The sunset in the mirror: a regulator for inequalities in the masses

Transcription

1 The sunset in the mirror: a regulator for inequalities in the masses Pierre Vanhove 2nd French Russian Conference Random Geometry and Physics Institut Henri Poincaré, Paris, Decembre 17-21, 2016 based on [arxiv: ], [arxiv: ], [arxiv: ] Spencer Bloch, Matt Kerr Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

2 The loop amplitudes In a perturbative treatement of scattering amplitudes in QFT A = A tree + g A 1 loop + + g L A L loop + It is a major conceptual and technical question in high-energy physics to understand the nature of the basis of integrals at loop orders Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

3 The loop amplitudes Integration by part considerations indicate the existence of a finite basis of (master) integral functions B(L) at each loop order [Petukhov-Smirnov, Lee] A L loop = i B(L) coeff i Integral i + Rational dimension of the basis at L 2 loop is not known Construction of the basis is still a major open question Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

4 The loop amplitudes For instance at one-loop order in D = 4 2ɛ dimensions, the basis of integral function is known for a long time [Bern,Dixon,Kosower] to be consisting of Boxes, triangles, bubble, tadpole integrals I = d D l (l 2 m 2 1 )((l + K 1) 2 m 2 2 )((l + K 1 + K 2 ) 2 m 2 3 )((l K 4) 2 m 2 4 ) d D l I = (l 2 m1 2)((l + K 1) 2 m2 2)((l + K 1 + K 2 ) 2 m3 2) d D l I = (l 2 m1 2)((l + K 1) 2 m2 2) Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

5 Feynman Integrals: parametric representation Typically form of a Feynman graph with L loops and n propagators I Γ = Ω Γ The domain of integration = {x i 0} P n 1 The integrand is the differential form Ω Γ = Γ(n LD 2 ) U n (L+1) D 2 n (U i m2 i x ( 1) j 1 x i F) n L D j dx 1 dx j dx n 2 U and F are the Symanzik polynomials [Itzykson, Zuber] j=1 U is of degree L and F of degree L + 1 in the x i Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

6 Feynman Integrals: numerical periods I Γ = Ω Γ UV and IR divergences treated by an analytic continuation in D Since the dimension of space-time only enters in the exponent Ω Γ = Γ(n LD 2 ) U n (L+1) D 2 (U i m2 i x i F) n L D 2 n ( 1) j 1 x j dx 1 dx j dx n j=1 We can perform a Laurent expansion in ɛ = (4 D)/2 I Γ = c i ɛ i i= 2L The c i are finite and are numerical period integrals Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

7 Feynman integrals: period integrals 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) Sunset in the mirror 21/10/ / 34

8 Periods according [Kontsevich, Zagier] [Kontsevich, Zagier] define : 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) Sunset in the mirror 21/10/ / 34

9 Periods of VMHS I Γ = Ω Γ We have Ω Γ H n 1 (P n 1 \{g(x i ) = 0}) But {g(x i ) = 0} and H n 1 (P n 1 \{g(x i ) = 0}) The Feynman integral are periods of the relative cohomology after performing the appropriate blow-ups [Bloch,Esnault,Kreimer] H n 1 ( P n 1 \{g(x i ) = 0}), ) Since Ω Γ varies when one changes the kinematic variables one needs to study familly of cohomology:variation of (mixed) Hodge structure Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

10 The triangle graph integral И Паниковский от правого конца прямой повел вверх волнистый перпендикуляр. [...] Тут Паниковский соединил обе линии третьей, так что на песке появилось нечто похожее на треугольник, и закончил: [...] Балаганов с уважением посмотрел на треугольник. Tout en parlant, il traça une perpendiculaire ondulée montant depuis l extrémité droite de sa ligne. [...] Panikovski réunit alors les deux lignes par une troisième qui formait sur le sable avec les deux autres comme une sorte de triangle et acheva: [...] Balaganov regarde le triangle avec respect. (Ilf and Petrov Golden Calf) Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

11 The triangle graph integral p 1 + p 2 + p 3 = 0; p 2 i 0 I = x 0 y 0 dxdy (p 2 1 x + p2 2 y + p2 3 )(xy + x + y) = D(z) ( p p 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 integral has branch cuts arising from the square root since D(z) is analytic Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

12 The triangle graph integral I = dxdy (p 2 1 x + p2 2 y + p2 3 )(xy + x + y) = {x = 0} {y = 0} {z = 0} The denominator is the quadric E = {(p1 2 x + p2 2 y + p2 3z)(xy + xz + yz) = 0} dxdy (p 2 1 x + p2 2 y + p2 3 )(xy + x + y) H := H2 (P 2 E, \( E ) ) Because we passed to the relative cohomology Because E = {[1, 0, 0], [0, 1, 0], [0, 0, 1]} we one need to blow-up these 3 points Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

