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Sections 14.2, 14.4 and 14.9.2 distinguish massless Standard Model fields, massive neutrino propagation and the Dirac–Majorana question used in a.

At an electron-positron collider tuned to the charmonium resonance, select the mixed leptonic tau lepton decaysand their charge conjugates. Require exactly two well-reconstructed oppositely charged tracks, one identified electron and one identified muon, little extra visible activity, visible energy below the beam total, and significant missing momentum pointing into the instrumented region. The multiple neutrinos prevent full reconstruction of each tau lepton, but the known initial four-momentum, lepton spectra and missing energy support a statistical signal extraction. Off-resonance data constrain nonresonant electron-positron annihilation into a tau lepton pair; that continuum is present at both beam energies.^[ref]

A resonant muon pair is one possible detector background: one muon can be misidentified as an electron, while radiation escaping along the beam or a badly measured track produces apparent missing energy. Particle-identification requirements and rejection of almost back-to-back tracks near the beam momentum suppress it. This background needs detector mismeasurement; an ideal measurement would distinguish it.

Above the open-charm threshold, $\psi (3770)\to D\bar{D}$ supplies an additional resonant background. Semileptonic decays of the two D mesons can give an electron, a muon and neutrinos; unreconstructed hadrons let some events pass the selection. This on-shell D meson-pair background is kinematically forbidden at $\psi (3686)$.

With the four-momenta in a, $q^2=-2E{E}^{\prime }(1-\cos \theta )$, $p_2\cdot q=m_p(E-E^{\prime })$, and $p_2\cdot p_1=m_{p}E$. Thus the measured beam energy $E$, scattered electron energy $E^{\prime }$ and scattering angle $\theta$ determineHere $Q^2$ measures the spacelike momentum transfer, $y$ is the fractional energy transfer, and the Bjorken scaling variable $x$ is the struck parton's longitudinal momentum fraction in the leading collinear description. The hadronic invariant mass satisfies $W^2=(p_2+q{)}^2=m_p^2+Q^2(1/x-1)$; large $Q^2$ and $W^2$ select deep inelastic scattering.

A beam spectrometer measures $E$ before a thin stationary hydrogen target. Tracking layers on both sides of a magnetic spectrometer measure the outgoing electron direction, charge and curvature; an electromagnetic calorimeter measures its energy. Matching the track to its particle shower, with downstream muon rejection where useful, separates the scattered electron from hadronic tracks. A downstream hadronic detector measures the hadronic activity and supports event-quality checks. The sketch in a shows these separate measurements.

The scattered electron provides a clean, single-particle reconstruction. Reconstructing the entire hadronic system requires summing many charged and neutral fragments, correcting unequal particle calorimeter response and losses through beam holes, and sometimes accounting for escaping neutrinos. Thus the leptonic definition is experimentally convenient. It is not a fundamental requirement of Lorentz invariance: exact reconstruction of $p_X$ gives the same $q=p_X-p_2$, and hadronic or combined reconstructions can be preferable in some kinematic regions.

For a selected lepton sample, subtract backgrounds and correct for detector acceptance, reconstruction and particle-identification efficiency, trigger efficiency and live time. Schematically,The first expression applies when the published signal definition is a specified fiducial region; the second additionally extrapolates by acceptance $A$. Resolution-induced migration across selections must be unfolded or included in a detector-response fit. Tau lepton reconstruction also needs the relevant branching fractions if only selected decay modes are counted. Initial- and final-state QED radiation and beam-energy spread change the measured line shape; extracting a Born-level or pole cross section requires the corresponding radiative corrections. In the electron channel, Bhabha scattering also has a non-annihilation $t$-channel contribution that must be treated.

For quarks, hadronization means the detector observes hadrons and jets, rather than free quarks. Correct the selection and any flavor tagging using fragmentation and decay modeling, including losses to heavy-hadron decays with neutrinos. In comparison with a pointlike lepton cross section, also include the three color charges per quark flavor and QCD corrections, schematically $N_c[1+{\alpha }_s/\pi +\cdots ]$ with $N_c=3$ at high energy. The color multiplicity is a physical production factor, not an extra factor by which to divide an already efficiency-corrected inclusive hadronic event count.

