the central hole. Thismay be because the accuracy of 𝑦 becomes worse toward the edge of the acceptance, as canbe seen from the sieve-slit data.
The angular resolution was expected to be worse by Al cell. The Al cell has a thicknessof ∼ 400 𝜇m, and the particles have an angle of 13.2◦ , so that it has an eﬀective thicknessof 400/sin 13.2◦ and the eﬀect of multiple scattering is signiﬁcant as mentioned in Sec. 4.4.3.
This eﬀect was estimated by the HRS Geant4 simulator. Materials on the HRS path wereincorporated to include the multiple scattering eﬀect. As a result of the simulation, it was
found that the case with the Al cell was worse than the case with carbon foil by Δ𝑥 0 =X (cm)83

X (cm)

Chapter 5 Results and Discussion

HRS-R

Y (cm)Y (cm)

Data

CountsCountsFigure 5.4 Result of Monte Carlo simulation for the sieve slit. The left ﬁgure shows thesieve pattern not including resolution. The right ﬁgure shows including resolution.

Simulation

Data

HRS-R
Simulation

X (cm)Y (cm)

Figure 5.5 Comparison between simulation and data of sieve-slit pattern. Figures show
the projection of the center hole proﬁle. Left and right show the x- and y-projections,
respectively.

5.0 × 10−4 rad and Δ𝑦 0 = 8.5 × 10−4 rad. Finally, the angle resolution with the tritium target
was estimated to be Δ𝑥 0 = 2.4 × 10−3 rad and Δ𝑦 0 = 1.7 × 10−3 rad.

5.2.2

Momentum resolution

The momentum resolution was also estimated with the HRS Geant4 since we did notacquire reference data for momentum resolution. In Sec. 9.2.2 of Ref. [66], results of theelastic scattering of electrons are described. In this study, it is reported that a resolution of
Δ𝑝/𝑝 = 1.1 × 10−4 was achieved in the acceptance region at 𝛿𝑝/𝑝 = 1.5%. The momentumChapter 5 Results and Discussion

84

resolution was estimated with the HRS Geant4 under the same conditions and was found to
be Δ𝑝/𝑝 = 1.1 × 10−4 . This agrees with the result of Ref. [66].
The momentum resolution under the same conditions as in the present experiment wasestimated with the HRS Geant4. The air and other materials were also taken into account for
estimating multiple scattering eﬀects. The momentum resolution for the scattered electrons
is shown in Fig. 5.6. The blue line shows Δ𝑝/𝑝 = 1.3 × 10−4 with 𝑧 = 0 and without material,
the purple line shows Δ𝑝/𝑝 = 1.8 × 10−4 with 𝑧 = target length and without material, and the
red line shows Δ𝑝/𝑝 = 4.7 × 10−4 with 𝑧 = target length and with material, which is the same
set up as in this experiment.
In actual analyses, a correction function was used to compensate for the energy loss, as
shown in Sec. 4.4.3. The red line shifts due to the energy loss. By correcting the energy
loss eﬀect, not only the momentum shift is corrected, but also the width is expected to
be improved. A Geant4 simulator that implements only the target was constructed, and thediﬀerence in momentum resolution between the cases with and without energy loss correction
was compared to estimate this eﬀect. The result is shown in Fig. 5.7. The blue line is theresult without correction, and the red line is that with the correction. The resolution without
energy loss correction is 150 keV. The angular dependence can be corrected, resulting in an
increase in the resolution to 100 keV.
The momentum resolution can be decomposed into three items: intrinsic resolution of
the spectrometer, energy straggling, and multiple scattering. The intrinsic resolution can becalculated as 1.8 × 10−4 × 2218 MeV = 400 keV. Moreover, the overall resolution is known to be
4.7 × 10−4 × 2218 MeV = 1040 keV; thus, the contribution of multiple scattering is calculated
to be 950 keV. The decomposition of the momentum resolution is shown in Tab. 5.2. The
resolution after correction is Δ𝑝/𝑝 = 4.7 × 10−4 as shown in the black line in the Fig. 5.6, which
has no large diﬀerence due to the large contribution of multiple scattering. The momentum
resolution of incident electrons was Δ𝑝/𝑝 ≈ 1.0 × 10−4 in FWHM.
Table 5.2 Decomposition of momentum resolution.

5.2.3

Δ𝑝 (w/o Ene. Cor.)Δ𝑝 (w/ Ene. Cor.)

Spectrometer

400 keV

400 keV

Straggling

150 keV

100 keV

Multi. Scat.

950 keV

950 keV

All1040 keV

1040 keV

Result85

Counts

Chapter 5 Results and Discussion

Δp/p
Figure 5.6 Momentum resolution estimated by the HRS Geant4. The blue line showswith 𝑧 = 0 and without material. The purple line shows with 𝑧 ∈ [−12.5, 12.5] and withoutmaterial. The red line shows with 𝑧 ∈ [−12.5, 12.5] and with material. The black line shows
after the energy loss correction for the red line.

The mass resolution was estimated using the HRS Geant4 and found to be 𝜎 = 1.4 ±0.1 MeV/c2 for Λ. This agrees with the experimental value of 𝜎 = 1.3 ± 0.1 MeV/c2 . In thesame way, the mass resolution for 𝑛𝑛Λ was found to be 𝜎 = 1.5 ± 0.2 MeV/c2 . The above
systematic error is mainly due to the error in the magnet setting described in Sec. 4.3.2.

5.3

Systematic uncertainties

5.3.1

Cross section

Systematic errors related to eﬃciencies have been discussed in Chap. 3 and Chap. 4. Thesesystematic errors are summarized in Tab. 5.3. Δ𝑁HYP will be discussed in Sec. 5.4.1. Table 5.3
shows common systematic errors for the events, and the total result is shown as a sum ofthese squares. The systematic errors due to ΔΩ𝐾 were calculated separately because they
were used for each event.

5.3.2

Mass accuracy

86Counts

Chapter 5 Results and DiscussionΔp (MeV)Figure 5.7 Momentum distribution before (blue) and after (red) energy correction.

