HyperInt - exact integration with hyperlogarithms

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2 HyperInt - exact integration with hyperlogarithms Erik Panzer erikpanzer@ihes.fr (CNRS, ERC grant , F. Brown) Institute des Hautes Études Scientifiques 35 Route de Chartres 9144 Bures-sur-Yvette France Higgs Centre for Theoretical Physics Edinburgh April 3th, 215

3 Feynman integrals: special functions and numbers Many (not all) FI are expressible via 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 and there special values, like multiple zeta values (MZV) ζ n1,...,n d = Li n1,...,n d (1,..., 1). 1 / 18

4 Feynman integrals: special functions and numbers Many (not all) FI are expressible via 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 and there special values, like multiple zeta values (MZV) ζ n1,...,n d = Li n1,...,n d (1,..., 1). Example (p1 2 = 1, p2 2 = z 2, p3 2 = 1 z 2 ) ( ) Φ = 252ζ 3 ζ ζ 3, ζ4 2 ( ) Φ = 4i Im [Li 2(z) + log(1 z) log z ] z z 1 / 18

5 Feynman integrals: special functions and numbers Many (not all) FI are expressible via 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 and there special values, like multiple zeta values (MZV) ζ n1,...,n d = Li n1,...,n d (1,..., 1). Example (p1 2 = 1, p2 2 = z 2, p3 2 = 1 z 2 ) ( ) Φ = 252ζ 3 ζ ζ 3, ζ4 2 ( ) Φ = 4i Im [Li 2(z) + log(1 z) log z ] z z Questions on MPL from FI Why? When? How? 1 / 18

6 Properties of MPL locally analytic (multivalued) functions non-analytic on divisors z k z k+1 z d {, 1, }, called alphabet at worst logarithmic divergences span an algebra (shuffle and stuffle) plenty of functional equations like Li 2 (1 z) = ζ 2 Li 2 (z) log(1 z) log(z) numerical evaluation solved iterated integrals: path-concatenation and monodromy formulas with rational prefactors: closed under differentiation and integration MPL are well-understood, suitable for computer algebra and therefore particularly useful representations of Feynman integrals. Several tools are available (for example HPL, Ginac, HyperInt). 2 / 18

7 Parametric integration

8 Schwinger parameters Scalar propagators (p 2 e + m 2 e) ae, sdd = e a e D/2 loops(g): Φ(G) = Γ(sdd) e Γ(a e) (, ) E Volume form: Ω = δ(1 α N ) e dα e ( ) Ω ψ sdd ψ D/2 αe ae 1 ϕ e E 3 / 18

9 Schwinger parameters Scalar propagators (p 2 e + m 2 e) ae, sdd = e a e D/2 loops(g): Φ(G) = Γ(sdd) e Γ(a e) (, ) E ( ) Ω ψ sdd ψ D/2 αe ae 1 ϕ e E Volume form: Ω = δ(1 α N ) e dα e, graph polynomials: ψ = U = α e ϕ = F = q 2 (T 1 ) α e + ψ T e / T e / F e F =T 1 T 2 m 2 eα e 3 / 18

10 Schwinger parameters Scalar propagators (p 2 e + m 2 e) ae, sdd = e a e D/2 loops(g): Φ(G) = Γ(sdd) e Γ(a e) (, ) E ( ) Ω ψ sdd ψ D/2 αe ae 1 ϕ e E Volume form: Ω = δ(1 α N ) e dα e, graph polynomials: ψ = U = α e ϕ = F = q 2 (T 1 ) α e + ψ T e / T e / F e F =T 1 T 2 m 2 eα e Example (D = 4 2ε, a e = 1) ψ = α 1 + α 2 + α 3 ϕ = p 2 1α 2 α 3 + p 2 2α 1 α 3 + m 2 α 3 (α 1 + α 2 + α 3 ) 3 p 1 p 2 G = 2 1 Φ(G) = Γ(1 + ε) (, ) E p 3 Ω ψ 1 2ε ϕ 1+ε 3 / 18

11 Schwinger parameters Scalar propagators (p 2 e + m 2 e) ae, sdd = e a e D/2 loops(g): Φ(G) = Γ(sdd) e Γ(a e) (, ) E ( ) Ω ψ sdd ψ D/2 αe ae 1 ϕ e E Volume form: Ω = δ(1 α N ) e dα e, graph polynomials: ψ = U = α e ϕ = F = q 2 (T 1 ) α e + ψ T e / T e / F e F =T 1 T 2 m 2 eα e Example (D = 4 2ε, a e = 1) ψ = α 1 + α 2 + α 3 ϕ = p 2 1α 2 α 3 + p 2 2α 1 α 3 + m 2 α 3 (α 1 + α 2 + α 3 ) 3 p 1 p 2 G = 2 1 Φ(G) = Γ(1 + ε) p 3 n= ε n n! (, ) E Ω ψ2 logn ψϕ ϕ 3 / 18

12 Integration with hyperlogarithms proposed by Brown, applications by Chavez & Duhr, Wißbrock, Anastasiou et. al. Idea: integrate out one variable after the other, f n = f n 1 dα n. 4 / 18

