We compute the color-singlet and color-octet nonrelativistic QCD (NRQCD) long-distance matrix elements for inclusive production of $P$-wave quarkonia in the framework of potential NRQCD. In this way, the color-octet NRQCD long-distance matrix element can be determined without relying on measured cross section data, which has not been possible so far. We obtain inclusive cross sections of $χ_{cJ}$ and $χ_{bJ}$ at the LHC, which are in good agreement with data. In principle, the formalism developed in this Letter can be applied to all inclusive production processes of heavy quarkonia.
We present a new approach for the prompt production of quarkonia which is based on the $k_T$-factorization method. The production of even C-parity quarkonia proceeds via the fusion of two (off-shell) gluons. Especially in the kinematics of the LHCb experiment these processes are thus expected to be a sensitive probe of the small-$x$ gluon distribution. We calculate the relevant off-shell matrix elements in terms of the light-front wave functions of the quarkonium states. We present our results for scalar and pseudoscalar charmonia and discuss photon transition form factors as well as cross sec…
Based on the potential nonrelativistic QCD formalism, we compute the nonrelativistic QCD long-distance matrix elements (LDMEs) for inclusive production of $S$-wave heavy quarkonia. This greatly reduces the number of nonperturbative unknowns and brings in a substantial enhancement in the predictive power of the nonrelativistic QCD factorization formalism. We obtain improved determinations of the LDMEs and find cross sections and polarizations of $J/ψ$, $ψ(2S)$, and excited $Υ$ states that agree well with LHC data. Our results may have important implications in pinning down the heavy quarkonium …
We compute NRQCD long-distance matrix elements that appear in the inclusive production cross sections of $P$-wave heavy quarkonia in the framework of potential NRQCD. The formalism developed in this work applies to strongly coupled charmonia and bottomonia. This makes possible the determination of color-octet NRQCD long-distance matrix elements without relying on measured cross section data, which has not been possible so far. We obtain results for inclusive production cross sections of $χ_{cJ}$ and $χ_{bJ}$ at the LHC, which are in good agreement with measurements.
In this work, we present a thorough analysis of scalar $P$-wave $χ_{Q0}$, $Q=c,b$ quarkonia electromagnetic form factors for the $γ^* γ^* \to χ_{Q0}$ couplings, as well as their hadroproduction observables in $k_\perp$-factorisation using the light-front potential approach for the quarkonium wave function. The electromagnetic form factors are presented as functions of photon virtualities. We discuss the role of the Melosh spin-rotation and relativistic corrections estimated by comparing our results with those in the standard nonrelativistic QCD (NRQCD) approach. Theoretical uncertainties of ou…
We develop a formalism for computing inclusive production cross sections of heavy quarkonia based on the nonrelativistic QCD and the potential nonrelativistic QCD effective field theories. Our formalism applies to strongly coupled quarkonia, which include excited charmonium and bottomonium states. Analogously to heavy quarkonium decay processes, we express nonrelativistic QCD long-distance matrix elements in terms of quarkonium wavefunctions at the origin and universal gluonic correlators. Our expressions for the long-distance matrix elements are valid up to corrections of order $1/N_c^2$. The…
Working in the nonrelativistic-QCD factorization framework at next-to-leading order in $α_s$, we fit the relevant color octet (CO) long-distance matrix elements (LDMEs) of the $ψ(2S)$ meson, $\langle {\cal O}^{ψ(2S)}({^1S}_0^{[8]})\rangle$, $\langle {\cal O}^{ψ(2S)}({^3S}_1^{[8]})\rangle$, and $\langle {\cal O}^{ψ(2S)}({^3P}_0^{[8]})\rangle$, to 1001 data points of upolarized and polarized $ψ(2S)$ inclusive hadroproduction. We do four different fits, with filters on polarization and low to middle transverse momentum $p_T$. We find that a successful description of the data is only possible with…
