Research highlights Shunzo Kumano
I have been working in theoretical hadron and nuclear physics. My studies are classified into
two categories. One is on properties of hadron resonances, and the other is on structure functions
of hadrons and nuclei. I explain several results from my past works. The detailed explanation
of my research history is given in my home page at https://research.kek.jp/people/kumanos/
(→ Research history).
* Properties of hadron resonances
Electromagnetic properties of unstable hadrons have been key quantities for developing
hadron models. Among them, we investigated properties of the ∆ resonance and exotic hadron
candidates, especially f
0
(980), a
0
(980), and Λ(1405).
∆ electromagnetic moments
The spectroscopy and static properties of hadrons played a central role in the development of
quark models. However, little precise information was available for baryons outside the ground
state octet. We investigated possible determination of ∆ electromagnetic moments in the pion-
nucleon bremsstrahlung [1]. In 1980’s, people were also interested in the electric quadrupole mo-
ment of ∆, in addition to the magnetic dipole moment, because ∆ was expected to be deformed
due to the tensor force in the gluon exchange potential b etween quarks. First, we calculated
the ∆ electromagnetic moments by the isobar model. The same dynamics which renormalize
the mass and provide the strong decay width, namely coupling to the open πN channel, also
renormalize the moments. These pionic contributions to the ∆ electromagnetic moments were
calculated and they were µ(∆
++
) = −0.4 + i 0.6 µ
p
and Q(∆
++
) = +0.2 + i 0.05 fm
2
. Next, we
predicted theoretically that accurate determination of the ∆ magnetic moment can be done by
the polarized pion-nucleon bremsstrahlung [1]. Based on this idea, experimentalists proposed
an experiment at the Paul Scherrer Institute, and they actually determined it as µ
∆
++
= 1.64 µ
p
[2]. On the other hand, we found that it is not possible to find the quadrupole moment because
its effects are very small on the cross sections.
[1] Pion-nucleon bremsstrahlung and ∆ electromagnetic moments,
L. Heller, S. Kumano, J. C. Martinez, E. J. Moniz, Phys. Rev. C 35 (1987) 718.
[2] Polarized-target asymmetry in pion-proton bremsstrahlung at 298 MeV,
A. Bosshard et al., Phys. Rev. Lett. 64 (1990) 2619.
Structure of exotic hadron candidates f
0
(980) and a
0
(980)
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 configurations
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 considered as exotic
hadron candidates, for example, because strong decay widths are too large to explain experi-
mental ones if they are ordinary q¯q hadrons with light u and d quarks [3]. They were considered
as s¯s, K
¯
K molecule, tetraquark (qq¯q¯q), or glueball (gg). In order to determine their structure,
we proposed to use the radiative decay of ϕ meson into these scalar mesons at ϕ factories by
calculating the decay widths in different configurations (ordinary q¯q mesons, s¯s, K
¯
K molecule,
tetraquark, or glueball) [4]. The radiative decay ϕ → Sγ [S = f
0
(980) or a
0
(980)] is the elec-
tric dipole decay. Since the electric dipole moment is proportional to the spatial separation,
the decay widths are sensitive to the internal structure of the scalar mesons. We showed that
1
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
substructures (q ¯q, qq¯q¯q, K
¯
K, glueball). Hence, we could discriminate among various models by
measuring these widths 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.
[3] Decay of mesons in flux-tube quark model,
S. Kumano and V. R. Pandharipande, Phys. Rev. D 38 (1988) 146.
[4] Scalar mesons in ϕ radiative decay: Their implications for spectroscopy and for studies of
CP violation at ϕ factories, F.E.Close, N.Isgur, and S.Kumano, Nucl. Phys. B 389 (1993) 513.
In subsequent studies, we developed this field in a different way, by using our knowledge on
high-energy hadron physics, for finding internal configurations of exotic hadron candidates. We
proposed to use hadron tomography by three-dimensional structure functions [5], constituent
counting rule of perturbative quantum chromodynamics (QCD) [5,6,7], and differences between
favored and disfavored fragmentation functions [8]. These are new ideas for examining exotic
hadron candidates, and experimental studies of these proposals will be done in future.