13 Integration with hyperlogarithms proposed by Brown, applications by Chavez & Duhr, Wißbrock, Anastasiou et. al. Idea: integrate out one variable after the other, f n = f n 1 dα n. Definition (Hyperlogarithms, Lappo-Danilevsky 1927) G(σ 1,..., σ }{{ w ; z) := } σ z dz 1 z 1 σ 1 z1 dz 2 z 2 σ 2 zw 1 dz w z w σ w 4 / 18

14 Integration with hyperlogarithms proposed by Brown, applications by Chavez & Duhr, Wißbrock, Anastasiou et. al. Idea: integrate out one variable after the other, f n = f n 1 dα n. Definition (Hyperlogarithms, Lappo-Danilevsky 1927) Strategy: G(σ 1,..., σ }{{ w ; z) := } σ z dz 1 z 1 σ 1 z1 1 Write f n 1 in terms of hyperlogarithms: f n 1 = σ,τ,k dz 2 z 2 σ 2 zw 1 dz w z w σ w G( σ; α n ) (α n τ) k λ σ,τ,k with σ and τ independent of α n. 4 / 18

15 Integration with hyperlogarithms proposed by Brown, applications by Chavez & Duhr, Wißbrock, Anastasiou et. al. Idea: integrate out one variable after the other, f n = f n 1 dα n. Definition (Hyperlogarithms, Lappo-Danilevsky 1927) Strategy: G(σ 1,..., σ }{{ w ; z) := } σ z dz 1 z 1 σ 1 z1 1 Write f n 1 in terms of hyperlogarithms: f n 1 = σ,τ,k dz 2 z 2 σ 2 zw 1 dz w z w σ w 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. 4 / 18

16 Integration with hyperlogarithms proposed by Brown, applications by Chavez & Duhr, Wißbrock, Anastasiou et. al. Idea: integrate out one variable after the other, f n = f n 1 dα n. Definition (Hyperlogarithms, Lappo-Danilevsky 1927) Strategy: G(σ 1,..., σ }{{ w ; z) := } σ z dz 1 z 1 σ 1 z1 1 Write f n 1 in terms of hyperlogarithms: f n 1 = σ,τ,k dz 2 z 2 σ 2 zw 1 dz w z w σ w 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 4 / 18

17 Integration with hyperlogarithms Example: massless triangle ( ) Φ = = = 1 z z Ω (α 1 + α 2 + α 3 )(α 2 α 3 + z zα 1 α 2 + (1 z)(1 z)α 2 α 3 ) Ω (zα 1 + α 2 )( zα 1 + α 2 ) log (α 1 + α 2 )(z zα 1 + α 2 ) (1 z)(1 z)α 1 α 2 ( 1 dα 2 α 2 + z 1 ) log (α 2 + 1)(α 2 + z z) α 2 + z (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 5 / 18

18 Integration with hyperlogarithms Example: massless triangle ( ) Φ = = = 1 z z Ω (α 1 + α 2 + α 3 )(α 2 α 3 + z zα 1 α 2 + (1 z)(1 z)α 2 α 3 ) Ω (zα 1 + α 2 )( zα 1 + α 2 ) log (α 1 + α 2 )(z zα 1 + α 2 ) (1 z)(1 z)α 1 α 2 ( 1 dα 2 α 2 + z 1 ) log (α 2 + 1)(α 2 + z z) α 2 + z (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 Remark MPL can be rewritten as hyperlogarithms and back: G( n d 1, σ d,..., n 1 1, σ 1 ; z) = ( 1) d Li n1,...,n d ( σ2 σ 1,, σ d σ d 1, z σ d ) 5 / 18

19 Linear reducibility We need that all partial integrals f n := f n 1 dα n = f dα 1 dα n (, ) n ( f = ) ψsdd D/2 ϕ sdd αe ae 1 e are hyperlogarithms in α n+1 with rational prefactors. 6 / 18

20 Linear reducibility We need that all partial integrals f n := f n 1 dα n = f dα 1 dα n (, ) n ( f = ) ψsdd D/2 ϕ sdd αe ae 1 e are hyperlogarithms in α n+1 with rational prefactors. Definition If this holds for some ordering e 1,..., e N of its edges, the Feynman graph G is called linearly reducible. 6 / 18

21 Linear reducibility We need that all partial integrals f n := f n 1 dα n = f dα 1 dα n (, ) n ( f = ) ψsdd D/2 ϕ sdd αe ae 1 e are hyperlogarithms in α n+1 with rational prefactors. Definition If this holds for some ordering e 1,..., e N of its edges, the Feynman graph G is called linearly reducible. condition on the polynomials ψ and ϕ only; independent of ε-order and expansion point (D, a) ε= 2N Z N sufficient criteria: polynomial reduction algorithms (Brown, Panzer) 6 / 18

22 Polynomial reduction Denote alphabets (divisors) by sets S of irreducible polynomials. Definition Let S denote a set of polynomials f = f e α e + f e linear in α e. Then with [f, g] e := f e g e f e g e, S e shall be the set of irreducible factors of Example (massless triangle) {f e, f e : f S} and {[f, g] e : f, g S}. S = {ψ, ϕ} = {α 1 + α 2 + α 3, α 2 α 3 + z zα 1 α 3 + (1 z)(1 z)α 1 α 2 } [ϕ, ψ] 3 = (α 1 + α 2 )(α 2 + α 1 z z) (1 z)(1 z)α 1 α 2 7 / 18

