8 Results

In this section, the main results of our analysis are summarized. These results have been obtained by varying the experimental quantities, e.g. the value of the top mass , , etc., and the theoretical parameters, e.g. the B-parameters, the strange quark mass , etc., according to their errors. Values and errors of the input quantities used in the following are reported in tables 2--4. We assume a Gaussian distribution for the experimental quantities and a flat distribution (with a width of 2) for the theoretical ones. The only exception is , taken from quenched lattice QCD calculations, for which we have assumed a Gaussian distribution, according to the results of ref. [32].

The theoretical predictions (, , etc.) depend on several fluctuating parameters. We have obtained their distributions numerically, from which we have calculated the central values and the errors reported below.

  
Table: Values of the fluctuating parameters used in the numerical analysis.

  
Table: Constants used in the numerical analysis.

  
Table: Values of the B-parameters, for operators renormalized at the scale GeV. The only exception is , which is the renormalization group invariant B-parameter. has been taken equal to , at any renormalization scale. The value reported in the table is GeV). Entries with a are educated guesses, the others are taken from lattice QCD calculations.

Using the values given in the tables and the formulae given in the previous sections, we have obtained the following results:

a)

The distribution for , obtained by comparing the experimental value of with its theoretical prediction, is given in fig. 3. As already noticed in refs. [7,8] and [20,21], large values of and favour , given the current measurement of . When the condition 160 MeV MeV is imposed (-cut), most of the negative solutions disappear, giving the dashed histogram of fig. 3, from which we estimate

  
Figure: Distributions of values for , , and , for GeV, using the values of the parameters given in tabs. 2--4. The solid histograms are obtained without using the information coming from -- mixing. The dashed ones use the information, assuming that 160 MeV MeV.

  
Figure: Contour plots in the -- plane. The solid, dashed and dotted contours contain , and of the generated events respectively. The contours are given by excluding or including the -cut. Similar results can be found in refs. [20,21].

  
Figure: Distributions of the events in the plane -- without and with the -cut. The corresponding contour plots are displayed below the Lego plots.

b)

A contour plot in the -- plane is given in fig. 4. It shows the current limits on the unitarity triangle defined in fig. 1.

c)

In fig. 5, several pieces of information on are provided. Lego plots of the distribution of the generated events in the -- plane are shown, without and with the -cut. In the same figure, the corresponding contour plots are displayed. One notices a very mild dependence of on . As a consequence, one obtains approximately the same prediction in the two cases (see also fig. 3). In the HV scheme the results are

and

 

whereas in the NDR scheme we obtain

and

 

By averaging the results given in eqs. (75) and (77), we obtain our best estimate

where the third error comes from the difference of the central values in the two schemes and gives an estimate of the uncertainty due to higher-order corrections.

A similar result has been obtained in ref. [14], using a different approach to the hadronic-matrix-element evaluation. They quote

for GeV. For this value of the top mass, the cancellation between penguin and electropenguin contributions is less effective, thus their prediction is significantly larger than ours. Actually the two predictions agree, once the difference in the top mass is taken into account. It is reassuring that theoretical predictions, obtained by using quite different approaches to matrix elements evaluation, are in good agreement.

On the basis of the latest analyses, it seems very difficult that is larger than . Theoretically, this may happen by taking the matrix elements of the dominant operators, and , much more different than it is usually assumed. One possibility, discussed in ref. [14], is to take and , instead of the usual values . To our knowledge, no coherent theoretical approach can accommodate such large values of .