Diﬀerential cross section measurement andphase shift analysis for Σ+p elastic scattering

Takuya Nanamura

Abstract
By investigating interactions between octet baryons, we can extend the nuclear force
based on the ﬂavor SU(3) symmetry and reveal the role of dynamics of quarks and gluons
in the short-range part of baryon-baryon interaction (BB interaction). In particular, it
is expected that ΣN interaction is strongly aﬀected by the Pauli principle at the quark
level (quark Pauli eﬀect). A larger repulsive core is expected to be observed in the 3 S1channel of the Σ+ p system than in the core of nuclear force. Unfortunately, the predicted
strengths of the repulsive force changes so much according to theoretical models. Reliable
experimental input is needed to understand the BB interactions. A scattering experimentis the most direct method to investigate the two-body BB interactions. However, hyperonnucleon scattering experiments have been diﬃcult for a long time due to a short lifetimeof hyperons, typically 10−10 s. The past scattering data have a large uncertainty due topoor statistics.
Recently, we successfully performed a novel Σ± p scattering experiment with highstatistics of more than a few thousands of the scattering events at the J-PARC Hadron
Experimental Facility by 2020 June. In this experiment, we measured the diﬀerential
cross sections for the Σ± p elastic scattering and Σ− p → Λn inelastic scattering. In thisthesis, the data taking and results of the Σ+ p elastic scattering is reported.
A liquid hydrogen target was used for both a Σ+ production target and Σ+ p scatteringtarget. In the former process, Σ+ particles were produced via the π + p → K + Σ+ reaction
induced by a high intensity π + beam with a momentum of 1.41 GeV/c. Momentum rangeof the produced Σ+ were from 0.44-0.80 GeV/c in the laboratory frame, where typicaldistances between two particles in the Σ+ p scatterings are 0.8-0.5 fm and they would
overlap each other. In total, 4.9 × 107 Σ+ production events were tagged. The recoil
proton from the Σ+ p scattering and proton from the Σ+ → pπ 0 decay were detectedwith a detector system CATCH, surrounding a target. The Σ+ p scattering events wereidentiﬁed by the kinematical consistency check method. In total, approximately 2400 Σ + p
elastic scattering events were identiﬁed.
The diﬀerential cross sections of the Σ+ p scattering in a momentum range 0.44 to 0.80
GeV/c were derived, with the small uncertainties of approximately 15% and a ﬁne angularstep of ∆ cos θCM = 0.1. Owing to precise data points and the simple representation ofthe Σ+ p system with respect to the ﬂavor SU(3) symmetry, we could reveal the strengthof interaction in the 3 S1 and 1 P1 states quantitatively with the phase shift analysis. The
phase shift of the 3 S1 state, where a large repulsive core was predicted owing to the quarkPauli eﬀect, was determined with a precision of approximately 4 degrees for each of threemomentum regions. The absolute values of the phase shift for the 3 S1 state range from20◦ < |δ3 S1 | < 35◦ in the present momentum range, which suggests the repulsive force isrelatively moderate. Since theoretical models describing BB interaction has been built on
little hyperon-proton scattering data, these data will be essential inputs in constructing
realistic BB interaction models.

Contents
1 Introduction
1.1 Baryon-Baryon interaction and importance of ΣN interaction . . . . . . .
1.1.1 Quark Pauli eﬀect in the quark cluster model (QCM) . . . . . . . .
1.2 Equation of state for neutron stars, hyperon puzzle, and three body force .
1.3 Theoretical Models for BB interaction . . . . . . . . . . . . . . . . . . . .
1.3.1 Nijmegen Extended-soft-core Model (ESC) . . . . . . . . . . . . . .
1.3.2 Chiral Eﬀective Field Theory (χEFT) . . . . . . . . . . . . . . . . .
1.3.3 BB interaction including QCM . . . . . . . . . . . . . . . . . . . .
1.3.4 Lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.5 Theoretical predictions for ΣN (I = 3/2,3 S1 ) interaction . . . . . .
1.4 Previous experimental studies on ΣN and Σ-nucleus interaction . . . . . .
1.4.1 Σ± p scattering experiment . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Σ hypernuclei and quasi-free Σ production . . . . . . . . . . . . . .
1.4.3 Σ− atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.4 Femtoscopy technique . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Present experiment: J-PARC E40 experiment . . . . . . . . . . . . . . . .
1.6 Thesis composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Experiment
2.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 J-PARC and Hadron Experimental Facility . . . . . . . . . . . . . . . . . .
2.3 K1.8 beam line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 K1.8 beam-line spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Hodoscopes in the K1.8 beam-line spectrometer . . . . . . . . . . .
2.4.2 Trackers in the K1.8 beam-line spectrometer . . . . . . . . . . . . .
2.5 KURAMA spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Detectors in the KURAMA spectrometer . . . . . . . . . . . . . . .
2.6 Liquid-hydrogen (LH2 ) target . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 CATCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Detectors of CATCH . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Data Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Analysis ﬂow and Monte Carlo simulation
40
3.1 Analysis ﬂow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1

4 Analysis-I:Analysis of spectrometers and CATCH
44+
4.1 π analysis using the K1.8 beam-line spectrometer . . . . . . . . . . . . . 44
4.1.1 BFT analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.2 Local Tracking for BCout (BCout Tracking) . . . . . . . . . . . . . 46
4.1.3 K1.8 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 K + analysis using the KURAMA spectrometer . . . . . . . . . . . . . . . . 48
4.2.1 SDCin Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2 SDCout Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.3 KURAMA Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.4 K + identiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Σ+ identiﬁcation with (π + , K + ) analysis . . . . . . . . . . . . . . . . . . . 56
4.4 CATCH for the recoil/decay protons . . . . . . . . . . . . . . . . . . . . . 59
4.4.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.2 CFT Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.3 Energy measurement with CFT and BGO . . . . . . . . . . . . . . 64
4.4.4 Particle identiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5 Evaluations of the system speciﬁcation by combined analysis . . . . . . . . 68
4.5.1 The vertex resolution evaluation of the spectrometer system using
the π + p → π + π + π − p reaction . . . . . . . . . . . . . . . . . . . . . 68
4.5.2 The momentum resolution evaluation of KURAMA using the π + pelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.5.3 The detection probability estimation for accidental proton by CATCH 71
5 Analysis-II:Identiﬁcation of the Σ+ p scattering events
5.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Kinematical consistency check method . . . . . . . . . . . . . . . . . . . .
5.2.1 Kinematical consistency check for the decay proton from Σ + → pπ 0
decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Kinematical consistency check for the decay proton of pp scattering
following the Σ+ → pπ 0 decay . . . . . . . . . . . . . . . . . . . . .
5.2.3 Kinematical consistency check for the recoil proton from the Σ + p
scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Cut conditions to select the Σ+ p scattering events . . . . . . . . . . . . . .
5.3.1 Vertex cut and closest distance cut . . . . . . . . . . . . . . . . . .
5.3.2 Missing mass cut to select the scattered Σ+ decay . . . . . . . . .
5.3.3 Kinematical cut for secondary pp scattering . . . . . . . . . . . . .
5.3.4 Kinematical cut for π + p elastic scattering . . . . . . . . . . . . . .
5.3.5 Σ+ p scattering identiﬁcation after all cuts . . . . . . . . . . . . . .
5.4 Estimation of the number of Σ+ p scattering events . . . . . . . . . . . . .

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6 Analysis III: Evaluations of factors to derive the diﬀerential cross sections
88
6.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Total track length of the incident Σ+ in the LH2 target . . . . . . . . . . . 88
6.3 Average eﬃciency of the Σ+ p scattering events including the detection andanalysis eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3.1 Detection eﬃciency of CATCH . . . . . . . . . . . . . . . . . . . . 90

2

6.3.2Average eﬃciency for the Σ+ p scattering events considering the realdetection eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7 Experimental result and discussion
96
7.1 Diﬀerential cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.2 Comparison with the past experiments and theoretical calculations . . . . 96
7.3 Phase-shift analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3.1 Speciﬁc expressions of diﬀerential cross section as a function ofphase shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3.2 Consideration of the ﬁxed phase shifts in the ﬁtting . . . . . . . . . 100
7.3.3 Fitting results and phase shifts of 3 S1 and 1 P1 states . . . . . . . . 101
8 Summary104

Acknowledgements

106

A Tables of the diﬀerential cross sections

112

B Derivation of the integrated cross sections for the Σ+ p elastic scatteringwith a ﬁne momentum step
114

3

Chapter 1
Introduction
1.1

Baryon-Baryon interaction and importance of ΣNinteraction

The nuclear force, interaction between nucleons (N N interaction), plays a very importantrole in forming elements and nuclear matter like neutron stars as they exist now. Owing
to a plenty of proton-proton and neutron-proton scattering experiments, the nuclear force
is well described: The nuclear force is attractive in the middle to long ranges (>1 fm)
and repulsive in the short range so called repulsive core. The behavior at the middle
and long ranges is explained by meson exchange model. In contrast, the origin of theshort-range repulsion is still unclear. In pioneering nuclear force research, the short-range
repulsion was treated phenomenologically [1] or was attributed to the ω exchange in aboson-exchange picture [2]. The eﬀect of antisymmetrization between quarks on shortrange interactions was ﬁrst considered using the quark cluster model (QCM) by Oka andYazaki [3, 4]. They reported that the short-range repulsion could be explained by thePauli principle between quarks (quark Pauli eﬀect) and the color-magnetic interactions.
In order to understand the role of quarks in the nuclear force, it is important toextend the nuclear force to baryon-baryon interactions (BB interactions) adding anotherspecies of quarks. One of simple extended systems is the octet baryons composed of u, d, squarks (n, p, Λ, Σ− , Σ0 , Σ+ , Ξ− , Ξ0 ). The latter six baryons include at least one s quark
and are called Hyperons (Y ). Because the mass of s quark is relatively similar to onesof u, d quarks, the BB interactions can be approximately treated under the ﬂavor SU(3)symmetry. The possibility of hyperon appearance in neutron stars (NSs) is discussed [5].
The BB interactions between the octet baryons are labeled by the multiplets as8 ⊗ 8 = 27 ⊕ 8s ⊕ 1 ⊕ 10∗ ⊕ 10 ⊕ 8a .

(1.1)

And the relationship between the isospin basis and the ﬂavor SU(3) basis for the BB
interaction channels with S = 0 and S = −1 are summarized in Table 1.1.
Although the 27-plet and the 10∗ -plet can be examined from the N N interactions,the other four multiplets can be studied only from hyperon-nucleon (Y N ) and hyperonhyperon (Y Y ) interactions. They could be expected to have diﬀerent features from theN N interaction in the short-range region, where interactions between quarks and gluons
should be considered. For example, the 8s -plet and the 10-plet are predicted to be a quite
large repulsive force due to a Pauli forbidden state at a quark level. In contrast to them,4

Table 1.1: Relationship between the isospin and ﬂavor SU(3) bases for the BB interaction
channels. Spin states S and a parity of the orbital angular momentum L are denoted as
2S+1(−1)L , such as singlet-even (1 even) and triplet-odd (3 odd).
1
3
strangeness BB channel (I)even or 3 odd
even or 1 odd
0−1

N N (I = 0)
N N (I = 1)–(27)(10∗ )–

ΛN (I = 12 )
ΣN (I = 21 )
ΣN (I = 32 )√1 [(8s ) + 3(27)]10√1 [3(8s ) − (27)]10√1 [−(8a ) + (10∗ )]2√1 [(8a ) + (10∗ )]
2

(27)

(10)

the 8a -plet and 1-plet are expected to have a weakly repulsive core and attractive pocket,
respectively. These were ﬁrstly predicted by QCM [6] and recently reproduced by a lattice
QCD simulations [7][8]. Therefore, revealing the natures of the ﬂavor SU(3) multiplets isimportant for the systematic understanding of the BB interactions. In the ΣN channels,ﬁve SU(3) multiplets except for the 1-plet are included. Especially, the Σ + p and Σ− nchannel are the pure I = 3/2 and their triplet-even and singlet-odd states are suitable toinvestigate states the 10-plet, whose 3 S1 state is expected to have a large repulsive core
originated from the quark Pauli eﬀect.

1.1.1

Quark Pauli eﬀect in the quark cluster model (QCM)

In the framework of QCM, the wave function for a baryon-baryon, (3q)-(3q) system, isexpressed as
X
A[ϕA (1, 2, 3)ϕB (4, 5, 6)χβ (R)],(1.2)
Φ(1, 2, · · · , 6) =β

where χ(R) is the relative wave function and A is the antisymmetrization operator forthe six constituent quarks. A and B specify the baryon species and the β represents
quantum numbers of the two-baryon channel. ϕ = ϕ(orb) ξ SF ξ C is the internal baryon
wave function, composed of the orbital part, the isospin-coupled basis of the spin-ﬂavor
SU(6) wave functions, which incorporates the Pauli principle in the baryon level, and thecolor-singlet wave function. For ϕorb
A (123), the product of three (0s) wave functions in
the harmonic oscillator potential is adopted. In the ground state of two-baryon relative
motion, six constituent quarks are in the (0s) state of the harmonic oscillator potential.
Figure 1.1 shows the schematic drawings of the quark Pauli eﬀect mechanism for the
3
S1 state in the Σ+ p system, where two u quarks have the same spin and color with ahigh probability. In this situation, at least one u quark must be excited to the higher
orbit in order to satisfy the Pauli principle. Because quark-Pauli forbidden states with(0s)6 conﬁguration are vanished due to the antisymmetrization, the χ(R) components
orthogonal to the ground state of the two-baryon relative motion increases depending on
the appearance probability of the forbidden states. This makes a node of the relative
wave function at short distance and the eﬀective repulsive force.
Historically, the short range core of α-α system had been explained by similar mechanism [9] [10], so to speak, “nucleon Pauli eﬀect” before the QCM for BB interaction waspresented. The quark Pauli eﬀect is analogous to that.
5

u

1S

To be excited

+

0S

p

s

u

u

u

u

d

Pauli principle
Violation

Figure 1.1: Rough sketch of the quark Pauli eﬀect in the Σ+ p system. Two u quarks
have the same spin and color with a high probability. In order to avoid violating Pauliprinciple, one u quarks should be excited, which result in the short range repulsive force.

1.2

Equation of state for neutron stars, hyperon puzzle, and three body force

The Y N and Y Y interactions and potentials of hyperons in a nuclear matter, UY , are also
important to discuss the structure of the neutron stars (NSs). Hyperons may appear in the
inner core of NSs at densities of about 2-3ρ0 , where ρ0 is the nuclear saturation density. Insuch a high density, the nucleon chemical potential is large enough to make the conversionof nucleons into hyperons energetically favourable. Not only the lightest hyperon, Λ, butalso hyperons with negative charge, Σ− and Ξ− , are important because the onset of
negative-charged hyperons permits the appearance of protons owing to charge neutrality.
According to the relativistic mean-ﬁeld models by J. Schaﬀner and I. N. Mishustin [11],the fraction of baryons and leptons in neutron star matter is calculated as shown in Figure
1.2. The Λ and Ξ− present at 2.3 ρ0 and 2.7 ρ0 , respectively. In contrast, Σ doesn’t appear
even at 10 ρ0 because of its repulsive single-particle potential (UΣ = +30 MeV). These
conversion relieves the Fermi pressure exerted by the baryons and makes the Equation ofstate (EOS) softer. Consequently, the predicted maximum mass of NS becomes smaller
than that without hyperon. Although this induction seems to be natural, the predictedmaximum masses of NS with hyperons are incompatible with observations, such as the
PSR J0740+6620 [12] which has the mass of 2.14−+0.10
0.09 M⊙ . This problem is called “hyperon
puzzle”. To solve it, a mechanism is necessary that could provide the additional repulsion
needed to make the EoS stiﬀer and therefore the maximum mass compatible with thecurrent observational limits.
One of the possible mechanisms that could provide such an additional repulsion is theinclusion of repulsive hyperonic three-body forces. D. Lonardoni et al. calculated theEOS of NS with the auxiliary ﬁeld diﬀusion Monte Carlo algorithm [13]. They indicatedthe importance of the three-body Y N interaction for hardening the EOS. Two types ofthe ΛN N force were constructed in order to reproduce the hyperon separation energies oflight or medium mass hypernuclei. Figure 1.3 shows the relation between the mass andradius of NS. The both types of three-body force make the EOS stiﬀer but the maximum
masses strongly depend on the type of three-body force: 1.36 M⊙ for type-(I) three-bodyforce and 2.09 M⊙ for type-(II) one, which has additional repulsive force from type-(I).

6Figure 1.2: Calculated fraction of baryons and leptons in neutron star matter for a RMF
calculation by J. Schaﬀner and I. N. Mishustin [11]Figure 1.3: Mass-radius relations of NS for pure nucleon matter (PNM) and hyperon
appearance cases [13].
In order to investigate a three-body force, both accurate understanding of the two-bodyforce and the high-resolution experimental data of hypernuclei are needed. Establishing
the experimental method for the Y N scattering with a small error is quite important to
get information on the former.

7

1.3

Theoretical Models for BB interaction

Many theoretical models have been constructed to describe the BB interactions. In
this section, typical models are brieﬂy introduced. In regard to the ΣN (I = 3/2,3 S1 )interaction, a large repulsive force due to quark Pauli eﬀect is expected to work. However,theoretical predictions for the strength of “large repulsive force” are diﬀerent from eachother.

1.3.1

Nijmegen Extended-soft-core Model (ESC)The Nijmegen soft core models and extended-soft-core models, such as NSC97[14] andESC08[15], are constructed based on the meson exchange picture between two baryons.
For example, ESC08 consists of (i) One-boson-exchange, in which pseudo scaler mesons (π,η, η ′ , K), vector mesons (ρ, ϕ, K ∗ , ω), scaler mesons (a0 (962), f0 (993), κ(800), f0 (760)),
and axial-vector-mesons (a1 , b1 , f1 , K1 , f1′ ) are considered, (ii) two pseudo scalar mesons
exchange and (iii) Meson-Pair-exchange. To explain the short range force, the ESC models
additionally include (iv) multiple-gluon-exchange (Pomeron and Odderon exchange) and(v) structural eﬀects due to the quark-core of the baryons, which is related to the quarkPauli eﬀect. In ESC models, the coupling constants of each particles, QCD cut-oﬀ masses
are treated as parameters. These parameters are determined by ﬁtting to abundant N N
scattering data below Tlab = 350 MeV and 52 Y N scattering data with constraints frominformation derived from hypernuclei and hypernuclear matter.

1.3.2

Chiral Eﬀective Field Theory (χEFT)The χEFT models has successed for the derivation of the N N interaction. Applying thepower counting to the eﬀective potential, the calculations can be improved systematicallyby going to higher orders in a perturbative expansion. In higher order calculations, many
body forces are also derived in a consistent way. For the N N interaction, χEFT hasreached up to the ﬁfth order (N4 LO) and the sixth order calculations have started [16].
For the Y N interaction, This method is applied up to the second order (next-to-leading
order, NLO [17][18]). The chiral Y N potential consists of pseudo scalar meson (π, K, η)
exchange and a series of contact interactions with an increasing number of derivatives,
which corresponds to the order of momentum. The coupling constants and coeﬃcientsof contact interactions (low energy constants, LEC) are the parameters in χEFT frameworks. As a basis of LECs, one can select partial wave contributions for each of the
ﬂavor SU(3) multiplets. For example, six LECs are considered at the 0-th order (LO)
1
10
8a
8s
(C127S0 , C310∗
S1 , C1 S0 , C3 S1 , C3 S1 , C1 S0 ). For the NLO calculation, additional three and thirteen LECs for the contributions of S − D transition and P waves are needed. These LECs
are determined by ﬁtting to experimental data. Based on available N N scattering data
and low energy Y N scattering data, LECs for all S wave contributions and higher-partial
wave contributions of only 27-plet and 10*-plet are determined.

1.3.3

BB interaction including QCM

FSS and fss2 [19] use QCM for describing the short-range interaction and an eﬀective
meson exchange potential for the middle and long range interactions. In these models,8

the quantum mechanical Hamiltonian of BB interaction is deﬁned as a Hamiltonian for
6-body mechanics as
H=

6 
X
i=1

6 
 X
X Sβ X PSβ X Vβ 
p2i
Uij +
Uij +
Uij , (1.3)
mi c +− TG +
UijCf + UijFB +
2mi
i<j
β
β
β
2

where UijCF is the conﬁnement potential of quarks , UijFB is the Fermi-Breit interaction related to quark-quark interaction and UijS , UijPS , UijV are eﬀective-meson exchange potentials,
in which the ﬂavor singlet and octet mesons are considered. The Quark cluster models
has four parameters: size parameter b deﬁning the internal harmonic oscillator potential for quarks, the up-down quark mass mud , the strength of the quark-gluon coupling
constant αS and the mass ratio of the strange to up-down quarks λ = ms /mud . For themeson exchange part, the quark-meson coupling constants of the singlet and octet mesonand singlet-octet mixing angle for each of the scaler, pseudo scaler and vector mesons areparameters. Mass of scaler mesons are also parameters within some appropriate ranges.
These parameters are determined by ﬁtting to N N scattering data (J ≤ 2, Tlab = 350
MeV) and low-energy Y N total cross sections.

1.3.4

Lattice QCD

Recently, a numerical QCD simulation called Lattice QCD was successfully carried out for
the BB interactions owing to developments of calculation techniques and great progressof computer performance. HAL QCD collaborations performed a large scale lattice QCD
calculation with the almost physical quark masses corresponding to (mπ , mK ) = (146, 525)
MeV and large volume (La)4 = (8.1fm)4 . Figure 1.4 shows the six calculated potentials
for baryon-baryon S wave interactions by a lattice QCD simulation. The strong repulsive
forces in 8s-plet and 10-plet and the attractive core in 1-plet are reproduced. Thesefeatures indicates the importance of the quark Pauli eﬀect for the short-range baryonbaryon interaction.

9

Figure 1.4: Calculated potentials for baryon-baryon S wave interactions by the HALQCD collaboration [20]. These are obtained from hyperon interaction potentials in thebaryon-basis in strangeness S = −2 sector by rotation assuming the exact ﬂavor SU (3)symmetry.
Although calculations for potentials of higher partial waves need huge computational
resources, they are expected to be realized with the supercomputer Fugaku.

1.3.5

Theoretical predictions for ΣN (I = 3/2,3 S1 ) interaction

δ3S1 [degree]

Theoretical treatments of the short-range interaction in the BB interactions had led toconsiderably diﬀerent results on the ΣN (I = 3/2, 3 S1 ) interaction. The momentum
dependence of the calculated phase shift of 3 S1 state is shown in Figure 1.5. fss2, which

ESC08a80
NSC97f

60
40

fss2

20
χEFT NLO19

0
−20
−40
−60
−80
0

0.1

0.2

0.3

0.4

0.5

0.6 0.7 0.8 0.9 1
Σ+ momentum [GeV/c]

Figure 1.5: Calculated phase shifts of 3 S1 state for the Σ+ p scattering as a function of
the incident Σ+ momentum.
10includes QCM in the short-range region, naturally predicts a considerably repulsive interaction [19]. However, the pure meson-exchange models, such as the NSC models [21, 14],which represent the short-range interaction from a heavy vector-meson exchange, predicted attractive interaction. In the Nijmegen ESC model, the additional short-range
interaction owing to Pomeron exchange is newly included to explain the repulsive nature of this channel [15, 22]. Recent calculations by χEFT predict repulsive interaction
for this channel [17, 18]. By the recent lattice QCD simulation in the almost physicalquark masses, the ΣN (I = 3/2, 3 S1 ) potential was demonstrated a repulsive core in the
short-range region without any attractive pocket in the middle-range region as shown the”10-plet” in Figure 1.4.
In summary, all of recent theoretical calculations predict repulsive interactions in theΣN (I = 3/2, 3 S1 ) channel. However, the strength of the repulsive interaction, i.e., thephase shift value of the 3 S1 channel, is quite diﬀerent from each other and it should bedetermined from experiments.

1.4

Previous experimental studies on ΣN andΣ-nucleus interaction

1.4.1

Σ± p scattering experimentA Y N scattering experiment is the most direct method to derive the Y N interaction as in
the case of the N N interaction. However, the Y N scattering experiment is quite diﬃcult
due to the short lifetime of hyperons and data are quite scarce.
For a low momentum region of around 0.15 GeV/c, where the S wave is dominant, the±Σ p scattering cross section was measured with a few tens percent error using a bubble
chamber [23].
Data for higher momentum range more than 0.4 GeV/c is desired in order to investigate short-range interactions and higher wave contributions. Accumulating enoughscattering events is generally more diﬃcult for the higher momentum because the crosssection decreases proportionately to 1/k 2 = (ℏ/pCM )2 . In past KEK E251 [24] and E289
[25] [26] experiments, a scintillation ﬁber (SCIFI) active target, which is a triggerable
imaging detector, was used in order to recognize the Σ± p scattering events and they
were accumulated eﬃciently. Nevertheless, statistics of the Σ ± p scatterings for 0.3-0.75[GeV/c] region were limited to less than 100 events and the errors of the diﬀerential crosssections were more than 50%, which were too large to restrict BB interaction models well.
Although there is one spin-observable measurement on the Σ + p channel in the KEK E452
experiment [27], no phase shift value has been obtained experimentally.

1.4.2

Σ hypernuclei and quasi-free Σ production

Since Y N scattering measurements are diﬃcult, information on the Y N interactions havebeen studied through hypernuclear data. Especially, this method has been succeeded forthe ΛN hypernuclei. For the ΣN interaction, there are important results for light Σ hypernuclei and medium-to-heavy Σ hypernuclei. The former is for the A = 4 hypernnuclei.
T. Nagae et al.[28] revealed that a clear peak below the Σ+ production threshold could
be observed in the 4 He(K − , π − ) reaction at 600 MeV/c, whereas in 4 He(K − , π + ) reaction11

no such peak was found as shown in Figure 1.6. This result meant only the T = 1/2
state was bound state because it can be produced only by the 4 He(K − , π − ) reaction. It
was caused by the large isospin dependence in the ΣN interaction: T = 1/2 state is
attractive and T = 3/2 state is strongly repulsive. Only 4Σ He state is observed as a boundΣ hypernuclear state.

Figure 1.6: Excitation energy spectra of 4 He(K − , π − ) and 4 He(K − , π + ) at 600 MeV/c
K − momentum measured at 4◦ in the BNL E905 experiment[28].
While medium-to-heavy Σ hypernuclei bound states have not been observed, Σ-nucleus
potential was investigated by analyzing the spectrum of the (π − , K + ) reaction on thevarious nuclear targets in KEK E438 experiment [29][30]. The missing mass spectrum of
the 28 Si(π − , K + ) reaction at the beam momentum of 1.2 GeV/c is shown in Figure 1.7. In
the analysis, the spin-isospin averaged potential was assumed to be Woods-Saxon (WS)type potential as
V0Σ + iW0Σ

(1.4)
UΣ =
1 + exp r−Rawhere V0Σ and W0Σ are the strength of the real and imaginary parts of the potential. The
best V0Σ and W0Σ were evaluated to be (V0Σ , W0Σ ) = (30MeV, −40MeV) by T. Harada and
Y. Hirabayashi [31]. This result indicates that the single particle potential of a Σ particle
in nuclei is repulsive.

