Flavour in particle physics 

Flavour quantum numbers 

Related quantum numbers 

Combinations 

Flavour mixing 
In nuclear physics and particle physics, isospin (I) is a quantum number related to the up and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions of baryons and mesons.
The name of the concept contains the term spin because its quantum mechanical description is mathematically similar to that of angular momentum (in particular, in the way it couples; for example, a protonneutron pair can be coupled either in a state of total isospin 1 or in one of 0^{ [1] }). But unlike angular momentum it is a dimensionless quantity, and is not actually any type of spin.
Etymologically, the term was derived from isotopic spin, a confusing term to which nuclear physicists prefer isobaric spin, which is more precise in meaning. Before the concept of quarks was introduced, particles that are affected equally by the strong force but had different charges (e.g. protons and neutrons) were considered different states of the same particle, but having isospin values related to the number of charge states.^{ [2] } A close examination of isospin symmetry ultimately led directly to the discovery and understanding of quarks, and to the development of Yang–Mills theory. Isospin symmetry remains an important concept in particle physics.
In the modern formulation, isospin (I) is defined as a vector quantity in which up and down quarks have a value of I = ^{1}⁄_{2}, with the 3rdcomponent (I_{3}) being +^{1}⁄_{2} for up quarks, and −^{1}⁄_{2} for down quarks, while all other quarks have I = 0. Therefore, for hadrons in general,^{ [3] }
where n_{u} and n_{d} are the numbers of up and down quarks respectively.
In any combination of quarks, the 3rd component of the isospin vector (I_{3}) could either be aligned between a pair of quarks, or face the opposite direction, giving different possible values for total isospin for any combination of quark flavours. Hadrons with the same quark content but different total isospin can be distinguished experimentally, verifying that flavour is actually a vector quantity, not a scalar (up vs down simply being a projection in the quantum mechanical zaxis of flavourspace).
For example, a strange quark can be combined with an up and a down quark to form a baryon, but there are two different ways the isospin values can combine – either adding (due to being flavouraligned) or cancelling out (due to being in opposite flavourdirections). The isospin 1 state (the ^{}Σ^{0}
_{}) and the isospin 0 state (the ^{}Λ^{0}
_{}) have different experimentally detected masses and halflives.
Isospin is regarded as a symmetry of the strong interaction under the action of the Lie group SU(2), the two states being the up flavour and down flavour. In quantum mechanics, when a Hamiltonian has a symmetry, that symmetry manifests itself through a set of states that have the same energy (the states are described as being degenerate ). In simple terms, that the energy operator for the strong interaction gives the same result when an up quark and an otherwise identical down quark are swapped around.
Like the case for regular spin, the isospin operator I is vectorvalued: it has three components I_{x}, I_{y}, I_{z} which are coordinates in the same 3dimensional vector space where the 3 representation acts. Note that it has nothing to do with the physical space, except similar mathematical formalism. Isospin is described by two quantum numbers: I, the total isospin, and I_{3}, an eigenvalue of the I_{z} projection for which flavor states are eigenstates, not an arbitrary projection as in the case of spin. In other words, each I_{3} state specifies certain flavor state of a multiplet. The third coordinate (z), to which the "3" subscript refers, is chosen due to notational conventions which relate bases in 2 and 3 representation spaces. Namely, for the spin^{1}⁄_{2} case, components of I are equal to Pauli matrices, divided by 2, and so I_{z} = ^{1}⁄_{2}τ_{3}, where
While the forms of these matrices are isomorphic to those of spin, these Pauli matrices only act within the Hilbert space of isospin, not that of spin, and therefore is common to denote them with τ rather than σ to avoid confusion.
Although isospin symmetry is actually very slightly broken, SU(3) symmetry is more badly broken, due to the much higher mass of the strange quark compared to the up and down. The discovery of charm, bottomness and topness could lead to further expansions up to SU(6) flavour symmetry, which would hold if all six quarks were identical. However, the very much larger masses of the charm, bottom, and top quarks means that SU(6) flavour symmetry is very badly broken in nature (at least at low energies) and assuming this symmetry leads to qualitatively and quantitatively incorrect predictions. In modern applications, such as lattice QCD, isospin symmetry is often treated as exact for the three light quarks (uds), while the three heavy quarks (cbt) must be treated separately.
Hadron nomenclature is based on isospin.^{ [4] }
Isospin was introduced as a concept in 1932, well before the 1960s development of the quark model. The man who introduced it, Werner Heisenberg,^{ [5] } did so to explain symmetries of the then newly discovered neutron (symbol n):
This behavior is not unlike the electron, where there are two possible states based on their spin. Other properties of the particle are conserved in this case. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, Eugene Wigner introduced the term "isospin" to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.^{ [6] }
Protons and neutrons were then grouped together as nucleons because they both have nearly the same mass and interact in nearly the same way, if the (much weaker) electromagnetic interaction is neglected. In particle physics, the near massdegeneracy of the neutron and proton points to an approximate symmetry of the Hamiltonian describing the strong interactions. It was thus convenient to treat them as being different states of the same particle.
