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\begin{document}
%\hfill BNL-xxx       % for bnl number

\title{CP symmetry and the strong interactions}
\author{Michael Creutz}
\affiliation{
Physics Department, Brookhaven National Laboratory\\
Upton, NY 11973, USA
}
%\email{creutz@bnl.gov}

\date{March 2003}                                            

\begin{abstract}
{ I discuss several aspects of CP non-invariance in the strongly
interacting theory of quarks and gluons.  I use a simple effective
Lagrangian technique to map out the region of quark masses where CP
symmetry is spontaneously broken.  I then turn to the possible
explicit CP violation arising from a complex quark mass.  After
summarizing the definition of the renormalized theory as a limit, I
argue that attempts to remove the CP violation by making the lightest
quark mass vanish are not well defined.  I close with some warnings
for lattice simulations.}
\end{abstract}

\pacs{
11.30.Er, 12.39.Fe, 11.15.Ha, 11.10.Gh
}% PACS, the Physics and Astronomy
                             % Classification Scheme.
\maketitle

\section{Introduction}

The SU(3) non-Abelian gauge theory of the strong interactions is quite
remarkable in that, once an arbitrary overall scale is fixed, the only
parameters are the quark masses.  Using only a few pseudoscalar meson
masses to fix these parameters, the non-Abelian gauge theory
describing quark confining dynamics is unique.  It has been known for
some time \cite{dashen} that, as the parameters are varied from their
physical values, exotic phenomena can occur, including spontaneous
breakdown of CP symmetry.

The possibility of a spontaneous CP violation is most easily
demonstrated in terms of an effective chiral Lagrangian.  In Section
\ref{review} I will review this model for the strong interactions with
three quarks, namely the up, down, and strange quarks.  This lays the
groundwork for the discussion in Section \ref{diagram} of where the CP
violating phase arises.  Section \ref{etaprime} discusses how heavier
states, most particularly the $\eta^\prime$ meson, enter without
qualitatively changing the structure.

Included among the mass parameters of the strong interactions is a
complex phase which, if present, explicitly violates CP symmetry.
This parameter appears to be extremely small \cite{dipole} since no
such violation is seen phenomenologically.  A puzzle for grand
unification asks why is CP violation small for the strong interactions
but not the weak \cite{pecceiquinn}.  It is sometimes suggested that a
massless up quark would solve this problem, and I turn to this issue
in section \ref{mup}.  There I argue that asking whether the up quark
mass vanishes is not physically meaningful.  For this I elucidate the
meaning of the continuum theory and the corresponding ambiguities in
defining the quark mass.  These issues remain even with the recently
discovered chirally symmetric lattice fermions.  Finally, Section
\ref{final} contains some brief remarks, including possible impacts of
the CP violating structures on lattice gauge simulations.

\section{The effective model}
\label{review}

A CP violating phase appears naturally in the simplest chiral sigma
model of interacting pseudoscalar mesons.  In this section I review
the basic model and the standard connections between the quark masses
and pseudoscalar meson masses.  Nothing in this section is new; I am
setting the stage for later discussion.

To be specific, consider the three flavor theory with its approximate
SU(3) symmetry.  Using three flavors simplifies the discussion,
although the CP violating phase can also be demonstrated for the two
flavor theory following the discussion in \cite{mymasspaper}.  I work
with the familiar octet of light pseudoscalar mesons $\pi_\alpha$ with
$\alpha=1\ldots8$.  In a standard way (see for example \cite{km}) I
consider an effective field theory defined in terms of the SU(3)
valued group element
\begin{equation}
\label{sigma}
\Sigma=\exp(i\pi_\alpha \lambda_\alpha/f_\pi)\in SU(3)
\end{equation}
Here the $\lambda_\alpha$ are the usual Gell-Mann matrices which
generate the flavor group and $f_\pi$ is a dimensional constant with a
phenomenological value of about 93 MeV.  I follow the normalization
convention that ${\rm Tr} \lambda_\alpha \lambda_\beta =
2\delta_{\alpha\beta}$.  The neutral pion and the eta meson will play
a special role in the later discussion; they are the coefficients of
the commuting generators
\begin{equation}
\lambda_3=\pmatrix{1&0&0\cr
0&-1&0\cr 0&0&0\cr}
\end{equation}
and 
\begin{equation}
\lambda_8={1\over \sqrt 3}\pmatrix{1&0&0\cr
0&1&0\cr 0&0&-2\cr},
\end{equation}
respectively.  In the chiral limit of vanishing quark masses, we model
the interactions of the eight massless Goldstone bosons with the
effective Lagrangian density
\begin{equation}
\label{kinetic}
L_0={f_\pi^2\over 4}{\rm Tr}(\partial_\mu \Sigma^\dagger \partial_\mu \Sigma)
\end{equation}
The non-linear constraint of $\Sigma$ onto the group SU(3) makes this
theory non-renormalizable.  It is to be understood only as the
starting point for an expansion of particle interactions in powers of
their momenta.  Expanding Eq.~(\ref{kinetic}) to second order in the
meson fields gives the conventional kinetic terms for our eight
mesons.

