Grand-unification Theory Atlas: Standard Model and Beyond

Aldo Deandrea; Francesco Sannino; Giacomo Cacciapaglia; Konstantinos Kollias

REVIEW 2 major objections 5 minor 2 cited by

Grand-unification Theory Atlas: Standard Model and Beyond

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper charts simple gauge theories with massless fermion families and shows that a free-energy degree count selects SU(5) Georgi-Glashow as the minimal three-generation grand-unified theory, with SO(10) close behind.

desk verdict Useful atlas, solid classification, but the SU(5) 'single-out' is a choice of compass, not a unique prediction. read the letter →

arxiv 2507.06368 v3 pith:EXSUPWSD submitted 2025-07-08 hep-ph hep-th

Giacomo Cacciapaglia , Aldo Deandrea , Konstantinos Kollias , Francesco Sannino This is my paper

classification hep-phhep-th

keywords grandunificationgaugetheoryatlasfreeenergyfermionfamiliesasymptoticfreedomdualitySO(10)SU(5)

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to make grand unification a matter of classification rather than taste. It defines an atlas of simple gauge theories whose fermion content consists only of families whose masses are forbidden by gauge invariance, subject to asymptotic freedom and a single non-abelian gauge group, with no light scalars. It then proposes a compass: the high-temperature free energy, which counts the gauge and fermionic degrees of freedom of each theory. For three fermion generations, that counter picks the SU(5) Georgi-Glashow model as the minimal grand-unified theory, with SO(10) spinorial matter as the runner-up. If the atlas and its compass are right, the standard model has a principled place among high-energy theories, and any beyond-standard-model gauge extension can be checked for whether it admits a grand-unified completion.

What carries the argument

The machinery has two parts. The first is the atlas itself: the three conditions that a theory must satisfy to be in it (one simple non-abelian gauge group; anomaly-free irreducible fermion families whose gauge-invariant mass terms are absent; asymptotic freedom of the gauge coupling), together with the division of families into chiral and pseudo-real types. The second is the compass: the free-energy counter $f_{\rm FE}=d_G+\frac{7}{8}\sum_f n_f^g\, n_f$, where $d_G$ is the dimension of the adjoint representation and $n_f$ is the number of Weyl spinors in a single family $f$. Asymptotic freedom sets a maximum number of family copies, $n_f^g<1/\xi_f$ with $\xi_f=\frac{2}{11C_G}\sum_{r\in f}T_r$, so the counter ranks only theories that can in principle reach arbitrarily high energies. An alternative counter $f_a$, derived from the a-function, gives more weight to gauge bosons and is used as a cross-check.

What would settle it

Run an independent enumeration of all asymptotically free simple gauge theories with three anomaly-free chiral fermion families and no gauge-invariant mass terms, computing $f_{\rm FE}$ for each; the central claim is false if any theory with $f_{\rm FE}<63.4$ exists, so the completeness of the paper's Table II is what must be checked.

Watch

Extended reading notes

Core claim

The central claim is that an ab-initio-defined atlas of simple gauge theories, navigated by a degree-of-freedom counter, contains the standard model's unification in a natural way. Specifically, among all asymptotically free simple gauge theories with three identical anomaly-free chiral fermion families and no light scalars, the high-temperature free energy $f_{\rm FE}=d_G+\frac{7}{8}\sum_f n_f^g\, n_f$ is minimized by SU(5) with a $10\oplus\bar 5$ family ($f_{\rm FE}=63.4$), followed by SO(10) with three copies of the 16 spinor ($f_{\rm FE}=87$). The paper further uses the atlas to define the 'dryland' of grand-unifiable gauge extensions of the standard model. Two applications are worked out: a magnetic-dual completion of the standard model is viable only for three generations, and the unique SU(5)×SU(N) extension allowed by the atlas is based on SU(8) with one family $56_{$A^{3}$}+2\times28_A+3\times8_F$, whose confined SU(3) sector could produce composite Higgs-like scalars and partial-compositeness partners.

Load-bearing premise

The ranking stands only if the ultraviolet theory is one simple gauge group with only fermion families whose masses are forbidden by gauge invariance and no light scalar fields; weaken any of these conditions and the free-energy ordering and dryland conclusions are not guaranteed.

