Isospin

 ( Baryons ) 

$I\_3\; =\; \backslash frac\{1\}\{2\}(n\_u\; -\; n\_d).$

In any combination of quarks, the 3rd component of the isospin vector (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  axis of flavour space).

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 flavour-aligned) or cancelling out (due to being in opposite flavour directions). The isospin-1 state (the ) and the isospin-0 state (the ) have different experimentally detected masses and half-lives.

Like the case for regular spin, the isospin operator I is vector space-valued: it has three components I _x, I _y, I _z, which are coordinates in the same 3-dimensional vector space where the 3 representation acts. Note that this vector space has nothing to do with the physical space, except similar mathematical formalism. Isospin is described by two : the total isospin, and ₃ an eigenvalue of the I _z projection for which flavor states are . In other words, each ₃ state specifies certain flavor state of a multiplet. The third coordinate (), to which the "3" subscript refers, is chosen due to notational conventions that relate bases in 2 and 3 representation spaces. Namely, for the spin- case, components of I are equal to Pauli matrices divided by 2, and so I _z =  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.

• Particles of total isospin are named and can be made by a combination of any three up or down quarks (but only up or down quarks).
• Particles of total isospin 1 can be made from two up quarks, two down quarks, or one of each:
• certain mesons further differentiated by total spin into (total spin 0) and (total spin 1)
• with an additional quark of higher flavour Sigma baryons
• Particles of total isospin can be made from:
• a single up or down quark together with an additional quark of higher flavour strange (), charm (D meson), or bottom (B meson)
• a single up or down quark together with two additional quarks of higher flavour Xi baryon
• an up quark, a down quark, and either an up or a down quark nucleons. Note that three identical quarks would be forbidden by the Pauli exclusion principle due to requirement of anti-symmetric wave function
• Particles of total isospin 0 can be made from
• a neutral quark-antiquark pair: $\backslash mathrm\{u\backslash bar\{u\}\}$ or $\backslash mathrm\{d\backslash bar\{d\}\}$) + c_2(\mathrm{s\bar{s}}) for an isospin-0 combination, as $\backslash frac\{1\}\{\backslash sqrt\{2\}\}(\backslash mathrm\{u\backslash bar\{u\}\; -\; d\backslash bar\{d\}\})$ yields $I\; =\; 1$|group=note}} eta mesons
• one up quark and one down quark, with an additional quark of higher flavour Lambda baryons
• anything not involving any up or down quarks

As further particles were discovered, they were assigned into isospin multiplets according to the number of different charge states seen: 2 doublets of kaon (, ), (, ), a triplet of Sigma baryons (, , ), a singlet Lambda baryon (), a quartet Delta baryons (, , , ), 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).

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 ) 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 is required (giving the projections ). Thus, all the Deltas were said to have isospin , and each individual charge had different (e.g. the was associated with ).

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 (), uud (), udd (), and ddd (); the difference in charge being difference in the charges of up and down quarks (+  e and −  e respectively); yet, they can also be thought of as the excited states of the nucleons.

This Yang–Mills theory describes interacting vector bosons, like the photon of electromagnetism. 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.

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 . The coupling to the would be the same as the self-coupling 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 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 rho-photon mixing and the concept of vector meson dominance, ideas which led to successful theoretical pictures of GeV-scale photon-nucleus scattering.

In the quark model, the isospin projection ( I₃) 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 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

and the (spin-up) neutron by

Here, $\backslash mathrm\{\backslash vert\; u\; \backslash rangle\}$ is the up quark flavour eigenstate, and $\backslash mathrm\{\backslash vert\; d\; \backslash rangle\}$ is the down quark flavour eigenstate, while $\backslash left\backslash vert\backslash uparrow\backslash right\backslash rangle$ and $\backslash left\backslash vert\backslash downarrow\backslash right\backslash rangle$ are the eigenstates of $S\_z$. 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 are given by:

$$\backslash begin\{align\}$$

 \vert \pi^+\rangle &= \vert \mathrm{u\overline {d}}\rangle \\
 \vert \pi^0\rangle &= \tfrac{1}{\sqrt{2}}\left(\vert \mathrm{u\overline {u}}\rangle - \vert \mathrm{d \overline{d}} \rangle \right) \\
 \vert \pi^-\rangle &= -\vert \mathrm{d\overline {u}}\rangle.
     

Although the discovery of the 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.

• (1980). 9780070320710, McGraw-Hill. . ISBN 9780070320710
• D. Griffiths (1987). 9780471603863, John Wiley & Sons. ISBN 9780471603863

• Chan–Paton factor

• i8 i Nuclear Structure and Decay Data - IAEA Nuclides' Isospin