Nonrelativistic QCD Research Articles

Tetraquarks made of sufficiently unequal-mass heavy quarks are bound in QCD

Tetraquarks, bound states composed of two quarks and two antiquarks, have been the subject of intense study but are challenging to understand from first principles. We apply variational and Green’s function Monte Carlo methods to compute tetraquark ground-state energies in potential nonrelativistic QCD using a wide range of color and spatial wave functions. We find no evidence for bound tetraquarks composed of equal-mass quarks and antiquarks. Conversely, we find clear evidence for the existence of bound tetraquarks for sufficiently unequal quark/antiquark mass ratios at all overall mass scales where our effective theory results are applicable. We predict the critical mass ratios for bound state formation and study tetraquark bound states’ spatial and color structure at leading order and next-to-leading order in potential nonrelativistic QCD. Published by the American Physical Society 2024

Quarkonium polarization in medium from open quantum systems and chromomagnetic correlators

We study the spin-dependent in-medium dynamics of quarkonia by using the potential nonrelativistic QCD (pNRQCD) and the open quantum system framework. We consider the pNRQCD Lagrangian valid up to the order rM0=r and r0M=1M in the double power counting. By considering the Markovian condition and applying the Wigner transformation upon the diagonal spin components of the quarkonium density matrix with the semiclassical expansion, we systematically derive the Boltzmann transport equation for quarkonia with polarization dependence in the quantum optical limit. Unlike the spin-independent collision terms governed by certain chromoelectric field correlators, new gauge invariant correlators of chromomagnetic fields determine the recombination and dissociation terms with polarization dependence at the order we are working. We also derive a Lindblad equation describing the in-medium transitions between spin-singlet and spin-triplet heavy quark-antiquark pairs in the quantum Brownian motion limit. The Lindblad equation is governed by new transport coefficients defined in terms of the chromomagnetic field correlators. Our formalism is generic and valid for both weakly coupled and strongly coupled quark gluon plasmas. It can be further applied to study spin alignment of vector quarkonia in heavy ion collisions. Published by the American Physical Society 2024

Kinetic energy and speed powers vn of a heavy quark inside S wave and P wave heavy-light mesons

The average values of the powers of |q→|n¯≡qn and |v→|n¯≡vn are commonly used physical quantities, where q→ and v→ are the three-dimensional momentum and velocity of a quark inside a meson, respectively. For example, the average kinetic energy μπ2=q2 is an important quantity in the heavy quark effective theory, and the expansion of vn is commonly used in the nonrelativistic quantum chromodynamics. In this paper, based on the instantaneous Bethe-Salpeter equation method, we calculate the average values qn and vn (n=1, 2, 3, 4) of a heavy quark inside S wave and P wave heavy-light mesons. We obtain the kinetic energy μπ2=0.455 GeV2 for the B meson, which is consistent with the experimental result 0.464±0.076 GeV2. For the Bs, D, and Ds mesons, the μπ2 are 0.530, 0.317, and 0.369 GeV2, respectively, and v2=0.0185, 0.0215, 0.121, and 0.140 for B, Bs, D, and Ds, respectively. We obtain some relationships, for example, q0−n(mS)≈q1−n(mS), q0+n(mP)≈q1+′n(mP′)>q1+n(mP)≈q2+n(mP), and qn(mS)<qn(mP) (m=1, 2, 3), etc. Published by the American Physical Society 2024

Factorization for J/ψ leptoproduction at small transverse momentum

Nonrelativistic Quantum Chromodynamics (NRQCD) breaks down in the region of low transverse momentum, where the transverse momentum of the produced quarkonium state is sensitive to multiple scattering with the incoming hadron and to soft gluon radiation. In this kinematic regime, the transverse-momentum-dependent (TMD) factorization framework is required, promoting the long-distance matrix elements (LDMEs) of NRQCD to the so-called TMD shape functions (TMDShFs), which encode both the soft gluon radiation and the formation of the heavy-quark bound state. In this work, we apply an effective-field theory approach (combining NRQCD and SCET) to the photon-gluon fusion process in inclusive J/ψ leptoproduction. We derive a factorization theorem for the cross section in terms of TMDShFs, compute these quantities at next-to-leading order, establish their evolution, and study their matching onto the corresponding LDMEs in the high-transverse-momentum region. Our results are particularly relevant to the Electron-Ion Collider, where J/ψ leptoproduction can be used to probe gluon transverse-momentum-dependent parton distribution functions (gluon TMDPDFs).

