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 talk we report on the recent progresses on IR logarithms resummation for the Thrust distribution in e^{+}e^{-} collisions. Using renormalisation group (RG) evolution in Laplace space, the resummation of logarithmically enhanced corrections is performed to next-to-next-to-leading logarithmic (NNLL) accuracy. To combine the resummed expressions with the fixed-order results, we derive the log(R)-matching and R-matching of the NNLL approximation to the fixed-order NNLO distribution.
We calculate the resummed distributions for the thrust in DIS in the limit T->1. Two variants of the thrust are considered: that normalised to Q/2, and that normalised to the energy in the current hemisphere. The results expanded to second order are compared to predictions from the Monte Carlo programs DISENT and DISASTER++. A prescription is given for matching the resummed expressions with the full fixed order calculation.
The thrust distribution in electron-positron annihilation is a classical precision QCD observable. Using renormalization group (RG) evolution in Laplace space, we perform the resummation of logarithmically enhanced corrections in the dijet limit, $T\to 1$ to next-to-next-to-leading logarithmic (NNLL) accuracy. We independently derive the two-loop soft function for the thrust distribution and extract an analytical expression for the NNLL resummation coefficient $g_3$. To combine the resummed expressions with the fixed-order results, we derive the $\log(R)$-matching and $R$-matching of the NNLL …
Infrared power corrections for the average thrust < T > in e+e- annihilation are analyzed in the framework of renormalon resummation, motivated by analogy with the skeleton expansion in QED and the BLM approach. Performing the ``massive gluon'' renormalon integral a renormalization scheme invariant result is obtained. We find that a major part of the discrepancy between the known next-to-leading order (NLO) calculation and experiment can be explained by resummation of higher order perturbative terms. This fact does not preclude the infrared finite coupling interpretation with a substantial 1/Q…
We obtain a prediction for the hadron-collider event-shape variable transverse thrust in which the terms enhanced in the dijet limit are resummed to next-to-next-to-leading logarithmic accuracy. Our method exploits universality properties made manifest in the factorized expression for the cross section and only requires one-loop calculations. The necessary two-loop ingredients are extracted using known results and existing numerical codes. Our technique is general and applicable to other observables as well.
Endpoint divergences in the convolution integrals appearing in next-to-leading-power factorization theorems prevent a straightforward application of standard methods to resum large logarithmic power-suppressed corrections in collider physics. We study the power-suppressed configuration of the thrust distribution in the two-jet region, where a gluon-initiated jet recoils against a quark-antiquark pair. With the aid of operatorial endpoint factorization conditions, we derive a factorization formula where the individual terms are free from endpoint divergences and can be written in terms of renor…
The lack of convergence of the convolution integrals appearing in next-to-leading-power (NLP) factorization theorems prevents the applications of existing methods to resum power-suppressed large logarithmic corrections in collider physics. We consider thrust distribution in the two-jet region for the flavour-nonsinglet off-diagonal contribution, where a gluon-initiated jet recoils against a quark-antiquark pair, which is power-suppressed. With the help of operatorial endpoint factorization conditions, we obtain a factorization formula, where the individual terms are free from endpoint divergen…
Infrared power corrections for Minkowskian QCD observables are analyzed in the framework of renormalon resummation, motivated by analogy with the skeleton expansion in QED and the BLM approach. Performing the ``massive gluon'' renormalon integral a renormalization scheme invariant result is obtained. Various regularizations of the integral are studied. In particular, we compare the infrared cutoff regularization with the standard principal value Borel sum and show that they yield equivalent results once power terms are included. As an example the average thrust < T > in e+e- annihilation is an…
On Thrust Resummation Ambiguities in $e^+e^-$ Annihilation into Hadrons
CERN（欧洲核子研究中心）；INFN, Sezione di Milano-Bicocca（意大利国家核物理研究所米兰-比科卡分部）；Università degli Studi di Milano-Bicocca（米兰比可卡大学）；Dipartimento di Fisica G. Occhialini, Università degli Studi di Milano-Bicocca（米兰比可卡大学G.奥奇亚利尼物理系）；Università degli Studi di Torino（都灵大学）；INFN, Sezione di Torino（意大利国家核物理研究所都灵分部）
When the allowed range of an observable grows order-by-order in perturbation theory, its perturbative expansion can have discontinuities (as in the $C$ parameter) or discontinuities in its derivatives (as in thrust or heavy jet mass) called Sudakov shoulders. We explore the origin of these logarithms using both perturbation theory and effective field theory. We show that for thrust and heavy jet mass, the logarithms arise from kinematic configurations with narrow jets and deduce the next-to-leading logarithmic series. The left-shoulder logarithms in heavy jet mass ($ρ$) of the form $r\ α_s^n \…
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.
