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 [1–72].
1. 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 [1,2]. 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
2
µµ
, the dimuon
transverse-momentum q
T
, the dimuon rapidity y in the center-of-momentum frame, and
the magnitude of the gluon transversity
T
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.
1
2. 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 2030%, 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
0
π
0
at KEKB [4].
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
0
π
0
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
1
and Θ
2
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
1
and Θ
2
, gravitational
densities of the pion were calculated. Then, we obtained the mass (energy) radius and
the mechanical (pressure and shear force) radius from Θ
2
and Θ
1
, respectively. They
were calculated as
r
2
mass
= 0.32 0.39 fm, whereas the mechanical radius was larger
r
2
mech
= 0.82 0.88 fm. This is the first report on the gravitational radius of a hadron
from actual experimental measurements [3,4]. It is interesting to find the possibility that
the gravitational mass and mechanical radii could be different from the experimental charge
radius
r
2
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.
3. Tensor-polarized structure function b
1
in standard convolution description of deuteron
Tensor-polarized structure functions b
14
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
1
and b
2
, and
they are related by the Callan-Gross-like relation in the Bjorken scaling limit. The other
functions b
3
and b
4
are higher-twist ones, which may not be easily accessed experimentally
at this stage.
In this work, we theoretically calculated b
1
in the standard convolution description for
2
the deuteron [5]. Two different theoretical models, a basic convolution description and a
virtual nucleon approximation, were used for calculating b
1
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
1
of the deuteron was expressed by the structure functions F
1
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
b
1
. Estimating the convolution expression for b
1
, 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
1
studies could shed light on this new field of hadron physics. In particular, detailed
experimental studies of b
1
will start at the Thomas Jefferson National Accelerator Facility.
In addition, there are possibilities to investigate tensor-polarized parton distribution func-
tions and b
1
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.
4. 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 experiments, 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 [6] 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
2
< 1 GeV
2
and invariant-mass squared W
2
< 4 GeV
2
, the reaction is
described by quasi-elastic scattering with nucleons and by nucleon resonances. At high
3
energies with Q
2
> 1 GeV
2
and W
2
> 4 GeV
2
, it is described by deep inelastic scattering
(DIS) with a nucleus in terms of quarks and gluons. In the region with Q
2
< 1 GeV
2
and W
2
> 4 GeV
2
, the reaction is expressed by Reggeons and Pomerons by noting the
partially conserved axial-vector current (PCAC) relation at Q
2
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.
5. 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
2
analysis of charged-hadron production data in electron-positron annihila-
tion and by the Hessian method for error estimation [31]. 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
2
. 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
2
, the
uncertainty estimation is very important for analyzing hadron-production data at small
Q
2
or p
T
(Q
2
, p
2
T
<< M
2
Z
) 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, re-
spectively, at the KEK and SLAC B factories, whereas other available e
+
e
measurements
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 [8]. Our results indicated that
the fragmentation functions are determined more accurately not only by the scaling vio-
lation 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
2
values. In particular, the electric and
weak charges are different depending on the quark type, so that a light-quark flavor sep-
aration 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-
4
tions [7]. 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 [29]. 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
0
(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
0
(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.
6. 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
1
and b
2
. The HERMES collaboration found much larger |b
1
|
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
1
at JLab (Thomas Jefferson National Accelerator Facility). There was also
an interesting indication in the HERMES result that finite antiquark tensor polarization
exists
dxb
1
(x) = [ 0.35±0.10 (stat)±0.18 (sys) ], which indicates a deviation from the sum
rule
dxb
1
(x) = 0 in Ref. [63]. 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 [9]. The tensor-p olarized PDFs at the HERMES kinematics (Q
2
=
2.5 GeV
2
) [23] were evolved to the ones at the larger-Q
2
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
Q
2
evolution, even if it were zero at a low Q
2
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.
5
7. 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
+
µ
n in the coming high-momentum b eam line of J-PARC [10].
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 1020 GeV. A clean signal of exclusive pion-induced Drell-Yan process can be
identified in the missing-mass spectrum of dimuon events with 24 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
+
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
s
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.
