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\title{Theory and Phenomenology of CP Violation}
\author{Thomas Mannel\address{Theretische Physik I, University of Siegen, \\
57068 Siegen, Germany}}
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\runtitle{Theory and Phenomenology of CP Violation}
\runauthor{Thomas Mannel}
\begin{document}
\begin{abstract}
\noindent
In this talk I summarize a few peculiar features of CP violation in the
Standard Model, focussing on the CP violation of quarks.
\vspace{1pc}
\end{abstract}
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\maketitle
\vspace*{-1.5cm}
\section{Introduction}
%
%
%
CP Violation and more generally the flavour structure observed in elementary-partlcle
interactions still remains one of the unexplained mysteries in high-energy physics.
In the framework of the standard model flavour mixing and CP violation is encoded
in the CKM matrix for the quarks and in the PMNS matrix for the leptons. However,
this is only a parametrization, in which CP violation appears through irreducble
phases in these matrices and which is up to now completely consistent with the
experimental facts, at least with what is found at accelerator experiments.
On the other hand, the CP violation in the standard model is a small effect. In
particular it is too small to create the observed matter-antimatter asymmetry of
the universe, for which CP violation is an indispensable ingredient. The standard
model is a local, Lorentz-covariant and causal quantum field theory, which means
that we shall assume strict CPT conservation. Keeping this in mind, let us recall
here the criteria established by Sakharov \cite{Sakharov}
for the appearance of a baryon-antibaryon
asymmetry:
\begin{itemize}
\item There have to be baryon Number violating interactions:
$$ {\cal L}(\Delta n_{\rm Bar} \neq 0) \neq 0 \, . $$
\item CP has to be violated in order to have different reaction rates for baryons and
antibaryons
$$
\Gamma(N \,\, \stackrel{{\cal L}(\Delta n_{\rm Bar} \neq 0)}{\longrightarrow} \,\, f ) \neq
\Gamma(\bar{N} \,\, \stackrel{{\cal L}(\Delta n_{\rm Bar} \neq 0)}{\longrightarrow } \,\, \bar{f} ) \, .
$$
\item The universe had to be out of thermal equilibrium, since in thermal
equilibrium CPT invariance is equivalent to CP invariance.
\end{itemize}
Although the standard model has all these ingredients, it turns out that the observed
matter-antimatter asymmetry cannot be accommodated by the standard model, and
one reason is that CP violation turns out to be too small.
Furthermore, CP violation (and with it the complete flavour structure) of the
Standard Model is quite peculiar and fully compatible with the data from particle
accelerators. In the following I will focus on the CP violation of quarks and
point out a few of these features which follow
from the parametrization with the CKM matrix and which have a specific
phenomenology and which are not easily reproduced by generic new physics models.
\section{CP Violation in the Standard Model}
%
%
%
In general, CP violation emerges in Lagrangian field theory through complex
coupling constants, the phases of which are irreducible\footnote{This means that
the phases cannot be removed by phase redefinitions of the fields.}.
Schematically this means
\begin{equation}
{\cal L} = \sum_i a_i {\cal O}_i + h.c. \qquad
(CP) \, {\cal O}_i (CP)^\dagger = {\cal O}_i^\dagger \, .
\end{equation}
In the Standard Model there is only a single irreducible phase which is induced
by the Yukawa couplings. It has become a general convention to define the phases
of the fields such that the irreducible imaginary part appears in the CKM Matrix
\begin{equation}
V_{\rm CKM} \neq V_{\rm CKM}^*
\end{equation}
In fact this is already one of the peculiarities of CP violation in the standard model,
where the unique source of CP violation only appears in the charged current couplings.
There are infinitely many possible parametrizations of the CKM matrix in terms of
three angles and a phase. One possibility, the PDG parametrization, may be written
as a product of three rotations and a phase matrix in the form
\begin{figure}[t]
\vspace*{-5mm}
\includegraphics[scale=0.6]{UTstd.pdf}
\vspace*{-1cm}
\caption{The standard unitarity triangle.} \vspace*{-0.7cm}
\label{fig1}
\end{figure}
{\tiny \begin{eqnarray*}
&U_{12} = \left[ \begin{array}{ccc}
c_{12} & s_{12} & 0 \\
-s_{12} & c_{12} & 0 \\
0 & 0 & 1 \end{array} \right] \,\, ,
\,
U_{13} = \left[ \begin{array}{ccc}
c_{13} & 0 & s_{13} \\
0 & 1 & 0 \\
-s_{13} & 0 & c_{13} \end{array} \right] \,\, ,
\\
& U_{23} = \left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & c_{23} & s_{23} \\
0 &-s_{23} & c_{23} \end{array} \right] \,\, , \,
U_\delta = \left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & e^{-i\delta_{13}} \end{array} \right] \,\, .
