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{\LARGE\bf ʿ��7ǯ�١�������ʸ}

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{\Huge\bf �˻Ҥ���Ӹ��ҳ˹�¤�ؿ���$Q^2$��¸��}

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{\Large ���������ر� ���طϸ���� ʪ�����칶}

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{\LARGE ����\ ����}
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{\Large\bf ʿ��7ǯ�١�������ʸ}

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{\Huge\bf �˻Ҥ���Ӹ��ҳ˹�¤�ؿ���$Q^2$��¸��}

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�˻Ҥ���Ӹ��ҳ˹�¤�ؿ��� $Q^2$ ��¸���ˤĤ���Ĵ�٤���
�����ι�¤�ؿ��� $Q^2$ ��¸���򵭽Ҥ����������Ȥ��ơ�
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) �������ȡ�
����˥ѡ��ȥ�Ʒ��θ��̤�ޤ᤿��������¸�ߤ��롣
�����ǡ��ܸ���Ǥ� brute-force ˡ�ȸƤФ����Ͳ�ˡ��
�Ѥ��� FORTRAN �ץ��������������
����飲�Ĥ��������ο��Ͳ�ˤĤ��Ƥθ����Ԥä���
���η�̡��˻ҹ�¤�ؿ��� $Q^2$ ��¸���ˤĤ��Ƥϡ�
$\alpha_s$ �κ��㼡�Τ� (LO) �η�̤���Ӥ��ơ�
�⼡������������줿��� (NLO) �η�̤Ͼ����� $Q^2$ �ΰ��
���褯�¸��ǡ�����Ƹ�������
�ޤ����ҳ˹�¤�ؿ��ˤĤ��Ƥϡ�
New Muon Collaboration (NMC) �� $Q^2$ ��¸����
�ؤ���¸��ǡ��� $F_2^A/F_2^D$ �Ȥ���Ӥ�Ԥ���
$Q^2$ ȯŸ�ˤ�����ѡ��ȥ�Ʒ��θ��̤ϡ�
������ $x$ �ΰ�����������Ȥ�Ƚ��������
�������ʤ��鸽�����Ǥ��θ��̤򸡽Ф���˻�äƤ��ʤ���
����ˡ��Ƕ� NMC �ˤ������줿 $Q^2$ ȯŸ��
���ҳ˰�¸�� $\partial (F^{Sn}_2/ F^C_2) / \partial \ln Q^2$ ��
�Ĥ��Ƥ���Ϥ�����
���η�̡������褽�� $x$ ��¸���ϸ��������Ƥ���
�ѡ��ȥ�ʬ�ۤθ��ҳ���Ǥ��Ѳ�
�� NLO �� DGLAP �������������Ǥ��뤳�Ȥ�ʬ���ä���
�ޤ���$Q^2$ ȯŸ�ˤ�����ѡ��ȥ�Ʒ��θ��̤⤽�θ��ҳ˰�¸�����
�ܤ�������Ǥ��뤳�Ȥ������줿��


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���Ҥ� $\alpha$ γ�Ҥ���ͤ����� Rutherford ���𤬡����Ҥ�������¤�Ȥ���
���ҳˤ�¸�ߤ����餫�ˤ����褦�ˡ�
����γ�Ҥ�������¤�ˤĤ���Ĵ�٤뤿���ͭ�Ѥʼ��ʤΰ�ĤȤ��ơ�
���λ�������ˤĤ���Ĵ�٤뤳�Ȥ��������롣
���Τ褦����ͳ���顢�˻Ҥ�������¤�ˤĤ���Ĵ�٤뤳�Ȥ���Ū�Ȥ��ơ�
�͡��ʻ������������Ӽ¸����椬�Ԥ��Ƥ�����

������Ρ����˹⥨�ͥ륮�����ŻҤ⤷���ϥߥ塼����ȳ˻Ҥ�
������������ˤĤ��ƹͤ��롣������ʬ�����Ѥϡ�
���Ĥ���Ω�ѿ� $Q^2=-q^2$ �� $x \equiv Q^2/2M\nu$ �δؿ�
�Ǥ��빽¤�ؿ� $F_1(x, Q^2)$ ����� $F_2(x, Q^2)$ ���Ѥ��ƽ�ɽ�����Ȥ��Ǥ��롣
��������$q$ �ϻ���ˤ�����ܹԣ�����ư�̡�
$\nu$ �ϥ��ͥ륮���ܹԡ������ $M$ �ϳ˻Ҽ��̤Ǥ��롣

1969 ǯ�� SLAC (Stanford Linear Accelerator Center) �ǤΡ�
�ۻҤ���ۻҤ�ɸŪ�Ȥ����ŻҤȤο�����������¸��ˤ�ꡢ
Bjorken ��������󥰤�����Ω�äƤ��뤳�Ȥ��¸�Ū�˼����줿 \cite{SLACSC}��
���� Bjorken ��������󥰤Ȥϡ�
��¤�ؿ��� $Q^2$ �ˤ�餺�� $x$ �Τߤδؿ���
ɽ�����Ȥ�������Ǥ��� \cite{BJSC}�� 
�����ʬ��ǽ���������� $Q^2$ �򤢤��Ƥ��äƤ⡢
����ʾ��������¤�������Ƥ��ʤ��Ȥ������Ȥ��̣����
�˻������˥ѡ��ȥ�ȸƤФ�빽¤������ʤ�����γ�Ҥ�¸�ߤ��Ƥ���
���Ȥ򼨤��Ƥ��롣

�������ʤ��顢���θ�ι��� $x$ �ΰ�ˤ錄�롢�����̩�ʼ¸��ˤ�ꡢ
��¤�ؿ��ˤ鷺���� $Q^2$ ��¸�������뤳�Ȥ���ǧ���줿��
���� ``��������󥰤��ˤ�" �ȸƤФ�븽�ݤ�
�ۻ������Υ������������롼��������Ф���
�ޤ����Υ��롼���󤬥�������$\cdot$ȿ���������Ф��������ơ�
�������������롼����䳤���������α���ʤ���Ƥ��뤿���
�����Ƥ���Ȳ��Ǥ��롣
���ʤ����$Q^2$ �򤢤���ʬ��ǽ�򤢤��Ƥ����Ȥ��Τ褦�ʤ�꾮������ư�̤����
�ѡ��ȥ󤬸����Ƥ��뤳�Ȥ���������ȹͤ����롣

���������䥰�롼�����
������ߺ��Ѥ� QCD (Quantum Chromodynamics, �̻ҿ��ϳ�) �ˤ�ä�
���Ҥ���롣
�����ơ���¤�ؿ����Τ� Lattice QCD �ʤɤ�����ư��Ū����ˡ�Ǥ���
�׻����뤳�Ȥ��Ǥ��ʤ��������Υ�������󥰤��ˤ�ϡ�
��ư��Ū QCD �ˤ�ä�ɾ�����뤳�Ȥ��Ǥ��롣����Ū�ʷ׻��ϡ�
��ư��Ū QCD �ˤ��Ƴ��
����� Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) ������
���Ѥ��ƹԤ����Ȥ��Ǥ��� \cite{DGLAP}��
�����������ϡ�����礭�ʱ�ư�̤���ĥѡ��ȥ󤬡������Ψ��
�ѡ��ȥ�����Ф��ơ���꾮������ư�̤���ĥѡ��ȥ�ˤʤ������
�׻����Ƥ�����ΤǤ��롣
��äơ����� DGLAP �������ο��Ͳ����ơ���������󥰤��ˤ졢
���ʤ����¤�ؿ��� $Q^2$ ��¸���ˤĤ��Ƹ��椹�뤳�Ȥϡ�
��ư��Ū QCD �θ��ڤȤ�����̣�����˰յ��Τ��뤳�ȤǤ��롣
����ˡ������������ϼ¸�����������θ���ˤ��������ˤ褯�Ѥ������ΤʤΤǡ�
������������򤯥���ԥ塼���ץ��������������¾�θ���Ԥ����ѤǤ���褦��
�������뤳�Ȥ�ͭ�ѤǤ��롣

�Ȥ����ǡ�ɸŪγ�Ҥ�˻Ҥ��鸶�ҳˤ��Ѥ�������ͤ��롣
����ȳ˻Ҥι�¤�ؿ����б����Ƹ��ҳˤι�¤�ؿ������뤳�Ȥ��Ǥ��롣
���θ��ҳ˹�¤�ؿ��ˤĤ��ơ�1983 ǯ�����˶�̣�������ݤ�ȯ�����줿��
����ϡ����ҳ˹�¤�ؿ� $F_2^A$ ���˻Ҥι�¤�ؿ� $F_2^N$ ��
�ۤʤäƤ���Ȥ�����Τǡ�
���ʤ�� $F_2^A(x,Q^2)/F_2^N(x,Q^2) \neq 1$ �Ǥ��롣
���θ��ݤ� European Muon Collaboration (EMC) �ˤ�ä�
ȯ�����줿�Τ� EMC ���̤ȸƤФ� \cite{AEMC}��
���θ��ݤ��������뤿����͡����Ϸ����󾧤���Ƥ��롣
�������Ϸ��Τʤ��˥ѡ��ȥ�Ʒ��Ȥ���
��Τ����� \cite{GLUE}--\cite{RGR}��
����ϡ��ۤʤ�˻Ҥ���Υѡ��ȥ󤬸ߤ�����ߺ��Ѥ򤷤�ͻ�礷�Ƥ���������
�ͤ����ΤǤ��ꡢ shadowing (�ƻ���) �ȸƤФ�뾮���� $x$ �ΰ��
���������Ϸ��Ǥ��롣
���θ��̤� $Q^2$ ȯŸ�β����Τʤ��ˤ����äƤ���ȹͤ���졢
���θ��̤� DGLAP �������˴ޤ᤿��������������Ƥ��� \cite{GLRMQ,PRQ}��
��äơ��ѡ��ȥ�Ʒ����Ѥ��Ƶ����븶�ҳ˹�¤�ؿ���
���ҳ���ѡ��ȥ�ʬ�ۤ� $Q^2$ ȯŸ��������ˤϡ�
�������������Ѥ��뤳�Ȥ�ɬ�פǤ��롣�������ʤ��顢
���ҳ˹�¤�ؿ��� $x$ ��¸���ˤĤ��ƤϤ��ʤ�ܤ������椬�ʤ���Ƥ���ˤ�
������餺�������������ο��Ͳ���������ʤɤˤĤ��ƤϤ��ޤ�
�ܤ�������Ϥʤ���Ƥ��ʤ���
��äơ����ҳ˹�¤�ؿ��� $Q^2$ ��¸���������������ǵ��ҤǤ���Τ�
�Ȥ������Ȥ϶�̣�Τ������Ǥ��롣
�äˡ�����μ¸��Ǥ��Τ褦�ʸ��̤򸡽Ф��뤳�Ȥ��Ǥ���ȹͤ�����Τǡ�
�������ˤ����ơ������������ο��Ͳ����뤳�ȤϽ��פʸ������Ǥ��롣
�ޤ����Ƕ� New Muon Collaboration (NMC) ��
��� $\partial [F^{Sn}_2/ F^C_2] / \partial \ln Q^2$ �ˤĤ��Ƥ����٤Τ褤
�ǡ����������Ƥ��� \cite{NMCSN-C}��
���������Ƥ��븶�ҳ���ѡ��ȥ�ʬ�ۤȾ嵭�Υѡ��ȥ�Ʒ��θ��̤�
���줿 $Q^2$ ȯŸ�������������η�̤������Ǥ���Τ��Ȥ������ˤĤ��Ƥ�
����ⶽ̣������

���Τ褦�ʾ�����դޤ��ơ��ܸ���Ǥϳ˻Ҥ���Ӹ��ҳˤι�¤�ؿ��� $Q^2$ ��¸��
�ˤĤ��ƿ��Ͳ�θ����Ԥä���������ʸ�ι����ϰʲ����̤�Ǥ��롣
�ޤ��裲�Ϥˤ����ơ�DGLAP �������ˤĤ��ƾܤ���������
³�����裳�ϤǸ��ҳ˹�¤�ؿ��� $Q^2$ ȯŸ�Ȥ��ơ�
�ѡ��ȥ�Ʒ��θ��̤�ޤ� $Q^2$ ȯŸ�������ˤĤ����������롣
����ˡ��裴�ϤǤ� brute-force ˡ�ȸƤФ����Ͳ�ˡ�ȡ������
�Ѥ��� FORTRAN �ץ������ˤĤ�����������
�裵�ϤǤ��Υץ��������Ѥ��Ʒ׻�������̤��͡��ʼ¸���̤Ȥ���Ӥ�Ԥ���
�����ƺǸ���裶�ϤǷ�����Ҥ٤롣


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�����Ǥ�Ҥ٤��褦�˳˻Ҥ�������¤�ˤĤ���Ĵ�٤뤿���ͭ�Ѥ���ˡ�Ȥ��ơ�
�����λ�������ˤĤ��ƹͤ��뤳�Ȥ��������롣�����ǡ�
�˻Ҥ�������¤�ˤĤ���Ĵ�٤뤿��ˡ��ŻҤ��ۻҤ�����������ˤĤ��ƹͤ��롣
�¸����ϤǤ��Żҡ��ۻ��������������ʬ�����Ѥϡ�

\begin{equation}
\frac{d^2\sigma}{d\Omega dE^{\prime}}
=\frac{\alpha^2}{4E^2\sin^4\frac{\theta}{2}}
\left[2W_1(\nu,Q^2)\sin^2\frac{\theta}{2}+W_2(\nu,Q^2)\cos^2\frac{\theta}{2}\right] 
\end{equation}

\begin{center}
\begin{tabular}{r l}
$E$ , $E^{\prime}$ : & �Ͼ��֤���ӽ����֤��ŻҤΥ��ͥ륮�� \\
$\theta$ : & ����� \\
$\nu$ : & ���ͥ륮���ܹ� $\nu=E-E^{\prime}$ \\
$Q^2$ : & ������ư�̰ܹԤ� $q$ �Ȥ��� $Q^2=-q^2$
\end{tabular}
\end{center}

\noindent
�Τ褦�˹�¤�ؿ� $W_1$ �� $W_2$ �ǽ�ɽ����� \cite{RGR,HM,GAUGE}��

�Ȥ����ǡ�$x=Q^2/2M\nu$ ��ͭ�¤��ݤ��ʤ��� 
$Q^2 \rightarrow \infty $, $\nu \rightarrow \infty$ ��
�˸¤�ͤ����Ȥ���$W_1$ �� $W_2$ ����
{
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MW_1(Q^2,\nu) & \rightarrow & F_1(x) \\
\nu W_2(Q^2,\nu) & \rightarrow & F_2(x) 
\end{eqnarray}
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\vspace{-0.5cm}
\noindent
�Τ褦�ˡ�$x$ �Τߤδؿ��Ȥ���ɽ�����Ȥ������Ȥ� Bjorken �ˤ�ä��󾧤��졢
����� Bjorken ��������󥰤ȸƤ� \cite{BJSC}��
�����Ƥ��� Bjorken ��������󥰤� SLAC ��
��� 1969 ǯ�˼¸�Ū�˳Τ����줿 \cite{SLACSC}��

$Q^2$ ���礭�����Ƥ����Ȥ������Ȥ�ʬ��ǽ�򤢤��뤳�Ȥ��������롣
��ä� $Q^2$ ��������ʬ��ǽ���㤤����ξ��ˤϡ�
�ۻҤ�������¤�򸫤뤳�ȤϤǤ��ʤ������Τ��ᡢ����β��۸��Ҥ�
�ۻ����Τ���ߺ��Ѥ��Ƥ��롣
������ $Q^2$ ���礭�����Ƥ����Ȥ䤬�Ʋ��۸��Ҥ��ۻҤ������ˤޤ�������ߡ�
�ۻҤϲ����졢�����֤ˤ����Ƥ�̵���Υϥɥ��󷲤���������롣
���Τ褦������������ˤʤ���ۻҤ�������¤�������Ϥ�롣
�����Ƥ���� $Q^2$ ���礭���������������٤������礭���ΰ��
���������ΰ�ȸƤӡ������ΰ�Ǥλ��������������ȸƤ֡�
Bjorken ��������󥰤μ�ĥ����Ȥ����ϡ����Τ褦�ʿ��������ΰ�ˤ����Ƥϡ�
$Q^2$ �򤢤���ʬ��ǽ�������Ƥ⡢����ʾ��������¤�������Ƥ��ʤ�
�Ȥ������ȤǤ��롣���ʤ���ۻ������ˤϹ�¤�Τʤ�����γ�Ҥ�¸�ߤ���
���������ΰ�Ǥϡ�
��ץȥ󤫤�β��۸��ҤϤ�������γ�Ҥ���ߺ��Ѥ򤷤Ƥ���Ȥ������ȤǤ��롣
��������γ�Ҥϥѡ��ȥ�ȸƤФ�롣

�Ȥ������ۻҤα�ư�̤�̵����Ȥ���Ϥˤ����Ƥϡ��ѿ� $x$ ��
�ۻҤ��Ф���ѡ��ȥ�λ��ı�ư������������뤳�Ȥ�������롣
���ʤ�����۸��Ҥ���ߺ��Ѥ���ѡ��ȥ�λ��ı�ư�̤� $P$ ��˻Ҥ�
��ư�̤Ȥ��� $xP$ �Ǥ��롣

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�˻������˹�¤������ʤ�����γ�Ҥ�¸�ߤ򼨤��� Bjorken ��������󥰤Ǥ��뤬��
���θ�ι��� $x$ �ΰ�ˤ錄�롢�����̩�ʼ¸��ˤ�ꡢ�鷺�����ˤ�Ƥ��뤳�Ȥ�
�狼�ä��ʿ� \ref{fig : fig2-1}�ˡ�
���θ��ݤ� ``��������󥰤��ˤ�" �ȸƤФ�롣�������ʤ��顢
����ϳ˻Ҥ�������¤��
�����������ѡ��ȥ��Ϸ��ˤ�����򤬸���Ǥ���Ȥ������Ȥ��̣����櫓�ǤϤʤ���
�˻������Υ������������롼����䳤���������α���ޤȤäƤ��뤳�Ȥ�
��������ȹͤ��뤳�Ȥ��Ǥ��롣
���ʤ�� $Q^2$ ���礭���ʤä�ʬ��ǽ�������ȡ����Τ褦�ʱ��������Ƥ���
�ȹͤ����롣
����ȡ�����Ū�ˤ� $Q^2$ ���礭���ʤ�ˤĤ��
�礭�� $x$ ����ĥѡ��ȥ󤬸��ꡢ
��꾮���� $x$ ����ĥѡ��ȥ������뤳�Ȥˤʤ롣
�� \ref{fig : fig2-1} �򸫤�ȳΤ��˾����� $x$ �Ǥ� $Q^2$ �����ä�
�Ȥ�˹�¤�ؿ������ä��Ƥ��ꡢ�礭�� $x$ �ˤ����Ƥ� $Q^2$ �����äˤ������ä�
��¤�ؿ��ϸ������Ƥ��뤳�Ȥ��狼�롣

