Infrared fixed point
Encyclopedia
In physics
, an infrared fixed point is a set of coupling constants, or other parameters that evolve from initial values at very high energies (short distance), to fixed stable values, usually predictable, at low energies (large distance). This usually involves the use of the renormalization group
, a mathematical apparatus for theoretically evolving physical systems from one scale to another.
Conversely, if the length-scale decreases and the physical parameters approach fixed values, then we have ultraviolet fixed points. The fixed points are generally independent of the initial values of the parameters over a large range of the initial values. This is known as "universality
".
of second order phase transition
s, the physical system approaches an infrared fixed point that is independent of the
initial short distance dynamics that defines the material. This determines the properties of the phase transition at the critical temperature, or critical point
. Observables, such as "critical exponents" usually depend only upon dimension of space, and are independent of the atomic or molecular constituents.
's coupling constant
gets closer to zero as the energy increases. This is an ultraviolet fixed point, associated with the phenomenon known as 'asymptotic freedom
'. This causes quark
s and gluon
s to behave as effectively free
noninteracting particles at high energies. This phenomenon was first anticipated by "Bjorken Scaling", and observed in electroproduction experiments. It was critical to the development of quantum chromodynamics
.
There is a remarkable infrared fixed point of the coupling constants that determine the masses of very heavy quarks. In the Standard Model
, quarks and leptons have "Yukawa couplings" to the Higgs boson
. These determine the mass of the particle. All of the quarks' and leptons' Yukawa couplings are small compared to the top quark
's Yukawa coupling. Yukawa couplings are not constants and their properties change depending on the energy scale at which they are measured, this is known as running
of the constants. The dynamics of Yukawa couplings are determined by the renormalization group equation
:
,
where
is the color
gauge coupling (which is a function of
and associated with asymptotic freedom
) and
is the Yukawa coupling. This equation describes how the Yukawa coupling changes with energy scale 
.
The Yukawa couplings of the up, down, charm, strange and bottom quarks, are small at the extremely high energy scale of grand unification,
GeV. The 
term can be neglected in the above equation. Solving, we then find that 
is increased slightly at the low energy scales at which the quark masses are generated by the Higgs, 
GeV.
On the other hand, solutions to this equation for large initial values
cause the rhs to quickly approach zero. This locks 
to the QCD coupling 
. This is known as a (quasi-infrared) fixed point of the renormalization group equation for the Yukawa coupling. No matter what the initial starting value of the coupling is, if it is sufficiently large it will reach this fixed point value, and the corresponding quark mass is predicted.
The value of the fixed point is fairly precisely determined in the Standard Model, leading to a predicted top quark mass of 230 GeV. If there is more than one Higgs doublet, the value will be reduced by Higgs mixing angle effects. The observed top quark mass is slightly lower, about 171 GeV (see top quark
).
In the minimal supersymmetric extension of the Standard Model (MSSM
), there are two Higgs doublets and the renormalization group equation for the top quark Yukawa coupling is slightly modified. This leads to a fixed point where the top mass is smaller, 170–200 GeV. Some theorists believe this is supporting evidence for the MSSM.
The "quasi-infrared fixed point" was proposed in 1981 by C. T. Hill, B. Pendleton and G. G. Ross. The prevailing view at the time was that the top quark mass would lie in a range of 15 to 26 GeV. The quasi-infrared fixed point has formed the basis of top quark condensation theories of electroweak symmetry breaking in which the Higgs boson is composite at extremely short distance scales, composed of a pair of top and anti-top quarks. Many authors have explored other aspects of infrared fixed points to understand the anticipated spectrum of Higgs bosons in multi-Higgs models.
Another example of an infrared fixed point is the Banks-Zaks fixed point
in which the coupling constant of a Yang-Mills theory evolves to a fixed large value. The beta-function vanishes, and the theory possesses a symmetry known as conformal symmetry
.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, an infrared fixed point is a set of coupling constants, or other parameters that evolve from initial values at very high energies (short distance), to fixed stable values, usually predictable, at low energies (large distance). This usually involves the use of the 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...
, a mathematical apparatus for theoretically evolving physical systems from one scale to another.
Conversely, if the length-scale decreases and the physical parameters approach fixed values, then we have ultraviolet fixed points. The fixed points are generally independent of the initial values of the parameters over a large range of the initial values. This is known as "universality
Universality
Universality may refer to:* Universality in physical science * Universality * Universality , meaning present in all places and all times* Universality...
".
Statistical Physics
In the statistical physicsStatistical physics
Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...
of second order phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
s, the physical system approaches an infrared fixed point that is independent of the
initial short distance dynamics that defines the material. This determines the properties of the phase transition at the critical temperature, or critical point
Critical point (thermodynamics)
In physical chemistry, thermodynamics, chemistry and condensed matter physics, a critical point, also called a critical state, specifies the conditions at which a phase boundary ceases to exist...
. Observables, such as "critical exponents" usually depend only upon dimension of space, and are independent of the atomic or molecular constituents.
Particle Physics
In particle physics the best known fixed point is that the strong interactionStrong interaction
In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...
