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 [52] and transversity evolu-
tion equations [51]. 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) [
51], as explained in the topic item 35. We supplied these Q
2
evolution
codes on our web, so that other scientists could use them for their own studies.
40. Flavor asymmetric antiquark distributions ¯u −
¯
d, (¯u +
¯
d)/2 − ¯s
Light antiquark distributions were expected to be flavor symmetric b ecause they are con-
sidered to be created mainly through p erturbative QCD splitting processes from a gluon
(g → q¯q). However, it became clear that they are not flavor symmetric from experiments.
The strange-quark distribution is about a half of the up and down antiquark distribu-
tions [(¯u +
¯
d)/2 ∼ ¯s] from neutrino-induced opposite-sign dimuon measurements. The
inequality ¯u ̸=
¯
d also became obvious from the NMC (New Muon Collaboration) find-
ing on the Gottfried-sum-rule violation and Fermilab Drell-Yan experiments. The flavor
asymmetric antiquark distribution ¯u −
¯
d, created in perturbative QCD, originates from
next-to-leading-order effects, so that it is much smaller than the NMC finding on ¯u −
¯
d in
the region Q
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
30