Feynman Integrals, Regulators and Elliptic Polylogarithms

Transcription

1 Feynman Integrals, Regulators and Elliptic Polylogarithms Pierre Vanhove Automorphic Forms, Lie Algebras and String Theory March 5, 2014 based on [arxiv: ] and work in progress Spencer Bloch, Matt Kerr Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

2 There are many technics to compute amplitudes in field theory On-shell (generalized) unitarity On-shell recursion methods twistor geometry, Graßmanian, Symbol,... Dual conformal invariance Infra-red behaviour (inverse soft-factors,...) amplitude relations, String theory,... These methods indicate that amplitudes have simpler structures than the diagrammatic from Feynman rules suggest The questions are: what are the basic building blocks of the amplitudes? Can the amplitudes be expressed in a basis of integrals functions? Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

3 Part I One-loop amplitudes Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

4 The one-loop amplitude In four dimensions any one-loop amplitude can be expressed on a basis of integral functions [Bern, Dixon,Kosower] = r c r Integral r + Rational terms Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

5 The one-loop amplitude The integral functions are the box, triangle, bubble, tadpole The integrals functions are given by dilogarithms and logarithms z Boxes, Triangles Li 2 (z) = 0 Bubble log(1 z) = z 0 log(1 t)d log t d log(1 t) Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

6 The one-loop amplitude This allows to characterize in a simple way one-loop amplitudes in various gauge theory Only boxes for N = 4 SYM for one-loop graph No triangle property of N = 8 SUGRA [Bern, Carrasco, Forde, Ita, Johansson; Bjerrum-Bohr, Vanhove] Only box for QED multi-photon amplitudes with n 8 photons [Badger, Bjerrum-Bohr, Vanhove] Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

7 Monodromies, periods Amplitudes are multivalued quantities in the complex energy plane with monodromies around the branch cuts for particle production They satisfy differential equation with respect to its parameters : kinematic invariants s ij, internal masses m i,... monodromies with differential equations : typical of periods integrals Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

8 A one-loop example I We consider the 3-mass triangle p 1 + p 2 + p 3 = 0 and p 2 i 0 I (p 2 1, p2 2, p2 3 ) = Which can be represented as I = x 0 y 0 R 1,3 d 4 l l 2 (l + p 1 ) 2 (l p 3 ) 2 dxdy (p1 2x + p2 2 y + p2 3 )(xy + x + y) Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

9 A one-loop example II and evaluated as D(z) I = ( p p2 4 + p4 3 (p2 1 p2 2 + p2 1 p2 3 + p2 2 p2 3 )) 1 2 z and z roots of (1 x)(p 2 3 xp2 1 ) + p2 2 x = 0 Single-valued Bloch-Wigner dilogarithm for z C\{0, 1} D(z) = Im(Li 2 (z)) + arg(1 z) log z The permutation of the 3 masses: {z, 1 z, 1 z, 1 1 z, 1 1 z, z this set is left invariant by the D(z) 1 z } The integral has branch cuts arising from the square root since D(z) is analytic Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

10 The triangle graph motive I I = x 0 y 0 dxdy (p1 2x + p2 2 y + p2 3 )(xy + x + y) The integral is defined over the domain = {[x, y, z] P 2, x, y, z 0} and the denominator is the quadric C := (p1 2 x + p2 2 y + p2 3z)(xy + xz + yz) Let L = {x = 0} {y = 0} {z = 0} and D = {x + y + z = 0} C dxdy (p 2 1 x + p2 2 y + p2 3 )(xy + x + y) H := H2 (P 2 D, L\(L C ) L, Q) We need to consider the relative cohomology because the domain is not in H 2 (P 2 D) because Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

11 The triangle graph motive II Since C = {[1, 0, 0], [0, 1, 0], [0, 0, 1]} one needs to perform a blow-up these 3 points. One can define a mixed Tate Hodge structure [Bloch, Kreimer] with weight W 0 H W 2 H W 4 H and grading gr W 0 H = Q(0), gr W 2 H = Q( 1)5, gr W 4 H = Q( 2) The Hodge matrix and unitarity Li 1 (z) 2iπ 0 Li 2 (z) 2iπ log z (2iπ) The construction is valid for all one-loop amplitudes in four dimensions Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

12 Part II Loop amplitudes Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

13 Feynman parametrization Typically form of the Feynman parametrization of a graph Γ A Feynman graph with L loops and n propagators I Γ 0 δ(1 n U n (L+1) D 2 x i ) (U i m2 i x i F) n L D 2 i=1 n dx i i=1 U and F are the Symanzik polynomials [Itzykson, Zuber] U is of degree L and F of degree L + 1 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

14 Feynman parametrization and world-line formalism Rewrite the integral as I Γ 0 δ(1 n i=1 x i) ( i m2 i x i F) n L D 2 n i=1 dx i U D 2 U = det Ω is the determinant of the period matrix of the graph The period matrix of integral of homology vectors v i on oriented loops C i Ω ij = v j C i Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

15 Feynman parametrization and world-line formalism Rewrite the integral as I Γ 0 δ(1 n i=1 x i) ( i m2 i x i F) n L D 2 n i=1 dx i U D 2 U = det Ω is the determinant of the period matrix of the graph ( ) T1 + T Ω 2(a) = 3 T 3 T 3 T 2 + T 3 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

16 Feynman parametrization and world-line formalism Rewrite the integral as I Γ 0 δ(1 n i=1 x i) ( i m2 i x i F) n L D 2 n i=1 dx i U D 2 U = det Ω is the determinant of the period matrix of the graph T 1 + T 2 T 2 0 Ω 3(b) = T 2 T 2 + T 3 + T 5 + T 6 T 3 0 T 3 T 3 + T 4 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

17 Feynman parametrization and world-line formalism Rewrite the integral as I Γ 0 δ(1 n i=1 x i) ( i m2 i x i F) n L D 2 n i=1 dx i U D 2 U = det Ω is the determinant of the period matrix of the graph T 1 + T 4 + T 5 T 5 T 4 Ω 3(c) = T 5 T 2 + T 5 + T 6 T 6 T 4 T 6 T 3 + T 4 + T 6 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

18 Feynman parametrization and world-line formalism Rewrite the integral as I Γ 0 δ(1 n i=1 x i) ( i m2 i x i F) n L D 2 n i=1 dx i U D 2 F = 1 r<s n k r k s G(x r, x s ; Ω) sum of Green s function G 1 loop (x r, x s ; L) = 1 2 x s x r (x r x s ) 2. T Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

19 Feynman parametrization and world-line formalism Rewrite the integral as I Γ 0 δ(1 n i=1 x i) ( i m2 i x i F) n L D 2 n i=1 dx i U D 2 F = 1 r<s n k r k s G(x r, x s ; Ω) sum of Green s function G 2 loop same line (x r, x s ; Ω 2 ) = 1 2 x s x r + T 2 + T 3 (x s x r ) 2 2 (T 1 T 2 + T 1 T 3 + T 2 T 3 ) G 2 loop diff line (x r, x s ; Ω 2 ) = 1 2 (x r + x s ) + T 3(x r + x s ) 2 + T 2 x 2 r + T 1 x 2 s 2(T 1 T 2 + T 1 T 3 + T 2 T 3 ) Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

20 Periods A Feynman graph with L loops and n propagators I Γ 0 δ(1 n U n (L+1) D 2 x i ) (U i m2 i x i F) n L D 2 i=1 n dx i i=1 [Kontsevich, Zagier] define periods are follows. P C is the ring of periods, is z P if Re(z) and Im(z) are of the forms R n f (x i ) g(x i ) n dx i < with f, g Z[x 1,, x n ] and is algebraically defined by polynomial inequalities and equalities. i=1 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

21 Periods [Kontsevich, Zagier] define periods are follows. P C is the ring of periods, is z P if Re(z) and Im(z) are of the forms R n f (x i ) g(x i ) n dx i < with f, g Z[x 1,, x n ] and is algebraically defined by polynomial inequalities and equalities. i=1 Problem for Feynman graphs {g(x i ) = 0} Generally the domain of integration is not closed Need to consider the relative cohomology and perform blow-ups [Bloch,Esnault,Kreimer] Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

22 Part III The banana graphs Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

23 the two-loop sunset integral We consider the two-loop sunset integral in two Euclidean dimensions given by I 2 d 2 l 1 d 2 l 2 R 4 (l m2 1 )(l2 2 + m2 2 )((l 1 + l 2 K ) 2 + m3 2) Related to D = 4 by dimension shifting formula [Tarasov; Baikov; Lee] Expression given in term of elliptic function [Laporta, Remiddi; Adams, Bogner, Weinzeirl] Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

24 the two-loop sunset integral The Feynman parametrisation is given by I 2 = x 0 y 0 dxdy (m 2 1 x + m2 2 y + m2 3 )(x + y + xy) K 2 xy = One has the remarkable representation D ω. I 2 = x I 0 ( tx) 3 K 0 (m i x) dx i=1 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

25 the two-loop sunset integral The Feynman parametrisation is given by I 2 = x 0 y 0 dxdy (m 2 1 x + m2 2 y + m2 3 )(x + y + xy) K 2 xy = The sunset integral is the integration of the 2-form ω D ω. ω = zdx dy + xdy dz + ydz dx A (x, y, z) H 2 (P 2 E K 2) The graph is based on the elliptic curve E K 2 : A (x, y, z) = 0 A (x, y, z) := (m 2 1 x + m2 2 y + m2 3 z)(xz + xy + yz) K 2 xyz. Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

26 the two-loop sunset integral The Feynman parametrisation is given by I 2 = x 0 y 0 dxdy (m 2 1 x + m2 2 y + m2 3 )(x + y + xy) K 2 xy = The sunset integral is the integration of the 2-form ω D ω. ω = zdx dy + xdy dz + ydz dx A (x, y, z) H 2 (P 2 E K 2) The domain of integration D is D := {[x : y : z] P 2 x 0, y 0, z 0} Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

27 the sunset graph mixed Hodge structure The elliptic curve intersects the domain of integration D D {A (x, y, z) = 0} = {[1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1]} We need to blow-up work in P 2 E K 2 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

28 the sunset graph mixed Hodge structure The domain of integration D H 2 (P 2 E K 2) because D Need to pass to the relative cohomology If P P 2 is the blow-up and ÊK 2 is the strict transform of E K 2 Then in P we have resolved the two problems D ÊK 2 = ; D H 2(P ÊK 2, h (h ÊK 2)) We have a variation (with respect to K 2 ) of Hodge structures H 2 K := H 2 (P 2 ÊK 2, h (h ÊK 2)) The sunset integral is a period of the mixed Hodge structure H 2 K 2 [Bloch, Esnault, Kreimer; Müller-Stach, Weinzeirl, Zayadeh] Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

29 Special values I 2 (K 2, m 2 i ) = x 0 y 0 dxdy (m 2 1 x + m2 2 y + m2 3 )(x + y + xy) K 2 xy Branch cut at the 3-particle threshold K 2 = (m 1 + m 2 + m 3 ) 2 so K 2 C\[(m 1 + m 2 + m 3 ) 2, [ Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

30 Special values At the special fibers K 2 = 0 the sunset equal a massive one-loop triangle graph I 2 (0, m 2 i ) 1 m 2 1 D(z) z z (1 x)(m3 2 m2 1 x) m2 2 x = m2 1 (x z)(x z) Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

31 Special values At the special fibers K 2 = 0 the sunset equal a massive one-loop triangle graph For equal masses m i = m then z = ζ 6 such that (ζ 6 ) 6 = 1 I 2 (0) 1 m 2 D(ζ 6 ) Im(ζ 6 ) Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

32 the elliptic curve of the sunset integral With all mass equal m i = m and t = K 2 /m 2 the integral reduces to I (t) = 1 m dxdy (x + y + 1)(x + y + xy) txy. E t : (x + y + 1)(x + y + xy) txy = 0. Special values At t = 0, t = 1 and t = + the elliptic curve factorizes. At t = 9 we have the 3-particle the threshold t C\[9, + [. Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

33 the elliptic curve of the sunset integral With all mass equal m i = m and t = K 2 /m 2 the integral reduces to I (t) = 1 m dxdy (x + y + 1)(x + y + xy) txy. E t : (x + y + 1)(x + y + xy) txy = 0. Family of elliptic curve surface with 4 singular fibers leads to a K 3 pencil E t E t f f X 1 (6) X 1 (6) {cusps} This is a universal family of X 1 (6) modular curves with a point of order 6 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

34 The sunset integral and the motive E t E t f f X 1 (6) X 1 (6) {cusps} Elliptic local system V on X 1 (6) with fibre Q 2 For s 1, s 2 h E t then the divisor s 1 s 2 on E t is of torsion of order 6 therefore H 2 t = Q(0) 3 Ĥ2 t 0 Q(0) Ĥ2 t H 1 (E t, Q( 1)) 0 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

35 the picard-fuchs equation of the sunset integral E t : (x + y + 1)(x + y + xy) txy = 0 The picard-fuchs operator is L t = d dt ( t(t 1)(t 9) d ) + (t 3) dt Acting on the integral we have L t I 2 (t) = D dβ = β 0 D We find using the Bessel integral representation 0 xi 0( tx)k 0 (x) 3 dx d dt ( t(t 1)(t 9) di (t) dt ) + (t 3)I (t) = 6 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

36 the sunset integral as an elliptic dilogarithm The Hauptmodul t is given by [Zagier; Stienstra] t = η(2τ) ( ) η(6τ) 5 η(3τ) η(τ) The period ϖ r and ϖ c, with q := exp(2iπτ(t)) are given by ϖ r η(τ)6 η(6τ) η(2τ) 3 η(3τ) 2 ; ϖ c = τ ϖ r Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

37 the sunset integral as an elliptic dilogarithm This leads to the solutions of the PF equation [Bloch, Vanhove] I 2 (t) = iπϖ r (t)(1 2τ) 6ϖ r (t) E (τ), π E (τ) = D(ζ 6) Im(ζ 6 ) 1 2i ( ) (Li 2 (q n ζ 6 )+Li 2 (q n ζ 2 6 ) Li 2 q n ζ 4 ( 6 Li 2 q n ζ 5 6) ) n 0 We have Li 2 (x) and not the Bloch-Wigner D(x) Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

38 the sunset integral as an elliptic dilogarithm This leads to the solutions of the PF equation [Bloch, Vanhove] I 2 (t) = iπϖ r (t)(1 2τ) 6ϖ r (t) E (τ), π E (τ) = 1 2 n 0 ψ 2 (n) n q n ψ 2 (n) is an odd mod 6 character { 1 for n 1 mod 6 ψ 2 (n) = 1 for n 5 mod 6 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

39 The sunset integral and the motive The integral is given by I 2 dxdy (t) = (x + y + 1)(x + y + xy) txy 0 0 The 2-form has only log-pole on E t and there is a residue 1-form I 2 (t) = periods + ϖ r ɛ 1 τ + ɛ 2, dτ (m,n) (0,0) ψ 2 (n)(ɛ 1 τ + ɛ 2 ) (m + nτ) 3 Character ψ : Lattice(E t ) S 1. Pairing ɛ 1, ɛ 2 = ɛ 2, ɛ 1 = 2iπ The amplitude integral is not the regulator map which involves a real projection r : K 2 (E t ) H 1 (E t, R) The amplitude is multivalued in t whereas the regulator is single-valued [Bloch, Vanhove] Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

