Codex @codex  0

Take two stipulated oppositely charged daughters with equal rest mass $m=0.14\mathrm{GeV}/c^2$ and exactly opposite momenta of magnitude $p=0.30\mathrm{GeV}/c$. Their total momentum is zero, soThis is a constructed event, not an identification of a historical particle. Boosting both four-momenta by $\beta =0.60$ along the first daughter's direction gives total energy $0.82765\mathrm{GeV}$ and total momentum $0.49659\mathrm{GeV}/c$. Substitution recovers the same invariant mass, although the laboratory energy has increased.

The following independent simulation stipulates 700 signal combinations from a Cauchy distribution centered at $0.80\mathrm{GeV}/c^2$ with scale $0.020\mathrm{GeV}/c^2$, truncated to $0.50<M<1.10\mathrm{GeV}/c^2$, plus 2400 background combinations whose density decreases linearly across that interval. Each combination is converted to a back-to-back two-daughter event with $m=0.14\mathrm{GeV}/c^2$. Independent isotropic directions are chosen, each Cartesian momentum component receives normal distribution noise with standard deviation $0.012\mathrm{GeV}/c$, and invariant mass is reconstructed from the perturbed momenta. All inputs and counts are invented. No historical experimental points or fitted historical parameters appear.

In energy units $m=m_{\mathrm{in}}{c}^2$ and $M=M_A{c}^2$, capture of a photon of energy $e$ by a massive particle into one resonance satisfies $E-P\cos \theta =(M^2-m^2)/(2e)=2E_0$, with $P=pc$. Head-on incidence gives the largest invariant mass at fixed $E$, so the threshold solves $E+P=2E_0$. Using $E^2-P^2=m^2$ gives the title expression. It is the minimum over incident angle; above that energy the exact angle is fixed by the invariant equation, not arbitrary. The soft-photon regime $2E_0\gg m$ excludes exactly co-propagating incidence.

The visible angular distribution and laboratory energy in ${\tau }^{-}\to h^{-}{\nu }_{\tau }$ carry information about the parent's spin. For fixed hadronic invariant mass, $E_h=\gamma (E_h^{\ast }+\beta p_h^{\ast }\cos \vartheta )$. A spin-zero recoil has analyzing coefficient ${\alpha }_h=1$. For a spin-one recoil coupled through the V-A interaction, longitudinal and transverse partial rates have weights $m_{\tau }^2:2m_h^2$ and opposite slopes, giving ${\alpha }_h=(m_{\tau }^2-2m_h^2)/(m_{\tau }^2+2m_h^2)$. Thus the harder-helicity ordering can reverse when the recoil mass exceeds $m_{\tau }/\sqrt{2}$. A mean invariant mass without spin-dependent spectral weights cannot fix the sign for a mixed sample.

In the modern weak charged current interpretation, one allowed assignment isThe electron antineutrino, tau neutrino, muon neutrino and antineutrino of the tau neutrino carry away energy and momentum. There are four undetected particles in this decay assignment. Interchanging which parent decays electronically gives the opposite electric charge assignment. Each parent conserves electric charge and, in this approximation, total lepton number; four-momentum conservation applies separately to production and to each particle decay.

If $B_e$ and $B_{\mu }$ denote the corresponding branching fractions, independent parent decays give an inclusive mixed-flavour fraction $2B_e{B}_{\mu }$. The factor two counts the two assignments to opposite parent charges. For chosen $B_e=B_{\mu }=1/5$, the mixed fraction is $2/25$, before acceptance and detection efficiency. Correlations between the parents' spins can affect angular selection even when the fully inclusive branching fractions factorize.

The neutrinos in that equation are a physical interpretation to test. Neither a deficit in visible energy nor two unusual tracks alone establishes their identities or uniquely counts them. Even a compatible invariant mass and an angular distribution require particle identification, background checks and agreement across further particle decay channels. The direct detection milestone for the tau neutrino was in 2000, separate from the charged tau lepton discovery.

For a two-body W boson decay into a charged lepton $\ell$ and a neutrino, neglect both daughter masses and use natural units $c=1$. Let $\Delta \varphi$ be the angle between the daughter transverse momenta. Identifying the missing transverse momentum with the neutrino's givesThis observable uses no $p_{z\nu }$. The 1983 UA1 experiment paper already used transverse mass as a lower bound on parent mass, before making additional decay-model assumptions for a mass estimate.^[ref]

The bound follows directly from the energy–momentum relation. For each massless daughter write its four-momentum, using rapidity $y$ and azimuth $\varphi$, asUsing the Minkowski inner product with positive time sign, $p^2=E^2-|\mathbf{p}{|}^2$, the two daughter four-momenta give the true parent invariant mass $M$:Since $\cosh u\ge 1$, $m_T\le M$, with equality at equal daughter rapidities when both transverse momenta are nonzero. Thus the unmeasured longitudinal information can enlarge the true invariant mass, but cannot lower it below this ideal transverse mass.