\end{eqnarray*} }
\noindent
where $s_{ij} = \sin \theta_{ij}$ etc. are the sines and cosines of the rotation angles.
The standard parametrization is obtained by
\begin{equation}
V_{\rm CKM} = U_{23} U_\delta^\dagger U_{13} U_\delta U_{12}
\end{equation}
In this parametrization large phases appear in $V_{td}$ and $V_{ub}$ which are
small matrix elements as far as their absolute value is concerned.
The unitarity of $V_{\rm CKM}$ is usually depicted by the unitarity triangle.
There are in total six unitarity relations (aside from the normalization relations
for the rows and columns), which may be depicted as traingles in the complex
plane. However, due to the hierarchy of the CKM matrix elements there are only
two triangles with comparable sides which coincide to leading order in the
Wolfenstein expansion. This triangle is shown in Fig.~\ref{fig1}
An invariant measure for the size of CP violation is the area of this triangle. In fact,
on can show that all traingles have to have the same area due to unitarity. The area
of all these triangles is proportional to the quantity
\begin{eqnarray}
{\rm Im} \Delta &=& {\rm Im} V_{ud} V_{td}^* V_{tb} V_{ub}^* \\ \nonumber
&=& c_{12} s_{12} c_{13}^2 s_{13} s_{23} c_{23} \sin \delta_{13}
\end{eqnarray}
It is interesting to note that the maximal possible value for this quantity for some
``optimized'' values of the angles and the phase is
\begin{equation}
\delta_{\rm max} = \frac{1}{6 \sqrt{3}} \sim 0.1
\end{equation}
while the value realized in nature is several orders smaller,
$\delta_{\rm exp} \sim 0.0001$. This quantifies the statement that CP violation is a
small effect.
Finally we note that CP violation vanishes in the case of degeneracies of up or
down quark masses, in which case one may rotate away the CP violating phase.
It has been noted by Jarlskog \cite{Jarlskog} that the following quantity
is a clear indication for the presence of CP violation:
\begin{eqnarray}
J &=& {\rm Det}([M_u \, , \, M_d]) \\ \nonumber
&=& 2 i {\rm Im} \Delta
(m_u-m_c) (m_u-m_t) (m_c-m_t) \\ \nonumber
&& \qquad \quad \times (m_d-m_s) (m_d-m_b) (m_s-m_b)
\end{eqnarray}
In second order in the weak interactions there is the possibility of flavour
oszillations. The box-diagramms shown in fig.~\ref{fig2} can mediate transitions
between $B_d \leftrightarrow \overline{B}_d$, $B_s \leftrightarrow \overline{B}_s$,
$K^0 \leftrightarrow \overline{K}^0$
and also between $D^) \leftrightarrow \overline{B}^0$
\begin{figure}
\vspace*{-5mm}
\includegraphics[scale=0.42]{BoxHorizColor.pdf} \,\,
\includegraphics[scale=0.43]{BoxVertColor.pdf}
\vspace*{-1cm}
\caption{The box diagramms mediating $\Delta F = 2$ transitions
in the standard model.} \vspace*{-0.5cm}
\label{fig2}
\end{figure}
For later use we note that the mixing amplitude for $B_d - \overline{B}_d$
mixing is proportional to the phase factor $\exp (2i\beta)$, while the phase
in $B_s - \overline{B}_s$ mixing is negligibly small in the standard model.
In the kaon system, the short distance contribution to mixing is also proportional
to $\exp (2i\beta)$, while the mixing in the up quark sector ($D - \overline{D}$
mixing) is heavily GIM suppressed.
