 Solution

ID: past-exam-of-the-advanced-quantum-field-theory-for-particle-physics-course-of-the-university-of-oxford/2021/2/a/iii/solution

The exchanged particle is now a gluon, and the incoming up quark mass replaces

m_{e}

Σ_{u} = m_{u}^{2} + m_{t}^{2}, F_{u} = t^{2} + u^{2} + 4 s Σ_{u} - 2 Σ_{u}^{2} .

The Dirac trace and spin average are unchanged. Normalize the fundamental gauge generators by

tr (T^{A} T^{B}) = δ^{A B} /2

. Averaging over the

N_{c}^{2}

initial color charge choices and summing the final color charge choices gives

C_{g} = \frac{1}{N _{c}^{2}} \sum_{A, B} tr (T^{A} T^{B}) tr (T^{A} T^{B}) = \frac{N _{c}^{2} - 1}{4 N _{c}^{2}} = \frac{2}{9} .

Replace the factor

N_{c} e^{4} Q_{e}^{2} Q_{t}^{2}

from part (ii) by

g_{s}^{4} C_{g}

. Therefore

\overline{∣ M_{g} ∣^{2}} = \frac{4 g _{s}^{4}}{9 s ^{2}} F_{u} .

Keeping

m_{u}

is allowed; if the usual negligible up quark mass approximation is wanted, simply set

m_{u} = 0

. The factor

1/ N_{c}^{2}

is essential: initial quark color charge states are averaged, whereas final ones are summed.

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