The sunset in the mirror: a regulator for inequalities in the masses - - PowerPoint PPT Presentation

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:1309.5865], [arXiv:1406.2664], [arXiv:1601.08181] Spencer Bloch, Matt Kerr

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 1 / 34

The loop amplitudes

In a perturbative treatement of scattering amplitudes in QFT A = Atree + g A1−loop + · · · + gL AL−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 21/10/2016 2 / 34

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]

AL−loop =

• i∈B(L)

coeffi Integrali + 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 21/10/2016 3 / 34

The loop amplitudes

For instance at one-loop order in D = 4 − 2ǫ 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 =

• dDℓ

(ℓ2 − m2

1)((ℓ + K1)2 − m2 2)((ℓ + K1 + K2)2 − m2 3)((ℓ − K4)2 − m2 4)

I⊲ =

• dDℓ

(ℓ2 − m2

1)((ℓ + K1)2 − m2 2)((ℓ + K1 + K2)2 − m2 3)

I◦ =

• dDℓ

(ℓ2 − m2

1)((ℓ + K1)2 − m2 2)

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 4 / 34

Feynman Integrals: parametric representation

◮ Typically form of a Feynman graph with L loops and n propagators

IΓ =

• ∆

ΩΓ

◮ The domain of integration ∆ = {xi 0} ⊂ Pn−1 ◮ The integrand is the differential form

ΩΓ = Γ(n−LD 2 ) Un−(L+1) D

2

(U

i m2 i xi − F)n−L D

2

n

• j=1

(−1)j−1xjdx1∧· · · dxj∧· · ·∧dxn

◮ U and F are the Symanzik polynomials [Itzykson, Zuber] ◮ U is of degree L and F of degree L + 1 in the xi

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 5 / 34

Feynman Integrals: numerical periods

IΓ =

• ∆

ΩΓ

◮ UV and IR divergences treated by an analytic continuation in D ◮ Since the dimension of space-time only enters in the exponent

ΩΓ = Γ(n−LD 2 ) Un−(L+1) D

2

(U

i m2 i xi − F)n−L D

2

n

• j=1

(−1)j−1xjdx1∧· · · dxj∧· · ·∧dxn

◮ We can perform a Laurent expansion in ǫ = (4 − D)/2

IΓ =

∞

• i=−2L

ci ǫi

◮ The ci are finite and are numerical period integrals

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 6 / 34

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 sij, internal masses mi, . . .

◮ monodromies with differential equations : typical of periods

integrals

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 7 / 34

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

• ∆∈Rn

f(xi) g(xi)

n

• i=1

dxi < ∞ with f, g ∈ Z[x1, · · · , xn] and ∆ is algebraically defined by polynomial inequalities and equalities.

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 8 / 34

Periods of VMHS

IΓ =

• ∆

ΩΓ

◮ We have ΩΓ ∈ Hn−1(Pn−1\{g(xi) = 0}) ◮ But ∂∆ ∩ {g(xi) = 0} ∅ and ∂∆ ∅

∆ Hn−1(Pn−1\{g(xi) = 0})

◮ The Feynman integral are periods of the relative cohomology after

performing the appropriate blow-ups [Bloch,Esnault,Kreimer] Hn−1(

• Pn−1\{g(xi) = 0}),

∆)

◮ Since ΩΓ 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 21/10/2016 9 / 34

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 21/10/2016 10 / 34

The triangle graph integral

p1 + p2 + p3 = 0; p2

i 0

I⊲ =

• x0

y0

dxdy (p2

1x + p2 2y + p2 3)(xy + x + y) =

D(z)

• p4

1 + p4 2 + p4 3 − (p2 1p2 2 + p2 1p2 3 + p2 2p2 3)

1

2

z and ¯ z roots of (1 − x)(p2

3 − xp2 1) + p2 2x = 0 ◮ Single-valued Bloch-Wigner dilogarithm for z ∈ C\{0, 1}

D(z) = ℑm(Li2(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 21/10/2016 11 / 34

The triangle graph integral

I⊲ =

• ∆

dxdy (p2

1x + p2 2y + p2 3)(xy + x + y)

∆ = {x = 0}∪{y = 0}∪{z = 0} The denominator is the quadric E⊲ = {(p2

1x + p2 2y + p2 3z)(xy + xz + yz) = 0}

dxdy (p2

1x + p2 2y + p2 3)(xy + x + y) ∈ H := H2(P2 − 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 21/10/2016 12 / 34

The triangle graph motive

We can then deduce the Hodge period matrix [Bloch, Kreimer]   1 −Li1(z) 2iπ −Li2(z) 2iπ log z (2iπ)2                    

◮ 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 I, I⊲ ∼ Li2 (z) = − z log t d log(1 − t) I◦ ∼ log(1 − z) = z d log(1 − t) x

