In particle physics, the Georgi–Glashow model[1] is a particular Grand Unified Theory (GUT) proposed by Howard Georgi and Sheldon Glashow in 1974. In this model, the Standard Model gauge groups SU(3) × SU(2) × U(1) are combined into a single simple gauge group SU(5). The unified group SU(5) is then thought to be spontaneously broken into the Standard Model subgroup below a very high energy scale called the grand unification scale.
Since the Georgi–Glashow model combines leptons and quarks into single irreducible representations, there exist interactions which do not conserve baryon number, although they still conserve the quantum number associated with the symmetry of the common representation. This yields a mechanism for proton decay, and the rate of proton decay can be predicted from the dynamics of the model. However, proton decay has not yet been observed experimentally, and the resulting lower limit on the lifetime of the proton contradicts the predictions of this model. Nevertheless, the elegance of the model has led particle physicists to use it as the foundation for more complex models which yield longer proton lifetimes, particularly SO(10) in basic and SUSY variants.
(For a more elementary introduction to how the representation theory of Lie algebras are related to particle physics, see the article Particle physics and representation theory.)
Also, this model suffers from the doublet–triplet splitting problem.
SU(5) acts on
C5
\wedgeC5
C2 ⊕ C3
\begin{matrix} \phi:&U(1) x SU(2) x SU(3)&\longrightarrow&S(U(2) x U(3))\subsetSU(5)\\ &(\alpha,g,h)&\longmapsto& \begin{pmatrix} \alpha3g&0\\ 0&\alpha-2h \end{pmatrix}\\ \end{matrix}
\{(\alpha,\alpha-3Id2,\alpha2Id3)|\alpha\inC,\alpha6=1\}\congZ6
SU(3) x SU(2) x U(1)/Z6
{stylewedge}0C5
C0 ⊗ C ⊗ C
{stylewedge}1C5\congC5
C5\congC2 ⊕ C3
C2
C | ||||
|
⊗ C2* ⊗ C
C2\congC2*
C3
C | ||||
|
⊗ C ⊗ C3
The second power
{stylewedge}2C5
{stylewedge}2(V ⊕ W)={stylewedge}2V2 ⊕ (V ⊗ W) ⊕ {stylewedge}2V2
C5
{stylewedge}pC5\cong({stylewedge}5-pC5)*
\wedgeC5
Similar motivations apply to the Pati–Salam model, and to SO(10), E6, and other supergroups of SU(5).
Owing to its relatively simple gauge group
SU(5)
\overline{5
10
\overline{5
10F=\begin{pmatrix} 0&u
c&u | |
1&d |
1\\ -u
c&u | |
2&d |
2\\ u
c&0&u | |
3&d |
3\\ -u1&-u2&-u3&0&eR\\ -d1&-d2&-d3&-eR&0 \end{pmatrix}
di
ui
c | |
d | |
i |
c | |
u | |
i |
\nu
e
eR
In addition to the fermions, we need to break
SU(3) x SUL(2) x UY(1) → SU(3) x UEM(1)
5
5H=(T1,T2,T
+,H | |
3,H |
0)T
H+
H0
Ti
The SM gauge fields can be embedded explicitly as well. For that we recall a gauge field transforms as an adjoint, and thus can be written as
a | |
A | |
\mu |
Ta
Ta
SU(5)
3 x 3
2 x 2
a | |
\begin{pmatrix}G | |
\mu |
a | |
T | |
3&0\\0&0\end{pmatrix} |
SU(3)
\begin{pmatrix}0&0\\0& | \sigmaa |
2 |
a | |
W | |
\mu\end{pmatrix} |
SU(2)
0 | |
NB | |
\mu\operatorname{diag}\left(-1/3, |
-1/3,-1/3,1/2,1/2\right)
U(1)
N
This explicit embedding can be found in Ref.[2] or in the original paper by Georgi and Glashow.
SU(5) breaking occurs when a scalar field (Which we will denote as
24H
\langle24H\rangle=v24\operatorname{diag}\left(-1/3,-1/3,-1/3,1/2,1/2\right)
Using the embedding from the previous section, we can explicitly check that
SU(5)
SU(3) x SU(2) x U(1)
[\langle24H\rangle,G\mu]=[\langle24H\rangle,W\mu]=[\langle24H\rangle,B\mu]=0
SU(5)
[SU(3) x SU(2) x U(1)Y]/\Z6.
Under this unbroken subgroup, the adjoint 24 transforms as
24 → (8,1)0 ⊕ (1,3)0 ⊕ (1,1)0 ⊕
(3,2) | ||||
|
⊕
(\bar{3},2) | ||||
|
The Standard Model's quarks and leptons fit neatly into representations of SU(5). Specifically, the left-handed fermions combine into 3 generations of
\overline{5
\begin{align} \overline{5
\ell
Since the homotopy group is
\pi | ||||
|
\right)=\Z
Because the electromagnetic charge is a linear combination of some SU(2) generator with, these monopoles also have quantized magnetic charges, where by magnetic, here we mean magnetic electromagnetic charges.
The minimal supersymmetric SU(5) model assigns a
\Z2
\Z2
As complex representations:
label | description | multiplicity | SU(5) rep | \Z2 | |
---|---|---|---|---|---|
1 | 24 | + | |||
1 | 5 | + | |||
1 | \overline{5 | + | |||
\overline{5 | 3 | \overline{5 | - | ||
10 | 3 | 10 | - | ||
(fractal) | 1 | - |
A generic invariant renormalizable superpotential is a (complex)
SU(5) x \Z2
\begin{matrix} \Phi2&&
A | |
\Phi | |
B |
B | |
\Phi | |
A |
\\[4pt] \Phi3&&
A | |
\Phi | |
B |
B | |
\Phi | |
C |
C | |
\Phi | |
A |
\\[4pt] Hd Hu&&{Hd
The first column is an Abbreviation of the second column (neglecting proper normalization factors), where capital indices are SU(5) indices, and and are the generation indices.
The last two rows presupposes the multiplicity of
Nc
Hu 10i 10j
c | |
N | |
j |
The vacua correspond to the mutual zeros of the and terms. Let's first look at the case where the VEVs of all the chiral fields are zero except for .
W=Tr\left[a\Phi2+b\Phi3\right]
The zeros corresponds to finding the stationary points of subject to the traceless constraint
Tr[\Phi]=0~.
2a\Phi+3b\Phi2=λ1 ,
Up to an SU(5) (unitary) transformation,
\Phi=\begin{cases} \operatorname{diag}(0,0,0,0,0)\\ \operatorname{diag}(
2a | , | |
9b |
2a | , | |
9b |
2a | , | |
9b |
2a | ,- | |
9b |
8a | |
9b |
)\\ \operatorname{diag}(
4a | , | |
3b |
4a | , | |
3b |
4a | ,- | |
3b |
2a | ,- | |
b |
2a | |
b |
) \end{cases}
The three cases are called case I, II, and III and they break the gauge symmetry into
SU(5), \left[SU(4) x U(1)\right]/\Z4
\left[SU(3) x SU(2) x U(1)\right]/\Z6
In other words, there are at least three different superselection sections, which is typical for supersymmetric theories.
Only case III makes any phenomenological sense and so, we will focus on this case from now onwards.
It can be verified that this solution together with zero VEVs for all the other chiral multiplets is a zero of the F-terms and D-terms. The matter parity remains unbroken (right up to the TeV scale).
The gauge algebra 24 decomposes as
\begin{pmatrix}(8,1)0\\(1,3)0\\(1,1)0\\(3,2)
|
\\(\bar{3},2) | ||||
|
\end{pmatrix}~.
This 24 is a real representation, so the last two terms need explanation. Both
(3,2) | ||||
|
(\bar{3},2) | ||||
|
(3,2) | ||||
|
(\bar{3},2) | ||||
|
(8,1)0,(1,3)0
(1,1)0
a\Phi2+b<\Phi>\Phi2~.
The sterile neutrinos, if any exist, would also acquire a GUT scale Majorana mass coming from the superpotential coupling .
Because of matter parity, the matter representations
\overline{5
It is the Higgs fields 5 and
\overline{5
5H | \bar{5}H | ||
\begin{pmatrix} (3,1)-\tfrac{1{3}}\\ (1,2)\tfrac{1{2}} \end{pmatrix} | \begin{matrix} \\\\\ ??? \end{matrix} | \begin{pmatrix} (\bar{3},1)\tfrac{1{3}}\\ (1,2)-\tfrac{1{2}} \end{pmatrix} |
The two relevant superpotential terms here are
5H \bar{5}H
\langle24\rangle5H \bar{5}H~.
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)−5/6 bifundamental representations, 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.
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+ boson which decays to a right-handed (or left-handed) positron and a left-handed (or right-handed) anti-down quark:
uL+uR\toX+\to
+ | |
e | |
R |
+\bar{d
uL+uR\toX+\to
+ | |
e | |
L |
+\bar{d
This process conserves weak isospin, weak hypercharge, and color. GUTs equate anti-color with having two colors,
\bar{g}\equivrb ,
Since SM states are regrouped into
SU(5)
Yd=
T | |
Y | |
e |
and Yu=
T | |
Y | |
u |
In particular this predicts
me,\mu\tau ≈ md,s,b
As mentioned in the above section the colour triplet of the
{5
lL\supset{5
SU(5)
24H ,
v24
{5
H
T
mT>1012 GeV
\mu
100 GeV
a
b :
\mu
mT
This is known as the doublet–triplet (DT) splitting problem: In order to be consistent we have to 'split' the 'masses' of
T
H ,
a
b~.
A review of the DT splitting problem can be found in.
As the SM the original Georgi–Glashow model proposed in 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
SU(5)
On the other hand, on can just parametrize the ignorance about neutrinos using the dimension 5 Weinbergoperator:
l{O}W=(\overline{5
Y\nu
3 x 3