A semi-numerical approach to one-loop multi-leg amplitudes p.1

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1 A semi-numerical approach to one-loop multi-leg amplitudes Gudrun Heinrich Universität Zürich UNIVERSITAS TURICENSIS MDCCC XXXIII LoopFest V, SLAC, A semi-numerical approach to one-loop multi-leg amplitudes p.1

2 The start of the LHC is round the corner! extracting the signal and studying properties of the Higgs boson/new particles will be an experimental challenge theoretical predictions should be as precise as possible A semi-numerical approach to one-loop multi-leg amplitudes p.2

3 event rates at the LHC process events per second QCD jets E T > 150 GeV 100 W eν 15 t t 1 Higgs, m H = 130 GeV 0.02 gluinos, m = 1 TeV enormous backgrounds! NLO predictions mandatory for reliable estimates, especially in the presence of kinematic cuts A semi-numerical approach to one-loop multi-leg amplitudes p.3

4 NLO wishlist for LHC (Les Houches workshop 05) process (V {Z, W, γ}) background to 1. pp V V jet t th, new physics 2. pp t t b b t th 3. pp t t + 2 jets t th 4. pp V V b b VBF H V V, t th, new physics 5. pp V V + 2 jets VBF H V V 6. pp V + 3 jets various new physics signatures 7. pp V V V SUSY trilepton A semi-numerical approach to one-loop multi-leg amplitudes p.4

5 NLO predictions for multi-particle processes most of the relevant background processes involve multi-particle final states with several mass scales for a partonic NLO calculation: need one-loop amplitudes with many legs and several mass scales: lengthy calculations A semi-numerical approach to one-loop multi-leg amplitudes p.5

6 NLO predictions for multi-particle processes most of the relevant background processes involve multi-particle final states with several mass scales for a partonic NLO calculation: need one-loop amplitudes with many legs and several mass scales: lengthy calculations main difficulties are numerical instabilities and enormous sizes of expressions ( long CPU times, debugging complex) A semi-numerical approach to one-loop multi-leg amplitudes p.5

7 NLO predictions for multi-particle processes most of the relevant background processes involve multi-particle final states with several mass scales for a partonic NLO calculation: need one-loop amplitudes with many legs and several mass scales: lengthy calculations main difficulties are numerical instabilities and enormous sizes of expressions ( long CPU times, debugging complex) virtual amplitudes are the stumbling block for automatisation A semi-numerical approach to one-loop multi-leg amplitudes p.5

8 NLO predictions for multi-particle processes most of the relevant background processes involve multi-particle final states with several mass scales for a partonic NLO calculation: need one-loop amplitudes with many legs and several mass scales: lengthy calculations main difficulties are numerical instabilities and enormous sizes of expressions ( long CPU times, debugging complex) virtual amplitudes are the stumbling block for automatisation not addressed in this talk: real radiation (general algorithms for arbitrary N well established at NLO) combination with parton shower talks of S. Frixione, Z. Nagy & Tuesday afternoon A semi-numerical approach to one-loop multi-leg amplitudes p.5

9 status NLO scattering processes involving maximally 4 particles (2 1, 2 2, 1 3): "standard" A semi-numerical approach to one-loop multi-leg amplitudes p.6

10 status NLO scattering processes involving maximally 4 particles (2 1, 2 2, 1 3): "standard" 2 3: only a few NLO results for hadronic collisions A semi-numerical approach to one-loop multi-leg amplitudes p.6

11 status NLO scattering processes involving maximally 4 particles (2 1, 2 2, 1 3): "standard" 2 3: only a few NLO results for hadronic collisions pp 3 jets pp V + 2 jets pp γγ jet pp t th, b bh pp t t jet p p W b b Beenakker, Bern, Brandenburg, Campbell, Dawson, De Florian, Del Duca, Dittmaier, Dixon, R.K. Ellis, Febres Cordero, Frixione, Giele, Glover, Kilgore, Kosower, Krämer, Kunszt, Maltoni, Miller, Nagy, Orr, Plümper, Reina, Signer, Spira, Trocsanyi, Uwer, Wackeroth, Weinzierl, Zerwas,... A semi-numerical approach to one-loop multi-leg amplitudes p.6

