The Standard Model prediction for the rare <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>K</mi><mi>L</mi></msub><mo stretchy="false">→</mo><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup></math> decay depends critically on the long-distance contribution coming from the exchange of two photons. Such a contribution can be computed using lattice QCD and an effective three-flavor theory including only the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>u</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>s</mi></math> quarks, provided terms falling as the inverse square of the omitted charm quark mass, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mo>/</mo><msubsup><mi>m</mi><mi>c</mi><mn>2</mn></msubsup></math>, are neglected. Because of the missing Glashow-Iliopoulos-Maiani cancellation, this three-flavor theory contains additional low-energy constants that depend on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>c</mi></msub></math>. Here we show how these constants can be determined from a practical four-flavor lattice QCD calculation performed on a small volume with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>u</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi></math> quark masses that are heavier than physical.

United States

En-Hung Chao

Norman Christ

Ceran Hu

Renormalizing a three-flavor lattice calculation of the two-photon contribution to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>K</mi><mi>L</mi></msub><mo stretchy="false">→</mo><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup></math>

We revisit an anomaly-free extension of the Standard Model viz. the gauged <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>L</mi><mi>μ</mi></msub><mo>−</mo><msub><mi>L</mi><mi>τ</mi></msub></math> model in the Dirac framework, where the local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mrow><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>−</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>τ</mi></mrow></msub></mrow></msub></mrow></math> symmetry breaks and gives rise to a new gauge boson <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Z</mi><mo>′</mo></msup></math> and corresponding gauge coupling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>g</mi><mrow><mi>μ</mi><mi>τ</mi></mrow></msub></math>. Three additional heavy vectorlike fermions, three light right-handed neutrinos and two heavy singlet scalars are added to complete the model framework for Dirac neutrinos. Another singlet vectorlike fermion is added with a new gauge charge, which serves as a viable dark matter (DM) candidate, and the correct relic abundance is obtained via the resonance effect. The resonance effect triggers early kinetic decoupling of the DM candidate, which significantly influences the cosmological sector of this study. The parameter space is considered after satisfying the current bounds on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>M</mi><msup><mi>Z</mi><mo>′</mo></msup></msub></math> and the gauge coupling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>g</mi><mrow><mi>μ</mi><mi>τ</mi></mrow></msub></math>. The influence of dark radiations coming from the additional light degrees of freedoms are studied in connection with the dark matter. After imposing the relevant theoretical and experimental constraints, the viable parameter space is found to be significantly restricted while remaining accessible to ongoing and near-future experiments. The framework provides a consistent setting in which neutrino masses, dark matter, and cosmological observables can be studied simultaneously. In particular, we identify viable regions of the parameter space that satisfy the constraints from neutrino masses, dark matter relic abundance, and cosmological observables.

India

Pritam Das

Cosmological signature and light dark matter in Dirac <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>−</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>τ</mi></mrow></msub></mrow></math> model

We examine whether nonlinear matter dependence in <math altimg='si24.svg'><mrow><mi>f</mi><mo>(</mo><mrow><mi>Q</mi><mo>,</mo><msub><mi>L</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></math>gravity can account for the observed late-time acceleration of the Universe. For this purpose, we consider <math altimg='si25.svg'><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>Q</mi><mo>,</mo><msub><mi>L</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow><mo linebreak='goodbreak'>=</mo><mi>a</mi><mi>Q</mi><mo linebreak='goodbreak'>+</mo><mn>2</mn><msubsup><mi>L</mi><mi>m</mi><mi>n</mi></msubsup><mo linebreak='goodbreak'>+</mo><mi>b</mi></mrow></math> and reconstruct the expansion history using <math altimg='si26.svg'><mrow><mi>H</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo linebreak='goodbreak'>=</mo><msub><mi>H</mi><mn>0</mn></msub><mtext>exp</mtext><mi>&#8289;</mi><mrow><mo>(</mo><mrow><mi>&#945;</mi><mi>z</mi><mo linebreak='badbreak'>+</mo><mi>&#946;</mi><msup><mrow><mi>z</mi></mrow><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></math>. The parameters of the Hubble parametrization are constrained through Bayesian analysis using the joint Cosmic Chronometer and DESI BAO datasets. The reconstructed evolution is consistent with the observational measurements and describes a smooth transition from past deceleration to the present accelerated phase. The model produces positive energy density and negative pressure at late times, leading to a quintessence-like cosmic evolution without entering the phantom regime. Its dynamical behaviour also provides testable departures from the spatially flat <math altimg='si12.svg'><mstyle mathvariant='normal'><mi>&#923;</mi></mstyle></math>CDM expansion history. These findings support nonlinear <math altimg='si24.svg'><mrow><mi>f</mi><mo>(</mo><mrow><mi>Q</mi><mo>,</mo><msub><mi>L</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></math> gravity as a viable framework for describing late-time cosmic acceleration.

S.N. Bayaskar

A.D. Vishwakarma

S.A.A. Qadeer

Azerbaijan

J. Praveen

S.C. Darunde

Dark energy in non-linear <math altimg='si23.svg'><mrow><mi>f</mi><mo>(</mo><mrow><mi>Q</mi><mo>,</mo><msub><mi>L</mi><mi>m</mi></msub></mrow><mo>)</mo></mrow></math> gravity with exponential parametrization of Hubble parameter

We present an extension of the Standard Model based upon the gauged <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mrow><msub><mi>L</mi><mi>μ</mi></msub><mo>−</mo><msub><mi>L</mi><mi>τ</mi></msub></mrow></msub></math> symmetry, featuring an ultralight vector boson that couples to the difference of muon and tau lepton numbers. This anomaly-free construction, whilst historically motivated by the muon’s anomalous magnetic moment, is here employed as an independent phenomenological framework for solar neutrino physics. Through kinetic mixing with the photon, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Z</mi><mo>′</mo></msup></math> boson generates an effective matter potential within the solar interior, thereby modifying neutrino flavor oscillations via the Mikheyev-Smirnov-Wolfenstein mechanism. Utilizing recent solar neutrino measurements from Borexino, SNO, and Super-Kamiokande, together with an up-to-date solar standard model, we constrain the boson mass to be below <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>5.7</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>17</mn></mrow></msup><mtext> </mtext><mtext> </mtext><mi>eV</mi></math>, positioning it as a potential ultralight dark matter candidate. The optimal model, corresponding to a coupling parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>o</mi></msub><mo>≈</mo><mo>−</mo><mn>2</mn></math>, yields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>χ</mi><mi>ν</mi><mn>2</mn></msubsup><mo>=</mo><mn>2.7</mn></math>, which compares favorably with the standard three-flavor scenario (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>χ</mi><mi>ν</mi><mn>2</mn></msubsup><mo>=</mo><mn>3.1</mn></math>). This best-fit configuration corresponds to an effective four-fermion coupling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>g</mi><msup><mi>Z</mi><mo>′</mo></msup><mn>2</mn></msubsup><mi>ϵ</mi><mo>/</mo><msubsup><mi>m</mi><msup><mi>Z</mi><mo>′</mo></msup><mn>2</mn></msubsup><mo>≈</mo><mn>6.6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>23</mn></mrow></msup><mtext> </mtext><mtext> </mtext><msup><mrow><mi>eV</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></math>, approximately 5.7 times the Fermi constant, which for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϵ</mi><mo>≈</mo><mn>1</mn></math> and a reference mediator mass of 1 eV is equivalent to a gauge coupling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>g</mi><msup><mi>Z</mi><mo>′</mo></msup></msub><mo>≈</mo><mn>8</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>12</mn></mrow></msup></math>. We furthermore present projections for forthcoming experiments. The Jiangmen Underground Neutrino Observatory (JUNO), which commenced operations in August 2025 and has already achieved world-leading precision in oscillation parameter measurements, together with the XLZD consortium, should substantially refine these constraints. Such ultralight particles may address long-standing discrepancies in galaxy simulations within the cold dark matter paradigm, and recent theoretical developments demonstrate their compatibility with nonthermal dark matter production mechanisms. Additionally, terrestrial long-baseline neutrino experiments provide complementary probes of this parameter space.

Portugal

Ilídio Lopes

Constraining ultralight vector bosons via solar neutrino oscillations: The gauged <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>L</mi><mi>μ</mi></msub><mo>−</mo><msub><mi>L</mi><mi>τ</mi></msub></math> model and prospects for JUNO and XLZD