Heisenberg's particular contribution was to note that the mathematical formulation of this symmetry was in certain respects similar to the mathematical formulation of spin, whence the name "isospin" derives. The neutron and the proton are assigned to the doublet (the spin^{1}⁄_{2}, 2, or fundamental representation) of SU(2). The pions are assigned to the triplet (the spin1, 3, or adjoint representation) of SU(2). Though there is a difference from the theory of spin: the group action does not preserve flavor (specifically, the group action is an exchange of flavour).
Similar to a spin ^{1}⁄_{2} particle, which has two states, protons and neutrons were said to be of isospin ^{1}⁄_{2}. The proton and neutron were then associated with different isospin projections I_{3} = +^{1}⁄_{2} and −^{1}⁄_{2} respectively.
Although the neutron does in fact have a slightly higher mass due to isospin breaking (this is now understood to be due to the difference in the masses of the up and down quarks and the effects of the electromagnetic interaction), the appearance of an approximate symmetry is useful even if it does not exactly hold; the small symmetry breakings can be described by a perturbation theory, which gives rise to slight differences between the neardegenerate states.
When constructing a physical theory of nuclear forces, one could simply assume that it does not depend on isospin, although the total isospin should be conserved.
These considerations would also prove useful in the analysis of mesonnucleon interactions after the discovery of the pions in 1947. The three pions (^{}π^{+}
_{}, ^{}π^{0}
_{}, ^{}π^{−}
_{}) could be assigned to an isospin triplet with I = 1 and I_{3} = +1, 0 or −1. By assuming that isospin was conserved by nuclear interactions, the new mesons were more easily accommodated by nuclear theory.
As further particles were discovered, they were assigned into isospin multiplets according to the number of different charge states seen: 2 doublets I = ^{1}⁄_{2} of K mesons (^{}K^{−}
_{}, ^{}K^{0}
_{}),(^{}K^{+}
_{}, ^{}K^{0}
_{}), a triplet I = 1 of Sigma baryons (^{}Σ^{+}
_{}, ^{}Σ^{0}
_{}, ^{}Σ^{−}
_{}), a singlet I = 0 Lambda baryon (^{}Λ^{0}
_{}), a quartet I = ^{3}⁄_{2} Delta baryons (^{}Δ^{++}
_{}, ^{}Δ^{+}
_{}, ^{}Δ^{0}
_{}, ^{}Δ^{−}
_{}), and so on.
The power of isospin symmetry and related methods comes from the observation that families of particles with similar masses tend to correspond to the invariant subspaces associated with the irreducible representations of the Lie algebra SU(2). In this context, an invariant subspace is spanned by basis vectors which correspond to particles in a family. Under the action of the Lie algebra SU(2), which generates rotations in isospin space, elements corresponding to definite particle states or superpositions of states can be rotated into each other, but can never leave the space (since the subspace is in fact invariant). This is reflective of the symmetry present. The fact that unitary matrices will commute with the Hamiltonian means that the physical quantities calculated do not change even under unitary transformation. In the case of isospin, this machinery is used to reflect the fact that the mathematics of the strong force behaves the same if a proton and neutron are swapped around (in the modern formulation, the up and down quark).
For example, the particles known as the Delta baryons – baryons of spin ^{3}⁄_{2} were grouped together because they all have nearly the same mass (approximately 1232 MeV/c^{2} ), and interact in nearly the same way.
They could be treated as the same particle, with the difference in charge being due to the particle being in different states. Isospin was introduced in order to be the variable that defined this difference of state. In an analogue to spin, an isospin projection (denoted I_{3}) is associated to each charged state; since there were four Deltas, four projections were needed. Like spin, isospin projections were made to vary in increments of 1. Hence, in order to have four increments of 1, an isospin value of ^{3}⁄_{2} is required (giving the projections I_{3} = ^{3}⁄_{2}, ^{1}⁄_{2}, −^{1}⁄_{2}, −^{3}⁄_{2}). Thus, all the Deltas were said to have isospin I = ^{3}⁄_{2} and each individual charge had different I_{3} (e.g. the ^{}Δ^{++}
_{} was associated with I_{3} = +^{3}⁄_{2}).
In the isospin picture, the four Deltas and the two nucleons were thought to simply be the different states of two particles. The Delta baryons are now understood to be made of a mix of three up and down quarks – uuu (^{}Δ^{++}
_{}), uud (^{}Δ^{+}
_{}), udd (^{}Δ^{0}
_{}), and ddd (^{}Δ^{−}
_{}); the difference in charge being difference in the charges of up and down quarks (+^{2}⁄_{3} e and −^{1}⁄_{3} e respectively); yet, they can also be thought of as the excited states of the nucleons.