The Flavour structure and the pattern of CP violation in the Standard Model
is quite peculiar. Here I list a few of these peculiarities:
%
\paragraph{Strong CP violation:} The QCD sector of the standard model can
also contain a CP violating interaction of the form
\begin{equation}
{\cal L}_{\rm strong\,\, CP} = \theta \frac{\alpha_s}{\pi}
{\rm Tr} \left\{\vec{E} \cdot \vec{B} \right\}
\end{equation}
where $\vec{E}$ and $\vec{B}$ are the chromo-electric and the chromo-magnetic
field strenghts. The electric dipole moment of the neutron tells us
that the coupling $\theta$ has to be extremely small. It has been suggested by
Pecchei and Quinn \cite{PQ} to introduce an additional symmetry to explain $\theta = 0$,
but in these scenarios a new light pseudoscalar particle, the so-called axion
appears for which we do not have any experimental evidence. It is fair to say
that the strong CP problem is still unsolved.
\noindent
\paragraph{Flavour Diagonal CP violation:} One of the most peculiar features
of CP violation in the standard model is the enormous suppression of flavour
diagonal CP violation. Assuming that the CKM matrix is the only source of CP
violation, it is easy to see that an electric dipole moment of a quark can be
induced only at the two loop level at least, evaluating a diagram as the one
shown in fig.~\ref{fig3}.
\begin{figure}
\vspace*{-5mm}
\begin{center}
\includegraphics[scale=0.42]{edm.pdf}
\end{center}
\vspace*{-1.5cm}
\caption{Feynman Diagram leading to an electric dipole moment for the up quark.}
\vspace*{-0.7cm}
\label{fig3}
\end{figure}
It has been shown by Shabalin \cite{Shabalin}
that the sum of all the two loop diagrams
lead to a vanishing electric dipole moment for the quarks, and a non-vanishing
contribution requires at least one more loop, which can be an additional gluon.
Hence naive counting leads to the following estimate of the electric dipole moment
of the neutron
\begin{equation}
d_e \sim e \, \frac{\alpha_s}{\pi} \, \frac{G_F^2}{(16 \pi^2)^2} \,
\frac{m_t^2}{M_W^2} \,
{\rm Im} \Delta \, \mu^3
\sim 10^{-32} {\rm e \, cm}
\end{equation}
where the factor $m_t^2/M_W^2$ originates from the GIM suppression,
$\Delta$ encodes the necesary CKM factors and $\mu$ is a typical hadronic
scale which we set to $\mu = 300$ MeV. This has to be compared to
the current experimental limit which is \cite{exp}
$$
d_{\rm exp} \le 3.0 \times 10^{-26} {\rm e \, cm}
$$
Similar statements hold for other flavour diagonal and CP violating observables,
and from the experimental side this strong suppression of these effects is
supported.
\paragraph{Strong Suppression of CP violation in the up-quark sector:}
The pattern of mixing and CP violation in the standard model is determined by
the GIM mechanism. This means in particular, that the effects for the up, charm
and the top quark are severely suppressed, since the mass differences
between the down-type quarks are small compared to the weak boson masses.
Furthermore, the mixing angles of the first and second generations into the third
are small, which means that charm physics is basically a ``two family'' problem.
Since CP violation with only two families is not possible through the CKM mechanism,
this results in a strong suppression of CP violation in the charm sector. Again this
fact seems to be supported by experiment.
\section{Phenomenology of CP Violation}
%
%
%
In the following we shall concentrate on CP violation in the decays of particles.
In general, decays are mediated by an effective interaction consisting of a sum
of local operators $O_i$
\begin{equation}
H_{\rm eff} = \frac{G_F}{\sqrt{2}} \sum_{i=1}^n C_i O_i
\end{equation}
where in the standard model $C_i$ are coefficients containing the CKM factors
(including possible phases) and the contributions from the QCD running from the
weak scale down to the typical scale of process. Furthermore, in the standard model
we have $n=10$, where the operators $i=1,2$ are called tree operators,
$i=3,..6$ are the QCD penguins and $i=7,...10$ are the electroweak penguins.
CP violation occurs through complex phases of the $C_i$ in the following way.
Consider an amplitude for a decay of a $B$ meson into some final
state $f$, with two contributions
\begin{equation}
A (B \to f) = \lambda_1 a_1 + \lambda_2 a_2
\end{equation}
where the $\lambda_i$ are complex coupling constants and $a_i$ are
hadronic matrix elements $\langle B | O_i | f \rangle $.
