Precise Predictions for Hadronic Collisions from On-Shell Methods Carola F. Berger CTP, MIT PHENO 2009, May 12th 2009
BlackHat and Sherpa BlackHat: Outline ● BlackHat and CFB, Zvi Bern, Lance Dixon, Fernando Febres Sherpa ● Outline Cordero, Darren Forde, Harald Ita, David Kosower, Introduction Daniel Maitre BlackHat BlackHat: arXiv:0902.2760, PRD78 (2008) 036003. Badger: JHEP 0901 (2009) 049. Forde: PRD75 (2007) 125019. CFB, Bern, Dixon, Forde, Kosower: PRD74 (2006) 036009. Summary Sherpa liaison (real emissions): Tanju Gleisberg Gleisberg et al, JHEP 0902 (2009) 007. Gleisberg, Krauss, Eur. Phys. J C53 (2008) 501. Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 2/15
Outline ■ Introduction Do we really need NLO? Outline ● BlackHat and ■ What is BlackHat? Sherpa ● Outline ◆ Terms with logarithms (dilogs, ...) from Introduction generalized unitarity ◆ Rational terms BlackHat ■ Physics Results – W + 3 jets Summary ⇒ Fernando’s Talk! Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 3/15
Precision Calculations Outline Introduction ● Precision Calculations ● The LHC Wishlists ● NLO Corrections to LHC Processes BlackHat Summary Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 4/15
Precision Calculations Outline Introduction ● Precision Calculations ● The LHC Wishlists ● NLO Corrections to LHC Processes BlackHat Summary Dissertori (CMS) Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 4/15
The (In)Famous Experimenters’ Wishlists Les Houches 2005 process wanted at NLO background to Outline ( V ∈ { Z, W, γ } ) Introduction t ¯ 1. pp → V V + jet tH , new physics ● Precision Calculations 2. pp → H + 2 jets H production by ● The LHC Wishlists vector boson fusion (VBF) ● NLO Corrections to LHC Processes tb ¯ 3. pp → t ¯ t ¯ b tH BlackHat 4. pp → t ¯ t ¯ t + 2 jets tH 5. pp → V V b ¯ VBF → H → V V , t ¯ b tH , new physics Summary 6. pp → V V + 2 jets VBF → H → V V 7. pp → V + 3 jets new physics 8. pp → V V V SUSY trilepton 2 → 3 , computed via standard methods, + one process 2 → 4 Bredenstein, Dittmaier, Denner, Pozzorini 2 → 4 , computed via on-shell methods (BlackHat and (partially) Rocket) Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 5/15
The (In)Famous Experimenters’ Wishlists Run II Monte Carlo Workshop 2001 Outline Single boson Diboson Triboson Heavy flavor t ¯ Introduction W + ≤ 5 j W W + ≤ 5 j W W W + ≤ 3 j t + ≤ 3 j ● Precision W + b ¯ W W + b ¯ W W W + b ¯ t ¯ b + ≤ 3 j b + ≤ 3 j b + ≤ 3 j t + γ + ≤ 2 j Calculations t ¯ W + c ¯ c + ≤ 3 j W W + c ¯ c + ≤ 3 j W W W + γγ + ≤ 3 j t + W + ≤ 2 j ● The LHC Wishlists t ¯ Z + ≤ 5 j ZZ + ≤ 5 j Zγγ + ≤ 3 j t + Z + ≤ 2 j ● NLO Corrections Z + b ¯ ZZ + b ¯ t ¯ b + ≤ 3 j b + ≤ 3 j W ZZ + ≤ 3 j t + H + ≤ 2 j to LHC Processes t ¯ c + ≤ 3 j c + ≤ 3 j ZZZ + ≤ 3 j b + ≤ 2 j Z + c ¯ ZZ + c ¯ tb ¯ BlackHat γ + ≤ 5 j γγ + ≤ 5 j b + ≤ 3 j γ + b ¯ γγ + b ¯ b + ≤ 3 j b + ≤ 3 j Summary c + ≤ 3 j c + ≤ 3 j γ + c ¯ γγ + c ¯ W Z + ≤ 5 j W Z + b ¯ b + ≤ 3 j W Z + c ¯ c + ≤ 3 j W γ + ≤ 3 j Zγ + ≤ 3 j Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 5/15
NLO Corrections to LHC Processes Outline Introduction ● Precision Calculations ● The LHC Wishlists ● NLO Corrections to LHC Processes BlackHat Summary Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 6/15
NLO Corrections to LHC Processes ■ Relevant processes all 2 → n ≥ 3 Outline Introduction ● Precision Calculations ● The LHC Wishlists ● NLO Corrections to LHC Processes BlackHat Summary Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 7/15
NLO Corrections to LHC Processes ■ Relevant processes all 2 → n ≥ 3 Outline ■ Real-virtual cancellations a solved problem, Introduction ● Precision automated Calculations ● The LHC Wishlists ● NLO Corrections to LHC Processes BlackHat Summary Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 7/15
