ch_LFT_improved.tex

\chapter{LFT: Improvement schemes}\label{ch:improve}


In LFT the correlation length $\xi$ is related to the mass $m$ of the lightest 
\index{correlation length}\index{improvement}
particle by
\begin{equation}
  \xi=\frac{1}{ma}.
\end{equation}
This can be seen from looking at the spectral decomposition of the
connected correlation function
\begin{equation}\label{eq:spectralDecomp}
  \ev{X(x)X(y)}_c=\sum_{n> 0} 
                   \left|\bra{0}X\ket{n}\right|^2
                    e^{-(E_n-E_0)\Delta t},
\end{equation}
which has been evaluated at $\vec{x}=\vec{y}=0$ for notational simplicity and
we define $\Delta t$ as the temporal distance between $x$ and $y$.
For large $\Delta t$ we find schematically
\begin{equation}
  \ev{X(x)X(y)}_c\sim e^{(E_1-E_0)\Delta t} =e^{-m\Delta t}=e^{-\Delta t/\xi}.
\end{equation}

Equation~\eqref{eq:spectralDecomp} measures how correlated the field 
$X$ at $x$ is with the field at $y$. In the continuum limit, $\xi$ will 
diverge since $m$ is constant, i.e. the system experiences a second-order 
phase transition. Correspondingly the correlation function increases to
maximum correlation. The divergence of $\xi$ implies a divergence in the
integrated autocorrelation time
\index{critical slowing down}\index{critical exponent!dynamical}
\begin{equation}\label{eq:criticalSlowingDown}
  \tau_{\text{int}, X} \sim \xi_X^z
\end{equation}
with {\it dynamical critical exponent} $z$ that depends on the algorithm 
details. For $z\neq 0$, \equatref{eq:criticalSlowingDown} implies a
deterioration in the efficacy of the updating process toward the
continuum limit; this is {\it critical slowing down}.

Due to critical slowing down, configuration generation becomes increasingly
expensive as one lowers the lattice spacing.
On the other hand, one wishes to minimize discretization error. One way to
proceed is as follows: There are multiple discretizations of a lattice action
that will lead to the same continuum limit action, so one can design a lattice
action where one suppresses the discretization effects by hand. Such actions
are examples of {\it improved actions}. They are in general slower to generate
configurations at a given $a$ than unimproved actions, but have the advantage
of being in some sense ``closer to the continuum limit" than the unimproved
actions at a given $N_\tau$.

One knows that the Wilson action has
$\order{a^2}$ discretization errors. If we add to the Wilson action some
terms to cancels the $\order{a^2}$ part out, we will have eliminated all 
discretization error
up to $\order{a^3}$. This is called an $\order{a^3}$ {\it improved action}
or an $\order{a^4}$ {\it action}. Ultimately, our goal is to calculate
expectation values, so we will generally also want to improve our 
lattice interpolators. 
%For instance suppose we find some $\order{a^p}$
%action $S_\text{lat}$ with $p\geq1$. Generically it is related to the 
%continuum action $S_{\text{cont}}$ by
%\begin{equation}
%  S_\text{lat}=S_\text{cont}+a^pS_p+\order{a^{p+1}},
%\end{equation}
%where $S_p$ is a correction term with dimension [fm$^{-p}$]. (Since masses are
%inverse lengths, this is often called {\it mass dimension}.) 
%\index{mass dimension} Similarly we will
%have for our interpolator $X_\text{lat}$ 
%\begin{equation}
%   X_\text{lat} = X_\text{cont}+a^qX_q +\order{a^{q+1}}
%\end{equation}
%with $q\geq1$. Expanding the exponential, one finds for the partition function
%\begin{equation}
%  Z_\text{lat}=Z_\text{cont}\left(1-a^p\ev{S_p}+\order{a^{p+1}}\right),
%\end{equation}
%from which it follows that
%\begin{equation}
%\ev{X_\text{lat}}=\ev{X_\text{cont}}+\order{a^p,a^q},
%\end{equation}
%i.e. the discretization error in the expectation value is controlled by both
%the action and the interpolator, whichever is worse.

