Master integrals without subdivergences

Transcription

1

2 Master integrals without subdivergences Joint work with Andreas von Manteuffel and Robert Schabinger Erik Panzer 1 (CNRS, ERC grant ) Institute des Hautes Études Scientifiques 35 Route de Chartres Bures-sur-Yvette France HOCTOOLS NNLO meeting NCSR Demokritos, Athens January 18th, erikpanzer@ihes.fr

3 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 [4, 5]. 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 0 k=1 such that f (x) is bounded on [0, 1]N i 3 Regulate all (now normal crossing) divergences at zero: 1 0 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) 0 1 / 20

4 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 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. 2 / 20

5 Example: Two-loop non-planar form factor (4 2ɛ) = 4(1 ɛ)(3 4ɛ)(1 4ɛ) ɛs 2 (6 2ɛ) (6 2ɛ) 10 65ɛ + 131ɛ2 74ɛ 3 ɛ 3 s 2 (4 2ɛ) ɛ + 355ɛ2 420ɛ ɛ 4 (1 2ɛ)ɛ 3 s 3 3 / 20

6 improving convergence via partial integration

7 Feynman integrals in Schwinger parameters Scalar propagators (pe 2 + me) 2 ae, sdd = e a e D/2 loops(g): Φ(G) = Γ(sdd) e Γ(a ψ sdd D/2 ϕ sdd αe ae 1 dα e δ(1 α N ) e) 0 e E 5 / 20

8 Feynman integrals in Schwinger parameters Scalar propagators (pe 2 + me) 2 ae, sdd = e a e D/2 loops(g): Φ(G) = Γ(sdd) e Γ(a ψ sdd D/2 ϕ sdd αe ae 1 dα e δ(1 α N ) e) 0 Graph polynomials: ψ = U = T e / T α e Example (arbitrary ε) e E ϕ = F = q 2 (T 1 ) α e + ψ e / F e F =T 1 T 2 m 2 eα e Γ(1 + ε) δ(1 α N ) dα 1 dα 2 dα 3 Φ(G) = (α α 2 + α 3 ) 1 2ε [ m 2 (α 1 + α 2 + α 3 ) + p1 2α 2 + p2 2α ] 1+ε 1 α 1+ε 3 3 p 1 p 2 D = 4 2ε a e = 1 sdd = 1 + ε G = 2 1 p 3 5 / 20

9 Feynman integrals in Schwinger parameters Scalar propagators (pe 2 + me) 2 ae, sdd = e a e D/2 loops(g): Φ(G) = Γ(sdd) e Γ(a ψ sdd D/2 ϕ sdd αe ae 1 dα e δ(1 α N ) e) 0 Graph polynomials: ψ = U = T e / T α e Example (expanded in ε 0) e E ϕ = F = q 2 (T 1 ) α e + ψ e / F e F =T 1 T 2 m 2 eα e Γ(1 + ε) δ(1 α N ) dα 1 dα 2 dα 3 Φ(G) = (α α 2 + α 3 ) 1 2ε [ m 2 (α 1 + α 2 + α 3 ) + p1 2α 2 + p2 2α ] 1+ε 1 α 1+ε 3 ε n δ(1 α N ) dα 1 dα 2 dα 3 = Γ(1 + ε) log n ψ2 n! ψϕ ϕ n=0 0 5 / 20

10 Divergences in Schwinger parameters 3 p 1 p 2 ψ = α 1 + α 2 + α 3 G = 2 1 p 3 ( ϕ = α 3 m 2 ψ + p1 2α 2 + p2 2α ) 1 In D = 4 2ε, sdd = 1 + ε such that 0 dα 3 diverges at the lower boundary when ε 0: Ω (α 1 + α 2 + α 3 ) 1 2ε [ m 2 (α 1 + α 2 + α 3 ) + p1 2α 2 + p2 2α ] 1+ε 1 α 1+ε 3 6 / 20

11 Divergences in Schwinger parameters 3 p 1 p 2 ψ = α 1 + α 2 + α 3 G = 2 1 p 3 ( ϕ = α 3 m 2 ψ + p1 2α 2 + p2 2α ) 1 In D = 4 2ε, sdd = 1 + ε such that 0 dα 3 diverges at the lower boundary when ε 0: Ω (α 1 + α 2 + α 3 ) 1 2ε [ m 2 (α 1 + α 2 + α 3 ) + p1 2α 2 + p2 2α ] 1+ε 1 α 1+ε Example (Regularization) Let φ := ϕ/α 3 = m 2 ψ + p2 2α 1 + p1 2α 2 and integrate by parts: Ω α ε 3 ψ 1 2ε ϕ 1+ε α3 ε = εψ 1 2ε ϕ 1+ε + 1 Ω α 3 =0 ε α3 ε ψ 1+2ε ϕ 1 ε α 3 = 1 [ ε Ωα 3 2ε 1 ψ 1 2ε ϕ 1+ε (1 + ε)α 3m 2 ] when ε < 0. ψ ϕ 3 6 / 20

