Computation of multi-leg amplitudes with NJet Valery Yundin Niels Bohr International Academy & Discovery Center in collaboration with Simon Badger, Benedikt Biedermann and Peter Uwer ACAT 2013, 16–21 May 2013, IHEP Beijing
NLO calculations NLO results provide more accurate predictions and theoretical uncertainties for multi-jet backgrounds in new physics searches. Hard process ingredients � � � � + σ NLO = � dσ B � dσ R � n + dσ V dσ S n +1 − dσ S n + n +1 n +1 n 1 n +1 n d 3 k ℓ n = 1 � � � 2 � Θ n-njet (2 π ) 4 δ ( P ) dσ V � M n ( ij → n ) (2 π ) 3 2 E ℓ 2ˆ s ℓ =1 QCD matrix elements � 2 = � � � � † A 1-loop × A tree � M n ( ij → n ) n n spin color 1 / N
Automated One-Loop Amplitudes General solutions to virtual corrections ◮ Helac-NLO [public] SM [arXiv:1110.1499] ◮ GoSam [public] SM, MSSM, UFO [arXiv:1111.2034] ◮ NJet [public] jets [arXiv:1209.0100] ◮ BlackHat [semi-public] V+jets, jets [arXiv:0803.4180+. . . ] ◮ MadLoop, SM, BSM [arXiv:1103.0621] ◮ Open Loops, QCD SM [arXiv:1111.5206] ◮ Recola, SM+EW [arXiv:1211.6316] ◮ Rocket, W+jets, WW+jets, t ¯ t +jet [arXiv:0805.2152+. . . ] ◮ MCFM ∗ [public] max. 2 → 3 [ http://mcfm.fnal.gov ] ◮ Feynman based approaches: VBFNLO [public], Denner et al., FeynCalc [public], Reina et al. . . . 2 / N ∗ MCFM collects the known analytic results for one-loop amplitudes
From NGluon to NJet NGluon : public C++ library for multi-parton primitive amplitudes via unitarity (now part of NJet ) [ arXiv:1011.2900] ◮ Efficient tree amplitudes using Berends-Giele recursion. ◮ Rational terms from massive loop cuts. ◮ Extraction of integral coefficients via Fourier projections. ◮ Everything is in 4 dimensions (except loop integrals). NJet: public C++ library for multi-parton matrix elements in massless QCD [ https://bitbucket.org/njet/njet ] [ arXiv:1209.0100] Features ◮ Full colour-summed amplitudes for up to 5 outgoing partons. ◮ Binoth Les Houches Accord interface for MC generators. 3 / N
Structure of One-Loop Amplitudes known scalar integrals � c 4; K 4 c 1 R { K 4 } pure rational rational terms coefficients ◮ Gauge theory amplitudes reduced to box topologies or simpler [Passarino,Veltman;Melrose] ◮ Isolate logarithms with cuts and exploit on-shell simplifications [Bern,Dixon,Kosower] 4 / N
Multi-Fermion Primitive Amplitudes NParton computes arbitrary multi-fermion primitives. 2 3 2 3 1 4 1 4 l 0 A [ m ] (1 ¯ l 0 A [ f ] (1 u , 2 ¯ u , 2 d , 3 ¯ d , 4 u , . . . ) u , 3 g , 4 g , . . . ) All primitives are separated into two classes ◮ With mixed fermion and gluon loop content ( l 0 = gluon) ◮ With internal fermion loops ( l 0 = quark) These two classes cover all partonic primitives in one loop QCD. 5 / N
Partial Amplitudes and Colour Summation Colour decomposition of an L-loop amplitude: � A ( L ) A ( L ) n ( { p i } ) = T c ( { a i } ) n ; c ( p 1 , . . . , p n ) � �� � c � �� � colour basis partial amplitudes Partial amplitudes → squared matrix elements � 2 = � � � � � � n ; c · C cc ′ · A (0) † A ( L ) n A (0) † A ( L ) � M n = n n ; c ′ hel col hel cc ′ Colour matrix � C cc ′ = T c ( { a i } ) T c ′ ( { a i } ) { a i } T c ( { a i } ) = T a 1 jk . . . δ lm . . . 6 / N