The heavy jet mass distribution in e+e- collisions is computed to next-to-next-to-next-to leading logarithmic (NNNLL) and next-to-next-to leading fixed order accuracy (NNLO). The singular terms predicted from the resummed distribution are confirmed by the fixed order distributions allowing a precise extraction of the unknown soft function coefficients. A number of quantitative and qualitative comparisons of heavy jet mass and the related thrust distribution are made. From fitting to ALEPH data, a value of alpha_s is extracted, alpha_s(m_Z)=0.1220 +/- 0.0031, which is larger than, but not in co…
We present the framework for obtaining precise predictions for the transverse momentum of hadrons with respect to the thrust axis in $e^+e^-$ collisions. This will enable a precise extraction of transverse momentum dependent (TMD) fragmentation functions from a recent measurement by the Belle Collaboration. Our analysis takes into account, for the first time, the nontrivial interplay between the hadron transverse momentum and the cut on the thrust event shape. To this end, we identify three different kinematic regions, derive the corresponding factorization theorems within Soft Collinear Effec…
The thrust distribution in e+e- annihilation is calculated exploiting its exponentiation property in the two-jet region t = 1-T << 1. We present a general method (DGE) to calculate a large class of logarithmically enhanced terms, using the dispersive approach in renormalon calculus. Dressed Gluon Exponentiation is based on the fact that the exponentiation kernel is associated primarily with a single gluon emission, and therefore the exponent is naturally represented as an integral over the running coupling. Fixing the definition of Lambda is enough to guarantee consistency with the exact expon…
We present preliminary results on the resummation of leading and next-to-leading logarithms for the thrust distribution in deep inelastic scattering. Our predictions, expanded to O(alpha_s^2), are compared to corresponding results from the Monte Carlo programs DISASTER++ and DISENT.
Resummed results are discussed for DIS event shape variable distributions involving the thrust defined using the thrust axis of the current hemisphere, the C parameter and the jet mass. The resummation of these variables requires the development of some techniques different to those applied before for the resummation of DIS event shapes. Including power corrections and matching a comparison with H1 data is presented.
We derive the transverse momentum dependent (TMD) factorization and resummation formula of the unpolarized transverse momentum distribution ($j_T$) for the single hadron production with the thrust axis in electron-positron collision. Two different kinematic regions are considered, including small transverse momentum limit $j_T \ll Q$, and joint transverse momentum and threshold limit $j_T \ll Q(1-z_h) \ll Q$, where $Q$ and $z_h$ are the hard scattering energy and the observed hadron momentum fraction. Using effective theory methods, we resum logarithms $\ln(Q/j_T)$ and $\ln(1-z_h)$ to all orde…
I review the treatment of high-energy QCD in Minkowski space, with an emphasis on factorization theorems as extensions of the operator product expansion. I discuss how the factorization properties of high-energy cross sections and amplitudes lead to evolution equations that resum large logarithms for two-scale problems.
In this paper we study the factorization and resummation of s-channel single top quark production in the Standard Model at both the Tevatron and the LHC. We show that the production cross section in the threshold limit can be factorized into a convolution of hard function, soft function and jet function via soft-collinear-effective-theory (SCET), and resummation can be performed using renormalization group equation in the momentum space resummation formalism. We find that in general, the resummation effects enhance the Next-to-Leading-Order (NLO) cross sections by about $3%-5%$ at both the Tev…
We investigate the factorization and resummation of t-channel single top (antitop) quark production in the SM at both the Tevatron and the LHC in SCET. We find that the resummation effects decrease the NLO cross sections by about 3% at the Tevatron and about 2% at the LHC. And the resummation effects significantly reduce the factorization scale dependence of the total cross section. The transfer momentum cut dependence and other matching scale dependencies are also discussed. We also show that when our numerical results for s- and t-channel single top production at the Tevatron are combined, i…
We discuss the procedure to resum large logarithms to all orders for DIS event shape variable distributions. Results are described for two variants of the thrust variable, both defined wrt the boson axis in the current hemisphere of the Breit frame, but with different normalisations.
