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Chiral symmetry breaking

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Chiral symmetry breaking

In particle physics, chiral symmetry breaking generally refers to the dynamical spontaneous breaking of a chiral symmetry associated with massless fermions. This is usually associated with a gauge theory such as quantum chromodynamics, the quantum field theory of the strong interaction, and it also occurs through the Brout-Englert-Higgs mechanism in the electroweak interactions of the Standard Model. This phenomenon is analogous to magnetization and superconductivity in condensed matter physics - where, for example, chiral symmetry breaking is the mechanism by which disordered 3D magnetic systems have a finite transition temperature. The basic idea was introduced to particle physics by Yoichiro Nambu, in particular, in the Nambu–Jona-Lasinio model, which is a solvable theory of composite bosons that exhibits dynamical spontaneous chiral symmetry when a 4-fermion coupling constant becomes sufficiently large. Nambu was awarded the 2008 Nobel prize in physics "for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics".

Massless fermions in 4 dimensions are described by either left or right-handed spinors that each have 2 complex components. These have spin either aligned (right-handed chirality), or counter-aligned (left-handed chirality), with their momenta. In this case the chirality is a conserved quantum number of the given fermion, and the left and right handed spinors can be independently phase transformed. More generally they can form multiplets under some symmetry group .

A Dirac mass term explicitly breaks the chiral symmetry. In quantum electrodynamics (QED) the electron mass unites left and right handed spinors forming a 4 component Dirac spinor. In the absence of mass and quantum loops, QED would have a chiral symmetry, but the Dirac mass of the electron breaks this to a single symmetry that allows a common phase rotation of left and right together, which is the gauge symmetry of electrodynamics. (At the quantum loop level, the chiral symmetry is broken, even for massless electrons, by the chiral anomaly, but the gauge symmetry is preserved, which is essential for consistency of QED.)

In QCD, the gauge theory of strong interactions, the lowest mass quarks are nearly massless and an approximate chiral symmetry is present. In this case the left- and right-handed quarks are interchangeable in bound states of mesons and baryons, so an exact chiral symmetry of the quarks would imply "parity doubling", and every state should appear in a pair of equal mass particles, called "parity partners". In the notation, (spin)‹The template Smallsup is being considered for deletion.› parity, a meson would therefore have the same mass as a parity partner meson.

Experimentally, however, it is observed that the masses of the pseudoscalar mesons (such as the pion) are much lighter than any of the other particles in the spectrum. The low masses of the pseudoscalar mesons, as compared to the heavier states, is also quite striking. The next heavier states are the vector mesons, , such as rho meson, and the scalars mesons and vector mesons are heavier still, appearing as short-lived resonances far (in mass) from their parity partners.

This is a primary consequence of the phenomenon of spontaneous symmetry breaking of chiral symmetry in the strong interactions. In QCD, the fundamental fermion sector consists of three "flavors" of light mass quarks, in increasing mass order: up u, down d, and strange s  (as well as three flavors of heavy quarks, charm c, bottom b, and top t ). If we assume the light quarks are ideally massless (and ignore electromagnetic and weak interactions), then the theory has an exact global chiral flavor symmetry. Under spontaneous symmetry breaking, the chiral symmetry is spontaneously broken to the "diagonal flavor SU(3) subgroup", generating low mass Nambu–Goldstone bosons. These are identified with the pseudoscalar mesons seen in the spectrum, and form an octet representation of the diagonal SU(3) flavor group.

Beyond the idealization of massless quarks, the actual small quark masses (and electroweak forces) explicitly break the chiral symmetry as well. This can be described by a chiral Lagrangian where the masses of the pseudoscalar mesons are determined by the quark masses, and various quantum effects can be computed in chiral perturbation theory. This can be confirmed more rigorously by lattice QCD computations, which show that the pseudoscalar masses vary with the quark masses as dictated by chiral perturbation theory, (effectively as the square-root of the quark masses).

The three heavy quarks: the charm quark, bottom quark, and top quark, have masses much larger than the scale of the strong interactions, thus they do not display the features of spontaneous chiral symmetry breaking. However bound states consisting of a heavy quark and a light quark (or two heavies and one light) still display a universal behavior, where the ground states are split from the parity partners by a universal mass gap of about (confirmed experimentally by the ) due to the light quark chiral symmetry breaking (see below).

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