pith. machine review for the scientific record. sign in

arxiv: 2605.08294 · v1 · submitted 2026-05-08 · ✦ hep-th · hep-lat· hep-ph

Recognition: no theorem link

Completely asymptotically free chiral theories with scalars

Francesco Sannino, Giacomo Cacciapaglia, Sophie Wagner

Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:44 UTC · model grok-4.3

classification ✦ hep-th hep-lathep-ph
keywords asymptotic freedomchiral gauge theoryGeorgi-Glashow modelBars-Yankielowicz modelscalar fieldsYukawa couplingsquartic couplingsbeta functions
0
0 comments X

The pith

Chiral gauge theories with scalars achieve complete asymptotic freedom for specific numbers of colors and fermion families.

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

The paper establishes conditions for complete asymptotic freedom in chiral gauge theories that include scalars, motivated by grand unified model building. It examines generalized Georgi-Glashow and Bars-Yankielowicz setups where a scalar transforms in the fundamental or adjoint representation of the gauge group, together with multiple chiral fermion families. The analysis tracks the one-loop renormalization group flows of the gauge, Yukawa, and scalar quartic couplings to identify parameter choices where every interaction remains asymptotically free. A reader would care because these models remain well-defined and perturbative at arbitrarily short distances without Landau poles, offering concrete UV-complete chiral theories.

Core claim

In generalised Georgi-Glashow and Bars-Yankielowicz chiral gauge theories augmented by a scalar in the fundamental or adjoint representation and multiple chiral fermion families, the gauge, Yukawa and quartic couplings can all be made asymptotically free when the number of colours and the multiplicities of vector-like and chiral families are chosen appropriately; the one-loop beta functions for all three types of interaction then remain negative.

What carries the argument

One-loop beta functions for the gauge coupling, Yukawa couplings between the scalars and fermions, and the scalar quartic self-coupling, whose simultaneous negativity ensures every coupling flows to zero in the ultraviolet.

If this is right

  • Both the fundamental and adjoint scalar representations admit complete asymptotic freedom for suitable discrete choices of colour number and family multiplicities.
  • All three classes of coupling—gauge, Yukawa and quartic—flow to zero at high energies when the conditions are met.
  • The models remain perturbative up to arbitrarily high scales, providing ultraviolet-complete chiral gauge theories.
  • The constructions generalize earlier chiral models by incorporating scalars while preserving ultraviolet freedom.

Where Pith is reading between the lines

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

  • These discrete parameter sets could serve as starting points for constructing larger grand-unified theories that remain asymptotically free after additional fields are added.
  • Higher-loop or non-perturbative checks, such as lattice simulations, could confirm whether the identified points survive beyond one-loop order.
  • The same beta-function balancing technique might be applied to other gauge groups or representations to enlarge the list of viable chiral models.

Load-bearing premise

The one-loop perturbative beta-function analysis remains valid and sufficient to guarantee ultraviolet freedom without non-perturbative effects or additional consistency conditions altering the flow for the chosen representations.

What would settle it

A two-loop computation of the beta functions at the specific values of colour number and family multiplicities that satisfy the one-loop conditions, if it reveals a positive beta function for any coupling, would show that complete asymptotic freedom fails.

Figures

Figures reproduced from arXiv: 2605.08294 by Francesco Sannino, Giacomo Cacciapaglia, Sophie Wagner.

