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A beta-decay spectrum gives the distribution of emitted electron or positron energies in beta decay. In ordinary three-body beta decay, the charged lepton shares the available energy with a neutrino and the recoiling daughter nucleus, producing a continuous spectrum.

For a simple allowed decay, its shape includes phase-space factors, an electron Coulomb correction and an endpoint set by the available energy. Schematically, $d\Gamma /dE\propto F(Z,E)pE(E_0-E{)}^2$ when the neutrino mass is neglected and $E$ is total electron energy. Nuclear matrix elements and forbiddenness can modify this expression. The continuum was central to understanding the missing energy carried by the neutrino.

For a massless incident neutrino and a stationary target electron, the invariant squared center-of-mass energy is $s=m_e^2+2m_e{E}_{\nu }$ in natural units. Production of a charged lepton of mass $m_{\ell }$ and a massless neutrino requires $s\ge m_{\ell }^2$, giving the displayed threshold. For the electron flavour the threshold is zero; for heavier charged leptons it is enhanced by the small stationary-target mass.

In normalization $Q=T_3+Y$, the left quark doublet has $Y=1/6$, right up/down singlets have $2/3,-1/3$, the left lepton doublet has $-1/2$, the right charged lepton has $-1$, and the Higgs doublet has $1/2$. The alternative convention $Q=T_3+\hat{Y}/2$ uses $\hat{Y}=2Y$. Up-type Yukawa interactions use the conjugate doublet of hypercharge $-1/2$.

A spin-zero parent decaying through a chiral weak charged current to a charged lepton and a massless neutrino cannot supply the helicities selected by a massless interaction. A charged-lepton mass insertion makes the amplitude proportional to $m_{\ell }$ and the decay width proportional to $m_{\ell }^2$. A spin-one W boson does not suffer this suppression.

A neutrino state selected by the charged lepton accompanying a weak charged current production or detection process. It is generally a coherent quantum superposition of mass eigenstates, rather than a state of definite propagating mass.

A lepton is a Standard Model fermion without color charge. The charged leptons have electric charge $-1$ and their corresponding neutrinos have charge zero. There are three generations; their antiparticles carry opposite electric charge and lepton number. Charged leptons and neutrinos have lepton number $+1$, with the corresponding antiparticles carrying $-1$.

For an unpolarized parent charged lepton undergoing a three-body leptonic V-A interaction decay, neglecting daughter masses and radiative corrections, the normalized charged-daughter energy distribution in the parent rest frame is $f(x)=2x^2(3-2x)$, where $x=2E_{\ell }^{\ast }/m_{\mathrm{parent}}$ lies in $[0,1]$. Direct integration gives ${\int }_0^{1}f(x)dx=1$ and ${\int }_0^{1}xf(x)dx=7/10$. A specified parent spin orientation also produces angular terms, so this one-variable law alone does not describe polarized samples.

How can the relative abundance of tracks and showers distinguish hypotheses? Consider an original teaching model, with chosen inputs throughout. Condition on a sample of $N$ selected events, each assigned exactly one of two labels: track or shower. Multitrack events and unclassified events lie outside this model. A shower label need not identify an electron; background can imitate it. Conversely, a true electron interaction need not receive a shower label.

Let $b$ be the background fraction among selected events. Among the selected signal events, let $f$ be the fraction whose associated charged lepton is an electron, with the remainder associated with a muon. These are fractions after event selection, rather than fractions of the incident neutrino particle flux. Define the conditional probabilitiesHere $a$ and $r$ describe classification within the selected sample. They are not measured historical detection efficiencies. The law of total probability gives the overall shower probabilityThis formula is the bridge between an interaction hypothesis and a visible label. It keeps beam contamination, background and mistaken classification conceptually distinct: genuine electron neutrino interactions belong in $f$, while non-signal events belong in $b$.

Assume independent random variables for the event labels, all with this same $q$. If $K$ counts shower labels, every specified pattern with $k$ showers has probability $q^k(1-q{)}^{N-k}$. Counting those patterns with a binomial coefficient yields the binomial distributionThe track count is $N-K$, so its expected value is $N(1-q)$. At fixed $N$, the two counts cannot fluctuate independently. The variance above measures sampling fluctuations under the stipulated model, not uncertainty in $a$, $r$, $b$ or $q_b$.

Choose $N=40$, $a=0.90$, $r=0.05$, $b=0.10$ and $q_b=0.40$. Compare a signal with only muon-associated interactions ($f=0$) against one with equal selected electron- and muon-associated interactions ($f=1/2$). These give $q=0.085$ and $q=0.4675$, respectively. Their expected shower counts are $3.4$ and $18.7$; their expected track counts are $36.6$ and $21.3$. A background-only model with the specified $q_b$ gives $16$ showers and $24$ tracks on average. These are expectations, so they need not be integers.

For a hypothetical observation $K=4$, the likelihood ratio of the two signal models isThe binomial coefficient cancels because the same hypothetical sample is used in both likelihoods. This ratio is not the probability that either hypothesis is true. A separate original check gives $\mathrm{Pr}(K\le 5)\simeq 0.880$, $4.57\times 10^{-6}$ and $1.44\times 10^{-4}$ for the muon-associated, equal-mixture and specified background-only models, respectively.

