Editing Georgi–Glashow model (section)

==Problems of the Georgi–Glashow model==
===Proton decay in SU(5)===
[[File:Proton decay GUT simple.svg|thumb|The most common source of [[proton decay]] in SU(5).  A left-handed and a right-handed [[up quark]] annihilate yielding an X<sup>+</sup> boson which decays to a [[positron]] and an anti-[[down quark]] of opposite handedness.]]

Unification of the Standard Model via an SU(5) group has significant phenomenological implications. Most notable of these is proton decay which is present in SU(5) with and without supersymmetry. This is allowed by the new vector bosons introduced from the [[adjoint representation]] of SU(5) which also contains the gauge bosons of the Standard Model forces. Since these new gauge bosons are in (3,2)<sub>−5/6</sub> [[bifundamental representation]]s, they violated baryon and lepton number. As a result, the new operators should cause protons to decay at a rate inversely proportional to their masses. This process is called dimension 6 proton decay and is an issue for the model, since the proton is experimentally determined to have a lifetime greater than the age of the universe. This means that an SU(5) model is severely constrained by this process.

As well as these new gauge bosons, in SU(5) models, the [[Higgs field]] is usually embedded in a '''5''' representation of the GUT group. The caveat of this is that since the Higgs field is an SU(2) doublet, the remaining part, an SU(3) triplet, must be some new field - usually called D or T. This new scalar would be able to generate proton decay as well and, assuming the most basic Higgs vacuum alignment, would be massless so allowing the process at very high rates.

While not an issue in the Georgi–Glashow model, a supersymmeterised SU(5) model would have additional proton decay operators due to the superpartners of the Standard Model fermions. The lack of detection of proton decay (in any form) brings into question the veracity of SU(5) GUTs of all types; however, while the models are highly constrained by this result, they are not in general ruled out.

==== Mechanism ====

In the lowest-order [[Feynman diagram]] corresponding to the simplest source of [[proton decay]] in SU(5), a left-handed and a right-handed [[up quark]] annihilate yielding an X<sup>+</sup> boson which decays to a right-handed (or left-handed) [[positron]] and a left-handed (or right-handed) anti-[[down quark]]:

:<math>\mathrm{u}_\mathsf{L} + \mathrm{u}_\mathsf{R} \to X^ + \to \mathrm{e}_\mathsf{R}^+ + \mathrm{\bar{d}}_\mathsf{L}\ ,</math>

:<math>\mathrm{u}_\mathsf{L} + \mathrm{u}_\mathsf{R} \to X^+\to \mathrm{e}_\mathsf{L}^+ + \mathrm{\bar{d}}_\mathsf{R} ~.</math>

This process conserves [[weak isospin]], [[weak hypercharge]], and [[color charge|color]]. GUTs equate anti-color with having two colors, <math>\ \bar{g} \equiv rb\ ,</math> and SU(5) defines left-handed normal leptons as "white" and right-handed antileptons as "black". The first vertex only involves fermions of the {{math|'''10'''}} representation, while the second only involves fermions in the {{math|'''5̅'''}} (or {{math|'''10'''}}), demonstrating the preservation of SU(5) symmetry.

===Mass relations===
Since SM states are regrouped into <math>SU(5)</math> representations their Yukawa matrices have the following relations:

:<math>Y_\mathrm{d} = Y_\mathrm{e}^\mathsf{T} \quad \mathsf{and} \quad Y_\mathrm{u} = Y_\mathrm{u}^\mathsf{T}</math>

In particular this predicts <math>m_{e,\mu\tau}\approx m_{d,s,b}</math> at energies close to the scale of unification. This is however not realized in nature.

===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 />

===Neutrino masses===

As the SM the original Georgi–Glashow model proposed in<ref name=GG /> does not include neutrino masses. However, since [[neutrino oscillation]] has been observed such masses are required. The solutions to this problem follow the same ideas which have been applied to the SM: One on hand on can include a <math>SU(5)</math> singulet which then can generate either Dirac masses or Majorana masses. As in the SM one can also implement the [[Seesaw mechanism|type-I seesaw mechanism]] which then generates naturally light masses.

On the other hand, one can just parametrize the ignorance about neutrinos using the dimension 5 Weinberg operator:
:<math>\mathcal{O}_{W}=(\overline{\mathbf{5}}_F \mathbf{5}_H)\frac{Y_\nu}{\Lambda}(\overline{\mathbf{5}}_F \mathbf{5}_H)+h.c.
</math>
with <math> Y_\nu</math> the <math> 3\times 3</math> [[Yukawa interaction|Yukawa matrix]] required for the mixing between flavours.