\documentclass[a4paper,12pt]{article}
\voffset-1cm
\usepackage{epsfig}
\usepackage{amssymb}
%\usepackage{feynmf}
%\usepackage{axodraw}

\setlength{\voffset}{-1cm}
\setlength{\evensidemargin}{0cm}
\setlength{\oddsidemargin}{0cm}
\setlength{\textwidth}{16.25cm}
\setlength{\textheight}{22.5cm}
\setlength{\floatsep}{0pt}
\setlength{\parskip}{1mm}

\setcounter{topnumber}{6}
\setcounter{bottomnumber}{6}
\setcounter{totalnumber}{10}
\renewcommand{\topfraction}{1.0}
\renewcommand{\bottomfraction}{1.0}
\renewcommand{\textfraction}{0.0}
\renewcommand{\baselinestretch}{1.2}

\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\br}{\begin{eqnarray}}
\newcommand{\er}{\end{eqnarray}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}
\newcommand{\bn}{\begin{enumerate}}
\newcommand{\en}{\end{enumerate}}
\newcommand{\bc}{\begin{center}}
\newcommand{\ec}{\end{center}}
\newcommand{\ul}{\underline}
\newcommand{\ol}{\overline}
\newcommand{\eps}{\epsilon}
\newcommand{\bs}{$B_s\rightarrow \mu^+\mu^-$~}
\newcommand{\cbs}{{\cal B}(B_s\rightarrow\mu^+\mu^-)}

\def\theequation{\arabic{section}.\arabic{equation}}

\newcommand{\lt}{\left}
\newcommand{\rt}{\right}
\newcommand{\no}{\nonumber}
\newcommand{\nn}{\nonumber\\}
\newcommand{\ds}{\displaystyle}
\newcommand{\e}{\epsilon}
\newcommand{\eq}[1]{(\ref{#1})}
\newcommand{\imag}{{\rm Im}\,}
\newcommand{\real}{{\rm Re}\,}
\newcommand{\mev}{\mbox{MeV}}
\newcommand{\gev}{\mbox{GeV}}
\newcommand{\ov}[1]{\overline{#1}}
\def\Ord{\buildrel{\scriptscriptstyle <}\over{\scriptscriptstyle\sim}}
\def\OOrd{\buildrel{\scriptscriptstyle >}\over{\scriptscriptstyle\sim}}


\newcommand{\gsim}{\lower.7ex\hbox{$\;\stackrel{\textstyle>}{\sim}\;$}}
\newcommand{\lsim}{\lower.7ex\hbox{$\;\stackrel{\textstyle<}{\sim}\;$}}
\newcommand{\hd}{{\bf h_d}}
\newcommand{\hu}{{\bf h_u}}
\newcommand{\hhu}{{\bf \hat{h}_u}}
\newcommand{\hhd}{{\bf \hat{h}_d}}
\newcommand{\Eg}{{\bf \hat{E}_g}}
\newcommand{\Eu}{{\bf \hat{E}_u}}
\newcommand{\Au}{{\bf \hat{A}_u}}
\newcommand{\Ad}{{\bf \hat{A}_d}}
\newcommand{\asos}{{\bf 1}}
\newcommand{\ULQ}{{\bf {\cal U}_L^Q}}
\newcommand{\URu}{{\bf {\cal U}_R^u}}
\newcommand{\URd}{{\bf {\cal U}_R^d}}
\newcommand{\Vckm}{{\bf {V}}}
\newcommand{\Md}{{\bf \hat{M}_d}}
\newcommand{\Mu}{{\bf \hat{M}_u}} 
\newcommand{\Rh}{{\bf \hat{R}}}
\newcommand{\Ro}{{\bf {R}}}
\newcommand{\Egt}{{\bf \tilde{E}_g}}
\newcommand{\Eut}{{\bf \tilde{E}_u}}
\newcommand{\Egb}{{\bf {E}_g}}
\newcommand{\Eub}{{\bf {E}_u}}
\newcommand{\DmK}{{\Delta M_K}}
\newcommand{\DmBd}{\Delta M_{B_d}}
\newcommand{\DmBs}{\Delta M_{B_s}}
\newcommand{\DmBq}{\Delta M_{B_q}}
\newcommand{\ek}{{\epsilon_K}}
\newcommand{\epe}{{\frac{\epsilon'}{\epsilon}}}
% ket and bra
\newcommand{\bra}[1]{\langle #1|}
\newcommand{\ket}[1]{|#1\rangle}
\newcommand{\braket}[2]{\langle #1|#2\rangle}


\begin{document}
\tolerance=100000

\begin{flushright}
FERMILAB-Pub-02/226-T\\[-1mm]
MC-TH/2002-05\\[-1mm]
hep-ph/yymmnnn\\[-1mm]
September 2002 
\end{flushright}

\bigskip\bigskip

\begin{center}
{\Large \bf Resummed Effective Lagrangian for Higgs-Mediated }\\[0.3cm] 
{\Large \bf FCNC Interactions in the CP-Violating MSSM}\\[1.7cm]

{{\large Athanasios Dedes}$^{\, a}$ 
{\large and Apostolos Pilaftsis}$^{\, b,c}$ }\\[0.5cm]
{\it $^a$Physikalisches Institut der Universit\"at Bonn,  Nu\ss allee 12, 
 D-53115 Bonn, Germany } \\[3mm]
{\it $^b$Fermilab, P.O. Box 500, Batavia IL 60510, U.S.A.}\\[3mm]
{\it $^c$Department of Physics and Astronomy, University of Manchester,\\ 
Manchester M13 9PL, United Kingdom }
\end{center}

\vspace*{1.cm}\centerline{\bf ABSTRACT}   
\vspace{0.1cm}\noindent   
We  derive    the  properly  resummed   effective     Lagrangian   for
Higgs-mediated flavour-changing neutral-current (FCNC) interactions in
the Minimal Supersymmetric Standard Model (MSSM), without resorting to
particular assumptions based on  the  squark-mass or the  quark-Yukawa
structure  of  the  theory.  In  our  derivation we  also  include the
possibility       of    explicit   CP      violation    through    the
Cabibbo--Kobayashi--Maskawa         mixing   matrix   and         soft
supersymmetry-breaking  mass  terms.  The  advantages  of our resummed
FCNC effective  ~~Lagrangian are  explicitly  demonstrated within  the
context  of   phenomenologically motivated scenarios.   We  obtain new
testable predictions in the large $\tan\beta$ regime of the theory for
CP-conserving  and CP-violating  observables related  to the $K$-  and
$B$-meson  systems,     such  as  $\Delta    M_{K,B}$,   $\epsilon_K$,
$\epsilon'/\epsilon$,  ${\cal B}(B_{d,s}  \to \ell^+\ell^-)$ and their
associated leptonic CP asymmetries.


\vspace*{\fill}
\newpage


\setcounter{equation}{0}
\section{Introduction}
\label{sec:intro}

The  appearance of  too large  FCNC  interactions of  Higgs bosons  to
fermionic  matter is  a  generic feature  of SU(2)$_L\times  $U(1)$_Y$
theories with two and more  Higgs doublets. Unless there is a symmetry
to forbid these Higgs-mediated FCNC  interactions to occur in the bare
Lagrangian of  the model~\cite{GW}, their  unsuppressed existence will
inevitably  lead to  predictions for  rare processes  in the  kaon and
$B$-meson systems  that often  violate experimental limits  by several
orders  of  magnitude~\cite{GW,CS}.   In  the minimal  realization  of
softly-broken   supersymmetry  (SUSY),   the   Minimal  Supersymmetric
Standard  Model  (MSSM),  the  holomorphicity  of  the  superpotential
prevents  the occurrence  of  Higgs-boson FCNCs  by  coupling the  one
Higgs-doublet superfield,  $\widehat{H}_1$, to the  down-quark sector,
and the other one,  $\widehat{H}_2$, to the up-quark sector.  However,
the above  holomorphic property of  the superpotential is  violated by
finite  radiative (threshold)  corrections due  to  soft SUSY-breaking
interactions~\cite{Banks,RH,CGNW}.   As a  consequence, Higgs-mediated
FCNCs reappear at the one-loop level, but are naturally suppressed for
low  and intermediate  values  of $\tan\beta  = \langle  \widehat{H}_2
\rangle  /  \langle   \widehat{H}_1  \rangle$,  i.e.\  for  $\tan\beta
\stackrel{<}{{}_\sim}  20$.  For larger  values of  $\tan\beta$, e.g.\
$\tan\beta \stackrel{>}{{}_\sim} 30$, the FCNCs partially overcome the
loop  suppression factor  $1/(16\pi^2)$ and  become phenomenologically
very  relevant~\cite{FCNC,KB}, especially for  the $K$-  and $B$-meson
systems.

Recently,  the topic  of  Higgs-boson FCNCs  in  the large-$\tan\beta$
limit        of       the     MSSM         has     received       much
attention~\cite{FCNC,Frank,KB,Isidori,Dedes1,ADKT,Ambrosio,Buras,MTW,IN}.
Several approaches have been  devised to implement the non-holomorphic
finite radiative corrections  into  the  phenomenological analysis  of
FCNC  processes,  such  as $K^0\bar{K}^0$  and $B^0\bar{B}^0$ mixings,
$B_s \to X_s  \gamma$   and $B_s\to  \ell^+\ell^-$.  In  most   cases,
however, the  suggested   approaches to threshold   radiative  effects
involve certain explicit  or implicit   assumptions pertinent to   the
squark-mass and the quark-Yukawa structures of the theory, such as the
dominance of the top quark in the FCNC transition amplitudes.  We term
the latter assumption  the $t$-quark dominance hypothesis.   Moreover,
some  of the approaches neglect  higher-order terms in the resummation
of threshold  corrections,  which   become  very important    in   the
large-$\tan\beta$ regime of the theory~\cite{LN}.

In this paper  we derive the effective Lagrangian  that properly takes
into  account the  resummation  of higher-order  threshold effects  on
Higgs-boson  FCNC interactions  to  fermions.  To  accomplish this  in
Section 2, we  also avoid the imposition of  particular assumptions on
the structure of the soft squark masses and the quark-Yukawa couplings
of  the  theory.  Moreover,  we  do  not  rely on  specific  kinematic
approximations   to   the   transition   amplitudes,   such   as   the
aforementioned  $t$-quark  dominance  hypothesis  in the  FCNC  matrix
elements.   In our  derivation of  the effective  Lagrangian,  we also
consider the possibility of  CP violation through two sources: (i)~the
Cabibbo-Kobayashi-Maskawa (CKM)  mixing matrix~\cite{CKM} and (ii)~the
soft  SUSY-violating  mass terms.   As  we  explicitly demonstrate  in
Section~3, our  resummed FCNC effective  Lagrangian gives rise  to new
testable  predictions  for   CP-conserving  as  well  as  CP-violating
observables related  to the  $K$- and $B$-meson  systems. In  the same
section,   we   also  qualitatively   discuss   the  implications   of
$\tan\beta$-enhanced       Higgs-mediated       interactions       for
$\epsilon'/\epsilon$ and  the decays $B_d  \to \phi K_S$ and  $B_d \to
J/\Psi  K_S$.  Section~4  is devoted  to our  numerical analysis  of a
number  of  $K$- and  $B$-meson  observables,  such  as $\Delta  M_K$,
$\epsilon_K$,   $\Delta   M_{B_{d,s}}$,   ${\cal   B}(   B_{d,s}   \to
\ell^+\ell^-  )$ and  their associated  leptonic CP  asymmetries.  Our
conclusions are summarized in Section~5.


\setcounter{equation}{0}
\section{Resummed FCNC effective Lagrangian}
\label{sec:sec2}

To elucidate our points, let  us  first consider the following  simple
form for the effective  Yukawa Lagrangian governing the Higgs-mediated
FCNC interactions in the $d$-quark sector:
\begin{eqnarray}
  \label{lag}
-{\cal L}_Y\ =\ \bar{d}_R^0\, \hd \Big[\, \Phi^{0*}_1\: +\:
\Phi^{0*}_2\, \Big( \Eg\: +\: \Eu\, \hu^\dagger \hu \Big)\,\Big]\, d_L^0\ +\
\Phi_2^0\, \bar{u}_R^0\, \hu\, u_L^0\ +\ {\rm H.c.} \,,
\end{eqnarray}
where $\Phi^0_{1,2}$ are  the   neutral components of the   two  Higgs
doublets\footnote{Throughout the paper, we follow the notation and the
CP-phase conventions of~\cite{CEPW}.}   and the superscript `0' on the
$d$-  and $u$-type  quarks denotes fields  in  the  interaction basis.
In~(\ref{lag}), $\hd$ and $\hu$  are $3\times 3$-dimensional down- and
up-quark Yukawa matrices, and
\begin{eqnarray}
  \label{Eg}
\Eg &=& \asos\, 
\frac{2 \alpha_s}{3\pi}\, m_{\tilde{g}}^* \mu^*\, I(m_{\tilde{Q}_L}^2,
m_{\tilde{d}_R}^2,|m_{\tilde{g}}|^2) \;, \\
  \label{Eu}
\Eu &=& \asos\, \frac{1}{16\pi^2}\,\mu^* A_U^*\, I(m_{\tilde{Q}_L}^2,
m_{\tilde{u}_R}^2,|\mu|^2) 
\end{eqnarray}
are finite non-holomorphic  radiative effects, where the loop integral
$I(x,y,z)$ is given by 
\begin{equation}
  \label{Ixyz}
I(x,y,z)\ =\ \frac{ xy \ln (x/y)\: +\: y z \ln (y/z)\: +\: x z \ln (z/x)}
{(x-y)\,(y-z)\,(x-z)}\ ,
\end{equation}
with $I(x,x,x) = 1/(2x)$.  To keep  things simple in the beginning, we
have assumed that the bilinear and trilinear soft SUSY-breaking masses
in~(\ref{Eg}) and~(\ref{Eu}),   $m_{\tilde{Q}_L}$,  $m_{\tilde{u}_R}$,
$m_{\tilde{d}_R}$   and  $A_U$,   are flavour-diagonal   and universal
already    at  the SUSY  scale.   The   consequences  of relaxing this
assumption will be discussed later on.

{}From~(\ref{lag}), we can easily  write down the effective Lagrangian
relevant to the effective $d$- and $u$-type quark masses:
\begin{eqnarray}
  \label{masslag}
-{\cal L}_{\rm mass}\ =\ \frac{v_1}{\sqrt{2}}\, \bar{d}_R^0\,  \hd \Big[\,
\asos\: +\: \tan\beta\, \Big(\,\Eg\: +\: \Eu\, 
\hu^\dagger \hu\, \Big)\, \Big]\, d_L^0\ +\
\frac{v_2}{\sqrt{2}}\, \bar{u}_R^0\, \hu\, u_L^0\ +\ {\rm H.c.}
\end{eqnarray}
Our next step is to redefine the quark fields as follows:
\begin{eqnarray}
  \label{rotation}
u_L^0\ =\ \ULQ\, u_L \,,\quad  d_L^0\ =\ \ULQ\, \Vckm\, d_L\,,\quad 
u_L^0\ =\ \URu\, u_R\,,\quad d_R^0\ =\ \URd\, d_R \,,
\end{eqnarray}
where  $\ULQ$, $\URu$, $\URd$  and $\Vckm$ are 3-by-3 unitary matrices
that relate  the weak   to mass  eigenstates of  quarks.    Evidently,
$\Vckm$   is     by     construction the      physical   CKM   matrix.
Substituting~(\ref{rotation}) into (\ref{masslag}) yields
\begin{eqnarray}
  \label{diago}
-{\cal L}_{\rm mass}  \!\!\!&=&\!\!\! 
\frac{v_1}{\sqrt{2}}\, \bar{d}_R\,  \URd^\dagger \hd \ULQ\,
\Big[\, \asos\: +\: \tan\beta\, \Big(\,\Eg\: + \: \Eu
|\hhu|^2\,\Big)\, \Big]\, \Vckm\, d_L\
+\ \frac{v_2}{\sqrt{2}}\, \bar{u}_R\, \hhu\, u_L\ +\ {\rm H.c.}
\nonumber \\[3mm] 
&=& \bar{d}_R\, \Md\, d_L\ +\ \bar{u}_R\, \Mu\, u_L\ +\ {\rm H.c.}\;,
\end{eqnarray}
where   $\Md$  and $\Mu$  are the    physical  $d$- and $u$-quark mass
matrices, respectively. Consistency of~(\ref{diago}) implies
\begin{eqnarray}
  \label{Mu}
\Mu &=& \frac{v_2}{\sqrt{2}}\, \hhu \;,\\
  \label{hd}
\URd^\dagger \hd \ULQ &=& \frac{\sqrt{2}}{v_1}\, \Md \Vckm^\dagger\,
\Rh^{-1}\;, 
\end{eqnarray}
with 
\begin{eqnarray}
  \label{Rhat}
\Rh \ =\  \asos\: +\: \Eg \tan\beta \: +\: \Eu \tan\beta\, |\hhu|^2  \;.
\end{eqnarray}
Observe that the matrix $\Rh$ cannot be zero, as  this would result in
massless $d$ quarks.

With the help  of~(\ref{rotation}) and~(\ref{hd}), we can now  express
our original  Yukawa  Lagrangian  (\ref{lag}) in   terms  of the  mass
eigenstates $d_{L,R}$ and $u_{L,R}$ in a resummed form:
\begin{eqnarray}
  \label{LYeff}
-{\cal L}_Y &=& \frac{\sqrt{2}}{v_1}\,  
\bar{d}_R\,  \Md \Vckm^\dagger \Rh^{-1} 
\Big[\, \Phi_1^{0*}\: +\: \Phi_2^{0*} \Big(\Eg\: +\: \Eu |\hhu|^2\Big)\, \Big]
\Vckm\, d_L\ +\ \Phi_2^0\, \bar{u}_R \hhu u_L\ +\ {\rm H.c.} \nonumber\\
&=& \frac{\sqrt{2}}{v_1}\,
\bigg( \Phi_1^{0*}\: -\: \frac{\Phi_2^{0*}}{\tan\beta}\, \bigg)\,
\bar{d}_R\,  \Md \Vckm^\dagger \Rh^{-1}\Vckm\, d_L\ +\
\frac{\sqrt{2}}{v_1}\, \frac{\Phi_2^{0*}}{\tan\beta}\, 
\bar{d}_R\,  \Md\, d_L\nonumber\\
&&+\ \Phi_2^0\, \bar{u}_R \hhu u_L\ +\ {\rm H.c.}
\end{eqnarray}
In deriving  the last equality  in (\ref{LYeff}), we have employed the
relation: $\Rh^{-1}     (\Eg   +  \Eu   |\hhu|^2  )\  =\     (\asos  -
\Rh^{-1})/\tan\beta$.
 
