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105-454: Experiments have confirmed that at high energy, the electromagnetic interaction and weak interaction unify into a single combined electroweak interaction . GUT models predict that at even higher energy , the strong and electroweak interactions will unify into one electronuclear interaction. This interaction is characterized by one larger gauge symmetry and thus several force carriers , but one unified coupling constant . Unifying gravity with

210-470: A = 1 , 2 , 3 {\displaystyle a=1,2,3} ) and B μ ν {\displaystyle B^{\mu \nu }} are the field strength tensors for the weak isospin and weak hypercharge gauge fields. L f {\displaystyle {\mathcal {L}}_{f}} is the kinetic term for the Standard Model fermions. The interaction of

315-456: 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

420-420: A Grand Unified Theory might actually be realized in nature. The two smallest irreducible representations of SU(5) are 5 (the defining representation) and 10 . (These bold numbers indicate the dimension of the representation.) 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

525-466: A covering map homomorphism from SU(4) to SO(6). In addition to the four families A i , B i , C i , and D i above, there are five so-called exceptional Dynkin diagrams G 2 , F 4 , E 6 , E 7 , and E 8 ; these exceptional Dynkin diagrams also have associated simply connected and centerless compact groups. However, the groups associated to the exceptional families are more difficult to describe than those associated to

630-464: A departure from the conventional particle physics paradigm, indicating a frontier in beyond-the-Standard-Model physics. Electroweak interaction In particle physics , the electroweak interaction or electroweak force is the unified description of two of the fundamental interactions of nature: electromagnetism (electromagnetic interaction) and the weak interaction . Although these two forces appear very different at everyday low energies,

735-735: A discrete commutative group . Given a (nontrivial) subgroup K ⊂ π 1 ( G ) {\displaystyle K\subset \pi _{1}(G)} of the fundamental group of some Lie group G {\displaystyle G} , one can use the theory of covering spaces to construct a new group G ~ K {\displaystyle {\tilde {G}}^{K}} with K {\displaystyle K} in its center. Now any (real or complex) Lie group can be obtained by applying this construction to centerless Lie groups. Note that real Lie groups obtained this way might not be real forms of any complex group. A very important example of such

840-404: A manually broken symmetry. Later around 1967, while investigating spontaneous symmetry breaking , Weinberg found a set of symmetries predicting a massless, neutral gauge boson . Initially rejecting such a particle as useless, he later realized his symmetries produced the electroweak force, and he proceeded to predict rough masses for the W and Z bosons . Significantly, he suggested this new theory

945-480: A non-compact dual. In addition the complex plane is also a Hermitian symmetric space; this gives the complete list of irreducible Hermitian symmetric spaces. The four families are the types A III, B I and D I for p = 2 , D III, and C I, and the two exceptional ones are types E III and E VII of complex dimensions 16 and 27. R , C , H , O {\displaystyle \mathbb {R,C,H,O} }   stand for

1050-431: 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

1155-452: A real group is the metaplectic group , which appears in infinite-dimensional representation theory and physics. When one takes for K ⊂ π 1 ( G ) {\displaystyle K\subset \pi _{1}(G)} the full fundamental group, the resulting Lie group G ~ K = π 1 ( G ) {\displaystyle {\tilde {G}}^{K=\pi _{1}(G)}}

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1260-460: A representation in terms of 4 × 4 quaternion unitary matrices which has a 16 dimensional real representation and so might be considered as a candidate for a gauge group. Sp(8) has 32 charged bosons and 4 neutral bosons. Its subgroups include SU(4) so can at least contain the gluons and photon of SU(3) × U(1) . Although it's probably not possible to have weak bosons acting on chiral fermions in this representation. A quaternion representation of

1365-445: 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 SU(5) and immediately carry the correct observed charges, is one of the first and most important reasons why people believe that

