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Flavour in particle physics v • d • e

Flavour quantum numbers: Lepton number: L Baryon number: B Strangeness: S Charm: C Bottomness: B' Topness: T Isospin: Iz or I Weak isospin: Tz Electric charge: Q Combinations: Hypercharge: Y Y=2(Q-Iz) Weak hypercharge: YW YW=2(Q-Tz) YW=B−L Related topics: CPT symmetry CKM matrix CP symmetry Chirality

In physics, and specifically, particle physics, isospin (isotopic spin, isobaric spin) is a quantum number related to the strong interaction and applies to the interactions of the neutron and proton. This term was derived from isotopic spin, but the term isotopic spin is confusing as two isotopes of nucleus have different amount of nucleons, besides, 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.

Contents 1 Symmetry 2 SU(2) 3 Relationship to flavour 4 Isospin symmetry of quarks 5 Weak isospin 6 Yang-Mills Theory 7 References

Isospin was introduced by Werner Heisenberg to explain several related symmetries:

• The mass of the neutron and the proton are almost identical: they are nearly degenerate, and 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.
• The mass of the pions which mediate the strong interaction between the nucleons are the same. In particular, the mass of the positive pion (and its antiparticle) is nearly identical to that of the neutral pion.

In quantum mechanics, when a Hamiltonian has a symmetry, that symmetry manifests itself through a set of states that have (almost) the same energy; that is, the states are degenerate. In particle physics, mass is the same thing as energy (since E = mc²), and so the near mass-degeneracy of the neutron and proton points at a symmetry of the Hamiltonian describing the strong interactions. The neutron does have a slightly higher mass: the mass degeneracy is not exact. The proton is charged, the neutron is not. However, here, as the case would be in general for quantum mechanics, the appearance of a symmetry can be imperfect, as it is perturbed by other forces, which give rise to slight differences between states.

Heisenberg's contribution was to note that the mathematical formulation of this symmetry was in certain respects similar to the mathematical formulation of spin, from 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 or fundamental representation) of SU(2). The pions are assigned to the triplet (the spin-1 or adjoint representation) of SU(2).

Just as is the case for regular spin, isospin is described by two numbers, I, the total isospin, and I₃, the component of the spin vector in a given direction. The proton and neutron both have I=1/2, as they belong to the doublet. The proton has I₃=+1/2 or 'isospin-up' and the neutron has I₃=−1/2 or 'isospin-down'. The pions, belonging to the triplet, have I=1, and π⁺, π⁰ and π⁻ have, respectively, I₃=+1, 0, −1.

The discovery and subsequent close 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 flavour 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, more general symmetry that contained isospin as a subset. 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). This immediately led to Gell-Mann's proposal of the existence of quarks. The quarks would belong to the fundamental representation of the flavour SU(3) symmetry, and it is from the fundamental rep and its conjugate (the quarks and the anti-quarks) that the higher representation (the mesons and baryons) could be assembled. In short, the theory of Lie groups and Lie algebras modelled the physical reality of particles in the most exceptional and unexpected way.

The discovery of the J/ψ meson and charm led to the expansion of flavour symmetry to SU(4), and the discovery of the upsilon meson (and the corresponding top and bottom quarks) led to the current SU(6) flavour symmetry. Isospin symmetry is just one little corner of this broader symmetry. There are strong theoretical reasons, confirmed by experiment, that lead one to believe that things stop there, and that there are no further quarks to be found.

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

and the (spin-up) neutron by

Here, is the up quark flavour eigenstate, and is the down quark flavour eigenstate, while and are the eigenstates of S_z. Although the above is the technically correct way of denoting a proton and neutron in terms of quark flavour and spin eigenstates, this is almost always glossed over, and these are more simply referred to as uud and udd.

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

The overline denotes, as usual, the complex conjugate representation of SU(2), or, equivalently, the antiquark.

Main article: weak isospin

The quarks also feel the weak interaction; however, the mass eigenstates of the strong interaction are not exactly the same as the eigenstates of the weak interaction. Thus, while there are still a pair of quarks u and d that take part in the weak interaction, they are not quite the same as the strong u and d quarks. The difference is given by a rotation, whose magnitude is called the Cabibbo angle or more generally, the CKM matrix.

Chen Ning Yang was aware that in the general theory of relativity the notion of absolute direction was not universally well defined. If a gyroscope is spinning very fast so that its y-axis is always pointing in a certain direction, and then it is slowly moved around a large loop in space, it comes back in a different orientation when in returns to its starting position. This is a fundamental propery of space-time, the curvature, and it determines the form of the gravitational interaction.

