Heavy Quarkonium Production through the Semi-Exclusive $e^+ e^-$ Annihilation Channels around the $Z^0$ Peak
Zhan Sun, Xing-Gang Wu, Gu Chen, Jun Jiang, Zhi Yang
备注：
28 pages, 20 figures, 15 tables. to be published in Phys.Rev.D. We are grateful for the anonymous referee's comments and suggestions that substantially improve the manuscript
Within the framework of the non-relativistic QCD, we present a detailed discussion on the heavy quarkonium production at the leading order in $α_s$ at a $e^+ e^-$ collider with the collision energy around the $Z^0$ peak. Quarkonia are produced through the semi-exclusive channels $e^{+}e^{-} \rightarrow |H_{Q\bar{Q}}\rangle +X$ with $X=Q\bar{Q}$ or $gg$, where $Q$ indicates a heavy quark (respectively $b$ or $c$). It is noted that in addition to the color-singlet 1S-level quarkonium states, the 2S and 1P color-singlet states and the color-octet $|(Q\bar{Q})[1^3S_1^{({\bf 8})}]g\rangle$ state also provide sizable contributions. The heavy quarkonium transverse momentum and rapidity distributions for the $e^+ e^-$ collision energy $E_{cm}=m_Z$ are presented. For both charmonium and bottomonium production via the $Z^0$ propagator, there is approximate "spin degeneracy" between the spin-triplet and spin-singlet quarkonium states. Uncertainties for the total cross sections are estimated by taking $m_c=1.50\pm0.15$ GeV and $m_b=4.90\pm0.15$ GeV. Around $E_{cm}=m_{Z}$, due to the $Z^0$-boson resonance effect, total cross sections for the channels via the $Z^0$-propagator become much larger than the channels via the virtual photon propagator. We conclude that, in addition to the $B$ factories as BaBar and Belle and the hadronic colliders as Tevatron and LHC, such a super $Z$-factory will present an excellent platform for studying the heavy quarkonium properties.
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Phenomenological studies of exclusive heavy-quarkonium electroproduction at NLO
Université Paris-Saclay, CNRS, IJCLab（巴黎萨克雷大学）；Centre for Cosmology, Particle Physics and Phenomenology (CP3), Université Catholique de Louvain（鲁汶天主教大学宇宙学、粒子物理与现象学中心）
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Exclusive processes at large momentum transfer Q factor into perturbatively calculable short-distance parts and long-distance hadronic wave functions. Usually, only contributions from the leading Fock states have to be included to leading order in 1/Q. We show that for exclusive decays of P-wave quarkonia the contribution from the next-higher Fock state |Q Qbar g> contributes at the same order in 1/Q. We investigate how the constituent gluon attaches to the hard process in order to form colour-singlet final-state hadrons and argue that a single additional long-distance factor is sufficient to …
In this paper, we calculate twist-2 light-cone distribution amplitudes (LCDAs) of the S-wave and P-wave quarkonia (namely $^1S_0$ state $η_{Q}$, $^3S_1$ state $J/ψ(Υ)$, $^1P_1$ state $h_{Q}$ and $^3P_J$ states $χ_{QJ}$ with $J=0,1,2$ and $Q=c,b$) to the next-leading order of the strong coupling $α_s$ and leading order of the velocity expansion $v$ in the non-relativistic QCD (NRQCD). We apply these LCDAs to some single quarkonium exclusive productions at large center-of-mass energy, such as $γ^*\to η_Qγ, χ_{QJ}γ~(J=0,1,2)$, $Z\to η_Qγ, χ_{QJ}γ~(J=0,1,2), J/ψ(Υ)γ, h_Qγ$ and $h\to J/ψγ$, by adop…
Using a four dimensional approach, we show that the singularities for small gluon momenta, which arise in the usual three dimensional treatment of the annihilation decay, disappear if all poles in the relative energy are taken into account correctly in the integration. We obtain an explicit formula for the decay width which involves a non locality originating from the kinetic energy. We calculate not only the familiar logarithmic dependence on the binding energy, but also the constant to be added to the logarithm. For positronium this differs from an earlier result in the literature and leads …
We study the large-order asymptotic behavior of the perturbation series for short-distance coefficients in the NRQCD factorization formulas for the decays J/psi --> e^+e^- and eta_c --> gamma gamma. The short-distance coefficients of the leading matrix elements are calculated to all orders in the large-N_f limit. We find that there is a universal Borel resummable renormalon associated with the cancellation of the Coulomb singularity in the short-distance coefficients. We verify that the ambiguities in the short-distance coefficients from the first infrared renormalon are canceled by ambiguitie…
