\documentstyle[aps,twocolumn]{revtex}
\begin{document}
\newcommand{\bb}{\begin{eqnarray}}
\newcommand{\ee}{\end{eqnarray}}
\wideabs{
\title{Is there still a strong CP problem?}
\author{H. Banerjee,} 
\address{BF-79 Salt Lake, Calcutta 700064}
\author{D. Chatterjee}
\address{Vijaygarh Jyotish Ray College, Calcutta 700032}
\author{and}
\author{P. Mitra}
\address{Saha Institute of Nuclear Physics, 1/AF Bidhannagar,
Calcutta 700064}
\date
\maketitle
\begin{abstract}{A cloud of mystery surrounds the r\^{o}le 
of a chiral U(1) phase in the quark mass in QCD. In operator formulation, 
parity is a symmetry and the phase is easily removed by a mere change in 
the representation of the Dirac $\gamma$ matrices. Moreover,  
these properties are also realized in
a Pauli-Villars regularized version of the theory.
On the other hand, in the {\it popular}
functional integral scenario, attempts to remove the chiral phase by a chiral
transformation are frustrated by a nontrivial Jacobian arising from
the fermion measure and the chiral phase seems to have the potential to
break parity. The nontrivial Jacobian is shown to be
the direct fallout of an unregularized
action, an essential ingredient in the popular scenario.  The Jacobian becomes trivial 
if one starts from the regularized action symmetric under parity in the 
presence of the chiral phase, which can now be removed with impunity by chiral
rotation. The mystery melts away leading to a solution of 
the strong CP problem.}
\end{abstract}
}
\section{Introduction}
The strong interactions do not violate parity in the manner of the weak
interactions: the gauge interactions in quantum chromodynamics involve vector
currents instead of chiral currents. Experiments too do not indicate
CP violation in the QCD sector. However, if one writes the Lagrangian
density as
\bb
{\cal L}&=&\bar\psi(i\gamma^\mu D_\mu-me^{i\theta'\gamma_5})\psi
-\frac{1}{4}~{\rm tr}~F_{\mu\nu}F^{\mu\nu}\nonumber\\
&&-n_f{g^2\theta\over
32\pi^2}~{\rm tr}~F_{\mu\nu}\tilde F^{\mu\nu},
\label{L}\ee
it contains the QCD vacuum angle $\theta$, which violates CP and, if
nonzero, may lead to experimentally detectable CP violating effects.
More important is the presence in the quark mass term of the chiral phase
$\theta'$, which comes from the electroweak sector and may be large ($\approx 1$).
It is considered unnatural that CP violation due to $\theta'$
should be cancelled by that due to the $\theta$ term. 
In the {\it popular} functional integral scenario, the phase $\theta'$
is supposed to be convertible into a $\theta$-like term
through anomalous chiral rotations of the quarks, so that there is
only one effective parameter $\bar\theta\equiv\theta-\theta'$
and according to {\it popular} methods of calculation \cite{peccei} the CP-violating
electric dipole moment of the neutron is supposed to be about $10^{-16}\bar\theta$
e-cm, to be compared with the experimental upper bound of $10^{-26}$ e-cm.
This is supposed to mean that $$|\bar\theta|{\stackrel{(?)}{<}}10^{-10},$$ 
requiring a staggering amount of fine-tuning.
This is the strong CP problem \cite{peccei}.
Various models have been proposed to avoid this supposed need
for fine tuning. But the strong
CP problem persists and is very much alive a quarter of a century after its birth
\cite{peccei}.

We seek to reanalyse the problem from first principles, and as the $\theta$
term is difficult to handle, 
we distinguish between $\theta$ and $\theta'$ and concentrate
on parity violation generated from the $\theta'$ term. 
We consider here a single flavour for simplicity. This means that $m$
in (\ref{L}) is a (real) number instead of a matrix. An extension can be made to
more quark flavours without difficulty.

