pith. sign in

arxiv: 2603.15710 · v3 · pith:I74HA2PInew · submitted 2026-03-16 · ✦ hep-ph

Toward a Special E₆to G(2) times SU(3)_A Embedding for Standard Model and Dark Matter and an E₇ Completion Proposal

Nicol\`o Masi This is my paper

Pith reviewed 2026-05-21 10:43 UTC · model grok-4.3

classification ✦ hep-ph
keywords E6 unificationG(2) hidden sectordark glueballsproton decay suppressiongauge coupling unificationE7 completionspecial embeddinghypercharge matching
0
0 comments X

The pith

A special non-regular E6 embedding into G(2) times SU(3) yields a hidden dark sector and recovers exact Standard Model hypercharges after an E7 uplift.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a unified framework using a special embedding of E6 that breaks first to G(2) times SU(3)_A. The G(2) factor provides a hidden strong sector whose gluons confine to form dark glueballs, while SU(3)_A serves as the parent of the electroweak group. A 650 representation scalar sector breaks these groups further to the Standard Model gauge group plus a U(1) from the E7 uplift to match hypercharges exactly. This construction naturally suppresses tree-level leptoquark couplings that cause proton decay in conventional GUTs and achieves one-loop gauge coupling unification. The dark matter sector remains secluded due to group-theoretic orthogonality, with cosmological history including diluted monopoles.

Core claim

The special (non-regular) embedding E6 to G(2) times SU(3)_A establishes the feature of darkness, naturally suppresses tree-level leptoquark couplings, and with an E7 uplift introducing U(1)_X recovers exact Standard Model hypercharges while allowing consistent one-loop gauge coupling unification using a minimal scalar sector from the 650 of E6.

What carries the argument

The special (non-regular) embedding E6 to G(2) times SU(3)_A that carries the argument for darkness and tree-level leptoquark suppression.

If this is right

  • G(2) gluons confine into heavy dark glueballs with parametrically suppressed communication with the SU(3)_A and U(1)_X sectors.
  • The scalar potential for the Higgs sectors yields the heavy gauge-bosons spectrum and a consistent one-loop running across intermediate scales.
  • Cosmological history including topological defects, inflation and reheating shows that monopole relics are naturally diluted.
  • Exotic states are organized in E7-derived vectorlike pairs and made ultraheavy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar non-regular embeddings could be explored in other exceptional groups to generate secluded dark sectors without additional ad hoc assumptions.
  • The ultraheavy exotics might leave imprints on early-universe gravitational wave signals if their mass scale is near the unification threshold.
  • Precision measurements of gauge couplings at future colliders could test the intermediate breaking scales predicted by the 650 representation.

Load-bearing premise

A minimal scalar sector organized around the 650 representation of E6 can implement the breaking steps G(2) to SU(3)_C and SU(3)_A to SU(2)_L times U(1)_A while permitting consistent one-loop running of the gauge couplings that satisfies E6 unification at high scales.

What would settle it

Direct computation of the one-loop beta functions failing to unify at a common scale or observation of light leptoquarks that mediate observable proton decay would falsify the claims.

Figures

Figures reproduced from arXiv: 2603.15710 by Nicol\`o Masi.

Figure 1
Figure 1. Figure 1: One-loop running of gauge couplings with exact piecewise solutions. The legend labels the inverse [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
read the original abstract