1470-427: A spontaneous electroweak symmetry breaking which explains why its mass would be heavy (see seesaw mechanism ). The next simple Lie group which contains the standard model is Here, the unification of matter is even more complete, since the irreducible spinor representation 16 contains both the 5 and 10 of SU(5) and a right-handed neutrino, and thus the complete particle content of one generation of

1575-501: A unique real form whose corresponding centerless Lie group is compact . It turns out that the simply connected Lie group in these cases is also compact. Compact Lie groups have a particularly tractable representation theory because of the Peter–Weyl theorem . Just like simple complex Lie algebras, centerless compact Lie groups are classified by Dynkin diagrams (first classified by Wilhelm Killing and Élie Cartan ). [REDACTED] For

1680-418: Is where   q f   {\displaystyle \ q_{f}\ } is the fermions' electric charges. The neutral weak current   J μ 3   {\displaystyle \ J_{\mu }^{3}\ } is where T f 3 {\displaystyle T_{f}^{3}} is the fermions' weak isospin. The charged current part of

1785-543: Is a cover of the quotient of G by a maximal compact subgroup H , and the compact one is a cover of the quotient of the compact form of G by the same subgroup H . This duality between compact and non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry. A symmetric space with a compatible complex structure is called Hermitian. The compact simply connected irreducible Hermitian symmetric spaces fall into 4 infinite families with 2 exceptional ones left over, and each has

1890-438: Is a pure vector quaternion (both of which are 4-vector bosons) then the interaction term is: 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

1995-413: Is a real simple Lie algebra, its complexification is a simple complex Lie algebra, unless L is already the complexification of a Lie algebra, in which case the complexification of L is a product of two copies of L . This reduces the problem of classifying the real simple Lie algebras to that of finding all the real forms of each complex simple Lie algebra (i.e., real Lie algebras whose complexification

2100-458: Is a simple Lie group. The most common definition is that a Lie group is simple if it is connected, non-abelian, and every closed connected normal subgroup is either the identity or the whole group. In particular, simple groups are allowed to have a non-trivial center, but R {\displaystyle \mathbb {R} } is not simple. In this article the connected simple Lie groups with trivial center are listed. Once these are known,

2205-429: Is again the spin group , but the latter again has a center (cf. its article). The diagram D 2 is two isolated nodes, the same as A 1 ∪ A 1 , and this coincidence corresponds to the covering map homomorphism from SU(2) × SU(2) to SO(4) given by quaternion multiplication; see quaternions and spatial rotation . Thus SO(4) is not a simple group. Also, the diagram D 3 is the same as A 3 , corresponding to

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2310-465: Is best described by breaking it up into several parts as follows. The kinetic term L K {\displaystyle {\mathcal {L}}_{K}} contains all the quadratic terms of the Lagrangian, which include the dynamic terms (the partial derivatives) and the mass terms (conspicuously absent from the Lagrangian before symmetry breaking) where the sum runs over all the fermions of

2415-582: 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

2520-517: 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 SU(5) and heavily constrains the other models. The lack of detected supersymmetry to date also constrains many models. Some GUT theories like SU(5) and SO(10) suffer from what

2625-480: Is not explained in the Standard Model of particle physics. While the description of strong and weak interactions within the Standard Model is based on gauge symmetries governed by the simple symmetry groups SU(3) and SU(2) which allow only discrete charges, the remaining component, the weak hypercharge interaction is described by an abelian symmetry U(1) which in principle allows for arbitrary charge assignments. The observed charge quantization , namely

2730-423: Is often referred to as Killing-Cartan classification. Unfortunately, there is no universally accepted definition of a simple Lie group. In particular, it is not always defined as a Lie group that is simple as an abstract group. Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether R {\displaystyle \mathbb {R} }

2835-415: Is possible due to the energy scale dependence of force coupling parameters in quantum field theory called renormalization group "running" , which allows parameters with vastly different values at usual energies to converge to a single value at a much higher energy scale. The renormalization group running of the three gauge couplings in the Standard Model has been found to nearly, but not quite, meet at

