\chapter{Renormalisation}
\label{chap2}
\label{chap:renorm}

\section{Symmetries and identities in QCD}
\label{sec:symm-ident}

It is a general feature of field theories that the existence of a
continuous symmetry implies some identity for physical quantities.  In
classical field theories this manifests itself in Noether's theorem,
linking every global, continuous symmetry with a conserved current. 
Thus, conservation of energy and momentum can be seen as a consequence
of translational invariance, and electromagnetic current
conservation a consequence of global gauge invariance.

In quantum field theory most of the interesting symmetries are expressed
through Ward identities or Slavnov--Taylor identities between
Green functions.  The most important of these are derived from the gauge
invariance of the theory.

\subsection{The generic Ward identity}
\label{sec:WI-generic}

The generic Ward identity can be derived as a direct consequence of the
invariance of Green functions under some local transformation.  Consider
a Green function
%
\beq
G(x_1, x_2,\ldots) = \langle\Op(x_1,x_2,\ldots)\rangle = 
\int\cald\Phi\Op[\Phi](x_1,x_2,\ldots)e^{-S[\Phi]}
\eeq

If we postulate that $G$ is invariant under some transformation of the
fields, parametrised by a continuous variable $\omega(x)$, we have
%
\begin{eqnarray}
0 = \frac{\delta G}{\delta\omega(x)} & = &
\int\cald\Phi\left( \frac{\delta\Op}{\delta\omega(x)}
-\Op\frac{\delta S}{\delta\omega(x)}\right) e^{-S[\Phi]} \\
& \Rightarrow & \left<\frac{\delta\Op}{\delta\omega(x)}\right>
- \left<\Op\frac{\delta S}{\delta\omega(x)}\right> = 0 
\label{eq:WI-generic}
\end{eqnarray}

By choosing different operators $\Op$ we can derive Ward identities
connecting Green functions with any number of external legs and operator
insertions.  Alternatively, they can be expressed in a compact form as
identities of the generating functionals.

\subsection{Slavnov--Taylor identities}
\label{sec:STI}

Consider the gauge fixed generating functional
%
\beq
e^{W[J]} = \int\cald A\det{\cal M} 
e^{-\int d^4x(\lagr^{YM}+\frac{1}{2\xi}F^2 + J\cdot A)}
\eeq
where $F$ is the gauge fixing functional and $\det{\cal M}$ is the
Faddeev--Popov determinant, cf.\ section \ref{sec:gauge-quantise}.
\(\cald A\det{\cal M}(A)\) is invariant under the gauge transformation
\(\delta A^a = D^{ab}\delta\alpha^b\), so we can transform the variables
in the integral.  This gives
%
\beq
e^{W[J]} = \int\cald A\det{\cal M} 
e^{-\int d^4x(\lagr_{YM}+\frac{1}{2\xi}F^2 + J\cdot A
+ \frac{1}{\xi}F{\cal M}\delta\alpha + J\cdot D\delta\alpha)}
\eeq
since \(\delta F = {\cal M}\delta\alpha\).

Taking $\delta\alpha$ to be the nonlocal transformation \(\delta\alpha =
{\cal M}^{-1}\delta\omega\), and expanding to the lowest order in
$\delta\omega$, we get
\beq
\int\cald A\det{\cal M} 
\int d^4x(\frac{F}{\xi}+J\cdot D{\cal M}^{-1})\delta\omega 
e^{-\int d^4x(\lagr + \frac{1}{2\xi}F^2 + J\cdot A)} = 0
\eeq
which implies
\beq
\frac{1}{\xi}F_a\left[\frac{\delta}{\delta J(x)}\right]e^{W[J]} = 
\int d^4y\left(J_aD_{bc}\left[\frac{\delta}{\delta J}\right]\right)(y)
G_{ca}(y,x) 
\eeq
where
\beq
{\cal M}_{bc}\left[\frac{\delta}{\delta J(y)}\right]G_{ca}(y,x)
= \delta_{ba}\delta^4(x-y)e^{W[J]}
\eeq

This gives us the Slavnov--Taylor identities for full, connected Green
functions.  The identities for the proper functions can be obtained by
observing that the action (with ghosts included) is invariant under the
Becchi--Rouet--Stora transformation
%
\begin{eqnarray}
\delta A^\mu_a(x) & = & D^\mu_{ab}(x)\eta_b(x)\delta\zeta \equiv 
sA\delta\zeta \label{BRST1} \\
\delta\bar{\eta_a}(x) & = & \frac{1}{\xi}F_a[A(x)]\delta\zeta \equiv
s\bar{\eta}\delta\zeta \\
\delta\eta_a(x) & = & -\frac{g}{2}f_{abc}\eta_b(x)\eta_c(x)\delta\zeta 
\equiv s\eta\delta\zeta \\
\delta\psi(x) & = & -gt_a\eta_a(x)\psi(x)\delta\zeta \equiv
s\psi\delta\zeta \\
\delta\psibar(x) & = & -g\psibar(x)t_a\eta_a(x)\delta\zeta \equiv
s\psibar\delta\zeta \label{BRST4} 
\end{eqnarray}

