Past exam of the particle physics course of the University of Oxford / 2021 (A11047W1)

Source paper: www.whatdotheyknow.com/request/oxford_physics_past_papers_2/response/2340696/attach/9/Physics%20Papers%202020.pdf?cookie_passthrough=1

The source is on bundle pages 34–41. Its cover gives Trinity Term 2021 despite the attachment's 2020 filename. These solutions use natural units $\hslash =c=1$ unless a conversion is written explicitly.

In a relativistic two-body decay, four-momentum conservation in the parent's rest frame requires ${\mathbf{p}}_1^{\ast }=-{\mathbf{p}}_2^{\ast }$ andFor fixed daughter masses, this fixes a unique $p^{\ast }$, independent of direction. In the Lorentz-invariant phase space integral, the Dirac delta distribution enforcing conservation of energy therefore performs the radial momentum integration. Only the solid angle remains to be integrated; the resulting $p^{\ast }$ can be taken outside that integral, even if the matrix element depends strongly on direction.

For three or more final particles, four-momentum conservation generally leaves continuously variable daughter energies and momenta, so the decay rate includes integrations over them, or equivalently over independent invariant masses in a Dalitz plot. Integrating over a variable daughter invariant mass is also necessary when an unstable daughter is treated with a finite decay width rather than a fixed mass.

Useful examples with no net quark flavor are the eta c meson ${\eta }_c=c\bar{c}$ and the eta b meson ${\eta }_b=b\bar{b}$. Examples with nonzero quark flavor are the kaon $K^{+}=u\bar{s}$ and the D meson $D^{+}=c\bar{d}$. These are pseudoscalar mesons: in the lowest quark model configuration the constituent spins combine to $S=0$, the relative orbital angular momentum is $L=0$, and the total angular momentum and parity are $J^P=0^{-}$. For the self-conjugate examples, meson parity and charge conjugation also gives $C=(-1{)}^{L+S}=+1$. Electrical neutrality alone would not imply zero net quark flavor; for example $K^0=d\bar{s}$ has strangeness.

Restoring reduced Planck constant, decay width and mean lifetime satisfy $\Gamma \tau =\hslash$. Thus the representative numbers correspond toThe mean lifetimes differ by at least $3.0\times 10^{10}$. Hidden-heavy-flavor quarkonium can undergo quark-antiquark annihilation into gluons, followed by hadronization. A $0^{-+}$ color-singlet state can annihilate into two gluons. A lightest open-flavor meson cannot simply annihilate through the strong interaction, which conserves net quark flavor. With no lighter accessible final state carrying that flavor, it must undergo a weak interaction changing a constituent's flavor. The small weak interaction amplitude, CKM matrix factors and available Lorentz-invariant phase space produce a much smaller decay width.

Qualification. A decay width of order tens of MeV is not a universal property of flavor-neutral pseudoscalar mesons. The neutral pion has $\Gamma \simeq 7.8\mathrm{eV}$ and decays predominantly by the electromagnetic interaction, a concrete counterexample. Its mass lies below the lightest allowed hadronic final state. The examples above illustrate the intended annihilation-versus-open-flavor comparison; the numerical scale depends on thresholds and selection rules.^[ref]

Assume the charmonium state annihilates through a virtual photon, with lepton universality and negligible higher-order corrections. For a vector meson of mass $M$, the partial decay width into a charged lepton pair contains both the relativistic two-body decay factor ${\beta }_{\ell }$ and a vector-current spin factor:^[ref] For example, contracting the vector polarization sum with the Dirac trace for the two final fermions gives a factor proportional to $M^2+2m_{\ell }^2$, explaining why retaining only ${\beta }_{\ell }$ is insufficient. Using $m_{\tau }=1.777\mathrm{GeV}$, $m_{\mu }=0.1057\mathrm{GeV}$ and $M=3.686\mathrm{GeV}$ givesThe supplied total decay width also gives ${\Gamma }_{\mu \mu }=2.352\mathrm{keV}$ and ${\Gamma }_{\tau \tau }\simeq 0.91\mathrm{keV}$. It cancels from the branching fraction ratio.

