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#2\end{minipage}\\[1mm]
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\begin{document}
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\begin{flushright}
Here we put the preprint numbers
%DCPT/08/38\\
%IPPP/08/19 \\
%arXiv:0804.1228 [hep-ph]
\end{flushright}
\vspace{1cm}
\begin{center}
{\large\sc {\bf
Flavour Les Houches Accord: Interfacing Flavour related Codes}}
\vspace{5.cm}
% {\sc
% F.~Mahmoudi$^{1,2}$%
% \footnote{
% email: mahmoudi@in2p3.fr
% }%
% , S.~Heinemeyer$^{3}$%
% \footnote{
% email: Sven.Heinemeyer@cern.ch
% }
% , \ldots
% }
% K.~Agashe$^{1}$,
% G.~Belanger$^{2}$,
% A.~Bharucha$^{3}$,
% F.~Boudjema$^{2}$,
% }%
% , S.~Kraml$^{5}$,
% , M.~Muhlleitner$^{7}$,
% W.~Reece$^{8}$,
% J.~Reuter$^{9}$,
% P.~Skands$^{10}$,
% P.~Slavich$^{11}$
%
\vspace*{0.5cm}
% {\sl
% $^1$Clermont Universit\'e, Universit\'e Blaise Pascal, Laboratoire de Physique Corpusculaire, \\
% BP 10448, F-63000 Clermont-Ferrand, France
%
% \vspace*{0.1cm}
% $^2$CNRS/IN2P3, UMR 6533, LPC, F-63177 Aubi\`ere Cedex, France
%
% \vspace*{0.1cm}
%
% $^3$Instituto de F\'isica de Cantabria (CSIC-UC), Santander, Spain
% }
%
%
% $^2$
%
% \vspace*{0.1cm}
%
% $^3$
%
% \vspace*{0.1cm}
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% \vspace*{0.1cm}
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% $^5$
%
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% \vspace*{0.1cm}
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% $^7$
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% \vspace*{0.1cm}
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% $^8$
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% \vspace*{0.1cm}
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% $^9$
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% \vspace*{0.1cm}
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% $^{10}$
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% \vspace*{0.1cm}
%
% $^{11}$
%
% }
\end{center}
\begin{abstract}
An accord specifying a unique set of conventions for flavour related
parameters and observables using the generic SLHA file structure is
defined. The Flavour Les Houches Accord (FLHA) defines relevant SM
masses, Wilson coefficients, form factors, decay tables, flavour observables,
etc. The accord provides a universal and model independent interface
between codes evaluating and/or using flavour related observables.
\end{abstract}
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\section{Introduction}
Advanced programs dedicated to the calculation of flavour related
observables (Wilson coefficients, branching ratios, mixing amplitudes,
RGE running including flavour effects, etc.) have
appeared~\cite{Mahmoudi:2007vz,Degrassi:2007kj},
along with an increasing number of refined approaches in the literature.
Flavour related observables are also implemented in many other
non-dedicated public codes as additional checks for the models under
investigation
\cite{Belanger:2008sj,Arbey:2009gu,Heinemeyer:1998yj,Lee:2003nta,Ellwanger:2005dv,Paige:2003mg}.
These quantities are subsequently often used by other codes, e.g.\ as
constraints on the parameter space of the model under consideration
\cite{Lafaye:2004cn,Bechtle:2004pc,deAustri:2006pe,Master3}.
At present, a small number of specialized interfaces exist between
various codes. Such tailor-made interfaces are not easily generalized
and are time-consuming to construct and test for each specific
implementation. A universal interface would clearly be an advantage
here.
%
A similar problem appeared some time ago in the context of Supersymmetry
(SUSY). The solution found is the SUSY Les Houches Accord
(SLHA)~\cite{slha1,slha2}, which is nowadays used frequently to exchange
information between SUSY related codes, such as soft SUSY-breaking
parameters, particle masses and mixings, branching ratios etc.
The SLHA is a robust solution, exchanging information between
different codes via ASCII files
for inputs and outputs.
The detailed structure of these files is described in \citeres{slha1,slha2}.
The goal of this article is to exploit the existing organizational structure
of the SLHA and use it to define an accord for the exchange of flavour
related quantities, the ``Flavour Les Houches Accord'' (FLHA). Briefly
stated, the purpose of this Accord is thus to present a set of generic
definitions for an input/output file structure which provides a
universal framework for interfacing flavour related calculation
programs. Furthermore, such a standard format will allow the users to
have a clear and well-structured result that can eventually be used for
different purposes.
The structure is set up in such a way that the SLHA and the FLHA can be
used together or independently.
Obviously, some of the SLHA entries, such as the Standard Model (SM) measured
values and CKM matrix elements are also needed for flavour observable
calculations. Therefore, a FLHA file can indeed contain a SLHA block if
necessary. For this reason and also for sake of clarity, the FLHA block
names start with ``\texttt{F}''. Also, in order to avoid any confusion,
the SLHA blocks are not modified and redefinition of SLHA blocks by
means of FLHA is not allowed. If a block needs to be extended to include
flavour requirements, a new ``\texttt{F}'' block is defined instead.
Note that different codes may have different implementations of how FLHA
input/output is \emph{technically} achieved. The details of how to
`switch on' FLHA input/output with a particular program should be
described in the manual of that program and are not covered here.
For the SLHA, libraries have been developed to permit an easy
implementation of the input/output routines~\cite{slha_io1}. In
principle these programs could be extended to include also the FLHA.
It should be noted that, while the SLHA was developed especially for the
case of SUSY, the FLHA is, at least in principle, model
independent. While it is possible to indicate the model used in a
specific block, the general structure for the information exchange can
be applied to any model.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conventions}
\label{sec:conventions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Standard Model Parameters}
\label{sec:smconv}
In general, the spectrum of the SM particles plays by
definition a crucial role in flavour physics.
Consequently, experimental measurements of masses and coupling constants
at the electroweak scale enter.
In the SLHA this block was defined as \texttt{SMINPUTS}.
This block is borrowed from SLHA as it is.
It is also important to note that all presently available experimental
determinations of, e.g., $\alpha_s$ and the running $b$ mass are based on
assuming the SM as the underlying theory, for natural reasons. When
extending the field content of the SM to that of a New
Physics Model (NPM), the \emph{same} measured results would be obtained
for \emph{different} values of these quantities, due to the different
underlying field content present in the NPM. However, since these
values are not known, all parameters contained in block
\texttt{SMINPUTS} should be the `ordinary' ones obtained from SM fits,
i.e.\ with no NPM corrections included. Any flavour code itself is then
assumed to convert these parameters into ones appropriate to an NPM
framework.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{CKM matrix}
The CKM matrix structure is also taken from SLHA2 as it is. The real and
imaginary parts of the $\overline{\rm{DR}}$ CKM matrix are given in \texttt{VCKM} and
\texttt{IMVCKM}, respectively. The format of the individual
entries is the same as for mixing matrices in the SLHA1. \\
{\color{red} To be expanded.}
{\color{\colorGoto}
\paragraph{(Goto):}
I propose to use the SLHA2 blocks \texttt{VCKMIN} and \texttt{UPMNSIN}
for input parameters (see my insertions below).
The blocks \texttt{(IM)VCKM Q=...} and \texttt{(IM)UPMNS Q=...} in SLHA2
are output blocks for $\overline{\rm{DR}}$ running parameters in the
MSSM Lagrangian.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Wilson coefficients}
\label{wcoeff}
The Wilson coefficients are classified according to the transition type $\Delta F$. The real and
imaginary parts are given in \texttt{FWCOEF} and \texttt{FIMWCOEF}, respectively.
The Wilson coefficients are to be given in the standard operator basis (see Appendix~\ref{app:operators}).
The different orders $C^{(k)}_i$ have to be given separately according to the following convention for the perturbative expansion:
\begin{equation}
C_i(\mu) = C^{(0)}_i(\mu) + \dfrac{\alpha_s(\mu)}{4 \pi} C^{(1)}_i(\mu) + \cdots
\end{equation}
The couplings should therefore not be included in the Wilson coefficients.\\
%
{\color{red} To be expanded.}
{\color{\colorGoto}
\paragraph{(Goto):}
I suggest to follow the expansion given in \citere{Buras:1999st}:
%
\begin{eqnarray}
C_{i}(\mu) &=&
C^{(0)}_{i}(\mu)
+ \dfrac{\alpha_s(\mu)}{4\pi} C^{(1)}_{i,s}(\mu)
+ \left( \dfrac{\alpha_s(\mu)}{4\pi} \right)^2 C^{(2)}_{i,s}(\mu)
\nonumber\\&&
+ \dfrac{\alpha(\mu)}{4\pi} C^{(1)}_{i,e}(\mu)
+ \dfrac{\alpha(\mu)}{4\pi}
\dfrac{\alpha_s(\mu)}{4\pi} C^{(2)}_{i,es}(\mu)
+ \cdots.
\label{eq:WCexpansion}
\end{eqnarray}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Definitions of the Interfaces}
In this section, the Flavour Les Houches Accord input and output files
are described. We concentrate here on the technical structure
only.
