Order Sum Rules Research Articles

Construction of symbols satisfying sum rules of order p using standard pairs

In this paper, using standard pairs, we present a method to construct symbols satisfying the sum rules of order [Formula: see text] for any given [Formula: see text]. It is shown that using matrix polynomial theory symbols, symmetric or non-symmetric, satisfying the sum rules of order [Formula: see text] can be constructed efficiently. The construction is illustrated using various examples.

Linear operators and construction of symmetric symbols satisfying sum rules of order p

ABSTRACT We present a new method to construct symmetric symbols corresponding to solvable multiscaling equations using linear operators. First, we define a linear operator corresponding to a symbol. We demonstrate that the existence of a solution to a multiscaling equation depends on the 1-eigenspace of the corresponding linear operator. Furthermore, we show how to construct symmetric symbols satisfying the sum rules of orders 1 and 2 by appropriately defining the linear operator on a basis. We also construct symmetric symbols satisfying sum rules of order 3 with degree 3. Several examples are provided to illustrate our theoretical results.

Decomposable pairs and construction of multiwavelets

We study the construction of refinable function vectors and multiwavelets from the perspective of decomposable pairs. There exists a pair of matrices known as the decomposable pair, associated with a refinement mask of the form H(ξ)=12∑k=0lHke−iξk, which gives the spectral information about H(ξ). We show that the existence of a solution to a matrix refinement equation is related to the dimension of 1-eigenspace of the square matrix in the decomposable pair. We also show that the existence of a refinement mask with sum rules of order 1 is related to the dimension of (−1)-eigenspace of the square matrix in the decomposable pair. A systematic procedure is developed to create refinement masks satisfying the sum rules of order 1 using decomposable pairs, such that the corresponding matrix refinement equation has a solution. Several examples are provided to illustrate the theoretical results obtained.

Standard pairs and construction of multiwavelets using refinement masks satisfying sum rules of order one

A multiwavelet is typically constructed starting from a vector-valued function satisfying a matrix refinement equation. The approximation order of such a refinable function vector is related to the sum rules of order [Formula: see text] satisfied by the corresponding refinement mask. A refinable function vector can be obtained using cascade algorithm by constructing a refinement mask which satisfies the sum rules of order [Formula: see text] A standard pair associated with a refinement mask gives information about its spectral properties. In this paper, we present a procedure for constructing refinement masks satisfying the sum rules of order [Formula: see text], starting from standard pairs. How this helps in the construction of asymmetric multiwavelets using standard pairs is illustrated through examples. A sufficient condition on a standard pair and a necessary and sufficient condition on a left standard pair are established so that the corresponding refinement mask satisfies the sum rules of order [Formula: see text].

Zc,b-like states from QCD Laplace sum rules at NLO

We review our results on Zc-like states [1] which we complete with the ones on Zb-like states by using relativistic QCD Laplace Sum Rules (LSR) within stability criteria and including Factorized Next-to-Leading Order (FNLO) Perturbative (PT) corrections and Lowest Order (LO) QCD condensates up to 〈G3〉. We emphasize the importance of PT radiative corrections for heavy quark sum rules in order to justify the use of the running heavy quark mass in the analysis. Our estimates are compiled in Tables 3 and 4. From our results, the observed Zcs(3983) state are good candidate for being Tcs tetramole (superposition of nearly degenerated molecules and tetraquark states having the same quatum numbers JPC and with almost the same couplings to the currents). The Zcs bump around 4100 MeV can be interpreted as a combination of D0⁎Ds1 and Ds0⁎D1 molecules. The physical states Zcs(4000) and Zcs(4220) found by LHCb are too low to be considered as the first radial excitations of Zcs(3983). For the future Zb, Zbs and Zbss, we suggest to scan the region around (10.3 ∼ 10.9) GeV while the 1st radial excitations are about 2.4 GeV above the ground states.

