Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 301 / 4

an internal electron line of momentum $r$ : $i (\neq r + m) / (r^{2} - m^{2} + i ϵ)$ ;
an internal photon line of momentum $k$ : $- i η_{μν} / (k^{2} + i ϵ)$ ;
an electron-photon vertex: $- i e γ^{μ}$ , with four-momentum conserved;
incoming and outgoing electrons: $u_{s} (p)$ and $\overline{u}_{s} (p)$ ;
incoming and outgoing positrons: $\overline{v}_{s} (p)$ and $v_{s} (p)$ along the oriented fermion chain;
incoming and outgoing photons: $ϵ_{μ}^{(λ)} (k)$ and $ϵ_{μ}^{(λ) *} (k)$ .

One integrates each undetermined loop momentum, includes a factor

- 1

for each closed fermion loop, and imposes overall momentum conservation. None of the following tree diagrams contains a loop.

i

Label

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

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} .

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

Label

e^{-} (p_{1}, s_{1}) + e^{+} (p_{2}, s_{2}) ⟶ γ (k_{1}, λ_{1}) + γ (k_{2}, λ_{2}) .

There are two electron-exchange diagrams, distinguished by which labelled outgoing photon is emitted first along the fermion line. They are the

t

- and

u

-channel topologies. Their sum is

i M = (- i e)^{2} \overline{v} (p_{2}) [γ^{ν} \frac{i ( \neq p _{1} - \neq k _{1} + m )}{( p _{1} - k _{1} ) ^{2} - m ^{2} + i ϵ} γ^{μ} + γ^{μ} \frac{i ( \neq p _{1} - \neq k _{2} + m )}{( p _{1} - k _{2} ) ^{2} - m ^{2} + i ϵ} γ^{ν}] u (p_{1}) ϵ_{μ}^{*} (k_{1}, λ_{1}) ϵ_{ν}^{*} (k_{2}, λ_{2}) .

The two terms are added because the external photons are identical bosons. Replacing either polarization by its momentum makes their sum vanish by the Ward identity.