Isospin

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Flavour in particle physics
Flavour quantum numbers: Isospin: I or I₃ Charm: C Strangeness: S Topness: T Bottomness: B′ Related quantum numbers: Baryon number: B Lepton number: L Weak isospin: T or T₃ Electric charge: Q X-charge: X Combinations: Hypercharge: Y Y = (B + S + C + B′ + T) Y = 2 (Q − I₃) Weak hypercharge: Y_W Y_W = 2 (Q − T₃) X + 2Y_W = 5 ( B − L) Flavour mixing CKM matrix PMNS matrix Flavour complementarity

In physics, and specifically, particle physics, isospin (isotopic spin, isobaric spin) is a quantum number related to the strong interaction. 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. 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.

Motivation for isospin

Combinations of three u, d or s-quarks forming baryons with spin-³⁄₂ form the baryon decuplet.

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 (although it was named by Eugene Wigner in 1937) 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. 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.

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. 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.

These considerations would also prove useful in the analysis of meson-nucleon interactions after the discovery of the pions in 1947. 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.

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. 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. 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.

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. In particle physics, the near mass-degeneracy of the neutron and proton points to an approximate symmetry of the Hamiltonian describing the strong interactions. 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.

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. 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). 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 spin-1, 3, or adjoint representation) of SU(2).

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.

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. 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 Eight-fold Way by Murray Gell-Mann, and was promptly recognized to correspond to the adjoint representation of SU(3). 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.

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 modern applications, such as lattice QCD, isospin symmetry is often treated as exact while the heavier quarks must be treated separately.

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.).

Baryon groups and isospin
Baryon group	I	I_z = +³⁄₂	I_z = +1	I_z = +¹⁄₂	I_z = 0	I_z = −¹⁄₂	I_z = −1	I_z = −³⁄₂
Deltas	³⁄₂	Δ++ (uuu)		Δ+ (uud)		Δ0 (udd)		Δ− (ddd)
Sigmas	1		Σ+ (uus)		Σ0 (uds)		Σ− (dds)
Charmed Sigmas	1		Σ++ c (uuc)		Σ+ c (udc)		Σ0 c (ddc)
Bottom Sigmas	1		Σ+ b (uub)		Σ0 b (udb)		Σ− b (ddb)
Nucleons	¹⁄₂			p+ (uud)		n0 (udd)
Xis	¹⁄₂			Ξ0 (uss)		Ξ− (dss)
Charmed Xis	¹⁄₂			Ξ+ c (usc)		Ξ0 c (dsc)
Double charmed Xis	¹⁄₂			Ξ++ cc (ucc)		Ξ+ cc (dcc)
Bottom Xis	¹⁄₂			Ξ0 b (usb)		Ξ− b (dsb)
Charmed bottom Xis	¹⁄₂			Ξ+ cb (ucb)		Ξ0 cb (dcb)
Double bottom Xis	¹⁄₂			Ξ0 bb (ubb)		Ξ− bb (dbb)
Lambdas	0				Λ0 (uds)
Charmed Lambdas	0				Λ+ c (udc)
Bottom Lambdas	0				Λ0 b (udb)
Omegas	0				Ω− (sss)
Charmed Omegas	0				Ω0 c (ssc)
Double charmed Omegas	0				Ω+ cc (scc)
Bottom Omegas	0				Ω− b (ssb)
Charmed bottom Omegas	0				Ω0 cb (scb)
Double bottom Omegas	0				Ω− bb (sbb)
Triple Charmed Omegas	0				Ω++ ccc (ccc}
Double charmed bottom Omegas	0				Ω+ ccb (ccb)
Charmed double bottom Omegas	0				Ω0 cbb (cbb)
Triple bottom Omegas	0				Ω− bbb (bbb)

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. 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

$\vert p\uparrow \rangle = \frac 1{3\sqrt 2}\left(\begin{array}{ccc} \vert duu\rangle & \vert udu\rangle & \vert uud\rangle \end{array}\right) \left(\begin{array}{ccc} 2 & -1 & -1\\ -1 & 2 & -1\\ -1 & -1 & 2 \end{array}\right) \left(\begin{array}{c} \vert\downarrow\uparrow\uparrow\rangle\\ \vert\uparrow\downarrow\uparrow\rangle\\ \vert\uparrow\uparrow\downarrow\rangle \end{array}\right)$

and the (spin-up) neutron by

$\vert n\uparrow \rangle = \frac 1{3\sqrt 2}\left(\begin{array}{ccc} \vert udd\rangle & \vert dud\rangle & \vert ddu\rangle \end{array}\right) \left(\begin{array}{ccc} 2 & -1 & -1\\ -1 & 2 & -1\\ -1 & -1 & 2 \end{array}\right) \left(\begin{array}{c} \vert\downarrow\uparrow\uparrow\rangle\\ \vert\uparrow\downarrow\uparrow\rangle\\ \vert\uparrow\uparrow\downarrow\rangle \end{array}\right)$

Here, $\vert u \rangle$ is the up quark flavour eigenstate, and $\vert d \rangle$ is the down quark flavour eigenstate, while $\vert\uparrow\rangle$ and $\vert\downarrow\rangle$ 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. 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:

$\vert \pi^+\rangle = \vert u\overline {d}\rangle$

$\vert \pi^0\rangle = \frac{1}{\sqrt{2}}\left(\vert u\overline {u}\rangle - \vert d \overline{d} \rangle \right)$

$\vert \pi^-\rangle = -\vert d\overline {u}\rangle$

Weak isospin

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 left-handed particles in all generations; for example, up and down quarks, top and bottom quarks, electrons and electron neutrinos. 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. 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 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 kaons. The coupling to the pions 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 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 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

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