[5] Tomography of exotic hadrons in high-energy exclusive processes,
H. Kawamura and S. Kumano, Phys. Rev. D 89 (2014) 054007.
[6] Determination of exotic hadron structure by constituent-counting rule for hard exclusive
processes, H. Kawamura, S. Kumano, and T. Sekihara, Phys. Rev. D 88 (2013) 034010.
[7] Constituent-counting rule in photoproduction of hyperon resonances,
Wen-Chen Chang, S. Kumano, and T. Sekihara, Phys. Rev. D 93 (2016) 034006.
[8] Proposal for exotic-hadron search by fragmentation functions,
M. Hirai, S. Kumano, M. Oka, and K. Sudoh, Phys. Rev. D 77 (2008) 017504.
Constituent counting rule for probing exotic hadron nature
Although there have been reports on exotic hadron candidates, 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 [5,6,7]. According to the perturbative QCD, exclusive high-energy hadron
reactions occur by hard gluon exchanges , and their cross sections are estimated by considering
hard quark and gluon propagators. It leads to the so-called constituent counting rule which
indicates, for example, that a two-body exclusive hadron reaction cross section a + b → c + d
scales like dσ/dt ∼ f(θ
cm
)/s
n−1
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 and t are Mandelstam variables.
First, we investigated internal structure of hyperons and their excited states by calculating
the cross section for exclusive processes, for example π
−
+ p → K
0
+ Λ(1405), which could be
measured at J-PARC. We suggested that the internal quark configuration of Λ(1405) should
be determined by the asymptotic scaling behavior of the cross section [6]. If it is an ordi-
nary three-quark baryon, the scaling of the cross section is s
8
dσ/dt =constant, whereas it is
s
10
dσ/dt =constant if Λ(1405) is a five-quark hadron. Second, we analyzed the JLab-CLAS
data on the photoproduction of hyperon resonances [7]. Especially, Λ(1405) could be considered
to be a
¯
KN molecule with the constituent number n = 5. We found that the current data are
not enough to conclude the numb ers of its constituent. However, it is interesting to find the
tendency of the energy dependence in the constituent number. Λ(1405) looked like a penta-
quark state at lower energies, but it became a three-quark one at high energies. If Λ(1405) is
a mixture of three-quark and five-quark states, the energy dependence could be valuable for
finding its composition and mixture.
2
* Structure functions
Since 1988, I have been investigating hadron and nuclear structure functions. In the follow-
ing, I explain some results.
Sum rule for the tensor-polarized structure function b
1
In spin-1 hadrons, there are new structure functions, in addition to the ones of the spin-1/2
nucleons, and one of them is the twist-2 structure function b
1
. It will be measured at JLab in
the middle of 2020’s. In our work, we derived a sum rule for b
1
by the parton model [9]. The
integral of b
1
over the Bjorken variable x was written in terms of tensor-polarized parton dis-
tribution functions (PDFs). Then, helicity amplitudes of elastic photon-hadron scattering were
expressed by electric monop ole 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, and we obtained
∫
dx b
1
(x) = − lim
t→0
5
24
t F
Q
(t) +
∑
i
e
2
i
∫
dx δ
T
¯q
i
(x),
where F
Q
(t) is the electric quadrupole form factor of the hadron, and δ
T
¯q
i
is the tensor-polarized
antiquark distribution. The first term vanishes, so that a finite sum of b
1
indicates a finite
tensor-polarized antiquark distribution. The vanishing first term comes from the fact that the
valence-quark number does not depend on the tensor polarization, whereas it depends on the
flavor in the Gottfried sum (1/3). This b
1
sum rule was investigated experimentally by the HER-
MES collaboration [10] and they obtained
∫
0.85
0.002
dx b
1
(x) =
1
2
[1.05±0.34(stat)±0.35(sys)]×10
−2
,
where the 1/2 factor is introduced so as to express b
1
per nucleon. It indicated an existence of
finite tensor-polarized antiquark distributions. In 2017, b
1
was calculated in a standard deuteron
model [11] and the obtained b
1
was very different from the HERMES data. It indicates a new
hadronic mechanism in the deuteron beyond the simple bound system of a proton and a neutron.
