Feynman integrals as Periods

Transcription

1 Feynman integrals as Periods Pierre Vanhove Amplitudes 2017, Higgs Center, Edinburgh, UK based on [arxiv: ], [arxiv: ], [arxiv: ] Spencer Bloch, Matt Kerr Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

2 Scattering amplitudes are essential tools to understand a variety of physical phenomena from gauge theory to classical and quantum gravity A convenient approach is to use modern unitarity methods for expanding the amplitude on a basis of integral functions A L loop = i B(L) coeff i Integral i + Rational What are the intrinsic properties of amplitudes of QFT in flat space? How much can we understand about the amplitudes without having to compute them? Generic constraints on the integral coefficients? [cf Tourkine s talk] What are the elements of the basis of integral functions? These questions are central to many talks at this meeting Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

3 -It s bigger on the inside! -Yes, the TARDIS is dimensionally transcendental In this talk we will present some approach to Feynman integral that makes a essential use of the algebraic geometric setup defined by the graphs This approach is an alternative to the IBP method described in [Lee] s and [Tancredi] s talk Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

4 Feynman Integrals: parametric representation Any Feynman integrals with L loops and n propagators I Γ = has the parametric representation L i=1 d D l i n i=1 d ν i i I Γ = Γ(ν LD 2 ) x i 0 U ν (L+1) D 2 (U i m2 i x i V) ν L D 2 δ(x n = 1) n i=1 dx i x ν i 1 i U and V are the Symanzik polynomials U is of degree L and V of degree L + 1 in the x i Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

5 What are the Symanzik polynomials? I Γ = Γ(ν LD 2 ) x i 0 U ν (L+1) D 2 (U i m2 i x i V) ν L D 2 δ(x n = 1) n i=1 dx i x ν i 1 i U = det Ω determinant of the period matrix of the graph Ω ij = C i v j ( x1 ) + x x1 + x Ω 2(a) =( 3 x 2 x x 3 x 2 + x ); Ω 3 3(b) = x 2 x 2 + x 3 + x 5 + x 6 x 3 0 x 3 x 3 + x 4 ( x1 ) + x 4 + x 5 x 5 x 4 Ω 3(c) = x 5 x 2 + x 5 + x 6 x 6 x 4 x 6 x 3 + x 4 + x 6 Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

6 What are the Symanzik polynomials? I Γ 0 δ(1 x n ) ( i m2 i x i V/U) n L D 2 n i=1 x ν i 1 i dx i U D 2 V/U = 1 r<s n k r k s G(x r /T r, x s /T s ; Ω) sum of Green s function G 1 loop (α r, α s ; L) = 1 2 α s α r (α r α s ) 2. T Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

7 Feynman integral and periods I Γ = Γ(ν LD 2 ) n Ω Γ ; Ω Γ := Uν (L+1) D 2 Φ Γ (x i ) ν L D 2 n 1 i=1 dx i x ν i 1 i The integrand is an algebraic differential form Ω Γ on the complement of the graph hypersurface Ω Γ H n 1 (P n 1 \X Γ ) X Γ := {Φ Γ (x) = 0, x P n 1 } The domain of integration is the simplex n with boundary n D n := {x 1 x n = 0} But n X Γ therefore n H n 1 (P n 1 \X Γ ) This is resolved by looking at the relative cohomology H (P n 1 \X Γ ; D n \D n X Γ ) Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

8 Feynman integral and periods D n and X Γ are separated by performing a series of iterated blowups of the complement of the graph hypersurface [Bloch, Esnault, Kreimer] The Feynman integral are periods of the relative cohomology after performing the appropriate blow-ups [Bloch, Esnault, Kreimer] and [Brown s talk] H n 1 ( P n 1 \ X F ; Dn \ Dn X Γ ) Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

9 Feynman integral and periods One can apply the construction and the discussion of the ɛ = (D D c )/2 expansion of the Feynman integral [Belkale, Brosnan; Bogner, Weinzierl] I Γ = c i ɛ i i n The periods are master integrals. The minimal number of master integral for this topology determined by the middle cohomology of the motive H ( P n 1 \ X F ; Dn \ Dn X Γ ) Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

10 Differential equation M(s ij, m i ) := H ( P n 1 \ X F ; Dn \ Dn X Γ ) Since Ω Γ varies when one changes the kinematic variables s ij one needs to study a variation of (mixed) Hodge structure Consequently the Feynman integral will satisfy a differential equation L Γ I Γ = S Γ The Picard-Fuchs operator will arise from the study of the variation of the differential in the cohomology when kinematic variables change The inhomogeneous terms arises from the extension Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

11 The sunset graph Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

12 The sunset integral We consider the sunset integral in two Euclidean dimensions I 2 = Ω ; 3 := {[x : y : z] P 2 x 0, y 0, z 0} 3 The sunset integral is the integration of the 2-form Ω = zdx dy + xdy dz + ydz dx (m 2 1 x + m2 2 y + m2 3 z)(xz + xy + yz) p2 xyz H2 (P 2 \X ) The sunset family of open elliptic curve (modular only for all equal masses) X = {(m 2 1 x + m2 2 y + m2 3 z)(xz + xy + yz) p2 xyz = 0} Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