12 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 = 0. 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 0 7 / 20

13 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 = 0. 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 0 7 / 20

14 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 = 0. 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 0 7 / 20

15 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 = 0. 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 7 / 20

16 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 = 0. 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 ( ) ) 7 / 20

17 Regularization of subdivergences Subdivergences (IR and UV) manifest as integrand λ ωγ 1 with ω γ ε=0 0, when some edges α e λ (e γ) get small jointly Well-known power counting: ω γ = { sdd(g/γ) if γ connects external legs (IR), sdd(γ) otherwise (UV). 8 / 20

18 Regularization of subdivergences Subdivergences (IR and UV) manifest as integrand λ ωγ 1 with ω γ ε=0 0, when some edges α e λ (e γ) get small jointly Well-known power counting: ω γ = { sdd(g/γ) if γ connects external legs (IR), sdd(γ) otherwise (UV). Suitable partial integration increments ω γ After finitely many steps, all ω γ ɛ=0 > 0 (no subdivergences) ε-expansion of integrand gives convergent integrals 8 / 20

19 Regularization of subdivergences Subdivergences (IR and UV) manifest as integrand λ ωγ 1 with ω γ ε=0 0, when some edges α e λ (e γ) get small jointly Well-known power counting: ω γ = { sdd(g/γ) if γ connects external legs (IR), sdd(γ) otherwise (UV). Suitable partial integration increments ω γ After finitely many steps, all ω γ ɛ=0 > 0 (no subdivergences) ε-expansion of integrand gives convergent integrals Representation in terms of primitive Feynman integrals, with shifted D and a e 8 / 20

20 Regularization of subdivergences Subdivergences (IR and UV) manifest as integrand λ ωγ 1 with ω γ ε=0 0, when some edges α e λ (e γ) get small jointly Well-known power counting: ω γ = { sdd(g/γ) if γ connects external legs (IR), sdd(γ) otherwise (UV). Suitable partial integration increments ω γ After finitely many steps, all ω γ ɛ=0 > 0 (no subdivergences) ε-expansion of integrand gives convergent integrals Representation in terms of primitive Feynman integrals, with shifted D and a e In practice, this creates too many terms. Thus use IBP! 8 / 20

21 Choose master integrals without subdivergences

22 Primitive (quasi-finite) master integrals Corollary (IBP, Euclidean kinematics) For any topology, one can choose the master integrals to be scalar and quasi-finite (free of subdivergences), given that one allows for shifted dimensions D + 2, D + 4,... and dots. 10 / 20

23 Primitive (quasi-finite) master integrals Corollary (IBP, Euclidean kinematics) For any topology, one can choose the master integrals to be scalar and quasi-finite (free of subdivergences), given that one allows for shifted dimensions D + 2, D + 4,... and dots. 1 available IBP tools have dimension-shifts [16] implemented [10] or are easily extended 2 general algorithm to find quasi-finite basis [17] 3 it suffices to guess enough quasi-finite integrals 10 / 20

24 Primitive (quasi-finite) master integrals Corollary (IBP, Euclidean kinematics) For any topology, one can choose the master integrals to be scalar and quasi-finite (free of subdivergences), given that one allows for shifted dimensions D + 2, D + 4,... and dots. 1 available IBP tools have dimension-shifts [16] implemented [10] or are easily extended 2 general algorithm to find quasi-finite basis [17] 3 it suffices to guess enough quasi-finite integrals Advantages: 1 ε-expansion of the integrand 2 directly suitable for numeric quadrature 3 No splitting into non-feynman integrals like sector decomposition 4 Empirically: Higher pole terms from subtopologies 10 / 20