Measurement of J/ψ and [formula omitted] production in p + p and p + d interactions at 120 GeV

We report the p+p and p+d differential cross sections measured in the SeaQuest experiment for J/ψ and ψ(2S) production at ▪ beam energy covering the forward x-Feynman (xF) range of 0.5<xF<0.9. The measured cross sections are in good agreement with theoretical calculations based on the nonrelativistic QCD (NRQCD) using the long-distance matrix elements deduced from a recent global analysis of proton- and pion-induced charmonium production data. The σψ(2S)/σJ/ψ cross section ratios are found to increase as xF increases, indicating that the qq¯ annihilation process has larger contributions in the ψ(2S) production than the J/ψ production. The σpd/2σpp cross section ratios are observed to be significantly different for the Drell-Yan process and J/ψ production, reflecting their different production mechanisms. We find that the σpd/2σpp ratios for J/ψ production at the forward xF region are sensitive to the d¯/u¯ flavor asymmetry of the proton sea, analogous to the Drell-Yan process. The transverse momentum (pT) distributions for J/ψ and ψ(2S) production are also presented and compared with data collected at higher center-of-mass energies.

Pseudoscalar heavy quarkonium production in heavy ion ultraperipheral collision

The inclusive production of pseudoscalar heavy quarkonia (ηc, ηb, and Bc) via photon-photon fusion in heavy ion ultraperipheral collision (UPC) are calculated to QCD next-to-leading order in the framework of nonrelativistic QCD (NRQCD). The total cross section of ηc produced in Pb-Pb UPC is 194 nb and 1275 nb at nucleon-nucleon c.m. energies SNN=5.52 TeV and 39.4 TeV, respectively. The cross sections for ηb and Bc mesons are more than two to three orders of magnitude smaller. We make a detailed phenomenological analysis on the ηc production; the uncertainties caused by the renormalization scale and the charm quark mass, the cross sections in other ultraperipheral nucleon-nucleon colliding systems, and the transverse momentum distribution are discussed. At the coming HL-LHC and future FCC, the heavy ion UPC opens another door of the study on the production of heavy quarkonium. Published by the American Physical Society 2024

Studies of Z boson decay into double ϒ mesons at the NLO QCD accuracy

In this paper, we employ the nonrelativistic quantum chromodynamics (QCD) factorization to conduct a comprehensive examination of the Z boson decay into a pair of ϒ mesons, achieving accuracy at the next-to-leading-order (NLO) in αs. Our calculations demonstrate that the quantum electrodynamics (QED) diagrams are indispensable in comparison to the pure QCD diagrams, and the implementation of QCD corrections markedly enhances the QCD results, whereas it substantially diminishes the QED results. To ensure consistency with the experimental methodology, we have taken into account the feed-down transitions originating from higher excited states, which exhibit significant relevance. Combining all the contributions, we arrive at the NLO prediction of BZ→ϒ(nS)+ϒ(mS)∼10−11, which is notably lower than the upper limits set by CMS. Published by the American Physical Society 2024

The distribution amplitude of the ηc-meson at leading twist from lattice QCD

Distribution amplitudes are functions of non-perturbative matrix elements describing the hadronization of quarks and gluons. Thanks to factorization theorems, they can be used to compute the scattering amplitude of high-energy processes. Recently, new ideas have allowed their computation using lattice QCD, which should provide us with a general, fully relativistic determination. We present the first lattice calculation of the ηc-meson distribution amplitude at leading twist. Starting from the relevant matrix element in discrete Euclidean space on a set of Nf = 2 CLS ensembles, we explain the method to connect to continuum Minkowski spacetime. After addressing several sources of systematic uncertainty, we compare to Dyson-Schwinger and non-relativistic QCD determinations of this quantity. We find significant deviations between the latter and our result even at small Ioffe times.

