A Relation Between Gluon Condensates from a Local QCD Sum Rule

In this paper, we investigate the semileptonic decays Bc→ηcℓν¯ℓ using the quantum chromodynamics (QCD) sum rules within the framework of Standard Model (SM). We further explore the potential to probe signatures of new physics (NP) beyond the SM through these decays. First, we derive the ξ-moments ⟨ξ2;ηcn⟩ of the ηc-meson leading-twist distribution amplitude ϕ2;ηc using the QCD sum rules within the background field theory. Considering contributions from the vacuum condensates up to dimension-six, the first two nonzero ξ-moments at the scale of 4 GeV are found to be ⟨ξ2;ηc2⟩=0.103−0.009+0.009 and ⟨ξ2;ηc4⟩=0.031−0.003+0.003. Using these moments, we then fix the Gegenbauer expansion series of ϕ2;ηc and apply it to compute the Bc→ηc transition form factors (TFFs) using QCD light cone sum rules. Second, we extrapolate those TFFs to physically allowable q2-range via a simplified series expansion, and we obtain Rηc|SM=0.308−0.062+0.084. Furthermore, we explore the potential impacts of various NP scenarios on Rηc. Specifically, we compute the forward-backward asymmetry AFB(q2), the convexity parameter CFτ(q2), and the longitudinal and transverse polarizations PL(q2) and PT(q2) for Bc→ηc transitions within both the SM and two types of NP scenarios. Our results contribute to a deeper understanding of Bc-meson semileptonic decays and provide insights into the search for the NP beyond the SM. Published by the American Physical Society 2025

In this work, we study the mass spectrum of the hidden-charm hybrid states with the JPC=0−+, 0++, 0−−, 1++, 1+−, 1−+, 1−−, 2−+ and 2++ via the quantum chromodynamics (QCD) sum rules in a consistent way. We calculate the vacuum condensates up to dimensions-6 by taking account of both the leading order and next-to-leading order contributions, and take the energy scale formula μ=MX/Y/Z2−(2Mc)2 to choose the suitable energy scales of the QCD spectral densities, it is the first time to explore the energy scale dependence of the QCD sum rules for the hidden-charm hybrid states.

Aimed at graduate students and researchers in theoretical physics, this book presents the modern theory of strong interaction: quantum chromodynamics (QCD). The book exposes various perturbative and nonperturbative approaches to the theory, including chiral effective theory, the problems of anomalies, vacuum tunnel transitions, and the problem of divergence of the perturbative series. The QCD sum rules approach is exposed in detail. A great variety of hadronic properties (masses of mesons and baryons, magnetic moments, form factors, quark distributions in hadrons, etc.) have been found using this method. The evolution of hadronic structure functions is presented in detail, together with polarization phenomena. The problem of jets in QCD is treated through theoretical description and experimental observation. The connection with Regge theory is emphasized. The book covers many aspects of theory which are not discussed in other books, such as CET, QCD sum rules, and BFKL.

We review the present understanding of the spin structure of protons and neutrons, the fundamental building blocks of nuclei collectively known as nucleons. The field of nucleon spin provides a critical window for testing Quantum Chromodynamics (QCD), the gauge theory of the strong interactions, since it involves fundamental aspects of hadron structure which can be probed in detail in experiments, particularly deep inelastic lepton scattering on polarized targets.QCD was initially probed in high energy deep inelastic lepton scattering with unpolarized beams and targets. With time, interest shifted from testing perturbative QCD to illuminating the nucleon structure itself. In fact, the spin degrees of freedom of hadrons provide an essential and detailed verification of both perturbative and nonperturbative QCD dynamics.Nucleon spin was initially thought of coming mostly from the spin of its quark constituents, based on intuition from the parton model. However, the first experiments showed that this expectation was incorrect. It is now clear that nucleon physics is much more complex, involving quark orbital angular momenta as well as gluonic and sea quark contributions. Thus, the nucleon spin structure remains a most active aspect of QCD research, involving important advances such as the developments of generalized parton distributions (GPD) and transverse momentum distributions (TMD).Elastic and inelastic lepton-proton scattering, as well as photoabsorption experiments provide various ways to investigate non-perturbative QCD. Fundamental sum rules—such as the Bjorken sum rule for polarized photoabsorption on polarized nucleons—are also in the non-perturbative domain. This realization triggered a vigorous program to link the low energy effective hadronic description of the strong interactions to fundamental quarks and gluon degrees of freedom of QCD. This has also led to advances in lattice gauge theory simulations of QCD and to the development of holographic QCD ideas based on the AdS/CFT or gauge/gravity correspondence, a novel approach providing a well-founded semiclassical approximation to QCD. Any QCD-based model of the nucleon’s spin and dynamics must also successfully account for the observed spectroscopy of hadrons. Analytic calculations of the hadron spectrum, a long sought goal of QCD research, have now being realized using light-front holography and superconformal quantum mechanics, a formalism consistent with the results from nucleon spin studies.We begin this review with a phenomenological description of nucleon structure in general and of its spin structure in particular, aimed to engage non-specialist readers. Next, we discuss the nucleon spin structure at high energy, including topics such as Dirac’s front form and light-front quantization which provide a frame-independent, relativistic description of hadron structure and dynamics, the derivation of spin sum rules, and a direct connection to the QCD Lagrangian. We then discuss experimental and theoretical advances in the nonperturbative domain—in particular the development of light-front holographic QCD and superconformal quantum mechanics, their predictions for the spin content of nucleons, the computation of PDFs and of hadron masses.