2940-400: 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. In 2020, physicist Juven Wang introduced a concept known as "ultra unification". It combines the Standard Model and grand unification, particularly for the models with 15 Weyl fermions per generation, without

3045-460: Is simple if its Lie algebra is simple . An important technical point is that a simple Lie group may contain discrete normal subgroups. For this reason, the definition of a simple Lie group is not equivalent to the definition of a Lie group that is simple as an abstract group . Simple Lie groups include many classical Lie groups , which provide a group-theoretic underpinning for spherical geometry , projective geometry and related geometries in

3150-457: Is termed a theory of everything. Some common mainstream GUT models are: Not quite GUTs: Note : These models refer to Lie algebras not to Lie groups . The Lie group could be [ SU ( 4 ) × SU ( 2 ) × SU ( 2 ) ] / Z 2 , {\displaystyle [{\text{SU}}(4)\times {\text{SU}}(2)\times {\text{SU}}(2)]/\mathbb {Z} _{2},} just to take

3255-462: Is the SU(5) theory together with some heavy bosons which act on the generation number. GUTs with four families / generations, O(16) : Again assuming 4 generations of fermions, the 128 particles and anti-particles can be put into a single spinor representation of O(16) . Symplectic gauge groups could also be considered. For example, Sp(8) (which is called Sp(4) in the article symplectic group ) has

Grand Unified Theory - Misplaced Pages Continue

3360-455: Is the given complex Lie algebra). There are always at least 2 such forms: a split form and a compact form, and there are usually a few others. The different real forms correspond to the classes of automorphisms of order at most 2 of the complex Lie algebra. Symmetric spaces are classified as follows. First, the universal cover of a symmetric space is still symmetric, so we can reduce to the case of simply connected symmetric spaces. (For example,

3465-458: Is the trivial subgroup. Every simple Lie group is semisimple. More generally, any product of simple Lie groups is semisimple, and any quotient of a semisimple Lie group by a closed subgroup is semisimple. Every semisimple Lie group can be formed by taking a product of simple Lie groups and quotienting by a subgroup of its center. In other words, every semisimple Lie group is a central product of simple Lie groups. The semisimple Lie groups are exactly

3570-517: Is the universal cover of the centerless Lie group G {\displaystyle G} , and is simply connected. In particular, every (real or complex) Lie algebra also corresponds to a unique connected and simply connected Lie group G ~ {\displaystyle {\tilde {G}}} with that Lie algebra, called the "simply connected Lie group" associated to g . {\displaystyle {\mathfrak {g}}.} Every simple complex Lie algebra has

3675-438: The 10 contains the six up-type quark components, the left-handed down-type quark color triplet, and the right-handed electron . This scheme has to be replicated for each of the three known generations of matter . It is notable that the theory is anomaly free with this matter content. The hypothetical right-handed neutrinos are a singlet of SU(5) , which means its mass is not forbidden by any symmetry; it doesn't need

3780-519: The Higgs field h {\displaystyle h} and its interactions with itself and the gauge bosons, where v {\displaystyle v} is the vacuum expectation value. The   L y   {\displaystyle \ {\mathcal {L}}_{y}\ } term describes the Yukawa interaction with the fermions, and generates their masses, manifest when

3885-539: The Pati–Salam model , predict the existence of magnetic monopoles . While GUTs might be expected to offer simplicity over the complications present in the Standard Model , realistic models remain complicated because they need to introduce additional fields and interactions, or even additional dimensions of space, in order to reproduce observed fermion masses and mixing angles. This difficulty, in turn, may be related to

3990-451: The Planck scale of 10 19 {\displaystyle 10^{19}} GeV)—and so are well beyond the reach of any foreseen particle hadron 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 of the following: Some GUTs, such as

4095-419: The T 3 component of weak isospin ( Q = T 3 + 1 2 Y W {\displaystyle Q=T_{3}+{\tfrac {1}{2}}\,Y_{\mathrm {W} }} ) that does not couple to the Higgs boson . That is to say: the Higgs and the electromagnetic field have no effect on each other, at the level of the fundamental forces ("tree level"), while any other combination of