We perform a complete study of prompt double $J/ψ$ hadroproduction at leading order in the nonrelativistic-QCD factorization framework by including all possible pairings of the $c\bar{c}$ Fock states ${}^1\!S_0^{[8]}$, ${}^3\!S_1^{[1,8]}$, and ${}^3\!P_J^{[1,8]}$ with $J=0,1,2$. We find that the ${}^1\!S_0^{[8]}$ and ${}^3\!P_J^{[8]}$ channels of $J/ψ$ and $ψ^\prime$ production and the ${}^3\!P_J^{[1]}$ and ${}^3\!S_1^{[8]}$ channels of $χ_{cJ}$ production, which have been overlooked so far, greatly dominate at large invariant masses and rapidity separations of the $J/ψ$ pair, and that their i…
Recent results on the hadroproduction of charm and beauty are discussed in the context of the current theoretical and experimental status. The areas covered include production of open charm and beauty particles and charm and beauty quarkonia. Experimental results from both fixed target and colliding beam experiments are presented and compared to theoretical predictions. Predictions based on QCD perturbation theory are generally successful in explaining the shapes of differential cross sections but underestimate the scale of the cross section by factors of 2-3. An exception is production of the…
Experimental data for $b\bar{b}$ quarkonia have been compared with the predictions of a variety of nonrelativistic quark models. It is found that a potential $a\sqrt{r} - b/r +$ const gives good agreement, while many others do not. Some implications of this observation are discussed.
We discuss the various mechanisms potentially at work in hadroproduction of heavy quarkonia in the light of computations of higher-order QCD corrections both in the Colour-Singlet (CS) and Colour-Octet (CO) channels and the inclusion of the contribution arising from the s-channel cut in the CS channel. We also discuss new observables meant to better discriminate between these different mechanisms.
The temperature dependent energy, mass and some of thermodynamic quantities of charmonium and bottomonium have been calculated by solving the radial Schr{ö}dinger equation with the extended Cornell potential at finite temperature using the Nikiforov-Uvarov method. Obtained results accord with experimental data and theoretical results of the previous studies. We have found the spectrum of quarkonia mass strictly decreases at high values of temperature.
A systematic study of a wide class of nonrelativistic models of $b\overline{b}$ quarkonia is described. It is found that the potential $V(r) = 0.706380(\sqrt{r} - \frac{0.460442}{r}) + 8.81715$ (all in GeV) with the $b$-quark mass $m_b = 4.80303$ GeV gives a satisfactory description of the experimental data below the threshold for strong decays ($ χ^2/DF = 6.5/7$). Limitations and implications of this observation are discussed.
We systematically calculate the relativistic corrections to prompt $J/ψ$ photoproduction and hadroproduction using the factorization formalism of nonrelativistic QCD. Specifically, we include the ${}^3\!S_1^{[1]}$ and ${}^3\!P_{J}^{[1]}$ color-singlet and the ${}^3\!S_1^{[8]}$, ${}^1\!S_{0}^{[8]}$, and ${}^3\!P_{J}^{[8]}$ color-octet channels as well as the effects due to the mixing between the ${}^3\!S_1^{[8]}$ and ${}^3\!D_1^{[8]}$ channels. We provide all the squared hard-scattering amplitudes in analytic form. Assuming the nonrelativistic-QCD long-distance matrix elements to satisfy the ve…
Hadroproductions of charmonium and bottomonium $χ_J$ in polarized $pp$ collisions at RHIC energies are studied. Two--spin asymmetries $A_{LL}^{χ_J}(pp)$ for these reactions are very good parameters for examining the polarized gluon distributions in a proton.