An observation is given that the field strength or the dual potential becomes a good coordinate in the low energy region and that its correlation is of short range if gluons condense in the vacuum in the form of color singlet. Taking the axial gauge the field strength formulation of Quantum Chromodynamics is given after separating the zero mode of the vector potential in order to take into account of the effect of the condensation. It turns out that unstable modes which are present in the spectrum of the vector potential are eliminated by the dual transformation and that the gluonic condensation gives a mass gap to the dual potential. As a dual Meissner effect the flux tube of the color electric field is formed. Numerical estimates of the string tension are given with a value of correct order of magnitude. The same mechanism gives a solution to the U(1) problem. The gluon condensation is finally shown by developing the leading logarithm series of the effective potential which enables us to find the minimum successively for small coupling constant.

The strong interactions between quarks are believed to be described by Quantum Chromodynamics (QCD), which is a non-abelian SU(3) gauge theory. It is known that QCD undergoes a deconfining phase transition at very high temperatures, that is, at low temperatures QCD is in a confined phase, at sufficient high temperatures it is in a unconfined phase. Also, quark confinement is believed to be due to the string formation. In this dissertation we studied SU(2) gauge theory using numerical methods of LGT, which will provide some insights about the properties of QCD because SU(2) is similar to SU(3). We measured the flux distributions of a q q pair at various temperatures in different volumes. We find that in the limit of infinite volumes the flux distribution is different in the two phases. In the confined phase strong evidence is found for the string formation, however, in the unconfined phase there is no string formation. On the other hand, in the limit of zero temperature and finite volumes we find clear signal for string formation in the large volume region, however, the string tension measured in intermediate volumes is due to finite volume effects, there is no intrinsic string formation. The color flux energies (action) of the q q pair are described by Michael sum rules. The original Michael sum rules deal with a static q q pair at zero temperature in infinite volumes. To check these sum rules with our flux data at finite temperatures, we present a complete derivation for the sum rules, then generalize them to account for finite temperature effects. We find that our flux data are consistent with the prediction of generalized sum rules. Our study elucidates the rich structures of QCD, and provides evidences for the quark confinement and string formation. This supports the belief that QCD is a correct theory for strong interactions, and quark confinement can be explained by QCD.

ALEPH Tau Spectral Functions and QCD

A gauge invariant and Lorentz-irreducible three-gluon conserved vector current is constructed, and the mass of the vector glueball is calculated in the framework of the Quantum Chromodynamics sum rules.

A new method of analyzing QCD sum rules employing the Maximum Entropy Method, is introduced and is applied successfully to the rho meson, the nucleon and the charmonium at finite temperature. This method enables us to directly obtain the spectral function from the sum rules, without having to introduce any model assumption about its functional form. In the nucleon sum rule, we show that the Gaussian sum rule successfully reproduces the ground state. Dependences on the interpolating field operator are discussed. Finite temperature effects for the charmonium sum rule are incorporated by changes of various gluonic condensates, extracted from lattice QCD. As a result, we find that both J/ψ and ηc dissolve into the continuum already at temperatures around 1.0 ∼ 1.2 Tc.

The consequences of the condensation of gluons are discussed by mean field approximation. Assuming <AµaAνb> to be non-zero, the contribution of the condensate is singled out from the equal time commutator and from the equation of motion. Gaµν become good coordinates and acquire mass. Green's functions are determined. Effects of weak external electric and magnetic sources are discussed in the linear approximation. The vacuum is shown to have the property of electric confinement and magnetic superconductivity.

Using recent values of the QCD (non-)perturbative parameters given in Table 1 and an estimate of the N3LO QCD perturbative contributions based on the geometric growth of the PT series, we re-use QCD spectral sum rules (QSSR) known to N2LO PT series and including all dimension-six NP condensate contributions in the full QCD theory, for improving the existing estimates of m¯c,b and fD(s),B(s) from the open charm and beauty systems. We especially study the effects of the subtraction point on “different QSSR data” and use (for the first time) the Renormalization Group Invariant (RGI) scale-independent quark masses in the analysis. The estimates [rigourous model-independent upper bounds within the SVZ framework] reported in Table 8: fD/fπ=1.56(5)[⩽1.68(1)], fB/fπ=1.58(5)[⩽1.80(3)] and fDs/fK=1.58(4)[⩽1.63(1)], fBs/fK=1.50(3)[⩽1.61(3.5)], which improve previous QSSR estimates, are in perfect agreement (in values and precisions) with some of the experimental data on fD,Ds and on recent lattice simulations within dynamical quarks. These remarkable agreements confirm both the success of the QSSR semi-approximate approach based on the OPE in terms of the quark and gluon condensates and of the Minimal Duality Ansatz (MDA) for parametrizing the hadronic spectral function which we have tested from the complete data of the J/ψ and ϒ systems. The values of the running quark masses m¯c(mc)=1286(66) MeV and m¯b(mb)=4236(69) MeV from MD,B are in good agreement though less accurate than the ones from recent J/ψ and ϒ sum rules.