12

Figure 1.7: Missing mass spectrum of the 28 Si(π − , K + ) reaction at the beam momentum of
1.2 GeV/c [29][30]. The DWIA calculation results with several parameters of Σ-nucleous
potential [31] were overlaid as lines.

1.4.3

Σ− atom

When a hadron with a negative charge is captured in an atomic orbit around a nucleus,
the hadron and nucleus can overlap because its orbital radii is much shorter than theordinary atomic states. Such exotic atoms are a probe to the strong interaction between
the hadron and nuclei at the surface of nuclei. The X-ray energy shift from the electromagnetic interaction gives information on the real part of the optical potential and the
X-ray width is relevant to the imaginary part. The Σ− atom data had been taken for
the 11 kinds of nuclei (O, Mg, Al, Si, S [32], C, Ca, Ti, Ba [33], W and Pb [34]). The
latest analysis [31] conﬁrmed that the Σ-nucleus potential having a repulsion inside thenuclear surface and an attraction outside the nucleus with a sizable absorption, which issimilar to the potential mentioned in the previous subsection, can explain simultaneously
the data of the Σ− atoms and 28 Si(π − , K + ) reaction.

1.4.4

Femtoscopy technique

Recently, the study of two-particle correlations in momentum space measured in ultrarelativistic p-p and p-Pb collisions at the LHC has provided access to the interaction
between particle pairs in vacuum. In this process, all hadrons originate from very small
space-time volumes and typical distances between two particles are about 1 fm. From theuncertainty principle, the interaction between baryon pairs can be investigated for relative
momentum up to 200 MeV/c. With this method, many channels of the BB interactions
are investigated, such as the pp, pΛ, ΛΛ [35], and pΞ− [36] interactions. For the ΣNinteractions, ALICE collaboration applied this technique to the pΣ0 channel [37], in whichthe Σ0 → Λγ decay can be reconstructed relatively easily by the ALICE detectors. At
present, the experimental precision is not enough to select the best theoretical model.
With the expected larger data samples to be obtained in LHC Run3 and Run4, lowmomentum ΣN interactions will be constrained from femtoscopic measurements in future.
13

1.5

Present experiment: J-PARC E40 experiment

The J-PARC E40 experiment [38] was proposed to conﬁrm predicted strong repulsive
force in the Σ+ p channel, which is related to the quark Pauli eﬀect, and to study the ΣNinteraction systematically. The purpose of this experiment is to measure the diﬀerential
cross sections of Σ+ p elastic, Σ− p elastic, and Σ− p → Λn inelastic scattering with highstatistics. Momentum range of the incident Σ± were from 0.45-0.85 GeV/c, where typicaldistances between two particles in the Σ± scatterings are 0.8-0.5 fm and they would
overlap each other. Since the large uncertainties of cross sections in previous experiments
were predominantly caused by the lack of statistics, high-statistics scattering experimentis essential to improve the data quality. More reliable data enables comparison withtheoretical calculations. For example, the diﬀerences of phase shifts shown in Figure 1.5
result in the diﬀerence of diﬀerential cross sections by approximately 30 %. In order todistinguish which theoretical calculation agrees with data, more precise data is needed.
High-statistics data further enables us to obtain the precise angular dependence of thediﬀerential cross sections. It reﬂects the contributions of higher partial waves, for which
there are few reliable experimental data. Moreover, exploiting high-quality diﬀerential
cross section data and the simple representation of Σ+ p interaction in the SU(3) ﬂavor
symmetry, a phase shift analysis on Σ+ p scattering were realized as later explained in
Chapter 7.
This experiment was performed at K1.8 beamline in the J-PARC hadron experimental
facility in the separated periods of June 2018, February-March 2019 (for Σ − p scattering
data taking), April 2019 and May-June 2020 (for Σ+ p scattering data taking).
To achieve the high-statistic Σ± p scattering experiment, production of many incident Σ+ particles are essential. In order to produce many incident Σ + particles via theπ ± p → K + Σ± reaction, a high intensity π + beam of approximately 1.0 × 107 Hz andlarge acceptance spectrometer for the outgoing K + were used. Additionally, so-called
CATCH system [39] surrounding the target was developed, which has enough rate tolerance against the high intensity beam condition and a large acceptance for recoil protons
from Σ+ p scattering (−1 < cos θCM < 0.5). Owing to these experimental features, ahigh-statistical hyperon-proton scattering experiment was realized.
Results on the Σ− p elastic scattering [40] and Σ− p → Λn inelastic scattering [41]have already been reported. Regarding the statistics, approximately 4500 Σ − p elastic
and 2300 Σ− p → Λn inelastic scattering events were identiﬁed. The derived diﬀerential
cross sections for Σ− p elastic and Σ− p → Λn inelastic scatterings are shown in Figure 1.8
and Figure 1.9, respectively. The statistical error of the 10% level was achieved even witha ﬁne angular step of d cos θ = 0.1 and the data compared to predictions of theoreticalmodels. These data will be reliable input to determine the 8 s -plet and 8a -plet interactions,as shown in Table 1.1.
In this thesis, whole of the Σ+ p scattering experiment as an important part of J-PARC
E40 experiment is described.

14-dσ/dΩ (mb/sr)

Differential cross sections of Σ p scattering8pΣ =470-550 (MeV/c)6pΣ =550-650 (MeV/c)

This work
E289

4

Chiral EFT
fss2
ESC08cdσ/dΩ (mb/sr)

20pΣ =650-750 (MeV/c)pΣ =750-850 (MeV/c)

6
4
2
0
−1

−0.5

0

0.5

−1
cosθ

−0.5

0

0.5

1
cosθ

Figure 1.8: Diﬀerential cross sections for the Σ− p elastic scattering obtained from the
J-PARC E40 experiment [40]. The black point represents the J-PARC E40 result.

Figure 1.9: Diﬀerential cross sections for the Σ− p → Λ elastic scattering obtained fromthe J-PARC E40 experiment [41]. The black point represents the J-PARC E40 result.

1.6

Thesis composition

The setup of present experiment is described in Chapter 2. The whole analysis ﬂow
to derive the diﬀerential cross sections for the Σ+ p scatterings is brieﬂy summarized in
15

Chapter 3. The analysis was divided into three stages, the analysis of spectrometers and
CATCH, identiﬁcation of the Σ+ p scattering events, and evaluation of factors to derive
the diﬀerential cross sections. These are described in Chapter 4, 5, and 6, respectively.
The obtained results are presented and ΣN (I = 3/2) interaction are discussed in Chapter
7. Finally, the Σ+ p scattering experiment is summarized in Chapter 8.

16Chapter 2
Experiment
2.1

OutlineThe Σ+ p scattering data taking as a part of the J-PARC E40 experiment was performedat the K1.8 beam line in the J-PARC Hadron Experimental Facility with π + beams of
1.41 GeV/c. Typical intensity of the π + beam was 2.0 × 107 particles per spill of 2.0 s
duration every 5.2 s.
A conceptual drawing of the experiment is shown in Figure 2.1. A liquid hydrogentarget (LH2 target) is used both for the incident Σ+ production and Σ+ p scattering. Σ+ p
scattering events can be identiﬁed by measuring the Σ+ momentum, the recoil proton’s
kinetic energy and the proton’s recoil angle. Σ+ particles are produced via the π + p →K + Σ+ reaction induced by π + beam. The momentum of each Σ+ can be calculated as
the missing momentum of the π + beam and outgoing K + , analyzed using two magnetic
spectrometers. The produced Σ+ particles can induce additional reactions, such as thedesired Σ+ p scattering. Using the detector system surrounding the LH 2 target, charged
particles involved in these reactions and decay products of Σ + are measured. In orderto produce many incident Σ+ particles, a high intensity π + beam and large acceptancespectrometer for the outgoing K + were used. The surrounding detector system, so-called
CATCH system [39] comprises a ﬁber tracker and calorimeter, which has enough rate
tolerance against the high intensity beam condition and a large acceptance for recoil
protons from Σ+ p scattering. Owing to these experimental features, a high-statistical
hyperon-proton scattering experiment was realized.

2.2

J-PARC and Hadron Experimental Facility

The Japan Proton Accelerator Research Complex (J-PARC) is a multi-purpose accelerator
facility located in Tokai village, Japan [42, 43]. Experiments related to not only thebasic science of particle and nuclear physics but also the materials and life sciences are
performed, using various types of secondary beams of neutrinos, muons, pions, kaons,protons, neutrons, and so on. An aerial photograph of J-PARC is shown in Figure 2.2.
J-PARC comprises three proton accelerators, a 400 MeV linac, a 3 GeV rapid cycling
synchrotron (RCS), and a 30 GeV main ring synchrotron (MR) and three experimental
facilities, the Neutrino Experimental Facility (NU), the Hadron Experimental Facility(HEF), and the Material and Life Science Facility (MLF).

17

Kinetic energy of recoil protonp2.

±p scattering

LH2 target±p scattering

Recoil angle of proton±±±consistency check
identification of the

decay

p

1. production: ±p K+
Momentum of +± reaction

K+

Figure 2.1: Conceptual drawing of the Σ+ p elastic scattering in the J-PARC E40 experiment. The initial Σ+ is produced by the π + p → K + Σ+ reaction. Then, it travels in
the LH2 target and may cause Σ+ p scattering. The Σ+ p elastic scattering can be kinematically identiﬁed by measuring the initial Σ + ’s momentum, the recoil proton’s kinetic
energy, and recoil angle.

Figure 2.2: Aerial photograph of J-PARC [44]. Three accelerators, linac, RCS, and MR
and three experimental facilities, NU, HEF, and MLF are shown. The construction of anaccelerator-driven transmutation facility is planned.
At the linac, negative hydrogen ions (H− ) are accelerated up to 400 MeV and they
are ﬁnally converted to protons with a charge-stripper foil and injected to RCS. RCS has
circumferences of 348.3 m and accelerates the injected protons up to 3 GeV at a repetition
rate of 25 Hz. Most of the RCS beam pulses are delivered to MLF and the rest of thepulses are injected to MR. In MLF, muon and neutron beams, which are produced bycolliding 3 GeV protons with carbon and mercury targets, are available. The protons
injected from RCS to MR are accelerated up to 30 GeV and delivered to NU or HEF with
diﬀerent extraction modes, fast extraction and slow extraction, respectively. In the fast
18

extraction mode, where all beam bunches are extracted to NU within a one-turn time
period of approximately 5 µs. Then neutrino and anti-neutrino beams are generated asa decay products of π ± , which are produced by colliding the extracted proton beam withgraphite target. They are used for the Tokai to Kamioka long-baseline neutrino oscillation
experiment, T2K. While in the slow extraction mode, the beam are gradually extracted
to HEF over 2s using the resonant extraction method . Beams extracted in one cycle orone-cycle extraction itself are called spill. The spill cycle were 5.2 s, of which 2 seconds
were on-beam.
The layout of HEF is shown in Figure 2.3. The extracted proton beam is bombarded tothe production target, T1 and secondary particles, such as p, π ± , K ± , K0L were produced.
The secondary particles are delivered to experimental areas through the secondary beamlines, K1.8, K1.8 BR, KL and K1.1. In the data-taking period for present experiment, theproduction target was a golden bar with a size of 15mm (W) × 6mm (H) × 66 mm (T).
Recently, new beam lines, high-p and COMET, were constructed, where primary proton
beams are available.

Figure 2.3: Layout of the hadron experimental facility[44]. K1.8, K1.8 BR, KL, and K1.1
are secondary beam lines. In high-p and COMET beam lines, primary proton beam isavailable for experiments.

2.3

K1.8 beam line

The K1.8 beam line [45] is a multi-purpose beam line, where varyous secondary hadronbeams up to 2.0 GeV/c, such as K ± , π ± , p and p̄ are available. In the present experiment,
we used the π + beam at 1.41 GeV/c with a beam intensity of 2 × 107 particles per spill.
Additionally, proton beams with various momenta between 0.45 and 0.85 GeV/c were19

used in order to collect pp scattering data for calibration. Figure 2.4 shows the schematic
drawing of the K1.8 beam line. This beam line comprises three sections, the upstream
section, mass separation section, and momentum analysis section.

Figure 2.4: Schematic drawing of the K1.8 beam line.

Upstream section
The upstream section is from D1 to IFV in Figure 2.4. Two dipole magnets (D1 and D2)
and two quadruple magnets (Q1 and Q2) are equipped. The momentum of the extracted
secondary beam particle is determined by the D1 magnet setting and the transported beam
is vertically focused at the intermediate focus (IF) point. At the IF point, a horizontal
slit (IFH) and a vertical slit (IFV), which are made of a 30-cm thick brass block, areequipped and they shapes beam proﬁles. Using IFH and IFV, the so-called cloud pions,which are produced by the decay of Ks0 , and the scattered pions from materials in the
upstream section are rejected.

Mass separation section
The mass separation section is from Q3 to MS2 in Figure 2.4. This section consists of
two-stage mass separation systems, the ﬁrst stage mass separation system (Q3 to D3)
and second stage mass separation system (Q7 to MS2). Electrostatic separators (ESS1and ESS2) are the key components of this section, which generate transverse electricﬁeld along the beam axis to separate particles of the same momentum by their massdiﬀerences. Each electrostatic separator generates an electric ﬁeld with a gap of 10 cm
between parallel electrode plates with a size of 30 cm in width and 6 m in length. The
cathode and anode plates are made of anodized aluminum and stainless steel, respectively.
20

In the present experiment, the gradient of the electric ﬁelds was set to 30 kV/cm for π +beam. For proton beam, only ESS1 was used and the gradient was set to 10 kV/cm.
The displacement of particle orbits due to ESS is corrected using correction magnets(CM1-CM4) equipped at the entrance and exit of the electrostatic separators. After ESS1
and ESS2, the beam is focused in the vertical direction at each mass slit (MS1 and MS2)
and momentum dispersion become large horizontally. The mass of beam particles can beselected by tuning the magnetic ﬁelds of correction magnets and mass slit positions. For
the present pion beam experiment, the settings of them were adjusted to let only pions
pass through the slit. A horizontal momentum slit (MOM) determines the momentum
bite of the secondary particles.

Momentum analysis section
The momentum analysis section called K1.8 beam-line spectrometer is placed at the endof K1.8 beam line, which is also the entrance of K1.8 experimental area. The section
comprises ﬁve magnets in a QQDQQ conﬁguration and counters and position detectors.
The details of this section are described in the next section.

2.4

K1.8 beam-line spectrometer

Figure 2.5 shows the schematic drawing of the K1.8 beam-line spectrometer. The ﬁveanalyzer magnets in a QQDQQ conﬁguration form the K1.8 beam-line spectrometer together with detectors upstream and downstream of the magnets. A plastic scintillator
hodoscope (BH1) and plastic scintillator ﬁber detector (BFT) were placed upstream ofthe magnets. In contrast, the drift chambers (BC3 and BC4) and another plastic scintillator hodoscope (BH2) were placed downstream of these magnets. The π + beam was
focused on the center of the LH2 target and its momentum was reconstructed event-byevent using the spatial information at BFT, BC3, and BC4, and the third-order transfer
matrix for the spectrometer. Figure 2.6 shows the distribution of the beam π + particles
at the center of the target.

21

BH2
BC3,4
Q13
K1.8 Beam-LineSpectrometer

Q12

D4
BH1
BFT

Q11

Q10

Counts

2400
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
0
−80

CountsFigure 2.5: Schematic drawing of the K1.8 beam-line spectrometer. The K1.8 beam-line
spectrometer comprises ﬁve analyzer magnets in a QQDQQ conﬁguration (D4 and Q10Q13), two hodoscopes (BH1 and BH2), and three tracking detectors (BFT, BC3, and
BC4).

3500
3000
2500
2000
1500
1000
500

−60

−40

−20

0
−80

0
20
40
60
80
Horizontal position (x) [mm]

−60

−40

−20

0

20
40
60
80
Vertical position (y) [mm]Figure 2.6: (Left) Horizontal and (Right) vertical distributions of the beam π + particles
at the center of the target.
The eﬀective areas and position resolutions of the detectors are summarized in Table
2.1. More detailed information on detectors is described in the following subsections. The
design value of the momentum resolution is ∆p/p = 3.3 × 10−4 (FWHM) with a position
resolution of 200 µm [46]. The central momentum is determined by the magnetic ﬁeldof the dipole magnet (D4). Using a high precision Hall probe [Digital Teslameter 151
22

(DTM-151)] installed in the D4 magnet, the magnetic ﬁeld was monitored. The typicalmagnetic ﬁeld for 1.41 GeV/c π + beam was 1.1738 T and observed deviation was lessthan 0.03 %.
Table 2.1: Eﬀective areas and position resolutions of detectors in the K1.8 beam-line
spectrometer.
Detector

type

role

BH1
BFT
BC3,4
BH2

plastic scintillator hodoscope
plastic scintillator ﬁber detector
drift chambers
plastic scintillator hodoscope(BFT hits selection)
trackertrackertrigger

2.4.1

eﬀective area
H × V [mm2 ]170 × 66
160 × 80
200 × 100
118 × 60

σpos
[mm]
–
0.19
0.20
–

Hodoscopes in the K1.8 beam-line spectrometer

The beam hodoscopes, BH1 and BH2, are usually used for beam particle identiﬁcation
by measuring a time-of-ﬂight for 10.4 m distance. However, in the present experiment,
the contaminations of other particle species in the π + beam were negligibly small. Thus,
BH1 was primarily used for selecting the BFT hits corresponding to the triggered event in
multiple-beam events. BH2 was used as the most important trigger counter determining
the origin of timing for all detectors.
BH1
BH1 was located upstream of the analyzer magnets. It comprises eleven segments ofplastic scintillators (Saint-Goban BC-420) with diﬀerent width of 8, 12, 16, and 20mm
as shown in the left side of Figure 2.7. The width of each segment is adjusted for the
single counting rate to be uniform. The total eﬀective area is 170mm (H) × 66mm
(V). Scintillation signals are read using PMTs (Hamamatsu H6524MOD) on the top andbottom of scintillator through acrylic light-guides. In order to stand a high-rate beam
condition, a three-stage booster was equipped by PMTs.
BH2
BH2 was located most downstream of the K1.8 beam-line spectrometer. As BH1, it
comprises eight segments of plastic scintillator (Saint-Goban BC-420) with diﬀerent widthof 7, 10, 35mm as shown in the right side of Figure 2.7. The total eﬀective area is 118
mm (H) × 60 mm (V). Signals are read from both vertical ends using PMTs (HPK
R9880U-110MOD) equipping a three-stage booster.

2.4.2

Trackers in the K1.8 beam-line spectrometer

In order to reconstruct the momentum of beam particle using the third-order transfer
matrix, an one-dimensional hit position at the entrance of the magnets and position andangle at the exit of the magnets were measured.
23

8
5

12

7

20

5

8
16

7

30

10

35

Acrylic
light guide
Acrylic
light guide

Beam

90

60

170

Beam

Scintillater
BC420
118

PMT
HPK H6524MOD

Scintillater
BC420

PMT
HPK
H9880U-101MOD

Figure 2.7: Schematic drawings of BH1 (Left) and BH2 (Right).
BFT
BFT was located between BH1 and Q10 to measure the horizontal position of the beam.
BFT has a two-layers conﬁguration (XX’) made of 320 scintillation ﬁbers (Kuraray SCSF78MJ) with a diameter of 1mm. Scintillation signals are read by a MPPC (Hamamatsu
S10362-11-100P) attached at the end of each ﬁber. The eﬀective area is 160 mm (H) ×80mm (V) and the position resolution was estimated to be 190 µm in σ. The pictures of
BFT and the MPPC card are shown in Figure 2.8.
BC3 and BC4
Two MWDC were located at the exit of the analyzer magnets to measure the beam trajectory, which means position and angles. Both chambers have a six layers conﬁguration(xx′ uu′ vv ′ ) with 3-mm wire spacing. u and v planes are tilted by ±15◦ , respectively. The
sense wire is gold-plated tungsten wire with 12.5 µm-diameter and the potential wire isa gold-plated copper beryllium wire with 75 µm-diameter. A set of two layers such asxx′ , uu′ , vv ′ is called as a pair plane. In the pair plane, wire positions are shifted by ahalf cell size of 1.5mm. Owing to this structure, the left/right ambiguity of each hit canbe solved. In the operation, a mixed gas of Ar(76%), iso-C 4 H10 (20%), and methylal(4%)
was ﬁlled in BC3 and BC4. Methylal prevents anode wires from sputtering due to beam
particles. A raw signal from each wire was read by Ampliﬁer Shaper Discriminator (ASD)
card attached on the chamber. A high voltage of −1.19 kV for sense wires and −1.21kV
for potential wires were applied.

2.5

KURAMA spectrometer

Outgoing charged particles produced at the LH2 target by the π + p reaction were analyzed
by the KURAMA spectrometer. Figure 2.9 shows the schematic drawing of the KURAMA
24

Figure 2.8: Pictures of BFT and the MPPC card for readout [47]. Edges of scintillatingﬁbers were ﬁxed to the ReadOut frame. By attaching the MPPC card to ReadOut framewith screws, MPPCs and scintillating ﬁbers were contacted.
spectrometer. The KURAMA spectrometer comprises a dipole magnet (KURAMA magnet), plastic scintillating ﬁber tracker (SFT), ﬁne segmented plastic scintillator hodoscopes(SCH, FHT1, and FHT2), an aerogel Cherenkov counter (SAC), three MWDCs (SDC1,
SDC2, and SDC3), and a plastic scintillator wall (TOF). The KURAMA magnet was
excited to 0.78 T at the central position. The trajectories of the charged particles in
the magnetic ﬁeld were reconstructed using the Runge-Kutta method [48]. Their momenta were obtained as the best momenta to reproduce the hit positions measured at
the tracking detectors. The time-of-ﬂight of the outgoing particle along a ﬂight path ofapproximately 3m distance was measured using TOF. The spectrometer acceptance for
K + in the π + p → K + Σ+ reaction was approximately 6.7%, and the survival ratio of K +was 65%. The large acceptance and short ﬂight length are advantages of the KURAMA
spectrometer for accumulating many Σ+ particles.

25

KURAMA
Spectrometer

TOF
FHT1,2

SDC2,3SCH
SAC

KURAMA
Magnet
SDC1SFT

LH2 target
CATCH

Figure 2.9: Schematic drawing of the KURAMA spectrometer. The KURAMA spectrometer comprised a dipole magnet (KURAMA manget), seven tracking detectors(SFT,
SCH, SDC1, SDC2, SDC3, FHT1, and FHT2) and two counters (SAC and TOF). Five
curves represent typical trajectory for expected particles. The blue curve corresponds
to 1.4 GeV/c π + beam and the other four red curves correspond to K + s following thekinematics of the π + p → K + Σ+ reaction.
The eﬀective areas and position resolutions of the detectors are summarized in Table
2.2. More detailed information on detectors is described in the following subsection. As
shown the blue curve in Figure 2.9, 1.4 GeV/c π + beam particles can pass through the
KURAMA spectrometer. Therefore, high-rate π + beam particles would hit detectors. To
operate the detectors stably and suppress triggers caused by unexpected beam interaction under such an experimental condition, countermeasures were applied to most of thedetectors. These countermeasures are also introduced in the following subsection. As for
the KURAMA magnet, its horizontal center was shifted from the beam axis and beams
passed through at the side region of the KURAMA magnet. K + s scattered to the left
side were detected by the KURAMA spectrometer.

2.5.1

Detectors in the KURAMA spectrometer

SFT
SFT is located most upstream of the KURAMA spectrometer. It comprises two independent frames, x frame and uv frame. As a tracker, it has three planes (xuv) and enoughhigh-rate tolerance. Information on hit positions of SFT is also used 3D-Matrix trigger.
The design of SFT-x frame shown in Figure 2.10 is similar to BFT. It has a twolayers conﬁguration (xx′ ) made of 512 scintillation ﬁbers (Kuraray SCSF-78MJ) with a26

Table 2.2: Eﬀective areas and position resolutions of detectors in the KURAMA spectrometer.
Detector

type

role

SFT
SDC1
SAC
SCH
FHT1
SDC2
SDC3
FHT2
TOF

ﬁber tracker
drift chamber
aerogel Cherenkov counter
scintillator hodoscope
scintillator hodoscope
drift chamber
drift chamber
scintillator hodoscope
scintillator wall

tracker, trigger
trackertriggertrigger, tracker
trackertrackertrackertrackertrigger, (tracker)

eﬀective areaσpos
H × V [mm2 ][mm]
280 × 160
0.18
380 × 2640.30
484 × 402–
673 × 450
6.0
196 × 450
0.581185 × 1185
0.40
1920 × 1280
0.40
256 × 450
0.58
1800 × 1800 40(H), 20(V)diameter of 1 mm. Scintillation signals are read by a MPPC (Hamamatsu S10362-11100P) attached at an end of each ﬁber. The eﬀective area and position resolution of SFTx plane are 256 mm (H) × 160 mm (V) and 180µm, respectively.

Figure 2.10: Schematic drawings of SFT x frame. The beam spot is roughly indicated as
red elipses.
The design of SFT-uv frame is shown in Figure 2.11. The uv layers are made of 960
scintillation ﬁbers (Kuraray SCSF-78MJ) with a diameter of 0.5 mm. u and v planes aretilted ±45◦ , respectively. Scintillation signals from three ﬁbers are read using one MPPC
(Hamamatsu S10362-11-100P) as shown in the right side of Figure 2.11. The eﬀective
area and position resolution of uv plane are 280 mm (H) × 160 mm (V) and 260 µm,
27respectively.

Figure 2.11: Schematic drawings of SFT uv frame. The beam spot is roughly indicatedas red elipses.
The beam spot is roughly indicated as red ellipses in Figure 2.10 and Figure 2.11. In
SFT-uv, there are aluminum support frame at all sides. Therefore, the position of SFT-uv
was determined to cover not only the acceptance for K + but also the all beam region. On
the other hand, there is no support frame at beam region for SFT-x and its position wasdetermined to cover the acceptance for the scattered particles.
SDC1
SDC1, a MWDC, was located at the entrance of the KURAMA magnet. It has a six
layers conﬁguration (vv ′ xx′ uu′ ), where u and v planes are tilted ±15◦ , respectively. The
conﬁguration of wires is shown in Figure 2.12. Each anode wire is surrounded by six
potential wires forming a hexagonal cell structure, called honeycomb structure and anodewire pitch is 6 mm. Anode wires are made of gold-plated tungsten-rhenium (Au-W/Re)
wires with a diameter of 20 µm. Potential and shield wires are made of gold-plated
aluminum wires with a diameter of 80 µm.
The gas and readout systems were common to BC3,4: A mixed gas of Ar(76%), isoC4 H10 (20%), and methylal(4%) was ﬁlled and a raw signal from each wire was read by
Ampliﬁer Shaper Discriminator (ASD) card attached on the chamber. In operation, ahigh voltage of −1.56 kV for potential wires and −1.6kV for shield wires were applied.
In order to avoid the high-rate beam particles, SDC1 was placed apart from the beam
center so as to detect only scattering events from 3 to 30 degrees.
SAC
In order to reject scattered π particles at a trigger level, a threshold type Cherenkov
counter with a refractive index of 1.1 was placed. The threshold index of Cherenkov
radiation as a function of particle momentum is shown in Figure 2.13. Figure 2.14 shows
the schematic drawing of SAC. SAC comprises four segments (rooms) with diﬀerent sizes,
28

Figure 2.12: Schematic drawings of SDC1 wire conﬁguration.

Index

which was designed taking into account the singles rate. Near the beam spot, SAC has ahole, which prevents SAC from making unwanted veto signals due to beam π + particles.
Cherenkov lights are detected by the Fine mesh PMTs (Hamamatsu R6682). Although
PMTs were in the magnetic ﬁelds of approximately 0.3 T, gains were maintained above
90%.
1.25
1.2

K+

1.15
1.1
1.05

π+

1
0.2

0.4

0.6

0.8

1

1.2

1.4 1.6 1.8
2
Momentum [GeV/c]

Figure 2.13: Threshold index of Cherenkov radiation as a function of particle momentum. Red and blue lines corresponds to scattered K + and π + , respectively. Expected
momentum regions of K + and π + were drawn with real lines. The refractive index of 1.1
is enough to discriminate π + and K + in this experiment.

SCH
A hodoscope was located at the entrance of the KURAMA magnet. Figure 2.15 shows theschematic drawing of SCH. SCH consists of 64 segments of plastic scintillators (EJ212)
with a size of 11.5 mm (H) × 450 mm (V) × 2 mm (T). Each segment is arranged in a
staggered relation with an overlap of 1 mm with each other. A wave length shifting ﬁber
29

Hole forbeam region
Beam center

Figure 2.14: Schematic drawings of SAC [49].
(Kuraray PSFY-11J) with a diameter of 1 mm is put in a hole on the scintillator surface.
Then the scintillation signals are read using a MPPC connected to an end of the ﬁber.
SCH was primarily a trigger counter used to make Matrix trigger. However, it was also
used as a tracker especially for particles near the beam spot, where many other trackers
were made insensitive.
Fiber Kuraray PSFY-11J

MPPC circuit board

434

450

Beam

Scintillator EJ-212

Frame window673

2

Cross-sectional view

1

11.5

1

Figure 2.15: Schematic drawings of SCH.
SDC2 and SDC3
Two MWDCs were located dounstream of the KURAMA magnet. Both chambers had
a four plane conﬁguration (xx′ yy ′ ) and honeycomb wire structure as SDC1. Anode wire
pitches of SDC2 and SDC3 are 9 mm and 20mm, respectively. The speciﬁcations of30

SDC2,3 wires are summarized in Table 2.3.
Table 2.3: Speciﬁcations of SDC2,3 wires.
Chamber
SDC2
SDC3

Anode
Potential
20 µm gold-plated W 100 µm gold-plated Al
30 µm gold-plated W 50 µm gold-plated CuBe

Shield100 µm gold-plated Al
50 µm gold-plated CuBe

A mixed gas of Ar (50%) and ethane (50%) was ﬁlled in SDC2 and SDC3 and kept
ﬂowing. A signal from each wire are processed by pre-ampliﬁer card, which is attached on
the chambers, and an ampliﬁer discriminator board. In SDC2 operation, a high voltage of−2.05 kV for potential wires and −1.5 kV for shield wires were applied. For SDC3, a highvoltage of −2.6 kV and −1.5 kV were applied to potential and shield wires, respectively.
Because SDC2 and SDC3 doesn’t have enough high-rate tolerance for beam particles,
the wires of SDC2 and SDC3 near the beam region were made insensitive by not applying
the operation voltage to the potential wires. Both SDC2 and SDC3 lost about 10% and
5% of their eﬀective area in the horizontal and vertical directions, respectively.
FHT1 and FHT2
Two sets of hodoscopes, which was used as additional trackers, was located just upstreamof SDC2 and downstream of SDC3 to cover horizontal insensitive area of SDC2 and SDC3.
Figure 2.16 shows the setup of SDC3 and FHT2. Each of FHT1 and FHT2 comprises two
same detectors, upper and downer one. Although the segment size is diﬀerent, structureof FHT is similar to SCH: Plastic scintillator (EJ212), which have a hole on the scintillator
surface, with a size of 6 mm (H) × 550 mm (V) × 2mm (T) is arranged in a staggered
relation with an overlap of 2mm with each other. A wave length shifting ﬁber (Kuraray
PSFY-11J) with a diameter of 1 mm is put in a hole on the scintillator surface and thescintillation signals are read using a MPPC connected to an end of the ﬁber. A detectorof FHT1 has 96 scintillator segments and the eﬀective area of 196 mm (H) × 450 mm (V).
On the other hand, A detector of FHT2 has 128 scintillator segments and the eﬀective
area of 256 mm (H) × 450 mm (V).
TOF
The plastic scintillator wall was located most downstream of KURAMA spectrometer.
Flight length of a scattered particle from the targets to TOF is approximately 3 m.
Figure 2.17 shows the schematic drawing of the TOF. TOF consists of 24 segments ofplastic scintillators (EJ200) with a size of 80 mm (H) × 1800 mm (V) × 30 mm (T).
The scintillators are arranged in a staggered relation with an overlap of 5 mm each other.
Scintillation signals are read by two PMTs (Hamamatsu H1945) attached on the bothvertical ends of the scintillator through light guides. To prevent beam particles frommaking unwanted triggers, special TOF segments were prepared. In the special TOF
segment, the vertical center region of 20-cm length was substituted by an acrylic bar.
Owing to this prescription, TOF became insensitive to beam particles, keeping suﬃcient
acceptance for scattered particles.

31Figure 2.16: Photo of SDC3 and FHT2 setup.

Acrylic

Figure 2.17: Schematic drawings of TOF. Six segments near the π + beam spot werechanged to the special segments, where the vertical center region of 20-cm length wassubstituted by an acrylic bar.
32

2.6

Liquid-hydrogen (LH2) target

Figure 2.18 shows the schematic drawings of the LH2 target system. In this system, thehydrogen target is cooled by a heat exchanger using a GM cyrocooler. Figure 2.19 shows
the drawings of the LH2 target container, which is mainly made from a Mylar sheet of
0.25 mm thickness. It is cylindrical container of diameter 40 mm and length 300 mm withhalf-sphere end-caps at both edges. A vaccum window around the target region is createdusing a CFRP cylinder of diameter 80 mm and thickness 1 mm. The target density wascontinuously measured via the monitored gas pressure. It was 0.070743(4) g/cm 3 .

Figure 2.18: Schematic drawings of the LH2 target system.

Figure 2.19: Drawings of the LH2 target container.

2.7

CATCH

Charged particles involved in Σ+ p scattering, such as the recoil proton and decay productsof Σ+ , were detected using detector system, CATCH [39]. Figure 2.20 shows the schematic
drawings of CATCH. CATCH surrounded the LH2 target and comprised a cylindricalﬁber tracker (CFT), BGO calorimeter (BGO), and plastic scintillator hodoscope (PiID).
The trajectory was reconstructed using CFT and the kinetic energy was measured by33

summing the energy deposits in CFT and BGO on the reconstructed trajectory. Using
the correlation of energy loss in CFT and total energy deposit in CATCH, protons canbe distinguished from π ± and e± .

CFT

BGO

LH2 target

PiID

Figure 2.20: Schematic drawings of CATCH, perspective view (Left) and cross-sectionalview (Right). The BGO and PiID segments overlapping the K + path to the KURAMA
spectrometer were removed.

2.7.1

Detectors of CATCH

CFT
A cylindrical ﬁber tracker with 400-mm length along the beam axis surrounds a LH 2
target and reconstructs tracks of charged particles from the target. CFT comprises eightcylindrical layers of plastic scintillation ﬁbers (Kuraray SCSF-78M) with a diameter of 1mm. Scintillation signals are read using a MPPC (Hamamatsu S10362-11-050P) contacted
at an end of each ﬁber. Radial distance and number of ﬁbers for each layer are summarizedin Table 2.4 and the schematic drawings of the ﬁber arrangements are shown in Figure
2.21. The ﬁbers are placed parallel to the beam axis in four layers, called ϕ layers. In theother four layers, called uv layers, ﬁbers are arranged in a spiral shape. This conﬁguration
enables the reconstruction of trajectories in three dimensions.
Table 2.4: Radial distance and number of ﬁbers for each layer of CFT.
ϕ layer
ϕ1
ϕ2
ϕ3
ϕ4Radial distance [mm] 54
64
74
84
Number of ﬁbers
584 692 800 910
uv layer
u1
v2
u3
v4
Radial distance [mm] 49
59
69
79
Number of ﬁbers
426 472 510 538
Tilt angle [degree]
37.6 42.8 47.3 51.1

34

Figure 2.21: Schematic drawings of the CFT ﬁber arrangements for so-called ϕ and uv
layers [39].
BGO
A calorimeter was placed outside around CFT to measure the kinetic energy of protons
from Σ+ p scattering by stopping it in the calorimeter. The size of each BGO crystal
is 400 mm (l) × 30 mm (w) × 25 mm (t). Scintillation signals are read using PMT(Hamamatsu H1 1934-100) attached at an end of each BGO crystals. In the present
experiment, the average singles rate of a BGO segment was 80 k/spill and a raw signal
from BGO has a relatively long decay time of 300 ns. In order to decompose the pile-up
events, the waveform readout is needed. The raw BGO signal is ﬁltered with a integral
circuit and shaped to the suitable waveform for the ﬂash ADC readout as shown in Figure
2.22. Then, the waveforms were recorded by a ﬂash ADC module (CAEN V1724) with asampling frequency of 33 MHz. From the recorded waveform, the pulse height informationis reconstructed by ﬁtting the data point with a template waveform.

Figure 2.22: Typical waveform of the BGO calorimeter after the shaping circuit [39]. The
black points show the ﬂash ADC data with a sampling rate of 33 MHz. The blue line
shows the reconstructed pulse shape by ﬁtting the data point with a template waveform.

PiID
A hodoscope was located most outside of CATCH to determine whether the charged
particles penetrated BGO or not. PiID comprises 32 plastic scintillators (EJ212) with asize of 400 mm (l) × 30 mm (w) × 15 mm (t). As SCH and FHT, each scintillator has a35

hole on the scintillator surface to put a wave length shifting ﬁber (Kuraray Y-11 (200)M)
with a diameter of 1 mm. Scintillation signals are read using a MPPC (Hamamatsu
S10362-11-100P) attached to an end of the ﬁber.

2.8

TriggerIn the data taking, the (π + , K + ) events were selected by the ﬁrst- and second- level
triggers. Figure 2.23 shows the diagram of the trigger system. From the logic signals
from the trigger counters, trigger signal was made using three FPGA-based modules,
HUL trigger [50], HUL martix trigger, and HUL mass trigger modules.
8

BH2
SAC
TOF-HT

HUL Trigger
Module8

HUL Matrix
Module

8

TOF

SCH

×8

HUL
Mass Trigger
ModuleL1 Accept

L2 Accept or Clear

L1 Request

SFT

MasterTriggerModuleTrigger

Readout
Busy

Figure 2.23: Diagram of the trigger system. BH2 × TOF × SAC × TOFlargedE logicalsignal were generated in the HUL trigger module. 2D- and 3D- matrix trigger signals
were generated by the HUL Matrix trigger module. The 2nd-level trigger is decided by
the HUL mass trigger module.
The ﬁrst-level πK trigger is deﬁned as
BH2 × TOF × SAC × TOFlargedE × 2DMtx × 3DMtx,(2.1)

where TOFlargedE is a veto signal using the pulse height of TOF. It could reject particles
with small β, that is, mainly protons with low momenta. 2DMtx and 3DMtx represent
two types of matrix triggers, in which combinations of hit segments of SCH, TOF, andSFT x plane were used to decide the trigger. Figure 2.24 shows the correlations of the
SCH and TOF hit segments. In the left ﬁgure of Figure 2.24, we can see the verticalstripe like event concentration around SCH segment 10. It corresponds to events frominteraction due to beam particles or beam particles themselves. Therefore, 2DMtx vetoregion was determined to reject these events. On the other hand, the 3DMtx accepted
region was determined based on the result of the K + identiﬁcation analysis. Because themomenta of the scattered K + were strongly correlated with the hit segment combinations,the matrix trigger was quite eﬀective to suppress the trigger rate. The trigger timing wasdetermined as the BH2 timing, by taking the coincidence of the BH2 and ﬁrst-level trigger
36

TOF segment

signals. The coincidence logical signals were created for each segment of BH2 and thensummed in order to prevent saturation of the OR signal due to the high intensity beam.
24
22
20
18
16
14
12
10
8
6
4
2
0
0

10

20

30

40

50
60
SCH segment

Figure 2.24: Correlations of the SCH and TOF hit segments for all events without matrix
triggers (Left) and K + events (Right). The region within blue lines in the left ﬁgure
indicates 2DMtx veto region. The 3DMtx accepted region is inside of the red lines in theright ﬁgure.

CountsIn addition to momentum information from the hit segment combinations, selecting thetime-of-ﬂight of the scattered particle enables us to identify K + more correctly. This is aprinciple of the second-level trigger called “mass trigger.” TOF timings, which correspondto the time-of-ﬂight between BH2 and TOF, were recorded by a high-resolution TDC in
the HUL mass trigger module and the second-level trigger was decided. The gate of themass trigger was optimized for each combination of SCH and TOF, which was acceptedby 3DMtx. A typical gate width was 4 ns. Figure 2.25 shows the TOF timing distribution
by particle species. Owing to the mass trigger, many background protons and pions wereprevented from making triggers.

104
103
102
10
1
180

185

190

195

×103
200
205
210
215
220
High-resolution TDC value [ch]Figure 2.25: TOF timing distribution, which corresponds to the time-of-ﬂight between
BH2 and TOF, for a combination of TOF and SCH hit segment. The red ﬁlled histogram
corresponds to K + event. The green, black, blue lines corresponds to π + , proton, andrelatively low momentum (<0.9 GeV/c) proton, respectively. The light blue vertical lines
represents the time-of-ﬂight gate to decide the mass trigger.

37

2.9

DAQ

DAQ gate

MTM

Trigger

Front-End PC
HUL RM
HUL HRTDC
HUL scaler

VME busXVB 601
VME RMQDC CAEN V792

Spill gate

VME bus

A network controlled system was adopted to the data acquisition (DAQ) system of K1.8
beam line. Figure 2.26 shows a diagram of the DAQ system in the K1.8 beam line. Data
from each detector was read out through each subsystem. In order to synchronize theevent between diﬀerent subsystems, a trigger managing system using a master trigger
module (MTM) and a receiver module (RM) were adopted. The trigger managing system distributed an event tag, which comprises spill number and event number, to eachsubsystem.

Main DAQ computer
Controller
Event builder
Event distributer
RecorderFE PC (BC3,4,SDC1)
HUL RM
HUL MHTDC

HUL RM
HUL MHTDC

Online monitor
Trigger

FE PC (CFT, PiID, FHT)
VME RM
VME EASIROC(fast)
L1 trigger

XVB 601
VME RMCAEN V1724 (×1)
CAEN V1724 (×2)
Optical controller

Ethernet cable

VME bus VME bus VME bus

FE PC(SDC2,3 ,SFT, Mst, Mtx)

Event tagBusy
Data transfer
Slow control

VME bus

Front-End PC
EASIROC BOARD

Figure 2.26: Schematic diagram of the DAQ system in the K1.8 beam line.
Regarding the readout modules, Hadron Universal Logic (HUL) modules [51] has beendeveloped recently. This FPGA-based modules has good extensibility by changing the
mezzanine cards. In the present experiment, HUL RM, HUL Multihit TDC (MHTDC),
and HUL High-resolution TDC (HRTDC) were extensively used. Pulse height information
on BH1, BH2, TOF, and SAC were recorded using CAEN V792 modules, which were
controlled by VME-CPU module (XVB 601). Timing information on BH1, BH2, TOF
were recorded using HUL HRTDC. MPPC signals were read by the two types of EASIROC
modules: EASIROC test board and VME-EASIROC module. Details of these EASIROC
38

modules are explained in Ref. [52] and [39], respectively. Timing information on BFT and
SCH were recorded by EASIROC test boards. Although signals from SFT were ampliﬁed,
shaped, and discriminated in EASIROC test boards, discriminated timing signals wererecorded by the HUL MHTDC modules. Both timing and pulse height information on
CFT, FHT, PiID were recorded using VME-EASIROC modules. The waveforms of signals
from BGO calorimeter were recorded using three ﬂash ADC modules, CAEN V1724,
controlled by VME-CPU module. In order to reduce the data transfer time, one module
was read by the VME-CPU module and the other two module were read by so-called
optical controller computer. Signals of all drift chambers were recorded using the HULMHTDC modules.
Digitized signals from each sub system were collected to the host computer. A part ofdata were distributed to the online monitor system to check the detector performances.
Typical DAQ performances and trigger rates in the data taking are summarized inTable 2.5.
Table 2.5: Typical DAQ performances and trigger rates in the data taking. For the events
rejected by the Level 2 trigger, the Level 2 decision time was the eﬀective DAQ dead time.
DAQ busy
30 µs
Level 2 decision time
4 µs
DAQ eﬃciency
82%
Data size
20.6 kB/event
Level 1 request
2.6 × 104 /spill
Level 1 accept
2.2 × 104 /spill
Level 2 clear
1.3 × 104 /spill
Level 2 accept
9 × 103 /spill
Level 2 eﬃciency
41%

2.10

Data SummaryThe Σ+ p scattering data taking of J-PARC E40 experiment was performed in April 2019and May-June 2020 with the same experimental setup. In each period, we collected
Σ+ p scattering data for approximately 240 hours of beam time. Pion On Target (POT)
in 2019 and 2020 run were 3.89 × 1012 and 3.65 × 1012 , respectively. Additionally, pp
scattering data using proton beams with various momenta between 0.45 and 0.85 GeV/cwere collected. The pp scattering data were used for energy calibration and estimation of
CATCH detection eﬃciency.

39

Chapter 3
Analysis ﬂow and Monte Carlosimulation
3.1

Analysis ﬂow

A conceptual drawing of the present experiment is shown again in Figure 3.1. Σ + p
scattering events can be identiﬁed by measuring the Σ+ momentum, the recoil proton’s
kinetic energy and the proton’s recoil angle.
Kinetic energy of recoil protonp2.

±p scattering

LH2 target±p scattering

Recoil angle of proton±±±consistency check
identification of the

decay

p

1. production: ±p K+
Momentum of +± reaction

K+

Figure 3.1: Conceptual drawing of the Σ+ p elastic scattering in the J-PARC E40 experiment same as Figure 2.1. The initial Σ+ is produced by the π + p → K + Σ+ reaction. Then,it travels in the LH2 target and may cause Σ+ p scattering. The Σ+ p elastic scattering canbe kinematically identiﬁed by measuring the initial Σ + ’s momentum, the recoil proton’s
kinetic energy, and recoil angle.
The whole analysis ﬂow to deriving the diﬀerential cross sections is summarized in
Figure 3.2. The analysis of the Σ+ p scattering events comprises three stages. In the ﬁrst
stage analysis, the spectrometer systems, K1.8 beam-line and KURAMA spectrometers,and CATCH were analyzed independently. From the analysis of spectrometer systems,
Σ+ production events were identiﬁed and the momentum of each Σ + was tagged. The
trajectories of charged particles involved in Σ+ p scattering were reconstructed and kinetic energy of protons were measured using CATCH. In the second stage analysis, theseinformations were combined and Σ+ p scattering events were identiﬁed by requiring the40kinematical consistency for the recoil proton. In the third stage, factors to derive the differential cross sections, such as the Σ+ total track length and eﬃciencies were evaluated.
Finally, diﬀerential scattering cross sections were obtained. The each stage of analysis is
described in Chapter 4, 5, and 6, respectively. The obtained diﬀerential cross sections
were shown in Chapter 7.
K1.8 beam-line
spectrometer
BFT hit

KURAMA
spectrometer

BC3,4 hit
Bcout
Tracking

K1.8
Tracking

Tracker hit
SDCin
Tracking

CATCH

TOF hit

SDCoutTracking

CFT hit

BGO hit

CFT
Tracking

KURAMA
Tracking

Energydepositcalculation

Particle
identification

Protonidentification

Momentumreconstructed ++ production

Momentumreconstructed K+

Momentum identification
reconstructed +Momentumreconstructed
2 proton

+p scattering
identification

Total flight length
evaluationcut
efficiency

Detection
efficiency

Efficiency
evaluation

Differential
cross sections

Figure 3.2: Analysis ﬂow of diﬀerential cross section derivation for Σ + p scattering.

3.2

Monte Carlo simulations

In the following analysis, a Monte Carlo simulation was performed for many purposes. In
the simulation, the dedicated event generator was implemented. By selecting the mode ofthe event generator, the corresponding events were generated among several implemented
reactions. The purposes and the implemented reactions are as follows.
1. Energy calibration of CATCH: The pp and π + p elastic scatterings were generated
to obtain the relation between the proton’s scattering angles and energy deposits
41

in CFT and BGO for protons with a wide momentum range. This relation wascompared to the real data for the CATCH energy calibration. The detail is describedin Subsection 4.4.3.
2. The ﬂight length evaluation of the Σ+ beam: The ﬂight length of Σ+ in the LH2
target was estimated by generating Σ+ particles with the realistic momentum andposition distribution. As the inputs of the Σ+ momentum and position, the analysis
results of the spectrometers were used. The detail is described in Section 6.2.
3. Evaluation of CATCH detection eﬃciency for protons: Protons with arbitrary angle
and energy were generated for evaluating the detection eﬃciency with CATCH. The
detail is described in Subsection 6.3.1.
4. Σ+ p scattering events simulation for background estimation and the overall eﬃciency
evaluation of CATCH: The π + p → K + Σ+ reaction and the sequential secondary
reactions in the LH2 target were generated. Figure 3.3 shows the ﬂow chart of theevent generation. As the secondary reactions, not only the Σ + p scattering but also
background reactions that were the reactions between the Σ + decay products andproton, such as the pp scattering following the Σ+ → pπ 0 decay, were implemented
to reproduce the background structure. In this simulation, the diﬀerential crosssection of the Σ+ p scattering was assumed to be 2.4 mb/sr with an isotropic angular
distribution. Moreover, the secondary backgrounds were generated based on theircross sections. Further, accidental coincidence events were generated based on theirmeasured probabilities described in Subsection 4.5.3.
The information about the generated particles by the event generator was transferredto the passage simulator based on Geant4 package [53]. In this simulator, the LH 2 target, all detectors and most materials of the KURAMA spectrometer and CATCH (theKURAMA magnet, the frames of CFT, and so on) were implemented. The interactions
between the generated particles and experimental materials were simulated, and the information about the energy deposit and hit position in the virtual detectors was recorded
considering the energy and position resolutions of the real detectors. The simulated
records were analyzed in the same analysis program as the real data to compare between
them.

42

startGenerate
π+p→Σ+K+

Σ stepΣ+ decay?

No

Yes
N(n or p), π+In targetNoΣ+p scat?

N,π+ step

Yes
Generate
Σ+’, recoil p

Np

Np,π+p scat?

π+pNo
Generate
N’ π, recoil p

Generate
π+’ N, recoil p

Generate
Σvirtual

No

Accidental?

Yes
Generate
Accidental π,pTo simulator
part

Figure 3.3: Flow chart of Σ+ p scattering and background generation in the event generator. The “step” means a series of procedure: Moving a particle to small displacementx + dx, the occurrences of decay or scattering were probabilistically judged and the energy loss in the displacement was calculated to iterate the particle momentum. Whensecondary N p and π + p scattering does not occur, a virtual Σ+ particle with a lifetime of
0 is generated at the decay point in order to reproduce mere Σ + decay events.

43

Chapter 4
Analysis-I:Analysis of spectrometers
and CATCH
In this chapter, independent analyses of the spectrometer systems for Σ + production
identiﬁcation and CATCH for proton detection are described. The Σ + particles wereidentiﬁed from the missing mass spectrum of the π + p → K + X reaction. The particle
identiﬁcation and momentum reconstruction for both the π + beam and outgoing K +were performed with the K1.8 beam-line spectrometer and the KURAMA spectrometer,
respectively. The momenta of the produced Σ+ particles were obtained from the momentaof π + and K + . The trajectories of the charged particles involved in Σ + p scattering, suchas the recoil proton from Σ+ p scattering and Σ+ → pπ 0 decay were reconstructed andkinetic energy of protons were measured using CATCH.

4.1

π + analysis using the K1.8 beam-line spectrometer

The momenta of the π + beam particles were analyzed by the K1.8 beam-line spectrometer with the 3rd-order transfer matrix method. The x position at the entrance of thespectrometer was measured with BFT. The position and direction at the exit of it wasreconstructed with two MWDCs (BC3, 4).
The contaminations of other particle species were negligibly small due to setup of
ESS ,CM magnets and MS slits, which can be conﬁrmed from the time-of ﬂight spectrum
between BH1 and BH2 shown in Figure 4.1. The time oﬀset of time-of-ﬂight was adjusted
to π + for each BH1 segment. The typical time-of-ﬂight resolution was 300 ps in σ. Because
the slewing correction using the correlation between the pulse height and the timing could
not be completely performed due to the signal pile-up under the high counting rate, thistime-of-ﬂight resolution was not the best value, while this resolution was enough to identify
the time diﬀerence of ∼ 1.9 ns between 1.41 GeV/c π + and K + .

44

Counts

×103
180
160
140
120
100
80
60
40
20
0
−5

−4

−3

−2

−1

0
1
2
3
4
5
BH1-BH2 Time-Of-Flight [ns]Figure 4.1: Time-of ﬂight distribution between BH1 and BH2; These had a ﬂight distanceof 10.4 m.

4.1.1

BFT analysis

BFT position [mm]

BFT measured the x position of beam particles upstream of the QQDQQ analyzer magnets. Because BFT had two layers conﬁguration (xx′ ) with an overlap, neighbor hits wereclustered. A typical time resolution of BFT was about 900 ps in σ under the 20M/spill
beam intensity and the time gate of ±6 ns was applied to select the triggered event. 96.8%of the triggered events had at least one clustered hit. In about 40% of the triggered events,
BFT had multi clustered hits including accidental hits. In order to reject accidental BFT
hits, the correlation between the BH1 hit segment and BFT hit position, which is shownin Figure 4.2, was used. Applying this selection rule, the eﬀective BFT multi clustered
events were reduced to 9.6%.
80
60
40
20
0
−20
−40
−60
−80
0

2

4

6

8
10
BH1 segment [ch]Figure 4.2: Correlation between BH1 hit segment and BFT hit position. When BFT had
multi clustered hits, Only BFT hits lying within red lines were accepted.
45

4.1.2

Local Tracking for BCout (BCout Tracking)The position and direction at the exit of the QQDQQ analyzer magnets were reconstructed using two MWDCs (BC3 and BC4). Both chambers had a six layers conﬁguration (xx′ uu′ vv ′ ) with 3-mm wire spacing. For a track candidate, which is the sets of hit
information, the straight line ﬁtting was performed to minimize the reduced chi-squareas follows;χ2 /ndf =12
 X − f (z ) 2
X
1
i
i
Hi
,
Nhit − 4 i=1
σi

(4.1)

Xi = wp ± dli (t),f (zi ) = x(zi ) cos αi + y(zi ) sin αi ,x(zi ) = x0 + u0 zi ,y(zi ) = y0 + v0 zi ,(1 (If a hit of i-th layer is included in the track candidate)Hi =0 (otherwise)
Nhit =

12
X

Hi ≥ 8,i=1

where, x0 , y0 , u0 and v0 are the parameters representing the horizontal and vertical positions and angles of the track. The tilt angle and z position of i-th hit plane is denoted as
αi and zi , respectively. The tilt angles αi of xx′ , uu′ , and vv ′ planes were 0◦ , +15◦ , and−15◦ , respectively. The local hit position in each layer, Xi was calculated by the positionof hit wire (wp) and drift length(dl) which was calculated from the drift time (t) and thedrift function. σi is the intrinsic resolution of each plane, typically 200 µm. When thelayers had multiple hit wires, all possible track candidates were ﬁtted in order to ﬁnd
the most probable track by sorting them according to the χ2 /ndf. In order to prevent
combinatorial explosion, a correlation between the hit wires and hit segment of BH2 wasused. The typical reduced chi-square distribution is shown in Figure 4.3. The threshold
was set to 10 in order to select good events.

46

Counts

105

104

103

0

5

10

15

20
25
30
χ2 of BC3 & 4 tracking

Figure 4.3: Distribution of the reduced chi-square of BCout tracking. The black and redlines show the total and single-track events, respectively.

4.1.3

K1.8 Tracking

The π + beam momentum was reconstructed by connecting the upstream horizontal position and downstream trajectory using the 3rd-order transfer matrix. The transfer matrixof the K1.8 beam-line spectrometer with QQDQQ analyzer magnets conﬁguration was
calculated using ORBIT code [54]. Horizontal position at the entrance xin was measured
using BFT and positions and angles at the exit, xout , yout , uout , and vout were analyzed
by BCout tracking. Using inverse transfer matrix M (for i-th order, denoted as M (i) ) ,xin can be written with xout , yout , uout , vout , and the momentum deviation from the central
momentum δ as
xin =

5
X
i=1

(1)

M1i ξi +X(2)

M1ij ξi ξj +

i,j

X

(3)

M1ijk ξi ξj ξk ,

(4.2)

i,j,k

ξ = (xout , uout , yout , vout , δ).
The equation (4.2) can be regarded as a cubic equation for δ and the beam momentum is obtained as a general solution of the equation. The momentum resolution of theK1.8 beam-line spectrometer σp /p in designed value is better than 1 × 10−3 [46]. The
reconstructed momentum distribution of π + beam is shown in Figure 4.4. The spread of
momentum distribution is mainly determined by the setting of the momentum slit in the
beam line.

47

Counts

×103
700
600
500
400
300
200
100
0
1.32

1.34

1.36

1.38

1.4

1.42 1.44 1.46 1.48
Momentum [GeV/c]

Figure 4.4: Momentum distribution of the π + beam analyzed by the K1.8 beam-line
spectrometer.

4.2

K + analysis using the KURAMA spectrometer

The outgoing tracks produced from the LH2 target by the π + p reaction were analyzed by
the KURAMA spectrometer. The trajectories of the outgoing particles in the magnetic
ﬁeld were traced with Runge-Kutta method [48], based on equation of motion deﬁned
by the initial parameters, namely, the momentum vector and the position at TOF. The
parameters were optimized in order to reproduce a set of hit positions measured by the
tracking detectors. This procedure is called “KURAMA Tracking”. In order to determine
the set of the hit information used by KURAMA Tracking, local straight line trackings
upstream and downstream of the KURAMA spectrometer were performed. These local
tracking are called “SDCin Tracking” and “SDCout Tracking”, respectively.
The setup of the KURAMA spectrometer is shown again in Figure 4.5. Not only
scattered particles but also π + beam particles pass through the KURAMA spectrometer.
Because the drift chambers cannot be operated stably under the high intensity beam of
2.0 × 107 /spill, three MWDCs (SDC1-3) were made insensitive to beam particles. SDC1
was apart from the beam center and insensitive to particles near the beam spot. SDC2and SDC3 had cross-shaped insensitive areas as shown in Figure 4.6. The horizontal
insensitive areas were covered by FHT1 and FHT2. However, the vertical insensitive were
not recovered. Therefore, vertical positions of particles near the beam height were notmeasured by the SDC2 and SDC3.

48

KURAMA
Spectrometer

TOF
FHT1,2

SDC2,3SCH
SAC

KURAMA
Magnet
SDC1SFT

LH2 target
CATCH

Figure 4.5: Schematic drawing of the KURAMA spectrometer same as Figure 2.9. Five
curves represent typical trajectory for expected particles. The blue curve corresponds
to 1.4 GeV/c π + beam and the other four red curves correspond to K + s following thekinematics of the π + p → K + Σ+ reaction. “SDCin Tracking” was performed using hits of
SFT, SDC1, and SCH. “SDCout tracking” use hit information on SDC2, SDC3, FHT1,
FHT2 and TOF.

Figure 4.6: Rough sketch of insensitive region in SDC2 or SDC3. The area surrounded by
red dotted lines indicates insensitive region of SDC and the beam particles passed througharound the center of cross-shape, where SDCs were completely insensitive. Although thehorizontal insensitive areas were covered by FHT, the vertical insensitive areas could not
be recovered.
Because K + with small scattering angles had a similar trajectory to beam particles,49

particles near the beam spot (in SDCin Tracking) and near the beam height (in SDCout
Tracking) should be analyzed as many as possible. For this purpose, looser tracking
scheme is applied to particles near the beam spot and height. The ﬂow of SDCin and
SDCout tracking were shown in Figure 4.7.

No

SDCin Hit

SDCout Hit

SFT
SDC1
SCH

SDC1, SDC2
FHT1, FHT2
TOF

SDC1
has hit?

NormalSDCoutTracking

Loose
SDCin
Tracking

Yes
NormalSDCin
Tracking

Yes

No
Beam
height
flag?

SDCin
Track

Loose
SDCoutTracking

No
Is Track
Found?

No
Terminate
Tracking

No
Near beamheight?

No

FinishTracking

Yes

Is Track
Found?

Yes

Beamheightflag

Yes

Is Track
Found?

SDCoutTrack
Terminate
Tracking

FinishTracking

Figure 4.7: Flow of SDCin and SDCout tracking.

4.2.1

SDCin Tracking

SDCin local tracking was performed using hits of SFT, SDC1, and SCH. The speciﬁcationsof each layer were summarized in Table 4.1. SDC1 was placed apart from the beam axis
and insensitive to particles near the beam spot. In addition to “normal tracking” similar to
BCout Tracking, “loose tracking” without straight line ﬁtting was additionally performed.
Normal SDCin Tracking
When particles hit SDC1, the straight line ﬁtting like BCout tracking can be performed.
Because SDC1 wires had honeycomb conﬁguration, electric ﬁeld around an anode wire
had a circular shape. Then, the hit position calculation from drift length needed the track50

Table 4.1: Speciﬁcations of each layer used for SDCin Tracking.
detector plane resolution σi [mm] tilt angle αi [degree]
note

i
1
2
3

SFT
SFT
SFT

u
v
x

0.26
0.26
0.18

45
-45
0

4-5
6-7
8-9

SDC1
SDC1
SDC1

vv ′
xx′
uu′

0.30
0.30
0.30

15
0
-15

10

SCH

x

6

0

xx′ clustered

scintillator

angle with respect to perpendicular direction to the chamber plane, which was determinedby iteration. The reduced chi-square in l-th iteration is described as follows,9 X
 X − f l (z ) 2 X
 cos θl (X l − f l (z l )) 2 
1
i
i
i
i
i
(χ /ndf) =Hi+Hi, (4.3)lNhit − 4 i=1,2,3,10
σi
σ
i
i=4
{z
}
|
{z
} |
2

l

for SDC1

for SFT, SCH

Xi = sp,lf (zi ) = xl (zi ) cos αi + y l (zi ) sin αi ,xl (zi ) = xl0 + ul0 zi ,y l (zi ) = y0l + v0l zi
 df l ilθi = arctan= arctan(ul0 cos αi + v0l sin αi )ldzi(
wpi + dli (ti ) cos θil−1 (fil−1 > wpi )lXi =wpi − dli (ti ) cos θil−1 (fil−1 < wpi )(wzi − dli (ti ) sin θil−1 (fil−1 > wpi )lzi =wzi + dli (ti ) sin θil−1 (fil−1 < wpi )(1 (If a hit of i-th layer is included in the track candidate)Hi =0 (otherwise)
Nhit =

10
X

Hi ≥ 5,i=1where, parameters to be determined are x0 , y0 , u0 , v0 . θ and wz indicate the track angle
and the z position of a hit wire, respectively. sp represents the position of the SFT and
SCH hit segment. σi is the intrinsic resolution of each plane, which is summarized inTable 4.1. In the ﬁrst iteration, the θ was set to 0. Iteration was carried out several times
until the reduced chi-square value converge, where the diﬀerence from the last iteration
became less than 1 × 10−3 . The typical reduced chi-square distribution is shown in theleft side of Figure 4.8 and the threshold was set to 50.
For the particles near the beam height at SDC2 and SDC3 position, loose SDCout51

tracking scheme should be applied. By extending the found track to SDC2 and SDC3
position, whether the particles were near the beam height or not were judged.
Loose SDCin Tracking
SCH and SFT-uv planes were sensitive to both beam and scattered particles whereas
beam particles did not hit SDC1 and SFT-x plane. However, scattered particles withsmall scattering angles could hit SFT-x. Therefore, particles hitting all SFT planes andSCH should be analyzed as a scattered particle. Since the straight line ﬁtting cannot
be applied to events with only four layer hits, the mere combination of SCH and SFT
hit information was treated as SDCin track. Because of small scattering angles, theseparticles would be near the beam height at SDC2 and SDC3 position.

4.2.2

SDCout Tracking

SDCout tracking was performed using hits in SDC2, SDC3, FHT1, FHT2 and TOF. The
speciﬁcations of each layer were summarized in Table 4.2. TOF was regarded as tracking
detecter having 2 layers conﬁguration (xy). Because SDC2 and SDC3 had the verticalinsensitive areas as explained in Figure 4.6, vertical position information (y) for particles
near the beam height could not be measured. Therefore, loose tracking scheme, tracking
in a zx-plane, was applied to events where normal tracking could not reconstruct any
track and a particle near the beam height was predicted by the SDCin Tracking result.

i

Table 4.2: Speciﬁcations of each layer used for SDCout Tracking.
detector plane resolution σi [mm] tilt angle αi [degree]
note

1-2
3-4

SDC2
SDC2

xx′
yy ′

0.40
0.40

0
90

5-6
7-8

SDC3
SDC3

yy ′
xx′

0.40
0.40

90
0

9
10

TOF
TOF

x
y

40.0
20.0

0
90

11-18

FHT1,2

xx′

0.58

0

scintillator
scintillator

Normal SDCout tracking
Firstly, the local straight tracking similar to SDCin normal tracking was performed. Because both of SDC2 and SDC3 had honeycomb-type wire conﬁguration, the tracking was
performed according to minimize the chi-square almost same as Equation (4.3).
The typical reduced chi-square distribution is shown in the right side of Figure 4.8.
The χ2 /ndf threshold was set to 50 in order to select good tracking events.

52

Loose SDCout Tracking (straight line tracking in the ZX plane)CountsCountsWhen a SDCout normal tracking cannot ﬁnd any track and a particle near the beamheight was predicted by the SDCin Tracking result, the local straight tracking in the ZX
plane was performed. ZX plane ﬁt is equivalent to the y0 = v0 = 0 and ndf = Nhit − 2case in equation (4.3). The χ2 /ndf threshold was also set to 50.

106

106
105

105

104

104

103
103
102
0

10

20

30

40

50

60

70 80 90 100
χ2 of SdcIn tracking

0

10

20

30

40

50

60 70 80 90 100
χ2 of SdcOut tracking

Figure 4.8: Reduced chi-square distribution of SDCin tracking (Left) and SDCout tracking(Right). The black and red lines show the total and single-track events, respectively.

4.2.3

KURAMA Tracking

In order to analyze the momentum of the scattered particle, the trajectories of the outgoing particles in the magnetic ﬁeld are traced with Runge-Kutta method based on equationof motion. The magnetic ﬁeld was calculated with ANSYS code [55]. The optimal momentum and trajectory of the outgoing particle were obtained by iterating ﬁve parameters(x, y, u, v, p), which represent the momentum vector and the position at TOF. The optimization is performed to reduce the following reduced chi-square as,
X  X hit − X track 2
1
i
i
Hi
,
(4.4)
Nhit − 5 iσi(
1 (If a hit of i-th layer is included in the SDCin/SDCout track).
Hi =,0 (otherwise)
X
Nhit =Hi ,χ2KURAMA /ndf =i=1

where Xihit and Xitrack are the hit position at a detector and the predicted position of
the reconstructed track. The convergence of the iteration was judged by the following
criterion, (χ2l+1 − χ2l )/χ2l < 10−3 , where χ2l represents the result of the l-th iteration. The
typical reduced chi-square distribution is shown in the Figure 4.9. The χ2 /ndf threshold
was set to 50. About 75 % of triggered events had at least one KURAMA track.

53

Counts

105

104

103

0

10

20

30

40

50

60 70 80 90 100
χ2 of Kurama tracking

Figure 4.9: Reduced chi-square distribution of KURAMA tracking. The black and redlines show the total and single-track events, respectively.
The momentum resolution of the KURAMA spectrometer was evaluated by the combined analysis of KURAMA and CATCH for the π + p elastic scattering as describe in
Subsection 4.5.2.

4.2.4

K + identiﬁcation

The mass of an outgoing particles was calculated using the KURAMA tracking result
and the measured time-of-ﬂight. From the mass, scattered K + can be identiﬁed. The
velocity β of the particle can be calculated using the path length Ltrack of the reconstructedtrajectory and the time-of-ﬂight between the target and TOF, t. The mass squared m2
for each outgoing particle was calculated as; p 2(1 − β 2 ),
(4.5)
m2 =
βc
Ltrackβ=.
ctFigure 4.10 shows the correlation between m2 and the reconstruct momentum of scatteredparticles. Basically, K + from the π + p → K + Σ+ reaction were identiﬁed by the m2 gate
as 0.15 < m2 [GeV2 /c4 ] < 0.40 and the momentum gate as 0.65 < p[GeV/c] < 1.05.
In the following analysis, the contributions of other reactions with K + , such as theπ + p → K + Λπ + reaction, were negligible because momenta of K + from these reactions
were enough small to be rejected by the matrix trigger (at a trigger level) and the momentum gate. However, there are backgrounds due to the accidental coincidence. In order toreject them, an additional event selection was applied.
Figure 4.11 shows the distributions of m2 by the diﬀerent number of BFT clusteredhits, which is related to a probability of accidental coincidence. A constant backgroundstructure in the m2 spectrum can be seen between the π + and the proton peaks. Thesebackground events were caused by the high-intensity π + beam. When multiple beam
particles including accidental coincidence hit the same segment of BH2 within a shorter
54

Momentum [GeV/c]

2

3

10

1.8
1.6
1.4
102

1.2
1
0.8

10

0.6
0.4
0.2
0

−0.4 −0.2

0

0.2

0.4 0.6 0.8 1 1.2 1.4
Mass squared m2 [(GeV/c2)2]

1

Figure 4.10: Correlation between the squared mass m2 and the reconstructed momentumof scattered particles. K + from the π + p → K + Σ+ reaction were identiﬁed in the redsurrounded region.

2500

Counts

10000

CountsCountstime interval than its pulse shape, BH2 sometimes recorded the timing of the accidental
particle instead of reacted particle. These fake BH2 timings irrelevant to the reaction
results in the miscalculation of time-of-ﬂight and therefore m2 . As shown in the center
ﬁgure in Figure 4.11, The background can be suppressed for the event with single BFT
clustered hit because such a event had low probability to have multiple beam particles.
On the other hand, it was enhanced for the BFT multi-cluster events as shown in theright ﬁgure in Figure 4.11.

8000
7000

8000
6000

2000

6000
1500

5000
4000

4000
2000

1000

3000
2000

500

1000
0
−0.4−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
Mass squared m2 [(GeV/c2)2]0−0.4−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
Mass squared m2 [(GeV/c2)2]0−0.4−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
Mass squared m2 [(GeV/c2)2]

Figure 4.11: Reconstructed mass squared distribution for all events (Left), BFT single
cluster events (Center) and BFT multi cluster event(Right). The momentum selection of
0.65 < p[GeV/c] < 1.05 was already applied. Blue and red lines represent distributions
without and with TOF dE/dx cut, respectively.
The energy loss in TOF was used to select K + -like events and suppress the constantbackground. The left side of Figure 4.12 shows the correlation between the momentum
and the dE/dx in TOF. The loci correspending to π + , K + and proton could be seen
depending on the diﬀerence of the energy deposit in each momentum. Because they55

8dE/dx [MeV/cm]

dE/dx [MeV/cm]

are not completely separate, this TOF dE/dx selection was not applied for BFT single
cluster events, where the background contamination was already small. In right sideof Figure 4.12, the correlation in the case of BFT multi cluster event with m2 gate of
0.15 < m2 < 0.40, a target of the TOF dE/dx selection, is displayed. There are someevents corresponding to proton and π + in spite of m2 gate. These background events were
rejected by the TOF dE/dx selection.

7
6

p

5
4

7
6

p

5
4

3

K

2

3

+

0.7

0.8

K+

2

π+

1
0

8

π+

1
0.9

0

1
1.1
1.2
Momentum [GeV/c]

0.7

0.8

0.9

1
1.1
1.2
Momentum [GeV/c]

Figure 4.12: Correlation between the momentum and the energy loss in TOF (Left) forall events and (Right) for BFT multi cluster event with m2 gate of 0.15 < m2 < 0.40.
The region between two red lines was selected as the K + region for the BFT multi cluster
event.
The m2 spectrum after applying the event selection is shown in Figure 4.13. Finally,K + was selected from the m2 gate of 0.15 < m2 [GeV2 /c4 ] < 0.40, as indicated by the
red solid lines in Figure 4.13 with an additional momentum gate of 0.65 < p[GeV/c] <1.05. The background structure was estimated as the shaded spectrum in Figure 4.13
by selecting the non-K + region in the TOF dE/dx analysis for multi-beam events. The
contamination fraction of the miscalculated events was estimated to be 10.0% for theselected K + events.

4.3

Σ+ identiﬁcation with (π +, K +) analysis

Σ+ particles were identiﬁed from the missing mass spectrum in the π + p → K + X reaction
using the reconstructed momentum of the π + beam and the outgoing K + . Information
regarding the vertex of the π + p → K + X reaction, which was determined as the closest point between the π + and K + tracks, was additionally used to select the reactions
occurring in the LH2 target.
The left side of Figure 4.14 shows the z-vertex distribution. The LH2 target can be
identiﬁed from −200 mm to 150 mm from the vertex image. For the Σ+ p scattering analysis, the −150 < z[mm] < 150 region shown with the red dotted lines in the left side of
Figure 4.14 was selected, considering CATCH acceptance. The right side of Figure 4.14
shows the correlation between the x- and y-vertices. In this plot, the detection of two
protons with CATCH was required to enhance the background events, owing to interactions between the π + beam and the target vessel. A horizontally wider beam causedsuch background events. x- and y-vertices are required within the red line in the right56

Counts

9000

π+

8000
7000
6000

p

K+

5000
4000
3000
2000
1000
0

−0.4 −0.2

0

0.2

0.4 0.6 0.8 1 1.2 1.4
Mass squared m2 [(GeV/c2)2]

Figure 4.13: Reconstructed m2 distribution after applying the event selection to rejectbackground contamination. The red shaded spectrum shows the estimated backgrounddue to the miscalculated time-of-ﬂight for multi-beam events. The K + and side-band
regions are indicated by the red solid and blue dashed lines, respectively. The sideband region was used to estimate the contribution of the miscalculated events in further
analyses.
side of Figure 4.14 to suppress the backgrounds. The vertex resolution of the spectrometers was evaluated using multi-particle events, such as the π + p → π + π + π − p reaction, bycomparing the vertex obtained from the spectrometer analysis with that obtained fromthe two other tracks measured by CATCH from the same reaction vertex, as describedin Subsection 4.5.1. The z-vertex resolution depends on the scattering angle θK + , andtypical resolutions are σz = 16 mm and 10 mm for θK + = 10◦ and 20◦ , respectively. The
x- and y-vertex resolutions were evaluated as σx = 2.6 mm and σy = 3.5 mm, respectively,
with a negligibly small angular dependence. These vertex resolutions were considered in
the simulation study to estimate the analysis cut eﬃciency.
The missing mass and missing momentum of the π + p → K + X reaction were calculated
as
q
tot 2
) − (pπ − pK )2 ,(4.6)
MX = (Eπtot + mp − EK
pX = pπ − pK ,
(4.7)
p
p
tot
where Eπtot = m2π + p2π and EK= m2K + p2K are total energies of π + and K + , respectively. The missing mass spectrum of the π + p → K + X reaction is shown in Figure 4.15.
A clear peak corresponding to Σ+ was identiﬁed. As mentioned in the previous subsection, there were misidentiﬁcation backgrounds in the K + selection owing to multiple beam
events. Their contribution was examined by selecting the sideband region of K + , as shownby the blue dashed lines in Figure 4.13 with a momentum gate of 0.65 < p[GeV/c] < 1.05.
The red shaded histogram in Figure 4.15 shows the missing mass spectrum for the sideband contribution. We selected Σ+ particles from 1.15 to 1.25 GeV/c2 , as shown by thearrows in Figure 4.15. Contamination was estimated to be 8.6% for Σ + selection. In total,
4.9 ×107 Σ+ particles were accumulated after subtracting the contamination.
57

Y vertex [mm]Counts

700
600
500

50
40
30
20

90
80
70

10

60

400

0

50

300

−10

40

−20

30

−30

20

−40

10

200
100
0
−600

−400

−200

0

200

400
600
Z vertex [mm]0−50−50 −40 −30 −20 −10 0 10 20 30 40 50
X vertex [mm]CountsFigure 4.14: Vertex distributions of the (π + , K + ) reaction. (Left) z-vertex distribution.
(Right) Correlation between the x- and y-vertices for events in which two protons weredetected with CATCH. The gray dotted lines show the envelopes of the target containerand vacuum window. The events inside the red-line region were selected to suppress thecontamination of the reaction at the target container.
×103
800
700
600
500
400
300
200
100
0
1

1.05

1.1

1.15

1.2

1.25 1.3 1.35 1.4
Missing Mass [GeV/c2]

Figure 4.15: Missing mass spectrum of the π + p → K + X reaction for the K + events(black open histogram) and side-band events of K + (red shaded histogram) to estimate
the eﬀect of the contamination of the miscalculated events under the K + region in the m2
spectrum. Σ+ events were selected by the 1.15 < MX [GeV/c2 ] < 1.25 gate, represented
by the arrows.
The reconstructed Σ+ momentum, as the missing momentum of the π + p → K + X reaction, is shown in Figure 4.16. This ranges from 0.44 to 0.85 GeV/c. In the Σ+ p scattering
analysis, the Σ+ events were categorized into three momentum ranges: the low- (0.44 <pΣ [GeV/c] < 0.55), middle- (0.55 < pΣ [GeV/c] < 0.65), and high-momentum (0.65 <pΣ [GeV/c] < 0.80) regions. The resolution of the Σ+ momentum was 6 × 10−3 GeV/c in
σ, which was determined predominantly by the momentum resolution of the KURAMA
58

Counts

spectrometer.
×103
500
400
300
200
100
0
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Missing Mommentum [GeV/c]

Figure 4.16: Σ+ momentum reconstructed as the missing momentum of the π + p → K + Σ+reaction. The red dotted lines show the boundaries of the three momentum regions: low(0.44 < pΣ [GeV/c] < 0.55), middle- (0.55 < pΣ [GeV/c] < 0.65), and high-momentum(0.65 < pΣ [GeV/c] < 0.80) regions.

4.4

CATCH for the recoil/decay protons

4.4.1

Outline

The charged particles involved in the Σ+ p scattering, that is, the recoil proton and decay
product of Σ+ , were detected using CATCH. The trajectory was reconstructed using CFT.
The kinetic energy was measured by summing the energy deposits in CFT (dECFT ) andBGO (EBGO ) along the trajectory. The measured energy is denoted as Emeas . Particle
identiﬁcation in CATCH was performed using the so-called dE-E method between dECFT ,corrected by the path length in CFT (dE/dx), and Emeas for each track. In this way, four
momentum vectors for protons were determined.

4.4.2

CFT Tracking

First of all, trajectories of charged particles were reconstructed using CFT. CFT has thefour ϕ layers in which ﬁbers are arranged parallel to the beam axis (described as Z axis)
and the four uv layers in which the ﬁbers are set in a spiral shape, as shown in Figure 4.17.
In the following, the coordinate axes are deﬁned as Figure 4.17 for the cylindrical shapeof CFT. In order to reconstruct three-dimensional trajectories with CFT, two stages oftracking were performed. The ﬁrst and second stages are called ”ϕ tracking” and ”uv
tracking”, respectively.

59

Figure 4.17: Structures of ﬁber arrangement for the straight and spiral layers of CFT and
the deﬁnition of the coordinates[39].
ϕ tracking
From the hit segment of i-th ϕ plane, the x and y coordinate of the hit position (Xi , Yi )can be calculated as,
Xi = ri cos ϕi , Yi = ri sin ϕi ,where ri is the designed radius of the i-th plane and ϕi is determined from the hit segmentof i-th layer. Using these information, the straight line tracking in the xy plane was
performed with hit positions of ϕ layers to minimize the reduced chi-square as follows,4 Y (r , ϕ ) − (AX (r , ϕ ) + B) 2
X
1
i i
i
i i
i
,
(4.8)
χ /ndf =Hi
Nhit − 2 i=1σi(
1 (If a hit of i-th ϕ layers is included in the track candidate)Hi =,0 (otherwise)2

Nhit =

4
X

Hi ≥ 3,i=1

where, A and B are the parameters to optimize the straight line in xy plane. If the hit
positions are close to y axis, Xi and Yi in the equation (4.8) are inverted. Although similar
inversion is actually considered in the uv tracking, treatment of only one conﬁguration,y = Ax + B and x = u0 z + x0 case, is simply described in the following. σi is theintrinsic resolution of each plane, typically 260 µm. Using the result of ϕ tracking, theϕ-coordinate of the hit position of the j-th uv plane, ϕj can be calculated from thecoordinate of intersection point (Xj , Yj ) between the circle x2 + y 2 = rj2 and half straightline y = Ax + B.
uv tracking
A given segment of the j-th layer of the uv layers was on a straight line in the side planeof a cylinder, zϕ-plane as shown in Figure 4.18.