The amplitude of the CP conjugate process is obtained by conjugating the
couplings $\lambda_i$, while the phases of the the hadronic matrix elements
remain the same due to CP invariance of strong interactions. The CP
asymmetry is given by
\begin{eqnarray}
{\cal A}_{CP} &\propto& \nonumber
\Gamma (B \to f) - \Gamma (\overline B \to \overline{f}) \\
&=& 2 \, {\rm Im} [\lambda_1 \lambda_2^* ] \,
{\rm Im} [a_1 a_2^*]
\end{eqnarray}
Thus a CP asymmetry requires imaginary parts of the coupling constants, but
also a {\em strong} phase difference between the hadronic matrix elements.
The neutral flavoured mesons can decay into a CP eigenstate. In this case we can
have a strong phase difference from a different origin. Since the two mass
eigenstates in the neutral flavoured meson systems ($B_d$, $B_s$, $K^0$ and $D^0$)
have different mass eigenvalues, the time evolution creates a phase such that
\begin{equation}
{\rm Im} [a_1 a_2^*] \sim \sin (\Delta m \, t)
\end{equation}
where $\Delta m$ is the mass difference between the two eigenstates.Thus there
will be a time dependent CP asymmetry which takes the form
\begin{equation}
{\cal A}_{CP} (t) = \frac{
C \, \cos(\Delta m \, t) - S \, \sin(\Delta m \, t)}
{\cosh(\Delta\Gamma \,t/2)
+ D \sinh (\Delta\Gamma \, t/2)}
\end{equation}
where the coefficients satisfy $ C^2 + S^2 +D^2 =1$.
The problem in obtaining quantitative predictions for the CP asymmetries lies in
our inability to perform a first principles calculation of the hadronic matrix elements.
Thus we have to refer to approximate methods which fall into three classes which
I shall briefly describe.
\vspace*{-3mm}
\paragraph{Flavour Symmetries:}
%
%
%
Isospin or more genrerally flavour $SU(3)$ may be used to relate matrix elements
for different processes \cite{FlavSym}.
While isospin is believed to be a good symmetry which is
broken only by electromagnetism and the mass difference between the up and the
down quark, the situation is worse for U- and V-spin which are the two other possible
choices of the $SU(2)$ subgroup of $SU(3)$, since the mass of the strange quark
is significantly higher than the one of the up and the down quark.
The breaking of flavour $SU(3)$ is not easily quantified. only for factorizable
amplitudes $SU(3)$ breaking is included by a factor of $f_\pi / f_K$ which also
give the approximate size of the effects which have to be expected. However, the
intrinsic uncertainties emerging from using flavour symmetries are hard to estimate.
Flavour symmetry arguments are often supplemented by arguments based on
diagram topologies: A rearrangement of color indices will result in a color-suppression
factor $1/N_C = 1/3$, an annihilation topology (i.e. a diagram where two quarks in
a meson have to annihilate) results in a suppression by a factor $f_M / m_M$, where
$f_M$ and $m_M$ are the decay constant and the mass of the meson $M$.
Furthermore, a contraction of quark lines involving a quark loop (penguin contraction)
involves a loop factor, the perturbative value of which is $1/(16 \pi^2)$ but is generally expected to be between this value and unity.
This approach, which is completely model independent,
allows us to have some semi-quantitative insight into rates and
CP asymmetries, however, in many applications the uncertainties of the order of tens
of percent are hard to estimate.
\paragraph{QCD Factorization and SCET:}
%
%
%
It has recently been pointed out that in the limit of infinite heavy quark mass
one may factorize the amplitudes of exclusive non-leptonic two body modes
as sketched in fig.~\ref{figFact}\cite{BBNS}.
Thus in the infinite mass limit the amplitude can be expressed in terms of
a non-perturbative (soft) form factor, the non-pertutbative wave functions
of the hadrons and a perturbatively calculable hard scattering kernel $T$.
This limit may be formulated as an effective field theory, the soft collinear
effective field theory (SCET) .
\begin{figure}
\includegraphics[scale=0.45]{BBNStheo.pdf}
\vspace*{-1cm}
\caption{Sketch of QCD factorization}
\label{figFact}
\vspace*{-1cm}
\end{figure}
The most interesting observation from QCD factorization is that the strong phases
in non-leptonic heavy hardon decays are perturbatively calculable to leading order,
and thus are predicted to be small: Strong phases are either of order $\alpha_s (m)$
or suppressed by powers of $1/m$. However, the quantitative agreement with data
is still not good, indicating sizable corrections of subleading orders in $1/m$. Although
the number of subleading non-perturbative parameters is large, QCD factorization still
provides a systematic approach to exclusive non-leptonic decays.