8. 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 rep orts 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 [11,17]. 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-
6
body exclusive hadron reaction cross section a + b c +d scales like /dt f(θ
cm
)/s
n1
with n = n
a
+ n
b
+ n
c
+ n
d
, where n
i
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
π
+ p K
0
+ Λ(1405) was estimated at the scattering angle θ = 90
in the center-of-
mass frame by using current experimental data [17]. In comparison, the cross section for
the ground-state Λ production π
+ p K
0
+ 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
8
/dt =constant, whereas it is s
10
/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 CLAS data on the photoproduction of hyp eron resonances, as
well as the available data for the ground state Λ and Σ
0
of the CLAS and SLAC-E84
collaborations, by considering constituent-counting rule suggested by perturbative QCD
[11]. From the analyses of the γ p K
+
Λ and K
+
Σ
0
reactions, we found that the
number of the elementary constituents is consistent with n
γ
= 1, n
p
= 3, n
K
+
= 2, and
n
Λ
= n
Σ
0
= 3. Then, the analysis was made for the photoproductions of the hyperon
resonances Λ(1405), Σ(1385)
0
, and Λ(1520), where Λ(1405) could be considered to be
a
¯
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 numb er,
especially for Λ(1405). Namely, Λ(1405) looked like a p enta-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.
9. 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 possible method to understand the hadron
7
structure is to investigate compositeness of bound-state systems.
In our studies, structure of the a
0
(980) and f
0
(980) resonances was investigated with the
a
0
(980)-f
0
(980) mixing intensity from the viewpoint of compositeness [12], 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
0
ππ by quark models with the
q¯q configuration [69] and also from experimental measurements on the radiative decay
ϕ f
0
γ [57] and the two-photon decay width (f
0
γγ), f
0
(980) could be considered as
a K
¯
K molecule.
For this purpose, we first formulated the a
0
(980)-f
0
(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
0
(980)-f
0
(980) mixing intensity
with the existing Flatte parameters from experiments, we found that many combinations
of the a
0
(980) and f
0
(980) Flatte parameters can reproduce the experimental value of the
a
0
(980)-f
0
(980) mixing intensity by BES. Next, from the same Flatte parameters, we also
calculated the K
¯
K compositeness for a
0
(980) and f
0
(980). Although the compositeness
with the correct normalization became complex in general for resonance states, we found
that the Flatte parameters for f
0
(980) imply large absolute value of the K
¯
K composite-
ness and the parameters for a
0
(980) led to small but nonnegligible absolute value of the
K
¯
K compositeness. Then, connecting the mixing intensity and the K
¯
K compositeness via
the a
0
(980)- and f
0
(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
¯
K
compositeness for the a
0
(980) and f
0
(980) resonances. Moreover, the experimental value
of the a
0
(980)-f
0
(980) mixing intensity implied that the a
0
(980) and f
0
(980) resonances
cannot be simultaneously K
¯
K molecular states. These two results suggested a new view-
point that f
0
(980) is mainly a K
¯
K state and a
0
(980) is an ordinary quark bound state,
although both states used to be considered as K
¯
K states.
10. 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” [8], 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
8
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. [8].
11. 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 [14], 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
2
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 e + h
¯
h is sensitive to the exotic signature by looking at the h
¯
h
invariant-mass dependence by taking light hadrons, h = f
0
(980) and a
0
(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 [14].
12. 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
9
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 [15], 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 er. 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
+
Σ
and π
Σ
+
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
¯
KN
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
0
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.
13. 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 [16] 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
10
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.
14. Numerical solution of Q
2
evolution equations for fragmentation functions
Semi-inclusive hadron-production processes 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
s
. 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
2
evolution) phenomena became clear for the
first time by these data and it became possible to probe the gluon fragmentation functions.
The Q
2
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
2
evolution code for their theoretical and experimental projects. The optimum
fragmentation functions were supplied at fixed Q
2
, usually small-Q
2
region where the
perturbative QCD could be applied, so that they should be evolved to different Q
2
scales
for their own experiments and theoretical model calculations. In our work, we explained a
numerical solution method and created a Q
2
evolution code for general users. The DGLAP
evolution equations are expressed by a differentiation of Q
2
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
2
evolution of the fragmentation functions
in the leading order (LO) and next-to-leading order (NLO) of α
s
[18]. 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.
