Transverse momentum (${\mathbf{p}}_T=(p_x,p_y)$)

Choose the beam as the $z$ direction. A particle's transverse momentum is its momentum projected onto the perpendicular plane; $p_T=\sqrt{p_x^2+p_y^2}$. A Lorentz boost along the beam changes energy and longitudinal momentum, while leaving transverse momentum unchanged. Large transverse momentum identifies motion substantially across the beam rather than merely along it.

Missing transverse momentum is the negative vector sum of reconstructed visible transverse momenta, with detector corrections specified. Under the ideal hypotheses of zero total incoming transverse momentum, complete visible reconstruction and exactly one invisible neutrino, momentum conservation givesThe neutrino is not directly seen. Even under these ideal hypotheses, its longitudinal momentum $p_{z\nu }$ is not measured. Beam remnants can carry substantial unobserved longitudinal momentum; the colliding quarks' longitudinal momenta are also unknown event by event. Imposing an assumed parent mass can constrain $p_{z\nu }$, but that imports a hypothesis rather than measuring it. Finite coverage, poorly reconstructed particle jets and other invisible particles can also generate or modify missing transverse momentum.

For a two-body W boson decay into a charged lepton $\ell$ and a neutrino, neglect both daughter masses and use natural units $c=1$. Let $\Delta \varphi$ be the angle between the daughter transverse momenta. Identifying the missing transverse momentum with the neutrino's givesThis observable uses no $p_{z\nu }$. The 1983 UA1 experiment paper already used transverse mass as a lower bound on parent mass, before making additional decay-model assumptions for a mass estimate.^[ref]

The bound follows directly from the energy–momentum relation. For each massless daughter write its four-momentum, using rapidity $y$ and azimuth $\varphi$, asUsing the Minkowski inner product with positive time sign, $p^2=E^2-|\mathbf{p}{|}^2$, the two daughter four-momenta give the true parent invariant mass $M$:Since $\cosh u\ge 1$, $m_T\le M$, with equality at equal daughter rapidities when both transverse momenta are nonzero. Thus the unmeasured longitudinal information can enlarge the true invariant mass, but cannot lower it below this ideal transverse mass.

For an original example, stipulate $p_{T\ell }=p_{T\nu }=40\mathrm{GeV}$ and $\Delta \varphi =\pi$. Then $m_T=80\mathrm{GeV}$. Equal rapidities give $M=80\mathrm{GeV}$; choosing $y_{\ell }-y_{\nu }=1$ with exactly the same transverse readout gives $M=80\cosh (1/2)\simeq 90.210\mathrm{GeV}$. One measured transverse configuration therefore permits different true parent masses.