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Andrew Hornig

Publications and source records attributed to Andrew Hornig.

At least 19 recordsLinked to original sources

$e^+ e^-$ angularity distributions at NNLL$^\prime$ accuracy

We present predictions for the $e^{+}e^{-}$ event shape angularities at NNLL$^{\prime}$ resummed and $\mathcal{O}(α_s^{2})$ matched accuracy and compare them to LEP data at center-of-mass energies $Q=91.2$ GeV and $Q=197$ GeV. We perform the resummation within the framework of Soft-Collinear Effective Theory, and make use of recent results for the two-loop angularity soft function. We determine the remaining NNLL$^{\prime}$ and $\mathcal{O}(α_s^2)$ ingredients from a fit to the EVENT2 generator, and implement a shape function with a renormalon-free gap parameter to model non-perturbative effects. Using values of the strong coupling $α_s(m_Z)$ and the universal non-perturbative shift parameter $Ω_1$ that are consistent with those obtained in previous fits to the thrust and $C$-parameter distributions, we find excellent agreement between our predictions and the LEP data for all angularities with $a\in[-1,0.5]$. This provides a robust test of the predictions of QCD, factorization, and the universal scaling of the non-perturbative shift across different angularities. Promisingly, our results indicate that current degeneracies in the $\{α_s(m_Z),Ω_1\}$ parameter space could be be alleviated upon fitting these parameters to experimental data for the angularity distributions.

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Transverse Vetoes with Rapidity Cutoff in SCET

We consider di-jet production in hadron collisions where a transverse veto is imposed on radiation for (pseudo-)rapidities in the central region only, where this central region is defined with rapidity cutoff. For the case where the transverse measurement (e.g., transverse energy or min $p_T$ for jet veto) is parametrically larger relative to the typical transverse momentum beyond the cutoff, the cross section is insensitive to the cutoff parameter and is factorized in terms of collinear and soft degrees of freedom. The virtuality for these degrees of freedom is set by the transverse measurement, as in typical transverse-momentum dependent observables such as Drell-Yan, Higgs production, and the event shape broadening. This paper focuses on the other region, where the typical transverse momentum below and beyond the cutoff is of similar size. In this region the rapidity cutoff further resolves soft radiation into (u)soft and soft-collinear radiation with different rapidities but identical virtuality. This gives rise to rapidity logarithms of the rapidity cutoff parameter which we resum using renormalization group methods. We factorize the cross section in this region in terms of soft and collinear functions in the framework of soft-collinear effective theory, then further refactorize the soft function as a convolution of the (u)soft and soft-collinear functions. All these functions are calculated at one-loop order. As an example, we calculate a differential cross section for a specific partonic channel, $q q' \to q q'$, for the jet shape angularities and show that the refactorization allows us to resum the rapidity logarithms and significantly reduce theoretical uncertainties in the jet shape spectrum.

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Non-global Structure of the O(α_s^2) Dijet Soft Function

High energy scattering processes involving jets generically involve matrix elements of light- like Wilson lines, known as soft functions. These describe the structure of soft contributions to observables and encode color and kinematic correlations between jets. We compute the dijet soft function to O(α_s^2) as a function of the two jet invariant masses, focusing on terms not determined by its renormalization group evolution that have a non-separable dependence on these masses. Our results include non-global single and double logarithms, and analytic results for the full set of non-logarithmic contributions as well. Using a recent result for the thrust constant, we present the complete O(α_s^2) soft function for dijet production in both position and momentum space.

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Analytic and Monte Carlo Studies of Jets with Heavy Mesons and Quarkonia

We study jets with identified hadrons in which a family of jet-shape variables called angularities are measured, extending the concept of fragmenting jet functions (FJFs) to these observables. FJFs determine the fraction of energy, z, carried by an identified hadron in a jet with angularity, τ_a. The FJFs are convolutions of fragmentation functions (FFs), evolved to the jet energy scale, with perturbatively calculable matching coefficients. Renormalization group equations are used to provide resummed calculations with next-to-leading logarithm prime (NLL') accuracy. We apply this formalism to two-jet events in e^+ e^- collisions with B mesons in the jets, and three-jet events in which a J/ψis produced in the gluon jet. In the case of B mesons, we use a phenomenological FF extracted from e^+ e^- collisions at the Z^0 pole evaluated at the scale μ= m_b. For events with J/ψ, the FF can be evaluated in terms of Non-Relativistic QCD (NRQCD) matrix elements at the scale μ=2 m_c. The z and τ_a distributions from our NLL' calculations are compared with predictions from monte carlo event generators. While we find consistency between the predictions for B mesons and the J/ψdistributions in τ_a, we find the z distributions for J/ψdiffer significantly. We describe an attempt to merge PYTHIA showers with NRQCD FFs that gives good agreement with NLL' calculations of the z distributions.

