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A conformal anomaly, scale anomaly, trace anomaly or Weyl anomaly is an anomaly, i.e. a quantum phenomenon that breaks the
conformal symmetry Conformal symmetry is a property of spacetime that ensures angles remain unchanged even when distances are altered. If you stretch, compress, or otherwise distort spacetime, the local angular relationships between lines or curves stay the same. Th ...
of the
classical theory Classical physics refers to physics theories that are non-quantum or both non-quantum and non-relativistic, depending on the context. In historical discussions, ''classical physics'' refers to pre-1900 physics, while ''modern physics'' refers to p ...
. In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
when we set
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
\hbar to zero we have only Feynman tree diagrams, which is a "classical" theory (equivalent to the
Fredholm theory In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given ...
of a
classical field theory A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called qua ...
). One-loop (''N''-loop)
Feynman diagrams 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 ...
are proportional to \hbar (\hbar^N). If a current is conserved classically (\hbar=0) but develops a divergence at loop level in quantum field theory (\propto \hbar), we say there is an anomaly. A famous example is the axial current anomaly where massless fermions will have a classically conserved axial current, but which develops a nonzero divergence in the presence of gauge fields. A scale invariant theory, one in which there are no mass scales, will have a conserved
Noether current Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mathem ...
called the "scale current." This is derived by performing scale transformations on the coordinates of space-time. The divergence of the scale current is then the trace of the stress tensor. In the absence of any mass scales the stress tensor trace vanishes (\hbar=0), hence the current is "classically conserved" and the theory is classically scale invariant. However, at loop level the scale current can develop a nonzero divergence. This is called the "scale anomaly" or "trace anomaly" and represents the generation of mass by quantum mechanics. It is related to the
renormalization group In theoretical physics, the renormalization group (RG) is a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying p ...
, or the "running of coupling constants," when they are viewed at different mass scales. While this can be formulated without reference to gravity, it becomes more powerful when general relativity is considered. A classically conformal theory with arbitrary background metric has an action that is invariant under rescalings of the background metric and other matter fields, called
Weyl transformation In theoretical physics, the Weyl transformation, named after German mathematician Hermann Weyl, is a local rescaling of the metric tensor: g_ \rightarrow e^ g_ which produces another metric in the same conformal class. A theory or an expressi ...
s. Note that if we rescale the coordinates this is a general coordinate transformation, and merges with general covariance, the exact symmetry of general relativity, and thus it becomes an unsatisfactory way to formulate scale symmetry (general covariance implies a conserved stress tensor; a "gravitational anomaly" represents a quantum breakdown of general covariance, and should not be confused with Weyl (scale) invariance). However, under Weyl transformations we do not rescale the coordinates of the theory, but rather the metric and other matter fields. In the sense of Weyl, mass (or length) are defined by the metric, and coordinates are simply scale-less book-keeping devices. Hence Weyl symmetry is the correct statement of scale symmetry when gravitation is incorporated and there will then be a conserved Weyl current. There is an extensive literature involving spontaneous breaking of Weyl symmetry in four dimensions, leading to a dynamically generate Planck mass together with inflation. These theories appear to be in good agreement with observational cosmology. A conformal quantum theory is therefore one whose path integral, or partition function, is unchanged by rescaling the metric (together with other fields). The variation of the action with respect to the background metric is proportional to the stress tensor, and therefore the variation with respect to a conformal rescaling is proportional to the trace of the stress tensor. As a result, the trace of the stress tensor must vanish for a conformally invariant theory. The trace of the stress tensor appears in the divergence of the Weyl current as an anomaly, thus breaking the Weyl (or Scale) invariance of the theory.


