Editing Georgi–Glashow model (section)

===Doublet-triplet splitting===

As mentioned in the above section the colour triplet of the <math>{\mathbf{5}}</math> which contains the SM Higgs can mediate dimension 6 proton decay. Since protons seem to be quite stable such a triplet has to acquire a quite large mass in order to suppress the decay. This is however problematic. For that consider the scalar part of the Greorgi-Glashow Lagrangian:
:<math> 
\mathcal L \supset{\mathbf{5}}_\mathrm{H}^\dagger(a+b\mathbf{24}_\mathrm{H} ){\mathbf{5}}_\mathrm{H} \overset{SSB}{\longrightarrow} (a+2bv_{24})T^\dagger T + (a-3bv_{24})H^\dagger H=m_\mathrm{T}^2 T^\dagger T - \mu^2 H^\dagger H</math>
We here have denoted the adjoint used to break <math>\ SU(5)\ </math> to the SM with <math>\ \mathbf{24}_H\ ,</math> {{math|T}} is VEV by <math>\ v_{24}\ </math> and <math>\ {\mathbf{5}}_\mathrm{H} = (T,H)^\mathsf{T}\ </math> the defining representation. which contains the SM Higgs <math>\ H\ </math> and the colour triplet <math>T</math> which can induce proton decay. As mentioned, we require <math>\ m_\mathrm{T} > 10^{12}\ \mathrm{GeV}\ </math> in order to sufficiently suppress proton decay. On the other hand, the <math>\ \mu\ </math> is typically of order <math>\ 100\ \mathrm{GeV}\ </math> in order to be consistent with observations. Looking at the above equation it becomes clear that one has to be very precise in choosing the parameters <math>\ a\ </math> and <math>\ b\ :</math> any two random parameters will not do, since then <math>\ \mu\ </math> and <math>\ m_\mathrm{T}\ </math> could be of the same order!

This is known as the [[doublet–triplet splitting problem|doublet–triplet (DT) splitting problem]]: In order to be consistent we have to 'split' the 'masses' of <math>\ T\ </math> and <math>\ H\ ,</math> but for that we need to fine-tune <math>\ a\ </math> and <math>\ b ~.</math>There are however some solutions to this problem (see e.g.<ref>{{cite journal |author1=Masiero, A. |author2=Nanopoulos, A. |author3=Tamvakis, K. |author4=Yanagida, T. |year=1982 |title=Naturally Massless Higgs Doublets in Supersymmetric SU(5) |journal=[[Physics Letters B]] |volume=115 |issue=5 |pages=380–384 |doi=10.1016/0370-2693(82)90522-6 |bibcode=1982PhLB..115..380M |url=https://cds.cern.ch/record/138184 }}</ref>) which can work quite well in [[Supersymmetry|SUSY]] models.

A review of the DT splitting problem can be found in.<ref name=ms />