QCD Sum Rule Studies of Heavy Quarkonium-like States
Robin Kleiv
备注：
PhD thesis (University of Saskatchewan, September 2013), based on arXiv:1206.6776, arXiv:1204.0049, arXiv:1304.7816, arXiv:1010.2971 and arXiv:0804.2195. This is an abridged version with links to preprints and published articles. Full version at http://hdl.handle.net/10388/ETD-2012-09-1319 . 124 pages, 18 figures
The research presented here uses QCD sum rules (QSR) to study exotic hadrons. There are several themes in this work. First is the use of QSR to predict the masses of exotic hadrons that may exist among the heavy quarkonium-like states. The second theme is the application of sophisticated loop integration methods in order to obtain more complete theoretical results. These in turn can be extended to higher orders in the perturbative expansion in order to predict the properties of exotic hadrons more accurately. The third theme involves developing a renormalization methodology for these higher order calculations. This research has implications for the $Y(3940)$, $X(3872)$, $Z_c^\pm\left(3895\right)$, $Y_b\left(10890\right)$, $Z_b^{\pm}(10610)$ and $Z_b^{\pm}(10650)$ particles, thereby contributing to the ongoing effort to understand these and other heavy quarkonium-like states.
We review the recent theoretical progress in heavy quarkonium spectroscopy within the boundstate theory based on perturbative QCD. New microscopic pictures of the heavy quarkonium systems are obtained.
In the framework of a specific scheme of the QCD sum rules for $S$-wave levels of the heavy quarkonium, one derives an expression, relating the energetic density of quarkonium states and universal characteristics in the heavy quarkonium physics, such as the difference between the masses of a heavy quark $Q$ and meson $(Q\bar q)$ and the number of heavy quarkonium levels below the threshold of $(Q\bar Q) \to (Q\bar q) + (\bar Q q)$ decay.
In the framework of a specific scheme of the QCD sum rules for $S$-wave levels of the heavy quarkonium, one derives expressions, relating the leptonic constants, the energetic density of quarkonium states and universal characteristics in the heavy quarkonium physics, such as the difference between the masses of heavy quark $Q$ and meson $(Q\bar q)$ and the number of heavy quarkonium levels below the threshold of $(Q\bar Q) \to (Q\bar q) + (\bar Q q)$ decay.
In this talk, I briefly review several models of the heavy quarkonium production at collider energies, and discuss the status of QCD factorization for these production models.
QCD nonrelativistic effective field theories (NREFT) are the modern and most suitable frame to describe heavy quarkonium properties. Here I summarize few relevant concepts and some of the interesting physical applications (spectrum, decays, production) of NREFT.
QCD nonrelativistic effective field theories (NREFT) are the modern and most suitable frame to describe heavy quarkonium properties. In this talk I summarize few relevant concepts and some interesting physical applications of NREFT.
Non-relativistic bound state theories for QED and QCD have become fairly mature and amenable to a textbook-level understanding and computation. In this talk we give an introductory review of the following subjects related to the recent computation of the heavy quarkonium spectrum using perturbative QCD: (1) Technological developments in higher-order computation, (2) Physics predictions, (3) Challenge towards analytic evaluation of the 3-loop static QCD potential.
Some recent progress and a personal selection of open problems in heavy quarkonium physics (spectroscopy, decay and production) inspired by the activity of the Quarkonium Working Group are reviewed.
We review the developments of QCD multipole expansion and its applications to hadronic transitions and some radiative decays of heavy quarkonia. Theoretical predictions are compsred with updated experimental results.
Using a specific scheme of the QCD sum rules, one derives an universal formula for the mass differences for the $nS$-levels of heavy qiarkonium. This relation does not depend on the flavours of the heavy quarks, composing the quarkonium, and it is in a good agreement with the experimental mass values for the $ψ$- and $Υ$-families.
We determine the hard-loop resummed propagator in an anisotropic QCD plasma in general covariant gauges and define a potential between heavy quarks from the Fourier transform of its static limit. We find that the potential exhibits angular dependence and that binding of very small quarkonium states is stronger than in an isotropic plasma.
