Feynman Integrals, Elliptic polylogarithms and mixed Hodge structures Pierre Vanhove New Geometric structures meeting Oxford University Septembre 23, 2014 based on [arXiv:1309.5865], [arXiv:1406.2664] and work in progress Spencer Bloch, Matt Kerr Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 1 / 37
There are many technics to compute amplitudes in field theory ◮ On-shell (generalized) unitarity ◮ On-shell recursion methods ◮ twistor geometry, Graßmanian, Symbol,. . . ◮ Dual conformal invariance ◮ Infra-red behaviour (inverse soft-factors, ...) ◮ amplitude relations, ◮ String theory,. . . These methods indicate that amplitudes have simpler structures than the diagrammatic from Feynman rules suggest The questions are: what are the basic building blocks of the amplitudes? Can the amplitudes be expressed in a basis of integrals functions? Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 2 / 37
Part I One-loop amplitudes Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 3 / 37
The one-loop amplitude ◮ In four dimensions any one-loop amplitude can be expressed on a basis of integral functions [Bern, Dixon,Kosower] � = c r Integral r + Rational terms r Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 4 / 37
The one-loop amplitude ◮ The integral functions are the box, triangle, bubble, tadpole ◮ The integrals functions are given by dilogarithms and logarithms � z Boxes , Triangles ∼ Li 2 ( z ) = − log ( 1 − t ) d log t 0 � z Bubble ∼ log ( 1 − z ) = d log ( 1 − t ) 0 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 4 / 37
The one-loop amplitude This allows to characterize in a simple way one-loop amplitudes in various gauge theory ◮ Only boxes for N = 4 SYM for one-loop graph ◮ No triangle property of N = 8 SUGRA [Bern, Carrasco, Forde, Ita, Johansson; Arkani-hamed, Cachazo, Kaplan; Bjerrum-Bohr, Vanhove] ◮ Only box for QED multi-photon amplitudes with n � 8 photons [Badger, Bjerrum-Bohr, Vanhove] Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 4 / 37
Monodromies, periods ◮ 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 & IHES) Elliptic polylogarithms 23/09/2014 5 / 37
A one-loop example I We consider the 3-mass triangle p 1 + p 2 + p 3 = 0 and p 2 i � 0 � d 4 ℓ I ⊲ ( p 2 1 , p 2 2 , p 2 3 ) = ℓ 2 ( ℓ + p 1 ) 2 ( ℓ − p 3 ) 2 R 1 , 3 Which can be represented as � dxdy I ⊲ = ( p 2 1 x + p 2 2 y + p 2 3 )( xy + x + y ) x � 0 y � 0 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 6 / 37
A one-loop example II and evaluated as D ( z ) I ⊲ = � � 1 p 4 1 + p 4 2 + p 4 3 − ( p 2 1 p 2 2 + p 2 1 p 2 3 + p 2 2 p 2 3 ) 2 z roots of ( 1 − x )( p 2 3 − xp 2 1 ) + p 2 z and ¯ 2 x = 0 ◮ Single-valued Bloch-Wigner dilogarithm for z ∈ C \{ 0 , 1 } D ( z ) = ℑ m ( Li 2 ( z )) + arg ( 1 − z ) log | z | ◮ The permutation of the 3 masses: { z , 1 − ¯ z , 1 z , 1 − 1 1 ¯ z z , 1 − z , − z } ¯ 1 − ¯ this set is left invariant by the D ( z ) ◮ The integral has branch cuts arising from the square root since D ( z ) is analytic Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 7 / 37
The triangle graph motive I � dxdy I ⊲ = ( p 2 1 x + p 2 2 y + p 2 3 )( xy + x + y ) x � 0 y � 0 The integral is defined over the domain ∆ = { [ x , y , z ] ∈ P 2 , x , y , z � 0 } and the denominator is the quadric C ⊲ := ( p 2 1 x + p 2 2 y + p 2 3 z )( xy + xz + yz ) Let L = { x = 0 } ∪ { y = 0 } ∪ { z = 0 } and D = { x + y + z = 0 } ∪ C ⊲ dxdy 3 )( xy + x + y ) ∈ H := H 2 ( P 2 − D , L \ ( L ∪ C ⊲ ) ∩ L , Q ) ( p 2 1 x + p 2 2 y + p 2 We need to consider the relative cohomology because the domain ∆ is not in H 2 ( P 2 − D ) because ∂∆ ∩ � ∅ Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 8 / 37
The triangle graph motive II Since ∂∆ ∩ C ⊲ = { [ 1 , 0 , 0 ] , [ 0 , 1 , 0 ] , [ 0 , 0 , 1 ] } one needs to perform a blow-up these 3 points. One can define a mixed Tate Hodge structure [Bloch, Kreimer] with weight W 0 H ⊂ W 2 H ⊂ W 4 H and grading gr W gr W 2 H = Q (− 1 ) 5 , gr W 0 H = Q ( 0 ) , 4 H = Q (− 2 ) The Hodge matrix and unitarity 0 0 1 0 0 − Li 1 ( z ) 2 i π 0 0 ( 2 i π ) 2 − Li 2 ( z ) 2 i π log z ◮ The construction is valid for all one-loop amplitudes in four dimensions Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 9 / 37
