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