QCD Factorization Formalism Research Articles

Refactorizing NRQCD short-distance coefficients in exclusive quarkonium production

In a typical exclusive quarkonium production process, when the center-of-mass energy, s , is much greater than the heavy quark mass m, large kinematic logarithms of s / m 2 will unavoidably arise at each order of perturbative expansion in the short-distance coefficients of the nonrelativistic QCD (NRQCD) factorization formalism, which may potentially harm the perturbative expansion. This symptom reflects that the hard regime in NRQCD factorization is too coarse and should be further factorized. We suggest that this regime can be further separated into “hard” and “collinear” degrees of freedom, so that the familiar light-cone approach can be employed to reproduce the NRQCD matching coefficients at the zeroth order of m 2 / s and order by order in α s . Taking two simple processes, exclusive η b + γ production in e + e − annihilation and Higgs boson radiative decay into ϒ, as examples, we illustrate how the leading logarithms of s / m 2 in the NRQCD matching coefficients are identified and summed to all orders in α s with the aid of Brodsky–Lepage evolution equation.

Nucleon electromagnetic form factors in QCD

The nucleon electromagnetic form factors are calculated in a light cone QCD sum rules framework using the most general form of the nucleon interpolating current. Using two models for the distribution amplitudes, we predict the form factors. The predictions are also compared with existing experimental data. It is shown that our results describe remarkably well the existing experimental data.

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.

Nucleon to delta electromagnetic transition form factors in lattice QCD

The electromagnetic nucleon to {delta} transition form factors are evaluated using two degenerate flavors of dynamical Wilson fermions and using dynamical sea staggered fermions with domain-wall valence quarks. The two subdominant quadrupole form factors are evaluated for the first time in full QCD to sufficient accuracy to exclude a zero value, which is taken as a signal for deformation in the nucleon-{delta} system. For the Coulomb quadrupole form factor the unquenched results begin to deviate from the quenched results at low q{sup 2} indicating that dynamical lattice results are closer to experiment. This can be taken as a first confirmation of the expected importance of pion cloud contributions on this quantity.

Pion form factors in holographic QCD

Using a holographic dual model of QCD, we compute the pion electromagnetic form factor F_pi(Q^2) in the spacelike momentum transfer region, as well as pion couplings to vector mesons g_rho^(n) pi pi. Spontaneous and explicit chiral symmetry breaking are intrinsic features of this particular holographic model. We consider variants with both ``hard-wall'' and ``soft-wall'' infrared cutoffs, and find that the F_pi(Q^2) data tend to lie closer to the hard-wall model predictions, although both are too shallow for large Q^2. By allowing the parameters of the soft-wall model (originally fixed by observables such as m_rho) to vary, one finds fits that tend to agree better with F_pi(Q^2). We also compute the pion charge radius for a variety of parameter choices, and use the values of f^(n)_rho, g_{rho^(n) pi pi} and m^(n)_rho to observe the saturation of F_pi(0) by rho poles.

Inclusive charm production inΥ(nS)decay

Based on the nonrelativistic QCD (NRQCD) factorization formalism, we calculate the inclusive charm production rate in $\ensuremath{\Upsilon}(nS)$ decay at leading order in the strong coupling constant ${\ensuremath{\alpha}}_{s}$ and the relative velocity $v$ of the $b$ quark in the quarkonium rest frame. The branching fractions for $\ensuremath{\Upsilon}(nS)$ to charm for $n=1$, 2, and 3 are all around 7%--9%. About 60% of the branching fraction into charm is from annihilation of the color-singlet $b\overline{b}$ pair into ${\ensuremath{\gamma}}^{*}\ensuremath{\rightarrow}c\overline{c}$. Most of the remaining branching fraction is from annihilation of the color-singlet $b\overline{b}$ pair decaying into $c\overline{c}gg$. We also compute the momentum distributions of the charm quark and charmed hadrons in the decays. The virtual-photon contribution to the charm-quark momentum distribution is concentrated at the end point while the $c\overline{c}gg$ contribution is broad. The momentum distributions for charmed hadrons are obtained by convolving the charm-quark momentum distribution with charm fragmentation functions. This makes the momentum distributions for charmed hadrons softer than that for the charm quark. This effect is particularly significant in the virtual-photon contribution.

Investigation of the bottomonium ground stateηbvia its inclusive charm decays

Based on the nonrelativistic QCD factorization formalism, we calculate the bottomonium ground state {eta}{sub b}, inclusive charm decays at the leading order in the strong coupling constant {alpha}{sub s}, and quarkonium internal relative velocity v. The inclusive charm pair production in {eta}{sub b} decay is mainly realized through the {eta}{sub b}{yields}ccg process, where the charm and anticharm quarks then dominantly hadronize into charmed hadrons. The momentum distributions of the final states are presented. In this work, we also calculate the J/{psi} inclusive production rate in the {eta}{sub b} decays, where the color-octet contribution is found to be very important. We expect this study may shed some light on finding {eta}{sub b} or knowing more about its nature.

