The impact of present Δm_{s} measurement on UT fit
Experimental Status (Winter 2006)
The CDF collaboration reported
the first 5σ measurement of Δm_{s},
which provides a test of the Standard Model through the comparison
to the indirect determination, obtained from a combined fit to all the remaining constraints
(V_{ub}/V_{cb}, Δ m_{d}, ε_{K}, sin2β, cos2β, α and γ).
Amplitude as a function of Δm_{s} from CDF

This comparison is a crucial test for the Standard Model and
it plays in the b → s sector the same role of
sin2β measurements in b → d sector few years ago. At this point,
the two informations do not show any significant evidence of disagreement
INDIRECT MEASUREMENT:


Pull distribution
DIRECT MEASUREMENT:

Δm_{s} = (18.6 ± 2.3) ps^{1}


Δm_{s} = (17.77 ± 0.12) ps^{1}


The amplitude method
The amplitude method has been introduced by H.G. Moser and A.Roussarie and is
described in Nucl. Instrum. Meth. A384 (1997) 491. with the aim of
setting limits on Δm_{s} and to combine results from different
analyses.
The method consists in modifying the equation describing
the probability that a B^{0} meson oscillates into a
B^{0} in the following way :
1 ± cos Δm_{s} t ⇒ 1 ± A cos Δm_{s} t.
A and σ_{A} are measured at fixed values of
Δm_{s} instead of Δm_{s} itself.
In case of a clear oscillation signal, at a given frequency, the amplitude should be
compatible with A = 1 at this frequency.
With this method it is easy to set a limit. The values of
Δm_{s} excluded at 95% C.L.
are those satisfying the condition A(Δm_{s}) +
1.645 σ_{A} (Δm_{s}) < 1.
Furthermore the sensitivity of the experiment can be defined as the value of
Δm_{s} corresponding to 1.645 σ_{A} (Δm_{s}) = 1
(taking A(Δm_{s}) = 0), namely supposing that the ``true'' value of
Δm_{s} is well above the measurable value.
For example we show, the combined result of LEP/SLD/CDF/D0 analyses (situation at Moriond06)
(see HFAG)
is shown in Figure (plot on the left) below
Amplitude as a function of Δm_{s} (HFAG WA, not including new CDF)

Δ log L^{infinity}(Δm_{s}).

The Likelihood ratio method
The 95% C.L. limit and the sensitivity, are useful to summarize the results of the
analysis. However to include Δm_{s} in a CKM fit and to determine
probability regions for the Unitarity Triangle parameters, continuous information
about the degree of exclusion of a given value of Δm_{s} is needed.
The loglikelihood values (it is the loglikelihood referenced to its value obtained for
Δm_{s}=infinity) can be easily deduced from A and σ_{A} using
the expressions :
The last two equations give the average loglikelihood value for
Δm_{s} corresponding to the true oscillation frequency
(mixing case) and for Δm_{s} being far from the oscillation
frequency (Δm_{s}Δm_{s}^{true} >> Γ/2,
nomixing case). Γ is here the full width at half maximum of the amplitude
distribution in case of a signal; typically Γ ~ 1/τ(B_{s}).
The Δ log L^{infinity}(Δm_{s}) plot from the world average is
shown in the above Figure (plot on the right)
The Likelihood Ratio R is defined as :
It has been shown (Yellow Book CERNEP/2003002
hepph/0304132 pages 182190)
that in the Bayesian approach the correct method to include this information
is the Likelihood ratio method. A similar method to incorporate results from mixing can be
found in hepph/9607469.