Heavy Quarkonium Research Articles

Three-loop matching coefficients for heavy flavor-changing currents and the phenomenological applications

Within the framework of non-relativistic QCD (NRQCD) factorization, we compute the matching coefficients between full Quantum Chromodynamics (QCD) and NRQCD for the heavy flavor-changing vector, axial-vector, scalar and pseudo-scalar currents up to next-to-next-to-next-to-leading order (N3LO). We accomplish the analytical expressions for the three-loop renormalization constants and the corresponding anomalous dimensions for all of the four NRQCD currents with two different heavy flavors. The three-loop QCD corrections to the matching coefficients turn out to be significantly larger than lower order corrections. By employing the scale relation, we approximately simulate the N3LO correction to the wave functions at the origin for the vector {B}_c^{ast } meson and the pseudo-scalar Bc meson from the known result for the equal-mass heavy quarkonium in potential NRQCD (pNRQCD). We find large cancellations at the third order between the matching coefficients and the wave functions at the origin, and obtain the convergent decay constants of {B}_c^{ast } and Bc up to N3LO. We present the complete perturbative QCD predictions for the decay constants, leptonic decay widths, and branching ratios of the beauty-charmed mesons.

Thermodynamic properties of heavy mesons in strongly coupled quark gluon plasma using the fractional of non-relativistic quark model

AbstractA generalized fractional derivative of the analytical exact iteration method is used, in which the two-body potential in strongly coupled quark–gluon plasma is devoted to solve theN-dimensional radial Schrödinger equation. The energy eigenvalues for any state (n,l) and mass spectra in theN-dimensional space have been investigated. The dissociation temperatures were computed in theN-dimensional space for different states of quarkonia. The effect of fractional-order parameter is investigated on the dissociation temperatures of heavy quarkonium masses such as charmonium and bottomonium and thermodynamic properties such as entropy, free energy, internal energy, and specific heat in the 3D and the higher-dimensional space. Also, the effect of dimensionality number on dissociation temperatures is discussed. A comparison with other recent works is displayed. We deduce that the fractional-order plays an essential role in 3D and higher-dimensional space.

New bound state solutions of diatomic molecules and mass spectrum of heavy quarkonium with improved modified Kratzer potential plus screened Coulomb potential model in the framework of three-dimensional nonrelativistic noncommutative phase-space symmetries

Approximate analytical solutions of the modified Schrodinger equation have been obtained with a combination of two proposed potentials, called the improved modified Kratzer potential plus screened Coulomb (IMKSC) potential, using the approximation scheme to the centrifugal term for any l-states. We applied the generalized Bopp’s shift method and standard perturbation theory. IMKSC is a superposition of modified Kratzer potential plus screened Coulomb (MKSC) potential, and new terms appear as a result of the effects of the noncommutativity properties of space and phase potential on the MKSC potential. The new bound state energy eigenvalues for some diatomic molecules were obtained in terms of the discrete atomic quantum numbers (j,n,s,m), the infinitesimal NC phase-space parameters, the dissociation energy, the equilibrium internuclear separation, the depth of the potential, and the range of the potential. Furthermore, we have shown that the NC-Hamiltonian operator is the sum of the Hamiltonian operator of the MKSC potential, the perturbed spin-orbit, and the modified Zeeman operator. The new mass spectra of heavy quarkonium systems are calculated using the new energy spectrum expression. It appears as a combination of corresponding values in the literature plus an infinitesimal part that describes the deformation phase-space.

Charmonium transport in the high- μB medium

We employ the transport model coupled with hydrodynamic equations to study the charmonium dissociation and regeneration in the quark-gluon plasma (QGP) with the large baryon chemical potential in Au-Au collisions at the energies of $\sqrt{{s}_{NN}}=$ 39, 14.5, and 7.7 GeV. The baryon chemical potential ${\ensuremath{\mu}}_{B}$ is encoded in both Debye mass characterizing the heavy-quark potential and also the equation of state of the bulk medium. After considering ${\ensuremath{\mu}}_{B}$ corrections in both heavy quarkonium and the QGP medium, we calculate the nuclear modification factor ${R}_{AA}$ of charmonium. We find the ${\ensuremath{\mu}}_{B}$ influence on charmonium production at $\sqrt{{s}_{NN}}$ = 39 and 14.5 GeV is negligible, while the ${R}_{AA}$ of charmonium is reduced considering ${\ensuremath{\mu}}_{B}$ influence at $\sqrt{{s}_{NN}}=7.7$ GeV Au-Au collisions, which is crucial for studying charmonium production in low energy and also fixed-target heavy-ion collisions.

