Simplified differential equations approach for NNLO calculations

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1 Simplified differential equations approach for NNLO calculations Costas. G. Papadopoulos INPP, NCSR Demokritos, Athens UCLA, June 19, 2015 Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

2 Perturbative QCD at NNLO What do we need for an NNLO calculation? p 1, p 2 p 3,..., p m+2 Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

3 Perturbative QCD at NNLO What do we need for an NNLO calculation? p 1, p 2 p 3,..., p m+2 σ NNLO + + m ( dφ m 2Re(M m (0) M m (2) ) + M m (1) 2) J m (Φ) ( )) dφ m+1 (2Re M (0) m+1 M(1) m+1 J m+1 (Φ) M (0) dφ m+2 2 J m+2 (Φ) m+1 m+2 m+2 VV RV RR RV + RR Talks by M. Czakon, Currie, Abelof, Boughezal, Somogyi,... STRIPPER,... Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

4 Perturbative QCD at NNLO What do we need for an NNLO calculation? p 1, p 2 p 3,..., p m+2 σ NNLO + + m ( dφ m 2Re(M m (0) M m (2) ) + M m (1) 2) J m (Φ) ( )) dφ m+1 (2Re M (0) m+1 M(1) m+1 J m+1 (Φ) M (0) dφ m+2 2 J m+2 (Φ) m+1 m+2 m+2 VV RV RR RV + RR Talks by M. Czakon, Currie, Abelof, Boughezal, Somogyi,... STRIPPER,... Radcor > LoopFest Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

5 OPP at two loops Write the OPP-type equation at two loops N (l 1, l 2 ; {p i }) D 1 D 2... D n = min(n,8) m=1 (l i1i2...im 1, l 2 ; {p i }) D i1 D i2... D im S m;n S m;n stands for all subsets of m indices out of the n ones Talk by G.Ossola Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

6 The new master formula N(q) D 0 D1 D m 1 = m 1 d(i 0 i 1 i 2 i 3 ) + d(q; i 0 i 1 i 2 i 3 ) D i i0 Di1 Di2 Di2 0<i 1<i 2<i 3 m 1 c(i 0 i 1 i 2 ) + c(q; i 0 i 1 i 2 ) D i i0 Di1 Di2 0<i 1<i 2 m 1 b(i 0 i 1 ) + b(q; i0 i 1 ) D i i0 D i1 0<i 1 m 1 a(i 0 ) + ã(q; i 0 ) D i i0 0 + rational terms CutTools, Samurai, Ninja,... ( ) ( m m c 4 + c ) + c 2 ( m 2 Talk by G.Ossola ) Cm 4 Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

7 Reduction at the integrand level Over the last few years very important activity to extend unitarity and integrand level reduction ideas beyond one loop J. Gluza, K. Kajda and D. A. Kosower, Towards a Basis for Planar Two-Loop Integrals, Phys. Rev. D 83 (2011) [arxiv: [hep-th]]. D. A. Kosower and K. J. Larsen, Maximal Unitarity at Two Loops, Phys. Rev. D 85 (2012) [arxiv: [hep-th]]. P. Mastrolia and G. Ossola, On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes, JHEP 1111 (2011) 014 [arxiv: [hep-ph]]. [arxiv: [hep-ph]]. S. Badger, H. Frellesvig and Y. Zhang, Hepta-Cuts of Two-Loop Scattering Amplitudes, JHEP 1204 (2012) 055 Y. Zhang, Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods, JHEP 1209 (2012) 042 [arxiv: [hep-ph]]. P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, Integrand-Reduction for Two-Loop Scattering Amplitudes through Multivariate Polynomial Division, arxiv: [hep-ph]. P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, Multiloop Integrand Reduction for Dimensionally Regulated Amplitudes, arxiv: [hep-ph]. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

8 Multivariate Division and Groebner Basis D. Cox, J. Little, D. O Shea Ideals, Varieties and Algorithms multivariate polynomial division Given any set of polynomials π i, the ideal I, f = i π ih i, we can define a unique Groebner basis up to ordering < g 1,..., g s > Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

9 Multivariate Division and Groebner Basis D. Cox, J. Little, D. O Shea Ideals, Varieties and Algorithms multivariate polynomial division Given any set of polynomials π i, the ideal I, f = i π ih i, we can define a unique Groebner basis up to ordering < g 1,..., g s > f = h 1 g h n g n + r Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

10 Multivariate Division and Groebner Basis D. Cox, J. Little, D. O Shea Ideals, Varieties and Algorithms multivariate polynomial division Given any set of polynomials π i, the ideal I, f = i π ih i, we can define a unique Groebner basis up to ordering < g 1,..., g s > f = h 1 g h n g n + r Strategy: Start with a set of denominators, pick up a 4d parametrisation, define an ideal, I =< D 1,..., D n >, even D 2 i Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

11 Multivariate Division and Groebner Basis D. Cox, J. Little, D. O Shea Ideals, Varieties and Algorithms multivariate polynomial division Given any set of polynomials π i, the ideal I, f = i π ih i, we can define a unique Groebner basis up to ordering < g 1,..., g s > f = h 1 g h n g n + r Strategy: Start with a set of denominators, pick up a 4d parametrisation, define an ideal, I =< D 1,..., D n >, even D 2 i Find the GB of I, G =< g 1,..., g s > Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

12 Multivariate Division and Groebner Basis D. Cox, J. Little, D. O Shea Ideals, Varieties and Algorithms multivariate polynomial division Given any set of polynomials π i, the ideal I, f = i π ih i, we can define a unique Groebner basis up to ordering < g 1,..., g s > f = h 1 g h n g n + r Strategy: Start with a set of denominators, pick up a 4d parametrisation, define an ideal, I =< D 1,..., D n >, even D 2 i Find the GB of I, G =< g 1,..., g s > Perform the division of an arbitrary polynomial N Express back g i in terms of D i N = h 1 g h n g s + v N = h 1 D h n D n + v Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

13 OPP at two loops Write the OPP-type equation at two loops N (l 1, l 2 ; {p i }) D 1 D 2... D n = min(n,8) m=1 (l i1i2...im 1, l 2 ; {p i }) D i1 D i2... D im S m;n i1i2...i m (l 1, l 2 ; {p i }) spurious ISP irreducible integrals D i1 D i2... D im Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

14 OPP at two loops Write the OPP-type equation at two loops N (l 1, l 2 ; {p i }) D 1 D 2... D n = min(n,8) m=1 (l i1i2...im 1, l 2 ; {p i }) D i1 D i2... D im S m;n i1i2...i m (l 1, l 2 ; {p i }) spurious ISP irreducible integrals D i1 D i2... D im ISP-irreducible integrals use IBPI to Master Integrals Libraries in the future: QCD2LOOP, TwOLOop Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

