\title{Lattice quantum chromodynamics}\label{chap:latticeqcd}
\author{Tetsuo Hatsuda}
\institute{Tetsuo Hatsuda \at Nishina Center, RIKEN, Saitama 351-0198, Japan,  \email{thatsuda@riken.jp}}
\maketitle
\abstract{
Concepts and applications of lattice quantum chromodynamics (LQCD) are introduced.
After discussing how to define quarks and gluons on the Euclidean hypercubic lattice,
 the strong coupling expansion  and the weak coupling expansions are reviewed
 to see the vital role played by the quantum fluctuations in QCD.
 Fundamental techniques for numerical LQCD simulations  such as the Markov Chain Monte Carlo method and the
  Hybrid Monte Carlo method are discussed in some details.  As
  examples of the high precision LQCD simulations,  numerical results of the heavy quark potential and the hadron masses
   are shown.  Recent LQCD results on the baryon-baryon interactions are briefly discussed.
   }

%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%

In this chapter, we introduce  lattice quantum chromodynamics (LQCD)
originally proposed by K. Wilson in 1974 \cite{Wilson:1974sk}. What makes the LQCD unique and powerful is that
it can allow first-principle,  gauge invariant and non-perturbative calculations of strongly interacting
quarks and gluons.  After the first numerical attempts by M. Creutz \cite{Creutz:1980zw},
 LQCD simulations have been extensively applied to study heavy quark potentials,
hadron masses,  hadronic matrix elements, QCD phase transition at finite temperature,
and so on.  In recent years, there are also progresses
in deriving the  baryon-baryon interactions, which are particularly relevant to
nuclear physics and astrophysics. Throughout this chapter, we will focus on the theoretical concepts, numerical techniques and some
applications to hadron masses and nuclear forces.  LQCD at finite temperature and/or baryon density will
not be covered.  The interested readers may want to consult the  following review articles for further details, or may even
want to run the open source codes or to use the open LQCD configurations.

\vspace{0.3cm}

 \noindent
 {\bf Review articles:}   Here is a list a few articles which are useful to learn more about LQCD.
 \begin{itemize}
\item Comprehensive review on QCD can be seen in \cite{Brambilla:2014jmp}.
\item The origin of LQCD is discussed in \cite{Wilson:2004de}.
\item The basic concepts  of LQCD are summarized in the monographs \cite{Creutz:1984mg,Rothe:1992nt}.
 \item Recent progresses of LQCD can be seen in the reviews \cite{Hoelbling:2014uea,Ukawa:2015eka} and references therein.
 \end{itemize}

\vspace{0.3cm}

 \noindent
 {\bf Open source codes:}
 It takes a lot of time and energy to develop the LQCD code from scratch. To lower the bar,
 several source codes for LQCD simulations have been released for public use.
  \begin{itemize}
\item Bridge++: \url{http://bridge.kek.jp/Lattice-code/index_e.html}
\item Lattice ToolKit: \url{http://nio-mon.riise.hiroshima-u.ac.jp/LTK/}
\item OpenQCD: \url{http://luscher.web.cern.ch/luscher/openQCD/index.html}
\item USQCD: \url{http://usqcd-software.github.io/}
\end{itemize}

\vspace{0.3cm}
 \noindent
{\bf LQCD configurations:}  Outputs of large scale LQCD simulations  is a {\it big data}
called "LQCD configurations".   Physicists can study various aspects of QCD by using these configurations.
 The International Lattice Data Grid (ILDG) is a project to share the configurations around the world.
  \begin{itemize}
 \item ILDG \url{http://plone.jldg.org/wiki/index.php/Main_Page}
\end{itemize}

%================================
  \subsection{Euclidean QCD action}
%================================
LQCD is formulated on the Euclidean spacetime lattice.
Observables in the Minkowski spacetime are obtained by the analytic continuation of the
imaginary-time $\tau$ to the real-time $t$.  In terms of the time evolution operator in quantum mechanics,
it corresponds to the continuation from the imaginary time evolution, $e^{-H \tau} $, to the real time
evolution, $e^{-iHt}$, where $H$ being the Hamiltonian of the system.  The functional integral representation of the
the Euclidean QCD partition function ${\cal Z}$  on a finite spatial box $L^3$  and the temperature $T$ is given by
\beq
\label{eq:Z-QCD}
{\cal Z}(T,V,J) = \int [dA d\bar{q} dq] e^{- \int_{0}^{1/T}  d\tau \int_{L^3} d^3 x \left( {\cal L}_{\rm QCD}^{\rm E} + J \Xi \right) }
\eeq
where the Euclidean QCD Lagrangian in terms of quarks $q^{\alpha=1,2,3}$ and gluons $A_{\mu=1,2,3,4}^{b=1, \cdots, 8}$
is given by
\beq
{\cal L}_{\rm QCD}^{\rm E} = \bar{q}^{\alpha} (\Gamma_{\mu}  D_{\mu}^{\alpha \beta} + m \delta^{\alpha \beta}) q^{\beta}
+ \frac{1}{4} G_{\mu \nu}^b G_{\mu \nu}^b,
\eeq
Here the Euclidean version of the $\gamma$-matrices, $\Gamma_\mu$, is defined in
 the Appendix [Four vectors and Dirac matrices].
The quark mass matrix in the flavor space ($u, d, s, \cdots$) is denoted by $m$ with the flavor indices suppressed.
 The color covariant derivative is defined by
\beq
D_{\mu}^{\alpha \beta} = \partial_{\mu} \delta^{\alpha \beta} + ig A_{\mu}^{\alpha \beta},
\eeq
with  $x_{\mu}= (\tau, \mathbf{x})$,  $\partial_{\mu} = (\partial_{\tau}, \nabla )$ and the $3 \times 3$ matrix field, $A_{\mu} = A_{\mu}^b t^b$.
The explicit form of the color SU(3) generators $t^{b=1, \cdots, 8}$ is given in the Appendix [SU$(N)$ algebra].
The field strength tensor is $G_{\mu \nu}=G_{\mu \nu}^b t^b$ with
 $G_{\mu \nu}^b= \partial_{\mu}A_{\nu}^b - \partial_{\nu} A_{\mu}^b -g f_{bcd} A_{\mu}^c A_{\nu}^d$.
 The arbitrary external fields (such as the external source of the quarks and gluons,
 external electroweak fields etc) are denoted by $J$, while the corresponding dynamical operators
  are denoted by $\Xi (A, \bar{q}, q)$.  The functional integration measure for the
 c-number gluons and the Grassmann-number quarks is defined by
 \beq
 [dA d\bar{q} dq] \equiv \prod_{x, {\rm color, spin, flavor}}  dA_{\mu}^b(x) d\bar{q}^{\alpha}(x)  dq^{\beta} (x).
\eeq
See Appendix [Gaussian and Grassmann integrals] for the examples of integration with these measures.
The temporal boundary condition of the gluon (quark) field is periodic (anti-periodic) due to its
c-number (Grassmann-number) character;  $A_{\mu}^b (\tau=0, \mathbf{x})=A_{\mu}^b (\tau=1/T, \mathbf{x})$,
$\bar{q}^{\alpha}(\tau=0,\mathbf{x})=-\bar{q}^{\alpha}(\tau=1/T,\mathbf{x})$, and
$q^{\beta}(\tau=0,\mathbf{x})=-q^{\beta}(\tau=1/T,\mathbf{x})$.    On the other hand, the spatial boundary conditions are
not constrained and can be taken to be either periodic or anti-periodic; the difference should disappear in the
thermodynamic limit, $L \rightarrow \infty$.
 Throughout this chapter, we take $T = 1/L $   to
  study hadrons and hadron-hadron interactions at zero temperature in the thermodynamic limit.

Further details of the functional integral formulation of the general many-body systems of fermions and bosons
can be seen in the textbook \cite{Negele:1988vy}.

%================================
 \subsection{Quantum fluctuations}
%================================
In weak coupling perturbation theory, one assumes that the QCD coupling $g$  is small and expand the partition function
 ${\cal Z}$ in terms of a power series of $g$.  Such a perturbative expansion in QCD is justified, however, only in  limited
 circumstances  such as at extreme high  temperature/density  or at very short distances. This is because  the
  renormalized QCD coupling (or often called the running coupling) becomes small only in the processes
  with the energy scale much above 1 GeV. This is called the asymptotic freedom.
 To calculate low-energy hadron properties below 1 GeV,
  we need to go beyond perturbation theory and to evaluate the functional integral with full quantum fluctuations.
 The lattice QCD provides a way to carry out this task numerically in a gauge invariant manner.

%%%%%%%%%%%%%%%%%%%%%%
\section{Lattice QCD: theoretical basis }
%%%%%%%%%%%%%%%%%%%%%%

%================================
\subsection{Wilson line}
%================================

% FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
\includegraphics[scale=0.6]{Chapter3-figures/wilson-line.eps}
 \end{center}
\caption{(a) The Wilson line in the Euclidean spacetime. (b)
 Basic quark bilinears with gauge invariance.}
\label{fig:wilson-line}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let us first start with the {\it Wilson line} which is defined on
  a path $P$ connecting the point $y$  and $x$ in the continuous Euclidean spacetime
   as shown in Fig.\ref{fig:wilson-line}(a).
By parametrizing the path in terms of a coordinate
   $z(s)$ with $z(s=0)= y$ and $z(s=1)=x$, the Wilson line reads
\beq
\label{eq:5.wilson-line}
U_P(x,y;A) & = &
{\rm P} \ {\rm exp} \left( ig \int_P dz_{\mu} A_{\mu} \right)
={\rm P} \ {\rm exp} \left( ig \int_0^1 ds\ \lambda_{\mu}
 A_{\mu} \right)
  \nonumber \\
  &  & \! \! \! \! \!
  \! \! \! \! \! \! \! \! \! \! \! \! \! \! = \sum_{n=0}^{\infty}
 \frac{(ig)^n }{n!} \int_0^1 ds_1 \int_0^1 ds_2 \cdot \cdot \cdot \int_0^1 ds_n \
 {\rm P}[\lambda \cdot A(s_1)\ \cdot \cdot \cdot \
  \lambda \cdot A(s_n)],
 \eeq
 where
 $\lambda_{\mu} = dz_{\mu}/ds$. The path ordered symbol ${\rm P}$ is
  necessary because $A_{\mu} = A_{\mu}^b t^b$ is a
  matrix in the color space.

  The Wilson line has the following  properties
   which can be proved from  the definition of $U_P$ (Exercise \ref{prob:1}) :
\begin{enumerate}
  \item[(i)]  It can be broken into parts at any arbitrary points on the path;
\beq
\label{eq:5.wilson-line-i}
U_P(x,y;A) = U_{P_2}(x,z(s);A) U_{P_1}(z(s),y;A).
\eeq
\item[(ii)] It satisfies a differential equation,
\beq
\label{eq:5.wilson-line-ii}
\frac{d}{ds} U_P(z(s),y;A)=
 \left[ ig \lambda(s)\cdot  A(z(s)) \right] \ U_P(z(s),y;A).
\eeq
\item[(iii)] Under the local gauge transformation
$A_{\mu}^V(x) = V(x)[A_{\mu}(x) V^{\dagger}(x) -(i/g) V(x) (\partial_{\mu}  V^{\dagger}(x))$,
 it transforms covariantly,
\beq
\label{eq:5.wilson-line-iii}
U_P(x,y;A) \rightarrow U_P(x,y;A^V) = V(x) U_P(x,y;A) V^{\dagger}(y).
\eeq
 \end{enumerate}

The Wilson line is useful to define the non-local and gauge-invariant objects.
In particular, the gauge-invariant quark bilinear
$\bar{q}(x) U_P(x,y;A) q(y)$, and the gauge-invariant {\it Wilson loop}
 ${\rm tr} \ U_P(x,x;A)$ turn out to be important building blocks to define
   the QCD action on the lattice.  Here ``tr" implies the trace
 over  color indices.

%===========================
\subsection{Lattice gluons}
%===========================

% FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
\includegraphics[scale=0.6]{Chapter3-figures/cube.eps}
 \end{center}
\caption{A hypercubic lattice in Euclidean spacetime with
 a lattice constant $a$ and the lattice size $L$.
 Quarks $q(n)$ (gluons $U_{\mu} (n)$ ) are
 defined on the sites (links).}
\label{fig:cube}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let us consider a four dimensional  hyper-cubic lattice
 with a lattice spacing $a$ and the four dimensional volume $L^4$.
 Each lattice site is specified by  $n_{\mu}$ corresponding to the
 Euclidean coordinates through $x_{\mu} = a n_{\mu}$ (see Fig.\ref{fig:cube}).
   The {\it link variable} (the shortest  Wilson line on the lattice) is an SU(3)
 matrix connecting the neighboring sites $n$ and $n + \hat{\mu}$,
\beq
\label{eq:5.wilson-link}
U_{\mu}(n) = {\rm exp} \left(  ig a A_{\mu} (n) \right) .
\eeq
 Here $\hat{\mu}$ implies a vector  pointing the
 direction of $\mu$ with a length $a$.
 Since it is the minimal
     Wilson line, we do not need the path ordering symbol ${\rm P}$.
   Also, any non-minimal Wilson line on the lattice  is represented by a
    product of link-variables.  For later purpose, we introduce the link variable pointing the
    opposite direction as  $U_{-\mu}(n+\hat{\mu}) = [U_{\mu}(n)]^{\dagger}$.

Let us now  define the smallest closed loop;
\beq\label{eq:5.wilson-plaq}
 U_{\mu \nu}(n) =
 U_{\nu}^{\dagger}(n) U_{\mu}^{\dagger}(n+ \hat{\nu} )
  U_{\nu}(n+ \hat{\mu} )U_{\mu}(n ).
\eeq
Under local gauge
 transformation (rotation under  arbitrary  SU(3) matrix $V(n)$), we have
\beq
 U_{\mu}(n) \rightarrow  V(n) U_{\mu}(n) V^{\dagger}(n+\hat{\mu}), \ \
 U_{\mu \nu}(n) \rightarrow  U_{\mu \nu}^V(n)=V(n) U_{\mu \nu}(n) V^{\dagger}(n),
 \eeq
which are the  direct consequence of Eq.(\ref{eq:5.wilson-line-iii}).