In first order, $\theta'$ appears to be involved in a pseudoscalar term, and
its matrix element has sometimes been thought \cite{crewther,aoki} to be a measure of
the CP violation produced by the term. However, this is illusory.
In theories like QCD with chirally invariant interactions, perturbation theory cannot
find any effect of such phases as they cancel at each vertex \cite{bcm}.
Thus the quark propagator can be written as
\bb
(\gamma^\mu p_\mu -me^{i\theta'\gamma_5})^{-1}%\nonumber\\
=e^{-i{\theta'\over 2}\gamma_5}
(\gamma^\mu p_\mu -m)^{-1} e^{-i{\theta'\over 2}\gamma_5},
\ee
and the chiral phases formally cancel at each gauge vertex because
\bb
e^{-i{\theta'\over 2}\gamma_5} \gamma^\mu e^{-i{\theta'\over 2}\gamma_5}=\gamma^\mu.
\ee
Moreover, in spite of the presence of the
chiral phase in the quark mass term, parity can be defined
so as to be conserved at the classical level.
This parity transformation, which leaves
the fermionic part of the Lagrangian $\int d^3x {\cal L}_\psi$ invariant
\cite{bcm}, involves the
usual parity operation for gauge fields, while the operation for fermions
includes a chiral rotation:
\bb
\bar\psi(x_0,\vec x)&\rightarrow &\bar\psi(x_0,-\vec x)
e^{i\theta'\gamma_5}\gamma^0\nonumber\\
\psi(x_0,\vec x)&\rightarrow &
\gamma^0e^{i\theta'\gamma_5}\psi(x_0,-\vec x).
\label{parity}\ee
The full mass term, with the chiral phase, is a scalar under this parity.

The question of physicality or unphysicality of $\theta'$ is intimately tied with
the chiral anomaly which arises from short distance singularities and may affect
the perturbative $\theta'$-independence as well as the
conservation of the charge generating the transformation (\ref{parity}).
It has to be examined carefully whether the apparent properties
of the theory survive regularization. 

The popular belief in the physicality of the $\theta'$ term and its equivalence with
a vacuum angle like $\theta$ arises mainly because of a nontrivial
Jacobian supposed to be produced by a spacetime independent chiral
transformation in the euclidean functional integral. For a chiral transformation
\bb
\bar\psi\rightarrow\bar\psi e^{i\alpha(x)\gamma_5},\quad
\psi\rightarrow e^{i\alpha(x)\gamma_5}\psi,
\label{chi}\ee
where $\alpha$ may depend on $x$, the Jacobian reads
\bb
J=%e^{i\int d^4x\sum_n\phi^\dagger_n\alpha(x)\gamma_5\phi_n}\equiv 
e^{i\int d^4x\alpha(x)X(x)},
\ee
with \cite{fujikawa}
\bb
X(x)={g^2\over 16\pi^2}{\rm tr} F_{\mu\nu}\tilde{F}_{\mu\nu}.
\ee
$X(x)$ is identified as the chiral anomaly \cite{fujikawa}
from the anomalous Ward identity
\bb
\langle\partial_\mu (\bar\psi(x)\gamma_\mu\gamma_5\psi(x))\rangle=
-2\langle\bar\psi m%e^{i\theta'\gamma_5} 
\gamma_5\psi\rangle+X(x),
\label{ward}\ee
which follows from the golden rule that the functional integral over
$\psi,\bar\psi$ cannot change under a chiral rotation of integration variables.

If $\alpha(x)=-\theta'/2$, the chiral phase gets removed from the mass term,
but the Jacobian causes a parity violating vacuum angle term to be added to the action:
\bb
Z^{[\theta']}&\equiv&\int d\psi d\bar\psi e^{-\int d^4x \bar\psi(\gamma^\mu 
D_\mu-me^{i\theta'\gamma_5})\psi}\nonumber\\
&=&Z^{[0]}e^{-i\theta'{g^2\over 32\pi^2}
\int d^4x {\rm tr} F_{\mu\nu}\tilde{F}_{\mu\nu}}.
\label{z}\ee

