Grand unification theory
Encyclopedia
The term Grand Unified Theory, often abbreviated as GUT, refers to any of several similar candidate models in particle physics
in which at high-energy, the three gauge interactions
of the Standard Model
which define the electromagnetic
, weak
, and strong
interactions
, are merged into one single interaction characterized by one larger gauge symmetry
and thus one unified coupling constant
. In contrast, the experimentally verified Standard Model of particle physics is based on three independent interactions, symmetries and coupling constants.
Models that do not unify all interactions using one simple Lie 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 new particles predicted by models of grand unification cannot be observed directly at particle colliders because their masses are expected to be of the order of the so-called GUT scale, which is predicted to be just a few orders of magnitude below the Planck scale
and thus far beyond the reach of currently foreseen collision experiments. Instead, effects of grand unification might be detected through indirect observations such as proton decay
, electric dipole moments of elementary particles, or the properties of neutrino
s. Some grand unified theories predict the existence of magnetic monopoles.
, all 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. 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.
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.
s of electron
s and proton
s 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.There are however certain constraints on the choice of particle charges from theoretical consistency, in particular anomaly cancellation
. The observed charge quantization, namely the fact that all known elementary particle
s carry electric charges which appear to be exact multiples of 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 interaction
s 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.
which contains the standard model
, and upon which the first Grand Unified Theory was based, is
.
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 (2009) matter particles fit nicely into three copies of the smallest group representation
s of
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
are 
and 
. In the standard assignment, the 
contains the charge conjugates of the right-handed down-type quark
color
triplet
and a left-handed lepton
isospin
doublet, while the
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 not contained in any of these representations, which can explain their relative heaviness (see seesaw mechanism
).
.
Here, the unification of matter is even more complete, since the irreducible
spinor
representation
contains both the 
and 
of 
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 15x15 matrix from the 10+5 representation of SU(5) and adding an extra row and collumn 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 it's new partner in each row and column. These pairs combine to create the familiar 16D Dirac spinor
matrices of SO(10).
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 (grassman-)Jordan algebra
, which has the symmetry group of one of the exceptional Lie groups (
) depending on the details.
Because they are fermions the anti-commutators of the Jordan algebra become commutators. It is known that
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 poses theoretical problems.
called renormalization group running
, which allows parameters with vastly different values at collider energies to converge at much higher energy scales.
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:
.
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 (March 2011). Also, most model builders simply assume supersymmetry
because it solves the hierarchy problem
—i.e., it stabilizes the electroweak Higgs mass
against radiative correction
s.
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
.
, including gravitation
, is termed a theory of everything
. Some common mainstream
GUT models are:
Not quite GUTs:
Note: These models refer to Lie algebra
s not to Lie group
s. 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 defect
s such as monopoles, cosmic strings, domain wall
s, and others. But none have been observed. Their absence is known as the monopole problem in cosmology
. Most GUT models also predict proton decay
, although not the Pati-Salam model
; current experiments still haven't detected proton decay. This experimental limit on the proton's lifetime pretty much rules out minimal SU(5).
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 quark
s with lepton
s, 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.
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.
s indicates that the Standard Model is incomplete and has led to renewed interest toward certain GUT such as
. 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.
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
GeV, which is slightly suggestive. This interesting numerical observation is called the gauge coupling unification, and it works particularly well if one assumes the existence of superpartner
s of the Standard Model particles. Still it is possible to achieve the same by postulating, for instance, that ordinary (non supersymmetric)
models break with an intermediate gauge scale, such as the one of Pati-Salam group.
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
in which at high-energy, the three gauge interactions
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
of the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
which define the electromagnetic
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
, weak
Weak interaction
Weak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...
, and strong
Strong interaction
In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...
interactions
Fundamental interaction
In particle physics, fundamental interactions are the ways that elementary particles interact with one another...
, are merged into one single interaction characterized by one larger gauge symmetry
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
and thus one unified coupling constant
Coupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
. In contrast, the experimentally verified Standard Model of particle physics is based on three independent interactions, symmetries and coupling constants.
