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# 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.

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 mp= 938 MeV, the bound quarks, with mu ≈ 2 MeV, md ≈ 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, fπ ≈ 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] mπ ≈ √v mq / fπ .

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.

References

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.

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..

Peskin, Michael; Schroeder, Daniel (1995). An Introduction to Quantum Field Theory. Westview Press. p. 670. ISBN 0-201-50397-2.

J Donoghue, E Golowich and B Holstein, Dynamics of the Standard Model, ( Cambridge University Press, 1994) ISBN 9780521476522 .

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.

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