The B LACK H AT ( 黑帽 ) Library for One-Loop Amplitudes David A. Kosower Institut de Physique Théorique, CEA–Saclay on behalf of the B LACK H AT Collaboration Z. Bern, L. Dixon, Fernando Febres Cordero, Stefan Höche, Harald Ita, DAK, Daniel Maître, Kemal Ozeren [ , ; , , , , & work in progress] ACAT 2013, Beijing, China May 17, 2013

Next-to-Leading Order in QCD Precision QCD requires at least NLO QCD at LO is not quantitative: large dependence on unphysical renormalization and factorization scales NLO: reduced dependence, first quantitative prediction NLO importance grows with increasing number of jets Applications to Multi-Jet Processes:  Measurements of Standard-Model distributions & cross sections  Estimating backgrounds in Searches Expect predictions reliable to 10–15% <5% predictions will require NNLO

Tree-level matrix elements for LO and real-emission terms known since ’80s …but we’ve improved efficiency since then ( CSW, BCFW ) Singular (soft & collinear) behavior of tree-level amplitudes, integrals, initial-state collinear behavior known since ’90s NLO parton distributions known since ’90s General framework for numerical programs known since ’90s Catani, Seymour (1996); Giele, Glover, DAK (1993); Frixione, Kunszt, Signer (1995) Automating real—virtual cancellation for general processes Gleisberg, Krauss; Seymour, Tevlin; Hasegawa, Moch, Uwer; Frederix, Gehrmann, Greiner (2008); Frederix, Frixione, Maltoni, Stelzer (2009) On-shell Methods: one-loop amplitudes W+2 jets ( MCFM )  W+3 jets  W+4 jets  W+5 jets  Bern, Dixon, DAK, Weinzierl (1997–8); B LACK H AT ; B LACK H AT Campbell, Glover, Miller (1997) Rocket Bottleneck: one-loop amplitudes Ingredients for NLO Calculations

On-Shell Methods Calculations of complete one-loop amplitudes Using only information from physical, on-shell states Avoids factorial explosion of Feynman diagrams & associated cancelations Reflects relative simplicity of answers

B LACK H AT On-shell methods  Numerical implementation Automated implementation  industrialization C++ framework for automated one-loop calculations: organization, integral basis, spinor products, residue extraction, tree ingredients, caching SHERPA for real subtraction, real emission, phase-space integration, and analysis

Groups with codes using on-shell methods – BlackHat – CutTools+HELAC-NLO: Ossola, Papadopoulos, Pittau, Actis, Bevilacqua, Czakon, Draggiotis, Garzelli, van Hameren, Mastrolia, Worek & their clients – Rocket: Ellis, Giele, Kunszt, Lazopoulos, Melnikov, Zanderighi – Samurai/GoSam: Mastrolia, Ossola, van Deurzen, Greiner, Luisoni, Mirabella, Peraro, Reiter, von Soden-Fraunhofen, Tramontano – Ngluon/NJets: Badger, Biedermann, & Uwer + Sattler & Yundin – MadLoop: Hirschi, Frederix, Frixione, Garzelli, Maltoni, & Pittau – Giele, Kunszt, Stavenga, Winter

N-Tuples Save files of generated events (B ORN, V IRTUAL, R EAL - S UBTRACTED, I NTEGRATED SUBTRACTION ) – R OOT format – Mild generation-level cuts (avoid IR-divergent regions) – Choice of jet algorithms & sizes Can (re-)analyze later – Change cuts – Change renormalization/factorization scales (modestly, but including functional form) – Estimate PDF uncertainties efficiently Pass on to experimenters – Detailed experimental cuts (e.g. R IVET [ at least for central values ]) – Compare distributions, not at event level

SHERPA B LACK H AT can be used in various frameworks (SHERPA, MadGraph, P OWHEG B OX — via B INOTH L ES H OUCHES A CCORD ) We’ve used SHERPA – Overall run management – Phase-space integration – Subprocess management – COMIX for Born and real-emission matrix elements Two phases: grid generation (once) & production running (parallel) MPI crucial for efficient grid generation (using tree-level matrix elements) with high multiplicities Production in parallel with or without MPI

Rescue System Subset of terms contains unphysical singularities at points in phase space — cancellation would be exact analytically but could lead to numerical instabilities Could have special expansions at each such point Instead, recalculate just unstable terms at higher precision – Criteria: IR cancellations; absence of excess-power terms; cancellation between cut and rational parts – Few % of events have (some) unstable terms – 10–20% of running time

W+4 Jets Scale variation reduced substantially at NLO Successive jet distributions fall more steeply Shapes of 4 th jet distribution unchanged at NLO — but first three are slightly steeper

Z+4 Jets

W+5 Jets Last jet shape is stable, harder jets have steeper spectrum at NLO

Jet Ratios Relaxation of kinematic restrictions leads to NLO corrections at large p T in V+3/V+2, otherwise stable Ratio is not constant as a function of p T — fits to α+β n will have α & β dependent on p Tmin

Extrapolations Let’s try extrapolating ratios to larger n We know the W+2/W+1 ratio behaves differently from W+n/W+(n − 1) ratios, because of kinematic constraints & missing processes (especially at LO) We could extrapolate from W+4/W+3 & W+3/W+2 — but with two points and two parameters, how meaningful is that? With the W+5/W+4 ratio, a linear fit (with excellent χ 2 /dof) makes the extrapolation meaningful: W − + 6 jets: 0.15 ± 0.01 pb W jets: 0.30 ± 0.03 pb

B LACK H AT ( 黑帽 ) Summary On-shell methods are maturing into the method of choice for QCD calculations for colliders Study of Standard-Model signals in widely-varying kinematic regimes gives confidence in our ability to understand backgrounds quantitatively at high multiplicity Extrapolations Processes with large jet multiplicity compared with data Supplied to experimenters via R OOT n-tuple files