Why are the controls essential? If the background-only model may choose any $q_b$, taking $q_b=K/N$ can fit the observed shower fraction. Counts alone then fail to distinguish it from a signal model. Independent background measurements and calibrated particle identification constrain that freedom. Changing the decay path additionally tests where the candidate-producing particles originate. The power of the experiment comes from combining apparatus interventions with event classification; the small binomial distribution model isolates only the latter part of that reasoning.

A neutrino label defined by the associated charged lepton in weak charged current production or detection. A produced flavour state can evolve through neutrino oscillation.

Let $v_H=⟨H⟩$ be the vacuum expectation value in the convention $M=v_H\lambda$. For canonically normalized fermion kinetic terms, physical charged lepton masses are the nonnegative values in the singular value decomposition of $M$, equivalently the square roots of the eigenvalues of $M^{†}M$.

In case (i),The opposite algebraic eigenvalues $y$ and $-y$ of the doublet block do not represent different physical masses. The unavoidable twofold degeneracy conflicts with the three distinct nonzero measured charged lepton masses, for any assignment of the three families.

In case (ii),One can fit the tau lepton mass by choosing $|v_H{y}_3|$, but the electron and muon remain massless. Thus neither exact dihedral group assignment with this single singlet complex scalar field reproduces the observed spectrum. Additional symmetry-breaking Yukawa interactions or other sources of mass are required; a basis change cannot remove the degeneracy or change the matrix rank.

Work in the W boson rest frame, taking the spin-quantization axis along its laboratory motion. Neglect charged-lepton and neutrino masses. The weak charged current produces a negative-helicity ${\ell }^{-}$ and a positive-helicity antineutrino. Their momenta are opposite, so both spin Vectors point opposite to the charged lepton's direction $\hat{\mathbf{n}}$; the final total spin projection along $\hat{\mathbf{n}}$ is $-1$.

Let ${\lambda }_W=+1,0,-1$ denote the parent spin projection along $z$. In the given probability amplitude, use $j=1$ and the rotation element $d_{-1,{\lambda }_W}^1(\theta )$. The supplied spin-one rotation elements giveThe transverse states have spin projections $\pm 1$; the longitudinal state has projection zero, not total spin zero. The original diagram below distinguishes momenta from spins and shows the corresponding normalized angular profiles.

Use leptonic decays: an isolated high transverse momentum positive electron or muon, together with substantial missing transverse momentum from the neutrino, provides a clean $W^{+}$ signature. The lepton transverse momentum has a Jacobian scale near $m_W/2$, and the lepton–missing-momentum transverse mass has an edge near $m_W$, broadened by width, recoil and resolution. Lepton identification, isolation and a prompt primary-vertex origin suppress Jets faking leptons and heavy-flavour decays. Missing-momentum quality requirements suppress mismeasured multijet events; a veto on a second oppositely charged lepton suppresses $Z\to {\ell }^{+}{\ell }^{-}$ when both are reconstructed. Hadronic $W$ decays have a much larger ordinary-QCD background.

The charged-current process iswith a $t$-channel $W^{-}$ emitted at the electron-to-neutrino vertex and absorbed at the antimuon-to-antineutrino vertex. It is not an $s$-channel Annihilation of the unlike-flavour charged leptons into one $W$.

At high momentum transfer the incident electron's left-chiral fraction tends to $4/5$. In the collision centre-of-mass frame, the incoming antimuon moves along $-z$ and its charged-current allowed right helicity has spin along $-z$. A rest-frame spin at Angle $\theta$ to $+z$ projects onto that spin direction with probability ${\sin }^2(\theta /2)$. Therefore, if ${\sigma }_0$ denotes the cross section for a pure left-helicity electron and fully selected antimuon spin,It vanishes at $\theta =0$ in the ultrarelativistic selection limit and is maximal at $\theta =\pi$. Finite-mass corrections can fill the suppressed orientation. The absolute normalization depends on beam energy and the exchanged-boson propagator; the spin-angle dependence is what is fixed here.

For an incoming ${\mu }^{-}$, a putative $e^{-}{\mu }^{-}\to {\nu }_e{\nu }_{\mu }$ process would change total electric charge from $-2$ to zero. It is forbidden: both negative charged leptons would have to emit a $W^{-}$, and a single exchanged charged boson cannot connect those two emission vertices. Ordinary elastic $e^{-}{\mu }^{-}\to e^{-}{\mu }^{-}$ scattering remains possible through $t$-channel photon or $Z$ exchange, shown in the figure, but this is not the charged-current reaction of part d. The neutral-current weak contribution is distinct from the electromagnetic one.

