Nonrelativistic QCD Matrix Elements Research Articles

Gluon TMDs from J/ψ production in longitudinally polarized deeply inelastic scattering

We investigate the feasibility of exploring the gluon transverse-momentum-dependent distribution functions (TMDs) inside a longitudinally polarized nucleon. We utilize quarkonium production via the color-octet mechanism combined with TMD formalism in semi-inclusive deeply inelastic scattering (SIDIS) at low transverse momentum as a tool to access polarized gluon TMDs. The corresponding cross-section of the process is expressed in terms of gluon TMDs and non-relativistic QCD matrix elements. We provide the expressions for the sin⁡2ϕ azimuthal asymmetries of J/ψ production in SIDIS with an unpolarized beam colliding on longitudinally polarized nucleon. The asymmetry receives a contribution from the time-reversal-odd gluon TMD h1L⊥g(x,pT2). The maximum possible asymmetry deduced from the positivity bound is sizable and could be measured. We also estimate the double longitudinal spin asymmetry ALL of J/ψ production using a spectator model result for g1Lg(x,pT2).

Next-to-leading-order QCD corrections to a vector bottomonium radiative decay into a charmonium

Within the framework of nonrelativistic QCD (NRQCD) factorization, we calculate the next-to-leading-order (NLO) perturbative corrections to the radiative decay Υ → ηc(χcJ) + γ. Both the helicity amplitudes and the helicity decay widths are obtained. It is the first computation for the processes involving both bottomonium and charmonium at two-loop accuracy. By employing the Cheng-Wu theorem, we are able to convert most of complex-valued master integrals (MIs) into real-valued MIs, which makes the numerical integration much efficient. Our results indicate the mathcal{O}left({alpha}_sright) corrections are moderate for ηc and χc2 production, and are quite marginal for χc0 and χc1 production. It is impressive to note the NLO corrections considerably reduce the renormalization scale dependence in both the decay widths and the branching fractions for χcJ, and slightly improve that for ηc. With the NRQCD matrix elements evaluated via the Buchmüller-Tye potential model, we find the decay width for ηc production is one-order-of-magnitude larger than χcJ production, which may provide a good opportunity to search for Υ → ηc + γ in experiment. In addition, the decay width for χc1 production is several times larger than those for χc0,2. Finally, we find the NLO NRQCD prediction for the branching fraction of Υ → χc1 + γ is only half of the lower bound of the experimental data measured recently by Belle. Moreover, there exists serious contradiction between theory and experiment for Υ → ηc + γ. The discrepancies between theory and experiment deserve further research efforts.

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.

O(αsv2) corrections to the hadronic decay of vector quarkonia

Within the nonrelativistic QCD (NRQCD) factorization framework, we compute the ${\mathcal O}(\alpha_s v^2)$ corrections to the hadronic decay rate of vector quarkonia, exemplified by $J/\psi$ and $\Upsilon$. Setting both the renormalization and NRQCD factorization scales to be $m_Q$, we obtain $\Gamma(J/\psi\to {\rm LH})= 0.0716\frac{\alpha_s^3}{m_c^2} \langle \mathcal{O}_1({}^3S_1)\rangle_{J/\psi} [1-1.19\alpha_s+(-5.32+3.03\alpha_s)\langle v^2\rangle_{J/\psi}]$ and $\Gamma(\Upsilon\to {\rm LH})= 0.0716\frac{\alpha_s^3}{m_b^2}\langle\mathcal{O}_1({}^3S_1)\rangle_{\Upsilon}[1-1.56\alpha_s+(-5.32+4.61\alpha_s)\langle v^2\rangle_{\Upsilon}]$. We confirm the previous calculation of $\mathcal{O}(\alpha_s)$ corrections on a diagram-by-diagram basis, with the accuracy significantly improved. For $J/\psi$ hadronic decay, we find that the ${\mathcal O}(\alpha_sv^2)$ corrections are moderate and positive, nevertheless unable to counterbalance the huge negative corrections. On the other hand, the effect of ${\mathcal O}(\alpha_sv^2)$ corrections for $\Upsilon(nS)$ is sensitive to the $\mathcal{O}(v^2)$ NRQCD matrix elements. With the appropriate choice of the NRQCD matrix elements, our theoretical predictions for the decay rates may be consistent with the experimental data for $\Upsilon(1S,2S)\to {\rm LH}$. As a byproduct, we also present the theoretical predictions for the branching ratio of $J/\psi(\Upsilon)\to 3\gamma$ accurate up to $\mathcal{O}(\alpha_s v^2)$.

Next-to-next-to-leading-order radiative corrections to e+e\u2212\u2192 \u03c7cJ + \u03b3 at B factory

Within the nonrelativistic QCD (NRQCD) factorization framework, we have computed the mathcal{O}left({alpha}_s^2right) corrections to the exclusive production of P-wave spin-triplet charmonia χcJ (J = 0, 1, 2) accompanied with a hard photon at B factory. For the first time, we have explicitly verified the validity of NRQCD factorization for exclusive P-wave quarkonium production to two-loop order. Unlike the χcJ electromagnetic decays, the mathcal{O}left({alpha}_s^2right) corrections are found to be smaller than the mathcal{O} (αs) corrections in all three channels e+e− → χc0,1,2 + γ. In particular, the mathcal{O}left({alpha}_s^2right) corrections appear moderate for χc1 + γ case, and marginal for χc0 + γ. Moreover, the predictions in next-to-next-to-leading order (NNLO) accuracy for the production rates of χc0,1 + γ are insensitive to the renormalization and factorization scales. All these features may indicate that perturbative expansion in these two channels exhibits a decent convergence behavior. By contrast, both the mathcal{O} (αs) and mathcal{O}left({alpha}_s^2right) corrections to the χc2 + γ production rate are sizable, which reduce the Born order cross section by one order of magnitude after including the NNLO perturbative corrections. Taking the values of the long-distance NRQCD matrix elements from nonrelativistic potential model, our prediction to χc1 + γ production rate is consistent with the recent Belle measurement. The NNLO predictions to the χc0,2 + γ production rates are much smaller than that for χc1 + γ, which seems to naturally explain why the e+e− → χc0,2 + γ channels have escaped experimental detection to date.

Relativistic corrections to light-cone distribution amplitudes of S-wave Bc mesons and heavy quarkonia

In collinear factorization, light-cone distribution amplitudes (LCDAs) are key ingredients to calculate the production rate of a hadron in high energy exclusive processes. For a doubly-heavy meson system (such as Bc, J/ψ, Y etc), the LCDAs contain perturbative scales that can be integrated out and then are re-factorized into products of perturbatively calculable distribution parts and non-relativistic QCD matrix elements. In this re-factorization scheme, the LCDAs are known at next-to-leading order in the strong coupling constant αs and at leading order in the velocity expansion. In this work, we calculate the mathcal{O}left({v}^2right) corrections to twist-2 LCDAs of S-wave Bc mesons. These results are applicable to heavy quarkonia like ηc,b, J/ψ and Y by setting mb = mc. We apply these relativistically corrected LCDAs to study their inverse moments and a few Gegenbauer moments which are important for phenomenological study. We point out that the relativistic corrections are sizable, and comparable with the next-to-leading order radiative corrections. These results for LCDAs are useful in future theoretical analyses of the productions of heavy quarkonia and Bc mesons.

Next-to-next-to-leading-order QCD corrections to χc0,2→γγ

We calculate the next-to-next-to-leading-order (NNLO) perturbative corrections to $P$-wave quarkonia annihilation decay to two photons, in the framework of nonrelativistic QCD (NRQCD) factorization. The order-$\alpha_s^2$ short-distance coefficients associated with each helicity amplitude are presented in a semi-analytic form, including the "light-by-light" contributions. With substantial NNLO corrections, we find disquieting discrepancy when confronting our state-of-the-art predictions with the latest \textsf{BESIII} measurements, especially fail to account for the measured $\chi_{c2}\to\gamma\gamma$ width. Incorporating the effects of spin-dependent forces would even exacerbate the situation, since it lifts the degeneracy between the nonperturbative NRQCD matrix elements of $\chi_{c0}$ and $\chi_{c2}$ toward the wrong direction. We also present the order-$\alpha_s^2$ predictions to $\chi_{b0,2}\to\gamma\gamma$, which await the future experimental test.

