When computed to next-to-leading order in perturbative QCD, the non-linear Balitsky-Kovchegov (BK) equation for the high-energy evolution of the dipole-hadron scattering appears to be unstable. We show that this instability can be avoided by using the rapidity of the dense hadronic target (instead of that of the dilute dipole projectile) as the evolution time. Using this variable, we construct a collinearly-improved version of the BK equation, where the dominant radiative corrections to the kernel -- those enhanced by double collinear logarithms -- are resummed to all orders.
We present a universal formalism for transverse momentum resummation in the view of soft-collinear effective theory (SCET), and establish the relation between our SCET formula and the well known Collins-Soper-Sterman's pQCD formula at the next-to-leading logarithmic order (NLLO). We also briefly discuss the reformulation of joint resummation in SCET.
We present a method of merging the exclusive LO-matched high energy resummation of High Energy Jets (HEJ) with the parton shower of Pythia which preserves the accuracy of the LO cross sections and the logarithmic accuracy of both resummation schemes across all of phase space. Predictions produced with this merging prescription are presented with comparisons to data from experimental studies and suggestions are made for further observables and experimental cuts which highlight the importance of both high energy and soft-collinear effects.
I discuss general unified formulas for resumming collinear and soft contributions to QCD hard scattering cross sections at large x. Expansions of the resummed cross sections to next-to-next-to-next-to-leading order are also shown along with applications of the formalism.
We consider Drell-Yan process in the threshold region $z\to 1$ where large logarithms appear due to soft-gluon radiations. We present a soft-collinear effective theory approach to re-sum these Sudakov-type logarithms following an earlier treatment of deep inelastic scattering, and the result is consistent with that obtained earlier through perturbative QCD factorization
The proton structure function F2 is studied in the low x regime using BFKL evolution. The next to leading logarithmic (NLL) analysis requires the inclusion of running coupling effects which lead to off-diagonal terms in the evolution kernel. An all-orders resummation is used to improve the collinear behavior of the NLL BFKL result. We emphasize the theoretical uncertainties that appear throughout the analysis and give a comparison to the combined HERA data.
The summation of the small x-corrections to hard-scattering QCD amplitudes by collinear factorisation method is reconsidered and the K-factor is derived in leading ln x approximation with a result differing from the corresponding expression by Catani and Hautmann (Nucl. Phys. B 427, 475, 1994). The significance of the difference is demonstrated in the examples of structure function F_L and of exclusive vector meson electroproduction. The formulation covers the channels of non-vanishing conformal spin n paving the way for new applications.
Scattering processes in laser backgrounds are degenerate to the emission of unobservable photons collinear with the laser. We identify processes and observables for which such degeneracies factorise and exponentiate, obtaining the leading-order intensity dependence of these inclusive observables at high laser intensity, correct to all orders in the fine-structure constant (all loops, all emissions). The results show an exponential intensity dependence distinct from that predicted by the Narozhny-Ritus conjecture on the high-intensity behaviour of quantum electrodynamics in strong fields.
The procedure to improve the convergence in transverse momentum space of the NLL BFKL kernel using a w-shift is revisited. An accurate approximation to this shift only depending on transverse momenta is presented. This approximation is based on a Bessel function of the first kind with argument depending on the strong coupling and a double logarithm of the ratio of transverse scales. A comparison between different renormalization schemes is also included.
We present a global fit to HERA data on the reduced cross section measured in electron-proton collisions in the region of small Bjorken-$x$: $x\le x_0=10^{-2}$ and moderate to high values of the virtuality $Q^2<Q^2_{max}=650$ GeV$^2$. The main dynamical ingredients in the fits are two recently proposed improved BK equations for the description of the small-$x$ evolution of the dipole scattering amplitude. These two new equations provide an all-order resummation of double collinear logarithms that arise beyond leading logarithmic accuracy. We show that a very good description of data is possibl…
We analyze transverse thrust in the framework of Soft Collinear Effective Theory and obtain a factorized expression for the cross section that permits resummation of terms enhanced in the dijet limit to arbitrary accuracy. The factorization theorem for this hadron-collider event-shape variable involves collinear emissions at different virtualities and suffers from a collinear anomaly. We compute all its ingredients at the one-loop order, and show that the two-loop input for next-to-next-to-leading logarithmic accuracy can be extracted numerically, from existing fixed-order codes.