We developed a unified framework based on a special (non-regular) embedding of the exceptional group $E_6$ in which the main stage of symmetry breaking chain realizes $E_6\to G(2) \times SU(3)_A$. The exceptional factor $G(2)$ plays the role of a hidden strong sector, while $SU(3)_A$ acts as an ancestor of the electroweak gauge group. A minimal scalar sector is organized around a \(\mathbf{650}\)-based \(E_6\) Higgs sector. Its components $(\mathbf7,\mathbf1)$ and $(\mathbf1,\mathbf8)$ implement the subsequent breaking steps $G(2)\to SU(3)_C$ and $SU(3)_A\to SU(2)_L\times U(1)_A$. The \emph{speciality} of this symmetry breaking establishes the feature of \textit{darkness}. Defining an hypercharge from the $t_{8}$ generator of $SU(3)_A$ is not sufficient to recover the exact Standard Model hypercharges, leading to the necessity of an $E_7$ uplift which introduces a proper additional $U(1)_X$ factor. The special embedding naturally suppresses tree-level leptoquark couplings that typically mediate proton decay in regular GUTs. The scalar potential for the Higgs sectors has been constructed, deriving the heavy gauge-bosons spectrum and presenting a consistent one-loop running of the gauge couplings across the intermediate scales, which is shown to satisfy an $E_6$ unification. The exotic states are organized in \(E_7\)-derived vectorlike pairs and are made ultraheavy. The $G(2)$ gluons ensemble confines into heavy dark glueballs with parametrically suppressed communication with $SU(3)_A$ and $U(1)_X$ sectors. Cosmological history is analyzed, including topological defects, inflation and reheating, demonstrating that monopole relics are naturally diluted. The resulting framework provides a minimal and internally consistent exceptional apparatus which includes the Standard Model and a dark matter sector which is secluded by the group-theoretic orthogonality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a unified framework using a special (non-regular) embedding of E6 into G(2) × SU(3)_A, with a minimal 650-dimensional scalar sector whose (7,1) and (1,8) components break G(2) to SU(3)_C and SU(3)_A to SU(2)_L × U(1)_A. An E7 uplift introduces a U(1)_X factor to recover exact SM hypercharges. The paper claims this embedding naturally suppresses tree-level leptoquark couplings, permits construction of the scalar potential with a derived heavy gauge boson spectrum, yields consistent one-loop gauge coupling running that satisfies E6 unification at high scales, organizes exotics into ultraheavy E7-derived vectorlike pairs, and produces dark glueballs from G(2) confinement as a secluded dark matter sector, with cosmological analysis showing natural dilution of monopoles.

Significance. If the spectrum and running claims hold without light states altering the beta functions, the work offers a group-theoretic mechanism to embed the SM plus a dark sector within exceptional groups, with built-in proton decay suppression and orthogonality-based seclusion of dark matter. The explicit construction of the scalar potential and inclusion of topological defects, inflation, and reheating add concrete elements that could be of interest to GUT model builders.

major comments (3)
  1. [scalar sector discussion (around the 650 Higgs)] The central unification claim rests on the light spectrum from the 650 representation. The text states that the (7,1) and (1,8) components implement the breaking steps G(2)→SU(3)_C and SU(3)_A→SU(2)_L×U(1)_A while other components are heavy, but provides no explicit decomposition of the full 650 under G(2)×SU(3)_A nor the resulting mass hierarchy. Without this, it is impossible to confirm that no additional light scalars modify the one-loop beta functions between the intermediate scales and the unification scale.
  2. [gauge coupling running and unification section] The manuscript asserts that a consistent one-loop running of the gauge couplings is presented across the intermediate scales and satisfies E6 unification. However, the beta-function coefficients for the G(2)×SU(3)_A, SU(3)_C×SU(3)_A, and lower phases are not tabulated, nor are the numerical values of the breaking scales or the resulting coupling trajectories shown with sufficient detail to allow independent verification. This is load-bearing for the unification statement.
  3. [E7 completion and exotic states] The E7 uplift is introduced to recover exact SM hypercharges via an additional U(1)_X. The vectorlike pairs are stated to be ultraheavy so they do not affect running below the E6 scale, but no mechanism or mass estimates are given to ensure all such states lie above the unification scale without residual contributions to the beta functions.
minor comments (2)
  1. [hypercharge definition] The definition of hypercharge from the t8 generator of SU(3)_A and its adjustment via U(1)_X could be made more explicit with generator normalizations.
  2. [unification discussion] The abstract and text refer to 'consistent one-loop running' and 'shown to satisfy E6 unification'; adding a short table of beta coefficients and scale values would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address each major comment below and will incorporate clarifications and additional details in the revised manuscript to improve verifiability while preserving the core claims of the framework.

read point-by-point responses

  1. Referee: [scalar sector discussion (around the 650 Higgs)] The central unification claim rests on the light spectrum from the 650 representation. The text states that the (7,1) and (1,8) components implement the breaking steps G(2)→SU(3)_C and SU(3)_A→SU(2)_L×U(1)_A while other components are heavy, but provides no explicit decomposition of the full 650 under G(2)×SU(3)_A nor the resulting mass hierarchy. Without this, it is impossible to confirm that no additional light scalars modify the one-loop beta functions between the intermediate scales and the unification scale.