It will be very illuminating to see how the FCNC part of (\ref{LYeff})
compares  with   the    literature,  e.g.\ with    that   obtained  in
Ref.~\cite{KB}.  To  this end,  let  us first  assume that $\asos +\Eg
\tan\beta \ne 0$ and decompose $\Rh^{-1}$ as follows:
\begin{eqnarray}
  \label{Rhdec}
\Rh^{-1}\ =\ \frac{\asos}{\asos+\Eg \tan\beta}\ +\ \chi_{\rm FC}\; ,  
\end{eqnarray}
where  $\chi_{\rm  FC}$ is  the  diagonal  matrix
\begin{equation}
  \label{chiFC}
\chi_{\rm FC}\ =\ -\,
\frac{\Eu\, |{\bf \hat{h}_u}|^2\, \tan\beta}
{(\asos\, +\, \Eg \tan\beta)\, \Rh}\ \; .  
\end{equation}
Making use  of the above  linear decomposition  of $\Rh^{-1}$  and the
unitarity  of the CKM  matrix in~(\ref{LYeff}),  the  FCNC part of our
resummed effective Lagrangian reads
\begin{eqnarray}
  \label{cbk}
-{\cal L}_{\rm FCNC} \!\!&=&\!\! 
 \frac{\sqrt{2}}{v_1}\, \bigg( \Phi_1^{0*}\: -\:
 \frac{\Phi_2^{0*}}{\tan\beta}\, \bigg)\, 
\bar{d}_{iR}\, m_{d_i}\,
\bigg(\, V_{ti}^* \chi_{\rm FC}^{(t)} V_{tj}\: +\: V_{ci}^* \chi_{\rm FC}^{(c)}
V_{cj}\: +\: V_{ui}^* \chi_{\rm FC}^{(u)} V_{uj}\, \bigg)  d_{jL}
\,,\nonumber\\ &&
\end{eqnarray}
where $\chi_{\rm FC}^{(u,c,t)}$ are the diagonal entries of $\chi_{\rm
FC}$  and   summation over $i,j   =  d,s,b$  is  understood.  The term
proportional    to    $\chi_{\rm  FC}^{(t)}$   gives    the top  quark
contribution,  which   is  the  result   of~\cite{KB}  and  subsequent
articles~\cite{Isidori,Ambrosio,Buras}.   However,  we  should  remark
here that the frequently-used approximation called above the top-quark
dominance hypothesis, which results  from  assuming that only  the top
quark dominates the FCNC   Higgs couplings to  $d$ quarks,  cannot  be
entirely  justified from   considerations    based only  on    minimal
flavour-violation models~\cite{KB,Isidori,Ambrosio,Buras},  for  which
the CKM  matrix is the only source  of flavour and  CP violations.  In
fact,  the  other  terms   in~(\ref{cbk})  and  especially   the   one
proportional   to $\chi_{\rm     FC}^{(c)}$ due  to  the   charm-quark
contribution becomes very relevant in the limit $\asos + \Eg \tan\beta
\to 0$.   Also, in the same  limit in~(\ref{cbk}),  the singularity in
$\chi_{\rm  FC}^{(t)}$   is canceled  against  the  singularities  of
$\chi_{\rm FC}^{(c)}$ and $\chi_{\rm  FC}^{(u)}$  as a result of   the
unitarity of $\Vckm$.  In this context,  we should note that the limit
$\asos  + \Eg \tan\beta  \to  0$ is not  attainable  before the theory
itself   reaches   a   non-perturbative regime.    Requiring  that all
$d$-quark Yukawa couplings are perturbative, we can estimate the lower
bound,  $|\asos   +  \Eg  \tan\beta|   \gsim 2.5\times  10^{-2}$,  for
$\tan\beta=50$.\footnote{To  obtain  this lower  limit, we simply take
the   trace of  the   square   of~(\ref{hd})  and   demand that  ${\rm
Tr}|\hhd|^2  <  3$, or equivalently  ${\rm  Tr} [ (\Rh^{-1})^\dagger V
|\Md|^2  V^\dagger \Rh^{-1} ] < 3  v_1^2/ 2$.  The latter implies that
$|\asos\: +\: \Eg \tan\beta|\ \gsim\ (m_s \tan\beta)/(\sqrt{3}\, m_t)\
=\ 5\times  10^{-4} \tan\beta$. Finally, it  is amusing to notice that
if $\Im  m \Eg =0$ and $\Re  e \Eg < 0$, an  additional upper bound on
$\tan\beta$, $\tan\beta \lsim   1/|\Eg|$, may  be derived.}   Although
$|\asos\: +\: \Eg \tan\beta|$  must not vanish in perturbation theory,
it  can  be  sufficiently  close  to  zero,   so that   the  $c$-quark
contribution becomes competitive with the $t$-quark one.
 
So far, we  have assumed that  the  radiatively-induced matrices $\Eg$
and   $\Eu$  in   the   effective  Yukawa   Lagrangian~(\ref{lag}) are
proportional to the unity matrix.  However, this assumption of flavour
universality is rather specific, as  it gets generally invalidated  by
the  mixing  of  the squark generations,   the  soft trilinear  Yukawa
couplings   and  renormalization-group  (RG)  running  of    the  soft
SUSY-breaking parameters from the unification to the low-energy scale.
In this  respect,  the minimal-flavour-violation  hypothesis, although
better motivated, should also be viewed  as a very particular way from
minimally departing from universality.

Nevertheless, given that threshold radiative effects  on the up sector
are  negligible, especially  for large  values  of  $\tan\beta$, it is
straightforward  to  derive   the   general resummed   form   for  the
Higgs-mediated  FCNC  effective  Lagrangian.   Starting  from  general
non-diagonal  matrices $\Egt$ and  $\Eut$ in~(\ref{lag}) and following
steps very  analogous to  those from~(\ref{lag})  to~(\ref{LYeff}), we
arrive  at  the same form as   in~(\ref{LYeff}) for  the resummed FCNC
effective Lagrangian, but with $\Eg$, $\Eu$ and $\Rh$ replaced by
\begin{eqnarray}
  \label{Epars} 
\Egb &=& \ULQ^\dagger \Egt \ULQ\,, \qquad\qquad 
\Eub\ =\ \ULQ^\dagger \Eut \ULQ\,, \\ 
  \label{Ro}
\Ro &=& \asos\: +\: \Egb \tan\beta\:  +\: \Eub \tan\beta\, |\hhu|^2\:
+\: \dots \,,
\end{eqnarray}
where   the  ellipses in~(\ref{Ro})  denote  additional   (generically
sub-dominant) threshold  effects~\cite{comment}.  Notice  that besides
the possibility of  squark non-universality, the unitary matrix $\ULQ$
in~(\ref{Epars}), which is only constrained by the relation $|\hhu|^2=
\ULQ^\dagger  \hu^\dagger \hu \ULQ$, introduces additional non-trivial
model dependence in the matrices $\Egb$,  $\Eub$ and $\Ro$.  Moreover,
even in   minimal  flavour-violating  scenarios,  $\Ro$   may  contain
additional  threshold  contributions    proportional  to $\ULQ^\dagger
\hd^\dagger \hd \ULQ$ induced by RG running of  the squark masses from
the unification to  the soft SUSY-breaking scale.  Those contributions
may also  be   resummed  individually  by taking  appropriately    the
hermitian   square   of   the  modified~(\ref{hd})  and   solving  for
$\ULQ^\dagger \hd^\dagger  \hd \ULQ$.  This  last step may involve the
use of  iterative  or other numerical  methods.    Thus, even in  this
general case with  a  non-universal squark  sector, the resummed  FCNC
couplings of the Higgs bosons to down-type  quarks can be successfully
parameterized in terms of a  well-defined set of parameters, i.e.\ the
soft  SUSY-breaking masses, the $u$-   and  $d$-quark masses, and  the
unitary matrices $\ULQ$ and $\Vckm$.

In  our last step in  deriving the resummed FCNC effective Lagrangian,
we  express  the Higgs fields $\Phi^0_{1,2}$  in   terms of their mass
eigenstates $H_{1,2,3}$ and the neutral would-be Goldstone boson $G^0$
in  the   presence  of  CP   violation~\cite{APLB}.    Following   the
conventions  of~\cite{CEPW}, we  relate the weak   to mass eigenstates
through the linear transformations:
\begin{eqnarray} 
  \label{CPtr}
\Phi_1^0 &=& \frac{1}{\sqrt{2}}\, \Big[\, O_{1i}\, H_i\ +\ i\, 
\Big( \cos\beta ~G^0\: -\: \sin\beta ~O_{3i}\, H_i\,\Big)\, \Big]\,,
  \nonumber \\[3mm]
\Phi_2^0 &=& \frac{1}{\sqrt{2}}\, \Big[\, O_{2i}\, H_i\ +\ i\,
\Big( \sin\beta ~G^0\: +\: \cos\beta ~O_{3i}\, H_i\,\Big)\, \Big] \,,
\end{eqnarray}
where $O$ is a 3-by-3 orthogonal matrix that accounts for CP-violating
Higgs-mixing  effects~\cite{PW}.   If we  substitute  the  weak  Higgs
fields $\Phi^0_{1,2}$ by virtue of~(\ref{CPtr}) into (\ref{LYeff}), we
obtain the properly resummed  effective Lagrangian for the diagonal as
well as off-diagonal Higgs interactions to $d$ quarks,
\begin{eqnarray}
  \label{master}
{\cal L}_{H_i\bar{d}   d'}\  =\ -\,  \frac{g_w}{2  M_W}\, \sum_{i=1}^3\,
H_i\:  \bar{d}\, \Big(\, \Md\, {{\bf g}}_{H_i\bar{d}  d' }^L\, P_L\ +\
{{\bf g}}_{H_i\bar{d}  d' }^R\, {\bf \hat{M}_{d'}} P_R\, \Big)\, d' \,,
\end{eqnarray}
where $P_{L\, (R)} = [1 -(+)\, \gamma_5 ]/2$ and
\begin{eqnarray}
  \label{couplings}
{{\bf g}}_{H_i\bar{d} d' }^L \!\!&=&\!\! 
\Vckm^\dagger\Ro^{-1}\Vckm\, \frac{O_{1i}}{\cos\beta}\ +\
\bigg(\asos\: -\: \Vckm^\dagger\Ro^{-1}\Vckm\,\bigg)\,\frac{O_{2i}}{\sin\beta}
\ -\ i\, \bigg( \asos\: -\: \frac{1}{\cos^2 \beta}\, 
\Vckm^\dagger\Ro^{-1}\Vckm\, \bigg)\, \frac{O_{3i}}{\tan\beta}\ ,
\nonumber\\[3mm]
{{\bf g}}_{H_i\bar{d} d' }^R \!\!&=&\!\!  \big(\,{{\bf g}}_{H_i\bar{d} d'
  }^L\,\big)^\dagger\, .
\end{eqnarray}
The $3\times 3$-dimensional  matrix  $\Ro$ in~(\ref{couplings}), which
resums all  radiative  threshold  effects,  is given    by~(\ref{Ro}).
Equation~(\ref{master}), along with~(\ref{couplings}), constitutes the
major result of the present  paper, which will  be extensively used in
our phenomenological discussions in Section~3.

Finally,  let us   summarize the  most   important  properties  of the
resummed effective Lagrangian~(\ref{master}):

\begin{itemize}

\item[(i)] The  FCNC  interactions in~(\ref{master}) are  described by
$\tan^2\beta$-enhanced terms that   are proportional to  $O_{1i}$  and
$O_{3i}$ and to  $\Vckm^\dagger  \Ro^{-1} \Vckm$ in~(\ref{couplings}).
These $\tan^2\beta$-enhanced FCNC   terms properly  take into  account
resummation,\footnote{In addition to the non-holomorphic contributions
we   have  been considering here,  there    are in general holomorphic
radiative effects on the $\Phi^0_1$ coupling to  $d$ quarks which have
an analogous matrix  structure,   i.e.\ ${\bf \varepsilon_g}    + {\bf
\varepsilon_u}  |\hhu|^2$.   These  additional holomorphic  terms  are
generally small, typically    of order $10^{-2}$, and   only  slightly
modify the form of the matrix  ${\bf R}$ to: ${\bf R}  = {\bf 1}\, +\,
({\bf 1} + {\bf \varepsilon_g}  + {\bf \varepsilon_u} |\hhu |^2 )^{-1}
(\Egb\,  +\,  \Eub |\hhu    |^2)\,  \tan\beta$.  Obviously,   such   a
modification does  not  alter   the  general form  of  the   effective
couplings in~(\ref{couplings}) and is beyond the one-loop order of our
resummation.  Therefore, these additional  small holomorphic terms can
be safely  neglected.}  squark-mass  non-universality and CP-violating
effects.

\item[(ii)] The resummation matrix ${\bf R}$  controls the strength of
the   Higgs-mediated FCNC  effects.   For  instance,  if  ${\bf R}$ is
proportional  to unity, then a  kind  of a Glashow--Iliopoulos--Maiani
(GIM)  cancellation mechanism~\cite{GIM}   becomes  operative and  the
Higgs-boson contributions  to all FCNC observables  vanish identically
in this case.  Furthermore, as  well as the  top quark,  the other two
lighter up-type quarks can also give significant contributions to FCNC
transition amplitudes, which  are naturally included in~(\ref{master})
through the resummation matrix ${\bf R}$.

\item[(iii)] In the CP-invariant   limit of the theory,  the effective
couplings ${{\bf  g}}_{H_i\bar{d} d' }^{L,R}$ are  either pure real or
pure imaginary numbers. Moreover, in   the limit $\Vckm \to {\bf  1}$,
the    effective    Lagrangian   (\ref{master})     of the    diagonal
Higgs-couplings to down quarks is in  excellent agreement with the one
presented in~\cite{CEPW}.

\item[(iv)] If  $M_{H^+} \sim M_{H_{2,3}}  \gg M_{H_1}$, one  can show
that  $O_{11}  \approx  \cos\beta$,  $O_{21}  \approx  \sin\beta$  and
$O_{31}\approx  0$.  In this  case, the  $H_1$-coupling to  $d$ quarks
becomes SM  like and so  $H_1$-mediated FCNC effects are  getting very
suppressed.  However, in the above decoupling limit of the two heavier
Higgs bosons, FCNC effects in  the $H_2$-$H_3$ sector do not decouple,
are $\tan\beta$-enhanced and can therefore be sizeable.


\end{itemize}

In the next section, we will closely study the above properties of the
$\tan\beta$-enhanced  FCNC effects  mediated by  Higgs  bosons on rare
processes  and CP asymmetries    related to  the   $K$-  and $B$-meson
systems.

 
\setcounter{equation}{0}
\section{Applications to $K$- and $B$-meson systems}


We   shall    now  analyze  the   impact    of our  resummed effective
Lagrangian~(\ref{master})   for  Higgs-mediated  FCNC  interactions on
representative   $K$- and  $B$-meson   observables.  For comprehensive
reviews   on  $K$- and   $B$-meson   physics,  we  refer    the reader
to~\cite{kaonrev,Neubert,Branco}.

\subsection{$\DmK$, $\ek$ and $\epsilon'/\epsilon$}

Our starting point is the effective Hamiltonian for the $\Delta S = 2$
interactions,
\begin{eqnarray}
  \label{DS2}
H_{\rm eff}^{\Delta S=2}\ =\ 
\frac{G_F^2}{16\pi^2}\, M_W^2\, \sum_i\, C_i(\mu)\, Q_i (\mu ) \;,
\end{eqnarray}
where $G_F=1.16639\times    10^{-5}~{\rm  GeV}^{-2}$ is     the  Fermi
constant, $C_i   (\mu)$ are  the  scale-dependent Wilson  coefficients
associated to the $\Delta S=2$  quark-dependent operators $Q_i$.  Note
that the CKM  matrix elements in~(\ref{DS2})  have  been absorbed into
the  Wilson coefficients.  The $\Delta  S=2$  operators  $Q_i$ may  be
summarized as follows:
\begin{eqnarray}
  \label{operbasis}
Q_1^{\rm VLL} &=&(\bar{s}\gamma_\mu P_L d)(\bar{s}\gamma^\mu P_L d)\,,
\qquad\!
Q_1^{\rm VRR}\ =\ (\bar{s}\gamma_\mu P_R d)(\bar{s}\gamma^\mu P_R d)\,,
\nonumber\\[3mm]
Q_1^{\rm LR} &=&(\bar{s}\gamma_\mu P_L d)(\bar{s}\gamma^\mu P_R d)\, ,
\qquad\
Q_2^{\rm LR}\ =\ (\bar{s} P_L d)(\bar{s} P_R d)\,,
\nonumber \\[3mm]
Q_1^{\rm SLL} &=&(\bar{s} P_L d)(\bar{s} P_L d)\, ,\qquad\qquad
Q_1^{\rm SRR}\ =\ (\bar{s} P_R d)(\bar{s} P_R d)\,,
\nonumber \\[3mm]
Q_2^{\rm SLL} &=&(\bar{s}\sigma_{\mu\nu} P_L d)(\bar{s}\sigma^{\mu\nu} 
P_L d)\, ,\quad
Q_2^{\rm SRR}\ =\ (\bar{s}\sigma_{\mu\nu} P_R d)(\bar{s}\sigma^{\mu\nu} 
P_R d) \;,
\end{eqnarray}
with  $\sigma_{\mu\nu}=\frac{1}{2}[\gamma_\mu,\gamma_\nu]$. Here, much
of our discussion and notation follows Ref.~\cite{Jager}.  

