Gamma Combination from the UTfit goup

We use the relevant post-Moriond 2012 HFAG averages as inputs.

The results of combination:

Parameter Full fit
\gamma [^{\circ}] -108.2 \pm 9.6 \text{ and } 71.7 \pm 9.0
\delta_{B}(DK) [^{\circ}] -64 \pm 11 \text{ and } 116 \pm 11
r_{B}(DK) 0.1016 \pm 0.0074
\delta_{B}(DK^{*}) [^{\circ}] -54 \pm 35 \text{ and } 125 \pm 35
r_{B}(DK^{*}) 0.123 \pm 0.056
\delta_{B}(D^{*}K) [^{\circ}] -51 \pm 14 \text{ and } 129 \pm 14
r_{B}(D^{*}K) 0.117 \pm 0.019




Full fit result for \,\gamma [^{\circ}]
-108.2 \pm 9.6 \text{ and } 71.7 \pm 9.0
95% prob:[-126, -91.] U [52.7, 89.1]
99% prob:[-135, -83.] U [44.4, 96.3]
EPS - PDF - PNG - JPG - GIF




Full fit result for \,\delta_{B}(DK) [^{\circ}]
-64 \pm 11 \text{ and } 116 \pm 11
95% prob:[-86.4, -44.9] U [93.7, 135.2]
99% prob:[-96.6, -37.1] U [83.3, 142.8]
EPS - PDF - PNG - JPG - GIF




Full fit result for \,r_{B}(DK)
0.1016 \pm 0.0074
95% prob:[0.0866, 0.1163]
99% prob:[0.079, 0.1238]
EPS - PDF - PNG - JPG - GIF




Full fit result for \,\delta_{B}(DK^{*}) [^{\circ}]
-54 \pm 35 \text{ and } 125 \pm 35
95% prob:[-180, -178.] U [-143, 2.5] U [36.6, 180]
99% prob:[-180, -165.] U [-160., 180]
EPS - PDF - PNG - JPG - GIF




Full fit result for \,r_{B}(DK^{*})
0.123 \pm 0.056
95% prob:[0, 0.21]
99% prob:[0, 0.263]
EPS - PDF - PNG - JPG - GIF




Full fit result for \,\delta_{B}(D^{*}K) [^{\circ}]
-51 \pm 14 \text{ and } 129 \pm 14
95% prob:[-83.2, -25.6] U [97.1, 154.3]
99% prob:[-106., -14.1] U [73.4, 166.1]
EPS - PDF - PNG - JPG - GIF




Full fit result for \,r_{B}(D^{*}K)
0.117 \pm 0.019
95% prob:[0.077, 0.152]
99% prob:[0.051, 0.168]
EPS - PDF - PNG - JPG - GIF

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.

The Gronau-London-Wyler (GLW) method (M. Gronau, D. Wyler, Phys. Rev. Lett. B {\bf 253} (1991) 483; M. Gronau, D. London, Phys. Rev. Lett. B {\bf 265} (1991) 172) 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.

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