Doubly hidden 0++ molecules and tetraquarks states from QCD at NLO

Motivated by the LHCb-group discovery of exotic hadrons in the range (6.2 ∼ 6.9) GeV, we present new results for the masses and couplings of 0++ fully heavy (Q‾Q)(QQ‾) molecules and (QQ)(QQ‾) tetraquaks states from relativistic QCD Laplace Sum Rule (LSR) within stability criteria where Next-to-Leading Order (NLO) Factorized (F) Perturbative (PT) corrections is included. As the Operator Product Expansion (OPE) usually converges for d ⩽ 6–8, we evaluated the QCD spectral functions at Lowest Order (LO) of PT QCD and up to 〈G3〉. We also emphasize the importance of PT radiative corrections for heavy quark sum rules in order to justify the use of the running heavy quark mass value in the analysis. We compare our predictions in Table 3 with the ones from ratio of Moments (MOM). The broad structure arround (6.2 ∼ 6.9) GeV can be described by the η‾cηc, J/ψ‾J/ψ and χc1‾χc1 molecules or/and Sc‾Sc, Ac‾Ac and Vc‾Vc tetraquarks lowest mass ground states. The narrow structure at (6.8 ∼ 6.9) GeV if it is a 0++ state can be a χc0‾χc0 molecules or/and its analogue Pc‾Pc tetraquark. The χc1‾χc1 predicted mass is found to be below the χc1χc1 threshold while for the beauty states, all of the estimated masses are above the ηbηb and ϒ(1S)ϒ(1S) threshold.

Zc -like spectra from QCD Laplace sum rules at NLO

We present a global analysis of the observed Z_c, Z_cs and future Z_css-like spectra using the inverse Laplace transform (LSR) version of QCD spectral sum rules (QSSR) within stability criteria. Integrated compact QCD expressions of the LO spectral functions up to dimension-six condensates are given. Next-to-Leading Order (NLO) factorized perturbative contributions are included. We re-emphasize the importance to include PT radiative corrections (though numerically small) for heavy quark sum rules in order to justify the (ad hoc) definition and value of the heavy quark mass used frequently at LO in the literature. We also demonstrate that, contrary to a na\"ive qualitative 1/N_c counting, the two-meson scattering contributions to the four-quark spectral functions are numerically negligible confirming the reliability of the LSR predictions. Our results are summarized in Tables III to VI. The Z_c(3900) and Z_cs(3983) spectra are well reproduced by the T_c(3900) and T_cs(3973) tetramoles (superposition of quasi-degenerated molecules and tetraquark states having the same quantum numbers and with almost equal couplings to the currents). The Z_c(4025) or Z_c(4040) state can be fitted with the D*_0D_1 molecule having a mass 4023(130) MeV while the Z_cs bump around 4.1 GeV can be likely due to the (D^*_s0D_1+ D^*_0D_s1) molecules. The Z_c(4430) can be a radial excitation of the Z_c(3900) weakly coupled to the current, while all strongly coupled ones are in the region (5634-6527) MeV. The double strange tetramole state T_css which one may identify with the future Z_css is predicted to be at 4064(46) MeV. It is remarkable to notice the regular mass-spliitings of the tetramoles due to SU(3) breakings M_{T_cs}-M_{T_c}= M_{T_css}-M_{T_cs= (73- 91) MeV.

Spectral sum rules for the Schrödinger equation

We study the sum rules of the form Z(s)=∑nEn−s, where En are the eigenvalues of the time-independent Schrödinger equation (in one or more dimensions) and s is a rational number for which the series converges. We have used perturbation theory to obtain an explicit formula for the sum rules up to second order in the perturbation and we have extended it non-perturbatively by means of a Padé-approximant. For the special case of a box decorated with one impurity in one dimension we have calculated the first few sum rules of integer order exactly; the sum rule of order one has also been calculated exactly for the problem of a box with two impurities. In two dimensions we have considered the case of an impurity distributed on a circle of arbitrary radius and we have calculated the exact sum rules of order two. Finally we show that exact sum rules can be obtained, in one dimension, by transforming the Schrödinger equation into the Helmholtz equation with a suitable density.

Doubly-hidden scalar heavy molecules and tetraquarks states from QCD at NLO

Alerted by the recent LHCb discovery of exotic hadrons in the range (6.2 -- 6.9) GeV, we present new results for the doubly-hidden scalar heavy $(\bar QQ) (Q\bar Q)$ charm and beauty molecules using the inverse Laplace transform sum rule (LSR) within stability criteria and including the Next-to-Leading Order (NLO) factorized perturbative and $\langle G^3\rangle$ gluon condensate corrections. We also critically revisit and improve existing Lowest Order (LO) QCD spectral sum rules (QSSR) estimates of the $({ \bar Q \bar Q})(QQ)$ tetraquarks analogous states. In the example of the anti-scalar-scalar molecule, we separate explicitly the contributions of the factorized and non-factorized contributions to LO of perturbative QCD and to the $\langle\alpha_sG^2\rangle$ gluon condensate contributions in order to disprove some criticisms on the (mis)uses of the sum rules for four-quark currents. We also re-emphasize the importance to include PT radiative corrections for heavy quark sum rules in order to justify the (ad hoc) definition and value of the heavy quark mass used frequently at LO in the literature. Our LSR results for tetraquark masses summarized in Table II are compared with the ones from ratio of moments (MOM) at NLO and results from LSR and ratios of MOM at LO (Table IV). The LHCb broad structure around (6.2 --6.7) GeV can be described by the $\overline{\eta}_{c}{\eta}_{c}$, $\overline{J/\psi}{J/\psi}$ and $\overline{\chi}_{c1}{\chi}_{c1}$ molecules or/and their analogue tetraquark scalar-scalar, axial-axial and vector-vector lowest mass ground states. The peak at (6.8--6.9) GeV can be likely due to a $\overline{\chi}_{c0}{\chi}_{c0}$ molecule or/and a pseudoscalar-pseudoscalar tetraquark state. Similar analysis is done for the scalar beauty states whose masses are found to be above the $\overline\eta_b\eta_b$ and $\overline\Upsilon(1S)\Upsilon(1S)$ thresholds.

