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particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, neutral particle oscillation is the transmutation of a particle with zero
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
into another neutral particle due to a change of a non-zero internal
quantum number In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can ...
, via an interaction that does not conserve that quantum number. Neutral particle oscillations were first investigated in 1954 by
Murray Gell-mann Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American physicist who received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles. He was the Robert Andrews Millikan Professor of Theoretical ...
and Abraham Pais. For example, a
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the atomic nucleus, nuclei of atoms. Since protons and ...
cannot transmute into an
antineutron The antineutron is the antiparticle of the neutron with symbol . It differs from the neutron only in that some of its properties have equal magnitude but opposite sign. It has the same mass as the neutron, and no net electric charge, but has ...
as that would violate the
conservation Conservation is the preservation or efficient use of resources, or the conservation of various quantities under physical laws. Conservation may also refer to: Environment and natural resources * Nature conservation, the protection and manageme ...
of
baryon number In particle physics, the baryon number is a strictly conserved additive quantum number of a system. It is defined as ::B = \frac\left(n_\text - n_\bar\right), where ''n''q is the number of quarks, and ''n'' is the number of antiquarks. Baryo ...
. But in those hypothetical extensions of the
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...
which include interactions that do not strictly conserve baryon number, neutron–antineutron oscillations are predicted to occur. Such oscillations can be classified into two types: * Particle–
antiparticle In particle physics, every type of particle is associated with an antiparticle with the same mass but with opposite physical charges (such as electric charge). For example, the antiparticle of the electron is the positron (also known as an antie ...
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 quantity which 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 Babar ( ur, ), also variously spelled as Baber, Babur, and Babor is a male given name of Pashto, and Persian origin, and a popular male given name in Pakistan. It is generally taken in reference to the Persian ''babr'' (Persian: ببر), meanin ...
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 the two states.


The 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 South Dakota (; Sioux: , ) is a U.S. state in the North Central region of the United States. It is also part of the Great Plains. South Dakota is named after the Lakota and Dakota Sioux Native American tribes, who comprise a large po ...
. Chlorine nuclei in the perchloroethylene absorb to produce argon via the reaction : \mathrm, which is essentially :\mathrm. 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


A special case: considering mixing only

:''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, described by a "mixing matrix" (e.g. the CKM or PMNS matricies).'' Let \,H_0\, be the Hamiltonian of the two-state system, and \;\left, 1 \right\rangle\; and \;\left, 2 \right\rangle\; be its orthonormal eigenvectors with
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
\,E_1\, and \,E_2\, respectively. Let \,\left, \Psi\left( t \right) \right\rangle\, be the state of the system at time \,t~. If the system starts as an energy eigenstate of \,H_0\;, i.e. say : \left, \Psi\left( 0 \right) \right\rangle = \left, 1 \right\rangle then, the time evolved state, which is the solution of the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
will be, : \left, \Psi \left( t \right) \right\rangle = \left, 1 \right\rangle e^ But this is physically same as \left, 1 \right\rangle as the exponential term is just a phase factor and does not produce a new state. In other words, energy eigenstates are stationary eigenstates, i.e. they do not yield physically new states under time evolution. In the basis \,\left\\;, \,H_0\, is diagonal. That is, : H_0 = \begin E_1 & 0 \\ 0 & E_2 \\ \end 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 W in H_0 such that the resultant Hamiltonian H is still
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
. Then, : W = \begin W_ & W_ \\ W_^* & W_ \\ \end where W_, W_ \in \mathbb and W_ \in \mathbb and, Then, the eigenvalues of H are, Since \,H\, is a general Hamiltonian matrix, it can be written as, : H = \sum\limits_^3 a_j \sigma_j = a_0 \sigma_0 + H' The following two results are clear: * \,\left , H'\right= 0\, : * \,^2 = I\, : With the following parametrization (this parametrization helps as it normalizes the eigenvectors and also introduces an arbitrary phase \phi making the eigenvectors most general) : \hat = \left( \sin\theta \cos\phi, \sin\theta \sin\phi, \cos\theta \right), and using the above pair of results the orthonormal eigenvectors of H' and hence of H are obtained as, Writing the eigenvectors of \,H_0\, in terms of those of \,H\, we get, Now if the particle starts out as an eigenstate of \,H_0\, (say, \,\left, 1 \right\rangle\,), that is, : \left, \Psi \left( 0 \right) \right\rangle = \left, 1 \right\rangle then under time evolution we get, : \left, \Psi\left( t \right) \right\rangle = e^ \left( \cos\frac\left, + \right\rangle e^ - \sin\frac\left, - \right\rangle e^ \right) which unlike the previous case, is distinctly different from \;\left, 1 \right\rangle ~. We can then obtain the probability of finding the system in state \;\left, 2 \right\rangle\; at time \,t\, as, which is called Rabi's formula. Hence, starting from one eigenstate of the unperturbed Hamiltonian \,H_0\;, the state of the system oscillates between the eigenstates of \,H_0\, with a frequency (known as
Rabi frequency The Rabi frequency is the frequency at which the probability amplitudes of two atomic energy levels fluctuate in an oscillating electromagnetic field. It is proportional to the Transition Dipole Moment of the two levels and to the amplitude (''not ...
), From the expression of P_(t) we can infer that oscillation will exist only if \;\left, W_ \^2 \ne 0 ~. \,W_\, is thus known as the coupling term as it couples the two eigenstates of the unperturbed Hamiltonian H_0 and thereby facilitates oscillation between the two. Oscillation will also cease if the eigenvalues of the perturbed Hamiltonian H are degenerate, i.e. \;E_+ = E_- ~. But this is a trivial case as in such a situation, the perturbation itself vanishes and H takes the form (diagonal) of H_0 and we're back to square one. Hence, the necessary conditions for oscillation are: * Non-zero coupling, i.e. \;\left, W_ \^2 \ne 0 ~. * Non-degenerate eigenvalues of the perturbed Hamiltonian \,H\,, i.e. \;E_+ \ne E_- ~.


