Symbolic integration of multiple polylogarithms

Transcription

1 Symbolic integration of multiple polylogarithms Erik Panzer Institute des Hautes E tudes Scientifiques Applications of Computer Algebra July 22nd, 215 Kalamata, Greece

2 Problem: Multiple integrals of rational functions log(1 + 1/(xz)) dx dy (1 + y)(1 + x + y + 1/z) =? 1 / 2

3 Problem: Multiple integrals of rational functions log(1 + 1/(xz)) dx dy (1 + y)(1 + x + y + 1/z) =? Sometimes such integrals are expressible as multiple polylogarithms (MPL) k z 1 1 Li n1,...,n d (z 1,..., z d ) = zk d d k n 1 <k 1 < <k d 1 kn d d ( n N d) and their special values, like multiple zeta values (MZV) ζ n = Li n (1,, 1). 1 / 2

4 Problem: Multiple integrals of rational functions log(1 + 1/(xz)) dx dy (1 + y)(1 + x + y + 1/z) = ζ 3 ζ 2 log(z) Li 1,2 (1, z) Li 3 ( z) Sometimes such integrals are expressible as multiple polylogarithms (MPL) k z 1 1 Li n1,...,n d (z 1,..., z d ) = zk d d k n 1 <k 1 < <k d 1 kn d d ( n N d) and their special values, like multiple zeta values (MZV) ζ n = Li n (1,, 1). 1 / 2

5 Problem: Multiple integrals of rational functions log(1 + 1/(xz)) dx dy (1 + y)(1 + x + y + 1/z) = ζ 3 ζ 2 log(z) Li 1,2 (1, z) Li 3 ( z) Sometimes such integrals are expressible as multiple polylogarithms (MPL) k z 1 1 Li n1,...,n d (z 1,..., z d ) = zk d d k n 1 <k 1 < <k d 1 kn d d ( n N d) and their special values, like multiple zeta values (MZV) Example ζ n = Li n (1,, 1). dα 1 dα 2 dα 3 dα 4 [α 1 + α 4 + (α 1 + α 4 + 1)(α 2 + α 3 )][(α 1 + α 2 )(α 3 + α 3 α 4 + α 4 ) + α 1 α 2 (α 3 + α 4 )] 1 = 6ζ 3 = 6 k 3 k=1 1 / 2

6 Motivation

7 Perturbative Quantum Field Theory? 2 / 2

8 Perturbative Quantum Field Theory 2 / 2

9 Perturbative Quantum Field Theory 2 / 2

10 Perturbative Quantum Field Theory 2 / 2

11 Perturbative Quantum Field Theory 2 / 2

12 Perturbative Quantum Field Theory 2 / 2

13 Perturbative Quantum Field Theory 2 / 2

14 Perturbative Quantum Field Theory Feynman graphs G 2 / 2

15 Perturbative Quantum Field Theory Feynman graphs G each Feynman graph represents a Feynman integral (FI), Φ(G) truncated sum G Φ(G) approximates the process 2 / 2

16 Perturbative Quantum Field Theory Feynman graphs G each Feynman graph represents a Feynman integral (FI), Φ(G) truncated sum G Φ(G) approximates the process very accurate measurements demand precise theoretical predictions Challenges: number of graphs & complexity of integrals 2 / 2

17 Some FI are expressible as MPL, depending on momenta and masses like ( ) Φ = 4i Im [Li 2(z) + log(1 z) log z ] z z and also including constants (like MZV) as in Φ ( ) = 6ζ 3. Many conjectures by PSLQ, for example ( ) Φ = 252ζ 3 ζ ζ 3, ζ4 2 3 / 2

18 Some FI are expressible as MPL, depending on momenta and masses like ( ) Φ = 4i Im [Li 2(z) + log(1 z) log z ] z z and also including constants (like MZV) as in Φ ( ) = 6ζ 3. Many conjectures by PSLQ, for example ( ) Φ = 252ζ 3 ζ ζ 3, ζ4 2 Questions 1 How to tell if a FI evaluates to MPL? What is the alphabet? 2 How to compute it explicitly? 3 / 2

19 Some FI are expressible as MPL, depending on momenta and masses like ( ) Φ = 4i Im [Li 2(z) + log(1 z) log z ] z z and also including constants (like MZV) as in Φ ( ) = 6ζ 3. Many conjectures by PSLQ, for example ( ) Φ = 252ζ 3 ζ ζ 3, ζ4 2 Questions 1 How to tell if a FI evaluates to MPL? What is the alphabet? 2 How to compute it explicitly in an efficient and automated way? 3 / 2

