Nonrelativistic QCD Research Articles

Non-relativistic and potential non-relativistic effective field theories for scalar mediators

Yukawa-type interactions between heavy Dirac fermions and a scalar field are a common ingredient in various extensions of the Standard Model. Despite of that, the non-relativistic limit of the scalar Yukawa theory has not yet been studied in full generality in a rigorous and model-independent way. In this paper we intend to fill this gap by initiating a series of investigations that make use of modern effective field theory (EFT) techniques. In particular, we aim at constructing suitable non-relativistic and potential non-relativistic EFTs of Yukawa interactions (denoted as NRY and pNRY respectively) in close analogy to the well known and phenomenologically successful non-relativistic QCD (NRQCD) and potential non-relativistic QCD (pNRQCD). The phenomenological motivation for our study lies in the possibility to explain the existing cosmological observations by introducing heavy fermionic dark matter particles that interact with each other by exchanging a light scalar mediator. A systematic study of this compelling scenario in the framework of non-relativistic EFTs (NREFTs) constitutes the main novelty of our approach as compared to the existing studies.

Inclusive diffractive heavy quarkonium photoproduction including quark subprocesses

The inclusive $J/\Psi$, $\Psi(2S)$ and $\Upsilon(1S)$ direct and resolved photoproduction are investigated by including the quark subprocesses in the framework of non-relativistic quantum chromodynamics (NRQCD). We find that the theoretical total cross section of heavy quarkonium productions are in good agreement with the data available at HERA, once the $\gamma q$, $qg$ and $qq$ subprocesses in the heavy quark pair productions are taken into account. The inclusive diffractive rapidity and transverse momentum distributions of $J/\Psi$, $\Psi(2S)$ and $\Upsilon(1S)$ in $pp$, $pPb$ and $PbPb$ collisions at LHC are also studied by our quark improved NRQCD model combined with the resolved pomeron model. We find that the contributions from the quark involved subprocesses can reach to $8\%$ in the rapidity distribution and $6\%$ in the transverse momentum distribution. The numerical results show that the contributions from quark involved subprocesses are significant in heavy quarkonium photoprodution.

Bc meson exclusive decays to a P -wave charmonium and a pion at NLO accuracy

In this paper, we calculate the next-to-leading order (NLO) quantum chromodynamics (QCD) corrections to the exclusive processes $B_c^+\to \chi_{cJ}(h_c)\pi^+$ in the framework of the nonrelativistic QCD (NRQCD) factorization formalism. The results show that NLO QCD corrections markedly enhance the branching ratios with $K$ factors of about 2.5. In combination with the study of $B_c^+\to J/\psi\pi^+$, we find that the NLO NRQCD prediction for the ratio of branching fractions $\frac{\mathcal{B}(B_c^+\to \chi_{c0}\pi^+)}{\mathcal{B}(B_c^+\to J/\psi \pi^+)}$ is then compatible with the experimental measurement.

Inclusive production of fully-charmed 1+− tetraquark at B factory * *The work of Y.-S. H., Y. J. and J.-Y. Z. is supported in part by the National Natural Science Foundation of China (11925506, 11875263, 12070131001 (CRC110by DFG and NSFC)). The work of F. F. is supported by the National Natural Science Foundation of China (11875318, 11505285) and by the Yue

Inspired by the recent discovery of the meson in the experiment, we investigate the inclusive production rate of the -odd fully-charmed tetraquarks associated with light hadrons at the factory within the nonrelativistic QCD (NRQCD) factorization framework. The short-distance coefficient is computed at the lowest order of velocity and . Employing two different kinds of phenomenological models to approximately estimate the long-distance NRQCD matrix element, we predict the rate for the inclusive production of the state and discuss the observation prospect of the experiment.

Bc → J/ψ tensor form factors at large momentum recoil

We present a next-to-leading order (NLO) relativistic correction to [Formula: see text] tensor form factors within nonrelativistic QCD (NRQCD). We also consider complete Dirac bilinears [Formula: see text] with [Formula: see text] matrices [Formula: see text] in the [Formula: see text] transition. The relation among different current form factors is given and it shows that symmetries emerge in the heavy bottom quark limit. For a phenomenological extension, we propose to extract the long-distance matrix elements (LDMEs) for [Formula: see text] meson from the recent HPQCD lattice data and the NLO form factors at large momentum recoil.

Double prompt J/ψ production at hadron colliders

In this paper, we give a brief review of recent theoretical studies of double prompt [Formula: see text] production at the FNAL Tevatron and the CERN LHC, to which both single- and double-parton-scattering processes contribute. In single parton scattering, we mainly present the results computed within the framework of non-relativistic quantum chromodynamics factorization combined with one of the two approaches to describe the parton inside the (anti)proton, the collinear parton model and the parton Reggeization approach. The contributions from double parton scattering calculated with different values of [Formula: see text] are also discussed. We conclude that, in double prompt [Formula: see text] hadroproduction, there remain some differences between theoretical predictions and experimental measurements.

NRQCD analysis of charmonium production with pion and proton beams at fixed-target energies

We present an analysis of hadroproduction of J∕ψ and ψ(2S) at fixed-target energies in the framework of non-relativistic QCD (NRQCD). Using both pion- and proton-induced data, a new determination of the color-octet long-distance matrix elements (LDMEs) is obtained. Compared with previous results, the contributions from the qq̄ and color-octet processes are significantly enhanced, especially at lower energies. A good agreement between the pion-induced J∕ψ production data and NRQCD calculations using the newly obtained LDMEs is achieved. We find that the pion-induced charmonium production data are sensitive to the gluon density of pions, and favor pion PDFs with relatively large gluon contents at large x.

Bottomonia production in p + p collisions under NRQCD formalism

In this work, we present the calculation of the production cross sections of bottomonia states using Non-Relativistic Quantum Chromodynamics (NRQCD) formalism. The direct production cross-section of a resonance can be factorised in terms of short distance Quantum Chromodynamics (QCD) cross sections and long distance matrix elements (LDMEs) under NRQCD. We use a large set of measured ϒ(nS) production data at Tevatron and LHC energies in both central and forward rapidity regions to extract the LDMEs with better precision. The feed down contributions from the higher states including the χb(3P) state are taken into account for the LDME extraction. The formalism provides a good description of the bottomonia data in wide transverse momentum range at different collision energies.

Toward accurate form factors for B -to-light meson decay from lattice QCD

We present the results of a lattice QCD calculation of the scalar and vector form factors for the unphysical $B_s\to\eta_s$ decay, over the full physical range of $q^2$. This is a useful testing ground both for lattice QCD and for our wider understanding of the behaviour of form factors. Calculations were performed using the highly improved staggered quark (HISQ) action on $N_f = 2 + 1 + 1$ gluon ensembles generated by the MILC Collaboration with an improved gluon action and HISQ sea quarks. We use three lattice spacings and a range of heavy quark masses from that of charm to bottom, all in the HISQ formalism. This permits an extrapolation in the heavy quark mass and lattice spacing to the physical point and nonperturbative renormalisation of the vector matrix element on the lattice. We find results in good agreement with previous work using nonrelativistic QCD $b$ quarks and with reduced errors at low $q^2$, supporting the effectiveness of our heavy HISQ technique as a method for calculating form factors involving heavy quarks. A comparison with results for other decays related by SU(3) flavour symmetry shows that the impact of changing the light daughter quark is substantial but changing the spectator quark has very little effect. We also map out form factor shape parameters as a function of heavy quark mass and compare to heavy quark effective theory expectations for mass scaling at low and high recoil. This work represents an important step in the progression from previous work on heavy-to-heavy decays ($b\to c$) to the numerically more challenging heavy-to-light decays.

Fragmentation functions for a quark into a spin-singlet quarkonium: Different flavor case

In the paper, we calculate the fragmentation functions for a quark to fragment into a spin-singlet quarkonium, where the flavor of the initial quark is different from that of the constituent quark in the quarkonium. The ultraviolet divergences in the phase space integral are removed through the operator renormalization under the modified minimal subtraction scheme. The fragmentation function $D_{q \to \eta_Q}(z,\mu_F)$ is expressed as a two-dimensional integral. Numerical results for the fragmentation functions of a light quark or a bottom quark to fragment into the $\eta_c$ are presented. As an application of those fragmentation functions, we study the processes $Z \to \eta_c+q\bar{q}g(q=u,d,s)$ and $Z \to \eta_c+b\bar{b}g$ under the fragmentation and the direct nonrelativistic QCD approaches.

Exploring J/ψ Production Mechanism at the Future Electron-Ion ColliderH. X. is supported by Guangdong Major Project of Basic and Applied Basic Research (Grant No. 2020B0301030008), and the National Natural Science Foundation of China (Grant Nos. 12022512 and 12035007), as well as the Science and Technology Program of Guangzhou (Grant No. 2019050001). J.-W.Q. is supported by the U.S. Department of Energy

We propose to use transverse momentum p T distribution of J/ψ production at the future Electron Ion Collider (EIC) to explore the production mechanism of heavy quarkonia in high energy collisions. We apply QCD and QED collinear factorization to the production of a pair at high p T, and non-relativistic QCD factorization to the hadronization of the pair to a J/ψ. We evaluate J/ψ p T-distribution at both leading and next-to-leading order in strong coupling, and show that production rates for various color-spin channels of a pair in electron-hadron collisions are very different from that in hadron-hadron collisions, which provides a strong discriminative power to determine various transition rates for the pair to become a J/ψ. We predict that the J/ψ produced in electron-hadron collisions is likely unpolarized, and the production is an ideal probe for gluon distribution of colliding hadron (or nucleus). We find that the J/ψ production is dominated by the color-octet channel, providing an excellent probe to explore the gluon medium in large nuclei at the EIC.

