Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 305 / 1

Use the Minkowski metric $g^{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ throughout. Taking the Hermitian conjugation of the massive Dirac equation and using the Dirac adjoint givesTaking the matrix transpose and multiplying by the charge conjugation matrix $C$ yieldsThus the charge conjugation of a Dirac field preserves the free massive Dirac equation:The unitarity of $C$ and ${\gamma }^0{C}^{-1}{\gamma }^0=-C^{-1}{\gamma }^{0T}$ also giveConsequentlyHere $\hat{C}$ is the unitary operator acting on the relativistic quantum field, whereas $C$ acts on Dirac spinor indices.

The bosonic annihilation operator $a^{\lambda }(p)$ removes a vector particle of four-momentum $p$ and polarization vector ${\epsilon }^{\mu ,\lambda }(p)$. The bosonic creation operator $c^{\lambda †}(p)$ creates its antiparticle with the same four-momentum and polarization label. The polarization vector describes the corresponding spin-one mode expansion of a free field.

Because charge conjugation acts through a unitary operator, it leaves the numerical polarization vector coefficients unchanged. Applying the specified charge conjugation to the operators givesComparison with the Hermitian conjugation of the original expansion therefore givesThe factor ${\eta }_C$ is the intrinsic charge-conjugation phase.

For a normal ordering of a fermion bilinear, the result of part (a) and the fermionic sign from interchanging the fields giveFor the Dirac current, $(C^{-1}{\gamma }^{\mu }C{)}^T=-{\gamma }^{\mu }$. For the axial current, the definition of ${\gamma }^5$ and the gamma matrix identities implyThus the charge conjugation of fermion bilinears givesThe Dirac current is odd under charge conjugation, and the axial current is even under charge conjugation. The normal ordering specifies the usual treatment of coincident-field contact terms.

For a real vector relativistic quantum field, $V^{\mu }=V^{\mu †}$. Its transformed field must also be a Hermitian operator, so the intrinsic charge-conjugation phase satisfies ${\eta }_C={\eta }_C^{\ast }$; combined with $|{\eta }_C|=1$, this givesThe quantum electrodynamics interaction is $-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$. Since the Dirac electromagnetic current is odd under charge conjugation, charge conjugation invariance of this interaction requires the photon field to be odd too:

Write $J_V^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$ for the Dirac current and $J_A^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$ for the axial current. Under parity symmetry in quantum field theory, the Dirac spinor phases cancel andThe extra sign for the axial current follows from ${\gamma }^0{\gamma }^5{\gamma }^0=-{\gamma }^5$. Combining these transformations with part (c), both currents transform under CP symmetry as $J_{V,A}^{\mu }(x)\mapsto -{\mathbb{P}}^{\mu }{}_{\nu }{J}_{V,A}^{\nu }(x_P)$. The vector relativistic quantum field transforms as $V^{\mu }(x)\mapsto {\eta }_C{\mathbb{P}}^{\mu }{}_{\nu }{V}^{\nu †}(x_P)$. The two parity matrices cancel in the tensor contraction, givingFor a real vector relativistic quantum field with ${\eta }_C=-1$, the interaction is invariant under CP symmetry. A complex vector relativistic quantum field instead maps to the conjugate-field interaction, which must be compared with its Hermitian conjugation partner.

The flavour-diagonal weak neutral current coupled to the real Z boson has real vector and axial current couplings. With ${\eta }_C(Z)=-1$, part (e) shows that this tree-level interaction is invariant under CP symmetry, although the simultaneous presence of the Dirac current and axial current generally violates charge conjugation and parity symmetry in quantum field theory separately. This conclusion concerns the tree-level weak neutral current, rather than every interaction or radiative effect in the Standard Model.

The weak charged current couples distinct quark flavours and involves the Cabibbo-Kobayashi-Maskawa matrix:Here $u_{Li}$ and $d_{Lj}$ are left-handed quark fields. A CP symmetry transformation exchanges the two conjugate-field operators; because it acts through a unitary operator, it does not complex-conjugate their numerical coefficients. Equality with the original interaction therefore requires a Cabibbo-Kobayashi-Maskawa matrix that can be made real by quantum field rephasings. With three generations, a physical CP-violating phase remains in general, measured by the Jarlskog invariant. Therefore the W boson interactions can violate CP symmetry. Merely seeing a complex coefficient is insufficient: the phase must survive quantum field rephasing, unlike the real two-generation Cabibbo angle rotation.