The decisive change at $3.770\mathrm{GeV}$ is crossing the open-charm D meson-pair threshold, approximately $3.73\mathrm{GeV}$. The strong interaction can now create a light quark-antiquark pair and rearrange $c\bar{c}$ into $D\bar{D}$, without annihilating the heavy pair. This increases the total decay width from hundreds of keV to tens of MeV. The leptonic decay partial decay width is additionally smaller because $\psi (3770)$ is predominantly a $1{}^3{D}_1$ state, whereas $\psi (3686)$ is predominantly $2{}^3{S}_1$; the short-distance annihilation overlap is suppressed for nonzero orbital angular momentum. A roughly hundredfold increase in total decay width together with an order-ten reduction in leptonic partial decay width explains the order-thousand reduction in branching fraction. Attributing all three orders of magnitude to the threshold alone would miss the second effect.^[ref]

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 $\psi (3686)$.

The three structures are the radiative cascadesIn the quark model, the J/psi meson and $\psi (2S)$ are respectively $1{}^3{S}_1$ and $2{}^3{S}_1$ charmonium states. Thus $L=0$, $S=1$, $J=1$, and meson parity and charge conjugation gives $J^{PC}=1^{--}$. The intermediate chi c mesons are $1{}^3{P}_J$: $L=1$, $S=1$, and addition of angular momentum permits $J=0,1,2$. ConsequentlyThese are electric dipole transitions, changing $L$ by one and preserving $S$ to leading order. A singlet $1{}^1{P}_1$ state has $J^{PC}=1^{+-}$ and is not another member of these spin-triplet cascades.

The primary photon energy is $(M_{2S}^2-M_{{\chi }_J}^2)/(2M_{2S})$, about $261$, $171$ and $128\mathrm{MeV}$ for $J=0,1,2$. The secondary photon remains more energetic even after its boost into the parent rest frame. Hence $m^2(J/\psi {\gamma }_h)=M_{{\chi }_J}^2$: with this quantity on the vertical axis, each resonance gives a horizontal band. A finite decay width and detector measurement resolution broaden each band; photon ordering removes the mirrored duplication of an unordered Dalitz plot.

The faint band beside the negative-slope diagonal boundary is $\psi (2S)\to J/\psi {\pi }^0$, followed by ${\pi }^0\to \gamma \gamma$. The neutral pion is predominantly $(u\bar{u}-d\bar{d})/\sqrt{2}$, a ${}^1{S}_0$ pseudoscalar meson with $J^{PC}=0^{-+}$. This production transition violates isospin, accounting for its small rate. To locate the band, set $X=m^2(J/\psi {\gamma }_l)$ and $Y=m^2(J/\psi {\gamma }_h)$. Four-momentum conservation givesFor the neutral pion, $m_{\gamma \gamma }^2=m_{{\pi }^0}^2$ is small, putting the line just inside the $m_{\gamma \gamma }^2=0$ boundary. The allowed $J/\psi \eta$ channel, with the eta meson also $0^{-+}$, produces a parallel line further inside, at $m_{\gamma \gamma }^2=m_{\eta }^2$. The positive-slope edge $X=Y$ instead comes from ordering the two photon energies; proximity to that edge alone would not identify the two-photon state.

Use the Minkowski metric $(+,-,-,-)$ and choose the incoming electron along $+z$, with the scattered electron in the $xz$ plane. In natural units, neglecting the electron mass, the four-momenta areThe momentum transfer entering the proton is thereforeThe outgoing hadronic system carries $p_X=p_2+q$. The energy transfer is $\nu =E-E^{\prime }$, while $\mathbf{q}$ need not point along the original beam.

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.

At leading order in QED, the electron exchanges one spacelike virtual photon with a charged parton inside the proton.

The two Feynman vertices contribute $e$ and $e{e}_q$, while the photon propagator contributes $1/q^2=-1/Q^2$. Hence the scattering amplitude scales as $e^2{e}_q/Q^2$ and its squared absolute value as $e^4{e}_q^2/Q^4$. Since the fine-structure constant is $\alpha =e^2/(4\pi )$, the electromagnetic coupling dependence is ${\alpha }^2/Q^4$. The numerical $4\pi$ in the stated differential scattering cross-section additionally uses the flux, Lorentz-invariant phase space, azimuthal integration and the convention defining the deep-inelastic structure function; it cannot be deduced from the propagator alone.