Quantum computing for heavy quarkonium spectroscopy

We report a first demonstration for the application of quantum computing to heavy quarkonium spectroscopy study. Based on a Cornell-potential model for the heavy quark and antiquark system, we show how this Hamiltonian problem can be formulated and solved with the VQE approach on the IBM cloud quantum computing platform. Errors due to a global depolarizing noise channel are corrected with a zero-noise extrapolation method, resulting in good agreement with the expected value. We also extend the calculation to excited states on a noiseless quantum simulator by orthogonalization with respect to the ground state.

Gluon gravitational form factors of protons from charmonium photoproduction* *Supported by the National Natural Science Foundation of China (12065014, 12047501), the Natural Science Foundation of Gansu province, China (22JR5RA266). We acknowledge the West Light Foundation of the Chinese Academy of Sciences (21JR7RA201)

Inspired by the recent near-threshold photoproduction measurements, we discuss gluon gravitational form factors (GFFs) and internal properties of the proton. This work presents a complete analysis of the proton gluon GFFs connecting the gluon part of the energy-momentum tensor and the heavy quarkonium photoproduction. In particular, a global fitting of the differential and total cross section experimental data is used to determine the gluon GFFs as functions of the squared momentum transfer t. Combined with the quark contributions to the D-term form factor extracted from the deeply virtual Compton scattering experiment, the total D-term is obtained to investigate their applications in describing the proton mechanical properties. These studies provide a unique perspective on investigating the proton gluon GFFs and important information for enhancing QCD constraints on the gluon GFFs.

Dirac sea effects on heavy quarkonia decay widths in magnetized matter: A field theoretical model of composite hadrons

We study the partial decay widths of charmonium (bottomonium) states to $D\bar D (B\bar B)$ mesons in magnetized (nuclear) matter using a field theoretic model of composite hadrons with quark (and antiquark) constituents. These are computed from the mass modifications of the decaying and produced mesons, including the nucleon Dirac sea effects within a chiral effective model. The Dirac sea contributions are observed to lead to an rise (drop) in the quark condensates as the magnetic field is increased, an effect called the (inverse) magnetic catalysis. In the presence of a magnetic field, there are mixings of spin 0 (pseudoscalar) and spin 1 (vector) states (PV mixing) The nucleon Dirac sea effects are observed to be significant and the anomalous magnetic moments (AMMs) of the nucleons are observed to play an important role. For $\rho_B$=0, there is observed to be magnetic catalysis (MC) without and with AMMs, whereas, for $\rho_B=\rho_0$, the inverse magnetic catalysis (IMC) is observed when the AMMs are taken into account, contrary to MC, when the AMMs are ignored. The magnetic field effects on the heavy quarkonium decay widths should have observable consequences on the production of the hidden and open heavy flavour mesons from ultra-relativistic peripheral heavy ion collision experiments, at RHIC and LHC, where the produced magnetic fields are extremely large.

Imaginary potential of heavy quarkonia from thermal fluctuations in rotating matter from holography

Imaginary potential of heavy quarkonia from thermal fluctuations in rotating matter from holography

Heavy hybrid decays to quarkonia

The decay rates of the XYZ exotics discovered in the heavy quarkonium sector are crucial observables for identifying the nature of these states. Based on the framework of nonrelativistic effective field theories, we calculate the rates of semi-inclusive decays of heavy quarkonium hybrids into conventional heavy quarkonia. We compute them at leading and subleading power in the inverse of the heavy-quark mass, extending and updating previous results. We compare our predictions with experimental data of inclusive decay rates for candidates of heavy quarkonium hybrids.

Ultraperipheral nuclear collisions as a source of heavy quarkonia

Heavy quarkonium production in ultraperipheral nuclear collisions (UPC) is described within the QCD dipole formalism. Realistic quarkonium wave functions in the $Q\overline{Q}$ rest frame are calculated by solving the Schr\"odinger equation with a subsequent Lorentz boost to high energy. We rely on several realistic $Q\overline{Q}$ potentials, which allow us to describe well the quarkonium masses and decay widths, as well as data on diffractive electroproduction of quarkonia on protons. Nuclear effects are calculated with the phenomenological dipole cross sections fitted to deep-inelastic scattering (DIS) data. The higher twist quark shadowing related to the lowest $Q\overline{Q}$ Fock component of the photon, as well as the leading twist gluon shadowing, related to higher components containing gluons, are included. The results for coherent and incoherent photoproduction of charmonia and bottomonia in UPC of heavy nuclei are in good accord with available data from the LHC. They can also be verified in future experiments at electron-ion colliders.