60

φ=2π

400mm

Expandingz=zminz=zmaxz0

φ=0
z=zminz=zmaxFigure 4.18: Schematic drawing of a ﬁber in the CFT uv layer. It is on a straight line in
the zϕ-plane, which is sideplane of a cylinder.
For example, in the case of u plane, the z position is expressed as follows;zmax −z0,j,seg
 400 ϕ + z0,j,seg
0
≤
ϕ
<
2π
·
2π
400


zj,seg (ϕ) =z−zmax
0,j,seg400
 (ϕ − 2π) + z0,j,seg
2π ·≤ ϕ < 2π ,2π400where zmax represents the z-coordinate of the far end point of CFT and z0,j,seg is theϕ-intersect of the straight line. The straight line tracking in xz plane was performed withhit positions of uv layers and ϕj calculated using the result of the ϕ tracking to minimize
the reduced chi-square as follows,4 X (r ) − (u z (ϕ ) + x ) 2
X
1
j j
0 j
j
0
,
(4.9)
Hj
Nhit − 2 j=1
σj(
1 (If a hit of j-th uv layers is included in the track candidate),Hj =0 (otherwise)χ2 /ndf =Nhit =

4X

Hj ≥ 3,j=1

where, x0 and u0 are the parameters. The vertical position at z = 0, y0 , and angle, v0 , ofthe track can be expressed as y0 = Ax0 + B, v0 = Au0 . In this way, the straight line in
the three-dimensional spacex − x0y − y0==z
u0
v0
was reconstructed.
Correction of ﬁber positions
The positions of CFT ﬁbers were slightly deviated from the design positions due to theworking accuracy of fabrication. The deviations of ﬁbers, dϕ(z), dr(z) for ϕ layer anddz(ϕ) and dr(ϕ) for uv layer, were estimated from the cosmic ray data by comparing the61

3dz [mm]dφ [degree]

designed position with the reconstructed position from the tracking. Because a cosmic
ray crossed CFT and hit 8 × 2 = 16 layers, the accuracy of the position estimation at eachlayer was improved from the normal tracking with eight layers. The typical distributionsof dϕ(z) and dz(ϕ) are shown in Figure 4.19. Figure 4.20 show those of dr(z) and dr(ϕ).

80

2

70

1
0

−2
−3
0

50

100

150

200

250 300 350 400
z in the φ 3 layer [mm]

60

4
3

60

2

50

1

50
40

0

40

−1

5

30

−1

30

−2

20

−3

10

−4

0

−5
0

20
10

50

100

150

200
250
300
350
φ in the uv 1 layer [degree]05

dr [mm]dr [mm]Figure 4.19: Distributions of the ﬁber position deviation dϕ(z) in the CFT ϕ3 layer (Left)and dz(ϕ) in the CFT uv1 layer (Right). dϕ(z) were corrected by 5th-order polynomialof z and dz(ϕ) were corrected by 1st-order polynomial of ϕ. The parameters of dz(ϕ)
correction were determined for each of ϕ = 10◦ section.

25

4
3

20

2

18

4
16

3
14

2

1
−1

10

−2
−3

5

−4
50

100

150

200

250 300 350 400
z in the φ 3 layer [mm]

12

1

15

0

−5
0

5

0

10

−1

8

−2

6

−3

4

−4

2

−5
0

0

50

100

150

200
250
300
350
φ in the uv 1 layer [degree]0

Figure 4.20: Distributions of the ﬁber position deviation dr(z) in the CFT ϕ3 layer (Left)and dr(ϕ) in the CFT uv3 layer (Right). dr(z) were corrected by 5th-order polynomialof z and dr(ϕ) were corrected by 1st-order polynomial of ϕ. The parameters of dr(ϕ)
correction were determined for each of ϕ = 10◦ section.
Tracking with position correction was performed following “1st tracking”, tracking
without correction, by replacing ri 7→ ri +dri (zi,1st ) and ϕi 7→ ϕi +dϕi (zi,1st ) in the equation(4.8), where zi,1st means the z position of i-th ϕ layer hit calculated from the result of
the 1st tracking. Using the result of 2nd ϕ tracking, 2nd uv tracking was performed byreplacing Xj (rj ) 7→ Xj,2nd (rj + drj (ϕj,2nd )) and zj (ϕj ) 7→ zj (ϕj,2nd ) + dzj (ϕj,2nd ) in theequation (4.9). The typical reduced chi-square distributions of ϕ tracking and uv tracking
are shown in Figure 4.21. The distribution became closer to natural χ2 distribution after
the position correction.
62

Counts

Counts

106
105

105

104
104

103

0

20

40

60

80

100 120 140 160 180 200
χ2 of CFT φ tracking

0

20

40

60

80

100 120 140 160 180 200
χ2 of CFT uv tracking

Figure 4.21: Reduced chi-square distribution of CFT-ϕ tracking (Left) and CFT-uv tracking (Right). The black and red lines show the result of the 1st and 2nd tracking, respectively. The thresholds were set to 150 for the 1st tracking and 30 for the 2nd tracking.
Angular resolution of CFT tracking

Counts

The angular resolution of CFT tracking was evaluated using the pp elastic scattering
data with proton beam. In the pp elastic scattering, opening angle of two proton θlab
is 90◦ in the non-relativistic limit. The actual θlab is slightly smaller than 90◦ owing
to relativistic eﬀect. Figure 4.22 shows the opening angle between two protons for pp
scattering with the beam momentumof 0.6 GeV/c. Because the spread of θ1 + θ2 (σθ1 +θ2 )√was expressed as σθ1 +θ2 = 2σθ , angular resolution of CFT tracking σθ was estimated tobe 1.5 degrees. This angular resolution of CFT was taken into account in the simulationfor Σ+ p scattering.

700
600
500
400
300
200
100
0
60

70

80

90
100
110
120
opening angle between 2protons

Figure 4.22: Opening angle between two protons for pp scattering with the beam momentum of 0.6 GeV/c.
The vertex resolutions of the CFT tracking were also studied by reconstructing thetarget container image from the crossing point between two tracks measured by CATCH.
The x and z distributions of the crossing point are shown in Figure 4.23. The z vertex
63counts

countsresolution was evaluated to be 2.7 mm and the x and y vertex resolutions were assumed
to be same with the detector structure and evaluated to be 2.1 mm.
1800
1600
1400

900
800
700

1200

600

1000

500

800

400

600

300

400

200

200

100

0
−30

0
−200 −150 −100 −50
050
100 150 200
Z position of the crossing point between π p [mm]

−20
−10
0
10
20
30
X position of the crossing point between π p [mm]Figure 4.23: x (Left) and z (Right) position distributions of the crossing point between
two tracks detected with CATCH. The arrow in left ﬁgure indicates the target vessel andthe one in right ﬁgure indicates the edge of the target.

4.4.3

Energy measurement with CFT and BGO

For each reconstructed CFT track, the information on ﬁber hits included in the track
was recorded. Measured energies with BGO segments near the CFT track were also
added to the CFT track. By summing up the energy deposits measured in the ﬁbers and
BGO segments along the track, the measured energy deposits of the charged particle wascalculated.
Energy calibration of BGO
Each BGO crystal equipped with a PMT. The waveform data of the shaped signal withan integral circuit were recorded using a ﬂash ADC with a sampling frequency of 33 MHz.
The pulse height and the timing of each waveform were obtained through the templateﬁtting method.
BGO energy calibration was performed with the pp elastic scattering data using 0.6
GeV/c proton beam. The energy of scattered protons has a one-to-one correlation withthe scattering angle θ because the pp elastic scatterings are two-body reactions. However, there were the target vessel and CFT before the BGO calorimeter. This eﬀect wasconsidered by the passage simulator based on Geant4 package [53] described in Section
3.2. The relation between the BGO pulse height and the energy deposit in BGO was
obtained by comparing the measured pulse height to the simulated energy deposit for thesame scattering angle θ, shown in Figure 4.24. The obtained correlation was ﬁtted with aphenomenological relation [56] between the photon yield and the energy deposit expressedby the following equationP HBGO = a × E − b × log

E + bb,(4.10)

where a and b are the parameters and P H and E correspond to the pulse height andthe energy deposit in the BGO calorimeter, respectively. The ﬁtted calibration curve is
64

BGO pulse height [ADCch]

20000

energy deposit in BGO [MeV]shown in Figure 4.25. The BGO energy calibration was performed for all BGO segments
using this method.

70

18000
16000

60

14000

50

12000

40

10000
8000

30

6000

20

4000
10

2000
0
0

10

20

30

40

50 60 70 80 90 100
θ of CFT tracking [degree]

200
180

250

simulation

160

200

140
120

150

100
80

100

60
40

50

20
0
0

0

10

20

30

40

50 60 70 80 90 100
θ of CFT tracking [degree]0

BGO pulse height [ADCch]

Figure 4.24: Angular dependences of the pulse height in the data (Left) and the simulated
energy deposit (Right) in the BGO calorimeter for the pp scattering events caused by 0.6
GeV/c proton beam.

20000
18000
16000
14000
12000
10000
8000
6000
4000
2000
0
0

20

40

60

80 100 120 140 160 180
energy deposit in BGO [MeV]

Figure 4.25: Energy calibration curve for BGO calorimeter.
Energy calibration of CFT
For each MPPC attached to a scintillating ﬁber in CFT, the information on the pulse
height was recorded by peak-hold ADC in VME-EASIROC modules. These ADC values
were normalized layer by layer using pp scattering data. The CFT energy calibrationfrom the normalized ADC values to the energy deposit was performed using the recoil
proton from the π + p elastic scattering with 1.41 GeV/c π + beam. Owing to the higher
beam momentum, the energy range of the recoil proton from the π + p elastic scattering
was wider than that from the pp elastic scattering data. The energy calibration procedure
for CFT was similar to one for BGO. The relation between the normalized CFT ADC
value and the energy deposit in CFT ﬁber was obtained by comparing the ADC value to65the simulated energy deposit for the same BGO energy loss shown in Figure 4.26. The
obtained correlation was ﬁtted with a function as;n −b × dE o
PHCFT = a 1 − exp,(4.11)
a4

energy deposit in uv2 layer [MeV]

normalized CFT pulse height [a.u.]where , a and b are the parameters corresponding to the eﬀective pixel numbers of MPPC
and photon numbers per energy deposit. By this parametrization, the saturation eﬀectof the MPPC is taken into account. The ﬁtted calibration curve is shown in Figure 4.27.

50

3.5
3

40

2.5
30

2
1.5

20

1
10

0.5
0
0

20

40

60

80 100 120 140 160 180 200
energy measured with BGO [MeV]4

simulation

3

35
30

2.5
25

2

20

1.5

15

1

10

0.5

5

0
0

0

40

3.5

20

40

60

80

100 120 140 160 180 200
energy deposit in BGO [MeV]0normalized CFT pulse height [a.u.]

Figure 4.26: BGO energy dependences of the pulse height (Left) and the simulated energy
deposit in the CFT ﬁber (Right) for the π + p elastic scattering events caused by 1.41 GeV/c
π + beam. The protons with the moderate kinetic energy were selected by 55 < θ < 80
gate.

4
3.5
3
2.5
2
1.5
1
0.5
0
0

0.5

1

1.5
2
2.5
3
3.5
4
energy deposit in uv2 layer [MeV]

Figure 4.27: Energy calibration function of CFT ﬁber for the 2nd uv layer
Energy resolution of CATCH
The energy resolution of the summed energy deposit in CFT and BGO, was evaluated fromthe pp elastic scattering data. For each proton track, the energy of the recoil proton can
66be calculated from the scattering angle θ obtained by CFT tracking. In this calculation,
the eﬀect of the energy loss in the LH2 target is considered by Bethe-Bloch’s formula.
The calculated kinetic energy of the recoil proton is deﬁned as Ecal (θ). The kinematicalconsistency ∆E for the pp scattering is deﬁned as∆E(pp scattering with p beam) = Emeas − Ecal (θ),
2mp p2p cos2 θ
Ecal (θ) =,(Eptot + mp )2 − p2p cos2 θ(4.12)(4.13)

countspwhere Emeas is the measured energy deposit with CFT and BGO. pp and Eptot = m2p + p2p
is the momentum and total energy of the beam proton. The spread of ∆E was the energy
resolution of CATCH including the energy resolutions of CFT and BGO, the angularresolution of CFT tracking, and the ﬂuctuation of the energy loss in the target. Figure
4.28 shows the ∆E distribution at Ecal = 100 MeV. At this point, σE ∼ 6 MeV andenergy dependence of the energy resolution was small. This energy resolution of CATCH
was taken into account in the simulation for the Σ+ p scattering.

3000
2500
2000
1500
1000
500
0
−100 −80 −60 −40 −20

020 40 60 80 100∆E(pp scattering) [MeV]

Figure 4.28: ∆E distribution of the pp scattering at Ecal (θ) = 100 MeV.

4.4.4

Particle identiﬁcation

Particle identiﬁcation in CATCH was performed by using the so-called dE-E method
between dECFT , corrected by the path length in CFT (dE/dx), and Emeas for each track.
Figure 4.29 shows the dE-E plot for the Σ+ production events. The locus deﬁned by
the two lines corresponds to the protons. The typical purity of a proton was 90% for
the selection gate. The other locus shows an approximately constant dE/dx distribution
with a branch toward the higher dE/dx value mainly corresponding to π + s. Most of π + s
penetrated BGO by losing only a part of the kinetic energy. Therefore, the only available
information for π + is the tracking information.
Low-energy protons, which stopped in CFT before reaching to BGO, were also identiﬁed by setting a dE/dx value larger than 2.7 MeV/mm in CFT. These protons were also
used for the Σ+ p scattering analysis.
67

CountsdE/dx in CFT [MeV/mm]

8
7

500

6
400

5

p
300

4
3

200

π

2

100

1
0
0

20

40

60

80

100 120 140 160 180 200
E+dE in CATCH [MeV]0

Figure 4.29: dE-E correlation between the energy loss in CFT and summed energy deposit
in CATCH. The two red lines show the selection region for protons.

4.5

Evaluations of the system speciﬁcation by combined analysis

The combination of the spectrometer system and CATCH enables detailed analysis ofcomplex reactions. These reactions can be used for evaluations for the position andmomentum resolution of the spectrometers. Additionally, a detection probability for an
accidental proton was estimated.

4.5.1

The vertex resolution evaluation of the spectrometer system using the π + p → π + π + π − p reaction

An event display of the π + p → π + π + π − p reaction is shown in Figure 4.30. For such areaction, the multi track can be reconstructed with CATCH in addition to the incident
π + beam and the outgoing π ± or proton analyzed with the spectrometers.

68

Run# 8171 Event# 2436
Event Display ZX plane

Event Display XY plane
Y [mm]

150
100

150

50
0

100

−50
−100

50

−150
−200

0

−100

0

100

200

300

400

300

400

Event Display ZY plane

150

−50

100
50

−100

0
−50

−150

−100
−150

−100

−50

0

50

100

150
X [mm]

−150
−200

−100

0

100

200

Figure 4.30: Event display of the π + p → π + π + π − p reaction, which is reconstructed using
track information from the spectrometers and CATCH. The pion-like tracks reconstructedwith CATCH are indicated by the blue lines and the incident π + and the outgoing proton
are indicated by the gray lines.

resolution of Z position [mm]The π + p → π + π + π − p reaction events were selected by the cut condition requiring
detections of both a proton with the KURAMA spectrometer and two pions with CATCH
or one pion with the KURAMA spectrometer and one proton and pion with CATCH. The
vertex resolutions of the spectrometer system can be evaluated by comparing the reaction
vertices reconstructed with the spectrometer system and CATCH because the vertex
resolutions of CATCH were comparable to or better than one of the spectrometer system.
The z vertex resolution depended on the scattering angle as shown in Figure 4.31. The
x and y vertex resolutions were evaluated as σx = 2.6 mm and σy = 3.5 mm and theangular dependences were negligible.

100
90
80
70
60
50
40
30
20
10
0
0

5

10

15
20
25
30
scattering angle θKURAMA [degree]

Figure 4.31: z position resolution of the spectrometer system as a function of scattering
angle.
69

4.5.2The momentum resolution evaluation of KURAMA using
the π + p elastic scattering

In the past experiments at the J-PARC K1.8 beam line, the momentum resolution evaluations of the spectrometers for scattered particle, the KURAMA and previous SKS spectrometer, were performed with empty target data. However, the LH 2 target and CATCH
used in the J-PARC E40 experiment cannot be moved easily. Therefore, the evaluationof the momentum resolution of KURAMA was performed using the π + p elastic scattering
kinematics. In this analysis, detection of the recoil proton with CATCH and the scattered π + with the KURAMA spectrometer was required. From the scattering angle ofthe proton measured with CFT, the momentum of scattered π + , pπcal , can be calculated.
The kinematical consistency ∆p for the π + p scattering is deﬁned as∆p = pKURAMA − pπcal ,(4.14)Countswhere pKURAMA is the momentum obtained by KURAMA tracking. The ∆p/p distribution
is shown in Figure 4.32. The spread of blue histogram corresponds to the momentum
resolution of the KURAMA tracking. It was evaluated as σp /p = 2.5 × 10−2 for 1.37
GeV/c π + .

1600
1400
1200
1000
800
600
400
200
0
−0.1 −0.08 −0.06 −0.04−0.02 0

0.02 0.04 0.06 0.08 0.1∆p/p

Figure 4.32: ∆p/p distribution for the momentum resolution evaluation for 1.37 GeV/c
π + . The spread of blue histogram corresponds to the momentum resolution of the KURAMA tracking. In the red histogram, pKURAMA was calculated from the scattering angle
θKURAMA and the momentum resolution was improved, although the angular resolutionof approximately 1◦ deteriorates the momentum resolution.
Once particle identiﬁcation for the outgoing particle was performed and the two bodyreaction was determined, the momentum of the outgoing particle can be calculated fromthe scattering angle θKURAMA . By applying this analysis method, the momentum resolution was improved as shown by the red histogram in Figure 4.32. After the contribution of
the angular resolution of approximately 1◦ was subtracted, the momentum resolution was
evaluated to be σp /p = 6.5×10−3 for 1.37 GeV/c π + . In the analysis of the π + p → K + Σ+reaction, the momentum of K + was calculated from the scattering angle θK + .
70

4.5.3The detection probability estimation for accidental proton
by CATCH

Total energy deposit in CATCH [MeV]In the π + + p reactions induced by 1.41 GeV/c π + beam, the inelastic scattering with pp̄
production are not allowed in terms of energy. If the outgoing proton was detected with
the KURAMA spectrometer, ideally CATCH could not detect the protons. Thus, thedetection probability of an accidental proton with CATCH can be estimated by eventsin which detection of both a proton from π + p scattering with KURAMA and anotherproton with CATCH were required. It was about 2% and the distributions of energy andthe angle of such protons shown in Figure 4.33 were also obtained. The energy and thescattering angle of about 40 % the accidental protons were consistent with the kinematicsof the π + p elastic scattering. These protons were derived from the π + p elastic scattering
caused by the accidental π + beam. The rest of accidental proton were from the reaction
between the target vessel and the accidental π + beam. The probability of accidental
coincidence and the energy and angle distributions of accidental proton were taken into
account in the simulation study to estimate the background of the Σ + p scattering events.

200

30

180
25

160
140

20

120
100

15

80
10

60
40

5

20
0
0

20

40

60

80 100 120 140 160 180
θ of accidental proton [degree]0

Figure 4.33: Correlation of the accidental proton’s energy and scattering angle. The
red dotted curve corresponds to the kinematics of the π + p elastic scattering. Protons
in the region inside red lines were derived from the π + p elastic scattering caused by theaccidental π + beam.

71

Chapter 5
Analysis-II:Identiﬁcation of the Σ+pscattering events
5.1

OutlineAs explained in Section 4, the momentum vector of the incident Σ + is reconstructed fromthe spectrometer information. In addition, the momentum vector of the recoil proton wasmeasured using CATCH. Combining these momentum vectors enables us to identify theΣ+ p scattering events by checking the kinematical consistency of the recoil proton between
the measured energy Emeas and the calculated energy Ecal from the recoil angle. Thissection describes the analysis of the event identiﬁcation method by kinematical consistency
check, especially for Σ+ p identiﬁcation, and background suppression and derivation of thenumbers of scattering events.

5.2

Kinematical consistency check method

A particular reaction involving Σ+ was identiﬁed by kinematical consistency check method.
In this procedure, a reaction to be analyzed was assumed. Under assumption that thereaction occurred, the kinetic energy Ecal , or momentum pcal of a proton was calculated
from initial Σ+ momentum vector analyzed with the spectrometers and the angle of the
proton reconstructed by CATCH. By comparing Ecal or pcal with the measured protonenergy Emeas or momentum pmeas using CATCH , the events for the reaction were identiﬁed as a peak in the kinematical consistency spectra. Kinematical consistency is deﬁned
as ∆E = Emeas − Ecal or ∆p = pmeas − pcal .

5.2.1

Kinematical consistency check for the decay proton from
Σ+ → pπ 0 decay

The most simple example of a kinematical consistency check is the Σ + → pπ 0 decay case.
This is also the easiest reaction to analyze because Σ+ decays into pπ 0 with a probabilityof 51 %. The schematic drawing of Σ+ → pπ 0 decay is shown in Figure 5.1.

72

K+

π+
Σ+

π0
θ

p

Figure 5.1: Schematic drawing of the Σ+ → pπ 0 decay. θ indicates the opening angle
between the Σ+ track reconstructed with the spectrometers and the proton track reconstructed with CATCH.
The kinematical consistency check for this reaction can be performed for events, inwhich at least one proton was detected with CATCH. The kinetic energy of the decay
proton can be calculated from the momentum of the Σ+ , pΣ , and the opening angle
between the Σ+ and proton track θ asq2
+ m2p − mp ,(5.1)
Edecayp,cal = pdecayp,cal√ApΣ cos θ + D
pdecayp,cal = tot 2,(5.2)
EΣ − p2Σ cos2 θm2Σ + m2p − m2π0
A=
,
2
2
2
D = (ApΣ cos θ)2 − (EΣtot − p2Σ cos2 θ)(m2p EΣtot − A2 ),pwhere mΣ , mp , and mπ0 are the mass of Σ+ , p, and π 0 , respectively. EΣtot = m2Σ + p2Σ isthe total energy of the incident Σ+ . The obtained ∆E spectrum is shown in Figure 5.2.
The clear peak can be seen around ∆E = 0. In this way, kinematical consistency checkfor a given reaction was performed.

73

Counts

3500
3000
2500
2000
1500
1000
500
0
−100 −80 −60 −40 −20 0
20 40 60 80 100∆ E for Σ+ → pπ0 Decay [MeV]

Figure 5.2: ∆E spectrum for the Σ+ → pπ 0 decay.

5.2.2

Kinematical consistency check for the decay proton of pp
scattering following the Σ+ → pπ 0 decay

The pp scattering following the Σ+ → pπ 0 decay shown in Figure 5.3 is the main background reaction in analyzing events with two proton in ﬁnal state. This reaction can be
identiﬁed by the kinematical consistency check for the decay proton.

K+

π+

Σ+

π0

θ
α
p

p

Figure 5.3: Schematic of the pp scattering following the Σ+ → pπ 0 decay. θ indicatesthe opening angle between the Σ+ track reconstructed with spectrometers and the decay
proton track. The momentum of the decay proton was reconstructed from the momentaof the two protons detected with CATCH.
The analysis for this reaction required the detection of two protons with CATCH.
Assuming the two protons were derived from this reaction, the momentum of the decay
74

0.2

Counts

∆P for pp scattering [GeV/c]

proton p can be reconstructed from the momenta of the two detected protons, p1 and p2 ,as p = p1 + p2 . The reconstructed momentum amplitude was regarded as pmeas . Because
the incident proton in this pp scattering is also the decay proton from the Σ+ → pπ 0
decay, the reconstructed momentum should satisfy the Σ + decay kinematics for a real
pp scattering event. This kinematical consistency check was performed by evaluating
the consistency between pmeas and the calculated momentum pcal , which is calculated
from equation (5.2) using the emission angle of the decay proton from the initial Σ + .
The opening angle of the two protons, α, was also veriﬁed because α is kinematicallyconstrained to be approximately 90◦ for the pp scattering event. The correlation between∆p and α is illustrated in the left side of Figure 5.4, where a clear event concentration
owing to pp scattering can be conﬁrmed. The right side of Figure 5.4 shows the ∆pspectrum with the opening angle gate.
70

0.15

60

700
600

0.1
50

500

0.05
40

400

0

−0.05

30

300

−0.1

20

200

−0.15

10

100

−0.2
0

0

20

40

60
80 100 120 140 160 180
opening angle between 2 proton [degree]

0−0.2 −0.15 −0.1 −0.05

00.05 0.1 0.15 0.2∆P for pp scattering [GeV/c]

Figure 5.4: (Left) Correlation of the ∆p and the opening angle between two detected
proton α and (Right) ∆p spectra for the pp scattering following Σ+ → pπ 0 decay. In the∆p spectrum, 79◦ < α < 97◦ gate was applied.

5.2.3

Kinematical consistency check for the recoil proton fromthe Σ+ p scattering

A schematic drawing of Σ+ p scattering in the LH2 target is shown in Figure 5.5. The Σ+ p
scattering event identiﬁcation was performed by a kinematical consistency check for therecoil proton indicated by the red arrow in Figure 5.5. The kinetic energy of the recoil
proton Ecal can be calculated from the momentum of the incident Σ+ and the recoil angleof the recoil proton θ as
Ecal =

2mp p2Σ cos2 θ.
(EΣtot + mp )2 − p2Σ cos2 θ(5.3)

After Σ+ p scattering, the scattered Σ+ decays mainly into nπ + or pπ 0 . For Σ+ p scattering
followed by Σ+ → nπ + decay, one proton of the recoil proton is in the ﬁnal state. Figure
5.6 shows the ∆E spectra for the events with one proton in the ﬁnal state. The left sideof Figure 5.6 is the spectrum without event selection to emphasize the Σ + p scattering
events and the right is the one with it. Even after applying the event selection, there
were substantial backgrounds. Like this, this recoil proton is severely contaminated by75

Figure 5.5: Schematic drawing of the Σ+ p scattering followed by the Σ+ → pπ 0 decay. θ
indicates the opening angle between the Σ+ track, which is reconstructed from the incoming π + and outroing K + using the spectrometers, and the recoil proton track measured
with CATCH.

16

×103

CountsCountsthe mere Σ+ → pπ 0 decay and other secondary background events. Therefore, we focus
on Σ+ p scattering followed by Σ+ → pπ 0 decay, in which two protons can be observedwith CATCH.