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\label{fig : fig2-1}
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���Τ褦�ʥ�������󥰤��ˤ�򵭽Ҥ���������
�Ȥ��ơ�Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) ������
������ \cite{DGLAP}������ϰʲ��Τ褦������ʬ�������Ǥ��롣

\begin{equation}
{\partial \over {\partial t}} \ \widetilde{q}_{_{NS}} \left({x,t}\right)\
=\ \int_{x}^{1}{dy \over y}\ 
\widetilde{P}_{_{NS}}\left({{x \over y}}\right)\ 
           q_{_{NS}} \left({y,t}\right)\ 
\label{eqn : DGLAPNS}
\end{equation}

\begin{equation}
{\partial \over {\partial t}} \ \widetilde{q}_i^{\ +} \left({x,t}\right)\
=\ \int_{x}^{1}{dy \over y}\ 
\left[\ \sum_j \widetilde{P}_{q_i^+ q_j^+}\left({{x \over y}}\right)\ 
           \widetilde{q}_j^{\ +} \left({y,t}\right)\ 
{}+\  \widetilde{P}_{q^+ g}\left({{x \over y}}\right)\ 
\widetilde{g}\left({y,t}\right)\ \right]
\label{eqn : DGLAPQI}
\end{equation}

\begin{equation}
{\partial \over {\partial t}} \ \widetilde{q}_{_{S}} \left({x,t}\right)\
=\ \int_{x}^{1}{dy \over y}\ 
\left[\ \sum_i \sum_j \widetilde{P}_{q_i^+ q_j^+}\left({{x \over y}}\right)\ 
           \widetilde{q}_j^{\ +} \left({y,t}\right)\ 
{}+\  N_f \widetilde{P}_{q^+ g}\left({{x \over y}}\right)\ 
\widetilde{g}\left({y,t}\right)\ \right]
\label{eqn : DGLAPQS}
\end{equation}

\begin{equation}
{\partial \over {\partial t}} \ \widetilde{g}\left({x,t}\right)\ 
=\ \int_{x}^{1}{dy \over y}\ 
\left[\ \widetilde{P}_{gq}\left({{x \over y}}\right)\ 
\widetilde{q}_{_{S}} \left({y,t}\right)\ 
+\ \widetilde{P}_{gg}\left({{x \over y}}\right)\ 
\widetilde{g}\left({y,t}\right)\ \right]
\label{eqn : DGLAPG}
\end{equation}

\noindent
������ $t$ �ϡ�

\begin{equation}
t \equiv -{2\over \beta_0}\ln 
\biggl[{\alpha_s(Q^2)\over \alpha_s(Q_0^2)}\biggr]
\label{eqn : tq2}
\end{equation}

\noindent
�����������ѿ��Ǥ��ꡢ$\alpha_s(Q^2)$ �϶�����ߺ��Ѥμ¸��������Ǥ��롣
�ޤ���$q_i(x,t)$, $\bar{q}_i(x,t)$�򤽤줾��
�ե졼�С� $i$ ����ĥ�������ʬ�ۤ����ȿ��������ʬ�ۤȤ��ơ�
$q_i^{\ +}=q_i+\bar{q}_i$ �Ǥ��ꡢ
����� $q_{_{NS}}$, $q_{_{S}}$, $g$ �Ϥ��줾��
�ե졼�С��󣱽Ź�ʬ�ۡ��ե졼�С����Ź�ʬ�ۤ���ӥ��롼����ʬ�ۤ�ɽ����
������ $\widetilde{f}(x)=xf(x)$ �Ǥ��ꡢ
$N_f$ �ϥե졼�С��ο��Ǥ��롣

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\epsfile{file=fig2-2.eps,width=13.0cm}
\end{center}
\caption{DGLAP �������ˤ�����ʬ���β��� (LO)}
\label{fig : fig2-2}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�����ǡ�$P(y/x)$ ��ʬ���ؿ��ȸƤФ���Τǡ�
�ä� (\ref{eqn : DGLAPQI}) $\sim$ (\ref{eqn : DGLAPG}) ��
�� $P_{ij}(y/x)$ �ϱ�ư���� $y$ ����ĥѡ��ȥ� $j$ ��ʬ�����ơ�
��ư���� $x$ ����ĥѡ��ȥ� $i$ �ˤʤ��Ψ��ɽ���Ƥ��롣
��äơ�$\alpha_s$ �κ��㼡 (Leading Order,  �ʸ� LO) �Τߤ�ͤ�����ˤϡ�
(\ref{eqn : DGLAPQI}) ����� (\ref{eqn : DGLAPQS}) ����
�����裱�प����裲���
���줾��� \ref{fig : fig2-2} �� (a), (b) ��ɽ����(\ref{eqn : DGLAPG}) ����
�����裱�प����裲��Ϥ��줾�� (c), (d) �β�����ɽ���Ƥ��롣
�ޤ���(\ref{eqn : DGLAPNS}) ���α��դϿ� \ref{fig : fig2-2} �� (a) �β�����ɽ���� 
��ä� LO �Τߤξ��ˤϡ�(a) �β�����
�ե졼�С����Ѥ�뤳�ȤϤʤ��Τǡ�

\begin{equation}
P_{q_i^+ q_j^+} = \delta_{ij} \ P_{_{NS}}
\end{equation}

\noindent
�Ǥ��롣��äơ�
���ξ��ˤ� (\ref{eqn : DGLAPQI}) ����� (\ref{eqn : DGLAPQS}) ����
����˴�ñ�ˡ�

\begin{eqnarray}
{\partial \over {\partial t}} \ \widetilde{q}_i^{\ +} \left({x,t}\right)
& = & \int_{x}^{1}{dy \over y}\ 
\left[\ \widetilde{P}_{_{NS}}\left({{x \over y}}\right)\ 
           \widetilde{q}_i^{\ +} \left({y,t}\right)\ 
{}+\  \widetilde{P}_{q^+ g}\left({{x \over y}}\right)\ 
\widetilde{g}\left({y,t}\right)\ \right] \\
{\partial \over {\partial t}} \ \widetilde{q}_{_{S}} \left({x,t}\right)
& = & \int_{x}^{1}{dy \over y}\ 
\left[\ \widetilde{P}_{_{NS}}\left({{x \over y}}\right)\ 
           \widetilde{q}_{_{S}} \left({y,t}\right)\ 
{}+\  N_f \widetilde{P}_{q^+ g}\left({{x \over y}}\right)\ 
\widetilde{g}\left({y,t}\right)\ \right]
\end{eqnarray}

\noindent
�Ƚ񤯤��Ȥ��Ǥ��롣
��������$q_i^+$ ʬ�ۤ˴ؤ��Ƥϡ����롼����ʬ�ۤ�������
�˥ե졼�С����Ź�ʬ�� $q_{_{S}}$ ���ޤޤ�Ƥ��뤿��ˡ�
����ʬ�ۤ�ñ�ȤǤϷ׻����뤳�ȤϤǤ�����
�ե졼�С����Ź�ʬ�� $q_{_{S}}$ ��Ʊ���˷׻����ʤ���Фʤ�ʤ���

�������������ˤ��С����� $Q^2$ �ˤ�����ѡ��ȥ�ʬ�ۤ��狼��С�
������Ȥ�Ǥ�դ� $Q^2$ �ǤΥѡ��ȥ�ʬ�ۤ����뤳�Ȥ��Ǥ��롣
���Τ��Ȥ��餳������������ $Q^2$ ȯŸ�������ȸƤФ�롣
���Τ褦�ˤ���������ѡ��ȥ�ʬ�ۤ� LO �ξ��ˤϡ�

\begin{equation}
F_2(x, Q^2) = \sum_i e_i^2 \widetilde{q}_i^{\ +}(x, Q^2)
\end{equation}

\noindent
�δط����Ѥ��ƹ�¤�ؿ� $F_2$ ��ľ�����Ȥ��Ǥ��롣������ $e_i$ ��
�ե졼�С� $i$ �Υ��������������Ų٤Ǥ��롣
��ä� DGLAP �������ǥѡ��ȥ�ʬ�ۤ� $Q^2$ ��¸����Ĵ�٤�С�
��¤�ؿ��� $Q^2$ ��¸��������졢�¸��ͤ���Ӥ��뤳�Ȥ��Ǥ��롣
�ޤ���¤�ؿ� $F_1$ �� $F_2$ �ˤϡ�LO �Ǥ�

\begin{equation}
F_2(x, Q^2)  =  2x F_1(x, Q^2)
\end{equation}

\noindent
�Ȥ����ط�������Ω�äƤ���Τǡ��ɤ��餫���������򰷤��Ф褤���Ȥ��狼�롣
���δط��� Callan-Gross �δط��Ȥ�Ф�롣

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
{\renewcommand{\thesection}{\it {\arabic{section}}}
\subsection{{\it $\alpha_s$ �ι⼡��δ�Ϳ}}}
\vspace{0.4cm}

�ܸ���Ǥ��ä� $\alpha_s$ �ι⼡��δ�Ϳ�����줿 DGLAP �������ˤĤ���
�⸦���Ԥä� \cite{HWW,NLO}������ϣ��Ĥ���ʬ�˴ޤޤ�Ƥ��롣
�ޤ��裱�ˤ� $\alpha_s$ ���Τ�Τ����äƤ��롣
���ʤ�� LO �ξ��ˤϡ�

\begin{equation}
\alpha_s^{LO}(Q^2) = \frac{4\pi}{\beta_0 \ln (Q^2/\Lambda^2)} 
\end{equation}

\noindent
�Ǥ��뤬���裱���ι⼡�� (Next-to-Leading Order, �ʸ� NLO) ��
��Ϳ��ޤ���ˤϡ�

\begin{equation}
\alpha_s^{NLO}(Q^2)=\frac{4\pi}{\beta_0 \ln(Q^2/\Lambda^2)}
\left[1-\frac{\beta_1 
\ln \ln(Q^2/\Lambda^2)}{\beta_0^2\ln(Q^2/\Lambda^2)}\right]   
\end{equation}

\noindent
�Ȥʤ롣�Ȥ����� NLO �δ�Ϳ��ͤ�����ˤϡ�������߽�����
���ꤷ�ʤ���Фʤ�ʤ���
����ʸ�Ǥ� modified minimal subtruction scheme ($\overline{MS}$) ��
�ƤФ��������Ѥ�����
�ޤ���$\beta_0$ �� $\beta_1$ �Ϥ��줾��ʲ��μ����������Ƥ��롣

\begin{equation}
\beta_0=\frac{11}{3}C_G-\frac{4}{3}T_R N_f  \ \ \ , \ \ \
\beta_1=\frac{34}{3}C^2_G-\frac{10}{3}C_G N_f-2C_F N_f 
\end{equation}

\noindent
��������

\begin{equation}
C_G=N_c \ \ \ , \ \ \ C_F=\frac{N_c^2-1}{2N_c} \ \ \ , \ \ \ T_R=\frac{1}{2} 
\end{equation}

\noindent
�Ǥ��ꡢ$N_f$ , $N_c$ �Ϥ��줾��ե졼�С��ο��ȥ��顼�ο� $(N_c=3)$ �Ǥ��롣

�裲�� DGLAP �����������äƤ��롣
���������δ�Ϳ�� (\ref{eqn : DGLAPNS}) $\sim$ (\ref{eqn : DGLAPG}) ����
�����Ѥ�����ʬ���ؿ���
�ʤ��˴ޤ�뤳�Ȥ��Ǥ��롣
���ʤ����(\ref{eqn : DGLAPNS}) $\sim$ (\ref{eqn : DGLAPG}) ���ˤ�����
ʬ���ؿ� $\widetilde{P}(x)$ �ϡ�

\begin{equation}
\widetilde{P}(x)=\widetilde{P}^{(0)}(x)
{}+{{\alpha_s(Q^2)} \over{2\pi}} R(x)
\end{equation}

\noindent
�Ȥʤ롣�����ǡ�$R(x)$ �ϡ�

\begin{equation}
R (x)\equiv \widetilde{P}^{(1)}(x)-{\beta_1\over 2\beta_0}
\widetilde{P}^{(0)}(x) 
\label{eqn : R}
\end{equation}

\noindent
���������롣
��������$P^{(0)}(x)$ �� LO ��ʬ���ؿ��� $P^{(1)}(x)$ �� NLO �δ�Ϳ�Ǥ��롣
���� (\ref{eqn : R}) ���α����裲��� $Q^2$ ������� (\ref{eqn : tq2}) ����
�������� $t$ ���Ѥ��Ƥ��뤳�Ȥˤ���ΤǤ��롣
LO ����� NLO ��ʬ���ؿ�����Ͽ A �˼�����

������裳�ˡ��ѡ��ȥ�ʬ�ۤ��鹽¤�ؿ��ˤ���Ȥ��ˤ� NLO �δ�Ϳ��
���äƤ��롣��¤�ؿ��ȥѡ��ȥ�ʬ�ۤδط��ϡ�

\begin{equation}
\widetilde{F}_{n}^{NS}(x,{Q}^{2})\ =\ 
\int_{x}^{1}{dy \over y}\ 
\widetilde{C}_{n}^{q}\left({{x \over y},{\alpha }_{s}}\right)\ 
\widetilde{q}_{n}^{NS}(y,{Q}^{2})
\label{eqn : QFNS}
\end{equation}

\begin{equation}
\widetilde{F}_{n, i}^{\ +} (x,{Q}^{2})\ =\ 
\int_{x}^{1}{dy \over y}\ 
\widetilde{C}_{n}^{q}\left({{x \over y},{\alpha }_{s}}\right)\ 
\widetilde{q}_{i}^{\ +} (y,{Q}^{2})
\ + \ 
\int_{x}^{1}{dy \over y}\ 
\widetilde{C}_n^{g}\left({{x \over y},{\alpha }_{s}}\right)\ 
\widetilde{g}(y,{Q}^{2})
\label{eqn : QFI}
\end{equation}

\noindent
��ɽ�����Ȥ��Ǥ��롣
�����ǡ�ź�� $n$ �� $F_1$, $F_2/x$, $F_3$ �Τ����줫��
������$i$ �ϥե졼�С���ɽ����
�����Ʒ����ؿ� $C_n^{q,g}$ �� NLO �δ�Ϳ�����äƤ��롣�����ؿ��ϡ�

\begin{eqnarray}
C_n^{q}(x,\alpha_s) & = & \delta(1-x)  + 
                {{\alpha_s}\over{4\pi}} B_n^{q}(x) + O(\alpha_s^2) \\
C_n^{g}(x,\alpha_s) & = &  
                {{\alpha_s}\over{4\pi}} B_n^{g}(x) + O(\alpha_s^2)
\end{eqnarray}

\noindent
�Ȥ������򤷤Ƥ��롣$B_n^{q}(x)$, $B_n^{g}(x)$ ����Ͽ B �˼�����
��äơ�LO �ξ��ˤϡ�

\begin{eqnarray}
\widetilde{F}_{n}^{NS}(x,{Q}^{2}) & = & \widetilde{q}_{n}^{NS}(x,{Q}^{2})
\label{eqn : QEFNS} \\
\widetilde{F}_{n, i}^{+} (x,{Q}^{2}) & = & \widetilde{q}_{i}^{\ +} (x,{Q}^{2})
\label{eqn : QEFI} 
\end{eqnarray}

\noindent
�Ȥʤäƹ�¤�ؿ��ϥ�������ʬ�ۤ��Τޤޤ�ɽ����뤬��
$Q^2$ ȯŸ�� NLO �δ�Ϳ��ޤ����ˤϡ���¤�ؿ���
ľ���Ȥ��� $B_n^{q}(x)$, $B_n^{g}(x)$ �δ�Ϳ������ʤ���Фʤ�ʤ���
�����Ƥ��θ���㤨�� $F_2$ ��

\begin{equation}
F_2(x, Q^2) = \sum_i e_i^2 {F}_{2, i}^{+}(x, Q^2)
\end{equation}

\noindent
�Ȥ��Ƶ��뤳�Ȥˤʤ롣

�ޤ����Τ��Ȥ��顢���� $Q_0^2$ �Ǥι�¤�ؿ���Ϳ����
��¤�ؿ����Ȥ� $Q^2$ ȯŸ��������ˤϡ�
�����;ʬ�ʹबʬ���ؿ������äƤ��롣
��¤�ؿ����Ф��롢NLO �δ�Ϳ��ޤ� DGLAP ��������
�ʲ��Τ褦��ɽ����롣

\begin{equation}
{\partial \over {\partial t}}
\ \widetilde F_{_{NS}}^{\pm}(x,t)\ = \ \int_x^1{dy\over y} \ 
\biggl [ \ 
\widetilde P_{_{NS}}^{(0)}\biggl({x\over y}\biggr)
        +{{\alpha_s(t)} \over{2\pi}} R_{q^\pm ,NS}\biggl({x\over y}\biggr) \ 
\biggr ] \ 
\widetilde F_{_{NS}}^{\pm}(y,t)
\end{equation}

\begin{equation}
{\partial \over {\partial t}}
\left({\begin{array}{c}{\widetilde {F}_i}^{+}\left({x,t}\right)\\
\widetilde {g}(x,t)\end{array}}\right)\ 
=\ \int_{x}^{1}{\frac{dy}{y}}\ 
\left[ {\widetilde {\bf P}}^{\left({0}\right)} \left(x \over y \right)
\ +\ {\frac{{\alpha }_{s}\left({t}\right)}{2\pi }}\ 
{\bf R} \left(x \over y \right) \right]
\ \left({\begin{array}{c}{\widetilde {F}_j}^{+}\left({y,t}\right)\\
\widetilde {g}(y,t)\end{array}}\right)
\label{eqn : 1STEP}
\end{equation}

\noindent
�����ǡ�$F_{_{NS}}^{\pm}$ �� $q \pm \bar{q}$ ���Υե졼�С��󣱽Ź��
��¤�ؿ���ɽ��������ˡ�

\begin{equation}
R_{q^\pm ,NS} (x) = \widetilde{P}_{q^\pm ,NS}^{(1)}(x)
-{\beta_1\over 2\beta_0}
\widetilde{P}_{_{NS}}^{(0)}(x)  
- {{\beta_0} \over 4} \widetilde B_n^{q} (x)
\end{equation}

\begin{equation}
{\bf R} (x) \ = \ 
{\widetilde {\bf P}}^{\left({1}\right)} (x)
\ -\ {\frac{{\beta }_{1}}{2{\beta }_{0}}}\ 
{\widetilde {\bf P}}^{\left({0}\right)} (x)
\ -\ {\frac{{\beta }_{0}}{4}}\ 
\widetilde {\bf D} ^{\left({1}\right)} (x)
\ +\ \widetilde {\bf E} (x)
\end{equation}

\noindent
�Ǥ��롣${\bf P}^{(0),(1)}(x)$, ${\bf D} ^{(1)}(x)$, ${\bf E}(x)$ ��
��Ͽ A �˼�����
(\ref{eqn : QEFNS}) ����� (\ref{eqn : QEFI}) ���Ǽ������褦�� LO �ξ��ˤϡ�
��¤�ؿ��ϥ�������ʬ�ۤ��������Τǡ����Τ褦��;ʬ�ʹ�ϸ���Ƥ��ʤ���
��äƤ��ξ��ˤϡ��դĤ��� LO �� DGLAP �������Ƿ׻����뤳�Ȥ��Ǥ��롣
�����ͤ˹�¤�ؿ����Ȥ� $Q^2$ ȯŸ��������ˡ��
�����ƥå׷����ȸƤ� \cite{HWW,NLO}���ѡ��ȥ�ʬ�ۤ� $Q^2$ ȯŸ������
���θ�ǹ�¤�ؿ���ľ����ˡ�򣲥��ƥå׷����ȸƤ֡�


\vfill\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{{\bf ���ҳ˹�¤�ؿ��� $Q^2$ ȯŸ}}
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
{\renewcommand{\thesection}{\it {\arabic{section}}}
\subsection{{\it EMC ����}}}
\vspace{0.4cm}

�˻Ҥȥ�ץȥ�ο�����������ˤ����ơ�ɸŪγ�Ҥ�˻Ҥ��鸶�ҳˤ��Ѥ���ȡ�
���ҳ˹�¤�ؿ������뤳�Ȥ��Ǥ��롣���θ��ҳ˹�¤�ؿ����˻Ҥι�¤�ؿ���
�������ʤ����Ȥ� EMC �ˤ�ä� 1983 ǯ��ȯ�����졢
EMC ���̤ȸƤФ�Ƥ��� (�� \ref{fig : fig3-1}) \cite{AEMC}��
���ʤ����$F_2^A$ , $F_2^D$ �򤽤줾�츶�ҳ˹�¤�ؿ������ۻҤ�
��¤�ؿ��Ȥ��ơ�

\begin{equation}
R_2 \equiv \frac{F_2^A(x,Q^2)}{F_2^D(x,Q^2)} \neq 1
\end{equation}