Models that do not unify all interactions using one simple Lie group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
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
Theory of everything
A theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle....
(TOE), rather than a GUT. Nevertheless, GUTs are often seen as an intermediate step towards a TOE.
The new particles predicted by models of grand unification cannot be observed directly at particle colliders because their masses are expected to be of the order of the so-called GUT scale, which is predicted to be just a few orders of magnitude below the Planck scale
Planck scale
In particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1019 GeV at which quantum effects of gravity become strong...
and thus far beyond the reach of currently foreseen collision experiments. Instead, effects of grand unification might be detected through indirect observations such as proton decay
Proton decay
In particle physics, proton decay is a hypothetical form of radioactive decay in which the proton decays into lighter subatomic particles, such as a neutral pion and a positron...
, electric dipole moments of elementary particles, or the properties of neutrino
Neutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...
s. Some grand unified theories predict the existence of magnetic monopoles.
, all 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. 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 groupSimple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
SU(5), was proposed by Howard Georgi
Howard Georgi
Howard Mason Georgi III, born January 6, 1947 in San Bernardino, California, is Harvard College Professor and Mallinckrodt Professor of Physics at Harvard University...
and Sheldon Glashow in 1974. The Georgi–Glashow model was preceded by the Semisimple Lie algebra Pati–Salam model by Abdus Salam
Abdus Salam
Mohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk Mohammad Abdus Salam, NI, SPk (Urdu: محمد عبد السلام, pronounced , (January 29, 1926– November 21, 1996) was a Pakistani theoretical physicist and Nobel laureate in Physics for his work on the electroweak unification of the...
and Jogesh Pati
Jogesh Pati
Jogesh C. Pati is an Indian American theoretical physicist at the University of Maryland, College Park.-Biography:...
, who pioneered the idea to unify gauge interactions.
The acronym GUT was first coined in 1978 by CERN researchers John Ellis
John Ellis (physicist)
Jonathan Richard Ellis FRS is a British theoretical physicist who is currently Clerk Maxwell Professor of Theoretical Physics at King's College London. After completing his secondary education at Highgate School, he attended Cambridge University, earning his Ph.D. in theoretical particle physics...
, Andrzej Buras, Mary K. Gaillard, and Dimitri Nanopoulos
Dimitri Nanopoulos
Dimitri Nanopoulos is a Greek physicist. He is one of the most regularly cited researchers in the world, cited more than 35,800 times over across a number of separate branches of science....
, 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 chargeElectric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
s of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s and proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s 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
Strong interaction
In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...
and weak
Weak interaction
Weak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...
interactions within the Standard Model is based on gauge symmetries
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
governed by the simple symmetry groups
Simple Lie group
In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...
SU(3) and SU(2) which allow only discrete charges, the remaining component, the weak hypercharge
Weak hypercharge
The weak hypercharge in particle physics is a conserved quantum number relating the electrical charge and the third component of weak isospin, and is similar to the Gell-Mann–Nishijima formula for the hypercharge of strong interactions...
interaction is described by an abelian symmetry
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
U(1) which in principle allows for arbitrary charge assignments.There are however certain constraints on the choice of particle charges from theoretical consistency, in particular anomaly cancellation
Gauge anomaly
In theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e...
. The observed charge quantization, namely the fact that all known elementary particle
Elementary particle
In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...
s carry electric charges which appear to be exact multiples of of the "elementary" charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
, has led to the idea that hypercharge
Hypercharge
In particle physics, the hypercharge Y of a particle is related to the strong interaction, and is distinct from the similarly named weak hypercharge, which has an analogous role in the electroweak interaction...
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 interaction
Fundamental interaction
In particle physics, fundamental interactions are the ways that elementary particles interact with one another...
s 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
- For an elementary introduction to how Lie algebraLie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s are related to particle physics, see the article Particle physics and representation theoryParticle physics and representation theoryIn physics, the connection between particle physics and representation theory is a natural connection, first noted by Eugene Wigner, between the properties of elementary particles and the representation theory of Lie groups and Lie algebras...