Editorial extensions

If this is right

The SU(5) Georgi-Glashow theory is the most economical grand-unified model with three generations under the stated assumptions, so minimality arguments in GUT model building should start from it, with SO(10) spinorial matter as the next candidate.
The same ordering survives in the supersymmetric extension, except that asymptotic freedom eliminates some families, so the atlas also points to SU(5) and SO(10) in the SUSY case.
Any standard-model gauge extension that cannot be embedded into a simple group belonging to the atlas is outside the grand-unifiable 'dryland' and would need extra assumptions or additional gauge factors to reach a unified description.
A magnetic-gauge completion of the standard model with more than three generations is not grand-unifiable within the atlas, which selects $N_g=3$ for this class of dual theories.
The single viable SU(5)×SU(N) extension found by the atlas is based on SU(8) with the family $56_{A^3}+2\times28_A+3\times8_F$; its confined SU(3) dynamics could provide composite scalars with standard-model Higgs quantum numbers and composite partners for the third generation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the ranking fixes $N_g=3$ as an input rather than deriving it; a full explanation of why there are three generations would need a separate dynamical mechanism, and the atlas is a selection rule only among theories that already have three families.
Beyond the paper: the $f_{\rm FE}$ gap between SU(5) and SO(10) is not huge, so including scalar sectors or other minimality measures could plausibly reorder the top two; the more robust conclusion from the atlas may be that the preferred GUTs form the pair {SU(5), SO(10)}.
Beyond the paper: the same decomposition algorithm used for SU(5)×SU(N) could be applied to other standard-model extensions such as Pati-Salam or trinification, producing a systematic map of which gauge structures are grand-unifiable.
Beyond the paper: if the SU(8) composite-Higgs scenario is realized, the new SU(3) confinement scale near the electroweak scale would imply new bound states around the TeV scale, which is a concrete collider signature that follows from the atlas but is not explored in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this paper's real contribution is the atlas itself—a systematic enumeration of simple gauge groups with anomaly-free, asymptotically free chiral families, with tables for SU(n) up to SU(17) that look correct and will be useful. The free-energy compass is a new organizing principle, and the dryland analysis is a genuinely new application. But the advertised 'single-out' of SU(5) is not robust: it holds for fFE, not for the a-function counter fa that the paper itself defines. With fa, SU(4) ranks below SU(5) at Ng=3, and that SU(4) model cannot contain the SM. The authors disclose this in the text (Table II and the paragraph after it), yet the abstract and the 'FINDING THE STANDARD MODEL' section still present fFE as singling out SU(5). That overstates the claim.

More broadly, the compass is a heuristic, not a derivation. Counting degrees of freedom is a sensible way to rank theories, but nothing forces that choice; Ng is an input, and the atlas assumptions (single simple gauge group, no light scalars, only fermions whose mass is forbidden by gauge invariance) are model-building decisions that exclude many candidate GUTs. The paper is candid about these limitations, which I appreciate. The classification arithmetic checks out—beta functions, anomaly cancellation, and the fFE/fa values are internally consistent.

The dryland part is the most original. Checking whether a given SM extension can be embedded in a simple GUT, with the SU(8) example, is a neat and testable-in-principle way to constrain model building, even though the SU(8) model itself relies on an extra strongly coupled sector that is speculative. The dual-SM discussion is also sensible, though it inherits the same assumptions.

Who benefits? Model builders and anyone mapping the space of chiral gauge theories. The paper won't change phenomenology on its own, but it provides a useful reference and a language for ranking GUT candidates.

My recommendation: send it to peer review at a good hep-ph journal, but ask the authors to revise the abstract to say 'under the free-energy compass' rather than 'the free energy singles out,' and to give the fa comparison equal prominence in the main text. With that, the paper is a solid contribution. I'd cite it if I worked on GUT model building.

Referee Report

2 major / 5 minor

Summary. The paper defines a 'Grand-Unification Theory Atlas' (GTA): simple non-abelian gauge theories in 4D with anomaly-free families of massless fermions whose mass is forbidden by gauge invariance, subject to asymptotic freedom. A 'free-energy compass' fFE = dG + (7/8)∑ n_g n_f is proposed to rank models, with a second counter fa from the a-function introduced for comparison. For Ng = 3 identical chiral generations, Table II gives SU(5) Georgi-Glashow as the minimal fFE model (63.4), followed by SO(10) (87). The atlas is then applied to two beyond-SM settings: gauge-dual completions of the SM, and SU(5)×SU(N) extensions, leading to a unique SU(8) model in the 'dryland'.