\noindent
�Ǥ��롣�����������ۻҤϼ�ͳ�ʳ˻Ҥȶ���Ǥ���ȹͤ��뤳�Ȥ��Ǥ��롣

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\hspace{-1.0cm}
\epsfile{file=fig3-1.eps,width=12.0cm}
\end{center}
\vspace{-6.0cm}
\caption{EMC ����}
\label{fig : fig3-1}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vfill\eject

�� \ref{fig : fig3-1} ����狼��褦�ˡ�
���ҳ���ˤ����� $F_2$ ���Ѳ��ϰʲ��Σ��Ĥ��ΰ��ʬ���뤳�Ȥ��Ǥ��롣

\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item
$x$ \raisebox{-1mm}{$\stackrel{\textstyle <}{\sim}$} $0.05$ ��
�ΰ�ˤ����� $R_2$ �� $1$ ��꾮�����ʤ롣����� shadowing �ȸƤ֡�
\item
$0.1$ \raisebox{-1mm}{$\stackrel{\textstyle <}{\sim}$} $x$ 
\raisebox{-1mm}{$\stackrel{\textstyle <}{\sim}$} $0.25$
���ΰ�Ǥ� $R_2 > 1$ �Ȥʤ롣
���θ��ݤ� shadowing ���Ф��� antishadowing �ȸƤФ�롣
\item
$x=0.25 \sim 0.30$ ���ΰ�� $R_2 \simeq 1$ ��
\item
$0.3$ \raisebox{-1mm}{$\stackrel{\textstyle <}{\sim}$} $x$ 
\raisebox{-1mm}{$\stackrel{\textstyle <}{\sim}$} $0.8$
���ΰ�Ǥ� $R_2 < 1$ ��
\item
$x \rightarrow 1$ �ˤĤ�� $R_2$ �� $1$ ��ۤ�ñĴ�����ä��롣
\end{enumerate}

\noindent
�����ơ������θ��ݤ��������뤿���
�͡����Ϸ����󾧤���Ƥ��롣
������ǡ������� $x$ �ΰ� (a) �ˤ����� shadowing �����������Ϸ��Ȥ��ơ�
�ѡ��ȥ�Ʒ�� (parton recombination) ������ \cite{GLUE}--\cite{RGR}��

��ư���� $x$ ����ĥѡ��ȥ�ν������α�ư�ΰ� $\Delta L$ ��
 $P$ ��˻Ҥα�ư�̤Ȥ��Ƥ��褽 $\Delta L \sim 1/xP$ �Ǥ��ꡢ$x$ ��
 �������ʤ�Фʤ�ۤ��礭���ʤäƤ�����
���α�ư�ΰ褬��������ʿ�ѳ˻Ҵֵ�Υ��ۤ���褦�ʾ����� $x$ ����ĥѡ��ȥ�ϡ�
¾�γ˻ҤΥѡ��ȥ�ȽŤʤ�礤����Ϥ伫ͳ�ʥѡ��ȥ�ȸ��ʤ����ȤϤǤ��ʤ��� 
�ѡ��ȥ�Ʒ��Ȥϡ������ξ����� $x$ ����ĥѡ��ȥ�ɤ�������ߺ��Ѥ���
ͻ�礹������Ǥ��롣���Τ褦�ʥѡ��ȥ�Ʒ�礬������ȡ�
������ $x$ ����ĥѡ��ȥ�ɤ�����ͻ�礷�Ƥ���ʬ�ۤ����뤿�ᡢ
�����ΰ�ǹ�¤�ؿ��⾮�����ʤ롣
��äƾ����� $x$ �ΰ�� $F_2^A/F_2^D < 1$ �Ȥʤꡢ
shadowing ��������ȹͤ����롣

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
{\renewcommand{\thesection}{\it {\arabic{section}}}
\subsection{{\it $Q^2$ ȯŸ�˴ؤ���ѡ��ȥ�Ʒ��θ���}}}
\vspace{0.4cm}

shadowing ���������뤿���Ƴ�����줿�ѡ��ȥ�Ʒ��ϡ�
$Q^2$ ȯŸ�β����Τʤ��ˤ����äƤ���ȹͤ��뤳�Ȥ��Ǥ������θ��̤�
���줿 $Q^2$ ȯŸ��������������Ƥ��� \cite{GLRMQ,PRQ}��
������������ $Q^2$ ��������ˤ�������������礭����ư���� $x$ ����ĥѡ��ȥ�
������ $x$ �Υѡ��ȥ��ʬ�����Ƥ���������
�׻����Ƥ��� DGLAP �������ˡ��� \ref{fig : fig3-2} �Τ褦��
������ $x$ ���ĥѡ��ȥ�ɤ�����ͻ�礷�Ƥ���������������Ȥ��ƴޤ�Ǥ��롣
��äơ��ѡ��ȥ�Ʒ��ˤ�������븶�ҳ˹�¤�ؿ���
���ҳ���ѡ��ȥ�ʬ�ۤ� $Q^2$ ��¸���ˤĤ��ƹͤ�����ˤϡ�
�������������Ѥ��Ʒ׻���Ԥ����Ȥ�ɬ�פǤ��롣

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\vspace*{-1.0cm}
\epsfile{file=fig3-2.eps,width=13.0cm}
\end{center}
\caption{�ѡ��ȥ�Ʒ��β���}
\label{fig : fig3-2}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

����Ū�ˤϡ��ޤ�����ͳ�ʳ˻���ˤ�����ѡ��ȥ�ʬ�ۤ��Ф��ơ����ҳ����
�ѡ��ȥ�ʬ�ۤ� $\delta \widetilde p^A(x,t)$ ���������������ȹͤ��롣
���ʤ����

\begin{equation}
\widetilde p^A(x,t)=\widetilde p(x,t)-\delta \widetilde p^A(x,t) 
\label{eqn : PA}
\end{equation}

\noindent
�Ǥ��롣�����������ҳ���ѡ��ȥ�ʬ�ۤϼ��̿� $A$ �ǳ�ä���ΤȤ������������
�����ơ���ͳ�ʳ˻���Υѡ��ȥ�ʬ�� $\widetilde p(x,t)$ ��
������ $\delta \widetilde p^A(x,t)$ �򤽤줾�� $Q^2$ ȯŸ������
�Ǹ�� (\ref{eqn : PA}) ���ˤ�ä� $Q^2$ ȯŸ�������ҳ���Υѡ��ȥ�ʬ�ۤ����롣

�˻���Υѡ��ȥ�ʬ�ۤ�ºݤˤϳ˻���ǤΥѡ��ȥ�Ʒ��θ��̤�
������ȹͤ����롣
����ϡ��˻���Ǥξ����� $x$ ����ĥѡ��ȥ�ɤ����νŤʤ�礤�ˤ�äƵ�����
ͻ������Ǥ��롣���θ��̤����줿 $Q^2$ ȯŸ�������ϰʲ��Τ褦�ˤʤ롣

\begin{eqnarray}
& & {\partial \over {\partial t}} \ 
\widetilde {q}_i^{\ +}\left({x,t}\right)\ =\ 
\int_{x}^{1}{dy \over y}\ 
\left[\ 
\widetilde {P}_{q^+, NS}\left({{x \over y}}\right)\ 
\widetilde q_i^{\ +} \left({y,t}\right)\ 
{}+\ 2C_{_F} T_R \widetilde F_{qq} \left({{x \over y}}\right)\ 
\widetilde q_{_S} \left({y,t}\right)\ 
\right.  \nonumber \\
& & \left.
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ 
{}+\  \widetilde {P}_{qg}\left({{x \over y}}\right)\ 
\widetilde g\left({y,t}\right)\ \right] \nonumber \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
{}-\ 2\ {K \over \pi{R_0}^{2}}\ {1 \over {Q}^{2}}\ 
\left\{{{(2\pi)^2 {\alpha }_s (t) 
\over {\rm N}_{\rm c}\left({{\rm N}_{\rm c}^{\rm 2}\rm -1}\right)}\ 
\left({{4 \over 15}{N}_{c}^{2}\ -\ {3 \over 5}}\right)\ 
{\left[{\widetilde g\left({x,t}\right)}\right]}^{2}}
\right.  \nonumber \\
& & 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ 
\left.
{}+\ \int_{x}^{1}{dy \over y}\ 
\widetilde {\bar P}_{qg}\left({{x \over y}}\right)\ 
\widetilde {G}_{HT}\left({y,t}\right) \ 
\right\} 
\label{eqn : PRQI} \\
& & \nonumber \\
& & {\partial \over {\partial t}} \ \widetilde {q}_{_S}\left({x,t}\right)\ =\ 
\int_{x}^{1}{dy \over y}\ 
\left[\ 
\left\{\ \widetilde {P}_{q^+, NS} \left({{x \over y}}\right)\  
{}+\ 2 N_f C_{_F} T_R \widetilde F_{qq} \left({{x \over y}}\right)\ \right\}\
\widetilde q_{_S} \left({y,t}\right)\ 
\right. \nonumber \\
& & \left.
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ 
{}+\  N_f \widetilde {P}_{qg}\left({{x \over y}}\right)\ 
\widetilde g\left({y,t}\right)\ \right] \nonumber \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
{}-\ 2 N_f\ {K \over \pi{R_0}^{2}}\ {1 \over {Q}^{2}}\ 
\left\{{{(2\pi)^2 {\alpha }_s (t) 
\over {\rm N}_{\rm c}\left({{\rm N}_{\rm c}^{\rm 2}\rm -1}\right)}\ 
\left({{4 \over 15}{N}_{c}^{2}\ -\ {3 \over 5}}\right)\ 
{\left[{\widetilde g\left({x,t}\right)}\right]}^{2}}
\right.  \nonumber \\
& &
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ 
\left.
{}+\ \int_{x}^{1}{dy \over y}\ 
\widetilde {\bar P}_{qg}\left({{x \over y}}\right)\ 
\widetilde {G}_{HT}\left({y,t}\right) \ 
\right\} 
\label{eqn : PRQS} \\
& & \nonumber \\
& & {\partial \over {\partial t}} \ 
\widetilde g\left({x,t}\right)\ =\ 
\int_{x}^{1}{dy \over y}\ 
\left[{\ \widetilde {P}_{gq^+}\left({{x \over y}}\right)\ 
\widetilde q_{_S}\left({y,t}\right)\ 
{}+\  \widetilde {P}_{gg}\left({{x \over y}}\right)\ 
\widetilde g\left({y,t}\right)\ }\right]  \nonumber \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \
{}-\ {K \over \pi{R_0}^{2}}\ {1 \over {Q}^{2}}\ 
\left\{{{8{\pi }^{2} \over {N}_{c}^{2}-1} \ 
\alpha_s (t) \ {C}_{G}^2 \ 
\int_{x}^{1}{dy \over y}\ 
{\left[{\widetilde g\left({y,t}\right)}\right]}^{2}}\right\}
\label{eqn : PRG}
\end{eqnarray}

\noindent
���������ѡ��ȥ�Ʒ��θ��̤� NLO ��Ʊ�� $\alpha_s$ ��
���������Ȥ������äƤ���Τǡ���Ȥ� DGLAP �������� NLO �δ�Ϳ��ޤ��
�׻��򤹤�ɬ�פ����롣
�����μ��α��դ��裱�����ʬ��ʬ�� NLO �δ�Ϳ��ޤ��̾�� DGLAP ������
�Ǥ��ꡢ����¾�ιब�ѡ��ȥ�Ʒ��θ��̤�ɽ����
��������(\ref{eqn : PRQI}), (\ref{eqn : PRQS}) ���������
�� \ref{fig : fig3-2} (a) �� (b) �β�����ɽ����(\ref{eqn : PRG}) ��
������� (c) �β�����ɽ����
�����ǡ�NLO �δ�Ϳ��ޤ�ʬ���ؿ� $P_{q_i^+q_j^+}$ ����

\begin{eqnarray*}
P_{q_i^+ q_j^+}^{(0)}(x) & = & \delta_{ij} \ P_{NS}^{(0)}(x) \\
P_{q_i^+ q_j^+}^{(1)}(x) & = & \delta_{ij} \ P_{q^+, NS}^{(1)}(x) + 
2 C_F T_R F_{qq}(x)
\end{eqnarray*}

\noindent
��ɽ���뤳�Ȥ򤢤����Ѥ��Ƥ��� (��Ͽ A ����)��
�����ꡢ$q_i^{\ +}$ ��ʬ�ۤ�׻�������ˡ�
���٤ƤΥե졼�С� $i$ ���Ф��� $q_i^{\ +}$ ��ɬ�פǤϤʤ��������¤Ǥ���
�ե졼�С����Ź�ʬ�� $q_{_{S}}$ ����Ʊ���˷׻�����Ф褤���Ȥ��狼�롣
������ $P_{_{NS}}^{(0)}(x)$, $P_{q^+, NS}^{(1)}(x)$ , $F_{qq}(x)$ ��
��Ͽ A �˼�����
���������˸���� ${\bar P}_{qg}(x)$ �ϡ�
 
\begin{equation}
\bar {P}_{qg}(x) = 2 ~(-2x+15x^2-30x^3+18x^4)
\end{equation}

\noindent
�Ǥ��롣����ˡ�$G_{HT}(x,t)$ �Ϲ⼡���롼����ʬ��
�ȸƤФ��ʬ�ۤǡ��� \ref{fig : fig3-2} (b) �˼������
�����Τʤ��˸���Ƥ���ʬ�ۤǤ��롣����ʬ�ۤ� $Q^2$ ȯŸ�ϡ�

\begin{equation}
{\partial \over {\partial t}}\ 
\widetilde {G}_{HT}\left({x,t}\right)\ =\ 
{8{\pi }^{2} \over {N}_{c}^{2}-1} \ 
\alpha_s(t)\ {C}_{G}^2\ 
\int_{x}^{1}{dy \over y}\ 
{\left[{\widetilde g\left({y,t}\right)}\right]}^{2}
\end{equation}

\noindent
���������˽�����
������������򤯤���ˤ� $\widetilde G_{HT}$ �ν��ʬ�ۤ�ɬ�פǤ��뤬��
����ʬ�ۤ���ĤΥ��롼�������ߺ��Ѥ˴ط����뤳�Ȥ��顢

\begin{equation}
\widetilde G_{HT}(x,Q_0^2)\ =\ K_{HT} [\widetilde g(x,Q_0^2)]^2
\label{eqn : HDG}
\end{equation}

\noindent
�Ȳ��ꤹ�롣������ $K_{HT}$ �ϼ�ͳ�ʥѥ�᡼���Ǥ��ꡢ
�����Ǥϻ���ʸ�� \cite{PRQ} �˽��� $K_{HT}=1.68$ �Ȥ�����

����� $\delta \widetilde p^A(x,t)$ �� $Q^2$ ȯŸ�� DGLAP ��������
���ҳ���Υѡ��ȥ�Ʒ��θ��̤����줿�������˽�����
����Ū���������η��ϰʲ��Τ褦�ˤʤ롣

\begin{eqnarray}
& &{\partial \over {\partial t}} \ 
\delta \widetilde {q}_i^{\ +,A}\left({x,t}\right)\ =\ 
\int_{x}^{1}{dy \over y}\ 
\left[\ 
\widetilde {P}_{q^+,\  NS}\left({{x \over y}}\right)\ 
\delta \widetilde q_i^{\ +,A} \left({y,t}\right)\ 
{}+\ 2C_{_F} T_R \widetilde F_{qq} \left({{x \over y}}\right)\ 
\delta \widetilde q_{_S}^{^{A}} \left({y,t}\right)\ 
\right. \nonumber \\
& & \left.
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
{}+\  \widetilde {P}_{qg}\left({{x \over y}}\right)\ 
\delta \widetilde g^{^{A}} \left({y,t}\right)\ \right] \nonumber \\
& & \ \ \ \ \ \ \ \ \ \ \ 
{}+\ 2 
\left({3 \over 2}R_A\overline{n}\ -\ {K \over \pi {R_0}^{2}}\right)\ 
{1 \over {Q}^{2}}\ \left\{{{(2\pi)^2 {\alpha }_s (t) 
\over {\rm N}_{\rm c}\left({{\rm N}_{\rm c}^{\rm 2}\rm -1}\right)}\ 
\left({{4 \over 15}{N}_{c}^{2}\ -\ {3 \over 5}}\right)\ 
{\left[{\widetilde g\left({x,t}\right)}\right]}^{2}}
\right.  \nonumber \\
& & 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.
{}+\ \int_{x}^{1}{dy \over y}\ 
\widetilde {\bar P}_{qg}\left({{x \over y}}\right)\ 
\widetilde {G}_{HT}\left({y,t}\right) \ 
\right\} \label{eqn : PRDQI} \\
& & \nonumber \\
& & {\partial \over {\partial t}} \ 
\delta \widetilde {q}_{_S}^{^{\ A}} \left({x,t}\right)\ =\ 
\int_{x}^{1}{dy \over y}\ 
\left[\ 
\left\{\ \widetilde {P}_{q^+, NS}\left({{x \over y}}\right)\  
{}+\ 2 N_f C_{_F} T_R \widetilde F_{qq} \left({{x \over y}}\right)\ \right\}\
\delta \widetilde q_{_S}^{^{A}} \left({y,t}\right)\ 
\right. \nonumber \\
& & \left.
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
{}+\  N_f \widetilde {P}_{qg}\left({{x \over y}}\right)\ 
\delta \widetilde g^{^{A}} \left({y,t}\right)\ \right] \nonumber \\
& & \ \ \ \ \ \ \ \ \ \ \ 
{}+\ 2 N_f
\left({3 \over 2}R_A\overline{n}\ -\ {K \over \pi {R_0}^{2}}\right)\ 
{1 \over {Q}^{2}}\ \left\{{{(2\pi)^2 {\alpha }_s (t) 
\over {\rm N}_{\rm c}\left({{\rm N}_{\rm c}^{\rm 2}\rm -1}\right)}\ 
\left({{4 \over 15}{N}_{c}^{2}\ -\ {3 \over 5}}\right)\ 
{\left[{\widetilde g\left({x,t}\right)}\right]}^{2}}
\right.  \nonumber \\ 
& & 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.
{}+\ \int_{x}^{1}{dy \over y}\ 
\widetilde {\bar P}_{qg}\left({{x \over y}}\right)\ 
\widetilde {G}_{HT}\left({y,t}\right) \ 
\right\}
\end{eqnarray}

\vfill\eject

\begin{eqnarray}
& & {\partial \over {\partial t}} \ 
\delta \widetilde {g}^{^{\ A}}\left({x,t}\right)\ =\ \int_{x}^{1}{dy \over y}\ 
\left[{\ 
\widetilde {P}_{g q^+}\left({{x \over y}}\right)\ 
\delta \widetilde {q}_{_S}^{^A}\left({y,t}\right)\ +\ 
\widetilde {P}_{gg}\left({{x \over y}}\right)\ 
\delta \widetilde {g}^{^{A}}\left({y,t}\right)\ }\right]
\ \ \ \ \ \ \nonumber \\
& & \ \ \ \ \ \ \ \ \ \ \
{}+\ \left({3 \over 2}R_A\overline{n}\ -\ {K \over \pi {R_0}^{2}}\right)\ 
{1 \over {Q}^{2}}\ \left\{{{8{\pi }^{2} \over {N}_{c}^{2}-1} \ 
\alpha_s (t) \ {C}_{G}^2 \ 
\int_{x}^{1}{dy \over y}\ 
{\left[{\widetilde g\left({y,t}\right)}\right]}^{2}}\right\}
\label{eqn : PRDG}
\end{eqnarray}