If we introduce source terms $K$ and $L$ for the operators $sA$ and
$s\eta$ respectively, performing the BRS transformation on the path
integral gives
%
\begin{eqnarray}
\lefteqn{\int\cald A\cald\eta\cald\bar{\eta}\cald\psi\cald\psibar
\int d^4x\left(JsA +
\bar{\omega}s\eta - \frac{1}{\xi}F\omega - g\bar{\sigma}\eta\psi +
g\psibar\eta\sigma\right)(x)} \nonumber \\
& & e^{-\int d^4y(\lagr^{YM}+\lagr^{GF}+\lagr^{FP})+J\cdot A +
\bar{\omega}\cdot\eta + \bar{\eta}\cdot\omega + \psibar\cdot\sigma +
\bar{\sigma}\cdot\psi} = 0 \\
& \Rightarrow & \int d^4x \bigg(J\frac{\delta}{\delta K} -
\bar{\omega}\frac{\delta}{\delta L} - \frac{1}{\xi}\omega
F[\frac{\delta}{\delta J}] \nonumber\\
& & -\bar{\sigma}\frac{\delta^2}{\delta\bar{\omega}\delta\bar{\sigma}} +
\frac{\delta^2}{\delta\sigma\delta\bar{\omega}}\bigg)(x)e^W = 0 \\
& \Rightarrow & \int d^4x \bigg(J\frac{\delta}{\delta K} -
\bar{\omega}\frac{\delta}{\delta L} - \frac{1}{\xi}\omega
F[\frac{\delta}{\delta J}] \nonumber \\
& & -\bar{\sigma}\frac{\delta^2}{\delta\bar{\omega}\delta\bar{\sigma}} +
\frac{\delta^2}{\delta\sigma\delta\bar{\omega}}\sigma\bigg)(x)
W[J,\omega,\bar{\omega},K,L,\sigma,\bar{\sigma}]
 = 0 
\end{eqnarray}
 
Performing a Legendre transformation on this gives us
%
\beqa
\lefteqn{\int d^4x 
\Bigg(\frac{\delta\Gamma}{\delta A}\frac{\delta\Gamma}{\delta K}
+ \frac{\delta\Gamma}{\delta\eta}\frac{\delta\Gamma}{\delta L} 
+ \frac{\delta\Gamma}{\delta\psi}\left[\frac{\delta^2\Gamma}
{\delta\eta\delta\psi}\right]^{-1}}\nonumber\\
& + & 
\left[\frac{\delta^2\Gamma}{\delta\psibar\delta\eta}\right]^{-1}
\frac{\delta\Gamma}{\delta\psibar} 
-\frac{1}{\xi}\frac{\delta\Gamma}{\delta\bar{\eta}}F[A]\Bigg) = 0
\label{eq:ST} 
\eeqa

We can also, independently of this, perform a change of variables
\(\bar{\eta}\to\bar{\eta}+\delta\bar{\eta}\), which yields
%
\begin{eqnarray}
\lefteqn{\left(\omega(x)-\Phi\frac{\delta}{i\delta K(x)}\right)
W[J,\omega,\bar{\omega},K,L] = 0} \\
& \Rightarrow & \Phi^\mu_{ab}\frac{\delta\Gamma}{\delta K^\mu_b(x)} +
\frac{\delta\Gamma}{\delta\bar{\eta}_a(x)} = 0 \label{eq:ST-aux}
\end{eqnarray}
where we have written
\beq
F_a(x) = \Phi^\mu_{ab}(x)A_{\mu b}(x)
\eeq

For example, taking the derivative wrt $A(y)$ and $\eta(z)$ of
(\ref{eq:ST}) gives the Slavnov--Taylor equation for the gluon
self-energy
%
\beq
\int d^4x\frac{\delta^2\tilde{\Gamma}}
{\delta A^b_\nu(y)\delta A^a_\mu(x)} \frac{\delta^2\tilde{\Gamma}}
{\delta\eta^c(z)\delta K^a_\mu(x)} = 0
\eeq
where \(\tilde{\Gamma}=\Gamma-(1/2\xi)F^2\) and
\beq
\partial^\mu_x \frac{\delta^2\tilde{\Gamma}}
{\delta\eta^c(z)\delta K^a_\mu(x)} 
+ \frac{\delta^2\tilde{\Gamma}}{\delta\eta^c(z)\delta\bar{\eta}^a(x)} = 0,
\eeq
follows from (\ref{eq:ST-aux}).

\section{Renormalisation and counterterms}
\label{sec:renorm-general}

\subsection{The principle of renormalisation}
\label{sec:renorm-intro}

The (bare) parameters that enter into the action are not, in general,
equal to those corresponding parameters that can be measured.  The
masses, kinetic terms and couplings will have contributions from
self-interactions of the fields beyond what can naively be read out of
the (bare) action.  They will have to be renormalised, in order that the
parameters we do put into the theory correspond to, or can be related
to, measurable quantities.  This is done by rewriting the theory in
terms of the physical or renormalised parameters.  In doing this, new
terms --- counterterms --- will appear in the action.  The theory is
{\em renormalisable} if
\begin{enumerate}
\item only a finite number of counterterms are required
\item all the counterterms are local and Lorentz invariant
\item the bare and renormalised theories have the same symmetries
\end{enumerate}