Production ofχc- andχb-mesons in high energy hadronic collisions

This paper is devoted to a phenomenological study of ${\ensuremath{\chi}}_{c,b}$-meson production in high energy hadronic collisions in the framework of nonrelativistic QCD (NRQCD). We analyze all available experimental data on ${\ensuremath{\chi}}_{c}$-meson production and extract nonperturbative NRQCD matrix elements by data fitting. It is shown that measured ${p}_{T}$ spectra of ${\ensuremath{\chi}}_{c}$-mesons is mainly formed by color singlet components, while the $\ensuremath{\sigma}({\ensuremath{\chi}}_{c2})/\ensuremath{\sigma}({\ensuremath{\chi}}_{c1})$ ratio depends strongly on color octet matrix elements; this ratio becomes a highly sensitive tool to study contributions of the different NRQCD expansion terms. Predictions for ${\ensuremath{\chi}}_{b}$-meson production cross sections, which are obtained using NRQCD scaling rules, are also given.

GENXICC2.1: An improved version of GENXICC for hadronic production of doubly heavy baryons

We present an improved version of GENXICC, which is a generator for hadronic production of the doubly heavy baryons $\Xi_{cc}$, $\Xi_{bc}$ and $\Xi_{bb}$ and has been raised by C.H. Chang, J.X. Wang and X.G. Wu [Comput. Phys. Commun. 177 (2007) 467; Comput. Phys. Commun. 181 (2010) 1144]. In comparison with the previous GENXICC versions, we update the program in order to generate the unweighted baryon events more effectively under various simulation environments, whose distributions are now generated according to the probability proportional to the integrand. One Les Houches Event (LHE) common block has been added to produce a standard LHE data file that contains useful information of the doubly heavy baryon and its accompanying partons. Such LHE data can be conveniently imported into PYTHIA to do further hadronization and decay simulation, especially, the color-flow problem can be solved with PYTHIA8.0.

Higher-order relativistic corrections to gluon fragmentation into spin-triplet S-wave quarkonium

We compute the relative-order-v^4 contribution to gluon fragmentation into quarkonium in the 3S1 color-singlet channel, using the nonrelativistic QCD (NRQCD) factorization approach. The QCD fragmentation process contains infrared divergences that produce single and double poles in epsilon in 4-2epsilon dimensions. We devise subtractions that isolate the pole contributions, which ultimately are absorbed into long-distance NRQCD matrix elements in the NRQCD matching procedure. The matching procedure involves two-loop renormalizations of the NRQCD operators. The subtractions are integrated over the phase space analytically in 4-2epsilon dimensions, and the remainder is integrated over the phase-space numerically. We find that the order-v^4 contribution is enhanced relative to the order-v^0 contribution. However, the order-v^4 contribution is not important numerically at the current level of precision of quarkonium-hadroproduction phenomenology. We also estimate the contribution to hadroproduction from gluon fragmentation into quarkonium in the 3PJ color-octet channel and find that it is significant in comparison to the complete next-to-leading-order-in-alpha_s contribution in that channel.

O(αsv2)correction toe+e−→J/ψ+ηcatBfactories

We investigate the ${\mathcal O}(\alpha_s v^2)$ correction to the $e^+e^-\to J/\psi+\eta_c$ process in the nonrelativistic QCD (NRQCD) factorization approach. Within some reasonable choices of the relative order-$v^2$ NRQCD matrix elements, we find that including this new ingredient of correction only mildly enhances the existing NRQCD predictions. We have also deduced the asymptotic expressions for the ${\mathcal O}(\alpha_s v^2)$ short-distance coefficients, and reconfirm the early speculation that at next-to-leading order in $\alpha_s$, the double logarithm of type $\ln^2 (s/m_c^2)$ appearing in various NRQCD short-distance coefficients is always associated with the helicity-suppressed channels.