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 13 / 34

The sunset graph

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 14 / 34

The sunset integral

We consider the sunset integral in two Euclidean dimensions I2

⊖ =

• ∆

Ω⊖; ∆ := {[x : y : z] ∈ P2|x 0, y 0, z 0}

◮ The sunset integral is the integration of the 2-form

Ω⊖ = zdx ∧ dy + xdy ∧ dz + ydz ∧ dx (m2

1x + m2 2y + m2 3z)(xz + xy + yz) − K 2xyz ∈ H2(P2 − EK 2) ◮ The sunset family of open elliptic curve (modular only for all equal

masses) EK 2 = {(m2

1x + m2 2y + m2 3z)(xz + xy + yz) − K 2xyz = 0}

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 15 / 34

The sunset geometry

S = {P1 = [1 : 0 : 0], P2 = [0 : 1 : 0], P3 = [0 : 0 : 1], Q1, Q2, Q3}

◮ Pi − Qi i = 1, 2, 3 are 2-torsion divisors ◮ The elliptic curve intersects the domain of integration ∆ ∩ EK 2 = S.

We need to blow-up P2 − EK 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 21/10/2016 16 / 34

The sunset mixed Hodge structure

◮ If P → P2 is the blow-up and ˆ

EK 2 is the strict transform of EK 2

◮ Hexagon h0 union of strict transform of ∂D = {xyz = 0} and the 3

P1 divisors

◮ Then in P we have resolved the two problems h = h0 − (h0 ∩ ˆ

EK 2) ˜ ∆ ∩ ˆ EK 2 = ∅; ˜ ∆ ∈ H2(P − ˆ EK 2, h) = H2(P − ˆ EK 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 21/10/2016 17 / 34

The sunset motive

We have the follow (short) sequence H1(G2

m, Q(2)) α

− → H1(E0

K 2, Q(2)) → H2(G2 m, E0 K 2; Q(2))

→ H2(G2

m, Q(2)) → 0.

with E0

K 2 = EK 2 − {P1, P2, P3, Q1, Q2, Q3} and P2 − h = G2 m ◮ Since Image(α) = spand log(X/Z), d log(Y/Z) ◮ Introducing the regulator

L2 X

Z , Y Z

• = F(P3) + F(Q2) − F(P2) − F(Q3)

F(x) = − x

x0

log X Z (y)

• d log y

◮ with the 2-torsion relations Qi = −Pi for i = 1, 2, 3

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 18 / 34

The sunset elliptic dilogarithm

The Feynman integral is given by the regulator [Bloch, Kerr,Vanhove] I⊖ ≡ i̟r π

• L2

X Z , Y Z

• + L2

Z X , Y X

• + L2

X Y , Z Y

• mod period

◮ ̟r is the elliptic curve period which is real on the line

0 < K 2 < (m1 + m2 + m3)2

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 19 / 34

The sunset elliptic dilogarithm

s

ϕ ϕ ϕ ϕ ϕ ϕ ϕ

2 3 4 5 6 1 1

P P P

12 23 31

q q q

1 2 3

E

Representing the ratio of the coordinates on the sunset cubic curve as functions on E⊖ ≃ C×/qZ X Z (x) = θ1(x/Q1)θ1(x/P3) θ1(x/P1)θ1(x/Q3) Y Z (x) = θ1(x/Q2)θ1(x/P3) θ1(x/P2)θ1(x/Q3) θ1(x) is the Jacobi theta function θ1(x) = q

1 8 x1/2 − x−1/2

i

• n1

(1 − qn)(1 − qnx)(1 − qn/x) .

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 20 / 34

The sunset elliptic dilogarithm

We find I⊖(s) ≡ i̟r π

• ˆ

E2 P1 P2

• + ˆ

E2 P2 P3

• + ˆ

E2 P3 P1

• mod periods

where ˆ E2(x) =

• n0

(Li2 (qnx) − Li2 (−qnx)) −

• n1

(Li2 (qn/x) − Li2 (−qn/x)) . An equivalent expression using elliptic multiple-polylogarithms has been given by [Adams, Bogner, Weinzeirl] (see as well [Brown,

Levin])

ELim,n(x, y; q) =

• j,k1

xj jm yk kn qjk =

• j1

xj jm Lin(qj)

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 21 / 34

The sunset elliptic dilogarithm

◮ The elliptic dilogarithm ˆ

E2(x) is not invariant under q-translation and transforms according ˆ E2(qx) = ˆ E2(x) − π2 2 + iπ log(x) ˆ E2(x/q) = ˆ E2(x) + π2 2 − iπ log(x/q) . This is because the Feynman integral we are studying is a multivalued

• function. Shifting the point P in C×/qZ changes the expression for I⊖

by a period of the elliptic curve EK 2

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 22 / 34

The sunset Picard-Fuchs equation

Setting s = (m1 + m2 + m3)2/K 2 the differential equation satisfied by the sunset integral is δs = s d

ds

• δ2

s + q1(s)δs + q0(s)

