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With $\Omega =\gamma B_x/2$, the Born rule applied to the supplied down-spin probability amplitude givesFor nonzero transverse field, the maxima occur atThe first maximum is reached at $\pi /{\omega }_R$; on resonance this is a Rabi pi pulse. The resonance envelope of a Rabi transition is a Lorentzian line shape, centred on ${\omega }_{\mathrm{res}}={\omega }_z$ under the orientation convention above. It has half its peak value at $|\omega -{\omega }_z|=|\Omega |$, so its full width at half maximum isIf resonance width is instead defined as the half-width, the ratio differs by a factor of two but has the same requested proportionality. The quality factor is large precisely in the regime where the rotating-wave approximation is justified. The envelope is obtained by choosing the best time separately at each drive frequency; a measurement at one fixed pulse duration has a different line profile.

The ground electron configuration of caesium has a single $6s$ electron, with $J=1/2$. Addition of angular momentum giving $F=3,4$ fixes the nuclear spin to $I=7/2$. With dimensionless angular momentum operators, write the Fermi contact interaction asThe two hyperfine levels therefore differ by $4A$. In the convention ${\mathbf{\mu }}_I=g_I{\mu }_B\mathbf{I}$ and $g_s\simeq 2$,This exposes a missing ingredient in a prediction from the supplied data: $Z$ and $g_I$ alone do not specify the valence electron contact density. Screening effect and penetration of the closed core strongly affect it. If $g_I$ instead denotes the total nuclear moment in units of ${\mu }_B$, it must first be divided by $I$ in the displayed convention.

An explicitly crude unscreened hydrogen-like ion model uses $|{\psi }_{6s}(0){|}^2=Z^3/(\pi 6^3{a}_0^3)$ and gives ${\nu }_0\simeq 5.76\times 10^{11}\mathrm{Hz}$, or ${\omega }_0\simeq 3.62\times 10^{12}{\mathrm{s}}^{-1}$. It describes neither the shielding nor the radial function of neutral caesium. Using a pure unit-charge $6s$ hydrogen orbital instead gives only $3.46\mathrm{MHz}$, illustrating why a unique useful estimate cannot follow from an unspecified effective charge.

For physical neutral ${}^{133}\mathrm{Cs}$, the independently known zero-magnetic field clock transition isThis measured frequency, used to define the second, is supplementary physical information, not a consequence of $Z=55$ and the rounded $g_I$ alone.^[ref] The direct electronic transition is in the microwave range and is a magnetic dipole transition, rather than a direct optical electric dipole transition. An optical implementation needs a two-field Raman transition.

Using the corrected weak-pulse Ramsey spectrum of part (b), the peak at $\delta =0$ has height $(\Omega {\tau }_p{)}^2$. The quantum interference fringes have spacing $2\pi /T$ in angular frequency, with zeros at $(2k+1)\pi /T$, and the sinc function envelope has its first zeros at $\delta =\pm 2\pi /{\tau }_p$. For $T\gg {\tau }_p$, the central peak has approximately full width at half maximum $\pi /T$ in angular frequency. The clock transition between the two $M_F=0$ states has no first-order Zeeman energy, although a quadratic one remains.

Let the upper-state population lifetime be $T_1=1/\Gamma$, and take the lower energy level to be stable. Spontaneous emission gives survival probability $e^{-\Gamma t}$, hence a field or excited-state probability amplitude proportional to $e^{-\Gamma t/2}{e}^{-i{\omega }_{0}t}$ for $t\ge 0$. Its Fourier transform at angular frequency $\omega$ is proportional toThe squared modulus therefore has the Lorentzian line shapeHere $L$ is normalized over the full mathematical frequency axis. The full width at half maximum in ordinary frequency is $1/(2\pi T_1)$. If both energy levels decay, their population decay rates add in the natural linewidth; pure dephasing adds to the amplitude-decay rate instead.

Impact-regime pressure broadening provides another example. Independent phase-randomizing collisions with rate ${\gamma }_c$ give optical correlation proportional to $e^{-{\gamma }_c|t|}$, whose Fourier transform is a Lorentzian line shape of full width at half maximum $2{\gamma }_c$. This assumes short, uncorrelated impacts: a general static distribution of perturbing fields need not give a Lorentzian line shape.

