# nLab GUT

Contents

### Context

#### Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

scalar bosons

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks ($q$)
up-typeup quark ($u$)charm quark ($c$)top quark ($t$)
down-typedown quark ($d$)strange quark ($s$)bottom quark ($b$)
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion ($u d$)
ρ-meson ($u d$)
ω-meson ($u d$)
f1-meson
a1-meson
strange-mesons:
ϕ-meson ($s \bar s$),
kaon, K*-meson ($u s$, $d s$)
eta-meson ($u u + d d + s s$)

charmed heavy mesons:
D-meson ($u c$, $d c$, $s c$)
J/ψ-meson ($c \bar c$)
bottom heavy mesons:
B-meson ($q b$)
ϒ-meson ($b \bar b$)
baryonsnucleons:
proton $(u u d)$
neutron $(u d d)$

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)

solitons

in grand unified theory

minimally extended supersymmetric standard model

superpartners

bosinos:

dark matter candidates

Exotica

auxiliary fields

# Contents

## Idea

### General idea

The standard model of particle physics asserts that the fundamental quantum physical fields and particles are modeled as sections of and connections on a vector bundle that is associated to a $G$-principal bundle, where the Lie group $G$ – called the gauge group – is the product of (special) unitary groups $G = SU(3) \times SU(2) \times U(1)$ (or rather a quotient of this by the cyclic group $Z/6$, see there) and where the representation of $G$ used to form the associated vector bundle looks fairly ad hoc on first sight.

A grand unified theory (“GUT” for short) in this context is an attempt to realize the standard model as sitting inside a conceptually simpler model, in particular one for which the gauge group is a bigger but simpler group $\hat{G}$, preferably a simple Lie group in the technical sense, which contains $G$ as a subgroup. Such a grand unified theory would be phenomenologically viable if a process of spontaneous symmetry breaking at some high energy scale – the “GUT scale” – would reduce the model back to the standard model of particle physics without adding spurious extra effects that would not be in agreement with existing observations in experiment.

The terminology “grand unified” here refers to the fact that such a single simple group $\hat{G}$ would unify the fundamental forces of electromagnetism, the weak nuclear force and the strong nuclear force in a way that generalizes the way in which the electroweak field already unifies the weak nuclear force and electromagnetism, and electromagnetism already unifies, as the word says, electricity and magnetism.

Since no GUT model has been fully validated yet (but see for instance Fong-Meloni 14), GUTs are models “beyond the standard model”. Often quantum physics “beyond the standard model” is expected to also say something sensible about quantum gravity and hence unify not just the three gauge forces but also the fourth known fundamental force, which is gravity. Models that aim to do all of this would then “unify” the entire content of the standard model of particle physics plus the standard model of cosmology, hence “everything that is known about fundamental physics” to date. Therefore such theories are then sometimes called a theory of everything.

(Here it is important to remember the context, both “grand unified” and “of everything” refers to aspects of presently available models of fundamental physics, and not to deeper philosophical questions of ontology.)

length scales in the observable universe (from cosmic scales, over fundamental particle-masses around the electroweak symmetry breaking to GUT scale and Planck scale):

graphics grabbed from Zupan 19

### The $SU(5)$-GUT (Georgi-Glashow)

The argument for the hypothesis of “grand unification” is fairly compelling if one asks for simple algebraic structures in the technical sense (simple Lie groups and their irreducible representations):

The exact gauge group of the standard model of particle physics is really a quotient group

$G_{SM} \;=\; \big( U(1) \times SU(2) \times SU(3) \big) / \mathbb{Z}_6 \,,$

where the cyclic group $\mathbb{Z}_6$ acts freely, hence exhibiting a subtle global twist in the gauge structure. This seemingly ad hoc group turns out to be isomorphic to the subgroup

$\underset{ \simeq \big( U(1) \times SU(2) \times SU(3) \big) / \mathbb{Z}_6 }{ \underbrace{ S \big( U(2) \times U(3)\big) }} \;\subset\; SU(5)$

of SU(5) (see Baez-Huerta 09, p. 33-36). The latter happens to be a simple Lie group, thus exhibiting the exact standard model Lie group as being “simply” a “(2+3)-breaking” of a simple Lie group.

