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     \preprint{\vbox{\it  \null\hfill\rm DOE/ER/41014-11-N97}\\\\}

     \title{On the Symmetry Constraints of CP Violations in QCD}
    
     \author{Chuan-Tsung Chan\thanks{E-mail address: 
             chan@alpher.npl.washington.edu}}

     \address{Department of Physics, Box 351560,
              University of Washington,
              Seattle, WA 98195-1560, USA}

     \date{\today}

     \maketitle

   \begin{abstract}

     \input{z1}

   \end{abstract}

     \pacs{}

     \narrowtext

   \section{Motivations}
     \label{sec:z2}

     \input{z2}

   \section{Introduction}
     \label{sec:z3}

     \input{z3}

   \section{Strong CP Violations in QCD}
     \label{sec:z4}

     \input{z4}

   \section{Symmetry Constraints of CP Violations in QCD}
     \label{sec:z5}
	
     \input{z5}

   \section{A Graphic Illustration of the Symmetry Constraints of 
            CP Violations in QCD}
     \label{sec:z6}

     \input{z6}

   \section{Why Strong CP Violations Are Small}
     \label{sec:z7}

     \input{z7}

   \section{Summary and Conclusion}
     \label{sec:z8}

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   \acknowledgments

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   \begin{references}

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   \end{references}

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\end{document}
      
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\begin{document}

  {\large \bf On the Symmetery Constraints of CP violations in QCD }

  \begin{enumerate}

    \item Motivations

    \item Introduction

    \item Strong CP violations in QCD

    \item Symmetry Constraints of CP Violations in QCD

          \begin{enumerate}

            \item Non--Perturbative Nature of CP Violations in QCD

            \item Chiral Limit and CP Violations in QCD

            \item $U_{A}(1)$ Anomaly Constraint of CP Violations in QCD

          \end{enumerate}

    \item A Graphic Illustration of the Symmetry Constraints of 
          CP Violations in QCD

    \item Why Strong CP Violations Are Small

    \item Summary and Conclusion

  \end{enumerate}

\end{document}
 \newpage
  
 \begin{figure}
    \vspace{8cm}
    \caption{Quark propagator in the presence of pseudo-mass term}
    \label{fig:Quark propagator}
 \end{figure}

 \begin{figure}
    \vspace{12cm}
    \caption{Quark EM moments in the presence of pseudo-mass term}
    \label{fig:Quark EM moments}
 \end{figure}

 \newpage

 \begin{figure}
    \vspace{10cm}
    \caption{The phase plane and CP torus of QCD}
    \label{fig:CP torus}
 \end{figure}

 \begin{figure}
    \vspace{10cm}
    \caption{The equivalent classes of QCD Lagrangian}
    \label{fig:equiv class}
 \end{figure}

 \newpage

 \begin{figure}
    \vspace{10cm}
    \caption{The chiral limit of the CP torus of QCD}
    \label{fig:chiral limit}
 \end{figure}

 \begin{figure}
    \vspace{10cm}
    \caption{The chiral symmetric CP torus of QCD}
    \label{fig:second limit}
 \end{figure}

%Strong CP violations in QCD

Having defined the strong CP problem within the framework of the standard
model, we shall write down the explicit definitions to set up proper notations
for further reference. For the sake of simplicity, we shall discuss a theory
with one quark flavor\footnote{The simplification is due to the fact that
strong CP violation is a flavor-singlet problem.}. The generalization to a
multi-flavor case can be found elsewhere \cite{QCD:C-T Chan}.