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Jet Shapes in Dijet Events at the LHC in SCET

We consider the class of jet shapes known as angularities in dijet production at hadron colliders. These angularities are modified from the original definitions in e+e- collisions to be boost invariant along the beam axis. These shapes apply to the constituents of jets defined with respect to either k_T-type (anti-k_T, C/A, and k_T) algorithms and cone-type algorithms. We present an SCET factorization formula and calculate the ingredients needed to achieve next-to-leading-log (NLL) accuracy in kinematic regions where non-global logarithms are not large. The factorization formula involves previously unstudied "unmeasured beam functions," which are present for finite rapidity cuts around the beams. We derive relations between the jet functions and the shape-dependent part of the soft function that appear in the factorized cross section and those previously calculated for e+e- collisions, and present the calculation of the non-trivial, color-connected part of the soft-function to O(α_s). This latter part of the soft function is universal in the sense that it applies to any experimental setup with an out-of-jet p_T veto and rapidity cuts together with two tagged jets and it is independent of the choice of jet (sub-)structure measurement. In addition, we implement the recently introduced soft-collinear refactorization to resum logarithms of the jet size, valid in the region of non-enhanced non-global logarithm effects. While our results are valid for all 2 \to 2 channels, we compute explicitly for the qq' \to qq' channel the color-flow matrices and plot the NLL resummed differential dijet cross section as an explicit example, which shows that the normalization and scale uncertainty is reduced when the soft function is refactorized. For this channel, we also plot the jet size R dependence, the p_T^{\rm cut} dependence, and the dependence on the angularity parameter a.

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A Soft-Collinear Mode for Jet Cross Sections in Soft Collinear Effective Theory

We propose the addition of a new "soft-collinear" mode to soft collinear effective theory (SCET) below the usual soft scale to factorize and resum logarithms of jet radii $R$ in jet cross sections. We consider exclusive 2-jet cross sections in $e^+e^-$ collisions with an energy veto $Λ$ on additional jets. The key observation is that there are actually two pairs of energy scales whose ratio is $R$: the transverse momentum $QR$ of the energetic particles inside jets and their total energy $Q$, and the transverse momentum $ΛR$ of soft particles that are cut out of the jet cones and their energy $Λ$. The soft-collinear mode is necessary to factorize and resum logarithms of the latter hierarchy. We show how this factorization occurs in the jet thrust cross section for cone and $k_T$-type algorithms at $O(α_s)$ and using the thrust cone algorithm at $O(α_s^2)$. We identify the presence of hard-collinear, in-jet soft, global (veto) soft, and soft-collinear modes in the jet thrust cross section. We also observe here that the in-jet soft modes measured with thrust are actually the "csoft" modes of the theory SCET$_+$. We dub the new theory with both csoft and soft-collinear modes SCET$_{++}$. We go on to explain the relation between the "unmeasured" jet function appearing in total exclusive jet cross sections and the hard-collinear and csoft functions in measured jet thrust cross sections. We do not resum logs that are non-global in origin, arising from the ratio of the scales of soft radiation whose thrust is measured at $Qτ/R$ and of the soft-collinear radiation at $2ΛR$. Their resummation would require the introduction of additional operators beyond those we consider here. The steps we outline here are a necessary part of summing logs of $R$ that are global in nature and have not been factorized and resummed beyond leading-log level previously.

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On Statistical Aspects of Qjets

The process by which jet algorithms construct jets and subjets is inherently ambiguous and equally well motivated algorithms often return very different answers. The Qjets procedure was introduced by the authors to account for this ambiguity by considering many reconstructions of a jet at once, allowing one to assign a weight to each interpretation of the jet. Employing these weighted interpretations leads to an improvement in the statistical stability of many measurements. Here we explore in detail the statistical properties of these sets of weighted measurements and demonstrate how they can be used to improve the reach of jet-based studies.

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Combining Higher-Order Resummation with Multiple NLO Calculations and Parton Showers in GENEVA

We extend the lowest-order matching of tree-level matrix elements with parton showers to give a complete description at the next higher perturbative accuracy in alpha_s at both small and large jet resolutions, which has not been achieved so far. This requires the combination of the higher-order resummation of large Sudakov logarithms at small values of the jet resolution variable with the full next-to-leading order (NLO) matrix-element corrections at large values. As a by-product this combination naturally leads to a smooth connection of the NLO calculations for different jet multiplicities. In this paper, we focus on the general construction of our method and discuss its application to e+e and pp collisions. We present first results of the implementation in the GENEVA Monte Carlo framework, where we employ N-jettiness as the jet resolution variable, combining its next-to-next-to-leading logarithmic resummation with fully exclusive NLO matrix elements, and PYTHIA8 as the backend for further parton showering and hadronization. For hadronic collisions, we take Drell-Yan production as an example to apply our construction. For e+e- -> jets, taking alpha_s(mZ) = 0.1135 from fits to LEP thrust data, together with the PYTHIA8 hadronization model, we obtain good agreement with LEP data for a variety of 2-jet observables.

