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In
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, chiral symmetry breaking is the
spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
of a
chiral symmetry A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particle, ...
– usually by a
gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations ( Lie grou ...
such as
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
, the
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
of the
strong interaction The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called th ...
. Yoichiro Nambu was awarded the 2008 Nobel prize in physics for describing this phenomenon ("for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics").


Overview


Quantum chromodynamics

Experimentally, it is observed that the masses of the octet of pseudoscalar
meson In particle physics, a meson ( or ) is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, ...
s (such as the
pion In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
) are much lighter than the next heavier
state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * ''Our S ...
s such as the octet of vector mesons, such as
rho meson Rho (uppercase Ρ, lowercase ρ or ; el, ρο or el, ρω, label=none) is the 17th letter of the Greek alphabet. In the system of Greek numerals it has a value of 100. It is derived from Phoenician letter res . Its uppercase form uses the sa ...
. This is a consequence of
spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
of chiral symmetry in a fermion sector of QCD with 3 flavors of light quarks, , , and  . Such a theory, for idealized massless quarks, has global 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). Beyond this idealization of massless quarks, the actual small
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...
masses also break the chiral symmetry explicitly as well (providing non-vanishing pieces to the divergence of chiral currents, commonly referred to as ''partially conserved axial currents'' CAC. The masses of the pseudoscalar meson octet are specified by an expansion in the quark masses which goes by the name of
chiral perturbation theory Chiral perturbation theory (ChPT) is an effective field theory constructed with a Lagrangian consistent with the (approximate) chiral symmetry of quantum chromodynamics (QCD), as well as the other symmetries of parity and charge conjugation.
. The internal consistency of this argument is further checked by lattice QCD computations, which allow one to vary the quark mass and confirm that the variation of the pseudoscalar masses with the quark masses is as dictated by
chiral perturbation theory Chiral perturbation theory (ChPT) is an effective field theory constructed with a Lagrangian consistent with the (approximate) chiral symmetry of quantum chromodynamics (QCD), as well as the other symmetries of parity and charge conjugation.
, effectively as the square-root of the quark masses. For the three heavy quarks: the
charm quark The charm quark, charmed quark or c quark (from its symbol, c) is the third-most massive of all quarks, a type of elementary particle. Charm quarks are found in hadrons, which are subatomic particles made of quarks. Examples of hadrons containin ...
,
bottom quark The bottom quark or b quark, also known as the beauty quark, is a third-generation heavy quark with a charge of −  ''e''. All quarks are described in a similar way by electroweak and quantum chromodynamics, but the bottom quark has exce ...
, and
top quark The top quark, sometimes also referred to as the truth quark, (symbol: t) is the most massive of all observed elementary particles. It derives its mass from its coupling to the Higgs Boson. This coupling y_ is very close to unity; in the Standard ...
, their masses, and hence the explicit breaking these amount to, are much larger than the QCD spontaneous chiral symmetry breaking scale. Thus, they cannot be treated as a small perturbation around the explicit symmetry limit.


Mass generation

Chiral symmetry breaking is most apparent in the mass generation of
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number). Until the 1960s, nucleons were ...
s from more elementary light
quarks A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All common ...
, accounting for approximately 99% of their combined mass as a
baryon In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classifie ...
. It thus accounts for most of the mass of all visible matter. For example, in the
proton A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' elementary charge. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton–electron mass ...
, of mass the
valence quark In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Ei ...
s, two
up quark The up quark or u quark (symbol: u) is the lightest of all quarks, a type of elementary particle, and a significant constituent of matter. It, along with the down quark, forms the neutrons (one up quark, two down quarks) and protons (two up quark ...
s with and one
down quark The down quark or d quark (symbol: d) is the second-lightest of all quarks, a type of elementary particle, and a major constituent of matter. Together with the up quark, it forms the neutrons (one up quark, two down quarks) and protons (two up ...
with only contribute about 9.4 MeV to the proton's mass. The source of the bulk of the proton's mass is quantum chromodynamics binding energy, which arises out of QCD chiral symmetry breaking.


Fermion condensate

The spontaneous symmetry breaking may be described in analogy to
magnetization In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or D ...
. A vacuum condensate of bilinear expressions involving the
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...
s in the
QCD vacuum In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type o ...
is known as the fermion condensate. It can be calculated as :\langle \bar^a_\mathsf \, q^b_\mathsf \rangle = v \, \delta^ ~, formed through nonperturbative action of QCD gluons, with This cannot be preserved under an isolated or rotation. The
pion decay constant In particle physics, the pion decay constant is the square root of the coefficient in front of the kinetic term for the pion in the low-energy effective action. It is dimensionally an energy scale and it determines the strength of the chiral symm ...
, may be viewed as a measure of the strength of the chiral symmetry breaking.


