Integrand-Reduction Techniques for NLO Calculations and Beyond
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1 Integrand-Reduction Techniques for NLO Calculations and Beyond Giovanni Ossola New York City College of Technology City University of New York (CUNY) RadCor-LoopFest UCLA, June 15-19, 2015 Based on work with GoSam Collaboration G. Cullen, H. van Deurzen, N. Greiner, G. Heinrich, G. Luisoni, P. Mastrolia, E. Mirabella, T. Peraro, J. Schlenk, J.F. von Soden-Fraunhofen, F. Tramontano and P. Mastrolia, E. Mirabella, T. Peraro Work in progress with: R. Frederix, V. Hirschi, U. Schubert, Z. Zhang Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
2 Introduction Outline Outline of this talk: 1 NLO Automation Recent developments in GoSam 2.0 Progress on Integrand-Reduction Techniques for one-loop multi-leg Ninja, a.k.a. Integrand Reduction via Laurent Expansion Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
3 Introduction Outline Outline of this talk: 1 NLO Automation Recent developments in GoSam 2.0 Progress on Integrand-Reduction Techniques for one-loop multi-leg Ninja, a.k.a. Integrand Reduction via Laurent Expansion 2 Physical applications at NLO precision Higgs Boson production in conjunction with jets and top-quark pairs Interfacing with Monte Carlo tools Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
4 Introduction Outline Outline of this talk: 1 NLO Automation Recent developments in GoSam 2.0 Progress on Integrand-Reduction Techniques for one-loop multi-leg Ninja, a.k.a. Integrand Reduction via Laurent Expansion 2 Physical applications at NLO precision Higgs Boson production in conjunction with jets and top-quark pairs Interfacing with Monte Carlo tools 3 From NLO towards NNLO (and higher orders) A glance beyond GoSam 2.0 Integrand Reduction via Multivariate Polynomial Division The Maximum-Cut theorem Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
5 NLO Automation σ NLO = n ( ) ( dσ B + dσ V + dσ A + dσ R dσ A) 1 n+1 Monte Carlo Tools (MC) One-Loop Programs (OLP) Tree-level Contributions Subtraction Terms Integration over phase-space The values of the Virtual Contributions (at each given phase-space point) Strategies to full NLO automation: MC controls the OLP via Binoth Les Houches Accord interface (BLHA) OLP is fully incorporated within the MC Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
6 The GoSam framework The GoSam Project Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
7 The GoSam framework The GoSam Project GoSam 1.0 GoSam Automated One-Loop Calculations with GoSam Cullen, Greiner, Heinrich, Luisoni, Mastrolia, G.O., Reiter, Tramontano Eur.Phys.J. C72 (2012) 1889 [arxiv: ] GOSAM-2.0: a tool for automated one-loop calculations within the Standard Model and beyond Cullen, van Deurzen, Greiner, Heinrich, Luisoni, Mastrolia, Mirabella, G.O., Peraro, Schlenk, von Soden-Fraunhofen, Tramontano Eur.Phys.J. C74 (2014) 3001 [arxiv: ] Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
8 The GoSam framework Virtual NLO: One-Loop master formula d d q N ( q) D i0 Di1... D n 1 = i d i Box i + i c i Triangle i + i b i Bubble i + i a i Tadpole i + R, 1) Generation: Compute the unintegrated amplitudes for all diagrams 2) Reduction: Extract all coefficients and rational terms 3) Master Integrals: Calculate the Master Integrals (scalar integrals) and combine with the coefficients There are several techniques available for Generation+Reduction and available codes to compute the one-loop Scalar Integrals Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
9 The GoSam framework Virtual corrections with GoSam 1 Algebraic Generation Amplitudes generated with Feynman diagrams Optimization: grouping of diagrams, smart caching Algebra in dimension d, different schemes Suited for QCD, EW, effective Higgs coupling and BSM models ( ) GoSam employs a Python wrapper which: generates analytic integrands from Feynman diagrams using QGRAF manipulates and simplifies them with FORM writes them into FORTRAN95 code for the reduction Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