Unlike the perturbative argument or the parity constructed above, this
functional integral
argument is thought to be robust because it takes the anomaly into account.
However, to settle the issue, it is necessary to consider a {\it regularized} theory.
We shall first show that a Pauli-Villars type regularization
is compatible with parity defined in (\ref{parity}). 
This means that $\theta'$ does {\it not}
cause any violation of parity even in the quantum theory. 
We shall then examine
the euclidean functional integral of the {\it regularized}
theory, where it will turn out that an $F\tilde F$ term is {\it not} generated
if the phase $\theta'$  is rotated away: the Jacobian is trivial in spite 
of the existence of the chiral anomaly, 
which is popularly thought to be responsible for CP violation \cite{peccei,aokiplus}. 
%The $\theta'$ term can thus again be seen not to cause any parity violation at all.
This provides a mechanism for solving the strong CP problem
in a natural way by setting $\theta=0$. 



\section{Parity in regularized QCD in presence of chiral phase}
The easiest proof of the unphysicality of $\theta'$ is in the operator 
formulation of quantum chromodynamics.
Instead of an anomalous chiral transformation of the quark fields,
a change of $\gamma$-matrices can also be used to remove the chiral phase $\theta'$.
Using the relation
\bb
\bar\psi=\psi^\dagger\gamma^0,
\ee
one can write the spinor part of Eq. (\ref{L})  as
\bb
{\cal L}_\psi^{[\theta']}=\psi^\dagger(i\gamma^0\gamma^\mu D_\mu-
m\gamma^0e^{i\theta'\gamma_5})\psi,
\ee
where $D_\mu,m,\theta'$ carry no spinorial indices. It can also be rewritten as
\bb
{\cal L}_\psi^{[\theta']}=\psi^\dagger(i\widetilde{\gamma^0}\widetilde{\gamma^\mu}
D_\mu-m \widetilde{\gamma^0})\psi,
\ee
where
\bb
\widetilde{\gamma^\mu}\equiv e^{-i\theta'\gamma_5/2}{\gamma^\mu}
e^{i\theta'\gamma_5/2}.
\ee
It is clear that the new
matrices satisfy the Dirac algebra and
also have the same hermiticity properties as their parent matrices.
Thus the chiral phase can be absorbed in a simple
redefinition of $\gamma$-matrices, and can have no physical effect.
The argument fails if there are Yukawa or other chirally noninvariant interactions.

It is to be noted that this argument does not go through directly in euclidean field
theory, where $\bar\psi$ is taken to be independent of $\psi$. One can
modify the argument so that it holds in euclidean spacetime.
However, it is Minkowski spacetime that one is finally interested in.

The argument skirts the issue of the chiral anomaly arising from
short distance singularities at the quantum level. We therefore need to
examine whether or not it is jeopardized in
the regularized theory.
In the generalized Pauli-Villars regularization,
the Lagrangian density has to be augmented to include some extra species:
\bb
{\cal L}_{\psi,~ reg}^{[0]}&=&
\bar\psi(i\gamma^\mu D_\mu-m
)\psi+\nonumber\\
&&\sum_j\sum_{k=1}^{|c_j|} \bar\chi_{jk}(i\gamma^\mu D_\mu-M_j
)\chi_{jk}.\label{PV}
\ee
Here the $\chi_{jk}$ are regulator spinor fields with 
fermionic or bosonic statistics, which determines 
the signs, positive or negative, of the integers $c_j$ \cite{faddeev};
the $c_j$-s have to satisfy relations
\bb
1+\sum_j c_j=0,\quad m^2+\sum_j c_jM_j^2=0 
\label{c}\ee
to cancel divergences.
The masses $M_j$ 
are supposed to be taken to infinity at the end of calculations.
This regularization is standard, but in the presence of the chiral phase $\theta'$ 
in the physical fermion mass term,
a chiral phase has to be provided to the regulator mass terms as well:
\bb
{\cal L}_{\psi,~ reg}^{[\theta']}&=&
\bar\psi(i\gamma^\mu D_\mu-m
e^{i\theta'\gamma_5})\psi+\nonumber\\
&&\sum_j\sum_{k=1}^{|c_j|} \bar\chi_{jk}(i\gamma^\mu D_\mu-M_j
e^{i\theta'\gamma_5})\chi_{jk}.\label{reg}
\ee
 
