Research History Shunzo Kumano

I have been working on theoretical hadron and nuclear physics. I explain my major research

results in order from the most recent one. References are cited from the publication list [1–74].

1. Gluon content of proton and deuteron at NICA SPD

The Spin Physics Detector (SPD) is a future multipurpose experiment foreseen to run at

the Nuclotron-based Ion Collider fAcility (NICA), which is currently under construction at

the Joint Institute for Nuclear Research (JINR, Dubna, Russia). The physics program of

the experiment is based on collisions of longitudinally and transversely polarized protons

and deuterons at

√

s up to 27 GeV and luminosity up to 10

−2

−1

. The SPD will

operate as a universal facility for comprehensive study of unpolarized and polarized gluon

content of the nucleon, using diﬀerent complementary probes such as: charmonia, open

charm, and prompt photon production processes. The purpose of this work [1] is to make

a thorough review of the physics objectives that can p otentially be addressed at the SPD,

underlining related theoretical aspects and discussing relevant experimental results when

available. Among diﬀerent pertinent phenomena, particular attention is drawn to the

study of the gluon helicity, gluon Sivers and Boer-Mulders functions in the nucleon, as

well as the gluon transversity distribution in the deuteron, via the measurement of speciﬁc

single and double spin asymmetries.

In this paper [1], S. Kumano contributed especially to the sections of “5.3 Gluon transver-

sity in deuteron” and “5.4 Tensor-polarized gluon distribution in deuteron”. Since the

NICA will have a polarized-deuteron beam, it is possible to investigate polarized structure

functions, which are speciﬁc to the spin-1 deuteron. There exists the gluon transversity

in the deuteron, whereas it does not exist in the spin-1/2 nucleon because the helicity ﬂip

∆s = 2 is not possible. The deuteron is a bound state of a proton and a neutron; however,

they cannot contribute directly to the gluon transversity of the deuteron. Therefore, the

gluon transversity is an appropriate observable to ﬁnd new hadronic physics beyond the

simple bound system of the nucleons. The tensor-polarized structure functions are addi-

tional structure functions in the spin-1 deuteron to the ones of the spin-1/2 nucleon. They

are valuable in shedding light on a new asp ect of high-energy spin physics. Since the con-

ventional convolution description does not agree with the existing HERMES data on the

tensor-polarized structure function b

[7], they could be good quantities to ﬁnd an exotic

hadronic eﬀect. These new observables will be investigated at NICA by direct-photon,

J/ψ, and other hadron production process with the polarized-deuteron b eam.

2. Transverse-momentum-dependent parton distribution functions for spin-1 hadrons

We showed possible transverse-momentum-dependent parton distribution functions (TMDs)

for spin-1 hadrons including twist-3 and 4 functions in addition to the leading twist-2

ones by investigating all the possible decomposition of a quark correlation function in the

Lorentz-invariant way 2. The Hermiticity and parity invariance were imposed in the de-

composition; however, the time-reversal invariance was not used due to an active role of

gauge links in the TMDs. Therefore, there exist time-reversal odd functions in addition

to the time-reversal even ones in the TMDs. We listed all the functions up to twist-4

level because there were missing terms associated with the lightcone vector n in previ-

ous works on the twist-2 part and there was no correlation-function study in the twist-3

and 4 parts for spin-1 hadrons. We showed that 40 TMDs exist in the tensor-polarized

spin-1 hadron in the twist 2, 3, and 4. Some expressions of twist-2 structure functions

are modiﬁed from previous derivations due to the new terms with n, and we found 30

new structure functions in the twist 3 and 4 in this work. Since time-reversal-odd terms

of the collinear correlation function should vanish after integrals over the partonic trans-

verse momentum, we obtained new sum rules for the time-reversal-odd structure functions,

∫

3LL

= 0. In addition, we indicated that new transverse-

momentum-dependent fragmentation functions exist in tensor-polarized spin-1 hadrons.

The TMDs are rare observables to ﬁnd explicit color degrees of freedom in terms of color

ﬂow, which cannot be usually measured because the color is conﬁned in hadrons. Fur-

thermore, the studies of TMDs enable not only to ﬁnd three-dimensional structure of

hadrons, namely hadron tomography including transverse structure, but also to provide

unique opportunities for creating interesting interdisciplinary physics ﬁelds such as gluon

condensates, color Aharonov-Bohm eﬀect, and color entanglement. The tensor structure

functions may not be easily measured in experiments. However, high-intensity facility

such as the Thomas Jeﬀerson National Accelerator Facility (JLab), the Fermilab Main

Injector, and future accelerators like electron-ion collider (EIC) may probe such observ-

ables. In addition, since the Nuclotron-based Ion Collider fAcility (NICA) focuses on

spin-1 deuteron structure functions, there is a possibility to study the details of polarized

structure functions of the deuteron at this facility.

3. Gluon transversity in polarized proton-deuteron Drell-Yan process

Nucleon spin structure functions have been investigated mainly by longitudinally-polarized

ones for ﬁnding the origin of the nucleon spin. Other types of spin structure functions are

transversely-polarized ones. In particular, quark transversity distributions in the nucleons

have very diﬀerent properties from the longitudinally-polarized quark distribution func-

tions, especially in scaling violation, because they are decoupled from the gluon transver-

sity, due to the fact that they are helicity-ﬂip, namely chiral-odd, distributions. Such

studies are valuable for ﬁnding not only the origin of the nucleon spin but also a signature

on physics b eyond the standard model, b ecause the electric dipole moment of the neu-

tron is prop ortional to the transversity distributions. Now, there is experimental progress

on the quark transversity distributions; however, there is no experimental information on

gluon transversity. In fact, the gluon transversity does not exist for the spin-1/2 nucleons

due to the helicity-conservation constraint. One needs a hadron with spin more than or

equal to one, so that the helicity ﬂip of two units is allowed. A stable spin-1 target is, for

example, the deuteron for studying the gluon transversity.

In our work, we proposed a possibility for ﬁnding the gluon transversity at hadron-

accelerator facilities, especially in the proton-deuteron Drell-Yan process with the linearly-

polarized deuteron, by showing theoretical formalism and numerical results [3,4]. In the

experiment, the information on the dimuon angular distribution is necessary in the ﬁnal

state; however, the proton beam does not have to be polarized. We showed the de-

pendencies of the Drell-Yan cross section on the dimuon-mass squared M

µµ

, the dimuon

transverse-momentum q

, the dimuon rapidity y in the center-of-momentum frame, and

the magnitude of the gluon transversity ∆

g. We also showed typical spin asymmetries in

the Drell-Yan process. The gluon transversity is not experimentally measured; however,

there are future experimental projects to measure them at Thomas Jeﬀerson National

Accelerator Facility (JLab) and Electron-Ion Collider (EIC). Therefore, much progress is

expected for the gluon transversity in the near future. On the other hand, independent

experiments are desirable at other experimental facilities, especially at hadron accelerator

facilities, to probe diﬀerent kinematical regions of the gluon transversity from the JLab and

EIC ones. There are available hadron facilities at Fermilab, J-PARC (Japan Proton Accel-

erator Research Complex), GSI-FAIR (Gesellschaft f¨ur Schwerionenforschung -Facility for

Antiproton and Ion Research), and NICA (Nuclotron-based Ion Collider fAcility). In addi-

tion, if the ﬁxed-deuteron target becomes possible at RHIC, Large Hadron Collider (LHC),

or EIC, there could be a possibility. We showed in our studies that the gluon transversity

could be investigated by the proton-deuteron Drell-Yan process with the linearly-polarized

deuteron. This experiment is under consideration in the Fermilab-E1039 project. Since

the internal spin-1/2 nucleons within the deuteron cannot contribute directly to the gluon

transversity, it could be a good observable to ﬁnd a new non-nucleonic component beyond

the simple bound system of nucleons in nuclei.

4. Gravitational form factors of hadrons

The nucleon spin used to be explained by a combination of three-quark spins in the nucleon

according to the basic quark model. However, it became clear that the quark contribution

to the nucleon spin is 20−30%, and the rest of spin should come from gluon-spin and

partonic orbital-angular-momentum (OAM) contributions. For ﬁnding the OAM part,

it became necessary to investigate three-dimensional structure of the nucleon, including

transverse structure in addition to the longitudinal one described by the Bjorken variable x.

This ﬁeld is called hadron tomography. It can be investigated by three-dimensional struc-

ture functions such as generalized parton distributions (GPDs), transverse-momentum-

dependent parton distributions, and generalized distribution amplitudes (GDAs).

In our work, we extracted the GDAs, which are s-t crossed quantities of the GPDs, from

cross-section measurements of hadron-pair production process γ

∗

γ → π

at KEKB [5,6].

This work was the ﬁrst attempt to obtain the GDAs and gravitational form factors from

the actual experimental data. The GDAs were expressed by a number of parameters and

they were determined from the data of γ

∗

γ → π

by including intermediate scalar- and

tensor-meson contributions to the cross section. Our results indicated that the depen-

dence of parton-momentum fraction z in the GDAs is close to the asymptotic one. The

timelike gravitational form factors Θ

and Θ

were obtained from our GDAs, and they

were converted to the spacelike ones by the dispersion relation. To be precise, they are

the form factors of energy-momentum tensors in QCD; however, they are often called

gravitational form factors in hadron physics. From the spacelike Θ

and Θ

, gravitational

densities of the pion were calculated. Then, we obtained the mass (energy) radius and

the mechanical (pressure and shear force) radius from Θ

and Θ

, respectively. They

were calculated as

√

⟨r

⟩

mass

= 0.32 ∼ 0.39 fm, whereas the mechanical radius was larger

√

⟨r

⟩

mech

= 0.82 ∼ 0.88 fm. This is the ﬁrst report on the gravitational radius of a

hadron from actual experimental measurements [6]. It is interesting to ﬁnd the possibility

that the gravitational mass and mechanical radii could b e diﬀerent from the experimental

charge radius

√

⟨r

⟩

charge

= 0.672 ± 0.008 fm for the charged pion.

For drawing a clear conclusion on the GDAs of hadrons, accurate experimental data are

needed, and it should be possible, for example, by future measurements of super-KEKB

and international linear collider (ILC). Accurate measurements will not only provide im-

portant information on hadron tomography but also possibly shed light on gravitational

physics in the quark and gluon level. Gravitational physics used to be considered as a

ﬁeld on macroscopic world. However, we showed that it is possible to investigate it in the

microscopic level in terms of fundamental particles of quarks and gluons. In future, we

expect much progress on origin of hadron masses and internal hadron pressures in terms

of quark and gluon degrees of freedom. This work together with our studies on the gluon

transversity was selected one of highlight research results of KEK in the annual report of

2019.

5. Tensor-polarized structure function b

in standard convolution description of deuteron

Tensor-polarized structure functions b

1−4

of a spin-1 hadron are additional observables

which do not exist for the spin-1/2 nucleons. They could probe novel aspects of the

internal hadron structure. Twist-2 tensor-polarized structure functions are b

and b

, and

they are related by the Callan-Gross-like relation in the Bjorken scaling limit. The other

functions b

and b

are higher-twist ones, which may not be easily accessed experimentally

at this stage.

In this work, we theoretically calculated b

in the standard convolution description for

the deuteron [7]. Two diﬀerent theoretical models, a basic convolution description and a

virtual nucleon approximation, were used for calculating b

and their results were com-

pared with the HERMES measurement. The convolution model means that the deuteron

structure function is evaluated by the nucleonic structure function convoluted with the

nucleon momentum distribution in the deuteron. Namely, the virtual-photon interaction

is split into two processes in the charged-lepton scattering with the deuteron. First, the

nucleon momentum distribution within the deuteron is calculated, and then the photon

interaction with a quark or an antiquark is calculated. The total eﬀect is expressed by a

convolution integral. The deuteron is mainly the S-wave bound state of proton and neu-

tron; however, there is a small admixture of D wave. From this convolution description, the

structure function b

of the deuteron was expressed by the structure functions F

for the

nucleons with the tensor-polarization combination of the nucleon’s lightcone-momentum

distributions. There were an S-D interference contribution and a purely D-wave one for

. Estimating the convolution expression for b

, we found large diﬀerences between our

theoretical results and the data. In the measured x range (x < 0.5), the experimental

magnitude was one-order larger than both theoretical estimates. Furthermore, there were

relatively large distributions even at large x (0.6 < x < 0.8). Because the HERMES

errors are large, we cannot draw a solid conclusion from this comparison. However, the

large diﬀerences indicated that a new hadron physics mechanism could be possibly needed

for their interpretation, although there are still some rooms to improve, for example, by

considering higher-twist eﬀects.

