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: