We extend our previous work on scattering amplitudes [1] to hadron collisions. We provide a general formalism for the computation of eikonal amplitudes squared for the radiation of soft and energy-ordered gluons off three hard-legs at finite-N$_c$ to any order in the perturbative expansion. Examples of three-hard legs processes include vector/Higgs boson production in association with a single hard jet and dijet production in DIS. Explicit expressions for the radiation of up to four gluons are provided as an illustration of the formalism.
We approach the issue of exponentiation of soft gauge boson corrections to scattering amplitudes from a path integral point of view. We show that if one represents the amplitude as a first quantized path integral in a mixed coordinate-momentum space representation, a charged particle interacting with a soft gauge field is represented as a Wilson line for a semi-infinite line segment, together with calculable fluctuations. Combining such line segments, we show that exponentiation in an abelian field theory follows immediately from standard path-integral combinatorics. In the non-abelian case, w…
We introduce an eikonal Monte Carlo model (running in conjunction with HERWIG) for simulating multiparticle production in hadron-hadron interactions. We compare our simulated data to the CDF Tevatron measurement of the underlying event activity in hard inelastic proton-antiproton scattering at sqrt(s)=1.8 TeV. By fixing the only free parameter in our model, the total hadron-hadron cross section, we find that our model describes the data better than either the HERWIG Underlying Event model or the Hard Multiparton model.
Most of the models of jet quenching generally assumes that a jet always travels in a straight eikonal path, which is indeed true for sufficiently hard jet but may not be a good one for moderate and low momentum jet. In this article an attempt has been made to relax part of this approximation for 2 --> 3 processes and found a (15-20)% suppression in the differential cross-section for moderately hard jets because of the noneikonal effects. In particular, for the process qq' --> qq'g in the centre of momentum frame the scattering with an angle wider than \pm 0.52π, is literally forbidden unlike t…
We apply factorization and eikonal methods from gauge theories to scattering amplitudes in gravity. We hypothesize that these amplitudes factor into an IR-divergent soft function and an IR-finite hard function, with the former given by the expectation value of a product of gravitational Wilson line operators. Using this approach, we show that the IR-divergent part of the n-graviton scattering amplitude is given by the exponential of the one-loop IR divergence, as originally discovered by Weinberg, with no additional subleading IR-divergent contributions in dimensional regularization.
Soft gluon exponentiation in non-abelian gauge theories can be described in terms of webs. So far this description has been restricted to amplitudes with two hard partons, where webs were defined as the colour-connected subset of diagrams. Here we generalise the concept of webs to the multi-leg case, where the hard interaction involves non-trivial colour flow. Using the replica trick from statistical physics we solve the combinatorial problem of non-abelian exponentiation to all orders. In particular, we derive an algorithm for computing the colour factor associated with any given diagram in t…
Hamiltonian eikonal model for multiple production in high energy hadron-hadron collisions is presented and worked out with the aim of providing a simple frame for various different observables. An important role is played by unitarity which is built in by construction in the Hamiltonian formulation. The eikonal approximation allows both a very effective simplification of the dynamics, and facilitates the discussion on the relevance of possible spatial inhomogeneities of the hadrons. The model is intended to describe only the hard interaction of the constituents, the structure of the incoming h…
New results are presented for total ${\rm pp}/ {\rm \bar p p}$ cross-sections, in the framework of our QCD based model (GGPS). This is an improved eikonal mini-jet model, where soft gluon radiation tames the fast energy rise normally present in mini-jet models. We discuss the variability in our predictions and provide a handy parametrization of our results for the LHC. We find that our model predictions span the range $σ_{tot}^{LHC}=100 ^{+10}_{-13} mb$. While this matches nicely with the range of most other models, it does not agree with recent ones which include a "hard" Pomeron, even though…
We apply recently developed path integral resummation methods to perturbative quantum gravity. In particular, we provide supporting evidence that eikonal graviton amplitudes factorize into hard and soft parts, and confirm a recent hypothesis that soft gravitons are modelled by vacuum expectation values of products of certain Wilson line operators, which differ for massless and massive particles. We also investigate terms which break this factorization, and find that they are subleading with respect to the eikonal amplitude. The results may help in understanding the connections between gravity …
I present results for two-loop soft anomalous dimensions, which are derived from dimensionally regularized diagrams with eikonal quark lines and control soft-gluon emission in hard-scattering processes. Detailed results for the UV poles of the eikonal integrals are shown for massive quarks, and the massless limit is also taken. The construction of soft anomalous dimensions at two-loops allows soft-gluon resummations at NNLL accuracy.
