Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 305 / 4 / b / Solution

Below the strange-quark threshold, the gauge-field and two-flavour quark terms are

L = - \frac{1}{4} G_{μν}^{a} G^{a μν} - \frac{1}{4} F_{μν} F^{μν} + \sum_{q = u, d} \overline{q} (i γ^{μ} D_{μ} - m_{q}) q,

where

D_{μ} = \partial_{μ} - i g_{s} G_{μ}^{a} T^{a} - i e Q_{q} A_{μ}, Q_{u} = \frac{2}{3}, Q_{d} = - \frac{1}{3} .

This is the low-energy Quantum chromodynamics plus electromagnetism Lagrangian for the light flavours.

Every renormalizable gauge-invariant term contains equal numbers of quark and antiquark fields. The common phase symmetry

q \mapsto e^{i β /3} q

is therefore automatic rather than imposed in constructing the Lagrangian. Its conserved charge is baryon number, making

U (1)_{B}

an accidental symmetry of the renormalizable low-energy theory.

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