10 The GoSam framework Virtual corrections with GoSam 1 Algebraic Generation Amplitudes generated with Feynman diagrams Suited for QCD, EW, effective Higgs coupling and BSM models ( ) 2 Flexibility in the Reduction Different reduction algorithms available at run-time Integrand-Level and/or Tensorial Reduction available Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
11 The GoSam framework Virtual corrections with GoSam 1 Algebraic Generation Amplitudes generated with Feynman diagrams Suited for QCD, EW, effective Higgs coupling and BSM models ( ) 2 Flexibility in the Reduction Different reduction algorithms available at run-time Integrand-Level and/or Tensorial Reduction available 3 Selection of codes for the evaluation of the Master Integrals OneLoop (van Hameren); QCDLoop (Ellis, Zanderighi) Golem95C (Binoth et al.); LoopTools (Hahn et al.) Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
12 The GoSam framework Virtual corrections with GoSam 1 Algebraic Generation Amplitudes generated with Feynman diagrams Suited for QCD, EW, effective Higgs coupling and BSM models ( ) 2 Flexibility in the Reduction Different reduction algorithms available at run-time Integrand-Level and/or Tensorial Reduction available 3 Selection of codes for the evaluation of the Master Integrals OneLoop (van Hameren); QCDLoop (Ellis, Zanderighi) Golem95C (Binoth et al.); LoopTools (Hahn et al.) 4 Fully interfaced within several Monte Carlo frameworks Modular Structure Ability to incorporate new ideas and techniques The GoSam framework is in continuous evolution Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
13 The GoSam framework Reduction algorithms within GoSam Ninja Default in GoSam 2.0, more Stable and Fast Integrand-Level Reduction + Laurent Expansion van Deurzen, Luisoni, Mastrolia, Mirabella, G.O., Peraro Golem95 Default Rescue System Tensorial Reduction Binoth, Guillet, Heinrich, Pilon, Reiter, von Soden-Fraunhofen Samurai Default in GoSam 1.0 d dimensional Integrand-Level Reduction Mastrolia, G.O., Reiter, Tramontano Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
14 The Ninja algorithm One-loop Integrand Decomposition N ( q) = + n 1 i<<m n 1 i<<k ijklm ( q) ijk ( q) n 1 h i,j,k,l,m n 1 h i,j,k D h + n 1 D h + i<j n 1 i<<l ij ( q) ijkl ( q) n 1 h i,j n 1 h i,j,k,l D h + n 1 n 1 D h + i ( q) i h i D h #1 What is the Functional Form of all the Residues ij...kl? They are polynomials in the components of q They have a universal, process-independent form parametrised by process-dependent coefficients #2 Extract all coefficients by sampling on the kinematic cuts Polynomial fitting allows to find all coefficients ( fit-on-the-cut approach) we only need Numerator Function evaluated on the cuts GO, Papadopoulos, Pittau (2007); Ellis, Giele, Kunszt, Melnikov (2008) Mastrolia, GO, Reiter, Tramontano (2010); Mastrolia, Mirabella, GO, Peraro (2012) Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
15 The Ninja algorithm One-loop Integrand Decomposition Ninja-style N ( q) = + n 1 i<<m n 1 i<<k ijklm ( q) ijk ( q) n 1 h i,j,k,l,m n 1 h i,j,k D h + n 1 D h + i<j n 1 i<<l ij ( q) ijkl ( q) n 1 h i,j n 1 h i,j,k,l D h + n 1 n 1 D h + i ( q) i h i D h #1 What is the Functional Form of all the Residues ij...kl? They are polynomials in the components of q They have a universal, process-independent form parametrised by process-dependent coefficients #2 Extraction of coefficients via Laurent Expansion If the numerator functions are known analytically, all coefficients can be extracted by performing a Laurent expansion with respect to the free parameters which appear in the solutions of the cuts Ninja Mastrolia, Mirabella, Peraro (2012) Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
16 The Ninja algorithm Integrand Reduction Ninja-style Mastrolia, Mirabella, Peraro (2012) The coefficients of the integrand can be extracted by performing a Laurent expansion with respect to one of the free parameters which appear in the solutions of the cut Advantages: Corrections at the coefficient level replace subtractions at the integrand level Lighter reduction algorithm where fewer coefficients are computed Quadruple-cut decoupled from triple-, double-, and single-cut. No more sampling on the cuts (no need of finding solutions for the cuts) Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
17 The Ninja algorithm Integrand Reduction Ninja-style Mastrolia, Mirabella, Peraro (2012) van Deurzen, Luisoni, Mastrolia, Mirabella, G.O., Peraro (2014) This method has been implemented in the library Ninja and interfaced with GoSam providing an improvement in the computational performance Series of selected processes involving massive particles, from six to eight legs Timings refer to full color-summed and helicity-summed amplitudes Intel Core i GHz compiled with ifort (*) Intel(R) Xeon(R) CPU 2.00 GHz compiled with gfortran Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
18 The Ninja algorithm Integrand Reduction Ninja-style Mastrolia, Mirabella, Peraro (2012) van Deurzen, Luisoni, Mastrolia, Mirabella, G.O., Peraro (2014) The coefficients of the integrand can be extracted by performing a Laurent expansion with respect to one of the free parameters which appear in the solutions of the cut This method has been implemented in the library Ninja and interfaced with GoSam providing an improvement in the computational performance Standalone and public version of the Ninja library Peraro (2014) Interfaced within FormCalc 8.4: FormCalc generates all numerator expansions needed by Ninja as an independent subroutine Hahn et al. (2014) Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
19 The Ninja algorithm Example: One-loop Bubbles via Laurent Expansion Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
20 The Ninja algorithm Example: One-loop Bubbles via Laurent Expansion Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
21 NLO Applications with GoSam GoSam NLO Pheno Results N. Greiner, S. Hoeche, G. Luisoni, M. Schoenherr, V. Yundin, J. Winter, Phenomenological analysis of Higgs boson production through gluon fusion in association with jets, arxiv: G. Luisoni, C. Oleari, F. Tramontano, Wbbj production at NLO with POWHEG+MiNLO, JHEP 1504 (2015) 161 M. J. Dolan, C. Englert, N. Greiner, M. Spannowsky, Further on up the road: hhjj production at the LHC, Phys.Rev.Lett. 112 (2014) G. Heinrich, A. Maier, R. Nisius, J. Schlenk, J. Winter, NLO QCD corrections to WWbb production with leptonic decays in the light of top quark mass and asymmetry measurements, JHEP 1406 (2014) 158 T. Gehrmann, N. Greiner, and G. Heinrich, Precise QCD predictions for the production of a photon pair in association with two jets, Phys.Rev.Lett. 111 (2013) N. Greiner, G. Heinrich, J. Reichel, and J. F. von Soden-Fraunhofen, NLO QCD corrections to diphoton plus jet production through graviton exchange, JHEP 1311 (2013) 028 H. van Deurzen, G. Luisoni, P. Mastrolia, E. Mirabella, G.O. and T. Peraro, NLO QCD corrections to Higgs boson production in association with a top quark pair and a jet, Phys.Rev.Lett. 111 (2013) G. Cullen, H. van Deurzen, N. Greiner, G. Luisoni, P. Mastrolia, E. Mirabella, G.O., T. Peraro, and F. Tramontano, NLO QCD corrections to Higgs boson production plus three jets in gluon fusion, Phys.Rev.Lett. 111 (2013) S. Hoeche, J. Huang, G. Luisoni, M. Schoenherr and J. Winter, Zero and one jet combined NLO analysis of the top quark forward-backward asymmetry, Phys.Rev. D88 (2013) G. Luisoni, P. Nason, C. Oleari