Group SU Research Articles

Letter To Feynman, Einstein, Wallace, Darwin, Maxwell and Mendeleev

The Roberts-Janet Nuclear Periodic Table has emerged recently. The inversion of the Periodic Table to accommodate spatial variation of atomic energy levels relative to the nucleus has subsequently been underwritten by Quantum Field Theory’s U (1) X SU (2) x SU (3) group symmetry and Clifford Algebra resulting in a one-toone mapping between the Roberts-Janet Table and The Quantum Mechanical Table. This manuscript attempts to show the over-arching nature of the Roberts-Janet Table epitomised by two cycles. The first of these is the role of causality within the lower half of the table in nucleosynthesis and cosmology whilst the second attempts to outline causality’s path in the upper half of the table in biochemical settings. The link between the cycles is the set of elements themselves; within theoretically an infinite group of elements as radioactivity is reignited having been extinguished temporarily in the ebb and flow of production and annihilation of white dwarfs, neutron stars and black holes. The current scientific landscape is outlined to create a platform from which to proceed. Various sizes of black hole production suggest a hierarchy of outcomes which produces a reignition of radioactivity and potentially a creation of other universes from the explosions of larger supermassive black holes as energies increase to the Planck scale resulting in periods of inflation and condensation that predate quark production. Universes could be superimposed on previous universes explaining why some supermassive black holes appear nearer than current theoretical models.

Counting twisted sheaves and S-duality

We suggest a construction of Tanaka-Thomas's Vafa-Witten invariants for étale gerbes over smooth projective surface using the moduli space of μr-gerbe twisted sheaves and Higgs sheaves. Twisted sheaves and their moduli space are naturally used to study the period-index theorem for the corresponding μr-gerbe in the Brauer group of the surface. Deformation and obstruction theory of twisted sheaves and Higgs sheaves behave like general sheaves and Higgs sheaves. We define the virtual fundamental class for the moduli space and the twisted Vafa-Witten invariants using virtual localization and Behrend function techniques. As an application, for the Langlands dual group SU(r)/Zr of SU(r) we define the SU(r)/Zr-Vafa-Witten invariants. We prove the S-duality conjecture of Vafa-Witten for the projective plane in the case of rank two and for K3 surface in the case of prime ranks.

Scotogenic neutrino masses with gauged matter parity and gauge coupling unification

Building up on previous work we propose a Dark Matter (DM) model with gauged matter parity and dynamical gauge coupling unification, driven by the same physics responsible for scotogenic neutrino mass generation. Our construction is based on the extended gauge group SU(3)c ⊗ SU(3)L ⊗ U(1)X ⊗ U(1)N, whose spontaneous breaking leaves a residual conserved matter parity, MP, stabilizing the DM particle candidates of the model. The key role is played by Majorana SU(3)L-octet leptons, allowing the successful gauge coupling unification and a one-loop scotogenic neutrino mass generation. Theoretical consistency allows for a plethora of new particles at the ≲ mathcal{O} (10) TeV scale, hence accessible to future collider and low-energy experiments.

The magic square and half-hypermultiplets in F-theory

Abstract In six-dimensional F-theory/heterotic string theory, half-hypermultiplets arise only when they correspond to particular quaternionic Kähler symmetric spaces, which are mostly associated with the Freudenthal–Tits magic square. Motivated by the intriguing singularity structure previously found in such F-theory models with a gauge group SU(6), SO(12), or E7, we investigate, as the final magical example, an F-theory on an elliptic fibration over a Hirzebruch surface of the non-split I6 type, in which the unbroken gauge symmetry is supposed to be Sp(3). We find significant qualitative differences between the previous F-theory models associated with the magic square and the present case. We argue that the relevant half-hypermultiplets arise at the E6 points, where half-hypermultiplets 20 of SU(6) would have appeared in the split model. We also consider the problem on the non-local matter generation near the D6 point. After stating what the problem is, we explain why this is so by using the recent result that a split/non-split transition can be regarded as a conifold transition.