In this thesis we present an introduction to Soft-Collinear Effective Theory, which can be used to prove (or disprove) factorization theorems to all orders in the strong coupling constant for some B decays into light and energetic particles. Specifically, the factorizable amplitudes for inclusive B->X_u l nu and exclusive B->gamma l nu are calculated in renormalization-group improved perturbation theory to first non-trivial order. Form factors encoding the exclusive decay amplitudes for B->P l nu and B->V l nu (P=light pseudoscalar meson, V=light vector meson) are studied and proved to be domi…
We review soft-collinear effective theory (SCET), and apply it to discuss quark electromagnetic form factor, then present the resumed transverse momentum distribution of Higgs-boson production via gluon fusion under this framework, where we derive a relatively full differential formula in transverse momentum $Q_T$ space like one which have been obtained by Dokshitzer-D'Yanov-Troyan (DDT) in perturbative Quantum Chromodynamics (pQCD). Furthermore, our above result can be generalized to even higher order. Comparing our formula with the integral formula of Collins-Soper-Sterman (CSS) in impact pa…
In this article we show how the resummation of infrared and collinear logarithms within Soft-Collinear Effective Theory (SCET) can be formulated in a way that makes it suitable for a Monte-Carlo implementation. This is done by applying the techniques developed for automated resummation using the branching formalism, which have resulted in the general resummation approach CAESAR/ARES. This work builds a connection between the two resummation approaches, and paves the way to automated resummation in SCET. As a case study we consider the resummation of the thrust distribution in electron-positron…
We derive a factorization theorem for the Higgs boson transverse momentum (p_T) and rapidity (Y) distributions at hadron colliders, using the Soft Collinear Effective Theory (SCET), for m_h>> p_T>> Λ_{QCD} where m_h denotes the Higgs mass. In addition to the factorization of the various scales involved, the perturbative physics at the p_T scale is further factorized into two collinear impact-parameter Beam Functions (iBFs) and an inverse Soft Function (iSF). These newly defined functions are of a universal nature for the study of differential distributions at hadron colliders. The additional f…
Within the framework of the dipole factorisation, we use a recent collinearly-improved version of the Balitsky-Kovchegov equation to fit the HERA data for inclusive deep inelastic scattering at small Bjorken $x$. The equation includes an all-order resummation of double and single transverse logarithms and running coupling corrections. Compared to similar equations previously proposed in the literature, this work makes a direct use of Bjorken $x$ as the rapidity scale for the evolution variable. We obtain excellent fits for reasonable values for the four fit parameters. We find that the fit qua…
We study the link between parton dynamics in the collinear limit and the logarithmically enhanced terms of the groomed jet mass distribution, for jets groomed with the modified mass-drop tagger (mMDT). While the leading-logarithmic (LL) result is linked to collinear evolution with leading-order splitting kernels, here we derive the NLL structure directly from triple-collinear splitting kernels. The calculation we present is a fixed-order calculation in the triple-collinear limit, independent of resummation ingredients and methods. It therefore constitutes a powerful cross-check of the NLL resu…
We present a master formula, with applicability conditions, which allows us to automate the resummation of infrared and collinear logarithms appearing in distributions of jet observables in QCD at next-to-leading logarithmic accuracy.
We explore similarities and differences between widely-used threshold resummation formalisms, employing electroweak boson production as an instructive example. Resummations based on both full QCD and soft-collinear effective theory (SCET) share common underlying factorizations and resulting evolution equations. The formalisms differ primarily in their choices of boundary conditions for evolution, in moment space for many treatments based on full QCD, and in momentum space for treatments based on soft-collinear effective theory. At the level of factorized hadronic cross sections, these choices …
We consider the quark beam function that describes collinear initial-state radiation that is constrained by a veto on reconstructed jets. As the veto is imposed on the transverse momenta of the jets, the beam function is subject to rapidity divergences, and we use the collinear-anomaly framework to extract the perturbative matching kernels to next-to-next-to-leading order (NNLO) in the strong-coupling expansion. Our calculation is based on a novel framework that automates the computation of beam functions in Mellin space and it provides the ingredients to extend jet-veto resummations for quark…
We parameterize the enhancement of threshold effects away from hadronic endpoint that arise due to the steeply falling nature of parton distribution functions, within the context of soft-collinear effective theory. This is accomplished in a process-independent way by directly linking the charac- teristic scale of soft and collinear radiation, λ, to the shape of the pdfs. This allows us quantify the power corrections to partonic threshold resummation as a function of the invariant mass and rapidity of the final state. In the context of SCET, being able to compute λin a process-independent manne…
We analyze soft and collinear gluon resummation effects at the N$^3$LL level for Standard Model Higgs boson production via gluon fusion $gg\to H$ and the neutral scalar and pseudoscalar Higgs bosons of the minimal supersymmetric extension at the N$^3$LL and NNLL level, respectively. We introduce refinements in the treatment of quark mass effects and subleading collinear gluon effects within the resummation. Soft and collinear gluon resummation effects amount to up to about 5% beyond the fixed-order results for scalar and pseudoscalar Higgs boson production.