Charmonia production in W → (cc̄)Ds(∗) decays

In this paper, production of charmonium state [Formula: see text] in exclusive [Formula: see text] decays is analyzed in the framework of both leading order Non-relativistic Quantum Chromodynamics (NRQCD) and light-cone (LC) expansion models. Analytical and numerical predictions for the branching fractions of these decays in both the approaches are given. The typical value of the branching fractions is [Formula: see text][Formula: see text][Formula: see text] and it turns out that the LC results are significantly larger than NRQCD ones (approximately two or four times increase depending on the quantum numbers of the final particles), so the effect of internal quark motion should be taken into account. Some rough estimates of color-octet contributions are presented and it is shown that these contributions could be comparable with color-singlet results.

Inclusive Hadroproduction of P-Wave Heavy Quarkonia in Potential Nonrelativistic QCD.

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.

Study on ηc2(ηb2) electromagnetic decay into double photons

Within the framework of nonrelativistic QCD (NRQCD) factorization formalism, we compute the helicity amplitude as well as the decay width of $\eta_{Q2}$ ($Q=c,b$) electromagnetic decay into two photons up to next-to-next-to-leading order (NNLO) in $\alpha_s$ expansion. For the first time, we verify the validity of NRQCD factorization for the D-wave quarkonium decay at NNLO. We find that the $\mathcal{O}(\alpha_s)$ and $\mathcal{O}(\alpha_s^2)$ corrections to the helicity amplitude are negative and moderate, nevertheless both corrections combine to suppress the leading-order prediction for the decay width significantly. By approximating the total decay width of $\eta_{Q2}$ as the sum of those for the hadronic decay and the electric $E1$ transition, we obtain the branching ratios ${\rm Br}(\eta_{c2}\to 2\gamma)\approx 5\times10^{-6}$ and ${\rm Br}(\eta_{b2}\to 2\gamma)\approx 4\times10^{-7}$. To explore the potential measurement on $\eta_{Q2}$, we further evaluate the production cross section of $\eta_{Q2}$ at LHCb at the lowest order in $\alpha_s$ expansion. With the kinematic constraint on the longitudinal rapidity $4.5>y>2$ and transverse momentum $P_T>(2-4)m_Q$ for $\eta_{Q2}$, we find the cross section can reach $2-50$ nb for $\eta_{c2}$, and $1-22$ pb for $\eta_{b2}$. Considering the integrated luminosity $\mathcal{L}=10\, {\rm fb}^{-1}$ at $\sqrt{s}=7$ TeV and $\sqrt{s}=13$ TeV, we estimate that there are several hundreds events of $pp\to \eta_{c2}\to 2\gamma$. Since the background is relatively clean, it is promising to reconstruct $\eta_{c2}$ through its electromagnetic decay. On the contrary, due to small branching ratio and production cross section, it is quite challenging to detect $\eta_{b2}\to 2\gamma$ at LHCb.

Quarkonium semiclassical transport in quark-gluon plasma: factorization and quantum correction

We study quarkonium transport in the quark-gluon plasma by using the potential nonrelativistic QCD (pNRQCD) effective field theory and the framework of open quantum systems. We argue that the coupling between quarkonium and the thermal bath is weak using separation of scales, so the initial density matrix of the total system factorizes and the time evolution of the subsystem is Markovian. We derive the semiclassical Boltzmann equation for quarkonium by applying a Wigner transform to the Lindblad equation and carrying out a semiclassical expansion. We resum relevant interactions to all orders in the coupling constant at leading power of the nonrelativistic and multipole expansions. The derivation is valid for both weakly coupled and strongly coupled quark-gluon plasmas. We find reaction rates in the transport equation factorize into a quarkonium dipole transition function and a chromoelectric gluon distribution function. For the differential reaction rate, the definition of the momentum dependent chromoelectric gluon distribution function involves staple-shaped Wilson lines. For the inclusive reaction rate, the Wilson lines collapse into a straight line along the real time axis and the distribution becomes momentum independent. The relation between the two Wilson lines is analogous to the relation between the Wilson lines appearing in the gluon parton distribution function (PDF) and the gluon transverse momentum dependent parton distribution function (TMDPDF). The centrality dependence of the quarkonium nuclear modification factor measured by experiments probes the momentum independent distribution while the transverse momentum dependence and measurements of the azimuthal angular anisotropy may be able to probe the momentum dependent one. We discuss one way to indirectly constrain the quarkonium in-medium real potential by using the factorization formula and lattice calculations. The leading quantum correction to the semiclassical transport equation of quarkonium is also worked out. The study can be easily generalized to quarkonium transport in cold nuclear matter, which is relevant for quarkonium production in eA collisions in the future Electron-Ion Collider.

