In physics, and specifically, particle physics, isospin (isotopic spin, isobaric spin) is a quantum number related to the strong interaction. In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them Quantum numbers describe values of conserved numbers in the dynamics of the Quantum system. In High energy physics, the lepton number is the number of Leptons minus the number of antileptons In Particle physics, the baryon number is an approximate conserved Quantum number of a system The charm Quark is a second-generation quark with an electric charge of +(2/3 e. The bottom quark is a third-generation Quark with a charge of − e. The top quark is the third- generation up-type Quark with a charge of +(2/3 e. The weak isospin in Particle physics is a quantum number relating to the Weak interaction, and parallels the idea of Isospin under the Strong Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Particle physics, the hypercharge (represented by Y) of a particle is related to the Strong interaction, and it should not be confused with similarly The weak hypercharge in Particle physics is a quantum number relating the Electrical charge and the third component of Weak isospin, and is similar In High energy physics, B−L (pronounced "bee minus ell" is the Baryon number minus the Lepton number. CPT symmetry is a fundamental symmetry of Physical laws under transformations that involve the inversions of charge, parity and In the Standard Model of Particle physics, the Cabibbo-Kobayashi-Maskawa matrix ( CKM matrix, quark mixing matrix, sometimes also called In Particle physics, CP violation is a violation of the postulated CP symmetry of the laws of physics A phenomenon is said to be chiral if it is not identical to its Mirror image (see Chirality) Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them Quantum numbers describe values of conserved numbers in the dynamics of the Quantum system. In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and This term was derived from isotopic spin, but the term isotopic spin is confusing as two isotopes of a nucleus have different numbers of nucleons; in contrast, rotations of isospin maintain the number of nucleons. Nuclear physicists prefer isobaric spin, which is more precise in meaning. Isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions of baryons and mesons. In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their Baryons are the family of Subatomic particles with a Baryon number of 1 In Particle physics, a meson is a strongly interacting Boson &mdashthat is a Hadron with integer spin. Isospin symmetry remains an important concept in particle physics, and a close examination of this symmetry historically led directly to the discovery and understanding of quarks and of the development of Yang-Mills theory. In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations

Motivation for isospin

Combinations of three u, d or s-quarks forming baryons with spin-³⁄₂ form the baryon decuplet. In Physics, the Eightfold Way is a term coined by American Physicist Murray Gell-Mann for a theory organizing subatomic Baryons

Combinations of three u, d or s-quarks forming baryons with spin-¹⁄₂ form the baryon octet

Isospin was introduced by Werner Heisenberg in 1932^[1] (although it was named by Eugene Wigner in 1937^[2]) to explain symmetries of the then newly discovered neutron:

• The mass of the neutron and the proton are almost identical: they are nearly degenerate, and both are thus often called nucleons. In Physics, the Eightfold Way is a term coined by American Physicist Murray Gell-Mann for a theory organizing subatomic Baryons Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the Eugene Paul "EP" Wigner ( Hungarian Wigner Pál Jenő) ( November 17, 1902 &ndash January 1, 1995) was a This article is a discussion of neutrons in general For the specific case of a neutron found outside the nucleus see Free neutron. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Physics a nucleon is a collective name for two Baryons the Neutron and the Proton. Although the proton has a positive charge, and the neutron is neutral, they are almost identical in all other respects.
• The strength of the strong interaction between any pair of nucleons is the same, independent of whether they are interacting as protons or as neutrons.