Following general structure for the SLHA~\cite{slha1,slha2} we assume
these definitions:
\begin{itemize}
\item All quantities with dimensions of energy (mass) are implicitly
understood to be in GeV (GeV$/c^2$).
\item Particles are identified by their PDG particle codes. See appendix
\ref{app:pdg} for lists of these, relevant for flavour observables.
\item The first character of every line is reserved for control and
comment statements. Data lines should have the first character empty.
\item In general, formatted output should be used for write-out, to
avoid ``messy-looking'' files, while a free format should be used on
read-in, to avoid misalignment etc.~leading to program crashes.
\item Read-in should be performed in a case-insensitive way, again to
increase stability.
\item The general format for all real numbers is the FORTRAN format
E16.8\footnote{E16.8:
a 16-character wide real number in scientific notation, whereof
8 digits are decimals, e.g., ``\texttt{-0.12345678E+000}''.}.
This large number of digits is used to avoid any possible numerical
precision issue, and since it is no more difficult for, e.g., the spectrum
calculator to write out such a number than a shorter version. For typed
input, it merely means that at least 16 spaces are reserved for the number,
but, e.g., the number \texttt{123.456} may be typed in ``as is''. See
also the example file in appendix \ref{app:example}.
\item A ``\texttt{\#}''
mark anywhere means that the rest of the line is intended as a comment
to be ignored by the reading program.
\item All input and output is divided into sections in the form of
``blocks''.
\item To clearly identify the blocks of the FLHA, the first letter of
the name of a block is an ``F'' (with the exception of the blocks
borrowed from SLHA).
\item A ``\texttt{BLOCK Fxxxx}''
(with the ``\texttt{B}'' being the first
character on the line) marks the beginning of entries belonging to
the block named ``\texttt{Fxxxx}''. E.g.\
``\texttt{BLOCK FMASS}'' marks that all following lines until the next
``\texttt{BLOCK}'' statement contain mass values, to be read
in a specific format, intrinsic to the \texttt{FMASS} block. The order
of blocks is arbitrary, except that input blocks should always come
before output blocks.
\item Further definitions can be found in Sect.~3 of \citere{slha1}.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
The following general structure for the FLHA file is proposed:
\begin{itemize}
\item \texttt{BLOCK FCINFO}:
Information about the flavour code used to prepare the FLHA file.
\item \texttt{BLOCK FMODSEL}:
Information about the underlying model used for the calculations.
This is the only place for ``model dependent'' information.
\item \texttt{BLOCK SMINPUTS}:
Measured values of SM parameters used for the calculations.
{\color{\colorGoto}
\item \texttt{BLOCK VCKMIN}:
Input parameters of the CKM matrix for the calculations.
\item \texttt{BLOCK UPMNSIN}:
Input parameters of the PMNS neutrino mixing matrix for the calculations.
}
\item \texttt{BLOCK VCKM}:
Real part of the CKM matrix elements.
\item \texttt{BLOCK IMVCKM}:
Imaginary part of the CKM matrix elements.
\item \texttt{BLOCK FMASS}:
Masses of quarks, mesons, hadrons, etc.
\item \texttt{BLOCK FLIFE}:
Lifetime (in seconds) of flavour related mesons, hadrons, etc.
\item \texttt{BLOCK FCONST}:
Decay constants.
\item \texttt{BLOCK FCONSTRAIO}:
Ratios of decay constants.
\item \texttt{BLOCK FBAG}:
Bag parameters.
\item \texttt{BLOCK FWCOEF}:
Real part of the Wilson coefficients.
\item \texttt{BLOCK FIMWCOEF}:
Imaginary part of the Wilson coefficients.
\item \texttt{BLOCK FOBS}:
Prediction of flavour observables.
\item \texttt{BLOCK FOBSERR}:
Theory error on the prediction of flavour observables.
\item \texttt{BLOCK FOBSSM}:
SM prediction for flavour observables.
\item \texttt{BLOCK FFORM}:
Form Factors.
\item \texttt{BLOCK FSHAPE}:
Shape factors.\\
\end{itemize}
%
More details on each block is given in the following.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FCINFO}}
Flavour code information, including the name and the version of the program:\\
\numentry{1}{Name of the flavour calculator}
\numentry{2}{Version number of the flavour calculator}
Optional warning or error messages can also be specified:\\
\numentry{3}{If this entry is present, warning(s) were produced by the
flavour calculator. The resulting file may still be used. The entry
should contain a description of the problem (string).}
\numentry{4}{If this entry is present, error(s) were produced by the
flavour calculator. The resulting file should not be used. The entry
should contain a description of the problem (string).}
This block is purely informative, and is similar to
\texttt{BLOCK SPINFO} in SLHA.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FMODSEL}}
This block is also purely informative and provides switches and options
for the model selection. The entries in this block should consist of an
index, identifying the particular switch in the listing below, followed
by at least one other integer number, specifying the option chosen.
Following the SLHA2 agreement, more than one realization in a given model is not allowed. For example in supersymmetry, CPV and RPV scenarios cannot be chosen at the same time.
Switches so far defined are: \\[5mm]
\numentry{0}{
\numentry{0}{Standard Model.}}\\
%
\numentry{1}{Two Higgs doublet models:\\
\numentry{1}{CP conserved\\
\numentry{0}{General\ldots}
\numentry{1}{Type I}
\numentry{2}{Type II}
\numentry{3}{Type III}
\numentry{4}{Type IV}}\\
%
\numentry{2}{CP violated\\
\numentry{0}{General\ldots}
\numentry{1}{Type I}
\numentry{2}{Type II}
\numentry{3}{Type III}
\numentry{4}{Type IV}}}\\
%
The definitions of the different 2HDM types are given in Appendix~\ref{app:2hdm}.\\
%
\numentry{2}{Supersymmetry\\
\numentry{1}{Choice of SUSY breaking model. \\
\numentry{0}{General MSSM}
\numentry{1}{(m)SUGRA model.}
\numentry{2}{(m)GMSB model.}
\numentry{3}{(m)AMSB model.}
\numentry{4}{other}
}\\
%
\numentry{3}{Choice of particle content. \\
\numentry{0}{MSSM.}
\numentry{1}{NMSSM.}
\numentry{2}{other}
}\\
%
\numentry{4}{R-parity violation. \\
\numentry{0}{R-parity conserved. }
\numentry{1}{R-parity violated. }
}
%
\numentry{5}{CP violation. \\
\numentry{0}{CP is conserved. }
\numentry{1}{CP is violated, but only by the standard CKM phase. All
other phases are assumed zero.}
\numentry{2}{CP is violated. Completely general CP phases allowed. }
}
%
\numentry{6}{Flavour violation. \\
\numentry{0}{No (SUSY) flavour violation. }
\numentry{1}{Quark flavour is violated. }
\numentry{2}{Lepton flavour is violated. }
\numentry{3}{Lepton and quark flavour is violated. }}}
%
The conventions for the second entry from 1 to 6 are the same as in
SLHA2. \bigskip \\
%
\numentry{3}{Alternative electroweak symmetry breaking models:\\
\numentry{1}{Technicolor}
\numentry{2}{Topcolor}
\numentry{3}{Little Higgs}
\numentry{4}{$\ldots$}}\\
%
\numentry{4}{Extradimensions\\
\numentry{1}{Universal flat extradimension}
\numentry{2}{Randall Sundrum}
\numentry{3}{$\ldots$}}\\
%
For the last two categories of models, additional sub-entries for further model specifications can be defined as well.\\
\\
{\color{red}
How far do we want to go in model definitions?
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK SMINPUTS}}
Measured SM parameters used for the flavour calculations. This block is
strictly identical to the SLHA \texttt{BLOCK SMINPUTS} and is reproduced
here for completeness. It should be noted that some programs have
hard-coded defaults for various of these parameters, hence only a subset
may sometimes be available as free inputs. The parameters are:\\[2mm]
\numentry{1}{$\alpha_\mathrm{em}^{-1}(m_{Z})^{\overline{\mathrm{MS}}}$.
Inverse electromagnetic coupling at the $Z$ pole in the
$\overline{\mathrm{MS}}$ scheme (with 5 active flavours).}
\numentry{2}{$G_F$. Fermi constant (in units of GeV$^{-2}$).}
\numentry{3}{$\alpha_s(m_{Z})^{\overline{\mathrm{MS}}}$. Strong coupling
at the $Z$ pole in the $\overline{\mathrm{MS}}$ scheme (with 5 active
flavours).}%
\numentry{4}{$m_Z$, pole mass.}
\numentry{5}{$m_b(m_b)^{\overline{\mathrm{MS}}}$. $b$ quark running mass
in the $\overline{\mathrm{MS}}$ scheme.}
\numentry{6}{$m_t$, pole mass.}
\numentry{7}{$m_\tau$, pole mass.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\color{\colorGoto}
\subsection*{\texttt{BLOCK VCKMIN}}
This block is strictly identical to the SLHA2 \texttt{BLOCK VCKMIN}.