Unraveling gluon TMDs in J/ψ and pion production at the EIC

We investigate the azimuthal asymmetries such as cos2ϕT and Sivers symmetry for J/ψ and π± production in electron-proton scattering, focusing on scenarios where the J/ψ and the pion are produced in an almost back-to-back configuration. The electron is unpolarized, while the proton can be unpolarized or transversely polarized. For the J/ψ formation, we use nonrelativistic QCD (NRQCD), while π± is formed due to parton fragmentation. In this kinematics, we utilize the transverse momentum-dependent factorization framework to calculate the cross sections and asymmetries. We consider both quark and gluon-initiated processes and show that the gluon contribution dominates. In this work, we used the generalized parton model (GPM) for the transverse momentum distribution (TMD) parametrizations and did not consider the effect of TMD evolution. We provide numerical estimates of the upper bounds on the azimuthal asymmetries, as well as employ a Gaussian parametrization for the gluon TMDs, within the kinematical region accessible by the upcoming Electron-Ion Collider (EIC). Published by the American Physical Society 2024

Photoproduction of fully charmed tetraquark at electron-ion colliders

In this work, we investigate the inclusive photoproduction of the C-odd, S-wave fully charmed tetraquark at electron-ion colliders within the nonrelativistic QCD (NRQCD) factorization framework, at the lowest order in velocity and αs. The value of the NRQCD long-distance matrix element is estimated from two phenomenological potential models. Our studies reveal that the photoproduction of the 1+− fully charmed tetraquark may be difficult to observe at HERA and the EicC; nevertheless, its observation prospects at the EIC appear to be bright. Published by the American Physical Society 2024

Producing fully-charmed tetraquarks via charm quark fragmentation in colliders

Within the framework of nonrelativistic QCD (NRQCD), we calculate the fragmentation function for a charm quark into an S-wave fully-charmed tetraquark, denoted as T4c. The charm-to-T4c fragmentation function is expressed as a sum of products of the perturbatively calculable short-distance coefficients and the nonperturbative long-distance matrix elements (LDMEs). The short-distance coefficients are ascertained through the perturbative matching procedure at lowest order in αs expansion. The LDMEs are approximated using the T4c four-body wave functions at the origin, which have been evaluated by various phenomenological potential models in literature. Incorporating the celebrated QCD factorization and the charm-to-T4c fragmentation function, we predict the T4c production rate at high transverse momentum pT regime in colliders. Both the differential distribution over pT and the integrated cross sections are predicted at the LHC. The cross sections for T4c states production can reach several femtobarns to several hundreds femtobarns, suggesting a substantial potential for T4c event production at the LHC. Additionally, we estimate for the photoproduction of T4c in electron-proton (ep) collisions. It is observed that the cross sections for these processes are moderate at the HERA and EIC, and relatively small at the EicC. Given the luminosities of these colliders, the prospect of detecting these fully-charmed tetraquarks at ep colliders is somewhat challenging.

Spin asymmetries for C -even quarkonium production as a probe of gluon distributions

Within the framework of transverse momentum dependent factorization in combination with nonrelativistic quantum chromodynamics (QCD), we study charmonium and bottomonium production in hadronic collisions. We focus on quarkonium states with even charge conjugation, for which the color-singlet production mechanism is expected to be also dominant in the small transverse momentum region, qT2≪4Mc,b2. It is shown that the distributions of linearly polarized gluons inside unpolarized, longitudinally, and transversely polarized protons contribute to the cross sections for scalar and pseudoscalar quarkonia in a very distinctive, parity-dependent way, whereas their effects on higher angular momentum states are strongly suppressed. We derive analytical expressions for single and double spin asymmetries, which would allow for the direct extraction of the gluon transverse momentum dependent distributions, mirroring the phenomenological studies of the Drell-Yan processes aimed at the extraction of their quark counterparts. By adopting Gaussian models for the gluon transverse momentum dependent distributions, which fulfill without saturating everywhere their positivity bounds, we provide numerical predictions for the transverse single-spin asymmetries. These observables could be measured at LHCSpin, the fixed target experiment planned at the LHC. Published by the American Physical Society 2024