If the number of signals is 𝑁 and the resolution is 𝜎, the mass accuracy from the statistics√
is expressed as 𝜎/ 𝑁. In this analysis, the resolution was estimated with the Gaussianfunction; thus, 𝑁 was also estimated with the Gaussian integral. The data points in Fig. 4.11
were ﬁtted with the Gaussian function. Here, the number of Λ’s in the region represented
by the Gaussian was 830, and the number of Σ0 ’s was 220. The accuracy from the statistics
was calculated to be Δ𝐵Λ = ±0.1 MeV. The accuracy becomes worse by the number of events
missed in the radiative tale. The number of 𝑝(𝑒, 𝑒 0 𝐾)Λ/Σ0 reactions in the whole region was
1899, which means that 45% events are included in the radiative tale. The systematic error of𝐵Λ from transfer matrix tuning was already studied in Ref. [30] and was estimated to be lessthan 0.1 MeV.
The uncertainty of the thickness of the Al cell causes the largest error in the mass measurement. The thickness of hydrogen cell and tritium cell have uncertainties of 25% and 7.6%,
respectively, as described in Sec. 4.4.3. If the thickness is estimated improperly, the energyΛloss cannot be corrected properly. We denote Δ𝑀𝐻
as the mass diﬀerence in Λ resulting

from the uncertainty in the thickness of the hydrogen cell, and Δ𝑀𝑇Λ as the mass diﬀerencein Λ resulting from the uncertainty in the thickness of the tritium cell. If the hydrogen cell
thickness is incorrectly estimated, there will be a shift in the peak mass of Λ in the setting of𝑀calib. . Since matrix tuning uses the mass of Λ, Σ0 as a reference, the momentum matrix is
distorted to reproduce the peak value of Λ, Σ0 . This distortion is reﬂected in the setting ofΛ𝑀phys. ; thus, there will be a deviation of Δ𝑀𝐻
in the tritium cell, even if the thickness of theChapter 5 Results and Discussion

87

Table 5.3 Relative errors for the eﬃciencies and correction factors that contributed to theuncertainty on the cross-section calculation in addition to the acceptance uncertainties.

Item

Relative errorΔ𝜖track

0.2%

Δ𝜖T

0.3%

Δ𝜖He

0.7%

Δ𝜖DAQ

0.1%

Δ𝜖ctime0.5%

Δ𝜖density

1.1%

Δ𝜖vertex

7.8%

Δ𝜖PID

6.9%

Δ𝜖𝜋

1.8%

Δ𝜖multi

1.3%

ΔΓint

8.5%

Δ𝑁𝑒

1.0%

Δ𝑁𝑇

0.9%

Δ𝑁HYP

6.8%

Side band estimation

1.9%

Total

15%

tritium cell is estimated correctly. In contrast, if the thickness of the tritium cell is incorrectly
estimated, an error Δ𝑀𝑇Λ will be caused by the error in the correction function. To prevent thedeterioration of the mass accuracy due to the above eﬀects, calibration was performed using
the hydrogen contamination in the tritium cell. Since the tritium cell contained several percent of hydrogen, the Λ peak could be observed by recalculating the missing mass assuming
the mass of hydrogen. This is shown in Fig. 5.8. Since there is a large amount of backgrounddue to the quasi free Λ coming from the tritium, ﬁtting was performed by representing the
BG as a polynomial and the signal as a Gaussian function. The function is shown by the redline in Fig. 5.8. The position of the Λ peak was −0.3 ± 0.3 MeV. Therefore, this value was usedΛ
to correct for the shift. The error in the accuracy resulting from Δ𝑀𝐻,𝑇
was estimated to be

Δ𝐵Λ = ±0.3 MeV.
Furthermore, a systematic error Δ𝑀 𝑛𝑛Λ exists due to the diﬀerence in kinematics between
Λ and 𝑛𝑛Λ. Since the above correction for the hydrogen in the tritium cell is based on theassumption of a hydrogen target, it is necessary to estimate how much the peak position can beshifted in the case of a tritium target. Figure 5.9 shows the simulation results for the case of atritium cell with a thickness of 428 𝜇m. In the ﬁgure, the peak position of Λ is 425±11 keV andChapter 5 Results and Discussion

88

Counts /(0.8 MeV/c2)

𝜒 2 = 60.4/57

𝑝 𝑒, 𝑒 ′ 𝐾 + Λ

Q.F.Λ
MX – mΛ (MeV/c2)
Figure 5.8 Missing mass distribution of 3 H(𝑒, 𝑒 0 𝐾 + )𝑋 reaction assuming hydrogen target.

that of 𝑛𝑛Λ is 513 ± 25 keV, indicating a signiﬁcant diﬀerence. The diﬀerence was estimatedto be Δ𝐵Λ = ±0.1 MeV by changing the cell thickness in the range of uncertainties. Thisis thought to be because the contribution from the momentum correction is stronger whenthe mass number is larger. Summarizing all the above uncertainties, the error of the massaccuracy was estimated to be 0.4 MeV.

5.4

Upper limit

5.4.1

Fitting function

Three response functions for accidental background, quasi-free Λ (QF) production, and𝑛𝑛Λ signal are necessary for the ﬁtting.
The shape of the accidental background was obtained from the mixed event analysis as
described in Sec. 5.1. The function is ﬂat around the threshold and does not make sharp
peaks as shown by the black points in Fig. 5.1. This was ﬁtted with a quadratic function in theﬁtting region. The scale was ﬁxed from the number of events; thus, this function including
the scale was ﬁxed when ﬁtting.
In general, QF is considered to rise at −𝐵Λ = 0 MeV and to increase monotonically. The line

Counts /(0.08 MeV/c2)

Chapter 5 Results and Discussion

89

sim: 𝑝 𝑒, 𝑒 ′ 𝐾 Λ
425 ± 11 keVCounts /(0.3 MeV/c2)

MX - MΛ (MeV/c2)

sim: 3H 𝑒, 𝑒 ′ 𝐾 𝑛𝑛Λ
513 ± 25 keV-BΛ (MeV)
Figure 5.9 Missing mass distributions for 𝑝(𝑒, 𝑒 0 𝐾 + )Λ and 3 H(𝑒, 𝑒 0+ )𝑛𝑛Λ reactions estimated with the HRS Geant4.

