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{\LARGE\bf ����7�N�x�@�C�m�_��}

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{\Huge\bf �j�q����ь��q�j�\���֐���$Q^2$�ˑ���}

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{\LARGE �[�R\ ���I}
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{\Large\bf ����7�N�x�@�C�m�_��}

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{\Huge\bf �j�q����ь��q�j�\���֐���$Q^2$�ˑ���}

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{\Large  �[�R\ ���I}

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�j�q����ь��q�j�\���֐��� $Q^2$ �ˑ����ɂ��Ē��ׂ��B
�����̍\���֐��� $Q^2$ �ˑ������L�q����������Ƃ��āA
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) �������ƁA
����Ƀp�[�g���Č����̌��ʂ��܂߂������������݂���B
�����ŁA�{�����ł� brute-force �@�ƌĂ΂�鐔�l��@��
�p���� FORTRAN �v���O�������쐻���A
�����Q�̕������̐��l���ɂ��Ă̌������s�����B
���̌��ʁA�j�q�\���֐��� $Q^2$ �ˑ����ɂ��ẮA
$\alpha_s$ �̍Œ᎟�̂� (LO) �̌��ʂƔ�r���āA
�������̕␳����ꂽ�ꍇ (NLO) �̌��ʂ͏����� $Q^2$ �̈��
���悭�����f�[�^���Č������B
�܂����q�j�\���֐��ɂ��ẮA
New Muon Collaboration (NMC) �� $Q^2$ �ˑ�����
�ւ�������f�[�^ $F_2^A/F_2^D$ �Ƃ̔�r���s���A
$Q^2$ ���W�ɂ�����p�[�g���Č����̌��ʂ́A
������ $x$ �̈�Œ��������Ƃ����������B
�������Ȃ��猻���_�ł��̌��ʂ����o����Ɏ����Ă��Ȃ��B
����ɁA�ŋ� NMC �ɂ�蓾��ꂽ $Q^2$ ���W��
���q�j�ˑ��� $\partial (F^{Sn}_2/ F^C_2) / \partial \ln Q^2$ ��
���Ă���͂����B
���̌��ʁA�����悻�� $x$ �ˑ����͌��ݓ����Ă���
�p�[�g�����z�̌��q�j���ł̕ω�
�� NLO �� DGLAP �������Ő����ł��邱�Ƃ����������B
�܂��A$Q^2$ ���W�ɂ�����p�[�g���Č����̌��ʂ����̌��q�j�ˑ������
�ڂ��������ł��邱�Ƃ������ꂽ�B


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���q�� $\alpha$ ���q���Փ˂����� Rutherford �U�����A���q�̓����\���Ƃ���
���q�j�̑��݂𖾂炩�ɂ����悤�ɁA
���闱�q�̓����\���ɂ��Ē��ׂ邽�߂̗L�p�Ȏ�i�̈�Ƃ��āA
���̎U���ߒ��ɂ��Ē��ׂ邱�Ƃ���������B
���̂悤�ȗ��R����A�j�q�̓����\���ɂ��Ē��ׂ邱�Ƃ�ړI�Ƃ��āA
�l�X�ȎU���̗��_����ю����������s���Ă����B

���̒��́A���ɍ��G�l���M�[�̓d�q�������̓~���[�I���Ɗj�q��
�[��e���U���ɂ��čl����B���̔����f�ʐς́A
�Q�̓Ɨ��ϐ� $Q^2=-q^2$ �� $x \equiv Q^2/2M\nu$ �̊֐�
�ł���\���֐� $F_1(x, Q^2)$ ����� $F_2(x, Q^2)$ ��p���ď����\�����Ƃ��ł���B
�������A$q$ �͎U���ɂ�����ڍs�S���^���ʁA
$\nu$ �̓G�l���M�[�ڍs�A����� $M$ �͊j�q���ʂł���B

1969 �N�� SLAC (Stanford Linear Accelerator Center) �ł́A
�z�q��d�z�q��W�I�Ƃ����d�q�Ƃ̐[��e���U�������ɂ��A
Bjorken �X�P�[�����O�����藧���Ă��邱�Ƃ������I�Ɏ����ꂽ \cite{SLACSC}�B
���� Bjorken �X�P�[�����O�Ƃ́A
�\���֐��� $Q^2$ �ɂ�炸�� $x$ �݂̂̊֐���
�\�����Ƃ��������ł��� \cite{BJSC}�B 
����͕���\�ɑ������� $Q^2$ �������Ă����Ă��A
����ȏ�̓����\���������Ă��Ȃ��Ƃ������Ƃ��Ӗ����A
�j�q�����Ƀp�[�g���ƌĂ΂��\���������Ȃ��_�󗱎q�����݂��Ă���
���Ƃ������Ă���B

�������Ȃ���A���̌�̍L�� $x$ �̈�ɂ킽��A��萸���Ȏ����ɂ��A
�\���֐��ɂ킸���� $Q^2$ �ˑ��������邱�Ƃ��m�F���ꂽ�B
���� ``�X�P�[�����O�̔j��" �ƌĂ΂�錻�ۂ�
�z�q�����̃N�H�[�N���O���[�I������o���A
�܂����̃O���[�I�����N�H�[�N$\cdot$���N�H�[�N�΂𐶐����āA
�N�H�[�N���O���[�I����C�N�H�[�N�̉_�ɕ����Ă��邽�߂�
�����Ă���Ɖ��߂ł���B
���Ȃ킿�A$Q^2$ �������ĕ���\�������Ă����Ƃ��̂悤�Ȃ�菬�����^���ʂ�����
�p�[�g���������Ă��邱�Ƃɑ�������ƍl������B

�N�H�[�N��O���[�I����
�������ݍ�p�� QCD (Quantum Chromodynamics, �ʎq�F�͊w) �ɂ����
�L�q�����B
�����āA�\���֐����̂� Lattice QCD �Ȃǂ̔�ۓ��_�I�ȕ��@�ł���
�v�Z���邱�Ƃ��ł��Ȃ����A���̃X�P�[�����O�̔j��́A
�ۓ��_�I QCD �ɂ���ĕ]�����邱�Ƃ��ł���B��̓I�Ȍv�Z�́A
�ۓ��_�I QCD �ɂ�蓱�o
����� Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) ������
��p���čs�����Ƃ��ł��� \cite{DGLAP}�B
���̕������́A���傫�ȉ^���ʂ����p�[�g�����A����m����
�p�[�g������o���āA��菬�����^���ʂ����p�[�g���ɂȂ�ߒ���
�v�Z���Ă������̂ł���B
����āA���� DGLAP �������̐��l�������߂āA�X�P�[�����O�̔j��A
���Ȃ킿�\���֐��� $Q^2$ �ˑ����ɂ��Č������邱�Ƃ́A
�ۓ��_�I QCD �̌��؂Ƃ����Ӗ��Ŕ��ɈӋ`�̂��邱�Ƃł���B
����ɁA���̕������͎�������ї��_�̌����ɂ����Ĕ��ɂ悭�p��������̂Ȃ̂ŁA
���̕������������R���s���[�^�v���O�������쐻���A���̌����҂����p�ł���悤��
���J���邱�Ƃ͗L�p�ł���B

�Ƃ���ŁA�W�I���q���j�q���猴�q�j�ɕς����ꍇ���l����B
����Ɗj�q�̍\���֐��ɑΉ����Č��q�j�̍\���֐��𓾂邱�Ƃ��ł���B
���̌��q�j�\���֐��ɂ��āA1983 �N�ɔ��ɋ����[�����ۂ��������ꂽ�B
����́A���q�j�\���֐� $F_2^A$ ���j�q�̍\���֐� $F_2^N$ ��
�قȂ��Ă���Ƃ������̂ŁA
���Ȃ킿 $F_2^A(x,Q^2)/F_2^N(x,Q^2) \neq 1$ �ł���B
���̌��ۂ� European Muon Collaboration (EMC) �ɂ����
�������ꂽ�̂� EMC ���ʂƌĂ΂� \cite{AEMC}�A
���̌��ۂ�������邽�߂ɗl�X�Ȗ͌^���񏥂���Ă���B
�����̖͌^�̂Ȃ��Ƀp�[�g���Č����Ƃ���
���̂����� \cite{GLUE}--\cite{RGR}�B
����́A�قȂ�j�q����̃p�[�g�����݂��ɑ��ݍ�p�����ėZ�����Ă����ߒ���
�l������̂ł���A shadowing (�e�U��) �ƌĂ΂�鏬���� $x$ �̈��
��������͌^�ł���B
���̌��ʂ� $Q^2$ ���W�̉ߒ��̂Ȃ��ɂ������Ă���ƍl�����A
���̌��ʂ� DGLAP �������Ɋ܂߂���������������Ă��� \cite{GLRMQ,PRQ}�B
����āA�p�[�g���Č�����p���ċ��߂��錴�q�j�\���֐���
���q�j���p�[�g�����z�� $Q^2$ ���W������ꍇ�ɂ́A
���̕�������p���邱�Ƃ��K�v�ł���B�������Ȃ���A
���q�j�\���֐��� $x$ �ˑ����ɂ��Ă͂��Ȃ�ڂ����������Ȃ���Ă���ɂ�
������炸�A���̕������̐��l����Ó����Ȃǂɂ��Ă͂��܂�
�ڂ��������͂Ȃ���Ă��Ȃ��B
����āA���q�j�\���֐��� $Q^2$ �ˑ��������̕������ŋL�q�ł���̂�
�Ƃ������Ƃ͋����̂���_�ł���B
���ɁA����̎����ł��̂悤�Ȍ��ʂ����o���邱�Ƃ��ł���ƍl������̂ŁA
�����_�ɂ����āA���̕������̐��l�������߂邱�Ƃ͏d�v�Ȍ����ۑ�ł���B
�܂��A�ŋ� New Muon Collaboration (NMC) ��
��� $\partial [F^{Sn}_2/ F^C_2] / \partial \ln Q^2$ �ɂ��Ă̐��x�̂悢
�f�[�^�������Ă��� \cite{NMCSN-C}�A
���ݓ����Ă��錴�q�j���p�[�g�����z�Ə�L�̃p�[�g���Č����̌��ʂ�
���ꂽ $Q^2$ ���W���������A���̌��ʂ�����ł���̂��Ƃ����_�ɂ��Ă�
�����������[���B

���̂悤�ȏ󋵂��ӂ܂��āA�{�����ł͊j�q����ь��q�j�̍\���֐��� $Q^2$ �ˑ���
�ɂ��Đ��l���̌������s�����B���̘_���̍\���͈ȉ��̒ʂ�ł���B
�܂���Q�͂ɂ����āADGLAP �������ɂ��ďڂ��������A
�����đ�R�͂Ō��q�j�\���֐��� $Q^2$ ���W�Ƃ��āA
�p�[�g���Č����̌��ʂ��܂� $Q^2$ ���W�������ɂ��Đ�������B
����ɁA��S�͂ł� brute-force �@�ƌĂ΂�鐔�l��@�ƁA�����
�p���� FORTRAN �v���O�����ɂ��Đ������A
��T�͂ł��̃v���O������p���Čv�Z�������ʂƗl�X�Ȏ������ʂƂ̔�r���s���B
�����čŌ�ɑ�U�͂Ō��_���q�ׂ�B


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���_�ł��q�ׂ��悤�Ɋj�q�̓����\���ɂ��Ē��ׂ邽�߂̗L�p�ȕ��@�Ƃ��āA
�����̎U���ߒ��ɂ��čl���邱�Ƃ���������B�����ŁA
�j�q�̓����\���ɂ��Ē��ׂ邽�߂ɁA�d�q�Ɨz�q�̔�e���U���ɂ��čl����B
�������n�ł̓d�q�E�z�q��e���U���̔����f�ʐς́A