Moreover, the gauge group-representation $V_{SM}$ that captures one generation of fundamental particles of the standard model of particle physics, which looks fairly ad hoc as a representation of $G_{SM}$ (e.g. Baez-Huerta 09, table 1), but as a representation of $SU(5)$ it is simply

$V_{SM} \simeq \Lambda \mathbb{C}^5$

(see Baez-Huerta 09, p. 36-41).

This leads to the $SU(5)$-GUT due to Georgi-Glashow 74

### D-Series GUTs

#### The $Spin(10)$-GUT (known as “$SO(10)$”)

There is a further inclusion $SU(5) \hookrightarrow$ Spin(10) into the spin group in 10 (Euclidean) dimensions (still a simple Lie group), and one generation of fundamental particles regarded as an $SU(5)$-representation $\Lambda \mathbb{C}^5$ as above extends to a spin representation (see Baez-Huerta 09, theorem 2). This has the aesthetically pleasing effect that under this identification the 1-generation rep $V_{SM}$ is now identified as

$V_{SM} \;\simeq\; \mathbf{16} \oplus \overline{\mathbf{16}}$

namely as the direct sum of the two (complex) irreducible representations of Spin(10), together being the Dirac representation of Spin(10).

The exact gauge group of the standard model of particle physics (see there) is isomorphic to the subgroup of Spin(9) $\subset$ Spin(10) which respects a splitting $\mathbb{H} \oplus \mathbb{H} \simeq_{\mathbb{R}} \mathbb{C} \oplus \mathbb{C}^3$ (Krasnov 19).

Again, this means that under the embedding of the gauge group $G_{SM}$ all the way into the simple Lie group Spin(10), its ingredients become simpler, not just in a naive sense, but in the technical mathematical sense of simple algebraic objects.

Discussion of SO(10) (i.e. Spin(10)) GUT-models with realistic phenomenology is in BLM 09 Malinský 09, Lavoura-Wolfenstein 10 Altarelli-Meloni 13 Dueck-Rodejohann 13 Ohlsson-Pernow 19 CPS 19.

slide grabbed from Malinský 09

Discussion of leptoquarks in $SO(10)$-models possibly explaining the flavour anomalies: AMM 19

#### The $Spin(11)$-GUT (known as “$SO(11)$”)

Models with Spin(11) (“SO(11)”) GUT group.

Specifically with gauge-Higgs unification in a Randall-Sundrum model-like 6d spacetime: Hosotani-Yamatsu 15, Furui-Hosotani-Yamatsu 16, Hosotani 17, Hosotani-Yamatsu 17

See the references below.

#### The $Spin(12)$-GUT (known as “$SO(12)$”)

Models with Spin(12) (“SO(12)”) GUT group.

Specifically with gauge-Higgs unification in a Randall-Sundrum model-like 6d spacetime: Nomura-Sato 08, Nomura 09, Chiang-Nomura-Sato 11)

See the references below.

#### The $Spin(16), Spin(18)$-GUT (known as “$SO(16), SO(18)$”)

Models with Spin(16) (“SO(16)”) GUT group.

See also di Lucio 11, p. 44 and see the references below.

Predicts fourth generation of fermions

### E-series GUTs

The most studied choices of GUT-groups $G$ are SU(5), Spin(10) (in the physics literature often referred to as SO(10)) and E6 (review includes Witten 86, sections 1 and 2).

It so happens that, mathematically, the sequence SU(5), Spin(10), E6 naturally continues (each step by consecutively adding a node to the Dynkin diagrams) with the exceptional Lie groups E7, E8 that naturally appear in heterotic string phenomenology (exposition is in Witten 02a) and conjecturally further via the U-duality Kac-Moody groups E9, E10, E11 that are being argued to underly M-theory. In the context of F-theory model building, also properties of the observes Yukawa couplings may point to exceptional GUT groups (Zoccarato 14, slide 11, Vafa 15, slide 11).

