Angular Ordering Research Articles

Corrections to angular ordering in multiple hadroproduction

One of the key ideas in describing the multiparticle production at high energies in the perturbative QCD is the angular ordering in successive soft-gluon emission. We analyse the angular distribution of particles in a jet, and investigate the corrections to the angular ordering. At a small angle from the jet direction, the angular ordering is exact at the next-to-leading order. At a large angle, the angular ordering overestimates the angular particle density by a factor (1−γln cos2(γ/2)), whereγ is the anomalous dimension of the multiplicity. TheO(γ) correction restores the boost invariance of the cross section.

QCD coherence in the structure function and associated distributions at small x★

We recall the origin of angular ordering in soft parton emission and show that at small x this coherent structure is masked in the structure function while it can be detected in the associated distributions. This is due to the fact that collinear singularities cancel completely in the structure function at fixed transverse momentum for x → 0. In this limit the dependence on the hard scale is lost, the angular ordered region becomes equivalent to the multi-Regge region in which all transverse momenta are of the same order, and one derives the BFKL equation. For the associated distributions at small x such a complete cancellation of collinear singularities does not hold in general, thus large singular contributions are neglected if angular ordering is replaced by multi-Regge phase space. The deduction of these features requires an analysis without any collinear approximations which is done by extending to small x the soft gluon factorization techniques typically used in the region of large x. Since the coherent structure of parton emission is the same in the small and large x regions, one can formulate a unified evolution equation for the structure function, a unified coherent branching and jet algorithm which allows the calculation of associated distributions in all x regions. Such a unified formulation, valid for all x, is presented and compared with usual treatments.

Evidence for gluon interference in hadronic Z decays

We present evidence for soft gluon interference, as required by QCD. This interference is expected to manifest itself in an angular ordering of the gluons radiated within a jet. Using hadronic decays of the Z boson in the L3 detector at LEP, we compare variables sensitive to such an angular ordering, namely the energy-energy correlation asymmetry and the newly introduced particle-particle correlation asymmetry, with the predictions of various parton shower models. Only those models which incorporate the expected interference agree with the data.

Coherence and small x cascade

The main features of coherence of perturbative QCD in the small x region are recalled. At the inclusive level cancellations of collinear singularities implie that angular ordering can be neglected and one recovers the BFKL structure function. In the associated distributions these cancellations are not effective and the coherent structure can not be neglected.

Fractal phase space as a diagnostic tool for high-energy multijet processes.

The fractal phase space introduced by the Lund group for the description of QCD multijet phenomena is discussed using a different choice of coordinates. We advocate these coordinates as a useful diagnostic tool in the analysis of complex event structures. Features of QCD such as color coherence or angular ordering, and the string effect, are described in this language.

On the properties of the color dipoles in QCD

In this paper I will undertake a detailed investigation of the local properties of the color dipoles, i.e. two color connected neighbours, in a QCD cascade. It is known that it is a good approximation (which incorporates in particular “the strong angular ordering conditions” corresponding to coherence) to treat the cascade as proceeding through repeated emission of gluons from such dipoles. The dipole motion reflects basically the motion of a Lund String, in which the gluons are treated as internal kink excitations on the string. Therefore a description of the local properties of the dipoles is basically a description of the behaviour of the String before fragmentation. My major findings are that the dipoles have a (logarithmic) size distribution which settles into a gaussian-like shape, that they show short-range correlations and that they have a multiplicity distribution similar to the blackbody radiation in QED. The dipoles in this description are not color neutral objects. The size of a dipole as used in this paper does, however, reflect the phase space region (at the given virtuality) inside which a color produced in the cascade is neutralised. It is interesting to note that the dipoles in a nonabelian field theory like QCD, which are close to “neutrality”, also behave very similar to the (neutral) photon field quanta in the abelian QED. The relative simplicity to describe the rather complex QCD cascades indicates that it may be worthwhile to try to quantise the theory in terms of such objects. I will use classical gain-loss considerations leading to first-order partial differential equations with simple solutions. I show the close relationship to the corresponding renormalisation group treatment in terms of Callan-Symanzik equations.

