The Pentabox Master Integrals with the Simplified Differential Equations approach

Transcription

1 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 / 36

2 Introduction At The gluon fusion cross section N3LO, there are five contributions:!! Triple! virtual! Real-virtual squared Double virtual real Double real virtual Triple real C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog, A. Lazopoulos and B. Mistlberger, arxiv: C.G.Papadopoulos (INPP) 5box / 36

3 Factorization Perturbative QCD Factorization Collins,Soper,Sterman Calculate Scattering probability Gluon emission probability Measure Long distance interactions Particle decay rates Divide et Impera Quantity of interest: Total interaction rate Convolution of short & long distance physics σ p1p2 X = dx 1dx 2 f p1,i (x 1, µ 2 F )fp2,j (x2, µ2 F ) ˆσ ij X (x 1x 2, µ 2 F }{{}}{{} ) i,j {q,g} long distance physics short distance physics Stefan Höche From amplitudes to experiments 7 QCD as a perturbative quantum field theory: Fixed-order calculations C.G.Papadopoulos (INPP) High-energy phenomenology group Athens / 22 C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

4 Taming the beast... From Feynman graphs... gg ng # FG ,485 34, ,405 10,525, ,449,225 C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

5 Taming the beast... From Feynman graphs... gg ng # FG ,485 34, ,405 10,525, ,449,225 The Dyson-Schwinger recursion Imagine a theory with 3- and 4- point vertices and just one field. to Dyson-Schwinger recursion! Helac-Phegas Then it is straightforward to write an equation that gives the amplitude for 1 n = gg ng 2 3 HEP -4 NCSR5Democritos # C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

6 Taming the beast... C.G.Papadopoulos (INPP) 5box / 36

7 Perturbative QCD at NNLO What do we need for an NNLO calculation? p 1, p 2 p 3,..., p m+2 C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

8 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 Antenna-S, Colorfull-S, STRIPPER, q T, N-jetiness A. Gehrmann-De Ridder, T. Gehrmann and M. Ritzmann, JHEP 1210 (2012) 047 P. Bolzoni, G. Somogyi and Z. Trocsanyi, JHEP 1101 (2011) 059 M. Czakon and D. Heymes, Nucl. Phys. B 890 (2014) 152 S. Catani and M. Grazzini, Phys. Rev. Lett. 98 (2007) R. Boughezal, C. Focke, X. Liu and F. Petriello, Phys. Rev. Lett. 115 (2015) no.6, C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

9 OPP at two loops coefficients of MI spurious terms 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 0 <i 1 <i 2 <i i0 Di1 Di2 Di2 3 m 1 c(i 0 i 1 i 2 ) + c(q; i 0 i 1 i 2 ) D i 0 <i 1 <i i0 D i1 D i2 2 m 1 b(i 0 i 1 ) + b(q; i 0 i 1 ) D i 0 <i i0 D i1 1 m 1 a(i 0 ) + ã(q; i 0 ) D i i0 0 + rational terms G. Ossola, C. G. Papadopoulos and R. Pittau, Nucl. Phys. B 763, 147 (2007) C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

10 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 i1i2...im (l 1, l 2 ; {p i }) D i1 D i2... D im S m;n i1 i 2...i m (l 1, l 2 ; {p i }) D i1 D i2... D im spurious ISP irreducible integrals C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

11 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 i1i2...im (l 1, l 2 ; {p i }) D i1 D i2... D im S m;n i1 i 2...i m (l 1, l 2 ; {p i }) D i1 D i2... D im spurious ISP irreducible integrals ISP-irreducible integrals use IBPI to Master Integrals Libraries in the future: QCD2LOOP, TwOLOop P. Mastrolia, T. Peraro and A. Primo, arxiv: [hep-ph]. J. Gluza, K. Kajda and D. A. Kosower, Phys. Rev. D 83 (2011) H. Ita, arxiv: [hep-th]. C. G. Papadopoulos, R. H. P. Kleiss and I. Malamos, PoS Corfu 2012 (2013) 019. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

12 IBPI: 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 Or numerical: SecDec, Weinzierl C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

13 IBPI: 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 Or numerical: SecDec, Weinzierl C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

14 IBPI: 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 F. V. Tkachov, Phys. Lett. B 100 (1981) 65. K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys. B 192 (1981) 159. IBP Laporta: FIRE, AIR, Reduze reduce these to MI MI computed, Feynman parameters, Mellin-Barnes, Differential Equations Or numerical: SecDec, Weinzierl C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

15 IBPI: 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 Mi 2 F [a 1,..., a N ] = d d kd d 1 l D a Da N N ( d d kd d {k µ, l µ, υ µ ) } l {k µ, l µ } D a = 0 Da N N IBP Laporta: FIRE, AIR, Reduze reduce these to MI S. Laporta, Int. J. Mod. Phys. A 15 (2000) 5087 C. Anastasiou and A. Lazopoulos, JHEP 0407 (2004) 046 C. Studerus, Comput. Phys. Commun. 181 (2010) 1293 A. V. Smirnov, Comput. Phys. Commun. 189 (2014) 182 MI computed, Feynman parameters, Mellin-Barnes, Differential Equations Or numerical: SecDec, Weinzierl C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