shape near the binding threshold may be aﬀected by the Λ𝑛 ﬁnal state interaction (FSI); however, this has not been conﬁrmed either in theory or through experimental results. Therefore,a linear function, which is the simplest assumption, was adopted in the present analysis. The
above linear function was convoluted with a Gaussian function assuming the experimental
resolution to consider the leakage of QF events below −𝐵Λ < 0 MeV.
For the 𝑛𝑛Λ signal, a template function was created. As can be seen from Fig. 3.6 of the𝑝(𝑒, 𝑒 0 𝐾 + )Λ/Σ0 reactions, the contribution of the radiative tale is signiﬁcant. Similarly, the𝑛𝑛Λ signal is also expected to have a large radiative tale. To obtain a shape that incorporates
the radiative tale and the experimental resolution, the HRS Geant4 simulation was used toobtain the spectrum of the 𝑛𝑛Λ response. The template function was obtained by ﬁtting tothis spectrum. This is shown in Fig. 5.10. In addition, to incorporate the natural width of𝑛𝑛Λ, the function was convoluted with a Breit-Wigner function of the natural width Γ. Thisis shown in Fig. 5.11. This function was used as a template function.
However, the function estimated by the HRS Geant4 has a slightly smaller number ofradiative tails than the experimental data. Since this diﬀerence may also exist in 𝑛𝑛Λ, thecontribution to the 𝑛𝑛Λ production cross section was estimated through a simple test. In thistest, the strength of the tale in the radiative region was adjusted to reproduce the experimental
spectrum of 𝑝(𝑒 , 𝑒 0 𝐾 + )Λ as shown in Fig. 5.12. The black line is the function originally obtainedby the HRS Geant4. The red line shows the result of adjusting the radiative region, where

90

counts (/0.3 MeV)

Chapter 5 Results and Discussion

𝜒 2 = 295.4/334−𝐵Λ (MeV)
Figure 5.10 𝑛𝑛Λ signal estimated with the HRS Geant4.

the number of events in the radiative tails is equal for the experimental data and simulation.
This change was also assumed to exist in the 𝑛𝑛Λ signal. The cross-sections were estimatedunder this condition, and the diﬀerence from the case of the original function was taken as
the systematic error. The eﬀect to the cross section was Δ𝑀HYP = ±6.8%. The diﬀerences in
the left side of the 𝑝(𝑒, 𝑒 0 𝐾 + )Λ peak between the HRS Geant4 and the experimental data werealso examined in the same test and found to be negligibly small.

5.4.2

Spectral ﬁt

A systematic error due to the binning eﬀect was large in the threshold region where thenumber of events was limited. The missing mass distribution for the 3 H(𝑒 , 𝑒 0 𝐾 + )X reaction
when the binning was changed are shown in Fig. 5.13. The upper left ﬁgure is the same as
Fig. 5.1, with binning at 2 MeV. The upper right ﬁgure shows the result when the splitting
is shifted by 1 MeV while maintaining the binning width. The lower left ﬁgure and thelower right ﬁgure show the results for the binning width of 4 MeV and 6 MeV, respectively.
Bin-dependent ﬁtting results in large systematic errors, as can be seen from these ﬁgures.
Therefore, ﬁtting was performed using the unbinned-maximum-likelihood method. The

Chapter 5 Results and Discussion91

Γ = 0 MeVΓ = 1 MeVΓ = 2 MeVΓ = 5 MeVΓ = 10 MeV−𝐵Λ (MeV)
Figure 5.11 Template functions for the 𝑛𝑛Λ signal. The functions with several widths ofBreit-Wigner distributions are shown.

Likelihood was maximized with the parameter 𝜃:𝐿(𝒙; 𝜽) =𝑁∏

𝑓 (𝑥 𝑖 , 𝜽),

(5.9)

𝑖=1

where 𝒙 = 𝑥1 , 𝑥 2 , ..., 𝑥 𝑁 denotes the cross-section data for each event. 𝑓 (𝑥 𝑖 , 𝜽) is the probability
density function (PDF). In this ﬁtting, the sum of QF distribution, accidental background,and 𝑛𝑛Λ signal were normalized. The 𝑠𝜎 statistical error is deﬁned as the point where thelikelihood diﬀerence from the minimum (Δ𝐿) is 𝑠 2 /2. However, since the vertical axis is notin a count but a cross section, correction is necessary. This correction was incorporated as aform of correction to the Hessean matrix that represents the error [68, 69].

5.4.3

Discussion

The upper panel of Fig. 5.14 shows the ﬁtting results assuming that the 𝑛𝑛Λ signal corresponds to the theoretical prediction (−𝐵Λ , Γ) = (0.25, 0.8) [19] and (0.55, 4.7) MeV [23],respectively. The red line is the assumed QF distribution, the black dotted line is the accidental background, and the blue line is the 𝑛𝑛Λ signal. The diﬀerential cross sections for the 𝑛𝑛Λsignal were obtained to be 11.2 ± 4.8(stat.)+4.1(sys.) nb/sr and 18.1 ± 6.8(stat.)+4.2(sys.) nb/sr,
−2.1
−2.9
respectively. In addition, the bottom ﬁgure in Fig. 5.14 shows the ﬁtting results when theCounts /(1 MeV/c2)

Chapter 5 Results and Discussion

92

data
𝑝 𝑒, 𝑒 ′ 𝐾 + Λ (sim)Λ

𝑝 𝑒, 𝑒 ′ 𝐾 + Σ0 (sim)Σ0

MX – mΛ (MeV/c2)
Figure 5.12 The correction for the radiative tale. Black line shows the ﬁtting result without
the modiﬁcation. Red line shows the ﬁtting result with the modiﬁcation.

