Reduction at the integrand level beyond NLO

Transcription

1 Reduction at the integrand level beyond NLO Costas G. Papadopoulos NCSR Demokritos, Athens, Greece & CERN, Geneva, Switzerland Corfu 2012, September 15, 2012 Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 1 / 22

2 LHC outstanding performance calls for serious theoretical work Precision becomes very important for most of the processes Fixed-order calculations need to advance Capitalize over previous achievements Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 2 / 22

3 Dyson-Schwinger Recursive Equations 1999 HELAC: The first code to calculate recursively tree-order amplitudes for (practically) arbitrary number of particles A. Kanaki and C. G. Papadopoulos, Comput. Phys. Commun. 132 (2000) 306 [arxiv:hep-ph/ ]. = Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 3 / 22

4 Dyson-Schwinger Recursive Equations 1999 HELAC: The first code to calculate recursively tree-order amplitudes for (practically) arbitrary number of particles 2000 PHEGAS: The first code to automatically produce phase-space mappings based on all FD C. G. Papadopoulos, Comput. Phys. Commun. 137 (2001) 247 [arxiv:hep-ph/ ]. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 3 / 22

5 Dyson-Schwinger Recursive Equations 1999 HELAC: The first code to calculate recursively tree-order amplitudes for (practically) arbitrary number of particles 2000 PHEGAS: The first code to automatically produce phase-space mappings based on all FD Including all SM, in both unitary and F-gauge, masses, CKM, unstable particle widths, complex mass scheme, etc. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 3 / 22

6 Dyson-Schwinger Recursive Equations 1999 HELAC: The first code to calculate recursively tree-order amplitudes for (practically) arbitrary number of particles 2000 PHEGAS: The first code to automatically produce phase-space mappings based on all FD Including all SM, in both unitary and F-gauge, masses, CKM, unstable particle widths, complex mass scheme, etc. For QCD color connection representation: revival of the t Hooft ideas ( 71) in the modern era. Citation Alert! Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 3 / 22

7 HELAC Color treatment M a 1,i 2,...,i k j 2,...,j k t a 1 i 1 j 1 M i 1,i 2,...,i k j 1,j 2,...,j k M i 1,i 2,...,i k j 1,j 2,...,j k = σ δ iσ1,j 1 δ iσ2,j 2... δ iσk,j k A σ gluons (i, j), quark (i, 0), anti-quark (0, j), other (0, 0) C σ,σ {i},{j} {i},{j} M i 1,i 2,...,i k j 1,j 2,...,j k 2 σ,σ A σc σ,σ A σ δ iσ1,j 1 δ iσ2,j 2... δ iσk,j k δ iσ 1,j 1 δ iσ 2,j 2... δ iσ k,j k = N m(σ,σ ) c Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 4 / 22

8 HELAC Color treatment Color-connection Feynman Rules (j,σ i ) g δ σ i j g (i,σ i ) (j,σ j ) g Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 5 / 22

9 HELAC tree order current version 2007 HELAC: A. Cafarella, C. G. Papadopoulos and M. Worek, Comput. Phys. Commun. 180 (2009) 1941 [arxiv: [hep-ph]]. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 6 / 22

10 HELAC tree order current version 2007 HELAC: Generate all subprocesses for pp, p p collisions, calculate cross sections, produce Les Houches accord file Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 6 / 22

11 HELAC tree order current version 2007 HELAC: Generate all subprocesses for pp, p p collisions, calculate cross sections, produce Les Houches accord file Very easy to use: just edit the user.inp file and then execute the command./run.sh # Compulsory information colpar 1 # colliding particles: 1=pp, 2=ppbar, 3=e+e- inist # initial state; enter 0 to sum over initial states finst # final state energy # collision energy (GeV) # For reference, here is the particle numbering: # ve e u d vm mu c s vt ta t b a z w+ w- g h chi f+ f- jet # # The respective antiparticles have a minus sign (for example: positron is - 2) # A jet in the final state is denoted by the number 100 # Enter here your additional commands if you wish to alterate the default values Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 6 / 22

12 HELAC tree order current version 2007 HELAC: Generate all subprocesses for pp, p p collisions, calculate cross sections, produce Les Houches accord file Very easy to use: just edit the user.inp file and then execute the command./run.sh Including kt-reweight for jet matching Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 6 / 22