I present an introduction to the NRQCD factorization method for calculating annihilation rates and inclusive production rates for heavy quarkonium. Using this method, annihilation decay rates and sufficiently inclusive cross sections are factored into long-distance NRQCD matrix elements and perturbative short-distance coefficients. I derive the velocity-scaling rules that are used to estimate the magnitudes of the nonperturbative NRQCD matrix elements and I describe perturbative matching methods for calculating their short-distance coefficients. Some simple applications for which NRQCD factori…
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Two-photon and two-gluon annihilation rates of $η_c$, $η'_c$, $η_b$ and $η'_b$ are estimated in the relativistic Salpeter method. By solving the full Salpeter equation with a well defined relativistic wave function, we estimate $M_{η_c}=2.979\pm 0.432$ GeV, $M_{η'_c}=3.566\pm 0.437$ GeV, $M_{η_b}=9.364\pm 1.120$ GeV and $M_{η'_b}=9.941\pm 1.112$ GeV. We calculated the transition amplitude using the Mandelstam formalism and estimate the decay widths: $Γ(η_c \to 2γ)=7.14\pm 0.95$ KeV, $Γ(η'_c \to 2γ)=4.44\pm 0.48$ KeV, $Γ(η_b \to 2γ)=0.384\pm 0.047$ KeV and $Γ(η'_b \to 2γ)=0.191\pm 0.025$ KeV. W…
A rigorous QCD analysis of the inclusive annihilation decay rates of heavy quarkonium states is presented. The effective-field-theory framework of nonrelativistic QCD is used to separate the short-distance scale of annihilation, which is set by the heavy quark mass $M$, from the longer-distance scales associated with quarkonium structure. The annihilation decay rates are expressed in terms of nonperturbative matrix elements of 4-fermion operators in nonrelativistic QCD, with coefficients that can be computed using perturbation theory in the coupling constant $α_s(M)$. The matrix elements are o…
We study the Central Exclusive Production (CEP) of chi_cJ and eta_c mesons at RHIC in proton-proton collisions. We consider the chi_cJ --> J/psi + gamma decay channels and, recalling that the J_z=0 suppression of the J=1,2 states can be compensated by their larger branchings to J/psi + gamma, present predictions of rates and distributions for chi_c(0,1,2) production. Particular attention is paid to the impact of p_t cuts applied to the outgoing protons, which can influence the relative rates significantly. The distribution in the azimuthal angle difference between the outgoing protons and the …
I review recent progress in the theory of gamma gamma annihilation into meson or baryon pairs at large energy, and of the process gamma* gamma* -> pi0 at large photon virtuality.
Dimensional regularization is incompatible with the standard covariant projection methods that are used to calculate the short-distance coefficients in inclusive heavy quarkonium production and annihilation rates. A new method is developed that allows dimensional regularization to be used consistently to regularize the infrared and ultraviolet divergences that arise in these perturbative calculations. We illustrate the method by calculating the leading color-octet terms and the leading color-singlet terms in the gluon fragmentation functions for arbitrary quarkonium states. We resolve a discre…
We study the exclusive decay of $1^{--}$ heavy quarkonium into one photon and two pions in the kinematic region, where the two-pion system has a invariant mass which is much smaller than the mass of heavy quarkonium. Neglecting effects suppressed by the inverse of the heavy quark mass, the decay amplitude can be factorized, in which the nonperturbative effect related to heavy quarkonium is represented by a non-relativistic QCD matrix element, and that related to the two pions is represented by a distribution amplitude of two gluons in the isoscalar pion pair. By taking the asymptotic form for …
We review recent progress in the description of heavy-quarkonium production in 2->2 processes at next-to-leading order in the factorization framework of nonrelativistic quantum chromodynamics. Specifically, we consider the production of prompt charmonium in association with a hadron jet or a prompt photon in two-photon collisions and exclusive double-charmonium production in e^+e^- annihilation.