\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}
$E$ , $E^{\prime}$ : & �n��Ԃ���яI��Ԃ̓d�q�̃G�l���M�[ \\
$\theta$ : & �U���p \\
$\nu$ : & �G�l���M�[�ڍs $\nu=E-E^{\prime}$ \\
$Q^2$ : & �S���^���ʈڍs�� $q$ �Ƃ��� $Q^2=-q^2$
\end{tabular}
\end{center}

\noindent
�̂悤�ɍ\���֐� $W_1$ �� $W_2$ �ŏ����\����� \cite{RGR,HM,GAUGE}�B

�Ƃ���ŁA$x=Q^2/2M\nu$ ��L���ɕۂ��Ȃ��� 
$Q^2 \rightarrow \infty $, $\nu \rightarrow \infty$ ��
�Ɍ����l�����Ƃ��A$W_1$ �� $W_2$ ���A
{
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\begin{eqnarray}
MW_1(Q^2,\nu) & \rightarrow & F_1(x) \\
\nu W_2(Q^2,\nu) & \rightarrow & F_2(x) 
\end{eqnarray}
\setcounter{equation}{\value{enumi}}}

\vspace{-0.5cm}
\noindent
�̂悤�ɁA$x$ �݂̂̊֐��Ƃ��ĕ\�����Ƃ������Ƃ� Bjorken �ɂ���Ē񏥂���A
����� Bjorken �X�P�[�����O�ƌĂ� \cite{BJSC}�B
�����Ă��� Bjorken �X�P�[�����O�� SLAC ��
��� 1969 �N�Ɏ����I�Ɋm���߂�ꂽ \cite{SLACSC}�B

$Q^2$ ��傫�����Ă����Ƃ������Ƃ͕���\�������邱�Ƃɑ�������B
����� $Q^2$ ������������\���Ⴂ�U���̏ꍇ�ɂ́A
�z�q�̓����\�������邱�Ƃ͂ł��Ȃ��B���̂��߁A�U���̉��z���q��
�z�q�S�̂Ƒ��ݍ�p���Ă���B
������ $Q^2$ ��傫�����Ă����Ƃ₪�ĉ��z���q�͗z�q�̓����ɂ܂œ��荞�݁A
�z�q�͉󂳂�A�I��Ԃɂ����Ă͖����̃n�h�����Q�����������B
���̂悤�Ȕ�e���U���ɂȂ�Ɨz�q�̓����\���������n�߂�B
�����Ă���� $Q^2$ ��傫�������A��e���x�̔��ɑ傫���̈��
�[��e���̈�ƌĂсA���̗̈�ł̎U����[��e���U���ƌĂԁB
Bjorken �X�P�[�����O�̎咣����Ƃ���́A���̂悤�Ȑ[��e���̈�ɂ����ẮA
$Q^2$ �������ĕ���\�𑝂��Ă��A����ȏ�̓����\���������Ă��Ȃ�
�Ƃ������Ƃł���B���Ȃ킿�z�q�����ɂ͍\���̂Ȃ��_�󗱎q�����݂��A
�[��e���̈�ł́A
���v�g������̉��z���q�͂��̓_�󗱎q�Ƒ��ݍ�p�����Ă���Ƃ������Ƃł���B
���̓_�󗱎q�̓p�[�g���ƌĂ΂��B

�Ƃ���ŗz�q�̉^���ʂ𖳌���Ƃ���n�ɂ����ẮA�ϐ� $x$ ��
�z�q�ɑ΂���p�[�g���̎��^���ʔ�ɑ������邱�Ƃ��������B
���Ȃ킿���z���q�Ƒ��ݍ�p����p�[�g���̎��^���ʂ� $P$ ���j�q��
�^���ʂƂ��� $xP$ �ł���B

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\subsection{{\it �X�P�[�����O�̔j��� DGLAP ������}}}
\vspace{0.4cm}

�j�q�����ɍ\���������Ȃ��_�󗱎q�̑��݂������� Bjorken �X�P�[�����O�ł��邪�A
���̌�̍L�� $x$ �̈�ɂ킽��A��萸���Ȏ����ɂ��A�킸���ɔj��Ă��邱�Ƃ�
�킩�����i�} \ref{fig : fig2-1}�j�B
���̌��ۂ� ``�X�P�[�����O�̔j��" �ƌĂ΂��B�������Ȃ���A
����͊j�q�̓����\����
�N�H�[�N�E�p�[�g���͌^�ɂ�闝�������ł���Ƃ������Ƃ��Ӗ�����킯�ł͂Ȃ��A
�j�q�����̃N�H�[�N���O���[�I����C�N�H�[�N�̉_���܂Ƃ��Ă��邱�Ƃ�
�N������ƍl���邱�Ƃ��ł���B
���Ȃ킿 $Q^2$ ���傫���Ȃ��ĕ���\�������ƁA���̂悤�ȉ_�������Ă���
�ƍl������B
����ƁA��̓I�ɂ� $Q^2$ ���傫���Ȃ�ɂ��
�傫�� $x$ �����p�[�g��������A
��菬���� $x$ �����p�[�g���������邱�ƂɂȂ�B
�} \ref{fig : fig2-1} ������Ɗm���ɏ����� $x$ �ł� $Q^2$ �̑�����
�Ƃ��ɍ\���֐����������Ă���A�傫�� $x$ �ɂ����Ă� $Q^2$ �������ɂ���������
�\���֐��͌������Ă��邱�Ƃ��킩��B

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\vspace*{-1.0cm}
\begin{center}
\epsfile{file=fig2-1.eps,width=10.0cm}
\end{center}
\vspace{-1.5cm}
\caption{�X�P�[�����O�̔j��}
\label{fig : fig2-1}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

���̂悤�ȃX�P�[�����O�̔j����L�q���������
�Ƃ��āADokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) ������
������ \cite{DGLAP}�A����͈ȉ��̂悤�Ȕ��ϕ��������ł���B

\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
������ $t$ �́A

\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
�Œ�`�����ϐ��ł���A$\alpha_s(Q^2)$ �͋������ݍ�p�̎��������萔�ł���B
�܂��A$q_i(x,t)$, $\bar{q}_i(x,t)$�����ꂼ��
�t���[�o�[ $i$ �����N�H�[�N���z����є��N�H�[�N���z�Ƃ��āA
$q_i^{\ +}=q_i+\bar{q}_i$ �ł���A
����� $q_{_{NS}}$, $q_{_{S}}$, $g$ �͂��ꂼ��
�t���[�o�[��P�d�����z�A�t���[�o�[�P�d�����z����уO���[�I�����z��\���B
������ $\widetilde{f}(x)=xf(x)$ �ł���A
$N_f$ �̓t���[�o�[�̐��ł���B

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

�����ŁA$P(y/x)$ �͕���֐��ƌĂ΂����̂ŁA
���� (\ref{eqn : DGLAPQI}) $\sim$ (\ref{eqn : DGLAPG}) ��
�� $P_{ij}(y/x)$ �͉^���ʔ� $y$ �����p�[�g�� $j$ �����򂵂āA
�^���ʔ� $x$ �����p�[�g�� $i$ �ɂȂ�m����\���Ă���B
����āA$\alpha_s$ �̍Œ᎟ (Leading Order,  �Ȍ� LO) �݂̂��l����ꍇ�ɂ́A
(\ref{eqn : DGLAPQI}) ����� (\ref{eqn : DGLAPQS}) ����
�E�ӑ�P������ё�Q����
���ꂼ��} \ref{fig : fig2-2} �� (a), (b) ��\���A(\ref{eqn : DGLAPG}) ����
�E�ӑ�P������ё�Q���͂��ꂼ�� (c), (d) �̉ߒ���\���Ă���B
�܂��A(\ref{eqn : DGLAPNS}) ���̉E�ӂ͐} \ref{fig : fig2-2} �� (a) �̉ߒ���\���A 
����� LO �݂̂̏ꍇ�ɂ́A(a) �̉ߒ���
�t���[�o�[���ς�邱�Ƃ͂Ȃ��̂ŁA

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

\noindent
�ł���B����āA
���̏ꍇ�ɂ� (\ref{eqn : DGLAPQI}) ����� (\ref{eqn : DGLAPQS}) ����
����ɊȒP�ɁA

\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
�Ə������Ƃ��ł���B
�������A$q_i^+$ ���z�Ɋւ��ẮA�O���[�I�����z�̕�����
�Ƀt���[�o�[�P�d�����z $q_{_{S}}$ ���܂܂�Ă��邽�߂ɁA
���̕��z��P�Ƃł͌v�Z���邱�Ƃ͂ł����A
�t���[�o�[�P�d�����z $q_{_{S}}$ �������Ɍv�Z���Ȃ���΂Ȃ�Ȃ��B

�����̕������ɂ��΁A���� $Q^2$ �ɂ�����p�[�g�����z���킩��΁A
��������ƂɔC�ӂ� $Q^2$ �ł̃p�[�g�����z�𓾂邱�Ƃ��ł���B
���̂��Ƃ��炱���̕������� $Q^2$ ���W�������ƌĂ΂��B
���̂悤�ɂ��ē�����p�[�g�����z�� LO �̏ꍇ�ɂ́A

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

\noindent
�̊֌W��p���č\���֐� $F_2$ �ɒ������Ƃ��ł���B������ $e_i$ ��
�t���[�o�[ $i$ �̃N�H�[�N�����d�ׂł���B
����� DGLAP �������Ńp�[�g�����z�� $Q^2$ �ˑ����𒲂ׂ�΁A
�\���֐��� $Q^2$ �ˑ����������A�����l�Ɣ�r���邱�Ƃ��ł���B
�܂��\���֐� $F_1$ �� $F_2$ �ɂ́ALO �ł�

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

\noindent
�Ƃ����֌W�����藧���Ă���̂ŁA�ǂ��炩��������������΂悢���Ƃ��킩��B
���̊֌W�� Callan-Gross �̊֌W�Ƃ�΂��B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
{\renewcommand{\thesection}{\it {\arabic{section}}}
\subsection{{\it $\alpha_s$ �̍������̊�^}}}
\vspace{0.4cm}

�{�����ł͓��� $\alpha_s$ �̍������̊�^����ꂽ DGLAP �������ɂ���
���������s���� \cite{HWW,NLO}�B����͂R�̕����Ɋ܂܂�Ă���B
�܂���P�ɂ� $\alpha_s$ ���̂��̂ɓ����Ă���B
���Ȃ킿 LO �̏ꍇ�ɂ́A

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

\noindent
�ł��邪�A��P���̍����� (Next-to-Leading Order, �Ȍ� NLO) ��
��^���܂ޏꍇ�ɂ́A

\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
�ƂȂ�B�Ƃ���� NLO �̊�^���l����ꍇ�ɂ́A���荞�ݏ�����
���肵�Ȃ���΂Ȃ�Ȃ��B
�{�_���ł� modified minimal subtruction scheme ($\overline{MS}$) ��
�Ă΂�鏈����p�����B
�܂��A$\beta_0$ �� $\beta_1$ �͂��ꂼ��ȉ��̎��Œ�`����Ă���B

\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
�������A

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

\noindent
�ł���A$N_f$ , $N_c$ �͂��ꂼ��t���[�o�[�̐��ƃJ���[�̐� $(N_c=3)$ �ł���B

��Q�� DGLAP �������ɓ����Ă���B
���������̊�^�� (\ref{eqn : DGLAPNS}) $\sim$ (\ref{eqn : DGLAPG}) ����
�`�͕ς����A����֐���
�Ȃ��Ɋ܂߂邱�Ƃ��ł���B
���Ȃ킿�A(\ref{eqn : DGLAPNS}) $\sim$ (\ref{eqn : DGLAPG}) ���ɂ�����
����֐� $\widetilde{P}(x)$ �́A

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

\noindent
�ƂȂ�B�����ŁA$R(x)$ �́A

\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
�Œ�`�����B
�������A$P^{(0)}(x)$ �� LO �̕���֐��� $P^{(1)}(x)$ �� NLO �̊�^�ł���B
���� (\ref{eqn : R}) ���̉E�ӑ�Q���� $Q^2$ �̑���� (\ref{eqn : tq2}) ����
��`����� $t$ ��p���Ă��邱�Ƃɂ����̂ł���B
LO ����� NLO �̕���֐��͕t�^ A �Ɏ����B

