Expand near a physical threshold

Let $\sqrt s=m_1+m_2+\epsilon$ with positive daughter masses and excess energy much smaller than either mass. At threshold both daughters are at rest in their common center-of-momentum frame. Just above it, their nonrelativistic kinetic energies add to $p^{*2}/(2\mu)$, where $\mu=m_1m_2/(m_1+m_2)$ is the reduced mass. This gives the leading square-root momentum opening. The same result follows by expanding both factors of the exact Källén function instead of assuming either daughter remains stationary.

The positivity of both masses matters. If one daughter is massless and the other has mass m, the exact momentum is $(s-m^2)/(2\sqrt s)$. With $\sqrt s=m+\epsilon$, its leading term is $\epsilon$, not the square root of excess energy. Applying the reduced-mass formula with zero reduced mass would therefore lose the leading behavior. A useful expansion states what is small, which quantities are held fixed, and which mass limits are excluded before discarding terms.

For a fixed target of mass $m_b$, incoming total energy gives $s=m_a^2+m_b^2+2m_bE_a$. For a final state with an accessible continuum above mass sum $M_f$, the minimum collision energy must satisfy both $\sqrt s\ge M_f$ and the incoming on-shell requirement $\sqrt s\ge m_a+m_b$. In an endothermic channel, $M_f\ge m_a+m_b$, solving $s=M_f^2$ gives the beam requirement. For a lighter final mass sum, the kinematic kinetic-energy threshold is zero, rather than the negative value a naive mass-sum substitution could suggest. At a strictly endothermic threshold, the final particles can all be at rest relative to each other in their center-of-momentum frame, while moving together in the laboratory to carry the initial momentum.

This distinction explains why simply paying the new particle's rest energy is insufficient in a fixed-target reaction. Some laboratory energy must remain in overall recoil. The mass-sum condition is necessary only for channels allowed by conserved charges and other exact selection rules; kinematics alone cannot make a forbidden process occur. For hadronic beams, thresholds for a chosen constituent subprocess also depend on available momentum fractions and remnants. State whether a calculation refers to an exclusive complete reaction or an idealized elementary collision.

For a regular short-range two-body channel, a higher outgoing orbital angular momentum suppresses the amplitude with additional momentum powers. Combining that amplitude behavior with the opening phase space produces the threshold power in the equation. Long-range Coulomb effects, a nearby pole, singular interactions, or an initial channel also at threshold can modify the law. It is an approximation with a dynamical domain, not a consequence of energy conservation alone. A fitted exponent should therefore be interpreted with the interaction and flux model stated.

Invariant-mass endpoints in multibody decays can constrain mass combinations even when one product is invisible. Their ideal positions follow from a constrained maximization, but detector smearing and selection can move an observed edge or remove events near it. A cascade introduces additional on-shell constraints and can produce an endpoint below the naive parent-minus-daughter mass difference. Before extracting a mass, identify the topology, invisible masses, and which intermediate states are actually on shell. A measured edge is evidence about that whole model rather than a mass measurement independent of it.