The double spin asymmetry for exclusive $J/ψ$ pair production in the polarized $p-p$ collisions at RHIC is investigated. Our study shows that the asymmetry in this process is measurable at RHIC-SPIN experiments in the near future, and, hence it can be used to determine the gluon distribution function $ΔG (x)$ in the polarized nucleon.
The cross section value of doubly charmonium production in e+e- annihilations have been estimated at interaction energy 10.6 GeV. The quarkonium wave function shape, as well as the nonzero value of charm quark have been taken into account.
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The ratio of the photon mediated production or annihilation rates of spin triplet and spin singlet heavy quarkonium states is computed to the next-to-next-to-leading logarithmic accuracy within the nonrelativistic renormalization group approach. The result is presented in analytical form and applied to the phenomenology of $t\bar{t}$, $b\bar{b}$ and $c\bar{c}$ systems. The use of the nonrelativistic renormalization group considerably improves the behaviour of the perturbative expansion and is crucial for accurate theoretical analysis. For bottomonium decays we predict $Γ(η_b(1S) \to γγ)=0.659\…
By employing the method of differential equations, we compute the various types of two-loop master integrals involved in CP-even heavy quarkonium exclusive production and decays. All the integrals presented in this paper can be casted into canonical forms and expressed in terms of Goncharov polylogarithms and Harmonic polylogarithms. These master integrals are frequently used in the calculation of NNLO corrections of the heavy quarkonium production processes, as $γ^*γ\rightarrow Q\bar{Q}$, $e^+e^-\rightarrow γ+ Q\bar{Q}$, and $H/Z^0\rightarrow γ+ Q\bar{Q}$, and decay processes. They are also a…
Software package for Monte-Carlo simulation of e+e- exclusive annihilation channels written in the C++ language for Linux/Solaris platforms has been developed. It incorporates matrix elements for several mechanisms of multipion production in a model of consequent two and three-body resonance decays. Possible charge states of intermediate and final particles are accounted automatically under the assumption of isospin conservation. Interference effects can be taken into acccount. Package structure allows adding new matrix elements written in a gauge-invariant form.
Exclusive proton-antiproton annihilation into two photons at large s (~10 GeV^2) and |t|,|u| ~ s can be described by a generalized parton picture analogous to the 'soft mechanism' in wide-angle real Compton scattering. The two photons are emitted in the annihilation of a single fast quark and antiquark. The matrix element describing the transition of the p-pbar system to a q-qbar pair can be related to the timelike proton elastic form factors as well as to the quark/antiquark distributions measured in inclusive deep-inelastic scattering. The reaction could be studied with the proposed 1.5-15 G…
Forward photoproduction of $J/ψ$ can be used to extract Generalized Parton Distributions(GPD's) of gluons. We analyze the process at twist-3 level and study relevant classifications of twist-3 gluon GPD's. At leading power or twist-2 level the produced $J/ψ$ is transversely polarized. We find that at twist-3 the produced $J/ψ$ is longitudinally polarized. Our study shows that in high energy limit the twist-3 amplitude is only suppressed by the inverse power of the heavy quark mass relatively to the twist-2 amplitude. This indicates that the power correction to the cross-section of unpolarized …
We calculate cross sections of a pair $B_c$ meson production on the basis of two-photon mechanism from electron-positron annihilation. We investigate the production cross sections in nonrelativistic approximation and with the account of relativistic corrections. Relativistic production amplitudes of S-wave pair pseudoscalar, vector and pseudoscalar+vector $B_c$-mesons are constructed on the basis of relativistic quark model. Numerical values of the production cross sections are obtained at different center-of-mass energies. The comparison of one-photon and two-photon annihilation contributions…
The hierarchy of long-distance matrix elements (MEs) for quarkonium production depends on their scaling with the velocity $v$ of the heavy quark in the bound state. Ranges for the velocities in various bound states and uncertainties of colour-singlet MEs are estimated in a quark-potential model. Different possibilities for the scaling with $v$ of the MEs are discussed; they depend on the actual values of $v$ and the QCD scale. As an application, J/psi polarization in e^+e^- annihilation is discussed. The first non-perturbative estimates of colour-octet MEs are presented and compared with pheno…
