ResultsSummer2010PreICHEP < UTfit

Fit results: Summer 2010 (pre-ICHEP)

Parameter	Input value	Full fit	SM Prediction
$\bar{\rho}$		$0.132 \pm 0.02$
$\bar{\eta}$		$0.358 \pm 0.012$
$\rho$		$0.135 \pm 0.021$
$\eta$		$0.367 \pm 0.013$
		$0.8095 \pm 0.0095$
$\lambda$	$0.2253 \pm 0.0011$	$0.22545 \pm 0.00065$
$\|V_{ub}\|$	$0.00376 \pm 0.0002$	$0.00364 \pm 0.00011$	$0.00355 \pm 0.00014$
$\|V_{cb}\|$	$0.04083 \pm 0.00045$	$0.04117 \pm 0.00043$	$0.04269 \pm 0.00099$
$\sin\theta_{12}$		$0.22545 \pm 0.00065$
$\sin\theta_{23}$		$0.04117 \pm 0.00043$
$\sin\theta_{13}$		$0.00364 \pm 0.00011$
$\delta$		$69.7 \pm 2.9$
$m_{b},{\rm {GeV}/c^{2}}$	$4.21 \pm 0.08$
$m_{c},{\rm {GeV}/c^{2}}$	$1.3 \pm 0.1$
$m_{t},{\rm {GeV}/c^{2}}$	$163.4 \pm 1.2$	$163.4 \pm 1.2$	$163.5 \pm 9.5$
$\Delta m_{s},{\rm ps^{-1}}$	$17.77 \pm 0.12$	$17.77 \pm 0.12$	$18.3 \pm 1.3$
$\Delta m_{d},{\rm ps^{-1}}$	$0.507 \pm 0.005$
$\Delta m_{K},10^{-15}{\rm ps^{-1}}$	$1.8 \pm 1.8$
$f_{B_{s}}$	$0.239 \pm 0.01$	$0.2359 \pm 0.0056$	$0.2349 \pm 0.0067$
$f_{B_{s}}/f_{B_{d}}$	$1.23 \pm 0.03$	$1.225 \pm 0.025$	$1.213 \pm 0.044$
$B_{B_{s}}/B_{B_{d}}$	$1.06 \pm 0.04$	$1.069 \pm 0.036$	$1.113 \pm 0.085$
$B_{B_{s}}$	$0.87 \pm 0.04$	$0.845 \pm 0.036$	$0.769 \pm 0.065$
$\alpha, [^{\circ}]$	$91.4 \pm 6.1$	$87.8 \pm 3.0$	$85.4 \pm 3.7$
$\beta, [^{\circ}]$		$22.42 \pm 0.74$	$25.2 \pm 1.6$
$\sin(2\beta)$	$0.654 \pm 0.026$	$0.705 \pm 0.018$	$0.771 \pm 0.036$
$\cos(2\beta)$	$0.87 \pm 0.13$	$0.71 \pm 0.018$	$0.639 \pm 0.043$
$2\beta+\gamma, [^{\circ}]$	$-90 \pm 56 \text{ and } 94 \pm 52$	$114.7 \pm 3.1$	$114.9 \pm 3.1$
$\gamma, [^{\circ}]$	$-106 \pm 11 \text{ and } 74 \pm 11$	$69.8 \pm 3.0$	$69.6 \pm 3.1$
$\|\varepsilon_{K}\|$	$0.00222994 \pm 1.04974 \times 10^{-5}$	$0.00222854 \pm 9.98004 \times 10^{-06}$	$0.00192 \pm 0.00018$
$B(B\rightarrow\tau\nu),10^{-4}$	$1.72 \pm 0.28$	$0.867 \pm 0.078$	$0.805 \pm 0.071$
$J_{cp}\times 10^{5}$		$3.09 \pm 0.11$

The fit results for all the nine CKM elements are

${\small V_{CKM}=\left(\begin{array}{ccc} 0.97425 \pm 0.00015 & 0.22549 \pm 0.00064 & (0.00364 \pm 0.00011)e^{i(-69.7 \pm 2.9)^\circ}\\ -(0.2253 \pm 0.00064)e^{i( 0.0348 \pm 0.0012)^\circ} & 0.97341 \pm 0.00015 & 0.04117 \pm 0.00043 \\ (0.00871 \pm 0.00019)e^{i(-22.46 \pm 0.73)^\circ} & -(0.04039 \pm 0.00043)e^{i( 1.089 \pm 0.038)^\circ} & 0.999145 \pm 1.8\times 10^{-5}\end{array}\right)}$

Full fit result for $\,\bar{\rho}$

$0.132 \pm 0.02$ 95% prob:[0.092, 0.171] 99% prob:[0.074, 0.190]
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Full fit result for $\,\bar{\eta}$