\paragraph{QCD (Light Cone) Sum Rules:}
%
%
%
QCD sum rules are a well established method for the estimate of hadronic
matrix elements. They rely on parton hadron duality and on the analyticity
of the amplitudes.
%\begin{figure}
% \includegraphics[scale=0.46]{QCDSR11.pdf}
% \includegraphics[scale=0.46]{QCDSR12.pdf}
% \vspace*{-1cm}
%\caption{QCD sum rules}
%\label{figQCDSR}
%\end{figure}
For the case of two-body non-leptonic $B$ decays involving light mesons
in the final state the so-called light cone sum rules are applied \cite{Khod},
where one
of the light mesons is represented by its light cone wave function, while the
decaying meson and the other light meson is interpolated by appropriate
currents. This type of sum rules has been formulated recently directly in SCET.
It is interesting to note that the results from light-cone QCD sum rules
quantitatively agree with the ones from QCD factorization. In particular,
the statement about weak phases is the same in QCD sum rules.
In summary, the main obstacle to calculate CP violation in hadronic decays
and thus to relate the measurements to the fundamental parameters
is the evaluation of the hadronic matrix elements, in particular of their weak
phases. Only in a few cases precise predictions are possible.
\section{CP Violation and ``New Physics''}
%
%
%
Currently there is no convincing idea which explains flavour. Even in grand unified
theories flavour is usually implemented by triplication of the spectrum, which will
not give any explanation.
Most scenarios of ``new physics'' involve additional degrees of freedom, in particular,
many models involve an extended Higgs sector. This in general implies additional
couplings which may carry irreducible phases, i.e. to additional sources of CP
violation. This additional CP violation generically also appears in flavour diagonal
processes, in contradiction with the observed suppression of the e.g. electric dipole
moments of particles. Furthermore, flavour changing neutral currents may appear,
in some models even at tree level.
The pattern of CP violation and flavour parametrized by the CKM Ansatz is very
special and in complete agreement with the observations in particle physics.
For this reason,
scenarios of ``new physics'' are thus formulated often with ``minimal flavour
violation'' (MFV) which means that any flavour structure can be reduced to the
Yukawa couplings and the CKM matrix. This
makes these models consistent with present observations
but does not explain the phenomenon of flavour.
\vspace*{-0.3cm}
\begin{thebibliography}{9}
\bibitem{Sakharov} A.~D.~Sakharov,
Pisma Zh.\ Eksp.\ Teor.\ Fiz.\ {\bf 5} (1967) 32.
%%CITATION = ZFPRA,5,32;%%
\bibitem{Jarlskog} C.~Jarlskog,
Phys.\ Rev.\ Lett.\ {\bf 55}, 1039 (1985).
%%CITATION = PRLTA,55,1039;%%
\bibitem{Shabalin}
E.~P.~Shabalin,
Sov.\ J.\ Nucl.\ Phys.\ {\bf 28}, 75 (1978).
%%CITATION = SJNCA,28,75;%%
\bibitem{exp}
C.~A.~Baker {\it et al.},
arXiv:hep-ex/0602020.
%%CITATION = HEP-EX 0602020;%%
\bibitem{PQ}
R.~D.~Peccei and H.~R.~Quinn, Phys.\ Rev.\ Lett.\ {\bf 38}, 1440 (1977).
%%CITATION = PRLTA,38,1440;%%
\bibitem{FlavSym} C.~W.~Chiang {\it et al.},
Phys.\ Rev.\ D {\bf 70}, 034020 (2004).
%%CITATION = HEP-PH 0404073;%%
\bibitem{BBNS}
M.~Beneke {\it et al.}.
Nucl.\ Phys.\ B {\bf 606}, 245 (2001).
\bibitem{Khod} A.~Khodjamirian,
Nucl.\ Phys.\ B {\bf 605}, 558 (2001).
%%CITATION = HEP-PH 0012271;%%
\end{thebibliography}
\end{document}