We extend our previous analysis on inclusive heavy quarkonia hadroproduction to the whole Upsilon(nS) (n=1,2,3) resonance family. We use a Monte Carlo framework with the colour-octet mechanism implemented in the PYTHIA event generator. We include in our study higher order QCD effects such as initial-state emission of gluons and Altarelli-Parisi evolution of final-state gluons. We extract some NRQCD colour-octet matrix elements relevant for Upsilon(nS) (n=1,2,3) hadroproduction from CDF data at the Fermilab Tevatron. Then we extrapolate to LHC energies to predict prompt bottomonia production ra…
Color-octet contributions to the associate J/psi+gamma hadroproduction are studied in detail, and found to be negligible, compared to the ordinary color-singlet contribution. Within the color-singlet model the J/psi+gamma production in the leading order is possible only through gluon-gluon fusion process. Therefore, the associate J/psi+gamma hadroproduction remains to be useful as a clean channel to probe the gluon distribution inside proton, to study heavy quarkonia production mechanism, and to find proton's spin structure.
We systematically calculate the relativistic corrections to the polarization variables of prompt $J/ψ$ photoproduction and hadroproduction using the factorization formalism of nonrelativistic QCD. Specifically, we include the ${}^3\!S_1^{[1]}$ and ${}^3\!P_{J}^{[1]}$ color-singlet and the ${}^3\!S_1^{[8]}$, ${}^1\!S_{0}^{[8]}$, and ${}^3\!P_{J}^{[8]}$ color-octet channels as well as the effects due to the mixing between the ${}^3\!S_1^{[8]}$ and ${}^3\!D_1^{[8]}$ channels. We provide all the squared hard-scattering amplitudes in analytic form. Assuming the nonrelativistic-QCD long-distance mat…
Structure of heavy quarkonia in a strong magnetic field
Advanced Science Research Center, Japan Atomic Energy Agency（日本原子力机构先进科学研究中心）；Research Center for Nuclear Physics, The University of Osaka（大阪大学核物理研究中心）
We study the radiative (E1 and M1) decays of P-wave quarkonia in a strong magnetic field based on the Lagrangian of potential nonrelativistic QCD. To investigate their properties, we implement a polarized wave function basis justified in the Paschen-Back limit. In a magnetic field stronger than the spin-orbit coupling, the wave functions of the P-wave quarkonia are drastically deformed by the Hadronic Paschen-Back effect. Such deformation leads to the anisotropy of the direction of decays from the P-wave quarkonia. The analytic formulas for the radiative decay widths in the nonrelativistic lim…
We consider a general scheme for calculating the leptonic constant of heavy quarkonium QQ-bar in the framework of nonrelativistic quantum chromodynamics, NRQCD, operating as the effective theory of nonrelativistic heavy quarks. We explore the approach of static potential in QCD, which takes into account both the evolution of effective charge in the three-loop approximation and the linearly raising potential term, which provides the quark confinement. The leptonic constants of bb-bar and cc-bar systems are evaluated by making use of two-loop anomalous dimension for the current of nonrelativisti…
Using the quark-antiquark interactions obtained in the framework of the bootstrap method we construct a potential model, investigate the possibility of describing of heavy quarkonia and calculate the bottomonium spectrum. The potential of the interaction was obtained as a nonrelativistic limit of the relativistic quark-antiquark amplitudes Q{\bar Q} -> Q\bar Q.