Use first-order time-dependent perturbation theory, setting $c_1=1$ in the equation for $c_2$ and retaining the generally complex matrix element $d=⟨{\psi }_2|{\hat{D}}_0|{\psi }_1⟩$. With $\delta ={\omega }_0-\omega$ and $\Sigma ={\omega }_0+\omega$, the probability amplitude isFor $t>0$, split the integral at the pulse maximum. For either $x=\delta$ or $x=\Sigma$,Thus the full first-order answer, without discarding the counter-rotating contribution, isAt long times it givesThis full expression is not exactly one squared Lorentzian line shape. To obtain the intended near-resonance spectral line profile, additionally assume $b,|\delta |\ll {\omega }_0$, so the pulse has a narrow positive-frequency lobe and $F(\Sigma ,t)$ is negligible. ThenThe weak-excitation condition near resonance is $|d|/(\hslash b)\ll 1$. The squared Lorentzian line shape occurs because the symmetric exponential pulse has a Lorentzian line shape amplitude Fourier transform, which is then squared to obtain probability.

At half the maximum, $[1+(\delta /b{)}^2{]}^2=2$, soThe full width at half maximum refers to varying the drive frequency through the positive-frequency resonance. For a concrete failure of the unqualified squared-Lorentzian line shape claim, set $b=\omega ={\omega }_0$ while keeping $|d|/(\hslash b)\ll 1$. The full first-order resonant probability is $36/25$ times the one-lobe prediction. Weak radiation alone therefore does not justify dropping the second Fourier transform lobe.

Let $\mathcal{R}=|r{|}^2$ be the single-mirror power reflectivity and ${\theta }_g$ the propagation angle inside the gas. Define $\lambda$ as the vacuum wavelength. Successive transmitted complex amplitudes differ by a round-trip factor $\mathcal{R}{e}^{i\delta }$, where, for real dielectric reflection coefficients whose product has positive sign,The cosine arises by comparing successive emerging rays on the same outgoing wavefront; it is not the literal zigzag distance $2d/\cos {\theta }_g$. Equivalently, the round-trip phase is $2k_{z}d$. If the source's $\theta$ is the angle in external air, Snell's law gives $n\sin {\theta }_g=\sin \theta$ andIf $\theta$ denotes the internal angle, simply set ${\theta }_g=\theta$.

For identical lossless mirrors, their power transmission is $1-\mathcal{R}$. In an energy-normalized amplitude convention this is $|t{|}^2=1-\mathcal{R}$; more general interface amplitudes require the corresponding wave impedance factors. The geometric series of transmitted amplitudes givesHenceThis is Fabry–Pérot transmission from multiple reflections.^[ref]

At normal incidence, for nondispersive $n$ and zero net reflection phase modulo $2\pi$, the resonance condition is $\delta =2\pi m$. The resonance frequencies and free spectral range areFor finite $\mathcal{R}<1$ the resonances have finite width, and transmission away from them is nonzero. Thus “only discrete frequencies” is an ideal high-reflectivity description of the selected resonances, not an exact statement about the support of the transmitted spectrum. For example, $\mathcal{R}=0.9$ at $\delta =\pi$ still gives $I_t/I_0=(0.1/1.9{)}^2\simeq 0.00277$.

At half the unit peak, $Q{\sin }^2(\delta /2)=1$. Measuring the phase displacement from a resonance yieldsThe last expression is the high-reflectivity approximation, with finesse $\pi \sqrt{\mathcal{R}}/(1-\mathcal{R})$. At $\mathcal{R}=0.9$, the full phase width is about $0.2109$ radians and $\Delta {\nu }_{\mathrm{FWHM}}/\Delta {\nu }_{\mathrm{FSR}}\simeq 0.03357$. At $\mathcal{R}=0.1$, the minimum transmission is $0.6694$, so the trace never reaches half the peak and this full width at half maximum is not defined. The exact half-maximum condition requires $\mathcal{R}\ge 3-2\sqrt{2}$.

For the absorbing Fabry–Pérot mirror phase and transmission, write $r=\sqrt{\mathcal{R}}{e}^{i{\varphi }_r}$ and let $\mathcal{T}$ be the power transmission of one mirror. Optical absorption makes $\mathcal{R}+\mathcal{T}<1$. The transmitted intensity becomesHere the phase formula is at normal incidence. The peak is reduced below unity, while a change in $\mathcal{R}$ changes the width. The complex reflection phase also moves the resonances: if ${\varphi }_r$ is approximately constant,If either $n$ or ${\varphi }_r$ depends on frequency, this is an implicit condition and the free spectral range need not be constant. No numerical shift is determined without the mirrors' optical response.

