Differential Equations for Feynman Integrals

for Feynman Integrals Ulrich Schubert Max-Planck-Institut für Physik Föhringer Ring 6, München January 18, 2015 based on work with M. Argeri, S. Di Vita, P. Mastrolia, E. Mirabella, J. Schlenk, L. Tancredi, V. Yundin arxiv:1401.2979, arxiv:1408.3107 Ulrich Schubert for Feynman Integrals 1 / 45

Table of Contents Introduction 1 Introduction 2 3 massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop 4 Ulrich Schubert for Feynman Integrals 2 / 45

Three Steps of a modern multi-loop and -leg calculation 1) Find integral basis for the process 2) Determine coefficients of the basis elements 3) Calculate elements (integrals) of the basis Ulrich Schubert for Feynman Integrals 3 / 45

Tools Integration-by-parts identities (IBP-Ids) Chetyrkin,Tkachov; Laporta Integrand Reduction Ossola, Papadopoulos, Pittau; Mastrolia, Ossola; Badger, Frellesvig, Zhang; Zhang; Mastrolia, Mirabella, Ossola, Peraro Generalized/Maximal Unitarity, Bern, Dixon, Dunbar, Kosower Kosower, Larsen; Caron-Huot; Carrasco, Johansson; Zhang, Kotikov;Remiddi; Gehrmann, Remiddi; Henn;Papadopoulos Ulrich Schubert for Feynman Integrals 4 / 45

one-loop two-loop graphs only planar planar and non-planar integral basis known determined case by case integrals known only for certain cases IR poles cancellation between cancellation between twoone-loop and tree level and one-loop and tree level appearing functions logs and dilogs logs, polylogarithms, generalized polylogs, elliptic functions and more?? a lot more exploring has to be done Ulrich Schubert for Feynman Integrals 5 / 45

1 Introduction 2 3 massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop 4 Ulrich Schubert for Feynman Integrals 6 / 45

Feynman integrals are functions of Mandelstam variables Internal and external masses Spacetime dimensions Facts Not all Feynman integrals are independent IBP-ids connect different Feynman integrals We can find an integral basis called master integrals Exploit this by Taking derivatives of the master integrals in respect to the kinematic invariants Reduce result back to master integrals Solve the obtained first order differential equation analytically Let s have a look at an easy example Ulrich Schubert for Feynman Integrals 7 / 45

The one-loop massless bubble Construct differential operator Apply derivative Reduce back to master integrals with IBP-ids Gives us the differential equation Ulrich Schubert for Feynman Integrals 8 / 45

First order differential equation x f (x, ɛ) = A(x, ɛ) f (x, ɛ) where ɛ is the dimensional regularization parameter D = 4 2ɛ. Properties of A(x, ɛ) Block triangular Rational in x and ɛ Satisfies integrability condition (for at least two invariants x and y) y A(x, ɛ) xa(y, ɛ) + [A(x, ɛ), A(y, ɛ)] = 0 Bottom-up Approach Solve each line in A(x, ɛ) bottom up Previously solved integrals will appear as inhomogenous parts of the next DEQ Henn Conjecture: We can find a basis such that x g(x, ɛ) = ɛã(x) g(x, ɛ) Makes integration simple But finding the basis is difficult Ulrich Schubert for Feynman Integrals 9 / 45

1 Introduction 2 3 massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop 4 Ulrich Schubert for Feynman Integrals 10 / 45

Change of Basis g(x, ɛ) = B(x, ɛ) f (x, ɛ) Â(x, ɛ) = B 1 (x, ɛ)a(x, ɛ)b(x, ɛ) B 1 (x, ɛ) xb(x, ɛ) Reformulate the Problem Find a change of basis which brings us to the canonical form Solving a DEQ for B(x, ɛ) xb(x, ɛ) = A(x, ɛ)b(x, ɛ) ɛb(x, ɛ)â(x) Can be as hard as initial problem Assumption Linear DEQ A(x, ɛ) = A 0(x) + ɛa 1(x) We have to solve xb(x) = A 0(x)B(x) Can be done with Magnus Theorem Ageri, Di Vita, Mastrolia, Mirabella, Schlenk, Tancredi, U.S. Ulrich Schubert for Feynman Integrals 11 / 45

