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 • “Simple” amplitudes: iterated integrals and multiple polylogarithms 0 d t functions are relatively well-understood. • At higher loops we cannot expect such simple integrals. curves, K3 surfaces, Calabi-Yau manifolds). 1 Feynman integrals for scattering amplitudes (e.g. N = 4 SYM) � z G ( a 1 , . . . , a n ; z ) = G ( a 2 , . . . , a n ; t ) t − a 1 • Algorithmic obstructions (identities, integration, …), but the • More complicated integrals involve interesting geometry (elliptic
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 More loops: complete intersection in a toric variety 3. Conclusion and further directions 3 Two loops: elliptic curve as intersection of two quadrics in P 3 Three loops: four-fold cover of P 1 × P 1 (K3)
Introduction
Two ways to fjnd a geometry Direct integration using Feynman parameters: • Constant means the integral is polylogarithmic • Residue of highest codimension: leading singularity • Jacobians may allow for more residues than propagators • Take residues around the poles of the propagators. Taking residues: d x some variety. 4 integrations � � d 4 ℓ 1 · · · d 4 ℓ L × Rational integrand − − − − − − − → ω × Polylogs γ Here γ is some integration contour and ω is a holomorphic form on 4 x 3 − g 2 x − g 3 → elliptic curve y 2 = 4 x 3 − g 2 x − g 3 . Example: ω = �
Momentum twistors y Conformal transformations Point x Dual momentum space 5 p a x a Dual coordinates: p a = x a + 1 − x a x a + 1 Momentum twistors: [Hodges (2009)] p a + 1 p a − 1 � � λ α Z = ∈ P 3 α ˜ x α ˙ λ ˙ x a − 1 α The twistor dictionary: Momentum twistor space P 3 Line L x = A x ∧ B x ( x − y ) 2 Four bracket � A x B x A y B y � ( x − y ) 2 = 0 Lines L x and L y intersect PGL( 4 ) transformations
The massive box integral x 4 separated. There are two confjgurations where all dual points are light-like • Two solutions characterized by cross transversal to L i . • External points: four skew lines L i Apply the twistor dictionary: [Hodges (2010)] Figure 1: The box integral x 3 x 2 x 1 (Some normalization) 6 � d 4 x ℓ ( 2 π ) 4 ( x ℓ − x 1 ) 2 ( x ℓ − x 2 ) 2 ( x ℓ − x 3 ) 2 ( x ℓ − x 4 ) 2 • ( x ℓ − x i ) 2 = 0 : Find a fjfth line x ℓ ratios κ and ˜ κ on P 1 . • Everything is manifestly conformal.
Traintrack integrals
Traintrack integrals Figure 2: The traintrack integral family • Hypersurfaces in weighted projective space • Two loops: elliptic curve • Three loops: K3 surface, four loops: Calabi-Yau threefold 7 Previously studied by [Bourjaily, He, McLeod, von Hippel, Wilhelm (2018)] : • Feynman parameters, direct integration
Two loops: building an elliptic curve x 1 x 2 x 3 Figure 3: Relationship between the endcap of the traintrack and the quadric. A quadric has two rulings (families of lines): • The lines within one ruling are skew. • Two lines from difgerent rulings intersect. 8 x ℓ Three skew lines L i = A i ∧ B i determine a quadric in P 3 : Q ( Z ) = � ZA 1 B 1 A 3 �� ZA 2 B 2 B 3 � − � ZA 1 B 1 A 3 �� ZA 2 B 2 A 3 �
Two loops: building an elliptic curve Figure 4: The double box integral 9 x ℓ L x ℓ R Build quadrics Q L and Q R for the left and the right loop respectively. Imposing ( x ℓ L − x ℓ R ) 2 = 0 means that the quadrics intersect. The intersection C of two quadrics in P 3 is an elliptic curve.
Elliptic curve: holomorphic form Q L Q R Here: 10 Take Poincaré residues to get a holomorphic form on C = Q L ∩ Q R : ω P 3 ω C = Res , ω P 3 = Z 0 d Z 1 d Z 2 d Z 3 ± ( permutations ) Q L , Q R Check the weight under rescaling Z → α Z : ω P 3 → α 4 ω P 3 , Q L → α 2 Q L , Q R → α 2 Q R . Elliptic curves are characterized by one modulus, the j -invariant. j = 256 ( z 2 − z + 1 ) 3 z 2 ( z − 1 ) 2 , where z depends on the quadrics Q L and Q R .