It now proves convenient to  decompose both the $K^0$-$\bar{K}^0$ mass
difference $\DmK$ and  the known  CP-violating mixing parameter  $\ek$
into a SM and a SUSY contribution:
\begin{eqnarray}
\Delta M_K &=& M_{K_L} \: -\: M_{K_S}\ =\ 
\Delta M_K^{\rm SM} + \Delta M_K^{\rm SUSY} \;,\nonumber\\ 
\epsilon_K &=& \epsilon_K^{\rm SM} + \epsilon_K^{\rm SUSY} \;.
\end{eqnarray}
To a good approximation, one has
\begin{eqnarray}
\Delta M_K^{\rm SM,\,SUSY} &=& 2\, \Re e\, 
\bra{\bar{K}^0}\,
H_{\rm eff}^{\Delta S=2}\, \ket{K^0}_{\rm SM,\,SUSY} \;, \\[3mm]
\epsilon_K^{\rm SM,\,SUSY} &=& \frac{{\rm exp}(i\pi/ 4)}{\sqrt{2}\, 
\Delta M_K}\ \Im m\,  \bra{\bar{K}^0}\, H_{\rm eff}^{\Delta S=2}\,
\ket{K^0}_{\rm SM,\, SUSY} \; .
\end{eqnarray}
The SUSY  contribution to  the matrix   element $\bra{\bar{K}^0}H_{\rm
eff}^{\Delta   S=2}   \ket{K^0}_{\rm   SUSY}$ may  be   written   down
as~\cite{Jager,Chankowski}
\begin{eqnarray}
  \label{Kme}
\bra{\bar{K}^0}\, H_{\rm eff}^{\Delta S=2}\, \ket{K^0}_{\rm SUSY} &=&
\frac{G_F^2}{12\pi^2}\, M_W^2 m_K F_K^2 \eta_2 \hat{B}_K\, \Big[\,
\bar{P}_1^{\rm VLL}\, \Big(\, C_1^{\rm VLL}\: +\:  C_1^{\rm VRR}
\,\Big)\nonumber \\[3mm]
&&+\: \bar{P}_1^{\rm LR}\, C_1^{\rm LR}\ 
+\ \bar{P}_2^{\rm LR}\, C_2^{\rm LR} \nonumber\\[3mm]
&&+\: \bar{P}_1^{\rm SLL}\, \Big(\, C_1^{\rm SLL}\: +\: C_1^{\rm SRR}\, \Big)\
+\ \bar{P}_2^{\rm SLL}\, 
\Big(\, C_2^{\rm SLL}\: +\: C_2^{\rm SRR}\, \Big)\, \Big]\,,\qquad 
\end{eqnarray}
where  $m_K=498$   MeV,   $F_K=160$~MeV  and   $\bar{P}$'s  are    the
next-to-leading order (NLO)  QCD  factors  that include  the  relevant
hadronic                                                        matrix
elements~\cite{Jager,Ciuchini,Misiak,Becirevic,Chankowski}.     At the
scale $\mu=2$ GeV, they are given by~\cite{Jager}
\begin{eqnarray}
  \label{Pbars}
\bar{P}_1^{\rm VLL} = 0.25\,,\quad
\bar{P}_1^{\rm LR} = -18.6\,, \quad
\bar{P}_2^{\rm LR} = 30.6\,, \quad
\bar{P}_1^{\rm SLL} = -9.3\,, \quad 
\bar{P}_2^{\rm SLL} = -16.6\;.
\end{eqnarray}
On   obtaining~(\ref{Pbars}),  we  have  used  the   numerical values:
$\eta_2=0.57$ and $\hat{B}_K=0.85 \pm 0.15$.

{}From studies in  the     CP-conserving MSSM with  minimal    flavour
violation~\cite{Chankowski,Isidori,Buras},  it is known that for large
values   of   $\tan\beta   \stackrel{>}{{}_\sim}   40$,   the dominant
contribution     to    $\bra{\bar{K}^0}H_{\rm    eff}^{\Delta     S=2}
\ket{K^0}_{\rm SUSY}$  comes  from  Higgs-mediated two-loop double
penguin (DP)   diagrams proportional to   $C_2^{\rm  LR}$.  Within the
framework     of   our    large-$\tan\beta$-resummed  FCNC   effective
Lagrangian~(\ref{master}) that   includes  CP  violation,  the  Wilson
coefficients due to DP graphs are found to be
\begin{eqnarray}
  \label{dpkaon}
C_1^{\rm SLL\,(DP)} &=& -\,\frac{16\pi^2 m_s^2}{\sqrt{2}\, G_F M_W^2}\,
\sum_{i=1}^3\, \frac{g_{H_i\bar{s}d}^L\, g_{H_i\bar{s}d}^L }{M_{H_i}^2} \;, 
\nonumber \\[3mm]
C_1^{\rm SRR\,(DP)} &=& -\,\frac{16\pi^2 m_d^2}{\sqrt{2}\, G_F M_W^2}\,
\sum_{i=1}^3\, \frac{g_{H_i\bar{s}d}^R\, g_{H_i\bar{s}d}^R }{M_{H_i}^2} \;,  
\nonumber \\[3mm]
C_2^{\rm LR\,(DP)} &=& -\,\frac{32\pi^2 m_s m_d}{\sqrt{2}\, G_F M_W^2}\,
\sum_{i=1}^3\, \frac{g_{H_i\bar{s}d}^L\, g_{H_i\bar{s}d}^R }{M_{H_i}^2} \;,
\end{eqnarray}
where   the  couplings         $g_{H_i\bar{s}d}^{L,R}$    are    given
in~(\ref{couplings}).    Note   that   the   DP  Wilson   coefficients
in~(\ref{dpkaon})   exhibit  a  $\tan^4\beta$-dependence and, although
being  two-loop suppressed,  they   become very significant for  large
values of $\tan\beta \stackrel{>}{{}_\sim} 40$. Analogous DP-graphs 
have been recently~\cite{DER} found to dominate the lepton flavour
violating B-decays such as $B_{s,d}\to l^+ l^{-'}$. In this case,
the second vertex in (\ref{dpkaon}) has
to be replaced by  $g_{H_i\bar{l}l'}^{L,R}$ which can straightforwardly
been derived by following our analysis in Section 2 with appropriate
replacements for the lepton sector.

In addition to the  aforementioned DP contributions due to Higgs-boson
exchange   graphs,  there  exist  relevant  one-loop  contributions to
$\bra{\bar{K}^0}H_{\rm eff}^{\Delta S=2}\ket{K^0}_{\rm SUSY}$ at large
$\tan\beta$: (i) the $t$-$H^\pm$ box contribution to $C_2^{\rm LR}$ of
the  two-Higgs-doublet-model   (2HDM)  type, and  (ii)  the   one-loop
chargino-stop box diagram contributing to  $C_1^{\rm SLL}$.  The first
contribution~(i)  becomes significant,  up   to  10\%,  only  in   the
kinematic region $M_{H^\pm}\approx   m_t$.  In this  case,  to a  good
approximation,     $C_1^{\rm          SLL}$         may    be    given
by~\cite{Chankowski,Isidori}
\begin{eqnarray}
  \label{2HDM}
C_2^{\rm LR\,(2HDM)} \approx -\,\frac{2 m_s m_d}{M_W^2}\, 
(V_{ts}^*V_{td})^2\, \tan^2\beta \;. 
\end{eqnarray}
Note that   the light-quark   masses contained  in~(\ref{dpkaon})  and
(\ref{2HDM})  are running  and are  evaluated   at the  top-quark mass
scale, i.e.\ $m_s(m_t)  \simeq 61$ MeV, $m_d (m_t)  \simeq 4$ MeV. The
second contribution~(ii)  becomes non-negligible only for small values
of  the $\mu$-parameter~\cite{Isidori,Chankowski},  i.e.\ for  $|\mu |
\stackrel{<}{{}_\sim} 200$~GeV.

In  view of  the above  discussion,  the kinematic parameter range  of
interest to us is:
\begin{eqnarray}
  \label{par}
M_{\rm SUSY},\; \mu\ \gg\ m_t\, , \qquad \tan\beta\ \gsim\ 40\,, 
\end{eqnarray}
including   the  case   $M_{H^\pm}  \approx    m_t$,   for which   the
Higgs-mediated   DP  effects   can    dominate  the  $K^0$-$\bar{K}^0$
transition amplitude.  Thus, taking also into account the sub-dominant
2HDM contribution~(\ref{2HDM}), formula~(\ref{Kme}) simplifies to
\begin{eqnarray}
  \label{Ksusy}
\bra{\bar{K}^0}\,H_{\rm eff}^{\Delta S=2}\,\ket{K^0}_{\rm SUSY}
\!\!&=&\!\! 
4.6\times 10^{-11} {\rm ~GeV}\ 
\bigg(\, \frac{F_K}{160 {\rm ~MeV}}\, \bigg)^2\,
\bigg(\,\frac{\eta_2}{0.57}\, \bigg)\,
\bigg(\,\frac{\hat{B}_K}{0.85}\, \bigg)\nonumber\\[3mm]
&&\hspace{-2.5cm} \times\,
\biggl [\, 30.6\, \biggl (\, C_2^{\rm  LR\,(DP)}\: +\: 
C_2^{\rm  LR\,(2HDM)} \,\biggr )\ -\ 9.3\, 
\biggl (\, C_1^{\rm  SLL\,(DP)}\: +\: 
C_1^{\rm  SRR\,(DP)}\, \biggr)\, \biggr ] \,.\qquad 
\end{eqnarray}
Observe  that the Wilson coefficients $C_1$  and $C_2$ contribute with
opposite       signs     to   $\bra{\bar{K}^0}\,H_{\rm    eff}^{\Delta
S=2}\,\ket{K^0}_{\rm  SUSY}$. Based   on~(\ref{Ksusy}), we will   give
numerical estimates of $\Delta M_K$ and $\epsilon_K$ in Section 4.

We  now     turn our  attention  to the     computation of  the direct
CP-violation parameter $\epsilon'/\epsilon$ in the kaon system, due to
CP-violating  Higgs-mediated FCNC interactions.  In the SM, unlike the
$Z$-penguin graphs, the  Higgs-penguin contribution to  $K\to \pi\pi$,
which is proportional to the operator
\begin{eqnarray}
 \label{qhiggs}
Q_H\ =\ (\bar{s} P_L d)\, \sum_{q=u,d,s}\, (\bar{q}q) \;,
\end{eqnarray}
has a  suppressed  Wilson coefficient proportional to  $m^2_q/M^2_H$,
where the  SM Higgs-boson mass $M_H$ is  subject into the experimental
bound: $M_H \stackrel{>}{{}_\sim} 114$~GeV.  One  might even think  of
the possibility that  the operator $Q_H$ in~(\ref{qhiggs}), which  has
enhanced Wilson  coefficients for $q=c,b$,  mixes with the gluonic and
electroweak  penguins,  as  well as  with  the  other  basis operators
in~(\ref{operbasis}).     However,  as    was  already pointed     out
in~\cite{Buchalla}, this is not the case,  and so the SM-Higgs penguin
effects remain negligible.

The  situation changes  drastically  in  the   MSSM with explicit   CP
violation, since the   Higgs-boson FCNC couplings to  down-type quarks
are substantially enhanced by $\tan^2\beta$-dependent terms, for large
values of  $\tan\beta$.    Furthermore, besides  the CKM  phase,   the
presence of complex  soft SUSY-breaking masses with large CP-violating
phases   may further   increase    the Higgs-boson   FCNC  effects  on
$\epsilon'/\epsilon$.  In fact, we note  that soft CP-odd phases could
even be   the   only   source~\cite{Banks}  to  account  for    direct
CP violation.

To reliably  estimate the new SUSY  effect on $\epsilon'/\epsilon$ due
to Higgs-boson  FCNC  interactions,   we  normalize  each   individual
contribution with respect to the dominant SM contribution arising from
the   operator $Q_6  = \sum_{q=u,d,s}   (\bar{s}   P_R q)(\bar{q}  P_L
d)$~\cite{EAP,Bosch}, with Wilson coefficient $y_6$, viz.\
\begin{eqnarray}
  \label{ee6}
\epe \ =\ \biggl ( \epe \biggr )_6\, \Big(\, 
\Omega_{\rm SM}\ +\ \Omega_{\rm SUSY}\ +\ 
\Omega_{\rm SUSY}^{\rm Higgs}\, \Big)\ .
\end{eqnarray}
In the MSSM with     minimal flavour violation, the  non-Higgs    SUSY
contribution  $\Omega_{\rm  SUSY}$ is   small~\cite{Guidice}.  Sizable
contributions  may  be obtained  if  one  relaxes the  assumptions  of
universality   and      CP    conservation   in         the     squark
sector~\cite{epeSUSY,Kagan:1999iq}.

Here, we compute  a novel contribution to $\epsilon'/\epsilon$, namely
the   quantity $\Omega_{\rm  SUSY}^{\rm Higgs}$ in~(\ref{ee6}),  which
entirely  originates   from  Higgs-boson exchange   diagrams  in   the
CP-violating    MSSM.     Based  on    our  resummed   FCNC  effective
Lagrangian~(\ref{master}), we    obtain  in   the  zero   strong-phase
approximation
\begin{eqnarray}
  \label{direct}
\Omega_{\rm SUSY}^{\rm Higgs} \!\!&=&\!\! 2\, \sum_{i=1}^3\, \sum_{q=u,d,s}\,
\frac{m_s\, m_q}{M_{H_i}^2} \nonumber\\[3mm]
&&\hspace{-1,3cm}
\times\, \Bigg[\, 
\frac{\Im m(g_{H_i\bar{s}d}^L)\, g_{H_i\bar{q}q}^S}{A^2\lambda^5\eta}\,
\Bigg(\, \frac{\bra{\pi^+\pi^-}(\bar{s}P_L d )(\bar{q}q) \ket{K^0}_0}{y_6\, 
\bra{\pi^+\pi^-} Q_6 \ket{K^0}_0}\ -\ \frac{1}{|\omega|}\, 
\frac{\bra{\pi^+\pi^-}(\bar{s}P_L d )(\bar{q}q) \ket{K^0}_2}{y_6\, 
\bra{\pi^+\pi^-} Q_6 \ket{K^0}_0}\, \Bigg)\nonumber \\[3mm]
&&\hspace{-1.3cm}
+\, \frac{\Re e (g_{H_i\bar{s}d}^L)\, g_{H_i\bar{q}q}^P}{A^2\lambda^5\eta}\,
\Bigg(\, \frac{\bra{\pi^+\pi^-}(\bar{s}P_L d )(\bar{q}\gamma_5q)
 \ket{K^0}_0}{y_6\, \bra{\pi^+\pi^-} Q_6 \ket{K^0}_0}\ -\ \frac{1}{|\omega|}\,
\frac{\bra{\pi^+\pi^-}(\bar{s}P_L d )(\bar{q}\gamma_5 q) \ket{K^0}_2}{y_6\, 
\bra{\pi^+\pi^-} Q_6 \ket{K^0}_0}\, \Bigg)\, \Bigg]\nonumber\\[3mm]
&&\hspace{-1.3cm} +\, 2 \sum_{i=1}^3
\sum_{q=u,d,s}\, \frac{m_d\, m_q}{M_{H_i}^2}\\[3mm] 
&&\hspace{-1.3cm}
\times\, \Bigg[\, \frac{\Im m(g_{H_i\bar{s}d}^R)\, 
g_{H_i\bar{q}q}^S}{A^2\lambda^5\eta}\, 
\Bigg(\, \frac{\bra{\pi^+\pi^-}(\bar{s}P_R d )(\bar{q}q) \ket{K^0}_0}{y_6\, 
\bra{\pi^+\pi^-} Q_6 \ket{K^0}_0}\ -\ \frac{1}{|\omega|}\, 
\frac{\bra{\pi^+\pi^-}(\bar{s}P_R d )(\bar{q}q) \ket{K^0}_2}{y_6\, 
\bra{\pi^+\pi^-} Q_6 \ket{K^0}_0}\, \Bigg)\nonumber\\[3mm]
&&\hspace{-1.3cm} +\,
\frac{\Re e (g_{H_i\bar{s}d}^R)\, g_{H_i\bar{q}q}^P}{A^2\lambda^5\eta}\,
\Bigg(\, \frac{\bra{\pi^+\pi^-}(\bar{s}P_R d )(\bar{q}\gamma_5q)
 \ket{K^0}_0}{y_6\, 
\bra{\pi^+\pi^-} Q_6 \ket{K^0}_0}\ -\ \frac{1}{|\omega|}\, 
\frac{\bra{\pi^+\pi^-}(\bar{s}P_R d )(\bar{q}\gamma_5 q) \ket{K^0}_2}{y_6\, 
\bra{\pi^+\pi^-} Q_6 \ket{K^0}_0}\, \Bigg)\, \Bigg]\, ,\nonumber
\end{eqnarray}
where the  subscripts 0,  2 adhered  to  the hadronic  matrix elements
denote the total isospin  $I$ of the  final states, and $A^2 \lambda^5
\eta$   is   a  CKM  combination in the Wolfenstein  parameterization,
which has the value $A^2 \lambda^5  \eta \approx 1.3\times 10^{-4}$ in
the  SM.   Furthermore, for $\Lambda_{\rm  QCD}  = 325$~MeV and $m_s =
150$ MeV, the SM  Wilson coefficient $y_6$  and the matrix element  of
$Q_6$    take  on   the values~\cite{Bosch}:    $y_6  =  -0.089$   and
$\bra{\pi^+\pi^-} Q_6 \ket{K^0}_0 = -0.35 {\rm ~GeV}^3$, respectively.
Also,   experimental analyses suggest   the  value $|\omega| = 0.045$,
approximately yielding the SM contribution  $\Omega_{\rm SM}\sim 1$ to
$\epsilon'/\epsilon$  in~(\ref{ee6}).   Finally,    the     parameters
$g^{S,P}_{H_i\bar{d}d}$      and  $g^{S,P}_{H_i\bar{u}u}$ that   occur
in~(\ref{direct}) are  the diagonal scalar and  pseudoscalar couplings
of  the $H_i$  bosons to $u$-  and  $d$-type quarks~\cite{CEPW}, whose
strengths  are normalized  to  the  SM  Higgs-boson coupling.    These
reduced $H_i$-couplings are given by
\begin{eqnarray}
  \label{HSPd}
g^S_{H_i\bar{d}d} \!&=&\! {\textstyle \frac{1}{2}}\, \big(\, 
g^L_{H_i\bar{d}d}\: +\:  g^R_{H_i\bar{d}d}\,\big)\,,\qquad
g^P_{H_i\bar{d}d} \  =\ {\textstyle \frac{i}{2}}\, \big(\, 
g^L_{H_i\bar{d}d}\: -\:  g^R_{H_i\bar{d}d}\,\big)\,,\qquad\\
  \label{HSPu}
g^S_{H_i\bar{u}u} \!&=&\! O_{2i}/\sin\beta\,, \qquad
g^P_{H_i\bar{u}u} \  =\ -O_{3i}\cot\beta\,,
\end{eqnarray}
where we  have neglected the  small radiative threshold effects in the
up sector.