Magnus theorem Starting from a first order differential equation x f (x) = A(x) f (x) The solution is given by the Magnus exponential f (x) = e Ω[A](x,x 0 ) f (x0) e Ω[A](x) f (x0) Ω[A](x) = Ω 1[A](x) = Ω 2[A](x) = 1 2 x x 0 dτ 1A(τ 1) x x 0 dτ 1 τ1 x 0 dτ 2[A(τ 1), A(τ 2)],... Ω n[a](x) n=1 Connected to the Dyson Series ) f (x) = (1 + P n(x) f (x0), P n(x) = n=1 P 1(x) = Ω 1(x) P 2(x) = Ω 2(x) + 1 2 Ω2 1(x),... x τn 1 dτ 1... dτ n A(τ 1)A(τ 2)...A(τ n) x 0 x 0 Ulrich Schubert for Feynman Integrals 12 / 45

Finding the canonical form with Magnus From a linear DEQ A(x, ɛ) = A 0(x) + ɛa 1(x) Magnus Theorem provides a basis change to the canonical form B(x) = e Ω[A 0](x) g(x, ɛ) = B(x) f (x, ɛ) Reducing DEQ Lee (2014) DEQ is rational in x Mosers algorithm reduces all poles to simple poles (if possible) k S n(ɛ) A(x, ɛ) = with x n being constant x x n n=1 Referred to as Fuchsian form If Eigenvalues of S n are linear in ɛ with integer coefficients We can shift them to multiples of ɛ A similarity transformation ɛ S = T 1 (ɛ)s(ɛ)t (ɛ) gives us the canonical form Ulrich Schubert for Feynman Integrals 13 / 45

Example: massive QED Sunrise Three master integrals Dimensionless variable s (1 x)2 = m2 x Linear and Fuchsian DEQ x f (x) = A(x, ɛ) f (x) 0 0 0 A(x, ɛ) = ɛ ɛ 1+6ɛ + 1+3ɛ + 1 2(1 x) 2(1+x) 1+x x 1 x 0 Let s apply Magnus and Lees algorithm 2ɛ x + 4ɛ 1 x 1 2ɛ 2(1 x) ɛ x 1 6ɛ 2(1+x) 1 + 2 x 1 x Ulrich Schubert for Feynman Integrals 14 / 45

QED Sunrise with Magnus Introduction Split ɛ 0 -part into diagonal and off-diagonal part and transform with diagonal part first A 0(x) = D 0(x) + N 0(x), dx D B 1 = e 0 (x) = 1 0 0 0 0 0 x 1 x 2 0 x (1 x) 2 Transformed DEQ Â(x) = B 1 1 A(x)B 1 B 1 1 xb 1 = 0 0 0 ɛ x 0 3ɛ x 6ɛ 1+x 1 2ɛ (x 1) 2 2ɛ x 0 ɛ x Ulrich Schubert for Feynman Integrals 15 / 45

QED Sunrise with Magnus Introduction Transform with off-diagonal part  0(x) = ˆN 0(x), Transformed DEQ II dx ˆN B 2 = e 0 (x) = Ã(x) = B 1 2 Â(x)B 2 B 1 2 xb 2 0 0 0 Ã(x) = ɛ x 0 g = B 1 2 B 1 1 f = 2ɛ + 5ɛ 1 x x 2ɛ x 1 0 0 0 1 1 1 x 0 0 1 6ɛ 2ɛ 6ɛ + 3ɛ 1+x 1 x x 1+x Masters for canonical basis are given by 1 0 0 1 x 0 (1 x)(x+1) x 0 0 2ɛ 2ɛ 1 x x x (x 1) 2 x f Ulrich Schubert for Feynman Integrals 16 / 45