In the naive continuum limit where $a \rightarrow 0$, we have
\beq
 \label{eq:5.wilson-plaq-cont}
 U_{\mu}(n) -1
   &=& iga A_{\mu}(n)  + O(a^2), \\
 \label{eq:5.wilson-plaq-cont-2}
 U_{\mu \nu}(n) -1
  &=& \exp \left( iga^2 t^b (G_{\mu \nu}^b(n) + O(a^3) ) \right) -1
  = ig a^2 G_{\mu \nu}(n)  + O(a^4), \\
{\rm tr} \left(    U_{\mu \nu}(n) -1 \right)
 &=&   {\rm tr} \left[  iga^2 t^b (G_{\mu \nu}^b(n) + O(a^3)) - \frac{1}{2} g^2 a^4 G_{\mu \nu}(n)^2 + O(a^5) \right] \nonumber \\
 &=& - \frac{1}{4} g^2 a^4 (G_{\mu \nu}^b(n))^2 + O(a^5) .
\eeq
Here  Eq.(\ref{eq:5.wilson-plaq-cont-2}) is obtained by
 using the Baker-Campbell-Hausdorff formula,
  ${\rm exp}X\cdot {\rm exp}Y =
  {\rm exp}(X + Y + [X,Y]/2 + \cdot \cdot \cdot )$.  (Exercise \ref{prob:2}).

Finally,
a  gluon action on the lattice, which reduces to the Yang-Mills
 action in the leading order of the  naive continuum limit ($a \rightarrow 0$), reads
\beq
\label{eq:5.wilson-action-cont}
 S_{\rm G}  &=& \beta \sum_{\rm Pl} \left[ 1 - \frac{1}{N_c} {\rm Re} \ {\rm tr} \ U_{\mu \nu}(n) \right] \\
 &=& \beta  a^4 \sum_n \sum_{\mu < \nu} \left[ 1 - \frac{1}{2N_c} {\rm tr}  \left( U_{\mu \nu}(n) +  U_{\mu \nu}^{\dagger}(n) \right) \right] \nonumber \\
 & = & \frac{1}{g^2} \sum_n \sum_{\mu \neq \nu} {\rm tr} \left[ 1 -U_{\mu \nu} (n) \right]
  \ \ \ \ \xrightarrow{a\rightarrow 0}   \frac{1}{4}\int d^4x  \ G_{\mu \nu}^b (x)^2 , \nonumber
 \eeq
where $\sum_{\rm Pl} $ is a sum over all non-oriented {\it plaquettes} (minimum square tile on the lattice with the area $a^2$).
Note that  $\beta \equiv \frac{2N_c}{g^2}$ with $N_c$ being the number of colors ($N_c=3$ for QCD)
 should not be  confused with the inverse temperature.
    The lattice gluon action is not unique in the sense
 that one may add arbitrary non-minimal terms which vanish
 in the continuum limit ($a\rightarrow 0$).

%============================
\subsection{Lattice fermions}
%============================

There exist three types of gauge invariant objects made of nearest neighbor fermions as shown in Fig.\ref{fig:wilson-line}(b);
\beq
\label{eq:5.nn-link}
\bar{q}(n) q(n), \ \
\bar{q}(n+\hat{\mu}) U_{\mu}(n) q(n), \ \
\bar{q}(n-\hat{\mu}) U_{-\mu}(n) q(n).
\eeq
Here one may put any $\gamma$-matrices  between $\bar{q}$ and $q$
 without spoiling the color gauge invariance.
A special combination of the above   terms is called the Wilson's fermion action
\beq
\label{eq:5.wilson-fermion}
\! \! \!
S_{\rm F}
 &=& a^4 \sum_n
  \left[
    m   \bar{q}(n)q(n) - \frac{1}{2a}
    \sum_{\pm \mu} \bar{q}(n+\hat{\mu}) {\Gamma}_{\mu} U_{\mu}(n) q(n)
    - \frac{r}{2a} \sum_{\pm \mu}   \left( \bar{q}(n+\hat{\mu}) U_{\mu}(n) q(n) - \bar{q}(n)q(n) \right)
  \right]  \nonumber \\
\label{eq:5.wilson-fermion2}
 & \equiv & a^4 \sum_{n',n} \bar{q}(n')
 \left( m \delta_{n',n} + D_{_{\rm W}}(n',n;U) \right) q(n)  \\
 \label{eq:5.wilson-fermion3}
 & & \xrightarrow[a \rightarrow 0] \ \  \int d^4x\  \bar{q} (x)  \left(\Gamma_{\mu} D_{\mu} + m - \frac{ar}{2} D_{\mu}^2 \right)   q(x),
\eeq
where the Wilson's Dirac operator in Eq.(\ref{eq:5.wilson-fermion2})  with the Wilson's parameter $r$ reads
\beq
\label{eq:5.wilson-dirac}
D_{\rm W}(n',n;U)
  = - \frac{1}{2a} \sum_{\pm \mu}
 \left[
 \delta_{n',n+\hat{\mu}} (r+ {\Gamma}_{\mu} ) U_{\mu}(n)
   - r \delta_{n',n}
 \right]      .
\eeq
 To take the continuum limit in Eq.(\ref{eq:5.wilson-fermion3}),
we use  the midpoint prescription,  $(f(x+a)-f(x-a))/2a = f'(x) + O(a^2)$
 and $(f(x+a)+f(x-a) -2f(x))/a^2 = f''(x) + O(a^2)$.
 One of the important properties of $ D_{\rm W}(n',n;U) $ is its $\Gamma_5$ Hermiticity (Exercise  \ref{prob:3}),
 \beq
 \label{eq:gamma5-hermite}
 \Gamma_5 D_{\rm W} \Gamma_5 = D_{\rm W}^{\dagger},
 \eeq
where $\Gamma_5$ is given in Appendix [Four vectors and Dirac matrices].  Note that the
 Hermitian conjugate is taken for color, spin and  spacetime.

The dispersion relation (relation between the energy and momentum)
for free fermion can be obtained from Eq.(\ref{eq:5.wilson-fermion}) by taking $U_{\mu}=1$ (or equivalently $g=0$)
and substituting  the Fourier transform,
$q(n) = \int_{-\pi/a}^{\pi/a} \frac{d^4p}{(2\pi)^4} e^{i p_{\mu} n_{\mu}} q(p)$.
 This leads to $S_{\rm F}^{\rm (free)}= \int \frac{d^4p}{(2\pi)^4} \bar{q}(-p) {\cal G}_{\rm F}^{-1} q(p)$ with the
 free fermion propagator (Exercise  \ref{prob:4}),
 \beq
 \label{eq:DF}
 {\cal G}_{\rm F} (p) &=& \frac{-i  \sum_{\mu} \bar{p}_{\mu} \Gamma_{\mu}  + m(p) }{\sum_{\mu} \bar{p}_{\mu}^2 + m^2(p)},  \\
 \label{eq:Mp}
 \ \ \ \ \bar{p}_{\mu} &=& \frac{1}{a} \sin (p_{\mu} a), \ \  m(p)  =  m(0) + \frac{r}{a}  \sum_{\mu}      \left( 1-\cos (p_{\mu} a) \right)  .
 \eeq
Since $\sin (p_{\mu} a)$ becomes zero
 for $p_{\mu}a$=$(0,0,0,0)$, $(\pi, 0,0,0)$,$ \cdot $
  $(0,\pi,\pi,\pi)$, $(\pi,\pi,\pi,\pi)$, there arise
  $2^4 =16$ degenerate fermions  if we take  $r=0$.
 This is called the fermion doubling problem on the
 lattice. In fact, there is a no-go theorem
 by  Nielsen and Ninomiya:
 The  fermion doubling always exists, if the free fermion action  on the lattice
 has (i) bilinearity in quark field,
  (ii) translational invariance,
   (iii) hermiticity (in the Minkowski spacetime),
    (iv) locality in spacetime, and (v) exact chiral symmetry.
Indeed, (i)-(v) are all satisfied for $r=0$.

The doubling in the dispersion relation in  the Minkowski spacetime is easily seen by Wick rotating
$p_4 \rightarrow i E$ in Eq.(\ref{eq:DF}).  As an illustration, let us take the case with massless fermion ($m(0)=0$)
in (1+1)-dimension.  Then, the zero of the denominator after the Wick rotation for small $a$ gives,
\beq
E^2(p) \simeq \left( \frac{1}{a} \sin (pa) \right)^2 + \left( \frac{r}{a}  (1-\cos(pa)) \right)^2 ,
\eeq
whose positive energy solution is plotted in Fig.\ref{fig:dispersion} for several values of $r$.
One finds that the unphysical massless pole at $pa=\pi$ is lifted up as $r$ increases.

% FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
\includegraphics[scale=0.75]{Chapter3-figures/dispersion.eps}
 \end{center}
\caption{The  dispersion relation for massless fermion in (1+1)-dimension on the lattice for different values of
 the Wilson's parameter $r$.  The linear dispersion in the continuum ($E=p$) is also shown for comparison.}
\label{fig:dispersion}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 In general, $r\neq 0$  leads to a mass splitting   of 16 fermions:
$    m(p)  \simeq  m(0)\  (^{\forall }p_{\mu} \rightarrow 0) $ and
$m(p)= m(0) +  \frac{2r}{a}N_{\pi}  \   (^{\exists }p_{\mu} \rightarrow \pi/a) $,
 where $N_{\pi}(=1,2,3,4)$ being the number of $\pi$'s
 in  $p_{\mu}a$.  This implies that we can select only one light
 fermion by choosing $m(0) \simeq 0$ and all the other
 15 fermions have masses of $O(1/a) $  for positive  $r$.
 A price to pay  is that the non-vanishing  $r$ breaks chiral symmetry
 explicitly for finite $a$, i.e. $\{ \gamma_5, D_{\rm W} \} \neq 0$ even for $m(0)=0$.
  Namely, the Nielsen-Ninomiya's no-go theorem is evaded by breaking the condition (v).

  Better way to  evade the no-go theorem
  is to break the condition (v)  in a way that the definition of
 chiral symmetry is modified.
 Suppose we consider a modified chiral rotation in the flavor space,
 \beq
 \label{eq:5.mod-chiral}
 q \rightarrow   {\rm e}^{-i \theta_{\rm {_A}} \hat{\Gamma}_5   } q,
  \ \ \bar{q} \rightarrow \bar{q} {\rm e}^{-i \theta_{\rm {_A}} \Gamma_5 }  \ \
   {\rm with} \ \  \hat{\Gamma}_5 =  \Gamma_5 (1-2 a D_{_{\rm GW}})  ,
\eeq
which  reduces to the standard axial rotation for $a\rightarrow 0$.
Here  $D_{\rm GW}$ is a generalized Dirac operator which is
 constructed so that
  $\bar{q} D_{\rm GW} q$ is  invariant under
 Eq.(\ref{eq:5.mod-chiral}) even for finite $a$;
\beq
 \label{eq:5.GW-1}
\Gamma_5 D_{\rm GW} + D_{\rm GW} \hat{\Gamma}_5 =0 ,
\eeq
or equivalently  $\{ \Gamma_5, D_{_{\rm GW}} \}
 = 2a D_{_{\rm GW}} \Gamma_5 D_{_{\rm GW}}$.
 This     is called the Ginsparg-Wilson relation.
  An explicit form of $D_{\rm GW}$ may be constructed as
\beq
 \label{eq:5.overlap-op}
 D_{\rm GW} =
 \frac{1}{2a} \left( 1+  \frac{X}{\sqrt{X^{\dagger}X}} \right)  \ \
 {\rm with} \ \
 X \equiv   D_{\rm W}^{(r=1)} - m_0  ,
\eeq
where $m_0 a$ being a dimensionless parameter of $O(1)$.
 Unlike the case of $m$ in the Wilson fermion,
 $m_0$  is not directly related to the physical fermion mass.
 Nevertheless,  if we choose the region $0< m_0 a < 2$,
  there exists
 an  exact massless mode for $N_{\pi}=0$ for finite $a$,
 and  other 15 modes have a large mass
 $(2/a)(2N_{\pi}-m_0 a)>0$.   (Exercise  \ref{prob:5}).

 Going back to the Wilson's fermion action, Eq.(\ref{eq:5.wilson-fermion}).
it  can be  conveniently rewritten  as
\beq
 \label{eq:5.wilson-fermion-lat}
S_{\rm F}
    & = & \sum_{n',n} \bar{\psi}(n') F(n',n;U) \psi(n) ,\\
\label{eq:5.wilson-fermion-op}
F(n',n;U) &=& \delta_{n'n}
- \kappa \sum_{\pm \mu} \delta_{n',n+\hat{\mu}}
 (r + {\Gamma}_{\mu}) U_{\mu}(n),
\eeq
where we have redefined the quark field
 as $\psi = a^{3/2} q /\sqrt{2\kappa }$ with
 $\kappa = [2(ma + 4r)]^{-1}$
 being the  {\it hopping parameter}.
  If the quark mass $m$ is large, $\kappa$ is
 small and the ``hopping" to the
  neighboring lattice site is suppressed.

%=================================
\subsection{Partition function on the lattice}
%=================================

The functional integration over quarks and gluons in continuum QCD
in Eq.(\ref{eq:Z-QCD}) is now transformed to the integration over quarks
  on each site and gluons on each link in lattice QCD.
 With Eq.(\ref{eq:5.wilson-action-cont}) and
   Eq.(\ref{eq:5.wilson-fermion-lat}), the partition function without the external field ($J=0$) reads
\beq
\label{eq:5.lattice-Z}
{\cal Z}  =  \int[dU d\bar{\psi}d\psi]   {\rm e}^{-  S_{\rm G}(U) - S_{\rm F}(\bar{\psi},\psi,U) }
    =  \int[dU]\ {\rm Det}\ F(U) \ {\rm e}^{- S_{\rm G}(U) }
    = \int[dU]\  \ {\rm e}^{- S_{\rm eff}(U) }
\eeq
To obtain the second equality, we have explicitly carried out the integration over the
 Grassmann variables, $\bar{\psi}$ and $\psi$,
 by using the formula in Appendix [Gaussian and Grassmann integrals]. Here ${\rm Det}$ implies the determinant in
  spacetime, color, flavor and spin degrees of freedom.
In the third equality, the exponent  is defined as
 \beq
\label{eq:Seff}
 S_{\rm eff}(U) \equiv S_{\rm G}(U) - {\rm ln Det} F(U).
 \eeq

 The integration over the group element $[dU] = \prod_{\mu,n} dU_{\mu}(n) $ can be defined through
  the {\it Haar measure} $dU$ which has the property,
  $ d(V_{\rm L}U V_{\rm R}^{\dagger})=dU$
  with  $V_{\rm L,R}$ being arbitrary group elements. Such a measure is
   unique for compact groups such as SU$(N)$.
 If we parametrize the group element as $U=\exp(i \theta_a t^a)$,  one can define the
 distance in the group space as $ds^2 = g_{ab}  (\theta) d\theta_a d\theta_b$, where the
 metric is given by $g_{ab}=  {\rm tr} (L_aL_b) =  {\rm tr} (R_a R_b) $ with
 \beq
 \label{eq:LR-form}
 L_a= -i U^{-1} ( {\partial} U/{\partial \theta_a}), \ \ \
 R_a= -i ({\partial}U/{\partial \theta_a} ) U^{-1}.
 \eeq
 Then the Haar measure can be   explicitly written as
  \beq
  \label{eq:Haar-measure}
   dU = {\cal N} \sqrt{ \det g} \ \prod_a d\theta_a,
  \eeq
  with an overall normalization factor ${\cal N}$.

