Regge-pole based descriptions of pion-electroproduction on nucleons have given a very good description of the longitudinal components of the cross sections. However, these very same models grossly underestimate the transverse components. A related problem appears in QCD-based scaling arguments that predict the predominance of longitudinal over transverse electroproduction of pions by terms $\propto Q^2$. However, data from JLAB, Cornell and DESY, covering a wide kinematical range $1 < Q^2 < 11\, {\rm GeV}^2$ and $2\, {\rm GeV} < W < 4\, {\rm GeV}$, do not show this expected behavior. We address here this issue of the transverse response in pion-electroproduction by considering the contributions of high-lying ($W > 2$ GeV) nucleon resonances to pion production. The coupling strengths and form factors are obtained through resonance-parton duality. We show that in a wide range of electron energies and four-momentum transfers such a model describes all the available data very well.
We present a simple empirical parameterization of the x- and t-dependence of generalized parton distributions at zero skewness, using forward parton distributions as input. A fit to experimental data for the Dirac, Pauli and axial form factors of the nucleon allows us to discuss quantitatively the interplay between longitudinal and transverse partonic degrees of freedom in the nucleon ("nucleon tomography"). In particular we obtain the transverse distribution of valence quarks at given momentum fraction x. We calculate various moments of the distributions, including the form factors that appea…
A phenomenological model for the nucleon structure functions is presented. Visualising the nucleon as a cavity filled with parton gas in thermal equilibrium and parametrizing the effects due to the finiteness of the nucleon volume, we obtain a good fit to the data on the unpolarized nucleon structure functions.
The observation of the violation of Lam-Tung relation in the $πN$ Drell-Yan process triggered many theoretical speculations. The TMD Boer-Mulders functions characterizing the correlation of transverse momentum and transverse spin for partons in unpolarized hadrons could nicely account for the violation. The COMPASS experiment at CERN will measure the angular distributions of dimuons from the unpolarized Drell-Yan process over a wide kinematic region and study the beam particle dependence. Significant statistics is expected from a successful run in 2015 which will bring further understanding of…
We predict a strong enhancement of multijet production in proton-nucleus collisions at collider energies, as compared to a naive expectation of a cross section $\propto A$. The study of the process would allow to measure, for the first time, the double parton distribution functions in a nucleon in a model independent way and hence to study both the longitudinal and the transverse correlations of partons.
QCD-based scaling arguments predict the predominance of longitudinal over transverse electroproduction of pions by terms $\propto Q^2$. However, data from JLAB, Cornell and DESY, covering a wide kinematical range $1 < Q^2 < 11\, {\rm GeV}^2$ and $2\, {\rm GeV} < W < 4\, {\rm GeV}$, do not show this expected behavior. At the same time standard descriptions of pion-electroproduction on nucleons have given a very good description of the longitudinal components of the cross sections. However, these very same models have failed grossly in describing the transverse component. We discuss here a commo…
The unpolarized, helicity and transversity parton distribution functions of the nucleon are studied within a convolution model where the bare nucleon is dressed by its virtual meson cloud. Using light-front time-ordered perturbation theory, the Fock states of the physical nucleon are expanded in a series involving a bare nucleon and two-particle, meson-baryon, states. The bare baryons and mesons are described with light-front wave functions (LFWFs) for the corresponding valence-parton components. Using a representation in terms of overlap of LFWFs, the role of the non-perturbative antiquark de…
In a Glauber Monte Carlo framework, taking account of the transverse spatial distribution of hard partons in the nucleon, we analyse the nuclear modification factor $R_{dAu}$ for $J/ψ$ in d+Au collisions with the EPS09 shadowing parametrization. After the influence of nucleon hard partons distribution is considered, a clearly upward correction is revealed for the dependence of $R_{dAu}$ on $N_{coll}$ in peripheral d+Au collisions, however, an unconspicuous correction is shown for the results versus $p_{T}$. The theoretical results are in good agreement with the experimental data from PHENIX.