4200-426: The general linear group is neither simple, nor semisimple . This is because multiples of the identity form a nontrivial normal subgroup, thus evading the definition. Equivalently, the corresponding Lie algebra has a degenerate Killing form , because multiples of the identity map to the zero element of the algebra. Thus, the corresponding Lie algebra is also neither simple nor semisimple. Another counter-example are

4305-561: The hierarchy problem —i.e., it stabilizes the electroweak Higgs mass against radiative corrections . 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

Grand Unified Theory - Misplaced Pages Continue

4410-457: The photon , are produced through the spontaneous symmetry breaking of the electroweak symmetry SU(2) × U(1) Y to U(1) em , effected by the Higgs mechanism (see also Higgs boson ), an elaborate quantum-field-theoretic phenomenon that "spontaneously" alters the realization of the symmetry and rearranges degrees of freedom. The electric charge arises as the particular linear combination (nontrivial) of Y W (weak hypercharge) and

4515-407: The special orthogonal groups in even dimension. These have the matrix − I {\displaystyle -I} in the center , and this element is path-connected to the identity element, and so these groups evade the definition. Both of these are reductive groups . A semisimple Lie group is a connected Lie group so that its only closed connected abelian normal subgroup

4620-488: The special unitary group , SU( r + 1) and as its associated centerless compact group the projective unitary group PU( r + 1) . B r has as its associated centerless compact groups the odd special orthogonal groups , SO(2 r + 1) . This group is not simply connected however: its universal (double) cover is the spin group . C r has as its associated simply connected group the group of unitary symplectic matrices , Sp( r ) and as its associated centerless group

4725-432: The weak interaction and hypercharge seem to meet at a common length scale called the GUT scale and equal approximately to 10 GeV (slightly less than the Planck energy of 10 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

4830-412: The 19th century, but its physical implications and mathematical structure are qualitatively different. 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 Such group symmetries allow the reinterpretation of several known particles, including the photon, W and Z bosons, and gluon, as different states of

4935-617: The 248 fermions in the lowest multiplet of E 8 , 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. 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. The unification of forces

5040-442: The Higgs field acquires a non-vanishing vacuum expectation value dictated by the potential of the previous section. As a result of this rewriting, the symmetry breaking becomes manifest. In the history of the universe, this is believed to have happened shortly after the hot big bang, when the universe was at a temperature 159.5 ± 1.5  GeV (assuming the Standard Model of particle physics). Due to its complexity, this Lagrangian

5145-421: The Higgs field acquires a nonzero vacuum expectation value, discussed next. The   y k i j   , {\displaystyle \ y_{k}^{ij}\ ,} for   k ∈ { u , d , e }   , {\displaystyle \ k\in \{\mathrm {u,d,e} \}\ ,} are matrices of Yukawa couplings. The Lagrangian reorganizes itself as

5250-516: The Higgs three-point and four-point self interaction terms, L H V {\displaystyle {\mathcal {L}}_{\mathrm {HV} }} contains the Higgs interactions with gauge vector bosons, L W W V {\displaystyle {\mathcal {L}}_{\mathrm {WWV} }} contains the gauge three-point self interactions, L W W V V {\displaystyle {\mathcal {L}}_{\mathrm {WWVV} }} contains

5355-515: The Lagrangian contain the interactions between the fermions and gauge bosons, where   e = g   sin ⁡ θ W = g ′   cos ⁡ θ W   . {\displaystyle ~e=g\ \sin \theta _{\mathrm {W} }=g'\ \cos \theta _{\mathrm {W} }~.} The electromagnetic current J μ e m {\displaystyle \;J_{\mu }^{\mathrm {em} }\;}

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5460-516: The Lagrangian is given by where   ν   {\displaystyle \ \nu \ } is the right-handed singlet neutrino field, and the CKM matrix M i j C K M {\displaystyle M_{ij}^{\mathrm {CKM} }} determines the mixing between mass and weak eigenstates of the quarks. L H {\displaystyle {\mathcal {L}}_{\mathrm {H} }} contains