16 IBPI: 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 Mi 2 F [a 1,..., a N ] = d d kd d 1 l D a Da N d d kd d l {k µ, l µ } 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. V. A. Smirnov, Phys. Lett. B 460 (1999) 397 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]]. Or numerical: SecDec, Weinzierl C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

17 IBPI: 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 Or numerical: SecDec, Weinzierl S. Borowka, G. Heinrich, S. P. Jones, M. Kerner, J. Schlenk and T. Zirke, Comput. Phys. Commun. 196 (2015) 470 S. Becker, C. Reuschle and S. Weinzierl, JHEP 1012 (2010) 013 C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

18 IBPI: The current approach Find a better IBP algorithm... Generating function technique, Baikov? F a1...a N = P. A. Baikov, Nucl. Instrum. Meth. A 389 (1997) 347 V. A. Smirnov and M. Steinhauser, Nucl. Phys. B 672 (2003) 199 K. J. Larsen and Y. Zhang, Phys. Rev. D 93 (2016) no.4, i=masters c (i) a 1...a N G i Baikov polynomial LZ construction Sector cut ( δ (k + p) 2 m 2) z=0 dz 1 z n=1 Cut with higher powers in denominator C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

19 IBPI: The current approach Find a better IBP algorithm... Generating function technique, Baikov? F a1...a N = c a (i) 1...a N G i i=masters Baikov polynomial LZ construction Sector cut ( δ (k + p) 2 m 2) z=0 dz 1 z n=1 Cut with higher powers in denominator C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

20 IBPI: The current approach Find a better IBP algorithm... Generating function technique, Baikov? F a1...a N = c a (i) 1...a N G i i=masters Baikov polynomial LZ construction Sector cut ( δ (k + p) 2 m 2) z=0 dz 1 z n=1 Cut with higher powers in denominator C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

21 IBPI: The current approach Find a better IBP algorithm... Generating function technique, Baikov? F a1...a N = c a (i) 1...a N G i i=masters Baikov polynomial LZ construction Sector cut ( δ (k + p) 2 m 2) z=0 dz 1 z n=1 Cut with higher powers in denominator C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

22 IBPI: The current approach Find a better IBP algorithm... Generating function technique, Baikov? F a1...a N = c a (i) 1...a N G i i=masters Baikov polynomial LZ construction Sector cut ( δ (k + p) 2 m 2) 1 dz z n=1 z=0 Cut with higher powers in denominator (3d 10) (3d 8) (3d 10) (3d 8) (d 3) F = F (d 4) 2 (p 2 ) F (d 4) 2 (p 2 ) (d 4) p 2 F11110 C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

23 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 b1,...,b n F [b 1,..., b 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 f not MI! 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]]. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

24 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 b1,...,b n F [b 1,..., b 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 f not MI! 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]]. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

25 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 b1,...,b n F [b 1,..., b 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 f not MI! 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]]. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

26 Differential Equations Approach Iterated Integrals K. T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831 Multiple Polylogarithms, Symbol algebra Goncharov Polylogarithms G (a n,..., a 1, x) = x with the special cases, G(x) = 1 and Shuffle algebra 0 1 dt G (a n 1,..., a 1, t) t a n G 0,... 0, x = 1 }{{} n! logn (x) n C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

27 Differential Equations Approach Iterated Integrals Multiple Polylogarithms, Symbol algebra Goncharov Polylogarithms 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]]. 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 C. Bogner and F. Brown Shuffle algebra G 0,... 0, x = 1 }{{} n! logn (x) n C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

28 Differential Equations Approach Iterated Integrals Multiple Polylogarithms, Symbol algebra Goncharov Polylogarithms G (a n,..., a 1, x) = x with the special cases, G(x) = 1 and Shuffle algebra 0 1 dt G (a n 1,..., a 1, t) t a n G 0,... 0, x = 1 }{{} n! logn (x) n G (a 1, a 2 ; x) G (b 1 ; x) = G (a 1, a 2, b 1 ; x) + G (a 1, b 1, a 2 ; x) + G (b 1, a 1, a 2 ; x) C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

29 The Simplified Differential Equations Approach C. G. Papadopoulos, JHEP 1407 (2014) 088 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 Factorizing external momenta dependence: x : (q 1 = xp 1, q 2 = p 12 xp 1,...) x (q 1 = p 1, q 2 = p 2,...) Now the integral as a function of x, 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 C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

30 The Simplified Differential Equations Approach C. G. Papadopoulos, JHEP 1407 (2014) 088 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 Factorizing external momenta dependence: x : (q 1 = xp 1, q 2 = p 12 xp 1,...) x (q 1 = p 1, q 2 = p 2,...) Now the integral as a function of x, 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 C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

