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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{2em}
{\sc
F.~Mahmoudi$^{1}$%
\footnote{
email: mahmoudi@in2p3.fr
}%
, S.~Heinemeyer$^{2}$%
\footnote{
email: Sven.Heinemeyer@cern.ch
}%
%, \ldots
% }
, A.~Arbey$^{3}$%
, A.~Bharucha$^{4}$, \\
T.~Goto$^{5}$%
, T. Hahn$^{6}$%
, U. Haisch$^{7}$%
, S.~Kraml$^{8}$%
, M.~Muhlleitner$^{9}$, \\
J.~Reuter$^{10}$%
, P.~Skands$^{11}$%
, P.~Slavich$^{12}$
%
}
\vspace*{0.5cm}
{\sl
$^1$Clermont Universit\'e, Universit\'e Blaise Pascal, CNRS/IN2P3,\\
LPC, BP 10448, F--63000 Clermont-Ferrand, France
\vspace*{0.1cm}
$^2$Instituto de F\'isica de Cantabria (CSIC-UC), Santander, Spain\\
\vspace*{0.1cm}
$^3$Universit\'e de Lyon, France; Universit\'e Lyon 1, F--69622; CRAL, Observatoire de Lyon,\\ F--69561 Saint-Genis-Laval;
CNRS, UMR 5574; ENS de Lyon, France
\vspace*{0.1cm}
$^4$IPPP, Department of Physics, University of Durham, Durham DH1 3LE, UK
\vspace*{0.1cm}
$^5$KEK Theory Center, Institute of Particle and Nuclear Studies,\\
KEK, Tsukuba, 305-0801 Japan
\vspace*{0.1cm}
$^6$Max-Planck-Institut f\"ur Physik, F\"ohringer Ring 6, D--80805 Munich, Germany
\vspace*{0.1cm}
$^7$Institut f\"ur Physik (WA THEP), Johannes Gutenberg-Universit\"at, \\
D--55099 Mainz, Germany
\vspace*{0.1cm}
$^8$Laboratoire de Physique Subatomique et de Cosmologie (LPSC), \\
UJF Grenoble 1, CNRS/IN2P3, 53 Avenue des Martyrs,
F--38026 Grenoble, France
\vspace*{0.1cm}
$^9$Institut f\"ur Theoretische Physik, Karlsruhe Institute of Technology, \\
D--76128 Karlsruhe, Germany
\vspace*{0.1cm}
$^{10}$University of Freiburg, Insitute of Physics, Hermann-Herder-Str. 3, \\
D--79104 Freiburg, Germany
\vspace*{0.1cm}
$^{11}$CERN PH-TH, Case 01600, CH-1211 Geneva 23, Switzerland
\vspace*{0.1cm}
$^{12}$LPTHE, 4, Place Jussieu, F--75252 Paris, France
}
\end{center}
\begin{abstract}
We present the Flavour Les Houches Accord (FLHA) which specifies a unique
set of conventions for flavour-related parameters and observables.
The FLHA uses the
generic SUSY Les Houches Accord (SLHA) file structure. It defines the
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}
In addition to the increasing number of refined approaches in the literature
for calculating flavour-related observables, advanced programs dedicated to
the calculation of such quantities, e.g. Wilson coefficients, branching ratios,
mixing amplitudes, renormalisation group equation (RGE) running including
flavour effects have recently been
developed~\cite{Mahmoudi:2007vz,Degrassi:2007kj,sufla}.
Flavour-related observables are also implemented by many other
non-dedicated public codes to provide additional checks for the models under
investigation
\cite{Belanger:2008sj,Arbey:2009gu,Heinemeyer:1998yj,Lee:2003nta,Ellwanger:2005dv,Paige:2003mg}.
The results are often subsequently 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 specialised interfaces exist between
the various codes. Such tailor-made interfaces are not easily generalised
and are time-consuming to construct and test for each specific
implementation. A universal interface would clearly be an advantage
here.
%
A similar problem arose some time ago in the context of Supersymmetry
(SUSY). The solution took the form of the SUSY Les Houches Accord
(SLHA)~\cite{Skands:2003cj,Allanach:2008qq}, which is nowadays frequently used
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, allowing information to be exchanged between
different codes via ASCII files.
The detailed structure of these input and output files is described in
\citeres{Skands:2003cj,Allanach:2008qq}.
While the first definitions, SLHA1~\cite{Skands:2003cj}, were
concerned with the simplest version of the MSSM, the
SLHA2~\cite{Allanach:2008qq} also contains definitions for the case of
complex parameters, $R$-parity violation, non-minimal flavor violation etc.
The goal of this article is to exploit the existing organisational structure
of the SLHA and use it to define an accord for the exchange of flavour
related quantities, which we refer to as the ``Flavour Les Houches Accord''
(FLHA). In brief, 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 programs. Furthermore, the
standardised format will provide the users with a clear and well-structured
result that could eventually be used for other 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 measured parameters in the
Standard Model (SM) and the Cabibbo-Kobayashi-Maskawa (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 the sake of clarity, the
new FLHA block
names start with ``\texttt{F}''. Also, in order to avoid any confusion,
the SLHA blocks are not modified or redefined in the FLHA.
If a block needs to be extended to meet the requirements of flavour physics, a
new ``\texttt{F}'' block is defined instead.
Note that different codes may \emph{technically} achieve the FLHA
input/output in different ways. The details of how to
`switch on' the FLHA input/output for 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{Hahn:2006nq}. In
principle these programs could be extended to include the FLHA as well.
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{Definitions of the interfaces}
In this section, the FLHA input and output files are described.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{General structure}
Following the general structure for the SLHA~\cite{Skands:2003cj,Allanach:2008qq} we assume
the following:
\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, a 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, this 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 and
should be ignored by the reading program.
\item Any 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 ``\texttt{F}''.
There are two exceptions to this rule: blocks borrowed from the
SLHA, which keep their original name, and blocks containing
imaginary parts, which start with ``\texttt{IMF}''.
\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}''. For instance,
``\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.
\item The order of the blocks is arbitrary.
\end{itemize}
Further definitions can be found in section~3 of \citere{Skands:2003cj}.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 where ``model dependent'' information can be found (in the case of SUSY, the SLHA \texttt{BLOCK MODSEL} is used instead).
\item \texttt{BLOCK SMINPUTS}:
Measured values of SM parameters used for the calculations.