25 Double box: Primitive master integrals (6 2ɛ) (6 2ɛ) (6 2ɛ) (6 2ɛ) (6 2ɛ) (6 2ɛ) (4 2ɛ) (4 2ɛ) 11 / 20

26 Double box: IBP reduction (4 2ɛ) (6 2ɛ) (6 2ɛ) = A 1 + A 2 + A 4 ɛ 2 (6 2ɛ) + A 5 ɛ 2 (6 2ɛ) + A 7 ɛ 3 (4 2ɛ) + A 3 ɛ 3 (6 2ɛ) + A 8 ɛ 4 (6 2ɛ) + A 6 ɛ 3 (4 2ɛ) 12 / 20

27 Linearly reducible Feynman graphs

28 Integration with hyperlogarithms following Brown [8] Applications by Chavez & Duhr [9], Wißbrock [1], Anastasiou et. al. [3, 2] To compute 0 f dα e where f is a rational linear combination of polylogarithms that depend rationally on α e : 1 Rewrite f using hyperlogarithms: f = w,σ,n G(w; α e ) (α e σ) n λ w,σ,n with constants λ w,σ,n w.r.t. α e. Implemented in the Maple code HyperInt [14], completely algebraic. Assumption: Finite integrals! 14 / 20

29 Integration with hyperlogarithms following Brown [8] Applications by Chavez & Duhr [9], Wißbrock [1], Anastasiou et. al. [3, 2] To compute 0 f dα e where f is a rational linear combination of polylogarithms that depend rationally on α e : 1 Rewrite f using hyperlogarithms: f = w,σ,n G(w; α e ) (α e σ) n λ w,σ,n with constants λ w,σ,n w.r.t. α e. 2 Construct an antiderivative αe F = f. Implemented in the Maple code HyperInt [14], completely algebraic. Assumption: Finite integrals! 14 / 20

30 Integration with hyperlogarithms following Brown [8] Applications by Chavez & Duhr [9], Wißbrock [1], Anastasiou et. al. [3, 2] To compute 0 f dα e where f is a rational linear combination of polylogarithms that depend rationally on α e : 1 Rewrite f using hyperlogarithms: f = w,σ,n G(w; α e ) (α e σ) n λ w,σ,n with constants λ w,σ,n w.r.t. α e. 2 Construct an antiderivative αe F = f. 3 Evaluate the limits 0 f dα e = lim F (α α e e) lim F (α e). α e 0 Implemented in the Maple code HyperInt [14], completely algebraic. Assumption: Finite integrals! 14 / 20

31 Linear reducibility Precondition: For all n < N, f n := [ n 1 e=1 0 dα e ] ψ sdd D/2 ϕ sdd e E α ae 1 e can be written as hyperlogarithms of α e over denominators that factor linearly in α e. Definition If this holds for some ordering e 1,..., e N of its edges, the Feynman graph G is called linearly reducible. 15 / 20

32 Linear reducibility Precondition: For all n < N, f n := [ n 1 e=1 0 dα e ] ψ sdd D/2 ϕ sdd e E α ae 1 e can be written as hyperlogarithms of α e over denominators that factor linearly in α e. Definition If this holds for some ordering e 1,..., e N of its edges, the Feynman graph G is called linearly reducible. Combinatorial condition on the polynomials ψ and ϕ only; independent of ε-order and expansion point (D, a) ε=0 2N Z N Polynomial reduction algorithms [8, 7] available (e.g. HyperInt) to check sufficient criteria for linear reducibility 15 / 20

33 Linearly reducible massless propagators all up to four loops [11]: MZV and maybe alternating sums all φ 4 -periods up to seven loops, except for K 3,4 = P 7,11 = Integrable with graphical functions, O. Schnetz [15]. Extremely efficient (graphs up to ten loops). Linearly reducible after change of variables. Does not give a multiple zeta value! 16 / 20

34 Linearly reducible 3-point graphs Off-shell massless three-point integrals (m e = 0 and p 2 1, p2 2, p2 3 0): All up to three loops [13] 17 / 20

35 Linearly reducible 3-point graphs Off-shell massless three-point integrals (m e = 0 and p 2 1, p2 2, p2 3 0): All up to three loops [13] All with vertex-width three [12] 17 / 20

36 Linearly reducible 4-point graphs Massless on-shell four-point graphs (m e = p 2 1 = = p2 4 = 0): All up to two loops [6] 18 / 20

37 Linearly reducible 4-point graphs Massless on-shell four-point graphs (m e = p 2 1 = = p2 4 = 0): All up to two loops [6] Minors of ladder-boxes (and some generalizations [12]) Also with up to two legs off-shell. 18 / 20

