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The approximate electron energy at which radiative and collisional energy-loss rates are equal in a given material. Below it, ionization increasingly dominates and multiplication of the electromagnetic shower ends. It is material dependent, not a universal particle constant.

A homogeneous calorimeter uses its showering volume as the active medium, such as a scintillating crystal. It avoids absorber-versus-active sampling fluctuations, but still has signal statistics, intrinsic shower fluctuations, noise and calibration nonuniformity. Its constant and noise terms need not be smaller than those of a sampling calorimeter.

The ideal two-shower event fraction is $B_e^2$, approximately $4\%$ in simple channel counting or $3.18\%$ using the measured tau lepton electronic branching fraction. The selected particle decay chain isThe electron and Positron each make a contained electromagnetic shower; muons penetrate, and generic hadrons do not deposit all their energy as a contained electromagnetic shower.

To estimate $B_e$, use lepton universality and neglect final masses and Quantum chromodynamics corrections. The electronic and muonic decay widths each carry weight one. The accessible inclusive hadronic channels have weights $3|V_{ud}{|}^2$ and $3|V_{us}{|}^2$, where three is the color multiplicity in a decay width. The charm quark is inaccessible in physical tau lepton hadronic decays. With $|V_{ud}{|}^2+|V_{us}{|}^2\simeq 1$, the total weight is $1+1+3=5$, givingThe 2021 measured value $B_e=0.1782$ instead gives $B_e^2=0.0317552$.^[ref] The difference reflects effects omitted by the simple free-channel estimate, especially the hadronic contribution to the total decay width.

There are two tau leptons per produced pair and two per selected pair, so the fraction of all produced tau leptons belonging to these two-shower events is also $B_e^2$. The unrestricted fraction of individual tau leptons decaying electronically is $B_e$; these are different selections. A real high-energy shower cut gives an additional acceptance factor: $N_{\mathrm{selected}}/N_{\mathrm{pairs}}=B_e^2{\epsilon }_{ee}$, which cannot be computed without the cut and detector response. No factor of two is needed for the unique $e^{+}{e}^{-}$ assignment.

The magnetic rigidity gives the momentum, the bending direction gives the sign of electric charge, and the calorimetric energy then gives the mass. For planar tracks perpendicular to a uniform magnetic field, the Lorentz force and circular motion giveTrace each track away from its production or decay vertex. With the magnetic field into the page, a positive track bends counterclockwise and a negative track bends clockwise. For example, a positive particle initially travelling left accelerates downwards. The known incoming negative pion fixes the sign convention experimentally. A radius alone does not distinguish electric charge signs.

For a calibrated total energy $E$, the relativistic energy-momentum relation givesIf the calorimeter instead supplies only kinetic energy $T$, solve $(T+m{c}^2{)}^2=p^2{c}^2+m^2{c}^4$ to obtainCompare with candidate pion, kaon, and proton masses, together with the electric charge sign and shower behavior. An electron makes an electromagnetic shower; a muon is penetrating; a hadron generally requires a hadronic calorimeter and its own calibration. If either energy escapes or the response is uncalibrated, simply substituting a deposited signal for $E$ need not identify the particle correctly. The absent figure prevents attaching identities to individual drawn tracks, but the requested identification procedure does not require its geometry.

The parton model treats a fast proton as an ensemble of approximately collinear, nearly free quarks and antiquarks. A hard deep inelastic scattering collision resolves one parton on a time scale short compared with its binding motion. The probabilities are parton distribution functions; the simplest approximation neglects intrinsic transverse momentum, quark masses and the proton mass in the hard collision. Quantum chromodynamics introduces logarithmic violations of Bjorken scaling, so the parton distribution functions depend on a resolution scale as well as a parton momentum fraction.

Write the incident electron and proton four-momenta as $k,P$, the scattered electron four-momentum as $k^{\prime }$, and set $q=k-k^{\prime }$. With the $(+---)$ Minkowski metric, the Bjorken scaling variable and the inelasticity areAn incident parton with four-momentum $\xi P$ emerges on shell, giving $(\xi P+q{)}^2\simeq 2\xi P\cdot q-Q^2=0$. Thus $\xi =x_B$: the observable Bjorken scaling variable measures the struck parton momentum fraction in this approximation.

For clarity, measure the scattering angle $\vartheta$ from the incoming electron direction, and denote the outgoing electron energy by $E_e^{\prime }$. Direct four-momentum products giveThe expression uses the known beam energies and two measured quantities. An electromagnetic calorimeter measures $E_e^{\prime }$; a tracking detector, the collision interaction vertex and the position of the electromagnetic shower determine $\vartheta$. Coverage close to the incoming electron direction matters at small $Q^2$. A hadronic calorimeter supplies an independent consistency check and rejects backgrounds. If the angle is instead measured from the incoming proton, its sine and cosine half-angle factors interchange.

At $p=600\mathrm{MeV}/c$, the velocities are $\beta =p/\sqrt{p^2+m^2{c}^2}$, giving approximately $1.0000$, $0.98485$ and $0.53874$ for electrons, muons and protons. The glass threshold is $\beta >1/1.55=0.64516$. Thus the first two produce Cherenkov radiation, with internal angleswhereas a primary proton at this momentum is below threshold.

If an optical arrangement transports those cone angles without changing them, a plane $L=2\mathrm{m}$ downstream would intersect rings of radiusThese are conditional geometric estimates, not the ray trace for a flat slab in air. The $0.1\mathrm{m}$ emission depth alone spreads the radius by order $0.12\mathrm{m}$, already larger than the approximately $0.062\mathrm{m}$ separation. The slab is one radiation length: electrons lose appreciable energy by bremsstrahlung and begin an electromagnetic shower, producing a broader and less clean pattern. Muons mainly ionize and retain a comparatively clean cone, with some Multiple Coulomb scattering. A below-threshold primary proton gives no direct cone, although energetic delta electrons and other secondary interactions can produce light.

There is an essential optical qualification. For a normal beam through a parallel-faced glass slab followed by air, the internal critical angle is ${\theta }_{\mathrm{crit}}=\arcsin (1/1.55)\simeq 40.18^{\circ }$. Both emission angles exceed it. Snell's law would require $\sin {\theta }_{\mathrm{air}}=1.55\sin {\theta }_C>1$, so the ideal photons undergo total internal reflection at the downstream face and do not form those direct downstream rings. Optical coupling, a prism or different surface geometry must be specified to extract them. The stated material parameters alone do not define a unique detected pattern; this flat-slab geometry is a concrete counterexample to silently assuming straight transmitted rays.

A large, optically clean water Cherenkov detector provides many protons, a transparent radiator and a relatively inexpensive homogeneous target. A surrounding array of inward-facing photomultipliers records hit times and charge. Reconstructing the vertex and cone intersections identifies track directions and visible energies; an outer active veto rejects entering particles. Water has refractive index near $1.33$, and the charged decay products are above its relevant thresholds.

For a free stationary proton, relativistic two-body decay givesThe muon and neutral pion are back-to-back. The pion decays essentially at the vertex into two gamma rays: $c{\tau }_{{\pi }^0}\simeq 2.4\times 10^{-8}\mathrm{m}$ before its modest Lorentz factor is included. Each gamma ray converts and initiates an electromagnetic shower, producing a diffuse ring. The muon gives a relatively sharp ring and, if it stops, a delayed Michel Positron on the $2.2\mathrm{microsecond}$ scale. The ideal topology therefore has one muon-like and two electron-like rings, although overlaps or a weak photon may merge or hide a ring.

Require the two-photon invariant mass to match $m_{{\pi }^0}$, the total reconstructed mass to match $m_p$, and small total momentum. Bound oxygen protons have Fermi motion and their hadronic products can rescatter, so those cuts need realistic nuclear corrections; the water target also contains free hydrogen protons. Install the detector deep underground to suppress cosmic-ray muons, with shielding and an active veto. Important irreducible backgrounds are atmospheric-neutrino interactions producing a muon and neutral pion, potentially with an undetected recoil nucleon.