Exact sum rules for heterogeneous spherical drums

We have obtained explicit integral expressions for the sums of inverse powers of the eigenvalues of the Laplacian on a unit sphere, in presence of an arbitrary variable density. The exact expressions for the sum rules are obtained by properly “renormalizing” the series, excluding the divergent contribution of the vanishing lowest eigenvalue. For a non–trivial example of a variable density we have applied our formulas to calculate the exact sum rules of order two and three, and we have verified these results calculating the sum rules numerically using the eigenvalues obtained with the Rayleigh–Ritz method.

A note on spectral properties of Hermite subdivision operators

In this paper we study the connection between the spectral condition of an Hermite subdivision operator and polynomial reproduction properties of the associated subdivision scheme. While it is known that in general the spectral condition does not imply the reproduction of polynomials, we here prove that a special spectral condition (defined by shifted monomials) is actually equivalent to the reproduction of polynomials. We further put into evidence that the sum rule of order ℓ>d associated with an Hermite subdivision operator of order d does not imply that the spectral condition of order ℓ is satisfied, while it is known that these two concepts are equivalent in the case ℓ=d.

From frame-like wavelets to wavelet frames keeping approximation properties and symmetry

For a given symmetric refinable mask obeying the sum rule of order n, an explicit method is suggested for the construction of mutually symmetric almost frame-like wavelet system providing approximation order n. A transformation based on the lifting scheme is described that allows to improve almost frame-like wavelets up to dual wavelet frames and preserve other properties. A direct method for the construction of dual wavelet frames providing approximation order n and mutual symmetry properties is also discussed. For an abelian symmetry group H, a technique providing the H-symmetry property for each wavelet function is given for the above three methods.

B0−B¯0 mixing: Matching to HQET at NNLO

We compute perturbative matching coefficients to the heavy quark effective theory (HQET) representation for the QCD effective local $\mathrm{\ensuremath{\Delta}}B=2$ Hamiltonian that determines the mass difference in the ${B}^{0}\text{\ensuremath{-}}{\overline{B}}^{0}$ system of states. We report on the results at next-to-next-to-leading order in the strong coupling constant for matching coefficients of two physical operators in HQET. Our results provide firm confirmation that the recent next-to-leading order sum rules analysis of the bag parameter ${B}_{q}$ is stable with regard of inclusion of higher order radiative corrections. As a byproduct of our calculation we give a fully analytical solution for the one-loop QCD-to-HQET matching problem: we present the explicit formulas for the renormalization of four-quark operators of the full bases in both QCD and HQET and the expressions for matching coefficients in a closed form.

Exact sum rules for quantum billiards of arbitrary shape

We have derived explicit expressions for the sum rules of order one of the eigenvalues of the negative Laplacian on two dimensional domains of arbitrary shape. Taking into account the leading asymptotic behavior of these eigenvalues, as given from Weyl’s law, we show that it is possible to define sum rules that are finite, using different prescriptions. We provide the explicit expressions and test them on a number of non trivial examples, comparing the exact results with precise numerical results.

Symmetric interpolatory dual wavelet frames

For any symmetry group $H$ and any appropriate matrix dilation we give an explicit method for the construction of $H$-symmetric refinable interpolatory refinable masks which satisfy sum rule of arbitrary order $n$. For each such mask we give an explicit technique for the construction of dual wavelet frames such that the corresponding wavelet masks are mutually symmetric and have the vanishing moments up to the order n. For an abelian symmetry group $H$ we modify the technique such that each constructed wavelet mask is $H$-symmetric.