#### The $E_6$-GUT

(…)

review in Britto 17

(…)

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## Properties

### Relation to proton decay

Many GUT models imply that the proton – which in the standard model of particle physics is a stable bound state (of quarks) – is in fact unstable, albeit with an extremely long mean liftetime, and hence may decay (e.g. KM 14). Experimental searches for such proton decay (see there for more) put strong bounds on this effect and hence heavily constrain or rule out many GUT models.

But in recent years it is claimed that there are in fact realistic $SU(5)$ GUT models that do not imply any proton decay, quite generically so for MSSM-models (Mütter-Ratz-Vaudrvange 16), but also for non-supersymmetric models ( Fornal-Grinstein 17, Fornal-Grinstein 18, in particular in gauge-Higgs grand unification such as Spin(11)- (“SO(11)”-) and Spin(12)- (“SO(12)”-) models: (Hosotani-Yamatsu 15, Furui-Hosotani-Yamatsu 16, Sec. 2.6 Hosotani 17, Section 6).

### Relation to neutrino masses

The high energy scale required by a seesaw mechanism to produce the experimentally observer neutrino masses happens to be about the conventional GUT scale, for review see for instance (Mohapatra 06).

I also noted at the same time that interactions between a pair of lepton doublets and a pair of scalar doublets can generate a neutrino mass, which is suppressed only by a factor $M^{-1}$, and that therefore with a reasonable estimate of $M$ could produce observable neutrino oscillations. The subsequent confirmation of neutrino oscillations lends support to the view of the Standard Model as an effective field theory, with M somewhere in the neighborhood of $10^{16} GeV$. (Weinberg 09, p. 15)

Detailed matching of parameters of non-supersymmetric $Spin(10)$-GUT to neutrino masses is discussed in Ohlsson-Pernow 19

### Relation to leptoquarks

Generically, GUT-theories predict the existence of leptoquarks (Murayama-Yanagida 92), possibly related to the apparently observed

### Occurrence in string phenomenology

Discussion of string phenomenology of intersecting D-brane models KK-compactified with non-geometric fibers such that the would-be string sigma-models with these target spaces are in fact Gepner models (in the sense of Spectral Standard Model and String Compactifications) is in (Dijkstra-Huiszoon-Schellekens 04a, Dijkstra-Huiszoon-Schellekens 04b):

A plot of standard model-like coupling constants in a computer scan of Gepner model-KK-compactification of intersecting D-brane models according to Dijkstra-Huiszoon-Schellekens 04b.

The blue dot indicates the couplings in $SU(5)$-GUT theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.

fundamental scales (fundamental physical units)

• speed of light$\,$ $c$

• Planck's constant$\,$ $\hbar$

• gravitational constant$\,$ $G_N = \kappa^2/8\pi$

• Planck scale

• Planck length$\,$ $\ell_p = \sqrt{ \hbar G / c^3 }$

• Planck mass$\,$ $m_p = \sqrt{\hbar c / G}$

• depending on a given mass $m$

• Compton wavelength$\,$ $\lambda_m = \hbar / m c$

• Schwarzschild radius$\,$ $2 m G / c^2$

• depending also on a given charge $e$

• Schwinger limit$\,$ $E_{crit} = m^2 c^3 / e \hbar$
• GUT scale

• string scale

• string tension$\,$ $T = 1/(2\pi \alpha^\prime)$

• string length scale$\,$ $\ell_s = \sqrt{\alpha'}$

• string coupling constant$\,$ $g_s = e^\lambda$

## References

### General

Original articles include

Textbook accounts:

Discussion with an eye towards supergravity unification:

A basic textbook account is in

and a detailed one is in

• Luca di Lucio, Aspects of Symmetry Breaking inGrand Unified Theories, 2011 (pdf)

Survey of arguments for the hypothesis of grand unification includes

Introduction to GUTs aimed more at mathematicians include

• Edward Witten, section 1 and 2 of Physics and geometry, Proceedings of the international congress of mathematicians, 1986 (pdf)