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, H can be written as, : H = M - \frac\Gamma = \begin M_ & M_ \\ M_^* & M_ \\ \end - \frac\begin \Gamma_ & \Gamma_ \\ \Gamma_^* & \Gamma_ \\ \end The eigenvalues of H are, The suffixes stand for Heavy and Light respectively (by convention) and this implies that \Delta m is positive. The normalized eigenstates corresponding to \mu_L and \mu_H respectively, in the natural basis \left\ \equiv \left\ are, p and q are the mixing terms. Note that ''these'' eigenstates are no longer orthogonal. Let the system start in the state \left, P \right\rangle. That is, : \left, P \left( 0 \right) \right\rangle = \left, P \right\rangle = \frac\left( \left, P_L \right\rangle + \left, P_H \right\rangle \right) Under time evolution we then get, : \left, P \left( t \right) \right\rangle = \frac\left( \left, P_L \right\rangle e^ + \left, P_H \right\rangle e^ \right) = g_+ \left( t \right) \left, P \right\rangle - \frac g_- \left( t \right) \left, \bar \right\rangle Similarly, if the system starts in the state \left, \bar \right\rangle, under time evolution we obtain, : \left, \bar(t) \right\rangle = \frac\left( \left, P_L \right\rangle e^ - \left, P_H \right\rangle e^ \right) = -\frac g_- \left( t \right)\left, P \right\rangle + g_+ \left( t \right) \left, \bar \right\rangle


CP violation as a consequence

If in a system \left, P \right\rangle and \left, \right\rangle represent CP conjugate states (i.e. particle-antiparticle) of one another (i.e. CP\left, P \right\rangle = e^ \left, \bar \right\rangle and CP\left, \bar \right\rangle = e^ \left, P \right\rangle), and certain other conditions are met, then
CP violation In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge symmetry) and P-symmetry ( parity symmetry). CP-symmetry states that the laws of physics should be t ...
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 \left\ decay to final states \left\, where the barred and the unbarred kets of each set are CP conjugates of one another. The probability of \left, P \right\rangle decaying to \left, f \right\rangle is given by, : \wp_ \left( t \right) = \left, \left\langle f , P\left( t \right) \right\rangle \^2 = \left, g_+ \left( t \right) A_f - \frac g_- \left( t \right) \bar_f \^2 , and that of its CP conjugate process by, : \wp_\left( t \right) = \left, \left\langle \bar , \bar \left( t \right) \right\rangle \^2 = \left, g_+ \left( t \right) \bar_\bar - \frac g_- \left( t \right) A_\bar \^2 If there is no CP violation due to mixing, then \left, \frac \ = 1. 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 \left, \bar \right\rangle starting from \left, P \right\rangle is given by, : \wp_ \left( t \right) = \left, \left\langle , P\left( t \right) \right\rangle \^2 = \left, \frac g_- \left( t \right) \^2 , and that of its CP conjugate process by, : \wp_ \left( t \right) = \left, \left\langle P , \bar\left( t \right) \right\rangle \^2 = \left, \frac g_- \left( t \right) \^2 . The above two probabilities are unequal if, Hence, the particle-antiparticle oscillation becomes a CP violating process as the particle and its antiparticle (say, \left, P \right\rangle and \left, \right\rangle respectively) are no longer equivalent eigenstates of CP.