20 Schwinger parameters Φ(G) = Graph polynomials: ψ = T (, ) E 1 ψ D/2 ( ψ ϕ α e e / T ) E h1 (G)D/2 δ(1 α N ) e ϕ = q 2 (T 1 ) α e + ψ e / F e F =T 1 T 2 dα e m 2 eα e 4 / 2

21 Schwinger parameters Φ(G) = Graph polynomials: Example ψ = T (, ) E 1 ψ D/2 ( ψ ϕ α e e / T ) E h1 (G)D/2 δ(1 α N ) e ϕ = q 2 (T 1 ) α e + ψ e / F e F =T 1 T 2 p 3 1 p 2 ψ = α 1 + α 2 + α 3 dα e m 2 eα e G = 2 1 p 3 ϕ = p1α 2 2 α 3 + p2α 2 1 α 3 + p3α 2 1 α 2 dα 2 dα 3 Φ(G) = ψϕ α1 =1 4 / 2

22 Schwinger parameters Φ(G) = Graph polynomials: Example ψ = T (, ) E 1 ψ D/2 ( ψ ϕ α e e / T ) E h1 (G)D/2 δ(1 α N ) e ϕ = q 2 (T 1 ) α e + ψ e / F e F =T 1 T 2 p 3 1 p 2 ψ = α 1 + α 2 + α 3 dα e m 2 eα e G = 2 1 p 3 ϕ = p1α 2 2 α 3 + p2α 2 1 α 3 + p3α 2 1 α 2 dα 2 dα 3 Φ(G) = ψϕ α1 =1 4 / 2

23 Schwinger parameters Φ(G) = Graph polynomials: Example ψ = T (, ) E 1 ψ D/2 ( ψ ϕ α e e / T ) E h1 (G)D/2 δ(1 α N ) e ϕ = q 2 (T 1 ) α e + ψ e / F e F =T 1 T 2 p 3 1 p 2 ψ = α 1 + α 2 + α 3 dα e m 2 eα e G = 2 1 p 3 ϕ = p1α 2 2 α 3 + p2α 2 1 α 3 + p3α 2 1 α 2 dα 2 dα 3 Φ(G) = ψϕ α1 =1 4 / 2

24 Schwinger parameters Φ(G) = Graph polynomials: Example ψ = T (, ) E 1 ψ D/2 ( ψ ϕ α e e / T ) E h1 (G)D/2 δ(1 α N ) e ϕ = q 2 (T 1 ) α e + ψ e / F e F =T 1 T 2 p 3 1 p 2 ψ = α 1 + α 2 + α 3 dα e m 2 eα e G = 2 1 p 3 ϕ = p1α 2 2 α 3 + p2α 2 1 α 3 + p3α 2 1 α 2 dα 2 dα 3 Φ(G) = ψϕ α1 =1 4 / 2

25 Schwinger parameters Φ(G) = Graph polynomials: Example ψ = T (, ) E 1 ψ D/2 ( ψ ϕ α e e / T ) E h1 (G)D/2 δ(1 α N ) e ϕ = q 2 (T 1 ) α e + ψ e / F e F =T 1 T 2 p 3 1 p 2 ψ = α 1 + α 2 + α 3 dα e m 2 eα e G = 2 1 p 3 ϕ = p1α 2 2 α 3 + p2α 2 1 α 3 + p3α 2 1 α 2 dα 2 dα 3 Φ(G) = ψϕ α1 =1 4 / 2

26 Schwinger parameters Φ(G) = Graph polynomials: Example ψ = T (, ) E 1 ψ D/2 ( ψ ϕ α e e / T ) E h1 (G)D/2 δ(1 α N ) e ϕ = q 2 (T 1 ) α e + ψ e / F e F =T 1 T 2 p 3 1 p 2 ψ = α 1 + α 2 + α 3 dα e m 2 eα e G = 2 1 p 3 ϕ = p1α 2 2 α 3 + p2α 2 1 α 3 + p3α 2 1 α 2 dα 2 dα 3 Φ(G) = ψϕ α1 =1 4 / 2

27 Schwinger parameters Φ(G) = Graph polynomials: Example ψ = T (, ) E 1 ψ D/2 ( ψ ϕ α e e / T ) E h1 (G)D/2 δ(1 α N ) e ϕ = q 2 (T 1 ) α e + ψ e / F e F =T 1 T 2 p 3 1 p 2 ψ = α 1 + α 2 + α 3 dα e m 2 eα e G = 2 1 p 3 ϕ = p1α 2 2 α 3 + p2α 2 1 α 3 + p3α 2 1 α 2 dα 2 dα 3 Φ(G) = ψϕ α1 =1 4 / 2

28 Schwinger parameters Φ(G) = Graph polynomials: Example ψ = T (, ) E 1 ψ D/2 ( ψ ϕ α e e / T ) E h1 (G)D/2 δ(1 α N ) e ϕ = q 2 (T 1 ) α e + ψ e / F e F =T 1 T 2 p 3 1 p 2 ψ = α 1 + α 2 + α 3 dα e m 2 eα e G = 2 1 p 3 ϕ = p1α 2 2 α 3 + p2α 2 1 α 3 + p3α 2 1 α 2 dα 2 dα 3 Φ(G) = ψϕ α1 =1 4 / 2

29 Some other definite integral representations hypergeometric functions p F q (expansion in a, b,... ) 2F 1 ( a, b c ) z = Appell s functions F 1, F 2, F 3, F 4 ( a, a ) c x F 3 y b, b = 1 1 v Phase-space integrals Γ(c) 1 t b 1 (1 t) c b 1 (1 zt) a dt Γ(b)Γ(c b) Γ(c) Γ(b)Γ(b )Γ(c b b ) u b 1 v b 1 (1 u v) c b b 1 (1 ux) a (1 vy) a dudv periods of moduli spaces, string amplitudes <t 1 < <t 5 <1 dt 1 dt 5 (1 t 1 )(1 t 2 )t 3 (t 4 t 2 )(t 5 t 3 )t 5 = 6 5 ζ2 2 = π4 3 5 / 2

30 Techniques

31 We can represent MPL as iterated path integrals in one dimension: ( G(, σ }{{ d,...,, σ } 1 ; z) = ( 1) d σ2 σ d Li }{{} n1,...,n d,,, σ 1 σ d 1 n d n 1 Hyperlogarithms [Poincaré 1884, Lappo-Danilevsky 1927] z dz z1 1 dz zw 1 2 dz w G(σ 1,..., σ w ; z) := z 1 σ 1 z 2 σ 2 z w σ w z σ d ) 6 / 2

32 We can represent MPL as iterated path integrals in one dimension: ( G(, σ }{{ d,...,, σ } 1 ; z) = ( 1) d σ2 σ d Li }{{} n1,...,n d,,, σ 1 σ d 1 n d n 1 Hyperlogarithms [Poincaré 1884, Lappo-Danilevsky 1927] z dz z1 1 dz zw 1 2 dz w G(σ 1,..., σ w ; z) := z 1 σ 1 z 2 σ 2 z w σ w locally analytic, multivalued functions on C \ {, σ 1,..., σ w } divergences at z σ i (or ) are logarithmic span an algebra via the shuffle product G(a, b; z) G(c; z) = G(a, b, c; z) + G(a, c, b; z) + G(c, a, b; z) path concatenation (analytic continuation) via coproduct z σ d ) 6 / 2

33 We can represent MPL as iterated path integrals in one dimension: ( G(, σ }{{ d,...,, σ } 1 ; z) = ( 1) d σ2 σ d Li }{{} n1,...,n d,,, σ 1 σ d 1 n d n 1 Hyperlogarithms [Poincaré 1884, Lappo-Danilevsky 1927] z dz 1 G(σ 1,..., σ w ; z) := G(σ 2,..., σ w ; z 1 ), G( z 1 σ 1 }{{} ; z) = logw (z) w! w locally analytic, multivalued functions on C \ {, σ 1,..., σ w } divergences at z σ i (or ) are logarithmic span an algebra via the shuffle product G(a, b; z) G(c; z) = G(a, b, c; z) + G(a, c, b; z) + G(c, a, b; z) path concatenation (analytic continuation) via coproduct z σ d ) 6 / 2

34 We can represent MPL as iterated path integrals in one dimension: ( G(, σ }{{ d,...,, σ } 1 ; z) = ( 1) d σ2 σ d Li }{{} n1,...,n d,,, σ 1 σ d 1 n d n 1 Hyperlogarithms [Poincaré 1884, Lappo-Danilevsky 1927] z dz 1 G(σ 1,..., σ w ; z) := G(σ 2,..., σ w ; z 1 ), G( z 1 σ 1 }{{} ; z) = logw (z) w! w locally analytic, multivalued functions on C \ {, σ 1,..., σ w } divergences at z σ i (or ) are logarithmic span an algebra via the shuffle product G(a, b; z) G(c; z) = G(a, b, c; z) + G(a, c, b; z) + G(c, a, b; z) path concatenation (analytic continuation) via coproduct Differential algebra, closed under integration Q[z, 1 z σ : σ Σ] lin Q {G( σ; z): σ Σ } z σ d ) 6 / 2

35 We can represent MPL as iterated path integrals in one dimension: ( G(, σ }{{ d,...,, σ } 1 ; z) = ( 1) d σ2 σ d Li }{{} n1,...,n d,,, σ 1 σ d 1 n d n 1 Hyperlogarithms [Poincaré 1884, Lappo-Danilevsky 1927] z dz 1 G(σ 1,..., σ w ; z) := G(σ 2,..., σ w ; z 1 ), G( z 1 σ 1 }{{} ; z) = logw (z) w! w locally analytic, multivalued functions on C \ {, σ 1,..., σ w } divergences at z σ i (or ) are logarithmic span an algebra via the shuffle product G(a, b; z) G(c; z) = G(a, b, c; z) + G(a, c, b; z) + G(c, a, b; z) path concatenation (analytic continuation) via coproduct Differential algebra, closed under integration Q[z, 1 z σ : σ Σ] lin Q {G( σ; z): σ Σ } z σ d ) { 1 σ τ : σ, τ Σ (σ τ) } 6 / 2

36 Multiple integration with hyperlogarithms Idea: Use Fubini s theorem to compute f n = f n 1 dα n = f dα 1 dα n. If f is simple enough (linearly reducible), each f n (α n+1,...) is a rational linear combination of hyperlogarithms in α n+1. 7 / 2

37 Multiple integration with hyperlogarithms Idea: Use Fubini s theorem to compute f n = f n 1 dα n = f dα 1 dα n. If f is simple enough (linearly reducible), each f n (α n+1,...) is a rational linear combination of hyperlogarithms in α n+1. 1 Write f n 1 in terms of hyperlogarithms: f n 1 = σ,τ,k G( σ; α n ) (α n τ) k λ σ,τ,k with σ and τ independent of α n. 7 / 2

38 Multiple integration with hyperlogarithms Idea: Use Fubini s theorem to compute f n = f n 1 dα n = f dα 1 dα n. If f is simple enough (linearly reducible), each f n (α n+1,...) is a rational linear combination of hyperlogarithms in α n+1. 1 Write f n 1 in terms of hyperlogarithms: f n 1 = σ,τ,k G( σ; α n ) (α n τ) k λ σ,τ,k with σ and τ independent of α n. 2 Construct an antiderivative αn F = f n 1. 7 / 2

39 Multiple integration with hyperlogarithms Idea: Use Fubini s theorem to compute f n = f n 1 dα n = f dα 1 dα n. If f is simple enough (linearly reducible), each f n (α n+1,...) is a rational linear combination of hyperlogarithms in α n+1. 1 Write f n 1 in terms of hyperlogarithms: f n 1 = σ,τ,k G( σ; α n ) (α n τ) k λ σ,τ,k with σ and τ independent of α n. 2 Construct an antiderivative αn F = f n 1. 3 Evaluate the limits f n := f n 1 dα n = lim F (α α n n) lim F (α n). α n 7 / 2

40 Problem Rewrite G( σ(α); z) as hyperlogs G( σ; α) with constant letters α σ =. 8 / 2

41 Problem Rewrite G( σ(α); z) as hyperlogs G( σ; α) with constant letters α σ =. dg( σ; z) = w i=1 G(, σ i, ; z) d log σ i σ i 1 σ i σ i+1 (σ := z, σ w+1 := ) Algorithm 1: weight recursion (example) α G (, α; 1) = 1 G( α; 1) α 8 / 2

42 Problem Rewrite G( σ(α); z) as hyperlogs G( σ; α) with constant letters α σ =. dg( σ; z) = w i=1 G(, σ i, ; z) d log σ i σ i 1 σ i σ i+1 (σ := z, σ w+1 := ) Algorithm 1: weight recursion (example) α G (, α; 1) = 1 [ ] G(; α) G( 1; α) α 8 / 2

43 Problem Rewrite G( σ(α); z) as hyperlogs G( σ; α) with constant letters α σ =. dg( σ; z) = w i=1 G(, σ i, ; z) d log σ i σ i 1 σ i σ i+1 (σ := z, σ w+1 := ) Algorithm 1: weight recursion (example) α G (, α; 1) = 1 [ ] G(; α) G( 1; α) α G (, α; 1) = G(, ; α) + G(, 1; α) 8 / 2

44 Problem Rewrite G( σ(α); z) as hyperlogs G( σ; α) with constant letters α σ =. dg( σ; z) = w i=1 G(, σ i, ; z) d log σ i σ i 1 σ i σ i+1 (σ := z, σ w+1 := ) Algorithm 1: weight recursion (example) α G (, α; 1) = 1 [ ] G(; α) G( 1; α) α G (, α; 1) = G(, ; α) + G(, 1; α) + ζ 2 8 / 2

45 Problem Rewrite G( σ(α); z) as hyperlogs G( σ; α) with constant letters α σ =. dg( σ; z) = w i=1 G(, σ i, ; z) d log σ i σ i 1 σ i σ i+1 (σ := z, σ w+1 := ) Algorithm 1: weight recursion (example) α G (, α; 1) = 1 [ ] G(; α) G( 1; α) α G (, α; 1) = G(, ; α) + G(, 1; α) + ζ 2 Algorithm 2: integration constants at α uses shuffles, Möbius transformations and path concatenation 8 / 2

46 Problem Rewrite G( σ(α); z) as hyperlogs G( σ; α) with constant letters α σ =. dg( σ; z) = w i=1 G(, σ i, ; z) d log σ i σ i 1 σ i σ i+1 (σ := z, σ w+1 := ) Algorithm 1: weight recursion (example) α G (, α; 1) = 1 [ ] G(; α) G( 1; α) α G (, α; 1) = G(, ; α) + G(, 1; α) + ζ 2 Algorithm 2: integration constants at α uses shuffles, Möbius transformations and path concatenation Corollary (algorithmic) If all σ i Q(α), then G( σ; σ ) is a hyperlogarithm in α with alphabet Σ α = {zeros and poles of σ i (α) σ j (α): i < j w + 1} 8 / 2

47 Example: massless triangle Φ ( ) = dα 2 dα 3 (1 + α 2 + α 3 )(α 2 α 3 + z zα 3 + (1 z)(1 z)α 2 ) 9 / 2

48 Example: massless triangle Φ ( ) = dα 2 dα 3 (1 + α 2 + α 3 )(α 2 α 3 + z zα 3 + (1 z)(1 z)α 2 ) Singularities of the original integrand: S = {ψ, ϕ}, i.e. at α 3 = σ i for σ 1 = 1 α 2 and σ 2 = α 2(1 z)(1 z) α 2 + z z 9 / 2

49 Example: massless triangle Φ ( ) = dα 2 dα 3 (1 + α 2 + α 3 )(α 2 α 3 + z zα 3 + (1 z)(1 z)α 2 ) Singularities of the original integrand: S = {ψ, ϕ}, i.e. at α 3 = σ i for σ 1 = 1 α 2 and σ 2 = α 2(1 z)(1 z) α 2 + z z After integrating α 1 from to, the integrand has singularities S 3 = { } 1 + α }{{ 2, α } 2, 1 z, 1 z, α }{{} 2 + z z, z + α }{{} 2, z + α }{{ 2 } σ 1 = σ 2 = σ 2 = σ 1 =σ 2 (pinch) 9 / 2

50 Example: massless triangle Φ ( ) = = dα 2 dα 3 (1 + α 2 + α 3 )(α 2 α 3 + z zα 3 + (1 z)(1 z)α 2 ) dα 2 (z + α 2 )( z + α 2 ) log (1 + α 2)(z z + α 2 ) (1 z)(1 z)α 2 Singularities of the original integrand: S = {ψ, ϕ}, i.e. at α 3 = σ i for σ 1 = 1 α 2 and σ 2 = α 2(1 z)(1 z) α 2 + z z After integrating α 1 from to, the integrand has singularities S 3 = { } 1 + α }{{ 2, α } 2, 1 z, 1 z, α }{{} 2 + z z, z + α }{{} 2, z + α }{{ 2 } σ 1 = σ 2 = σ 2 = σ 1 =σ 2 (pinch) 9 / 2

51 Example: massless triangle Φ ( ) = = dα 2 dα 3 (1 + α 2 + α 3 )(α 2 α 3 + z zα 3 + (1 z)(1 z)α 2 ) dα 2 (z + α 2 )( z + α 2 ) log (1 + α 2)(z z + α 2 ) (1 z)(1 z)α 2 Singularities of the original integrand: S = {ψ, ϕ}, i.e. at α 3 = σ i for σ 1 = 1 α 2 and σ 2 = α 2(1 z)(1 z) α 2 + z z After integrating α 1 from to, the integrand has singularities S 3 = { } 1 + α }{{ 2, α } 2, 1 z, 1 z, α }{{} 2 + z z, z + α }{{} 2, z + α }{{ 2 } σ 1 = σ 2 = σ 2 = σ 1 =σ 2 (pinch) With the same logic, predict the possible singularities after dα 2 : S 3,2 = {z, z, 1 z, 1 z, z z, z z 1} 9 / 2

52 Example: massless triangle Φ ( ) = = 1 z z dα 2 dα 3 (1 + α 2 + α 3 )(α 2 α 3 + z zα 3 + (1 z)(1 z)α 2 ) ( dα2 α 2 + z dα ) 2 log (α 2 + 1)(α 2 + z z) α 2 + z (1 z)(1 z)α 2 Singularities of the original integrand: S = {ψ, ϕ}, i.e. at α 3 = σ i for σ 1 = 1 α 2 and σ 2 = α 2(1 z)(1 z) α 2 + z z After integrating α 1 from to, the integrand has singularities S 3 = { } 1 + α }{{ 2, α } 2, 1 z, 1 z, α }{{} 2 + z z, z + α }{{} 2, z + α }{{ 2 } σ 1 = σ 2 = σ 2 = σ 1 =σ 2 (pinch) With the same logic, predict the possible singularities after dα 2 : S 3,2 = {z, z, 1 z, 1 z, z z, z z 1} 9 / 2

53 Example: massless triangle Φ ( ) = dα 2 dα 3 (1 + α 2 + α 3 )(α 2 α 3 + z zα 3 + (1 z)(1 z)α 2 ) = 2 Li 2(z) 2 Li 2 ( z) + [log(1 z) log(1 z)] log(z z) z z Singularities of the original integrand: S = {ψ, ϕ}, i.e. at α 3 = σ i for σ 1 = 1 α 2 and σ 2 = α 2(1 z)(1 z) α 2 + z z After integrating α 1 from to, the integrand has singularities S 3 = { } 1 + α }{{ 2, α } 2, 1 z, 1 z, α }{{} 2 + z z, z + α }{{} 2, z + α }{{ 2 } σ 1 = σ 2 = σ 2 = σ 1 =σ 2 (pinch) With the same logic, predict the possible singularities after dα 2 : S 3,2 = {z, z, 1 z, 1 z, z z, z z 1} 9 / 2

54 Definition (Polynomial reduction [Brown]) Let S denote a set of polynomials, then S e are the irreducible factors of {lead e (f ), f αe= : f S } and {[f, g] e : f, g S}. Lemma (approximation of Landau varieties) If the singularities of F are cointained in S, then the singularities of F dα e are contained in S e. Goal: bounds as tight as possible 1 / 2

55 Definition (Polynomial reduction [Brown]) Let S denote a set of polynomials, then S e are the irreducible factors of {lead e (f ), f αe= : f S } and {[f, g] e : f, g S}. Lemma (approximation of Landau varieties) If the singularities of F are cointained in S, then the singularities of F dα e are contained in S e. Goal: bounds as tight as possible This gives only very coarse upper bounds. For example, z z 1 is spurious: It drops out in S 2,3 S 3,2 = {z, z, 1 z, 1 z, z z} because S 2,3 = {z, z, 1 z, 1 z, z z, z z z z}. Improvements Fubini algorithm [Brown]: intersect over different orders Compatibility graphs [Brown, Panzer] 1 / 2

56 Compatibility graphs Keep track of compatibilities C ( S 2) between polynomials: start with the complete graph ψ ϕ in S e, only take resultants [f, g] e for compatible {f, g} C in C e, only pairs [f, g] e [g, h] e become compatible 11 / 2

57 Compatibility graphs Keep track of compatibilities C ( S 2) between polynomials: start with the complete graph ψ ϕ in S e, only take resultants [f, g] e for compatible {f, g} C in C e, only pairs [f, g] e [g, h] e become compatible 1 z 1 z (S, C) 3 = 1+α 2 z z +α 2 z +α 2 z +α 2 11 / 2

58 Compatibility graphs Keep track of compatibilities C ( S 2) between polynomials: start with the complete graph ψ ϕ in S e, only take resultants [f, g] e for compatible {f, g} C in C e, only pairs [f, g] e [g, h] e become compatible 1 z 1 z (S, C) 3 = 1+α 2 z z +α 2 z +α 2 z +α 2 z zα 1 + α 2 and α 1 + α 2 not compatible no resultant 1 z z in (S, C) 3,2 11 / 2

59 Problem for multiple integrals If some σ i (α) σ j (α) does not factorize linearly in α, the transformation to G( ; α) introduces algebraic letters. Definition If for some order of variables (edges), all S 1,...,k are linear in α k+1, then S (the Feynman graph G with S = {ψ, ϕ}) is called linearly reducible. Write O(S) = Q[ α, f 1 : f S] and MPL(S) = O(S) linq {iterated integrals of d log(f ) s (f S)}. Lemma (algorithmic) If S is linearly reducible and f MPL(S), then f dα 1 α N MPL(S 1,...,N ) Q C. 12 / 2

60 MPL have plenty of relations, like G(, α; 1) = G(, ; α) G(, 1; α) ζ 2 13 / 2

61 MPL have plenty of relations, like Li 2 ( 1/α) = 1 2 log2 (α) + Li 2 ( α) ζ 2 13 / 2

62 MPL have plenty of relations, like Lemma Li 2 ( 1/α) = 1 2 log2 (α) + Li 2 ( α) ζ 2 All hyperlogarithms G( σ; α), α σ =, are linearly independent over C(α). 13 / 2

