Nonrelativistic QCD Matrix Elements Research Articles

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.

Spin correlations and velocity scaling in color-octet nonrelativistic QCD matrix elements

We compute spin-dependent decay matrix elements for S-wave charmonium and bottomonium in lattice nonrelativistic quantum chromodynamics (NRQCD). Particular emphasis is placed upon the color-octet matrix elements, since the corresponding production matrix elements are expected to appear in the dominant contributions to the production cross sections at large transverse momenta. We use three slightly different versions of the heavy-quark lattice Green's functions in order to minimize the contributions that scale as powers of the ultraviolet cutoff. The lattice matrix elements that we calculate obey the hierarchy that is suggested by the velocity-scaling rules of NRQCD.

Comparison of the color-evaporation model and the nonrelativistic QCD factorization approach in charmonium production

We compare the color-evaporation model (CEM) and nonrelativistic QCD (NRQCD) factorization predictions for inclusive quarkonium production. Using the NRQCD factorization formulas for quarkonium production and for perturbative QQ-bar production, we deduce relationships that are implied by the CEM between the nonperturbative NRQCD matrix elements that appear in the factorization formula for quarkonium production. These relationships are at odds with the phenomenological values of the matrix elements that have been extracted from the Tevatron data for charmonium production at large transverse momentum. A direct comparison of the CEM and NRQCD factorization predictions with the CDF charmonium production data reveals that the CEM fits to the data are generally unsatisfactory, while the NRQCD factorization fits are generally compatible with the data. The inclusion of k_T smearing improves the CEM fits substantially, but significant incompatibilities remain. The NRQCD factorization fits to the chi_c data indicate that multiple gluon radiation is an essential ingredient in obtaining the correct shape of the cross section as a function of p_T.

Soft, collinear, and nonrelativistic modes in radiative decays of very heavy quarkonium

We analyze the end-point region of the photon spectrum in semi-inclusive radiative decays of very heavy quarkonium $(m{\ensuremath{\alpha}}_{\mathrm{s}}^{2}\ensuremath{\gg}{\ensuremath{\Lambda}}_{\mathrm{QCD}}).$ We discuss the interplay of the scales arising in the soft-collinear effective theory, m, $m(1\ensuremath{-}{z)}^{1/2},$ and $m(1\ensuremath{-}z)$ for z close to 1, with the scales of heavy quarkonium systems in the weak coupling regime, m, $m{\ensuremath{\alpha}}_{\mathrm{s}},$ and $m{\ensuremath{\alpha}}_{\mathrm{s}}^{2}.$ For $1\ensuremath{-}z\ensuremath{\sim}{\ensuremath{\alpha}}_{\mathrm{s}}^{2}$ only collinear and (ultra)soft modes are seen to be relevant, but the recently discovered soft-collinear modes show up for $1\ensuremath{-}z\ensuremath{\ll}{\ensuremath{\alpha}}_{\mathrm{s}}^{2}.$ The S- and P-wave octet shape functions are calculated. When they are included in the analysis of the photon spectrum of the $\ensuremath{\Upsilon}(1S)$ system, the agreement with data in the end-point region becomes excellent. The nonrelativistic QCD matrix elements $〈{1}^{3}{S}_{1}|{O}_{8}{(}^{1}{S}_{0})|{1}^{3}{S}_{1}〉$ and $〈{1}^{3}{S}_{1}|{O}_{8}{(}^{3}{P}_{J})|{1}^{3}{S}_{1}〉$ are also obtained.

Exclusive photoproduction of a heavy vector meson in QCD

The process of exclusive heavy vector meson photoproduction, gamma p -> V p, is studied in the framework of QCD factorization. The mass of the produced meson, V = Upsilon or J/Psi, provides a hard scale for the process. We demonstrate, that in the heavy quark limit and at the one-loop order in perturbation theory, the amplitude factorizes in a convolution of a perturbatively calculable hard-scattering amplitude with the generalized parton densities and the nonrelativistic QCD matrix element <O_1>_V. We evaluate the hard scattering amplitude at one-loop order and compare the data with theoretical predictions using an available model for generalized parton distributions.

