Chiral symmetry breaking

In particle physics, chiral symmetry breaking is an example of spontaneous symmetry breaking affecting the chiral symmetry of a gauge theory such as Quantum Chromodynamics, the quantum field theory of the strong interactions.

Overview

The principal and manifest consequence of this symmetry breaking is the generation of 99% of the mass of nucleons, and hence the bulk of all visible matter, out of very light quarks.^[1] For example, for the proton, of mass m_p= 938 MeV, the bound quarks, with m_u ≈ 2 MeV, m_d ≈ 5 MeV, only contribute about 9 MeV to its mass, the bulk of it arising out of QCD chiral symmetry breaking, instead.^[2]

Yoichiro Nambu was awarded the 2008 Nobel prize in physics for his understanding of this phenomenon.

The origin of the symmetry breaking may be described as an analog to magnetization, the fermion condensate (vacuum condensate of bilinear expressions involving the quarks in the QCD vacuum),

$\langle \bar{q}^a_R q^b_L \rangle = v \delta^{ab} ~,$

formed through nonperturbative action of QCD gluons, with v ≈ −(250 MeV)³. It is clear that this cannot be preserved under an isolated L or R rotation. The pion decay constant, f_π ≈ 93 MeV, may be viewed as a measure of the strength of the chiral symmetry breaking.^[3]

For two light quarks, u and d, the symmetry of the QCD Lagrangian called chiral symmetry, and denoted as $U(2)_L \times U(2)_R$, can be decomposed into

$SU(2)_L \times SU(2)_R \times U(1)_V \times U(1)_A ~.$

The quark condensate spontaneously breaks the $SU(2)_L \times SU(2)_R$ down to the diagonal vector subgroup SU(2)_V, known as isospin. The resulting effective theory of baryon bound states of QCD (which describes protons and neutrons), then, has mass terms for these, disallowed by the original linear realization of the chiral symmetry, but allowed by the nonlinear (spontaneously broken) realization thus achieved as a result of the strong interactions.^[4]

The Nambu-Goldstone bosons corresponding to the three broken generators are the three pions, charged and neutral. More precisely, because of small quark masses which make this chiral symmetry only approximate, the pions are Pseudo-Goldstone bosons instead, with a nonzero, but still atypically small mass,^[5] m_π ≈ √v m_q / f_π .

For three quarks, u, d, s, instead, the flavor-chiral symmetries likewise decompose to

$SU(3)_L \times SU(3)_R \times U(1)_V \times U(1)_A$.

The chiral symmetry broken is now the nondiagonal part of the respective $SU(3)_L \times SU(3)_R$; so, then, eight axial generators, corresponding to the eight light pseudoscalar mesons.

The remaining eight unbroken vector generators constitute the manifest standard "Eightfold Way" flavor symmetries.

Pseudo-Nambu−Goldstone bosons

Pseudo-Goldstone bosons arise in a quantum field theory with both spontaneous and explicit symmetry breaking. These two very different breakings are, in general, not known to be causally connected to each other: They are two independent phenomena which, in principle, occur separately, in theories where they happen to do, at different energy scales.

The controlling approximate symmetries, if they were exact, would be spontaneously broken (hidden), and would thus engender massless Nambu-Goldstone bosons, in the absence of explicit breaking. The additional explicit symmetry breaking at a smaller scale, as a perturbation, gives these bosons a small mass. The properties of these pseudo-Goldstone bosons can normally be thus found by an expansion around the (exactly) symmetric theory in terms of the explicit symmetry-breaking parameters.

Quantum chromodynamics (QCD), the theory of strong particle interactions, provides the best known example in nature, through its chiral symmetry breaking; also see the article on the QCD vacuum for details. Experimentally, it is observed that the masses of the octet of pseudoscalar mesons (such as the pion) are much lighter than the next heavier states, e.g., the octet of vector mesons (such as the rho meson).

In QCD, this is interpreted as a consequence of spontaneous symmetry breaking of chiral symmetry in a sector of QCD with 3 flavors of light quarks, u, d, and s. Such a theory, for idealized massless quarks, has global SU(3) × SU(3) chiral flavor symmetry. Under SSB, this is spontaneously broken to the diagonal flavor SU(3) subgroup, generating eight Nambu-Goldstone bosons, which are the pseudoscalar mesons transforming as an octet representation of this flavor SU(3).

In actual full QCD, the small quark masses further break the chiral symmetry explicitly as well. The masses of the actual pseudoscalar meson octet are found by an expansion in the quark masses which goes by the name of chiral perturbation theory. The internal consistency of this argument is further checked by lattice QCD computations, which allow one to vary the quark mass and check that the variation of the pseudoscalar masses with the quark masses is as dictated by chiral perturbation theory, effectively as the square-root of the quark masses.

For the three heavier quarks, c, b, and t their masses, and hence the explicit breaking these amount to, are much larger than the QCD spontaneous chiral symmetry breaking scale, so they cannot be treated as a small perturbation around the explicit symmetry limit.

See also

• Spontaneous symmetry breaking
• Goldstone bosons
• QCD vacuum
• Little Higgs

References

1. Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics, (Oxford 1984) ISBN 978-0198519614; Wilczek, F. (1999). "Mass Without Mass I: Most of Matter". Physics Today 52 (11): 11–11. Bibcode:1999PhT....52k..11W. doi:10.1063/1.882879.
2. This is a rough formal wisecrack. The actual chiral limit of the nucleon mass is about 880 MeV, cf. Procura, M.; Musch, B.; Wollenweber, T.; Hemmert, T.; Weise, W. (2006). "Nucleon mass: From lattice QCD to the chiral limit". Physical Review D 73 (11). arXiv:hep-lat/0603001. Bibcode:2006PhRvD..73k4510P. doi:10.1103/PhysRevD.73.114510..
3. Peskin, Michael; Schroeder, Daniel (1995). An Introduction to Quantum Field Theory. Westview Press. p. 670. ISBN 0-201-50397-2.
4. J Donoghue, E Golowich and B Holstein, Dynamics of the Standard Model, ( Cambridge University Press, 1994) ISBN 9780521476522 .
5. Gell-Mann, M.; Renner, B. (1968). "Behavior of Current Divergences under SU_{3}×SU_{3}". Physical Review 175 (5): 2195. Bibcode:1968PhRv..175.2195G. doi:10.1103/PhysRev.175.2195.

• Gell-Mann, M.; Lévy, M. (1960), "The axial vector current in beta decay", Il Nuovo Cimento 16: 705–726, doi:10.1007/BF02859738 online copy