23 Polynomial reduction Denote alphabets (divisors) by sets S of irreducible polynomials. Definition Let S denote a set of polynomials f = f e α e + f e linear in α e. Then with [f, g] e := f e g e f e g e, S e shall be the set of irreducible factors of Example (massless triangle) {f e, f e : f S} and {[f, g] e : f, g S}. S = {ψ, ϕ} = {α 1 + α 2 + α 3, α 2 α 3 + z zα 1 α 3 + (1 z)(1 z)α 1 α 2 } [ϕ, ψ] 3 = (zα 1 + α 2 )( zα 1 + α 2 ) S 3 = {α 1 + α 2, zα 1 + α 2, zα 1 + α 2, z zα 1 + α 2, α 1, α 2, 1 z, 1 z} 7 / 18

24 Polynomial reduction Denote alphabets (divisors) by sets S of irreducible polynomials. Definition Let S denote a set of polynomials f = f e α e + f e linear in α e. Then with [f, g] e := f e g e f e g e, S e shall be the set of irreducible factors of Example (massless triangle) {f e, f e : f S} and {[f, g] e : f, g S}. S = {ψ, ϕ} = {α 1 + α 2 + α 3, α 2 α 3 + z zα 1 α 3 + (1 z)(1 z)α 1 α 2 } [ϕ, ψ] 3 = (zα 1 + α 2 )( zα 1 + α 2 ) S 3 = {α 1 + α 2, zα 1 + α 2, zα 1 + α 2, z zα 1 + α 2, α 1, α 2, 1 z, 1 z} Lemma If the singularities of F are cointained in S, then the singularities of F dα e are contained in S e. 7 / 18

25 Polynomial reduction Corollary (linear reducibility) If all S k := (S k 1 ) k are linear in α k+1, then any MPL F with alphabet in S integrates to a MPL F n e=1 dα e with alphabet in S n. Example (massless triangle) S = {ψ, ϕ} = {α 1 + α 2 + α 3, α 2 α 3 + z zα 1 α 3 + (1 z)(1 z)α 1 α 2 } S 3 = {α 1 + α 2, zα 1 + α 2, zα 1 + α 2, z zα 1 + α 2, α 1, α 2, 1 z, 1 z} S 3,2 = {z, z, 1 z, 1 z, z z, z z 1} 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}. 8 / 18

26 Polynomial reduction Corollary (linear reducibility) If all S k := (S k 1 ) k are linear in α k+1, then any MPL F with alphabet in S integrates to a MPL F n e=1 dα e with alphabet in S n. Example (massless triangle) S = {ψ, ϕ} = {α 1 + α 2 + α 3, α 2 α 3 + z zα 1 α 3 + (1 z)(1 z)α 1 α 2 } S 3 = {α 1 + α 2, zα 1 + α 2, zα 1 + α 2, z zα 1 + α 2, α 1, α 2, 1 z, 1 z} S 3,2 = {z, z, 1 z, 1 z, z z, z z 1} 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}. There are more sophisticated (and much more powerful) polynomial reduction alorithms (Fubini and several variants of compatibility graphs). 8 / 18

27 Linearly reducible graphs

28 Linear reducibility: Schwinger parameters Definition (vertex-width 3 (Brown), 3-constructible (Schnetz)) The class of 3-point graphs including star, triangle and closed under and Example 9 / 18

29 Linear reducibility: Schwinger parameters Definition (vertex-width 3 (Brown), 3-constructible (Schnetz)) The class of 3-point graphs including star, triangle and closed under and Example 9 / 18

30 Linear reducibility: Schwinger parameters Definition (vertex-width 3 (Brown), 3-constructible (Schnetz)) The class of 3-point graphs including star, triangle and closed under and Example 9 / 18

31 Linear reducibility: Schwinger parameters Definition (vertex-width 3 (Brown), 3-constructible (Schnetz)) The class of 3-point graphs including star, triangle and closed under and Example 9 / 18

32 Linear reducibility: Schwinger parameters Definition (vertex-width 3 (Brown), 3-constructible (Schnetz)) The class of 3-point graphs including star, triangle and closed under and Example 9 / 18

33 Linear reducibility: Schwinger parameters Definition (vertex-width 3 (Brown), 3-constructible (Schnetz)) The class of 3-point graphs including star, triangle and closed under and Example 9 / 18

34 Linear reducibility: Schwinger parameters Definition (vertex-width 3 (Brown), 3-constructible (Schnetz)) The class of 3-point graphs including star, triangle and closed under and Example Linearly reducible: 1 3-constructible massless propagators (Brown) 9 / 18

35 Linear reducibility: Schwinger parameters Definition (vertex-width 3 (Brown), 3-constructible (Schnetz)) The class of 3-point graphs including star, triangle and closed under and Linearly reducible: 1 3-constructible massless propagators (Brown) 9 / 18

36 Linear reducibility: Schwinger parameters Definition (vertex-width 3 (Brown), 3-constructible (Schnetz)) The class of 3-point graphs including star, triangle and closed under and Linearly reducible: 1 3-constructible massless propagators (Brown) 2 all 4-loop massless propagators (Panzer) 9 / 18

37 Linear reducibility: Schwinger parameters Definition (vertex-width 3 (Brown), 3-constructible (Schnetz)) The class of 3-point graphs including star, triangle and closed under and Linearly reducible: 1 3-constructible massless propagators (Brown) 2 all 4-loop massless propagators (Panzer) 3 all massless off-shell 3-point with 2 loops (Chavez & Duhr) 9 / 18

38 Linear reducibility: Schwinger parameters Definition (vertex-width 3 (Brown), 3-constructible (Schnetz)) The class of 3-point graphs including star, triangle and closed under and Linearly reducible: 1 3-constructible massless propagators (Brown) 2 all 4-loop massless propagators (Panzer) 3 all massless off-shell 3-point with 3 loops (Panzer) 9 / 18

39 Linear reducibility: Schwinger parameters Definition (vertex-width 3 (Brown), 3-constructible (Schnetz)) The class of 3-point graphs including star, triangle and closed under and Linearly reducible: 1 3-constructible massless propagators (Brown) 2 all 4-loop massless propagators (Panzer) 3 all massless off-shell 3-point with 3 loops (Panzer) 4 massless on-shell 4-point up to 2 loops (Lüders) 9 / 18

40 Linear reducibility: Forest functions Theorem (Panzer) All 3-constructible massless 3-point graphs evaluate to MPL. 1 / 18

41 All minors of ladder boxes (with p 2 1 = p2 2 = ) evaluate to MPL. 1 / 18 Linear reducibility: Forest functions Theorem (Panzer) All 3-constructible massless 3-point graphs evaluate to MPL. Minors of ladder boxes are closed under the operations v 1 v 4 v 1 v 4 v 1 v v e,,v e v 2 v 3 v 2 v 3 v 3 v 2 v 3 v 2 v 4 e v 3 Example v v 4 v 1 v 4 v 1 v 4 v 1 v 4 v 2 v 3 v 2 v 3 v 2 v 3 v 2 v 3 Theorem (Panzer)

42 HyperInt

43 HyperInt open source Maple program integration of hyperlogarithms transformations of arguments (functional equations) polynomial reduction graph polynomials 11 / 18

44 HyperInt open source Maple program integration of hyperlogarithms transformations of arguments (functional equations) polynomial reduction graph polynomials symbolic computation of constants (no numerics) 11 / 18

45 HyperInt Example open source Maple program integration of hyperlogarithms transformations of arguments (functional equations) polynomial reduction graph polynomials symbolic computation of constants (no numerics) > 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 / 18

46 HyperInt: propagator > E := [[2,1],[2,3],[2,5],[5,1],[5,3],[5,4],[4,1],[4,3]]: > psi := graphpolynomial(e): > phi := secondpolynomial(e, [[1,1], [3,1]]): > add((epsilon*log(psiˆ5/phiˆ4))ˆn/n!,n=..2)/psiˆ2: > hyperint(eval(%,x[8]=1), [seq(x[n],n=1..7)]): > collect(fibrationbasis(%), epsilon); ( 254ζ ζ 5 2ζ 2 ζ 5 196ζ ζ ) ε 2 ( + 28ζ ζ ) 7 ζ3 2 ε + 2ζ ζ2 2ζ 3 12 / 18

47 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])); (Hlog (z, [1]) Hlog (zz, []) Hlog (z, []) Hlog (zz, [1]) + Hlog (zz, [, 1]) Hlog (zz, [1, ]) + Hlog (z, [1, ]) Hlog (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} 13 / 18

48 HyperInt: MGEW s s 2 t 1 s 1 t 2 1 Linearly reducible representation: 1 1 F (2) (2,2) (α) = dx 1 dx 2 p (x 1, α)p (x 2, α)g (2) (2,2) (x 1, x 2 ) ( ) G (2) (2,2) (x x1 1 x 2 1, x 2 ) = θ(x 1 x 2 ) ln 1 x 1 x ( 2 ) 1 p ɛ (x, γ) = x + 1/(α 1) 1 1 ɛ x α/(α 1) 2 14 / 18

49 HyperInt: MGEW s s 2 t 1 s 1 t 2 Lemma 1 2 Linearly reducible representation: 1 1 F (2) (2,2) (α) = dx 1 dx 2 p (x 1, α)p (x 2, α)g (2) (2,2) (x 1, x 2 ) ( ) G (2) (2,2) (x x1 1 x 2 1, x 2 ) = θ(x 1 x 2 ) ln 1 x 1 x ( 2 ) 1 p ɛ (x, γ) = x + 1/(α 1) 1 1 ɛ x α/(α 1) All gluon-exchange graphs with two Wilson lines are linearly reducible (can be integrated directly) and give MPL with alphabet {A, 1 A} (A := α 2 ). > web2([2,1]); 4ζ 3 + Hlog (,,, [A]) 2Hlog (, 1,, [A]) + 2ζ 2 Hlog (, [A]) (1 A) 2 14 / 18

50 HyperInt: MGEW s Angles between pairs (β i, β j ) of Wilson lines: γ ij = 2β i β j β 2 i β 2 j = α ij 1 α ij F (n) (D) φ (n) D (x i; ɛ) = 1 1 n k=1 n 1 [dx k p ɛ (x k, γ k )] φ (n) D (x; ɛ) dy k (1 y k ) 1+2ɛ y 1+2kɛ k Θ D [{x k, y k }] 15 / 18

