Two-Loop Integrand Decomposition Into Master Integrands And Surface - PowerPoint PPT Presentation

Two-Loop Integrand Decomposition Into Master Integrands And Surface Terms Matthieu Jaquier Physics Institute University of Freiburg Loopfest XV Buffalo NY 15 − 17 th August 2016 Based on work with S. Abreu, Z. Bern, F. Febres-Cordero, H. Ita, B. Page and M. Zeng. Matthieu Jaquier (Physics Institute Freiburg) Two-loop Integrand Decomposition August 11, 2016 1 / 19

Introduction LHC era High-luminosity run of the LHC will narrow down experimental errors substantially. Need to provide NNLO predictions for many processes. Matthieu Jaquier (Physics Institute Freiburg) Two-loop Integrand Decomposition August 11, 2016 2 / 19

Introduction Progress in NNLO phenomenology Di-photon: [Catani, Cieri, de Florian, Ferrera, Grazzini 11; Campbell, Ellis, Li, Williams 16] Dijet: [Currie, Gehrmann-De Ridder, Gehrmann, Glover, Pires, Wells 14] W+J: [Boughezal, Focke, Liu, Petriello 15] Z+J: [Gehrmann-De Ridder, Gehrmann, Glover, Huss, Morgan 15; Boughezal, Campbell, Ellis, Focke, Giele, Liu, Petriello 15] H+J: [Chen, Cruz-Martinez, Gehrmann, Glover, MJ 16; Caola, Melnikov, Schulze 15; Boughezal, Focke, Giele, Liu, Petriello 15] tt: [Czakon, Fiedler, Heymes, Mitov 16] WW: [Gehrmann, Grazzini, Kallweit, Maierh¨ ofer, v. Mannteuffel, Pozzorini, Rathlev, Tancredi 14; Caola, Melnikov, R¨ ontsch, Tancredi 15] ZZ: [Cascioli, Gehrmann, Grazzini, Kallweit, Maierh¨ ofer, v. Mannteuffel, Pozzorini, Rathlev, Tancredi, Weihs 14; Grazzini, Kallweit, Rathlev 15; Caola, Melnikov, R¨ ontsch, Tancredi 16] ZH: [Ferrera, Grazzini, Tramontano 14; Campbell, Ellis, Williams 16] Z γ , W γ : [Grazzini, Kallweit, Rathlev, Torre 14] HH: [de Florian, Grazzini, Hanga, Kallweit, Lindert, Maierh¨ ofer, Mazzitelli, Rathlev 16] Can we go beyond 2 → 2? Multiscale processes? 5-point amplitudes [Badger, Frellesvig, Zhang 15; Gehrmann, Henn, Lo Presti 15] 6-point amplitudes [Dunbar, Perkins 2016; Badger, Mogull, Peraro 16] Matthieu Jaquier (Physics Institute Freiburg) Two-loop Integrand Decomposition August 11, 2016 3 / 19

Introduction Current limitations ✑ Main bottleneck: two loop contribution Feynman diagrams Many diagrams Large cancellations ↓ Process specific Feynman Integrals Tensor reduction [Tarasov 96; Anastasiou, Glover, Oleari 99] IBP identities [Tkachov, Chetyrkin 81] ⇒ Few master integrals Useful for several processes Differential equations [Gehrmann, Remiddi 01] Matthieu Jaquier (Physics Institute Freiburg) Two-loop Integrand Decomposition August 11, 2016 4 / 19