����ɑ�R�ɁA�p�[�g�����z����\���֐��ɂ���Ƃ��ɂ� NLO �̊�^��
�����Ă���B�\���֐��ƃp�[�g�����z�̊֌W�́A

\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
�ƕ\�����Ƃ��ł���B
�����ŁA�Y�� $n$ �� $F_1$, $F_2/x$, $F_3$ �̂����ꂩ��
�����A$i$ �̓t���[�o�[��\���B
�����ČW���֐� $C_n^{q,g}$ �� NLO �̊�^�������Ă���B�W���֐��́A

\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
�Ƃ����`�����Ă���B$B_n^{q}(x)$, $B_n^{g}(x)$ �͕t�^ B �Ɏ����B
����āALO �̏ꍇ�ɂ́A

\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
�ƂȂ��č\���֐��̓N�H�[�N���z���̂܂܂ŕ\����邪�A
$Q^2$ ���W�� NLO �̊�^���܂߂�ꍇ�ɂ́A�\���֐���
�����Ƃ��� $B_n^{q}(x)$, $B_n^{g}(x)$ �̊�^�����Ȃ���΂Ȃ�Ȃ��B
�����Ă��̌�ŗႦ�� $F_2$ ���A

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

\noindent
�Ƃ��ċ��߂邱�ƂɂȂ�B

�܂����̂��Ƃ���A���� $Q_0^2$ �ł̍\���֐���^����
�\���֐����g�� $Q^2$ ���W������ꍇ�ɂ́A
����ɗ]���ȍ�������֐��ɓ����Ă���B
�\���֐��ɑ΂���ANLO �̊�^���܂� DGLAP ��������
�ȉ��̂悤�ɕ\�����B

\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
�����ŁA$F_{_{NS}}^{\pm}$ �� $q \pm \bar{q}$ �^�̃t���[�o�[��P�d����
�\���֐���\���B����ɁA

\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
�ł���B${\bf P}^{(0),(1)}(x)$, ${\bf D} ^{(1)}(x)$, ${\bf E}(x)$ ��
�t�^ A �Ɏ����B
(\ref{eqn : QEFNS}) ����� (\ref{eqn : QEFI}) ���Ŏ������悤�� LO �̏ꍇ�ɂ́A
�\���֐��̓N�H�[�N���z�ɓ������̂ŁA���̂悤�ȗ]���ȍ��͌���Ă��Ȃ��B
����Ă��̏ꍇ�ɂ́A�ӂ��� LO �� DGLAP �������Ōv�Z���邱�Ƃ��ł���B
���̗l�ɍ\���֐����g�� $Q^2$ ���W��������@��
�P�X�e�b�v�`���ƌĂ� \cite{HWW,NLO}�A�p�[�g�����z�� $Q^2$ ���W������
���̌�ō\���֐��ɒ������@���Q�X�e�b�v�`���ƌĂԁB


\vfill\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{{\bf ���q�j�\���֐��� $Q^2$ ���W}}
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}

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

�j�q�ƃ��v�g���̐[��e���U���ɂ����āA�W�I���q���j�q���猴�q�j�ɕς���ƁA
���q�j�\���֐��𓾂邱�Ƃ��ł���B���̌��q�j�\���֐����j�q�̍\���֐���
�������Ȃ����Ƃ� EMC �ɂ���� 1983 �N�ɔ�������A
EMC ���ʂƌĂ΂�Ă��� (�} \ref{fig : fig3-1}) \cite{AEMC}�B
���Ȃ킿�A$F_2^A$ , $F_2^D$ �����ꂼ�ꌴ�q�j�\���֐��A�d�z�q��
�\���֐��Ƃ��āA

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

\noindent
�ł���B�������A�d�z�q�͎��R�Ȋj�q�Ƌߎ��ł���ƍl���邱�Ƃ��ł���B

%%% 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} ����킩��悤�ɁA
���q�j���ɂ����� $F_2$ �̕ω��͈ȉ��̂T�̗̈�ɕ����邱�Ƃ��ł���B

\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item
$x$ \raisebox{-1mm}{$\stackrel{\textstyle <}{\sim}$} $0.05$ ��
�̈�ɂ����� $R_2$ �� $1$ ��菬�����Ȃ�B����� shadowing �ƌĂԁB
\item
$0.1$ \raisebox{-1mm}{$\stackrel{\textstyle <}{\sim}$} $x$ 
\raisebox{-1mm}{$\stackrel{\textstyle <}{\sim}$} $0.25$
�̗̈�ł� $R_2 > 1$ �ƂȂ�B
���̌��ۂ� shadowing �ɑ΂��� antishadowing �ƌĂ΂��B
\item
$x=0.25 \sim 0.30$ �̗̈�� $R_2 \simeq 1$ �B
\item
$0.3$ \raisebox{-1mm}{$\stackrel{\textstyle <}{\sim}$} $x$ 
\raisebox{-1mm}{$\stackrel{\textstyle <}{\sim}$} $0.8$
�̗̈�ł� $R_2 < 1$ �B
\item
$x \rightarrow 1$ �ɂ�� $R_2$ �� $1$ ���z���P���ɑ�������B
\end{enumerate}

\noindent
�����āA�����̌��ۂ�������邽�߂�
�l�X�Ȗ͌^���񏥂���Ă���B
���̒��ŁA������ $x$ �̈� (a) �ɂ����� shadowing ���������͌^�Ƃ��āA
�p�[�g���Č��� (parton recombination) ������ \cite{GLUE}--\cite{RGR}�B

�^���ʔ� $x$ �����p�[�g���̏c�����̉^���̈� $\Delta L$ ��
 $P$ ���j�q�̉^���ʂƂ��Ă��悻 $\Delta L \sim 1/xP$ �ł���A$x$ ��
 �������Ȃ�΂Ȃ�قǑ傫���Ȃ��Ă����B
���̉^���̈悪�c�����̕��ϊj�q�ԋ������z����悤�ȏ����� $x$ �����p�[�g���́A
���̊j�q�̃p�[�g���Əd�Ȃ荇���A���͂⎩�R�ȃp�[�g���ƌ��Ȃ����Ƃ͂ł��Ȃ��B 
�p�[�g���Č����Ƃ́A�����̏����� $x$ �����p�[�g���ǂ��������ݍ�p����
�Z������ߒ��ł���B���̂悤�ȃp�[�g���Č������N����ƁA
������ $x$ �����p�[�g���ǂ������Z�����Ă��̕��z�����邽�߁A
���̗̈�ō\���֐����������Ȃ�B
����ď����� $x$ �̈�� $F_2^A/F_2^D < 1$ �ƂȂ�A
shadowing ���N����ƍl������B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
{\renewcommand{\thesection}{\it {\arabic{section}}}
\subsection{{\it $Q^2$ ���W�Ɋւ���p�[�g���Č����̌���}}}
\vspace{0.4cm}

shadowing ��������邽�߂ɓ������ꂽ�p�[�g���Č����́A
$Q^2$ ���W�̉ߒ��̂Ȃ��ɂ������Ă���ƍl���邱�Ƃ��ł��A���̌��ʂ�
���ꂽ $Q^2$ ���W��������������Ă��� \cite{GLRMQ,PRQ}�B
���̕������� $Q^2$ ��������ɂ��������A���傫���^���ʔ� $x$ �����p�[�g����
������ $x$ �̃p�[�g���ɕ��򂵂Ă����ߒ���
�v�Z���Ă��� DGLAP �������ɁA�} \ref{fig : fig3-2} �̂悤��
������ $x$ �����p�[�g���ǂ������Z�����Ă����ߒ���␳���Ƃ��Ċ܂�ł���B
����āA�p�[�g���Č����ɂ�蓾���錴�q�j�\���֐���
���q�j���p�[�g�����z�� $Q^2$ �ˑ����ɂ��čl����ꍇ�ɂ́A
���̕�������p���Čv�Z���s�����Ƃ��K�v�ł���B

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

��̓I�ɂ́A�܂��A���R�Ȋj�q���ɂ�����p�[�g�����z�ɑ΂��āA���q�j����
�p�[�g�����z�� $\delta \widetilde p^A(x,t)$ �����␳���󂯂�ƍl����B
���Ȃ킿�A

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

\noindent
�ł���B�������A���q�j���p�[�g�����z�͎��ʐ� $A$ �Ŋ��������̂Ƃ��Ē�`�����B
�����āA���R�Ȋj�q���̃p�[�g�����z $\widetilde p(x,t)$ ��
�␳�� $\delta \widetilde p^A(x,t)$ �����ꂼ�� $Q^2$ ���W�����A
�Ō�� (\ref{eqn : PA}) ���ɂ���� $Q^2$ ���W�������q�j���̃p�[�g�����z�𓾂�B

�j�q���̃p�[�g�����z�����ۂɂ͊j�q���ł̃p�[�g���Č����̌��ʂ�
�󂯂�ƍl������B
����́A�j�q���ł̏����� $x$ �����p�[�g���ǂ����̏d�Ȃ荇���ɂ���ċN����
�Z���ߒ��ł���B���̌��ʂ���ꂽ $Q^2$ ���W�������͈ȉ��̂悤�ɂȂ�B

\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
�������A�p�[�g���Č����̌��ʂ� NLO �Ɠ��� $\alpha_s$ ��
�I�[�_�[�Ƃ��ē����Ă���̂ŁA���Ƃ� DGLAP �������� NLO �̊�^���܂߂�
�v�Z������K�v������B
�����̎��̉E�ӂ̑�P���̐ϕ������� NLO �̊�^���܂ޒʏ�� DGLAP ������
�ł���A���̑��̍����p�[�g���Č����̌��ʂ�\���B
�������A(\ref{eqn : PRQI}), (\ref{eqn : PRQS}) �̕␳����
�} \ref{fig : fig3-2} (a) �� (b) �̉ߒ���\���A(\ref{eqn : PRG}) ��
�␳���� (c) �̉ߒ���\���B
�����ŁANLO �̊�^���܂ޕ���֐� $P_{q_i^+q_j^+}$ ���A

\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
�ƕ\���邱�Ƃ������ɗp���Ă��� (�t�^ A �Q��)�B
������A$q_i^{\ +}$ �̕��z���v�Z����ꍇ�ɁA
���ׂẴt���[�o�[ $i$ �ɑ΂��� $q_i^{\ +}$ �͕K�v�ł͂Ȃ��A���̘a�ł���
�t���[�o�[�P�d�����z $q_{_{S}}$ ���������Ɍv�Z����΂悢���Ƃ��킩��B
������ $P_{_{NS}}^{(0)}(x)$, $P_{q^+, NS}^{(1)}(x)$ , $F_{qq}(x)$ ��
�t�^ A �Ɏ����B
�␳���̒��Ɍ���� ${\bar P}_{qg}(x)$ �́A
 
\begin{equation}
\bar {P}_{qg}(x) = 2 ~(-2x+15x^2-30x^3+18x^4)
\end{equation}

\noindent
�ł���B����ɁA$G_{HT}(x,t)$ �͍����O���[�I�����z
�ƌĂ΂�镪�z�ŁA�} \ref{fig : fig3-2} (b) �Ɏ������
�ߒ��̂Ȃ��Ɍ���Ă��镪�z�ł���B���̕��z�� $Q^2$ ���W�́A

\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
�̕������ɏ]���B
���̕��������������߂ɂ� $\widetilde G_{HT}$ �̏������z���K�v�ł��邪�A
���̕��z����̃O���[�I���̑��ݍ�p�Ɋ֌W���邱�Ƃ���A

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

\noindent
�Ɖ��肷��B������ $K_{HT}$ �͎��R�ȃp�����[�^�ł���A
�����ł͎Q�l���� \cite{PRQ} �ɏ]�� $K_{HT}=1.68$ �Ƃ����B