[9] A sum rule for the spin dependent structure function b
1
(x) for spin one hadrons,
F. E. Close and S. Kumano, Phys. Rev. D 42 (1990) 2377.
[10] Measurement of the tensor structure function b
1
of the deuteron,
A. Airapetian et al., Phys. Rev. Lett. 95 (2005) 242001.
[11] Tensor-polarized structure function b
1
in standard convolution description of deuteron,
W. Cosyn, Yu-Bing Dong, S. Kumano, and M. Sargsian, Phys. Rev. D 95 (2017) 074036.
Anomalous dimensions of chiral-odd structure function h
1
The nucleon spin structure was investigated mainly for longitudinal polarizations; however,
transverse spin structure also needed to be understood. One of important transverse structure
functions is the leading-twist structure function h
1
, which is also called the quark transversity
distribution ∆
T
q. The quark transversity distribution 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 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
(MS) scheme [12]. In order to study h
1
in QCD, we needed to introduce a set of local op er-
ators O
νµ
1
···µ
n
= S
n
ψ i γ
5
σ
νµ
1
iD
µ
2
· · · iD
µ
n
ψ (n = 1, 2, ...). The bare op erator 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/ϵ 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. We calculated all the Feynman diagrams in the two-loop level by using
the Feynman gauge and the MS scheme for obtaining the anomalous dimensions for h
1
. Because
3
of these studies, it became possible to investigate h
1
in the NLO level. We also provided a useful
code for calculating this Q
2
evolution [13].
[12] Two-loop anomalous dimensions for the structure function h
1
,
S. Kumano and M. Miyama, Phys. Rev. D 56 (1997) R2504.
[13] Numerical solution of Q
2
evolution equation for the transversity distribution ∆
T
q,
M. Hirai, S. Kumano, and M. Miyama, Comput. Phys. Commun. 111 (1998) 150.
Flavor asymmetric antiquark distributions
Light antiquark distributions were exp ected to be flavor symmetric because they are consid-
ered to be created mainly through perturbative QCD splitting processes from a gluon (g → q¯q).
However, it became clear that they are not flavor symmetric from experiments. The strange-
quark distribution is about a half of the up and down antiquark distributions [(¯u+
¯
d)/2 ∼ ¯s] from
neutrino-induced opposite-sign dimuon measurements. The inequality ¯u ̸=
¯
d also became obvi-
ous from the New-Muon-Collaboration (NMC) finding 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. Therefore, the flavor asymmetric antiquark distributions should come
mainly from a nonperturbative mechanism.
As such a nonperturbative mechanism, we investigated meson-cloud effects on the antiquark
distributions of the nucleon. In the pion-cloud model, there exists a momentum cutoff parameter
in the πNN form factor. Fixing this parameter by the distribution (¯u +
¯
d)/2 − ¯s, we predicted
the flavor asymmetric distribution ¯u−
¯
d theoretically [14]. We found that the order of magnitude
of the NMC finding on ¯u −
¯
d can be understood by this pion-cloud contribution. The 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. In 1998, a
paper was written for summarizing theoretical and experimental investigations, such as historic
background, perturbative QCD effects, various hadron models, past experiments, and future
prospects, on the Gottfried sum rule and the ¯u −
¯
d distribution [15].
[14] πNN form factor for explaining sea quark distributions in the nucleon,
S. Kumano, Phys. Rev. D 43 (1991) 59 & 3067.
[15] Flavor asymmetry of anti-quark distributions in the nucleon,
S. Kumano, Phys. Rept. 303 (1998) 183.
Global analyses of parton distribution functions and fragmentation functions
One of major achievements is on nuclear parton distribution functions (NPDFs). 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 sections, accurate NPDFs
are needed. 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
/σ
A
′
DY
. We proposed this global χ
2
method on the NPDFs for the first time in the 2001 paper [16]. The analyses were done in the
leading order (LO) [16,17] and next-to-leading order (NLO) [18] 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 [17,18], so that we can discuss the NLO improvement on the determination. 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. Codes were provided for calculating the NPDFs and their
uncertainties at given x and Q
2
in the LO and NLO. The 2007 version of our NPDFs was widely
4
used as one of the standard NPDFs for a long time. In the similar way, we studied global
analyses of longitudinally polarized PDFs [19] and fragmentation functions [20], and created
useful codes for calculating them at given x and Q
2
.