Normally, the ( CP conserving ) QCD Lagrangian is given by
\begin{eqnarray}
       { \cal L }_{QCD} &\equiv&           \bar{\psi} i \deb \psi
                                  + \massq \bar{\psi}        \psi
                                  + \frac{1}{4}  G^2               \\
     \mbox{where} \hspace{2cm}
                  \deb  &\equiv& ( \m \partial_\mu
                                    + i g_s  B_\mu^a \frac{ \lambda^a }{2} \m )
                                  \cdot \gamma^\mu
\end{eqnarray}
   The meanings of various symbols are:
      \begin{eqnarray*}
      \psi                  &:& \mbox{quark field }                          \\
      \bar{\psi}            &:& \mbox{Dirac adjoint of the quark field, }
                                \bar{\psi} \equiv {\psi}^{\dagger} \gamma_0  \\
      B_\mu^a               &:& \mbox{gluon field, } a = 1,..,8              \\
      \frac{\lambda^a}{2}   &:& \mbox{generators of the color } SU(3)
                                \mbox{ gauge group, } a = 1,..,8             \\
      G_{\mu\nu}            &:& \mbox{gluonic tensor field, } G_{\mu\nu} \equiv
                               [\m \partial_\mu + i g_s B_\mu \m,
                                \m \partial_\nu + i g_s B_\nu \m],           \\
                            & & B_\mu \equiv B_\mu^a \frac{\lambda^a}{2},
                                \m \m G^2 \equiv  G_{\mu\nu} G^{\mu\nu}      \\
      g_s                   &:& \mbox{strong coupling constant in QCD}         
     \end{eqnarray*}
 
       To calculate various correlation functions in the quantum theory, it is
     useful to define the QCD generating functional ( denoted by Z ):
     \begin{equation}
      Z[ \m \zeta, \bar{\zeta}, J_\mu \m ] \equiv
         \frac{1}{N} \int [ D \psi ] [ D \bar{\psi} ] [ DB_\mu ]
        e^ { iS_{QCD} + \bar{\zeta} \psi + \bar{\psi} \zeta + J_\mu B^\mu }
     \label{eq:gen}
     \end{equation}
     where the QCD action 
     \begin{equation}
      S_{QCD} \equiv \int d^4 x \m { \cal L }_{QCD}
     \end{equation}

      The normalization constant for the generating functional is
     \begin{equation}
      N \equiv  \int [D\psi] [D\bar{\psi}] [DB_{\mu}] e^ {iS_{QCD}}
     \end{equation}
      such that
     \begin{equation}
      Z[ \m \zeta=0, \m \bar{\zeta}=0, \m J_\mu=0 \m ] = 1
     \end{equation}

    Notice that in the generating functional ( Eq.~\ref{eq:gen} ), the fermion 
    field $\psi, \bar{\psi}$ and the gluon field $B_\mu^a$ are dummy 
    variables; only the external source fields $\zeta, \bar{\zeta}, J_\mu$ 
    specify the physical ground state ( QCD vacuum ) of the theory. 
    Therefore, we can redefine these dummy variables freely without changing
    the physical contents of the theory. 
    In particular, we can perform a $U_A (1)$ chiral rotation on the fermion 
    field:
    \begin{eqnarray}
      \psi \rightarrow { \psi '} &\equiv& e^{i \theta \gamma_5} \psi
      \label{eq:chiral} \\
      \mbox{or \hspace{2cm} } { \psi '}_i &=&
      [\m  \cos \theta ( I )_{ij}
      + i  \sin \theta ( {\gamma_5} )_{ij} \m ] \m {\psi}_j
    \end{eqnarray}

    While it is clear that the quark mass term  $\m {\massq} \bar{\psi} \psi \m$
    transforms into $\m \massq \bar{\psi} e^{ 2 i \theta \gamma_5 } \psi \m$
    under a $U_A (1)$ chiral rotation, it is not a trivial task to show that
    a $U_A (1)$ chiral rotation is not an unitary transformation ( $U_A (1)$
    anomaly ) and a Jacobian associated with this change of variables in the 
    functional space has to be implemented. 
    As shown by Fujikawa \cite{QCD:Fuji}, the fermionic measure in the
    functional integral
    $[ D \psi ] [ D \bar{\psi} ]$ transforms, under a $U_A (1)$ chiral
    rotation,
    \begin{equation}
     [ D \psi ] [ D \bar{\psi} ] \rightarrow [ D \psi ] [ D \bar{\psi} ]
      \m e^{ i \frac{ g_s^2 2 \theta }{ 32 \pi^2 } \int d^4 x G \tilde G
              ( x ) }
    \end{equation}

    Hence, we generate a new ( but equivalent ) QCD Lagrangian with two extra
    terms ( together with a change of the chiral phases of the external fermion
    source fields $\zeta$, $\bar \zeta$ ):
    \begin{equation}
     \massq \m \bar{\psi}   \m ( e^{i 2 \theta \gamma_5} - 1 ) \m \psi \m 
                            \approx 
   i \massq \m \sin 2 \theta \m \bar{\psi} \gamma_5 \psi,  \hspace{1.5cm}
     \frac{ g_s^2 2 \theta }{ 32 \pi^2 } G \tilde G  \nonumber
    \end{equation}

