Scattering amplitudes of QCD string in the worldline formalism

I review the derivation of large-N QCD meson scattering amplitudes in the Regge regime, where the effective theory of long strings applies in d = 4. A special attention is payed to the reparametrization path integral which plays a crucial role in the consistency of off-shell amplitudes. I show how the linear Reggeon trajectory is obtained for QCD string in the mean-field approximation, which turns out to be exact for the Nambu-Goto string, and discuss the interrelation with perturbative QCD.     

QCD/string holographic mapping and high-energy scattering amplitudes

We find a one-to-one mapping between low-energy string dilaton states in AdS bulk and high-energy glueball states on the corresponding boundary. This holographic mapping leads to a relation between bulk and boundary scattering amplitudes. From this relation and the dilaton action we find the appropriate momentum scaling for high-energy QCD amplitudes at fixed angles.     

Introduction to quantum chromodynamics at hadron colliders

A basic introduction to the application of QCD at hadron colliders is presented. I briefly review the phenomenological and theoretical origins of QCD, and then discuss factorization and infrared safety, parton distributions, the computation of hard scattering amplitudes and applications of perturbative QCD.     

Color coherence and parton-hadron duality in a model of effective string amplitudes

We derive and discuss a model of effective string amplitudes exhibiting the properties of color coherence and parton-hadron duality. The parameters on which this model depends, the running coupling and the running slope, are constrained by the compatibility with QCD and by the consistency of string quantization. This approach could provide a better theoretical foundation to the phenomenologically successful color dipole model and some hints about the effective string theory of QCD.     

Path integral over reparametrizations: Lévy flights versus random walks

We investigate the properties of the path integral over reparametrizations (or the boundary value of the Liouville field in string theory). Discretizing the path integral, we apply the Metropolis–Hastings algorithm to numerical simulations of a proper (subordinator) stochastic process and find that typical trajectories are not Brownian but rather have discontinuities of the type of Lévy's flights. We study a fractal structure of these trajectories and show that their Hausdorff dimension is zero. We confirm thereby previous results on QCD scattering amplitudes by analytical and numerical calculations. We also perform Monte Carlo simulations of the path integral over reparametrization in the effective string ansatz for a circular Wilson loop and discuss their subtleties associated with the discretization of Douglas' functional.     

qqq→qqq Reaction in Quark-Gluon Matter

Thermalization of quark matter and antiquark matter can be influenced by the scattering processes from quark-quark-antiquark to quark-quark-antiquark in quark-gluon matter, and the scattering amplitudes of such processes are the basis of studying thermalization. According to the perturbative QCD, a Fortran code for deriving individual squared amplitudes of all the qqq→qqq scattering Feynman diagrams and interference terms between different diagrams at order α⁴_s is written and corresponding squared amplitudes and interference terms are derived.     

Duality and multi-gluon scattering

For the six-gluon scattering process we give explicit and simple expressions for the amplitude and its square. To achieve this we use an analogy with string theories to identify a unique procedure for writing the multi-gluon scattering amplitudes in terms of a sum of gauge invariant dual sub-amplitudes multiplied by an appropriate color (Chan-Paton) factor. The sub-amplitudes defined in this way are invariant under cyclic permutations, satisfy powerful identities which relate different non-cyclic permutations and factorize in the soft gluon limit, the two-gluon collinear limit and on multi-gluon poles. Also, to leading order in the number of colors these sub-amplitudes sum incoherently in the square of the full matrix element. The results contained here are important for Monte Carlo studies of multi-processes at hadron colliders as well as for understanding the general structure of QCD.     

Leading logarithmic approximation in QCD

The leading logarithmic approximation (LLA) for the scattering amplitudes in QCD is reviewed. The double-logarithmic asymptotics of scattering amplitudes is obtained as a solution to nonlinear evolution equations in the infrared cutoff. The DGLAP equation describes an evolution of parton distributions with increasing parton virtuality. The evolution of the amplitudes with respect to the scale in the longitudinal subspace is given by the BFKL equation. The gluon and quarks in QCD lie on the Regge trajectories calculable in perturbation theory. Mesons and baryons are composite states of Reggeized quarks. Similarly the Pomeron and Odderon are colorless ground states of Reggeized gluons. In the case of multicolor QCD, the Reggeon field theory in LLA is completely integrable. The Reggeon interactions in QCD are derived from a gauge-invariant effective action. In particular, next-to-leading corrections to the BFKL equation in QCD and in supersymmetric gauge models are obtained in this way.     

Scattering of massive open strings in pure spinor

In Park (2008) [4], it was proposed that the D-brane geometry could be produced by open string quantum effects. In an effort to verify the proposal, we consider scattering amplitudes involving massive open superstrings. The main goal of this paper is to set the ground for two-loop “renormalization” of an oriented open superstring on a D-brane and to strengthen our skill in the pure spinor formulation of a superstring, an effective tool for multi-loop string diagrams. We start by reviewing scattering amplitudes of massless states in the 2D component method of the NSR formulation. A few examples of massive string scattering are worked out. The NSR results are then reproduced in the pure spinor formulation. We compute the amplitudes using the unintegrated form of the massive vertex operator constructed by Berkovits and Chandia (2002) [15]. We point out that it may be possible to discover new Riemann type identities involving Jacobi ?-functions by comparing a NSR computation and the corresponding pure spinor computation.     

