Dimensional transmutation
Encyclopedia
In particle physics
, dimensional transmutation is a physical mechanism that transforms a dimensionless parameter into a dimensionful parameter.
In classical field theory
such as gauge theory
in four-dimensional spacetime
, the coupling constant
is a dimensionless constant. However, logarithmic divergences in one-loop diagrams imply that this "constant" actually depends on the typical energy scale of the processes under considerations. The "running" is determined by the beta-function
and renormalization group
.
Consequently, the strength of the interaction may be described by a dimensionful parameter, namely the energy scale where the interaction strength reaches the value 1. In the case of quantum chromodynamics
, this energy scale is called the QCD scale and its value 150 MeV replaces the original dimensionless coupling constant.
This is a fancy way of saying that the conformal symmetry
of the classical theory is anomalously broken, thereby setting up a mass scale. See conformal anomaly
.
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
, dimensional transmutation is a physical mechanism that transforms a dimensionless parameter into a dimensionful parameter.
In classical field theory
Classical field theory
A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics ....
such as gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
in four-dimensional spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
, the coupling constant
Coupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
is a dimensionless constant. However, logarithmic divergences in one-loop diagrams imply that this "constant" actually depends on the typical energy scale of the processes under considerations. The "running" is determined by the beta-function
Beta-function
In theoretical physics, specifically quantum field theory, a beta function β encodes the dependence of a coupling parameter, g, on the energy scale, \mu of a given physical process....
and renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
.
Consequently, the strength of the interaction may be described by a dimensionful parameter, namely the energy scale where the interaction strength reaches the value 1. In the case of quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
, this energy scale is called the QCD scale and its value 150 MeV replaces the original dimensionless coupling constant.
This is a fancy way of saying that the conformal symmetry
Conformal symmetry
In theoretical physics, conformal symmetry is a symmetry under dilatation and under the special conformal transformations...
of the classical theory is anomalously broken, thereby setting up a mass scale. See conformal anomaly
Conformal anomaly
Conformal 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 with arbitrary background metric, has an action that is invariant under rescalings of the background metric...
.