QED Sunrise with Lee Introduction Focus only on 2x2 system of the Sunrise, since the rest of the system is already ɛ-factorized System is Fuchsian Shift all Eigenvalues to multiples of ɛ x 1 : { 1 2ɛ, 2 + 2ɛ} x 0 : {1 + ɛ, 1 + 2ɛ} x 1 : {0, 1 6ɛ} x : {1 + ɛ, 1 + 2ɛ} A balance transformation will shift one eigenvalue by +1 the other one by 1 B(P, x 1, x 2 x) = P + c x x2 x x 1 P, P = u(λ 1)v (λ 2), P = 1 P After four transformation we brought all Eigenvalues in the right form x 1 : { 2ɛ, 2ɛ} x 0 : {ɛ, 2ɛ} x 1 : {0, 6ɛ} x : {ɛ, 2ɛ} Ulrich Schubert for Feynman Integrals 17 / 45

QED Sunrise with Lee Introduction The last step is to find a similarity transformation which brings us to the canonical form For each pole matrix we solve A(x, ɛ) A(x, µ) T (ɛ, µ) = T (ɛ, µ) ɛ µ Finally we arrive at ( 5 x x g = ɛ 6 + 2 1+x 1 x 6 3 + 2 x 1+x 1 x 2 x 2 1 x 2 x Combining all transformation we find for our masters ( ) (1 x)(1+x) 1 x x x g = f Reminder: BMagnus = 0 (x 1)2 x ) g 1 0 0 0 (1 x)(x+1) x 0 0 1 x x (x 1) 2 x Ulrich Schubert for Feynman Integrals 18 / 45

1 Introduction 2 3 massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop 4 Ulrich Schubert for Feynman Integrals 19 / 45

Solving DEQ in the matrix approach The canonical form xg(x, ɛ) = ɛã(x)g(x, ɛ) implies ( ) g(x, ɛ) = 1 + ɛ n P n(x) g(x 0, ɛ) P n(x) = n=1 x Ã(x) determines the types of functions which will appear If Ã(x) is Fuchsian τn 1 dτ 1... dτ n Ã(τ 1)Ã(τ2)...Ã(τn) x 0 x 0 Ã(x) = k n=1 S n x x n with x n being constant then g(x, ɛ) can be written in terms of Goncharov Polylogarithms Ulrich Schubert for Feynman Integrals 20 / 45

Goncharov Polylogarithm Introduction Goncharov Basic definition G(a 1; x) = G(a 1,..., a n; x) = x 0 x 0 dt t a 1 G( 0 n; x) = 1 n! logn (x) n is the weight of the Polylog Connections to other functions: dt t a 1 G(a 2,..., a n ; t) Logarithm: G(a 1,..., a 1; x) = 1 n! logn (1 x a 1 ) classical Polylog G( 0 n, 1; x) = Li n(x) Harmonic Polylogs (HPLs) Remiddi, Vermaseren G( a; x) = ( 1) p H( a, x) with a i { 1, 0, 1} Two-dimensional harmonic polylogarithms Gehrmann, Remiddi G( a; x) = ( 1) p H( a, x) with a i {0, 1, y, 1 y} Ulrich Schubert for Feynman Integrals 21 / 45

Properties Introduction Invariant under rescaling for k C G(k a; k x) = G( a; x) Shuffle relations G( a; x)g( b; x) = c a b G( c; x) where a b means taking every permutation of the elements in a and b such that the individual ordering of a and b is kept e.g. G(a 1; x)g(b 1, b 2; x) = G(a 1, b 1, b 2; x) + G(b 1, a 1, b 2; x) + G(b 1, b 2, a 1; x) Many more known/unknown relations Led to development of Symbol and Coproduct Goncharov, Spradlin, Vergu, Volovich; Duhr Ulrich Schubert for Feynman Integrals 22 / 45

1 Introduction 2 3 massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop 4 Ulrich Schubert for Feynman Integrals 23 / 45