15 The one loop paradigm basis of scalar integrals: G. Passarino and M. J. G. Veltman, Nucl. Phys. B 160 (1979) 151. Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B 425 (1994) 217 [arxiv:hep-ph/ ]. A = d i1 i 2 i 3 i 4 + c i1 i 2 i 3 + b i1 i 2 + a i1 + R a, b, c, d cut-constructible part R rational terms A = I {0,1,,m 1} µ (4 d)d d q (2π) d NI ( q) i I D i ( q) D 0, C 0, B 0, A 0, scalar one-loop integrals: t Hooft and Veltman QCDLOOP Ellis & Zanderighi ; OneLOop A. van Hameren Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

16 OPP at two loops - Rational terms Rational terms ( l 1 l 1 + l (2ε) 1, l 2 l 2 + l (2ε) 2, l 1,2 l (2ε) 1,2 = 0 l (2ε) 1 ) 2 ( = µ11, { } l (4) 1, l (4) 2 Welcome: I = I prime ideal l (2ε) 2 ) 2 = µ22, l (2ε) S. Badger, talk in Amplitudes l (2ε) 2 = µ 12 { l (4) 1, l (4) 2, µ 11, µ 22, µ 12 } Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

17 OPP at two loops - Rational terms Rational terms ( l 1 l 1 + l (2ε) 1, l 2 l 2 + l (2ε) 2, l 1,2 l (2ε) 1,2 = 0 l (2ε) 1 ) 2 ( = µ11, { } l (4) 1, l (4) 2 Welcome: I = I prime ideal l (2ε) 2 ) 2 = µ22, l (2ε) S. Badger, talk in Amplitudes l (2ε) 2 = µ 12 { l (4) 1, l (4) 2, µ 11, µ 22, µ 12 } R 2 terms: hopefully à la 1 loop Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

18 Rational Terms Numerically treat D = 4 2ɛ, means 4 1 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 i 0<i 1 [ ] m 1 d(i 0 i 1 i 2 i 3 ; q 2 ) + d(q; i 0 i 1 i 2 i 3 ; q 2 ) D i i i 0,i 1,i 2,i 3 [ c(i0 i 1 i 2 ; q 2 ) + c(q; i 0 i 1 i 2 ; q 2 ) ] m 1 m 1 [ ] m 1 b(i 0 i 1 ; q 2 ) + b(q; i 0 i 1 ; q 2 ) D i i i 0,i 1 i i 0,i 1,i 2 Di m 1 [ a(i0 ; q 2 ) + ã(q; i 0 ; q 2 ) ] m 1 m 1 Di + P(q) Di Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

19 Rational Terms Numerically treat D = 4 2ɛ, means 4 1 Expand in D-dimensions? N(q) = m 1 i 0<i 1<i 2<i 3 m 1 i 0<i 1<i 2 i 0<i 1 [ ] m 1 d(i 0 i 1 i 2 i 3 ; q 2 ) + d(q; i 0 i 1 i 2 i 3 ; q 2 ) Di i i 0,i 1,i 2,i 3 [ c(i0 i 1 i 2 ; q 2 ) + c(q; i 0 i 1 i 2 ; q 2 ) ] m 1 m 1 [ b(i 0 i 1 ; q 2 ) + b(q; ] m 1 i 0 i 1 ; q 2 ) Di i i 0,i 1 m 1 [ a(i0 ; q 2 ) + ã(q; i 0 ; q 2 ) ] m 1 i 0 i i 0 i i 0,i 1,i 2 Di m 1 D i + P(q) i D i m 2 i m 2 i q 2 Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

20 Rational Terms Numerically treat D = 4 2ɛ, means 4 1 Expand in D-dimensions? N(q) = m 1 i 0<i 1<i 2<i 3 m 1 i 0<i 1<i 2 i 0<i 1 [ ] m 1 d(i 0 i 1 i 2 i 3 ; q 2 ) + d(q; i 0 i 1 i 2 i 3 ; q 2 ) Di i i 0,i 1,i 2,i 3 [ c(i0 i 1 i 2 ; q 2 ) + c(q; i 0 i 1 i 2 ; q 2 ) ] m 1 m 1 [ b(i 0 i 1 ; q 2 ) + b(q; ] m 1 i 0 i 1 ; q 2 ) Di i i 0,i 1 m 1 [ a(i0 ; q 2 ) + ã(q; i 0 ; q 2 ) ] m 1 i 0 i i 0 i i 0,i 1,i 2 Di m 1 D i + P(q) i D i m 2 i m 2 i q 2 Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

21 Rational Terms 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 NNLO Radcor-Loopfest, LA, / 39

22 Rational Terms - R 2 A different source of Rational Terms, called R 2, can also be generated from the ɛ-dimensional part of N(q) N( q) = N(q) + Ñ( q2, ɛ; q) R 2 1 (2π) 4 d n q Ñ( q 2, ɛ; q) D 0 D1 D 1 m 1 (2π) 4 d n q R 2 q = q + q, γ µ = γ µ + γ µ, ḡ µ ν = g µν + g µ ν. New vertices Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

numerically the numerators Ni 6(q), N5 i (q),... with the values of the loop momentum q provided by CutTools generates all inequivalent partitions of 6,5,4,3.

23 The one-loop calculation in a nutshell The computation of pp(p p) e + ν eµ ν µb b involves up to six-point functions. A(q) = N(6) i (q) + N(5) i (q) + N(4) i (q) + N(3) i (q) + D i0 D i1 D i5 D i0 D i1 D i4 D i0 D i1 D i3 D i0 D i1 D i2 }{{}}{{}}{{}}{{} In order to apply the OPP reduction, HELAC evaluates numerically the numerators Ni 6(q), N5 i (q),... with the values of the loop momentum q provided by CutTools generates all inequivalent partitions of 6,5,4,3... blobs attached to the loop, and check all possible flavours (and colours) that can be consistently running inside hard-cuts the loop (q is fixed) to get a n + 2 tree-like process The R 2 contributions (rational terms) are calculated in the same way as the tree-order amplitude, taking into account extra vertices MadGraph, RECOLA, OpenLoops, Talks by S.Uccirati, V.Hirschi,... Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

24 Towards a full NNLO solution From Feynman Diagrams to recursive equations: taming the n!. Dyson-Schwinger Recursive Equations Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

25 Towards a full NNLO solution From Feynman Diagrams to recursive equations: taming the n!. The Dyson-Schwinger recursion Dyson-Schwinger Recursive Equations Imagine a theory with 3- and 4- point vertices and just one field. A. Kanaki and C. G. Papadopoulos, Comput. Phys. Commun. 132 (2000) 306 [arxiv:hep-ph/ ]. Then it is straightforward to write an equation that gives the amplitude for 1 n F. A. Berends and W. T. Giele, Nucl. Phys. B 306 (1988) 759. F. Caravaglios and M. Moretti, Phys. Lett. B 358 (1995) 332. = Unfortunately not so much on the second line! HEP - NCSR Democritos S. Weinzierl Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

26 Integrand reduction at two loops Amplitude level (explicit color, helicity) Using t Hooft-Veltman scheme, d = 4 ME Fully automated Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