and F. Tramontano, HW/HZ + 0 and 1 jet at NLO with the POWHEG BOX interfaced to GoSam and their merging within MiNLO, JHEP 1310 (2013) 083 M. Chiesa, G. Montagna, L. Barze, M. Moretti, O. Nicrosini, F. Piccinini and F. Tramontano, Electroweak Sudakov Corrections to New Physics Searches at the CERN LHC, Phys.Rev.Lett. 111 (2013) T. Gehrmann, N. Greiner, and G. Heinrich, Photon isolation effects at NLO in gamma gamma + jet final states in hadronic collisions, JHEP 1306, 058 (2013) H. van Deurzen, N. Greiner, G. Luisoni, P. Mastrolia, E. Mirabella, G.O., T. Peraro, J. F. von Soden-Fraunhofen, and F. Tramontano, NLO QCD corrections to the production of Higgs plus two jets at the LHC, Phys. Lett. B 721, 74 (2013) G. Cullen, N. Greiner, and G. Heinrich, Susy-QCD corrections to neutralino pair production in association with a jet, Eur. Phys. J. C 73, 2388 (2013) N. Greiner, G. Heinrich, P. Mastrolia, G. O., T. Reiter and F. Tramontano, NLO QCD corrections to the production of W+ W- plus two jets at the LHC, Phys. Lett. B 713, 277 (2012) Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
22 NLO Applications with GoSam Interfaces with external MC GoSam + MadGraph+MadDipole+MadEvent ad-hoc interface [Greiner] GoSam + Sherpa fully automated [Luisoni, Schönherr, Tramontano] GoSam + Powheg fully automated apart from phase space generation [Luisoni, Nason, Oleari, Tramontano] GoSam + Herwig++/Matchbox fully automated [Bellm, Gieseke, Greiner, Heinrich, Plätzer, Reuschle, von Soden-Fraunhofen] GoSam + mg5 amc@nlo fully automated [van Deurzen, Frederix, Hirschi, Luisoni, Mastrolia, G.O.] Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
23 NLO Applications with GoSam pp H+Jets in Gluon Fusion (GF) Large Top-Mass Approximation (m t ) : the Higgs coupling to gluons, mediated by a top-quark loop, can be described by an effective operator new Feynman rules: vertices involving the Higgs field and up to four gluons Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
24 NLO Applications with GoSam Higher Rank Extension XSamurai Theoretical Challenge: Effective Hgg coupling leads to numerators with rank r larger than the number n of the denominators, i.e. r n + 1 The form of the integrand-level identity for the numerator was extended Mastrolia, Mirabella, Peraro (2012) The decomposition of any one-loop n-point amplitude in terms of master integrals (MIs) acquires new contributions M one-loop n = A n + δa n A n corresponds to the standard decomposition (r n) δa n enters only if r n + 1 The extended integrand decomposition has been implemented in the samurai library Mastrolia, GO, Reiter, Tramontano (2010) van Deurzen et al. (2012) Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
25 NLO Applications with GoSam Higher Rank Extension XSamurai Theoretical Challenge: Effective Hgg coupling leads to numerators with rank r larger than the number n of the denominators, i.e. r n + 1 The form of the integrand-level identity for the numerator was extended Mastrolia, Mirabella, Peraro (2012) The extended integrand decomposition has been implemented in the samurai library Mastrolia, GO, Reiter, Tramontano (2010) van Deurzen et al. (2012) Also the tensorial reduction of Golem95C was extended to support higher ranks applied in BSM physics (i.e. spin two particles) Guillet, Heinrich, von Soden-Fraunhofen (2014) Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
![4 [Campbell, Ellis, Williams] van Deurzen, Greiner, Luisoni, Mastrolia, Mirabella, G.O.](/docs-images/83/87075996/images/26-1.jpg ", Peraro, von Soden-Fraunhofen, Tramontano (2013) (also appeared in Handbook of LHC Higgs Cross")
26 NLO Applications with GoSam Warm-up: Higgs + 2 jets in NLO Results obtained with GoSam+Sherpa Agreement with MCFM v6.4 [Campbell, Ellis, Williams] van Deurzen, Greiner, Luisoni, Mastrolia, Mirabella, G.O., Peraro, von Soden-Fraunhofen, Tramontano (2013) (also appeared in Handbook of LHC Higgs Cross Sections: 3. Higgs Properties ) Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
