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.
Contents
- CP violation
- The solar neutrino problem
- A special case considering mixing only
- The general case considering mixing and decay
- CP violation as a consequence
- CP violation through decay only
- CP violation through mixing only
- CP violation through mixing decay interference
- An alternative classification
- Direct CP violation
- Indirect CP violation
- Neutrino Oscillation
- Neutrino mass splitting
- Length scale of the system
- Which then is the real particle
- The mixing matrix a brief introduction
- References
Such oscillations can be classified into two types:
K0
–
K0
oscillation,
B0
–
B0
oscillation,
D0
–
D0
oscillation).
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.
Note that it is the probability of detecting any one of the two particles that oscillates as a function of distance traveled (or, as a function of time of flight). It is not that the particle itself transmutes periodically as it travels. What propagates is a mixed state and on detection we measure the energy (mass) associated with any one of its pure states; and what oscillates is the probability of obtaining a particular energy (mass) as a result of measurement on the mixed state.
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 quantity which is conserved. However, in 1964 Cronin and Fitch reported CP violation in the neutral Kaon system. 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. Direct CP violation in the
B0
–
B0
system was reported by both the labs by 2005.
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 solar neutrino problem
The pp chain in the sun produces an abundance of
ν
e. In 1968, Raymond 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, USA. Chlorine nuclei in the perchloroethylene absorb
ν
e to produce argon via the reaction
which is essentially
The experiment collected argon for several months. Because 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
ν
e (produced in the sun) can transform into some other neutrino species (
ν
μ or
ν
τ), 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
ν
e 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.
A special case: considering mixing only
Let
Let
If the system starts as an energy eigenstate of
then, the time evolved state, which is the solution of the Schrödinger equation
will be,
But this is physically same as
In the basis
It can be shown, that oscillation between states will occur if and only if off-diagonal terms of the Hamiltonian are non-zero.
Hence let us introduce a general perturbation
and,
Then, the eigenvalues of
Since
The following two results are clear:
With the following parametrization (this parametrization helps as it normalizes the eigenvectors and also introduces an arbitrary phase
and using the above pair of results the orthonormal eigenvectors of
Writing the eigenvectors of
Now if the particle starts out as an eigenstate of
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
which is called Rabi's formula. Hence, starting from one eigenstate of the unperturbed Hamiltonian
From the expression of
Oscillation will also cease if the eigenvalues of the perturbed Hamiltonian
Hence, the necessary conditions for oscillation are:
The 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,
The eigenvalues of
The suffixes stand for Heavy and Light respectively (by convention) and this implies that
The normalized eigenstates corresponding to
Let the system start in the state
Under time evolution we then get,
Similarly, if the system starts in the state
CP violation as a consequence
If in a system
CP violation through decay only
Consider the processes where
The probability of
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 (as a function of time) of observing
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,
CP violation through mixing-decay interference
Let
and,
From the above two quantities, it can be seen that even when there is no CP violation through mixing alone (i.e.
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,
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.
Neutrino Oscillation
Considering a strong coupling between two flavor eigenstates of neutrinos (for example,
ν
e–
ν
μ,
ν
μ–
ν
τ, etc.) and a very weak coupling between the third (that is, the third does not affect the interaction between the other two), equation (6) gives the probability of a neutrino of type
where,
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.
Neutrino mass splitting
With three flavors of neutrinos, there are three mass splittings:
But only two of them are independent (i.e.
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
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
Equation (13) indicates that an appropriate length scale of the system is the oscillation wavelength
CP violation through mixing only
The 1964 paper by Christenson et al. 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:
Since the two pion decay is much faster than the three pion decay,
where,
Writing
where,
Since
CP violation through decay only
The
K0
L and
K0
S have two modes of two pion decay:
π0
π0
or
π+
π−
. Both of these final states are CP eigenstates of themselves. We can define the branching ratios as,
Experimentally,
In other words, direct CP violation is observed in the asymmetry between the two modes of decay.
CP violation through mixing-decay interference
If the final state (say
π+
π−
), then there are two different decay amplitudes corresponding to two different decay paths:
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:
The neutral Kaon system adds a subtle twist to the old question, 'What is a particle?' Kaons are typically produced by the strong interactions, in eigenstates of strangeness (
K0
and
K0
), but they decay by the weak interactions, as eigenstates of CP (K1 and K2). Which, then, is the 'real' particle? If we hold that a 'particle' must have a unique lifetime, then the 'true' particles are K1 and K2. But we need not be so dogmatic. In practice, it is sometimes more convenient to use one set, and sometimes, the other. The situation is in many ways analogous to polarized light. Linear polarization can be regarded as a superposition of left-circular polarization and right-circular polarization. If you imagine a medium that preferentially absorbs right-circularly polarized light, and shine on it a linearly polarized beam, it will become progressively more left-circularly polarized as it passes through the material, just as a
K0
beam turns into a K2 beam. But whether you choose to analyze the process in terms of states of linear or circular polarization is largely a matter of taste.
The mixing matrix - a brief introduction
If the system is a three state system (for example, three species of neutrinos
ν
e–
ν
μ–
ν
τ, three species of quarks
d
–
s
–
b
), then, just like in the two state system, the flavor eigenstates (say
In case of leptons (neutrinos for example) the transformation matrix is the PMNS matrix, and for quarks it is the CKM matrix.
N.B. The three familiar neutrino species
ν
e–
ν
μ–
ν
τ 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.