NLO Corrections to LHC Processes ■ Relevant processes all 2 → n ≥ 3 Outline ■ Real-virtual cancellations a solved problem, Introduction ● Precision automated Calculations ● The LHC Wishlists ■ Bottleneck: 1-loop virtual amplitudes ● NLO Corrections to LHC Processes It took 11 years to go from 5-gluon 1-loop amplitudes to 6 gluons! BlackHat Summary Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 7/15
NLO Corrections to LHC Processes ■ Relevant processes all 2 → n ≥ 3 Outline ■ Real-virtual cancellations a solved problem, Introduction ● Precision automated Calculations ● The LHC Wishlists ■ Bottleneck: 1-loop virtual amplitudes ● NLO Corrections to LHC Processes It took 11 years to go from 5-gluon 1-loop amplitudes to 6 gluons! BlackHat ■ New methods based on (generalized) unitarity and Summary recursion ⇒ new codes: BlackHat, Rocket (D-dim unitarity), CutTools/OneLOop (D-dim unitarity at integrand level) Rocket: Ellis, Giele, Kunszt, Melnikov, Zanderighi. CutTools/OneLOop: van Hameren, Ossola, Papadopoulos, Pittau Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 7/15
One-Loop Decomposition Outline I I I 4 2 3 Introduction BlackHat ● One-Loop Decomposition ● Generalized Unitarity Any n -leg (massless) one-loop amplitude expressible ● Tree Level ● Proof at Tree-Level in terms of scalar box, triangle and bubble integrals: ● Rational Terms from Recursion ● Rational Terms - A = c 4 I 4 + c 3 I 3 + c 2 I 2 + rational D-dim Unitarity With massive partons there are additionally I 1 Summary (tadpoles) We know the integrals, the task is to determine the coefficients Bern, Dixon, Dunbar, Kosower Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 8/15
Generalized Unitarity 1 � d 4 l c 4 I 4 = c 4 l 2 ( l − K 1 ) 2 ( l − K 2 ) 2 ( l − K 3 ) 2 Outline 1 1 Introduction P 2 + iδ + ( P 2 ) P 2 + iε = BlackHat ● One-Loop Box integrals have unique leading singularity ⇒ Decomposition ● Generalized generalized unitarity Unitarity ● Tree Level ● Proof at Tree-Level ● Rational Terms � d 4 lδ + ( l 2 ) δ + (( l − K 1 ) 2 ) c 4 ∆ LS I 4 = from Recursion ● Rational Terms - D-dim Unitarity × δ + (( l − K 2 ) 2 ) δ + (( l − K 3 ) 2 ) Summary × A tree ( l ) × A tree ( l ) × A tree ( l ) × A tree ( l ) 1 2 3 4 c 4 = A tree ( l sol ) × A tree ( l sol ) × A tree ( l sol ) × A tree ( l sol ) 1 2 3 4 Tree graphs on shell Trees “recycled” into loops Britto, Cachazo, Feng Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 9/15
On-Shell Recursion Relations at Tree Level A n Outline Introduction 2 p = 0 BlackHat Complex continue (shift) spinors and momenta: ● One-Loop Decomposition ● Generalized p i → p i ( z ) p j → p j ( z ) Unitarity ● Tree Level p i + p j → p i + p j ● Proof at Tree-Level ● Rational Terms from Recursion Momentum conservation is maintained, momenta ● Rational Terms - on-shell ( p i ( z ) 2 = p j ( z ) 2 = 0 ). D-dim Unitarity Summary ^ ^ n−1 i n j 1 A n A A = R L 2 2 p = 0 Britto, Cachazo, Feng Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 10/15
Proof at Tree-Level Propagators and thus amplitudes are now functions of the complex parameter: Outline 1 /P 2 1 /P 2 → l...j...m ( z ) Introduction l...j...m 1 BlackHat l...j...m ( z ) A − h � � A h A ( z ) = L ( z ) R ( z ) ● One-Loop P 2 Decomposition ● Generalized l,m h Unitarity ● Tree Level ● Proof at Tree-Level ● Rational Terms from Recursion ● Rational Terms - D-dim Unitarity Summary Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 11/15
Proof at Tree-Level Propagators and thus amplitudes are now functions of the complex parameter: Outline 1 /P 2 1 /P 2 → l...j...m ( z ) Introduction l...j...m 1 BlackHat l...j...m ( z ) A − h � � A h A ( z ) = L ( z ) R ( z ) ● One-Loop P 2 Decomposition ● Generalized l,m h Unitarity ● Tree Level If A ( z → ∞ ) → 0 - Cauchy’s theorem z ● Proof at Tree-Level ● Rational Terms from Recursion 1 dz � ● Rational Terms - z A ( z ) = 0 D-dim Unitarity C 2 πi C A ( z ) Summary � A (0) = − Res z z = z α poles α 1 A − h � � A h = L ( z α ) R ( z α ) P 2 l...j...m poles α h Britto, Cachazo, Feng, Witten Carola F. Berger PHENO 2009, May 12th 2009 Precise Predictions for Hadronic Collisions, BlackHat - 11/15
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