. Mao Zeng 29 October 2018 Institute for Theoretical Physics, ETH Zurich . 1 The high-multiplicity frontier for two-loop QCD
• Background • Numerical unitarity for 2-loop amplitudes • Difgerential equations at high multiplicity • Future outlook 2 Outline
• Background • Numerical unitarity for 2-loop amplitudes • Difgerential equations at high multiplicity • Future outlook 2 Outline
• Background • Numerical unitarity for 2-loop amplitudes • Difgerential equations at high multiplicity • Future outlook 2 Outline
• Background • Numerical unitarity for 2-loop amplitudes • Difgerential equations at high multiplicity • Future outlook 2 Outline
. . Phys. Rev. Lett. 119, 142001, arXiv:1703.05273, S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, B. Page, MZ . . Phys. Rev. D. 97, 116014, arXiv:1712.03946, S. Abreu, F. Febres Cordero, H. Ita, B. Page, MZ . . arXiv:1807.11522, Samuel Abreu, Ben Page, MZ 3 Main references
. Background
• Perturbative QCD essential for predictions (PDFs, fixed-order, resummation / parton showers) • NNLO needed for percent-level accuracy. explosion of 2 2 calculations. (amplitudes + subtractions) • Beginning to break the 2 3 barrier! 4 Precision QCD • ∼ 140 ħb − 1 of data from LHC Run 2. ⇒ Precision measurements and BSM searches. . =
• Perturbative QCD essential for predictions (PDFs, fixed-order, resummation / parton showers) • NNLO needed for percent-level accuracy. explosion of 2 2 calculations. (amplitudes + subtractions) • Beginning to break the 2 3 barrier! 4 Precision QCD • ∼ 140 ħb − 1 of data from LHC Run 2. ⇒ Precision measurements and BSM searches. . =
• Perturbative QCD essential for predictions (PDFs, fixed-order, resummation / parton showers) • NNLO needed for percent-level accuracy. (amplitudes + subtractions) • Beginning to break the 2 3 barrier! 4 Precision QCD • ∼ 140 ħb − 1 of data from LHC Run 2. ⇒ Precision measurements and BSM searches. . = → explosion of 2 → 2 calculations.
• Perturbative QCD essential for predictions (PDFs, fixed-order, resummation / parton showers) • NNLO needed for percent-level accuracy. (amplitudes + subtractions) 4 Precision QCD • ∼ 140 ħb − 1 of data from LHC Run 2. ⇒ Precision measurements and BSM searches. . = → explosion of 2 → 2 calculations. • Beginning to break the 2 → 3 barrier!
production. 5 NNLO 2 → 3 processes • pp → 3 j : constrains strong coupling constant α s . • pp → H + 2 j : gluon-fusion background for VBF Higgs g g m t → ∞ H t g g • Many more: V + 2 j , V + V ′ + j , t ¯ t + j . . .
• Master integrals: analytic / numerical evaluation • Phenomenology: need sophisticated IR subtraction. • Loop integrand: too many Feynman diagrams • Integral reduction / IBP: explosion of analytic complexity, 5 kinematic scales Degree- d polynomial in n variables: d n n terms. 6 Challenges for 2 → 3 at two loops
• Phenomenology: need sophisticated IR subtraction. • Loop integrand: too many Feynman diagrams • Integral reduction / IBP: explosion of analytic Degree- d polynomial in n variables: n terms. • Master integrals: analytic / numerical evaluation 6 Challenges for 2 → 3 at two loops complexity, ≥ 5 kinematic scales ( ) d + n
• Loop integrand: too many Feynman diagrams • Integral reduction / IBP: explosion of analytic Degree- d polynomial in n variables: n terms. • Phenomenology: need sophisticated IR subtraction. 6 Challenges for 2 → 3 at two loops complexity, ≥ 5 kinematic scales ( ) d + n • Master integrals: analytic / numerical evaluation
• Loop integrand: too many Feynman diagrams • Integral reduction / IBP: explosion of analytic Degree- d polynomial in n variables: n terms. • Phenomenology: need sophisticated IR subtraction. 6 Challenges for 2 → 3 at two loops complexity, ≥ 5 kinematic scales ( ) d + n • Master integrals: analytic / numerical evaluation
. Numerical unitarity for 2-loop amplitudes
7 Example: NLO pp high-multiplicity limit! Polynomial complexity, faster than analytic results in [Bern, Dixon, Febres Cordero, Hoeche, Ita, Kosower, Maitre, Ozeren, 2013] 5 j (BlackHat & Sherpa) . l 5 j W Madgraph, NJet, OpenLoops, Recola … Hugely successful at one loop, "NLO revolution". BlackHat, GoSam, HELAC-1Loop/CutTools, Forde, Ita, Kosower, Maitre, 2008 … Berger, Bern, Dixon, Febres Cordero, Giele, Kunszt, Melnikov, 2008 Ellis, Giele, Kunszt, 2007 Ossola, Papadopoulos, Pittau, 2006 Figure 1: arXiv:0803.4180 Numerical unitarity: one loop
Hugely successful at one loop, "NLO revolution". Figure 1: arXiv:0803.4180 Ossola, Papadopoulos, Pittau, 2006 Ellis, Giele, Kunszt, 2007 Giele, Kunszt, Melnikov, 2008 Berger, Bern, Dixon, Febres Cordero, Forde, Ita, Kosower, Maitre, 2008 … BlackHat, GoSam, HELAC-1Loop/CutTools, Madgraph, NJet, OpenLoops, Recola … [Bern, Dixon, Febres Cordero, Hoeche, Ita, Kosower, Maitre, Ozeren, 2013] Polynomial complexity, faster than analytic results in high-multiplicity limit! 7 Numerical unitarity: one loop Example: NLO pp → W + 5 j → l ¯ ν + 5 j (BlackHat & Sherpa) .
