We study the bottomonium spectrum using a relativistic potential model in the momentum space. This model is based on a complete one gluon exchange interaction with a momentum dependent screening factor to account for the effects due to virtual pair creation that appear close to the decay thresholds. The overall model does not make use of nonrelativistic approximations. We fit well established bottomonium states below the open charm threshold and predict the rest of the spectrum up to $\approx 11200$ MeV and $J^{PC}=3^{--}$. Uncertainties are treated rigorously and propagated in full to the parameters of the model using a Monte Carlo to identify if which deviations from experimental data can be absorbed into the statistical uncertainties of the models and which can be related to physics beyond the $b\bar{b}$ picture, guiding future research. We get a good description of the spectrum, in particular the Belle measurement of the $η_b(2S)$ state and the $Υ(10860)$ and $χ_b(3P)$ resonances.
We propose the study of unpolarized transverse momentum dependent gluon parton distributions as well as the effect of linearly polarized gluons on transverse momentum and rapidity distributions of $J/ψ$ and $Υ$ production within the framework of transverse momentum dependent factorization employing color evaporation model (CEM) in unpolarized proton-proton collision. We estimate the transverse momentum and rapidity distributions of $J/ψ$ and $Υ$ at LHCb, RHIC and AFTER energies using TMD evolution formalism.
Since effective potentials derived from Feynman diagrams are naturally given in momentum space, we formulate the non-relativistic Coulomb problem entirely in momentum representation. We give momentum wave functions for all quantum numbers in one-dimensional integrals, even though they can be evaluated. Angular momentum decomposed Green's functions are then compactly represented. We apply this formalism to investigate the next to next leading order charm effects on 1S bottomonium level shift. Our one insertion results are given completely in analytic form and numerically agree with previous res…
We study the possibility to probe the unpolarized and linearly polarized transverse momentum- dependent gluon distributions in unpolarized pp collision in charmonium and bottomonium produc- tion, employing non-relativistic QCD (NRQCD) based color octet model within transverse momentum dependent (TMD) factorization framework. The transverse momentum (p T ) and rapidity distributions of J/ψ and Υ(1S) at LHCb, RHIC and AFTER energies are estimated. Significant modulations in transverse momentum spectrum of quarkonium in the low p T region is obtained when contributions of linearly polarized gluon…
On a cubic lattice the zero-momentum meson states have one of 20 possible Lambda^{PC} combinations where Lambda labels the irreducible representation of the octahedral group. Each continuum bottomonium state with specific J^{PC} quantum numbers is contained within one or more of the lattice Lambda^{PC} states. In this work, bottomonium quark-antiquark operators are constructed for all 20 lattice Lambda^{PC} combinations which allows many continuum high angular momentum states to be accessible as ground states of their associated lattice channels. From a dynamical simulation, realistic results …
Recently it has been put forward that linearly polarized gluons inside unpolarized protons affect the transverse momentum distribution of final state particles in hadronic collisions. They lead to a characteristic modulation of the differential cross section in Higgs production and to azimuthal asymmetries in, for instance, heavy quark pair production. Here we study the effect on charmonium and bottomonium production in hadronic collisions, such as at LHCb and at the proposed fixed target experiment AFTER at LHC. We focus mainly on the scalar and pseudoscalar quarkonia, eta_c, chi_{c0}, eta_b,…
Using Non-Relativistic QCD (NRQCD), we study heavy quark mass dependence of S-wave and P-wave bottomonium correlators for 0.42Tc <= T <= 2.09Tc and study spectral functions of S-wave bottomonium states moving in a thermal bath at these temperatures using Maximum Entropy Method with NRQCD kernel. For the studied momentum range, the energy of moving states shows quadratic momentum-dependence and the width of moving states does not show significant changes as the momentum of bottomonium is increased. Also, we find that in correlator ratios, the temperature effect is larger than the effect caused …
In this article, we investigate the structures of the pseudoscalar charmonium and bottomonium in the framework of the coupled rainbow Schwinger-Dyson equation and ladder Bethe-Salpeter equation with the confining effective potential (infrared modified flat bottom potential). As the current masses are very large, the dressing or renormalization for the $c$ and $b$ quarks are tender, however, mass poles in the timelike region are absent. The Euclidean time fourier transformed quark propagator has no mass poles in the timelike region which naturally implements confinement. The Bethe-Salpeter wave…
The coulomb plus linear (Cornell) potential is used to investigate the mass spectrum of bottomonium. Gaussian wave function is used in position and momentum space to estimate values of potential and kinetic energies, respectively. Based on our calculations, we study newly observed ${{\mathit Υ}{(10860)}}$ as an admixture of $4^{3}S_{1}$ with $4^{3}D_{1}$, and ${{\mathit Υ}{(10753)}}$ as an admixture of $6^{3}S_{1}$ with $4^{3}D_{1}$ also, we try to assign ${{\boldsymbol Υ}{(11020)}}$ as a pure $4^{3}D_{1}$ bottomonium state. We also study the Regge trajectories in the $(J,M^{2})$ and $(n_r,M^{…
The recently obtained solutions of Dirac equation in the confining SU(3)-Yang-Mills field in Minkowski spacetime are applied to describe the energy spectra of quarkonia (charmonium and bottomonium). The nonrelativistic limit is considered for the relativistic effects to be estimated in a self-consistent way and it is shown that the given effects are extremely important for both the energy spectra and the confinement mechanism.
