SO(10) (physics)
Encyclopedia
In particle physics
, one of the grand unified theories (GUT) is based on the SO(10) Lie group
. (The Lie group
involved is not really the special orthogonal group SO(10), but rather its double cover Spin(10); but calling it SO(10) is the standard convention.)
Before SU(5)
, Harald Fritzsch and Peter Minkowski and independently Howard Georgi
found that all the matter contents are incorporated into a single representation, spinorial 16 of SO(10). (Historical note: the before in the previous sentence is misleading: Georgi found the SO(10) theory a few hours before finding SU(5) at the end of 1973.)
If the hypercharge
is contained within SU(5), this is the conventional Georgi-Glashow model
, with the
as the matter fields, the 
as the electroweak Higgs field and the 
within the 
as the GUT Higgs field. The superpotential
may then include renormalizable terms of the form
; 
; 
, 
and 
. The first three are responsible to the gauge symmetry breaking at low energies and give the Higgs
mass, and the latter two give the matter particles masses and their Yukawa couplings to the Higgs.
There is another possible branching, under which the hypercharge is a linear combination of an SU(5) generator and χ. This is known as flipped SU(5)
.
Another important subgroup is either
or 
depending upon whether or not the left-right symmetry
is broken, yielding the Pati-Salam model
, whose branching rule is
) OR (a 126H AND a 
)) ).
Let's say we choose a 54H. When this Higgs field acquires a GUT scale VEV, we have a symmetry breaking to
, i.e. the Pati-Salam model
with a Z2 left-right symmetry
.
If we have a 45H instead, this Higgs field can acquire any VEV in a two dimensional subspace without breaking the standard model. Depending on the direction of this linear combination, we can break the symmetry to SU(5)×U(1), the Georgi-Glashow model
with a U(1) (diag(1,1,1,1,1,-1,-1,-1,-1,-1)), flipped SU(5)
(diag(1,1,1,-1,-1,-1,-1,-1,1,1)), SU(4)×SU(2)×U(1) (diag(0,0,0,1,1,0,0,0,-1,-1)), the minimal left-right model (diag(1,1,1,0,0,-1,-1,-1,0,0)) or SU(3)×SU(2)×U(1)×U(1) for any other nonzero VEV.
The choice diag(1,1,1,0,0,-1,-1,-1,0,0) is called the Dimopoulos-Wilczek mechanism aka the missing VEV mechanism and it is proportional to B−L.
The choice of a 16H and a
breaks the gauge group down to the Georgi-Glashow SU(5). The same comment applies to the choice of a 126H and a 
.
It is the combination of BOTH a 45/54 and a 16/
or 126/
which breaks SO(10) down to the Standard Model
.
s. See doublet-triplet splitting problem
.
Among the solutions for it is the Dimopoulos-Wilczek mechanism, or the choice of diag(0,0,0,1,1,0,0,0,-1,-1) of <45>. Unfortunately, this is not stable once the 16/
or 126/
sector interacts with the 45 sector.
representations φ and a Yukawa coupling
(see double seesaw mechanism) or add the Yukawa interaction 
or add the nonrenormalizable coupling 
. See seesaw mechanism
.
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...
, one of the grand unified theories (GUT) is based on the SO(10) 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...
. (The 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...
involved is not really the special orthogonal group SO(10), but rather its double cover Spin(10); but calling it SO(10) is the standard convention.)
Before SU(5)
Georgi-Glashow model
In particle physics, the Georgi–Glashow model is a particular grand unification theory proposed by Howard Georgi and Sheldon Glashow in 1974. In this model the standard model gauge groups SU×SU×U are combined into a single simple gauge group -- SU...
, Harald Fritzsch and Peter Minkowski and independently 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...
found that all the matter contents are incorporated into a single representation, spinorial 16 of SO(10). (Historical note: the before in the previous sentence is misleading: Georgi found the SO(10) theory a few hours before finding SU(5) at the end of 1973.)
Important subgroups
It has the branching rules to [SU(5)×U(1)χ]/Z5.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 contained within SU(5), this is the conventional Georgi-Glashow model
Georgi-Glashow model
In particle physics, the Georgi–Glashow model is a particular grand unification theory proposed by Howard Georgi and Sheldon Glashow in 1974. In this model the standard model gauge groups SU×SU×U are combined into a single simple gauge group -- SU...