We study the nucleon's partonic angular momentum (AM) content at peripheral transverse distances $b = \mathcal{O}(M_π^{-1})$, where the structure is governed by chiral dynamics. We compute the nucleon form factors of the energy-momentum tensor in chiral effective field theory (ChEFT) and construct the transverse densities of AM at fixed light-front time. In the periphery the spin density is suppressed, and the AM is predominantly orbital. In the first-quantized representation of ChEFT in light-front form, the field-theoretical AM density coincides with the quantum-mechanical orbital AM density…
We construct a model for inelastic neutrino- and electron-nucleon scattering cross sections using effective leading order parton distribution functions with a new scaling variable $ξ_w$. Non-perturbative effects are well described using the $ξ_w$ scaling variable, in combination with multiplicative $K$ factors at low $Q^2$.Our model describes all inelastic charged lepton-nucleon scattering (including resonance) data (HERA/NMC/BCDMS/SLAC/JLab) ranging from very high $Q^2$ to very low $Q^2$ and down to the photo-production region. The model describes existing inelastic neutrino-nucleon scatterin…
It is shown that the latest results from NMC and E665 on the $F_2^A(x)/F_2^{D}(x)$ obtained in the shadowing region, bring new evidence of the universal $A$ -- dependence of distorsions of a free nucleon structure function by nuclear medium. The universality holds in the entire $x$ -range and can be explained as a saturation of the distortions of the parton distributions in a four-nucleon system.
Using heavy baryon chiral perturbation theory we study the leading chiral corrections to the complete set of nucleon generalized parton distributions (GPDs). We compute the leading quark mass and momentum transfer dependence of the moments of nucleon GPDs through the nucleon off-forward twist-2 matrix elements. These results are then applied to get insight on the GPDs and their impact parameter space distributions.
We report progress on the development of Perturbative Integral Representation Ansätze to compute the nucleon Faddeev Wave function, using an explicit quark-diquark picture. Our formalism is able to handle non-pointlike diquark and to mimic the strong, dynamical correlations observed when solving the Faddeev Equation. We then project the wave function in order to compute the leading-twist Parton Distribution Amplitude.
Spin-dependent quark densities, matrix elements of specific density operators in proton states of definite spin-polarization, indicate that the nucleon may harbor an infinite variety of non-spherical shapes. We show that these matrix elements are closely related to specific transverse momentum dependent parton distributions accessible in the angular dependence of the semi-inclusive processes electron plus proton goes to electron plus pion plus anything, and the Drell-Yan reaction proton plus proton goes to a lepton anti-lepton pair plus anything. New measurements or analyses would allow the di…
We describe the architecture and functionalities of a C++ software framework, coined PARTONS, dedicated to the phenomenology of Generalized Parton Distributions. These distributions describe the three-dimensional structure of hadrons in terms of quarks and gluons, and can be accessed in deeply exclusive lepto- or photo-production of mesons or photons. PARTONS provides a necessary bridge between models of Generalized Parton Distributions and experimental data collected in various exclusive production channels. We outline the specification of the PARTONS framework in terms of practical needs, ph…
Valence double parton distribution functions of the nucleon are evaluated in the framework of a simple model, where the conservation of the longitudinal momentum is taken into account. The leading-order DGLAP QCD evolution from the low quark-model scale to higher renormalization scales is carried out via the Mellin moments of the distributions. Results of the valence quark correlation function show that in general the double distributions cannot be approximated as a product of the single-particle distributions.
Generalized parton distributions have been introduced in recent years as a suitable theoretical tool to study the structure of the nucleon. Unifying the concepts of parton distributions and hadronic form factors, they provide a comprehensive framework for describing the quark and gluon structure of the nucleon. In this review their formal properties and modeling are discussed, summarizing the most recent developments in the phenomenological description of these functions. The status of available data is also presented.
A brief summary on the Wigner rotation effect in the understanding of the proton spin ``crisis" related with the Ellis-Jaffe sum rule violation, and on the proton-neutron isospin symmetry breaking explanation of the Gottfried sum rule violation. The proton spin could be fully provided by the spin sum of quarks if one considers the Wigner rotation effect in the measured quark helicity distributions.
We review the current status of spin-averaged and spin-dependent parton distribution functions (PDFs) of the nucleon. After presenting the formalism used to fit PDFs in modern global data analyses, we discuss constraints placed on the PDFs by specific data types. We give representative examples of unpolarized and polarized PDFs and their errors, and list open questions in global QCD fitting. Finally, we anticipate how future facilities, with fixed-target and collider experiments, may impact our knowledge of PDFs and reduce their uncertainties.