    Authors: We agree that an explicit decomposition of the 650 under G(2) × SU(3)_A, together with a clear discussion of the resulting mass hierarchy from the scalar potential, would strengthen the presentation and allow independent confirmation that only the (7,1) and (1,8) components remain light. In the revised manuscript we will add the full branching rules and explain how the potential minimization generates the desired hierarchy, thereby confirming that no additional light scalars enter the beta functions between the intermediate and unification scales. revision: yes

  2. Referee: [gauge coupling running and unification section] The manuscript asserts that a consistent one-loop running of the gauge couplings is presented across the intermediate scales and satisfies E6 unification. However, the beta-function coefficients for the G(2)×SU(3)_A, SU(3)_C×SU(3)_A, and lower phases are not tabulated, nor are the numerical values of the breaking scales or the resulting coupling trajectories shown with sufficient detail to allow independent verification. This is load-bearing for the unification statement.

    Authors: We acknowledge that tabulating the one-loop beta-function coefficients for each successive symmetry phase and supplying explicit numerical values for the breaking scales together with the coupling trajectories would facilitate verification. The revised version will include a dedicated table of beta coefficients for the G(2)×SU(3)_A, SU(3)_C×SU(3)_A and lower stages, along with numerical results and a figure illustrating the running from the intermediate scales up to the E6 unification point. revision: yes

  3. Referee: [E7 completion and exotic states] The E7 uplift is introduced to recover exact SM hypercharges via an additional U(1)_X. The vectorlike pairs are stated to be ultraheavy so they do not affect running below the E6 scale, but no mechanism or mass estimates are given to ensure all such states lie above the unification scale without residual contributions to the beta functions.

    Authors: We will expand the E7 completion section to specify the mechanism that renders the vectorlike exotic pairs ultraheavy. In the revision we will discuss how higher-dimensional operators allowed by the E7 structure generate masses parametrically above the unification scale, and we will provide order-of-magnitude estimates showing that residual contributions to the beta functions below this scale remain negligible. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained against external benchmarks

full rationale

The paper constructs an explicit scalar potential for the 650 representation, derives the heavy gauge boson spectrum from the stated VEVs of the (7,1) and (1,8) components, and presents one-loop beta-function running that meets at a high scale under the listed light spectrum. No equation reduces by construction to a fitted parameter renamed as a prediction, no self-citation supplies a uniqueness theorem that forbids alternatives, and the unification statement is not shown to be equivalent to its input assumptions via the paper's own algebra. The framework is therefore internally consistent on its stated assumptions without load-bearing circular steps.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 2 invented entities

The framework depends on the existence of the special embedding, viability of the 650 Higgs for the stated breakings, and choice of scales to achieve unification; several group-theoretic assumptions and postulated heavy states are introduced without independent falsifiable handles in the abstract.

free parameters (1)
  • intermediate symmetry breaking scales
    Scales separating G(2)→SU(3)_C and SU(3)_A→electroweak stages are implicitly chosen to produce consistent one-loop unification.
axioms (2)
  • domain assumption Existence and properties of a special non-regular embedding E6 → G(2) × SU(3)_A
    Central to the symmetry breaking chain and darkness feature described in the abstract.
  • domain assumption The 650 representation permits a minimal scalar sector realizing the required breakings
    Invoked to implement G(2)→SU(3)_C and SU(3)_A→SU(2)_L×U(1)_A.
invented entities (2)
  • U(1)_X factor from E7 uplift no independent evidence
    purpose: To recover exact Standard Model hypercharges
    Introduced because t8 generator of SU(3)_A alone is insufficient.
  • Dark glueballs from G(2) gluons no independent evidence
    purpose: Secluded dark matter candidates
    Confined states of the hidden strong sector with suppressed communication.