At low charged-lepton energy, collisional ionization and excitation dominate over radiation. A slowing Positron also eventually annihilates, usually yielding photons. Low-energy photons lose energy through Compton scattering and, at sufficiently low energy, photoelectric absorption; pair production is kinematically unavailable below its threshold.

The electron critical energy $E_c$ is the energy where mean bremsstrahlung and collisional losses are equal. With $-(dE/dx{)}_{\mathrm{rad}}\simeq E/X_0$ and a slowly varying collisional loss,Above this energy, radiation and pair creation multiply shower particles efficiently. Below it, charged particles mainly deposit their energy by collisions and the shower dies out. Critical energy is material dependent; it is not a new photon-production threshold.

In the binary branching model of an electromagnetic shower, after $n$ ideal equal-sharing generations, the model has $N_n=2^n$ particles of energy $E_n=E_0/2^n$, at depth $n{X}_0$. Stop the branching when $E_n\le E_c$. ThusThe continuous estimate is $13.29X_0$. At 13 generations the particles still have approximately $12.2\mathrm{MeV}$, whereas at 14 they have approximately $6.10\mathrm{MeV}$. The ideal branching model therefore suggests roughly $13$–$14X_0$ before multiplication stops.

This is not an accurate containment criterion. Real conversion and radiation distances fluctuate, bremsstrahlung energy sharing is broad rather than always one half, some energetic branches penetrate beyond the average, and particles below critical energy still have stopping and absorption ranges. The simplified stopping depth is closer to its own model's shower-maximum scale than a depth containing all deposited energy. A calorimeter needs additional longitudinal depth to control leakage and event-to-event tails. These effects explain why approximately $25X_0$ can be required even though the equal-sharing estimate is smaller. The toy model does not predict the realistic shower maximum or leakage fraction quantitatively.

The principal passive-material requirement is a short radiation length in centimetres, normally obtained using a dense, high-atomic-number absorber such as lead or tungsten. It produces and contains an electromagnetic shower in a compact detector. Mechanical construction, transverse spread, cost, radiation tolerance and inactive gaps also affect the choice, but the absorber is selected for shower formation rather than signal production.

Liquid-argon gaps are a suitable active sampling medium. Ionization electrons drift in an electric field and induce a measurable charge signal. Advantages include radiation tolerance, stable response and useful spatial segmentation. Disadvantages include cryogenic infrastructure, high-voltage and purity requirements, signal collection times and electronic-noise control. Only the active fraction is measured directly, so fluctuations of energy shared between absorber and active gaps introduce a sampling contribution to the resolution. A high active fraction and fine sampling help, at the price of construction complexity and additional channels.

For independent contributions the parametrization meansThe stochastic coefficient $A$ includes fluctuations in the sampled shower fraction, the finite number of signal quanta, and intrinsic shower fluctuations. A number of detected quanta proportional to $E$ has fluctuations proportional to $\sqrt{E}$, giving relative resolution proportional to $E^{-1/2}$. The noise coefficient $B$ describes an approximately fixed energy-equivalent readout uncertainty, including electronic noise and, for specified running conditions, Pile-up contributions. The constant term $C$ represents residual fractional nonuniformity, calibration and response variations, imperfect correction for inactive material, and leakage effects over the fitted range. The simple three-term model is an approximation; leakage and Pile-up need not follow these powers exactly.

A homogeneous calorimeter uses the showering material itself as the active medium, for example a scintillating crystal. It removes the absorber-versus-active sampling fluctuation and can have a much smaller stochastic term, but it does not eliminate signal statistics or intrinsic shower fluctuations. Its noise and constant terms are not automatically smaller: they depend on light collection, calibration, temperature, radiation damage, electronics and geometry. Homogeneous detectors can offer excellent energy resolution and compact construction, but large crystal volumes can be costly and fine longitudinal segmentation difficult. Sampling detectors often provide economical large coverage and flexible segmentation with robust absorbers, at the cost of sampling fluctuations and inactive material. If energy is expressed in GeV, $A$ has units ${\mathrm{GeV}}^{1/2}$, $B$ has units GeV and $C$ is dimensionless.

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

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

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 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.

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.