Spectral Sum Rules Research Articles

Some remarks on compositeness of * *Supported in part by the National Nature Science Foundations of China (12335002, 12375078, 11975028). H.Q. Zheng and Z. Xiao are also Supported by "the Fundamental Research Funds for the Central Universities"

Recently, the LHCb experimental group found an exotic state from the process. A key question is whether it is just a molecule or may have a confined tetraquark ingredient. To investigate this, different methods were used, including a two-channel ( and ) K-matrix unitarization and a single-channel Flatté-like parametrization method analyzed utilizing the pole counting rule and spectral density function sum rule. These analyses demonstrated that is a molecular state, although the possibility that there may exist an elementary ingredient cannot be excluded, according to an approximate analysis of its production rate.

Formula omitted] from QCD Laplace sum rule

We present a review of our recent works on heavy-light exotic states, with new compact integrated expressions of the spectral functions at lowest order (LO) of perturbative (PT) QCD and up to d=6 condensates of the Operator Product Expansion (OPE). Including higher order (HO) PT corrections, which we have estimated by assuming the factorization of the molecule/four-quark spectral functions, we evaluate masses and couplings with the Laplace sum rules (LSR) and finite energy (FESR) QCD spectral sum rules (QSSR) approaches. Our optimal results based on stability criteria are summarized and compared with experimental candidates and some LO QSSR results. We conclude that, our predicted spectra, do not confirm that the X(5568) is a pure molecule or a four-quark state. The masses of the XZ observed states are compatible with (almost) pure JPC=1+±,0++ molecule or/and four-quark states. The SU3 corrections on the meson masses are tiny: ≤10% (respectively ≤3%) for the c (respectively b)-quark channel but can be large for the couplings (≤20%). We expect that the resonance near the D−K+ threshold can be originated from the 0+(D−K+) molecule and/or D−K+ scattering. The X0(2900), XJ(3150) (if J = 0), X1(2900) and the XJ(3350) (if J = 1) can emerge from a minimal mixing model between tetramoles and first radial excitation. The Zc(3900,4025,4040,4430) and Zcs(3983) spectra are well reproduced by the molecules and tetramoles candidates, or radial excitation.

On the nature of X(3960)

A near-threshold enhancement in the D^{+}_{s}D^{-}_{s} system, dubbed as X(3960), is observed by the LHCb collaboration recently. A combined analysis on chi _{c0}(3930)~(rightarrow D^+ D^-), X(3960)~(rightarrow D^{+}_{s}D^{-}_{s}), and X(3915)~(rightarrow J/psi omega ) is performed using both a K-matrix approach of D_{(s)}bar{D}_{(s)} four-point contact interactions and a model of Flatté-like parameterizations. The use of the pole counting rule and spectral density function sum rule indicate, under current statistics, that this D^{+}_{s}D^{-}_{s} near-threshold state has probably the mixed nature of a cbar{c} confining state and D^{+}_{s}D^{-}_{s} continuum.

X(6900) peak could be a molecular state

The analyses of the LHCb data on $X(6900)$ found in di-$J/\psi$ and $J/\psi \psi(3686)$ systems are performed using a momentum-dependent {Flatt\'{e}}-like parameterization. The use of the pole counting rule and spectral density function sum rule give consistent conclusions that $X(6900)$ may not be a molecule of $J/\psi \psi(3686)$. Nevertheless it is still possible that $X(6900)$ be a molecule of higher states, such as $J/\psi\psi(3770)$, $\chi_{c0}\chi_{c2}$, etc.

Cumulant expansion in the Holstein model: Spectral functions and mobility

We examine the range of validity of the second-order cumulant expansion (CE) for the calculation of spectral functions, quasiparticle properties, and mobility of the Holstein polaron. We devise an efficient numerical implementation that allows us to make comparisons in a broad interval of temperature, electron-phonon coupling, and phonon frequency. For a benchmark, we use the dynamical mean-field theory which gives, as we have recently shown, rather accurate spectral functions in the whole parameter space, even in low dimensions. We find that in one dimension, the CE resolves well both the quasiparticle and the first satellite peak in a regime of intermediate coupling. At high temperatures, the charge mobility assumes a power law $\ensuremath{\mu}\ensuremath{\propto}{T}^{\ensuremath{-}2}$ in the limit of weak coupling and $\ensuremath{\mu}\ensuremath{\propto}{T}^{\ensuremath{-}3/2}$ for stronger coupling. We find that, for stronger coupling, the CE gives slightly better results than the self-consistent Migdal approximation (SCMA), while the one-shot Migdal approximation is appropriate only for a very weak electron-phonon interaction. We also analyze the atomic limit and the spectral sum rules. We derive an analytical expression for the moments in CE and find that they are exact up to the fourth order, as opposed to the SCMA where they are exact to the third order. Finally, we analyze the results in higher dimensions.

QCD spectral sum rules 2022

We present a compact review of the status of QCD spectral sum rules until 2022. We emphasize the recent progresses for determining the QCD input parameters (αs, running quark masses, quark and gluon condensates) where their correlations have been taken into account. Some selected phenomenological uses of the sum rules in different channels (light and heavy quarks, gluonia/glueballs, hybrids and four-quark states) are briefly reviewed and commented. The estimate of the 1−+ light hybrid mass is revised which confirms the hybrid nature of the π1(1600) but not the π1(2050).

