the two visible four-momenta written in the rest frame of the respective parent particles: $$\begin{aligned}
p_{\cPq_1} &= \left(\frac{m_{\PSq}^2+m_{\cPq_1}^2-m_{\chiz_1}^2}{2m_{\PSq}},\; \frac{{\ensuremath{M_\Delta}\xspace}}{2}\,\hat{u}_{\cPq_1}\right),\\
p_{\cPq_2} &= \left(\frac{m_{\PSq}^2+m_{\cPq_2}^2-m_{\chiz_1}^2}{2m_{\PSq}},\; \frac{{\ensuremath{M_\Delta}\xspace}}{2}\,\hat{u}_{\cPq_2}\right).\end{aligned}$$ where $\hat{u}_{\cPq_i}$ (${i} = 1,2$) is a unit three-vector and $m_\cPq{}_i$ represents the mass corresponding to the megajet, e.g., the top-quark mass for $\PSQt \to \cPqt \PSGczDo$. Here we have parameterized the magnitude of the three-momenta by the mass scale ${\ensuremath{M_\Delta}\xspace}$, where $${\ensuremath{M_\Delta}\xspace}^2 \equiv \frac{\left[m_{\PSq}^2-(m_\cPq+m_{\chiz_1})^2\right]\left[m_{\PSq}^2-(m_\cPq-m_{\chiz_1} )^2\right]}{m_{\PSq}^2}.$$ In the limit of massless megajets we then have ${\ensuremath{M_\Delta}\xspace}= (m_{\PSq}^2
- m_{\chiz_1}^2)/m_{\PSq}$ and the four-momenta reduce to $$\begin{aligned}
p_{\cPq_1} &= \frac{{\ensuremath{M_\Delta}\xspace}}{2}(1,\; \hat{u}_{\cPq_1}),\\
p_{\cPq_2} &= \frac{{\ensuremath{M_\Delta}\xspace}}{2}(1,\; \hat{u}_{\cPq_2}).\end{aligned}$$

The razor variable ${\ensuremath{M_R}\xspace}$ is defined in terms of the momenta of the two megajets by $$\label{eq:MRstar}
 {\ensuremath{M_R}\xspace}\equiv
 \sqrt{(\abs{\vec{p}_{\cPq_{1}}}+\abs{\vec{p}_{\cPq_{2}}})^2 -(p^{\cPq_1}_z+p^{\cPq_2}_z)^2}.$$ where $\vec{p}_{\cPq_i}$ is the momentum of megajet $\cPq_i$ (${i} = 1,2$) and $p^{\cPq_i}_z$ is its component along the beam direction.

For massless megajets, ${\ensuremath{M_R}\xspace}$ is invariant under a longitudinal boost. It is always possible to perform a longitudinal boost to a razor frame where $p^{\cPq_1}_z+p^{\cPq_2}_z$ vanishes, and ${\ensuremath{M_R}\xspace}$ becomes just the scalar sum of the megajet three-momenta added in quadrature. For heavy particle production near threshold, the three-momenta in this razor frame do not differ significantly from the three-momenta in the actual parent particle 