The angle \gamma of the CKM triangle can be measured comparing V_{cb} and V_{ub} mediated transitions in B \rightarrow D^{(*)}K^{(*)} decays. The decays proceed through the following diagrams:
Gamma From Trees 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 D^{0} decay:

  • a singly Cabibbo-suppressed CP eigenstate, like D^{0}\rightarrow h^+h^- for Gronau-London-Wyler (GLW) method;
  • a doubly Cabibbo-suppressed flavor eigenstate, like D^{0}\rightarrow K^+\pi^- for Atwood-Dunietz-Soni (ADS) method;
  • a Cabibbo-allowed self-conjugate 3-body state, like D^{0} \rightarrow K_{S}\pi^{+}\pi^{-} for Giri-Grossman-Soffer-Zupan (GGSZ) method.
Generally, the observables of the methods also depend on the amplitude ratio r_{B}\equiv\frac{A( b\to u)}{A( b\to c)} and the relative CP conserving phase\delta_{B} between the two amplitudes. These parameters depend on the B 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 B decay to D^{0} K, where D^{0} and \bar D^{0} decay to CP-even or CP-odd eigenstates. The D modes normally used are:
  • CP+: K^{+}K^{-}, \pi^{+}\pi^{-};
  • CP-: K_S\pi^0, \phi K_S, \eta K_S, \rho K_S, and \omega K_{S}.
For the normalization, B^{+} \rightarrow \bar D^{0} K^{+}, with \bar D^0 \rightarrow K^+ \pi^- is also reconstructed.

The four observables for this method are formed in the following way:

R_{CP^{\pm}}=\frac{\Gamma(B^{+}\rightarrow D^0_{\pm}K^+)+\Gamma(B^-\rightarrow D^0_{\pm} K^{-})}{\Gamma(B^{+}\rightarrow D^0 K^+)+\Gamma(B^-\rightarrow \bar D^0 K^{-})}=1+r_{B}^2\pm 2 r_{B}\cos\gamma\cos\delta_{B},

A_{CP^{\pm}}=\frac{\Gamma(B^{+}\rightarrow D^{0}_{\pm} K^{+})-\Gamma(B^{-}\rightarrow D^{0}_{\pm} K^{-})}{\Gamma(B^{+}\rightarrow D^{0}_{\pm} K^{+})+\Gamma(B^{-}\rightarrow D^{0}_{\pm} K^{-})}=\frac{\pm 2 r_{B} \sin\gamma\sin\delta_{B}}{R_{CP^{\pm}}}.

This set can provide an information on \gamma, \delta_{B}, and r_{B} 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), \gamma is measured from the study of B\rightarrow DK decays, where D mesons decay into non CP eigenstate final states. The suppression of b\rightarrow u transition with respect to the b \rightarrow c one is partly overcome by the study of decays of the B meson in final states which can proceed in two ways: either through a favored b \rightarrow c B decay followed by a doubly-Cabibbo-suppressed D decay, or through a suppressed b \rightarrow u B decay followed by a Cabibbo-favored D decay.

Neglecting D-mixing effects, which in the SM give very small corrections to \g\ and do not affect the r_{B} measurement, the measured ratios R_{ADS} and A_{ADS} are related to the B and D mesons' decay parameters through the following relations:

R_{\rm ADS}=\frac{\Gamma(B^{+}\rightarrow [\bar f]_{D^{0}} K^{+})+\Gamma(B^{-}\rightarrow [f]_{D^{0}} K^{-})}{\Gamma(B^{+}\to[f]_{D^{0}} K^{+})+\Gamma(B^{-}\to[\bar f]_{D^{0}} K^{-})}=(r_{B}^2+r_{D}^2+2 r_{B} r_{D} k_{D}k_{B}\cos\gamma\cos\delta),

A_{\rm ADS}=\frac{\Gamma(B^{+}\rightarrow [\bar f]_{D^{0}} K^{+})-\Gamma(B^{-}\rightarrow [f]_{D^{0}} K^{-})}{\Gamma(B^{+}\to[f]_{D^{0}} K^{+})+\Gamma(B^{-}\to[\bar f]_{D^{0}} K^{-})}=(r_{B}^2+r_{D}^2+2 r_{B} r_{D} k_{D}k_{B}\sin\gamma\sin\delta)/R_{\rm ADS},

with:

r_{D}^2 \equiv\frac{\Gamma(D^0\to f)}{\Gamma(D^{0}\to \bar f)}=\frac{\int dm\, A_{\rm DCS}(m)}{\int dm\, A_{\rm CA}(m)},

k_{D} e^{i \delta_{D}}= \frac{\int dm\, A_{\rm DCS}(m)A_{\rm CA}e^{i\delta(m)}}{\sqrt{\int dp\, A_{\rm DCS}^2(p)\int dp\, A_{\rm CA}^2(p)}},

In case of the B\to D^{0} K analysis with D^{0}\to K\pi\pi^0 we use the following ratios:

R^{\pm}=\frac{\Gamma(B^{\pm}\rightarrow [\bar f]_{D^{0}} K^{\pm})}{\Gamma(B^{\pm}\to[f]_{D^{0}} K^{\pm})}=(r_{B}^2+r_{D}^2+2 r_{B} r_{D} k_{D}k_{B}\cos(\gamma\pm\delta)),

The used observables are connected to the "classical" R_{\rm ADS} and A_{\rm ADS} set by simple relations: R_{\rm ADS}=\frac{R^{+}+R^{-}}{2} and A_{\rm ADS}=\frac{R^{-}-R^{+}}{R^{-}+R^{+}}.

The values of k_{D} and \delta_D are taken from our study of charm mixing or the CLEO-c collaboration results. The ratio r_D 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 B decay to D^{0} K, where D^{0} and \bar D^{0} decay K_S^{0}\pi^{+}\pi^{-};

The four observables for this method are formed in the following way:

x_{\pm}=r_{B}\cos(\gamma\pm\delta_{B}),

y_{\pm}=r_{B}\sin(\gamma\pm\delta_{B}),
 
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