Towards an understanding of the static structure of suspensions of ionic rod-like macromolecules

Structural properties of charged rod-like systems are analyzed in order to understand the onset of angular correlations in a concentration regime where the centres of mass separation between the rods is comparable to their elongation L. This study includes Monte Carlo simulations and first-order perturbation theories. The angular dependent correlation function g(12) and the static structure factor S M (k) are expanded in terms of rotational invariants and the first five coefficients are evaluated. In particular, from the g(220; r) coefficient, referred to a space-fixed frame, the angular ordering parameter in the system is calculated. As intermolecular potential, we use a dumb-bell model with a screened Coulomb form between two interaction centres on each rod, which includes a cylindrical hard core part to avoid overlap on increasing the rod concentration. The results of perturbation theories are obtained according to the mean potential, and reference averaged Mayer potential (RAM), the angular median pot...

Monte Carlo simulation of general hard processes with coherent QCD radiation

In this paper we extend our previous work on the simulation of coherent soft-gluon radiation to hard collisions that involve incoming as well as outgoing coloured partons. Existing simulations correctly sum the leading collinear singularities for initial- and final-state radiation, and in some cases the leading infrared contributions from outgoing partons, but not those for incoming (or the interference between incoming and outgoing). Asymptotically, however, the leading infrared and collinear contributions are comparable, the bulk of gluon emission occuring in the soft region. Furthermore, a correct treatment of leading infrared terms is necessary for the inclusive cancellation of singularities in the Sudakov form factor. We show how such a treatment ,ay be formulated in terms of an angular ordering procedure applicable to all hard processes. We then describe a new Monte Carlo program which incorporates this procedure, together with other new features such as azimuthal correlations due to gluon polarization and interference. The program is designed as a general-purpose event generator, simulating hard lepton-lepton, lepton-hadron and hadron-hadron scattering in a single package. Simulation of soft hadronic collisions and underlying events is also included. We present the predictions of the program for a wide variety of processes, and compare them with analytical results and experimental data.

Next-to-leading order corrections to multiplicities

It is desirable to develop a systematic and practical method in perturbative QCD to analyze the semi-hard processes in the fragmentation of jets. In this paper, an integral equation with angular ordering is introduced to analyze the multiplicities of hadrons in high-energy jet productions. In order to demonstrate how the method works, the method is applied in the next-to-leading order to the energy dependence of the multiplicity and to the ratio of multiplicities in a gluon jet and in a quark jet. The results agree with the calculations done by Mueller by a different method.

The Angular Ordering in Soft-Gluon Emission

The way to evaluate multi-parton cross-sections systematically is discussed. In the leading-double-log approximation in QCD, the successive emission of soft gluons is at successively smaller angles. The angular ordering, however, is violated in the next-to-leading order.

Monte Carlo approach to QCD jets in the next-to-leading-logarithmic approximation.

We present a Monte Carlo model for the parton generation in the next-to-leading-logarithmic (NLL) approximation. This model provides us with the natural introduction of cross sections for nontrivial hard scatterings and it enables us to determine the scheme for the fundamental parameter \ensuremath{\Lambda} in QCD. The implementation of the NLL effects into the Monte Carlo approach is studied extensively. The stochastic treatment requires the positivity of functions for hard scatterings, two-body decays, and three-body decays. This leads to the condition of angular ordering which restricts the phase space. We apply this model to ${e}^{+}$${e}^{\mathrm{\ensuremath{-}}}$ annihilation and incorporate clean three-jet events into the QCD cascade.

A comparative study of coherent and non-coherent parton shower evolution

We present a new algorithm for the probalistic evolution of jet cascades, based on the leading log approximation to perturbative QCD, augmented by coherence studies. Compared to previous models, emphasis is put on expressing kinematical variables in terms of Lorentz invariants and on matching on to the three-jet matrix element in e +e − annihilation. A systematic study is made of various assumptions in the model, such as the implementation of kinematics, the choice of Q 2 scale and the inclusion of angular ordering. In particular, we address the issue of how to find experimental evidence for coherence effects.

THREE-BODY DECAY FUNCTIONS IN QCD JETS

Jet evolution based on perturbative QCD is studied taking into account the next-to-leading-order effect. The 3-body decay functions are required to be positive to generate the jet evolution in a Monte Carlo simulation and their magnitude is to be small enough to assure the perturbative expansion. We present a description of the angular ordering which enables us to obtain decay functions for gluon→3 gluons and quark→ quark+2 gluons which are positive and of sizable magnitude.