����� $\delta \widetilde p^A(x,t)$ �� $Q^2$ ���W�� DGLAP ��������
���q�j���̃p�[�g���Č����̌��ʂ���ꂽ�������ɏ]���B
��̓I�ȕ������̌`�͈ȉ��̂悤�ɂȂ�B

\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
�������A�����Ă��錴�q�j�̎��ʐ��� $A$ �Ƃ��� $R_A=R_1 A^{1/3}$ �ł���A
�����ł͊ȒP�� $R_1=R_0=1.0$ fm �Ƃ����B
�܂��A$\bar{n}$ �͌��q�j���ɂ�����j�q�����x�ł���B
������A$R_A \bar{n} \propto A^{1/3}$ �ƂȂ�A����̓p�[�g���Č�����
���ʂ��A���q�j���̃p�[�g���ǂ������d�Ȃ荇���x�����ɔ�Ⴕ�A
����ɂ��ꂪ�c�����̊j�q���Ɣ�Ⴕ�Ă��邱�ƂɋN������B
�܂��A�萔 $K$ �� $A=1$ �Ƃ����Ƃ��ɕ␳����������悤�� $K=9/8$ �Ƃ����B

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

�Ƃ���ŁA�p�[�g���Č����Ƃ����ꍇ�ɂ̓O���[�I���ǂ����̍Č����̑��ɁA
���ۂɂ̓O���[�I���ƃN�H�[�N��N�H�[�N�ƃN�H�[�N
�̍Č����ɂ��Ă��l���Ȃ���΂Ȃ�Ȃ��B
�������A�} \ref{fig : fig3-3} �ɂ���悤�ɁA
�p�[�g���Č������N����悤�ȏ����� $x$ �̈�ɂ����ẮA
�O���[�I���̕��z�ɑ΂��ăN�H�[�N�̕��z�͔��ɏ������̂ŁA
������ $Q^2$ ���W�Ɍ����Ă���̂�
�O���[�I���ǂ����̍Č����ł���ƍl���Ă悢�B
���������āA�O���[�I���ƃN�H�[�N�̗����̍Č�������舵���̂�
���ɕ��G�ł��邱�Ƃ���A���i�K�ɂ����Ă̓O���[�I���ǂ����̍Č����̂�
���l���Ă���B

\vfill\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{{\bf ���l�v�Z}}
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}

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

��X�̃O���[�v�ł́A Laguerre ��������p�������l��@�ɂ��A
DGLAP �������̐��l���ɂ��Č�����i�߂Ă��� \cite{LAG}�B
���̕��@�̓p�[�g�����z�� Laguerre �������œW�J����
�v�Z��������̂ŁALaguerre �������̊֐��`���p�[�g�����z��
�`�Ɨގ����Ă��邽�߂Ɏ����������A
SUN--IPX ���g�p���Ă킸�����b�Ƃ������ɒZ��
�v�Z���ԂŌv�Z���x�̗ǂ����ʂ𓾂邱�Ƃ��ł���B
�������A���Ƀt���[�o�[��P�d���̏ꍇ�A
�傫�� $x$ �̈����ɏ����� $x$ �̈�ł��v�Z���x�������Ă��܂��B
�܂�������Ƃ��Ă���A�p�[�g���Č����̌��ʂƂ��Ĕ���`�ȕ␳����
�܂� $Q^2$ ���W�������͉������Ƃ��ł����A
����ɏ����� $x$ �̈�œ��قȂӂ�܂������鏉�����z��
�������Ƃ��ł��Ȃ��Ȃǂ̌��_������B

�����Ŗ{�����ł͑��̐��l��@�Ƃ̔�r�Ƃ����ړI�����˂āA
���ϕ��������̐��l��@�Ƃ��Ă͍ł��ȒP�� brute-force �@�ƌĂ΂����@��
�p���Đ��l���̌������s�����B 
brute-force �@�Ƃ� $x$ ����� $t$ �� $N_x$, $N_t$ �̏������X�e�b�v��
���ꂼ�ꕪ�����A$t$ �ł̔�������� $x$ �ł̐ϕ���

{
\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
�̂悤�ɒu�������Đ��l�v�Z���s�����̂ł���B
�������A$\Delta t_j = t_{j+1} - t_j$, $\Delta x_k = x_k - x_{k-1}$ �ł���B
��Ƃ��ăt���[�o�[��P�d���̏ꍇ���l���Ă݂�΁A

$$
{\partial \over {\partial t}} \ \widetilde q_{_{NS}}(x,t)\ 
= \ \int_x^1{dy\over y} \ 
\widetilde P_{_{NS}}\biggl({x\over y}\biggr) \ 
\widetilde q_{_{NS}}(y,t)
$$
$$
\Downarrow
$$
\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
�ƂȂ�A���� $Q^2=Q_0^2 \ (t_0=0)$ �ɂ�����p�[�g�����z��^����΁A
���̎����J��Ԃ��p���邱�Ƃɂ��C�ӂ� $t$ ���Ȃ킿�C�ӂ� $Q^2$ �ɂ�����
�p�[�g�����z�𓾂邱�Ƃ��ł���B
���̐��l��@�͔��ɊȒP�ȕ��@�ł��邪�A��荂���v�Z���x�̌��ʂ𓾂悤��
����Ȃ�A �X�e�b�v�̐� $N_x$ , $N_t$ ����ɑ傫�����K�v������B
�܂����̂��߂��Ȃ蒷���v�Z���Ԃ�v����B
�v�Z���x�ɂ��Ă� 4.3 �߂ŏڂ����q�ׂ�B

�Ƃ���ŁA
���񈵂��p�[�g���Č����̌��ʂ��A������ $x$ �̈�Ɍ����Ă��邱�Ƃ�A
����̎����ł�菬���� $x$ �ł̕��z�����肳��邱�Ƃ��\�z����邱�ƂȂǂ���A
�����_�ł́A������ $x$ �̈�ɋ���������B����ď����� $x$ �̈�
�Ő��x�ǂ��v�Z���s�����߂� $x$ �����̂܂� $N_x$ �ɓ���������
�̂ł͂Ȃ��A$\log_{10} x$ �� $N_x$ �ɓ��������Čv�Z���s�����B
�������A���̏ꍇ�ɂ͌v�Z���� $x$ �̍ŏ��l�����߂Ȃ���΂Ȃ�Ȃ��B

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

�O�߂ŏq�ׂ����l��@��p���� FORTRAN �v���O�������쐻�����B
���łɏq�ׂ��悤�� DGLAP �������́A���_�Ǝ����̂ǂ���̌����ɂ�
�p��������ɏd�v�ȕ������ł���B
����āA�{�����ł̂ݗp����v���O�����Ƃ��Ăł͂Ȃ��A
�l�X�Ȍ����҂����p�ł���悤�Ɍ��J���邱�Ƃ�ړI�Ƃ��A
�ėp���̍����v���O�������쐻�����B
����쐻�����v���O�����͈ȉ��̗l�Ȃ��̂ł���B

\vspace{0.5cm}
\begin{center}
\begin{tabular}{r l}
�v���O������ : & BF1 \\
�v���O�������� : & FORTRAN 77 \\
�v���O�����s�� : & 2439 �s \\
�ΏۂƂȂ������ : & Doksitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) ������ \\
 & �p�[�g���Č����̌��ʂ��܂� $Q^2$ ���W������ \\
�v�Z�\�ȕ��z�֐� : &  $xq_{_{NS}}(x,Q^2)$, $xq_{_{S}}(x,Q^2)$, 
$xq_i^{\ +}(x,Q^2)$, $xg(x,Q^2)$\\
�v�Z�\�ȍ\���֐� : & $xF_{_{NS}}(x,Q^2)$, $xF_{_S}(x,Q^2)$, $xF_i^{+} (x,Q^2)$ \\
���l��@ : & brute-force �@ \\
�������z : & MRS(G), ���q�j���p�[�g�����z \cite{SKF2} \\
& (���̏������z���t�@�C�����ǂݍ��݂��\) \\
�v�Z���� : & �� 5 �� (NLO nonsinglet case) \\
& �� 60 �� (NLO singlet case) \\
& �� 80 �� (NLO singlet with Parton Recombinations case) \\
& (AlphaSever 2100 4/200 �ł̌v�Z����) \\
\end{tabular}
\end{center}

\bigskip

\noindent
�Ȃ��A�\���֐��̌v�Z�ɂ��ẮA2.3 �߂ŏq�ׂ�
�\���֐����̂��̂� $Q^2$ ���W������P�X�e�b�v�`���ƁA
�p�[�g�����z�� $Q^2$ ���W�����āA���ꂩ��\���֐����v�Z����Q�X�e�b�v�`����
�Q�̌`���Ōv�Z���邱�Ƃ��ł���悤�ɂ����B
�������A�{�����ɂ����Ă͏������z�Ƃ��ăp�[�g�����z���^������̂ŁA
�Q�X�e�b�v�`����p���Čv�Z���s�����B
�p�[�g�����z�̏������z���� (\ref{eqn : QFNS}), (\ref{eqn : QFI}) ����
�\���֐����v�Z���A���ꂩ��P�X�e�b�v�`��
�Ōv�Z���邱�Ƃ��\�ł��邪�A���̂悤�ȕ��@�ł̌v�Z�͍s���Ă��Ȃ��B
���̗��R�́A�P�X�e�b�v�`���̏ꍇ�ɂ͕���֐��ɂ���ɗ]���ȍ�
���܂܂�Ă���̂ŁA�v�Z���Ԃ������Ȃ�A����Ɍv�Z���x�������邽�߂ł���B
�������A�P�X�e�b�v�`���ƂQ�X�e�b�v�`���̂Q�̕��@�Ōv�Z���s���A
�v�Z�덷�������΂����̌��ʂ���v���邱�Ƃ͊m�F�����B

���̃v���O���������s����Ƃ��ɂ́A�ȉ��� 21 �̃p�����[�^��
�t�@�C�� \#10 ������͂��Ȃ���΂Ȃ�Ȃ��B

\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
�܂��������z�̓v���O�����̍Ō�̕����Ɉȉ��̂悤�Ȋ֐���
�^�����Ă���B
\vspace{0.5cm}
\begin{center}
\begin{tabular}{r l}
QNS0 : & �t���[�o�[��P�d�����z \\
QS0 : & �t���[�o�[�P�d�����z \\
G0 : & �O���[�I�����z \\
QSA0 : & ���q�j���ɂ�����t���[�o�[�P�d�����z \\
GA0 : & ���q�j���ɂ�����O���[�I�����z \\
QI0 : & $q_i^+$ ���z \\
QIA0 : & ���q�j���ɂ����� $q_i^+$ ���z
\end{tabular}
\end{center}

\bigskip

\noindent
�����ŗ^���Ă��� MRS(G) �Ȃǂ̏������z�ł͂Ȃ��A
���̏������z��p����ꍇ�ɂ́A�����̊֐������������邩�A
�������� Input=2 �Ƃ��āA
�t�@�C�� \#13, \#14, \#15, \#16, \#17 ����ǂݍ��܂���΂悢�B
�������AHMRS-B, Duke-Owens I (valence quark), CCFR ($F_3$), ���q�j
���t���[�o�[��P�d�����z \cite{SKF2}, Qiu \cite{PRQ} �̏������z��
�R�����g���Ƃ��ė^���Ă���B

���̃v���O�����́A�O�ɂ��q�ׂ��悤�ɑ��̌����҂̂��߂Ɍ��J���邱�Ƃ�
�ړI�Ƃ��č쐻���ꂽ�B�����ŁA�C���^�[�l�b�g��ʂ���
�ȉ��� anonymous FTP �� WWW �œ���ł���悤�ɂ����B
�t�@�C������ bf1.fort77.gz �ł���B

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
�Ȃ��A���̃v���O�����Ɋւ��邳��ɏڂ���������
�Q�l���� \cite{MK} �łȂ���Ă���B


