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\vspace{1.5in}

\centerline{\fone On the Strong CP-Problem without Axion Field}
\vspace{ 0.3in}

\centerline{\ftwo G\'abor Etesi}
\vspace{0.1in}

\centerline{Alfr\'ed R\'enyi Institute of Mathematics,}
\centerline{Hungarian Academy of Sciences}
\centerline{Re\'altanoda u. 13-15, Budapest, H-1053 Hungary}
\centerline{\tt e-mail: etesi@math-inst.hu}
\vspace{0.2 in}

\begin{abstract}
In this letter we present a topological
observation related to the CP problem in non-Abelian gauge theories, based
on significant differences between space-times and their Wick-rotation.

We point out that models containing stationary black holes the 
problem of CP breaking does not appear.
\end{abstract}
\vspace{0.1in}

\centerline{Keywords: {\it CP-problem, instantons, black holes}}
\centerline{PACS numbers: 11.15, 11.30.E, 04.20.G, 04.70}
\vspace{0.1in}

\pagestyle{myheadings}
\markright{G. Etesi: On the Strong CP-Problem}
\section{Introduction}
The famous solution of the long-standing $U(1)$-problem in the Standard 
Model via instanton effects was presented by 't Hooft about two
decades ago \cite{tho}. This solution demonstrated that {\it instantons},
i.e. finite-action solutions of the {\it Euclidean} Yang--Mills-equations
discovered by Belavin {\it et al.} \cite{bpst} should be taken seriously
in gauge theories. Another problem arose in these models over 
{\it flat} space-times, however: if instantons really exist, they induce a
CP-violating so-called $\theta$-term in the effective Yang--Mills action.
But according to fine experimental results, such a CP-violation does not
exists. 

The most accepted solution of this problem is the so-called {\it
Peccei--Quinn mechanism} \cite{pec}. A consequence of this mechanism is
the existence of a light particle, the so-called {\it axion}. However,
this particle has not been observed yet.

In this paper we point out if we consider a gauge theory over a
space-time containing black holes, the causal structure of this
space-time yields that the $\theta$-problem for an observer outside the
black hole does not occur
while the (possibly) $U(1)$-like problems can be solved exactly the same
way as in the flat case. This is because the vacuum structure of a {\it
Lorentzian} gauge theory over general relativistic stationary space-times 
is significantly different comparing to the flat $\R^4$ case while the
instanton structure of the corresponding {\it Euclidean} gauge theory is
in general similar to that of a Wick-rotated theory over flat $\R^4$.
 
First, let us summarize the vacuum structure of a gauge theory over 
Minkowski space-time. Consider a gauge theory over flat $\R^4$
given by the usual action (we fix the gauge coupling to be $1$).
\begin{equation}
S(A)=-{1\over 8\pi^2}\int\limits_{\R^4}\tr\left(F_A\wedge *F_A\right).
\label{ym}
\end{equation}
where $*$ is the Hodge-operator
induced by the flat metric on $\R^4$ (note that gravity is not coupled to 
the gauge field via Einstein's equations).
Without loss of generality we choose the gauge group
to be $G=SU(2)$. The Yang--Mills equations read as follows:
\begin{equation}
\dd_AF_A=0,\:\:\:\:\:\dd_A*F_A=0.
\label{yme}
\end{equation}
The gauge field A,  i.e. an $\su$-valued one-form over $\R^4$, 
is a vacuum field if and only if $F_A=0$, i.e. its field strength or
curvature is equal to zero. Such gauge fields can be written in the form 
$A=f^{-1}\dd f$, where $f: \R^4\rightarrow SU(2)$ is a smooth function.

But by the existence of a global temporal gauge over $\R^4$ and the
stationarity of the flat metric on $\R^4$ it is enough to consider
the restriction of $f$ to a space-like submanifold of Minkowski
space-time,
i.e. $f: \R^3\rightarrow SU(2)$. Minkowski space-time is
asymptotically flat as well, hence there is a point $i^0$ called 
space-like infinity. This point represents the ``infinity of space''
hence can be added to $\R^3$ completing it to the
three-sphere $\R^3\cup\{i^0\}=S^3$. It is well-known that vacuum fields
(possibly after a null-homotopic gauge-transformation around $i^0$) can be
extended to
the whole $S^3$ hence classical vacua are classified by maps $f:
S^3\rightarrow SU(2)$. It is well known that such maps up to homotopy are
classified by $\pi_3 (SU(2))=\Z$. This phenomenon can be interpreted as 
classical vacua are separated from each other by barriers of finite height
i.e. it is impossible to travel only through vacuum states between two
vacua of different winding numbers. 