Future b

studies could shed light on this new ﬁeld of hadron physics. In particular, detailed

experimental studies of b

will start at the Thomas Jeﬀerson National Accelerator Facility.

In addition, there are possibilities to investigate tensor-polarized parton distribution func-

tions and b

at Fermi National Accelerator Laboratory and a future electron-ion collider.

Therefore, further theoretical studies are needed for understanding the tensor structure of

the spin-1 deuteron, including a new mechanism to explain the large diﬀerences between

the current data and our theoretical results.

6. Uniﬁed model of neutrino-nucleus reactions

Neutrino oscillation experiments have been done for ﬁnding new physics beyond the stan-

dard model. Since nuclear targets are used in the exp eriments, for example water in the

T2K project, a precise description of neutrino-nucleus reactions plays a key role in ad-

dressing fundamental questions such as the leptonic CP violation and the neutrino mass

hierarchy through analyzing data on neutrino oscillation experiments. The neutrino en-

ergy relevant to the neutrino-nucleus reactions spans a broad range. Accordingly, the

dominant reaction mechanism varies across the energy region from quasi-elastic scatter-

ing through nucleon resonance excitations to deep inelastic scattering. This corresponds

to transitions of the eﬀective degree of freedom for theoretical description from nucleons

through meson-baryon to quarks.

The main purpose of our review [8] was to report our recent eﬀorts towards a uniﬁed

description of the neutrino-nucleus reactions over the wide energy range, and recent overall

progress in the ﬁeld was also sketched. Starting with an overview of the current status of

neutrino-nucleus scattering experiments, we formulated the cross section to be commonly

used for the reactions over all the energy regions. At low energies with the momentum-

transfer squared Q

< 1 GeV

and invariant-mass squared W

< 4 GeV

, the reaction is

described by quasi-elastic scattering with nucleons and by nucleon resonances. At high

energies with Q

> 1 GeV

and W

> 4 GeV

, it is described by deep inelastic scattering

(DIS) with a nucleus in terms of quarks and gluons. In the region with Q

< 1 GeV

and W

> 4 GeV

, the reaction is expressed by Reggeons and Pomerons by noting the

partially conserved axial-vector current (PCAC) relation at Q

→ 0. We also noted the

quark-hadron duality in connecting both resonance and DIS regions. These theoretical

models were checked by experimental measurements of electron scattering, and the axial-

vector terms were introduced for the neutrino reactions. A description of the neutrino-

nucleon reactions followed and, in particular, a dynamical coupled-channels model for

meson productions in and beyond the ∆(1232) region was discussed in detail. We then

discussed the neutrino-nucleus reactions, putting emphasis on our theoretical approaches.

We started the discussion with electroweak processes in few-nucleon systems studied with

the correlated Gaussian method. Then, we describ ed quasi-elastic scattering with nuclear

spectral functions, and meson productions with a ∆-hole model. Nuclear modiﬁcations of

the parton distribution functions determined through a global analysis were also discussed.

Finally, we discussed issues to be addressed for future developments.

7. Global analyses of fragmentation functions

Fragmentation functions indicate probabilities to ﬁnd hadrons created from parent quarks

or a gluon, and they are essential, for example, for calculating hadron-production cross

sections in high-energy reactions in order to study the origin of nucleon spin, properties

of quark-gluon plasma, and signatures beyond the standard model. In our studies, frag-

mentation functions and their uncertainties were determined for pion, kaon, and proton

by a global χ

analysis of charged-hadron production data in electron-positron annihila-

tion and by the Hessian method for error estimation [33]. It is especially important that

the uncertainties of the fragmentation functions were estimated in this analysis for the

ﬁrst time. The results indicated that the fragmentation functions, especially gluon and

light-quark fragmentation functions, had large uncertainties at small Q

. We found that

determination of the fragmentation functions is improved in next-to-leading-order (NLO)

analyses for the pion and kaon in comparison with leading-order ones. Such a NLO im-

provement was not obvious in the proton. Since the uncertainties are large at small Q

, the

uncertainty estimation is very important for analyzing hadron-production data at small

or p

, p

<< M

) in lepton scattering and hadron-hadron collisions. A code is

available for general users for calculating obtained fragmentation functions.

In 2013, accurate measurements were reported by the Belle and BaBar collaborations

for the fragmentation functions at the center-of-mass energies of 10.52 GeV and 10.54

GeV, respectively, at the KEK and SLAC B factories, whereas other available e

−

mea-

surements were mostly done at higher energies, mainly at the Z mass of 91.2 GeV. We

reported our global analysis of the fragmentation functions especially to show impacts of

the B-factory measurements on the fragmentation function determination [10]. Our re-

sults indicated that the fragmentation functions are determined more accurately not only

by the scaling violation but also by high-statistical nature of the Belle and BaBar data.

We also explained how the ﬂavor dependence of quark fragmentation functions and the

gluon function are separated by using measurements at diﬀerent Q

values. In particular,

the electric and weak charges are diﬀerent depending on the quark type, so that a light-

quark ﬂavor separation also became possible in principle due to the precise data at both

√

s ≃ 10.5 GeV and 91.2 GeV.

Next, we performed the ﬁrst iterative Monte Carlo (IMC) analysis of fragmentation func-

tions [9]. The IMC method eliminates potential bias in traditional analyses based on

single ﬁts introduced by ﬁxing parameters not well constrained by the data and provides

a statistically rigorous determination of uncertainties. Our analysis revealed speciﬁc fea-

tures of fragmentation functions using the new IMC methodology and those obtained from

previous analyses, especially for light quarks and for strange-quark fragmentation to kaons.

Third, we proposed that fragmentation functions should be used to identify exotic hadrons

by using properties of favored- and disfavored-fragmentation functions [31]. The favored

fragmentation means a fragmentation from a quark or an antiquark which exists in a

hadron as a constituent in a quark model, and the disfavored means a fragmentation from

a sea quark. As an example, fragmentation functions of the scalar meson f

(980) were

investigated. It was pointed out that the second moments and functional forms of the

u- and s-quark fragmentation functions could distinguish the tetraquark structure from

q¯q. By the global analysis of f

(980)-production data in electron-positron annihilation, its

fragmentation functions and their uncertainties were determined. It was found that the

current available data are not suﬃcient to determine its internal structure, while precise

data in future should be able to identify exotic quark conﬁgurations.

8. Tensor-polarization asymmetry in proton-deuteron Drell-Yan process

Tensor-polarized parton distribution functions are new quantities in spin-one hadrons such

as the deuteron, and they could probe new quark-gluon dynamics in hadron and nuclear

physics. In charged-lepton deep inelastic scattering (DIS), they are studied by the twist-

two structure functions b

and b

. The HERMES collaboration found much larger |b

values than a naive theoretical expectation based on the standard deuteron model. The

situation should be signiﬁcantly improved in the near future by an approved experiment

to measure b

at JLab (Thomas Jeﬀerson National Accelerator Facility). There was also

an interesting indication in the HERMES result that ﬁnite antiquark tensor polarization

exists

∫

dxb

(x) = [ 0.35±0.10 (stat)±0.18 (sys) ], which indicates a deviation from the sum

rule

∫

dxb

(x) = 0 in Ref. [65]. This ﬁnite tensor polarization could play an important role

in solving a mechanism on tensor structure in the quark-gluon level. The tensor-polarized

antiquark distributions are not easily determined from the charged-lepton DIS such as the

HERMES and JLab measurements; however, they can be measured in a proton-deuteron

Drell-Yan process with a tensor-polarized deuteron target.

In our work, we estimated the tensor-polarization asymmetry for a possible Fermilab

Main-Injector experiment E1039 by using optimum tensor-polarized PDFs to explain the

HERMES measurement [11]. The tensor-polarized PDFs at the HERMES kinematics

= 2.5 GeV

) [25] were evolved to the ones at the larger-Q

Fermilab region. We found

that the asymmetry is typically a few percent. If it is measured, it could probe new hadron

physics, and such studies could create an interesting ﬁeld of high-energy spin physics. In

addition, we found that a signiﬁcant tensor-polarized gluon distribution should exist due to

evolution, even if it were zero at a low Q

scale. The tensor-polarized gluon distribution

has never been observed, so that it is an interesting future project. Our formalism is valid

not only for the Fermilab experiment but also for any other hadron facilities such as J-

PARC, GSI-FAIR, and NICA. Furthermore, if the ﬁxed-deuteron target becomes possible

at RHIC, Large Hadron Collider (LHC), or EIC, our studies can be used.

9. GPD experiment by using pion-induced exclusive Drell-Yan process at J-PARC

Generalized parton distributions (GPDs) encoding multidimensional information of hadron

partonic structure appear as the building blocks in a factorized description of hard exclusive

reactions. The nucleon GPDs have been accessed by deeply virtual Compton scattering

and deeply virtual meson production with lepton beam. A complementary probe with

hadron beam is the exclusive pion-induced Drell-Yan process.

In our work, we discussed recent theoretical advances on describing this process in terms

of nucleon GPDs and pion distribution amplitudes. In the framework of the J-PARC E50

experiment, we addressed the feasibility of measuring the exclusive pion-induced Drell-

Yan process π

−

p → µ

−

n in the coming high-momentum b eam line of J-PARC [12].

Detailed simulations on signal reconstruction eﬃciency as well as on rejection of the most

severe random background channel were performed for the pion beam momentum in the

range of 10−20 GeV. A clean signal of exclusive pion-induced Drell-Yan process can be

identiﬁed in the missing-mass spectrum of dimuon events with 2−4 fb

−1

integrated lu-

minosity. The statistics accuracy is adequate for discriminating between the predictions

from two current GPD modelings. The realization of this measurement will represent not

only a new approach of accessing nucleon GPDs and pion distribution amplitudes (DAs)

in the timelike pro cess, but also a novel test of the factorization of an exclusive Drell-Yan

process associated with timelike virtuality and the universality of GPDs in spacelike and

timelike processes. Since both inclusive and exclusive Drell-Yan events could be measured

simultaneously, the data could reveal interesting features in the transition from inclusive

Drell-Yan to the semi-exclusive and exclusive limits. The pion pole in the GPD

E is

expected to give a dominant contribution in the cross sections at small |t|. The cross

sections at this pion-pole dominance region will provide a unique opportunity to access

the pion timelike form factor other than the approach of e

−

annihilation process. The

input cross section used for the exclusive Drell-Yan events in this work was the prediction

within the factorization approach using the partonic hard scattering in the leading order

of α

convoluted with the leading-twist pion DAs and nucleon GPDs. Realization of such

measurement at J-PARC will provide a new test of perturbative QCD descriptions of a

novel class of hard exclusive reactions. It is complementary to the JLab experiment in the

sense that the J-PARC experiment will probe a diﬀerent x region (x = 0.1-0.3) from the

JLab one x = 0.3-0.6. It will also oﬀer the possibility of experimentally accessing nucleon

GPDs at large timelike virtuality. In 2019, a LoI (Letter of Intent) was submitted on this

experiment by joining the J-PARC-E50 project, whose main topic is on charmed-baryon

spectroscopy, in the high-momentum beamline.

10. Exotic hadron structure by constituent-counting rule for hard exclusive processes

A basic quark model indicates that baryons consist of three quarks (qqq) and mesons of a

quark-antiquark pair (q¯q). Hadrons with other compositions, such as qq¯q¯q and qqqq¯q, are

exotic hadrons. After 2004, there have been several reports on exotic hadron candidates.

However, it is not easy to conﬁrm their exotic nature by global observables like masses,

decay widths, and spins. Therefore, we proposed to use high-energy hadron reactions

and perturbative QCD for ﬁnding internal structure of the exotic candidates to clarify

the exotic nature [13,19]. According to perturbative QCD, exclusive high-energy hadron

reactions occur by hard gluon exchanges. Therefore, their cross sections are estimated by

considering hard quark and gluon propagators together with other kinematical factors. It

leads to the so-called constituent counting rule which indicates, for example, that a two-

body exclusive hadron reaction cross section a + b → c + d scales like dσ/dt ∼ f(θ

)/s

n−1

with n = n

+ n

, where n

is the number of elementary constituents participate

in the reaction and s is the center-of-mass energy squared. This theoretical prediction had

been conﬁrmed by BNL and JLab experiments.