14

600
500

12
10

400

8

300

6

200

4
100

2
0
−100 −80 −60 −40 −20

0−100 −80 −60 −40 −20 0
20 40 60 80 100
DeltaE for Σp scattering [MeV]0
20 40 60 80 100∆E for Σp scattering [MeV]

Figure 5.6: ∆E spectra for the Σ+ p scattering followed by the Σ+ → nπ + decay. In
the left ﬁgure, the event selection to emphasize Σ+ p scattering events are not applied.
The substantial backgrounds are seen even in the right ﬁgure after applying the event
selection.
In the following analysis, the detection of two protons with CATCH is required andthese events are called “two-proton events”. The ∆E spectrum for two-proton events isshown in the left side of Figure 5.7. In the analysis of two-proton events, the proton withthe smaller |∆E| value was regarded as the recoil proton. In the left side of Figure 5.7, apeak structure can be identiﬁed around ∆E = 0 without any further selection of the Σ+ p76

scattering.
CountsCounts

all

2500

40000
35000
30000

2000

25000
1500

20000

1000

15000

Σp
pp
decay
decay+accidental π pdecay+accidental others

simulation

10000
500
0
−100 −80 −60 −40 −20

5000
0−100 −80 −60 −40 −20

020 40 60 80 100∆E for Σp scattering [MeV]0
20 40 60 80 100∆E for Σp scattering [MeV]

Figure 5.7: ∆E spectra for the two-proton events without cuts to select the Σ + p scattering
events for data (Left) and simulation (Right). As shown by the arrows, a peak structure
was identiﬁed around ∆E = 0. In the simulation (right), the contributions of the assumed
reactions are additionally shown, reaction-by-reaction.
The ∆E spectrum was compared with that of a Monte Carlo simulation. As the
background contamination in the two-proton events, the four cases shown in Figure 5.8
were considered in the simulation. Figure 5.8 (a) shows the pp scattering following Σ+ →pπ 0 decay. In the case shown in Figure 5.8 (b), the Σ+ → pπ 0 decay ﬁnally produces
a proton and e+ e− pair, where the e+ or e− are misidentiﬁed as a proton by CATCH.
The misidentiﬁcation is caused by e± with a large energy deposit in CFT as mentioned
in Subsection 4.4.4. Because the misidentiﬁcation was reproduced in the simulation, we
can estimate this background contribution by generating mere Σ + decay events in thesimulation. The two other reactions are accidental coincidences of Σ + production anddiﬀerent reactions of the LH2 target or target vessel, shown in Figure 5.8 (c) and (d),respectively, induced by the accidental π + beam. To reproduce accidental backgrounds,
the probability of accidental coincidence and the distributions of the energy and angleof the accidental protons were estimated as described in Subsection 4.5.3. From thisanalysis, the probabilities for types (c) and (d) were obtained as approximately 0.8%
and 1.2% for the number of Σ+ production events, respectively. These probabilities are
considered in the simulations. The right side of Figure 5.7 shows the ∆E spectrum
obtained by analyzing the simulation, considering these backgrounds. The simulated
spectrum consistently reproduced that of the real data.

77

Figure 5.8: Schematics of four types of backgrounds in the two-proton events, (a) thepp scattering following the Σ+ → pπ 0 decay, (b) the combination of a proton from the
Σ+ → pπ 0 decay and misidentiﬁed e± , (c) the combination of a proton from the Σ+ → pπ 0
decay and a proton from the elastic π + p scattering in the LH2 target caused by theaccidental π + , and (d) the combination of a proton from the Σ+ → pπ 0 decay and a
proton from the scattering between the accidental π + and the target vessel.

5.3

Cut conditions to select the Σ+p scattering events

To reduce backgrounds from the ∆E spectrum, shown in Figure 5.7, additional cuts
regarding the spatial and kinematical information obtained from the two detected protons
were applied. The detailed procedure is described in the following subsections. Table 5.1
summarizes the survival ratios for Σ+ p scattering events and the considered four types ofbackground events. Finally, more than 90% of the background events were eliminated,while maintaining approximately half of the Σ+ p scattering events.
Table 5.1: Survival ratios of the Σ+ p scattering and background events after applying
cuts as described in each subsection. (a)-(d) types correspond to the background types
in Figure 5.8.
Σ+ p (a) type (b) type (c) type (d) type
Cuts in 5.3.1
89.9% 69.3%
74.8%
55.7%
53.5%
Cuts in 5.3.2
86.7% 68.2%
8.3%
13.0%
5.7%
Cuts in 5.3.3
69.8% 12.8%
7.6%
11.3%
4.7%
Cuts in 5.3.4 (All cuts) 54.5%
7.1%
5.5%
0.4%
1.6%

78

5.3.1Vertex cut and closest distance cut

Y vertex [mm]Counts

The spatial consistency between two proton tracks can be used to select Σ + p scattering
events. The vertex of Σ+ p scattering, which was calculated as the closest point between
the incident Σ+ and recoil proton tracks, should be inside the target. Accordingly, the2scattering vertex (xscat , yscat , zscat ) must be x2scat + yscat< 252 mm2 , and |zscat | < 170 mm
as shown in the Figure 5.9.

500
400
300

50
40

30

30
20
10

25
20

0
−10

200

−20
−30

100

−40
0
−250 −200 −150 −100 −50

050

100 150 200 250
Z vertex [mm]

15
10
5

0
−50
−50 −40 −30 −20 −10 0 10 20 30 40 50
X vertex [mm]Figure 5.9: Reconstructed Σ+ p scattering vertex distribution of two-proton events in theanalysis. The region inside the red lines are accepted.

Y vertex [mm]CountsSimilarly, the decay vertex (xdecay , ydecay , zdecay ), which was obtained from the decay
proton and scattered Σ+ , was required to be −30 < xdecay < 25 mm, |ydecay | < 30 mm,
and |zdecay | < 180 mm as shown in Figure 5.10. The momentum vector of the scattered′Σ+ , denoted as Σ+ , can be kinematically calculated from the recoil angle of the proton.
′
The trajectory of Σ+ is reconstructed from the momentum vector and the scattering
vertex.

400
350
300

50

24

40

22

30

20
18

20

16

250

10

14

200

0

12

−10

10

−20

8

150
100

−30

6
4

50

−40

0
−250 −200 −150 −100 −50

0−50−50 −40 −30 −20 −10 0 10 20 30 40 50
X vertex [mm]050

100 150 200 250
Z vertex [mm]2

Figure 5.10: Reconstructed Σ+ → pπ 0 decay vertex distribution of two-proton events inthe analysis. The region inside the red lines are accepted.

79

The closest distances between Σ+ and the proton tracks at the scattering and decay
points also reﬂect spatial consistency. The simulated distributions of the closest distancesat the two vertices are shown in Figure 5.11. The closest distances at the scattering anddecay points were required to be less than 20 mm and 25 mm, respectively. These cuts
can reduce the background events by 30–50% while maintaining approximately 90% of
the Σ+ p scattering events, as shown in the second row of Table 5.1.
allCountsCountsallselected

Σp
pp
decay

103

selected

pp
decay

103

decay+accidental π pdecay+accidental π pdecay+accidental others

decay+accidental others

simulation

simulation

102

Σp

102

10
10
0

5

10

15 20 25 30 35 40 45 50
closest distance between p and Σ+ [mm]

0

5

10

15 20 25 30 35 40 45 50
closest distance between p and Σ+’ [mm]Figure 5.11: Simulated distributions of the closest distances at the scattering (left) anddecay points (right). The dotted lines indicate the boundaries of selected regions: 20 and
25 mm at the scattering and decay points, respectively..

5.3.2

Missing mass cut to select the scattered Σ+ decay′

Assuming that Σ+ p scattering is followed by Σ+ → pπ 0 decay, the missing mass of the′
Σ+ → pX reaction, MX , should be the mass of π 0 . The squared missing mass (MX2 )distribution shows a π 0 peak and a broad distribution toward the negative region, as
shown in Figure 5.12. The broad distribution in the negative region was mainly attributed
to accidental backgrounds. The event forming the π 0 peak comes not only from Σ+ p
scattering, but also from pp scattering following the Σ+ → pπ 0 decay, because bothreactions have the same ﬁnal state of ppπ 0 , originating from the initial Σ+ . The green
shaded spectrum in Figure 5.12 shows the MX2 distribution for pp scattering, selected ∆pand α gate described in the next subsection. The cut condition MX2 > 0, shown by thedotted line in Figure 5.12, was determined to include the green spectrum. The survival
ratios estimated from the Monte Carlo simulation are listed in the third row of Table 5.1.

5.3.3

Kinematical cut for secondary pp scattering

The pp scattering following the Σ+ → pπ 0 decay, shown in Figure 5.8 (a), can be identiﬁedby the kinematical consistency check for the decay proton as described in Subsection 5.2.2.
Events from this reaction were rejected using the correlation between ∆p and α, shownin Figure 5.13. To reject the identiﬁed pp scattering events, the cut region deﬁned as the±2σ areas for ∆p and α was deﬁned, as shown by the red box in Figure 5.13. Although
this cut condition rejected approximately 20% of the Σ + p scattering events, the signalto-noise (S/N) ratio was improved by cutting approximately 80% of the secondary pp
scattering events.

80Counts

selected

1000
800
600
400
200
0
−0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0
0.05 0.1
M2X for Σ+’ → pX [GeV2/c 4]

′

0.2
45

0.15
40

0.1

35

0.05

30

0

25

−0.05

20
15

−0.1

10

−0.15
−0.2
0

Counts∆p for pp scattering [GeV/c]

Figure 5.12: Squared missing mass distribution of the Σ + → pX reaction. The green
shaded spectrum shows the pp scattering following the Σ+ → pπ 0 decay, which can beselected using the kinematical consistency and opening angle of the two detected protons,
as explained in the Subsection 5.3.3. Here, MX2 > 0 is selected.

5

20

40

60
80 100 120 140 160 180
opening angle between 2proton [degree]0

Figure 5.13: Correlation between ∆p and the opening angle α of two detected protons
from real data. The events inside the red-line region were rejected as the secondary pp
scattering background.

5.3.4

Kinematical cut for π + p elastic scattering

Finally, the rejection of the accidental coincidence of π + p elastic scattering is described.
The recoil proton by π + p elastic scattering has a kinematical correlation between therecoil angle and energy. Figure 5.14 shows the correlation between the angle of protons
with respect to the central axis of CFT and the energy. The locus corresponding to theπ + p kinematics, indicated by the red dotted line, can be conﬁrmed. The events inside
the red-line region in Figure 5.14 were rejected as the protons from elastic π + p scattering.
81

200

25

180
160

CountsEmeas of the ‘‘recoil proton’’ [MeV]This cut reduced almost all of the type (c) background in Figure 5.8.

20

140
120

15

100
80

10

60
40

5

20
0
0

10

20

30

40
50
60
70
80
90
θlab of the ‘‘recoil proton’’ [degree]0

Figure 5.14: Correlation between the angle of protons with respect to the central axis of
CFT, θlab , and Emeas of the recoil proton. The red dotted line indicates the kinematicsof π + p elastic scattering induced by the 1.41-GeV/c π + beam. When calculating thekinematics, the energy loss in the LH2 target was considered. The events inside the redline region were rejected as the protons from elastic π + p scattering in response to theaccidental π + .

5.3.5

Σ+ p scattering identiﬁcation after all cuts

The ∆E spectrum for two-proton events after applying all cuts is shown in the left sideof the Figure 5.15. The S/N ratio in the peak region of −20 < ∆E[MeV] < 20 was 1.78,
signiﬁcantly improved from that of Figure 5.7. The evaluation of the S/N ratio is basedon the ﬁtting results of the ∆E spectra explained in the next section. The simulated
spectrum after the same cuts, which is shown in the right side of Figure 5.15, agreed
with the data. The analysis eﬃciency for the Σ+ p scattering events was estimated to be
54.5%, and the rejection factors of the four background sources were higher than 90%, assummarized in Table 5.1.

82

Counts

Counts

4500

400

4000

350

3500
300

3000

250

2500

200

all
Σp
pp
decay
decay+accidental π pdecay+accidental others

simulation

2000

150

1500

100

1000

50

500

0
−100 −80 −60 −40 −20

0−100 −80 −60 −40 −20

020 40 60 80 100∆E for Σp scattering [MeV]0
20 40 60 80 100∆E for Σp scattering [MeV]

Figure 5.15: ∆E spectra for the two-proton events with cuts to select the Σ+ p scattering
events for (left) data and (right) simulation. In the simulation (right), the contributionsof the assumed reactions are shown reaction-by-reaction.

5.4

Estimation of the number of Σ+p scattering events

To estimate the number of Σ+ p scattering events and survival background events, the∆E spectrum for the Σ+ p scattering was ﬁtted with the sum of the simulated spectra
for both the Σ+ p scattering and background reactions. Fitting was performed for the∆E spectrum at each scattering angle independently to correctly reproduce the angulardependence of the background contribution. In order to constrain the contribution of the
pp scattering events in this ﬁtting, the ∆p spectrum for the pp scattering kinematics was
simultaneously ﬁtted. The ﬁtting method was the maximum log-likelihood method, inwhich parameters were optimized to minimize the negative log likelihood described as− log Lp = − log Lp,∆E − log Lp,∆p ,
N
bin 

X
data
− log Lp,∆E =fi,∆E − Ni,∆E · log fi,∆E ,

(5.4)
(5.5)

i=1
sim,Σp
sim,a
sim,b
sim,c
sim,d
fi,∆E = CΣ+ p Ni,∆E+ Ca Ni,∆E+ Cb Ni,∆E+ Cc Ni,∆E+ Cd Ni,∆E
,
N
bin 

X
data
− log Lp,∆p =fi,∆p − Ni,∆p· log fi,∆p ,

(5.6)

i=1
sim,Σp
sim,a
sim,b
sim,c
sim,d
fi,∆p = CΣ+ p Ni,∆p+ Ca Ni,∆p+ Cb Ni,∆p+ Cc Ni,∆p+ Cd Ni,∆p,where CΣ+ p , Ca , Cb , Cc , Cd were the parameters corresponding to the scale factors of simulated distributions. Ni,∆E and Ni,∆p are counts of the i-th bin of the histograms and thesubscripts a − d correspond to the type of backgrounds in Figure 5.8.
The cut condition for the ∆p spectrum, where the pp scattering rejection cut describedin subsection 5.3.3 was not applied, was diﬀerent from that for the ∆E spectrum inorder to emphasize the contribution of the pp scattering events. The ﬁtting for only the∆E spectra was also examined in order to study the systematic diﬀerences due to thebackground estimation and ﬁtting procedure. The uncertainty due to the bin size of thespectra was also estimated by iterating the same procedure for the spectra with diﬀerent
sets of bin sizes. The diﬀerence in the estimated Σ+ p scattering events in these ﬁttings is
considered as the systematic uncertainty.
83

Counts

The ﬁtting results of ∆E for all measurements within detector acceptance are shownin Figure 5.16, 5.17, and 5.18. A ﬁtting result of ∆p spectra for the low momentum regionis also shown in Figure 5.19.

-1.00 <cosθCM < -0.90-0.90 <cosθCM < -0.80-0.80 <cosθCM < -0.70-0.70 <cosθCM < -0.60-0.60 <cosθCM < -0.50-0.50 <cosθCM < -0.40-0.40 <cosθCM < -0.30-0.30 <cosθCM < -0.20-0.20 <cosθCM < -0.10-0.10 <cosθCM < 0.00

0.00 <cosθCM < 0.10

0.10 <cosθCM < 0.20

060 0.20 <cosθCM < 0.30

0.30 <cosθCM < 0.4030
20Counts10060
40Counts20
060
40Counts2040
200−100 −50

0

50 100 −50∆E [MeV]0data allΣpsimulation sum
decay
pp
decay+accidental πpdecay+accidental others

50 100∆E [MeV]Figure 5.16: ∆E spectra at each scattering angle of Σ+ in the low-momentum region(0.44 < pΣ [GeV/c] < 0.55). The data points with error bars show the experimental data.
Simulated spectra for the assumed reactions are also shown and the red histogram showing
the sum of these spectra.
As shown in these spectra, the S/N ratio of the ∆E spectra worsened in the forward angular region because of the limited acceptance of the low-energy recoil proton.
Therefore, we set the maximum scattering angle for each incident Σ + momentum region:cos θCM = 0.4 for the low momentum region, cos θCM = 0.5 for the middle momentum
region, and cos θCM = 0.6 for the high momentum region. The estimated numbers of
Σ+ p scattering events for each incident Σ+ momentum region are shown in Figure 5.20
as a function of cos θCM . The error bars and boxes represent statistical and systematic
errors, respectively. The systematic error derived from the uncertainty in the backgroundestimation, as previously discussed. A total of approximately 2400 Σ + p scattering eventswere identiﬁed.

84

Counts

15-1.00 <cosθCM < -0.90-0.90 <cosθCM < -0.80-0.80 <cosθCM < -0.70-0.70 <cosθCM < -0.60

0
20 -0.60 <cosθCM < -0.50-0.50 <cosθCM < -0.40-0.40 <cosθCM < -0.30-0.30 <cosθCM < -0.20-0.10 <cosθCM < 0.00

0.00 <cosθCM < 0.10

0.10 <cosθCM < 0.20

0.30 <cosθCM < 0.40

0.40 <cosθCM < 0.50

10Counts515
10Counts50
20 -0.20 <cosθCM < -0.10
1510Counts50
20 0.20 <cosθCM < 0.30
15
10
5
0
−100 −50

0

50 100 −50∆E [MeV]0

50 100 −50∆E [MeV]0

50 100∆E [MeV]Figure 5.17: ∆E spectra at each scattering angle of Σ+ in the middle-momentum region(0.55 < pΣ [GeV/c] < 0.65). The legends are the same as those in Figure 5.16.

85

6-0.90 <cosθCM < -0.80-0.80 <cosθCM < -0.70-0.70 <cosθCM < -0.60-0.60 <cosθCM < -0.50-0.50 <cosθCM < -0.40-0.40 <cosθCM < -0.30-0.30 <cosθCM < -0.20

0-0.20 <cosθCM < -0.10
15-0.10 <cosθCM < 0.00

0.00 <cosθCM < 0.10

0.10 <cosθCM < 0.20

0.30 <cosθCM < 0.40

0.40 <cosθCM < 0.50

0.50 <cosθCM < 0.60

Counts-1.00 <cosθCM < -0.90

4Counts2010Counts5

10Counts500.20 <cosθCM < 0.30
15
10
5
0
−100 −50

0

50 100 −50∆ E [MeV]

0

50 100 −50∆ E [MeV]

0

50 100 −50∆ E [MeV]

0

50 100∆ E [MeV]

Figure 5.18: ∆E spectra at each scattering angle of Σ+ in the high-momentum region(0.65 < pΣ [GeV/c] < 0.80). The legends are the same as those in Figure 5.16.

86

40

-0.90 <cosθCM < -0.80-0.80 <cosθCM < -0.70-0.70 <cosθCM < -0.60

0100 -0.60 <cosθCM < -0.50-0.50 <cosθCM < -0.40-0.40 <cosθCM < -0.30-0.30 <cosθCM < -0.20-0.20 <cosθCM < -0.10-0.10 <cosθCM < 0.00

0.00 <cosθCM < 0.10

0.10 <cosθCM < 0.20

0.20 <cosθCM < 0.30

0.30 <cosθCM < 0.40

Counts-1.00 <cosθCM < -0.903020Counts

10

Counts50

0100Counts50

080data allΣp60

simulation sum
decay

40

pp
decay+accidental πp200decay+accidental others−0.2 −0.1 0

0.1 0.2 −0.2
∆p [GeV/c]00.2∆p [GeV/c](a) 0.44 < p (GeV/c) < 0.55

200(b) 0.55 < p (GeV/c) < 0.65

70

100

− 0.5

0

0.5

1
cosθCM45(c) 0.65 < p (GeV/c) < 0.80

35

50

30

40

25

30

20
15

20

50

50

40

60

150

0
−1

80

Counts250

CountsCountsFigure 5.19: ∆p spectra at each scattering angle of Σ+ in the low-momentum region(0.44 < pΣ [GeV/c] < 0.55). The legends are the same as those in Figure 5.16.

10

10

5

0
−1

0
−1

− 0.5

0

0.5

1
cosθCM

− 0.5

0

0.5

1
cosθCM

Figure 5.20: Estimated numbers of the Σ+ p scattering events for each scattering angle
and momentum region of Σ+ : (a) the low momentum region (0.44 < pΣ [GeV/c] < 0.55),(b) the middle momentum region (0.55 < pΣ [GeV/c] < 0.65), and (c) the high momentum
region (0.65 < pΣ [GeV/c] < 0.80). The error bars and boxes indicate the statistical andsystematic errors, respectively.

87

Chapter 6
Analysis III: Evaluations of factors toderive the diﬀerential cross sections
6.1

OutlineFor deriving the diﬀerential cross sections, several values should be evaluated for eachscattering angle and Σ+ momentum region. Therefore, these values are denoted as afunction of pΣ and cos θCM , such that N (pΣ , cos θCM ) represents the number of scattering
events. The diﬀerential cross section was calculated as follows:
N (pΣ , cos θCM )
dσ
=
,
dΩ
ρ · NA · L(pΣ ) · ε̄(pΣ , cos θCM ) · ∆Ω

(6.1)

where ρ and NA represent the density of the LH2 target, 0.071 g/cm3 , and Avogadro’s
number, respectively. L(pΣ ) is the total ﬂight length of the incident Σ+ in the LH2 target.
ε̄ represents the eﬃciency of the Σ+ p scattering event averaged for the vertex position. ∆Ωrepresents a constant solid angle of ∆Ω = 2π∆ cos θCM . The following sections describe
the evaluation of each factor.

6.2

Total track length of the incident Σ+ in the LH2
targetIn ordinary scattering experiments, the expression ρ · NAvo · t · Nbeam is used for the
luminosity, where t and Nbeam represent a target thickness and number of beam particles,
respectively. However, this evaluation is inappropriate in this experiment because theincident Σ+ was produced in the LH2 target, and primarily decayed inside the target. The
direct measurement of the Σ+ track length, event-by-event, is also diﬃcult because of thelimited acceptance for the decay proton. However, the total track length of the incident
Σ+ can be reliably evaluated using a Monte Carlo simulation. Information regarding
the production vertices and momentum vectors of all identiﬁed incident Σ + particles was
obtained from the spectrometer analysis. Σ+ particles with measured momenta weregenerated at the production points in the passage simulator described in Section 3.2.
The ﬂight length of Σ+ was subsequently summed until Σ+ decayed or exited the target.
Figure 6.1 shows the estimated Σ+ track length distribution. The total track lengths
for each momentum region L(pΣ ) were obtained by integrals of these histograms. The88background contribution in the Σ+ identiﬁcation was also estimated from the sideband
event in the m2 distribution. Table 6.1 summarizes the estimated total track lengths after
subtracting background contributions.

Figure 6.1: Distribution of estimated Σ+ track lengths for the three momentum regions.

Table 6.1: Estimated Σ+ total track lengths for the three momentum regions: lowmomentum region (0.44 < pΣ [GeV/c] < 0.55), middle-momentum region (0.55 <pΣ [GeV/c] < 0.65), and high-momentum region (0.65 < pΣ [GeV/c] < 0.80). The values
in the row of “All events” include the contributions from the miscalculated background
events. The background contributions are listed in the row of “Sideband BG”. By subtracting “sideband BG” from “All events”, the Σ+ track length was calculated and islisted in the row labeled “Σ+ ”.
RegionLow
MiddleHigh77
All events [cm]3.69 × 10 1.13 × 10 6.70 × 106
Sideband BG [cm] 0.27 × 107 0.12 × 107 0.86 × 106
Σ+ [cm]3.42 × 107 1.00 × 107 5.84 × 106

In this procedure, the simulation inputs of the vertex point and momentum vector
contained uncertainties owing to the resolution and systematic errors of the spectrometers.
This may have caused uncertainties in the estimated track length. The error of the total
track length derived from the resolutions was at most 1%. If the Σ + momentum wassystematically shifted due to the accuracy of the spectrometers, the track length would
be aﬀected. The momentum accuracy of the spectrometers depended on the scattering
angle of K + . In the worst case, the analyzed momentum of Σ+ was expected to be 13
MeV/c larger. Even if the momenta of all Σ+ were shifted +13 MeV/c systematically, theestimated total track length would change in 2-3 %. These uncertainty is considerably
smaller than other uncertainties, such as the statistical errors shown in Figure 5.20.

89

Average eﬃciency of the Σ+p scattering events
including the detection and analysis eﬃciency

6.3

The average eﬃciency ε̄ for the Σ+ p scattering events, including the detection and analysis eﬃciencies, was evaluated by analyzing the simulated data with the same analyzer
program for the real data. Here, the diﬀerence in the detection eﬃciency of CATCH for
protons between the simulation and real data should be considered. In the following, theevaluation of the CATCH eﬃciency was ﬁrstly discussed. Then, the average eﬃciency ε̄
is obtained by correcting the diﬀerence between the simulation and real data.

6.3.1

Detection eﬃciency of CATCH

300
3000

250
2500

200
2000

kinetic energy [MeV]

kinetic energy [MeV]

The detection eﬃciency of CATCH includes the geometrical acceptance, tracking eﬃciency of CFT, and the energy measurement eﬃciency for protons. They depend on theangle in the laboratory frame θlab , the kinetic energy E, and z-vertex position z. The
eﬃciencies were evaluated based on the pp scattering data with seven beam momenta
between 0.45 and 0.85 GeV/c. In the real pp scattering data, (θlab , E) region were constrained by the kinematics, as shown in the left side of Figure 6.2. Therefore, the protoneﬃciency is also estimated with a Monte Carlo simulation, where protons with arbitrary
angles and energies can be generated.
300

900
800

250

700

200

150

1500

100

1000

100

50

500

50

0
0

0

0
0

600
500

150

400
300
200
100

20

40

60

80

100

120 140 160 180
θlab of proton [deg]

20

40

60

80

100

120 140 160 180
θlab of proton [deg]0Figure 6.2: Correlations between the angle and energy of proton used for the eﬃciency
evaluation. The left is the real pp scattering data and the right is the simulation.
The procedures for the eﬃciency evaluation using the pp scattering data are as follows:
1. For the eﬃciency estimation, at least one proton among the two protons in the ﬁnal
state must be detected by CATCH. From the kinematics, the momentum (p′ (θ′ )) of
the detected proton with the recoil angle θ′ was calculated. The scattering vertex(xscat , yscat , zscat ) was also reconstructed as the closest point between the beam andrecoil proton tracks.
2. The momentum vector of the other proton was obtained as p = pbeam − p′ (θ′ ),where the pbeam was the proton beam momentum analyzed by the K1.8 beam-line
spectrometer. From the momentum vector p, the angle and kinetic energy of thesecond proton can be predicted; they are denoted as θ and E, respectively.
90

3. The CATCH eﬃciency was estimated by checking whether the predicted track andenergy were measured or not. The tracking and energy measurement eﬃciencies
were derived separately.

simulation

0.9
0.8
0.7
0.6
0.5

300

1

0.9

250

0.8
0.7

200

0.6

150

0.5

0.4

0.4

100

0.3
0.2

0.3
0.2

50

0.1
0
0

efficiency

data

1

kinetic energy [MeV]

Energy measurement efficiency

First, energy measurement eﬃciency εBGO (θ, E, zscat ) is explained. In this case, we checked
whether the measured energy for the predicted track agreed with the predicted E within
40 MeV. The obtained εBGO (θ, E, zscat ) at θ = 37◦ is shown as the red points in the left sideof Figure 6.3 as an example. This eﬃciency was compared with the eﬃciency estimatedfrom the simulation. As shown in the left side of Figure 6.3, the simulation-based eﬃciency accurately reproduced the data-based eﬃciency. Therefore, the simulation-based
eﬃciency for the energy measurement was used for further analysis to cover the entire
θ, E, and zscat regions. The eﬃciency was evaluated for each lattice in the (θ, E, z) spacewith a step of (∆θ, ∆E, ∆z) = (2.5◦ , 5 MeV, 30 mm). For a given particle with (θ, E, z),a eﬃciency is calculated by the linear interpolation. The right side of Figure 6.3 showsthe energy measurement eﬃciency map for 0 < z[mm] < 30 section, near the center of
the target.