On the other hand if $f_1$, $f_2$
are vacua of winding numbers $n_1$, $n_2$ respectively, there is a gauge
transformation $g: S^3\rightarrow SU(2)$ of winding number $n_2-n_1$
satisfying $gf_1=f_2$. Hence we can see that equivalence of
vacua suggested by the {\it dynamics} of the theory (i.e. the {\it
homotopy equivalence} of maps $f: S^3\rightarrow SU(2)\simeq S^3$) is
different from the equivalence of vacua forced by the {\it symmetry} of
the gauge theory (i.e. the {\it gauge equivalence} of the above maps).
This discrepancy-phenomenon is related to $\R^4$ only, as we will see
in a moment. Over more general space-times the two notions of equivalence
are the same.   

To avoid this discrepancy between the notion of dynamical and
symmetry-equivalence of vacua, we proceed as follows. Suppose
we have constructed the Hilbert space ${\cal H}_{\R^4}$ of the
corresponding quantum gauge theory. If $\vert n\rangle\in{\cal H}_{\R^4}$
denotes the  quantum vacuum state belonging to a classical vacuum $f$
of winding number $n$, the simplest way to construct a state which is
invariant (up to phase) under both dynamical (i.e. homotopy) and symmetry 
(i.e. gauge) transformation is to introduce the state
\begin{equation}
\vert\theta\rangle 
:=\sum\limits_{n=-\infty}^{\infty}e^{in\theta}\vert n\rangle ,\:\:\:\:
\theta\in U(1).
\label{teta}
\end{equation}
These states are called ``$\theta$-vacua''. From the physical point of
view, the introduction of $\theta$-vacua is also necessary because the
vacuum states of different winding numbers can be joined semi-classically
i.e. by a tunneling process induced by non-trivial instantons of the
corresponding {\it Euclidean} gauge theory. 

Indeed, as it is well known \cite{che}\cite{kak}, the instanton number of
an instanton is an
element $k\in H^4(S^4, \Z )\simeq\Z$ (here $S^4$ is the one-point 
conformal compactification of the Euclidean flat $\R^4$. Note that the
notion of ``instanton number'' comes from a very different
compactification comparing to the derivation of vacuum-winding numbers).
If two vacua, $\vert n_1\rangle$, $\vert n_2\rangle$ ($n_1, n_2\in
\pi_3(S^3)\simeq\Z$) are given then there is an instanton of instanton
number $n_2-n_1\in H^4(S^4,\Z )\simeq\Z$ tunneling between them in
temporal gauge
\cite{che}\cite{kak}. But the value of $\theta$ cannot be changed in
any
order of perturbation, hence this implies that tunnelings induce
the effective term
\[{\theta\over 8\pi^2}\int\limits_{\R^4}\tr\left( F_A\wedge F_A\right)\]
in action (\ref{ym}). But it is not difficult to see that such a term
violates the parity symmetry $P$ of the theory. 
\section{The Lorentzian Schwarzschild Manifold}
Now we would like to simply mimic the above machinery for certain general
relativistic space-times. Let $(M, g)$ be {\it one outer} globally
hyperbolic {\it region} of a maximally analytically extended, stationary,
asymptotically flat space-time satisfying the averaged null energy
condition (for
details see \cite{wal} while our chosen energy condition is defined in 
\cite{fri}). Consider a theory (\ref{ym}) (\ref{yme}), defined over $(M,
g)$. We may assume that $g$ is a solution of the coupled
Einstein--Yang--Mills equations, satisfying the above
conditions. We will
see later that the particular choice of the metric is immaterial hence for
the sake of simplicity we will assume that $(M,g)=(M_-, g_-)$ is nothing
but one outer region of the (maximally extended)
Lorentzian Schwarzschild manifold. This region $M_-$
of this space-time is homeomorphic to the space $S^2\times\R^2$ while the
metric is given by 
\[
\dd s^2=-\left(1-{2m\over r}\right)\dd t^2+\left(1-{2m\over
r}\right)^{-1}\dd r^2+
r^2(\dd\Theta^2+\sin^2\Theta \dd\phi^2),\]
\begin{equation}
r\in (2m,\infty ),\:\:\:t\in\R ,\:\:\: \Theta\in [0,\pi ],   
\:\:\:\phi\in [0,2\pi )
\label{schwarz1}
\end{equation}
 and $m>0$ is a real parameter (``mass''). Since there is a natural 
embedding 
\[S^2\times \R^2=\R^4\setminus\R\subset\R^4\]
it is easily seen that for $m\rightarrow 0$ this metric can be extended to
the whole $\R^4$ as the flat metric.