The parameters are:\\[2mm]
\numentry{1}{$\lambda$.}
\numentry{2}{$A$.}
\numentry{3}{$\bar{\rho}$.}
\numentry{4}{$\bar{\eta}$.}
We use the PDG definition, Eq.~(11.4) of \citere{Amsler:2008zzb}, which
is exact to all orders in $\lambda$.
\subsection*{\texttt{BLOCK UPMNSIN}}
This block is strictly identical to the SLHA2 \texttt{BLOCK UPMNSIN}.
The parameters are:\\[2mm]
\numentry{1}{$\theta_{12}$.}
\numentry{2}{$\theta_{23}$.}
\numentry{3}{$\theta_{13}$.}
\numentry{4}{$\delta$.}
\numentry{5}{$\alpha_1$.}
\numentry{6}{$\alpha_2$.}
We use the PDG parametrization, Eq.~(13.30) of \citere{Amsler:2008zzb}.
All the angles and phases should be given in radians.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FMASS}}
Mass spectrum for the involved particles. This is an extension of the
SLHA \texttt{BLOCK MASS} which contained only pole masses. Here we
specify additional information concerning the renormalization scheme as
well as the scale at which the masses are given. The standard for each
line in the block should correspond to the following FORTRAN format
\begin{center}
\texttt{(1x,I9,3x,1P,E16.8,0P,3x,I2,3x,1P,E16.8,0P,3x,\#',1x,A)},
\end{center}
where the first nine digit integer should be the PDG code of a particle,
followed by a double precision number for its mass. The next integer
corresponds to the renormalization scheme, and finally the last double
precision number points to the energy scale (0 if not relevant).
The schemes are defined as follows:\\
\numentry{0}{pole}
\numentry{1}{$\overline{\mathrm{MS}}$}
\numentry{2}{$\overline{\mathrm{DR}}$}
\numentry{3}{1S}
\numentry{4}{kin}
\numentry{5}{\ldots}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FLIFE}}
Lifetimes of mesons and hadrons in seconds. The standard for each line
in the block should correspond to the FORTRAN format
\begin{center}
\texttt{(1x,I9,3x,1P,E16.8,0P,3x,'\#',1x,A)},
\end{center}
where the first 9--digit integer should be the PDG code of a particle
and the double precision number its lifetime.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FCONST}}
Decay constants in GeV. The standard for each line in the block should
correspond to the FORTRAN format
\begin{center}
\texttt{(1x,I9,3x,I2,3x,1P,E16.8,0P,3x,'\#',1x,A)},
\end{center}
where the first nine digit integer should be the PDG code of a particle,
the second integer the number of the decay constant (in the case the
decay requires more than one input parameter, such as in $\eta$--$\eta'$ mixings), and the double precision number its decay constant. \\
{\color{red} We should come up with a clearer definition for the decay constant numbering for the cases where more than one is available.}
{\color{\colorGoto}
\paragraph{(Goto):}
I suggest to define the decay constant $f_P$ of a pseudoscalar meson $P$
as:
%
\begin{equation}
\langle 0 | \bar{q}\gamma^\mu \gamma_5 Q | P(p) \rangle = -i f_P p^\mu,
\end{equation}
%
for $q\neq Q$ quark contents ($P=\pi^\pm$, $K$, $D$, $B$), therefore
$f_\pi$ is about 130MeV and not 92MeV.
The minus sign comes from the difference of the conventions for
$\gamma_5$ used here and the textbook \citere{Donoghue:DynamicsofSM},
where ``left'' is written as ``$1+\gamma_5$''.
For $\pi^0$, $\eta$ and $\eta'$, I prefer to define
%
\begin{eqnarray}
\frac{1}{\sqrt{2}}
\langle 0 |
\bar{u} \gamma^\mu \gamma_5 u
- \bar{d} \gamma^\mu \gamma_5 d
| \pi^0(p) \rangle
&=&
-i f_\pi p^\mu,
\\
\frac{1}{\sqrt{2}}
\langle 0 |
\bar{u} \gamma^\mu \gamma_5 u
+ \bar{d} \gamma^\mu \gamma_5 d
| \eta^{(\prime)}(p) \rangle
&=&
-i f_{\eta^{(\prime)}}^{q} p^\mu,
\\
\langle 0 |
\bar{s} \gamma^\mu \gamma_5 s
| \eta^{(\prime)}(p) \rangle
&=&
-i f_{\eta^{(\prime)}}^{s} p^\mu,
\end{eqnarray}
%
assuming isospin symmetry.
Other possible choice for $\eta$ and $\eta'$ may be:
%
\begin{eqnarray}
\frac{1}{\sqrt{6}}
\langle 0 |
\bar{u} \gamma^\mu \gamma_5 u
+ \bar{d} \gamma^\mu \gamma_5 d
- 2 \bar{s} \gamma^\mu \gamma_5 s
| \eta^{(\prime)}(p) \rangle
&=&
-i f_{\eta^{(\prime)}}^{8} p^\mu,
\\
\frac{1}{\sqrt{3}}
\langle 0 |
\bar{u} \gamma^\mu \gamma_5 u
+ \bar{d} \gamma^\mu \gamma_5 d
+ \bar{s} \gamma^\mu \gamma_5 s
| \eta^{(\prime)}(p) \rangle
&=&
-i f_{\eta^{(\prime)}}^{1} p^\mu,
\end{eqnarray}
%
Also, the following matrix elements are defined:
%
\begin{eqnarray}
(m_q + m_Q)
\langle 0 | \bar{q} \gamma_5 Q | P(p) \rangle &=& i h_P,
\\
(m_u + m_d)
\frac{1}{\sqrt{2}}
\langle 0 |
\bar{u} \gamma_5 u
- \bar{d} \gamma_5 d
| \pi^0(p) \rangle
&=&
i h_\pi,
\\
(m_u + m_d)
\frac{1}{\sqrt{2}}
\langle 0 |
\bar{u} \gamma_5 u
+ \bar{d} \gamma_5 d
| \eta^{(\prime)}(p) \rangle
&=&
i h_{\eta^{(\prime)}}^{q},
\\
2 m_s
\langle 0 |
\bar{s} \gamma_5 s
| \eta^{(\prime)}(p) \rangle
&=&
i h_{\eta^{(\prime)}}^{s}.
\end{eqnarray}
%
$h$'s may be unnecessary except for $\eta$ and $\eta'$ since they can be
written in terms of other quantities as $h_\pi = m_\pi^2 f_\pi$ etc..
$h_{\eta^{(\prime)}}^{q,s}$ do not satisfy relations of this kind due to
the contributions of anomaly terms.
Decay constants of a vector meson $V$, whose quark content is $\Bar{q}Q$
(such as $\rho^\pm$ and $K^*$), are defined by the following matrix
elements.
%
\begin{eqnarray}
\langle 0 | \Bar{q}\gamma^\mu Q | V(p) \rangle
&=&
m_V f_V \epsilon^\mu,
\\
\langle 0 | \Bar{q} \sigma^{\mu\nu} Q | V(p) \rangle
&=&
i f^T_V ( p^\nu \epsilon^\mu - p^\mu \epsilon^\nu ),
\end{eqnarray}
%
where $\epsilon^\mu$ is the polarization vector of $V$.
$f_{\rho,\omega,\phi}$ in the ``ideal mixing'' limit are defined as:
%
\begin{eqnarray}
\frac{1}{\sqrt{2}}
\langle 0 |
\Bar{u}\gamma^\mu u - \Bar{d}\gamma^\mu d
| \rho^0(p) \rangle
&=&
m_{\rho} f_{\rho} \epsilon^\mu,
\\
\frac{1}{\sqrt{2}}
\langle 0 |
\Bar{u}\gamma^\mu u + \Bar{d}\gamma^\mu d
| \omega(p) \rangle
&=&
m_{\omega} f_{\omega} \epsilon^\mu,
\\
\langle 0 |
\Bar{s} \gamma^\mu s
| \phi(p) \rangle
&=&
m_{\phi} f_{\phi} \epsilon^\mu.
\end{eqnarray}
%
$f^T_{\rho,\omega,\phi}$ are also defined with the same flavor
combinations.
It is possible to define decay constants of $\omega$ and $\phi$ as
%
\begin{eqnarray}
\frac{1}{\sqrt{2}}
\langle 0 |
\Bar{u}\gamma^\mu u + \Bar{d}\gamma^\mu d
| \omega(\phi)(p) \rangle
&=&
m_{\omega(\phi)} f_{\omega(\phi)}^{q} \epsilon^\mu,
\\
\langle 0 |
\Bar{s} \gamma^\mu s
| \omega(\phi)(p) \rangle
&=&
m_{\omega(\phi)} f_{\omega(\phi)}^{s} \epsilon^\mu,
\end{eqnarray}
%
or
%
\begin{eqnarray}
\frac{1}{\sqrt{6}}
\langle 0 |
\Bar{u}\gamma^\mu u + \Bar{d}\gamma^\mu d
- 2 \Bar{s}\gamma^\mu s
| \omega(\phi)(p) \rangle
&=&
m_{\omega(\phi)} f_{\omega(\phi)}^{8} \epsilon^\mu,
\\
\frac{1}{\sqrt{3}}
\langle 0 |
\Bar{u}\gamma^\mu u + \Bar{d}\gamma^\mu d
+ \Bar{s} \gamma^\mu s
| \omega(\phi)(p) \rangle
&=&
m_{\omega(\phi)} f_{\omega(\phi)}^{1} \epsilon^\mu.