Partial Amplitudes and Colour Summation Colour decomposition of a 1-loop amplitude: Partial amplitudes are linear combinations of primitive amplitudes. � A (1) a k ; c A [ m ] + N f b k ; c A [ f ] n ; c = n n k Partial-Primitive decomposition for gluons and q ¯ q + gluons: ◮ Tree level: Kleiss-Kuijf basis of ( n − 2)! primitives ◮ One-loop: a basis of ( n − 1)! primitives. [Kleiss,Kuijf], [Bern,Dixon,Dunbar,Kosower] Partial-Primitive decomposition for multi-quark case: No analytic formula. Reconstruct partials using diagram matching. [Ellis,Kunszt,Melnikov,Zanderighi], [Ita,Ozeren], [NJet] 7 / N
Generic Partial-Primitive decomposition Outline of the algorithm 1. Generate all diagrams’ topologies for the amplitude A n 2. Write primitives P i as combinations of colour-stripped diagrams K i using matching matrix M ij 3. Invert the system to get partial amplitudes in terms of independent set of primitives ˆ P ˆ N pri � � Q cj ˆ A n = T c ( { a i } ) P j c j =1 Ensure linearly independent set by capturing all relations between color-ordered diagrams. 8 / N
Number of primitives in tree, mixed and fermion loop amplitudes N [0] N [ m ] N [ f ] N [0] N [ m ] N [ f ] Process Process pri pri pri pri pri pri 4 g 2 3 3 5 g 6 12 12 uu + 2 g 2 6 1 uu + 3 g 6 24 6 uudd 1 4 1 uuddg 3 16 3 N [0] N [ m ] N [ f ] N [0] N [ m ] N [ f ] Process Process pri pri pri pri pri pri 6 g 24 60 60 7 g 120 360 360 uu + 4 g 24 120 33 uu + 5 g 120 720 230 uudd + 2 g 12 80 13 uudd + 3 g 60 480 75 uuddss 4 32 4 uuddssg 20 192 20 N [0] N [ m ] N [ f ] Process pri pri pri 8 g 720 2520 2520 uu + 6 g 720 5040 1800 uudd + 4 g 360 3360 712 uuddss + 2 g 120 1344 263 uuddsscc 30 384 65 9 / N
Desymmetrized amplitudes Squared amplitudes are totally symmetric over final state gluons � 2 = � � � � � 2 � A ( x, g 1 , . . . , g n , y ) � A ( x, σ { g 1 , . . . , g n } , y ) Gluon phase space integration is a symmetric operator � � F ( g 1 , . . . , g n ) d PS n = F ( σ { g 1 , . . . , g n } ) d PS n Could replace squared amplitudes with something simpler � � � 2 d PS n = � � A dsym ( g 1 , . . . , g n ) d PS n � A x → n ( g ) � � � 2 , � A dsym ( g 1 , . . . , g n ) = n ! where � A x → n ( g ) P n ∈ σ { g 1 , . . . , g n } P n Example: ��� b ��� b ( x 2 y + x 2 z + xy 2 + xz 2 + y 2 z + yz 2 ) dx dy dz = 6 x 2 y dx dy dz a a 10 / N
Desymmetrized gluonic amplitudes Special non-symmetric gluon colour sums ◮ Contain significantly fewer loop primitives ◮ Give original full colour sums after symmetrization � d PS n A (0) † · C n ! × ( n +1)! / 2 · A (1) σ V gg → n ( g ) = � d PS n A (0) † · C dsym n ! × ( n +1) · A (1) , dsym = ( n − 2)! n ! / 2 reduction of time per point 1 gg → 3 g gg → 4 g gg → 5 g Standard sum 0.22 s 6.19 s 171.31 s De-symmetrized 0.07 s 0.50 s 2.76 s Speedup × 3 × 12 × 60 1 Where n is the number of final state gluons 11 / N
