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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
% }
% K.~Agashe$^{1}$,
, A.~Arbey$^{3}$
% G.~Belanger$^{2}$,
, A.~Bharucha$^{4}$, \\
% F.~Boudjema$^{2}$,
% }%
T.~Goto$^{4}$
, T. Hahn$^{5}$
, U. Haisch$^{6}$
, S.~Kraml$^{7}$
, M.~Muhlleitner$^{8}$, \\
% , W.~Reece$^{8}$,
, J.~Reuter$^{9}$
, P.~Skands$^{10}$
, P.~Slavich$^{11}$
%
}
\vspace*{0.5cm}
{\sl
$^1$Clermont Universit\'e, Universit\'e Blaise Pascal, CNRS/IN2P3,\\
LPC, BP 10448, 63000 Clermont-Ferrand, France
\vspace*{0.2cm}
$^2$Instituto de F\'isica de Cantabria (CSIC-UC), Santander, Spain\\
\vspace*{0.2cm}
$^3$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*{0.2cm}
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}
\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 using 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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
Advanced programs dedicated to the calculation of flavour related
observables (Wilson coefficients, branching ratios, mixing amplitudes,
renormalization group equation (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 exists 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 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, 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 the FLHA
input/output is \emph{technically} achieved. The details of how to
`switch on' the 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 the 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, in blocks
\texttt{VCKMIN} and \texttt{UPMNSIN}. The real and
imaginary parts of the $\overline{\rm{DR}}$ CKM matrix can also be given
in \texttt{VCKM} and \texttt{IMVCKM}, respectively. The format of the
individual entries is the same as for the mixing matrices in the SLHA1.
Analogous blocks are defined for the neutrino sector, called
\texttt{UPMNS} and \texttt{IMUPMNS}.\\
%\htr{NM: To be expanded?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Wilson coefficients}
\label{wcoeff}
The real and imaginary parts of the Wilson coefficients are given in \texttt{FWCOEF}
and \texttt{IMFWCOEF}, 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
\cite{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}
The couplings should therefore not be included in the Wilson coefficients.\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 the general structure for the SLHA~\cite{slha1,slha2} 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, 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 and
to 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. The order
of the 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}.\\
\bigskip
\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.
\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 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 flavour related 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 FFORM}:
Form Factors.\\
\end{itemize}
%
More details on each block are 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 the SLHA.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK MODSEL}}
%\htr{SH: following Peter's suggestion}\\
This block provides switches and options for the model selection.
% We follow excactly the SLHA2 format, but extend it to allow for more flexibility.
The SLHA2 \texttt{BLOCK MODSEL} is extended to allow for more flexibility \cite{xxx} and here we follow exactly this newly extended format.
%The entries in this block consist of an index, followed by another index
%or possibly followed by a string
%minimal type of model, while the string can give more precise
%definitions (for non-SUSY models). The second index can define a special
%particle content of the model.
\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:\\
\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{31}{THDM}
\snumentry{99}{other model. This choice requires a string given in the
entry \ttt{99}}
}\\
%
\numentry{3}{(Default=0) Choice of particle content, only used for SUSY models. The defined switches are:\\
\snumentry{0}{MSSM}
\snumentry{1}{NMSSM}
\snumentry{2}{...}
}\\
%
\numentry{4}{(Default=\ttt{0}) R-parity violation. Switches defined are:\\
\snumentry{0}{R-parity conserved. This corresponds to the SLHA1.}
\snumentry{1}{R-parity violated.
% The blocks defined in section~\ref{sec:rpv} should be present.
}
}\\
%
\numentry{5}{(Default=\ttt{0}) CP violation. Switches defined are:\\
\snumentry{0}{CP is conserved. No information even 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{31}{defines the type of THDM, is used only if entry~\ttt{1} is
given as~31, otherwise it is ignored.\\
\snumentry{1}{type I}
\snumentry{2}{type II}
\snumentry{3}{type III}
\snumentry{4}{type IV}
}
%
\numentry{99}{a string that defines other models, is used only if
entry~\ttt{1} is given as~99, otherwise it is ignored.}
%\numentry{1}{
%\numentry{-99}{Non-SUSY Model (\texttt{MODSEL} entry 2 must be present)}
%\numentry{-1}{SM}
%\numentry{0}{General MSSM}
%\numentry{1}{(m)SUGRA model}
%\numentry{2}{(m)GMSB model}
%\numentry{3}{(m)AMSB model}
%\numentry{4}{\ldots}
%}
%An extended and updated list will be available at the FLHA web page.\\
%
%\numentry{2}{
%String specifying model name. Mandatory when \texttt{MODSEL}(1) = -99, ignored
%otherwise (but can of course still be used for information purposes for
%other \texttt{MODSEL}(1) values). The FLHA web site will provide a list
%of which ones have so far been oficially defined.}\\
%
%\numentry{3}{(Default=0) Choice of particle content
% (ignored when \texttt{MODSEL}(1) = -99)\\
%\numentry{0}{MSSM (this corresponds to SLHA1).}
%\numentry{1}{NMSSM (this corresponds to SLHA2).}\\
%}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK SMINPUTS}}
The block \texttt{BLOCK SMINPUTS} contains the 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.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}}
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 renormalization 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 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}}
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 9--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.}
}
More details and definitions can be found in appendix~\ref{app:decayconst}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FCONSTRATIO}}
The block \texttt{BLOCK FCONSTRATIO} contains 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,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 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}}
The block \texttt{BLOCK FBAG} contains the 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, and the double
precision number its Bag parameter.
So far no normalization etc.\ has been defined, which at this stage has
to be taken care of by the user. An unambiguous definition will be given
elsewhere.
%\\
%\htr{NM: We should provide the list of the main B parameters (similarly to
% the decay constants) and propose an implementation in the block.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}, respecting the conventions
given in section \ref{wcoeff}. % and appendix \ref{app:operators}.
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 correspond to a unique identifier
for any possible Wilson coefficient. 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}.
\begin{table}[!t]
\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 \\
$\sum_q$ & & 07 & $\sum_l$ & & 17\\
\hline
\end{tabular}
\caption{PDG codes and two-digit number identifications of quarks and leptons.\label{tab:pdgcodes}}
\end{center}
\end{table}
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}.
\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c||c|c|}
\hline
operator & two-digit number &
operator & two-digit number \\
\hline
$P_L$ & 01 & $P_L T_a$ & 11 \\
$P_R$ & 02 & $P_R T_a$ & 12 \\
$\gamma^\mu$ & 03 & $\gamma^\mu T_a$ & 13 \\
$\sigma^{\mu\nu}$ & 04 & $\sigma^{\mu\nu} T_a$ & 14 \\
$\gamma^\mu \gamma^\nu \gamma^\rho$ & 05 &
$\gamma^\mu \gamma^\nu \gamma^\rho T_a$ & 15 \\
$\gamma^\mu P_L$ & 31 & $\gamma^\mu T_a P_L$ & 41 \\
$\gamma^\mu P_R$ & 32 & $\gamma^\mu T_a R_R$ & 42 \\
$\sigma^{\mu\nu} P_L$ & 33 & $\sigma^{\mu\nu} T_a P_L$ & 43 \\
$\sigma^{\mu\nu} P_R$ & 34 & $\sigma^{\mu\nu} T_a P_R$ & 44 \\
$\gamma^\mu \gamma^\nu \gamma^\rho P_L$ & 35 &
$\gamma^\mu \gamma^\nu \gamma^\rho T_a P_L$ & 45 \\
$\gamma^\mu \gamma^\nu \gamma^\rho P_R$ & 36 &
$\gamma^\mu \gamma^\nu \gamma^\rho T_a P_R$ & 46 \\
$F_{\mu\nu}$ & 22 & $G_{\mu\nu}^a$ & 21\\
\hline
\end{tabular}
\caption{Two-digit number definitions for the operators.\label{tab:opcodes}}
\end{center}
\end{table}
Here $T^a$ ($a = 1 \ldots 8$) denote the $SU(3)_C$ generators, and
$P_{L,R} = \frac{1}{2} (1 \mp \gamma_5)$.
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~4141.