Quantum chromodynamics

In
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the study 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 of ...
in the chiral limit, the classical theory has no
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
scale so there is a conformal symmetry. Naively, we would expect that the proton is nearly massless because the quark kinetic energy and potential energy cancel by the relativistic
virial theorem In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by a conservative force (where the work done is independent of path), with ...
. However, in the quantum case the symmetry is broken by a conformal anomaly. This introduces a scale, the scale at which
colour confinement In quantum chromodynamics (QCD), color confinement, often simply called confinement, is the phenomenon that color-charged particles (such as quarks and gluons) cannot be isolated, and therefore cannot be directly observed in normal conditions b ...
occurs and determines the masses of
hadron In particle physics, a hadron is a composite subatomic particle made of two or more quarks held together by the strong nuclear force. Pronounced , the name is derived . They are analogous to molecules, which are held together by the electri ...
s, and the phenomenon of
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 ...
. Beside the anomaly (believed to contribute to about 20% of the proton mass), the rest can be attributed to the light
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 nucleus, atomic nuclei ...
s sigma terms (i.e., the fact that quark have small non-zero masses that are not associated with the trace anomaly) believed to contribute to about 17%, and the quark and gluon energies believed to contribute to about 29% and 34% of the proton mass, respectively. Hence QCD, via the trace anomaly, quark and gluon energies and sigma terms, is responsible for more than 99% of the mass of ordinary
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic pa ...
in the Universe, the
Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the Mass generation, generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles ...
directly contributing only less than one percent via mostly the u quark, d quark and electron masses.


Coleman–Weinberg potentials

Sidney Coleman Sidney Richard Coleman (7 March 1937 – 18 November 2007) was an American theoretical physicist noted for his research in high-energy physics. Life and work Sidney Coleman grew up on the Far North Side of Chicago. In 1957, he received h ...
and Erick Weinberg showed how
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 o ...
of electroweak interactions involving a fundamental Higgs scalar could occur via Feynmans loops. Moreover, the authors showed how to "improve" the results of their calculation using the
renormalization group In theoretical physics, the renormalization group (RG) is a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying p ...
. In fact, the Coleman–Weinberg mechanism can be traced entirely to the renormalization group running of the quartic Higgs coupling, \lambda. The resulting Coleman–Weinberg potential is proportional to the associated \beta-function, while the trace anomaly is given by \beta(\lambda)/\lambda, hence the Coleman–Weinberg potential can be viewed as arising directly from the trace anomaly. It has been conjectured that all mass in nature is generated by trace anomalies, hence by quantum mechanics alone.


String theory

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 ...
is not classically scale invariant since it is defined with a massive "string constant". In string theory, conformal symmetry on the
worldsheet In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind as a direct generalization of the world line concept for a point particle in special an ...
is a local Weyl symmetry. There is also a potential gravitational anomaly in two dimensions and this anomaly must therefore cancel if the theory is to be consistent. The required cancellation of the gravitational anomaly implies that the spacetime dimensionality must be equal to the
critical dimension In the renormalization group analysis of phase transitions in physics, a critical dimension is the dimensionality of space at which the character of the phase transition changes. Below the lower critical dimension there is no phase transition. ...
which is either 26 in the case of
bosonic string theory Bosonic string theory is the original version of string theory, developed in the late 1960s. It is so called because it contains only bosons in the spectrum. In the 1980s, supersymmetry was discovered in the context of string theory, and a new ve ...
or 10 in the case of
superstring theory Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. 'Superstring theory' is a shorthand for supersymmetric string t ...
. This case is called ''critical string theory''. There are alternative approaches known as '' non-critical string theory'' in which the space-time dimensions can be less than 26 for the bosonic theory or less than 10 for the superstring ''i.e.'' the four-dimensional case is plausible within this context. However, some intuitive postulates like flat space being a valid background, need to be given up.


See also

*
Anomaly (physics) In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physics, a classical anomaly is the failure of a symm ...
*
Charge (physics) In physics, a charge is any of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges correspond to the time-invariant generators of a symmetry group, and specificall ...
*
Central charge In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other element ...
* Anomalous scaling dimension *
Dimensional transmutation In particle physics, dimensional transmutation is a physical mechanism providing a linkage between a dimensionless parameter and a dimensionful parameter.Cao, Tian Yu. From Current Algebra to Quantum Chromodynamics: A Case for Structural Realism' ...


References

{{String theory topics , state=collapsed Anomalies (physics) Conformal field theory Quantum chromodynamics Renormalization group String theory