more-than-MHV amplitudes in QCD Simon Badger 18th March 2016 - PowerPoint PPT Presentation

more-than-MHV amplitudes in QCD Simon Badger 18th March 2016 MHV@30, Fermilab, 16th-19th March 2016

helicity amplitudes MHV “UHV” n ) n + -n - -6 -4 -2 2 4 6 [L. Dixon] vanish at tree-level and to all loops in super-symmetric theories

Modelling hadron collisions New physics backgrounds Hadronisation P a S r t h o o n PDF w e Hadronisation r S P h a o r Keeping theory predictions in t w o e n r line with experimental data Hadronisation S P h next frontier: NNLO 2 → 3 a o r w t o e n r Parton Shower PDF Hadronisation Determination of SM parameters

spurious singularities 10 6 numerical D-dimensional analytic generalised unitarity numerical 10 5 vs 10 4 analytic computation with finite integrals basis #points (e.g. Bern, Dixon, Kosower 10 3 [hep-ph/9302280]) 10 2 removing spurious poles also simplifies coefficients ⇒ faster 10 1 (~100x in this case) NJet gg → 3 g 10 0 − 16 − 14 − 12 − 10 − 8 − 6 − 4 − 2 0 accuracy (digits) need to switch to quadruple precision evaluation applications of loop amplitudes in NNLO computations more intensive

one-loop finite amplitudes in QCD all multiplicity expressions! • Off-shell currents for one-loop amplitudes to O ( ✏ 0 ) [Mahlon (1993)] • all-plus ↔ N = 4 D-dimensional unitarity cuts [Bern, Dixon, Dunbar, Kosower (1996)] • BCFW recursion for O ( ✏ 0 ) [Bern, Dixon, Kosower (2005)] • CSW(MHV) rules for D-dimensional integrands [Elvang, Freedman, Kiermaier (2012)]

outline • d-dimensional generalised unitarity ⇒ multi-loops from trees • two-loop all-plus amplitudes • local integrands in d-dimensional amplitudes

automated one-loop amplitudes solving on-shell conditions requires complex momenta ⇒ factorise residues into tree amplitudes Unitarity: double cuts [BDDK ’94] [triple cuts BDK ’97] Integrand reduction [OPP ’05] Generalized unitarity: ∆ 3 = − quadruple cuts [BCF ’04] D-dim. generalized unitarity [GKM ’08] triple cuts [e.g. Forde ’07] ∆ i ( k, p ) Z multi-scale X A = X kinematic algebra (propagators) i A = (rational) i (integral) i k i performed i numerically find complex contour to isolate explicitly remove poles integral coefficient

multi-loop amplitudes from trees Integrand reduction via Maximal unitarity polynomial division [Kosower, Larsen, Johansson, Caron-Huot, Zhang, Søgaard] [Mastrolia, Ossola, SB, Frellesvig, Zhang, Mirabella, Peraro, Malamos, Kleiss, Papadopolous, Verheyen, Feng, Huang] e.g. IBPs ∆ i ( k, p ) Z X X A = A = (rational) i (integral) i (propagators) i k i i IBPs must be free of [Gluza, Kosower, Kajda 1009.0472] [Schabinger 1111.4220] doubled propagator MI [Ita 1510.05626] [Larsen, Zhang 1511.01071]

a toolbox for multi-loop integrands six-dimensional momentum twistors spinor-helicity [Hodges (2009)] ∆ i ( k, p ) Z X A = (propagators) i k i generalised unitarity integrand reduction cuts colour/kinematics C ( ∆ i ) ∆ i X Q D α A = S i BCJ relations i

multi-loop integrand reduction [Mastrolia, Ossola 1107.6041] [SB, Frellesvig, Zhang 1202.2019] [Zhang 1205.5707] [Mastrolia, Mirabella, Ossola, Peraro 1205.7087] ISP monomials c T ; α 1 ... α n ( k 1 · p 5 ) α 1 ( k 2 · p 2 ) α 2 . . . ( k 1 · ω 2 ) α m . . . µ α n X ∆ T ( { k } ) = 12 { α } extra-dimensional ISPs spurious directions rational coefficients µ ij = − k − 2 ✏ · k − 2 ✏ i j

integrand reduction � � ∆ c ; T 0 � � A (0) Y X ∆ c ; T � � = − i Q l ∈ T 0 /T D l � � � � i T 0 cut cut on-shell the numerators can be written as products of fix basis of monomials in tree-level amplitudes irreducible scalar products via polynomial division (Gröbner basis) integrand parameterisations not unique - freedom in the choices of ISP monomials

six-dimensional trees six dimensional spinor helicity [Cheung, O’Connell 0902.0981] (one-loop applications [Bern at al. 1010.0494] [Davies 1108.0398]) dimensional reduction to using additional scalar amplitudes 4 − 2 ✏ 6D is a convenient way to manage all helicities simultaneously · d ) = i A (1 a ˙ a , 2 b ˙ b , 3 d ˙ d , 4 d ˙ st h 1 a 2 b 3 c 4 d i [1 ˙ a 2 ˙ b 3 ˙ c 4 ˙ d ]

General colour decompositions [Dixon, Del Duca, Maltoni (1999)] Inserting the DDM � m − → σ decomposition σ | σ | = m into colour dressed ( a ) ( cuts leads to a compact loop � σ 2 − → decomposition σ 1 , σ 2 | σ 1 ∪ σ 2 | = m σ 1 m ( b ) ( σ 3 σ 3 � − → + σ 2 σ 2 σ 1 , σ 2 , σ 3 | σ 1 ∪ σ 2 ∪ σ 3 | = m m σ 1 σ 1 ( a ) ( general tree-level DDM colour bases including fermions [Johansson, Ochirov arXiv:1507.00332]

non-planar from planar e.g. [Bern, Carrasco, Dixon, Johansson, Roiban 1201.5366] A 4 (1 , 2 , 3 , 4) = s 13 A 4 (1 , 3 , 2 , 4) see talks by Carrasco and Johansson s 12 factorization st i ) � A 3 (1 , 2 , � P 12 ) A 3 ( P 12 , 3 , 4) = Res s 12 =0 ( A 4 (1 , 2 , 3 , 4)) = s 13 A 4 (1 , 3 , 2 , 4) � s 12 =0 � � st i ) 4) = � ( k 1 − P 123 ) 2 � � ( k 1 + k 2 + p 3 ) 2 ◆� ◆◆ � ✓ ✓ ✓ ◆ ✓ ◆ ✓ � � st i ) ( k 1 − P 123 ) 2 ∆ = + ∆ ∆ − ∆ � � � � � � cut cut

applications: all-plus amplitudes in QCD

one-loop 4pt all-plus + + ⇢ ✓ ◆ � D s � 2 [ µ 2 = h 12 ih 23 ih 34 ih 41 i { � st } · 11 ] I + + + + D s � 2 � st = + O ( ✏ ) h 12 ih 23 ih 34 ih 41 i 6 + + [Bern, Kosower (1991)]

one-loop 4pt single minus + - ( ✓ ◆ ✓ ◆ [ µ 2 = ~ c · 11 ] , I [ µ 11 ] , I + + ✓ ◆ ✓ ◆ [ µ 11 ] , I [ µ 11 ] , I ) � � � � [ µ 11 ] , I [ µ 11 ] I ( D s � 2)[24] 2 u , s 2 t 2 u 2 , t 2 ( s � u ) , s 2 ( t � u ) ⇢ st � , � t ( t � u ) , � s ( s � u ) ~ c = [12] h 23 ih 34 i [41] u 2 u 2 tu su [Bern, Morgan (1995)] + - ( D s � 2)[24] 2 u = 6 + O ( ✏ ) [12] h 23 ih 34 i [41] + + [Bern, Kosower (1991)]

one-loop 5pt all-plus + + ( ✓ ◆ + [ µ 2 = ~ c · 11 ] , I + + ) ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ [ µ 2 [ µ 2 [ µ 2 [ µ 2 [ µ 2 11 ] , I 11 ] , I 11 ] , I 11 ] , I 11 ] I ( D s � 2) c = h 12 ih 23 ih 34 ih 45 ih 51 i tr 5 (1234) { � s 12 s 23 s 34 s 45 s 51 , ~ s 12 s 23 tr − (1345) , s 23 s 34 tr − (2451) , s 34 s 45 tr − (3512) , s 45 s 51 tr − (4123) , s 51 s 12 tr − (5234) } [Bern, Morgan (1995)] + + = ( D s � 2) s 12 s 23 + s 23 s 34 + s 34 s 45 + s 45 s 51 + s 51 s 12 + tr 5 (1234) + + O ( ✏ ) 6 h 12 ih 23 ih 34 ih 45 ih 51 i + + [Bern, Dixon, Kosower (1993)]