I⊖(s) s = Y⊖(s) +

3

• i=1

log(m2

i )νi(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⊖ ∈ H2(P2 − EK 2) satisfies a 2nd order differential equation

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 23 / 34

The sunset Picard-Fuchs equation

The inhomogeneous term arising from the boundary of ∆ is composed by

◮ the Yukawa coupling

Y⊖(s) =

• Ω⊖ ∧ ∇ d

ds Ω⊖

◮ log-term from the integration between the punctures on the elliptic

curve

v q q p q p

1 1 2 2 3 3

p v v v v v

1 2 3 5 6 4

Es

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 24 / 34

The sunset Picard-Fuchs equation

For the all equal mass case this equation takes the simple form δs = s d

ds

• δ2

s +

2s(9s − 5) (s − 1)(9s − 1)δs + 3s(3s − 1) (s − 1)(9s − 1) −1 s I⊖(s)

• =

6 (9s − 1)(s − 1)

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 25 / 34

The sunset Gromov-Witten invariants

The holomorphic period around s(= 1/t) = 0 π0 =

• ϕ0

Ω⊖ =

• m0

sm

• b1+b2+b2=m

mb1

1 mb2 2 mb3 3

• m!

b1!b2!b3! 2 and the logarithmic Mahler measure defined by π0 = d

dsR0

R0 = iπ−

• |x|=|y|=1

log(s−1−(m2

1x+m2 2y+m2 3)(x−1+y−1+1)) d log xd log y

(2πi)2 . The sunset Feynman integral leads to Gromov-Witten numbers I⊖(s) = −π0    3R3

0 +

• ℓ1+ℓ2+ℓ3=ℓ>0

(ℓ1,ℓ2,ℓ3)∈N3\(0,0,0)

ℓ(1 − ℓR0)Nℓ1,ℓ2,ℓ3

3

• i=1

mℓi

i eℓR0

    .

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 26 / 34

The sunset Gromov-Witten invariants

The local Gromov-Witten numbers Nℓ1,ℓ2,ℓ3 can be expressed in terms

• f the virtual integer number of degree ℓ rational curves by

Nℓ1,ℓ2,ℓ3 =

• d|ℓ1,ℓ2,ℓ3

1 d3 n ℓ1

d , ℓ2 d , ℓ3 d .

ℓ (100)

k>0

(k00) (110) (210) (111) (310) (220) (211) (221) Nℓ 2 2/k 3 −2 6 −1/4 −4 10 nℓ 2 −2 6 −4 10 ℓ (410) (320) (311) (510) (420) (411) (330) (321) (222) Nℓ −2/27 −1 −189/4 nℓ −1 −48

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 27 / 34

The sunset Gromov-Witten invariants

For the all equal masses case m1 = m2 = m3 = 1, the mirror map is Q = eR0 = −q

• n1

(1 − qn)nδ(n); δ(n) := (−1)n−1 −3 n

• ,

where −3

n

• = 0, 1, −1 for n ≡ 0, 1, 2 mod 3.

The local Gromov-Witten numbers

Nℓ 6 = 1, −7 8, 28 27, −135 64 , 626 125, −751 54 , 14407 343 , −69767 512 , 339013 729 , −827191 500 , 8096474 1331 , −367837 16 , 195328680 2197 , −137447647 392 , 4746482528 3375 , −23447146631 4096 , 115962310342 4913 , − 574107546859 5832 , 2844914597656 6859 , −1410921149451 800 , 10003681368433 1323 , . . .

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 28 / 34

Mirror Symmetry

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 29 / 34

The sunset mirror symmetry

Why is this all happening?

◮ The sunset elliptic curve is embedded into a singular

compactification X0 of the local Hori-Vafa 3-fold Y := {1−s(m2

1x +m2 2y +m2 3)(1+x−1+y−1)+uv = 0} ⊂ (C∗)2×C2 , ◮ The GW numbers are computed for the local mirror symmetry of a

semi-stably degenerating a family of elliptically-fibered Calabi-Yau 3-folds Xz0 → X0

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 30 / 34

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 H3(Xz0) Heven(X◦

Q0) ,

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, m1, m2, m3) → Q(K 2, m1, m2, m3) = eR0

Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 31 / 34

The sunset Yukawa coupling

◮ The Yukawa coupling of the non-compact CY X

Yijk =

• X

˜ Ω ∧ ∇δiδjδk ˜ Ω

◮ descends to the local Yukawa of the sunset elliptic curve

Y loc

0ij ∝ Y⊖ =

• Ω⊖ ∧ ∇ d

ds Ω⊖ =

1 2iπ ∂2R1 ∂Ri

0∂Rj ◮ 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 21/10/2016 32 / 34

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 21/10/2016 33 / 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 21/10/2016 34 / 34