The rules of the game

The hypothetical world of this analysis is:

• three experiments ($A$, $B$ and $C$) took data in an $\rm {e^+e^-}$ collider, at different energies and with different sensitivity to the $H$ particle production (`$H$ stands for hypothetical...'). The experiments reported 0 candidates and no background was expected (this is a minor approximation to simplify the formulae: we have seen how the background and its uncertainty may be treated).
• The beam energy was 0.09 and 0.1 in arbitrary units (you may think of TeV) and the kinematical factor which suppresses the production near threshold (and eventually takes into account efficiencies, tagging, etc.) is chosen somehow `arbitrarily' to be $\beta^3$ factor, where $\beta$ is the velocity of the pair produced particles.
• Cross-section and integrated luminosity are summarized into a sensitivity factor $k$, such that the expected number of events is
  $$\lambda = k \,\beta^3 = k\, \left(1-\frac{m^2}{E_b^2}\right)^{3/2}\,.$$

• We also have other pieces of information on $H$: two indirect determinations are characterized by a Gaussian likelihood, and each of them would allow a Gaussian determination of the mass, if one considered that this could be uniformly distributed from $-\infty$ to $+\infty$ (see Section ).
• The five datasets are considered to be independent.
• The prior of the scientific community about the value of the mass has changed in recent years, due not only to negative results, but also to theoretical progress:
  • essentially, once there was uncertainty even in the order of magnitude, i.e. $f(\ln m)\approx k$, yielding $f(m)\approx 1/m$; as a conservative position, one could still stick to this position;
  • at present, many think that ${\cal O}(m)\approx 0.1$-$0.2$; this state of uncertainty can be modelled by a uniform distribution over the range of interest.