All the $\gamma_5$-s in (\ref{reg}) can be removed by changing to the representation
$\widetilde{\gamma^\mu}$ because the same chiral phase has been
chosen for all $j,k$: 
\bb
{\cal L}_{\psi,~ reg}^{[\theta']}&=&
\psi^\dagger\widetilde{\gamma^0}(i\widetilde{\gamma^\mu} D_\mu-m
)\psi+\nonumber\\
&&\sum_j\sum_{k=1}^{|c_j|} \chi_{jk}^\dagger\widetilde{\gamma^0}
(i\widetilde{\gamma^\mu} D_\mu-M_j
)\chi_{jk}.
\ee
This is no more parity violating than (\ref{PV}).
This proves parity conservation and 
the unphysicality of $\theta'$ in 
the regularized theory.  
The parity transformation (\ref{parity}) translates to the usual parity
in terms of the $\widetilde{\gamma}$ representation. 

If (\ref{reg}) were modified by changing the
chiral phases in the regulator sector, it would no longer respect the natural extension 
of (\ref{parity}) to the regulators mentioned above. Such a choice
of phases would be equivalent to choosing nonzero phases for the regulators in the
absence of a phase in the physical mass term and could introduce parity violation
{\it by hand} (cf. \cite{measure})
just as an artificial breaking of rotational
symmetry is induced by a lattice regularization.



\section{Trivial Jacobian for chiral rotation in regularized functional integral}
The preceding arguments for the unphysicality of $\theta'$
are lucid and persuasive enough to suggest that something is seriously amiss
in the anomaly argument leading
to (\ref{z}) in the popular approach. 
It is easy to identify what it is: neither the LHS nor the RHS in (\ref{ward})  
is regularized.
Even the term $X(x)$, representing the anomaly, is calculated \cite{fujikawa}
through an {\it a posteriori} regularization. The genesis of this flaw resides in the
{\it unregularized action} from which the popular approach starts.
In this section, we start instead from the regularized Lagrangian density
(\ref{PV}).
The measure of integration now includes the Pauli-Villars fields: 
\bb d\mu=
d\psi d\bar\psi\prod_{jk} d\chi_{jk} d\bar\chi_{jk},
\ee
The fermionic functional integral is defined by
\bb
Z_{reg}^{[0]}\equiv\int d\mu e^{-\int d^4x {\cal L}_{\psi,~ reg}^{[0]}}
\ee

If the chiral transformation (\ref{chi}) is extended to the regulators, 
the Jacobian factors corresponding to the different $\chi_{jk},\bar\chi_{jk}$
come with powers $c_j/|c_j|$
by virtue of the fermionic or bosonic statistics, 
while $\psi,\bar\psi$ obey fermionic statistics. Altogether,
\bb
J_{reg}=e^{i(1+\sum_jc_j)\int d^4x \alpha(x)X(x)}=1,
\ee
because $1+\sum_j c_j=0$  \cite{fujikawa}.
Thus in the regularized framework the Jacobian is trivial 
and the anomalous axial Ward identity is represented as
\bb
\langle\partial_\mu J_{\mu 5~reg}\rangle=
-2m\langle\bar\psi%e^{i\theta'\gamma_5}
\gamma_5\psi\rangle
-2\sum_{jk}M_j\langle\bar\chi_{jk}%e^{i\theta'\gamma_5}
\gamma_5\chi_{jk}\rangle,
\label{Ward}\ee
where
\bb
 J_{\mu 5~reg}=
\bar\psi(x) \gamma_\mu\gamma_5\psi(x)+
\sum_{jk}\bar\chi_{jk}(x) \gamma_\mu
\gamma_5\chi_{jk}(x).
\ee
Both sides of the Ward identity (\ref{Ward}) are regularized and well-defined.
Note that divergent pieces in the physical mass term are cancelled exactly
by those in the regulator terms. As the regularization is removed,
it is these latter terms which yield the standard chiral anomaly \cite{brown}:
\bb
-2\sum_{jk}\lim_{M_j\to\infty}M_j\langle\bar\chi_{jk}
\gamma_5\chi_{jk}\rangle_{reg}={g^2\over 16\pi^2}{\rm tr} F_{\mu\nu}\tilde{F}_{\mu\nu}.
\label{anomaly}\ee