$0.358 \pm 0.012$ 95% prob:[0.332, 0.383] 99% prob:[0.321, 0.396]
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Full fit result for $\,\bar{\rho} - \bar{\eta}$


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Full fit result for $\,\rho$

$0.135 \pm 0.021$ 95% prob:[0.095, 0.175] 99% prob:[0.076, 0.195]
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Full fit result for $\,\eta$

$0.367 \pm 0.013$ 95% prob:[0.341, 0.393] 99% prob:[0.329, 0.406]
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Full fit result for $\,A$

$0.8095 \pm 0.0095$ 95% prob:[0.791, 0.83] 99% prob:[0.782, 0.839]
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Fit Input for $\,\lambda$

$0.2253 \pm 0.0011$ 95% prob:[0.2231, 0.2275] 99% prob:[0.2218, 0.2285]
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Full Fit result for $\,\lambda$

$0.22545 \pm 0.00065$ 95% prob:[0.2242, 0.2268] 99% prob:[0.2236, 0.2274]
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Fit Input for $\,\|V_{ub}\|$

$0.00376 \pm 0.0002$ 95% prob:[0.00340, 0.00428] 99% prob:[0.00327, 0.00463]
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Full Fit result for $\,\|V_{ub}\|$

$0.00364 \pm 0.00011$ 95% prob:[0.00342, 0.00386] 99% prob:[0.00332, 0.00399]
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SM Fit prediction for $\,\|V_{ub}\|$

$0.00355 \pm 0.00014$ 95% prob:[0.00327, 0.00385] 99% prob:[0.00313, 0.00401]
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Compatibility Plot for $\,\|V_{ub}\|$


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Fit Input for $\,\|V_{cb}\|$

$0.04083 \pm 0.00045$ 95% prob:[0.03995, 0.04177] 99% prob:[0.03955, 0.04217] U [0.04219, 0.04233]
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Full Fit result for $\,\|V_{cb}\|$

$0.04117 \pm 0.00043$ 95% prob:[0.04037, 0.04209] 99% prob:[0.03995, 0.04247]
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SM Fit prediction for $\,\|V_{cb}\|$

$0.04269 \pm 0.00099$ 95% prob:[0.04069, 0.0447] 99% prob:[0.03971, 0.04563]
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Compatibility Plot for $\,\|V_{cb}\|$


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Full fit result for $\,\sin\theta_{12}$

$0.22545 \pm 0.00065$ 95% prob:[0.2242, 0.2268] 99% prob:[0.2236, 0.2274]
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Full fit result for $\,\sin\theta_{23}$

$0.04117 \pm 0.00043$ 95% prob:[0.04033, 0.04209] 99% prob:[0.03993, 0.04251]
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Full fit result for $\,\sin\theta_{13}$

$0.00364 \pm 0.00011$ 95% prob:[0.00342,0.003867] 99% prob:[0.00332,0.00399]
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Full fit result for $\,\delta$

$69.7 \pm 2.9$ 95% prob:[63.9, 75.7] 99% prob:[61.1, 78.6]
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Fit Input for $\,m_{t},{\rm {GeV}/c^{2}}$

Gaussian likelihood used $163.4 \pm 1.2$
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Full Fit result for $\,m_{t},{\rm {GeV}/c^{2}}$

$163.4 \pm 1.2$ 95% prob:[161, 165.7] 99% prob:[159.9, 166.9]
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SM Fit prediction for $\,m_{t},{\rm {GeV}/c^{2}}$

$163.5 \pm 9.5$ 95% prob:[144.7, 183.1] 99% prob:[137.6, 194.2]
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Compatibility Plot for $\,m_{t},{\rm {GeV}/c^{2}}$


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Fit Input for $\,\Delta m_{s},{\rm ps^{-1}}$

Gaussian likelihood used $17.77 \pm 0.12$
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Full Fit result for $\,\Delta m_{s},{\rm ps^{-1}}$

$17.77 \pm 0.12$ 95% prob:[17.5, 18.0] 99% prob:[17.4, 18.1]
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SM Fit prediction for $\,\Delta m_{s},{\rm ps^{-1}}$

$18.3 \pm 1.3$ 95% prob:[15.9, 20.9] 99% prob:[14.8, 22.3]
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Compatibility Plot for $\,\Delta m_{s},{\rm ps^{-1}}$


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Fit Input for $\,f_{B_{s}}$

Gaussian likelihood used $0.239 \pm 0.01$
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Full Fit result for $\,f_{B_{s}}$

$0.2359 \pm 0.0056$ 95% prob:[0.2252, 0.2477] 99% prob:[0.22, 0.2537]
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SM Fit prediction for $\,f_{B_{s}}$

$0.2349 \pm 0.0067$ 95% prob:[0.2221, 0.2491] 99% prob:[0.217, 0.2571]
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Compatibility Plot for $\,f_{B_{s}}$


