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 ...
, dimensional transmutation is a physical mechanism providing a linkage between a
dimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
parameter and a
dimensionful
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quantities (such as length, mass, time, and electric current) and units of measur ...
parameter.
In
classical field theory
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
, such as
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 groups) ...
in four-dimensional
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
, the
coupling constant is a dimensionless constant. However, upon quantization, logarithmic divergences in
one-loop diagrams of perturbation theory imply that this "constant" actually depends on the typical
energy scale
In physics, length scale is a particular length or distance determined with the precision of at most a few orders of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot aff ...
of the processes under considerations, called the
renormalization group
In theoretical physics, the term renormalization group (RG) refers to 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 ...
(RG) scale. This "running" of the coupling is specified by the
beta-function
In theoretical physics, specifically quantum field theory, a beta function, ''β(g)'', encodes the dependence of a coupling parameter, ''g'', on the energy scale, ''μ'', of a given physical process described by quantum field theory.
It is ...
of the renormalization group.
Consequently, the interaction may be characterised by a dimensionful parameter , namely the value of the RG scale at which the coupling constant diverges. In the case of
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 ...
, this energy scale is called the
QCD scale
In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
, and its value 220 MeV supplants the role of the original dimensionless coupling constant in the form of the logarithm (at one-loop) of the ratio and . Perturbation theory, which produced this type of running formula, is only valid for a (dimensionless) coupling ≪ 1. In the case of QCD, the energy scale is an
infrared cutoff
In theoretical physics, cutoff (AE: cutoff, BE: cut-off) is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculat ...
, such that implies , with the RG scale.
On the other hand, in the case of theories such as QED, is an
ultraviolet cutoff
In theoretical physics, cutoff (AE: cutoff, BE: cut-off) is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculat ...
, such that implies .
This is also a way of saying that the
conformal symmetry
In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry ...
of the classical theory is
anomalously broken upon quantization, thereby setting up a mass scale. See
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 wi ...
.
Quantum field theory
Renormalization group
{{quantum-stub