Figure 1
Figure 1. Figure 1: CAF regions in the parameter space (N, p) for (a) the GG model and (b) the BY model, with one scalar in the fundamental representation. The dark blue area represents the fixed-flow CAF region, bounded above by the b0 = 0 condition (dashed black line) and below by the b0 − d ′ 2 + √ k ′ = 0 condition (dotted black line) for the GG model. The light blue region highlights the CAF region off fixed flow, bounde… view at source ↗
Figure 2
Figure 2. Figure 2: CAF regions in the (N, p) parameter space of the GG model with one fundamental scalar for Ng = {1, 3, 5, 6}. The dark blue area indicates the CAF region on fixed flow, which is bounded above by the b0 = 0 condition (dashed black line) and below by the b0 − c1 > 0 condition (dotted black line). The light blue area highlights the CAF region off the fixed flow, bounded strictly between b0 = 0 (above) and k = … view at source ↗
Figure 3
Figure 3. Figure 3: CAF regions in the (N, p) parameter space of the BY model with one fundamental scalar for Ng = {1, 2, 3, 4}. The dark blue area indicates the CAF region on fixed flow, which is bounded above by the b0 = 0 condition (dashed black line). The light blue area highlights the CAF region off fixed flow, bounded strictly between b0 = 0 (above) and k = 0 (dash-dotted black line, below). The striped region indicates… view at source ↗
Figure 4
Figure 4. Figure 4: Root structure of the fourth-degree polynomial for the GG model, when (a) all couplings are on fixed flow and (b) the Yukawa coupling is off fixed flow. The shaded regions denote the presence of four complex roots (white) and two real and two complex roots (light green). The dashed black line marks the b0 = 0 condition, underneath which the gauge coupling is asymptotically free. We can now perform the comp… view at source ↗
Figure 5
Figure 5. Figure 5: CAF region for the GG model with adjoint scalar when the Yukawa coupling αy is off fixed flow (light blue). The dashed black line illustrates the b0 = 0 condition, Eq. (5.6), while the solid black line the ∆ = 0 condition for αy off fixed flow, Eq. (A.66). fixed flow. Interestingly, CAF models only exist for N ≥ 7 and p ≥ 32, where no solution exists for the SU(5) GUT model with one generation. The analysi… view at source ↗
Figure 6
Figure 6. Figure 6: Properties of the GG model with Ng = 3 chiral families. Panel (a) shows the root structure of the fourth-degree polynomial with all couplings on fixed flow. The shaded regions denote four complex roots (white) or two real and two complex roots (light green). Panel (b) depicts the CAF region with Yukawa coupling αy off fixed flow (light blue). The dashed black line illustrates the b0 = 0 condition, Eq. (5.6… view at source ↗
Figure 7
Figure 7. Figure 7: Properties of the GG model with adjoint scalar when Ng = {5, 6, 7}. Panel (a) shows the root structure of the fourth-degree polynomial with all couplings on fixed flow. The shaded regions denote four complex roots (white) or two real and two complex roots (light green). The remaining panels (b,c,d) depict the CAF regions with all couplings on fixed flow (dark blue) and for the Yukawa coupling αy off fixed … view at source ↗
Figure 8
Figure 8. Figure 8: CAF regions in the (N, p) parameter space for the BY model with adjoint scalar for Ng = {1, 2, 3, 4}. The dark blue area indicates the CAF region on the fixed flow, which is bounded above by the b0 = 0 condition (dashed black line) in Eq. (5.8) and below by the ∆ = 0 condition (black dash dotted line) with αy on fixed flow in Eq. (A.75). The light blue area highlights the region of CAF with the Yukawa coup… view at source ↗
read the original abstract

We provide the conditions for complete asymptotic freedom for chiral gauge theories including scalars, as motivated by grand unified models. These are generalised Georgi-Glashow and Bars-Yankielowicz theories that feature a scalar field transforming either in the fundamental or in the adjoint of the gauge group. In both scenarios, we consider the addition of multiple chiral fermion families. We systematically analyse the interplay between gauge, Yukawa, and quartic couplings required for all interactions to remain asymptotically free at short distances. We find that for both scalar representations, complete asymptotic free models can be obtained for a specific number of colours and multiplicity of vector-like and chiral families.

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 / 1 minor

Summary. The paper analyzes renormalization-group beta functions for gauge, Yukawa, and scalar quartic couplings in chiral SU(N_c) gauge theories with scalars transforming in the fundamental or adjoint representation (generalized Georgi-Glashow and Bars-Yankielowicz models). It adds multiple chiral fermion families plus vector-like pairs and identifies specific discrete values of N_c together with the multiplicities of vector-like and chiral families for which all three classes of couplings remain asymptotically free.