27 Integrand reduction at two loops Amplitude level (explicit color, helicity) Using t Hooft-Veltman scheme, d = 4 ME Fully automated MG5 amc> add process p p > e+ ve j j 0 MG5 amc> add process p p > e+ ve j j 1 MG5 amc> add process p p > e+ ve j j 2 Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

28 Master Integrals: The current approach m independent momenta l loops, N = l(l + 1)/2 + lm scalar products basis composed by D 1... D N, allows to express all scalar products D i = ({k, l} + p i ) 2 M 2 i F [a 1,..., a N ] = d d kd d l {k µ, l µ } d d kd d 1 l D a Da N N ( {k µ, l µ, υ µ ) } D a = 0 Da N N IBP Laporta: FIRE, AIR, Reduze reduce these to MI MI computed, Feynman parameters, Mellin-Barnes, Differential Equations Z. Bern, L. J. Dixon and D. A. Kosower, Phys. Lett. B 302 (1993) 299. T. Gehrmann and E. Remiddi, Nucl. Phys. B 580 (2000) 485 [hep-ph/ ]. J. M. Henn, Phys. Rev. Lett. 110 (2013) 25, [arxiv: [hep-th]]. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

29 Master Integrals: The current approach m independent momenta l loops, N = l(l + 1)/2 + lm scalar products basis composed by D 1... D N, allows to express all scalar products D i = ({k, l} + p i ) 2 M 2 i F [a 1,..., a N ] = d d kd d l {k µ, l µ } d d kd d 1 l D a Da N N ( {k µ, l µ, υ µ ) } D a = 0 Da N N IBP Laporta: FIRE, AIR, Reduze reduce these to MI MI computed, Feynman parameters, Mellin-Barnes, Differential Equations Z. Bern, L. J. Dixon and D. A. Kosower, Phys. Lett. B 302 (1993) 299. T. Gehrmann and E. Remiddi, Nucl. Phys. B 580 (2000) 485 [hep-ph/ ]. J. M. Henn, Phys. Rev. Lett. 110 (2013) 25, [arxiv: [hep-th]]. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

30 Master Integrals: The current approach m independent momenta l loops, N = l(l + 1)/2 + lm scalar products basis composed by D 1... D N, allows to express all scalar products D i = ({k, l} + p i ) 2 M 2 i F [a 1,..., a N ] = d d kd d l {k µ, l µ } d d kd d 1 l D a Da N N ( {k µ, l µ, υ µ ) } D a = 0 Da N N IBP Laporta: FIRE, AIR, Reduze reduce these to MI MI computed, Feynman parameters, Mellin-Barnes, Differential Equations Z. Bern, L. J. Dixon and D. A. Kosower, Phys. Lett. B 302 (1993) 299. T. Gehrmann and E. Remiddi, Nucl. Phys. B 580 (2000) 485 [hep-ph/ ]. J. M. Henn, Phys. Rev. Lett. 110 (2013) 25, [arxiv: [hep-th]]. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

31 Master Integrals: The current approach m independent momenta l loops, N = l(l + 1)/2 + lm scalar products basis composed by D 1... D N, allows to express all scalar products D i = ({k, l} + p i ) 2 M 2 i F [a 1,..., a N ] = d d kd d l {k µ, l µ } d d kd d 1 l D a Da N N ( {k µ, l µ, υ µ ) } D a = 0 Da N N IBP Laporta: FIRE, AIR, Reduze reduce these to MI MI computed, Feynman parameters, Mellin-Barnes, Differential Equations Z. Bern, L. J. Dixon and D. A. Kosower, Phys. Lett. B 302 (1993) 299. T. Gehrmann and E. Remiddi, Nucl. Phys. B 580 (2000) 485 [hep-ph/ ]. J. M. Henn, Phys. Rev. Lett. 110 (2013) 25, [arxiv: [hep-th]]. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

32 Master Integrals: The current approach m independent momenta l loops, N = l(l + 1)/2 + lm scalar products basis composed by D 1... D N, allows to express all scalar products D i = ({k, l} + p i ) 2 M 2 i F [a 1,..., a N ] = d d kd d l {k µ, l µ } d d kd d 1 l D a Da N N ( {k µ, l µ, υ µ ) } D a = 0 Da N N IBP Laporta: FIRE, AIR, Reduze reduce these to MI MI computed, Feynman parameters, Mellin-Barnes, Differential Equations Z. Bern, L. J. Dixon and D. A. Kosower, Phys. Lett. B 302 (1993) 299. T. Gehrmann and E. Remiddi, Nucl. Phys. B 580 (2000) 485 [hep-ph/ ]. J. M. Henn, Phys. Rev. Lett. 110 (2013) 25, [arxiv: [hep-th]]. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

33 Differential Equations Approach Iterated Integrals K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831 Polylogarithms, Symbol algebra A. B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Phys. Rev. Lett. 105 (2010) C. Duhr, H. Gangl and J. R. Rhodes, JHEP 1210 (2012) 075 [arxiv: [math-ph]]. C. Bogner and F. Brown G (a n,..., a 1, x) = x with the special cases, G(x) = 1 and 0 1 dt G (a n 1,..., a 1, t) t a n G 0,... 0, x = 1 }{{} n! logn (x) n Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

34 Differential Equations Approach Iterated Integrals K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831 Polylogarithms, Symbol algebra A. B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Phys. Rev. Lett. 105 (2010) C. Duhr, H. Gangl and J. R. Rhodes, JHEP 1210 (2012) 075 [arxiv: [math-ph]]. C. Bogner and F. Brown G (a n,..., a 1, x) = x with the special cases, G(x) = 1 and 0 1 dt G (a n 1,..., a 1, t) t a n G 0,... 0, x = 1 }{{} n! logn (x) n Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

35 Differential Equations Approach The integral is a function of external momenta, so one can set-up differential equations by differentiating and using IBP p µ j p µ i G[a 1,..., a n ] C a 1,...,a n G[a 1,..., a n] Find the proper parametrization; Bring the system of equations in a form suitable to express the MI in terms of GPs m f (ε, {x i }) = εa m ({x i }) f (ε, {x i }) m A n n A m = 0 [A m, A n ] = 0 J. M. Henn, Phys. Rev. Lett. 110 (2013) 25, [arxiv: [hep-th]]. Boundary conditions: expansion by regions or regularity conditions. B. Jantzen, A. V. Smirnov and V. A. Smirnov, Eur. Phys. J. C 72 (2012) 2139 [arxiv: [hep-ph]]. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

36 Differential Equations Approach The integral is a function of external momenta, so one can set-up differential equations by differentiating and using IBP p µ j p µ i G[a 1,..., a n ] C a 1,...,a n G[a 1,..., a n] Find the proper parametrization; Bring the system of equations in a form suitable to express the MI in terms of GPs m f (ε, {x i }) = εa m ({x i }) f (ε, {x i }) m A n n A m = 0 [A m, A n ] = 0 J. M. Henn, Phys. Rev. Lett. 110 (2013) 25, [arxiv: [hep-th]]. Boundary conditions: expansion by regions or regularity conditions. B. Jantzen, A. V. Smirnov and V. A. Smirnov, Eur. Phys. J. C 72 (2012) 2139 [arxiv: [hep-ph]]. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