In these proceedings we discuss recent progress in nucleon structure using lattice QCD simulations at or near the physical value of the pion mass. Main focus will be given in observables such as the nucleon axial charge and the first moments of parton distributions, for both the valence and sea quark contributions, and discuss their implications on the spin content of the nucleon. We will will also report developments on the evaluation of the gluon momentum fraction, which contributes significantly to the nucleon spin.
We reply to the Comment of Brodsky and Gardner on our paper "New limits on intrinsic charm in the nucleon from global analysis of parton distributions" [Phys. Rev. Lett. 114, 082002 (2015)]. We address a number of incorrect claims made about our fitting methodology, and elaborate how global QCD analysis of all available high-energy data provides no evidence for a large intrinsic charm component of the nucleon.
We discuss the general formalism for the calculation in light-front quark models of the fully unintegrated, off-diagonal quark-quark correlator of the nucleon, parametrized in terms of generalized transverse momentum dependent parton distributions (GTMDs). By taking specific limits or projections, these GTMDs yield various transverse-momentum dependent and generalized parton distributions, thus providing a unified framework to simultaneously model different observables. The corresponding distributions in impact-parameter space are the Wigner functions which provide multidimensional images of t…
We review the status of our understanding of nucleon structure based on the modelling of different kinds of parton distributions. We use the concept of generalized transverse momentum dependent parton distributions and Wigner distributions, which combine the features of transverse-momentum dependent parton distributions and generalized parton distributions. We revisit various quark models which account for different aspects of these parton distributions. We then identify applications of these distributions to gain a simple interpretation of key properties of the quark and gluon dynamics in the…
We briefly discuss the current status of lattice QCD simulations and review selective results on nucleon observables focusing on recent developments in the lattice QCD evaluation of the nucleon form factors and radii, parton distribution functions and their moments, and the neutron electric dipole moment. Nucleon charges and moments of parton distribution functions are presented using simulations generated at physical values of the quark masses, while exploratory studies are performed for the parton distribution functions and the neutron electric dipole moment at heavier than physical value of…
Backward photoproduction of $ω$ off a proton target is investigated in a Reggeized model where the $u$-channel nucleon Reggeon is constructed from the nucleon Born terms in a gauge invariant way. The $t$-channel meson exchanges are considered as a background. While the $N_α$ trajectory of the nucleon Reggeon reproduces the overall shape of NINA data measured at the Daresbury Laboratory in the range of $u$-channel momentum transfer squared $-1.7<u<0.02$ GeV$^2$ and energies at $E_γ=2.8$, 3.5 and 4.7 GeV, a possibility of parton contributions is searched for through the nucleon isoscalar form fa…
This work presents the first calculation in lattice QCD of three moments of spin-averaged and spin-polarized generalized parton distributions in the proton. It is shown that the slope of the associated generalized form factors decreases significantly as the moment increases, indicating that the transverse size of the light-cone quark distribution decreases as the momentum fraction of the struck parton increases.
At large N_c the nucleon can be viewed as a soliton of the effective chiral lagrangian. This picture of nucleons allows a consistent nonperturbative calculation of the leading-twist parton distributions at a low normalization point. We derive general formulae for the polarized and unpolarized distributions (singlet and non-singlet) in the chiral quark-soliton model. The consistency of our approach is demonstrated by checking the baryon number, isospin and total momentum sum rules, as well as the Bjorken sum rule. We present numerical estimates of the quark and antiquark distributions and find …
We present a systematic study of generalized transverse-momentum dependent parton distributions (GTMDs). By taking specific limits or projections, these GTMDs yield various transverse-momentum dependent and generalized parton distributions, thus providing a unified framework to simultaneously model different observables. We present such simultaneous modeling by considering a light-cone wave function overlap representation of the GTMDs. We construct the different quark-quark correlation functions from the 3-quark Fock components within both the light-front constituent quark model as well as wit…
We point out that charge symmetry violation in both the valence and sea quark distributions of the nucleon has a non-perturbative source. We calculate this non-perturbative charge symmetry violation using the meson cloud model, which has earlier been successfully applied to both the study of SU(2) flavour asymmetry in the nucleon sea and quark-antiquark asymmetry in the nucleon. We find that the charge symmetry violation in the valence quark distribution is well below 1%, which is consistent with most low energy tests but significantly smaller than the quark model prediction about 5%-10%. Our …
We argue that generalized parton distributions (GPDs), accessible in hard exclusive processes, carry information about the spatial distribution of forces experienced by quarks and gluons inside hadrons. This way the measurements of hard exclusive processes open a possibility for direct "measurements" of strong forces in different parts of nucleons and nuclei. Also such studies open a venue for addressing questions of the properties of the quark (gluon) matter inside hadrons and nuclei. We give a simple example of relations between GPDs and properties of "nuclear matter" in finite nuclei.