27 NLO Applications with GoSam from H+2 to H+3 jets Huge Theoretical and Computational Challenges More than 10,000 diagrams Higher-Rank terms Rank-7 hexagons The calculation of H + 3 jets required lots of work and theoretical progress Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
28 NLO Applications with GoSam from H+2 to H+3 jets Improvements in the reduction Upgrade Reduction Algorithms to include Higher-Rank Terms (effective couplings) Implementation of the corresponding Higher-Rank Master Integrals Ninja reduction based on Integrand reduction via Laurent expansion Interface for Ninja implemented in GoSam Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
29 NLO Applications with GoSam from H+2 to H+3 jets Improvements in the reduction Improvements in the code generation Diagrams with identical sets of denominators are summed algebraically during generation New optimization strategy via FORM 4.0 Numerical polarization vectors Reduced code size Parallelization of diagram generation Reduction of generation time Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
30 NLO Applications with GoSam from H+2 to H+3 jets Improvements in the reduction Improvements in the code generation Improvements in the Monte Carlo see talk of Nicolas Greiner In the original paper, all cross sections obtained with a hybrid setup GoSam+ Sherpa for Born and of the Virtual contributions MadGraph+MadDipole+MadEvent for reals/subtraction/integrated dipoles Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
31 NLO Applications with GoSam from H+2 to H+3 jets Improvements in the reduction Improvements in the code generation Improvements in the Monte Carlo see talk of Nicolas Greiner Finally... a first result for Higgs + 3 jets in GF dσ/dp T,j[pb/GeV] 10 1 LHC 8 TeV cteq6me pdf 10 2 anti-kt: R=0.5, pt > 20 GeV, η < st jet LO st jet NLO 2 nd jet LO 2 nd jet NLO rd jet LO 3 rd jet NLO p T,j[GeV] NLO/LO dσ/dpt,h[pb/gev] 10 2 LO NLO 10 3 LHC 8 TeV cteq6me pdf anti-kt: R=0.5, pt > 20GeV, η < pt,h[gev] Cullen, van Deurzen, Greiner, Luisoni, Mastrolia, Mirabella, G.O., Peraro, Tramontano (2013) Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
32 NLO Applications with GoSam Higgs + 3 jets NLO: MC analyses An updated analysis appeared in Physics at TeV Colliders: Standard Model Working Group Report [arxiv: ] dσ/dpt,h[pb/gev] Ratio H + 3 jets: Higgs transverse momentum LHC 8 TeV CT10nlo pdf anti-kt: R=0.4, pt > 30GeV, η < 4.4 H+2 jets NLO H+3 jets NLO pt,h[gev] PRELIMINARY dσ/dpt,h[pb/gev] NLO/LO H + 3 jets: Higgs transverse momentum LO (cteq6l1 PDFs) NLO (CT10nlo PDFs) LHC 8 TeV anti-kt: R=0.4, pt > 30GeV, η < 4.4 PRELIMINARY pt,h[gev] Cullen, van Deurzen, Greiner, Huston, Luisoni, Mastrolia, Mirabella, G.O., Peraro,Tramontano, Yundin, Winter (2014) Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
33 NLO Applications with GoSam Higgs + 3 jets NLO: MC analyses New results with GoSam + Sherpa!! Greiner, Hoeche, Luisoni, Schoenherr, Yundin, Winter, Phenomenological analysis of Higgs boson production through gluon fusion in association with jets, Talk of Nicolas Greiner on Monday arxiv: Ratio σtot [pb] Total inclusive cross section with gluon fusion cuts at 8 TeV H+1 jet LO H+1 jet NLO H+2 jets LO H+2 jets NLO r 2/1 r 3/2 r 4/3 H+3 jets LO H+3 jets NLO GoSam+Sherpa αs(ĥ T 2 ) 3 αs(mh) 2 αs(ĥ T 2 ) 5 αs(mh)5 Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
![NLO Applications with GoSam pp Ht t + 1 jet @ NLO First Application of GoSam/Ninja + Sherpa σ [pb] 0.18 0.16 0.](/docs-images/83/87075996/images/34-0.jpg "14 3 GA = m T T,t m T,t H p T = T,i m T,H + p T,j LO NLO LO NLO µ 0 µ 0 µ 0 µ 0 = 2xGA T = 2xGA T = H T = H T 0.12 0.1 0.")
