Master’s Thesis

Feasibility study for measurement ofbeta-decay rates of 𝚲 hypernuclei

Kento Kamada

Department of Physics
Graduate School of Science
Tohoku University
2021

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ii

Abstract
It has been discussed that the baryon properties can be modified in nuclear medium.
Experimentally, only the EMC effect indicates the baryon modification in nuclear matter,but its mechanism has not been clearly explained. We consider that the baryon
modification can be clarified by measuring the beta decay of Λ hypernucleus. We expectthat a Λ is swelled since u and d quark wavefunctions are affected by the meson field ina nucleus. If it is true, when Λ’s beta decay takes place, the spatial overlap of the initialstate s quark and the final state u quark wavefunctions may be reduced comparing withthat in the free space. This change of the baryon structure can be detected by measuring
the beta decay rate of a Λ in a nucleus. According to a calculation by Quark Meson
Coupling model, the structure change makes the axial charge g ! of a Λ reduced by 10%
at maximum in the nuclear density of 0.17 fm#$ , and then the Λ’s beta decay rate
decreases by 20% at maximum.
To clearly measure a change of the beta decay rate by baryon modification, the othereffects (nuclear many-body effects and hadronic effects) should be small. Since theseeffects are expected to be small for light nuclei, we will use %&He hypernucleus for thebeta decay measurement. We plan to carry out an experiment at the J-PARC K1.1
beamline, in which %&He will be produced via the 'Li (π( , K ( ) %'Li reaction. Since we wantto measure the beta decay rate within 4.5% accuracy, the branching ratio of the beta decay
and the lifetime of %&He should be measured within 4% and 2% accuracy, respectively. In
the case of the %&He lifetime, it was previously measured with 4% accuracy in the KEK
E462 experiment by using the same (π( , K ( ) reaction. We will use almost the same setup
as the KEK E462 experiment and achieve 2% accuracy. In the case of the branching ratio,
no measurement has been carried out yet. Since the branching ratio of the Λ’s beta decay( ~ 8.0 × 10#) ) is much smaller than that of the Λ ’s main decay modes ( Λ →nπ* and Λ → pπ# ), huge background from π* and π# is expected to be a severe problem
for measuring the beta decay events. The background counts should be of the order of 1%of the beta decay electron counts. We designed detectors around the target. A 'Li target
is surrounded by plastic counters, lucite Cerenkov counters and a BGO calorimeter. The
plastic counters identify charged particles, the lucite Cerenkov counters distinguish thebeta decay electron from other charged particles and the BGO calorimeter measures theenergy of the beta decay electron. We made simulations to develop backgroundsuppression methods and show feasibility of the beta decay measurement by using
GEANT4 simulation code. In the analysis in a BGO calorimeter, by selecting one-clusterihit events as the beta decay electron, 97% of π* and 92.8% of π# can be rejected. In theanalysis in the plastic counter, by selecting a MIP peak region in the energy loss, 88% of
π* and 99.9% of π# can be rejected. In addition, 95.5% of π# and 90% of π* are rejectedby the Lucite Cerenkov counter. Via the background suppression study, the backgroundrate per beta decay electron signal is reduced down to 4%.
Using the yield and the accuracy of the KEK E462 experiment, it is estimated that thebranching ratio and the lifetime can be measured with 4% and 2% statical errors in 1400
hours and 120 hours of beam time, respectively. The background will be significantly
suppressed and give about 3% systematic error. Therefore, this beta decay experiment isfound to be feasible. Most of the present results have been reported as a Letter-Of-Intent
to J-PARC. In near future, we will make more realistic simulation considering the energy
resolution of the BGO.

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Contents
1 Introduction ............................................................................................................... １
1.1 Modification of Λ’s beta decay in nuclear matter..................................................１
1.2 Change of gA by QMC model ................................................................................３
1.3 Λ’s weak decay .......................................................................................................５

2 Experiment for measuring the Λ’s beta-decay rate ............................................... ８
2.1 Modification of Λ in !"He ........................................................................................８
2.2 Production of !"He at J-PARC K1.1 beamline ........................................................９2.2.1 Via the (π! , K ! ) or (K " , π" ) reaction? ..................................................................... １０
2.2.2 K1.1 beamline spectrometer .................................................................................... １０
2.2.3 SKS spectrometer .................................................................................................... １２
2.2.4 #Li target ................................................................................................................. １２

2.3 Apparatus around the target for measuring ....................................................... １３
2.4 Detectors around the Li target ........................................................................... １４
2.4.1 BGO calorimeter ..................................................................................................... １４
2.4.2 Plastic counter ......................................................................................................... １５
2.4.3 Lucite Cerenkov counter ......................................................................................... １５

2.5 Setup for measuring the lifetime ....................................................................... １６

3 Background reduction study for the branching ratio ........................................ １８
3.1 Principles of background suppression and simulations ..................................... １８
3.2 π# and π$ background study ............................................................................. ２０
3.2.1 Cluster analysis in BGO counter ............................................................................. ２１
3.2.2 Analysis in the plastic counter ................................................................................. ２２
3.2.3 Analysis in lucite Cerenkov counter ........................................................................ ２５
3.2.4 Correlation between hit positions of detectors ........................................................ ２５
3.2.5 Photon detector around the SKS spectrometer ....................................................... ２７
3.2.6 Spectrum of beta decay electron .............................................................................. ２７

3.3 Optimization of BGO counter ........................................................................... ２９3.3.1 Optimization of the number of segments of BGO .................................................. ２９3.3.2 Optimization of the thickness of BGO .................................................................... ３１

3.4 Optimization of the photon veto counter in SKS .............................................. ３２iii

3.5 Study of other background................................................................................. ３４
3.5.1 Nonmesonic weak decay of Λ ................................................................................... ３４
3.5.2 Reduction of nonmesonic weak decay of Λ .............................................................. ３４
3.5.3 µ" from three body decay of Λ ............................................................................... ３７
3.5.4 Quasi-free Λ ............................................................................................................. ３８
3.5.5 Reduction of quasi-free Λ ........................................................................................ ４３

3.6 Effects from beam π% ......................................................................................... ４７

4 Feasibility of the beta decay experiment ............................................................. ４９
4.1 Yield and accuracy of branching ratio of beta decay.......................................... ４９
4.2 Yield and accuracy of lifetime ............................................................................ ５０

5 Summary and future prospect.............................................................................. ５１
Appendix A................................................................................................................. ５３
A.1 Beta decay theory .............................................................................................. ５３
A.2 Beta decay model under V−A ........................................................................... ５５
A.3 g & and g ' values ................................................................................................ ５６

Appendix B ................................................................................................................. ５７
B.1 J-PARC .............................................................................................................. ５７
B.2 J-PARC Hadron Experimental Facility ............................................................. ５８
B.3 J-PARC K1.1 beamline...................................................................................... ５８

Appendix C ................................................................................................................. ６０
C.1 Λ → nγ decay rate in nuclear matter .................................................................. ６０
C.2 Cluster analysis in the BGO calorimeter .......................................................... ６１
C.3 Analysis in the plastic counters ......................................................................... ６１
C.4 Analysis of the total energy deposit in BGO ..................................................... ６２
C.5 Summary ........................................................................................................... ６２iv

List of Figures
1.1 Schematic view of the beta decay of a Λ baryon in the free space and in a nucleus... 2
1.2 Change of the axial vector charge g ! of a Λ in nuclear density ρ predicted by QMC
model .......................................................................................................................... 3
1.3 Decay branching ratios of a Λ in the free space .......................................................... 5
1.4 Kinetic energy spectra of π# and π* from %&He calculated by Motoba ...................... 6
1.5 The total energy of neutron-proton pair and neutron-neutron pair from %&He weak
decay measured in the KEK E462 experiment ......................................................... 6
1.6 The kinetic energy spectra of the beta decay electron from Λ → pe# ν<+ decay ......... 7
2.1 The quenching effect of the axial vector charge g ! for beta decay of various lightnuclei ......................................................................................................................... 8
2.2 Experimental setup for measuring the beta decay rate at J-PARC K1.1 beamline .... 9
2.3 The excitation energy spectrum of the %'Li by the (π( , K ( ) reaction measured in the
KEK E462 experiment ............................................................................................ 11
2.4 The excitation energy spectrum of the %'Li by the (K # , π# ) reaction ...................... 11
2.5 Schematic view of the setup for measuring the branching ratio .............................. 14
2.6 A schematic view of the expected setup around the target for the lifetime
measurement .......................................................................................................... 17
2.7 The decay time spectra for %&He and ,-%C measured in the KEK E462 experiment ... 17
3.1 An illustration of events of the beta decay electron as well as π* and π# from the Λ’smain decay modes .................................................................................................... 19
3.2 Apparatus around the target in the present GEANT4 simulation ............................ 20
3.3 The number of clusters in BGO for the beta decay electron, π* and π# from %&He
decay in the simulation ........................................................................................... 21
3.4 Examples of simulated hit patterns in BGO segments for the beta decay electron, π*
and π# events ............................................................................................................ 21
3.5 The total energy deposit in the plastic counter for the beta decay electron, π* andπ# in the simulation ................................................................................................. 22
3.6 Schematic view of the detectors around the Li target .............................................. 23
3.7 Simulated 2D plot of the total energy deposit and the path length in the plastic counterfor π* and the beta decay electron ............................................................................ 24
3.8 The energy deposit per path length in the plastic counter for the beta decay electron,
π* and π# in the simulation ...................................................................................... 24v

3.9 Schematic views of a beta decay electron event and a misidentified π* event in thesimulation ............................................................................................................... 26
3.10 Difference of the z positions between points A and C for the beta decay electron andπ* in the simulation ................................................................................................ 26
3.11 The expected photon veto counters to be installed around the SKS magnet ......... 27
3.12 Simulated spectrum of the beta decay electron energy and the remaining background
events after all the background reduction analysis .................................................. 28
3.13 Simulated background rate for the beta decay electron as a function of the numberof BGO segments ................................................................................................... 29
3.14 An example of the hit pattern of beta decay electron, π* and π# in the case of 225
BGO segments in the simulation ........................................................................... 30
3.15 An example of the hit pattern of beta decay electron, π* and π# in the case of 49
BGO segments in the simulation ........................................................................... 30
3.16 The number of clusters in the BGO for the beta decay electron, π* and π# in the caseof 49 segments (simulation) ................................................................................... 30
3.17 Simulated background rate of the beta decay electron as a function of the BGO
thickness ................................................................................................................. 31
3.18 An example of the simulated hit pattern in the BGO calorimeter for π# in the caseof 10 cm and 30cm BGO thickness ........................................................................ 32
3.19 The relation between the detection rate of 100 MeV photon and the thickness of thelead (5 mm ~ 15 mm) in the case of one-layer structure (simulation) ................... 33
3.20 Measured angle between neutron-proton pair and neutron-neutron pair in the KEK
E462 experiment .................................................................................................... 34
3.21 The number of clusters in BGO for Λp → np and Λn → nn in the simulation ...... 35
3.22 The total energy deposit in the plastic counter for Λp → np and Λn → nn in thesimulation ............................................................................................................... 35
3.23 The energy deposit per path length in the plastic counter for Λp → np and Λn → nn
after selecting 0.7 − 3 MeV as the beta decay electron in the total energy deposit(simulation) ............................................................................................................ 36
3.24 Simulated energy spectra in BGO of remaining Λp → np and Λn → nn
events ...................................................................................................................... 36
3.25 Calculated spectra of the kinetic energy and the velocity of muons in the Λ → pe# ν<.
decay ..................................................................................................................... 37
3.26 Missing mass spectra of production of %'Li and quasi-free Λ (Q-F Λ) .................. 38
3.27 Schematic view of quasi-free Λ (Q-F Λ) production ............................................. 38vi