13 HELAC tree order current version 2007 HELAC: Generate all subprocesses for pp, p p collisions, calculate cross sections, produce Les Houches accord file Very easy to use: just edit the user.inp file and then execute the command./run.sh Including kt-reweight for jet matching Latest: W + 5 jets at LHC Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 6 / 22

14 NLO Tree-order integrated over m-body phase space + SF in D=4 Virtual corrections integrated over m-body phase space + SF in D=4 Real corrections subtracted integrated over (m+1)-body phase space + SF in D=4 I- integrated over m-body phase space + SF in D=4 KP- integrated over m-body phase space + SF integrated over SF, in D=4 Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 7 / 22

15 The HELAC-NLO adventure Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

16 The HELAC-NLO adventure 2006 OPP: The method that enables us to think seriously about NLO calculations. Based on previous work by Bern, Dixon, Kosower, Britto, Cachazo, Feng. Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B 425 (1994) 217 [arxiv:hep-ph/ ]. R. Britto, F. Cachazo and B. Feng, Nucl. Phys. B 725 (2005) 275 [arxiv:hep-th/ ]. Complete framework: numerical (fast) & algebraic (stable) G. Ossola, C. G. Papadopoulos and R. Pittau, Nucl. Phys. B 763 (2007) 147 [arxiv:hep-ph/ ]. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

17 The HELAC-NLO adventure A = + + m 1 i 0 <i 1 <i 2 <i 3 d(i 0i 1i 2i 3)D 0(i 0i 1i 2i 3) m 1 i 0 <i 1 <i 2 c(i 0i 1i 2)C 0(i 0i 1i 2) m 1 i 0 <i 1 b(i 0i 1)B 0(i 0i 1) m 1 + a(i 0)A 0(i 0) i 0 + rational terms Algebra & Integrals Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

18 The HELAC-NLO adventure N(q) = m 1 i 0 <i 1 <i 2 <i 3 m 1 A N(q) Di [ d(i 0i 1i 2i 3) + d(q; ] m 1 i 0i 1i 2i 3) D i i i 0,i 1,i 2,i 3 i 0 <i 1 <i 2 [c(i 0i 1i 2) + c(q; i 0i 1i 2)] m 1 i 0 <i 1 m 1 [ ] m 1 b(i 0i 1) + b(q; i 0i 1) D i i i 0,i 1 i 0 [a(i 0) + ã(q; i 0)] m 1 i i 0 D i Solving for known values of the loop momentum q m 1 i i 0,i 1,i 2 D i Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

19 The HELAC-NLO adventure R 1 : the rational terms from the reduction itself Let s go back to the integrand N(q) A( q) = D 0 D 1 D m 1 Insert the expression for N(q) we know all the coefficients N(q) = m 1 i 0<i 1<i 2<i 3 [ ] m 1 d + d(q) D i + i i 0,i 1,i 2,i 3 Finally rewrite all denominators using D i D i = Z i, with Z i m 1 i 0<i 1<i 2 [c + c(q)] (1 q2 D i ) m 1 i i 0,i 1,i 2 D i + Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

20 The HELAC-NLO adventure A( q) = m 1 d(i 0i 1i 2i 3) + d(q; i0i 1i 2i 3) D i 0 <i 1 <i 2 <i i0 Di1 Di2 Di3 3 m 1 c(i 0i 1i 2) + c(q; i 0i 1i 2) D i 0 <i 1 <i i0 Di1 Di2 2 m 1 b(i 0i 1) + b(q; i 0i 1) D i 0 <i i0 D i1 1 m 1 a(i 0) + ã(q; i 0) D i i0 0 m 1 i i 0 m 1 i i 0,i 1 Z i Zi m 1 i i 0,i 1,i 2 Zi m 1 i i 0,i 1,i 2,i 3 Zi The rational part is produced, after integrating over d n q, by the q 2 dependence in Z i ) Zi (1 q2 D i Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