The last condition (which must be satisfied order by order in
perturbation theory) leads to severe restrictions on the form the
counterterms can take, since it means the Slavnov--Taylor identities
must be satisfied order by order for gauge theories.  It turns out that
this can indeed be done.  The Slavnov--Taylor identities guarantee the
gauge invariance of the theory, in particular the universality of the
coupling constant.  If we write the counterterms as
%
\begin{eqnarray}
\Delta\lagr & = & (Z_3-1)\half\Tr[(\partial_\mu A_\nu-\partial_\nu
A_\mu)^2] \nonumber \\
& & - g(Z_1-1)\Tr(\partial_\mu A_\nu-\partial_\nu A_\mu)[A_\mu,A_\nu])
\nonumber \\
& & + (Z_4-1)\frac{g^2}{2}Tr([A_\mu,A_\nu][A_\mu,A_\nu]) \\
& & + (\tilde{Z}_3-1)(-\bar{\eta}_a\partial^2\eta_a)
+ g(\tilde{Z}_1-1)f_{abc}A^a_\mu\partial_\mu\bar{\eta}_b\eta_c \nonumber \\
& & + (Z_2-1)\psibar\dslash\psi + (Z_2\frac{m_0}{m}-1)m\psibar\psi
\nonumber \\
& & + ig(Z_{1F}-1)\psibar\gamma_\mu A_\mu\psi \nonumber
\end{eqnarray}
then
\beq
\frac{Z_4}{Z_1} = \frac{Z_1}{Z_3} = \frac{\tilde{Z}_1}{\tilde{Z}_3} =
\frac{Z_{1F}}{Z_2}
\eeq
follows from the Slavnov--Taylor identities and guarantees that the
renormalised coupling \(g_R = Z_gg = Z_1{Z_3}^{-3/2}g\) enters in the
same way everywhere in the renormalised action.  If we define the
renormalised fields
\beqa
A^R_\mu & = & Z_3^{-1/2}A^0_\mu \\
\psi^R & = & Z_2^{-1/2}\psi^0 \\
\eta^R & = & \tilde{Z_3}^{-1/2}\eta^0
\eeqa
the renormalised action takes the same form as the bare action, but with
the renormalised fields and parameters rather than the bare ones
entering. 

In most quantum field theories, divergences will arise when evaluating
Green functions perturbatively.  Since the renormalised Green functions
and parameters must be finite, this means that the bare parameters are
infinite (or zero), and the infinities are absorbed into the
counterterms.  If this is to happen in a well-defined way, the theory
must be regularised --- which means that a momentum cutoff is imposed
to make all quantities finite.  In continuum calculations, this is
normally done by dimensional regularisation, while the lattice
automatically provides a momentum cutoff of $\pi/a$.  A set of
renormalisation conditions is then applied to the regularised Green
functions.  In dimensional regularisation, the most common scheme is
the modified minimum subtraction scheme $\bar{MS}$, where only the
divergent part and some associated terms are subtracted off.  Although
it is possible to implement an analogous minimal subtraction in lattice
theories as well, momentum schemes, where the Green functions at some
set of momenta are related to the tree level Green
functions, are more appropriate and more commonly used.

The renormalisation constants will in general depend on all the
defining  parameters of the theory --- coupling constants and masses
--- plus a scale that enters through the renormalisation.  It is
possible, and sometimes desirable, to choose a scheme where the
dependence on for instance the masses is eliminated.  The standard
minimal subtraction scheme is such a mass-independent, but most lattice
schemes are not, and it is considerably more complicated to disentangle
the mass dependence in such schemes.

\subsection{Renormalisation group and the QCD $\beta$-function}
\label{sec:betafunc}

Consider a Green function 
\beqa
\lefteqn{
{\cal G}_0^{(n)}(p_1,\ldots,p_{n_\psi},q_1,\ldots,q_{n_A};g_0,m_0)}
\nonumber \\
& = &
Z_2^{n_\psi/2} Z_3^{n_A/2} 
{\cal G}_R^{(n)}(p_1,\ldots,p_{n_\psi},q_1,\ldots,q_{n_A};g_R,m,\mu)
\eeqa
and the associated proper Green function 
\beqa
\lefteqn{
\Gamma_0^{(n)}(p_1,\ldots,p_{n_\psi},q_1,\ldots,q_{n_A};g_0,m_0)}
\nonumber\\ 
& = & Z_2^{-n_\psi/2} Z_3^{-n_A/2} 
\Gamma_R^{(n)}(p_1,\ldots,p_{n_\psi},q_1,\ldots,q_{n_A};g_R,m,\mu) 
\eeqa
with $n_\psi$ external fermion legs and $n_A$ external gauge legs, in a
mass-in\-de\-pen\-dent renormalisation scheme (so that we can simplify
things by ignoring the masses).  Since $\Gamma_0$ does
obviously not depend on the renormalisation scale $\mu$, we have
%
\beq
0 = \mu\dd{\mu}\Gamma_0 
 = \left[\mu\dd{\mu} + \beta(g)\dd{g} - n_A\gamma_A -
n_\psi\gamma_\psi\right]\Gamma_R^{(n)}
\label{eq:RGE}
\eeq
which is the renormalisation group equation, with
\beqa
\beta & = & \mu\dfdx{g}{\mu} \label{eq:beta-def}\\
\gamma_A & = & Z_3^{-1/2}\mu\dd{\mu}Z_3^{1/2} \\
\gamma_\psi & = & Z_2^{-1/2}\mu\dd{\mu}Z_2^{1/2}
\eeqa