Introduction Status at NLO Significant improvements due to on-shell methods: Fully automatised computations. All necessary master integrals known. Amplitudes assembled from on-shell and gauge-invariant pieces. (Tree amplitudes ← recursion [Berends, Giele 88; Britto, Cachazo, Feng, Witten 05] ) → Feynman diagrams avoided! Implementations: Blackhat [Bern, Dixon, Febres Cordero, H¨ oche, Ita, Kosower, Maˆ ıtre, Ozeren 13] NJET [Badger, Biedermann, Uwer, Yundin 12] OpenLoops [Cascioli, Maierh¨ ofer, Pozzorini 12] MadGraph [Alwall, Frederix, Frixione, Hirschi, Maltoni, Mattelaer, Shao, Stelzer, Torrielli, Zaro 14] GoSam [Cullen, v. Deurzen, Greiner, Heinrich, Luisoni, Mastrolia, Mirabella, Ossola, Peraro, Schlenk, v, Soden-Fraunhofen, Tramontano 14] ✑ Can the same be done at two loops? ✒ [Badger, Bobadilla, Caron-Huot, Frelleswig, Johansson, Kosower, Larsen, Mastrolia, Ossola, Primo, Zhang, . . . ] Matthieu Jaquier (Physics Institute Freiburg) Two-loop Integrand Decomposition August 11, 2016 5 / 19

Introduction Goal Find parametrisation j c j t j i c i t i k c k t k ( ℓ ) � master ( ℓ ) + � surface ( ℓ ) � � � d D ℓ d D ℓ I i 1 ... i n = = . ρ 1 . . . ρ n ρ 1 . . . ρ n [Ossola, Papadopoulos, Pittau 06; Bern, Dixon, Kosower] The coefficients c k can be determined on the cut [Bern, Dixon, Kosower 06] , ρ 1 ρ 2 ρ n � c k t k ( ℓ ) = ( ℓ ) . k ρ 3 Parametrisation in terms of integrands of master integrals and terms vanishing upon integration. Matthieu Jaquier (Physics Institute Freiburg) Two-loop Integrand Decomposition August 11, 2016 6 / 19

Integrand decomposition at one loop Adapted coordinates Parametrise the loop momentum in terms of inverse propagators ρ i and (depending on the number of propagators available) additional transverse variables α : D p D t ℓ µ = � � r i v µ i + α a n µ [ van Neerven , Vermaseren 84; see also The analytic S − matrix . ] a a =1 i =1 r i = − 1 ( ρ i + m 2 i − q 2 i ) − ( ρ i − 1 + m 2 i − 1 − q 2 � � i − 1 ) . 2 With r i = ( ℓ · p i ) and α i = ( ℓ · n i ). Putting propagators on shell easily implemented as ρ i → 0. Matthieu Jaquier (Physics Institute Freiburg) Two-loop Integrand Decomposition August 11, 2016 7 / 19

Integrand decomposition at one loop Adapted coordinates We have D +1 loop variables, ρ 0 , . . . , ρ D p , α 1 , . . . , α D t , and one constraint, � D p D t � 2 0 = ℓ 2 − m 2 � � α 2 a − m 2 0 − ρ 0 = + 0 − ρ 0 = c ( ρ, α ) . r i v i a =1 i =1 c ( ρ, α ) = 0 defines the physical momentum space. Tensor terms are given by algebraic functions: t µ 1 ...µ n ℓ µ 1 . . . ℓ µ n = � ( α a ) k a ( ρ i ) k i , a , i with k a and k i bounded by QCD power counting. ⇒ Parametrisation of the integrand together with the scalar one. Matthieu Jaquier (Physics Institute Freiburg) Two-loop Integrand Decomposition August 11, 2016 8 / 19

Integrand decomposition at one loop Parametrisation of the Integrand What are the master integrals? ֒ → Solve integration-by-part (IBP) identities: � u µ t ( ℓ ) d D ℓ � � = 0 , (2 π ) D ∂ µ ρ 1 . . . ρ n ⇒ integrand parametrisation: All integrands = master integrands + surface terms (IBP’s) [Ita 15] , where the surface terms vanish upon integration. Automatically performs the reduction to master integrals → advantage for numerical computation. Keep surface terms only during intermediate steps of the computation. Matthieu Jaquier (Physics Institute Freiburg) Two-loop Integrand Decomposition August 11, 2016 9 / 19

