Past exam of the particle physics course of the University of Oxford / 2021 / 6 / d / Solution

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

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

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