binding energy is varied while keeping the width of this theoretical line shape. The horizontal
axis shows the peak position of the 𝑛𝑛Λ signal. The black line shows the 1𝜎 statistical error,
and the systematic error is indicated by the selection symbol. Although an excess event canbe found around −5 < −𝐵Λ < 5 MeV, the statistical signiﬁcance was not greater than 3𝜎. Here,a sharp drop in cross section was observed in the −𝐵Λ > 10 MeV region. This is considered
because the slope of the QF distribution becomes steeper due to the presence of an excess
near −𝐵Λ ∼ 0 MeV, and the cross section of the 𝑛𝑛Λ signal was reduced in −𝐵 > 10 MeV tocompensate for this. In addition, the ﬁt range is limited to −𝐵Λ < 20 MeV because the QF
distribution cannot be reproduced by the linear function above −𝐵Λ > 20 MeV. Moreover,systematic errors that are not considered in the present analysis are expected in the region
where −𝐵Λ is large because the 𝑛𝑛Λ signal has a large tail on the high mass side.
Since no signiﬁcant peak were found, the 90%-conﬁdence level (C.L.) of the upper limit ofthe diﬀerential cross section was obtained. The statistical error of 90% C.L. was calculated asfollows∫𝑥∫ 0∞
0

𝑔(𝑥)𝑑𝑥𝑔(𝑥)𝑑𝑥= 0.90,(5.10)

where 𝑔(𝑥) denotes the Gaussian function. The ﬁnal upper limit is the sum of the statisticalerror and the systematic error. The upper limit assuming (−𝐵Λ , Γ) = (0.25, 0.8), (0.55, 4.7) MeV
is 21 nb/sr and 31 nb/sr, respectively. In addition, to obtain results independent of theChapter 5 Results and Discussion

9330

dσ/dΩ (nb/sr /2 MeV)dσ/dΩ (nb/sr /2 MeV)

h_CS

25
20
15
10
5
0
-100

-50

0

50

100

150

−𝐵Λ (MeV)dσ/dΩ (nb/sr /6 MeV)dσ/dΩ (nb/sr /4 MeV)

−𝐵Λ (MeV)−𝐵Λ (MeV)−𝐵Λ (MeV)

Figure 5.13 The change in histograms for 3 H(𝑒, 𝑒 0 𝐾 + )𝑛𝑛Λ reaction due to binning. The
upper left ﬁgure is the same as the one shown in Fig. 5.1. The upper right ﬁgure shows thesame binning width but with the binning internally shifted by 1 MeV. The lower left ﬁgureand the lower right ﬁgure show the histogram when the bin width is doubled and tripled,
respectively.

theoretical predictions, two-dimensional scans of −𝐵Λ and Γ were performed. The results areshown in Fig. 5.15. The horizontal axis shows the peak position of the 𝑛𝑛Λ signal and thevertical axis shows the natural width.
In the current results, a signiﬁcant peak exceeding 3𝜎 was not found. The reason that thepeak structure could not be observed may be due to the small cross section or the broad
width of the resonance. In contrast, there are some events around 𝑛𝑛Λ threshold over the
background but we cannot draw deﬁnitive conclusions due to the low statistics. It is desiredthat further experiments would conﬁrm this structure. Schäfer [24, 25] suggests the existenceof the resonance is highly depend on the model of Λ𝑛 interactions. It is calculated that thephysical resonance appears with in the NSC97f and Chiral EFT (NLO) but not in the Alexander
B and Chiral EFT (LO). In contrast, Afnan’s calculations show that there are no resonance inΓ = 0.8 MeV−𝐵Λ (MeV)dσ/dΩ (nb/sr/2 MeV)(–BΛ, Γ) = (0.25, 0.8) MeV

94

dσ/dΩ (nb/sr)

dσ/dΩ (nb/sr)

dσ/dΩ (nb/sr/2 MeV)

Chapter 5 Results and Discussion(–BΛ, Γ) = (0.55, 4.7) MeV

Γ = 4.7 MeV−𝐵Λ (MeV)

Figure 5.14 The results of the spectral ﬁt assuming the 𝑛𝑛Λ signal by theoretical calculations. The top ﬁgures show the ﬁtting results assuming (−𝐵Λ , Γ) = (0.25, 0.8) MeV [19]and (0.5, 4.7) MeV [23], respectively. The blue, red and black dotted lines show the 𝑛𝑛Λsignal, QF and accidental background, respectively. The black solid line shows the sum of
the functions. The bottom ﬁgures show the results of the cross section as a function of −𝐵Λwhen the natural widths are assumed. The 90% C.L. upper limits are shown in blue solidline.

any interaction. No Λ𝑛 scattering data exist so far, and the Λ𝑛 interaction is highly uncertain.
If the results of the cross section calculations exist, we can expect to impose limitations onthese interactions and calculation methods. However, there are no theoretical work on the
production cross section of the 𝑛𝑛Λ signal in this reaction. It is strongly interesting to comparewith the calculation.

95

Belyaevdσ/dΩ (nb/sr)

Γ (MeV)

Chapter 5 Results and Discussion

Schäfer
Kamada−𝐵Λ (MeV)
Figure 5.15 The 90% C.L. of the upper limit of the 𝑛𝑛Λ cross section when the spectral ﬁt
was performed by changing 𝐵Λ and natural width.

