Elliptic dilogarithms and two-loop Feynman integrals

Transcription

1 Elliptic dilogarithms and two-loop Feynman integrals Pierre Vanhove 12th Workshop on Non-Perturbative Quantum Chromodynamics IAP, Paris June, based on work to appear with Spencer Bloch Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

2 Explicit amplitude computations display rather unexpectedly simple structures allowing to compute many more processes than expected On-shell recursion methods twistor geometry, Graßmanian, Symbol,... Dual conformal invariance amplitude relations... All these simplifications hints on simpler structures than the diagrammatic from Feynman rules suggest, and lead to the use of a basis of integrals for expressing the amplitudes What are the basic functions entering the expressions for the amplitudes? Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

3 Monodromies, periods Amplitudes are multivalued quantities : they satisfy monodromy properties when going around the branch cuts for particle production As well the amplitude satisfy differential equation with respect to its parameters : kinematic invariants s ij, internal masses m i,... monodromies and differential equation are typical of periods Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

4 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 inequalities and equalities. Typically form of the Feynman parametrization of a graph i=1 A Feynman graph with L loops and n edges I graph 0 δ(1 n i=1 x i ) Un (L+1) D 2 F n L D 2 n dx i i=1 U is homogenous of degree L in the x i and F is homogenous of degree L + 1 in x i Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

5 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 inequalities and equalities. Typically form of the Feynman parametrization of a graph i=1 Problem for Feynman graphs {g(x i ) = 0} : needs to blow-up the intersection points As well generaly the domain of integration is not closed This leads to the notion of generalized periods Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

6 Part I Tree-level amplitudes Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

7 Tree-level amplitudes in QCD Tree-level amplitudes in gauge theory are decomposed into color (n 1)!/2 color-ordered sub-amplitudes A(1,..., n) σ S n 1 /Z 2 Tr (λ a1 λ aσ(2) λ aσ(n)) A(1, σ(2,..., n)) λ a are generators in the fundamental representation A(1, σ(2,..., n)) are the color ordered amplitudes Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

8 Tree-level amplitudes in QCD The color ordered amplitudes are not all independent For instance they satisfy the photon decoupling identity A n (1, σ(2,..., n)) = 0 σ S n 1 There was the question of the independent amplitudes and their relations Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

9 Tree-level amplitudes in QCD X X X X X = α 0 X X X X X X X X X 0 1 We evaluate them by considering the α 0 limit of open string amplitudes on the disc. PSL(2, R) invariance z 1 = 0, z n 1 = 1 and z n = +. (3 marked points) A(σ(1,..., n)) = f (x i x j ) x σ(1) < <x σ(n) 1 i<j n (x i x j ) 2α k i k j d n 3 x The function f (x j ) does not have branch cut but has poles. Depends on the polarisation of the external states. Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

10 Monodromies from contour deformation Contour deformation gives monodromy relations between the ordered amplitudes The monodromy relations lead to a set of linear system of equations relating different ordering of the external states σ S n 2 S[σ(2,..., n 1) β(2,..., n 1)] k1 A n (n, σ(2,..., n 1), 1) = 0 [Bern, Carrasco, Johansson; Bjerrum-bohr, Damgaard, Vanhove; Stieberger; Bjerrum-bohr, Damgaard, Søndergaard, Vanhove] Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

11 Momentum kernel Contour deformation gives monodromy relations between the ordered amplitudes n-2 n i-1 i i+1 This leads to an object named momentum kernel S S[i 1,..., i k j 1,..., j k ] p := k k (p k it + θ(t, q) k it k iq ) t=1 q>t θ(t, q) = 1 if (i t i q )(j t j q ) < 0 and 0 otherwise Exists as well in string theory with α 0 as sin α (...)/α [Bjerrum-bohr, Damgaard, Vanhove; Bjerrum-Bohr, Damgaard, Feng, Sondergaard; Bjerrum-bohr, Damgaard, Søndergaard, Vanhove] Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

12 Tree-level Gravity and Gauge amplitudes From closed string heterotic string setup we have points on the sphere holomorphic factorization z α k i k j x α 2 k i k j y α 2 k i k j gives product of ordered disc integrations with relative ordered contour of integrations C x and C y M(1,..., n) = d n 3 x d n 3 y C x C y 1 i<j n (x i x j ) α ki k j 2 (y i y j ) α k i k j 2 f (x ij )g(y ij ) [Kawai,Lewellen, Tye; Tye, Zhang; Bjerrum-Bohr, Damgaard, Søndergaard, Vanhove] Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

13 Tree-level Gravity and Gauge amplitudes From closed string heterotic string setup we have points on the sphere holomorphic factorization z α k i k j x α 2 k i k j y α 2 k i k j gives product of ordered disc integrations with relative ordered contour of integrations C x and C y The gauge theory and gravity amplitudes take a similar forms A YM n = A vector S A scalar M Grav n = A vector S A vector These relations are generic and independent of any precise parametrisation of the tree-level amplitudes [Kawai,Lewellen, Tye; Tye, Zhang; Bjerrum-Bohr, Damgaard, Søndergaard, Vanhove] Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

