Non-global Logarithms Research Articles

Calculating the angle between jet axes

We study the angle between i) the standard jet axis, ii) the axis of a jet which has been groomed using soft drop, with reduced sensitivity to soft radiation, iii) the jet axis obtained with the winner-take-all recombination scheme, which is insensitive to soft radiation at leading power. We calculate the distributions for these angles at next- to-leading logarithmic accuracy, including non-global logarithms. The angle between the standard and groomed jet axis directly probes soft wide-angle radiation, leading to a novel factorization formula. This angle is also very sensitive to nonperturbative physics, which is directly connected to nonperturbative contribution to the rapidity anomalous dimension for transverse momentum distributions. Comparing our predictions to Pythia we find good agreement, and we foresee applications to jet substructure in proton-proton and heavy ion collisions.

Hadron-in-jet production at partonic threshold

We consider the longitudinal momentum distribution of hadrons inside jets in proton-proton collisions. At partonic threshold large double logarithmic corrections arise which need to be resummed to all orders. We develop a factorization formalism within SCET that allows for the joint resummation of threshold and jet radius logarithms. We achieve next-to-leading logarithmic NLL′ accuracy by including non-global logarithms in the leading-color approximation. Overall, we find that the threshold resummation leads to a sizable enhancement of the cross section and a reduced QCD scale dependence, suggesting that the all-order resummation can be important for the reliable extraction of fragmentation functions in global analyses when jet substructure data is included.

The soft drop groomed jet radius at NLL

We present results for the soft drop groomed jet radius Rg at next-to-leading logarithmic accuracy. The radius of a groomed jet which corresponds to the angle between the two branches passing the soft drop criterion is one of the characteristic observables relevant for the precise understanding of groomed jet substructure. We establish a factorization formalism that allows for the resummation of all relevant large logarithms, which is based on demonstrating the all order equivalence to a jet veto in the region between the boundaries of the groomed and ungroomed jet. Non-global logarithms including clustering effects due to the Cambridge/Aachen algorithm are resummed to all orders using a suitable Monte Carlo algorithm. We perform numerical calculations and find a very good agreement with Pythia 8 simulations. We provide theoretical predictions for the LHC and RHIC.

Resummation of boson-jet correlation at hadron colliders

We perform a precise calculation of the transverse momentum ( {overrightarrow{q}}_T ) distribution of the boson+jet system in boson production events. The boson can be either a photon, W, Z or Higgs boson with mass mV, and {overrightarrow{q}}_T is the sum of the transverse momenta of the boson and the leading jet with magnitude qT = | {overrightarrow{q}}_T |. Using renormalization group techniques and soft-collinear effective theory, we resum logarithms log (Q/qT) and log R at next-to- leading logarithmic accuracy including the non-global logarithms, where Q and R are respectively the hard scattering energy and the radius of the jet. Specifically, we investigate two scenarios of {p}_T^J ≲ mV or {p}_T^J ≳ mV in Z +jet events, and we examine the qT distributions with different jet radii and study the effect of non-global logarithms. In the end we compare our theoretical calculations with Monte Carlo simulations and data from the LHC.

Probing transverse-momentum distributions with groomed jets

We present the transverse momentum spectrum of groomed jets in di-jet events for e+e− collisions and semi-inclusive deep inelastic scattering (SIDIS). The jets are groomed using a soft-drop grooming algorithm which helps in mitigating effects of non-global logarithms and underlying event. At the same time, by reducing the final state hadronization effects, it provides a clean access to the non-perturbative part of the evolution of transverse momentum dependent (TMD) distributions. In SIDIS experiments we look at the transverse momentum of the groomed jet measured w.r.t. the incoming hadron in the Breit frame. Because the final state hadronization effects are significantly reduced, the SIDIS case allows to probe the TMD parton distribution functions. We discuss the sources of non-perturbative effects in the low transverse momentum region including novel (but small) effects that arise due to grooming. We derive a factorization theorem within SCET and resum any large logarithm in the measured transverse momentum up to NNLL accuracy using the ζ-prescription as implemented in the artemide package and provide a comparison with simulations.

The jet shape at NLL\u2032

The jet shape is the fraction of the jet energy within a cone r centered on the jet axis. We calculate the jet shape distribution at next-to-leading logarithmic accuracy plus next-to-leading order (NLL′), accounting for logarithms of both the jet radius R and the ratio r/R. This is the first phenomenological study that takes the recoil of the jet axis due to soft radiation into account, which is needed to reach this accuracy, but complicates the calculation of collinear radiation and requires the treatment of rapidity logarithms and non-global logarithms. We present numerical results, finding good agreement with ATLAS and CMS measurements of the jet shape in an inclusive jet sample, pp → jet + X, for different kinematic bins. The effect of the underlying event and hadronization are included using a simple one-parameter model, since they are not part of our perturbative calculation.