The momentum distribution of the neutron in ,-C measured by,C (e, e/ p) reaction ............................................................................................. 39
3.29 π( n → K ( Λ reaction in the Lab and the CM systems ............................................. 40
3.30 Calculated momentum distributions of Q-F Λ for x, y, z directions ...................... 41
3.31 Calculated distributions of Λ/ s generation poins for x, y, z directions ................... 41
3.32 The excitaion energy of calculated Q-F Λ .............................................................. 42
3.33 Calculated momentum distributions of Q-F Λ for x, y, z directions after selecting−5 MeV ≤ E+0 ≤ 5 MeV ....................................................................................... 42
3.34 Calculated distributions of Λ/ s generation points for x, y ,z directions after selecting−5 MeV ≤ E+0 ≤ 5 MeV ...................................................................................... 42
3.35 The hit pattern and the number of clusters in BGO for Q-F Λ in the simulation ... 43
3.36 The total energy deposit in the plastic counter for Q-F Λ in the simulation .......... 44
3.37 The energy deposit per path length in the plastic counter for Q-F Λ after selecting
0.7 − 3 MeV as the beta decay electron in the total energy deposit ...................... 44
3.38 Simulated energy spectrum in BGO of remaining Q-F Λ events .......................... 45
3.39 The theta segment number of the center of one-cluster in the BGO ..................... 46
3.40 BGO counting rate with more than 1 MeV as a function of segment theta number inthe simulation ......................................................................................................... 47
3.41 Fraction of the number of BGO segments with more than 103 counts (n) out of all
the BGO segments of 225 (N) as a function of the energy threshold in thesimulation ............................................................................................................. 48
3.42 Fraction of the number of BGO segments with more than 103 counts (n) out of all
the BGO segments of 225 (N) as a function of the number of BGO segments in thesimulation ............................................................................................................. 48
3.28

B.1 A schematic view of J-PARC .................................................................................. 57
B.2 A schematic view of the J-PARC Hadron Experimental Facility ............................ 58
B.3 A schematic view of the K1.1 beamline at the J-PARC Hadron Experimental.
Facility .................................................................................................................... 59
C.1 A diagram of Λ → nγ decay .................................................................................... 60
C.2 The number of clusters in the BGO for Λ → nγ, π* and π# in the simulation ........ 61
C.3 The total energy deposit in the plastic counter for Λ → nγ , π* and π# in thesimulation ................................................................................................................. 61
C.4 The total energy deposit in the BGO for Λ → nγ, π* and π# in the simulation ...... 63
C.5 Simulated energy spectrum of the photon from Λ → nγ decay in the simulation .... 63vii

List of Tables
1.1 Total and partial weak decay rates of %&He hypernucleus shown in the unit of the decayrate of Λ in the free space Γ% ....................................................................................... 6
3.1 Momenta and branching ratios of beta decay electron and those of π# and π* frommain decay modes for %&He ....................................................................................... 20
3.2 The reduction rates of detectors and correlation method (Δz) for the 𝜋 * and 𝜋 #background events through the analyses with each detector and the correlation method(Δz) ............................................................................................................................28
4.1 Expected yield of the %&He and its beta decay events estimated by the %&He yieldmeasured in the KEK E462 experiment and a GEANT4 simulation ........................ 49
4.2 Expected yield of the %&He and proton events from their nonmesonic weak decay andthe expected lifetime accuracy, estimated in comparison with the KEK E462
experiment ................................................................................................................ 50

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1 Introduction
1.1 Modification of 𝚲’s beta decay in nuclear matter
Since a nucleon is made of quarks and gluons with meson cloud, it is natural to consider
that the properties and structure of nucleon may be modified in a nucleus via theinteractions between the quarks in the nucleon and the meson fields spread over the
nucleus. The only clear evidence for nucleon modification in nuclear matter is the EMC
effect. The EMC effect indicates change of the momentum distribution of quarks in the
nucleon in nuclear matter, but the mechanism of the change has not been clarified very
well [1]. There are theoretical models which predict nucleon modification in a nucleus,
but it is difficult to find which theory is correct because experimental data suggesting thebaryon modification is quite limited other than the EMC effect. Since there are manybody effects in a nucleus, it was thought difficult to separate the effect of baryon
modification from other effects.
We thought of an idea that baryon modification via the meson field may be clearly
detected by using the Λ hypernucleus. Λ is the baryon composed of u, d and s quarks, andit has been used to study the properties of the deeply inner part of the nucleus. This isbecause a Λ baryon is free from Pauli blocking from nucleons, and it can stay in the 0s
orbit in the nucleus. In this motivation, our group is now preparing for an experiment(E63) to measure the magnetic moment of a Λ in a nucleus at J-PARC (Japan Proton
Accelerator Research Complex). Another possible probe for baryon modification is toinvestigate weak decay properties of a Λ in a nucleus. It has been discussed that the sizeof nucleon may be swelled since the spatial distribution of u and d quark wavefunctions
may be spread due to their interaction with the meson field in the nucleus. Here, Figure
1.1 shows a schematic view of a Λ in the free space and in the nucleus. As shown in
Figure 1.1, it is expected that the spatial distribution of u and d quarks is spread and thatof s quarks is unchanged because the coupling of s quarks with the meson field is muchsmaller than that of u and d quarks. When the beta decay of Λ takes place in the nucleus,
the spatial overlap between the u quark wavefunction and the unchanged s quark
wavefunction is expected to decrease. This change of the nucleon structure can bemeasured as a reduction of the axial charge g ! and then the beta decay rate of Λ in a
nucleus. The measurements for g ! /g 1 of a Λ in the free space was taken place, and gave
g ! /g 2 = −0.718 ± 0.015 [2]. However, there is no experimental data on Λ’s beta decay
in a nucleus.

１

As described in Section 1.2, according to a calculation by Quark Meson Coupling model(QMC model), the axial charge is predicted to be reduced by 10 % at maximum, and thus
the beta decay rates of Λ is predicted to be reduced by 20% at maximum in a nucleus [3].
If the modification of the baryon structure in a nucleus is clearly measured, we will beable to understand how and why the hierarchical structure between the baryon and thequark is separated. In addition, the baryon modification may give a clue to solve theproblem of “Hyperon puzzle” for the neutron star mass. If the baryon modification takes
place in nuclear matter, the effect of modification should be considered in the neutron starmass calculation. The study of the baryon modification is quite important to develop
nuclear physics based on QCD.
This thesis describes our study on how to measure Λ′s beta decay rates.

Figure 1.1: Schematic view of the beta decay of a 𝚲 baryon in the free space (left) and ina nucleus (right). When the beta decay takes place, an s quark in the 𝚲 changes to a uquark. In the nucleus, the spatial distribution of u and d quark wavefunctions may be
spread due to the meson filed, and that of s quarks is unchanged. The overlap of the u ands quarks are expected to reduce, and this change of the baryon structure can be measured
via the beta decay rate.

２

1.2 Change of 𝐠 𝐀 by QMC model
Quark Meson Coupling Model (QMC model) [3] was developed by P. A. M Guichon
and A. W. Thomas to describe nuclear properties based on the MIT bag model. QMC
model predicts modification of the bound baryon structure via the interactions between u,
d quarks in a baryon and mesons in a nucleus. QMC model considers the couplings of the𝜎, 𝜔 and 𝜌 mesons to the u and d quarks in the baryon, but not to the s quarks. Due to thecoupling with the meson filed, the effective mass of quarks in the nucleus is changed, andit makes the axial vector charge changed.
Figure 1.2 shows the relation between the change of the axial vector charge g ! and thenuclear density. Three lines correspond to the different coupling constants of theinteraction between quarks and the meson field estimated by three different theoreticalmethods. As shown in Figure 1.2, the g ! value decreases for larger densities. Since thesaturation density of nucleus is 0.17 cm#$ , the reduction rate of g ! is expected to be about
9 ± 1.6%.

Figure 1.2: Change of the axial vector charge g ! of a Λ in nuclear density ρ predicted by
QMC model. In the nuclear matter density ~ 0.17 fm#$ , the reduction rate is 9 ± 1.6% [3].

３In general, the beta decay rate, Γ3+45 , is written in terms of the vector charge g 1 , the axial
vector charge g ! , and nuclear matrix elements as𝛤6789 ∝ 𝑔:- 𝑀;- + 𝑔<- 𝑀=>,

as described in Appendix A. This relation is also valid for Λ ’s beta decay in a
hypernucleus. M?- and M@A
are the matrix elements of Fermi transition and Gamov-Tellertransition. Here, we consider that M?- and M@Avalues are the same as those of a Λ in thefree space. Then the beta decay rate is given by𝛤6789 ∝ 𝑔:- + 3𝑔<- .
When Λ’s beta decay in a nucleus occurs only via Gamov-Teller transition as
𝛤6789 ∝ 3𝑔<- ,the beta decay rate decreases by about 20%. When Λ’ s beta decay in a 0( nucleus occurs
in combinations of Fermi and Gamov-Teller transition,
𝛤6789 ∝ 𝑔:- + 3𝑔<- ,the beta decay rate decreases by about 12%.

４

1.3 𝚲’s weak decay
Figure 1.3 shows the decay branching ratios of a Λ in the free space [2]. As shown in
Figure 1.3, the Λ’s main decay modes are mesonic weak decay of Λ → nπ* and Λ → pπ# .
The branching ratio of the beta decay is ~ 8.0 × 10#) , and it is much smaller than that ofmasonic weak decay (0.997).

Figure 1.3: Decay branching ratios of a Λ in the free space [2].
Table 1.1 shows total and partial weak decay rates of %&He hypernucleus shown in theunit of the decay rate of Λ in the free space. ΓB$ , ΓB% and ΓCD are decay widths of mesonicweak decay and nonmesonic weak decay, respectively. Γ% and Γ4E4 are the sum of the
decay widths in the free space and in %&He, respectively. As shown in Table 1.1, the Λ’smain decay modes are greatly suppressed in the nucleus in comparison with in the freespace. This is because the momentum of a proton from Λ’s mesonic decay of ~ 100 MeV/c
is smaller than the fermi momentum of ~ 260 MeV/c, and this decay mode is Pauliblocked. Since the mesonic decay width in %&He is ΓB$ + ΓB% ~ 0.54, we estimate theeffect from Pauli blocking to be ~ 0.6. In addition, we can assume that Λ’s beta decay in&%He is also suppressed by a factor of ~ 0.6, because the proton momentum after the Λ’sbeta decay (0 −160 MeV/c) and the nucleon momenta after the Λ’s mesonic weak decay(~ 100 MeV/c) are similar. Therefore, we estimate that the branching ratio of the beta
decay in %&He is 8.0 × 10#) × 0.6 = 0.00048.
The momentum of π* and π# from %&He decay was theoretically calculated by Motoba
as shown in Figure 1.4 (a), (b) [12]. In the case of π* , the momentum distribution has apeak at 105.4 MeV/c and maximum momentum is 106.1 MeV/c. In the case of π# , themomentum distribution has a peak at 100 MeV/c and maximum momentum is 102.4
MeV/c. Figure 1.5 shows the total energy of the neutron-proton pair and the neutronneutron pair from %&He nonmesonic weak decay (Λp → np and Λn → nn) measured in the５Table 1.1: Total and partial weak decay rates of %&He hypernucleus shown in the unit ofthe decay rate of Λ in the free space Γ% .

Figure 1.4 (a), (b): Kinetic energy spectra of π$ and π# (MeV) from %&He calculated by
Motoba [12]. π* momentum distribution has a peak at 105.4 MeV/c and maximum
momentum is 106.1 MeV/c. π# momentum distribution has a peak at 100 MeV/c andmaximum momentum is 102.4 MeV/c.

Figure 1.5 (a), (b): The total energy of neutron-proton pair and neutron-neutron pair from&%He weak decay measured in the KEK E462 experiment [13].

６KEK E462 experiment [13]. The sum energy distributes in the range of 60 − 220 MeV,
and the momentum of nucleon is calculated to be < 470 MeV/c. Energy widely distributes
because of Fermi motion.
In addition, we calculate the electron kinetic energy from Λ → pe# ν<+ , and the energy
distribution distributes in 0 − 163 MeV as shown in Figure 1.6.

Figure 1.6: The kinetic energy spectra of the beta decay electron from Λ → pe# ν<+ decay.
The kinetic energy distributes in 0 − 163 MeV.

７

2 Experiment for measuring the Λ’s beta-decay rate
2.1 Modification of 𝚲 in 𝚲𝟓𝐇𝐞
According to the calculation of the QMC model, the axial charge g ! of Λ in nuclear
matter density may decrease by 10% at maximum. We want to measure g ! reduction by
separating other effects from baryon modification, but it seems difficult. In the beta decayof ordinary nuclei, nuclear many-body effects and hadronic effects such as mesonexchange current are known to reduce the beta decay rate of Gamov-Teller transition.
Such reduction is ascribed to quenching of the effective g ! value in nuclei [14]. Figure
2.1 shows the quenching effects of g ! in light nuclei [15]. The quenching effects becomelarger in heavier nuclei.

