A Grand Unified Theory (GUT) is a model in
particle physics in which, at high energy,
the three gauge interactions of the Standard
Model which define the electromagnetic, weak,
and strong interactions, or forces, are merged
into one single force. This unified interaction
is characterized by one larger gauge symmetry
and thus several force carriers, but one unified
coupling constant. If Grand Unification is
realized in nature, there is the possibility
of a grand unification epoch in the early
universe in which the fundamental forces are
not yet distinct.
Models that do not unify all interactions
using one simple group as the gauge symmetry,
but do so using semisimple groups, can exhibit
similar properties and are sometimes referred
to as Grand Unified Theories as well.
Unifying gravity with the other three interactions
would provide a theory of everything (TOE),
rather than a GUT. Nevertheless, GUTs are
often seen as an intermediate step towards
a TOE.
The novel particles predicted by GUT models
are expected to have masses around the GUT
scale—just a few orders of magnitude below
the Planck scale—and so will be well beyond
the reach of any foreseen particle collider
experiments. Therefore, the particles predicted
by GUT models will be unable to be observed
directly and instead the effects of grand
unification might be detected through indirect
observations such as proton decay, electric
dipole moments of elementary particles, or
the properties of neutrinos. Some GUTs, such
as the Pati-Salam model, predict the existence
of magnetic monopoles.
GUT models which aim to be completely realistic
are quite complicated, even compared to the
Standard Model, because they need to introduce
additional fields and interactions, or even
additional dimensions of space. The main reason
for this complexity lies in the difficulty
of reproducing the observed fermion masses
and mixing angles which may be related to
an existence of some additional family symmetries
beyond the conventional GUT models. Due to
this difficulty, and due to the lack of any
observed effect of grand unification so far,
there is no generally accepted GUT model.
== History ==
Historically, the first true GUT which was
based on the simple Lie group SU(5), was proposed
by Howard Georgi and Sheldon Glashow in 1974.
The Georgi–Glashow model was preceded by
the semisimple Lie algebra Pati–Salam model
by Abdus Salam and Jogesh Pati, who pioneered
the idea to unify gauge interactions.
The acronym GUT was first coined in 1978 by
CERN researchers John Ellis, Andrzej Buras,
Mary K. Gaillard, and Dimitri Nanopoulos,
however in the final version of their paper
they opted for the less anatomical GUM (Grand
Unification Mass). Nanopoulos later that year
was the first to use the acronym in a paper.
== Motivation ==
The fact that the electric charges of electrons
and protons seem to cancel each other exactly
to extreme precision is essential for the
existence of the macroscopic world as we know
it, but this important property of elementary
particles is not explained in the Standard
Model of particle physics. While the description
of strong and weak interactions within the
Standard Model is based on gauge symmetries
governed by the simple symmetry groups SU(3)
and SU(2) which allow only discrete charges,
the remaining component, the weak hypercharge
interaction is described by an abelian symmetry
U(1) which in principle allows for arbitrary
charge assignments. The observed charge quantization,
namely the fact that all known elementary
particles carry electric charges which appear
to be exact multiples of ⅓ of the "elementary"
charge, has led to the idea that hypercharge
interactions and possibly the strong and weak
interactions might be embedded in one Grand
Unified interaction described by a single,
larger simple symmetry group containing the
Standard Model. This would automatically predict
the quantized nature and values of all elementary
particle charges. Since this also results
in a prediction for the relative strengths
of the fundamental interactions which we observe,
in particular the weak mixing angle, Grand
Unification ideally reduces the number of
independent input parameters, but is also
constrained by observations.
Grand Unification is reminiscent of the unification
of electric and magnetic forces by Maxwell's
theory of electromagnetism in the 19th century,
but its physical implications and mathematical
structure are qualitatively different.
== Unification of matter particles ==
=== SU(5) ===
SU(5) is the simplest GUT. The smallest simple
Lie group which contains the standard model,
and upon which the first Grand Unified Theory
was based, is
S
U
(
5
)
⊃
S
U
(
3
)
×
S
U
(
2
)
×
U
(
1
)
{\displaystyle SU(5)\supset SU(3)\times SU(2)\times
U(1)}
.Such group symmetries allow the reinterpretation
of several known particles as different states
of a single particle field. However, it is
not obvious that the simplest possible choices
for the extended "Grand Unified" symmetry
should yield the correct inventory of elementary
particles. The fact that all currently known
matter particles fit nicely into three copies
of the smallest group representations of SU(5)
and immediately carry the correct observed
charges, is one of the first and most important
reasons why people believe that a Grand Unified
Theory might actually be realized in nature.
The two smallest irreducible representations
of SU(5) are 5 and 10. In the standard assignment,
the 5 contains the charge conjugates of the
right-handed down-type quark color triplet
and a left-handed lepton isospin doublet,
while the 10 contains the six up-type quark
components, the left-handed down-type quark
color triplet, and the right-handed electron.
This scheme has to be replicated for each
of the three known generations of matter.
It is notable that the theory is anomaly free
with this matter content.
The hypothetical right-handed neutrinos are
a singlet of SU(5), which means its mass is
not forbidden by any symmetry; it doesn't
need a spontaneous symmetry breaking which
explains why its mass would be heavy. (see
seesaw mechanism).
=== SO(10) ===
The next simple Lie group which contains the
standard model is
S
O
(
10
)
⊃
S
U
(
5
)
⊃
S
U
(
3
)
×
S
U
(
2
)
×
U
(
1
)
{\displaystyle SO(10)\supset SU(5)\supset
SU(3)\times SU(2)\times U(1)}
.Here, the unification of matter is even more
complete, since the irreducible spinor representation
16 contains both the 5 and 10 of SU(5) and
a right-handed neutrino, and thus the complete
particle content of one generation of the
extended standard model with neutrino masses.
This is already the largest simple group which
achieves the unification of matter in a scheme
involving only the already known matter particles
(apart from the Higgs sector).
Since different standard model fermions are
grouped together in larger representations,
GUTs specifically predict relations among
the fermion masses, such as between the electron
and the down quark, the muon and the strange
quark, and the tau lepton and the bottom quark
for SU(5) and SO(10). Some of these mass relations
hold approximately, but most don't (see Georgi-Jarlskog
mass relation).
The boson matrix for SO(10) is found by taking
the 15 × 15 matrix from the 10 + 5 representation
of SU(5) and adding an extra row and column
for the right-handed neutrino. The bosons
are found by adding a partner to each of the
20 charged bosons (2 right-handed W bosons,
6 massive charged gluons and 12 X/Y type bosons)
and adding an extra heavy neutral Z-boson
to make 5 neutral bosons in total. The boson
matrix will have a boson or its new partner
in each row and column. These pairs combine
to create the familiar 16D Dirac spinor matrices
of SO(10).
=== Extended Grand Unified Theories ===
Non-chiral extensions of the Standard Model
with vectorlike split-multiplet particle spectra
which naturally appear in the higher SU(N)
GUTs considerably modify the desert physics
and lead to the realistic (string-scale) grand
unification for conventional three quark-lepton
families even without using supersymmetry
(see below). On the other hand, due to a new
missing VEV mechanism emerging in the supersymmetric
SU(8) GUT the simultaneous solution to the
gauge hierarchy (doublet-triplet splitting)
problem and problem of unification of flavor
can be found.GUTs with four families / generations,
SU(8): Assuming 4 generations of fermions
instead of 3 makes a total of 64 types of
particles. These can be put into 64 = 8 +
56 representations of SU(8). This can be divided
into SU(5) × SU(3)F × U(1) which is the
SU(5) theory together with some heavy bosons
which act on the generation number.
GUTs with four families / generations, O(16):
Again assuming 4 generations of fermions,
the 128 particles and anti-particles can be
put into a single spinor representation of
O(16).
=== Symplectic groups and quaternion representations
===
Symplectic gauge groups could also be considered.
For example, Sp(8) (which is called Sp(4)
in the article symplectic group) has a representation
in terms of 4 × 4 quaternion unitary matrices
which has a 16 dimensional real representation
and so might be considered as a candidate
for a gauge group. Sp(8) has 32 charged bosons
and 4 neutral bosons. Its subgroups include
SU(4) so can at least contain the gluons and
photon of SU(3) × U(1). Although it's probably
not possible to have weak bosons acting on
chiral fermions in this representation. A
quaternion representation of the fermions
might be:
[
e
+
i
e
¯
+
j
v
+
k
v
¯
u
r
+
i
u
r
¯
+
j
d
r
+
k
d
r
¯
u
g
+
i
u
g
¯
+
j
d
g
+
k
d
g
¯
u
b
+
i
u
b
¯
+
j
d
b
+
k
d
b
¯
]
L
{\displaystyle {\begin{bmatrix}e+i{\overline
{e}}+jv+k{\overline {v}}\\u_{r}+i{\overline
{u_{r}}}+jd_{r}+k{\overline {d_{r}}}\\u_{g}+i{\overline
{u_{g}}}+jd_{g}+k{\overline {d_{g}}}\\u_{b}+i{\overline
{u_{b}}}+jd_{b}+k{\overline {d_{b}}}\\\end{bmatrix}}_{L}}
A further complication with quaternion representations
of fermions is that there are two types of
multiplication: left multiplication and right
multiplication which must be taken into account.
It turns out that including left and right-handed
4 × 4 quaternion matrices is equivalent to
including a single right-multiplication by
a unit quaternion which adds an extra SU(2)
and so has an extra neutral boson and two
more charged bosons. Thus the group of left-
and right-handed 4 × 4 quaternion matrcies
is Sp(8) × SU(2) which does include the standard
model bosons:
S
U
(
4
,
H
)
L
×
H
R
=
S
p
(
8
)
×
S
U
(
2
)
⊃
S
U
(
4
)
×
S
U
(
2
)
⊃
S
U
(
3
)
×
S
U
(
2
)
×
U
(
1
)
{\displaystyle SU(4,H)_{L}\times H_{R}=Sp(8)\times
SU(2)\supset SU(4)\times SU(2)\supset SU(3)\times
SU(2)\times U(1)}
If
ψ
{\displaystyle \psi }
is a quaternion valued spinor,
A
μ
a
b
{\displaystyle A_{\mu }^{ab}}
is quaternion hermitian 4 × 4 matrix coming
from Sp(8) and
B
μ
{\displaystyle B_{\mu }}
is a pure imaginary quaternion (both of which
are 4-vector bosons) then the interaction
term is:
ψ
a
¯
γ
μ
(
A
μ
a
b
ψ
b
+
ψ
a
B
μ
)
{\displaystyle {\overline {\psi ^{a}}}\gamma
_{\mu }\left(A_{\mu }^{ab}\psi ^{b}+\psi ^{a}B_{\mu
}\right)}
=== Octonion representations ===
It can be noted that a generation of 16 fermions
can be put into the form of an octonion with
each element of the octonion being an 8-vector.
If the 3 generations are then put in a 3x3
hermitian matrix with certain additions for
the diagonal elements then these matrices
form an exceptional (Grassmann-) Jordan algebra,
which has the symmetry group of one of the
exceptional Lie groups (F4, E6, E7 or E8)
depending on the details.
ψ
=
[
a
e
μ
e
¯
b
τ
μ
¯
τ
¯
c
]
{\displaystyle \psi ={\begin{bmatrix}a&e&\mu
\\{\overline {e}}&b&\tau \\{\overline {\mu
}}&{\overline {\tau }}&c\end{bmatrix}}}
[
ψ
A
,
ψ
B
]
⊂
J
3
(
O
)
{\displaystyle [\psi _{A},\psi _{B}]\subset
J_{3}(O)}
Because they are fermions the anti-commutators
of the Jordan algebra become commutators.
It is known that E6 has subgroup O(10) and
so is big enough to include the Standard Model.
An E8 gauge group, for example, would have
8 neutral bosons, 120 charged bosons and 120
charged anti-bosons. To account for the 248
fermions in the lowest multiplet of E8, these
would either have to include anti-particles
(and so have baryogenesis), have new undiscovered
particles, or have gravity-like (spin connection)
bosons affecting elements of the particles
spin direction. Each of these possess theoretical
problems.
=== Beyond Lie groups ===
Other structures have been suggested including
Lie 3-algebras and Lie superalgebras. Neither
of these fit with Yang–Mills theory. In
particular Lie superalgebras would introduce
bosons with the wrong statistics. Supersymmetry
however does fit with Yang–Mills. For example,
N=4 Super Yang Mills Theory requires an SU(N)
gauge group.
== Unification of forces and the role of supersymmetry
==
The unification of forces is possible due
to the energy scale dependence of force coupling
parameters in quantum field theory called
renormalization group running, which allows
parameters with vastly different values at
usual energies to converge to a single value
at a much higher energy scale.The renormalization
group running of the three gauge couplings
in the Standard Model has been found to nearly,
but not quite, meet at the same point if the
hypercharge is normalized so that it is consistent
with SU(5) or SO(10) GUTs, which are precisely
the GUT groups which lead to a simple fermion
unification. This is a significant result,
as other Lie groups lead to different normalizations.
However, if the supersymmetric extension MSSM
is used instead of the Standard Model, the
match becomes much more accurate. In this
case, the coupling constants of the strong
and electroweak interactions meet at the grand
unification energy, also known as the GUT
scale:
Λ
GUT
≈
10
16
GeV
{\displaystyle \Lambda _{\text{GUT}}\approx
10^{16}\,{\text{GeV}}}
.It is commonly believed that this matching
is unlikely to be a coincidence, and is often
quoted as one of the main motivations to further
investigate supersymmetric theories despite
the fact that no supersymmetric partner particles
have been experimentally observed. Also, most
model builders simply assume supersymmetry
because it solves the hierarchy problem—i.e.,
it stabilizes the electroweak Higgs mass against
radiative corrections.