Significance. If the classification is taken as given, the paper provides a clean, transparent enumeration of asymptotically free chiral and pseudo-real families for the classical groups, with explicit representation-theoretic data in the supplemental tables. The arithmetic of fFE and the beta-function bounds is straightforward and internally consistent, and the acknowledgment of limitations (Ng as input, no scalar sector, pseudo-real families deferred) is honest. The main significance is as a reference atlas and an organizing criterion for model comparison. Its central physical message, however, is conditional: the 'selection' of SU(5) is a property of the fFE compass, not an ab initio prediction, and the alternative fa counter already shows different rankings in the paper's own Table II.

major comments (2)

[Abstract; 'FINDING THE STANDARD MODEL'; Eq. (4); Table II] The central claim that 'the free energy singles out SU(5)' is not robust to the choice of compass. As Eq. (4) and Table II show, the alternative counter fa ranks the SU(4) model 10S + 8×4F at fa = 4.51, below SU(5) at fa = 4.82. Although the text states this explicitly ('As expected, fa gives larger weights... only surpassed by SU(4)'), the abstract and Outlook present fFE as selecting SU(5) without the caveat that this is one of two degree-of-freedom counters considered. Since no physical argument is given for preferring fFE over fa, the selection claim should be framed as 'minimal under the fFE compass' or, if SM embeddability is imposed, that further condition should be stated as part of the selection rule. This is a load-bearing interpretive point, not an arithmetic error.
['WHICH GAUGE EXTENSION?', Eq. (9) and surrounding text] The uniqueness claim for the SU(8) dryland model ('after analyzing all chiral families for N ≥ 2, we found a unique viable model') is presented without the supporting enumeration. The reader cannot verify that no other SU(N+5) chiral family satisfies the three-generation and SM-embedding requirements, and the paper does not provide the search algorithm, bounds on N, or a repository of the enumeration. Since this section is one of the two advertised applications of the atlas, the completeness of the dryland claim needs explicit support, either in the main text or in the supplemental material.

minor comments (5)

[Introduction] 'chromomagnetic' appears to be a typo for 'chromodynamic' in the description of strong interactions.
[Eq. (2) and Table I] The notation for the number of copies of a family, variously typeset as n f g, n_g^f, and n_g, is confusing; a single consistent symbol such as n_g^{(f)} should be defined once.
[Supplemental Material, after Table S-1] 'integer Dunkin index' should read 'integer Dynkin index'.
[Supplemental Material, Eq. (15)] The inequality display in Eq. (15) is poorly formatted and hard to parse; it should be rewritten with clear parentheses.
[Supplementary Tables S-4 through S-19] The 'condition' columns are informative but the ranges such as '2≤ x≤ 5' often refer to x without a definition immediately visible in the table heading; adding a legend would improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the SU(5) minimum is a direct evaluation of the fixed fFE counter over an independently enumerated atlas; the alternative fa ranking is a disclosed conditionality, not a self-referential reduction.

full rationale

The paper's central selection claim is not circular in the technical sense. The free-energy counter fFE is defined in Eq. (2) as dG + (7/8)Σ n_g^f n_f, a standard high-temperature free-energy count that contains no SU(5)-specific input; the ranking in Table II is obtained by evaluating this fixed function over the independently enumerated set of anomaly-free, asymptotically free chiral families. The Ng = 3 input is explicit and restricts the comparison, but it does not encode the SU(5) representation content. The alternative a-function counter fa in Eq. (4) is also defined and evaluated, and the paper explicitly discloses that fa ranks the SU(4) model below SU(5); the text immediately notes that this SU(4) model cannot contain the Standard Model gauge symmetry. This makes the 'single-out' conditional on the choice of fFE as the compass, which is a robustness or underdetermination concern, not a circular derivation. The self-citations present (free-energy references, the a-function expression, and the gauge-duality constructions) are used for standard formulas or prior duality frameworks; the atlas enumeration and the fFE arithmetic are self-contained and would not change if those citations were removed. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors' prior work is invoked to forbid alternative compasses. The central result therefore has independent content beyond its inputs, and the paper's own disclosure of the fa ordering prevents the claim from reducing to a hidden fit.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central claim rests on the definition of the GTA: simple gauge group, anomaly-free asymptotic-freedom, massless fermion families only, and on the choice of the free-energy counter fFE. None of these are derived from first principles; they are modeling choices. There are no fitted numerical parameters. The only invented entity is the speculative SU(3) composite sector in the SU(8) dryland example, which is not needed for the main claim.