Figure 2.1: The quenching effect of the axial vector charge g ! for beta decay of various
light nuclei. The reduction of g ! becomes larger for heavier nuclei, but that of the s-shell
nuclei is around ~ 5% [14,15].
As shown in Figure 2.1, the quenching effect become larger for the heavier nuclei.
However, we can see that the quenching effect is small (~ 5%) for the s-shell (A ≤4) nuclei. Thus, the quenching effect is expected to be about 5% for %&He hypernucleus.
For this reason, we propose an experiment to measure the beta decay rate of %&He.
The beta decay rate Γ3+45 is described by the branching ratio of the beta decay
BR 3+45 and the lifetime of τ% in %&He as Γ3+45 = BR 3+45 /τ% . BR 3+45 is obtained from thenumber of beta decay events from produced %&He hypernuclei. Considering the available
beamtime at J-PARC, we expect that BR 3+45 can be measured within 4% accuracy andτ% can be measured within 2% accuracy. In our proposed experiment, we hope to confirm
the reduction of Γ3+45 (~ 20% at maximum) with more than 3σ confidence level. This
goal will be achieved by measuring BR 3+45 , τ% within 4% and 2% accuracy, respectively.
In the case of the %&He lifetime, it was previously measured with 4% accuracy in the KEK
E462 experiment [5,6,13]. We will use a similar setup as the KEK E462 experiment and８achieve 2% accuracy, as described later. In the case of the branching ratio, no
measurement has been carried out yet.

2.2 Production of 𝚲𝟓𝐇𝐞 at J-PARC K1.1 beamline
D4

Q10
Q11

Figure 2.2: Experimental setup for measuring the beta decay rate at J-PARC K1.1
beamline. 1.1 GeV/c π( beams are provided to the target, and the scattered K ( ’s are
identified by the SKS spectrometer.
We plan to carry out measurements of the beta decay rate at the J-PARC K1.1 beamline
because the K1.1 beamline is suitable for producing S = −1 hypernuclei. The K1.1
beamline is the secondary beamline which maximumly provides π( beam with momenta
up to 1.1 GeV/c (see the detailed information in Appendix B).
Figure 2.2 shows the experimental setup planed at the K1.1 beamline. The %&He
hypernucleus will be produced via the 'Li (π( , K ( ) %'Li reaction with a 'Li target. The
produced %'Li hypernucleus is unbound and immediately decays as %'Li → %&He + p, andthen the %&He hypernucleus is produced. The production of %&He is identified by the
missing mass method. The missing mass is calculated as
-

fff⃗
ffff⃗
𝑀FGH = b(𝐸I + 𝑀 − 𝐸J )- − d𝑃
I − 𝑃J h .

９

fff⃗
Here, EB , EK are the total energies of the beam π( and the scattered K ( , respectively. P
B,((
fff⃗K are the momentum vectors of the beam π and the scattered K , respectively. M is thePmass of the target nucleus, 'Li. The momentum of beam π( is measured by the K1.1
beamline spectrometer, and that of the scattered K ( is measured by the SKS spectrometer.
Figure 2.3 shows the excitation energy spectrum of %'Li by the (π( , K ( ) reaction [16]. (b)and (c) show the spectra in coincidence with pions and protons. The ground state of %'Li
is clearly seen.

2.2.1 Via the (𝛑$ , 𝐊 $ ) or (𝐊 % , 𝛑% ) reaction?
&
(
(
# #
%He can be produced by the (π , K ) or the (K , π ) reactions. Figure 2.4 shows the# #excitation energy spectrum of the (K , π ) reaction obtained by an old experiment [17].

As shown in Figure 2.4, the ground state of %'Li is clearly seen, but there are peaks of theunbound states at about 8 MeV and 18 MeV. These states correspond to the orbital
angular momentum transfer ΔL = 0 , and are called “substitutional state”. The
substitutional state is likely to be produced in small momentum transfer reactions such as(K # , π# ) reaction (q < 150 MeV/c with 0.8−1.1 GeV/c beam momentum and 0° − 10°scattering angle). By using large momentum transfer reactions such as (π( , K ( ) reaction(q ~ 300 MeV/c), a large orbital angular momentum transfer ΔL ≥ 2 is given to thehypernucleus, and production of substitutional states are greatly suppressed. According
to the KEK E419 experiment in which the %'Li hypernucleus is produced, the mass
resolution was 4.2 MeV using a 13.7 g/cm$ -thick MLi target [18]. In our proposed
experiment, we will use almost the same thickness of 'Li target considering the yield of&# #%He (described later). In the case of the (K , π ) reaction, with the mass resolution of ~4 MeV, about 10% of the excited substitutional state at 8 MeV contaminates the groundstate peak. In the case of the (π( , K ( ) reaction, the substitutional states are not populated.
In addition, in the case of (K # , π# ) reaction, π# from the beam K # decay causes
misidentification as (K # , π# ) reaction. Therefore, we decided to use the (π( , K ( )reaction for the %&He hypernucleus production.

2.2.2 K1.1 beamline spectrometer
The K1.1 beamline spectrometer is composed of D4, Q10 and Q11 magnets. At theentrance of D4 magnet, a scintillating fiber detector, BFT-U, is installed. At the exit of
the beamline, a drift chamber, BC3, and a fiber detector, BFT-D2, are installed. The beam
track is reconstructed by three hit points on BC3, BFT-U and BFT-D2. In addition, plastic
counter hodoscopes, BH1 and BH2, are installed at the entrance of D4 and the exit of thebeamline, respectively. Time-of-flight (TOF) of a particle is measured by flight time from

１０

BH1 to BH2. A beam π( can be distinguished from other particles and the momentum ofπ( is measured. The momentum resolution is expected to be 0.042 % (FWHM) [19].

Figure 2.3: The excitation energy spectrum of %'Li by the (π( , K ( ) reaction measured in
the KEK E462 experiment [16]. (b) and (c) show the spectra in coincidence with pions
and protons. The ground state of %'Li is clearly seen in any of the coincidence spectra.

Figure 2.4: The excitation energy spectrum of the %'Li by the (K # , π# ) reaction. The
ground state of %'Li is clearly seen, but substitutional states at 8 MeV and 18 MeV
excitation are more prominent [17].

１１2.2.3 SKS spectrometer
The SKS spectrometer is composed of a superconducting dipole magnet (SKS magnet),
three drift chambers (SDC2, SDC3, SDC4), time of flight detectors (TOF) and aerogel
Cerenkov counters (AC). TOF identifies K ( by using flight time from BH2 to TOF. AC
separates scattered K ( from beam π( . SDC2-4 record scattered particle tracks. Via thesedevices, a scattered K ( can be distinguished from other particles and the mentum of
K ( is measured. The SKS covers a wide range of scattering angle (-15° ~ 15° ), which islarger than the angular range (-10° ~ 10° ) of the scattered K ( in the (π( , K ( ) reaction.
The momentum acceptance of the SKS spectrometer is 0.7 − 1.1 GeV/c, and theacceptance of SKS is about 100 msr.

2.2.4 𝟔𝐋𝐢 target
To achieve ~ 4 MeV mass resolution of hypernucleus and clearly see the ground statepeak of %'Li in (π( , K ( ) reaction, we will use a 90% enriched 14 g/cm$ 'Li target with acylindrical shape. We expect that a length of 'Li target is 30cm and diameter is 3 cm tocover diameter of beam π( . The target is packed in a laminated plastic bag filled with Ar
gas.

１２

2.3 Apparatus around the target for measuring
the branching ratio of the beta decay
After the production of %&He, the beta decay electron from the hypernucleus should bemeasured. We want to measure the beta decay electron as many as possible by thedetectors surrounding the Li target. When measuring the beta decay electron, there is aproblem of the background from Λ’s other decay modes. As shown in Table 1.1, theexpected background is as follows.
・ Mesonic weak decay
Λ → pπ# , π# pn → nn (BR = 0.4)
・ Mesonic weak decay
Λ → nπ* , π* → γγ(BR = 0.2)
・ Nonmesonic weak decay Λn → nn and Λp → np (BR = 0.4)
We must distinguish the beta decay electron from the other particles such as π# , p, γ andn. The beta decay electron is a charged particle with a velocity β~1. The background canbe rejected by using this character. We made a rough design of the detectors around the
Li target as shown in the followings.
The detectors are composed of
1. Plastic counters
to separate charged particles and noncharged particles,
2. Lucite Cerenkov counters (index n = 1.5)
to detect charged particle with β > 0.67, and3. BGO calorimeter
to measure the energy of the beta decay electron.
The geometry of each detector is shown in Figure 2.5. The Li target is surrounded by
the plastic counters and the lucite Cerenkov counters, and then the BGO calorimeter. As
shown in Figure 2.5, the beta decay electron from %&He hypernucleus produced via the(π( , K ( ) reaction is emitted from the Li target. The beta decay electron is identified by
the plastic counters and the lucite Cerenkov counters, and then the energy is measured in
the BGO calorimeter. We expect that the number of the beta decay electrons is counted
in the final energy spectrum in the BGO. The detailed information of each detector isshown in the next sections.

１３

Figure 2.5: Schematic view of the setup for measuring the branching ratio. A 'Li target issurrounded by plastic counters, lucite Cerenkov counters and a BGO calorimeter. The
BGO calorimeter has two holes at upstream and downstream for the beam and thescattered K ( . The beta decay electron from %&He is separated from the other particles by
the plastic counters and the lucite Cerenkov detectors, and then the energy is measured
by the BGO.

2.4 Detectors around the 𝐋𝐢 target
2.4.1 BGO calorimeter
The BGO calorimeter measures the total beta decay electron energy which distributes 0− 163 MeV. The beta decay electron produces an electromagnetic shower in the BGO.
The Moliere radius and the depth of the electromagnetic shower are roughly estimated tobe ~ 2 X* = 2.2 cm and ~ 20 X* = 22 cm. Here, X* ≈1.1 cm is the radiation length of
BGO. The target is surrounded by spherically shaped segmented BGO counters. We
assume the width of BGO crystal is 7 cm and the thickness is 20 cm (7 cm × 7 cm × 20
cm crystal) to accept most of the electromagnetic shower from the beta decay electron.
The BGO calorimeter is divided by the θ and ϕ directions, and the number of the BGO

１４

segments is 15 ( θ ) × 15 ( ϕ ) = 225. These dimensions will be further optimized
considering the preparation time and the cost. The signal is read out by a PMT which is
attached at the outer end of each BGO crystal.

2.4.2 Plastic counter
The Li target is surrounded by the circumferentially segmented plastic counters. Its
length is 60 cm (the same as the BGO inner diameter) to detect the beta decay electrons
as many as possible. It has a 5 mm thickness and a 6 mm width. By analyzing the energy
deposit in the plastic counters, we can measure the velocity of the particles from theenergy loss per path length (dE/dx). Since the cross section of each plastic scintillator rod
is small, we will use MPPCs for reading out a signal. MPPCs are attached at both ends ofthe plastic and the hit position is measured via the timing difference and the segment
number. Namely, the z hit position is obtained from the timing difference and the x, y hit
positions are obtained from the segment number with a hit. We expect that the time
resolution of the plastic counters and the position resolution along the beam direction to
be 100 ps and 20 mm in rms, respectively.

2.4.3 Lucite Cerenkov counter
The Li target and the plastic counters are surrounded by circumferentially segmented
lucite Cerenkov counters (index n = 1.5). Its length is 60 cm (the same as the BGO innerdiameter and the length of the plastic counters). It has a 5 mm thickness and a 6 mm width.
By using the lucite Cerenkov counter, the beta decay electron can be distinguished fromother charged particles such as pions and protons from the hypernucleus decay (βB$ ≤0.6 for PB$ ≤ 101 MeV/c, and βN ≤ 0.4 for PN ≤ 400 MeV/c ). MPPCs are attached atboth ends of each lucite Cerenkov counter. We expect that the efficiency is more than
99% for electrons. According to a GEANT4 simulation (described later), themisidentification rate for the charged pions and protons due to δ ray emission in the target
is found to be ~ 4%.

