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 Incoming links: Bhabha scattering

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 301 / 4 / i / Solution 2026-09-28

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e^{-} (p_{1}, s_{1}) + e^{+} (p_{2}, s_{2}) ⟶ e^{-} (p_{3}, s_{3}) + e^{+} (p_{4}, s_{4}) .

(1)

This is Bhabha scattering. There are two diagrams:

the $s$ -channel annihilation topology $e^{-} e^{+} \to γ^{*} (p_{1} + p_{2}) \to e^{-} e^{+}$ ;
the $t$ -channel exchange topology in which the electron and positron lines exchange $γ^{*} (p_{1} - p_{3})$ .

With

s = (p_{1} + p_{2})^{2}

and

t = (p_{1} - p_{3})^{2}

, a consistent external-fermion ordering gives

i M_{s} i M_{t} i M = [\overline{v} (p_{2}) (- i e γ^{μ}) u (p_{1})] \frac{- i η _{μν}}{s} [\overline{u} (p_{3}) (- i e γ^{ν}) v (p_{4})], = - [\overline{u} (p_{3}) (- i e γ^{μ}) u (p_{1})] \frac{- i η _{μν}}{t} [\overline{v} (p_{2}) (- i e γ^{ν}) v (p_{4})], = i M_{s} + i M_{t} .

(2)

The displayed relative minus sign is the fermionic sign from putting the external fermion operators into the common chosen order.

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SPEAR psi energy-scan experiment 2026-10-18

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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.

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