31 The Simplified Differential Equations Approach C. G. Papadopoulos, JHEP 1407 (2014) 088 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 Factorizing external momenta dependence: x : (q 1 = xp 1, q 2 = p 12 xp 1,...) x (q 1 = p 1, q 2 = p 2,...) Now the integral as a function of x, 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 C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

32 The Simplified Differential Equations Approach 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 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 111 = c Γ 1 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. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

33 The Simplified Differential Equations Approach 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 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 111 = c Γ 1 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. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

34 The Simplified Differential Equations Approach 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 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 111 = c Γ 1 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. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

35 The Simplified Differential Equations Approach 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 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. How far we can go with the Simplified Differential Equations approach? C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

36 The Simplified Differential Equations Approach The two-loop 3-off-shell-legs triangle C.G.Papadopoulos (INPP) 5box / 36

37 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 C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

38 The Simplified Differential Equations Approach The regularized 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 C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

39 The Simplified Differential Equations Approach The regularized 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 Five-point one-loop integral with up to one off-shell leg at O(ε) C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

40 Simplified differential equations approach simple parametrization of external momenta based on Triangle rule: Criterion for the x parametrization xp 1 p 12 p 12 xp 1 Figure : Required parametrization for off mass-shell triangles after possible pinching of internal line(s). DE in one parameter: addressing problems with many scales Boundary terms straightforwardly obtained by the DE itself, based on one-scale MI Expressions in terms of GP s straightforwardly obtained by expanding the DE in ε C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

41 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. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

42 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, C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

43 Two-loop, four-point, two off-shell legs Planar topologies P 12 : { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 111m10111, 11101m111}, P 13 : { , , , , , , , , , , , , , , , , , , , , , , , , , , , , m }, P 23 : { , , , , , , , , , , , , , , , , , , , , , , , , , , , 111m10111}. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

44 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. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

45 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}. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

46 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 }. }, C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

47 Two-loop, four-point, two off-shell legs Example G P ( x, {sij }, 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, JHEP 1501 (2015) 072 C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

48 5box - one leg off-shell: all families C. G. Papadopoulos, D. Tommasini and C. Wever, arxiv: [hep-ph]. Figure : The three planar pentaboxes of the families P 1 (left), P 2 (middle) and P 3 (right) with one external massive leg. Figure : The five non-planar families with one external massive leg. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

49 5box - one leg off-shell: P1 p(q 1 )p (q 2 ) V (q 3 )j 1 (q 4 )j 2 (q 5 ), q 2 1 = q 2 2 = 0, q 2 3 = M 2 3, q 2 4 = q 2 5 = 0. xp 1 p 1234 p 4 xp 2 p 123 xp 12 Figure : The parametrization of external momenta in terms of x for the planar pentabox of the family P 1. All external momenta are incoming. s 12 := p 2 12, s 23 := p 2 23, s 34 := p 2 34, s 45 := p 2 45 = p 2 123, s 51 := p 2 15 = p 2 234, q 2 1 = q2 2 = q2 4 = q2 5 = 0 q2 3 = (s 45 s 12 x) (1 x) q 2 12 = s 12x 2 q 2 23 = s 45 (1 x) + s 23 x q 2 34 = (s 34 s 12 (1 x)) x q 2 45 = s 45 q 2 51 = s 51x C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

50 5box - one leg off-shell: P1 xp1 p1234 xp 1 p 1234 xp1 p1234 p4 xp2 p123 xp12 xp2 p123 xp12 p4 xp 2 p 123 xp 12 p 4 xp 1 p 1234 p 4 xp 1 p 1234 p 4 xp1 p 1234 p 4 xp 2 p 123 xp 12 xp 2 p 123 xp 12 xp 2 p 123 xp 12 Figure : The five-point Feynman diagrams, besides the pentabox itself in Figure 4, that are contained in the family P 1. All external momenta are incoming. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

51 5box - one leg off-shell: P1 G P 1 a 1 a 11 (x, s, ɛ) := e 2γ E ɛ d d k 1 d d k 2 1 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 1 + p 1234 ) 2a 5 k 2a 6 2 (k 2 xp 1 ) 2a 7 (k 2 xp 12 ) 2a 8 (k 2 p 123 ) 2a 9 (k 2 p 1234 ) 2a 10 (k 1 + k 2 ) 2a 11, P 1 : { , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 111m , m1111, , , , , m0111, , 010m , , , , , , 111m , m1101, , 1110m101011, , 111m , , m1111, 111m , , m0111, , m1111, 111m }, Choosing m= 1 or 2 C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