\item \texttt{BLOCK VCKMIN}:
Input parameters of the CKM matrix in the Wolfenstein parameterisation.
\item \texttt{BLOCK UPMNSIN}:
Input parameters of the PMNS neutrino mixing matrix in the PDG parameterisation.
\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 UPMNS}:
Real part of the PMNS matrix elements.
\item \texttt{BLOCK IMUPMNS}:
Imaginary part of the PMNS matrix elements.
\item \texttt{BLOCK FMASS}:
Masses of quarks, mesons, hadrons, etc.
\item \texttt{BLOCK FLIFE}:
Lifetime (in seconds) of mesons, hadrons, etc.
\item \texttt{BLOCK FCONST}:
Decay constants.
\item \texttt{BLOCK FCONSTRATIO}:
Ratios of decay constants.
\item \texttt{BLOCK FBAG}:
Bag parameters.
\item \texttt{BLOCK FWCOEF}:
Real part of the Wilson coefficients.
\item \texttt{BLOCK IMFWCOEF}:
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 FPARAM}:
process dependent variables and parameters.\\
\end{itemize}
%
More details on each block are given in the following.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Definition of the blocks}
The FLHA input and output files are described 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 the SLHA.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK MODSEL}}
This block provides switches and options for the model selection in
the case of SUSY. It is taken over directly from SLHA2.
However, to permit the definition of non-SUSY models also the
\texttt{BLOCK FMODSEL} will be defined (see below). In case of
ambiguities the entries of \texttt{BLOCK MODSEL} are used.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FMODSEL}}
This block provides switches and options for the model selection in case
of non-SUSY models. It is similar to the
SLHA2 \texttt{BLOCK MODSEL}, but extended to allow more
flexibility.
\numentry{1}{Choice of SUSY breaking model or indication of other
model. By default, a
minimal type of model will always be assumed. Possible
values are (note: in the case of SUSY \texttt{BLOCK MODSEL} should be
used instead):\\
\snumentry{-1}{SM}
\snumentry{0}{General MSSM simulation}
\snumentry{1}{(m)SUGRA model}
\snumentry{2}{(m)GMSB model}
\snumentry{3}{(m)AMSB model}
\snumentry{4}{...}
\snumentry{3O}{General THDM}
\snumentry{31}{THDM type I}
\snumentry{32}{THDM type II}
\snumentry{33}{THDM type III}
\snumentry{34}{THDM type IV}
\snumentry{35}{...}
\snumentry{99}{other model. This choice requires a string given in the
entry \ttt{99}}
}\\
%
%
\numentry{5}{(Default=\ttt{0}) CP violation. Switches defined are:\\
\snumentry{0}{CP is conserved. No information on the CKM phase
is used.}
\snumentry{1}{CP is violated, but only by the standard CKM
phase. All other phases are assumed zero.}
\snumentry{2}{CP is violated. Completely general CP phases
allowed.}
}\\
%
\numentry{6}{(Default=\ttt{0}) Flavour violation. Switches defined are:\\
\snumentry{0}{No flavour violation.
}
\snumentry{1}{Quark flavour is violated.}
\snumentry{2}{Lepton flavour is violated.}
\snumentry{3}{Lepton and quark flavour is violated.}
}
%
\numentry{99}{a string that defines other models is used only if
entry~\ttt{1} is given as~\ttt{99}, otherwise it is ignored.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK SMINPUTS}}
In general, the spectrum of the SM particles plays a crucial role in flavour physics.
Consequently, experimental measurements of masses and coupling constants
at the electroweak scale are required.
In the SLHA the block containing these quantities is
\texttt{SMINPUTS}. This block is borrowed from SLHA as it is and is
reproduced here for completeness.
It is also important to note that all presently available experimental
determinations of quantities, e.g. $\alpha_s$ or the running bottom
quark mass, are clearly based on
the assumption that the SM is the underlying theory. When
extending the field content of the SM to that of a New
Physics (NP) Model, 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 NP. However, since these
values are not known, all parameters contained in the block
\texttt{SMINPUTS} should be the `ordinary' ones obtained from SM fits,
i.e.\ with no NP corrections included. Any flavour code itself is then
assumed to convert these parameters into ones appropriate to an NP
framework.
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}{$m_b(m_b)^{\overline{\mathrm{MS}}}$, bottom quark running mass
in the $\overline{\mathrm{MS}}$ scheme.}
\numentry{6}{$m_t$, top-quark pole mass.}
\numentry{7}{$m_\tau$, tau pole mass.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 parameterisation, Eq.~(13.30) of \citere{Amsler:2008zzb}.
All the angles and phases should be given in radians.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FMASS}}
The block \texttt{BLOCK FMASS} contains the mass spectrum for the
involved particles. It is an addition to the
SLHA \texttt{BLOCK MASS} which contained only pole masses and to the
SLHA \ttt{BLOCK SMINPUTS} which contains quark masses.
If a mass is given in two blocks the block \ttt{FMASS} overrules the
other blocks.
In \ttt{FMASS} we
specify additional information concerning the renormalisation scheme as
well as the scale at which the masses are given and thus allow for
larger flexibility. 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 renormalisation scheme, and finally the last double
precision number points to the energy scale (0 if not relevant).
An additional comment can be given after \texttt{\#}.
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}}
The block \texttt{BLOCK FLIFE} contains the 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 nine-digit integer should be the PDG code of a particle
and the double precision number its lifetime.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FCONST}}
The block \texttt{BLOCK FCONST} contains the 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, and the double precision number its decay constant.
The decay constants for the most commonly used mesons with several decay constants are defined as:\\[0.3cm]
%
\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.}
}
The definitions of the decay constants ($f$, $h$, etc.)~and more
details can be found in Appendix~\ref{app:decayconst}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FCONSTRATIO}}
The block \texttt{BLOCK FCONSTRATIO} contains the ratios of decay constants, which often have less uncertainty than the decay constants themselves. 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,I2,3x,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,
the third and fourth integers the numbers of the decay constants, which
correspond 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 Q= \ldots}}
The block \texttt{BLOCK FBAG Q= \ldots} contains the bag parameters at the scale \texttt{Q}. 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, and the double
precision number its bag parameter.