Dirac gaugino from grand gauge-Higgs unification

Abstract We show that models of the Dirac gaugino can naturally be embedded into a kind of grand unified theory (GUT), the grand gauge-Higgs unification (gGHU) model, with the gauge group $SU(5)\times SU(5)/\mathbb {Z}_2$ on an $S^1/\mathbb {Z}_2$ orbifold. The supersymmetric gGHU is known to possess a light chiral adjoint supermultiplet after the GUT breaking, thanks to the exchange symmetry of two SU(5) groups. Identifying the “predicted” adjoint fermion with the Dirac partner of the gaugino, we argue that the supersoft term, responsible for the Dirac gaugino mass, can be obtained from the supersymmetric Chern–Simons (CS)-like term in the gGHU setup. Although the latter term does not respect the exchange symmetry, we propose a novel way to introduce its breaking effect within a consistent orbifold construction. We also give a concrete setup of fermion field contents (bulk and boundary-localized fermions) that induce the requisite CS-like term, and calculate its coefficient from the bulk profile of chiral fermion zero modes. Our gGHU setup may be regarded as an extra-dimensional realization of the Goldstone gaugino scenario that was proposed before as a solution to the problem of adjoint scalar masses.

Vector boson dark matter from trinification

We show how trinification models based on the gauge group SU(3)C× SU(3)L× SU(3)R realized near the TeV scale can provide naturally a variety of dark matter (DM) candidates. These models contain a discrete T parity which may remain unbroken even after spontaneous symmetry breaking. The lightest T-odd particle, which could be a fermion, a scalar, or a gauge boson, can constitute the dark matter of the universe. This framework naturally admits a doublet-singlet fermionic DM, a singlet scalar DM, or a vector boson DM. Here we develop the vector boson DM scenario wherein the DM couples off-diagonally with the usual fermions and vector-like fermions present in the theory. We show consistency of this framework with dark matter relic abundance and direct detection limits as well as LHC constraints. We derive upper limits of 900 GeV on the vector gauge boson DM mass and 4.5 TeV on the vector-like quark masses. We also show the consistency of spontaneous gauge symmetry breaking down to Standard Model times an extra U(1) while preserving the T-parity.

Quantum gravity at the fifth root of unity

We consider quantum transition amplitudes, partition functions and observables for 3D spin foam models within SU(2) quantum group deformation symmetry, where the deformation parameter is postulated to be a complex fifth root of unity. By considering fermionic cycles through the foam, we couple this SU(2) quantum group with the same deformation of SU(3) so that we have quantum numbers linked with spacetime symmetry and charge gauge symmetry in the computation of observables. In this case, the amplitudes have a polytope interpretation. The generalization to higher-dimensional Lie groups SU(N), G2 and E8 is suggested.

Constant‐Roll Inflation in the Generalized SU(2) Proca Theory

AbstractThe generalized SU(2) Proca theory (GSU2P) is a variant of the well known generalized Proca theory where the vector field belongs to the Lie algebra of the SU(2) group of global transformations under which the action is made invariant. New interesting possibilities arise in this framework because of the existence of new interactions of purely non‐Abelian character and new configurations of the vector field resulting in spatial spherical symmetry and the cosmological dynamics being driven by the propagating degrees of freedom. We study the 2D phase space of the system that results when the cosmic triad configuration is employed in the Friedmann–Lemaitre–Robertson–Walker background and find an attractor curve whose attraction basin both covers almost all the allowed region and does not include a Big‐Bang singularity. Such an attractor curve corresponds to a primordial inflationary solution that has the following characteristic properties: 1) it is a de Sitter solution whose Hubble parameter is regulated by a generalized version of the SU(2) group coupling constant, 2) it is constant‐roll including, as a limiting case, the slow‐roll variety, 3) a number of e‐folds is easily reached, 4) it has a graceful exit into a radiation dominated period powered by the canonical kinetic term of the vector field and the Einstein–Hilbert term. The free parameters of the action are chosen such that the tensor sector of the theory is the same as that of general relativity at least up to second‐order perturbations, thereby avoiding the presence of ghost and Laplacian instabilities in the tensor sector as well as making the gravity waves propagate at light speed. This is a proof of concept of the interesting properties that could be found in this scenario when the coupling constants be replaced by general coupling functions and more terms be discovered in the GSU2P.

