Beta function (physics): Difference between revisions

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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|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|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.

The value of the quasi-fixed point is fairly precisely determined in the Standard Model, leading to a predicted [[top quark]] mass of 230 &nbsp;GeV. The observed top quark mass of 174 GeV is slightly lower than the standard model prediction by about 30% which suggests there may be more Higgs doublets beyond the single standard model Higgs boson.

The observed top quark mass of 174 GeV is slightly lower

than the standard model prediction by about 30% which suggests there may be

more Higgs doublets beyond the single standard model Higgs boson

=== Minimal Supersymmetric Standard Model ===

{{Main|Minimal Supersymmetric Standard Model#Gauge-Coupling Unification}}

Renomalization group studies in the Minimal Supersymmetric Standard Model (MSSM) of grand unification and the Higgs-Yukawa fixed points were very encouraging that the theory was on the right track. So far, however, no evidence of the predicted MSSM particles has emerged in experiment at the [[Large Hadron Collider]].

of grand unification and the Higgs-Yukawa fixed points were very encouraging that

the theory was on the right track.

So far, however, no evidence of the predicted MSSM particles

has emerged in experiment at the [[Large Hadron Collider]].

==See also==

*[[InfraredBanks–Zaks fixed point]]

*[[Callan-Symanzik equation]]

*[[Quantum triviality]]

*[[Infrared fixed point]]

*[[Banks–Zaks fixed point|Banks-Zaks fixed point]]

==References==