Differential spectra in observables that resolve additional soft or collinear QCD emissions exhibit Sudakov double logarithms in the form of logarithmic plus distributions. Important examples are the total transverse momentum $q_T$ in color-singlet production, N-jettiness (with thrust or beam thrust as special cases), but also jet mass and more complicated jet substructure observables. The all-order logarithmic structure of such distributions is often fully encoded in differential equations, so-called (renormalization group) evolution equations. We introduce a well-defined technique of distrib…
Using methods of effective field theory, we derive the first all-order factorization theorem for the Higgs-boson production cross section with a jet veto, imposed by means of a standard sequential recombination jet algorithm. Like in the case of small-q_T resummation in Drell-Yan and Higgs production, the factorization is affected by a collinear anomaly. Our analysis provides the basis for a systematic resummation of large logarithms log(m_H/p_T^veto) beyond leading-logarithmic order. Specifically, we present predictions for the resummed jet-veto cross section and efficiency at next-to-next-to…
We present a unified derivation of the resummation of Sudakov logarithms, directly from the factorization properties of cross sections in which they occur. We rederive in this manner the well-known exponentiation of leading and nonleading logarithmic enhancements near the edge of phase space for cross sections such as deeply inelastic scattering, which are induced by an electroweak hard scattering. For QCD hard-scattering processes, such as heavy-quark production, we show that the resummation of nonleading logarithms requires in general mixing in the space of the color tensors of the hard scat…
Factorizing the cross section for single hadron production in $e^+e^-$ annihilations is a highly non trivial task when the transverse momentum of the outgoing hadron with respect to the thrust axis is taken into account. We work in a scheme that allows to factorize the $e^+e^- \to HX$ cross section as a convolution of a calculable hard coefficient and a Transverse Momentum Dependent (TMD) fragmentation function. The result, differential in $z_h$, $P_T$ and thrust, will be given to all orders in perturbation theory and explicitly computed to Next to Leading Order (NLO) and Next to Leading Log (…
We apply Soft-Collinear Effective Theory to prove at leading power in Lambda_QCD/m_b a factorization formula for the radiative leptonic decay B -> gamma l nu. Large logarithms entering the hard-scattering kernel are systematically resummed by a two-step perturbative matching procedure.
We study the resummation of singular distributions for electroweak-boson production at large transverse momentum in hadronic collisions. We describe the factorization properties of the cross section near partonic threshold and construct the resummed cross section at next-to-leading logarithmic accuracy in moment space in terms of soft gluon anomalous dimensions. We present full analytical results for the expansion of the resummed cross section up to next-to-next-to-leading order. Our results can be applied to W, Z, and virtual gamma production at hadron colliders.
Resummation and power-corrections play a crucial role in the phenomenology of event-shape variables like the thrust T. Previous investigations showed that the perturbative contribution to the average thrust is dominated by gluons of small invariant mass, of the order of 10% of Q, where Q is the center-of-mass energy. The effect of soft gluons is also important, leading to a non-perturbative 1/Q correction. These conclusions are based on renormalon analysis in the single dressed gluon (SDG) approximation. Here we analyze higher moments of the thrust distribution using a similar technique. We fi…
We discuss the production of heavy colored paricles at the Large Hadron Collider (LHC) through gluon-gluon fusion process. A factorization theorem is obtained for this process using Soft Collinear Effective Theory. Our factorization theorem does not depend on any assumptions regarding the physics above the mass of the heavy colored particle. In this sense it is universal. The matching coefficient at the heavy particle mass scale depends however on the unkown physics above that scale and thus it is model dependent. Due to the large mass of the heavy colored particle, i.e., the hard scale and ne…
Factorization theorems allow to separate out the universal, non-perturbative content of the hadronic cross section from its perturbative part, which can be computed in perturbative QCD, up to the desired order. In this paper, we derive a rigorous proof of factorization of the $e^+ e^- \to h\,X$ cross section, sensitive to the transverse momentum of the detected hadron with respect to the thrust axis, in a completely general framework, based on the Collins-Soper-Sterman approach. The results are explicitly computed to NLO-NLL accuracy and subsequently generalized to all orders in perturbation t…
Jet broadening is an event-shape variable probing the transverse momenta of particles inside jets. It has been measured precisely in e+e- annihilations and is used to extract the strong coupling constant. The factorization of the associated cross section at small values of the broadening is afflicted by a collinear anomaly. Based on an analysis of this anomaly, we present the first all-order expressions for jet-broadening distributions, which are free of large perturbative logarithms in the two-jet limit. Our formulae reproduce known results at next-to-leading logarithmic order but also extend…