Thus, isospin was introduced as a concept well before the development in the 1960's of the quark model which provides our modern understanding. In Physics, the quark model is a classification scheme for Hadrons in terms of their valence quarks, i

The nucleons, baryons of spin ¹⁄₂, were grouped together because they both have nearly the same mass and interact in nearly the same way. In Physics a nucleon is a collective name for two Baryons the Neutron and the Proton. Thus, it was convenient to treat them as being different states of the same particle. Since a spin ¹⁄₂ particle has two states, the two were said to be of isospin ¹⁄₂. The proton and neutron were then associated with different isospin projections I_z = +¹⁄₂ and −¹⁄₂ respectively. When constructing a physical theory of nuclear forces, one could then simply assume that it does not depend on isospin. The nuclear force (or nucleon-nucleon interaction or residual strong force) is the force between two or more Nucleons It is responsible for

These considerations would also prove useful in the analysis of meson-nucleon interactions after the discovery of the pions in 1947. In Particle physics, a meson is a strongly interacting Boson &mdashthat is a Hadron with integer spin. In Particle physics, pion (short for pi meson) is the collective name for three Subatomic particles, and. The three pions (π⁺, π⁰, π⁻) could be assigned to an isospin triplet with I = 1 and I_z = +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: a doublet I = ¹⁄₂ of K mesons, a triplet I = 1 of Σ baryons, a single I = 0 Λ, four I = ³⁄₂ Δ baryons, and so on. This multiplet structure was combined with strangeness in Murray Gell-Mann's Eightfold Way, ultimately leading to the quark model and quantum chromodynamics. Murray Gell-Mann (born September 15, 1929) is an American Physicist who received the 1969 Nobel Prize in physics for his work In Physics, the quark model is a classification scheme for Hadrons in terms of their valence quarks, i Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the

Modern understanding of isospin

Observation of the light baryons (those made of up, down and strange quarks) lead us to believe that some of these particles are so similar in terms of their strong interactions that they can be treated as different states of the same particle. Baryons are the family of Subatomic particles with a Baryon number of 1 The up quark is a particle described by the Standard Model theory of Physics. The down quark is a first-generation Quark with a charge of -(1/3 e. The strange quark is a second- generation Quark with a charge of &minus(1/3 e and a strangeness of &minus1 In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and In the modern understanding of quantum chromodynamics, this is because up and down quarks are very similar in mass, and have the same strong interactions. Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the Particles made of the same numbers of up and down quarks have similar masses and are grouped together. For examples, the particles known as the Delta baryons — baryons of spin ³⁄₂ made of a mix of three up and down quarks — are grouped together because they all have nearly the same mass (approximately 1,232 MeV/c²), and interact in nearly the same way. The Delta baryons are relatively light ( Baryons made of only up (u and down (d Quarks of Isospin 3/2 and spin 3/2 whose ground state parity In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin

However, because the up and down quarks have different charges (²⁄₃ e and −¹⁄₃ e respectively), the four Deltas also have different charges (Δ⁺⁺ (uuu), Δ⁺ (uud), Δ⁰ (udd), Δ⁻ (ddd)). These Deltas could be treated as the same particle and the difference in charge being due to the particle being in different states. Isospin was devised as a parallel to spin to associate an isospin projection (denoted I_z or I₃) to each charged state. Since there were four Deltas, four projections were needed. Because isospin was modeled on spin, the isospin projections were made to vary in increments of 1 and to have four increments of 1, you needed an isospin value of ³⁄₂ (giving the projections I_z = 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_z (e. g. the Δ⁺⁺ was associated with I_z = +³⁄₂).

After the quark model was elaborated, it was noted that the isospin projection was related to the up and down quark content of particles. The relation is I_z = ¹⁄₂(N_u − N_d) where N_u and N_d are the number of up and down quarks respectively.

In the isospin picture, the four Deltas and the two nucleons were thought to be the different states of two particles. In the quark model, the Deltas can be thought of as the excited states of the nucleons.