• John Baez, John Huerta, The algebra of grand unified theories, Bull. Am. Math. Soc. 47:483-552,2010 (arXiv:0904.1556)

• Vivian Anthony Britto, A Mathematical Construction of an $E_6$ Grand Unified Theory, Munich 2017 (pdf, pdf)

### Proton (non-)decay

Discussion of experimental bounds on proton decay in GUTs includes

• Helena Kolešová, Michal Malinský, Proton lifetime in the minimal $SO(10)$ GUT and its implications for the LHC, Phys. Rev. D 90, 115001 (2014) (arXiv:1409.4961)

Claim that proton decay may be entirely avoided:

• Andreas Mütter, Michael Ratz, Patrick K.S. Vaudrevange, Grand Unification without Proton Decay (arXiv:1606.02303)

(claims that many string theory and supergravity models have this property)

• Bartosz Fornal, Benjamin Grinstein, $SU(5)$ Unification without Proton Decay, Physics Review Letters (arXiv:1706.08535)

• Bartosz Fornal, Benjamin Grinstein, Grand Unified Theory with a Stable Proton, Int. J. Mod. Phys. A 33 (2018) 1844013 (arXiv:1808.00953)

Claim that threshold corrections can considerably alter (decrease) proton decay rate predictions in non-supersymmetric GUTs:

• Joydeep Chakrabortty, Stephen F. King, Rinku Maji, Unification, Proton Decay and Topological Defects in non-SUSY GUTs with Thresholds (arXiv:1901.05867)

### Realistic models and phenomenology

Discussion of phenomenologically viable GUT-models (compatible with experiment and the standard model of particle physics):

#### $SO(10)$-GUT

The idea of “SO(10)” (Spin(10)) GUT originates with

Review:

• Michal Malinský, 35 years of GUTs - where do we stand?, 2009 (pdf)

• Stefano Bertolini, Luca Di Luzio, Michal Malinský, Intermediate mass scales in the non-supersymmetric SO(10) grand unification: a reappraisal, Phys. Rev. D80:015013, 2009 (arXiv:0903.4049)

for non-superymmetric models:

• L. Lavoura and Lincoln Wolfenstein, Resuscitation of minimal $SO(10)$ grand unification, Phys. Rev. D 48, 264 (doi:10.1103/PhysRevD.48.264)

• Guido Altarelli, Davide Meloni, A non Supersymmetric SO(10) Grand Unified Model for All the Physics below $M_{GUT}$ (arXiv:1305.1001)

• Alexander Dueck, Werner Rodejohann, Fits to $SO(10)$ Grand Unified Models (arXiv:1306.4468)

• Chee Sheng Fong, Davide Meloni, Aurora Meroni, Enrico Nardi, Leptogenesis in $SO(10)$ (arXiv:1412.4776)

(in view of leptogenesis)

• Tommy Ohlsson, Marcus Pernow, Fits to Non-Supersymmetric SO(10) Models with Type I and II Seesaw Mechanisms Using Renormalization Group Evolution (arXiv:1903.08241)

• Mainak Chakraborty, M.K. Parida, Biswonath Sahoo, Triplet Leptogenesis, Type-II Seesaw Dominance, Intrinsic Dark Matter, Vacuum Stability and Proton Decay in Minimal SO(10) Breakings (arXiv:1906.05601)

Results indicating non-SUSY $SO(10)$ as self sufficient theory for neutrino masses, baryon asymmetry, dark matter, vacuum stability of SM scalar potential, origin of three gauge forces, and observed proton stability.

