Integrand reduction for five-parton two-loop scattering amplitudes in QCD High Precision for Hard Processes Christian Brønnum-Hansen in collaboration with Simon Badger, Bayu Hartanto, and Tiziano Peraro 1/15 Christian Brønnum-Hansen High Precision for Hard Processes
some desired processes at NNLO QCD pp → H + 2 jets pp → γγ + jet pp → 3 jets Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report 2/15 Christian Brønnum-Hansen High Precision for Hard Processes
some desired processes at NNLO QCD pp → H + 2 jets pp → γγ + jet pp → 3 jets Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report 2/15 Christian Brønnum-Hansen High Precision for Hard Processes
some desired processes at NNLO QCD pp → H + 2 jets pp → γγ + jet pp → 3 jets Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report measurement of strong coupling from jet ratio pp → 3 j pp → 2 j α S ( m Z ) = 0 . 1148 ± 0 . 0014 (exp.) ± 0 . 0018 (PDF) ± 0 . 0050 (theory) CMS Collaboration @ 7 TeV: arXiv:1304.7498 2/15 Christian Brønnum-Hansen High Precision for Hard Processes
all-plus sector including non-planar contribution long known • A two-loop five-gluon helicity amplitude in QCD Badger, Frellesvig, and Zhang 2013 • A complete two-loop, five-gluon helicity amplitude in Yang-Mills theory Badger, Mogull, Ochirov, and O’Connell 2015 • Analytic form of the two-loop planar five-gluon all-plus helicity amplitude in QCD Gehrmann, Henn, and Lo Presti 2015 • Local integrands for two-loop all-plus Yang-Mills amplitudes Badger, Mogull, and Peraro 2016 3/15 Christian Brønnum-Hansen High Precision for Hard Processes
all-plus sector including non-planar contribution long known • A two-loop five-gluon helicity amplitude in QCD Badger, Frellesvig, and Zhang 2013 • A complete two-loop, five-gluon helicity amplitude in Yang-Mills theory Badger, Mogull, Ochirov, and O’Connell 2015 • Analytic form of the two-loop planar five-gluon all-plus helicity amplitude in QCD Gehrmann, Henn, and Lo Presti 2015 • Local integrands for two-loop all-plus Yang-Mills amplitudes Badger, Mogull, and Peraro 2016 several recent results • A first look at two-loop five-gluon amplitudes in QCD Badger, CBH, Hartanto, and Peraro 2017 • Planar two-loop five-gluon amplitudes from numerical unitarity Abreu, Febres-Cordero, Ita, Page, and Zeng 2017 • Planar two-loop five-parton amplitudes from numerical unitarity Abreu, Febres-Cordero, Ita, Page, and Sotnikov 2018 • Two-loop five-point massless QCD amplitudes within the IBP approach Chawdhry, Lim, and Mitov 2018 3/15 Christian Brønnum-Hansen High Precision for Hard Processes
leading colour gluon contribution � tr ( T a σ (1) · · · T a σ (5) ) A (2) (1 , 2 , 3 , 4 , 5) = σ ∈ S 5 / Z 5 × A (2) ( σ (1) , σ (2) , σ (3) , σ (4) , σ (5)) (1) 4/15 Christian Brønnum-Hansen High Precision for Hard Processes
leading colour gluon contribution � tr ( T a σ (1) · · · T a σ (5) ) A (2) (1 , 2 , 3 , 4 , 5) = σ ∈ S 5 / Z 5 × A (2) ( σ (1) , σ (2) , σ (3) , σ (4) , σ (5)) (1) colour-ordered amplitude � � � ∆ T ( { k } , { p } ) A (2) (1 , 2 , 3 , 4 , 5) = � (2) α ∈ T D α T 4/15 Christian Brønnum-Hansen High Precision for Hard Processes
leading colour gluon contribution � tr ( T a σ (1) · · · T a σ (5) ) A (2) (1 , 2 , 3 , 4 , 5) = σ ∈ S 5 / Z 5 × A (2) ( σ (1) , σ (2) , σ (3) , σ (4) , σ (5)) (1) colour-ordered amplitude � � � ∆ T ( { k } , { p } ) A (2) (1 , 2 , 3 , 4 , 5) = � (2) α ∈ T D α T sum over planar topologies 4/15 Christian Brønnum-Hansen High Precision for Hard Processes
leading colour gluon contribution � tr ( T a σ (1) · · · T a σ (5) ) A (2) (1 , 2 , 3 , 4 , 5) = σ ∈ S 5 / Z 5 × A (2) ( σ (1) , σ (2) , σ (3) , σ (4) , σ (5)) (1) irreducible numerator colour-ordered amplitude � � � ∆ T ( { k } , { p } ) A (2) (1 , 2 , 3 , 4 , 5) = � (2) α ∈ T D α T sum over planar topologies 4/15 Christian Brønnum-Hansen High Precision for Hard Processes
leading colour gluon contribution � integrand reduction tr ( T a σ (1) · · · T a σ (5) ) A (2) (1 , 2 , 3 , 4 , 5) = Ossola, Papadopoulos, Pittau, σ ∈ S 5 / Z 5 Mastrolia, Badger, Frellesvig, × A (2) ( σ (1) , σ (2) , σ (3) , σ (4) , σ (5)) Zhang, Peraro, Mirabella, (1) irreducible numerator . . . (2005-) colour-ordered amplitude � � � ∆ T ( { k } , { p } ) A (2) (1 , 2 , 3 , 4 , 5) = � (2) α ∈ T D α T sum over planar topologies 4/15 Christian Brønnum-Hansen High Precision for Hard Processes
