MEKS: A program for computation of inclusive jet cross sections at hadron colliders☆
Introduction
Production of particle jets in high-energy collisions is a cornerstone process of the physics program at the CERN Large Hadron Collider (LHC) and Fermilab Tevatron collider [1], [2], [3], [4], [5], [6]. Historically, observation of final-state jets formed by hadrons in collisions confirmed the asymptotic freedom of strong interactions. In modern experiments, measurements of inclusive jet production at and colliders have reached unprecedented precision. They serve both for exacting tests of perturbative quantum chromodynamics (PQCD) and for searches for hypothetical new interactions at the highest energy scales attained. Within PQCD, measurements of single-jet production cross sections at the Tevatron Run-2 constrain the QCD coupling constant [7] and parton distribution functions (PDF) in the proton [8], [9], [10], [11]. Jet production has unique sensitivity to the momentum distribution of gluons with large momentum fractions , which is not available in other scattering processes [12]. Invariant mass distributions of dijets [13], dijet angular distributions [14], [15], and other jet observables at the LHC [6], [16], [17] are examined to search for quark compositeness and heavy particle resonances. All these analyses depend on reliable theoretical computations which evolve to stay on par with experimental developments.
From the experimental point of view, jet production has an advantage of very high statistics and a drawback of sizable systematic errors associated with the complexities of jet reconstruction. On the theory side, predictions for jet observables are known to next-to-leading order (NLO) only [18], [19], [20], [21], [22], even though the NNLO computation is currently in progress. Theoretical uncertainties due to the QCD scale dependence and the fixed-order model for the jet algorithm are comparable to the experimental errors. Some phenomenological studies also include partial next-to-next-to-leading order (NNLO) contributions to jet cross sections obtained by threshold resummation [23].
This paper describes MEKS, a program for predicting probabilities of observation of a single jet or of jet pairs that are accompanied by arbitrary final states, and . An early numerical code (EKS) for the NLO calculation of single inclusive jet and dijet distributions was developed by S.D. Ellis, Z. Kunszt and D.E. Soper in the 1990’s [18] based on the subtraction method. The MEKS program is based on the original EKS calculation that has been augmented by new elements to boost its stability, flexibility, and efficiency. The NLOJET++ program [20], [21] is another independent NLO calculation of inclusive one and two jet cross sections, while FastNLO [24], [25] and APPLGRID [26] programs provide fast interpolation of NLOJET++ cross sections in the kinematical bins of already published experimental measurements.1 Besides these fixed-order calculations, POWHEG combines the NLO jet production cross sections with leading-logarithm QCD showering effects [27].
There are several reasons to release the new MEKS program now. Precision calculations for these processes are challenging because of the rapid falloff of the cross sections with the jet’s transverse momentum () and rapidity (). In a typical experimental data set, jet cross sections vary by up to 6–9 orders of magnitude. In addition, large numerical cancellations occur between some and contributions due to the presence of QCD singularities. MEKS shares the same matrix elements with the EKS code, but its numerical implementation was deeply revised compared to the original EKS in order to achieve quick and accurate integration.
Given the importance of the inclusive jet cross sections for the PDF fits and QCD physics program, it is essential to have at least two completely independent programs that could be compared to validate the predictions. EKS and NLOJET++ are two such independent programs. They perform the base NLO calculation, while APPLGRID and FASTNLO are not independent on their own, but interpolate tables of NLO cross sections computed by NLOJET++.
However, the original EKS program was written in an age of limited computational resources. It relies on a computational scheme that incorporated a fast, but less sophisticated Monte Carlo integration algorithm. The fluctuating EKS output was smoothed after integration by a trial function that was adjusted for each jet observable. This implementation was sufficient for moderately precise calculations, but it made the original EKS program difficult to use when high precision and parallel computations became necessary.
The MEKS program includes a modern Monte Carlo integration module, convenient output, and parallel computation features. It does not require adjustable smoothing of the output cross sections as the old EKS did. The user can control the accuracy simply by changing the number of Monte Carlo events.