96

Chapter 6

Summary
The existence of the 𝑛𝑛Λ state was claimed by the HypHI collaboration at GSI. However,considering that 3Λ H with 𝐼 = 0 is a weakly bound state, it is diﬃcult for the 𝑛𝑛Λ state with
𝐼 = 1 to be bound. For example, the existence of a bound state (−𝐵Λ < 0) is considered diﬃcult
in theoretical calculations using the models that explains the currently observed few-body
hypernuclei. In contrast, some papers suggest the existence of a resonant state (−𝐵Λ > 0), but
the results strongly depend on the Λ𝑛 interaction, which is currently uncertain.
We performed a search experiment for the 𝑛𝑛Λ state using the 3 H(𝑒, 𝑒 0 𝐾 + )𝑋 reaction at JLabin 2018. We used a 4.32-GeV/c electron beam at CEBAF in JLab. In the electro-productionof hypernuclei, high mass accuracy can be achieved by using 𝑝(𝑒 , 𝑒 0 𝐾 + )Λ/Σ0 reactions as
reference. It is also possible to explore a wide range of binding energy from bound toresonant state. Scattered electrons and 𝐾 + were measured with two spectrometers, HRS-L
and HRS-R. The central momenta of the electron and 𝐾 + were 2.22 GeV/c and 1.82 GeV/c,
respectively, and the central angles of HRSs were set at 13.2◦ with respect to the electronbeam.
To estimate some physical quantities, such as acceptance and resolution, we developed aMonte Carlo simulator (HRS Geant4) for this experiment. The HRS Geant4 was successful toreproduce the experimental data. The estimated mass resolution for Λ is 1.4 ± 0.1 MeV/c2 ,which agrees with the experimental value of 1.3±0.1 MeV/c2 . The mass resolution for the 𝑛𝑛Λstate was also estimated to be 1.5 ± 0.2 MeV/c2 by using the simulation. The diﬀerential crosssection of the 3 H(𝛾∗ , 𝐾 + )X was calculated by estimating eﬃciencies and correction factors
with the experimental data and the HRS Geant4.
The obtained missing-mass spectrum for the 3 H(𝑒 , 𝑒 0 𝐾 + )𝑛𝑛Λ reaction shows some excess
around the 𝑛𝑛Λ threshold. We performed a spectral ﬁt to this distribution to evaluate theexcess above expected background events quantitatively. The Breit-Wigner (BW) function forthe decay width was convoluted with a response function estimated by the HRS Geant4 to
obtain the functional form of the 𝑛𝑛Λ signal. Typical theoretical predictions that show theChapter 6 Summary

97

small and large decay widths were used for the BW function, which are (−𝐵Λ , Γ) = (0.25, 0, 8)
and (0.55, 4.7) MeV. The unbinned maximum likelihood was used for the spectral ﬁt since thestatistics near the threshold was limited. As a result, no peak that was statistically signiﬁcantabove 3𝜎 was observed. The cross-section upper limits at the 90%-conﬁdence level for the
3 H(𝛾 ∗ , 𝐾 + )𝑛𝑛Λ reaction were obtained to be 21 nb/sr and 31 nb/sr, when (−𝐵Λ , Γ) = (0.25, 0.8)

and (0.55, 4.7) MeV were assumed. In addition, the cross-section upper limit with various𝐵Λ and Γ other than the above ﬁxed assumptions were obtained to provide data that are
independent on the theoretical predictions.
Currently, no other experimental or theoretical information is available for the 𝑛𝑛Λ production cross section. The present data is the ﬁrst result for the upper limit of the crosssection. This has a signiﬁcant impact on the existence of 𝑛𝑛Λ. In the theoretical calculations,the existence of 𝑛𝑛Λ resonant states strongly depends on the Λ𝑛 interactions and calculation
methods. Schäfer [24, 25] suggests the resonance peak exists in the NSC97f and Chiral EFT(NLO) but not in the Alexander B and Chiral EFT (LO). The present data will constrain theΛ𝑛 interaction models by comparing with theoretical calculations. Therefore, theoretical
calculations on the cross section are strongly desirable.
In this experiment, we did not observe clear signals of the 𝑛𝑛Λ state reported by the HypHI
collaboration. The next 𝑛𝑛Λ search experiment is currently being prepared and will be carriedout in the spring of 2022 by the HypHI collaboration. Furthermore, a new experiment at JLabcan also be designed based on the present result. Experimental attempts such as the above
new experiments are indispensable to solve the existence puzzle of the 𝑛𝑛Λ state.

98

Acknowledgment
I would like to express my sincere gratitude to my advisor, Prof. Gogami, for his exceptional
cooperation. He has always given me a lot of time for my research and has accompanied me
for many discussions on minor to major issues. Thanks to his generous spirit, I was able toconsult him easily and proceed with my research. It would have been diﬃcult for me to write
this paper without his help. In particular, it would have been impossible to publish the paperin three years. Meeting him in graduate school and having the opportunity to do researchwith him has been a great treasure in my career.
I would also like to thank Prof. Nakamura of Tohoku University for his great support.
During our daily meetings, I received a lot of ideas and made considerable progress in myresearch. Due to the nature of our research, most of our meetings were held in the evening,
but he spent a lot of time on the research of JLab, including me. Because of the atmospherein the group created by Prof. Nakamura, I was able to enjoy my research activities.
I am indebted to Prof. Nagae since I entered the graduate school. He was always concerned
about the progress of my research and gave me advice on a wide range of topics, including
my research and my career path. The comments he gave me at various times helped me a lot
when I was stuck in my research.
I thank all the spokespersons for this experiment for their exceptional help. Prof. Tang,
Prof. Reinhold, Prof. Markowitz, Prof. Urciuoli, and Prof. Garibaldi provided me with a great
number of discussions that enabled me to complete my research despite the time diﬀerence.
I would also like to thank them for their assistance during my visit to JLab in the U.S.
I also want to thank Mr. Itabashi, Dr. Bishun, Mr. Okuyama, and Prof. Nagao for conducting
the research analysis together. I believe that we were able to create a better paper by conducting
numerous analyses and cross-checking together. I have been able to grow as a scientist by
working with such talented people.
I wish to thank Mr. Toyama, Mr. Akiyama, Mr. Kaneta, and all the other members of ELS
at Tohoku University. Listening to your presentations every month has motivated me to do
my own research. In addition, thank you very much for accepting me when I visited Tohoku
University.

Acknowledgment

99

I would like to express my appreciation to Prof. Miyagawa for his cooperation in thetheoretical aspect of my research. He took time out of his busy schedule for me and patiently
guided me because I was not familiar with the theory. I also received theoretical guidance inthe elementary process from Schäfer and Bydẑovský in a conference in Czech Republic.
I also received a lot of support from the staﬀ of the Nuclear hadron physics laboratory at
Kyoto University. Prof. Naruki was especially helpful for my master’s thesis. What I learned
during this time will be useful in my future career. Prof. Murakami, Prof. Zenihiro, andProf. Dozono also gave me advice which I was able to utilize in my research. I also enjoyedtalking with them about miscellaneous things other than research, and I was able to enjoy
my research activities. The "laissez-faire" attitude of the laboratory suited me very well. I
was free to do the research I wanted to do, and I could ask questions and discuss with theprofessors if I got stuck in my research. This atmosphere helped me to acquire the ability totake action.
I would also like to thank everyone who helped us complete our experiments at JLab.
Conducting experiments with tritium was very challenging, but we were able to complete
our research thanks to their help.
Finally, I would like to thank my family for raising me so far.