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Fit Input for $\,f_{B_{s}}/f_{B_{d}}$

Gaussian likelihood used $1.23 \pm 0.03$
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Full Fit result for $\,f_{B_{s}}/f_{B_{d}}$

$1.225 \pm 0.025$ 95% prob:[1.175, 1.275] 99% prob:[1.151, 1.299]
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SM Fit prediction for $\,f_{B_{s}}/f_{B_{d}}$

$1.213 \pm 0.044$ 95% prob:[1.13, 1.303] 99% prob:[1.083, 1.085] U [1.093, 1.352]
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Compatibility Plot for $\,f_{B_{s}}/f_{B_{d}}$


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Fit Input for $\,B_{B_{s}}/B_{B_{d}}$

Gaussian likelihood used $1.06 \pm 0.04$
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Full Fit result for $\,B_{B_{s}}/B_{B_{d}}$

$1.069 \pm 0.036$ 95% prob:[0.997, 1.141] 99% prob:[0.963, 1.179]
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SM Fit prediction for $\,B_{B_{s}}/B_{B_{d}}$

$1.113 \pm 0.085$ 95% prob:[0.96, 1.279] 99% prob:[0.893, 1.3]
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Compatibility Plot for $\,B_{B_{s}}/B_{B_{d}}$


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Fit Input for $\,B_{B_{s}}$

Gaussian likelihood used $0.87 \pm 0.04$
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Full Fit result for $\,B_{B_{s}}$

$0.845 \pm 0.036$ 95% prob:[0.775, 0.919] 99% prob:[0.738, 0.954]
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SM Fit prediction for $\,B_{B_{s}}$

$0.769 \pm 0.065$ 95% prob:[0.648, 0.915] 99% prob:[0.608, 0.999]
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Compatibility Plot for $\,B_{B_{s}}$


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Fit Input for $\,\alpha, [^{\circ}]$

$91.4 \pm 6.1$ 95% prob:[81, 102.] U [161., 169] 99% prob:[76.8, 108.] U [157., 171.]
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Full Fit result for $\,\alpha, [^{\circ}]$

$87.8 \pm 3.0$ 95% prob:[82.1, 93.8] 99% prob:[79.2, 96.2]
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SM Fit prediction for $\,\alpha, [^{\circ}]$

$85.4 \pm 3.7$ 95% prob:[78.3, 93.2] 99% prob:[74.5, 96.2]
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Compatibility Plot for $\,\alpha, [^{\circ}]$


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Full Fit result for $\,\beta, [^{\circ}]$

$22.42 \pm 0.74$ 95% prob:[20.9, 23.9] 99% prob:[20.2, 24.7]
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SM Fit prediction for $\,\beta, [^{\circ}]$

$25.2 \pm 1.6$ 95% prob:[22.3, 28.6] 99% prob:[21.3, 30.2]
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Fit Input for $\,\sin(2\beta)$

$0.654 \pm 0.026$ 95% prob:[0.601, 0.708] 99% prob:[0.574, 0.735]
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Full Fit result for $\,\sin(2\beta)$

$0.705 \pm 0.018$ 95% prob:[0.669, 0.742] 99% prob:[0.651, 0.762]
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SM Fit prediction for $\,\sin(2\beta)$

$0.771 \pm 0.036$ 95% prob:[0.706, 0.844] 99% prob:[0.68, 0.872] U [0.875, 0.878]
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Compatibility Plot for $\,\sin(2\beta)$


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Fit Input for $\,\cos(2\beta)$

$0.87 \pm 0.13$ 95% prob:[0.44, 0.99] 99% prob:[0.12, 0.99]
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Full Fit result for $\,\cos(2\beta)$

$0.71 \pm 0.018$ 95% prob:[0.672, 0.745] 99% prob:[0.649, 0.76]
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SM Fit prediction for $\,\cos(2\beta)$

$0.639 \pm 0.043$ 95% prob:[0.544, 0.712] 99% prob:[0.507, 0.731]
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Compatibility Plot for $\,\cos(2\beta)$


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Fit Input for $\,2\beta+\gamma, [^{\circ}]$

$-90 \pm 56 \text{ and } 94 \pm 52$ 95% prob:[-166, 166.] 99% prob:[-179, 179]
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Full Fit result for $\,2\beta+\gamma, [^{\circ}]$

$114.7 \pm 3.1$ 95% prob:[108.3, 120.7] 99% prob:[105.6, 123.4]
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SM Fit prediction for $\,2\beta+\gamma, [^{\circ}]$

$114.9 \pm 3.1$ 95% prob:[108.6, 120.9] 99% prob:[105.8, 123.6]
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Compatibility Plot for $\,2\beta+\gamma, [^{\circ}]$


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Fit Input for $\,\gamma, [^{\circ}]$

$-106 \pm 11 \text{ and } 74 \pm 11$ 95% prob:[-128, -85.] U [52.1, 94.4] 99% prob:[-139, -75.] U [41.4, 104.]
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Full Fit result for $\,\gamma, [^{\circ}]$

$69.8 \pm 3.0$ 95% prob:[63.9, 75.7] 99% prob:[61, 78.5]
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SM Fit prediction for $\,\gamma, [^{\circ}]$

$69.6 \pm 3.1$ 95% prob:[63.4, 75.6] 99% prob:[60.5, 78.8]
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Compatibility Plot for $\,\gamma, [^{\circ}]$


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Fit Input for $\,\|\varepsilon_{K}\|$

$0.00222994 \pm 1.04974\times 10^{-5}$ 95% prob:[0.00220745, 0.00224944] 99% prob:[0.00219845, 0.00225644]
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Full Fit result for $\,\|\varepsilon_{K}\|$

$0.00222854 \pm 9.98004\times 10^{-06}$ 95% prob:[0.00220858, 0.0022485] 99% prob:[0.0021986, 0.00225848]
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SM Fit prediction for $\,\|\varepsilon_{K}\|$

$0.00192 \pm 0.00018$ 95% prob:[0.00157, 0.00230] 99% prob:[0.00141, 0.00252]
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Compatibility Plot for $\,\|\varepsilon_{K}\|$


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Fit Input for $\,B(B\rightarrow\tau u),10^{-4}$

Gaussian likelihood used $1.72 \pm 0.28$
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Full Fit result for $\,B(B\rightarrow\tau u),10^{-4}$

$0.867 \pm 0.078$ 95% prob:[0.721, 1.031] 99% prob:[0.661, 1.127]
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SM Fit prediction for $\,B(B\rightarrow\tau u),10^{-4}$

$0.805 \pm 0.071$ 95% prob:[0.674, 0.958] 99% prob:[0.619, 1.051]
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Compatibility Plot for $\,B(B\rightarrow\tau u),10^{-4}$


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Full fit result for $\,J_{cp}\times 10^{-5}$

$3.09 \pm 0.11$ 95% prob:[2.87, 3.30] 99% prob:[2.77, 3.42]
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Tree level Fit

In principle, the presence of New Physics might affect the result of the UT analysis, changing the functional dependencies of the experimental quantities upon ρ and η. On the contrary, two constraints now available, are almost unchanged by the presence of NP: |Vub/Vcb| from semileptonic B decays and the UT angle γ from B → D(*)K decays. As usual from this fit one can gets predictions for each observable related to the Unitarity Triangle. This set of values is the minimal requirement that each model describing New Physics has to satisfy in order to be taken as a realistic description of physics beyond the Standard Model.

Parameter	Input value	Full fit
$\bar{\rho}$		$0.111 \pm 0.07$
$\bar{\eta}$		$0.381 \pm 0.03$
$\rho$		$0.114 \pm 0.071$
$\eta$		$0.391 \pm 0.031$
		$0.804 \pm 0.01$
$\lambda$		$0.22535 \pm 0.00065$
$\alpha, [^{\circ}]$		$83 \pm 10$
$\beta, [^{\circ}]$		$23.0 \pm 1.4$
$\sin(2\beta)$		$0.72 \pm 0.035$
$\gamma, [^{\circ}]$	$-106 \pm 11 \text{ and } 74 \pm 11$	$73 \pm 11$

The fit results for all the nine CKM elements are

${\small V_{CKM}=\left(\begin{array}{ccc} 0.97426 \pm 0.00015 & 0.22535 \pm 0.00065 & (0.00376 \pm 0.0002)e^{i(-73.8 \pm 9.4)^\circ}\\ -(0.2252 \pm 0.00065)e^{i(0.0365 \pm 0.0028)^\circ} & 0.97345 \pm 0.00015 & 0.04083 \pm 0.00045 \\ (0.00896 \text{ and } 0.01081 \pm 0.0006)e^{i(-22.9 \pm 1.4)^\circ} & -(0.03979 \pm 0.00052)e^{i(1.163 \pm 0.084)^\circ} & 0.99916 \pm 1.8\times 10^{-5}\end{array}\right)}$

Full fit result for $\,\bar{\rho}$

$0.111 \pm 0.07$ 95% prob:[0.000, 0.230] 99% prob:[0.000, 0.298]
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Full fit result for $\,\bar{\eta}$

$0.381 \pm 0.03$ 95% prob:[0.311, 0.446] 99% prob:[0.264, 0.484]
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Full fit result for $\,\bar{\rho} - \bar{\eta}$


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Full fit result for $\,\rho$

$0.114 \pm 0.071$ 95% prob:[0.000, 0.236] 99% prob:[0.000, 0.305]
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Full fit result for $\,\eta$

$0.391 \pm 0.031$ 95% prob:[0.320, 0.457] 99% prob:[0.275, 0.494]
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Full fit result for $\,A$

$0.804 \pm 0.01$ 95% prob:[0.7850, 0.8250] 99% prob:[0.7760, 0.8360]
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Full fit result for $\,\lambda$

$0.22535 \pm 0.00065$ 95% prob:[0.22410, 0.22670] 99% prob:[0.22350, 0.22740]
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Full fit result for $\,\alpha, [^{\circ}]$

$83 \pm 10$ 95% prob:[63.9, 103.] 99% prob:[60.0, 115.]
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Full fit result for $\,\beta, [^{\circ}]$

$23.0 \pm 1.4$ 95% prob:[20.2, 26.5] 99% prob:[18.6, 29.0]
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Full fit result for $\,\sin(2\beta)$

$0.72 \pm 0.035$ 95% prob:[0.651, 0.802] 99% prob:[0.609, 0.851]
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Fit Input for $\,\gamma, [^{\circ}]$

$-106 \pm 11 \text{ and } 74 \pm 11$ 95% prob:[-128, -85.] U [52.1, 94.4] 99% prob:[-139, -75.] U [41.4, 104.]
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Full Fit result for $\,\gamma, [^{\circ}]$

$73 \pm 11$ 95% prob:[52.1, 94.3] 99% prob:[41.3, 104.]
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Universal Unitarity Triangle (UUT) Fit

It is possible to generalize the full UTfit beyond the Standard Model to all those NP models characterized by Minimal Flavour Violation, i.e. having quark mixing ruled only by the Standard Model CKM couplings ( http://arxiv.org/abs/hep-ph/0007085). In fact, in this case no additional weak phases are generated and several observables entering into the Standard Model fit (the tree-level processes and the measurement of angles through the use of time dependent CP asymmetries) are not affected by the presence of New Physics. The only sizable effect we are sensitive to is a shift of the Inami-Lim function of the top contribution in meson mixing. This means that in general εK and Δmd cannot be used in a common SM and MFV framework. Also the ratio Δmd/Δms cannot be used in general, as Δms can get additional NP contributions at large tanβ. So, simply removing the information related to εK, Δmd and Δms from the full UTfit, one can obtain a more precise determination of the Universal Unitarity Triangle, which is a common starting point for the Standard Model and any MFV model.

Parameter	Input value	Full fit
$\bar{\rho}$		$0.143 \pm 0.03$
$\bar{\eta}$		$0.342 \pm 0.015$
$\rho$		$0.146 \pm 0.031$
$\eta$		$0.351 \pm 0.016$
		$0.807 \pm 0.01$
$\lambda$	$0.2253 \pm 0.0011$	$0.22545 \pm 0.00065$
$\alpha, [^{\circ}]$	$91.4 \pm 6.1$	$90.8 \pm 4.8$
$\beta, [^{\circ}]$		$21.73 \pm 0.74$
$\sin(2\beta)$	$0.654 \pm 0.026$	$0.688 \pm 0.018$
$\gamma, [^{\circ}]$	$-106 \pm 11 \text{ and } 74 \pm 11$	$67.3 \pm 4.8$

The fit results for all the nine CKM elements are

${\small V_{CKM}=\left(\begin{array}{ccc} 0.97425 \pm 0.00015 & 0.22545 \pm 0.00065 & (0.00352 \pm 0.00011)e^{i(-67.7 \pm 4.4)^\circ}\\ -(0.2253 \pm 0.00065)e^{i(0.033 \pm 0.0016)^\circ} & 0.97342 \pm 0.00015 & 0.04099 \pm 0.00046 \\ (0.00854 \pm 0.00031)e^{i(-21.66 \pm 0.73)^\circ} & -(0.04024 \pm 0.00045)e^{i(1.04 \pm 0.048)^\circ} & 0.999154 \pm 1.8\times 10^{-5}\end{array}\right)}$

Full fit result for $\,\bar{\rho}$

$0.143 \pm 0.03$ 95% prob:[0.082, 0.204] 99% prob:[0.050, 0.238]
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Full fit result for $\,\bar{\eta}$

$0.342 \pm 0.015$ 95% prob:[0.310, 0.372] 99% prob:[0.293, 0.388]
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Full fit result for $\,\bar{\rho} - \bar{\eta}$


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Full fit result for $\,\rho$

$0.146 \pm 0.031$ 95% prob:[0.084, 0.209] 99% prob:[0.051, 0.244]
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Full fit result for $\,\eta$

$0.351 \pm 0.016$ 95% prob:[0.319, 0.382] 99% prob:[0.301, 0.398]
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Full fit result for $\,A$

$0.807 \pm 0.01$ 95% prob:[0.7870, 0.8280] 99% prob:[0.7780, 0.8390]
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Fit Input for $\,\lambda$

$0.2253 \pm 0.0011$ 95% prob:[0.2231, 0.2275] 99% prob:[0.2218, 0.2285]
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Full Fit result for $\,\lambda$

$0.22545 \pm 0.00065$ 95% prob:[0.22410, 0.22670] 99% prob:[0.22360, 0.22750]
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Fit Input for $\,\alpha, [^{\circ}]$

$91.4 \pm 6.1$ 95% prob:[81.0, 102.] U [161., 169.] 99% prob:[76.8, 108.] U [157., 171.]
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Full Fit result for $\,\alpha, [^{\circ}]$

$90.8 \pm 4.8$ 95% prob:[81.2, 100.] 99% prob:[76.4, 106.]
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Full fit result for $\,\beta, [^{\circ}]$

$21.73 \pm 0.74$ 95% prob:[20.2, 23.2] 99% prob:[19.6, 24.0]
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Fit Input for $\,\sin(2\beta)$

$0.654 \pm 0.026$ 95% prob:[0.601, 0.708] 99% prob:[0.574, 0.735]
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Full Fit result for $\,\sin(2\beta)$

$0.688 \pm 0.018$ 95% prob:[0.651, 0.725] 99% prob:[0.633, 0.745]
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Fit Input for $\,\gamma, [^{\circ}]$

$-106 \pm 11 \text{ and } 74 \pm 11$ 95% prob:[-128, -85.] U [52.1, 94.4] 99% prob:[-139, -75.] U [41.4, 104.]
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Full Fit result for $\,\gamma, [^{\circ}]$

$67.3 \pm 4.8$ 95% prob:[57.5, 77.1] 99% prob:[52.0, 82.2]
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New Physics Fit

The fit presented here is meant to constrain the NP contributions to |Δ F|=2 transitions by using the available experimental information on loop-mediated processes In general, NP models introduce a large number of new parameters: flavour changing couplings, short distance coefficients and matrix elements of new local operators. The specific list and the actual values of these parameters can only be determined within a given model. Nevertheless mixing processes are described by a single amplitude and can be parameterized, without loss of generality, in terms of two parameters, which quantify the difference of the complex amplitude with respect to the SM one. Thus, for instance, in the case of $B^0_q-\bar{B}^0_q$ mixing we define $C_{B_q} \, e^{2 i \phi_{B_q}} = \frac{\langle B^0_q|H_\mathrm{eff}^\mathrm{full}|\bar{B}^0_q\rangle} {\langle B^0_q|H_\mathrm{eff}^\mathrm{SM}|\bar{B}^0_q\rangle}\,, \qquad (q=d,s),$ where $H_\mathrm{eff}^\mathrm{SM}$ includes only the SM box diagrams, while $H_\mathrm{eff}^\mathrm{full}$ also includes the NP contributions. In the absence of NP effects, $C_{B_q}=1$ and $\phi_{B_q}=0$ by definition. In a similar way, one can write $C_{\epsilon_K} = \frac{\mathrm{Im}[\langle K^0|H_{\mathrm{eff}}^{\mathrm{full}}|\bar{K}^0\rangle]} {\mathrm{Im}[\langle K^0|H_{\mathrm{eff}}^{\mathrm{SM}}|\bar{K}^0\rangle]}\,,\qquad C_{\Delta m_K} = \frac{\mathrm{Re}[\langle K^0|H_{\mathrm{eff}}^{\mathrm{full}}|\bar{K}^0\rangle]} {\mathrm{Re}[\langle K^0|H_{\mathrm{eff}}^{\mathrm{SM}}|\bar{K}^0\rangle]}\,. \label{eq:ceps}$ Concerning $\Delta m_K$ , to be conservative, we add to the short-distance contribution a possible long-distance one that varies with a uniform distribution between zero and the experimental value of $\Delta m_K$ .

The experimental quantities determined from the $B^0_q-\bar{B}^0_q$ mixings are related to their SM counterparts and the NP parameters by the following relations:

$\Delta m_d^\mathrm{exp} = C_{B_d} \Delta m_d^\mathrm{SM} \,,\; \\ \sin 2 \beta^\mathrm{exp} = \sin (2 \beta^\mathrm{SM} + 2\phi_{B_d})\,,\; \\ \alpha^\mathrm{exp} = \alpha^\mathrm{SM} - \phi_{B_d}\,, \\ \Delta m_s^\mathrm{exp} = C_{B_s} \Delta m_s^\mathrm{SM} \,,\; \\ \phi_s^\mathrm{exp} = (\beta_s^\mathrm{SM} - \phi_{B_s})\,,\; \\ \Delta m_K^\mathrm{exp} = C_{\Delta m_K} \Delta m_K^\mathrm{SM} \,,\; \\ \epsilon_K^\mathrm{exp} = C_{\epsilon_K} \epsilon_K^\mathrm{SM} \,,\; \\$

in a self-explanatory notation.

