Diagonal Basis Research Articles

Conformal anomaly of super Wilson loop

Classically supersymmetric Wilson loop on a null polygonal contour possesses all symmetries required to match it onto non-MHV amplitudes in maximally supersymmetric Yang–Mills theory. However, to define it quantum mechanically, one is forced to regularize it since perturbative loop diagrams are not well defined due to presence of ultraviolet divergences stemming from integration in the vicinity of the cusps. A regularization that is adopted by practitioners by allowing one to use spinor helicity formalism, on the one hand, and systematically go to higher orders of perturbation theory is based on a version of dimensional regularization, known as Four-Dimensional Helicity scheme. Recently it was demonstrated that its use for the super Wilson loop at one loop breaks both conformal symmetry and Poincaré supersymmetry. Presently, we exhibit the origin for these effects and demonstrate how one can undo this breaking. The phenomenon is alike the one emerging in renormalization group mixing of conformal operators in conformal theories when one uses dimensional regularization. The rotation matrix to the diagonal basis is found by means of computing the anomaly in the Ward identity for the conformal boost. Presently, we apply this ideology to the super Wilson loop. We compute the one-loop conformal anomaly for the super Wilson loop and find that the anomaly depends on its Grassmann coordinates. By subtracting this anomalous contribution from the super Wilson loop we restore its interpretation as a dual description for reduced non-MHV amplitudes which are expressed in terms of superconformal invariants.

Generation of polarization entangled photon pairs at telecommunication wavelength using cascaded χ^(2) processes in a periodically poled LiNbO_3 ridge waveguide

We report the generation of high-purity correlated photon-pairs and polarization entanglement in a 1.5 μm telecommunication wavelength-band using cascaded χ((2)):χ((2)) processes, second-harmonic generation (SHG) and the following spontaneous parametric down conversion (SPDC), in a periodically poled LiNbO(3) (PPLN) ridge-waveguide device. By using a PPLN module with 600%/W of the SHG efficiency, we have achieved a coincidence-to-accidental ratio (CAR) higher than 4000 at 7.45×10(-5) of the mean number of the photon-pair per pulse. We also demonstrated that the maximum reach of the CAR was truly dark-count-limited by the single-photon detectors used here. This indicates that the fake (noise) photons were negligibly small in this system, even though the photon-pairs, the Raman noise photons, and the pump photons were in the same wavelength band. Polarization entangled photon pairs were also generated by constructing a Sagnac-loop-type interferometer which included the PPLN module and an optical phase-difference compensator to observe maximum entanglement. We achieved two-photon interference visibilities of 99.6% in the H/V basis and 98.7% in the diagonal basis. The peak coincidence count rate was approximately 50 counts per second at 10(-3) of the mean number of the photon-pair per pulse.

KÄHLER MODULI INFLATION AND WMAP7

Inflationary potentials are investigated for specific models in type IIB string theory via flux compactification. As concrete models, we investigate several cases where the internal spaces are weighted projective spaces. The models we consider have two, three, or four Kähler moduli. The Kähler moduli play a role of inflaton fields and we consider the cases where only one of the moduli behaves as the inflaton field. For the cases with more than two moduli, we choose the diagonal basis for the expression of the Calabi–Yau volume, which can be written down as a function of four-cycle. With the combination of multiple moduli, we can express the multi-dimensional problem as an effective one-dimensional problem. In the large volume scenario, the potentials of these three models turn out to be of the same type. By taking the specific limit of the relation between the moduli and the volume, the potentials are reduced to simpler ones which induce inflation. For the case of two Kähler moduli, we exclude the potential as an inflationary model because the moduli might not be stable during inflation. As a toy model, we first consider the simple potential. We calculate the slow roll parameters ϵ, η and ξ for each inflationary potential. Then, we check whether the potentials give reasonable spectral indices ns and their running αs's by comparing with the recently released seven-year WMAP data. For both models, we see reasonable spectral indices for the number of e-folding 47<Ne<61. Conversely, by inserting the observed seven-year WMAP data, we see that the potential of the toy model gives requisite number of e-folds while the potential of the Kähler moduli gives much smaller number of e-folding. Finally, we see that two models do not produce reasonable values of the running of the spectral index.