100

Appendix A

Summary of theoretical study

A.1

Bound state

B. W. Downs and R. H. Dalitz (1959) [12]The ﬁrst theoretical calculation for a three-body system including hyperons was reported by
Downs and Dalitz in 1958, only six years after the discovery of the hyperon. The calculations
were carried out by the variational method. As a trial function, the following exponentialfunction with 6 parameters was used.
𝜓 = (𝑒 −𝑎𝑟1 + 𝑥𝑒 −𝑏𝑟1 )(𝑒 −𝑎𝑟2 + 𝑥𝑒 −𝑏𝑟2 )(𝑒 −𝑎3 𝑟3 + 𝑦𝑒 −𝑏3 𝑟3 ),(A.1)

where 𝑟1 and 𝑟2 are the distances between individual nucleons and the Λ particle, and 𝑟3 isthe distance between two nucleons. They concluded that it is diﬃcult to bound the 𝑁 𝑁Λsystem with I = 1.

K. Miyagawa et al. (1995) [13]Miyagawa et al. used the Faddeev method to calculate the 𝑁 𝑁Λ three-body system. The
Nĳmegen89 potential was used as the YN interaction. The main feature of this calculationis that it incorporates the Λ-Σ conversion. The binding energy of the Hypertriton for (𝐼, 𝐽) =(0, 1/2) was successfully explained without heavily relying on 𝑁 𝑁 interactions (Nĳmegen93,
Paris, Bonn B). Furthermore, the interaction due to Λ-Σ conversion was found to be attractive
and played an important role in the binding of hypertriton.
In contrast, the same method was applied to the 𝐼 = 1 state. The interactions used were
the Nĳmegen89 YN interaction and the Bonn B NN interaction, which successfully explained
the hypertriton. As a result of this calculation, there was no bound state for both 𝐽 = 1/2 and𝐽 = 3/2. Therefore, it was examined by scaling the entire YN interaction to see whether abound state would appear. For (𝐼, 𝐽) = (1, 1/2), the bound state appeared at 1.2 times, and forAppendix A Summary of theoretical study

101(𝐼, 𝐽) = (1, 3/2), the bound state did not appear even at 1.32 times. Furthermore, the eﬀect ofthe Λ-Σ conversion was suggested to be an attractive force for (𝐼, 𝐽) = (1, 1/2) and a repulsive
force for (𝐼, 𝐽) = (1, 3/2).

E. Hiyama et al. (2014) [14]Hiyama et al. performed variational calculations following the report on the observationof the 𝑛𝑛Λ bound state at GSI in 2013. The NN interaction is AV8 potential, and the YN
interaction is NSC97f, which incorporates Λ𝑁-Σ𝑁 coupling. The advantage of using theabove interactions is that the binding energies of 3Λ H, 4Λ H, 4Λ He are well reproduced at
0.19, 2.33, 2.28 MeV. Even with these interactions, the 𝑛𝑛Λ bound state does not exist.
Therefore, by modifying some interactions, a way for the 𝑛𝑛Λ state to be bound while
maintaining the above binding energy was explored. First, the tensor components of the
Λ𝑁-Σ𝑁 coupling were scaled. Then, the bound state appeared after scaling by 1.2. However,the binding energy of the 3Λ H was found to be over bound at −𝐵Λ = 0.72 MeV, and the excitedstate of 𝐽 = 3/2, which was not conﬁrmed in the experiment, was also bound. When T=1,
the 1 𝑆0 component of 𝑛𝑛 was scaled, and it was found that 𝑛𝑛Λ was bound by a factor of
1.35. However, since di-neutron (𝑛𝑛) is bound by a factor of 1.13, the 𝑛𝑛Λ state could not beinterpreted by the present nuclear physics.

A. Gal and H. Garcilazo (2014)
Gal and Garcilazo discussed the 𝑛𝑛Λ bound state, and compared the experimental resultson the current few-body systems (Λ𝑝 scattering, 3Λ H, 4Λ H). As a result, they found that the 𝑛𝑛Λbound state cannot be explained by the current understanding of hypernuclei. In particular,
Faddeev calculations that incorporate Λ𝑁-Σ𝑁 coupling were performed for the comparison
with 4Λ H. For the YN interaction, the G-matrix 0𝑠 𝑁 0𝑠𝑌 eﬀective interaction developed by
Akaishi [70] et al. based on the NSC97 was used. Even under this condition, 𝑛𝑛Λ was
unbound, and even when the interaction was adjusted to be bound, dineutron (nn) wasbound ﬁrst. This calculation does not incorporate the eﬀect of the 𝑁 𝑁Λ three-body force
which is not directly related to the Λ𝑁-Σ𝑁 conversion. However, even if other three-bodyforces are considered, such as those involving Σ(1385), the eﬀect of the attractive force on the𝑛𝑛Λ state is still considered to be small compared to that on the 3Λ H, and its existence would
be still diﬃcult.

H. Garcilazo and A. Valcarce (2007, 2014) [16, 17]
Garcilazo et al. used the chiral constituent quark model to derive interactions. In thismodel, a baryon is represented as a three-body cluster of constituent quarks, and its massis generated by spontaneous breaking of chiral symmetry. Here, Λ𝑁-Σ𝑁 conversion isAppendix A Summary of theoretical study

102

incorporated, which includes not only the S-wave but also the D-wave tensor component.
The Faddeev equation was solved to calculate the 𝑛𝑛Λ bound state using this interaction. As
a result of the calculation, it was found that the bound state of 𝑛𝑛Λ is diﬃcult. In contrast, forsimplicity, Faddeev calculations were also performed for separate potentials containing only
S waves, but the results did not signiﬁcantly change.

