The angle
of the CKM triangle can be measured comparing
and
mediated transitions in
decays. The decays proceed through the following diagrams:
These diagrams are practically free from the New Physics contribution.
There are three methods to extract relevant information, each of them deals with its own
decay:
- a singly Cabibbo-suppressed CP eigenstate, like for Gronau-London-Wyler (GLW) method;
- a doubly Cabibbo-suppressed flavor eigenstate, like for Atwood-Dunietz-Soni (ADS) method;
- a Cabibbo-allowed self-conjugate 3-body state, like for Giri-Grossman-Soffer-Zupan (GGSZ) method.
Generally, the observables of the methods also depend on the amplitude ratio
and the relative
conserving phase
between the two amplitudes. These parameters depend on the
decay under investigation.
Gronau London Wyler method:
The Gronau-London-Wyler (GLW) method (
M. Gronau D. Wyler Phys.Lett. B265 (1991) 172;
M. Gronau, D. London, Phys.Lett. B253 (1991) 483) is based on the reconstruction of the
decay to
, where
and
decay to
-even or
-odd eigenstates. The
modes normally used are:
For the normalization,
, with
is also reconstructed.
The four observables for this method are formed in the following way:
This set can provide an information on
,
, and
with an 8-fold ambiguity for the phases.
Atwood Dunitz Soni Method:
In the ADS method,
D. Atwood, I. Dunietz and A. Soni, Phys. Rev. Lett. 78, 3257 (1997),
is measured from the study of
decays, where
mesons decay into non
eigenstate final states. The suppression of
transition with respect to the
one is partly overcome by the study of decays of the
meson in final states which can proceed in two ways: either through a favored
decay followed by a doubly-Cabibbo-suppressed
decay, or through a suppressed
decay followed by a Cabibbo-favored
decay.
Neglecting
-mixing effects, which in the SM give very small corrections to \g\ and do not affect the
measurement, the measured ratios
and
are related to the
and
mesons' decay parameters through the following relations:
with:
In case of the
analysis with
we use the following ratios:
The used observables are connected to the "classical"
and
set by simple relations:
and
.
The values of
and
are taken from
our study of charm mixing or the
CLEO-c collaboration results. The ratio
has been measured in different experiments and we take the average value from
PDG.
Giri Grossman Soffer Zupan (GGSZ) method:
The Giri Grossman Soffer Zupan (GGSZ), also called Dalitz method (
A. Giri, Y. Grossman, A. Soffer and J. Zupan, Phys. Rev. D 68, 054018 (2003)) is based on the reconstruction of the
decay to
, where
and
decay
;
The four observables for this method are formed in the following way: