Geometry of Feynman Integrals in Twistor Space Based on - - PowerPoint PPT Presentation

Geometry of Feynman Integrals in Twistor Space

Based on arXiv:2005.08771 in collaboration with Cristian Vergu. Building on work with Jacob Bourjaily, Andrew McLeod, Matt von Hippel, Matthias Wilhelm.

Matthias Volk September 10, 2020

Niels Bohr Institute

Context

Feynman integrals for scattering amplitudes (e.g. N = 4 SYM)

• “Simple” amplitudes: iterated integrals and multiple

polylogarithms G(a1, . . . , an; z) = z dt t − a1 G(a2, . . . , an; t)

• Algorithmic obstructions (identities, integration, …), but the

functions are relatively well-understood.

• At higher loops we cannot expect such simple integrals.
• More complicated integrals involve interesting geometry (elliptic

curves, K3 surfaces, Calabi-Yau manifolds).

1

Questions to address

• How to attach a geometry to a Feynman integral?

(Parametric representations, momentum twistor space, …)

• What are the properties this geometry?

(Invariants, moduli space, …)

• Where to go next?

2

Table of contents

• 1. Introduction
• 2. Traintrack integrals

Two loops: elliptic curve as intersection of two quadrics in P3 Three loops: four-fold cover of P1 × P1 (K3) More loops: complete intersection in a toric variety

• 3. Conclusion and further directions

3

Introduction

Two ways to fjnd a geometry

Direct integration using Feynman parameters:

• d4ℓ1 · · · d4ℓL × Rational integrand

integrations

− − − − − − − →

• γ

ω × Polylogs Here γ is some integration contour and ω is a holomorphic form on some variety. Example: ω =

dx

• 4x3−g2x−g3 → elliptic curve y2 = 4x3 − g2x − g3.

Taking residues:

• Take residues around the poles of the propagators.
• Jacobians may allow for more residues than propagators
• Residue of highest codimension: leading singularity
• Constant means the integral is polylogarithmic

4

Momentum twistors

xa+1 xa xa−1 pa+1 pa pa−1 y

Dual coordinates: pa = xa+1 − xa Momentum twistors: [Hodges (2009)] Z =

• λα

xα ˙

α˜

λ ˙

α

• ∈ P3

The twistor dictionary: Dual momentum space Momentum twistor space P3 Point x Line Lx = Ax ∧ Bx (x − y)2 Four bracket AxBxAyBy (x − y)2 = 0 Lines Lx and Ly intersect Conformal transformations PGL(4) transformations

5

The massive box integral

• d4xℓ

(2π)4 (Some normalization) (xℓ − x1)2(xℓ − x2)2(xℓ − x3)2(xℓ − x4)2

xℓ x1 x2 x3 x4

Figure 1: The box integral

Apply the twistor dictionary: [Hodges (2010)]

• External points: four skew lines Li
• (xℓ − xi)2 = 0: Find a fjfth line

transversal to Li.

• Two solutions characterized by cross

ratios κ and ˜ κ on P1.

• Everything is manifestly conformal.

There are two confjgurations where all dual points are light-like separated.

6

Traintrack integrals

Traintrack integrals

Figure 2: The traintrack integral family

Previously studied by [Bourjaily, He, McLeod, von Hippel, Wilhelm (2018)]:

• Feynman parameters, direct integration
• Hypersurfaces in weighted projective space
• Two loops: elliptic curve
• Three loops: K3 surface, four loops: Calabi-Yau threefold

7

Two loops: building an elliptic curve

xℓ x1 x2 x3

Figure 3: Relationship between the endcap of the traintrack and the quadric.

Three skew lines Li = Ai ∧ Bi determine a quadric in P3: Q(Z) = ZA1B1A3ZA2B2B3 − ZA1B1A3ZA2B2A3 A quadric has two rulings (families of lines):

• The lines within one ruling are skew.
• Two lines from difgerent rulings intersect.

8

Two loops: building an elliptic curve

xℓL xℓR

Figure 4: The double box integral

Build quadrics QL and QR for the left and the right loop respectively. Imposing (xℓL − xℓR)2 = 0 means that the quadrics intersect. The intersection C of two quadrics in P3 is an elliptic curve.

9

Elliptic curve: holomorphic form

Take Poincaré residues to get a holomorphic form on C = QL ∩ QR: ωC = Res

QL,QR

ωP3 QLQR , ωP3 = Z0 dZ1 dZ2 dZ3 ± (permutations) Check the weight under rescaling Z → αZ: ωP3 → α4ωP3, QL → α2QL, QR → α2QR. Elliptic curves are characterized by one modulus, the j-invariant. Here: j = 256(z2 − z + 1)3 z2(z − 1)2 , where z depends on the quadrics QL and QR.

10

Elliptic curve: comparison

Previous work by [Bourjaily, McLeod, Spradlin, von Hippel, Wilhelm (2017)]:

• Direct integration (Feynman parameters):
• d4ℓ1d4ℓ2 × Rational integrand

int.

− − → ∞ dx

• P4(x)

× Polylogs

• Elliptic curve defjned by y2 = P4(x) with complicated j-invariant.

Here: elliptic curve as intersection of two quadrics with j-invariant j = 256(z2 − z + 1)3 z2(z − 1)2 . Result The two j-invariants agree.

11

Three loops: K3 surface

xℓ x1 x2

Figure 5: The three-loop traintrack integral

Geometry in twistor space:

• As before: two quadrics QL and QR
• Lines L1 and L2 associated to x1 and x2
• Line Lℓ parameterized by two points P1 ∈ L1 and P2 ∈ L2
• Bezout’s theorem: Lℓ intersects QL and QR in two points each.

12

Three loops: K3 surface

Figure 6: The geometry of the three-loop traintrack integral

Geometry in twistor space:

• As before: two quadrics QL and QR
• Lines L1 and L2 associated to x1 and x2
• Line Lℓ parameterized by two points P1 ∈ L1 and P2 ∈ L2
• Bezout’s theorem: Lℓ intersects QL and QR in two points each.

13

Three loops: K3 surface

Where is the K3 surface?

• We can freely choose P1 ∈ L1 and P2 ∈ L2 while satisfying all
• constraints. Thus, the leading singularity is two-dimensional.
• For chosen P1 and P2, there are 2 × 2 = 4 choices for the

intersection points of Lℓ with QL and QR.

• Thus, we have a four-fold cover of P1 × P1.

Aside: elliptic curve in P2

• Pick four points in P1.
• Set three of them to {0, 1, ∞} and call the last one λ.
• Legendre form: y2 = x(x − 1)(x − λ)
• The curve is a double cover of P1 branched over four points.

14

K3 surface: branching

Branching occurs when the line Lℓ is tangent to QL or QR. Lℓ is tangent if the following equations are fulfjlled: ∆L ≡ QL(P1, P2)2 − QL(P1, P1)QL(P2, P2) = 0 ∆R ≡ QR(P1, P2)2 − QR(P1, P1)QR(P2, P2) = 0 We get two curves ∆L and ∆R of bi-degree (2, 2) in P1 × P1:

• They are themselves elliptic curves.
• Two branches of the surface over each curve
• Bezout: eight intersection points (only one branch)

15

K3 surface: characteristics

Points in P1 × P1 Branches P1 × P1 − ∆L ∪ ∆R 4 ∆L ∪ ∆R − ∆L ∩ ∆R 2 ∆L ∩ ∆R 1 We compute the Euler characteristic using surgery: χ = 4 ×

• χ(P1 × P1) − χ(∆L ∪ ∆R)
• + 2 × [χ(∆L ∪ ∆R) − χ(∆L ∩ ∆R)]

+ 1 × χ(∆L ∩ ∆R) Using for example χ(∆L) = χ(∆R) = 0 we get χ = 24 as required.

16

K3 surface: characteristics

Dimension of the moduli space: 11 (Picard rank ρ = 9) Holomorphic form: ωK3 = ωP1ωP1 √∆L √∆R Nikulin involutions and automorphisms: [Nikulin (1979)]

• Number of fjxpoints gives bounds on Picard rank
• In this case: ρ ≥ 9

Open questions: How to compare to the hypersurface in weighted projective space? What are the invariants?

17

Four and more loops

xℓ1 x1 x2 xℓ2 x3 x4

Figure 7: The four-loop traintrack integral Figure 8: The geometry of the four-loop traintrack integral

18

Four and more loops

We can build a Calabi-Yau as a complete intersection in a toric variety:

• Use combinatorial description to compute Hodge numbers

[Batyrev, Borisov (1994); …]

• Three-fold at four loops: h11 = 12, h12 = 28 and χ = −32

(computed with PALP [Kreuzer, Skarke (2004); …])

• Problem: codimension in the embedding space grows with the

number of loops General traintrack integral with L loops Calabi-Yau (L − 1)-fold in a toric variety of dimension 2(L − 1).

19

Conclusion and further directions

Summary

Leading singularity of the traintrack integrals:

• Two loops: elliptic curve as the intersection of two quadrics in P3
• Three loops: four-fold cover of P1 × P1
• More loops: Calabi-Yau complete intersection

Good properties of momentum twistor space:

• Intersections of lines are easier than quadratic equations.
• No extra (unphysical) parameters
• Dual-conformal symmetry manifest

20

Further directions

Supersymmetrization:

• Amplitudes in N = 4 SYM are superconformal
• Formulate the intersections in terms of δ-functions (invariant

under PSL(4))

• Replace P3 by P3|4 and the δ-functions by supersymmetric

versions. More complicated diagrams:

• Fishnet-type N × M box graphs: also Calabi-Yau
• Massive internal propagators

21

Even further directions

How does the geometry degenerate?

• Polylogarithmic limits, e.g. ladder diagrams [Ussyukina, Davydychev

(1993); Broadhurst (1993)]

• Basso-Dixon integrals [Basso, Dixon (2017)]

How to deal with non-planar integrals? The role of difgerent parametrizations?

• Feynman parameters, Baikov representation, …
• Isogenies, e.g. [Bogner, Müller-Stach, Weinzierl (2019)]
• Which changes of variables are allowed?

Calabi-Yau mirror symmetry (for example as in [Bloch, Kerr, Vanhove (2016);

Bönisch, Fischbach, Klemm, Nega, Safari (2020); …])? 22

Thank you.

22

Elliptic curve: j-invariant

Construct an invariant in four steps:

• 1. Pencil of quadrics: λLQL + λRQR with [λL : λR] ∈ P1.
• 2. Think of QL and QR as 4 × 4 symmetric matrices and compute

det(λLQL + λRQR) ∼ #λ4

L + ˜

#λ3

LλR + · · · .

• 3. Compute the cross ratio of the four roots λi = [λi

L : λi R]:

z = 1234 1324, ij = det(λi, λj). Permutations of the λi send z → z′ ∈ { 1

z , 1 − z, . . .}.

• 4. The j-invariant is the permutation-invariant combination

j = 256(z2 − z + 1)3 z2(z − 1)2 .