Fit radiatively corrected visible cross sections across the Z boson resonance. The line shape determines $M_Z$, the total decay width ${\Gamma }_Z$, and pole normalizationsThe first extraction uses the fitted total and visible partial decay widths. Assuming lepton universality, ${\Gamma }_e={\Gamma }_{\mu }={\Gamma }_{\tau }={\Gamma }_{\ell }$, an absolute leptonic pole cross section gives ${\Gamma }_{\ell }={\Gamma }_Z\sqrt{M_Z^2{\sigma }_{\ell }^0/(12\pi )}$. The hadronic-to-leptonic ratio gives $R_{\ell }={\Gamma }_{\mathrm{had}}/{\Gamma }_{\ell }$. Subtracting every visible channel then yieldsWithout lepton universality, separately fitted visible leptonic widths replace $3{\Gamma }_{\ell }$, with sufficient independent information to disentangle their products with ${\Gamma }_e$.

A second, useful peak-normalization extraction first determines the invisible-to-leptonic ratio. Becauseone obtainsThis ratio extraction does not require the measured line width. Multiplication by an independently predicted leptonic partial decay width, or by ${\Gamma }_Z/(R_{\mathrm{inv}}+R_{\ell }+3)$ from the scan, converts it to ${\Gamma }_{\mathrm{inv}}$. Alternatively dividing $R_{\mathrm{inv}}$ by the predicted single-species ratio ${\Gamma }_{\nu }/{\Gamma }_{\ell }$ gives the number of light active neutrinos. These are two standard uses of line-shape information, with different choices of theoretical input; if both use the same fitted ${\Gamma }_{\ell }$, they are algebraically equivalent and statistically correlated, not independent determinations.

Statistical uncertainties in the reference-process counts give fluctuations in the collider luminosity at individual scan points, propagating into normalization and fitted line width. They decrease with additional reference events. A common multiplicative luminosity-scale systematic largely cancels in $R_{\ell }$ and the normalized line shape, but shifts every absolute peak cross section. It therefore does not cancel from either invisible-width extraction. For a fractional upward luminosity shift $\epsilon$, ${\sigma }_{\mathrm{had}}^0$ shifts by $-\epsilon$, and at fixed $R_{\ell },M_Z,{\Gamma }_Z$,Thus overestimating luminosity increases the inferred invisible contribution. Energy-dependent systematic errors can also distort the line shape and its total decay width. The covariance matrix must retain correlations between scan points and experiments, including common theoretical uncertainty in the Bhabha scattering calculation.

In vacuum the electron-flavour survival probability depends only on $|U_{ei}{|}^2$ and the mass-squared differences. Complex conjugation of the lepton mixing matrix therefore leaves it unchanged:Thus a reactor disappearance measurement cannot search for the intrinsic CP violation phase in the standard three-neutrino model. Sensitivity to that phase calls for suitable appearance channels and, usually, comparison of neutrino and antineutrino beams. Matter can produce differences between propagation of neutrinos and antineutrinos; these are not by themselves intrinsic CP violation, and are small for the proposed short reactor baseline.

In the pion rest frame, momentum conservation makes the muon and neutrino momenta opposite, with common magnitude $p$. For a massless neutrino, $E_{\nu }=pc$; energy conservation and the relativistic energy-momentum relation giveSubstitute $pc=m_{\pi }{c}^2-E_{\mu }$ into the second equation. The $E_{\mu }^2$ terms cancel, leavingFor completeness, the complementary relativistic two-body decay energy is $E_{\nu }=(m_{\pi }^2-m_{\mu }^2)c^2/(2m_{\pi })$. The result is a total energy, including the muon rest energy, rather than its kinetic energy.

Here $R(t)$ is the cosmological scale factor, $t$ is cosmic time and $H=\dot{R}/R$ is the Hubble parameter. With $R$ a length, $k=+1,0,-1$ labels positively curved, flat and negatively curved spatial sections. This $k$ is unrelated to Question 1's specific orbital energy. The density $\rho$ is mass-equivalent energy density: $\rho c^2$ has energy-per-volume units. In the displayed Friedmann equation $\Lambda$ has units ${\mathrm{s}}^{-2}$; the geometric cosmological constant is ${\Lambda }_{\mathrm{geom}}=\Lambda /c^2$. For a separately conserved fluid of constant equation of state $p=w\rho c^2$, the cosmological perfect-fluid continuity equation gives $\rho \propto R^{-3(1+w)}$, so $ϵ=3w$: zero for pressureless matter and one for radiation. With several fluids, sum their densities in the expansion equation and evolve each separately.