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
{\renewcommand{\thesection}{\it {\arabic{section}}}
\subsection{{\it �v�Z���x}}}
\vspace{0.4cm}

4.1 �߂ŏq�ׂ��悤�ɁA���̐��l��@�̌v�Z���x�̓X�e�b�v�̐� $N_x$ �� $N_t$ ��
�ˑ����Ă���B
���̂��߂ɁA$N_x$ , $N_t$ �ƌv�Z���x�̊֌W�ɂ��Ē��ׂ��B
�����ł͏������z
�Ƃ��� HMRS-B \cite{HMRS} �ƌĂ΂�镪�z��p���āA
$Q_0^2$ ���� $Q^2=200 \ GeV^2$ �܂� NLO �̊�^����ꂽ
�������� $Q^2$ ���W�������B
�������A�t���[�o�[�̐��� $N_f=4$ �� $x$ �̍ŏ��l�� $x=10^{-4}$ �Ƃ����B
HMRS-B �� $Q_0^2=4.0 \ GeV^2$ �ŁA

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

\noindent
�̂悤�Ȋ֐��`�ŗ^�����Ă���A�e�X�̕��z�֐��ɂ��Ẵp�����[�^��
���\�Ɏ������B�������A$\Lambda=0.19 \ GeV$ �ł���B
���̕��z�ɂ�� $q_{_{NS}}$, $q_{_{S}}$, g ��
�������z��} \ref{fig : fig4-1} �Ɏ����B

\begin{table}[htbp]
\caption{HMRS-B �̃p�����[�^}
\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{�������z HMRS-B}
\label{fig : fig4-1}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�܂��̓t���[�o�[��P�d�����z�̏ꍇ�ł� $N_t$ �ˑ����ɂ��Ē��ׂ��B
���Ȃ킿�A$N_x$ �� $N_x=1000$ ��
�Œ肵�� $N_t$ �� $N_t=10$, 20, 50, 500 �Ƃ��Čv�Z�����B
���ʂ͐} \ref{fig : fig4-2a} �ł���B���̐}���A$N_t$ ��ς��Ă�
�قƂ�Ǖ��z�͕ω����Ă��Ȃ����Ƃ��킩��B
$N_t=500$ �̕��z�ɑ΂��� $N_t=10$, 20, 50 �̕��z��
�덷�� $10^{-4} < x < 0.5$ �͈̔͂ł�
���ꂼ�� 1\%, 0.6\%, 0.2\% �ȓ��ł���A
$x=0.8$ �ł� 4\%, 2\%, 0.8\% �ł���B
���� $N_x$ �ˑ����ɂ��Ă����l�� $N_t$ �� $N_t=50$ �ŌŒ肵�āA
$N_x$ �� $N_x=100$, 300, 1000, 4000 �Ƃ��Čv�Z���A
���ꂼ��A�_���A��_�j���A�j���A�����Ő} \ref{fig : fig4-2b} �Ɏ������B
$N_x=4000$ �̕��z�ɑ΂��� $N_x=100$, 300, 1000 �̕��z�̌덷�́A
$10^{-4} < x < 0.5$ �͈̔͂ł��ꂼ�� 9\%, 3\%, 0.7\% �ȓ��ŁA
����� $x=0.8$ �ł� 24\%, 8\%, 2\% �ł���B
�����̌��ʂ��
�t���[�o�[��P�d�����z�̏ꍇ�ɂ� $N_t \geq 50$ ����� $N_x \geq 1000$ ��
����΁A$10^{-4} < x < 0.5$ �� 1\% �ȓ��̌v�Z���x�𓾂邱�Ƃ��ł���B

�����ăt���[�o�[�P�d�����z�̏ꍇ�����l�ɒ��ׂ��B
�܂� $N_x$ �� $N_x=1000$ �ŌŒ肵�A$N_t=20$, 50, 200, 1000 �Ƃ���
�v�Z�����B���ʂ͐} \ref{fig : fig4-3a} �ł���A���ꂼ��� $N_t$ �ɑ΂���
�_���A��_�j���A�j���A�����Ŏ������B
$N_t=1000$ �̕��z�ɑ΂��� $N_t=20$, 50, 200 �̕��z��
�덷�� $10^{-4} < x < 0.9$ �͈̔͂ł��ꂼ�� 5\%, 3\%, 0.4\% �ł���B
������ $N_x$ �ɂ��Ă� $N_t$ �� $N_t=200$ �ŌŒ肵�A
$N_x=100$, 300, 1000, 4000 �Ƃ��Čv�Z�����B
���ʂ͐} \ref{fig : fig4-3b} �Ɏ����B
���̏ꍇ�� $N_x=4000$ �̕��z�ɑ΂��� $N_x=100$, 300, 1000 �̕��z��
�덷�� $10^{-4} < x < 0.5$ �͈̔͂�
���ꂼ�� 8\%, 3\%, 0.7\% �ȓ��ł��� $x=0.8$ �� 24\%, 8\%, 2\% �ł���B

�Ō�ɃO���[�I�����z�ɂ��Ē��ׂ��B
�܂� $N_x$ �� $N_x=1000$ �ŌŒ肵�A$N_t=20$, 50, 200, 1000 �Ƃ���
�v�Z���āA�} \ref{fig : fig4-4a} �ɂ��ꂼ��_���A��_�j���A�j���A�����Ŏ������B
$N_t=1000$ �̕��z�ɑ΂��� $N_t=20$, 50, 200 �̕��z��
�덷�� $10^{-4} < x < 0.6$ �͈̔͂�
���ꂼ�� 5\%, 2\%, 0.4\% �ȓ��ł���B
������ $N_t$ �� $N_t=200$ �ŌŒ肵 $N_x$ �� $N_x=100$, 300, 1000, 4000 ��
���Čv�Z�������̂�} \ref{fig : fig4-4b} ��
���ꂼ��_���A��_�j���A�j���A�����Ŏ������B
���̏ꍇ�� $N_x=4000$ �̕��z�ɑ΂���$N_x=100$, 300, 1000 �̕��z��
�덷�� $10^{-4} < x < 0.6$ �͈̔͂�
���ꂼ�� 25\%, 8\%, 2\% �ȓ��ł���B���� $N_x=1000$ ��
�ꍇ�ɂ� $10^{-4} < x < 0.2$ �͈̔͂Ɍ���� 1\% �ȓ��ł���B
�t���[�o�[�P�d�����z�ƃO���[�I�����z�̌��ʂ��݂�ƁA
$N_t \geq  200$ , $N_x \geq 1000$ ��
����� $10^{-4} < x < 0.5$ �͈̔͂� 1\% �ȓ��A$0.5 < x < 0.8$ ��
�͈͂ł� 2\% �ȓ��̃t���[�o�[�P�d�����z�ƁA$10^{-4} < x < 0.2$ ��
�͈͂� 1\% �ȓ��A$0.2 < x < 0.6$ �͈̔͂ł� 2\% �ȓ��̌v�Z���x��
�O���[�I�����z�𓾂邱�Ƃ��ł���B

�ȏ�̂悤�Ȍv�Z���x�̊m�F���A�p�[�g���Č����̌��ʂ�
�܂� $Q^2$ ���W�������ɂ��Ă��s���A�قړ��l�̌��ʂ𓾂��B
����� LO �̕������Ɋւ��ẮA����ɏ��Ȃ� $N_t$, $N_x$ �ł������̌v�Z���x
�𓾂邱�Ƃ��ł����B
�Ƃ���ŁA�ǂ̕��z�ɂ����Ă� $t$ �����̃X�e�b�v�̐� $N_t$ �́A$x$ ������
�X�e�b�v�̐� $N_x$ �ɔ�ׂď��Ȃ����ł悢���Ƃ��킩��B
����͂��Ƃ��� Bjorken �X�P�[�����O���ߎ��I�ɐ��藧���Ă���A
$Q^2$ �ˑ������Ȃ킿 $t$ �ˑ����͂����܂ł����ꂩ��̂킸���Ȃ���ł���
�Ƃ������Ƃ��痝���ł���B
�܂� $x$ ���傫���̈�ɂ����Ă͌v�Z���x�����Ⴍ�Ȃ��Ă���B
����́A���z���̂̒l�����ɏ������Ȃ��Ă��邱�Ƃ�A$\log_{10} x$ ��
���������Ă��邽�߂ɑ傫�� $x$ �ł̌v�Z�_�����Ȃ��Ȃ��Ă��邱�Ƃɂ��B
���̓_�� $x$ �����̕����� $\log_{10} x$ �ł͂Ȃ��� $x$ �œ��������Ă����΁A
������x���������ł��낤�B
�������O�ɏq�ׂ��悤�Ɍ��i�K�ł͂ނ���
������ $x$ �̈�ɂ�����p�[�g�����z�̂ӂ�܂��ɋ���������̂ŁA
���̓_�ɂ��Ă͂���ȏ㌾�y���Ȃ��B

DGLAP �������A���� NLO �̊�^���܂ނ��̂̕���֐��͔���
���G�Ȏ��ɂȂ��Ă���B����ɗl�X�ȕ��z���v�Z�ł���悤�ɂ����̂�
�v���O�����͂��Ȃ蕡�G�Ȃ��̂ɂȂ����B
����ăv���O�����ɊԈႢ���Ȃ����𒲂ׂ邽�߂ɑ��̌����Ƃ̔�r���s�����B
�܂��͂��߂� HMRS \cite{HMRS} �ɂ�� $Q^2$ �ˑ����̌��ʂƂ̔�r���s���A
���̌��� $x > 0.01$ �͈̔͂� 2\% �ȓ��ň�v���邱�Ƃ��킩�����B
�܂��ACTEQ \cite{CTEQ} �� $Q^2$ �ˑ����̌��ʂƂ̔�r�ł�
�� \% �ȓ��ň�v�����B
����� Laguerre ��������p�������@�ɂ�� $Q^2$ ���W�̌��ʂƂ�
��r�ł� 1\% �ȓ��ň�v�����B
�����̌��ʂ��A�v���O�����ɊԈႢ�͂Ȃ��ƍl������B

\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{$N_t$ �ƌv�Z���x�̊֌W (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{$N_x$ �ƌv�Z���x�̊֌W (nonsinglet)}
\label{fig : fig4-2b}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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}

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{
\setcounter{enumi}{\value{figure}}
\addtocounter{enumi}{1}
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\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{$N_t$ �ƌv�Z���x�̊֌W (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{$N_x$ �ƌv�Z���x�̊֌W (singlet)}
\label{fig : fig4-3b}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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}

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{
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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{$N_t$ �ƌv�Z���x�̊֌W (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{$N_x$ �ƌv�Z���x�̊֌W (gluon)}
\label{fig : fig4-4b}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\vfill\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{{\bf �v�Z���ʂƍl�@}}
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\vspace{0.7cm}

�O�͂ŏq�ׂ� brute-force �@�ɂ�� FORTRAN �v���O������p����
�v�Z���s���A�l�X�Ȏ������ʂƂ̔�r���s�����B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
{\renewcommand{\thesection}{\it {\arabic{section}}}
\subsection{{\it �j�q�\���֐�}}}
\vspace{0.4cm}

�܂��͊j�q�\���֐��� $Q^2$ �ˑ����ɂ��čŋ߂̎����Ƃ̔�r��
�s�����B
�����ł͏������z�Ƃ��āAMRS(G) \cite{MRSG} �ƌĂ΂����̂�p�����B
���̕��z�� $Q_0^2=4.0 \ GeV^2$ �ŁA

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

\noindent
�Ƃ����֐��`�ŗ^�����Ă���A���ꂼ��̃p�����[�^�͉��\�̒ʂ�ł���B
�܂� $\Lambda=0.255 \ GeV$ �ł���B

\begin{table}[htbp]
\caption{MRS(G) �̃p�����[�^}
\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
�������A$S$ �͊C�N�H�[�N�̕��z�֐�
�ł���A$S=2(\bar{u}+\bar{d}+\bar{s}+\bar{c})$ �ł���B
���̏������z�́A���ł� 3.2 �߂Ő} \ref{fig : fig3-3} �Ɏ������B