The analytic continuation of the operator product expansion of the scattering amplitude allows to compute the heavy-quarkonium hadron total cross section. The energy dependence of the Upsilon and Upsilon' cross sections with a proton is discussed.
Using a specific scheme of the QCD sum rules for heavy quarkonium in the leading approximation over the inverse heavy quark mass, one gets the estimate of the difference between the masses of the heavy meson and heavy quark $\bar Λ= 0.59\pm 0.02$ GeV.
Quarkonium production at low transverse momentum in hadron collisions can be used to extract Transverse-Momentum-Dependent(TMD) gluon distribution functions, if TMD factorization holds there. We show that TMD factorization for the case of P-wave quarkonium with $J^{PC}=0^{++}, 2^{++}$ holds at one-loop level, but is violated beyond one-loop level. TMD factorization for other P-wave quarkonium is also violated already at one-loop.
We report on a recent QCD based research on hybrid mesons containing $c\bar c$ or $b\bar b$ quarks. We present results for the spectrum, the decay widths to heavy quarkonium, and the role of mixing with the latter. We point out that mixing with heavy quarkonium provides a potentially large source of spin symmetry breaking. We identify candidates to hybrid mesons among the so called XYZ states in the charmonium and bottomonium spectrum.
The value of quarkonium wave function at the origin is an important quantity while studying many physical problems concerning a heavy quarkonium. This is because that it is widely used to evaluate the production and decay amplitudes of the heavy quarkonium within the effective filed theory framework, e.g., the non-relativistic QCD (NRQCD). In this paper, the value of the Schr${\rm \ddot{o}}$dinger radial wave function or its first nonvanishing derivative at zero quark-antiquark separation, i.e., $|(|c\bar{c})[n]\rangle$-, $|(|b\bar{c})[n]\rangle$-, and $|(b\bar{b})[n]\rangle$-quarkonium, have …
In this talk I review the calculation of the third order corrections to the heavy quarkonium spectrum in the nonrelativistic effective theory framework and its application to the phenomenology of top quark threshold production.
A calculation of the melting temperatures of heavy quarkonium states with the holographic potential was introduced in a previous work. In this paper, we consider the holographic potential at sub-leading order, which permits finite coupling corrections to be taken into account. It is found that this correction lowers the dissociation temperatures of heavy quarkonium.
We calculate the heavy quarkonium production in double pomeron exchange processes in perturbative QCD by using two-gluon exchange model. For the $P$-wave $χ_J$ productions, we find $χ_1$ and $χ_2$ production amplitudes which vanish in the forward scattering limit. We also calculate direct $J/ψ(Υ)+γ$ production in the same approach, and these direct contributions are much smaller than the feeddown contributions from the $P$-wave states.
By employing the differential equations, we compute analytically the elliptic sectors of two-loop master integrals appearing in the NNLO QCD corrections to CP-even heavy quarkonium exclusive production and decays, which turns out to be the last and toughest part in the relevant calculation. The integrals are found can be expressed as Goncharov polylogarithms and iterative integrals over elliptic functions. The master integrals may be applied to some other NNLO QCD calculations about heavy quarkonium exclusive production, like $γ^*γ\rightarrow Q\bar{Q}$, $e^+e^-\rightarrow γ+ Q\bar{Q}$,~and~$H/…
The hybrid quarkonium states are studied using the Born-Oppenheimer expansion. The first step in this expansion is the determination of the energy levels of the gluons in the presence of a static quark-antiquark pair as a function of the quark-antiquark separation. The spectrum of such gluonic excitations is determined from first principles using lattice QCD.
I review some recent progress and open problems in the calculation of heavy-quarkonium inclusive decay widths into light particles in the framework of QCD non-relativistic effective field theories.
FDCHQHP is a Fortran package to calculate the transverse momentum (p_t) distribution of yield and polarization for heavy quarkonium hadroproduction at next-to-leading-order (NLO) based on non-relativistic QCD(NRQCD) framework. It contains the complete color-singlet and color-octet intermediate states in present theoretical level, and is available to calculate different polarization parameters in different frames. As the LHC running now and in the future, it supplies a very useful tool to obtain theoretical prediction on the heavy quarkonium hadroproduction.