For each measured $x,Q^2$ bin form the reduced quantityPlot this against $Q^2$ at several fixed $x$ values, normally using a logarithmic $Q^2$ axis. Bjorken scaling predicts an approximately horizontal curve at each $x$, or equivalently collapse of different-$Q^2$ data onto one $F_2(x)$ curve. Raw cross sections do not collapse because their explicit kinematic factors vary. QCD predicts logarithmic scaling violations, so exact independence is only the ideal parton model limit. The stated extraction also assumes negligible longitudinal deep-inelastic structure function, as in the leading massless spin-half parton approximation.

This labelled part supplies a definition rather than a separate task. Its notation is interpreted as an incoherent charge-squared sum: the photon couples to a quark with electric charge $e{e}_q$, and the squared scattering amplitude weights its contribution by $e_q^2$. To make the antiquark convention explicit, letThen, restricting the displayed example to three active light quark flavors, the deep-inelastic structure function isThus the source's sum must either include antiquarks as separate partons or define each $q^p$ to include its antiquark distribution. It must not silently omit the sea quarks, especially in the small-$x$ limit used below.

Assume approximate charge symmetry of nucleon parton distributions, the interchange of up quarks and down quarks under the isospin rotation carrying the proton into the neutron:Corrections from the up quark-down quark mass difference and the electromagnetic interaction are neglected. With the proton-only shorthand in d,If charm is active, add the same $4x(c^p+{\bar{c}}^p)/9$ to each; heavier flavors are treated similarly. These formulae use the same $Q^2$ and factorization convention for all parton distribution functions.

Subtracting the deep-inelastic structure functions in e givesThus at moderate and large $x$, the charge-weighted up quark content of the proton exceeds its down quark content. In the valence quark dominated regime this is consistent with the $uud$ valence structure, compared with $udd$ for the neutron. The larger $2/3$ up quark electric charge makes the proton scatter more strongly. This observation alone does not establish $u^p(x)=2d^p(x)$ pointwise: the valence-number sum rules constrain integrals, not every $x$ bin.

If the small-$x$ observation means equality to good relative accuracy, it supports dominance by approximately charge-symmetric sea quarks, with $U\simeq D$ and negligible relative valence excess. Gluon splitting creates quark-antiquark pairs, and the leading strong interaction treats light quark flavors similarly. It does not follow that every sea quark flavor has exactly the same distribution at every $x$.

Logical qualification. Equality of the limiting values alone is weaker than equality of their ratio. For example $U=2$, $D=1$, $S=0$ near $x=0$ gives $F_2^p=x$ and $F_2^n=2x/3$: both tend to zero, although $U\ne D$ and $F_2^n/F_2^p=2/3$. The usual inference about a symmetric sea needs $F_2^n/F_2^p\to 1$ or equivalent information about relative differences, rather than merely coincident zero limits.

A mass eigenstate $|{\nu }_i⟩$ has a definite neutrino mass $m_i$ and acquires a definite propagation phase at fixed momentum. An interaction eigenstate is the neutrino flavor selected by a weak charged current: ${\nu }_e$ is produced or detected together with an electron, and ${\nu }_{\mu }$ together with a muon. The two bases are related by a unitary matrix, here the real rotation matrix $U$.

Inverting the supplied transformation gives $|{\nu }_e⟩=c|{\nu }_1⟩+s|{\nu }_2⟩$ and $|{\nu }_{\mu }⟩=-s|{\nu }_1⟩+c|{\nu }_2⟩$, where $c=\cos \theta$, $s=\sin \theta$. If $\theta =0$, these bases coincide: each produced neutrino flavor propagates as one mass eigenstate, and there is no neutrino oscillation, even if the two masses differ. Conversely, equal masses also prevent vacuum neutrino oscillation, even when a nonzero basis-rotation angle is written down.

Write $\bar{\chi }=({\chi }_1+{\chi }_2)/2$ and $\delta \chi ={\chi }_2-{\chi }_1$. Since the supplied rotation matrix is real and orthogonal, $U^{-1}=U^T$. Direct matrix multiplication gives the flavor-basis HamiltonianThe common term $\bar{\chi }I$ produces only an overall quantum phase. The off-diagonal elements $H_{e\mu }^{\prime }=H_{\mu e}^{\prime }=(\delta \chi /2)\sin 2\theta$ couple the two interaction eigenstates in the Schrödinger equation. Flavor change requires both $\delta \chi \ne 0$ and $\sin 2\theta \ne 0$. For ultrarelativistic neutrinos in vacuum, $\delta \chi \simeq \Delta m^2/(2E)$, so a nonzero squared-mass difference is needed. Observable neutrino oscillation also requires coherent production and propagation of the two components and a resolvable nonzero relative phase; special propagation times can return the initial flavor even when these Hamiltonian conditions hold.