pith-pipeline@v0.9.0 · 5949 in / 1823 out tokens · 67287 ms · 2026-05-21T10:43:08.048113+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

160 extracted references · 160 canonical work pages · 1 internal anchor

  1. [1]

    Thomson,Modern particle physics

    M. Thomson,Modern particle physics. Cambridge University Press, 2013

    work page 2013

  2. [2]

    Bertone,Particle Dark Matter: Observations, Models and Searches

    G. Bertone,Particle Dark Matter: Observations, Models and Searches. Cambridge University Press,

    work page

  3. [3]

    Available: http://qut.eblib.com.au/patron/FullRecord.aspx?p=542829 33

    [Online]. Available: http://qut.eblib.com.au/patron/FullRecord.aspx?p=542829 33

    work page

  4. [4]

    Profumo,An introduction to particle dark matter

    S. Profumo,An introduction to particle dark matter. World Scientific Publishing Europe Ltd, 2017

    work page 2017

  5. [5]

    Mitsou,Infrared non-local modifications of general relativity

    E. Mitsou,Infrared non-local modifications of general relativity. Springer, 2016

    work page 2016

  6. [6]

    Eleftherios,Modifications of Einstein’s theory of gravity at large distances

    P. Eleftherios,Modifications of Einstein’s theory of gravity at large distances. Springer, 2015. [Online]. Available: http://link.springer.com/book/10.1007/978-3-319-10070-8

    work page doi:10.1007/978-3-319-10070-8 2015

  7. [7]

    Modified Gravity and Cosmology,

    T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, “Modified Gravity and Cosmology,”Phys. Rept., vol. 513, pp. 1–189, 2012

    work page 2012

  8. [8]

    Trinification frome 6 symmetry breaking,

    K. Babu, “Trinification frome 6 symmetry breaking,”JHEP, 2023

    work page 2023

  9. [9]

    A realistic theory ofe 6 unification through novel intermediate stages,

    K. S. Babu, “A realistic theory ofe 6 unification through novel intermediate stages,”JHEP, 2024

    work page 2024

  10. [10]

    Special grand unification,

    N. Yamatsu, “Special grand unification,”Prog. Theor. Exp. Phys., vol. 2017, p. 061B01, 2017

    work page 2017

  11. [11]

    Non-abelian dark sectors and their collider signatures,

    L. Forestell, D. E. Morrissey, and K. Sigurdson, “Non-abelian dark sectors and their collider signatures,” Phys. Rev. D, vol. 95, p. 015032, 2017

    work page 2017

  12. [12]

    Non-Abelian Dark Forces and the Relic Densities of Dark Glueballs,

    ——, “Non-Abelian Dark Forces and the Relic Densities of Dark Glueballs,”Phys. Rev., vol. D95, no. 1, p. 015032, 2017

    work page 2017

  13. [13]

    Hidden SU(N) Glueball Dark Matter,

    A. Soni and Y . Zhang, “Hidden SU(N) Glueball Dark Matter,”Phys. Rev. D, vol. 93, no. 11, p. 115025, 2016

    work page 2016

  14. [14]

    The resurgence of the G(2) group for the strong sector and the emergence of dark matter,

    N. Masi, “The resurgence of the G(2) group for the strong sector and the emergence of dark matter,” Nuclear Physics B, vol. 1004, p. 116562, 2024

    work page 2024

  15. [15]

    An exceptional G(2) extension of the Standard Model from the correspondence with Cayley–Dickson algebras automorphism groups,

    ——, “An exceptional G(2) extension of the Standard Model from the correspondence with Cayley–Dickson algebras automorphism groups,”Scientific Reports, vol. 11, no. 1, nov 2021. [Online]. Available: https://doi.org/10.1038%2Fs41598-021-01814-1

    work page 2021

  16. [16]

    SU(n) gauge theories in four dimensions: exploring the approach to n=∞ ,

    B. Lucini and M. Teper, “SU(n) gauge theories in four dimensions: exploring the approach to n=∞ ,” 2001

    work page 2001

  17. [17]