Recent developments in heavy quarknoium physics are reviewed in brief, including (i) nonrelativistic QCD(NRQCD), (ii) the importance of color-octet $(Q\bar{Q})$ components in the decay and the production of a physical heavy quarkonium state, (iii) limitation of the NRQCD factorization and (iv) the double humps in the $ππ$ spectrum in the decay $Υ(3S) \to Υ(1S) ππ$.
By using conformable fractional of the Nikiforov-Uvarov (CF-NU) method, the radial Schrodinger equation is analytically solved. The energy eigenvalues and corresponding functions are obtained, in which the dependent temperature potential is employed. The effect of fraction-order parameter is studied on heavy-quarkonium masses such as charmonium and bottomonium in a hot QCD medium in the 3D and the higher dimensional space. A comparison is studied with recent works. We conclude that the fractional-order plays an important role in a hot QCD medium in the 3D and higher-dimensional space.
Using the non-relativisitc reduction of Coulomb gauge QCD we compute spectrum of the low mass hybrid mesons containing a heavy quark-antiquark pair. The gluon degrees of freedom are treated in the mean field approximation calibrated to the gluelump spectrum. We discuss the role of the non-abelian nature of the QCD Coulomb interaction in the ordering of the spin-parity levels.
We compute the complete leading-log terms of the next-to-next-to-next-to-leading-order corrections to potential NRQCD. As a by-product we obtain the leading logs at $O(mα_s^5)$ in the heavy quarkonium spectrum. These leading logs, when $Λ_{QCD} \ll mα_s^2$, give the complete $O(mα_s^5 \ln α_s)$ corrections to the heavy quarkonium spectrum.
We compute the total cross section of a heavy quarkonium on a hadron target in leading twist QCD, including target mass corrections. Our method relies on the analytical continuation of the operator product expansion of the scattering amplitude, obtained long ago by Bhanot and Peskin. The cross section has a simple partonic form, which allows us to investigate the phenomenology of J/psi and Upsilon dissociation by both pions and protons.
We report on recent theory progress in understanding the production of heavy quarkonium in heavy-ion collisions based on the in-medium heavy-quark potential extracted from lattice QCD simulations. On the one hand, the proper in-medium potential allows us to study the spectral properties of heavy quarkonium in thermal equilibrium, from which we estimate the $ψ^\prime$ to $J/ψ$ ratio in heavy-ion collisions. On the other hand, the potential provides a central ingredient in the description of the real-time evolution of heavy-quarkonium formulated in the open-quantum-systems framework.
An introduction to the recent developments in perturbative heavy quarkonium physics is given. Covered are the effective theories NRQCD, pNRQCD and vNRQCD, the threshold expansion, the role of renormalons, and applications to bottom and top quark physics.
Heavy quarkonium production in the framework of the nonrelativistic quantum chromodynamics and leading order of the parton Reggeization approach at the Tevatron and LHC Colliders is discussed. The new results, which are reviewed in this report, include the comparison with recent data on chi_{c1,c2} production from ATLAS Collaboration and special discussion of heavy quarkonium polarization issues in the considered framework.
In this paper, we calculate the mass spectra of heavy quarkonium by using matrix Numerov's method to make the predictions of F and G states for further experiments. The method gives a very reasonable result which is in a good agreement with other methods and with recently published theoretical data. From the yielded wave functions we calculate the root mean square radius r_ms and \b{eta} coefficients of heavy quarkonium
We study the masses and the leptonic decay widths of heavy quarkonium in the dual Ginzburg-Landau (DGL) theory of QCD, in which the abelian monopole condensation plays an essential role for color confinement. The effect of color screening due to the light quark pair creation is introduced by the infrared momentum cutoff in the gluon propagator. We find that the color screening effect is important to reproduce the properties of the heavy quarkonium, in particular the leptonic decay width. We also discuss the underlying systematics of the heavy quark spectroscopy in the DGL theory.
Heavy quarkonium decays and transitions are discussed in the framework of non-relativistic effective field theories. Emphasis is put on the matching procedure in the non-perturbative regime. Some exact results valid for the magnetic dipole couplings are discussed.
Based on the factorization formalism of nonrelativistic quantum chromodynamics (NRQCD), we calculate the production cross section for the charmonium and the bottomonium produced by the hard photoproduction processes and fragmentation processes in relativistic heavy ion collisions. It is shown that the existing experimental data on heavy quarkonium production at the Large Hadron Collider (LHC) can be described in the framework of the NRQCD formalism, and the phenomenological values of matrix elements for color-singlet and color-octet components give the main contribution. The numerical results …
The process of exclusive electroproduction of vector quarkonium (EEQ), $e p\to epV$, is per se an interesting topic in studies of quarkonium production mechanism, QCD description of diffractive interaction and nucleon structure. We investigate this process in the framework of nonrelativistic QCD and QCD collinear factorization at the next-to-leading order QCD accuracy. The perturbative convergence behavior is discussed in a large range of photon virtuality $Q^2$. The $J/ψ$ large-$Q^2$ electroproduction data at HERA can be well explained, and the $Υ$ differential production rate is predicted. T…
We compute the c \bar{b} spectrum from a first principle Salpeter equation obtained in a preceding paper. For comparison we report also the heavy-light quarkonium spectrum and the hyperfine separations previously presented only in a graphical form. Notice that all results are parameter free.
In this talk, we present the QCD factorization formula for heavy quarkonium production at large $p_T$ with factorized leading-power and next-to-leading power contributions in the $1/p_T$ expansion. We show that the leading order analytical calculations in this QCD factorization approach can reproduce effectively the full next-to-leading order numerical results derived using non-relativistic QCD (NRQCD) factorization formalism. We demonstrate that the next-to-leading power contributions are crucial to the description of the channels that are the most relevant for the rate as well as polarizatio…
We explore polarized heavy quarkonium production using the color evaporation model at leading order. We present the polarized to total yield ratio as a function of center of mass energy and rapidity in $p+p$ collisions. At energies far above the $Q \overline Q$ production threshold, we find charmonium and bottomonium production to be longitudinally polarized ($J_z=0$). The quarkonium states are also longitudinally polarized at central rapidity, becoming transversely polarized ($J_z=\pm1$) at the most forward rapidities.
We use Ward identities of broken scale invariance to infer the amount of scalar glueball--$\bar{q}q$ meson mixing from the ratio of quark and gluon condensates in the QCD vacuum. Assuming dominance by a single scalar state, as suggested by a phase-shift analysis, we find a mixing angle $γ\sim 36^{\circ}$, corresponding to near-maximal mixing of the glueball and $\bar{s}s$ components.
Study of the polarized heavy quarkonium production in recently proposed QCD factorization formalism requires knowledge of a large number of input fragmentation functions (FFs) from a single parton or a heavy quark-antiquark pair to a polarized heavy quarkonium. In this paper, we calculate these FFs at the input scale in terms of nonrelativistic QCD (NRQCD) factorization. We derive all relevant polarized NRQCD long-distance matrix elements based symmetries and propose a self-consistent scheme to define them in arbitrary $d$ dimensions. We then calculate polarized input FFs contributed from all …
Using a specific scheme of the QCD sum rules, one derives the relation for leptonic constants of $nS$-wave heavy qiarkonia, so $f^2_{n_1}/f^2_{n_2}= n_2/n_1$, that does not depend on the quarkonium content, and it is in a good agreement with the experimental values of constants for the $ψ$- and $Υ$-families.
I review heavy quarkonium physics in view of recent experimental results. In particular, I discuss new results on spin singlet states, photon and hadronic transitions, D-states and discovery of yet unexplained narrow X(3872) state.
We study some general aspects of the formalism of potential NRQCD (pNRQCD), an effective field theory that deals with ultrasoft degrees of freedom in Heavy Quarkonium systems. Specific attention is paid to its effective Lagrangian that it is displayed at the present level of accuracy.
There are many outstanding discrepancies comparing the predictions of perturbative QCD and measurements of the rate of production and decay of heavy quark systems. The problems include the $J/ψ\to ρπ$ puzzle, leading charmed particle effects, the anomalous behavior of the heavy quark sea components of structure functions, anomalous nuclear target effects, and the large rates observed for single and double quarkonium production at large $x_F$ and large $p_T$. I argue that these anomalies may be associated with nonperturbative effects in the higher Fock structure of hadron wavefunctions.