    Several comments are in order:

    (1) These two terms carry the same quantum numbers and both are odd under
    parity ( P ) and time reversal ( T ) transformations. We shall refer to the
    former as a quark pseudomass term, and the latter as a gluon anomaly term. 
    We emphasize that these are the lowest dimensional CP violating operators
    that one can write down in the QCD Lagrangian, consistent with basic 
    requirements\footnote{These requirements include: (1)
    Hermiticity, (2) Lorentz invariance, and (3) gauge invariance.} of a 
    relativistic quantum field theory.

    (2) We can generalize the discussion to a ( generally CP violating ) QCD
    Lagrangian with an arbitrary quark pseudomass term $ \massq \bar{\psi} 
    e^{i \thetaq \gamma_5} \psi$ and a gluon anomaly term $\frac{ g_s^2 
    \thetag }{ 32 \pi^2 } G \tilde G$.
    \begin{equation}
       { \cal L }_{QCD; \thetag, \thetaq} \equiv 
                                        \bar{\psi} i \deb                 \psi
                               + \massq \bar{\psi} e^{i \thetaq \gamma_5} \psi
                               + \frac{1}{4}  G^2 
                               + \frac{ g_s^2\thetag }{ 32 \pi^2 } G \tilde G
    \end{equation}     
    It can be shown that, under a $U_A (1)$ chiral rotation
    ( Eq.~\ref{eq:chiral} ), both $\thetaq$ and $\thetag$ change by $2 \theta$.
    Therefore, the difference
    \begin{equation}
    \thetabar \equiv \thetag - \thetaq
    \end{equation}
    is an invariant of the $U_A (1)$ chiral rotation, which can be used to
    label the classes of equivalent QCD Lagrangians. Since the physical
    observables are independent of the representations of the generating
    functional, we conclude that any CP violating observable has to be 
    proportional to the $U_A (1)$ invariant chiral phase $\thetabar$.

    (3) The $\thetabar$ parameter, being a difference between two chiral phases,
    is an angular variable with period $2 \pi$ ( in the case of one quark
    flavor ). Consequently, any physical observable has to be a periodic
    function of the $\thetabar$ parameter.

    (4) In the muti-flavor case of QCD, the number of quark chiral phases is
    equal to the number of quark flavors. Nevertheless, since we can perform a
    $U_A (1)$ chiral rotation independently on each flavor, there is still only
    one physical parameter which characterizes the strength of strong CP 
    violations.
    In that case, the $\thetabar$ parameter is defined as 
    \begin{equation}
    \thetabar \equiv \thetag - \sum_j \thetaq_j
    \end{equation}
    where $j$ is the flavor index for light quarks.

%Symmetry constraints of CP violations in QCD 
                                      
 The previous discussions seem to suggest that there are close relationships 
 between the $U_A(1)$ chiral symmetry and the CP violations in QCD. 
 Indeed, it is necessary that the chiral symmetry is broken both spontaneously 
 and explicitly so that strong CP violations are possible. 
 We shall refer to these relations as symmetry constraints and discuss their 
 meanings and implications in this section.

  (1) Non--Perturbative Nature of CP Violations in QCD

   Given the fact that the gluon anomaly term $G \tilde G$ can be
   written as a total divergence of the Chern-Simon current $K_{\mu}$,
   \begin{eqnarray}
                     K_\mu &\equiv&
                     \frac{g_s^2}{16 \pi^2} \sum \epsilon_{\mu \nu \rho \sigma}
                     B^{a \nu} \{ \partial^\rho B^{a \sigma} +
                     \frac{1}{3}   f_{abc}      B^{b \rho}   B^{c \sigma} \} \\
                     \partial^\mu K_\mu &=& \frac{g_s^2}{32 \pi^2} G \tilde G
   \end  {eqnarray}
    it is not too surprising that this term has no effect on a perturbative
    calculation of any CP violating observable. 
    Since a potential modification of the perturbative expansion caused by the
    gluon anomaly term can only be related to the gluonic propagator in the 
    Feynman rules, it turns out such a contribution to the gluonic propagator
    is zero, as can be verified by a direct calculation.