We proposed to use hard exclusive production of an exotic hadron for ﬁnding its internal

quark-gluon conﬁguration by the constituent-counting rule in perturbative QCD. Esp e-

cially, we investigated internal structure of hyperons and their excited states by using the

high-momentum pion beam at J-PARC. First, the cross section for the exclusive process

−

+ p → K

+ Λ(1405) was estimated at the scattering angle θ = 90

◦

in the center-of-

mass frame by using current experimental data [19]. In comparison, the cross section for

the ground-state Λ production π

−

+ p → K

+ Λ was also shown. We suggested that

the internal quark conﬁguration of Λ(1405) should be determined by the asymptotic scal-

ing behavior of the cross section. If it is an ordinary three-quark baryon, the scaling of

the cross section is s

dσ/dt =constant, whereas it is s

dσ/dt =constant if Λ(1405) is a

ﬁve-quark hadron, where s and t are Mandelstam variables. Such a measurement will be

possible, for example, by using the high-momentum beamline at J-PARC. In addition,

another exclusive process γ + p → K

+ Λ(1405) could be investigated at LEPS and JLab

for ﬁnding the nature of Λ (1405). We indicated that the constituent-counting rule could

be used as a valuable observable in determining internal structure of exotic hadrons by

high-energy exclusive processes, where quark-gluon degrees of freedom explicitly appear.

Furthermore, it is interesting to investigate the transition from hadron degrees of freedom

to quark-gluon ones for exclusive exotic-hadron production processes.

Second, we analyzed the JLab-CLAS data on the photoproduction of hyperon resonances,

as well as the available data for the ground state Λ and Σ

of the CLAS and SLAC-

E84 collaborations, by considering constituent-counting rule suggested by perturbative

QCD [13]. From the analyses of the γ p → K

Λ and K

reactions, we found that the

number of the elementary constituents is consistent with n

= 1, n

= 3, n

= 2, and

= n

= 3. Then, the analysis was made for the photoproductions of the hyperon

resonances Λ(1405), Σ(1385)

, and Λ(1520), where Λ(1405) could be considered to be

KN molecule and hence its constituent number could be ﬁve. However, we found

that the current data are not enough to conclude the numbers of their constituent. It is

necessary to investigate the higher-energy region at

√

s > 2.8 GeV experimentally beyond

the energy of the available CLAS data for counting the number of constituents clearly. We

also mentioned that our results indicate energy dependence in the constituent number,

especially for Λ(1405). Namely, Λ(1405) looked like a penta-quark state at lower energies,

but it became a three-quark one at high energies. If an excited hyperon is a mixture of

three-quark and ﬁve-quark states, the energy dependence of the scaling behavior could be

valuable for ﬁnding its composition and mixture. We expect to have much progress in

future on the internal structure of exotic hadron candidates by using high-energy hadron

reactions, as a new ﬁeld on exotic hadrons.

11. Compositeness of exotic hadron candidates

In recent years, there are a number of reports on exotic hadron candidates. They are theo-

retically described by ordinary hadrons (q¯q, qqq), exotic conﬁgurations, hadron molecules,

or their mixtures. These models contain parameters which are adjusted to explain ex-

perimental observables, so that their descriptions are not necessarily appropriate ways

to understand their internal structure. As a p ossible metho d to understand the hadron

structure is to investigate compositeness of bound-state systems.

In our studies, structure of the a

(980) and f

(980) resonances was investigated with the

(980)-f

(980) mixing intensity from the viewpoint of compositeness [14], which corre-

sponds to the amount of two-bo dy states composing resonances as well as bound states.

If it is one, the hadron is a bound state of a hadron molecule, and it is an ordinary hadron

described by the basic quark mo del if the compositeness is zero. For example, since it

is not possible to explain the strong decay width of f

→ ππ by quark models with the

q¯q conﬁguration [71] and also from experimental measurements on the radiative decay

ϕ → f

γ [59] and the two-photon decay width (f

→ γγ), f

(980) could be considered as

a K

K molecule.

For this purpose, we ﬁrst formulated the a

(980)-f

(980) mixing intensity as the ratio of

two partial decay widths of a parent particle, in the same manner as the recent analysis in

BES (Beijing Spectrometer) experiments. Calculating the a

(980)-f

(980) mixing intensity

with the existing Flatte parameters from experiments, we found that many combinations

of the a

(980) and f

(980) Flatte parameters can reproduce the experimental value of the

(980)-f

(980) mixing intensity by BES. Next, from the same Flatte parameters, we also

calculated the K

K compositeness for a

(980) and f

(980). Although the compositeness

with the correct normalization became complex in general for resonance states, we found

that the Flatte parameters for f

(980) imply large absolute value of the K

K composite-

ness and the parameters for a

(980) led to small but nonnegligible absolute value of the

K compositeness. Then, connecting the mixing intensity and the K

K compositeness via

the a

(980)- and f

(980)-K

K coupling constants, we established a relation between them.

As a result, a small mixing intensity indicated a small value of the product of the K

compositeness for the a

(980) and f

(980) resonances. Moreover, the experimental value

of the a

(980)-f

(980) mixing intensity implied that the a

(980) and f

(980) resonances

cannot be simultaneously K

K molecular states. These two results suggested a new view-

point that f

(980) is mainly a K

K state and a

(980) is an ordinary quark bound state,

although both states used to be considered as K

K states.

12. Achievements of B factories and their fragmentation-function measurements

The B factories at KEK and SLAC had investigated breaking of particle-antiparticle asym-

metry and found new particles by using abundantly-produced B mesons, τ particles, and

charmed mesons. At these facilities, signiﬁcant achievements were done by ﬁnding the

symmetry breading between B meson and anti-B meson and conﬁrming the Kobayashi-

Masukawa theory. In addition, they had important ﬁndings on exotic hadrons and precise

measurements on fragmentation functions in hadron physics. In this report “The Physics

of the B Factories” [15], these achievements were summarized as sections on accelerator fa-

cilities, analysis tools and methods, Kobayashi-Maskawa mechanism and its conﬁrmation,

hadron spectroscopy, fragmentation functions, and so on. This report was press released

from KEK on July 11, 2014 as the “50 years since the discovery of CP symmetry break-

ing: Joint report of Belle and BaBar experiments for conﬁrming the Kobayashi-Maskawa

theory”.

S. Kumano contributed especially to the explanations on fragmentation functions in this

report. The fragmentation functions were measured mainly in the Z-mass region by the

electron-positron annihilation at Large Electron-Positron Collider (LEP) and SLAC Large

Detector (SLD). However, the Belle and BaBar collaborations reported very-precise data

on the fragmentation functions in 2013 for light hadrons (π, K, p/¯p). Furthermore, these

data covered the wide kinematical region of the energy fraction z, which is the ratio

of produced-hadron energy to the half of the center-of-mass energy, so that it became

possible to determine the fragmentation functions accurately. These B-factory energies

of about 10 GeV are much smaller than the Z-mass energy region of LEP and SLD,

so that gluon fragmentation functions should be determined for the ﬁrst time through

the scaling violation. These studies contributed to other ﬁelds of physics, such as on

properties of quark-gluon plasma in heavy-ion collisions and on origin of nucleon spin in

polarized-proton reactions, because the fragmentation functions are necessary quantities

for describing cross sections of semi-inclusive hadron productions. In addition, chiral-odd

fragmentation-function measurements at the B factories made it possible to ﬁnd the chiral-

odd parton distributions and transverse-spin physics for clarifying the origin of the nucleon

spin. The details of these B factory achievements were explained in Ref. [

15].

13. Tomography of exotic hadrons in high-energy exclusive processes with GPDs and GDAs

There are a number of reports on exotic hadrons in recent years; however, it is not clear

whether they are really exotic ones from global observables such as masses and decay

widths. As a new approach, we investigated the possibility of determining internal struc-

ture of exotic hadrons by using high-energy reaction processes [16], where quarks and

gluons are appropriate degrees of freedom. In particular, it should be valuable to inves-

tigate the high-energy exclusive processes which include generalized parton distributions

(GPDs) and generalized distribution amplitudes (GDAs). The GPDs and GDAs contain

momentum distributions of partons and form factors. We found that the exotic nature ap-

pears in momentum distributions of quarks as suggested by the constituent-counting rule

and in the form factors associated with exotic hadron sizes and the number of constituents.

First, we showed that the valence-quark distributions of tetraquark (n = 4) and pentaquark

(n = 5) hadrons shift to smaller-x regions from the distributions of the pion (n = 2) and the

nucleon (n = 3). Here, n is the number of valence quarks. The x-dependent distributions

were determined by the number of valence quarks and the momentum fraction carried

by quarks. Second, the large-Q

behavior of form factors is given by the constituent

counting rule of perturbative QCD, so that exotic nature should be found by looking at

the high-momentum region of the hadron form factors. Since exotic hadrons are unstable

particles, they cannot be used as ﬁxed targets and the spacelike GPDs cannot be measured

experimentally, although transition GPDs to exotic hadrons could probe such signatures.

However, exotic hadrons can be investigated by timelike processes, as we proposed that

these exotic signatures should be found in exclusive production processes of exotic hadrons

from γ

∗

γ in electron-positron annihilation. For example, the GDAs contain information

on a time-like form factor of the energy-momentum tensor of a hadron h. We showed that

the cross section of eγ → e + h

h is sensitive to the exotic signature by looking at the h

invariant-mass dependence by taking light hadrons, h = f

(980) and a

(980). From such

GDA measurements, the tomography of exotic hadrons will become possible, for example,

by Belle and BaBar experiments and by future linear collider [16].

14. Determination of Λ(1405) compositeness from its radiative decay

Nucleons, N

∗

resonances, and other baryons are generally described by the quark model

with the qqq composition. However, the mass of Λ(1405) is much diﬀerent from experi-

mental measurements, so that it is considered as a

KN molecule state. Since new kaonic

nuclei have been investigated at J-PARC and other facilities, the internal structure of

Λ(1405) should be clariﬁed because it is possibly the simplest bound system of

KN. In

our studies, the radiative decay of Λ(1405) was investigated from the viewpoint of compos-

iteness [17], which corresponds to the amount of two-body states composing resonances

as well as bound states. This radiative decay is an E1 transition, which is described by

the matrix element of the electric dipole operator e⃗r. It indicates that a diﬀuse molecular

state or a compact quark-bound state could be distinguished by this E1 radiative decay.

In particular, we investigated the compositeness of Λ(1405) by this decay.

For a

KN(I = 0) bound state without couplings to other channels, we established a rela-

tion b etween the radiative decay width and the compositeness. Especially, the radiative

decay width of the bound state is proportional to the compositeness. Applying the formu-

lation to Λ(1405), we observed that the decay to Λγ is dominated by the K

−

p component

inside Λ(1405), because in this decay π

−

and π

−

strongly cancel with each other

and the πΣ component can contribute to the Λγ decay only through the slight isospin

breaking. This means that the decay Λ(1405) → Λγ is suitable for the study of the

component in Λ(1405). Fixing the Λ(1405)-πΣ coupling constant from the usual decay of

Λ(1405) → πΣ, we showed a relation between the absolute value of the

KN compositeness

for Λ(1405) and the radiative decay width of Λ(1405) → Λγ and Σ

γ, and we found that

large decay width to Λγ implies large

KN compositeness for Λ(1405). The comp ositeness

of πΣ was estimated about 0.19, so that the πΣ molecular composition is relatively small.

By using the “experimental” data on the radiative decay widths based on an isobar-model

ﬁtting of the K

−

p atom data, we estimated the

KN compositeness for Λ(1405). We also

discussed the pole-position dependence of our relation on the Λ(1405) radiative decay

width and the eﬀects of the two-pole structure for Λ(1405). For a precise determination

of the compositeness, we need to have an accurate measurement of the radiative decay

width, which should be possible at J-PARC.