Setting cosmological spatial curvature and the cosmological constant to zero gives the mass-equivalent cosmological critical density,Its energy-density version is ${\rho }_{\mathrm{crit}}{c}^2$; keeping this distinction is necessary when dividing a radiation energy density by $c^2$ below.

In an Einstein-de Sitter universe, $H^2\propto R^{-3}$ integrates to $R\propto t^{2/3}$ and $H=2/(3t)$. With $1+z=R_0/R$,Here $z_1>z_2$ are the two epoch redshifts relative to today. This is elapsed cosmic time, not comoving distance divided by $c$.

For collisionless preservation of a Planck spectrum, use frequency redshift as well as photon-number conservation. Let $f=R/R_i$, so a frequency bin $d\nu$ now corresponds to $d{\nu }_i=fd\nu$ and ${\nu }_i=f\nu$ initially. Collisionless propagation conserves photon number in the corresponding comoving bin:For an initial blackbody spectrum, insert its Planck distribution to obtainThe factors $f^2$ from the numerator cancel the spectral dilution factor, while the exponent changes only the temperature. Integrating gives photon number density proportional to $T^3\propto R^{-3}$ and radiation energy density proportional to $T^4\propto R^{-4}$. Conservation of total photon number alone would not guarantee a blackbody spectrum shape; the frequency-bin mapping and absence of injection or spectral redistribution are essential.

For the numerical density comparisons, the supplied Hubble constant is $H_0=2.33312\times 10^{-18}{\mathrm{s}}^{-1}$. Taking $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ and $c=2.99792458\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$ givesThe radiation specified here is cosmic microwave background photons only. Adding a relic neutrino density would change the photon-only matter-radiation equality result. The rounded present matter and vacuum fractions sum to one without this small photon contribution; the requested pairwise density comparisons do not require silently renormalizing them.

For the final unlabelled flight-time request, insert the case-(c) value $1+z_c=(0.7/{\Omega }_{\gamma 0}{)}^{1/4}=10.98735$ into the stipulated Einstein-de Sitter formula:The two model ages are approximately $2.486\times 10^8$ and $2.478\times 10^5$ years. This uses exactly the instructed matter-only model with the same $H_0$. It is not an exact age difference in the matter-plus-vacuum cosmology; a high-redshift matter-era normalization using $H_0\sqrt{{\Omega }_{m0}}$ would be a different approximation.

core-collapse supernovae follow loss of pressure support in the evolved core of a massive star. electron capture and nuclear dissociation contribute to collapse; formation of a hot compact object launches a shock that can stall as dissociation and neutrino losses drain it. neutrino heating and multidimensional fluid motions can revive the shock, but some collapses fail to produce a normal bright explosion. The remnant may be a neutron star or black hole. The neutrino burst detected from SN 1987A provides direct evidence for the core-collapse energy channel: the water-Cherenkov signal is a particle-interaction measurement, requiring detector-response and accidental-background checks rather than an optical light-curve inference.^[ref]

Type Ia supernovae instead involve thermonuclear destruction of a degenerate carbon-oxygen white dwarf. They leave expanding ejecta without an ordinary surviving neutron-star core. The ignition channel need not have a unique total mass or binary history. Radioactive nickel/cobalt decay powers much of the later luminosity, and empirical light-curve corrections make these events useful distance indicators. Spectral type is not identical to explosion mechanism: hydrogen-rich Type II events are core collapse, but stripped-envelope Ib/c events can also be core collapse despite lacking conspicuous hydrogen.

The ejecta first expand nearly freely while swept-up mass is small. A strong shock heats ambient gas, and once swept-up material dominates, an approximately adiabatic remnant approaches the Sedov-Taylor scalingSubsequent radiative cooling makes a dense shell and ends that energy-conserving approximation. Thermal X-ray lines probe shocked gas and nucleosynthetic ejecta; radio and nonthermal X-rays trace relativistic particles and magnetic fields. shock acceleration makes remnants plausible contributors to Galactic cosmic rays. Expansion, ambient-density structure, cooling and cosmic-ray energy losses complicate energy estimates; a bright synchrotron shell alone is not proof of a specific accelerated-particle composition. Supernovae return metals and mechanical energy to the interstellar medium and therefore connect stellar death to the feedback cycle.

