Irreducible Representations Of SU Research Articles

Gysin sequences and SU(2)‐symmetries of C∗‐algebras

Motivated by the study of symmetries of C*-algebras, as well as by multivariate operator theory, we introduce the notion of an SU(2)-equivariant subproduct system of Hilbert spaces. We analyse the resulting Toeplitz and Cuntz-Pimsner algebras and provide results about their topological invariants through Kasparov's bivariant K-theory. In particular, starting from an irreducible representation of SU(2), we show that the corresponding Toeplitz algebra is equivariantly KK-equivalent to the algebra of complex numbers. In this way, we obtain a six term exact sequence of K-groups containing a noncommutative analogue of the Euler class.

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.

The invariant space of multi-Higgs doublet models

In a model with more than one scalar doublet, the parameter space encloses both physical and unphysical information. Invariant theory provides a detailed description of the counting and characterization of the physical parameter space. The Hilbert series for the 3HDM is computed for the first time using partition analysis, in particular Omega calculus, giving rise to the possibility of a full description of its physical parameters. A rigorous counting of the physical parameters is given for the full class of models with N scalar doublets as well as a decomposition of the Lagrangian into irreducible representations of SU(N). For the first time we derive a basis-invariant technique for counting parameters in a Lagrangian with both basis-invariant redundancies and global symmetries.

Designing locally maximally entangled quantum states with arbitrary local symmetries

One of the key ingredients of many LOCC protocols in quantum information is a multiparticle (locally) maximally entangled quantum state, aka a critical state, that possesses local symmetries. We show how to design critical states with arbitrarily large local unitary symmetry. We explain that such states can be realised in a quantum system of distinguishable traps with bosons or fermions occupying a finite number of modes. Then, local symmetries of the designed quantum state are equal to the unitary group of local mode operations acting diagonally on all traps. Therefore, such a group of symmetries is naturally protected against errors that occur in a physical realisation of mode operators. We also link our results with the existence of so-called strictly semistable states with particular asymptotic diagonal symmetries. Our main technical result states that the Nth tensor power of any irreducible representation of SU(N) contains a copy of the trivial representation. This is established via a direct combinatorial analysis of Littlewood-Richardson rules utilising certain combinatorial objects which we call telescopes.

Scalar anomaly cancellation reveals the hidden superalgebraic structure of the quantum chiral SU(2/1) model of leptons and quarks

At the classical level, the SU(2/1) superalgebra offers a natural description of the elementary particles: leptons and quarks massless states, graded by their chirality, fit the smallest irreducible representations of SU(2/1). Our new proposition is to pair the left/right space-time chirality with the superalgebra chirality and to study the model at the one-loop quantum level. If, despite the fact that they are non-Hermitian, we use the odd matrices of SU(2/1) to minimally couple an oriented complex Higgs scalar field to the chiral Fermions, novel anomalies occur. They affect the scalar propagators and vertices. However, these undesired new terms cancel out, together with the Adler-Bell-Jackiw vector anomalies, because the quarks compensate the leptons. The unexpected and striking consequence is that the scalar propagator must be normalized using the anti-symmetric super-Killing metric and the scalar-vector vertex must use the symmetric d_aij structure constants of the superalgebra. Despite this extraordinary structure, the resulting Lagrangian is actually Hermitian.

Two-loop beta function for complex scalar electroweak multiplets

We present the general form of the renormalizable four-point interactions of a complex scalar field furnishing an irreducible representation of SU(2), and derive a set of algebraic identities that facilitates the calculation of higher-order radiative corrections. As an application, we calculate the two-loop beta function for the SM extended by a scalar multiplet, and provide the result explicitly in terms of the group invariants. Our results include the evolution of the Higgs-portal couplings, as well as scalar “minimal dark matter”. We present numerical results for the two-loop evolution of the various couplings.