0+ XTZ states from QCD spectral sum rules

We review our results in [1] for the masses and couplings of Tccqq′ (JP=0+) states from (inverse) QCD Laplace sum rule (LSR), their ratios R and double ratio of sum rules (DRSR) within stability criteria and including Factorized Next-to-Leading Order (FNLO) Perturbative (PT) corrections and Lowest Order (LO) QCD condensates up to 〈G3〉. We show that combining R and DRSR can provide more precise results. Calibrated to the observed Xc(3872) and Tcc1+(3875), R combined with DRSR lead to a more precise prediction of MTcc0+=3883(3)MeV. In a similar way, calibrated to the new prediction of Tcc0+R⊕DRSR lead to the improved mass predictions: MTccs¯u¯0+=3927(6)MeV and MTccs¯s¯0+=3993(11)MeV. We extend our analysis to the bottom sector and compare our results with the ones from different LSR predictions and some other determinations (lattice, quark and potential models, …) in the literature.

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.

DK and BK-like spectra from Laplace sum rule at NLO

Encouraged by the agreement, with the recent LHCb data on the D−K+ invariant mass from B→D+D−K+ decay, of our results for the masses of the 0+ and 1− open charm (c¯d¯)(us) tetraquarks and molecules states from QCD spectral sum rules within stability criteria, which we review here, we extend our analysis to the b-quark channel. We find, in the 0+ case the lowest mass MBK=5195(15)MeV with fBK=8.3(2.4)keV and three (almost) degenerate states having respectively the masses MSS=5702(60)MeV, MAA=5661(75)MeV and MB⁎K⁎=5720(71)MeV and couplings fSS=22.2(2.3)keV, fAA=30.1(3.1)keV and fB⁎K⁎=26.5(2.8)keV, from which we can associate a scalar tetramole with MTM0=5694(69)MeV and fTM0=26.5(2.7)keV. In the spin 1 case, we find four (almost) degenerate states associated with a tetramole having MTM1=5700(81)MeV and fTM1=16.2(2.6)keV. For the first radial excitation of the BK molecule, we have M(BK)1=6265(146)MeV and f(BK)1=22.8(3.2)keV. For the remaining states, we associate a scalar and vector tetramoles having respectively M(TM0)1=7439(314)MeV, f(TM0)1=74.7(8.4)keV and M(TM1)1=7544(345)MeV, f(TM1)1=33.0(6.7)keV.

Multiple Magnetorotons and Spectral Sum Rules in Fractional Quantum Hall Systems.

We study, numerically, the charge neutral excitations (magnetorotons) in fractional quantum Hall systems, concentrating on the two Jain states near quarter filling, ν=2/7 and ν=2/9, and the ν=1/4 Fermi-liquid state itself. In contrast to the ν=1/3 states and the Jain states near half filling, on each of the two Jain states ν=2/7 and ν=2/9 the graviton spectral densities show two, instead of one, magnetoroton peaks. The magnetorotons have spin 2 and have opposite chiralities in the ν=2/7 state and the same chirality in the ν=2/9 state. We also provide a numerical verification of a sum rule relating the guiding center spin s[over ¯] with the spectral densities of the stress tensor.

Measurement of topological order based on metric-curvature correspondence

A unified expression for topological invariants was proposed recently to describe the topological order in Dirac models belonging to any dimension and symmetry class. We uncover a correspondence between the curvature function that integrates to this unified topological invariant and the quantum metric that measures the distance between properly defined many-body Bloch states in momentum space. Based on this metric-curvature correspondence, a time-resolved and angle-resolved photoemission spectroscopy experiment is proposed to measure the violation of the spectral sum rule caused by a pulse electric field to detect the quantum metric, from which the topological properties of the system may be extracted.

Some remarks on X(6900) * *Supported in part by National Nature Science Foundations of China (11975028, 10925522) and China Postdoctoral Science Foundation (2020M680500)

The analysis of the LHCb data on found in the di- system was performed using a momentum-dependent Flatté-like parameterization. The use of the pole counting rule and spectral density function sum rule provides consistent evidence that both confining and molecular states are possible. Alternatively, the nature of cannot be distinguished if only the di- experimental data with current statistics are available. Nevertheless, we found that the lowest state in the di- system likely has the same quantum numbers as , and is probably not interpreted as a molecular state.

New universality classes of the non-Hermitian Dirac operator in QCD-like theories

In non-Hermitian random matrix theory there are three universality classes for local spectral correlations: the Ginibre class and the nonstandard classes ${\mathrm{AI}}^{\ifmmode\dagger\else\textdagger\fi{}}$ and ${\mathrm{AII}}^{\ifmmode\dagger\else\textdagger\fi{}}$. We show that the continuum Dirac operator in two-color QCD coupled to a chiral U(1) gauge field or an imaginary chiral chemical potential falls in class ${\mathrm{AI}}^{\ifmmode\dagger\else\textdagger\fi{}}$ (${\mathrm{AII}}^{\ifmmode\dagger\else\textdagger\fi{}}$) for fermions in pseudoreal (real) representations of SU(2). We introduce the corresponding chiral random matrix theories and verify our predictions in lattice simulations with staggered fermions, for which the correspondence between representation and universality class is reversed. Specifically, we compute the complex eigenvalue spacing ratios introduced recently. We also derive novel spectral sum rules.