assumptions (5)

domain assumption Any mass term that is not forbidden by a local symmetry exists and is as large as possible.
Used to justify restricting matter to fermions whose mass is forbidden by gauge invariance (Introduction, third paragraph). This is a naturalness-style assumption, not a proven principle.
domain assumption Below the Planck scale, the theory of Nature features Weyl fermions and at least one gauge group, with no light scalars.
Defines the content of the atlas (Introduction, third paragraph). Excludes scalar fields and possible vector-like fermions.
ad hoc to paper The gauge symmetry is a single non-abelian simple Lie group.
Condition 1 in defining the GTA (Section 1). Multiple gauge factors and abelian factors are excluded, which is a restriction not derived from first principles.
domain assumption Asymptotic freedom is required for the gauge coupling.
Condition 3 in the GTA; used to define viable high-energy theories and to bound the number of families via Eq. (5).
domain assumption The high-temperature free energy of the theory is approximated by the free-gas expression fFE = dG + (7/8) sum n_f (Eq. 2).
This is the proposed 'compass'; it ignores interactions and is not derived from the full dynamics. It is a counting device, not a thermodynamic result for the interacting theory.

invented entities (1)

Additional strongly coupled SU(3) gauge sector in the SU(8) dryland model
purpose: In the dryland example, a new strongly coupled SU(3) factor confines near the electroweak scale, producing composite scalars and baryons that could serve as the Higgs and partial compositeness partners.
This is a speculative model-building proposal (Section 'WHICH GAUGE EXTENSION?'), not a prediction with an external observable independent of the model.

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Cite this review

Pith. "Pith review of Grand-unification Theory Atlas: Standard Model and Beyond." pith.science (2026). https://pith.science/paper/EXSUPWSD

@misc{pith2026250706368,
  author       = {Pith},
  title        = {Pith review of: Grand-unification Theory Atlas: Standard Model and Beyond},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EXSUPWSD}},
  note         = {Machine review of arXiv:2507.06368}
}

read the original abstract

Under a reasonable set of ab-initio assumptions, we define and chart the atlas of simple gauge theories with families of fermions whose masses are forbidden by gauge invariance. We propose a compass to navigate the atlas based on counting degrees of freedom. When searching for Grand-unification Theories with three matter generations, the free energy singles out the SU(5) Georgi-Glashow model as the minimal one, closely followed by SO(10) with spinorial matter. The atlas also defines the dryland of grand-unifiable gauge extensions of the standard model. We further provide examples relevant for gauge dual completions of the standard model as well as extensions by an additional SU(N) gauge symmetry.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The asymptotically-free gauge theories
hep-th 2025-07 conditional novelty 7.0 of 10

All asymptotically-free gauge theories with purely fermionic matter in 4D are classified by finite tables in which at most two Dynkin labels are nonzero and none exceeds four.
Comment to "The asymptotically-free gauge theories"
hep-th 2025-07 accept novelty 2.0 of 10

A comment showing that the Gripaios-Nguyen classification of asymptotically-free gauge theories misses anomaly cancellation constraints and prior classifications, and overstates its novelty.

Reference graph

Works this paper leans on

66 extracted references · 43 canonical work pages · cited by 2 Pith papers

[1]

In this letter we consider SU( N), SO( N), Sp(2N) and exceptional gauge groups

One non-abelian simple Lie group as gauge sym- metry. In this letter we consider SU( N), SO( N), Sp(2N) and exceptional gauge groups

work page
[2]

By a fam- ily we mean an irreducible set of massless fermions so that a mass term is forbidden by gauge invari- ance [25]

Anomaly-free fermionic matter families. By a fam- ily we mean an irreducible set of massless fermions so that a mass term is forbidden by gauge invari- ance [25]

work page
[3]

Asymptotic freedom in the gauge coupling, which is the only renormalizable interaction. arXiv:2507.06368v2 [hep-ph] 21 Jul 2025 2 We will restrict our considerations tod = 4 space-time and renormalizable theories, where asymptotic freedom allows the theory to be in principle viable at arbitrarily high energies. Each non-equivalent theory is character- ize...

arXiv 2025
[4]

Yang and R

C.-N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev. 96, 191 (1954)

work page 1954
[5]

J. C. Pati and A. Salam, Lepton Number as the Fourth Color, Phys. Rev. D 10, 275 (1974), [Erratum: Phys.Rev.D 11, 703–703 (1975)]

work page 1974
[6]

Georgi and S

H. Georgi and S. L. Glashow, Unity of All Elementary Particle Forces, Phys. Rev. Lett.32, 438 (1974)

1974
[7]

Note also that the only model in the list that cannot contain the SM gauge symmetry is based on SU(4)

with fFE = 87. Note also that the only model in the list that cannot contain the SM gauge symmetry is based on SU(4). The same ranking is valid in the SUSY case, where the SU(4) model is excluded for loss of asymptotic free- dom. As expected, fa gives larger weights to the gauge part, so that the SU(5) model is only surpassed by SU(4), which has a very si...

work page
[8]

S. L. Glashow, Partial Symmetries of Weak Interactions, Nucl. Phys. 22, 579 (1961)

work page 1961

Show all 66 references

[9]

Weinberg, A Model of Leptons, Phys

S. Weinberg, A Model of Leptons, Phys. Rev. Lett.19, 1264 (1967)

1967
[10]

Salam, Weak and Electromagnetic Interactions, Conf

A. Salam, Weak and Electromagnetic Interactions, Conf. Proc. C 680519, 367 (1968)

1968
[11]

Salam and J

A. Salam and J. A. Strathdee, Supergauge Transforma- tions, Nucl. Phys. B 76, 477 (1974)

1974
[12]

Ferrara, J

S. Ferrara, J. Wess, and B. Zumino, Supergauge Multiplets and Superfields, Phys. Lett. B 51, 239 (1974)

1974
[13]

Fritzsch and P

H. Fritzsch and P . Minkowski, Unified Interactions of Lep- tons and Hadrons, Annals Phys. 93, 193 (1975)

1975
[14]

Gursey, P

F. Gursey, P . Ramond, and P . Sikivie, A Universal Gauge Theory Model Based on E6, Phys. Lett. B 60, 177 (1976)

1976
[15]

M. Ito, S. Kuwakino, N. Maekawa, S. Moriyama, K. Taka- hashi, K. Takei, S. Teraguchi, and T. Yamashita, E6 grand unified theory with three generations from heterotic string, Phys. Rev. D 83, 091703 (2011), arXiv:1012.1690 [hep-ph]

2011 arXiv
[16]

Zee, Maximal local symmetry, Phys

A. Zee, Maximal local symmetry, Phys. Lett. B 99, 110 (1981)

1981
[17]

Abel and F

S. Abel and F. Sannino, Radiative symmetry breaking from interacting UV fixed points, Phys. Rev. D96, 056028 (2017), arXiv:1704.00700 [hep-ph]

2017 arXiv
[18]

Fabbrichesi, C

M. Fabbrichesi, C. M. Nieto, A. Tonero, and A. Ugolotti, Asymptotically safe SU(5) GUT, Phys. Rev. D 103, 095026 (2021), arXiv:2012.03987 [hep-ph]

2021 arXiv
[19]

D. F. Litim and F. Sannino, Asymptotic safety guaranteed, JHEP 12, 178, arXiv:1406.2337 [hep-th]

arXiv
[20]

D. F. Litim, M. Mojaza, and F. Sannino, Vacuum stability of asymptotically safe gauge-Yukawa theories, JHEP 01, 081, arXiv:1501.03061 [hep-th]

arXiv
[21]

Sannino,αs at LHC: Challenging asymptotic freedom, in Workshop on high-precision alpha s measurements: from LHC to FCC-ee (2015) pp