In what follows, we will look at some improvement strategies, guided by 
Chapter 10 of \cite{degrand_lattice_2006} and Chapter 9 of
\cite{gattringer_quantum_2010}, where there is also more discussion to 
be found. Improvement schemes are rather technically
involved, so I will not be able to verify all details carefully.

\section{Symanzik improvement}\label{sec:symanzikImprovement}

The Symanzik improvement scheme \cite{symanzik_continuum_1983,
symanzik_continuum_1983-1,curci_symanziks_1983,luscher_shell_1985} 
is a systematic way to\index{improvement!Symanzik}
improve lattices actions. Gattringer and Lang summarize the strategy as
carrying out the following steps:
\begin{enumerate}
  \item Start with a discretized version of your quantity $Q$.
  \item Identify correction terms using the continuum theory, keeping in mind
        what it allowed given the symmetries of $Q$, and organize them according
        to their mass dimension.
  \item Add discretized versions of the correction terms, multiplied with
        suitable coefficients, to the discretized version of $Q$ so that 
        lattice artifacts vanish up to the desired order.
\end{enumerate}
There is often more than one way to pick correction terms.

We begin by improving the gauge action. In the continuum theory, the gauge
action is proportional to the dimension-four operator
\begin{equation}
  O_4 = \tr F_{\mu\nu}F_{\mu\nu}.
\end{equation}
Our first correction term is at $\order{a^2}$, and there are no dimension-five
operators anyway, so we write the three dimension-six operators\footnote{You 
can convince yourself this is all there is by keeping track of the
dimensions of the comprising operators, for example $D_\mu$ has mass dimension
one and $F_{\mu\nu}$ therefore has mass dimension two, and by demanding gauge
and Lorentz invariance. In the end you need two $D_\mu$'s and two 
$F_{\mu\nu}$'s, and there are three different ways to contract their 
Lorentz indices.}
\begin{equation}\begin{aligned}
  O_{6a}&=\sum_{\mu\nu}\tr D_\mu F_{\mu\nu} D_\mu F_{\mu\nu},\\
  O_{6b}&=\sum_{\mu\nu\rho}\tr D_\mu F_{\nu\rho} D_\mu F_{\nu\rho},\\
  O_{6c}&=\sum_{\mu\nu\rho}\tr D_\mu F_{\mu\rho} D_\nu F_{\nu\rho}.
\end{aligned}\end{equation}
If we expand the plaquette in the lattice spacing, all of the above
operators will appear:
\begin{equation}\label{eq:trPlaqExpansion}
  \tr \plaq = N_c + \frac{1}{2}a^4O_4+a^6\sum_i r_iO_{6i} + ...,
\end{equation}
where $i\in\{\text{a, b, c}\}$.

\begin{figure}
  \centering
  \includegraphics[width=0.5\linewidth]{figs/luescherWeisz.pdf}
  \caption{Lattice dimension-six operators. The top is the rectangle, the
           bottom-left is the parallelogram, and the bottom-right is the 
           chair. Light grey lines are to guide the eye.}
  \label{fig:dim6operators}
\end{figure}
To improve our gauge action thus requires at least three dimension-six
lattice operators, which should cancel the three dimension-six operators
appearing in \equatref{eq:trPlaqExpansions}. 
The simplest are the rectangle ``rt", the parallelogram
``pg", and the chair ``ch", which are depicted in 
\figref{fig:dim6operators}. Their multiplicities are listed in
\tabref{tab:dim6operators}. An improvement on the Wilson action could
then be written as
\begin{equation}
  S_\text{LW}=\frac{\beta}{3}\sum_{x,i} c_i \Re\tr\left(\id-U_i(x)\right),
\end{equation}
where $i\in\{\text{pl, rt, pg, ch}\}$, with ``pl" standing for plaquette,
and $U_i$ is the corresponding lattice operator, summed over
all orientations. In their original paper,
L\"uscher and Weisz chose the normalization
\begin{equation}\label{eq:LWnorm}
  c_\text{pl}+8c_\text{rt}+8c_\text{pg}+16c_\text{ch}=1.
\end{equation}