In this brief review, I summarize the current theoretical knowledge of heavy quarkonium inclusive decays, with emphasis on recent progress made in the framework of QCD effective field theories. In appendix, I list the imaginary parts of the matching coefficients of the dimension 6 and dimension 8 NRQCD four-fermion operators as presently known.
The nuclear suppression of heavy quarkonium production at low transverse momentum in pA collisions in high energy scatterings is investigated in the small-x factorization formalism. A universal suppression is found in the large Nc limit between the two formalisms to describe the heavy quarkonium production: the non-relativistic QCD (NRQCD) and the color-evaporation model (CEM). This provides an important probe to the saturation momentum at small-x in big nucleus. We also comment on the phenomenological applications of our results.
We review a number of results for the spectrum and inclusive decays of heavy quarkonium systems which can be derived from QCD under well controlled approximations. They essentially follow from the hierarchy of scales in these systems, which can be efficiently exploited using non-relativistic effective field theories. In particular, we discuss under which conditions non-relativistic potential models emerge as effective theories of QCD.
In this article, we study the radiative transitions among the vector and scalar heavy quarkonium states with the covariant light-front quark model. In calculations, we observe that the radiative decay widths are sensitive to the constituent quark masses and the shape parameters of the wave-functions, and reproduce the experimental data with suitable parameters.
By employing the method of differential equations, we compute the various types of two-loop master integrals involved in CP-even heavy quarkonium exclusive production and decays. All the integrals presented in this paper can be casted into canonical forms and expressed in terms of Goncharov polylogarithms and Harmonic polylogarithms. These master integrals are frequently used in the calculation of NNLO corrections of the heavy quarkonium production processes, as $γ^*γ\rightarrow Q\bar{Q}$, $e^+e^-\rightarrow γ+ Q\bar{Q}$, and $H/Z^0\rightarrow γ+ Q\bar{Q}$, and decay processes. They are also a…
We compute corrections of relative order v^4 to the rates for the decays of ^1S_0 heavy quarkonium into two photons and into light hadrons and for the decays of ^3S_1 heavy quarkonium into a lepton pair and into light hadrons. In particular, we compute the coefficients of the decay operators that have the same quantum numbers as the heavy quarkonium. We also confirm previous calculations of the order-v^2 corrections to these rates. We find that the v expansion converges well for the decays of ^1S_0 heavy quarkonium and for the decay of ^3S_1 heavy quarkonium into a lepton pair. Large higher-or…
Soft-gluon resummation within the framework of heavy quarkonium hadroproduction is considered. A few selected cases are studied in detail. A sizeable increase of the cross sections with respect to the next-to-leading order predictions with central factorization/renormalization scale choice can generally be observed. Improvements in the dependence of the cross sections on the two scales, especially when they are kept equal, are also found.
In this paper we solved Schrödinger equation with Song-Lin potential by using Asymptotic Iteration Method (AIM). We obtained spin-averaged energy levels of charmonium and bottomonium via AIM. Obtained results agree well with available experimental data and other theoretical studies.
We study radiative decays of heavy $Q{\bar Q}$ states, both for $Q=c$ and $Q=b$, using an effective Lagrangian approach which exploits spin symmetry for such states. We use existing data on radiative quarkonium transitions to predict some unmeasured decay rates. We also discuss how these modes can be useful to understand the structure of X(3872).
Encouraged by recent developments in AdS/QCD models for light quark system, we study heavy quarkonium in the framework of the AdS/QCD models. We calculate the masses of $c\bar c$ vector meson states using the AdS/QCD models at zero and at finite temperature. Among the models adopted in this work, we find that the soft wall model describes the low-lying heavy quark meson states at zero temperature relatively well. At finite temperature, we observe that once the bound state is above $T_c$, its mass will increase with temperature until it dissociates at a temperature of around $494 {\rm MeV}$. It…
Quarkonium suppression in heavy ion collisions is a potential signature of the formation of the quark-gluon plasma. After a very brief review of the J/psi result at CERN, we restrict our discussion to the effects of the high-energy multiple scattering of the quark pair in the colliding nuclei.