\end{eqnarray}
An implementation in \texttt{BLOCK FCONST} may be:
\\[2mm]
\numentry{321}{$K^+$.\\
\numentry{1}{$f_K$ in GeV.}
\numentry{11}{$h_K$ in GeV$^3$.}
}
\numentry{221}{$\eta$.\\
\numentry{1}{$f_{\eta}^q$ in GeV.}
\numentry{2}{$f_{\eta}^s$ in GeV.}
\numentry{11}{$h_{\eta}^q$ in GeV$^3$.}
\numentry{12}{$h_{\eta}^s$ in GeV$^3$.}
}
\numentry{213}{$\rho(770)^+$.\\
\numentry{1}{$f_{\rho}$ in GeV.}
\numentry{11}{$f^T_{\rho}$ in GeV.}
}
\numentry{223}{$\omega(782)$.\\
\numentry{1}{$f_{\rho}^{q}$ in GeV.}
\numentry{2}{$f_{\rho}^{s}$ in GeV.}
\numentry{11}{$f^{T,q}_{\rho}$ in GeV.}
\numentry{12}{$f^{T,s}_{\rho}$ in GeV.}
}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FCONSTRATIO}}
The ratio of decay constants, which often have less uncertainties. The
ratios are specified by the two PDG codes in the form
f(code1)/f(code2). The standard for each line in the block should
correspond to the FORTRAN format
\begin{center}
\texttt{(1x,I9,3x,I9,3x,1P,E16.8,0P,3x,'\#',1x,A)},
\end{center}
{\color{\colorGoto}
\begin{center}
\texttt{(1x,I9,3x,I9,3x,I2,3x,1P,E16.8,0P,3x,'\#',1x,A)},
\end{center}
}
where the two nine digit integers should be the two PDG codes of
particles,
{\color{\colorGoto}
the third integer the number of the decay constant, which corresponds to
the second index of the entry in \texttt{BLOCK FCONST},
}
and the double precision number the ratio of the decay
constants.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FBAG}}
Bag parameters. The standard for each line in the block should
correspond to the FORTRAN format
\begin{center}
\texttt{(1x,I9,3x,I2,3x,1P,E16.8,0P,3x,'\#',1x,A)},
\end{center}
where the
first nine digit integer should be the PDG code of a particle, the
second integer the number of the Bag parameter (in the case several Bag
parameters are needed, e.g. for neutral meson mixings), and the double
precision number its Bag parameter. \\
{\color{red} We should come up with a clearer definition for the B parameter numbering for the cases where more than one is available.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FWCOEF Q= \ldots \, M= \ldots}}
Real part of the Wilson coefficients at the scale \texttt{Q} and for the model \texttt{M}, respecting the conventions given in section \ref{wcoeff}. The Wilson coefficients can be provided either separately for the new physics contributions and SM contributions, or as a total contribution of both new physics and SM, depending on the code generating them. To avoid any confusion, in the \texttt{M} area it must be specified whether the given Wilson coefficients correspond to the SM contributions, new physics contributions or sum of them, using the following definitions for \texttt{M}:\\
\numentry{0}{SM}
\numentry{1}{NPM}
\numentry{2}{SM+NPM}
New Physics (NPM) model is the model specified in the \texttt{BLOCK FMODSEL}.
The entries in \texttt{BLOCK FWCOEF} should consist of, first the transition type (see Appendix~\ref{app:operators}) followed by the transition sub-type number in the
second place: \\
\numentry{1}{$\Delta F = 1$\\
\numentry{1}{$b \leftrightarrow s$}
\numentry{2}{$b \leftrightarrow d$}
\numentry{3}{$s \leftrightarrow d$}
\numentry{4}{$c \leftrightarrow u$}
\numentry{5}{\ldots}}
\numentry{2}{$\Delta F = 2$\\
\numentry{1}{$bb \leftrightarrow ss$}
\numentry{2}{$bb \leftrightarrow dd$}
\numentry{3}{$ss \leftrightarrow dd$}
\numentry{4}{$cc \leftrightarrow uu$}
\numentry{5}{$db \leftrightarrow cu$}
\numentry{6}{\ldots}}
\numentry{3}{leptonic\\
\numentry{1}{$\mu \leftrightarrow e$}
\numentry{2}{$\tau \leftrightarrow \mu$}
\numentry{3}{$\tau \leftrightarrow e$}
\numentry{4}{\ldots}}
%
The next entries consist of the number of the Wilson coefficient
followed by the order at which they are computed. For the Wilson coefficient numbers, the following convention is adopted:
\begin{itemize}
\item Wilson coefficients are given positive numbers when they correspond to the normal operators described in appendix~\ref{app:operators},
\item Wilson coefficients are given negative numbers for the inverted chirality operators ($C'$ or $\tilde C$ coefficients).
\end{itemize}
%
\noindent The standard for each line in the block should thus correspond to the
FORTRAN format
\begin{center}
\texttt{(1x,I2,3x,I2,3x,I2,3x,I1,3x,1P,E16.8,0P,3x,'\#',1x,A)},
\end{center}
where the double precision number is the real part of the Wilson coefficient.
Note that there can be several such blocks for different scales \texttt{Q} or
model types (e.g. SM, NPM). \\
{\color{red} Here we give the Wilson coefficients order by order. Is this sufficient, or should we accommodate also the total resummed values? If so, how to do it?
}
{\color{\colorGoto}
\paragraph{(Goto):}
I propose to extend the order number to two-digit.
FOTRAN format for this block is:
\begin{center}
\texttt{(1x,I2,3x,I2,3x,I2,3x,I2,3x,1P,E16.8,0P,3x,'\#',1x,A)},
\end{center}
and the fourth index corresponds to each term in
(\ref{eq:WCexpansion}):\\[2mm]
\numentry{0}{$C^{(0)}_{i}(\mu)$.}
\numentry{1}{$C^{(1)}_{i,s}(\mu)$.}
\numentry{2}{$C^{(2)}_{i,s}(\mu)$.}
\numentry{10}{$C^{(1)}_{i,e}(\mu)$.}
\numentry{11}{$C^{(2)}_{i,es}(\mu)$.}
\numentry{99}{total (or \texttt{-1} can be used).}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FIMWCOEF Q= \ldots M= \ldots}}
Imaginary part of the Wilson coefficients at the scale \texttt{Q}, for the model \texttt{M}.
The structure is exactly the same as for \texttt{BLOCK FWCOEF}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FOBS}}
Flavour observables. The structure of this block is based on the decay
table in SLHA format. The decay is defined by the PDG number of the
parent, the type of the observable, the value of the observable, the
number of daughters and PDG IDs of the daughters.\\
The types of the observables are defined as follows:\\
\numentry{1}{Branching ratio}
\numentry{2}{Ratio of the branching ratio to the SM value}
\numentry{3}{Asymmetry -- CP}
\numentry{4}{Asymmetry -- isospin}
\numentry{5}{Asymmetry -- forward-backward}
\numentry{6}{Asymmetry -- lepton-flavour}
\numentry{7}{Mixing}
\numentry{8}{\ldots}
%
The standard for each line in the block should correspond to the FORTRAN
format
\begin{center}
\texttt{(1x,I9,3x,I2,3x,1P,E16.8,0P,3x,I1,3x,I9,3x,I9,3x,\ldots,3x,'\#',1x,A)},
\end{center}
where the first nine digit integer should be the PDG code of the parent
decaying particle, the second integer the type of the observable, the
double precision number the value of the observable, the next interger
the number of daughters, and the following nine digit integers the PDG
codes of the daughters. It is strongly advised to give the descriptive
name of the observable as comment.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FOBSERR}}
Theoretical error for flavour observables, with the structure similar to
\texttt{BLOCK FOBS} where the double precision number for the value of
the observable is replaced by two double precision numbers for the minus
and plus uncertainties (which is particularly helpful for asymmetric
errors).
In a similar way, for every block, a corresponding error block with the
name \texttt{BLOCK FnameERR} can be defined.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FOBSSM}}
The SM values of the flavour observables in the same format as in
\texttt{BLOCK FOBS}. The given SM values may be very helpful as a
comparison reference.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FFORM}}
Form factors related to decays are given by defining the decay as in
\texttt{BLOCK FOBS}, but replacing the type of the observable by the
number of the form factor. It is essential here to describe the variable
in the comment area.
If the form factors depend on additional parameters (e.g. $q^2$ in $B \to K^* \mu^+ \mu^-$ process) this dependence is not described by the accord, but can be specified as a comment.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FSHAPE}}
Shape factors related to decays are given in a format identical to
\texttt{BLOCK FFORM}. Again it is essential to describe the variable in
the comment area.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
Interplay of collider and flavour physics is entering a new area with
the start up of the LHC and in the future more and more programs will be
interfaced in order to exploit a maximum amount of information from both
collider and flavour data. In this direction, an accord will play a
crucial role. The present accord specifies a unique set of conventions
in ASCII file format for most commonly investigated flavour related
observables and provides a universal framework for interfacing different
programs.