１５

2.5 Setup for measuring the lifetime
Figure 2.6 shows a schematic view of the apparatus for measuring the lifetime of&%He [20]. The experimental setup is similar to that of the KEK E462 experiment. A block'of the Li target is surrounded by the timing counter T1 and T2L / T2R. The target in thisexperiment is different from the target for measuring the branching ratio. It has a 3 cm
thickness and a 10 cm length. T1 is made of 20 pieces of 2 mm-wide 50 mm-long 5 mm
thick plastic counter, and the signal of each piece is read out by MPPCs attached at bothends. THL and THR are made of 80 pieces 3 mm-thick plastic counters of 2 mm wide
and 80 mm long, read out via MPPC’s at both ends. T1 measures the timing of the beampion and T2L or T2R measure the timing of the proton from nonmesonic weak decay of&%He. In addition, THL, THR, T2L and T2R measure the hit position of the proton. BFT-

D and T1 also measure the position of the beam pion. The time difference between theT1 hit and the T2L or T2R hit is used to calculate the decay time of hypernuclei after
correcting for the pion flight time between T1 and the reaction point, and the proton flighttime between T2L or T2R and the reaction point. We will use 2 mm-wide plastic counter
to improve the resolution of the hit positions. We expect that the time resolution of T1
and T2L / T2R is better than 50 ps in rms, and the resolution of the hit position is ~ 2
mm. This position resolution corresponds to the resolution of the decay time of ~ 10 ps.
T2L and T2R are made of 20 pieces of 4 mm-wide 150 mm-long 5 mm-thick plastic
counter, each of which is read out via MPPC’s at both ends. Data of the E462 experiment
for the difference of timing between the beam 𝜋 ( and the decay proton at the reaction
point in the target is plotted in Figure 2.7 (left for %&He and right for ,-%C data) [16]. This
spectrum is called “decay time spectrum”. By fitting the decay time spectrum, the lifetimeof %&He is obtained. In the E462 experiment, the accuracy for measuring the lifetime isabout 4% for %&He . In our proposed experiment, we want to achieve 2 % accuracy for 120
hours (5 day) beamtime (will be described in the chapter 4).

１６

Figure 2.6: A schematic view of the expected set up around the target for the lifetime
measurement [20]. The lifetime of a Λ is measured by the timing difference of thetiming counters, T1 and T2.

Figure 2.7: The decay time spectra for %&He (left) and ,-%C (right) measured in the KEK
E462 experiment [16]. The upper figures show “response function” of the spectrum
measured with the prompt reaction of (π( , pp). The lower figures show decay time spectra
for %&He and ,-%C hypernuclei via the (π( , K ( ) reaction measured with protons emitted via
nonmesonic weak decay of %&He .

１７

3 Background reduction study for the branching ratio
To achieve 4.5% accuracy of measuring the beta decay rate of Λ, the background from
Λ’s decay modes should be suppressed down to the order of 1% of the beta decay electron
signal.

3.1 Principles of background suppression and simulations
In Chapter 2, we discussed the apparatus around the Li target for measuring the beta
decay rate of the %&He hypernucleus and reducing the background from the Λ’s decay. The
following detectors are introduced in Chapter 2.
・ The BGO calorimeter (225 BGO segments and 7 cm × 7 cm × 20 cm crystal)→ Measure the energy of the beta decay electron.
・ The plastic counter (30 segments with 60 cm length, 5 mm thickness and 6 mm width)→ Identify charged particles (distinguish the beta decay electron from π* )・ The lucite Cerenkov counter (30 segments with 60 cm length, 5 mm thickness and 6
mm width)→ Identify charged particles with velocity β > 0.67 (distinguish the beta decay
electron from π# and proton from weak decay of %&He).
Figure 3.1 shows an illustration of events when the beta decay electron, as well as π*
and π# from the Λ’s main decay (background), are produced. We expect that the beta
decay electron produces one-cluster hit in the BGO calorimeter. π* mainly decays to
2γ ( π* → 2γ), and is expected to produce two-cluster hit in the BGO. When π# enters
the BGO, π# is expected to interact with the nuclei in BGO by 𝜋 # absorption mainly via
π# "pn" → nn. Emitted neutrons produce hits in many segments.
In the case of π* , when one photon from the π* → 2γ decay escapes from the BGO
upstream or downstream hole, the other photon produces one-cluster hit in the BGO.
Since the solid angle of the upstream or downstream hole is 30.7 msr and 89.8 msr,
respectively, the probability of the photon-leak event is 100 × (39.8 + 89.8) × 10#$ /4𝜋 ≈ 1 (%). In addition, when one photon from π* converts to e# e( , it is identified as
the beta decay electron by the plastic counters. The conversion rate of γ → e# e( in theplastic and the target is estimated to be about 4% by calculating the cross section of paircreation of photon. Considering the branching ratio of π* decay of %&He (0.2) and thebranching ratio of the beta decay (0.00048), the estimated background rate per beta decay
electron signal is 0.2 × 0.01 × 0.04 ÷ 0.00048 = 0.16 (16%).

１８

Figure 3.1: An illustration of events of the beta decay electron (signal) as well as π* andπ# from the Λ’s main decay modes (background). See text for details.
In the case of π# , the remaining events depend on the misidentification rate by δ rays in
the lucite Cerenkov counters. Thus, we studied it by a GEANT4 simulation (describedlater)
Since we need to reduce the background to the order of 1% of the beta decay electron,
we have developed various methods of reducing background using GEANT4 simulation
code (GEANT4 is a platform developed by CERN for simulating tracks of particles inmaterials by using Monte Carlo methods).
In Chapter 3, we describe simulations for the Λ’s main decay mode of mesonic weak
decay and then non-mesonic weak decay. We also made a simulation for optimization of
the BGO detector.

１９3.2 𝛑𝟎 𝐚𝐧𝐝 𝛑% background study
Figure 3.2 shows a geometry of setup around the target in GEANT4. The Li target issurrounded by the plastic scintillators (red), the lucite Cerenkov counters (blue) and the
BGO calorimeter (cyan). A red plate installed at the BGO downstream hole is a photon
veto counter to detect leak photon from π* → γγ. In this simulation, beta decay electron,

Figure 3.2: Apparatus around the target in the present GEANT4 simulation. The Li target
is surrounded by the plastic scintillators (red), lucite Cerenkov counters (blue) and the
BGO counters (cyan). Red plate installed at the BGO downstream hole is a photon vetocounter.
π* and π# are produced according to their branching ratios and momenta. The branchingratio of π* and π# are taken from the previous experiment (KEK E462) as listed in Table
1.1. Their branching ratios and momenta we used in the GEANT4 simulation are listed
in Table 3.1.
Table 3.1: Momenta and branching ratios of beta decay electron and those of π$ andπ# from main decay modes for %&He.
Particle
Momentum (MeV/c)
BR (in !"He)

Beta decay electron

π# ( Λ → nπ# )

π$ ( Λ → pπ$ )0 − 1630.00048

105.4
0.4

100
0.2

２０

3.2.1 Cluster analysis in BGO counter
Background from π* and π# can be greatly rejected via cluster analysis in BGO. Figure
3.3 shows the number of clusters in BGO, and Figure 3.4 shows an example of the BGO
hit pattern showing the BGO segments with more than 1 MeV energy loss. A cluster is
identified when segments with more than 1 MeV adjoin each other, or a single segment
has energy loss over 1 MeV. As shown in Figure 3.4, a beta decay electron produces onecluster. A π* mainly produces two-cluster since π* mainly decays to 2 γ . In thissimulation, Dalitz decay mode π* → e# e( γ is also considered, and this mode produces
two or three clusters. One cluster of π* is caused by one γ escape from the BGO upstream
or downstream hole. A π# produces many clusters by many neutrons and nuclear γ-rays
from interaction between π# and nuclei in BGO. It was found that, by selecting onecluster events as beta decay electron, 97% of π* and 92.8% of π# events can be rejected.

Figure 3.3: The number of clusters in BGO for beta decay electron, π* and π# from %&He
decay in the simulation. The beta decay electron produces one cluster, π* produces two
clusters by π* → γγ, and π# produces several clusters due to neutron emission from BGO.

Figure 3.4: Examples of simulated hit patterns in BGO segments for the beta decay
electron, π* and π# events.

２１

3.2.2 Analysis in the plastic counter
Analysis of energy deposit in the plastic counters plays an important role in reducing
background. Figure 3.5 shows total energy loss of beta decay electron, π# and π$ from&%He decay. As shown in Figure 3.5, the beta decay electron shows a peak of MIP, while#the π from Λ decay with momenta lower than 100 MeV/c makes a peak at larger energy
deposit. The energy loss of π* widely distributes. The low energy (< 1 MeV) part comesfrom photoelectric effect and Compton scattering, and the high energy (>1 MeV) partcomes from e# e( pair creation. According to the Bethe-Bloch equation, energy loss ofelectron (- dE/dx) e is about 2 MeV g -1 cm2. We expect that the energy loss per length inplastic is about 2 MeV g -1 cm2 × 0.5 g cm -3 = 1 MeV cm -1. To distinguish beta decay
electron from π* using this energy loss spectrum, we calculate energy deposit per path
length in the plastic counter.

Figure 3.5:
The total energy deposit in the plastic
counter for the beta decay electron,
π* and π# in the simulation.
Beta decay electron distribution has apeak of MIP. π* distributes widely due
to interactions of γ from π* in theplastic counters.
Figure 3.6 shows a schematic view of the detectors around the target. Point A, B, C, D
are the generation point in the Li target, the position where the particle enters the plastic
counter, the position where the particle exits the plastic counter and the hit position of
BGO. The point B and C are calculated as the intersections of the line AD and the innerand outer surfaces of the plastic counters, and then the length of BC (path length) is
calculated. In this simulation, we assumed the resolution of the generation point A is
σ0 = 1.2 mm, σO = 2.6 mm and σP = 22 mm. These values were measured in the E13２２experiment at J-PARC as the resolution of the vertex point determined from the beam
track and the scattered particle track [21]. The E13 experiment was previously carriedout at the J-PARC K1.8 beamline, and production events of hypernuclei via the(K # ,π# ) reaction are identified by the beamline spectrometer and the SKS spectrometer.
The beta decay experiment will be caried out at the J-PARC K1.1 beamline, but the
setup is almost the same as the E13 experiment. Therefore, we used the experimental
data of the E13 experiment. Figure 3.7 shows the 2D plot of the path length and thetotal energy deposit. In the case of the beta decay electron, the total energy deposit ofbeta decay electron is proportial to the path length due to the constant -dE/dx from a
MIP passing in the plastic counter. In the case of π* , the total energy deposit has ahigher energy component than the energy deposit by a MIP. This is because of multiple
charged particles from pair production. We found that the ratio of the total energy
deposit to the path length (the energy deposit per path length) can distinguish beta
decay electron from π* . Figure 3.8 shows the energy deposit per path length.
The π* distribution has two peaks corresponding to either of e# or e( , and to bothe# and e( . By this calculation, separation of π* and beta decay electron becomes better
than comparing of the total energy deposit. After selecting 0.7 − 3.0 MeV as the beta
decay electron in the total energy deposit (around a MIP peak), we select 0.12 − 0.25
MeV/mm in the energy deposit per path length as the beta decay electron. Finally, 88%of π* and 99.9% of π# events are reduced.

Figure 3.6: Schematic view of the detectors around the Li target. Beta decay electron is
produced at A. Then the electron hits the inner and outer surfaces of the plastic counterat B, C and hits BGO at D.

２３

Figure 3.7: Simulated 2D plot of the total energy deposit and the path length in the plastic
counter for π* and the beta decay electron. Left figure shows π* events and right figure
shows the beta decay electron events.

Figure 3.8:
The energy deposit per path length inplastic counter for the beta decay
electron, π* and π# in the simulation.
Electron and π# can be clearly
separated. π* distribution has two
peaks from an electron and a positron.

２４

3.2.3 Analysis in lucite Cerenkov counter
Since the index of lucite is n =1.5, lucite Cerenkov counter can distinguish β > 1/n =0.67 from those with β < 0.67. We found that 99.8 % of beta decay electron events canbe detected by the lucite Cerenkov counters, and π# and π* events are rejected by about
95.5% and 90%, respectively. In addition, 4.5% of π# events are found to emit δ rays andthey are misidentified as beta decay electron.

3.2.4 Correlation between hit positions of detectors
When one photon from π* decay escapes from the BGO hole, the other photon may
become background for beta decay electron. The other photon produces electromagnetic
shower in BGO, and charged particles ( e( and e# ) sometimes escape from BGO. If theescaping charged particles enter the plastic counters and the lucite Cerenkov counters,
these particles are misidentified as beta decay electron.
Figure 3.9 shows schematic views of a beta decay electron event and a misidentified
event of π* . Points O, A, B are the generation point, and the hit points in the plastic
counter and BGO. Point C is intersection of the line OB and the plastic counter. With
respect to the point C, the z position is determined by timing difference of MPPCs
attached on both ends of the plastic counter, and x, y positions are determined by thesegment number divided in φ direction. Regarding the point B, the hit position isdetermined by the cluster center in BGO. To calculate the cluster center, energy deposits
in hit segments are weighted. Finally, the difference of z position between A and C (Δz)
is calculated. As shown in Figure 3.10, in the case a beta decay electron event, Δz
distributes around 0 because a track of beta decay electron is a straight line. In the caseof misidentified π* , Δz widely distributes since charged particles which are produced
from electromagnetic shower in BGO and escape back from BGO hits the plastic counter.
By selecting −90 mm < Δz < 90 mm as beta decay electron, 20 % of π* events are
rejected.