15. Report on future nuclear physics in Japan: Nucleon-structure physics

In the 21st century, world’s leading accelerator facilities such as J-PARC, KEKB, and

RIBF were completed in Japan. Together with RCNP and ELPH, it became possible to

investigate diverse aspects of hadron and nuclear physics. In addition, there was signiﬁcant

progress on performance of super-computers. On the other hand, there are major accelera-

tor facilities in the world such as CERN-LHC, CERN-COMPASS, RHIC, Fermilab, JLab,

and GSI, and many Japanese scientists participate in these facility experiments. These

experimental projects cover diverse ﬁelds of hadron and nuclear physics. Nuclear physi-

cists devote to their own projects and there is a tendency that they may not pay attention

to developments on other ﬁelds. In addition, more than 20 years had passed for the J-

PARC and RIBF since the early planning stage, so that physics projects of these facilities

should be re-examined. Therefore, by the proposal of the Japanese Nuclear Physics Exec-

utive Committee, we wrote a report on plans on future nuclear physics projects [18] and

showed possible direction of nuclear physics in Japan. This report covered a wide range of

hadron and nuclear physics on unstable nuclei, precision nuclear physics, strangeness nu-

clear physics, low-energy hadron physics, high-energy heavy-ion physics, nucleon structure,

fundamental physics with nuclei, and computational nuclear physics.

Within this report, S. Kumano contributed to the nucleon-structure section. We ex-

plained proton-spin puzzle, QCD factorization and parton distribution functions (PDFs),

and lepton-proton and proton-proton scattering experiments and their global analyses for

determining polarized PDFs. Transverse spin physics and higher-twist eﬀects were also

discussed. The proton-spin composition was shown in a color-gauge invariant way. For

ﬁnding the origin of nucleon spin, the contribution from partonic orbital angular momenta

should be determined by measuring three-dimensional structure functions. Furthermore,

theoretical hadron models and lattice QCD results were summarized on the structure

functions. Finally, we introduced future experimental projects, CERN-COMPASS, RHIC,

Fermilab, KEKB, JLab, EIC, and J-PARC, on nucleon-structure physics.

16. Numerical solution of Q

evolution equations for fragmentation functions

Semi-inclusive hadron-production pro cesses are becoming important in high-energy hadron

reactions. They are used for investigating properties of quark-hadron matters in heavy-ion

collisions, for ﬁnding the origin of nucleon spin in polarized lepton-nucleon and nucleon-

nucleon reactions, and possibly for ﬁnding exotic hadrons. For describing the hadron-

production cross sections in high-energy reactions, fragmentation functions are essential

quantities. A fragmentation function indicates the probability of producing a hadron from

a parton in the leading order of the running coupling constant α

. In 2013, the Belle and

BaBar collaborations reported very precise experimental data on the fragmentation func-

tions, which were much more accurate than the Large Electron-Positron Collider (LEP)

and SLAC Large Detector (SLD). The LEP and SLD groups measured the fragmentation

functions at the Z-mass region, whereas the Belle and BaBar measurements were at 10.5

GeV. It means that the scaling violation (Q

evolution) phenomena became clear for the

ﬁrst time by these data and it became possible to probe the gluon fragmentation functions.

The Q

dependence is described by the standard DGLAP (Dokshitzer-Gribov-Lipatov-

Altarelli-Parisi) evolution equations, which are often used in theoretical and experimental

analyses of the fragmentation functions and in calculating semi-inclusive cross sections.

The DGLAP equations are complicated integro-diﬀerential equations, which cannot be

solved in an analytical method. On the other hand, there was a strong need from scientists

to use a Q

evolution code for their theoretical and experimental projects. The optimum

fragmentation functions were supplied at ﬁxed Q

, usually small-Q

region where the

perturbative QCD could be applied, so that they should be evolved to diﬀerent Q

scales

for their own experiments and theoretical model calculations. In our work, we explained a

numerical solution method and created a Q

evolution code for general users. The DGLAP

evolution equations are expressed by a diﬀerentiation of Q

and integrals of splitting

functions multiplied by the fragmentation functions over the energy fraction z, which is

deﬁned by the ratio of the produced-hadron energy over the half of the center-of-mass

energy.

In this work, a simple method was employed for solving the evolution equations by using

the Euler method and the Gauss-Legendre quadrature for evaluating integrals, and a

useful code was provided for calculating the Q

evolution of the fragmentation functions

in the leading order (LO) and next-to-leading order (NLO) of α

[20]. The renormalization

scheme is MS in the NLO evolution. Our evolution code was explained for using it in one’s

studies on the fragmentation functions.

17. Test of CDF dijet anomaly within the standard model

In April of 2011, the Fermilab reported that the CDF (Collider Detector at Fermilab) col-

laboration discovered a new signature beyond the standard model by the proton-antiproton

collision with the 1.96 TeV energy at Tevatron, and their work was published in Physical

Review Letters 106 (2011) 171801. Observing produced W and 2 jets in the proton-

antiproton collision, they found an unexplained peak in the cross section as the function

of two-jet invariant mass. In order to conﬁrm that this CDF result is a new discovery, we

needed to show that such a peak cannot be reproduced with the standard model. In the

W - and jet-production cross section, various parton distribution distributions (PDFs) are

involved in the initial proton and antiproton, so that their accurate momentum distribu-

tions should be understood.

This dijet anomaly was investigated within the standard model by considering eﬀects of

the PDFs on various processes: W +dijet, Z+dijet, W W , ZW , and top production. Since

the anomalous peak existed in the dijet-mass region of 140 GeV with the p¯p center-of-

mass energy

√

s=1.96 TeV, a relevant momentum fraction x of partons is roughly 0.1. In

this x region, recent HERMES semi-inclusive charged-lepton scattering experiment indi-

cated that the strange-quark distribution could be very diﬀerent from a conventional one

s ≃ 0.4(¯u +

d)/2, which has been used for many years, based on opposite-sign dimuon

measurements in neutrino-induced deep inelastic scattering. We investigated eﬀects of

such variations in the strange-quark distribution s(x) on the anomaly [21]. We found that

distributions of W +dijets and other process are aﬀected by the strange-quark modiﬁca-

tions in wide dijet-mass regions including the 140 GeV one. Since the CDF anomaly was

observed in the shoulder region of the dijet-mass distribution, a slight modiﬁcation of the

distribution shape could explain at least partially the CDF excess. Therefore, it is im-

portant to consider such eﬀects within the standard model for judging whether the CDF

anomaly indicated new physics beyond the standard model. We also showed modiﬁcation

eﬀects of the strange-quark distribution in the LHC (Large Hadron Collider) kinematics,

where cross sections are sensitive to a smaller-x region of s(x).

On the other hand, the strange-quark distribution itself contains interesting physics. We

expect that the major part of s(x) should be produced in perturbative-QCD process of

the gluon splitting (g → s¯s). However, there could be a nonperturbative-QCD mechanism

to produce s(x) which is separate from the Q

-evolution process, and it is called intrinsic

strange. Because it has never been identiﬁed experimentally, it is an interesting topic by

itself. Our paper was accepted in Physical Review D [21] two days after our submission,

which indicated an importance and urgency of our work. Later, the anomalous peak

disappeared by the CDF reanalysis. However, our theoretical studies are valid in any case

and are independent from the existence of the original dijet anomaly.

18. Structure of Λ

(2940)

by its strong and electromagnetic decays

A number of exotic hadrons were reported by the Belle and BaBar collaborations in heavy-

quark hadron systems. One of them is Λ

(2940)

, which is considered as a molecular state

of nucleon and D

∗

. However, there is a possibility of the D-wave excitation of Λ

(2286)

In order to ﬁnd its internal structure, we proposed to use its strong and electromagnetic

decays. First, we studied the radiative decay Λ

(2940)

→ Λ

(2286)

γ by the hadron

molecule picture and predicted its decay width [24]. We formulated the decay process

by considering ND

∗

loops by paying attention to the proper gauge invariance. There

was a large contribution in the radiation from the internal proton, and the other terms,

radiations from D

∗

, Λ

, and Λ

∗

vertices, were relatively small. In the molecular

scenario, the Λ

(2940)

baryon was described by a superposition of |pD

∗0

⟩ and |nD

∗+

⟩

components with the explicit admixture expressed by the mixing angle θ (|Λ

(2940)

⟩ =

cos θ |pD

∗0

⟩ + sin θ |nD

∗+

⟩). The calculated radiative-decay widths displayed a sizable

sensitivity to the mixing angle θ and to the scale parameter Λ. Especially, the cancellation

between the contributions of the radiations from internal hadrons resulted in a rather

pronounced θ-dep endence. This eﬀect could provide a stringent constraint on the role of

the two molecular components pD

∗0

and nD

∗+

in the Λ

(2940)

resonance. The decay

width also depended much on the momentum cutoﬀ parameter Λ at the Λ

∗

vertex.

We showed our decay width as the functions of θ and Λ

∗

. For example, it was 84 keV

for θ = 10

◦

and Λ = 1 GeV. Possible future measurements of the radiative decay width

could provide further insights into the structure of the Λ

(2940)

state.

Second, we investigated the strong three-bo dy decays Λ

(2940)

→ Λ

(2286)

−

(2286)

by considering the same molecular composition [22]. In our calculation,

we employed the extended SU(4) chiral Lagrangians to describe the interaction terms

contained in L

πD

∗

and L

πBB

′

. Therefore, the necessary couplings

πD

∗

and

πBB

′

were well determined. We showed the explicit contributions from the two-step processes

(2940)

→ Σ

−

→ Λ

(2286)

+ π

−

, Λ

(2940)

→ Σ

→ Λ

(2286)

+ π

−

(2940)

→ Σ

→ Λ

(2286)

+ π

, and Λ

(2940)

→ ρ

(2286)

→ Λ

(2286)

−

. In particular, the contribution Λ

(2940)

→ Σ

π → Λ

(2286)

+ ππ was the

largest, and the intermediate ρ term was essentially negligible. The charged decay mode

involving π

−

was less than two times larger than the neutral π

mode. The strong

decay widths were shown as the functions of the mixing angle θ and cutoﬀ parameter Λ.

For example, it was 4.9 MeV for θ = 10

◦

and Λ = 1 GeV. Our results for the three-body

decay widths presented another test for the molecular interpretation of the Λ

(2940)

. We

expect that future measurements of the radiative and strong decay widths will clarify the

internal structure of Λ

(2940)

in comparison with our theoretical predictions.

19. Clustering properties in nuclear structure functions

Many nuclei are described by the shell model, and their density distributions are shown

by absolute value squared of wave functions. However, there are nuclei which cannot be

easily described by the shell model with a limited number of model energy levels. These

nuclei exist around the mass-number region of 10 and heavy unstable-nucleus region. They

have cluster structure within nuclei. Recently, there were measurements on deep inelastic

electron scattering measurements on a nucleus with the cluster structure, and we had

much progress on structure functions and parton distribution functions for nuclei with

clusters. An experimental group of the Thomas Jeﬀerson National Accelerator Facility

(JLab) measured the structure function F

for the beryllium-9 nucleus, and they showed

the gradient of the nuclear modiﬁcation with respect to the Bjorken-scaling variable x

(|d(F

)/dx|) as a function of average nuclear density. Although the slopes of the

He,

He, and

C nuclei were along the smooth line, the

Be slope was much diﬀerent, which

looked like an “anonymous” result.

In our studies, we pointed out that the phenomenon comes from the clustering eﬀect in

the

Be nucleus [23]. For understanding this anomalous nuclear eﬀect for the beryllium-9

nucleus, clustering aspects were studied in the structure functions by using momentum

distributions calculated in antisymmetrized or fermionic molecular dynamics (AMD or

FMD) and also in a simple shell model for comparison. According to the AMD, the

Be nucleus consists of two α-like clusters with a surrounding neutron. The clustering

produced high-momentum comp onents in nuclear wave functions, which aﬀected nuclear

modiﬁcations of the structure functions. We investigated whether clustering features could

appear in the structure function F

Be along with studies for other light nuclei. We

found that nuclear modiﬁcations of F

are similar in both AMD and shell models within our

simple convolution description although there are slight diﬀerences in

Be. It indicated that

the anomalous

Be result should be explained by a diﬀerent mechanism from the nuclear

binding and Fermi motion. If nuclear-modiﬁcation slopes d(F

)/dx are shown by the

maximum local densities, the

Be anomaly can be explained by the AMD picture, namely

by the clustering structure, whereas it certainly cannot be described in the simple shell

model. This fact suggested that the large nuclear modiﬁcation in

Be should be explained

by large densities in the clusters. In our work, we considered that nuclear structure

functions F

consist of a mean conventional part and a remaining part depending on the

maximum local density. The ﬁrst mean part did not show a signiﬁcant cluster eﬀect on

. This is because of the average over the angle, although the local clusters exist in

the nucleus. However, we proposed that the remaining part could explain the anonymous

JLab slope, and it is associated with high densities created by the cluster formation in

Be. The JLab measurement is possibly the ﬁrst signature of clustering eﬀects in high-

energy nuclear reactions. A responsible physics could be an internal nucleon modiﬁcation,

which is caused by the high densities due to the cluster conﬁguration, and/or a short-

range correlation between nucleons. The clustering aspect of nuclear structure functions

is an unexplored topic which is interesting for future investigations, and this project was

proposed at JLab (JLab PAC-35 proposal, PR12-10-008).