Isospin symmetry

In quantum mechanics, when a Hamiltonian has a symmetry, that symmetry manifests itself through a set of states that have the same energy; that is, the states are degenerate. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes This article refers to physical states having the same energy In particle physics, the near mass-degeneracy of the neutron and proton points to an approximate symmetry of the Hamiltonian describing the strong interactions. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them The neutron does have a slightly higher mass due to isospin breaking; this is due to the difference in the masses of the up and down quarks and the effects of the electromagnetic interaction. However, the appearance of an approximate symmetry is still useful, since the small breakings can be describe by a perturbation theory, which gives rise to slight differences between the near-degenerate states. This article describes perturbation theory as a general mathematical method

SU(2)

Heisenberg's 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. In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin To be precise, the isospin symmetry is given by the invariance of the Hamiltonian of the strong interactions under the action of the Lie group SU(2). In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n The neutron and the proton are assigned to the doublet (the spin-¹⁄₂, 2, or fundamental representation) of SU(2). In Quantum mechanics, a doublet is a quantum state of a system with a spin of 1/2 such that there are two allowed values of the spin component −1/2 and +1/2 In Representation theory of Lie groups and Lie algebras a fundamental representation is an irreducible finite-dimensional representation of a semisimple The pions are assigned to the triplet (the spin-1, 3, or adjoint representation) of SU(2). In Physics, '''spin''' is the Angular momentum intrinsic to a body as opposed to Orbital angular momentum, which is the motion of its Center of mass In Mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its

Just as is the case for regular spin, isospin is described by two quantum numbers, I, the total isospin, and I_z, the component of the spin vector in some direction. Quantum numbers describe values of conserved numbers in the dynamics of the Quantum system.

Relationship to flavor

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. In Particle physics, a meson is a strongly interacting Boson &mdashthat is a Hadron with integer spin. Baryons are the family of Subatomic particles with a Baryon number of 1 In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their 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. In Particle physics, a kaon (/ˈkeɪɒn/ also called K-meson and denoted) is any one of a group of four Mesons distinguished by the fact that they The larger symmetry was named the Eight-fold Way by Murray Gell-Mann, and was promptly recognized to correspond to the adjoint representation of SU(3). In Physics, the Eightfold Way is a term coined by American Physicist Murray Gell-Mann for a theory organizing subatomic Baryons Murray Gell-Mann (born September 15, 1929) is an American Physicist who received the 1969 Nobel Prize in physics for his work Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n To better understand the origin of this symmetry, Gell-Mann proposed the existence of up, down and strange quarks which would belong to the fundamental representation of the SU(3) flavor symmetry. In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle.

Although isospin symmetry is 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, but the very large masses of these quarks makes such symmetries almost useless. In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their In Particle physics, flavour or flavor (see spelling differences) is a Quantum number of Elementary particles related to their Special Unit 2In Mathematics, the special unitary group of degree n, denoted SU( n) is the group of n × n In modern applications, such as lattice QCD, isospin symmetry is often treated as exact while the heavier quarks must be treated separately. In Physics, lattice quantum chromodynamics (lattice QCD is a theory of Quarks and Gluons formulated on a space-time lattice.

Quark content and isospin

Up and down quarks each have isospin I = ¹⁄₂, and isospin z-components (I_z) of ¹⁄₂ and −¹⁄₂ respectively. All other quarks have I = 0. A composite particle made of quarks must have I_z equal to the sum of the I_z of its quarks and I less than or equal

• Particles of isospin ³⁄₂ can only be made by a mix of three u and d quarks (Δ's).
• Particles of isospin 1 are made of a mix of two u quarks and d quarks (Σ's, π's, ρ's, etc. ).
• Particles of isospin ¹⁄₂ can be made of a mix of three u and d quarks (N's) or from one u or d quark (Ξ's, K's, D's, etc. )
• Particles of isospin 0 can be made of one u and one d quark (Λ's, η's, ω's, etc. ), or from no u or d quarks at all (Ω's, φ's, etc. ).