Supernovae are explosive stellar events with different energy sources and progenitors; a remnant is the expanding ejecta and shocked environment left afterward. Their spectra classify the observed event, while explosion mechanisms give a related but distinct physical classification. Lack of hydrogen absorption defines TypeI spectroscopically; TypeII shows hydrogen. TypeIa typically shows silicon features, while stripped-envelope core-collapse events can be TypeIb or Ic.

In core collapse, an evolved massive star's central core loses support as nuclear processing ceases to supply stable pressure support, electron captures reduce electron pressure and photodisintegration consumes energy. Collapse can form a neutron star or black hole. A shock launched near core bounce loses energy, so a successful explosion requires additional processes, often modeled through neutrino heating aided by multidimensional flow; some collapses fail to eject a bright ordinary supernova. The compact remnant and neutrino emission distinguish this channel physically from a thermonuclear disruption.

TypeIa explosions involve thermonuclear burning of a carbon-oxygen white dwarf in a binary-related evolutionary channel. Both near-Chandrasekhar and sub-Chandrasekhar routes are studied; no single progenitor path is forced by the spectral classification alone. Radioactive nickel and cobalt decay can power much of the later optical emission. Empirical light-curve standardization makes these events useful distance indicators, but dust, population differences and calibration must be controlled; they are not intrinsically identical candles.

Initially ejecta may expand nearly freely, with radius $R_s\sim v_{\mathrm{ej}}t$, until swept-up external mass becomes dynamically important. The shock heats the surrounding gas and a reverse shock heats ejecta. For an approximately spherical, adiabatic explosion of fixed energy $E$ into a uniform density ${\rho }_0$, dimensional analysis gives the Sedov-Taylor stage:The dimensionless coefficient $\xi$ depends on the equation of state; no value is needed without specifying that model. For a strong nonrelativistic shock in a monatomic gas, an equilibrated post-shock temperature scale is $k_{B}T\simeq 3\mu m_p{v}_s^2/16$. Collisionless electron and ion temperatures may differ.

Later radiative losses can produce a dense cooled shell. Idealized pressure-driven and momentum-conserving snowplough models then have different expansion exponents; the environment and continuing interior pressure determine which description applies. For uniform ambient density and conserved shell momentum, $M_{\mathrm{swept}}{\dot{R}}_s\propto R_s^3{\dot{R}}_s$ gives $R_s\propto t^{1/4}$. Real remnants can be asymmetric and encounter nonuniform winds or clouds.

Thermal X-ray emission traces shocked plasma, optical lines trace cooling gas, and radio synchrotron emission traces relativistic electrons and magnetic fields. Shocks can accelerate cosmic rays and alter the energy budget. Ejecta abundances, a surviving compact object, pulsar-powered nebulae and the remnant's dynamics help reconstruct the explosion, but present morphology alone does not uniquely identify the progenitor.

Supernovae enrich the interstellar medium, stir and heat galactic gas and can drive outflows or trigger local compression. Their feedback couples small stellar scales to galaxy evolution. Quantifying that effect requires accounting for cooling and the collective environment rather than assigning all explosion energy to a permanently retained galactic reservoir.

Use the supplied atomic masses consistently. Their mass defect per four hydrogen atoms is $4(1.0078)-4.0026=0.0286$ atomic mass units. The conversion efficiency isThe atomic-mass bookkeeping includes the associated positron-annihilation energy; an exact photon-heating budget also subtracts energy carried away by neutrinos. With the stated ten-percent fuel fraction and constant present solar luminosity, the main sequence lifetime estimate isThe much longer timescale explains why stellar nuclear fusion can power the Sun throughout its long main-sequence history.

Two major suppressions are missing from the one-femtometre rectangular estimate. First, the Coulomb barrier extends far beyond nuclear distances and is not rectangular. A semiclassical integral through it produces the Gamow factor $\exp (-2\pi \eta )$, with $\eta ={\alpha }_{\mathrm{fs}}\sqrt{\mu c^2/(2E)}$; at the illustrative thermal energy it is already of order $10^{-11}$. A thermal average additionally weights the high-energy tail and is not the same as evaluating a single thermal energy.

Second, two protons cannot simply form a stable bound diproton. The proton–proton chain starts with a weak-interaction conversion producing a deuteron, a Positron and a neutrino. The weak transition probability supplies a further very large suppression. Thus the quoted tiny fusion probability is not a pure one-dimensional barrier transmission probability with universal energy-independent meaning.