CP violation through mixing-decay interference

Let \left, f \right\rangle be a final state (a CP eigenstate) that both \left, P \right\rangle and \left, \bar \right\rangle can decay to. Then, the decay probabilities are given by, : \begin \wp_ \left( t \right) &= \left, \left\langle f , P\left( t \right) \right\rangle \^2 \\ &= \left, A_f \^2 \frac \left[ \left( 1 + \left, \lambda_f \^2 \right) \cosh\left( \fract \right) + 2\operatorname\left( \lambda_f \right) \sinh\left( \fract \right) + \left( 1 - \left, \lambda_f \^2 \right) \cos\left( \Delta mt \right) + 2\operatorname\left( \lambda_f \right) \sin\left( \Delta mt \right) \right] \\ \end and, :\begin \wp_\left( t \right) &= \left, \left\langle f , \bar\left( t \right) \right\rangle \^2 \\ &= \left, A_f \^2 \left, \frac \^2 \frac \left[ \left( 1 + \left, \lambda_f \^2 \right) \cosh\left( \fract \right) + 2\operatorname\left( \lambda_f \right) \sinh\left( \fract \right) - \left( 1 - \left, \lambda_f \^2 \right) \cos\left( \Delta mt \right) - 2\operatorname\left( \lambda_f \right) \sin\left( \Delta mt \right) \right] \\ \end From the above two quantities, it can be seen that even when there is no CP violation through mixing alone (i.e. \left, q/p \ = 1) and neither is there any CP violation through decay alone (i.e. \left, \bar_f/A_f \ = 1) and thus \left, \lambda_f \ = 1, the probabilities will still be unequal provided, 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 flavor eigenstates of neutrinos (for example, –, –, etc.) and a very weak coupling between the third (that is, the third does not affect the interaction between the other two), equation () gives the probability of a neutrino of type \alpha transmuting into type \beta as, : P_ \left( t \right) = \sin^2\theta \sin^2\left( \fract \right) where, E_+ and E_- are energy eigenstates. 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: : \begin \left( \Delta m^2 \right)_ &= ^2 - ^2 \\ \left( \Delta m^2 \right)_ &= ^2 - ^2 \\ \left( \Delta m^2 \right)_ &= ^2 - ^2 \end But only two of them are independent, because \left( \Delta m^2 \right)_ + \left( \Delta m^2 \right)_ + \left( \Delta m^2 \right)_ = 0~. This implies that two of the three neutrinos have very closely placed masses. Since only two of the three \Delta m^2 are independent, and the expression for probability in equation () is not sensitive to the sign of \Delta m^2 (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

Equation () indicates that an appropriate length scale of the system is the oscillation wavelength \lambda_\text. We can draw the following inferences: * If x/\lambda_\text \ll 1, then P_ \simeq 0 and oscillation will not be observed. For example, production (say, by radioactive decay) and detection of neutrinos in a laboratory. * If x/\lambda_\text \simeq n, where n is a whole number, then P_ \simeq 0 and oscillation will not be observed. * In all other cases, oscillation will be observed. For example, x/\lambda_\text \gg 1 for solar neutrinos; x \sim \lambda_\text for neutrinos from nuclear power plant detected in a laboratory few kilometers away.