52 C.G.Papadopoulos = (s (INPP) (s s ) + s s + s (s 5box s )) 2 + 4s s s (s + s s ) QCD@LHC / 36 5box P1 - DE x G = M ({ s ij }, ε, x ) G x G = M G M = TMT 1 + ( x T) T 1 G = TG (M D ) IJ = δ IJ M II (ε = 0), I, J = G S 1 G, S = exp ( ) dx M D and M S 1 (M M D ) S C M IJ = N IJ (ε) IJ;ijk ε k 1 1 (x l i ) j + Letters (20): i=1 j=1 k=0 j=0 k=0 C IJ;jk ε k x j. 0, 1, 1 s 34 s 51 s 12, s 45 s 45 s 23, s 45 s 23 s 12, s 51 s 12, s 45 s 12, 1 s 34 s 12, 1 + s 23 s 12, s 45 s 23 +s 45 +s 51, s 45 s 34 +s 45, s 12 s 23 2s 12 s 45 s 12 s 51 s 23 s 34 +s 34 s 45 s 45 s 51 ± 1, 2s 12 (s 23 s 45 s 51 ) s 12 s 23 s 12 s 45 s 12 s 51 s 23 s 34 +s 34 s 45 s 45 s 51 ± 2, 2s 12 (s 23 s 45 s 51 ) s 12 s 23 s 12 s 51 s 23 s 34 +s 34 s 45 s 45 s 51 ± 1, 2s 12 (s 23 +s 34 s 51 ) s 12 s 45 ± 3 s 12 s 34 +s 12 s 45, s 45 s 12 +s 23,

53 C.G.Papadopoulos = (s (INPP) (s s ) + s s + s (s 5box s )) 2 + 4s s s (s + s s ) QCD@LHC / 36 5box P1 - DE x G = M ({ s ij }, ε, x ) G x G = M G M = TMT 1 + ( x T) T 1 G = TG (M D ) IJ = δ IJ M II (ε = 0), I, J = G S 1 G, S = exp ( ) dx M D and M S 1 (M M D ) S C M IJ = N IJ (ε) IJ;ijk ε k 1 1 (x l i ) j + Letters (20): i=1 j=1 k=0 j=0 k=0 C IJ;jk ε k x j. 0, 1, 1 s 34 s 51 s 12, s 45 s 45 s 23, s 45 s 23 s 12, s 51 s 12, s 45 s 12, 1 s 34 s 12, 1 + s 23 s 12, s 45 s 23 +s 45 +s 51, s 45 s 34 +s 45, s 12 s 23 2s 12 s 45 s 12 s 51 s 23 s 34 +s 34 s 45 s 45 s 51 ± 1, 2s 12 (s 23 s 45 s 51 ) s 12 s 23 s 12 s 45 s 12 s 51 s 23 s 34 +s 34 s 45 s 45 s 51 ± 2, 2s 12 (s 23 s 45 s 51 ) s 12 s 23 s 12 s 51 s 23 s 34 +s 34 s 45 s 45 s 51 ± 1, 2s 12 (s 23 +s 34 s 51 ) s 12 s 45 ± 3 s 12 s 34 +s 12 s 45, s 45 s 12 +s 23,

54 C.G.Papadopoulos = (s (INPP) (s s ) + s s + s (s 5box s )) 2 + 4s s s (s + s s ) QCD@LHC / 36 5box P1 - DE x G = M ({ s ij }, ε, x ) G x G = M G M = TMT 1 + ( x T) T 1 G = TG (M D ) IJ = δ IJ M II (ε = 0), I, J = G S 1 G, S = exp ( ) dx M D and M S 1 (M M D ) S C M IJ = N IJ (ε) IJ;ijk ε k 1 1 (x l i ) j + Letters (20): i=1 j=1 k=0 j=0 k=0 C IJ;jk ε k x j. 0, 1, 1 s 34 s 51 s 12, s 45 s 45 s 23, s 45 s 23 s 12, s 51 s 12, s 45 s 12, 1 s 34 s 12, 1 + s 23 s 12, s 45 s 23 +s 45 +s 51, s 45 s 34 +s 45, s 12 s 23 2s 12 s 45 s 12 s 51 s 23 s 34 +s 34 s 45 s 45 s 51 ± 1, 2s 12 (s 23 s 45 s 51 ) s 12 s 23 s 12 s 45 s 12 s 51 s 23 s 34 +s 34 s 45 s 45 s 51 ± 2, 2s 12 (s 23 s 45 s 51 ) s 12 s 23 s 12 s 51 s 23 s 34 +s 34 s 45 s 45 s 51 ± 1, 2s 12 (s 23 +s 34 s 51 ) s 12 s 45 ± 3 s 12 s 34 +s 12 s 45, s 45 s 12 +s 23,

