Editing Beta function (physics) (section)

===Standard Model Higgs–Yukawa Couplings===
{{Main|Infrared fixed point}}

In the [[Standard Model]], quarks and leptons have "[[Yukawa interaction|Yukawa coupling]]s" to the [[Higgs boson]]. These determine the mass of the particle. Most all of the quarks' and leptons' Yukawa couplings are small compared to the [[top quark]]'s Yukawa coupling. These Yukawa couplings change their values depending on the energy scale at which they are measured, through   ''[[Renormalization group|running]]''. The dynamics of Yukawa couplings of quarks are determined by the [[Exact renormalization group equation|renormalization group equation]]:

<math>\mu \frac{\partial}{\partial\mu} y  \approx \frac{y}{16\pi^2}\left(\frac{9}{2}y^2 - 8 g_3^2\right)</math>,

where <math>g_3</math> is the [[color charge|color]] [[gauge theory|gauge]] coupling (which is a function of  <math>\mu</math> and associated with [[asymptotic freedom]])  and <math>y</math> is the Yukawa coupling. This equation describes how the Yukawa coupling changes with energy scale <math>\mu</math>.

The Yukawa couplings of the up, down, charm, strange and bottom quarks, are small at the extremely high energy scale of [[Grand Unified Theory|grand unification]], <math> \mu \approx 10^{15} </math> GeV.  Therefore, the <math>y^2</math> term can be neglected in the above equation. Solving, we then find that <math>y</math>  is increased slightly at the low energy scales at which the quark masses are generated by the Higgs, <math> \mu \approx 100 </math> GeV.

On the other hand, solutions to this equation for large initial values <math>y</math> cause the ''rhs'' to quickly approach smaller values as we descend in energy scale. The above equation then locks <math>y</math> to the QCD coupling <math>g_3</math>. This is known as the (infrared) quasi-fixed point of the renormalization group equation for the Yukawa coupling.<ref>{{cite journal|last1=Pendleton|first1=B.|last2=Ross|first2=G.G.|title=Mass and Mixing Angle Predictions from Infrared Fixed points|journal=Phys. Lett.|date=1981|volume=B98|issue=4 |page=291|doi=10.1016/0370-2693(81)90017-4|bibcode = 1981PhLB...98..291P }}</ref><ref>{{cite journal|last1=Hill|first1=C.T.|title=Quark and Lepton masses from Renormalization group fixed points|journal=Phys. Rev.|date=1981|volume=D24|issue=3 |page=691|doi=10.1103/PhysRevD.24.691|bibcode = 1981PhRvD..24..691H }}</ref> No matter what the initial starting value of the coupling is, if it is sufficiently large it will reach this quasi-fixed point value, and the corresponding quark mass is predicted.