We derive an all order resummation formula for the deeply virtual Compton scattering (DVCS) amplitude, which takes into account soft gluon exchanges in the non-singlet quark coefficient function. We identify the ladder diagrams responsible in light-like gauge for [alphaS log^2 (x +/- xi)]^n contributions. The resummed series results in a simple closed expression.
Starting from the first renormalized factorization theorem for a process described at subleading power in soft-collinear effective theory, we discuss the resummation of Sudakov logarithms for such processes in renormalization-group improved perturbation theory. Endpoint divergences in convolution integrals, which arise generically beyond leading power, are regularized and removed by systematically rearranging the factorization formula. We study in detail the example of the $b$-quark induced $h\toγγ$ decay of the Higgs boson, for which we resum large logarithms of the ratio $M_h/m_b$ at next-to…
Soft-collinear effective theory is used to prove factorization of the B->gamma+l+nu decay amplitude at leading power in Lambda/m_b, including a demonstration of the absence of non-valence Fock states and of the finiteness of the convolution integral in the factorization formula. Large logarithms entering the hard-scattering kernel are resummed by performing a two-step perturbative matching onto the low-energy effective theory, and by solving evolution equations derived from the renormalization properties of the leading-order B-meson light-cone distribution amplitude. As a byproduct, the evolut…
The bilocal expansion of Borel transform provides an efficient way of Borel resummation with low order perturbations in QCD. Its applications to the heavy quark pole mass, static potential, and lattice calculation are reviewed.
We describe the resummation of singular distributions at partonic threshold in heavy quark production, generalizing results for Drell-Yan and other electroweak hard-scattering processes.
It is shown that amplitude-based, exact resummation tames the un-cancelled IR divergences at O(alpha_s^2) in initial state radiation in QCD with massive quarks. Implications for precision predictions for LHC physics are discussed.
We present the theory of QED x QCD resummation and its interplay with shower/matrix element matching in precision LHC physics scenarios. We illustrate the theory using single heavy gauge boson production at hadron colliders.
I give a pedagogical introduction to resummation and power corrections, using the thrust variable in electron-positron annihilation as an example, followed by an discussion of issues of predictability in perturbative QCD.
I present results for the resummation of soft and collinear gluon contributions to QCD hard-scattering cross sections at two loops. This requires the calculation of UV and IR divergences through two loops and the construction of soft anomalous dimensions that control resummation at NNLL. Explicit results are provided for processes that involve massless and/or massive quarks, including single top and top pair production.
These lectures provide an introduction to Soft-Collinear Effective Theory. As applications we discuss resummation for (i) soft-photon effects in QED, (ii) event-shape variables and (iii) jet cross sections.
We study applications of QCD soft-gluon resummations to electroweak annihilation cross sections. We focus on a formalism that allows to resum logarithmic corrections arising near partonic threshold and at small transverse momentum simultaneously.
We show that the resummation of large radiative corrections in QCD processes can be performed both in covariant gauge and in axial gauge. We extend the resummation technique to inclusive processes, concentrating on deeply inelastic scattering, Drell-Yan production, and inclusive heavy meson decays in end-point regions. Leading and next-to-leading logarithms produced by radiative corrections are organized into Sudakov factors, whose expressions are the same as those derived from the conventional Wilson-loop approach. The gauge invariance of the resummation in the end-point region is demonstrate…
These lectures provide an introduction to Soft-Collinear Effective Theory. After discussing the expansion of Feynman diagrams around the high-energy limit, the effective Lagrangian is constructed, first for a scalar theory, then for QCD. The underlying concepts are illustrated with the Sudakov form factor, i.e. the quark vector form factor at large momentum transfer. We then apply the formalism in two examples: We perform soft gluon resummation as well as transverse-momentum resummation for the Drell-Yan process using renormalization group evolution in SCET, and we derive the infrared structur…
We investigate the resummation effects for the SM Higgs and vector boson associated production at the LHC with a jet veto in soft-collinear effective theory using "collinear anomalous" formalism. We calculate the jet vetoed invariant mass distribution and the cross section for this process at Next-to-Next-to-Leading-Logarithmic level, which are matched to the QCD Next-to-Leading Order results, and compare the differences of the resummation effects with different jet veto $p_{T}^{\rm veto}$ and jet radius $R$. Our results show that both resummation enhancement effects and the scale uncertaintie…
I present results for the resummation of soft-gluon contributions to QCD hard-scattering cross sections at next-to-next-to-leading logarithm accuracy. A key ingredient is the calculation of two-loop soft anomalous dimensions for the partonic processes. Explicit expressions and applications are provided for processes that involve massless partons and/or massive quarks.