Comparing Quantum Gravity Models: String Theory, Loop Quantum Gravity, and Entanglement Gravity versus SU(∞)-QGR

In a previous article we proposed a new model for quantum gravity (QGR) and cosmology, dubbed SU(∞)-QGR. One of the axioms of this model is that Hilbert spaces of the Universe and its subsystems represent the SU(∞) symmetry group. In this framework, the classical spacetime is interpreted as being the parameter space characterizing states of the SU(∞) representing Hilbert spaces. Using quantum uncertainty relations, it is shown that the parameter space—the spacetime—has a 3+1 dimensional Lorentzian geometry. Here, after a review of SU(∞)-QGR, including a demonstration that its classical limit is Einstein gravity, we compare it with several QGR proposals, including: string and M-theories, loop quantum gravity and related models, and QGR proposals inspired by the holographic principle and quantum entanglement. The purpose is to find their common and analogous features, even if they apparently seem to have different roles and interpretations. The hope is that this exercise provides a better understanding of gravity as a universal quantum force and clarifies the physical nature of the spacetime. We identify several common features among the studied models: the importance of 2D structures; the algebraic decomposition to tensor products; the special role of the SU(2) group in their formulation; the necessity of a quantum time as a relational observable. We discuss how these features can be considered as analogous in different models. We also show that they arise in SU(∞)-QGR without fine-tuning, additional assumptions, or restrictions.

Generalization of Spin Operators

The study shall look to the group of generators SU(4). From these generators, a new group spin operator will be constructed. We will classify these groups into right handed groups and left handed groups. These two groups will satisfy all the properties of Pauli spin operators Sx, Sy and Sz with respect to the frame xyz. The analysis shows that the number of groups spin operators depends on the order of the group. This leads us to construct the theorem which defines the number of the groups spin operators. The analysis also leads to two kinds of frames: left handed frame (LHF) and right handed frame (RHF). The right handed operators will act on the RHF, and left hand operators act on the LHF. The study shall discuss the notion of spin squeezing for pure spin 3/2 system by using our new frames and new spin operators. It will show that our calculation is equivalent to the calculation by using Pauli spin operators.

Phase diagram and Higgs phases of three-dimensional lattice SU(N_{c}) gauge theories with multiparameter scalar potentials.

We consider three-dimensional lattice SU(N_{c}) gauge theories with degenerate multicomponent (N_{f}>1) complex scalar fields that transform under the fundamental representation of the gauge SU(N_{c}) group and of the global U(N_{f}) invariance group, interacting with the most general quartic potential compatible with the global (flavor) and gauge (color) symmetries. We investigate the phase diagrams, identifying the low-temperature Higgs phases and their global and gauge symmetries, and the critical behaviors along the different transition lines. In particular, we address the role of the quartic scalar potential, which determines the Higgs phases and the corresponding symmetry-breaking patterns. Our study is based on the analysis of the minimum-energy configurations and on numerical Monte Carlo simulations. Moreover, we investigate whether some of the transitions observed in the lattice model can be related to the behavior of the renormalization-group flow of the continuum field theory with the same symmetries and field content around its stable charged fixed points. For N_{c}=2, numerical results are consistent with the existence of charged critical behaviors for N_{f}>N_{f}^{★}, with 20<N_{f}^{★}<40.

Gravitational waves from the phase transition in the B-LSSM

Based on the gauge symmetry group SU(3)C ⨂ SU(2)L ⨂ U(1)Y ⨂ U(1)B–L, the minimal supersymmetric extension of the SM with local B-L gauge symmetry(B-LSSM) has been introduced. In this model, we study the Higgs masses with the one-loop zero temperature effective potential corrections. Besides, the finite temperature effective potentials connected with two U(1)B-L Higgs singlets are deduced specifically. Then we can obtain the gravitational wave spectrums generated from the strong first-order phase transition. In the B-LSSM, with the fine-tuned parameter regions, we can obtain the strength parameter αθ ~ 0.14 and the ratio of speed to Hubble rate β/Hn ~ 5 at nucleation temperature, and then obtain observable gravitational wave signals. The gravitational wave signals can be as strong as h2ΩGW ~ 10–9, which may be detectable in the future experiments.