Recap: of the canonical DEQ ( ) g(x, ɛ) = 1 + ɛ n P n(x) g(x 0, ɛ) n=1 We can obtain g(x 0, ɛ) by Known limits Taking the limit x x 0 to a known function Fix the boundary constant by matching the solution to the known function lim p s 0 p 2 1 s = p 1 p 2 Pseudo-thresholds has physical and unphysical divergences Physical divergences correspond to particle thresholds Unphysical divergences are absent in the final result Demanding their absence gives relations between the boundary constants of the master integrals Leftover constants must be provided (usually elementary integrals) Ulrich Schubert for Feynman Integrals 24 / 45

massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop 1 Introduction 2 3 massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop 4 Ulrich Schubert for Feynman Integrals 25 / 45

massive QED vertex at two-loop massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop (1) Two-loop QED vertices [Bonciani, Mastrolia and Remiddi 03] Bonciani, Mastrolia, Remiddi 17 MI s for all relevant topologies p1 p2 p1 k1 p2 k2 s (1 x)2 m2 = x The f s obey an -linear DE [ADVMMSST 14] p12 p12 p1 p12 f1 = 2 T1 f2 = 2 T2 f3 = 2 T3 f4 = 2 T4 f5 = 2 T5 T1 T2(s) p1 T3(s) p1 T4 p1 T5(s) f6 = 2 T6 f7 = 2 T7 f8 = 3 T8 f9 = 3 T9 f10 = 2 T10 f11 = 3 T11 f12 = 3 T12 f13 = 2 T13 f14 = 3 T14 f15 = 4 T15 p12 p2 p2 p2 p2 f16 = 4 T16 f17 = 4 T17 p1 T6(s) p1 T7 p1 T8(s) p1 T9(s) p1 T10(s) After getting rid of A 0, the g s obey a canonical DE p2 T11(s) p2 T12(s) p1 p2 T13(s) p1 p2 T14(s) p2 T15(s) @ x g(, x) = Â1(x) g(, x) Â 1 (x) = M 1 x + M 2 1 + x + M 3 1 x [(k1 +k2) 2 ] p2 p2 M1 = 0 B @ T16(s) T17(s 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 5 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 4 0 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 1 0 0 0 1 2 3 0 0 3 3 0 0 0 0 0 0 0 0 0 0 1 1 2 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 3 0 0 0 0 3 3 0 0 0 0 0 1 0 0 1 1 0 2 2 0 0 0 0 2 2 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 2 1 0 2 1 0 2 0 2 0 0 2 2 2 0 0 0 0 1 1 0 3 2 0 6 2 0 0 4 4 4 1 C A M2 = 0 B @ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 C A M3 = 0 B @ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 4 0 0 4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 C A S. Di Vita (MPI Munich) Magnus and Dyson Series for MI s LoopFest XIII (NYC) 11 / 17 Ulrich Schubert for Feynman Integrals 26 / 45

Introduction massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop ( ) Bonciani, Remiddi, P.M. (2013) T AdVMMSST (2014) p1 M 2 = 1 2, M 1 = 5 ] [1 2 2 H(0,x), (1 x) M0 = 19 [ ] 2 + ζ(2) + 2 1 [ζ(2) 5H(0, x)+2h( 1, 0, x)] (1 x) + 2 [ ] 1 H(0, 0,x)+ (1 x) (1 x) 1 [ζ(2)h(0,x) (1 + x) +H(0, 0, 0,x)]. [ ] N 2 a N 1 a N0 a [ ] [ ] [ (p2 k1) ] +H(0, 0, 0,x)]. ( = 1 8 + 1 [ x + 1 ], ( 16 x = 9 [ 2+x + 1 ] 1 [ 4+x 1 ] 1 H(0,x)+ 32 x 8 x (1 x) H(0,x), ( = 63 32 + ζ(2) 2 + 63 [( 1+ 16 ) 64 63 ζ(2) x + 1 ] ζ(2) x (1 x) 1 [ 32 + 9x 16 9 ] (16 + ζ(2)) ζ(2) H(0,x)+ H(0,x) x 4(1 x) 4(1 + x) H(0,x) 1 [ 2 1 4 x 4 ] H(0, 0,x)+ 1 [ 4+x 1 ] (1 x) 4 x 8 H( 1,0,x) (1 x) + 1 [ ] 1 4 (1 x) 1 H(0, 0, 0,x). ( (1 + x) g (0) 12 =0, g (1) 12 =0, g (2) 12 =0, g (3) 12 = H(0, 0, 0; x) ζ2h(0; x), g (4) 12 = 2H( 1, 0, 0, 0; x) +2H(0, 1, 0, 0; x) +2H(0, 0, 1, 0; x) g (0) 13 =0, g (1) 13 =0, 3H(0, 0, 0, 0; x) 4H(0, 1, 0, 0; x) +ζ2( 2H( 1, 0; x) +6H(0, 1; x) H(0, 0; x)) + 2 ζ3 H(0; x)+ ζ4 4, g (2) ζ2 13 =H(0, 0; x)+3 2, g (3) 13 = 2H( 1, 0, 0; x) 2H(0, 1, 0; x)+4h(0, 0, 0; x)+4h(1, 0, 0; x) + ζ2( 6H( 1; x)+2h(0;x) 3log2) ζ3 4, g (4) 13 =4H( 1, 1, 0, 0; x) +4H( 1, 0, 1, 0; x) 8H( 1, 0, 0, 0; x) p2 p1 p2 8H( 1, 1, 0, 0; x) +4H(0, 1, 1, 0; x) 8H(0, 1, 0, 0; x) 8H(0, 0, 1, 0; x) +10H(0, 0, 0, 0; x) +12H(0, 1, 0, 0; x) 8H(1, 1, 0, 0; x) 8H(1, 0, 1, 0; x) +16H(1, 0, 0, 0; x) 1 +16H(1, 1, 0, 0; x) +12Li4 2 + log4 2 +2ζ2 (12 log 2 H( 1; x) 2 +12log2H(1;x)+6H( 1, 1; x) 2H( 1, 0; x) 8H(0, 1; x) +H(0, 0; x) 12 H(1, 1; x)+4h(1, 0; x)+3log 2 2 ) 2 ζ3(5 H( 1; x)+4h(0;x)+11h(1;x)) T T 47 ζ4, 4 Ulrich Schubert for Feynman Integrals 27 / 45

massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop Non-planar massless box Tausk (1999); Anastasiou, Gehrmann, Oleari, Remiddi, Tausk (2) Two-loop non-planar box [Tausk 99; Anastasiou, Gehrmann, Oleari, Remiddi, Tausk 00] p1 p3 12 MI s for the crossed topology k1 k1 k2 p2 x = s t, s > 0, t < 0, s > t p2 p4 p1 p1 p3 p1 p3 The f s obey an -linear DE [ADVMMSST 14] p12 Ta(s) p1 p2 Te(s) p2 Tb(s) p1 p3 p2 p4 Tf (s,t) p2 p4 Tc(s,t) p1 p2 Tg(s,t p3 p4 p2 p4 Td(s,t) [(k2 +p1) 2 ] After getting rid of A 0, the g s obey a canonical DE @ x g(, x) = Â1(x) g(, x) Â 1 (x) = M 1 x + M 2 1 x M1 = 0 B @ 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 0 0 2 0 0 0 0 0 0 0 0 3 2 3 0 1 0 0 0 0 0 0 0 1 1 2 0 0 0 0 2 0 0 0 0 0 2 0 1 1 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 6 9 2 0 4 2 18 12 12 1 1 2 3 9 21 4 3 2 3 12 6 18 0 0 2 4 4 1 C A f1 = 2 sta(s), f2 = 2 tta(t), f3 = 2 uta(u), f4 = 3 st b(s), f5 = 3 sttc(s, t), f6 = 3 sutc(s, u), f7 = 4 ut d(s, t), f8 = 4 st d(t, u), f9 = 4 tt d(u, s), f10 = 4 s 2 Te(s), f11 = 4 3 h stut f (s, t) 2 s 2 Ta(s)+t 2 Ta(t)+u 2 Ta(u) 4 s (4 + 1) i 4 4 u 2 T d(s, t)+s 2 T d(t, u)+t 2 T d(u, s), f12 = 4 3 h sttg(s, t) 2 s 2 Ta(s)+t 2 Ta(t)+u 2 Ta(u) 8 u (4 + 1) i 4 4 u 2 T d(s, t)+s 2 T d(t, u)+t 2 T d(u, s), M2 = 0 B @ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 0 3 1 0 0 0 0 0 0 0 0 3 2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 1 2 0 0 0 0 2 0 0 0 0 2 1 2 0 1 2 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 6 9 2 0 4 2 18 12 12 1 1 2 21 9 27 4 6 2 4 12 6 24 1 1 0 4 4 S. Di Vita (MPI Munich) Magnus and Dyson Series for MI s LoopFest XIII (NYC) 12 / 17 Ulrich Schubert for Feynman Integrals 28 / 45 1 C A

massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop Higgs+Jet at two-loop Gehrmann, Remiddi; Di Vita, Mastrolia, Yundin, U.S. T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 T 11 T 12 T 13 (k1 + p2) 2 T 14 T 15 T 16 T 17 T 18 Ulrich Schubert for Feynman Integrals 29 / 45

massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop Higgs+Jet at two-loop Gehrmann, Remiddi; Di Vita, Mastrolia, Yundin, U.S. dimensionless variables x = s mh 2 y = t m 2 h ɛ-linear basis x f (x, y, ɛ) = (A1,0(x, y) + ɛa 1,1(x, y)) f (x, y, ɛ) y f (x, y, ɛ) = (A2,0(x, y) + ɛa 2,1(x, y)) f (x, y, ɛ) Canonical form with Magnus x g(x, y, ɛ) = ɛâ 1(x, y) g(x, y, ɛ) y g(x, y, ɛ) = ɛâ2(x, y) g(x, y, ɛ) Log-form Â(x, y) = M 1log(x) + M 2log(1 x) + M 3log(y) + M 4log(1 y) +M 5log( x + y x ) + M 1 x y 6log( ) 1 x x Â(x, y) = Â 1(x, y) y Â(x, y) = Â 2(x, y) Alphabet {x, 1 x, y, 1 y, x + y, 1 x y} Ulrich Schubert for Feynman Integrals 30 / 45

Boundary condition Introduction massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop Four pseudo-thresholds x 1 (s m 2 H), y 1 (t m 2 H), y x (u m 2 H), y 1 x (u 0) Demanding their absence at the matrix level leads to relations between the boundary constants of different integrals, After solving all relations only one input integral is needed Ulrich Schubert for Feynman Integrals 31 / 45

massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop Higgs+Jet at two-loop Gehrmann, Remiddi; Di Vita, Mastrolia, Yundin, U.S. Ulrich Schubert for Feynman Integrals 32 / 45

massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop Higgs+Jet at two-loop Gehrmann, Remiddi; Di Vita, Mastrolia, Yundin, U.S. Ulrich Schubert for Feynman Integrals 33 / 45

massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop Higgs+Jet at three-loop Di Vita, Mastrolia, Yundin, U.S. (2014) (3) Higgs + 1Jet 3-loop ladder [Mastrolia, Schubert, Yundin, DV... work in progress!] taken from Pierpaolo s slides at Amplitudes 2014 Ulrich Schubert for Feynman Integrals 34 / 45

Higgs+Jet at three-loop Introduction massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop Di Vita, Mastrolia, Yundin, U.S. ɛ-linear basis x f (x, y, ɛ) = (A1,0(x, y) + ɛa 1,1(x, y)) f (x, y, ɛ) y f (x, y, ɛ) = (A2,0(x, y) + ɛa 2,1(x, y)) f (x, y, ɛ) Canonical form with Magnus x g(x, y, ɛ) = ɛâ 1(x, y) g(x, y, ɛ) Log-form y g(x, y, ɛ) = ɛâ 2(x, y) g(x, y, ɛ) Â(x, y) = M 1log(x) + M 2log(1 x) + M 3log(y) + M 4log(1 y) +M 5log( x + y x ) + M 1 x y 6log( ) 1 x x Â(x, y) = Â 1(x, y) y Â(x, y) = Â 2(x, y) Alphabet {x, 1 x, y, 1 y, x + y, 1 x y} same as in the two-loop case! Ulrich Schubert for Feynman Integrals 35 / 45

1 Introduction 2 3 massive QED vertex at two-loop 2 2 non-planar massless box Higgs+Jet at two-loop Higgs+Jet at three-loop 4 Ulrich Schubert for Feynman Integrals 36 / 45