37 Differential Equations Approach The integral is a function of external momenta, so one can set-up differential equations by differentiating and using IBP p µ j p µ i G[a 1,..., a n ] C a 1,...,a n G[a 1,..., a n] Find the proper parametrization; Bring the system of equations in a form suitable to express the MI in terms of GPs m f (ε, {x i }) = εa m ({x i }) f (ε, {x i }) m A n n A m = 0 [A m, A n ] = 0 J. M. Henn, Phys. Rev. Lett. 110 (2013) 25, [arxiv: [hep-th]]. Boundary conditions: expansion by regions or regularity conditions. B. Jantzen, A. V. Smirnov and V. A. Smirnov, Eur. Phys. J. C 72 (2012) 2139 [arxiv: [hep-ph]]. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

38 Differential Equations Approach J. M. Henn, K. Melnikov and V. A. Smirnov, arxiv: [hep-ph]. T. Gehrmann, A. von Manteuffel and L. Tancredi, arxiv: [hep-ph]. p p p 3 p 4 S = (q 1 + q 2 ) 2 = (q 3 + q 4 ) 2, T = (q 1 q 3 ) 2 = (q 2 q 4 ) 2, U = (q 1 q 4 ) 2 = (q 2 q 3 ) 2 ; S M 2 3 = (1 + x)(1 + xy), T M 2 3 = xz, M 2 4 M 2 3 = x 2 y. d f (x, y, z; ɛ) = ɛ d Ã(x, y, z) f (x, y, z; ɛ) 15 Ã = Ã αi log(α i ) i=1 α = {x, y, z, 1 + x, 1 y, 1 z, 1 + xy, z y, 1 + y(1 + x) z, xy + z, 1 + x(1 + y z), 1 + xz, 1 + y z, z + x(z y) + xyz, z y + yz + xyz}. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

39 The Simplified Differential Equations Approach C. G. Papadopoulos, arxiv: [hep-ph]. Making the whole procedure systematic (algorithmic) and straightforwardly expressible in terms of GPs. Introduce one parameter G (x) = d d k 1 iπ d/2 (k 2 ) (k + x p 1 ) 2 (k + p 1 + p 2 ) 2... (k + p 1 + p p n) 2 Now the integral becomes a function of x, which allows to define a differential equation with respect to x, schematically given by x G (x) = 1 x G (x) + xp 2 1 G x G and using IBPI we obtain, for instance for the one-loop 3 off-shell legs ( ) ( ) m 1 xg x G 021 = 1 x d 4 G x m 3 /m ( ) ( ) + d 3 1 m 1 m 3 x 1 1 G101 G 110 x m 3 /m 1 x Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

40 The Simplified Differential Equations Approach C. G. Papadopoulos, arxiv: [hep-ph]. Making the whole procedure systematic (algorithmic) and straightforwardly expressible in terms of GPs. Introduce one parameter G (x) = d d k 1 iπ d/2 (k 2 ) (k + x p 1 ) 2 (k + p 1 + p 2 ) 2... (k + p 1 + p p n) 2 Now the integral becomes a function of x, which allows to define a differential equation with respect to x, schematically given by x G (x) = 1 x G (x) + xp 2 1 G x G and using IBPI we obtain, for instance for the one-loop 3 off-shell legs ( ) ( ) m 1 xg x G 021 = 1 x d 4 G x m 3 /m ( ) ( ) + d 3 1 m 1 m 3 x 1 1 G101 G 110 x m 3 /m 1 x Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

41 The Simplified Differential Equations Approach C. G. Papadopoulos, arxiv: [hep-ph]. Making the whole procedure systematic (algorithmic) and straightforwardly expressible in terms of GPs. Introduce one parameter G (x) = d d k 1 iπ d/2 (k 2 ) (k + x p 1 ) 2 (k + p 1 + p 2 ) 2... (k + p 1 + p p n) 2 Now the integral becomes a function of x, which allows to define a differential equation with respect to x, schematically given by x G (x) = 1 x G (x) + xp 2 1 G x G and using IBPI we obtain, for instance for the one-loop 3 off-shell legs ( ) ( ) m 1 xg x G 021 = 1 x d 4 G x m 3 /m ( ) ( ) + d 3 1 m 1 m 3 x 1 1 G101 G 110 x m 3 /m 1 x Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

42 The Simplified Differential Equations Approach The integrating factor M is given by and the DE takes the form, d = 4 2ε, M = x (1 x) 4 d 2 ( m 3 + m 1 x) 4 d 2 x MG 1 ( 111 = c Γ ε (1 x) 1+ε ( m 3 + m 1 x) 1+ε ( m1 x 2) ε ( m3 ) ε) Integrating factors ɛ = 0 do not have branch points DE can be straightforwardly integrated order by order GPs. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

43 The Simplified Differential Equations Approach The integrating factor M is given by and the DE takes the form, d = 4 2ε, M = x (1 x) 4 d 2 ( m 3 + m 1 x) 4 d 2 x MG 1 ( 111 = c Γ ε (1 x) 1+ε ( m 3 + m 1 x) 1+ε ( m1 x 2) ε ( m3 ) ε) Integrating factors ɛ = 0 do not have branch points DE can be straightforwardly integrated order by order GPs. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