２５

Figure 3.9: Schematic views of a beta decay electron event and a misidentified π* eventin the simulation. Charged particles ( 𝑒 ± ) from the electromagnetic shower produced by
the photon from π* decay hit the plastic counters, and they are misidentified as beta decay
electron.

Figure 3.10: Difference of the z positions between the points A and C for the beta decay
electron and π* in the simulation. In the case of the beta decay electron, the z difference
distributes around 0 due to a straight track. In the case of the π* , the z difference widelydistributes because charged particles from the electromagnetic shower in the BGO hit theplastic counters.

２６

3.2.5 Photon detector around the SKS spectrometer
By the analysis described above, the remaining backgrounds of π* and π# are 12.6 %and 0.16 % of the beta decay electron signal, respectively. The π# events are rejected well,
but the reduction of π* is not enough. This is because when one photon escapes from the
BGO holes in upstream or downstream, one cluster is produced in BGO by the otherphoton. To reduce such photon leak events, we plan to install photon veto counters aroundthe SKS magnet. In this simulation, photon veto counters are installed at the end of thedownstream BGO hole. By the simulation, 70% of remaining π* events are rejected by
vetoing leak photon detection events. Figure 3.11 shows the expected photon veto counter.
The materials are lead and plastic scintillator. The photon converts to e# e( in the lead,and they are detected by the plastic counter. We consider the structure of the veto countermade of one layer or two layers. Detailed information and the simulation study for thephoton veto counters are shown later.

Figure 3.11: The expected photon veto counters to be installed around the SKS magnet.
Left shows the detector of one-layer structure (Pb + Plastic), and right shows the detectorof two-layers structure.

3.2.6 Spectrum of beta decay electron
Table 3.2 shows the reduction rates of detectors and correlation method (Δz) for the π*
and π# background events through the analyses with each detector and the correlation
method (Δz). By all the analysis described above, the remaining background is 0.16%(π# ) + 3.6% (π* ) ≈ 4% of the beta decay electrons. This value is almost the same as therequired statical error of the beta decay measurement.
Figure 3.12 shows a simulated energy spectrum of beta decay electron (blue) and theremaining background (red).

２７

Table 3.2: The reduction rates of detectors and correlation method (Δz) for the π* and π#
background events through the analyses with each detector and the correlation method(Δz).

Figure 3.12: Simulated spectrum of the beta decay electron energy (blue) and theremaining background events (red) after all the background reduction analysis (see text).
In our simulation, the background is estimated by the ideal conditions such as fine BGO
segments and a large BGO thickness. To show the feasibility of the experiment, thenumber of segments and the thickness of BGO should be optimized in order to reduce
preparation time and cost.

２８

3.3 Optimization of BGO counter
The BGO detectors should be optimized considering the preparation time and the cost.
Since BGO crystal is expensive, we studied effects of the background rates by changing
the number of BGO segments and the thickness of each BGO crystal.

3.3.1 Optimization of the number of segments of BGO
Figure 3.13 shows the relation between the background rate and the number of BGO
segments. Here, the background rate was estimated from the number of events which
remained after all the background suppression analysis described before. As shown in
Figure 3.13, the background rate increases when the number of BGO segments is lessthan 100. This is because the cluster analysis does not work well for the less segments.
Figure 3.14 shows an example of the hit pattern in BGO in the case of 225 segments, andFigure 3.15 shows the case of 49 segments. With respect to the beta decay electron and
π* , the number of clusters does not change by changing the number of segments.
However, the number of clusters for π# changed. In the case of 225 segments, π# mostly
produces two or more clusters. In the case of 49 segments, π# produces one-cluster.
Figure 3.16 shows the number of clusters in the case of 49 segments. As shown in Figure
3.16, the events of the beta decay electron and π* producing one-cluster events do notincrease. However, in the case of π# , one-cluster events greatly increase as 7.8 % (225segments) → 31 % (49 segments). Therefore, more π# ′s cannot be distinguished from thebeta decay electron, and the background from π# increases under less BGO segments.
Finally, we found that the number of BGO segments should be larger than 100.

Figure 3.13: Simulated background rate for the beta decay electron as a function of thenumber of BGO segments. The background rate increases when the number of BGO
segments is less than 100 due to worse performance of the cluster analysis.

２９Figure 3.14: An example of the hit pattern of beta decay electron, π* and π# in the caseof 225 BGO segments in the simulation.

Figure 3.15: An example of the hit pattern of beta decay electron, π* and π# in the caseof 49 BGO segments in the simulation.

Figure 3.16: The number of clusters in the BGO for the beta decay electron, π* and π# in
the case of 49 segments (simulation). In the case of the beta decay electron and π* , onecluster events do not increase. In the case of π# , one-cluster events greatly increase as
7.8 % (225 segments) → 31 % (49 segments).

３０3.3.2 Optimization of the thickness of BGO
Figure 3.17 shows the relation between the background rate and the thickness of BGO.
Here, the background rate was estimated from the number of events which remained after
all the background suppression analysis described before. As shown in Figure 3.17, thebackground rate increases for the thickness less than 20 cm. Figure 3.18 shows anexample of the hit pattern in BGO of π# with a 10 cm thickness and a 30 cm thickness.
As shown in Figure 3.18, the number of clusters decreases by reducing the BGO thickness(30 cm → 10 cm). Since the reaction rate of neutron is proportial to the BGO thickness,
neutrons from π# produce fewer clusters in the case of 10 cm BGO thickness than 30 cm
thickness. Finally, we found that the BGO thickness should be more than 20 cm.

Figure 3.17: Simulated background rate of the beta decay electron as a function of the
BGO thickness. The background rate increases for the BGO thickness less than 10 cm
due to reduction of the neutron reaction rate.

３１

Figure 3.18: An example of the simulated hit pattern in the BGO calorimeter for π# in
the case of 10 cm and 30 cm BGO thickness. Due to reduction of the neutron reaction
rate, the number of clusters decrease as 3 clusters → 1 cluster.

3.4 Optimization of the photon veto counter in SKS
In the previous π* , π# simulation, we installed photon veto counters which cover all thesolid angle of the BGO downstream hole. However, in reality, the photon veto counters
made of plastic and lead sheets as shown in Figure 3.11 are installed on the wall of theentrance of SKS magnet and on the pole face of the SKS magnet. In addition, the realdetection rate of photon is not 100%. Since the shape and the size of the counters dependon the installed position, we optimized only the thickness of the photon veto counter. In
the simulation, we assumed that the maximum thickness is 20 mm because the acceptanceof the SKS magnet for the scattered kaons decreases by the photon veto counter. The
thickness of the plastic counter is assumed to be 5 mm, and MPPCs to read a signal areattached at both ends of the plastic. We studied the detection rate of photon by changing
the thickness of lead. Figure 3.19 shows the relation between the detection rate of 100
MeV photon and the thickness of lead (5mm ~ 15mm) in the case of one-layer structure.
As shown in Figure 3.19, the detection rate saturates at about 60% over 10 mm. Therefore,
the thickness should be more than 5 mm (plastic) + 10 mm (lead) = 15 mm if photon
detector has one- layer. In the case of two-layers structure, we studied 3, 4, 5 mm-thicklead (16 mm, 18 mm, 20 mm thick in total). The detection rates are found to be 64 %,56% and 47%, respectively. In conclusion, detection rate of photon detector is 64 % at
maximum with the photon veto counters of two-layers structure. In near future, we will

３２

study 3, 4-layers structure, and more realistic design will be made depending on the
installed position.

Figure 3.19: The relation between the detection rate of 100 MeV photon and the thicknessof the lead (5mm ~ 15mm) in the case of one-layer structure (simulation).

３３

3.5 Study of other background3.5.1 Nonmesonic weak decay of 𝚲
Since a Λ is in a nucleus, non-mesonic weak decay (NMWD) may become background.
NMWD in %&He was measured in the KEK E462 experiment, and the total kinetic energy
of np and nn pair are shown in Figure 1.5. Figure 3.20 (a) and (b) show the angle between
the nucleon pair. The angle between neutron and proton distributes around π (rad). In thepresent GEANT4 simulation, the total kinetic energy of np pair is randomly selected in
the range of 100 − 160 MeV (see Section 1.3), and the pair particles are produced backto back from the target. The number of nn and np pairs is determined according to themeasured branching ratio shown in Table 1.1.

Figure 3.20 (a), (b): Measured angle between neutron-proton pair and neutron-neutronpair in the KEK E462 experiment. The angle distribute at almost π (rad). Thus, we regard
the angle between two nucleons as 180° (back to back in the simulation) [13].

3.5.2 Reduction of nonmesonic weak decay of 𝚲
At first, the background of NMWD is reduced via the cluster analysis in BGO. Figure
3.21 shows the number of clusters in BGO for Λp → np and Λn → nn events. As shownin Figure 3.21, both Λp → np and Λn → nn produce several clusters. By selecting onecluster events, 61% of Λp → np and 67% of Λn → nn events are rejected.

３４

Figure 3.21: The number of clusters in BGO for Λp → np and Λn → nn in thesimulation. Both Λp → np and Λn → nn produce several clusters.
Further reduction of background can be made by the analysis with the plastic counters.
Figure 3.22 shows the energy deposit in the plastic counters for Λp → np and Λn →nn. Both energy loss spectra widely distribute. In addition, Figure 3.23 shows the energy
deposit per path length in the plastic counters. After selecting 0.7 − 3.0 MeV as the beta
decay electron in the total energy loss spectrum in the plastic counter, 99.8% of Λp → np
and 98% of Λn → nn events are rejected by selecting 0.12 − 0.27 MeV/mm as the beta
decay electron.
In addition, 99.5% of Λp → np and 95.4 % of Λn → nn events are rejected by the
analysis in the lucite Cerenkov counters since the velocities of most of the nucleons from
NMWD are small (β < 0.4).

Figure 3.22: The total energy deposit in the plastic counter for Λp → np and Λn → nn inthe simulation. In both cases, the energy distributes widely, and they can be rejected wellby selecting the MIP region for the beta decay electron.

３５

Figure 3.23: The energy deposit per path length in the plastic counter for Λp → np andΛn → nn after selecting 0.7 − 3 MeV as the beta decay electron in the total energy
deposit (simulation).
More analyses are carried out, and then Figure 3.24 shows the total energy spectrum of
the remaining np and nn events, respectively. The rate of nn and np events to the beta
decay electron are 4 % and 2%, respectively. The energy distributes in the region lessthan 40 MeV in both Λp → np and Λn → nn, as shown in Figure 3.24. Remaining events
can be rejected further by cutting events under 40 MeV, but beta decay electron events
are also cut. We plan to determine the energy threshold by considering beta decay electronand other backgrounds.

Figure 3.24: Simulated energy spectra in BGO of remaining Λp → np and Λn → nn
events. The rate of background (Λp → np + Λn → nn) to the beta decay electron signal
is 4% + 2% = 6%.

３６

3.5.3 𝛍% from three body decay of Λ
Since the branching ratio of Λ → pµ# ν| . is 20% of that of the beta decay, muons may
become background. Figure 3.25 shows calculated kinetic energy and velocity
distribution of muons in the Λ → pµ# ν| . decay. As shown in Figure 3.25, the kinetic
energy distributes in 0 − 75 MeV, and the velocity distributes in 0 − 0.8. 74% of muons
are rejected by the lucite Cerenkov counter. We studied the muon background in the
GEANT4 simulation as well as other background and found that 99.4 % of muon events
are rejected. The rate of muon to the beta decay electron signal becomes 20% × 0.006 =
0.12%. In conclusion, muon does not become a problem for the beta decay experiment.

Figure 3.25: Calculated spectra of the kinetic energy and the velocity of muons in theΛ → pµ# ν| . decay. The kinetic energy distributes in 0 − 75 MeV, and the velocity
distributes in 0 − 0.8.

３７

3.5.4 Quasi-free 𝚲
As shown in Figure 3.26 quasi-free Λ (Q-F Λ) production events contaminate the groundstate of %'Li in the missing mass spectrum, and they may produce background for the Λ in&
%He . In the KEK E462 experiment, events with excitation energy −5 MeV ≤ E+0 ≤5 MeV are identified as the production of %'Li ground state. Therefore, we simulated
events momentum of Q-F Λ whose E+0 meets −5 MeV ≤ E+0 ≤ 5 MeV in GEANT4.