15. 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-
11
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 [19]. 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
2
-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 [19] 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.
16. Structure of Λ
c
(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
c
(2940)
+
, which is considered as a molecular state
of nucleon and D
. However, there is a possibility of the D-wave excitation of Λ
c
(2286)
+
.
In order to find its internal structure, we proposed to use its strong and electromagnetic
decays. First, we studied the radiative decay Λ
c
(2940)
+
Λ
c
(2286)
+
γ by the hadron
molecule picture and predicted its decay width [22]. 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
+
c
, and Λ
+
c
ND
vertices, were relatively small. In the molecular
scenario, the Λ
c
(2940)
+
baryon was described by a superposition of |pD
0
and |nD
+
components with the explicit admixture expressed by the mixing angle θ (|Λ
c
(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
12
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
c
(2940)
+
resonance. The decay
width also depended much on the momentum cutoff parameter Λ at the Λ
+
c
ND
vertex.
We showed our decay width as the functions of θ and Λ
+
c
ND
. 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
c
(2940)
+
state.
Second, we investigated the strong three-body decays Λ
c
(2940)
+
Λ
c
(2286)
+
π
+
π
,
Λ
c
(2286)
+
π
0
π
0
by considering the same molecular composition [20]. In our calculation,
we employed the extended SU(4) chiral Lagrangians to describe the interaction terms
contained in L
πD
BB
h
and L
πBB
. Therefore, the necessary couplings g
πD
BB
h
and g
πBB
were well determined. We showed the explicit contributions from the two-step processes
Λ
c
(2940)
+
Σ
++
c
π
Λ
c
(2286)
+
+ π
+
π
, Λ
c
(2940)
+
Σ
0
c
π
+
Λ
c
(2286)
+
+ π
+
π
,
Λ
c
(2940)
+
Σ
+
c
π
0
Λ
c
(2286)
+
+ π
0
π
0
, and Λ
c
(2940)
+
ρ
0
Λ
c
(2286)
+
Λ
c
(2286)
+
+
π
+
π
. In particular, the contribution Λ
c
(2940)
+
Σ
c
π Λ
c
(2286)
+
+ ππ was the
largest, and the intermediate ρ term was essentially negligible. The charged decay mode
involving π
+
π
was less than two times larger than the neutral π
0
π
0
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
c
(2940)
+
. We
expect that future measurements of the radiative and strong decay widths will clarify the
internal structure of Λ
c
(2940)
+
in comparison with our theoretical predictions.
17. 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
2
for the beryllium-9 nucleus, and they showed
the gradient of the nuclear modification with respect to the Bjorken-scaling variable x
(|d(F
A
2
/F
D
2
)/dx|) as a function of average nuclear density. Although the slopes of the
3
He,
4
He, and
12
C nuclei were along the smooth line, the
9
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
9
Be nucleus [21]. 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
9
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
2
of
9
Be along with studies for other light nuclei. We
found that nuclear modifications of F
2
are similar in both AMD and shell models within our
13
simple convolution description although there are slight differences in
9
Be. It indicated that
the anomalous
9
Be result should be explained by a different mechanism from the nuclear
binding and Fermi motion. If nuclear-modification slopes d(F
A
2
/F
D
2
)/dx are shown by the
maximum local densities, the
9
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
9
Be should be explained
by large densities in the clusters. In our work, we considered that nuclear structure
functions F
A
2
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
F
2
. 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
9
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 asp ect 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).
18. 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-
1
2
nucleon. There
are eight structure functions, F
1
, F
2
, g
1
, g
2
, b
1
, b
2
, b
3
, and b
4
, in charged-lepton scattering
from a spin-one hadron, whereas four of them (F
1
, F
2
, g
1
, g
2
) 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 [
28].
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
1
[23] 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 exp erimental data, so that theoretical-
model calculations could be tested and also for submitting a new experimental proposal.
The structure functions b
1
and b
2
are described by tensor-polarized quark and antiquark
distributions δ
T
q and δ
T
¯q. Using HERMES data on the b
1
structure function for the
deuteron, we made an analysis of extracting the distributions δ
T
q and δ
T
¯q in a simple
x-dependent functional form [23]. By imposing the sum rule
dx b
1
(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.
