Grand Unified Theory

Historically, the first true GUT, which was based on the simple Lie group , 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 also in 1974, who pioneered the idea to unify gauge interactions. The acronym GUT was first coined in 1978 by CERN researchers John Ellis, Andrzej Buras, Mary K. Gaillard, and Dimitri Nanopoulos, however in the final version of their paper they opted for the less anatomical GUM (Grand Unification Mass). Nanopoulos later that year was the first to use the acronym in a paper. == Motivation ==

The fact that the electric charges of electrons and protons seem to cancel each other exactly to extreme precision is essential for the existence of the macroscopic world as we know it, but this important property of elementary particles is not explained in the Standard Model of particle physics. While the description of strong and weak interactions within the Standard Model is based on gauge symmetries governed by the simple symmetry groups and which allow only discrete charges, the remaining component, the weak hypercharge interaction is described by an abelian symmetry which in principle allows for arbitrary charge assignments. The observed charge quantization, namely the postulation that all known elementary particles carry electric charges which are exact multiples of one-third 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 field theory of electromagnetism in the 19th century, but its physical implications and mathematical structure are qualitatively different. == Unification of matter particles ==

SU(5) s, 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. 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 . Such group symmetries allow the reinterpretation of several known particles, including the photon, W and Z bosons, and gluon, as different states of a single particle field. However, it is not obvious that the simplest possible choices for the extended "Grand Unified" symmetry should yield the correct inventory of elementary particles. The fact that all currently known matter particles fit perfectly into three copies of the smallest group representations 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 (the defining representation) and . (These bold numbers indicate the dimension of the representation.) 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 a singlet of , which means its mass is not forbidden by any symmetry; it doesn't need a spontaneous electroweak symmetry breaking which explains why its mass would be heavy (see seesaw mechanism). SO(10) , 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 : \rm 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 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 that 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 and . Some of these mass relations hold approximately, but most don't (see Georgi-Jarlskog mass relation). The boson matrix for is found by taking the matrix from the representation of 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 . E6 In some forms of string theory, including E8 × E8 heterotic string theory, the resultant four-dimensional theory after spontaneous compactification on a six-dimensional Calabi–Yau manifold resembles a GUT based on the group E6. Notably E6 is the only exceptional simple Lie group to have any complex representations, a requirement for a theory to contain chiral fermions (namely all weakly-interacting fermions). Hence the other four (G2, F4, E7, and E8) can't be the gauge group of a GUT. Extended Grand Unified Theories Non-chiral extensions of the Standard Model with vectorlike split-multiplet particle spectra which naturally appear in the higher SU(N) GUTs considerably modify the desert physics and lead to the realistic (string-scale) grand unification for conventional three quark-lepton families even without using supersymmetry (see below). On the other hand, due to a new missing VEV mechanism emerging in the supersymmetric SU(8) GUT the simultaneous solution to the gauge hierarchy (doublet-triplet splitting) problem and problem of unification of flavor can be argued. GUTs with four families / generations, SU(8): Assuming 4 generations of fermions instead of 3 makes a total of types of particles. These can be put into representations of . This can be divided into which is the theory together with some heavy bosons which act on the generation number. GUTs with four families / generations, O(16): Again assuming 4 generations of fermions, the 128 particles and anti-particles can be put into a single spinor representation of . Symplectic groups and quaternion representations Symplectic gauge groups could also be considered. For example, (which is called in the article symplectic group) has a representation in terms of quaternion unitary matrices which has a dimensional real representation and so might be considered as a candidate for a gauge group. has 32 charged bosons and 4 neutral bosons. Its subgroups include so can at least contain the gluons and photon of . 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} + j\ v + k\ \overline{v} \\ u_r + i\ \overline{u}_\mathrm{\overline r} + j\ d_\mathrm{r} + k\ \overline{d}_\mathrm{\overline r} \\ u_g + i\ \overline{u}_\mathrm{\overline g} + j\ d_\mathrm{g} + k\ \overline{d}_\mathrm{\overline g} \\ u_b + i\ \overline{u}_\mathrm{\overline b} + j\ d_\mathrm{b} + k\ \overline{d}_\mathrm{\overline b} \\ \end{bmatrix}_\mathrm{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 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 quaternion matrices is which does include the Standard Model bosons: : \mathrm{ SU(4,\mathbb{H})_L\times \mathbb{H}_R = Sp(8)\times SU(2) \supset SU(4)\times SU(2) \supset SU(3)\times SU(2)\times U(1) } If \psi is a quaternion valued spinor, A^{ab}_\mu is quaternion hermitian matrix coming from and B_\mu is a pure vector 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)\ Octonion representations It can be noted that a generation of 16 fermions can be put into the form of an octonion with each element of the octonion being an 8-vector. If the 3 generations are then put in a 3x3 hermitian matrix with certain additions for the diagonal elements then these matrices form an exceptional (Grassmann) Jordan algebra, which has the symmetry group of one of the exceptional Lie groups (, , , or ) 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 \mathrm{J}_3(\mathbb{O})\ Because they are fermions the anti-commutators of the Jordan algebra become commutators. It is known that has subgroup and so is big enough to include the Standard Model. An 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 , 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 possesses 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 incorrect statistics. Supersymmetry, however, does fit with Yang–Mills. == 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. == Neutrino masses ==