\noindent
�����������äƤ��븶�ҳˤμ��̿��� $A$ �Ȥ��� $R_A=R_1 A^{1/3}$ �Ǥ��ꡢ
�����Ǥϴ�ñ�� $R_1=R_0=1.0$ fm �Ȥ�����
�ޤ���$\bar{n}$ �ϸ��ҳ���ˤ�����˻ҿ�̩�٤Ǥ��롣
�����ꡢ$R_A \bar{n} \propto A^{1/3}$ �Ȥʤꡢ����ϥѡ��ȥ�Ʒ���
���̤������ҳ���Υѡ��ȥ�ɤ������Ťʤ�礦�ٹ礤�����㤷��
����ˤ��줬�������γ˻ҿ������㤷�Ƥ��뤳�Ȥ˵������롣
�ޤ������ $K$ �� $A=1$ �Ȥ����Ȥ��������ब�ä���褦�� $K=9/8$ �Ȥ�����

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\vspace*{-2.5cm}
\epsfile{file=fig3-3.eps,width=13.0cm}
\end{center}
\vspace{-8.0cm}
\caption{�ѡ��ȥ�ʬ�� MRS(G)}
\label{fig : fig3-3}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�Ȥ����ǡ��ѡ��ȥ�Ʒ��Ȥ������ˤϥ��롼����ɤ����κƷ���¾�ˡ�
�ºݤˤϥ��롼����ȥ��������䥯�������ȥ�������
�κƷ��ˤĤ��Ƥ�ͤ��ʤ���Фʤ�ʤ���
���������� \ref{fig : fig3-3} �ˤ���褦�ˡ�
�ѡ��ȥ�Ʒ�礬������褦�ʾ����� $x$ �ΰ�ˤ����Ƥϡ�
���롼�����ʬ�ۤ��Ф��ƥ���������ʬ�ۤ����˾������Τǡ�
����� $Q^2$ ȯŸ�˸����Ƥ���Τ�
���롼����ɤ����κƷ��Ǥ���ȹͤ��Ƥ褤��
�������äơ����롼����ȥ���������ξ���κƷ����갷���Τ�
����ʣ���Ǥ��뤳�Ȥ��顢���ʳ��ˤ����Ƥϥ��롼����ɤ����κƷ��Τ�
��ͤ��Ƥ��롣

\vfill\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{{\bf ���ͷ׻�}}
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
{\renewcommand{\thesection}{\it {\arabic{section}}}
\subsection{{\it brute-force ˡ}}}
\vspace{0.4cm}

�桹�Υ��롼�פǤϡ� Laguerre ¿�༰���Ѥ������Ͳ�ˡ�ˤ�ꡢ
DGLAP �������ο��Ͳ�ˤĤ��Ƹ����ʤ�Ƥ��� \cite{LAG}��
������ˡ�ϥѡ��ȥ�ʬ�ۤ� Laguerre ¿�༰��Ÿ������
�׻��򤹤��Τǡ�Laguerre ¿�༰�δؿ������ѡ��ȥ�ʬ�ۤ�
����������Ƥ��뤿��˼�«��®����
SUN--IPX ����Ѥ��Ƥ鷺�����äȤ�������û��
�׻����֤Ƿ׻����٤��ɤ���̤����뤳�Ȥ��Ǥ��롣
���������ä˥ե졼�С��󣱽Ź�ξ�硢
�礭�� $x$ �ΰ�����˾����� $x$ �ΰ�Ǥ��׻����٤�����Ƥ��ޤ���
�ޤ���������Ȥ��Ƥ��롢�ѡ��ȥ�Ʒ��θ��̤Ȥ������������������
�ޤ� $Q^2$ ȯŸ�������ϲ򤯤��Ȥ��Ǥ�����
����˾����� $x$ �ΰ���ðۤʤդ�ޤ��򤹤���ʬ�ۤ�
�������Ȥ��Ǥ��ʤ��ʤɤη��������롣

�������ܸ���Ǥ�¾�ο��Ͳ�ˡ�Ȥ���ӤȤ�����Ū���ͤơ�
����ʬ�������ο��Ͳ�ˡ�Ȥ��ƤϺǤ��ñ�� brute-force ˡ�ȸƤФ����ˡ��
�Ѥ��ƿ��Ͳ�θ����Ԥä��� 
brute-force ˡ�Ȥ� $x$ ����� $t$ �� $N_x$, $N_t$ �Ĥξ��������ƥåפ�
���줾��ʬ�䤷��$t$ �Ǥ���ʬ����� $x$ �Ǥ���ʬ��

{
\setcounter{enumi}{\value{equation}}
\addtocounter{enumi}{1}
\setcounter{equation}{0}
\renewcommand{\theequation}{\arabic{section}.\theenumi\alph{equation}}
\begin{eqnarray}
{\frac{df(x,t)\ }{dt}} \ \ & \Longrightarrow  & \ \  
\frac{f(x_i , t_{j+1})-f(x_i , t_j)}{\Delta t_j} \\
\int dx f(x,t) \ \ & \Longrightarrow & \ \ 
\sum_{k=1}^{N_x} \Delta x_k f(x_k , t_j)
\end{eqnarray}
\setcounter{equation}{\value{enumi}}
}

\noindent
�Τ褦���֤������ƿ��ͷ׻���Ԥ���ΤǤ��롣
��������$\Delta t_j = t_{j+1} - t_j$, $\Delta x_k = x_k - x_{k-1}$ �Ǥ��롣
��Ȥ��ƥե졼�С��󣱽Ź�ξ���ͤ��Ƥߤ�С�

$$
{\partial \over {\partial t}} \ \widetilde q_{_{NS}}(x,t)\ 
= \ \int_x^1{dy\over y} \ 
\widetilde P_{_{NS}}\biggl({x\over y}\biggr) \ 
\widetilde q_{_{NS}}(y,t)
$$
$$
\Downarrow
$$
\begin{equation}
{\widetilde{q}}_{_{NS}}({x}_{i},{t}_{j+1})\ 
=\ {\widetilde{q}}_{_{NS}}({x}_{i},{t}_{j})\ 
+\ \Delta {t}_{j}\ \sum\limits_{k=i}^{{N}_{x}} 
{\Delta {x}_{k} \over {x}_{k}}
\ {\widetilde{P}}_{_{NS}}\left({{{x}_{i} \over {x}_{k}}}\right)\ 
{\widetilde{q}}_{_{NS}}({x}_{k},{t}_{j})
\end{equation}

\noindent
�Ȥʤꡢ���� $Q^2=Q_0^2 \ (t_0=0)$ �ˤ�����ѡ��ȥ�ʬ�ۤ�Ϳ����С�
���μ��򷫤��֤��Ѥ��뤳�Ȥˤ��Ǥ�դ� $t$ ���ʤ��Ǥ�դ� $Q^2$ �ˤ�����
�ѡ��ȥ�ʬ�ۤ����뤳�Ȥ��Ǥ��롣
���ο��Ͳ�ˡ�����˴�ñ����ˡ�Ǥ��뤬�����⤤�׻����٤η�̤����褦��
����ʤ顢 ���ƥåפο� $N_x$ , $N_t$ �������礭�����ɬ�פ����롣
�ޤ����Τ��ᤫ�ʤ�Ĺ���׻����֤��פ��롣
�׻����٤ˤĤ��Ƥ� 4.3 ��Ǿܤ����Ҥ٤롣

�Ȥ����ǡ�
���󰷤��ѡ��ȥ�Ʒ��θ��̤��������� $x$ �ΰ�˸����Ƥ��뤳�Ȥ䡢
����μ¸��Ǥ�꾮���� $x$ �Ǥ�ʬ�ۤ�¬�ꤵ��뤳�Ȥ�ͽ�ۤ���뤳�Ȥʤɤ��顢
�������Ǥϡ������� $x$ �ΰ�˶�̣�����롣��äƾ����� $x$ �ΰ�
�������ɤ��׻���Ԥ������ $x$ �򤽤Τޤ� $N_x$ �Ĥ���ʬ�䤹��
�ΤǤϤʤ���$\log_{10} x$ �� $N_x$ �Ĥ���ʬ�䤷�Ʒ׻���Ԥä���
�����������ξ��ˤϷ׻����� $x$ �κǾ��ͤ���ʤ���Фʤ�ʤ���

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
{\renewcommand{\thesection}{\it {\arabic{section}}}
\subsection{{\it FORTRAN �ץ������}}}
\vspace{0.4cm}

����ǽҤ٤����Ͳ�ˡ���Ѥ��� FORTRAN �ץ����������������
���Ǥ˽Ҥ٤��褦�� DGLAP �������ϡ������ȼ¸��Τɤ���θ���ˤ�
�Ѥ��������˽��פ��������Ǥ��롣
��äơ��ܸ���ǤΤ��Ѥ���ץ������Ȥ��ƤǤϤʤ���
�͡��ʸ���Ԥ����ѤǤ���褦�˸������뤳�Ȥ���Ū�Ȥ���
�������ι⤤�ץ����������������
������������ץ������ϰʲ����ͤʤ�ΤǤ��롣

\vspace{0.5cm}
\begin{center}
\begin{tabular}{r l}
�ץ������̾ : & BF1 \\
�ץ��������� : & FORTRAN 77 \\
�ץ������Կ� : & 2439 �� \\
�оݤȤʤ������� : & Doksitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) ������ \\
 & �ѡ��ȥ�Ʒ��θ��̤�ޤ� $Q^2$ ȯŸ������ \\
�׻���ǽ��ʬ�۴ؿ� : &  $xq_{_{NS}}(x,Q^2)$, $xq_{_{S}}(x,Q^2)$, 
$xq_i^{\ +}(x,Q^2)$, $xg(x,Q^2)$\\
�׻���ǽ�ʹ�¤�ؿ� : & $xF_{_{NS}}(x,Q^2)$, $xF_{_S}(x,Q^2)$, $xF_i^{+} (x,Q^2)$ \\
���Ͳ�ˡ : & brute-force ˡ \\
���ʬ�� : & MRS(G), ���ҳ���ѡ��ȥ�ʬ�� \cite{SKF2} \\
& (¾�ν��ʬ�ۤ�ե��������ɤ߹��ߤ���ǽ) \\
�׻����� : & �� 5 ʬ (NLO nonsinglet case) \\
& �� 60 ʬ (NLO singlet case) \\
& �� 80 ʬ (NLO singlet with Parton Recombinations case) \\
& (AlphaSever 2100 4/200 �Ǥη׻�����) \\
\end{tabular}
\end{center}

\bigskip

\noindent
�ʤ�����¤�ؿ��η׻��ˤĤ��Ƥϡ�2.3 ��ǽҤ٤�
��¤�ؿ����Τ�Τ� $Q^2$ ȯŸ�����룱���ƥå׷����ȡ�
�ѡ��ȥ�ʬ�ۤ� $Q^2$ ȯŸ�����ơ����줫�鹽¤�ؿ���׻����룲���ƥå׷�����
���Ĥη����Ƿ׻����뤳�Ȥ��Ǥ���褦�ˤ�����
���������ܸ���ˤ����ƤϽ��ʬ�ۤȤ��ƥѡ��ȥ�ʬ�ۤ�Ϳ������Τǡ�
�����ƥå׷������Ѥ��Ʒ׻���Ԥä���
�ѡ��ȥ�ʬ�ۤν��ʬ�ۤ��� (\ref{eqn : QFNS}), (\ref{eqn : QFI}) ����
��¤�ؿ���׻��������줫�飱���ƥå׷���
�Ƿ׻����뤳�Ȥ��ǽ�Ǥ��뤬�����Τ褦����ˡ�Ǥη׻��ϹԤäƤ��ʤ���
������ͳ�ϡ������ƥå׷����ξ��ˤ�ʬ���ؿ��ˤ����;ʬ�ʹ�
���ޤޤ�Ƥ���Τǡ��׻����֤�Ĺ���ʤꡢ����˷׻����٤�����뤿��Ǥ��롣
�������������ƥå׷����ȣ����ƥå׷����Σ��Ĥ���ˡ�Ƿ׻���Ԥ���
�׻�����������Ф����η�̤����פ��뤳�Ȥϳ�ǧ������

���Υץ�������¹Ԥ���Ȥ��ˤϡ��ʲ��� 21 �ĤΥѥ�᡼����
�ե����� \#10 �������Ϥ��ʤ���Фʤ�ʤ���

\noindent 
\begin{tabbing}
 (99) \= IORDER   \= = 1, \= structure function \= $xF_1(x,Q^2)$; \kill
 (1) \> IOUT      \> = 1, \> 
             {write $x$ and $xq_{_{NS}}(x)$ [or $xF_{_{NS}}(x)$] 
              at fixed $Q^2$ (=Q2) in the file \#11;} \\
     \>          \> = 2, \> 
             {write $Q^2$ and $xq_{_{NS}}(Q^2)$ [or $xF_{_{NS}}(Q^2)$] 
              at fixed $x$ (=XX) in the file \#11;} \\ 
     \>          \> = 3, \> 
             {$x$, $xq_{_S}(x)$ [or $xF_{_{S}}(x)$], and $xg(x)$;} \\ 
     \>          \> = 4, \> 
             {$Q^2$, $xq_{_S}(Q^2)$ [or $xF_{_{S}}(Q^2)$], and $xg(Q^2)$;} \\
     \>          \> = 5, \>
             {$x$, $xq_i^+(x)$ [or $xF_i^+(x)$],
                     $xq_{_S}(x)$ [or $xF_{_{S}}(x)$], and $xg(x)$;} \\ 
     \>          \> = 6, \> 
             {$Q^2$, $xq_i^+(Q^2)$ [or $xF_i^+(Q^2)$],
                     $xq_{_S}(Q^2)$ [or $xF_{_{S}}(Q^2)$], and $xg(Q^2)$.} \\
 (2) \> INPUT    \> = 1, \> 
                        {parton distributions in the nucleon;} \\
     \>          \> = 2, \> 
                        {parton distributions in a nucleus.} \\ 
     \>          \>     \> Note: If MODIFY=2 and INPUT=2 are chosen,
                           parton distributions \\
     \>          \>     \> both in a nucleus and in the nucleon should
                           be supplied. \\ 
 (3) \> IREAD   \> = 1, \> give initial distribution(s) 
                                  in function subroutine(s); \\
     \>         \> = 2, \> read initial distribution(s) from data file(s). \\
 (4) \> MODIFY  \> = 1, \> 
                        {DGLAP $Q^2$ evolution;} \\
     \>         \> = 2, \> 
                        {$Q^2$ evolution 
                         with parton recombinations.} \\ 
 (5) \> INDIST  \> = 1, \> 
                        {do not write initial distribution(s);} \\ 
     \>         \> = 2, \> 
                        {write initial distribution(s) in the file \#12;} \\ 
     \>         \> = 3, \> 
                        {write initial distribution(s) in the file \#12
                         without calculating evolution.} \\ 
 (6) \> IORDER  \> = 1, \> 
                        {leading order in $\alpha_ s$;} \\
     \>         \> = 2, \> 
                        {next-to-leading order.} \\ 
 (7) \> ITYPE   \> = 1, \> structure function \> $xF_1(x,Q^2)$; \\
     \>         \> = 2, \>                    \> $~~F_2(x,Q^2)$; \\
     \>         \> = 3, \>                    \> $xF_3(x,Q^2)$; \\
     \>         \> = 4, \> 
                        {quark distribution $xq(x,Q^2)$.} \\
     \>         \>      \>
                         ITYPE determines the output distribution type. \\
\end{tabbing}

%\vfill\eject

\begin{tabbing}
 (99) \= IORDER   \= = 1, \= structure function \= $xF_1(x,Q^2)$; \kill
 (8) \> ISTEP   \> = 1, \> 
                         one-step evolution: \=
                             $xq(Q_0^2) \ \rightarrow \ xq(Q^2)$
                         [or $xF(Q_0^2) \ \rightarrow \ xF(Q^2)$], \\
     \>         \>      \>     \> {$xg(Q_0^2) \ \rightarrow \ xg(Q^2)$;} \\
     \>         \>      \> 
                     note: one-step $xF_{1,S}$ evolution is not supplied; \\ 
     \>         \> = 2, \> 
                         two-step evolution: \=
                             $xq(Q_0^2) \ \rightarrow \ xq(Q^2)
                                        \ \rightarrow \ xF(Q^2)$, \\
     \>         \>      \>     \> {$xg(Q_0^2) \ \rightarrow \ xg(Q^2)$.} \\
 (9) \> IMORP   \> = 1, \> 
                        {$q_i-\bar q_i$ type distribution;} \\
     \>         \> = 2, \> 
                        {$q_i+\bar q_i$ type distribution.} \\
 (10) \> Q02     \> 
                {= initial $Q^2$ ($\equiv Q_0^2$ in GeV$^2$) 
                   at which an initial distribution is supplied.} \\
(11) \> Q2      \> 
                {= $Q^2$ to which the distribution is evolved 
                                        ($Q^2\ne Q_0^2$).} \\
(12) \> DLAM    \> 
                {= QCD scale parameter $\Lambda$ in GeV.} \\
(13) \> NF      \>  
                {= number of quark flavors (NF=3 or 4).} \\
(14) \> XX      \>  
                {= $x$ at which $Q^{2}$ dependent
                   distributions are written (IOUT=2, 4, or 6 case).} \\
(15) \> NX      \>  
                {= number of $x$ steps (NX$<$5000).} \\
(16) \> NT      \>  
                {= number of $t$ steps (NT$<$5000).} \\
(17) \> NSTEP  \> 
                {= number of $x$ steps or $t$ steps 
                           for writing output distribution(s).} \\      
(18) \> NXMIN   \> 
                {= $log_{10}$(minimum of $x$)
                   $[0<min(x)= 10^{^{NXMIN}}<XX]$.} \\
(19) \> NA      \> 
                {= mass number of a nucleus (NA=1 in the nucleon case).} \\
(20) \> RFM     \> 
                {= $R_0$=$R_1$ (fm).} \\
(21) \> DKHT    \> 
                {= $K_{HT}$ in $G_{HT}(x)$.} \\
     \>         \> 
            {\ \ ($G_{HT}(x)$ is a higher dimensional gluon distribution).} \\
\end{tabbing}

\noindent
�ޤ����ʬ�ۤϥץ������κǸ����ʬ�˰ʲ��Τ褦�ʴؿ���
Ϳ�����Ƥ��롣
\vspace{0.5cm}
\begin{center}
\begin{tabular}{r l}
QNS0 : & �ե졼�С��󣱽Ź�ʬ�� \\
QS0 : & �ե졼�С����Ź�ʬ�� \\
G0 : & ���롼����ʬ�� \\
QSA0 : & ���ҳ���ˤ�����ե졼�С����Ź�ʬ�� \\
GA0 : & ���ҳ���ˤ����륰�롼����ʬ�� \\
QI0 : & $q_i^+$ ʬ�� \\
QIA0 : & ���ҳ���ˤ����� $q_i^+$ ʬ��
\end{tabular}
\end{center}

\bigskip

\noindent
������Ϳ���Ƥ��� MRS(G) �ʤɤν��ʬ�ۤǤϤʤ���
¾�ν��ʬ�ۤ��Ѥ�����ˤϡ������δؿ���񤭴����뤫��
�⤷���� Input=2 �Ȥ��ơ�
�ե����� \#13, \#14, \#15, \#16, \#17 �����ɤ߹��ޤ���Ф褤��
��������HMRS-B, Duke-Owens I (valence quark), CCFR ($F_3$), ���ҳ�
��ե졼�С��󣱽Ź�ʬ�� \cite{SKF2}, Qiu \cite{PRQ} �ν��ʬ�ۤ�
������ʸ�Ȥ���Ϳ���Ƥ��롣

���Υץ������ϡ����ˤ�Ҥ٤��褦��¾�θ���ԤΤ���˸������뤳�Ȥ�
��Ū�Ȥ��ƺ������줿�������ǡ����󥿡��ͥåȤ��̤���
�ʲ��� anonymous FTP �� WWW ������Ǥ���褦�ˤ�����
�ե�����̾�� bf1.fort77.gz �Ǥ��롣

ftp.cc.saga-u.ac.jp/pub/paper/riko/quantum1/

http://www.cc.saga-u.ac.jp/saga-u/riko/physics/quantum1/structure.html

\noindent
�ʤ������Υץ������˴ؤ��뤵��˾ܤ���������
����ʸ�� \cite{MK} �Ǥʤ���Ƥ��롣


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
{\renewcommand{\thesection}{\it {\arabic{section}}}
\subsection{{\it �׻�����}}}
\vspace{0.4cm}