14
The tensor-polarized PDFs were defined by unpolarized partons in a polarized hadron,
so that their Q
2
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
2
evolution by fixing it at the HERMES-data
average Q
2
= 2.5 GeV
2
. 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.
19. 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) [25]. 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 number 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 [24]. 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 =
σ
NA
/(A σ
NN
)]. 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
15
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.
20. 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 [32]. The new data
set included JLab, Hermes, and Compass measurements on spin asymmetry A
1
for the
neutron and deuteron in lepton scattering. Our new analysis also utilized the double-spin
asymmetry for π
0
production in polarized pp collisions, A
π
0
LL
, 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
v
(x) at large x and ¯q(x) at x 0.1. The Phenix π
0
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
d
1
in comparison with the Compass ones at x 0.05.
We investigated impact of π
0
-production data at Relativistic Heavy Ion Collider (RHIC)
and future E07-011 experiment for the structure function g
1
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 [
26]. By global analyses of polarized
lepton-nucleon scattering and the π
0
-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
0
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
1
and RHIC π
0
data. The JLab-E07-011 measurements for g
d
1
should
be also able to reduce the gluon uncertainty, and the reduction was comparable to the one
by RUN-5 π
0
-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
d
1
. We found that the JLab-E07-011
data are accurate enough to probe the NLO gluonic term in g
1
. Both RHIC and JLab
data contribute to better determination of the polarized gluon distribution in addition to
improvement on polarized quark and antiquark distributions.
16
21. 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 p ossible
by using the newly-constructed high-momentum beamline from 2020.
In Ref. [27], possible J-PARC projects were explained in high-energy hadron physics, par-
ticularly by using the 3050 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 dep endence 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].
22. 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 1020% 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-
17
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
A
2
/F
A
2
and Drell-Yan cross-section ratios σ
A
DY
A
DY
[30]. The analyses were done
in the leading order (LO) and next-to-leading order (NLO) of running coupling constant
α
s
. 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
A
2
/F
D
2
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
2
data at x < 0.05 and there was almost no nuclear modification at x = 0.1 due to the
Drell-Yan data σ
pA
pD
, 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
D
2
/F
p
2
in the analysis. A code was provided
for calculating the NPDFs and their uncertainties at given x and Q
2
in the LO and NLO.
23. HERMES effect
From charged-lepton deep inelastic scattering measurements, two nuclear structure func-
tions F
A
1
and F
A
2
are measured. By virtual-photon polarizations, F
A
1
is associated with the
transverse polarization and F
A
2
is with b oth transverse and longitudinal polarizations. Re-
moving the transverse part from F
A
2
, we obtain the longitudinal structure function F
A
L
. Al-
though it should vanish (F
A
L
= 0) in the scaling limit Q
2
, the longitudinal-transverse
structure function ratio R = F
A
L
/F
A
1
is finite in Q
2
regions of actual experiments. In the
1980’s, nuclear modifications on F
A
2
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 [37].
Using a convolution description of nuclear structure functions, we derived the nuclear mod-
ification of the longitudinal-transverse structure function ratio R(x, Q
2
) 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
2
). 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
A
L
of a nucleus
is described by both nucleonic structure functions F
N
L
and F
N
1
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
18
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
2
and disappear at large Q
2
. The nuclear modification effects
are dominated by the binding and Fermi-motion effects contained implicitly in the con-
volution expression. Since the mixture occurs 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.
24. Nuclear effects and possible explanations on the NuTeV weak-mixing sin
2
θ
W
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
W
. This
angle was accurately measured by collider experiments as sin
2
θ
W
= 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
2
θ
W
=
0.2277 ± 0.0013 (stat) ± 0.0009 (syst), was significantly different from a global analysis
of other data, sin
2
θ
W
= 0.2227 ± 0.0004. This is called the NuTeV weak-mixing-angle
anomaly. Since the mixing angle sin
2
θ
W
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
2
θ
W
in the NuTeV experiment.