In the limit of heavy quark mass, the production cross section and polarization of quarkonia can be calculated in perturbative QCD. We study the $p_\perp$-averaged production of charmonium states in $πN$ collisions at fixed target energies. The data on the relative production rates of $\jp$ and $χ_J$ is found to disagree with leading twist QCD. The polarization of the $\jp$ indicates that the discrepancy is not due to poorly known parton distributions nor to the size of higher order effects ($K$-factors). Rather, the disagreement suggests important higher twist corrections, as has been surmise…
recently proposed strictly phenomenological static quark-antiquark potential belonging to the generality $V(r)=-Ar^{-α}+κr^β+V_{0}$ is tested with heavy quarkonia in the context of the shifted large N-expansion method. This nonrelativistic potential model fits the spin-averaged mass spectra of the $c\bar{c},$ $b\bar{b}$ and $c% \bar{b}$ quarkonia within a few ${\rm MeV}$ and also the five experimentally known leptonic decay widths of the $c\bar{c}$ and $b% \bar{b}$ vector states. Further, we compute the hyperfine splittings of the bottomonium spectrum as well as the fine and hyperfine splittin…
In this work, we present a thorough analysis of $η_c(1S,2S)$ quarkonia hadroproduction in $k_\perp$-factorisation in the framework of the light-front potential approach for the quarkonium wave function. The off-shell matrix elements for the $g^* g^* η_c(1S,2S)$ vertices are derived. We discuss the importance of taking into account the gluon virtualities. %We discuss the role of relativistic corrections, the Melosh spin-rotation and theoretical uncertainties by comparing our results with the standard NRQCD approach. We present the transverse momentum distributions of $η_c$ for several models of…
The fragmentation functions D_{a -> H}(x,mu^2) of a heavy hadron H, with mass m_H, satisfy the phase-space constraint D_{a - > H}(x,mu^2)=0 for x < m_H^2/mu^2, which is violated by the naive mu^2 evolution equations. Using appropriately generalized mu^2 evolution equations, we reconsider the inclusive hadroproduction of prompt J/psi mesons with high transverse momenta in the framework of the factorization formalism of nonrelativistic quantum chromodynamics, and determine the resulting shifts in the values of the leading colour-octet matrix elements, which are fitted to data from the Fermilab T…
In the fractional nonrelativistic potential model, the decomposition of heavy quarkonium in a hot magnetized medium is investigated. The analytical solution of the fractional radial Schrodinger equation for the hot-magnetized interaction potential is displayed by using the conformable fractional Nikiforov-Uvarov method. Analytical expressions for the energy eigenvalues and the radial wave function are obtained for arbitrary quantum numbers. Next, we study the charmonium and bottmonium binding energies for different magnetic field values in the thermal medium. The effect of the fractional param…
Two complementary approaches to the theory of heavy quarkonia are discussed. The nonrelativistic potential models give amazingly accurate predictions, but lack a theoretical justification. The expansion in powers of $v/c$ is theoretically very acceptable, but is not as good in giving numerical predictions. The importance of combining these two approaches is stressed.
Applying the nonrelativistic quantum chromodynamics factorization formalism to the $Υ(1S,2S,3S)$ hadroproduction, a complete analysis on the polarization parameters $λ_θ$, $λ_{θϕ}$, $λ_ϕ$ for the production are presented at QCD next-to-leading order. With the long-distance matrix elements extracted from experimental data for the production rate and polarization parameter $λ_θ$ of $Υ$ hadroproduction, our results provide a good description for the measured parameters $λ_{θϕ}$ and $λ_ϕ$ in both the helicity and the Collins-Soper frames. In our calculations the frame invariant parameter $\tildeλ$…
I review recent applications of the open quantum system framework in the understanding of quarkonium suppression in heavy-ion collisions, which has been used as a probe of the quark-gluon plasma for decades. The derivation of the Lindblad equations for quarkonium in both the quantum Brownian motion and the quantum optical limits and their semiclassical counterparts is explained. The hierarchy of time scales assumed in the derivation is justified from the separation of energy scales in nonrelativistic effective field theories of QCD. Physical implications of the open quantum system approach are…