$\beta$ can be calculated in perturbation theory, and to two loop order
in a SU($N$) gauge theory with $N_f$ flavours, it is
\beqa
\beta(g) & = & -b_0 g^3 - b_1 g^5 + \Op(g^7) \label{eq:beta-expand} \\
b_0 & = & \frac{11N-2N_f}{3(16\pi^2)} \label{eq:b0-def} \\
b_1 & = & \frac{34N^2-10NN_f-3N_f(N^2-1)/N}{3(16\pi^2)^2} \label{eq:b1-def}
\eeqa

The coefficients $b_0$ and $b_1$ are scheme independent, as can be
easily shown.  If we call the renormalised coupling in one scheme $g$
and in another scheme $g_1$, then we can expand $g_1$ in powers of $g$
\beq
g_1 = f(g) = g(1+a_1g^2 + \Op(g^4))
\eeq
This gives
\beqa
\beta_1(g_1) & = & \mu\dd{\mu}f(g) = \mu\dfdx{g}{\mu}\dfdx{f}{g}
\nonumber\\
& = & \beta(g)(1+3a_1g^2+\Op(g^4)) \nonumber\\
& = & -b_0g^3(1+3a_1g^2) - b_1g^5 + \Op(g^7) \nonumber\\
& = & -b_0g_1^3 - b_1g_1^5+\Op(g_1^7)
\eeqa

Integrating (\ref{eq:beta-def}), and using (\ref{eq:beta-expand}), we
find that
\beq
g^2(\mu) = \frac{g_0^2}{1\!+\!g_0^2b_0\ln(\mu^2\!\!/\mu_0^2)
\!+\!g_0^2\frac{b_1}{b_0}\ln(1\!\!+\!\!g_0^2\ln(\mu^2\!\!/\!\mu_0^2)
\!+\!g_0^2\frac{b_1}{b_0}\ln(b_0g_0^2))}
\eeq
where $g_0=g(\mu_0)$.  This can be rewritten in terms of a
renormalisation group invariant scale parameter $\Lambda$,
\beq
g^2(\mu) = \frac{1}{b_0\ln(\mu^2/\Lambda^2)
+\frac{b_1}{b_0}\ln\ln(\mu^2/\Lambda^2)}
\eeq
where the scale parameter $\Lambda$ is
\beq
\Lambda = \mu e^{-\frac{1}{2b_0g^2(\mu)}}
\left(b_0g^2(\mu)\right)^{-\frac{b1}{2b_0^2}}
\label{eq:Lambda_QCD}
\eeq
Since $g$ is not a scheme-independent quantity, $\Lambda$ will also be
different for different renormalisation schemes.

\subsection{Lattice perturbation theory}
\label{sec:LPT}

Most of the discussion in the previous sections carries over directly
to lattice gauge theories.  In particular, if we add a discretised
version of $\lagr^{GF}$ and $\lagr^{FP}$, as defined in
(\ref{eq:gfix-lagr}) and (\ref{eq:fp-lagr}) to the Wilson action
$S_\Box+S^{WF}$, this can be shown to be invariant under BRS
transformations defined in analogy to (\ref{BRST1})-(\ref{BRST4}), and
the effective action $\Gamma$ satisfies the Slavnov--Taylor identities
(\ref{eq:ST}) and (\ref{eq:ST-aux}).  This, and the renormalisability
of lattice QCD to all orders in pertrubation theory, is worked out in
detail by Reisz in \cite{Reisz}. 


Since numerical simulations are always performed using the bare fields
and parameters, the renormalisation of the fields and couplings entering
into the action will not be an issue when computing hadronic quantities.
 However, as will be explained below, the composite operators involved
in hadronic physics also need to be renormalised, and their
renormalisation constants have usually been computed perturbatively.

It turns out that perturbation series in \(\alpha^{LAT} = g_0^2/4\pi =
3/2\pi\beta\), at least at the one-loop order, often give results in
very poor agreement with experimental or non-perturbative
determinations of the same quantities.  This should not be a surprise,
since $\alpha^{LAT}$ is a bare quantity and perturbation series in the
continuum are always expressed in terms of the renormalised coupling. 
If we re-express the perturbation series in terms of a renormalised
coupling at some appropriate scale, it will be more well-behaved, as
Lepage and Mackenzie \cite{LM} have shown.

\section{Composite operators}
\label{sec:op-renorm}

In hadronic physics, we are interested in Green functions involving
local operators composed of several quark and gluon fields: operators
for creating and destroying mesons or baryons, electroweak currents,
${\rm K^0}$ -- $\bar{\rm K^0}$ mixing operators, etc.  These operators
can be introduced into the partition function by coupling them to
auxiliary sources, generating Green functions with operator insertions
by taking derivatives with regard to the sources.  Simple power counting
will reveal that these insertions affect the degree of divergence of
diagrams, possibly leading to new divergences which must be cancelled by
new counterterms.\footnote{Naively, one could expect that the
renormalisation of the fields the operators are composed from is
sufficient to cancel those divergences.  However, this only renormalises
the contributions from insertions of the kind $\Phi(x)\Phi(y)$, which
involve 2 independent momenta, while a composite operator of the type
$\Phi_a(x)\Phi_b(x)$ involves only one momentum.  Alternatively, it can
be argued that the composite operators (those whose renormalisation
matters, at least) only appear in effective theories --- eg, when the
electroweak sector of the standard model is separated off, electroweak
matrix elements are computed from Green functions containing current
operators, bilinear in the quark fields.}  This will require
new renormalisation conditions to be imposed on Green functions
containing these operators.