A useful distinction for the next part is that $H_{ee}^{\prime }$ is an energy, not a time-dependent survival probability amplitude. The latter is the $ee$ element of $S(t)=\exp (-i{H}^{\prime }t)$.

Starting with the electron neutrino state in a, the survival probability amplitude isTaking its squared absolute value givesThis derives the supplied survival probability from the evolution operator rather than from a Hamiltonian entry. It also shows why the two written forms agree. For the stipulated phase and $\theta =22.5^{\circ }$, ${\sin }^{2}2\theta =1/2$, so at $L=79\mathrm{km}$,The phase decreases monotonically from $7.9$ to $0.79$ over the open interval $1<E<10$. Differentiating gives $d{P}_{ee}/dE=[7.9/(2E^2)]\sin (15.8/E)$. The interior maxima occur at $7.9/E=n\pi$ and the interior minima at $7.9/E=(n+1/2)\pi$. Enumerating those phases givesThe minimum just above $1\mathrm{MeV}$ must not be lost by rounding the lower endpoint. Neither excluded endpoint is an additional extremum.

For solar neutrinos propagating in the same vacuum model, the vastly longer baseline makes the oscillations extremely fine in energy, with the same bounds $1/2$ and $1$. The figure uses a deliberately schematic enlargement of the oscillation spacing to show those bounds. A $1\mathrm{MeV}$ detector energy resolution averages very many periods, so $⟨{\cos }^2\phi ⟩=1/2$ andThe detector averages event rates weighted by source spectrum, interaction cross section and response; the displayed constant follows when these vary slowly across each tiny oscillation period. Reactor production physically gives electron antineutrinos. In this two-flavor vacuum approximation their survival probability is identical; the positive matter-potential discussion in d concerns solar neutrinos.

For a positive electron neutrino matter potential and the normal ordering implicit in positive $K$, choose $0\le {\theta }_m\le \pi /2$ by continuous diagonalization of the matter Hamiltonian. With $D=[(\cos 2\theta -K\rho {)}^2+{\sin }^{2}2\theta {]}^{1/2}$, the quadrant is fixed byAt high density $\cos 2{\theta }_m\to -1$, soTaking a principal-value arctangent alone and concluding ${\theta }_m\to 0$ would select the wrong branch. The production state is the second matter propagation eigenstate, rather than literally the vacuum mass eigenstate while still inside the dense medium.

Assume the solar density decreases slowly enough for adiabatic matter-enhanced neutrino conversion, and neglect subsequent Earth matter effects. The state follows ${\nu }_{2m}$ continuously into the vacuum mass eigenstate ${\nu }_2$. Since $⟨{\nu }_e|{\nu }_2⟩=\sin \theta$,This is substantially below the phase-averaged vacuum result $0.75$. There is no interference between two populated vacuum mass eigenstates in this ideal adiabatic limit. More generally, after phase averaging, $P_{ee}=\frac{1}{2}+\frac{1}{2}\cos 2\theta \cos 2{\theta }_m$ at production if the evolution is adiabatic. In the high-density limit, a nonadiabatic level-jump probability $P_c$ instead gives $P_{ee}=(1-P_c){\sin }^2\theta +P_c{\cos }^2\theta$. Thus the supplied core-density condition alone does not guarantee the numerical Earth result: the usual solar adiabatic assumption is also needed. These are the stipulated two-flavor parameters, not a precision prediction for the measured three-flavor solar spectrum.

The chiral projectors $P_{L,R}=(1\mp {\gamma }^5)/2$ obey ${\gamma }^{\mu }{P}_L=P_R{\gamma }^{\mu }$, while the Dirac adjoints are ${\bar{\psi }}_L=\bar{\psi }{P}_R$ and ${\bar{\psi }}_R=\bar{\psi }{P}_L$. Consequently the cross terms vanish and the vector current isThe vector interaction preserves fermion chirality at a vertex and couples equally to left and right components. This statement about the vertex does not assert that a fermion mass term preserves chirality. Under parity, left and right components are exchanged; their equal coefficients allow a parity-invariant contraction with a true vector field.