Jet veto resummation with jet rapidity cuts

Jet vetoes are widely used in experimental analyses at the LHC to distinguish different hard-interaction processes. Experimental jet selections require a cut on the (pseudo)rapidity of reconstructed jets, |ηjet| ≤ ηcut. We extend the standard jet-pT (jet-veto) resummation, which implicitly works in the limit ηcut → ∞, by incorporating a finite jet rapidity cut. We also consider the case of a step in the required pTcut at an intermediate value of |η| ≃ 2.5, which is of experimental relevance to avoid the increased pile-up contamination beyond the reach of the tracking detectors. We identify all relevant parametric regimes, discuss their factorization and resummation as well as the relations between them, and show that the phenomenologically relevant regimes are free of large nonglobal logarithms. The ηcut dependence of all resummation ingredients is computed to the same order to which they are currently known for ηcut → ∞. Our results pave the way for carrying out the jet-veto resummation including a sharp cut or a step at ηcut to the same order as is currently available in the ηcut → ∞ limit. The numerical impact of the jet rapidity cut is illustrated for benchmark qoverline{q} and gg initiated color-singlet processes at NLL′+NLO. We find that a rapidity cut at high ηcut = 4.5 is safe to use and has little effect on the cross section. A sharp cut at ηcut = 2.5 can in some cases lead to a substantial increase in the perturbative uncertainties, which can be mitigated by instead using a step in the veto.

Factorized Groomed Jet Mass Distribution in Inclusive Jet Processes

We consider the factorized groomed jet mass distribution in inclusive jet processes using modified mass drop tagger (mMDT), corresponding to soft drop with the angular exponent $\beta =0$. A grooming procedure is implemented rather than tagging in the sense that grooming always returns a groomed jet, while tagging dose not return a jet when a single particle remains after tagging. We find that the grooming procedure makes the jet mass distribution infrared safe and only ultraviolet divergences appear in each factorized part. The groomed jet mass distributions are investigated in a wide range of the jet mass considering various limits on the jet mass variable $\rho = M_J^2/(p_T^JR)^2$ and the grooming cut $y_c$. Appropriate effective theories in different kinematic regions are employed to resum large logarithms, in which the analysis in the region $\rho \sim y_c \ll 1$ is included due to the different type of factorization. The analytic computation of the factorized groomed jet mass distribution is presented by resumming the large logarithms in the jet mass, and $y_c$. Numerically, the effect of the resummation is notably enhanced, compared with the calculation at next-to-leading order, and nonglobal logarithms are estimated to be small.

Transverse momentum spectra at threshold for groomed heavy quark jets

We present the transverse momentum spectrum for a heavy hadron at threshold in a groomed jet initiated by a heavy quark. The cross section is doubly differential in the energy fraction of an identified heavy hadron in the jet and its transverse momentum measured with respect to the groomed (recoil free) jet axis. The grooming is implemented using a soft-drop grooming algorithm and helps us in mitigating the effects of Non-Global logarithms and pile up. For the particular case of a B meson, we identify two distinct regimes of the transverse momentum spectrum and develop an EFT within the formalisms of Soft Collinear Effective Theory (SCET) and Heavy Quark Effective Theory (HQET) for each of these regions. We show how each region can be matched smoothly into the other to provide a prediction for the perturbative transverse momentum spectrum. The EFT also predicts the scaling behavior of the leading non-perturbative power corrections and implements a simple shape function to account for hadronization. We work in the threshold region where the heavy hadron carries most of the energy of the jet since in this regime, we have a very good discriminating power between heavy quark and gluon initiated jets. We observe that the shape of the spectrum is independent of the energy of the jet over a large range of transverse momentum. We propose that this spectrum can be used as a probe of evolution for heavy quark TMD fragmentation function. At the same time, it can be treated as a jet substructure observable for probing Quark-Gluon Plasma (QGP).

Non-global logarithms in jet and isolation cone cross sections

Starting from a factorization theorem in effective field theory, we derive a parton-shower equation for the resummation of non-global logarithms. We have implemented this shower and interfaced it with a tree-level event generator to obtain an automated framework to resum the leading logarithm of non-global observables in the large-Nc limit. Using this setup, we compute gap fractions for dijet processes and isolation cone cross sections relevant for photon production. We compare our results with fixed-order computations and LHC measurements. We find that naive exponentiation is often not adequate, especially when the vetoed region is small, since non-global contributions are enhanced due to their dependence on the veto-region size. Since our parton shower is derived from first principles and based on renormalization-group evolution, it is clear what ingredients will have to be included to perform resummations at subleading logarithmic accuracy in the future.