Neutral kaon oscillation and decay


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. \left, K^0 \right\rangle and \left, \bar^0 \right\rangle being the strangeness eigenstates (with eigenvalues +1 and −1 respectively), the energy eigenstates are, : \begin \left, K_^0 \right\rangle &= \frac \left(\left, K^0 \right\rangle + \left, \bar^0 \right\rangle\right) \\ \left, K_2^0 \right\rangle &= \frac\left( \left, K^0 \right\rangle - \left, \bar^0 \right\rangle \right) \end 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 \left, K_^0 \right\rangle 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. * \left, K_2^0 \right\rangle having a CP eigenvalue −1, can decay only to three pions and never to two. Since the two pion decay is much faster than the three pion decay, \left, K_^0 \right\rangle was referred to as the short-lived Kaon \left, K_S^0 \right\rangle, and \left, K_2^0 \right\rangle as the long-lived Kaon \left, K_L^0 \right\rangle. The 1964 experiment showed that contrary to what was expected, \left, K_L^0 \right\rangle could decay to two pions. This implied that the long lived Kaon cannot be purely the CP eigenstate \left, K_2^0 \right\rangle, but must contain a small admixture of \left, K_^0 \right\rangle, thereby no longer being a CP eigenstate. Similarly, the short-lived Kaon was predicted to have a small admixture of \left, K_2^0 \right\rangle. That is, : \begin \left, K_L^0 \right\rangle &= \frac \left( \left, K_2^0 \right\rangle + \varepsilon \left, K_1^0 \right\rangle \right) \\ \left, K_S^0 \right\rangle &= \frac \left( \left, K_1^0 \right\rangle + \varepsilon \left, K_2^0 \right\rangle \right) \end where, \varepsilon is a complex quantity and is a measure of departure from CP invariance. Experimentally, \left, \varepsilon \ = \left( 2.228 \pm 0.011 \right)\times 10^. Writing \left, K_^0 \right\rangle and \left, K_2^0 \right\rangle in terms of \left, K^0 \right\rangle and \left, \bar^0 \right\rangle, we obtain (keeping in mind that m_ > m_) the form of equation (): : \begin \left, K_L^0 \right\rangle &= \left( p\left, K^0 \right\rangle - q\left, \bar^0 \right\rangle \right) \\ \left, K_S^0 \right\rangle &= \left( p\left, K^0 \right\rangle + q\left, \bar^0 \right\rangle \right) \end where, \frac = \frac. Since \left, \varepsilon \\ne 0, condition () is satisfied and there is a mixing between the strangeness eigenstates \left, K^0 \right\rangle and \left, \bar^0 \right\rangle giving rise to a long-lived and a short-lived state.


CP violation through decay only

The and have two modes of two pion decay: or . Both of these final states are CP eigenstates of themselves. We can define the branching ratios as, : \begin \eta_ &= \frac = \frac = \frac \\ pt \eta_ &= \frac = \frac = \frac \end. Experimentally, \eta_ = \left( 2.232 \pm 0.011 \right) \times 10^ and \eta_ = \left( 2.220 \pm 0.011 \right) \times 10^. That is \eta_ \ne \eta_, implying \left, A_/\bar_ \ \ne 1 and \left, A_/\bar_ \ \ne 1, and thereby satisfying condition (). 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 f_) is a CP eigenstate (for example ), then there are two different decay amplitudes corresponding to two different decay paths: : \begin K^0 &\to f_ \\ K^0 &\to \bar^0 \to f_ \end. 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 mixing matrix - a brief introduction

If the system is a three state system (for example, three species of neutrinos , three species of quarks ), then, just like in the two state system, the flavor eigenstates (say \left, \right\rangle, \left, \right\rangle, \left, \right\rangle ) are written as a linear combination of the energy (mass) eigenstates (say \left, \psi_1 \right\rangle, \left, \psi_2 \right\rangle, \left, \psi_3 \right\rangle ). That is, : \begin \left, \right\rangle \\ \left, \right\rangle \\ \left, \right\rangle \\ \end = \begin \Omega_ & \Omega_ & \Omega_ \\ \Omega_ & \Omega_ & \Omega_ \\ \Omega_ & \Omega_ & \Omega_ \\ \end\begin \left, \psi_1 \right\rangle \\ \left, \psi_2 \right\rangle \\ \left, \psi_3 \right\rangle \\ \end . In case of leptons (neutrinos for example) the transformation matrix is the PMNS matrix, and for quarks it is the CKM matrix. 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.


See also

* CKM matrix *
CP violation In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge symmetry) and P-symmetry ( parity symmetry). CP-symmetry states that the laws of physics should be t ...
*
CPT symmetry Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and T ...
*
Kaon KAON (Karlsruhe ontology) is an ontology infrastructure developed by the University of Karlsruhe and the Research Center for Information Technologies in Karlsruhe. Its first incarnation was developed in 2002 and supported an enhanced version of ...
* PMNS matrix * Neutrino oscillation * Rabi cycle


Footnotes


References

{{DEFAULTSORT:Neutral Particle Oscillation Particle physics Standard Model