Isospin symmetry of quarks

In the framework of the Standard Model, the isospin symmetry of the proton and neutron are reinterpreted as the isospin symmetry of the up and down quarks. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles The up quark is a particle described by the Standard Model theory of Physics. The down quark is a first-generation Quark with a charge of -(1/3 e. Technically, the nucleon doublet states are seen to be linear combinations of products of 3-particle isospin doublet states and spin doublet states. That is, the (spin-up) proton wave function, in terms of quark-flavour eigenstates, is described by

^[3]

and the (spin-up) neutron by

^[3]

Here, is the up quark flavour eigenstate, and is the down quark flavour eigenstate, while and are the eigenstates of S_z. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system The up quark is a particle described by the Standard Model theory of Physics. The down quark is a first-generation Quark with a charge of -(1/3 e. 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. Note also that the derivation above is assumes exact isospin symmetry and is modified by SU(2)-breaking terms.

Similarly, the isopsin symmetry of the pions are given by:

Weak isospin

Main article: weak isospin

Isospin is similar to, but should not be confused with weak isospin. The weak isospin in Particle physics is a quantum number relating to the Weak interaction, and parallels the idea of Isospin under the Strong The weak isospin in Particle physics is a quantum number relating to the Weak interaction, and parallels the idea of Isospin under the Strong Briefly, weak isospin is the gauge symmetry of the weak interaction which connects quark and lepton doublets of left-handed particles in all generations; for example, up and down quarks, top and bottom quarks, electrons and electron neutrinos. The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature Isospin connects only up and down quarks, acts on both chiralities (left and right) and is a global (not a gauge) symmetry.

Gauged isospin symmetry

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. Chen-Ning Franklin Yang ( (born October 1, 1922) is a Chinese -born American Physicist who worked on Statistical mechanics Robert L Mills ( April 15, 1927 - October 27, 1999) was a Physicist, specializing in Quantum field theory, the theory 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. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations

This Yang-Mills theory describes interacting vector bosons, like the photon of electromagnetism. Yang-Mills is a Gauge theory of Quantum field theory based on the SU(N group. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena Unlike the photon, the SU(2) gauge theory would contain self-interacting gauge bosons. The condition of gauge invariance suggests that they have zero mass, just as in electromagnetism. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations

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. In Particle physics, a kaon (/ˈkeɪɒn/ also called K-meson and denoted) is any one of a group of four Mesons distinguished by the fact that they The coupling to the pions would be the same as the self-coupling of the vector bosons to themselves. In Particle physics, pion (short for pi meson) is the collective name for three Subatomic particles, and.

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. Jun John Sakurai (桜井 純 Sakurai Jun) ( January 31 1933 – November 1 1982) was a Japanese American particle physicist The rho mesons were discovered a short time later, and were quickly identified as Sakurai's vector bosons. In Particle physics, a rho meson is a short-lived Hadronic particle that is an Isospin triplet whose three states are denoted as, and. 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 rho-photon mixing and the concept of vector meson dominance, ideas which led to successful theoretical pictures of GeV-scale photon-nucleus scattering.

Although the discovery of the quarks led to reinterpretation of the rho meson as a vector bound state of a quark and an antiquark, it is sometimes still useful to think of it as the gauge boson of a hidden local symmetry^[4]

References

• Claude Itzykson and Jean-Bernard Zuber, Quantum Field Theory (1980) McGraw-Hill Inc. In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. New York. ISBN 0-07-032071-3

• David Griffiths, Introduction to Elementary Particles (1987) John Wiley & Sons Inc. New York. ISBN 0-471-60386-4

1. ^ "Über den Bau der Atomkerne" (Zietschrift für Physik 77: 1-11)
2. ^ Physical Review 51: 106-119
3. ^ ^{a b} Greiner, Walter; Müller, Berndt (1989). Quantum mechanics: symmetries. Springer Verlag, p. 279. ISBN 3540580808.  
4. ^ Bando et al. , "Is the ρ Meson a Dynamical Gauge Boson of Hidden Local Symmetry?" (1985) PRL54 1215.

• Particle Physics Booklet - Particle Data Group CERN ( Version July 2000 )