Now we prove the existence of a global temporal gauge on $M_-$. Since the
above embedding of $M_-$ into $\R^4$ is compatible with the metric, 
the restriction of a temporal
gauge transformation $h: \R^4\rightarrow SU(2)$ onto $M_-$ implies the
existence of such gauge over $M_-$ as well. Let $A$ be a vacuum gauge
field on $M_-$. 
Being $M_-$ simply connected, similarly to the flat $\R^4$ case, such
fields
are classified by functions $f:M_-\rightarrow SU(2)$ ($A=f^{-1}\dd f$).
However in temporal gauge it is enough to consider functions restricted
to a space-like submanifold $S$ of $M_-$ (by global hyperbolicity this is 
a Cauchy-surface as well). It is not difficult to see by (\ref{schwarz1})
that $S$ is homeomorphic to $S^2\times\R^+$. However, we can exploit the
fact that region $M_-$ is asymptotically flat, hence there is a point
$i^0$ representing space-like infinity and $S$ can be completed to
$S^2\times\R^+\cup\{ i^0\}$. But this space is nothing but the
three-ball $B^3$. After possibly null-homotopic gauge transformations
around $i^0$, each $f$ can be extended to a smooth function $f:
B^3\rightarrow SU(2)$. 

If for topological spaces $(X, x_0)$, $(Y, y_0)$ and
continuous maps $f: X\rightarrow Y$ obeying $f(x_0)=y_0$, $[X, 
Y]_0$ denotes the set of homotopy-equivalent maps, then
homotopically different vacua over $M_-$ are classified by the set
\[[B^3, SU(2)]_0=1.\]
This space contains only one element, since $B^3$ is contractible (for
$X=S^3$ we get $[S^3, SU(2)]_0=\pi_3(S^3)=\Z$ as we know). 
But this means that on one outer region of the Schwarzschild space-time 
each vacua are homotopy-equivalent in $SU(2)$ gauge theory! 

This result can be explained
from a different point of view as well. Since the outer part $M_-$ is
globally hyperbolic, the space-like submanifold $S$ forms a
Cauchy-surface for $M_-$. Hence, if we know the initial values of two
gauge
fields, $A_1$, $A_2$ on $S$, by using the field equations we know their
values over the whole {\it outer} space-time $M_-$. This implies that
the values of the fields $A_1$ and $A_2$ ``beyond'' the event horizon 
in a moment are irrelevant for an observer outside the black hole
since to determine the fields over the whole outer region it is
enough to know them on $S$. But we just proved that every vacuum
fields are homotopic restricted to $S$. Roughly speaking, homotopical
differences between vacua ``can be swept'' into the Schwarzschild black
hole. 

Hence we can see that in this model at the classical level at least,
the already mentioned dynamical (i.e. homotopical) and symmetry- (i.e.
gauge) equivalences of classical vacuum states are the same. Consider
the Hilbert space ${\cal H}_{S^2\times\R^2}$ of a quantum gauge theory
outside the black hole. We wish to construct $\theta$-vacuum states in
this space repeating (\ref{teta}). Using our result we can write
\[
\vert\theta\rangle
=\sum\limits_{n=-\infty}^{\infty}e^{in\theta}\vert n\rangle
=e^{i\theta\cdot 0}\vert 0\rangle =\vert 0\rangle .\]
Hence this space does not contain $\theta$-vacuum states.

