  The variable \sqrt{s}_{min} was originally proposed in arXiv:0812.1042 as a
model-independent, global and fully inclusive measure of the new physics mass
scale in missing energy events at hadron colliders. In the original incarnation
of \sqrt{s}_{min}, however, the connection to the new physics mass scale was
blurred by the effects of the underlying event, most notably initial state
radiation and multiple parton interactions. In this paper we advertize two
improved variants of the \sqrt{s}_{min} variable, which overcome this problem.
First we show that by evaluating the \sqrt{s}_{min} variable at the RECO level,
in terms of the reconstructed objects in the event, the effects from the
underlying event are significantly diminished and the nice correlation between
the peak in the \sqrt{s}_{min}^{(reco)} distribution and the new physics mass
scale is restored. Secondly, the underlying event problem can be avoided
altogether when the \sqrt{s}_{min} concept is applied to a subsystem of the
event which does not involve any QCD jets. We supply an analytic formula for
the resulting subsystem \sqrt{s}_{min}^{(sub)} variable and show that its peak
exhibits the usual correlation with the mass scale of the particles produced in
the subsystem. Finally, we contrast \sqrt{s}_{min} to other popular inclusive
variables such as H_T, M_{Tgen} and M_{TTgen}. We illustrate our discussion
with several examples from supersymmetry, and with dilepton events from top
quark pair production.