This theory is invariant under parity and charge conjugation.  These
operators are represented by simple transformations
\begin{equation}
\begin{plain}
\eqalign{
&P:\ \Sigma\ \rightarrow\ \Sigma^{-1}\cr
&CP:\ \Sigma\ \rightarrow\ \Sigma^{*}\cr
}
\end{plain}
\end{equation}
where the operation $*$ refers to complex conjugation.  The eight
meson fields are pseudoscalars.  The neutral pion and the eta meson
are both even under charge conjugation.

With massless quarks, the underlying quark-gluon theory has a chiral
symmetry under
\begin{equation}
\begin{plain}
\eqalign{
&\psi_L\rightarrow \psi_L g_L\cr
&\psi_R\rightarrow \psi_R g_R\cr
}
\end{plain}
\end{equation}   
Here $(g_L,g_L)$ is in $(SU(3)\times SU(3))$ and $\psi_{L,R}$
represent the chiral components of the quark fields, with flavor
indices understood.  This symmetry is expected to be broken
spontaneously to a vector SU(3) via a vacuum expectation value for
$\overline \psi_L \psi_R$.  This motivates the sigma model through the
identification
\begin{equation}
\langle 0 | \overline \psi_L \psi_R | 0 \rangle \leftrightarrow v \Sigma
\label{vec}
\end{equation}
The quantity $v$, of dimension mass cubed, characterizes the strength
of the spontaneous breaking of this symmetry.  Thus our effective
field transforms under the chiral symmetry as
\begin{equation}
\Sigma\rightarrow g_L^\dagger \Sigma g_R
\end{equation} 
Our initial Lagrangian density is the simplest non-trivial expression
invariant under this symmetry.  

The quark masses break the chiral symmetry explicitly.  From the
analogy in Eq.~(\ref{vec}), these are introduced through a 3 by 3 mass
matrix $M$ appearing in a potential term added to the Lagrangian
density
\begin{equation}
L= L_0 - v {\rm Re\ Tr}(\Sigma M)
\end{equation} 
Here $v$ is the same dimensionful factor appearing in Eq.~(\ref{vec}).
The chiral symmetry of our starting theory shows the physical
equivalence of a given mass matrix $M$ with a rotated matrix
$g_R^\dagger M g_L$.  Using this freedom we can put the mass matrix
into a standard form.  I will assume it is diagonal with increasing
eigenvalues
\begin{equation}
M=\pmatrix{ 
m_u & 0 & 0 \cr
0 & m_d & 0 \cr
0 & 0 & m_s \cr
}
\end{equation} 
representing the up, down, and strange quark masses.  Note that this
matrix has both singlet and octet parts under the vector flavor
symmetry
\begin{equation}
M={m_u+m_d+m_s\over 3}
+{m_u-m_d\over 2}\ \lambda_3+{m_u+m_d-2 m_s\over 2\sqrt 3}\ \lambda_8
\end{equation} 

In general the mass matrix can still be complex.  The chiral symmetry
allows us to move phases between the masses, but the determinant of
$M$ is invariant and physically meaningful.  Under charge conjugation
the mass term would only be invariant if $M=M^*$.  If $|M|$ is not
real, then its phase is the famous CP violating parameter usually
associated with topological structure in the gauge fields.  For the
moment I take all quark masses as real.  Since I am looking for
spontaneous CP violation, I consider the case where there is no
explicit CP violation.