On the experimental side, the  latest world average  result for $\Re e
(\epsilon'/\epsilon)$ is~\cite{Nir} 
\begin{equation}
  \label{eprimexp}
\Re e\bigg(\, \frac{\epsilon'}{\epsilon}\,\bigg)\ =\  (1.66 \pm
0.16)\times 10^{-3}\; ,
\end{equation}
at the 1-$\sigma$ confidence level.  In the  light of the experimental
result~(\ref{eprimexp})  and the  discussion   given   above,  we  may
conservatively require that
\begin{eqnarray}
  \label{boundepe}
|\Omega_{\rm SUSY}^{\rm Higgs}|\ \lsim\ 1 \;.
\end{eqnarray}
The   biggest contribution in  the  sum  over quarks in~(\ref{direct})
comes   from the  $d$-quark   and  exhibits  the qualitative  scaling
behaviour
\begin{eqnarray}
  \label{eprimapprox}
\Omega_{\rm SUSY}^{\rm Higgs}\ \simeq\ \frac{2 m_s m_d}{M_H^2} \,
\frac{\tan^3\beta}{|\omega|}\, \times\, {\cal O}(1)\;.
\end{eqnarray}
For    instance,    for     $\tan\beta=50$    and     $M_H=200$   GeV,
(\ref{eprimapprox})  gives $\Omega_{\rm SUSY}^{\rm  Higgs}  \simeq 0.1
\times {\cal O}(1)$.  Obviously, such a contribution is, in principle,
non-negligible, but very sensitively  depends on the actual values  of
the new hadronic matrix elements:
\begin{eqnarray}
  \label{QLRSP}
(Q^{L,R}_{S,P})_I\ =\ \bra{\pi^+\pi^-}\,
(\bar{s}P_{L,R} d )\,(\bar{q}(1,\gamma_5) q)\, \ket{K^0}_I \;,
\end{eqnarray}
with $I=0,2$.  A detailed calculation of  the hadronic matrix elements
$(Q^{L,R}_{S,P})_I$  in~(\ref{QLRSP})  will be   given elsewhere.   In
Section~4, however, we will  present  numerical estimates of   $\Delta
M_K$ and $\epsilon_K$ within the context of generic soft SUSY-breaking
models.



\subsection{$\Delta M_{B_q}$, $B_q\to \ell^+\ell^-$ and associated 
  CP asymmetries}

We start our discussion of a set of observables in the $B$-meson
system by first analyzing the $B^0_q$-$\bar{B}^0_q$ mass difference,
$\DmBq$ with $q={s,d}$, in the CP-violating MSSM at large $\tan\beta$.
In the applicable limit of equal $B$-meson lifetimes, $\DmBq$ may be
written as the modulus of a sum of a SM and a SUSY term:
\begin{eqnarray} 
\DmBq\ =\ 2\, 
| \bra{\bar{B}_q^0}\, H_{\rm eff}^{\Delta B=2}\, \ket{B^0_q}_{\rm SM}\ + \
\bra{\bar{B}_q^0}\, H_{\rm eff}^{\Delta B=2}\, \ket{B^0_q}_{\rm SUSY}|\;,
\end{eqnarray}
where  the effective  $\Delta  B=2$  Hamiltonian $H_{\rm  eff}^{\Delta
B=2}$ may be obtained from the $\Delta S=2$ one stated in~(\ref{DS2}),
after  making the  obvious replacements:  $s\to b$ and  $d\to q$, with
$q=d,s$.  Proceeding as in Section~3.1, we  arrive at analogous closed
expressions for the SUSY  contributions to the $\Delta B=2$ transition
amplitudes:
\begin{eqnarray}
  \label{Bsusy}
\bra{\bar{B}^0_d}\,H_{\rm eff}^{\Delta B=2}\,
\ket{B^0_d}_{\rm SUSY} \!\!&=&\!\!  1711 {\rm ~ps}^{-1} 
\Bigg(\,\frac{\hat{B}_{B_d}^{1/2}\, F_{B_d}}{230 {\rm ~MeV}}\, \Bigg)^2\,
\Bigg(\,\frac{\eta_B}{0.55}\,\Bigg) \nonumber\\[3mm] 
&&\hspace{-2.7cm}
\times\, \Big[\, 0.88\, \Big(\,
C_2^{\rm  LR\, (DP)}\: +\: C_2^{\rm  LR\, (2HDM)}\, \Big)\ -\
0.52\, \Big(\, C_1^{\rm  SLL\,(DP)}\: +\: 
C_1^{\rm  SRR\,(DP)}\, \Big)\, \Big]\;, \nonumber \\[3mm]
\bra{\bar{B}^0_s}\, H_{\rm eff}^{\Delta B=2}\, \ket{B^0_s}_{\rm SUSY} 
\!\!&=&\!\! 
2310 {\rm ~ps}^{-1} \Bigg(\,
\frac{\hat{B}_{B_s}^{1/2}\, F_{B_s}}{265 {\rm ~MeV}}\, \Bigg)^2
\Bigg(\,\frac{\eta_B}{0.55}\,\Bigg) \nonumber\\[3mm] 
&&\hspace{-2.7cm}
\times\, \Big[\, 0.88\, \Big(\,
C_2^{\rm  LR\, (DP)}\: +\: C_2^{\rm  LR\, (2HDM)}\, \Big)\ -\
0.52\, \Big(\, C_1^{\rm  SLL\,(DP)}\: +\: 
C_1^{\rm  SRR\,(DP)}\, \Big)\, \Big]\;.\qquad\quad
\end{eqnarray}
In deriving~(\ref{Bsusy}),  we   have  also  substituted the    values
determined   in~\cite{Jager,Ciuchini,Misiak,Becirevic,Chankowski}  for
the NLO-QCD factors, along with their hadronic  matrix elements at the
scale $\mu=4.2$~GeV:
\begin{eqnarray}
\bar{P}_1^{\rm LR}\ =\ -0.58 \;,\qquad
\bar{P}_2^{\rm LR}\ =\ 0.88 \;,\qquad
\bar{P}_1^{\rm SLL}\ =\ -0.52 \;,\qquad
\bar{P}_2^{\rm SLL}\ =\ -1.1 \;.
\end{eqnarray}
Moreover,  the   corresponding Wilson   coefficients         appearing
in~(\ref{Bsusy})  may  be recovered  from those  in~(\ref{dpkaon}) and
(\ref{2HDM}), after performing the quark replacements mentioned above.

Another observable, which  is enhanced  at  large $\tan\beta$, is  the
pure             leptonic          decay              of           $B$
mesons~\cite{FCNC,Frank,KB,Isidori,Dedes1,ADKT,Ambrosio,Buras,MTW},  
$\bar{B}^0_q
\to   \ell^+  \ell^-$,   with  $\ell =     \mu$,   $\tau$.  Neglecting
contributions proportional to the  lighter quark masses $m_{d,s}$, the
relevant effective Hamiltonian for $\Delta B=1$ FCNC transitions, such
as $b\to q \ell^+ \ell^-$ with $q = d,s$, is given by
\begin{eqnarray}
  \label{DB1}
H_{\rm eff}^{\Delta B=1}\ =\ -\,2\,\sqrt{2}\, G_F\, V_{tb}V_{tq}^*\,
\Big(\, C_S\, {\cal O}_S\ +\ C_P\, {\cal O}_P\ +\ C_{10}\, {\cal O}_{10}
\Big)\;,
\end{eqnarray}
where 
\begin{eqnarray}
{\cal O}_S &=& \frac{e^2}{16\pi^2}\, m_b\, 
(\bar{q} P_R b)\, (\bar{\ell}\ell) \;, \nonumber \\[2mm]
{\cal O}_P &=& \frac{e^2}{16\pi^2}\, m_b\, (\bar{q} P_R b)\, 
(\bar{\ell}\gamma_5 \ell) \;,\nonumber \\[2mm]
{\cal O}_{10} &=& \frac{e^2}{16\pi^2}\,  (\bar{q}\gamma^\mu P_L b)\,
 (\bar{\ell}\gamma_\mu \gamma_5 \ell) \;.
\end{eqnarray}
Employing our resummed FCNC effective Lagrangian~(\ref{master}), it is
not difficult   to  compute  the     Wilson
coefficients $C_S$ and $C_P$ at large values of $\tan\beta$:
\begin{eqnarray}
  \label{CSCP}
C_S &=& \frac{2 \pi m_\ell}{\alpha_{\rm em}}\, 
\frac{1}{V_{tb} V_{tq}^*}\, \sum_{i=1}^3\, \frac{g_{H_i\bar{q}b}^R\, 
g_{H_i\bar{\ell}\ell}^S}{M_{H_i}^2} \ ,\nonumber\\[3mm]
C_P &=& i\, \frac{2 \pi m_\ell}{\alpha_{\rm em}}\, \frac{1}{V_{tb} V_{tq}^*}\,
\sum_{i=1}^3\, \frac{g_{H_i\bar{q}b}^R\, 
g_{H_i\bar{\ell}\ell}^P}{M_{H_i}^2} \ ,
\end{eqnarray}
while  $C_{10}=-4.221$ is the   leading SM  contribution.  In  analogy
to~(\ref{HSPd}), the reduced scalar  and pseudoscalar  Higgs couplings
to charged  leptons $g_{H_i\bar{\ell}\ell}^{S,P}$  in~(\ref{CSCP}) are
given by
\begin{equation}
g_{H_i \bar{\ell} \ell}^S\ =\ \frac{O_{1i}}{\cos\beta}\ , \qquad
g_{H_i \bar{\ell} \ell}^P\ =\ -\,\tan\beta\, O_{3i} \; ,
\end{equation}
where non-holomorphic vertex effects on the  leptonic sector have been
omitted as being negligibly small.


With the approximations mentioned  above, the branching ratio  for the
$\bar{B}^0_q$  meson decay  to   $\ell^+ \ell^-$  acquires the  simple
form~\cite{Frank}
\begin{eqnarray}
  \label{Bll}
{\cal B}(\bar{B}^0_q \to \ell^+ \ell^-) &=& \\
&&\hspace{-2cm}
\frac{G_F^2 \alpha_{\rm em}^2}{16\pi^3}\, M_{B_q} \tau_{B_q}\, 
|V_{tb}V_{tq}^*|^2\, \sqrt{1-\frac{4 m_\ell^2}{M_{B_q}^2}}\
\Bigg[\, \Bigg(\,1-\frac{4 m_\ell^2}{M_{B_q}^2} \Bigg)\, |F^q_S|^2\ +\
|F_P^q\: +\: 2 m_\ell F_A^q|^2\, \Bigg] \;,\nonumber
\end{eqnarray}
where $\tau_{B_q}$ is the total lifetime of the $B_q$ meson and
\begin{eqnarray}
  \label{FSP}
F_{S,P}^q\ =\ -\,\frac{i}{2}\, M_{B_q}^2 F_{B_q}\, 
\frac{m_b}{m_b+m_q}\, C_{S,P}\;,\qquad 
F_A^q\ =\ -\,\frac{i}{2}\,F_{B_q}\, C_{10} \;.
\end{eqnarray}
In our numerical estimates, we  ignore the contribution from $C_{10}$,
as being  subdominant in the  region of  large $\tan\beta$, i.e.\  for
$\tan\beta \stackrel{>}{{}_\sim} 40$,  where all Higgs-particle masses
are well below the   TeV scale, The   SM  predictions as well  as  the
current experimental bounds   pertinent to  ${\cal  B}(\bar{B}^0_d \to
\ell^+ \ell^-)$ can be read off from Table~1 in~\cite{Dedes2}.
  
In the CP-violating  MSSM, an equally  important class of  observables
related to ${\cal B}(\bar{B}^0_{d,s}  \to \ell^+ \ell^-)$~\cite{IN} is
the  one probing possible  CP asymmetries  that can take  place in the
same leptonic $B$-meson decays.  The leptonic  CP asymmetries may shed
even light on the CP nature of possible new-physics effects, as the SM
prediction for these   observables turns out to  be  dismally small of
order $10^{-3}$~\cite{Liao}.   This SM result is  a consequence of the
fact that the CP-violating phase in $B^0$-$\bar{B}^0$-mixing parameter
$q/p$  is opposite to   the one entering the  ratio  of the amplitudes
$\bar{{\cal        A}}_{L(R)}(\bar{B}^0_{d,s}      \to      l^+_{L(R)}
l^-_{L(R)})/{\cal A}_{L(R)} ({B}^0_{d,s} \to l^+_{L(R)}  l^-_{L(R)})$,
such  that the net  CP-violating   effect on the observable  parameter
$\lambda_{L(R)}  = (q/p)    (\bar{{\cal  A}}_{L(R)}/{\cal  A}_{L(R)})$
almost cancels out.
 
There are two  possible time-dependent CP asymmetries associated  with
leptonic $B$-meson decays that are physically allowed:
\begin{eqnarray}
  \label{ACPlept}
{\cal A}_{\rm CP}^{(B_q^0\to l_L^+ l_L^-)}\ =\
\frac{\int_0^{\infty} dt \, \Gamma(B_q^0(t)\to l_L^+l_L^-)\: 
-\: \int_0^{\infty}  dt \, \Gamma(\bar{B}_q^0(t)\to l_R^+l_R^-)}
{\int_0^{\infty} dt \, \Gamma(B_q^0(t)\to l_L^+l_L^-)\: 
+\: \int_0^{\infty} dt \, \Gamma(\bar{B}_q^0(t)\to l_R^+l_R^-)} 
\end{eqnarray}
and  ${\cal   A}_{\rm    CP}^{(B_q^0 \to   l_R^+   l_R^-)}$,  with  $L
\leftrightarrow R$.  Under the assumption  that $q/p$ is a pure phase,
one finds~\cite{Liao}
\begin{eqnarray}
 \label{bmumuass}
{\cal   A}_{\rm CP}^{(B_q^0\to  l_L^+ l_L^-)}\  =\ -\,\frac{2 x_q\,\Im m
\lambda_q} {(2\, +\, x_q^2)\: +\:  x_q^2\, |\lambda_q|^2}\ ,\qquad
{\cal A}_{\rm  CP}^{(B_q^0\to l_R^+ l_R^-)}\ =\ -\, \frac{2  
x_q\,\Im m \lambda_q}{(2\,+\,x_q^2)\,|\lambda_q|^2\: +\: x_q^2 } \ ,\quad
\end{eqnarray} 
where $x_q = \Delta M_{B_q}/\Gamma_{B_q}$ and
\begin{eqnarray}
  \label{lq}
\lambda_q\ =\ \frac{M^{q*}_{12}}{|M^q_{12}|}\ 
\Bigg(\,\frac{V_{tb}V_{tq}^*}{V^*_{tb}V_{tq}}\,\Bigg)\
\frac{\beta_l\,C_S \: +\: C_P\: +\: 2m_l C_{10}/(m_b M_{B_q})}{
\beta_l\,C_S^*\: -\: C_P^*\: -\: 2m_l C_{10}/(m_b M_{B_q})} \ .
\end{eqnarray}
In (\ref{lq}), $\beta_l =  (1-4 m_l^2/M_{B_q}^2)^{1/2}$, $M^q_{12}$ is
the dispersive part  of the $B^0_q$-$\bar{B}^0_q$ matrix  element, and
$C_{S,P}$ are Wilson  coefficients given in~(\ref{CSCP}).  The miximal
value that the  leptonic CP asymmetries in~(\ref{bmumuass}) can  reach
is ${\cal A}_{\rm CP }^{\rm max}=1/\sqrt{2+x_q^2}$ and is obtained for
$\Im    m\lambda_q  = |\lambda_q  |^2$.    {}From current experimental
data~\cite{PDG},  one  may    extract    the values  $x_d=0.76$    and
$x_s\stackrel{>}{{}_\sim} 19$ at the 95\%   CL, which leads to  ${\cal
A}_{\rm CP }^{\rm max}(B_s) \approx 5\%$ and  ${\cal A}_{\rm CP }^{\rm
max}(B_d) \approx 62\%$.

Within the framework of   our resummed FCNC effective  Lagrangian,  we
also improve earlier calculations~\cite{Liao,IN} of the CP asymmetries
by    including    $B^0_q$-$\bar{B}^0_q$  mixing    effects    through
$M^{q*}_{12}/|M^q_{12}|$  in~(\ref{lq}).   According  to our  standard
approach  of splitting the amplitude   into a SM  and a  MSSM part, we
obtain for the SM part
\begin{eqnarray}
M_{12}^{q*\, ({\rm SM})}\ =\ -\,\frac{G_F^2 M_W^2}{12\pi^2}\ M_{B_q} \eta_B 
\hat{B}_{B_q} F_{B_q}^2\, (V_{tq}V_{tb}^*)^2\, S_{tt} \;,
\end{eqnarray}
where $S_{tt}\approx  2.38$  is     the   value  of   the     dominant
$m_t$-dependent loop function  for a  top-pole mass  $m_t =  175$ GeV.
The    SUSY   contribution   to     $M_{12}^{q*\,   ({\rm  SUSY})}   =
\bra{\bar{B}_d^0}\, H_{\rm eff}^{\Delta B=2}\, \ket{B^0_d}_{\rm SUSY}$
may be obtained from (\ref{Bsusy}). 