We calculate the probability that the rapidity gaps in diffractive processes survive both eikonal and enhanced rescattering. We present arguments that enhanced rescattering, which violates soft-hard factorization, is not very strong. Accounting for NLO effects, there is no reason to expect that the black disc regime is reached at the LHC. We discuss the predictions for the survival of the rapidity gaps for exclusive Higgs production at the LHC.
We perform a case study of the behavior of gluon radiation beyond the soft approximation, using as an example the Drell-Yan production cross section at NNLO. We draw a careful distinction between the eikonal expansion, which is in powers of the soft gluon energies, and the expansion in powers of the threshold variable $1 - z$, which involves important hard-collinear effects. Focusing on the contribution to the NNLO Drell-Yan K-factor arising from real-virtual interference, we use the method of regions to classify all relevant contributions up to next-to-leading power in the threshold expansion…
A two-component (soft + hard) model (TCM) of hadron production in high-energy nuclear collisions is applied to ensemble-mean $p_t$ (denoted by $\bar p_t$) data for $p$-$p$, $p$-Pb and Pb-Pb collisions from the relativistic heavy ion collider (RHIC) and large hadron collider (LHC). This $\bar p_t$ TCM is directly related to a recently-published TCM for the charge-multiplicity $n_{ch}$ and collision-energy dependence of $p_t$ spectrum data from $p$-$p$ collisions. Multiplicity dependence of the $p$-$p$ spectrum hard component observed in the previous study is consistent with similar behavior for…
An impact parameter representation for soft gluon radiation is applied to obtain both the initial decrease of the total cross-section ($σ_{tot}$) for proton-proton collisions as well as the later rise of $σ_{tot}$ with energy for both $pp$ and $p\bar{p}$. The non-perturbative soft part of the eikonal includes only limited low energy gluon emission and leads to the initial decrease in the proton-proton cross- section. On the other hand, the rapid rise in the hard, perturbative jet part of the eikonal is tamed into the experimentally observed mild increase by soft gluon radiation whose maximum e…
We compute the full-color two-loop five-gluon amplitude for the all-plus helicity configuration. In order to achieve this, we calculate the required master integrals for all permutations of the external legs, in the physical scattering region. We verify the expected divergence structure of the amplitude, and extract the finite hard function. We further validate our result by checking the factorization properties in the collinear limit. Our result is fully analytic and valid in the physical scattering region. We express it in a compact form containing logarithms, dilogarithms and rational funct…
We study the soft limit of one-loop QCD amplitudes and we derive the process-independent factorization formula that controls the singular behaviour in this limit. This is obtained from the customary eikonal factorization formula valid at tree (classical) level by introducing a generalized soft-gluon current that embodies the quantum corrections. We compute the explicit expression of the soft-gluon current at one-loop order. It contains purely non-abelian correlations between the colour charges of each pair of hard-momentum partons in the matrix element. This leads to colour correlations betwee…
Non-linear effects in hadronic interactions are treated by means of enhanced pomeron diagrams, assuming that pomeron-pomeron coupling is dominated by soft partonic processes. It is shown that the approach allows to resolve a seeming contradiction between realistic parton momentum distributions, measured in deep inelastic scattering experiments, and the energy behavior of total proton-proton cross section. Also a general consistency with both ``soft'' and ``hard'' diffraction data is demonstrated. An important feature of the proposed scheme is that the contribution of semi-hard processes to the…
We study the factorization of soft and collinear singularities in dimensionally-regularized fixed-angle scattering amplitudes in massless gauge theories. Our factorization is based on replacing the hard massless partons by light-like Wilson lines, and defining gauge-invariant jet and soft functions in dimensional regularization. In this scheme the factorized amplitude admits a powerful symmetry: it is invariant under rescaling of individual Wilson-line velocities. This symmetry is broken by cusp singularities in both the soft and the eikonal jet functions. We show that the cancellation of thes…