21 The HELAC-NLO adventure The Extra Integrals are of the form I (n;2l) s;µ 1 µ r d n q q 2l q µ1 q µr D(k 0 ) D(k s ), where D(k i ) ( q + k i ) 2 m 2 i, k i = p i p 0 These integrals: - have dimensionality D = 2(1 + l s) + r - contribute only when D 0, otherwise are of O(ɛ) Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

22 The HELAC-NLO adventure Expand in D-dimensions? D i = D i + q 2 N(q) = m 1 i 0 <i 1 <i 2 <i 3 m 1 i 0 <i 1 <i 2 m 1 i 0 <i 1 m 1 i 0 [ ] m 1 d(i 0i 1i 2i 3; q 2 ) + d(q; i 0i 1i 2i 3; q 2 ) Di i i 0,i 1,i 2,i 3 [ ] m 1 c(i 0i 1i 2; q 2 ) + c(q; i 0i 1i 2; q 2 ) Di i i 0,i 1,i 2 [ b(i 0i 1; q 2 ) + b(q; ] m 1 i 0i 1; q 2 ) Di i i 0,i 1 [ ] m 1 a(i 0; q 2 ) + ã(q; i 0; q 2 ) i i 0 m 1 D i + P(q) i D i m 2 i m 2 i q 2 Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

23 The HELAC-NLO adventure Polynomial dependence on q 2 b(ij; q 2 ) = b(ij) + q 2 b (2) (ij), c(ijk; q 2 ) = c(ijk) + q 2 c (2) (ijk). q2 d n q = [m iπ2 D i D 2i + m 2j (p i p j ) 2 ] + O(ɛ), j 2 3 d n q q 2 D i D j D k = iπ2 2 + O(ɛ), d n q q 4 D i D j D k D l = iπ2 6 + O(ɛ). Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

24 The HELAC-NLO adventure Furthermore, by defining D (m) (q, q 2 ) m 1 i 0<i 1<i 2<i 3 the following expansion holds D (m) (q, q 2 ) = [ d(i 0 i 1 i 2 i 3 ; q 2 ) + d(q; ] m 1 i 0 i 1 i 2 i 3 ; q 2 ) Di, i i 0,i 1,i 2,i 3 m q (2j 4) d (2j 4) (q), j=2 where the last coefficient is independent on q d (2m 4) (q) = d (2m 4). Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

25 The HELAC-NLO adventure In practice, once the 4-dimensional coefficients have been determined, one can redo the fits for different values of q 2, in order to determine b (2) (ij), c (2) (ijk) and d (2m 4). R 1 = i 96π 2 d (2m 4) i 32π 2 m 1 i 32π 2 b (2) (i 0 i 1 ) i 0 <i 1 m 1 G. Ossola, C. G. Papadopoulos and R. Pittau,arXiv: [hep-ph] c (2) (i 0 i 1 i 2 ) i 0 <i 1 <i 2 (m 2i0 + m 2i1 (p i 0 p i1 ) 2 ). 3 Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

26 The HELAC-NLO adventure A different source of Rational Terms, called R 2, can also be generated from the ɛ-dimensional part of N(q) R 2 1 (2π) 4 N( q) = N(q) + Ñ( q2, ɛ; q) d n q Ñ( q 2, ɛ; q) D 0 D1 D 1 m 1 (2π) 4 q = q + q, γ µ = γ µ + γ µ, ḡ µ ν = g µν + g µ ν. UV-behavior: only up to 4-vertices (beyond one loop?). Alternatives: GKMZ-approach, Blackhat d n q R 2 Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

27 The HELAC-NLO adventure 2006 OPP: The method that enables us to think seriously about NLO calculations CutTools: Reduction at the integrand level + rational terms R 1 G. Ossola, C. G. Papadopoulos and R. Pittau, JHEP 0803 (2008) 042 [arxiv: [hep-ph]]. G. Ossola, C. G. Papadopoulos and R. Pittau, JHEP 0805 (2008) 004 [arxiv: [hep-ph]]. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

28 The HELAC-NLO adventure 2006 OPP: The method that enables us to think seriously about NLO calculations CutTools: Reduction at the integrand level + rational terms R 1 OneLOop: One-loop scalar integrals in dimensional regularization (UV+IR) including complex masses Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

29 The HELAC-NLO adventure 2006 OPP: The method that enables us to think seriously about NLO calculations CutTools: Reduction at the integrand level + rational terms R 1 OneLOop: One-loop scalar integrals in dimensional regularization (UV+IR) including complex masses 2008 HELAC-1LOOP: Based on HELAC to produce virtual one-loop amplitudes Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

30 The HELAC-NLO adventure 2006 OPP: The method that enables us to think seriously about NLO calculations CutTools: Reduction at the integrand level + rational terms R 1 OneLOop: One-loop scalar integrals in dimensional regularization (UV+IR) including complex masses 2008 HELAC-1LOOP: Based on HELAC to produce virtual one-loop amplitudes A. van Hameren, C. G. Papadopoulos and R. Pittau, JHEP 0909 (2009) 106 [arxiv: [hep-ph]]. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

31 The HELAC-NLO adventure 2006 OPP: The method that enables us to think seriously about NLO calculations CutTools: Reduction at the integrand level + rational terms R 1 OneLOop: One-loop scalar integrals in dimensional regularization (UV+IR) including complex masses 2008 HELAC-1LOOP: Based on HELAC to produce virtual one-loop amplitudes 2009 HELAC-Dipoles: Based on HELAC to automatically produce Catani-Seymour dipoles, I-operator, KP-operator, arbitrary masses M. Czakon, C. G. Papadopoulos and M. Worek, JHEP 0908 (2009) 085 [arxiv: [hep-ph]]. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 8 / 22

32 HELAC-NLO Complete software for NLO-QCD at LHC: LO: highly automated Virtual: very efficient Real: KP- and I-operator contributions also very efficient, Real-subtracted: quite efficient taking into account the current theoretical developments Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 9 / 22

33 HELAC-NLO Complete software for NLO-QCD at LHC: LO: highly automated Virtual: very efficient Real: KP- and I-operator contributions also very efficient, Real-subtracted: quite efficient taking into account the current theoretical developments Speed, stability, efficiency issues under control Improvements in PS for real corrections under investigation (KALEU from A. van Hameren). Towards an order of magnitude improvement! Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 9 / 22

34 HELAC-NLO Complete software for NLO-QCD at LHC: LO: highly automated Virtual: very efficient Real: KP- and I-operator contributions also very efficient, Real-subtracted: quite efficient taking into account the current theoretical developments Speed, stability, efficiency issues under control Improvements in PS for real corrections under investigation (KALEU from A. van Hameren). Towards an order of magnitude improvement! Provide NLO calculator for all processes 2 n with 6-7 particles attached to the loop Providing a library of preconfigured setups for all (6-7 particles attached to the loop, up to 12 particles overall) processes of interest for LHC Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 9 / 22

35 HELAC-NLO pp t tb b: proof-of-concept G. Bevilacqua, M. Czakon, C. G. Papadopoulos, R. Pittau and M. Worek, JHEP 0909 (2009) 109 [arxiv: [hep-ph]]. M. Worek, JHEP 1202 (2012) 043 [arxiv: [hep-ph]]. pp t t + j + j: one of the most advanced G. Bevilacqua, M. Czakon, C. G. Papadopoulos and M. Worek, Phys. Rev. Lett. 104 (2010) [arxiv: [hep-ph]]. G. Bevilacqua, M. Czakon, C. G. Papadopoulos and M. Worek, Phys. Rev. D 84 (2011) [arxiv: [hep-ph]]. pp W + W b b including (t t): finite width and complex masses G. Bevilacqua, M. Czakon, A. van Hameren, C. G. Papadopoulos and M. Worek, JHEP 1102 (2011) 083 [arxiv: [hep-ph]]. pp t tt t: BSM phenomenology G. Bevilacqua and M. Worek, JHEP 1207 (2012) 111 [arxiv: [hep-ph]]. pp t t + j PS : interfacing with Parton Shower at NLO (POWHEG) A. Kardos, C. Papadopoulos and Z. Trocsanyi, Phys. Lett. B 705 (2011) 76 [arxiv: [hep-ph]]. pp t t + H PS M. V. Garzelli, A. Kardos, C. G. Papadopoulos and Z. Trocsanyi, Europhys. Lett. 96 (2011) [arxiv: [hep-ph]]. pp t t + Z/W ± PS M. V. Garzelli, A. Kardos, C. G. Papadopoulos and Z. Trocsanyi, arxiv: [hep-ph]. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 10 / 22