Now we prove that the particular choice of the metric $g$ and the gauge
group $SU(2)$ was not important. First, it is not hard to see that $SU(2)$
can be replaced by any compact Lie group $G$ (compactness is necessary
because this implies $\pi_2(G)=0$ and this is used in extension of fields
to $i^0$).

By a recently proved global uniqueness theorem on outer globally
hyperbolic regions of stationary, asymptotically flat space-times
satisfying averaged null energy condition \cite{ete1} we see that for
every
such metric if space-time contains one black hole the outer region is
homeomorphic to $S^2\times\R^2$. This means that Schwarzschild space-time
is a typical member of the class of these space-times from the topological
point of view. 
(Moreover, we managed to prove a similar topological
classification for many-black hole space-times obeying the above
conditions and the triviality of gauge-vacua also follows for these more 
general space-times.)

The natural question arises: are there instanton solutions in the
corresponding Wick-rotated theory? What is the physical relevance of these
solutions?
\section{The Euclidean Schwarzschild Manifold}
Next we turn our attention to the semi-classical sector of the $SU(2)$
gauge theory described above. First, consider the Wick rotation
$(M_+,g_+)$ of Schwarzschild space-time $(M_-, g_-)$ given by $t\mapsto
i\tau$:
\[\dd s^2=\left(1-{2m\over r}\right)\dd\tau^2+\left(1-{2m\over
r}\right)^{-1}\dd r^2+
r^2(\dd\Theta^2+\sin^2\Theta \dd\phi^2),\]
Here
\begin{equation}
r\in (2m,\infty ),\:\:\:\tau\in [0,8\pi m),\:\:\: \Theta\in [0,\pi ],
\:\:\:\phi\in [0,2\pi )
\label{schwarz2}
\end{equation} 
It is well-known \cite{wal} that this metric can be realized as a smooth,
complete, maximally analytically extended metric on $M_+=S^2\times\R^2$ if
and only if $\tau$ is periodic with period $8\pi m$. In this picture 
$\tau$ plays the role of the angular variable while $r$ is the radial
coordinate of the polar coordinate system of $\R^2$. Note that although
$M_-$ and $M_+$ are actually homeomorphic they are endowed with metrics
in a
different way. The periodicity of imaginary time (i.e. the existence of a
finite
temperature, namely the Hawking-temperature, in the original theory)
causes many difficulties. 
First, this implies if $m\rightarrow 0$ in (\ref{schwarz2}), we
recover the
flat metric on $\R^3$ only, not on $\R^4$ since $\tau\in [0, 8\pi m)$.

Since $M_+$ is a four dimensional spin manifold equipped with an Einstein
metric $g_+$, the projection of its Levi--Civit\'a connection to one $\su$
component of $\so =\su_+\oplus\su_-$ produces a self-dual connection
on $(M_+, g_+)$ \cite{ete2}:
\begin{equation}
A_+(\tau, r, \Theta ,\phi)={1\over 2}\sqrt{1-{2m\over r}}\dd\Theta\:\ii
+{1\over 2}\sqrt{1-{2m\over r}}\sin\Theta\:\dd\phi\:\jj +{1\over
2}\left(\cos\Theta
\dd\phi -{m\over r^2}\dd\tau\right)\kk .
\label{insztanton}
\end{equation}
We have used the identification $\su\simeq {\rm Im}\:\HH$ given by
$\{\sigma_1 ,\sigma_2 ,\sigma_3\}\mapsto\{\ii , \jj ,\kk\}$.
It is not difficult to calculate the action of this connection that is
equal to $1$ hence this is an instanton field in the Wick-rotated theory.
We would like to transform it into $\tau$-temporal gauge in order to
understand its relation to vacua. This is
obstructed by the fact that the embedding $M_+\subset\R^4$ is not
compatible with the metric $g_+$. More explicitly, we can see that a
gauge transformation which transforms the instanton (\ref{insztanton})
into temporal gauge ought to take the form
\[h(\tau, r, \Theta, \phi )={\rm exp}\:\kk\left( B(r, \Theta, \phi     
)-{2m\tau\over r^2}\right) ,\]
where $B$ is an arbitrary function. The map $h$ must be periodic in
$\tau$ as well but it is impossible.