Significance. If the reported parameter sets are both anomaly-free and correctly yield negative beta-function coefficients at one loop (and remain stable under higher-order corrections), the work supplies concrete, fully perturbative UV-complete chiral gauge theories. Such examples are useful for GUT model building because they eliminate Landau poles while preserving chirality. The systematic treatment of the coupled gauge-Yukawa-quartic system is a positive feature.

major comments (3)
  1. The manuscript does not verify that the reported (N_c, family-multiplicity) combinations satisfy gauge-anomaly cancellation. For chiral SU(N_c) theories the cubic anomaly coefficient A_3 must vanish independently of the sign of the gauge beta function; non-zero A_3 renders the quantum theory inconsistent regardless of perturbative asymptotic freedom. This check is load-bearing for the central claim that the listed models are viable.
  2. Explicit one-loop beta-function expressions for the gauge, Yukawa, and quartic couplings are not displayed, nor are the numerical values of the beta-function coefficients that lead to the quoted (N_c, family) solutions. Without these, the specific numbers cannot be reproduced or checked for algebraic or numerical errors.
  3. The analysis is performed at one-loop order. For complete asymptotic freedom to be robust, the paper should at least comment on the stability of the negative beta-function signs under two-loop corrections, especially for the quartic and Yukawa sectors where higher-order terms can be sizable.
minor comments (1)
  1. A compact table listing the allowed (N_c, n_vector-like, n_chiral) triples for each scalar representation would improve readability and allow immediate comparison with anomaly-cancellation conditions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating the revisions we plan to implement.

read point-by-point responses

  1. Referee: The manuscript does not verify that the reported (N_c, family-multiplicity) combinations satisfy gauge-anomaly cancellation. For chiral SU(N_c) theories the cubic anomaly coefficient A_3 must vanish independently of the sign of the gauge beta function; non-zero A_3 renders the quantum theory inconsistent regardless of perturbative asymptotic freedom. This check is load-bearing for the central claim that the listed models are viable.

    Authors: We appreciate the referee for emphasizing this crucial consistency requirement. The parameter sets we report were chosen to be anomaly-free, but we did not explicitly display the A_3 calculations. We will add a dedicated paragraph (or short subsection) that recalls the standard formula for the cubic anomaly coefficient in SU(N_c) and verifies its vanishing for each of the listed (N_c, vector-like, chiral-family) combinations. revision: yes

  2. Referee: Explicit one-loop beta-function expressions for the gauge, Yukawa, and quartic couplings are not displayed, nor are the numerical values of the beta-function coefficients that lead to the quoted (N_c, family) solutions. Without these, the specific numbers cannot be reproduced or checked for algebraic or numerical errors.

    Authors: We agree that full transparency requires the explicit expressions. Although the beta functions were derived from the standard one-loop formulas, the manuscript presented only the resulting conditions rather than the intermediate coefficients. We will insert the complete one-loop beta-function expressions for the gauge, Yukawa, and scalar quartic couplings, together with the numerical coefficient values that yield the reported solutions, either in the main text or in a new appendix. revision: yes

  3. Referee: The analysis is performed at one-loop order. For complete asymptotic freedom to be robust, the paper should at least comment on the stability of the negative beta-function signs under two-loop corrections, especially for the quartic and Yukawa sectors where higher-order terms can be sizable.