Figure 3.26: Missing mass spectra of production of %'Li and quasi-free Λ (Q-F Λ). Q-F Λ
events contaminate the ground state of %'Li.
Figure 3.27 shows the schematic view of Q-F Λ production. Beam π( hits a neutron in'
Li, and then K ( and Q-F Λ are produced. In this reaction, we expect that 'Li is composedof neutron and the residual core R ( &Li ), and they have fermi momenta 𝐏𝐧 and 𝐏𝐑 ,respectively. We used the momentum distribution of the neutron in ,-C measured by,C (e, e/ p) reaction as shown in Figure 3.28 [22].

Figure 3.27: Schematic view of quasi-free Λ (Q-F Λ) production.

３８

Figure 3.28: The momentum distribution of the neutron in ,-C measured by,C (e, e/ p) reaction [22]. The residual core has the same momentum as the neutron.
Conservations of energies and momenta are written as follows.
𝑷𝝅 + 𝑷𝒏 + 𝑷𝑹 = 𝑷𝑹 + 𝑷𝜦 + 𝑷𝑲
𝐸I + 𝐸Y + 𝐸Z = 𝐸Z + 𝐸[ + 𝐸\

(i)
(ii)

By subtracting 𝐏𝐑 , E] (we expect that R is not affected in this reaction), the equations(i), (ii) are given as
𝑷𝝅 + 𝑷 𝒏 = 𝑷 𝜦 + 𝑷 𝑲
𝐸I + 𝐸Y = 𝐸[ + 𝐸\

(i)
(ii).

In this calculation, we adjusted the mass of the neutron to satisfy EC + E] = M^ , whereM^ is the mass of 'Li. The effective mass of neutron MC∗ is given as

𝑀Y∗ = • 𝑀`- + 𝑀Z- − 2𝑀` b𝑀Z- + 𝒑-𝑹 .

To calculate the momentum of Λ in π( n → K ( Λ, we consider the CM system first. Here,Figure 3.29 shows π( n → K ( Λ reaction in Lab and CM systems, respectively.

３９

Lab system

CM system

Figure 3.29: π( n → K ( Λ reaction in the Lab and the CM systems. Considering the CM
system, the momentum of the beam π( and the neutron are the same, and the scatteredK ( and the Q-F Λ also have the same momentum.
Considering the CM system, the momenta of the beam π( and the neutron are the same.
K ( and Q-F Λ also have the same momentum, and their momenta are calculated to be-

𝑀- + 𝑀J- − 𝑀[|𝑃[ | = |𝑃\ | = •‚
ƒ − 𝑀J- .
2𝑀
MK and M% are the mass of K ( and Λ. M is the total energy of the beam π( and the neutronin the CM system. M is the invariant mass under the Lorentz transformation. It is givenas𝑀- = (𝐸I + 𝐸Y )- − (𝑷𝝅 + 𝑷𝒏 )- = 𝑀I- + 𝑀Y∗ - + 2(𝐸I 𝐸Y − 𝑷𝝅 ∙ 𝑷𝒏 ).
MB and MC are the mass of π( and the neutron. We expect that |𝐏𝛑 | is 1.1 GeV/c (K1.1
beamline), and 𝐏𝐧 has a random direction with fermi momentum of |𝐏𝐧 |. Converting theCM system to the Lab system by using beta of CM system 𝜷`b ,𝜷`b =𝑷𝝅 + 𝑷𝒏
𝐸Y + 𝐸I

the momentum of K ( and Λ are calculated. The reaction points of π( n → K ( Λ are
uniformly generated in the Li target, and the direction of 𝐏𝐧 are also uniformly generated.
After calculating, we selected the events of K ( which escaped from the BGO downstream
hole to reproduce (π( , K ( ) reaction in our calculation.
Figure 3.30, 3.31 shows the calculated momentum of Q-F Λ and the reaction point for x,y, z directions. As shown in Figure 3.30, momenta of x,y direction widely distributes, andthat of z distributes over 400 MeV/c. As shown in Figure 3.31, x, y position distributes

４０

Figure 3.30: Calculated momentum distributions of Q-F Λ for x, y, z directions.
Momentum of x and y direction distributes around 0 MeV/c, and momentum of z
distributes more than 400 MeV/c.

Figure 3.31: Calculated distributions of Λ/ s generation poins for x, y, z directions. z
position concentrates at 200 mm.
around 0 mm, and z position concentrates at 200 mm (near BGO downstream hole).
However, we have to consider the Q-F Λ whose excitation energy −5 MeV ≤ E+0 ≤5 MeV, we calculate E+0 via
𝐸7c = b𝑀[- + 𝑀Z- + 2𝐸[ 𝐸Z − 2𝑷𝜦 ∙ 𝑷𝑹 − (𝑀[ + 𝑀Z − 𝐵[ ).
Figure 3.32 shows the excitation energy E+0 of calculated Q-F Λ. We can see that E+0
distributes widely, and it has peak at 0 MeV. We selected −5 MeV ≤ E+0 ≤ 5 MeV, andobtained momenta and reaction points of Q-F Λ which are misidentified as the bound stateΛ. Then, we generated these eventd in GEANT4 and simulated effects from Q-F Λ on

４１

mesurering the beta decay electron. Figure 3.33, 3.34 shows calculated momentum of QF Λ and reaction point for x, y, z directions after selecting −5 MeV ≤ E+0 ≤ 5 MeV.

Figure 3.32: The excitaion energy of calculated Q-F Λ. The energy distributes widelyand concentrates at 0 MeV.

Figure 3.33: Calculated momentum distributions of Q-F Λ for x, y, z directions after
selecting −5 MeV ≤ E+0 ≤ 5 MeV.

Figure 3.34: Calculated distributions of Λ/ s generation points for x, y ,z directions after
selecting −5 MeV ≤ E+0 ≤ 5 MeV.

４２

3.5.5 Reduction of quasi-free 𝚲At first, the background of Q-F Λ is reduced via the cluster analysis in BGO. Figure 3.35
shows the hit pattern and the number of clusters in BGO. By selecting one-cluster events,
82% of Q-F Λ events are rejected.

Figure 3.35: The hit pattern and the number of clusters in BGO for Q-F Λ in thesimulation. Q-F Λ produces several clusters.
In addition, 57% of Q-F Λ events are rejected by the analysis in the lucite Cerenkov
counters. This reduction rate is worse in comparison with that of Λ in %&He. This is because
π# from mesonic weak of Q-F Λ decay has a faster velocity than that of Λ in %&He.
Further reduction of background can be made by the analysis with the plastic counters.
Figure 3.36 shows the energy deposit in the plastic counters. The energy loss spectrum
widely distributes. In addition, Figure 3.37 shows the energy deposit per path length in
the plastic counters. After selecting 0.7 − 3.0 MeV as the beta decay electron in the totalenergy loss spectrum in the plastic counter, 98.1% of Q-F Λ events are rejected by
selecting 0.12 − 0.27 MeV/mm as the beta decay electron.

４３

Figure 3.36: The total energy deposit in the plastic counter for Q-F Λ in the simulation.
The energy distributes widely.

Figure 3.37: The energy deposit per path length in the plastic counter for Q-F Λ after
selecting 0.7 − 3 MeV as the beta decay electron in the total energy deposit (simulation).
More analyses are carried out, and then Figure 3.38 shows the total energy spectrum of
the remaining Q-F Λ events. The rate of Q-F Λ events to the beta decay electron is 11.9%.
However, as shown in Figure 3.38 the energy distributes in the region more thanmaximum kinetic energy of beta decay electron (163 MeV), thus remaining events canbe rejected further by selecting the energy less than 163 MeV. By this selection, 16% ofremaining Q-F Λ events are rejected. This corresponds to 7% background rate for the beta
decay electron.

４４

Figure 3.38: Simulated energy spectrum in BGO of remaining Q-F Λ events. The rate of
the background to the beta decay electron signal is 14.3%. By selecting the energy lessthan 163 MeV, the rate is reduced down to 10.5%.
To reduce more remaining events, we used the theta segment number which is the centerof one-cluster in the BGO as shown in Figure 3.39. We can see that the center theta
segment number distributes around #14 (downstream of BGO) due to the high z
momentum of Q-F Λ. Thus, if events with high theta number hit is rejected, moreremaining events can be rejected. However, this selection also cuts the beta decay electronwith uniform theta number distribution. When #14 hit events are rejected as the Q-F Λ
hit, the background rate is reduced down to 5.9% (6.7% beta decay electron events are
rejected, and the statical error become 3.8% (no cut) → 3.99%). Furthermore, when #13
and #14 hit events are rejected, the background rate is reduced down to 5.5% (13.4% beta
decay electron events are rejected, and the statical error become 3.99% → 4.14%).
Therefore, #13 and #14 reduction is the limit of Q-F Λ considering the goal of staticalerror of the beta decay electron (4%). In conclusion, we found that Q-F Λ events do not
become a severe problem by rejecting high theta number hit in the BGO. In near future,
we will simulate Q-F Λ by considering the mass resolution of hypernucleus.

４５

Figure 3.39: The theta segment number of the center of one-cluster in the BGO.

４６

3.6 Effects from beam 𝛑$
Since the intensity of beam π( is high (~10M /s), a lot of particles from interactions ofbeam π( and the Li target are expected to enter the BGO crystal in a high counting rate.
BGO is an inorganic scintillator with a large atomic number, and the time response isquite slow (the decay time constant scintillation is ~ 1 µs). Thus, the BGO detector cannot
operate correctly at a rate more than 105 counts per sec. Therefore, we studied beam π(effects on the BGO counting rate by a GEANT4 simulation. Figure 3.40 shows the countsof each of the BGO segments with more than 1 MeV energy loss for various theta
positions when 105 π( beam particles are injected to the target. The number of BGO
segments are 225. When 105 π( ′s are produced in this simulation, more than 103 counts
in a BGO segment correspond to 105 counts per sec in a real experiment. As shown in
Figure 3.40, most of BGO segments have counts over 103.

Figure 3.40: BGO counting rate with more than 1 MeV as a function of segement theta
number in the simulation.
Figure 3.41 shows the fraction of the number of BGO segments with more than 103
counts (n) out of all the BGO segments of 225 (N) as a function of the energy thresholdin the simulation. We found that the fraction can be reduced to 24% when the energy
threshold is 10 MeV. Figure 3.42 shows fraction of the BGO segments with more than
103 counts (n) out of all the BGO segments of 225 (N) as a function of the total numberof BGO segments in the simulation. We found that the fraction can be reduced to 15%

４７when the number of segments is 400. In conclusion, with 10 MeV energy threshold and
400 BGO segments, 15 % BGO segments have more than 103 counts. These segments
should be replaced by fast response scintillator in the real experiment.

Figure 3.41: Fraction of the number of BGO segments with more than 103 counts (n) out
of all the BGO segments of 225 (N) as a function of the energy threshold in the simulation.

Figure 3.42: Fraction of the number of BGO segments with more than 103 counts (n) out
of all the BGO segments of 225 (N) as a function of the number of BGO segments in thesimulation.

４８

4 Feasibility of the beta decay experiment
4.1 Yield and accuracy of branching ratio of beta decay
Via the background suppression study in Chapter 3, the background rate from the Λ’smain decay mode can be reduced down to 4% (mesonic) + 6% (nonmesonic) + 5.5% (QF) = 15.5% of the beta decay signal with the cut of 30 % beta decay electron events. if
the remaining background can be subtracted within 30% accuracy, the systematic errorof the background for the beta decay electron become ~ 5% (30% is roughly estimated).
In order to measure the beta-decay branching ratio with a statistical error of 4% (the sameas the expected systematic error of ~ 4% from the background), more than 625 counts ofthe beta decay electron events should be measured. Table 4.1 shows the yield of the KEK
462 experiment and the expected yield of the proposed beta decay experiment. The E462
experiment was carried out with 2.5 × 10,- π( on a 3.7g/cm$ 'Li target, and 45653 %&He
events were obtained. Since our beta decay experiment will use the same setup for the K (spectrometer as the KEK E462 experiment, we can estimate the number of %&He eventsfrom the E462 data. Since 30% beta decay electron events will be cut away in the data
analysis, according to the GEANT4 simulation described above, we estimate the beta
decay electron efficiency is 0.7. Considering these factors, we found that if 29 × 10,π( enter a 14 g/cm$ 'Li target, 673 counts of the beta decay electron can be obtained.
Then the statical error will be ~3.8%. When we assume that the number of π( is3.0 × 10M /spill and the accelerator operation cycle is 5.2 second, the total beam time of
5.5 × 10M second (~1400 hours) is needed.
Table 4.1: Expected yield of the %&He and its beta decay events estimated by the %&He yieldmeasured in the KEK E462 experiment and a GEANT4 simulation.