Elliptic curve: comparison 0 The two j -invariants agree. Result Previous work by [Bourjaily, McLeod, Spradlin, von Hippel, Wilhelm (2017)] : d x int. 11 • Direct integration (Feynman parameters): � ∞ � d 4 ℓ 1 d 4 ℓ 2 × Rational integrand − − → × Polylogs � P 4 ( x ) • Elliptic curve defjned by y 2 = P 4 ( x ) with complicated j -invariant. Here: elliptic curve as intersection of two quadrics with j -invariant j = 256 ( z 2 − z + 1 ) 3 z 2 ( z − 1 ) 2 .
Three loops: K3 surface x 1 x 2 Figure 5: The three-loop traintrack integral Geometry in twistor space: 12 x ℓ • As before: two quadrics Q L and Q R • Lines L 1 and L 2 associated to x 1 and x 2 • Line L ℓ parameterized by two points P 1 ∈ L 1 and P 2 ∈ L 2 • Bezout’s theorem: L ℓ intersects Q L and Q R in two points each.
Three loops: K3 surface Figure 6: The geometry of the three-loop traintrack integral Geometry in twistor space: 13 • As before: two quadrics Q L and Q R • Lines L 1 and L 2 associated to x 1 and x 2 • Line L ℓ parameterized by two points P 1 ∈ L 1 and P 2 ∈ L 2 • Bezout’s theorem: L ℓ intersects Q L and Q R in two points each.
Three loops: K3 surface Where is the K3 surface? constraints. Thus, the leading singularity is two-dimensional. 14 • We can freely choose P 1 ∈ L 1 and P 2 ∈ L 2 while satisfying all • For chosen P 1 and P 2 , there are 2 × 2 = 4 choices for the intersection points of L ℓ with Q L and Q R . • Thus, we have a four-fold cover of P 1 × P 1 . Aside: elliptic curve in P 2 • Pick four points in P 1 . • Set three of them to { 0 , 1 , ∞} and call the last one λ . • Legendre form: y 2 = x ( x − 1 )( x − λ ) • The curve is a double cover of P 1 branched over four points.
• They are themselves elliptic curves. K3 surface: branching • Two branches of the surface over each curve • Bezout: eight intersection points (only one branch) 15 Branching occurs when the line L ℓ is tangent to Q L or Q R . L ℓ is tangent if the following equations are fulfjlled: ∆ L ≡ Q L ( P 1 , P 2 ) 2 − Q L ( P 1 , P 1 ) Q L ( P 2 , P 2 ) = 0 ∆ R ≡ Q R ( P 1 , P 2 ) 2 − Q R ( P 1 , P 1 ) Q R ( P 2 , P 2 ) = 0 We get two curves ∆ L and ∆ R of bi-degree ( 2 , 2 ) in P 1 × P 1 :
K3 surface: characteristics 2 1 We compute the Euler characteristic using surgery: 4 Branches 16 Points in P 1 × P 1 P 1 × P 1 − ∆ L ∪ ∆ R ∆ L ∪ ∆ R − ∆ L ∩ ∆ R ∆ L ∩ ∆ R χ ( P 1 × P 1 ) − χ (∆ L ∪ ∆ R ) � � χ = 4 × + 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.
K3 surface: characteristics Holomorphic form: • Number of fjxpoints gives bounds on Picard rank Open questions: How to compare to the hypersurface in weighted projective space? What are the invariants? 17 Dimension of the moduli space: 11 (Picard rank ρ = 9 ) ω P 1 ω P 1 ω K 3 = √ ∆ L √ ∆ R Nikulin involutions and automorphisms: [Nikulin (1979)] • In this case: ρ ≥ 9
Four and more loops x 1 x 2 x 3 x 4 Figure 7: The four-loop traintrack integral Figure 8: The geometry of the four-loop traintrack integral 18 x ℓ 1 x ℓ 2
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); …] • Problem: codimension in the embedding space grows with the number of loops General traintrack integral with L loops 19 • Three-fold at four loops: h 11 = 12 , h 12 = 28 and χ = − 32 (computed with PALP [Kreuzer, Skarke (2004); …] ) Calabi-Yau ( L − 1 ) -fold in a toric variety of dimension 2 ( L − 1 ) .
Conclusion and further directions
Summary Leading singularity of the traintrack integrals: • More loops: Calabi-Yau complete intersection • Intersections of lines are easier than quadratic equations. • No extra (unphysical) parameters • Dual-conformal symmetry manifest 20 • Two loops: elliptic curve as the intersection of two quadrics in P 3 • Three loops: four-fold cover of P 1 × P 1 Good properties of momentum twistor space:
Further directions Supersymmetrization: versions. More complicated diagrams: 21 • Amplitudes in N = 4 SYM are superconformal • Formulate the intersections in terms of δ -functions (invariant under PSL( 4 ) ) • Replace P 3 by P 3 | 4 and the δ -functions by supersymmetric • Fishnet-type N × M box graphs: also Calabi-Yau • Massive internal propagators
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