 The followings are some examples of the  SU($N$)  group integration, which can be proved
 by the invariant property of the Haar measure (except for the first one which determines the
 normalization of the measure) (Exercise \ref{prob:6}):
 \beq
 \label{eq:group-int-0}
  & &   \int dU \  1  =  1 , \\
 \label{eq:group-int-1}
 & &   \int dU \ U_{ij}  =  0, \\
 \label{eq:group-int-2}
 & &    \int  dU \ U_{ij} U_{k\ell}^{\dagger}  =  \frac{1}{N} \delta_{i\ell} \delta_{jk} , \\
 \label{eq:group-int-4}
 & &  \int dU \ U_{ij} U_{k\ell}  U_{i'j'}^{\dagger} U_{k'\ell'}^{\dagger}
 = \frac{ \delta_{ij'} \delta_{ji'} \delta_{k\ell'} \delta_{\ell k'} + {(j' \leftrightarrow \ell',  i' \leftrightarrow k' ) }}{N^2-1}
   -  \frac{\delta_{ij'} \delta_{jk'} \delta_{k\ell'} \delta_{\ell i'} + { (j' \leftrightarrow \ell',  i' \leftrightarrow k' )}  }{N}, \\
 \label{eq:group-int-B}
 & &  \int dU \ U_{i_1 j_1} \cdots U_{i_N j_N} =
 \frac{1}{N!} \epsilon_{i_1 \cdots i_N}    \epsilon_{j_1 \cdots j_N} .
 \eeq

Similar to the statistical systems such as the Ising model,
 observables are obtained by averaging over the statistical weight  as
 \beq
 \langle {\cal O} \rangle = \frac{1}{{\cal Z}}  \int[dU]\ {\cal O}(U) \ {\rm e}^{- S_{\rm eff}(U) }.
 \eeq
    Due to the gauge invariance of the Haar  measure,
 gauge non-invariant quantities have vanishing expectation values
  (Elitzer's theorem).  For example, consider the expectation value of the link variable,
 \beq
 \label{eq:elitzer}
 \langle U_{\mu} (n) \rangle
 = \frac{1}{{\cal Z}}  \int [dU] U_{\mu} (n)  \ {\rm e}^{- S_{\rm eff}(U) }  = V(n)  \langle U_{\mu} (n) \rangle ,
\eeq
where we have made a change of variable, $U_{\mu}(n) \rightarrow V(n) U_{\mu}(n)$ with $V(n)$
being the $SU(N)$ matrix, and used the  gauge invariance of the Haar measure as well as ${\rm Det}\ F(U)$ and
$S_{\rm G}(U)$.  Since Eq.(\ref{eq:elitzer}) must be true for arbitrary $V(n)$, we have $ \langle U_{\mu} (n) \rangle =0$.

Some examples of the non-vanishing  observables are shown  in Fig.\ref{fig:correlation}:
 (a) and (b) correspond to the mesic and baryonic correlations, respectively,  while (c) is a correlation related to the
baryon-baryon interactions.  The filled circles are the spacetime points where the quarks and anti-quarks
are created or absorbed. Each  line with arrow indicates the quark propagator $F^{-1}(n,n';U)$
connecting two spacetime points $n$ and $n'$.  Thus, the explicit forms of the
 mesic and baryonic correlations are
 \beq
 \label{eq:correlation-M}
C_{\rm M}(n,n')
 &=& \frac{1}{{\cal Z}}  \int [dU] \ F_{\alpha \beta}^{-1}(n,n';U) F_{\beta \alpha}^{-1}(n',n;U)  \ {\rm e}^{- S_{\rm eff}(U) } ,\\
\label{eq:correlation-B}
C_{\rm B}(n,n')
 &=& \frac{1}{{\cal Z}}  \int [dU] \ \epsilon_{\alpha \beta \gamma} \epsilon_{\alpha' \beta' \gamma'}
 F_{\alpha \alpha'}^{-1}(n,n';U) F_{\beta \beta'}^{-1}(n,n';U) F_{\gamma \gamma'}^{-1}(n,n';U)  \ {\rm e}^{- S_{\rm eff}(U) } ,
  \eeq
where all the color indices are contracted so that $C_{\rm M}$ and $C_{\rm B}$ are gauge invariant.
 Other quantum numbers such as spin and flavor  associated with $F^{-1}$ are
 not shown explicitly.
  Spacetime, spin and flavor dependences of  $C_{\rm M}(n,n')$ in (a) and $C_{\rm B}(n,n')$ in (b)
  have all the information on the hadronic states in various different channels, while
  $C_{\rm BB}(n,m,n',m')$   in (c) has the information on baryon-baryon interactions.

% FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[t]
%\vspace{-1cm}
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
\includegraphics[scale=0.2]{Chapter3-figures/correlations.eps}
 \end{center}
%\vspace{-1cm}
\caption{
(a) Single meson correlation representing the propagation of a  meson created at point $n'$ and absorbed at point $n$.
(b)  Single baryonic correlation representing the propagation of a  baryon created at point $n'$ and absorbed at point $n$.
(c)  Two baryon correlation which contains the information on baryon-baryon interaction.}
\label{fig:correlation}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%===========================================
\subsection{Strong coupling expansion and quark confinement}
%===========================================

% FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
\includegraphics[scale=0.55]{Chapter3-figures/wilson-L.eps}
 \end{center}
\caption{A rectangular  Wilson loop with the
 temporal (spatial) size ${\cal T}$ ($R$).}
\label{fig:wilson-L}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

One of the remarkable properties of QCD is the   confinement of quarks inside
hadrons.  Simplest setup to see this phenomena is to consider
the potential $V(R)$ between  an infinitely heavy quark $Q$ and an anti-quark $\bar{Q}$ with
a fixed separation $R$.
It corresponds to Fig.\ref{fig:wilson-L} and can be written as
\beq
 \label{eq:5.wilson-loop-lattice}
\langle W(C) \rangle
 & = & \langle {\rm tr}\ \prod_{{\rm link}\in C}  U_{\mu}(n) \rangle , \\
 \label{eq:5.wilson-loop-asym}
 & \propto  &
  {\rm e}^{-V(R) {\cal T} } \simeq
 \exp \left[ -\left( K R + b + \frac{c}{R} + \cdots \right) {\cal T} \right] ,
\eeq
where we  have taken a limit
${\cal T} \gg R \rightarrow \infty$ in   Eq.(\ref{eq:5.wilson-loop-asym}).
 Remembering the fact that the real time $t$ and the imaginary time $\tau$ are
 related as  $\tau=it$, the exponential falloff of  $\langle W(C) \rangle$ in $\tau$ implies the
  temporal oscillation in $t$, and its $R$-dependent coefficient is nothing but the
 interaction  energy between $Q$ and $\bar{Q}$.

In Eq.(\ref{eq:5.wilson-loop-asym}),
   $K > 0$  implies
 the existence of a string-like
linear confining potential.
 It also implies
 the area law  of the Wilson loop
 $\langle W(C) \rangle \sim \exp(-K{\cal A})$
 where ${\cal A}= R \times {\cal T} $ is
  the area inside the path $C$.
    In full QCD where pair creation of light quarks
 are allowed,
  the linear rising potential becomes eventually
    flat at long distances due to the breaking of the string,
  $Q\bar{Q} \rightarrow (Q\bar{q})(q\bar{Q})$.

  To make the analysis simple, let us now consider the
  SU$(N_c)$ Yang-Mills theory
  without light quarks: This is called the quenched
  approximation and corresponds to take $F(U) =1$.
In this case,  the Wilson loop can be evaluated analytically
 in the strong coupling regime ($g \rightarrow \infty$).
First of all,  $S_{\rm G}$ is proportional to $1/g^2$,
so that  one can make an expansion,
$\exp (-S_{\rm G}) = 1 - S_{\rm G}+ S_{\rm G}^2 /2 + \cdots $
 and finds (Exercise \ref{prob:7})
\beq
\label{eq:5.wilson-strong}
\langle W(C) \rangle = \frac{1}{\cal Z}
 \int [dU] \ {\rm tr} \prod_{{\rm link}\in C} U_{\mu}(n) \
 \sum_{\ell =0}^{\infty} \frac{1}{\ell !} (-S_{\rm G})^{\ell}.
\eeq

Only the first three integrals, Eqs.(\ref{eq:group-int-0},\ref{eq:group-int-1},\ref{eq:group-int-2}),
 are necessary to extract the leading contribution to
 $\langle W(C) \rangle $  in the strong coupling.
   Key observation is that all the $U$'s
  from the Wilson loop and $U^{\dagger}$'s
 from $(-S_{\rm G})^{\ell}$ should be paired in the leading order of $1/g^2$
 in   Eq.(\ref{eq:5.wilson-strong}).
  This means that the area inside the Wilson loop is tiled up
 with minimum number of plaquettes as shown in
  Fig.\ref{fig:strong-c}. All the structures other than
  the minimal surface are higher orders in  $1/g^2$.

% FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[t]
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
\includegraphics[scale=0.65]{Chapter3-figures/strong-c.eps}
 \end{center}
\caption{A minimum surface in which the Wilson loop is tiled
   up by the fundamental plaquettes in the
 strong coupling limit.}
\label{fig:strong-c}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 In the evaluation of the numerator of
 Eq.(\ref{eq:5.wilson-strong}),
 each plaquette has a contribution $1/g^2$.
 Also each  integration on the link
   gives a factor $1/N_c$
   and the contraction of the color indices gives a factor $N_c$
 on each  site.
  On the other hand,
 ${\cal Z}$  (the denominator of Eq.(\ref{eq:5.wilson-strong})
 is   unity in the leading order.
    Thus, one arrives at the formula in the lowest order
 of the strong coupling expansion,
\beq
 \label{eq:5.strong-estimate}
\frac{1}{N_c} \langle W(C) \rangle
& & \xrightarrow[g^2\rightarrow \infty]
 \  \frac{1}{N_c} \cdot
\left(\frac{1}{g^2} \right)^{N_{\rm plaq}} \cdot
\left(\frac{1}{N_c} \right)^{N_{\rm link}} \cdot N_c^{N_{\rm site}}  , \nonumber \\
& & \ \ \ \ \ \ \ \ \ \ \ =\left(\frac{1}{N_c g^2} \right)^{\frac{R {\cal T}}{a^2}}
= \exp \left( - \frac{\ln N_c g^2}{a^2} R {\cal T} \right) ,
\eeq
where we have used a relation, $N_{\rm link}-N_{\rm site}+1
 = N_{\rm plaq}$ and
$N_{\rm plaq} a^2 = R {\cal T} ={\cal A}$.
 Since it shows the area law, \index{area law}
  the confinement is naturally obtained in the strong coupling
  with the linear rising potential,
\beq
 \label{eq:5.linear-potential}
V(R) = K R \ \ {\rm with} \ \ K = \frac{1}{a^2} \ln (N_c g^2).
\eeq

 If we consider higher orders of the strong coupling expansion,
   ``rough" surfaces should be taken
 into account.
 Nevertheless, the confining feature is
 stable against small perturbations in $1/g^2$.
  In fact, there exits a theorem
  that,  for sufficiently large $g$, the strong coupling expansion
  converges and shows confinement for all compact gauge groups
  in all spacetime dimensions.

  A question here is that whether the real world
  corresponds to the strong coupling region discussed above.
  The answer is no;  the real world corresponds to  the weak coupling regime.
  For compact QED (quantum electrodynamics
  formulated in terms of the  U(1) link variable),
  the confinement  $K>0$ in the strong coupling regime
   changes to  $K=0$ in the weak coupling regime.
  On the other hand,  in QCD in four spacetime dimensions
 with $N_c=3$,   the confinement feature is expected to persist even in the weak coupling regime.
 Indeed, there are  strong evidences for this statement
  from  LQCD simulations.
  Its analytic proof, however,  is  still missing and is being one of the
   most challenging problems in mathematical  physics.

%===============================================
\subsection{Weak coupling expansion and continuum limit}
%===============================================

  Lattice QCD can be regarded as an effective field theory
   with an ultraviolet (UV)  cutoff in the coordinate space.
   The gauge coupling  $g$ is then interpreted as
  a bare coupling defined at the scale $a$ where
  quantum fluctuations with the wave length shorter than
  $a$ are integrated out.  In non-Abelian gauge theories,
   it can be shown that $g(a)$ decreases logarithmically as $a$ decreases
   unless the number of matter fields is not too large.
    This is called the {\it asymptotic freedom}, and is essential for taking
    the continuum limit ($a \rightarrow 0$) to remove the
    lattice artifact.

 For simplicity, let us consider the case with massless fermions, where
 observables
   ${\cal O}$ such as the string tension and the hadron masses  depend only
   on the coupling $g$ and the  regularization scale $a$.
   Then,  from the dimensional ground, one can  write
     \beq
\label{eq:5.O-dim}
 {\cal O} (g(a),a) = a^{-d} X(g(a)),
 \eeq
 where $d$ is the mass-dimension of ${\cal O}$ and $X$ is a dimensionless
 function of $g$.  The $a$-independence of the observable implies
 \beq
\label{eq:5.O-RG}
 a\frac{d{\cal O}}{da}
  =
  \left( a \frac{\partial}{\partial a}
 - \beta_{_{\rm LAT}}  \frac{\partial}{\partial g} \right) {\cal O}(g(a), a)  = 0 ,
\ \   \beta_{_{\rm LAT}}(g) =  -a \frac{dg(a)}{da} .
\eeq
By integrating the first equation in Eq.(\ref{eq:5.O-RG}),  we find
\beq
\label{eq:5.lat-F}
 X(g) =
 \exp \left( -d \int^g \frac{dg'} {\beta_{_{\rm LAT}}(g')} \right) .
 \eeq
Suppose that
 the beta-function can be expanded in terms of $g$ for small $a$:
$\beta_{_{\rm LAT}}(g)   =
  - \beta_0 g^3 - \beta_1  g^5 + \cdot \cdot \cdot $. Here
  $\beta_0$ and $\beta_1$ can be shown to be independent of the
   regularization scheme and are known to be
\beq
\beta_0= \frac{1}{(4\pi)^2} \left( \frac{11N_c}{3}-\frac{2N_f}{3} \right),
\ \ \  \beta_1= \frac{1}{(4\pi)^4} \left( \frac{34N_c^2}{3}-\frac{38N_f}{3} \right),
\eeq
           for QCD with $N_c$ colors and     $N_f$ fermions.

% FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}[t]
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
%\includegraphics[scale=0.6]{as-scale.eps}
\includegraphics[scale=0.30]{Chapter3-figures/crossover.eps}
 \end{center}
\caption{Crossover behavior of the dimensionless
 string tension $Ka^2$ from the strong coupling
 regime  $\beta=2N_c/g^2 \rightarrow 0$ to the weak coupling
  (asymptotic scaling) regime  $\beta=2N_c/g^2 \rightarrow \infty$
  for SU($N_c=2$) Yang-Mills theory.  The figure is adapted from
  \cite{Creutz:1980zw}.
  }
\label{fig:as-scale}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

By integrating  the second equation in Eq.(\ref{eq:5.O-RG}) with the above expansion of
$\beta_{_{\rm LAT}}(g)$, one finds
 \beq
\label{eq:5.a-vs-L}
 a = \Lambda_{_{\rm LAT}}^{-1} \cdot
 \exp  \left( -\frac{1}{2\beta_0 g^2}  \right)  \cdot
  (\beta_0 g^2)^{-\frac{\beta_1}{2 \beta_0^2}}
 \cdot (1 + O(g^2)).
 \eeq
Here  $ \Lambda_{_{\rm LAT}}$ is called the scale parameter on the lattice, and
   $g(a)$ can be expressed in terms of $a$ and $ \Lambda_{_{\rm LAT}}$ ((Exercise \ref{prob:8}),
 \beq
\label{eq:5.lat-running-g}
\frac{1}{g^2(a)} = \beta_0 \ln \left( \frac{1} {a^2  \Lambda_{_{\rm LAT}}^2} \right)
+ \frac{\beta_1}{\beta_0} \ln \ln \left( \frac{1}{a^2  \Lambda_{_{\rm LAT}}^2} \right)
+ \cdot \cdot \cdot .
\eeq
 This is the asymptotic freedom
 in which $g(a)$ decreases as $a$ decreases.  This also justified the
 assumption that the beta-function can be expanded by $g(a)$ for small $a$.

Direct way to extract the actual value of $ \Lambda_{_{\rm LAT}}$  is to
 carry out  numerical simulations of
 a certain physical quantity (such as the string tension)
 and compare the result with the experimental value.
 For example, the string tension, which has mass-dimension
 two ($d=2$) should  behave as
\beq
\label{eq:5.lat-K}
K a^2 = C_K \exp \left( - \frac{1}{\beta_0 g^2} \right)
(\beta_0 g^2)^{-\beta_1 /\beta_0^2} = C_K  \Lambda_{_{\rm LAT}}^2 ,
\eeq
with $C_K$ being a dimensionless numerical constant independent
 of $g$.  As  can be seen from this example,
 the functional form of the physical quantities
 for $g\sim 0$  is  severely constrained.
This is called the {\em asymptotic scaling} which
 is used to check whether the system is
  close enough to the continuum limit.
Shown in Fig.\ref{fig:as-scale} is a historic numerical study, which shows
 a crossover of $Ka^2$  from the strong coupling regime to the weak coupling regime
 in SU(2) Yangs-Mills theory.

%===============================
\subsection{Running coupling}
%===============================

% FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
%\includegraphics[scale=0.6]{as-scale.eps}
\includegraphics[scale=0.45]{Chapter3-figures/running-g.eps}
 \end{center}
\caption{Perturbative running coupling $\frac{1}{4\pi}\bar{g}^2(\lambda)$ as a function of $\lambda^{-1}$.
In the short distance limit ($\lambda =1$), the running coupling coincides with the bare coupling $\bar{g}(\lambda=1)=g(a)$,
while, in the long distance regime ($\lambda \ll 1$), the running coupling grows. }
\label{fig:running-g}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let us now consider an observable ${\cal O}$ which depends
not only on $g(a)$ and $a$ but also on some external dimensionful parameter.
For concreteness, we consider the heavy quark potential $V(R; g(a),a)$ in the
quenched approximation.  Since it has the dimension of energy, one may write
\beq
V(R,g(a),a) = R^{-1} \tilde{V} (R/a, g(a)).
\eeq
Then the cutoff independence of the observable, $a \frac{d}{da}V(R,g(a),a)=0$,  leads to
\beq
\left( \lambda \frac{\partial}{\partial \lambda} + \beta_{_{\rm LAT}} \frac{\partial}{\partial g} \right) \tilde{V} (\lambda,g)=0,
\label{eq:RG-R}
\eeq
where we have introduced
a dimensionless  scaling parameter $\lambda$ through $R=\lambda a$.

]The solution of the {\it renormalization group equation}, Eq.(\ref{eq:RG-R}), reads
\beq
\label{eq:RG-sol}
 \tilde{V} (\lambda,g(a)) = \tilde{V} (1, \bar{g}(\lambda)).
 \eeq
 Here $\bar{g}(\lambda)$ is called the {\it running coupling} which is  a solution of
\beq
\lambda \frac{d\bar{g}}{d\lambda}  = - \beta_{_{\rm LAT}} (\bar{g}(\lambda)),
\eeq
with the boundary condition, $\bar{g}(\lambda=1)= g(a)$.
One can show that  Eq.(\ref{eq:RG-sol})  satisfies Eq.(\ref{eq:RG-R}) explicitly by applying the partial derivatives
 or more generally by the method of characteristics in Appendix [Method of characteristics].
Then, we eventually arrive at the formula
\beq
V(R,g(a), a) = \frac{a}{R} V (a, \bar{g}(R/a), a) .
\eeq
If $R$ is  in the interval,  $a < R \ll  \Lambda_{_{\rm LAT}}^{-1}$, the running coupling $\bar{g}$ is small enough, so that
one may use the perturbative expansion of $\beta_{_{\rm LAT}}$ to obtain
\beq
\bar{g}^2(R/a) \simeq \frac{g^2(a)}{1-2\beta_0 g^2(a) \ln (R/a)} =  \frac{1}{2\beta_0 \ln (1/(R \Lambda_{_{\rm LAT}}) ) }.
\label{eq:running-R}
\eeq
In Fig.\ref{fig:running-g}, the behavior of $\frac{1}{4\pi}\bar{g}^2(\lambda)$  as a function of $\lambda^{-1}$ is shown.
The bare coupling $\frac{1}{4\pi}g^2(a)$ appears as the boundary condition at shortest distance $\lambda=R/a=1$,
while $\frac{1}{4\pi} \bar{g}^2(\lambda)$ grows  as $\lambda$ increases.
The latter implies that the strong interaction has anti-screening feature.

For $R$ sufficiently close to $a$, one may evaluate the potential by using perturbation theory as
$V (a, \bar{g}(R/a), a)  = - C_{{\rm F}} \frac{\bar{g}(R/a)^2}{4\pi a}$, so that we finally obtain
\beq
\label{eq:RG-Coulomb}
V(R,g(a),a) \simeq - C_{{\rm F}} \frac{\bar{g}^2(R/a)}{4\pi R} \ \ \ \ \ (a < R \ll  \Lambda_{_{\rm LAT}}^{-1}),
\eeq
where $C_{{\rm F}}=4/3$ for $N_c=3$ is given in Appendix [SU($N$) algebra].
Eq.(\ref{eq:RG-Coulomb}) is nothing but the Coulomb potential with the running coupling constant.
Note that the left hand side is $a$-independent, while the right hand side has
logarithmic $a$-dependence through $\bar{g}$.  This is due to the use of perturbation theory;
such a logarithmic $a$-dependence  is cancelled by the next-to-leading order term.
Note also that the similar analysis can be done for any other observables.
The QCD thermal pressure at finite temperature $P(T,g(a),a)$ is a typical example, in which
the weak coupling analysis (i.e. the description by  quark-gluon plasma picture)
is justified  under the condition  $  \Lambda_{_{\rm LAT}} \ll T < a^{-1}$.

%%%%%%%%%%%%%%%%%%%
\section{Lattice QCD:  numerical simulations}
%%%%%%%%%%%%%%%%%%%

Suppose we have a lattice having $N_{\rm s}$ ($N_{\tau}$) number of sites
in each spatial (temporal) direction. Then
 the total number of links is
   $  N_{\rm s}^3 \times N_{\tau}  \times 4$.
 Therefore the total  number of gluon integrations $\int [dU]$
 for a moderate lattice size  $N_{\rm s} = N_{\tau} =32$ reads
\beq
\label{eq:5.lattice-size}
 ( N_{\rm s}^3 \times N_{\tau} \times 4 )_{\rm links} \times 8_{\rm
 color} \sim 3 \times 10^7 .
\eeq
This is hopelessly a large number for standard methods of numerical
integration.  In this case, the Monte Carlo (MC) integration, which is
a statistical way to evaluate the integral, plays a powerful role. For
rapidly varying integrand, the MC integration should be supplemented
by the importance sampling to have better accuracy, in which the
rapidly varying part is sampled more than the slowly varying part.

 %==========================
\subsection{Importance sampling}
\label{ss:IS}
%==========================

     Let us consider the general partition function ${\cal Z}= \int [d\phi] \exp(-S(\phi)) $,
    with some c-number field $\phi$  and try to calculate  an observable ${\cal O} $ by
  \beq
  \langle {\cal O} \rangle = \frac{1}{{\cal Z}} \int [d \phi] \ {\cal O} (\phi) e^{-S(\phi)}.
  \eeq
  The basic procedure of the MC integration
  with the {\it importance sampling} consists of two steps:
  \begin{enumerate}
  \item[(I)]   Generate a set of field configurations, $ \{  \phi^{(1)}, \phi^{(2)}, \cdots, \phi^{(N)} \}$,
with $\phi^{(n)}$  being arranged to appear with
 a probability in ``equilibrium", $W_{\rm eq}[\phi]={\cal Z}^{-1} \exp (-S(\phi))$.
 \item[(II)]  The field configurations thus generated
 are  used to calculate the expectation value,
 \beq
\label{eq:5.obs-average}
\langle {\cal O} \rangle = \frac{1}{N}\sum_{n=1}^N {\cal O}^{(n)}
\pm \sqrt{\frac{\sigma^2}{N}}, \ \ \ \ \
\sigma^2 =  \frac{1}{N-1} \sum_{n=1}^N \langle {\cal O}^{(n)} - \langle {\cal O} \rangle \rangle^2 ,
\eeq
with ${\cal O}^{(n)}={\cal O}(\phi^{(n)})$.
\end{enumerate}

 %=========================================
\subsection{Markov chain Monte Carlo (MCMC)}
%==========================================

 For large  number of integration variables such as Eq.(\ref{eq:5.lattice-size}),
 it  is essential to develop an appropriate scheme to carry out  Step (I) in Sec. \ref{ss:IS}.
 The {\it Markov chain Monte Carlo (MCMC)}  method is one of such schemes.

Let us consider a chain of configurations  generated successively starting from an
initial configuration,
\beq
\label{eq:M-chain}
 \phi_{0} \rightarrow  \phi_{1} \rightarrow \phi_{2}
 \rightarrow \cdots \rightarrow \phi_{i} \rightarrow  \phi_{i+1}
\rightarrow \cdots ,
\eeq
where the ``{\it update}" of the $i$-th configuration ($\phi$)  to
the $(i+1)$-th configulation ($\phi'$)  is governed by the conditional probability
$P$ or equivalently  the transition matrix $\mathbf{T}$,
\beq
\label{eq:M-transition}
P(\phi \rightarrow \phi') = (\mathbf{T})_{\phi \phi'}
\eeq
which has the property, $\sum_{\phi'} P(\phi \rightarrow \phi')=1$.
Eq.(\ref{eq:M-chain}) generated by Eq.(\ref{eq:M-transition}) is called the {\it Markov chain} since
it is governed by the Markov process where the conditional probability depends only on the
neighbouring pair. The probability distribution $W[\phi]$ (with the properties, $W[\phi] \ge 0$ and
$\sum_{\phi} W[\phi]=1$) is updated successively by Eq.(\ref{eq:M-transition}),
\beq
\label{eq:5.MC-update}
W'[\phi' ] =  \sum_\phi  W[\phi] P(\phi \rightarrow \phi').
\eeq

If  the Markov chain is {\it irreducible} (any $\phi$ and $\phi'$ are connected with each other) and  {\it aperiodic}
(absence of $\phi$ which appears periodically) \footnote{Rigorous definitions are as follows. (i) The Markov chain is said to be
irreducible if one can find a finite positive integer $n (< \infty)$  such that  $(\mathbf{T}^n)_{\phi \phi'} > 0 $ for all $\phi$ and $\phi'$.
(ii)  The period, $d(\phi)$, is defined by the greatest common divisor  of the set of positive integers $n (\ge 1)$
such that  $(\mathbf{T}^n)_{\phi \phi} > 0 $ is satisfied. If $d(\phi)=1$ for all $\phi$, the Markov chain is
said to be aperiodic \cite{Haggstrom:2002}.},  there exists a theorem that the Markov chain
 has a unique equilibrium distribution $W_{\rm eq}$ satisfying
\beq
\label{eq:transition-W}
W_{\rm eq}[\phi']  =  \sum_\phi W_{\rm eq}[\phi] P (\phi \rightarrow \phi'),
\eeq
and it can be reached by $\mathbf{T}^{\infty}$ starting from arbitrary initial distribution.
For a heuristic proof of this theorem, see Exercise \ref{prob:9}. For mathematical proof,  see  \cite{Haggstrom:2002}.

As is easily seen, a  sufficient but not necessary condition for $P$ to lead $W_{\rm eq}$  is the {\it detailed balance}:
\beq
\label{eq:5.det-balance}
W_{\rm eq}[\phi] P (\phi \rightarrow \phi' )  =
 W_{\rm eq}[\phi'] P (\phi' \rightarrow \phi ).
\eeq
There also exits specific algorithm without the detailed balance in MCMC  \cite{SuwaTodo:2010}.