Tests of quark–hadron duality in τ-decays

An exhaustive number of QCD finite energy sum rules for [Formula: see text]-decay together with the latest updated ALEPH data is used to test the assumption of global duality. Typical checks are the absence of the dimension d = 2 condensate, the equality of the gluon condensate extracted from vector or axial vector spectral functions, the Weinberg sum rules, the chiral condensates of dimensions d = 6 and d = 8, as well as the extraction of some low-energy parameters of chiral perturbation theory. Suitable pinched linear integration kernels are introduced in the sum rules in order to suppress potential quark–hadron duality violations and experimental errors. We find no compelling indications of duality violations in hadronic [Formula: see text]-decay in the kinematic region above s [Formula: see text] 2.2 GeV2 for these kernels.

Approximation order and approximate sum rules in subdivision

Several properties of stationary subdivision schemes are nowadays well understood. In particular, it is known that the polynomial generation and reproduction capability of a stationary subdivision scheme is strongly connected with sum rules, its convergence, smoothness and approximation order. The aim of this paper is to show that, in the non-stationary case, exponential polynomials and approximate sum rules play an analogous role of polynomials and sum rules in the stationary case. Indeed, in the non-stationary univariate case we are able to show the following important facts: (i) reproduction of N exponential polynomials implies approximate sum rules of order N; (ii) generation of N exponential polynomials implies approximate sum rules of order N, under the additional assumption of asymptotical similarity and reproduction of one exponential polynomial; (iii) reproduction of an N-dimensional space of exponential polynomials and asymptotical similarity imply approximation order N; (iv) the sequence of basic limit functions of a non-stationary scheme reproducing one exponential polynomial converges uniformly to the basic limit function of the asymptotically similar stationary scheme.

Predictions for the Dirac Phase in the Neutrino Mixing Matrix and Sum Rules

Using the fact that the neutrino mixing matrix U = U†eUν, where Ue and Uv result from the diagonalisation of the charged lepton and neutrino mass matrices, we analyse the sum rules which the Dirac phase δ present in U satisfies when Uv has a form dictated by, or associated with, discrete symmetries and Ue has a “minimal” form (in terms of angles and phases it contains) that can provide the requisite corrections to Uv, so that reactor, atmospheric and solar neutrino mixing angles θ13, θ23 and θ12 have values compatible with the current data. The following symmetry forms are considered: i) tri-bimaximal (TBM), ii) bimaximal (BM) (or corresponding to the conservation of the lepton charge L' = Le — Lμ — Lτ (LC)), iii) golden ratio type A (GRA), iv) golden ratio type B (GRB), and v) hexagonal (HG). We investigate the predictions for 5 in the cases of TBM, BM (LC), GRA, GRB and HG forms using the exact and the leading order sum rules for cos δ proposed in the literature, taking into account also the uncertainties in the measured values of sin2 θ12, sin2 θ23 and sin2 θ13. This allows us, in particular, to assess the accuracy of the predictions for cos δ based on the leading order sum rules and its dependence on the values of the indicated neutrino mixing parameters when the latter are varied in their respective 3σ experimentally allowed ranges.

Multirate systems with shortest spline-wavelet filters

Motivated by the need of short FIR filters for perfect-reconstruction multirate systems, the main objective of this paper is to derive the shortest filters for such filter banks with M channels, for any integer M⩾2, based on the M-dilated refinement sequence pm of the mth order cardinal B-spline. By imposing the additional constraint of ℓth order sum rule on the M-dual low-pass sequence am, the smoothness property of the M-dual scaling function, along with its corresponding analysis wavelets, is studied, and consequently yielding the ℓth order vanishing moment for each of the M−1 synthesis (spline) wavelets. Several illustrative examples and tables of the filter systems are also included in this paper.

Leptogenesis in an [formula omitted] golden ratio flavour model

In this paper we discuss a minor modification of a previous SU(5)×A5 flavour model which exhibits at leading order golden ratio mixing and sum rules for the heavy and the light neutrino masses. Although this model could predict all mixing angles well it fails in generating a sufficient large baryon asymmetry via the leptogenesis mechanism. We repair this deficit here, discuss model building aspects and give analytical estimates for the generated baryon asymmetry before we perform a numerical parameter scan. Our setup has only a few parameters in the lepton sector. This leads to specific constraints and correlations between the neutrino observables. For instance, we find that in the model considered only the neutrino mass spectrum with normal mass ordering and values of the lightest neutrino mass in the interval 10–18 meV are compatible with the current data on the neutrino oscillation parameters. With the introduction of only one NLO operator, the model can accommodate successfully simultaneously even at 1σ level the current data on neutrino masses, on neutrino mixing and the observed value of the baryon asymmetry.