13 The triangle graph motive We can then deduce the Hodge period matrix [Bloch, Kreimer] Li 1 (z) 2iπ 0 Li 2 (z) 2iπ log z (2iπ) The construction is valid for all one-loop amplitudes in four dimensions The finite part of these integral functions are given by dilogarithms and logarithms z I, I Li 2 (z) = log t d log(1 t) I log(1 z) = 0 x Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34 0 z d log(1 t)

14 The sunset graph Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

15 The sunset integral We consider the sunset integral in two Euclidean dimensions I 2 = Ω ; := {[x : y : z] P 2 x 0, y 0, z 0} 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) K 2 xyz H2 (P 2 E K 2) The sunset family of open elliptic curve (modular only for all equal masses) E K 2 = {(m 2 1 x + m2 2 y + m2 3 z)(xz + xy + yz) K 2 xyz = 0} Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

16 The sunset geometry S = {P 1 = [1 : 0 : 0], P 2 = [0 : 1 : 0], P 3 = [0 : 0 : 1], Q 1, Q 2, Q 3 } P i Q i i = 1, 2, 3 are 2-torsion divisors The elliptic curve intersects the domain of integration E K 2 = S. We need to blow-up P 2 E K 2 For generic graph the difficulty is the structure at infinity of the intersection of the poles of the integrand of the Feynman integral and the (blown-up) domain of integration Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

17 The sunset mixed Hodge structure If P P 2 is the blow-up and ÊK 2 is the strict transform of E K 2 Hexagon h 0 union of strict transform of D = {xyz = 0} and the 3 P 1 divisors Then in P we have resolved the two problems h = h 0 (h 0 ÊK 2) ÊK 2 = ; H 2 (P ÊK 2, h) = H2 (P ÊK 2, h) The sunset integral is a period of this (mixed) Hodge structure I = Ω, When varying K 2 we have a family of elliptic curves and an associated variation of Hodge structures [Bloch, Esnault, Kreimer; Müller-Stach, Weinzeirl, Zayadeh] Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

18 The sunset motive We have the follow (short) sequence H 1 (G 2 m, Q(2)) α H 1 (E 0 K 2, Q(2)) H 2 (G 2 m, E 0 K 2 ; Q(2)) H 2 (G 2 m, Q(2)) 0. with E 0 K 2 = E K 2 {P 1, P 2, P 3, Q 1, Q 2, Q 3 } and P 2 h = G 2 m Since Image(α) = span d log(x/z ), d log(y /Z ) Introducing the regulator L 2 { X Z, Y Z } = F(P3 ) + F(Q 2 ) F(P 2 ) F(Q 3 ) x ( ) X F(x) = log x 0 Z (y) d log y with the 2-torsion relations Q i = P i for i = 1, 2, 3 Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

19 The sunset elliptic dilogarithm The Feynman integral is given by the regulator [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 elliptic curve period which is real on the line 0 < K 2 < (m 1 + m 2 + m 3 ) 2 Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

20 The sunset elliptic dilogarithm 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 Representing the ratio of the coordinates on the sunset cubic curve as functions on E C /q Z X Z (x) = θ 1(x/Q 1 )θ 1 (x/p 3 ) θ 1 (x/p 1 )θ 1 (x/q 3 ) θ 1 (x) is the Jacobi theta function θ 1 (x) = q 8 1 x 1/2 x 1/2 i n 1 Y Z (x) = θ 1(x/Q 2 )θ 1 (x/p 3 ) θ 1 (x/p 2 )θ 1 (x/q 3 ) (1 q n )(1 q n x)(1 q n /x). Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

21 The sunset elliptic dilogarithm We find I (s) iϖ r π ( ( ) P1 Ê 2 + P Ê2 2 ( P2 P 3 ) + Ê2 ( P3 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)). An equivalent expression using elliptic multiple-polylogarithms has been given by [Adams, Bogner, Weinzeirl] (see as well [Brown, Levin]) ELi m,n (x, y; q) = j,k 1 x j y k j m k n qjk = x j j m Li n(q j ) j 1 Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

22 The sunset elliptic dilogarithm The elliptic dilogarithm Ê2(x) is not invariant under q-translation and transforms according Ê 2 (qx) = Ê2(x) π2 2 Ê 2 (x/q) = Ê2(x) + π2 2 + iπ log(x) iπ log(x/q). This is because the Feynman integral we are studying is a multivalued function. Shifting the point P in C /q Z changes the expression for I by a period of the elliptic curve E K 2 Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

