Conclusions

The initial motivation of this paper was didactic, i.e. how to perform a sceptical combination of results by MCMC using a convenient program, after having got a better insight of the problem by Bayesian network (this is the name also used for the graphical models we have encountered here). The choice of the physics case was fortuitous, having been recently personally interested in the charged kaon mass and having learned thus about `apparent' disagreements between the most accurate measurements. However, it is clear that this paper is far from attempting to give a definite answer, for which not only a `statistical'²⁶but also a serious phenomenological analysis should be required. For example in Ref. [5] there are interesting hints on not well understood high order corrections [33,34] and it would be interesting to investigate if the question has been settled down in the meanwhile and what should be the effect on the published mass values, or whether and how its uncertain value should contribute to the overall uncertainty.

The result of this analysis is $\left.m_{k^\pm}\right\vert _{I} = 493.677\pm 0.013\,\mbox{MeV}\,,$ where $I$ stands for all the conditions referred in section 6.2 (probability is always conditional probability and hence so are also pdf's and moments of distributions). The result seems in practical perfect agreement with the PDG one reminded in Fig. 1. But, first, the $f(m\,\vert\,I)$ estimated by sampling is not trivial and definitely far from Gaussian (see solid thick line of Fig. 15), yielding e.g. the following probability intervals (not “C.L.'s”!):

$$\begin{eqnarray*} P(493.650 \le\, m_{k^\pm}/\mbox{MeV}\, \le 493.697\,\vert\,I)... ... m_{k^\pm}/\mbox{MeV} \, \le 493.678\,\vert\,I) & = & 50\,\%\,. \end{eqnarray*}$$

Second, even if the numerical results coincide, this agreement is just due to a compensation of two effects in the PDG analysis which go into apposite directions:

• a weighted average of all nine individual results (see Tab. 2), with the final `error' scaled according to the $\sqrt {\chi ^2/\nu }$ prescription, would have lead to $493.664\pm 0.011\,$ MeV, that is 13 keV lower than that reported by the PDG [3];
• however, the analysis was not performed on the nine individual results of Tab. 2, but on the six ones of Tab. 1, where the precise result $493.631\pm 0.007\,$ MeV of Ref. [8] had been `weakened' by the other three because of the $\sqrt {\chi ^2/\nu }$ scaling prescription already applied by the authors. For this reason the overall result went up to $493.677\pm 0.013\,$MeV, hence producing a bias of $+13\,$keV, that is of the same size of the quote `error'.

The latter point is the surprising novelty of this work, and it deserves another paper [35] and perhaps further investigation to check if other, perhaps more important results are affected by such a bias too.

It is a pleasure to tank Andrea Messina, Enrico Franco and Paolo Gauzzi for discussions on the subject and comments on the manuscript.