The Sun contains of order $10^{57}$ nucleons in a hot dense core and offers enormously many encounters over astronomical times. A very small reaction probability per encounter can therefore supply its luminosity while consuming only a small fraction of the fuel per unit time. The small weak reaction rate is also crucial for the long stellar lifetime; a larger core temperature increases burning and triggers expansion and cooling.

Work first in units $c=1$. In the neutron rest frame, let $s_{p\nu }=(p_p+p_{\nu }{)}^2$. From $p_n-p_e=p_p+p_{\nu }$,The electron's minimum possible total energy is its rest energy. This configuration has the electron at rest while the proton and massless antineutrino move back to back with equal momentum magnitudes. Their common momentum satisfies a two-body energy balance for the remaining mass $m_n-m_e$:For the maximum electron energy, minimize $s_{p\nu }$. Since the massless antineutrino has nonnegative energy, $s_{p\nu }\ge m_p^2$, with the boundary reached as its energy tends to zero. ThenAt this limiting endpoint the electron and proton have equal and opposite momenta. The proton recoils; it is not at rest. A massless neutrino has no rest frame, so “neutrino at rest” must be replaced by the precise soft-energy limit $E_{\nu }\to 0$. The electron's maximum kinetic energy is the displayed total energy minus $m_e{c}^2$, rather than the same number. These endpoint configurations lie on phase-space boundaries; their kinematic existence does not imply a nonzero event density exactly at the boundary.

The charged meson decays proceed through a virtual $W^{+}$:The first quark vertex has CKM factor $V_{ud}$, while the second has $V_{us}$. In the simple Cabibbo description these are approximately $\cos {\theta }_C$ and $\sin {\theta }_C$, so the strange amplitude carries a suppression $\tan {\theta }_C$ relative to the nonstrange one, or ${\tan }^2{\theta }_C\simeq 0.053$ in their corresponding squared weak-coupling factors.

For a charged pseudoscalar and a negligibly massive neutrino, the leading leptonic energy width has the formin natural units. The helicity factor $m_{\mu }^2$ is common to these two muon channels, while the threshold factor is much less favorable for the pion. Using the usual masses, its value is about $0.182$ for the pion and $0.910$ for the charged kaon: the pion's threshold factor alone is only about one fifth as large. Including the explicit $m_P$ factor gives a kinematic ratio about $0.0564$. The pseudoscalar decay constants $f_P$ also matter, so Lorentz-invariant phase space and CKM factors alone do not determine the full lifetime ratio.^[ref]

For $K^{+}\to {\pi }^{+}{\pi }^0$, one tree topology has the spectator $u$ unchanged and $\bar{s}\to \bar{u}+W^{+\ast }$ followed by $W^{+\ast }\to u\bar{d}$. The spectator $u$ can combine with $\bar{d}$ to form ${\pi }^{+}$, and the $u\bar{u}$ component supplies ${\pi }^0$. This carries the strangeness-changing weak factor $V_{us}^{\ast }{V}_{ud}$. Hadronization, other quark rearrangements and strong final-state interactions determine the hadronic amplitude; the diagram is not a complete precision width calculation. The small pion–muon energy release suppresses the pion leptonic channel kinematically, whereas the strange weak vertex suppresses the kaon channels through Cabibbo suppression.

The first step of the proton–proton chain isHere $d$ is the deuteron. A weak charged current converts one proton into a neutron, while the resulting proton–neutron pair becomes bound. The Q value follows from the difference of initial and final rest energies:This includes the emitted Positron's rest energy in the final-state mass and excludes its subsequent electron-positron annihilation. The nuclear binding energy of the deuteron makes this reaction possible although the isolated conversion $p\to n+e^{+}+{\nu }_e$ is energetically forbidden. The released energy is shared by final-state kinetic energies, including the escaping neutrino.