Because of the trivial Jacobian associated with a chiral rotation, we see
that the fermionic functional integral
\bb
Z_{reg}^{[\theta']}\equiv\int d\mu e^{-\int d^4x {\cal L}_{\psi,~ reg}^{[\theta']}}
\ee
can be converted to the integral with $\theta'$ rotated away:
\bb
Z_{reg}^{[\theta']}=Z_{reg}^{[0]},
\ee
{\it i.e.,} the regularized theories with and without $\theta'$ are equivalent.
In other words, the chiral phase $\theta'$ is completely unphysical.
%\hfill{$\Box$}

It is of some interest to note that in the original work 
of Peccei and Quinn (see \cite{peccei}) 
the result (\ref{z}) was motivated not from a nontrivial Jacobian 
as in the current popular
approach \cite{fujikawa} but from an assumed nonzero value of the spacetime
integral of the divergence of the axial current. In the regularized approach,
however,
\bb
\int d^4x\langle\partial_\mu J_{\mu 5~reg}\rangle=0,
\ee
irrespective of whether or not the gauge field has an instanton-like
configuration, subject only to the proviso that for massless quarks
one takes the zero mass limit only at the end of calculations \cite{hb}. In the present
context, however, where one is concerned with the physicality of a nonzero
chiral phase $\theta'$ in the quark mass, the latter condition is obviously
irrelevant. The above equation follows directly from (\ref{Ward})
and the observation that each term on its RHS is independent of the mass
%as the integral of the RHS of (\ref{Ward}) vanishes 
\cite{brown}:
\bb
{\partial\over\partial m^2}[2m\int d^4x\langle\bar\psi\gamma_5\psi\rangle_{reg}]=0.
\ee
The spacetime integral of the RHS of (\ref{anomaly}), {\it viz.}, %is twice the winding
%number $\nu$ of the gauge field configuration:
\bb
\int d^4x {g^2\over 16\pi^2}{\rm tr} F_{\mu\nu}\tilde{F}_{\mu\nu}=2\nu,
\ee
is cancelled by the spacetime integral \cite{hb} of the first term on the RHS
of (\ref{Ward}), {\it i.e.,}
\bb
-\int d^4x [2m\langle\bar\psi\gamma_5\psi\rangle_{reg}]=2(n_--n_+),
\ee
by virtue of the index theorem
\bb
n_+-n_-=\nu.
\ee



The unphysicality of the chiral phase $\theta'$ proved earlier in operator
formulation thus stands vindicated in the framework of functional integrals
in the euclidean metric.
The lesson of the above exercise in the Pauli-Villars scheme is that a
regularized axial Ward identity signals a trivial Jacobian and conversely,
a non-trivial Jacobian signals an unregularized version of the Ward identity.
The chiral anomaly arises in different ways in regularized and
unregularized approaches, but is the same in both cases. 

\section{Conclusion}
Let us summarize the proofs presented in this letter.
One proof works in Minkowski spacetime and
involves a change of $\gamma$-matrices.  As is widely known, no
representation of $\gamma$-matrices is more sacred than others, and if something
can be removed by a mere change of representation, it cannot be physically
observable. Another way of understanding this is to see that while the usual parity
operation involves $\gamma^0$, the presence of a chiral phase in the
mass term changes it to $\widetilde{\gamma^0}$.
This symmetry transformation involves a chiral transformation of the 
fermion fields, which alerts us to the possibility of an anomaly. 
We have used a generalized Pauli-Villars regularization 
to explore this possibility. It does not vitiate or modify 
the above observations, showing that there is no difficulty.
A different approach involves a chiral transformation in which the chiral
phases in the quark and regulator mass terms are rotated away.
The measure of the regularized euclidean functional integral
does not change, and no $F\tilde F$ term gets generated,
so that the theories with and without $\theta'$ are equivalent.