An axial vector must couple to the axial current to obtain the analogous parity-invariant interaction. Since ${\gamma }^5{\psi }_L=-{\psi }_L$ and ${\gamma }^5{\psi }_R={\psi }_R$,The relative sign changes; the overall sign is conventional. A pure axial interaction can itself conserve parity when $A_{\mu }$ transforms as an axial vector. Simply contracting the original vector current with an axial-vector field would instead produce a parity-odd interaction.

For a massless fermion, chirality is fixed by helicity. For a massless antifermion, the field-chirality label is opposite to its physical helicity: a left-chiral field creates negative-helicity particles and positive-helicity antiparticles. At $E\gg m$, wrong-chirality amplitudes are suppressed by $m/E$, so chirality-preserving vector vertices conserve helicity along a physical fermion line up to these corrections.

For electron-positron annihilation into a muon pair through a vector mediator, each annihilating or created pair therefore has opposite physical helicities. Writing each pair in the order (negative particle, positive particle), the nonzero possibilities areHere $L,R$ refer to physical helicity along each particle's own momentum, not to the sign of its spin projected on a single common axis. The incoming pair has total spin projection $m=\pm 1$ along the electron beam; the outgoing pair has $m^{\prime }=\pm 1$ along the muon direction. Same-helicity pairs vanish in the strictly massless vector-current limit.

Assume unpolarized beams, negligible electron and muon masses, a single scattering s-channel spin-one mediator with equal left and right vector couplings, and leading-order QED. Z boson exchange and radiative corrections are omitted. The angular factors are the Wigner D-matrix elements $d_{m{m}^{\prime }}^1({\theta }^{\star })$, with $m,m^{\prime }=\pm 1$. Up to common phases and normalization they are $(1+\cos {\theta }^{\star })$ for equal pair-spin projections and $(1-\cos {\theta }^{\star })$ for opposite projections. The four orthogonal helicity channels add incoherently, givingWith $s$ the squared center-of-mass energy, the normalized differential scattering cross-sections areThe second expression includes the solid angle Jacobian: a distribution per angle is different from one per $\cos {\theta }^{\star }$. Integration gives $\sigma =4\pi {\alpha }^2/(3s)$.

Parity reverses all momenta but leaves the spin axial vector unchanged, so it interchanges $L\leftrightarrow R$. The $LR\to LR$ and $RL\to RL$ amplitudes have equal magnitudes, as do $LR\to RL$ and $RL\to LR$. Thus every configuration has the same rate as its parity transform, proving parity conservation. The summed distribution is also forward-backward symmetric. Strictly, parity reverses both the beam and outgoing momenta, leaving their relative angle unchanged; forward-backward reflection by itself is not the definition of parity.

Assume the same left and right couplings $g_l,g_r$ at the electron and muon vertices. This lepton universality assumption is necessary to describe both vertices using just two numbers. Define $a=|g_l{|}^2$, $b=|g_r{|}^2$ and $z=\cos {\theta }^{\star }$. The two same-chirality channels have squared coupling weights $a^2,b^2$ and angular factor $(1+z{)}^2$; the other two each have weight $ab$ and factor $(1-z{)}^2$. Hence, for a common positive normalization $C$,If $|g_l/g_r|\ne 1$, then $a\ne b$, so a forward-backward term survives. At the amplitude level, parity interchanges the $g_l^2$ and $g_r^2$ channels; their unequal squared magnitudes show directly that the interaction violates parity. Equivalently the chiral current contains both vector current and axial current terms, with coefficients $(g_r+g_l)/2$ and $(g_r-g_l)/2$; unequal magnitudes preclude a pure parity eigen-current.

The odd-in-$z$ term is a useful signature within these assumptions, but it should not itself be called a parity-odd scalar: the unpolarized relative angle is unchanged when all momenta are reversed. If the two species have independent couplings, the coefficient is instead proportional to $(a_e-b_e)(a_{\mu }-b_{\mu })$, and a parity-violating coupling at only one vertex need not produce an unpolarized forward-backward asymmetry.