The New Charm-Strange Resonances in the D−K+ Channel

We evaluate the masses and decay constants of the 0+ and 1− open-charm (c¯d¯)(us) tetraquarks and molecular states from QCD spectral sum rules (QSSR) by using QCD Laplace sum rule (LSR). This method takes into account the stability criteria where the factorised perturbative NLO corrections and the contributions of quark and gluon condensates up to dimension-6 in the OPE are included. We confront our results with the D−K+ invariant mass recently reported by LHCb from B+ → D+(D−K+) decays. We expect that the resonance near the D−K+ threshold can be originated from the 0++(D−K+) molecule and/or D−K+ scattering. The X0(2900) scalar state and the resonance XJ(3150) (if J = 0) can emerge from a minimal mixing model, with a tiny mixing angle θ0 ≃ (5.2 ± 1.9)0, between a scalar Tetramole (TM0) (superposition of nearly degenerated hypothetical molecules and compact tetraquarks states with the same quantum numbers), having a mass MTM0=2743(18)MeV, and the first radial excitation of the D−K+ molecule with mass M(DK)1 = 3678(310) MeV. In an analogous way, the X1(2900) and the XJ(3350) (if J = 1) could be a mixture between the vector Tetramole (TM1), with a mass MTM1=2656(20)MeV, and its first radial excitation having a mass MTM1=4592(141)MeV with an angle θ0 ≃ (9.1 ± 0.6)0. A (non)-confirmation of these statements requires experimental findings of the quantum numbers of the resonances at 3150 and 3350 MeV.

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.

X0,1(2900) and (D−K+) invariant mass from QCD Laplace sum rules at NLO

We revisit, improve and complete some recent estimates of the 0+ and 1− open charm (c¯d¯)(us) tetraquarks and the corresponding molecules masses and decay constants from QCD spectral sum rules (QSSR) by using QCD Laplace sum rule (LSR) within stability criteria where the factorised perturbative NLO corrections and the contributions of quark and gluon condensates up to dimension-6 in the OPE are included. We confront our results with the D−K+ invariant mass recently reported by LHCb from B+→D+(D−K+) decays. We expect that the bump near the D−K+ threshold can be originated from the 0++(D−K+) molecule and/or D−K+ scattering. The prominent X0 (2900) scalar peak and the bump XJ(3150) (if J=0) can emerge from a minimal mixing model, with a tiny mixing angle θ0≃(5.2±1.9)0, between a scalar Tetramole (TM0) (superposition of nearly degenerated hypothetical molecules and compact tetraquarks states with the same quantum numbers) having a mass MTM0 = 2743(18) MeV and the first radial excitation of the D−K+ molecule with mass M(DK)1=3678(310) MeV. In an analogous way, the X1 (2900) and the XJ(3350) (if J=1) could be a mixture between the vector Tetramole(TM1) with a mass MTM1=2656(20) MeV and its first radial excitation having a mass M(TM1)1=4592(141) MeV with an angle θ1≃(9.1±0.6)0. A (non)-confirmation of the previous minimal mixing models requires an experimental identification of the quantum numbers of the bumps at 3150 and 3350 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.

Classical spectral sum rules and “half-naked” electron effects in radiation from relativistic electrons in external field

Two properties of the radiation emitted by a relativistic electron in an external field, in the classical approximation, are presented in details: 1) spectral sum rules and their relationship with the sum rules for oscillator strengths in atomic physics. Filtered sum rules can be used to remove infrared or ultraviolet divergences. 2) “half-naked electron” effects like the Landau-Pomeranchuk-Migdal and Ternovskij-Shul'ga-Fomin effects. These properties are best understood when introducing the apparent transverse velocity.

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We extract (for the first time) the correlated values of the running masses m‾c and m‾b from MBc using QCD Laplace sum rules (LSR) within stability criteria where perturbative (PT) expressions at N2LO and non-perturbative (NP) gluon condensates at LO are included. Allowing the values of m‾c,b(m‾c,b) to move inside the enlarged range of recent estimates from charmonium and bottomium sum rules (Table 1) obtained using similar stability criteria, we deduce: m‾c(m‾c)=1286(16) MeV and m‾b(m‾b)=4202(8) MeV. Combined with previous estimates (Table 2), we deduce a tentative QCD Spectral Sum Rules (QSSR) average: m‾c(m‾c)=1266(6) MeV and m‾b(m‾b)=4197(8) MeV where the errors come from the precise determinations from J/ψ and ϒ sum rules. As a result, we present an improved prediction of fBc=371(17) MeV and the tentative upper bound fBc(2S)≤139(6) MeV, which are useful for a further analysis of Bc-decays.