(r5) GammaFromTrees < UTfit

Gamma Combination from the UTfit goup

Input parameters and combination results: Winter 2012 (post-Moriond12)

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]
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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]
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Full fit result for $\,r_{B}(DK)$

$0.1016 \pm 0.0074$ 95% prob:[0.0866, 0.1163] 99% prob:[0.079, 0.1238]
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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]
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Full fit result for $\,r_{B}(DK^{*})$

$0.123 \pm 0.056$ 95% prob:[0, 0.21] 99% prob:[0, 0.263]
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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]
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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

Methods used:

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

conserving phase $\delta_{B}$ 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. 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

decay to $D^{0} K$ , where $D^{0}$ and $\bar D^{0}$ decay to

-even or

-odd eigenstates. The

modes normally used are:

: $K^{+}K^{-}$ , $\pi^{+}\pi^{-}$ ;
: $K_S\pi^0$ , $\phi K_S$ , $\eta K_S$ , $\rho K_S$ , 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:

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

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