51 HyperInt: MGEW s Angles between pairs (β i, β j ) of Wilson lines: γ ij = 2β i β j β 2 i β 2 j = α ij 1 α ij F (n) (D) φ (n) D (x i; ɛ) = 1 1 n k=1 n 1 [dx k p ɛ (x k, γ k )] φ (n) D (x; ɛ) dy k (1 y k ) 1+2ɛ y 1+2kɛ k Θ D [{x k, y k }] Conjecturally, subtracted webs are expressible in terms of (A := α 2 ) [ 1 M n,k,l = ds 1 + s 1 ] k (s + 1)(s + A) ln ln l α s + A A + s A s lnn s + 1, which themselves are MPL over the alphabet {A, 1 A}. > Msq(2,,); 2Hlog (A, [,, ]) 4ζ 3 8ζ 2 Hlog (A, [1]) 4Hlog (A, [1,, ]) +4ζ 2 Hlog (A, []) + 8Hlog (A, [1, 1, ]) 4Hlog (A, [, 1, ]) 15 / 18

52 Divergences in Schwinger parameters 3 p 1 p 2 ψ = α1 + α 2 + α 3 G = 2 1 ϕ = p 2 1 α 2α 3 + p 2 2 α 1α 3 + m 2 α 3 (α 1 + α 2 + α 3 ) p 3 16 / 18

53 Divergences in Schwinger parameters 3 p 1 p 2 ψ = α1 + α 2 + α 3 G = 2 1 p 3 ϕ = p 2 1 α 2α 3 + p 2 2 α 1α 3 + m 2 α 3 (α 1 + α 2 + α 3 ) = α 3 ϕ In D = 4 2ε, subdivergence dα 3 α 3 at ε = : ( ) Ω Φ D = Γ(1 + ε) ψ 1 2ε ϕ 1+ε α3 1+ε 16 / 18

54 Divergences in Schwinger parameters 3 p 1 p 2 ψ = α1 + α 2 + α 3 G = 2 1 p 3 ϕ = p 2 1 α 2α 3 + p 2 2 α 1α 3 + m 2 α 3 (α 1 + α 2 + α 3 ) = α 3 ϕ In D = 4 2ε, subdivergence dα 3 α 3 at ε = : ( ) Ω Φ D = Γ(1 + ε) ψ 1 2ε ϕ 1+ε α3 1+ε Regularization: integrate by parts Ω α ε 3 ψ 1 2ε ϕ 1+ε α3 1+ε = εψ 1 2ε ϕ 1+ε + 1 α 3 = ε = 1 [ Ωα 3 2ε 1 ε ψ 1 2ε ϕ 1+ε ψ Ω α3 ε ψ 1+2ε ϕ 1 ε α 3 (1 + ε)α 3m 2 ϕ ] 16 / 18

55 Divergences in Schwinger parameters 3 p 1 p 2 ψ = α1 + α 2 + α 3 G = 2 1 p 3 ϕ = p 2 1 α 2α 3 + p 2 2 α 1α 3 + m 2 α 3 (α 1 + α 2 + α 3 ) = α 3 ϕ In D = 4 2ε, subdivergence dα 3 α 3 at ε = : ( ) Φ D = ( 2 1 ε ) Φ D+2 ( ) 2m2 ε Φ D+2 Regularization: integrate by parts Ω α ε 3 ψ 1 2ε ϕ 1+ε α3 1+ε = εψ 1 2ε ϕ 1+ε + 1 α 3 = ε = 1 [ Ωα 3 2ε 1 ε ψ 1 2ε ϕ 1+ε ψ ( ) Ω α3 ε ψ 1+2ε ϕ 1 ε α 3 (1 + ε)α 3m 2 ϕ ] 16 / 18

56 Regularization of subdivergences Theorem Every Euclidean Feynman integral is a finite linear combination of Feynman integrals without subdivergences. 17 / 18

57 Regularization of subdivergences Theorem Every Euclidean Feynman integral is a finite linear combination of Feynman integrals without subdivergences. only the original Feynman graph occurs, with shifted D and a e preserves linear reducibility! decomposition computable by partial integrations allows for numeric integration reproves: coefficients of ε-expansion are periods (Bogner & Weinzierl) 17 / 18

58 Regularization of subdivergences Theorem Every Euclidean Feynman integral is a finite linear combination of Feynman integrals without subdivergences. only the original Feynman graph occurs, with shifted D and a e preserves linear reducibility! decomposition computable by partial integrations allows for numeric integration reproves: coefficients of ε-expansion are periods (Bogner & Weinzierl) Corollary (IBP, Euclidean kinematics) One can choose master integrals to be scalar and free of subdivergences, given that one allows for shifted dimensions D + 2, D + 4,... and dots. 17 / 18

59 Summary polynomial reduction lots of linearly reducible graphs with non-trivial kinematics few easily recognizable, infinite families 18 / 18

60 Summary polynomial reduction lots of linearly reducible graphs with non-trivial kinematics few easily recognizable, infinite families HyperInt 18 / 18

61 Summary polynomial reduction lots of linearly reducible graphs with non-trivial kinematics few easily recognizable, infinite families HyperInt analytic regularization in terms of Feynman graphs 18 / 18

