Constrain an invisible decay without inventing its momentum

If the initial total transverse momentum is negligible and all visible final-state momenta are correctly included, conservation gives $\mathbf p_T^{\mathrm{miss}}=-\sum_{\mathrm{visible}}\mathbf p_T$. This equals the vector sum of the invisible particles' transverse momenta. It does not identify how many invisible particles were present or how that vector should be split between them. Its magnitude is a transverse-momentum quantity; historical names involving missing energy do not supply an independent measurement of their energies or invariant masses.

The ideal equality requires a detector model. Limited angular coverage, unmeasured low-momentum activity, calibration errors, pileup, and a badly reconstructed energetic object can also create apparent imbalance. A complete event reconstruction includes the relevant soft contribution and recoil, with uncertainties and quality requirements. A large missing vector is therefore a candidate signature of invisible momentum, not a proof of a particular invisible particle. When using it in a decay constraint, justify which visible system and invisible system belong to the same parent.

For a visible and invisible system from one parent, write their energy and longitudinal momentum in terms of transverse energies and true rapidities. Their full invariant mass contains $\cosh(y_v-y_i)$ multiplying the product of transverse energies. Since this hyperbolic cosine is at least one, replacing it by one gives the smallest parent mass consistent with those transverse momenta and masses. The result is the two-system transverse mass, which differs from the single-particle coordinate quantity introduced earlier.

Equality requires equal visible and invisible rapidities, assuming their transverse energies are nonzero. If both systems are massless, the formula reduces to $m_T^2=2p_Tq_T(1-\cos\Delta\varphi)$. A zero invisible-mass hypothesis can still provide a conservative lower value when the true invisible mass is positive, but its endpoint interpretation changes. Several invisible particles from different parents generally cannot be assigned to one two-body decay just by using their total missing vector. A different topology needs different constraints.

Knowing an invisible transverse vector supplies two momentum components. A massless invisible particle still has an unknown longitudinal component, with energy then fixed by its mass shell. Supplying a known parent mass adds a constraint that can yield two solutions rather than one. Detector fluctuations can even give no real solution to an ideal on-shell equation. Selecting one root by a heuristic is an additional reconstruction prescription and must be evaluated for bias; it is not an implication of momentum conservation alone.

For a correctly modeled on-shell two-system decay, a transverse-mass distribution ends at the parent mass, but population near the endpoint depends on dynamics and acceptance. Finite parent width, measurement resolution, extra invisible particles, or wrong assignments can produce values beyond a simple nominal endpoint. An analysis should distinguish these effects from a breakdown of the kinematic inequality. The bound concerns the true event and its correct masses; a histogram of reconstructed estimates contains additional uncertainty and selection physics.