We study inclusive processes involving two heavy quarkonia in nonrelativisitic QCD (NRQCD) and demonstrate that, in the presence of two P-wave Fock states, NRQCD factorization breaks down, leaving uncanceled infrared singularities. As phenomenologically important examples, we consider the decay $Υ\to χ_{cJ}+X$ via $b\bar{b}({}^3P_{J_b}^{[8]})\to c\bar{c}({}^3P_J^{[1]})+gg$ and the production process $e^+e^-\to J/ψ+χ_{cJ}+X$ via $e^{+}e^{-}\to c\bar{c}({}^3P_{J_1}^{[8]})+c\bar{c}({}^3P_J^{[1]})+g$. We infer that such singularities will appear for double quarkonium hadroproduction at next-to-lea…
We present the first complete study of $Υ$ and prompt $J/ψ$ production from single-parton scattering, including the complete $\mathcal{O}(α_S^6)$ color-singlet contribution, the $\mathcal{O}(α_S^2α^2)$ electroweak contribution, the complete nonrelativistic S-wave and P-wave color-octet contribution as well as the feeddown contribution. Our study was motivated by the recent evidence reported by D0 Collaboration of prompt $J/ψ$ and $Υ$ simultaneous production at the Tevatron. With our complete evaluation, we are able to refine the determination of the double parton scattering contribution made b…
A model based on a nonrelativistic potential supplemented by spin dependent terms of the Breit-Fermi type is described. This model reproduces the known masses of the $\bar{b}b$ quarkonia within 1MeV and the known masses of the $\bar{c}c$ quarkonia within 5MeV. Many predictions are made, some of them, e.g. for the fine splittings of the high $L$ states of the $\bar{b}b$ quarkonia and of the mass spectrum of the $\bar{b}c$ quarkonia, significantly different from previous predictions made by other authors.
Exclusive photoproduction of heavy quarkonia at HERA is analyzed in the framework of the Regge-eikonal approach together with the nonrelativistic bound state formalism. Total and differential cross-sections for the process $γ+p\to (Q\bar{Q})_{1S}+p$ are calculated. The model predicts cross-sections of Exclusive Double Diffractive Events (EDDE) at TeVatron and LHC.
In the framework of QCD sum rules one uses a scheme, allowing one to apply the conditions of both nonrelativistic heavy quark motion inside mesons and independence of nonsplitting nS-state density on the heavy quark flavours. In the leading order an analitic expression is derived for leptonic constants of both heavy quarkonia and heavy mesons with a single heavy quark. The expression allows one explicitly to determine scaling properties of the constants.
Within the framework of nonrelativistic quark-antiquark Cornell potential model formalism, we study the annihilation of heavy quarkonia. We determine their annihilation widths resulting into digluon, dilepton, $3γ$, $3g$ and $γgg$ and compare our findings with the available theoretical results and experimental data. We also provide the charge radii and absolute square of radial Schrödinger wave function at zero quark-antiquark separation for heavy quarkonia and $B_c$ mesons.
We update the study of the QCD corrections to direct J/psi and Upsilon hadroproduction in association with a photon in the QCD-based approach of the Colour-Singlet (CS) Model. After comparison with the recent full next-to-leading-order (NLO) computation for this process, we provide an independent confirmation to the inclusive case that NLO QCD corrections to quarkonium-production processes whose LO exhibits a non-leading P_T behaviour can be reliably computed at mid and large P_T by considering only the real emission contributions accompanied with a kinematical cut. In turn, we evaluate the le…
Production of heavy quark antiquark systems in high energy heavy ion collisions must involve relativistic momentum components in a quantum mechanical approach. If the color forces are screened in a deconfining medium, one can define the analog of a formation or separation time by an overlap integral in the nonrelativistic bound state rest frame. This time parameter has some interesting properties which depend on the momentum spectrum of the initial quarks. Consequences of these properties for the phenomenology of deconfinement signals are discussed.