\begin{table}
\begin{tabularx}{\linewidth}{LR} \hline\hline
         Operator type & Number elements per site\\\hline
         Rectangle & $3\times2\times2=12$ \\
         Parallelogram & $4\times2\times2=16$ \\
         Chair & $6\times4\times2=48$\\
        \hline\hline 
\end{tabularx}
\caption{Dimension-six operator multiplicities in \dimens{4}. Loops that differ only by
         how they close, i.e. CW vs CCW, are considered equal. We compute
         each multiplicity with $N_L\times 2\times 2$, where $N_L$ counts
         the number of orientations starting at that site winding CCW.
         For example if the right leg of the rectangle is in the $x$-direction,
         possible orientations span the $x$-$y$, $x$-$z$, and $x$-$t$ planes.
         The first factor 2 in the above formula comes because, using the same
         starting site, one can also construct a CW path winding to the right.
         The final factor 2 comes from backward orientations.}
\label{tab:dim6operators}
\end{table}

To fix the coefficients, one chooses an improvement condition, for example
that the zero-temperature static potential gives
\begin{equation}
  V(r)\approx\frac{1}{r}
\end{equation}
up to $\order{a^4,g^2a^2}$. This yields 
\begin{equation}
  c_\text{pg}=0~~~~\text{and}~~~~c_\text{rt}+c_\text{ch}=-\frac{1}{12}.
\end{equation}
Furthermore since there are the most chair operators, it is convenient
to pick $c_\text{ch}=0$. According to the normalization condition
\equatref{eq:LWnorm}, one thus finds
\begin{equation}
  c_\text{pl}=\frac{5}{3},~~~~c_\text{rt}=-\frac{1}{12},~~~~
    c_\text{pg}=c_\text{ch}=0,
\end{equation}
which is called the {\it L\"uscher-Weisz} action~\cite{luscher_shell_1985}
\index{action!L\"uscher-Weisz} or {\it tree-level Symanzik} action.
\index{action!tree-level Symanzik}

Next we turn to an improvement of the Wilson fermionic action. Now the leading
correction is $\order{a}$, and owing to $\bar{\psi}\psi$ being dimension-three,
there is now the possibility for dimension-five operators. Demanding that
our operators satisfy the symmetries of the Wilson action, e.g.
charge conjugation $\qconj$ and parity $\parity$, we end up in the
continuum with 
\begin{equation}\begin{aligned}
  O_{5a}&=\bar\psi \sigma_{\mu\nu}F_{\mu\nu}\psi, \\
  O_{5b}&=\bar\psi\left(\Dright_\mu\Dright_\mu
                          +\Dleft_\mu\Dleft_\mu\right)\psi,\\ 
  O_{5c}&=m \tr F_{\mu\nu}F_{\mu\nu},\\
  O_{5d}&=m\bar\psi\left(\overrightarrow{\slashed{D}\strut}
                         -\overleftarrow{\slashed{D}\strut}\right)\psi,\\
  O_{5e}&=m^2 \bar\psi \psi,
\end{aligned}\end{equation}
where
\begin{equation}
  \sigma_{\mu\nu}\equiv\frac{1}{2i}[\gamma_\mu,\gamma_\nu].
\end{equation}
This list of operators can be reduced by using the field equation
\begin{equation}
 (\slashed{D}+m)\psi=0,
\end{equation}
from which it follows
\begin{equation}
  O_{5a}-O_{5b}+2O_{5e}=0~~~~\text{and}~~~~O_{5d}+2O_{5e}=0,
\end{equation}
i.e. $O_{5b}$ and $O_{5d}$ are linearly dependent on the other operators,
and can therefore be eliminated. Of the remaining operators, $O_{5c}$ and
$O_{5e}$ are, up to the factor $m$, present in the original action, so
they can be absorbed in the original action by redefining the bare
parameters $m$ and $g$.