To lowest order the pseudoscalar meson masses appear on expanding the
mass term quadratically in the meson fields.  This generates an
effective mass matrix for the eight mesons
\begin{equation}
{\cal M}_{\alpha\beta}\ \propto\ {\rm Re\ Tr}\ \lambda_\alpha\lambda_\beta M
\end{equation}
The isospin breaking up-down mass difference gives this matrix an off
diagonal piece mixing the $\pi_0$ and the $\eta$
\begin{equation}
{\cal M}_{3,8}\ \propto\ m_u-m_d
\end{equation}
The eigenvalues of this matrix give the standard mass relations
\begin{equation}
\label{mesons}
\begin{plain}
\eqalign{
 m_{\pi_0}^2 \propto\ & {2\over 3} \bigg(m_u+m_d+m_s
-\sqrt{m_u^2+m_d^2+m_s^2-m_um_d-m_um_s-m_dm_s}\bigg)\cr
m_{\pi_+}^2= &\  m_{\pi_-}^2\propto m_u+m_d\cr
m_{K_+}^2= &\ m_{K_-}^2\propto m_u+m_s\cr
m_{K_0}^2= &\ m_{\overline K_0}^2\propto m_d+m_s\cr
m_{\eta}^2 \propto &\ {2\over 3} \bigg(m_u+m_d+m_s
+\sqrt{m_u^2+m_d^2+m_s^2-m_um_d-m_um_s-m_dm_s}\bigg)\cr
}
\end{plain}
\end{equation}  
Here I label the mesons with their conventional names.

Redundancies in these relations test the validity of the model.  For
example, comparing two expressions for the sum of the three quark
masses
\begin{equation}
{2(m_{\pi_+}^2+m_{K_+}^2+m_{K_0}^2)
\over 3(m_{\eta}^2+m_{\pi_0}^2)} \sim 1.07
\end{equation}  
suggests the symmetry should be good to a few percent.  Further ratios
of meson masses then give estimates for the ratios of the quark
masses \cite{km,weinberg,leutwyler}.  For one such combination, look at
\begin{equation}
{m_u\over m_d}=
{
m_{\pi^+}^2+m_{K_+}^2-m_{K_0}^2\over
m_{\pi^+}^2-m_{K_+}^2+m_{K_0}^2}\sim 0.66
\end{equation}
This particular combination is polluted by electromagnetic effects;
another combination partially cancels such while ignoring small
$m_um_d/m_s$ corrections
\begin{equation} 
{m_u\over
m_d}= { 2m_{\pi^0}^2-m_{\pi^0}^2+m_{K_+}^2-m_{K_0}^2\over
m_{\pi^+}^2-m_{K_+}^2+m_{K_0}^2}\sim 0.55
\label{updown}
\end{equation} 
Later I will comment on a third combination for this ratio.  For the
strange quark, one can take
\begin{equation}
{2 m_s\over m_u+m_d}=
{
m_{K_+}^2+m_{K_0}^2-m_{\pi^+}^2\over
m_{\pi^+}^2
}\sim 26
\end{equation}

\section{Spontaneous CP violation}
\label{diagram}
So far all this is standard.  Now I vary the quark masses and look for
interesting phenomena.  In particular, I want to find spontaneous
breaking of the CP symmetry.  Normally the $\Sigma$ field fluctuates
around the identity in SU(3).  However, for some values of the quark
masses this ceases to be true.  When the vacuum expectation of
$\Sigma$ deviates from the identity, some of the meson fields will
acquire expectation values.  As they are pseudoscalars, this
necessarily involves a breakdown of parity.