34 NLO Applications with GoSam pp Ht t + 1 NLO First Application of GoSam/Ninja + Sherpa σ [pb] GA = m T T,t m T,t H p T = T,i m T,H + p T,j LO NLO LO NLO µ 0 µ 0 µ 0 µ 0 = 2xGA T = 2xGA T = H T = H T Two different mass scales: Higgs and Top 51 hexagons in the gluon-gluon channel Timing/PS-point: qq 0.2 sec gg 2.5 sec GoSam/Ninja+Sherpa µ/µ 0 van Deurzen, Luisoni, Mastrolia, Mirabella, G.O., Peraro (2013) Updated and improved analysis under way Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
35 NLO Applications with GoSam pp t NLO First Application of GoSam + mg5 amc@nlo dσ/dp T top (pb/gev) 1e-05 1e-06 NLO GoSam+MG5aMC LO MG5aMC NLO/LO p T top (GeV) van Deurzen, Frederix, Hirschi, Luisoni, Mastrolia, G.O. (work in progress) Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
36 NLO Applications with GoSam pp t NLO First Application of GoSam + mg5 amc@nlo dσ/dp T top (pb/gev) 1e-05 1e-06 GoSam+MG5aMC GoSam+Sherpa MG5aMC/Sherpa p T top (GeV) van Deurzen, Frederix, Hirschi, Luisoni, Mastrolia, G.O. (work in progress) Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
37 NNLO and beyond what about higher loops? Problem: at two loops (and higher), it is not available a Standard Complete Basis of Master Integrals Gluza, Kosower, Kajda; Feng, Zeng, Huang, Zhou; Søgaard, Zhang; von Manteuffel, Schabinger; [... ] The most common (and successful) approach to NNLO relies on: Amplitude generation via Feynman diagrams Reduction to a minimal set of MIs using Integration-by-Parts Identities Computation of the MIs (analytic or numerical) Differential Equations Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
38 NNLO and beyond what about higher loops? Problem: at two loops (and higher), it is not available a Standard Complete Basis of Master Integrals The most common (and successful) approach to NNLO relies on: Amplitude generation via Feynman diagrams Reduction to a minimal set of MIs using Integration-by-Parts Identities Computation of the MIs (analytic or numerical) Differential Equations Plenty of progress in the field (several talks at UCLARadCor-LoopFest) This is the Time of NNLO calculations Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
39 NNLO and beyond what about higher loops? Problem: at two loops (and higher), it is not available a Standard Complete Basis of Master Integrals The most common (and successful) approach to NNLO relies on: Amplitude generation via Feynman diagrams Reduction to a minimal set of MIs using Integration-by-Parts Identities Computation of the MIs (analytic or numerical) Differential Equations Plenty of progress in the field (several talks at UCLARadCor-LoopFest) The path to full automation is still long and difficult Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
40 NNLO and beyond A glimpse at the use of GoSam beyond NLO 1 GoSam 2.0 has been already used within NNLO calculations, for the evaluation of the real-virtual contributions: V. Del Duca, C. Duhr, G. Somogyi, F. Tramontano, Z. Trocsanyi, Higgs boson decay into b-quarks at NNLO accuracy, JHEP 1504 (2015) 036 J. Gao, H.X. Zhu, Top-quark forward-backward asymmetry in e+eannihilation at NNLO in QCD, Phys.Rev.Lett. 113 (2014) J. Gao, H.X. Zhu, Electroweak prodution of top-quark pairs in e+eannihilation at NNLO in QCD: the vector contributions, Phys.Rev. D90 (2014) Towards GoSam 2-Loops Greiner, Heinrich, Jones, Kerner, Luisoni, Mastrolia, Schlenk, Zirke Extension of the GoSam generator to produce all two-loop Feynman diagrams for any proceess: algebra done automatically (FORM), uses projectors on tensor structures, depicts all contributing diagrams as output on file. Interfaces to Reduze, LiteRed, FIRE in progress. Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
41 NNLO and beyond Approximate NNLO: Hard Functions at NLO with GoSam Broggio, Ferroglia, Pecjak, Yang The partonic cross section for top pair + Higgs production receives potentially large corrections from soft gluon emission The resummation of soft emission corrections can be carried out by means of effective field theory methods hard functions and soft functions The calculation of the NLO hard function requires the evaluation of one loop amplitudes obtained separating out the various color components By default GoSam provides squared amplitudes summed over colors to build the hard functions we need to combine color decomposed (complex) amplitudes This required ad-hoc modifications of the GoSam code [see talk of A. Ferroglia at SCET 2015] Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