\begin{figure}
  \centering
  \includegraphics[width=0.5\linewidth]{figs/clover.pdf}
  \caption{The ``clover" sum $Q_{\mu\nu}(x)$ used in the discretization of the
           field strength $\hat{F}_{\mu\nu}$ in \equatref{eq:cloverField}.
           The point $x$ is in the middle of the squares.}
           
  \label{fig:clover}
\end{figure}

Taken altogether, all that remains for improvement is the operator $O_{5a}$,
so that the $\order{a}$ improved Wilson fermion action becomes
\begin{equation}\label{eq:cloverAction}
  S_\text{clover}=S_\text{Wilson}+\frac{c_\text{SW}}{2}
               \,a^5\sum_{x;\,\mu<\nu}\bar{\psi}(x)
                  \sigma_{\mu\nu}\hat{F}_{\mu\nu}\psi(x),
\end{equation}
with the real {\it Sheikholeslami-Wohlert} coefficient $c_\text{SW}$
\cite{sheikholeslami_improved_1985} and $\hat{F}_{\mu\nu}$ being the
\index{Sheikholeslami-Wohlert}
following discretization of the field strength tensor:
\begin{equation}\label{eq:cloverField}
  \hat{F}_{\mu\nu}(x)\equiv
    -\frac{i}{8a^2}\left(Q_{\mu\nu}(x)-Q_{\nu\mu}(x)\right)
\end{equation}
where $Q_{\mu\nu}(x)$ is the plaquette sum (see \figref{fig:clover})
\begin{equation}
  Q_{\mu\nu}(x)\equiv U_{\mu\nu}(x)+U_{\nu,-\mu}(x)
                 +U_{-\mu,-\nu}(x)+U_{-\nu,\mu}(x).
\end{equation}
Due to the shape of $Q_{\mu\nu}$, the latter term in 
\equatref{eq:cloverAction} is called the {\it clover term}, and the
action is said to be {\it clover-improved}.\index{action!clover-improved}
This discretization of $F_{\mu\nu}$ is chosen because it is more
symmetric than the simple plaquette discretization.
The coefficient $c_\text{SW}$ can be calculated perturbatively; one finds
\cite{sheikholeslami_improved_1985}
\begin{equation}\label{eq:SWcoeff}
  c_\text{SW}=1+0.2659 g^2+\order{g^4}.
\end{equation} 

To round out our discussion of Symanzik improvement, we now turn to the
improvement of some interpolators. The hitherto discussed action improvements
suffice to improve on-shell quantities, e.g. hadron masses, at $\order{a}$. 
For these on-shell quantities, only eigenstates of the Hamiltonian
contribute. However it is not enough for off-shell quantities such as
correlators, which have off-diagonal contributions, and their interpolators
need to be improved. We will discuss now two examples, the isovector axial
current $A_\mu^a$ and the pseudoscalar density $P^a$,
\begin{equation}\begin{aligned}
A_\mu^a&=\frac{1}{2}\bar{\psi}\gamma_\mu\gamma_5\sigma^a\psi,\\
P^a&=\frac{1}{2}\bar{\psi}\gamma_5\sigma^a\psi.
\end{aligned}\end{equation}
The dimension-four operators needed to improve $A_\mu^a$ are
\begin{equation}\begin{aligned}
  O_{4a,\mu}^a&=\frac{1}{2}\bar{\psi}\gamma_5\sigma_{\mu\nu}
                 \left(\Dright_\nu-\Dleft_\nu\right)\sigma^a\psi,\\
  O_{4b,\mu}^a&=\frac{1}{2}\partial_\mu
                  \left(\bar{\psi}\gamma_5\sigma^a\psi\right)\\
  O_{4c,\mu}^a&=\frac{m}{2}\bar{\psi}\gamma_\mu\gamma_5\sigma^a\psi.
\end{aligned}\end{equation}
Again, as before, we can apply the Dirac equation to show $O_{4a,\mu}$ is
linearly dependent on the other two, and again $O_{4c,\mu}$ is equal to
the original current up to the factor $m$, so it can be absorbed
in a redefinition. The improved current is therefore
\begin{equation}
  A_{\text{imp},\mu}^a=A_\mu^a+c_A a\hat{\partial}_\mu P^a,
\end{equation}
where $c_A\in\R$ and $\hat{\partial}$ is the symmetric difference
discretization of the derivative. From perturbation theory one finds
\begin{equation}
  c_A=-0.00756g^2+\order{g^4}.
\end{equation}
With this discussion we see that the
operator $O_{4c,\mu}$ can be absorbed into a redefinition of $P^a$, i.e.
\begin{equation}
  P_\text{imp}^a=P^a.
\end{equation}