Temperley–Lieb, Brauer and Racah algebras and other centralizers of 𝔰𝔲(2)

In the spirit of the Schur–Weyl duality, we study the connections between the Racah algebra and the centralizers of tensor products of three (possibly different) irreducible representations of s u ( 2 ) \mathfrak {su}(2) . As a first step we show that the Racah algebra always surjects onto the centralizer. We then offer a conjecture regarding the description of the kernel of the map, which depends on the irreducible representations. If true, this conjecture would provide a presentation of the centralizer as a quotient of the Racah algebra. We prove this conjecture in several cases. In particular, while doing so, we explicitly obtain the Temperley–Lieb algebra, the Brauer algebra and the one-boundary Temperley–Lieb algebra as quotients of the Racah algebra.

Spin networks and cosmic strings in 3+1 dimensions

Spin networks, the quantum states of discrete geometry in loop quantum gravity, are directed graphs whose links are labeled by irreducible representations of SU(2), or spins. Cosmic strings are 1-dimensional topological defects carrying distributional curvature in an otherwise flat spacetime. In this paper we prove that the classical phase space of spin networks coupled to cosmic strings may obtained as a straightforward discretization of general relativity in 3+1 spacetime dimensions. We decompose the continuous spatial geometry into 3-dimensional cells, which are dual to a spin network graph in a unique and well-defined way. Assuming that the geometry may only be probed by holonomies (or Wilson loops) located on the spin network, we truncate the geometry such that the cells become flat and the curvature is concentrated at the edges of the cells, which we then interpret as a network of cosmic strings. The discrete phase space thus describes a spin network coupled to cosmic strings. This work proves that the relation between gravity and spin networks exists not only at the quantum level, but already at the classical level. Two appendices provide detailed derivations of the Ashtekar formulation of gravity as a Yang–Mills theory and the distributional geometry of cosmic strings in this formulation.

Relaxed hypermultiplet in four dimensional N=2 conformal supergravity

Superconformal matter multiplets play a crucial role in the construction of Poincare supergravity invariants. Off-shell multiplets allow for construction of general matter couplings in supergravity. In Nucl. Phys. B214 (1983) 519-531, relaxed hypermultiplet was constructed in rigid supersymmetry which on coupling with the real scalar multiplet allowed for an off-shell formulation of the rigid hypermultiplet. In this paper, we extend the relaxed hypermultiplet to conformal supergravity. For consistency with the superconformal algebra, we find that the fields have to be allowed to transform in a non-canonical way under SU(2) symmetry. We find suitable field redefinitions to obtain fields which are irreducible representations of SU(2) R-symmetry and present the full non-linear transformation rule.

Baryon-baryon interactions and spin-flavor symmetry from lattice quantum chromodynamics

Lattice quantum chromodynamics is used to constrain the interactions of two octet baryons at the SU(3) flavor-symmetric point, with quark masses that are heavier than those in nature (equal to that of the physical strange quark mass and corresponding to a pion mass of $\approx 806~\tt{MeV}$). Specifically, the S-wave scattering phase shifts of two-baryon systems at low energies are obtained with the application of L\"uscher's formalism, mapping the energy eigenvalues of two interacting baryons in a finite volume to the two-particle scattering amplitudes below the relevant inelastic thresholds. The values of the leading-order low-energy scattering parameters in the irreducible representations of SU(3) are consistent with an approximate SU(6) spin-flavor symmetry in the nuclear and hypernuclear forces that is predicted in the large-$N_c$ limit of QCD. The two distinct SU(6)-invariant interactions between two baryons are constrained at this value of the quark masses, and their values indicate an approximate accidental SU(16) symmetry. The SU(3) irreducible representations containing the $NN~({^1}S_0)$, $NN~({^3}S_1)$ and $\frac{1}{\sqrt{2}}(\Xi^0n+\Xi^-p)~({^3}S_1)$ channels unambiguously exhibit a single bound state, while the irreducible representation containing the $\Sigma^+ p~({^3}S_1)$ channel exhibits a state that is consistent with either a bound state or a scattering state close to threshold. These results are in agreement with the previous conclusions of the NPLQCD collaboration regarding the existence of two-nucleon bound states at this value of the quark masses.