A nonrelativistic Hamiltonian with plausible spin dependent corrections is proposed for the quarkonia below their respective strong decay thresholds. With only six free parameters this model reproduces the nine known masses of the bottomonia within about 1 MeV, the six known masses of the charmonia within a few MeV and the five known leptonic decay widths of the ${}^3S_1$ states within about {20 %}. The model is then used to predict the masses of the remaining 43 quarkonia (some of them for the first time) and of the leptonic decay widths of the two ${}^1S_0(\bar{b}c)$ states. Comparison with …
The heavy quarkonia (Charmonium \(c\bar{c}\) and Bottomonium \(b\bar{b}\)) are investigated in the framework of the instantaneous BS-equation (Salpeter equation). We parametrize confinement alternatively by a linearly rising scalar or a vector interaction kernel and take into account the one-gluon-exchange (OGE) interaction in the instantaneous approximation. Mass spectra as well as leptonic, two-photon, E1 and M1 decay widths are calculated. Our results show that a reasonable description of the experimental data can be obtained with both spin structures for the confining kernel. The relativis…
The energy dependence of the total hadroproduction cross section of pseudo-scalar quarkonia is computed via matching Next-to-Leading Order (NLO) Collinear-Factorisation (CF) results with resummed higher-order corrections, proportional to $α_s^{n}\ln^{n-1}(1/z)$, to the CF hard-scattering coefficient, where $z=M^2/\hat{s}$ with $M$ and $\hat{s}$ being the quarkonium mass and the partonic center-of-mass energy squared. The resummation is performed using High-Energy Factorisation (HEF) in the Doubly-Logarithmic (DL) approximation, which is a subset of the leading logarithmic $\ln (1/z)$ approxima…
We compute derived quantities for various values of the model parameter of the Cornell potential model for the S-wave heavy quarkonia with radial quantum numbers n=1, 2, and 3. Our results can be used to determine leading and relative-order-v^2 nonrelativistic quantum chromodynamics matrix elements for S-wave charmonia and bottomonia such as psi(2S), eta_c(2S), and Upsilon(nS) for n=1, 2, and 3. These matrix elements will be essential ingredients for resumming relativistic corrections to processes involving those S-wave heavy quarkonium states.
We study the Single-Parton-Scattering (SPS) production of double quarkonia (J/psi+J/psi, J/psi+Upsilon, and Upsilon+Upsilon) in pp and pp(bar) collisions at the LHC and the Tevatron as measured by the CMS, ATLAS, LHCb, and D0 experiments in the Colour-Evaporation Model (CEM), based on the quark-hadron-duality, including Next-to-Leading Order (NLO) QCD corrections up to alpha_s^5. To do so, we also perform the first true NLO --up to alpha_s^4-- study of the p_T-differential cross section for single-quarkonium production. This allows us to fix the non-perturbative CEM parameters at NLO accuracy …
The hadroproduction of the radially excited heavy-quarkonium states $ψ(2S)$ and $Υ(3S)$ at high energies is studied in the parton reggeization approach and the factorization formalism of nonrelativistic QCD at lowest order in the strong-coupling constant $α_s$ and the relative heavy-quark velocity $v$. A satisfactory description of the $ψ(2S)$ transverse-momentum ($p_T$) distributions measured by ATLAS, CMS, and LHCb at center-of-mass energy $\sqrt{S}=7$ TeV is obtained using the color-octet long-distance matrix elements (LDMEs) extracted from CDF data at $\sqrt{S}=1.96$ TeV. The importance of…
We study the polarization observables of J/psi hadroproduction at next-to-leading order within the factorization formalism of nonrelativistic quantum chromodynamics. We complete the present knowledge of the relativistic corrections by also providing the contribution due to the intermediate ^3P_J^{[8]} color-octet states at this order, which turns out to be quite significant. Exploiting the color-octet long-distance matrix elements previously extracted through a global fit to experimental data of unpolarized J/psi production, we provide theoretical predictions in the helicity and Collins-Soper …
We investigate the associated production of heavy quarkonia, with angular-momentum quantum numbers ^{2S+1}L_J = ^1S_0, ^3S_1, ^1P_1, ^3P_J (J = 0, 1, 2), and photons, Z bosons, and W bosons in photon-photon, photon-hadron, and hadron-hadron collisions within the factorization formalism of nonrelativistic quantum chromodynamics providing all contributing partonic cross sections in analytic form. In the case of photoproduction, we also include the resolved-photon contributions. We present numerical results for the processes involving J/psi and chi_{cJ} mesons appropriate for the Fermilab Tevatro…