Probing transverse-momentum dependent evolution with groomed jets

We propose an observable which involves measuring the properties (transverse momentum ph⊥ and energy fraction zh) of an identified hadron inside a groomed jet. The jet is identified with an anti-kT/CA algorithm and is groomed by implementing the modified mass drop procedure with an energy cut-off parameter, zcut. The transverse momentum of the hadron inside the jet is measured with respect to the groomed jet axis. We obtain a factorization theorem in the framework of Soft Collinear Effective Theory (SCET), to define a Transverse Momentum Dependent Fragmenting Jet Function (TMDFJF). The TMDFJF is factorized into collinear and collinear soft modes by matching onto SCET+. We resum large logarithms in EJ/ph⊥, where EJ is the ungroomed jet energy, to NLL accuracy and apply this formalism for computing the shape of the ph⊥ distribution of a pion produced in an e+ + e− collision. We observe that the introduction of grooming makes this observable insensitive to non-global logarithms and particularly sensitive to non-perturbative physics of the transverse momentum dependent evolution at low values of ph⊥, which can be probed in the variation of the cut-off parameter, zcut, of the groomer. We discuss how this observable can be used to distinguish between non-perturbative models that describe universal TMD evolution and provide a window into the three dimensional structure of hadrons.

Resummation of non-global logarithms and the BFKL equation

We consider a ‘color density matrix’ in gauge theory. We argue that it systematically resums large logarithms originating from wide-angle soft radiation, sometimes referred to as non-global logarithms, to all logarithmic orders. We calculate its anomalous dimension at leading- and next-to-leading order. Combined with a conformal transformation known to relate this problem to shockwave scattering in the Regge limit, this is used to rederive the next-to-leading order Balitsky-Fadin-Kuraev-Lipatov equation (including its nonlinear generalization, the so-called Balitsky-JIMWLK equation), finding perfect agreement with the literature. Exponentiation of divergences to all logarithmic orders is demonstrated. The possibility of obtaining the evolution equation (and BFKL) to three-loop is discussed.

Factorization and resummation for groomed multi-prong jet shapes

ABSTRACTObservables which distinguish boosted topologies from QCD jets are playing an increasingly important role at the Large Hadron Collider (LHC). These observables are often used in conjunction with jet grooming algorithms, which reduce contamination from both theoretical and experimental sources. In this paper we derive factorization formulae for groomed multi-prong substructure observables, focusing in particular on the groomed D2 observable, which is used to identify boosted hadronic decays of electroweak bosons at the LHC. Our factorization formulae allow systematically improvable calculations of the perturbative D2 distribution and the resummation of logarithmically enhanced terms in all regions of phase space using renormalization group evolution. They include a novel factorization for the production of a soft subjet in the presence of a grooming algorithm, in which clustering effects enter directly into the hard matching. We use these factorization formulae to draw robust conclusions of experimental relevance regarding the universality of the D2 distribution in both e+e− and pp collisions. In particular, we show that the only process dependence is carried by the relative quark vs. gluon jet fraction in the sample, no non-global logarithms from event-wide correlations are present in the distribution, hadronization corrections are controlled by the perturbative mass of the jet, and all global color correlations are completely removed by grooming, making groomed D2 a theoretically clean QCD observable even in the LHC environment. We compute all ingredients to one-loop accuracy, and present numerical results at next-to-leading logarithmic accuracy for e+e− collisions, comparing with parton shower Monte Carlo simulations. Results for pp collisions, as relevant for phenomenology at the LHC, are presented in a companion paper [1].

High-energy evolution to three loops

The Balitsky-Kovchegov equation describes the high-energy growth of gauge theory scattering amplitudes as well as nonlinear saturation effects which stop it. We obtain the three-loop corrections to the equation in planar mathcal{N} = 4 super Yang-Mills theory. Our method exploits a recently established equivalence with the physics of soft wide-angle radiation, so-called non-global logarithms, and thus yields at the same time the threeloop evolution equation for non-global logarithms. As a by-product of our analysis, we develop a Lorentz-covariant method to subtract infrared and collinear divergences in crosssection calculations in the planar limit. We compare our result in the linear regime with a recent prediction for the so-called Pomeron trajectory, and compare its collinear limit with predictions from the spectrum of twist-two operators.

Resumming double non-global logarithms in the evolution of a jet

We consider the Banfi-Marchesini-Smye (BMS) equation which resums ‘non-global’ energy logarithms in the QCD evolution of the energy lost by a pair of jets via soft radiation at large angles. We identify a new physical regime where, besides the energy logarithms, one has to also resum (anti)collinear logarithms. Such a regime occurs when the jets are highly collimated (boosted) and the relative angles between successive soft gluon emissions are strongly increasing. These anti-collinear emissions can violate the correct time-ordering for time-like cascades and result in large radiative corrections enhanced by double collinear logs, making the BMS evolution unstable beyond leading order. We isolate the first such a correction in a recent calculation of the BMS equation to next-to-leading order by Caron-Huot. To overcome this difficulty, we construct a ‘collinearly-improved’ version of the leading-order BMS equation which resums the double collinear logarithms to all orders. Our construction is inspired by a recent treatment of the Balitsky-Kovchegov (BK) equation for the high-energy evolution of a space-like wavefunction, where similar time-ordering issues occur. We show that the conformal mapping relating the leading-order BMS and BK equations correctly predicts the physical time-ordering, but it fails to predict the detailed structure of the collinear improvement.