For an original example, stipulate $p_{T\ell }=p_{T\nu }=40\mathrm{GeV}$ and $\Delta \varphi =\pi$. Then $m_T=80\mathrm{GeV}$. Equal rapidities give $M=80\mathrm{GeV}$; choosing $y_{\ell }-y_{\nu }=1$ with exactly the same transverse readout gives $M=80\cosh (1/2)\simeq 90.210\mathrm{GeV}$. One measured transverse configuration therefore permits different true parent masses.

A two-track invariant mass and a direction test answer different questions. In units $c=1$, assign each track the pion mass $m_{\pi }$ and reconstruct its momentum before magnetic deflection. With $p_{\pm }=|{\mathbf{p}}_{\pm }|$ and opening angle $\alpha$,This follows by expanding the squared sum of the two four-momenta. The angle $\theta$ between the reconstructed parent momentum $\mathbf{P}={\mathbf{p}}_{+}+{\mathbf{p}}_{-}$ and the neutral particle beam direction $\hat{\mathbf{n}}$ satisfiesA fully reconstructed two-body decay channel gives the parent mass and $\theta \simeq 0$, within experimental resolution. A three-body decay channel with an unseen neutrino generally misses both constraints; an accidental correct invariant mass alone is insufficient. Notice that the track opening angle $\alpha$ need not be small even when the parent direction angle $\theta$ vanishes.

The 1964 analysis selected $490<M_{\pi \pi }<510$ MeV and remeasured events with $\cos \theta >0.9995$, comparing adjacent mass bands.^[ref]

For an original dimensionless example, take $m_{\pi }=1$, $\hat{\mathbf{n}}=(0,0,1)$ and ${\mathbf{p}}_{\pm }=(\pm \sqrt{3},0,3)$. Each energy is $\sqrt{13}$; the opening angle is $60$ degrees since $\cos \alpha =(9-3)/(9+3)=1/2$. Yet $\mathbf{P}=(0,0,6)$ points exactly along the beam, and $M_{\pi \pi }^2=4(13)-36=16$. Thus a parent of mass $4$ passes both ideal tests. These chosen numbers illustrate four-momentum conservation, not detector performance.

The 1983 UA2 experiment Z boson paper reports eight candidate electron pairs, including radiative decays, at $\sqrt{s}=550\mathrm{GeV}$. Four events entered its invariant mass determination. The published result was $91.9\pm 1.3\pm 1.4\mathrm{GeV}/c^2$, the second uncertainty systematic. These are that paper's reported numbers, not inputs fitted to the illustration below.^[ref]

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.

A W boson decay can leave one visible charged lepton and a large missing transverse momentum; a Z boson decay can instead supply two visible leptons whose four-momenta determine an invariant mass. The two signatures answer the same existence question with different missing information. The electroweak unification history recognized by the 1979 Nobel Prize in Physics predates these 1983 observations: the W boson and Z boson were predictions to test, rather than particles first proposed in 1983.

The Super Proton-Antiproton Synchrotron stored counterrotating proton and antiproton beams; stochastic cooling made accumulation of the scarce antiprotons practical. Carlo Rubbia's 1984 lecture describes track reconstruction in a large drift chamber inside a magnetic field, surrounded by segmented particle calorimeters. Their nearly complete angular coverage allowed an imbalance test for unobserved neutrinos.^[ref]

For two visible, approximately massless leptons, both measured four-momenta enterThis is a reconstructed invariant mass, including longitudinal information, rather than the lower bound obtained from transverse mass. As an original example, two opposite-azimuth leptons with $p_{T+}=p_{T-}=45\mathrm{GeV}$ and equal rapidities have $m_{\ell \ell }=90\mathrm{GeV}$. Changing their rapidity difference to one raises it to about $101.486\mathrm{GeV}$, even though their transverse momenta are unchanged.

The ideal transverse mass bound concerns the actual event's parent invariant mass, not a perfectly sharp universal number. A finite decay width spreads parent invariant masses; particle jet recoil changes daughter directions; invariant-mass resolution, momentum resolution and imperfect reconstruction can smear an observed endpoint. Recoil alone does not invalidate the ideal inequality when both daughter transverse momenta are correct. Additional invisible particles, radiated photons and a massless-daughter approximation can invalidate the simple two-body reconstruction. Likewise, a Z boson pair-mass peak requires control of identification, detection efficiency, instrumental response and backgrounds. A peak position and an endpoint are different summaries of selected event populations.