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Combining Higher-Order Resummation with Multiple NLO Calculations and Parton Showers in the GENEVA Monte Carlo Framework

We discuss the GENEVA Monte Carlo framework, which combines higher-order resummation (NNLL) of large Sudakov logarithms with multiple next-to-leading-order (NLO) matrix-element corrections and parton showering (using PYTHIA8) to give a complete description at the next higher perturbative accuracy in alpha_s at both small and large jet resolution scales. Results for e+e- -> jets compared to LEP data and for Drell-Yan production are presented.

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Qjets: A Non-Deterministic Approach to Tree-Based Jet Substructure

Jet substructure is typically studied using clustering algorithms, such as kT, which arrange the jets' constituents into trees. Instead of considering a single tree per jet, we propose that multiple trees should be considered, weighted by an appropriate metric. Then each jet in each event produces a distribution for an observable, rather than a single value. Advantages of this approach include: 1) observables have significantly increased statistical stability; and, 2) new observables, such as the variance of the distribution, provide new handles for signal and background discrimination. For example, we find that employing a set of trees substantially reduces the observed fluctuations in the pruned mass distribution, enhancing the likelihood of new particle discovery for a given integrated luminosity. Furthermore, the resulting pruned mass distributions for (background) QCD jets are found to be substantially wider than that for (signal) jets with intrinsic mass scales, e.g. jets containing a W decay. A cut on this width yields a substantial enhancement in significance relative to a cut on the standard pruned jet mass alone. In particular the luminosity needed for a given significance requirement decreases by a factor of two relative to standard pruning.

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Double Non-Global Logarithms In-N-Out of Jets

We derive the leading non-global logarithms (NGLs) of ratios of jet masses m_{1,2} and a jet energy veto Λdue to soft gluons splitting into regions in and out of jets. Such NGLs appear in any exclusive jet cross section with multiple jet measurements or with a veto imposed on additional jets. Here, we consider back-to-back jets of radius R produced in e^+e^- collisions, found with a cone or recombination algorithm. The leading NGLs are of the form α_s^2 \ln^2(Λ/m_{1,2}) or α_s^2\ln^2(m_1/m_2). Their coefficients depend both on the algorithm and on R. We consider cone, \kt, anti-\kt, and Cambridge-Aachen algorithms. In addition to determining the full algorithmic and R dependence of the leading NGLs, we derive new relations among their coefficients. We also derive to all orders in α_s a factorized form for the soft function S(k_L,k_R,Λ) in the cross section σ(m_1,m_2,Λ) in which dependence on each of the global logs of μ/k_L, μ/k_R and μ/Λdetermined by the renormalization group are separated from one another and from the non-global logs. The same kind of soft function, its associated non-global structure, and the algorithmic dependence we derive here will also arise in exclusive jet cross sections at hadron colliders, and must be understood and brought under control to achieve precise theoretical predictions.

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Subtractions for SCET Soft Functions

We present a method to calculate the soft function in Soft-Collinear Effective Theory to NLO for N-jet events, defined with respect to arbitrarily complicated observables and algorithms, using a subtraction-based method. We show that at one loop the singularity structure of all observable/algorithm combinations can be classified as one of two types. Type I jets include jets defined with inclusive algorithms for which a jet shape is measured. Type II jets include jets found with exclusive algorithms, as well as jets for which only the direction and energy are measured. Cross sections that are inclusive over a certain region of phase space, such as the forward region at a hadron collider, are examples of Type II jets. We show that for a large class of measurements the required subtractions are already known analytically, including traditional jet shape measurements at hadron colliders. We demonstrate our method by calculating the soft functions for the case of jets defined in eta-phi space with an out-of-jet pT cut and a rapidity cut on the jets, as well as for the case of 1-jettiness.

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On the effectiveness of threshold resummation away from hadronic endpoint

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 manner allows us to determine the correct scale for threshold resummation after integration with the pdfs, without any additional procedure.

