 Solution

ID: past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2017/3/b/solution

Use

Re (e^{ia} - 1) = cos a - 1 = - 2 sin^{2} (a /2)

. With

k = 1/4

, the source's

Δ_{ij} = 2 k (ϕ_{i} - ϕ_{j})

is precisely

(ϕ_{i} - ϕ_{j}) /2 = (m_{i}^{2} - m_{j}^{2}) L / (4 E)

c_{12}^{2} c_{13}^{2}

s_{12}^{2} c_{13}^{2}

, and

s_{13}^{2}

. Therefore

P_{ee} = 1 - c_{13}^{4} sin^{2} (2 θ_{12}) sin^{2} Δ_{12} - sin^{2} (2 θ_{13}) [c_{12}^{2} sin^{2} Δ_{13} + s_{12}^{2} sin^{2} Δ_{23}] .

Neither

θ_{23}

nor the Dirac phase appears in vacuum electron-flavour survival. The same vacuum survival expression applies to reactor electron antineutrinos. A sufficiently large matter effect would require a different propagation calculation, but is negligible at the short reactor baselines used for measuring

θ_{13}

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