Finding the Canonical Form Introduction But first steps have been done: No algorithm for finding the canonical form Magnus Exponential DEQ has to be linear in ɛ Magnus series might not converge B(x) = e Ω[A 0](x) Reducing DEQ Not all DEQ are reducible to a Fuchsian Form k S n Ã(x) = x x n n=1 Related to the negative answer of the 21st Hilbert Problem Eigenvalues of S n have to be linear in ɛ Ulrich Schubert for Feynman Integrals 37 / 45

Equal-mass sunrise Introduction Laporta, Remiddi; Bloch, Vanhove; Adams, Bogner, Weinzierl m p m m DEQ is known to evaluate to elliptic functions Easy to find a DEQ s f = A(s, ɛ) f which is Fuchsian and linear in ɛ = (D 4) 0 0 0 0 A(s, ɛ) = 1 ɛ 1 + ɛ 0 0 s 0 0 0 0 0 0 1 2 ɛ 1 + ɛ 0 0 0 0 + 1 ɛ 2ɛ 0 0 s m 2 0 0 0 0 0 ɛ 2 3 + 3 ɛ 1 + 2ɛ 4 2 2 0 0 0 0 + 1 0 0 0 0 s 9m 2 0 2ɛ 1 2ɛ 4 ɛ 1 0 + 1 ɛ 1 4 2 2 Let s apply Magnus and Lees algorithm Ulrich Schubert for Feynman Integrals 38 / 45

Magnus Introduction Split ɛ 0 -part into diagonal and off-diagonal part and construct transformation A 0 = D 0 + N 0 0 0 0 0 0 D 0 = 1 0 0 s 0 0 1 0 s 9m 2 0 0 0 1 1 + 1 s s m 2 s 9m 2 Transformed DEQ  =, B 1 = e 0 0 0 0 ɛ s m 2 ɛ 2ɛ s dx D0 = 0 0 s m 2 0 2ɛ 2ɛ 36 32 s s 9m 2 s s m 2 0 4mɛ (s 9m 2 ) 2 Transform with the ɛ 0 part of  3m2 (1+2ɛ) 18(2ɛm4 +m 4 ) 2(s 9m 2 ) 2 (s 9m 2 ) 3 0 0 0 0 0 0 0 0 ˆN 0 = 0 0 0 36 s 32 s m 2 0 0 18m4 (s 9m 2 ) 3 3m2 2(s 9m 2 ) 2 0 This matrix is not nilpotent Dyson/Magnus Series will not converge ɛ 2ɛ s s m 2 1 0 0 0 1 0 0 0 s 0 0 1 0 s 9m 2 0 0 0 8 9 s m 2 s Ulrich Schubert for Feynman Integrals 39 / 45

Lees Algorithm Introduction Eigenvalues of pole matrices s 0 : {0, 0, 1 + ɛ, 1 + ɛ} s m 2 : {0, 0, 1 + 2ɛ, 2ɛ} s 9m 2 : {0, 0, 0, 2ɛ} s : {0, 1 + ɛ, 1 + ɛ, 1 + 2ɛ} Use balance transformation to shift eigenvalues B(P, x 1, x 2 x) = P + c x x2 x x 1 P, P = u(λ 1)v (λ 2), P = 1 P After two transformations we arrive at s 0 : {0, 0, ɛ, ɛ} s m 2 : {0, 0, 1 + 2ɛ, 2ɛ} s 9m 2 : {0, 0, 0, 2ɛ} s : {0, 1 + ɛ, ɛ, 2ɛ} Eigenvalues 1 + 2ɛ and 1 + ɛ can not be balanced since there eigenvectors are orthogonal to each other Both results indicate that the ɛ 0 -part is related to the elliptic functions Ulrich Schubert for Feynman Integrals 40 / 45

Conclusion Introduction New ideas stimulated the method -Work at matrix level -Bring DEQ in canonical form g = ɛa(x) g Magnus Series finds a canonical basis for a DEQ which is linear in ɛ Has been applied to a variety of known and unknown processes -QED Vertex at two-loop -2 2 massless box -Higgs+jet at two-loop -Ladder topology for Higgs+jet at three loop More are coming so stay tuned Ulrich Schubert for Feynman Integrals 41 / 45