's 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...
gets closer to zero as the energy increases. This is an ultraviolet fixed point, associated with the phenomenon known as 'asymptotic freedom
Asymptotic freedom
In physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become arbitrarily weak at energy scales that become arbitrarily large, or, equivalently, at length scales that become arbitrarily small .Asymptotic freedom is a feature of quantum...
'. This causes quark
Quark
A quark is an 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 nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
s and gluon
Gluon
Gluons are elementary particles which act as the exchange particles for the color force between quarks, analogous to the exchange of photons in the electromagnetic force between two charged particles....
s to behave as effectively free
Free particle
In physics, a free particle is a particle that, in some sense, is not bound. In classical physics, this means the particle is present in a "field-free" space.-Classical Free Particle:The classical free particle is characterized simply by a fixed velocity...
noninteracting particles at high energies. This phenomenon was first anticipated by "Bjorken Scaling", and observed in electroproduction experiments. It was critical to the development 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...
.
There is a remarkable infrared fixed point of the coupling constants that determine the masses of very heavy quarks. In the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
, quarks and leptons have "Yukawa couplings" to the Higgs boson
Higgs boson
The Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...
. These determine the mass of the particle. All of the quarks' and leptons' Yukawa couplings are small compared to the top quark
Top quark
The top quark, also known as the t quark or truth quark, is an elementary particle and a fundamental constituent of matter. Like all quarks, the top quark is an elementary fermion with spin-, and experiences all four fundamental interactions: gravitation, electromagnetism, weak interactions, and...
's Yukawa coupling. Yukawa couplings are not constants and their properties change depending on the energy scale at which they are measured, this is known as running
Running
Running is a means of terrestrial locomotion allowing humans and other animals to move rapidly on foot. It is simply defined in athletics terms as a gait in which at regular points during the running cycle both feet are off the ground...
of the constants. The dynamics of Yukawa couplings are determined by the renormalization group equation
Renormalization group equation
Renormalization group equation may refer to:* Beta-function* Callan–Symanzik equation* Exact renormalization group equation...
:
,where
is the colorColor charge
In particle physics, color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics . Color charge has analogies with the notion of electric charge of particles, but because of the mathematical complications of QCD,...
gauge coupling (which is a function of
and associated with asymptotic freedomAsymptotic freedom
In physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become arbitrarily weak at energy scales that become arbitrarily large, or, equivalently, at length scales that become arbitrarily small .Asymptotic freedom is a feature of quantum...
) and
is the Yukawa coupling. This equation describes how the Yukawa coupling changes with energy scale .The Yukawa couplings of the up, down, charm, strange and bottom quarks, are small at the extremely high energy scale of grand unification,
GeV. The term can be neglected in the above equation. Solving, we then find that is increased slightly at the low energy scales at which the quark masses are generated by the Higgs, GeV.On the other hand, solutions to this equation for large initial values
cause the rhs to quickly approach zero. This locks to the QCD coupling . This is known as a (quasi-infrared) fixed point of the renormalization group equation for the Yukawa coupling. No matter what the initial starting value of the coupling is, if it is sufficiently large it will reach this fixed point value, and the corresponding quark mass is predicted.The value of the fixed point is fairly precisely determined in the Standard Model, leading to a predicted top quark mass of 230 GeV. If there is more than one Higgs doublet, the value will be reduced by Higgs mixing angle effects. The observed top quark mass is slightly lower, about 171 GeV (see top quark
Top quark
The top quark, also known as the t quark or truth quark, is an elementary particle and a fundamental constituent of matter. Like all quarks, the top quark is an elementary fermion with spin-, and experiences all four fundamental interactions: gravitation, electromagnetism, weak interactions, and...
).
In the minimal supersymmetric extension of the Standard Model (MSSM
Minimal Supersymmetric Standard Model
The Minimal Supersymmetric Standard Model is the minimal extension to the Standard Model that realizes supersymmetry, although non-minimal extensions do exist. Supersymmetry pairs bosons with fermions; therefore every Standard Model particle has a partner that has yet to be discovered...
), there are two Higgs doublets and the renormalization group equation for the top quark Yukawa coupling is slightly modified. This leads to a fixed point where the top mass is smaller, 170–200 GeV. Some theorists believe this is supporting evidence for the MSSM.
The "quasi-infrared fixed point" was proposed in 1981 by C. T. Hill, B. Pendleton and G. G. Ross. The prevailing view at the time was that the top quark mass would lie in a range of 15 to 26 GeV. The quasi-infrared fixed point has formed the basis of top quark condensation theories of electroweak symmetry breaking in which the Higgs boson is composite at extremely short distance scales, composed of a pair of top and anti-top quarks. Many authors have explored other aspects of infrared fixed points to understand the anticipated spectrum of Higgs bosons in multi-Higgs models.
Another example of an infrared fixed point is the Banks-Zaks fixed point
Banks-Zaks fixed point
In quantum chromodynamics with massless flavors, if the number of flavors, Nf, is sufficiently small , the theory can flow to an interacting conformal fixed point of the renormalization group...
in which the coupling constant of a Yang-Mills theory evolves to a fixed large value. The beta-function vanishes, and the theory possesses a symmetry known as conformal symmetry
Conformal symmetry
In theoretical physics, conformal symmetry is a symmetry under dilatation and under the special conformal transformations...
.