Exclusive quarkonium pair production in electron-positron collisions is studied in non-relativistic QCD. The obtained cross section for J/ψ+ η_c production in the leading order is confronted against the recent measurements by the Belle Collaboration at KEKB. It is shown that a large renormalization K-factor is necessary to explain the experimental data. We point out that the J^{PC}=0^{-+} nature of the hadronic systems that are assigned to be η_c should be tested by the triple angular distributions in terms of the scattering angle, and, polar and azimuthal angles of J/ψinto leptons. We further…
We analyze the exclusive charmonium $J/ψ+η_c$ pair production in $e^+e^-$ annihilation using the nonfactorized perturbative QCD and the light-front quark model(LFQM) that goes beyond the peaking approximation. We effectively include all orders of higher twist terms in the leading order of QCD coupling constant and compare our nonfactorized analysis with the usual factorized analysis in the calculation of the cross section. We also calculate the quark distribution amplitudes, the Gegenbauer moments, and the decay constants for $J/ψ$ and $η_c$ mesons using our LFQM. Our nonfactorized result enha…
A completely relativistic quark model in the Bethe-Salpter framework is employed to calculate the exclusive production ratio of the neutral over charged Kaon pair in $e^+e^-$ annihilation continuum region for center of mass energies smaller than the $J/Ψ$ mass. The valence quark charge plays the key rôle. The cancellation of the diagrams for the same charge case (in $K_S + K_L$) and the non-cancellation of the diagrams for the different charge case (in $K^-+K^+$) lead to the ratio as $(m_s-m_d)^2/M_{Kaon}^2 \sim 1/10$.
The light-front approach is a relativistic quark model and offers many insights to the internal structures of the hadronic bound states. In this study, we apply the covariant light-front approach to ground-state heavy quarkonium. The pesudoscalar and vector meson decay constants are discussed. We present a detailed study of two-photon annihilation P \to γγand magnetic dipole transition V \to P γprocesses. The numerical predictions of the light-front approach are consistent with the experimental data and those in other approaches. The relations of the light-front approach with the other methods…
The intermediate quarkonium state $χ_2(^3P_2)$ in electron-positron annihilation to proton and antiproton as well as in antiproton-proton annihilation to electron and positron can produce backward-forward asymmetry, when populated through two photon exchange. We use the dispersion relation method, which permits to express the asymmetry in terms of partial widths of quarkonium decay. The asymmetry dependence on the center of mass energy in the range near the resonance is presented. The comparison with a similar effect in these reactions with the neutral $Z$-boson in the intermediate state is gi…
Partial decay widths and branching fractions are calculated for the exclusive decays of the Standard Model Higgs boson into a pair of vector quarkonium states, $ H\to J/ψJ/ψ, H\to ΥΥ, H\to J/ψϕ, H\to J/ψΥ$, with relativistic corrections due to quark motion in mesons taken into account.
The processes of P-wave Quarkonium exclusive decays to two mesons are investigated, in which the final state vector mesons with various polarizations are considered separately. In the calculation, the initial heavy quarkonia are treated in the framework of non-relativistic quantum chromodynamics, whereas for light mesons, the light cone distribution amplitudes up to twist-3 are employed. It turns out that the higher twist contribution is significant and provides a possible explanation for the observation of the hadron helicity selection rule violated processes $χ_{c1}\rightarrow ϕϕ,ωω$ by the …
We study the exclusive double-photon annihilation processes, $e^+ e^- \to γγ^\ast\to γV^0$ and $e^+ e^- \to γ^\ast γ^\ast \to V^0_a V^0_b,$ where the $V^0_i$ is a neutral vector meson produced in the forward kinematical region: $s \gg -t$ and $-t \gg Λ_{\rm QCD}^2$. We show how the differential cross sections $\frac{dσ}{dt}$, as predicted by QCD, have additional falloff in the momentum transfer squared $t$ due to the QCD compositeness of the hadrons, consistent with the leading-twist fixed-$θ_{\rm CM} $ scaling laws. However, even though they are exclusive channels and not associated with the …