The Paschos-Wolfenstein relation R
= (σ
νN
NC
σ
¯νN
NC
)/(σ
νN
CC
σ
¯νN
CC
) = 1/2 sin
2
θ
W
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
v
and d
v
, 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 [38]. Next, using charge and baryon-number conservations
for nuclei, we discussed an effect of nuclear modification difference between u
v
and d
v
on
the sin
2
θ
W
through the modified Paschos-Wolfenstein relation [33]. 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
v
and d
v
, 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.
25. 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
19
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
3
,
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
2
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
A
3
/F
D
3
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
3
, g
4
, and g
5
in neutrino reactions. Using these functions
as well as g
1
and g
2
, we should be able to establish the nucleon spin structure. The
polarized valence-quark distributions should be determined by the structure function g
3
(or g
5
notation in some researchers). Furthermore, the quark spin content is directly ob-
tained by measuring the structure function g
1
, whereas analysis of charged-lepton data
have uncertainties.
26. 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 [39], 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
2
< 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
2
part of ρ contributes to the structure function g
1
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
2
(1 GeV
2
). 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.
20
Furthermore, we studied the relation between the ratio of the proton-deuteron (pd) Drell-
Yan cross section to the proton-proton (pp) one
(T )
σ
pd
/2∆
(T )
σ
pp
and the flavor asymme-
try in polarized light-antiquark distributions [42]. 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 imp ortant 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.
27. 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
2
analysis method
[40]. 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 1020% for
medium-size nuclei, so that we decided to determine such mo difications 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
A
=
V
+ A
2/3
σ
S
). Then, the cross
section per nucleon is expressed as σ
A
/A = σ
V
+ σ
S
/A
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
s
by a χ
2
analysis. The parton distributions were provided at Q
2
=1
GeV
2
with a number of parameters, which were determined by a χ
2
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
2
=1 GeV
2
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”.
21
28. 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 [42,43,44], 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
[44]. However, the number reduced to 22 after integrating the cross section over the
virtual-photon transverse momentum
Q
T
or after taking the limit Q
T
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 [43]. 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
Q
T
-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
1
, 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 [42].
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
1
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
2
and that
it could be used for determining the vector- and tensor-polarized antiquark distributions.
29. 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 [36,41].
Since the measurement of the polarized structure function g
1
for the nucleon by the EMC,
the origin of the nucleon spin has been investigated. However, it was not clear how an-
22
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 [41]. 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
1
data of the proton, deuteron, and
3
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
2
= 1 GeV
2
by
parameters, and then spin asymmetry A
1
was calculated theoretically. From the fit to the
asymmetry data A
1
, 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
[36]. In the χ
2
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.
30. Anomalous dimensions of the structure function h
1
and its Q
2
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
1
, 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
2
evolution of h
1
was known
only in the leading order (LO) of α
s
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
1
in the minimal
subtraction scheme [48]. In order to study h
1
in perturbative Quantum Chromodynamics
(QCD), we needed to introduce a set of local operators O
νµ
1
···µ
n
= S
n
ψ i γ
5
σ
νµ
1
iD
µ
2
· · ·
iD
µ
n
ψ trace terms (n = 1, 2, ...). The bare operator is defined by O
n
B
= Z
O
n
O
n
R
with the
renormalized one O
n
R
. Anomalous dimensions for O
n
are given by these renormalization
constants as γ
O
n
= µ (ln Z
O
n
)/∂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 singularities in the renormalization constants for
calculating the anomalous dimensions. Due to the chiral-odd nature, the gluon transversity
does not exist in the nucleon, as mentioned in the topic item 1. We calculated all the
23
Feynman diagrams in the two-loop level by using the Feynman gauge and the minimal
subtraction (MS) scheme for obtaining the anomalous dimensions for h
1
. Because of our
studies, it became possible to investigate h
1
in the NLO level.