%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}
The third 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}
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 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, 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}{NPM}
\numentry{2}{SM+NPM}
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 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}}
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 (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 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 FOBSFIT}}
%
% The block \texttt{BLOCK FOBSFIT} contains the fitted values of the flavour
% observables with the same structure as in \texttt{BLOCK FOBS}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\texttt{BLOCK FFORM}}
The block \texttt{BLOCK FFORM} contains the 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. The dependence on $q^2$ can be specified as a comment.
A more unambigous definition will be given elsewhere.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection*{\texttt{BLOCK FSHAPE}}
%
% The block \texttt{BLOCK FSHAPE} contains the 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.
% A more unambigous definition will be given elsewhere.
%\\
%\htr{NM: Do we want to define a ``miscellaneous'' block containing form
% factors, shape factors, etc, instead of defining separate blocks for
% each?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
The interplay of collider and flavour physics is entering a new era 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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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'.
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|}
%\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 & 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 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 the 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{Effective Operators}
\label{app:operators}
%xxx
Here we give a list of the most relevant effective operators together
with their unique two number identifier.
\begin{align}
O_1 &= (\bar{s} \gamma_{\mu} T^a P_L c)
(\bar{c} \gamma^{\mu} T_{a} P_L b) &: 03040405~4141 ,\nonumber\\[4mm]
O_2 &= (\bar{s} \gamma_{\mu} P_L c)
(\bar{c} \gamma^{\mu} P_L b) &: 03040405~3131 ,\nonumber\\[3mm]
O_3 &= (\bar{s} \gamma_{\mu} P_L b)
{\displaystyle\sum_q} (\bar{q} \gamma^{\mu} q)
&: 03050707~3103 ,\nonumber\\[1mm]
O_4 &= (\bar{s} \gamma_{\mu} T^a P_L b)
{\displaystyle\sum_q} (\bar{q} \gamma^{\mu} T_{a} q)
&: 03050707~4113 ,\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)
&: 03050707~3505,\\[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)
&: 03050707~4515 ,\nonumber\\[1mm]
O_7 &= (O_{\gamma})= \dfrac{e}{16\pi^2} \left[ \bar{s} \sigma^{\mu \nu}
(m_b P_R) b \right] F_{\mu \nu} &: 03050000~3422,\nonumber\\[2mm]
O_8 &= (O_g)= \dfrac{g}{16\pi^2} \left[ \bar{s} \sigma^{\mu \nu}
(m_b P_R) T_{a} b \right] G_{\mu \nu}^a &: 03050000~4421.\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 ($P=\pi^\pm$, $K$, $D$, $B$), 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 $\eta'$, 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}
%
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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%NNLO,''
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%
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%
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% %$s^{)} \to \ell^{+} \ell^{-}$ and $\bar{B} \to K \ell^{+} \ell^{-}$,''
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% [hep-ph/0104284].
% %%CITATION = PHRVA,D64,074014;%%
%
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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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% [arXiv:0907.5376].
% %%CITATION = ARXIV:0907.5376;%%
%
% \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,''
% JHEP {\bf 0404} (2004) 071
% [arXiv:hep-ph/0312090].
% %%CITATION = JHEPA,0404,071;%%
%
% \bibitem{Gabbiani:1996hi}
% F.~Gabbiani, E.~Gabrielli, A.~Masiero and L.~Silvestrini,
% %``A complete analysis of FCNC and CP constraints in general SUSY
% %extensions
% %of the standard model,''
% Nucl.\ Phys.\ B {\bf 477} (1996) 321
% [arXiv:hep-ph/9604387].
% %%CITATION = NUPHA,B477,321;%%
%
% \bibitem{Buras:2001ra}
% A.~J.~Buras, S.~Jager and J.~Urban,
% %``Master formulae for Delta(F) = 2 NLO-QCD factors in the standard
% %model and
% %beyond,''
% Nucl.\ Phys.\ B {\bf 605} (2001) 600
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% %%CITATION = NUPHA,B605,600;%%
%
% \bibitem{Kitano:2000fg}
% R.~Kitano and Y.~Okada,
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% Phys.\ Rev.\ D {\bf 63} (2001) 113003
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% %%CITATION = PHRVA,D63,113003;%%
%
% \bibitem{Kitano:2002mt}
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% %%CITATION = PHRVA,D66,096002;%%
\end{thebibliography}
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 ``$P$''.\\
\\
\htr{NM: 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}...
}