The MEKS output is produced in the form of two-dimensional differential cross sections (, ) after integration over the unobserved momentum components using the VEGAS method from the CUBA2.1 library [28]. Other types of output distributions can be implemented with the help of a customizable interface. The Monte Carlo integration is automatically optimized to improve the speed of the calculation. The generated events are written into finely binned two-dimensional histograms that can be rebinned into any set of coarse bins of a given experiment at the stage of the user’s final analysis. This format is different from the FastNLO format, which provides the cross sections in the coarse bins that are taken from pre-existing experimental publications.
The paper provides a user guide for the MEKS program. Another goal was to systematically compare the MEKS and FastNLO codes and document their input settings that bring them into excellent agreement. The analyses that use the NLO jet calculations can be terse about the specific theoretical inputs that were assumed, such as the exact definitions of the jet algorithm and QCD scales. However, the NLO theoretical uncertainties are currently comparable to experimental errors, and it is no longer acceptable to use the NLO jet calculations as “black boxes”. We attempted to rectify these omissions and spell out the settings of the QCD scales and other parameters that are important for the description of the current data. We also illustrate the range of theoretical uncertainties that arise when these parameters are varied.
This document is structured as follows. Section 2 reviews theoretical prerequisites for the calculation of single inclusive jet and dijet cross sections at NLO. Section 3 describes the structure of the program, its inputs and outputs, installation, and running. Section 4 considers the performance of the program and summarizes its benchmark comparison against FastNLO. Section 5 contains the conclusion.
Section snippets
Factorized cross sections and measurement functions
A jet reveals itself by tracks and calorimeter energy depositions left by final-state hadrons in a collider detector. Numerous particles comprise a typical jet, and their detailed distribution is complicated. Nevertheless the probability for producing the whole jet can be deduced with high confidence from the cross section for production of the partons (quarks or gluons) that are the jet’s progenitors. Each contributing parton-level cross section can be computed in PQCD. It is included into the
Algorithm
The goal of the MEKS program is to calculate double-differential cross sections for single-inclusive jet or dijet production at hadron colliders up to NLO in QCD. Its basic algorithm is shown in Fig. 1. All executables, input files, and output files are stored in the subdirectory data/. The main computation is carried out in an executable jetbin, which performs Monte Carlo integration of fully differential NLO cross sections using the VEGAS algorithm provided by the CUBA library [28]. The input
Performance and benchmark comparison
Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7 compare our representative numerical results with the ones provided by FastNLO 1.0 [24] for the distributions of single-inclusive jets, the invariant mass distributions of dijets, and (in the case of D0 Run-2) the angular distributions () of dijets. Kinematical bins of the Tevatron () [1], [2], [3], [4] and LHC () [6], [17] measurements, and CTEQ6.6 central PDFs [37] were used. For this benchmark comparison, we use the Midpoint
Conclusion
In conclusion, this document describes the upgraded EKS program (MEKS) that provides a fast and stable NLO calculation of double-differential cross sections for single-inclusive jet and dijet production at hadron colliders. The new program uses the VEGAS Monte Carlo sampling and the EKS function to generate weighted events and fill them into finely binned two-dimensional histograms for a later analysis. It also includes a user interface to add new jet observables, which is advantageous compared
Acknowledgments
This work was supported by the U.S. DOE Early Career Research AwardDE-SC0003870 and by Lightner-Sams Foundation; by the U.S. Department of Energy under Grant No. DE-FG02-96ER40969; by the U.S. National Science Foundation under Grant No. PHY-0855561; by the National Science Council of Taiwan under Grant Nos. NSC-98-2112-M-133-002-MY3 and NSC-101-2112-M-133-001-MY3. PMN appreciates helpful discussions of benchmark comparison with J. Huston, J. Rojo, and M. Wobisch. PMN and DES also benefited from
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This paper and its associated computer program are available via the Computer Physics Communication homepage on ScienceDirect (http://www.sciencedirect.com/science/journal/00104655).