When renormalising composite operators, it must be taken into account
that they can mix with each other. Consider, for example, an operator
$\Op_1$ of dimension $n$.  If there are other operators
\(\Op_2,\ldots,\Op_m\) with the same quantum numbers and with dimension
$d \leq n$ (which must be compensated with factors $a^{d-n}$), these
will in general mix with $\Op_1$, so that the physical Green functions
are linear combinations of the Green functions containing
\(\Op_1,\ldots,\Op_m\).  The renormalised operator will then be
%
\beq
\Op^R_1 = \sum_{j=1}^m Z_{1j}\Op_j
\eeq
and similarly for the other operators.  Alternatively, the operator can
be redefined to subtract off the divergences from the lower-dimensional
operators. 

%\subsection{Perturbative renormalisation}

All these operators can be renormalised in perturbation theory the same
way as the fields and parameters in the action, by evaluating Green
functions with operator insertions to $n$-loop order and imposing
renormalisation conditions.  But sometimes higher-order or
non-perturbative contributions to the renormalisation are significant,
even when using a tadpole-improved scheme.  It would therefore be
useful to have a non-perturbative renormalisation procedure.  Two such
procedures are described below.


\subsection{Chiral Ward identities and current renormalisation constants}
\label{sec:WI}

We define the various lattice vector currents
%
\begin{eqnarray}
V^{La}_{\mu}(x) & = & \psibar(x)\gamma_{\mu}\half\lambda^a\psi(x) \\
V^{PSa}_{\mu}(x) & = & \half
\{\psibar(x+\hat{\mu})\gamma_{\mu}\half\lambda^a U_{\mu}(x)\psi(x) +
{\rm h.c.} \}\\
V^{Ca}_{\mu}(x) & = &
\half\{\psibar(x)(\gamma_{\mu}-r)U_{\mu}(x)\half\lambda^a
\psi(x+\hat{\mu}) \nonumber\\
 & & \mbox{ }+
\psibar(x+\hat{\mu})(\gamma_{\mu}+r)U^{\dag}_{\mu}(x)
\half\lambda^a\psi(x) \} \\
% & & \mbox{}+ (x
%\rightarrow x+\hat{\mu})\} \\
V^{CI\,a}_{\mu}(x) & = & V^{Ca}_{\mu}(x) +\frac{r}{2}\sum_{\rho}
\partial_{\rho}(\psibar(x)\sigma_{\rho\mu}\half\lambda^a\psi(x)),
\end{eqnarray}
%
the axial currents
%
\begin{eqnarray}
A^{La}_{\mu}(x) & = & \psibar(x)\gamma_{\mu}\gamma_5\half\lambda^a\psi(x) \\
A^{P\!Sa}_{\mu}(x) & = & \half
\{\psibar(x+\mu)\gamma_{\mu}\gamma_5\half\lambda^a U_{\mu}(x)\psi(x) +
{\rm h.c.} \},
\end{eqnarray}
%
and the pseudoscalar and scalar densities
%
\begin{eqnarray}
P^{a}(x) & = & \psibar(x)\gamma_5\half\lambda^a\psi(x) \\
S^{a}(x) & = & \psibar(x)\half\lambda^a\psi(x)
\end{eqnarray}

These are all flavour-mixing (non-singlet) currents; $\lambda^a$ is a
flavour SU(3) generator. The superscript `PS' here denotes `point
split', while `L' denotes `local'.  In the continuum limit, these
operators are related to the physical operators by multiplicative
renormalisation constants $Z_V^L, Z_V^{P\!S}, Z_A^L$ etc, so that
$V_{\mu} = Z_V^L V^L_{\mu}$ etc.  The currents $V^{C}_{\mu}$ and
$V^{CI}_{\mu}$ are both conserved, therefore their renormalisation
constants are both 1, while the partial conservation of the local and
point-split vector currents and axial currents guarantee that their
renormalisation constants are finite.  $Z_P$ and $Z_S$ are both infinite,
but as we shall see, the Ward identities guarantee that the ratio
$Z_P/Z_S$ remains finite.

$V^{CI}_{\mu}$ is also improved according to the prescription in
section \ref{sec:symanzik}. Thus the renormalisation constants for
$V^{L}_{\mu}$ and $V^{P\!S}_{\mu}$ can be easily determined by
evaluating 
% 
\begin{equation} \label{ratio-CI}
Z_{V}^{L,P\!S} = \frac{\bra\alpha\mi V^{CI}_{\mu}(x)
\mi\beta\ket}{\bra\alpha\mi V^{L,P\!S}_{\mu}(x) \mi\beta\ket}
\end{equation} 
%
between arbitrary hadron states \(\bra\alpha\mi\) and \(\mi\beta\ket\).
Alternatively, they may be determined from  
% 
\begin{equation}
\label{ratio-3pt} 
Z_{V}^{L,P\!S} = \frac{\sum_{\vec{x}}\bra P^{\dag}(0)P(\vec{x},T)\ket}  
{\sum_{\vec{x},\vec{y}} \bra P^{\dag}(0) V^{L,P\!S}_4(\vec{y},t)
P(\vec{x},T) \ket}
\end{equation}

