Relativistic collision mass threshold ($W\ge c^2{\sum }_i{\mu }_i$)

If the total incoming four-momentum is timelike and has centre-of-momentum energy $W$, production of particles with rest masses ${\mu }_i$ requires $W\ge {\sum }_i{\mu }_i{c}^2$. In the centre-of-momentum frame, total final energy is a sum of $\sqrt{{\mu }_i^2{c}^4+c^2|{\mathbf{p}}_i{|}^2}$, each at least its rest energy. For massive products equality occurs with all at rest in that frame. A single product requires equality of the entire invariant mass, not just an inequality; a massless product cannot have nonzero energy and be at rest.

For an incident particle of rest mass $m_1$ and total energy $E_1$ hitting a stationary target of mass $m_2>0$, four-momentum conservation gives $W^2=m_1^2{c}^4+m_2^2{c}^4+2m_2{c}^2{E}_1$. Applying the relativistic collision mass threshold gives the displayed necessary condition. The incoming energy must also satisfy $E_1\ge m_1{c}^2$; other conserved quantum numbers or reaction dynamics can impose further restrictions.

For a photon of energy $h\nu$ hitting a stationary target of mass $m_2$, the incoming invariant energy satisfies $W^2=m_2^2{c}^4+2m_2{c}^{2}h\nu$. The massive multi-product threshold is the displayed value when ${\sum }_i{\mu }_i>m_2$. If the algebraic bound is nonpositive, the lower limit of allowed positive frequencies can be zero. A single final product instead fixes the frequency by invariant-mass equality, and a final massless particle can make a nominal threshold an infimum rather than an attained value.

// Target: classical-mechanics.bigb

For a photon incident on a stationary Electron, producing an additional electron and positron leaves three massive particles in the final state. Four-momentum conservation gives $W^2=m_e^2{c}^4+2m_e{c}^2{E}_{\gamma }$ for the centre-of-momentum energy. Its minimum is $3m_e{c}^2$, so the incident threshold is $4m_e{c}^2$. At threshold all final particles are at rest in their centre-of-momentum frame, which moves at $4c/5$ relative to the target laboratory frame. This differs from a heavy-target threshold because the electron target's recoil cannot be neglected.

A photon hitting a resting mass $m$ has squared total energy invariant $m^2{c}^4+2m{c}^{2}E$. Producing a larger mass $M$ and an outgoing photon requires this invariant to exceed $M^2{c}^4$ if the photon has positive energy. The displayed threshold is therefore the limiting infimum as the final photon becomes soft, not an attained physical state with a zero-energy photon. Above threshold the outgoing photon energy in the centre-of-momentum frame is positive.