Attempts have been made to promote isospin from a global to a local symmetry. In 1954, Chen Ning Yang and Robert Mills suggested that the notion of protons and neutrons, which are continuously rotated into each other by isospin, should be allowed to vary from point to point. To describe this, the proton and neutron direction in isospin space must be defined at every point, giving local basis for isospin. A gauge connection would then describe how to transform isospin along a path between two points.
This Yang–Mills theory describes interacting vector bosons, like the photon of electromagnetism. Unlike the photon, the SU(2) gauge theory would contain selfinteracting gauge bosons. The condition of gauge invariance suggests that they have zero mass, just as in electromagnetism.
Ignoring the massless problem, as Yang and Mills did, the theory makes a firm prediction: the vector particle should couple to all particles of a given isospin universally. The coupling to the nucleon would be the same as the coupling to the kaons. The coupling to the pions would be the same as the selfcoupling of the vector bosons to themselves.
When Yang and Mills proposed the theory, there was no candidate vector boson. J. J. Sakurai in 1960 predicted that there should be a massive vector boson which is coupled to isospin, and predicted that it would show universal couplings. The rho mesons were discovered a short time later, and were quickly identified as Sakurai's vector bosons. The couplings of the rho to the nucleons and to each other were verified to be universal, as best as experiment could measure. The fact that the diagonal isospin current contains part of the electromagnetic current led to the prediction of rhophoton mixing and the concept of vector meson dominance, ideas which led to successful theoretical pictures of GeVscale photonnucleus scattering.
The discovery and subsequent analysis of additional particles, both mesons and baryons, made it clear that the concept of isospin symmetry could be broadened to an even larger symmetry group, now called flavor symmetry. Once the kaons and their property of strangeness became better understood, it started to become clear that these, too, seemed to be a part of an enlarged symmetry that contained isospin as a subgroup. The larger symmetry was named the Eightfold Way by Murray GellMann, and was promptly recognized to correspond to the adjoint representation of SU(3). To better understand the origin of this symmetry, GellMann proposed the existence of up, down and strange quarks which would belong to the fundamental representation of the SU(3) flavor symmetry.
In the quark model, the isospin projection (I_{3}) followed from the up and down quark content of particles; uud for the proton and udd for the neutron. Technically, the nucleon doublet states are seen to be linear combinations of products of 3particle isospin doublet states and spin doublet states. That is, the (spinup) proton wave function, in terms of quarkflavour eigenstates, is described by^{ [2] }
and the (spinup) neutron by
Here, is the up quark flavour eigenstate, and is the down quark flavour eigenstate, while and are the eigenstates of . Although these superpositions are the technically correct way of denoting a proton and neutron in terms of quark flavour and spin eigenstates, for brevity, they are often simply referred to as "uud" and "udd". The derivation above assumes exact isospin symmetry and is modified by SU(2)breaking terms.
Similarly, the isospin symmetry of the pions are given by:
Although the discovery of the quarks led to reinterpretation of mesons as a vector bound state of a quark and an antiquark, it is sometimes still useful to think of them as being the gauge bosons of a hidden local symmetry.^{ [7] }
Isospin is similar to, but should not be confused with weak isospin. Briefly, weak isospin is the gauge symmetry of the weak interaction which connects quark and lepton doublets of lefthanded particles in all generations; for example, up and down quarks, top and bottom quarks, electrons and electron neutrinos. By contrast (strong) isospin connects only up and down quarks, acts on both chiralities (left and right) and is a global (not a gauge) symmetry.
In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks. Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classified as fermions because they have halfinteger spin.
In particle physics, mesons are hadronic subatomic particles composed of an equal number of quarks and antiquarks, usually one of each, bound together by strong interactions. Because mesons are composed of quark subparticles, they have a meaningful physical size, a diameter of roughly one femtometer (1×10^{−15} m), which is about 1.2 times the size of a proton or neutron. All mesons are unstable, with the longestlived lasting for only a few hundredths of a microsecond. Heavier mesons decay to lighter mesons and ultimately to stable electrons, neutrinos and photons.
In chemistry and physics, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines an isotope's mass number.
A proton is a subatomic particle, symbol ^{}p^{} or ^{}p^{+}
_{}, with a positive electric charge of +1e elementary charge and a mass slightly less than that of a neutron. Protons and neutrons, each with masses of approximately one atomic mass unit, are collectively referred to as "nucleons".