13 The differential operator: from the period Consider the integral which is the same integral as the sunset one with a different cycle of integration π 0 (p 2 ) := Ω x = y =1 This is the cut integral of Tancredi talk. Changing the domain of integration realises the cut The other period is π 1 (s) = log(s) π 0 (s) + ϖ 1 (s) with ϖ 1 (s) analytic. They are the two periods of the elliptic curve By definition they are annihilated by the Picard-Fuchs operator L PF π 0 = L PF π 1 = 0 Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

14 The differential operator: from the period The integral is the analytic period of the elliptic curve around p 2 π 0 (p 2 ) := 1 ( ) n! 2 3 (p 2 ) n+1 m 2n i n 1!n 2!n 3! i n 0 i=1 n 1 +n 2 +n 3 =n From the series expansion we can deduce the Picard-Fuch differential operator L π 0 (p 2 ) = 0 With this method one easily derives the PF at all loop order for the all equal mass banana and show the order(pf)=loop order Gives for the 3-loop banana 2 unequal masses PF of order 4, and order 6 for 3 different masses [Vanhove, to appear] The PF has maximal unipotent monodromy we can reconstruct all the periods using Frobenius method Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

15 The differential equation By general consideration we know that since the integrand is a top form we have L Γ I Γ = dβ Γ = β Γ = S Γ 0 n n Writing the differential equation as δ s := s d ds s = 1/p2 ( ) ( ) δ 2 1 s + q 1 (s)δ s + q 0 (s) s I (s) = Y + 3 log(mi 2 )c i(s) i=1 The right-hand side are the tadpole graphs [Tancredi s Talk] Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

16 The differential equation The works from [del Angel,Müller-Stach] and [Doran, Kerr] on the rank of the D-module system of differential equations imply that Y is the Yukawa coupling for the residue form ω = Res X (Ω ) = dx Ω 1 (X ) P(x) Y (s) := X ω s dds ω = 2s2 4 i=1 µ i 4s i m2 i i=1 (µ2 i s 1) The Yukawa coupling is the Wronskian of the Picard-Fuchs operator and only depends on the form of the Picard-Fuchs operator ( ) π0 (s) π Y = det 1 (s) s d ds π 0(s) s d ds π 1(s) So far all we got can be deduced from the graph polynomial, and the associated Picard-Fuchs operator. Applies to all banana graphs Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

17 The differential equation The logarithmic terms arise more specific to the open elliptic curve geometry v 3 q 3 v 4 v2 p 1 q 2 p p 2 3 v 1 E s v 5 q 1 v 6 The mass dependent log-terms come from derivative of partial elliptic integrals on globally well-defined algebraic 0-cycles arising from the punctures on the elliptic curve [Bloch, Kerr,Vanhove] c i (s) = d ds qi+1 q i ω ; c 1 (s) + c 2 (s) + c 3 (s) = 0 They are rational function by construction. Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

18 Solving the differential equation 4 ϕ 0 q 3 3 ϕ P 0 31 ϕ 5 0 P q 23 1 ϕ ϕ 2 0 ϕ 1 ϕ P 12 q 2 E s The integral divided by a period of the elliptic curve is a (regulator) function defined on punctured torus [Bloch, Kerr,Vanhove] I iϖ r π ( { X L 2 Z, Y } { Z + L 2 Z X, Y } { X + L 2 X Y, Z }) Y mod period ϖ r is the real period on 0 < p 2 < (m 1 + m 2 + m 3 ) 2 Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

19 Solving the differential equation Representing the ratio of the coordinates on the sunset cubic curve as functions on E C /q Z X Z (x) = θ 1(x/x(Q 1 ))θ 1 (x/x(p 3 )) θ 1 (x/x(p 1 ))θ 1 (x/x(q 3 )) θ 1 (x) is the Jacobi theta function θ 1 (x) = q 1 8 x 1/2 x 1/2 i Y Z (x) = θ 1(x/x(Q 2 ))θ 1 (x/x(p 3 )) θ 1 (x/x(p 2 ))θ 1 (x/x(q 3 )) (1 q n )(1 q n x)(1 q n /x). n 1 P 1 = [1, 0, 0]; Q 1 = [0, m 2 3, m2 2 ]; x(p 1)x(Q 1 ) = 1 P 2 = [0, 1, 0]; Q 1 = [ m 2 3, 0, m2 1 ]; x(p 1)x(Q 1 ) = 1 P 3 = [0, 0, 1]; Q 1 = [ m 2 2, m2 1, 0]; x(p 1)x(Q 1 ) = 1 Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