In our   last (but not  least)  application of our  resummed effective
Lagrangian~(\ref{master})  to     the   $B$-meson    system,  we  will
qualitatively discuss the CP asymmetries related to the time evolution
and decay  of the $B^0_q$ and  $\bar{B}^0_q$  mesons into non-leptonic
final states $f_{\rm CP}$ of definite CP.  The time-dependence of this
category of exclusive CP asymmetries is determined by~\cite{ACP}
\begin{equation}
  \label{AfCP}
{\cal A}_{f_{\rm CP}}(t)\ =\ 
-\, C_{f_{\rm CP}} \cos (\DmBq t)\ +\ S_{f_{\rm CP}}
\sin (\DmBq t) \;,
\end{equation}
where  the CP-violating factors $C_{f_{\rm  CP}}$ and $S_{f_{\rm CP}}$
are given by
\begin{equation}
  \label{CSfCP}
C_{f_{\rm CP}}\ =\
\frac{1\: -\: |\lambda_{f_{\rm CP}}|^2}{1\: +\: 
|\lambda_{f_{\rm CP}}|^2}\ ,\qquad 
S_{f_{\rm CP}}\ =\ \frac{2 \Im m  \lambda_{f_{\rm CP}}}
{1\: +\: |\lambda_{f_{\rm CP}}|^2} \ .
\end{equation}
The CP-violating parameter  $\lambda_{f_{\rm CP}}$ in~(\ref{CSfCP}) is
defined as
\begin{eqnarray}
  \label{lam}
\lambda_{f_{\rm CP}}\ =\ 
\sqrt {\frac{M_{12}^* - \frac{i}{2}\Gamma_{12}^*}{M_{12} -
\frac{i}{2}\Gamma_{12}}}\ 
\frac{\bra{f_{\rm CP}}\, {\cal H}\, \ket{\bar{B}^0}}
{\bra{f_{\rm CP}}\,{\cal H}\,\ket{{B}^0}}\ ,
\end{eqnarray}
where $M_{12}$ and  $\Gamma_{12}$  are the  dispersive and  absorptive
parts of  the $B^0_q \to \bar{B}^0_q$  transition amplitude, with $q =
d,s$.  Experimentally, it is  $M_{12} \gg  \Gamma_{12}$, which enables
one to approximate $\lambda_{f_{\rm CP}}$ in~(\ref{lam}) as
\begin{equation}
  \label{lamappr}
\lambda_{f_{\rm CP}}\ \approx\ e^{-2i\phi_q}\, \frac{\bar{A}_q}{A_q},
\end{equation}
where 
\begin{equation}
  \label{phiM}
-\, 2\phi_q\ =\ {\rm arg} (M^*_{12})\ =\ {\rm arg}\, \bra{\bar{B}_q^0}\,
H_{\rm eff}^{\Delta B=2} \ket{B^0_q}\; ,
\end{equation}
and $A_q (\bar{A}_q) = \bra{f_{\rm CP}}\, {\cal H}\, \ket{B^0_q
(\bar{B}^0_q)}$.

In the SM, one finds  that $\phi^{\rm SM}_d = \beta  = {\rm arg}\,  [-
V_{cd}V^*_{cb}/(V_{td}   V^*_{tb})]$     and      $\phi^{\rm     SM}_s
\stackrel{<}{{}_\sim} \lambda^2   \approx 0.05$. On the other  hand,
experimental data for the decays $B_d\to J/\Psi K_S$ (or $\bar{b} \to
\bar{s}c\bar{c}$ at the quark level)  imply that $\sin 2\phi_d = 0.734
\pm 0.054$~\cite{Nir}.  According   to our  standard procedure, we
write $\phi_d$  as  a sum of two   contributions: $\phi_d  = \phi^{\rm
SM}_d +  \phi_d^{\rm  MSSM}$,  where   the MSSM  contribution  can  be
determined  from~(\ref{Bsusy}).  In the same vein~\cite{Gronau,Worah},
we  split the decay amplitude  $A_d$  and its CP-conjugate counterpart
$\bar{A}_d$ into a  (dominant) SM part,  $A_d^{\rm  SM}$, and an  MSSM
part, $A_d^{\rm MSSM}$.  We also  introduce the ratio $r_d = |A_d^{\rm
MSSM}|/|A_d^{\rm SM}|$ and  denote the  corresponding weak and  strong
phases as $\theta^{\rm SM,\, MSSM}_d$ and $\delta^{\rm SM,\, MSSM}_d$,
respectively.  Then, the MSSM contribution to the CP-violating factors
$C_{f_{\rm CP}}$ and $S_{f_{\rm CP}}$ in~(\ref{CSfCP}) may be computed
by
\begin{eqnarray}
  \label{CS}
C_{f_{\rm CP}} &=& 2 r_d \sin(\theta_d^{\rm SM}-\theta_d^{\rm MSSM})\,
\sin(\delta_d^{\rm SM}-\delta_d^{\rm MSSM})\;, \\
S_{f_{\rm CP}} &=& \sin [2(\phi_d + \theta_d^{\rm SM})]\: +\:
2 r_d \cos [2 ( \phi_d + \theta_d^{\rm SM})]\, \sin(\theta_d^{\rm SM}
- \theta_d^{\rm MSSM})\, 
\cos(\delta_d^{\rm SM} -\delta_d^{\rm MSSM}) \;.\nonumber 
\end{eqnarray}
Even    though we have  concentrated   on  the  $B^0_d$ meson, a  very
analogous discussion can be followed for the $B^0_s$-meson decays.

The  CP-violating  observable~(\ref{AfCP})  can ideally be   tested at
$B$-factories by analyzing  the decays $B_d  \to J/\Psi  K_S$ and $B_d
\to \phi  K_S,\ \eta'  K_S$, which  correspond at the  quark  level to
$\bar{b} \to    \bar{s}   c  \bar{c}$   and  $\bar{b}\to   \bar{s}   s
\bar{s}$.    All  these decays    measure  almost the    same value of
$\phi_d^{\rm SM} = \beta$, since  electroweak corrections to the ratio
of the  decay amplitudes in~(\ref{lamappr}) are small~\cite{Worah} and
$\theta_d^{\rm   SM}(J/\Psi    K)  <    \theta_d^{\rm  SM}    (\phi K)
\stackrel{<}{{}_\sim} {\cal  O}  (\lambda^2)$.  As  a consequence, one
finds a small  difference between the  CP-violating factors $S_{J/\Psi
K}$ and $S_{\phi K}$  in  the SM, i.e.\   $D_{CP}  = |S_{J/\Psi  K}  -
S_{\phi   K}| \stackrel{<}{{}_\sim}   0.05$.    However, the  recently
reported value,
\begin{equation}
  \label{DCP} 
D_{CP}\   =\ |S_{J/\Psi K}  - S_{\phi   K}|\ =\  1.12\, \pm\, 0.41 \; ,
\end{equation}    
from the BaBar and Belle  collaborations shows a $2.7\sigma$ deviation
from the SM prediction (for more details  see \cite{Hiller,Nir}).  One
might then  raise the   question whether such   a  large deviation  in
$D_{CP}$ could be naturally explained within  the MSSM.  Here, we will
not perform  an    extensive  quantitative analysis of     this issue.
Instead, we wish to point out that $\tan\beta$-enhanced Higgs-mediated
FCNC interactions may potentially account for the observed large value
of   $D_{CP}$ in  the CP-violating MSSM.     In particular, within the
framework of our resummed effective Lagrangian~(\ref{master}), we will
argue below that  non-zero values for  $\theta_d^{\rm MSSM}$ and $r_d$
at a  level  required for explaining   $D_{CP}$ are still possible  in
(\ref{CS}).
   