0.1

50

100

150

0
0

200
250
ETp,cal [MeV]

20

40

60

80

100

120 140 160 180
θlab of Proton [deg]0Figure 6.3: (Left) Energy measurement eﬃciency as a function of the kinetic energy at
the scattering angle of θlab = 37◦ . The red circles and black squares represent the dataand simulation, respectively. The eﬃciencies for the various z positions are written. The
simulated eﬃciency reproduces the data well. (Right) Energy measurement eﬃciency map
for 0 < z[mm] < 30 section evaluated by the simulation.
Hereafter, the tracking eﬃciency of CFT εCFT (θ, E, z) is described. This was evaluatedby checking whether a track with the predicted direction was detected. Therefore, theeﬀect of detector acceptance was also included in εCFT . The energy dependence of the
tracking eﬃciencies estimated from the simulation and pp scattering data is shown by theblack and red points in the left side of Figure 6.4, respectively. Because CFT tracking
required at least six layer hits, the eﬃciency decreased sharply at low energies. This
energy dependence of the eﬃciency can be phenomenologically represented by the Fermi
function for both the data and simulation. The eﬃciency was then formulated as follows:
εCFT (θ, E, z) =εmax (θ, z),E−Ehalf (θ)
1 + exp
d(θ)

(6.2)

where εmax (θ, z), d(θ) and Ehalf (θ) are parameters representing the maximum eﬃciency,
diﬀusion, and kinetic energy with half eﬃciency, respectively. These parameters were
91

1
0.8
0.6
simulation

300

1

0.9

250

0.8
0.7

200

0.6

150

0.5
0.4

0.4

100

data
Fermi function fitted to sim

0.2

0.3
0.2

50

0.1

Fermi function fitted to data

00efficiency

kinetic energy [MeV]

CFT efficiency

determined for each (θ, z) area with a step of (∆θ, ∆z) = (5◦ , 30 mm) by ﬁtting Eq. (6.2)
to the estimated eﬃciency, as indicated by the solid red line in the left side of Figure 6.4.
The realistic eﬃciency is slightly lower than that of the simulation, typically by 10%. Thisdiﬀerence is attributed to the geometrical eﬀect of the ﬁber placement in the uv layersof CFT. There are ineﬀective regions for tracks with scattering angles of approximately
45◦ , owing to the zigzag ﬁber conﬁguration in the uv layers. In addition, both the kinetic
energy with half eﬃciency Ehalf and diﬀusion parameter d were slightly larger than thosein the simulation. These diﬀerences indicated that the realistic amount of material in
the experimental setup was larger than that considered in the simulation. It was diﬃcult
to incorporate the real spiral ﬁber conﬁguration into the CFT uv layers and the missing
amount of material within the simulation. Therefore, the data-based eﬃciency for CFT
tracking was used for the analysis of the cross section. The right side of Figure 6.4 showsthe eﬃciency map for 0 < z[mm] < 30 section as a function of θ and E.

20

40

60

80

0
0

100 120 140 160 180 200
ETp,cal [MeV]

20

40

60

80

100

120 140 160 180
θlab of Proton [deg]0Figure 6.4: (Left) CFT tracking eﬃciency as a function of the kinetic energy at the
scattering angle of θlab = 54◦ . The red circles and black squares represent the data andsimulation, respectively. The red and black curves are the ﬁt functions obtained for thedata and simulation. (Right) Data-based tracking eﬃciency map used for analysis, which
was calculated from Eq. (6.2) and parameters were determined by the ﬁtting to the realdata. It is for 0 < z[mm] < 30 section.
The obtained eﬃciency map was checked by deriving the diﬀerential cross sections forthe calibration pp scattering data with proton beam. It was calculated as follows;
1dσ(cosθCM ) =dΩρ · NA · ∆z · Nbeam ∆Ω1,εBGO (θlab , Ecal , zscat ) · εCFT (θlab , Ecal , zscat )events(6.3)
3
where ρ and NA represent the density of the LH2 target, 0.071 g/cm , and Avogadro’s
number, respectively. In the derivation, each z section with 30-mm length was regarded
as the target. Therefore, target thickness ∆z was 3 cm. Nbeam was counted based on the
beam tracking result: Whether a beam particle passed through the z section or not. The
BGO and CFT eﬃciencies were corrected for each detected particle. Figure 6.5 shows
typical derived diﬀerential cross sections for the pp scattering induced by the proton beam,
which agreed with the reference values to within 5%, except for the acceptance edge.
X

N (cos θCM )

92

4

6
4
3

2

2

1

1

6

−0.5

0

0.5

0
−1

1

cosθCM,p

pp= 0.55 GeV/c (149 MeV)

5
4
3

6

0.5

1

cosθCM,p

pp= 0.60 GeV/c (170 MeV)

3
2
1
−0.5

0

0.5

0
−1

1

cosθCM,p

pp= 0.65 GeV/c (200 MeV)

5
4

6

2

2

1

1
0

0.5

0
−1

1

cosθCM,p

0

0.5

1

cosθCM,p

pp= 0.75 GeV/c (263 MeV)

4
3

−0.5

−0.5

5

3

0
−1

0

4

1

6

−0.5

5

2
0
−1

pp= 0.50 GeV/c (125 MeV)

5

3d σ/d Ω [mb/sr]d σ/d Ω [mb/sr]d σ/d Ω [mb/sr]

5

0
−1

d σ/d Ω [mb/sr]

pp= 0.45 GeV/c (95 MeV)d σ/d Ω [mb/sr]d σ/d Ω [mb/sr]

6

−0.5

0

0.5

1

cosθCM,pFigure 6.5: Typical diﬀerential cross sections of pp scattering derived from proton beamdata. The red points represents the derived ones, and the black dotted line shows thecalculated ones by ESC96 using calculator on [57].
To estimate the eﬃciency uncertainty at the acceptance edge, the possible lowest andhighest CFT tracking eﬃciencies were also made by changing the parameters within areasonable range, i.e., changing d within 20% and Ehalf within 4 MeV. The validity of themargin of eﬃciency deﬁned by the lowest and highest cases was additionally veriﬁed using
another calibration reaction. This calibration reaction was that of pp scattering following
the Σ+ → pπ 0 decay, which described in Subsection 5.2.2. From the data analysis, theangular distribution of the recoil proton was obtained, shown by the red points in Figure 6.6. This angular distribution was compared with a Monte Carlo simulation, including
the secondary pp scattering process with a realistic angular distribution. While analyzing
the simulated data, the data-based CFT tracking eﬃciencies for the lowest and highest
cases were considered. The angular distribution in the data was sandwiched between thetwo distributions estimated using the highest and lowest eﬃciencies, as shown by the blueand green points in Figure 6.6. In the next subsection, the detection eﬃciency for Σ + p
scattering was corrected using these two eﬃciencies, and the diﬀerence was considered tobe the systematic uncertainty.

93

Counts

data

1800

simulation with the highest efficiency

1600

simulation with the lowest efficiency

1400
1200
1000
800
600
400
200
0
−1 −0.8 −0.6 −0.4 −0.2

0

0.2

0.4

0.6

0.8 1
cosθCM,pFigure 6.6: Angular distribution of the protons from the pp scattering following the
Σ+ → pπ 0 decay events. The red points represent the data. The green and blue points
show the simulation results with the eﬃciency correction using the possible lowest andhighest CFT tracking eﬃciencies, respectively. The simulations are normalized by thecounts at 0 < cos θCM,p < 0.1.

6.3.2

Average eﬃciency for the Σ+ p scattering events considering the real detection eﬃciency

The average eﬃciency ε̄ for the Σ+ p scattering events, including the detection and analysis eﬃciencies, was evaluated by analyzing the simulated data with the same analyzer
program for the real data. ε̄ is deﬁned as follows:
Nanalyzed (pΣ , cos θCM )Ngenerated,Σ+ id (pΣ , cos θCM )Ndetected (pΣ , cos θCM )Nanalyzed (pΣ , cos θCM )=·Ngenerated,Σ+ id (pΣ , cos θCM ) Ndetected (pΣ , cos θCM )= ε̄detect (pΣ , cos θCM ) · ε̄ana (pΣ , cos θCM ).

ε̄(pΣ , cos θCM ) =

(6.4)

(6.5)

Ngenerated,Σ+ id in Equation (6.4) represents the number of generated Σ+ p scattering eventsin the simulation. Σ+ identiﬁcation from missing mass analysis in the analyzer program isrequired. Nanalyzed represents the number of identiﬁed two-proton Σ+ p scattering events
that satisfy all cut conditions for Σ+ p scattering. The eﬀect of the branching ratio of the
Σ+ → pπ 0 decay is also included in Nanalyzed . ε̄ is factorized by the detection eﬃciency in
CATCH, εdetect , and the analysis cut eﬃciency, εana , from the second equation. Ndetected
represents the number of two-proton events in the Σ+ identiﬁcation events. The diﬀerence
simin the CFT tracking eﬃciency between the data, εdata
CFT , and simulation, εCFT , was correctedby changing ε̄detect as follows:Ndetected
X εdata (θp , Ecal,p , zscat ) · εdata (θp , Ecal,p , zdecay ) 
1
1
2
2
CFT
CFT
,
ε̄detect 7→sim
sim
Ngenerated,Σ+ id events εCFT (θp1 , Ecal,p1 , zscat ) · εCFT (θp2 , Ecal,p2 , zdecay )
1

94

(6.6)

(a) 0.44 < p (GeV/c) < 0.55

0.12
0.1

0.14

(b) 0.55 < p (GeV/c) < 0.65

0.12
0.1

average efficiency

0.14

average efficiency

average efficiency

where the eﬃciency correction for both protons was considered. The analysis cuts are
explained in Section 5.3; and the analysis eﬃciency for the entire angular region is summarized in Table 5.1. Here, the analysis eﬃciency was estimated for each scattering angle.
The evaluated eﬃciencies are shown in Figure 6.7. The vertical error represents the difference between the lowest and highest possible CFT tracking eﬃciencies, as mentioned
in the previous subsection. The angular dependence of the eﬃciency can be understoodfrom the kinetic energies of protons. The eﬃciency decreases for the forward scattering
angle, because the tracking eﬃciency also decreases for recoil protons with a lower kinetic
energy. Similarly, the kinetic energy of the decay proton decreases for the backward angle,
and the eﬃciency therefore decreases for the backward angle. The errors in the eﬃciency
were considered as systematic errors in deriving diﬀerential cross sections.
0.14
0.12
0.1

0.08

0.08

0.08

0.06

0.06

0.06

0.04

0.04

0.04

0.02

0.02

0.02

0
­1

0
­1

­0.5

0

0.5

1
cosθCM

­0.5

0

0.5

1
cosθCM(c) 0.65 < p (GeV/c) < 0.80

0
­1

­0.5

0

0.5

1
cosθCM

Figure 6.7: Average eﬃciencies for the Σ+ p scattering for each scattering angle and momentum region. The vertical error represents the diﬀerence in the obtained eﬃciency
using the two possible lowest and highest CFT tracking eﬃciencies.

95

Chapter 7
Experimental result and discussion
7.1

Diﬀerential cross sections

The diﬀerential cross section was calculated using Eq. (6.1). The obtained diﬀerential
cross sections for the three incident Σ+ momentum regions are shown as black circles in
Figure 7.1. The mean momenta of the three momentum regions are 0.50, 0.59, and 0.71
GeV/c, respectively. The error bars and boxes of the data points represent statisticaland systematic uncertainties, respectively. The systematic error was estimated as thequadratic sum of the error from the background estimation, average eﬃciency, and Σ +total ﬂight length. The data has small uncertainty typically less than 20% and a ﬁneangular step of ∆ cos θ = 0.1. The values of the diﬀerential cross sections and theiruncertainties are summarized in Tables A.1, A.2, and A.3 in Appendix A.

7.2

Comparison with the past experiments and theoretical calculations

The diﬀerential cross sections obtained from present experiments are shown as black circles
in Figure 7.1. For the lower two momentum regions, past measurements at KEK PS areplotted with red boxes [24] and blue triangles [26] in Figure 7.1. The data quality inthe present experiment was improved signiﬁcantly. Thus, a meaningful comparison withtheories has become possible.
Theoretical calculations were overlaid as lines in Figure 7.1. The blue dotted and dotdashed lines show the calculations from the FSS and fss2 models [19], which include the
QCM in the short-range region and naturally incorporate the quark Pauli eﬀect. Therefore, the comparison with these models is important for the veriﬁcation of the quark Paulieﬀect, which is an important motivation for this experiment. The diﬀerence between thetwo models is mainly caused by the diﬀerent size parameter, which deﬁnes the harmonic
oscillator wave functions of quarks in the baryons and determines the strength of thequark Pauli eﬀect. The FSS model, using the larger size parameter, predicts a strong
repulsive interaction, which increases the diﬀerential cross section. However, the predictions by FSS and fss2 are much larger than the present data, indicating that the repulsive
forces in FSS and fss2 are too large and unrealistic. After the parameter tuning of thesemodels using our precise data, the QCM picture itself will be validated.
The green solid and black-dashed lines show the predictions from the Nijmegen NSC97f
96dσ /dΩ [mb/sr]

8(a) 0.44 < p (GeV/c) < 0.55

FSS
fss2
NSC97f
χEFT NLO13 (cutoff 600MeV)

6

χEFT NLO19 (cutoff 600MeV)
ESC08 (0.55 GeV/c)
E289 data (0.35<p(GeV/c)<0.75)

4dσ /dΩ [mb/sr]

208E251 data (0.3<p(GeV/c)<0.6)
present work(c) 0.65 < p (GeV/c) < 0.80(b) 0.55 < p (GeV/c) < 0.65

642
0
−1 −0.8 −0.6 −0.4 −0.2 0

0.2 0.4 0.6 0.8 −1 −0.8 −0.6 −0.4 −0.2 0
cosθCM

0.2 0.4 0.6 0.8 1
cosθCM

Figure 7.1: Derived diﬀerential cross sections of the Σ+ p scattering for the three momentum regions. The error bars and boxes show the statistical and systematic uncertainties,
respectively. The red boxes and blue triangles represent the data of past measurements,
KEK E251 [24] and KEK E289 [26], respectively. The blue dotted and dot-dashed lines
show the calculations from FSS and fss2 [19], respectively. The green-solid lines andblack-dashed lines show the calculations from the Nijmegen NSC97f [14] and ESC08 [15]models, respectively. The orange and red dot-dashed lines show the calculations from theχEFT NLO models [17] [18].
[14] and ESC08 [15] models, respectively, based on the boson-exchange picture. Historically, in the Nijmegen models, it has been diﬃcult to describe the repulsive nature of theΣN (I = 3/2,3 S1 ) channel. Although NSC97f agrees well with our data in terms of thediﬀerential cross sections, it predicts an attractive Σ + p interaction, which does not agreewith the current common understanding of the ΣN interaction. In ESC08, additional repulsive eﬀects, including the quark picture, are considered by making an eﬀective Pomeron
potential as the sum of a pure Pomeron exchange and a Pomeron-like representation of
the Pauli repulsion. Subsequently, ESC08 predicts a moderate repulsive force within thischannel. Although there were sizable discrepancies between our data and ESC08, especially in the middle-momentum region, ESC08 was closer to present data than fss2. Thissuggests that the size of the repulsive force used in ESC08 is relatively reasonable.
The orange and red dot-dashed lines show the calculations using the χEFT models
extended to the Y N sector (NLO13 [17] and NLO19 [18], respectively), which use diﬀerent
sets of LECs. In both cases, a cutoﬀ value of 600 MeV was used. The LECs are essential
parameters of the χEFT models, representing the short-range part of the interaction, andshould be determined from the experimental data. At present, the LECs for S waves have97

been determined based on existing hyperon-proton scattering data in the low-momentum
region. However, the LECs for P waves have not been well-constrained owing to thelack of experimental data for the momentum region around the present data. At present,
χEFT predicts much larger cross sections, especially in the higher-momentum region.
LECs for P waves in the χEFT models will be determined using our data, which may
make at least angular dependences closer to our data.
For the ﬁrst time, we presented precise data for the Σ+ p channel in the highermomentum range than 0.4 GeV/c. Currently, no theoretical model can reproduce ourdata consistently for the three momentum regions. This was mainly because of the lackof precise data. Therefore, our data are essential inputs for improving these theoretical
calculations in order to become realistic BB interaction models.

7.3

Phase-shift analysis

Extracting the contribution of the 3 S1 states from the diﬀerential cross section data isimportant to study the repulsive nature of Σ+ p system related to the quark Pauli eﬀect.
For this purpose, a phase-shift analysis was performed based on a general formulation of
the scattering problem in quantum mechanics. This was the ﬁrst application to hyperonnucleon scattering data, whereas precise phase-shift analysis has been performed for the
N N scattering data to derive the phase shifts for each partial wave [58].

7.3.1

Speciﬁc expressions of diﬀerential cross section as a function of phase shifts

Similarly to the N N scattering case [59], an asymptotic form of the wave function for theΣ+ p scattering can be written as follows:es′
ikz s′
ψm
ξ m′ +′ (r) ∼ e

ikr X

r

′

s,s
s
ξm
Mm,m
′ (θ, ϕ)

(7.1)

s,m

where k represents the wavenumber of relative motion in the CM system deﬁned as k =spCM /ℏ, ξmdenotes the spin state with spin quantum number s and the projection on thes,s′quantization axis m. Mm,m′ are the matrix elements of the spin-1/2 spin-1/2 scattering
amplitude with a polar angle θ and azimuthal angle ϕ. By the partial-wave decomposition,
the matrix element becomes
s,s′
Mm,m
′ (θ, ϕ) =

L+s
X X

′
J+s
X

p
′
4π(2L′ + 1)YLm −m (θ, ϕ)

L J=|L−s| L′ =|J−s′ |′× CL×s (J, m′ , m′ − m, m)CL′ ×s′ (J, m′ , 0, m′ )iL −L

hL, s| S mat − 1 |L′ , s′ i
2ik
(7.2)

where the CL×s are Clebsch-Gordan coeﬃcients deﬁned as
CL×s (J, mJ , mL , ms ) = hLmL sms |LsJmJ i ,98and YLm is a spherical harmonic, and S mat is the scattering matrix. The diﬀerential crosssection for scattering of an unpolarized incident particle on an unpolarized target I0 is
expressed by the matrix elements as follows:
1 0,0 2 1 1,1 2 1 1,1 2 1 1,1 2 1 1,1 2 1 1,1 2
I0 = |M0,0| + |M1,1 | + |M0,0 | + |M0,1 | + |M1,0 | + |M1,−1 | .
4
2
4
2
2
2

(7.3)

The explicit formulae of the matrix elements as a function of the partial wave amplitudes
h’s for a general angular momentum L can be found in Ref. [60]. The speciﬁc expressions
up to D wave (L ≤ 2) are described as follows: 3 cos2 θ − 1 0,0= h1 S0 + 3h1 P1 cos θ + 5h1 D2 ×
M0,0
,
(7.4)
2
√


2 3 S1 − 3 D 1   3
3
1,1
M1,1 = h3 S1 −h+h3 P2 + h3 P1 cos θ
2
2 √ 2

5
1
2 3 S1 −3 D1  3 cos2 θ − 1
h
×
,
(7.5)
+ 2h3 D3 + h3 D2 + h3 D1 −
2
2
2
2
√
3
3
1,1
M0,0
= (h3 S1 + 2h S1 − D1 ) + (2h3 P2 + h3 P0 ) cos θ√ 3 3
3 cos2 θ − 1,(7.6)+ (3h3 D3 + 2h3 D1 + 2h S1 − D1 ) ×
2


3
3
1,1
M0,1
= − √ h3 P2 + √ h3 P1 × (− sin θ)
2 2
2 2

4
5
1
1 3 3 + − √ h3 D3 + √ h3 D2 + √ h3 D1 − √ h S1 − D1 × (−3 cos θ sin θ), (7.7)
3 2
6 2
2 2
2
 1

1
1,1
M1,0 = √ h3 P2 − √ h3 P0 × (− sin θ)
2
2
 1
1
1 3 S1 − 3 D 1 + √ h 3 D3 − √ h 3 D1 − √ h× (−3 cos θ sin θ),
(7.8)
2
2
2
1
5
1
1 3 3 1,1
M1,−1=
h3 D3 − h3 D2 + h3 D1 − √ h S1 − D1 × (3 sin2 θ),
(7.9)
6
12
4
2 2
where partial wave amplitudes h’s were deﬁned as(1(cos(2ϵ̄1 ) exp(2iδ̄2s+1 LJ ) − 1) (3 S1 and 3 D1 cases)
h2s+1 LJ = 2ik
1(exp(2iδ̄2s+1 LJ ) − 1)(otherwise)
2ik
1
3
3
sin(2ϵ̄1 ) exp(iδ̄3 S1 + iδ̄3 D1 ).
h S1 − D 1 =2k(7.10)(7.11)

δ̄2s+1 LJ and ϵ̄1 are the bar-phase shifts and mixing parameter for the 3 S1 −3 D1 mixing. In aprecise sense, bar-phase shifts are diﬀerent from commonly-used nuclear bar-phase shifts,
in which Coulomb eﬀects were subtracted. Because the energies of the Σ + p scattering are
suﬃciently high and the data for very-forward angle is absent, the Coulomb eﬀects might
be negligible. Therefore, the bar-phase shifts were equated with the nuclear bar-phaseshifts and called merely “phase shifts δ” in the following.
In this way, the diﬀerential cross sections can be represented as a function of the phaseshifts, scattering angle θCM , and momentum pCM in the CM system. Here, the tripleteven and singlet-odd states of the Σ+ p system are expressed by the 10-plet of the ﬂavor
SU(3) multiplets and the other states are expressed by the 27-plet as shown in Table 1.1.
99

Therefore, this function is denoted by I0 (θCM , pCM , δ[27] , δ[10] ). In present analysis, partial
waves up to D are considered. The phase shifts for the 27-plet are taken up to ﬁve spinstates: δ[27] = {δ1 S0 , δ3 P2 , δ3 P1 , δ3 P0 , δ1 D2 }. For the 10-plet, ﬁve phase shifts and a mixing
parameter for 3 S1 -3 D1 mixing, that is, δ[10] = {δ3 S1 , δ1 P1 , δ3 D3 , δ3 D2 , δ3 D1 , ϵ1 }, are included.

7.3.2

Consideration of the ﬁxed phase shifts in the ﬁtting

The function I0 (θCM , pCM , δ[27] , δ[10] ) has 11 phase-shift parameters. Because a ﬁtting with
11 free parameters results in a large ambiguity for each parameter, the number of freeparameters should be reduced by ﬁxing some of them to reasonable values.
The phase shifts δ[27] can be reliably constrained because δ[27] becomes identical tothe phase shifts in the N N (I = 1) channel in the limit of ﬂavor SU(3) symmetry. In thislimit, δ[27] can be obtained from the phase shifts of pp scattering for the corresponding
momentum. However, in reality, δ[27] in the Σ+ p scattering should be slightly diﬀerent
from that in the pp scattering, owing to the breaking of ﬂavor symmetry. In fact, alltheories (FSS, fss2, ESC, and NSC97f) predict smaller 1 S0 phase shifts in Σ+ p scatteringthan those in pp scattering. However, the diﬀerence between the theoretical predictionsof δ[27] is small because these models are also constrained by pp scattering data. In thisanalysis, the eﬀect of the uncertainty in δ[27] was examined using three diﬀerent sets ofδ[27] . The phase-shift values of pp scattering and theoretical predictions in ESC16 [22]and NSC97f [14] were used in this study.
In contrast, the phase shifts δ[10] are unique to the ΣN (I = 3/2) channel, and thesephase shifts should be determined from the ﬁtting. The theoretically uncertain two phaseshifts δ3 S1 and δ1 P1 , representing the short-range interaction, were regarded as free parameters. For the remaining phase shifts, namely δ3 D3 , δ3 D2 , δ3 D1 , and ϵ1 , the variation among
the theoretical models is rather small because the pion-exchange mechanism is expected
to be dominant for the momentum range of present experiment. Therefore, these phaseshifts were ﬁxed at the theoretical values as an approximation.
In summary, the two phase-shift parameters, δ3 S1 and δ1 P1 , were obtained by ﬁtting
the diﬀerential cross sections with the function I0 (θCM , pCM , δ[27] , δ[10] ). To study thediﬀerence of the results due to the ﬁxed parameter sets, ﬁtting was performed for three
conditions with diﬀerent ﬁxed parameter sets.
A δ[27] was ﬁxed at values taken from the pp scattering. δ3 D3 , δ3 D2 , δ3 D1 , and ϵ1 were
ﬁxed at 0.
B δ[27] was ﬁxed at values from the ESC16 or NSC97f models. δ3 D3 , δ3 D2 , δ3 D1 , and ϵ1
were ﬁxed at 0.
C δ[27] was ﬁxed at values from the ESC16 or NSC97f models. δ3 D3 , δ3 D2 , δ3 D1 , and ϵ1
were ﬁxed at the values from ESC16 or NSC97f.
Although the sign of δ3 S1 is expected to be negative, as predicted by recent theoreticalmodels including ESC16, numerical ﬁttings with a positive δ3 S1 are possible, which canbe roughly understood from the fact that diﬀerential cross sections would be proportional
to (sin δ0 /k)2 in the S-wave limit. To investigate the negative and positive δ3 S1 cases, two
diﬀerent sets of ﬁxed parameters from ESC16 and NSC97f, respectively, were examined.
The ﬁxed phase shifts are presented in Table 7.1.