.
SU(5)
SU(5) is the simplest GUT. The smallest simple Lie groupSimple Lie group
In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...
which contains the standard model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
, and upon which the first Grand Unified Theory was based, is
.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 (2009) matter particles fit nicely into three copies of the smallest group representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
s of
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
are and . In the standard assignment, the contains the charge conjugates of the right-handed down-type quarkQuark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
color
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
triplet
Triplet state
A spin triplet is a set of three quantum states of a system, each with total spin S = 1 . The system could consist of a single elementary massive spin 1 particle such as a W or Z boson, or be some multiparticle state with total spin angular momentum of one.In physics, spin is the angular momentum...
and a left-handed lepton
Lepton
A lepton is an elementary particle and a fundamental constituent of matter. The best known of all leptons is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties. Two main classes of leptons exist: charged leptons , and neutral...
isospin
Isospin
In physics, and specifically, particle physics, isospin is a quantum number related to the strong interaction. This term was derived from isotopic spin, but the term is confusing as two isotopes of a nucleus have different numbers of nucleons; in contrast, rotations of isospin maintain the number...
doublet, while the
contains the six up-type quarkQuark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
components, the left-handed down-type quark
Quark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
color
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
triplet, and the right-handed electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
. This scheme has to be replicated for each of the three known generations of matter
Generation (particle physics)
In particle physics, a generation is a division of the elementary particles. Between generations, particles differ by their quantum number and mass, but their interactions are identical....
. It is notable that the theory is anomaly free
Gauge anomaly
In theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e...
with this matter content.
The hypothetical right-handed neutrinos are not contained in any of these representations, which can explain their relative heaviness (see seesaw mechanism
Seesaw mechanism
In theoretical physics, the seesaw mechanism is a mechanism within grand unification theory, and in particular in theories of neutrino masses and neutrino oscillation, where it can be used to explain the smallness of observed neutrino masses relative to those of quarks and leptons.There are several...
).
SO(10)
The next simple Lie group which contains the standard model is.Here, the unification of matter is even more complete, since the irreducible
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
contains both the and of and a right-handed neutrino, and thus the complete particle content of one generation of the extended standard modelStandard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
with neutrino masses
Neutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...
. This is already the largest simple group
Simple Lie group
In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...
which achieves the unification of matter in a scheme involving only the already known matter particles (apart from the Higgs sector
Higgs mechanism
In particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....
).
Since different standard model fermions are grouped together in larger representations, GUTs specifically predict relations among the fermion masses, such as between the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
and the down quark
Down quark
The down quark or d quark is the second-lightest of all quarks, a type of elementary particle, and a major constituent of matter. It, along with the up quark, forms the neutrons and protons of atomic nuclei...
, the muon
Muon
The muon |mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...
and the strange quark
Strange quark
The strange quark or s quark is the third-lightest of all quarks, a type of elementary particle. Strange quarks are found in hadrons, which are subatomic particles. Example of hadrons containing strange quarks include kaons , strange D mesons , Sigma baryons , and other strange particles...
, and the tau lepton
Tau lepton
The tau , also called the tau lepton, tau particle or tauon, is an elementary particle similar to the electron, with negative electric charge and a spin of . Together with the electron, the muon, and the three neutrinos, it is classified as a lepton...
and the bottom quark
Bottom quark
The bottom quark, also known as the beauty quark, is a third-generation quark with a charge of − e. Although all quarks are described in a similar way by the quantum chromodynamics, the bottom quark's large bare mass , combined with low values of the CKM matrix elements Vub and Vcb, gives it a...
for SU(5) and SO(10). Some of these mass relations hold approximately, but most don't (see Georgi-Jarlskog mass relation
Georgi-Jarlskog mass relation
In grand unified theories of the SU or SO type, there is a mass relation predicted between the electron and the down quark, the muon and the strange quark and the tau lepton and the bottom quark called the Georgi-Jarlskog mass relations. The relations were formulated by Howard Georgi and Cecilia...