38 Linearly reducible graphs with masses Examples: 19 / 20

39 Summary Master integrals can be chosen to be primitive existing tools for IBP-reduction can be used or extended potential for direct numeric integration Extends exact parametric integration to divergent integrals many examples of linearly reducible graphs with non-trivial kinematics Maple TM implementation: HyperInt [14] arbitrary ε-order, D ε=0 2N, tensors, a e = n e + εν e So far clear for Euclidean kinematics only; possible extension to more general kinematics and phase-space integrals to be investigated 20 / 20

40 Thank you.

41 4-point recursions Start with the box and repeat, in any order: Appending a vertex: v 1 v 4 v 1 v 4 G = G = e v 2 v 3 v 2 v 3 v 3 or v 1 v 4 v 2 v 3 e v 4 2 / 15

42 4-point recursions Start with the box and repeat, in any order: Appending a vertex: v 1 v 4 v 1 v 4 G = G = e v 2 v 3 v 2 v 3 v 3 or v 1 v 4 v 2 v 3 e v 4 Adding an edge: v 1 v 4 G = v 1 v 4 G = v 2 v 3 v 2 v 3 e 2 / 15

43 4-point recursions Start with the box and repeat, in any order: Appending a vertex: v 1 v 4 v 1 v 4 G = G = e v 2 v 3 v 2 v 3 v 3 or v 1 v 4 v 2 v 3 e v 4 Adding an edge: v 1 v 4 G = v 1 v 4 G = v 2 v 3 v 2 v 3 e 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 2 / 15

44 HyperInt > E := [[1,2],[2,3],[3,4],[4,1],[5,1],[5,2],[5,3],[5,4]]: > psi := graphpolynomial(e): > phi := secondpolynomial(e, [[1,1], [3,1]]): > sdd := nops(e)-(1/2)*4*(4-2*epsilon): > f := series(psiˆ(-2+epsilon+sdd)*phiˆ(-sdd), epsilon=0): > f := add(coeff(f,epsilon,n)*epsilonˆn,n=0..2): > z := [x[1],x[2],x[6],x[5],x[3],x[4],x[7],x[8]]: > hyperint(eval(f,z[-1]=1), z[1..-2]): > collect(fibrationbasis(%), epsilon); 3 / 15

45 HyperInt > E := [[1,2],[2,3],[3,4],[4,1],[5,1],[5,2],[5,3],[5,4]]: > psi := graphpolynomial(e): > phi := secondpolynomial(e, [[1,1], [3,1]]): > sdd := nops(e)-(1/2)*4*(4-2*epsilon): > f := series(psiˆ(-2+epsilon+sdd)*phiˆ(-sdd), epsilon=0): > f := add(coeff(f,epsilon,n)*epsilonˆn,n=0..2): > z := [x[1],x[2],x[6],x[5],x[3],x[4],x[7],x[8]]: > hyperint(eval(f,z[-1]=1), z[1..-2]): > collect(fibrationbasis(%), epsilon); ( 254ζ ζ 5 200ζ 2 ζ 5 196ζ ζ ) ε 2 ( + 28ζ ζ ) 7 ζ3 2 ε + 20ζ 5. 5 ζ2 2ζ 3 3 / 15

46 Recursion of forest functions Example (D = 6 and a e = 1) v 4 ; z F v 1 v 2 v 3 = z3 0 F v v 4 v 2 v 3 ; z 12, z 14, z 3, z 4 dz 3 4 / 15

47 Recursion of forest functions 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 / 15

48 Recursion of forest functions 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 / 15

49 Recursion of forest functions 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 = 0 z4 0 F v 1 v 4 ; z 12, z 14, z 3, z 4 dz 4 v 2 v 3 4 / 15

50 Recursion of forest functions 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 = 0 1 ln z 12(z 3 + z 14 )(z 4 + z 14 ) z 12 z 14 z 14 Q 4 / 15

51 Recursion of forest functions 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 v 4 ; z F v 1 v 2 v ln z 12(z 3 + z 14 )(z 4 + z 14 ) z 12 z 14 z 14 Q = 1 z12 Q 2 F 0 v 1 v 4 v 2 v 3 ; z 12 x, z 14, z 3, z 4 xdx 4 / 15

52 Recursion of forest functions Example (kinematics: s = (p 1 + p 2 ) 2 and x = (p 1 + p 4 ) 2 /s) sφ ( ) = = = dz 12 z 12 + x dz 12 z 12 + x 0 0 z 12 dz 3 dz 4 [z 12 (1 + z 3 + z 4 ) + z 4 z 3 ] 2 0 dz 3 (1 + z 3 )(z 12 + z 3 ) dz 12 ln z 12 (z 12 + x)(z 12 1) = π2 + ln 2 x 2(1 + x) 5 / 15