44 The Simplified Differential Equations Approach G 111 = c Γ (m 1 m 3 )x I x 1 = (1 x) ɛ ( ( m1) ε + ( m3) ε + ( m1) ε ( m3) ε) x1 I = ε 2 ( ( m1) ε ( m3) ε) ( ) ( m3 x1g, 1 ( m1) ε ( m3) ε) ( ( ) ( )) m3 m3 G, 1 G, x m 1 m 1 m 1 + ε ( + ( m1) ε ( m3) ε) ( ( ) ( ) ( m3 m3 m3 G, 1 G, x G, m3 ) ( m3, 1 G, m3 )) (, x + x1 2G(0, 1, x) ( m1) ε m1 m1 m1 m1 m1 m1 ( + 2G 0, m3 ) ( ) (, x ( m1) ε m3 + 2G, 1, x ( m1) ε m3 + G, m3 ) ( ), 1 ( m1) ε m3 G, x log(1 x) ( m1) ε m1 m1 m1 m1 m1 ( ) ( ) ( m3 m3 m3 ) 2G, x log(x) ( m1) ε + 2 log(1 x) log(x) ( m1) ε 2 ( m3) ε G, 1, x ( m3) ε G,, 1 m1 m1 m1 m1 ( ( m1) ε ( m3) ε) ( ) ( ( ) ) ( ) ) m3 m3 G, 1 G, x log(1 x) + ( m3) ε m3 G, x log(1 x) m1 m1 m1 ( ( + ε ( m1) ε ( m3) ε) ( ( ) ( m3 m3 G, x G, m3 ) ( ) ( m3 m3, 1 G, 1 G, m3 ) ( m3, x G, m3, m3 ), 1 m1 m1 m1 m1 m1 m1 m1 m1 m1 ( m3 + G, m3, m3 )), x + 1 ( ( 2 x1 ( m1) ε ( m3) ε) ( ) ( ( m3 G, 1 log 2 m3 (1 x) + 2G, m3 )), x m1 m1 m1 m1 m1 m1 ( ) ( m3 + G, x 4 log 2 ( (x) 2 ( m1) ε ( m3) ε) ( m3 G, m3 ) (, ( m3) ε ( m1) ε) ( ) ) m3 G, 1 log(1 x) m1 m1 m1 m1 ( ( ) ( ) m3 + 2 G, m3, m3, 1 ( m1) ε m3 + G, m3, 1 log(1 x) ( m1) ε 2 log(1 x) log 2 (x) 4G(0, 0, 1, x) m1 m1 m1 m1 m1 ( + 4G 0, 0, m3 ) (, x 2G(0, 1, 1, x) + 4G 0, m3 ) (, 1, x 2G 0, m3, m3 ) ( ) ( m3 m3, x + 2G, 0, 1, x 2G, 0, m3 ), x m1 m1 m1 m1 m1 m1 m1 ( m3 ) ( m3 ) ( m3) ε G,,, 1 ( m3) ε G,, 1 log(1 x) + log 2 (1 x) log(x) m1 m1 m1 m1 m1 ( + 4G(0, 1, x) log(x) 4G 0, m3 ) ( ) ( m3 m3, x log(x) 4G, 1, x log(x) + 2G, m3 ) ))), x log(x) m1 m1 m1 m1 It is straightforward to see that by taking now the limit x 1, as described above, we Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

45 The Simplified Differential Equations Approach The two-loop 3-off-shell-legs triangle Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

46 The Simplified Differential Equations Approach We are interested in G The DE involves also the MI G , so we have a system of two coupled DE, as follows: x f (x) = A 3 (2 3ε)(1 x) 2ε x 1+ε (m 1 x m 3 ) 2ε 2ε(2ε 1) + m 1 ε(1 x) 2ε (m 1 x m 3 ) 2ε g(x) 2ε 1 x g(x) = A 3 (3ε 2)(3ε 1)( m 1) 2ε (1 x) 2ε 1 x 3ε (m 1 x m 3 ) 2ε 1 2ε 2 +(2ε 1)(3ε 1)(1 x) 2ε 1 (m 1 x m 3 ) 2ε 1 f (x) where f (x) M G and g (x) M G , M = (1 x) 2ε x ε+1 (m 1 x m 3 ) 2ε and M = x ε Solve sequentially in ε expansion Reproduce limit ε 0 Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

47 The Simplified Differential Equations Approach The singularity at x = 0 is proportional to x 1+ε and can easily be integrated by the following decomposition x dt t 1+ε x F (t) = F (0) dt t 1+ε x + dt = F (0) xε x ε + F (t) F (0) dt t 0 F (t) F (0) t ε t ( 1 + ε log (t) ε2 log 2 (t) +... ) Reproduce correctly boundary term x = 0 Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

48 The Simplified Differential Equations Approach General setup m: number of denominators x G m+1 = H ({s ij }, ɛ; x) G m+1 + m 0 = 3 in the case of two loops m m m 0 R ({s ij }, ɛ; x) G m, Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

49 The Simplified Differential Equations Approach General setup m: number of denominators x G m+1 = H ({s ij }, ɛ; x) G m+1 + m 0 = 3 in the case of two loops x M = MH x (MG m+1 ) = M m m m m 0 R ({s ij }, ɛ; x) G m, m m 0 R ({s ij }, ɛ; x) G m. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

50 The Simplified Differential Equations Approach General setup m: number of denominators m x G m+1 = H ({s ij }, ɛ; x) G m+1 + R ({s ij }, ɛ; x) G m, m m 0 m 0 = 3 in the case of two loops x M = MH x (MG m+1 ) = M m m m 0 R ({s ij }, ɛ; x) G m. m M R ({s ij }, ɛ; x) G m =: x 1+β i ɛ Ĩ (i) sin ({s ij}, ɛ) + Ĩreg ({s ij }, ɛ; x). m m 0 i withcostas. β being G. Papadopoulos typically rational numbers. NNLO Radcor-Loopfest, LA, / 39

51 The Simplified Differential Equations Approach MG m+1 = C({s ij }, ɛ) + i x β i ɛ x β i ɛ Ĩ (i) sin ({s ij}, ɛ) + dx Ĩ reg ({s ij }, ɛ; x ), 0 Integrating factors M rational functions of x in the limit ɛ 0 Sufficient condition DE solvable in terms of GPs. All re-summed parts at x 0 fully determined by the one-scale MI involved in the system Two-point integrals, two three-point integrals and double one-loop integrals homogenous differential equations. C({s ij }, ɛ) = 0: no independent calculation of boundary terms needed. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

52 The Simplified Differential Equations Approach MG m+1 = C({s ij }, ɛ) + i x β i ɛ x β i ɛ Ĩ (i) sin ({s ij}, ɛ) + dx Ĩ reg ({s ij }, ɛ; x ), 0 Integrating factors M rational functions of x in the limit ɛ 0 Sufficient condition DE solvable in terms of GPs. All re-summed parts at x 0 fully determined by the one-scale MI involved in the system Two-point integrals, two three-point integrals and double one-loop integrals homogenous differential equations. C({s ij }, ɛ) = 0: no independent calculation of boundary terms needed. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

53 The Simplified Differential Equations Approach MG m+1 = C({s ij }, ɛ) + i x β i ɛ x β i ɛ Ĩ (i) sin ({s ij}, ɛ) + dx Ĩ reg ({s ij }, ɛ; x ), 0 Integrating factors M rational functions of x in the limit ɛ 0 Sufficient condition DE solvable in terms of GPs. All re-summed parts at x 0 fully determined by the one-scale MI involved in the system Two-point integrals, two three-point integrals and double one-loop integrals homogenous differential equations. C({s ij }, ɛ) = 0: no independent calculation of boundary terms needed. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

54 The Simplified Differential Equations Approach MG m+1 = C({s ij }, ɛ) + i x β i ɛ x β i ɛ Ĩ (i) sin ({s ij}, ɛ) + dx Ĩ reg ({s ij }, ɛ; x ), 0 Integrating factors M rational functions of x in the limit ɛ 0 Sufficient condition DE solvable in terms of GPs. All re-summed parts at x 0 fully determined by the one-scale MI involved in the system Two-point integrals, two three-point integrals and double one-loop integrals homogenous differential equations. C({s ij }, ɛ) = 0: no independent calculation of boundary terms needed. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