In this work we explore the non-singular bouncing scenarios in the framework of <math altimg='si9.svg'><mrow><mi>f</mi><mo>(</mo><mi>Q</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>)</mo></mrow></math> gravity, where the gravitational action depends on the non-metricity scalar ( <italic>Q</italic>) and matter Lagrangian density <math altimg='si10.svg'><mrow><mo>(</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>)</mo></mrow></math>. Using the functional form <math altimg='si11.svg'><mrow><mi>f</mi><mrow><mo>(</mo><mi>Q</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>)</mo></mrow><mo linebreak='goodbreak'>=</mo><mo>&#8722;</mo><mi>&#945;</mi><mi>Q</mi><mo linebreak='goodbreak'>+</mo><mn>2</mn><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo linebreak='goodbreak'>+</mo><mi>&#946;</mi></mrow></math>, two different bouncing scenario models are analyzed in a homogeneous, isotropic and flat FLRW universe. The cosmological evolution is analyzed through the behavior of the Hubble parameter, deceleration parameter, energy density, pressure, and equation of state parameter. Both models exhibit a successful bounce characterized by a smooth transition from a contracting universe ( <italic>H</italic>&#8239;&lt;&#8239;0) to an expanding phase ( <italic>H</italic>&#8239;&gt;&#8239;0) without encountering any initial singularity <math altimg='si12.svg'><mrow><mo>(</mo><mi>H</mi><mo linebreak='goodbreak'>=</mo><mn>0</mn><mo>)</mo></mrow></math>. The EOS parameter exhibits phantom crossing behavior, while the energy conditions reveal NEC and SEC violation around the bounce, consistent with bouncing cosmology requirements. Stability analysis through metric perturbations and adiabatic index confirms stable late-time cosmic evolution. The study suggests that <math altimg='si9.svg'><mrow><mi>f</mi><mo>(</mo><mi>Q</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>)</mo></mrow></math> theory offers a promising geometrical background for understanding non-singular early universe dynamics and bouncing scenarios.

Vinod Kumar Bhardwaj

Saudi Arabia

Meraj Ali Khan

Shyam Das

Saibal Ray

Exploring the scenarios of non-singular bouncing cosmology in <math altimg='si9.svg'><mrow><mi>f</mi><mo>(</mo><mi>Q</mi><mo>,</mo><msub><mi mathvariant='script'>L</mi><mi>m</mi></msub><mo>)</mo></mrow></math> gravity

We study a dynamical realization of the low-scale Inverse Seesaw mechanism in which the approximate $\textit{B}$ – $\textit{L}$ symmetry is gauged and spontaneously broken. Anomaly cancellation requires additional chiral fermions, one of which becomes a stable dark matter candidate after symmetry breaking, while another remains massless and contributes to dark radiation. Focusing on the regime of feeble gauge interactions, we compute the dark matter relic abundance produced via the freeze-in mechanism through the $\textit{B}$ – $\textit{L}$ gauge boson and identify the parameter space consistent with cosmological and laboratory constraints. We show that the same region naturally avoids thermalization of heavy neutral leptons, preserving the viability of ARS leptogenesis. The interplay between dark matter production, dark radiation constraints, and leptogenesis requirements leads to a predictive scenario where future cosmological surveys and intensity-frontier experiments such as SHiP can probe significant portions of the viable parameter space.

Spain

Enrique Fernández-Martínez

Brazil

Ana Foguel

Italy

Xabier Marcano

Germany

Daniel Naredo-Tuero

Vsevolod Syvolap

Denmark

Kevin Urquía-Calderón

Leptogenesis and dark matter in an inverse Seesaw from gauged $\textit{B}$ − $\textit{L}$ breaking

For over 30 years, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo>∼</mo><mn>20</mn><mo>%</mo></mrow></math> deficit, now exceeding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>5</mn><mi>σ</mi></math>, has persisted between measured and predicted neutrino capture rates on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>Ga</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>71</mn></mrow></mmultiscripts></mrow></math>, as observed in radioactive source experiments (namely GALLEX, SAGE, and more recently BEST) using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>Cr</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>51</mn></mrow></mmultiscripts></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>Ar</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>37</mn></mrow></mmultiscripts></mrow></math>. This long-standing discrepancy, referred to as the gallium anomaly, has posed a significant challenge to our understanding of both experimental methods and theoretical predictions. In this Letter, we revisit the theoretical calculation of the neutrino capture cross section by moving beyond the standard treatment of the leptonic wave functions, revealing limitations in the commonly used factorization approach based on the detailed balance principle. Incorporating phenomenologically constrained Gamow-Teller transition densities, able to correctly reproduce the precisely measured half-life of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>Ge</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>71</mn></mrow></mmultiscripts></mrow></math>, we find that the revised cross section can be significantly reduced, potentially resolving the gallium anomaly without invoking new physics.

