OneLoop Amplitudes with Black Hat David A Kosower

One-Loop Amplitudes with Black. Hat David A. Kosower Institut de Physique Théorique, CEA–Saclay on behalf of the Black. Hat Collaboration Carola Berger, Z. Bern, L. Dixon, Fernando Febres Cordero, Darren Forde, Harald Ita, DAK, Daniel Maître 43 rd Rencontres de Moriond, La Thuile, Aosta March 8– 15, 2008

Hunting for Anomalies One-Loop Amplitudes with Black. Hat , Moriond in La Thuile, March 12, 2008

…Avoiding Backgrounds One-Loop Amplitudes with Black. Hat , Moriond in La Thuile, March 12, 2008

Physics Backgrounds at the LHC • Many automated tools available to produce LO predictions (treelevel matrix elements): MADGRAPH; Comp. HEP; AMEGIC++; Anastasiou, Dixon, Melnikov, & Petriello ALPGEN; HELAC; O’MEGA – • Used as black boxes by experimenters QCD at LO is not quantitative – renormalization scale enters into the definition of the coupling – physical quantities are independent of it – computations to fixed order are not LO: large dependence, no quantitative prediction • NLO: reduced dependence, first quantitative prediction • NNLO: precision prediction • One-Loop Amplitudes with Black. Hat , Moriond in La Thuile, March 12, 2008

NLO Technology • Ingredients for W +n-jet computations 2 → W +n-parton tree-level amplitudes (LO term) known since the ’ 80 s – 2 → W +(n+1)-parton tree-level amplitudes known since the ’ 80 s – 2 → W +n-parton one-loop amplitudes only n=4 (known for 10 years) – Bern, Dixon, DAK, Weinzierl (1997– 8); Kunszt, Signer, Trocsanyi (1994); Campbell, Glover, Miller (1997) Singular (soft & collinear) behavior of tree-level amplitudes – & their integrals over phase space known for > 10 years – Initial-state singular behavior known for > 10 years – NLO parton distributions known for > 10 years • Framework for general numerical programs known for > 10 years • Mature technology • Giele, Glover, DAK (1993); Frixione, Kunszt, Signer (1995); Catani, Seymour (1996); Gleisberg, Kraus (2007) Bottleneck One-Loop Amplitudes with Black. Hat , Moriond in La Thuile, March 12, 2008

Wish Lists Process (V = Z, W , γ) Background to pp → V Vjet pp → H + 2 jets , new physics H production via vector boson fusion pp → + 2 jets pp → V V H (via VBF) → V V, pp → V V+ 2 jets H (via VBF) → V V pp → V + 3 jets, V + 4 jets pp → V VV One-Loop Amplitudes with , new physics , SUSY, and other new physics SUSY (trilepton) Black. Hat , Moriond in La Thuile, March 12, 2008

Why Feynman Diagrams Won’t Get You There Huge number of diagrams in calculations of interest — factorial growth • 2 → 6 jets: 34300 tree diagrams, ~ 2. 5 ∙ 107 terms ~2. 9 ∙ 106 1 -loop diagrams, ~ 1. 9 ∙ 1010 terms • But answers often turn out to be very simple • Vertices and propagators involve gauge-variant off-shell states • Each diagram is not gauge invariant — huge cancellations of gauge-noninvariant, redundant, parts in the sum over diagrams Simple results should have a simple derivation — Feynman ( attr ) • Is there an approach in terms of physical states only? • One-Loop Amplitudes with Black. Hat , Moriond in La Thuile, March 12, 2008

New Technologies: On-Shell Methods Use only information from physical states • Use properties of amplitudes as calculational tools • Factorization → on-shell recursion relations – Unitarity → unitarity method Known integral basis: – Underlying field theory → integral basis – • Formalism Unitarity On-shell Recursion Pre-2008: analytic calculations: through six-gluon, all-n MHV • Other groups: Ossola, Papadopoulos, Pittau; Ellis, Giele, Kunszt, Melnikov; Anastasiou, Britto, Feng • One-Loop Amplitudes with Black. Hat , Moriond in La Thuile, March 12, 2008

Intelligent Automation • To date: bespoke calculations Need industrialization automation • Want numerical results: numerical stability • Do analysis analytically • Do algebra numerically • One-Loop Amplitudes with Black. Hat , Moriond in La Thuile, March 12, 2008

Black. Hat Written in C++ • Framework for automated one-loop calculations • Organization in terms of integral basis (boxes, triangles, bubbles) – Assembly of different contributions – Library of functions (spinor products, integrals, residue extraction) • Tree amplitudes (ingredients) • Caching • • Thus far: implemented gluon amplitudes One-Loop Amplitudes with Black. Hat , Moriond in La Thuile, March 12, 2008

Outline of a Calculation • • • Unitarity freezes all propagators in box integrals: coefficients are given by a product of tree amplitudes with complex momenta (Britto, Cachazo, Feng) — compute them numerically Unitarity leaves one degree of freedom in triangle integrals: coefficients are the residues at ∞ (Forde): compute these numerically using discrete Fourier projection Two degrees of freedom in bubbles: two-dimensional discrete Fourier projection Physical poles: on-shell recursion numerically (complex momenta) Spurious poles (‘cut completion’): locations numerically, functional form from each integral known analytically Integral functions known analytically One-Loop Amplitudes with Black. Hat , Moriond in La Thuile, March 12, 2008

Numerics and Examples Usually ordinary double precision is sufficient for stability • For exceptional points (1 -2%), use quad precision • • Check MHV amplitudes (− −++…+), against all-n analytic expressions • Check other six-point amplitudes One-Loop Amplitudes with Black. Hat , Moriond in La Thuile, March 12, 2008

Summary • On-shell methods are method of choice for QCD calculations for colliders • First light for automated one-loop calculations One-Loop Amplitudes with Black. Hat , Moriond in La Thuile, March 12, 2008