To explore this possibility, I concentrate on the lightest meson from
Eq.~(\ref{mesons}), the $\pi_0$.  From Eq.~(\ref{mesons}) we can
calculate the product of the $\pi_0$ and $\eta$ masses
\begin{equation}
\label{prod}
m_{\pi_0}^2m_\eta^2
\ \propto\ m_um_d+m_um_s+m_dm_s.
\end{equation}
Whenever
\begin{equation}
\label{boundary}
m_u={-m_sm_d\over m_s+m_d}
\end{equation}
the lowest order chiral relation gives a vanishing $\pi_0$ mass.  For
increasingly negative up-quark masses, our simple expansion around
vanishing pseudoscalar meson fields fails.  The vacuum is then no
longer represented by $\Sigma$ fluctuating around the unit matrix.
Instead it fluctuates about an SU(3) matrix of form
\begin{equation}
\Sigma=\pmatrix{
e^{i\phi_1}&0&0\cr
0&e^{i\phi_2}&0\cr
0&0&e^{-i\phi_1-i\phi_2}\cr
}
\end{equation}
where the phases satisfy
\begin{equation}
m_u \sin(\phi_1)=m_d\sin(\phi_2)=-m_s\sin(\phi_1+\phi_2)
\end{equation}
There are two minimum action solutions, differing by flipping the
signs of these angles.  The transition is a continuous one, with
$\Sigma$ going smoothly to the identity as the boundary given by
Eq.~(\ref{boundary}) is approached.

In the new vacuum the pseudoscalar meson fields acquire expectation
values.  As the neutral pion is CP odd, we spontaneously break this
symmetry.  This will have various experimental consequences, for
example eta decay into two pions becomes allowed since a virtual third
pion can be absorbed by the vacuum.  Fig.~(\ref{fig:phasediagram})
shows the inferred phase diagram as a function of the up and down
quark masses.  Chiral rotations insure a symmetry under the flipping
of the signs of both quark masses.

\begin{figure*}
\centering
\includegraphics[width=3in]{fig1.eps}
\caption{\label{fig:phasediagram} The phase diagram of quark-gluon
dynamics as a function of the two lightest quark masses.  The shaded
region exhibits spontaneous CP breaking.  The diagonal lines with
$m_u=\pm m_d$ trace where we have three degenerate pions due to
isospin symmetry.  The neutral pion mass vanishes on the boundary of
the CP violating phase.}
\end{figure*}

At first sight the appearance of such a phase at negative up quark
mass seems surprising.  Naively in perturbation theory the sign of a
fermion mass can be rotated away by a redefinition $\psi\rightarrow
\gamma_5 \psi$.  However this rotation is anomalous, making the sign
of the quark mass observable.  A more general complex phase in the
mass would also have physical consequences, i.e.  explicit CP
violation.  With real quark masses the underlying Lagrangian is CP
invariant, but there exists a large region where the ground state
spontaneously breaks this symmetry.

Vafa and Witten \cite{vafawitten} argued on rather general conditions
that CP could not be spontaneously broken in the strong interactions.
However their argument makes positivity assumptions on the path
integral measure.  When a quark mass is negative, the fermion
determinant need not be positive for all gauge configurations; in this
case the assumptions fail.

The possible existence of this phase was anticipated on the lattice
some time ago by Aoki \cite{aoki}.  For the one flavor case he found
this parity breaking phase with Wilson lattice gauge fermions.  He
went on to discuss also two flavors, finding both flavor and parity
symmetry breaking.  This case is now regarded as a lattice artifact of
Wilson fermions.  For a review of these issues see \cite{myreview}.

In conventional discussions of CP non-invariance in the strong
interactions \cite{witten} appears a phase $e^{i\theta}$ appearing on
tunneling between topologically distinct gauge field configurations.
The famous U(1) anomaly formally allows us to move this phase into the
determinant of the quark mass matrix.  After then rotating all phases
into the up quark, we see that our spontaneous breaking of CP is
occurring at an angle $\theta=\pi$.  It does not, however, occur for
up quark masses greater than a non-zero minimum value.  There exists a
finite region with $\theta=\pi$ that does not undergo this symmetry
breaking.  The chiral model indicates a smooth behavior in the quark
mass when it is in the vicinity of zero.  Indeed, from the effective
Lagrangian point of view, the real and imaginary parts of the quark
mass are completely independent parameters.  The absence of
experimental evidence for strong CP violation suggests that the
imaginary part of the quark mass matrix vanishes, but says nothing
about the real part.