    Since we can always perform chiral rotations to shift the strong CP
    violating phases into the gluon anomaly term ( i.e., $\thetaq=0, \m 
    \thetabar=\thetag$ ), and any physical observable should not depend on the 
    particular representation we choose to do a calculation, we conclude that
    the strong CP violation has to be a purely nonperturbative effect. 
    However, it remains to see how a calculation of CP violating observables 
    based on the quark pseudomass term leads to the same conclusion, if we 
    insist on the reparametrization invariance of the CP violating observables.
    This is a problem, because it seems that the presence of a quark pseudomass
    term will lead to a modification of the quark propagator, hence generate 
    contributions to the CP violating observables in perturbative calculations.
    Apparently, such an effect is in contradiction to our previous
    observation.

    Indeed, if we perform a calculation of a quark electric dipole moment (
    denoted as qEDM ) using a perturbative expansion on both fine structure 
    constant ${e^2}/{\hbar c}$ and strong coupling constant $g_s$, we do find a
    finite contribution to a tensor structure which corresponds to a qEDM, with
    the strength proportional to the quark chiral phase $\sin \thetaq$, see 
    Fig.~\ref{fig:Quark EM moments}.
    Nevertheless, we should be careful not to interpret this result as a 
    physical observable. As we discussed before, $\thetaq$, by itself,
    is a representation-dependent parameter and CP violating physical
    observables should depend only on $\thetabar$. What goes wrong here?

    The answer is that there are other representation-dependent chiral phases
    we should include in the extraction of a physical observable associated
    with a chirally covariant tensor. 
    For example, in the case of a particle EDM, the relevant tensor 
    $i \sigma_{\mu \nu } \gamma_5$ is mixed with the tensor associated with the 
    anomalous magnetic moment ( denoted as AMM ) $\sigma_{\mu \nu}$ under a 
    $U_A (1)$ chiral rotation on the particle field. 
    Besides that, the same $U_A(1)$ chiral rotation also causes the mass of the
    particle to develope a chiral phase, 
    $M \rightarrow M e^{i \alpha \gamma_5}$, which can be viewed as a mixing 
    between $I$ and $i \gamma_5$ tensors, see Fig.~\ref{fig:Quark propagator}. 
    We need to subtract the relative phases, $ \arctan ( \m \frac { \mbox{EDM} }
    { \mbox{AMM} } \m )$ and $\alpha$, in order to define a 
    representation-independent answer for the physical observables. 
    It can be verified that in the calculation of the qEDM, both $ \arctan ( \m
    \frac { \mbox{EDM} }{ \mbox{AMM} } \m ) $ and $\alpha$ are
    equal to $\thetaq$; hence there is no quark EDM from the perturbative
    calculation in any representation of the QCD generating
    functional\footnote{This point has been emphasized by E.P. Shabalin 
    \cite{CP:Shabalin}. However, the argument in support of the conclusion in
    these works seems unclear to the present author.}.

    With a suitable generalization, one can convince oneself that this
    conclusion holds true to all orders in QCD, i.e., the inclusions of any
    higher order loops does not affect the conclusion.  
    One can also apply the same argument to the bound states of quarks, i.e.,
    hadrons, and show that any CP violating observables have to come from the 
    nonperturbative contributions of QCD\footnote{By perturbative 
    contributions, we mean those arising from an expansion of any physical 
    observable in a power series of the strong coupling constant $g_s$.
    A purely nonperturbative observable has zero coefficients to all order in 
    its power series expansion. For example, $\langle \bar q q \rangle \propto
    e^{ - 1/{g^2_s} }$.}.

    Since the nonperturbative property of QCD can be characterized in terms of
    the spontaneous chiral symmetry breaking, which is manifested by the
    presence of vacuum condensates, we can rephrase the conclusion in the
    following way:

    If there is no spontaneous chiral symmetry breaking in QCD, there is no
    strong CP violation, even with a nonzero $\thetabar$.