62 Summary polynomial reduction lots of linearly reducible graphs with non-trivial kinematics few easily recognizable, infinite families HyperInt analytic regularization in terms of Feynman graphs Schwinger parameters are not always optimal: sometimes Φ(G) is MPL, but not linearly reducible 18 / 18

63 Thank you! Questions?

64 Forest polynomials Definition Spanning forest polynomial Φ A,B := F separate the vertices A and B. e / F α e over 2-forests F which f 12 := Φ {1,2},{3,4} f 3 := Φ {3},{1,2,4} f 14 := Φ {1,4},{2,3} f 4 := Φ {4},{1,2,3} Example v v 4 v 2 v 3 ψ = α 1 + α 2 + α 3 + α 4 f 12 = α 2 α 4 f 14 = α 1 α 3 f 3 = α 2 α 3 f 4 = α 3 α 4 ϕ = F = (p 1 + p 2 ) 2 f 12 + (p 1 + p 4 ) 2 f 14 + p 2 3f 3 + p 2 4f 4 1 / 15

65 Restricting forest polynomials Definition F G (z) := R E + ψ D/2 G δ (4) ( f ψ z ) e E Example (a 1 = a 2 = a 3 = a 4 = 1) α ae 1 e dα e ( R 4 + R + ) 1 v 1 v 4 (D = 4) z 4 3 z 4 F 1 3 ; z 2 = z 12 [z v 2 v 12 (z 14 + z 3 + z 4 ) + z 3 z 4 ] 3 2 (D = 6) }{{} Q 2 / 15

66 Appending a vertex G = v 1 v 2 v 3 v 4 G = v 1 v 4 e v 2 v 3 v 3 or v 1 v 2 v 3 Using (f 12, f 14, f 3, f 4, ψ ) = (f 12, f 14, f 3, f 4 + α e ψ, ψ) where x = α e, F G (z) = z4 F G (z 12, z 14, z 3, z 4 x) x ae 1 dx v 4 e v 4 3 / 15

67 Appending a vertex G = v 1 v 2 v 3 v 4 G = v 1 v 4 e v 2 v 3 v 3 or v 1 v 4 v 2 v 3 Using (f 12, f 14, f 3, f 4, ψ ) = (f 12, f 14, f 3, f 4 + α e ψ, ψ) where x = α e, F G (z) = z4 F G (z 12, z 14, z 3, z 4 x) x ae 1 dx e v 4 Example (D = 6 and a e = 1) v 4 ; z F v 1 v 2 v 3 = z3 F v v 4 v 2 v 3 ; z 12, z 14, z 3, z 4 dz 3 3 / 15

68 Appending a vertex G = v 1 v 2 v 3 v 4 G = v 1 v 4 e v 2 v 3 v 3 or v 1 v 4 v 2 v 3 Using (f 12, f 14, f 3, f 4, ψ ) = (f 12, f 14, f 3, f 4 + α e ψ, ψ) where x = α e, F G (z) = z4 F G (z 12, z 14, z 3, z 4 x) x ae 1 dx e v 4 Example (D = 6 and a e = 1) v 1 v 4 z3 z F ; z 12 dz 3 = [z 12 (z 14 + z 3 + z 4 ) + z 3 z 4 ] 2 = z 3 (z 14 + z 4 ) Q v 2 v 3 3 / 15

69 Appending a vertex G = v 1 v 2 v 3 v 4 G = v 1 v 4 e v 2 v 3 v 3 or v 1 v 4 v 2 v 3 Using (f 12, f 14, f 3, f 4, ψ ) = (f 12, f 14, f 3, f 4 + α e ψ, ψ) where x = α e, F G (z) = z4 F G (z 12, z 14, z 3, z 4 x) x ae 1 dx e v 4 Example (D = 6 and a e = 1) v 1 v 4 z3 z F ; z 12 dz 3 = [z 12 (z 14 + z 3 + z 4 ) + z 3 z 4 ] 2 = z 3 (z 14 + z 4 ) Q v 2 v 3 v 4 ; z F v 1 v 2 v 3 = z4 F v 1 v 4 ; z 12, z 14, z 3, z 4 dz 4 v 2 v 3 3 / 15

70 Appending a vertex G = v 1 v 2 v 3 v 4 G = v 1 v 4 e v 2 v 3 v 3 or v 1 v 4 v 2 v 3 Using (f 12, f 14, f 3, f 4, ψ ) = (f 12, f 14, f 3, f 4 + α e ψ, ψ) where x = α e, F G (z) = z4 F G (z 12, z 14, z 3, z 4 x) x ae 1 dx e v 4 Example (D = 6 and a e = 1) v 1 v 4 z3 z F ; z 12 dz 3 = [z 12 (z 14 + z 3 + z 4 ) + z 3 z 4 ] 2 = z 3 (z 14 + z 4 ) Q v 2 v 3 v 4 ; z F v 1 v 2 v 3 = 1 ln z 12(z 3 + z 14 )(z 4 + z 14 ) z 12 z 14 z 14 Q 3 / 15