55 C.G.Papadopoulos = (s (INPP) (s s ) + s s + s (s 5box s )) 2 + 4s s s (s + s s ) QCD@LHC / 36 5box P1 - DE x G = M ({ s ij }, ε, x ) G x G = M G M = TMT 1 + ( x T) T 1 G = TG (M D ) IJ = δ IJ M II (ε = 0), I, J = G S 1 G, S = exp ( ) dx M D and M S 1 (M M D ) S C M IJ = N IJ (ε) IJ;ijk ε k 1 1 (x l i ) j + Letters (20): i=1 j=1 k=0 j=0 k=0 C IJ;jk ε k x j. 0, 1, 1 s 34 s 51 s 12, s 45 s 45 s 23, s 45 s 23 s 12, s 51 s 12, s 45 s 12, 1 s 34 s 12, 1 + s 23 s 12, s 45 s 23 +s 45 +s 51, s 45 s 34 +s 45, s 12 s 23 2s 12 s 45 s 12 s 51 s 23 s 34 +s 34 s 45 s 45 s 51 ± 1, 2s 12 (s 23 s 45 s 51 ) s 12 s 23 s 12 s 45 s 12 s 51 s 23 s 34 +s 34 s 45 s 45 s 51 ± 2, 2s 12 (s 23 s 45 s 51 ) s 12 s 23 s 12 s 51 s 23 s 34 +s 34 s 45 s 45 s 51 ± 1, 2s 12 (s 23 +s 34 s 51 ) s 12 s 45 ± 3 s 12 s 34 +s 12 s 45, s 45 s 12 +s 23,

56 5box P1 - DE M IJ = N IJ (ε) i=1 j=1 k=0 C IJ;ijk ε k (x l i ) j j=0 k=0 C IJ;jk ε k x j. x 1 dt (t a n) 2 G (a n 1,..., a 1, t) 0 x 0 dt t m G (a n 1,..., a 1, t) Fuchsian N IJ (ε) = n J (ε) /n I (ε), G I n I (ε) G I M IJ = i=1 j=1 k=0 C IJ;ijk ε k (x l i ) j j=0 k=0 C IJ;jk ε k x j. G (I K i ) G, M (M x K i K i M) (I K i ) 1 i = 1, 2, 3 ( ) 19 M a x G = ε G (x l a=1 a) C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

57 5box P1 - DE Fuchsian N IJ (ε) = n J (ε) /n I (ε), G I n I (ε) G I M IJ = i=1 j=1 k=0 C IJ;ijk ε k (x l i ) j j=0 k=0 C IJ;jk ε k x j. G (I K i ) G, M (M x K i K i M) (I K i ) 1 i = 1, 2, 3 ( ) 19 M a x G = ε G (x l a=1 a) C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

58 5box P1 - DE Fuchsian N IJ (ε) = n J (ε) /n I (ε), G I n I (ε) G I M IJ = i=1 j=1 k=0 C IJ;ijk ε k (x l i ) j j=0 k=0 C IJ;jk ε k x j. G (I K i ) G, M (M x K i K i M) (I K i ) 1 i = 1, 2, 3 M (ε = 0) contains (x l i ) 2 and x 0 { dx(m (ε = 0))IJ I, J 69, 74 (K 1 ) IJ = 0 I, J = 69, 74 { dx(m (ε = 0))IJ I, J 74 (K 2 ) IJ = (K 3 ) IJ = 0 I, J = 74 dx(m (ε = 0)) IJ M.A. Barkatou and E.Pflügel, Journal of Symbolic Computation, 44 (2009),1017 ( ) 19 M a x G = ε G (x l a=1 a) C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

59 5box P1 - DE Fuchsian N IJ (ε) = n J (ε) /n I (ε), G I n I (ε) G I M IJ = i=1 j=1 k=0 C IJ;ijk ε k (x l i ) j j=0 k=0 C IJ;jk ε k x j. G (I K i ) G, M (M x K i K i M) (I K i ) 1 i = 1, 2, 3 ( ) 19 M a x G = ε G (x l a=1 a) C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

60 5box P1 - solution Solution: G = ε 2 b ( 2) 0 + ε 1 ( G am ab ( 2) 0 + b ( 1) ) 0 + ε 0 ( G ab M am b b ( 2) 0 + G am ab ( 1) 0 + b (0) ) 0 ( + ε Gabc M am b M c b ( 2) 0 + G ab M am b b ( 1) 0 + G am ab (0) ) 0 +b(1) 0 + ε 2 ( G abcd M am b M c M d b ( 2) 0 + G abc M am b M c b ( 1) 0 + Gab M am b b (0) 0 + G am ab (1) ) 0 +b(2) 0 b (k) 0, k = 2,..., 2 representing the x-independent boundary terms in the limit x = 0 at order εk 2 G ε k k+2 b (k) x 0 n logn (x) + subleading terms. k= 2 n=0 G a,b,... = G (l a, l b,... ; x) with a, b, c, d = 1,..., 19. Uniform transcendental: UT multi- vs one-parameter DE M a depend on kinematics, but eigenvalues not: (x l a) naε, n a positive integers, x l a. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