Bag parameters for $B_d^0-\bar{B}_d^0$ mixing in
$\overline{\mathrm{MS}}$ scheme are listed as:
\\
\numentry{511}{$B_d^0$\\
\numentry{1}{$B_1^{\overline{\mathrm{MS}}}(Q)$}
\numentry{2}{$B_2^{\overline{\mathrm{MS}}}(Q)$}
\numentry{3}{$B_3^{\overline{\mathrm{MS}}}(Q)$}
\numentry{4}{$B_4^{\overline{\mathrm{MS}}}(Q)$}
\numentry{5}{$B_5^{\overline{\mathrm{MS}}}(Q)$}
}
%
We define the bag parameters in $\overline{\mathrm{MS}}$ scheme with the
renormalization scale $Q$.
Normalizations can be found in Appendix~\ref{app:bagpar}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FWCOEF Q= \ldots}}
The block \texttt{BLOCK FWCOEF Q= \ldots} contains the real part of the
Wilson coefficients at the scale \texttt{Q}.
The Wilson coefficients are calculated for a set of operators (see Appendix~\ref{app:operators}
for a list of the most relevant effective operators).
The different orders $C^{(k)}_i$ have to be given separately according
to the following convention for the perturbative expansion:
%
\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}
The couplings should therefore not be included in the Wilson coefficients.
The entries in \texttt{BLOCK FWCOEF} should consist of two integers
defining the fermion structure of the operator and the operator
structure itself. These two numbers are not thought to give a full
representation including normalisation etc.\ of the operator, but
merely correspond to a unique identifier
for any possible Wilson coefficient.
Consequently, the user has to take care that a consistent
normalisation including prefactors etc.\ is indeed fulfilled.
The most relevant examples are
listed in Appendix~\ref{app:operators}.
As an example, for the operator $O_1$,
\begin{align}
O_1 &= (\bar{s} \gamma_{\mu} T^a P_L c)
(\bar{c} \gamma^{\mu} T^a P_L b)
\label{O1}
\end{align}
the definition of the two numbers is given as follows.
The appearing fermions are encoded by a two-digit number originating
from their PDG code, where no difference is made between particles and
antiparticles, as given in Table~\ref{tab:pdgcodes}.
%
Correspondingly, the first integer number defining $O_1$, containing the
fermions $\bar s c \bar c b$, is given by~03040405.
The various operators are defined in Table~\ref{tab:opcodes}.
Correspondingly, the second integer number defining $O_1$, containing
the operators $\gamma_\mu T^a P_L \, \gamma^\mu T^a P_L$ is given by~6161.
A few more rules are needed for an unambiguous definition.\\[-2em]
\begin{itemize}
\item
If an operators appears without fermions (as it is possible, e.g., for
$F_{\mu\nu}$) it should appear right-most, so that the encoded fermions
correspond to the left-most operators.\\[-2em]
\item
In the case of a possible ambiguity, for instance
$O_1 = (\bar s \gamma_\mu T^a P_L c) (\bar c \gamma^\mu T^a P_L b)$
corresponding to 03040405~6161 and
$O_1 = (\bar c \gamma_\mu T^a P_L b) (\bar s \gamma^\mu T^a P_L c)$
corresponding to 04050304~6161
the ``smaller'' number, i.e.\ in this case 03040405~6161 should be
used.\\[-1.5em]
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[!ht]
\begin{center}
\begin{tabular}{|c|c|c||c|c|c|}
\hline
name & PDG code & two-digit number &
name & PDG code & two-digit number \\
\hline
$d$ & 1 & 01 & $e$ & 11 & 11 \\
$u$ & 2 & 02 & $\nu_e$ & 12 & 12 \\
$s$ & 3 & 03 & $\mu$ & 13 & 13 \\
$c$ & 4 & 04 & $\nu_\mu$ & 14 & 14 \\
$b$ & 5 & 05 & $\tau$ & 15 & 15 \\
$t$ & 6 & 06 & $\nu_\tau$ & 16 & 16 \\
$\sum_q q$ & & 07 & $\sum_l l$ & & 17\\
$\sum_q q Q_q$ & & 08 & $\sum_l l Q_l$ & & 18\\
\hline
\end{tabular}
\caption{PDG codes and two-digit number identifications of quarks and
leptons. The summations are over active fermions.\label{tab:pdgcodes}}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{-0.5em}
\begin{table}[!htb]
\begin{center}
\begin{tabular}{|c|c||c|c||c|c|}
\hline
operator & number &
operator & number &
operator & number \\
\hline
$1$ & 30 & $T^a$ & 50 & $\delta_{ij}$ & 70 \\
$P_L$ & 31 & $P_L T^a$ & 51 & $P_l \delta_{ij}$ & 71 \\
$P_R$ & 32 & $P_R T^a$ & 52 & $P_R \delta_{ij}$ & 72 \\
$\gamma^\mu$ & 33 & $\gamma^\mu T^a$ & 53 &
$\gamma^\mu \delta_{ij}$ & 73 \\
$\gamma_5$ & 34 & $\gamma_5 T^a$ & 54 &
$\gamma_5 \delta_{ij}$ & 74 \\
$\sigma^{\mu\nu}$ & 35 & $\sigma^{\mu\nu} T^a$ & 55 &
$\sigma^{\mu\nu} \delta_{ij}$ & 75 \\
$\gamma^\mu \gamma^\nu \gamma^\rho$ & 36 &
$\gamma^\mu \gamma^\nu \gamma^\rho T^a$ & 56 &
$\gamma^\mu \gamma^\nu \gamma^\rho \delta_{ij}$ & 76 \\
$\gamma^\mu \gamma_5$ & 37 &
$\gamma^\mu \gamma_5 T^a$ & 57 &
$\gamma^\mu \gamma_5 \delta_{ij}$ & 77 \\
$\gamma^\mu P_L$ & 41 & $\gamma^\mu T^a P_L$ & 61 &
$\gamma^\mu \delta_{ij} P_L$ & 81 \\
$\gamma^\mu P_R$ & 42 & $\gamma^\mu T^a P_R$ & 62 &
$\gamma^\mu \delta_{ij} P_R$ & 82 \\
$\sigma^{\mu\nu} P_L$ & 43 & $\sigma^{\mu\nu} T^a P_L$ & 63 &
$\sigma^{\mu\nu} \delta_{ij} P_L$ & 83 \\
$\sigma^{\mu\nu} P_R$ & 44 & $\sigma^{\mu\nu} T^a P_R$ & 64 &
$\sigma^{\mu\nu} \delta_{ij} P_R$ & 84 \\
$\gamma^\mu \gamma^\nu \gamma^\rho P_L$ & 45 &
$\gamma^\mu \gamma^\nu \gamma^\rho T^a P_L$ & 65 &
$\gamma^\mu \gamma^\nu \gamma^\rho \delta_{ij} P_L$ & 85 \\
$\gamma^\mu \gamma^\nu \gamma^\rho P_R$ & 46 &
$\gamma^\mu \gamma^\nu \gamma^\rho T^a P_R$ & 66 &
$\gamma^\mu \gamma^\nu \gamma^\rho \delta_{ij} P_R$ & 86 \\
$F_{\mu\nu}$ & 22 & $G_{\mu\nu}^a$ & 21 & & \\
\hline
\end{tabular}
\caption{Two-digit number definitions for the operators.