Since Majorana masses of the right-handed neutrino are forbidden by symmetry, 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. These predictions are independent of the Georgi–Jarlskog mass relations, wherein some GUTs predict other fermion mass ratios. == Proposed theories ==

Several theories have been proposed, but none is currently universally accepted. An even more ambitious theory that includes all fundamental forces, including gravitation, is termed a theory of everything. Some common mainstream GUT models are: • Pati–Salam model – • Georgi–Glashow model – ; and Flipped – • model; and Flipped – • model; and Trinification – • minimal left-right model – • 331 model – • chiral color Note: These models refer to Lie algebras not to Lie groups. The Lie group could be [\mathrm{SU}(4) \times \mathrm{SU}(2) \times \mathrm{SU}(2)] / \Z_2, just to take a random example. The most promising candidate is . (Minimal) 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 . They are the minimal left-right model, , flipped and the Pati–Salam model. The GUT group contains , but models based upon it are significantly more complicated. The primary reason for studying models comes from heterotic string theory. GUT models generically predict the existence of topological defects such as monopoles, cosmic strings, domain walls, and others. But none have been observed. Their absence is known as the monopole problem in cosmology. Many GUT models also predict proton decay, although not the Pati–Salam model. As of now, proton decay has never been experimentally observed. The minimal experimental limit on the proton's lifetime pretty much rules out minimal and heavily constrains the other models. The lack of detected supersymmetry to date also constrains many models. {{Gallery | Dimension 6 proton decay mediated by the  boson \ (3,2)_{-\tfrac{5}{6}}\ in GUT | class1= | alt1= | Dimension 6 proton decay mediated by the  boson \ (3,2)_{\tfrac{1}{6}}\ in flipped GUT | class2= | alt2= | Dimension 6 proton decay mediated by the triplet Higgs \ T(3,1)_{-\tfrac{1}{3}} and the anti-triplet Higgs \ \bar{T} (\bar{3},1)_{\tfrac{1}{3}} in GUT | class3= | alt3= }} Some GUT theories like and suffer from what is called the doublet-triplet problem. These theories predict that for each electroweak Higgs doublet, there is a corresponding colored Higgs triplet field with a very small mass (many orders of magnitude smaller than the GUT scale here). In theory, unifying quarks with leptons, the Higgs doublet would also be unified with a Higgs triplet. Such triplets have not been observed. They would also cause extremely rapid proton decay (far below current experimental limits) and prevent the gauge coupling strengths from running together in the renormalization group. Most GUT models require a threefold replication of the matter fields. As such, they do not explain why there are three generations of fermions. Most GUT models also fail to explain the little hierarchy between the fermion masses for different generations. == Ingredients ==

A GUT model consists of a gauge group which is a compact Lie group, a connection form for that Lie group, a Yang–Mills action for that connection given by an invariant symmetric bilinear form over its Lie algebra (which is specified by a coupling constant for each factor), a Higgs sector consisting of a number of scalar fields taking on values within real/complex representations of the Lie group and chiral Weyl fermions taking on values within a complex rep of the Lie group. The Lie group contains the Standard Model group and the Higgs fields acquire VEVs leading to a spontaneous symmetry breaking to the Standard Model. The Weyl fermions represent matter. == Current evidence ==

The discovery of neutrino oscillations indicates that the Standard Model is incomplete, but there is currently no clear evidence that nature is described by any Grand Unified Theory. Neutrino oscillations have 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. However, minimum proton lifetimes from research (at or exceeding the ~ year range) have ruled out simpler GUTs and most non-SUSY models. The maximum upper limit on proton lifetime (if unstable), is calculated at  years for SUSY models and  years for minimal non-SUSY GUTs. The gauge coupling strengths of QCD, the weak interaction and hypercharge seem to meet at a common length scale called the GUT scale and equal approximately to  GeV (slightly less than the Planck energy of  GeV), which is somewhat suggestive. This interesting numerical observation is called the gauge coupling unification, and it works particularly well if one assumes the existence of superpartners of the Standard Model particles. Still, it is possible to achieve the same by postulating, for instance, that ordinary (non supersymmetric) models break with an intermediate gauge scale, such as the one of Pati–Salam group. == See also ==