12 status NLO scattering processes involving maximally 4 particles (2 1, 2 2, 1 3): "standard" 2 3: only a few NLO results for hadronic collisions 2 3 for e + e collisions: e + e 4 jets e + e ν νh e + e e + e H e + e ν νγ e + e t th e + e ZHH, ZZH γ γ t th Bern, Dixon, Kosower, Weinzierl, Signer, Nagy, Trocsanyi, Campbell, Cullen, Glover, Miller, Denner, Dittmaier, Roth, Weber, Bélanger, Boudjema, Fujimoto, Ishikawa, Kaneko, Kato, Kurihara, Shimizu, Yasui, Jegerlehner, Tarasov, Chen, Ren-You, Wen-Gan, Hui, Yan-Bin, Hong-Sheng, Pei-Yun,... A semi-numerical approach to one-loop multi-leg amplitudes p.6

13 status NLO scattering processes involving maximally 4 particles (2 1, 2 2, 1 3): "standard" 2 3: only a few NLO results for hadronic collisions 2 4: e + e 4 fermions (complete EW corrections) Denner, Dittmaier, Roth, Wieders 02/05 e + e HHν ν GRACE group (Boudjema et al.) 10/05 A semi-numerical approach to one-loop multi-leg amplitudes p.6

14 status NLO scattering processes involving maximally 4 particles (2 1, 2 2, 1 3): "standard" 2 3: only a few NLO results for hadronic collisions 2 4: e + e 4 fermions (complete EW corrections) Denner, Dittmaier, Roth, Wieders 02/05 e + e HHν ν GRACE group (Boudjema et al.) 10/05 no complete σ NLO for 2 4 processes at the LHC known! 6-gluon amplitude Berger, Bern, Del Duca, Dixon, Dunbar, Forde, Kosower; Ellis, Giele Zanderighi A semi-numerical approach to one-loop multi-leg amplitudes p.6

15 multi-particle processes at NLO main problem with "conventional" approaches beyond N = 4 external legs: tensor reduction à la Passarino-Veltman leads to severe numerical instabilities (vanishing inverse Gram determinants) analytical representation in terms of a fixed set of Logs, Dilogs not convenient in all phase space regions (e.g. large cancellations between Dilogs) "Polylogarithmic mess" (L. Dixon) sheer complexity of expressions A semi-numerical approach to one-loop multi-leg amplitudes p.7

16 example 4-point integrals (boxes): known analytically invariants s, t, m q 6-point integrals (hexagons): 10 invariants, one non-linear constraint for analytic representation: reduce to integrals with less than 5 legs A semi-numerical approach to one-loop multi-leg amplitudes p.8

17 algebraic reduction integrals with less legs + from reduction of tensor rank and number of legs at the same time non-trivial tensor structure scalar 6-point function = 6 i=1 b i... i reduction until set of basis integrals is reached basis integrals: 2-, 3- and 4-point functions known A = C 4 + C 3 + C 2 A semi-numerical approach to one-loop multi-leg amplitudes p.9

18 new approaches new analytical methods (unitarity cuts, bootstrap,... ) avoid Feynman diagrams! talk of C. Berger A semi-numerical approach to one-loop multi-leg amplitudes p.10

19 new approaches new analytical methods (unitarity cuts, bootstrap,... ) avoid Feynman diagrams! talk of C. Berger semi-numerical methods (tensor reduction and/or integration of loop integrals partly numerically ) see also talks of S. Dittmaier, W. Giele A semi-numerical approach to one-loop multi-leg amplitudes p.10

20 new approaches new analytical methods (unitarity cuts, bootstrap,... ) avoid Feynman diagrams! talk of C. Berger semi-numerical methods (tensor reduction and/or integration of loop integrals partly numerically ) see also talks of S. Dittmaier, W. Giele fully numerical methods talks of A. Daleo, E. de Doncker; see also D. Soper, Z. Nagy A semi-numerical approach to one-loop multi-leg amplitudes p.10

