Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2021 / 2 / a / i

Solution

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

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

and

u = (p_{1} - p_{4})^{2}

. The external on shell relations are

p_{1}^{2} = p_{2}^{2} = m_{e}^{2}

and

p_{3}^{2} = p_{4}^{2} = m_{t}^{2}

p_{3} + p_{4} = p_{1} + p_{2}

, gives

s + t + u = 3 p_{1}^{2} + p_{2}^{2} + p_{3}^{2} + p_{4}^{2} + 2 p_{1} \cdot (p_{2} - p_{3} - p_{4}) = p_{1}^{2} + p_{2}^{2} + p_{3}^{2} + p_{4}^{2} = 2 (m_{e}^{2} + m_{t}^{2}) .

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