４９

4.2 Yield and accuracy of lifetime
The lifetime of a Λ in %&He is measured by the same method as the KEK E462 experiment
which measured the %&He lifetime with 4.0% statical accuracy. As written in Chapter 2,
the lifetime of a Λ is measured by two timing counters around the Li target. One measuresa timing of beam π( just upstream of the target, and the other surrounding the target
measures that of a proton from nonmesonic weak decay of %&He. To measure the decay
time spectrum of the proton from Λ’s non mesonic weak decay (Λp → np) within a 2.0%
accuracy, as shown in Table 4.2, we need to measure the four times larger number ofprotons than the E462 experiment, assuming that the resolution of the timing counters
and the accuracy of the track length measurement are the same or better than E462. Since,
the branching ratio of Λp → np is much larger than that of the beta decay electron, thislifetime measurement does not need a long beamtime. If we request 120 hours beamtime,4768 protons events can be detected, which satisfies the statistical error less than 2.0%.
Table 4.2: Expected yields of the %&He and proton events from their nonmesonic weak
decay and the expected lifetime accuracy, estimated in comparison with the KEK E462
experiment.

５０

5 Summary and future prospect
The possible modification of baryon structure in nuclear matter by the interaction
between the quarks in the baryon and the meson field in nuclear matter has been discussed
for a long time. Only the EMC effect indicates experimental evidence for baryon
modification in nuclear matter, but its mechanism has not been clearly explained. To
measure the modification caused by the meson fields, we propose an experiment formeasuring the beta decay of Λ hypernuclei. The QMC (Quark Meson Coupling) model
predicts that the axial charge g ! of a Λ reduces by 10% at maximum in nuclear matter,and the beta decay of a Λ reduces by 20% at maximum. To separate the baryon
modification effects from nuclear many-body effects and hadronic effects, we plan to use
a light hypernucleus, %&He . We expect that %&He hypernuclei will be produced via the'Li(π( , K ( ) %'Li, %'Li → %&He + p reaction at J-PARC K1.1 beamline, and the beta decayrate will be determined with a 4.5% accuracy, by measuring the branching ratio and thelifetime determined with 4% and 2%, respectively.
In the measurement for the Λ’s weak decay lifetime, we will use the same setup as the
KEK E462 experiment. We expect that the lifetime will be measured with less than 2%
accuracy by measuring the time difference between the beam pion and the weak decay
proton.
In the measurement of the branching ratio, a huge background of π* and π# from the
Λ’s main decay modes should be reduced down to the order of 1% of the beta decay
electron signal. For this purpose, we designed apparatus around the 'Li target made ofplastic and lucite Cerenkov counters together with a BGO 4π calorimeter and studied how
to remove the background by simulation using the GEANT4 code. In the analysis for the
BGO calorimeter, by selecting one-cluster hit events as the beta decay electron, 97% of
π* and 92.8% of π# can be rejected. In the analysis for the plastic counter, by selecting a
MIP peak region in the energy loss, 88% of π* and 99.9% of π# can be rejected. In
addition, 95.5% of π# and 90% of π* events are rejected by the lucite Cerenkov counter.
Via all the background suppression study, the background rate for the beta decay electron
is found to be reduced down to 4%.
In the case of nonmesonic weak decay, Λp → np and Λn → nn events are found to be
reduced well by the plastic counters and the lucite Cerenkov counters due to the low
momentum of emitted nn and np. Finally, Λp → np and Λn → nn events are reduced to
2% and 4% of the beta decay electron signal.

５１

In the case of quasi-free Λ (Q-F Λ), the background rate for the beta decay electron is
reduced down to 11.9%. In addition, by the cut of total energy loss in the BGO and thehigh theta number hit, the rate become 5.5%.
Roughly estimating, the systematic error of the background for the beta decay electron
become ~ 5% by subtracting the background within 30% accuracy.
Furthermore, we made simulations for optimization of the number of the BGO segments
and the BGO thickness in order to reduce the preparing cost and the time. We found thatthe number of BGO segments should be larger than 100 because the analysis of cluster
does not work well in the fewer BGO segments and the thickness should be larger than
20 cm, since more π# events produce one cluster in the BGO for the smaller BGO
thickness due to the reduction of the reaction rate of neutrons.
In addition, in order to detect leak photons from π* → γγ through the BGO downstream
hole, the simulation for the photon veto counter to be installed around the SKS magnet
was made. We found that by using two-layers structure ((5 mm lead and 5 mm
plastic)× 2), 64 % leak photon can be detected.
We also estimated the yield and accuracy for measuring the beta decay rate, and 1400
and 120 hours beamtime are found to be needed to achieve statistical errors of 4% and
2% for measuring the branching ratio and the lifetime. Therefore, we found that this beta
decay experiment is found to be feasible.
In near future, we will make a more realistic background reduction simulationconsidering the energy resolution of BGO counters. Most of the present results have beenreported in Letter-Of-Intent submitted to J-PARC [20]. After the detailed studies of thedetectors by further simulations, we will write and submit a proposal of the experiment
to the J-PARC PAC.

５２

Appendix A
Beta decay
A.1 Beta decay theory
The theory of the beta decay was constructed by Fermi in 1934. The probability of thebeta decay 𝑤 is given as
𝑤 =

2𝜋
2 𝑑𝑛
Š𝑀𝑖𝑓 Š
.
ℏ
𝑑𝐸𝑒This is called Fermi’s golden role. 𝑀de is the matrix element of the initial state and thefinal state of the nucleus and 𝑑𝑛/𝑑𝐸7 is the density of states of the electron. The
probability of the beta decay with the electron energy in the range 𝐸7 ~ 𝐸7 + 𝑑𝐸7 is givenas(/( 01)* 0

𝑤 𝑑𝐸. = 23+ ℏ, 5 + 𝐹 (𝑍, 𝑝) 𝐸. (𝐸6 − 𝐸. )2 0𝐸.2 − 𝑚.2 𝑑𝐸. ,where g is the coupling constant. F (Z, p) is a factor of an effect from the Coulomb
potential, and 𝐸* is the sum of the Q value and the electron mass.
The decay probability 𝜆 is calculated by integrating as--

𝑔- Š𝑀de Š
𝑔- Š𝑀de Š- “𝐸 (𝐸)𝜆=Ž𝐹(𝑍, 𝑝) 𝐸7 * − 𝐸7𝑓,
7 − 𝑚7 𝑑𝐸7 =$ M $2𝜋 $ ℏM 𝑐 $g& 2𝜋 ℏ 𝑐
f%

f%

𝑓= Ž

g&

𝐹(𝑍, 𝑝) 𝐸7 (𝐸* − 𝐸7 )- “𝐸7- − 𝑚7- 𝑑𝐸7 .

The half-life t is given as
𝑡=

𝑙𝑜𝑔2
𝑙𝑜𝑔2
=
𝜆
𝑔- Š𝑀 Š,de𝑓
2𝜋 $ ℏM 𝑐 $∴ 𝑓𝑡 =

2𝜋 $ ℏM 𝑐 $𝑔- Š𝑀de Š

５３- 𝑙𝑜𝑔2 .

This 𝑓𝑡 value is often used in the beta decay. By measuring the half-life 𝑡 and by
calculating 𝑓, and if we know the matrix element 𝑀de , the coupling constant of the weak
decay g is obtained. To calculate the matrix element, Hamiltonian density 𝐻h should beconsidered. At first, Fermi gave
𝐺h i
𝐻h =
•𝑙 (𝑥)𝑁ij (𝑥) + 𝑁i (𝑥)𝑙ij (𝑥) .
√2
𝐺h is the coupling constant of the beta decay. 𝑙 i and 𝑁ij are the lepton and nucleon
currents given as
𝑙 i = 𝜓<7 (𝑥)𝛾i 𝜓k (𝑥), 𝑁ij = 𝜓<l (𝑥)𝛾i 𝜓Y (𝑥).
Therefore, Hamiltonian density is
𝐻h =

='
√-

(𝜓<7 (𝑥)𝛾 i 𝜓k (𝑥)𝜓<l (𝑥)𝛾i 𝜓Y (𝑥) + 𝜓<Y (𝑥)𝛾 i 𝜓l (𝑥)𝜓<k (𝑥)𝛾i 𝜓7 (𝑥)).

In the 𝜓<7 (𝑥)𝛾 i 𝜓k (𝑥)𝜓<l (𝑥)𝛾i 𝜓Y (𝑥) part, 𝜓<7 ,𝜓<l and 𝜓k represent the generation of anelectron, a proton and an anti-neutrino respectively. 𝜓Y represents extinction of a neutron.
This part corresponds to 𝛽 # decay (𝑛 → 𝑝𝑒 # 𝜈̅7 ).
The 𝜓<Y (𝑥)𝛾 i 𝜓l (𝑥)𝜓<k (𝑥)𝛾i 𝜓7 (𝑥) part corresponds to 𝛽 ( decay (𝑝 → 𝑛𝑒 ( 𝜈7 ) andelectron capture (𝑝𝑒 # → 𝑛𝜈7 ).
The matrix element 𝑀de is given by
𝑀de = Ž 𝑑 $ 𝑥 ¦𝑓Š𝜓<7 (𝑥)𝛾 i 𝜓k (𝑥)𝜓<l (𝑥)𝛾i 𝜓Y (𝑥) + 𝜓<Y (𝑥)𝛾 i 𝜓l (𝑥)𝜓<k (𝑥)𝛾i 𝜓7 (𝑥)Š𝑖¨= 𝑀; .
Here, |𝑖⟩ and |𝑓⟩ are the initial state and the final state of the nucleus, respectively. In thistransition, the spin of the nucleus 𝐼 does not change (Δ 𝐼 = 0). In addition, the angularmomentum 𝐽 and the parity 𝜋 also do not change. This beta decay is called “Fermi
transition”. If the spin of the nucleus changes in the beta decay, 𝑀; becomes 0. Fermi’s
Hamiltonian density above cannot describe the beta decay with Δ 𝐼, Δ 𝐽 ≠ 0 and thechange of the parity. Later, the beta decay with changes of the spin and the angularmomentum was found, and it is called “Gamov-Teller transition”. Therefore, theHamiltonian density including both Fermi transition and Gamov-Teller transition wereneeded. Fermi’s Hamiltonian is composed of a scalar product of two vector currents tosatisfy invariance under the Lorentz transformation. The lepton current and nucleon
current should be four-vectors under the Lorentz transformation. For the types of thecurrents in the Hamiltonian, various types of currents such as 𝜓<𝜓 (Scalar), 𝜓<𝛾 i 𝜓

５４(Vector), 𝜓<𝛾 i 𝛾 k 𝜓 (Tensor), 𝜓<𝛾 & 𝜓 (pseudoscalar), 𝜓<𝛾 i 𝛾 & 𝜓 (Axial vector) (called as S,
V, T, P, A, respectively) are allowed. Considering the invariance under the Lorentz
transformation, 𝐻h is expected to be made of coupling of S-S, V-V, T-T, P-P, S-P, V-A,
A-A. Which current combination is correct was determined by experiments.

A.2 Beta decay model under V−A
By experiments, the lepton and nucleon currents in the beta decay was found to be
described as V – A. The Fermi’s beta decay is described by Vector part of the nucleon
current, and the Gamov-Teller’s beta decay is described by Axial vector part of thenucleon current. Therefore, Hamiltonian density is given by linear combination of thevector and the axial vector nucleon currents as𝐺h𝐻h =-d𝜓<l 𝛾i 𝜓Y h(𝜓<7 (𝐶: + 𝐶:/ 𝛾& )𝛾 i 𝜓k ) + d𝜓<l 𝛾i 𝛾& 𝜓Y h(𝜓<7 (𝐶< + 𝐶</ 𝛾& )𝛾 i 𝛾 & 𝜓k )√2
+ ℎ. 𝑐 °,where h.c. means Hermitian conjugate. Since the neutrino mass is regarded as 0 and thehelicity of neutrino is negative,
𝐶: = 𝐶:/ ,𝐶< = 𝐶</1
1(1 − 𝛾& )𝜓k = 𝜓k ,(1 + 𝛾& )𝜓k = 0 .
2
2
Hamiltonian density is given by

𝐻h =

𝐺h
√2

-(𝜓<7 𝛾 i (1 − 𝛾& )𝜓k )d𝜓<l 𝛾i (𝐶: − 𝛾& 𝐶< )𝜓Y h + ℎ. 𝑐. °.