$T^a$ ($a = 1 \ldots 8$) denote the $SU(3)_C$ generators,
$P_{L,R} = \frac{1}{2} (1 \mp \gamma_5)$, and
$(T^a)_{ij} (T^a)_{kl} = \frac{1}{2} (\delta_{il} \delta_{kj}
- 1/N_c \, \delta_{ij} \delta_{kl})$, where
$i,j,k,l$ are colour indices.
\label{tab:opcodes}}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The third index corresponds to each term in Eq.~(\ref{eq:WCexpansion}):\\[2mm]
\numentry{00}{$C^{(0)}_{i}(Q)$}
\numentry{01}{$C^{(1)}_{i,s}(Q)$}
\numentry{02}{$C^{(2)}_{i,s}(Q)$}
\numentry{10}{$C^{(1)}_{i,e}(Q)$}
\numentry{11}{$C^{(2)}_{i,es}(Q)$}
\numentry{99}{total}
The information about the order is given by a two-digit number $xy$, where
$x$ indicates $\mathcal{O}(\alpha^x)$ and $y$ indicates
$\mathcal{O}(\alpha_s^y)$, and 0 indicates $C_i^{(0)}$.
The Wilson coefficients can be provided either via separate new
physics and SM contributions, or as a total contribution
of both new physics and SM, depending on the code generating them. To
avoid any confusion, the fourth entry must specify whether the given
Wilson coefficients correspond to the SM contributions, new physics
contributions or to the sum of them, using the following definitions:\\
\numentry{0}{SM}
\numentry{1}{NP}
\numentry{2}{SM+NP}
The new Physics model is the model specified in the \texttt{BLOCK FMODSEL}.
\noindent The standard for each line in the block should thus correspond to the
FORTRAN format
\begin{center}
\texttt{(1x,I8,1x,I4,3x,I2,3x,I1,3x,1P,E16.8,0P,3x,'\#',1x,A)},
\end{center}
where the eight-digit integer specifies the fermion content, the
four-digit integer the operator structure, the two-digit integer the
order at which the Wilson coefficients are calculated followed by the
one-digit integer specifying the model, and finally the double precision
number gives the real part of the Wilson coefficient.
Note that there can be several such blocks for different scales \texttt{Q}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK IMFWCOEF Q= \ldots}}
The block \texttt{BLOCK IMFWCOEF} contains the imaginary part of the
Wilson coefficients at the scale \texttt{Q}.
The structure is exactly the same as for the \texttt{BLOCK FWCOEF}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FOBS}}
The block \texttt{BLOCK FOBS} contains the 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 integer
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}}
The block \texttt{BLOCK FOBSERR} contains the 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.
In a similar way, for every block \texttt{Fname}, a corresponding error block with the
name \texttt{BLOCK FnameERR} can be defined.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FOBSSM}}
The block \texttt{BLOCK FOBSSM} contains 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 FPARAM}}
The block \texttt{BLOCK FPARAM} contains process dependent variables, such as form factors, shape functions,... for a specific decay.
The decay should be defined 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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
The interplay of collider and flavour physics is entering a new era with
the start-up of the LHC. In the future more and more programs will be
interfaced in order to exploit maximal information from both
collider and flavour data. Towards this end, an accord will play a
crucial role. The accord presented 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 studies
related to flavour physics.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}
We thank the organizers of the Les Houches workshop 2009, where
this work was started, for their hospitality and support.
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'.
The work of S.H.\ was partially supported by CICYT (grant FPA 2007--66387).
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 baryons 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||c|c|}
\hline
Name & PDG code & Name & PDG 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{Two-Higgs Doublet Model \label{app:2hdm}}
The charged Higgs boson couplings to fermions in the four types of Two-Higgs Doublet Model can be expressed as
\begin{equation}
\begin{aligned}
H^+D\bar{U}:\qquad
&\frac{ig}{2\sqrt{2}m_W}V_{UD}\left[\lambda^Um_U\left(1-\gamma^5\right)
-\lambda^Dm_D\left(1+\gamma^5\right)\right]\\
H^+\ell^-\bar{\nu}_\ell:\qquad
&-\frac{ig}{2\sqrt{2}m_W}\lambda^\ell m_\ell\left(1+\gamma^5\right),
\end{aligned}
\end{equation}
where $U$, $D$ and $\ell$ stand, respectively, for the up-type quarks, the down-type quarks and the leptons. The conventions used for the different Two-Higgs Doublet Model (2HDM) types, corresponding to different Yukawa couplings are given in Table~\ref{tab:yukawas}.
%
\begin{table}[!h]
\centering
\begin{tabular*}{0.7\columnwidth}{@{\extracolsep{\fill}}cccc}
\hline
Type & $\lambda^U$ & $\lambda^D$ & $\lambda^\ell$ \\
\hline
I & $\cot\beta$ & $\cot\beta$ & $\cot\beta$ \\
II & $\cot\beta$ & $-\tan\beta$ & $-\tan\beta$ \\
III & $\cot\beta$ & $-\tan\beta$ & $\cot\beta$ \\
IV & $\cot\beta$ & $\cot\beta$ & $-\tan\beta$ \\
\hline
\end{tabular*}
\caption{Charged Higgs Yukawa coupling coefficients $\lambda^f$ in the
$Z_2$-symmetric types of the 2HDM. \label{tab:yukawas}}
\end{table}%
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{Effective Operators}
\label{app:operators}
%
Here we give a list of the most relevant effective operators together
with their unique two-number identifier.