one-loop 5pt all-plus ∆ (1) (1 + , 2 + , 3 + , . . . , n + ) = D s � 2 h 12 i 4 µ 2 11 ∆ (1) , [ N =4] (1 − , 2 − , 3 + , . . . , n + ) [Bern, Dixon, Dunbar, Kosower (1996)] + + ( + ✓ ◆ [ µ 3 = ~ c · 11 ] , I + + ) ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ ✓ ◆ [ µ 2 [ µ 2 [ µ 2 [ µ 2 [ µ 2 11 ] , I 11 ] , I 11 ] , I 11 ] , I 11 ] I ( D s � 2) c = h 12 ih 23 ih 34 ih 45 ih 51 i { 2tr 5 (1234) , s 12 s 23 , s 23 s 34 , s 34 s 45 , s 45 s 51 , s 51 s 12 } ~

two-loop 4pt all-plus (drop Parke-Taylor [Bern, Dixon, Kosower (2000)] pre-factors from now on....) + + 0 1 ⇣ ⌘ ⇣ ⌘ s +( l 1 + l 2 ) 2 A [ F 1 ] , I = { s 2 t, t 2 s, st } · { I [ F 1 ] , I [ F 2 + F 3 ] } @ s + + µ 11 µ 22 + ( µ 11 + µ 22 ) 2 + 2( µ 11 + µ 22 ) µ 12 + 16( µ 2 � � F 1 = ( D s � 2) 12 � µ 11 µ 22 ) , dimension shifting F 2 = 4( D s � 2)( µ 11 + µ 22 ) µ 12 , numerators F 3 = ( D s � 2) 2 µ 11 µ 22 .

two-loop 5pt all-plus [SB, Frellesvig, Zhang (2013)] + + ( + ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ X = + ∆ + ∆ + ∆ ∆ cyclic + + ⌘ ) ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ ⇣ + ∆ + ∆ + ∆ + ∆ = s 12 s 23 s 45 ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ ∆ { s 34 s 45 s 15 , tr + (1345) } · { I [ F 1 ] , I [ F 1 ] } tr 5 = { − s 34 s 2 45 tr + (1235) ⇣ ⌘ ⇣ ⌘ ∆ } · { I [ F 1 ] } tr 5 = { s 12 s 23 s 34 s 45 s 15 ⇣ ⌘ ⇣ ⌘ } · { I [ F 1 ] } ∆ tr 5 = − s 12 tr + (1345) ⇣ ⌘ ∆ { s 23 , 1 } · { 2 s 13 ⇣ ⌘ ⇣ ⌘ s 45 +( l 1 + l 2 ) 2 s 45 +( l 1 + l 2 ) 2 [ F 2 + F 3 ] , I [(2 k 1 · ! )( F 2 + F 3 )] } I s 45 s 45 = { − ( s 45 − s 12 )tr + (1345) ⇣ ⌘ ⇣ ⌘ s 45 +( l 1 + l 2 ) 2 } · { I [ F 2 + F 3 ] } ∆ s 45 2 s 13 ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ [ F 3 ( l 1 + l 2 ) 2 ] , = ~ c · { I [ F 2 ] , I [ F 3 ] , I ∆ ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ [ F 3 ( k 1 · 3)( k 2 · 3)] , I [ F 3 ( k 1 · 3)] , I [ F 3 ( k 2 · 3)] , . . . } I

two-loop 5pt all-plus ∆ (2) (1 + , 2 + , 3 + , . . . , n + ) = D s � 2 h 12 i 4 F 1 ∆ (2) , [ N =4] (1 − , 2 − , 3 + , . . . , n + ) + (1-loop) 2 all genuine two-loop topologies related to N =4 MHV non-planar cuts via BCJ A (2) 5 (1 + , 2 + , 3 + , 4 + , 5 + )= " ◆ [SB, Mogull, Ochirov, O’Connell (2015)] ✓ 1 ✓ ◆ ✓ ◆ + 1 ✓ ◆ X I C 2 ∆ + ∆ 2 ∆ σ ∈ S 5 ! ✓ ◆ ✓ ◆ ✓ ◆ +1 + 1 + ∆ 2 ∆ 2 ∆ ◆ complete BCJ numerator ✓ ✓ ◆ ✓ ◆ ✓ ◆ 1 + 1 + 1 + C 4 ∆ 2 ∆ 2 ∆ representation ! ✓ ◆ ✓ ◆ + 1 − ∆ 4 ∆ [Mogull, O’Connell (2015)] ◆ !# ✓ ✓ ◆ ✓ ◆ ✓ ◆ 1 + 1 + 1 + C 4 ∆ 2 ∆ 2 ∆