21 Golem our approach: semi-numerical method implemented in program GOLEM (General One-Loop Evaluator for Matrix elements) [Binoth, Guffanti, Guillet, GH, Karg, Kauer, Pilon, Reiter,... ] A semi-numerical approach to one-loop multi-leg amplitudes p.11

22 Golem our approach: semi-numerical method implemented in program GOLEM (General One-Loop Evaluator for Matrix elements) [Binoth, Guffanti, Guillet, GH, Karg, Kauer, Pilon, Reiter,... ] main features: use algebraic reduction until singularities can be easily isolated (dim. reg.) inverse Gram determinants can be completely avoided by choosing a convenient set of (non-scalar) basis integrals ("stop reduction before it generates problems") valid for massless and massive particles and for (in principle) arbitrary number of legs A semi-numerical approach to one-loop multi-leg amplitudes p.11

23 The algorithm diagram generation (QGRAF, FeynArts) A = i Cµ 1...µ r i I µ1...µ r (FORM) reduction to basis integrals semi-numerically algebraically A = Lorentz tensors form factors form factors = i K i Ii master numerical evaluation reduction to analytic functions phase space integration A semi-numerical approach to one-loop multi-leg amplitudes p.12

24 treatment of basis integrals, example N=4 I n+2 4 (1 j 1 j 1, j 2 j 1, j 2, j 3 ), I n+4 4 (1 j 1 ) numerical no B > Λ 2 yes analytic direct evaluation algebraic reduction I n 2 (1), I n 3 (1), I 6 4(1) Numerical value A semi-numerical approach to one-loop multi-leg amplitudes p.13

25 special features manifestly shift invariant formulation basis integrals can contain Feynman parameters in the numerator ("tensor integrals") way to avoid inverse Gram determinants IR poles only in 3-point (and 2-point) functions easily isolated proven that higher dimensional integrals I n+2m N (m > 0) are absent for all N 5 form factors: algebraic simplifications already built in A semi-numerical approach to one-loop multi-leg amplitudes p.14

26 manifest shift invariance I n, µ 1...µ r N (a 1,..., a r ) = d n k q a µ q µ r a r i π n/2 (q1 2 m2 1 + iδ)... (q2 N m2 N + iδ) q i = k + r i, r j r j 1 = p j, N j=1 p j = 0 p 4 q 3 q 2 p 3 p 2 q N q 1 p N 2 p 1 advantages of this representation: combinations q i = k + r i appear naturally (e.g. fermion propagators) manifestly shift invariant formulation by Lorentz tensors defined in terms of difference vectors µ i j = rµ i rµ j and g µν p N 1 p N A semi-numerical approach to one-loop multi-leg amplitudes p.15

27 Lorentz structure and form factors I n, µ 1...µ r N (a 1,..., a r ; S) = [ l1 {µ1 µ r ] l r } {a 1 a r } AN,r l 1...,l r (S) + + l 1 l r S l 1 l r 2 S l 1 l r 4 S [ ] {µ1 µ g l r } 1 l r 2 {a 1 a r } BN,r l 1...,l r 2 (S) [ ] {µ1 µ g g l r } 1 l r 4 {a 1 a r } CN,r l 1...,l r 4 (S) important: more than two metric tensors g µν never occur! in a gauge where rank nb of legs reason: for N 6 : I n,µ 1...µ r N (a 1,..., a r ; S) = j S C µ 1 ja 1 I n,µ 2...µ r N 1 (a 2,..., a r ; S \ {j}) A semi-numerical approach to one-loop multi-leg amplitudes p.16

