Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 12C / c

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- Solution c

Solution

Momentum conservation is

P_{A} + P_{B} = P_{C} + P_{D}

. The diagonal bilinears are

P_{a}^{2} = m_{a}^{2} c^{2}

. The six off-diagonal ones are

2 P_{A} \cdot P_{B} = s - (m_{A}^{2} + m_{B}^{2}) c^{2},

2 P_{A} \cdot P_{C} = (m_{A}^{2} + m_{C}^{2}) c^{2} - t, 2 P_{A} \cdot P_{D} = (m_{A}^{2} + m_{D}^{2}) c^{2} - u,

2 P_{B} \cdot P_{C} = (m_{B}^{2} + m_{C}^{2}) c^{2} - u, 2 P_{B} \cdot P_{D} = (m_{B}^{2} + m_{D}^{2}) c^{2} - t,

and

2 P_{C} \cdot P_{D} = s - (m_{C}^{2} + m_{D}^{2}) c^{2} .

These express every bilinear invariant through the Mandelstam variables and masses. Expanding

s + t + u

and using momentum conservation cancels all mixed products, leaving

s + t + u = (m_{A}^{2} + m_{B}^{2} + m_{C}^{2} + m_{D}^{2}) c^{2} .

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