[16] Determination of nuclear parton distributions,
M. Hirai, S. Kumano, and M. Miyama, Phys. Rev. D 64 (2001) 034003.
[17] Nuclear parton distribution functions and their uncertainties,
M. Hirai, S. Kumano, and T.-H. Nagai, Phys. Rev. C 70 (2004) 044905.
[18] Determination of nuclear parton distribution functions and their uncertainties in next-to
-leading order, M. Hirai, S. Kumano, and T.-H. Nagai, Phys. Rev. C 76 (2007) 065207.
[19] Global analyses of polarized PDFs: Y. Goto et al., Phys. Rev. D 62 (2000) 034017;
M. Hirai, S. Kumano, and N. Saito, Phys. Rev. D 69 (2004) 054021; D 74 (2006) 014015;
M. Hirai and S. Kumano, Nucl. Phys. B 813 (2009) 106.
[20] Global analyses of fragmentation functions: M. Hirai, S. Kumano, T.-H. Nagai, and
K. Sudoh, Phys. Rev. D 75 (2007) 094009; M. Hirai, H. Kawamura, S. Kumano, and
K. Saito, PTEP 2016 (2016) 113B04; N. Sato et al., Phys. Rev. D 94 (2016) 114004.
Gravitational form factors of a hadron
Since gravitational interactions are too weak to be measured in microscopic systems, the
measurement of the gravitational form factors used to be considered impossible for hadrons and
nuclei. However, we know generalized parton distributions (GPDs) and generalized distribution
amplitudes (GDAs) contain matrix elements of energy-momentum tensor, which is a source of
gravity within a hadron. Here, the GDAs are s-t crossed quantities of the GPDs, so that they
could be called timelike GPDs. The spacelike GPDs are measured in virtual Compton scattering
at lepton-accelerator facilities, but it is also possible to investigate them by using exclusive
hadron reactions at hadron-accelerator facilities [21]. We extracted the gravitational form factors
for the first time from actual experimental measurements [22]. 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. The GDAs were expressed by a number of
parameters and they were determined from the data of γ
∗
γ → π
0
π
0
. 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. 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
[22]. 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. Gravitational physics used to be considered as a field on macroscopic world.
However, we showed that it is p ossible to investigate it in the microscopic level in terms of
fundamental particles of quarks and gluons. In future, we expect much progress on origin of
hadron masses and internal hadron pressures in terms of quark and gluon degrees of freedom.
This work together with our studies on the gluon transversity [23] was selected one of highlight
research results of KEK in the annual report of 2019.
[21] GPDs at hadron-accelerator facilities: S. Kumano, M. Strikman, and K. Sudoh,
Phys. Rev. D 80 (2009) 074003; T. Sawada et al., Phys. Rev. D 93 (2016) 114034.
[22] Hadron tomography by generalized distribution amplitudes in pion-pair production process
