Moment Sum Rules Research Articles

Neutron scattering sum rules, symmetric exchanges, and helicoidal magnetism in MnSb2O6

MnSb$_{2}$O$_{6}$ is based on the noncentrosymmetric $P321$ space group with magnetic Mn$^{2+}$ ($S={5/2}$, $L\approx 0$) spins ordering below $T_{\mathrm{N}}=12$ K in a helicoidal structure. The ground state magnetic structure, expected to be built and originate from 7 Heisenberg exchange constants, has been shown to be coupled to the underlying crystallographic chirality with polar domain switching being reported. We apply neutron spectroscopy to extract these symmetric exchange constants. Given the high complexity of the magnetic exchange network, crystallographic structure and complications fitting linear spin-wave models, we take advantage of multiplexed neutron instrumentation to use the first moment sum rule of neutron scattering to estimate the 7 exchange constants. We then use these parameters to calculate the low-energy spin-waves in the N\'eel state to reproduce the neutron response without strong antisymmetric coupling. Using Green's response functions, the stability of long-wavelength excitations in the context of proposed magnetic structures is then discussed. The results show the presence of strong exchange constants for the chiral exchange pathways and illustrate an underlying coupling between crystallographic and magnetic ``chirality" through predominantely symmetric exchange.

Nonequilibrium spectral moment sum rules of the Holstein–Hubbard model

We derive a general procedure for evaluating the nth derivative of a time-dependent operator in the Heisenberg representation and employ this approach to calculate the zeroth to third spectral moment sum rules of the retarded electronic Green’s function and self-energy for a system described by the Holstein–Hubbard model allowing for arbitrary spatial and time variation of all parameters (including spatially homogeneous electric fields and parameter quenches). For a translationally invariant (but time-dependent) Hamiltonian, we also provide sum rules in momentum space. The sum rules can be applied to various different phenomena like time-resolved angle-resolved photoemission spectroscopy and benchmarking the accuracy of numerical many-body calculations. This work also corrects some errors found in earlier work on simpler models.

Searching for fully-heavy tetraquark states in QCD moment sum rules

In this talk, we briefly report the investigations of the mass spectra for the ccc¯c¯, bbb¯b¯, bcb¯c¯ and ccb¯b¯ tetraquark states by using the QCD moment sum rule method. The calculations for the fully-charm ccc¯c¯ tetraquarks have been successfully predicted the existence of di-J/ψ resonances including X(6900) in LHCb's observation. The quantum numbers for these resonance structures are also suggested. More efforts are still needed in both theoretical and experimental aspects to study the properties of these fully-heavy tetraquark states. They may be observed at facilities such as LHCb, CMS and RHIC in the future.

Mass spectra for the ccb¯b¯ and bbc¯c¯ tetraquark states

We have studied the masse spectra for the $cc\overline{b}\overline{b}$ and $bb\overline{c}\overline{c}$ tetraquark states with quantum numbers ${J}^{P}={0}^{\ifmmode\pm\else\textpm\fi{}},{1}^{\ifmmode\pm\else\textpm\fi{}}$, and ${2}^{+}$. We systematically construct the interpolating currents with various spin-parity quantum numbers and calculate their two-point correlation functions in the framework of QCD moment sum rule method. Our calculations show that the masses are about 12.3--12.4 GeV for the positive parity $cc\overline{b}\overline{b}$ tetraquark ground states with ${J}^{P}={0}^{+},{1}^{+},{2}^{+}$, while 12.8--13.1 GeV for the negative parity channels with ${J}^{P}={0}^{\ensuremath{-}},{1}^{\ensuremath{-}}$. The mass predictions for the positive parity $cc\overline{b}\overline{b}$ ground states are lower than the ${B}_{c}{B}_{c}$ threshold, implying that these tetraquarks can only decay via weak interaction and thus are expected to be stable and narrow.

Deep inelastic scattering on the quark-gluon plasma

We provide an interpretation of the structure functions of a thermal medium such as the quark-gluon plasma in terms of the scattering of an incoming electron on the medium via the exchange of a spacelike photon. We then focus on the deep-inelastic scattering (DIS) regime, and formulate the corresponding moment sum rules obeyed by the structure functions. Accordingly, these moments are given by the thermal expectation value of twist-two operators, which is computable from first principles in lattice QCD for the first few moments. We also show how lattice QCD calculations can be used to probe how large the photon virtuality needs to be in order for the Bjorken scaling of structure functions to set in. Finally, we provide the parton-model interpretation of the structure functions in the Bjorken limit and test its consistency. As in DIS on the proton, the kinematic variable x is proportional to the longitudinal momentum carried by the partons, however x ranges from zero to infinity. Choosing the parton momentum parametrization to be xT u where u is the fluid four-velocity and T its temperature in the rest frame, the parton distribution function for a plasma of non-interacting quarks is proportional to x log(1 + e−x/2).