5565-442: The Lie group PSp( r ) = Sp( r )/{I, −I} of projective unitary symplectic matrices. The symplectic groups have a double-cover by the metaplectic group . D r has as its associated compact group the even special orthogonal groups , SO(2 r ) and as its associated centerless compact group the projective special orthogonal group PSO(2 r ) = SO(2 r )/{I, −I}. As with the B series, SO(2 r ) is not simply connected; its universal cover

5670-488: 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. 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 SO(10) . One of

5775-491: The Lie groups whose Lie algebras are semisimple Lie algebras . The Lie algebra of a simple Lie group is a simple Lie algebra. This is a one-to-one correspondence between connected simple Lie groups with trivial center and simple Lie algebras of dimension greater than 1. (Authors differ on whether the one-dimensional Lie algebra should be counted as simple.) Over the complex numbers the semisimple Lie algebras are classified by their Dynkin diagrams , of types "ABCDEFG". If L

5880-410: The Pati–Salam model. The GUT group E 6 contains SO(10) , but models based upon it are significantly more complicated. The primary reason for studying E 6 models comes from E 8 × 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

5985-567: The contraction of the 4-gradient with the Dirac matrices , defined as and the covariant derivative (excluding the gluon gauge field for the strong interaction ) is defined as Here   Y   {\displaystyle \ Y\ } is the weak hypercharge and the   T j   {\displaystyle \ T_{j}\ } are the components of the weak isospin. The L h {\displaystyle {\mathcal {L}}_{h}} term describes

6090-475: 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. Also, most model builders simply assume supersymmetry because it solves

6195-752: The discovery of the W and Z gauge bosons in proton–antiproton collisions at the converted Super Proton Synchrotron . In 1999, Gerardus 't Hooft and Martinus Veltman were awarded the Nobel prize for showing that the electroweak theory is renormalizable . After the Wu experiment in 1956 discovered parity violation in the weak interaction , a search began for a way to relate the weak and electromagnetic interactions . Extending his doctoral advisor Julian Schwinger 's work, Sheldon Glashow first experimented with introducing two different symmetries, one chiral and one achiral, and combined them such that their overall symmetry

6300-561: The electromagnetic and weak force . It is thought that the required temperature of 10  K has not been seen widely throughout the universe since before the quark epoch, and currently the highest human-made temperature in thermal equilibrium is around 5.5 × 10  K (from the Large Hadron Collider ). Sheldon Glashow , Abdus Salam , and Steven Weinberg were awarded the 1979 Nobel Prize in Physics for their contributions to

6405-402: The electron and the down quark , the muon and the strange quark , and the tau lepton and the bottom quark for SU(5) and SO(10) . Some of these mass relations hold approximately, but most don't (see Georgi-Jarlskog mass relation ). The boson matrix for SO(10) is found by taking the 15 × 15 matrix from the 10 + 5 representation of SU(5) and adding an extra row and column for

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6510-419: The electronuclear interaction would provide a more comprehensive theory of everything (TOE) rather than a Grand Unified Theory. Thus, GUTs are often seen as an intermediate step towards a TOE. The novel particles predicted by GUT models are expected to have extremely high masses—around the GUT scale of 10 16 {\displaystyle 10^{16}} GeV (just three orders of magnitude below

6615-411: The electroweak interactions is divided into four parts before electroweak symmetry breaking becomes manifest, The L g {\displaystyle {\mathcal {L}}_{g}} term describes the interaction between the three W vector bosons and the B vector boson, where W a μ ν {\displaystyle W_{a}^{\mu \nu }} (

6720-436: The existence of family symmetries beyond the conventional GUT models. Due to this and the lack of any observed effect of grand unification so far, there is no generally accepted GUT model. Models that do not unify the three interactions using one simple group as the gauge symmetry but do so using semisimple groups can exhibit similar properties and are sometimes referred to as Grand Unified Theories as well. Historically,