The number of flavour related codes is growing constantly, while the
connection between results from flavour physics and high $p_T$ physics
becomes more relevant to disentangle the underlying physics model.
Using the
lessons learnt from the SLHA, we hope the FLHA will prove useful for
flavour physics related studies\ldots
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}
The work of S.H.\ was partially supported by CICYT (grant FPA 2007--66387).
Work supported in part by the European Community's Marie-Curie Research
Training Network under contract MRTN-CT-2006-035505
`Tools and Precision Calculations for Physics Discoveries at Colliders'.
{\color{\colorGoto}
The work of T.\ G.\ is supported in part by the Grant-in-Aid for Science
Research, Japan Society for the Promotion of Science, No.\ 20244037.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\appendix
\section{The PDG Particle Numbering Scheme \label{app:pdg}}
Listed in the tables below are the PDG codes for the SM quarks and
mesons. Codes for other particles may be found in
\cite{Amsler:2008zzb}. \\[1cm]
\begin{table}[!h]
\vspace{-2ex}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Name & Code \\
\hline
$d$ & 1\\
$u$ & 2\\
$s$ & 3\\
$c$ & 4\\
$b$ & 5\\
$t$ & 6\\
\hline
\end{tabular}
\caption{PDG codes for SM quarks.}
\end{center}
\end{table}
\vspace*{1cm}
\begin{table}[!h]
\vspace{-2ex}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Name & Code & Name & Code \\
\hline
$\pi^0$ & 111 & $D^+$ & 411 \\
$\pi^+$ & 211 & $D^0$ & 421 \\
$\rho(770)^0$ & 113 & $D_s^+$ & 431 \\
$\rho(770)^+$ & 213 & $D_s^{*+}$ & 433 \\
$\eta$ & 221 & $B^0$ & 511 \\
$\eta^\prime(958)$& 331 & $B^+$ & 521 \\
$\omega(782)$ & 223 & $B^{*0}$ & 513 \\
$\phi(1020)$ & 333 & $B^{*+}$ & 523 \\
$K_L^0$ & 130 & $B_s^0$ & 531 \\
$K_S^0$ & 310 & $B_s^{*0}$ & 533 \\
$K^0$ & 311 & $B_c^+$ & 541 \\
$K^+$ & 321 & $B_c^{*+}$ & 543 \\
$K^{*0}(892)$ & 313 & $J/\psi(1S)$ & 443 \\
$K^{*+}(892)$ & 323 & $\Upsilon(1S)$ & 553 \\
$\eta_c(1S)$ & 441 & $\eta_b(1S)$ & 551 \\
\hline
\end{tabular}
\caption{PDG codes for most commonly considered mesons.}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Effective Operators \label{app:operators}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$\Delta F = 1$}
The relevant Wilson coefficients for $b \leftrightarrow s$ decays are given in the standard operator basis \cite{Chetyrkin:1997gb}. For the cases of other quark transitions one should
make obvious replacements of the quark fields.
\begin{align}
O_1 &= (\bar{s} \gamma_{\mu} T^a P_L c)
(\bar{c} \gamma^{\mu} T_{a} P_L b)\;,\nonumber\\[4mm]
O_2 &= (\bar{s} \gamma_{\mu} P_L c)
(\bar{c} \gamma^{\mu} P_L b)\;,\nonumber\\[3mm]
O_3 &= (\bar{s} \gamma_{\mu} P_L b)
{\displaystyle\sum_q} (\bar{q} \gamma^{\mu} q)\;,\nonumber\\[1mm]
O_4 &= (\bar{s} \gamma_{\mu} T^a P_L b)
{\displaystyle\sum_q} (\bar{q} \gamma^{\mu} T_{a} q)\;,\nonumber\\[1mm]
O_5 &= (\bar{s} \gamma_{\mu_1}\gamma_{\mu_2}\gamma_{\mu_3} P_L b)
{\displaystyle\sum_q} (\bar{q} \gamma^{\mu_1}\gamma^{\mu_2}
\gamma^{\mu_3} q)\;,\\[1mm]
O_6 &= (\bar{s} \gamma_{\mu_1}\gamma_{\mu_2}\gamma_{\mu_3} T^a P_L b)
{\displaystyle\sum_q} (\bar{q} \gamma^{\mu_1}\gamma^{\mu_2}
\gamma^{\mu_3} T_{a} q)\;,\nonumber\\[1mm]
O_7 &= (O_{\gamma})= \dfrac{e}{16\pi^2} \left[ \bar{s} \sigma^{\mu \nu}
(m_s P_L + m_b P_R) b \right] F_{\mu \nu}\;,\nonumber\\[2mm]
O_8 &= (O_g)= \dfrac{g}{16\pi^2} \left[ \bar{s} \sigma^{\mu \nu}
(m_s P_L + m_b P_R) T_{a} b \right] G_{\mu \nu}^a\;.\nonumber
\end{align}
%
Here $T^a$ ($a = 1 \ldots 8$) denote the $SU(3)_C$ generators, and
$P_{L,R} = \frac{1}{2} (1 \mp \gamma_5)$.\\
\\
The relevant operators for $b \to s \ell \bar{\ell}$ decays are \cite{Bobeth:1999mk,Bobeth:2001sq}:
\begin{align}
O_9 &= (O_V)= \dfrac{e^2}{16\pi^2} (\bar{s} \gamma_{\mu} P_L b) \,
(\bar{\ell} \gamma^{\mu} \ell)\;,\nonumber\\[1mm]
O_{10} &= (O_A)= \dfrac{e^2}{16\pi^2} (\bar{s} \gamma_{\mu} P_L b) \,
(\bar{\ell} \gamma^{\mu} \gamma_5 \ell) \;, \\
O_{83} &= (O_S)= \frac{e^2}{16 \pi^2} m_b (\bar{s} P_R b) (\bar{\ell}\ell)\;,\nonumber\\
O_{80} &= (O_P)= \frac{e^2}{16 \pi^2} m_b (\bar{s} P_R b) (\bar{\ell} \gamma_5 \ell)\;. \nonumber
\end{align}
Note that the numbers ``83'' and ``80'' correspond respectively to the decimal ASCII code for ``$S$'' and ``$P$''.\\
\\
{\color{red}
Here we use the standard operator basis which is used for $b \to s \gamma$ calculations. $O_{\gamma}$ and $O_g$ are called $O_7$ and $O_8$ respectively and are commonly used and numbered in this way in the literature. The question is how to write and to number the electroweak penguin operators in this basis (in Buras notations, i.e. traditional basis, they are called $O_{7 \cdots 10 \cdots}$).
}
{\color{\colorGoto}
\paragraph{(Goto):}
For the definitions of the operator basis and the Wilson coefficients,
I would like to ask experts on QCD corrections (Uli Haisch?) to provide
us with the ``best'' choice.
My questions/comments are the following.
%
\begin{itemize}
\item
How should we define the overall normalization of the Wilson
coefficients?
In the example given in Appendix D, we see $C_2^{(0)}(m_t?)=1$.
On the other hand, for example, $C_2^{c(0)}=-1$ is used in
\citere{Bobeth:1999mk}.
Furthermore, is it a good way to factor out CKM matrix elements even
when New Physics contributions (which are independent of the CKM matrix
in principle) are assumed to exist?
\item
Very naively, we can write the followoing 80 $b\to s$ four-quark
operators:
%
\begin{eqnarray}
\mathcal{O}_{VLL}^{q,1} &=&
(\bar{s} \gamma^\mu P_L b) (\bar{q} \gamma_\mu P_L q),
\\
\mathcal{O}_{VLR}^{q,1} &=&
(\bar{s} \gamma^\mu P_L b) (\bar{q} \gamma_\mu P_R q),
\\
\mathcal{O}_{SLL}^{q,1} &=&
(\bar{s} P_L b) (\bar{q} P_L q),
\\
\mathcal{O}_{TLL}^{q,1} &=&
(\bar{s} \sigma^{\mu\nu} P_L b) (\bar{q} \sigma_{\mu\nu} P_L q),
\\
\mathcal{O}_{VLL}^{q,8} &=&
(\bar{s} \gamma^\mu T^a P_L b) (\bar{q} \gamma_\mu T^a P_L q),
\\
\mathcal{O}_{VLR}^{q,8} &=&
(\bar{s} \gamma^\mu T^a P_L b) (\bar{q} \gamma_\mu T^a P_R q),
\\
\mathcal{O}_{SLL}^{q,8} &=&
(\bar{s} T^a P_L b) (\bar{q} T^a P_L q),
\\
\mathcal{O}_{TLL}^{q,8} &=&
(\bar{s} \sigma^{\mu\nu} T^a P_L b) (\bar{q} \sigma_{\mu\nu} T^a P_L
q),
\\
&&
q=u,\,d,\,s,\,c,\,b,
\end{eqnarray}
%
and their mirror images ($P_L \Leftrightarrow P_R$).