This should give a precise estimate provided the effect of the
off-diagonal matrix elements $\bra P_m \mi V^{L,P\!S}_4 \mi P_n \ket$
can be neglected, since, if we write
%
\begin{eqnarray}
C^{3pt} & = & \sum_{\vec{x},\vec{y}}\left\langle
P^{\dagger}(0)V_4(\vec{y},t)P(x,T)\right\rangle \nonumber\\
 & = & \sum_{\vec{x},\vec{y}}\sum_{m,n}\frac{1}{2E_m}\frac{1}{2E_n}
\bra 0\mi P^{\dagger}(0)\mi P_m\ket\bra P_m\mi V_4(\vec{y},t)\mi
P_n\ket \nonumber \\
& & \mbox{ }\bra P_n\mi P(\vec{x},T)\mi 0\ket \nonumber\\
& = & \sum_{m,n}\frac{C_m}{2E_m}\sum_{\vec{y}}\bra P_m\mi
V_4(0)e^{-i(\vec{p}_m-\vec{p}_n)\vec{y}}e^{-(E_m-E_n)t}\mi P_n\ket
\frac{C_n}{2E_n}e^{-E_{n}T} \nonumber \\
& = & \sum_{m,n}\frac{C_mC_n}{(2E_m)(2E_n)}\bra P_m\mi V_4(0)\mi P_n\ket
e^{-E_{n}T}e^{-(E_m-E_n)t} 
\label{eq:ZV-ratio-derive}
\end{eqnarray}
and
\beq
C^{2pt} = \sum_{\vec{x}}\left\langle
P^{\dagger}(0)P(\vec{x},T)\right\rangle =
\sum_{n}\frac{C_n^2}{2E_n}e^{-E_nT} 
\eeq
%
then, since
\(\bra P_n\mi V_4(0)\mi P_n\ket = 2E_n/Z_V\) for degenerate mesons, it
follows that \(C^{3pt}=C^{2pt}/Z_V\) if we ignore the off-diagonal
elements.  For non-degenerate mesons, we have instead
\(C^{3pt}=C^{2pt}/2Z_V\), since in this case \( \bra P_n\mi V_4(0)\mi
P_n\ket = E_n/Z_V \). 

\subsubsection{Ward identities with the standard Wilson action}
\label{sec:WI-wilson}

For the axial case, there is no conserved current or other ``easy'' way
of determining the renormalisation constants, but they can be obtained
using chiral Ward Identities.  
If we define the transformations
%
\begin{equation}
 \delta_A \psi(x) = i\alpha^a(x)\half\lambda^a\gamma_5\psi(x)
\hspace{1cm} 
 \delta_A \psibar(x) = i\alpha^a(x)\psibar(x)\half\lambda^a\gamma_5,
\end{equation}
%
equation~(\ref{eq:WI-generic}), with \(\omega(x) = \alpha^a(x)\),  
becomes\footnote{The Ward identities that naturally come out of this
are expressed in terms of the point-split currents and the forward
lattice derivative.  However, if we replace the forward by the
symmetric derivative and point-split by local currents, the resulting
Ward identities only differ by terms of $O(a^2)$.} 
%
\begin{eqnarray}
\lefteqn{i\left\langle\frac{\delta
O(x_1,x_2,\cdots)}{\delta\alpha^a(x)}\right\rangle =
\left\bra\partial_{x}^{\mu} A^a_{\mu} O(x_1,x_2,\cdots) \right\ket} 
\nonumber \\
 & & - \left\bra O(x_1,x_2,\cdots)[\psibar(x)\{\half\lambda^a,M_0\}\gamma_5\psi(x) +
X^a(x)]\right\ket \label{eq:WI-chiral}
\end{eqnarray}
%
where $M_0$ is the bare mass matrix and $X^a(x)$ is the chiral variation
of the Wilson term of the action.  This is a dimension-5 operator that
is equal to zero at tree level in the continuum limit, but it mixes with
\(\partial_\mu A^a_{\mu}\) and \(\psibar\lambda^a\gamma_5\psi\), so we
define, following \cite{Boch},
%
\beq
\bar{X}^a = X^a + \{\half\lambda^a,\bar{M}\}P^a +
(Z_A-1)\partial_{\mu}A^a_{\mu}
\eeq
Requiring that \(\bra\alpha\mi\bar{X}^a\mi\beta\ket=0\) for all on-shell
\(\bra\alpha\mi\) and \(\mi\beta\ket\) determines
%
\beq
\rho = \frac{m}{Z_A} = \frac{M_0+\bar{M}}{Z_A} = \frac{\bra 0\mi\partial_4
A^{a}_4\mi P^a\ket}{2\bra 0\mi P^a\mi P^a\ket} \label{eq:rhodef}
\eeq
where $\rho, m$ and $\bar{M}$ are in general matrices in flavour space.
If we set \(\Op(y,0) = A^{Lb}_{\nu}(y) V^{Lc}_{\rho}(0)\) and assume all
quark masses are degenerate, (\ref{eq:WI-chiral}) can be reordered to yield
\cite{Boch}
%
\beqa
\lefteqn{\partial_x^\mu\bra\hat{A}^a_{\mu}(x)\hat{A}^b_{\nu}(y)
\hat{V}^c_{\rho}(0)\ket =} \nonumber \\
 & & \bra\bar{X}^a(x)\hat{A}^b_{\nu}(y)\hat{V}^c_{\rho}(0)\ket
 + \bra\psibar(x)\{\half\lambda^a,m\}\gamma_5\psi(x)
  \hat{A}^b_{\nu}(y)\hat{V}^c_{\rho}(0)\ket \nonumber\\
 & + & if^{abd}\delta(x-y)\frac{Z_A}{Z_V}
\bra\hat{V}^d_{\nu}(y)\hat{V}^c_{\rho}(0)\ket \nonumber \\
 & + & if^{acd}\delta(x)\frac{Z_V}{Z_A}
\bra\hat{A}^b_\nu(y)\hat{A}^d_\rho(0)\ket 
\eeqa
where $\hat{A}=Z_A^L A^L$ and $\hat{V}=Z_V^L V^L$ are the operators with the
correct continuum limit.
The continuum Ward identity
\beqa
\lefteqn{\partial_x^\mu\bra A^a_\mu(x)A^b_\nu(y)V^c_\rho(0)\ket = }
\nonumber\\
 & &\bra\psibar(x)\{\half\lambda^a,m\}\gamma_5\psi(x)A^b_\nu(y)V^c_\rho(0)\ket
 \nonumber \\
& + & if^{abd}\bra V^d_{\nu}(y)V^c_{\rho}(0)\ket 
 +  if^{acd}\delta(x)\bra{A}^b_\nu(y)A^d_\rho(0)\ket 
\eeqa
is recovered if
%
\begin{eqnarray}
\label{WI-ZA}
\lefteqn{\bra\xi^{La}(x) A^{Lb}_{\nu}(y) V^{Lc}_{\rho}(0)\ket =} \nonumber \\
 & & \mbox{ }i f^{abd}\frac{Z_V^L}{{Z_A^L}^2}
\delta(x-y)\bra V^{Ld}_{\nu}(y)V^{Lc}_{\rho}(0)\ket 
\nonumber \\
 & + & 
 i f^{acd}\frac{1}{Z_V^L}\delta(x)\bra A^{Lb}_{\nu}(y)A^{Ld}_{\rho}(0)\ket
\end{eqnarray}
where
\beq
\xi^{La}(x) = \partial^{\mu}A^{La}_{\mu}(x) - 2\rho P^a(x)
\eeq