The Influence of Deformation Phase-Space on Spectra of Heavy Quarkonia in Improved Energy Potential at Finite Temperature Model of Shrodinger Equation Via the Generalized Boob’s Shift Method and Standard Perturbation Theory

In this work, we obtain solutions of the deformed Schrödinger equation (DSE) with improved internal energy potential at a finite temperature model in a 3-dimensional nonrelativistic noncommutative phase-space (3D-NRNCPS) symmetries framework, using the generalized Bopp’s shift method in the case of perturbed nonrelativistic quantum chromodynamics (pNRQCD). The modified bound state energy spectra are obtained for the heavy quarkonium system such as charmonium cc- and bottomonium bb- at finite temperature. It is found that the perturbative solutions of the discrete spectrum are sensible to the discreet atomic quantum numbers (j,l,s,m) of the ( QQ- (Q=c,b)) state, the parameters of internal energy potential (T,αs(T), mD (T),β,c), which are the Debye screening mass mD (T), the running coupling constant αs(T) the critical temperature β, the free parameter c in addition to noncommutativity parameters (Θ,θ-). The new Hamiltonian operator in 3D-NRNCPS symmetries is composed of the corresponding operator in commutative phase-space and three additive parts for spin-orbit interaction, the new magnetic interaction, and the rotational Fermi-term. The obtained energy eigenvalues are applied to obtain the mass spectra of heavy quarkonium systems (cc- and bb-). The total complete degeneracy of the new energy levels of the improved internal energy potential changed to become equal to the new value 3n2 in 3D-NRNCPS symmetries instead of the value n2 in the symmetries of 3D-NRQM. Our non-relativistic results obtained from DSE will possibly be compared with the Dirac equation in high-energy physics.

Exclusive production of quarkonia pairs in collinear factorization framework

In this paper, we analyze the exclusive photoproduction of heavy quarkonia pairs in the collinear factorization framework. We evaluate the amplitude of the process for the $J/\ensuremath{\psi}\ensuremath{-}{\ensuremath{\eta}}_{c}$ quarkonia pair in the leading order of the strong coupling ${\ensuremath{\alpha}}_{s}$, and express it in terms of generalized parton distributions (GPDs) of gluons in the proton. We make numerical estimates in the kinematics of the Electron Ion Collider and find that, in the photoproduction regime, when the virtuality of the photon is much smaller than the quarkonia mass, the cross section of the process is sufficiently large for experimental studies. We demonstrate that the study of this channel can complement existing studies of gluon GPDs from other channels.

Hadronic structure on the light front. III. The Hamiltonian, heavy quarkonia, spin, and orbit mixing

This is the third paper on hadronic light-front wave functions. We derive a light-front Hamiltonian from first principles, using the key features of the QCD vacuum at low resolution. In the first approximation, it gives transverse oscillator and longitudinal harmonic modes, and yields the correct Regge trajectories. For heavy quarkonia, we compare its spectrum to that obtained from the usual Schr\"odinger equation in the rest frame. We use the same approach for light quarks, and investigate the role of confinement and chiral symmetry breaking in the quark-antiquark sector. We then study spin-spin and spin-orbit mixing, resulting in e.g. quadrupole moments of vector mesons. For the light mesons, we show how to extend the famed 't Hooft interaction to the light front, which solves the U(1) problem and helps produce a light pion. We use the ensuing light front wave functions to derive the pertinent parton distribution functions, parton amplitudes, and low energy constants.

Dissociation of heavy quarkonium states in a rapidly varying strong magnetic field

Various heavy quarkonium states can be employed as a testing ground for the magnetic field created and later possibly persisted in heavy ion collision (HIC) experiments. In this paper, we have shown the modification in the binding energies, and ionization potentials categorically for different species of quarkonia in the medium in the presence of a magnetic field. For the study, we have considered quarkonium states which are well described by the real-valued effective potential in the thermal medium which is evolving by respecting the hydrodynamical power law. Furthermore while addressing the issue of their survival, we have not ignored the possibility of transitions between various bound-to-bound and bound-to-unbound states for different charmonium and bottomonium states. Our analysis shows that this phenomenon can be considered as a benchmark for understanding the presence, strength and persistence of the magnetic field which is expected to be produced through the high-energy nucleus–nucleus collisions.