53 Massless φ 4 theory: primitive sixth roots of unity P 7,11 = P 7,11 is not linearly reducible: After integrating ten variables, denominator d 10 = 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 1 1 2α 2 α 3 α 2 1 α 2 3α 2 1. Changing 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 10 = (α 2 + α 3)(α 2 + α 2 α 4 α 1)(α 1α 4 + α 2 + α 2 α 4 + α 3α 4) factors linearly and α 1, α 3, α 4 can be integrated (α 2 = 1). The final integrand is HPL(α 1 )/(1 α 1 + α1 2 ) and gives not a multiple zeta value, but a polylogarithm at sixth roots of unity. 6 / 15

54 3 P(P7,11 ) ( ) = Im ζ 9 Li ζ 7 Li ζ3( 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 +10 Li 3,7 ) ) Abbreviation: Li n1,...,n r := Li n1,...,n r (e iπ/3 ) 7 / 15

55 References

56 Bibliography I [1] J. Ablinger, J. Blümlein, A. Hasselhuhn, S. Klein, C. Schneider, and F. Wißbrock. Massive 3-loop ladder diagrams for quarkonic local operator matrix elements. Nucl. Phys. B, 864(1):52 84, November URL: S , arxiv: , doi: /j.nuclphysb [2] C. Anastasiou, C. Duhr, F. Dulat, F. Herzog, and B. Mistlberger. Real-virtual contributions to the inclusive Higgs cross-section at N 3 LO. JHEP, 2013(12):88, December arxiv: , doi: /jhep12(2013) / 15

57 Bibliography II [3] C. Anastasiou, C. Duhr, F. Dulat, and B. Mistlberger. Soft triple-real radiation for Higgs production at N 3 LO. JHEP, 2013(7):1 78, jul arxiv: , doi: /jhep07(2013)003. [4] T. Binoth and G. Heinrich. An automatized algorithm to compute infrared divergent multiloop integrals. Nucl. Phys. B, 585(3): , October arxiv:hep-ph/ , doi: /s (00) [5] T. Binoth and G. Heinrich. Numerical evaluation of multiloop integrals by sector decomposition. Nucl. Phys. B, 680(1 3): , March arxiv:hep-ph/ , doi: /j.nuclphysb / 15

58 Bibliography III [6] C. Bogner and M. Lüders. Multiple polylogarithms and linearly reducible Feynman graphs. ArXiv e-prints, February arxiv: [7] F. C. S. Brown. On the periods of some Feynman integrals. ArXiv e-prints, October arxiv: [8] F. C. S. Brown. The Massless Higher-Loop Two-Point Function. Commun. Math. Phys., 287(3): , May arxiv: , doi: /s / 15

59 Bibliography IV [9] F. Chavez and C. Duhr. Three-mass triangle integrals and single-valued polylogarithms. JHEP, 11:114, November arxiv: , doi: /jhep11(2012)114. [10] R. N. Lee. Presenting LiteRed: a tool for the Loop InTEgrals REDuction. December arxiv: [11] E. Panzer. On the analytic computation of massless propagators in dimensional regularization. Nucl. Phys. B, 874(2): , September URL: S , arxiv: , doi: /j.nuclphysb / 15

60 Bibliography V [12] E. Panzer. Feynman integrals and hyperlogarithms. PhD thesis, Humboldt-Universität zu Berlin, to appear. URL: panzer/paper/phd.pdf. [13] E. Panzer. On hyperlogarithms and Feynman integrals with divergences and many scales. JHEP, 2014(3):71, March arxiv: , doi: /jhep03(2014) / 15

61 Bibliography VI [14] E. Panzer. Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals. Computer Physics Communications, 188: , March arxiv: , doi: /j.cpc [15] O. Schnetz. Graphical functions and single-valued multiple polylogarithms. ArXiv e-prints, February arxiv: [16] O. V. Tarasov. Connection between Feynman integrals having different values of the space-time dimension. Phys. Rev. D, 54(10): , November arxiv:hep-th/ , doi: /physrevd / 15

62 Bibliography VII [17] E. von Manteuffel, A. Panzer and R. M. Schabinger. A quasi-finite basis for multi-loop Feynman integrals. December arxiv: / 15