In 1954, Yang and Mills suggested that the notion of protons and neutrons, which are continuously rotated into each other by isospin, also changes under transport around closed loops. The idea was that a proton moving along a closed loop would come back as a quantum superposition of a proton and a neutron. To describe this, a notion of proton and neutron direction in isospin space must be defined at every point, a local basis for isospin.

In terms of this basis, (B_p(x),B_n(x)) are the probability amplitudes that a particle at x whose trajectory we are following is a proton or a neutron. The basis is arbitrary, so it should be possible to change basis at every point independently. An x-dependent change of basis multiplies the amplitudes (B_p(x),B_n(x)) by an x-dependent SU(2) matrix.

A fundamental condition in the Yang-Mills theory is that redefining the proton and neutron by an arbitrary isospin matrix at every different position should be possible. This condition is called gauge invariance, an idea introduced by Hermann Weyl to describe electromagnetism in the language of General Relativity. In any basis, moving in a direction V should mix up the B's by an infinitesimal unitary matrix, which is the identity plus an amount linearly proportional to the direction components.

The quantity iA_k is an infinitesimal isospin generator, which means that it is antihermitian. Because hermitian matrices are more familiar to physicists, the physics literature introduces the factor of i, to make A hermitian.

The change in proton/neutron character of a particle that is transported along a closed loop is given by the product over each infinitesimal element of the quantity (1 + iA_kdx^k) in order, which when the dx are infinitesimal can be rewritten as an exponential:

Where a path ordered exponential, the quantity on the right hand side, is really defined by this expression. The total change when a particle is returned to the starting point is known as the wilson loop in physics and the holonomy in mathematics.

When the basis is rotated, the quantities A are different, because the transport from one point to another depends on the basis for the two points. The change in the A matrix comes from two independent effect. First is just the change which would happen under a global isospin rotation:

The second is peculiar to gauge theory, and is a generalization of the usual gauge transformations of electrodynamics. Since A rotates a particle traveling from a point to a nearby point, if the g matrix varies between the two points, this variation changes the A by an amount proportional to the derivative.

The generalization of the electric and magnetic field tensor of electromagnetism can be found by considering the wilson loop around an infinitesimal closed curve starting at x the traveling in a little square in the dx^μ,dx^ν directions. Since this loop starts and ends at the same point, it does not care about the basis at any other point. For an infinitesimal curve:

and, up to a scaling factor which determines the scale of A, which is also the amount by which A rotates a proton and so is the strength of the coupling, the Lagrangian for Yang Mills theory is the usual lagrangian for electrodynamics with the nonabelian F:

The theory describes interacting vector bosons, like the photon of electromagnetism. Unlike the photon, there are tree level interactions which describe the scattering of the vector bosons with each other. 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. If a vector particle is a Yang Mills field for the isospin symmetry, it would couple to all strongly interacting particles universally. The coupling to the proton/neutron 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 to identify with the field A. 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, it would interact with the proton and the neutron the same as it does with the kaons. 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 correctly described by a gauge theory, 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 vector meson dominance, ideas which led to successful theoretical pictures of GeV scale photon-nucleus scattering.

After the discovery of the standard model, much of these successes were forgotten. In the standard model, the proton and neutron are composite, and the quarks are the particles which transform into each other under isospin. The only truly fundamental gauge symmetry of the quarks mixes the left-handed up and down quarks with each other, but not the right-handed quarks. This symmetry is the fundamental SU(2)_L weak isospin, which is not the same as isospin since it only acts on half of the field. The gauge theory of the gluons is fundamental too, as is the electric field. But because the up and down quark masses are neither infinitesimal nor exactly the same, isospin is not even an exact global symmetry, let alone a gauge symmetry.

Nevertheless, a minority of researchers developed the idea further. The lightest axial vector meson A₁(1200MeV) was suggested to be the gauge boson for the axial vector current, the conserved current associated with chiral quark symmetry. This symmetry is spontaneously broken by the quark chiral condensate. Other researchers in the 1980s suggested that the regge trajectory of vector mesons corresponding to the rhos and the A's are a tower of higher effective gauge symmetries of the strong interactions, somehow related to one another. This was the idea of hidden local symmetry.

These ideas, long marginalized, were fully vindicated in recent years by work in string theory. An effective string description of confining gauge theories was constructed and this string description not only explains why the rho should interact as a vector meson, but also why the rho comes with a scalar partner which gives it mass by the Higgs mechanism. It further explained the occurrence of the tower of hidden local symmetries, and predicts that all the tower interacts with gauge-like interactions. These ideas are the subject of active ongoing research.

• Claude Itzykson and Jean-Bernard Zuber, Quantum Field Theory (1980) McGraw-Hill Inc. 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