28 schematic overview I n,µ 1...µ r N N 6 no yes r-1, N-1 Form factors N = 5 A 5,r, B 5,r, C 5,r no N = 4 no A 4,r, B 4,r, C 4,r N = 3 A 3,r, B 3,r no N 2 I n 2 I n 2 (1 j 1 j 1, j 2 ), I n 3 (1 j 1 j 1, j 2 j 1, j 2, j 3 ), I n+2 3 (1 j 1 ) I n+2 4 (1 j 1 j 1, j 2 j 1, j 2, j 3 ), I n+4 4 (1 j 1 ) A semi-numerical approach to one-loop multi-leg amplitudes p.17

29 basis integrals I n 3 (j 1,..., j r ) = Γ 3 n Z I n+2 3 (j 1 ) = Γ 2 n Z I n+2 4 (j 1,..., j r ) = Γ 3 n Z I n+4 4 (j 1 ) = Γ 2 n Z and scalar I3 n, In+2 3, I n+2 4, I n+4 4 3Y i=1 3Y i=1 4Y i=1 4Y i=1 dz i δ(1 dz i δ(1 dz i δ(1 dz i δ(1 important: r max = 3 l=1 3X z j1... z jr z l ) ( 1 2 z S z iδ)3 n/2 l=1 3X z j1 z l ) ( 1 2 z S z iδ)2 n/2 l=1 4X z j1... z jr z l ) ( 1 2 z S z iδ)3 n/2 4X z j1 z l ) ( 1 2 z S z iδ)2 n/2 l=1 alternatives for evaluation: 1. direct numerical evaluation (contour deformation) 2. further algebraic reduction to scalar integrals A semi-numerical approach to one-loop multi-leg amplitudes p.18

30 algebraic reduction: example N=4, r=1 algebraic reduction re-introduces inverse Gram determinants 1/B I n+2 4 (l; S) = 1 B 8 < : b l I n+2 4 (S) X S 1 jl I3 n (S \ {j}) 1 2 j S X j S\{l} 9 = b j I3 n (l; S \ {j}) ; B = Det(G)/Det(S) ( 1) N+1 S ij = (r i r j ) 2 m 2 i m2 j G ij = 2 r i r j A semi-numerical approach to one-loop multi-leg amplitudes p.19

31 algebraic reduction: example N=4, r=1 algebraic reduction re-introduces inverse Gram determinants 1/B I n+2 4 (l; S) = 1 B 8 < : b l I n+2 4 (S) X S 1 jl I3 n (S \ {j}) 1 2 j S X j S\{l} 9 = b j I3 n (l; S \ {j}) ; B = Det(G)/Det(S) ( 1) N+1 S ij = (r i r j ) 2 m 2 i m2 j G ij = 2 r i r j 1/B does not pose a problem in most regions of phase space allows fast evaluation use numerical method only in regions where 1/B becomes small A semi-numerical approach to one-loop multi-leg amplitudes p.19

32 numerical behaviour I 6 4 (z 3z 4 ) and I 6 4 (z 3z 2 4 ) (occur in the reduction of a rank 6 hexagon) exceptional kinematics for x 0 solid line: numerical evaluation, only partial algebraic reduction dashed: numerical behaviour of analytic representation after full algebraic reduction A semi-numerical approach to one-loop multi-leg amplitudes p.20

33 Applications gg W W l ν l ν Binoth, Ciccolini, Kauer, Krämer gg HHH Binoth, Karg A semi-numerical approach to one-loop multi-leg amplitudes p.21

34 Applications q q q q q q A (k 1,..., k 6 ) = g6 1 (4π) n/2 s { A2 ɛ 2 + A } 1 ɛ + A 0 + O(ɛ) A semi-numerical approach to one-loop multi-leg amplitudes p.21

35 Applications q q q q q q A (k 1,..., k 6 ) = sample point g6 1 (4π) n/2 s { A2 ɛ 2 + A } 1 ɛ + A 0 + O(ɛ) k k 0 k x k y k z k k k k k k A semi-numerical approach to one-loop multi-leg amplitudes p.21

36 q q q q q q hexagon: Re(P 2 ) Im(P 2 ) Re(P 1 ) Im(P 1 ) Re(P 0 ) Im(P 0 ) A semi-numerical approach to one-loop multi-leg amplitudes p.22

