A Medley of Potpourri: Grand Unified Theory

Friday, May 1, 2015

Grand Unified Theory

From Wikipedia, the free encyclopedia

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 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 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.^[1] Some grand unified theories predict the existence of magnetic monopoles.

As of 2012, 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 group

SU(5)

, was proposed by Howard Georgi and Sheldon Glashow in 1974.^[2] The Georgi–Glashow model was preceded by the Semisimple Lie algebra Pati–Salam model by Abdus Salam and Jogesh Pati,^[3] 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^[4] they opted for the less anatomical GUM (Grand Unification Mass). Nanopoulos later that year was the first to use^[5] the acronym in a paper.^[6]

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.^{[note 1]} The observed charge quantization, namely the fact that all known elementary particles carry electric charges which appear to be exact multiples of 1/3 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

Schematic representation of fermions and bosons in

SU(5)

GUT showing

5 + 10

split in the multiplets. Neutral bosons (photon, Z-boson, and neutral gluons) are not shown but occupy the diagonal entries of the matrix in complex superpositions

SU(5)[edit]

The pattern of weak isospins, weak hypercharges, and strong charges for particles in the SU(5) model, rotated by the predicted weak mixing angle, showing electric charge roughly along the vertical. In addition to Standard Model particles, the theory includes twelve colored X bosons, responsible for proton decay.

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

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 (2009) 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 not contained in any of these representations, which can explain their relative heaviness (see seesaw mechanism).

SO(10)

The pattern of weak isospin, W, weaker isospin, W', strong g3 and g8, and baryon minus lepton, B, charges for particles in the SO(10) Grand Unified Theory, rotated to show the embedding in E₆.

The next simple Lie group which contains the standard model is

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

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 \times 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)

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) \times SU(3) F \times U(1)

which is the

SU(5)

theory together with some heavy bosons which act on the generation number.

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 \times 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) \times 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:

\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 \times 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 \times 4

quaternion matrcies is

Sp(8) \times SU(2)

which does include the standard model bosons:

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)

\psi

is a quaternion valued spinor,

A^{ab}_\mu

is quaternion hermitian

4 \times 4

matrix coming from

Sp(8)

and

B_\mu

is a pure imaginary quaternion (both of which are 4-vector bosons) then the interaction term is:

\overline{\psi^{a}} \gamma_\mu\left( A^{ab}_\mu\psi^b + \psi^a B_\mu \right)

E8 and 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 (grassman-) Jordan algebra, which has the symmetry group of one of the exceptional Lie groups (F₄, E₆, E₇ or E₈) depending on the details.

\psi=\begin{bmatrix} a & e & \mu \\ \overline{e} & b & \tau \\ \overline{\mu} & \overline{\tau} & c \end{bmatrix}

[\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 poses 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.^[7]

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)

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:

\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 (Apr 2015). Also, most model builders simply assume supersymmetry because it solves the hierarchy problem—i.e., it stabilizes the electroweak Higgs mass against radiative corrections.^{[citation needed]}

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 such 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:

minimal left-right model — $SU(3) C \times SU(2) L \times SU(2) R \times U(1) B-L$
Georgi–Glashow model — $SU(5)$
SO(10)
Flipped $SU(5)$ — $SU(5) \times U(1)$
Pati–Salam model — $SU(4) \times SU(2) \times SU(2)$
Flipped $SO(10)$ — $SO(10) \times U(1)$

Trinification — $SU(3) \times SU(3) \times SU(3)$
$SU(6)$
$E 6$
331 model
chiral color

Not quite GUTs:

Note: These models refer to Lie algebras not to Lie groups. The Lie group could be

[SU(4) \times SU(2) \times SU(2)]/ Z 2

, just to take a random example.

The most promising candidate is

SO(10)

.^{[citation needed]} (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

E 8 \times E 8

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.
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)

Proton Decay. These graphics refer to the X bosons and Higgs bosons.
Dimension 6 proton decay mediated by the X boson $(3,2)_{-\frac{5}{6}}$ in $SU(5)$ GUT
Dimension 6 proton decay mediated by the X boson $(3,2)_{\frac{1}{6}}$ in flipped $SU(5)$ GUT
Dimension 6 proton decay mediated by the triplet Higgs $T (3,1)_{-\frac{1}{3}}$ and the anti-triplet Higgs $\bar{T} (\bar{3},1)_{\frac{1}{3}}$ in $SU(5)$ GUT

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 basically 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

As of 2012, there is still no hard evidence that nature is described by a Grand Unified Theory. Moreover, since we have no idea which Higgs particle has been observed, the smaller electroweak unification is still pending.^[8] 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.

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, 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 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.

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