In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is the mechanism of interaction between subatomic particles that is responsible for the radioactive decay of atoms. The weak interaction participates in nuclear fission, and the theory describing its behaviour and effects is sometimes called quantum flavourdynamics (QFD). However, the term QFD is rarely used, because the weak force is better understood by electroweak theory (EWT).
In particle physics, a pion is any of three subatomic particles: ^{}π^{0}
_{}, ^{}π^{+}
_{}, and ^{}π^{−}
_{}. Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more generally, the lightest hadrons. They are unstable, with the charged pions ^{}π^{+}
_{} and ^{}π^{−}
_{} decaying after a mean lifetime of 26.033 nanoseconds, and the neutral pion ^{}π^{0}
_{} decaying after a much shorter lifetime of 84 attoseconds. Charged pions most often decay into muons and muon neutrinos, while neutral pions generally decay into gamma rays.
In physical sciences, subatomic particles are smaller than atoms. They can be composite particles, such as the neutron and proton; or elementary particles, which according to the standard model are not made of other particles. Particle physics and nuclear physics study these particles and how they interact. The concept of a subatomic particle was refined when experiments showed that light could behave like a stream of particles as well as exhibiting wavelike properties. This led to the concept of sub atomic particles to reflect that quantumscale particles behave like both particles and waves. Another concept, the uncertainty principle, states that some of their properties taken together, such as their simultaneous position and momentum, cannot be measured exactly. The wave–particle duality has been shown to apply not only to photons but to more massive particles as well.
In particle physics, annihilation is the process that occurs when a subatomic particle collides with its respective antiparticle to produce other particles, such as an electron colliding with a positron to produce two photons. The total energy and momentum of the initial pair are conserved in the process and distributed among a set of other particles in the final state. Antiparticles have exactly opposite additive quantum numbers from particles, so the sums of all quantum numbers of such an original pair are zero. Hence, any set of particles may be produced whose total quantum numbers are also zero as long as conservation of energy and conservation of momentum are obeyed.
In particle physics, the baryon number is a strictly conserved additive quantum number of a system. It is defined as
In particle physics, the hyperchargeY of a particle is a quantum number conserved under the strong interaction. The concept of hypercharge provides a single charge operator that accounts for properties of isospin, electric charge, and flavour. The hypercharge is useful to classify hadrons; the similarly named weak hypercharge has an analogous role in the electroweak interaction.
In quantum mechanics, a singlet state usually refers to a system in which all electrons are paired. The term 'singlet' originally meant a linked set of particles whose net angular momentum is zero, that is, whose overall spin quantum number . As a result, there is only one spectral line of a singlet state. In contrast, a doublet state contains one unpaired electron and shows splitting of spectral lines into a doublet; and a triplet state has two unpaired electrons and shows threefold splitting of spectral lines.
In physics, the eightfold way is an organizational scheme for a class of subatomic particles known as hadrons that led to the development of the quark model. American physicist Murray GellMann and Israeli physicist Yuval Ne'eman both proposed the idea in 1961. The name comes from GellMann's 1961 paper and is an allusion to the Noble Eightfold Path of Buddhism.
The QCD vacuum is the vacuum state of quantum chromodynamics (QCD). It is an example of a nonperturbative vacuum state, characterized by nonvanishing condensates such as the gluon condensate and the quark condensate in the complete theory which includes quarks. The presence of these condensates characterizes the confined phase of quark matter.
The nuclear force is a force that acts between the protons and neutrons of atoms. Neutrons and protons, both nucleons, are affected by the nuclear force almost identically. Since protons have charge +1 e, they experience an electric force that tends to push them apart, but at short range the attractive nuclear force is strong enough to overcome the electromagnetic force. The nuclear force binds nucleons into atomic nuclei.
In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Eightfold Way, the successful classification scheme organizing the large number of lighter hadrons that were being discovered starting in the 1950s and continuing through the 1960s. It received experimental verification beginning in the late 1960s and is a valid effective classification of them to date. The model was independently proposed by physicists Murray GellMann, who dubbed them "quarks" in a concise paper, and George Zweig, who suggested "aces" in a longer manuscript. André Petermann also touched upon the central ideas from 1963 to 1965, without as much quantitative substantiation. Today, the model has essentially been absorbed as a component of the established quantum field theory of strong and electroweak particle interactions, dubbed the Standard Model.
In particle physics, chiral symmetry breaking is the spontaneous symmetry breaking of a chiral symmetry – usually by a gauge theory such as quantum chromodynamics, the quantum field theory of the strong interaction. Yoichiro Nambu was awarded the 2008 Nobel prize in physics for describing this phenomenon.
The Delta baryons are a family of subatomic particle made of three up or down quarks.
In particle physics, isospin multiplets are families of hadrons with approximately equal masses. All particles within a multiplet, have the same spin, parity, and baryon numbers, but differ in electric charges.