55 The Simplified Differential Equations Approach MG m+1 = C({s ij }, ɛ) + i x β i ɛ x β i ɛ Ĩ (i) sin ({s ij}, ɛ) + dx Ĩ reg ({s ij }, ɛ; x ), 0 Integrating factors M rational functions of x in the limit ɛ 0 Sufficient condition DE solvable in terms of GPs. All re-summed parts at x 0 fully determined by the one-scale MI involved in the system Two-point integrals, two three-point integrals and double one-loop integrals homogenous differential equations. C({s ij }, ɛ) = 0: no independent calculation of boundary terms needed. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

56 The Simplified Differential Equations Approach MG m+1 = C({s ij }, ɛ) + i x β i ɛ x β i ɛ Ĩ (i) sin ({s ij}, ɛ) + dx Ĩ reg ({s ij }, ɛ; x ), 0 Integrating factors M rational functions of x in the limit ɛ 0 Sufficient condition DE solvable in terms of GPs. All re-summed parts at x 0 fully determined by the one-scale MI involved in the system Two-point integrals, two three-point integrals and double one-loop integrals homogenous differential equations. C({s ij }, ɛ) = 0: no independent calculation of boundary terms needed. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

57 The Simplified Differential Equations Approach When the DE are coupled x Gm+1 = H ({s ij }, ɛ; x) G m+1 + m m m 0 R ({s ij }, ɛ; x) G m, M D : x M D = M D H D, where H D is the diagonal part of H. H =: M D (H H D ) M 1 D of the reduced system of DE is then a strictly triangular matrix at order ɛ 0 and the system becomes effectively uncoupled. Problem: In very few specific cases, C x 2+β i ɛ appears in the matrix H, Solution: x 1/x back to x 1+β i ɛ in the inhomogeneous part of the DE. more details talk by C. Wever Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

58 The Simplified Differential Equations Approach When the DE are coupled x Gm+1 = H ({s ij }, ɛ; x) G m+1 + m m m 0 R ({s ij }, ɛ; x) G m, M D : x M D = M D H D, where H D is the diagonal part of H. H =: M D (H H D ) M 1 D of the reduced system of DE is then a strictly triangular matrix at order ɛ 0 and the system becomes effectively uncoupled. Problem: In very few specific cases, C x 2+β i ɛ appears in the matrix H, Solution: x 1/x back to x 1+β i ɛ in the inhomogeneous part of the DE. more details talk by C. Wever Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

59 The Simplified Differential Equations Approach When the DE are coupled x Gm+1 = H ({s ij }, ɛ; x) G m+1 + m m m 0 R ({s ij }, ɛ; x) G m, M D : x M D = M D H D, where H D is the diagonal part of H. H =: M D (H H D ) M 1 D of the reduced system of DE is then a strictly triangular matrix at order ɛ 0 and the system becomes effectively uncoupled. Problem: In very few specific cases, C x 2+β i ɛ appears in the matrix H, Solution: x 1/x back to x 1+β i ɛ in the inhomogeneous part of the DE. more details talk by C. Wever Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

60 The Simplified Differential Equations Approach When the DE are coupled x Gm+1 = H ({s ij }, ɛ; x) G m+1 + m m m 0 R ({s ij }, ɛ; x) G m, M D : x M D = M D H D, where H D is the diagonal part of H. H =: M D (H H D ) M 1 D of the reduced system of DE is then a strictly triangular matrix at order ɛ 0 and the system becomes effectively uncoupled. Problem: In very few specific cases, C x 2+β i ɛ appears in the matrix H, Solution: x 1/x back to x 1+β i ɛ in the inhomogeneous part of the DE. more details talk by C. Wever Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

61 The Simplified Differential Equations Approach When the DE are coupled x Gm+1 = H ({s ij }, ɛ; x) G m+1 + m m m 0 R ({s ij }, ɛ; x) G m, M D : x M D = M D H D, where H D is the diagonal part of H. H =: M D (H H D ) M 1 D of the reduced system of DE is then a strictly triangular matrix at order ɛ 0 and the system becomes effectively uncoupled. Problem: In very few specific cases, C x 2+β i ɛ appears in the matrix H, Solution: x 1/x back to x 1+β i ɛ in the inhomogeneous part of the DE. more details talk by C. Wever Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

62 Two-loop, four-point, two off-shell legs p(q 1 )p (q 2 ) V 1 ( q 3 )V 2 ( q 4 ), q 2 1 = q 2 2 = 0, q 2 3 = M 2 3, q 2 4 = M 2 4. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

63 Two-loop, four-point, two off-shell legs p(q 1 )p (q 2 ) V 1 ( q 3 )V 2 ( q 4 ), q 2 1 = q 2 2 = 0, q 2 3 = M 2 3, q 2 4 = M 2 4. q 1 = xp 1, q 2 = xp 2, q 3 = p 123 xp 12, q 4 = p 123, p 2 i = 0, s 12 := p 2 12, s 23 := p 2 23, q := p 2 123, Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

64 Two-loop, four-point, two off-shell legs p(q 1 )p (q 2 ) V 1 ( q 3 )V 2 ( q 4 ), q 2 1 = q 2 2 = 0, q 2 3 = M 2 3, q 2 4 = M 2 4. q 1 = xp 1, q 2 = xp 2, q 3 = p 123 xp 12, q 4 = p 123, p 2 i = 0, s 12 := p 2 12, s 23 := p 2 23, q := p 2 123, S = (q 1 + q 2 ) 2 T = (q 1 + q 3 ) 2 S = s 12 x 2, T = q (s 12 + s 23 )x, M 2 3 = (1 x)(q s 12 x), M 2 4 = q. U = (q 1 + q 4 ) 2 : S + T + U = M M2 4. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

65 Two-loop, four-point, two off-shell legs xp1 p123 xp2 xp1 xp1 p123 xp2 p123 xp12 p123 xp12 p123 p123 xp12 xp2 Figure : The parametrization of external momenta for the three planar double boxes of the families P 12 (left), P 13 (middle) and P 23 (right) contributing to pair production at the LHC. All external momenta are incoming. xp1 p123 xp2 xp1 xp1 p123 xp2 p123 xp12 p123 xp12 p123 xp2 p123 xp12 Figure : The parametrization of external momenta for the three non-planar double boxes of the families N 12 (left), N 13 (middle) and N 34 (right) contributing to pair production at the LHC. All external momenta are incoming. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

66 Two-loop, four-point, two off-shell legs Triangle rule: xp 1 p 12 p 12 xp 1 Figure : Required parametrization for off mass-shell triangles after possible pinching of internal line(s). more details talk by C. Wever Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

