= Z pair mass and W transverse-mass limits
{c}
For two visible, approximately massless <leptons>, both measured <four-momenta> enter
$$
m_{\ell\ell}^2=(p_{\ell^+}+p_{\ell^-})^2
=2p_{T+}p_{T-}\bigl[\cosh(y_+-y_-)-\cos\Delta\phi\bigr].
$$
This is a reconstructed <invariant mass>, including longitudinal information, rather than the lower bound obtained from <transverse mass>. As an original example, two opposite-azimuth <leptons> with $p_{T+}=p_{T-}=45\,\mathrm{GeV}$ and equal <rapidities> have $m_{\ell\ell}=90\,\mathrm{GeV}$. Changing their <rapidity> difference to one raises it to about $101.486\,\mathrm{GeV}$, even though their <transverse momenta> are unchanged.
\Image[/w-transverse-mass-and-z-pair-readout.png]
{title=One visible lepton plus inferred transverse imbalance versus two reconstructed lepton four-momenta}
{description=Original functional cross sections and invented massless examples in natural units. Solid blue arrows represent visible charged leptons; orange patches represent calorimeter energy deposits. The dashed arrow is inferred missing transverse momentum and supplies no direct neutrino hit or longitudinal measurement. No historical events, mass spectra or fitted discovery data are reproduced.}
The ideal <transverse mass> bound concerns the actual event's parent <invariant mass>, not a perfectly sharp universal number. A finite <decay width> spreads parent <invariant masses>; <particle jet> recoil changes daughter directions; <invariant-mass resolution>, momentum resolution and imperfect reconstruction can smear an observed endpoint. Recoil alone does not invalidate the ideal inequality when both daughter <transverse momenta> are correct. Additional invisible <particles>, radiated <photons> and a massless-daughter approximation can invalidate the simple two-body reconstruction. Likewise, a <Z boson> pair-mass peak requires control of identification, <detection efficiency>, instrumental response and backgrounds. A peak position and an endpoint are different summaries of selected event populations.
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