We consider the radiation of three soft gluons in a generic process for multiparton hard scattering in QCD. In the soft limit the corresponding scattering amplitude has a singular behaviour that is factorized and controlled by a colorful soft current. We compute the tree-level current for triple soft-gluon emission from both massless and massive hard partons. The three-gluon current is expressed in terms of maximally non-abelian irreducible correlations. We compute the soft behaviour of squared amplitudes and the colour correlations produced by the squared current. The radiation of one and two…
We discuss different resummations of large logarithms that arise in hard-scattering cross sections of quarks and gluons in regions of large and small x. The large-x logarithms are typically dominant near threshold for the production of a specified final state. These soft and collinear gluon corrections produce large enhancements of the cross section for many processes, notably top quark and Higgs production, and typically the higher-order corrections reduce the factorization and renormalization scale dependence of the cross section. The small-x logarithms are dominant in the regime where the m…
We consider the radiation of three soft gluons in a generic process for multiparton hard scattering in QCD. In the soft limit the corresponding scattering amplitude has a singular behaviour that is factorized and controlled by a colorful soft current. We compute the tree-level current for triple soft-gluon emission from both massless and massive hard partons. The three-gluon current is expressed in terms of maximally non-abelian irreducible correlations. We compute the soft behaviour of squared amplitudes and the colour correlations produced by the squared current. The radiation of one and two…
In the present thesis I discuss the hard spectator interaction amplitude in $B\toππ$ at NLO i.e. at $\mathcal{O}(α_s^2)$. This special part of the amplitude, whose LO starts at $\mathcal{O}(α_s)$, is defined in the framework of QCD factorization. QCD factorization allows to separate the short- and the long-distance physics in leading power in an expansion in $\lqcd/m_b$, where the short-distance physics can be calculated in a perturbative expansion in $α_s$. Compared to other parts of the amplitude hard spectator interactions are formally enhanced by the hard collinear scale $\sqrt{\lqcd m_b}$…
We calculate the classical gluon field of a fast projectile passing through a dense medium. We show that this allows us to calculate both the initial state gluon production in proton-nucleus collisions and the final state gluon radiation off a hard parton produced in nucleus-nucleus collisions. This unified description of these two phenomena makes the relation between the saturation scale $Q_s$ and the transport coefficient $\hat q$ more transparent. Also, we discuss the validity of the eikonal approximation for gluon propagation inside the nucleus in proton-nucleus collisions at RHIC energy.
We present a computation of elastic energy loss of hard partons traversing the bulk hydrodynamical medium created in ultrarelativistic heavy-ion collisions. The model is based on perturbative Quantum Chromodynamics (pQCD) cross sections for $2\to 2$ processes in which a hard incoming parton is assumed to interact with a thermal parton from the medium. We model the interactions of this type in a Monte-Carlo framework to account properly for exact energy-momentum conservation, non-eikonal parton propagation, parton conversion reactions and the possibility to create additional hard recoiling part…
We present a new parton model approach for hadron-hadron interactions and, in particular, for the initial stage of nuclear collisions at very high energies (RHIC, LHC and beyond). The most important aspect of our approach is a self-consistent treatment, using the same formalism for calculating cross sections and particle production, based on an effective, QCD-inspired field theory, where many of the inconsistencies of presently used models will be avoided. In addition, we provide a unified treatment of soft and hard scattering, such that there is no fundamental cutoff parameter any more defini…