57 topologies, 425 irreducible numerators. examples: pentabox, maximal topology bubble insertion, maximal topology divergent cut Abreu, Febres Cordero, Ita, Jaquier, and Page 2017 (one-loop) 2 5/15 Christian Brønnum-Hansen High Precision for Hard Processes
reconstruct integrand from d -dimensional unitarity cuts � � � � ∆ = Cut (3) 6/15 Christian Brønnum-Hansen High Precision for Hard Processes
reconstruct integrand from d -dimensional unitarity cuts � � � � ∆ = Cut (3) generalised unitarity cuts Bern, Rozowsky, Yan, Dixon, Kosower, de Freitas, Wong, . . . 1997- 6/15 Christian Brønnum-Hansen High Precision for Hard Processes
reconstruct integrand from d -dimensional unitarity cuts � � � � ∆ = Cut (3) finite field reconstruction Peraro 2016 generalised unitarity cuts Bern, Rozowsky, Yan, Dixon, Kosower, de Freitas, Wong, . . . 1997- 6/15 Christian Brønnum-Hansen High Precision for Hard Processes
reconstruct integrand from d -dimensional unitarity cuts � � � � ∆ = Cut (3) finite field reconstruction Peraro 2016 generalised unitarity cuts Bern, Rozowsky, Yan, Dixon, Kosower, Berends-Giele recursion & de Freitas, Wong, . . . 1997- six-dimensional spinor-helicity Cheung, O’Connell 2009 6/15 Christian Brønnum-Hansen High Precision for Hard Processes
reconstruct integrand from d -dimensional unitarity cuts � � � � ∆ = Cut (3) finite field reconstruction Peraro 2016 momentum twistors Hodges 2009 generalised unitarity cuts Bern, Rozowsky, Yan, Dixon, Kosower, Berends-Giele recursion & de Freitas, Wong, . . . 1997- six-dimensional spinor-helicity Cheung, O’Connell 2009 6/15 Christian Brønnum-Hansen High Precision for Hard Processes
split loop momentum in parallel and perpendicular components k i = k � , i + k ⊥ , i (4) 7/15 Christian Brønnum-Hansen High Precision for Hard Processes
split loop momentum in parallel and perpendicular components k i = k � , i + k ⊥ , i (4) k � , i = � a ij p j , a ij = a ij ( k i · p j ) 7/15 Christian Brønnum-Hansen High Precision for Hard Processes
split loop momentum in parallel and perpendicular components k i = k � , i + k ⊥ , i (4) k � , i = � a ij p j , k [4] ⊥ , i + k [ − 2 ǫ ] a ij = a ij ( k i · p j ) ⊥ , i 7/15 Christian Brønnum-Hansen High Precision for Hard Processes
split loop momentum in parallel and perpendicular components k i = k � , i + k ⊥ , i (4) k � , i = � a ij p j , k [4] ⊥ , i + k [ − 2 ǫ ] a ij = a ij ( k i · p j ) ⊥ , i ⊥ , i = � b ij ω j , k [4] b ij = b ij ( k i · ω j ) 7/15 Christian Brønnum-Hansen High Precision for Hard Processes
split loop momentum in parallel and perpendicular components k i = k � , i + k ⊥ , i (4) k � , i = � a ij p j , k [4] ⊥ , i + k [ − 2 ǫ ] a ij = a ij ( k i · p j ) ⊥ , i ⊥ , i = � b ij ω j , k [4] b ij = b ij ( k i · ω j ) relation to extra-dimensional ISPs µ ij = − k [ − 2 ǫ ] · k [ − 2 ǫ ] ⊥ , i ⊥ , j = k i · k j − k � , i · k � , j − k [4] ⊥ , i · k [4] (5) ⊥ , j 7/15 Christian Brønnum-Hansen High Precision for Hard Processes
going back to the all-plus case � � � � tr + (1235)( k 1 + p 5 ) 2 + s 12 s 34 s 45 ∆ ∝ F ( d s , µ ij ) (6) 8/15 Christian Brønnum-Hansen High Precision for Hard Processes
going back to the all-plus case � � � � tr + (1235)( k 1 + p 5 ) 2 + s 12 s 34 s 45 ∆ ∝ F ( d s , µ ij ) (6) � � µ 11 µ 22 + ( µ 11 + µ 22 ) 2 + 2 µ 12 ( µ 11 + µ 22 ) F ( d s , µ ij ) = ( d s − 2) � � µ 2 + 16 12 − µ 11 µ 22 d s is spin dimension, FDH results obtained for d s = 4 8/15 Christian Brønnum-Hansen High Precision for Hard Processes
going back to the all-plus case � � � � tr + (1235)( k 1 + p 5 ) 2 + s 12 s 34 s 45 ∆ ∝ F ( d s , µ ij ) (6) � � µ 11 µ 22 + ( µ 11 + µ 22 ) 2 + 2 µ 12 ( µ 11 + µ 22 ) F ( d s , µ ij ) = ( d s − 2) � � µ 2 + 16 12 − µ 11 µ 22 d s is spin dimension, FDH results obtained for d s = 4 same pattern for the other contributing topologies 8/15 Christian Brønnum-Hansen High Precision for Hard Processes
construction of a simple “no- µ ” basis � � m α ij ∆ T = c i (7) j i m j ∈ S 9/15 Christian Brønnum-Hansen High Precision for Hard Processes
rational coefficient in external kinematics construction of a simple “no- µ ” basis � � m α ij ∆ T = c i (7) j i m j ∈ S 9/15 Christian Brønnum-Hansen High Precision for Hard Processes
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