Shung-Ichi Ando et al. (2015) [18]Ando et al. calculated the 𝑛𝑛Λ state by solving the coupled integral equation using pionlesseﬀective ﬁeld theory (Z𝜋 EFT) of the leading order. The Lagrangian of this model incorporates
three body forces as the contact terms. As a result of this calculation, the possibility of theexistence of Eﬃmov states was suggested. However, since the experimental data to determine
the parameters of this three-body force are very scarce, no quantitative discussion has beenpresented.

A.2

Resonant state

V. B. Belyaev et al. (2008) [19]Belyaev et al. were the ﬁrst to discuss resonant states in 𝑛𝑛Λ three-body systems. The
resonant state was searched for by calculating the zeros of the three-body Jost function. The
model proposed by Nemura et al. [20] was used as the interaction. This model is based on the
Minnesota potential used for NN potential, and its parameters were determined to reproduce
the 𝐴 = 3, 4 Λ-hypernuclei. There are three Λ𝑛 potentials which were proposed by Nemura
et al., and they are denoted A, B, and C. Figure 1.4 shows the potential of the S-state under
the assumption of these potentials. In all of them, there is a pocket of weak attraction. The
results are shown in Tab. 1.1. It can be seen that there is a broad resonance in A and B, while
there is no resonance in C. The results highly depend on how the Λ𝑁 potential. These do
not incorporate the Λ𝑁-Σ𝑁 channel, which may result in a narrower width and closer to thethreshold. When this potential was scaled, a bound state appeared at 1.5 times the potential.

I. R. Afnan and B. F. Gibson (2015) [21]Afnan et al. performed a Faddeev calculation using the separate potential. The separable
potential of a two-body S-wave is described as follows
𝑉(𝑘, 𝑘 0) =1
𝑘 2 + 𝛽2

𝐶1
𝑘 02 + 𝛽 2

,

(A.2)

where 𝐶 and 𝛽 are the parameters determined by the scattering length and eﬀective rangeof the Nĳmegen model D potential, respectively. The calculations were performed with

Appendix A Summary of theoretical study

and without the tensor force, respectively. These models do not incorporate the Λ𝑁-Σ𝑁
conversion and the three-body force. When the calculations were performed without tensor
force, the result was 𝐸 = −0.154 − 𝑖0.753 MeV and no resonant state existed. However, bymultiplying the strength of 1 𝑆0 , 3 𝑆1 components by 1.07, a resonant state appeared, andby multiplying it by 1.35, a bound state appeared. A similar calculation was performed
with various Λ𝑛 interactions including tensor forces, and no signiﬁcant diﬀerences wereobserved. Figure A.2 shows the results of the calculations with Nĳmegen model D, Chiral
EFT, Nĳmegen NSC97f, and Julich. The calculation result was −0.107 − 𝑖0.622 MeV, and noresonant state existed. However, by multiplying 1 𝑆0 , 3 𝑆1 by 1.05, a resonant state appeared,

AMIN F. GIBSON

Re[E]

103

and by multiplying by 1.25, a bound state
appeared.
PHYSICAL REVIEW C 92, 054608 (2015)

Im[E]

0.0

1S , 3S
0
1

0.0

Mod D
Chiral
NSC97fJulich04

3D AG
1

0.1

0.0
Re[E]

0.1

0.2

FIG. 2. (Color online) Trajectory of the resonance pole as one
sonance pole as one varies thestrength
FigureA.1 The pole trajectory of 𝑛𝑛Λ state taken from Refs. [21]. The trajectories show
varies the strength of the n interaction. In this case we use thecase we have used the N N and N
the dependenceof theinteraction.
same
nn Λ𝑛
potential
given in Eq. (14) for all four curves. The n
I of Ref. [11] with a tensor interaction
potentials correspond to Yamaguchi fits as given in Table II with Mod
D for Nijmegen model D, Chiral for chiral ( = 600), NSC97f for
Nijmegen NSC97f, and Jülich04 for the Jülich one boson exchange
V, the singlet n scattering
length et al.
potential.
(In[22]
scaling the n potential we have insured that no twoI. Filikhin(2016)and the effective range becomes
body bound state is formed. For example, for Mod D the scaling of
was used as the Λ𝑁 potential and MT-I-III semi-realistic NN potential was used
loser to the values NSC97f
coming fromthe potential by s = 1.35 moved the two-body pole closest to the realen in Table II. For
thethresholdas the 𝑁 𝑁 potential.
As akresultof thiscalculation,rather wide range of resonant statesaxis from= −0.315
i fm tok = −0.164 ia fm.)o s = 1.350, the singlet scattering
inand
the2.919
𝐸 = fm,0.2 − 𝑖1 MeV range was found. In addition, the following were perturbatively
ould be −4.922 fm
by p scattering data
unless thereas three-body forces.
incorporated
depending on the scattering lengths and effective ranges of the
metry breaking. Clearly ours is notn potential. In Fig. 2 we plot four different
trajectories for thee actual answer will depend on a
(A.3)
𝑉3𝑏 𝑓 (𝜌) = −𝛿 exp (−𝛼𝜌2 )𝑆(𝑠Λ , 𝑠 𝑁 𝑁 )resonance pole. In each case we use the same nn potential given
n and nn interactions.
in Eq. (14). For
the n interaction we use the 1 S0 and 3 S1
ertainty in our results,
we𝜌consid2 = 𝑥 2 + 𝑦 2 and 𝑥, 𝑦 are mass scaled
whereis the hyper-radius:𝜌
Jacobi coordinates. In(i.e., no tensor force) that fit the effective range and scattering
the NN and N interactions ineither thechiralmodel orexchange
addition,) is afrom
parameterthatdepends
onmesonthe spin,and model.
the following two cases
Λ , 𝑠 𝑁 𝑁length11] with a tensor force
in the𝑆(𝑠NThe
Yamaguchi
fits
to
these
potentials
are
given
in
Table
II
sonance pole for s = 1.0 lies atand are: Mod D for a fit to Nijmegen model D [5], Chiral for a
MeV with an eigenvalue
fit to chiral ( = 600) [8], and NSC97f for a fit to Nijmegen
NSC97f [6], and Jülich04 for the Jülich one boson exchangei.
(19)
potential [7]. We first observe that all the curves are similar
ength scale s of 5% will produce
in that they trace the same shape as one varies the strength of
0.455 i MeV, while an increase in
the n interaction. However, the starting points for the fource a threshold bound state in the
curves corresponding to s = 1.0 are all different, suggesting
lot the trajectory of the resonancethat a pole at a given energy will require a different scaling.