    Authors: We acknowledge that one-loop results constitute a necessary but not automatically sufficient condition. We will add a brief discussion paragraph noting the possible size of two-loop corrections in the Yukawa and quartic sectors and stating that a dedicated two-loop analysis lies outside the scope of the present work. The one-loop search nevertheless identifies the candidate models that can be examined at higher order in future studies. revision: partial

Circularity Check

0 steps flagged

No circularity: conditions obtained by direct solution of beta-function equations

full rationale

The paper derives the reported values of N_c and family multiplicities by solving the coupled perturbative beta-function equations for the gauge, Yukawa, and quartic couplings in the chosen scalar representations. These outputs are not defined in terms of themselves, not obtained by fitting a subset of data and relabeling the fit as a prediction, and do not rely on load-bearing self-citations or imported uniqueness theorems. The derivation remains self-contained against the standard one-loop beta-function formulae and the explicit representation content; anomaly cancellation is an independent consistency requirement outside the scope of the circularity analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard perturbative QFT assumptions and group-theoretic representation choices; no new entities or fitted parameters are introduced.

axioms (2)
  • domain assumption Perturbative beta functions accurately capture the ultraviolet behavior of the theory.
    Standard assumption in asymptotic-freedom studies; invoked implicitly when claiming complete freedom from the flow equations.
  • domain assumption The chosen fermion and scalar representations yield anomaly-free theories.
    Required for consistency of chiral gauge theories; not explicitly verified in the abstract.

pith-pipeline@v0.9.0 · 5398 in / 1267 out tokens · 42200 ms · 2026-05-12T01:44:26.216341+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · 1 internal anchor

  1. [1]

    Partial Symmetries of Weak Interactions

    S. L. Glashow. “Partial Symmetries of Weak Interactions” . In: Nucl. Phys. 22 (1961), pp. 579–

    work page 1961

  2. [2]

    doi: 10.1016/0029-5582(61)90469-2

    work page doi:10.1016/0029-5582(61)90469-2

  3. [3]

    A model of leptons

    Steven Weinberg. “A Model of Leptons” . In: Phys. Rev. Lett. 19 (1967), pp. 1264–1266. doi: 10.1103/PhysRevLett.19.1264

    work page doi:10.1103/physrevlett.19.1264 1967

  4. [4]

    Broken symmetries, massless particles and gauge fields

    Peter W. Higgs. “Broken symmetries, massless particles and gauge fields” . In: Phys. Lett. 12 (1964), pp. 132–133. doi: 10.1016/0031-9163(64)91136-9

    work page doi:10.1016/0031-9163(64)91136-9 1964

  5. [5]

    Spontaneous Symmetry Breakdown without Massless Bosons

    Peter W. Higgs. “Spontaneous Symmetry Breakdown without Massless Bosons” . In: Phys. Rev. 145 (1966), pp. 1156–1163. doi: 10.1103/PhysRev.145.1156

    work page doi:10.1103/physrev.145.1156 1966

  6. [6]

    Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC

    Georges Aad et al. “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC” . In: Phys. Lett. B 716 (2012), pp. 1–29. doi: 10.1016/j.physletb.2012.08.020. arXiv: 1207.7214 [hep-ex]

    work page internal anchor Pith review doi:10.1016/j.physletb.2012.08.020 2012

  7. [7]

    Higgs mass and vacuum stability in the Standard Model at NNLO

    Giuseppe Degrassi, Stefano Di Vita, Joan Elias-Miro, Jose R. Espinosa, Gian F. Giudice, Gino Isidori, and Alessandro Strumia. “Higgs mass and vacuum stability in the Standard Model at NNLO” . In: JHEP 08 (2012), p. 098. doi: 10 . 1007 / JHEP08(2012 ) 098. arXiv: 1205 . 6497 [hep-ph]

    work page 2012

  8. [8]

    Standard Model Vacuum Stability and Weyl Consistency Conditions

    Oleg Antipin, Marc Gillioz, Jens Krog, Esben Mølgaard, and Francesco Sannino. “Standard Model Vacuum Stability and Weyl Consistency Conditions” . In: JHEP 08 (2013), p. 034. doi: 10.1007/JHEP08(2013)034. arXiv: 1306.3234 [hep-ph]

    work page doi:10.1007/jhep08(2013)034 2013

  9. [9]

    Unity of All Elementary Particle Forces

    H. Georgi and S. L. Glashow. “Unity of All Elementary Particle Forces” . In: Phys. Rev. Lett. 32 (1974), pp. 438–441. doi: 10.1103/PhysRevLett.32.438

    work page doi:10.1103/physrevlett.32.438 1974

  10. [10]