Back up slides Ulrich Schubert for Feynman Integrals 42 / 45

x f = A(x) S 1(ɛ) f with A(x) = x + 1 + S0(ɛ) + S1(ɛ) x x 1 Eigenvalues of the pole matrices: S 1(ɛ) : {0, 0, 3 6ɛ} S 0(ɛ) : {0, ɛ, 1 + 2ɛ} S 1(ɛ) : {0, 1 2ɛ, 2 + 2ɛ} S (ɛ) : {0, ɛ, 1 + 2ɛ} with S (ɛ) = S 1(ɛ) S 0(ɛ) S 1(ɛ) use balance transformations B(P, x 1, x 2 x) = P + c x x 2 x x 1 P to make all Eigenvalues ɛ S 1(ɛ) : {0, 0, 6ɛ} S0(ɛ) : {0, ɛ, 2ɛ} S 1(ɛ) : {0, 2ɛ, 2ɛ} S (ɛ) : {0, ɛ, 2ɛ} Find the similarity transformation which brings us to the canonical form through solving S i(ɛ) T = T S i(µ) ɛ µ Ulrich Schubert for Feynman Integrals 43 / 45

Our new masters are g = T f (1 2ɛ)(2 3ɛ)(1 3ɛ) 0 0 T = 2(4ɛ 1) + 2(3ɛ 1) ɛ + 8ɛ + 2ɛ 4(1 4ɛ) + 4ɛ 2xɛ 4(1 12ɛ) (x 1) 2 x 1 x x 1 x (x 1) 2 x 1 4ɛ(2 3ɛ) + 2ɛ(2 3ɛ) 2(12ɛ2 11ɛ+2) x 1 12ɛ 2 +11ɛ 2 + 9ɛ2 +9ɛ 2 (x 1) 2 x 1 which satisfy a canonical DEQ g = ɛã(x) 0 0 0 Ã(x) = 1 3 x + 1 2 6 24 0 0 0 + 1 x 0 0 0 2 4 15 1 2 7 (x 1) 2 1 x 1 After Integrating and fixing the boundary we find non uniform transcendental(ut) masters f 2 = 9ɛ2 8 + ɛ3 ( 1 4 G(0; x) + 117 16 ) ( + ɛ 4 13 8 G(0; x) 3 G( 1, 0; x) G(0, 0; x) 2 ( )) 3105 4π 2 1 2 G(1, 0; x) + 1 96 canonical form does not imply UT masters Boundary conditions can spoil UT + 6ɛ 2(2 17ɛ) 10xɛ x 2(36ɛ2 27ɛ+2) x 1 2 + 29ɛ 39ɛ 2 0 0 0 1 2 2 16 1 0 2 Ulrich Schubert for Feynman Integrals 44 / 45

massive Sunrise with Lee Starting from DEQ A(x) = Introduction 0 0 0 0 ɛ x 1 ɛ x 0 0 ɛ 1 2ɛ x 2ɛ x 9 ɛ ɛ 2(x 9) 2(x 1) x 1 0 0 2ɛ 1 x 9 2ɛ 1 + 2ɛ 1 4(x 9) 2x + 3(2ɛ+1) 4(x 1) with x = s m 2 Transformation generated by shifting eigenvalues twice 1 0 0 0 0 B = 1 x+1 0 x 2x 0 0 1 0 0 0 Resulting DEQ Â = (x+1)ɛ 2x 1 x 4 x 9 2ɛ 1 + ɛ 1 + 1 x 1 x x 9 0 0 0 0 ɛ x 1 0 0 10ɛ 2ɛ + 8ɛ 9(x 9) x 1 9x 2ɛ 2ɛ 9x 9(x 9) ɛ 2 ɛ + 9ɛ 20ɛ2 9x 2(x 1) 18(x 9) 5ɛ2 9x 2ɛ + 5ɛ + 5(8ɛ+9) x 1 9x 9(x 9) 8ɛ 9 ɛ 9(x 9) 9x + 160ɛ2 108ɛ+81 + 8ɛ2 4ɛ 3 36(x 9) 4(x 1) 20 2 9(x 9) 9x 4 4 9x 9(x 9) 9 20ɛ + 2ɛ 1 + 11ɛ 9(x 9) x 1 9x Ulrich Schubert for Feynman Integrals 45 / 45