Bottomonium sequential suppression and strong heavy-quark potential in heavy-ion collisions

We employ the time-dependent Schrödinger equation with different complex potentials to study the bottomonium sequential suppression in Pb-Pb collisions at sNN=2.76 TeV and 5.02 TeV and Au-Au collisions at sNN=200 GeV. Both color screening effect and the random scatterings with thermal partons are considered in the real and imaginary parts of the heavy-quark potentials. As the real part of the heavy-quark potential is between the free energy F(T,r) and the internal energy U(T,r) of heavy quarkonium, we parametrize different potentials with a function of F and U to evolve the bottomonium wave packages in the medium. We find that when the real part of the potential is close to U(T,r), it can explain well the pattern of bottomonium sequential suppression where their nuclear modification factors satisfy the relation RAA(1s)>RAA(2s)>RAA(3s) observed in experiments. In the other limit of F(T,r), bottomonium wave packages tend to expand due to weak attracted force, which results in evident transitions from ϒ(2s) to ϒ(3s) components and does not satisfy the sequential suppression pattern. We suggest that the bottomonium sequential suppression can be a probe of strong heavy-quark potential in the medium.

Gauge Invariance of Non-Abelian Field Strength Correlators: The Axial Gauge Puzzle.

Many transport coefficients of the quark-gluon plasma and nuclear structure functions can be written as gauge invariant correlation functions of non-Abelian field strengths dressed with Wilson lines. We discuss the applicability of axial gauge n·A=0 to calculate them. In particular, we address issues that appear when one attempts to trivialize the Wilson lines in the correlation functions by gauge fixing. We find it is always impossible to completely remove the gauge fields n·A in Wilson lines that extend to infinity in the n direction by means of gauge transformations. We show how the obstruction appears in an explicit example of a perturbative calculation, and we also explain it more generally from the perspective of the path integral that defines the theory. Our results explain why the two correlators that define the heavy quark and quarkonium transport coefficients, which are seemingly equal in axial gauge, are actually different physical quantities of the quark-gluon plasma and have different values. Furthermore, our findings provide insights into the difference between two inequivalent gluon parton distribution functions.

Note on rare Z-boson decays to double heavy quarkonia* *Supported in part by the National Natural Science Foundation of China (11575175, 12047502) and National Research and development Program of China (2020YFA0406400)

Within the standard model, we have investigated rare Z-boson decays into double heavy quarkonia, and , with V and P denoting vector and pseudoscalar quarkonia, respectively. It is assumed that the leading-order QCD diagrams would give the dominant contributions to these processes, and the corresponding branching fractions, for instance, , have been estimated to be approximately in literature. However, these decays could also happen through electromagnetic transitions and , with the virtual photon transforming into V. Interestingly, the smallness of the vector quarkonium mass can give rise to a large factor relative to the QCD contributions, which thus counteracts the suppression from the electromagnetic coupling. We systematically include these two types of contributions in our calculation to predict branching fractions for these decays. Particularly, owing to the virtual photon effects, it is found that will be significantly enhanced, up to .

