Infrared fixed point


Infrared fixed point

In physics, an infrared fixed point is a set ofcoupling 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 aregenerally independent of the initial values of theparameters over a large range of theinitial values. This is known as "universality."

tatistical Physics

In the statistical physics of secondorder phase transitions, the physical system approachesan infrared fixed point that is independent of theinitial short distance dynamics that defines the material. This determines the properties of the phase transition atthe critical temperature, or critical point.Observables, such as "critical exponents" usually dependonly upon dimension of space, and are independent of the atomic ormolecular constituents.

Particle Physics

In particle physics the best known fixed point isthe approach of the strong interaction QCD couplingconstant to zero as the energy increases. This anultraviolet fixed point, associated with the phenomenonknown as asymptotic freedom. This causes quarks and gluonsto behave as effectively free, or noninteracting, particles at highenergies. This phenomenon was first anticipated by "Bjorken Scaling," and observed in electroproduction experiments,and was critical to the development of the theory of stronginteractions known as Quantum Chromodynamics.

There is a remarkable infrared fixed point ofthe 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 quark and lepton Yukawa couplings are small compared to the top quark Yukawa coupling. Yukawa couplings are not constants and their properties change depending on the energy scale atwhich they are measured. The dynamics of Yukawa couplings are determined by the renormalization group equation:

mu frac{partial}{partialmu} y approx frac{y}{16pi^2}left(frac{9}{2}y^2 - 8 g_3^2 ight),

where g_3 is the color gauge coupling (which is a function of mu and associated with asymptotic freedom) and y is the Yukawa coupling. This equation describes howthe Yukawa coupling changes with energy scale mu.

The Yukawa couplings of the up, down, charm, strange and bottom quarks, are small at the extremely high energy scale of grand unification, mu approx 10^{15} GeV. The y^2 term canbe neglected in the above equation. Solving, we then find that y is increasedslightly at the low energy scales at which the quark masses are generated by the Higgs, mu approx 100 GeV.

On the other hand, solutions to thisequation for large initial values y cause the "rhs" to quickly approach zero. This locks y to the QCD coupling g_3. This is known as a (quasi-infrared) fixed point of the renormalization group equation for the Yukawacoupling. 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 bereduced by Higgs mixing angle effects. The observedtop quark mass is slightly lower, about 171 GeV (see
Top quark).In the minimal supersymmetric extension of the Standard Model (the 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 viewat the time was that the top quark mass would lie in a rangeof 15 to 26 GeV. The quasi-infrared fixed point has formed the basis of top quark condensation theories of electroweak symmetry breaking in whichthe Higgs boson is composite at "extremely" short distancescales, composed of a pair of top and anti-top quarks. Manyauthors have explored other aspects of infrared fixed points tounderstand 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 theoryevolves to a fixed large value. The beta-function vanishes, andthe theory possesses a symmetry known as conformal symmetry.

See also

* Top quark
* cutoff scale


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