40 The sunset integral and the motive The integral is given by I 2 (t) = 0 0 dxdy (x + y + 1)(x + y + xy) txy The 2-form has only log-pole on E t and there is a residue 1-form I 2 (t) = periods + ϖ r ɛ 1 τ + ɛ 2, dτ (m,n) (0,0) ψ 2 (n)(ɛ 1 τ + ɛ 2 ) (m + nτ) 3 The regulator is an Eichler integral i I 2 (t) = periods + ϖ r τ (m,n) (0,0) ψ 2 (n)(τ x) (m + nx) 3 dx Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

41 The sunset integral and the motive The integral is given by I 2 (t) = 0 0 dxdy (x + y + 1)(x + y + xy) txy The 2-form has only log-pole on E t and there is a residue 1-form I 2 (t) = periods + ϖ r ɛ 1 τ + ɛ 2, dτ (m,n) (0,0) ψ 2 (n)(ɛ 1 τ + ɛ 2 ) (m + nτ) 3 The regulator is an Eichler integral I 2 (t) = periods + ϖ 1 (m,n) (0,0) ψ 2 (n) n 2 (m + nτ) Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

42 three-loop banana graph: integral We look at the 3-loop banana graph in D = 2 dimensions The Feynman parametrisation is given by I 2 (m Q i; K 2 ) = x i 0 1 (m i=1 m2 i x i)(1 + 3 i=1 x 1 i ) K 2 3 i=1 dx i x i Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

43 three-loop banana graph: differential equation The geometry of the 3-loop banana graph is a K 3 surface (Shioda-Inose family for Γ 1 (6) +3 ) with Picard number 19 and discriminant of Picard lattice is 6 3 (m4 2 + mi 2 x i)(1 + i=1 3 i=1 x 1 i ) 3 x i t i=1 3 x i = 0 i=1 The all equal mass case with t = K 2 /m 2 satisfies the Picard-Fuchs equation [vanhove] (t 2 (t 4)(t 16) d 3 dt 3 + 6t(t2 15t + 32) d 2 dt 2 + (7t 2 68t + 64) d )J dt + t 4 2 (t) = 4! Q One miracle is that this picard-fuchs operator is the symmetric square of the picard-fuchs operator for the sunset graph [Verrill] Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

44 three-loop banana graph: solution It is immediate to use the Wronskian method to solve the differential equation [Bloch, Kerr, Vanhove] m 2 I 2 (t) = Q 40π2 log(q) ϖ 1 (τ) ( ) 48ϖ 1 (τ) 24Li 3 (τ, ζ 6 ) + 21 Li 3 (τ, ζ 2 6 ) + 8Li 3(τ, ζ 3 6 ) + 7Li 3(τ, 1) with Li 3 (τ, z) [Zagier; Beilinson, Levin] Li 3 (τ, z) := Li 3 (z) + (Li 3 (q n z) + Li 3 (q n z 1) ) n 1 ( 1 12 log(z) log(q) log(z)2 1 ) 720 (log(q))3. Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

45 three-loop banana graph: solution Which can be written using as an Eisenstein series m 2 I 2 Q (t(τ)) = ϖ 1(τ) ( 4(log q) where ψ 3 (n) is an even mod 6 character n 0 ψ 3 (n) n 3 ) 1 + q n 1 q n Arising from the regulator for Sym 2 H 1 (E (t)) with dz = (ɛ 1 τ + ɛ 2 ) 2 ϖ 1 dz 2, m,n dτdz 2 ψ 3 (n) (m + nτ) 4 Again this is given by the Eichler integral i (x τ) 2 ψ 3 (n) period + ϖ 1 (m + nx) 4 dx τ m,n Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