\section{Tadpole improvement}\index{improvement!tadpole}

\begin{figure}
  \centering
  \includegraphics[width=\linewidth]{figs/averageLink.png}
  \caption{Expectation value of the trace of a link, evaluated in the
           Landau gauge. Results are calculated by MCMC and in first-order
           perturbation theory, expanding in two ``good" parameters
           (see Ref.~\cite{lepage_viability_1993}) and the lattice
           coupling. These were carried out for quenched configurations
           with $\beta\geq5.7$. Image taken from 
           Ref.~\cite{lepage_viability_1993}.}
  \label{fig:averageLink}
\end{figure}

In the early 1990's, physicists working in LFT encountered
a somewhat troubling problem: Many quantities calculated perturbatively
in the bare coupling constant $g\equiv g_\text{lat}$ turned out to differ
substantially from MCMC results, even for rather small lattice spacing.
An example of the failure was calculated by Lepage and Mackenzie
\cite{lepage_viability_1993}, and is shown in \figref{fig:averageLink}.
In this paper, the authors argue that $g_\text{lat}$ is a poor expansion
parameter, and they give two alternative couplings that converge better.

\begin{figure}
  \centering
  \includegraphics[width=0.5\linewidth]{figs/tadpole.png}
  \caption{A tadpole diagram contributing to the fermion self-energy.}
  \label{fig:tadpole}
\end{figure}

More pertinent to this discussion of action and interpolator improvement
is the origin of this mismatch, which they also investigate. They argue as
follows: In lattice perturbation theory one can expand the link variable as
\begin{equation}
  U_\mu(x)=\exp\left[igaA_\mu^a(x)T^a\right]
          =\id+iagA_\mu^a(x)+\order{a^2g^2}.
\end{equation}
Superficially it seems as though we always get some power of $ag$, but for
tadpole diagrams, their divergence in $a$ exactly cancels the power of $a$ 
coming from the vertex. These diagrams therefore are suppressed by a
power of $g$ only. An example tadpole is shown in \figref{fig:tadpole}.
These tadpoles come from the UV part of loop integrals. To obtain a
better behaved perturbative expansion, one could imagine factoring out
(integrating out) this UV contribution, so that the perturbative expansion
would instead read
\begin{equation}
  U_\mu(x)\approx u_0\left(\id+iagA^{\text{IR},a}_\mu(x)T^a\right),
\end{equation}
with the {\it tadpole factor} $u_0$ and with the gauge field only 
including\index{tadpole factor}
IR modes. Here $u_0$ has to be a constant\footnote{Otherwise we are not
guaranteed that $U_\mu(x)$ will work correctly as a gauge connection.}, 
and it represents the average value of the link.

In \secref{sec:symanzikImprovement}, improvement coefficients were
determined using perturbation theory. One can imagine that our coefficients
will be better determined taking the influence of tadpoles into account.
This is called {\it tadpole improvement}, and to employ it, we need to
determine $u_0$\footnote{In general $u_0$ depends on the parameters of the
theory.}. One could imagine extracting it from the average link; to do
this one must fix to a particular gauge, since a single link is a
gauge-dependent quantity that would therefore otherwise vanish. One
can also measure it from gauge-invariant constructions such as
\begin{equation}
  u_0=\left(\frac{1}{N_c}\ev{\tr \plaq}\right)^{1/4}.
\end{equation}
Tadpole improvement then amounts to replacing all links $U_\mu$ in lattice
expressions with $U_\mu/u_0$.