Heavy quarkonium production is a powerful implement to study the strong interaction dynamics and QCD theory. Fragmentation is the dominant production mechanism for heavy quarkonia with large transverse momentum. With the large heavy quark mass, the relative motion of the heavy quark pair inside a heavy quarkonium is effectively nonrelativistic and it is also well-known that their fragmentation functions can be calculated in the perturbative QCD framework. Here, we analytically calculate the process-independent fragmentation functions for a gluon to split into the spin-singlet and -triplet $S$-…
Quenched lattice data for the quark-antiquark interaction (in terms of heavy quark free energies) in the color singlet channel at finite temperatures are fitted and used within the nonrelativistic Schroedinger equation formalism to obtain binding energies and scattering phase shifts for the lowest eigenstates in the charmonium and bottomonium systems in a hot gluon plasma. The partial dissociation rate due to the Bhanot-Peskin process is calculated using different assumptions for the gluon distribution function, including free massless gluons, massive gluons, and massive damped gluons. It is d…
We investigate the implications of spin symmetry for heavy quarkonia production and decay. We first compare spin symmetry predictions for charmonia and bottomonia radiative transitions with data and find they agree quite well. We next use spin symmetry along with nonrelativistic QCD power counting to determine the leading order alignment of P and D-wave orthoquarkonia that are produced through gluon fragmentation. Finally, we discuss a mechanism which may resolve the current factor of 30 discrepancy between theoretical predictions and experimental measurements of prompt $ψ'$ production at the …
We present a rigorous next-to-leading order analysis of J/psi yield and polarization within the factorization theorem of nonrelativistic QCD (NRQCD). To the orders considered, this framework depends on three free parameters, the color-octet long-distance matrix elements. We extract their values in a global fit to inclusive J/psi production data from various hadroproduction, photoproduction, two-photon scattering and electron-positron annihilation experiments. We show that this fit is constrained and stable and describes all data sufficiently well. We then make predictions for J/psi polarizatio…
In this work, we calculate the nonrelativistic asymptotic behavior of the amplitudes of $q\overline{q}\rightarrow 2g \rightarrow q\overline{q}$ in the leading order of $α_s$ (LO-$α_s$) with $q\overline{q}$ in the $^3P_J$ channels. In the practical calculation we take the momenta of quarks and antiquarks on-shell and expand the amplitudes on the three-momentum of the quarks and antiquarks to order 6 and get three nonzero terms. The imaginary parts of the first term and the second term are the old. The real parts of the results have IR divergence. When applying the results to the heavy quarkonia…
The mass spectra of charmonia, bottomonia and B_c mesons are calculated in the framework of the QCD-motivated relativistic quark model based on the quasipotential approach. The dynamics of heavy quarks and antiquarks is treated fully relativistically without application of the nonrelativistic v^2/c^2 expansion. The known one-loop radiative corrections to the heavy quark potential are taken into account perturbatively. The heavy quarkonium masses are calculated up to rather high orbital and radial excitations (L=5, n_r=5). On this basis the Regge trajectories are constructed both in the total a…
We report on recent progress in testing the factorization formalism of nonrelativistic quantum chromodynamics (NRQCD) at next-to-leading order (NLO) for $J/ψ$ yield and polarization. We demonstrate that it is possible to unambiguously determine the leading color-octet long-distance matrix elements (LDMEs) in compliance with the velocity scaling rules through a global fit to experimental data of unpolarized $J/ψ$ production in $pp$, $p\bar{p}$, $ep$, $γγ$, and $e^+e^-$ collisions.Three data sets not included in the fit, from hadroproduction and from photoproduction in the fixed-target and colli…