All the measured observables can be written as a function of these NP parameters and the SM ones ρ and η, and additional parameters such as masses, form factors, and decay constants.

Click on the parameter name to jump to the corresponding plot

Parameter	Input value	Full fit
$\bar{\rho}$		$0.135 \pm 0.04$
$\bar{\eta}$		$0.374 \pm 0.026$
$\rho$		$0.138 \pm 0.041$
$\eta$		$0.384 \pm 0.027$
		$0.804 \pm 0.01$
$\lambda$	$0.2253 \pm 0.0011$	$0.22535 \pm 0.00065$
$C_{B_{d}}$		$0.95 \pm 0.14$
$\phi_{B_{d}}, [^{\circ}]$		$-3.1 \pm 1.7$
$C_{B_{s}}$		$0.95 \pm 0.095$
$\phi_{B_{s}}, [^{\circ}]$		$-68.2 \pm 7.6 \text{ and } -20.3 \pm 7.7$
$C_{\epsilon_{K}}$		$1.05 \pm 0.12$
$A_{SL_{d}}$	$-0.0005 \pm 0.0056$	$-0.0028 \pm 0.0024$
$A_{SL_{s}}$	$-0.0017 \pm 0.0091$	$-0.0044 \pm 0.0014$

The fit results for all the nine CKM elements are

${\small V_{CKM}=\left(\begin{array}{ccc} 0.97427 \pm 0.00015 & 0.22535 \pm 0.00065 & (0.00377 \pm 0.00021)e^{i(-70.0 \pm 5.6)^\circ}\\ -(0.22525 \pm 0.00065)e^{i(0.0358 \pm 0.0025)^\circ} & 0.97345 \pm 0.00015 & 0.04082 \pm 0.00045 \\ (0.00869 \pm 0.00039)e^{i(-23.3 \pm 1.3)^\circ} & -(0.04007 \pm 0.00044)e^{i(1.138 \pm 0.076)^\circ} & 0.99916 \pm 1.8\times 10^{-5}\end{array}\right)}$

Full fit result for $\,\bar{\rho}$

$0.135 \pm 0.04$ 95% prob:[0.069, 0.223] 99% prob:[0.040, 0.262]
EPS - PDF - PNG - JPG - GIF

Full fit result for $\,\bar{\eta}$

$0.374 \pm 0.026$ 95% prob:[0.322, 0.433] 99% prob:[0.297, 0.471]
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Full fit result for $\,\bar{\rho} - \bar{\eta}$


EPS - PDF - PNG - JPG - GIF

Full fit result for $\,\rho$

$0.138 \pm 0.041$ 95% prob:[0.070, 0.228] 99% prob:[0.041, 0.269]
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Full fit result for $\,\eta$

$0.384 \pm 0.027$ 95% prob:[0.330, 0.444] 99% prob:[0.305, 0.482]
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Full fit result for $\,A$

$0.804 \pm 0.01$ 95% prob:[0.7840, 0.8240] 99% prob:[0.7750, 0.8350]
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Fit Input for $\,\lambda$

$0.2253 \pm 0.0011$ 95% prob:[0.2231, 0.2275] 99% prob:[0.2218, 0.2285]
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Full Fit result for $\,\lambda$

$0.22535 \pm 0.00065$ 95% prob:[0.22410, 0.22670] 99% prob:[0.22350, 0.22750]
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Full fit result for $\,C_{B_{d}}$

$0.95 \pm 0.14$ 95% prob:[0.70, 1.27] 99% prob:[0.59, 1.51]
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Full fit result for $\,\phi_{B_{d}}, [^{\circ}]$

$-3.1 \pm 1.7$ 95% prob:[-7., 0.1] 99% prob:[-10, 2.1]
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2D for $\,\Phi_{B_{d}} - C_{B_{d}}$


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2D for $\,\Phi_{B_{d}} - A^{NP}_{d}/A^{SM}_{d}$


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Full fit result for $\,C_{B_{s}}$

$0.95 \pm 0.095$ 95% prob:[0.776, 1.162] 99% prob:[0.706, 1.295]
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Full fit result for $\,\phi_{B_{s}}, [^{\circ}]$

$-68.2 \pm 7.6 \text{ and } -20.3 \pm 7.7$ 95% prob:[-81, -51] U [-38, -6.] 99% prob:[-85, -1.]
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2D for $\,\Phi_{B_{s}} - C_{B_{s}}$


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2D for $\,\Phi_{B_{s}} - A^{NP}_{s}/A^{SM}_{s}$


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Full fit result for $\,C_{\epsilon_{K}}$

$1.05 \pm 0.12$ 95% prob:[0.82, 1.34] 99% prob:[-0.8, -0.8] U [0.70, 1.60]
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Fit Input for $\,A_{SL_{d}}$

