Jet Function with a Jet Algorithm in SCET

The jet function for the factorized cross section $e^+e^-$ into dijets is given as a function of the jet invariant mass s and with a generic jet algorithm at $\mathcal{O}(\alpha_s)$. We demonstrate the results using the Sterman-Weinberg algorithm and show that the jet function is independent of the energy fraction $\beta$ of the soft radiation. The anomalous dimension has the same form with and without the cone half-angle $\delta$. The dependence of the finite part of the jet function on the cone angle is given.Comment: 10 pages, 5 figures, journal versio

Jet Regions from Event Shapes and the N-Jet Soft Function at Hadron Colliders

The N-jettiness event shape divides phase space into N+2 regions, each containing one jet or beam. These jet regions are insensitive to the distribution of soft radiation and, with a geometric measure for N-jettiness, have circular boundaries. We give a factorization theorem for the cross section which is fully differential in the mass of each jet, and compute the corresponding soft function at next-to-leading order (NLO). For N-jettiness, all ingredients are now available to extend NLO cross sections to resummed predictions at next-to-next-to-leading logarithmic order.Comment: 3 pages, 2 figures, part of PANIC 2011 proceeding

Jet production at hadron colliders

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 101-104).Hadronic jets feature in many final states of interest in modern collider experiments. They form a significant Standard Model background for many proposed new physics processes and also probe QCD interactions at several different scales. At high energies incoming protons produce beam jets. Correctly accounting for the beam and central jets is critical to precise understanding of hadronic final states at the Large Hadron Collider. We study jet cross sections as a function of the shape of both beam and central jets. This work focuses on measuring jet mass but our methods can be applied to other jet shape variables as well. Measuring jet mass introduces additional scales to the collision process and these scales produce large logarithms that need to be resummed. Factorizing the cross section into hard, jet, beam, and soft functions enables such resummation. We begin by studying jet production at e + e- collisions in order to focus on the effects of jet algorithms. These results can be carried over to the more complicated case of hadron collisions. We use the Sterman-Weinberg algorithm as a specific example and derive an expression for the quark jet function. Turning to hadron colliders, we show how the N-jettiness event shape divides phase space into N +2 regions, each containing one central or beam jet. Thus, N-jettiness works as a jet algorithm. Using a geometric measure gives central jets with circular boundaries. We then give a factorization theorem for the cross section fully differential in the mass of each jet, and compute the corresponding soft function at next-to-leading order (NLO). We use a method of hemisphere decomposition, which can also be applied to calculate N-jet soft functions defined with other jet algorithms. Our calculation of the N-jettiness soft function provides the final missing ingredient to extend NLO cross sections to resunmmed predictions at next-to-next-to-leading logarithmic order. We study the production of an exclusive jet together with a Standard Model Higgs boson. Based on theoretical reasons and agreement between our calculation and data from the ATLAS collaboration, we argue that our results for the jet mass spectrum are a good approximation also for inclusive jet production and other hard processes.by Teppo T. Jouttenus.Ph.D

Interacting electrons on a quantum ring: exact and variational approach

We study a system of interacting electrons on a one-dimensional quantum ring using exact diagonalization and the variational quantum Monte Carlo method. We examine the accuracy of the Slater-Jastrow -type many-body wave function and compare energies and pair distribution functions obtained from the two approaches. Our results show that this wave function captures most correlation effects. We then study the smooth transition to a regime where the electrons localize in the rotating frame, which for the ultrathin quantum ring system happens at quite high electron density.Comment: 19 pages, 10 figures. Accepted for publication in the New Journal of Physic

Fully-Unintegrated Parton Distribution and Fragmentation Functions at Perturbative k_T

We define and study the properties of generalized beam functions (BFs) and fragmenting jet functions (FJFs), which are fully-unintegrated parton distribution functions (PDFs) and fragmentation functions (FFs) for perturbative k_T. We calculate at one loop the coefficients for matching them onto standard PDFs and FFs, correcting previous results for the BFs in the literature. Technical subtleties when measuring transverse momentum in dimensional regularization are clarified, and this enables us to renormalize in momentum space. Generalized BFs describe the distribution in the full four-momentum k_mu of a colliding parton taken out of an initial-state hadron, and therefore characterize the collinear initial-state radiation. We illustrate their importance through a factorization theorem for pp -> l^+ l^- + 0 jets, where the transverse momentum of the lepton pair is measured. Generalized FJFs are relevant for the analysis of semi-inclusive processes where the full momentum of a hadron, fragmenting from a jet with constrained invariant mass, is measured. Their significance is shown for the example of e^+ e^- -> dijet+h, where the perpendicular momentum of the fragmenting hadron with respect to the thrust axis is measured.Comment: Journal versio

A general method for the resummation of event-shape distributions in e⁺ e− annihilation

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

The Quark Beam Function at NNLL

In hard collisions at a hadron collider the most appropriate description of the initial state depends on what is measured in the final state. Parton distribution functions (PDFs) evolved to the hard collision scale Q are appropriate for inclusive observables, but not for measurements with a specific number of hard jets, leptons, and photons. Here the incoming protons are probed and lose their identity to an incoming jet at a scale \mu_B << Q, and the initial state is described by universal beam functions. We discuss the field-theoretic treatment of beam functions, and show that the beam function has the same RG evolution as the jet function to all orders in perturbation theory. In contrast to PDF evolution, the beam function evolution does not mix quarks and gluons and changes the virtuality of the colliding parton at fixed momentum fraction. At \mu_B, the incoming jet can be described perturbatively, and we give a detailed derivation of the one-loop matching of the quark beam function onto quark and gluon PDFs. We compute the associated NLO Wilson coefficients and explicitly verify the cancellation of IR singularities. As an application, we give an expression for the next-to-next-to-leading logarithmic order (NNLL) resummed Drell-Yan beam thrust cross section.Comment: 54 pages, 9 figures; v2: notation simplified in a few places, typos fixed; v3: journal versio

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 \Lambda due 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 \alpha_s^2 \ln^2(\Lambda/m_{1,2}) or \alpha_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 \alpha_s a factorized form for the soft function S(k_L,k_R,\Lambda) in the cross section \sigma(m_1,m_2,\Lambda) in which dependence on each of the global logs of \mu/k_L, \mu/k_R and \mu/\Lambda determined 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.Comment: 19 pages, 10 figures. v2: minor edits, additional discussion in Introduction. v3: version published in JHE

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 \tau_a to one-loop order (O(alpha_s)) and resum a subset of the large logarithms of \tau_a needed for next-to-leading logarithmic (NLL) accuracy for both cone and kT-type jets. We compare our predictions for the resummed \tau_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.Comment: 62 pages plus 21 pages of Appendices, 13 figures, uses JHEP3.cls. v2: corrections to finite parts of NLO jet functions, minor changes to plots, clarified discussion of power corrections. v3: Journal version. Introductory sections significantly reorganized for clarity, classification of logarithmic accuracy clarified, results for non-Mercedes-Benz configurations adde