20 Solving the differential equation Since and L 2 { X Z, Y Z } x ( ) X = log x 0 Z (y) d log y log(θ 1 (x)) d log x = (Li 1 (q n x) + Li 1 (q n /x) + cste) d log(x) n 1 = n 1(Li 2 (q n x) Li 2 (q n /x)) + cste log(x) Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

21 Solving the differential equation I (s) iϖ r π ( ( ) x(p1 ) Ê 2 + x(p 2 ) Ê2 ( ) x(p2 ) + x(p 3 ) Ê2 ( )) x(p3 ) x(p 1 ) mod periods where Ê 2 (x) = n 0 (Li 2 (q n x) Li 2 ( q n x)) n 1 (Li 2 (q n /x) Li 2 ( q n /x)). For the all equal mass case x(p k ) = e 2iπk/6 are sixth root of unity An equivalent expression using elliptic multiple-polylogarithms has been given by [Adams, Bogner, Weinzeirl] (see as well [Brown, Levin; Adams talk]) The three-loop banana all equal mass is elliptic 3-logarithms because the geometry is the symmetric square of the sunset curve [Bloch, Kerr, Vanhove] Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

22 The sunset integral as an Eichler For the all equal mass case we have family of elliptic curve with 4 singular fiber which by [Beauville] classification is a modular family X H/Γ 1 (6) We can pullback the residue 1-form from the elliptic curve to H giving [Bloch,Vanhove] i I 2 (t) = periods + ϖ r dx(τ x) = periods ϖ r 12 (2πi) 2 τ ( 1)n n 2 n Z (m,n) (0,0) sin πn 3 ( 1) n sin 2πn 3 1 q n sin πn 3 + sin 2πn 3 (m + nτ) 3 Similar result at 3 loops the elliptic 3-log is the Eichler integral of a weight 4 Eisenstein series [Bloch, Kerr, Vanhove] Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

23 Mirror Symmetry Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

24 The sunset Gromov-Witten invariants Around 1/s = p 2 = the sunset Feynman integral has the expansion I (s) = π 0 3R3 0 + l 1 +l 2 +l 3 =l>0 (l 1,l 2,l 3 ) N 3 \(0,0,0) l(1 lr 0 )N l1,l 2,l 3 3 N l1,l 2,l 3 are local genus zero Gromov-Witten numbers N l1,l 2,l 3 = d l 1,l 2,l 3 1 d 3 n l 1 d, l 2 d, l 3 d. the Kaehler parameter is the logarithmic Mahler measure d log xd log y log Q = iπ log(φ (x, y)/(xy)) (2πi) 2. x = y =1 i=1 m l i i Q l. Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

25 The sunset Gromov-Witten invariants Around 1/s = p 2 = the sunset Feynman integral has the expansion I (s) = π 0 3R3 0 + l 1 +l 2 +l 3 =l>0 (l 1,l 2,l 3 ) N 3 \(0,0,0) l(1 lr 0 )N l1,l 2,l 3 3 N l1,l 2,l 3 are local genus zero Gromov-Witten numbers N l1,l 2,l 3 = d l 1,l 2,l 3 1 d 3 n l 1 d, l 2 d, l 3 d. For the all equal masses case m 1 = m 2 = m 3 = 1, the mirror map is Q = q ( ) 3 (1 q n ) nδ(n) ; δ(n) := ( 1) n 1 n n 1 i=1 m l i i Q l. Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

26 The sunset mirror symmetry The sunset elliptic curve is embedded into a singular compactification X 0 of the local Hori-Vafa 3-fold Y := {1 s(m 2 1 x +m2 2 y +m2 3 )(1+x 1 +y 1 )+uv = 0} (C ) 2 C 2, We have an isomorphism of A- and B-model Z-variation of Hodge structure H 3 (X z0 ) H even (X Q 0 ), and taking (the invariant part of) limiting mixed Hodge structure on both sides yields the number of independent periods given by the surviving periods One can map the computation to the one of [Huang, Klemm, Poretschkin] who studied elliptically fibered CY 3-fold with a based given by a toric del Pezzo surface and get the expansion at infinity Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26

27 Outlook We have described a systematic way of deriving the differential operators for the general multiloop sunset graphs The inhomogeneous terms arise from total derivative going back and forth because parametric and Feynman representation we hope this can help with the integration by part method They are many interesting useful connection to mathematics i) The connection to the work on mirror symmetry gives a natural way of understanding the basis of master integral ii) [Broadhurst] shows that banana graph at p 2 = 1 gives L-functions values in the critical band. They are realisation of [Deligne] s conjectures on periods [Bloch, Kerr, Vanhove] iii) The graph polynomials and the differential equation are sometime known to mathematician cf. [v. Straten et al.] classification of Fano varieties iv) Interesting connection to Gamma class conjecture of [Golyshev, Zagier] Pierre Vanhove (IPhT) Feynman integrals, Periods and Motives 12/07/ / 26