%
For $q=b$, $\mathcal{O}_{VLL}^{1,b}$ and
$\mathcal{O}_{VLL}^{8,b}$ are equivalent (with Fierz rearrangement in
four dimension).
Also, among four operators $\mathcal{O}_{SLL}^{b,1}$,
$\mathcal{O}_{SLL}^{b,8}$, $\mathcal{O}_{TLL}^{b,1}$ and
$\mathcal{O}_{TLL}^{b,8}$, only two are linearly independent.
Same relations apply for $q=s$ and mirror images.
Therefore, we have 68 linearly independent four-quark operators for
$b\to s$.
In the SM, only six (tree and QCD penguins) or ten ($+$ electroweak
penguins) combinations are relevant.
However, for example, all of them may be generated by SUSY box diagrams,
in principle.
The numbering scheme for ten SM operators has to be defined anyway.
Scheme for mirror images is ready (with minus sign).
How about for remaining 24?
Should we define a basis for them and assign numbers, or leave
undefined?
%
\item
Signs of the dipole operators $O_{7(\gamma)}$ and $O_{8(g)}$ (or their
coefficients) depend on the sign convention in the gauge covariant
derivatives.
Here, it is assumed that QCD and QED covariant derivatives are taken as
$\partial_\mu + i g T_a G^a_\mu$ and $\partial_\mu + i e Q A^a_\mu$
for a quark with electric charge $Q$ ($Q=+\frac{2}{3}$ for up-type and
$-\frac{1}{3}$ for down-type), and that $g>0$, $e>0$,
$F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$,
$G^a_{\mu\nu}=\partial_\mu G^a_\nu - \partial_\nu G^a_\mu + \cdots$
and $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^\mu,\gamma^\nu]$,
I guess.
Is this correct?
In any case, we should show our convention explicitly.
%
\item
Semileptonic operators are, again naively,
%
\begin{eqnarray}
\mathcal{O}_{VLL}^{\ell} &=&
(\bar{s} \gamma^\mu P_L b)(\bar{\ell} \gamma_\mu P_L \ell),
\\
\mathcal{O}_{VLR}^{\ell} &=&
(\bar{s} \gamma^\mu P_L b)(\bar{\ell} \gamma_\mu P_R \ell),
\\
\mathcal{O}_{SLL}^{\ell} &=&
(\bar{s} P_L b)(\bar{\ell} P_L \ell),
\\
\mathcal{O}_{SLR}^{\ell} &=&
(\bar{s} P_L b)(\bar{\ell} P_R \ell),
\\
\mathcal{O}_{TLL}^{\ell} &=&
(\bar{s} \sigma^{\mu\nu} P_L b)(\bar{\ell} \sigma_{\mu\nu} P_L \ell),
\end{eqnarray}
%
and their mirror images for charged leptons, and
%
\begin{eqnarray}
\mathcal{O}_{VLL}^{\nu} &=&
(\bar{s} \gamma^\mu P_L b)(\bar{\nu} \gamma_\mu P_L \nu),
\\
\mathcal{O}_{VRL}^{\nu} &=&
(\bar{s} \gamma^\mu P_R b)(\bar{\nu} \gamma_\mu P_L \nu),
\end{eqnarray}
%
for neutrinos.
Operators listed in (3) are equivalent to $\mathcal{O}_{VLL}^{\ell}$,
$\mathcal{O}_{VLR}^{\ell}$, $\mathcal{O}_{SLL}^{\ell}$ and
$\mathcal{O}_{SLR}^{\ell}$.
We should assign numbers for remaining ones.
Also lepton flavors should be distinguished:
The operators (coefficients) for $\ell=e$, $\mu$ and $\tau$ have to be
given different numbers.
How about neutrinos?
Mass basis or flavor basis?
%
\item
There is another issue about the dipole and semileptonic operators.
In \citere{Bobeth:2003at}, these operators are divided by
$\frac{\alpha_s}{4\pi}$.
Which normalization should we take?
\end{itemize}
I propose to use three-digit (and a minus sign) numbering system for the
operators of this class in the blocks \texttt{F(IM)WCOEF}.
Assignments looks like:
\\
\numentry{1--99}{Four-quark operators.}
\numentry{101}{``$O_{7}$'' or ``$O_{\gamma}$''.}
\numentry{102}{``$O_{8}$'' or ``$O_{g}$''.}
\numentry{21x}{Semileptonic operators for $\ell=e$.}
\numentry{22x}{Semileptonic operators for $\ell=\mu$.}
\numentry{23x}{Semileptonic operators for $\ell=\tau$.}
\numentry{24x}{Semileptonic operators for $\ell=\nu_e$.}
\numentry{25x}{Semileptonic operators for $\ell=\nu_\mu$.}
\numentry{26x}{Semileptonic operators for $\ell=\nu_\tau$.}
Last digit \texttt{x} in semileptonic operators stands for spinor
structure.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$\Delta F = 2$}
The vector, scalar and tensor operators relevant for $bb \leftrightarrow
ss$ oscillations are \cite{Buras:2002vd,Virto:2009wm}:
\begin{align}
O_1& = (\bar s \gamma_\mu P_L b) (\bar s \gamma^\mu P_L b)\;, \nonumber\\[3mm]
O_2& = (\bar s P_L b) (\bar s P_L b)\;, \nonumber\\[3mm]
O_3& = (\bar s T^a P_L b) (\bar s T_a P_L b)\;, \nonumber\\[3mm]
O_4& = (\bar s P_L b) (\bar s P_R b)\;, \\[3mm]
O_5& = (\bar s T^a P_L b) (\bar s T_a P_R b)\;, \nonumber\\[3mm]
O_6& = (\bar s \gamma_\mu P_L b) (\bar s \gamma^\mu P_R b)\;, \nonumber\\[3mm]
O_7& = (\bar s \sigma_{\mu\nu} P_L b) (\bar s \sigma^{\mu\nu} P_L b)\;. \nonumber
\end{align}
For the cases of other mixings one should make obvious replacements of the quark fields.\\
\\
{\color{red}
To be completed?\\
\\
Do we need to give also leptonic operators? How far do we want to go?
}
{\color{\colorGoto}
\paragraph{(Goto):}
There are two popular $\Delta F=2$ operator bases.
In \citere{Virto:2009wm,Gabbiani:1996hi} where SUSY contributions are
considered, the operator basis is taken as:
%
\begin{eqnarray}
\mathcal{O}_{1} &=&
(\bar{s}^\alpha \gamma_\mu P_L d^\alpha)
(\bar{s}^\beta \gamma^\mu P_L d^\beta ),
\\
\mathcal{O}_{2} &=&
(\bar{s}^\alpha P_L d^\alpha)
(\bar{s}^\beta P_L d^\beta ),
\\
\mathcal{O}_{3} &=&
(\bar{s}^\alpha P_L d^\beta )
(\bar{s}^\beta P_L d^\alpha),
\\
\mathcal{O}_{4} &=&
(\bar{s}^\alpha P_L d^\alpha)
(\bar{s}^\beta P_R d^\beta ),
\\
\mathcal{O}_{5} &=&
(\bar{s}^\alpha P_L d^\alpha)
(\bar{s}^\beta P_R d^\beta ),
\end{eqnarray}
%
and mirror images of $\mathcal{O}_{1,2,3}$.
$\alpha$ and $\beta$ are color indices.
On the other hand, in \citere{Buras:2001ra,Buras:2002vd} where QCD
corrections are calculated, a different basis is chosen:
%
\begin{eqnarray}
Q_{1}^{\mathrm{VLL}} &=&
(\bar{s}^\alpha \gamma_\mu P_L d^\alpha)
(\bar{s}^\beta \gamma^\mu P_L d^\beta ),
\\
Q_{1}^{\mathrm{LR}} &=&
(\bar{s}^\alpha \gamma_\mu P_L d^\alpha)
(\bar{s}^\beta \gamma^\mu P_R d^\beta ),
\\
Q_{2}^{\mathrm{LR}} &=&
(\bar{s}^\alpha P_L d^\alpha )
(\bar{s}^\beta P_R d^\beta),
\\
Q_{1}^{\mathrm{SLL}} &=&
(\bar{s}^\alpha P_L d^\alpha)
(\bar{s}^\beta P_L d^\beta ),
\\
Q_{2}^{\mathrm{SLL}} &=&
(\bar{s}^\alpha \sigma_{\mu\nu} P_L d^\alpha)
(\bar{s}^\beta \sigma^{\mu\nu} P_L d^\beta ),
\end{eqnarray}
%
where $\sigma^{\mu\nu}=\frac{1}{2}[\gamma^\mu,\gamma^\nu]$ (so we may
have to put minus sign to $Q_{2}^{\mathrm{SLL}}$ if we take a definition
$\sigma^{\mu\nu}=\frac{i}{2}[\gamma^\mu,\gamma^\nu]$).
In either case, the way of color contraction and appearance of
$\gamma_5$ (or $P_{L,R}$) look ``traditional''-like rather than
``standard'' ones in $\Delta F=1$ case.