The description  of the  aforementioned  non-leptonic $\Delta B  =  1$
$B$-meson decays is based on the effective Hamiltonian
\begin{eqnarray}
  \label{db1}
H_{\rm eff}^{\Delta B=1} \!\!\!&=&\!\!
\frac{G_F}{\sqrt{2}}\, V_{tb}V_{ts}^*\, \sum_{i=1}^{10}\,\Big(\,
C_i {\cal O}_i\, +\, C_{7\gamma} {\cal O}_{7\gamma}\, 
+\, C_{8 g} {\cal O}_{8 g}\,\Big)\: +\:
\frac{G_F}{\sqrt{2}}\, \sum_{q=s,c}\, \Big(\,
C_S^q {\cal O}_S^q\, +\, C_P^q {\cal O}_P^q\, \Big)\;,\nonumber\\
&&
\end{eqnarray}
which is enlarged to include Higgs-penguin interactions in addition to
electroweak  penguins. The standard    form for the operators   ${\cal
O}_{i}$,  ${\cal  O}_{7\gamma}$,    ${\cal   O}_{8g}$  may   be  found
in~\cite{Wyler}.  The  new Higgs-mediated operators ${\cal O}^q_{S,P}$
in~(\ref{db1}) are given by
\begin{equation}
  \label{OqSP}
{\cal O}^q_S\ =\ (\bar{s} P_R b)\, (\bar{q} q) \;,\qquad\quad
{\cal O}^q_P\ =\ (\bar{s} P_R b)\, (\bar{q}\gamma_5 q)\,,
\end{equation}
with $q = s, c$.  Notice that we  have neglected operators of the form
$(\bar{s} P_L    b)  \,   (\bar{q}q)$  and    $(\bar{s} P_L    b)   \,
(\bar{q}\gamma_5 q)$,  since their respective Wilson  coefficients are
suppressed by extra factors $m_s/m_b$ with respect to those considered
in~(\ref{db1}).  In the region of large $\tan\beta$ values, the Wilson
~~~coefficients   $C^q_{S,P}$  associated with  ${\cal O}^q_{S,P}$ are
dominated   by Higgs-boson  exchange    graphs.   {}From our  resummed
effective Lagrangian~(\ref{master}), we obtain (with $q = s,c$)
\begin{eqnarray}
C_S^q &=& -\,2\, m_q m_b\, \sum_{i=1}^3\, 
   \frac{g_{H_i\bar{s}b}^R\, g_{H_i\bar{q}q}^S}{M_{H_i}^2} \ , \\[2mm]
C_P^q &=& -\,2 i\, m_q m_b\, \sum_{i=1}^3\,
 \frac{g_{H_i\bar{s}b}^R\, g_{H_i\bar{q}q}^P}
{M_{H_i}^2} \ .
\end{eqnarray}
In   the  large-$\tan\beta$ regime,   these    coefficients scale  like
$C_{S,P}^s \propto  m_s \tan^3\beta$    and  $C_{S,P}^c \propto    m_c
\tan\beta$. Since it is $C_{S,P}^c \ll C^s_{S,P}$ for $\tan\beta 
\stackrel{>}{{}_\sim} 40$, this allows us to carry out the
crude approximations:
\begin{eqnarray}
  \label{thetas}
\theta_d^{\rm MSSM} (J/\Psi K_S)\ \ll\
\theta_d^{\rm MSSM} (\phi K_S) \ \sim\ {\rm arg}\, \Bigg( 
\sum_{i=1}^3 \frac{g_{H_i\bar{s}b}^R\,
g_{H_i\bar{s}s}^{S,P}}{M_{H_i}^2}\,\Bigg)\ =\ {\cal O}(1) \;.
\end{eqnarray}
 

The above large hierarchy in~(\ref{thetas})  for the MSSM weak  phases
of the two decay amplitudes, $A(B_d\to J/\Psi K_S)$ and $A(B_d\to \phi
K_S)$, is  very  important,  as it  can   be sufficient by  itself  to
explain,   through~(\ref{CS}),   the   large     value  of    $D_{CP}$
in~(\ref{DCP}).    Nevertheless,  there   can  easily exist additional
factors that  may conspire to  obtain a  large $D_{CP}$ value.   To be
specific, the amplitude   $A(B_d\to  J/\Psi K_S)$ arises at   the tree
level in    the SM, so one  expects   to be  the  dominant  one, i.e.\
$r_d^{J/\Psi  K} \sim  0$.  In such   a case, CP-violating effects  on
$S_{J/\Psi   K}$ can  only   come  from the process-independent  phase
$\phi_{d}^{\rm  MSSM}$ due to the  $B^0$-$\bar{B}^0$ mixing and, given
that $\theta_d^{\rm SM}$ is small for  both the above decay modes, the
phase $\phi_{d}^{\rm  MSSM}$  almost  cancels  in $D_{CP}$ against   a
similar contribution present  in $S_{\phi K}$.  However, the amplitude
for the decay $B_d \to \phi K_S$  is induced at  the one-loop level in
the SM,    and its value~\cite{Wyler}  (within   the QCD factorization
scheme~\cite{Beneke})  becomes comparable,    at   large  values    of
$\tan\beta$,  to  the  amplitude  generated  by  the  operators ${\cal
O}^q_{S,P}$.   In this case, one   might expect that $r_d^{\phi K}\sim
1$, which can also lead to large deviations in $D_{CP}$ of the desired
order.  An extensive quantitative  treatment of these  asymmetries, in
conjunction with the decay $B_d \to \eta' K_S$, is beyond the scope of
the present article and may be  given elsewhere.  Instead, in the next
section, we    will    present  numerical  predictions     for $\Delta
M_{B_{d,s}}$,  ${\cal   B}  (B^0_q  \to  \ell^+\ell^-   )$  and  their
associated leptonic CP asymmetries.



\setcounter{equation}{0} 
\section{Numerical estimates}

In  this section,  we shall  numerically   analyze the  impact of  the
$\tan^2\beta$-enhanced  FCNC  interactions  on  a number  of  $K$- and
$B$-meson  observables which were discussed in  detail in Sections 3.1
and 3.2, such  as  $\DmK$, $\epsilon_K$,  $\DmBd$, $\DmBs$, $B_d   \to
\tau^+ \tau^-$, $B_s \to \mu^+ \mu^-$ and their associated leptonic CP
asymmetries.   For   our  illustrations,  we   consider   two  generic
low-energy soft SUSY-breaking scenarios, (A) and (B).

In scenario~(A), the squark masses are taken to be universal and $\Eg$
and  $\Eu$   are  proportional  to  the  unity    matrix at the   soft
SUSY-breaking scale $M_{\rm SUSY}$.  The CP-conserving version of this
scenario has frequently  been  discussed in the literature  within the
context of minimal flavour-violation models, see e.g.~\cite{KB}.

In  scenario~(B) we assume the existence  of  a mass hierarchy between
the first two  generation of squarks  and the third generation, namely
the first two generations are degenerate and  can be much heavier than
the third  one.  In addition,   although not mandatory, we assume  for
simplicity    that    the  model-dependent   unitary   matrix   $\ULQ$
in~(\ref{Epars}) equals the  unity matrix~${\bf 1}$.   As a  result of
this choice, $\Egb$  and   $\Eub$ become  diagonal matrices   in  this
scenario. Clearly, in the limit, in which  all squarks are degenerate,
scenario~(B) coincides with (A).

In Fig.~\ref{fig1} we give  a schematic representation of the  generic
mass spectrum that  will be assumed in  our  numerical analysis.  More
explicitly,  we fix the charged  Higgs-boson  $M_{H^+}$  to the  value
$200$~GeV.  Since the effect of the SU(2)$_L$ gauginos $\widetilde{W}$
on the resummation matrix ${\bf R}$~\cite{MTW,IN} can be significantly
reduced  for $m_{\tilde{W}} \ll  \mu$, we ignore  this contribution by
considering  the relatively low value  $m_{\tilde{W}} \approx 2m_t \ll
M_{\rm SUSY}$ in our numerical estimates, with $m_t = 175$~GeV. As can
be  seen from Fig.~\ref{fig1}, the  third-generation  soft squark mass
$m_{\tilde{t}}$, the $\mu$-parameter,  the gluino mass $m_{\tilde{g}}$
and the trilinear soft  Yukawa coupling $A_U$  are set, for simplicity
reasons, to the common  soft SUSY-breaking scale $M_{\rm SUSY}$, which
is typically taken to be 1~TeV.

The soft squark masses of the other two  generations are assumed to be
equal to $m_{\tilde{q}}$  in our generic framework.   To account for a
possible  hierarchical   difference   between   the     mass    scales
$m_{\tilde{t}}$ and   $m_{\tilde{q}}$,   we introduce  the   so-called
hierarchy factor $\rho$, such that $m_{\tilde{q}} = \rho m_{\tilde{t}}
= \rho M_{\rm SUSY}$.  As has been mentioned  above, models of minimal
flavour violation  correspond to scenario~(A) with $\rho  = 1$.  As we
will see in detail in  Sections 4.1 and  4.2, the predictions for  the
$K$- and $B$-meson FCNC observables crucially depend  on the values of
the  hierarchy factor $\rho$.   Equally important modifications in the
predictions are obtained for different values of the soft CP-violating
phases $\phi_g  = {\rm arg}\,  (m_{\tilde{g}})$ and $\phi_{A_U} = {\rm
arg}\, (A_U)$.  In addition,  the $K$- and $B$-meson FCNC  observables
exhibit a non-trivial dependence on  the CKM phase $\delta_{\rm CKM}$,
which is  varied independently in our  figures.  

Although we primarily use $\tan\beta = 50$  and $M_{H^+} = 0.2$~TeV as
inputs in our   numerical analysis, approximate predictions  for other
values of  the   input parameters may be   obtained  by  rescaling the
numerical estimates by a factor
\begin{equation}
  \label{scale}  
x_{\cal O}\ =\ \bigg(\,\frac{\tan\beta}{50}\,\bigg)^n\:
\times\: \bigg(\,\frac{0.2~{\rm TeV}}{M_{H^+}}\,\bigg)^k\,,
\end{equation}
where the integers  $n$ and $k$ depend  on the FCNC  observable ${\cal
O}$ under  study.  Such a rescaling proves  to be  fairly accurate for
$\tan\beta    \stackrel{>}{{}_\sim}          40$      and     $M_{H^+}
\stackrel{>}{{}_\sim} 150$~GeV, which  is the kinematic  region of our
interest.



\subsection{$\DmK$ and $|\epsilon_K|$}

The SM effects on $\DmK$ and $|\epsilon_K|$ were extensively discussed
in the literature~\cite{NH}, so we will not dwell upon this issue here
as well. Instead, we assume that the SM explains well the experimental
results for the above two observables~\cite{PDG}:
\begin{eqnarray}
  \label{expDmK}
\Delta M_K^{\rm exp} &=& (3.490 \pm 0.006)\times 10^{-12} ~{\rm MeV} \\
  \label{expek}
|\epsilon^{\rm exp}_K| &=& (2.282 \pm 0.017)\times 10^{-3}\;.
\end{eqnarray}
Given the  significant  uncertainties in the  calculation  of hadronic
matrix element,  however, our approach  will be to constrain  the soft
SUSY-breaking  parameters  by conservatively  requiring  that  $\Delta
M_K^{\rm  SUSY}$ and $|\epsilon_K^{\rm SUSY}|$ do  not  exceed in size
the SM  predictions.\footnote{Both   $\DmK$  and  $|\epsilon_K|$ place
important constraints on the    $\rho$-$\eta$ plane of  the  unitarity
triangle.  The so-derived limits can be used to constrain new physics.
In this case, a global fit of all the relevant FCNC observables to the
unitarity triangle might be  more  appropriate.  We intend  to address
this issue in a future~work.}

To start with, we display in  Fig.~\ref{fig2} numerical values for the
Higgs-boson DP  effects on $\DmK$ and  $|\epsilon_K|$  as functions of
the gluino   phase ${\rm arg}\,   m_{\tilde{g}}$, where  the hierarchy
factor $\rho$ and the  phase $\phi_{A_U} =  {\rm arg}\, (A_{t,b})$  of
the soft SUSY-breaking trilinear Yukawa  couplings assume the discrete
values: $(\rho,\ \phi_{A_U})  = (1,\ 0^\circ),\ (10,\  0^\circ),\ (1,\
90^\circ),\  (10,\  90^\circ),  (1,\  180^\circ),\  (10,\ 180^\circ)$.
According to our  CP-phase  conventions~\cite{CEPW}, $\mu$   is always
taken be positive, while the CKM phase $\delta_{\rm CKM}$ is chosen to
its   maximal  value   $90^\circ$.     The  subdominant one-loop  2HDM
contribution  coming    from   $W^\pm$-$H^\mp$  box   graphs     [cf.\
(\ref{2HDM})] has also been indicated by  an arrow in Fig.~\ref{fig2}.
Predictions for  $M_{H^+}$ and $\tan\beta$   values  other than  those
shown in Fig.~\ref{fig1} may be  approximately obtained by multiplying
the   numerical     estimates   by  a      factor    $x_{\cal   O}   =
(\tan\beta/50)^4\times (0.2~{\rm    TeV}/M_{H^+})^2$.   We observe  in
Fig.~\ref{fig2} that the resulting  values for $\Delta M_K^{\rm SUSY}$
can exceed the  experimental error  in  (\ref{expDmK}) by an order  of
magnitude,  for   $\rho=10$ and   $|\phi_{A_U}|,\   |\phi_{\tilde{g}}|
\stackrel{>}{{}_\sim}     90^\circ$.        For   the   same   inputs,
$|\epsilon_K^{\rm SUSY}|$      takes on values     comparable to   the
experimentally measured one (\ref{expek}).  Here, we should stress the
fact that universal squark-mass scenarios  corresponding to $\rho = 1$
can  still predict sizeable   effects on $\epsilon_K$.   This non-zero
result  should be  contrasted with  the  one of the  gluino-squark box
contributions  to   $\DmK$ and $|\epsilon_K|$~\cite{Nilles}   which do
vanish in  the   limit of   strictly degenerate    squarks  due  to  a
SUSY-GIM-cancellation mechanism.

In our case of Higgs-mediated FCNC observables, however, the situation
is slightly different.  As we have discussed in Section~2, the size of
the  FCNC effects is  encoded in the  flavour structure  of the 3-by-3
resummation matrix $\Ro$. To be more specific, the Higgs-mediated FCNC
interactions    in~(\ref{master})       are    described    by     the
$\tan\beta$-enhanced terms proportional to
\begin{eqnarray}
  \label{VRV}
(\Vckm^\dagger\, \Ro^{-1}\, \Vckm)_{dd'}\  =\ V^*_{ud}\, R^{-1}_u\, V_{ud'}\:
+\: V^*_{cd}\, R^{-1}_c\, V_{cd'}\: +\: V^*_{td}\, R^{-1}_t\, V_{td'} \;,
\end{eqnarray}
where $d$ and $d'$ collectively denote all  down-type quarks.  For the
parameters adopted in~Fig.~\ref{fig1}, the quantities $R^{-1}_{u,c,t}$
may  be  simplified further to\footnote{For $\rho\stackrel{>}{{}_\sim}
10$,  the first  two   equations   in~(\ref{express}) may  be   better
approximated by replacing $\rho \to \rho/\sqrt{2}$.}
\begin{eqnarray}
  \label{express}
R^{-1}_u &\approx& \bigg[\, 1\: +\: \bigg(\, 
\frac{\alpha_s}{3\pi\rho^2}\, e^{-i\phi_{\tilde{g}}}\: +\: 
\frac{|h_u|^2}{32\pi^2\rho^2}\, 
e^{-i\phi_{A_U}}\,\bigg)\tan\beta\, \bigg]^{-1}\;, \nonumber \\[3mm]
R^{-1}_c &\approx& \bigg[\, 1\: +\: \bigg(\,
\frac{\alpha_s}{3\pi\rho^2}\, e^{-i\phi_{\tilde{g}}}\: +\: 
\frac{|h_c|^2}{32\pi^2\rho^2}\, e^{-i\phi_{A_U}}\,\bigg)\,\tan\beta\,
\bigg]^{-1}\;, \nonumber\\[3mm]
R^{-1}_t &=& \bigg[\, 1\: +\: \bigg(\, \frac{\alpha_s}{3\pi}\, 
e^{-i\phi_{\tilde{g}}}\: 
+\: \frac{|h_t|^2}{32\pi^2}\,
e^{-i\phi_{A_U}}\,\bigg)\,\tan\beta\,\bigg]^{-1}\;. 
\end{eqnarray}
Then,  from~(\ref{express}), it is easy to  see  that the off-diagonal
elements of  $\Vckm^\dagger \Ro^{-1} \Vckm$  increase if $\phi_{A_U},\
\phi_{\tilde{g}}   =   \pm  \pi$   and so   the    effective couplings
$g^{L,R}_{H_i\bar{d}    d'}$,  thereby   giving    rise   to  enhanced
predictions.   This   is a very   generic feature  which  is reflected
in~Fig.~\ref{fig2} and, as we will see in Section 4.2, also holds true
for our numerical estimates of $B$-meson FCNC observables.

Neglecting the small Yukawa couplings of the first two generations and
making use of the unitarity of $\Vckm$, we find for $d\neq d'$
\begin{eqnarray}
  \label{VRVapprox}
(\Vckm^\dagger \Ro^{-1} \Vckm)_{dd'}\ \propto\
V_{td}^*\, V_{td'}\, \Bigg[\, \frac{\alpha_s}{3\pi}\,e^{-i\phi_{\tilde{g}}}\,
\Bigg(\,\frac{1}{\rho^2}\: -\: 1\,\Bigg)\ -\ 
\frac{|h_t|^2}{32\pi^2}\,e^{-i\phi_{A_U}}\, \Bigg]\, \tan\beta\;. 
\end{eqnarray}
If $\rho=1$, the dominant FCNC  effect originates from the second term
in  the    square  brackets of~(\ref{VRVapprox}),   provided   $|1   +
\frac{\alpha_s}{3\pi}\, e^{-i\phi_{\tilde{g}}} \tan\beta | \gg 5\times
10^{-4}\,   \tan\beta$ (see also  footnote~2).  If  $\rho \gg 1$, then
gluino  corrections become dominant;  they  are  larger  by a   factor
$\frac{a_s}{3\pi}/\frac{1}{32\pi^2}\simeq  3.6$.  However, between the
low and high $\rho$-regime, there is an  intermediate value of $\rho$,
where $(\Vckm^\dagger \Ro^{-1} \Vckm)_{dd'}$  does exactly  vanish for
$d \neq d'$, and so the effective couplings $g^{L,R}_{H_i\bar{d} d'}$.
In this  case, one has $R_u^{-1}=R_c^{-1}=R_t^{-1}$ in (\ref{express}),
implying that $\Ro$ is  proportional to the unity  matrix. Then, it is
$(\Vckm^\dagger   \Ro^{-1}    \Vckm)_{dd'} = 0$,  as   a   result of a
GIM-cancellation mechanism due to the unitarity of  the CKM matrix. We
call  such a point in the  parameter space  {\em GIM operative point}.
The   $\rho$ value, for  which the  GIM-cancellation mechanism becomes
fully    operative,      may           easily      be       determined
from~(\ref{VRVapprox}),~i.e.\
\begin{eqnarray}
  \label{rgim}
\rho^2_{\rm GIM}\ =\ \bigg(\, 1\: +\: \frac{3|h_t|^2}{32\pi\alpha_s}\,
 e^{i(\phi_g-\phi_{A_U})}\, \bigg)^{-1} \;. 
\end{eqnarray}
For  the  MSSM  parameter  space   under  study,  there  is  always  a
GIM-operative value for the hierarchy factor $\rho$, iff $\phi_{A_U} -
\phi_{\tilde{g}} = 0$ or $\pm \pi$.  If $\phi_{A_U} - \phi_{\tilde{g}}
= 0$,  we have $\rho_{\rm GIM} <  1$, whereas it is  $\rho_{\rm GIM} >
1$, for  $\phi_{A_U} - \phi_{\tilde{g}}  = \pi$.  In fact,  the second
case is realized in Fig.~\ref{fig3} for $\rho_{\rm GIM} \approx 1.22$,
where  $\Delta M_K^{\rm  SUSY}$ and  $|\epsilon_K^{\rm  SUSY}|$ vanish
independently  of  the  value  of  the CKM  phase.   Here,  we  should
emphasize the fact that the  value of $\rho_{\rm GIM}$ does not depend
on  the  FCNC  observable  under  consideration and  is  in  excellent
agreement  with the  one determined  by~(\ref{rgim}).  Because  of the
above  flavour-universal property  of $\rho_{\rm  GIM}$, one  may even
face the  very unusual possibility of discovering  SUSY at high-energy
colliders, without accompanying such  a discovery with any new-physics
signal in low-energy $K$- and $B$-meson FCNC observables.

It is   now instructive to  gauge  the relative size of  the different