Scaling test to estimate the accuracy loss Sources of accuracy loss ◮ Accumulation of rounding errors – negligible ◮ Catastrophic large cancellations – significant in certain kinematic regions (small Gram determinants, etc) Large cancellation 1 . 111111115495439 A C − B − 1 . 111111112345678 C ∼ 1 = 0 . 00000000 314976100000000 � �� � � �� � If C → 0 then A → B lost new tail In finite precision machine arithmetic the tail is zero-extended. 12 / N
Scaling test to determine accuracy Scaling test ◮ Evaluate the amplitude several times using different “scaled” units (for instance: 1 × GeV, 1 . 33 × GeV, etc). ◮ Use known dimension of the amplitudes to scale them back to a common unit (GeV). ◮ The difference between obtained values is an error estimate. Why it works? 1 . 111111115495439 − 1 . 111111112345678 A 1 = 0 . 00000000314976100000000 × 1 . 33 1 . 111111118228751 ✚ ❩ 43 ← round-off ✚ ❩ ❩ ✚ × 1 . 33 − 1 . 111111113913578 ✚ 86 ← round-off ❩ × 1 . 33 = 0 . 00000000431517300000000 A 2 = 0 . 00000000314976131386861 � �� � difference 13 / N
Testing the scaling test Scaling Test vs Analytic Formulae 5 ⋅ 10 3 + + + + + + − + + + + + − − + + + + 4 ⋅ 10 3 Number of Events 3 ⋅ 10 3 2 ⋅ 10 3 1 ⋅ 10 3 0 ⋅ 10 0 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Accuracy � � � � 2( A NGluon − A scaled A NGluon − A analytic NGluon ) log − log A NGluon + A scaled A analytic NGluon Reliable, but essentially statistical. A safety margin of 2 digits is advised. 14 / N
Scaling test of 5 jet amplitudes Left: 7 gluon squared amplitude. Right: 4 quarks + 3 gluons. ε -2 ε -2 gg → 5g dd → dd + 3g ε -1 ε -1 10 3 10 3 ε 0 ε 0 ε -2 quad ε -2 quad Number of events Number of events ε -1 quad ε -1 quad ε 0 quad ε 0 quad 10 2 10 2 10 1 10 1 10 0 10 0 -15 -10 -5 0 -15 -10 -5 0 Accuracy Accuracy Thick lines – double precision. Thin lines – fixed with quadruple precision. 15 / N
Evaluation times Full colour and helicity sum time per point [clang, Xeon 3.30 GHz]. process T sd [s] T 4 dig. [s] (%) process T sd [s] T 4 dig. [s] (%) 4 g 0.030 0.030 (0.00) 5 g 0.22 0.22 (0.22) uu +2 g 0.032 0.032 (0.00) uu +3 g 0.34 0.35 (0.06) uudd 0.011 0.011 (0.00) uudd + g 0.11 0.11 (0.00) uuuu 0.022 0.022 (0.00) uuuu + g 0.22 0.22 (0.03) process T sd [s] T 4 dig. [s] (%) process T sd [s] T 4 dig. [s] (%) 6 g 6.19 6.81 (1.37) 7 g 171.3 276.7 (8.63) uu +4 g 7.19 7.40 (0.38) uu +5 g 195.1 241.2 (3.25) uudd +2 g 2.05 2.06 (0.08) uudd +3 g 45.7 48.8 (0.88) uuuu +2 g 4.08 4.15 (0.21) uuuu +3 g 92.5 101.5 (1.29) uuddss 0.38 0.38 (0.00) uuddssg 7.9 8.1 (0.23) uudddd 0.74 0.74 (0.00) uuddddg 15.8 16.2 (0.29) uuuuuu 2.16 2.17 (0.02) uuuuuug 47.1 48.6 (0.41) All times include two evaluations for the scaling test. 16 / N
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