6825-399: 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

6930-469: The fermions might be: A further complication with quaternion representations of fermions is that there are two types of multiplication: left multiplication and right multiplication which must be taken into account. It turns out that including left and right-handed 4 × 4 quaternion matrices is equivalent to including a single right-multiplication by a unit quaternion which adds an extra SU(2) and so has an extra neutral boson and two more charged bosons. Thus

7035-493: 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 10~10 year range) have ruled out simpler GUTs and most non-SUSY models. The maximum upper limit on proton lifetime (if unstable), is calculated at 6×10 years for SUSY models and 1.4×10 years for minimal non-SUSY GUTs. The gauge coupling strengths of QCD,

7140-415: 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. 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

7245-476: The first true GUT, which was based on the simple Lie group SU(5) , was proposed by Howard Georgi and Sheldon Glashow in 1974. The Georgi–Glashow model was preceded by the semisimple Lie algebra Pati–Salam model by Abdus Salam and Jogesh Pati 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

7350-450: The gauge bosons and the fermions are through the gauge covariant derivative , where the subscript j sums over the three generations of fermions; Q , u , and d are the left-handed doublet, right-handed singlet up, and right handed singlet down quark fields; and L and e are the left-handed doublet and right-handed singlet electron fields. The Feynman slash D / {\displaystyle D\!\!\!\!/} means

7455-414: 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. A GUT model consists of a gauge group which is a compact Lie group ,

7560-515: The gauge four-point self interactions,   L Y   {\displaystyle \ {\mathcal {L}}_{\mathrm {Y} }\ } contains the Yukawa interactions between the fermions and the Higgs field, Simple Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups . The list of simple Lie groups can be used to read off

7665-425: The gauge group of a GUT. 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

7770-492: The group of left- and right-handed 4 × 4 quaternion matrices is Sp(8) × SU(2) which does include the standard model bosons: If ψ {\displaystyle \psi } is a quaternion valued spinor,   A μ a b   {\displaystyle \ A_{\mu }^{ab}\ } is quaternion hermitian 4 × 4 matrix coming from Sp(8) and   B μ   {\displaystyle \ B_{\mu }\ }

7875-451: The hypercharge and the weak isospin must interact with the Higgs. This causes an apparent separation between the weak force, which interacts with the Higgs, and electromagnetism, which does not. Mathematically, the electric charge is a specific combination of the hypercharge and T 3 outlined in the figure. U(1) em (the symmetry group of electromagnetism only) is defined to be the group generated by this special linear combination, and

7980-475: The infinite (A, B, C, D) series of Dynkin diagrams, a connected compact Lie group associated to each Dynkin diagram can be explicitly described as a matrix group, with the corresponding centerless compact Lie group described as the quotient by a subgroup of scalar matrices. For those of type A and C we can find explicit matrix representations of the corresponding simply connected Lie group as matrix groups. A r has as its associated simply connected compact group

8085-485: The infinite families, largely because their descriptions make use of exceptional objects . For example, the group associated to G 2 is the automorphism group of the octonions , and the group associated to F 4 is the automorphism group of a certain Albert algebra . See also E 7 + 1 ⁄ 2 . with fixed volume. The following table lists some Lie groups with simple Lie algebras of small dimension. The groups on

8190-459: 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. Several theories have been proposed, but none is currently universally accepted. An even more ambitious theory that includes all fundamental forces, including gravitation ,

8295-505: The list of simple Lie algebras and Riemannian symmetric spaces . Together with the commutative Lie group of the real numbers, R {\displaystyle \mathbb {R} } , and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension . Many commonly encountered Lie groups are either simple or 'close' to being simple: for example,