67 Two-loop, four-point, two off-shell legs Planar topologies G P 12 d a (x, s, ɛ) := e 2γ E ɛ d d k1 d k2 1 1 a 9 iπ d/2 iπ d/2 k 2a 1 1 (k 1 + xp 1 ) 2a 2 (k 1 + xp 12 ) 2a 3 (k 1 + p 123 ) 2a 4 1 k 2a 5 2 (k 2 xp 1 ) 2a 6 (k 2 xp 12 ) 2a 7 (k 2 p 123 ) 2a 8 (k 1 + k 2 ) 2a 9, G P 13 d a (x, s, ɛ) := e 2γ E ɛ d d k1 d k2 1 1 a 9 iπ d/2 iπ d/2 k 2a 1 1 (k 1 + xp 1 ) 2a 2 (k 1 + xp 12 ) 2a 3 (k 1 + p 123 ) 2a 4 1 k 2a 5 2 (k 2 xp 1 ) 2a 6 (k 2 p 12 ) 2a 7 (k 2 p 123 ) 2a 8 (k 1 + k 2 ) 2a 9, G P 23 d a (x, s, ɛ) := e 2γ E ɛ d d k1 d k2 1 1 a 9 iπ d/2 iπ d/2 k 2a 1 1 (k 1 + xp 1 ) 2a 2 (k 1 + p 123 xp 2 ) 2a 3 (k 1 + p 123 ) 2a 4 1 k 2a 5 2 (k 2 p 1 ) 2a 6 (k 2 + xp 2 p 123 ) 2a 7 (k 2 p 123 ) 2a 8 (k 1 + k 2 ) 2a 9, Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

68 Two-loop, four-point, two off-shell legs Planar topologies P 12 : { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 111m10111, 11101m111}, P 13 : { , , , , , , , , , , , , , , , , , , , , , , , , , , , , m }, P 23 : { , , , , , , , , , , , , , , , , , , , , , , , , , , , 111m10111}. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

69 Two-loop, four-point, two off-shell legs Non-planar topologies G N 12 d a (x, s, ɛ) := e 2γ E ɛ d d k1 d k2 1 1 a 9 iπ d/2 iπ d/2 k 2a 1 1 (k 1 + xp 1 ) 2a 2 (k 1 + xp 12 ) 2a 3 (k 1 + p 123 ) 2a 4 1 k 2a 5 2 (k 2 xp 1 ) 2a 6 (k 2 p 123 ) 2a 7 (k 1 + k 2 + xp 2 ) 2a 8 (k 1 + k 2 ) 2a 9, G N 13 d a (x, s, ɛ) := e 2γ E ɛ d d k1 d k2 1 1 a 9 iπ d/2 iπ d/2 k 2a 1 1 (k 1 + xp 1 ) 2a 2 (k 1 + xp 12 ) 2a 3 (k 1 + p 123 ) 2a 4 1 k 2a 5 2 (k 2 xp 12 ) 2a 6 (k 2 p 123 ) 2a 7 (k 1 + k 2 + xp 1 ) 2a 8 (k 1 + k 2 ) 2a 9, G N 34 d a (x, s, ɛ) := e 2γ E ɛ d d k1 d k2 1 1 a 9 iπ d/2 iπ d/2 k 2a 1 1 (k 1 + xp 1 ) 2a 2 (k 1 + xp 12 ) 2a 3 (k 1 + p 123 ) 2a 4 1 k 2a 5 2 (k 2 xp 1 ) 2a 6 (k 2 p 123 ) 2a 7 (k 1 + k 2 + xp 12 p 123 ) 2a 8 (k 1 + k 2 ) 2a 9. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

70 Two-loop, four-point, two off-shell legs Non-planar topologies N 12 : { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1m , , 0m , , 1m , 1m1111m11}, N 13 : { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , m , , m , 0m , 00111m111, , , , , m }, N 34 : { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 11m010111, 110m10111, 11001m111, , m , 011m10111, 01101m111, , , , , 111m10111}. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

71 Two-loop, four-point, two off-shell legs GP-indices { I (P 12 ) = 0, 1, }, q, s 12 s 12 q, q, 1 s 23 q s 23 q, 1 + s 23 s 12, s 12 s 12 + s 23 { } I (P 13 ) = q 0, 1,, s 12 + s 23 q q(q s 23 ),, ξ, ξ +, s 12 s 12 q s 23 q 2 (q + s 12 )s 23, { q I (P 23 ) = 0, 1,, 1 + s 23 q q,,, q s } 23, s 12 s 12 q s 23 s 12 + s 23 s 12 ξ ± = qs 12 ± qs 12 s 23 ( q + s 12 + s 23 ) qs 12 s 12 s 23. I (N 12 ) = I (P 23 ), { s12 I (N 34 ) = I (P 12 ) I (P 23 ), s 12 + s 23 q s 23 q I (N 13 ) = I (P 23 ) { ξ, ξ +, 1 +, q2 qs 23 s 12 s 23, s qs 23 + s 12 s 23 s 12 (q s 23 ) s 12 (s 12 + s 23 ) q s 12 + q q + s 23 }. }, Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

72 Two-loop, four-point, two off-shell legs Example { G P (x, s, ɛ) = A 3 (ɛ) 1 x 2 s 12 ( q + x(q s 23 )) 2 2ɛ ( ( ) ( ) q q ɛ 3 GP ; x + 2 GP ; x s 12 q s GP(0; x) GP(1; x) + log ( s 12 ) + 9 ) + 1 ( ( ) ( ) q q 4 4ɛ 2 18 GP ; x 36 GP ; x s 12 q s 23 ( ) ( ) ( ) ) q q s23 q 8 GP 0, ; x + 16 GP 0, ; x + 8 GP + 1, ; x + s 12 q s 23 s 12 q s ( ( ( ) ) ( ( ) ) ) q q 9 GP 0, ; x + GP(0, 1; x) 4 GP 0, 0, ; x + GP(0, 0, 1; x) + ɛ s 12 s ( GP ( 0, 0, 1, ξ ; x ) + GP (0, 0, 1, ξ +; x) ) ( ) } q q (q s 23 ) 2 GP 0, 0,, q s 23 q 2 s 23 (q + s 12 ) ; x +. A 3 (ɛ) = e 2γ E ɛ Γ(1 ɛ)3 Γ(1 + 2ɛ). Γ(3 3ɛ) C. G. Papadopoulos, D. Tommasini and C. Wever, arxiv: [hep-ph]. Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

73 The Simplified Differential Equations Approach Summary & Outlook One-loop up to 5-point at order ɛ: 6 scales, GP-weight 3(4) (look forward for pentaboxes) Two-loop triangles and 4-point MI Double boxes with two external off-shell legs (more than 100 MI) P12 P13 P23 N12 N13 N34 topologies completed and tested! Completing the list of all MI with arbitrary off-shell legs (m = 0). Talk by Tancredi Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

74 The Simplified Differential Equations Approach Summary & Outlook One-loop up to 5-point at order ɛ: 6 scales, GP-weight 3(4) (look forward for pentaboxes) Two-loop triangles and 4-point MI Double boxes with two external off-shell legs (more than 100 MI) P12 P13 P23 N12 N13 N34 topologies completed and tested! Completing the list of all MI with arbitrary off-shell legs (m = 0). Talk by Tancredi Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