).
The boson matrix for SO(10) is found by taking the 15x15 matrix from the 10+5 representation of SU(5) and adding an extra row and collumn 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 it's new partner in each row and column. These pairs combine to create the familiar 16D Dirac spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
matrices of SO(10).
E8 and Octonion Representations
It can be noted that a generation of 16 fermions can be put into the form of an OctonionOctonion
In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
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 (grassman-)Jordan algebra
Jordan algebra
In abstract algebra, a Jordan algebra is an algebra over a field whose multiplication satisfies the following axioms:# xy = yx # = x ....
, which has the symmetry group of one of the exceptional Lie groups (
) depending on the details.Because they are fermions the anti-commutators of the Jordan algebra become commutators. It is known that
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 BaryogenesisBaryogenesis
In physical cosmology, baryogenesis is the generic term for hypothetical physical processes that produced an asymmetry between baryons and antibaryons in the very early universe, resulting in the substantial amounts of residual matter that make up the universe today.Baryogenesis theories employ...
), have new undiscovered particles, or have gravity-like (Spin connection
Spin connection
In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the Levi-Civita connection...
) bosons affecting elements of the particles spin direction. Each of these poses theoretical problems.
Unification of forces and the role of supersymmetry
The unification of forces is possible due to the energy scale dependence of parameters in quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
called renormalization group running
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
, which allows parameters with vastly different values at collider energies to converge at much higher energy scales.
The renormalization group
Renormalization group
In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
running of the three gauge couplings in the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
has been found to nearly, but not quite, meet at the same point if the hypercharge
Hypercharge
In particle physics, the hypercharge Y of a particle is related to the strong interaction, and is distinct from the similarly named weak hypercharge, which has an analogous role in the electroweak interaction...
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
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
extension MSSM
Minimal Supersymmetric Standard Model
The Minimal Supersymmetric Standard Model is the minimal extension to the Standard Model that realizes supersymmetry, although non-minimal extensions do exist. Supersymmetry pairs bosons with fermions; therefore every Standard Model particle has a partner that has yet to be discovered...
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:
.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
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
despite the fact that no supersymmetric partner particles have been experimentally observed (March 2011). Also, most model builders simply assume supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
because it solves the hierarchy problem
Hierarchy problem
In theoretical physics, a hierarchy problem occurs when the fundamental parameters of some Lagrangian are vastly different than the parameters measured by experiment. This can happen because measured parameters are related to the fundamental parameters by a prescription known as renormalization...
—i.e., it stabilizes the electroweak Higgs mass
MSSM Higgs Mass
The MSSM Higgs Mass is a prediction of the Minimal Supersymmetric Standard Model. Because the Higgs boson has not yet been found, despite extensive searches, the MSSM has to go to great lengths to make the Higgs sufficiently heavy....
against radiative correction
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
s.
Neutrino masses
Since MajoranaMajorana fermion
In physics, a Majorana fermion is a fermion which is its own anti-particle. The term is used in opposition to Dirac fermion, which describes particles that differ from their antiparticles...
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
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
in those models. In supersymmetric
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
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
Neutrino oscillation
Neutrino oscillation is a quantum mechanical phenomenon predicted by Bruno Pontecorvowhereby a neutrino created with a specific lepton flavor can later be measured to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates...
) via the seesaw mechanism
Seesaw mechanism
In theoretical physics, the seesaw mechanism is a mechanism within grand unification theory, and in particular in theories of neutrino masses and neutrino oscillation, where it can be used to explain the smallness of observed neutrino masses relative to those of quarks and leptons.There are several...
.
Proposed theories
Several such theories have been proposed, but none is currently universally accepted. An even more ambitious theory that includes all fundamental forcesFundamental interaction
In particle physics, fundamental interactions are the ways that elementary particles interact with one another...
, including gravitation
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
, is termed a theory of everything
Theory of everything
A theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle....
