Past exam of the particle physics course of the University of Oxford / 2021 / 5

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.