14 Part II Two-loop amplitudes Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

15 two-loop integrals We consider the two-loop sunset integral in two dimensions given by I 2 d 2 l 1 d 2 l 2 R 4 (l 2 1 m2 1 )(l2 2 m2 2 )((l 1 + l 2 K 2 ) m 2 3 ) Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

16 two-loop integrals 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) K2 xy. We are again the setup of [Kontsevich, Zagier] The sunset integral is D ω with the 2-form ω = zdx dy + xdy dz + ydz dx A (x, y, z) H 2 (P 2 E) The graph is based on the elliptic curve E : A (x, y, z) = 0 A (x, y, z) := (m 2 1x + m 2 2y + m 2 3z)(xz + xy + yz) K 2 xyz. Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

17 two-loop integrals 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) K2 xy. We are again the setup of [Kontsevich, Zagier] The sunset integral is D ω with the 2-form ω = zdx dy + xdy dz + ydz dx A (x, y, z) H 2 (P 2 E) The domain of integration is D := {[x : y : z] P 2 x 0, y 0, z 0} Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

18 the sunset graph motive I 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 Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

19 the sunset graph motive II The domain of integration D H 2 (P 2 E) because D Need to pass to the relative cohomology If P P 2 is the blow-up and Ê is the strict transform of E (here Ê E) Hexagon h union of strict transform of D = {xyz = 0} and the 3 divisors. Then in P we have resolved the two problems D Ê = ; D H 2 (P Ê, h (h Ê)) The sunset integral is a period of H 2 (P Ê, h (h Ê)) [Bloch, Esnault, Kreimer; Müller-Stach, Weinzeirl, Zayadeh] Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

20 the elliptic curve of the sunset integral With all mass equal m i = m the integral is reduced to with t = K 2 /m 2 I (t) = 0 0 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 = K 2 /m 2 C\[9, + [. Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

21 the elliptic curve of the sunset integral With all mass equal m i = m the integral is reduced to with t = K 2 /m 2 I (t) = 0 0 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 = K 2 /m 2 C\[9, + [. Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

22 the elliptic curve of the sunset integral With all mass equal m i = m the integral is reduced to with t = K 2 /m 2 I (t) = 0 0 dxdy (x + y + 1)(x + y + xy) txy. E t : (x + y + 1)(x + y + xy) txy = 0. Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

23 the elliptic curve of the sunset integral With all mass equal m i = m the integral is reduced to with t = K 2 /m 2 I (t) = 0 0 dxdy (x + y + 1)(x + y + xy) txy. E t : (x + y + 1)(x + y + xy) txy = 0. We have a family of elliptic curve defining a K 3 surface with 4 singular fibers in [Beauville] classification. This is a universal family of X 1 (6) modular curves with a point of order 6 Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

24 the picard-fuchs equation of the sunset integral E t : (x + y + 1)(x + y + xy) txy = 0 Since H 1 (E t ) is generated by dx/y and d(dx/y)/dt, then exist p 0 (t), p 1 (t), p 2 (t) Z[t] such that p 0 (t) d2 dt 2 ( ) dx + p 1 (t) d y dt ( ) dx + p 2 (t) y ( ) dx = dβ y The homogeneous picard-fuchs operator is L t = d ( t(t 1)(t 9) d ) + (t 3) dt dt Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

25 the picard-fuchs equation of the sunset integral E t : (x + y + 1)(x + y + xy) txy = 0 The homogeneous picard-fuchs operator is L t = d ( t(t 1)(t 9) d ) + (t 3) dt dt Acting on the integral we have L t I 2 (t) = D dβ = β 0 D We find that (recovering the result of [Laporta, Remiddi]) ( d t(t 1)(t 9) di ) (t) + (t 3)I (t) = 6 dt dt Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18

26 the sunset integral as an elliptic dilogarithm Evaluating the integrals leads to (for details see my talk at String Math 2013) I 2 (t) α 1 ϖ 1 (t) + α 2 ϖ 2 (t) + ϖ 2 (t) E 1 (q t ), where ϖ 1 (t), ϖ 2 (t) are the complex and real periods and E 1 (q) is defined by E 1 (q) I 2 (0) 1 (Li 2 (q n ζ 6 ) + Li 2 (q n ζ 2 2i 6) Li 2 (q n ζ 4 6) Li 2 (q n ζ 5 6 )) n 0 (ζ 6 ) 6 = 1 and q = exp(2iπτ(t)) with τ(t) = ϖ 1 (t)/ϖ 2 (t) Notice that we have Li 2 and not the Bloch-Wigner D function The function is multivalued and convergent Pierre Vanhove (IPhT & IHES) Elliptic dilogarithms 10/06/ / 18