Let ${\theta }^{\star }$ be the angle from the incoming up quark, along the proton direction, to the outgoing charged antilepton in the W boson rest frame. The initial and final spin projections in ii are $-1$ and $+1$. The Wigner small-d matrix therefore givesAfter squaring and normalizing over $c=\cos {\theta }^{\star }$, the charged-lepton angular distribution isIts integral is one since ${\int }_{-1}^1(1-c{)}^{2}dc=8/3$. The charged antilepton is preferentially emitted toward the antiproton. Measuring the angle from the incoming down antiquark instead replaces $c$ by $-c$ and produces $(1+c{)}^2$; the axis convention is part of the answer. Per unit solid angle the normalized density is $3(1-\cos {\theta }^{\star }{)}^2/(16\pi )$.

Neglect the participating quark and lepton masses, so that the weak charged current selects a left-helicity $u$ and a right-helicity $\bar{d}$. Choose $+z$ along the proton beam in the partonic centre-of-momentum frame. Then the up quark moves along $+z$ with spin projection $-1/2$, while the down antiquark moves along $-z$ with helicity $+1/2$ and also has spin projection $-1/2$ along $+z$. ThusThe collinear two-body initial state has no orbital projection about its collision axis. In this approximation the produced W boson therefore has the indicated transverse polarization; its longitudinal spin state is absent. The unpolarized hadron beams contain other constituent spin states, but these do not participate in this massless charged-current subprocess.

In the final state the Positron has helicity $+1/2$ and the electron neutrino has helicity $-1/2$. Along the outgoing charged-lepton direction their combined angular-momentum projection is $+1$, because the oppositely moving neutrino has spin projection $+1/2$ on that axis. Finite masses, noncollinear production and sea contributions modify the ideal configuration.

Initial-state radiation of gluons gives the W boson nonzero transverse momentum. Its transverse Lorentz boost then changes the charged-lepton $p_T$, washing out the sharp $M_W/2$ edge and permitting events above it. The recoil must be modelled with quantum chromodynamics and constrained by measured hadronic activity. Independently, the finite W boson decay width spreads its actual invariant mass, also smearing the fixed-mass endpoint. Either is a theoretical broadening mechanism; neither is a detector-resolution effect.

Take an on-shell W boson of fixed mass $M_W$, zero transverse recoil, negligible charged-lepton mass, and unrestricted angular acceptance. In its rest frame the charged lepton has momentum magnitude $p_0=M_W/2$. A longitudinal Lorentz boost to the laboratory preserves transverse momentum, soBoth angular roots contribute to the same $p_T$. Set $x=p_T/p_0$ and $a=\sqrt{1-x^2}$. The change-of-variable Jacobian determinant has $|dc/d{p}_T|=x/(p_{0}a)$, and the two-root angular weight is $\frac{3}{8}[(1-a{)}^2+(1+a{)}^2]=\frac{3}{4}(2-x^2)$. ConsequentlyThis is the ideal W-decay Jacobian peak. The divergence at the endpoint is integrable; it is a change-of-variable effect, not an infinite decay probability. Reversing the polar axis does not change this transverse momentum density, because the sum of the two roots removes the term odd in $c$.

The useful modes are $\boxed{W^{+}\to e^{+}{\nu }_e\text{and}{W}^{+}\to {\mu }^{+}{\nu }_{\mu }}$ and their charge conjugates. An electron gives an isolated track and localized electromagnetic shower; a muon gives an isolated penetrating track. Both allow a well-calibrated charged-lepton transverse momentum and a missing transverse momentum measurement for the neutrino.

The tau lepton decays before detection, usually adding further neutrinos. Its visible momentum is only part of the original tau momentum, so the simple two-body endpoint is lost. For the direct electron and muon modes, the transverse massalso has an ideal endpoint at $M_W$ and avoids needing the unmeasured neutrino longitudinal momentum. Fits to this distribution or the charged-lepton $p_T$ spectrum determine $M_W$.

In a proton-proton collision, associate the reconstructed neutral B meson with the accompanying opposite-side bottom hadron. A direct semileptonic decay $b\to c{\ell }^{-}\bar{\nu }$ tags a $b$ on that side and hence an initial $\bar{b}$, or $B^0=d\bar{b}$, on the signal side. The opposite charged lepton tags the conjugate assignment. Charged kaons from the opposite-side decay chain and same-side fragmentation tracks can supply additional statistical flavour tagging information; the common CP final state alone does not reveal the production flavour.

A magnetic spectrometer with precise silicon vertex tracking, charged-particle identification and muon or electron identification is suitable. A forward geometry can exploit the boosted bottom-hadron pairs at the LHC. It reconstructs displaced decay vertices, measures the signed tagging tracks, and infers the signal proper time from $t\simeq L{m}_B/|{\mathbf{p}}_B|$. The supplied lifetime corresponds to a rest-frame mean flight length $c\tau \simeq 0.45\mathrm{mm}$, enlarged by the boost.

Even a perfect detector cannot remove physical mistags. An opposite-side neutral bottom meson can oscillate before its semileptonic decay, reversing the lepton charge relative to its production flavour. A cascade $b\to c\to {\ell }^{+}$ produces the charge opposite to a direct $b\to {\ell }^{-}$ decay. Additional bottom pairs or unrelated fragmentation particles can also make the chosen association incorrect. For mistag probability $w$, the measured asymmetry is diluted by $D=1-2w$; this is why the tagging calibration is part of a CP measurement.