== Neutrino masses ==
Since Majorana masses of the right-handed
neutrino are forbidden by SO(10) symmetry,
SO(10) GUTs predict the Majorana masses of
right-handed neutrinos to be close to the
GUT scale where the symmetry is spontaneously
broken in those models. In supersymmetric
GUTs, this scale tends to be larger than would
be desirable to obtain realistic masses of
the light, mostly left-handed neutrinos (see
neutrino oscillation) via the seesaw mechanism.
== Proposed theories ==
Several theories have been proposed, but none
is currently universally accepted. An even
more ambitious theory that includes all fundamental
forces, including gravitation, is termed a
theory of everything. Some common mainstream
GUT models are:
Not quite GUTs:
Note: These models refer to Lie algebras not
to Lie groups. The Lie group could be [SU(4)
× SU(2) × SU(2)]/Z2, just to take a random
example.
The most promising candidate is SO(10). (Minimal)
SO(10) does not contain any exotic fermions
(i.e. additional fermions besides the Standard
Model fermions and the right-handed neutrino),
and it unifies each generation into a single
irreducible representation. A number of other
GUT models are based upon subgroups of SO(10).
They are the minimal left-right model, SU(5),
flipped SU(5) and the Pati–Salam model.
The GUT group E6 contains SO(10), but models
based upon it are significantly more complicated.
The primary reason for studying E6 models
comes from E8 × E8 heterotic string theory.
GUT models generically predict the existence
of topological defects such as monopoles,
cosmic strings, domain walls, and others.
But none have been observed. Their absence
is known as the monopole problem in cosmology.
Many GUT models also predict proton decay,
although not the Pati–Salam model; proton
decay has never been observed by experiments.
The minimal experimental limit on the proton's
lifetime pretty much rules out minimal SU(5)
and heavily constrains the other models. The
lack of detected supersymmetry to date also
constrains many models.
Proton Decay. These graphics refer to the
X bosons and Higgs bosons.
Some GUT theories like SU(5) and SO(10) suffer
from what is called the doublet-triplet problem.
These theories predict that for each electroweak
Higgs doublet, there is a corresponding colored
Higgs triplet field with a very small mass
(many orders of magnitude smaller than the
GUT scale here). In theory, unifying quarks
with leptons, the Higgs doublet would also
be unified with a Higgs triplet. Such triplets
have not been observed. They would also cause
extremely rapid proton decay (far below current
experimental limits) and prevent the gauge
coupling strengths from running together in
the renormalization group.
Most GUT models require a threefold replication
of the matter fields. As such, they do not
explain why there are three generations of
fermions. Most GUT models also fail to explain
the little hierarchy between the fermion masses
for different generations.
== Ingredients ==
A GUT model consists of a gauge group which
is a compact Lie group, a connection form
for that Lie group, a Yang–Mills action
for that connection given by an invariant
symmetric bilinear form over its Lie algebra
(which is specified by a coupling constant
for each factor), a Higgs sector consisting
of a number of scalar fields taking on values
within real/complex representations of the
Lie group and chiral Weyl fermions taking
on values within a complex rep of the Lie
group. The Lie group contains the Standard
Model group and the Higgs fields acquire VEVs
leading to a spontaneous symmetry breaking
to the Standard Model. The Weyl fermions represent
matter.
== Current status ==
There is currently no hard evidence that nature
is described by a Grand Unified Theory. The
discovery of neutrino oscillations indicates
that the Standard Model is incomplete and
has led to renewed interest toward certain
GUT such as SO(10). One of the few possible
experimental tests of certain GUT is proton
decay and also fermion masses. There are a
few more special tests for supersymmetric
GUT. However, minimum proton lifetimes from
research (at or exceeding the 1034-1035 year
range) have ruled out simpler GUTs and most
non-SUSY models. The maximum upper limit on
proton lifetime (if unstable), is calculated
at 6 x 1039 years for SUSY models and 1.4
x 1036 years for minimal non-SUSY GUTs.The
gauge coupling strengths of QCD, the weak
interaction and hypercharge seem to meet at
a common length scale called the GUT scale
and equal approximately to 1016 GeV (slightly
less than the Planck energy of 1019 GeV),
which is somewhat suggestive. This interesting
numerical observation is called the gauge
coupling unification, and it works particularly
well if one assumes the existence of superpartners
of the Standard Model particles. Still it
is possible to achieve the same by postulating,
for instance, that ordinary (non supersymmetric)
SO(10) models break with an intermediate gauge
scale, such as the one of Pati–Salam group.
== See also ==
Paradigm shift
Classical unified field theories
X and Y bosons
B − L quantum number
== Notes