We had been gradually working towards building a comprehensive quarkonia suppression formalism to explain all 3 dependencies of quarkonium suppression obtained from heavy-ion collision experiments. We present here the improved version of quarkonia suppression framework. It assumes bottomonia produced in the early stage which dissociates due to color screening, gluonic dissociation, and collisional damping in addition to the shadowing as an initial state effect. The QGP medium formed in the collisions is assumed to evolve under ($3+1$)-dimensional relativistic viscous hydrodynamics which is mod…
We report predictions for the suppression and elliptic flow of the $Υ(1S)$, $Υ(2S)$, and $Υ(3S)$ as a function of centrality and transverse momentum in ultra-relativistic heavy-ion collisions. We obtain our predictions by numerically solving a Lindblad equation for the evolution of the heavy-quarkonium reduced density matrix derived using potential nonrelativistic QCD and the formalism of open quantum systems. To numerically solve the Lindblad equation, we make use of a stochastic unraveling called the quantum trajectories algorithm. This unraveling allows us to solve the Lindblad evolution eq…
In this work we calculate the mass spectrum, weak decay constants, two photon decay widths, and two gluon decay widths of ground (1S), and radially excited (2S,3S,...) states of pseudoscalar charmoniuum and bottomonium such as $η_{c}$ and $η_{b}$, as well as the mass spectrum and leptonic decay constants of ground state (1S), excited (2S,1D,3S,2D,4S,and 3D) states of vector charmonium and bottomonium such as $J/ψ$, and $Υ$, using the formulation of Bethe-Salpeter equation under covariant Instantaneous Ansatz (CIA). Our results are in good agreement with data (where ever available) and other mo…
We study the temperature dependence of bottomonium for temperatures in the range 0.4 Tc < T < 2.1 Tc, using non-relativistic dynamics for the bottom quark and full relativistic lattice QCD simulations for Nf=2 light flavors. We consider the behaviour of the correlators in Euclidean space, we analyze the associated spectral functions and we study the dependence on the momentum. Our results are amenable to a successful comparison with effective field theories. They help build a coherent picture of the behaviour of bottomonium in the plasma, consistent which the current LHC results.
Based on the non-relativistic QCD factorization formalism, we calculate the bottomonium ground state, $η_b$, inclusive charm decays at the leading order in the strong coupling constant $α_s$ and quarkonium internal relative velocity $v$. The inclusive charm pair production in $η_b$ decay is mainly realized through $η_b \to c \bar{c} g$ process, where the charm and anti-charm quarks then dominantly hadronize into charmed hadrons. The momentum distributions of the final states are presented. In this work, we also calculate the $J/ψ$ inclusive production rate in the $η_b$ decays, where the color-…
We discuss, how to study $I = 0$ quarkonium resonances decaying into pairs of heavy-light mesons using static potentials from lattice QCD. These static potentials can be obtained from a set of correlation functions containing both static and light quarks. As a proof of concept we focus on bottomonium with relative orbital angular momentum $L = 0$ of the $\bar{b} b$ pair corresponding to $J^{P C} = 0^{- +}$ and $J^{P C} = 1^{- -}$. We use static potentials from an existing lattice QCD string breaking study and compute phase shifts and $\mbox{T}$ matrix poles for the lightest heavy-light meson-m…
I review recent calculations of the suppression of bottomonium states in heavy ion collisions. A non-relativistic potential is used which is complex valued. This allows one to extract the binding energies and decay widths of the ground and excited states of bottomonium as a function of the typical plasma particle momentum and momentum-space anisotropy. The decay widths determined are used as input and integrated over space-time taking into account the dynamical evolution of the typical particle momentum and momentum-space anisotropy. The suppression of Upsilon(1s), Upsilon(2s), Upsilon(3s), ch…
The Regge behaviour of the solutions of a Dirac hamiltonian describing a heavy quark-light quark system in high orbital angular momentum states is analyzed. It is found that the solutions of a scalar confining potential are physically admissible while those of a vector confining potential are not. It is argued that with a Dirac hamiltonian a scalar confining potential is preferred over a vector confining potential for any value of the orbital angular momentum.