63 MPL have plenty of relations, like Lemma Li 2 ( 1/α) = 1 2 log2 (α) + Li 2 ( α) ζ 2 All hyperlogarithms G( σ; α), α σ =, are linearly independent over C(α). The recursive algorithm (differentiation & integration & limits) solves Problem: Bases for MPL Given some MPL G( σ( α), z( α)) or Li n ( z( α)) whose arguments ( σ, z or z) are rational functions of variables α 1,..., α n, write it in the basis σ 1,..., σ n G( σ 1 (α 2,..., α n ); α 1 )G( σ 2 (α 3,..., α n ); α 2 ) G( σ n ; α n ). 13 / 2

64 MPL have plenty of relations, like Lemma Li 2 ( 1/α) = 1 2 log2 (α) + Li 2 ( α) ζ 2 All hyperlogarithms G( σ; α), α σ =, are linearly independent over C(α). The recursive algorithm (differentiation & integration & limits) solves Problem: Bases for MPL Given some MPL G( σ( α), z( α)) or Li n ( z( α)) whose arguments ( σ, z or z) are rational functions of variables α 1,..., α n, write it in the basis σ 1,..., σ n G( σ 1 (α 2,..., α n ); α 1 )G( σ 2 (α 3,..., α n ); α 2 ) G( σ n ; α n ). completely symbolic (no numerics) dependens on order of the variables α 1,..., α n in general not the shortest or simplest representation allows for symbolic verification of MPL identities 13 / 2

65 HyperInt

66 Maple Example Manual.ws open source: polynomial reduction integration of hyperlogarithms transformations of MPL to G( ; α 1 ) G( ; α N )-basis symbolic computation of constants (MZV and alternating sums) Feynman graph polynomials > read "HyperInt.mpl": > hyperint(polylog(2,-x)*polylog(3,-1/x)/x,x=..infinity): > fibrationbasis(%); 8 7 ζ3 2 computes Li 2 ( x) Li 3 ( 1/x)dx = 8 7 ζ / 2

67 Sometimes a linearly reducible order is obvious, like for <t 1 < <t 5 <1 dt 1 dt 5 (1 t 1 )(1 t 2 )t 3 (t 4 t 2 )(t 5 t 3 )t 5 = 6 5 ζ2 2 = π4 3 > hyperint(1/(1-t1)/(1-t2)/t3/(t4-t2)/(t5-t3)/t5, [t1=..t2,t2=..t3,t3=..t4,t4=..t5,t5=..1]): > fibrationbasis(%); 6 5 ζ2 2 In complicated cases, one first computes a polynomial reduction to check if a linearly reducible order exists. Both, polynomial reduction and integration can be parallelized manually. Φ = ζ (ζ 3,5,3 ζ 3,5 ζ 3 ) ζ2 3ζ 5 ( ) + 896ζ ζ 3, ζ 3ζ ζ 8 15 / 2

68 HyperInt: triangle Graph polynomials: > E:=[[1,2],[2,3],[3,1]]: > M:=[[3,1],[1,z*zz],[2,(1-z)*(1-zz)]]: > psi:=graphpolynomial(e): > phi:=secondpolynomial(e,m): Integration: > hyperint(eval(1/psi/phi,x[3]=1),[x[1],x[2]]): > factor(fibrationbasis(%,[z,zz])); (G (z; 1) G (zz; ) G (z; ) G (zz; 1) + G (zz;, 1) G (zz; 1, ) + G (z; 1, ) G (z;, 1))/(z zz) Polynomial reduction: > L[{}]:=[{psi,phi}, {{psi,phi}}]: > cgreduction(l): > L[{x[1],x[2]}][1]; { 1 + z, 1 + zz, zz + z} 16 / 2

69 Linearly reducible Feynman graphs

70 Linearly reducible families (fixed loop order) 1 all 4 loop massless propagators [Panzer] 17 / 2

71 Linearly reducible families (fixed loop order) 1 all 4 loop massless propagators [Panzer] 2 all 3 loop massless off-shell 3-point [Chavez & Duhr, Panzer] 17 / 2

72 Linearly reducible families (fixed loop order) 1 all 4 loop massless propagators [Panzer] 2 all 3 loop massless off-shell 3-point [Chavez & Duhr, Panzer] 3 all 2 loop massless on-shell 4-point [Lüders] 17 / 2