from discrete Fourier • Integrand decomposition (ansatz): Ossola-Papadopoulos-Pittau. Integrand = scalar masters + surface / spurious terms • Fixing coeffjcients in decomposition: On cut surface, integrand factorizes into tree amplitudes (Berends-Giele recursion) Figure 2: arXiv:0803.4180 Fix n coeffjcients from n sample points. Inversion of linear system transform 8 Overview of one-loop numerical unitarity
from discrete Fourier • Integrand decomposition (ansatz): Ossola-Papadopoulos-Pittau. Integrand = scalar masters + surface / spurious terms integrand factorizes into tree amplitudes (Berends-Giele recursion) Figure 2: arXiv:0803.4180 Fix n coeffjcients from n sample points. Inversion of linear system transform 8 Overview of one-loop numerical unitarity • Fixing coeffjcients in decomposition: On cut surface,
• Integrand decomposition (ansatz): Ossola-Papadopoulos-Pittau. Integrand = scalar masters + surface / spurious terms integrand factorizes into tree amplitudes (Berends-Giele recursion) Figure 2: arXiv:0803.4180 Fix n coeffjcients from n sample points. Inversion of linear system transform 8 Overview of one-loop numerical unitarity • Fixing coeffjcients in decomposition: On cut surface, from discrete Fourier
• High-precision floating point for direct calculation 3. Effjcient and stable numerical fitting of integral 1. OPP-like minimal ansatz: Masters + surface terms. • Produce master coeffjcients w/o external IBP reduction • No doubled propagators except in a few topologies subleading poles: [Abreu, Frebres Cordero, Ita, Jaquier, Page, 2017] 2. Versatility: Berends-Giele recursion allows any vertices coeffjcients (top to bottom) & regulator dependence • Finite-field arithmetic for functional reconstruction 9 Design goals of 2-loop numerical unitarity
• High-precision floating point for direct calculation 3. Effjcient and stable numerical fitting of integral 1. OPP-like minimal ansatz: Masters + surface terms. • Produce master coeffjcients w/o external IBP reduction • No doubled propagators except in a few topologies subleading poles: [Abreu, Frebres Cordero, Ita, Jaquier, Page, 2017] 2. Versatility: Berends-Giele recursion allows any vertices coeffjcients (top to bottom) & regulator dependence • Finite-field arithmetic for functional reconstruction 9 Design goals of 2-loop numerical unitarity
• High-precision floating point for direct calculation 3. Effjcient and stable numerical fitting of integral 1. OPP-like minimal ansatz: Masters + surface terms. • Produce master coeffjcients w/o external IBP reduction • No doubled propagators except in a few topologies subleading poles: [Abreu, Frebres Cordero, Ita, Jaquier, Page, 2017] 2. Versatility: Berends-Giele recursion allows any vertices coeffjcients (top to bottom) & regulator dependence • Finite-field arithmetic for functional reconstruction 9 Design goals of 2-loop numerical unitarity
• High-precision floating point for direct calculation 3. Effjcient and stable numerical fitting of integral 1. OPP-like minimal ansatz: Masters + surface terms. • Produce master coeffjcients w/o external IBP reduction • No doubled propagators except in a few topologies subleading poles: [Abreu, Frebres Cordero, Ita, Jaquier, Page, 2017] 2. Versatility: Berends-Giele recursion allows any vertices coeffjcients (top to bottom) & regulator dependence • Finite-field arithmetic for functional reconstruction 9 Design goals of 2-loop numerical unitarity
• High-precision floating point for direct calculation • Finite-field arithmetic for functional reconstruction 1. OPP-like minimal ansatz: Masters + surface terms. • Produce master coeffjcients w/o external IBP reduction • No doubled propagators except in a few topologies subleading poles: [Abreu, Frebres Cordero, Ita, Jaquier, Page, 2017] 2. Versatility: Berends-Giele recursion allows any vertices coeffjcients (top to bottom) & regulator dependence 9 Design goals of 2-loop numerical unitarity 3. Effjcient and stable numerical fitting of integral
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