Machine-learning the Sato–Tate conjecture

We apply some of the latest techniques from machine-learning to the arithmetic of hyperelliptic curves. More precisely we show that, with impressive accuracy and confidence (between 99 and 100 percent precision), and in very short time (matter of seconds on an ordinary laptop), a Bayesian classifier can distinguish between Sato–Tate groups given a small number of Euler factors for the L-function. Our observations are in keeping with the Sato-Tate conjecture for curves of low genus. For elliptic curves, this amounts to distinguishing generic curves (with Sato–Tate group SU(2)) from those with complex multiplication. In genus 2, a principal component analysis is observed to separate the generic Sato–Tate group USp(4) from the non-generic groups. Furthermore in this case, for which there are many more non-generic possibilities than in the case of elliptic curves, we demonstrate an accurate characterisation of several Sato–Tate groups with the same identity component. Throughout, our observations are verified using known results from the literature and the data available in the LMFDB. The results in this paper suggest that a machine can be trained to learn the Sato–Tate distributions and may be able to classify curves efficiently.

Gravitational quantum states as finite representations of the Lorentz group

A manifestly Lorentz-covariant formulation of loop quantum gravity (LQG) is given in terms of finite-dimensional representations of the Lorentz group. The formulation accounts for discrete symmetries, such as parity and time-reversal, and it establishes a link with Wigner classification of particles. The resulting quantum model can be seen as LQG-like with the internal SU(2) ⊗ SU(2) group and it is free of the Immirzi parameter, while the scalar constraint is just the Euclidean part.

Standard model gauge fields localized on non-Abelian vortices in six dimensions

Abstract A brane-world SU(5) grand unified theory model with global non-Abelian vortices is constructed in six-dimensional spacetime. We find a solution with a vortex associated to SU(3) separated from another vortex associated to SU(2). This 3–2 split configuration achieves a geometric Higgs mechanism for SU(5) → SU(3) × SU(2) × U(1) symmetry breaking. A simple deformation potential induces a domain wall between non-Abelian vortices, leading to a linear confining potential. The confinement stabilizes the vortex separation moduli, and ensures the vorticities of the SU(3) and SU(2) groups are identical. This dictates the equality of the numbers of fermion zero modes in the fundamental representation of SU(3) (quarks) and SU(2) (leptons), leading to quark/lepton generations. The standard model massless gauge fields are localized on the non-Abelian vortices thanks to a field-dependent gauge kinetic function. We perform fluctuation analysis with an appropriate gauge fixing and obtain a four-dimensional effective Lagrangian of unbroken and broken gauge fields at quadratic order. We find that SU(3) × SU(2) × U(1) gauge fields are localized on the vortices and exactly massless. Complications in analyzing the spectra of gauge fields with the nontrivial gauge kinetic function are neatly worked out by a vector-analysis-like method.

Muon g \u2212 2 from millicharged hidden confining sector

We provide a novel explanation to the muon g − 2 excess with new physics contributions at the two-loop level. In this scenario, light millicharged particles are introduced to modify the photon vacuum polarization that contributes to muon g − 2 at one additional loop. The muon g − 2 excess can be explained with the millicharged particle mass mχ around 10 MeV and the product of the multiplicity factor and millicharge squared of Nχε2 ∼ 10−3. The minimal model faces severe constraints from direct searches at fixed-target experiments and astrophysical observables. However, if the millicharged particles are also charged under a hidden confining gauge group SU(Nχ) with a confinement scale of MeV, hidden-sector hadrons are unstable and can decay into neutrinos, which makes this scenario consistent with existing constraints. This explanation can be well tested at low-energy lepton colliders such as BESIII and Belle II as well as other proposed fixed-target experiments.