61 5box P1 - solution Solution: G = ε 2 b ( 2) 0 + ε 1 ( G am ab ( 2) 0 + b ( 1) ) 0 + ε 0 ( G ab M am b b ( 2) 0 + G am ab ( 1) 0 + b (0) ) 0 ( + ε Gabc M am b M c b ( 2) 0 + G ab M am b b ( 1) 0 + G am ab (0) ) 0 +b(1) 0 + ε 2 ( G abcd M am b M c M d b ( 2) 0 + G abc M am b M c b ( 1) 0 + Gab M am b b (0) 0 + G am ab (1) ) 0 +b(2) 0 b (k) 0, k = 2,..., 2 representing the x-independent boundary terms in the limit x = 0 at order εk 2 G ε k k+2 b (k) x 0 n logn (x) + subleading terms. k= 2 n=0 G a,b,... = G (l a, l b,... ; x) with a, b, c, d = 1,..., 19. Uniform transcendental: UT multi- vs one-parameter DE M a depend on kinematics, but eigenvalues not: (x l a) naε, n a positive integers, x l a. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

62 5box - boundary terms Resummed G res = lim x 0 G = j c j x i 0+jɛ + d j x i 0+1+jɛ + O(x i 0+2 ), DE: using the above and equating terms x i+jɛ, linear equations for c i and d i bottom-up: MI with homogeneous DE treated exactly MI needing special treatment (20) Expansion by regions (11) Shifted boundary point (6) Extraction from known integrals (3) C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

63 5box - boundary terms Resummed G res = lim x 0 G = j c j x i 0+jɛ + d j x i 0+1+jɛ + O(x i 0+2 ), DE: using the above and equating terms x i+jɛ, linear equations for c i and d i bottom-up: MI with homogeneous DE treated exactly MI needing special treatment (20) Expansion by regions (11) Shifted boundary point (6) Extraction from known integrals (3) C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

64 5box - boundary terms Resummed G res = lim x 0 G = j c j x i 0+jɛ + d j x i 0+1+jɛ + O(x i 0+2 ), DE: using the above and equating terms x i+jɛ, linear equations for c i and d i bottom-up: MI with homogeneous DE treated exactly MI needing special treatment (20) Expansion by regions (11) Shifted boundary point (6) Extraction from known integrals (3) C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

65 5box - boundary terms Resummed G res = lim x 0 G = j c j x i 0+jɛ + d j x i 0+1+jɛ + O(x i 0+2 ), DE: using the above and equating terms x i+jɛ, linear equations for c i and d i bottom-up: MI with homogeneous DE treated exactly MI needing special treatment (20) Expansion by regions (11) {( ), ( ), ( ), ( ), ( ), ( ), Shifted boundary point (6) ( ), (111m ), (111000m1111), ( ), (111001m0111)}. : {( ), ( ), ( ), ( ), ( )} (s 12 s 34 + s 51 )/s 12 : {( )} Extraction from known integrals (3) G (x, s 12, s 34, s 51 ) = G (x = 1, s 12, s 23, s 45 ), G (x, s 12, s 34, s 51 ) = G (x = 1, s 12, s 23, s 45 ), G 111m (x, s 12, s 34, s 51 ) = G 111m (x = 1, s 12, s 23, s 45 ), s 12 = x2 s 12, s 23 = xs 51, s 45 = xs 12 + xs 34 + x 2 s 12. (1) C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

66 5box - boundary terms Resummed G res = lim x 0 G = j c j x i 0+jɛ + d j x i 0+1+jɛ + O(x i 0+2 ), DE: using the above and equating terms x i+jɛ, linear equations for c i and d i bottom-up: MI with homogeneous DE treated exactly MI needing special treatment (20) Expansion by regions (11) Shifted boundary point (6) Extraction from known integrals (3) Systematic approach: combining information from the expansion by regions technique (asy2) and the DE itself Mellin-Barnes, XSummer C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

67 5box - on-shell All planar one-shell 5box by taking the limit x 1. x = 1 corresponds to l 2 G = n 2 n+2 ε n 1 i! c(n) i log i (1 x) i=0 with M 2 the residue matrix at x = 1 and G trunc G reg (x = 1) ( G x=1 = I M ) 2 M2 2 G trunc C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

68 5box - on-shell All planar one-shell 5box by taking the limit x 1. x = 1 corresponds to l 2 with M 2 the residue matrix at x = 1 and c (n) i = M 2 c (n 1) i 1 i 1 G reg = n 2 ε n c (n) 0. characteristic polynomial: x 61 (1 + x) 9 (2 + x) 4 ( (1 x) 2ε 1 ) ( (1 x) ε 1 ) G = G reg + ( 2ε) X + ( ε) Y X = ε n X (n) Y = ε n Y (n). n 1 n 1 ( 1) n M 2 n = M 2 2 ( 2 n 1 1 ) + M 2 ( 2 n 1 2 ), n 1. minimal polynomial: x(x + 1)(x + 2) G trunc G reg (x = 1) ( G x=1 = I M ) 2 M2 2 G trunc C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

69 5box - on-shell All planar one-shell 5box by taking the limit x 1. x = 1 corresponds to l 2 G = n 2 n+2 ε n 1 i! c(n) i log i (1 x) i=0 with M 2 the residue matrix at x = 1 and ( ) (1 x) 2ε 1 G = G reg + ( 2ε) X + ( ) (1 x) ε 1 ( ε) Y G trunc G reg (x = 1) ( G x=1 = I M ) 2 M2 2 G trunc C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