Precision calculations of heavy-to-light form factors are essential to sharpen our understanding towards the strong interaction dynamics of the heavy-quark system and to shed light on a coherent solution of flavor anomalies. We briefly review factorization properties of heavy-to-light form factors in the framework of QCD factorization in the heavy quark limit and discuss the recent progress on the QCD calculation of $B \to π$ form factors from the light-cone sum rules with the $B$-meson distribution amplitudes. Demonstration of QCD factorization for the vacuum-to-$B$-meson correlation function…
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…
The resummation for the event-shape variable jet broadening is extended to next-to-next-to-leading logarithmic accuracy by computing the relevant jet and soft functions at one-loop order and the collinear anomaly to two-loop accuracy. The anomaly coefficient is extracted from the soft function and expressed in terms of polylogarithmic as well as elliptic functions. With our results, the uncertainty on jet-broadening distributions is reduced significantly, which should allow for a precise determination of the strong coupling constant from the existing experimental data and provide a consistency…
Renormalization-group methods in soft-collinear effective theory are used to perform the resummation of large perturbative logarithms for deep-inelastic scattering in the threshold region x->1. The factorization theorem for the structure function F_2(x,Q^2) for x->1 is rederived in the effective theory, whereby contributions from the hard scale Q^2 and the jet scale Q^2(1-x) are encoded in Wilson coefficients of effective-theory operators. Resummation is achieved by solving the evolution equations for these operators. Simple analytic results for the resummed expressions are obtained directly i…
In this work, we complete the calculation of the soft part of the two-loop integrated jet thrust distribution in e+e- annihilation. This jet mass observable is based on the thrust cone jet algorithm, which involves a veto scale for out-of-jet radiation. The previously uncomputed part of our result depends in a complicated way on the jet cone size, r, and at intermediate stages of the calculation we actually encounter a new class of multiple polylogarithms. We employ an extension of the coproduct calculus to systematically exploit functional relations and represent our results concisely. In con…
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…
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 discuss the inclusion of running coupling effects in perturbative small x evolution equations. We show that a running coupling BFKL-like x-evolution equation is fully compatible, up to higher twist corrections, with the standard factorized perturbative evolution of parton distributions. We then use this result, combined with the well-known Airy asymptotics, to prove that the oscillations which are present in the running-coupling BFKL solution do not affect the associated splitting functions, which instead remain smooth in the small x limit. This allows us to give a prescription to include r…
We study the fragmentation function of the gluon to color-octet $^3S_1$ heavy quark-antiquark pair using the soft gluon factorization (SGF) approach, which expresses the fragmentation function in a form of perturbative short-distance hard part convoluted with one-dimensional color-octet $^3S_1$ soft gluon distribution (SGD). The short distance hard part is calculated to next-to-leading order in $α_s$ and a renormalization group equation for the SGD is derived. By solving the renormalization group equation, threshold logarithms are resummed to all orders in perturbation theory. The comparison w…
From a detailed analysis of cone-jet cross sections in effective field theory, we obtain novel factorization theorems which separate the physics associated with different energy scales present in such processes. The relevant low-energy physics is encoded in Wilson lines along the directions of the energetic particles inside the jets. This multi-Wilson-line structure is present even for narrow-cone jets due to the relevance of small-angle soft radiation. We discuss the renormalization-group equations satisfied by these operators. Their solution resums all logarithmically enhanced contributions …
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…
We present predictions for the thrust distribution in hadronic decays of the Higgs boson at the next-to-leading order and the approximate next-to-next-to-leading order. The approximate NNLO corrections are derived from a factorization formula in the soft/collinear phase-space regions. We find large corrections, especially for the gluon channel. The scale variations at the lowest orders tend to underestimate the genuine higher order contributions. The results of this paper is therefore necessary to control the perturbative uncertainties of the theoretical predictions. We also discuss on possibl…
Explicit applications of factorization theorems for processes at hadron colliders near the hadronic endpoint have largely focused on simple final states with either no jets (e.g., Drell-Yan) or one inclusive jet (e.g., deep inelastic scattering and prompt photon production). Factorization for the former type of process gives rise to a soft function that depends on timelike momenta, whereas the soft function for the latter type depends on null momenta. We derive in soft-collinear effective theory a factorization theorem that allows for an arbitrary number of jets, where the jets are defined wit…
The resummation of threshold logarithms for direct photon production cross sections in hadronic collisions is presented. The resummation is based on the factorization properties of the cross section and is formulated at next-to-leading logarithmic or higher accuracy. Full analytical and numerical results for the next-to-next-to-leading order expansion of the resummed cross section are given. A substantial reduction of factorization scale dependence is observed. A comparison to experimental results from the E-706 and UA-6 experiments is presented.