Which should we take, or something else?
In addition, there is an issue of normalization.
In \citeres{Virto:2009wm,Buras:2002vd}, Wilson coefficients are defined
as
%
\begin{itemize}
\item \citere{Virto:2009wm}:
\begin{equation}
\mathcal{H}_{\mathrm{eff}}^{\Delta F=2} =
\sum C_i \mathcal{O}_i,
\end{equation}
\item \citere{Buras:2002vd}:
\begin{equation}
\mathcal{H}_{\mathrm{eff}}^{\Delta B=2} =
\frac{G_F^2 M_W^2}{16\pi^2}
(V_{tb}^* V_{ts})^2
\sum C_i Q_i.
\end{equation}
\end{itemize}
%
This normalization should be defined in a consistent way with
$\Delta F=1$ operators.
}
{\color{\colorGoto}
\subsection{$\Delta LF = 1$}
\paragraph{(Goto):}
In \citere{Kitano:2000fg}, the following basis is used for $\tau\to\mu$
dipole and four-lepton operators:
%
\begin{eqnarray}
\lefteqn{
\mathcal{L}(\tau^+\to \mu^+ \mu^+ \mu^-)
}\nonumber\\ &=&
- \frac{4G_F}{\sqrt{2}}
\left[
m_{\tau}
\left(
A_R \bar{\tau} \sigma^{\mu\nu} P_L \mu
+
A_L \bar{\tau} \sigma^{\mu\nu} P_R \mu
\right) F_{\mu\nu}
+
\sum_{i=1}^{6} g_{i} \mathcal{O}_{i}(\tau^+\to \mu^+ \mu^+ \mu^-)
\right],
\\
\lefteqn{
\mathcal{L}(\tau^+\to \mu^+ e^+ e^-)
}\nonumber\\ &=&
- \frac{4G_F}{\sqrt{2}}
\left[
m_{\tau}
\left(
A_R \bar{\tau} \sigma^{\mu\nu} P_L \mu
+
A_L \bar{\tau} \sigma^{\mu\nu} P_R \mu
\right) F_{\mu\nu}
+
\sum_{i=1}^{10} \lambda_{i} \mathcal{O}_{i}(\tau^+\to \mu^+ e^+ e^-)
\right],
\end{eqnarray}
%
where
%
\begin{eqnarray}
\mathcal{O}_{1}(\tau^+\to \mu^+ \mu^+ \mu^-) &=&
(\bar{\tau} P_L \mu) (\bar{\mu} P_L \mu),
\\
\mathcal{O}_{2}(\tau^+\to \mu^+ \mu^+ \mu^-) &=&
(\bar{\tau} P_R \mu) (\bar{\mu} P_R \mu),
\\
\mathcal{O}_{3}(\tau^+\to \mu^+ \mu^+ \mu^-) &=&
(\bar{\tau} \gamma^\mu P_R \mu) (\bar{\mu} \gamma_\mu P_R \mu),
\\
\mathcal{O}_{4}(\tau^+\to \mu^+ \mu^+ \mu^-) &=&
(\bar{\tau} \gamma^\mu P_L \mu) (\bar{\mu} \gamma_\mu P_L \mu),
\\
\mathcal{O}_{5}(\tau^+\to \mu^+ \mu^+ \mu^-) &=&
(\bar{\tau} \gamma^\mu P_R \mu) (\bar{\mu} \gamma_\mu P_L \mu),
\\
\mathcal{O}_{6}(\tau^+\to \mu^+ \mu^+ \mu^-) &=&
(\bar{\tau} \gamma^\mu P_L \mu) (\bar{\mu} \gamma_\mu P_R \mu),
\end{eqnarray}
%
\begin{eqnarray}
\mathcal{O}_{1}(\tau^+\to \mu^+ e^+ e^-) &=&
(\bar{\tau} P_L \mu) (\bar{e} P_L e),
\\
\mathcal{O}_{2}(\tau^+\to \mu^+ e^+ e^-) &=&
(\bar{\tau} P_L \mu) (\bar{e} P_R e),
\\
\mathcal{O}_{3}(\tau^+\to \mu^+ e^+ e^-) &=&
(\bar{\tau} P_R \mu) (\bar{e} P_L e),
\\
\mathcal{O}_{4}(\tau^+\to \mu^+ e^+ e^-) &=&
(\bar{\tau} P_R \mu) (\bar{e} P_R e),
\\
\mathcal{O}_{5}(\tau^+\to \mu^+ e^+ e^-) &=&
(\bar{\tau} \gamma^\mu P_L \mu) (\bar{e} \gamma_\mu P_L e),
\\
\mathcal{O}_{6}(\tau^+\to \mu^+ e^+ e^-) &=&
(\bar{\tau} \gamma^\mu P_L \mu) (\bar{e} \gamma_\mu P_R e),
\\
\mathcal{O}_{7}(\tau^+\to \mu^+ e^+ e^-) &=&
(\bar{\tau} \gamma^\mu P_R \mu) (\bar{e} \gamma_\mu P_L e),
\\
\mathcal{O}_{8}(\tau^+\to \mu^+ e^+ e^-) &=&
(\bar{\tau} \gamma^\mu P_R \mu) (\bar{e} \gamma_\mu P_R e),
\\
\mathcal{O}_{9}(\tau^+\to \mu^+ e^+ e^-) &=&
(\bar{\tau} \sigma^{\mu\nu} P_L \mu) (\bar{e} \sigma_{\mu\nu} e),
\\
\mathcal{O}_{10}(\tau^+\to \mu^+ e^+ e^-) &=&
(\bar{\tau} \sigma^{\mu\nu} P_R \mu) (\bar{e} \sigma_{\mu\nu} e).
\end{eqnarray}
%
We can adopt above basis as
%
\\
\numentry{1}{
$(\bar{\tau} \gamma^\mu P_L \mu) (\bar{\mu} \gamma_\mu P_L \mu)$,}
\numentry{2}{
$(\bar{\tau} \gamma^\mu P_L \mu) (\bar{\mu} \gamma_\mu P_R \mu)$,}
\numentry{3}{ $(\bar{\tau} P_L \mu) (\bar{\mu} P_L \mu)$,}
\numentry{4}{
$(\bar{\tau} \gamma^\mu P_L \mu) (\bar{e} \gamma_\mu P_L e)$,}
\numentry{5}{
$(\bar{\tau} \gamma^\mu P_L \mu) (\bar{e} \gamma_\mu P_R e)$,}
\numentry{6}{ $(\bar{\tau} P_L \mu) (\bar{e} P_L e)$,}
\numentry{7}{ $(\bar{\tau} P_L \mu) (\bar{e} P_R e)$,}
\numentry{8}{
$(\bar{\tau} \sigma^{\mu\nu} P_L \mu) (\bar{e} \sigma_{\mu\nu} e)$,}
\numentry{101}{
$ m_\tau \bar{\tau} \sigma^{\mu\nu} P_L \mu F_{\mu\nu}$.}
%
Since the QED gauge covariant derivative is taken as
$\partial_\mu - i e Q A^a_\mu$ with $Q=-1$ for the electron in
\citere{Kitano:2000fg}, the sign of the dipole term may have to be
flipped, in accordance with the conventions in the quark sector.
It is also defined that
$\sigma^{\mu\nu}=\frac{i}{2}[\gamma^\mu,\gamma^\nu]$ in
\citere{Kitano:2000fg}.
More generally, if we need slots for operators like $(\bar{\tau}
\gamma^\mu P_L \mu) (\bar{\tau} \gamma_\mu P_L \tau)$, numbers $<100$
can be used for four-lepton operators in a similar way to the $\Delta
F=1$ four-quark operators.
For semihadronic operators relevant for $\tau^+\to \mu^+ \pi^0$
etc., the operator basis may be analogously defined as:
\\
\numentry{2q4}{
$(\bar{\tau} \gamma^\mu P_L \mu) (\bar{q} \gamma_\mu P_L q)$,}
\numentry{2q5}{
$(\bar{\tau} \gamma^\mu P_L \mu) (\bar{q} \gamma_\mu P_R q)$,}
\numentry{2q6}{ $(\bar{\tau} P_L \mu) (\bar{q} P_L q)$,}
\numentry{2q7}{ $(\bar{\tau} P_L \mu) (\bar{q} P_R q)$,}
\numentry{2q8}{
$(\bar{\tau} \sigma^{\mu\nu} P_L \mu) (\bar{q} \sigma_{\mu\nu} q)$,}
%
where \texttt{q=1,2,3} for $q=u,d,s$, respectively.
However, a different basis is used in \citere{Kitano:2002mt}...