Two-quark model

For two light quarks (the
up quark The up quark or u quark (symbol: u) is the lightest of all quarks, a type of elementary particle, and a significant constituent of matter. It, along with the down quark, forms the neutrons (one up quark, two down quarks) and protons (two up quark ...
and the
down quark The down quark or d quark (symbol: d) is the second-lightest of all quarks, a type of elementary particle, and a major constituent of matter. Together with the up quark, it forms the neutrons (one up quark, two down quarks) and protons (two up ...
) the QCD Lagrangian provides insight: The ''
chiral symmetry A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particle, ...
'' of the QCD Lagrangian describes invariance with respect to a
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
U(2)_\mathsf \times U(2)_\mathsf~. This symmetry group amounts to :\mathrm(2)_\mathsf \times \mathrm(2)_\mathsf \times \mathrm(1)_\mathsf \times \mathrm(1)_\mathsf ~. The quark condensate induced by non-perturbative
strong interaction The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called th ...
s spontaneously breaks the ~\mathrm(2)_\mathsf \times \mathrm(2)_\mathsf~ down to the diagonal vector subgroup   called ''
isospin In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions ...
''. The resulting effective theory of baryon bound states of QCD (which describes protons and
neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the atomic nucleus, nuclei of atoms. Since protons and ...
s), then, has mass terms for these, disallowed by the original linear realization of the chiral symmetry, but allowed by the spontaneously broken nonlinear realization thus achieved as a result of the
strong interaction The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called th ...
s. The Nambu– Goldstone bosons corresponding to the three broken generators are the three
pions In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
, charged and neutral. The next section outlines how a small explicit breaking in the Lagrangian gives these three
pion In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
s a small mass.


Pseudo-Goldstone bosons

Pseudo-Goldstone bosons arise in a
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
with ''both''
spontaneous Spontaneous may refer to: * Spontaneous abortion * Spontaneous bacterial peritonitis * Spontaneous combustion * Spontaneous declaration * Spontaneous emission * Spontaneous fission * Spontaneous generation * Spontaneous human combustion * Spontan ...
and
explicit symmetry breaking In theoretical physics, explicit symmetry breaking is the breaking of a symmetry of a theory by terms in its defining equations of motion (most typically, to the Lagrangian or the Hamiltonian) that do not respect the symmetry. Usually this term i ...
, simultaneously. These two types of symmetry breaking typically occur separately, and at different energy scales, and are not thought to be predicated on each other. In the absence of explicit breaking,
spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
would engender massless
Nambu–Goldstone boson In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu in part ...
s for the exact spontaneously broken chiral symmetries. The chiral symmetries discussed, however, are only approximate symmetries in nature, given their ''small'' explicit breaking. The explicit symmetry breaking occurs at a smaller energy scale. The properties of these pseudo-Goldstone bosons can normally be calculated using
chiral perturbation theory Chiral perturbation theory (ChPT) is an effective field theory constructed with a Lagrangian consistent with the (approximate) chiral symmetry of quantum chromodynamics (QCD), as well as the other symmetries of parity and charge conjugation.
, expanding around the exactly symmetric theory in terms of the explicit symmetry-breaking parameters. In particular, the computed mass must be small,


Three-quark model

For three light quarks, the
up quark The up quark or u quark (symbol: u) is the lightest of all quarks, a type of elementary particle, and a significant constituent of matter. It, along with the down quark, forms the neutrons (one up quark, two down quarks) and protons (two up quark ...
,
down quark The down quark or d quark (symbol: d) is the second-lightest of all quarks, a type of elementary particle, and a major constituent of matter. Together with the up quark, it forms the neutrons (one up quark, two down quarks) and protons (two up ...
, and
strange quark The strange quark or s quark (from its symbol, s) is the third lightest of all quarks, a type of elementary particle. Strange quarks are found in subatomic particles called hadrons. Examples of hadrons containing strange quarks include kaons ( ...
, the flavor-chiral symmetries extending those discussed above also decompose, to Gell-Mann's :\mathrm(3)_\mathsf \times \mathrm(3)_\mathsf \times \mathrm(1)_\mathsf \times \mathrm(1)_\mathsf ~. The chiral symmetry generators spontaneously broken comprise the
coset In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
space ~ ( \mathrm(3)_\mathsf \times \mathrm(3)_\mathsf ) / \mathrm(3)_\mathsf ~. This
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
is not a group, and consists of the eight axial generators, corresponding to the eight light
pseudoscalar meson In high-energy physics, a pseudoscalar meson is a meson with total spin 0 and odd parity (usually notated as Pseudoscalar mesons are commonly seen in proton-proton scattering and proton-antiproton annihilation, and include the pion (), ...
s, the nondiagonal part of \mathrm(3)_\mathsf \times \mathrm(3)_\mathsf ~. The remaining eight unbroken vector subgroup generators constitute the manifest standard "Eightfold Way" flavor symmetries,  .