It is not difficult to understand why $\theta'$ was so long thought to
lead to CP violation. The usual functional integral formulation
chose to ignore the regularization of the action and settled for a
regularization of the measure, which led to a $\theta'$-dependent Jacobian and
added to the $F\tilde F$ term.
A regularized action 
ensures that the functional integrals are well-defined and
yields a trivial Jacobian. 
It may be noted that it is not enough to regularize the action: one has to
see that parity violation is not introduced by hand through a bad
choice of phases in the regulator sector.

Experimental signatures of CP violation in the strong interactions are lacking.
CP violation is usually believed to get a contribution from the chiral phase
$\theta'$ of the quark mass term as well as the $\theta$ term in the gauge sector
and there was therefore a question of fine tuning
between the $\theta'$ and $\theta$ terms.
This was the strong CP problem. 
The present work demonstrates that the $\theta'$ term produces no
CP violation. Thus one may forget $\theta'$ and fine tuning. 
What can one say about the $\theta$ term?
This term changes sign under the standard parity transformation of gauge
fields, and there does not appear to be any redefinition of parity which
can restore the symmetry. However, this term is known to be an exact
divergence, and does not have any effect in perturbation theory. Its effect
has usually been estimated \cite{crewther} by conversion to a $\theta'$ term. Since such
a conversion is now seen to be impossible in a regularized theory, this method
has to be recognized to be meaningless. 
One may look for more reliable ways of
estimating CP violating quantities %like the electric dipole moment of the neutron 
on the basis of a $\theta$ term.
Meanwhile, even without knowing how parity-violating $\theta$ is, one can 
solve the strong CP problem simply by setting $\theta=0$ in
the $\theta F\tilde F$ term. This is no doubt a special value, but as
this choice increases the symmetry of the action, it deserves to be called a
{\it natural} choice.
%Of course, if future experiments do indicate parity violation in the strong interactions,
%it could be explained as being
%due to a nonzero $\theta F\tilde F$ term. Even the chiral phase
%could be used to manufacture parity violation through Yukawa interactions of
%quarks, for our proofs of the unphysicality of $\theta'$ apply only to
%chirally invariant interactions.



\begin{thebibliography}{99}

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R. Peccei and H. Quinn, Phys. Rev. Letters {\bf 38}, 1440 (1977). 
\bibitem{crewther} R.J. Crewther, P. Di Vecchia, G. Veneziano and
E. Witten, Phys.  Lett. {\bf 88B}, 123 (1979).
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{\bf 63}, 1125 (1989)
\bibitem{bcm} H. Banerjee, D. Chatterjee and P. Mitra, 
"Is there a strong CP problem?", 
SINP/TNP/90-5 (1990); incorporated into Sec. V of
%H. Banerjee, D. Chatterjee and P. Mitra,
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\bibitem{fujikawa} K. Fujikawa, Phys. Rev. {\bf D21}, 2848 (1980).
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\bibitem{faddeev} See, for example, L. Faddeev and A. Slavnov, {\it Gauge
Fields}, Benjamin-Cummings, Reading (1980).
\bibitem{measure} P. Mitra, %"Understanding CP violation in lattice QCD", 

%"Chiral phase in fermion measure and the resolution of the strong CP problem", 

\bibitem{brown} L. Brown, R. Carlitz and C. Lee,
Phys Rev. {\bf D16}, 417 (1977).
\bibitem{hb} H. Banerjee,  in "Quantum Field Theory: A twentieth century
profile" ed. A. N. Mitra,
(Indian National Science Academy, New Delhi, 2000) 

\end{thebibliography}
\end{document}