DP-induced Wilson coefficients in~(\ref{dpkaon}).  For simplicity, let
us take $\rho=1$.  Then, each individual DP-induced Wilson coefficient
in~(\ref{dpkaon}) may be approximately given by
\begin{eqnarray}
  \label{sums}
\sum_{i=1}^3\,(g^L_{H_i\bar{s}d})^2 &\approx&  
(\chi_{\rm FC}^{(t)})^2\, (V_{ts}^*V_{td})^2\ \sum_{i=1}^3
\, \frac{O_{1i}^2 -O_{3i}^2+2\, i\, O_{1i} O_{3i}}{M_{H_i}^2} \ , 
\nonumber \\[3mm]
\sum_{i=1}^3\,(g^R_{H_i\bar{s}d})^2 &\approx&  
(\chi_{\rm FC}^{(t)*})^2\, (V_{ts}^*V_{td})^2\, \sum_{i=1}^3
\, \frac{O_{1i}^2 -O_{3i}^2-2\, i\, O_{1i} O_{3i} }{M_{H_i}^2} \ , 
\nonumber \\[3mm]
\sum_{i=1}^3\, (g^L_{H_i\bar{s}d}\, g^R_{H_i\bar{s}d}) &\approx&  
|\chi_{\rm FC}^{{(t)}}|^2\, (V_{ts}^*V_{td})^2\, \sum_{i=1}^3
\, \frac{O_{1i}^2 +O_{3i}^2 }{M_{H_i}^2} \ , 
\end{eqnarray}
where $\chi_{\rm FC}^{{(i)}}$ are the  diagonal entries of the  matrix
$\chi_{\rm    FC}$     defined   in~(\ref{chiFC}).    For     $M_{H^+}
\stackrel{>}{{}_\sim} 180$~GeV, CP-violation and Higgs-mixing  effects
start  to   decouple  from    the   lightest $H_1$-sector~\cite{CEPW}.
Moreover,  in  the region  $\tan\beta \stackrel{>}{{}_\sim}  40$,  the
$\Phi_2$-component in the  $H_2$- and  $H_3$-boson mass-eigenstates is
very suppressed.  As a consequence of the latter, we obtain
\begin{eqnarray}
  \label{sum1}
\sum_{i=1}^3\ \frac{O_{1i}^2\: -\: O_{3i}^2}{M_{H_i}^2} &=&
\Big(\, O^2_{11}\: -\: O^2_{31}\,\Big)\, 
\Bigg(\,\frac{1}{M^2_{H_1}}\: -\: \frac{1}{M^2_{H_{23}}}\,\Bigg)\ +\
{\cal O}\Bigg(\,\frac{M^2_{H_2} - M^2_{H_3}}{M^2_{H_{23}}}\,\Bigg)\,,\\[3mm]
  \label{sum2}
\sum_{i=1}^3\ \frac{O_{1i}\, O_{3i}}{M_{H_i}^2} &=&
O_{11}\,O_{31}\, \Bigg(\,\frac{1}{M^2_{H_1}}\: -\:
\frac{1}{M^2_{H_{23}}}\,\Bigg)\ +\ 
{\cal O}\Bigg(\,\frac{M^2_{H_2} - M^2_{H_3}}{M^2_{H_{23}}}\,\Bigg)\,, 
\end{eqnarray}
where $M^2_{H_{23}} = \frac{1}{2}\, ( M^2_{H_2} + M^2_{H_3} )$ and the
orthogonality of the $O$ matrix has been  used.  Since it is $O_{11},\
O_{31} \ll 1$ in the kinematic region of our interest, then on account
of~(\ref{sum1})   and~(\ref{sum2})    and   for    maximal  CKM  phase
$\delta_{\rm CKM}  =  90^\circ$, the dominant  contribution to $\Delta
M_K^{\rm SUSY}$  and $\epsilon_K^{\rm  SUSY}$  comes from the last  DP
expression in~(\ref{sums}), namely from the Wilson coefficient $C^{\rm
LR\,  (DP)}_2$ in~(\ref{dpkaon}),  despite  the additional suppression
factor $m_d/m_s \approx 1/10$  with respect to $C^{\rm SLL\, (DP)}_1$.
If $\delta_{\rm CKM}= 0$,  $\Delta M_K^{\rm SUSY}$ still receives  its
largest    contribution  from     $C^{\rm   LR\,   (DP)}_2$,     while
$|\epsilon_K^{\rm SUSY}|$  is dominated   by the first   DP expression
in~(\ref{sums}),  i.e.\ from  $C^{\rm  SLL\, (DP)}_1$;  the  second DP
expression  in~(\ref{sums}) is  very   suppressed with respect to  the
first one   by   two   powers  of   the  ratio   $m_d/m_s$.     {}From
Fig.~(\ref{fig2}),  we also  see that  in  addition to   the CKM phase
$\delta_{\rm CKM}$, the   soft SUSY-breaking CP-phases, such as  ${\rm
arg}\,m_{\tilde{g}}$  and ${\rm arg}\,  (A_U)$, may  also give rise by
themselves  to  enhancements   of    $|\Delta M_K^{\rm  SUSY}|$    and
$|\epsilon_K^{\rm     SUSY}|$ even up   to  one   order of  magnitude.
Analogous remarks and observations  also hold  true for the  $B$-meson
FCNC observables which are to be discussed in the next section.

Finally, we  should comment on the  fact that the 2HDM contribution by
itself due to $C^{\rm   LR\,(2HDM)}_2$ in~(\ref{2HDM}) can  only  give
rise   to the undetectably  small  numerical values, $|\Delta M_K^{\rm
2HDM}|=5\times  10^{-17}$ MeV and  $|\epsilon_K^{\rm 2HDM}|  = 5\times
10^{-6}$ (indicated  by   an  arrow   in  the  Fig.~\ref{fig2}),   for
$\delta_{\rm CKM}=90^\circ$.


\subsection{$\Delta M_{B_q}$, $B_q\to \ell^+\ell^-$ and associated leptonic 
  CP asymmetries}

In this  section, we will present numerical  estimates for a number of
$B$-meson FCNC observables, such     as the mass   difference  $\Delta
M_{B_q}$, the branching  ratio for  $B_q\to  \ell^+\ell^-$ and the  CP
asymmetries  associated with  the  $B$-meson   leptonic decays.    The
current  experimental  status  of       these  observables   is     as
follows~\cite{PDG}:
\begin{eqnarray}
  \label{expBd}
\Delta M_{B_d} & = & 0.489\: \pm\: 0.008\ {\rm ps}^{-1}\,,\\
  \label{expBs}
\Delta M_{B_s} & > & 13.1\ {\rm ps}^{-1}\,,\\
  \label{expBsmu}
{\cal B}( B_s \to \mu^+\mu^- ) & < & 2.0\times 10^{-6}
\end{eqnarray}
and~\cite{Ligeti}
\begin{equation}
  \label{expBdtau}
{\cal B}( B_d \to \tau^+ \tau^- ) \ <\ 0.015\, .
\end{equation}
Future experiments at an upgraded  phase of the Tevatron collider  may
reach higher sensitivity to ${\cal B}( B_s \to \mu^+\mu^- )$ up to the
$10^{-8}$-level~\cite{Anikeev,ADKT,Dedes1}.

Let us  start our  discussion  by numerically analyzing the  $B$-meson
mass differences $\Delta M_{B_d}$ and $\Delta M_{B_s}$. As in the case
of  the $K$-meson observables, we use  the same  input values as those
shown in Fig.~\ref{fig1},  i.e.\  $M_{\rm SUSY}  = 1$~TeV,  $M_{H^+} =
0.2$~TeV  and $\tan\beta  =  50$.  Then,  Fig.~\ref{fig4} displays the
combined, SM and   Higgs-DP,  contributions to   $\Delta  M_{B_d}$ and
$\Delta M_{B_s}$  as functions   of the   gluino phase  ${\rm   arg}\,
(m_{\tilde{g}})$, for  $\delta_{\rm CKM}   = 90^\circ$  and  different
choices of hierarchy factor $\rho$ and $\phi_{A_U}$.  Note that the SM
contributions alone for $\delta_{\rm CKM} = 90^\circ$ are displayed by
horizontal dashed  lines.  Even though the  SM predictions for $\Delta
M_{B_{d,s}}$ may  adequately describe by  themselves the  experimental
values in~(\ref{expBd})  and (\ref{expBs}), they cannot yet decisively
exclude  possible new-physics   contributions   due to  the   inherent
uncertainties in the calculation  of hadronic matrix elements such  as
those  induced  by   SUSY    Higgs-mediated FCNC     interactions.  In
particular, we observe in Fig.~\ref{fig4} that SM and Higgs-DP effects
may   add constructively  or   destructively  to the mass  differences
$\Delta M_{B_{d,s}}$.   Similar features are found in~Fig.~\ref{fig5},
where the  SM and SUSY Higgs-DP contributions  to $\Delta M_{B_d}$ and
$\Delta M_{B_s}$ are plotted  versus the hierarchy factor  $\rho$, for
discrete  values  of $\delta_{\rm CKM}$ and  $\phi_{A_U}$.   In our SM
CKM-phase convention~\cite{PDG},   unlike   the  CKM matrix    element
$V_{ts}$,   the  matrix  element   $V_{td}$   is  very   sensitive  to
$\delta_{\rm  CKM}$  values.   As  a result,  the  SM  predictions for
$\Delta M_{B_d}$ strongly depend on the selected value of $\delta_{\rm
CKM}$, as can be seen from~Fig.~\ref{fig5}.

In  Fig.~\ref{fig6},  we exhibit numerical    values for the branching
ratios  ${\cal   B}  (\bar{B}^0_s \to    \mu^+\mu^-)$  and  ${\cal  B}
(\bar{B}^0_s  \to  \tau^+\tau^-)$  as functions  for  the gluino phase
${\rm arg}\, (m_{\tilde{g}})$, for $M_{\rm  SUSY} = 1$~TeV, $M_{H^+} =
0.2$~TeV,  $\tan\beta = 50$,  and $\delta_{\rm CKM} = 90^\circ$, where
$\rho$ and  $\phi_{A_U}$  are varied discretely.   Since the branching
ratios are   driven by Higgs-penguin  effects in  the region  of large
$\tan\beta$, predictions for other inputs of $\tan\beta$ and $M_{H^+}$
may be easily estimated by rescaling the numerical  values by a factor
$x_{\cal   O}   = (\tan\beta/50  )^6\times   (0.2~TeV/M_{H^+})^4$, for
$\tan\beta      \stackrel{>}{{}_\sim}   40$.  Thus,    confronting the
predictions for   ${\cal    B}  (\bar{B}^0_s  \to \mu^+\mu^-)$    with
experiment data   in~(\ref{expBsmu}),    combined   bounds  on     the
$\tan\beta$-$M_{H^+}$ plane  may be obtained for a  given set  of soft
SUSY-breaking parameters. As we see in~Fig.~\ref{fig6}, these combined
bounds become even more  restrictive  for large gluino phases,   ${\rm
arg}\,   m_{\tilde{g}} \stackrel{>}{{}_\sim}   90^\circ$, in agreement
with our  discussions  in    Section~4.1.  

However, there is  an additional factor  that may crucially affect our
predictions  for the branching  ratios of the  decays $\bar{B}^0_s \to
\mu^+\mu^-$ and $\bar{B}^0_s  \to \tau^+\tau^-$,  namely the hierarchy
parameter $\rho$.  As we show in Fig.~\ref{fig7}, even for the extreme
choice of   a gluino phase  ${\rm   arg}\, m_{\tilde{g}} = 180^\circ$,
${\cal B} (\bar{B}^0_d \to \mu^+\mu^-)$ and ${\cal B} (\bar{B}^0_s \to
\mu^+\mu^-)$ can get very suppressed for a specific value of $\rho$ in
certain soft SUSY-breaking scenarios that  can realize a GIM-operative
point in their  parameter space.  As we  detailed in Section~4.1, this
phenomenon occurs for the universal value of $\rho  = \rho_{\rm GIM} =
1.22$,  when $\phi_{A_U} - \phi_{\tilde{g}}  =  0$ or $\pm 180^\circ$.
As    can   be seen  in    Fig.~\ref{fig7},  the predicted  values for
$\bar{B}^0_s \to \mu^+\mu^-$ and $\bar{B}^0_s \to \tau^+\tau^-$, where
$\phi_{A_U} = 0^\circ$, confirm the above observation.

Let us now investigate the size of the  CP asymmetries in the leptonic
$B_d$-meson decays  in  the CP-violating  MSSM;  the corresponding  CP
asymmetries for the $B_s$  meson are experimentally constrained to  be
rather small, less   than 5\% (see also discussion  after~(\ref{lq})).
In Fig.~\ref{fig8}, we display numerical values for the CP asymmetries
${\cal A}_{\rm CP}^{(B^0_d \to \mu^+_L\mu^-_L  )}$ and ${\cal  A}_{\rm
CP}^{(B^0_d \to \mu^+_R\mu^-_R )}$  as  functions of the  gluino phase
${\rm arg}\, (m_{\tilde{g}})$, for $M_{\rm  SUSY} = 1$~TeV, $M_{H^+} =
0.2$~TeV,  $\tan\beta = 50$, and  $\delta_{\rm  CKM} = 90^\circ$.   As
usual, we independently vary the parameters $\rho$ and $\phi_{A_U}$ to
take on the discrete values $\rho = 1,\ 10$ and $\phi_{A_U} = 0^\circ,
90^\circ$ and $180^\circ$.   We find that if the $B^0_d$-$\bar{B}^0_d$
mixing is consistently taken into  account, the typical size of ${\cal
A}_{\rm  CP}^{(B^0_d  \to   \mu^+_L\mu^-_L  )}$   and   ${\cal A}_{\rm
CP}^{(B^0_d \to  \mu^+_R\mu^-_R )}$ does   not  exceed 0.7\% and  3\%,
respectively. If $B^0_d$-$\bar{B}^0_d$ mixing is  not included, the CP
asymmetries  can reach  slightly higher  values up to   1.2\% and 6\%,
respectively.   The apparent  reason for   the   smallness of  the  CP
asymmetries is due to the occurrence of  an accidental cancellation in
the  sum $C_S +  C_P$ in~(\ref{lq}) at  large $\tan\beta$, as the muon
velocity is $\beta_\mu \approx 1$.

Having gained  some insight  from  the  above exercise,  one may  seek
alternative ways  to enhance  the di-muon asymmetries   ${\cal A}_{\rm
CP}^{(B_d^0\to \mu_{L,(R)}^+ \mu_{L,(R)}^-)}$.  To this end, the first
attempt would be to  suppress  the effect of the   $B_d^0-\bar{B}_d^0$
mixing  by considering smaller   $\tan\beta$ values, e.g.\  $\tan\beta
\stackrel{<}{{}_\sim} 10$. In this intermediate region of $\tan\beta$,
the above cancellation in  the sum $C_S + C_P$  does not occur due  to
non-trivial CP-violating  Higgs-mixing    effects  and  so    the   CP
asymmetries  ${\cal  A}_{\rm       CP}^{(B_d^0\to        \mu_{L,(R)}^+
\mu_{L,(R)}^-)}$ can be significantly increased. To get an idea of the
magnitude of   the  CP asymmetries   in this  case,   we  consider the
so-called  CPX   scenario  introduced   in~\cite{CPX}   to    maximize
CP-violating effects in the    lightest Higgs sector of an   effective
MSSM. In the CPX scenario,  the $\mu$-parameter and the soft trilinear
Yukawa coupling $A_U$ are set by the relations:  $\mu= 4 M_{\rm SUSY}$
and $A_U=  2  M_{\rm SUSY}$.  Thus,  for   $M_{H^+}=0.15$ TeV, $M_{\rm
SUSY}=1$  TeV, $\tan\beta=7$,    $\rho=10$, $\phi_{A_U}=45^\circ$  and
$\phi_{\tilde{g}}=0^\circ$,  we  find that  CP-violating Higgs-penguin
effects can give rise to the CP asymmetries:
\begin{equation}
  \label{CPX1}
{\cal A}_{\rm CP}^{(B_d^0\to \mu_L^+ \mu_L^-)}\ \approx\ -9~\%\,,\qquad 
{\cal A}_{\rm CP}^{(B_d^0\to \mu_R^+ \mu_R^-)}\ \approx\ -37~\%
\end{equation} 
where ${\cal B}(B_d\to \mu^+ \mu^-)=3.6  \times 10^{-10}$, which is an
order   of  magnitude larger   than the   SM prediction~\cite{Dedes2}.
Another  variant of  the  CPX   scenario of equally   phenomenological
importance        utilizes      the     parameters:    $\tan\beta=10$,
$\phi_{A_U}=45^\circ$  and  $\phi_{\tilde{g}}=90^\circ$,  with $M_{\rm
SUSY} = 1$~TeV. In this case, we obtain
\begin{equation}
  \label{CPX2}
{\cal A}_{\rm CP}^{(B_d^0\to \mu_L^+ \mu_L^-)}\ \approx\ 11~\%\,,\qquad
{\cal A}_{\rm CP}^{(B_d^0\to \mu_R^+ \mu_R^-)}\ \approx\ 43~\% 
\end{equation}
and  ${\cal B}(B_d\to \mu^+ \mu^-)=4.1 \times   10^{-9}$, which is two
orders of magnitude above the SM  prediction.  The above scenario seem
to pass all the experimental constraints, including those deduced from
LEP2 analyses of direct Higgs searches~\cite{CPX,Hreview}. Moreover, a
possible observation   of  a non-zero CP  asymmetry  in  the  leptonic
di-muon channel will  constitute the harbinger  for new physics at $B$
factories.  At this stage, it is important to comment on the fact that
the numerical values stated in~(\ref{CPX1}) and~(\ref{CPX2}) should be
viewed as crude  estimates, since  they are  obtained entirely  on the
basis of our  resummed FCNC effective Lagrangian~(\ref{master}) at the
intermediate   $\tan\beta$ regime.    However,  in    this  region  of
$\tan\beta$, we expect  additional one-loop  effects to start  getting
relevant, such as supersymmetric
 $Z$-penguin  and box diagrams.  Even  though
our initial  estimates given above  appear to yield rather encouraging
results,  a complete study of the  leptonic $B$-meson branching ratios
and the respective CP asymmetries for all  values of $\tan\beta$ needs
be performed.

In  the  case of $\tau$-lepton  CP  asymmetries, the velocities of the
decayed  $\tau$-leptons $\beta_\tau$ is  roughly 0.5, so one naturally
gets an appreciably higher value for  the expression $\beta_\tau\, C_S
+ C_P \approx 0.5 C_P$ in~(\ref{lq}).  As a  result, larger values for
the  $\tau$-lepton    CP    asymmetries   are  expected.  Indeed,   in
Fig.~\ref{fig9}, we display numerical  predictions for ${\cal  A}_{\rm
CP}^{(B^0_d \to   \tau^+_L\tau^-_L )}$ and  ${\cal A}_{\rm CP}^{(B^0_d
\to  \tau^+_R\tau^-_R    )}$ versus the    gluino  phase  ${\rm arg}\,
(m_{\tilde{g}})$,  for the same values  of the input  parameters as in
Fig.~\ref{fig8}. Then, the CP  asymmetries ${\cal A}_{\rm  CP}^{(B^0_d
\to  \tau^+_L\tau^-_L )}$  and      ${\cal A}_{\rm CP}^{(B^0_d     \to
\tau^+_R\tau^-_R )}$ can be as high as 9\% and 36\%, respectively.

We conclude this section with some general remarks. In addition to the
$K$- and $B$-meson observables we  have been studying here, there is a
large number of  other FCNC observables which have to  be studied in a
full-fledged combined analysis. For  example, a central r\^ole in such
a    global    analysis    plays    the    decay    $B_s    \to    X_s
\gamma$~\cite{CGNW,Bsgamma},  which will enable  us to  delineate more
accurately the CP-conserving/CP-violating soft SYSY-breaking parameter
space  favoured  by  low-energy  FCNC observables.   Finally,  in  our
numerical analysis,  we have concentrated on  scenarios that minimally
depart  from  the  minimal  flavour-violation assumption  through  the
presence  of diagonal, but  non-universal squark  masses. In  the most
general case,  however, the squark mass matrices  and consequently the
resummation matrix $\Ro$ of the radiative threshold effects may not be
diagonal.   Such   low-energy  realizations  with   off-diagonal  soft
squark-mass matrices  can still be treated exactly  within the context
of   our  resummed   FCNC   effective  Lagrangian~(\ref{master}),   by
appropriately considering  non-trivial quark-squark CKM-like matrices,
such as the 3-by-3 unitary matrices $\ULQ$ and $\URd$ in~(\ref{hd}).



\setcounter{equation}{0} 
\section{Conclusions}

We have derived  the proper general form for  the effective Lagrangian
of  Higgs-mediated   FCNC  interactions  to   $d$-type  quarks,  where
large-$\tan\beta$  radiative  threshold  effects  have  been  resummed
consistently (cf.~(\ref{master}) and (\ref{couplings})).  Our properly
resummed   FCNC  effective  Lagrangian   is  free   from  pathological
singularities,  which  mainly  emanate  from the  top-quark  dominance
hypothesis  frequently  adopted  in   the  literature,  and  has  been
appropriately   generalized   to   include  effects   of   squark-mass
