Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 4 / c / ii

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- Solution ii

Solution

At energies well below

Λ_{QCD}

, collect the pion fields into

U (x) = exp (\frac{i π ^{a} ( x ) τ ^{a}}{f _{π}}), U \mapsto LU R^{†} .

The leading chiral perturbation theory Lagrangian is

L_{χ} = \frac{f _{π}^{2}}{4} Tr (\partial_{μ} U^{†} \partial^{μ} U) + \frac{f _{π}^{2} B _{0}}{2} Tr (M U^{†} + U M^{†}) + \dots,

where

M = diag (m_{u}, m_{d})

is treated as a symmetry-breaking spurion. The derivative and quark-mass expansion organizes the omitted terms.

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