The maximum entropy analytic continuation (MEAC) and ring polymer molecular dynamics (RPMD) methods provide complementary approaches to the calculation of real time quantum correlation functions. RPMD becomes exact in the high temperature limit, where the thermal time betavariant Planck's over 2pi tends to zero and the ring polymer collapses to a single classical bead. MEAC becomes most reliable at low temperatures, where betavariant Planck's over 2pi exceeds the correlation time of interest and the numerical imaginary time correlation function contains essentially all of the information that is needed to recover the real time dynamics. We show here that this situation can be exploited by combining the two methods to give an improved approximation that is better than either of its parts. In particular, the MEAC method provides an ideal way to impose exact moment (or sum rule) constraints on a prior RPMD spectrum. The resulting scheme is shown to provide a practical solution to the "nonlinear operator problem" of RPMD, and to give good agreement with recent exact results for the short-time velocity autocorrelation function of liquid parahydrogen. Moreover these improvements are obtained with little extra effort, because the imaginary time correlation function that is used in the MEAC procedure can be computed at the same time as the RPMD approximation to the real time correlation function. However, there are still some problems involving long-time dynamics for which the RPMD+MEAC combination is inadequate, as we illustrate with an example application to the collective density fluctuations in liquid orthodeuterium.

We develop a reliable parameter-free analytic continuation method for quantum\nmany-body calculations. Our method is based on a kernel grid, a causal spline,\na regularization using the second-derivative roughness penalty, and the L-curve\ncriterion. We also develop the L-curve averaged deviation to estimate the\nprecision of our analytic continuation. To deal with statistically obtained\ndata more efficiently, we further develop a bootstrap-averaged analytic\ncontinuation method. In the test using the exact imaginary-frequency Green's\nfunction with added statistical error, our method produces the spectral\nfunction that converges systematically to the exact one as the statistical\nerror decreases. As an application, we simulate the two-orbital Hubbard model\nfor various electron numbers with the dynamical-mean field theory in the\nimaginary time and obtain the real-frequency self-energy with our analytic\ncontinuation method, clearly identifying a non-Fermi liquid behavior as the\nelectron number approaches the half filling from the quarter filling. Our\nanalytic continuation can be used widely and it will facilitate drawing clear\nconclusions from imaginary-time quantum many-body calculations.\n

In the framework of a semiclassical expansion for quantum chromodynamics (QCD) in the instanton liquid background, the correlation function of the 0++ scalar glueball current is given. Besides the pure classical and quantum contributions, the contributions arising from the interactions between the classical instanton fields and quantum gluons are taken into account as well. Instead of the usual zero-width approximation for the resonance, the Breit–Wigner form for the spectral function of the finite-width resonance is adopted. The family of the Laplace sum rules for the scalar glueball in QCD with and without light quarks are studied. A consistency between the subtracted and unsubtracted sum rules is very well justified, and the values of the mass, decay width and the coupling to the corresponding current for the 0++ resonance in which the glueball fraction is dominant, are obtained.

The electric [Formula: see text] and magnetic [Formula: see text] Compton polarizabilities of both the charged and the neutral pion are of fundamental interest in the low-energy sector of quantum chromodynamics (QCD). Pion polarizabilities affect the shape of the [Formula: see text] Compton scattering angular distribution at back scattering angles and [Formula: see text] absolute cross sections. Theory derivations are given for the [Formula: see text] Compton scattering differential cross section, dispersion relations, and sum rules in terms of the polarizabilities. We review experimental charged and neutral polarizability studies and theoretical predictions. The [Formula: see text] polarizabilities were deduced from DESY Crystal Ball [Formula: see text] data, but with large uncertainties. The charged pion polarizabilities were deduced most recently from (1) radiative pion Primakoff scattering [Formula: see text] at CERN COMPASS, (2) two-photon pion pair production [Formula: see text] at SLAC Mark II, and (3) radiative pion photoproduction [Formula: see text] from the proton at MAMI in Mainz. A stringent test of chiral perturbation theory (ChPT) is possible based on comparisons of precision experimental charged pion polarizabilities with ChPT predictions. Only the CERN COMPASS charged pion polarizability measurement has acceptably small uncertainties. Its value [Formula: see text] agrees well with the two-loop ChPT prediction [Formula: see text], strengthening the identification of the pion with the Goldstone boson of chiral symmetry breaking in QCD.