Similarly, setting $O(y,0) = S^b(y) P^c(0)$ we obtain
%
\begin{eqnarray}
\label{WI-PS}
\lefteqn{\bra\xi^{La}(x) S^b(y) P^c(0)\ket =} \nonumber \\
 & - & d^{abd}\frac{Z_P}{Z_A^L Z_S}
\delta(x-y)\bra P^d(y)P^c(0)\ket 
\nonumber \\
 & - & d^{acd}\frac{Z_S}{Z_A^L Z_P}\delta(x)\bra S^b(y)S^d(0)\ket
\end{eqnarray}
%
where $d^{abc}$ is defined by
%
\[
\{ \lambda^a, \lambda^b \} = 2d^{abc}\lambda^c + \frac{4}{N_f}\delta^{ab}
\]
%
This equation can be used to obtain the ratio of the scalar and
pseudoscalar density renormalisation constants $Z_S/Z_P$.

\subsubsection{Ward identities with the improved action}
\label{sec:WI-clover}

The derivation of the Ward identities for the SW action is practically
identical to the derivation for the Wilson action.  However, further
complications are brought in by the improved operators.  Bilinear
improved operators can be obtained \cite{Heatlie} by replacing
\( \psibar(x) \Gamma \psi(x) \)
with
\( \psibar(x)(1+\frac{ra}{4}(\Dslashbak+m_0)) \Gamma
(1-\frac{ra}{4}(\Dslash-m_0)) \equiv
\psibar(x)\tilde{\Gamma}\psi(x) \).