Study of the isoscalar scalar bcu¯d¯ tetraquark Tbc with lattice QCD

We present a lattice QCD study of the elastic S-wave DB¯ scattering in search of tetraquark candidates with explicitly exotic flavor content bcu¯d¯ in the isospin I=0 and JP=0+ channel. We use four lattice QCD ensembles with dynamical u/d, s, and c quark fields generated by the MILC collaboration. A nonrelativistic QCD Hamiltonian, including improvement coefficients up to O(αsv4), is utilized for the bottom quarks. For the rest of the valence quarks we employ a relativistic overlap action. Five different valence quark masses are utilized to study the light quark mass dependence of the DB¯ scattering amplitude. The finite volume energy spectra are extracted following a variational approach. The elastic DB¯ scattering amplitudes are extracted employing Lüscher’s prescription. The light quark mass dependence of the continuum extrapolated amplitudes suggests an attractive interaction between the B¯ and D mesons. At the physical pseudoscalar meson mass (Mps=Mπ) the DB¯ scattering amplitude has a subthreshold pole corresponding to a binding energy of −39(−6+4)(−18+8) MeV with respect to the DB¯ threshold. The critical Mps at which the DB¯ scattering length diverges and the system becomes unbound corresponds to Mps*=2.94(15)(5) GeV. This result can hold significant experimental relevance in the search for a bound scalar Tbc tetraquark, which could well be the next “doubly heavy” bound tetraquark to be discovered with only weak decay modes. Published by the American Physical Society 2024

Production of Polarized $${J/}\psi$$ Mesons within Nonrelativistic QCD and Generalized Parton Model

Production of Polarized $${J/}\psi$$ Mesons within Nonrelativistic QCD and Generalized Parton Model

FKS subtraction for quarkonium production at NLO

We extend the local infrared-divergence subtraction formalism, originally proposed by Frixione, Kunszt and Signer (FKS), to calculate short-distance (differential) cross section for any inclusive process involving a quarkonium particle in non-relativistic QCD (NRQCD) factorisation at next-to-leading order (NLO) accuracy in the strong coupling constant αs. The new formulas are generally applicable to the production of an S- or P-wave quarkonium state in association with any number of elementary particles. The main new ingredients derived in this paper are the local and integrated soft counterterms for the colour-singlet and colour-octet P-wave bound states. It, therefore, paves the way to the automation of the NLO calculations for heavy quarkonium inclusive and associated production processes.

Off-lightcone Wilson-line operators in gradient flow

Off-lightcone Wilson-line operators are constructed using local operators connected by time-like or space-like Wilson lines, which ensure gauge invariance. Off-lightcone Wilson-line operators have broad applications in various contexts. For instance, space-like Wilson-line operators play a crucial role in determining quasi-distribution functions (quasi-PDFs), while time-like Wilson-line operators are essential for understanding quarkonium decay and production within the potential non-relativistic QCD (pNRQCD) framework. In this work, we establish a systematic approach for calculating the matching from the gradient-flow scheme to the MS¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\ extrm{MS}} $$\\end{document} scheme in the limit of small flow time for off-lightcone Wilson-line operators. By employing the one-dimensional auxiliary-field formalism, we simplify the matching procedure, reducing it to the matching of local current operators. We provide one-loop level matching coefficients for these local current operators. For the case of hadronic matrix element related to the quark quasi-PDFs, we show at one-loop level that the finite flow time effect is very small as long as the flow radius is smaller than the physical distance z, which is usually satisfied in lattice gradient flow computations. Applications include lattice gradient flow computations of quark/gluon quasi-PDFs, gluonic correlators related to quarkonium decay and production in pNRQCD, and spin-dependent potentials in terms of chromoelectric and chromomagnetic field insertions into a Wilson loop.

