Neutral particle oscillation


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. Neutral particle oscillations were first investigated in 1954 by Murray Gell-mann and Abraham Pais.
For example, a neutron cannot transmute into an antineutron as that would violate the conservation of baryon number. But in those hypothetical extensions of the Standard Model which include interactions that do not strictly conserve baryon number, neutron–antineutron oscillations are predicted to occur. There is a project to search for neutron–antineutron oscillations using ultracold neutrons.
Such oscillations do regularly occur for other neutral particles, and are classified into two types:
  • Particle–antiparticle oscillation.
  • Flavor oscillation.
In those cases where 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

CP violation

After the striking evidence for parity violation provided by Wu et al. in 1957, it was assumed that CP is the symmetry that is conserved. However, in 1964 Cronin and Fitch reported CP violation in the neutral kaon system. They observed the long-lived KL undergoing decays into two pions thereby violating CP conservation.
In 2001, CP violation in the system was confirmed by the BaBar and the Belle experiments. Direct CP violation in the system was reported by both the labs by 2005.
The Kaon#Oscillation| and the systems can be studied as two state systems, considering the particle and its antiparticle as two states of a single particle.

Solar neutrino problem

The pp chain in the sun produces an abundance of. In 1968, R. Davis et al. first reported the results of the Homestake experiment. Also known as the Davis experiment, it used a huge tank of perchloroethylene in Homestake mine, South Dakota. Chlorine nuclei in the perchloroethylene absorb 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 can transform into some other neutrino species, 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 collaboration, which measured both flux and the total neutrino flux.
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

Special case that only considers mixing

Let be the Hamiltonian of the two-state system, and and be its orthonormal eigenvectors with eigenvalues and respectively.
Let be the state of the system at time.
If the system starts as an energy eigenstate of, for example, say
then the time evolved state, which is the solution of the Schrödinger equation
will be
But this is physically same as since the exponential term is just a phase factor: It does not produce an observable new state. In other words, energy eigenstates are stationary eigenstates, that is, they do not yield observably distinct new states under time evolution.
Define to be a basis in which the unperturbed Hamiltonian operator,, is diagonal:
It can be shown, that oscillation between states will occur if and only if off-diagonal terms of the Hamiltonian are not zero.
Hence let us introduce a general perturbation imposed on such that the resultant Hamiltonian is still Hermitian. Then
where and and
The eigenvalues of the perturbed Hamiltonian,, then change to and, where
Since is a general Hamiltonian matrix, it can be written as
where

is a real unit vector in 3 dimensions in the direction of,,
and
are the Pauli spin matrices.

The following two results are clear:
  • is a unit vector and hence
  • The Levi-Civita symbol is antisymmetric in any two of its indices and hence.
With the following parametrization
and using the above pair of results the orthonormal eigenvectors of and consequently those of are obtained as
Writing the eigenvectors of in terms of those of we get
Now if the particle starts out as an eigenstate of , that is
then under time evolution we get
which unlike the previous case, is distinctly different from.
We can then obtain the probability of finding the system in state at time as
which is called Rabi's formula. Hence, starting from one eigenstate of the unperturbed Hamiltonian, the state of the system oscillates between the eigenstates of with a frequency,
From equation , for, we can conclude that oscillation will exist only if. So is known as the coupling term as it connects the two eigenstates of the unperturbed Hamiltonian and thereby facilitates oscillation between the two.
Oscillation will also cease if the eigenvalues of the perturbed Hamiltonian are degenerate, i.e.. But this is a trivial case as in such a situation, the perturbation itself vanishes and takes the form of and we're back to square one.
Hence, the necessary conditions for oscillation are:
  • Non-zero coupling, i.e..
  • Non-degenerate eigenvalues of the perturbed Hamiltonian, i.e..

    General case: considering mixing and decay

If the particle under consideration undergoes decay, then the Hamiltonian describing the system is no longer Hermitian. Since any matrix can be written as a sum of its Hermitian and anti-Hermitian parts, can be written as,
where-
and
and are Hermitian. Hence and
CPT conservation implies
The eigenvalues of are
The suffixes stand for Heavy and Light respectively and this implies that is positive.
The normalized eigenstates corresponding to and respectively, in the natural basis are
and are the mixing terms. Note that these eigenstates are no longer orthogonal.
Let the system start in the state.That is
Under time evolution we then get
Similarly, if the system starts in the state, under time evolution we obtain

CP violation as a consequence

If in a system and represent CP conjugate states of one another, and certain other conditions are met, then CP violation can be observed as a result of this phenomenon. Depending on the condition, CP violation can be classified into three types:

CP violation through decay only

Consider the processes where decay to final states, where the barred and the unbarred kets of each set are CP conjugates of one another.
The probability of decaying to is given by,
and that of its CP conjugate process by,
If there is no CP violation due to mixing, then.
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.

CP violation through mixing only

The probability of observing starting from is given by,
and that of its CP conjugate process by,
The above two probabilities are unequal if
Hence, the particle-antiparticle oscillation becomes a CP violating process as the particle and its antiparticle are no longer equivalent eigenstates of CP.

CP violation through mixing-decay interference

Let be a final state that both and can decay to. Then, the decay probabilities are given by,
and,
From the above two quantities, it can be seen that even when there is no CP violation through mixing alone and neither is there any CP violation through decay alone and thus the probabilities will still be unequal, provided that
The last terms in the above expressions for probability are thus associated with interference between mixing and decay.

An alternative classification

Usually, an alternative classification of CP violation is made:

Specific cases

Neutrino oscillation

Considering a strong coupling between two pairs of flavor eigenstates of neutrinos and a very weak coupling between either pair and the excluded third, equation gives the probability of a neutrino of type transmuting into type as
where, and are energy eigenstates.
The above can be written as
where,
, i.e. the difference between the squares of the masses of the energy eigenstates,
Proof

where is the momentum with which the neutrino was created.
Now, and.
Hence,
where

Thus, a coupling between the energy 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.

Neutrino mass splitting

With three flavors of neutrinos, there are three mass splittings:
But only two of them are independent, because.
This implies that two of the three neutrinos have very closely placed masses. Since only two of the three are independent, and the expression for probability in equation is not sensitive to the sign of , it is not possible to determine the neutrino mass spectrum uniquely from the phenomenon of flavor oscillation. That is, any two out of the three can have closely spaced masses.
Moreover, since the oscillation is sensitive only to the differences of the masses, direct determination of neutrino mass is not possible from oscillation experiments.