F. Sannino,αs at LHC: Challenging asymptotic freedom, in Workshop on high-precision alpha s measurements: from LHC to FCC-ee (2015) pp. 11–19, arXiv:1511.09022 [hep-ph]

2015 arXiv
[22]

G. M. Pelaggi, F. Sannino, A. Strumia, and E. Vigiani, Naturalness of asymptotically safe Higgs, Front. in Phys. 5, 49 (2017), arXiv:1701.01453 [hep-ph]

2017 arXiv
[23]

G. F. Giudice, G. Isidori, A. Salvio, and A. Strumia, Soft- ened Gravity and the Extension of the Standard Model up to Infinite Energy, JHEP02, 137, arXiv:1412.2769 [hep-ph]

arXiv
[24]

C. Pica, T. A. Ryttov, and F. Sannino, Conformal Phase Diagram of Complete Asymptotically Free Theories, Phys. Rev. D 96, 074015 (2017), arXiv:1605.04712 [hep-th]

2017 arXiv
[25]

A. V . Bednyakov and A. I. Mukhaeva, Asymptotic safety in the Litim-Sannino model at four loops, Phys. Rev. D 109, 065030 (2024), arXiv:2312.12128 [hep-th]

2024 arXiv
[26]

T. P . Cheng, E. Eichten, and L.-F. Li, Higgs Phenomena in Asymptotically Free Gauge Theories, Phys. Rev. D9, 2259 (1974)

1974
[27]

D. J. E. Callaway, Triviality Pursuit: Can Elementary Scalar Particles Exist?, Phys. Rept. 167, 241 (1988)

1988
[28]

Holdom, J

B. Holdom, J. Ren, and C. Zhang, Stable Asymptotically Free Extensions (SAFEs) of the Standard Model, JHEP 03, 028, arXiv:1412.5540 [hep-ph]

arXiv
[29]

Kobayashi, S

T. Kobayashi, S. Raby, and R.-J. Zhang, Searching for real- istic 4d string models with a Pati-Salam symmetry: Orb- ifold grand unified theories from heterotic string compact- ification on a Z(6) orbifold, Nucl. Phys. B 704, 3 (2005), arXiv:hep-ph/0409098

2005 arXiv
[30]

Eichten and J

E. Eichten and J. Preskill, Chiral Gauge Theories on the Lattice, Nucl. Phys. B 268, 179 (1986)

1986
[31]

For vector-like theories, a complete classification was first presented in [55]
[32]

Bajc and F

B. Bajc and F. Sannino, Asymptotically Safe Grand Unifi- cation, JHEP 12, 141, arXiv:1610.09681 [hep-th]

arXiv
[33]

Cacciapaglia, A

G. Cacciapaglia, A. S. Cornell, C. Cot, and A. Deandrea, Minimal SU(5) asymptotic grand unification, Phys. Rev. D 104, 075012 (2021), arXiv:2012.14732 [hep-th]

2021 arXiv
[34]

Chen and Y.-C

C.-M. Chen and Y.-C. Chung, On F-theory E6 GUTs, JHEP 03, 129, arXiv:1010.5536 [hep-th]

arXiv
[35]

Appelquist, A

T. Appelquist, A. G. Cohen, M. Schmaltz, and R. Shrock, New constraints on chiral gauge theories, Phys. Lett. B 459, 235 (1999), arXiv:hep-th/9904172

1999 arXiv
[36]

Appelquist, Z.-y

T. Appelquist, Z.-y. Duan, and F. Sannino, Phases of chiral gauge theories, Phys. Rev. D61, 125009 (2000), arXiv:hep- ph/0001043

2000
[37]

Complex anomaly-free irreducible representations exists for n≥ 5 [56, 57], however they always violate asymptotic freedom
[38]

Witten, An SU(2) Anomaly, Phys

E. Witten, An SU(2) Anomaly, Phys. Lett. B117, 324 (1982)

1982
[39]

Wang, X.-G

J. Wang, X.-G. Wen, and E. Witten, A New SU(2) Anomaly, J. Math. Phys.60, 052301 (2019), arXiv:1810.00844 [hep-th]

2019 arXiv
[40]

Cacciapaglia, K

G. Cacciapaglia, K. Kollias, F. Sannino, in preparation
[41]

Seiberg, Exact results on the space of vacua of four- dimensional SUSY gauge theories, Phys