75 The Simplified Differential Equations Approach Summary & Outlook One-loop up to 5-point at order ɛ: 6 scales, GP-weight 3(4) (look forward for pentaboxes) Two-loop triangles and 4-point MI Double boxes with two external off-shell legs (more than 100 MI) P12 P13 P23 N12 N13 N34 topologies completed and tested! Completing the list of all MI with arbitrary off-shell legs (m = 0). Talk by Tancredi Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

76 The Simplified Differential Equations Approach Summary & Outlook One-loop up to 5-point at order ɛ: 6 scales, GP-weight 3(4) (look forward for pentaboxes) Two-loop triangles and 4-point MI Double boxes with two external off-shell legs (more than 100 MI) P12 P13 P23 N12 N13 N34 topologies completed and tested! Completing the list of all MI with arbitrary off-shell legs (m = 0). Talk by Tancredi Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

77 The Simplified Differential Equations Approach Summary & Outlook Get DE in one parameter, that always go to the argument of GPs, all weights being independent of x, therefore no limitation on the number of scales (multi-leg). Boundary conditions, namely the x 0 limit, extracted from the DE itself All coupled systems of DE (up to fivefold) satisfying the decoupling criterion, i.e. solvable order by order in ɛ Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

78 The Simplified Differential Equations Approach Summary & Outlook Get DE in one parameter, that always go to the argument of GPs, all weights being independent of x, therefore no limitation on the number of scales (multi-leg). Boundary conditions, namely the x 0 limit, extracted from the DE itself All coupled systems of DE (up to fivefold) satisfying the decoupling criterion, i.e. solvable order by order in ɛ Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

79 The Simplified Differential Equations Approach Summary & Outlook Get DE in one parameter, that always go to the argument of GPs, all weights being independent of x, therefore no limitation on the number of scales (multi-leg). Boundary conditions, namely the x 0 limit, extracted from the DE itself All coupled systems of DE (up to fivefold) satisfying the decoupling criterion, i.e. solvable order by order in ɛ Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

80 NNLO in the future The NNLO automation to come In a few years the new wish list should be completed pp t t, pp W + W, pp W /Z + nj, pp H + nj,... Virtual amplitudes: Reduction at the integrand level IBP Master Integrals Virtual-Real & Real-Real STRIPPER, M. Czakon, Phys. Lett. B 693 (2010) 259 [arxiv: [hep-ph]]. Gabor Somogyi, Zoltan Trocsanyi D. Kosower, A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover R. Boughezal, X. Liu and F. Petriello, Phys. Rev. D 91 (2015) 9, [arxiv: [hep-ph]]. A HELAC-NNLO framework? HELAC-LO: 1999 HELAC-NLO: 2009 HELAC-NNLO:? Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

81 NNLO in the future The NNLO automation to come In a few years the new wish list should be completed pp t t, pp W + W, pp W /Z + nj, pp H + nj,... Virtual amplitudes: Reduction at the integrand level IBP Master Integrals Virtual-Real & Real-Real STRIPPER, M. Czakon, Phys. Lett. B 693 (2010) 259 [arxiv: [hep-ph]]. Gabor Somogyi, Zoltan Trocsanyi D. Kosower, A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover R. Boughezal, X. Liu and F. Petriello, Phys. Rev. D 91 (2015) 9, [arxiv: [hep-ph]]. A HELAC-NNLO framework? HELAC-LO: 1999 HELAC-NLO: 2009 HELAC-NNLO:? Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

82 NNLO in the future The NNLO automation to come In a few years the new wish list should be completed pp t t, pp W + W, pp W /Z + nj, pp H + nj,... Virtual amplitudes: Reduction at the integrand level IBP Master Integrals Virtual-Real & Real-Real STRIPPER, M. Czakon, Phys. Lett. B 693 (2010) 259 [arxiv: [hep-ph]]. Gabor Somogyi, Zoltan Trocsanyi D. Kosower, A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover R. Boughezal, X. Liu and F. Petriello, Phys. Rev. D 91 (2015) 9, [arxiv: [hep-ph]]. A HELAC-NNLO framework? HELAC-LO: 1999 HELAC-NLO: 2009 HELAC-NNLO:? Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

83 NNLO in the future The NNLO automation to come In a few years the new wish list should be completed pp t t, pp W + W, pp W /Z + nj, pp H + nj,... Virtual amplitudes: Reduction at the integrand level IBP Master Integrals Virtual-Real & Real-Real STRIPPER, M. Czakon, Phys. Lett. B 693 (2010) 259 [arxiv: [hep-ph]]. Gabor Somogyi, Zoltan Trocsanyi D. Kosower, A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover R. Boughezal, X. Liu and F. Petriello, Phys. Rev. D 91 (2015) 9, [arxiv: [hep-ph]]. A HELAC-NNLO framework? HELAC-LO: 1999 HELAC-NLO: 2009 HELAC-NNLO:? Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

84 NNLO in the future The NNLO automation to come In a few years the new wish list should be completed pp t t, pp W + W, pp W /Z + nj, pp H + nj,... Virtual amplitudes: Reduction at the integrand level IBP Master Integrals Virtual-Real & Real-Real STRIPPER, M. Czakon, Phys. Lett. B 693 (2010) 259 [arxiv: [hep-ph]]. Gabor Somogyi, Zoltan Trocsanyi D. Kosower, A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover R. Boughezal, X. Liu and F. Petriello, Phys. Rev. D 91 (2015) 9, [arxiv: [hep-ph]]. A HELAC-NNLO framework? HELAC-LO: 1999 HELAC-NLO: 2009 HELAC-NNLO:? Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

85 NNLO in the future The NNLO automation to come In a few years the new wish list should be completed pp t t, pp W + W, pp W /Z + nj, pp H + nj,... Virtual amplitudes: Reduction at the integrand level IBP Master Integrals Virtual-Real & Real-Real STRIPPER, M. Czakon, Phys. Lett. B 693 (2010) 259 [arxiv: [hep-ph]]. Gabor Somogyi, Zoltan Trocsanyi D. Kosower, A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover R. Boughezal, X. Liu and F. Petriello, Phys. Rev. D 91 (2015) 9, [arxiv: [hep-ph]]. A HELAC-NNLO framework? HELAC-LO: 1999 HELAC-NLO: 2009 HELAC-NNLO:? Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

86 NNLO in the future NNLO: the next frontier. These are the voyages at Radcor-Loopfest conferences. Its five-year mission: to explore strange new subtraction schemes, to seek out new methods to calculate Master Integrals and new tools to perform Integrand Reduction, to boldly go where no man has gone before. Thanks! Costas. G. Papadopoulos NNLO Radcor-Loopfest, LA, / 39

The Pentabox Master Integrals with the Simplified Differential Equations approach

The Pentabox Master Integrals with the Simplified Differential Equations approach Costas G. Papadopoulos INPP, NCSR Demokritos Zurich, August 25, 2016 C.G.Papadopoulos (INPP) 5box QCD@LHC 2016 1 / 36 Introduction