AGrand Unified Theory (GUT) is anymodel inparticle physics that merges theelectromagnetic,weak, andstrongforces (the threegauge interactions of theStandard Model) into a single force at highenergies. Although thisunified force has not been directly observed, many GUT models theorize its existence. If the unification of these three interactions is possible, it raises the possibility that there was agrand unification epoch in thevery early universe in which these threefundamental interactions were not yet distinct.
Experiments have confirmed that at high energy, the electromagnetic interaction and weak interaction unify into a single combinedelectroweak interaction.[1] GUT models predict that at evenhigher energy, the strong and electroweak interactions will unify into one electronuclear interaction. This interaction is characterized by one largergauge symmetry and thus severalforce carriers, but one unifiedcoupling constant. Unifyinggravity with the electronuclear interaction would provide a more comprehensivetheory 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 theGUT scale of1016 GeV/c2 (only three orders of magnitude below thePlanck scale of1019 GeV/c2)—and so are well beyond the reach of any foreseenparticle 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 thePati–Salam model, predict the existence ofmagnetic monopoles.
While GUTs might be expected to offer simplicity over the complications present in theStandard Model, realistic models remain complicated because they need to introduce additional fields and interactions, or even additional dimensions of space, in order to reproduce observedfermion masses and mixing angles. This difficulty, in turn, may be related to the existence[clarification needed] offamily 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 onesimple group as the gauge symmetry but do so usingsemisimple groups can exhibit similar properties and are sometimes referred to as Grand Unified Theories as well.
Historically, the first true GUT, which was based on thesimple Lie groupSU(5), was proposed byHoward Georgi andSheldon Glashow in 1974.[3] TheGeorgi–Glashow model was preceded by thesemisimple Lie algebra Pati–Salam model byAbdus Salam andJogesh Pati also in 1974,[4] who pioneered the idea to unify gauge interactions.
The acronym GUT was first coined in 1978 by CERN researchersJohn Ellis,Andrzej Buras,Mary K. Gaillard, andDimitri Nanopoulos, however in the final version of their paper[5] they opted for the less anatomical GUM (Grand Unification Mass). Nanopoulos later that year was the first to use[6] the acronym in a paper.[7]
The fact that theelectric charges ofelectrons andprotons 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 ofstrong and weak interactions within the Standard Model is based on gauge symmetries governed by thesimple symmetry groupsSU(3) andSU(2) which allow only discrete charges, the remaining component, theweak hypercharge interaction is described by anabelian symmetryU(1) which in principle allows for arbitrary charge assignments.[note 1] The observedcharge quantization, namely the postulation that all knownelementary 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, theweak 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 byMaxwell's field theory of electromagnetism in the 19th century, but its physical implications and mathematical structure are qualitatively different.
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SU(5) is the simplest GUT. The smallest simple Lie group which contains theStandard 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 smallestgroup representations ofSU(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 smallestirreducible representations ofSU(5) are5 (the defining representation) and10. (These bold numbers indicate the dimension of the representation.) In the standard assignment, the5 contains thecharge conjugates of the right-handeddown-type quarkcolortriplet and a left-handedleptonisospindoublet, while the10 contains the sixup-type quark components, the left-handed down-type quarkcolor triplet, and the right-handedelectron. This scheme has to be replicated for each of the three knowngenerations of matter. It is notable that the theory isanomaly free with this matter content.
The hypotheticalright-handed neutrinos are a singlet ofSU(5), 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[clarification needed] (seeseesaw mechanism).
The next simple Lie group which contains the Standard Model is
Here, the unification of matter is even more complete, since theirreduciblespinorrepresentation16 contains both the5 and10 ofSU(5) and a right-handed neutrino, and thus the complete particle content of one generation of the extendedStandard Model withneutrino masses. This is already the largestsimple group that achieves the unification of matter in a scheme involving only the already known matter particles (apart from theHiggs 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 thedown quark, themuon and thestrange quark, and thetau lepton and thebottom quark forSU(5) andSO(10). Some of these mass relations hold approximately, but most don't (seeGeorgi-Jarlskog mass relation).
The boson matrix forSO(10) is found by taking the15 × 15 matrix from the10 +5 representation ofSU(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 Diracspinor matrices ofSO(10).
In some forms ofstring theory, including E8 × E8heterotic string theory, the resultant four-dimensional theory after spontaneouscompactification on a six-dimensionalCalabi–Yau manifold resembles a GUT based on the groupE6. Notably E6 is the onlyexceptional simple Lie group to have anycomplex representations, a requirement for a theory to contain chiral fermions (namely all weakly-interacting fermions). Hence the other four (G2,F4,E7, andE8) can't be the gauge group of a GUT.[citation needed]
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 usingsupersymmetry (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.[8]
GUTs with four families / generations, SU(8): Assuming 4 generations of fermions instead of 3 makes a total of64 types of particles. These can be put into64 =8 +56 representations ofSU(8). This can be divided intoSU(5) × SU(3)F × U(1) which is theSU(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, the128 particles and anti-particles can be put into a single spinor representation ofO(16).