Part II Loop amplitudes Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 10 / 37
Feynman parametrization ◮ Typically form of the Feynman parametrization of a graph Γ ◮ A Feynman graph with L loops and n propagators � ∞ n U n −( L + 1 ) D n � � 2 I Γ ∝ δ ( 1 − x i ) dx i ( U � i x i − F ) n − L D i m 2 0 2 i = 1 i = 1 ◮ U and F are the Symanzik polynomials [Itzykson, Zuber] ◮ U is of degree L and F of degree L + 1 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 11 / 37
Feynman parametrization and world-line formalism ◮ Rewrite the integral as δ ( 1 − � n � n � ∞ i = 1 x i ) i = 1 dx i I Γ ∝ ( � F ) n − L D D i x i − � i m 2 U 0 2 2 ◮ U = det Ω is the determinant of the period matrix of the graph ◮ The period matrix of integral of homology vectors v i on oriented � loops C i Ω ij = v j C i Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 12 / 37
Feynman parametrization and world-line formalism ◮ Rewrite the integral as � ∞ δ ( 1 − � n � n i = 1 x i ) i = 1 dx i I Γ ∝ ( � i x i − � F ) n − L D D i m 2 U 0 2 2 ◮ U = det Ω is the determinant of the period matrix of the graph � T 1 + T 3 � T 3 Ω 2 ( a ) = T 3 T 2 + T 3 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 12 / 37
Feynman parametrization and world-line formalism ◮ Rewrite the integral as δ ( 1 − � n � n � ∞ i = 1 x i ) i = 1 dx i I Γ ∝ ( � i x i − � F ) n − L D D i m 2 U 0 2 2 ◮ U = det Ω is the determinant of the period matrix of the graph T 1 + T 2 T 2 0 Ω 3 ( b ) = T 2 T 2 + T 3 + T 5 + T 6 T 3 0 T 3 T 3 + T 4 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 12 / 37
Feynman parametrization and world-line formalism ◮ Rewrite the integral as δ ( 1 − � n � n � ∞ i = 1 x i ) i = 1 dx i I Γ ∝ ( � i x i − � F ) n − L D D i m 2 U 0 2 2 ◮ U = det Ω is the determinant of the period matrix of the graph T 1 + T 4 + T 5 T 5 T 4 Ω 3 ( c ) = T 5 T 2 + T 5 + T 6 T 6 T 4 T 6 T 3 + T 4 + T 6 Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 12 / 37
Feynman parametrization and world-line formalism ◮ Rewrite the integral as δ ( 1 − � n � n � ∞ i = 1 x i ) i = 1 dx i I Γ ∝ ( � i x i − � F ) n − L D D i m 2 U 0 2 2 F = � ◮ � 1 � r < s � n k r · k s G ( x r , x s ; Ω ) sum of Green’s function ( x r − x s ) 2 G 1 − loop ( x r , x s ; L ) = − 1 2 | x s − x r | + 1 . 2 T Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 13 / 37
Feynman parametrization and world-line formalism ◮ Rewrite the integral as δ ( 1 − � n � n � ∞ i = 1 x i ) i = 1 dx i I Γ ∝ ( � i x i − � F ) n − L D D i m 2 U 0 2 2 F = � ◮ � 1 � r < s � n k r · k s G ( x r , x s ; Ω ) sum of Green’s function ( x s − x r ) 2 same line ( x r , x s ; Ω 2 ) = − 1 2 | x s − x r | + T 2 + T 3 G 2 − loop 2 ( T 1 T 2 + T 1 T 3 + T 2 T 3 ) 2 ( x r + x s ) + T 3 ( x r + x s ) 2 + T 2 x 2 r + T 1 x 2 diff line ( x r , x s ; Ω 2 ) = − 1 G 2 − loop s 2 ( T 1 T 2 + T 1 T 3 + T 2 T 3 ) Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 13 / 37
Periods ◮ A Feynman graph with L loops and n propagators � ∞ n n U n −( L + 1 ) D � � 2 I Γ ∝ δ ( 1 − x i ) dx i ( U � i x i − F ) n − L D i m 2 0 2 i = 1 i = 1 ◮ [Kontsevich, Zagier] define periods are follows. P ∈ C is the ring of periods, is z ∈ P if ℜ e ( z ) and ℑ m ( z ) are of the forms � n f ( x i ) � dx i < ∞ g ( x i ) ∆ ∈ R n i = 1 with f , g ∈ Z [ x 1 , · · · , x n ] and ∆ is algebraically defined by polynomial inequalities and equalities. Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 14 / 37
Periods ◮ [Kontsevich, Zagier] define periods are follows. P ∈ C is the ring of periods, is z ∈ P if ℜ e ( z ) and ℑ m ( z ) are of the forms � n f ( x i ) � dx i < ∞ g ( x i ) ∆ ∈ R n i = 1 with f , g ∈ Z [ x 1 , · · · , x n ] and ∆ is algebraically defined by polynomial inequalities and equalities. ◮ Problem for Feynman graphs ∂∆ ∩ { g ( x i ) = 0 } � ∅ ◮ Generally the domain of integration is not closed ∂∆ � ∅ ◮ Need to consider the relative cohomology and perform blow-ups [Bloch,Esnault,Kreimer] Pierre Vanhove (IPhT & IHES) Elliptic polylogarithms 23/09/2014 14 / 37
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