Simulating NNLO QCD corrections for processes with giant K factors Sebastian Sapeta LPTHE, UPMC, CNRS, Paris in collaboration with Gavin Salam and Mathieu Rubin 1 HP 2 .3rd, Florence, 14-17 September 2010 1 M.Rubin, G.P.Salam and SS, arXiv:1006.2144 [hep-ph] Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 1 / 13
The problem of giant K factors ◮ Z+j at the LHC H T , jets = � p t , Z p t , hardest jet all jets p t , j 10 4 LO LO LO 10 3 d σ /dV [fb / 100 GeV] 10 2 10 pp, 14 TeV 1 anti-k t , R=0.7 p t,j1 > 200 GeV, Z → e + e - MCFM 5.7, CTEQ6M 10 -1 250 500 750 1000 250 500 750 1000 250 500 750 1000 V = p t,Z [GeV] V = p t,j1 [GeV] V = H T,jets [GeV] � �� � Z LO: g q Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 2 / 13
The problem of giant K factors ◮ Z+j at the LHC H T , jets = � p t , Z p t , hardest jet all jets p t , j 10 4 LO LO LO NLO NLO NLO 10 3 d σ /dV [fb / 100 GeV] 10 2 10 pp, 14 TeV 1 anti-k t , R=0.7 p t,j1 > 200 GeV, Z → e + e - MCFM 5.7, CTEQ6M 10 -1 250 500 750 1000 250 500 750 1000 250 500 750 1000 V = p t,Z [GeV] V = p t,j1 [GeV] V = H T,jets [GeV] � �� � � �� � ` α ew α 2 ´ s ln 2 p t , j 1 / M Z O α ew α 2 ` ´ s O q Z Z NLO: g g g q g Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 2 / 13
What do we have and what is missing? ◮ The large K factor for the Z+jet comes from the new “dijet type” topologies that appear at NLO q q Z Z g g g g Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 3 / 13
What do we have and what is missing? ◮ The large K factor for the Z+jet comes from the new “dijet type” topologies that appear at NLO q q Z Z g g g g ◮ though formally NLO diagrams for Z+jet, these are in fact leading contributions to p t , j 1 and H T spectra ◮ this raises doubts about the accuracy of these predictions ◮ need for subleading contributions for Z+jet, in this case NNLO Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 3 / 13
What do we have and what is missing? ◮ The large K factor for the Z+jet comes from the new “dijet type” topologies that appear at NLO q q Z Z g g g g ◮ though formally NLO diagrams for Z+jet, these are in fact leading contributions to p t , j 1 and H T spectra ◮ this raises doubts about the accuracy of these predictions ◮ need for subleading contributions for Z+jet, in this case NNLO Z+j at NNLO = Z+3j tree + Z+2j 1-loop + Z+j 2-loop � �� � Z+2j at NLO Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 3 / 13
What do we have and what is missing? ◮ The large K factor for the Z+jet comes from the new “dijet type” topologies that appear at NLO q q Z Z g g g g ◮ though formally NLO diagrams for Z+jet, these are in fact leading contributions to p t , j 1 and H T spectra ◮ this raises doubts about the accuracy of these predictions ◮ need for subleading contributions for Z+jet, in this case NNLO Z+j at NNLO = Z+3j tree + Z+2j 1-loop + Z+j 2-loop � �� � Z+2j at NLO ◮ 2-loop part ◮ we need it to cancel IR and collinear divergences from Z+2j at NLO result ◮ it will have the topology of Z+j at LO so it will not contribute much to the cross sections with giant K-factor Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 3 / 13
The basic idea How to cancel the infrared and collinear singularities? Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 4 / 13
The basic idea How to cancel the infrared and collinear singularities? ◮ use unitarity to simulate the divergent part of 2-loop diagrams Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 4 / 13
The basic idea How to cancel the infrared and collinear singularities? ◮ use unitarity to simulate the divergent part of 2-loop diagrams LoopSim procedure output: input: LoopSim all n − k final state event with n final particle events state particles (equivalently all k-loop events) Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 4 / 13