71 Adding an edge v 1 v 4 G = v 1 v 4 G = v 2 v 3 v 2 v 3 e 4 / 15

72 Adding an edge v 1 v 1 v 4 v 4 G = G = e v 2 v 3 v 2 v 3 Dodgson-identities between spanning forest polynomials: ( ) = f 12 (f 14 + f 3 + f 4 ) + f 3 f 4 = Q(f ) = ψ Φ {1,2},{3},{4} 4 / 15

73 Adding an edge v 1 v 1 v 4 v 4 G = G = e v 2 v 3 v 2 v 3 Dodgson-identities between spanning forest polynomials: ( ) = f 12 (f 14 + f 3 + f 4 ) + f 3 f 4 = Q(f ) = ψ Φ {1,2},{3},{4} z12 F G (z) [ = Q ae+sdd D x D/2 ae 1 Q D/2 sdd F G ]z dx 12 =z 12 x 4 / 15

74 Adding an edge v 1 v 4 G = v 1 v 4 G = v 2 v 3 v 2 v 3 e z12 F G (z) [ = Q ae+sdd D x D/2 ae 1 Q D/2 sdd F G ]z dx 12 =z 12 x Example (D = 6 and a e = 1) v 1 v 4 F ; z = 1 z12 Q 2 F v 2 v 3 = z 12 z 14 Q 2 + z 12 z 14 Q 2 Li 2 [ ln Q ( z3 z 4 Q v 1 v 4 v 2 v 3 ; z 12 x, z 14, z 3, z 4 xdx ln (z ( )] 14 + z 3 )(z 14 + z 4 ) z 3 z 4 z 14 (z 14 + z 3 + z 4 ) Li z3 z 4 (z 14 z 12 ) 2 z 14 Q ) + z 12 Q 2 ln z 14 z 3 z 4 z 12 (z 14 + z 3 )(z 14 + z 4 ) ln(z 3z 4 /Q) Q(z 14 + z 3 + z 4 ) 4 / 15

75 Kinematics from forest functions Corollary Φ(G) = ϕ = F = (p 1 + p 2 ) 2 f 12 + (p 1 + p 4 ) 2 f 14 + p 2 3f 3 + p 2 4f 4 Γ(sdd) F G (z) Ω e Γ(a [ e) (p1 + p 2 ) 2 z 12 + (p 1 + p 4 ) 2 z 14 + p3 2z 3 + p2 4 z sdd 4] Example (kinematics: s = (p 1 + p 2 ) 2 and u = (p 1 + p 4 ) 2 ) Φ ( ) = = dz 12 sz 12 + u z 12 dz 3 dz 4 [z 12 (1 + z 3 + z 4 ) + z 4 z 3 ] 2 dz 12 ln z 12 (sz 12 + u)(z 12 1) = π2 + ln 2 (s/u) 2(s + u) 5 / 15

76 Compatibility graph of box-ladders z 12 z 14 z 12 +z 3 Q z 12 +z 4 z 14 +z 3 +z 4 z 14 +z 3 z 14 +z 4 6 / 15

77 Two-loop non-planar form factor: regularization (4 2ɛ) = 4(1 ɛ)(3 4ɛ)(1 4ɛ) ɛs 2 (6 2ɛ) 1 65ɛ + 131ɛ2 74ɛ 3 ɛ 3 s 2 (6 2ɛ) ɛ + 355ɛ2 42ɛ ɛ 4 (1 2ɛ)ɛ 3 s 3 (4 2ɛ) 7 / 15

78 Divergences in Schwinger parameters In D = 4 2ε dimensions, Φ = Γ(2 + 3ε) Ω ϕ 2+3ε ψ 4ε where ϕ = α 4 ( ) + α 5 ( ) + α 4 α 5 ( ) vanishes at α 4 = α 5 =. Rescale α 4 λα 4 and α 5 λα 5, so ϕ λ ϕ and ψ ψ: Ωψ 4ε Ωψ 4ε ϕ 2+3ε = ϕ 2+3ε δ(α 4 + α 5 λ) dλ 8 / 15

79 Divergences in Schwinger parameters In D = 4 2ε dimensions, Φ = Γ(2 + 3ε) Ω ϕ 2+3ε ψ 4ε where ϕ = α 4 ( ) + α 5 ( ) + α 4 α 5 ( ) vanishes at α 4 = α 5 =. Rescale α 4 λα 4 and α 5 λα 5, so ϕ λ ϕ and ψ ψ: Ωψ 4ε ϕ 2+3ε = dλ ψ 4ε Ω δ(α 4 + α 5 1) λ 1+3ε ϕ 2+3ε 8 / 15

80 Divergences in Schwinger parameters In D = 4 2ε dimensions, Φ = Γ(2 + 3ε) Ω ϕ 2+3ε ψ 4ε where ϕ = α 4 ( ) + α 5 ( ) + α 4 α 5 ( ) vanishes at α 4 = α 5 =. Rescale α 4 λα 4 and α 5 λα 5, so ϕ λ ϕ and ψ ψ: Ωψ 4ε ( 1 ( ) ϕ 2+3ε = dλ ψ Ω δ(α 4 + α 5 1) 3ε) 4ε λ 3ε λ ϕ 2+3ε 8 / 15

