 Solution

ID: past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2017/6/c/solution

The weak charged current couples an up-type quark to the rotated down field

d_{i L}^{'} = \sum_{j} V_{ij} d_{j L}

. A neutral current is flavour universal within equal-charge quarks. Rotating it into the mass basis therefore gives

\sum_{i} \overset{ˉ}{d}_{i L}^{'} γ^{μ} d_{i L}^{'} = \sum_{j, k} \overset{ˉ}{d}_{j L} γ^{μ} (V^{†} V)_{jk} d_{k L} = \sum_{j} \overset{ˉ}{d}_{j L} γ^{μ} d_{j L} .

The off-diagonal terms vanish by CKM unitarity, so no tree-level flavour-changing neutral current is generated. In the two-generation description, the orthogonal strange-charm partner completes the rotation whose single down-strange combination alone would have produced an off-diagonal neutral coupling. The complete multiplet supplies the cancellation.

At loop level, a flavour-changing amplitude has the structure

A_{a \to b} \propto \sum_{i} V_{ai} V_{bi}^{*} F (m_{i}^{2} / m_{W}^{2}), a \neq = b .

If all internal masses were equal,

F

would be common and

\sum_{i} V_{ai} V_{bi}^{*} = 0

. Subtracting an arbitrary common value makes the suppression explicit:

A_{a \to b} \propto i \sum V_{ai} V_{bi}^{*} [F (m_{i}^{2} / m_{W}^{2}) - F (m_{ref}^{2} / m_{W}^{2})] .

This is the Glashow-Iliopoulos-Maiani mechanism. Unequal masses leave small loop-level flavour-changing neutral currents, so the statement that they are never observed would be too strong: the mechanism suppresses them and forbids them at tree level. The cancellation requires complex CKM products; the rounded table of absolute values alone must not be summed as if every phase were positive.

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