The terms involving $m_0$ can be eliminated using the equations of
motion
%
\begin{equation}
(\Dslash+m_0)S(x,y) = \delta(x-y) \hspace{1cm} S(x,y)(\Dslashbak-m_0) =
-\delta(x-y) 
\end{equation}
%
where $S$ is the quark propagator.  Inserting this into the LHS of
(\ref{WI-ZA}), we get
%
\begin{eqnarray*}
\lefteqn{\bra P^a(x)A^{Lb}_{\nu}(y)V^{Lc}_{\rho}(0)\ket} \\[2ex]
 & = &
\bra\psibar(x)\,\tilde{\Gamma}^a_5\,\psi(x)\psibar(y)\,
 \tilde{\Gamma}^b_{A\nu}\,\psi(y)
 \psibar(0)\,\tilde{\Gamma}^c_{V\rho}\,\psi(0)\ket \\[2ex]
 & = & - \bra\Tr S(0,x)\,\tilde{\Gamma}^a_5\,S(x,y)\,\tilde{\Gamma}^b_{A\nu}\,
S(y,0)\,\tilde{\Gamma}^c_{V\rho}\ket \nonumber\\
 & & \mbox{} 
- \bra\Tr S(y,x)\,\tilde{\Gamma}^a_5\,
S(x,0)\,\tilde{\Gamma}^c_{V\rho}\,S(0,y)\,\tilde{\Gamma}^c_{A\nu}\ket \\[2ex]
 & = & - \bra\Tr S(0,x)\,\hat{\Gamma}^a_5\,S(x,y)\,\hat{\Gamma}^b_{A\nu}\,
S(y,0)\,\hat{\Gamma}^c_{V\rho}\ket \nonumber\\
 & & \mbox{} 
- \bra\Tr S(y,x)\,\hat{\Gamma}^a_5\,
S(x,0)\,\hat{\Gamma}^c_{V\rho}\,S(0,y)\,\hat{\Gamma}^c_{A\nu}\ket \\
 & & \mbox{} + \frac{ra}{2}if^{acd}\delta(x) \bra\Tr
S(0,y)\,\hat{\Gamma}^b_{A\nu}\,S(y,0)\,\hat{\Gamma}^d_{A\rho}\ket \\
 & & \mbox{} 
+ \frac{ra}{2}if^{abd}\delta(x-y) \bra\Tr
S(0,y)\,\hat{\Gamma}^d_{V\nu}\,S(y,0)\,\hat{\Gamma}^c_{V\rho}\ket \\
 & & \mbox{} + \mbox{\rm terms at } y=0
\end{eqnarray*}
%
where 
\[
\hat{\Gamma} \equiv (1+\frac{ra}{2}\Dslashbak)\Gamma(1-\frac{ra}{2}\Dslash)
\]

So if we sum over all $x$ and spatial $\vec{y}$ (keeping
$y \neq 0$) in eq.~(\ref{WI-ZA}), we obtain
%
\begin{eqnarray}
\label{WI-ZA-int}
\lefteqn{2\rho\sum_{x,\vec{y}}\bra P^a(x)A^{Lb}_{\nu}(y)V^{Lc}_{\rho}(0)\ket}
\nonumber \\
& = & -i (\frac{Z_V^L}{{Z_A^L}^2}-\rho ra) 
f^{abd}\sum_{\vec{y}}\bra V^{Ld}_{\nu}(y) V^{Lc}_{\rho}(0)\ket \nonumber \\
 & & -i (\frac{1}{Z_V^L}-\rho ra)
f^{acd}\sum_{\vec{y}}\bra A^{Lb}_{\nu}(y)A^{Ld}_{\rho}(0)\ket 
\end{eqnarray}

Similarly, (\ref{WI-PS}) becomes
%
\begin{eqnarray}
\label{WI-PS-int}
\lefteqn{2\rho\sum_{x,\vec{y}}\bra P^a(x)S^b(y)P^c(0)\ket}
\nonumber \\
& = & (\frac{Z_P}{Z_A^L Z_S}-\rho ra) 
d^{abd}\sum_{\vec{y}}\bra P^d(y) P^c(0)\ket \nonumber \\
 & & (\frac{Z_S}{Z_A^L Z_P}-\rho ra)
d^{acd}\sum_{\vec{y}}\bra S^b(y)S^d(0)\ket 
\end{eqnarray}


\subsection{Renormalisation with quark Green functions}
\label{sec:quarkGF-renorm}

Another, more generally applicable method for non-perturbative
renormalisation of operators is by imposing conditions on matrix
elements of the operators between quark Green functions
\cite{quarkGF-renorm}.  This method requires gauge fixing, since the
quark Green functions are gauge dependent quantities, but has the
advantage that the scheme is defined in the continuum as well as on the
lattice, so that matching to commonly used continuum schemes is
straightforward.  The conditions imposed are also of the same kind as
you would naturally choose in lattice perturbation theory.

More precisely, the renormalised Green function ${\cal G}^R_\Op(pa)$
of the operator $\Op$ computed between between quark states $\mid
pa\rangle$ at a chosen momentum is fixed to its tree-level value
\beq
{\cal G}^R_\Op(pa) = Z_\Op(\mu a)\langle pa\mid\Op\mid pa\rangle|_{p^2=\mu^2} = 
\langle p\mid\Op\mid p\rangle_0
\eeq
In practice, this is done by fixing the renormalised proper Green
function $\Gamma_\Op(pa)$ to 1
\beq
Z_\Op(\mu a)Z_2^{-1}(\mu a)\Gamma_\Op(pa)|_{p^2=\mu^2} = 1
\label{eq:quarkGF-renorm} 
\eeq
where
\beq
\Gamma_\Op(pa) = 
\frac{1}{12}\Tr\left(S(pa)^{-1}{\cal G}_\Op(pa)S(pa)^{-1}P_\Op\right)
\label{eq:op-GF}
\eeq
$P_\Op$ is a projector onto the tree-level operator.  The quark
field renormalisation constant $Z_2$ can be defined, for instance by
requiring that the quark propagator equals the tree-level propagator
%
\beq
Z_2(\mu) = \left.
\frac{\Tr\left(-i\sum_\nu\gamma_\nu\sin(p_\nu a)S^{-1}(pa)\right)}
{4\sum_\nu\sin^2(p_\nu a)}\right|_{p^2=\mu^2} \label{eq:Zquark-def}
\eeq
or by requiring that $Z_V^C$, computed in this scheme, is equal to 1.
