We present recent next-to-leading order (NLO) results in perturbative QCD
obtained using the BlackHat software library. We discuss the use of n-tuples to
separate the lengthy matrix-element computations from the analysis process. The
use of n-tuples allows many analyses to be carried out on the same phase-space
samples, and also allows experimenters to conduct their own analyses using the
original NLO computation.Comment: Talk given at ACAT 2013, Beijing, China, May 16--21, 2013; 6 pages, 2
  figures; added reference

Bern, Z.

Cordero, F. Febres

Dixon, L. J.

Hoeche, S.

Ita, H.

Kosower, D. A.

Maitre, D.

Ozeren, K. J.

Journal of Physics : Conference Series

English

arXiv

We present recent next-to-leading order (NLO) results in perturbative QCD obtained using the BLACKHAT software library. We discuss the use of n-tuples to separate the lengthy matrix-element computations from the analysis process. The use of n-tuples allows many analyses to be carried out on the same phase-space samples, and also allows experimenters to conduct their own analyses using the original NLO computation

Bern, Zvi

Dixon, Lance

Febres Cordero, Fernando

Höche, Stefan

Ita, Harald

Kosower, David A.

Maître, Daniel

Ozeren, Kemal

FreiDok plus

The BlackHat library for one-loop amplitudes

UNT (University of North Texas) Digital Library

UCLA/13/TEP/109 SB/F/xxx-13 SLAC-PUB-15760IPPP-13-85 Saclay–IPhT–T13/229The BlackHat Library for One-Loop AmplitudesZ. Bern1, L. J. Dixon2, F. Febres Cordero3, S. Höche4, H. Ita5,D. A. Kosower6,∗, D. Mâıtre7, K. J. Ozeren81,8Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547, USA2,4SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA3Departamento de F́ısica, Universidad Simón Boĺıvar, Caracas 1080A, Venezuela5Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, D–79104 Freiburg, Germany6Institut de Physique Théorique, CEA–Saclay, F–91191 Gif-sur-Yvette cedex, France7Institute for Particle Physics Phenomenology, University of Durham, Durham DH1 3LE, UKAbstract. We present recent next-to-leading order (NLO) results in perturbative QCD obtained usingthe BlackHat software library. We discuss the use of n-tuples to separate the lengthy matrix-elementcomputations from the analysis process. The use of n-tuples allows many analyses to be carried out onthe same phase-space samples, and also allows experimenters to conduct their own analyses using theoriginal NLO computation.1. IntroductionExperimental studies of Standard-Model processes at the Large Hadron Collider (LHC), whetheras standard candles, as backgrounds to new physics, or as part of a program of precisionmeasurements, require a theoretical counterpart. The dominant contributions to theoreticalpredictions arise from perturbative quantum chromodynamics (QCD). In QCD, unlike the casefor other components of the Standard Model, leading-order (LO) calculations, which rely solelyon tree-level amplitudes in QCD, do not offer a quantitatively reliable prediction. This problemis a result of the uncompensated dependence of the prediction on the renormalization andfactorization scales. At LO this dependence arises only via the running coupling and theevolving parton distribution functions. Next-to-leading order (NLO) calculations in QCD curethis problem, and are thus the the first order in perturbation theory to provide a quantitativelyreliable estimate of backgrounds due to Standard-Model processes [1, 2].Recent years have seen remarkable progress in NLO calculations, thanks to advances incomputing the one-loop matrix elements which lie at their core [3, 4, 5, 6, 7, 8, 9, 10]. TheBlackHat collaboration has developed its eponymous software [11] to implement computationsof one-loop amplitudes in a numerical approach. For multiplicities beyond W+ 2-jet production,only tree-level amplitudes and integrals are implemented in computer code directly from analyticformulas. The library implements so-called on-shell methods, which rely only on information∗PresenterWork supported in part by US Department of Energy under contract DE-AC02-76SF00515.present in amplitudes with on-shell external states in order to arrive at values for more complexamplitudes. This keeps the relative simplicity — compared to the factorial complexity one mighthave feared from a Feynman-diagrammatic expansion — manifest at every stage of a calculation.This simplification is particularly evident in high-multiplicity calculations, for which the frontierhas moved from W+ 2-jet production [4, 12, 13] a decade and a half ago to W+ 5-jet productionearlier this year [14].2. N-TuplesWhile numerical one-loop libraries have made previously-unattainable NLO calculations feasible,they are not the only ingredient needed. In addition to the tree-level matrix elements for thebasic process, we also need tree-level matrix elements with an additional parton emission: a gluonemitted, or a gluon splitting to a quark–antiquark pair. We use the COMIX [15] library withinSHERPA [16] to compute these matrix elements. Different contributions to an NLO calculationhave