100

Table 7.1: δ[27] and δ[10] for the pp scattering and Σ+ p scattering in ESC16 [22] andpp
NSC97f [14], respectively. The units of pΣ and pCM are [GeV/c]. Elab(unit: [MeV])
represents the kinetic energy of the beam proton in the pp scattering in which pCM isequal to that of the Σ+ p scattering. The units of phase shifts are [◦ ].

pΣ
pCM
pp
Elab
δ 1 S0δ 3 P2
δ 3 P1
δ 3 P0
δ1 D2
δ 3 S1δ 1 P1
δ3 D3
δ3 D2
δ3 D1
ϵ1

7.3.3

low
pp
ESC16 NSC97f
0.496
0.50
0.50
0.214
0.216
0.216
87.6
–
–
27.9
19.1
20.2
9.92
6.76
6.44
−12.2 −13.2
−13.3
10.5
7.19
8.10
3.24
3.38
3.25
–
(−27.9) (21.9)–(8.33)(12.2)
–
1.14
1.42
–
−3.53
−3.23
–
1.35
1.48
–
−5.04
−1.65

mid
pp
ESC16 NSC97f
0.59
0.60
0.60
0.253
0.257
0.257
122.1
–
–
19.5
10.8
11.8
12.7
8.50
8.02
−15.5 −16.9
−17.1
7.29
3.59
4.49
4.71
4.99
5.02
–
(−32.6) (28.5)–(8.45)(13.71)
–
1.59
2.29
–
−4.87
−4.23
–
0.69
1.30
–
−5.24
0.11

high
pp
ESC16 NSC97f
0.71
0.70
0.70
0.303
0.297
0.297
173.7
–
–
10.4
2.80
3.71
14.9
9.82
9.04
−19.5 −20.8
−21.0
2.15
−0.92
−0.23
6.41
6.61
6.99
–
(−36.6) (35.01)–(7.46)(13.7)
–
1.93
3.18
–
−6.42
−5.31
–
−0.70
0.41
–
−5.14
1.87

Fitting results and phase shifts of 3 S1 and 1 P1 states

The ﬁtting results in the three momentum regions are shown in Figure 7.2, 7.3, and7.4. In all momentum regions, reasonable reduced χ2 values of approximately one wereobtained. The momentum dependencies of the obtained δ3 S1 and δ1 P1 values are plotted in
Figure 7.5. The absolute values of δ3 S1 for the low-, middle-, and high-momentum regions
were (28.3±1.5±2.1)◦ , (23.4±2.0±3.0)◦ , and (32.5±2.5±2.5)◦ , respectively. The former
error comes from the ﬁtting error and the latter reﬂects the eﬀect of the diﬀerent ﬁxed
parameters. If the sign is assumed to be negative, the momentum dependence of δ3 S1 is
consistent with the ESC16 model, suggesting that the repulsive force is moderate as in
the ESC models, as discussed in section 7.2. In contrast, the obtained δ1 P1 values deviateconsiderably in the range of −5◦ < δ1 P1 < 25◦ depending on the conditions. Although
the results of δ1 P1 are ambiguous, they may support the predictions of fss2, ESC, and
NSC97f, in which the interaction of the 1 P1 state in the Σ+ p system is weakly attractive.
By exploiting high-quality diﬀerential cross section data and the simple representationof Σ+ p interaction in the ﬂavor SU(3) symmetry, we succeeded to determine the δ3 S1 value
with a small error less than 4 degrees. This result will impose a strong constraint on thesize of the repulsive force in the Σ+ p systems.

101

5
4

dσ/dΩ [mb/sr]

dσ/dΩ [mb/sr]

6

A δ3S <01

B δ3S <01

C δ3S <0

6
5
4

1

1

B δ3S >01

C δ3S >0
1

3

3

2

2

1

1

0
−1 −0.8 −0.6 −0.4 −0.2

A δ3S >0

0

0.2

0.4

0.6

0
−1 −0.8 −0.6 −0.4 −0.2

0.8 1
cosθCM

0

0.2

0.4

0.6

0.8 1
cosθCM

654

dσ/dΩ [mb/sr]

dσ/dΩ [mb/sr]

Figure 7.2: Diﬀerential cross sections as a function of the scattering angle together with thecalculated angular distribution in the phase-shift analyses for the low-momentum region(0.44 < pΣ [GeV/c] < 0.55) in (Left) the negative δ3 S1 case and (Right) the positive case.
Three lines in each graph show the ﬁtting results with three diﬀerent ﬁtting conditions
A, B, and C, as described in the text. The typical χ2 /ndf is 4.4/11.

A δ3S <01

B δ3S <01

C δ3S <0

6
5
4

1

1

B δ3S >01

C δ3S >0
1

3

3

2

2

1

1

0
−1 −0.8 −0.6 −0.4 −0.2

A δ3S >0

0

0.2

0.4

0.6

0
−1 −0.8 −0.6 −0.4 −0.2

0.8 1
cosθCM

0

0.2

0.4

0.6

0.8 1
cosθCM

Figure 7.3: Diﬀerential cross section as a function of the scattering angle together withthe calculated angular distribution in the phase-shift analysis for the middle-momentum
region (0.55 < pΣ [GeV/c] < 0.65). The ﬁtting conditions are the same as in Figure 7.2.
The typical χ2 /ndf is 14.0/11.

102

5
4

dσ/dΩ [mb/sr]

dσ/dΩ [mb/sr]

6

A δ3S <01

B δ3S <01

C δ3S <06
5
4

1

A δ3S >01

B δ3S >01

C δ3S >0
1

3

3

2

2

1

1

0
−1 −0.8 −0.6 −0.4 −0.2

0

0.2

0.4

0.6

0
−1 −0.8 −0.6 −0.4 −0.2

0.8 1
cosθCM

0

0.2

0.4

0.6

0.8 1
cosθCM

A δ3S <01
A δ3S >080

δ1P1 [degree]δ3S1 [degree]

Figure 7.4: Diﬀerential cross section as a function of the scattering angle together withthe calculated angular distribution in the phase-shift analysis for the high-momentum
region (0.65 < pΣ [GeV/c] < 0.80). The ﬁtting conditions are the same as in Figure 7.2.
The typical χ2 /ndf is 11.0/13.

1

B δ3S <01
B δ3S >01
C δ3S <01
C δ3S >01
ESC16
NSC97f
fss2

60
40
20
0

1

B δ3S <01
B δ3S >01
C δ3S <01
C δ3S >01
ESC16
NSC97f
fss2

20
10
0

−20

−10

−40
−60

−20

−80
0

A δ3S <01
A δ3S >0

30

0.1

0.2

0.3

0.4

0.5

−30
0

0.6 0.7 0.8 0.9 1
Σ+ momentum [GeV/c]

0.1

0.2

0.3

0.4

0.5

0.6 0.7 0.8 0.9 1
Σ+ momentum [GeV/c]

Figure 7.5: Obtained phase shifts δ3 S1 and δ1 P1 as a function of the incident momentum.
The black-dashed, green-solid, and blue-dotted lines represent the calculated phase shiftsof ESC16 [22], NSC97f [14], and fss2 [19], respectively.

103

Chapter 8
Summary
For the systematic understanding of the baryon-baryon interactions between octet baryons,revealing the nature of ﬂavor SU(3) multiplets is important. Among them, the 10-plet ispredicted to be considerably repulsive due to the Pauli eﬀect in the quark level, whichis related to the origin of the repulsive core in the nuclear force. The ΣN (I = 3/2)
channel is one of the best channels to study the repulsive nature in the 10-plet. However,
the theoretical predictions for strengths of the repulsive force diﬀer from each other andthe reliable experimental inputs are needed to point out the strength. Hyperon-proton
scattering experiment is one of the most direct methods to study the Y N interaction, as
in the case of the N N interaction. Although it was experimentally diﬃcult due to short
lifetime of hyperons for a long time, we successfully performed novel high-statistics Σ ± p
scattering experiment at J-PARC (J-PARC E40). In this experiment, we measured thediﬀerential cross sections of not only the Σ+ p elastic scattering but also the Σ− p elastic
and Σ− p → Λn inelastic scatterings. In this thesis, the data taking and detailed resultsof the Σ+ p elastic scattering experiment to investigate the ΣN (I = 3/2) interaction werereported.
The experiment was performed at the K1.8 beam line in the J-PARC Hadron Experimental Facility in the separated periods of April 2019 and May-June 2020 for the datacollection of the Σ+ p scattering. The π + beam, having a high intensity of approximately
2 × 107 /spill and the central momentum of 1.41 GeV/c, was used to produce many Σ+particles inside the LH2 target via the π + p → K + Σ+ reaction. The produced Σ+ was
momentum-tagged from the beam π + and outgoing K + , which were analyzed using thebeam-line and forward spectrometers, respectively. In total, 4.9 × 107 Σ+ were tagged andthese Σ+ , traveling inside the LH2 target, were regarded as the incident particle for Σ+ p
scattering. The recoil proton from Σ+ p scattering were detected using the surrounding
detector system CATCH. Σ+ p scattering events were identiﬁed by checking the consistency for the recoil proton between the measured energy and the calculated energy fromkinematics of the Σ+ p scattering. In total, approximately 2400 Σ+ p elastic scattering
events were identiﬁed in the momentum range 0.44 – 0.80 GeV/c.
The diﬀerential cross sections of the Σ+ p scattering were derived for the three separatedmomentum regions. Their uncertainties were typically less than 20% with an angular stepof ∆ cos θCM = 0.1, and the data quality was drastically improved. The angular dependences of the obtained diﬀerential cross sections are rather isotropic for the present angularregions of −0.8 < cos θCM < 0.6, especially for the low momentum. The obtained valuesof the diﬀerential cross sections are around 2 mb/sr, which are not so large as predicted104

by the fss2 and FSS based on the QCM in the short-range region [19], which naturally
incorporate the quark Pauli eﬀect. The Nijmegen ESC models [15][22], which include themoderate repulsive force by the Pomeron eﬀect and phenomenological treatment of thequark Pauli eﬀect, were close to the data. χEFT predicts much larger cross sections, especially at the higher momentum region. We expect that our data will be used to specify
the LECs especially for P -waves in the χEFT models [17][18].
Owing to precise data points and the simple representation of the Σ + p system withrespect to the multiplets of the BB interaction, we derived the phase shifts of the 3 S1
and 1 P1 states by performing the phase-shift analysis for the obtained diﬀerential crosssections. This is the ﬁrst-time application to hyperon-nucleon scattering data. In thisanalysis, the less uncertain phase shifts were ﬁxed to the reasonable values and two important phase shifts, δ3 S1 and δ1 P1 , were determined by the ﬁtting. The δ3 S1 , which is closely
related to the strength of repulsive force due to the quark Pauli eﬀect was determinedwith a small error of approximately 4 degrees for each of the momentum regions. The
absolute values of δ3 S1 ranges 20◦ to 35◦ for the present momentum range. If the sign isassumed to be negative, the momentum dependence of the δ3 S1 is rather consistent with
ESC16, suggesting that the repulsive force is moderate like ESC16. Because theoreticalmodels describing BB interaction has been built on little hyperon-proton scattering data,
our data will be an essential input for making them realistic.

105

Acknowledgements
Owing to many helps of numerous people, I could ﬁnish the data taking of J-PARC
E40 experiment and ﬁnally summarize results in this thesis. I would like to express mygratitudes here.
First of all, I’m very grateful to my supervisor, Prof. Tomofumi Nagae. He wasone of the ﬁrst person to teach me nuclear physics and the joy of experiment when I
was an undergraduate student. After I got into graduate school, he led me into aninteresting research ﬁeld, strangeness nuclear physics. I learned many things in working
for his project, the spectroscopic study of Ξ hypernuclei at J-PARC. After I changed myresearch theme to Σ± p scattering experiment, he watched over me and gave me a lot ofsound advice.
I also deeply appreciate Prof. Koji Miwa, who is the spokes person of the J-PARC
E40 experiment. He gave me a precious opportunity to conduct the Σ + p data taking
and analyze the Σ+ p scattering channel, which is the ﬂagship channel of the J-PARC
E40 experiment. He managed the whole of experiment and got the funding resources asa spokes person. Moreover, he was always at the forefront of all stages of experiment:
R&D, preparation, data taking, and analysis. The paper for the Σ − p elastic scatteringand the Σ− p → Λn inelastic scattering were written by him. In this way, he showed me
an ideal images of a researcher. He made many essential comments on the analysis andwriting of the published academic paper and this thesis.
I am thankful to “Research Group for Hadron nuclear physics” in Advanced Science
Research Center of Japan Atomic Energy Agency (JAEA-ASRC) for accepting me as a
granted student for four years and a research assistant for one year. Without this ﬁnancial
support, I could not survive my long doctoral course of six years. Prof. Hiroyuki Sako, thegroup manager, provided this research enviroment. Prof. Kiyoshi Tanida and Dr. Yudai
Ichikawa gave me advice on many things from daily life in Tokai village to research as
host researchers. The research group for hadron nuclear physics have a close relation shipwith “Advanced Theoretical Physics group” in JAEA-ASRC. Especially, Prof. Makoto
Oka, who is the pioneer of the quark cluster model, always encouraged me. Prof. Toshiki
Maruyama gave me a hand-me-down car, which made my quality of life in Tokai village
much better.
I would like to thank all the K1.8 beam line members and collaborators of J-PARC
E40 experiment. We help each other to carry out many experiments in the beam line.
Their help was essential in J-PARC E40 experiment and I learned many things about
detectors and experimental systems through joining other experiments. Among them, I
want to express my special thanks to E40 core members, shift members in the Σ + p datataking in 2020 May-June, and students of Prof. K. Miwa.
I discussed many things about experiments, analysis, and writing of the paper inweekly meetings and e-mails with J-PARC E40 core members, Prof. K. Miwa, Prof.
106

Toshiyuki Takahashi, Prof. Ryotaro Honda, Dr. Yuya Akazawa, Dr. Shuhei Hayakawa,Dr. Mifuyu Ukai, Dr. Takeshi O Yamamoto, Mr. Yoshiyuki Nakada, Ms. Norina Fujioka,
Mr. Suharu Hoshino, Mr. Kunpei Matsuda, Mr. Ryohei Nagatomi, Mr. Tomomasa
Kitaoka, Ms. Tamao Sakao, Mr. Shunsuke Wada, Mr. Taito Morino and Mr. Kazuma
Ohashi. Discussions with them were very fruitful.
The latter half of the Σ+ p data taking was performed in 2020 May-June, when there
was the severe movement restriction due to COVID-19. Therefore, shift members, Prof.
K. Miwa, Prof. T. Takahashi, Prof. K. Tanida, Dr. Y. Akazawa, Dr. Manami Fujita,
Dr. S. H. Hayakawa, Dr. Y. Ichikawa, Dr. M. Ukai , Dr. T. O. Yamamoto, Mr. Yuji
Ishikawa, Ms. Honoka Kanauchi, Mr. T. Kitaoka, Mr. S. Wada, and Ms. T. Sakao,
had to work almost without holidays for a month. Owing to their dedicated help, we
successfully ﬁnished the data taking.
For recent 10 years, Prof. K. Miwa’s students, Prof. R. Honda, Mr. Yuuki Matsumoto,Dr. Y. Akazawa, Mr. Takehiro Shiozaki, Mr. Kazuya Kobayashi, Mr. Michihiko Ikeda,
Mr. Shotarou Ozawa, Ms. N. Fujioka, Mr. S. Hoshino, Mr. K. Matsuda, and Mr.
T. Kitaoka developed detectors and readout systems for J-PARC E40 experiment andcompiled them into a very good master’s thesis. On the basis of their excellent work, wewere able to conduct the experiment and I was able to get the results.
I also thank the staﬀ of the J-PARC accelerator and the Hadron Experimental Facility
for their support in providing the beam during beam time.
I want to express my gratitude to all the members of “Experimental Nuclear andHadronic Physics”(NH) group in Kyoto University, including past constituent members.
The staﬀs, Prof. T. Nagae, Prof. Megumi Naruki, Prof. Juzo Zenihiro, Dr. Toshiyuki
Gogami, Dr. Masanori Dozono, Dr. Nastuki, Tomida, Prof. Takahiro Kawabata, Prof.
Hiroyuki Fujioka, Prof. Masayuki Niiyama, and Dr. Tetsuya Murakami, run the groupand cared about my research. Among seniors who have already graduated, Mr. Shunsuke
Kanatsuki and Dr. Hiroyuki Ekawa taught me many things as the seniors of K1.8 beamline members. I have worked with members who got into graduate school after me, Mr.
Masaya Ichikawa, Ms. Ami Koshikawa, Ms. Sakiko Ashikaga, Dr. Kazuki Suzuki, Mr.
Eiichi Umezaki, Mr. Takeshi Harada, Mr. Tsubasa Ohashi, Mr. Kengo Ebata, and Mr.
Shuji Takahashi for experiments at J-PARC or related test experiments. I learned a lot
from them, but at the same time I wish I had been helpful to them.
Finally, I am grateful to my parents, Masanori Nanamura and Hiromi Nanamura, for
not only emotional but also ﬁnancial supports during my doctoral course.

Funding resources
J-PARC E40 experiment was supported by JSPS KAKENHI Grant Number 23684011,
15H00838, 15H05442, 15H02079 and 18H03693. This experiment was also supported by
Grants-in-Aid Number 24105003 and 18H05403 for Scientiﬁc Research from the Ministryof Education, Culture, Science and Technology (MEXT), Japan.

Credits of published paper
This thesis is based on the content of the following academic paper;
T. Nanamura, K. Miwa, et al.,107

Measurement of diﬀerential cross sections for Σ+ p elastic scattering in the
momentum range 0.44-0.80 GeV/c
Progress of Theoretical and Experimental Physics, Volume 2022, Issue 9, September
2022, 093D01 (DOI: 10.1093/ptep/ptac101)
This academic paper is an Open Access article distributed under the terms of theCreative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/).

108

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111

Appendix A
Tables of the diﬀerential crosssections
The values of the diﬀerential cross sections and their uncertainties are summarized inTable A.1, A.2, A.3.
Table A.1: Data table of diﬀerential cross sections for Σ+ p scattering in the lowmomentum region (0.44 < pΣ [GeV/c] < 0.55). The systematic error was estimated asa quadratic sum of the error from the background estimation (BG), averaged eﬃciency(eﬀ), and Σ+ total ﬂight length (L).
cos θCMdσ/dΩstat.
syst. (Total) syst. (BG) syst. (eﬀ) syst. (L)[mb/sr] [mb/sr][mb/sr][mb/sr][mb/sr][mb/sr]−0.85± 0.05
2.22
±0.46
±0.40
±0.23
±0.32
±0.04
−0.75± 0.05
2.31
±0.32
±0.33
±0.17
±0.28
±0.05
−0.65± 0.05
2.12
±0.26
±0.29
±0.18
±0.22
±0.04
±0.34
±0.17
±0.04
−0.55± 0.05
2.00
±0.29
±0.38
±0.41
±0.13
±0.04
−0.45± 0.05
1.93
±0.27
±0.43
−0.35± 0.05
2.40
±0.22
±0.22
±0.15
±0.15
±0.05
−0.25± 0.05
1.92
±0.19
±0.23
±0.18
±0.15
±0.04
−0.15± 0.05
2.22
±0.19
±0.31
±0.25
±0.17
±0.04
−0.05± 0.05
2.22
±0.19
±0.26
±0.16
±0.21
±0.04
0.05± 0.05
1.86
±0.21
±0.25
±0.15
±0.20
±0.04
0.15± 0.05
2.54
±0.23
±0.37
±0.12
±0.35
±0.05
0.25± 0.05
2.84
±0.29
±0.58
±0.17
±0.55
±0.06
±0.24
±0.86
±0.06
0.35± 0.05
3.02
±0.51
±0.90

112

Table A.2: Data table of diﬀerential cross sections for Σ+ p scattering in the middlemomentum region (0.55 < pΣ [GeV/c] < 0.65).
cos θCMdσ/dΩstat.
syst. (Total) syst. (BG) syst. (eﬀ) syst. (L)[mb/sr] [mb/sr][mb/sr][mb/sr][mb/sr][mb/sr]−0.75± 0.05
1.24
±0.38
±0.22
±0.17
±0.13
±0.02
−0.65± 0.05
0.90
±0.38
±0.15
±0.13
±0.07
±0.02
−0.55± 0.05
1.73
±0.32
±0.20
±0.18
±0.08
±0.03
±0.04
±0.04
±0.02
−0.45± 0.05
1.24
±0.28
±0.06
−0.35± 0.05
1.35
±0.27
±0.10
±0.08
±0.05
±0.03
−0.25± 0.05
1.79
±0.28
±0.09
±0.07
±0.05
±0.04
−0.15± 0.05
1.42
±0.23
±0.07
±0.06
±0.02
±0.03
−0.05± 0.05
1.07
±0.21
±0.10
±0.10
±0.02
±0.02
0.05± 0.05
1.70
±0.29
±0.10
±0.08
±0.05
±0.03
0.15± 0.05
1.11
±0.29
±0.11
±0.09
±0.05
±0.02
0.25± 0.05
2.34
±0.37
±0.16
±0.05
±0.15
±0.05
0.35± 0.05
2.00
±0.39
±0.22
±0.11
±0.19
±0.04
0.45± 0.05
2.25
±0.53
±0.47
±0.23
±0.40
±0.05

Table A.3: Data table of diﬀerential cross sections for Σ+ p scattering in the highmomentum region (0.65 < pΣ [GeV/c] < 0.80).
cos θCMdσ/dΩstat.
syst. (Total) syst. (BG) syst. (eﬀ) syst. (L)[mb/sr][mb/sr][mb/sr][mb/sr] [mb/sr][mb/sr]−0.85± 0.05
0.81
±0.52
±0.21
±0.19
±0.09
±0.01
−0.75± 0.05
0.58
±0.39
±0.41
±0.40
±0.04
±0.01
±0.19
±0.06
±0.02
−0.65± 0.05
1.26
±0.42
±0.20
−0.55± 0.05
0.68
±0.34
±0.12
±0.12
±0.02
±0.01
−0.45± 0.05
1.08
±0.33
±0.09
±0.08
±0.02
±0.02
±0.05
±0.02
±0.02
−0.35± 0.05
1.35
±0.31
±0.06
−0.25± 0.05
1.29
±0.35
±0.22
±0.21
±0.02
±0.02
±0.09
±0.01
±0.02
−0.15± 0.05
1.58
±0.44
±0.10
−0.05± 0.05
2.08
±0.40
±0.21
±0.20
±0.06
±0.03
±0.22
±0.06
±0.02
0.05± 0.05
1.50
±0.37
±0.23
0.15± 0.05
2.40
±0.47
±0.26
±0.23
±0.10
±0.04
±0.15
±0.06
±0.03
0.25± 0.05
2.02
±0.45
±0.17
0.35± 0.05
1.47
±0.43
±0.30
±0.29
±0.06
±0.02
±0.30
±0.11
±0.03
0.45± 0.05
1.90
±0.44
±0.32
0.55± 0.05
1.70
±0.52
±0.20
±0.09
±0.18
±0.03

113

Appendix B
Derivation of the integrated crosssections for the Σ+p elastic scatteringwith a ﬁne momentum stepIn the same way as the diﬀerential cross sections, the integrated cross section for the each
momentum was calculated directly by the formulaσ−1<cos θCM <0.5 (pΣ ) =N (pΣ ).
ρ · NA · L(pΣ ) · ε̄(pΣ )(B.1)

In this analysis, only the way of projecting ∆E spectrum was diﬀerent from the casefor diﬀerential cross sections. Because the events not divided by the scattering angle,
data with ﬁne momentum steps of 0.2 GeV/c could be obtained. Figure B.1 shows theﬁtting results of the ∆E spectra in the evaluation of N (pΣ ). Figure B.2, Figure B.3
and Figure B.4 shows the evaluated number of Σ+ p scattering events, averaged eﬃciency,
and total ﬂight length of Σ+ as a functions of momentum, respectively. The obtained
integrated cross sections are shown in Figure B.5. In this ﬁgure, the integrated crosssections calculated from the diﬀerential cross sections are also plotted. The values of theintegrated cross sections and their uncertainty are summarized in Table B.1.

114

Counts200

0.44 <p< 0.46

0.46 <p< 0.48

0.48 <p< 0.50

0.50 <p< 0.52

0.52 <p< 0.54

0.54 <p< 0.56

0.56 <p< 0.58

0.58 <p< 0.60

0.60 <p< 0.62

0.62 <p< 0.64

0.64 <p< 0.66

0.66 <p< 0.68

0.68 <p< 0.70

0.70 <p< 0.72

0.72 <p< 0.74

0.74 <p< 0.76

0.76 <p< 0.78

0.78 <p< 0.80

150
100

Counts50
0100Counts

50

0
30
20

Counts10020Counts100Σpsimulation sum
decay
pp
decay+accidental πpdecay+accidental others

15
10
5
0

−50

0

50 100 −50∆E [MeV]0

50 100∆E [MeV]Figure B.1: ∆E spectra for the each ﬁne-divided momentum region.

115

counts

600
500
400
300
200
100
0

0.45 0.5 0.55 0.6

0.65 0.7

0.75 0.8 0.85pΣ [GeV/c]average efficiency

Figure B.2: Evaluated number of Σ+ p scattering N (pΣ ) for each momentum.

0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0

0.45

0.5

0.55

0.6

0.65

0.7

0.75 0.8pΣ [GeV/c]

Figure B.3: Averaged eﬃciency ε̄(pΣ ) for each momentum.

116

Total Length [cm]

×103

9000
8000
7000
6000
5000
4000
3000
2000
1000
0.45 0.5 0.55 0.6

0.65 0.7

0.75 0.8 0.85pΣ [GeV/c]

σ-1<cosθ<0.5 [mb]Figure B.4: Total ﬂight length of Σ+ (L(pΣ )) for each momentum.

30
25
20
15
10
5
0

0.45 0.5 0.55 0.6

0.65 0.7

0.75 0.8 0.85pΣ [GeV/c]

Figure B.5: Integrated cross sections σ−1<cos θCM <0.5 (pΣ ) for the Σ+ p scattering. The blue
point represents the integrated cross sections calculated from the diﬀerential cross sections.
The error bars and boxes show the statistical and systematic uncertainties, respectively.

117

Table B.1: Data table of the integrated cross section of the Σ + p elastic scattering. The
systematic error was estimated as a quadratic sum of the error from the backgroundestimation (BG), averaged eﬃciency (eﬀ), and Σ+ total ﬂight length (L).
pΣ
σ−1<cos θCM <0.5 stat. syst. (Total) syst. (BG) syst. (eﬀ) syst. (L)
GeV/c
[mb]
[mb]
[mb]
[mb]
[mb]
[mb]
0.45± 0.01
18.1
±2.8
±5.9
±4.9
±3.3
±0.4
0.47± 0.01
21.9
±1.6
±4.6
±3.4
±3.1
±0.4
0.49± 0.01
23.9
±1.3
±3.7
±2.5
±2.7
±0.5
0.51± 0.01
19.8
±1.3
±2.4
±1.7
±1.7
±0.5
±1.1
±1.5
±0.4
0.53± 0.01
18.9
±1.3
±1.9
0.55± 0.01
17.7
±1.6
±1.3
±0.7
±1.1
±0.4
0.57± 0.01
15.6
±1.7
±0.8
±0.3
±0.7
±0.3
0.59± 0.01
14.8
±2.0
±1.0
±0.5
±0.8
±0.3
0.61± 0.01
10.7
±1.9
±0.6
±0.3
±0.5
±0.2
0.63± 0.01
11.3
±2.2
±0.8
±0.6
±0.4
±0.2
0.65± 0.01
15.3
±2.4
±0.8
±0.4
±0.6
±0.3
0.67± 0.01
14.8
±2.9
±1.2
±1.1
±0.5
±0.3
0.69± 0.01
15.7
±3.1
±1.1
±0.8
±0.8
±0.3
0.71± 0.01
14.8
±3.0
±1.1
±0.9
±0.5
±0.3
±0.6
±0.5
±0.3
0.73± 0.01
17.2
±3.4
±0.8
0.75± 0.01
13.1
±3.1
±1.1
±0.9
±0.5
±0.3
±1.3
±0.8
±0.3
0.77± 0.01
15.5
±4.4
±1.6
0.79± 0.01
15.1
±4.3
±0.9
±0.7
±0.5
±0.3

118