�܂��ŏ��ɍ\���֐� $F_3$ �� $Q^2$ �ˑ����ɂ��Ē��ׂ��B
���̍\���֐� $F_3$ ��
�~���[�I����d�q��p�����[��e���U���ɂ����Ă͂łĂ��Ȃ����A
�j���[�g���m��p�����ア���ݍ�p���֌W���锽���ɂ����āA
�p���e�B���j��邱�ƂɋN�����Č���Ă���B
$F_3$ �́A

{
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\renewcommand{\theequation}{\arabic{section}.\theenumi\alph{equation}}
\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
�ƕ\����A�����̕��ς͉��N�H�[�N���z $[xu_v+xd_v]$ �ŗ^������B
�������ANLO �̊�^������ꍇ�ɂ� 2.3 �߂ŏq�ׂ��悤�ɁA
���ꂼ��̃p�[�g�����z���\���֐��ɂȂ����Ă��� $F_3$ ��
�v�Z���Ȃ���΂Ȃ�Ȃ��B

%%% 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{�\���֐� $F_3$ �� CDHSW �̎������ʂƂ̔�r}
\label{fig : fig5-1}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

�����ŏ������z�� MRS(G) �̉��N�H�[�N���z $[xu_v+xd_v]$ ��p�����B
������ $Q_0^2=4.0 \ GeV^2$ ���� $Q^2=200 \ GeV^2$ �����
�t�� $Q^2=0.6 \ GeV^2$ �܂ŁA$x=0.225$, 0.045, 0.55 ��
�R�_�ł� $Q^2$ �ˑ����� LO �̕������� NLO �̊�^����ꂽ
������ (�Ȍ�P�� NLO) �Ōv�Z���s�����B
�������A�t���[�o�[�̐��� $N_f=4$ �Ƃ��A����� $N_t=50$ , $N_x=1000$ �Ƃ����B
���̌��ʂ� CDHSW �̎������� \cite{BCDHSW} ��} \ref{fig : fig5-1} �Ɏ������B
������ NLO �̊�^����ꂽ���ʂŔj���� LO �݂̂̌��ʂł���B
$Q_0^2=4.0 \ GeV^2$ �ɂ����Ă��ł� LO �� NLO �̌��ʂ��قȂ��Ă��邪�A
����́A�O�ɏq�ׂ��悤��
�p�[�g�����z����\���֐��ɂ���Ƃ��̌W���֐��ɂ� NLO �̊�^��
�����Ă��Ă��邽�߂ł���B
���̌��ʂɂ��΁ANLO �̊�^����ꂽ���ʂ͎����l���悭�Č�����
���邱�Ƃ��킩��B�܂� NLO �̊�^�͏����� $x$ �� $Q^2$ �ő傫��
�����Ă���B

�����č\���֐� $F_2$ �� $Q^2$ �ˑ����ɂ��Ē��ׂ��B
�z�q�̍\���֐� $F_2^p$  �́ALO �� $N_f=4$ �̏ꍇ�ɂ́A

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

\noindent
�ŕ\�����B�������A���ۂɂ͂��ׂẴt���[�o�[�ɂ��� $q_i^{\ +}$ ��
���߂�K�v�͂Ȃ��A$q_i^{\ +}$ ��
�v�Z�ɕK�v�� $q_{_{S}}=u^{+}+d^{+}+s^{+}+c^{+}$ ��p���āA

{
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\renewcommand{\theequation}{\arabic{section}.\theenumi\alph{equation}}
\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
�̂悤�ɂQ�� $q_i^{\ +}$ �������v�Z���邱�Ƃɂ�� $F_2^p$ ��
���߂邱�Ƃ��ł���B�����ł� (\ref{eqn : F2PB}) ���̗p�����B
������ $F_3$ �̏ꍇ�Ɠ��l�ɁA NLO �̏ꍇ�ɂ̓p�[�g�����z���A

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

\noindent
�̂悤�� (\ref{eqn : QFI}) ����p���č\���֐��ɂȂ����Ă���A

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

\noindent
�Ƃ��� $F_2^p$ ���v�Z���Ȃ���΂Ȃ�Ȃ��B

�����Ōv�Z�����̂́A$x=0.0013$, 0.013, 0.13, 0.45 �̂S�_�ł� $Q^2$ �ˑ���
�ł���A���ꂼ�� LO, NLO, �p�[�g���Č����̌��ʂ�
�܂ނ��� (Parton Recombination, �Ȍ� PR) ��
�R�̕������ł��ꂼ��� $Q^2$ �ˑ����̌v�Z���s�����B
�������A$N_t=200$, $N_x=1000$ �Ƃ����B
�Q�l���� \cite{PRQ} �ł� NLO �̊�^��
�܂܂�Ă��Ȃ� DGLAP �������Ƀp�[�g���Č����̌��ʂ����Ă���B
�������Ȃ���A3.2 �߂ɂ����ĊȒP�ɏq�ׂ��悤�ɁA�p�[�g���Č�����
���ʂ� NLO �Ɠ��� $\alpha_s$ �̃I�[�_�[�œ����Ă��Ă���A���̌��ʂ�
�傫�������Ă� NLO �̊�^�Ɠ����x���A�������� NLO �̊�^�̕����傫���Ȃ��Ă���B
����āAPR �ɂ� NLO �̊�^�������Ɋ܂߂�K�v������A�Ȍ� PR �Ƃ���
�ꍇ�ɂ� NLO �̊�^���܂߂Čv�Z���Ă���B
�����āA���̂悤�ɂ��Čv�Z�������ʂ� 
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) �̎������ʂƔ�r�����B
���ꂪ�} \ref{fig : fig5-2} �ł���B
���̐}�ɂ��� Fermilab-E665 �̌��ʂ������� NLO �̌��ʂ͎����l����ɗǂ�
�Č����Ă��邱�Ƃ��킩��B
���� Fermilab-E665 �Ƃ̂���́A���̂悤�Ȕ��ɏ����� $Q^2$ �̗̈�ł�
�ۓ��_�I QCD ���K�p�ł��Ȃ�����ł���ƍl�����A
�����ł͔�ۓ��_�I�ȕ��@��p���Ȃ���΂Ȃ�Ȃ��B

�Ƃ���Ō��ݐ��E�ō��G�l���M�[��
�d�q�E�z�q������ł��� HERA (Hadron Electron Ring Accelerator) ��
����ꂽ�f�[�^ (ZEUS ����� H1 �̃f�[�^) �́A���܂ő����Ă��Ȃ�����
���ɏ����� $x$ �̈�ɋy��ł���B
���̂悤�� $x$ �̈�̓p�[�g���Č����̊�^���傫�������Ă���
�ƍl������̈�Ȃ̂ŁA���̂悤�Ȍ��ʂ����o�ł���̂ł͂Ȃ����Ǝv����B
�������Ȃ���A�p�[�g���Č����̌��ʂ� 3.2 �߂�
�q�ׂ��悤�� $A^{1/3}$ �ɔ�Ⴗ��̂ŁA�j�q���̃p�[�g���Č����̌��ʂ�
�傫�Ȍ��q�j�̂���ɔ�ׂĔ��ɏ������B
����Đ} \ref{fig : fig5-2} �����Ă����炩�Ȃ悤�Ɍ����_�ł̎������x�ł́A
���̂悤�Ȍ��ʂ����o���Ɏ����Ă��Ȃ��B

%%% 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{�\���֐� $F_2$ �̎������ʂƂ̔�r}
\label{fig : fig5-2}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1.0cm}

\vfill\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
{\renewcommand{\thesection}{\it {\arabic{section}}}
\subsection{{\it ���q�j�\���֐�}}}
\vspace{0.4cm}

���Ɍ��q�j�\���֐��� $Q^2$ �ˑ����ɂ��Ē��ׂ��B
�܂��v���O�����̊m�F�̂��߂ɁA
�������z�Ƃ��ĎQ�l���� \cite{PRQ} �ō̗p����Ă�����̂�p����
�v�Z���s�����B���̌��ʁA�쐻�����v���O������
�Q�l���� \cite{PRQ} �ŗ^�����Ă��錋�ʂ��m����
�Č����邱�Ƃ��m�F�����B
�Ƃ���ł��̏������z�́A������ $x$ �̈�ɂ�����
�p�[�g���Č����̌��ʂ��A�����ȒP�Ɋ܂߂邱�Ƃɂ���ē����镪�z�ł���B
����āA���̕��z�� shadowing �̗̈�݂̂�
�l���� $x$ ���z�ł���A���� $x$ �̈�� $x=1$ �t�߂ł̂ӂ�܂��Ȃǂ�
�Č�����Ă��Ȃ��B
�����ŁA���q�j���̃p�[�g�����z�� $x$ �ˑ����ɂ��Ă�
�Q�l���� \cite{SKF2} �ɏڂ����������Ȃ���Ă���̂ŁA
������̏������z��p���� ${}^{40}{\rm Ca}$ �̌��q�j
�̍\���֐��ɂ��Ē��ׂ��B
���̏������z�̓p�[�g���Č����� $Q^2$ ���X�P�[�����O�Ƃ����Q�̖͌^��
�g�ݍ��킹�Čv�Z���ꂽ���̂ŁA������ $x$ �̈悩��
�傫�� $x$ �̈�ɂ킽���� �����f�[�^���Č�����B
�p�[�g���Č����ɂ��Ă͑�R�͂Ő��������B
$Q^2$ ���X�P�[�����O�͌^�Ƃ́A���q�j���ɂ����Ă͊j�q���ߐڂ��A���ɂ�
�d�Ȃ肠�����Ƃ���A�����߂̃T�C�Y�����R�Ȋj�q�ɔ�ׂđ傫���Ȃ�A

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

\noindent
�̂悤�Ɍ��q�j�\���֐��Ɗj�q�\���֐��� $Q^2$ �ɂ��ꂪ������ƍl����
�͌^�ł���B
���̖͌^�ɂ��Ă͎Q�l���� \cite{RGR,RESCALE} �ɏڂ����q�ׂ��Ă���B
���̏������z�� $Q^2=0.8 \ GeV^2$ �ŁA

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

\noindent
�̊֐��`�ŗ^�����Ă���B�������A���̏������z�ɂ́A
KMRS-B0 \cite{KMRSB0} �� MRS-1 \cite{MRS1} ��
���z�����ƂɌv�Z���ꂽ�Q��ނ̂��̂�����B
�����ł� KMRS-B0 �����Ƃɂ��Čv�Z���ꂽ�A�j�q�� ${}^{40}{\rm Ca}$ ��
�������z��p�����B
���̕��z�̊e�X�̃p�����[�^�͕\ \ref{tab : SK} �Ɏ����B
������ $S=2(\bar{u}+\bar{d}+\bar{s})$, $q_v=u_v+d_v$ �ł���A
����� $\Lambda=0.2 \ GeV$ �ł���B
�܂��A���߂ŗp���邽�߂� MRS-1 �����Ƃ�
�v�Z���ꂽ�j�q, ${}^{118}{\rm Sn}$, ${}^{12}{\rm C}$ �ɂ��Ă�
�p�����[�^���������B

\begin{table}[t]
\caption{���q�j���p�[�g�����z�̃p�����[�^}
\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}