γ
∗
γ → π
0
π
0
and gravitational form factors for pion, S. Kumano, Qin-Tao Song, and
O. V. Teryaev, Phys. Rev. D 97 (2018) 014020.
5
[23] Gluon transversity in polarized proton-deuteron Drell-Yan process,
S. Kumano and Qin-Tao Song, Phys. Rev. D 101 (2020) 054011; 094013.
Transverse-momentum-dependent parton distribution functions for spin-1 hadrons
We showed possible transverse-momentum-dependent parton distribution functions (TMDs)
for spin-1 hadrons including twist-3 and 4 functions in addition to the leading twist-2 ones
by investigating all the possible decomposition of a quark correlation function in the Lorentz-
invariant way [24]. The Hermiticity and parity invariance were imposed in the decomposition;
however, the time-reversal invariance was not used due to an active role of gauge links in the
TMDs. Therefore, there exist time-reversal odd functions in addition to the time-reversal even
ones in the TMDs. We listed all the functions up to twist-4 level because there was no study in
the twist-3 and 4 parts for spin-1 hadrons. We showed that 40 TMDs exist in the tensor-polarized
spin-1 hadron in the twist 2, 3, and 4. Among them, we found 30 new structure functions in the
twist 3 and 4 in this work. Since time-reversal-odd terms of the collinear correlation function
should vanish after integrals over the partonic transverse momentum, we obtained new sum
rules for the time-reversal-odd structure functions,
∫
d
2
k
T
h
LT
=
∫
d
2
k
T
h
LL
=
∫
d
2
k
T
h
3LL
= 0.
In addition, we indicated that new transverse-momentum-dependent fragmentation functions
exist in tensor-polarized spin-1 hadrons. The TMDs are rare observables to find explicit color
degrees of freedom in terms of color flow, which cannot be usually measured because the color is
confined in hadrons. Furthermore, the studies of TMDs enable not only to find three-dimensional
structure of hadrons, namely hadron tomography including transverse structure, but also to
provide unique opportunities for creating interesting interdisciplinary physics fields such as gluon
condensates, color Aharonov-Bohm effect, and color entanglement.
Next, in the similar way to the Wandzura-Wilczek relation and the Burkhardt-Cottingham
sum rule for the polarized structure functions g
1
and g
2
, we showed in Ref. [25] that an analogous
twist-2 relation and a sum rule exist for the tensor-polarized parton distribution functions f
1LL
and f
LT
, where f
1LL
is a twist-2 function and f
LT
is a twist-3 one. Namely, the twist-2 part
of f
LT
is expressed by an integral of f
1LL
(or b
1
) and the integral of the function f
2LT
=
(2/3)f
LT
−f
1LL
over x vanishes. If the parton-model sum rule for f
1LL
(b
1
) is applied by assuming
vanishing tensor-polarized antiquark distributions, another sum rule also exists for f
LT
itself.
These relations should be valuable for studying tensor-polarized distribution functions of spin-
1 hadrons and for separating twist-2 components from higher-twist terms. In deriving these
relations, we indicate that four twist-3 multiparton distribution functions F
LT
, G
LT
, H
⊥
LL
, and
H
T T
exist for tensor-polarized spin-1 hadrons.
Furthermore, we derived relations among the tensor-polarized distribution functions and
twist-3 multiparton distribution functions from the equation of motion for quarks [26]. We
found
(1) a relation for the twist-3 PDF f
LT
, the trasverse-momentum moment PDF f
(1)
1LT
, and the
multiparton distribution functions F
G,LT
and G
G,LT
,
(2) a relation for the twist-3 PDF e
LL
, the twist-2 PDF f
1LL
, and the multiparton distribution
function H
⊥
G,LL
.
Then, the Lorentz-invariance relation was obtained for relating f
LT
, f
(1)
1LT
, f
1LL
, and F
G,LT
. In
these derivations, we also found new relations among the multiparton distribution functions
defined by the field tensor [F
D,LT
(x, y), G
D,LT
(x, y), H
⊥
D,LL
(x, y), H
D,T T
(x, y)] and the ones
defined by the covariant derivative [F
G,LT
(x, y), G
G,LT
(x, y), H
⊥
G,LL
(x, y), H
G,T T
(x, y)]. These
relations are valuable in constraining the distribution functions and learning about multiparton
correlations in spin-1 hadrons.
6
[24] Transverse-momentum-dependent parton distribution functions up to twist 4 for spin-1
hadrons, S. Kumano and Qin-Tao Song, Phys. Rev. D 103 (2021) 014025.
[25] Twist-2 relation and sum rule for tensor-polarized parton distribution functions of spin-1
hadrons, S. Kumano, Qin-Tao Song, J. High Energy Phys. 09 (2021) 141, 1-22.
[26] Equation-of-motion and Lorentz-invariance relations for tensor-polarized parton distribution
functions of spin-1 hadrons, S. Kumano, Qin-Tao Song, Phys. Lett. B 826 (2022) 136908, 1-5.
7