37 q q q q q q hexagon: Re(P 2 ) Im(P 2 ) Re(P 1 ) Im(P 1 ) Re(P 0 ) Im(P 0 ) CPU time 0.21 sec (rel. accuracy better than 10 4 ) A semi-numerical approach to one-loop multi-leg amplitudes p.22

38 q q q q q q hexagon: Re(P 2 ) Im(P 2 ) Re(P 1 ) Im(P 1 ) Re(P 0 ) Im(P 0 ) CPU time 0.21 sec (rel. accuracy better than 10 4 ) phase space points investigated for various kinematic regions A semi-numerical approach to one-loop multi-leg amplitudes p.22

39 q q q q q q hexagon: Re(P 2 ) Im(P 2 ) Re(P 1 ) Im(P 1 ) Re(P 0 ) Im(P 0 ) CPU time 0.21 sec (rel. accuracy better than 10 4 ) phase space points investigated for various kinematic regions numerical evaluation of basis integrals (B < Λ 2 ) for about 1% of phase space points (Λ GeV) A semi-numerical approach to one-loop multi-leg amplitudes p.22

40 q q q q q q hexagon: Re(P 2 ) Im(P 2 ) Re(P 1 ) Im(P 1 ) Re(P 0 ) Im(P 0 ) CPU time 0.21 sec (rel. accuracy better than 10 4 ) phase space points investigated for various kinematic regions numerical evaluation of basis integrals (B < Λ 2 ) for about 1% of phase space points (Λ GeV) numerical evaluation slower but not a problem for 1% of phase space these regions often excluded by experimental cuts A semi-numerical approach to one-loop multi-leg amplitudes p.22

41 Summary and Outlook NLO calculations are reaching a new level of automatisation important tools for the LHC A semi-numerical approach to one-loop multi-leg amplitudes p.23

42 Summary and Outlook NLO calculations are reaching a new level of automatisation important tools for the LHC GOLEM is one contribution to the toolbox numerically robust valid for massless and massive external/internal particles no principal limitation on number of legs optionally more analytic or more numerical branch A semi-numerical approach to one-loop multi-leg amplitudes p.23

43 Summary and Outlook NLO calculations are reaching a new level of automatisation important tools for the LHC GOLEM is one contribution to the toolbox numerically robust valid for massless and massive external/internal particles no principal limitation on number of legs optionally more analytic or more numerical branch modular if more compact expressions for amplitudes can be achieved without using Feynman diagrams: very welcome! real radiation under construction combination with parton shower planned A semi-numerical approach to one-loop multi-leg amplitudes p.23

44 backup slides A semi-numerical approach to one-loop multi-leg amplitudes p.24

45 phase space effects enhanced by cuts A semi-numerical approach to one-loop multi-leg amplitudes p.25

46 N=5 rank 2 example I n, µ 1µ 2 5 (a 1, a 2 ; S) = g µ 1 µ 2 B 5,2 (S) + B 5,2 (S) = 1 2 A 5,2 l 1 l 2 (S) = j S j S l 1,l 2 S b j I n+2 4 (S \ {j}) ( S 1 j l1 b l2 + S 1 j l 2 b l1 µ 1 l 1 a 1 µ 2 l 2 a 2 A 5,2 l 1 l 2 (S) ) 2 S 1 l 1 l 2 b j + b j S {j} 1 l 1 l 2 I4 n+2 (S \ {j}) + 1 [ S 1 j l 2 2 S {j} 1 k l 1 + j S k S\{j} S 1 j l 1 S {j} 1 k l 2 ] I n 3 (S \ {j, k}) A semi-numerical approach to one-loop multi-leg amplitudes p.26

47 N=6 rank 1 example I n, µ 6 (a; S) = j S C µ j a In 5 (S \ {j}) = j,l S µ la S 1 lj k S b {j} k B {j,k} I n+2 4 (S \ {j, k}) + m S\{j,k} b {j,k} m I3 n (S \ {j, k, m}) A semi-numerical approach to one-loop multi-leg amplitudes p.27