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.
Such oscillations do regularly occur for other neutral particles, and are classified into two types: Particle–antiparticle oscillation (for example, oscillation). Flavor oscillation (for example, 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 (charge conjugation-parity) 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 (with ) undergoing decays into two pions (with ) 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 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 (it was deep underground to eliminate background from cosmic rays), 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 (produced in the sun) can transform into some other neutrino species ( or ), 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 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
Caution: "mixing" discussed in this article is not the type obtained from mixed quantum states. Rather, "mixing" here refers to the superposition of "pure state" energy (mass) eigenstates, prescribed by a "mixing matrix" (e.g. the CKM or PMNS matricies).
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:
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is a unit vector and hence The Levi-Civita symbol is antisymmetric in any two of its indices ( and in this case) and hence .
With the following parametrization (this parametrization helps as it normalizes the eigenvectors and also introduces an arbitrary phase making the eigenvectors most general)
and using the above pair of results the orthonormal eigenvectors of and consequently those of are obtained as
where and
Writing the eigenvectors of in terms of those of we get
Now if the particle starts out as an eigenstate of (say, ), 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 (known as Rabi frequency),
From equation (6), 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 (diagonal) 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(s) 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 (symmetry) implies
and
Proof Let Operator changes a particle to its antiparticle. That is
and
CPT conservation implies that the Hamiltonian and hence and are invariant under the following transformation:
and
is an anti-Unitary operator and satisfies the relation
hence
and similarly for the diagonal elements of
Hermiticity of and also implies that their diagonal elements are real.
The eigenvalues of are
where and satisfy
The suffixes stand for Heavy and Light respectively (by convention) and this implies that is positive.
The normalized eigenstates corresponding to and respectively, in the natural basis are
where and
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
where
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 (i.e. particle-antiparticle) of one another (i.e. and ), 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,
where,
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 (as a function of time) 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 (say, and respectively) are no longer equivalent eigenstates of CP.
CP violation through mixing-decay interference Let be a final state (a CP eigenstate) that both and can decay to. Then, the decay probabilities are given by,
and,
where,
From the above two quantities, it can be seen that even when there is no CP violation through mixing alone (i.e. ) and neither is there any CP violation through decay alone (i.e. ) 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:
Direct CP violation Direct CP violation is defined as, In terms of the above categories, direct CP violation occurs in CP violation through decay only. Indirect CP violation 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
Neutrino oscillation
Considering a strong coupling between two pairs of flavor eigenstates of neutrinos (for example, –, –, etc.) and a very weak coupling between either pair and the excluded third (that is, the third does not affect the interaction between the other two), 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, is the speed of light in vacuum, is the distance traveled by the neutrino after creation, is the energy with which the neutrino was created, and is the oscillation wavelength.
Proof
where is the momentum with which the neutrino was created.
Now, and .
Hence,
where
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.
Neutrino mass splitting With three flavors of neutrinos, there are three mass splittings:
But only two of them are independent, because . For solar neutrinos For atmospheric neutrinos
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 (as sine squared is independent of the sign of its argument), 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 squares) of the masses, direct determination of neutrino mass is not possible from oscillation experiments.
Length scale of the system