The Markov chain Eq.(\ref{eq:M-chain})  takes
certain {\it thermalization time} to reach equilibration.
Also, the nearby  field configurations  are strongly correlated  during the {\it autocorrelation time}.
To calculate the actual average in Eq.(\ref{eq:5.obs-average}), we then
need to discard non-themalized configurations and also
thin out the configurations to avoid the autocorrelations.
This is schematically shown for an observable ${\cal O}$ in Fig.\ref{fig:auto}.
The thermalization time can be estimated by monitoring the behavior of ${\cal O}$  under
successive update, while the autocorrelation time can be estimated by
calculating the correlations of ${\cal O}$ for different configurations.

% FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
%\includegraphics[scale=0.6]{as-scale.eps}
\includegraphics[scale=0.37]{Chapter3-figures/auto.eps}
 \end{center}
\caption{Schematic illustration on the behavior of ${\cal O}$ under successive update starting from
certain initial configuration. Blue crosses correspond to ${\cal O}^{(n)}$ to be used for the actual average in
Eq.(\ref{eq:5.obs-average}).}
\label{fig:auto}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%===============================
\subsection{Hybrid Monte Carlo (HMC)}
%===============================

 Most widely used method   for
  generating configurations in LQCD
  is  the hybrid Monte Carlo (HMC) method \cite{Duane:1987de} and its variations.
 The basic procedure of the HMC can be summarized as follows:
 First, we rewrite the partition function by introducing a conjugate momentum field $\pi$,
 so that ${\cal Z}$ is transformed to a phase space functional integral,
\beq
{\cal Z}= \int [d\phi]\ e^{-S(\phi)}= \int [d\Phi] \  e^{- H(\Phi)}, \ \  \ \ \
H(\Phi) = \frac{1}{2} \pi^2 + S(\phi) ,
\eeq
where $\Phi\equiv(\phi,\pi)$ and $[d\Phi]\equiv [d\phi d\pi]$.
Then we follow the steps below:
 \begin{enumerate}
 \item[1.]   Start with arbitrary chosen initial configuration, $\phi$.
 \item[2.]  Generate $\pi$ with the Gaussian distribution,
 \beq
 P_{\rm G}(\pi) \propto \exp(-\pi^2/2).
 \eeq
\item[3.]  Evolve $\Phi $ under transition probability $P_{\rm H}$ with the {\it reversibility condition},
\beq
\label{eq:reversible}
P_{\rm H}(\Phi \rightarrow \Phi') = P_{\rm H}(\Phi'_r \rightarrow \Phi_r),   \ \ \  \Phi_r \equiv (\phi,-\pi).
\eeq
\item[4.] Accept the configuration $\Phi'$ with the probability,
\beq
\label{eq:MET-test}
P_{\rm A} (\Phi \rightarrow \Phi')  = {\rm min.} \{ 1, e^{-\Delta H} \} ,
\eeq
where  $\Delta H=H(\Phi')- H(\Phi)$.  This is called the {\it Metropolis test}  \cite{Metropolis_1953}.
\item[5.] If the new configuration $\Phi'$ is accepted, go to Step 2 with $\phi'$.
 Otherwise, keep the original $\phi$ and go to Step 2.
\end{enumerate}

The above procedure satisfies the detailed balance  Eq.(\ref{eq:5.det-balance})
with $W_{\rm eq}[\phi]=\exp(-S(\phi))$.
In fact, the Step 4  satisfies the detail balance in phase space (Exercise \ref{prob:10}),
\beq
\label{eq:DT-balance}
e^{-H(\Phi)} P_{\rm A} (\Phi\rightarrow \Phi') = e^{-H(\Phi')} P_{\rm A} (\Phi' \rightarrow \Phi) .
\eeq
Then, we have
\beq
e^{-S(\phi)} P (\phi \rightarrow \phi' )
&=& e^{-S(\phi)} \int [d\pi d\pi'] P_{\rm G}(\pi)  P_{\rm H}(\Phi \rightarrow \Phi') P_{\rm A} (\Phi \rightarrow \Phi') , \nonumber \\
&=& \int [d\pi d\pi']\ e^{-H(\Phi)}  P_{\rm H}(\Phi \rightarrow \Phi') P_{\rm A} (\Phi \rightarrow \Phi') ,\nonumber \\
&=& \int [d\pi d\pi']\ e^{-H(\Phi')}  P_{\rm H}(\Phi \rightarrow \Phi') P_{\rm A} (\Phi' \rightarrow \Phi) ,\nonumber \\
&=& \int [d\pi d\pi']\ e^{-H(\Phi')}  P_{\rm H}(\Phi'_r \rightarrow \Phi_r)  P_{\rm A} (\Phi' \rightarrow \Phi) ,\nonumber \\
&=& \int [d\pi d\pi']\ e^{-H(\Phi')}  P_{\rm H}(\Phi' \rightarrow \Phi ) P_{\rm A} (\Phi' \rightarrow \Phi) ,\nonumber \\
&=& e^{-S(\phi')} \int [d\pi d\pi'] P_{\rm G}(\pi')  P_{\rm H}(\Phi' \rightarrow \Phi)  P_{\rm A} (\Phi' \rightarrow \Phi)
= e^{-S(\phi')}  P (\phi' \rightarrow \phi ) ,
\eeq
where we have used Eq.(\ref{eq:reversible}) to obtain the 4th line, and also used $H(\Phi)=H(\Phi_r)$
 to obtain the 5th line.

Note that $P_{\rm H}$ can be chosen to be any transition probability as long as it satisfies
Eq.(\ref{eq:reversible}).  In practice, the deterministic procedure based on the Molecular Dynamics (MD)
evolution along the  ``computer"  time $s$ is useful:
\beq
\label{eq:MD}
\frac{d}{ds}
\left(
\begin{array}{cc}
 \phi  \\
  \pi
\end{array}
\right)
=
\left(
\begin{array}{cc}
 0 &  1   \\
 -1  & 0
\end{array}
\right)
\left(
\begin{array}{cc}
  {\delta H (\phi,\pi)}/{\delta \phi}   \\
  {\delta H (\phi,\pi)}/{\delta \pi}
\end{array}
\right) =
\left(
\begin{array}{cc}
 \pi  \\
  -  {\delta S (\phi)}/{\delta \phi}
\end{array}
\right) ,
\eeq
which leads to
\beq
P_{\rm H}(\Phi \rightarrow \Phi')  = \delta (\Phi' - \Phi(s)),
\eeq
 on the phase space trajectories, $\Phi = \Phi(0) \rightarrow \Phi(s)$.

If we do not introduce the MD before the Metropolis test $P_{\rm A}$,
the procedure is essentially the MCMC with the Metropolis test.  It
becomes, however, very slow for non-local action such as
Eq.(\ref{eq:Seff}) where $S_{\rm G}(U)$ is local in spacetime while
${\rm Ln Det} F(U)$ is non-local.  The MD is a nice way to evolve the
whole variables on the lattice at once.  The computer time $s$ needs
to be discretized with a step size $\varepsilon$, which brings
inevitable numerical error in MD. However, the Metropolis test in Step
4 eliminates such error so that no extrapolation to $\epsilon=0$ is
required in HMC.

 There are numerical algorithms in MD to satisfy the reversibility and
 preserve the phase space area exactly for finite $\varepsilon$.  The
 {\it leapfrog integrator} is one of such algorithms widely used in
 LQCD (see Appendix [Leapfrog integrator in molecular dynamics]).
 Since this conserves the Hamiltonian with $0(\varepsilon^2$)
 accuracy, the acceptance rate in Step 4 can be kept high.

 In LQCD simulations, we need to treat the unitary matrices $U_{\mu}(n)$ as dynamical variables, i.e.
  the MD should be performed on the SU($N_c$)  group manifold.
 The appropriate choice of the conjugate momentum would be the element of the Lie algebra,
  $ P_l =  R_l^a t^a = -i  (d{U_l}/ds) U_l^{-1} $ (see Eq.(\ref{eq:LR-form})) where
  we have abbreviated the link index $n$ and site index $\mu$  as $l$ for simplicity.
 This  leads to the
  equation of motion for $U_l$,
\beq
\label{eq:EOM-U}
\frac{d U_l}{ds}= i P_l U_l .
\eeq
The effective Hamiltonian is naturally written as
\beq
\label{eq:EOM-H}
H= {\rm tr} \sum_{l} P_l^2 + S_{\rm eff}(U) ,
\eeq
 where ${\rm tr}$ is over color indices with the normalization given in Eq.(\ref{eq:tt}).
 Then the  time-parameter independence  $\frac{dH}{ds}=0$ leads to the equation of motion for $P_l$
 (Exercise \ref{prob:11}),
  \beq
 \label{eq:EOM-P}
 \frac{dP_l}{ds} = - i  \sum_{i,j} t^a \left( t^a U_l \right)_{ij} \frac{\partial S_{\rm eff}(U)}{\partial (U_l)_{ij}}.
\eeq
In the actual simulations, the ${\rm ln Det}F(U) $ part of the effective action is treated by
introducing a set of bosonic variables (pseudofermions) through the identity,
\beq
{\rm Det} F = ({\rm Det} F^{-1})^{-1} = \int [d\chi^* d\chi] \ \exp \left( -\sum_{IJ} \chi^*_{I} F_{IJ}^{-1} \chi_J \right),
\eeq
where $I$ and $J$ stand for all possible internal and spacetime indices carried by $F$.
For further details of HMC (and its variations) with pseudofermions,
consult the recent review \cite{Schaefer:2012tq} and references therein.

%===========================
\subsection{Error estimate}
%===========================

There are two kinds of errors  in the data obtained from  LQCD simulations.\\

\noindent {\bf Systematic errors:} \\
 They are related to the lattice spacing $a$, the lattice volume $L^3$,
  and the quark masses ($m$).  During
  the continuum extrapolation ($a\rightarrow 0$) and the thermodynamic extrapolation ($L \rightarrow \infty$)
  under the  guidance of  the asymptotic scaling for small $a$
 and the finite size scaling for  large $L$, some systematic errors are brought in.
 Also,  one often needs to make extrapolation to the physical quark mass by using
 lattice data  with heavier  quark masses. This  also brings some
 systematic errors.\\

 \noindent
{\bf Statistical error:} \\
It originates from the importance sampling.  A very useful procedure to estimate such error
 commonly used in LQCD is the {\it jackknife resampling method}. (The name
  originates from the ``jackknife"  which is an easy and  portable  tool  for general purposes).
  Let us consider the mean and the unbiased variance of a certain quantity $ {\cal O}$,
  \beq
   \langle {\cal O} \rangle = \frac{1}{N} \sum_{n=1}^N {\cal O}^{(n)}  \pm \sqrt{\frac{\sigma^2({\cal O} )}{N}}, \ \ \ \
   \sigma^2({\cal O}) = \left( \frac{N}{N-1} \right)  \frac{1}{N} \sum_{n=1}^N ( {\cal O}^{(n)}  - \langle {\cal O} \rangle )^2,
  \eeq
where the factor $\frac{N}{N-1}$ is called the Bessel's correction.
 The jackknife samples are obtained by
 \beq
  {\cal O}^{(n)}_J= \frac{1}{N-1} \sum_{n' \neq n}  {\cal O}^{(n')}      \ \ \ \  (n=1, \cdots, N).
 \eeq
 If we need to make a quick estimate of the
  mean and the variance of a function $f({\cal O})$, we have
 \beq
\label{eq:jack}
 \langle f({\cal O}_J)\rangle = \frac{1}{N} \sum_{n=1}^{N} f( {\cal O}^{(n)}_J) \pm  \sqrt{\frac{\sigma_J^2(f)}{N}}, \ \ \ \
  \sigma_J^2(f) = (N-1) \sum_{n=1}^N ( f( {\cal O}^{(n)}_J) -  \langle f( {\cal O}_J)\rangle)^2.
\eeq
For $f( {\cal O})={\cal O}$, we recover the original  mean and variance;
 $\langle  {\cal O}_J \rangle = \langle  {\cal O} \rangle$ and  $\sigma_J^2({\cal O})=\sigma^2({\cal O})$. (Exercise \ref{prob:12}).
One can generalize this procedure by dividing $N$ into $N_b=N/n_b$
with the bin-size $n_b$ and create the $N_b$ jackknife samples.
 Eq.(\ref{eq:jack})  corresponds to the case with $n_b=1$.

%===========================
\subsection{Heavy quark potential}
%===========================

 As one of the examples of the accurate  inter-quark potential obtained from LQCD simulations,
 we  show in Fig.\ref{fig:QQbar-pot}  the
 dimensionless ${\rm Q}\bar{\rm Q}$ potential
$[V(R)-V(R_0)]\times R_0$
 as a function of $R/R_0$ extracted from the calculation of the
   Wilson loop in the quenched approximation.
 $R$ is a distance between the heavy quarks
  and $R_0$ is called the {\it Sommer scale}   defined by
 $  \left. R^2 \frac{dV(R)}{dR} \right|_{R=R_0}=1.65$.

 Simulations with different lattice couplings $6/g^2$
  correspond to different lattice spacings $a$.
 The latter can be fixed, e.g., by
 taking a phenomenological value
  $R_0\simeq 0.5\ {\rm fm}$.
    The lattice spacings in the physical uniit in the figure are
   $a=0.094\ {\rm fm}$ (squares: $\beta=6/g^2=6.0$),
    $a= 0.069\ {\rm fm}$ (circles: $\beta=6.2$)
  and  $a=0.051\ {\rm fm}$ (triangles: $\beta=6.4$).
  Since there exists no appreciable $a$ dependence of the potential,
 the system is already close enough to the continuum limit.

 Fig.\ref{fig:QQbar-pot} clearly shows that the
  heavy quark potential has a linear confining part
   at long distance and an attractive Coulombic part
    at short distance. The LQCD results agree
     not only qualitatively but also quantitatively
  with an empirical linear + Coulomb potential
  (the Cornell potential)    shown by the solid line,
   $V(r) = Kr - b/r + {\rm const}$ with $b=0.295$.

% FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
%\includegraphics[scale=0.6]{as-scale.eps}
\includegraphics[scale=0.5]{Chapter3-figures/QQbar-pot.eps}
 \end{center}
\caption{A dimensionless ${\rm Q}\bar{\rm Q}$ potential as a
 function of a dimensionless quark$-$anti-quark
 separation $R/R_0$ with $R_0$ being the Sommer scale.
 Different  symbols correspond to different lattice couplings
  $g(a)$    and hence different lattice spacings.
  The solid line shows an empirical Cornell potential.
  The figure is adapted from \cite{Bali:2000gf}.
  }
\label{fig:QQbar-pot}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%============================
\subsection{Masses of light hadrons}
%============================

 Meson masses and baryon masses can be calculated with high accuracy
 by LQCD simulations with dynamical quarks.  The starting point is the
 hadronic correlation functions $C_{\rm H=M,B} (n,n')$ in
 Eqs.(\ref{eq:correlation-M},\ref{eq:correlation-B}) integrated over
 the spatial coordinates, $\mathbf{n}$ and
 $\mathbf{n}'$,
\beq \label{eq:5.D-tau} C_{\rm H}(\tau)
 = \sum_{\mathbf{n}, \mathbf{n}'} C_{\rm H}
 (n,n') \xrightarrow[\tau \rightarrow \infty] \ |Z_{\rm H}|^2 {\rm
 e}^{-M_{\rm H} \tau} ,
\eeq
where $\tau= (n_4-n_4')a$ is the temporal
 distance between the source at $n'$ and the sink at $n$, and $M_{\rm
 H}$ ($Z_{\rm H}$) corresponds to the mass (the pole residue) of a
 lightest bound state in each channel.  If the temporal extent of the
 lattice is infinite, one can extract the hadron mass from the
 formula, $M_{\rm H}= -(1/\tau) \ln C_{\rm
 H}(\tau)|_{\tau \rightarrow \infty} $.  In practice, the {\it
 effective mass} defined below is more useful, \beq aM_{\rm H}^{\rm
 eff}(\tau) = \ln \left( \frac{C_{\rm H}(\tau)}{C_{\rm H}(\tau+a)
 } \right) .
\eeq
The asymptotic plateau of the effective mass at large $\tau$
     corresponds to the hadron mass.  In actual simulations, the
     temporal extent is limited ($0 \le \tau/a \le N_{\tau}$), so that
     the exponential damping of Eq.(\ref{eq:5.D-tau}) is replaced by
     $C_{\rm H} \rightarrow \exp[-M_{\rm H} \tau ] \pm \exp [-M_{\rm
     H} (N_{\tau} a - \tau)]$ where $+ (-)$ for the periodic
     (anti-periodic) boundary condition.

 Shown in the left panel of Fig.\ref{fig:hadron-mass} is the
 dimensionless effective masses ($a M_{\rm H}^{\rm eff}$) against
 $\tau/a=(n_4-n_4')$ \cite{Durr:2008zz}.  Data points are the
 effective masses for the pion $(\pi)$, the kaon $(K)$, the nucleon
 ($N$), the cascade baryon ($\Xi$) and the omega baryon $(\Omega$)
 calculated on the lattice with $a \simeq 0.085$ fm and the pion mass
 $M_{\pi} \simeq 190$ MeV.  Reasonable plateau above $\tau/a > 9$ can
 be seen within the error bars.

 Shown in the right panel of Fig.\ref{fig:hadron-mass} is the
 $M_{\pi}^2$-dependence of the $N$ and $\Omega$ masses for three
 different values of $a$ \cite{Durr:2008zz}.  The crosses are the
 values extrapolated to the continuum limit and to the physical pion
 mass.  The $N$ and $\Omega$ masses predicted from LQCD and
 corresponding experimental numbers are
\beq
\label{eq:mass-LQCD}
M_N^{\rm LQCD} &=& 0.936(25)(22) \ {\rm GeV}, \ \ \ M_\Omega^{\rm
LQCD} =1.676(20)(15)\ {\rm GeV}, \\ M_N^{\rm exp.} &=& 0.939 \ {\rm
GeV}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M_\Omega^{\rm exp.} =1.672 \
{\rm GeV}.
\eeq
  Note that the numbers in the first (second) parenthesis in
 Eq.(\ref{eq:mass-LQCD}) represent the statistical (systematic) errors
 on the last digits.

% FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
%\includegraphics[scale=0.6]{as-scale.eps}
\includegraphics[scale=0.29]{Chapter3-figures/effective-mass.eps} \ \
\includegraphics[scale=0.28]{Chapter3-figures/NO_scaling.eps}
 \end{center}
\caption{(Left) The effective masses of hadrons against the
 temporal separation between the source and the sink.
(Right)  Hadron masses under the changes of the (pion mass)$^2$ as well as the lattice spacing $a$.
The figures are taken from \cite{Durr:2008zz}
  }
\label{fig:hadron-mass}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Shown in Fig.\ref{fig:pn-mass} are the high precision numerical
 results of the hadron mass splittings obtained by the QCD+QED lattice
 simulations with dynamical $u,d,s,c$ quarks \cite{Borsanyi:2014jba}.
 The horizontal lines are the experimental values and the grey shaded
 regions represent the experimental errors.  Red dots are the lattice
 results with their uncertainties denoted by the vertical error bars.
 The neutron-proton mass differences from numerical simulations and
 the corresponding experimental numbers are
\beq
(M_n-M_p)^{\rm LQCD+QED} &=&\Delta N = 1.51(16)(23)\   {\rm MeV}, \\
(M_n-M_p)^{\rm exp.} &=& 1.29\   {\rm MeV}.
\eeq

% FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
%\includegraphics[scale=0.6]{as-scale.eps}
\includegraphics[scale=0.2]{Chapter3-figures/pn-mass.eps}
 \end{center}
\caption{Mass splittings in channels that are stable under the strong and electromagnetic interactions.
$\Delta N=M_n-M_p$, $\Delta \Sigma = M_{\Sigma^-}- M_{\Sigma^+}$,
$\Delta \Xi =M_{\Xi^-}- M_{\Xi^+}$,
$\Delta D= M_{D^{\pm}} - M_{D^0}$,
$\Delta \Xi_{cc} = \Delta \Xi_{cc}^{++} - \Delta \Xi_{cc}^{+}$,
$\Delta CG = \Delta N-\Delta \Sigma + \Delta \Xi$.
The figure is taken from
\cite{Borsanyi:2014jba}.
  }
\label{fig:pn-mass}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%

Since all hadrons are composite particles of quarks and gluons, there
are numerous excited states \cite{RPP}.  To extract the properties of
the excited hadrons from LQCD, only looking at the asymptotic form as
shown in Eq.(\ref{eq:5.D-tau}) is not sufficient, and more
sophisticated methods such as the maximal entropy method (MEM) and the
variational method are required.  Interested readers should consult
the reviews \cite{Asakawa:2000tr,Fodor:2012gf} and references there
in.

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Lattice QCD and nuclear force}
%%%%%%%%%%%%%%%%%%%%%%%%

Understanding of the nuclear force from QCD is one of the most
 challenging problems in nuclear physics.  Below the pion production
 threshold, the notion of the $NN$ potential (either in the coordinate
 space or in the momentum space) has been known to be useful, since it
 can be used not only to describe the two-body system but also to
 study nuclear many-body problems through ab-initio
 calculations \cite{this_book}.

  Several high precision phenomenological $NN$ forces have been
  constructed to reproduce the neutron-proton and proton-proton
  scattering data (about 4500 data points) with a $\chi^2/{\rm
  dof} \sim 1$. However, they have typically 20-40 fitting parameters:
  e.g. the CD Bonn potential, AV18 potential and N$^3$LO chiral
  effective field theory have 38, 40, and 24 parameters,
  respectively \cite{Machleidt:2007ms}.  If one tries to extend these
  to hyperon-nucleon and hyperon-hyperon interactions, the task
  becomes extremely tough since the number of parameters increases and
  the scattering data are scarce.

  Under this situation, it is highly desirable to study
  baryon-baryon interactions from first principle LQCD simulations,
  where all the hadronic interactions are controlled only by the QCD
  coupling $g$ and the quark mass $m$ whose values are pretty well
  determined at present by the precision QCD
  simulations \cite{Aoki:2013ldr}.

 The finite volume method (FVM), a theoretical framework to study
  hadron-hadron interactions from LQCD, was first proposed by
  L\"{u}scher \cite{luescher}: For two hadrons in a finite box with a
  spatial size $L^3$, an exact relation between the energy spectra in
  the box and the elastic scattering phase shift can be derived.  If
  the range of the hadronic interaction $R_{\rm QCD}$ is sufficiently
  smaller than the size of the box $R_{\rm QCD}<L/2$, the behavior of
  the the equal-time Bethe-Salpeter amplitude (or more precisely the
  Nambu-Bethe-Salpeter (NBS) amplitude) $\psi (\mathbf{r})$ in the
  interval $R_{\rm QCD} < | \mathbf{r} | < L/2 $ has sufficient
  information to relate the phase shift and the energy shift
$\Delta E=M_{\rm HH}-2M_{\rm H}$.

The HAL QCD method was proposed as another theoretical framework to
 study the hadron-hadron interactions from LQCD by Ishii, Aoki and
 Hatsuda \cite{Ishii:2006ec} and was further developed by HAL QCD
 Collaboration \cite{HALQCD:2012aa}.  The starting point is the same
 equal-time NBS amplitude $\psi (\mathbf{r})$: Instead of looking at
 the amplitude outside the range of the interaction, the internal
 region $ |\mathbf{r} | < R_{\rm QCD}$ is considered and an
 energy-independent non-local potential $U(\mathbf{r}, \mathbf{r}')$
 is deduced from $\psi (\mathbf{r})$.  Since
 $U(\mathbf{r}, \mathbf{r}')$ in QCD is spatially localized due to the
 confinement of quarks and gluons, it is affected only weakly by the
 finite lattice volume. Physical quantities such as the scattering
 phase shifts, bound state spectra, and resonance energies can be
 calculated by solving the integro-differential differential equation
 satisfied by $\psi (\mathbf{r})$ with $U(\mathbf{r}, \mathbf{r}')$.

  Recently, a detailed comparison between the FVM and the HAL QCD
  method has been carried out: Although they agree with each other
  quite accurately for non-resonant pion-pion scattering, large signal
  to noise ratio inherent in the effective mass $\Delta E^{\rm
  eff}(\tau)$ for baryon-baryon scatterings prevents FVM to extract
  scattering observables \cite{Iritani:2015dhu}.  Therefore, we will
  focus on the HAL QCD method below.

%=============================================
\subsection{Master equation for baryon-baryon interaction}
%=============================================

Let us consider the baryon-baryon correlation in
Fig.\ref{fig:correlation}(c) and define the equal-time NBS amplitude
$\psi_{\ell}(\mathbf{r},\tau)$ from its large $\tau$ behavior:
\beq
C_{\rm BB}(\mathbf{r}, \tau)
=\sum_{\mathbf{n}',\mathbf{m}'} \left. C_{\rm
BB}(n,m,n',m') \right|_{n_4=m_4,n_4'=m_4'} \rightarrow \sum_{\ell}
a_{\ell} \psi_{\ell} (\mathbf{r},\tau) e^{-E_{\ell} \tau} ,
\eeq
where
$\mathbf{r}=(\mathbf{n}-\mathbf{m}) a$, $\tau=(n_4-n_4')a$, and
$\psi_{\ell} (\mathbf{r},\tau)$ being the NBS wave function for
$\ell$-th scattering state on the lattice.  For large lattice size,
$E_{\ell}$ is very dense, so that it is impossible to identify each
level.  This causes a fatal problem in FVM as mentioned above.  On the
other hand, if we define $ C_{\rm BB}(\mathbf{r};\tau)= {\cal
R}(\mathbf{r},\tau) e^{-2M_{\rm B} \tau}$, the following
integro-differential equation can be derived below the inelastic
threshold ($\tau > M_{\pi}^{-1}$),
\beq
  \left\{
  \frac1{4M_{\rm B}}\frac{\partial^2}{\partial \tau^2}
  -\frac{\partial}{\partial \tau}
  - H_0
  \right\}
  {\cal R}(\mathbf{r},\tau)
  =
  \int d^3 r'
  U(\mathbf{r}, \mathbf{r}')
  {\cal R}(\mathbf{r}', \tau),
  \label{eq.tdep}
\eeq
 with  $H_0= - \nabla^2/M_{\rm B}$.
 This is the master equation which has the
correct information of the S-matrix and hence the scattering phase shift for
elastic $BB$ scatterings \cite{HALQCD:2012aa}.

  If we further focus on the
  energies  much below  the inelastic threshold,
   the velocity expansion
   of $U(\mathbf{r},\mathbf{r}')$ in terms of its  non-locality can be adopted.
   In fact, the potential with hermiticity,
   rotational invariance, parity symmetry,   and time-reversal invariance may be expanded as \cite{okubo}
\begin{eqnarray}
\label{eq:U-del}
  U(\mathbf{r},\mathbf{r}')    &=& V(\mathbf{r}, \mathbf{v}) \delta(\mathbf{r}-\mathbf{r}'), \\
 V(\mathbf{r}, \mathbf{v})    & =&
   \underbrace{V_{\rm C}(r) + V_{\rm T}(r) S_{12}}_{\rm LO}
  + \underbrace{V_{\rm LS}(r) {\mathbf{L}} \cdot {\mathbf{S}}}_{\rm NLO}
  +\underbrace{{O}(\mathbf{v}^2)}_{{\rm N}^2{\rm LO}}
  + \cdots ,
   \label{eq:V-pot}
\end{eqnarray}
   where $\mathbf{v} = \mathbf{p}/(M_{\rm B}/2) $, $\mathbf{L} = \mathbf{r} \times \mathbf{p} $,
   $\mathbf{p} = -i \nabla$ and
   $S_{12}=3(\mathbf{\sigma}_1 \cdot \mathbf{r})(\mathbf{\sigma}_2 \cdot \mathbf{r})/r^2 - \mathbf{\sigma}_1 \cdot \mathbf{\sigma}_2$.
     The central potential $V_{\rm C}$ and
  the tensor potential $V_{\rm T}$ are classified as
  the leading order (LO) potentials since they
  are of $O(\mathbf{v}^0)$. The next-to-leading (NLO) potential of
  $O(\mathbf{v})$ is the spin-orbit potential  $V_{\rm LS}(r)$.