4.1 ��ǽҤ٤��褦�ˡ����ο��Ͳ�ˡ�η׻����٤ϥ��ƥåפο� $N_x$ �� $N_t$ ��
��¸���Ƥ��롣
���Τ���ˡ�$N_x$ , $N_t$ �ȷ׻����٤δط��ˤĤ���Ĵ�٤���
�����ǤϽ��ʬ��
�Ȥ��� HMRS-B \cite{HMRS} �ȸƤФ��ʬ�ۤ��Ѥ��ơ�
$Q_0^2$ ���� $Q^2=200 \ GeV^2$ �ޤ� NLO �δ�Ϳ�����줿
�������� $Q^2$ ȯŸ��������
���������ե졼�С��ο��� $N_f=4$ �� $x$ �κǾ��ͤ� $x=10^{-4}$ �Ȥ�����
HMRS-B �� $Q_0^2=4.0 \ GeV^2$ �ǡ�

\begin{equation}
xp(x) = N x^{\eta_1} (1-x)^{\eta_2} (1 + \gamma x) \\
\end{equation}

\noindent
�Τ褦�ʴؿ�����Ϳ�����Ƥ��ꡢ�ơ���ʬ�۴ؿ��ˤĤ��ƤΥѥ�᡼����
��ɽ�˼���������������$\Lambda=0.19 \ GeV$ �Ǥ��롣
����ʬ�ۤˤ�� $q_{_{NS}}$, $q_{_{S}}$, g ��
���ʬ�ۤ�� \ref{fig : fig4-1} �˼�����

\begin{table}[htbp]
\caption{HMRS-B �Υѥ�᡼��}
\begin{center}
\begin{tabular}{c c c | c c c | c c c | c c c | c c c} \hline
& & & & $N$ & & & $\eta_1$ & & & $\eta_2$ & & & $\gamma$ & \\
\hline
& $x(u_v+d_v)$ & & & 0.5469 & & & 0.237 & & & 4.07 & & & 23.8 \\
& $xd_v$ & & & 0.6957 & & & 0.426 & & & 4.82 & & & 6.32 \\
& $2x(\bar{u}+\bar{d}+\bar{s})$  & & & 5.25 & & & 0.401 & & & 9.75 &
 & & 0.0 \\
& $xg$  & & & 2.855 & & & 0.0 & & & 5.1 & & & 0.0 \\
\hline
\end{tabular}
\end{center}
\label{tab : HMRS}
\end{table}

\noindent

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\vspace*{-1.0cm}
\epsfile{file=fig4-1.eps,width=13.0cm}
\end{center}
\vspace{-8.0cm}
\caption{���ʬ�� HMRS-B}
\label{fig : fig4-1}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�ޤ��ϥե졼�С��󣱽Ź�ʬ�ۤξ��Ǥ� $N_t$ ��¸���ˤĤ���Ĵ�٤���
���ʤ����$N_x$ �� $N_x=1000$ ��
���ꤷ�� $N_t$ �� $N_t=10$, 20, 50, 500 �Ȥ��Ʒ׻�������
��̤Ͽ� \ref{fig : fig4-2a} �Ǥ��롣���οޤ�ꡢ$N_t$ ���Ѥ��Ƥ�
�ۤȤ��ʬ�ۤ��Ѳ����Ƥ��ʤ����Ȥ��狼�롣
$N_t=500$ ��ʬ�ۤ��Ф��� $N_t=10$, 20, 50 ��ʬ�ۤ�
������ $10^{-4} < x < 0.5$ ���ϰϤǤ�
���줾�� 1\%, 0.6\%, 0.2\% ����Ǥ��ꡢ
$x=0.8$ �Ǥ� 4\%, 2\%, 0.8\% �Ǥ��롣
���� $N_x$ ��¸���ˤĤ��Ƥ�Ʊ�ͤ� $N_t$ �� $N_t=50$ �Ǹ��ꤷ�ơ�
$N_x$ �� $N_x=100$, 300, 1000, 4000 �Ȥ��Ʒ׻�����
���줾�졢���������������������������ǿ� \ref{fig : fig4-2b} �˼�������
$N_x=4000$ ��ʬ�ۤ��Ф��� $N_x=100$, 300, 1000 ��ʬ�ۤθ����ϡ�
$10^{-4} < x < 0.5$ ���ϰϤǤ��줾�� 9\%, 3\%, 0.7\% ����ǡ�
����� $x=0.8$ �Ǥ� 24\%, 8\%, 2\% �Ǥ��롣
�����η�̤��
�ե졼�С��󣱽Ź�ʬ�ۤξ��ˤ� $N_t \geq 50$ ����� $N_x \geq 1000$ ��
����С�$10^{-4} < x < 0.5$ �� 1\% ����η׻����٤����뤳�Ȥ��Ǥ��롣

³���ƥե졼�С����Ź�ʬ�ۤξ���Ʊ�ͤ�Ĵ�٤���
�ޤ� $N_x$ �� $N_x=1000$ �Ǹ��ꤷ��$N_t=20$, 50, 200, 1000 �Ȥ���
�׻���������̤Ͽ� \ref{fig : fig4-3a} �Ǥ��ꡢ���줾��� $N_t$ ���Ф���
���������������������������Ǽ�������
$N_t=1000$ ��ʬ�ۤ��Ф��� $N_t=20$, 50, 200 ��ʬ�ۤ�
������ $10^{-4} < x < 0.9$ ���ϰϤǤ��줾�� 5\%, 3\%, 0.4\% �Ǥ��롣
������ $N_x$ �ˤĤ��Ƥ� $N_t$ �� $N_t=200$ �Ǹ��ꤷ��
$N_x=100$, 300, 1000, 4000 �Ȥ��Ʒ׻�������
��̤Ͽ� \ref{fig : fig4-3b} �˼�����
���ξ��� $N_x=4000$ ��ʬ�ۤ��Ф��� $N_x=100$, 300, 1000 ��ʬ�ۤ�
������ $10^{-4} < x < 0.5$ ���ϰϤ�
���줾�� 8\%, 3\%, 0.7\% ����Ǥ��� $x=0.8$ �� 24\%, 8\%, 2\% �Ǥ��롣

�Ǹ�˥��롼����ʬ�ۤˤĤ���Ĵ�٤���
�ޤ� $N_x$ �� $N_x=1000$ �Ǹ��ꤷ��$N_t=20$, 50, 200, 1000 �Ȥ���
�׻����ơ��� \ref{fig : fig4-4a} �ˤ��줾�����������������������������Ǽ�������
$N_t=1000$ ��ʬ�ۤ��Ф��� $N_t=20$, 50, 200 ��ʬ�ۤ�
������ $10^{-4} < x < 0.6$ ���ϰϤ�
���줾�� 5\%, 2\%, 0.4\% ����Ǥ��롣
������ $N_t$ �� $N_t=200$ �Ǹ��ꤷ $N_x$ �� $N_x=100$, 300, 1000, 4000 ��
���Ʒ׻�������Τ�� \ref{fig : fig4-4b} ��
���줾�����������������������������Ǽ�������
���ξ��� $N_x=4000$ ��ʬ�ۤ��Ф���$N_x=100$, 300, 1000 ��ʬ�ۤ�
������ $10^{-4} < x < 0.6$ ���ϰϤ�
���줾�� 25\%, 8\%, 2\% ����Ǥ��롣�ä� $N_x=1000$ ��
���ˤ� $10^{-4} < x < 0.2$ ���ϰϤ˸¤�� 1\% ����Ǥ��롣
�ե졼�С����Ź�ʬ�ۤȥ��롼����ʬ�ۤη�̤�ߤ�ȡ�
$N_t \geq  200$ , $N_x \geq 1000$ ��
����� $10^{-4} < x < 0.5$ ���ϰϤ� 1\% ���⡢$0.5 < x < 0.8$ ��
�ϰϤǤ� 2\% ����Υե졼�С����Ź�ʬ�ۤȡ�$10^{-4} < x < 0.2$ ��
�ϰϤ� 1\% ���⡢$0.2 < x < 0.6$ ���ϰϤǤ� 2\% ����η׻����٤�
���롼����ʬ�ۤ����뤳�Ȥ��Ǥ��롣

�ʾ�Τ褦�ʷ׻����٤γ�ǧ�򡢥ѡ��ȥ�Ʒ��θ��̤�
�ޤ� $Q^2$ ȯŸ�������ˤĤ��Ƥ�Ԥ����ۤ�Ʊ�ͤη�̤�������
����� LO ���������˴ؤ��Ƥϡ�����˾��ʤ� $N_t$, $N_x$ �Ǥ�Ʊ���η׻�����
�����뤳�Ȥ��Ǥ�����
�Ȥ����ǡ��ɤ�ʬ�ۤˤ����Ƥ� $t$ �����Υ��ƥåפο� $N_t$ �ϡ�$x$ ������
���ƥåפο� $N_x$ ����٤ƾ��ʤ����Ǥ褤���Ȥ��狼�롣
����Ϥ�Ȥ�� Bjorken ��������󥰤����Ū������Ω�äƤ��ꡢ
$Q^2$ ��¸�����ʤ�� $t$ ��¸���Ϥ����ޤǤ⤽�줫��Τ鷺���ʤ���Ǥ���
�Ȥ������Ȥ�������Ǥ��롣
�ޤ� $x$ ���礭���ΰ�ˤ����ƤϷ׻����٤�����㤯�ʤäƤ��롣
����ϡ�ʬ�ۼ��Τ��ͤ����˾������ʤäƤ��뤳�Ȥ䡢$\log_{10} x$ ��
��ʬ�䤷�Ƥ��뤿����礭�� $x$ �Ǥη׻��������ʤ��ʤäƤ��뤳�Ȥˤ�롣
�������� $x$ ������ʬ��� $\log_{10} x$ �ǤϤʤ��� $x$ ����ʬ�䤷�Ƥ����С�
�������ٲ�褵���Ǥ�������
���������˽Ҥ٤��褦�˸��ʳ��ǤϤष��
������ $x$ �ΰ�ˤ�����ѡ��ȥ�ʬ�ۤΤդ�ޤ��˶�̣������Τǡ�
�������ˤĤ��ƤϤ���ʾ���ڤ��ʤ���

DGLAP ���������ä� NLO �δ�Ϳ��ޤ��Τ�ʬ���ؿ�������
ʣ���ʼ��ˤʤäƤ��롣������͡���ʬ�ۤ�׻��Ǥ���褦�ˤ����Τ�
�ץ������Ϥ��ʤ�ʣ���ʤ�Τˤʤä���
��äƥץ������˴ְ㤤���ʤ�����Ĵ�٤뤿���¾�θ���Ȥ���Ӥ�Ԥä���
�ޤ��Ϥ���� HMRS \cite{HMRS} �ˤ�� $Q^2$ ��¸���η�̤Ȥ���Ӥ�Ԥ���
���η�� $x > 0.01$ ���ϰϤ� 2\% ����ǰ��פ��뤳�Ȥ��狼�ä���
�ޤ���CTEQ \cite{CTEQ} �� $Q^2$ ��¸���η�̤Ȥ���ӤǤ�
�� \% ����ǰ��פ�����
����� Laguerre ¿�༰���Ѥ�����ˡ�ˤ�� $Q^2$ ȯŸ�η�̤Ȥ�
��ӤǤ� 1\% ����ǰ��פ�����
�����η�̤�ꡢ�ץ������˴ְ㤤�Ϥʤ��ȹͤ����롣

\vfill\eject
\renewcommand{\thefigure}{\arabic{section}.\arabic{figure}}

{
\setcounter{enumi}{\value{figure}}
\addtocounter{enumi}{1}
\setcounter{figure}{0}
\renewcommand{\thefigure}{\arabic{section}.\theenumi\alph{figure}}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vspace*{-3.0cm}
\hspace{-2.0cm}
\epsfile{file=fig4-2a.eps,width=11.0cm}
\end{center}
\vspace{-4.0cm}
\caption{$N_t$ �ȷ׻����٤δط� (nonsinglet)}
\label{fig : fig4-2a}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vspace*{-1.5cm}
\hspace{-2.0cm}
\epsfile{file=fig4-2b.eps,width=11.0cm}
\end{center}
\vspace{-4.0cm}
\caption{$N_x$ �ȷ׻����٤δط� (nonsinglet)}
\label{fig : fig4-2b}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{figure}{\value{enumi}}
}

\vfill\eject

{
\setcounter{enumi}{\value{figure}}
\addtocounter{enumi}{1}
\setcounter{figure}{0}
\renewcommand{\thefigure}{\arabic{section}.\theenumi\alph{figure}}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vspace*{-3.0cm}
\hspace{-2.0cm}
\epsfile{file=fig4-3a.eps,width=11.0cm}
\end{center}
\vspace{-4.0cm}
\caption{$N_t$ �ȷ׻����٤δط� (singlet)}
\label{fig : fig4-3a}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vspace*{-1.5cm}
\hspace{-2.0cm}
\epsfile{file=fig4-3b.eps,width=11.0cm}
\end{center}
\vspace{-4.0cm}
\caption{$N_x$ �ȷ׻����٤δط� (singlet)}
\label{fig : fig4-3b}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{figure}{\value{enumi}}
}

\vfill\eject

{
\setcounter{enumi}{\value{figure}}
\addtocounter{enumi}{1}
\setcounter{figure}{0}
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%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vspace*{-3.0cm}
\hspace{-2.0cm}
\epsfile{file=fig4-4a.eps,width=11.0cm}
\end{center}
\vspace{-4.0cm}
\caption{$N_t$ �ȷ׻����٤δط� (gluon)}
\label{fig : fig4-4a}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vspace*{-1.5cm}
\hspace{-2.0cm}
\epsfile{file=fig4-4b.eps,width=11.0cm}
\end{center}
\vspace{-4.0cm}
\caption{$N_x$ �ȷ׻����٤δط� (gluon)}
\label{fig : fig4-4b}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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}


\vfill\eject
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{{\bf �׻���̤ȹͻ�}}
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\vspace{0.7cm}

���ϤǽҤ٤� brute-force ˡ�ˤ�� FORTRAN �ץ��������Ѥ���
�׻���Ԥ����͡��ʼ¸���̤Ȥ���Ӥ�Ԥä���

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
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\subsection{{\it �˻ҹ�¤�ؿ�}}}
\vspace{0.4cm}

�ޤ��ϳ˻ҹ�¤�ؿ��� $Q^2$ ��¸���ˤĤ��ƺǶ�μ¸��Ȥ���Ӥ�
�Ԥä���
�����ǤϽ��ʬ�ۤȤ��ơ�MRS(G) \cite{MRSG} �ȸƤФ���Τ��Ѥ�����
����ʬ�ۤ� $Q_0^2=4.0 \ GeV^2$ �ǡ�

\begin{equation}
xp(x) = a_1 x^{a_2} (1+a_3 \sqrt{x}+a_4 x) (1-x)^{a_5}
\end{equation}

\noindent
�Ȥ����ؿ�����Ϳ�����Ƥ��ꡢ���줾��Υѥ�᡼���ϲ�ɽ���̤�Ǥ��롣
�ޤ� $\Lambda=0.255 \ GeV$ �Ǥ��롣

\begin{table}[htbp]
\caption{MRS(G) �Υѥ�᡼��}
\begin{center}
\begin{tabular}{c c c | c c c | c c c | c c c | c c c | c c c} \hline
& & & & $a_1$ & & & $a_2$ & & & $a_3$ & & & $a_4$ & & & $a_5$ &\\
\hline 
& $xu_v+xd_v$ & & & 2.704 & & & 0.593 & & & $-0.76$ & & & 4.20 & 
& & 3.96 & \\
& $xd_v$ & & & 0.2513 & & & 0.335 & & & 8.63 & & & 0.32 & & & 4.41 & \\
& $xS$ & & & 1.74 & & & $-0.067$ & & & $-3.45$ & & & 10.3 & & & 10.1 & \\
& $xg$ & & & 1.51 & & & $-0.301$ & & & $-4.14$ & & & 10.1 & & & 6.06 & \\
\hline
\end{tabular}
\end{center}
\end{table}

\noindent
��������$S$ �ϳ�����������ʬ�۴ؿ�
�Ǥ��ꡢ$S=2(\bar{u}+\bar{d}+\bar{s}+\bar{c})$ �Ǥ��롣
���ν��ʬ�ۤϡ����Ǥ� 3.2 ��ǿ� \ref{fig : fig3-3} �˼�������

�ޤ��ǽ�˹�¤�ؿ� $F_3$ �� $Q^2$ ��¸���ˤĤ���Ĵ�٤���
���ι�¤�ؿ� $F_3$ ��
�ߥ塼������ŻҤ��Ѥ���������������ˤ����ƤϤǤƤ��ʤ�����
�˥塼�ȥ�Τ��Ѥ����夤��ߺ��Ѥ��ط�����ȿ���ˤ����ơ�
�ѥ�ƥ����ˤ�뤳�Ȥ˵������Ƹ���Ƥ��롣
$F_3$ �ϡ�

{
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\begin{eqnarray}
xF_3^{\nu{\rm N}}(x) & = & [xu_v+xd_v]+2[x\bar{s}(x)-x\bar{c}(x)] \\
xF_3^{\bar{\nu}{\rm N}}(x) & = & [xu_v+xd_v]-2[x\bar{s}(x)-x\bar{c}(x)] 
\end{eqnarray}
\setcounter{equation}{\value{enumi}}
}

\noindent
��ɽ���졢������ʿ�Ѥϲ���������ʬ�� $[xu_v+xd_v]$ ��Ϳ�����롣
��������NLO �δ�Ϳ���������ˤ� 2.3 ��ǽҤ٤��褦�ˡ�
���줾��Υѡ��ȥ�ʬ�ۤ�¤�ؿ��ˤʤ����Ƥ��� $F_3$ ��
�׻����ʤ���Фʤ�ʤ���

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\vspace*{-3.0cm}
\hspace{-2.0cm}
\epsfile{file=fig5-1.eps,width=13.0cm}
\end{center}
\vspace{-4.0cm}
\caption{��¤�ؿ� $F_3$ �� CDHSW �μ¸���̤Ȥ����}
\label{fig : fig5-1}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�����ǽ��ʬ�ۤ� MRS(G) �β���������ʬ�� $[xu_v+xd_v]$ ���Ѥ�����
������ $Q_0^2=4.0 \ GeV^2$ ���� $Q^2=200 \ GeV^2$ �����
�դ� $Q^2=0.6 \ GeV^2$ �ޤǡ�$x=0.225$, 0.045, 0.55 ��
�����Ǥ� $Q^2$ ��¸���� LO ���������� NLO �δ�Ϳ�����줿
������ (�ʸ�ñ�� NLO) �Ƿ׻���Ԥä���
���������ե졼�С��ο��� $N_f=4$ �Ȥ�������� $N_t=50$ , $N_x=1000$ �Ȥ�����
���η�̤� CDHSW �μ¸���� \cite{BCDHSW} ��� \ref{fig : fig5-1} �˼�������
������ NLO �δ�Ϳ�����줿��̤������� LO �Τߤη�̤Ǥ��롣
$Q_0^2=4.0 \ GeV^2$ �ˤ����Ƥ��Ǥ� LO �� NLO �η�̤��ۤʤäƤ��뤬��
����ϡ����˽Ҥ٤��褦��
�ѡ��ȥ�ʬ�ۤ��鹽¤�ؿ��ˤ���Ȥ��η����ؿ��ˤ� NLO �δ�Ϳ��
���äƤ��Ƥ��뤿��Ǥ��롣
���η�̤ˤ��С�NLO �δ�Ϳ�����줿��̤ϼ¸��ͤ�褯�Ƹ�����
���뤳�Ȥ��狼�롣�ޤ� NLO �δ�Ϳ�Ͼ����� $x$ �� $Q^2$ ���礭��
�����Ƥ��롣