Threshold resummation in [formula omitted] decays

We compute the QCD form factor resumming threshold logarithms in B→Xc+l+νl decays to next-to-leading logarithmic approximation. We present an interpolation formula including soft as well as collinear effects softened by the non-vanishing charm mass which, in the N-space, reads: σN=exp∫01dy[(1−y)N−1−1]{1y∫Q2y21+ρQ2y2y+ρA[ρ;αS(k2)]dk2k2+1yD[αS(Q2y21+ρ)]+(1y−1y+ρ)Δ[αS(Q2y2y+ρ)]+1y+ρB[αS(Q2y2y+ρ)]}, where y=(mXc2−mc2)/(Q2−mc2), ρ=mc2/(Q2−mc2) and Q=EXc+|p→Xc|. The function A(ρ;αS) describes soft radiation collinearly enhanced off the charm quark and reduces to the standard double-logarithmic function A(αS) in the massless limit ρ→0. The function B(αS) describes hard collinear emission off the charm quark, being the standard jet function. D(αS) and Δ(αS) describe soft radiation not collinearly enhanced off the beauty and the charm quark respectively.

Hadronic form factors in lattice QCD at small and vanishing momentum transfer

The introduction of partially twisted boundary conditions allows weak and electromagnetic form factors to be evaluated at specified values of the hadronic momenta (and hence momentum transfers) in lattice simulations. We present and demonstrate this technique for the computation of the K->pi semileptonic form factor at zero momentum transfer and for the electromagnetic form factor of the pion at arbitrarily small momentum transfers. These exploratory computations are carried out in full QCD with 3 flavours of sea quarks, but with only two values of m_u=m_d which limits our ability to perform the chiral extrapolations. The results should therefore be viewed primarily as a demonstration of the feasibility of the method. For the K-> pi form factor we compare the new technique to the conventional approach and for the pion form factor we assess our results for very small momentum transfer with the help of chiral perturbation theory.

Double-Charm Productione+e−→J/ψ+cc¯atBFactories with Next-to-Leading-Order QCD Corrections

The inclusive J/psi production in e;{+}e;{-}-->J/psicc at B factories is one of the most challenging open problems in heavy quarkonium physics. The observed cross section of this double-charm production process is larger than existing leading order (LO) QCD predictions by a factor of 5. In the nonrelativistic QCD (NRQCD) factorization formalism, we calculate the next-to-leading order (NLO) QCD virtual and real corrections to this process, and find that these corrections can substantially enhance the cross section with a K factor of about 1.8. We further take into account the feeddown contributions from higher charmonium states [mainly the psi(2S) as well as chi_{cJ}] and the two-photon contributions, and find that the discrepancy between theory and experiment can be largely removed.

N-to-ΔAxial Transition Form Factors from Lattice QCD

We evaluate the $N$ to $\ensuremath{\Delta}$ axial transition form factors in lattice QCD with no dynamical sea quarks, with two degenerate flavors of dynamical Wilson quarks, and using domain wall valence fermions with three flavors of staggered sea quarks. We predict the ratio ${C}_{5}^{A}({q}^{2})/{C}_{3}^{V}({q}^{2})$ relevant for parity violating asymmetry experiments and verify the off-diagonal Goldberger-Treiman relation.

Resummation of large endpoint corrections to color-octetJ/ψphotoproduction

An unresolved problem in $J/\ensuremath{\psi}$ phenomenology is a systematic understanding of the differential photoproduction cross section, $d\ensuremath{\sigma}/dz[\ensuremath{\gamma}+p\ensuremath{\rightarrow}J/\ensuremath{\psi}+X$], where $z={E}_{\ensuremath{\psi}}/{E}_{\ensuremath{\gamma}}$ in the proton rest frame. In the nonrelativistic QCD (NRQCD) factorization formalism, fixed-order perturbative calculations of color-octet mechanisms suffer from large perturbative and nonperturbative corrections that grow rapidly in the endpoint region, $z\ensuremath{\rightarrow}1$. In this paper, NRQCD and soft collinear effective theory are combined to resum these large corrections to the color-octet photoproduction cross section. We derive a factorization theorem for the endpoint differential cross section involving the parton distribution function and the color-octet $J/\ensuremath{\psi}$ shape functions. A one-loop matching calculation explicitly confirms our factorization theorem at next-to-leading order. Large perturbative corrections are resummed using the renormalization group. The calculation of the color-octet contribution to $d\ensuremath{\sigma}/dz$ is in qualitative agreement with data. Quantitative tests of the universality of color-octet matrix elements require improved knowledge of shape functions entering these calculations as well as resummation of the color-singlet contribution which accounts for much of the total cross section and also peaks near the endpoint.

