In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, a chiral anomaly is the
anomalous nonconservation of a
chiral
Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek language, Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is dist ...
current. In everyday terms, it is analogous to a sealed box that contained equal numbers of left and right-handed
bolts, but when opened was found to have more left than right, or vice versa.
Such events are expected to be prohibited according to classical
conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of mass-energy, conservation of linear momen ...
s, but it is known there must be ways they can be broken, because we have evidence of
charge–parity non-conservation ("CP violation"). It is possible that other imbalances have been caused by breaking of a ''chiral law'' of this kind. Many physicists suspect that the fact that the observable universe contains
more matter than antimatter is caused by a chiral anomaly. Research into
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 quant ...
laws is a major endeavor in particle physics research at this time.
Informal introduction
The chiral anomaly originally referred to the anomalous
decay rate
Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is conside ...
of the
neutral
Neutral or neutrality may refer to:
Mathematics and natural science Biology
* Neutral organisms, in ecology, those that obey the unified neutral theory of biodiversity
Chemistry and physics
* Neutralization (chemistry), a chemical reaction in ...
pion
In particle physics, a pion (, ) or pi meson, denoted with the Greek alphabet, Greek letter pi (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 ...
, as computed in the
current algebra of the
chiral model. These calculations suggested that the decay of the pion was suppressed, clearly contradicting experimental results. The nature of the anomalous calculations was first explained in 1969 by
Stephen L. Adler[
] and
John Stewart Bell
John Stewart Bell (28 July 1928 – 1 October 1990) was a physicist from Northern Ireland and the originator of Bell's theorem, an important theorem in quantum mechanics, quantum physics regarding hidden-variable theory, hidden-variable theor ...
&
Roman Jackiw.
[
] This is now termed the Adler–Bell–Jackiw anomaly of
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
. This is a symmetry of classical
electrodynamics
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
that is violated by quantum corrections.
The Adler–Bell–Jackiw anomaly arises in the following way. If one considers the classical (non-quantized) theory of
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
coupled to massless
fermion
In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s (electrically charged
Dirac spinor
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain comb ...
s solving the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
), one expects to have not just one but two
conserved current
In physics a conserved current is a current, j^\mu, that satisfies the continuity equation \partial_\mu j^\mu=0. The continuity equation represents a conservation law, hence the name.
Indeed, integrating the continuity equation over a volume V, ...
s: the ordinary electrical current (the
vector current), described by the Dirac field
as well as an
axial current When moving from the classical theory to the quantum theory, one may compute the quantum corrections to these currents; to first order, these are the
one-loop Feynman diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
s. These are famously divergent, and require a
regularization
Regularization may refer to:
* Regularization (linguistics)
* Regularization (mathematics)
* Regularization (physics)
* Regularization (solid modeling)
* Regularization Law, an Israeli law intended to retroactively legalize settlements
See also ...
to be applied, to obtain the
renormalized amplitudes. In order for the
renormalization
Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
to be meaningful, coherent and consistent, the regularized diagrams must obey the same symmetries as the zero-loop (classical) amplitudes. This is the case for the vector current, but not the axial current: it cannot be regularized in such a way as to preserve the axial symmetry. The axial symmetry of classical electrodynamics is broken by quantum corrections. Formally, the
Ward–Takahashi identities of the quantum theory follow from the
gauge symmetry
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 under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
of the electromagnetic field; the corresponding identities for the axial current are broken.
At the time that the Adler–Bell–Jackiw anomaly was being explored in physics, there were related developments in
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
that appeared to involve the same kinds of expressions. These were not in any way related to quantum corrections of any sort, but rather were the exploration of the global structure of
fiber bundle
In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a pr ...
s, and specifically, of the
Dirac operator
In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Ham ...
s on
spin structure
In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathemati ...
s having
curvature form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let ''G'' be a Lie group with Lie algebra ...
s resembling that of the
electromagnetic tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...
, both in four and three dimensions (the
Chern–Simons theory
The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who intr ...
). After considerable back and forth, it became clear that the structure of the anomaly could be described with bundles with a non-trivial
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
, or, in physics lingo, in terms of
instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ...
s.
Instantons are a form of
topological soliton; they are a solution to the ''classical'' field theory, having the property that they are stable and cannot decay (into
plane wave
In physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
s, for example). Put differently: conventional field theory is built on the idea of a
vacuum
A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
– roughly speaking, a flat empty space. Classically, this is the "trivial" solution; all fields vanish. However, one can also arrange the (classical) fields in such a way that they have a non-trivial global configuration. These non-trivial configurations are also candidates for the vacuum, for empty space; yet they are no longer flat or trivial; they contain a twist, the instanton. The quantum theory is able to interact with these configurations; when it does so, it manifests as the chiral anomaly.
In mathematics, non-trivial configurations are found during the study of
Dirac operator
In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Ham ...
s in their fully generalized setting, namely, on
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s in arbitrary dimensions. Mathematical tasks include finding and classifying structures and configurations. Famous results include the
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space ...
for Dirac operators. Roughly speaking, the symmetries of
Minkowski spacetime
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a s ...