Invisible Higgs decays from Higgs-graviscalar mixing

We recompute the invisible Higgs decay width arising from Higgs-graviscalar mixing in the Arkani-Hamed, Dimopoulos, Dvali model, comparing the original derivation in the nondiagonal mass basis to that in a diagonal mass basis. The results obtained are identical (and differ by a factor of 2 from the original calculation) but the diagonal-basis derivation is pedagogically useful for clarifying the physics of the invisible width from mixing. We emphasize that both derivations make it clear that a direct scan in energy for a process such as $WW\ensuremath{\rightarrow}WW$ mediated by Higgs plus graviscalar intermediate resonances would follow a single Breit-Wigner form with total width given by ${\ensuremath{\Gamma}}^{\mathrm{tot}}={\ensuremath{\Gamma}}_{h}^{\mathrm{SM}}+{\ensuremath{\Gamma}}_{\mathrm{invisible}}$. We also compute the additional contributions to the invisible width due to direct Higgs to graviscalar-pair decays. We find that the invisible width due to the latter is relatively small, unless the Higgs mass is comparable to or larger than the effective extra-dimensional Planck mass.

New theoretical approaches for correlated systems in nonequilibrium

We review recent developments in the theory of interacting quantum many-particle systems that are not in equilibrium. We focus mainly on the nonequilibrium generalizations of the flow equation approach and of dynamical mean-field theory (DMFT). In the nonequilibrium flow equation approach one first diagonalizes the Hamiltonian iteratively, performs the time evolution in this diagonal basis, and then transforms back to the original basis, thereby avoiding a direct perturbation expansion with errors that grow linearly in time. In nonequilibrium DMFT, on the other hand, the Hubbard model can be mapped onto a time-dependent self-consistent single-site problem. We discuss results from the flow equation approach for nonlinear transport in the Kondo model, and further applications of this method to the relaxation behavior in the ferromagnetic Kondo model and the Hubbard model after an interaction quench. For the interaction quench in the Hubbard model, we have also obtained numerical DMFT results using quantum Monte Carlo simulations. In agreement with the flow equation approach they show that for weak coupling the system relaxes to a "prethermalized" intermediate state instead of rapid thermalization. We discuss the description of nonthermal steady states with generalized Gibbs ensembles.

A new family symmetry for [formula omitted] GUTs

We argue that the projective special linear group PSL 2 ( 7 ) , also known as Σ ( 168 ) , has unique features which make it the most suitable discrete family symmetry for describing quark and lepton masses and mixing in the framework of SO ( 10 ) type unified models. In such models flavon fields in the sextet representation of PSL 2 ( 7 ) play a crucial role both in obtaining tri-bimaximal neutrino mixing as well as in generating the third family charged fermion Yukawa couplings. In preparation for physical applications, we derive the triplet representation of PSL 2 ( 7 ) in the basis S , T , U , V where S , T , U are the familiar triplet generators of S 4 in the diagonal charged lepton basis where T is diagonal. We also derive an analogous basis for the real sextet representation and identify the vacuum alignments which lead to tri-bimaximal neutrino mixing and large third family charged fermion Yukawa couplings.

Empirical neutrino mass matrix related to up-quark Masses

According to an idea that mass spectra of quarks and leptons originate in vacuum expectation values of 0(3) flavor (3 x 3)s = 1 + 5 (gauge singlet) scalars, a neutrino mass matrix related to up-quark and charged lepton masses is derived: Mi'v = MeMR−1MTe with Mr ∝ MeMu½ + Mu½ Me. In order to obtain the lepton mixing matrix, we must know forms of Me and Mu in the ‘e-basis’ which is defined as a diagonal basis of Me. By analogy with Mu = VTDUV in the d-basis (Du = diag(mu,mc,mt) and V is the CKM mixing matrix), we assume a form Mu½ = VT(δ)DU½ V(δ) in the e-basis. When we use the observed values of |VUS|, |Vcb| and |Vub| for the form V(δ) (δ is free), our neutrino mass matrix can give a nearly tribi-maximal mixing, i.e. |U13| = 0.00010, sin2 2θ23 = 0.9997, and tan2θ12 = 0.5125 at δ = π. We consider that this is not accidental and this result should be taken seriously. For more details, see Y.K, Phys.Lett. B665 (2008) 227 (arXiv:0804.4267 [hep-ph]).

Half-BPS SU(N) correlators in 𝒩 = 4 SYM

In this note we study half-BPS operators in N=4 super Yang-Mills for gauge group SU(N) at finite N. In particular we elaborate on the results of hep-th/0410236, providing an exact formula for the null basis operators algorithmically constructed there. For gauge groups U(N) and SU(N) we show that this basis is dual to the basis of multi-trace operators with respect to the two point function. We use this to extend the results of hep-th/0611290 concerning factorisation and probabilities from U(N) to SU(N). We also give a construction for a separate diagonal basis of the SU(N) operators in terms of the higher Hamiltonians of the complex matrix model reduction of this sector.