Then,
|𝑀de

|-

$

= ±Ž 𝑑 𝑥 ²𝑓±𝐺h√2-(𝜓<7 𝛾 i (1 − 𝛾& )𝜓k )d𝜓<l 𝛾i (𝐶: − 𝛾& 𝐶< )𝜓Y h + ℎ. 𝑐. °±𝑖³±-

Š𝑀de Š = 𝐶:- 𝑀;- + 𝐶<- 𝑀=>.

Therefore, the decay rate is-𝐺h- Š𝑀de Š
𝐺h- (𝐶:- 𝑀;- + 𝐶<- 𝑀=>)
𝛤6789 =
𝑓=𝑓.
2𝜋 $ ℏM 𝑐 $2𝜋 $ ℏM 𝑐 $５５-Here, the axial vector charge 𝑔< and the vector charge 𝑔: are defined as𝑔< = 𝐶< 𝐺h ,𝑔: = 𝐶: 𝐺hFinally, the decay rate is written as
𝐺h- (𝐶:- 𝑀;- + 𝐶<- 𝑀=>)
𝑔:- 𝑀;- + 𝑔<- 𝑀=>
𝛤6789 =
𝑓=𝑓,
2𝜋 $ ℏM 𝑐 $2𝜋 $ ℏM 𝑐 $and therefore,
𝛤6789 ∝ 𝑔:- 𝑀;- + 𝑔<- 𝑀=>.

A.3 𝐠 𝐕 and 𝐠 𝐀 values
In the quark and lepton level, g 1 and g ! are g 1 = 1 and g ! = −1, respectively. In the
hadronic level, a quark in the nucleon is affected via the strong interaction, and in fact,
the axial charge g ! is changed in the nucleon. In contrast, the vector charge g 1 isunchanged, and not affected by the hadronic effects. It is called Conserved Vector Current(CVC). The measurements for g ! and g 1 of neutron in the free space were made andfound to be g ! /g 1 = −1.2724 ± 0.0023 [2].
The measurements for g ! and g 1 of neutron in the nucleus was also made. The g ! valueof neutron was found to be quenched in a nucleus as shown in Figure 2.1. For a lightnucleus, $H, quenching effect is ~ 5%, but g 1 is unchanged due to CVC.

５６

Appendix B
J-PARC K1.1 beamline
B.1 J-PARC
J-PARC (Japan Proton Accelerator Research Complex) is a facility for studying nuclear
and particle physics and material and life science. At J-PARC, by using high intensity
proton beams, various secondary particles (ex. muon, neutron, neutrino, pion, and kaon)
can be produced, and various experiments are carried out. Figure B.1 shows a schematic
view of J-PARC. Linac (50 mA) accelerates protons to 400 MeV and Rapid Cycling
Synchrotron (RCS) (333 µA) accelerates protons to 3 GeV. Accelerated protons aretransported to Material and Life Science Facility (MLF) and Main Ring (MR). They areaccelerated to 30 GeV by MR (15 µA) and transported to Hadron Experimental Facility.

Figure B.1: A schematic view of J-PARC [23]. See text for details.

５７

B.2 J-PARC Hadron Experimental Facility
Figure B.2 shows a schematic view of the J-PARC Hadron Experimental Facility. Greenshows the current beamlines, and orange shows beamlines planned to be constructed. At
the Hadron Experimental Facility, 30 GeV protons are bombarded on the T1 target, thenπ and K mesons are produced, and they are transported to K1.8, K1.8BR and KL
beamlines. A part of the protons are also transported to High-p and COMET beamlines.

Figure B.2: A schematic view of the J-PARC Hadron Experimental Facility [24].
Accelerated protons are bombarded to the T1 target, and 𝜋 meson, K meson and otherparticles are produced and transported to secondary beamlines.

B.3 J-PARC K1.1 beamline
The K1.1 beamline is optimized for production of Λ and Σ hyperons (S = −1). It will
be constructed in near future in the current facility and then moved to the extended partof the facility. At the K1.1 beamline, the E63 experiment will be carried out. The E63
experiment will measure possible modification of the g-factor of a Λ in a nucleus, and the
motivation is similar to the proposed beta decay experiment. Figure B.3 shows theoverview of the K1.1 beamline. This beamline provides 1.05 GeV/c π( beam suitablefor Λ hypernuclear production with the expected intensity of ~ 3.0 × 10M π( /spill.
However, produced particles from the T1 target include not only π( but also othersecondary particles and impurity particles. Impurity particles are produced due to decay

５８

of secondary particles and interactions between beam particles and materials in thebeamline. To increase purity of π( , there are several devices such as IF, ESS, CM andMS.
The momentum of secondary beam particles is selected by dipole magnets (D1, D3) inthe beamline. Intermediate focus point (IF) is designed to make the secondary beam
particles focus but impurity particles not focus. After IF, two sets of electrostatic
separators (ESS) are installed, and they separate particles with different velocities byelectronic field. Compensation Magnets (CM) are installed in front of and behind each
ESS, and they change focusing point of the beam particles. After ESS and CM, particles
with a specific velocity, consequently with a specific mass, are selected by Mass Slit (MS).
Finally, high purity π( beam is transported to the K1.1 beamline spectrometer.

Figure B.3 A schematic view of the K1.1 beamline at J-PARC Hadron Experimental
Facility [25].

５９Appendix C
The modification of 𝚲 → 𝒏𝜸 decay rate
C.1 𝚲 → 𝒏𝜸 decay rate in nuclear matter
QMC model predicts that spatial distribution of u and d quarks is modified in a nucleus.
This modification affects not only the beta decay rate but also the decay rate of Λ → nγ.
Figure C.1 shows the diagram of Λ → nγ decay. When this decay occurs, an s quarkchanges to a u quark and a photon is emitted. We expect the decay rate of Λ → nγ may
also decrease. Although a calculation has not been done yet, we propose a measurementof the Λ → nγ decay rate together with the beta decay rate measurement.

Figure C.1: A diagram of Λ → nγ decay. In this decay, an s quarks changes to a u quark,and a photon is emitted. Since the decay rate of Λ → nγ depends on overlap of the initialstate s quark wavefunction and the final state u quark wavefunction, the decay rate may
decrease for the same reason as the beta decay.
The energy of the emitted photon from Λ → nγ is calculated to be about 159.4 MeV
considering the binding energy of Λ in %&He of ~ 3.1 MeV [26]. We studied backgroundrate for Λ → nγ by GEANT4 simulation. In our simulation, a photon with 159.4 MeV is
produced in random directions from the target, and π* and π# from %&He mesonic decay
are produced as background in the same way as the background study for the beta decay
electron measurement.

６０C.2 Cluster analysis in the BGO calorimeter
When one photon from Λ → nγ enters the BGO calorimeter, it is expected to produce
one cluster. Figure C.2 shows the number of clusters in BGO. As shown here, Λ → nγ
produces one cluster in BGO, and π* and π# produce two and several clusters,
respectively. By selecting one cluster events, 97% of π* and 92.8% of π# events are
rejected.

Figure C.2 The number of clusters in the BGO for Λ → nγ, π* and π# in the simulation.

C.3 Analysis in the plastic counters
Figure C.3 shows the total energy deposit in the plastic counter. As shown in Figure C.3,
in the case of Λ → nγ and π* , the energy deposit mostly distributes under 0.2 MeV
because a photon passes through. The energy distribution of π# has a peak at 3 MeV. By

Figure C.3 : The total energy depositin the plastic counter for Λ → nγ, π*
and π# in the simulation. In the caseof Λ → nγ and π* , energy deposit isalmost 0 due to non-charged particle.
Energy deposit for π# has a peak at 3
MeV.
６１

selecting < 0.2 MeV events as Λ → nγ, 96.5% of π# events can be rejected.

C.4 Analysis of the total energy deposit in BGO
Figure C.4 shows the total energy deposit of Λ → nγ, π* and π# in BGO after the
analyses described above. As shown in Figure C.4, the Λ → nγ spectrum shows a clear
peak at 159.4 MeV. In the case of π* , the total energy distributes at 30 − 140 MeV. Thisis because one photon from π* → 2γ enters BGO and the other photon escapes from BGO.
In the case of π# , the energy widely distributes due to interaction between neutrons fromπ# and nuclei in BGO. Via the background reduction above, Λ → nγ events can be
distinguished from π* and π# events by selecting energy around a 159.4 MeV peak. By
selecting events with 143 MeV −159.4 MeV in BGO as Λ → nγ, 99.6% of π* and 99.9%
of π# events are rejected. Figure C.5 shows a total energy spectrum of Λ → nγ (blue) andπ* +π# (red). Via the analysis of the detectors, we can separate Λ → nγ from backgroundwell. The background rate from π* and π# is about 0.24% of the Λ → nγ signal. The
reduction rate of Λ → nγ events is about 15%. Statical error of the branching ratio of Λ →nγ is expected to be about 3.4%.

C.5 Summary
We studied accuracy for measuring the Λ → nγ decay rate. Via the simulation, thebackground rate for the Λ → nγ signal and the statical error of the branching ratio arefound to be 0.24% and 3.4%, respectively. Therefore, measurements for Λ → nγ together
with beta decay is found to be feasible. In addition, the measurement of the Λ → nγ decayrate is easier than that of the beta decay, if the BGO energy resolution is sufficient toseparate the Λ → nγ peak at 159.4 MeV from the π* spectrum up to 140 MeV.

６２

Figure C.4: The total energy deposit in the BGO for Λ → nγ, π* and π# in the simulation.
The Λ → nγ spectrum shows a clear peak at 159.4 MeV. For π* and π# , the BGO energy
spectrum widely distributes. Right bottom figure shows the energy of Λ → nγ (blue) andthe main decay modes (red).

Figure C.5: Simulated energy spectrum of the photon from Λ → nγ decay in thesimulation. A clear photon peak from Λ → nγ is seen, and the background level isextremely low. The background rate from π* and π# for the Λ → nγ signal is about
0.24%.

６３

Reference[1] D. F. Geesaman, K. Saito, A. W. Tohmas, Ann. Rev. Nucl. Part. Sci. 45 (1995) 337.
[2] M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018)[3] P. A. M. Guichon, A.W. Thomas, Phys. Lett. B773 (2017) 332.
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[26] Satoshi N. Nakamura, EPJ Web of Conferences, 66 01020 (2014).

６４

謝

辞

本研究を遂⾏するにあたり、多くの⽅々からご指導頂来ましたことに、⼼から感謝申し上げます。
指導教員である⽥村裕和教授には、物理学の基礎から、使⽤する検出器・シミュレーションの⽅法について、丁寧なご指導を賜り、多いに⽣かす事ができました。深夜のメールや突然の部屋への訪問においても、⼼よくお時間を取って熱⼼にアドバイスをして下さいました。誠にありがとうございました。これ
からも、実験のための proposal やその準備等において、これまで同様ご迷惑をおかけするかもしれませんが、何卒よろしくお願いいたします。
JAEA の藤⽥真奈美博⼠研究員には、４年次の頃より、GEANT4 シミュレー
ションを始めるにあたって、プログラミングに疎かった私にコードの書き⽅から丁寧に教えて下さいました。藤⽥⽒には感謝しても仕切れません。誠にありがとうございました。
中村哲教授、三輪浩司准教授、早川修平助教におかれては、多くの助⾔とご指導を賜り、⼤変お世話になりました。誠にありがとうございました。
東海村の J-PARC ハドロン実験施設への出張やミーティングの際には、JAEAの⾕⽥聖⽒、⼭本剛史⽒、市川裕⼤⽒、KEK の⾼橋俊⾏⽒、鵜養美冬⽒、東北⼤学の⽯川勇⼆⽒、叶内萌⾹⽒、京都⼤学の後神利志⽒、原⽥健志⽒、東京⼯業⼤学の滝鷹介⽒、また J-PARC スタッフの皆様には、数々の助⾔を頂き、検出器・ハイパー核について丁寧に教えて下さいました。⼤変ありがとうございました。
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GPPU 試験合格のための極意を教えていただき、その結果無事合格することができました。感謝申し上げます。また和⽥俊輔⽒、北岡智真⽒、⼤浦⽂也⽒、
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最後に、本課程まで応援し続けてくれた家族、友⼈、本当にありがとうございました。
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令和４年 2 ⽉ 10 ⽇
鎌⽥

６５

健⼈