Next, by calculating the inverse Mellin transformation of the obtained anomalous dimen-
sions, the Q
2
evolution can be described by integrodifferential equations in the DGLAP
(Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) form. We investigated numerical solution of
the DGLAP Q
2
evolution equation for the transversity distribution
T
q or the structure
function h
1
[45]. 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
2
into small
steps, we solved the integrodifferential equation by the Euler method in the variable Q
2
and by the Simpson method in the variable x. We provided a FORTRAN program for
the Q
2
evolution and devolution of the transversity distribution
T
q or h
1
. Using the
program, we showed the LO and NLO evolution results of the valence-quark distribution
T
u
v
+
T
d
v
, the singlet distribution
i
(∆
T
q
i
+
T
¯q
i
), and the flavor-asymmetric dis-
tribution
T
¯u
T
¯
d. They were compared with the longitudinal evolution results [46]
to show unique features of the transversity evolution by finding differences. For example,
Q
2
variations were much smaller in the flavor-nonsinglet transversity distribution than
the one for the longitudinally-polarized distribution as the Q
2
is increased. In addition,
perturbative effects on the flavor asymmetric distribution
T
¯u
T
¯
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.
31. Nuclear structure functions F
A
2
by a parton model
Nuclear modifications of the structure function F
2
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
A
2
and nuclear parton distribution functions as an unified description
from small x to large x [54,55,56]. As this parton model, we used a Q
2
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
2
rescaling in this work, within F
A
2
. 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
2
(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 [54,55]. 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
A
2
(x)/F
D
2
(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
2
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.
24
Then, we investigated gluon distributions in the nuclei C and Sn by this parton model [56].
We obtained shadowing in the nuclear gluon distributions in the small x region (x < 0.02)
due to the recombinations and depletion [typically G
A
(x)/G
N
(x) 0.9] in the medium x
region (0.2 < x < 0.6). The ratio G
A
(x)/G
N
(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
Sn
(x)/G
C
(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.
32. Parton distributions in nuclei by a parton model
Using the parton model explained in the previous topic item 31, we predicted new phenom-
ena in nuclear parton distribution functions. First, we showed that finite flavor-asymmetric
antiquark distributions are possible [53], 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 [53]. Our results
suggested that the nuclear modification in the tungsten is a 2–10 % effect on the NMC
u d distribution. In the b eginning of 2020’s, Drell-Yan measurements will be rep orted
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
A
3
and valence-quark distributions
were investigated [51], 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
A
3
/F
D
3
at small x could be
useful in discriminating among different models, which produce similar shadowing behav-
ior in the structure function F
2
. This difference could be tested experimentally by neutrino
scattering measurements and also by π productions in charged-lepton scattering.
Third, Q
2
evolution of nuclear structure functions was investigated in order to under-
stand NMC measurements on the tin and carbon nucleus ratio of F
A
2
[49]. The F
2
was
evolved by using the leading-order DGLAP, next-to-leading-order DGLAP, and parton-
recombination equations. The NMC experimental result [F
Sn
2
/F
C
2
]/∂[ln Q
2
] ̸= 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
2
evolution with the
parton recombinations was used. It indicated that high-twist effects need to be understood
accurately.
Forth, we investigated effects on J suppression in heavy-ion collisions. Originally, the
J suppression was discussed by assuming that nuclear modifications are identical for
25
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 suppression about 510%.
33. 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
0
(980) and a
0
(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 the ϕ factories by calculating the decay widths in different
configurations (ordinary q¯q mesons, s¯s, K
¯
K molecule, tetraquark, or glueball) [57]. The
radiative decay ϕ Sγ [S = f
0
(980) or a
0
(980)] is the electric dipole decay. Since the
electric dipole moment is proportional to the spatial separation, the decay widths are sensi-
tive to the internal structure of the scalar mesons. We showed that ϕ radiative decays into
scalar mesons [f
0
(980), a
0
(980)] could provide important clues on the internal structures
of these mesons. The radiative decay widths varied widely depending on the substruc-
tures (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 valuable 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 av-
erage K
¯
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
0
(980) and a
0
(980)
could be investigated at the ϕ factories. We also found that the decay ϕ Sγ K
0
¯
K
0
γ
is not strong enough to pose a significant background problem for studying CP violation
via ϕ K
0
¯
K
0
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.
34. Numerical solution of Q
2
evolution equations on structure functions
The DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) evolution equations are inte-
grodifferential equations, which describe Q
2
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 [52,58] and by the
Euler’s (or “Brute-force”) method [45,46,50].
