(r12) GammaFromTrees < UTfit

Gamma Combination from the UTfit goup

Input parameters and combination results: Summer 2012 (post-ICHEP12)

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

The results of combination:

Parameter	Full fit
$\gamma [^{\circ}]$	$-107.7 \pm 9.4 \text{ and } 72.6 \pm 9.4$
$\delta_{B}(DK) [^{\circ}]$	$-63 \pm 11 \text{ and } 117 \pm 11$
$r_{B}(DK)$	$0.1011 \pm 0.0078$
$\delta_{B}(DK^{*}) [^{\circ}]$	$-90.55 \pm 0.75 \text{ and } -54.5 \pm 34.6 \text{ and } 125.75 \pm 34.35$
$r_{B}(DK^{*})$	$0.123 \pm 0.057$
$\delta_{B}(D^{*}K) [^{\circ}]$	$-51 \pm 14 \text{ and } 128 \pm 14$
$r_{B}(D^{*}K)$	$0.117 \pm 0.019$
$\delta_{B0}(DK^{0}) [^{\circ}]$	$-57 \pm 48 \text{ and } 78 \pm 2 \text{ and } 126 \pm 43$
$r_{B0}(DK^{0})$	$0.256 \pm 0.059$

Full fit result for $\,\gamma [^{\circ}]$

$-107.7 \pm 9.4 \text{ and } 72.6 \pm 9.4$ 95% prob:[-126, -90.] U [53.4, 89.8] 99% prob:[-135, -82.] U [44.9, 97.4]
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Full fit result for $\,\delta_{B}(DK) [^{\circ}]$

$-63 \pm 11 \text{ and } 117 \pm 11$ 95% prob:[-85.9, -44.3] U [94.4, 136.1] 99% prob:[-96.1, -36.4] U [83.9, 143.5]
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Full fit result for $\,r_{B}(DK)$

$0.1011 \pm 0.0078$ 95% prob:[0.086, 0.1161] 99% prob:[0.0783, 0.1238]
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Full fit result for $\,\delta_{B}(DK^{*}) [^{\circ}]$

$-90.55 \pm 0.75 \text{ and } -54.5 \pm 34.6 \text{ and } 125.75 \pm 34.35$ 95% prob:[-180, -177.] U [-144., 2.5] U [35.4, 180] 99% prob:[-180, -165.] U [-161., 180]
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Full fit result for $\,r_{B}(DK^{*})$

$0.123 \pm 0.057$ 95% prob:[0, 0.21] 99% prob:[0, 0.263]
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Full fit result for $\,\delta_{B}(D^{*}K) [^{\circ}]$

$-51 \pm 14 \text{ and } 128 \pm 14$ 95% prob:[-83, -25.6] U [96.6, 154.4] 99% prob:[-106., -13.9] U [72.7, 166.2]
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Full fit result for $\,r_{B}(D^{*}K)$

$0.117 \pm 0.019$ 95% prob:[0.076, 0.152] 99% prob:[0.051, 0.168]
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Full fit result for $\,\delta_{B0}(DK^{0}) [^{\circ}]$

$-57 \pm 48 \text{ and } 78 \pm 2 \text{ and } 126 \pm 43$ 95% prob:[-180, -162.] U [-155., 8.6] U [28.5, 177.2] 99% prob:[-180, 18.5] U [18.6, 178.1]
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Full fit result for $\,r_{B0}(DK^{0})$

$0.256 \pm 0.059$ 95% prob:[0.12, 0.369] 99% prob:[0.011, 0.406]
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The predictions of the observables coming from the full fit:

Parameter	Full fit
$R_{CP}^{+}(DK)$	$0.986 \pm 0.014$
$A_{CP}^{+}(DK)$	$0.169 \pm 0.019$
$R_{CP}^{-}(DK)$	$1.033 \pm 0.012$
$A_{CP}^{-}(DK)$	$-0.163 \pm 0.018$
$R_{CP}^{+}(DK^{*})$	$0.989 \pm 0.035$
$A_{CP}^{+}(DK^{*})$	$0.157 \pm 0.094$
$R_{CP}^{-}(DK^{*})$	$1.032 \pm 0.043$
$A_{CP}^{-}(DK^{*})$	$-0.002 \pm 0.166$
$R_{CP}^{+}(D^{*}K)$	$1.048 \pm 0.028$
$A_{CP}^{+}(D^{*}K)$	$-0.158 \pm 0.036$
$R_{CP}^{-}(D^{*}K)$	$0.978 \pm 0.025$
$A_{CP}^{-}(D^{*}K)$	$0.173 \pm 0.037$
$R_{ADS}(DK,K\pi)$	$0.0154 \pm 0.0014$
$A_{ADS}(DK,K\pi)$	$-0.536 \pm 0.09$
$R_{ADS}(DK^{*},K\pi)$	$0.016 \pm 0.013$
$A_{ADS}(DK^{*},K\pi)$	$-0.36 \pm 0.26$
$R_{ADS}(D^{*}K,K\pi,\pi^{0})$	$0.0136 \pm 0.0038$
$A_{ADS}(D^{*}K,K\pi,\pi^{0})$	$0.59 \pm 0.17$
$R_{ADS}(D^{*}K,K\pi,\gamma)$	$0.0192 \pm 0.0053$
$A_{ADS}(D^{*}K,K\pi,\gamma)$	$-0.38 \pm 0.15$
$x^{+}{DK}$	$-0.0961 \pm 0.0092$
$y^{+}{DK}$	$-0.016 \pm 0.032$
$x^{-}{DK}$	$0.073 \pm 0.013$
$y^{-}(DK)$	$0.071 \pm 0.012$
$x^{+}(DK^{*})$	$-0.091 \pm 0.052$
$y^{+}(DK^{*})$	$-0.023 \pm 0.064$
$x^{-}(DK^{*})$	$0.053 \pm 0.051$
$y^{-}(DK^{*})$	$0.067 \pm 0.066$
$x^{+}(D^{*}K)$	$0.106 \pm 0.02$
$y^{+}(D^{*}K)$	$0.039 \pm 0.033$
$x^{-}(D^{*}K)$	$-0.064 \pm 0.025$
$y^{-}(D^{*}K)$	$-0.095 \pm 0.027$

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:

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

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

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 (I. Dunietz, Phys. Rev. Lett. B 270 (1991) 75; Phys. Rev. Lett. D 52 (1995) 3048; 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

mesons decay into non

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

meson in final states which can proceed in two ways: either through a favored $b \rightarrow c$

decay followed by a doubly-Cabibbo-suppressed

decay, or through a suppressed $b \rightarrow u$

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 $r_{B}$ measurement, the measured ratios $R_{ADS}$ and $A_{ADS}$ are related to the

and

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 $A_{\rm ADS}=\frac{R^{-}-R^{+}}{R^{-}+R^{+}}$ .