Constrained invariant mass distributions in cascade decays. The shape of the "m_(qll)-threshold" and similar distributions

Considering the cascade decay $D\to c C \to c b B \to c b a A$ in which $D,C,B,A$ are massive particles and $c,b,a$ are massless particles, we determine for the first time the shape of the distribution of the invariant mass of the three massless particles $m_{abc}$ for the sub-set of decays in which the invariant mass $m_{ab}$ of the last two particles in the chain is (optionally) constrained to lie inside an arbitrary interval, $m_{ab} \in [ m_{ab}^\text{cut min}, m_{ab}^\text{cut max}]$. An example of an experimentally important distribution of this kind is the ``$m_{qll}$ threshold'' -- which is the distribution of the combined invariant mass of the visible standard model particles radiated from the hypothesised decay of a squark to the lightest neutralino via successive two body decay,: $\squark \to q \ntlinoTwo \to q l \slepton \to q l l \ntlinoOne $, in which the experimenter requires additionally that $m_{ll}$ be greater than ${m_{ll}^{max}}/\sqrt{2}$. The location of the ``foot'' of this distribution is often used to constrain sparticle mass scales. The new results presented here permit the location of this foot to be better understood as the shape of the distribution is derived. The effects of varying the position of the $m_{ll}$ cut(s) may now be seen more easily.