70 5box - numerical checks O(3, 000) GPs for all 74 MI Directly computed by using GiNaC All invariants negative (Euclidean): perfect agreement with SecDec O(10) secs. HyperInt analytic extraction of imaginary parts before numerics: increasing efficiency by O(100) Our physical region result published awaiting cross-checks when other calculations become available for 5boxes. Computing time expected after analytic extraction of imaginary parts O(10) secs for all 74 MI. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

71 5box - numerical checks O(3, 000) GPs for all 74 MI Directly computed by using GiNaC J. Vollinga and S. Weinzierl, Comput. Phys. Commun. 167 (2005) 177 All invariants negative (Euclidean): perfect agreement with SecDec O(10) secs. HyperInt analytic extraction of imaginary parts before numerics: increasing efficiency by O(100) Our physical region result published awaiting cross-checks when other calculations become available for 5boxes. Computing time expected after analytic extraction of imaginary parts O(10) secs for all 74 MI. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

72 5box - numerical checks O(3, 000) GPs for all 74 MI Directly computed by using GiNaC All invariants negative (Euclidean): perfect agreement with SecDec O(10) secs. HyperInt analytic extraction of imaginary parts before numerics: increasing efficiency by O(100) Our physical region result published awaiting cross-checks when other calculations become available for 5boxes. Computing time expected after analytic extraction of imaginary parts O(10) secs for all 74 MI. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

73 5box - numerical checks O(3, 000) GPs for all 74 MI Directly computed by using GiNaC All invariants negative (Euclidean): perfect agreement with SecDec O(10) secs. HyperInt analytic extraction of imaginary parts before numerics: increasing efficiency by O(100) E. Panzer, Comput. Phys. Commun. 188 (2014) 148 Our physical region result published awaiting cross-checks when other calculations become available for 5boxes. Computing time expected after analytic extraction of imaginary parts O(10) secs for all 74 MI. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

74 5box - numerical checks O(3, 000) GPs for all 74 MI Directly computed by using GiNaC All invariants negative (Euclidean): perfect agreement with SecDec O(10) secs. HyperInt analytic extraction of imaginary parts before numerics: increasing efficiency by O(100) Our physical region result published awaiting cross-checks when other calculations become available for 5boxes. Computing time expected after analytic extraction of imaginary parts O(10) secs for all 74 MI. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

75 5box - numerical checks O(3, 000) GPs for all 74 MI Directly computed by using GiNaC All invariants negative (Euclidean): perfect agreement with SecDec O(10) secs. HyperInt analytic extraction of imaginary parts before numerics: increasing efficiency by O(100) Our physical region result published awaiting cross-checks when other calculations become available for 5boxes. Computing time expected after analytic extraction of imaginary parts O(10) secs for all 74 MI. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

76 Discussion 1 SDE: proven reliable and efficient: evolving 2 IBP: better understanding 3 DE canonical form 4 Complete massless MI with up to 8 denominators, at least 3 of-shell legs 5 Including internal masses 6 Feynman parametrization - MB vs DE: pros and cons 7 Integrand reduction at two loops: implementation C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

77 Discussion 1 SDE: proven reliable and efficient: evolving 2 IBP: better understanding 3 DE canonical form 4 Complete massless MI with up to 8 denominators, at least 3 of-shell legs 5 Including internal masses 6 Feynman parametrization - MB vs DE: pros and cons 7 Integrand reduction at two loops: implementation C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

78 Discussion 1 SDE: proven reliable and efficient: evolving 2 IBP: better understanding 3 DE canonical form O. Gituliar and V. Magerya, arxiv: [hep-ph]. Christoph Meyer, Loops and Legs Complete massless MI with up to 8 denominators, at least 3 of-shell legs 5 Including internal masses 6 Feynman parametrization - MB vs DE: pros and cons 7 Integrand reduction at two loops: implementation C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

79 Discussion 1 SDE: proven reliable and efficient: evolving 2 IBP: better understanding 3 DE canonical form 4 Complete massless MI with up to 8 denominators, at least 3 of-shell legs 5 Including internal masses 6 Feynman parametrization - MB vs DE: pros and cons 7 Integrand reduction at two loops: implementation C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

80 Discussion 1 SDE: proven reliable and efficient: evolving 2 IBP: better understanding 3 DE canonical form 4 Complete massless MI with up to 8 denominators, at least 3 of-shell legs 5 Including internal masses 6 Feynman parametrization - MB vs DE: pros and cons 7 Integrand reduction at two loops: implementation C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

81 Discussion 1 SDE: proven reliable and efficient: evolving 2 IBP: better understanding 3 DE canonical form 4 Complete massless MI with up to 8 denominators, at least 3 of-shell legs 5 Including internal masses 6 Feynman parametrization - MB vs DE: pros and cons 7 Integrand reduction at two loops: implementation C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