20. Structure-function projections and optimum tensor-polarized PDFs for spin-1 deuteron

Spin structure of a spin-one hadron is interesting as a future research topic because there

exist new tensor structure functions which do not appear in the spin-

nucleon. There

are eight structure functions, F

, F

, g

, b

, and b

, in charged-lepton scattering

from a spin-one hadron, whereas four of them (F

, F

, g

) exist in the spin-1/2 nucleon.

These additional structure functions vanish if internal constituents are in the S state, and

they are related to tensor structure in spin-one hadrons. Studies of such tensor structure

will open a new ﬁeld of high-energy spin physics.

In our work, the projection operators were derived for the structure functions of a spin-

one hadron by using combination of its momentum, polarization, and spin vectors [30].

They are useful in theoretical calculations because the structure functions need to be

extracted from a calculated hadron tensor W

µν

in theoretical models, for example, the

nuclear convolution model for the spin-1 deuteron.

Next, optimum tensor-polarized parton distribution functions (PDFs) were determined

from HERMES experimental results on b

[25] for understanding the tensor structure in

terms of quark and gluon degrees of freedom. Prior to this work, it was not clear how

the HERMES measurements were related to the tensor-polarized PDFs. We determined

optimum tensor-polarized PDFs for explaining the experimental data, so that theoretical-

model calculations could be tested and also for submitting a new experimental proposal.

The structure functions b

and b

are described by tensor-polarized quark and antiquark

distributions δ

q and δ

¯q. Using HERMES data on the b

structure function for the

deuteron, we made an analysis of extracting the distributions δ

q and δ

¯q in a simple

x-dependent functional form [25]. By imposing the sum rule

∫

dx b

(x) = 0 suggested

by the parton model, the optimum distributions were proposed for the tensor-polarized

valence and antiquark distribution functions from the analysis of the HERMES data. A

ﬁnite tensor polarization was obtained for antiquarks if we impose a constraint that the

ﬁrst moments of tensor-polarized valence-quark distributions vanish. It is interesting to

investigate a physics mechanism to create a ﬁnite tensor-polarized antiquark distribution.

The tensor-polarized PDFs were deﬁned by unpolarized partons in a polarized hadron,

so that their Q

scale evolution is given by the usual unpolarized DGLAP equations.

However, the HERMES data are not accurate enough to probe the scale dependence and

the data analysis was done by assuming no Q

evolution by ﬁxing it at the HERMES-data

average Q

= 2.5 GeV

. We made the determined tensor-polarized PDFs available for

general public. By this work, it became possible to investigate practically the deuteron

tensor structure at high energies. In fact, the JLab experiment (Letter of Intent to JLab

PAC-37) was submitted by using our tensor PDFs, and this experiment will be realized

by the middle of 2020’s.

21. Possible studies of GPDs and color transparency at hadron accelerator facilities

At the stage of 2008, only possibilities for measuring the generalized parton distributions

(GPDs) were to use the virtual Compton scattering or hadron-production process at lep-

ton accelerator facilities. We proposed that it is possible to investigate them at hadron

accelerator facilities, such as J-PARC (Japan Proton Accelerator Research Complex) fa-

cility and GSI-FAIR (Gesellschaft f¨ur Schwerionenforschung -Facility for Antiproton and

Ion Research) [27]. We considered a novel class of hard branching hadronic processes

a + b → c + d + e, where hadrons c and d have large and nearly opposite transverse

momenta and large invariant energy which is a ﬁnite fraction of the total invariant en-

ergy, for investigating the GPDs. We showed that a numb er of GPDs can be investigated

in hadron facilities such as J-PARC and GSI-FAIR. In this work, the GPDs for the nu-

cleon and for the N → ∆ transition were studied in the reaction N + N → N + π + B,

where N, π, and B are a nucleon, a pion, and a baryon (nucleon or ∆), respectively,

with a large momentum transfer between B (or π) and the incident nucleon. In particular,

the Efremov-Radyushkin-Brodsky-Lepage (ERBL) region of the GPDs can be measured in

such exclusive reactions. We estimated the cross section of the processes N+N → N+π+B

by using current models for relevant GPDs and information about large angle πN reac-

tions. We found that it is feasible to measure these cross sections at the high-energy

hadron facilities, and to get novel information about the nucleon structure, for example,

contributions of quark orbital angular momenta to the nucleon spin. The advantages of

using the hadron reactions are that cross sections are generally larger than lepton reactions

and that a speciﬁc kinematical region, so called the Efremov-Radyushkin-Brodsky-Lepage

region −ξ < x < ξ (x is the momentum fraction, ξ is the skewedness parameter), can be

measured. If this experiment is realized, it will extend projects of the hadron accelera-

tor facilities and the GPDs will be investigated from diﬀerent viewpoint and kinematical

regions.

Next, we proposed a possible study on color transparency by using a high-momentum

pion beam [26]. It is valuable to investigate hadron interactions in nuclear medium for

understanding fundamental hadron reactions and for applications to high-energy nuclear

reactions. The cross section is dominated by a small hadron component in a reaction with

large-momentum transfer. This small hadron passes through the hadron medium without

much interactions, which is called color transparency (CT). The color transparency T is

deﬁned as the ratio of cross sections for nucleon-nucleus and nucleon-nucleon reactions [T =

/(A σ

)]. As the hard scale of the reaction, for example the proton-beam energy,

becomes larger, the transparency is expected to become larger. We demonstrated that

hard branching 2 → 3 particle processes with nuclei provide an eﬀective way to determine

the momentum transfers needed for eﬀects of point-like conﬁgurations to dominate large

angle 2 → 2 processes, by showing the transparency as the functions of the pion-beam

energy and nuclear mass. In contrast with previously proposed approaches, the discussed

reaction allows the eﬀects of the transverse size of conﬁgurations to be decoupled from

eﬀects of the space-time evolution of these conﬁgurations. It can be applied to a much

broader range of two-body processes than the original method including meson-meson

scattering (ππ, πK, KK) where one expects an earlier onset of the CT regime than for

meson (baryon)-baryon scattering. One could also look for the onset of CT in the meson-

baryon (πN, Λπ, ...) and baryon-baryon (pN, p∆, ...) scattering. Studies with beams of

energies in the 20-200 GeV range appear to be optimal for these purposes.

22. Determination of polarized PDFs with JLab and RHIC-Spin measurements

In order to understand the origin of the nucleon spin, we need to determine contributions

from the quark and gluon spins by determining polarized parton distribution functions

from experimental measurements. In this work, a global analysis was performed within

the next-to-leading order in Quantum Chromodynamics (QCD) to determine p olarized

parton distributions with new experimental data in spin asymmetries [34]. The new data

set included JLab, Hermes, and Compass measurements on spin asymmetry A

for the

neutron and deuteron in lepton scattering. Our new analysis also utilized the double-spin

asymmetry for π

production in polarized pp collisions, A

, measured by the Phenix

collaboration. The uncertainties of the polarized PDFs were estimated by the Hessian

method. Because of these new data, uncertainties of the polarized PDFs were reduced. In

particular, the JLab, Hermes, and Compass measurements were valuable for determining

∆d

(x) at large x and ∆¯q(x) at x ∼ 0.1. The Phenix π

data signiﬁcantly reduced the

uncertainty of ∆g(x). Furthermore, we discussed a possible constraint on ∆g(x) at large

x by using the Hermes data on g

in comparison with the Compass ones at x ∼ 0.05.

We investigated impact of π

-production data at Relativistic Heavy Ion Collider (RHIC)

and future E07-011 experiment for the structure function g

of the deuteron at the Thomas

Jeﬀerson National Accelerator Facility (JLab) on studies of nucleonic spin structure, es-

pecially on the polarized gluon distribution function [28]. By global analyses of polarized

lepton-nucleon scattering and the π

-production data, polarized parton distribution func-

tions were determined and their uncertainties were estimated by the Hessian method. Two

types of the gluon distribution function were investigated. One was a positive distribution

and the other was a node-type distribution which changes sign at x ∼ 0.1. Although the

RHIC π

data seemed to favor the node type for ∆g(x), it was diﬃcult to determine a

precise functional form from the current data. However, it was interesting to ﬁnd that

the gluon distribution ∆g(x) is positive at large x (> 0.2) due to constraints from the

scaling violation in g

and RHIC π

data. The JLab-E07-011 measurements for g

should

be also able to reduce the gluon uncertainty, and the reduction was comparable to the one

by RUN-5 π

-production data at RHIC. The reduction was caused mainly by the error

correlation between polarized antiquark and gluon distributions and by a next-to-leading-

order (NLO) gluonic eﬀect in the structure function g

. We found that the JLab-E07-011

data are accurate enough to probe the NLO gluonic term in g

. Both RHIC and JLab

data contribute to better determination of the polarized gluon distribution in addition to

improvement on polarized quark and antiquark distributions.

23. High-energy hadron physics at J-PARC

Nuclear structure and reactions are described by nucleons and other hadron degrees of

freedom, and this ﬁeld became one of established ones. The underlying fundamental

theory of strong interaction is QCD. However, it cannot be solved in nonperturbative

regions especially at ﬁnite densities, and it is sometimes diﬃcult to understand hadronic

and nuclear structure and reactions as many-body systems in terms of quarks and gluons.

In this situation, the J-PARC has projects to create new particles and quark-hadron

matters by changing the ﬂavor and hadron density. It is the most-intense hadron-beam

facility in the multi-GeV high-energy region. By using secondary beams of kaons, pions,

and others as well as the primary-beam proton, the facility could cover a wide range of

hadron physics from strongly interacting many-body systems with an extended hadronic

degree of freedom, strangeness, to new forms of hadrons and hadronic matters. At the

ﬁrst stage of the J-PARC operation, hadron topics are mainly on strangeness nuclear

physics such as hypernuclei and kaonic nuclei. Then, the studies could be extended to

exotic hadron searches, chiral dynamics in nuclear medium, structure functions, and hard

exclusive processes. With major upgrades of the facility, extensive studies could be done

for the nucleon spin and heavy-ion physics. In particular, new projects became possible

by using the newly-constructed high-momentum beamline from 2020.

In Ref. [29], possible J-PARC projects were explained in high-energy hadron physics, par-

ticularly by using the 30−50 GeV primary proton beam. Such proton reactions are in

a limit region where the perturbative QCD can be applied, so that such projects have

fundamental importance in understanding QCD physics from a perturbative region to a

nonperturbative one. There are proposed experiments on charm-production and Drell-Yan

processes as well as single spin asymmetries for investigating quark and gluon structure

of the nucleons and nuclei. Parton-energy loss could be studied in the Drell-Yan pro-

cesses. There is also a proposal on hadron-mass modiﬁcations in a nuclear medium by

using the proton beam. These topics include ﬂavor dependence of antiquark distributions,

transverse-momentum-dependent distributions such as the Boer-Mulders and Sivers func-

tions, and polarized exclusive reactions. In addition, possible topics are transition from

hadron to quark degrees of freedom by elastic pp scattering, color transparency by (p, 2p),

short-range correlation in nuclear force by (p, 2pN), tensor structure functions for spin-1

hadrons, fragmentation functions, and generalized parton distributions although proposals

are not written on these projects. If proton-beam polarization will be attained, it is possi-

ble to investigate details of nucleon spin structure. The J-PARC is an excellent accelerator

facility to investigate diverse hadron projects, and we expand the projects to various ﬁelds

in future [S. Kumano, Nucl. Phys. A782 (2007) 442; AIP Conf. Proc. 1056 (2008) 444;

J. Phys. Conf. Ser. 312 (2011) 032005].

24. Determination of nuclear parton distribution functions in the next-to-leading order

High-energy heavy-ion reactions have been investigated at RHIC and LHC for ﬁnding

properties of quark-gluon plasma from their cross sections. For describing the cross sec-

tions, accurate nuclear parton distribution functions (NPDFs) are needed. In addition,

for neutrino oscillation measurements, they also need accurate PDFs for the oxygen nu-

cleus because systematic errors are dominated by the neutrino-nucleus interaction part.

