Decay width ($\Gamma$)

In units with $\hslash =1$, the inverse lifetime of an unstable particle. A neutral meson mixing effective Hamiltonian operator incorporates decay through its anti-Hermitian part $-i\Gamma /2$.

The sum of the partial decay widths over mutually exclusive final channels. It determines the unstable state's mean lifetime through $\tau =\hslash /{\Gamma }_{\mathrm{tot}}$. A new open channel increases the total width even when an existing channel's partial width remains unchanged.

The normalized difference of two decay rates. An asymmetry between different final states is not automatically a test of CP violation; the physical meaning depends on which rates are compared.

The contribution of a specified decay channel to the total decay width. In an exponential-decay description, its probability rate is ${\lambda }_i={\Gamma }_i/\hslash$, the total width is $\Gamma ={\sum }_i{\Gamma }_i=\hslash /\tau$, and its branching fraction is ${\Gamma }_i/\Gamma$. Channels must be specified consistently, including any inclusive treatment of unresolved radiation. A partial decay width has energy units rather than inverse-time units.

For an unstable particle undergoing exponential decay in its rest frame, the decay waiting time has an exponential distribution with mean $\tau$. Its probability of surviving to time $t\ge 0$ is $e^{-t/\tau }$, and its total decay width is $\Gamma =\hslash /\tau$. In units $\hslash =1$, $\tau =1/\Gamma$.

The mean lifetime of an unstable particle measured in its rest frame, using elapsed proper time. In the exponential decay model, survival to proper time $\tau$ has probability $e^{-\tau /{\tau }_0}$. For constant Lorentz factor $\gamma$, time dilation gives laboratory mean lifetime $\gamma {\tau }_0$. A mean lifetime is a statistical mean, rather than a fixed decay time.

For a freely moving unstable particle of fixed speed $v=\beta c$ and proper mean lifetime ${\tau }_0$, the mean travelled distance before decay is ${\ell }_d=v\gamma {\tau }_0$. Survival beyond distance $L$ has probability $e^{-L/{\ell }_d}$, since elapsed proper time is $L/(\gamma v)$. For rest mass $m$ and three-momentum magnitude $p$, ${\ell }_d=p{\tau }_0/m$.

A branching fraction is the probability that a decay occurs through a particular mutually exclusive channel, equal to its partial decay width divided by the total decay width. Inclusive channel fractions sum to one. Rounding displayed fractions can make their numerical sum differ from one.

An inclusive branching fraction sums many exclusive final states. A clean reconstruction mode can be experimentally attractive without dominating the total decay rate. For example, the J/psi has useful electron-pair and muon-pair decays, but predominantly decays to hadrons inclusively. Detector acceptance, branching fraction and background rejection jointly determine the usefulness of a channel.^[ref]

The invariant decay amplitude is the transition matrix element for one incoming particle and a specified on-shell final state, with the overall momentum delta function factored out. For covariantly normalized states, write the connected S-matrix element as $i(2\pi {)}^4{\delta }^4(p_i-p_f)\mathcal{M}(i\to f)$. Its overall phase does not affect the decay width. The rate is $\Gamma =(2M_i{)}^{-1}{\sum }_{\mathrm{spins}}\int d{\Phi }_f|\mathcal{M}{|}^2$, with the initial spin average included when the initial state is unpolarized. For a spin-zero parent and an angle-independent two-body amplitude, the two-body Lorentz-invariant phase space gives $\Gamma =|\mathbf{k}|\sum |\mathcal{M}{|}^2/(8\pi M_i^2)$. In leptonic pion decay, contracting the axial hadronic matrix element with the chiral weak charged current makes this amplitude proportional to the charged-lepton mass, proving helicity suppression.

An absorptive phase arises from physical intermediate states or rescattering in a decay amplitude. For a pair of CP symmetry conjugate reactions with otherwise symmetric dynamics, this phase is the same in both amplitudes, whereas a weak phase changes sign. Their differing transformation laws allow interference to produce a direct CP asymmetry.

An unpolarized decay of a massive spin-one particle averages the squared amplitude over three parent spin states. Its polarization sum is $-{\eta }_{\mu \nu }+p_{\mu }{p}_{\nu }/m^2$. For a conserved external current the longitudinal $p_{\mu }{p}_{\nu }$ term drops out, but this does not replace the spin average by $1/2$: the physical massive parent still has three states.

