A Mathematica based Version of the CKMfitter Package

14 Chapter 3. CKMfitter
The quantity minimized in the fit is
χ 2 = −2 ln L(ymod) , (3.1)
with the likelihood function L(ymod), defined by a product of contributions of two
types:
L(ymod) = Lexp(xexp − xtheo(ymod)) · Ltheo(yQCD) . (3.2)
The experimental likelihood Lexp depends on the experimental measurements xexp,
which are gaussian distributed in general, and their theoretical predictions xtheo,
which are functions of the model parameters ymod. The theoretical likelihood Ltheo
describes the knowledge on the QCD parameters yQCD ∈ {ymod}, where the theoretical
uncertainties σsyst are considered to define allowed ranges:
[yQCD − σsyst, yQCD + σsyst] . (3.3)
In the Rfit scheme, the theoretical likelihoods Ltheo(i) do not contribute to the χ 2
of the fit, as long as the yQCD take on values within their.
3.2 Fit Metrology
is determined with
In a first step, the global minimum of Equation (3.1), χ2 min,global
respect to all Nmod parameters. Due to the experimental and theoretical systematics,
this absolute minimal value does in general not correspond to a unique ymod location.
In a second step, a selected subspace of interest of the parameter space, e. g. a =
{¯ρ, ¯η} is scanned, to determin the local χ2-minimum χ2 min,local (a) for each fixed point
of a grid in the parameter space a, with respect to the remaining parameters. The
offset-corrected χ2 is calculated as follows:
∆χ 2 (a) = χ 2 min,local (a) − χ2 min,global
where its minimum is equal to zero by construction.
, (3.4)
Finally, a confidence level (CL) for a is obtained using the well-known PROB function
from the CERN Program Library [18]:
1 − CL = P rob � ∆χ 2 �
(a), Ndof
=
which assumes gaussian statistics.
1
√ 2 Ndof Γ (Ndof /2)
� ∞
χ 2 (ymod)
(3.5)
e −t/2 t Ndof /2−1 dt , (3.6)