There exist 10−20% nuclear modiﬁcations for medium nuclei. It is important to under-

stand modiﬁcation mechanisms not only for ﬁnding physics of nuclear PDFs but also for

applications to the high-energy nuclear reactions. For example, neutrino experimental-

ists ask us to calculate neutrino cross sections within 5% accuracy for future oscillation

experiments intended to measure the CP violation in the lepton sector.

The NPDFs were determined by global analyses of experimental data on structure-function

ratios F

′

and Drell-Yan cross-section ratios σ

/σ

′

[32]. The analyses were done

in the leading order (LO) and next-to-leading order (NLO) of running coupling constant

. It was successful in explaining the data from the deuteron to a large lead nucleus. The

uncertainties of the determined NPDFs were estimated by the Hessian method in both LO

and NLO, so that we can discuss the NLO improvement on the determination. We found

slight NLO improvements for the antiquark and gluon distributions at small x (= 0.01 −

0.001); however, they were not signiﬁcant at larger x. Valence-quark distributions were

well determined, and antiquark distributions were also determined at x < 0.1. However,

the antiquark distributions had large uncertainties at x > 0.2. The gluon modiﬁcations

could not be ﬁxed. The valence-quark distributions were determined well from the F

measurements at x > 0.3, and they were constrained at small x by the baryon-number

conservation and charge conservation. The antiquark distributions were determined from

the F

data at x < 0.05 and there was almost no nuclear modiﬁcation at x = 0 .1 due to the

Drell-Yan data σ

/σ

, and they had large errors in the larger-x region. Although the

advantage of the NLO analysis, in comparison with the LO one, is generally the sensitivity

to the gluon distributions, gluon uncertainties were almost the same in the LO and NLO.

It was because scaling-violation data are not accurate enough to determine precise nuclear

gluon distributions. Modiﬁcations of the PDFs in the deuteron were also discussed by

including data on the proton-deuteron ratio F

in the analysis. A code was provided

for calculating the NPDFs and their uncertainties at given x and Q

in the LO and NLO.

25. HERMES eﬀect

From charged-lepton deep inelastic scattering measurements, two nuclear structure func-

tions F

and F

are measured. By virtual-photon polarizations, F

is associated with the

transverse polarization and F

is with both transverse and longitudinal polarizations. Re-

moving the transverse part from F

, we obtain the longitudinal structure function F

. Al-

though it should vanish (F

= 0) in the scaling limit Q

→ ∞, the longitudinal-transverse

structure function ratio R = F

is ﬁnite in Q

regions of actual experiments. In the

1980’s, nuclear modiﬁcations on F

were studied extensively, and they are known as the

EMC (European Muon Collaboration) eﬀect. Therefore, scientists started thinking about

a p ossible nuclear modiﬁcation in the ratio R in the 1990’s. In 2000, the HERMES col-

laboration reported the existence of a nuclear modiﬁcation in the longitudinal-transverse

structure function ratio R, so that it used to be called a HERMES eﬀect.

We claimed that such a nuclear eﬀect should exist in the medium and large x regions [39].

Using a convolution description of nuclear structure functions, we derived the nuclear mod-

iﬁcation of the longitudinal-transverse structure function ratio R(x, Q

) at medium and

large x. We found that the conventional convolution description of the nuclear structure

functions leads to the nuclear modiﬁcation of the transverse-longitudinal structure function

ratio R(x, Q

). The physical origin of the modiﬁcation is the transverse-longitudinal ad-

mixture of the nuclear structure functions due to the nucleon momentum transverse to the

virtual photon momentum. Namely, the longitudinal structure function F

of a nucleus

is described by both nucleonic structure functions F

and F

convoluted with nucleon

momentum distributions in the nucleus. For example, electron-nucleon scattering cross

sections and structure functions are described by taking the virtual-photon-momentum

direction as the z axis and the nucleon is at rest or moves in the −z direction. How-

ever, nucleons in a nucleus move in any spacial directions due to Fermi motion, so that

the transverse and longitudinal structure functions mix. The mixing eﬀects we found

were moderate at small Q

and disappear at large Q

. The nuclear modiﬁcation eﬀects

are dominated by the binding and Fermi-motion eﬀects contained implicitly in the con-

volution expression. Since the mixture o ccurs by the Fermi motion, such eﬀects appear

especially at large x. Although a later HERMES reanalysis seemed to deny the originally

reported modiﬁcation at small x, our theoretical results are valid and independent from

the original HERMES ﬁnding at small x. We hope that such nuclear eﬀects will be found

experimentally in future.

26. Nuclear eﬀects and possible explanations on the NuTeV weak-mixing sin

anomaly

Neutral currents contain not only isospin currents, which act on left-hand components, but

also electromagnetic currents which act on both left- and right-handed ones. The mixing

fraction of these currents is called the weak-mixing angle or Weinberg angle θ

. This

angle was accurately measured by collider experiments as sin

= 0.2227 ±0.0004 at the

stage of 2002. However, the NuTeV collaboration reported anomalously large weak mixing

angle in comparison with the standard-model prediction. The NuTeV result, sin

0.2277 ± 0.0013 (stat) ± 0.0009 (syst), was signiﬁcantly diﬀerent from a global analysis

of other data, sin

= 0.2227 ± 0.0004. This is called the NuTeV weak-mixing-angle

anomaly. Since the mixing angle sin

is one of important physics quantities, it is

important to ﬁnd a reason for the diﬀerence. Neutrino and antineutrino charged- and

neutral-current events were analyzed for extracting sin

in the NuTeV experiment.

The Paschos-Wolfenstein relation R

−

= (σ

νN

− σ

¯νN

)/(σ

νN

− σ

¯νN

) = 1/2 − sin

was

used for its determination, and this relation was derived for the isoscalar nucleon.

Since the NuTeV target was the iron instead of the isoscalar nucleon, various correction

factors needed to be considered to this Paschos-Wolfenstein relation. In addition, this

relation was obtained by assuming the isospin symmetry in the parton distribution func-

tions (PDFs) of the neutron to relate them to the PDFs of the proton. We showed the

correction terms to the Paschos-Wolfenstein relation from isospin breaking in the PDFs,

nuclear correction diﬀerence between u

and d

, ﬁnite distributions of s(x) − ¯s(x) and

c(x) − ¯c(x), and neutron-excess eﬀects, and we indicated that the NuTeV anomaly could

originate from these corrections [40]. Next, using charge and baryon-number conservations

for nuclei, we discussed an eﬀect of nuclear modiﬁcation diﬀerence between u

and d

the sin

through the modiﬁed Paschos-Wolfenstein relation [35]. These modiﬁcations

had been assumed to be identical, but it could be diﬀerent. According to the analysis, the

eﬀect could partially explain the NuTeV anomaly; however, such a diﬀerence cannot be

determined from the current experimental measurements. We need further studies about

the nuclear eﬀect on the NuTeV deviation. Even at the stage of 2020, uncertainties are

very large for the nuclear modiﬁcation diﬀerence between u

and d

, isospin violation ef-

fects on the PDFs, s(x) − ¯s(x), and c(x) − ¯c(x), so that the origin of the NuTeV anomaly

is not solved undoubtedly.

27. Studies of nucleonic and nuclear structure functions at neutrino factories

Neutrino interactions are weak, so that their cross sections are generally small and measure-

ment errors are large. For future developments of precise neutrino physics, a high-intensity

neutrino factory is necessary and we contributed to this project. We investigated possible

hadron-structure studies at the high-energy neutrino factory by using our experiences on

nucleon structure functions [S. Kumano, Nucl. Phys. Proc. Suppl. 112 (2002) 42; AIP

Conf. Proc. 721 (2004) 29]. In particular, possible studies were discussed in connection

with recent studies of the PDFs (parton distribution functions) in the nucleons and nuclei.

In neutrino-nucleon deep inelastic scattering, there exists a new structure function F

which does not exist in the charged-lepton scattering, due to the additional axial vector

current. Since this structure function is expressed by valence-quark distributions in the

nucleon, neutrino scattering data played an important role in determining them. The

nuclear modiﬁcations of the valence-quark distributions are found at medium and large x

from the modiﬁcation of F

in charged-lepton scattering; however, it is not easy to ﬁnd

them at small x although there are some constraints due to baryon-number and charge

conservations. The neutrino reactions should be valuable for ﬁnding nuclear modiﬁcation

of valence-quark distributions at small x if structure function ratios F

are measured

for various nuclei. Nuclear shadowing mechanism should be tested by ﬁnding such valence

modiﬁcations.

Next, possible studies of polarized PDFs at the neutrino factory were discussed. In

charged-lepton scattering, it is diﬃcult to separate the contribution of valence quarks

from the antiquark one; however, it could be done in neutrino scattering. There are new

polarized structure functions g

, g

, and g

in neutrino reactions. Using these functions

as well as g

and g

, we should be able to establish the nucleon spin structure. The

polarized valence-quark distributions should be determined by the structure function g

(or g

notation in some researchers). Furthermore, the quark spin content is directly ob-

tained by measuring the structure function g

, whereas analysis of charged-lepton data

have uncertainties.

28. Flavor asymmetry in polarized antiquark distributions

Antiquark distributions of the nucleon are produced mainly through the perturbative

splitting process g → q¯q. Then, up- and down-quark masses are small, so that the ¯u

and

d antiquark distributions are considered to be the same. However, it is known that

there exists ﬂavor dependence in the unpolarized light-antiquark distributions. It was not

obvious whether there is ﬂavor dependence in polarized light-antiquark distributions. For

understanding the origin of nucleon spin and nucleon structure in general including spin,

we needed to know the non-perturbative mechanisms to create the ﬂavor dependence.

The ﬂavor asymmetry was investigated in the polarized light-antiquark distributions by a

meson-cloud model [41], which was used for explaining the ﬂavor-asymmetry phenomena

in unpolarized antiquark distributions. The existence of “meson clouds” is known, for

example, from the negative experimental mean-square radius (⟨r

⟩ < 0) of the neutron.

Since the pion is a scalar particle, it does not contribute directly to the polarized asym-

metry ∆¯u − ∆

d. We calculated spin-1 ρ meson contributions to ∆¯u − ∆

d by considering

that the virtual photon from a charged lepton interacts with the ρ meson. We pointed

out that the g

part of ρ contributes to the structure function g

of the proton in addition

to the ordinary longitudinally polarized distributions in ρ. This kind of contribution be-

came important at medium x (> 0.2) with small Q

(∼1 GeV

). Including N → ρN and

N → ρ∆ splitting processes, we obtained the polarized-ρ eﬀects on the light-antiquark

ﬂavor asymmetry in the proton. The results showed ∆

d excess over ∆¯u, which is very dif-

ferent from some theoretical predictions. Our model could be tested by future experiments

by RHIC-Spin and COMPASS collaborations.

Furthermore, we studied the relation between the ratio of the proton-deuteron (pd) Drell-

Yan cross section to the proton-proton (pp) one ∆

(T )

/2∆

(T )

and the ﬂavor asymme-

try in polarized light-antiquark distributions [44]. Using our formalism of the polarized pd

Drell-Yan process, we showed that the diﬀerence between the pp and pd cross sections is

valuable for ﬁnding not only the ﬂavor asymmetry in longitudinally polarized antiquark

distributions but also the one in transversity distributions. It is especially important that

we pointed out the possibility of measuring the ﬂavor asymmetry in the transversity dis-

tributions because it cannot be found in W production processes and inclusive lepton

scattering due to the chiral-odd property.

29. Determination of parton distribution functions in nuclei

In order to describe high-energy nuclear reactions, we need to have accurate nuclear parton

distribution functions (PDFs). They have important applications for understanding prop-

erties of quark-gluon plasma and nuclear corrections to neutrino oscillation experiments.

However, there were not serious studies on determination of the nuclear PDFs such as the

CTEQ, GRV, and MRST analyses of the nucleonic PDFs.

