Editing Georgi–Glashow model (section)

==Breaking SU(5)==
SU(5) breaking occurs when a [[Scalar field theory|scalar field]] (Which we will denote as <math>\mathbf{24}_H</math>), analogous to the [[Higgs field]] and transforming in the [[adjoint endomorphism|adjoint]] of SU(5), acquires a [[vacuum expectation value]] (vev) proportional to the [[weak hypercharge]] generator 
:<math>\langle \mathbf{24}_H\rangle=v_{24}\operatorname{diag}\left(-1/3, -1/3, -1/3, 1/2, 1/2\right)</math>.
When this occurs, SU(5) is [[spontaneous symmetry breaking|spontaneously broken]] to the [[subgroup]] of SU(5) commuting with the group generated by ''Y''.

Using the embedding from the previous section, we can explicitly check that <math>SU(5)</math> is indeed equal to <math>SU(3)\times SU(2)\times U(1)</math> by noting that <math>[\langle \mathbf{24}_H\rangle,G_\mu]=[\langle \mathbf{24}_H\rangle,W_\mu]=[\langle \mathbf{24}_H\rangle,B_\mu]=0</math>. Computation of similar commutators further shows that all other <math>SU(5)</math> gauge fields acquire masses.
 
To be precise, the unbroken subgroup is actually
:<math>[SU(3)\times SU(2)\times U(1)_Y]/\Z_6.</math>

Under this unbroken subgroup, the adjoint '''24''' transforms as
:<math>\mathbf{24}\rightarrow (8,1)_0\oplus (1,3)_0\oplus (1,1)_0\oplus (3,2)_{-\frac{5}{6}}\oplus (\bar{3},2)_{\frac{5}{6}}</math>
to yield the [[gauge boson]]s of the Standard Model plus the new [[X and Y bosons]]. See [[restricted representation]].

The Standard Model's [[quark]]s and [[lepton]]s fit neatly into representations of SU(5). Specifically, the left-handed [[fermion]]s combine into 3 generations of <math>\ \overline{\mathbf{5}} \oplus\mathbf{10}\oplus\mathbf{1} ~.</math> Under the unbroken subgroup these transform as
:<math>\begin{align}
\overline{\mathbf{5}} &\to (\bar{3},1)_{\tfrac{1}{3}}\oplus (1,2)_{-\tfrac{1}{2}} && \mathrm{d}^\mathsf{c} \mathsf{~ and ~} \ell \\
\mathbf{10} &\to (3,2)_{\tfrac{1}{6}}\oplus (\bar{3},1)_{-\tfrac{2}{3}}\oplus (1,1)_1 && q, \mathrm{u}^\mathsf{c} \mathsf{~ and ~} \mathrm{e}^\mathsf{c} \\
\mathbf{1}  &\to (1,1)_0 && \nu^\mathsf{c}
\end{align} </math> 
to yield precisely the left-handed [[fermion]]ic content of the Standard Model where every [[Generation (particle physics)|generation]] {{math|d}}{{sup|c}}, {{math|u}}{{sup|c}}, {{math|e}}{{sup|c}}, and {{mvar|ν}}{{sup|c}} correspond to anti-[[Standard Model#Particles of Ordinary Matter|down-type quark]], anti-[[Standard Model#Particles of Ordinary Matter|up-type quark]], anti-[[Standard Model#Particles of Ordinary Matter|down-type lepton]], and anti-[[Standard Model#Particles of Ordinary Matter|up-type lepton]], respectively. Also, {{mvar|q}} and <math>\ell</math> correspond to quark and lepton. Fermions transforming as '''1''' under SU(5) are now thought to be necessary because of the evidence for [[neutrino oscillation]]s, unless a way is found to introduce an infinitesimal [[Majorana fermion|Majorana]] coupling for the left-handed neutrinos.

Since the [[homotopy group]] is
:<math>\pi_2\left(\frac{SU(5)}{[SU(3)\times SU(2)\times U(1)_Y]/\Z_6}\right)=\Z</math>,
this model predicts [['t Hooft–Polyakov monopole]]s.

Because the electromagnetic charge {{math|Q}} is a linear combination of some SU(2) generator with {{sfrac|{{math|Y}}|2}}, these monopoles also have quantized magnetic charges {{math|Y}}, where by ''magnetic'', here we mean magnetic electromagnetic charges.