Master integrals for massless three-loop form factors: One-loop and two-loop insertions

The three-loop form factors in massless QCD can be expressed as a linear combination of master integrals. Besides a number of master integrals which factorise into products of one-loop and two-loop integrals, one finds 16 genuine three-loop integrals. Of these, six have the form of a bubble insertion inside a one-loop or two-loop vertex integral. We compute all master integrals with these insertion topologies.

Sudakov resummations at higher orders

We summarize our recent results on the resummation of hard-scattering coefficient functions and on-shell form factors in massless perturbative QCD. The threshold resummation has been extended to the fourth logarithmic order for deep-inelastic scattering, Drell-Yan lepton pair production and Higgs production via gluon-gluon fusion. The leading six infrared pole terms have been derived to all orders in the strong coupling constant for the photon-quark-quark and the (heavy-top) Higgs-gluon-gluon form factors. These results have many implications, most notably they lead to a new best estimate for the Higgs production cross section at the LHC.

Nucleon form factors in QCD

We calculate the electromagnetic and the axial form factors of the nucleon within the framework of light cone sum rules (LCSR) to leading order in QCD and including higher twist corrections. In particular we motivate a certain choice for the interpolating nucleon field. We find that a simple model of the nucleon distribution amplitudes which deviate from their asymptotic shape, but much less compared to the QCD sum rule estimates, allows one to describe the data remarkably well.

How sensitive are high- electron spectra at RHIC to heavy quark energy loss?

In nucleus–nucleus collisions, high- p T electron spectra depend on the medium modified fragmentation of their massive quark parents, thus giving novel access to the predicted mass hierarchy of parton energy loss. Here we calculate these spectra in a model, which supplements the perturbative QCD factorization formalism with parton energy loss. In general, we find—within large errors—rough agreement between theory and data on the single inclusive electron spectrum in pp, its nuclear modification factor R A A e , and its azimuthal anisotropy v 2 e . However, the nuclear modification factor depends on the relative contribution of charm and bottom production, which we find to be affected by large perturbative uncertainties. In order for electron measurements to provide a significantly more stringent test of the expected mass hierarchy, one must then disentangle the b - and c -decay contributions, for instance by reconstructing the displaced decay vertices.

Next-to-Leading-Order QCD Correction toe+e−→J/ψ+ηcats=10.6 GeV

One of the most challenging open problems in heavy quarkonium physics is the double charm production in e+e- annihilation at B factories. The measured cross section of e+e- --> J/psi + eta(c) is much larger than leading order (LO) theoretical predictions. With the nonrelativistic QCD factorization formalism, we calculate the next-to-leading order (NLO) QCD correction to this process. Taking all one loop self-energy, triangle, box, and pentagon diagrams into account, and factoring the Coulomb-singular term into the cc bound state wave function, we get an ultraviolet and infrared finite correction to the cross section of e+e- --> J/psi + eta(c) at sqrt[s] = 10:6 GeV. We find that the NLO QCD correction can substantially enhance the cross section with a K factor (the ratio of NLO to LO) of about 1.8-2.1; hence, it greatly reduces the large discrepancy between theory and experiment.

Generalized parton distributions and structure functions from full lattice QCD

We present here the latest results from the QCDSF collaboration for (moments of) structure functions and generalized form factors in full QCD with N f = 2 O ( a ) -improved Wilson fermions based on simulations closer to the chiral and continuum limit.

Publisher's Note: Pion form factor in QCD: From nonlocal condensates to next-to-leading-order analytic perturbation theory [Phys. Rev. D 70, 033014 (2004)

Received 22 September 2004DOI:https://doi.org/10.1103/PhysRevD.70.079906©2004 American Physical Society

Double Diffractive Process Factorization from ηc and γ Associated Production at Tevatron and LHC**The project supported in part by National Natural Science Foundation of China under Grant No. 10175074, the Natural Science Foundation of Hebei Province of China under Grant No. 103143, and the Youth Foundation of North China Electric Power University under Grant No. 060214

We present a study of associated and double diffractive production in collision based on Ingelman-Schlein model, and the framework of non-relativistic QCD factorization formalism for quarkonia production. The prediction of and is more reliable than production, because the associated and production is a pure color-octet process, and the dominant contribution comes from color octet subprocess, which is related to the color octet matrix element of of by the heavy quark spin symmetry and the large production data. We find that the ratio of diffractive to inclusive cross sections is independent of the values of color octet matrix elements, but is sensitive to the gluon factor of the Pomeron and renormalized Pomeron flux factors. So experimental measurement of this ratio can give us more information of the nature of Pomeron and test the assumption of hard diffractive factorization in hadron-hadron collisions.