8400-958: The mixed gauge-gravitational anomaly . This proposal can also be understood as coupling the Standard Model (as quantum field theory) to the Beyond the Standard Model sector (as TQFTs or CFTs being dark matter ) via the discrete gauged B − L topological force. In either TQFT or CFT scenarios, the implication is that a new high-energy physics frontier beyond the conventional 0-dimensional particle physics relies on new types of topological forces and matter. This includes gapped extended objects such as 1-dimensional line and 2-dimensional surface operators or conformal defects, whose open ends carry deconfined fractionalized particle or anyonic string excitations. Understanding and characterizing these gapped extended objects requires mathematical concepts such as cohomology , cobordism , or category into particle physics. The topological phase sectors proposed by Wang signify

8505-482: The necessity of right-handed sterile neutrinos, by adding new gapped topological phase sectors or new gapless interacting conformal sectors consistent with the nonperturbative global anomaly cancellation and cobordism constraints (especially from the mixed gauge-gravitational anomaly , such as a Z / 16 Z class anomaly, associated with the baryon minus lepton number B − L and the electroweak hypercharge Y). Gapped topological phase sectors are constructed via

8610-489: The ones with non-trivial center are easy to list as follows. Any simple Lie group with trivial center has a universal cover , whose center is the fundamental group of the simple Lie group. The corresponding simple Lie groups with non-trivial center can be obtained as quotients of this universal cover by a subgroup of the center. An equivalent definition of a simple Lie group follows from the Lie correspondence : A connected Lie group

8715-417: The particles have essentially just been rotated, in the ( W 3 , B ) plane, by the angle θ W . This also introduces a mismatch between the mass of the Z and the mass of the W particles (denoted as m Z and m W , respectively), The W 1 and W 2 bosons, in turn, combine to produce the charged massive bosons W : The Lagrangian for

8820-408: 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

8925-479: 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

9030-401: The real numbers, complex numbers, quaternions , and octonions . In the symbols such as E 6 for the exceptional groups, the exponent −26 is the signature of an invariant symmetric bilinear form that is negative definite on the maximal compact subgroup. It is equal to the dimension of the group minus twice the dimension of a maximal compact subgroup. The fundamental group listed in

9135-577: The relevant field ( A , {\displaystyle A,} Z , {\displaystyle Z,} W ± {\displaystyle W^{\pm }} ) and f  by the structure constants of the appropriate gauge group. The neutral current   L N   {\displaystyle \ {\mathcal {L}}_{\mathrm {N} }\ } and charged current   L C   {\displaystyle \ {\mathcal {L}}_{\mathrm {C} }\ } components of

9240-430: The resultant four-dimensional theory after spontaneous compactification on a six-dimensional Calabi–Yau manifold resembles a GUT based on the group E 6 . Notably E 6 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 ( G 2 , F 4 , E 7 , and E 8 ) can't be

9345-518: 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) . In some forms of string theory , including E 8  × E 8 heterotic string theory ,

9450-407: 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,

9555-400: The sense of Felix Klein 's Erlangen program . It emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry. These exceptional groups account for many special examples and configurations in other branches of mathematics, as well as contemporary theoretical physics . As a counterexample,

9660-413: The so-called " special linear group " SL( n , R {\displaystyle \mathbb {R} } ) of n by n matrices with determinant equal to 1 is simple for all odd n  > 1, when it is isomorphic to the projective special linear group . The first classification of simple Lie groups was by Wilhelm Killing , and this work was later perfected by Élie Cartan . The final classification

9765-436: 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 64 types of particles. These can be put into 64 = 8 + 56 representations of SU(8) . This can be divided into SU(5) × SU(3) F × U(1) which

9870-449: The symmetry described by the U(1) em group is unbroken, since it does not directly interact with the Higgs. The above spontaneous symmetry breaking makes the W 3 and B bosons coalesce into two different physical bosons with different masses – the Z boson, and the photon ( γ ), where θ W is the weak mixing angle . The axes representing