M. Cadeddu

N. Cargioli

F. Dordei

L. Ferro

C. Giunti

M. Pitzalis

Possible Solution to the Gallium Anomaly Moving beyond the Leptonic Wave-Function Factorization

In the conventional gauged <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo>−</mo><mi>L</mi></math> extension of the standard model, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo>−</mo><mi>L</mi></math> charge of the singlet scalar <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>χ</mi></math>, responsible for the breaking of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mrow><mi>B</mi><mo>−</mo><mi>L</mi></mrow></msub></math> symmetry, is taken to be 2, such that it can anchor type-I seesaw by giving Majorana masses to the right-handed neutrinos, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>ν</mi><mi>R</mi></msub></math>. In this paper, we consider instead the cases <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>χ</mi><mo>∼</mo><mn>3</mn></math> or 4 under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo>−</mo><mi>L</mi></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>ν</mi><mi>R</mi></msub></math> may not acquire any Majorana mass and neutrinos are Dirac fermions. We then consider a vectorlike fermion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math> with two units of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo>−</mo><mi>L</mi></math> charge, which becomes a good candidate for dark matter, either Dirac for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>χ</mi><mo>∼</mo><mn>3</mn></math> or Majorana for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>χ</mi><mo>∼</mo><mn>4</mn></math>. In both cases, spontaneous <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo>−</mo><mi>L</mi></math> breaking can induce a strong first-order phase transition, producing stochastic gravitational waves (GW), which can be tested at GW experiments. Moreover, the presence of light <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>ν</mi><mi>R</mi></msub></math>s gives rise to an additional contribution to the effective number of relativistic degrees of freedom, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi><msub><mi>N</mi><mi>eff</mi></msub></math>, providing complementary constraints from current and upcoming CMB observations.

Ernest Ma

Partha Kumar Paul

Narendra Sahu

Naturally small Dirac neutrino mass and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo>−</mo><mi>L</mi></math> dark matter

We consider three families of semiclassical string solutions on AdS × $\textit{S}$ /$\textit{ℤ}$ orbifold backgrounds which correspond to ‘twisted’ single-trace operators in the SU(2), SU(3) and SL(2) sectors, respectively, of the dual 4d $$ \mathcal{N} $$ = 2 quiver gauge theory. The leading quantum correction to the classical energy of each solution is computed, and is matched, despite the lack of maximal supersymmetry, to the corresponding results from finite-size corrections to the twisted Bethe ansatz in the continuum $\textit{J}$ → $\textit{∞}$ limit and the associated Landau-Lifshitz low-energy effective theory. The $\textit{ℤ}$ quotient allows the string to have fractional winding branches $$ \mu =\overset{\sim }{m}+\frac{n}{L} $$ along orbifolded directions. As such, stability is a property of the full winding branch $\textit{μ}$, rather than of the individual twisted sectors $\textit{n}$. Restricting to the purely fractional branch $$ \left(\overset{\sim }{m}=0\right) $$ , we obtain new stable configurations which were absent in the analogous semiclassical string solutions on AdS × $\textit{S}$ . The energies of these stable states connect smoothly to those of states in relevant BPS sectors as $\textit{μ}$ → 0.

United Kingdom

Carlos Martínez

On quantum corrections to semiclassical strings on AdS5 × $\textit{S}$ 5/$\textit{ℤ}$ $\textit{L}$ orbifold backgrounds

A four-dimensional gauge-gravity unification based on local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>S</mi><mi>L</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></math> symmetry is developed in a universal Yang-Mills-type setting, which, however, appears dynamically consistent only in the symmetry-broken phase. In the exact symmetry limit, the theory may only be formulated in a premetric framework, where the accompanying tetrad multiplets, though promoted to dynamical fields, do not yet satisfy the conventional invertibility conditions. An ordinary Einstein-Cartan spacetime geometry emerges only in the broken postsoldering phase, in which the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>L</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></math> tetrad multiplets are treated as constrained dynamical fields selecting a neutral internal symmetry branch. This realizes the breaking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>L</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mspace linebreak="goodbreak"></mspace><mi>S</mi><mi>L</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>×</mo><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math>, thereby lifting all noncompact internal directions, while the surviving neutral tetrad is, as usual, associated with the gravitational field. A special ghost-free curvature-squared Lagrangian provides a consistent quadratic sector for the spin connection, propagating only admissible connection modes: the massless <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> vector fields together with massive axial-vector and pseudoscalar multiplets. The Einstein-Cartan linear curvature term is argued to arise radiatively from fermion loops, thereby relating the gravitational scale to the same <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>L</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></math>-covariant matter sector that defines the unified gauge coupling. Finally, the matter sector points to a deeper elementarity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>L</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></math> spinors, identified with preon constituents whose bound states form the observed quarks and leptons. Anomaly matching between preons and composites singles out <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>=</mo><mn>8</mn></math>. The chain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>L</mi><mo stretchy="false">(</mo><mn>16</mn><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mi>S</mi><mi>L</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>×</mo><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">)</mo></math> then naturally yields three composite quark-lepton families, while filtering out extraneous heavy states.

Georgia

J.L. Chkareuli

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>L</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></math> Yang-Mills theories: Direct internal forces and emerging gravity