Integrand decomposition at one loop IBP generating vectors Generic vector fields u µ yield IBP’s with doubled propagators. This can be avoided by choosing vectors satisfying [Gluza, Kajda, Kosower 11] u µ = ( f i ρ i , u a ) [ Ita 15; Larsen , Zhang 15] ( u µ ∂ µ ) ρ i = f i ( ℓ ) ρ i ∀ ρ i ⇒ Then, � u µ t ( ℓ ) � = ( ∂ µ u µ ) t ( ℓ ) + u µ ∂ µ t ( ℓ ) t ( ℓ ) u µ ∂ µ ρ j � − ∂ µ ρ 1 . . . ρ 2 ρ 1 . . . ρ n ρ 1 . . . ρ n ρ 1 . . . ρ n j . . . ρ n j = ( ∂ µ u µ ) t ( ℓ ) + u µ ∂ µ t ( ℓ ) t ( ℓ ) f j � − . ρ 1 . . . ρ n ρ 1 . . . ρ n ρ 1 . . . ρ n j Impose u µ ∂ µ c ( ρ, α ) = 0 to stay inside the physical momentum space. Matthieu Jaquier (Physics Institute Freiburg) Two-loop Integrand Decomposition August 11, 2016 10 / 19

Integrand decomposition at one loop IBP generating vectors at one loop 3 Types of IBP generators: Horizontal ( ρ i = 0), vertical ( α a = 0) and mixed. u µ Horizontal [ ab ] ∂ µ = α a ∂ b − α b ∂ a Generic topologies Vertical u µ ∂ µ = � Links different topologies i f i ρ i ∂ i u µ ∂ µ = � i f i ρ i ∂ i + � Mixed a g a α a ∂ a Degenerate on-shell PS How to find the integrand decomposition? Write down all monomials in α compatible with power counting. Act with the IBP generating vectors → surface terms. The master integrands are in the complement. This reproduces the well-known one-loop results for all topologies. Matthieu Jaquier (Physics Institute Freiburg) Two-loop Integrand Decomposition August 11, 2016 11 / 19

Integrand decomposition at two loops Two loop topologies Planar: two sets of one-loop para’s: p t p ˜ ˜ N − 1 ρ 0 , . . . , ρ D p , α 1 , . . . , α D t , c ( ρ, α ) p 1 ρ 0 ρ 0 , . . . , ˜ ˜ ρ ˜ D p , ˜ α 1 , . . . , ˜ α ˜ D t , ˜ c ( ρ, α ) p 2 ρ 1 ρ 0 ˆ ρ 2 p 3 ˜ p 3 ρ 2 ˜ Additional constraint from central ρ 1 ˜ ˜ p 2 propagator: ρ 0 ˜ p 1 ˜ c ( ρ, ˜ ˆ ρ, α, ˜ α ) = 0 p b p N − 1 Works for non-planar and higher loops as well. Matthieu Jaquier (Physics Institute Freiburg) Two-loop Integrand Decomposition August 11, 2016 12 / 19

Surface terms Two loop IBP generating vectors The generating vectors must satisfy u µ ∂ µ { c , ˜ c } = 0. Thus they are made of combinations of one loop generating vectors. Imposing furthermore u µ ∂ µ ˆ c ( ρ, ˜ ρ, α, ˜ α ) = 0 singles out the following combinations: Two loop rotations: u µ [ abc ] = ∂ [ a | u µ | bc ] , Diagonal rotations: u diag ,µ = u µ u µ [ ab ] + ˜ [ ab ] , [ ab ] Crossed rotations: u µ [ ab ][ cd ] = (˜ u [ cd ] ˆ c ) u µ [ ab ] − ( u [ ab ] ˆ c )˜ u µ [ cd ] . Additional IBP generators for: Nonplanar Degenerate phase space D dimensions Matthieu Jaquier (Physics Institute Freiburg) Two-loop Integrand Decomposition August 11, 2016 13 / 19