infrared divergences: in the virtual corrections, they are present explicitly, in the formof poles in the dimensional regulator ǫ; in the real-emission corrections, they arise in phase-space integrals over squared matrix elements. These divergences cancel in physical quantitiessuch as cross sections and distributions of infrared-safe observables. In order to make thiscancellation manifest in an operationally-tractable manner, one subtracts local approximationsto the squared real-emission matrix elements, and adds back the analytic integrals of these localapproximations. These integrals have manifest singularities, again in the form of poles in ǫ,which cancel those in the virtual corrections. The local subtraction terms and the computationof the appropriate subtraction integrals (using analytic formulas) are also performed by theCOMIX library. We thus have four types of contributions to integrate numerically: Born (B),virtual corrections (V), integrated subtraction (I), and subtracted real emission (R).A typical calculation at high multiplicity will involve dozens of subprocesses (two-quark,four-quark, six-quark, etc. in all possible crossings). Each must be integrated over final-statephase space, subject to cuts simulating the experimental ones. We use SHERPA to manage thesubprocesses and perform the required numerical integrations. We use the FastJet library [17]to apply a jet algorithm to phase-space configurations of partons. The different contributionsspan a large numerical range; to focus computer time on the largest contributions, it is helpfulto further split up each of the four types (B,V,I,R) into parts, corresponding to leading andsubleading color, or to different initial states, etc.This subdivision does exact a penalty, in the form of the increased complexity of managinga calculation, tracking all the different parts, and ensuring that each part was run correctly.In a physics study conducted using stand-alone NLO codes such as NLOJET++ [18], one wouldwrite code for a set of observables, and obtain distributions in those observables by runningthe complete code which computes both squared matrix elements and observables. Computingnew observables as a physics study evolves; varying renormalization and factorization scales;computing uncertainties due to our imprecise knowledge of parton distribution functions (PDFs),would all require rerunning the code from scratch. This would force recomputation of the short-distance matrix elements, which are much more expensive to compute than observables. Itwould also force us to confront the subprocesses’ and subparts’ management complexity aneweach time.It would simplify the calculation to separate the computation of the squared matrix elements(along with the generation of phase-space configurations) from the computation of observables.We do this by saving generated phase-space configurations, along with their weights, in a set ofRoot [19] n-tuple files. The weights are obtained by combining the squared matrix elementswith the phase-space measure, parton-distribution values, and appropriate Jacobian factors.In addition, we include a set of auxiliary coefficients which allow the recomputation of theweights using different renormalization and factorization scales, or different PDF sets. The filesare quite voluminous, especially for higher multiplicities, so the compression offered by Rootyields a significant reduction compared to a naive binary format both in disk-space usage andin transmission times. A lightweight analysis code scans the resulting files, and computes crosssections or histograms of observables. This code can be rerun as needed for new observables orfor estimating scale-sensitivity of observables, at relatively low cost in computer time.The use of n-tuple files has a bonus: we can hand over sets of n-tuple files to our experimentercolleagues, who can then run their own analyses on the events they contain. This avoids theexperimenters having to learn how to run a somewhat complicated set-up, and avoids themhaving to adopt our code or framework for performing analyses. The analyses are of parton-level fixed-order events, so that only distributions [of infrared-safe observables] can be comparedin a meaningful way to experimental data. (The n-tuples we have computed to date are notsuitable as input to existing NLO-matched parton-shower codes.)3. Vector Boson+Jets100 200 300 400 50010-510-410-310-210-1dσ / dpT    [ pb / GeV ]LONLO100 200 300 400 500First  Jet  pT  [ GeV ]0.511.522.533.5 LO / NLO100 200 300 400 500100 200 300 400 500Second  Jet  pT  [ GeV ]100 200 300 400 500100 200 300 400 500Third  Jet  pT  [ GeV ]100 200 300 400 500100 200 300 400 500Fourth  Jet  pT  [ GeV ]100 200 300 400 50010-510-410-310-210-1100 200 300 400 500Fifth Jet pT  [ GeV ]0.511.522.533.5NLO scale dependenceBlackHat+SherpaLO scale dependencepTjet  >  25 GeV,  | ηjet |  <  3 ETe  >  20 GeV,   | ηe |   <  2.5ETν > 20 GeV,   MTW  > 20 GeVR   =   0.5   [anti-kT]√s   =  7 TeVµR  =  µF  =  HT^ ’  / 2 W- + 5 jets + XFigure 1. The pT