,
Lorentz invariance
In a relativistic theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation. A Lorentz scalar may be generated from, e.g., the scalar product of vectors, or by contracting tensors. While ...
,
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
s, Dirac operators and the U(1)xSU(2)xSU(3)
fiber bundle
In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a pr ...
s can be taken to be a special case of a far more general setting in
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
; the exploration of the various possibilities accounts for much of the excitement in theories such as
string theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
; the richness of possibilities accounts for a certain perception of lack of progress.
The Adler–Bell–Jackiw anomaly is seen experimentally, in the sense that it describes the decay of the
neutral pion
In particle physics, a pion (, ) or 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 g ...
, and specifically, the
width of the decay of the neutral pion into two
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
s. The neutral pion itself was discovered in the 1940s; its decay rate (width) was correctly estimated by J. Steinberger in 1949. The correct form of the anomalous divergence of the axial current is obtained by Schwinger in 1951 in a 2D model of electromagnetism and massless fermions. That the decay of the neutral pion is suppressed in the
current algebra analysis of the
chiral model is obtained by Sutherland and Veltman in 1967. An analysis and resolution of this anomalous result is provided by Adler
[ and Bell & Jackiw][ in 1969. A general structure of the anomalies is discussed by Bardeen in 1969.
The ]quark model
In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks that give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Eig ...
of the pion indicates it is a bound state of a quark and an anti-quark. However, the quantum number
In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system.
To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantu ...
s, including parity and angular momentum, taken to be conserved, prohibit the decay of the pion, at least in the zero-loop calculations (quite simply, the amplitudes vanish.) If the quarks are assumed to be massive, not massless, then a chirality
Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable fro ...
-violating decay is allowed; however, it is not of the correct size. (Chirality is not a constant of motion In mechanics, a constant of motion is a physical quantity conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a ''mathematical'' constraint, the natural consequence of the equations of motion, rather tha ...
of massive spinors; they will change handedness as they propagate, and so mass is itself a chiral symmetry-breaking term. The contribution of the mass is given by the Sutherland and Veltman result; it is termed "PCAC", the partially conserved axial current.) The Adler–Bell–Jackiw analysis provided in 1969 (as well as the earlier forms by Steinberger and Schwinger), do provide the correct decay width for the neutral pion.
Besides explaining the decay of the pion, it has a second very important role. The one loop amplitude includes a factor that counts the grand total number of leptons that can circulate in the loop. In order to get the correct decay width, one must have exactly three generations of quarks, and not four or more. In this way, it plays an important role in constraining the Standard model
The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
. It provides a direct physical prediction of the number of quarks that can exist in nature.
Current day research is focused on similar phenomena in different settings, including non-trivial topological configurations of the electroweak theory
In particle physics, the electroweak interaction or electroweak force is the unified description of two of the fundamental interactions of nature: electromagnetism (electromagnetic interaction) and the weak interaction. Although these two forc ...
, that is, the sphaleron
A sphaleron ( "slippery") is a static (time-independent) solution to the electroweak field equations of the Standard Model of particle physics, and is involved in certain hypothetical processes that violate baryon and lepton numbers. Such proces ...
s. Other applications include the hypothetical non-conservation of baryon number
In particle physics, the baryon number (B) is an additive quantum number of a system. It is defined as
B = \frac(n_\text - n_),
where is the number of quarks, and is the number of antiquarks. Baryons (three quarks) have B = +1, mesons (one q ...
in GUTs and other theories.
General discussion
In some theories of fermion
In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s with 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 particl ...
, the quantization may lead to the breaking of this (global) chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved. The non-conservation happens in a process of tunneling from one vacuum
A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
to another. Such a process is called an instanton
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ...
.
In the case of a symmetry related to the conservation of a fermionic particle number, one may understand the creation of such particles as follows. The definition of a particle is different in the two vacuum states between which the tunneling occurs; therefore a state of no particles in one vacuum corresponds to a state with some particles in the other vacuum. In particular, there is a Dirac sea
The Dirac sea is a theoretical model of the electron vacuum as an infinite sea of electrons with negative energy, now called '' positrons''. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-en ...
of fermions and, when such a tunneling happens, it causes the energy level
A quantum mechanics, quantum mechanical system or particle that is bound state, bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical mechanics, classical pa ...
s of the sea fermions to gradually shift upwards for the particles and downwards for the anti-particles, or vice versa. This means particles which once belonged to the Dirac sea become real (positive energy) particles and particle creation happens.
Technically, in the path integral formulation
The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or ...
, an anomalous symmetry is a symmetry of the action
Action may refer to:
* Action (philosophy), something which is done by a person
* Action principles the heart of fundamental physics
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video gam ...
, but not of the measure and therefore ''not'' of the generating functional
:
of the quantized theory ( is Planck's action-quantum divided by 2). The measure consists of a part depending on the fermion field