particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...

, dimensional transmutation is a physical mechanism providing a linkage between a

dimensionless Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...

parameter and a dimensionful parameter.Cao, Tian Yu.
From Current Algebra to Quantum Chromodynamics: A Case for Structural Realism
'.

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, 2010. 163. In

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 ...

, 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 under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...

in four-dimensional

spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

, the

coupling constant 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 tw ...

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 This list compares various energies in joules (J), organized by order of magnitude. Below 1 J 1 to 105 J 106 to 1011 J 1012 to 1017 J 1018 to 1023 J Over 1024 J SI multiples See also * Conversion of units of e ...

of the processes under considerations, called 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 ...

(RG) scale. This "running" of the coupling is specified by the

beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^ ...

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 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 ...

, 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 tw ...

, 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 In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) ...

, such that implies , with the RG scale. On the other hand, in the case of theories such as QED, is an

ultraviolet cutoff Ultraviolet radiation, also known as simply UV, is electromagnetic radiation of wavelengths of 10–400 nanometers, shorter than that of visible light, but longer than X-rays. UV radiation is present in sunlight and constitutes about 10% of the ...

, such that implies . This is also a way of saying that 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 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. In quantum field theory when we set Planck constant \hbar to zero we have only ...

References

Quantum field theory Renormalization group {{quantum-stub