As a simple example we present the tree-level, tadpole-improved, one-flavor
clover Lagrangian density. We rescale the action by multiplying by 
the hopping parameter
\begin{equation}
  2\kappa=\frac{1}{m+4/a},
\end{equation}
rescaling the fermion fields by the factor $\sqrt{m+4/a}$, and
rescaling the link variables by the tadpole factor to get
\begin{equation}\begin{aligned}
  \Lagr_\text{clover}^\text{tad}=~&\bar{\psi}(x)\psi(x)\\
    &-\frac{\kappa}{u_0}\frac{1}{a}\bar{\psi}(x)
    \sum_{\mu=\pm1}^{\pm4}\left(\id-\gamma_\mu\right)U_\mu(x)
           \delta(x+a\hat{\mu}-y)\,\psi(y)\\
    &+\,\frac{\kappa}{u_0}\frac{c_\text{SW}}{u_0^3}
               \,a^5\sum_{\mu<\nu}\bar{\psi}(x)
                  \sigma_{\mu\nu}\hat{F}_{\mu\nu}\psi(x).
\end{aligned}\end{equation}
Using this Lagrangian in a simulation is exactly like using the original 
clover Lagrangian in a simulation; one just has a new hopping parameter
$\kappa'=\kappa/u_0$. From \equatref{eq:SWcoeff}, one finds the
tree-level, tadpole-improved Sheikholeslami-Wohlert coefficient to be
simply
\begin{equation}
  c_{SW}=\frac{1}{u_0^3}.
\end{equation}
One can improve beyond tree-level; for details I refer the reader
to Ref.~\cite{degrand_lattice_2006}.



\section{Fat links}\index{link!fat}

Typically the gauge connection between two neighboring sites $x$ and $y$
on the lattice is just a single link $U(x,y)$, which is in some sense the 
most local connection imaginable\footnote{Sometimes this to-be-replaced
link $U$ is called a {\it thin link}\index{link!thin}.}. 
One can also relax this locality, so that 
the gauge connection contains information from a larger region around 
$x$ and $y$; for example the connection could depend on a general sum, 
including many paths connecting $x$ and $y$. Let us call
this sum $\Sigma(x,y)$. Then the gauge connection could be $V(x,y)$,
where $V$ is chosen by extremizing\footnote{You may notice a dagger sneaked in.
This reflects the fact that the fat link is replacing $U$ as a gauge connection,
and hence it needs to transform the same way under a local gauge transformation.
In other words, $\Sigma$ needs to be ``pointing" in the same direction.}
 $\tr V\Sigma^\dagger$. These gauge
connections are called {\it fat links} \cite{blum_improving_1997}. 
We will encounter fat links in \secref{sec:HISQ}, where they are employed to reduce
taste-breaking effects, and indeed one of the reasons to use a fat link
is to correct hadron spectra. It should not surprise you that fat links
modify particle spectra, since they amount to a change of the lattice 
propagator.

Replacing a link by some weighted sum connecting $x$ and $y$ is consistent
with gauge invariance and hypercube symmetries of the lattice. These are
smeared gauge links. Replacing a link by a weighted average that is displaced
in directions perpendicular to the direction of the connection is a
discretization of higher order derivatives $\partial^2_\nu A_\mu$
\cite{blum_improving_1997} with $\mu\neq\nu$. For example the use of a
fat link might amount to a change
\begin{equation}
  U_\mu\to \left[1+\frac{\epsilon a^2 \Delta^2}{n}\right]^n U_\mu,
\end{equation} 
where $n$ is a small integer and $\epsilon$ is a smearing parameter
\cite{lepage:1997id}. This changes the gluon propagator by a factor
$\exp[-\epsilon a^2q^2]$ in momentum space, i.e. UV modes are 
significantly attenuated. One then sees a connection to Symanzik improvement:
By choosing $\epsilon$ carefully, one can adjust, e.g., 
$\order{a^2}$ corrections.

\begin{figure}
  \centering
  \includegraphics[width=0.6\linewidth]{figs/linkConstructs.pdf}
  \caption{Example three-link, five-link, and seven-link constructs that
           go into the AsqTad fat links. Light grey lines guide the eye.}
  \label{fig:AsqTadConstructs}
\end{figure}