We study single-particle inclusive (1PI) distributions in top-quark pair production at hadron colliders, working in the highly boosted regime where the top-quark p_T is much larger than its mass. In particular, we derive a novel factorization formula valid in the small-mass and soft limits of the differential partonic cross section. This provides a framework for the simultaneous resummation of soft gluon corrections and small-mass logarithms, and also an efficient means of obtaining higher-order corrections to the differential cross section in this limit. The result involves five distinct one-…
We study the transverse momentum dependent factorization for single spin asymmetries in Drell-Yan and semi-inclusive deep inelastic scattering processes at one-loop order. The next-to-leading order hard factors are calculated in the Ji-Ma-Yuan factorization scheme. We further derive the QCD resummation formalisms for these observables following the Collins-Soper-Sterman method. The results are expressed in terms of the collinear correlation functions from initial and/or final state hadrons coupled with the Sudakov form factor containing all order soft-gluon resummation effects. The scheme inde…
The $B$-meson light-cone distribution amplitude is an important non-perturbative quantity arising in the factorization of the amplitudes for many exclusive decays of $B$ mesons, such as $B^-\toγ\,\ell^-\barν$. We reconsider the renormalization-group (RG) equation satisfied by this function and present its solution at next-to-leading order (NLO) in RG-improved perturbation theory in Laplace space and, for the first time, in momentum space and the so-called diagonal (or dual) space. Since the information needed to describe the $B$ decay processes at leading order in $Λ_{\rm QCD}/m_b$ is most dir…
We study the transverse momentum resummation for dijet correlation in hadron collisions based on the Collins-Soper-Sterman formalism. The complete one-loop calculations are carried out in the collinear factorization framework for the differential cross sections at low imbalance transverse momentum between the two jets. Important cross checks are performed to demonstrate that the soft divergences cancelled out between different diagrams, and in particular, those associated with final state jets. The leading and sub-leading logarithms are identified. All order resummation is derived following th…
We present a novel method for resummation of event shapes to next-to-next-to-leading-logarithmic (NNLL) accuracy. We discuss the technique and describe its implementation in a numerical program in the case of e^+e^- collisions where the resummed prediction is matched to NNLO. We reproduce all the existing predictions and present new results for oblateness and thrust major.
In this paper we study the transverse energy spectrum for the Drell-Yan process. The transverse energy is measured within the central region defined by a (pseudo-) rapidity cutoff. Soft-collinear effective theory (SCET) is used to factorize the cross section and resum large logarithms of the rapidity cutoff and ratios of widely separated scales that appear in the fixed order result. We develop a framework which can smoothly interpolate between various regions of the spectrum and eventually match onto the fixed order result. This way a reliable calculation is obtained for the contribution of th…
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…
We present a factorization framework that takes into account the production of heavy quarks through gluon splitting in the thrust distribution for e+ e- --> hadrons. The explicit factorization theorems and some numerical results are displayed in the dijet region where the kinematic scales are widely separated, which can be extended systematically to the whole spectrum. We account for the necessary two-loop matrix elements, threshold corrections, and include resummation up to N3LL order. We include nonperturbative power corrections through a field theoretical shape function, and remove the O(La…
I present some comments on the partonic interpretation of the HERA data on the proton structure function. The effects of the resummation of the leading and next-to-leading $\ln x$-contributions are discussed. A new factorization scheme, in which these resummation effects are absorbed into a steep redefinition of the gluon density, is introduced and its (possible) interpretation and phenomenological relevance are suggested.
