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

Separating variables givesThereforeFor $b>0$, define the strong-coupling scaleThenIt decreases logarithmically toward zero in the ultraviolet and grows toward the infrared, becoming nonperturbative when $\mu$ approaches ${\Lambda }_{\mathrm{QCD}}$.

For $b>0$, the theory is asymptotically free: it is weakly coupled at high energies and strongly coupled in the infrared. For $b<0$, the coupling decreases toward the infrared but grows toward a finite ultraviolet Landau pole; such a theory is infrared free and requires an ultraviolet completion before that pole.

LetRunning each coupling down from the common value ${\alpha }_G$ givesSubtracting the $i=1,2$ equations determines $L$, and eliminating it from the $i=3,2$ difference giveswhich is the claimed gauge coupling unification relation.

Below the strange-quark threshold, the gauge-field and two-flavour quark terms arewhereThis 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 symmetryis 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.

For $m_u=m_d=0$ and electromagnetism neglected, the classical flavour symmetry isup to finite quotients. The chiral anomaly breaks $U(1{)}_A$ quantum mechanically, while $U(1{)}_V=U(1{)}_B$ remains. The quark condensatespontaneously breaksSmall quark masses and electromagnetism explicitly break parts of this approximate symmetry.

Introduce the linear sigma model fieldA suitable renormalizable potential iswith a small symmetry-breaking source proportional to the quark-mass matrix when desired. A vacuum $⟨\Sigma ⟩=vI$ is fixed precisely by $L=R$, so the unbroken group is the diagonal isospin $SU(2{)}_V$. The three broken axial generators produce the pion triplet ${\pi }^{\pm },{\pi }^0$, which become pseudo-Goldstone bosons when $m_u,m_d$ are restored.

At energies well below ${\Lambda }_{\mathrm{QCD}}$, collect the pion fields intoThe leading chiral perturbation theory Lagrangian iswhere $M=\mathrm{diag}(m_u,m_d)$ is treated as a symmetry-breaking spurion. The derivative and quark-mass expansion organizes the omitted terms.

Exact isospin symmetry would make the proton and neutron degenerate. Their observed mass difference is generated by explicit isospin breaking from $m_d-m_u$ together with electromagnetic effects. The approximate symmetry explains why the splitting is small compared with the nucleon mass, while chiral perturbation theory parametrizes it through symmetry-breaking operators and low-energy constants; symmetry alone does not predict its numerical value.

The strange quark is light enough compared with typical hadronic scales to permit an approximatechiral expansion. It adds kaons and the eta to the pseudo-Goldstone multiplet, although convergence is poorer because $m_s$ is substantially larger than $m_u,m_d$.

The charm quark is too heavy for the same light-flavour chiral expansion. At low energies it is integrated out; processes containing charm are instead treated using heavy-quark expansions or other effective theories. Thus the analysis extends approximately to $s$, but not to $c$ as another pseudo-Goldstone-producing light flavour.