Ground-state properties of a Tonks-Girardeau gas in a split trap

We determine the exact many-body properties of a bosonic Tonks-Girardeau gas confined in a harmonic potential with a tunable $\ensuremath{\delta}$-function barrier at the trap center. This is done by calculating the reduced single-particle density matrix, the pair-distribution function, and the momentum distribution of the gas as a function of barrier strength and particle number. With increasing barrier height we find that the ground-state occupation in a diagonal basis diverges from the $\sqrt{N}$ behavior that is expected for the case of a simple harmonic trap. In fact, the scaling of the occupation number depends on whether one has an even or odd number of particles. Since this quantity is a measure of the coherence of our sample we show how the odd-even effect manifests itself in both the momentum distribution of the Bose gas and interference fringe visibility during free temporal evolution.

Covariant methods for calculating differential cross sections and polarization characteristics of reactions

A covariant method for calculating reaction amplitudes in the diagonal spin basis is presented. In this basis, amplitudes represent the spin kinematics of the interacting particles in the simplest and most adequate manner. A matrix is obtained such that differential cross sections for polarized particles and amplitude combinations that are required to calculate various polarization characteristics of reactions can be expressed in an arbitrary basis in terms of the amplitudes calculated in one basis.

Time-localized projectors in string field theory with anE-field

We extend the analysis of hep-th/0409063 to the case of a constant electric field turned on the worldvolume and on a transverse direction of a D-brane. We show that time localization is still obtained by inverting the discrete eigenvalues of the lump solution. The lifetime of the unstable soliton is shown to depend on two free parameters: the b-parameter and the value of the electric field. As a by-product, we construct the normalized diagonal basis of the star algebra in $B_{\mu\nu}$-field background.

Diagonal Bases in Orlik-Solomon Type Algebras

To encode an important property of the “no broken circuit bases” of the Orlik- Solomon-Terao algebras, Szenes has introduced a particular type of bases, the so called ”diagonal basis.” We prove that this definition extends naturally to a large class of algebras, the so called χ- algebras. Our definitions make also use of an “iterative residue formula” based on the matroidal operation of contraction. This formula can be seen as the combinatorial analogue of an iterative residue formula introduced by Szenes. As an application we deduce nice formulas to express a pure element in a diagonal basis.

THE NEUTRINOLESS DOUBLE β DECAY AND THE NEUTRINO MASS HIERARCHY

Recently the evidence of the neutrinoless double β (0νββ) decay has been announced. This means that neutrinos are Majorana particles and their mass hierarchy is forced to certain patterns in the diagonal basis of charged lepton mass matrix. We estimate the magnitude of 0νββ decay in the classification of the neutrino mass hierarchy patterns as Type A, m1,2 ≪ m3, Type B, m1 ~ m2 ≫ m3, and Type C, m1 ~ m2 ~ m3, where mi is the ith generation neutrino absolute mass. The data of 0νββ decay experiment suggest that the neutrino mass hierarchy pattern should be Type B or C. Type B predicts a small magnitude of 0νββ decay which is just edge of the allowed region of experimental value in 95% c.l., where Majorana CP phases should be in a certain parameter region. Type C can induce the suitably large amount of 0νββ decay which is consistent with the experimental data, where overall scale of degenerate neutrino mass plays a crucial role, and its large value can induce the large 0νββ decay in any parameter regions of Majorana CP phases.

The diagonal spin basis and calculation of processes involving polarized particles

The review of developed by the authors new techniques for covariant calculation of matrix elements in QED, the so-called formalism of "Diagonal Spin Basis" (DSB), is presented. In DSB spin 4-vectors of in- and out- fermions are expressed just in terms of their 4-momenta. In this approach the little Lorentz group, common for the initial and final states,is realized. This brings the spin operators of in- and out-particles to coincidence. The developed approach is valid both for massive fermions and for massless ones. There occur no problems with accounting for spin flip amplitudes in it. Just 4-momenta of particles participating in reactions are required in it to construct the mathematical apparatus for calculations of matrix elements. We apply this formalism to the next processes: 1) M\"oller and Bhabha bremsstrahlung ($e^{\pm}e^- \to e^{\pm}e^- \gamma$) in the ultrarelativistic limit when initial particles and photon are helicity polarized; 2) Compton back-scattering of photons of intensive circularly polarized laser wave on a beam of longitudinally polarized ultrarelativistic electrons ($e+n\gamma_0 \to e+\gamma $); 3) $e^+e^-$-pair production by a hard photon in simultaneous collision with several laser beam photons ($\gamma+n \gamma_0 \to e^+ + e^-$); 4) Bethe-Heitler process in the case of a linearly polarized photon emission by an electron with account for proton recoil and form factors; 5) the reaction $ep \to ep \gamma$ with proton polarizability being taken into account in the kinematics when proton bremsstrahlung dominates; 6) orthopositronium 3-photon annihilation ($e^+e^{-} \to 3 \gamma$).

