In particle physics, **neutral particle oscillation** is the transmutation of a particle with zero electric charge into another neutral particle due to a change of a non-zero internal quantum number via an interaction that does not conserve that quantum number. For example, a neutron cannot transmute into an antineutron as that would violate the conservation of baryon number. But recently theoretical proposals of neutron–antineutron oscillations appeared.^{[1]}^{[2]}^{[3]}

Such oscillations can be classified into two types:

- Particle–antiparticle oscillation (for example,
K0
–
K0
oscillation,
B0
–
B0
oscillation, D0–D0 oscillation
^{[4]}). - Flavor oscillation (for example, ν e– ν μ oscillation).

In case the particles decay to some final product, then the system is not purely oscillatory, and an interference between oscillation and decay is observed.

# History and motivation

After the striking evidence for parity violation provided by Wu et al. in 1957, it was assumed that CP (charge conjugation-parity) is the quantity which is conserved.^{[5]} However, in 1964 Cronin and Fitch reported CP violation in the neutral Kaon system.^{[6]} They observed the long-lived K2 (CP = −1) undergoing two pion decays (CP = (−1)(−1) = +1), thereby violating CP conservation.

In 2001, CP violation in the B0–B0 system was confirmed by the BaBar and the Belle experiments.^{[7]}^{[8]} Direct CP violation in the B0–B0 system was reported by both the labs by 2005.^{[9]}^{[10]}

The K0–K0 and the B0–B0 systems can be studied as two state systems considering the particle and its antiparticle as the two states.

The pp chain in the sun produces an abundance of νe. In 1968, Raymond Davis et al. first reported the results of the Homestake experiment.^{[11]}^{[12]} Also known as the Davis experiment, it used a huge tank of perchloroethylene in Homestake mine (it was deep underground to eliminate background from cosmic rays), South Dakota, USA. Chlorine nuclei in the perchloroethylene absorb νe to produce argon via the reaction

which is essentially

The experiment collected argon for several months. Because the neutrino interacts very weakly, only about one argon atom was collected every two days. The total accumulation was about one third of Bahcall's theoretical prediction.

In 1968, Bruno Pontecorvo showed that if neutrinos are not considered massless, then νe (produced in the sun) can transform into some other neutrino species (νμ or ντ), to which Homestake detector was insensitive. This explained the deficit in the results of the Homestake experiment. The final confirmation of this solution to the solar neutrino problem was provided in April 2002 by the SNO (Sudbury Neutrino Observatory) collaboration, which measured both νe flux and the total neutrino flux.^{[14]} This 'oscillation' between the neutrino species can first be studied considering any two, and then generalized to the three known flavors.

# Description as a two-state system

* Caution*:

*"mixing" here does not refer to quantum mixed states, but rather to the pure state superposition of energy (mass) eigenstates that is described by a so-called "mixing matrix".*

then, the time evolved state, which is the solution of the Schrödinger equation

will be,^{[15]}

It can be shown, that **oscillation between states will occur if and only if off-diagonal terms of the Hamiltonian are non-zero**.

and,

The following two results are clear:

then under time evolution we get,^{[16]}

Hence, the necessary conditions for oscillation are:

- Non-zero coupling, i.e. .
- Non-degenerate eigenvalues of the perturbed Hamiltonian , i.e. .

Under time evolution we then get,

# CP violation as a consequence

and that of its CP conjugate process by,

Now, the above two probabilities are unequal if,

.

Hence, the decay becomes a CP violating process as the probability of a decay and that of its CP conjugate process are not equal.

and that of its CP conjugate process by,

The above two probabilities are unequal if,

and,

The last terms in the above expressions for probability are thus associated with interference between mixing and decay.

Usually, an alternative classification of CP violation is made:^{[20]}

Indirect CP violation is the type of CP violation that involves mixing. In terms of the above classification, indirect CP violation occurs through mixing only, or through mixing-decay interference, or both.

# Specific cases

The above can be written as,

Thus, a coupling between the energy (mass) eigenstates produces the phenomenon of oscillation between the flavor eigenstates. One important inference is that **neutrinos have a finite mass, although very small**. Hence, their speed is not exactly the same as that of light but slightly lower.

With three flavors of neutrinos, there are three mass splittings:

Moreover, since the oscillation is sensitive only to the differences (of the squares) of the masses, direct determination of neutrino mass is not possible from oscillation experiments.

- If , then and oscillation will not be observed. For example, production (say, by radioactive decay) and detection of neutrinos in a laboratory.
- If , where is a whole number, then and oscillation will not be observed.
- In all other cases, oscillation will be observed. For example, for solar neutrinos; for neutrinos from nuclear power plant detected in a laboratory few kilometers away.

The 1964 paper by Christenson et al.^{[6]} provided experimental evidence of CP violation in the neutral Kaon system. The so-called long-lived Kaon (CP = −1) decayed into two pions (CP = (−1)(−1) = 1), thereby violating CP conservation.

These two are also CP eigenstates with eigenvalues +1 and −1 respectively. From the earlier notion of CP conservation (symmetry), the following were expected:

- Because has a CP eigenvalue of +1, it can decay to two pions or with a proper choice of angular momentum, to three pions. However, the two pion decay is a lot more frequent.
- having a CP eigenvalue −1, can decay only to three pions and never to two.

The K0L and K0S have two modes of two pion decay: π0π0 or π+π−. Both of these final states are CP eigenstates of themselves. We can define the branching ratios as,^{[20]}

In other words, direct CP violation is observed in the asymmetry between the two modes of decay.

CP violation can then result from the interference of these two contributions to the decay as one mode involves only decay and the other oscillation and decay.

# Which then is the "real" particle?

The above description refers to flavor (or strangeness) eigenstates and energy (or CP) eigenstates. But which of them represents the "real" particle? What do we really detect in a laboratory? Quoting David J. Griffiths:^{[21]}

# The mixing matrix - a brief introduction

In case of leptons (neutrinos for example) the transformation matrix is the PMNS matrix, and for quarks it is the CKM matrix.^{[24]}

N.B. The three familiar neutrino species νe–νμ–ντ are flavor eigenstates, whereas the three familiar quarks species d–s–b are energy eigenstates.

The off diagonal terms of the transformation matrix represent coupling, and unequal diagonal terms imply mixing between the three states.

The transformation matrix is unitary and appropriate parameterization (depending on whether it is the CKM or PMNS matrix) is done and the values of the parameters determined experimentally.