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A stationary particle of rest mass $m$ absorbing a photon of energy $E$ acquires total invariant massThis follows from four-momentum conservation, since the incoming photon has momentum magnitude $E/c$. If the system splits into two equal particles, the largest kinematically permitted daughter rest mass is $M/2$, attained when both are at rest in the centre-of-momentum frame. Their common laboratory speed at threshold is $cE/(m{c}^2+E)$. Above this daughter-mass threshold there is insufficient centre-of-momentum energy; below it, relative kinetic energy is available.

The incident four-momentum has energy $E$ and spatial momentum $p=\gamma mv$, while the stationary target has energy $m{c}^2$ and zero spatial momentum. By the relativistic energy-momentum relation, $c^2{p}^2=E^2-m^2{c}^4$. For the fixed-target threshold for production of equal-mass particles, the squared total centre-of-momentum energy isIn the centre-of-momentum frame, each of the final $N$ particles has energy at least its rest energy $m{c}^2$. Four-momentum conservation therefore requires $W\ge Nm{c}^2$, and thusFor the physically relevant $N\ge 2$, this is equivalent to $\gamma \ge (N^2-2)/2$. The speed is an increasing function of $\gamma$, soConsequentlyEquality is the kinematic threshold at which all final particles are at rest in their common centre-of-momentum frame. It shows why a collision with a stationary target needs more incident energy than merely the additional rest energy: some laboratory energy must remain as collective motion. A one-particle final state of the specified mass, with no other outgoing energy, is impossible because its invariant energy would be $m{c}^2$, whereas the two-particle initial state has invariant energy at least $2m{c}^2$.

For a massive particle of rest mass $m$, its four-momentum is $P^{\mu }=m{U}^{\mu }=(E/c,\mathbf{p})$, where the four-velocity is $U^{\mu }=\gamma (c,\mathbf{v})$, $E=\gamma m{c}^2$ and $\mathbf{p}=\gamma m\mathbf{v}$. With metric signature $(-,+,+,+)$, $P\cdot P=-m^2{c}^2$. A photon has four-momentum $(E/c,(E/c)\hat{\mathbf{n}})$, a future-pointing null vector. Four-momentum conservation states that the total incoming and outgoing four-momenta agree for an isolated interaction; this includes both energy and momentum conservation in every inertial frame.

Take the incoming photon direction as $x$. Its energy is $\hslash \omega =\mu m{c}^2$, so the total initial four-momentum isThe invariant mass available after photon absorption is therefore $M=m\sqrt{1+2\mu }$. In the centre-of-momentum frame, the two equal daughters have opposite momenta. Their total energy is at least their combined rest energy $2\alpha m{c}^2$, with equality precisely when both daughters have zero momentum in that frame. Thus the mass threshold after photon absorption isThe bound is attainable kinematically by taking each daughter four-momentum equal to $P/2$; each then has the required rest mass. It immediately gives ${\alpha }_{\max }\to 1/2$ as $\mu \to 0$.

At this threshold both daughters move with the centre-of-momentum frame. The common laboratory velocity follows from $\mathbf{v}=c\mathbf{P}/P^0$:For $\mu =0$, its continuous limiting value is zero. At smaller allowed daughter rest masses, some of the centre-of-momentum energy appears as their relative kinetic energy, so their four-momenta need not be $P/2$.

The result grows linearly with the squared centre-of-momentum energy, $\sigma \propto G_F^{2}s$. This growth eventually violates partial-wave unitarity and signals the breakdown of the pointlike Fermi interaction. At energies comparable to $M_W$ or $M_Z$, the full gauge-boson propagators must replace the contact interaction; the renormalizable electroweak theory then softens the high-energy behavior.

In the Breit-Wigner resonance cross section, $E$ is the total centre-of-momentum energy, $E_0$ the resonance energy, and $p$ either incident particle's center-of-momentum momentum magnitude. The total decay width is $\Gamma =\hslash /\tau$, while ${\Gamma }_i$ and ${\Gamma }_f$ are partial widths into the entrance and specified exit channels. Their ratios to $\Gamma$ are the corresponding branching fractions. The factor $g$ averages over incident spin states and sums over the resonance's magnetic substates.

In natural units, $\hslash =c=1$, the resonant partial-wave cross-sectional scale is an area proportional to the squared reduced de Broglie wavelength, $1/p^2$. Equivalently a partial-wave scattering amplitude carries one factor $1/p$, and its squared modulus supplies $1/p^2$. In SI area units the same scale is $(\hslash /p{)}^2$; if $p$ is expressed in energy units, multiply the final inverse-energy-squared answer by $(\hslash c{)}^2$.

For unpolarized spin-half incident muons, the unpolarized resonance spin factor isHence $g_Z=3/4$ and $g_H=1/4$. The outgoing tau spins do not supply another independent multiplicity here: their allowed final states are already included in ${\Gamma }_f$.

Near a narrow resonance, treat the smooth prefactor $p^{-2}$ and the partial widths as constant across the line. At the peak the denominator is ${\Gamma }^2/4$. Half its height occurs when $(E-E_0{)}^2+{\Gamma }^2/4={\Gamma }^2/2$, namely at $E=E_0\pm \Gamma /2$. Thus the full width at half maximum is $\Gamma$ within the Breit-Wigner narrow-line approximation. A varying prefactor or convolution with the beam-energy distribution can modify an observed line shape.

The production amplitude has a muon pair joined to a Higgs propagator through their Yukawa coupling, followed by the bottom-quark pair at a second Yukawa vertex. The separate $H\to b\bar{b}$ graph is a tree-level Higgs–fermion vertex. There are no tree-level $Hgg$ or $H\gamma \gamma$ vertices in the renormalizable Standard Model: their lowest nonzero graphs are loops. A quark triangle, dominated by the top contribution, produces two gluons. Charged-fermion triangles contribute to two photons; charged-W loops also contribute and are dominant in the Standard Model amplitude. Photon permutations and the W contact topology belong to the full amplitude; the diagram shows these topologies rather than claiming one triangle alone is the entire gauge-invariant W contribution.^[ref]

Use the $+---$ metric signature, with the time component of four-momentum equal to $E/c$. The incident particle has energy $\gamma m{c}^2$ and momentum of magnitude $mc\sqrt{{\gamma }^2-1}$. Adding the stationary target's four-momentum gives total laboratory quantitiesThe centre-of-momentum energy $W$ follows from the invariant Minkowski norm of the total four-momentum:At the relativistic collision mass threshold, the two final particles have zero relative velocity in their centre-of-momentum frame. Their total energy there is therefore $2\alpha m{c}^2$. Four-momentum conservation requires $W\ge 2\alpha m{c}^2$, soThe condition $\alpha >3/2$ ensures ${\gamma }_{\min }>1$. This is the minimum required Lorentz factor; larger values allow final kinetic energy in the centre-of-momentum frame. Equivalently, the incident laboratory kinetic energy at threshold is $({\alpha }^2-9/4)m{c}^2$.

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.

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.