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Bhabha scattering is electron-positron elastic scattering $e^{-}{e}^{+}\to e^{-}{e}^{+}$. At tree level it receives an annihilation diagram in the $s$ channel and a photon-exchange diagram in the $t$ channel.

Both $s$-channel annihilation and $t$-channel gluon exchange are now allowed. Their spin sum factors are the Bhabha scattering factors with $e\mapsto g$. Their color charge tensors, with the external indices of Solution, areEach squared term has $K_{ss}=K_{tt}=(N_c^2-1)/(4N_c^2)$. The cross term instead has the color averageTo evaluate it, use the SU(N) fundamental completeness relationThis follows by completing the orthogonal basis of traceless Hermitian matrices with the identity. Since $\mathrm{tr}{T}^b=0$, it gives ${\sum }_a{T}^a{T}^b{T}^a=-T^b/(2N_c)$, and henceMultiplying the two squared Bhabha scattering terms by $K_{ss}$ and its cross term by $K_{st}$ givesFor the SU(3) gauge group, this isThe negative color charge overlap reverses the sign of the QED interference. Since $st<0$, this cross contribution is positive in the physical region. No additional spinor calculation is needed; all the new information is in the SU(N) fundamental completeness relation.

The two Bhabha scattering tree-level Feynman diagrams are the two panels of Solution with a photon mediator: $s$-channel annihilation with internal momentum $p_1+p_2$, and $t$-channel exchange with internal momentum $p_1-p_3$. Each interaction vertex preserves the electron species. There is no independent $u$-channel Feynman diagram, since the outgoing electron and Positron are distinguishable.

Removing the color charge factor from Solution and replacing $g$ by $e$ givesCrossing symmetry interchanges $s$ and $t$ for the exchange Feynman diagram, givingSquaring the sum of the two scattering amplitudes also produces the interference term $2\mathrm{Re}\overline{{\mathcal{M}}_s{\mathcal{M}}_t^{\ast }}$. The given third term must be this cross term: it has both propagator denominators $st$, whereas a squared individual Feynman diagram has $s^2$ or $t^2$. With the correct relative fermionic sign, its value is $4e^4{u}^2/(st)$. ThusThe coefficient of the last displayed fraction is positive, but $s>0$ and $t<0$ in the physical scattering region, so this interference is destructive. The relative sign originates in the two possible external fermion contractions; changing an external-state phase changes both scattering amplitudes together.^[ref]

There is one tree-level Feynman diagram: annihilation of the up quark and its antiquark into an $s$-channel gluon, followed by production of the down quark and its antiquark. A gluon interaction vertex preserves quark flavor, so a $t$-channel line joining the incoming up quark to the outgoing down quark is impossible.

Let $i,j,k,l$ label the initial quark, initial antiquark, final quark, final antiquark color charges. Up to an irrelevant common sign, the scattering amplitude isThe Mandelstam variables are $s=(p_1+p_2{)}^2$, $t=(p_1-p_3{)}^2$, $u=(p_1-p_4{)}^2$, with $s+t+u=0$ in the zero-mass limit. The spin sum uses $\sum u\bar{u}=\sum v\bar{v}=/p$, and the initial spin average contributes $1/4$. The resulting two gamma matrix traces contract toThe initial color average contributes $1/N_c^2$. Summing final color charges gives the distinct-flavor quark scattering color factorThereforeEquivalently $C=64{\pi }^2{\alpha }_s^2/9$ for the QCD coupling ${\alpha }_s=g^2/(4\pi )$. There is no initial gluon spin average: the gluon is internal.

LabelThis is Bhabha scattering. There are two diagrams:

With $s=(p_1+p_2{)}^2$ and $t=(p_1-p_3{)}^2$, a consistent external-fermion ordering givesThe displayed relative minus sign is the fermionic sign from putting the external fermion operators into the common chosen order.

For the resonant contribution only, subtract nonresonant Bhabha scattering and its interference before interpreting a fitted area. The integrated narrow vector resonance area follows from the Breit-Wigner resonance cross section, with $J=1$, $s_{e^{-}}=s_{e^{+}}=1/2$, and $p\simeq M/2$, becomesAcross a narrow scattering resonance, treat the prefactor and partial decay widths as constant and extend the integral over $E-M$ to the real line. Since ${\int }_{-\infty }^{\infty }du/(u^2+a^2)=\pi /a$,A normalized beam-energy response convolves the line shape without changing this full area. It can completely change the observed peak height and apparent width. If a uniform acceptance $ϵ$ is not already unfolded, use ${\mathcal{I}}_{ee}={\mathcal{I}}_{ee}^{\mathrm{obs}}/ϵ$. The distinction between $B_{ee}^2$ for this exclusive final state and $B_{ee}$ for an inclusive final state is essential.

At the Z boson pole, the accessible primary fermion pairs are $e^{+}{e}^{-}$, ${\mu }^{+}{\mu }^{-}$, ${\tau }^{+}{\tau }^{-}$, the three ${\nu }_i{\bar{\nu }}_i$ channels, and $u\bar{u},d\bar{d},s\bar{s},c\bar{c},b\bar{b}$. The top quark is too heavy. The charged channels also have photon exchange, and the electronic channel has a substantial nonresonant Bhabha scattering contribution. On-shell W boson pairs are above threshold; they are not primary $Z$ decays.