Integrating the distribution in d, use ${\int }_{-1}^1(1+z^2)dz=8/3$ and ${\int }_0^{1}zdz-{\int }_{-1}^{0}zdz=1$. ThereforeSubtract backgrounds and correct the measured forward and backward yields for detector acceptance, charge misidentification and selection efficiency; radiative effects and other mediators must be removed or included in a fit. With $t=\sqrt{4A_{\mathrm{fb}}/3}$ and $r=|g_l/g_r|$, inversion givesA single unpolarized measurement determines the ratio up to $r\leftrightarrow 1/r$. A polarized-beam observable or another signed chiral asymmetry is needed to choose which coupling is larger. Values outside $0\le A_{\mathrm{fb}}\le 3/4$ would contradict this ideal universal single-mediator model, apart from experimental uncertainties.

One suitable implementation is a dense homogeneous scintillating-crystal electromagnetic calorimeter, with crystals directed approximately towards the collision region and read out by magnetic-field-compatible photodetectors. An incident electron emits bremsstrahlung photons; these produce electron-Positron pairs, which radiate again. This electromagnetic shower develops over successive radiation lengths until the daughter energies approach the critical energy of an electromagnetic shower, below which ionization dominates. The deposited energy creates scintillation light, and calibrated integrated detector signals estimate the incident energy.

A depth of roughly $25$ radiation lengths contains most of a high-energy electromagnetic shower. Small transverse cells resolve its lateral profile, distinguish nearby deposits and permit position reconstruction. Crystal material, optical transmission, gain and electronics must remain stable under the LHC radiation exposure. Fast shaping, timing, segmentation and radiation-tolerant readout permit operation at high collision rates; collision pileup and particle showers from nearby particles require reconstruction and calibration. A dense crystal array offers good intrinsic energy resolution because it measures almost all the shower deposition rather than only the sampled fraction.

Independent fluctuations add as variances. A useful parametrization of the electromagnetic calorimeter energy resolution isThe convention makes $a,b,c$ dimensionless when $E$ is inserted in GeV. The three terms describe different mechanisms.

The stochastic term comes from fluctuations in the electromagnetic shower, scintillation production and detected photoelectron counts. If the number of detected photoelectrons is $N=kE$, Poisson distribution gives ${\sigma }_N=\sqrt{N}$, hence ${\sigma }_E/E=1/\sqrt{kE}\propto E^{-1/2}$. A sampling electromagnetic calorimeter also has fluctuations in the fraction deposited in its active material, contributing to the same approximate scaling. A homogeneous detector suppresses that sampling contribution.

The noise term represents approximately fixed absolute electronics and dark-current noise for a chosen reconstruction window. Additional collision pileup fluctuations can behave similarly over a specified operating range. A nearly constant ${\sigma }_E$ gives ${\sigma }_E/E\propto E^{-1}$.

The constant term describes residual fractional cell-calibration errors, gain drifts, temperature dependence, nonuniform light collection, dead material, and imperfect correction for longitudinal and transverse leakage. These effects often scale approximately with the incident energy, so they approach a constant fractional error. For electrons, upstream bremsstrahlung and conversion of the radiated photons spread the deposit over multiple cells; material mapping and clustering are essential. Leakage and material losses can also produce non-Gaussian tails, so the three-term parametrization describes a resolution core rather than every feature of the response. Noise dominates at low energy, stochastic fluctuations at intermediate energy, and the constant term eventually limits performance.

Use a clean Z boson sample with final-state radiation, $Z\to e^{+}{e}^{-}\gamma$, and independently calibrated electron four-momenta. For measured $P=p_{e^{+}}+p_{e^{-}}$ and measured photon direction ${\hat{n}}_{\gamma }$, write $p_{\gamma }=E_{\gamma }(1,{\hat{n}}_{\gamma })$. The known Z boson mass suppliesComparing the reconstructed photon energy with this prediction constrains its scale versus energy, detector position and conversion status. In practice fit the $e^{+}{e}^{-}\gamma$ invariant mass distribution, allowing for the physical Z boson decay width, detector response and backgrounds, instead of assigning every event exactly $M_Z$. Select final-state-radiation topologies: an initial-state-radiation photon is not part of the Z boson decay and generally does not satisfy the same mass constraint. The calibration applies over the populated photon energy range; extrapolation to still higher energies needs a validated response model.