Classical observables from coherent-spin amplitudes

The quantum field-theoretic approach to classical observables due to Kosower, Maybee and O’Connell provides a rigorous pathway from on-shell scattering amplitudes to classical perturbation theory. In this paper, we promote this formalism to describe general classical spinning objects by using coherent spin states. Our approach is fully covariant with respect to the massive little group SU(2) and is therefore completely synergistic with the massive spinor-helicity formalism. We apply this approach to classical two-body scattering due gravitational interaction. Starting from the coherent-spin elastic-scattering amplitude, we derive the classical impulse and spin kick observables to first post-Minkowskian order but to all orders in the angular momenta of the massive spinning objects. From the same amplitude, we also extract an effective two-body Hamiltonian, which can be used beyond the scattering setting. As a cross-check, we rederive the classical observables in the center-of-mass frame by integrating the Hamiltonian equations of motion to the leading order in Newton’s constant.

On Reductants of Two Groups

In this paper we consider the reductant of the dihedral group Dn, consisting of a set of axial symmetries, and the sphere S2 as a reductant of the group SU(2,C) ∼= S3 (the group of unit quaternions). By introducing the Sabinin’s multiplication on the reductant of Dn, we get a quasigroup with unit

A self-consistent framework of topological amplitude and its SU(N) decomposition

We propose a systematic theoretical framework for the topological amplitudes of the heavy meson decays and their SU(N) decomposition. In the framework, the topologies are expressed in invariant tensors and classified into tree- and penguin-operator-induced diagrams according to which four-quark operators, tree or penguin, being inserted into their effective weak vertexes. The number of possible topologies contributing to one type of decay can be counted by permutations and combinations. The Wigner-Eckhart theorem ensures the topological amplitudes under flavor symmetry are the same for different decay channels. By decomposing the four-quark operators into irreducible representations of SU(N) group, one can get the SU(N) irreducible amplitudes. Taking the D → PP decay (P denoting a pseudoscalar meson) with SU(3)F symmetry as an example, we present our framework in detail. The linear correlation of topologies in the SU(3)F limit is clarified in group theory. It is found there are only nine independent topologies in all tree- and penguin-operator-induced diagrams contributing to the D → PP decays in the Standard Model. If a large quark-loop diagram, named TLP, is assumed, the large ∆ACP and the very different D0→ K+K− and D0→ π+π− branching fractions can be explained with a normal U-spin breaking. Moreover, our framework provides a simple way to analyze the SU(N) breaking effects. The linear SU(3)F breaking and the high order U-spin breaking in charm decays are re-investigated in our framework, which are consistent with literature. Analogous to the degeneracy and splitting of energy levels, we propose the concepts of degeneracy and splitting of topologies to describe the flavor symmetry breaking effects in decay. As applications, we analyze the strange-less D decays in SU(3)F symmetry breaking into Isospin symmetry and the charm-less B decays in SU(4)F symmetry breaking into SU(3)F symmetry.

Majorana fermion dark matter in minimally extended left-right symmetric model

We present a minimal extension of the left-right symmetric model based on the gauge group SU(3)c× SU(2)L× SU(2)R× U(1)B−L× U(1)X, in which a vector-like fermion pair (ζL and ζR) charged under the U(1)B−L× U(1)X symmetry is introduced. Associated with the symmetry breaking of the gauge group SU(2)R× U(1)B−L× U(1)X down to the Standard Model (SM) hypercharge U(1)Y, Majorana masses for ζL,R are generated and the lightest mass eigenstate plays a role of the dark matter (DM) in our universe by its communication with the SM particles through a new neutral gauge boson “X”. We consider various phenomenological constraints of this DM scenario, such as the observed DM relic density, the LHC Run-2 constraints from the search for a narrow resonance, and the perturbativity of the gauge couplings below the Planck scale. Combining all constraints, we identify the allowed parameter region which turns out to be very narrow. A significant portion of the currently allowed parameter region will be tested by the High-Luminosity LHC experiments.