A Mathematica based Version of the CKMfitter Package

22 Chapter 4. A Mathematica based Version of the CKMfitter Package
4.5 Theory Packages
Theory packages provide the full physical information on the fit variables. Since no
global variables are used in Mathematica, all variables are initialized only in the context
of their theory package. For each observable, the theoretical expression xtheo
is defined and the partial derivatives ∂xtheo/∂ymod with respect to all model parameters
ymod are symbolically calculated. Finally, all expressions are written into
a list named “theory”, which is loaded from the analysis notebook. Theory packages
are stored in a binary format (.mx), that is optimized for input by Mathematica.
It is possible to define an observable in different theoretical frameworks, e. g. the B 0 -
¯B 0 oscillations in the Standard Model or in a New Physics model (see Chapter 6).
Thus, a theory package can include several theory lists, one for each theoretical
framework or possible different parametrization. Different theory lists in the same
theory package are distinguished by version labels, which have to be specified in the
datacard.
It is advantageous for the fit to store all expressions and derivatives in their simplest
form. This can be obtained using the intrinsic Mathematica functions Simplify and
FullSimplify. For complex expressions, this may lead to a very high CPU time
consumption during the theory package development. More detailed information on
the development of theory packages are given in a tutorial, which accrued during
this work. It is available in Appendix E.4.
In the following, four important theory packages, which are related to this work, are
introduced.
4.5.1 CKMmatrix
The theory package CKMmatrix is the basis of the Mathematica based CKMfitter
package and was coded by Jérôme Charles. It contains all relevant definitions
of CKM matrix and Unitarity Triangle parameters in the exact Wolfenstein
parametrization, e. g. the real and imaginary parts of the CKM matrix elements,
Re (Vij) and Im (Vij), and the UT angles α, β, γ.
An important object is the wolfCKM function. This is a list of rules and provides
the real and imaginary parts of all CKM matrix elements Re (Vij) and Im (Vij).
Its argument is the order of the expansion into Wolfenstein parameters, where “∞”
means the exact expressions up to all orders.
The CKMmatrix package needs to be sourced from each theory package, where the
Wolfenstein parameters are used or the wolfCKM function is evaluated.