Scattering amplitudes in open superstring theory

This review is concerned with scattering amplitudes in open superstring theories. In particular, we introduce two different formalisms to compute tree level amplitudes – the Ramond Neveu Schwarz‐ (RNS‐) and the Pure Spinor (PS‐) formalism. The RNS approach proves to be flexible in describing compactifications from ten to four flat spacetime dimensions. We solve the technical problems due to the underlying interacting conformal field theory on the worldsheet. This is exploited to extract phenomenologically relevant scattering amplitudes of gluons and quarks as well as production‐ and decay rates of massive vibration modes which have already been identified as virtual exchange particles at the massless level. In case of a TeV string scale, string specific signatures in parton collisions might be observed at the LHC experiment in the near future and constitute the first experimental evidence for string theory. These statements apply to a wide class of string vacua and therefore bypass the so‐called landscape problem of string theory. The PS formalism allows for a manifestly supersymmetric treatment of scattering amplitudes in ten spacetime dimensions with sixteen supercharges. We introduce a family of superfields which arises in tree amplitudes of massless open string states and can be naturally identified with diagrams made of cubic vertices. We firstly achieve a compact superspace representation of multiparticle field theory amplitudes and moreover express the complete n point superstring amplitude as a minimal linear combination of partial field theory amplitudes and hypergeometric functions. The latter carry the stringy effects and are analyzed from different perspectives.     

Hard exclusive meson production and nonforward parton distributions

We present an analysis of twist-2, leading order QCD amplitudes for hard exclusive leptoproduction of mesons in terms of double/nonforward parton distribution functions. After reviewing some general features of nonforward nucleon matrix elements of twist-2 QCD string operators, we propose a phenomenological model for quark and gluon nonforward distribution functions. The corresponding QCD evolution equations are solved in the leading logarithmic approximation for flavor nonsinglet distributions. We derive explicit expressions for hard exclusive , , and neutral vector meson production amplitudes and discuss general features of the corresponding cross sections. Received: 12 November 1997 / Published online: 26 February 1998     

High-energy zero-norm states and symmetries of string theory

High-energy limit of zero-norm states in the old covariant first quantized spectrum of the 26D open bosonic string, together with the assumption of a smooth behavior of string theory in this limit, are used to derive infinitely many linear relations among the leading high-energy, fixed-angle behavior of four-point functions of different string states. As a result, ratios among all high-energy scattering amplitudes of four arbitrary string states can be calculated algebraically and the leading order amplitudes can be expressed in terms of that of four tachyons as conjectured by Gross in 1988. A dual calculation can also be performed and equivalent results are obtained by taking the high-energy limit of Virasoro constraints. Finally, we compute all high-energy scattering amplitudes of three tachyons and one massive state at the leading order by saddle-point approximation to verify our results.     

Non-linear QCD evolution with improved triple-pomeron vertices

In a previous publication, we have constructed a set of non-linear evolution equations for dipole scattering amplitudes in QCD at high energy, which extends the Balitsky–JIMWLK hierarchy by including the effects of fluctuations in the gluon number. In doing so, we have relied on the color dipole picture, and we have neglected the non-locality of the two-gluon exchanges in the relation between scattering amplitudes and dipole densities. In this Letter, we relax the latter approximation, and thus restore the correct structure of the ‘triple-pomeron vertex’ which describes the splitting of one BFKL pomeron into two in the terms responsible for fluctuations. The ensuing triple-pomeron vertex coincides with the one previously derived by Braun and Vacca within perturbative QCD. The evolution equations can be recast in a Langevin form, but with a multivariable noise term with off-diagonal correlations. Our equations are shown to be equivalent with the modified version of the JIMWLK equation recently proposed by Mueller, Shoshi, and Wong.     

On the origin of scaling violations in fragmentation functions

We show that the scaling violations observed in the fragmentation functions of neutrino scattering can be mostly of a kinematical origin and not necessarily from a QCDQ ²-evolution. Due to confinement effects, the application of such evolution equations is nontrivial at non-asymptotic energies, but these QCD effects can be taken into account by a dynamical simulation of the parton branching processes followed by a string fragmentation method.     

Extremal black holes as fundamental strings

We show that polarization-dependent string-string scattering provides new evidence for the identification of the Dabholkar-Harvey (DH) string solution with the heterotic string itself. First, we construct excited versions of the DH solution which carry arbitrary left-moving waves yet are annihilated by half the supersymmetries. These solutions correspond in a natural way to Bogomolny-bound-saturating excitations of the ground state of the heterotic string. When the excited string solutions are compactified to four dimensions, they reduce to Sen's family of extremal black hole solutions of the toroidally compactified heterotic string. We then study the large impact parameter scattering of two such string solutions. We develop methods which go beyond the metric on moduli space approximation and allow us to read off the subleading polarization-dependent scattering amplitudes. We find perfect agreement with heterotic string tree amplitude predictions for the scattering of corresponding string states. Taken together, these results clearly identify the string states responsible for Sen's extremal black hole entropy. We end with a brief discussion of implications for the black hole information problem.     

The QCD scattering amplitude from area behaved Wilson loops

We explicitly construct the dominant saddle-point trajectory in the sum-over-path representation of meson scattering amplitudes in large N QCD for area-behaved Wilson loops and show that it dominates in the Regge regime. The graphic representation of the leading trajectory is very similar to the diagrams widely used to illustrate meson scattering.     

On-shell methods in perturbative QCD

We review on-shell methods for computing multi-parton scattering amplitudes in perturbative QCD, utilizing their unitarity and factorization properties. We focus on aspects which are useful for the construction of one-loop amplitudes needed for phenomenological studies at the Large Hadron Collider.     

Solution of the multi-Reggeon compound state problem in multicolor QCD

We study the properties of the color-singlet compound states of Reggeized gluons in multicolor QCD using their relation with non-compact two-dimensional Heisenberg spin magnets. Applying the methods of integrable models, we calculate their spectrum and discuss the application of the obtained results to high-energy asymptotics of the scattering amplitudes in perturbative QCD.