36 What do we need at NNLO? Virtual: n particle two-loop amplitudes Real-Virtual: n + 1 particle real times one-loop Real-Real: n + 2 particle real Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 11 / 22

37 What do we need at NNLO? Virtual: n particle two-loop amplitudes Real-Virtual: n + 1 particle real times one-loop Real-Real: n + 2 particle real Recent work on two-loop amplitudes P. Mastrolia and G. Ossola, JHEP 1111 (2011) 014 [arxiv: [hep-ph]]. D. A. Kosower and K. J. Larsen, Phys. Rev. D 85 (2012) [arxiv: [hep-th]]. H. Johansson, D. A. Kosower and K. J. Larsen, arxiv: [hep-th]. S. Badger, H. Frellesvig and Y. Zhang, JHEP 1204 (2012) 055 [arxiv: [hep-ph]]. S. Badger, H. Frellesvig and Y. Zhang, JHEP 1208 (2012) 065 [arxiv: [hep-ph]]. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 11 / 22

38 Two-loop amplitudes Reduction at the integrand level Master Integrals ostas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 12 / 22

39 Two-loop amplitudes Reduction at the integrand level Master Integrals Generic two-loop graph: igraph R. H. P. Kleiss, I. Malamos, C. G. Papadopoulos and R. Verheyen, arxiv: [hep-ph]. l 1 l 2 l 1 + p 1 l 1 + l 2 + p 8 l 2 + p 4 l 1 + p 2 l 1 + p 3 l 1 + l 2 + p 9 l 2 + p 7 l 2 + p 5 l 2 + p 6 D(l 1 + p i ), D(l 2 + p j ), D(l 1 + l 2 + p k ) Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 12 / 22

40 Two-loop amplitudes The general strategy consists in finding function x j x j (l 1, l 2 ) n 1 j=1 x j D(l 1 + p j ) + n 1 +n 2 j=n 1 +1 x j D(l 1 + l 2 + p j ) + n j=n 1 +n 2 +1 x j D(l 2 + p j ) = 1. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 13 / 22

41 Two-loop amplitudes The general strategy consists in finding function x j x j (l 1, l 2 ) n 1 j=1 x j D(l 1 + p j ) + n 1 +n 2 j=n 1 +1 Let us go a step back at one loop x j D(l 1 + l 2 + p j ) + n j=n 1 +n = T 1 (q)d 1 + T 2 (q)d T n (q)d n x j D(l 2 + p j ) = 1. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 13 / 22

42 Two-loop amplitudes The general strategy consists in finding function x j x j (l 1, l 2 ) n 1 j=1 x j D(l 1 + p j ) + n 1 +n 2 j=n 1 +1 Let us go a step back at one loop Constant terms: T j (q) = x j x j D(l 1 + l 2 + p j ) + n j=n 1 +n = T 1 (q)d 1 + T 2 (q)d T n (q)d n x j D(l 2 + p j ) = 1. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 13 / 22

43 Two-loop amplitudes The general strategy consists in finding function x j x j (l 1, l 2 ) n 1 j=1 x j D(l 1 + p j ) + n 1 +n 2 j=n 1 +1 Let us go a step back at one loop Constant terms: T j (q) = x j n q 2 x j + 2q µ x j D(l 1 + l 2 + p j ) + n j=n 1 +n = T 1 (q)d 1 + T 2 (q)d T n (q)d n j=1 n x j = 0, j=1 n x j p µ j + j=1 n x j µ j = 1. j=1 n x j p µ j = 0, j=1 n x j µ j = 1 j=1 x j D(l 2 + p j ) = 1. ostas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 13 / 22

44 Two-loop amplitudes The general strategy consists in finding function x j x j (l 1, l 2 ) n 1 j=1 x j D(l 1 + p j ) + n 1 +n 2 j=n 1 +1 Let us go a step back at one loop Constant terms: T j (q) = x j n q 2 x j + 2q µ x j D(l 1 + l 2 + p j ) + n j=n 1 +n = T 1 (q)d 1 + T 2 (q)d T n (q)d n j=1 n x j = 0, j=1 solution exists for n = 6 d = 4 n x j p µ j + j=1 n x j µ j = 1. j=1 n x j p µ j = 0, j=1 n x j µ j = 1 j=1 x j D(l 2 + p j ) = 1. ostas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 13 / 22