    Hierarchy of Interactions in Unified Gauge Theories

    H. Georgi, Helen R. Quinn, and Steven Weinberg. “Hierarchy of Interactions in Unified Gauge Theories” . In: Phys. Rev. Lett. 33 (1974), pp. 451–454. doi: 10.1103/PhysRevLett.33.451

    work page doi:10.1103/physrevlett.33.451 1974

  11. [11]

    Unified Interactions of Leptons and Hadrons

    H. Fritzsch and Peter Minkowski. “Unified Interactions of Leptons and Hadrons” . In: Annals Phys. 93 (1975), pp. 193–266. doi: 10.1016/0003-4916(75)90211-0

    work page doi:10.1016/0003-4916(75)90211-0 1975

  12. [12]

    Lepton Number as the Fourth Color

    Jogesh C. Pati and Abdus Salam. “Lepton Number as the Fourth Color” . In: Phys. Rev. D 10 (1974). [Erratum: Phys.Rev.D 11, 703–703 (1975)], pp. 275–289. doi: 10.1103/PhysRevD.10. 275

    work page doi:10.1103/physrevd.10 1974

  13. [13]

    Grand Unified Theories and Proton Decay

    Paul Langacker. “Grand Unified Theories and Proton Decay” . In: Phys. Rept. 72 (1981), p. 185. doi: 10.1016/0370-1573(81)90059-4

    work page doi:10.1016/0370-1573(81)90059-4 1981

  14. [14]

    Asymptotically Safe Grand Unification

    Borut Bajc and Francesco Sannino. “Asymptotically Safe Grand Unification” . In: JHEP 12 (2016), p. 141. doi: 10.1007/JHEP12(2016)141. arXiv: 1610.09681 [hep-th]

    work page doi:10.1007/jhep12(2016)141 2016

  15. [16]

    Higgs Phenomena in Asymptotically Free Gauge Theories

    T. P. Cheng, E. Eichten, and Ling-Fong Li. “Higgs Phenomena in Asymptotically Free Gauge Theories” . In: Phys. Rev. D 9 (1974), p. 2259. doi: 10.1103/PhysRevD.9.2259

    work page doi:10.1103/physrevd.9.2259 1974

  16. [17]

    Asymptotically Free Gauge Theories - I

    D. J. Gross and Frank Wilczek. “Asymptotically Free Gauge Theories - I” . In: Phys. Rev. D 8 (1973), pp. 3633–3652. doi: 10.1103/PhysRevD.8.3633

    work page doi:10.1103/physrevd.8.3633 1973

  17. [18]

    Stable Asymptotically Free Extensions (SAFEs) of the Standard Model

    Bob Holdom, Jing Ren, and Chen Zhang. “Stable Asymptotically Free Extensions (SAFEs) of the Standard Model” . In: Journal of High Energy Physics 2015.3 (Mar. 2015). issn: 1029-8479. doi: 10.1007/jhep03(2015)028. url: http://dx.doi.org/10.1007/JHEP03(2015)028. 25

    work page doi:10.1007/jhep03(2015)028 2015

  18. [19]

    Softened Gravity and the Extension of the Standard Model up to Infinite Energy

    Gian F. Giudice, Gino Isidori, Alberto Salvio, and Alessandro Strumia. “Softened Gravity and the Extension of the Standard Model up to Infinite Energy” . In: JHEP 02 (2015), p. 137. doi: 10.1007/JHEP02(2015)137. arXiv: 1412.2769 [hep-ph]

    work page doi:10.1007/jhep02(2015)137 2015

  19. [20]

    Conformal Phase Diagram of Com- plete Asymptotically Free Theories

    Claudio Pica, Thomas A. Ryttov, and Francesco Sannino. “Conformal Phase Diagram of Com- plete Asymptotically Free Theories” . In: Phys. Rev. D 96.7 (2017), p. 074015. doi: 10.1103/ PhysRevD.96.074015. arXiv: 1605.04712 [hep-th]

    work page arXiv 2017

  20. [21]

    Triviality Pursuit: Can Elementary Scalar Particles Exist?