Here we present the AsqTad\index{action!AsqTad}
action~\cite{naik_-shell_1989,orginos_variants_1999,lepage_flavor-symmetry_1999}, a staggered
fermion action with nearest- and third-nearest-neighbor interactions
\begin{equation}
  \Delta_\mu-\frac{a^2}{6}\Delta_\mu^3.
\end{equation}
The smeared links are
\begin{equation}\label{eq:asqtad}
  V_\mu=~c_1U_\mu
           +c_NU^{(N)}_\mu
           +\sum_{\nu\neq\mu}\left[c_3U_{\mu\nu}^{(3)}
           +\sum_{\substack{\rho\neq\mu\\ \rho\neq\nu}}\left(c_5U_{\mu\nu\rho}^{(5)}
           +\sum_{\substack{\sigma\neq\mu\\ \sigma\neq\nu\\ \sigma\neq\rho}} c_7U^{(7)}_{\mu\nu\rho\sigma}\right)
            +c_LU^{(L)}_{\mu\nu}\right],
\end{equation}
where the link constructs are given by
\begin{equation}\begin{aligned}
  U_{\mu}^{(N)}(x)
       &=U_\mu(x)U_\mu(x+a\hat{\mu})U_\mu(x+2a\hat{\mu}),\\
  U_{\mu\nu}^{(3)}(x)
       &=U_\nu(x)U_\mu(x+a\hat{\nu})U_\nu^\dagger(x+a\hat{\mu}),\\
  U_{\mu\nu\rho}^{(5)}(x)
       &=U_\nu(x)U_{\mu\rho}^{(3)}(x+a\hat{\nu})U_\nu^\dagger(x+a\hat{\mu}),\\
  U_{\mu\nu\rho\sigma}^{(7)}(x)
       &=U_\nu(x)U_{\mu\rho\sigma}^{(5)}(x+a\hat{\nu})
         U_\nu^\dagger(x+a\hat{\mu}),\\
  U_{\mu\nu}^{(L)}(x)
       &=U_\nu(x)U_{\mu\nu}^{(3)}(x+a\hat{\nu})U_\nu^\dagger(x+a\hat{\mu}).
\end{aligned}\end{equation}
The first 3-link construct represents a third-nearest-neighbor interaction 
called the {\it Naik} term.
The last 5-link construct $U^{(L)}_{\mu\nu}$, a 5-link staple that is restricted
to a plane, is called the {\it LePage}\index{LePage term} term.
Examples of these constructs are shown in \figref{fig:AsqTadConstructs}.
The coefficients are
\begin{equation}
c_1=\frac{1}{8},~~~c_3=\frac{1}{16},~~~c_5=\frac{1}{64},~~~c_7=\frac{1}{384},
~~~c_L=-\frac{1}{16},~~~c_{\rm Naik}=-\frac{1}{24}.
\end{equation}
When $c_L=c_{\rm Naik}=0$, this smear is called {\it Fat7}\index{link!Fat7}.


\section{APE smearing}\index{link!APE}

This is yet another technique to replaces a link with one that is ``more
typical" given its neighbors. The APE smear~\cite{APE:1987ehd} step replaces
a link with the link plus its staple sum times some constant $\epsilon$.
This is not generically in $\SU(N_c)$ due to the staple sum, so this gets
projected back to $\SU(N_c)$.

\section{Highly improved staggered fermions}\label{sec:HISQ}
\index{action!HISQ}

Highly improved staggered quarks (HISQ) were introduced in 2007 by
Follana {\it et al.} \cite{follana_highly_2007}. This paper summarizes also
some of the history behind staggered quarks, and is written in a generally
friendly way; therefore I encourage the reader to have a look at it.

\index{taste!exchange}
Taste breaking can be thought of through {\it taste exchange}, where
one quark changes its taste by exchanging a virtual gluon with momentum
$p=\pi/a$; a quark with low enough momentum can thereby be pushed into
another corner of the Brillouin zone. See \figref{fig:treeLevelTasteExchange}
for a tree-level diagram.
\begin{figure}[t]
  \centering
  \includegraphics{figs/treelevel.pdf}
  \caption{A low-energy quark absorbing a gluon with momentum $\zeta\pi/a$
           can become a quark of another taste. Here $\zeta$ is a vector
           labelling some unphysical corner of the Brillouin zone. Image
           taken from Ref. \cite{follana_highly_2007}.
           }