Emergence of heavy quark spin symmetry breaking in heavy quarkonium decays

Heavy-quark spin symmetry (HQSS) implies that in the direct decay of a heavy quarkonium with spin $S$, only lower lying heavy quarkonia with the same spin $S$ can be produced. However, this selection rule, expected to work very well in the $b$-quark sector, can be overcome if multiquark intermediate states are involved in the decay chain, allowing for transitions to the final-state heavy quarkonia with a different spin ${S}^{\ensuremath{'}}$. In particular, the measured decays $\mathrm{\ensuremath{\Upsilon}}(10860)\ensuremath{\rightarrow}\ensuremath{\pi}{Z}_{b}^{(\ensuremath{'})}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}\mathrm{\ensuremath{\Upsilon}}(nS)$ ($n=1$, 2, 3) and $\mathrm{\ensuremath{\Upsilon}}(10860)\ensuremath{\rightarrow}\ensuremath{\pi}{Z}_{b}^{(\ensuremath{'})}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}{h}_{b}(mP)$ ($m=1$, 2) appear to have nearly equal strengths, which is conventionally explained by a simultaneous presence of both ${S}_{b\overline{b}}=0$ and ${S}_{b\overline{b}}=1$ components in the wave functions of the ${Z}_{b}\mathrm{s}$ in equal shares. Meanwhile, the destructive interference between the contributions of the ${Z}_{b}$ and ${Z}_{b}^{\ensuremath{'}}$ to the decay amplitude for a $\ensuremath{\pi}\ensuremath{\pi}{h}_{b}$ final state kills the signal to zero in the strict HQSS limit. In this paper, we discuss how the HQSS violation needs to be balanced by the narrowness of the ${Z}_{b}^{(\ensuremath{'})}$ states in the physical case to allow for equal transition strengths into final states with different total heavy quark spins, and how spin symmetry is restored as a result of a subtle interplay of the scales involved, when the mass of a heavy quark becomes infinite. Moreover, we demonstrate how similar branching fractions of the decays into $\ensuremath{\pi}\ensuremath{\pi}{h}_{b}$ and $\ensuremath{\pi}\ensuremath{\pi}\mathrm{\ensuremath{\Upsilon}}$ can be obtained and how the mentioned HQSS breaking can be reconciled with the dispersive approach to the $\ensuremath{\pi}\ensuremath{\pi}/K\overline{K}$ interaction in the final state and matched with the low-energy chiral dynamics in both final states.

Heavy quark potential and LQCD based quark condensate at finite magnetic field

In the present work, we have studied heavy quarkonia potential in hot and magnetized quark gluon plasma. Inverse magnetic catalysis (IMC) effect is incorporated within the system through the magnetic field modified Debye mass by modifying the effective quark masses. We have obtained the real and imaginary part of the heavy quark potential in this new scenario. After the evaluation of the binding energy and the decay width we comment about the dissociation temperatures of the heavy quarkonia in presence of magnetic field.

Production of double heavy quarkonia at super Z factory

Within the color singlet model, we calculate the exclusive production of double charmonia, double bottomonia, and double B_c mesons at future super Z factory. The two heavy quarkonia or B_c’s are either two S-wave Fock states (^1S_0, ~^3S_1), or one S-wave and one P-wave states (^1P_1,~^3P_J~(J=0,1,2)). The top three Z^0 propagated channels in cross sections for double charmonia are sigma {(J/psi +h_c)}_{Z^0}=(454.0^{+39.1}_{-59.2})times 10^{-5} fb, sigma {(eta _c+chi _{c2})}_{Z^0}=(284.2^{+22.2}_{-35.3})times 10^{-5} fb, and sigma {(eta _c+chi _{c0})}_{Z^0}=(137.0^{+9.9}_{-16.4})times 10^{-5} fb. For double bottomonia, they are sigma {(eta _b+Upsilon )}_{Z^0}=(428.6^{+67.9}_{-60.9})times 10^{-4} fb, sigma {(Upsilon +chi _{b2})}_{Z^0}=(230.6^{+23.6}_{-22.2})times 10^{-4} fb, and sigma {(Upsilon +Upsilon )}_{Z^0}=(119.9^{+18.4}_{-16.6})times 10^{-4} fb. For double B_c mesons, they are sigma {(B_c^{*+}+B_c^{*-})}_{Z^0}=(1150^{+87}_{-84})times 10^{-3} fb, sigma {(B_c^{*+}+chi _{bc2}^-)}_{Z^0}=(1133^{-16}_{+22})times 10^{-3} fb, and sigma {(eta _{bc}^++B_c^{*-})}_{Z^0}=(634.6^{+47.8}_{-45.8})times 10^{-3} fb. Here the uncertainties come from the varying masses of constituent heavy quarks, which bring up to 20% corrections. The cross sections of double B_c mesons are roughly one order of magnitude larger than those of the double bottomonia, and two orders of magnitude larger than those of the double charmonia. To make it helpful for experimental study, we present the total cross sections sigma as functions of CM energy sqrt{s}, sigma as functions of the renormalization scale mu , the angle distributions dsigma /dcostheta , and the p_T distributions dsigma /dp_{t}. We also find that the initial state radiation can bring about 30–40% suppresions when 1%m_Z energy is losing, and cross sections can increase by about 2–3 times or decrease by an order of magnitude when adopting different potential models which becomes the major source of uncertainty. The numerical results show that it might be not optimistic for the experimental observation, but it is still far from excluded at the FCC-ee and also the CEPC running in the Z factory mode.