Heavy-light mesons

Mesons containing a heavy quark, such as charm (
D meson The D mesons are the lightest particle containing charm quarks. They are often studied to gain knowledge on the weak interaction. The strange D mesons (Ds) were called "F mesons" prior to 1986. Overview The D mesons were discovere ...
) or beauty, and a light anti-quark (either up, down or strange), can be viewed as systems in which the light quark is "tethered" by the gluonic force to the fixed heavy quark, like a ball tethered to a pole. The chiral symmetry breaking then causes the s-wave ground states (0^-,1^-) (spin^) to be split from p-wave parity partner excited states (0^+,1^+) by a common "mass gap," \Delta M . In 1993
William A. Bardeen William Allan Bardeen (born September 15, 1941 in Washington, Pennsylvania) is an American theoretical physicist at Fermi National Accelerator Laboratory. He is the son of John Bardeen and Jane Maxwell Bardeen. Biography After graduating from Co ...
and
Christopher T. Hill Christopher T. Hill (born June 19, 1951) is an American theoretical physicist at the Fermi National Accelerator Laboratory who did undergraduate work in physics at M.I.T. (B.S., M.S., 1972), and graduate work at Caltech (Ph.D., 1977, Murray Gel ...
studied the properties of these systems implementing both the heavy quark symmetry and the chiral symmetries of light quarks in a Nambu–Jona-Lasinio model approximation. This described the phenomenon and gave an estimate of the mass gap of ~ \Delta M \approx 338 \text~ (see eq.(37) and discussion below in the Bardeen-Hill paper) which would be zero if the chiral symmetry breaking was turned off. The excited states of non-strange, heavy-light mesons are usually short-lived resonances due to the principal strong decay mode \mathrm(0^+,1^+) \rightarrow \mathrm + \mathrm(0^-,1^-) ~, and are therefore hard to observe. In their paper, however, the authors remarked that though the results were approximate, the charm-strange excited mesons ~ \mathrm(0^+,1^+) ~ could be abnormally narrow (long-lived) since the principal decay mode, ~ \mathrm(0^+,1^+) \rightarrow \mathrm + \mathrm(0^-,1^-) ~, could be kinematically suppressed (or altogether blocked) owing to the mass of the
kaon KAON (Karlsruhe ontology) is an ontology infrastructure developed by the University of Karlsruhe and the Research Center for Information Technologies in Karlsruhe. Its first incarnation was developed in 2002 and supported an enhanced version of ...
(). These could then be readily observed. In 2003 the \; \mathrm^*_\mathrm(2317) \; was discovered by the
BaBar Babar ( ur, ), also variously spelled as Baber, Babur, and Babor is a male given name of Pashto, and Persian origin, and a popular male given name in Pakistan. It is generally taken in reference to the Persian ''babr'' (Persian: ببر), meanin ...
collaboration, and was seen to be surprisingly narrow, with a mass gap above the \; \mathrm \; of \; \Delta M \approx 348 \text within a few percent of the Bardeen–Hill model prediction. Bardeen, Eichten and Hill immediately recognized that this was, indeed, the parity partner of the ground state, and predicted numerous observable decay modes, many of which have been subsequently confirmed by experiments. Similar predictions are expected in the \; \mathrm \; system (a strange and anti-beauty quark) and heavy-heavy-light baryons.


See also

*
Conformal anomaly A conformal anomaly, scale anomaly, trace anomaly or Weyl anomaly is an anomaly, i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory. A classically conformal theory is a theory which, when placed on a surface ...
* Little Higgs


Footnotes


References

* * {{cite journal , last1=Bernstein , first1=J. , last2=Gell-Mann , first2=M. , last3=Michel , first3=L. , year=1960 , title=On the renormalization of the axial vector coupling constant in β-decay , journal= Il Nuovo Cimento , volume=16 , issue=3 , pages=560–568 , doi=10.1007/BF02731920 , s2cid=119424935 Quantum field theory Quantum chromodynamics Mathematical physics Asymmetry