The temporal intensity is proportional to the squared field modulus:Its half-maximum points satisfy $2a{t}^2=\ln 2$, so the intensity full width at half maximum isWith the chosen $e^{-i\omega t}$ convention, the instantaneous angular frequency is minus the time derivative of the field's argument. HenceA localized Gaussian requires $\mathrm{Re}\Gamma =a>0$. It is positively chirped precisely when $d\omega /dt=2\mathrm{Im}\Gamma >0$, or $b>0$; $b=0$ gives no chirp and $b<0$ gives negative chirp.

Put $b=\Delta {\omega }_b>0$ and $s=t-t_m$. At the peak time, $e^{-ip\Delta \omega t_m}=1$, so the exact comb sum becomesFor many occupied normal modes, $b\gg \Delta \omega$, replace the sum by an integral with density $1/\Delta \omega$. Setting $x=\omega -{\omega }_0$ and evaluating the Fourier transform of a Gaussian givesThus the isolated-optical pulse approximation hasThe optical pulse has a Gaussian function envelope and no chirp. Its carrier angular frequency is ${\omega }_0$, and its light intensity full width at half maximum is $2\sqrt{2\ln 2}/b$. Successive envelope maxima have different carrier-envelope phases, consistent with b.

Approximation in the printed assertion. A discrete optical frequency comb gives a train, not one exact isolated Gaussian pulse. Applying the Poisson summation formula to the ideal sum over all integers instead givesNear $t_m$, the $j=m$ term dominates when $bT\gg 1$. The other terms are exponentially small but not identically zero. A single Gaussian function used globally would decay between $0$ and $T$, whereas the exact light intensity has equal maxima there; this supplies a concrete reason the isolated form cannot be an exact global identity. The extension to negative frequencies is the usual narrow-positive-band analytic-field approximation when ${\omega }_0\gg b$.

Astronomical seeing is the atmospheric angular blurring of a point-source image, usually reported in arcseconds. Operationally it is commonly characterized by the full width at half maximum of a long-exposure stellar point spread function. The full observed width can also contain telescope and instrumental contributions; the atmospheric seeing is the turbulence contribution.

A detector too slow to resolve a femtosecond field can measure its integrated nonlinear response instead. Split the pulse, delay one replica, overlap both in a thin phase-matched crystal, and select the sum-frequency signal. For negligible depletion and sufficiently broad phase matching, the cross-generated field is proportional to $A(t)A(t-\tau )$; integrated signal gives the second-harmonic intensity autocorrelationA noncollinear geometry can reject each replica's individual second harmonic. The optical response supplies the fast gate while the detector integrates slowly. Crystal thickness, group-velocity mismatch, spectral acceptance, beam overlap and delay calibration limit resolution.

For a Gaussian pulse specified by its intensity profile, the autocorrelation's full width at half maximum is $\sqrt{2}$ times the intensity full width at half maximum. For a sech-squared intensity profile the ratio is about $1.543$. These are model-dependent conversion factors. An intensity autocorrelation is symmetric in delay and loses phase information; different pulses can share it. An interferometric autocorrelation retains rapid carrier fringes but still does not generally reconstruct an arbitrary complex field uniquely.

frequency-resolved optical gating also resolves the gated signal's spectrum, giving a two-dimensional delay-frequency record. For second-harmonic frequency-resolved optical gating,where $\omega$ is measured relative to the signal's carrier. An iterative retrieval seeks an envelope whose predicted record matches the measurement. Alternatively, polarization-gating frequency-resolved optical gating uses the optical Kerr effect to gate the probe by a delayed replica's intensity, with signal envelope proportional to $A(t)|A(t-\tau ){|}^2$. The original FROG work introduced nonlinear optical gating plus spectral resolution to characterize arbitrary femtosecond pulses.^[ref]

Absolute wave phase and absolute time origin remain undetermined by these intensity records. The SHG record also cannot distinguish $A(t)$ from $A^{\ast }(-t)$, so its time-direction ambiguity remains. Retrieval agreement, an independently measured spectrum, adequate signal dynamic range and gate-response calibration are practical consistency checks. A pulse train that changes between shots can yield an averaged record that no single pulse faithfully represents. Thus nonlinear gating solves the detector-speed problem but does not remove inverse problem ambiguities or experimental limitations.