Total Moment Sum Rule for Magnets in the Vicinity of Quantum Critical Point

It is known that the longitudinal and transverse excitation modes can exist in the vicinity of a quantum critical point in the ordered phase of quantum magnetic systems. The total moment sum rule for such systems is derived on the basis of the extended spin-wave theory, where both longitudinal and transverse magnetic excitations are taken into account. The sum rule is resolved into elastic, one-magnon, and two-magnon components. The formulation is applicable to spin systems with the longitudinal mode, such as $S=1$ systems with single-ion anisotropy of easy-plane type and spin dimer systems. The result helps us analyze and understand measured data of inelastic neutron scattering.

Form invariance of the moment sum-rules for jellium with the addition of short-range terms in the pair-potential

We find the first three (even) structure factor moments for a (non-quantum) one-component Jellium made of particles living in three dimensions and interacting with a Coulomb pair-potential plus a short-range term with either a finite range or decaying exponentially fast at large distances. Starting from the hierarchical Born-Green-Yvon equations we show that they are all form invariant respect to the addition of the short-range term. We discuss the relevance of the present study to interpret the failure of the moment sum-rules of ionic-liquids at criticality.

QCD parameters, [formula omitted] and [formula omitted] from relativistic heavy quark sum rules

We report results of our recent works [S. Narison, Int. J. Mod. Phys. A33 (2018) no. 10, 1850045; S. Narison. Addendum: Int. J. Mod. Phys. A33 (2018) no. 10, 1850045; S. Narison, Phys. Lett. B802 (2020) 135221.] where the correlations between the c,b-quark running masses m‾c,b, the gluon condensate 〈αsG2〉 and the QCD coupling αs in the MS‾-scheme from an analysis of the charmonium and bottomium spectra and the Bc-meson mass. We use optimized ratios of relativistic Laplace sum rules (LSR) evaluated at the μ-subtraction stability point where higher orders PT and D≤6−8-dimensions nonperturbative condensates corrections are included. We obtain [S. Narison, Int. J. Mod. Phys. A33 (2018) no. 10, 1850045; S. Narison. Addendum: Int. J. Mod. Phys. A33 (2018) no. 10, 1850045.] αs(2.85)=0.262(9) and αs(9.50)=0.180(8) from the (pseudo)scalar Mχ0c(0b)−Mηc(b) mass-splittings at μ=2.85(9.50) GeV. The most precise result from the charm channel leads to αs(Mτ)=0.318(15) and αs(MZ)=0.1183(19)(3) in excellent agreement with the world average: αs(MZ)=0.1181(11) [For a review, see e.g: S. Bethke, Nucl. Part. Phys. Proc. 282–284 (2017)149; C. Patrignani et al. (Particle Data Group), Chin. Phys. C40, 100001 (2016) and 2017 update.]. Updated results from a global fit of the (axial-)vector and (pseudo)scalar channels using Laplace and Moments sum rules @ N2LO [S. Narison, Int. J. Mod. Phys. A33 (2018) no. 10, 1850045; S. Narison. Addendum: Int. J. Mod. Phys. A33 (2018) no. 10, 1850045.] combined with the one from MBc [S. Narison, Phys. Lett. B802 (2020) 135221.] lead to the new tentative QCD spetral sum rules (QSSR) average:m‾c(m‾c)|average=1266(6) MeV and m‾b(m‾b)|average=4196(8) MeV. The values of the gluon condensate 〈αsG2〉 from the (axial)-vector charmonium channels combined with previous determinations in Table 1, leads to the new QSSR average: [S. Narison, Int. J. Mod. Phys. A33 (2018) no. 10, 1850045; S. Narison. Addendum: Int. J. Mod. Phys. A33 (2018) no. 10, 1850045.] 〈αsG2〉|average=(6.35±0.35)×10−2 Ge4. Our results clarify the (apparent) discrepancies between different estimates of 〈αsG2〉 from J/ψ sum rule but also shows the sensitivity of the sum rules on the choice of the μ-subtraction scale. As a biproduct, we deduce the Bc-decay constants fBc=371(17) MeV and fBc(2S)≤139(6) MeV.

Charmonia and bottomonia in asymmetric magnetized hot nuclear matter

We investigate the mass-shift of P-wave charmonium ( , ), and S and P-wave bottomonium ( , , , and ) states in magnetized hot asymmetric nuclear matter using the unification of QCD sum rules (QCDSR) and the chiral model. Within QCDSR, we use two approaches, i.e., the moment sum rule and the Borel sum rule. The magnetic field induced scalar gluon condensate and the twist-2 gluon operator calculated in the chiral ) model are utilised in QCD sum rules to calculate the in-medium mass-shift of the above mesons. The attractive mass-shift of these mesons is observed, which is more sensitive to magnetic field in the high density regime for charmonium, however less so for bottomonium. These results may be helpful to understand the decay of higher quarkonium states to the lower quarkonium states in asymmetric heavy ion collision experiments.

Formula omitted], 〈αsG2〉 and αs from Heavy Quarkonia

Explicit studies [S. Narison, Int. J. Mod. Phys. A33 (2018) no. 10, 1850045; S. Narison, addendum (to appear in Int. J. Mod. Phys. A).] of the correlations between the c, b-quark running masses m‾c,b, the gluon condensate 〈αsG2〉 and the QCD coupling αs in the MS‾-scheme from a global fit of charmonium and bottomium spectra are reviewed. We use optimized ratios of relativistic Laplace sum rules (LSR) evaluated at the μ-subtraction stability point where higher orders PT and D≤6−8-dimensions non-perturbative condensates corrections are included. Our results clarify the (apparent) discrepancies between different estimates of 〈αsG2〉 from J/ψ sum rule but also shows the sensitivity of the sum rules on the choice of the μ-subtraction scale which does not permit a high-precision estimate of m‾c,b from the alone vector channel. Updated results from a global fit of the (axial-)vector and (pseudo)scalar channels using Laplace and Moments sum rules @ N2LO lead to: m‾c(m‾c)|average=1264(6)MeV and m‾b(m‾b)|average=4190(8)MeV. The values of the gluon condensate 〈αsG2〉 from the (axial)-vector charmonium channels combined with previous determinations in Table 1, leads to the new sum rule average: 〈αsG2〉|average=(6.35±0.35)×10−2GeV4. The (pseudo)scalar mass-splittings Mχ0c(0b)−Mηc(b) allow to predict the QCD coupling αs at two different scales: αs(2.85)=0.262(9) for the charm and αs(9.50)=0.180(8) for the bottom. The strongest one from Mχ0c−Mηc leads to: αs(Mτ)=0.318(15)⇝αs(MZ)=0.1183(19)(3) in good agreement with the world average: αs(MZ)=0.1181(11) [For a review, see e.g: S. Bethke, Nucl. Part. Phys. Proc. 282–284 (2017) 149; PDG, C. Patrignani et al. (Particle Data Group), Chin. Phys. C40, 100001 (2016) and 2017 update.].

Influence of spin and charge fluctuations on spectra of the two-dimensional Hubbard model

The influence of spin and charge fluctuations on spectra of the two-dimensional fermionic Hubbard model is considered using the strong coupling diagram technique. Infinite sequences of diagrams containing ladder inserts, which describe the interaction of electrons with these fluctuations, are summed, and obtained equations are self-consistently solved for the ranges of Hubbard repulsions , temperatures and electron concentrations with t the intersite hopping constant. For all considered U the system exhibits a transition to the long-range antiferromagnetic order at . At the same time no indication of charge ordering is observed. Obtained solutions agree satisfactorily with results of other approaches and obey moments sum rules. In the considered region of the U-T plane, the curve separating metallic solutions passes from at the highest temperatures to U = 2t at for half-filling. If only short-range fluctuations are allowed for the remaining part of this region is occupied by insulating solutions. Taking into account long-range fluctuations leads to strengthening of maxima tails, which transform a part of insulating solutions into bad-metal states. For low T, obtained results allow us to trace the gradual transition from the regime of strong correlations with the pronounced four-band structure and well-defined Mott gap for to the Slater regime of weak correlations with the spectral intensity having a dip along the boundary of the magnetic Brillouin zone due to an antiferromagnetic ordering for . For and doping leads to the occurrence of a pseudogap near the Fermi level, which is a consequence of the splitting out of a narrow band from a Hubbard subband. Obtained spectra feature waterfalls and Fermi arcs, which are similar to those observed in hole-doped cuprates.