Associate production J/ψ + γ at the Large Hadron Collider and Electron–Ion Collider

AbstractCharmonium production associated with a photon in coherent hadron–hadron interactions at high‐energy colliders is a suitable observable to test the partonic density function in the proton and in nuclei. We employ nonrelativistic quantum chromodynamics to investigate the transverse momentum distribution and total cross sections of J/ψ + γ production at the Large Hadron Collider (LHC) and the electron‐ion collider (EIC). Our results show that discrimination can be made to test the parameters for the matrix elements operators, which varies by a factor of 3. A higher transverse momentum region is more sensitive for pPb collisions at the LHC, although the EIC kinematic region is also favorable at low transverse momentum.

ϒelectroproduction at HERA, EIC, and LHeC within the nonrelativistic QCD framework

Based on the nonrelativistic QCD framework, we study the $\Upsilon$ production in semi-inclusive deep inelastic electron-proton scattering (SIDIS) at HERA, EIC, and LHeC, with the main aim of assessing the viability of observing $\Upsilon$ electroproduction at the three colliders. The color-octet (CO) contributions are found to have a crucial effect on both the integrated and differential cross sections, serving to further establish the significance of the CO mechanism. By setting the kinematic cuts to $p_{t,\Upsilon}^{\star 2}>1~\textrm{GeV}^{2}$, $W>50$ GeV, $2<Q^2<100~\textrm{GeV}^2$, and $0.3<z<0.9$, only a few electroproduced $\Upsilon$ events can be generated at HERA, partially accounting for its lack of measured $\Upsilon$ electroproduction. However, under the same cut conditions, the EIC and LHeC can accumulate about $8.7\times10^{2}$ and $3.7\times10^{4}$ reconstructed $\Upsilon$ events in one operation year, respectively, which manifestly indicates the prospect of detecting the $\Upsilon$-related SIDIS processes at the two forthcoming $ep(eA)$ colliders.

Bethe-Salpeter amplitudes of Upsilons

Based on lattice non-relativistic QCD (NRQCD) studies we present results for Bethe-Salpeter amplitudes for $\Upsilon(1S)$, $\Upsilon(2S)$ and $\Upsilon(3S)$ in vacuum as well as in quark-gluon plasma. Our study is based on 2+1 flavor $48^3 \times 12$ lattices generated using the Highly Improved Staggered Quark (HISQ) action and with a pion mass of $161$ MeV. At zero temperature the Bethe-Salpeter amplitudes follow the expectations based on non-relativistic potential models. At non-zero temperatures, the interpretation of Bethe-Salpeter amplitudes turns out to be more nuanced, but consistent with our previous lattice QCD study of excited Upsilons in quark-gluon plasma.

An Effective Theory of Quarkonia in QCD Matter

The problem of quarkonium production in heavy ion collisions presents a set of unique theoretical challenges – from the relevant production mechanism of J/ψs and ϒs to the relative significance of distinct cold and hot nuclear matter effects in the observed attenuation of quarkonia. In these proceedings we summarize recent work on the generalization of non-relativistic Quantum Chromodynamics (NRQCD) to include Glauber gluon interactions in strongly interacting matter. This new effective theory provides for the first time a universal microscopic description of the in-medium interaction of heavy quarkonia, consistently applicable to a range of phases such as cold nuclear matter, dense hadron gas, and quark-gluon plasma. It is an important step forward in understanding the common trends in proton-nucleus and nucleus-nucleus data on quarkonium suppression. We derive explicitly the lowest order (LO) and next-to-leading order (NLO) interaction terms in the Lagrangian and show the connection of the LO result to existing phenomenology.

Inclusive photoproduction at the CEPC * *Supported by the National Natural Science Foundation of China (11975242) and the Key Research Program of Frontier Sciences, CAS, (Y7292610K1)

Inclusive photoproduction at the future Circular-Electron-Positron-Collider (CEPC) is studied, using the non-relativistic quantum chromodynamics (NRQCD) factorization formalism. Including the contributions from both direct and resolved photons, we present different distributions for the production. Our results suggest that there will be considerable events, implying that well measurements of the photoproduction can be performed to further study heavy quarkonium physics at electron-positron colliders, in addition to hadron colliders. This supplemental study is very important for clarifying the current situation regarding the heavy quarkonium production mechanism.