Neutral particle oscillations
Encyclopedia
In particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, neutral particle oscillation is the transmutation of a neutral particle with nonzero internal quantum number
Quantum number
Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...

s into its antiparticle
Antiparticle
Corresponding to most kinds of particles, there is an associated antiparticle with the same mass and opposite electric charge. For example, the antiparticle of the electron is the positively charged antielectron, or positron, which is produced naturally in certain types of radioactive decay.The...

. These oscillations and the associated mixing of particles gives insight into the realization of discrete parts of the Poincaré group
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...

, ie, parity
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...

 (P), charge conjugation (C) and time reversal invariance (T).

The phenomenon

Neutral particles such as the kaon
Kaon
In particle physics, a kaon is any one of a group of four mesons distinguished by the fact that they carry a quantum number called strangeness...

, neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...

, bottom quark
Bottom quark
The bottom quark, also known as the beauty quark, is a third-generation quark with a charge of − e. Although all quarks are described in a similar way by the quantum chromodynamics, the bottom quark's large bare mass , combined with low values of the CKM matrix elements Vub and Vcb, gives it a...

 mesons or neutrino
Neutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...

s have internal quantum numbers called flavour
Flavour (particle physics)
In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry...

. This means that the particle and antiparticle are different. If both particle and antiparticle can decay into the same final state, then it is possible for the decay and its time reversed process to contribute to oscillations—
A  →  F  →  B  →  F  →  A  →  ...

where A is the particle, B is the antiparticle, and F is the common set of particles into which both can decay. The example of the neutral kaon
Kaon
In particle physics, a kaon is any one of a group of four mesons distinguished by the fact that they carry a quantum number called strangeness...

 is pictured here.

Such a process is actually connected to the mass renormalization of the states A and B in quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

. However, under certain circumstances it can be tackled through a simpler quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 model which neglects these intermediate multi-particle quantum states and concentrates only on the states A and B.

Quantum mechanical model

Consider a state . Its time evolution is governed by the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

, H, through the action of the evolution operator  on . The 2×2 matrix Hamiltonian can be written as

where the Hamiltonian can be decomposed into a mass matrix M and a decay width matrix Γ, both of which are 2×2 Hermitian matrices. We introduce the notation MAB  =  |MAB| e and ΓAB  =  |ΓAB| ei(α+β).

A and B are both flavour eigenstates. Oscillations mix these states, and the mass eigenstates are the states which propagate without mixing, ie, the eigenvectors of H.

CPT symmetry

The action of the discrete spacetime symmetries
Spacetime symmetries
Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems, spacetime symmetries finding ample application in the study of exact solutions of Einstein's field...

 are

If the Hamiltonian is CPT symmetric, then (CPT)H(CPT)−1 = H. The transformation properties above imply that . Then , so a test of CPT symmetry is that the masses and the decay widths of the particle and the antiparticle are equal. This is a major class of experimental tests of CPT symmetry
CPT symmetry
CPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity, and time simultaneously.-History:...

.

Any 2×2 matrix can be written in the form , where I is the identity matrix, σi are the Pauli matrices
Pauli matrices
The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...

 and u is a unit vector. With CPT symmetry, the diagonal elements of H are equal, so

where φ is a complex angle. H is diagonalized by rotating u into a unit vector in the z-direction. The eigenvectors and eigenvalues are

where the plus signs are for the state and minus, for . A change in the phase convention, changes the definition of the eigenstates, but not the eigenvalues. By appropriate choice of this phase, the angle φ can always be set equal to zero, so that the eigenstates are orthogonal.

See also

  • Kaon
    Kaon
    In particle physics, a kaon is any one of a group of four mesons distinguished by the fact that they carry a quantum number called strangeness...

    s, B-Bbar oscillation
    B-Bbar oscillation
    Neutral B meson oscillations is one of the manifestations of the neutral particle oscillation, a fundamental prediction of the Standard Model of particle physics. It is the phenomenon of B mesons changing between their matter and antimatter forms before their decay...

    s and neutrino oscillation
    Neutrino oscillation
    Neutrino oscillation is a quantum mechanical phenomenon predicted by Bruno Pontecorvowhereby a neutrino created with a specific lepton flavor can later be measured to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates...

    s
  • CP violation
    CP violation
    In particle physics, CP violation is a violation of the postulated CP-symmetry: the combination of C-symmetry and P-symmetry . CP-symmetry states that the laws of physics should be the same if a particle were interchanged with its antiparticle , and left and right were swapped...

     and CPT symmetry
    CPT symmetry
    CPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity, and time simultaneously.-History:...

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