Doubly hidden-charm/bottom QQQQ tetraquark states

We study the mass spectra for the cccc and bbbb tetraquark states by developing a moment sum rule method. Our results show that the bbbb tetraquarks lie below the threshold of ηb(1S) ηb(1S ). They are probably stable and very narrow. The masses for the doubly hidden-charm states cccc are higher than the spontaneous dissociation thresholds of two charmonium mesons. We suggest to search for such states in the J/Ψ/J/Ψ and Ψc(1S) Ψc(1S) channels.

The moment sum-rules for ionic liquids at criticality

We discuss the first three well known moment charge–charge sum-rules for a general ionic liquid. For the special symmetric case of the Restricted Primitive Model, Das, Kim, and Fisher (2011) has recently discovered, through Monte Carlo simulations, that the Stillinger–Lovett or second-moment sum-rule fails at criticality. We critically discuss a possible explanation for this unexpected behavior. On the other hand the fourth-moment sum-rule turns out to be able to account for the results of the simulations at criticality.

Influence of spin-orbit coupling on the magnetic dipole termTα

The influence of the spin-orbit coupling (SOC) on the magnetic dipole term $T_{\alpha}$ is studied across a range of systems in order to check whether the $T_{\alpha}$ term can be eliminated from analysis of x-ray magnetic circular dichroism spectra done via the spin moment sum rule. Fully relativistic Korringa-Kohn-Rostoker (KKR) Green function calculations for Co monolayers and adatoms on Cu, Pd, Ag, Pt, and Au (111) surfaces were performed to verify whether the sum over magnetic dipole terms $T_{x}+T_{y}+T_{z}$ is zero and whether the angular dependence of the \ta\ term goes as $3\cos^{2}\theta-1$. It follows that there are circumstances when the influence of the SOC on $T_{\alpha}$ cannot be neglected even for 3$d$ atoms, where the SOC is nominally small. The crucial factor appears to be the dimensionality of the system: for 3$d$ adatoms, the influence of SOC on $T_{\alpha}$ can be significant while for monolayers it is always practically negligible. Apart from the dimensionality, hybridization between adatom and substrate states is also important: small hybridization enhances the importance of the SOC and vice versa.

Local moment approach as a quantum impurity solver for the Hubbard model

The local moment approach (LMA) has presented itself as a powerful semi-analytical quantum impurity solver (QIS) in the context of the dynamical mean-field theory (DMFT) for the periodic Anderson model and it correctly captures the low energy Kondo scale for the single impurity model, having excellent agreement with the Bethe ansatz and numerical renormalization group results. However, the most common correlated lattice model, the Hubbard model, has not been explored well within the LMA+DMFT framework beyond the insulating phase. Here in our work, within the framework we attempt to complete the phase diagram of the single band Hubbard model at zero temperature. Our formalism is generic to any particle filling and can be extended to finite temperature. We contrast our results with another QIS, namely the iterated perturbation theory (IPT) and show that the second spectral moment sum-rule improves better as the Hubbard interaction strength grows stronger in LMA, whereas it severely breaks down after the Mott transition in IPT. We also show that, in the metallic phase, the low-energy scaling of the spectral density leads to universality which extends to infinite frequency range at infinite correlation strength (strong-coupling). At large interaction strength, the off half-filling spectral density forms a pseudogap near the Fermi level and filling-controlled Mott transition occurs as one approaches the half-filling. Finally we study optical properties and find universal features such as absorption peak position governed by the low-energy scale and a doping independent crossing point, often dubbed as the "isosbestic point" in experiments.