Symplectic gauge groups could also be considered. For example,Sp(8) (which is calledSp(4) in the articlesymplectic group) has a representation in terms of4 × 4 quaternion unitary matrices which has a16 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 includeSU(4) so can at least contain the gluons and photon ofSU(3) × 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:
A further complication withquaternion 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-handed4 × 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-handed4 × 4 quaternion matrices isSp(8) × SU(2) which does include the Standard Model bosons:
If is a quaternion valued spinor, is quaternion hermitian4 × 4 matrix coming fromSp(8) and 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 anoctonion 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 (F4,E6,E7, orE8) depending on the details.
Because they are fermions the anti-commutators of the Jordan algebra become commutators. It is known thatE6 has subgroupO(10) and so is big enough to include the Standard Model. AnE8 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 ofE8, these would either have to include anti-particles (and so havebaryogenesis), 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 includingLie 3-algebras andLie superalgebras. Neither of these fit withYang–Mills theory. In particular Lie superalgebras would introduce bosons with incorrect[clarification needed] statistics.Supersymmetry, however, does fit with Yang–Mills.
The unification of forces is possible due to the energy scale dependence of forcecoupling parameters inquantum field theory calledrenormalization group "running", which allows parameters with vastly different values at usual energies to converge to a single value at a much higher energy scale.[2]
Therenormalization 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 thehypercharge is normalized so that it is consistent withSU(5) orSO(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 extensionMSSM 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 thegrand 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 thehierarchy problem—i.e., it stabilizes the electroweakHiggs mass againstradiative corrections.[9]
SinceMajorana masses of the right-handed neutrino are forbidden bySO(10) symmetry,SO(10) GUTs predict the Majorana masses of right-handed neutrinos to be close to theGUT scale where the symmetry isspontaneously 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 (seeneutrino oscillation) via theseesaw mechanism. These predictions are independent of theGeorgi–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 includesall fundamental forces, includinggravitation, is termed a theory of everything. Some common mainstream GUT models are:[citation needed]
Note: These models refer toLie algebras not toLie groups. The Lie group could be just to take a random example.
The most promising candidate isSO(10).[10][11](Minimal)SO(10) does not contain anyexotic fermions (i.e. additional fermions besides the Standard Model fermions and the right-handed neutrino), and it unifies each generation into a singleirreducible representation. A number of other GUT models are based upon subgroups ofSO(10). They are the minimalleft-right model,SU(5),flippedSU(5) and the Pati–Salam model. The GUT groupE6 containsSO(10), but models based upon it are significantly more complicated. The primary reason for studyingE6 models comes fromE8 × E8heterotic string theory.
GUT models generically predict the existence oftopological defects such asmonopoles,cosmic strings,domain walls, and others. But none have been observed. Their absence is known as themonopole problem incosmology. Many GUT models also predictproton 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 minimalSU(5) and heavily constrains the other models. The lack of detected supersymmetry to date also constrains many models.
Some GUT theories likeSU(5) andSO(10) suffer from what is called thedoublet-triplet problem. These theories predict that for each electroweak Higgs doublet, there is a correspondingcolored Higgs triplet field with a very small mass (many orders of magnitude smaller than the GUT scale here). In theory, unifying quarks withleptons, 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 thelittle hierarchy between the fermion masses for different generations.
A GUT model consists of agauge group which is acompact Lie group, aconnection form for that Lie group, aYang–Mills action for that connection given by aninvariantsymmetric bilinear form over its Lie algebra (which is specified by a coupling constant for each factor), aHiggs sector consisting of a number of scalar fields taking on values within real/complexrepresentations of the Lie group and chiralWeyl fermions taking on values within a complex rep of the Lie group. The Lie group contains theStandard Model group and the Higgs fields acquireVEVs leading to aspontaneous symmetry breaking to the Standard Model. The Weyl fermions represent matter.
The discovery ofneutrino 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 asSO(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. However, minimum proton lifetimes from research (at or exceeding the 1034~1035 year range) have ruled out simpler GUTs and most non-SUSY models.[12]The maximum upper limit on proton lifetime (if unstable), is calculated at6×1039 years for SUSY models and1.4×1036 years for minimal non-SUSY GUTs.[13]
Thegauge coupling strengths of QCD, theweak interaction andhypercharge seem to meet at a common length scale called theGUT scale and equal approximately to 1016 GeV (slightly less than thePlanck energy of 1019 GeV), which is somewhat suggestive. This interesting numerical observation is called thegauge coupling unification, and it works particularly well if one assumes the existence ofsuperpartners 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.
Part I: Particles, Strings, and Cosmology; Part II: Themes in Unification.
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