42 NNLO and beyond Integrand-Level Reduction at Higher Orders Can we build alternative approaches for multi-loop reduction by exploiting the multi-pole structure of scattering amplitudes? To pursue this challenging task, we need a better understanding of integrand decomposition and unitarity beyond one loop Multiloop Integrand Reduction Mastrolia, G.O. (2011); Badger, Frellersvig, Kleiss, Malamos, Papadopoulos, Verheyen; Zhang; Mastrolia, G.O., Mirabella, Peraro; Feng, Huang. see also Maximal Unitarity Talk of K. Larsen Kosower, Larsen (2011); Johansson, Kosower, Larsen; Søgaard, Zhang; Johansson, Kosower, Larsen, Søgaard. Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
43 NNLO and beyond Integrand-Level Reduction at Higher Orders Let s consider a two-loop integral with n denominators: N(q, k) dq dk D 1 D 2... D n As in the one-loop case, we want to construct an identity for the integrands: n n n n N(q, k) = i1,...,i 8 (q, k) D h i1,i 2 (q, k) i 1 <<i 8 h i 1,...,i 8 i 1 <<i 2 A(q, k) = n i1,...,i8(q, k) + D i1 D i2... D i8 i 1 <<i 8 n i1,...,i7(q, k) D i1 D i2... D i7 i 1 <<i 7 Which terms appear in the above expressions? What is the general form of the residues i1,...,i m? h i 1,i 2 D h n i1,i2(q, k) D i1 D i2 i 1 <<i 2 Can we detect the Master Integrals at the Integrand Level? Work in collaboration with P. Mastrolia, E. Mirabella, T. Peraro, U. Schubert Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
44 NNLO and beyond Integrand Reduction via Multivariate Polynomial Division Using the language of Algebraic Geometry... Set of Multivariate Polynomials {ω i (z)} where z = (z 1, z 2,...) { } Ideal J = ω 1 (z),..., ω s (z) J = i h i (z) ω i (z) Multivariate Polynomial Division of a function F (z) modulo {ω 1,..., ω s } F (z) = i h i (z) ω i (z) + R(z) h i (z) and R(z) are not unique Gröbner Basis {g 1 (z),..., g r (z)} It exists (Buchberger s algorithm) and generates the ideal J Provides a unique R(z) Hilberts Nullstellensatz V (J ) set of common zeros of J Weak Nullstellensatz: V (J ) = O 1 J Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
45 NNLO and beyond Integrand Reduction via Multivariate Polynomial Division...Integrand Reduction via Multivariate Polynomial Division I i1 i n = Ideal: J i1 i n = D i1,..., D in N i1 i n (z) D i1 (z) D in (z) Gröbner basis G i1 i n : same zero as the denominators Multivariate division of N i1 i n modulo G i1 i n N i1 i n (z) = Γ i1 i n + i1 i n (z) The quotient Γ i1 i n can be expressed in terms of denominators n Γ i1 i n = N i1 i κ 1i κ+1 i n (z)d iκ (z) κ=1 Which provides the Recursive Formula n I i1 i n = I i1 i κ 1i κ+1i n + i 1 i n D i1 D in κ=1 Giovanni Ossola (City Tech) RadCor-LoopFest Mastrolia, 2015 G.O., Mirabella, JunePeraro 16, 2015 (2012) 28 / 31
46 NNLO and beyond Integrand Reduction via Multivariate Polynomial Division Integrand Reduction at Arbitrary Loops: a Theorem Let s look at the on-shell conditions, and impose D 1 = D 2 =... = D n = 0 1) There are no solutions the diagram is reducible The integrand with n denominators can be expressed in terms of integrands with (n 1) denominators The diagram is fully reducible in terms of lower point functions i.e. six-point functions at one-loop Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
47 NNLO and beyond Integrand Reduction via Multivariate Polynomial Division Integrand Reduction at Arbitrary Loops: a Theorem Let s look at the on-shell conditions, and impose D 1 = D 2 =... = D n = 0 1) There are no solutions reducible 2) The cut has solutions there is a residue Divide the numerator modulo the Gröbner basis of the n-ple cut (a set of polynomials vanishing on the same on-shell cuts of the denominators) The remainder of the division is the residue of the n-ple cut. The quotients generate integrands with (n 1) denominators. Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
48 NNLO and beyond Integrand Reduction via Multivariate Polynomial Division Integrand Reduction at Arbitrary Loops: a Theorem Let s look at the on-shell conditions, and impose D 1 = D 2 =... = D n = 0 1) There are no solutions reducible 2) The cut has solutions residue 3) Finite number of solutions n s Maximum Cut First term in the integrand decomposition i.e. four-point function at one-loop in 4-dim its residue is a univariate polynomial parametrized by n s coefficients the corresponding residue can always be reconstructed at the cut the residue is determined as in the previous case, via MPD Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
