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\newcommand{\URu}{{\bf {\cal U}_R^u}}
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\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{\small    
We derive the general resummed effective Lagrangian for Higgs-mediated
flavour-changing  neutral-current  (FCNC) interactions  in the Minimal
Supersymmetric Standard  Model (MSSM), without resorting to particular
assumptions that rely 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.   Finally, based on our  resummed
FCNC effective Lagrangian, we can  identify configurations in the soft
supersymmetry-breaking   parameter space, for  which    a  kind of   a
Glashow--Iliopoulos--Maiani-cancellation mechanism  becomes  operative
and hence all Higgs-mediated, $\tan\beta$-enhanced effects on $K$- and
$B$-meson FCNC observables vanish.  }


\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   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}.   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
relevant~\cite{FCNC,BK}, 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,BK,CGNW,FCNCCP,Frank,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, $B
\to X_s \gamma$ 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.

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  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     qualitatively   discuss   the implications     of
$\tan\beta$-enhanced  Higgs-mediated     interactions for   the direct
CP-violation  parameter  $\epsilon'/\epsilon$  in   the  kaon  system.
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.   In  particular,  based on our
resummed FCNC  effective     Lagrangian, we  are   able  to   identify
configurations in the soft supersymmetry-breaking parameter space, for
which a  kind of  a Glashow--Iliopoulos--Maiani-cancellation mechanism
(GIM)~\cite{GIM} becomes operative in the Higgs--$d$-quark sector.  As
a result, all Higgs-mediated, $\tan\beta$-enhanced effects on $K$- and
$B$-meson FCNC observables  vanish.  Finally, Section~5 summarizes our
conclusions.


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

In  this  section,  we  derive  the general  form  for  the  effective
Lagrangian  of Higgs-mediated  FCNC interactions  in  the CP-violating
MSSM.    For   this   purpose,   we  also   consistently   resum   the
$\tan\beta$-enhanced  radiative   effects  on  the   $d$-quark  Yukawa
couplings~\cite{CGNW}.  First, we  analyze a simple soft SUSY-breaking
model    based    on     the    assumption    of    minimal    flavour
violation~\cite{BK,Isidori,Ambrosio,Buras},  where the  CKM  matrix is
the only source of flavour and CP violations. We find that even within
this minimal  framework, the usually  neglected $c$-quark contribution
to  Higgs-mediated  FCNC  interactions   may  be  competitive  to  the
$t$-quark one in certain regions  of the parameter space. After having
gained some insight from the  above considerations, we then extend our
resummed  effective Lagrangian  approach  to more  general cases  that
include a  non-universal or hierarchical  squark sector as well  as CP
violation originating  from the CKM matrix and  the soft SUSY-breaking
parameters.

Before discussing the  most general  case,  let us  first consider the
following simple  form  for the effective Yukawa  Lagrangian governing
the    Higgs-mediated    FCNC     interactions    in    the      quark
sector~\cite{FCNC,BK}:
\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~\cite{RH,FCNC,BK}
\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. 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
 assume 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
 at   the   soft  SUSY-breaking scale $M_{\rm    SUSY}$.    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_R^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_2}\,
\Big( \tan\beta\,\Phi_1^{0*}\: -\: \Phi_2^{0*} \Big)\,
\bar{d}_R\,  \Md \Vckm^\dagger \Rh^{-1}\Vckm\, d_L\ +\
\frac{\sqrt{2}}{v_2}\, \Phi_2^{0*}\, \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 is very illuminating to see how the FCNC part of (\ref{LYeff})
compares  with   the    literature,  e.g.\ with    that   obtained  in
Ref.~\cite{BK}.  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_2}\, \Big( \tan\beta\,\Phi_1^{0*}\: -\: 
\Phi_2^{0*} \Big)\, \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{BK}   and  subsequent
articles~\cite{Isidori,Ambrosio,Buras}.   However, we  should   remark
here that the frequently-used  top-quark
dominance  approximation,   cannot be
justified   from considerations    based only   on  minimal
flavour-violation models~\cite{BK,Isidori,Ambrosio,Buras}.  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 $\asos + \Eg   \tan\beta \to 0$. In this
 limit, 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 $\Re  e \Eg < 0$, a  perturbative upper bound on
$\tan\beta$, $\tan\beta \stackrel{<}{{}_\sim} 1/|\Eg|$, may be derived
beyond  the tree level.}  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. 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,    the  $3\times
3$-dimensional matrix $\Ro$ may generally 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 $M_{\rm SUSY}$.  These contributions can 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   more  generic  case  of 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 3-by-3 soft squark mass
matrices $m_{\tilde{Q}_L}$, $m_{\tilde{u}_R}$, $m_{\tilde{d}_R}$,  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 $O_{ij}$ 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 general 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  $\tan\beta$-enhanced  finite radiative  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.}    non-universality in the  squark  sector  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   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  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 
suppressed.  Instead, the FCNC couplings of the heavy $H_{2,3}$ bosons
to $d$-type quarks retain their  $\tan\beta$-enhanced strength in  the
above kinematic region.

