Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 305 / 1

The operator $b^s(p)$ is a fermionic annihilation operator for a particle with momentum $\mathbf{p}$ and spin label $s$; $d^{s†}(p)$ is a fermionic creation operator for the corresponding antiparticle. The c-number Dirac spinors $u^s(p)$ and $v^s(p)$ solve the positive-energy particle and antiparticle equations respectively. They supply the spinor wavefunctions in the mode expansion of a Dirac field; they are not creation or annihilation operators.

Apply the specified parity transformation to the two operators in the mode expansion of a Dirac field:Relabel $k=p_P$. The momentum sum is unchanged, $p\cdot x=k\cdot x_P$, and the spinor identities give $u^s(p)={\gamma }^0{u}^s(k)$ and $v^s(p)=-{\gamma }^0{v}^s(k)$. The two minus signs in the antiparticle term cancel. ThereforeTaking the Hermitian conjugate and using $({\gamma }^0{)}^{†}={\gamma }^0$, $({\gamma }^0{)}^2=1$ in the Dirac adjoint yieldsThe unit-modulus intrinsic-parity phase cancels from all neutral fermion bilinears.

Use the printed gamma matrix conventions throughout, in particular ${\gamma }^5=-i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$. The transpose identity for ${\gamma }^0$ gives $C{\gamma }^0=-{\gamma }^{0}C$, and its Hermitian conjugate gives the corresponding identity for $C^{†}$. Thus the Dirac adjoint of the charge-conjugated field isReordering the anticommuting components of the fermion bilinear supplies a minus sign:Invariance for arbitrary fields therefore requires $C^{†}C=I$, so the charge-conjugation matrix is a unitary matrix. This is the required normalization constraint; it is not a bosonic reordering calculation.

For a spinor matrix $\Gamma$, the same calculation and the given $C^{-1}=C^T$ giveWith $V^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$ and $A^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$, the supplied transpose identities implyConsequently the vector current is C-odd and the axial current is C-even: $V^{\mu }\mapsto -V^{\mu }$ and $A^{\mu }\mapsto A^{\mu }$.

For parity, put $\Lambda =\mathrm{diag}(1,-1,-1,-1)$. The gamma matrix identities ${\gamma }^0{\gamma }^{\mu }{\gamma }^0={\Lambda }^{\mu }{}_{\nu }{\gamma }^{\nu }$ and ${\gamma }^0{\gamma }^5{\gamma }^0=-{\gamma }^5$ giveThus $V^0$ is P-even and $\mathbf{V}$ P-odd, whereas $A^0$ is P-odd and $\mathbf{A}$ P-even. The axial current transforms as a pseudovector.

Keep the color matrix intact: ${\mathcal{A}}_{\mu }=A_{\mu }^a{T}^a$. Under parity, invariance of its contraction with the vector current requiresUnder charge conjugation, the fermionic reordering in part (b) transposes the color matrix as well as the spinor matrix. For a fixed color matrix $M$, $\bar{\psi }M{\gamma }^{\mu }\psi$ becomes $-\bar{\psi }{M}^T{\gamma }^{\mu }\psi$. Hence invariance of the interaction requiresThis is not a rule that changes the sign of every individual color component: transposition also acts on the generators of the fundamental representation.

Write the gauge field strength as ${\mathcal{F}}_{\mu \nu }={\partial }_{\mu }{\mathcal{A}}_{\nu }-{\partial }_{\nu }{\mathcal{A}}_{\mu }+ig[{\mathcal{A}}_{\mu },{\mathcal{A}}_{\nu }]$. Since $[A^T,B^T]=-[A,B{]}^T$, the charge conjugation rule gives ${\mathcal{F}}_{\mu \nu }\mapsto -{\mathcal{F}}_{\mu \nu }^T$. Including the coordinate reflection in CP symmetry,For generators normalized by $\mathrm{Tr}(T^a{T}^b)=\kappa {\delta }^{ab}$, the color contraction equals ${\kappa }^{-1}\mathrm{Tr}({\mathcal{F}}_{\mu \nu }{\mathcal{F}}_{\rho \sigma })$. The two charge-conjugation minus signs cancel, and transposition reverses the order inside the trace without changing it. The Levi-Civita symbol acquires $\det \Lambda =-1$ under the remaining reflection. ThusThe Yang-Mills theta term is CP-odd; a generic fixed nonzero $\theta$ produces CP violation. Setting $\theta =0$ removes this source. Although the density is a total derivative, it can affect the quantum theory through nontrivial topological sectors; the local transformation test is not rendered irrelevant by that fact.

An experimentally observed example is $K_L\to {\pi }^{+}{\pi }^{-}$, a decay of the long-lived neutral kaon into a CP-even two-pion state. Without CP violation, the long-lived state would be CP-odd and this decay would be forbidden.

For the CKM matrix mechanism of CP violation, three quark generations are the minimum. Rephasings remove every complex phase for one or two generations, while three generations leave one physical CP-violating phase. This count concerns weak quark mixing. The Yang-Mills theta term in part (c) is a separate possible source of CP violation and does not require three generations.