³���ƹ�¤�ؿ� $F_2$ �� $Q^2$ ��¸���ˤĤ���Ĵ�٤���
�ۻҤι�¤�ؿ� $F_2^p$  �ϡ�LO �� $N_f=4$ �ξ��ˤϡ�

\begin{equation}
F_2^p = \frac{4}{9} (\widetilde{u}^{\ +}+\widetilde{c}^{\ +})
+\frac{1}{9} (\widetilde{d}^{\ +}+\widetilde{s}^{\ +})
\end{equation}

\noindent
��ɽ����롣���������ºݤˤϤ��٤ƤΥե졼�С��ˤĤ��� $q_i^{\ +}$ ��
����ɬ�פϤʤ���$q_i^{\ +}$ ��
�׻���ɬ�פ� $q_{_{S}}=u^{+}+d^{+}+s^{+}+c^{+}$ ���Ѥ��ơ�

{
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\begin{eqnarray}
F_2^p & = & \frac{4}{9} \ \widetilde{q}_{_{S}} 
- \frac{1}{3} \ (\widetilde{d}^{\ +}+\widetilde{s}^{\ +}) \\
or \ \ F_2^p & = & \frac{1}{9} \ \widetilde{q}_{_{S}} 
+ \frac{1}{3} \ (\widetilde{u}^{\ +}+\widetilde{c}^{\ +})
\label{eqn : F2PB}
\end{eqnarray}
\setcounter{equation}{\value{enumi}}
}

\noindent
�Τ褦�ˣ��Ĥ� $q_i^{\ +}$ ������׻����뤳�Ȥˤ�� $F_2^p$ ��
���뤳�Ȥ��Ǥ��롣�����Ǥ� (\ref{eqn : F2PB}) ����Ѥ�����
������ $F_3$ �ξ���Ʊ�ͤˡ� NLO �ξ��ˤϥѡ��ȥ�ʬ�ۤ�

$$
q_i^{\ +}  \ \ \rightarrow \ \ F_{2,q_i^{\ +}}
$$

\noindent
�Τ褦�� (\ref{eqn : QFI}) �����Ѥ��ƹ�¤�ؿ��ˤʤ����Ƥ��顢

\begin{equation}
F_2^p = \frac{1}{9} F_{2,S} + \frac{1}{3} (F_{2,u^+}+F_{2,c^+})
\end{equation}

\noindent
�Ȥ��� $F_2^p$ ��׻����ʤ���Фʤ�ʤ���

�����Ƿ׻������Τϡ�$x=0.0013$, 0.013, 0.13, 0.45 �Σ����Ǥ� $Q^2$ ��¸��
�Ǥ��ꡢ���줾�� LO, NLO, �ѡ��ȥ�Ʒ��θ��̤�
�ޤ��� (Parton Recombination, �ʸ� PR) ��
���Ĥ��������Ǥ��줾��� $Q^2$ ��¸���η׻���Ԥä���
��������$N_t=200$, $N_x=1000$ �Ȥ�����
����ʸ�� \cite{PRQ} �Ǥ� NLO �δ�Ϳ��
�ޤޤ�Ƥ��ʤ� DGLAP �������˥ѡ��ȥ�Ʒ��θ��̤�����Ƥ��롣
�������ʤ��顢3.2 ��ˤ����ƴ�ñ�˽Ҥ٤��褦�ˡ��ѡ��ȥ�Ʒ���
���̤� NLO ��Ʊ�� $\alpha_s$ �Υ������������äƤ��Ƥ��ꡢ���θ��̤�
�礭���򸫤Ƥ� NLO �δ�Ϳ��Ʊ���٤����⤷���� NLO �δ�Ϳ�������礭���ʤäƤ��롣
��äơ�PR �ˤ� NLO �δ�Ϳ��Ʊ���˴ޤ��ɬ�פ����ꡢ�ʸ� PR �Ȥ���
���ˤ� NLO �δ�Ϳ��ޤ�Ʒ׻����Ƥ��롣
�����ơ����Τ褦�ˤ��Ʒ׻�������̤� 
SLAC \cite{SLAC,RWDATA} ($x=0.14$, 0.45), 
BCDMS \cite{BCDMS} ($x=0.14$, 0.45), 
EMC \cite{EMC,RWDATA} ($x=0.125$, 0.45), 
NMC \cite{NMC} ($x=0.0125$, 0.14), 
Fermilab-E665 \cite{E665} ($x=0.0012$, 0.012), 
ZEUS \cite{ZEUS} ($x=0.016$, 0.014, 0.11), 
H1 \cite{H1} ($x=0.013$, 0.013, 0.13) �μ¸���̤���Ӥ�����
���줬�� \ref{fig : fig5-2} �Ǥ��롣
���οޤˤ��� Fermilab-E665 �η�̤������ NLO �η�̤ϼ¸��ͤ������ɤ�
�Ƹ����Ƥ��뤳�Ȥ��狼�롣
���� Fermilab-E665 �ȤΤ���ϡ����Τ褦�����˾����� $Q^2$ ���ΰ�Ǥ�
��ư��Ū QCD ��Ŭ�ѤǤ��ʤ�����Ǥ���ȹͤ���졢
�����Ǥ�����ư��Ū����ˡ���Ѥ��ʤ���Фʤ�ʤ���

�Ȥ����Ǹ��������ǹ⥨�ͥ륮����
�Żҡ��ۻҲ�®��Ǥ��� HERA (Hadron Electron Ring Accelerator) ��
����줿�ǡ��� (ZEUS ����� H1 �Υǡ���) �ϡ����ޤ�¬���Ƥ��ʤ��ä�
���˾����� $x$ �ΰ�˵ڤ�Ǥ��롣
���Τ褦�� $x$ �ΰ�ϥѡ��ȥ�Ʒ��δ�Ϳ���礭�������Ƥ���
�ȹͤ������ΰ�ʤΤǡ����Τ褦�ʸ��̤򸡽ФǤ���ΤǤϤʤ����Ȼפ��롣
�������ʤ��顢�ѡ��ȥ�Ʒ��θ��̤� 3.2 ���
�Ҥ٤��褦�� $A^{1/3}$ �����㤹��Τǡ��˻���Υѡ��ȥ�Ʒ��θ��̤�
�礭�ʸ��ҳˤΤ������٤����˾�������
��äƿ� \ref{fig : fig5-2} �򸫤Ƥ����餫�ʤ褦�˸������Ǥμ¸����٤Ǥϡ�
���Τ褦�ʸ��̤򸫽Ф��˻�äƤ��ʤ���

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\vspace*{-3.0cm}
\hspace{-2.0cm}
\epsfile{file=fig5-2.eps,width=13.0cm}
\end{center}
\vspace{-4.0cm}
\caption{��¤�ؿ� $F_2$ �μ¸���̤Ȥ����}
\label{fig : fig5-2}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1.0cm}

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\subsection{{\it ���ҳ˹�¤�ؿ�}}}
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���˸��ҳ˹�¤�ؿ��� $Q^2$ ��¸���ˤĤ���Ĵ�٤���
�ޤ��ץ������γ�ǧ�Τ���ˡ�
���ʬ�ۤȤ��ƻ���ʸ�� \cite{PRQ} �Ǻ��Ѥ���Ƥ����Τ��Ѥ���
�׻���Ԥä������η�̡����������ץ�����ब
����ʸ�� \cite{PRQ} ��Ϳ�����Ƥ����̤�Τ���
�Ƹ����뤳�Ȥ��ǧ������
�Ȥ����Ǥ��ν��ʬ�ۤϡ������� $x$ �ΰ�ˤ�����
�ѡ��ȥ�Ʒ��θ��̤򡢤�����ñ�˴ޤ�뤳�Ȥˤ�ä�������ʬ�ۤǤ��롣
��äơ�����ʬ�ۤ� shadowing ���ΰ�Τߤ�
�ͤ��� $x$ ʬ�ۤǤ��ꡢ��� $x$ �ΰ�� $x=1$ �ն�ǤΤդ�ޤ��ʤɤ�
�Ƹ�����Ƥ��ʤ���
�����ǡ����ҳ���Υѡ��ȥ�ʬ�ۤ� $x$ ��¸���ˤĤ��Ƥ�
����ʸ�� \cite{SKF2} �˾ܤ������椬�ʤ���Ƥ���Τǡ�
������ν��ʬ�ۤ��Ѥ��� ${}^{40}{\rm Ca}$ �θ��ҳ�
�ι�¤�ؿ��ˤĤ���Ĵ�٤���
���ν��ʬ�ۤϥѡ��ȥ�Ʒ��� $Q^2$ �ꥹ������󥰤Ȥ������Ĥ��Ϸ���
�Ȥ߹�碌�Ʒ׻����줿��Τǡ������� $x$ �ΰ褫��
�礭�� $x$ �ΰ�ˤ錄�ä� �¸��ǡ�����Ƹ����롣
�ѡ��ȥ�Ʒ��ˤĤ��Ƥ��裳�Ϥ�����������
$Q^2$ �ꥹ��������Ϸ��Ȥϡ����ҳ���ˤ����Ƥϳ˻Ҥ����ܤ������ˤ�
�Ťʤꤢ�����Ȥ��顢�Ĥ�����Υ���������ͳ�ʳ˻Ҥ���٤��礭���ʤꡢ

\begin{equation}
F_2^A(x, Q^2) = F_2^N(x, \ \xi_A Q^2)
\end{equation}

\noindent
�Τ褦�˸��ҳ˹�¤�ؿ��ȳ˻ҹ�¤�ؿ��� $Q^2$ �ˤ��줬������ȹͤ���
�Ϸ��Ǥ��롣
�����Ϸ��ˤĤ��Ƥϻ���ʸ�� \cite{RGR,RESCALE} �˾ܤ����Ҥ٤��Ƥ��롣
���ν��ʬ�ۤ� $Q^2=0.8 \ GeV^2$ �ǡ�

\begin{equation}
xp = A x^{\eta_1} (1-\beta x)^{\eta_2} (1+\gamma x) \\
\end{equation}

\noindent
�δؿ�����Ϳ�����Ƥ��롣�����������ν��ʬ�ۤˤϡ�
KMRS-B0 \cite{KMRSB0} �� MRS-1 \cite{MRS1} ��
ʬ�ۤ��Ȥ˷׻����줿������Τ�Τ����롣
�����Ǥ� KMRS-B0 ���Ȥˤ��Ʒ׻����줿���˻Ҥ� ${}^{40}{\rm Ca}$ ��
���ʬ�ۤ��Ѥ�����
����ʬ�ۤγơ��Υѥ�᡼����ɽ \ref{tab : SK} �˼�����
������ $S=2(\bar{u}+\bar{d}+\bar{s})$, $q_v=u_v+d_v$ �Ǥ��ꡢ
����� $\Lambda=0.2 \ GeV$ �Ǥ��롣
�ޤ���������Ѥ��뤿��� MRS-1 ���Ȥ�
�׻����줿�˻�, ${}^{118}{\rm Sn}$, ${}^{12}{\rm C}$ �ˤĤ��Ƥ�
�ѥ�᡼���⼨������

\begin{table}[t]
\caption{���ҳ���ѡ��ȥ�ʬ�ۤΥѥ�᡼��}
\begin{center}
\begin{tabular}{c | c | c | c | c | c | c} \hline
\multicolumn{2}{c | }{} &
Nucleon & ${}^{40}{\rm Ca}$ & Nucleon & ${}^{12}{\rm C}$ &
${}^{118}{\rm Sn}$ \\
\hline           
\multicolumn{2}{c | }{INPUT} &
KMRS-B0 & KMRS-B0 & \ \  MRS-1 \ \ & \ \ MRS-1 \ \ & \ \ MRS-1 \ \  \\
\hline
& $A$        & 1.967  & 1.840  & 2.712  & 2.628  & 2.427   \\
& $\eta_1$     & 0.501  & 0.472  & 0.508  & 0.495  & 0.477   \\
$xq_v$ & $\eta_2$     & 3.89   & 4.06   & 3.36   & 3.61   & 3.86    \\
& $\gamma$   & 9.27   & 9.33   & 2.89   & 3.07   & 3.61    \\
& $\beta$ & 1.0    & 0.984  & 1.0    & 0.974  & 0.944   \\
\hline
& $A$        & 2.079  & 1.773  & 2.678  & 2.331  & 1.977   \\
& $\eta_1$     & 0.705  & 0.656  & 0.741  & 0.700  & 0.662   \\
$xd_v$ & $\eta_2$     & 4.77   & 5.23   & 4.55   & 5.36   & 6.15    \\
& $\gamma$   & 2.39   & 3.01   & 0.477  & 1.26   & 2.38    \\
& $\beta$    & 1.0    & 0.956  & 1.0    & 0.936  & 0.875   \\
\hline
& $A$        & 2.058  & 6.423  & 0.7604 & 0.6571 & 2.050   \\
& $\eta_1$   & 0.294  & 0.600  & 0.0683 & 0.0437 & 0.394   \\
$xS$ & $\eta_2$     & 11.2   & 8.13   & 4.13   & 5.62   & 4.69    \\
& $\gamma$   & 7.95   & $-0.568$ & $-0.591$ & 3.11   & $-0.642$  \\
& $\beta$    & 1.0    & 1.0    & 1.0    & 1.0    & 1.0     \\
\hline
& $A$        & 16.11  & 179.2  & 16.21  & 81.14  & 531.7   \\
& $\eta_1$   & 0.839  & 1.95   & 0.786  & 1.53   & 2.44    \\
$xg$ & $\eta_2$     & 5.29   & 7.32   & 5.06   & 6.35   & 8.02    \\
& $\gamma$ & $-0.597$ & $-0.619$ & $-0.598$ & $-0.620$ & 
$-0.625$  \\
& $\beta$    & 1.0    & 1.0    & 1.0    & 1.0    & 1.0     \\
\hline
\end{tabular}
\end{center}
\label{tab : SK}
\end{table}

����ʬ�ۤ��Ѥ��� $x=0.0085$, 0.035, 0.125, 0.45 �Σ���
�Ǥ� ${}^{40}{\rm Ca}$ �ι�¤�ؿ��� $Q^2$ ��¸���� LO, NLO, PR ��
���Ĥ��������Ƿ׻���Ԥ���NMC �μ¸���� \cite{NMCA} ����Ӥ�����
�������ե졼�С����� $N_f=3$ �ǡ�$N_t=200$, $N_x=1000$ �Ȥ�����
���η�̤�� \ref{fig : fig5-3a} $\sim$ \ref{fig : fig5-3d} �˼�����
�����������Ǽ������Τ� ${}^{40}{\rm Ca}$ �ι�¤�ؿ����Τ�ΤǤϤʤ���
���ۻҤι�¤�ؿ��Ȥ��� $F_2^{Ca}/F_2^D$ �� $Q^2$ ��¸���Ǥ��롣
�����οޤ��ѡ��ȥ�Ʒ��θ��̤ϳΤ��ˤ�꾮���� $x$ �ˤ�����
�礭�������Ƥ��뤳�Ȥ��狼�롣
�ޤ���3.2 ��� (\ref{eqn : PRQI}) $\sim$ (\ref{eqn : PRG}) ��
�� (\ref{eqn : PRDQI}) $\sim$ (\ref{eqn : PRDG}) ���ˤ��С�
���θ��̤� $1/Q^2$ �ΰ��Ҥ���Ĺ⼡�ĥ����ȤȸƤФ���ΤǤ��ꡢ
��äƾ����� $Q^2$ ���礭�������Ƥ���ȹͤ����롣
����˥ѡ��ȥ�Ʒ����̾�� DGLAP �������ǹͤ��Ƥ�������Ȥ�
ȿ�Фβ�����ͤ��Ƥ��뤿�ᡢ�����������������Ǥ��롣
�� \ref{fig : fig5-3a} ��ߤ�ȡ��Τ��˾����� $Q^2$ �ˤ�����¾��������
�ˤ���̤���٤Ʒ������������ʤäƤ��뤳�Ȥ��狼�롣
�������ʤ��鸽�����������Ƥ���¸��ǡ��������˶��� $Q^2$ ��
�ϰϤ˸¤��Ƥ��ꡢ�ʤ����ļ¸����٤⤳�Τ褦�ʸ��̤򸡽Ф���
�ۤɤˤ��ɤ��ʤ���
��ä� PR �η�̤� NLO �η�̤Τɤ��餬����ɤ��¸��ǡ�����Ƹ����Ƥ��뤫��
�Ȥ��ä����ȤϤ����Ǥϵ����Ǥ��ʤ���
$F_2$ �Τ������˴ؤ��ƤϺ���� HERA �μ¸��ˡ��ޤ����������������롼����
ʬ�ۤξܺ� \cite{SKGLUE}��
�Ĥ��Ƥ� RHIC (Relativistic Heavy Ion Collider) �Ǥμ¸��˴��Ԥ�������

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\epsfile{file=fig5-3a.eps,width=11.0cm}
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\caption{���ҳ˹�¤�ؿ��� $Q^2$ ��¸�� (x=0.0085)}
\label{fig : fig5-3a}
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\caption{���ҳ˹�¤�ؿ��� $Q^2$ ��¸�� (x=0.035)}
\label{fig : fig5-3b}
\end{figure}
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\caption{���ҳ˹�¤�ؿ��� $Q^2$ ��¸�� (x=0.125)}
\label{fig : fig5-3c}
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\caption{���ҳ˹�¤�ؿ��� $Q^2$ ��¸�� (x=0.45)}
\label{fig : fig5-3d}
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\subsection{{\it $Q^2$ ȯŸ�ˤ����븶�ҳ˰�¸�� \ 
$\partial [F^{Sn}_2 / F^C_2] / \partial \ln Q^2$}}}
\vspace{0.4cm}

�Ƕ� NMC �ˤ�� ${}^{118}{\rm Sn}$ �� ${}^{12}{\rm C}$ �ι�¤�ؿ���
�� $R_2^{Sn,C} \equiv F^{Sn}_2(x,Q^2) / F^C_2(x,Q^2)$ �� $Q^2$ ��¸��
�ˤĤ��ƾܤ����¸��ǡ���������졢
$\partial R_2^{Sn,C}(x,Q^2) / \partial \ln Q^2 \neq 0$ �Ȥ���
��̤������줿 \cite{NMCSN-C}��
���η�̤ϸ��ҳˤθ��̤� $Q^2$ ��¸�������뤳�Ȥ򼨤�����
��̣������̤Ǥ��롣�����ǡ������������ FORTRAN �ץ���������Ѥ���
���η׻���Ԥ����¸���̤Ȥ���Ӥ�Ԥä� \cite{KMSN-C}��
���ʬ�ۤ������Ʊ������ʸ�� \cite{SKF2} ��
��� ${}^{118}{\rm Sn}$ �� ${}^{12}{\rm C}$ ��
��Τ��Ѥ��������줾��Υѡ��ȥ�ʬ�ۤδؿ�������ӥѥ�᡼���Ϥ��Ǥ�
����Ǽ�������$Q_0^2$ �� $\Lambda$ ���ͤ�
����Ʊ�� $Q_0^2=0.8 \ GeV^2$ , $\Lambda=0.2 \ GeV$ �Ǥ��롣
${}^{118}{\rm Sn}$ ��ʬ�ۤˤĤ��Ƥϥѥ�᡼����Ϳ�����Ƥ��ʤ��ä��Τǡ�
����ʸ�� \cite{SKF2} �ǻ��Ѥ��줿�ץ������ˤ�꿷����
�ѥ�᡼����׻����Ƥ������Ѥ�����

�Ȥ����ǡ����� $\partial R_2^{Sn,C}(x,Q^2) / \partial \ln Q^2$ �η׻���
��ˡ�Ǥ��뤬���ޤ��� ${}^{118}{\rm Sn}$ ����� ${}^{12}{\rm C}$ �Ρ�
�׻���Ԥ� $Q^2=Q_1^2$ �Ǥι�¤�ؿ��ȡ�
������������ $\delta Q^2$ ���������� $Q^2$ �Ǥι�¤�ؿ���
�׻��򺣲�������� FORTRAN �ץ������ǹԤ���
�����ꡢbrute-force ˡ�����Τǡ�

\begin{equation}
\left.\frac{\partial R_2^{Sn,C}(x,Q^2)}{\partial \ln Q^2}
\right|_{Q^2=Q_1^2} 
\ \approx \ 
\frac{R_2^{Sn,C}(x,Q_1^2)-R_2^{Sn,C}(x,Q_1^2-\delta Q^2)}
{[\ln Q_1^2 - \ln (Q_1^2-\delta Q^2)]}
\end{equation}

\noindent
�Ȥ������뤳�Ȥ��Ǥ��롣�ܸ���Ǻ�������
�ץ������� $t$ �� $N_t$ �ĤΥ��ƥåפ�ʬ�䤷�Ʒ׻���
�ԤäƤ���Τǡ�$Q^2=Q_1^2$ �ǤΥѡ��ȥ�ʬ�ۤȤȤ��
���Σ����ƥå����ǤΥѡ��ȥ�ʬ�ۤ���Ϥ���褦�ˤ���
���Υѡ��ȥ�ʬ�ۤ��Ѥ��Ʒ׻�����
��¤�ؿ��� $Q^2=Q_1^2-\delta Q^2$ �ǤΤ�ΤȤ�����

�ޤ���$Q_1^2=5.0 \ GeV^2$ �Ȥ��ơ�LO, NLO, PR �Σ��Ĥ�ȯŸ�������Ƿ׻���
�Ԥä�����̤�� \ref{fig : fig5-4} �˼�����
�����������������������Ϥ��줾�� LO, NLO, PR �Ƿ׻�������ΤǤ��ꡢ
�¸��ǡ����� NMC �Τ�ΤǤ��� \cite{NMCSN-C}��
���η�̤򸫤�ȡ������褽�Τդ�ޤ��Ȥ��Ƥϼ¸��ͤȰ��פ��Ƥ��뤳�Ȥ��狼�롣
��̣�������Ȥ� PR �Ƿ׻��������ˤϼ¸��ͤȰۤʤä���̤�����줿��
���������η�̤ϥѡ��ȥ�Ʒ��θ��̤��������ʤ��Ȥ������Ȥ�
��̣�����ΤǤϤʤ����ѡ��ȥ�Ʒ���
���̤˴ޤޤ�Ƥ���⼡���롼����ʬ�ۤ�
����ͤȤ��� (\ref{eqn : HDG}) ���Τ褦�˲��ꤷ������
���� $K_{HT}$ �ϸ��ʳ��ǤϤɤ����٤��礭���Ǥ��뤫�Ϥ褯�狼��ʤ����ᡢ
����ʸ�� \cite{PRQ} �ˤ�� $K_{HT}=1.68$ �Ȳ��ꤷ�Ƥ��롣
��äƺ��夳�Τ褦�ʥѥ�᡼���˴ؤ��뤯�路������򤷤Ƥ���ɬ�פ����롣
�ޤ������ʬ�ۤ� $Q_0^2=0.8 \ GeV^2$ �Ȥ�侮���� $Q^2$ ��Ϳ�����Ƥ���
���Ȥ⸶���ȹͤ����롣����Ǥ�Ҥ٤�����
�ѡ��ȥ�Ʒ��θ��̤Ϲ⼡�ĥ����ȤȸƤФ����̤Ǥ��ꡢ
������ $Q^2$ �ΰ���礭�������Ƥ��롣

�Ȥ����ǡ�NMC �μ¸��ǡ����� $Q_1^2$  ���ͤ�����ǤϤʤ���
$x=0.01$ �ն�Ǥ� �� $GeV^2$ �ǡ�
$x=0.1$ �ն�Ǥ� $10 \sim 20 \ GeV^2$ ���٤Ǥ��롣
���Τ��ᡢNLO �� $Q_1^2$ �� $Q_1^2=2.0$, 5.0, 20.0 $GeV^2$ ��
�Ѥ��Ʒ׻��򤷤Ƥߤ������η�̤�� \ref{fig : fig5-5} �˼�����
���η�̤򸫤�� $x=0.01$ �ն�Ǥ� $Q^2=2.0 \ GeV^2$ ��
�׻���̤ȡ�$x=0.1$ �ն�Ǥ� $Q^2=20.0 \ GeV^2$ �η�̤�
���褯���פ��Ƥ��뤳�Ȥ��狼�롣
������������Ǥ�ޤ������� $x$ �ΰ�Ǥ�侮�����ǤƤ��롣

�����β��Ϥη�̤��ʬ���뤳�Ȥϡ�
$\partial R_2^{Sn,C}(x,Q^2) / \partial \ln Q^2 \neq 0$ �Ȥ�����̤�
�ѡ��ȥ�Ʒ��� $Q^2$ �ꥹ������󥰤Ȥ��ä��ѡ��ȥ��Ϸ��������
�������뤳�Ȥ��Ǥ����ޤ� NLO �Ǥη׻���̤����ʤ��ɤ��¸��ͤ�Ƹ�
���뤳�Ȥ��顢����ˤϥѡ��ȥ�Ʒ��ˤ�� $Q^2$ ȯŸ����
���⡢���ʬ�ۤˤ�����
���ҳˤθ��̤� $x$ ��¸�����礭�������Ƥ��뤳�Ȥ��狼�롣
�Ȥ����ǿ� \ref{fig : fig5-5} �ˤ�ꡢ
���ΤȤ��Ƽ¸��ͤ���㴳�������ͤ��ǤƤ��뤳�Ȥ򼨤�������
$Q_0^2$ �ˤ�����ѡ��ȥ�ʬ�ۤ� $x$ ��¸���˷�̤��ɤ����ٰ�¸���Ƥ���Τ���
���η����ˤĤ���Ĵ�٤Ƥߤ�������Ū�ˤϡ��ޤ�¾��ʬ�ۤϤ��Τޤޤǳ���������
�ν��ʬ�ۤ�ư����¸§���ˤ�ʤ��褦�ˤ��ʤ��� shadowing ��
�礭���ȤäƷ׻���ľ��������ʬ�ۤ���ʬ�ۤȤ��Ʒ׻���Ԥä���
����� $x \approx 0.03$ �ն�Ǥ��¸��ͤ˰��פ����̤�����줿��
³���ƥ��롼����ˤĤ��Ƥ�Ʊ�ͤΤ��Ȥ�Ԥ��ȡ���̤ϵդ˼¸���̤Ȥ�
���줬�礭���ʤä����Ѳ����礭���Ȥ��Ƥϳ�����������ʬ�ۤ��Ѳ����Ф���
����Ҵ��Ǥ��뤳�Ȥ��狼�ä���
�����Τ��Ȥ��顢���θ��ݤ�����ˤϡ����ʬ�ۤȤ��Ƥθ��ҳ���ѡ��ȥ�ʬ�ۡ�
�ä˳���������ʬ�ۤΤ������̩�ʸ��椬���פǤ���ȹͤ����롣

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{center}
\vspace*{-4.5cm}
\hspace{-1.5cm}
\epsfile{file=fig5-4.eps,width=17.0cm}
\end{center}
\vspace{-4.0cm}
\caption{$\partial R_2^{Sn,C}(x,Q^2) / \partial \ln Q^2$ \ \ 
(LO, NLO, PR)}
\label{fig : fig5-4}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[p]
\begin{center}
\vspace*{-4.5cm}
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\epsfile{file=fig5-5.eps,width=17.0cm}
\end{center}
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\caption{$\partial R_2^{Sn,C}(x,Q^2) / \partial \ln Q^2$ \ \
 ($Q^2=2.0$, 5.0, 20.0 $GeV^2$)}
\label{fig : fig5-5}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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�˻Ҥ���Ӹ��ҳ˹�¤�ؿ��� $Q^2$ ��¸���ˤĤ���
������� $Q^2$ ȯŸ�������ο��Ͳ�θ����Ԥä���
��Ĥ��̾��Ѥ����� DGLAP �������Ǥ��ꡢ
�⤦��Ĥ� DGLAP �������˥ѡ��ȥ�Ʒ��θ��̤����줿��ΤǤ��롣
���κ� $t$ ����� $x$ �� $N_t$, $N_x$ �ĤΥ��ƥåפˤ��줾��ʬ�䤷�Ʒ׻���
�Ԥ� brute-force ˡ���Ѥ��� FORTRAN �ץ����������������
���Υץ������ϼ㴳�׻����֤�Ĺ�������뤬��NLO �δ�Ϳ�����줿
�ե졼�С��󣱽Ź�ξ��� $N_t \geq 50$ , $N_x \geq 1000$ ���١�
�ե졼�С����Ź�ξ��� $N_t \geq 200$ , $N_x \geq 1000$ ���٤�
���С�$10^{-4} < x < 0.8$ ���ϰϤ� 2\% ����η׻����٤����뤳�Ȥ�
�Ǥ����������¾�θ���Ԥ����ѤǤ���褦�ˡ����󥿡��ͥåȤ��̤���
����Ǥ���褦�ˤ�����

�����Ƥ��Υץ��������Ѥ��ơ��˻Ҥι�¤�ؿ� $F_3$ ����� $F_2$ �ˤĤ��Ƥ�
�׻��򤪤��ʤ����͡��ʼ¸�
�ǡ��� (CDHSW, SLAC, BCDMS, EMC, NMC, Fermilab-E665, ZEUS, H1) �Ȥ�
��Ӥ�Ԥä������η�̡�
LO �η�̤���Ӥ��� NLO �η�̤Ͼ����� $Q^2$ �ΰ��
����ɤ� $Q^2$ ��¸���Υǡ�����Ƹ�������
�������ʤ��顢�ѡ��ȥ�Ʒ��� $Q^2$ ȯŸ���̤ˤĤ��Ƥ�
�¸��������ǧ�����˻�äƤ��ʤ���
³���Ƹ��ҳ˹�¤�ؿ��η׻���Ԥ���NMC �� $Q^2$ ��¸����
�ؤ���¸��ǡ��� $F_2^A/F_2^D$ �Ȥ���Ӥ�Ԥä���
���η�̡�$Q^2$ ȯŸ�ˤ�����ѡ��ȥ�Ʒ��θ��̤ϡ�
������ $x$ �ΰ褪��Ӿ����� $Q^2$ �ΰ�����������Ȥ�Ƚ����������
�������Ǥ��θ��̤򸡽Ф���˻�äƤ��ʤ���
�������˴ؤ��Ƥϡ����� $Q^2$ �ΰ���Ϥ롢
���٤��ɤ��¸���̤����������뤳�Ȥ˴��Ԥ�������

�Ǹ�� $\partial R_2^{Sn,C}(x,Q^2) / \partial \ln Q^2$ �η׻���Ԥä���
���η�̡������褽�� $x$ ��¸���ϸ��������Ƥ���
���ҳ���ѡ��ȥ�ʬ�ۤ� NLO �� DGLAP �������������Ǥ��뤳�Ȥ�ʬ���ä���
������Ф��ѡ��ȥ�Ʒ��θ��̤�
�ޤ�� $Q^2$ ȯŸ�������Ƿ׻���Ԥä����ˤ�
�¸��ͤȰۤʤ��̤�����줿��
����������ϥѡ��ȥ�Ʒ��ˤ���������������ʤ��Ȥ������Ȥ�
�������˰�̣����ΤǤϤʤ�������� $Q^2$ ȯŸ�ˤ�����ѡ��ȥ�Ʒ���
���̤ˤ� $K_{HT}$ �Τ褦�ʡ��ޤ��ɤ��狼�äƤ��ʤ��ѥ�᡼����
¸�ߤ��뤫��Ǥ��롣���塢������٤��ɤ��¸���̤�������С�
���Τ褦�����˴ؤ���ܤ���������ǽ�Ȥʤ�Ǥ�������
����ˡ����ʬ�ۤ�Ϳ�����Ƥ��� $Q^2$ ����侮�������뤳�Ȥ⸶��
�ȹͤ����롣
��������NLO �η�̤����ʤ�¸��ͤ� $x$ ��¸����Ƹ����뤳�Ȥ��顢
�ѡ��ȥ�ʬ�ۤθ��ҳ���ˤ����� $x$ ��¸�����Ѳ���
����礭���ƶ����Ƥ��ꡢ$Q^2$ ȯŸ�ˤ�����ѡ��ȥ�Ʒ��θ��̤�
���ޤ��礭���ϸ����Ƥ��ʤ��ȹͤ��뤳�Ȥ��Ǥ��롣
��äƤ��θ��̤Τ����̩������ˤϡ����ҳ���ѡ��ȥ�ʬ�ۤ� $x$ ��¸����
�ؤ���ܤ������椬���פǤ���ȹͤ����롣


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��Ƴ�����Ǥ��뷧��ӻ��������ˤϡ�������������Σ�ǯ�֤ˤ錄�ꡢ
���ڤʤ���Ƴ�򤤤���������
�ޤ����̻�ʪ���عֺ¤�Ĺë��ȶ�������������ꡢ���Ȱ����ؤ�
��������ᡢ�������θ�������ξ��������ᡢ������͵�ᡢ���Ŀ�ͺ��
�ˤϡ�¿���ˤ錄�뤴�����򤤤���������
�����ơ�����μ�һᡢ���²��ɻ�ˤ��������������ǯ���Σ�ǯ�֤ˤ錄�ꡢ
��Ʊ����ԤȤ����͡��ʵ����򤷤Ƥ��������������Ƥ������򤤤���������
�嵭�������ˡ������餪��򿽤��夲�롣
�ޤ������ͷ׻��˺ݤ��׻�������Ѥ����Ƥ���������
�����س�ʪ�����楻�󥿡��˴��դ��롣
���θ����ʿ����ǯ�ƴ��ˡ��֥�å��إ֥��ƹ��Ω�����ˤ�������ʬŪ��
�Ԥ�줿���������Ǥ��ںߤ˴ؤ��Ʒк�Ū�ʱ������Ӥ����ä�
���������� A. J. Baltz ��˴��դ��롣����θ����ν��ʤ������ʸ����
�˴ؤ��Ƥ�ʸ���ʲʳظ��������C (�����ֹ� 06640406) �α�����������


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\subsection*{��ϿA \ \ \ ʬ���ؿ�}
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���㼡 (LO) ��ʬ���ؿ��ϡ�

\begin{eqnarray}
P_{q_i^+ q_j^+}^{(0)}(x) & = & \delta_{ij} \ P_{_{NS}}^{(0)}(x) 
                    = \delta_{ij} \ C_F\ \left[\ {{1+x^2} \over {(1-x)_+}}
                    \ + \ {3 \over 2} \ \delta (1-x)\ \right] \\
P_{q_i^+ g}^{(0)}(x) &=& 2T_R \left[\ x^2 \ + \ (1-x)^2\ \right] \\
P_{gq}^{(0)}(x) &=& C_F \ {{1+(1-x)^2} \over x} \\
P_{gg}^{(0)}(x) &=& 2C_G \left[\ {x \over {(1-x)_+}} \ + \
                        {{1-x} \over x} \ + \ x(1-x) \right. \nonumber \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left. \mbox{} 
                     + \left( {{11}\over{12}} 
                     - {1 \over 3}{{N_fT_R} \over {C_G}}\right)
                    \ \delta (1-x) \ \right]
\end{eqnarray}

\noindent
��Ϳ�����롣�����ǡ�$1/(1-x)_+$ �ϰʲ��Τ褦���������롣

\begin{equation}
\int_0^1 dz {{f(z)} \over {(1-z)_+}} \ = \ 
\int_0^1 dz {{f(z)-f(1)} \over {1-z}}
\end{equation}

\noindent
����������ʬ�ϰϤ� $0 \le z \le 1$ ���������Ƥ��뤳�Ȥ����դ�ɬ�פǤ��롣
�ºݤ�DGLAP �������Τʤ��Ǥ���ʬ�ϰϤ� $x \le z \le 1$ ��Ϳ�����Ƥ��ꡢ
���ΤȤ��ϡ�

\begin{equation}
\int_x^1 dz {{f(z)} \over {(1-z)_+}} \ = \ 
\int_x^1 dz {{f(z)-f(1)} \over {1-z}}+f(1) \ln(1-x)
\end{equation}

\noindent
�Ȥʤ롣NLO ��ʬ���ؿ��ϻ���ʸ�� \cite{HWW,NLO} ��Ϳ�����Ƥ��ꡢ

\begin{eqnarray}
P_{q_i^+ q_j^+}^{(1)}(x) &=&  \delta_{ij} \ P_{q^+,NS}^{(1)}(x) \ + \
                              2 C_F T_R F_{qq}(x)
\label{eqn : P1QQ} \\
P_{q_i^+ g}^{(1)}(x) &=& - \ C_G T_R F_{qg}^1(x)
                       \ - \ C_F T_R F_{qg}^2(x) \\
P_{gq}^{(1)}(x) &=& - \ C_F^2 F_{gq}^1(x)
                      \ - \ C_F C_G F_{gq}^2(x)
                      \ - \ C_F T_R N_f F_{gq}^3(x) \\
P_{gg}^{(1)}(x) &=&      C_G T_R N_f F_{gg}^1(x)
                   \ + \ C_F T_R N_f F_{gg}^2(x)
                   \ + \ C_G^2 F_{gg}^3(x)
\label{eqn : P1GG}
\end{eqnarray}

\noindent
�Ǥ��롣�ե졼�С��󣱽Ź�� NLO ��ʬ���ؿ��� $q \pm \bar{q}$ ����
ʬ�� $\displaystyle{q_{_{NS}}=\sum_i a_i (q_i \pm \bar{q}_i)}$ ���Ф��ơ�

\begin{eqnarray}
P_{q^\pm, NS}^{(1)}(x)&=&C_F^2\biggl\{P_F(x) \pm P_A(x)+\delta(1-x)
\biggl[{3\over8}-{1\over2}\pi^2
+\zeta(3)-8\widetilde{S}(\infty)\biggr]\biggr\}  \nonumber \\
&+&{1\over2}C_FC_A\bigg\{P_G(x) \mp P_A(x)+\delta(1-x)
\biggl[{17\over12}+{11\over9}
\pi^2-\zeta(3)+8\widetilde{S}(\infty)\biggr]\biggr\} \nonumber \\
&+&C_FT_RN_f\biggl\{P_{N_F}(x)-\delta(1-x)
\biggl({1\over6}+{2\over9}\pi^2\biggr)
\biggr\}  
\end{eqnarray}

\noindent
��Ϳ�����롣
�����ǡ�$P_F(x)$, $P_G(x)$, $P_{N_F}(x)$, $P_A(x)$ ��
����ʸ�� \cite{NLO} �ˤ�ꡢ

\begin{eqnarray}
P_F(x) &=& -2{{1+x^2} \over {1-x}}\ln x\ln(1-x)-
\biggl({3\over {1-x}}+2x\biggr)\ln x \nonumber \\
& & \mbox{}-{1\over2}(1+x)\ln^2x -5(1-x) \\
P_G(x) &=& {1+x^2\over (1-x)_+}\bigg[\ln^2x+{11 \over3}\ln x+{67\over9}
-{1\over3}\pi^2\biggr]  \nonumber \\
& & \mbox{}+2(1+x)\ln x+{40\over 3}(1-x) \\
P_{N_F}(x) &=& {2\over 3}\Biggl[{1+x^2\over (1-x)_+}\biggl(
-\ln x-{5\over 3}\biggr)-2(1-x)\Biggr] \\
P_A(x) &=& 2{1+x^2\over 1+x}\int_{x/(1+x)}^{1/(1+x)}{dz \over z}
\ln {{1-z} \over z}+2(1+x)\ln x+4(1-x)
\end{eqnarray}