The basic idea How to cancel the infrared and collinear singularities? ◮ use unitarity to simulate the divergent part of 2-loop diagrams LoopSim procedure output: input: LoopSim all n − k final state event with n final particle events state particles (equivalently all k-loop events) ◮ notation: n LO ¯ – simulated 1-loop ¯ n ¯ n LO – simulated 2-loop and simulated 1-loop n NLO ¯ – simulated 2-loop and exact 1-loop Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 4 / 13
The basic idea How to cancel the infrared and collinear singularities? ◮ use unitarity to simulate the divergent part of 2-loop diagrams LoopSim procedure output: input: LoopSim all n − k final state event with n final particle events state particles (equivalently all k-loop events) ◮ notation: ¯ n LO – simulated 1-loop ¯ n ¯ n LO – simulated 2-loop and simulated 1-loop n NLO ¯ – simulated 2-loop and exact 1-loop ◮ this will work very well for the processes with large K factors e.g. � � �� α 2 s σ ¯ n NLO = σ NNLO 1 + O , K NNLO � K NLO ≫ 1 K NNLO Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 4 / 13
The LoopSim method: ¯ n LO, ¯ n ¯ n LO etc. Input event 1 2 4 beam 3 Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 5 / 13
The LoopSim method: ¯ n LO, ¯ n ¯ n LO etc. Input event Attributed emission seq. 1 1 2 2 jet clustering 4 4 beam 3 3 ◮ jet clustering ij → k is reinterpreted as the splitting k → ij Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 5 / 13
The LoopSim method: ¯ n LO, ¯ n ¯ n LO etc. Input event Attributed emission seq. Born particle id. 1 1 1 2 2 2 jet clustering 4 4 4 beam 3 3 3 ◮ jet clustering ij → k is reinterpreted as the splitting k → ij Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 5 / 13
The LoopSim method: ¯ n LO, ¯ n ¯ n LO etc. Input event Attributed emission seq. Born particle id. 1 1 1 2 2 2 jet clustering 4 4 4 beam 3 3 3 Output 1−loop event 2nd output 1−loop event Output 2−loop event (loop over beam) ◮ jet clustering ij → k is reinterpreted as the splitting k → ij ◮ weight of an event ∼ ( − 1) number of loops Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 5 / 13
The LoopSim method: ¯ n LO, ¯ n ¯ n LO etc. Input event Attributed emission seq. Born particle id. 1 1 1 2 2 2 jet clustering 4 4 4 beam 3 3 3 Output 1−loop event 2nd output 1−loop event Output 2−loop event (loop over beam) ◮ jet clustering ij → k is reinterpreted as the splitting k → ij ◮ weight of an event ∼ ( − 1) number of loops ◮ � all weights = 0 (unitarity) [Bloch, Nordsieck and Kinoshita, Lee, Nauenberg] Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 5 / 13
The LoopSim method: ¯ n LO, ¯ n ¯ n LO etc. Input event Attributed emission seq. Born particle id. 1 1 1 2 2 2 jet clustering 4 4 4 beam 3 3 3 Output 1−loop event 2nd output 1−loop event Output 2−loop event (loop over beam) ◮ jet clustering ij → k is reinterpreted as the splitting k → ij ◮ weight of an event ∼ ( − 1) number of loops ◮ � all weights = 0 (unitarity) [Bloch, Nordsieck and Kinoshita, Lee, Nauenberg] ◮ beware: the loops above are just a shortcut notation! Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 5 / 13
Including exact loops E n , l – input event with n final state particles and l loops U b – operator producing event with b Born particles and l loops l U b – operator generating all necessary loop diagrams at given order ∀ Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 6 / 13
Including exact loops E n , l – input event with n final state particles and l loops U b – operator producing event with b Born particles and l loops l U b – operator generating all necessary loop diagrams at given order ∀ How to introduce exact loop contributions? Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 6 / 13
Including exact loops E n , l – input event with n final state particles and l loops U b – operator producing event with b Born particles and l loops l U b – operator generating all necessary loop diagrams at given order ∀ How to introduce exact loop contributions? U b ∀ ( E n , 0 ) ◮ generate all diagrams from the tree level event Sebastian Sapeta (LPTHE, Paris) Simulating NNLO QCD corrections for processes with giant K factors 6 / 13
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