Properties of the quark-antiquark-gluon Fock state in theηcmeson

In this paper, the twist-3 distribution amplitude of the quark-antiquark-gluon Fock state in the ${\ensuremath{\eta}}_{c}$ meson is studied. To calculate the moments of this distribution amplitude, QCD sum rules are applied. Using the results of the calculation, the model of this distribution amplitude is built. In addition, nonrelativistic QCD matrix elements that determine the properties of the quark-antiquark-gluon Fock state are determined. In particular, the following properties have been calculated: probability amplitude to find the quark-antiquark pair in the color-octet state, fraction of momentum carried by gluon, relative velocity of the color-octet quark-antiquark pair.

$ \mathcal{O}\left( {{\alpha_s}{v^2}} \right) $ correction to pseudoscalar quarkonium decay to two photons

We investigate the $O(\alpha_s v^2)$ correction to the process of pseudoscalar quarkonium decay to two photons in nonrelativistic QCD (NRQCD) factorization framework. The short-distance coefficient associated with the relative-order $v^2$ NRQCD matrix element is determined to next-to-leading order in $\alpha_s$ through the perturbative matching procedure. Some technical subtleties encountered in calculating the ${O(\alpha_s)$ QCD amplitude are thoroughly addressed.

Color-singlet relativistic correction to inclusiveJ/ψproduction associated with light hadrons atBfactories

We study the first-order relativistic correction to the associated production of $J/\ensuremath{\psi}$ with light hadrons at $B$ factory experiments at $\sqrt{s}=10.58\text{ }\text{ }\mathrm{GeV}$, in the context of nonrelativistic QCD (NRQCD) factorization. We employ a strategy for NRQCD expansion that slightly deviates from the orthodox doctrine, in that the matching coefficients are not truly of a ``short-distance'' nature, but explicitly depend upon physical kinematic variables rather than partonic ones. Our matching method, with validity guaranteed by the Gremm-Kapustin relation, is particularly suited for the inclusive quarkonium production and decay processes with involved kinematics, exemplified by the process ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}J/\ensuremath{\psi}+gg$ considered in this work. Despite some intrinsic ambiguity affiliated with the order-${v}^{2}$ NRQCD matrix element, if we choose its value as what has been extracted from a recent Cornell-potential-model-based analysis, including the relative order-${v}^{2}$ effect is found to increase the lowest-order prediction for the integrated $J/\ensuremath{\psi}$ cross section by about 30%, and exert a modest impact on $J/\ensuremath{\psi}$ energy, angular and polarization distributions except near the very upper end of the $J/\ensuremath{\psi}$ energy. The order-${v}^{2}$ contribution to the energy spectrum becomes logarithmically divergent at the maximum of $J/\ensuremath{\psi}$ energy. A consistent analysis may require that these large end-point logarithms be resummed to all orders in ${\ensuremath{\alpha}}_{s}$.

Production of theX(3872)at the Tevatron and the LHC

We predict the differential cross sections for production of the $X(3872)$ at the Tevatron and the Large Hadron Collider from both prompt QCD mechanisms and from decays of $b$ hadrons. The prompt cross section is calculated using the nonrelativistic QCD (NRQCD) factorization formula. Simplifying assumptions are used to reduce the nonperturbative parameters to a single nonrelativistic QCD matrix element that is determined from an estimate of the prompt cross section at the Tevatron. For $X(3872)$ with transverse momenta greater than about 4 GeV, the predicted cross section is insensitive to the simplifying assumptions. We also discuss critically a recent analysis that concluded that the prompt production rate at the Tevatron is too large by orders of magnitude for the $X(3872)$ to be a weakly bound charm-meson molecule. We point out that if charm-meson rescattering is properly taken into account, the upper bound is increased by orders of magnitude and is compatible with the observed production rate at the Tevatron.

Leading twist distribution amplitudes ofP-wave nonrelativistic mesons

This paper is devoted to the study of the leading twist distribution amplitudes of $P$-wave nonrelativistic mesons. It is shown that at the leading order approximation in relative velocity of quark-antiquark pair inside the mesons these distribution amplitudes can be expressed through one universal function. As an example, the distribution amplitudes of $P$-wave charmonia mesons are considered. Within QCD sum rules the model for the universal function of $P$-wave charmonia mesons is built. In addition, relations are found between the moments of the universal function and the nonrelativistic QCD matrix elements that control relativistic corrections to any amplitude involving $P$-wave charmonia. Our calculation shows that characteristic size of these corrections is of order of $\ensuremath{\sim}30%$.