BRST quantisation of the N = 2 string in a real spacetime structure

We study the N = 2 string with a real structure on the (2,2) spacetime, using BRST methods. Several new features emerge. In the diagonal basis, the operator exp ( λ ∫ z J tot), which is associated with the moduli for the U(1) gauge field on the world-sheet, is local and it relates the physical operators in the NS and R sectors. However, the picture-changing operators are non-invertible in this case, and physical operators in different pictures cannot be identified. The three-point interactions of all physical operators lead to three different types of amplitudes, which can be effectively described by the interactions of a scalar NS operator and a bosonic spinorial R operator. In the off-diagonal bases for the fermionic currents, the picture-changing operators are invertible, and hence physical operators in different pictures can be identified. However, now there is no local operator exp ( λ ∫ z J tot)that relates the physical operators in different sectors. The physical spectrum is thus described by one scalar NS operator and one spinorial R operator. The NS and R operators give rise to different types of three-point amplitudes, and thus cannot be identified.

Diagonal spin basis and a spinor method for calculation of matrix elements

The pure-state wave functions of massive spin-1/2 particles are found in the diagonal spin basis. These results are used in a version of the covariant method of the calculation of the matrix elements for the interaction of polarized particles.

Stochastic dynamics of individual quantum systems: Stationary rate equations.

An open quantum system is usually characterized by a reduced ensemble density matrix, the dynamics of which is governed by a generalized Master equation. Transforming this equation of motion into the instantaneous diagonal basis of the corresponding reduced density matrix, we can separate the coherent and incoherent part of the dynamics: The coherent dynamics is incorporated in the time development of the diagonal basis states, while the coupling to the reservoirs leads to simple rate equations. Interpreting these rate equations as a stochastic point process allows one to simulate the stochastic time evolution (random telegraph signals, ``quantum jumps'') of single-quantum systems. The diagonal representation can be considered as a generalization of the dressed-state picture of open quantum systems. Numerical simulations (``quantum Monte Carlo'') allow one to derive various dynamical properties (including correlation functions) of single-quantum systems. This concept is applied to different two- and three-level scenarios (\ensuremath{\lambda} and \ensuremath{\nu} configuration), and its limitations are discussed.

Aspects of the μ-dependence in the effective hamiltonian for the ΔS = 1 transitions

We present the standard effective hamiltonian for ΔS = 1 transitions in the basis of the multiplicatively renormalized operators (diagonal basis). In this basis the μ-dependence of the relevant Wilson coefficient functions is very transparent. This allows us to investigate in a simple manner the related uncertainty in the standard phenomenological analyses which combine the μ-dependent QCD coefficient functions with the μ-independent matrix elements as obtained in the quark or bag models. In particular we show that the commonly used relations between operators which follow from the valence quark approximation are in variance with the μ-dependence predicted by QCD. The related uncertainty in the calculation of ε′/ε amounts roughly to 50%.

Representation and properties of para-Bose oscillator operators. II. Coherent states and the minimum uncertainty states

The energy, position, and momentum eigenstates of a para-Bose oscillator system were considered in paper I. Here we consider the Bargmann or the analytic function description of the para-Bose system. This brings in, in a natural way, the coherent states ‖z;α〉 defined as the eigenstates of the annihilation operator ?. The transformation functions relating this description to the energy, position, and momentum eigenstates are explicitly obtained. Possible resolution of the identity operator using coherent states is examined. A particular resolution contains two integrals, one containing the diagonal basis ‖z;α〉〈z;α‖ and the other containing the pseudodiagonal basis ‖z;α〉〈−z;α‖. We briefly consider the normal and antinormal ordering of the operators and their diagonal and discrete diagonal coherent state approximations. The problem of constructing states with a minimum value of the product of the position and momentum uncertainties and the possible α dependence of this minimum value is considered.

Quantum-mechanical treatment of collision-induced dissociation

A close-coupling technique for calculating quantum-mechanical probabilities of collision-induced dissociation (CID) of a diatomic molecule by an atom is presented. The internal Hamiltonian (of the diatomic) is first diagonalized in a discrete, square-integrable basis. The lowest several of the resulting discrete eigenstates approximate the true bound states and the remaining (pseudocontinuum) states represent the true continuum. Next, the stationary collision wavefunction is expanded in the diagonal basis to obtain a discrete set of close-coupled equations, which are integrated numerically by standard procedures. The method is applied to a collinear model in which the diatomic is bound by a Morse potential and the interaction is a repulsive exponential. The total CID probabilities appear to be converged to 1% or 2% in most cases. Vibrational ’’enhancement’’ of CID is observed in this model. A very general problem associated with the use of the exponential interaction in conjunction with a binding potential which supports a continuum is discussed.