���̕��z��p���� $x=0.0085$, 0.035, 0.125, 0.45 �̂S�_
�ł� ${}^{40}{\rm Ca}$ �̍\���֐��� $Q^2$ �ˑ����� LO, NLO, PR ��
�R�̕������Ōv�Z���s���ANMC �̎������� \cite{NMCA} �Ɣ�r�����B
�������t���[�o�[���� $N_f=3$ �ŁA$N_t=200$, $N_x=1000$ �Ƃ����B
���̌��ʂ�} \ref{fig : fig5-3a} $\sim$ \ref{fig : fig5-3d} �Ɏ����B
�����������Ŏ������̂� ${}^{40}{\rm Ca}$ �̍\���֐����̂��̂ł͂Ȃ��A
�d�z�q�̍\���֐��Ƃ̔� $F_2^{Ca}/F_2^D$ �� $Q^2$ �ˑ����ł���B
�����̐}���p�[�g���Č����̌��ʂ͊m���ɂ�菬���� $x$ �ɂ�����
�傫�������Ă��邱�Ƃ��킩��B
�܂��A3.2 �߂� (\ref{eqn : PRQI}) $\sim$ (\ref{eqn : PRG}) ��
�� (\ref{eqn : PRDQI}) $\sim$ (\ref{eqn : PRDG}) ���ɂ��΁A
���̌��ʂ� $1/Q^2$ �̈��q���������c�C�X�g�ƌĂ΂����̂ł���A
����ď����� $Q^2$ �ő傫�������Ă���ƍl������B
����Ƀp�[�g���Č����͒ʏ�� DGLAP �������ōl���Ă���ߒ��Ƃ�
���΂̉ߒ����l���Ă��邽�߁A���̕␳���̕����͕��ł���B
�} \ref{fig : fig5-3a} ���݂�ƁA�m���ɏ����� $Q^2$ �ɂ����đ��̕�����
�ɂ�錋�ʂƔ�ׂČX�����������Ȃ��Ă��邱�Ƃ��킩��B
�������Ȃ��猻���_�œ����Ă�������f�[�^�͔��ɋ��� $Q^2$ ��
�͈͂Ɍ����Ă���A�Ȃ����������x�����̂悤�Ȍ��ʂ����o����
�قǂɂ͗ǂ��Ȃ��B
����� PR �̌��ʂ� NLO �̌��ʂ̂ǂ��炪���ǂ������f�[�^���Č����Ă��邩�A
�Ƃ��������Ƃ͂����ł͋c�_�ł��Ȃ��B
$F_2$ �̂��̓_�Ɋւ��Ă͍���� HERA �̎����ɁA�܂��C�N�H�[�N�E�O���[�I��
���z�̏ڍ� \cite{SKGLUE}��
���Ă� RHIC (Relativistic Heavy Ion Collider) �ł̎����Ɋ��҂������B

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%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vspace*{-3.0cm}
\hspace{-2.0cm}
\epsfile{file=fig5-3a.eps,width=11.0cm}
\end{center}
\vspace{-3.8cm}
\caption{���q�j�\���֐��� $Q^2$ �ˑ��� (x=0.0085)}
\label{fig : fig5-3a}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vspace*{-1.5cm}
\hspace{-2.0cm}
\epsfile{file=fig5-3b.eps,width=11.0cm}
\end{center}
\vspace{-3.8cm}
\caption{���q�j�\���֐��� $Q^2$ �ˑ��� (x=0.035)}
\label{fig : fig5-3b}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vfill\eject

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vspace*{-3.0cm}
\hspace{-2.0cm}
\epsfile{file=fig5-3c.eps,width=11.0cm}
\end{center}
\vspace{-3.8cm}
\caption{���q�j�\���֐��� $Q^2$ �ˑ��� (x=0.125)}
\label{fig : fig5-3c}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\vspace*{-1.5cm}
\hspace{-2.0cm}
\epsfile{file=fig5-3d.eps,width=11.0cm}
\end{center}
\vspace{-3.8cm}
\caption{���q�j�\���֐��� $Q^2$ �ˑ��� (x=0.45)}
\label{fig : fig5-3d}
\end{figure}
%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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{\renewcommand{\thesection}{\it {\arabic{section}}}
\subsection{{\it $Q^2$ ���W�ɂ����錴�q�j�ˑ��� \ 
$\partial [F^{Sn}_2 / F^C_2] / \partial \ln Q^2$}}}
\vspace{0.4cm}

�ŋ� NMC �ɂ�� ${}^{118}{\rm Sn}$ �� ${}^{12}{\rm C}$ �̍\���֐���
�� $R_2^{Sn,C} \equiv F^{Sn}_2(x,Q^2) / F^C_2(x,Q^2)$ �� $Q^2$ �ˑ���
�ɂ��ďڂ��������f�[�^�������A
$\partial R_2^{Sn,C}(x,Q^2) / \partial \ln Q^2 \neq 0$ �Ƃ���
���ʂ������ꂽ \cite{NMCSN-C}�B
���̌��ʂ͌��q�j�̌��ʂ� $Q^2$ �ˑ��������邱�Ƃ���������
�����[�����ʂł���B�����ŁA����쐻���� FORTRAN �v���O���������p����
���̌v�Z���s���A�������ʂƂ̔�r���s���� \cite{KMSN-C}�B
�������z�͑O�߂Ɠ����Q�l���� \cite{SKF2} ��
��� ${}^{118}{\rm Sn}$ �� ${}^{12}{\rm C}$ ��
���̂�p�����B���ꂼ��̃p�[�g�����z�̊֐��`����уp�����[�^�͂��ł�
�O�߂Ŏ������B$Q_0^2$ �� $\Lambda$ �̒l��
�O�ߓ��l $Q_0^2=0.8 \ GeV^2$ , $\Lambda=0.2 \ GeV$ �ł���B
${}^{118}{\rm Sn}$ �̕��z�ɂ��Ă̓p�����[�^���^�����Ă��Ȃ������̂ŁA
�Q�l���� \cite{SKF2} �Ŏg�p���ꂽ�v���O�����ɂ��V����
�p�����[�^���v�Z���Ă�����g�p�����B

�Ƃ���ŁA���� $\partial R_2^{Sn,C}(x,Q^2) / \partial \ln Q^2$ �̌v�Z��
���@�ł��邪�A�܂��A ${}^{118}{\rm Sn}$ ����� ${}^{12}{\rm C}$ �́A
�v�Z���s�� $Q^2=Q_1^2$ �ł̍\���֐��ƁA
����������� $\delta Q^2$ ���������� $Q^2$ �ł̍\���֐���
�v�Z������쐻���� FORTRAN �v���O�����ōs���B
������Abrute-force �@�̗v�̂ŁA

\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
�Ƃ��ē��邱�Ƃ��ł���B�{�����ō쐻����
�v���O������ $t$ �� $N_t$ �̃X�e�b�v�ɕ������Čv�Z��
�s���Ă���̂ŁA$Q^2=Q_1^2$ �ł̃p�[�g�����z�ƂƂ���
���̂P�X�e�b�v�O�ł̃p�[�g�����z���o�͂���悤�ɂ��A
���̃p�[�g�����z��p���Čv�Z����
�\���֐��� $Q^2=Q_1^2-\delta Q^2$ �ł̂��̂Ƃ����B

�܂��A$Q_1^2=5.0 \ GeV^2$ �Ƃ��āALO, NLO, PR �̂R�̔��W�������Ōv�Z��
�s�����B���ʂ�} \ref{fig : fig5-4} �Ɏ����B
�����œ_���A�����A�j���͂��ꂼ�� LO, NLO, PR �Ōv�Z�������̂ł���A
�����f�[�^�� NMC �̂��̂ł��� \cite{NMCSN-C}�B
���̌��ʂ�����ƁA�����悻�̂ӂ�܂��Ƃ��Ă͎����l�ƈ�v���Ă��邱�Ƃ��킩��B
�����[�����Ƃ� PR �Ōv�Z�����ꍇ�ɂ͎����l�ƈقȂ������ʂ�����ꂽ�B
���������̌��ʂ̓p�[�g���Č����̌��ʂ��������Ȃ��Ƃ������Ƃ�
�Ӗ�������̂ł͂Ȃ��B�p�[�g���Č�����
���ʂɊ܂܂�Ă��鍂���O���[�I�����z��
�����l�Ƃ��� (\ref{eqn : HDG}) ���̂悤�ɉ��肵�����A
���� $K_{HT}$ �͌��i�K�ł͂ǂ̒��x�̑傫���ł��邩�͂悭�킩��Ȃ����߁A
�Q�l���� \cite{PRQ} �ɂ�� $K_{HT}=1.68$ �Ɖ��肵�Ă���B
����č��ケ�̂悤�ȃp�����[�^�Ɋւ��邭�킵�����������Ă����K�v������B
�܂��A�������z�� $Q_0^2=0.8 \ GeV^2$ �Ƃ�⏬���� $Q^2$ �ŗ^�����Ă���
���Ƃ������ƍl������B�O�߂ł��q�ׂ����A
�p�[�g���Č����̌��ʂ͍����c�C�X�g�ƌĂ΂����ʂł���A
������ $Q^2$ �̈�ő傫�������Ă���B

�Ƃ���ŁANMC �̎����f�[�^�� $Q_1^2$  �̒l�����ł͂Ȃ��A
$x=0.01$ �t�߂ł� �� $GeV^2$ �ŁA
$x=0.1$ �t�߂ł� $10 \sim 20 \ GeV^2$ ���x�ł���B
���̂��߁ANLO �� $Q_1^2$ �� $Q_1^2=2.0$, 5.0, 20.0 $GeV^2$ ��
�ς��Čv�Z�����Ă݂��B���̌��ʂ�} \ref{fig : fig5-5} �Ɏ����B
���̌��ʂ������ $x=0.01$ �t�߂ł� $Q^2=2.0 \ GeV^2$ ��
�v�Z���ʂƁA$x=0.1$ �t�߂ł� $Q^2=20.0 \ GeV^2$ �̌��ʂ�
���悭��v���Ă��邱�Ƃ��킩��B
�������A����ł��܂������� $x$ �̈�ł�⏬�����łĂ���B

�����̉�͂̌��ʂ�蕪���邱�Ƃ́A
$\partial R_2^{Sn,C}(x,Q^2) / \partial \ln Q^2 \neq 0$ �Ƃ������ʂ�
�p�[�g���Č����� $Q^2$ ���X�P�[�����O�Ƃ������p�[�g���͌^�̘g����
�������邱�Ƃ��ł��A�܂� NLO �ł̌v�Z���ʂ����Ȃ�ǂ������l���Č�
���邱�Ƃ���A����ɂ̓p�[�g���Č����ɂ�� $Q^2$ ���W����
�����A�������z�ɂ�����
���q�j�̌��ʂ� $x$ �ˑ������傫�������Ă��邱�Ƃ��킩��B
�Ƃ���Ő} \ref{fig : fig5-5} �ɂ��A
�S�̂Ƃ��Ď����l�����኱�������l���łĂ��邱�Ƃ����������A
$Q_0^2$ �ɂ�����p�[�g�����z�� $x$ �ˑ����Ɍ��ʂ��ǂ̒��x�ˑ����Ă���̂��A
���̌X���ɂ��Ē��ׂĂ݂��B��̓I�ɂ́A�܂����̕��z�͂��̂܂܂ŊC�N�H�[�N
�̏������z���^���ʕۑ�����j��Ȃ��悤�ɂ��Ȃ��� shadowing ��
�傫���Ƃ��Čv�Z�������A���̕��z���������z�Ƃ��Čv�Z���s�����B
����� $x \approx 0.03$ �t�߂ł������l�Ɉ�v���錋�ʂ�����ꂽ�B
�����ăO���[�I���ɂ��Ă����l�̂��Ƃ��s���ƁA���ʂ͋t�Ɏ������ʂƂ�
���ꂪ�傫���Ȃ����B�ω��̑傫���Ƃ��Ă͊C�N�H�[�N�̕��z�̕ω��ɑ΂���
���q���ł��邱�Ƃ��킩�����B
�����̂��Ƃ���A���̌��ۂ̗����ɂ́A�������z�Ƃ��Ă̌��q�j���p�[�g�����z�A
���ɊC�N�H�[�N���z�̂���ɐ����Ȍ������d�v�ł���ƍl������B