An interesting special case occurs when the up and down quarks have
the same magnitude but opposite sign for their masses, i.e.
$m_u=-m_d$.  In this situation it is illuminating to rotate the minus
sign into the phase of the strange quark.  Then the up and down quark
are degenerate, and we have restored an exact vector $SU(2)$ flavor
symmetry.  The excitation spectrum will show three degenerate pions,
but they will not be massless due to what might be thought of a vacuum
condensate of eta particles.

\section{Including the $\eta^\prime$}
\label{etaprime}

The above discussion was entirely in terms of the light pseudoscalar
mesons that become Goldstone bosons in the chiral limit.  One might
wonder how higher states can influence this phase structure.  Of
particular concern is the $\eta^\prime$ meson associated with the
anomalous $U(1)$ symmetry present in the classical quark-gluon
Lagrangian.  Non-perturbative processes, including topologically
non-trivial gauge field configurations, are well known to generate a
mass for this particle.  I will now argue that while this state can
shift masses due to mixing with the lighter mesons, it does not make a
qualitative difference in the existence of a phase with spontaneous CP
violation.

The easiest way to introduce the $\eta^\prime$ into the effective
theory is to promote the group element $\Sigma$ to an element of
$U(3)$ via an overall phase factor.  Thus I generalize
Eq.~(\ref{sigma}) to
\begin{equation}
\Sigma=\exp(i\pi_\alpha \lambda_\alpha/f_\pi+i\eta^\prime/f_\pi)
\in U(3)
\end{equation}
Our starting kinetic Lagrangian in Eq.~(\ref{kinetic}) would have this
particle also be massless.  One way to fix this deficiency is to mimic the
anomaly with a term proportional to the determinant of $\sigma$
\begin{equation}
L_0={f_\pi^2\over 4}{\rm Tr}(\partial_\mu \Sigma^\dagger \partial_\mu \Sigma)
-C|\Sigma|.
\end{equation}
The parameter $C$ parameterizes the strength of the anomaly in the
$U(1)$ factor.

Now if we include the mass term exactly as before, additional mixing occurs
between the $\eta^\prime$, the $\pi_0$, and the $\eta$.  The
corresponding mixing matrix takes the form
\begin{equation}
\pmatrix{
m_u+m_d & {m_u-m_d \over \sqrt 3} & m_u-m_d \cr
{m_u-m_d \over \sqrt 3} & {m_u+m_d+4m_s \over 3} 
          & {m_u+m_d-2m_s\over \sqrt 3}\cr
 m_u-m_d& {m_u+m_d-2m_s\over \sqrt 3} & m_a \cr
}
\end{equation}
where $m_a$ characterizes the contribution of the non-perturbative
physics to the $\eta^\prime$ mass.  This should have a value of order
the strong interaction scale; in particular, it should be large
compared to at least the up and down quark masses.  The two by two
matrix in the upper left of this expression is exactly what is
diagonalized to find the neutral pion and eta masses in
Eq.~(\ref{mesons}).

The boundary of the CP violating phase occurs where the determinant of
this matrix vanishes.  This modifies Eq.~(\ref{prod}) to
\begin{equation}
m_{\pi_0}^2m_\eta^2 m_{\eta^\prime}^2
\ \propto\ 
M (m_um_d+m_um_s+m_dm_s)
-m_u(m_d-m_s)^2-m_d(m_u-m_s)^2-m_s(m_u-m_d)^2
\end{equation}
The boundary shifts slightly from the earlier result, but still passes
through the origin leaving Fig.~(\ref{fig:phasediagram}) qualitatively
unchanged.

\section{Can the up quark be massless?}
\label{mup}
A oft proposed solution to the strong CP problem
\cite{leutwyler0, cohen, fleming}
asks whether $m_u=0$.  From the effective Lagrangian point of view,
this appears to be an artificial setting of two parameters to zero,
the real and imaginary parts of the quark mass.  It is only the
imaginary part that should vanish for CP to be a good symmetry, at
least when the up quark mass is larger than the value giving
spontaneous breaking.