We present results on the quark unpolarized, helicity and transversity parton distributions functions of the nucleon. We use the quasi-parton distribution approach within the lattice QCD framework and perform the computation using an ensemble of twisted mass fermions with the strange and charm quark masses tuned to approximately their physical values and light quark masses giving pion mass of 260 MeV. We use hierarchical probing to evaluate the disconnected quark loops. We discuss identification of ground state dominance, the Fourier transform procedure and convergence with the momentum boost.…
We calculate the photon production cross section arising from the hard scattering of partons in nucleon-nucleon collisions by taking into account the intrinsic parton transverse momentum distribution and the next-to-leading-order contributions. As first pointed out by Owens, the inclusion of the intrinsic transverse momentum distribution of partons leads to an enhancement of photon production cross section in the region of photon transverse momenta of a few GeV/c for nucleon-nucleon collisions at a center-of-mass energy of a few tens of GeV. The enhancement increases as $\sqrt{s}$ decreases. S…
The occurrence of the Bloom-Gilman local duality in the low-order moments of the nucleon structure function is investigated for values of the squared four-momentum transfer Q**2 between ~ 0.5 and 10 (GeV/c)**2. At variance with previous analyses truncated Cornwall-Norton moments, limited to the nucleon-resonance production regions, are considered. The role played by target-mass corrections is illustrated, showing that target-mass effects are necessary (but not sufficient) for producing the observed Bloom-Gilman duality of the nucleon structure function. The possibility of a local duality betwe…
The high rates of multi-parton interactions at the LHC can provide a unique opportunity to study the multi-parton structure of the hadron. To this purpose high energy collisions of protons with nuclei are particularly suitable. The rates of multi-parton interactions depend in fact both on the partonic multiplicities and on the distributions of partons in transverse space, which produce different effects on the cross section in pA collisions, as a function of the atomic mass number A. Differently with respect to the case of multi-parton interactions in pp collisions, the possibility of changing…
We present results on the axial and the electromagnetic form factors of the nucleon, as well as, on the first moments of the nucleon generalized parton distributions using maximally twisted mass fermions. We analyze two N_f=2+1+1 ensembles having pion masses of 210 MeV and 354 MeV at two values of the lattice spacing. The lattice scale is determined using the nucleon mass computed on a total of 18 N_f=2+1+1 ensembles generated at three values of the lattice spacing, $a$. The renormalization constants are evaluated non-perturbatively with a perturbative subtraction of ${\cal O}(a^2)$-terms. The…
A detailed study of nucleon parton distribution functions is performed within a radiative Next-to-Next-to Leading Order (NNLO) parton model whose low-scale input is rigidly described by wave functions which include quarks and dressing meson-cloud. The light-front Hamiltonian dynamics fixes the three-quark wave functions and the meson cloud is introduced by means of high-order Fock component in time-ordered perturbation theory in the infinite momentum frame. Non-strange as well as strange mesons-baryon fluctuations are considered ($π$, $ρ$, $ω$, $K$ and $K^*$, together with $N$, $Δ$, $Λ$ and $Σ…
The relevance of measuring Generalized Parton Distributions (GPDs) for few nucleon systems is illustrated. An approach which permits to calculate the GPDs of hadrons made of composite constituents by proper convolutions is described. The application of the method to the nucleon target, assumed to be made of composite constituents is reviewed. Calculations of GPDs for few nucleon systems are summarized, with special emphasis to the $^3$He target.