    The glueball spectrum from an anisotropic lattice study,

    C. J. Morningstar and M. Peardon, “The glueball spectrum from an anisotropic lattice study,”Phys. Rev. D, vol. 60, p. 034509, 1999

    work page 1999

  18. [18]

    Exceptional confinement in G(2) gauge theory,

    K. Holland, P. Minkowski, M. Pepe, and U.-J. Wiese, “Exceptional confinement in G(2) gauge theory,” Nuclear Physics B, vol. 668, pp. 207–236, 2003

    work page 2003

  19. [19]

    Greensite,An Introduction to the Confinement Problem, ser

    J. Greensite,An Introduction to the Confinement Problem, ser. Lecture Notes in Physics. Springer, 2011, vol. 821

    work page 2011

  20. [20]

    G2 gauge theories,

    A. Maas and B. H. Wellegehausen, “G2 gauge theories,”PoS, vol. LATTICE2012, p. 080, 2012

    work page 2012

  21. [21]

    Klein-gordon geon,

    D. J. Kaup, “Klein-gordon geon,”Physical Review, vol. 172, pp. 1331–1342, 1968

    work page 1968

  22. [22]

    Systems of self-gravitating particles in general relativity and the concept of an equation of state,

    R. Ruffini and S. Bonazzola, “Systems of self-gravitating particles in general relativity and the concept of an equation of state,”Physical Review, vol. 187, pp. 1767–1783, 1969

    work page 1969

  23. [23]

    Boson stars: Gravitational equilibria of self-interacting scalar fields,

    M. Colpi, S. L. Shapiro, and I. Wasserman, “Boson stars: Gravitational equilibria of self-interacting scalar fields,”Physical Review Letters, vol. 57, pp. 2485–2488, 1986

    work page 1986

  24. [24]

    Dynamical boson stars,

    S. L. Liebling and C. Palenzuela, “Dynamical boson stars,”Living Reviews in Relativity, vol. 15, p. 6, 2012

    work page 2012

  25. [25]

    Maximal subgroups of the classical groups,

    E. B. Dynkin, “Maximal subgroups of the classical groups,”Trudy Moskov. Mat. Obshch., vol. 1, pp. 39–166, 1952, english transl.: Amer. Math. Soc. Transl. (6) (1957) 245–378. 34

    work page 1952

  26. [26]

    Semisimple subalgebras of semisimple lie algebras,

    E. Dynkin, “Semisimple subalgebras of semisimple lie algebras,”Mathematics of the USSR Sbornik, vol. 30, pp. 349–462, 1957

    work page 1957

  27. [27]

    Finite-dimensional lie algebras and their representations for unified model building,

    N. Yamatsu, “Finite-dimensional lie algebras and their representations for unified model building,” arXiv e-prints, 2015

    work page 2015

  28. [28]

    Group theory for unified model building,

    R. Slansky, “Group theory for unified model building,”Physics Reports, vol. 79, pp. 1–128, 1981

    work page 1981

  29. [29]

    E 6 as a unifying gauge symmetry,

    Q. Shafi, “E 6 as a unifying gauge symmetry,”Phys. Lett. B, vol. 79, pp. 301–304, 1978

    work page 1978

  30. [30]

    A universal gauge theory model based on E6,

    F. G¨ursey, P. Ramond, and P. Sikivie, “A universal gauge theory model based on E6,”Physics Letters B, vol. 60, pp. 177–180, 1976

    work page 1976

  31. [31]

    Quark-lepton symmetry and mass scales in an E6 unified gauge model,

    Y . Achiman and B. Stech, “Quark-lepton symmetry and mass scales in an E6 unified gauge model,” Physics Letters B, vol. 77, pp. 389–393, 1978

    work page 1978

  32. [32]

    ”low-energy phenomenology of superstring inspirede6 models

    J. L. Hewett and T. G. Rizzo, “”low-energy phenomenology of superstring inspirede6 models”,”Physics Reports, vol. 183, pp. 193–381, 1989

    work page 1989

  33. [33]

    Georgi,Lie Algebras in Particle Physics: From Isospin to Unified Theories, 2nd ed