For a pseudoscalar meson of mass $M$ decaying into a pseudoscalar meson of mass $m$ and two massless leptons through a weak charged current, the transverse massless leptonic current selects $f_{+}(s)$. With hadronic current normalized as $(p+k{)}^{\mu }{f}_{+}(s)+(p-k{)}^{\mu }{f}_{-}(s)$ and flavour coefficient $V$,The Källén function has mass dimension four, and the coefficient has mass dimension $-7$. This decay rate assumes the stated form-factor normalization and no additional isospin factor. It follows by the integrated massless leptonic tensor and the pion momentum change of variable in the parent's centre-of-momentum frame.

For two outgoing particles the Lorentz-invariant phase space is $d{\Phi }_2=(2\pi {)}^4{\delta }^{(4)}(p-k-k^{\prime })d^{3}k/((2\pi {)}^{3}2E_k)d^3{k}^{\prime }/((2\pi {)}^{3}2E_{k^{\prime }})$. With massless daughters in their center-of-mass frame, $d{\Phi }_2=d\Omega /(32{\pi }^2)$ and $\int d{\Phi }_2=1/(8\pi )$. This fixes normalization factors in decay widths and scattering rates.

A display of three-body final states in two independent pairwise squared invariant masses. For a parent of mass $M$ decaying into daughters of masses $m_1,m_2,m_3$, four-momentum conservation gives $s_{12}+s_{13}+s_{23}=M^2+m_1^2+m_2^2+m_3^2$. A narrow intermediate two-body resonance fixes one pairwise invariant mass and appears as a band.

At fixed $s=s_{13}$, use $E_2^{\ast }=(M^2-m_2^2-s)/(2\sqrt{s})$ and $E_3^{\ast }=(s+m_3^2-m_1^2)/(2\sqrt{s})$ in the 13-pair rest frame. The endpoint values of its relative cosine give the displayed Dalitz plot boundary. The momenta are fixed by the Källén function; $(m_1+m_3{)}^2\le s\le (M-m_2{)}^2$. Integrating $s_{23}^{+}-s_{23}^{-}$ gives the invariant-coordinate area, whose phase-space normalization still contains $M^{-2}$.

After integrating the overall orientation of a scalar-parent three-body decay, the Lorentz-invariant phase space is constant in two pairwise squared invariant masses. The two-body factorization $d{\Phi }_3=dsd{\Phi }_2(M;\sqrt{s},m_2)d{\Phi }_2(\sqrt{s};m_1,m_3)/(2\pi )$ and the relative-angle Jacobian give the displayed density. The event density includes $|\mathcal{M}{|}^2/(2M)$ and need not be flat: a resonance produces bands. Identical-particle counting supplies its usual global symmetry factor.

For three massive daughters with a smooth nonzero decay amplitude at the decay threshold, the integrated Lorentz-invariant phase space vanishes as $Q^2$, where $Q$ is the parent's mass minus the sum of daughter masses. The spatial Dirac delta function leaves six independent momentum coordinates. Rescaling each by $Q^{1/2}$ contributes $Q^3$, while the conservation of energy Dirac delta function contributes $Q^{-1}$. The factors $1/(2E)$ are smooth nonzero constants at the decay threshold. Thus the decay width also scales as $Q^2$ when the decay amplitude has a nonzero constant term. A vanishing decay amplitude, a massless daughter, or a threshold singularity can change this law. In three spatial dimensions the same argument gives $Q^{(3N-5)/2}$ for $N$ massive daughters.

For one daughter of mass $m$ and two massless daughters, set $r=m^2/M^2$, where $M$ is the parent mass. In natural units, integrating over all directions and all kinematically allowed final momenta givesThis follows by splitting the final state into a massless spectator and a two-particle cluster of invariant squared mass $s$, and using $d{\Phi }_3=d{\Phi }_2(M;\sqrt{s},0)d{\Phi }_2(\sqrt{s};m,0)ds/(2\pi )$. The expression counts phase space alone; a physical decay rate must weight it with the squared decay amplitude. Near threshold the dimensionless factor vanishes as $(1-r{)}^3/3$.

For two distinct massless final particles in their centre-of-momentum frame, integrating the energy-momentum delta function gives $d{\Phi }_2=d\Omega /(32{\pi }^2)$. A massless two-particle initial state has flux $2s$, hence $d\sigma /d\cos \theta =\overline{|\mathcal{M}{|}^2}/(32\pi s)$. Average only over the initial spin states physically present.

For two distinct massless final particles, $d{\Phi }_2/d\Omega =1/(32{\pi }^2)$ in the centre-of-mass frame. The invariant initial flux is $4p_1\cdot p_2=2s$. Their ratio gives the displayed differential cross-section. Equivalently the denominator is $128{\pi }^2{p}_1\cdot p_2$. Initial spins are averaged and final spins and colors summed when defining an inclusive unpolarized cross-section; averaging final colors would change the observable.