Non-global and rapidity logarithms in narrow jet broadening

We derive an all-order factorization theorem for the narrow jet broadening event shape, a measure of the transverse momentum in jet events. This is a non-global observable which receives logarithmically enhanced contributions associated with the large rapidity difference between soft and collinear radiation and which is also sensitive to soft recoil effects. Our work is the first factorization analysis of an observable of this type and we show that with regard to the non-global nature, the rapidity logarithms do not constitute an essential complication since they can be tied to the jet function, which is the same as for global observables. As a consequence, the leading non-global logarithms in narrow jet broadening are encoded in the same overall factor relevant for the hemisphere soft function and light jet mass.

Fragmentation to a jet in the large z limit

We consider the fragmentation of a parton into a jet with small radius $R$ in the large $z$ limit, where $z$ is the ratio of the jet energy to the mother parton energy. In this region of phase space, large logarithms of both $R$ and $1-z$ can appear, requiring resummation in order to have a well defined perturbative expansion. Using soft-collinear effective theory, we study the fragmentation function to a jet (FFJ) in this endpoint region. We derive a factorization theorem for this object, separating collinear and collinear-soft modes. This allows for the resummation using renormalization group evolution of the logarithms $\ln R$ and $\ln(1-z)$ simultaneously. We show results valid to next-to-leading logarithmic order for the global Sudakov logarithms. We also discuss the possibility of non-global logarithms that should appear at two-loops and give an estimate of their size.

The asymptotic form of non-global logarithms, black disc saturation, and gluonic deserts

We develop an asymptotic perturbation theory for the large logarithmic behavior of the non-linear integro-differential equation describing the soft correlations of QCD jet measurements, the Banfi-Marchesini-Smye (BMS) equation. This equation captures the late-time evolution of radiating color dipoles after a hard collision. This allows us to prove that at large values of the control variable (the non-global logarithm, a function of the infra-red energy scales associated with distinct hard jets in an event), the distribution has a gaussian tail. We compute the decay width analytically, giving a closed form expression, and find it to be jet geometry independent, up to the number of legs of the dipole in the active jet. Enabling the asymptotic expansion is the correct perturbative seed, where we perturb around an anzats encoding formally no real emissions, an intuition motivated by the buffer region found in jet dynamics. This must be supplemented with the correct application of the BFKL approximation to the BMS equation in collinear limits. Comparing to the asymptotics of the conformally related evolution equation encountered in small-x physics, the Balitisky-Kovchegov (BK) equation, we find that the asymptotic form of the non-global logarithms directly maps to the black-disc unitarity limit of the BK equation, despite the contrasting physical pictures. Indeed, we recover the equations of saturation physics in the final state dynamics of QCD.

Factorization for the light-jet mass and hemisphere soft function

Many collider observables suffer from non-global logarithms not captured by standard resummation techniques. Classic examples are the light-jet mass event shape in the limit of small mass and the related hemisphere soft function. We derive factorization formulas for both of these and explicitly demonstrate that they capture all logarithms present at NNLO. These formulas achieve full scale separation and provide the basis for all-order resummations. A characteristic feature of non-global observables is that the soft radiation is driven by multi-Wilson-line operators, and the ones arising here map onto those relevant for the case of narrow-cone jet cross sections. Numerically, the contributions of non-global logarithms to resummed hemisphere-mass event shapes are sizeable.

Factorization for jet radius logarithms in jet mass spectra at the LHC

To predict the jet mass spectrum at a hadron collider it is crucial to account for the resummation of logarithms between the transverse momentum of the jet and its invariant mass $m_J$. For small jet areas there are additional large logarithms of the jet radius $R$, which affect the convergence of the perturbative series. We present an analytic framework for exclusive jet production at the LHC which gives a complete description of the jet mass spectrum including realistic jet algorithms and jet vetoes. It factorizes the scales associated with $m_J$, $R$, and the jet veto, enabling in addition the systematic resummation of jet radius logarithms in the jet mass spectrum beyond leading logarithmic order. We discuss the factorization formulae for the peak and tail region of the jet mass spectrum and for small and large $R$, and the relations between the different regimes and how to combine them. Regions of experimental interest are classified which do not involve large nonglobal logarithms. We also present universal results for nonperturbative effects and discuss various jet vetoes.