Spherical HarmonicsYlm(θ,ϕ): Positive and Negative Integer Representations ofsu(1,1)forl-mandl+m

The azimuthal and magnetic quantum numbers of spherical harmonicsYlm(θ,ϕ)describe quantization corresponding to the magnitude andz-component of angular momentum operator in the framework of realization ofsu(2)Lie algebra symmetry. The azimuthal quantum numberlallocates to itself an additional ladder symmetry by the operators which are written in terms ofl. Here, it is shown that simultaneous realization of both symmetries inherits the positive and negative(l-m)- and(l+m)-integer discrete irreducible representations forsu(1,1)Lie algebra via the spherical harmonics on the sphere as a compact manifold. So, in addition to realizing the unitary irreducible representation ofsu(2)compact Lie algebra via theYlm(θ,ϕ)’s for a givenl, we can also representsu(1,1)noncompact Lie algebra by spherical harmonics for given values ofl-mandl+m.

Emergence of a newSU(4)symmetry in the baryon spectrum

M. Denissenya, ∗ L. Ya. Glozman, † and M. Pak ‡ Institut fur Physik, FB Theoretische Physik, Universitat Graz, Universitatsplatz 5, 8010 Graz, Austria Recently a large degeneracy of J = 1 mesons, that is larger than the SU(2)L × SU(2)R × U(1)A symmetry of the QCD Lagrangian, has been discovered upon truncation of the near-zero modes from the valence quark propagators. It has been found that this degeneracy represents the SU(4) group that includes the chiral rotations as well as the mixing of leftand right-handed quarks. This symmetry group turnes out to be a symmetry of confinement in QCD. Consequently, one expects that the same symmetry should persist upon the near-zero mode removal in other hadron sectors as well. It has been shown that indeed the J = 2 mesons follow the same symmetry pattern upon the low-lying mode elimination. Here we demonstrate the SU(4) symmetry of baryons once the near-zero modes are removed from the quark propagators. We also show a degeneracy of states belonging to different irreducible representations of SU(4). This implies a larger symmetry, that includes SU(4) as a subgroup.

Flavor structure of the excited baryon spectra from lattice QCD

Excited state spectra are calculated using lattice QCD for baryons that can be formed from $u$, $d$ and $s$ quarks, namely the $N$, $\Delta$, $\Lambda$, $\Sigma$, $\Xi$ and $\Omega$ families of baryons. Baryonic operators are constructed from continuum operators that transform as irreducible representations of SU(3)$_F$ symmetry for flavor, SU(4) symmetry for Dirac spins of quarks and O(3) symmetry for orbital angular momenta. Covariant derivatives are used to realize orbital angular momenta. Using the operators, we calculate matrices of correlation functions in order to extract excited states. The resulting lattice spectra have bands of baryonic states with well-defined total spins up to $J=7/2$. Each state can be assigned a dominant flavor symmetry and the counting of states of each flavor and spin reflects $SU(6) \times O(3)$ symmetry for the lowest negative-parity and positive-parity bands. States with strong hybrid content are identified through the dominance of chromo-magnetic operators.

Unification of 3-3-1 models

A 3-3-1 model is constructed for three families that can be embedded into a single SU(8) unified model. Assuming appropriate branching rules and symmetry-breaking pattern, a complete fermion content is found within irreducible representations of SU(8), where light Standard Model fermions, heavy 3-3-1 fermions and superheavy fermions may be distinguished.

Orthogonal multiplet bases in SU(N c ) color space

We develop a general recipe for constructing orthogonal bases for the calculation of color structures appearing in QCD for any number of partons and arbitrary Nc. The bases are constructed using hermitian gluon projectors onto irreducible subspaces invariant under SU(Nc). Thus, each basis vector is associated with an irreducible representation of SU(Nc). The resulting multiplet bases are not only orthogonal, but also minimal for finite Nc. As a consequence, for calculations involving many colored particles, the number of basis vectors is reduced significantly compared to standard approaches employing overcomplete bases. We exemplify the method by constructing multiplet bases for all processes involving a total of 6 external colored partons.