non-universality, as well as CP-violation effects originating from the
CKM-mixing  matrix  and the  complex  soft  SUSY-breaking masses.   In
particular, our resummed effective  Lagrangian can be applied to study
Higgs-mediated  FCNC  effects   in  more  general  soft  SUSY-breaking
scenarios, beyond  those that have  already been discussed  within the
restricted framework  of models with minimal  flavour violation. Also,
an approach  to resumming radiative threshold  effects, very analogous
to the one developed in Section~2, can straightforwardly be applied to
see-saw  SUSY  models,  so  as  to  properly  describe  Higgs-mediated
lepton-flavour-violating interactions.
 
Within the  context of generic  soft SUSY-breaking scenarios,  we have
analyzed a number  of $K$- and $B$-meson observables,  such as $\Delta
M_{K,B}$,  $\epsilon_K$,  $\epsilon'/\epsilon$,  ${\cal  B}(B_{s,d}\to
\ell^+\ell^-)$, and  their associated leptonic  asymmetries, which are
enhanced  by  Higgs-boson  FCNC   interactions  for  large  values  of
$\tan\beta$.   In  the  same  context,  we  have  also  discussed  the
consequences  of $\tan\beta$-enhanced  Higgs-mediated FCNC  effects on
the decays $B_d \to \phi K_S$  and $B_d \to J/\Psi K_S$.  Moreover, we
find  that the  predictions crucially  depend  on the  choice of  soft
CP-violating phases  in a given set of  soft SUSY-breaking parameters.
For example,  for certain  values of the  gluino and stop  phases, the
predictions can reach and even exceed the current experimental limits,
whereas for other values of the  CP-odd phases the FCNC effects can be
reduced  dramatically, as a  result of  a kind  of a  GIM cancellation
which can be operative in the latter case.


Based on our resummed effective Lagrangian, the above analysis can now
be extended to a vast  number of other $K$- and $B$-meson observables.
Therefore,  further  dedicated  studies  need  be  performed  in  this
direction.  The obtained predictions  are then expected to sensitively
affect other low- and high-energy observables, such as measurements of
electron and neutron electric dipole moments and Higgs-boson searches,
as well  as studies on cosmological electroweak  baryogenesis and dark
matter.  It would be very interesting to determine to which degree the
emerging  CP-violating  MSSM  framework  with  CP-mixed  Higgs  bosons
mediating  $\tan\beta$-enhanced   interactions  to  matter   could  be
potentially   responsible    for   all   the    present   and   future
CP-conserving/CP-violating FCNC effects observed in Nature.
 
\subsection*{Acknowledgements}
We  wish to thank Ulrich Nierste  and Alexander Kagan for illuminating
discussions.  A.D.    acknowledges financial   support from the   CERN
Theory   Division      and   the  Network      RTN    European Program
HPRN-CT-2000-00148 ``Physics Across   the  Present  Energy   Frontier:
Probing the Origin of Mass.''

\newpage

\begin{thebibliography}{99}


\bibitem{GW} S.L. Glashow and S. Weinberg, Phys.\ Rev.\ {\bf D15}
(1977) 1958.

\bibitem{CS} T.P. Cheng and M. Sher, Phys.\ Rev.\ {\bf D35} (1987)
3484; G.C.~Branco, A.J.~Buras and J.M.~Gerard, Nucl.\ Phys.\ {\bf
B259} (1985) 306.


\bibitem{Banks} T.~Banks, Nucl.\ Phys.\ {\bf B303} (1988) 172; E.~Ma,
 Phys.\ Rev.\ {\bf D39} (1989) 1922.

\bibitem{RH}  R. Hempfling, Phys.\ Rev.\ {\bf  D49} (1994) 6168; 
 L.J.  Hall, R.  Rattazzi and U.  Sarid, Phys.\ Rev.\ {\bf D50} (1994)
 7048; T.  Blazek, S.   Raby and S.  Pokorski,  Phys.\ Rev.\ {\bf D52}
 (1995)  4151;  M.   Carena,   M.    Olechowski,  S.    Pokorski   and
 C.E.M. Wagner,  Nucl.\ Phys.\ {\bf  B426} (1994)  269; S. Heinemeyer,
 W. Hollik and G. Weiglein, Phys.\ Lett.\ {\bf B440} (1998) 296; Eur.\
 Phys.\ J.   {\bf C9} (1999)     343; F.  Borzumati,  G.  Farrar,   N.
 Polonsky and   S.~Thomas,  Nucl.\  Phys.\   {\bf  B555}   (1999)  53;
 J.R. Espinosa and R.J. Zhang, Nucl.\ Phys.\ {\bf B586} (2000)~3.

\bibitem{CGNW} M.~Carena, D.~Garcia, U.~Nierste and C.E.~Wagner,
 Nucl.\ Phys.\ {\bf B577} (2000) 88.

\bibitem{FCNC} C.S.  Huang  and Q.-S. Yan,  Phys.\  Lett.\  {\bf B442}
(1998) 209;  S.R.~Choudhury  and   N.~Gaur,  Phys.\ Lett.\  {\bf B451}
(1999) 86; C.S.   Huang, W.  Liao  and Q.-S.  Yuan,  Phys.\ Rev.\ {\bf
D59}  (1999) 011701; C.~Hamzaoui,   M.~Pospelov and M.~Toharia, Phys.\
Rev.\   {\bf   D59}   (1999) 095005;   P.H.      Chankowski and  Lucja
Slawianowska, Phys.\  Rev.\ {\bf D63}  (2001) 054012; C.S.   Huang, W.
Liao,  Q.-S.  Yuan and   S.-H.   Zhu, Phys.\   Rev.\ {\bf D63}  (2001)
114021;  M.~Carena,  D.~Garcia, U.~Nierste   and  C.~E.~Wagner, Phys.\
Lett.\ {\bf  B499} (2001)  141. 

\bibitem{Frank} C.~Bobeth, T.~Ewerth, F.~Kr\"uger and J.~Urban,
 Phys.\ Rev.\ {\bf D64} (2001) 074014.

\bibitem{KB} K.S. Babu and C. Kolda, Phys.\ Rev.\ Lett.\ {\bf 84}
  (2000) 228.

\bibitem{Isidori} G.~Isidori and A.~Retico, JHEP {\bf 0111} (2001)
001; hep-ph/0208159.

\bibitem{Dedes1} A.~Dedes, H.K.~Dreiner and U.~Nierste, Phys.\ Rev.\
  Lett.\ {\bf 87} (2001) 251804; A.~Dedes, H.K.~Dreiner, U.~Nierste
  and P.~Richardson, hep-ph/0207026.

\bibitem{ADKT} R.~Arnowitt, B.~Dutta, T.~Kamon and M.~Tanaka, Phys.\
Lett.\ {\bf B538} (2002) 121; S.~Baek, P.  Ko and W.Y. Song,
hep-ph/0208112.

\bibitem{Ambrosio} G.~D'Ambrosio, G.F.~Giudice, G.~Isidori and A.~Strumia,
  hep-ph/0207036.

\bibitem{Buras} A.J.~Buras, P.H.~Chankowski, J.~Rosiek and L.~Slawianowska,
  hep-ph/0207241.

\bibitem{MTW} J.K. Mizukoshi, X. Tata and Y. Wang, hep-ph/0208078.

\bibitem{IN} T. Ibrahim and P. Nath, hep-ph/0208142. 

\bibitem{LN} {}From a Feynman--diagrammatic point of view, resummation
of  radiative   threshold  corrections  is   equivalent  to  resumming
wave-function graphs  in the  on-shell renormalization scheme.   For a
detailed discussion  on this  issue, see H.E.   Logan and  U. Nierste,
Nucl.\ Phys.\ {\bf B586} (2000) 39 and \cite{CGNW}.

\bibitem{CKM} N.~Cabibbo, Phys.\ Rev.\ Lett.\ {\bf 10} (1963) 531;
 M.~Kobayashi and T.~Maskawa,  Prog.\ Theor.\  Phys.\ {\bf 49}  (1973)
 652.

\bibitem{CEPW} M. Carena, J. Ellis, A. Pilaftsis and C.E.M. Wagner,
 Nucl.\ Phys.\ {\bf B586} (2000) 92; A.~Pilaftsis, hep-ph/0207277 (to
 appear in Nuclear Physics B).

\bibitem{comment} For example, gaugino-Higgsino mixing effects studied
 in~\cite{MTW}  and~\cite{IN} can be  straightforwardly included in our
 formalism.

\bibitem{APLB} A. Pilaftsis, Phys.\  Rev.\ {\bf D58} (1998) 096010;
  Phys.\ Lett.\ {\bf B435} (1998) 88.

\bibitem{PW} A.~Pilaftsis and C.E.M.  Wagner, Nucl.\ Phys.\ {\bf B553}
 (1999) 3; D.A.  Demir, Phys.\ Rev.\ {\bf D60} (1999) 055006; S.Y.
 Choi, M.  Drees and J.S.  Lee, Phys.\ Lett.\ {\bf B481} (2000) 57;
 G.L. Kane and L.-T. Wang, Phys.\ Lett.\ {\bf B488} (2000) 383; T.
 Ibrahim and P.~Nath, Phys.\ Rev.\ {\bf D63} (2001) 035009;
 hep-ph/0204092; S.Y.  Choi, K.  Hagiwara and J.S.~Lee, Phys.\ Rev.\
 {\bf D64} (2001) 032004; S. Heinemeyer, Eur.\ Phys.\ J. {\bf C22}
 (2001) 521; M.  Carena, J.  Ellis, A.  Pilaftsis and C.E.M.  Wagner,
 Nucl.\ Phys.\ {\bf B625} (2002) 345; M.~Boz, Mod.\ Phys.\ Lett.\ {\bf
 A17} (2002) 215; S.W.  Ham, S.K.  Oh, E.J. Yoo, C.M. Kim and D. Son,
 hep-ph/0205244.

\bibitem{GIM} S.L. Glashow, J.  Iliopoulos and L. Maiani, Phys.\ Rev.\
  {\bf D2} (1970) 1285.

\bibitem{kaonrev} For reviews on kaon physics, see, 
  E.A.~Paschos and U.~T\"urke, Phys.\ Rept.\ {\bf 178} (1989) 145; 
  W.~Grimus, Fortschr.\ Phys.\ {\bf 36} (1988) 201; R.  Decker,
  Fortschr.\ Phys.\ {\bf 37} (1989) 657; L.~Lavoura, Annals Phys.\
  {\bf 207} (1991) 428; B. Weinstein and L. Wolfenstein, Rev.\ Mod.\
  Phys.\ {\bf 65} (1993) 1113.

\bibitem{Neubert} For pedagogical introductions to CP violation in
  the  $B$-meson system, see, M.  Neubert, Int.\ J.\ Mod.\ Phys.\ {\bf
  A11} (1996) 4173; A.J.~Buras, hep-ph/9806471.

\bibitem{Branco} G.C.~Branco, L.~Lavoura and J.P.~Silva,
  {\it  Oxford, UK: Clarendon (1999)}, p. 511.

\bibitem{Jager} A.J.~Buras, S.~Jager and J.~Urban, Nucl.\ Phys.\ {\bf
 B605} (2001) 600.

\bibitem{Chankowski} A.J.~Buras, P.H.~Chankowski, J.~Rosiek and 
 L.~Slawianowska, Nucl.\ Phys.\ {\bf B619} (2001) 434.

\bibitem{Ciuchini} M.~Ciuchini {\it et al.}, JHEP {\bf 0107} (2001) 013.
 
\bibitem{Misiak} A.J.~Buras, M.~Misiak and J.~Urban, Nucl.\ Phys.\ 
  {\bf B586} (2000) 397.
 
\bibitem{DER}
A.~Dedes, J.~Ellis and M.~Raidal, arXiv:hep-ph/0209207.


\bibitem{Becirevic} D.~Becirevic, V.~Gimenez, G.~Martinelli,
  M.~Papinutto and J.~Reyes, Nucl.\ Phys.\ Proc.\ Suppl.\  
  {\bf 106} (2002) 385.
 
\bibitem{Buchalla} G.~Buchalla, A.J.~Buras and M.K.~Harlander, 
  Nucl.\ Phys.\ {\bf B337} (1990) 313.


\bibitem{EAP} T. Hambye, G.O. Kohler, E.A. Paschos, P.H. Soldan
 and W.A. Bardeen, Phys.\ Rev.\ {\bf D58} (1998) 014017;
 T.~Hambye, G.O.~Kohler, E.A.~Paschos and P.H.~Soldan,
 Nucl.\ Phys.\ {\bf B564} (2000) 391. 

\bibitem{Bosch} S.~Bosch, A.J.~Buras, M.~Gorbahn, S.~Jager, M.~Jamin, 
  M.E.~Lautenbacher and L.~Silvestrini, Nucl.\ Phys.\ {\bf B565} (2000) 3.

\bibitem{Guidice} E.~Gabrielli and G.F.~Giudice, Nucl.\ Phys.\ 
  {\bf B433} (1995) 3.

\bibitem{epeSUSY} G.~Colangelo and G.~Isidori,
  JHEP {\bf 9809}  (1998) 009;
  A.J.~Buras, G.~Colangelo, G.~Isidori, A.~Romanino and L.~Silvestrini,
  Nucl.\ Phys.\ {\bf B566} (2000) 3.

\bibitem{Kagan:1999iq} A.L.~Kagan and M.~Neubert, Phys.\ Rev.\ Lett.\ 
 {\bf 83} (1999) 4929; A.J.~Buras, P.~Gambino, M.~Gorbahn, S.~Jager
 and  L.~Silvestrini, Nucl.\ Phys.\ {\bf B592} (2001) 55;
 C.H.~Chen, Phys.\ Lett.\ {\bf B541} (2002) 155.

\bibitem{Nir} Y.~Nir, hep-ph/0208080.

\bibitem{Dedes2} See second reference in~\cite{Dedes1}.

\bibitem{Liao}
C.S.~Huang and W.~Liao, Phys.\ Lett.\ {\bf B525} (2002) 107; Phys.\
Lett.\ {\bf B538} (2002) 301; P.H.~Chankowski and L.~Slawianowska,
Acta Phys.\ Polon.\ {\bf B32} (2001) 1895.

\bibitem{PDG}
K.~Hagiwara {\it et al.}  [Particle Data Group Collaboration], Phys.\
Rev.\ {\bf D66} (2002) 010001.

\bibitem{ACP} For a detailed discussion, see, e.g.~\cite{Branco,Nir}.

\bibitem{Gronau} M.~Gronau, Phys.\ Lett.\ {\bf B300} (1993) 163.
 
\bibitem{Worah} Y.~Grossman and M.P.~Worah, Phys.\ Lett.\ 
  {\bf B395} (1997) 241.

\bibitem{Hiller} G.~Hiller, hep-ph/0207356. 

\bibitem{Wyler} E.~Lunghi and D.~Wyler, Phys.\ Lett.\ {\bf B521} (2001) 320.

\bibitem{Beneke} M.~Beneke, G.~Buchalla, M.~Neubert and C.T.~Sachrajda,
Phys.\ Rev.\ Lett.\  {\bf 83} (1999) 1914.
 
\bibitem{NH} For instance, see, S.~Herrlich and U.~Nierste, Phys.\
Rev.\ {\bf D52} (1995) 6505.

\bibitem{Nilles}  See,  for  example, J.F.~Donoghue,  H.P.~Nilles  and
D.~Wyler, Phys.\ Lett.\ {\bf  B128} (1983) 55; J.M.~Gerard, W.~Grimus,
A.~Raychaudhuri and G.~Zoupanos, Phys.\ Lett.\ {\bf B140} (1984) 349.

\bibitem{Ligeti} Y. Grossman, Z. Ligeti and E. Nardi, Phys.\ Rev.\
{\bf D55} (1997) 2768.

\bibitem{Anikeev}
K.~Anikeev {\it et al.}, arXiv:hep-ph/0201071.


\bibitem{CPX} M. Carena, J. Ellis, A. Pilaftsis and C.E.M. Wagner,
Phys.\ Lett.\ {\bf B495} (2000) 155.

\bibitem{Hreview} For a recent review on Higgs phenomenology, see
M. Carena and H.E. Haber, hep-ph/0208209.

\bibitem{Bsgamma} S.~Bertolini, F.~Borzumati, A.~Masiero and
G.~Ridolfi, Nucl.\ Phys.\ {\bf B353} (1991) 591; D.A.~Demir and
K.A.~Olive, Phys.\ Rev.\ {\bf D65} (2002) 034007; \dots

\end{thebibliography}

\newpage

%%%%%%%%%%%%%%%%%%%%% FIGURES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[p]
\hspace*{3.5cm}\psfig{figure=hier.eps}
\caption{{\it  Schematic  representation  of the   SUSY mass  spectrum
considered in  our    numerical analysis,  where  $m_{\tilde  q}$  and
$m_{\tilde  t}$  denote   the  masses of   the  first   two  and third
generations of   squarks, respectively.  The  hierarchy factor $\rho$,
the   phase $\phi_{\tilde{g}}$ of   the   gluino mass, and  the phases
$\phi_{A_{t,b}}$ of the soft SUSY  breaking trilinear couplings,  with
$\phi_{A_t} = \phi_{A_b}  = \phi_{A_U}$, are varied independently (see
also discussion in the text).}}\label{fig1}
\end{figure}

%%%

\begin{figure}[p]
%\hspace*{3.5cm}
\psfig{figure=DmKeK_CKM90.eps,height=6in,angle=-90}
\caption{{\it SUSY Higgs-DP contributions  to $\epsilon_K$ and $\Delta
M_K$ given in units of $10^{-12}$ MeV  and $10^{-3}$, respectively, as
functions   of the gluino   phase  ${\rm arg}\,  (m_{\tilde{g}})$, for
$M_{\rm   SUSY} = 1$~TeV,  $M_{H^+}  = 0.2$~TeV, $\tan\beta  = 50$ and
$\delta_{\rm CKM}  =  90^\circ$.   As  is  shown  above, the different
curves  are  obtained      for   selected  values  of  $\rho$      and
$\phi_{A_U}$. The  size of the  2HDM  effect alone on  $\epsilon_K$ is
indicated by an arrow.}}\label{fig2}
\end{figure}

%%%

\begin{figure}[p]
%\hspace*{3.5cm}
\psfig{figure=GIM_rho.eps,height=5.9in,angle=-90}
\caption{{\it SUSY  Higgs-DP contributions to $\epsilon_K$ and $\Delta
M_K$ given in units of $10^{-12}$  MeV and $10^{-3}$, respectively, as
functions of the hierarchy factor $\rho$,  for $M_{\rm SUSY} = 1$~TeV,
$M_{H^+}   =   0.2$~TeV,    $\tan\beta =    50$,   and  ${\rm arg}\,
m_{\tilde{g}} = 180^\circ$, where the values of $\delta_{\rm CKM}$ and
$\phi_{A_U}$ are varied discretely.}}\label{fig3}
\end{figure}

%%%

\begin{figure}[p]
%\hspace*{3.5cm}
\psfig{figure=DmB_CKM90.eps,height=6.4in,angle=-90}
\caption{{\it SM and SUSY Higgs-DP contributions to $\Delta M_{B_d}$ and
$\Delta M_{B_s}$ as functions of the gluino phase ${\rm arg}\,
(m_{\tilde{g}})$, for $M_{\rm SUSY} = 1$~TeV, $M_{H^+} = 0.2$~TeV,
$\tan\beta = 50$ and $\delta_{\rm CKM} = 90^\circ$, where the
hierarchy factor $\rho$ and $\phi_{A_U}$ are varied independently as
shown above. The SM contributions alone for $\delta_{\rm CKM} = 90^\circ$
are displayed by horizontal dashed lines.}}\label{fig4}
\end{figure}

%%%

\begin{figure}[p]
%\hspace*{3.5cm}
\psfig{figure=GIM_DmB_rho.eps,height=5.9in,angle=-90}
\caption{{\it SM and SUSY Higgs-DP contributions to $\Delta M_{B_d}$
and $\Delta M_{B_s}$ versus the hierarchy factor $\rho$, for $M_{\rm
SUSY} = 1$~TeV, $M_{H^+} = 0.2$~TeV, $\tan\beta = 50$, and ${\rm
arg}\, m_{\tilde{g}} = 180^\circ$, where $\delta_{\rm CKM}$ and
$\phi_{A_U}$ obtain discrete values as shown above. Also shown are the
SM effects alone for different choices of the CKM phase $\delta_{\rm
CKM}$ (horizontal dashed lines).}}\label{fig5}
\end{figure}


%%%

\begin{figure}[p]
%\hspace*{3.5cm}
\psfig{figure=Bll_CKM90.eps,height=6.4in,angle=-90}
\caption{{\it SUSY Higgs-penguin contributions to ${\cal B}
(\bar{B}^0_s \to \mu^+\mu^-)$ and ${\cal B} (\bar{B}^0_d \to
\tau^+\tau^-)$ versus the gluino phase ${\rm arg}\, (m_{\tilde{g}})$,
for $M_{\rm SUSY} = 1$~TeV, $M_{H^+} = 0.2$~TeV, $\tan\beta = 50$, and
$\delta_{\rm CKM} = 90^\circ$, where $\rho$ and $\phi_{A_U}$ are
varied discretely.}}\label{fig6}
\end{figure}


%%%

\begin{figure}[p]
%\hspace*{3.5cm}
\psfig{figure=GIM_Bll_rho.eps,height=6.4in,angle=-90}
\caption{{\it SUSY Higgs-penguin contributions to ${\cal B}
(\bar{B}^0_s \to \mu^+\mu^-)$ and ${\cal B} (\bar{B}^0_d \to
\tau^+\tau^-)$ as functions of the hierarchy factor $\rho$, for
$M_{\rm SUSY} = 1$~TeV, $M_{H^+} = 0.2$~TeV, $\tan\beta = 50$, and
${\rm arg}\, m_{\tilde{g}} = 180^\circ$, where $\delta_{\rm CKM}$ and
$\phi_{A_U}$ take discrete values.}}\label{fig7}
\end{figure}

%%%

\begin{figure}[p]
%\hspace*{3.5cm}
\psfig{figure=Asymmetries_Bmumu_CKM90.eps,height=6.4in,angle=-90}
\caption{{\it Numerical values for the CP asymmetries ${\cal A}_{\rm
CP}^{(B^0_d \to \mu^+_L\mu^-_L )}$ and ${\cal A}_{\rm
CP}^{(B^0_d \to \mu^+_R\mu^-_R )}$ as functions of the gluino
phase ${\rm arg}\, (m_{\tilde{g}})$, for $M_{\rm SUSY} = 1$~TeV,
$M_{H^+} = 0.2$~TeV, $\tan\beta = 50$, and $\delta_{\rm CKM} =
90^\circ$, where $\rho$ and $\phi_{A_U}$ are varied
discretely. Also shown is the prediction for the CP asymmetries
without including $B^0_d$-$\bar{B}^0_d$ mixing.}}\label{fig8}
\end{figure}


%%%

\begin{figure}[p]
%\hspace*{3.5cm}
\psfig{figure=Asymmetries_Btautau_CKM90.eps,height=6.4in,angle=-90}
\caption{{\it Numerical estimates of the CP asymmetries ${\cal A}_{\rm
CP}^{(B^0_d \to \tau^+_L\tau^-_L )}$ and ${\cal A}_{\rm
CP}^{(B^0_d \to \tau^+_R\tau^-_R )}$ versus the gluino phase
${\rm arg}\, (m_{\tilde{g}})$, for $M_{\rm SUSY} = 1$~TeV, $M_{H^+} =
0.2$~TeV, $\tan\beta = 50$, and $\delta_{\rm CKM} = 90^\circ$, where
$\rho$ and $\phi_{A_U}$ take discrete values as shown
above.}}\label{fig9}
\end{figure}


\end{document}