. Some common mainstream
Mainstream
Mainstream is, generally, the common current thought of the majority. However, the mainstream is far from cohesive; rather the concept is often considered a cultural construct....
GUT models are:
- minimal left-right model —
- Georgi–Glashow model —
- SO(10)SO(10) (physics)In particle physics, one of the grand unified theories is based on the SO Lie group....
- Flipped SU(5)Flipped SU(5)The Flipped SU model is a Grand Unified Theory theory first contemplated by Stephen Barr in 1982, and by Dimitri Nanopoulos and others in 1984...
—
- Pati-Salam modelPati-Salam modelIn physics, the Pati–Salam model is a Grand Unification Theory was proposed in 1974 by nobel laureate Abdus Salam and Jogesh Pati. The unification is based on there being four quark color charges, dubbed red, green, blue and violet , instead of the conventional three, with the new "violet" quark...
—
- flipped SO(10) —
- Trinification —
- SU(6)
- E6
- 331 model331 modelThe 331 model in particle physics offers an explanation of why there must exist three families of quarks and leptons. One curious feature of the Standard Model is that the anomaly cancels exactly, for each quark-lepton family, of which we know three...
- chiral colorChiral colorIn particle physics phenomenology, chiral color is a speculative model which extends quantum chromodynamics , the generally accepted theory for the strong interactions of quarks...
Not quite GUTs:
- Technicolor modelsTechnicolor (physics)Technicolor theories are models of physics beyond the standard model that address electroweak symmetry breaking, the mechanism through which elementary particles acquire masses...
- Little HiggsLittle HiggsIn particle physics, little Higgs models are based on the idea that the Higgs boson is a pseudo-Goldstone boson arising from some global symmetry breaking at a TeV energy scale...
- PreonPreonIn particle physics, preons are postulated "point-like" particles, conceived to be subcomponents of quarks and leptons. The word was coined by Jogesh Pati and Abdus Salam in 1974...
s - String theoryString theoryString theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
- M-theoryM-theoryIn theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...
- Loop quantum gravityLoop quantum gravityLoop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...
- Causal dynamical triangulation theory
Note: These models refer to Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s not to Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s. 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
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
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)
Flipped SU(5)
The Flipped SU model is a Grand Unified Theory theory first contemplated by Stephen Barr in 1982, and by Dimitri Nanopoulos and others in 1984...
and the Pati-Salam model
Pati-Salam model
In physics, the Pati–Salam model is a Grand Unification Theory was proposed in 1974 by nobel laureate Abdus Salam and Jogesh Pati. The unification is based on there being four quark color charges, dubbed red, green, blue and violet , instead of the conventional three, with the new "violet" quark...
. 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 defect
Topological defect
In mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a quantum field theory homotopically distinct from the vacuum solution; it can be proven to exist because the boundary conditions entail the existence of...
s such as monopoles, cosmic strings, domain wall
Domain wall
A domain wall is a term used in physics which can have one of two distinct but similar meanings in magnetism, optics, or string theory. These phenomena can all be generically described as topological solitons which occur whenever a discrete symmetry is spontaneously broken.-Magnetism:In magnetism,...
s, and others. But none have been observed. Their absence is known as the monopole problem in cosmology
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
. Most GUT models also predict proton decay
Proton decay
In particle physics, proton decay is a hypothetical form of radioactive decay in which the proton decays into lighter subatomic particles, such as a neutral pion and a positron...
, although not the Pati-Salam model
Pati-Salam model
In physics, the Pati–Salam model is a Grand Unification Theory was proposed in 1974 by nobel laureate Abdus Salam and Jogesh Pati. The unification is based on there being four quark color charges, dubbed red, green, blue and violet , instead of the conventional three, with the new "violet" quark...
; current experiments still haven't detected proton decay. This experimental limit on the proton's lifetime pretty much rules out minimal SU(5).
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
Color charge
In particle physics, color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics . Color charge has analogies with the notion of electric charge of particles, but because of the mathematical complications of QCD,...