%=============================================
\subsection{Baryon-baryon interaction in flavor SU(3) limit}
%=============================================

 % FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{Chapter3-figures/Vc_27_npa2.eps}\ \ \ \ \ \ \
\includegraphics[scale=0.55]{Chapter3-figures/Vc_10s_npa2.eps}\\
\includegraphics[scale=0.55]{Chapter3-figures/Vc_1_npa2.eps}\ \ \ \ \ \ \
\includegraphics[scale=0.55]{Chapter3-figures/Vt_10s_npa2.eps}
 \end{center}
\caption{The baryon-baryon potentials from LQCD simulations in the flavour
SU(3) limit with several different masses of pseudo-scalar meson.
The figures are taken from \cite{Inoue:2011ai}.
  }
\label{fig:NN-potential}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 To obtain qualitative understandings of the nuclear force from QCD,
  the $S$-wave interaction between octet baryons
 in the flavour SU(3) limit  would be a good starting point.
 In this case,  two baryon states with a given angular momentum
are labelled by the irreducible flavour multiplets,
\begin{equation}
 {\bf 8} \otimes {\bf 8}
 = \underbrace{{\bf 27} \oplus {\bf 8}_s \oplus {\bf 1}}_{\mbox{symmetric}} ~
  \oplus \underbrace{{\bf 10}^* \oplus {\bf 10} \oplus {\bf 8}_a}_{\mbox{anti-symmetric}} \ .
\end{equation}
Here ``symmetric" and ``anti-symmetric" stand for the symmetry under the
flavour exchange of two baryons.
For the system in the orbital S-wave, the Pauli principle between two baryons imposes
${\bf 27}$, ${\bf 8}_s$ and ${\bf 1}$ to be spin singlet  ($^1S_0$) while
${\bf 10}^*$, ${\bf 10}$ and ${\bf 8}_a$ to be spin triplet ($^3S_1$).
Since there are no mixings among different multiplets in the SU(3) limit,
one may define the corresponding potentials as
 \beq
^1S_0 \ &:& \  V^{({\bf 27})}(r), \ V^{({\bf 8}_s)}(r), \ V^{({\bf 1})}(r),
\\
^3S_1 \ &:& \ V^{({\bf 10}^*)}(r), \ V^{({\bf 10})}(r), \ V^{({\bf 8}_a)}(r) ~.
\eeq
The diagonal potential ($B_1B_2 \rightarrow B_1 B_2)$ and
 the off-diagonal potentials ($B_1B_2 \rightarrow B_3 B_4$) in the particle basis,
 are obtained by  suitable combinations
of $V^{(\alpha)}(r)$ with $\alpha={\bf 27},{\bf 8}_s,{\bf 1},{\bf10}^*,{\bf 10},{\bf 8}_a$.

% FIG %%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.25]{Chapter3-figures/phase-shift.eps}\ \ \ \
\includegraphics[scale=0.27]{Chapter3-figures/neutron-matter.eps}
 \end{center}
\caption{(Left) Phase shifts of the NN scattering as a
function of energy in the laboratory frame, extracted from
LQCD data at the pion mass 469 MeV in the flavor-SU(3) limit.
 The black and gray dashed lines are the results of the partial wave
analysis (PWA) of the experimental data. (Right) Ground state energy per neutron for the
pure neutron matter as a function of the Fermi momentum.
 The APR with black dotted line (black solid line)   corresponds to the empirical equation of state without (with) the
phenomenological three nucleon force \cite{Akmal:1998cf}.
 \cite{Akmal:1998cf}.
The figures are taken from \cite{Inoue:2013nfe}.
  }
\label{fig:NN-phase_shift}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Shown in Fig.\ref{fig:NN-potential}  are the results of the exploratory study of the potentials
obtained by LQCD simulations on the lattice  with $a\simeq 0.12$ fm,
$L\simeq 4$ fm and 3 degenerate flavours  \cite{Inoue:2011ai}.  Corresponding pion mass ranges from
 469 MeV to 1171 MeV.
\begin{itemize}
\item  The upper left panel is the
central potential $V^{({\bf 27})}(r)$ to which the  $^1S_0$ nucleon-nucleon potential belongs.
 It  has a repulsive core at short distance and an attractive pocket at intermediate distance.
As the pion mass decreases,  the repulsive core gets stronger and the attractive tail gets
 longer.
\item As shown in the lower left panel, the structure of the potential is quite different for $V^{({\bf 1})}(r)$ to which the
  flavour singlet $H$ dibaryon (composed of $uuddss$) belongs.   There is no repulsive core and the attraction
  increases as the pion mass decreases.  Such a feature is consistent with the
   notion of  the quark Pauli principle previous discussed in phenomenological quark models \cite{Oka-Fujiwara}.
\item The upper and lower right panels of  Fig.\ref{fig:NN-potential}  are the
central potential and the tensor potential of $V^{({\bf 10}^*)}(r)$, respectively. The
$^3S_1$ nucleon-nucleon potential belongs to this channel. The central part has a similar
structure as the $^1S_0$ channel, while the tensor part has strong attraction and grows rapidly  as the
 pin mass decreases.  The latter aspect is qualitatively consistent with the one-pion-exchange picture at
 long distances.
 \end{itemize}

 Shown in  Fig.\ref{fig:NN-phase_shift} (left) is  the nucleon-nucleon scattering phase shifts
  in the $^1S_0$ channel and $^3S_1$ channel obtained by using the
   potentials, $V^{({\bf 27})}(r)$ and $V^{({\bf 10}^*)}(r)$,  for the pion mass 469 MeV \cite{Inoue:2013nfe}.
   The qualitative feature of the
   phase shift in the $^1S_0$ channel is similar to the experimental one denoted by the
   black solid  line, despite the fact that pion mass in the simulation is  more than 3 times
   heavier than the physical value. In the  $^3S_1$ channel, the deuteron bound state is
   not formed yet due to heavy pion mass, so that the phase shift starts from 0 at zero energy
    in contrast the the experimental one denoted by the black dashed line.
    There is however a tendency that the attraction in the $^3S_1$ channel is larger than
   the $^1S_0$ channel even for the heavy pion mass.
  Shown in  Fig.\ref{fig:NN-phase_shift} (right) is  the energy per particle $E/A$ as a function of the
  fermi momentum $k_{\rm F}$ for pure neutron matter calculated by using the Brueckner-Hartree-Fock method
  with the neutron-neutron potential in the $^1S_0$ channel in Fig.\ref{fig:NN-phase_shift} (left).
 As the pion mass decreases, the equation of state becomes stiffer due to the growth of the repulsive  core.
 The APR with black dotted line (black solid line)   corresponds to the empirical equation of state without (with) the
phenomenological three nucleon force  \cite{Akmal:1998cf}.

 We note that calculations of the baryon-baryon interactions  with
  (2+1)-flavour LQCD on a large volume ($L \simeq 8.2 {\rm fm}$, $a\simeq 0.085$ fm) at
   nearly the physical quark mass ($m_{\pi}\simeq146$ MeV, $m_{K}\simeq$ 525 MeV)
   are underway  \cite{Doi:2015oha}.

%%%%%%%%%%%%%
\section{Exercises}
%%%%%%%%%%%%%

\begin{prob} \label{prob:1}
Prove the properties of the Wilson line, Eqs.(\ref{eq:5.wilson-line-i}), (\ref{eq:5.wilson-line-ii}), and (\ref{eq:5.wilson-line-iii}).
\end{prob}

\begin{prob}\label{prob:2}
Derive the expression on $U_{\mu \nu}(n)-1$ in
Eq.(\ref{eq:5.wilson-plaq-cont-2}) by using the Baker-Campbell-Hausdorff formula.
\end{prob}

\begin{prob}\label{prob:3}
Show the $\Gamma_5$ Hermiticity of the Dirac operator in Eq.(\ref{eq:gamma5-hermite}).
\end{prob}

\begin{prob}\label{prob:4}
Derive  the free fermion propagator on the lattice in the momentum representation,
Eqs.(\ref{eq:DF}) and (\ref{eq:Mp}).
\end{prob}

\begin{prob}\label{prob:5}
Analyze the dispersion relation of the free fermion associated with the
Dirac operator, $D_{\rm GW}$ in Eq.(\ref{eq:5.overlap-op}).
\end{prob}

\begin{prob}\label{prob:6}
Derive the group integration formulas, Eq.(\ref{eq:group-int-1})- Eq.(\ref{eq:group-int-B}), by taking appropriate
contractions of the color indices.
\end{prob}

\begin{prob}\label{prob:7}
Derive the formula for the Wilson loop in the strong coupling limit Eq.(\ref{eq:5.wilson-strong}) by using the group integration formulas
Eqs.(\ref{eq:group-int-0})-(\ref{eq:group-int-2}).
\end{prob}

\begin{prob}\label{prob:8}
Derive the lattice coupling $g(a)$ as a function of the lattice spacing $a$, Eq.(\ref{eq:5.lat-running-g}).
\end{prob}

\begin{prob}\label{prob:9}
Show the convergence of $W[\phi]$ to the equilibrium distribution $W_{\rm eq}[\phi]$
under the Markov process by introducing the distance $D= \sum_{\phi} \left| W[\phi] -W_{\rm eq}[\phi] \right| $
 and by studying its behavior under update.
\end{prob}

\begin{prob}\label{prob:10}
Prove that the Metropolis test in Eq.(\ref{eq:MET-test})
satisfies the detailed balance Eq.(\ref{eq:DT-balance}).
\end{prob}

\begin{prob}\label{prob:11}
Derive the equation of motion for $P_l$ in Eq.(\ref{eq:EOM-P})
from Eq.(\ref{eq:EOM-U}) and  Eq.(\ref{eq:EOM-H}).
\end{prob}

\begin{prob}\label{prob:12}
Prove that the jackknife average and variance for $f({\cal O})={\cal O}$ reduce to the
standard mean and unbiased variance, respectively.
\end{prob}

\begin{prob}\label{prob:13}
Prove that  the leapfrog integrator satisfies the reversibility Eq.(\ref{eq:reversible}) exactly.
Also prove that  the leapfrog integrator preserves the phase space area exactly by
evaluate the Jacobian, $d\phi' d\pi'= J d\phi d\pi $.
\end{prob}

%%%%%%%%%%%%%%%%
\begin{acknowledgement}
%%%%%%%%%%%%%%%%
%If you want to include acknowledgments of assistance and the like at the end of an individual chapter
%please use the \verb|acknowledgement| environment -- it will automatically render Springer's preferred layout.
The author thanks Takumi Doi and Atsushi Nakamura for useful comments and information on various aspects of
LQCD simulations.
He also thank the members of HAL QCD Collaboration for  fruitful discussions on the hadron-hadron interactions
on the lattice.
This work was supported in part by MEXT SPIRE and JICFuS and also by RIKEN iTHES Project.
\end{acknowledgement}
%

\section{Appendix}
\addcontentsline{toc}{section}{Appendix}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection*{\center{Four vectors and Dirac matrices}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the (3+1)-dimensional  Minkowski spacetime, coordinates, derivatives
and four vectors with $\mu=0,1,2,3$ are
\beq
x^{\mu} = (t, \mathbf{x}),
\ \ \partial^{\mu}   =  (\partial_t, -\nabla),
\ \  A^{\mu} =(A^0, \mathbf{A}).
\label{eq:B.vector-M}
\eeq
In the 4-dimensional Euclidean space,
 we define the corresponding vectors  for $\mu=4,1,2,3$ as
 \beq
(x_{\mu})^{\rm E} = (\tau=it, \mathbf{x}),
\ \ (\partial_{\mu})^{\rm E}   =  (\partial_\tau = -i \partial_{t}, \nabla),
\ \  (A_{\mu})^{\rm E}  =(A_4=iA^0,\mathbf{A}).
\label{eq:B.vector-E}
\eeq

In the (3+1)-dimensional  Minkowski spacetime with the
 metric $g^{\mu \nu}={\rm diag}(1,-1,-1,-1)$,
  the Dirac matrices satisfy the following relations for $\mu=0,1,2,3$,
\beq
\{ \gamma^{\mu}, \gamma^{\nu} \} = 2 g^{\mu \nu},
\ \ \left( \gamma^{\mu} \right)^{\dagger} = \gamma^0 \gamma^{\mu} \gamma^0,
\ \ \gamma^5 &=& i \gamma^0 \gamma^1 \gamma^2 \gamma^3
 = \gamma_5 = \left( \gamma_5 \right)^{\dagger}.
\label{eq:B.gamma-min}
\eeq
In the standard Dirac representation, we have
\beq
\gamma^0 =
\left(  \begin{array}{cc}
        1   & 0   \\
        0   & -1  \\
        \end{array}  \right) , \ \
\gamma^j =
\left(  \begin{array}{cc}
        0           & \sigma_j   \\
        -\sigma_j   & 0          \\
        \end{array}  \right) , \ \
\gamma^5 =
\left(  \begin{array}{cc}
        0   & 1   \\
        1   & 0   \\
        \end{array}  \right)  ,
\eeq
where $\sigma_j$ are the Pauli matrices;
$
%\beq
\sigma_1 =
\left(  \begin{array}{cc}
        0   & 1   \\
        1   & 0   \\
        \end{array}  \right) ,
\sigma_2 =
\left(  \begin{array}{cc}
        0           & -i   \\
        i           & 0    \\
        \end{array}  \right) ,
\sigma_3 =
\left(  \begin{array}{cc}
        1   & 0    \\
        0   & -1   \\
        \end{array}  \right)  .
%\label{eq:B.Pauli-matrix}
%\eeq
$

In the 4-dimensional Euclidean space  with the metric
 $\delta_{\mu \nu}={\rm diag}(1,1,1,1)$, we define
  the Euclidean Dirac matrices as
 \beq
  \Gamma_{\mu}  \equiv
 \left(\gamma_4= \gamma^0, -i \mathbf{\gamma} \right),
 \ \ \Gamma_{-\mu} \equiv  - \Gamma_{\mu},
  \ \ {\rm and}   \ \ \Gamma_5 \equiv \gamma^5,
  \eeq
 which satisfy the relations,
 \beq
 \{ \Gamma_{\mu}, \Gamma_{\nu} \}  = 2 \delta_{\mu \nu},
\ \ \Gamma_{\mu}^{\dagger} = \Gamma_{\mu} ,
\ \ \ \ ({\rm for} \ \mu=1,2,3,4,5)
\label{eq:B.gamma-lat}
\eeq

%%%%%%%%%%%%%%%%%%%%%
\subsubsection*{\center{SU$(N)$ algebra}}
%%%%%%%%%%%%%%%%%%%%%

Let ${\cal T}^a$ ($a=1, \cdots , N^2-1$) are the
 Hermitian generators of the SU$(N)$ group. They satisfy
  the Lie algebra  \index{Lie algebra}
\beq
\left[  {\cal T}^a , {\cal T}^b \right] = i f_{abc} {\cal T}^c ,
\eeq
where $f_{abc}$ is the structure constant \index{structure constants}
 being totally anti-symmetric in its indices.
 $({\cal T}^b)^2$ commutes with every generator ${\cal T}^a$
  and   is called the quadratic Casimir operator.