, with the
as the matter fields, the as the electroweak Higgs field and the within the as the GUT Higgs field. The superpotentialSuperpotential
Superpotential is a concept from particle physics' supersymmetry.-Example of superpotentiality:Let's look at the example of a one dimensional nonrelativistic particle with a 2D internal degree of freedom called "spin"...
may then include renormalizable terms of the form
; ; , and . The first three are responsible to the gauge symmetry breaking at low energies and give the HiggsHiggs
The term Higgs appears in:* Alan Higgs, English businessman and philanthropist* Sir Derek Higgs, an English business leader and merchant banker* Eric Sidney Higgs, English archaeologist*Griffin Higgs...
mass, and the latter two give the matter particles masses and their Yukawa couplings to the Higgs.
There is another possible branching, under which the hypercharge is a linear combination of an SU(5) generator and χ. This is known as 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...
.
Another important subgroup is either
or depending upon whether or not the left-right symmetryLeft-right symmetry
Left–right symmetry is a general principle in physics which holds that valid physical laws must not produce a different result for a motion that is left-handed than motion that is right-handed...
is broken, yielding 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...
, whose branching rule is
Spontaneous symmetry breaking
The symmetry breaking of SO(10) is usually done with a combination of (( a 45H OR a 54H) AND ((a 16H AND a) OR (a 126H AND a )) ).Let's say we choose a 54H. When this Higgs field acquires a GUT scale VEV, we have a symmetry breaking to
, i.e. the Pati-Salam modelPati-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...
with a Z2 left-right symmetry
Left-right symmetry
Left–right symmetry is a general principle in physics which holds that valid physical laws must not produce a different result for a motion that is left-handed than motion that is right-handed...
.
If we have a 45H instead, this Higgs field can acquire any VEV in a two dimensional subspace without breaking the standard model. Depending on the direction of this linear combination, we can break the symmetry to SU(5)×U(1), the Georgi-Glashow model
Georgi-Glashow model
In particle physics, the Georgi–Glashow model is a particular grand unification theory proposed by Howard Georgi and Sheldon Glashow in 1974. In this model the standard model gauge groups SU×SU×U are combined into a single simple gauge group -- SU...
with a U(1) (diag(1,1,1,1,1,-1,-1,-1,-1,-1)), 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...
(diag(1,1,1,-1,-1,-1,-1,-1,1,1)), SU(4)×SU(2)×U(1) (diag(0,0,0,1,1,0,0,0,-1,-1)), the minimal left-right model (diag(1,1,1,0,0,-1,-1,-1,0,0)) or SU(3)×SU(2)×U(1)×U(1) for any other nonzero VEV.
The choice diag(1,1,1,0,0,-1,-1,-1,0,0) is called the Dimopoulos-Wilczek mechanism aka the missing VEV mechanism and it is proportional to B−L.
The choice of a 16H and a
breaks the gauge group down to the Georgi-Glashow SU(5). The same comment applies to the choice of a 126H and a .It is the combination of BOTH a 45/54 and a 16/
or 126/ which breaks SO(10) down to the 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...
.
The electroweak Higgs and the doublet-triplet splitting problem
The electroweak Higgs doublets come from an SO(10) 10H. Unfortunately, this same 10 also contains triplets. The masses of the doublets have to be stabilized at the electroweak scale, which is many orders of magnitude smaller than the GUT scale whereas the triplets have to be really heavy in order to prevent triplet-mediated 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...
s. See doublet-triplet splitting problem
Doublet-triplet splitting problem
In particle physics, the doublet-triplet problem is a problem of some Grand Unified Theories, such as SU, SO, E_6. Grand unified theories predict Higgs bosons arise from representations of the unified group that contain other states, in particular, states that are triplets of color...
.
Among the solutions for it is the Dimopoulos-Wilczek mechanism, or the choice of diag(0,0,0,1,1,0,0,0,-1,-1) of <45>. Unfortunately, this is not stable once the 16/
or 126/ sector interacts with the 45 sector.Matter
The matter representations come in three copies (generations) of the 16 representation. The Yukawa coupling is 10H 16f 16f. This includes a right-handed neutrino. We can either include three copies of singletSinglet
A pair of spin-1/2 particles can be combined to form one of three states of total spin 1 called the triplet, or a state of spin 0 which is called the singlet. In theoretical physics, a singlet usually refers to a one-dimensional representation...
representations φ and a Yukawa coupling
(see double seesaw mechanism) or add the Yukawa interaction or add the nonrenormalizable coupling . See seesaw mechanismSeesaw 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...
.