We determined the optimum nuclear PDFs for the ﬁrst time by using a χ

analysis method

[42]. Since it was the ﬁrst attempt, the analysis method was developed including the initial

functional form of the Bjorken variable x. Nuclear modiﬁcations are about 10−20% for

medium-size nuclei, so that we decided to determine such modiﬁcations from the nucleonic

PDFs, which were relatively-well determined from other analyses, instead of the nuclear

PDFs directly. The nuclear modiﬁcations depend on the variable x. There are negative

shadowing corrections at small x, positive antishadowing ones at x ≃ 0.1, negative nuclear-

binding ones, and positive nucleon Fermi-motion ones at x > 0.7. To approximate these

x-dependent eﬀects, quadratic or cubic functions of x with the factor 1/(1 −x) are used in

the analysis. About the mass number (A) dependence, we had the following consideration.

Nuclear reaction cross sections are generally described by the volume term proportional

to A and the surface one with the factor A

2/3

(σ

= Aσ

+ A

2/3

). Then, the cross

section per nucleon is expressed as σ

/A = σ

+ σ

1/3

, so that the 1/A

1/3

dependence

is expected in the nuclear PDFs.

Using these x- and A-dependent functionals, we determined the nuclear PDFs in the

leading order of α

by a χ

analysis. The parton distributions were provided at Q

GeV

with a number of parameters, which were determined by a χ

analysis of the data on

nuclear structure functions. Although valence-quark distributions in the medium-x region

were relatively well determined, the small-x distributions depended slightly on the assumed

functional form. It was diﬃcult to determine the antiquark distributions at medium x and

gluon distributions in the whole-x region. From the analysis, we proposed the parton

distributions at Q

=1 GeV

for nuclei from deuteron to heavy ones with the mass number

A ∼ 208. They were provided either analytical expressions or computer subroutines for

practical usage. Our studies should be important for understanding the physics mechanism

of the nuclear modiﬁcation and also for applications to heavy-ion reactions. This kind of

nuclear parametrization should also aﬀect existing parametrization studies in the nucleon

because “nuclear” data are partially used for obtaining the optimum distributions in the

“nucleon”.

30. Polarized proton-deuteron Drell-Yan processes and polarized parton distributions

Formalisms of high-energy polarized proton-proton reactions were investigated in details

and they are basics of performing RHIC-Spin experiments. High-energy hadron reactions

with spin-1 hadrons had never been investigated in connection with polarized structure

functions speciﬁc to spin-1 nature. In 1990’s, polarized deuteron acceleration was studied

among BNL accelerator scientists as a next project of the RHIC spin; however, nobody

knew what kind of new spin observables are possible. In this situation, we proposed a

general formalism for the structure functions which can be investigated in the polarized

Drell-Yan processes with spin-1/2 and spin-1 hadrons [44,45,46], namely in the polarized

proton-deuteron Drell-Yan processes.

Because of the spin-1 nature, there are new structure functions which cannot be stud-

ied in the proton-proton reactions. Imposing Hermiticity, parity conservation, and time-

reversal invariance, we found that 108 structure functions exist in the Drell-Yan processes

[46]. However, the number reduced to 22 after integrating the cross section over the

virtual-photon transverse momentum

⃗

or after taking the limit Q

→ 0. There are 11

new structure functions in addition to the 11 ones in the Drell-Yan processes of spin-1/2

hadrons. The additional structure functions are associated with the tensor structure of

the spin-1 hadron, and they could be measured by quadrupole spin asymmetries.

Then, we analyzed the polarized Drell-Yan processes with spin-1/2 and spin-1 hadrons in

a parton model [45]. Quark and antiquark correlation functions were expressed in terms

of possible combinations of Lorentz vectors and pseudovectors with the constrains of Her-

miticity, parity conservation, and time-reversal invariance. Then, the analysis indicated

that there are only four structure functions in the

⃗

-integrated case. They are unpo-

larized, longitudinally-polarized, transversity, tensor-polarized structure functions. The

tensor-structure functions were related to the tensor polarized distribution b

, and it does

not exist in the proton-proton reactions. The Drell-Yan processes have an advantage over

the lepton reaction in the sense that the antiquark tensor polarizations could be extracted

easily [44].

The deuteron acceleration was not realized in the RHIC-Spin project; however, such a

project could provide a new opportunity in a next generation high-energy spin physics.

In fact, the proton-deuteron Drell-Yan experiment is under consideration in the Fermilab-

E1039 project with a ﬁxed polarized-deuteron target at the stage of 2020. Furthermore,

our formalism can be used at other accelerator facilities such as GSI-FAIR, NICA, and

possibly EIC if a ﬁxed-target experiment is possible. The b

experiment will start at JLab

by the middle of 2020’s, and the proton-deuteron reactions are complementary to this JLab

experiment in the sense that it probes a diﬀerent kinematical region of x and Q

and that

it could be used for determining the vector- and tensor-polarized antiquark distributions.

31. Determination of polarized parton distribution functions

There were relatively-established unpolarized parton distribution functions (PDFs) for

the nucleon at the stage of 2000, as they had been investigated extensively by a few the-

ory groups such as the CTEQ (Coordinated Theoretical-Experimental Project on QCD).

However, studies of polarized PDFs were at a premature stage, so that we attempted to de-

termine optimum longitudinally-polarized PDFs by an analysis of world data on polarized

charged-lepton scattering from the polarized nucleon [38,43].

Since the measurement of the polarized structure function g

for the nucleon by the EMC,

the origin of the nucleon spin has been investigated. However, it was not clear how an-

tiquark and gluon spins contributed to the nucleon spin. We determined the polarized

parton distribution functions by using the data from the longitudinally-polarized deep

inelastic scattering experiments [43]. The studies were done by creating a Japanese col-

laboration, which is called Asymmetry Analysis Collaboration (AAC), among theoretical

and experimental researchers in high-energy spin physics. It was intended to propose the

optimum polarized PDFs by the analysis of g

data of the proton, deuteron, and

He.

A new parametrization of the polarized PDFs was adopted by taking into account the

positivity and the counting rule. The polarized PDFs were given at Q

= 1 GeV

parameters, and then spin asymmetry A

was calculated theoretically. From the ﬁt to the

asymmetry data A

, the polarized distribution functions of u- and d-valence quarks, sea

quarks, and gluon were obtained. The results indicated that the quark spin content was

∆Σ =0.20 and 0.05 in the leading order (LO) and the next-to-leading-order (NLO) MS

scheme, respectively. However, if x dependence of the sea-quark distribution was ﬁxed at

small x by “perturbative QCD” and Regge theory, it became ∆Σ = 0.24 ∼ 0.28 in the

NLO. The small-x behavior could not be uniquely determined by the existing data, which

indicated the importance of future experiments. From our analysis, we proposed one set

of LO distributions and two sets of NLO ones as the longitudinally-polarized parton dis-

tribution functions. The gluon-spin contribution was positive and the antiquark one was

a small negative value.

Then, errors of the determined polarized PDFs were estimated by the Hessian method

[38]. In the χ

analysis, an error matrix was obtained and it was used for calculating error

bands of the polarized PDFs. As a result, the polarized valence-quark distributions were

relatively well determined, the polarized gluon distribution had a large error. Therefore,

even ∆g = 0 could be possible by considering the error. In this way, the gluon polariza-

tion could not be ﬁxed by the charged-lepton DIS data, so that we may reply on RHIC

measurements. We made an AAC code for calculating the determined polarized PDFs,

and it has been used as one of standard parametrization models for the polarized PDFs.

32. Anomalous dimensions of the structure function h

and its Q

evolution

The nucleon spin structure was investigated mainly for longitudinal polarizations; how-

ever, transverse spin structure also needed to be understood. One of important transverse

structure functions is the structure function h

, which is also called the quark transversity

distribution, and it is a leading-twist (twist 2) function. The quark transversity distribu-

tion is associated with the quark spin-ﬂip amplitude, so that it is a chiral-odd distribution.

Although there were experimental plans to measure it, the Q

evolution of h

was known

only in the leading order (LO) of α

at the time of 1996.

Since the scaling violation is usually calculated by including, at least, next-to-leading-order

(NLO) terms, we studied two-loop anomalous dimensions for h

in the minimal subtrac-

tion (MS) scheme [50]. In order to study h

in perturbative Quantum Chromodynamics

(QCD), we needed to introduce a set of local operators O

νµ

···µ

= S

ψ i γ

νµ

· · ·

ψ− trace terms (n = 1, 2, ...). The bare operator is deﬁned by O

= Z

with the

renormalized one O

. Anomalous dimensions for O

are given by these renormalization

constants as γ

= µ ∂(ln Z

)/∂µ. Dimensional regularization and Feynman gauge were

used for calculating the two-loop contributions. In dimensional regularization with the

dimension d = 4 −ϵ, we tried to ﬁnd 1/ϵ singularities in the renormalization constants for

calculating the anomalous dimensions. Due to the chiral-odd nature, the gluon transver-

sity does not exist in the nucleon, as mentioned in the topic item 3. We calculated all the

Feynman diagrams in the two-loop level by using the Feynman gauge and the MS scheme

for obtaining the anomalous dimensions for h

. Because of our studies, it became possible

to investigate h

in the NLO level.

Next, by calculating the inverse Mellin transformation of the obtained anomalous dimen-

sions, the Q

evolution can be described by integrodiﬀerential equations in the DGLAP

(Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) form. We investigated numerical solution of

the DGLAP Q

evolution equation for the transversity distribution ∆

q or the structure

function h

[47]. The LO and NLO evolution equations were studied. The renormaliza-

tion scheme was MS or MS in the NLO case. Dividing the variables x and Q

into small

steps, we solved the integrodiﬀerential equation by the Euler method in the variable Q

and by the Simpson method in the variable x. We provided a FORTRAN program for

the Q

evolution and devolution of the transversity distribution ∆

q or h

. Using the

program, we showed the LO and NLO evolution results of the valence-quark distribution

∆

+ ∆

, the singlet distribution

∑

(∆

+ ∆

¯q

), and the ﬂavor-asymmetric dis-

tribution ∆

¯u − ∆

d. They were compared with the longitudinal evolution results [48]

to show unique features of the transversity evolution by ﬁnding diﬀerences. For example,

variations were much smaller in the ﬂavor-nonsinglet transversity distribution than

the one for the longitudinally-polarized distribution as the Q

is increased. In addition,

perturbative eﬀects on the ﬂavor asymmetric distribution ∆

¯u − ∆

d were conspicuous

in the large-x region, in comparison with the ones for the longitudinally-polarized one,

by using the developed codes. Using these results, we predicted various spin asymmetries

possibly in the RHIC-Spin project.

33. Nuclear structure functions F

by a parton model

Nuclear modiﬁcations of the structure function F

are known as the EMC (European

Muon Collaboration) eﬀect, and various mechanisms contribute to them depending on the

Bjorken scaling variable x. We investigated a parton model which describes the nuclear

structure functions F

and nuclear parton distribution functions as an uniﬁed description

from small x to large x [56,57,58]. As this parton model, we used a Q

rescaling model

with parton-recombination eﬀects. The average separation of nucleons in a nucleus is

about 2.2 fm, which is almost equal to the nucleon diameter. This fact indicates that a

quark conﬁnement radius could change due to possible fusions of two nucleons in a nucleus,

and it could result in the scale change, Q

rescaling in this work, within F

. In addition,

the conﬁnement radius of a parton becomes larger than the average nucleon separation

at small x, so that partons in diﬀerent nucleons could interact with each other. This

phenomena is called parton recombinations. We used a parton model which has these two

mechanisms for describing the nuclear structure functions.

First, we calculated nuclear structure functions F

(x) from small x (= 10

−3

) to large

x (= 0.9) in this parton model in order to compare them with experimental data of SLAC,

EMC, NMC (New Muon Collaboration), and Fermilab-E665 [56,57]. As a result, we

obtained a reasonably good agreement with the experimental data in the region (0.005 <

x < 0.8). In the large x region, the ratio F

(x)/F

(x) (> 1) was explained by quark-gluon

recombinations, which produced results similar to those by the nucleon Fermi motion. In

the medium x region, the EMC eﬀect was mainly due to the Q

rescaling mechanism in

our model. In the small x region, shadowing eﬀects were obtained through modiﬁcations

in gluon distributions. However, our shadowing eﬀects at very small x (< 0.01) were very

sensitive to the input gluon distribution of the nucleon.

Then, we investigated gluon distributions in the nuclei C and Sn by this parton model [58].