23 The sunset Picard-Fuchs equation Setting s = (m 1 + m 2 + m 3 ) 2 /K 2 the differential equation satisfied by the sunset integral is δ s = s d ds ( ) δ 2 I (s) s + q 1 (s)δ s + q 0 (s) = Y (s) + s 3 log(mi 2 )ν i(s) This is the Picard-Fuchs equation associated with the variation of Hodge structure i.e. derivable from I = Ω and the fact that Ω H 2 (P 2 E K 2) satisfies a 2nd order differential equation i=1 Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

24 The sunset Picard-Fuchs equation The inhomogeneous term arising from the boundary of is composed by the Yukawa coupling Y (s) = Ω d Ω ds log-term from the integration between the punctures on the elliptic curve 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 Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

25 The sunset Picard-Fuchs equation For the all equal mass case this equation takes the simple form δ s = s d ds ( δ 2 s + 2s(9s 5) (s 1)(9s 1) δ s + ) ( 3s(3s 1) 1 ) (s 1)(9s 1) s I (s) = 6 (9s 1)(s 1) Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

26 The sunset Gromov-Witten invariants The holomorphic period around s(= 1/t) = 0 π 0 = Ω = ϕ 0 m 0 s m b 1 +b 2 +b 2 =m m b 1 1 mb 2 2 mb 3 3 ( m! b 1!b 2!b 3! ) 2 and the logarithmic Mahler measure defined by π 0 = d ds R 0 R 0 = iπ log(s 1 (m1 2 x+m2 2 y+m2 3 )(x 1 +y 1 d log xd log y +1)) (2πi) 2. x = y =1 The sunset Feynman integral leads to Gromov-Witten numbers 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 m l i i e lr 0 i=1. Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

27 The sunset Gromov-Witten invariants The local Gromov-Witten numbers N l1,l 2,l 3 can be expressed in terms of the virtual integer number of degree l rational curves by 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. l (100) k>0 (k00) (110) (210) (111) (310) (220) (211) (221) N l 2 2/k / n l l (410) (320) (311) (510) (420) (411) (330) (321) (222) N l / /4 n l Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

28 The sunset Gromov-Witten invariants For the all equal masses case m 1 = m 2 = m 3 = 1, the mirror map is Q = e R 0 = q ( ) 3 (1 q n ) nδ(n) ; δ(n) := ( 1) n 1, n n 1 where ( ) 3 n = 0, 1, 1 for n 0, 1, 2 mod 3. The local Gromov-Witten numbers N l 6 = 1, 7 8, 28 27, , , , , , , , , , , , , , , , , 4913, , Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

29 Mirror Symmetry Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

30 The sunset mirror symmetry Why is this all happening? 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, The GW numbers are computed for the local mirror symmetry of a semi-stably degenerating a family of elliptically-fibered Calabi-Yau 3-folds X z0 X 0 Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

31 The sunset mirror symmetry Iritani s quantum Z-variation of Hodge structure on the even cohomology of the Batyrev mirror X of X allows to compare the asymptotic Hodge theory of this B-model to that of the mirror (elliptically fibered) A-model Calabi-Yau X 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 relation between regulator periods and local Gromov-Witten numbers The computation of the GW numbers uses the mirror map (K 2, m 1, m 2, m 3 ) Q(K 2, m 1, m 2, m 3 ) = e R 0 Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

32 The sunset Yukawa coupling The Yukawa coupling of the non-compact CY X Y ijk = Ω δi δ j δ k Ω X descends to the local Yukawa of the sunset elliptic curve Y0ij loc Y = Ω d Ω = 1 2 R 1 ds 2iπ R0 i Rj 0 The same construction applies to the 3-loop banana graph (4-fold CY) and the 4-loop banana graph (5-old CY). Polylogarithm are not enough from 4-loop At higher-loop loop the geometry is more intricate but we could expect more connection between Gromov-Witten prepotential and (massive) quantum field theory Feynman integrals. Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

33 Pavel Florensky I have the pleasure to announce the publication by Zone Sensible of the first foreign translation of Mnimosti v Geometrii by Paul Florensky Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34

34 Pavel Florensky Уже в элементарном курсе аналитической геометрии, учащийся сплошь и рядом сталкивается с мнимыми образами, но, не будучи в состоянии дать им конкретно - воззрительное содержание, принужден трактовать в высшей степени обобщающие термины, вроде например «мнимой точки», чисто - формально, тогда как на то и существует геометрия, чтобы знанию не быть оторванным от пространственного созерцания. Dans son cours élémentaire de géométrie analytique, l étudiant rencontre sporadiquement les imaginaires, mais n étant pas en état de leur donner un contenu concrètement visuel, il est forcé de traiter d une manière purement formelle de termes généralisants à l extrême, comme par exemple le «point imaginaire», alors que c est justement pour cette raison qu existe la géométrie : afin que la science ne soit pas détachée de l intuition spatiale. Pierre Vanhove (IPhT) Sunset in the mirror 21/10/ / 34