The conversion from a measured beam peak to a particle resonance mass is fixed, but the missing graph supplies neither the peak position nor its width. Let $T_R$ be the incoming pion kinetic energy at the peak and $\Delta T$ the full width at half maximum after allowing for instrumental resolution. With a stationary proton and natural units,Thus the peak invariant mass isThe width must also be transformed to the centre-of-mass frame. To first order about the peak,For a broad feature, transform the two half-height positions separately: if they are $T_{-}$ and $T_{+}$, then the corresponding width in invariant mass is $\sqrt{(m_p+m_{\pi }{)}^2+2m_p{T}_{+}}-\sqrt{(m_p+m_{\pi }{)}^2+2m_p{T}_{-}}$. An energy-dependent numerator or width can also displace the observed maximum from the resonance pole parameter.

The low-energy ${\pi }^{+}p$ channel is normally dominated by ${\Delta }^{++}(1232)$, a Delta baryon with $Q=2$, $B=1$ and $I=3/2$. As an external reference example, not a reading of the absent plot, $M=1232\mathrm{MeV}$ and $\Gamma =117\mathrm{MeV}$ give $T_R\simeq 190\mathrm{MeV}$, $\Delta T\simeq 154\mathrm{MeV}$ and $\tau \simeq 5.63\times 10^{-24}\mathrm{s}$.^[ref] The actual graphical estimates remain unavailable. In particular, using the laboratory width directly in $\tau =\hslash /\Delta T$ misses the factor $m_p/M$.

A constant-width Breit-Wigner resonance cross section is the simplest nonrelativistic curve; a P-wave scattering width and relativistic kinematics explain important expected distortions. A comparison to specific observed deviations cannot be made without the missing plot.

Write $E$ for the centre-of-mass frame kinetic energy, $\mu =m_p{m}_{\pi }/(m_p+m_{\pi })$ for the reduced mass, and $k$ for the relative wave number. The nonrelativistic relations areFor an isolated particle resonance of spin $J$, averaging over the initial proton spin and using the zero pion spin givesFor the nearly elastic ${\Delta }^{++}$ state, $J=3/2$ and ${\Gamma }_{\pi p}\simeq \Gamma$, soFreezing $k$ at $k_R$ gives a symmetric Lorentzian centered at $E_R$, with full width at half maximum $\Gamma$ and height $8\pi /k_R^2$. This is the local curve to sketch. If the displayed variable is the beam kinetic energy $T$, its width is $\Delta T\simeq (m_p+m_{\pi })\Gamma /m_p$ within this nonrelativistic approximation. Keeping $k(E)$ changes the prefactor and hence the symmetry. Extending a constant-width approximation all the way to $E=0$ produces a spurious divergence, so it is not a valid threshold model.

The pion has intrinsic parity $-1$ and the proton has $+1$. The positive parity and $J=3/2$ of the Delta baryon require odd orbital angular momentum, with the lowest choice $l=1$. The finite-range partial-wave threshold law gives ${\Gamma }_{\pi p}(k)\propto k^{2l+1}=k^3$ near threshold. Inserting this in the elastic numerator makes ${\sigma }_{\mathrm{el}}\propto k^4$ away from the pole as $k\to 0$, rather than diverging. Equivalently the elastic scattering phase shift has ${\delta }_1=O(k^3)$, and ${\sin }^2{\delta }_1/k^2=O(k^4)$. Away from threshold, form factors and other channels modify the simple cubic width.

At the illustrative peak of part (d), the incident pion kinetic energy is comparable to its rest energy, so a nonrelativistic pion treatment is already inaccurate. The exact relative momentum iswith $\sqrt{s}=\sqrt{(m_p+m_{\pi }{)}^2+2m_{p}T}$. Relative to the symmetric local estimate, this changes the horizontal conversion, peak normalization and tails. The illustrative $1232\mathrm{MeV}$ peak has $p_{\ast }\simeq 227\mathrm{MeV}/c$, giving $8\pi (\hslash c{)}^2/(p_{\ast }c{)}^2\simeq 190\mathrm{mb}$ before small inelastic effects.

Finally, nonresonant partial waves can interfere with the resonant scattering amplitude, higher particle resonances and inelastic scattering channels can enter, and instrumental resolution can broaden a measured peak. These are physically distinct possible deviations; the missing data do not justify claiming any particular one was actually observed. Both the missing graphical overlay and its data-specific explanation remain source gaps.

Temporal coherence measures correlation between an optical field and a delayed copy. For a stationary scalar complex field with carrier convention $e^{-i\omega t}$,The power spectrum $S(\omega )\ge 0$ and the field autocorrelation form a Fourier transform pair, the Wiener-Khinchin theorem:A Michelson interferometer supplies delay $\tau =\Delta L/c$. With equal arm intensities and matched polarization, its interferometric visibility is $|g^{(1)}(\tau )|$; unequal intensities or mode mismatch lower visibility independently of temporal coherence. The relevant spectrum is the optical field spectrum, not the low-frequency spectrum of detector intensity fluctuations.