We obtained shadowing in the nuclear gluon distributions in the small x region (x < 0.02)

due to the recombinations and depletion [typically G

(x)/G

(x) ∼ 0.9] in the medium x

region (0.2 < x < 0.6). The ratio G

(x)/G

(x) became large at x > 0.6 due to gluon

fusions from diﬀerent nucleons. The ratio in the medium-large x region was very sensitive

to the momentum cutoﬀ for leak-out partons in our model. Comparisons with the NMC

data on G

(x)/G

(x) indicated that more accurate experimental data are needed for

the nuclear gluon distributions. By our studies, the general features of nuclear structure

functions were understood in the wide x region from small x (∼ 0.001) to large x (∼ 0.9)

by the parton model with the rescaling and recombination eﬀects.

34. Parton distributions in nuclei by a parton model

Using the parton model explained in the previous topic item 33, we predicted new phenom-

ena in nuclear parton distribution functions. First, we showed that ﬁnite ﬂavor-asymmetric

antiquark distributions are possible [55], even if the antiquark distributions were ﬂavor

symmetric (¯u −

d = 0) in the nucleon, due to the parton recombinations. In order to test

the NMC ﬁnding of ﬂavor asymmetry u −d in the nucleon, existing Drell-Yan data for the

tungsten target were often used. However, we have to be careful in comparing nuclear data

with the nucleon ones. We investigated whether there exists a signiﬁcant nuclear modi-

ﬁcation of the u − d distribution in the parton-recombination model. In neutron-excess

nuclei such as the tungsten, there exist more d-valence quarks than u-valence quarks, so

that more d quarks are lost than u quarks due to parton recombinations [55]. Our results

suggested that the nuclear modiﬁcation in the tungsten is a 2–10 % eﬀect on the NMC

u − d distribution. In the beginning of 2020’s, Drell-Yan measurements will be reported

from the Fermilab experiments with various nuclear targets, so that the nuclear ¯u −

d will

become a popular topic for ﬁnding its generation mechanism in the near future.

Second, nuclear modiﬁcations of structure function F

and valence-quark distributions

were investigated [53], particularly focusing on the small-x region to shed light on dif-

ferent shadowing mechanisms. Namely, this work was intended to distinguish between

two typical shadowing models, the parton-recombination model and the vector-meson-

dominance (VMD) model. We found that the nuclear modiﬁcations of the valence-quark

distributions tend to increase at small x in the recombination model, and it was opposite

to the one by the VMD model. Namely, we found that these models predict completely

diﬀerent behavior at small x. Therefore, studies of the ratio F

at small x could be

useful in discriminating among diﬀerent models, which produce similar shadowing behav-

ior in the structure function F

. This diﬀerence could be tested experimentally by neutrino

scattering measurements and also by π productions in charged-lepton scattering.

Third, Q

evolution of nuclear structure functions was investigated in order to under-

stand NMC measurements on the tin and carbon nucleus ratio of F

[51]. The F

was

evolved by using the leading-order DGLAP, next-to-leading-order DGLAP, and parton-

recombination equations. The NMC experimental result ∂[F

]/∂[ln Q

] ̸= 0 could

be essentially understood by the diﬀerence of parton distributions in the tin and carbon

nuclei. However, there were some discrepancies from the data if the Q

evolution with the

parton recombinations was used. It indicated that high-twist eﬀects need to be understood

accurately.

Forth, we investigated eﬀects on J/ψ suppression in heavy-ion collisions. Originally, the

J/ψ suppression was discussed by assuming that nuclear modiﬁcations are identical for

both quark and gluon distributions. However, if both modiﬁcations are diﬀerent in central

and peripheral regions of nuclei or nuclear collisions, they should be properly taken into

account. We estimated that such eﬀects, namely the diﬀerence of the gluonic EMC eﬀect

from the quark one, can contribute to the J/ψ suppression about 5−10%.

35. Scalar meson substructure at ϕ factories

According to the basic quark model by Gell-Mann and Zweig, mesons and baryons consist

of quark compositions q¯q and qqq, respectively, and hadrons with other quark conﬁgura-

tions are called exotic hadrons. Scalar mesons below 1 GeV had been a persistent problem

in hadron spectroscopy until recently. The scalar mesons, f

(980) and a

(980), are consid-

ered as exotic hadron candidates, for example, because strong decay widths are too large to

explain experimental ones if they are ordinary q¯q hadrons with light u and d quarks. They

were considered as s¯s, K

K molecule, tetraquark (qq¯q¯q), or glueball (gg). At the stage of

1993 when this work was done, there were future experimental facility possibilities of ϕ

factory at Frascati and KEK.

In order to determine their structure, we proposed to use the radiative decay of ϕ meson

into these scalar mesons at ϕ factories by calculating the decay widths in diﬀerent conﬁgu-

rations (ordinary q¯q mesons, s¯s, K

K molecule, tetraquark, or glueball) [59]. The radiative

decay ϕ → Sγ [S = f

(980) or a

(980)] is the electric dipole decay. Since the electric dipole

moment is proportional to the spatial separation, the decay widths are sensitive to the

internal structure of the scalar mesons. We showed that ϕ radiative decays into scalar

mesons [f

(980), a

(980)] could provide important clues on the internal structures of these

mesons. The radiative decay widths varied widely depending on the substructures (q¯q,

qq¯q¯q, K

K, glueball). Hence, we could discriminate among various models by measuring

these widths at the ϕ factories. The understanding of these meson structures is valu-

able not only in hadron spectroscopy but also in nuclear physics in connection with the

widely-used but little-understood σ meson. Actually, our calculations indicated that the

decay width is large [BR(4 × 10

−5

)] if S is a loose K

K bound state, because the average

K separation is large. If S were a glueball, although such a possibility is denied at the

stage of 2020 by lattice QCD, the decay width was smaller [BR(10

−5

− 10

−6

)]. In this

way, we showed that the internal structure of the scalar mesons f

(980) and a

(980) could

be investigated at the ϕ factories. We also found that the decay ϕ → Sγ → K

γ is

not strong enough to pose a signiﬁcant background problem for studying CP violation

via ϕ → K

at the ϕ factories. Later, the radiative decay widths were measured, and

they, together with other experimental measurements, indicated that the scalar mesons

are tetraquark hadrons or K

K molecules.

36. Numerical solution of Q

evolution equations on structure functions

The DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) evolution equations are inte-

grodiﬀerential equations, which describe Q

variations of structure functions. They can-

not be easily solved, especially if complicated higher-order perturbative QCD corrections

are included in splitting functions. However, they are important in practical applications

such as in theoretical model calculations of the structure functions and comparisons with

and analysis of experimental measurements. We investigated numerical solutions of the

DGLAP evolution equations by using the Laguerre-polynomial method [54,60] and by the

Euler’s (or “Brute-force”) method [47,48,52].

First, this Laguerre-polynomial method was studied by expanding parton distribution

functions and the splitting functions in terms of the orthogonal Laguerre polynomials.

The Laguerre method is considered to be a very eﬃcient method for solving the DGLAP

equations, and we extended this method for spin-dependent problems. As a result, accurate

numerical solutions were obtained in the region 0.05< x <0.5 with merely 10 polynomials

and it took about a few seconds by the SUN-IPX in 1994. This method was also applicable

to the small-x (x ≈ 0.01) region if we take large number of Laguerre expansion coeﬃcients

(N=20∼30), and typical accuracy at x=0.01 was about 10% (3%) in the spin-dependent

(spin-independent) case. At intermediate x (0.05 < x < 0.5), the accuracy was much

better. However, they were worse at large x (x > 0.8) where structure functions are very

small so that it may not be serious problem practically. Because the 10% inaccuracy at

x = 0.01 ∼ 0.02 changed the integral

∫

0.01

dxg

(x) only 0.2%, the Laguerre method can be

used to investigate spin-dependent structure functions at small x, as long as it is not too

small (x < 0.01). Therefore, although the accuracies are not good in the very small and

very large x regions, it can be used as an eﬀective and accurate method in the “practical”

x region.

Second, the Euler’s method was used for solving the DGLAP and Mueller-Qiu evolution

equations for the nucleons and nuclei. In this method, the variables x and Q

were simply

divided into small steps to calculate the diﬀerentiations and integrals, and it can improve

the precision issue at small and large x of the Laguerre method. Numerical results indicated

that the accuracy is better than 2% in the region 10

−4

< x < 0.8 if more than two-

hundred Q

steps and more than one-thousand x steps are taken. The numerical solution

was discussed in detail, and evolution results were compared with Q

dependent data in

CDHSW, SLAC, BCDMS, EMC, NMC, Fermilab-E665, ZEUS, and H1 experiments. We

provided a FORTRAN program for Q

evolution (and “devolution”) of nonsinglet-quark,

singlet-quark, q

+ ¯q

, and gluon distributions (and corresponding structure functions) in

the nucleon and in nuclei.

These studies were extended to the longitudinally polarized [48] and transversity evolu-

tion equations [47]. In the longitudinal-polarization studies, Q

variations of the polarized

structure function g

and the spin asymmetry A

were investigated in both LO and NLO for

specifying the NLO eﬀects. At that time, analysis groups obtained g

often by neglecting

the Q

dependence in the asymmetry A

. However, we pointed out that it is inappropriate

because clear Q

dependence existed especially in the small Q

region (Q

< 2 GeV

). For

a precise determination of g

, the Q

dependence of A

needs to be taken into account

properly. Furthermore, we investigated the numerical solution for the transversity distri-

butions (h

or ∆

q) [47], as explained in the topic item 32. We supplied these Q

evolution

codes on our web, so that other scientists could use them for their own studies.

37. Flavor asymmetric antiquark distributions ¯u −

d, (¯u +

d)/2 − ¯s

Light antiquark distributions were expected to be ﬂavor symmetric because they are con-

sidered to be created mainly through perturbative QCD splitting processes from a gluon

(g → q ¯q). However, it became clear that they are not ﬂavor symmetric from experiments.

The strange-quark distribution is about a half of the up and down antiquark distribu-

tions [(¯u +

d)/2 ∼ ¯s] from neutrino-induced opposite-sign dimuon measurements. The

inequality ¯u ̸=

d also became obvious from the NMC (New Muon Collaboration) ﬁnd-

ing on the Gottfried-sum-rule violation and Fermilab Drell-Yan experiments. The ﬂavor

asymmetric antiquark distribution ¯u −

d, created in perturbative QCD, originates from

next-to-leading-order eﬀects, so that it is much smaller than the NMC ﬁnding on ¯u −

d in

the region Q

≥ 4 GeV

. Therefore, the ﬂavor asymmetric antiquark distributions should

come mainly from nonperturbative mechanisms.

As such a nonperturbative mechanism, we investigated meson-cloud eﬀects on the anti-

quark distribution of the nucleon. First, the parameter in the pion-cloud model was ﬁxed

by the ﬂavor asymmetric antiquark distribution (¯u +

d)/2 − ¯s for predicting the SU(2)

ﬂavor asymmetric distribution ¯u −

d. In the pion-cloud model, there exists a momentum

cutoﬀ parameter in the πNN form factor. This cutoﬀ was determined by the experimental

information on (¯u +

d)/2 − ¯s, actually a typical parametrization (HMRS) on antiquark

distributions, and we found the cutoﬀ Λ

of about 0.7 GeV for a monopole πNN form

factor. A typical πNN form factor with Λ

∼0.6 GeV in quark models could be consis-

tent with this result; however, it is softer than the πNN form factor with Λ

∼1 GeV

widely used in nuclear physics. In particular, it is much softer than the hard cutoﬀ of

1.4 GeV used in one-pion-exchange potentials. Then, ﬁxing the cutoﬀ parameter by the

distribution (¯u +

d)/2 − ¯s, we predicted SU(2) ﬂavor asymmetric distribution ¯u −

d theo-

retically [64]. In 1991, the NMC indicated that the ¯u distribution is diﬀerent from

d from

the Gottfried-sum-rule violation. I found that the pionic contribution to the deviation

from the Gottfried sum rule is around −0.04 (

∫

dx(¯u −

d) = −0.06) [63,64]. This value

corresponds to about 1/2 of the the discrepancy found by the NMC, so that the order of

magnitude of the NMC ﬁnding on ¯u −

d can be understood by the pion-cloud model, in

general meson-cloud models. This negative contribution was due to an excess of

d over ¯u

in π

and it was partly cancelled by a positive contribution due to an excess of ¯u over

in an extra