  (2) Chiral Limit and CP Violations in QCD

    Another important constraint on the strong CP violations has to do with a
    nonvanishing quark mass term in the QCD Lagrangian. It was first pointed
    out by Peccei and Quinn \cite{CP:PQ} that there is no strong CP violation 
    if there exists a massless quark in the QCD Lagrangian.

    This constraint can be understood in terms of the functional integral 
    formalism.
    Since a quark chiral phase $\thetaq$ is ill-defined if the quark mass is
    zero, we can take advantage of this fact to rotate away ( via an $U_A (1)$
    chiral transformation ) the gluon chiral phase $\thetag$, so that any QCD
    Lagrangian with a massless quark is equivalent to a CP conserving one.
    Therefore, it is necessary to have all quark masses finite to generate a 
    CP violating observable from QCD.

    The point of this argument is that we need to have a well-defined chirally
    covariant phase, in addition to the gluon chiral phase $\thetag$, such 
    that, after the reduction of the irrelevant degrees of freedom by 
    reparametrization invariance, we are left with a chirally invariant CP 
    violating parameter, e.g., $\thetabar$. 
    For instance, we can replace the quark mass term by a higher dimensional 
    operator which violates the chiral symmetry explicitly,
    e.g., $\bar q \sigma_{\mu \nu} e^{i \beta \gamma_5} q G^{\mu \nu}$
    \cite{NEDM:KW}. 
    In this case, even though the quark is massless, we still can have a CP
    violating observable; proportional to the chirally invariant phase 
    $\thetag - \beta$.
    Consequently, the second constraint can be phrased as follows:
    
    If there is no explicit chiral symmetry breaking, there is no strong CP
    violation in QCD.

  (3) $U_{A}(1)$ Anomaly Constraint of CP Violations in QCD

    The third constraint was first discussed by M.A. Shifman, A.I. Vainshtein 
    and  V.I. Zakharov \cite{CP:SVZ}, and then rediscovered by S. Aoki, 
    A. Gocksch,  A.V. Manohar and S.R. Sharpe \cite{CP:Aoki}.
       They pointed out that, in a diagrammatical language, the strong CP 
       violations only contribute to physical observables through the 
       internal fermion loops with a pseudo--mass insertion\footnote{Therefore,
       it is impossible to calculate the effect of strong CP violations in a
       quenched lattice calculation \cite{CP:Aoki}.}.
       Such a diagram is a manifestation of the anomalous Ward
       identity associated with the flavor singlet axial current and has a 
       close relationship with the $U_{A}(1)$ anomaly in QCD.
       This connection has been examined in the context of chiral perturbation
       theory by S. Aoki and T. Hatsuda \cite{CP:A&H}, and H-Y Cheng 
       \cite{CP:H-Y Cheng}.
    Essentially, this constraint requires that the chiral anomaly provides a 
    solution to the $U_A(1)$ problem \cite{CP:SVZ} \cite{UA1:tHooft}.
       If this is not the case, then there is no strong CP violation.
       In a hadronic calculation, this constraint implies that CP violating 
       observables should be proportional to the difference between $m_\pi^2$
       and $m_{\eta'}^2$ \cite{CP:A&H} \cite{CP:H-Y Cheng}. 
       A quantitative realization of this constraint in QCD implies that CP 
       violating observables should be proportional to the anomalous gluon 
       condensate $\cond{G \tilde G}$ \cite{NEDM:C-T Chan}.

   It should come as no surprise that these three constraints are not
   independent since ultimately both the anomalous gluon condensate 
   $\cond{G \tilde G}$ and the quark chiral radius \cite{QCD:C-T Chan}
   \begin{equation}
    R_q^2 \equiv { \left[ \m   \cond{ q          \bar q } \m \right] }^2 +
                 { \left[ \m i \cond{ q \gamma_5 \bar q } \m \right] }^2 
   \end{equation}
   can be related to the QCD scale $\Lambda_{QCD}$. 
   Indeed, through the use of a generalized anomalous Ward identity, we can 
   prove that $\cond{G \tilde G}$ is propotional to the product of $m_q, R_q$
   and $\sin \thetabar$ \cite{QCD:C-T Chan}.

   To summarize, the bottom line of the study of symmetry constraints is that
   we need three finite QCD parameters:
   $m_q, R_q$ and $\sin \thetabar$ to generate a CP violating observable from
   QCD.