For a single Lorentzian optical line of angular-frequency full width at half maximum $\Delta \omega$, use the narrow-line detuning modelSuch a model describes a phase-diffusing monochromatic source. An $e^{-1}$ correlation time is $2/\Delta \omega$, and its coherence length is $2c/\Delta \omega$. The Lorentzian extension over all detunings approximates a narrow optical line with $\Delta \omega \ll {\omega }_0$.

For two equal, mutually incoherent ideal monochromatic spectral lines,Here $\bar{\omega }=({\omega }_1+{\omega }_2)/2$ and $\Delta {\omega }_{12}={\omega }_2-{\omega }_1$. Visibility has zeros and revivals, so the overall spectral span alone does not determine a unique decay time. Giving each line a common finite Lorentzian linewidth multiplies this beating correlation by the corresponding exponential envelope.

These two examples distinguish irreversible envelope decay from cancellation and revival of discrete spectral components. The line profiles are explicit models, not claimed measurements of a particular source.

A spectral line has finite width when the emitted or absorbed field loses coherence, or when different emitters have different transition frequencies. It is useful to distinguish homogeneous spectral broadening of each emitter from inhomogeneous spectral broadening across the ensemble.

Finite lifetime and dephasing. For an upper-state population lifetime $\tau$ and a stable lower state, the radiating amplitude decays as $e^{-t/(2\tau )}$. Its Fourier spectrum is Lorentzian, with full width at half maximumMore generally, an optical coherence time $T_2$ gives $\Delta {\omega }_{\mathrm{hom}}=2/T_2$. Both level lifetimes and pure dephasing can contribute to $1/T_2$. Distinguishing population lifetime from amplitude coherence time prevents the usual factor-of-two error.

Thermal motion. To first order in velocity, $\delta \nu ={\nu }_0{v}_{\parallel }/c$. A Maxwell-Boltzmann velocity distribution therefore produces a Gaussian profile, withThis is inhomogeneous doppler broadening: different velocity classes have different resonance frequencies. A distribution of bulk velocities can add further broadening, while a common bulk velocity shifts the line.

Interactions with the surroundings. Collisions can interrupt the dipole wave phase and perturb the energy levels, producing pressure broadening and shifts. In the impact approximation, approximately exponential wave phase relaxation produces a Lorentzian contribution; its rate increases with perturber density times an appropriate velocity-averaged collision cross section. Slowly varying electric microfields can instead produce non-Lorentzian Stark profiles. Neutral perturbers, charged perturbers and resonance interactions need not produce the same line shape. These mechanisms and the Gaussian thermal limit are distinguished in NIST's atomic-spectroscopy compendium.^[ref]

Finite observation time limits coherence; transit-time broadening also occurs for particles crossing a narrow optical beam. Strong resonant excitation causes power broadening: for an ideal lifetime-limited two-level transition, $\Delta \omega =\Gamma \sqrt{1+s}$ with population decay rate $\Gamma$ and saturation parameter $s$. Unresolved fine structure, hyperfine structure, isotope components, or spatially varying Zeeman and Stark shifts may look like a single broadened feature even though several frequencies are present.

The observed spectrum is additionally convolved with the instrument response. A Gaussian Doppler contribution convolved with Lorentzian homogeneous broadening gives a Voigt profile; their full widths do not generally add in quadrature. An informative experiment varies temperature, perturber pressure and excitation intensity separately, calibrates the spectrometer response with a narrower reference, and checks optical thickness or self-absorption. Extrapolation to weak excitation and low pressure helps isolate the lifetime contribution, while a Doppler-free technique is needed if velocity spread still dominates.

Let $I_0$ be the intensity from either fibre alone at the observation point, assuming equal output intensities, matching polarizations and negligible drift. The screen samples the relative phase $\Phi (X)$ between the two outputs. For an ideal monochromatic laser, first-order optical coherence is complete andThe interference fringes are approximately equally spaced in the small-angle far field, with a slowly varying diffraction envelope. If the output separation is $a$ and the screen distance is $D$, the interference fringe spacing is approximately ${\lambda }_{0}D/a$. Equal phases at the two input ends do not ensure a central bright fringe: the extra fibre contributes carrier phase $2\pi n_{\mathrm{eff}}(l_2-l_1)/{\lambda }_0$ in addition to the free-space optical path difference.