    David J. E. Callaway. “Triviality Pursuit: Can Elementary Scalar Particles Exist?” In: Phys. Rept. 167 (1988), p. 241. doi: 10.1016/0370-1573(88)90008-7

    work page doi:10.1016/0370-1573(88)90008-7 1988

  21. [22]

    Phase structure of complete asymptotically free SU( Nc) theories with quarks and scalar quarks

    Frederik F. Hansen, Tadeusz Janowski, Kasper Langæble, Robert B. Mann, Francesco Sannino, Tom G. Steele, and Zhi-Wei Wang. “Phase structure of complete asymptotically free SU( Nc) theories with quarks and scalar quarks” . In: Phys. Rev. D 97.6 (2018), p. 065014. doi: 10.1103/ PhysRevD.97.065014. arXiv: 1706.06402 [hep-ph]

    work page arXiv 2018

  22. [23]

    Selection toolkit for standard and non-standard GUTs

    Giacomo Cacciapaglia, Aldo Deandrea, Konstantinos Kollias, and Francesco Sannino. “Selection toolkit for standard and non-standard GUTs” . In: Phys. Rev. D 113 (Apr. 2026), p. 075043. doi: doi.org/10.1103/395g-tl4h. arXiv: 2507.06368 [hep-ph]

    work page doi:10.1103/395g-tl4h 2026

  23. [24]

    New constraints on chiral gauge theories

    Thomas Appelquist, Andrew G. Cohen, Martin Schmaltz, and Robert Shrock. “New constraints on chiral gauge theories” . In: Phys. Lett. B 459 (1999), pp. 235–241. doi: 10 . 1016 / S0370 - 2693(99)00616-4. arXiv: hep-th/9904172

    work page arXiv 1999

  24. [25]

    Phases of chiral gauge theories

    Thomas Appelquist, Zhi-yong Duan, and Francesco Sannino. “Phases of chiral gauge theories” . In: Phys. Rev. D 61 (2000), p. 125009. doi: 10 . 1103 / PhysRevD . 61 . 125009. arXiv: hep - ph/0001043

    work page arXiv 2000

  25. [26]

    Is There a c Theorem in Four-Dimensions?

    John L. Cardy. “Is There a c Theorem in Four-Dimensions?” In: Phys. Lett. B 215 (1988), pp. 749–752. doi: 10.1016/0370-2693(88)90054-8

    work page doi:10.1016/0370-2693(88)90054-8 1988

  26. [27]

    H., Jedamzik, K., & Pogosian, L

    Nicola Andrea Dondi, Vladimir Prochazka, and Francesco Sannino. “Conformal Data of Funda- mental Gauge-Yukawa Theories” . In:Phys. Rev. D 98 (2018), p. 045002. doi: 10.1103/PhysRevD. 98.045002. arXiv: 1712.05388 [hep-th]

    work page doi:10.1103/physrevd 2018

  27. [28]

    Composite Quarks and Leptons as Solutions of Anomaly Constraints

    Itzhak Bars and Shimon Yankielowicz. “Composite Quarks and Leptons as Solutions of Anomaly Constraints” . In:Phys. Lett. B 101 (1981), pp. 159–165. doi: 10.1016/0370-2693(81)90664-X

    work page doi:10.1016/0370-2693(81)90664-x 1981

  28. [29]

    Dynamics from symmetries in chiral SU (N ) gauge theories

    Stefano Bolognesi, Kenichi Konishi, and Andrea Luzio. “Dynamics from symmetries in chiral SU (N ) gauge theories” . In: JHEP 09 (2020), p. 001. doi: 10.1007/JHEP09(2020)001 . arXiv: 2004.06639 [hep-th]

    work page doi:10.1007/jhep09(2020)001 2020

  29. [30]