73 Linearly reducible families (infinite) 3-constructible graphs [Brown, Schnetz, Panzer], Example 18 / 2

74 Linearly reducible families (infinite) 3-constructible graphs [Brown, Schnetz, Panzer], Example 18 / 2

75 Linearly reducible families (infinite) 3-constructible graphs [Brown, Schnetz, Panzer], Example 18 / 2

76 Linearly reducible families (infinite) 3-constructible graphs [Brown, Schnetz, Panzer], Example 18 / 2

77 Linearly reducible families (infinite) 3-constructible graphs [Brown, Schnetz, Panzer], Example 18 / 2

78 Linearly reducible families (infinite) 3-constructible graphs [Brown, Schnetz, Panzer], Example 18 / 2

79 Linearly reducible families (infinite) 3-constructible graphs [Brown, Schnetz, Panzer], Example 18 / 2

80 Linearly reducible families (infinite) 3-constructible graphs [Brown, Schnetz, Panzer], Theorem [Panzer] All ɛ-coefficients of these graphs (off-shell) are MPL over the alphabet {z, z, 1 z, 1 z, z z, 1 z z, 1 z z, z z z z}. 18 / 2

81 Linearly reducible families (infinite) 3-constructible graphs [Brown, Schnetz, Panzer], Theorem [Panzer] All ɛ-coefficients of these graphs (off-shell) are MPL over the alphabet {z, z, 1 z, 1 z, z z, 1 z z, 1 z z, z z z z}. minors of ladder-boxes ( 2 legs massive) Theorem [Panzer] All ɛ-coefficients of these graphs are MPL. For the massless case, the alphabet is just {x, 1 + x} for x = s/t. 18 / 2

82 Linear reducibility: Forest functions Minors of ladder boxes are closed under the operations,, Example Theorem All minors of ladder boxes (with p 2 1 = p2 2 = ) evaluate to MPL. 19 / 2

83 Summary

84 hyperlogarithms are very suitable for computer algebra 2 / 2

85 hyperlogarithms are very suitable for computer algebra Maple implementation HyperInt 2 / 2

86 hyperlogarithms are very suitable for computer algebra Maple implementation HyperInt polynomial reduction 2 / 2

87 hyperlogarithms are very suitable for computer algebra Maple implementation HyperInt polynomial reduction requires linear reducibility! Counterexample: dx dy xyz + (1 + x + y)(x + y + xy) 2 / 2

88 hyperlogarithms are very suitable for computer algebra Maple implementation HyperInt polynomial reduction requires linear reducibility! Counterexample: dx dy xyz + (1 + x + y)(x + y + xy) possible extension: algebraic functions in the differential forms (root-valued letters) 2 / 2

89 hyperlogarithms are very suitable for computer algebra Maple implementation HyperInt polynomial reduction requires linear reducibility! Counterexample: dx dy xyz + (1 + x + y)(x + y + xy) possible extension: algebraic functions in the differential forms (root-valued letters) Holy grail: Infer analytic structure of a FI from combinatorial structure, i.e. by looking at the graph 2 / 2

90 hyperlogarithms are very suitable for computer algebra Maple implementation HyperInt polynomial reduction requires linear reducibility! Counterexample: dx dy xyz + (1 + x + y)(x + y + xy) possible extension: algebraic functions in the differential forms (root-valued letters) Holy grail: Infer analytic structure of a FI from combinatorial structure, i.e. by looking at the graph extremely difficult 2 / 2

91 hyperlogarithms are very suitable for computer algebra Maple implementation HyperInt polynomial reduction requires linear reducibility! Counterexample: dx dy xyz + (1 + x + y)(x + y + xy) possible extension: algebraic functions in the differential forms (root-valued letters) Holy grail: Infer analytic structure of a FI from combinatorial structure, i.e. by looking at the graph extremely difficult so far, only very few results 2 / 2

92 hyperlogarithms are very suitable for computer algebra Maple implementation HyperInt polynomial reduction requires linear reducibility! Counterexample: dx dy xyz + (1 + x + y)(x + y + xy) possible extension: algebraic functions in the differential forms (root-valued letters) Holy grail: Infer analytic structure of a FI from combinatorial structure, i.e. by looking at the graph extremely difficult so far, only very few results but some infinite families 2 / 2

93 hyperlogarithms are very suitable for computer algebra Maple implementation HyperInt polynomial reduction requires linear reducibility! Counterexample: dx dy xyz + (1 + x + y)(x + y + xy) possible extension: algebraic functions in the differential forms (root-valued letters) Holy grail: Infer analytic structure of a FI from combinatorial structure, i.e. by looking at the graph extremely difficult so far, only very few results but some infinite families 2 / 2

94 hyperlogarithms are very suitable for computer algebra Maple implementation HyperInt polynomial reduction requires linear reducibility! Counterexample: dx dy xyz + (1 + x + y)(x + y + xy) possible extension: algebraic functions in the differential forms (root-valued letters) Holy grail: Infer analytic structure of a FI from combinatorial structure, i.e. by looking at the graph extremely difficult so far, only very few results but some infinite families Thank you! 2 / 2