First, this Laguerre-polynomial method was studied by expanding parton distribution
26
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=2030), 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
1
0.01
dxg
1
(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
2
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
2
steps and more than one-thousand x steps are taken. The numerical solution
was discussed in detail, and evolution results were compared with Q
2
dependent data in
CDHSW, SLAC, BCDMS, EMC, NMC, Fermilab-E665, ZEUS, and H1 experiments. We
provided a FORTRAN program for Q
2
evolution (and “devolution”) of nonsinglet-quark,
singlet-quark, q
i
+ ¯q
i
, and gluon distributions (and corresponding structure functions) in
the nucleon and in nuclei.
These studies were extended to the longitudinally polarized [46] and transversity evolu-
tion equations [45]. In the longitudinal-polarization studies, Q
2
variations of the polarized
structure function g
1
and the spin asymmetry A
1
were investigated in both LO and NLO for
specifying the NLO effects. At that time, analysis groups obtained g
1
often by neglecting
the Q
2
dependence in the asymmetry A
1
. However, we pointed out that it is inappropriate
because clear Q
2
dependence existed especially in the small Q
2
region (Q
2
< 2 GeV
2
). For
a precise determination of g
1
, the Q
2
dependence of A
1
needs to be taken into account
properly. Furthermore, we investigated the numerical solution for the transversity distri-
butions (h
1
or
T
q) [45], as explained in the topic item 30. We supplied these Q
2
evolution
codes on our web, so that other scientists could use them for their own studies.
35. 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 b ecame 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
27
the region Q
2
4 GeV
2
. 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
1
of about 0.7 GeV for a monopole πNN form
factor. A typical πNN form factor with Λ
1
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
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 [62]. 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) [61,62]. 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
¯
d
in an extra π
in the πN∆ process.
Then, various possible reasons were discussed for the Gottfried-sum-rule violation, includ-
ing significant contributions from valence quarks at very small x and SU(2)
f
breaking
in the sea. We suggested various experiments, which could give a direct measurement of
¯u
¯
d, including neutrino scattering and Drell-Yan measurements [47,53,59,60]. A paper
was written for summarizing theoretical and experimental investigations, such as historic
background, perturbative QCD effects, various hadron models, past experiments, and fu-
ture prospects, on the Gottfried sum rule and the ¯u
¯
d distribution [47].
36. Sum rule of tensor structure function b
1
In spin-1 hadrons, there are new structure functions, in addition to the ones (F
1
, F
2
, g
1
,
g
2
) which exist for the spin-1/2 nucleons, associated with its tensor structure. In 1989, the
new structure functions were named as b
1
, b
2
, b
3
, and b
4
in charged-lepton deep inelastic
scattering from a spin-one hadron such as the deuteron. Among them, twist-two functions
are related by the Callan-Gross type relation b
2
= 2xb
1
in the Bjorken scaling limit. The
functions b
3
and b
4
are higher-twist structure functions, so that we first investigated the
twist-2 structure function b
1
.
In our work, we derived a sum rule for b
1
. From the parity and time-reversal invariances,
the only global electromagnetic observable for a spin-1 hadron is the electric quadrupole
moment, which does not exist for the spin-1/2 nucleons. Therefore, we studied the b
1
sum rule by speculating that it could be related to the electric quadrupole moment. The
integral of b
1
over the Bjorken variable x was written in terms of tensor-polarized par-
ton distribution functions (PDFs). Then, helicity amplitudes of elastic photon-hadron
scattering were expressed by electric monopole and quadrupole form factors and also by
the tensor-polarized PDFs. Through the tensor-polarized PDFs, the b
1
integral was then
related to the electric quadrupole form factor. In this way, if antiquark distributions are
28
not tensor polarized, the sum became
dxb
1
(x) = lim
t0
(5t/12) F
Q
(t) = 0 [63], where
F
Q
(t 0) is the electric dipole moment of a spin-1 hadron. Furthermore, if the antiquark
distributions are tensor polarized, we obtained the sum
dxb
1
(x) = lim
t0
(5/12)tF
Q
(t) +
(1/18)
dx [8δ
T
¯u(x)+ 2δ
T
¯
d(x)+δ
T
s(x)