Hence we have presented an instanton which has no $\tau$-asymptotics
and cannot be transformed into temporal gauge because of periodicity.
These observations suggest that (\ref{insztanton}) is {\it not} related to
any tunneling-phenomenon in the Lorentzian theory. From the Lorentzian
point of view this observation is supported by the non-existence of
homotopically different vacuum states and $\theta$-vacua. 

In light of the above results, we see that for outer observers in
Schwarzschild space-time the notion
of $\theta$ vacua do not exist while instantons occur in the Euclidean
theory, but they cannot connect together vacuum states essentially due to
the Hawking-temperature of the Schwarzschild black hole. {\it Hence in
this gauge theory an effective instanton-induced CP-breaking $\theta$-term
in the action does not appear for observers outside the black hole.} We
can say that $\theta =0$ is produced by the causal structure of space-time
instead of an axion field. 

Since this mechanism is purely topological, the distance of a black hole
is not relevant; the situation is similar to charge quantization by
distant Dirac-monopoles.
 
By elements of Donaldson theory, it is possible to prove that
there are lot of instantons in the Euclidean Schwarzschild metric.
Moreover, their action is always an integer \cite{ete2}. Hence, in
spite of the vacuum structure, the
instanton sector of the $SU(2)$ gauge theory over Schwarzschild
space-time is similar to the flat case summarized in the Introduction.

Again, we would like to say something on whether the existence of
instantons and discreteness of their action is a particular
phenomenon of the Schwarzschild metric or not.
 
By general properties of self-duality equations on four-manifolds we
may expect that for a typical four-manifold with a typical metric the
moduli space of instantons is not empty and the values of their action
form a discrete set in $\R$. For a class of Riemannian manifolds it is
proved in \cite{ete2}. 

\section{Concluding Remarks}
In this letter we have studied the relationship between vacuum tunnelings
and instantons in gauge theories over stationary space-times containing
black holes. 

We have found a remarkable deviation in vacuum structure of these theories
comparing to the flat $\R^4$ case although the instanton-structure of
these theories seem to be identical to that of the flat theory. These
observations suggest that connection between tunnelings and instantons is
not evident in general gauge theories. A similar problem was
studied in \cite{bit} in two dimensional $\sigma$-models; the authors
proved the instanton structure of this model is more complicated than
vacuum sector. In spite of these properties authors were able to
interpret instantons as tunnelings in the $\sigma$-model.

In our model the situation is similar. In light of properties of vacua
and instantons in the $SU(2)$ gauge theory over Schwarzschild space-time,
one is naturally forced to reject the instanton-induced tunneling in
these theories. We have mentioned general results which prove that we
also have to face this phenomenon in general gauge theories over general
asymptotically flat, stationary space-times containing black
holes. 

As a straightforward consequence, we have seen that the problem of 
$\theta$-vacua and CP-problem can be solved in these models. But it would
be important to know whether $U(1)$-like problems occur in these models or
not.

But at this point one can exploit the fact that instanton-structure of
general gauge theories is similar to that of theories over flat $\R^4$.
The $U(1)$-problem of the Standard Model is solved by the observation that
the anomalous axial-vector-current is not
divergence-free because instantons always have a ``tail in infinity''
hence this current cannot be adjusted into a full-divergence. This is due
to a deep property of instantons: like holomorphic functions, self-dual
gauge fields cannot be altered on a compact set without destroying
self-duality; hence they always have a nonzero value in infinity, the
anomalous axial-vector-current cannot be conserved.

This property of instantons remains valid also in this case hence---at
least by first look---instantons solve the $U(1)$-problem in the same way
as in flat case.

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\bibitem{kak} M. Kaku: Quantum Field Theory, Oxford University
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\bibitem{bit} K.M. Bitar, S. Chang, G. Grammer Jr., J.D. Stack, 
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\end{thebibliography}
\end{document}

