The impact of present Δms bound on UT fit

Amplitude Method
Likelihood Ratio Method
Experimental Status (Winter 2005)

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 Δs and to combine results from different analyses.
The method consists in modifying the equation describing the probability that a B0 meson oscillates into a B0 in the following way : 1 ± cos Δms t ⇒ 1 ± A cos Δms t. A and σA are measured at fixed values of Δms instead of Δms 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 Δms excluded at 95% C.L. are those satisfying the condition A(Δms) + 1.645 σA (Δms) < 1. Furthermore the sensitivity of the experiment can be defined as the value of Δms corresponding to 1.645 σA (Δms) = 1 (taking A(Δms) = 0), namely supposing that the ``true'' value of Δms is well above the measurable value.
For example we show, the combined result of LEP/SLD/CDF analyses (situation at ICHEP04) (see HFAG) is shown in Figure (plot on the left) below

dms world average amplitude

Amplitude as a function of Δms

amplitude vs. dms

Δ log Linfinity(Δms).

The Likelihood ratio method

The 95% C.L. limit and the sensitivity, are useful to summarize the results of the analysis. However to include Δms 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 Δms is needed.
The log-likelihood values (it is the log-likelihood referenced to its value obtained for Δms=infinity) can be easily deduced from A and σA using the expressions :
formula of likelihood ratio I
The last two equations give the average log-likelihood value for Δms corresponding to the true oscillation frequency (mixing case) and for Δms being far from the oscillation frequency (|Δms-Δmstrue| >> Γ/2, no-mixing case). Γ is here the full width at half maximum of the amplitude distribution in case of a signal; typically Γ ~ 1/τ(Bs). The Δ log Linfinity(Δms) plot from the world average is shown in the above Figure (plot on the right)
The Likelihood Ratio R is defined as :
formula of likelihood ratio II
It has been shown (Yellow Book CERN-EP/2003-002 hep-ph/0304132 pages 182-190) 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 hep-ph/9607469.

Experimental Status (Winter 2005)

Nowdays, a combined experimental bound is available from SLD/LEP and Tevatron (CDF and D0) experiments. This bound gives a direct information that can be compared to the indirect determination, obtained from a combined fit to all the remaining constraints (|Vub/Vcb|, Δ md, εK, sin2β, cos2β, α and γ). 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 evidence of disagreement (the hint of a signal from the experimental bound is near the Standard Model expectation).

indirect determination of Δms

Compatibility plot for Δms

Pull distribution
Δms = 22.2 ± 3.1 ps-1
Δms > 14.5 ps-1 @ 95 % C.L.

As a reference for the future measurements, one can use the compatibility plot from the indirect determination to quantify the agreement between the value of Δ ms and the rest of theoretical and experimental results entering into the UT analysis. Using the present data, we find a sensitivity to new physics of:

3 σ @ 31 ps-1
5 σ @ 38 ps-1

total determination of Δms

Compatibility plot for Δms

Pull Distribution
Δms = 19.1 ± 2.0 ps-1

One can also add the experimental bound to the fit to improve the present knowledge on Δ ms. This gives a reference for the future experiments in terms of the present knowledge of Δms and sensitivity to New Physics:

3 σ @ 28 ps-1
5 σ @ 35 ps-1

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