    Some exact results in chiral gauge theories

    Csaba Csáki, Hitoshi Murayama, and Ofri Telem. “Some exact results in chiral gauge theories” . In: Phys. Rev. D 104.6 (2021), p. 065018. doi: 10.1103/PhysRevD.104.065018 . arXiv: 2104. 10171 [hep-th]

    work page doi:10.1103/physrevd.104.065018 2021

  30. [31]

    Probing the dynamics of chiralSU(N) gauge theories via generalized anomalies

    Stefano Bolognesi, Kenichi Konishi, and Andrea Luzio. “Probing the dynamics of chiral SU (N ) gauge theories via generalized anomalies” . In: Phys. Rev. D 103.9 (2021), p. 094016. doi: 10. 1103/PhysRevD.103.094016. arXiv: 2101.02601 [hep-th]

    work page arXiv 2021

  31. [32]

    Strong anomaly and phases of chiral gauge theories

    Stefano Bolognesi, Kenichi Konishi, and Andrea Luzio. “Strong anomaly and phases of chiral gauge theories” . In: JHEP 08 (2021), p. 028. doi: 10 . 1007 / JHEP08(2021 ) 028. arXiv: 2105 . 03921 [hep-th]

    work page 2021

  32. [33]

    Phases of confining SU(5) chiral gauge theory with three gen- erations

    Yang Bai and Daniel Stolarski. “Phases of confining SU(5) chiral gauge theory with three gen- erations” . In: JHEP 03 (2022), p. 113. doi: 10.1007/JHEP03(2022)113 . arXiv: 2111.11214 [hep-th]. 26

    work page doi:10.1007/jhep03(2022)113 2022

  33. [34]

    The Z2 anomaly in some chiral gauge theories

    Stefano Bolognesi, Kenichi Konishi, and Andrea Luzio. “The Z2 anomaly in some chiral gauge theories” . In: JHEP 08 (2023), p. 125. doi: 10.1007/JHEP08(2023)125 . arXiv: 2307.03822 [hep-th]

    work page doi:10.1007/jhep08(2023)125 2023

  34. [35]

    Dynamical symmetry breaking in Georgi-Glashow chiral-gauge theories

    Hao-Lin Li, Álvaro Pastor-Gutiérrez, Shahram Vatani, and Ling-Xiao Xu. “Dynamical symmetry breaking in Georgi-Glashow chiral-gauge theories” . In: JHEP 12 (2025), p. 020. doi: 10.1007/ JHEP12(2025)020. arXiv: 2507.21208 [hep-th]

    work page arXiv 2025

  35. [36]

    Confinement without symmetry breaking in chiral gauge theories

    Haolin Li, Álvaro Pastor-Gutiérrez, and Shahram Vatani. “Confinement without symmetry breaking in chiral gauge theories” . In: (Mar. 2026). arXiv: 2603.19355 [hep-th]

    work page arXiv 2026

  36. [37]

    Asymptotically safe and free chiral theories with and without scalars

    Esben Mølgaard and Francesco Sannino. “Asymptotically safe and free chiral theories with and without scalars” . In: Phys. Rev. D 96.5 (2017), p. 056004. doi: 10.1103/PhysRevD.96.056004 . arXiv: 1610.03130 [hep-ph]

    work page doi:10.1103/physrevd.96.056004 2017

  37. [38]

    Group Theory of the Spontaneously Broken Gauge Symmetries

    Ling-Fong Li. “Group Theory of the Spontaneously Broken Gauge Symmetries” . In: Phys. Rev. D 9 (1974), pp. 1723–1739. doi: 10.1103/PhysRevD.9.1723

    work page doi:10.1103/physrevd.9.1723 1974

  38. [39]

    Asymptotic Freedom: An Approach to Strong Interactions

    H. David Politzer. “Asymptotic Freedom: An Approach to Strong Interactions” . In: Phys. Rept. 14 (1974), pp. 129–180. doi: 10.1016/0370-1573(74)90014-3

    work page doi:10.1016/0370-1573(74)90014-3 1974

  39. [40]