The generalized parton distributions (GPDs) offer a window on three-dimensional imaging of the nucleon, providing understanding of how the fundamental properties of the nucleon, such as its mass and spin, arise from the underlying quark and gluon degrees of freedom. In this work, we present the first lattice calculation of the nucleon isovector helicity GPD at physical pion mass, using an $a \approx 0.09$ fm lattice ensemble with 2+1+1 flavors of highly improved staggered quarks generated by MILC Collaboration. We perform the GPD calculation in Breit frame using averaged nucleon boost momentum…
Based on the chiral perturbation theory, we investigate the low-energy dynamics of nucleon parton distributions. We show that in different regions of the momentum fraction $x$ the chiral expansion is significantly different. For nucleon parton distributions these regions are characterized by $x\sim1$, $x\sim m_π/M_N$ and $x\sim (m_π/M_N)^2$. We derive extended counting rules for each region and obtain model-independent results for the nucleon parton distributions down to $x\gtrsim m^2_π/M^2_N \approx 10^{-2}$.
We estimate the energy density pile-up at mid-rapidity in central Pb+Pb collisions from 2 - 200 GeV/nucleon. The energy density is decomposed into hadronic and partonic contributions. A detailed analysis of the collision dynamics in the framework of a microscopic transport model shows the importance of partonic degrees of freedom and rescattering of leading (di)quarks in the early phase of the reaction for lab-energies > 30 GeV/nucleon. In Pb+Pb collisions at 160 GeV/nucleon the energy density reaches up to 4 GeV/fm^3, 95% of which are contained in partonic degrees of freedom.
We present a comprehensive study of the lowest moments of nucleon generalized parton distributions in N_f=2+1 lattice QCD using domain wall valence quarks and improved staggered sea quarks. Our investigation includes helicity dependent and independent generalized parton distributions for pion masses as low as 350 MeV and volumes as large as (3.5 fm)^3, for a lattice spacing of 0.124 fm. We use perturbative renormalization at one-loop level with an improvement based on the non-perturbative renormalization factor for the axial vector current, and only connected diagrams are included in the isosi…
We present the $x$-dependent nucleon and pion gluon distribution from lattice QCD using the pseudo-PDF approach, on lattice ensembles with $2+1+1$ flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration. We use clover fermions for the valence action and momentum smearing to achieve pion boost momentum up to 2.56~GeV on three lattice spacings $a\approx 0.9, 0.12$ and 0.15~fm and three pion masses $M_π\approx 220$, 310 and 690~MeV. We compare our pion and preliminary nucleon gluon results with the determination by global fits.
Recent calculations of charge symmetry violation(CSV) in the valence quark distributions of the nucleon have revealed that the dominant symmetry breaking contribution comes from the mass associated with the spectator quark system.Assuming that the change in the spectator mass can be treated perturbatively, we derive a model independent expression for the shift in the parton distributions of the nucleon. This result is used to derive a relation between the charge and flavor asymmetric contributions to the valence quark distributions in the proton, and to calculate CSV contributions to the nucle…
We present a review of recent applications of nonlocal chiral effective theory to hadron structure studies. Starting from a nonlocal meson--baryon effective chiral Lagrangian, we show how the introduction of a correlation function representing the finite extent of hadrons regularizes the meson loop integrals and introduces momentum dependence in vertex form factors in a gauge invariant manner. We apply the framework to the calculation of nucleon electromagnetic form factors, unpolarized and polarized parton distributions, as well as transverse momentum dependent distributions and generalized p…
We present the first lattice calculation of the nucleon isovector unpolarized generalized parton distribution (GPD) at the physical pion mass using a lattice ensemble with 2+1+1 flavors of highly improved staggered quarks (HISQ) generated by MILC Collaboration, with lattice spacing $a\approx 0.09$~fm and volume $64^3\times 96$. We use momentum-smeared sources to improve the signal at nucleon boost momentum $P_z \approx 2.2$ GeV, and report results at nonzero momentum transfers in $[0.2,1.0]\text{ GeV}^2$. Nonperturbative renormalization in RI/MOM scheme is used to obtain the quasi-distribution…
We present the first lattice-QCD calculation of the nucleon isovector unpolarized parton distribution functions (PDFs) at the physical-continuum limit using Large-Momentum Effective Theory (LaMET). The lattice results are calculated using ensembles with multiple sea pion masses with the lightest one around 135~MeV, 3 lattice spacings $a\in[0.06,0.12]$~fm, and multiple volumes with $M_πL$ ranging 3.3 to 5.5. We perform a simultaneous chiral-continuum extrapolation to obtain RI/MOM renormalized nucleon matrix elements with various Wilson-link displacements in the continuum limit at physical pion…
While the total orbital angular momentum (OAM) of a definite quark flavor in a longitudinally-polarized nucleon can be obtained through a sum rule involving twist-two generalized parton distribution (GPDs), its distribution as a function of parton momentum in light-front coordinates is more complicated to define and measure because it involves intrinsically twist-three effects. In this paper, we consider two different parton OAM distributions. The first is manifestly gauge invariant, and its moments are local operators and calculable in lattice QCD. We show that it can potentially be measured …
Parton convolution models have been used extensively in describing the sea quarks in the nucleon and explaining quark distributions in nuclei (the EMC effect). From effective field theory point of view, we construct the parton convolution formalism which has been the underlying conception of all convolution models. We explain the significance of scheme and scale dependence of the auxiliary quantities such as the pion distributions in a nucleon. As an application, we calculate the complete leading nonanalytic chiral contribution to the isovector component of the nucleon sea.
