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ID: past-exam-of-the-advanced-quantum-field-theory-for-particle-physics-course-of-the-university-of-oxford/2024/3/c/solution

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Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2024 / 3 / c / Solution by

Codex  0 2026-10-25

Use the Feynman rules and Feynman gauge convention from part a. Put

s = q^{2}

,

t = r_{t}^{2}

,

u = r_{u}^{2}

, the Mandelstam variables, and define the three-boson tensor with incoming four-momenta

(q, - k_{1}, - k_{2})

:

V^{λ μν} = g^{λ μ} (q + k_{1})^{ν} + g^{μν} (k_{2} - k_{1})^{λ} - g^{ν λ} (q + k_{2})^{μ} .

The contributions, with external Dirac spinors included and both external gauge boson propagators amputated, are

M_{t}^{ab μν} M_{u}^{ab μν} M_{s}^{ab μν} M^{ab μν} = - g^{2} \overset{v}{ˉ} (p_{2}) t^{b} t^{a} γ^{ν} \frac{\neq r _{t} + m}{t - m ^{2} + i 0} γ^{μ} u (p_{1}), = - g^{2} \overset{v}{ˉ} (p_{2}) t^{a} t^{b} γ^{μ} \frac{\neq r _{u} + m}{u - m ^{2} + i 0} γ^{ν} u (p_{1}), = - i g^{2} ϵ^{ab c} \overset{v}{ˉ} (p_{2}) t^{c} γ_{λ} u (p_{1}) \frac{V ^{λ μν}}{s + i 0}, = M_{t}^{ab μν} + M_{u}^{ab μν} + M_{s}^{ab μν} .

For example,

i M_{t}

contains

(+ i g)^{2} i = - i g^{2}

. In the annihilation channel, the current vertex, propagator, and three-boson vertex give

(+ i g) (- i) g = g^{2}

, so division by the defining overall

i

gives

- i g^{2}

. These relative signs are essential for the cancellation in part d. The antisymmetric colour factor makes the last expression symmetric under simultaneously exchanging

(a, μ, k_{1})

and

(b, ν, k_{2})

, as required for identical outgoing gauge bosons. The on shell external Dirac spinors obey

(\neq p_{1} - m) u (p_{1}) = 0

,

\overset{v}{ˉ} (p_{2}) (\neq p_{2} + m) = 0

.

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