%%% Figure %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[p]
\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}
\hspace{-1.5cm}
\epsfile{file=fig5-5.eps,width=17.0cm}
\end{center}
\vspace{-4.0cm}
\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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�j�q����ь��q�j�\���֐��� $Q^2$ �ˑ����ɂ���
�Q��ނ� $Q^2$ ���W�������̐��l���̌������s�����B
��͒ʏ�p������ DGLAP �������ł���A
������� DGLAP �������Ƀp�[�g���Č����̌��ʂ���ꂽ���̂ł���B
���̍� $t$ ����� $x$ �� $N_t$, $N_x$ �̃X�e�b�v�ɂ��ꂼ�ꕪ�����Čv�Z��
�s�� brute-force �@��p���� FORTRAN �v���O�������쐻�����B
���̃v���O�����͎኱�v�Z���Ԃ͒��������邪�ANLO �̊�^����ꂽ
�t���[�o�[��P�d���̏ꍇ�� $N_t \geq 50$ , $N_x \geq 1000$ ���x�A
�t���[�o�[�P�d���̏ꍇ�� $N_t \geq 200$ , $N_x \geq 1000$ ���x��
���΁A$10^{-4} < x < 0.8$ �͈̔͂� 2\% �ȓ��̌v�Z���x�𓾂邱�Ƃ�
�ł����B����ɑ��̌����҂����p�ł���悤�ɁA�C���^�[�l�b�g��ʂ���
����ł���悤�ɂ����B

�����Ă��̃v���O������p���āA�j�q�̍\���֐� $F_3$ ����� $F_2$ �ɂ��Ă�
�v�Z�������Ȃ��A�l�X�Ȏ���
�f�[�^ (CDHSW, SLAC, BCDMS, EMC, NMC, Fermilab-E665, ZEUS, H1) �Ƃ�
��r���s�����B���̌��ʁA
LO �̌��ʂƔ�r���� NLO �̌��ʂ͏����� $Q^2$ �̈��
���ǂ� $Q^2$ �ˑ����̃f�[�^���Č������B
�������Ȃ���A�p�[�g���Č����� $Q^2$ ���W���ʂɂ��Ă�
�������x���ŔF�߂���Ɏ����Ă��Ȃ��B
�����Č��q�j�\���֐��̌v�Z���s���ANMC �� $Q^2$ �ˑ�����
�ւ�������f�[�^ $F_2^A/F_2^D$ �Ƃ̔�r���s�����B
���̌��ʁA$Q^2$ ���W�ɂ�����p�[�g���Č����̌��ʂ́A
������ $x$ �̈您��я����� $Q^2$ �̈�Œ��������Ƃ������������A
�����_�ł��̌��ʂ����o����Ɏ����Ă��Ȃ��B
���̓_�Ɋւ��ẮA�L�� $Q^2$ �̈�ɓn��A
���x�̗ǂ��������ʂ����㓾���邱�ƂɊ��҂������B

�Ō�� $\partial R_2^{Sn,C}(x,Q^2) / \partial \ln Q^2$ �̌v�Z���s�����B
���̌��ʁA�����悻�� $x$ �ˑ����͌��ݓ����Ă���
���q�j���p�[�g�����z�� NLO �� DGLAP �������Ő����ł��邱�Ƃ����������B
����ɑ΂��p�[�g���Č����̌��ʂ�
�܂� $Q^2$ ���W�������Ōv�Z���s�����ꍇ�ɂ�
�����l�ƈقȂ錋�ʂ�����ꂽ�B
����������̓p�[�g���Č����ɂ��`�����������Ȃ��Ƃ������Ƃ�
�������ɈӖ�����̂ł͂Ȃ��B����� $Q^2$ ���W�ɂ�����p�[�g���Č�����
���ʂɂ� $K_{HT}$ �̂悤�ȁA�܂��ǂ��킩���Ă��Ȃ��p�����[�^��
���݂��邩��ł���B����A��萸�x�̗ǂ��������ʂ�������΁A
���̂悤�ȓ_�Ɋւ���ڂ����������\�ƂȂ�ł��낤�B
����ɁA�������z�̗^�����Ă��� $Q^2$ ����⏬�������邱�Ƃ�����
�ƍl������B
�������ANLO �̌��ʂ����Ȃ�����l�� $x$ �ˑ������Č����邱�Ƃ���A
�p�[�g�����z�̌��q�j���ɂ����� $x$ �ˑ����̕ω���
���傫���e�����Ă���A$Q^2$ ���W�ɂ�����p�[�g���Č����̌��ʂ�
���܂�傫���͌����Ă��Ȃ��ƍl���邱�Ƃ��ł���B
����Ă��̌��ʂ̂�萸���ȗ����ɂ́A���q�j���p�[�g�����z�� $x$ �ˑ�����
�ւ���ڂ����������d�v�ł���ƍl������B


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\section*{{\bf �ӎ�}}
\addcontentsline{toc}{section}{\protect\numberline{\S}{�ӎ�}}
\vspace{0.7cm}

�w�������ł���F��r�O�������ɂ́A���m�O���ے��̂Q�N�Ԃɂ킽��A
���؂Ȃ��w���������������B
�܂��A�ʎq�����w�u���̒��J��Ƌ����A�͖�G������A�Y�ƈ�ȑ�w��
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�ɂ́A����ɂ킽�邲�����������������B
�����āA���ї��q���A������h���ɂ͔��m�O���ے��P�N���̂P�N�Ԃɂ킽��A
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��L�̕��X�ɁA�S���炨���\���グ��B
�܂��A���l�v�Z�ɍۂ��v�Z�@���g�p�����Ă���������
����w�j���������Z���^�[�Ɋ��ӂ���B
���̌����͕����V�N�Ċ��ɁA�u���b�N�w�u���č������������ɂ����ĕ����I��
�s��ꂽ���A�����ł̑؍݂Ɋւ��Čo�ϓI�ȉ�������т����b��
���������� A. J. Baltz ���Ɋ��ӂ���B�����̌�����̏o�Ȃ���ј_���쐻
�Ɋւ��Ă͕����ȉȊw��������C (�ۑ�ԍ� 06640406) �̉������󂯂��B


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\section*{{\bf �t�^}}
\addcontentsline{toc}{section}{\protect\numberline{\S}{�t�^}}

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\vspace{0.3cm}
\subsection*{�t�^A \ \ \ ����֐�}
\addcontentsline{toc}{subsection}{\protect\numberline{A}{����֐�}}
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}
\vspace{0.4cm}

�Œ᎟ (LO) �̕���֐��́A

\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
�ŗ^������B�����ŁA$1/(1-x)_+$ �͈ȉ��̂悤�ɒ�`�����B

\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
�������A�ϕ��͈͂� $0 \le z \le 1$ �Œ�`����Ă��邱�Ƃɒ��ӂ��K�v�ł���B
���ۂ�DGLAP �������̂Ȃ��ł͐ϕ��͈͂� $x \le z \le 1$ �ŗ^�����Ă���A
���̂Ƃ��́A

\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
�ƂȂ�BNLO �̕���֐��͎Q�l���� \cite{HWW,NLO} �ɗ^�����Ă���A

\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
�ł���B�t���[�o�[��P�d���� NLO �̕���֐��� $q \pm \bar{q}$ �^��
���z $\displaystyle{q_{_{NS}}=\sum_i a_i (q_i \pm \bar{q}_i)}$ �ɑ΂��āA

\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
�ŗ^������B
�����ŁA$P_F(x)$, $P_G(x)$, $P_{N_F}(x)$, $P_A(x)$ ��
�Q�l���� \cite{NLO} �ɂ��A

\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
�Œ�`����Ă���B
$\zeta$ �֐��� $\displaystyle{\zeta (k)=\sum_{n=1}^\infty {1 \over n^k}}$ ��
��`�������̂� $\zeta(3)=1.2020569...$ �ł���B
�܂��A$\widetilde S(\infty)$ �� $\zeta$ ��
����� $\widetilde S (\infty) =-{5 \over 8}\zeta(3)$ �̂悤�ɒ�`�����B
����ɁA$P_A(x)$ �̒��̐ϕ��� Spence �֐� $S(x)$ ��p���ĕ\�����Ƃ��ł��āA

\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
�ł���B������ Spence �֐��́A

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

\noindent
�ɂ���Ē�`�����֐��ł���B�������A$0 \le x \le 2$ �͈̔͂ɂ����ẮA

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

\noindent
�̂悤�ɏ������Ƃ��ł���̂ŁA�v�Z�ł͂����p���Ă���B
�����ɂ���Ă� $-S(x)$ �� Spence �֐��ƌĂԏꍇ������̂Œ��ӂ��K�v�ł���B

(\ref{eqn : P1QQ}) $\sim$ (\ref{eqn : P1GG}) ����
������ $F_{qq}$, $F_{qg}$, $F_{gq}$, $F_{gg}$ �͈ȉ��̂悤��
��`����Ă���B

\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
�����ŁA$I_x$ �� Spence �֐��ł��� [$I_x=S(x)$] �A$J_x$ ��

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

\noindent
�Œ�`�����B

�P�X�e�b�v�`���� $Q^2$ ���W������ (\ref{eqn : 1STEP}) �ɂ�����A
�s�� ${\bf P}^{(0),(1)}$, ${\bf D}$, ${\bf E}$ �́A

\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
�Œ�`�����B�����ōs�� ${\bf D}$ �Ɍ���Ă��� $B_n^q(x)$, $B_n^g(x)$ ��
�t�^ B �ŗ^����B�s�� ${\bf E}$ �̍s��v�f�́A

\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
�Œ�`����Ă���B�����ŁA

\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
�ł���B�㎮�� $A$ �� $B$ �́A

\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
�ŗ^������B������ (\ref{eqn : AA}) ���� $(17/3+170x/3-40x^2/3)$ ��
�����́A�Q�l���� \cite{HWW} ��
�����Ă� $(17+170x/3-40x^2/3)$ �ƂȂ��Ă���B
���������̕����Ƃ̔�r�̌��ʁA���ł��邱�Ƃ��킩��C�������B

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.3cm}
\subsection*{�t�^B \ \ \ �W���֐�}
\addcontentsline{toc}{subsection}{\protect\numberline{B}{�W���֐�}}
\renewcommand{\theequation}{B.\arabic{equation}}
\setcounter{equation}{0}
\vspace{0.4cm}

�W���֐��� 2.3 �߂Ŏ������悤�ɁA

\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
�ł���B�����Ō���Ă��� $B_n^{q}$, $B_n^{g}$ �́A
�P�X�e�b�v�`���̕����� (\ref{eqn : 1STEP}) �̕���֐��̍s�� ${\bf D}$ ��
�������̂Ɠ���ł���B$B_n^{q}$ �́A

\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
�ł���B$B_1^{q}$, $B_2^{q}$, $B_3^{q}$ �͂��ꂼ��A
$F_1$, $F_2/x$, $F_3$ �ɑΉ�������̂ł���A$F_q(x)$ �́A

\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
�ƒ�`�����B�������A�����Ō���� $[\ln (1-x)/(1-x)]_+$ �́A

\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
�Œ�`����A�t�^ A �Ō���� $1/(1-x)_+$ �Ɠ��l�ɁA
�ϕ��͈͂� $0 \le z \le 1$ �Œ�`����Ă��邱�Ƃɒ��ӂ��K�v�ł���B
�P�X�e�b�v�`���̔��W�������ŗp������ꍇ�ɂ�
�ϕ��͈͂� $x \le z \le 1$ �ƂȂ�̂ŁA���̏ꍇ�ɂ́A

\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
�ƂȂ�B

�����āA$B_n^{g}$ �́A

\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
�ł���B����������l��$B_1^{g}$, $B_2^{g}$ �͂��ꂼ��A
$F_1$, $F_2/x$ �ɑΉ�������̂ł���B$F_3$ �ɑΉ����� $B_3^{g}$ ��
���݂��Ȃ��̂́A$F_3$ �� $Q^2$ ���W�̌v�Z�ɂ����Ă̓O���[�I��
�̕��z�������Ă��Ȃ�����ł���B������ $F_g(x)$ �́A

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

\noindent
�ł���B


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

\begin{thebibliography}{}

\vspace{0.7cm}

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