We summarize our recent results on small x resummation in full QCD with n_f quark flavours and discuss their phenomenological impact in the extraction of parton distributions from present day structure function data and their extrapolation to the kinematics relevant for future colliders such as the LHC.
We build a computer code that fully automates the resummation of jet-observable distributions at next-to-leading logarithmic accuracy. As an application we present results for a jet shape in hadronic dijet production.
The new relativistic Coulomb-like threshold resummation $S$-factor in quantum chromodynamics is obtained. The consideration is given within the framework of quasipotential approach in quantum field theory which is formulated in relativistic configuration representation for two particles of unequal masses.
The resummation of logarithmically-enhanced terms to all perturbative orders is a prerequisite for many studies of QCD final-states. Until now such resummations have always been performed by hand, for a single observable at a time. In this letter we present a general `master' resummation formula (and applicability conditions), suitable for a large class of observables. This makes it possible for next-to-leading logarithmic resummations to be carried out automatically given only a computer routine for the observable. To illustrate the method we present the first next-to-leading logarithmic resu…
We perform a systematic comparison of soft-gluon resummation in SCET and in a standard "direct" QCD (dQCD) approach, both from an analytical and a phenomenological point of view. In particular, we concentrate on Higgs boson production in gluon-gluon fusion at a hadron collider. Using specifically a saddle point argument, we arrive at a detailed understanding of the quantitative differences between these two approaches.
We present a study on inclusive emissions of a double $Λ_c$ or of a $Λ_c$ plus a light-flavored jet system as probe channels in the semi-hard regime of QCD. Our formalism relies on the so-called hybrid high-energy/collinear factorization, where the standard collinear description is supplemented by the $t$-channel resummation à la BFKL of energy logarithms up to the next-to-leading accuracy. We make use of the JETHAD modular interface, suited to the analysis of different semi-hard reactions, employing the novel KKSS19 parameterization for the description of parton fragmentation into $Λ_c$ baryo…
Motivated by current searches for electroweak superpartners at the Large Hadron Collider, we present precision predictions for pair production of such particles in the framework of the Minimal Supersymmetric Standard Model. We make use of various QCD resummation formalisms and match the results to pure perturbative QCD computations. We study the impact of scale variations and compare our results to predictions obtained by means of traditionally used Monte Carlo event generators.
We investigate the resummation of large logarithmic perturbative corrections to hadron production in electron-positron annihilation and semi-inclusive deep-inelastic scattering. We find modest, but significant, enhancements of hadron multiplicities in the kinematic regimes accessible in present high-precision experiments. Our results are therefore relevant for the determination of hadron fragmentation functions from data for these processes.
We present calculations of next-to-leading order and resummed QCD corrections for semi-inclusive deep-inelastic scattering and single-inclusive e+e- annihilation. The resummation is performed to next-to-leading logarithmic accuracy. Knowing these QCD corrections is important in order to extract parton distribution functions and fragmentation functions from present and future data for these reactions. We present comparisons of our results to recent data by the COMPASS, HERMES, Belle, and BaBar experiments.
We present an innovative method to resum infrared and collinear logarithms appearing in distributions of jet observables in QCD. The method, based on a general master formula with applicability conditions, allows resummations at next-to-leading logarithmic accuracy in an automated way. As a sample application we present resummed results in hadronic dijet events.
It is argued why thermodynamic approximations in terms of dressed propagators may, at larger coupling strength, be better behaved than perturbative results, and why in hot QCD the hard thermal loop approximation of the thermodynamic potential cannot be expected to work close to the phase transition.
Singular factors originating from the QCD factorisation of scattering amplitudes in soft and collinear limits play a prominent role in both organising and computing high-order perturbative contributions to hard-scattering cross sections. In this talk, we start from the factorisation structure of scattering amplitudes in the collinear limit, and we introduce collinear functions that have a process-independent structure. These collinear functions, which are defined at the fully-differential level, can then be integrated over the appropriate observable-dependent phase space to compute logarithmic…
We compute the QCD form factor resumming threshold logarithms in B --> X_c + l + nu_l 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.
We discuss the resummation of distributions that are singular at the elastic limit of partonic phase space (partonic threshold) in QCD hard-scattering cross sections, such as heavy quark production. We show how nonleading soft logarithms exponentiate in a manner that depends on the color structure within the underlying hard scattering. This result generalizes the resummation of threshold singularities for the Drell-Yan process, in which the hard scattering proceeds through color-singlet annihilation. We illustrate our results for the case of heavy quark production by light quark annihilation, …
I review recent theoretical progress in the resummation of small x contributions to the evolution of parton distributions, in view of its potential significance for accurate phenomenology at future colliders. I show that a consistent perturbative resummation of collinear and energy logs is now possible, and necessary if one wishes to use recent NNLO results in the HERA kinematic region.