For two photons, four-momentum conservation and their vanishing rest masses give the diphoton invariant massTaking a logarithmic differential,For small mutually independent measurement errors, propagation of uncertainty consequently givesPrinted-formula correction. The source's left-hand side, if it means $(\Delta M/M{)}^2$ as its description suggests, is missing the factor $1/4$ on the right. Its right-hand side instead equals $\mathrm{Var}(M^2)/M^4$ to first order. A concrete counterexample is a fixed opening angle and an exactly known $E_2$: because $M\propto \sqrt{E_1}$, a small $1\%$ fractional error in $E_1$ produces a $0.5\%$ fractional error in $M$, not $1\%$. Likewise, interpreting $\Delta M^2$ literally as a small change in $M^2$ would make the unsquared left-hand side incompatible with the squared terms. Correlated errors require the covariance terms as well.

The space angle is determined by the two polar angles and azimuthal separation:At ${\theta }_1={\theta }_2=45^{\circ }$ and $\Delta \varphi =180^{\circ }$, $\cos \alpha =0$ and $\alpha =90^{\circ }$. The angular coefficient is therefore one. Equal contribution of opening-angle uncertainty and the combined two independent $1\%$ photon energy errors requiresFor independent equally precise single-photon polar directions, ${\sigma }_{\alpha }^2={\sigma }_{{\theta }_1}^2+{\sigma }_{{\theta }_2}^2$ at this geometry, soAn azimuthal error contributes only at second order about $\Delta \varphi =\pi$. The corresponding invariant-mass resolution from the two energy errors alone is $0.707\%$; with an equally large angular contribution it is $1.00\%$. The common missing factor of four cancels when equating the two contributions, so the angular estimates remain valid despite the printed error.

Use a longitudinally segmented electromagnetic calorimeter with fine transverse position readout. The electromagnetic shower centroids in two separated depth layers define a line, providing photon pointing without assuming the longitudinal collision position. Combine this information with the reconstructed primary collision vertex from charged-particle track reconstruction. The two lines from that vertex to the shower positions then determine the opening angle.

For an order-of-magnitude requirement, a single-direction precision of $10\mathrm{mrad}$ corresponds to a transverse position uncertainty of $15\mathrm{mm}$ over a $1.5\mathrm{m}$ vertex-to-shower lever arm, provided the vertex is known sufficiently well. Centroid reconstruction across finely segmented cells can reach this scale. At high collision pileup, identifying the correct primary vertex is essential: using the wrong vertex can dominate the angular error even if the shower position is accurately measured. Converted photons offer an additional pointing constraint from the reconstructed $e^{+}{e}^{-}$ conversion tracks, but conversion is not required by the method described here.

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.

Count a reference process whose fiducial cross section is accurately calculable. At an electron-positron collider, small-angle Bhabha scattering, dominated by $t$-channel photon exchange, is abundant and provides a standard choice. Require coincident high-energy electron and Positron deposits on opposite sides of the interaction point, with consistent back-to-back directions, and impose precisely specified angular and energy cuts. ThenThe theoretical prediction must include radiation and other relevant electroweak effects under the same event definition. Finite angular resolution, alignment, beam position, live time, trigger efficiency and background contamination enter the uncertainty budget. For small scattering angle $\theta$, $t\simeq -s{\theta }^2/4$ and the leading differential scattering cross-section behaves approximately as $d\sigma /d\theta \propto {\theta }^{-3}$; the inner fiducial angle must therefore be particularly accurately known. Binning these counts in time can also measure the instantaneous collider luminosity.

A compact silicon-tungsten sampling electromagnetic calorimeter can serve as the small-angle luminosity monitor. Alternate dense tungsten absorber plates with finely segmented silicon sensor planes. Incident electrons and Positrons create electromagnetic showers in the absorber; their charged daughters create electron-hole pairs by ionization in depleted silicon. An electric field collects the charge, and the summed calibrated signal estimates shower energy.

Radial and azimuthal sensor segmentation gives precise shower positions and hence polar angles using the measured distance to the interaction point. Multiple layers measure shower development and improve the centroid estimate. Opposite-side coincidence, beam-time synchronization and an energy requirement reject beam backgrounds. Mechanical survey, alignment and edge-response calibration define the small-angle fiducial boundary: high position precision matters especially because the Bhabha scattering rate changes steeply with angle. The monitor's overall energy resolution need not match the central precision crystal detector; it must accurately reconstruct its own fiducial selection and efficiency.

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.