Resummation of relativistic corrections toe+e−→J/ψ+ηc

We present a new calculation, in the nonrelativistic QCD (NRQCD) factorization formalism, of the relativistic corrections to the double-charmonium cross section $\ensuremath{\sigma}[{e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}J/\ensuremath{\psi}+{\ensuremath{\eta}}_{c}]$ at the energy of the Belle and BABAR experiments. In comparison with previous work, our calculation contains several refinements. These include the use of the improved results for the nonperturbative NRQCD matrix elements, the resummation of a class of relativistic corrections, the use of the vector-meson-dominance method to calculate the fragmentation contribution to the pure QED amplitude, the inclusion of the effects of the running of $\ensuremath{\alpha}$, and the inclusion of the contribution that arises from the interference between the relativistic corrections and the corrections of next-to-leading order in ${\ensuremath{\alpha}}_{s}$. We also present a detailed estimate of the theoretical uncertainty. We conclude that the discrepancy between the theoretical prediction for $\ensuremath{\sigma}[{e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}J/\ensuremath{\psi}+{\ensuremath{\eta}}_{c}]$ and the experimental measurements has been resolved.

Improved determination of color-singlet nonrelativistic QCD matrix elements forS-wave charmonium

We present a new computation of $S$-wave color-singlet nonrelativistic QCD matrix elements for the $J/\ensuremath{\psi}$ and the ${\ensuremath{\eta}}_{c}$. We compute the matrix elements of leading order in the heavy-quark velocity $v$ and the matrix elements of relative order ${v}^{2}$. Our computation is based on the electromagnetic decay rates of the $J/\ensuremath{\psi}$ and the ${\ensuremath{\eta}}_{c}$ and on a potential model that employs the Cornell potential. We include relativistic corrections to the electromagnetic decay rates, resumming a class of corrections to all orders in $v$, and find that they significantly increase the values of the matrix elements of leading order in $v$. This increase could have important implications for theoretical predictions for a number of quarkonium decay and production processes. The values that we find for the matrix elements of relative order ${v}^{2}$ are somewhat smaller than the values that one obtains from estimates that are based on the velocity-scaling rules of nonrelativistic QCD.

NRQCD factorization for twist-2 light-cone wave-functions of charmonia

We show that the twist-2 light-cone wave-functions of η c and J / ψ can be factorized with nonrelativistic QCD (NRQCD) at the one-loop level, where the nonperturbative effects are represented by NRQCD matrix elements. In quarkonium at rest the c - or c ¯ -quark moves with a small velocity v . The factorization is achieved by expanding the small velocity v . To the leading order of v , the twist-2 light-cone wave-functions of η c and J / ψ can be factorized as the product of a perturbative function and an NRQCD matrix element. The perturbative function is calculated at the one-loop level and is free from any soft divergences. Our results can be used for exclusive productions of quarkonia, which are studied in an experiment with two B -factories.

NRQCD factorization and universality of NRQCD matrix elements

The approach of nonrelativistic QCD (NRQCD) factorization was proposed to study inclusive production of a quarkonium. It is widely used and successful. However, a recent study of gluon fragmentation into a quarkonium at two-loop level shows that the factorization is broken. It is suggested that the color-octet NRQCD matrix elements should be modified by adding a gauge link to restore the factorization. The modified matrix elements may have extra soft-divergences at one-loop level which the unmodified cannot have, and this can lead to a violation of the universality of these matrix elements. In this letter, we examine in detail the NRQCD factorization for inclusive quarkonium production in e + e − annihilation at one-loop level. Our results show that the factorization can be made without the modification of NRQCD matrix elements and it can also be made for relativistic corrections. It turns out that the suggested gauge link will not lead to nonzero contributions to color-octet NRQCD matrix elements at one-loop level and at any order of v. Therefore the universality holds at least at one-loop level.