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two-Higgs Doublet Model \label{app:2hdm}}
The conventions used for the different Two-Higgs Doublet Model (2HDM) types, corresponding to different charged Higgs Yukawa couplings are given in Table~\ref{tab:yukawas}.\\
%
\begin{table}[!h]
\centering
\begin{tabular*}{0.7\columnwidth}{@{\extracolsep{\fill}}cccc}
\hline
Type & $\lambda^U$ & $\lambda^D$ & $\lambda^L$ \\
\hline
I & $-\tan\beta$ & $-\tan\beta$ & $-\tan\beta$ \\
II & $\cot\beta$ & $-\tan\beta$ & $-\tan\beta$ \\
III & $-\tan\beta$ & $-\tan\beta$ & $\cot\beta$ \\
IV & $\cot\beta$ & $-\tan\beta$ & $\cot\beta$ \\
\hline
\end{tabular*}
\caption{Charged Higgs Yukawa coupling coefficients $\lambda^f$ in the
$Z_2$-symmetric types of 2HDM. The superscripts U, D and L stand respectively for
the up-type quarks, the down-type quarks and the leptons.\label{tab:yukawas}}
\end{table}%
\noindent The notation and meaning of the different types vary in the literature. Sometimes type Y (III) and type X
(IV) are used. In supersymmetry, type III usually refers to the general model encountered when the $Z_2$ symmetry
of the tree-level type II model is broken by higher order corrections.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Example \label{app:example}}
An example of a FLHA file is provided below. \\
{\footnotesize
\begin{verbatim}
Block FCINFO # Program information
1 SUPERISO # flavor calculator
2 2.8_beta # version number
Block FMODSEL # Model selection
2 1 0 # Supersymmetry general MSSM
Block SMINPUTS # Standard Model inputs
1 1.27839951e+02 # alpha_em^(-1)
2 1.16570000e-05 # G_Fermi
3 1.17200002e-01 # alpha_s(M_Z)
4 9.11699982e+01 # m_Z(pole)
5 4.19999981e+00 # m_b(m_b)
6 1.72399994e+02 # m_top(pole)
7 1.77699995e+00 # m_tau(pole)
24 1.27000000e+00 # m_c(m_c)
Block FMASS # Mass spectrum in GeV
#PDG code mass scheme scale particle
3 1.05000000e-01 1 2.00000000e+00 # s
5 4.68000000e+00 3 0 # b
211 1.39600000e-01 0 0 # pi+
313 8.91700000e-01 0 0 # K*
321 4.93700000e-01 0 0 # K+
421 1.86484000e+00 0 0 # D0
431 1.96849000e+00 0 0 # D_s+
521 5.27950000e+00 0 0 # B+
531 5.36630000e+00 0 0 # B_s
Block FLIFE # Lifetime in sec
#PDG code lifetime particle
211 2.60330000e-08 # pi+
321 1.23800000e-08 # K+
431 5.00000000e-13 # D_s+
521 1.63800000e-12 # B+
531 1.42500000e-12 # B_s
Block FCONST # Decay constant in GeV
#PDG code number decay constant particle
431 1 2.41000000e-01 # D_s+
521 1 2.00000000e-01 # B+
531 1 2.45000000e-01 # B_s
Block FCONSTRATIO # Ratio of decay constant
#PDG code1 code2 ratio comment
321 211 1.18900000e+00 # f_K/f_pi
Block FBAG # Bag parameters
#PDG code number B-parameter particle
511 1 1.26709794e+00 # B_d
531 1 1.23000000e+00 # B_s
Block FFORM # Form Factors in GeV
# ParentPDG number value NDA ID1 ID2 ID3 ... comment
521 1 4.60000000e-01 3 421 -15 16 # Delta(w) in B+->D0 tau nu
521 2 1.02600000e+00 3 421 -15 16 # G(1) in B+->D0 tau nu
521 3 1.17000000e+00 3 421 -15 16 # rho^2 in B+->D0 tau nu
521 1 3.10000000e-01 2 313 22 # T1(B->K*)
Block FSHAPE # Shape factors
# ParentPDG number value NDA ID1 ID2 ID3 ... comment
5 1 5.80000000e-01 2 3 22 # C (b->s gamma)
Block FWCOEF Q= 1.60846e+02 M= 2
#Effective Wilson coefficients in the standard basis
# type sub nb order real part
1 1 2 0 1.00000000e+00
1 1 7 0 -1.82057567e-01
1 1 8 0 -1.06651571e-01
1 1 1 1 2.33177662e+01
1 1 4 1 5.29677461e-01
1 1 7 1 1.35373179e-01
1 1 8 1 -6.94496405e-01
1 1 1 2 3.08498153e+02
1 1 2 2 4.91587899e+01
1 1 3 2 -7.01872509e+00
1 1 4 2 1.25624440e+01
1 1 5 2 8.76122785e-01
1 1 6 2 1.64273022e+00
1 1 7 2 7.05439463e-01
1 1 8 2 -4.65529650e+00
Block FWCOEF Q= 2.34384e+00 M= 2
#Effective Wilson coefficients in the standard basis
# type sub nb order real part
1 1 1 0 -8.47809531e-01
1 1 2 0 1.06562816e+00
1 1 3 0 -1.34214747e-02
1 1 4 0 -1.29110603e-01
1 1 5 0 1.36343067e-03
1 1 6 0 2.88022278e-03
1 1 7 0 -3.73787589e-01
1 1 8 0 -1.80398551e-01
1 1 1 1 1.52422776e+01
1 1 2 1 -2.13433897e+00
1 1 3 1 9.52880033e-02
1 1 4 1 -4.81776851e-01
1 1 5 1 -2.10727176e-02
1 1 6 1 -1.22929476e-02
1 1 7 1 2.14544819e+00
1 1 8 1 -5.16870265e-01
1 1 7 2 1.98785400e+01
Block FOBS # Flavor observables
# ParentPDG type value NDA ID1 ID2 ID3 ... comment
5 1 2.97350499e-04 2 3 22 # BR(b->s gamma)
521 4 8.25882011e-02 2 313 22 # Delta0(B->K* gamma)
531 1 3.46978963e-09 2 13 -13 # BR(B_s->mu+ mu-)
521 1 1.09699841e-04 2 -15 16 # BR(B_u->tau nu)
521 2 9.96640362e-01 2 -15 16 # R(B_u->tau nu)
431 1 4.81251996e-02 2 -15 16 # BR(D_s->tau nu)
431 1 4.96947301e-03 2 -13 14 # BR(D_s->mu nu)
521 1 6.96556180e-03 3 421 -15 16 # BR(B+->D0 tau nu)
521 11 2.97261612e-01 3 421 -15 16 # BR(B+->D0 tau nu)/BR(B+-> D0 e nu)
321 11 6.45414388e-01 2 -13 14 # BR(K->mu nu)/BR(pi->mu nu)
321 12 9.99985822e-01 2 -13 14 # R_l23
Block FOBSERR # Theoretical error for flavor observables at 68% C.L.
# ParentPDG type -ERR +ERR NDA ID1 ID2 ID3 ... comment
5 1 0.30000000e-04 0.30000000e-04 2 3 22 # BR(b->s gamma)
Block FOBSSM # SM prediction for flavor observables
# ParentPDG type value NDA ID1 ID2 ID3 ... comment
5 1 2.97350499e-04 2 3 22 # BR(b->s gamma)
\end{verbatim}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%CITATION = CPHCB,175,290;%%
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F.~E.~Paige, S.~D.~Protopopescu, H.~Baer and X.~Tata,
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%reactions,''
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%%CITATION = HEP-PH/0312045;%%
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R.~Lafaye, T.~Plehn and D.~Zerwas,
%``SFITTER: SUSY parameter analysis at LHC and LC,''
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%%CITATION = HEP-PH/0404282;%%
\bibitem{Bechtle:2004pc}
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%observables using an iterative method,''
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%%CITATION = CPHCB,174,47;%%
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%%CITATION = JHEPA,0605,002;%%
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%%CITATION = NUPHA,B520,279;%%
\bibitem{Bobeth:1999mk}
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%l-),''
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%%CITATION = NUPHA,B574,291;%%
\bibitem{Bobeth:2001sq}
C.~Bobeth, T.~Ewerth, F.~Kruger and J.~Urban,
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%%CITATION = PHRVA,D64,074014;%%
\bibitem{Buras:2002vd}
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%``$\Delta M_{d,s}, B^0{d,s} \to \mu^{+} \mu^{-}$ and $B \to X_{s} \gamma$ in
%supersymmetry at large $\tan\beta$,''
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%%CITATION = NUPHA,B659,3;%%
\bibitem{Virto:2009wm}
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%Hamiltonian in the MSSM,''
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%%CITATION = ARXIV:0907.5376;%%
\bibitem{Amsler:2008zzb}
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%%CITATION = PHLTA,B667,1;%%
{\color{\colorGoto}
\bibitem{Buras:1999st}
A.~J.~Buras, P.~Gambino and U.~A.~Haisch,
%``Electroweak penguin contributions to non-leptonic Delta(F) = 1
%decays at
%NNLO,''
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%%CITATION = NUPHA,B570,117;%%
\bibitem{Bobeth:2003at}
C.~Bobeth, P.~Gambino, M.~Gorbahn and U.~Haisch,
%``Complete NNLO QCD analysis of anti-B --> X/s l+ l- and higher order
%electroweak effects,''
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%%CITATION = JHEPA,0404,071;%%
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\end{thebibliography}
\end{document}
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