For a finite linewidth, the time-averaged interference signal becomesHere $g^{(1)}$ may be defined for the slowly varying field envelope, with the carrier phase already included in $\Phi$. A realistic laser therefore gives reduced interferometric visibility; for $V=1/2$, its extrema are $I_0$ and $3I_0$.

For a Lorentzian line shape of frequency full width at half maximum $\Delta \nu$, temporal optical coherence givesThe relevant index is the group index. Using the intended nondispersive approximation $n_g\simeq n_{\mathrm{eff}}=2$,This delayed-fibre linewidth measurement assumes the loss of contrast comes from linewidth. Unequal output intensities, polarization mismatch or mechanical phase fluctuations also lower interferometric visibility. If dispersion matters, the stated phase index alone does not specify the necessary group index.

An unstable isolated level has survival amplitude $a(t)=\theta (t)e^{-i{E}_{0}t/\hslash }{e}^{-\Gamma t/(2\hslash )}$, because its survival probability falls as $e^{-\Gamma t/\hslash }$. Its energy-domain amplitude is proportional toTaking the modulus squared gives the denominator of the Breit-Wigner resonance cross sectionAt $E-E_0=\pm \Gamma /2$ the resonant factor is half its peak, so $\Gamma$ is the full width at half maximum. exponential decay is the isolated-level approximation; energy-dependent thresholds or overlapping levels need a more general lineshape.

Continuum amplitudes. The leading quantum electrodynamics diagram is electron-positron annihilation into a virtual photon, followed by conversion to a muon pair. Replacing the outgoing muons with a quark-antiquark pair gives the quark-level hadronic diagram. The first two panels below show these processes.

The Feynman rules give, up to signs and factors of $i$,The two vertices supply $e^2$, and the photon propagator supplies $1/s$. Each external current scales as $\sqrt{s}$, so the complete amplitude is of order $e^2$, and its squared magnitude is of order $e^4\propto {\alpha }^2$. The two-body relativistic scattering cross-section supplies the remaining $1/s$ from flux and phase space. Integrating the angular distribution yields ${\sigma }_{\mu \mu }=4\pi {\alpha }^2/(3s)$. Keeping only the squared propagator while ignoring the external currents would incorrectly predict $1/s^2$.

A quark of charge $Q_{f}e$ contributes $N_c{Q}_f^2$ times the massless muon rate. With $u,d,s,c,b$ open and $N_c=3$, the hadronic R ratio isThis is the requested massless leading-QED continuum approximation. Strong corrections, physical hadron thresholds, resonance contributions and Z exchange are separate effects; the pure photon formula is not a complete high-energy electroweak cross section throughout the entire stated range.

Efficiency normalization. Read the original PDF plot as a continuum of about $0.8\mathrm{nb}$ and a maximum around $1.7\mathrm{nb}$ at $E\simeq 10.58\mathrm{GeV}$. The supplied conversion constant givesHence near the peak,If $C$ multiplies the displayed cross sections, the required scale isHere $ϵ$ is the implied detection efficiency under the common-scale assumption. Using continuum readings $0.75$–$0.85\mathrm{nb}$ gives approximately $C=3.4$–$3.8$, illustrating the graph-reading precision. The nearby open-bottom threshold also limits the precision of the ideal continuum approximation.

Resonance quantum numbers and widths. Direct formation through a single photon requiresThese are the quantum numbers of the electromagnetic current. This concerns single-photon formation; associated production or two-photon reactions need not obey this restriction.

For a Breit-Wigner resonance cross section, subtract the continuum before finding the full width at half maximum. The plotted excess is about $0.9\mathrm{nb}$, so its half-height occurs at a total displayed cross section near $1.25\mathrm{nb}$, not $0.85\mathrm{nb}$. The crossings are roughly $10.57$ and $10.59\mathrm{GeV}$, with a separation around $0.02$–$0.025\mathrm{GeV}$. ThusFor spin-one formation from unpolarized electrons and positrons, $g=3/4$ and $p\simeq M/2$. With ${\Gamma }_{\mathrm{had}}\simeq \Gamma$, the peak resonance cross section and branching fractions relation becomesThe corrected excess is $C(1.7-0.8)\simeq 3.2\mathrm{nb}$. Taking $\Gamma \simeq 24\mathrm{MeV}$ as a representative graph reading givesThus the plot supportsThe width and electronic partial width are estimates from the printed graph under the prescribed Breit-Wigner and efficiency assumptions; these are not precision world-average values. A scale correction changes the peak height and inferred partial width, but not the peak position or width.