N. Seiberg, Exact results on the space of vacua of four- dimensional SUSY gauge theories, Phys. Rev. D 49, 6857 (1994), arXiv:hep-th/9402044

1994 arXiv
[42]

Seiberg, Electric - magnetic duality in supersymmetric nonAbelian gauge theories, Nucl

N. Seiberg, Electric - magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435, 129 (1995), arXiv:hep-th/9411149

1995 arXiv
[43]

J. L. Cardy, Is There a c Theorem in Four-Dimensions?, Phys. Lett. B 215, 749 (1988)

1988
[44]

in the SM: at high energies, the dual gauge symme- try is SU(2Ng− 3), where the duality is only possible for Ng = 3 (SU(3)), Ng = 4 (SU(5)) and Ng = 5 (SU(7)). Does the GTA allow us to select one high-energy theory out of these three, hence fixing the number of generations in ...
[45]

N. A. Dondi, V . Prochazka, and F. Sannino, Conformal Data of Fundamental Gauge-Yukawa Theories, Phys. Rev. D 98, 045002 (2018), arXiv:1712.05388 [hep-th]

2018 arXiv
[46]

Bars and S

I. Bars and S. Yankielowicz, Composite Quarks and Lep- tons as Solutions of Anomaly Constraints, Phys. Lett. B 101, 159 (1981)

1981
[47]

B. C. Allanach, B. Gripaios, and J. Tooby-Smith, Semisim- ple extensions of the Standard Model gauge algebra, Phys. Rev. D 104, 035035 (2021), [Erratum: Phys.Rev.D 106, 019901 (2022)], arXiv:2104.14555 [hep-th]

2021 arXiv
[48]

Maekawa and J

N. Maekawa and J. Sato, Duality of a supersymmetric standard model without R-parity, Prog. Theor. Phys. 96, 979 (1996), arXiv:hep-ph/9511395

1996 arXiv
[49]

Mojaza, M

M. Mojaza, M. Nardecchia, C. Pica, and F. Sannino, Dual of QCD with One Adjoint Fermion, Phys. Rev. D 83, 065022 (2011), arXiv:1101.1522 [hep-th]. 6

2011 arXiv
[50]

Sannino, The Standard Model is Natural as Mag- netic Gauge Theory, Mod

F. Sannino, The Standard Model is Natural as Mag- netic Gauge Theory, Mod. Phys. Lett. A 26, 1763 (2011), arXiv:1102.5100 [hep-ph]

2011 arXiv
[51]

Antipin, M

O. Antipin, M. Mojaza, C. Pica, and F. Sannino, Magnetic Fixed Points and Emergent Supersymmetry, JHEP06, 037, arXiv:1105.1510 [hep-th]

arXiv
[52]

Cacciapaglia and F

G. Cacciapaglia and F. Sannino, Charting standard model duality and its signatures, Phys. Rev. D111, 035013 (2025), arXiv:2407.17281 [hep-ph]

2025 arXiv
[53]

Maekawa, Duality of a supersymmetric standard model, Prog

N. Maekawa, Duality of a supersymmetric standard model, Prog. Theor. Phys. 95, 943 (1996), arXiv:hep- ph/9509407

1996
[54]

Vecchi, A dangerous irrelevant UV-completion of the composite Higgs, JHEP 02, 094, arXiv:1506.00623 [hep- ph]

L. Vecchi, A dangerous irrelevant UV-completion of the composite Higgs, JHEP 02, 094, arXiv:1506.00623 [hep- ph]

arXiv
[55]

C. T. Hill, P . A. N. Machado, A. E. Thomsen, and J. Turner, Scalar Democracy, Phys. Rev. D 100, 015015 (2019), arXiv:1902.07214 [hep-ph]

2019 arXiv
[56]

Georgi, Why unify?, Nature 288, 649 (1980)

H. Georgi, Why unify?, Nature 288, 649 (1980)

1980
[57]

Cacciapaglia, C

G. Cacciapaglia, C. Pica, and F. Sannino, Fundamental Composite Dynamics: A Review, Phys. Rept.877, 1 (2020), arXiv:2002.04914 [hep-ph]

2020 arXiv
[58]