• Nobuchika Okada, Digesh Raut, Qaisar Shafi, Inflation, Proton Decay, and Higgs-Portal Dark Matter in $SO(10) \times U(1)_\pri$ (arXiv:1906.06869)

• V. Suryanarayana Mummidi, Ketan M. Patel, Leptogenesis and fermion mass fit in a renormalizable $SO(10)$ model (arXiv:2109.04050)

• Archana Anandakrishnan, B. Charles Bryant, Stuart Raby, LHC Phenomenology of $SO(10)$ Models with Yukawa Unification II (arXiv:1404.5628)

• Ila Garg, New minimal supersymmetric $SO(10)$ GUT phenomenology and its cosmological implications (arXiv:1506.05204)

#### $SO(11)$-GUT

Discussion for Spin(11) GUT group (“SO(11)”):

#### $SO(12)$-GUT

Discussion for Spin(12) GUT group (“SO(12)”):

• S. Rajpoot and P. Sithikong, Implications of the $SO(12)$ gauge symmetry for grand unification, Phys. Rev. D 23, 1649 (1981) (doi:10.1103/PhysRevD.23.1649)

• Takaaki Nomura, Joe Sato, Standard(-like) Model from an $SO(12)$ Grand Unified Theory in six-dimensions with $S^2$ extra-space, Nucl.Phys.B811:109-122, 2009 (arXiv:0810.0898)

• Takaaki Nomura, Physics beyond the standard model with $S^2$ extra-space, 2009 (pdf, pdf)

• Cheng-Wei Chiang, Takaaki Nomura, Joe Sato, Gauge-Higgs unification models in six dimensions with $S^2/\mathbb{Z}_2$ extra space and GUT gauge symmetry (arXiv:1109.5835)

#### $SO(16)$- and $Spin(18)$-GUT

Discussion for Spin(16) and Spin(18)? GUT group (“SO(16)” and “SO(18)?”):

#### $E_8$-GUT

• Alfredo Aranda, Francisco J. de Anda, Stephen F. King, Exceptional Unification of Families and Forces (arXiv:2005.03048)

• Alfredo Aranda, Francisco J. de Anda, Complete $E_8$ Unification in 10 Dimensions (arXiv:2007.13248)

• George Manolakos, Gregory Patellis, George Zoupanos, $\mathcal{N}=1$ trinification from dimensional reduction of $N=1$, 10D $E_8$ over $SU(3)/U(1)\times U(1) \times \mathbb{Z}_3$ and its phenomenological consequences (arXiv:2009.07059)

• Francisco J. de Anda, Alfredo Aranda, António P. Morais, Roman Pasechnik, Gauge couplings evolution from the Standard Model, through Pati-Salam theory, into $E_8$ unification of families and forces (arXiv:2011.13902)

• Alfredo Aranda, Francisco J. de Anda, António P. Morais, Roman Pasechnik, A Different Take on $E_8$ Unification (arXiv:2107.05421)

• Alfredo Aranda, Francisco J. de Anda, António P. Morais, Roman Pasechnik, Sculpting the Standard Model from low-scale Gauge-Higgs-Matter $E_8$ Grand Unification in ten dimensions (arXiv:2107.05495)

### Heterotic string phenomenology

The historical origin of all string phenomenology is the top-down GUT-model building in heterotic string theory due to

Review and exposition:

#### The $E_8 \times E_8$-heterotic string

The following articles claim the existence of exact realization of the gauge group and matter-content of the MSSM in heterotic string theory on orbifolds (not yet checking Yukawa couplings):

A computer search through the “landscape” of Calabi-Yau varieties showed severeal hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as SU(5)-GUTs):

general computational theory:

The resulting database of heterotic line bundle models is here:

Review includes

Computation of metrics on these Calabi-Yau compactifications (eventually needed for computing their induced Yukawa couplings) is started in

This “heterotic standard model” has a “hidden sector” copy of the actual standard model, more details of which are discussed here:

The issue of moduli stabilization in these kinds of models is discussed in

Principles singling out heterotic models with three generations of fundamental particles are discussed in:

Discussion of non-supersymmetric: GUT models:

• Alon E. Faraggi, Viktor G. Matyas, Benjamin Percival, Classification of Non-Supersymmetric Pati-Salam Heterotic String Models (arXiv:2011.04113)

• Carlo Angelantonj, Ioannis Florakis, GUT Scale Unification in Heterotic Strings (arXiv:1812.06915)

#### The $SemiSpin(32)$-heterotic string

Discussion of string phenomenology for the SemiSpin(32)-heterotic string (see also at type I phenomenology):