82 Discussion 1 SDE: proven reliable and efficient: evolving 2 IBP: better understanding 3 DE canonical form 4 Complete massless MI with up to 8 denominators, at least 3 of-shell legs 5 Including internal masses 6 Feynman parametrization - MB vs DE: pros and cons 7 Integrand reduction at two loops: implementation C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

83 Summary 1 Understanding QFT and provide precise calculations for analysis of experimental data 2 NLO revolution: plethora of highly automated codes/software 3 LHC physics benefits: unprecedented 4 Moving beyond NLO: NNLO and N3LO 5 NNLO revolution: ante portas? C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

84 Summary 1 Understanding QFT and provide precise calculations for analysis of experimental data 2 NLO revolution: plethora of highly automated codes/software 3 LHC physics benefits: unprecedented 4 Moving beyond NLO: NNLO and N3LO 5 NNLO revolution: ante portas? C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

85 Summary 1 Understanding QFT and provide precise calculations for analysis of experimental data 2 NLO revolution: plethora of highly automated codes/software 3 LHC physics benefits: unprecedented 4 Moving beyond NLO: NNLO and N3LO 5 NNLO revolution: ante portas? C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

86 Summary 1 Understanding QFT and provide precise calculations for analysis of experimental data 2 NLO revolution: plethora of highly automated codes/software 3 LHC physics benefits: unprecedented 4 Moving beyond NLO: NNLO and N3LO 5 NNLO revolution: ante portas? C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

87 Summary 1 Understanding QFT and provide precise calculations for analysis of experimental data 2 NLO revolution: plethora of highly automated codes/software 3 LHC physics benefits: unprecedented 4 Moving beyond NLO: NNLO and N3LO 5 NNLO revolution: ante portas? C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

88 C.G.Papadopoulos (INPP) 5box / 36

89 Backup Slides C.G.Papadopoulos (INPP) 5box / 36

90 Two-loop, four-point, two off-shell legs Physical region S > ( M 23 + M 2 4 ) 2, T < 0, U < 0, M 2 3 > 0, M 2 4 > 0, q 2 = TU M2 3 M2 4 S > 0, q s 12 x > 1, > 1, xs 12 > q, q > 0. s 23 { s 23 < 0, s 12 + s 23 > q, q > 0 x > 1, s 23 > 0, s 12 + s 23 < q, s 12 > q/x. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

91 Two-loop, four-point, two off-shell legs Analytic continuation Feynman propagator D D + iɛ C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

92 Two-loop, four-point, two off-shell legs Analytic continuation Feynman propagator D D + iɛ s ij (s 12, s 23 and q in the present study) and the parameter x, s ij s ij + iδ sij η, x x + iδ x η, with η 0. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

93 Two-loop, four-point, two off-shell legs Analytic continuation Feynman propagator D D + iɛ s ij (s 12, s 23 and q in the present study) and the parameter x, s ij s ij + iδ sij η, x x + iδ x η, with η 0. δ sij and δ x are determined as follows: 1) Input data: G P ( ( 1 + x)( q + s 12 x)) 1 2ɛ (1 x) 1 2ɛ (1 xs 12 /q) 1 2ɛ ( q) 1 2ɛ C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

94 Two-loop, four-point, two off-shell legs Analytic continuation Feynman propagator D D + iɛ s ij (s 12, s 23 and q in the present study) and the parameter x, s ij s ij + iδ sij η, x x + iδ x η, with η 0. δ sij and δ x are determined as follows: 1) Input data: G P ( ( 1 + x)( q + s 12 x)) 1 2ɛ (1 x) 1 2ɛ (1 xs 12 /q) 1 2ɛ ( q) 1 2ɛ 2) Second graph polynomial: F C. Bogner and S. Weinzierl, Int. J. Mod. Phys. A 25 (2010) 2585 [arxiv: [hep-ph]]. in terms of s ij and x, should acquire a definite-negative imaginary part in the limit η 0. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36

95 Two-loop, four-point, two off-shell legs Analytic continuation Feynman propagator D D + iɛ s ij (s 12, s 23 and q in the present study) and the parameter x, s ij s ij + iδ sij η, x x + iδ x η, with η 0. δ sij and δ x are determined as follows: 1) Input data: G P ( ( 1 + x)( q + s 12 x)) 1 2ɛ (1 x) 1 2ɛ (1 xs 12 /q) 1 2ɛ ( q) 1 2ɛ 2) Second graph polynomial: F C. Bogner and S. Weinzierl, Int. J. Mod. Phys. A 25 (2010) 2585 [arxiv: [hep-ph]]. in terms of s ij and x, should acquire a definite-negative imaginary part in the limit η 0. E. Panzer, Comput. Phys. Commun. 188 (2014) 148 [arxiv: [hep-th]]. C.G.Papadopoulos (INPP) 5box QCD@LHC / 36