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 = 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 21/10/2016 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] � A L − loop = coeff i Integral i + Rational i ∈ B ( L ) ◮ 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 − 2 ǫ dimensions, the basis of integral function is known for a long time [Bern,Dixon,Kosower] to be consisting of Boxes, triangles, bubble, tadpole integrals � d D ℓ I � = ( 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 − m 2 1 )(( ℓ + K 1 ) 2 − m 2 2 )(( ℓ + K 1 + K 2 ) 2 − m 2 3 ) � d D ℓ I ◦ = ( ℓ 2 − m 2 1 )(( ℓ + K 1 ) 2 − m 2 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 � I Γ = Ω Γ ∆ ◮ The domain of integration ∆ = { x i � 0 } ⊂ P n − 1 ◮ The integrand is the differential form n U n −( L + 1 ) D Ω Γ = Γ ( n − LD � 2 (− 1 ) j − 1 x j dx 1 ∧ · · · � 2 ) dx j ∧ · · · ∧ dx n ( U � i x i − F ) n − L D i m 2 2 j = 1 ◮ U and F are the Symanzik polynomials [Itzykson, Zuber] ◮ U is of degree L and F of degree L + 1 in the x i Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 5 / 34

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 n U n −( L + 1 ) D Ω Γ = Γ ( n − LD � 2 (− 1 ) j − 1 x j dx 1 ∧ · · · � 2 ) dx j ∧ · · · ∧ dx n ( U � i x i − F ) n − L D i m 2 2 j = 1 ◮ We can perform a Laurent expansion in ǫ = ( 4 − D ) / 2 ∞ � c i ǫ i I Γ = i =− 2 L ◮ The c i 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 s ij , internal masses m i , . . . ◮ 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 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) Sunset in the mirror 21/10/2016 8 / 34

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] � P n − 1 \{ g ( x i ) = 0 } ) , � H n − 1 ( ∆ ) ◮ 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 21/10/2016 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 p 2 p 1 + p 2 + p 3 = 0 ; i � 0 � dxdy D ( z ) I ⊲ = 3 )( xy + x + y ) = � � 1 ( p 2 1 x + p 2 2 y + p 2 x � 0 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 y � 0 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 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 � dxdy I ⊲ = ∆ = { x = 0 } ∪ { y = 0 } ∪ { z = 0 } ( p 2 1 x + p 2 2 y + p 2 3 )( xy + x + y ) ∆ The denominator is the quadric E ⊲ = { ( p 2 1 x + p 2 2 y + p 2 3 z )( xy + xz + yz ) = 0 } dxdy 3 )( xy + x + y ) ∈ H := H 2 ( P 2 − E ⊲ , ∆ \ ( ∆ ∪ E ⊲ ) ∩ ∆ ) ( p 2 1 x + p 2 2 y + p 2 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]     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 ◮ 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 ) 0 � z I ◦ ∼ log ( 1 − z ) = d log ( 1 − t ) 0 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 � I 2 ∆ := { [ x : y : z ] ∈ P 2 | x � 0 , y � 0 , z � 0 } ⊖ = Ω ⊖ ; ∆ ◮ The sunset integral is the integration of the 2-form zdx ∧ dy + xdy ∧ dz + ydz ∧ dx 3 z )( xz + xy + yz ) − K 2 xyz ∈ H 2 ( P 2 − E K 2 ) Ω ⊖ = ( m 2 1 x + m 2 2 y + m 2 ◮ The sunset family of open elliptic curve (modular only for all equal masses) E K 2 = { ( m 2 1 x + m 2 2 y + m 2 3 z )( xz + xy + yz ) − K 2 xyz = 0 } Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 15 / 34

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 21/10/2016 16 / 34

The sunset mixed Hodge structure ◮ If P → P 2 is the blow-up and ˆ E K 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 ∩ ˆ E K 2 ) ∆ ∩ ˆ ˜ ∆ ∈ H 2 ( P − ˆ ˜ E K 2 , h ) = H 2 ( P − ˆ E K 2 , h ) ∨ E K 2 = ∅ ; ◮ The sunset integral is a period of this (mixed) Hodge structure � � Ω ⊖ , ˜ I ⊖ = ∆ ◮ 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 m , Q ( 2 )) α H 1 ( G 2 → 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 . K 2 = E K 2 − { P 1 , P 2 , P 3 , Q 1 , Q 2 , Q 3 } and P 2 − h = G 2 with E 0 m ◮ Since Image ( α ) = span � d log ( X / Z ) , d log ( Y / Z ) � ◮ Introducing the regulator � X � Z , Y L 2 = F ( P 3 ) + F ( Q 2 ) − F ( P 2 ) − F ( Q 3 ) Z � X � � x F ( x ) = − log Z ( y ) d log y x 0 ◮ with the 2-torsion relations Q i = − P i for i = 1 , 2 , 3 Pierre Vanhove (IPhT) Sunset in the mirror 21/10/2016 18 / 34