Highlights of previous research 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.

[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.

6