Antipin, M

O. Antipin, M. Redi, A. Strumia, and E. Vigiani, Accidental Composite Dark Matter, JHEP 07, 039, arXiv:1503.08749 [hep-ph]

arXiv
[59]

Contino, A

R. Contino, A. Podo, and F. Revello, Composite Dark Mat- ter from Strongly-Interacting Chiral Dynamics, JHEP 02, 091, arXiv:2008.10607 [hep-ph]

2008 arXiv
[60]

Eichten and F

E. Eichten and F. Feinberg, Comment on Tumbling Gauge Theories, Phys. Lett. B 110, 232 (1982). SUPPLEMENTAL MATERIAL Sp(2N) pseudo-real families and the Witten anomalies The well-known SU(2) Witten anomaly [32] admits a natural generalization to the symplectic groups Sp(2 n), ...

1982
[61]

D. D. Dietrich and F. Sannino, Conformal window of SU(N) gauge theories with fermions in higher dimen- sional representations, Phys. Rev. D 75, 085018 (2007), arXiv:hep-ph/0611341

2007 arXiv
[62]

Eichten, K

E. Eichten, K. Kang, and I.-G. Koh, Anomaly Free Complex Representations in SU(N), J. Math. Phys. 23, 2529 (1982)

1982
[63]

Gripaios and K

B. Gripaios and K. Le Nguyen Nguyen, Varieties of four-dimensional gauge theories, JHEP 12, 041, arXiv:2409.15430 [hep-th]

arXiv
[64]

Okubo and Y

S. Okubo and Y. Tosa, Further study of global gauge anomalies of simple groups, Physical Review D 40, 1925 (1989)

1989
[65]

Okubo and H

S. Okubo and H. Zhang, Global gauge anomaly of clas- sical groups in even dimension, in Perspectives on Particle Physics (1989) pp. 321–337

1989
[200]

The SU(N) representations are the fundamental (F), the two-index symmetric (S), and the two-index antisymmetric (A) and three- index (A3) antisymmetric

The models are listed with increasing free energy fFE. The SU(N) representations are the fundamental (F), the two-index symmetric (S), and the two-index antisymmetric (A) and three- index (A3) antisymmetric. it would deem the theory e ffective, as some couplings would diverge ...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

In this letter we consider SU( N), SO( N), Sp(2N) and exceptional gauge groups

One non-abelian simple Lie group as gauge sym- metry. In this letter we consider SU( N), SO( N), Sp(2N) and exceptional gauge groups

work page

[2] [2]

By a fam- ily we mean an irreducible set of massless fermions so that a mass term is forbidden by gauge invari- ance [25]

Anomaly-free fermionic matter families. By a fam- ily we mean an irreducible set of massless fermions so that a mass term is forbidden by gauge invari- ance [25]

work page

[3] [3]

Asymptotic freedom in the gauge coupling, which is the only renormalizable interaction. arXiv:2507.06368v2 [hep-ph] 21 Jul 2025 2 We will restrict our considerations tod = 4 space-time and renormalizable theories, where asymptotic freedom allows the theory to be in principle viable at arbitrarily high energies. Each non-equivalent theory is character- ize...

arXiv 2025

[4] [4]

Yang and R

C.-N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev. 96, 191 (1954)

work page 1954

[5] [5]

J. C. Pati and A. Salam, Lepton Number as the Fourth Color, Phys. Rev. D 10, 275 (1974), [Erratum: Phys.Rev.D 11, 703–703 (1975)]

work page 1974

[6] [6]

Georgi and S

H. Georgi and S. L. Glashow, Unity of All Elementary Particle Forces, Phys. Rev. Lett.32, 438 (1974)

1974

[7] [7]

Note also that the only model in the list that cannot contain the SM gauge symmetry is based on SU(4)

with fFE = 87. Note also that the only model in the list that cannot contain the SM gauge symmetry is based on SU(4). The same ranking is valid in the SUSY case, where the SU(4) model is excluded for loss of asymptotic free- dom. As expected, fa gives larger weights to the gauge part, so that the SU(5) model is only surpassed by SU(4), which has a very si...

work page

[8] [8]

S. L. Glashow, Partial Symmetries of Weak Interactions, Nucl. Phys. 22, 579 (1961)

work page 1961