• Kang-Sin Choi, Stefan Groot Nibbelink, Michele Trapletti, Heterotic $SO(32)$ model building in four dimensions, JHEP 0412:063, 2004 (arXiv:hep-th/0410232)

• Hans-Peter Nilles, Saul Ramos-Sanchez, Patrick K.S. Vaudrevange, Akin Wingerter, Exploring the $SO(32)$ Heterotic String, JHEP 0604:050, 2006 (arXiv:hep-th/0603086)

• Saul Ramos-Sanchez, Towards Low Energy Physics from the Heterotic String, Fortsch. Phys. 10:907-1036, 2009 (arXiv:0812.3560)

• Naoki Yamatsu, String-Inspired Special Grand Unification, Progress of Theoretical and Experimental Physics, Volume 2017, Issue 10, 1 (arXiv:1708.02078, doi:10.1093/ptep/ptx135)

• Jihn E. Kim, Grand unfication models from $SO(32)$ heterotic string (arXov:2008.00367)

• Hajime Otsuka, $SO(32)$ heterotic line bundle models, (arXiv:1801.03684)

### In type II string theory

Computer scan of Gepner model-compactifications in relation to GUT-models is in

Realization of GUTs in the context of M-theory on G2-manifolds and possible resolution of the doublet-triplet splitting problem is discussed in

• Edward Witten, Deconstruction, $G_2$ Holonomy, and Doublet-Triplet Splitting, (arXiv:hep-ph/0201018)

• Bobby Acharya, Krzysztof Bozek, Miguel Crispim Romao, Stephen F. King, Chakrit Pongkitivanichkul, $SO(10)$ Grand Unification in M theory on a $G_2$ manifold (arXiv:1502.01727)

Discussion of GUTs in F-theory includes

### In Connes-Lott models

Discussion of GUTs within Connes-Lott models:

### Exotica: Leptoquarks, $Z'$-bosons, etc.

Topological defects can play considerable role to constrain the non-SUSY and SUSY GUTs:

• Joydeep Chakrabortty, Rinku Maji, Sunando Kumar Patra, Tripurari Srivastava, Subhendra Mohanty, Roadmap of left-right models based on GUTs (arXiv:1711.11391, Phys.Rev. D97 (2018) no.9, 095010.)

Relation to Z'-bosons:

• S. Sahoo, The prediction of mass of $Z'$-boson in an $SO(10)$-based model, Indian J. Phys. 80 (2), 191-194 (2006) (pdf)

Relation to leptoquarks and flavour anomalies:

• H. Murayama, T. Yanagida, A viable $SU(5)$ GUT with light leptoquark bosons, Mod.Phys.Lett. A7 (1992) 147-152 (arXiv:315898, doi:10.1142/S0217732392000070)

• Damir Bečirević, Ilja Doršner, Svjetlana Fajfer, Nejc Košnik, Darius A. Faroughy, Olcyr Sumensari, Scalar leptoquarks from GUT to accommodate the $B$-physics anomalies, Phys. Rev. D 98, 055003 (2018) (arXiv:1806.05689)

• Ufuk Aydemir, Tanumoy Mandal, Subhadip Mitra, A single TeV-scale scalar leptoquark in SO(10) grand unification and B-decay anomalies (arXiv:1902.08108)

• Julian Heeck, Daniele Teresi, Pati-Salam explanations of the B-meson anomalies, JHEP 12 (2018) 103 (arXiv:1808.07492)

• Julian Heeck, Daniele Teresi, Pati-Salam and lepton universality in B decays (arXiv:1905.05211)

• Michal Malinský, Lepton non-universality in $B$-decays in the minimal leptoquark gauge model (arXiv:1906.09174)

• Shyam Balaji, Michael A. Schmidt, A unified $SU(4)$ theory for the $R_D^{(\ast)}$ and $R_K^{(\ast)}$ anomalies (arXiv:1911.08873)

Modeled on branes:

Last revised on October 2, 2021 at 02:23:07. See the history of this page for a list of all contributions to it.