A tau lepton pair gives two approximately back-to-back narrow low-multiplicity cones, usually one or three charged tracks per cone, with overall charge zero and missing energy and momentum from neutrinos. Opposite hemispheres can show electron, muon or hadronic tags. A decay vertex or track impact-parameter information helps establish the short tau lepton flight. Reject Bhabha scattering and dimuon events by their nearly full visible beam energy, particle identification and little missing momentum; reject ordinary quark multijets by their broader high-multiplicity activity and larger visible invariant masses. Control two-photon backgrounds with visible transverse momentum, forward activity and event geometry. Selection efficiencies depend on decay mode and must be calibrated.

Collider luminosity is the proportionality factor between a process cross section and its interaction rate: $dN/dt=\mathcal{L}\sigma$ before efficiency corrections. It has units of inverse area per time. Integrated collider luminosity is ${\mathcal{L}}_{\mathrm{int}}=\int \mathcal{L}(t)dt$, with inverse-area units, and gives $N=\sigma {\mathcal{L}}_{\mathrm{int}}$ for a constant cross section and unit efficiency.

At an $e^{+}{e}^{-}$ collider, count small-angle Bhabha scattering in a dedicated forward detector. Its calculable quantum electrodynamics cross section in a precisely defined angular acceptance provides $\mathcal{L}=N/(ϵ\sigma \Delta t)$, after background and radiative corrections. The inner angular boundary is especially important because the rate is sharply forward-peaked.

At a collider with two proton beams, a van der Meer scan calibrates a visible interaction monitor by separating the beams horizontally and vertically while recording the overlap-dependent rate. Known bunch populations and the measured overlap give an absolute collider luminosity and hence the monitor's visible cross section. Its rate then tracks collider luminosity during normal running. Correlations between transverse coordinates require a nonfactorizing scan analysis rather than blindly multiplying two one-dimensional profiles.

For the Pole forward-backward asymmetry for charged leptons, integrating the normalized distribution over the two hemispheres cancels the even term in the numerator. Since ${\int }_{-1}^1(1+z^2)dz=8/3$ and ${\int }_0^{1}zdz-{\int }_{-1}^{0}zdz=1$,Within this single-$Z$, tree-level model, a nonzero asymmetry requires unequal left/right coupling magnitudes, demonstrating parity violation in the vertices. For $g_L=g_R$ the odd angular term vanishes and $A_{FB}=0$; equal magnitudes with opposite signs also give zero. This inference is process-specific: a forward-backward asymmetry is not universally a parity-odd observable, since reversing both beam and outgoing momenta leaves their relative Angle unchanged. Purely electromagnetic Bhabha scattering is a concrete parity-conserving process with a forward-peaked, asymmetric distribution.

For the supplied ${\sin }^2{\theta }_W=0.23153$,The measurement differs by $0.0188-0.01620=0.00260$, or $0.79$ times its quoted uncertainty of $0.0033$, so it is consistent with the prediction. Statistical fluctuations alone preclude exact equality; photon exchange, initial-state radiation and electroweak radiative corrections also modify the idealized pure-resonance prediction and provide physical reasons for small differences.

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.

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.

Tag an energetic initial-state radiation photon in $e^{+}{e}^{-}\to \gamma Z^{(\ast )}\to \gamma +\text{invisible}$, observing a single-photon signature with missing energy and missing momentum. At known squared center-of-mass energy $s$, the leading single-photon recoil invariant mass isAbove the Z boson mass, radiative-return events cluster at $m_{\mathrm{recoil}}\simeq M_Z$. At or near the pole the full off-shell recoil spectrum is fitted instead. Folding the Breit-Wigner resonance cross section with the calculable QED radiation kernel gives a predicted tagged rate proportional to ${\Gamma }_e{\Gamma }_{\mathrm{inv}}$ for fixed total decay width. A fit with the known luminosity and visible line-shape parameters therefore determines ${\Gamma }_{\mathrm{inv}}$ directly from invisible events, rather than subtracting visible widths. Comparing identically selected radiative-return events with visible Z boson decays can instead constrain ${\Gamma }_{\mathrm{inv}}/{\Gamma }_{\mathrm{vis}}$ while cancelling much of the luminosity normalization.

The detector needs efficient single-photon triggering, good photon energy and angular resolution, reliable timing, and nearly hermetic charged-particle and particle calorimeter coverage. Forward vetoes reject radiative Bhabha scattering with escaped electrons; additional-photon vetoes reject $\gamma \gamma$ events with a lost photon. Require sufficient transverse photon momentum that a recoiling visible system could not all disappear down a beam hole. Timing rejects cosmic rays and beam backgrounds. Detector inefficiencies and beam holes must be measured, since visible final states with lost particles imitate the signal. The irreducible $e^{+}{e}^{-}\to {\nu }_e{\bar{\nu }}_e\gamma$ contribution from $t$-channel W boson exchange and its interference must be included when isolating the Z boson invisible decay width.

At SPEAR, counterrotating electron and Positron beams collided inside a magnetic particle detector. Finer steps in stored-beam energy resolved a sharp production peak. A magnetometer and measured orbit positions monitored the energy settings; Bhabha scattering normalized the cross sections. The paper attributed the observed width to beam-energy spread and radiation effects, reporting an upper limit on intrinsic width. Its muon identification system was not functioning: the non-electron two-body excess was not conclusively a muon pair.^[ref]

A scan must compare normalized production rates, since more selected events can result from more beam encounters or different detection efficiency. A proposed diagnostic is to revisit energies on either side of the peak while monitoring the energy calibration and event selection. Such neighboring settings test the local production background; their event totals alone are not an independent measurement of the intrinsic decay width.