Higgs triplet field with a very small mass (many orders of magnitude smaller than the GUT scale here). In theory, unifying quark
Quark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
s with lepton
Lepton
A lepton is an elementary particle and a fundamental constituent of matter. The best known of all leptons is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties. Two main classes of leptons exist: charged leptons , and neutral...
s, 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 basically consists of a gauge group which is a compact Lie group, a connection formConnection form
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms....
for that Lie group, a Yang-Mills action for that connection given by an invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
symmetric bilinear form
Symmetric bilinear form
A symmetric bilinear form is a bilinear form on a vector space that is symmetric. Symmetric bilinear forms are of great importance in the study of orthogonal polarity and quadrics....
over its Lie algebra (which is specified by a coupling constant
Coupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
for each factor), a Higgs sector
Higgs sector
In particle physics, the Higgs sector is the collection of quantum fields and/or particles that are responsible for the Higgs mechanism i.e. for the spontaneous symmetry breaking. The word "sector" refers to a subgroup of the total set of fields and particles...
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
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
to the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
. The Weyl fermions represent matter.
Current status
, there is still no hard evidence that nature is described by a Grand Unified Theory. Moreover, since the Higgs particle has not yet been observed, the smaller electroweak unification is still pending. The discovery of neutrino oscillationNeutrino oscillation
Neutrino oscillation is a quantum mechanical phenomenon predicted by Bruno Pontecorvowhereby a neutrino created with a specific lepton flavor can later be measured to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates...
s indicates that the Standard Model is incomplete and has led to renewed interest toward certain GUT such as
. One of the few possible experimental tests of certain GUT is proton decayProton decay
In particle physics, proton decay is a hypothetical form of radioactive decay in which the proton decays into lighter subatomic particles, such as a neutral pion and a positron...
and also fermion masses. There are a few more special tests for supersymmetric GUT.
The gauge coupling strengths of QCD
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
, the weak interaction
Weak interaction
Weak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...
and hypercharge
Hypercharge
In particle physics, the hypercharge Y of a particle is related to the strong interaction, and is distinct from the similarly named weak hypercharge, which has an analogous role in the electroweak interaction...
seem to meet at a common length scale called the GUT scale and equal approximately to
GeV, which is slightly suggestive. This interesting numerical observation is called the gauge coupling unification, and it works particularly well if one assumes the existence of superpartnerSuperpartner
In particle physics, a superpartner is a hypothetical elementary particle. Supersymmetry is one of the synergistic theories in current high-energy physics which predicts the existence of these "shadow" particles....
s of the Standard Model particles. Still it is possible to achieve the same by postulating, for instance, that ordinary (non supersymmetric)
models break with an intermediate gauge scale, such as the one of Pati-Salam group.See also
- Paradigm shiftParadigm shiftA Paradigm shift is, according to Thomas Kuhn in his influential book The Structure of Scientific Revolutions , a change in the basic assumptions, or paradigms, within the ruling theory of science...
- Classical unified field theoriesClassical unified field theoriesSince the 19th century, some physicists have attempted to develop a single theoretical framework that can account for the fundamental forces of nature – a unified field theory. Classical unified field theories are attempts to create a unified field theory based on classical physics...
- X and Y bosonsX and Y bosonsIn particle physics, the X and Y bosons are hypothetical elementary particles analogous to the W and Z bosons, but corresponding to a new type of force predicted by the Georgi–Glashow model, a grand unified theory.-Details:The X and Y bosons couple quarks to leptons, allowing violation of the...
- B-LB-LIn high energy physics, B − L is the difference between the baryon number and the lepton number .-Details:...
quantum number
Further reading
- Stephen HawkingStephen HawkingStephen William Hawking, CH, CBE, FRS, FRSA is an English theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity...
, A Brief History of TimeA Brief History of TimeA Brief History of Time is a popular science book written by renown physicist Stephen Hawking and first published by the Bantam Dell Publishing Group in 1988. It became a best-seller and has sold more than 10 million copies...
, includes a brief popular overview.