    H. Georgi,Lie Algebras in Particle Physics: From Isospin to Unified Theories, 2nd ed. Westview Press, 1999

    work page 1999

  34. [34]

    Baryon and lepton nonconserving processes,

    S. Weinberg, “Baryon and lepton nonconserving processes,”Phys. Rev. Lett., vol. 43, p. 1566, 1979

    work page 1979

  35. [35]

    Operator analysis of nucleon decay,

    F. Wilczek and A. Zee, “Operator analysis of nucleon decay,”Phys. Rev. Lett., vol. 43, p. 1571, 1979

    work page 1979

  36. [36]

    Search for proton decay via p→e +π0 and p→µ +π0 in 0.31 megaton·years exposure of the super-kamiokande water cherenkov detector,

    K. Abeet al., “Search for proton decay via p→e +π0 and p→µ +π0 in 0.31 megaton·years exposure of the super-kamiokande water cherenkov detector,”Physical Review D, vol. 95, p. 012004, 2017

    work page 2017

  37. [37]

    Review of particle physics,

    P. D. Group, R. L. Workmanet al., “Review of particle physics,”Progress of Theoretical and Experi- mental Physics, vol. 2024, p. 083C01, 2024

    work page 2024

  38. [38]

    Inflationary universe: A possible solution to the horizon and flatness problems,

    A. H. Guth, “Inflationary universe: A possible solution to the horizon and flatness problems,”Physical Review D, vol. 23, pp. 347–356, 1981

    work page 1981

  39. [39]

    Chaotic inflation,

    A. D. Linde, “Chaotic inflation,”Physics Letters B, vol. 129, pp. 177–181, 1983

    work page 1983

  40. [40]

    A new inflationary universe scenario,

    A. Linde, “A new inflationary universe scenario,”Phys. Lett. B, vol. 108, p. 389, 1982

    work page 1982

  41. [41]

    Baryogenesis without grand unification,

    M. Fukugita and T. Yanagida, “Baryogenesis without grand unification,”Physics Letters B, vol. 174, pp. 45–47, 1986

    work page 1986

  42. [42]

    Baryogenesis via leptogenesis: Spontaneous b and l violation,

    P. Fileviez P´erez, C. Murgui, and A. D. Plascencia, “Baryogenesis via leptogenesis: Spontaneous b and l violation,”Phys. Rev. D, vol. 104, p. 055007, Sep 2021. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.104.055007

    work page doi:10.1103/physrevd.104.055007 2021

  43. [43]

    Leptogenesis in the universe,

    C. S. Fong, E. Nardi, and A. Riotto, “Leptogenesis in the universe,”Adv. High Energy Phys., vol. 2012, p. 158303, 2012

    work page 2012

  44. [44]

    Unity of all elementary-particle forces,

    H. Georgi and S. L. Glashow, “Unity of all elementary-particle forces,”Physical Review Letters, vol. 32, pp. 438–441, 1974

    work page 1974

  45. [45]

    Unified interactions of leptons and hadrons,

    H. Fritzsch and P. Minkowski, “Unified interactions of leptons and hadrons,”Annals of Physics, vol. 93, pp. 193–266, 1975

    work page 1975

  46. [46]

    G. G. Ross,Grand Unified Theories. Westview Press, 1984

    work page 1984

  47. [47]

    R. N. Mohapatra,Unification and Supersymmetry: The Frontiers of Quark-Lepton Physics, 3rd ed. Springer, 2003. 35

    work page 2003

  48. [48]

    Lower limits on proton lifetime from cosmology and supergravity,

    J. Ellis, D. V . Nanopoulos, and K. A. Olive, “Lower limits on proton lifetime from cosmology and supergravity,”Physics Letters B, vol. 112, pp. 459–463, 1982

    work page 1982

  49. [49]

    & Knops, H

    M. Serdaro ˇglu, “An E6 gauge field theory model,”Physica A: Statistical Mechanics and its Applications, vol. 114, no. 1, pp. 271–277, 1982. [Online]. Available: https: //www.sciencedirect.com/science/article/pii/0378437182902953

    work page arXiv 1982

  50. [50]