We present a field theoretical description of quarkyonic matter consisting of quark, nucleon and ghost fields coupling to mesonic degrees of freedom. The ghosts are present to cancel over-counting of nucleon states that are Pauli blocked by the quark Fermi sea. Such a theory becomes an effective field theory of nucleons at low baryon density, and as such will reproduce nucleonic matter phenomenology. This theory can accommodate chiral symmetry restoration and the dynamical generation of a shell of nucleons at the Fermi surface. It is valid for finite temperature and density. In such a theory, …
We present results on the lower moments of the nucleon generalized parton distri butions within lattice QCD using two dynamical flavors of degenerate twisted mass fermions. Our simulations are performed on lattices with three different values of the lattice spacings, namely $a=0.089$ fm, $a=0.070$ fm and $a=0.056$ fm, allowing the investigation of cut-off effects. The volume dependence is examined using simulations on two lattices of spatial length $L=2.1$ fm and $L=2.8$ fm. The simulations span pion masses in the range of 260-470 MeV. Our results are renormalized non-perturbatively and the va…
We discuss the first moments of the parity-even Generalized Parton Distributions (GPDs) in a nucleon, corresponding to six (generalized) vector form factors. We evaluate these fundamental properties of baryon structure at low energies, utilizing the methods of covariant Chiral Perturbation Theory in the baryon sector (BChPT). Our analysis is performed at leading-one-loop order in BChPT, predicting both the momentum and the quark-mass dependence for the three (generalized) isovector and (generalized) isoscalar form factors, which are currently under investigation in lattice QCD analyses of bary…
Parton distribution and correlation functions describe the relation between a hadron and the quarks and gluons (or collectively, the partons) within it, and carry rich information on hadron's partonic structure that cannot be calculated by QCD perturbation theory. In this talk, I will review what lattice QCD can and cannot do for calculating the parton distribution and correlation functions, and the new ideas and efforts around the world to explore nucleon structure from lattice QCD calculations by combining the strength of both lattice QCD and perturbative QCD in such a way that is complement…
We show that exclusive production of a pair of high transverse momentum photons in pion-nucleon collisions can be systematically studied in QCD factorization approach if the photon's transverse momentum $q_T$ with respect to the colliding pion is much greater than $Λ_{\rm QCD}$. We demonstrate that the leading power non-perturbative contributions to the scattering amplitudes of this exclusive process are process-independent and can be systematically factorized into universal pion's distribution amplitudes (DAs) and nucleon's generalized parton distributions (GPDs), which are convoluted with co…
A simple model of a composite nucleon is developed in which a fermion and a boson, representing quark and diquark constituents of the nucleon, form a bound state owing to a contact interaction. Photon and pion couplings to the quark provide vertex functions for the photon and pion interactions with the composite nucleon. By a suitable choice of cutoff parameters of the model, realistic electromagnetic form factors are obtained. When a pseudoscalar pion-quark coupling is used, the pion-nucleon coupling is predominantly pseudovector. A virtual photopion amplitude is considered in which there are…
We specify conditions under which the nucleon-nucleon interaction, based on the t-channel meson-exchange mechanism, is equivalent to an interaction generated via an s-channel exchange of six-quark bags. The duality is possible provided the alternation of zeros and poles of the non-dispersive part of D function takes place in the normalization where the imaginary part of D is non-negative and the CDD poles are the only poles of D.