Why the lower Upsilon states are narrow. In electromagnetic production and decay of vector quarkonium, every $nS$ state can annihilate as $b\bar{b}\to {\gamma }^{\ast }\to e^{+}{e}^{-}$. The electronic width probes the wavefunction at the origin, ${\Gamma }_{ee}\propto {\alpha }^2{Q}_b^2|{\psi }_{nS}(0){|}^2/M_n^2$. The radial overlaps differ, but the same short-distance annihilation mechanism operates, so electronic widths remain of comparable order.

The hadronic mechanism changes sharply at $2m_B\simeq 10.560\mathrm{GeV}$. The first three Upsilon mesons lie below it and cannot produce an on-shell B-meson pair. Their leading purely strong annihilation channel is $b\bar{b}\to ggg$, followed by hadronization: a color-singlet pair cannot annihilate to one gluon, and two gluons cannot match its negative charge-conjugation parity. Three-gluon annihilation carries three powers of ${\alpha }_s$ in the rate and requires quark-antiquark overlap. Electromagnetic annihilation to light quarks and, for excited states, transitions to lower bottomonium plus pions or photons also contribute.

The fourth state lies above the pair threshold and can instead undergo ${\rm Y}(4S)\to B\bar{B}$ through light-pair creation, keeping the b and anti-b in separate outgoing mesons. The final panel depicts this quark-line topology; it represents a nonperturbative strong decay, not a free on-shell b quark radiating a real gluon. For two pseudoscalar B mesons the pair is in a P wave to give $1^{-}$. Although close to threshold, this open strong channel has a much larger rate than the annihilation channels. Consequently a total width of tens of MeV can coexist with an electronic width below a keV; below threshold the total widths are smaller by hundreds or more, while their electronic widths remain of similar order.

Hyperfine estimate. For two spin-half constituents,The meson hyperfine mass splitting therefore raises the vector mass above the pseudoscalar mass. To infer its coefficient from only one measured mass, adopt the intended simple constituent ansatz that the spin-independent baseline is $m_b+m_q=5.31\mathrm{GeV}$. ThenFor the spin-one partner,The estimate is the same for $\bar{b}u$ and $\bar{b}d$ with the supplied equal light-quark masses. The splitting is below $m_{\pi }$, so the strong decay $B^{\ast }\to B\pi$ is closed. An electromagnetic transition is open and dominates over weak decay:It preserves electric charge, giving the corresponding charged or neutral B meson; the photon energy is approximately $40\mathrm{MeV}$, slightly reduced by recoil. Strictly, the statement that the mass merely includes a spin-spin term does not fix $A$: an independent spin-independent binding energy would leave the baseline unknown. The numerical prediction above explicitly uses the additive constituent-mass approximation.

Write $C_R=4R/(1-R{)}^2$, keeping this coefficient distinct from the finesse $\mathcal{F}$. The transmission has maxima at $\delta =2\pi m$. At a displacement $ϵ$ from a maximum, its ratio to peak transmission is $[1+C_R{\sin }^2(ϵ/2){]}^{-1}$. Half maximum therefore requiresThis is the half-width in wave phase; the full width at half maximum is $2{\delta }_{1/2}$. Successive resonances are $2\pi$ apart, so the exact ideal-model result isFor narrow peaks, $(1-R)/(2\sqrt{R})\ll 1$, the small-angle approximation to the inverse sine givesThus the formula requested in the paper is the high-reflectivity approximation, not an exact identity at arbitrary $R$. Ordinary half-maximum peaks exist only when $(1-R)/(2\sqrt{R})\le 1$, or $R\ge 3-2\sqrt{2}$; below that threshold the transmission minimum never reaches half the peak. The prefactor $T^2/(1-R{)}^2$ cancels from this relative-width calculation.

Let ${\lambda }_1=589.0\mathrm{nm}$, ${\lambda }_2=589.6\mathrm{nm}$ and $\Delta \lambda =0.6\mathrm{nm}$. Requiring the separation to be at least $100$ times the measured full width at half maximum means a width no larger than $0.006\mathrm{nm}$, or a resolving power of approximately $589.3/0.006\simeq 9.8\times 10^4$.