The radiation registered in the stated solar luminosity includes two Positron annihilations but excludes the two escaping neutrinos. Thus the visible energy per completed helium-producing chain isThe signs in this energy ledger matter: adding the neutrino energy would overestimate the energy available to sustain the measured radiation. Using $1\mathrm{MeV}=1.602176634\times 10^{-13}$ J, the chain rate and hydrogen-consumption rate areWith a year taken as $365.25$ days, the stated elapsed time is $1.4516496\times 10^{17}$ s. Consequently,At the same fixed luminosity, the remaining time would beOnly about $5.93\%$ of the stipulated initial reservoir has been consumed under these assumptions. This is a calculation for an entirely accessible hydrogen reservoir with constant luminosity. It is not a prediction of the Sun's remaining main-sequence lifetime: only part of the hydrogen participates in core burning, and stellar structure and luminosity evolve.

For the first generation the flavor-diagonal weak neutral current permitsEach quark channel has three possible matching color charges. The weak charged current permitsThe two charges have equal partial decay widths to charge-conjugate channels to the accuracy needed here. Lepton universality and color multiplicity in a decay width give one unit of width for a massless lepton pair and three for a quark pair, apart from quark mixing and radiative corrections.

If only the first generation is included, the branching fractions are $1/(1+3)=1/4$ to the lepton channel and $3/4$ to quarks. For the actual accessible Standard Model channels, there are three lepton units and two accessible up-type quark rows, $u$ and $c$, each with three colors. The top quark is too heavy. Unitarity of the CKM matrix sums over accessible down-type partners, giving nine units in total:These fractions concern a charged lepton together with its appropriate neutrino; “lepton pair” does not mean two charged leptons in a $W$ decay. The Z boson has different vector and axial couplings for different fermions, so the same simple equal-unit rule cannot be transferred to all its partial decay widths.

An extra active generation with the same weak neutral current couplings would increase the Z boson total decay width if any new particle–antiparticle pair were light enough. A simple sufficient kinematic condition for closing each such two-body channel isThe corresponding within-generation W boson channels close whenIf both members of each doublet already exceed $m_Z/2$, these two sum constraints follow automatically. Mixing with lighter generations can also open charged-current channels involving one heavy and one light particle, depending on the mixing amplitudes; that information is not specified.

These are threshold estimates, not precise confidence limits. Close to threshold, two-body decay phase space suppresses a new partial decay width, and a quantitative lower mass bound requires the experimental width uncertainties and the fermion's couplings. The given observations yield no upper mass bound. A sterile neutrino with no standard active coupling would not obey the stated $Z$-width inference, whereas the question specifies the same quantum numbers as the existing generations.

The new light active leptons open one additional $W\to \sigma N$ channel and two $Z$ channels. At the stated equal-partial-width approximation,The quoted $501$ MeV is the total invisible width into the three known neutrino flavors, not the width into one flavor. The invisible Z width per active neutrino fixes this interpretation. The notation in the numerical input is ambiguous; the Standard Model normalization fixes the new active-neutrino contribution to $501/3=167$ MeV. Adding the new charged-lepton width therefore givesThis interpretation is consistent with the usual determination of the effective number of active neutrinos from the invisible width divided by the predicted width of a single species.^[ref]

Keep the original light-fermion production and observed-decay partial decay widths, masses and prefactors fixed. New leptonic channels alter their branching fractions through the total widths. From the peak formula in d,Consequently,The relatively small change is a cancellation between the increase in the two total widths; each separately changes substantially more. This assumes that the new quarks do not contribute and that adding leptons leaves the old partial decay widths unchanged.

If the source's $501$ MeV were instead imposed literally as the width of a single neutrino flavor, the algebra would give ${\Gamma }_Z^{\prime }=3080$ MeV and $R^{\prime }/R=1.23401$. That would be a different numerical premise, inconsistent with the Standard Model interpretation of the supplied invisible-width value. It is recorded to make the normalization issue explicit, rather than silently treating the three-flavor sum as a one-flavor partial decay width.

For a measurement of the cross-section ratio with standard deviation $0.019R$, the separation of the fourth-lepton prediction from the ordinary prediction isThus the peak-ratio model predicts only about a one-standard-deviation shift in this observable. In contrast, the direct Z boson width changes by $167+84=251$ MeV, givingUnder these assumptions the direct width measurement is overwhelmingly more sensitive. These are separate expected displacements, using the supplied experimental uncertainties and fixed theoretical inputs; they are not a combined statistical significance. The $501/3$ single-neutrino normalization and the peak-energy assumption are the same as in e. If the alternative literal $501$ MeV per neutrino premise were used, the corresponding displacements would instead be $12.3$ and $254$ standard deviations, illustrating why that normalization must be stated.