Uncertain constraints

where ,, ... are the constraint parameters, functions of many quantities; the most crucial of those determined experimentally are indicated on the left hand side of the equations. Note that the order of and is exchanged with respect to Eqs. (3)-(4) of Ref. [1]. This is because, since the present information concerning is of different quality with respect to the other quantities, the constraint needs a more careful treatment than the others, and it will be introduced after -. This is also the reason why appears explicitly in the constraint .

In the ideal case all parameters are perfectly known,
and the constraints would give curves in the - plane.
For example, would give a circle of radius .
In other terms, due to this constraint all
points of the circumference would be appear to us likely,
__unless__
there is any other experimental piece of information
(or theoretical prejudice) to assign a different weight to
different points. In the ideal case
the p.d.f. describing our beliefs
in the and values would be

(1) |

In the real case itself is not perfectly known. There are values which are more likely, and values which are less likely, classified by the p.d.f. . This means that, e.g. for the parameter , we deal with an infinite number of circles, each having its weight . It follows that the points of the (, ) plane get different weights.

The probability theory teaches us how to evaluate
taking into account all possible values of :

(2) |

(3) |

- First, we rely on the central limit theorem, assuming to be Gaussian for all parameters (but with no constraint on the shape of !).
- Expected value, standard (deviation) uncertainty and correlation are basically obtained by the usual propagation (see Ref. [5] for a similar application).
- Non linear effects in the propagation have been taken into account up to second order, using formulae of Ref. [6] (see this paper for the practical modeling of uncertainties and treatment of asymmetric cases).
- The correlation between and has been taken into account building a bi-variate .

As input quantities, Table 1 of Ref. [1] is used, combining
properly (i.e. quadratically) the standard deviations.
For example, for
expressed in MeV we obtain
, where
stands for the standard deviation
of a uniform distribution of half width 20 MeV.
Note that, contrary to what some authors critical
about Ref. [1] (and the related papers and presentations
to conferences) think, I have the impression that my colleagues
tend to make slightly conservative assessments
of uncertainties. This feeling that I had *a priori*
discussing with them is somewhat confirmed
*a posteriori* by the excellent overlap of the partial
inference by each constraint (as it will be shown in
Fig. 9) and self-consistency
between input parameters and values coming out of
the inference obtained without the their contribution
(see Ref. [1]).

The following results are obtained in terms of expected values and
standard deviations (``
''):

(4) | |||

(5) | |||

(6) | |||

(7) | |||

(8) |

Only the correlation between and is relevant, and all others can be neglected, being below the 10% level (even that between and , related by , is negligible, being only +7%). The radii of the circles given by and are and , respectively. These radii have the meaning of the sides of the unitarity triangle opposite to and , respectively, provided by and alone.