distributions of the leading five jets in W− + 5-jet production, ordered indecreasing pT from left to right. In the upper panels, the blue (dashed) line shows the LOprediction, and the black (solid) line, the NLO one. The lower panels show the ratio of thecentral LO prediction to the NLO one, along with the LO scale variation band in orange-brown(hatched) and the NLO scale variation band in gray (solid).As an example of recent studies we have performed using the BlackHat library runningunder SHERPA’s control, and using n-tuple files, we show the distribution of the leading five jetsin an NLO computation [14] of W− + 5-jet+X production in fig. 1. Successive jet distributionsfall more steeply. This reflects the fact that moving out on the distribution of, say, the third jetallows the fourth and fifth jet to remain at low pT, thus forcing up the center-of-mass energy ofthe short-distance partonic collision less than moving out on the distribution of the fifth jet. Asthe center-of-mass energy grows, the short-distance cross section decreases rapidly, both becauseof direct dependence in the squared matrix element and flux factors, and because the partondistributions decrease as well.For the leading through next-to-softest jet (the fourth jet in this case), the NLO distributionsfall somewhat more steeply, as reflected in the upward tilt of the LO/NLO ratio in the bottompanels of fig. 1. This continues a pattern seen at lower multiplicity, for example in the W + 4-jetcalculation [20].The scale-variation bands shrink dramatically in the NLO calculation as compared to the LOone. This fulfills one of the primary motivations for pursuing this calculation: we have achieved a10–15% level of reliability, within expected experimental uncertainties for this high-multiplicityprocess.1 2 3 4 5 6 70.180.200.220.240.260.280.300.32Jet production ratioW− LOW− NLOW+ LOW+ NLOFigure 2. Ratios of W + n- to W+ (n−1)-jet production at LO and NLO. The uppermost setof points shows the ratios for W+ production at LO; the next set, for W− production at LO;the third, for W+ production at NLO; and the bottommost, W− production at NLO.There is an added bonus, if we consider ratios of cross sections for W + n- to W+ (n−1)-jetproduction. This “jet-production” ratio [21] is sometimes called the “Berends” or “staircase”ratio. We should expect the W + 1-jet cross section to behave quite differently from crosssections with higher multiplicity, because some subprocesses (for example, gg-initiated ones) aremissing at LO, and also because of kinematic constraints on phase space both at LO and atNLO. This in turn means that the W + 2- to W + 1-jet ratio should be expected to behavequite differently than higher-multiplicity ratios. There are also kinematic constraints evident inW + 2-jet production at LO, but these are largely relaxed at NLO; so we might expect to seesigns of regularity or universality appearing with the W + 3- to W + 2-jet ratio at NLO. Weconsider the ratios of total cross sections with the cuts as shown in the leftmost panel of fig. 1.If we plot these ratios, as shown in fig. 2, regularity is exactly what we see: the ratios for W + 3-to W + 2-jet, W + 4- to W + 3-jet, and W + 5- to W + 4-jet production all lie on straight lines,both at LO and NLO. The former is a bit surprising, but it is the latter that is most significantfor applications to LHC physics. It allows us to extrapolate linearly, and make a prediction forW + 6-jet production. In this extrapolation, the W + 5-jet calculation plays a crucial role inestablishing the reasonableness of a linear extrapolation. We find (see ref. [14] for details),W− + 6 jets : 0.15± 0.01 pb ,W+ + 6 jets : 0.30± 0.03 pb , (1)using the same cuts as shown in fig. 1. The uncertainties are estimated using a Monte-Carloprocedure with synthetic data sets.AcknowledgmentsThis research was supported by the US Department of Energy under contracts DE–AC02–76SF00515 and DE–FG02–13ER42022. DAK’s research is supported by the European ResearchCouncil under Advanced Investigator Grant ERC–AdG–228301. DM’s work was supported bythe Research Executive Agency (REA) of the European Union under the Grant Agreementnumber PITN–GA–2010–264564 (LHCPhenoNet). SH’s work was partly supported by a grantfrom the US LHC Theory Initiative through NSF contract PHY–0705682. This research usedresources of Academic Technology Services at UCLA, and of the National Energy ResearchScientific Computing Center, which is supported by the Office of Science of the U.S. Departmentof Energy under Contract No. DE–AC02–05CH11231.References[1] Z. Bern et al. 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The BlackHat Library for One-Loop Amplitudes

Z Bern

L J Dixon

F Febres Cordero

S Höche

H Ita

D A Kosower

D Maître

K J Ozeren

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Journal of Physics Conference Series

https://digital.library.unt.edu/ark:/67531/metadc865491/m2/1/high_res_d/1122508.pdf

The BlackHat Library for One-Loop Amplitudes

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