Doubly heavy hadron production in ultraperipheral collision

We study the double heavy baryon ΞQQ′ and tetraquark TQQ production through photon-photon and photon-gluon fusion via ultraperipheral collisions at the LHC and Future Circular Collider (FCC) within the framework of nonrelativistic QCD factorization formalism. Various ion-ion collisions are taken into account, two cc(bb)-diquark configurations ([cc(bb),S31−3¯] and [cc(bb),S10−6]) and four bc-diquark configurations ([bc,S31−3¯], [bc,S31−6], [bc,S10−3¯], and [bc,S10−6]) are considered in the calculation. Numerical results indicate that the [cc,S31−3¯] diquark provides dominant contribution for Ξcc (Tcc) production, and a considerable number of Ξcc (Tcc) can be produced. Because the event topologies for ultraperipheral collision are very clear, the background from various QCD interactions can be suppressed, hence the experimental investigation for Ξcc and Tcc is feasible. The productions for Ξbc/bb are also discussed, leaving only a slight possibility for Ξbc through photon-gluon fusion with ultraperipheral collisions at the FCC. Published by the American Physical Society 2024

Bottomonium suppression from the three-loop QCD potential

We compute the suppression of bottomonium in the quark-gluon plasma using the three-loop QCD static potential. The potential describes the spin-averaged bottomonium spectrum below threshold with a less than 1% error. Within potential nonrelativistic quantum chromodynamics and an open quantum systems framework, we compute the evolution of the bottomonium density matrix. The values of the quarkonium transport coefficients are obtained from lattice QCD measurements of the bottomonium in-medium width and thermal mass shift; we additionally include for the first time a vacuum contribution to the dispersive coefficient γ. Using the three-loop potential and the values of the heavy quarkonium transport coefficients, we find that the resulting bottomonium nuclear modification factor is consistent with experimental observations, while at the same time reproducing the lattice measurements of the in-medium width. Published by the American Physical Society 2024

Decay constants of cb¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ c\\overline{b} $$\\end{document} mesons involving the ten heavy flavor-changing currents at N3LO QCD

Within the nonrelativistic QCD (NRQCD) framework, we complete the three-loop calculations of the NRQCD renormalization constants and the matching coefficients, for the heavy flavor-changing temporal vector, spatial-spatial tensor, spatial-temporal axial-tensor currents, which are coupled to the P-wave cb¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ c\\overline{b} $$\\end{document} mesons. We further study the ten decay constants for the cb¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ c\\overline{b} $$\\end{document} mesons (Bc, Bc∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {B}_c^{\\ast } $$\\end{document}, Bc0∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {B}_{c0}^{\\ast } $$\\end{document}, Bc1) coupled with the ten heavy flavor-changing currents involving (pseudo-)scalar, (axial-)vector and (axial-)tensor up to the next-to-next-to-next-to-leading order (N3LO) of αs. We obtain the six ratios of decay constants by approximating them to the corresponding ratios of matching coefficients. We find the N3LO perturbative QCD results for the six ratios of decay constants have good convergence and weak scale-dependence. We finally predict the hierarchical relationship among the ten decay constants for the cb¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ c\\overline{b} $$\\end{document} mesons coupled with the ten currents.

NLO QCD corrections to the Bc -pair hadroproduction

The Bc meson pair, including pairs of pseudoscalar states and vector states, productions in proton-proton collisions are investigated at the next-to-leading order (NLO) accuracy in the nonrelativistic quantum chromodynamics factorization formalism. The corresponding cross sections at the Large Hadron Collider (LHC) with s=14 TeV are evaluated. Numerical results indicate that the NLO corrections are substantial, and even dominate over the leading order contributions. Considering the predicted cross sections are sizable, the Bc-pair production is expected to be observable at the high-luminosity LHC experiment. Published by the American Physical Society 2024