  \label{fig:treeLevelTasteExchange}
\end{figure}
One goal of HISQ actions, then, is to suppress these processes, which is
done by smearing. Another way to look at taste violations is that they
are a lattice artifact, and so like any other lattice artifact, you would
like to remove its leading order contributions to improve the
approach to the continuum limit. Along this vein, another improvement
made by HISQ is to reduce lattice artifacts from the derivative discretization.

The basic strategy of HISQ can be summarized as:
\begin{enumerate}
  \item Improve finite difference derivatives by 
        \vspace{-1mm}
        \begin{equation*}
         \Delta_\mu\to\Delta_\mu-\frac{a^2}{6}(1+\epsilon)\Delta_\mu^3,
        \end{equation*}
        where $\epsilon$ depends on charm physics. If there are no quarks
        heavier than a strange in the system, $\epsilon=0$, and this is
        exactly the Naik term.\index{link!Naik} The parameter $\epsilon$ is called
        the {\it Naik} $\epsilon$.
  \item Find a smear $U_\mu\to\mathcal{F}_\mu U_\mu$ that vanishes
        for links carrying momentum $\pi/a$. We start with
        Fat7.\index{link!Fat7}
  \item Multiple smearing reduces mass splittings more, so do that.
  \item Remove new $\mathcal{O}(a^2)$ errors introduced by Fat7.
  \item Smearing enhances some one-loop taste exchange processes,
        which can be suppressed by reunitarizing.
\end{enumerate}
Overall the HISQ smear is then
\begin{equation}\label{eq:HISQsmear}
\mathcal{F}^{\rm HISQ}\equiv
\mathcal{F}^{\rm Fat7}_{\rm corr}\mathcal{U}\mathcal{F}^{\rm Fat7},
\end{equation}
where $\mathcal{F}^{\rm Fat7}_{\rm corr}$ has been corrected for the
errors referenced in step 4, and $\mathcal{U}$ is the reunitarization
operator needed for step 5. The HISQ action uses link variables smeared
by \equatref{eq:HISQsmear}, and its kinetic term uses improved
Naik discretization.

To judge how well the HISQ action does, the authors of
Ref.~\cite{follana_highly_2007} calculated several one-loop taste
exchange processes (that fall into several broad categories indicated
in \figref{fig:oneLoopTasteExchange}) and determined their coefficients, 
$d$. \tabref{tab:hisq} compares the suppression of taste exchange
to other smearing programs. Before HISQ, AsqTad was a popular dynamical
fermion action, and one sees that, measured in this way, taste effects
in the HISQ action are reduced by an order of magnitude.
\begin{figure}[t]
  \centering
  \includegraphics{figs/oneloop.pdf}
  \caption{Some example one-loop processes contributing to taste exchange.
           Image taken from Ref. \cite{follana_highly_2007}.
           }
  \label{fig:oneLoopTasteExchange}
\end{figure}
\begin{table}[t]
\begin{tabularx}{\linewidth}{lCCCC} \hline\hline
        & \multicolumn{3}{c}{Unimproved} & Improved \\
         & ASQTAD & HISQ & HYP & ASQTAD \\ \hline
         Avg $d$,$\tilde{d}$ & 0.23 & 0.02 & 0.02 & 0.13\\
        \hline\hline 
\end{tabularx}
\caption{Average coefficients for the taste processes indicated
         in \figref{fig:oneLoopTasteExchange} for a few
         types of lattice actions. Columns indicate whether the
         gluons are improved. Excerpt from Table II of
         Ref. \cite{follana_highly_2007}.}
\label{tab:hisq}
\end{table}


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