\noindent
���������Ƥ��롣
$\zeta$ �ؿ��� $\displaystyle{\zeta (k)=\sum_{n=1}^\infty {1 \over n^k}}$ ��
���������Τ� $\zeta(3)=1.2020569...$ �Ǥ��롣
�ޤ���$\widetilde S(\infty)$ �� $\zeta$ ��
��ä� $\widetilde S (\infty) =-{5 \over 8}\zeta(3)$ �Τ褦���������롣
����ˡ�$P_A(x)$ �������ʬ�� Spence �ؿ� $S(x)$ ���Ѥ���ɽ�����Ȥ��Ǥ��ơ�

\begin{equation}
\int_{x/(1+x)}^{1/(1+x)}{dz \over z} \ln {{1-z} \over z} = 
S\left({x\over{1+x}}\right)  - 
S\left({1\over{1+x}}\right)  - 
{1\over 2}  \left[\ \ln^2 {1\over{1+x}}  -  
                     \ln^2 {x\over{1+x}} \ \right]
\end{equation}

\noindent
�Ǥ��롣������ Spence �ؿ��ϡ�

\begin{equation}
S(x) \ = \ \int_x^1 dz \ {{\ln z}\over{1-z}}
\end{equation}

\noindent
�ˤ�ä���������ؿ��Ǥ��롣��������$0 \le x \le 2$ ���ϰϤˤ����Ƥϡ�

\begin{equation}
S(x) \ = \ - \ \sum_{k=1}^\infty {{(1-x)^k} \over {k^2}}
\end{equation}

\noindent
�Τ褦�˽񤯤��Ȥ��Ǥ���Τǡ��׻��ǤϤ�����Ѥ��Ƥ��롣
ʸ���ˤ�äƤ� $-S(x)$ �� Spence �ؿ��ȸƤ־�礬����Τ����դ�ɬ�פǤ��롣

(\ref{eqn : P1QQ}) $\sim$ (\ref{eqn : P1GG}) ����
������ $F_{qq}$, $F_{qg}$, $F_{gq}$, $F_{gg}$ �ϰʲ��Τ褦��
�������Ƥ��롣

\begin{eqnarray}
{F}_{qq}(x)&=&{\frac{20}{9}}{\frac{1}{x}}-2+6x
-{\frac{56}{9}}{x}^{2}+\left({1+5x+{\frac{24}{9}}{x}^{2}}\right)
{\ln}x-\left({1+x}\right){\ln}^{2}x \\
{F}_{qg}^{1}(x)&=&-{\frac{40}{9}}{\frac{1}{x}}+4-50x
+{\frac{436}{9}}{x}^{2}]
-\left({2+16x+{\frac{88}{3}}{x}^{2}}\right){\ln}x \nonumber \\
& &\mbox{}+(2+4x){\ln}^{2}x
+8x\left({1-x}\right)\ln\left({1-x}\right) \nonumber \\
& &\mbox{}+\left({2-4x+4{x}^{2}}\right)\left[{{\ln}^{2}\left({1-x}\right)
-{\frac{1}{6}}{\pi }^{2}}\right]-
4\left({1+2x+2{x}^{2}}\right){J}_{x} \\
{F}_{qg}^{2}(x)&=&-14+29x-20{x}^{2}
+\left({-3+4x-8{x}^{2}}\right){\ln}x \nonumber \\
& &\mbox{}-\left({1-2x+4{x}^{2}}\right){\ln}^{2}x-
8x\left({1-x}\right)\ln\left({1-x}\right) \nonumber \\
& &\mbox{}+\left({2-4x+4{x}^{2}}\right)\left[{-{\ln}^{2}\left({1-x}\right)
+2{\ln}x\ln\left({1-x}\right)+{\frac{1}{3}}{\pi }^{2}}\right] \\
{F}_{gq}^{1}(x)&=&{\frac{5}{2}}+{\frac{7}{2}}x-
\left({2+{\frac{7}{2}}x}\right){\ln}x+
\left({1-{\frac{1}{2}}x}\right){\ln}^{2}x \nonumber \\
& &\mbox{}+\left({{\frac{6}{x}}-6+5x}\right)\ln\left({1-x}\right)
+\left({{\frac{2}{x}}-2+x}\right){\ln}^{2}\ \left({1-x}\right) \\
{F}_{gq}^{2}(x)&=&-{\frac{1}{x}}-{\frac{19}{9}}
-{\frac{37}{9}}x-{\frac{44}{9}}{x}^{2}
+\left({12+5x+{\frac{8}{3}}{x}^{2}}\right){\ln}x \nonumber \\
& &\mbox{}-\left({2+x}\right){\ln}^{2}x
-{\frac{1}{3}}\left({{\frac{22}{x}}-22
+17x}\right)\ln\left({1-x}\right) \nonumber \\
& &\mbox{}-\left({{\frac{2}{x}}-2+x}\right)\left[{{\ln}^{2}\left({1-x}\right)
-2{\ln}x\ln\left({1-x}\right)-{\frac{1}{6}}{\pi }^{2}}\right] \nonumber \\
& &\mbox{}+\left({{\frac{4}{x}}+4+2x}\right)\ {J}_{x} \\
{F}_{gq}^{3}(x)&=&{\frac{40}{9}}{\frac{1}{x}}-{\frac{40}{9}}+
{\frac{32}{9}}x
+{\frac{4}{3}}\left({{\frac{2}{x}}-2+x}\right)\ln\left({1-x}\right)
\end{eqnarray}

\vfill\eject

\begin{eqnarray}
{F}_{gg}^{1}(x)&=&-{\frac{20}{9}}{\frac{1}{{\left({1-x}\right)}_{+}}}-
{\frac{2}{9}}\left({{\frac{23}{x}}-29+19x-23{x}^{2}}\right) \nonumber \\
& &\mbox{}-{\frac{4}{3}}\left({1+x}\right){\ln}x
-{\frac{4}{3}}\delta\left({1-x}\right) \\
{F}_{gg}^{2}(x)&=&{\frac{4}{3x}}-16+8x+{\frac{20}{3}}{x}^{2}-
\left({6+10x}\right){\ln}x \nonumber \\
& &\mbox{}-2\left({1+x}\right){\ln}^{2}x-\delta\left({1-x}\right) \\
{F}_{gg}^{3}(x)&=&{\frac{67}{9}}{\frac{1}{{\left({1-x}\right)}_{+}}}
-{\frac{8}{9}}-{\frac{1}{18}}x-\left({{\frac{47}{6}}
-{\frac{25}{6}}x+{\frac{44}{3}}{x}^{2}}\right){\ln}x \nonumber \\
& &\mbox{}+\left({{\frac{1}{1-x}}+{\frac{1}{1+x}}
+{\frac{5}{4}}+6x-2{x}^{2}}\right){\ln}^{2}x \nonumber \\
& &\mbox{}+\left({-{\frac{11}{2x}}+{\frac{7}{2}}
+{\frac{13}{4}}x}\right)\ln\left({1-x}\right)
+{\frac{1}{4}}\left({-{\frac{5}{x}}+5+2x}\right)
{\ln}^{2}\left({1-x}\right) \nonumber \\
& &\mbox{}-\left({{\frac{4}{1-x}}+{\frac{4}{x}}-5+7x-4{x}^{2}}\right)
{\ln}x\ln\left({1-x}\right)+\left({1+x}\right){I}_{x} \nonumber \\
& &\mbox{}+4\left({{\frac{1}{1+x}}-{\frac{1}{x}}-2-x-{x}^{2}}\right){J}_{x}
-{\frac{1}{3}}{\pi }^{2}\left({{\frac{1}{{\left({1-x}\right)}_{+}}}
+{\frac{1}{x}}-1+2x-{x}^{2}}\right) \nonumber \\
& &\mbox{}
+\delta\left({1-x}\right)\left[{-4\widetilde{S}\left({\infty}\right)
+{\frac{1}{2}}\zeta \left({3}\right)+{\frac{8}{3}}}\right]
\end{eqnarray}

\noindent
�����ǡ�$I_x$ �� Spence �ؿ��Ǥ��� [$I_x=S(x)$] ��$J_x$ ��

\begin{equation}
J_x = -{\frac{1}{12}}{\pi}^2 - \ln x\ln (1+x) + S(1+x)
\end{equation}

\noindent
���������롣

�����ƥå׷����� $Q^2$ ȯŸ������ (\ref{eqn : 1STEP}) �ˤ����롢
���� ${\bf P}^{(0),(1)}$, ${\bf D}$, ${\bf E}$ �ϡ�

\begin{equation}
{\bf P}^{(0),(1)} \left({x}\right)\ = \left({\begin{array}{cc}
{\displaystyle \sum_j} P_{q_i^+ q_j^+}^{(0),(1)}
\left({x}\right)&P_{q_i^+ g}^{(0),(1)} \left({x}\right) \\
P_{gq}^{(0),(1)} \left({x}\right) 
{\displaystyle \sum_j}&P_{gg}^{(0),(1)}
\left({x}\right)
\end{array}}\right)
\end{equation}

\begin{equation}
{\bf D}^{\left({1}\right)}\left({x}\right)\ = \rm 
\ \left({\begin{array}{cc}B_n^q(x)&B_n^g(x) \\
0&0\end{array}}\right)
\end{equation}

\begin{equation}
{\bf E}\left({x}\right)\ = \rm 
\ \left({\begin{array}{cc}{E}_{qq}
\left({x}\right)&{E}_{qg}\left({x}\right) \\
{E}_{gq}\left({x}\right)&{E}_{gg}
\left({x}\right)\end{array}}\right)
\end{equation}

\noindent
���������롣�����ǹ��� ${\bf D}$ �˸���Ƥ��� $B_n^q(x)$, $B_n^g(x)$ ��
��Ͽ B ��Ϳ���롣���� ${\bf E}$ �ι������Ǥϡ�

\begin{eqnarray}
{E}_{qq}\left({x}\right)&=&-{E}\left({{\gamma}_{gq}{C}^{g}}\right) \\
{E}_{qg}\left({x}\right)&=&{E}\left({{\gamma}_{qq}{C}^{g}}\right)
-{E}\left({{\gamma}_{qg}{C}^{q}}\right)
-{E}\left({{\gamma}_{gg}{C}^{g}}\right) \\
{E}_{gq}\left({x}\right)&=&{E}\left({{\gamma}_{gq}{C}^{q}}\right) \\
{E}_{gg}\left({x}\right)&=&-{E}_{qq}\left({x}\right)
\end{eqnarray}

\noindent
���������Ƥ��롣�����ǡ�

\begin{eqnarray}
{E}\left({{\gamma}_{gq}{C}^{g}}\right)/2{C}_{F}{T}_{R}&=&-{\frac{2}{3x}}
+{\frac{20}{3}}-{\frac{2x}{3}}
-{\frac{16{x}^{2}}{3}}
+\left({1+5x-{\frac{4}{3}}{x}^{2}}\right){\ln}x \nonumber \\
& &+\left({1+x}\right)\left({{\ln}^{2}\ x-2{\ln}x\ln\left({1-x}\right)
+2{I}_{x}}\right) \nonumber \\
& &+\left({-{\frac{4}{3x}}-1+x
+{\frac{4}{3}}{x}^{2}}\right)\ln\left({1-x}\right) \\
{E}\left({{\gamma}_{qq}{C}^{g}}\right)/{C}_{F}{T}_{R}&=&-7+10x
-\left({1-16x+32{x}^{2}}\right){\ln}x
-\left({1-2x+4{x}^{2}}\right){\ln}^{2}x \nonumber \\
& &+\left({6-12x+16{x}^{2}}\right){\ln}x\ln\left({1-x}\right) \nonumber \\
& &+\left({5-36x+32{x}^{2}}\right)\ln\left({1-x}\right) \nonumber \\
& &-4\left({1-2x+2{x}^{2}}\right)\left[{{\ln}^{2}\left({1-x}\right)
-{\frac{{\pi }^{2}}{6}}}\right] \nonumber \\
& &-2\left({1-2x+4{x}^{2}}\right){I}_{x}
\end{eqnarray}

\vfill\eject

\begin{eqnarray}
{E}\left({{\gamma}_{qg}{C}^{q}}\right)/{C}_{F}{T}_{R}&=&5-12x+16{x}^{2}
+\left({1+8x-12{x}^{2}}\right){\ln}x \nonumber \\
& &+\left({1-2x+4{x}^{2}}\right)\left[{-{\ln}^{2}x
+2{\ln}x\ln\left({1-x}\right)}\right] \nonumber \\
& &+\left({7-16x+12{x}^{2}}\right)\ln\left({1-x}\right) \nonumber \\
& &+\left({1-2x+2{x}^{2}}\right)\left[{-2{\ln}^{2}\left({1-x}\right)
+{\frac{2}{3}}{\pi }^{2}}\right] \nonumber \\
& &+2\left({1-2x}\right){I}_{x} \\
{E}\left({{\gamma}_{gq}{C}^{q}}\right)
/{\frac{1}{2}}{C}_{F}^{2}&=&3+6x
-3x{\ln}x+\left({2-x}\right){\ln}^{2}x \nonumber \\
& &+\left({{\frac{12}{x}}-16+7x}\right)\ln\left({1-x}\right) \nonumber \\
& &+2\left({{\frac{2}{x}}-2+x}\right)\left[{-{\ln}^{2}\left({1-x}\right)
+{\frac{1}{3}}{\pi }^{2}}\right] \nonumber \\
& &-2\left({2-x}\right){\ln}x\ln\left({1-x}\right)
+2\left({{\frac{4}{x}}-2+x}\right){I}_{x} \\
{E}\left({{\gamma}_{gg}{C}^{g}}\right)&=&{C}_{G}{T}_{R}A
+{\frac{4}{3}}{T}_{R}^{2}{N}_{f}B 
\end{eqnarray}

\noindent
�Ǥ��롣�弰�� $A$ �� $B$ �ϡ�

\begin{eqnarray}
A&=&-{\frac{4}{3x}}+18+{\frac{154}{3}}x-{\frac{193}{3}}{x}^{2}
+\left({{\frac{17}{3}}+{\frac{170}{3}}x
-{\frac{40}{3}}{x}^{2}}\right){\ln}x \nonumber \\
& &+2\left({1+4x}\right)\left({{\ln}^{2}x
+2{I}_{x}}\right)-\left({{\frac{8}{3x}}+{\frac{5}{3}}+{\frac{122}{3}}x
-{\frac{136}{3}}{x}^{2}}\right)\ln\left({1-x}\right) \nonumber \\
& &-4\left({1-2x+2{x}^{2}}\right)\left[{{\ln}^{2}\left({1-x}\right)
-{\frac{{\pi }^{2}}{6}}}\right]
-8x\left({3-x}\right){\ln}x\ln\left({1-x}\right) 
\label{eqn : AA} \\
B&=&-1+8x-8{x}^{2}
-\left({1-2x+2{x}^{2}}\right)\ln\left({{\frac{x}{1-x}}}\right)
\end{eqnarray}

\noindent
��Ϳ�����롣������ (\ref{eqn : AA}) ���� $(17/3+170x/3-40x^2/3)$ ��
��ʬ�ϡ�����ʸ�� \cite{HWW} ��
�����Ƥ� $(17+170x/3-40x^2/3)$ �ȤʤäƤ��롣
������¾��ʸ���Ȥ���Ӥη�̡�����Ǥ��뤳�Ȥ��狼�꽤��������

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
\subsection*{��ϿB \ \ \ �����ؿ�}
\addcontentsline{toc}{subsection}{\protect\numberline{B}{�����ؿ�}}
\renewcommand{\theequation}{B.\arabic{equation}}
\setcounter{equation}{0}
\vspace{0.4cm}

�����ؿ��� 2.3 ��Ǽ������褦�ˡ�

\begin{eqnarray*}
C_n^{q}(x,\alpha_s) & = & \delta(1-x)  + 
                {{\alpha_s}\over{4\pi}} B_n^{q}(x) + O(\alpha_s^2) \\
C_n^{g}(x,\alpha_s) & = &  
                {{\alpha_s}\over{4\pi}} B_n^{g}(x) + O(\alpha_s^2)
\end{eqnarray*}

\noindent
�Ǥ��롣�����Ǹ���Ƥ��� $B_n^{q}$, $B_n^{g}$ �ϡ�
�����ƥå׷����������� (\ref{eqn : 1STEP}) ��ʬ���ؿ��ι��� ${\bf D}$ ��
������Τ�Ʊ��Ǥ��롣$B_n^{q}$ �ϡ�

\begin{eqnarray}
B_1^{q}(x)&=&{C_F \over 2}[F_q(x)-4x]  \\
B_2^{q}(x)&=&C_F F_q(x) \\
B_3^{q}(x)&=&C_F [F_q(x)-2-2x]
\end{eqnarray}

\noindent
�Ǥ��롣$B_1^{q}$, $B_2^{q}$, $B_3^{q}$ �Ϥ��줾�졢
$F_1$, $F_2/x$, $F_3$ ���б������ΤǤ��ꡢ$F_q(x)$ �ϡ�

\begin{eqnarray}
F_q(x) &= &-{3 \over 2}{1+x^2 \over (1-x)_+}
        +{1\over 2}(9+5x)-2{1+x^2 \over 1-x} \ln x \nonumber \\
 &  & \mbox{}+2(1+x^2)\biggl[ {\ln (1-x) \over 1-x} \biggr]_+ 
        -\delta (1-x)(9+{2 \over 3}\pi^2)
\end{eqnarray}

\noindent
���������롣�������������Ǹ���� $[\ln (1-x)/(1-x)]_+$ �ϡ�

\begin{equation}
\int_0^1 dz f(z) \biggl[ {\ln (1-z) \over 1-z} \biggr]_+ 
= \int_0^1 dz [f(z)-f(1)] {\ln (1-z) \over 1-z}
\end{equation}

\noindent
��������졢��Ͽ A �Ǹ���� $1/(1-x)_+$ ��Ʊ�ͤˡ�
��ʬ�ϰϤ� $0 \le z \le 1$ ���������Ƥ��뤳�Ȥ����դ�ɬ�פǤ��롣
�����ƥå׷�����ȯŸ���������Ѥ�������ˤ�
��ʬ�ϰϤ� $x \le z \le 1$ �Ȥʤ�Τǡ����ξ��ˤϡ�

\begin{equation}
\int_x^1 dz f(z) \biggl[ {\ln (1-z) \over 1-z} \biggr]_+ 
= \int_x^1 dz [f(z)-f(1)] {\ln (1-z) \over 1-z} + f(1) \frac{1}{2} \ln^2(1-x)
\end{equation}

\noindent
�Ȥʤ롣

³���ơ�$B_n^{g}$ �ϡ�

\begin{eqnarray}
B_1^{g}(x)&=&2T_R [F_g(x)- 4x(1-x)] \\ 
B_2^{g}(x)&=&4T_R F_g(x) 
\end{eqnarray}

\noindent
�Ǥ��롣�������Ʊ�ͤ�$B_1^{g}$, $B_2^{g}$ �Ϥ��줾�졢
$F_1$, $F_2/x$ ���б������ΤǤ��롣$F_3$ ���б����� $B_3^{g}$ ��
¸�ߤ��ʤ��Τϡ�$F_3$ �� $Q^2$ ȯŸ�η׻��ˤ����Ƥϥ��롼����
��ʬ�ۤ����äƤ��ʤ�����Ǥ��롣������ $F_g(x)$ �ϡ�

\begin{equation}
F_g(x)=  (1-2x+2x^2) \ln \biggl( {{1-x} \over x} \biggr)
                           +8x(1-x)-1 
\end{equation}

\noindent
�Ǥ��롣


\vfill\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\addcontentsline{toc}{section}{\protect\numberline{\S}{����ʸ��}}

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\vspace{0.7cm}

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\end{thebibliography}


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