In physics, an infrared fixed point is a set of coupling constants, or other parameters, that evolve from arbitrary initial values at very high energies (short distance) to fixed, stable values, usually predictable, at low energies (large distance).[1] This usually involves the use of the renormalization group, which specifically details the way parameters in a physical system (a quantum field theory) depend on the energy scale being probed.
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.
In the statistical physics of second order phase transitions, the physical system approaches an infrared fixed point that is independent of theinitial 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.
In the Standard Model, quarks and leptons have "Yukawa couplings" to the Higgs boson which determine the masses of the particles. Most 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:
| \mu | \partial |
| \partial\mu |
yq ≈
| yq | \left( | |
| 16\pi2 |
| 9 | |
| 2 |
| 2 | |
| y | |
| q |
-8
| 2\right) , | |
| g | |
| 3 |
where
g3
\mu
yq
q\in\{u,b,t\}~.
\mu~.
A more complete version of the same formula is more appropriate for the top quark:
| \mu | \partial |
| \partial\mu |
yt ≈
| yt | \left( | |
| 16 \pi2 |
| 9 | |
| 2 |
| 2 | |
| y | |
| t |
-8
| 2- | |
| g | |
| 3 |
| 9 | |
| 4 |
| 2 | |
| g | |
| 2 |
-
| 17 | |
| 20 |
| 2 | |
| g | |
| 1 |
\right) ,
where is the weak isospin gauge coupling and is the weak hypercharge gauge coupling. For small or near constant values of and the qualitative behavior is the same.
The Yukawa couplings of the up, down, charm, strange and bottom quarks, are small at the extremely high energy scale of grand unification,
\mu ≈ 1015GeV~.
| 2 | |
| y | |
| q |
yq
\mu ≈ 125 GeV~.
On the other hand, solutions to this equation for large initial values typical for the top quark
yt
yt
g3~.
m ≈ 220 GeV~.
The renormalization group equation for large values of the top Yukawa coupling was firstconsidered in 1981 by Pendleton & Ross,[4] and the "infrared quasi-fixed point" was proposed by Hill.[5] 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 emerged in 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.[6]
While the value of the quasi-fixed point is determined in the Standard Model of about
m ≈ 220 GeV~,
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 value. The beta-function vanishes, and the theory possesses a symmetry known as conformal symmetry.[9]