While phenomenology, i.e. Eq.~(\ref{updown}), seems to suggest that
the up quark is not massless, there remains a lot of freedom in
extracting that ratio from the pseudoscalar meson masses.  From
Eq.~(\ref{mesons}), the sum of the $\eta$ and $\pi_0$ masses squared
should be proportional to the sum of the three quark masses.
Subtracting off the neutral kaon mass should leave just the up quark.
Thus motivated, look at
\begin{equation}
{m_u\over m_d}=
{
 3(m_{\eta}^2+m_{\pi_0}^2)/2-2m_{K_0}^2
\over
 m_{\pi^+}^2-m_{K_+}^2+m_{K_0}^2
}
\sim-0.8
\end{equation}
Thus even the sign of the up quark mass is ambiguous.  Attempts to
extend the naive quark mass ratio estimates to higher orders in the
chiral expansion have shown that there are fundamental ambiguities in
the definition of the quark masses \cite{km}.  But speculations on a
vanishing up quark mass continue, so it is interesting to ask if this
has physical meaning.  In this section I investigate precisely what is
meant by a quark mass, and what $m_u=0$ would mean.

If two quark masses were to vanish simultaneously, then we would have
exactly massless pions, Goldstone bosons for the resulting exact
flavored chiral symmetry.  Here I concentrate on whether the concept
of a single massless quark has any meaning.  While I could carry along
the baggage of the heavier quarks, let me simplify the discussion and
consider the theory reduced to a single flavor of quark.  I will
conclude that the question of whether $m_u$ could vanish is not well
posed.

Because renormalization is required, the concept of an ``underlying
basic Lagrangian'' does not exist in quantum field theory.  Instead
there are some basic underlying symmetries, and the continuum theory
is defined in terms of those and a few renormalized parameters.  In
practice this must be carried out as a limiting process on a cutoff
version of the theory.  As the lattice is the only well understood
non-perturbative cutoff, it provides the most natural framework for
such a definition.  But any regulator must accomodate the known chiral
anomalies, and thus there must be chiral symmetry breaking terms in
the cutoff theory.  These chiral breaking effects come in many guises.
With a Pauli-Villars scheme, there is a heavy regulator field.  With
dimensional regularization the anomaly is hidden in the fermionic
measure.  For Wilson lattice gauge theory there is the famous Wilson
term.  With domain wall fermions there is a residual mass from a
finite fifth dimension.  With overlap fermions things are hidden in a
combination of the measure and a certain non-uniqueness of the
operator.  I will return to this last case shortly.

The renormalization process tunes all relevant bare parameters as a
function of the cutoff to fix a set of renormalized quantities.  In
the case of the strong interactions, the bare gauge coupling is driven
to zero by asymptotic freedom.  Its cutoff dependence is absorbed into
an overall scale via the phenomenon of dimensional transmutation
\cite{cw}.  The only other parameters of the strong interactions are
the quark masses.  For these one inputs a few particle masses to
finally determine the continuum theory uniquely.  For the three flavor
theory the most natural observables to fix these parameters are the
pseudoscalar meson masses.

In the one flavor theory there are no Goldstone bosons, but massive
mesons and baryons should exist.  I need some physical parameter with
which to carry out the renormalization of the quark mass.  For this
purpose I choose the ratio of the lightest boson mass to the lightest
baryon mass.  As both are expected to be stable, this precludes any
ambiguity from particle widths.  Calling the lightest boson the $\eta$
and the baryon $p$ I define
\begin{equation}
r={m_\eta \over m_p}
\end{equation} 
I expect to be able to adjust this parameter via the quark mass, which
should be tuned to give the desired value.  It should be possible to
give this ratio any value throughout the range from $r=0$ at the
boundary of the above CP violating phase to $2/3$ in the heavy quark
limit.

With the lattice cutoff in place, I can in principle determine this
ratio given any values for the bare quark mass and bare coupling.  For
pedagogy, let me trade these parameters for the lattice spacing $a$
and the quark mass in lattice units, $m_q a$.  Both of these
quantities go to zero in the continuum limit.  The renormalization
prescription is to select a desired value of $r$ and follow the
contour with this value in the $(a,m_q)$ plane towards the origin.
This process is sketched in Fig.~(\ref{limit}).  Perturbative
divergences in the bare quark mass appear in the fact that these
contours approach the origin with zero slope.

\begin{figure*}
\centering
\includegraphics[width=4in]{fig2.eps}
\caption{
\label{limit}
Defining the continuum limit.  For one flavor strong interactions I
consider the ratio of the lightest boson to lightest baryon masses as
my renormalized parameter.  With the cutoff in place, we flow towards
the origin along a curve of constant renormalized quantity.  Below the
contour where this ratio vanishes lies the region of spontaneous CP
violation.  }
\end{figure*} 