%
\subsection{Effective operators for $b\to s$ transition}
%
Effective operators relevant to the $b\to s$ transition are\\
%
\begin{align}
O_1 &= O(03040405~6161)= (\bar{s} \gamma_{\mu} T^a P_L c)
(\bar{c} \gamma^{\mu} T^a P_L b),\nonumber\\[4mm]
O_2 &= O(03040405~4141)= (\bar{s} \gamma_{\mu} P_L c)
(\bar{c} \gamma^{\mu} P_L b),\nonumber\\[3mm]
O_3 &= O(03050707~4133)= (\bar{s} \gamma_{\mu} P_L b)
{\displaystyle\sum_q} (\bar{q} \gamma^{\mu} q),\nonumber\\[1mm]
O_4 &= O(03050707~6153)= (\bar{s} \gamma_{\mu} T^a P_L b)
{\displaystyle\sum_q} (\bar{q} \gamma^{\mu} T^a q),\nonumber\\[1mm]
O_5 &= O(03050707~4536)= (\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 &= O(03050707~6556)= (\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(0305~4422)= \dfrac{e}{16\pi^2} \left[ \bar{s} \sigma^{\mu \nu}
(m_b P_R) b \right] F_{\mu \nu},\nonumber\\[2mm]
O_8 &= O(0305~6421)= \dfrac{g}{16\pi^2} \left[ \bar{s} \sigma^{\mu \nu}
(m_b P_R) T^a b \right] G_{\mu \nu}^a.\nonumber
\end{align}
\subsection{Effective operators for neutral meson mixings}
Effective operators for $B_d^0-\bar{B}_d^0$ mixing are
%
\begin{align}
O(01050105~4141) &=
(\bar{d} \gamma_\mu P_L b)
(\bar{d} \gamma^\mu P_L b),
\nonumber\\[2mm]
O(01050105~3131) &=
(\bar{d} P_L b)
(\bar{d} P_L b),
\nonumber\\[2mm]
O(01050105~7171) &=
(\bar{d}_i P_L b^j)
(\bar{d}_j P_L b^i),
\nonumber\\[2mm]
O(01050105~3132) &=
(\bar{d} P_L b)
(\bar{d} P_R b),
\nonumber\\[2mm]
O(01050105~7172) &=
(\bar{d}_i P_L b^j)
(\bar{d}_j P_R b^i),
\end{align}
%
and those with opposite chiralities.
Operators for $B_s^0-\bar{B}_s^0$, $K^0-\bar{K}^0$ and
$D^0-\bar{D}^0$ mixings are defined in the same way.
\subsection{Effective operators for lepton flavor violations}
We give a list of effective operators for $\Delta LF=1$ lepton flavor
violating processes as follows.\\
%
\begin{itemize}
\item
$\mu\to e$ transitions
($\mu\to e\,\gamma$, $\mu\to e\,e\,e$ and $\mu-e$ conversion in a muonic atom):\\
%
\begin{align}
O(1311~4322) &=
m_\mu (\bar{\mu} \sigma^{\mu\nu} P_L e)
F_{\mu\nu},
\nonumber\\[5mm]
%%
O(13111111~3131) &=
(\bar{\mu} P_L e)
(\bar{e} P_L e),
\nonumber\\[2mm]
O(13111111~4141) &=
(\bar{\mu} \gamma_{\mu} P_L e)
(\bar{e} \gamma^{\mu} P_L e),
\nonumber\\[2mm]
O(13111111~4142) &=
(\bar{\mu} \gamma_{\mu} P_L e)
(\bar{e} \gamma^{\mu} P_R e),
\nonumber\\[5mm]
%%
O(13110101~3131) &=
(\bar{\mu} P_L e)
(\bar{d} P_L d),
\nonumber\\[2mm]
O(13110101~3132) &=
(\bar{\mu} P_L e)
(\bar{d} P_R d),
\nonumber\\[2mm]
O(13110101~4141) &=
(\bar{\mu} \gamma_\mu P_L e)
(\bar{d} \gamma^\mu P_L d),
\nonumber\\[2mm]
O(13110101~4142) &=
(\bar{\mu} \gamma_\mu P_L e)
(\bar{d} \gamma^\mu P_R d),
\nonumber\\[2mm]
O(13110101~4343) &=
(\bar{\mu} \sigma^{\mu\nu} P_L e)
(\bar{d} \sigma_{\mu\nu} P_L d),
\nonumber\\[5mm]
%%
O(13110202~3131) &=
(\bar{\mu} P_L e)
(\bar{u} P_L u),
\nonumber\\[2mm]
O(13110202~3132) &=
(\bar{\mu} P_L e)
(\bar{u} P_R u),
\nonumber\\[2mm]
O(13110202~4141) &=
(\bar{\mu} \gamma_\mu P_L e)
(\bar{u} \gamma^\mu P_L u),
\nonumber\\[2mm]
O(13110202~4142) &=
(\bar{\mu} \gamma_\mu P_L e)
(\bar{u} \gamma^\mu P_R u),
\nonumber\\[2mm]
O(13110202~4343) &=
(\bar{\mu} \sigma^{\mu\nu} P_L e)
(\bar{u} \sigma_{\mu\nu} P_L u),
\nonumber\\[5mm]
%%
O(13110303~3131) &=
(\bar{\mu} P_L e)
(\bar{s} P_L s),
\nonumber\\[2mm]
O(13110303~3132) &=
(\bar{\mu} P_L e)
(\bar{s} P_R s),
\nonumber\\[2mm]
O(13110303~4141) &=
(\bar{\mu} \gamma_\mu P_L e)
(\bar{s} \gamma^\mu P_L s),
\nonumber\\[2mm]
O(13110303~4142) &=
(\bar{\mu} \gamma_\mu P_L e)
(\bar{s} \gamma^\mu P_R s),
\nonumber\\[2mm]
O(13110303~4343) &=
(\bar{\mu} \sigma^{\mu\nu} P_L e)
(\bar{s} \sigma_{\mu\nu} P_L s).
\label{eq:Op-mu-e}
\end{align}
%
We also define operators with opposite chiralities by replacing $P_L$
and $P_R$ with each other in (\ref{eq:Op-mu-e}).