81 Divergences in Schwinger parameters In D = 4 2ε dimensions, Φ = Γ(2 + 3ε) Ω ϕ 2+3ε ψ 4ε where ϕ = α 4 ( ) + α 5 ( ) + α 4 α 5 ( ) vanishes at α 4 = α 5 =. Rescale α 4 λα 4 and α 5 λα 5, so ϕ λ ϕ and ψ ψ: Ωψ 4ε { Ωψ 4ε 4 ϕ 2+3ε = 1 ψ ϕ 2+3ε 3 ψ λ 2 + 3ε } 1 ϕ 3ε ϕ λ λ=1 8 / 15

82 Divergences in Schwinger parameters In D = 4 2ε dimensions, Φ = Γ(2 + 3ε) Ω ϕ 2+3ε ψ 4ε where ϕ = α 4 ( ) + α 5 ( ) + α 4 α 5 ( ) vanishes at α 4 = α 5 =. Rescale α 4 λα 4 and α 5 λα 5, so ϕ λ ϕ and ψ ψ: Ωψ 4ε { Ωψ 4ε 4 ϕ 2+3ε = 1 ψ ϕ 2+3ε 3 ψ λ 2 + 3ε } 1 ϕ 3ε ϕ λ λ=1 Numerator monomials correspond to squared propagators: ( Φ D ( ) = 1 3ε Φ D Φ D+2 ( ) ) 8 / 15

83 Sector decomposition Subdivergences (IR and UV) of Feynman integrals result in (higher order) poles in ɛ (dimensional regularization) and obstruct both analytical and numerical evaluation. Standard solution: Sector decomposition. 1 Split original integral into several sectors: f (x) dx 1 dx N = f (x) dx 1 dx N i S i 2 Change of variables in each sector (monomialise): ( N ) 1 f (x) dx 1 dx N = x β k 1 k dx k S i k=1 such that f (x) is bounded on [, 1]N i 3 Regulate all (now normal crossing) divergences at zero: 1 f i (x) x β 1 k dx k f i (x) = 1 β f (x) x k =1 1 1 x β k β dx k xk f i (x) 9 / 15

84 Undesirable features of sector decomposition Many different terms (sectors and subtraction terms) to consider Spurious structures (cancellation between sectors): An individual sector is more complicated than the total sum. 1 1 ln(1 + x) x(1 + x) dx = 1 2 ζ(2) 1 2 ln2 (2) ln(1 + x) x(1 + x) dx = 1 2 ζ(2) ln2 (2) Changes of variables are different in each sector and can destroy linear reducibility, a property allowing for analytical evaluation Idea Avoid sector decomposition and write the integral as a linear combination of subdivergence-free ( primitive or quasi-finite ) Feynman integrals. 1 / 15

85 Double box: IBP reduction to primitive master integrals (4 2ɛ) (6 2ɛ) (6 2ɛ) = A 1 + A 2 + A 3 ɛ 2 (6 2ɛ) + A 4 ɛ 2 (6 2ɛ) + A 5 ɛ 3 (4 2ɛ) + A 6 ɛ 3 (6 2ɛ) + A 7 ɛ 4 (6 2ɛ) + A 8 ɛ 3 (4 2ɛ) 11 / 15

86 Linear reducibility: Massive examples 12 / 15

87 Periods of cocommutative graphs The Feynman period of log. div. graphs depends on the renormalization scheme/point. Cocommutative graphs are an exception. Example (wheels in wheels) P ( ) = 72ζ ζ 7 P = 72ζ 3 3 P + = 48ζ 3 ζ 5 4ζ ζ 9 13 / 15

88 Massless φ 4 theory: primitive sixth roots of unity P 7,11 = is not linearly reducible! Tenth denominator: d 1 = α 2 α 2 4α 1 + α 2 α 2 4α 3 α 1 α 2 α 3 α 4 + α 2 2α 4 α 1 + α 2 2α 4 α 3 2α 2 α 2 3α 4 α 2 2α 2 3 2α 2 2α 3 α 1 2α 2 α 2 3α 1 α 2 3α 2 4 2α 2 3α 4 α 1 α 2 2α 2 1 2α 2 α 3 α 2 1 α 2 3α 2 1. Change variables: α 3 = α 3 α 1 α 1 +α 2 +α 4, α 4 = α 4 (α 2 + α 3 ) and α 1 = α 1 α 4, d 1 = (α 2 + α 3)(α 2 + α 2 α 4 α 1)(α 1α 4 + α 2 + α 2 α 4 + α 3α 4) Final result: not a multiple zeta value, instead MPL at e iπ/3 14 / 15

89 3 P(P7,11 ) ( ) = Im ζ 9 Li ζ 7 Li 4 +24ζ3( Li 2,3 7ζ 3 Li 2 ) 9675 π3 ζ 3, (36 Im Li 2,2,2,5 +27 Li 2,2,3,4 +9 Li 2,2,4,3 +9 Li 2,3,2,4 +3 Li 2,3,3,3 ) 43ζ 3 (Li 2,3,3 +3 Li 2,2,4 ) π Im ( Li 8 ζ Li 2, π2 Im Li 6 ζ Li 3,8 ) Li 4, Li 5,6 ( Li 2, Li 6 ζ Li 4, Li 3,6 ) ( + π ζ ζ ,7 215 Re(3 Li 4,6 +1 Li 3,7 ) ) Abbreviation: Li n1,...,n r := Li n1,...,n r (e iπ/3 ) 15 / 15