    Extra gauge bosons in e6,

    D. London and J. L. Rosner, “Extra gauge bosons in e6,”Phys. Rev. D, vol. 34, pp. 1530–1546, Sep

    work page

  51. [51]

    Available: https://link.aps.org/doi/10.1103/PhysRevD.34.1530

    [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.34.1530

    work page doi:10.1103/physrevd.34.1530

  52. [52]

    Grand unified theories,

    S. Raby, “Grand unified theories,”Rept. Prog. Phys., vol. 79, p. 036901, 2006

    work page 2006

  53. [53]

    Unification and supersymmetry,

    R. N. Mohapatra, “Unification and supersymmetry,”Springer, 1986

    work page 1986

  54. [54]

    Grand unified theories,

    P. D. Group, “Grand unified theories,”Prog. Theor. Exp. Phys., 2025

    work page 2025

  55. [55]

    Fermion masses and coupling unification in E6 : Life in the desert,

    B. Stech and Z. Tavartkiladze, “Fermion masses and coupling unification in E6 : Life in the desert,”Phys. Rev. D, vol. 70, p. 035002, Aug 2004. [Online]. Available: https: //link.aps.org/doi/10.1103/PhysRevD.70.035002

    work page doi:10.1103/physrevd.70.035002 2004

  56. [56]

    Generation symmetry and E6 unification,

    B. e. a. Stech, “Generation symmetry and E6 unification,”Phys. Rev. D, vol. 77, p. 076009, Apr 2008. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.77.076009

    work page doi:10.1103/physrevd.77.076009 2008

  57. [57]

    Mass of the Higgs boson in the trinification subgroup of E6,

    B. Stech, “Mass of the Higgs boson in the trinification subgroup of E6,”Phys. Rev. D, vol. 86, p. 055003, Sep 2012. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.86.055003

    work page doi:10.1103/physrevd.86.055003 2012

  58. [58]

    Trinification from E6 symmetry breaking,

    K. S. Babu, B. Bajc, and V . Susic, “Trinification from E6 symmetry breaking,”Journal of High Energy Physics, vol. 2023, no. 7, Jul. 2023. [Online]. Available: http://dx.doi.org/10.1007/JHEP07(2023)011

    work page doi:10.1007/jhep07(2023)011 2023

  59. [59]

    Dark matter in E 6 Grand unification,

    J. Schwichtenberg, “Dark matter in E 6 Grand unification,”JHEP, vol. 02, p. 016, 2018

    work page 2018

  60. [60]

    The physics of heavy Z’ gauge bosons,

    P. Langacker, “The physics of heavy Z’ gauge bosons,”Reviews of Modern Physics, vol. 81, pp. 1199–1228, 2009

    work page 2009

  61. [61]

    Phenomenology of an E6 inspired extension of the Standard Model: Higgs sector,

    S. Bhattacharyya and A. Datta, “Phenomenology of an E6 inspired extension of the Standard Model: Higgs sector,”Phys. Rev. D, vol. 105, p. 075021, Apr 2022. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.105.075021

    work page doi:10.1103/physrevd.105.075021 2022

  62. [62]

    Proton stability in grand unified theories, in strings, and in branes,

    P. Nath and P. Fileviez Perez, “Proton stability in grand unified theories, in strings, and in branes,”Phys. Rept., vol. 441, pp. 191–317, 2007

    work page 2007

  63. [63]

    Radiative corrections as the origin of spontaneous symmetry breaking,

    S. Coleman and E. Weinberg, “Radiative corrections as the origin of spontaneous symmetry breaking,” Phys. Rev. D, vol. 7, p. 1888, 1973

    work page 1973

  64. [64]

    On the representations of the semisimple lie groups. II. the exceptional groups,

    G. E. Baird and L. C. Biedenharn, “On the representations of the semisimple lie groups. II. the exceptional groups,”Journal of Mathematical Physics, vol. 5, pp. 1723–1733, 1964

    work page 1964

  65. [65]

    Lieart 2.0 – a mathematica application for lie algebras and representation theory,