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Jet Shapes and Jet Algorithms in SCET

Jet shapes are weighted sums over the four-momenta of the constituents of a jet and reveal details of its internal structure, potentially allowing discrimination of its partonic origin. In this work we make predictions for quark and gluon jet shape distributions in N-jet final states in e+e- collisions, defined with a cone or recombination algorithm, where we measure some jet shape observable on a subset of these jets. Using the framework of Soft-Collinear Effective Theory, we prove a factorization theorem for jet shape distributions and demonstrate the consistent renormalization-group running of the functions in the factorization theorem for any number of measured and unmeasured jets, any number of quark and gluon jets, and any angular size R of the jets, as long as R is much smaller than the angular separation between jets. We calculate the jet and soft functions for angularity jet shapes τ_a to one-loop order (O(alpha_s)) and resum a subset of the large logarithms of τ_a needed for next-to-leading logarithmic (NLL) accuracy for both cone and kT-type jets. We compare our predictions for the resummed τ_a distribution of a quark or a gluon jet produced in a 3-jet final state in e+e- annihilation to the output of a Monte Carlo event generator and find that the dependence on a and R is very similar.

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Factorization of Boosted Multijet Processes for Threshold Resummation

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 with respect to a jet algorithm, together with any number of non-strongly interacting particles. We find the soft function in general depends on the null components of the soft momenta inside the jets and on a timelike component of the soft momentum outside of the jets. This generalizes and interpolates between the soft functions for the cases of no jets and one inclusive jet. We verify consistency of our factorization theorem to order alpha_s for any number of jets. While in this paper we demonstrate consistency only near the hadronic endpoint, we keep the kinematics general enough (in particular allowing for nonzero boost) to allow for an extension to partonic threshold resummation away from the hadronic endpoint.

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Consistent Factorization of Jet Observables in Exclusive Multijet Cross-Sections

We demonstrate the consistency at the next-to-leading-logarithmic (NLL) level of a factorization theorem based on Soft-Collinear Effective Theory (SCET) for jet shapes in e+e- collisions. We consider measuring jet observables in exclusive multijet final states defined with cone and k_T-type jet algorithms. Consistency of the factorization theorem requires that the renormalization group evolution of hard, jet, and soft functions is such that the physical cross-section is independent of the factorization scale mu. The anomalous dimensions of the various factorized pieces, however, depend on the color representation of jets, choice of jet observable, the number of jets whose shapes are measured, and the jet algorithm, making it highly nontrivial to satisfy the consistency condition. We demonstrate the intricate cancellations between anomalous dimensions that occur at the NLL level, so that, up to power corrections that we identify, our factorization of the jet shape distributions is consistent for any number of quark and gluon jets, for any number of jets whose shapes are measured or unmeasured, for any angular size R of the jets, and for any of the algorithms we consider. Corrections to these results are suppressed by the SCET expansion parameter lambda (the ratio of soft to collinear or collinear to hard scales) and in the jet separation measure 1/t^2 = tan^2(R/2)/tan^2(psi/2), where psi is the angular separation between jets. Our results can be used to calculate a wide variety of jet observables in multijet final states to NLL accuracy.

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Factorization for generic jet production

Factorization is the central ingredient in any theoretical prediction for collider experiments. We introduce a factorization formalism that can be applied to any desired observable, like event shapes or jet observables, for any number of jets and a wide range of jet algorithms in leptonic or hadronic collisions. This is achieved by using soft-collinear effective theory to prove the formal factorization of a generic fully-differential cross section in terms of a hard coefficient, and generic jet and soft functions. In this formalism, whether a given observable factorizes in the usual sense, depends on whether it is inclusive enough, so the jet functions can be calculated perturbatively. The factorization formula for any such observable immediately follows from our general result, including the precise definition of the jet and soft functions appropriate for the observable in question. As examples of our formalism, we work out several results in two-jet production for both e+e- and pp collisions. For the latter, we also comment on how our formalism allows one to treat underlying events and beam remnants.

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Effective Predictions of Event Shapes: Factorized, Resummed, and Gapped Angularity Distributions

Using soft-collinear effective theory (SCET), which provides a unified framework for factorization, resummation of logarithms, and incorporation of universal nonperturbative functions in hard-scattering QCD cross-sections, we present a new prediction of angularity distributions in e+e- annihilation. Angularities tau_a are an infinite class of event shapes which vary in their sensitivity to the substructure of jets in the final state, controlled by a continuous parameter a<2. We calculate angularity distributions for all a<1 to first order in the strong coupling alpha_s and resum large logarithms in these distributions to next-to-leading logarithmic (NLL) accuracy. Our expressions for the next-to-leading order (NLO) O(alpha_s) partonic jet and soft functions in the factorization theorem for angularity distributions are given for the first time. We employ a model for the nonperturbative soft function with a gap parameter which cancels the renormalon ambiguity in the partonic soft function. We explore the relation between the SCET approach to resummation and past approaches in QCD, and discuss the advantages of the effective theory approach. In addition, we draw from the NLO calculations of the jet and soft functions an intuitive lesson about how factorization breaks down in the effective theory as a->1.

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