Dynamical and structural symmetries for the highest Landau levels on the AdS 2

Abstract The aim of the paper is to use the recurrence relations with respect to both indices of the associated Legendre functions for the extraction of the Dirac quantization condition and dynamical symmetry group U(1, 1) corresponding to the highest Landau levels on the hyperbolic plane with uniform magnetic field B. Irreducible representations of the su(2) algebra are obtained by the ladder differential operators which change B by 1/2 unit and mode number by one unit. Two different classes of the irreducible representations of SU(1, 1) with the even and odd boson numbers 2B − 1/2 are extracted for the Bargmann indices 1/4 and 3/4, respectively. Finally, we show that shape invariance symmetry is realized by the ladder operators which shift only the magnetic field B by 1/2 unit.

The Simultaneous Coupling Scheme for SU(3)

We present a simple method for calculating the Clebsch-Gordan coefficients for the tensor product of two unitary irreducible representations of SU(3). The calculation given here is an application and extension of the simultaneous coupling scheme proposed in 2010 for the rotation group SU(2).

Unitary irreducible representations of \documentclass[12pt]{minimal}$\mathrm{SL(2,\mathbb {C})}$ SL (2,C) in discrete and continuous \documentclass[12pt]{minimal}$\mathrm{SU(1,1)}$ SU (1,1) bases

We derive the matrix elements of generators of unitary irreducible representations of \documentclass[12pt]{minimal}\begin{document}$\mathrm{SL(2,\mathbb {C})}$\end{document} SL (2,C) with respect to basis states arising from a decomposition into irreducible representations of SU(1,1). This is done with regard to a discrete basis diagonalized by \documentclass[12pt]{minimal}\begin{document}$J^3$\end{document}J3 and a continuous basis diagonalized by \documentclass[12pt]{minimal}\begin{document}$K^1$\end{document}K1, and for both the discrete and continuous series of SU(1,1). For completeness, we also treat the more conventional SU(2) decomposition as a fifth case. The derivation proceeds in a functional/differential framework and exploits the fact that state functions and differential operators have a similar structure in all five cases. The states are defined explicitly and related to SU(1,1) and SU(2) matrix elements.

Matrix Inflation and its String Theory Origins

Motivated by the dynamics of N coincident D3 branes in some specific flux compactifications, we construct an inflationary model in which inflation is driven by three N × N hermitian matrices Φi,i = 1,2, 3, hence the name Matrix Inflation, or M-flation for short. We show that one can consistently restrict the classical dynamics to a sector in which the Φi are proportional to the N × N irreducible representation of SU(2). In this sector our model behaves like an effective inflaton field, which takes super-Planckian field values, and 3N2 − 1 isocurvature fields. These may have the observational effects such as production of iso-curvature perturbations on cosmic microwave background. Moreover, the existence of these extra scalars provides us with a natural preheating mechanism and exit from inflation. Due to the super-Planckian excursions of the canonical effective inflaton, the model is capable of producing a considerable amount of gravity waves that can be probed by future CMB polarization experiments. Furthermore, the fine-tunings associated with unnaturally small couplings in the chaotic type inflationary scenarios are removed. We also show that even if the cutoff of the theory is lowered by the square of number of species, one can still use the effective field theory approach to justify the absence of higher dimensional operators.

M-flation: inflation from matrix valued scalar fields

We propose an inflationary scenario, M-flation, inwhich inflation is driven by three N × N hermitian matricesΦi, i = 1, 2, 3. The inflation potential of our model, whichis strongly motivated from string theory, is constructed fromΦi and their commutators. We show that one can consistentlyrestrict the classical dynamics to a sector in which the Φiare proportional to the N × N irreducible representations ofSU(2). In this sector our model effectively behaves as anN-flation model with 3N2 number of fields and the effective inflaton field has a super-Planckian field value. Furthermore, the fine-tunings associated with unnaturally small couplings in the chaotic type inflationary scenarios are removed.Due to the matrix nature of the inflaton fieldsthere are 3N2−1 extra scalar fields in the dynamics.These have the observational effects such as production of iso-curvatureperturbations on cosmic microwave background.Moreover, the existence of these extra scalars provides us with anatural preheating mechanism and exit from inflation. As theeffective inflaton field can traverse super-Planckian distances in thefield space, the model is capable of producing a considerable amountof gravity waves that can be probed by future CMB polarizationexperiments such as PLANCK, QUIET and CMBPOL.