45 Two-loop amplitudes Linear terms T (q) = P 1 (q), count tensor structures: 1, q µ, q µ q ν, q 2 q µ. There are, for d = 4, therefore = 19 independent tensor structures. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 14 / 22

46 Two-loop amplitudes Linear terms T (q) = P 1 (q), count tensor structures: 1, q µ, q µ q ν, q 2 q µ. There are, for d = 4, therefore = 19 independent tensor structures. In d dimensions, tensor up to rank k, N(d, k) number of independent tensor structures ( d 1 + k N(d, k) = k ) k+1 ( d 1 + p + p p=0 ). (1) In the table below we give the results for various ranks and dimensionalities. k d = Values of N(d, k) Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 14 / 22

47 Two-loop amplitudes The OPP- miracle is that the OPP equation works with only 10(6) different coefficients all c (1) i 5 1 = D i (q)(c (0) i + c (1) i ɛ i (q)) i=1 being equal! rank deficient problems Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 15 / 22

48 Two-loop amplitudes The OPP- miracle is that the OPP equation works with only 10(6) different coefficients all c (1) i 5 1 = D i (q)(c (0) i + c (1) i ɛ i (q)) i=1 being equal! rank deficient problems Back to two loops: igraphs can be denoted by the triplet (n 1, n 2, n 3 ), n = n 1 + n 2 + n 3 n 1,2,3 4 (= d), n 1 + n 2 + n 3 11 (= 2d + 3). Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 15 / 22

49 Linear terms x i = a i + j b ij (l 1 t j ) + j c ij (l 2 t j ) T (d) = (4d d + 2)/2 n d = 6 d = 5 d = 4 d = 3 d = 2 d = T (d) Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 16 / 22

50 Quadratic terms x i = a i + j b ij (l 1 t j ) + j c ij (l 2 t j ) + j k d ijk (l 1 t j )(l 1 t k ) + T (d) = 4d 3 /3 + 10d d/3 2 (2) n d = 4 d = 3 d = T (d) Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 17 / 22

51 Cubic terms x i = a i + j b ij (l 1 t j ) + + j k g ijkl (l 1 t j )(l 1 t k )(l 1 t l ) + T (d) = 2d 4 /3 + 22d 3 /3 + 71d 2 /6 + d/6 + 1 n d = 6 d = 5 d = 4 d = / / / / / / / / / / / / / / / /2876 T (d) Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 18 / 22

52 OPP at two loops 1 = D i R i + D i D j R ij + D i D j D k R ijk + ostas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 19 / 22

53 OPP at two loops 1 = D i R i + D i D j R ij + D i D j D k R ijk + Reducible scalar products RSP give rise to terms with higher powers of D i, D RSP D D ostas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 19 / 22

54 OPP at two loops 1 = D i R i + D i D j R ij + D i D j D k R ijk + Reducible scalar products RSP give rise to terms with higher powers of D i, D RSP D D Parametrizing the residue functions with irreducible scalar products ISP R = D ISP ostas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 19 / 22

55 OPP at two loops 1 = D i R i + D i D j R ij + D i D j D k R ijk + Reducible scalar products RSP give rise to terms with higher powers of D i, D RSP D D Parametrizing the residue functions with irreducible scalar products ISP R = D ISP Solving the master equation 1 = 1 ostas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 19 / 22

56 OPP at two loops 1 = D i R i + D i D j R ij + D i D j D k R ijk + Reducible scalar products RSP give rise to terms with higher powers of D i, D RSP D D Parametrizing the residue functions with irreducible scalar products ISP R = D ISP Solving the master equation Rational terms R 1 + R 2? 1 = 1 Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 19 / 22

57 NNLO in the future The NLO revolution The NLO revolution : NLO Drell-Yan [Altarelli, Ellis & Martinelli] Gavin Salam (LPTHE, Paris) pqcd for LHC ICHEP 2010, July / 30 Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 20 / 22