    R. Feger, T. W. Kephart, and R. J. Saskowski, “Lieart 2.0 – a mathematica application for lie algebras and representation theory,”Computer Physics Communications, vol. 257, p. 107490, 2020. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0010465520302290

    work page 2020

  66. [66]

    Weinberg,The Quantum Theory of Fields

    S. Weinberg,The Quantum Theory of Fields. Vol. 2: Modern Applications. Cambridge University Press, 1996

    work page 1996

  67. [67]

    Masset, R

    M. Pepe, “Confinement and the center of the gauge group,”Nuclear Physics B - Proceedings Supplements, vol. 153, no. 1, pp. 207–214, Mar 2006. [Online]. Available: http://dx.doi.org/10.1016/j. nuclphysbps.2006.01.045

    work page doi:10.1016/j 2006

  68. [68]

    Exceptional deconfinement in gauge theory,

    M. Pepe and U.-J. Wiese, “Exceptional deconfinement in gauge theory,”Nuclear Physics B, vol. 768, no. 1-2, pp. 21–37, Apr 2007. [Online]. Available: http://dx.doi.org/10.1016/j.nuclphysb.2006.12.024 36

    work page doi:10.1016/j.nuclphysb.2006.12.024 2007

  69. [69]

    Grand unified theories and proton decay,

    P. Langacker, “Grand unified theories and proton decay,”Physics Reports, vol. 72, pp. 185–385, 1981

    work page 1981

  70. [70]

    Infrared singularities and massive fields,

    T. Appelquist and J. Carazzone, “Infrared singularities and massive fields,”Phys. Rev. D, vol. 11, pp. 2856–2861, 1975

    work page 1975

  71. [71]

    Phenomenology of exotic particles in e(6) theories,

    R. W. Robinett and J. L. Rosner, “Phenomenology of exotic particles in e(6) theories,”Phys. Rept., vol. 89, pp. 223–322, 1982

    work page 1982

  72. [72]

    Low-energy phenomenology of superstring-inspiredE 6 models,

    J. L. Hewett and T. G. Rizzo, “Low-energy phenomenology of superstring-inspiredE 6 models,”Phys. Rept., vol. 183, pp. 193–381, 1989

    work page 1989

  73. [73]

    R. A. Bertlmann,Anomalies in Quantum Field Theory. Oxford University Press, 1996

    work page 1996

  74. [74]

    E6tensors: A mathematica package for e6 tensors,

    T. Deppisch, “E6tensors: A mathematica package for e6 tensors,”Computer Physics Communications, vol. 213, pp. 130–135, 2017. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S0010465516302818

    work page 2017

  75. [75]

    Electroweak higgs potentials and vacuum stability,

    M. Sher, “Electroweak higgs potentials and vacuum stability,”Phys. Rept., vol. 179, p. 273, 1989

    work page 1989

  76. [76]

    Investigating the near-criticality of the higgs boson,

    D. e. a. Buttazzo, “Investigating the near-criticality of the higgs boson,”JHEP, vol. 12, p. 089, 2013

    work page 2013

  77. [77]

    The cp-conserving two-higgs-doublet model: The approach to the decoupling limit,

    J. F. Gunion and H. E. Haber, “The cp-conserving two-higgs-doublet model: The approach to the decoupling limit,”Phys. Rev. D, vol. 67, p. 075019, 2003

    work page 2003

  78. [78]

    The standard model as an effective field theory,

    I. Brivio and M. Trott, “The standard model as an effective field theory,”Phys. Rept., vol. 793, pp. 1–98, 2019

    work page 2019

  79. [79]

    Phenomenology of higgs triplet model,

    E. J. Chun, K. Kim, and J. Lee, “Phenomenology of higgs triplet model,”Phys. Rev. D, vol. 66, p. 073003, 2002

    work page 2002

  80. [80]

    Heavy higgs bosons in the type-ii seesaw model,

    Y . Cai, T. Han, T. Li, and R.-J. Zhang, “Heavy higgs bosons in the type-ii seesaw model,”Phys. Rev. D, vol. 96, p. 035027, 2017

    work page 2017

Showing first 80 references.