 For $N=2$, $f_{abc}$ reduces to the  anti-symmetric
 tensor $\epsilon_{ijk}$ with  $\epsilon_{123}=1$.
For $N=3$, the non-vanishing components of $f_{abc}$ read
$f_{123}=1, f_{147}=-f_{156}=f_{246}=f_{257}=f_{345}=-f_{367}=1/2,
f_{458}=f_{678}= {\sqrt{3}}/{2}$.

  In the fundamental representation,
 ${\cal T}^a$ is written by the $N \times N$ matrices $t^a$ as
\beq
t^a = \frac{1}{2} \lambda_a,
\eeq
 where $\lambda_a$ for $N=2$ reduce to  the Pauli matrices
 $\sigma_i$, while  those for $N=3$ reduce to the Gell-Mann matrices.

 Some useful relations of $t^a$ for general $N$ are
\beq
\label{eq:tt}
{\rm tr} (t^a t^b) = \frac{1}{2} \delta_{ab},
\ \ \ t_{ij}^a t_{kl}^b  =  \frac{1}{2} (\delta_{il}\delta_{jk} -\frac{1}{N} \delta_{ij}\delta_{kl} ),
\ \ \ (t^a t^a)_{ij} = C_{{\rm F}} \delta_{ij},
\ \ \ {\rm with} \ C_{{\rm F}}=\frac{N^2-1}{2N}.
\eeq

In the adjoint representation, \index{adjoint representation}
 ${\cal T}^a$ is written by  $(N^2-1) \times (N^2-1)$ matrices $T^a$ as
\beq
(T^a)_{bc} = -i f_{abc} ,
\eeq
which satisfy the relations
\beq
  {\rm tr} (T^a T^b) &=& N \delta_{ab},
  \ \ \  (T^a T^a)_{bc} = C_{{\rm A}} \delta_{bc},
  \ \ \  {\rm with} \ C_{{\rm A}}= N.
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection*{\center{Gaussian and Grassmann integrals}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Basic Gaussian and Grassmann integrals are
\beq
 \label{eq:C.gauss-x}
& &\int_{-\infty}^{+\infty} \frac{dx}{\sqrt{2\pi}}\ {\rm e}^{-ax^2/2}
    =\frac{1}{\sqrt{a}}, \\
 \label{eq:C.gauss-z}
& & \int \frac{dz^* dz}{2 \pi i}\ {\rm e}^{-b |z|^2} = \frac{1}{b}, \\
 \label{eq:C.gauss-xi}
& & \int d\bar{\xi} d\xi\  {\rm e}^{- c \bar{\xi} \xi} = c .
\eeq
Here $x$ ($z$) is a real (complex) number, while $\bar{\xi}$ and
 $\xi$ are anti-commuting Grassmann numbers ($\{ \xi, \bar{\xi} \} =0$,
 and  $\xi^2 = \bar{\xi}^2=0$).
 $a$ and $b$ are assumed to be real and positive numbers, while $c$ is an
  arbitrary complex number.
 Eq.(\ref{eq:C.gauss-z}) can be shown by rewriting the integral
 in terms of the real and imaginary parts of $z$ or in terms of the
 polar coordinates of $z$.   Eq.(\ref{eq:C.gauss-xi})
  can be shown by noting that
   ${\rm e}^{-c\bar{\xi}\xi} = 1- c \bar{\xi}{\xi}$
   and $ \int d\xi = \partial/\partial \xi $
   (integral = derivative) for Grassmann variables.

Generalization of the above results to the case of multiple variables
 is straightforward.  For $x =(x_1, \cdots, x_n)$,
  $z =(z_1, \cdots, z_n)$, $\xi =(\xi_1, \cdots, \xi_n)$,
 and $\bar{\xi} =(\bar{\xi}_1, \cdots, \bar{\xi}_n)$ with
  $\{ \xi_k, \xi_l \} = \{ \bar{\xi}_k, \bar{\xi}_l \}
   = \{ \xi_k, \bar{\xi}_l \} =0$,  we have
 \beq
 \label{eq:C.gauss-xn}
& &\int \prod_{l=1}^n \frac{dx_l}{ \sqrt{2\pi} }\
 {\rm e}^{-\frac{1}{2} x A x} = \frac{1}{\sqrt{{\rm Det}\ A}}, \\
 \label{eq:C.gauss-zn}
& & \int \prod_{l=1}^n \frac{dz_l^* dz_l}{2 \pi i}\
{\rm e}^{- z^* B z} = \frac{1}{{\rm Det}\ B}, \\
 \label{eq:C.Z-grass-xin}
& & \int \prod_{l=1}^n d\bar{\xi}_l d\xi_l \
  {\rm e}^{- \bar{\xi}C \xi} = {\rm Det}\ C .
\eeq
Here $A$ is a non-singular and real-symmetric matrix whose
 eigenvalues $a_l$ satisfy $a_l > 0$ for all $l$.
 $B$ is a non-singular complex matrix whose complex eigenvalues $b_l$
  obtained by the
  biunitary transformation ($U B V^{\dagger}$)
  satisfy ${\rm Re}\ b_l > 0$ for all $l$.
 $C$ is an arbitrary complex matrix with no conditions.
 Note that $B$ and $C$ do not have to be Hermitian matrices.
 In field theories, the label ``$l$" summarizes all possible indices including
  spin, flavor, color, spacetime points etc. and
  ``${\rm Det}$" denotes the determinant for all these indices.

%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection*{\center{Method of characteristics}}
%%%%%%%%%%%%%%%%%%%%%%%%%%

We need to construct  a general  solution of the
following partial differential equation,
\beq
 \left( \lambda \frac{\partial}{\partial \lambda} + \beta(g) \frac{\partial}{\partial g} \right) f(\lambda,g)=0.
 \label{eq:MCA-1}
 \eeq
 For this purpose, we introduce the running coupling $\bar{g}(\lambda)$ through
 $\lambda d\bar{g}/d\lambda = -\beta(\bar{g})$ whose formal solution reads
 \beq
 \lambda = \exp \left( - \int_g^{\bar{g}(\lambda) }  \frac{dg'}{\beta(g')}  \right) .
 \label{eq:MCA-2}
  \eeq
 Then the solution of Eq.(\ref{eq:MCA-1}) can be written as
 \beq
 f(\lambda, g) = f (1, \bar{g} (\lambda)).
 \eeq
 This can be explicitly checked  by applying the partial derivative on both sides,
 \beq
  \lambda \partial_{\lambda}  f(\lambda,g)&=&
  \lambda (\partial_{\lambda} \bar{g})   (\partial_{\bar{g}} f) = - \beta (\bar{g})  (\partial_{\bar{g}} f) , \nonumber \\
  \beta \partial_g f(\lambda,g) &=&
  \beta(g) \left. (\partial \bar{g}/\partial g) \right|_{\lambda}  (\partial_{\bar{g}} f) = \beta (\bar{g})  (\partial_{\bar{g}} f) .
 \eeq
 where we have used the  relation $\partial \bar{g}/\partial g = \beta(\bar{g})/\beta(g)$
  obtained from   Eq.(\ref{eq:MCA-2}).

 In general, the first-order partial differential equation (PDE)  can be transformed to
 a set of ordinary   differential equations (ODEs) and can  be solved by  {\it the method of characteristics}.
 As an illustration,  let us consider the following PDE,
 \beq
  a(t,x) \partial_t u(t,x) + b(t,x) \partial_x f(t,x) = c(t,x).
\label{eq:MCA-3}
 \eeq
This is equivalent to the coupled ODEs,
\beq
\frac{d \bar{t}}{ds}  = a (\bar{t}, \bar{x}) , \ \
\frac{d \bar{x}}{ds}  = b (\bar{t}, \bar{x}) , \ \
\frac{d f(\bar{t},\bar{x} ) } {ds}  = a (\bar{t}, \bar{x}) ,
\label{eq:MCA-4}
\eeq
where $s$ parametrizes the  "flow" of the coordinates. $(\bar{t}(s), \bar{x}(s))$.
This is called the {\it characteristic curve} as shown in Fig.\ref{fig:char-c}.

\begin{figure}[t]
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
%\includegraphics[scale=0.6]{as-scale.eps}
\includegraphics[scale=0.35]{Chapter3-figures/char-c.eps}
 \end{center}
\caption{Schematic illustration of the characteristic curve.}
\label{fig:char-c}
\end{figure}

The function $f$ can be obtained by integrating the last equation of Eq.(\ref{eq:MCA-4})
on the  characteristic curve from the initial point $(t_{\rm i},x_{\rm i})$ to the final point $(t_{\rm f},x_{\rm f})\equiv (t,x)$,
\beq
f(t,x)= f (t_{\rm i}, x_{\rm i}) + h (t, x, t_{\rm i}, x_{\rm i})
\label{eq:MCA-6}
\eeq
where  $h$ stands for an integration of the known function $c(\bar{t},\bar{x})$ on the
characteristic curve.  Eq.(\ref{eq:MCA-6}) implies  that the desired function at $(t,x)$ is obtained
essential by a "pullback" of the point to $(t_{\rm i},x_{\rm i})$ along the characteristic curve.
It is a straightforward exercise  to generalize the above derivation to the system with more coordinates,
$(t, \mathbf{x})$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsubsection*{\center{Leapfrog integrator in molecular dynamics}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let us start with a Tayler expansion of the field $\phi$:
\beq
\label{eq:LF-phi}
\phi(s+\varepsilon) &=& \phi(s) + \varepsilon \dot{\phi}(s) + \frac{\varepsilon^2}{2} \ddot{\phi}(s) + O(\varepsilon^3), \nonumber \\
&=&  \phi(s) + \varepsilon {\pi}(s) + \frac{\varepsilon^2}{2} \dot{\pi}(s) + O(\varepsilon^3), \nonumber  \\
&=& \phi(s) + \varepsilon {\pi}(s+\varepsilon/2) + O(\varepsilon^3),
\eeq
where we have used the equation of motion, $\dot{\phi}(s) \equiv d\phi(s)/ds = \pi(s)$.
To evaluate ${\pi}(s+\varepsilon/2)$, we take the midpoint prescription which does not have $O(\varepsilon^2)$ error,
 \beq
 \label{eq:LF-pi}
{\pi}(s+\varepsilon/2)&=&{\pi}(s-\varepsilon/2) + \varepsilon \dot{\pi}(s) + O(\varepsilon^3) \nonumber \\
&=&  {\pi}(s-\varepsilon/2)   - \varepsilon \frac{\delta S(\phi)}{\delta \phi(s)} + O(\varepsilon^3) .
\eeq
Eqs.(\ref{eq:LF-phi}) and (\ref{eq:LF-pi})  give a procedure to move the molecular dynamics one-step forward,
$(\phi(s), \pi(s-\varepsilon/2))  \rightarrow (\phi(s+\varepsilon), \pi(s+\varepsilon/2))$.
The initial and final steps need to receive special care,
\beq
\pi(\varepsilon/2) = \pi(0) - \frac{1}{2} \varepsilon \frac{\delta S(\phi)}{\delta \phi(s)} + O(\varepsilon^2), \ \ \
\pi(s_{\rm f}) =\pi(s_{\rm f}-\varepsilon/2) -  \frac{1}{2} \varepsilon \frac{\delta S(\phi)}{\delta \phi(s_{\rm f})} +
O(\varepsilon^2),
 \eeq
 which have only $O(\epsilon^2)$ accuracy.  An illustration of this leapfrog integrator is shown in Fig.(\ref{fig:LF}).
 Since the initial and final steps introduce  $O(\varepsilon^2)$ error irrespective of the
length of the MD trajectory, and  the intermediate steps introduce
$O(\varepsilon^3) \times \varepsilon^{-1} =O(\varepsilon^2)$ error as a whole,  one finds
 $\Delta H = O(\varepsilon^2)$ after one MD trajectory before the Metropolis test.

The leapfrog integrator satisfies the reversibility and symplectic property, which can be checked explicitly
by using the above definitions (Exercise \ref{prob:13}).

 \begin{figure}[t]
\begin{center}
%\framebox[74mm]{\rule[-26mm]{0mm}{52mm}}
%\includegraphics[scale=0.6]{as-scale.eps}
\includegraphics[scale=0.3]{Chapter3-figures/leapfrog.eps}
 \end{center}
\caption{The leapfrog integrator.}
\label{fig:LF}
\end{figure}

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\section{Outlook}

We consider achieving a test accuracy of $63.80\%$ given the size and characteristics of the dataset after 10 epochs of training already a great success, but we could not evaluate the models for their full potential within the scope of this project. With more computational resources available, the next obvious step would be to increase the size of the dataset to include all augmentations while also including more turtles again by lowering the number of minimum images per turtle. Then, it would also be interesting to conduct a more thorough hyperparameter search, which was not feasible for us. For example, we only used Adam with standard parameters as optimizer because it is computationally efficient and requires little memory \citep{Kingma2014}. The EfficientNet model family has been trained with Hinton's unpublished RMSProp \citep{Tan2019}. Further, we only reported categorical accuracy as model evaluation metric. In the original Zindi competition, mean average precision at top 5 (MAP@5) has been suggested for this problem, probably because of the many classes to consider. Another interesting approach could be a less aggressive downsampling of the images. As proposed in the original competition, we scaled the images down to a resolution of $224\times224$, exactly the resolution that is also suggested as input for the EfficientNetV2-B0. As already discussed earlier, EfficientNet is a model family and B0 the smallest one in the ensemble. By scaling up the image resolution, it would not only be possible to include more information in the images themselves, but also to employ a larger and stronger model.

Further, we manually designed the augmentation. Doing so effectively is usually dataset-specific and requires expert knowledge in the image domain, as different techniques work best for different domains. \citeauthor{Cubuk2018} have proposed their AutoAugment image augmentation procedure in \citeyear{Cubuk2018}, in which a search space of augmentations is explored by a controller RNN to find the best procedures given the data at hand.

The competition on Zindi is not yet over at the time of writing this documentation, but the top-5 leaderboard scores already achieve a MAP@5 of more than $98\%$. We are excited to see which procedure ultimately works best on this problem.