    Mass and Mixing Angle Predictions from Infrared Fixed Points

    B. Pendleton and Graham G. Ross. “Mass and Mixing Angle Predictions from Infrared Fixed Points” . In:Phys. Lett. B 98 (1981), pp. 291–294. doi: 10.1016/0370-2693(81)90017-4

    work page doi:10.1016/0370-2693(81)90017-4 1981

  40. [41]

    Reduction in the Number of Coupling Parameters

    W. Zimmermann. “Reduction in the Number of Coupling Parameters” . In: Commun. Math. Phys. 97 (1985), p. 211. doi: 10.1007/BF01206187

    work page doi:10.1007/bf01206187 1985

  41. [43]

    A Simple Proof of Descartes’s Rule of Signs

    Xiaoshen Wang. “A Simple Proof of Descartes’s Rule of Signs” . In: The American Mathematical Monthly 111.6 (2004), pp. 525–526. doi: 10.2307/2972804

    work page doi:10.2307/2972804 2004

  42. [44]

    Introducing RGBeta: a Mathematica package for the evaluation of renormalization group β-functions

    Anders Eller Thomsen. “Introducing RGBeta: a Mathematica package for the evaluation of renormalization group β-functions” . In: Eur. Phys. J. C 81.5 (2021), p. 408. doi: 10 . 1140 / epjc/s10052-021-09142-4 . arXiv: 2101.08265 [hep-ph]

    work page arXiv 2021

  43. [45]

    Dual of QCD with One Adjoint Fermion

    Matin Mojaza, Marco Nardecchia, Claudio Pica, and Francesco Sannino. “Dual of QCD with One Adjoint Fermion” . In: Phys. Rev. D 83 (2011), p. 065022. doi: 10.1103/PhysRevD.83.065022 . arXiv: 1101.1522 [hep-th]

    work page doi:10.1103/physrevd.83.065022 2011

  44. [46]

    The Standard Model is Natural as Magnetic Gauge Theory

    Francesco Sannino. “The Standard Model is Natural as Magnetic Gauge Theory” . In: Mod. Phys. Lett. A 26 (2011), pp. 1763–1769. doi: 10.1142/S0217732311036279. arXiv: 1102.5100 [hep-ph]

    work page doi:10.1142/s0217732311036279 2011

  45. [47]

    Charting standard model duality and its signa- tures

    Giacomo Cacciapaglia and Francesco Sannino. “Charting standard model duality and its signa- tures” . In: Phys. Rev. D 111.3 (2025), p. 035013. doi: 10.1103/PhysRevD.111.035013 . arXiv: 2407.17281 [hep-ph]

    work page doi:10.1103/physrevd.111.035013 2025

  46. [48]

    Orientifold theory dynamics and symmetry break- ing

    Francesco Sannino and Kimmo Tuominen. “Orientifold theory dynamics and symmetry break- ing” . In: Phys. Rev. D 71 (2005), p. 051901. doi: 10.1103/PhysRevD.71.051901 . arXiv: hep- ph/0405209

    work page doi:10.1103/physrevd.71.051901 2005

  47. [49]

    Conformal window of SU(N) gauge theories with fermions in higher dimensional representations

    Dennis D. Dietrich and Francesco Sannino. “Conformal window of SU(N) gauge theories with fermions in higher dimensional representations” . In: Phys. Rev. D 75 (2007), p. 085018. doi: 10.1103/PhysRevD.75.085018. arXiv: hep-ph/0611341

    work page doi:10.1103/physrevd.75.085018 2007

  48. [50]

    Gellert, ed

    W. Gellert, ed. The VNR concise encyclopedia of mathematics . New York: Van Nostrand Rein- hold, 1977. isbn: 0442226462. 27 A Detailed CAF analysis A.1 CAF Analysis from Solving ODE’s This approach considers coupled ODE’s governing the beta functions, which admit analytic solutions. A.1.1 Y ukawa Coupling The RG equation, which governs the Yukawa coupling...

    work page 1977