Radiative decay ofΥinto a scalar glueball

We study the radiative decay of $\Upsilon$ into a scalar glueball $\Upsilon \to \gamma G_s$ using QCD factorization. We find that for this process the non-perturbative effects can be factorized into a matrix element well defined in non-relativistic QCD (NRQCD) and the gluon distribution amplitude. The same NRQCD matrix element appears also in leptonic decay of $\Upsilon$ and therefore can be determined from data. In the asymptotic limit the gluon distribution amplitude is known up to a normalization constant. Using a QCD sum-rule calculation for the normalization constant, we obtain $Br(\Upsilon \to \gamma G_s)$ to be in the range $(1\sim 2)\times 10^{-3}$. We also discuss some of the implications for $\Upsilon \to \gamma f_i$ decays. Near future data from CLEO-III can provide crucial information about scalar glueball properties.

Corrections for two photon decays of χc0 and χc2 and color octet contributions

Using the fact that the c-quark inside a charmonium moves with a small velocity v in the charmonium rest-frame, one can employ an expansion in v to study decays of charmonia and results at the leading order for χ c0,2 → γγ exist in the literature. We study corrections at the next-to-leading order in the framework of nonrelativistic QCD(NRQCD) factorization. The study presented here is different than previous approaches where χ c0,2 is taken as a bound-state of a c c ̄ pair and a nonrelativistic wave-function is used for the pair. We find that the corrections are consist not only of relativistic corrections, but also of corrections from Fock state components of χ c0,2 in which the c c ̄ pair is in a color-octet state. For χ c2 there is also a contribution from a Fock state component in which the pair is in a F-wave state. We determine the factorization formula for decay widths in the form of NRQCD matrix elements representing nonperturbative effects related to χ c0,2 , and calculate the perturbative coefficients at tree-level. Because the NRQCD matrix elements are unknown, a detailed prediction for the decay χ c0,2 → γγ cannot be made, but the effect of these corrections can be determined at certain level. Estimations show that the effect is significant and cannot be neglected.

Exclusive decay of 1−− heavy quarkonium into photon and two pions

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 the distribution amplitude and by using chiral perturbative theory we are able to obtain numerical predictions for the decay. Numerical results show that the decay of $\jpsi$ can be observed at BEPC and at CESR. Experiment observation of this process in this kinematic region at BEPC and CESR can provide information about how gluons are converted into the two pions and may supply a unique approach to study $ I = 0$ s-wave $\pi \pi$ scattering.

NRQCD matrix elements in polarization ofJ/ψproduced frombdecay

We present the non-relativistic QCD (NRQCD) prediction for the polarization of the J-Psi produced in b to J-Psi + X, as well as the helicity-summed production rate. We propose that these observables provide a means of measuring the three most important color-octet NRQCD matrix elements involved in J-Psi production. Anticipating the measurement of the polarization parameter alpha, we determine its expected range given current experimental bounds on the color-octet matrix elements.

Testing quarkonium production with photoproducedJ/ψ+γ

I compute the leading color-octet contributions to the process $\gamma + p \to J/\psi + \gamma (+ X)$ within the non-relativistic QCD (NRQCD) factorization formalism. In the color-singlet model, $J/\psi + \gamma$ can only be produced when the photon interacts through its structure function, while the color-octet mechanism allows for production of $J/\psi + \gamma$ via direct photon-gluon fusion. Resolved photon processes can be easily be distinguished from direct photon processes by examining the fraction of the incident photon energy carried away by the $J/\psi$ in the event. Therefore, this process provides a conclusive test of the color-octet mechanism. $J/\psi + \gamma$ production is particularly sensitive to the NRQCD matrix element which figures prominently in the fragmentation production of $J/\psi$ at large $p_{\perp}$ in hadron colliders. I also examine the predictions of the color evaporation model (CEM) of quarkonium production and find that this process can easily discriminate between the NRQCD factorization formalism and the CEM.

QCD radiative corrections to the leptonic decay rate of the Bc meson.

The QCD radiative corrections to the leptonic decay rate of the ${\mathit{B}}_{\mathit{c}}$ meson are calculated using the formalism of nonrelativistic QCD (NRQCD) to separate short-distance and long-distance effects. The ${\mathit{B}}_{\mathit{c}}$ decay constant is factored into a sum of NRQCD matrix elements each multiplied by a short-distance coefficient. The short-distance coefficient for the leading matrix element is calculated to order ${\mathrm{\ensuremath{\alpha}}}_{\mathit{s}}$ by matching a perturbative calculation in full QCD with the corresponding perturbative calculation in NRQCD. This short-distance correction decreases the leptonic decay rate by approximately 15%.