46 Part IV Higher-loop banana amplitudes Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

47 Higher-order bananas I The integral we have been discussing are given by In 2 = x i 0 1 (1 + n i=1 x i)(1 + n i=1 x 1 i ) t n i=1 dx i x i They are given by the following Bessel integral representation I 2 n(t) = 2 n 1 0 x I 0 ( tx)k 0 (x) n dx The Bessel function K 0 (x) satisfies a differential equation that implies that ( n+1 ( p r (x) x d ) ) r K 0 (x) n = 0 dx r=0 This implies that the banana integral satisfies a differential equation Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

48 Higher-order bananas II ( n+1 r=0 ( p r (t) t d ) ) r I 2 dt n(t) = 0 Because p r (x) are polynomials in x 2 without constant terms [Borwein,Salvy] then one can factorizes a (td/dt) 2 operator and the differential operator for the n 1-loop banana graph is of order the number of loops ( n 1 r=0 ( q r (t) t d ) ) r I 2 dt n(t) = S n + S n log(t) Since the banana integral is regular at t = 0 then S n = 0 and the differential equation is Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

49 Higher-order bananas III ( n 1 r=0 ( q r (t) t d ) ) r I 2 dt n(t) = S n with [vanhove] q n 1 (t) = t n 2 n 2 +η(n) (t (n 2i) 2 ) q n 2 (t) = n 1 2 q 0 (t) = t n. i=0 dq n 1 (t) dt with η(n) = 0 if n 1 mod 2 and 1 if n 0 mod 2. The inhomogeneous term S n = n! Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

50 Higher-order bananas one-loop banana bubble (t 2) f (t) + t(t 4) f (1) (t) = 2! Solution are logarithms I 2 (m 1, m 2, K 2 ) = log(z+ ) log(z ) z ± and are roots and discriminant of the equation (m 2 1 x + m2 2 )(1 + x) K 2 x = m 2 1 (x z+ )(x z ) = 0 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

51 Higher-order bananas four-loop banana bubble (t 5) f (t) + (3t 5)(5t 57)f (1) (t) ( ) + 25 t t t 450 f (2) (t) ( ) + 10 t t t t f (3) (t) +t 2 (t 25)(t 1)(t 9)f (4) (t) = 5! Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

52 Higher-order bananas five-loop banana bubble ( ) (t 6) f (t) + 31 t t f (1) (t) ( ) + 90 t t t 6912 f (2) (t) ( ) + 65 t t t t f (3) (t) ( + 15 t t t t 2) f (4) (t) +t 3 (t 36)(t 4)(t 16)f (5) (t) = 6! Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

53 outlook Connection between the Feynman parametrization and the world-line formalism : Possible treatment of field theory graph a la string theory Amplitudes are multivalued functions given by motivic periods. The Hodge matrix corresponds to unitarity (discontinuity, cut). Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

54 outlook At special values t = 1 the integrals are pure periods related to values of L-functions in the critical band : Deligne s conjecture [Zagier] I Q (1) = 4π 15 L(K 3, 2) The new form for the L(K 3, s) [Peeters, Top, van der Vlugt] f (τ) = η(τ)η(3τ)η(5τ)η(15τ) ( q m2 +mn+4n 2 ) S 3 (15, ) 15 m,n Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36

55 outlook At special values t = 1 the integrals are pure periods related to values of L-functions in the critical band : Deligne s conjecture [Zagier] I Q (1) = 4π 15 L(K 3, 2) The new form for the L(K 3, s) [Peeters, Top, van der Vlugt] f (τ) = η(τ)η(3τ)η(5τ)η(15τ) ( q m2 +mn+4n 2 ) S 3 (15, ) 15 m,n But there is an obstruction to modularity at higher-loop [Brown] At higher-loop order the integral are not elliptic polylogarithm as seen on the Picard-Fuchs equation Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 5/03/ / 36