\item[(v)]  The  one-loop resummed effective Lagrangian~(\ref{master})
captures the major  bulk  of the one-loop radiative  effects~\cite{LN}
for  large      values   of  $\tan\beta$,   e.g.\     for   $\tan\beta
\stackrel{>}{{}_\sim} 40$, and for a  soft SUSY-breaking scale $M_{\rm
SUSY}$  much  higher  than   the   electroweak scale~\cite{CGNW}.   In
addition, (\ref{master})  is  only valid in   the limit  in which  the
four-momentum of the $d$ quarks and Higgs bosons  in the external legs
is  much smaller  than    $M_{\rm  SUSY}$.  This   last   condition is
automatically   satisfied  in our   computations   of low-energy  FCNC
observables.


\end{itemize}

In the next section,  we  will study  in detail  the  phenomenological
consequences 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, and $C_i   (\mu)$ 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 the
  $\bar{P}$'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 $\tan\beta \stackrel{>}{{}_\sim} 40$.

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,
induced by 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  (CL).  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,BK,FCNCCP,Frank,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  $C_P$ in
the region  of  large values of $\tan\beta$:\footnote{Our  approach to
Higgs-mediated FCNC  effects    presented here  may   be  extended  to
consistently account for charged-lepton flavour violation in $B$-meson
decays, such  as  $B_{s,d}\to \ell^+ \ell'^{-}$~\cite{DER}, where  the
effective     off-diagonal        Higgs-lepton-lepton        couplings
$g_{H_i\bar{\ell}\ell'}^{S,P}$    can    be derived    by  following a
methodology very analogous to the one described in Section~2.}
\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 the next section, we will present numerical  estimates for the $K$-
and  $B$-meson FCNC observables,  based  on  the analytic  expressions
derived above.



\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{BK}.

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 elements, 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  $\Ro$.  Since $\Ro$  is diagonal for the scenarios
(A)  and     (B)     under consideration,   we    can     expand   the
$\tan\beta$-enhanced FCNC terms    $(\Vckm^\dagger\,        \Ro^{-1}\,
\Vckm)_{dd'}$ in~(\ref{master}) as follows:
\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, with
$d\neq d'$.    For  the   parameters  adopted  in~Fig.~\ref{fig1}, the
quantities $R^{-1}_{u,c,t}$  can 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_{\tilde{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
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
$x_{\cal O} =  (\tan\beta/50 )^6\times (0.2~{\rm TeV}/M_{H^+})^4$, 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.

As  we have already  mentioned, the  observables $\Delta  M_{B_s}$ and
${\cal  B}(\bar{B}_s^0\to \mu^+  \mu^-)$ exhibit  a  different scaling
behaviour  with  respect to  $M_{H^+}$  and  $\tan\beta$, through  the
scaling  factor $x_{\cal  O}$  in~(\ref{scale}).  Once  the above  two
kinematic  parameters  are  fixed   to  some  input  values,  the  two
observables  $\Delta  M_{B_s}$   and  ${\cal  B}(\bar{B}_s^0\to  \mu^+
\mu^-)$  are  then  rather  correlated  to  each  other,  since  their
dependences on the  $\mu$-parameter and $A_U$ are very  similar in the
CP-invariant  minimal flavour-violating  case  $\rho =1$~\cite{Buras}.
As  one  can  see  from  Figs.~\ref{fig4}--\ref{fig7},  our  numerical
analysis agrees well with the above result for $\rho =1$.  However, we
also observe that the correlation  is practically lost for $\rho > 1$,
e.g.\  at  the  GIM-operative  points,  and/or  by  the  inclusion  of
CP-violating effects, as $\Delta M_{B_s}$ and ${\cal B}(\bar{B}_s^0\to
\mu^+ \mu^-)$ have different dependences on the soft CP-odd phases.

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: $\tan\beta=10$, $\rho  =   10$,
$\phi_{A_U}=45^\circ$  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
appears to pass  all   the experimental constraints,   including those
deduced      from      LEP2     analyses      of     direct      Higgs
searches~\cite{CPX,Hreview}.   Most interestingly,      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 $\tan\beta$   would  be
highly preferable.

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 considered in
a combined  full-fledged analysis. For  example, the decay $B  \to X_s
\gamma$~\cite{Bsgamma1,Bsgamma2}  plays  a  central r\^ole  in  such a
global  analysis, because  it  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 matrix $\ULQ$ in~(\ref{hd}).


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

We  have derived  the 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 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
$\tan\beta$.  We  have found 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 by one or  even two orders  of
magnitude.   Most  remarkably, we     have  been able    to   identify
configurations  in  the soft  SUSY-breaking  parameter space,  such as
$\rho_{\rm GIM}$, where a kind of a GIM-cancellation mechanism becomes
fully operative [cf.~(\ref{rgim})] and, as a result of the latter, all
Higgs-mediated, $\tan\beta$-enhanced  effects  on $K$-  and  $B$-meson
FCNC observables are completely absent.

Based on our resummed effective Lagrangian, one may now carry over the
present  analysis  to  a  vast  number of  other  $K$-  and  $B$-meson
observables.  Evidently,  further dedicated studies  need be performed
in this direction.  We  expect the obtained predictions 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,  and  Herbi  Dreiner   for  a  critical  reading  of  the
manuscript.  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.'' A.P.   thanks the Fermilab  Theory Group for  warm hospitality
and support.

\newpage

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\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.in,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}