The etalon resolving power and order overlap constraints are particularly clear in frequency. Put $\Delta \nu =c|1/{\lambda }_1-1/{\lambda }_2|$. At normal incidence, the free spectral range is ${\nu }_{\mathrm{FSR}}=c/(2d)$ and the instrumental width is $\delta \nu ={\nu }_{\mathrm{FSR}}/\mathcal{F}$. ThusAvoiding ambiguity between the doublet and adjacent interference orders requires a free spectral range exceeding the doublet separation:Together these giveUsing the narrow-peak finesse from the previous part, $\pi \sqrt{R}/(1-R)>100$ gives $\boxed{R\gtrsim 0.9691}$, with a practical margin above that limiting value. At the exact limit, the gap between orders vanishes, so equality is not a sensible operating point.

One feasible ideal design is $R=0.99$, $\mathcal{F}\simeq 313$, and $d=0.14018\mathrm{mm}$. It has an approximate wavelength free spectral range of $1.24\mathrm{nm}$ and instrumental width $0.0040\mathrm{nm}$ near the doublet. The doublet spacing is about $151$ widths. Even the spacing to the nearest replica of the other line exceeds $100$ widths in this example. If that latter separation is also required in general, impose the stronger condition $\min (\Delta \nu ,{\nu }_{\mathrm{FSR}}-\Delta \nu )\ge 100{\nu }_{\mathrm{FSR}}/\mathcal{F}$, rather than just the two basic inequalities.

Such mirror reflectivities and submillimetre gaps are plausible for a high-quality Fabry–Pérot etalon. Flatness, parallelism, coating loss, angular spread, mechanical stability and source width limit the achieved finesse. A wavelength width of $0.004\mathrm{nm}$ corresponds to a gap variation of order $d\delta \lambda /\lambda \simeq 1\mathrm{nm}$, so the ideal coating-limited estimate demands stringent uniformity. The paper expressly excludes intrinsic source broadening.

The first panel is the fixed etalon's transmission versus wavelength, a comb of resonances. It does not by itself have one peak at each sodium line. The second panel shows the two peaks obtained by scanning the etalon phase or gap and calibrating the scan to wavelength. A fixed-gap output spectrum is instead the incident spectrum multiplied by the transmission function. This distinction is needed to interpret a doublet measurement correctly.

Choose two hypothetical environments A and B of one chemical element, with line centers ${\mu }_A=284\mathrm{eV}$ and ${\mu }_B=286\mathrm{eV}$. Model each line by a normalized Gaussian function with common standard deviation $\sigma$, and stipulate integrated count weights $w_A=3/4$, $w_B=1/4$. On the binding energy axis $E$, the normalized count density isAll energies in this expression use electronvolts. Each component's integrated area equals its count weight, and $\int I(E)dE=1$. There is no background, statistical counting fluctuation, or additional emission feature in this stipulated model. These choices deliberately isolate photoelectron peak overlap and resolution.

Direct integration, or the law of total expectation applied to the component label, gives the centroidThe centroid lies between the chemical environments and is not either component's binding energy. Expanding $(E-\bar{E}{)}^2$ within each component, the odd cross term integrates to zero. The resulting variance isFor $\sigma =0.25\mathrm{eV}$ the variance is $0.8125{\mathrm{eV}}^2$ and the two maxima are separated. For $\sigma =1.00\mathrm{eV}$ it is $1.7500{\mathrm{eV}}^2$ and there is one maximum. Each Gaussian's full width at half maximum is $2\sqrt{2\ln 2}\sigma$, approximately $0.589$ and $2.355\mathrm{eV}$ respectively. The chemical separation remains $2\mathrm{eV}$ in both cases. The number of visible maxima therefore does not equal the number of chemical components.

One interpretation of this change is additional independent Gaussian instrument blur. The convolution of probability densities adds the variances, so broadening $\sigma =0.25$ to $1$ requires response standard deviation $\sqrt{1-0.25^2}=\sqrt{15}/4\simeq 0.968246\mathrm{eV}$. A normalized response with zero mean preserves the centroid and integrated counts. Real line shapes and backgrounds can depart from this model, and separating hidden components is an inverse problem; a plausible fit does not alone establish uniqueness.

Finally, count weights describe detection. In a simple response model, let $n_i$ denote the number of emitting sites sampled in environment $i$, $s_i$ its effective photoionization cross section, and $T_i$ the probability that its emitted electron contributes to the selected measured line. ThenThe common factor $C$ includes exposure. Here $T_i$ includes escape from the sample, analyzer transmission and detection. Count weights equal site fractions only if the relevant response factors are equal or corrected. They are not automatically molecular mole fractions; different molecules can supply different numbers of the selected emitting sites. In an original example, equal site numbers with $s_A{T}_A=3s_B{T}_B$ give the plotted $3:1$ count ratio despite a $1:1$ site ratio.