Wavefunctions of a relativistic heavy quark-light quark $(Q,q)$ system described by a Dirac hamiltonian are analyzed. By assuming that the confinement potential is a Lorentz scalar (S), the slope of the Isgur-Wise function is calculated at zero recoil point. The result obtained is $ξ^\prime(1)=-0.93\pm 0.05.$ This means that the solutions are perfectly consistent. If relativistic corrections in the light quark wave functions are included the result is $ξ^\prime(1)=-1.01\pm 0.04.$ From heavy-light data this suggest that if relativistic effects are considered, scalar confinement is reliable in l…
The low-energy QCD predictions to be tested by the DIRAC experiment are revised. The experimental method, the setup characteristics and capabilities, along with first experimental results are reported. Preliminary analysis shows good detector performance: alignment error via $Λ$ mass measurement $m_Λ= 1115.6 MeV/c^2$ with $σ= 0.92 MeV/c^2$, $p π^-$ relative momentum resolution $σ_Q \approx 2.7 MeV/c$, and evidence for $π^+ π^-$ low momentum Coulomb correlation.
We compute the momentum-transfer dependence of the proton Pauli form factor $F_{2}$ in the endpoint overlap model. We find the model correctly reproduces the scaling of the ratio of $F_{2}$ with the Dirac Form factor $F_{1}$ observed at the Jefferson Laboratory. The calculation uses the leading-power, leading twist Dirac structure of the quark light-cone wave function, and the same endpoint dependence previously determined from the Dirac form factor $F_{1}$. There are no parameters and no adjustable functions in the endpoint model's prediction for $F_{2}$. The model's predicted ratio $F_{2}(Q^…
Relativistic equations of Bethe-Salpeter type for hadron structure are most conveniently formulated in momentum space. The presence of confining interactions causes complications because the corresponding kernels are singular. This occurs not only in the relativistic case but also in the nonrelativistic Schrödinger equation where this problem can be studied more easily. For the linear confining interaction the singularity reduces to one of Cauchy principal value form. Although this singularity is integrable, it still makes accurate numerical solutions difficult. We show that this principal val…
While lattice QCD allows for reliable results at small momentum transfers (large quark separations), perturbative QCD is restricted to large momentum transfers (small quark separations). The latter is determined up to a reference momentum scale $Λ$, which is to be provided from outside, e.g. from experiment or lattice QCD simulations. In this article, we extract $Λ_{\overline{\textrm{MS}}}$ for QCD with $n_f=2$ dynamical quark flavors by matching the perturbative static quark-antiquark potential in momentum space to lattice results in the intermediate momentum regime, where both approaches are…
We study the charmonium spectrum using a complete one gluon exchange approach based on a phenomenological relativistic $q\bar{q}$ potential model with Dirac spinors in momentum space. We use phenomenological screening factors to include vacuum quantum effects. Our formulation does not rely on nonrelativistic approximations. We fit the lowest-lying charmonia (below the $D\bar{D}$ threshold) and predict the higher-lying resonances of the spectrum. In general, we reproduce the overall structure of the charmonium spectrum and, in particular, we can reasonably describe the $X(3872)$ resonance mass …
We investigate the relevant gluon-momentum region for confinement in lattice QCD on $16^4$ at $β$=5.7, 5.8 and 6.0, based on the Fourier expansion. We find that the string tension $σ$, i.e., the confining force, is almost unchanged even after removing the high-momentum gluon component above 1.5GeV in the Landau gauge. In fact, the confinement property originates from the low-momentum gluon component below 1.5GeV, which is the upper limit to contribute to $σ$. In the relevant region, smaller gluon momentum component is more important for confinement. Next, we develop a manifestly gauge-covarian…
We obtain the canonical and symmetrical Belinfante energy-momentum tensors of Dirac--Kähler's fields. It is shown that the traces of the energy-momentum tensors are not equal to zero. We find the canonical and Belinfante dilatation currents which are not conserved, but a new conserved dilatation current is obtained. It is pointed out that the conformal symmetry is broken. The canonical quantization is performed and the propagator of the massless fields in the first-order formalism is found.
We calculate the elliptic flow of bottomonia produced in Pb$\,+\,$Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV. We consider temperature-dependent decay widths for the anisotropic escape of various bottomonium states and observe that the transverse momentum dependence of bottomonia elliptic flow provides a tomographic information about the QGP fireball at different stages of its evolution. For the space-time evolution of the fireball, we employ simulation results from the 3+1D quasiparticle anisotropic hydrodynamic model. We find that our results for transverse momentum dependence of bottomoni…
We consider the dynamics of a relativistic Dirac particle constrained to move in the interior of a twisted tube by confining boundary conditions, in the approximation that the curvature of the tube is small and slowly varying. In contrast with the nonrelativistic theory, which predicts that a particle's spin does not change as the particle propagates along the tube, we find that the angular momentum eigenstates of a relativistic spin-1/2 particle may behave nontrivially. For example, a particle with its angular momentum initially polarized in the direction of propagation may acquire a nonzero …