49 NNLO and beyond Integrand Reduction via Multivariate Polynomial Division Integrand Reduction at Arbitrary Loops: a Theorem Let s look at the on-shell conditions, and impose D 1 = D 2 =... = D n = 0 1) There are no solutions reducible 2) The cut has solutions residue 3) Finite number of solutions n s Maximum Cut Mastrolia, G.O., Mirabella, Peraro Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
50 NNLO and beyond Integrand Reduction via Multivariate Polynomial Division Examples of Applications of MPD 1 Reconstruct all residues in the OPP and d-dimensional One-Loop Integrand Decomposition N 5 i1 i A = k D 0... D n 1 D i1 D ik k=1 {i 1,,i k } Take a rank-k polynomial in z (x 1, x 2, x 3, x 4, µ 2 ) as N (z) N (z) = j α j z j1 1 z j2 2 z j3 3 z j4 4 z j5 5 Apply the recursion formula: n I i1 i n = I i1 i κ 1i κ+1i n + i 1 i n D i1 D in κ=1 Read the residues: i1 i 5 = c 0, i1 i 4 = c 0 + c 1 x 4 + µ 2 (c 2 + c 3 x 4 + µ 2 c 4 ),... Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
51 NNLO and beyond Integrand Reduction via Multivariate Polynomial Division Examples of Applications of MPD 1 Reconstruct all residues in the OPP and d-dimensional One-Loop Integrand Decomposition 2 Examples of Two-loop topologies in N = 4 SYM Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
52 NNLO and beyond Integrand Reduction via Multivariate Polynomial Division Examples of Applications of MPD 1 Reconstruct all residues in the OPP and d-dimensional One-Loop Integrand Decomposition 2 Examples of Two-loop topologies in N = 4 SYM 3 Two-loop diagrams in QED and QCD processes Two-loop QED corrections to the photon self energy Two-loop diagrams entering the QCD corrections to gg H in the heavy top mass approximation Mastrolia, Mirabella, GO, Peraro Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
53 Conclusions/Outlook Conclusions/Outlook At present, there is a great variety of methods available for NLO scattering amplitudes Multi-Purpose NLO Computational Tools Generalized Unitarity, Integrand Reduction methods, Recursion Relations, Improved Tensorial techniques revolutionized the calculation of NLO multi-leg amplitudes Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
54 Conclusions/Outlook Conclusions/Outlook At present, there is a great variety of methods available for NLO scattering amplitudes Multi-Purpose NLO Computational Tools Generalized Unitarity, Integrand Reduction methods, Recursion Relations, Improved Tensorial techniques revolutionized the calculation of NLO multi-leg amplitudes OLP have been incorporated in the MC tools to generate theoretical prediction for a large variety of processes: high multiplicities, several scales, effective vertices... NLO automation has been fully achieved Several NLO analyses under way Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
55 Conclusions/Outlook Conclusions/Outlook At present, there is a great variety of methods available for NLO scattering amplitudes Multi-Purpose NLO Computational Tools Generalized Unitarity, Integrand Reduction methods, Recursion Relations, Improved Tensorial techniques revolutionized the calculation of NLO multi-leg amplitudes OLP have been incorporated in the MC tools to generate theoretical prediction for a large variety of processes: high multiplicities, several scales, effective vertices... NLO automation has been fully achieved Several NLO analyses under way In this talk, I described some applications of Integrand Reduction methods At NLO, Integrand reduction via Laurent (available with Ninja) provided a stable and reliable reduction for GoSam GoSam is in the process of being upgraded to be used in NNLO calculations (generation of NNLO virtual, numerical evaluation of real-virtual,... ) Integrand Reduction can be extended at Arbitrary Loops (Maximum-Cut Theorem). Not ready yet to be competitive beyond NLO. Will we ever achieve at NNLO the same degree of automation of the NLO? Which new ideas will play a role in the process? Giovanni Ossola (City Tech) RadCor-LoopFest 2015 June 16, / 31
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