Sum rules of the Bjorken-Uraltsev type in the Bakamjian-Thomas relativistic quark model

The Bakamjian-Thomas relativistic quark model, describing hadrons with a fixed number of constituents, yields in the heavy quark limit of QCD covariant Isgur-Wise functions and satisfies the whole tower of lowest moment sum rules (Bjorken-Uraltsev type sum rules). We first recall, as well as earlier results, the new formalism presented in our recent papers on Lorentz representations, which provide an elegant framework for the analysis of this model in the heavy quark limit and stress the results which have been already obtained in this direction. Then, we give some very explicit demonstrations of the fact that the Bakamjian-Thomas framework satifies the sum rules by considering simple cases of Isgur-Wise functions. In addition to the specific Bjorken and Uraltsev sum rules, an important sum rule that involves only heavy mesons with light cloud $j^P = {1 \over 2}^-$ and their radial excitations is demonstrated. This latter sum rule is phenomenologically interesting because it constrains the derivatives of the radially excited Isgur-Wise functions at zero recoil. On the other hand, we recall the limitations of the Bakamjian-Thomas scheme. At finite mass, current matrix elements with the current coupled to the heavy quark are no longer covariant, and higher moment sum rules that hold in the heavy quark limit of QCD are not satisfied.

Aspects of QCD current algebra on a null plane

Consequences of QCD current algebra formulated on a light-like hyperplane are derived for the forward scattering of vector and axial–vector currents on an arbitrary hadronic target. It is shown that current algebra gives rise to a special class of sum rules that are direct consequences of the independent chiral symmetry that exists at every point on the two-dimensional transverse plane orthogonal to the lightlike direction. These sum rules are obtained by exploiting the closed, infinite-dimensional algebra satisfied by the transverse moments of null-plane axial–vector and vector charge distributions. In the special case of a nucleon target, this procedure leads to the Adler–Weisberger, Gerasimov–Drell–Hearn, Cabibbo–Radicati and Fubini–Furlan–Rossetti sum rules. Matching to the dispersion-theoretic language which is usually invoked in deriving these sum rules, the moment sum rules are shown to be equivalent to algebraic constraints on forward S-matrix elements in the Regge limit.

On the nontrivial wave-vector dependence of the elastic modulus of glasses

Recent theoretical models for the vibrations in glasses assume that the complex elastic modulus depends on frequency but not on the wave vector, $q$. This assumption translates in a simple $q$ dependence of the dynamic structure factor, which can be experimentally tested. Following the suggestion of a recent paper [U. Buchenau, Phys. Rev. E 90, 062319 (2014)], we present here a new analysis, performed in $q$ space, of inelastic x-ray scattering data of supercooled silica. The outcome of the analysis is compared to the more common approach in the frequency domain and indicates that the mentioned theoretical assumption is consistent with the data only below the boson peak frequency. At higher frequencies it gives rise to a breakdown of the classical second moment sum rule. This violation arises from the underlying assumption of the presence of a single excitation in the spectra. A comparison with the vibrational dynamics of $\ensuremath{\alpha}$-cristobalite suggests, on the contrary, that in the terahertz frequency domain the inelastic spectrum of the glass gains contributions from both acousticlike and opticlike modes. A microscopic theory of the vibrations in glasses cannot neglect the medium range order in their structure, which gives rise to dispersion curves within a pseudo-Brillouin zone.

Charmonia decay constants from the QCD lattice and QCD sum rules

Using lattice QCD and QCD sum rules we compute the lowest state charmonia JPC=0−+(ηc), 1−(J/ψ), and 1+−(hc) decay constants. For calculating the decay constant of J/ψ we use both the vector and tensor currents and compare the results. Lattice QCD results are obtained from the unquenched (Nf=2) simulations using twisted mass QCD at four lattice spacings and taking the continuum limit. In the QCD sum rule calculation we apply the moment sum rules. We also comment the phenomenological implications of calculated charmonia decay constants in ηc→γγ decay, and B→Xcc‾K decays.

Fourth Moment Sum Rule for the Charge Correlations of a Two-Component Classical Plasma

We consider an ionic fluid made with two species of mobile particles carrying either a positive or a negative charge. We derive a sum rule for the fourth moment of equilibrium charge correlations. Our method relies on the study of the system response to the potential created by a weak external charge distribution with slow spatial variations. The induced particle densities, and the resulting induced charge density, are then computed within density functional theory, where the free energy is expanded in powers of the density gradients. The comparison with the predictions of linear response theory provides a thermodynamical expression for the fourth moment of charge correlations, which involves the isothermal compressibility as well as suitably defined partial compressibilities. The familiar Stillinger-Lovett condition is also recovered as a by-product of our method, suggesting that the fourth moment sum rule should hold in any conducting phase. This is explicitly checked in the low density regime, within the Abe-Meeron diagrammatical expansions. Beyond its own interest, the fourth-moment sum rule should be useful for both analyzing and understanding recently observed behaviours near the ionic critical point.