Below the strange-quark threshold, the gauge-field and two-flavour quark terms are
$$
\mathcal L
=-\frac14G_{\mu\nu}^aG^{a\mu\nu}
-\frac14F_{\mu\nu}F^{\mu\nu}
+\sum_{q=u,d}\overline q
\left(i\gamma^\mu D_\mu-m_q\right)q,
$$
where
$$
D_\mu=\partial_\mu-ig_sG_\mu^aT^a-ieQ_qA_\mu,
\qquad
Q_u=\frac23,\quad Q_d=-\frac13.
$$
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
$$
q\mapsto e^{i\beta/3}q
$$
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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-305/4/b/solution