%
\item
$\tau\to \mu$ transitions
($\tau\to \mu\,\gamma$, $\tau\to \mu\,\mu\,\mu$, $\tau\to \mu\, e^+\,e^-$ and $\tau\to\mu\,\mathrm{hadrons}$):
%
\begin{align}
O(1513~4322) &=
m_\tau (\bar{\tau} \sigma^{\mu\nu} P_L \mu)
F_{\mu\nu},
\nonumber\\[5mm]
%%
O(15131313~3131) &=
(\bar{\tau} P_L \mu)
(\bar{\mu} P_L \mu),
\nonumber\\[2mm]
O(15131313~4141) &=
(\bar{\tau} \gamma_{\mu} P_L \mu)
(\bar{\mu} \gamma^{\mu} P_L \mu),
\nonumber\\[2mm]
O(15131313~4142) &=
(\bar{\tau} \gamma_{\mu} P_L \mu)
(\bar{\mu} \gamma^{\mu} P_R \mu),
\nonumber\\[5mm]
%%
O(15131111~3131) &=
(\bar{\tau} P_L \mu)
(\bar{e} P_L e),
\nonumber\\[2mm]
O(15131111~3132) &=
(\bar{\tau} P_L \mu)
(\bar{e} P_R e),
\nonumber\\[2mm]
O(15131111~4141) &=
(\bar{\tau} \gamma_\mu P_L \mu)
(\bar{e} \gamma^\mu P_L e),
\nonumber\\[2mm]
O(15131111~4142) &=
(\bar{\tau} \gamma_\mu P_L \mu)
(\bar{e} \gamma^\mu P_R e),
\nonumber\\[2mm]
O(15131111~4343) &=
(\bar{\tau} \sigma^{\mu\nu} P_L \mu)
(\bar{e} \sigma_{\mu\nu} P_L e),
\nonumber\\[5mm]
%%
O(15130101~3131) &=
(\bar{\tau} P_L \mu)
(\bar{d} P_L d),
\nonumber\\[2mm]
O(15130101~3132) &=
(\bar{\tau} P_L \mu)
(\bar{d} P_R d),
\nonumber\\[2mm]
O(15130101~4141) &=
(\bar{\tau} \gamma_\mu P_L \mu)
(\bar{d} \gamma^\mu P_L d),
\nonumber\\[2mm]
O(15130101~4142) &=
(\bar{\tau} \gamma_\mu P_L \mu)
(\bar{d} \gamma^\mu P_R d),
\nonumber\\[2mm]
O(15130101~4343) &=
(\bar{\tau} \sigma^{\mu\nu} P_L \mu)
(\bar{d} \sigma_{\mu\nu} P_L d),
\nonumber\\[5mm]
%%
O(15130202~3131) &=
(\bar{\tau} P_L \mu)
(\bar{u} P_L u),
\nonumber\\[2mm]
O(15130202~3132) &=
(\bar{\tau} P_L \mu)
(\bar{u} P_R u),
\nonumber\\[2mm]
O(15130202~4141) &=
(\bar{\tau} \gamma_\mu P_L \mu)
(\bar{u} \gamma^\mu P_L u),
\nonumber\\[2mm]
O(15130202~4142) &=
(\bar{\tau} \gamma_\mu P_L \mu)
(\bar{u} \gamma^\mu P_R u),
\nonumber\\[2mm]
O(15130202~4343) &=
(\bar{\tau} \sigma^{\mu\nu} P_L \mu)
(\bar{u} \sigma_{\mu\nu} P_L u),
\nonumber\\[5mm]
%%
O(15130303~3131) &=
(\bar{\tau} P_L \mu)
(\bar{s} P_L s),
\nonumber\\[2mm]
O(15130303~3132) &=
(\bar{\tau} P_L \mu)
(\bar{s} P_R s),
\nonumber\\[2mm]
O(15130303~4141) &=
(\bar{\tau} \gamma_\mu P_L \mu)
(\bar{s} \gamma^\mu P_L s),
\nonumber\\[2mm]
O(15130303~4142) &=
(\bar{\tau} \gamma_\mu P_L \mu)
(\bar{s} \gamma^\mu P_R s),
\nonumber\\[2mm]
O(15130303~4343) &=
(\bar{\tau} \sigma^{\mu\nu} P_L \mu)
(\bar{s} \sigma_{\mu\nu} P_L s).
\label{eq:Op-tau-mu}
\end{align}
%
Operators with opposite chiralities are also defined.
%
\item
We define operators for $\tau\to e$ transitions by replacing the muon
(code 13) and the electron (code 11) fields with each other in
(\ref{eq:Op-tau-mu}).
%
\end{itemize}
Effective operators for $\Delta LF >1$ leptonic tau decay
$\tau^+\to \mu^-\,e^+\,e^+$ are
%
\begin{align}
O(15111311~3131) &=
(\bar{\tau} P_L e)
(\bar{\mu} P_L e),
\nonumber\\[2mm]
O(15111311~4141) &=
(\bar{\tau} \gamma_\mu P_L e)
(\bar{\mu} \gamma^\mu P_L e),
\nonumber\\[2mm]
O(15111311~4142) &=
(\bar{\tau} \gamma_\mu P_L e)
(\bar{\mu} \gamma^\mu P_R e),
\end{align}
%
and those with opposite chiralities.
Operators for $\tau^+\to e^-\,\mu^+\,\mu^+$ are defined by the
replacement of the muon and the electron fields.
For the processes in which both lepton and quark flavors are violated,
such as $\tau\to \mu\,K$, $B^0\to \mu\,\bar{e}$ and so on, relevant
operators are as follows.
%
\begin{align}
O(03011513~3131) &=
(\bar{s} P_L d)
(\bar{\tau} P_L \mu),
\nonumber\\[2mm]
O(03011513~3132) &=
(\bar{s} P_L d)
(\bar{\tau} P_R \mu),
\nonumber\\[2mm]
O(03011513~4141) &=
(\bar{s} \gamma_\mu P_L d)
(\bar{\tau} \gamma^\mu P_L \mu),
\nonumber\\[2mm]
O(03011513~4142) &=
(\bar{s} \gamma_\mu P_L d)
(\bar{\tau} \gamma^\mu P_R \mu),
\nonumber\\[2mm]
O(03011513~4342) &=
(\bar{s} \sigma^{\mu\nu} P_L d)
(\bar{\tau} \sigma_{\mu\nu} P_L \mu).
\end{align}
%
Definitions of operators with opposite chiralities and/or different
quark/lepton flavor combinations are straightforward.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Decay constants}
\label{app:decayconst}
%
The decay constant $f_P$ of a pseudoscalar meson $P$ can be defined 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 ($P=\pi^\pm$, $K$, $D$, $B$).