When the lattice spacing is non-zero, we expect artifacts to vary
between different lattice prescriptions.  In particular, the precise
locations of the constant $r$ contours will vary between different
formulations.  Holding the bare quark mass at zero will cross a
variety of $r$ contours, with none obviously favored as the origin is
approached.  Different cutoff schemes will give different continuum
limits for $m_u=0$, and thus asking that the up quark mass vanishes
is physical.

With two or more degenerate flavors there will be one special contour
where the lightest meson does represent a Goldstone boson.  With the
Wilson fermion formulation, the quark mass axis is represented by the
hopping parameter.  As this cutoff breaks chiral symmetry, the
critical hopping parameter, where the meson mass vanishes, is
renormalized away from its value in the continuum limit.

Recently there has been considerable progress with lattice fermion
formulations that preserve a remnant of exact chiral symmetry
\cite{myreview}.  With such, the two flavor theory will have the $r=0$
contour preserved as the $m_q=0$ axis.  However, for the one flavor
theory, this axis will still be expected to cut through various values
of $r$.  An interesting question is whether as we take the lattice
spacing to zero along this axis, some physical value of $r$ will be
picked out as special and corresponding to vanishing quark mass.  That
this is unlikely follows from the non-uniqueness of these chiral
lattice operators.  For example, the overlap operator \cite{overlap}
is constructed by a projection process from the conventional Wilson
lattice operator.  The latter has a mass parameter which is to be
chosen in a particular domain.  On changing this parameter, we still
have a good symmetry in the sense of the Ginsparg-Wilson relation
\cite{gw}, but the contours of constant $r$ in Fig.~(\ref{limit}) will
be expected to shift around.  Thus the horizontal axis is not expected
to select one contour as special.  Again, holding $m_u=0$ is not
expected to give a unique continuum theory.

\section{Final remarks}
\label{final}

While I have been exploring rather unphysical regions in parameter
space, these observations do raise some issues for practical lattice
calculations of hadronic physics.  Current simulations are done at
relatively heavy values for the quark masses.  This is because the
known fermion algorithms tend to converge rather slowly at light quark
masses.  Extrapolations by several tens of MeV are needed to reach
physical quark masses, and these extrapolations tend to be made in the
context of chiral perturbation theory.  The presence of a CP violating
phase quite near the physical values for the quark masses indicates a
strong variation in the vacuum state with a rather small change in the
up quark mass; indeed, less than a 10 MeV change in the traditionally
determined up quark mass can drastically change the low energy
spectrum.  Most simulations consider degenerate quarks, and chiral
extrapolations so far have been quite successful.  But some
quantities, namely certain baryonic properties \cite{thomas}, do seem
to require rather strong variations as the chiral limit is approached.
These effects and the strong dependence on the up quark mass may be
related.

Another issue is the validity of current simulation algorithms with
non-degenerate quarks.  With an even number of degenerate flavors the
fermion determinant is positive and can contribute to a measure for
Monte Carlo simulations.  With light non-degenerate quarks the
positivity of this determinant is not guaranteed.  Indeed, the CP
violation can occur only when the fermions contribute large phases to
the path integral.  Current algorithms for dealing with non-degenerate
quarks take a root of the determinant with multiple flavors.  In this
process any possible phases are dropped.  Such an algorithm is
incapable of seeing any of the CP violating phenomena discussed here.
This point may not be too serious in practice since the up and down
quarks are nearly degenerate and the strange quark is fairly heavy.
But these issues should serve as a warning that things might not work
as well as we want.

\section*{Acknowledgements}
I have benefited from lively discussions with many colleagues, in
particular F. Berruto, T. Blum, H. Neuberger and G. Senjanovic.  This
manuscript has been authored under contract number DE-AC02-98CH10886
with the U.S.~Department of Energy.  Accordingly, the U.S. Government
retains a non-exclusive, royalty-free license to publish or reproduce
the published form of this contribution, or allow others to do so, for
U.S.~Government purposes.

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\end{document}