9975-560: The symmetry extension (in contrast to the symmetry breaking in the Standard Model's Anderson-Higgs mechanism ), whose low energy contains unitary Lorentz invariant topological quantum field theories (TQFTs), such as 4-dimensional noninvertible, 5-dimensional noninvertible, or 5-dimensional invertible entangled gapped phase TQFTs. Alternatively, Wang's theory suggests there could also be right-handed sterile neutrinos, gapless unparticle physics, or some combination of more general interacting conformal field theories (CFTs) , to together cancel

10080-509: The symmetry group of one of the exceptional Lie groups ( F 4 , E 6 , E 7 , or E 8 ) depending on the details. Because they are fermions the anti-commutators of the Jordan algebra become commutators. It is known that E 6 has subgroup O(10) and so is big enough to include the Standard Model. An E 8 gauge group, for example, would have 8 neutral bosons, 120 charged bosons and 120 charged anti-bosons. To account for

10185-399: The system. These fields are the weak isospin fields W 1 , W 2 , and W 3 , and the weak hypercharge field B . This invariance is known as electroweak symmetry . The generators of SU(2) and U(1) are given the name weak isospin (labeled T ) and weak hypercharge (labeled Y ) respectively. These then give rise to the gauge bosons that mediate the electroweak interactions –

10290-400: The table below is the fundamental group of the simple group with trivial center. Other simple groups with the same Lie algebra correspond to subgroups of this fundamental group (modulo the action of the outer automorphism group). Simple Lie groups are fully classified. The classification is usually stated in several steps, namely: One can show that the fundamental group of any Lie group is

10395-783: The theory (quarks and leptons), and the fields   A μ ν   , {\displaystyle \ A_{\mu \nu }\ ,}   Z μ ν   , {\displaystyle \ Z_{\mu \nu }\ ,}   W μ ν −   , {\displaystyle \ W_{\mu \nu }^{-}\ ,} and   W μ ν + ≡ ( W μ ν − ) †   {\displaystyle \ W_{\mu \nu }^{+}\equiv (W_{\mu \nu }^{-})^{\dagger }\ } are given as with X {\displaystyle X} to be replaced by

10500-469: The theory models them as two different aspects of the same force. Above the unification energy , on the order of 246  GeV , they would merge into a single force. Thus, if the temperature is high enough – approximately 10   K – then the electromagnetic force and weak force merge into a combined electroweak force. During the quark epoch (shortly after the Big Bang ), the electroweak force split into

10605-404: The three W bosons of weak isospin ( W 1 , W 2 , and W 3 ), and the B boson of weak hypercharge, respectively, all of which are "initially" massless. These are not physical fields yet, before spontaneous symmetry breaking and the associated Higgs mechanism . In the Standard Model , the observed physical particles, the W and Z bosons , and

10710-618: The unification of the weak and electromagnetic interaction between elementary particles , known as the Weinberg–Salam theory . The existence of the electroweak interactions was experimentally established in two stages, the first being the discovery of neutral currents in neutrino scattering by the Gargamelle collaboration in 1973, and the second in 1983 by the UA1 and the UA2 collaborations that involved

10815-493: The universal cover of a real projective plane is a sphere.) Second, the product of symmetric spaces is symmetric, so we may as well just classify the irreducible simply connected ones (where irreducible means they cannot be written as a product of smaller symmetric spaces). The irreducible simply connected symmetric spaces are the real line, and exactly two symmetric spaces corresponding to each non-compact simple Lie group G , one compact and one non-compact. The non-compact one

10920-410: Was renormalizable. In 1971, Gerard 't Hooft proved that spontaneously broken gauge symmetries are renormalizable even with massive gauge bosons. Mathematically, electromagnetism is unified with the weak interactions as a Yang–Mills field with an SU(2) × U(1) gauge group , which describes the formal operations that can be applied to the electroweak gauge fields without changing the dynamics of

11025-399: Was unbroken. This did not yield a renormalizable theory , and its gauge symmetry had to be broken by hand as no spontaneous mechanism was known, but it predicted a new particle, the Z boson . This received little notice, as it matched no experimental finding. In 1964, Salam and John Clive Ward had the same idea, but predicted a massless photon and three massive gauge bosons with

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