For $\pi^0$, $\eta$ and $\eta'$, we 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}
%
In addition, 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}
%
The parameters $h_P$ 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 polarisation 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}
\section{Bag parameters}
\label{app:bagpar}
We define the bag parameters $B_{1,2,3,4,5}$ for $B_d^0-\bar{B}_d^0$
mixing matrix elements as follows.
%
\begin{align}
\langle B_d^0 |
%% O(01050105~4141)
(\bar{d} \gamma_\mu P_L b)
(\bar{d} \gamma^\mu P_L b)
| \bar{B}_d^0 \rangle &=
\frac{2}{3} m_{B_d^0}^2 f_{B_d^0}^2 B_1,
\nonumber\\
\langle B_d^0 |
%% O(01050105~3131)
(\bar{d} P_L b)
(\bar{d} P_L b)
| \bar{B}_d^0 \rangle &=
-\frac{5}{12} m_{B_d^0}^2 f_{B_d^0}^2
\left( \frac{m_{B_d^0}}{m_b + m_d} \right)^2 B_2,
\nonumber\\
\langle B_d^0 |
%% O(01050105~7171)
(\bar{d}_i P_L b^j)
(\bar{d}_j P_L b^i)
| \bar{B}_d^0 \rangle &=
\frac{1}{12} m_{B_d^0}^2 f_{B_d^0}^2
\left( \frac{m_{B_d^0}}{m_b + m_d} \right)^2 B_3,
\nonumber\\
\langle B_d^0 |
%% O(01050105~3132)
(\bar{d} P_L b)
(\bar{d} P_R b)
| \bar{B}_d^0 \rangle &=
\frac{1}{2} m_{B_d^0}^2 f_{B_d^0}^2
\left( \frac{m_{B_d^0}}{m_b + m_d} \right)^2 B_4,
\nonumber\\
\langle B_d^0 |
%% O(01050105~7172)
(\bar{d}_i P_L b^j)
(\bar{d}_j P_R b^i)
| \bar{B}_d^0 \rangle &=
\frac{1}{6} m_{B_d^0}^2 f_{B_d^0}^2
\left( \frac{m_{B_d^0}}{m_b + m_d} \right)^2 B_5.
\label{eq:bag-BdBd}
\end{align}
%
As for the renormalization scheme for the four-quark operators, quark
masses and $B_i$, we take the $\overline{\mathrm{MS}}$ scheme.
Bag parameters for $B_s^0-\bar{B}_s^0$, $K^0-\bar{K}^0$ and
$D^0-\bar{D}^0$ mixings are defined in the same way.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Example}
\label{app:example}
%
\htr{NM, SH: not updated yet}\\
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}
}
% \htb{
% \subsection*{Goto's comments concerning effective operators}
% Let me first decompose the effective Lagrangian as follows
% (btw, we have to decide which should be used: Lagrangian or
% Hamiltonian).
% %
% \begin{equation}
% \mathcal{L}_{\mathrm{eff}} =
% A
% \sum_{ijkl,xy}
% V_{ijkl}
% C_{ijkl,xy}
% P_{ijkl,xy}
% O_{ijkl,xy}.
% \end{equation}
% %
% \begin{itemize}
% \item $i,j,k,l$ are two-digit numbers listed in Table 1 and $x,y$ are
% those in Table 2.
% \item $O_{ijkl,xy}$ is the operator which is constructed from the
% elements in Tables 1 and 2 only:
% $O_{03050000,3422}=(\bar{s}\sigma^{\mu\nu}P_R b) F_{\mu\nu}$, for
% example.
% \item $P_{ijkl,xy}$ is an operator dependent prefactor like
% $e m_b/(4\pi)^2$.
% \item $V_{ijkl}$ is the CKM matrix factor which should be independent of
% the Dirac/color structure of the operator.
% \item $A$ is an overall factor such as $4G_F/\sqrt{2}$, which should be
% common to all the Wilson coefficients in a single input/output file.
% \item $C_{ijkl,xy}$ is the Wilson coefficient whose value is given in
% the block (IM)FWCOEF.
% \end{itemize}
% %
% I suggest to make additional blocks for $P_{ijkl,xy}$, $V_{ijkl}$ and
% $A$ for flexibility.
% Coefficients in several references can be encoded in the following way.
% %
% \begin{itemize}
% \item
% Ref.~\cite{Chetyrkin:1996vx} ($b\to s \gamma$): $A=-4G_F/\sqrt{2}$.
% %
% \begin{displaymath}
% \begin{array}{|ccccc|}
% \hline
% ijkl & xy & V & P & C \\
% \hline
% 03040405 & 4141 & -V_{ts}^* V_{tb} & 1 & C_1 \\
% & 1111 & & 1 & C_2 \\
% \hline
% 03050707 & 3103 & -V_{ts}^* V_{tb} & 1 & C_3 \\
% & 4113 & & 1 & C_4 \\
% & 3505 & & 1 & C_5 \\
% & 4515 & & 1 & C_6 \\
% \hline
% 03050000 & 3422 & -V_{ts}^* V_{tb} & \frac{e m_b}{16\pi^2} & C_7 \\
% & 4421 & & \frac{g_s m_b}{16\pi^2} & C_8 \\
% \hline
% \end{array}
% \end{displaymath}
% %
% \item Ref.~\cite{Bobeth:1999mk} ($b\to s l^+ l^-$): $A=4G_F/\sqrt{2}$.
% %
% \begin{displaymath}
% \begin{array}{|ccccc|}
% \hline
% ijkl & xy & V & P & C \\
% \hline
% 03020205 & 4141 & V_{us}^* V_{ub} & 1 & C_1^c \\
% & 1111 & & 1 & C_2^c \\
% \hline
% 03040405 & 4141 & V_{cs}^* V_{cb} & 1 & C_1^c \\
% & 1111 & & 1 & C_2^c \\
% \hline
% 03050707 & 3103 & V_{ts}^* V_{tb} & 1 & C_3^t - C_3^c \\
% & 4113 & & 1 & C_4^t - C_4^c \\
% & 3505 & & 1 & C_5^t - C_5^c \\
% & 4515 & & 1 & C_6^t - C_6^c \\
% \hline
% 03050000 & 3422 & V_{ts}^* V_{tb} & \frac{e m_b}{g_s^2} & C_7^t - C_7^c \\
% & 4421 & & \frac{m_b}{g_s } & C_8^t - C_8^c \\
% \hline
% 03051717 & 3103 & V_{ts}^* V_{tb} & \frac{e^2}{g_s^2} & C_9^t - C_9^c \\
% & 31?? & & \frac{e^2}{g_s^2} & C_{10}^t - C_{10}^c \\
% \hline
% \end{array}
% \end{displaymath}
% %
% The ``??'' in the last entry should be the index number for
% $\gamma^\mu \gamma_5$, which is not defined in Table 2.