Gaussian likelihood used $-0.0005 \pm 0.0056$
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Full Fit result for $\,A_{SL_{d}}$

$-0.0028 \pm 0.0024$ 95% prob:[-0.0077, 0.00129] 99% prob:[-0.0096, 0.00431]
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Fit Input for $\,A_{SL_{s}}$

Gaussian likelihood used $-0.0017 \pm 0.0091$
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Full Fit result for $\,A_{SL_{s}}$

$-0.0044 \pm 0.0014$ 95% prob:[-0.0072, -0.0015] 99% prob:[-0.0087, 0.00004]
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Explanation of the page content

In this page we present the results obtained for a set of interesting UT parameters in the framework of the Standard Model and some New Physics Models using all the available experimental and theoretical inputs which are available. Inputs to this analysis consist of a large body of both experimental measurements and theoretically determined parameters. All the analyses presented here rely on the several measurements: |Vub/Vcb|, Δmd, Δms, and the measurements of CP-violating quantities in the kaon (εK) and in the B sectors with the measurements of α (using ππ, ρρ and πρ modes), γ (using D K, DK*, D*K modes), 2β + γ (using Dπ(ρ) modes), and sin2β and cos 2β from B0 → J/ψKS and B0 → J/ψK* respectively. Among the theoretical parameters, Lattice QCD calculations play a central role.

The results are presented in a summary table and in a series of probability density functions. The tables contain three entries per variable : the input ("direct") value, the output value and the prediction ("indirect determination") for this variable in a given model.

The indirect determination of a particular quantity obtained performing the Unitarity Triangle fit in a given Model, including all the available constraints except from the direct measurement of the parameter of interest, gives a prediction of the quantity based on formulas which are valid in that given Model. The interest of this procedure is to quantify the agreement of all the measured quantities by the comparison between indirect parameter determinations and their direct experimental/theortical determinations. Let's consider for example the Standard Model. The comparison between these predictions and a direct measurements can thus quantify the agreement of the single measurement with the overall fit and possibly reveal new physics phenomena.

For some of the quantities we present the so called COMPATIBILITY PLOTS. In Unitarity Triangle fits based on a χ2 minimization, a conventional evaluation of compatibility stems automatically from the value of the χ2 at its minimum. The compatibility between constraints in the Bayesian approach is simply done by comparing two different p.d.f.’s.

Let us consider, for instance, two p.d.f.’s for a given quantity obtained from the Unitarity Triangle fit, f(x1), and from a direct measurement, f(x2): their compatibility is evaluated by constructing the p.d.f. of the difference variable, x2 − x1, and by estimating the distance of the most probable value from zero in units of standard deviations. The latter is done by integrating this p.d.f. between zero and the most probable value and converting it into the equivalent number of standard deviations for a Gaussian distribution. The advantage of this approach is that no approximation is made on the shape of p.d.f.’s. In the following analysis, f(x1) is the p.d.f. predicted by the Unitarity Triangle fit while the p.d.f of the measured quantity, f(x2), is taken Gaussian for simplicity. The number of standard deviations between the measured value, x2(meas.) ± σ(x2), and the predicted value (distributed according to f(x1)) is plotted as a function of x2(meas.) (x-axis) and σ(x2) (y-axis). The compatibility between x1 and x2 can be then directly estimated on the plot, for any central value and error of the measurement of x2.

The color code indicates the compatibility between direct and indirect determinations, given in terms of standard deviations, as a function of the measured value and its experimental uncertainty. The crosses indicate the direct world average measurement values.

Treatement of Lattice parameters in the fits.

In the Unitarity Triangle fits the non perturbative QCD parameters enter in the expressions of several contraints : $\Delta m_s, \Delta m_d, \epsilon_K, B \rightarrow \tau \nu,$ . Let's consider schematically the dependence of these observable in terms of the non perturbative QCD parameters :

$\Delta m_s \propto f^2_{Bs} {B_{Bs}} \\ \Delta m_d \propto f^2_{Bd} {B_{Bd}} = \frac{f^2_{Bs}}{f^2_{Bs}/f^2_{Bd}} \times \frac{B_{Bs}}{B_{Bs}/B_{Bd}} \\ Br(B \rightarrow \tau \nu) \propto f_{Bd}^2 = \frac{f_{Bs}^2}{f^2_{Bs}/f^2_{Bd}} \\ \epsilon_K \propto B_K \\$ .

We decide to express these observable in terms of five LQCD parameters

$f_{Bs}, \quad B_{Bs}, \quad \frac{f_{Bs}}{f_{Bd}} , \quad \frac{{B_{Bs}}}{{B_{Bd}}}, \quad B_K$

The reason of this choice is to maxime the parameters on the Bs sector and parameters which are mostly uncorrelated. To this set of five parameters we should add the non perturbative parameters entering in the expression of $V_{ub}$ (exclusive) and $V_{cb}$ .

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