# Neutral particle oscillation

In particle physics, neutral particle oscillation is the transmutation of a neutral particle with nonzero internal quantum numbers into its antiparticle. These oscillations and the associated mixing of particles gives insight into the realization of discrete parts of the Poincaré group, i.e., parity (P), charge conjugation (C) and time reversal invariance (T).

## The phenomenon

Oscillation of a neutral kaon into its antiparticle through the decay K0  →  ππ, where the final state can also be reached by the decay of the antiparticle.

Neutral particles such as the kaon, neutron, bottom quark mesons or neutrinos have internal quantum numbers called flavour. This means that the particle and antiparticle are different. If both particle and antiparticle can decay into the same final state, then it is possible for the decay and its time reversed process to contribute to oscillations—

A  →  F  →  B  →  F  →  A  →  ...

where A is the particle, B is the antiparticle, and F is the common set of particles into which both can decay. The example of the neutral kaon is pictured here.

Such a process is actually connected to the mass renormalization of the states A and B in quantum field theory. However, under certain circumstances it can be tackled through a simpler quantum mechanics model which neglects these intermediate multi-particle quantum states and concentrates only on the states A and B.

## Quantum mechanical model

Consider a state $|\psi (t)\rangle =a(t)|A\rangle +b(t)|B\rangle$. Its time evolution is governed by the Hamiltonian, H, through the action of the evolution operator $U(t)=e^{iHt}$ on $|\psi (0)\rangle$. The 2×2 matrix Hamiltonian can be written as

$H = \begin{pmatrix}H_{AA} & H_{AB}\\H_{BA} & H_{BB}\end{pmatrix} = M-\frac i2\Gamma,$

where the Hamiltonian can be decomposed into a mass matrix M and a decay width matrix Γ, both of which are 2×2 Hermitian matrices. We introduce the notation MAB  =  |MAB| e and ΓAB  =  |ΓAB| ei(α+β).

A and B are both flavour eigenstates. Oscillations mix these states, and the mass eigenstates are the states which propagate without mixing, i.e., the eigenvectors of H.

### CPT symmetry

The action of the discrete spacetime symmetries are

$C|A\rangle=-|B\rangle, P|A\rangle=-|A\rangle,\ \mathrm{and}\ T|A\rangle=+|A\rangle.\,$

If the Hamiltonian is CPT symmetric, then $(CPT)H(CPT)^{-1}=H^{\dagger}$. The transformation properties above imply that $CPT|A\rangle=|B\rangle$. Then $\langle A|H|A\rangle=\langle B|H|B\rangle$, so a test of CPT symmetry is that the masses and the decay widths of the particle and the antiparticle are equal. This is a major class of experimental tests of CPT symmetry.

Any 2×2 matrix can be written in the form $E_0 I+E\mathbf u\cdot\mathbf{\sigma}$, where I is the identity matrix, σi are the Pauli matrices and u is a unit vector. With CPT symmetry, the diagonal elements of H are equal, so

$u = \begin{pmatrix}\sin\phi\\ \cos\phi\\ 0\end{pmatrix},\qquad e^{i\phi} = \frac{M^*_{AB}-i\Gamma^*_{AB}/2}{M_{AB}-i\Gamma_{AB}/2} = e^{i\alpha} \sqrt{\frac{|M_{AB}|+\frac12|\Gamma_{AB}| e^{-i(\beta+\pi/2)}}{|M_{AB}|+\frac12|\Gamma_{AB}| e^{i(\beta+\pi/2)}}},$

where φ is a complex angle. H is diagonalized by rotating u into a unit vector in the z-direction. The eigenvectors and eigenvalues are

$\left|1,2\right\rangle = \frac1{\sqrt2} (\left|A\right\rangle\pm e^{i\phi}\left|B\right\rangle),$
$E_{1,2} = M_{AA}-\frac12\Gamma_{AA}\pm e^{i\alpha}\left|{|M_{AB}|}+\frac12{|\Gamma_{AB}|} e^{i(\beta+\pi/2)}\right|,$

where the plus signs are for the state $|1\rangle$ and minus, for $|2\rangle$. A change in the phase convention, $|B\rangle\rightarrow e^{-i\theta}|B\rangle$ changes the definition of the eigenstates, but not the eigenvalues. By appropriate choice of this phase, the angle φ can always be set equal to zero, so that the eigenstates are orthogonal.