% %
% \item Ref.~\cite{Okada:1999zk} ($\mu\to e e e$): $A=-4G_F/\sqrt{2}$.
% %
% \begin{displaymath}
% \begin{array}{|ccccc|}
% \hline
% ijkl & xy & V & P & C \\
% \hline
% 13111111 & 0101 & 1 & 1 & g_1 \\
% & 0202 & & 1 & g_2 \\
% & 3232 & & 1 & g_3 \\
% & 3131 & & 1 & g_4 \\
% & 3231 & & 1 & g_5 \\
% & 3132 & & 1 & g_6 \\
% \hline
% 13110000 & 3322 & 1 & m_\mu & A_R \\
% & 3422 & & m_\mu & A_L \\
% \hline
% \end{array}
% \end{displaymath}
% %
% \item Ref.~\cite{Ciuchini:1998ix} ($K-\bar{K}$ mixing): $A=-1$.
% %
% \begin{displaymath}
% \begin{array}{|ccccc|}
% \hline
% ijkl & xy & V & P & C \\
% \hline
% 01030103 & 3131 & 1 & 1 & C_1 \\
% & 3232 & & 1 & \bar{C}_1 \\
% & 0101 & & 1 & C_2 + \frac{1}{3}C_3 \\
% & 1111 & & 1 & 2 C_3 \\
% & 0202 & & 1 & \bar{C}_2 + \frac{1}{3}\bar{C}_3 \\
% & 1212 & & 1 & 2 \bar{C}_3 \\
% & 0102 & & 1 & C_4 + \frac{1}{3}C_5 \\
% & 1112 & & 1 & 2 C_5 \\
% \hline
% \end{array}
% \end{displaymath}
% %
% Here, I have used the identity
% $
% (\bar{d}^\alpha P_L s_\beta)(\bar{d}^\beta P_L s_\alpha)
% =
% 2 (\bar{d} P_L T^a s)(\bar{d} P_L T^a s)
% + \frac{1}{3}(\bar{d}^\alpha P_L s_\alpha)(\bar{d}^\beta P_L s_\beta)
% $,
% although I do not know if this conversion is valid for the higher order
% terms.
% If the color contraction like
% $(\bar{d}^\alpha P_L s_\beta)(\bar{d}^\beta P_L s_\alpha)$ is allowed
% as Uli suggested, the encoding will be more straightforward.
% \end{itemize}
% }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
\bibitem{Mahmoudi:2007vz}
F.~Mahmoudi,
Comput.\ Phys.\ Commun.\ {\bf 178} (2008) 745
[arXiv:0710.2067];
%%CITATION = CPHCB,178,745;%%
Comput.\ Phys.\ Commun.\ {\bf 180} (2009) 1579
[arXiv:0808.3144];
%%CITATION = CPHCB,180,1579;%%
Comput.\ Phys.\ Commun.\ {\bf 180} (2009) 1718.\\
%%CITATION = CPHCB,180,1718;%%
Code website: \url{http://superiso.in2p3.fr}.
\bibitem{Degrassi:2007kj}
G.~Degrassi, P.~Gambino and P.~Slavich,
%``SusyBSG: a fortran code for BR[B -> Xs gamma] in the MSSM with Minimal
%Flavour Violation,''
Comput.\ Phys.\ Commun.\ {\bf 179} (2008) 759
[arXiv:0712.3265].\\
%%CITATION = CPHCB,179,759;%%
Code website: \url{http://slavich.web.cern.ch/slavich/susybsg}.
\bibitem{sufla} P. Paradisi, talk given at ``Interplay of Collider and Flavour Physics, 3rd general meeting'', CERN, December 2009.
\bibitem{Belanger:2008sj}
G.~B\'elanger, F.~Boudjema, A.~Pukhov and A.~Semenov,
%``Dark matter direct detection rate in a generic model with micrOMEGAs2.1,''
Comput.\ Phys.\ Commun.\ {\bf 180} (2009) 747
[arXiv:0803.2360].\\
%%CITATION = CPHCB,180,747;%%
Code website: \url{http://wwwlapp.in2p3.fr/lapth/micromegas}.
\bibitem{Arbey:2009gu}
A.~Arbey and F.~Mahmoudi,
%``SuperIso Relic: A program for calculating relic density and flavour physics
%observables in Supersymmetry,''
Comput.\ Phys.\ Commun.\ {\bf 181} (2010) 1277
[arXiv:0906.0369].\\
%%CITATION = ARXIV:0906.0369;%%
Code website: \url{http://superiso.in2p3.fr/relic}.
\bibitem{Heinemeyer:1998yj}
S.~Heinemeyer, W.~Hollik and G.~Weiglein,
%``FeynHiggs: a program for the calculation of the masses of the neutral
%CP-even Higgs bosons in the MSSM,''
Comput.\ Phys.\ Commun.\ {\bf 124} (2000) 76
[hep-ph/9812320];
% S.~Heinemeyer, W.~Hollik and G.~Weiglein,
%``The Masses of the neutral CP - even Higgs bosons in the MSSM: Accurate
%analysis at the two loop level,''
Eur.\ Phys.\ J.\ C {\bf 9} (1999) 343
[arXiv:hep-ph/9812472].\\
%%CITATION = CPHCB,124,76;%%
Code website: \url{http://www.feynhiggs.de}.
%\cite{Heinemeyer:1998np}
\bibitem{Lee:2003nta}
J.~S.~Lee, A.~Pilaftsis, M.~S.~Carena, S.~Y.~Choi, M.~Drees, J.~R.~Ellis and C.~E.~M.~Wagner,
%``CPsuperH: A computational tool for Higgs phenomenology in the minimal
%supersymmetric standard model with explicit CP violation,''
Comput.\ Phys.\ Commun.\ {\bf 156} (2004) 283
[hep-ph/0307377].\\
%%CITATION = CPHCB,156,283;%%
Code website: \url{http://www.hep.man.ac.uk/u/jslee/CPsuperH.html}.
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U.~Ellwanger and C.~Hugonie,
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\end{thebibliography}
\end{document}