Appendix A Summary of theoretical study

104

were attempted.
𝑆(1/2, 1)=1
𝑆(1/2, 0)𝑆(1/2, 1)= 1/3
𝑆(1/2, 0)

(A.4)
(A.5)

However, it was not possible to create a 𝑛𝑛Λ bound state while maintaining the bound energyof 3Λ H.

H. Kamada et al. (2016) [23]YN Nĳmegen potential (NĳmYN89) was used as the YN interaction and NN Nĳmegenpotential (Nĳm93) as the NN interaction. Λ𝑁-Σ𝑁 coupling was also incorporated. The
Faddeev equation was calculated by using these interactions. Partial waves were incorporated
up to 𝐽𝑚𝑎𝑥 = 4 or 5. Thus, the resonant states of 𝐸 = 0.25 − 𝑖0.40 MeV were found. Moreover,when the YN interaction was changed to NSC97f, the result of 𝐸 = 0.60 ± 0.05 − 𝑖(0.25 ±0.05) MeV was obtained. Furthermore, the YN potential of NSC97f was scaled, and thebound state appeared by multiplying it by 1.2.

M. Schäfer et al. (2020,2021) [24, 25]
The pionless eﬀective ﬁeld theory (Z𝜋EFT) of the leading order was used as an interaction.
The s-wave contact interaction was included as the two-body and three-body force. These
parameters were determined to ﬁt the experimental data and theoretical calculations for the
A=3, 4 system. For the calculation of bound states, the three-body Schrödinger equation
was solved using the stochastic variational method (SVM). In addition, two independent
methods—the inverse analytic continuation in the coupling constant method (IACCC) andthe complex scaling method (CSM)—were used to enable the calculation of the resonant
states. Thus, no 𝑛𝑛Λ bound state was found, but there existed a resonant state depending on
the interaction. The interaction models NSC97f and Chiral EFT (NLO) showed the existenceof resonant states, while Alexander B and Chiral EFT (LO) showed no resonant states. The
pole position of the resonant state diﬀers depending on the model, but it exists in the energy
range of 𝐸 ≤ 0.3 and the width range of 1.16 ≤ Γ ≤ 2.00 MeV.

105

Appendix B

Determination of the magnetic settingof the HRS Geant4
As described in Sec. 4.3.2, the quadrupole magnetic ﬁeld setting of the HRS Geant4 wasadjusted to satisfy the following,
𝐷𝐿 : 𝑄1𝐿 : 𝑄2𝐿 : 𝑄3𝐿 = 𝐷𝑅 : 𝑄1𝑅 : 𝑄2𝑅 : 𝑄3𝑅 .

(B.1)

Q2 and Q3 are identical quadrupole magnets, and the ratio of their magnetic ﬁelds is ﬁxed.
Therefore, the optimal magnetic ﬁeld setting was searched by varying the magnetic ﬁeld of Q1
and Q2. Comparisons were made between experimental data and simulations, and residuals
were obtained. As examples, the results of the search for the momentum distribution and𝑦 0 are shown in Fig. B.1 and B.2. In particular, Fig. B.2 shows that the distributions diﬀer
signiﬁcantly depending on the magnetic ﬁeld of Q1 and Q2. In Fig. B.1, the distributions also
show a slight change. The red line is the region that we adopted as the systematic error inthis study, as mentioned in the text. The distributions used for the search are the momentum
distributions and the angular distributions at the VDC (𝑥 0 , 𝑦 0), and the angular distributions
at the target (𝑥 0𝑡 𝑔 , 𝑦𝑡0 𝑔 ). The residuals were taken for all of these, and the results are shownin Fig. 4.10. A comparison of each distribution in the best setting is shown in Fig. B.3. We
found that the momentum distributions agree well when the momentum cut (|𝛿𝑝/𝑝| < 4%)
was applied. The momentum acceptance and resolution for varying magnetic ﬁeld of Q1 andQ2 are also shown in Fig. B.4 and B.5, respectively.

Q2

Figure B.1 The momentum acceptance of HRS-L and HRS-R. The left ﬁgure shows experimental data. The right ﬁgures show thesimulation result with changing the magnetic ﬁeld of Q1 and Q2.

Q1

Data

Sim.

Appendix B Determination of the magnetic setting of the HRS Geant4
106

Q2

Figure B.2 The 𝑦 0 distribution at VDC. Purple dots show experimental results. Colored histograms show the simulation results.
Green shows the events of 𝑝(𝑒, 𝑒 0 𝐾 + )Λ reaction and blue shows the events of quasi-free Λ.

Q1Appendix B Determination of the magnetic setting of the HRS Geant4
107

x’

x’

y’

y’

x’tg

x’tg

y’tg

y’tg

Figure B.3 The comparison between experimental result and simulation. Purple dots show experimental results. Coloredhistograms show the simulation results. Green shows the events of 𝑝(𝑒 , 𝑒 0 𝐾 + )Λ reaction and blue shows the events of quasi-free Λ.

pe’

HRS-R (K+)

pe’

HRS-L (e’)

Appendix B Determination of the magnetic setting of the HRS Geant4
108

109

Q1

Figure B.4 The simulation results of momentum acceptance.

Q2

Appendix B Determination of the magnetic setting of the HRS Geant4

Appendix B Determination of the magnetic setting of the HRS Geant4

110
Δp/p

Q1

Q2
Figure B.5 The simulation results of momentum resolution. The axis shows the ratio withrespect to the most consistent magnetic ﬁeld setting. See the text about the red line.

111

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