58 NNLO in the future The NLO revolution The NLO revolution : NLO high-p t photoproduction [Aurenche et al] 1988: NLO b b, t t [Nason et al] 1993: dijets, Vj [JETRAD, Giele, Glover & Kosower] Gavin Salam (LPTHE, Paris) pqcd for LHC ICHEP 2010, July / 30 Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 20 / 22

59 NNLO in the future The NLO revolution The NLO revolution : NLO Wb b [MCFM: Ellis & Veseli] 2000: NLO Zb b [MCFM: Campbell & Ellis] 2001: NLO 3j [NLOJet++: Nagy] 2007: NLO t tj [Dittmaier, Uwer & Weinzierl 07] Gavin Salam (LPTHE, Paris) pqcd for LHC ICHEP 2010, July / 30 Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 20 / 22

60 NNLO in the future The NLO revolution The NLO revolution (W/Z+3j, ttbb, ttjj) : NLO W +3j [Rocket: Ellis, Melnikov & Zanderighi] [unitarity] 2009: NLO W +3j [BlackHat: Berger et al] [unitarity] 2009: NLO t tb b [Bredenstein et al] [traditional] 2009: NLO t tb b [HELAC-NLO: Bevilacqua et al] [unitarity] 2009: NLO q q b bb b [Golem: Binoth et al] [traditional] 2010: NLO t tjj [HELAC-NLO: Bevilacqua et al] [unitarity] 2010: NLO Z+3j [BlackHat: Berger et al] [unitarity] Gavin Salam (LPTHE, Paris) pqcd for LHC ICHEP 2010, July / 30 Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 20 / 22

61 NNLO in the future The NLO revolution The NNLO revolution to come ostas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 20 / 22

62 NNLO in the future The NLO revolution The NNLO revolution to come In a few years a new wish list pp t t, pp W + W, pp W /Z + nj, pp H + nj ostas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 20 / 22

63 NNLO in the future The NLO revolution The NNLO revolution to come In a few years a new wish list pp t t, pp W + W, pp W /Z + nj, pp H + nj Virtual amplitudes: Reduction at the integrand level IBP ostas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 20 / 22

64 NNLO in the future The NLO revolution The NNLO revolution to come In a few years a new wish list pp t t, pp W + W, pp W /Z + nj, pp H + nj Virtual amplitudes: Reduction at the integrand level IBP Master Integrals (SecDec) (e.g. Sophia Borowka talk) ostas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 20 / 22

65 NNLO in the future The NLO revolution The NNLO revolution to come In a few years a new wish list pp t t, pp W + W, pp W /Z + nj, pp H + nj Virtual amplitudes: Reduction at the integrand level IBP Master Integrals (SecDec) (e.g. Sophia Borowka talk) Virtual-Real (e.g. Aude Gehrmann-De Ridder talk) I. Bierenbaum, M. Czakon and A. Mitov, Nucl. Phys. B 856 (2012) 228 [arxiv: [hep-ph]]. ostas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 20 / 22

66 NNLO in the future The NLO revolution The NNLO revolution to come In a few years a new wish list pp t t, pp W + W, pp W /Z + nj, pp H + nj Virtual amplitudes: Reduction at the integrand level IBP Master Integrals (SecDec) (e.g. Sophia Borowka talk) Virtual-Real (e.g. Aude Gehrmann-De Ridder talk) I. Bierenbaum, M. Czakon and A. Mitov, Nucl. Phys. B 856 (2012) 228 [arxiv: [hep-ph]]. Real-Real (e.g. Aude Gehrmann-De Ridder talk) M. Czakon, Phys. Lett. B 693 (2010) 259 [arxiv: [hep-ph]]. ostas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 20 / 22

67 HELAC team Current G. Bevilacqua M. Czakon M. Garzelli A. van Hameren A. Kardos J. Malamos C.G. Papadopoulos R. Pittau Z. Trocsanyi M. Worek Contributors A. Kanaki A. Cafarella P. Draggiotis G. Ossola Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 21 / 22

68 Back-up slides For those who are interested in more details... Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 22 / 22

69 Back-up slides For those who are interested in more details have a drink with me. Costas G. Papadopoulos (NCSR-D & CERN) NNLO Corfu 22 / 22