Loop-Tree Duality Method

Transcription

1 Numerical Implementation of the Loop-Tree Duality Method IFIC Sebastian Buchta in collaboration with G. Rodrigo,! P. Draggiotis, G. Chachamis and I. Malamos 24. July 2015

2 Outline 1.Introduction! 2.A new method for higher order calculations: Loop-Tree Duality! 3.Numerical Implementation

3 Introduction When calculating NLO (NNLO) cross-sections one needs to consider the tree- and loop-contributions separately. Especially loops with many external legs prove to be challenging.! Considerable progress has already been made in order to attack this problem: OPP- Method, Unitarity Methods, Mellin-Barnes Representation, Sector Decomposition.! The advantage of these methods is that they made possible what was impossible before, but still a lot of effort has to be put in to cancel infrared singularities among real and virtual corrections. Additional difficulties arise from threshold singularities that lead to numerical instabilities.! The Loop-Tree Duality (LTD) method aims towards a combined treatment of treeand loop-contributions. Therefore the Loop-Tree Duality method casts the virtual corrections in a form that closely resembles the real ones.

4 p 1 q 1 Loop-Tree Duality at one loop p 2 q N [Catani, Gleisberg, Krauss, Rodrigo, Winter 08] l 1 q 2 Z NY p N L (1) (p 1,p 2,...,p N )= G F (q i ) `1 i=1 p 3 with G F (q i )= q 2 i 1 m 2 i + i0 Feynman propagator and q i = `1 + p p i = `1 + k i Internal momenta Z Z and `1 = i dd`1 (2 ) d Work carried out in! dimensional regularization!

5 Loop-Tree Duality at one loop L (N) Directly apply the residue theorem for! l 1,0 complex energy components of the! loop momenta! C L Z NY X ResIm{qi,0}<0 L (1) (p 1,p 2,...,p N )= 2 i G F (q j ) ~`1 j=1 Selects the poles with negative imaginary part and positive energy! (Duality beyond one-loop: [Bierenbaum, Catani, Draggiotis, Rodrigo 10],! Duality with higher order poles: [Bierenbaum, Buchta, Draggiotis, Malamos, Rodrigo 12])

6 Loop-Tree Duality at one loop X Z`1 NY L (1) (p 1,p 2,...,p N )= (qi ) G D (q i ; q j ) j=1 j6=i p 1 corresponds to p 2 δ(q p i i ) p i+1 q q i q = N 1 (q qi m2 i+1 i0 ηp i+1 p N i=1 p 3 p i+2

7 Loop-Tree Duality at one loop 2 i + (q 2 i m 2 i ) 1 q 2 j m 2 j i0 (q j q i ) X Z`1 NY L (1) (p 1,p 2,...,p N )= (qi ) G D (q i ; q j ) j=1 j6=i p 1 corresponds to p 2 δ(q p i i ) p i+1 q q i q = N 1 (q qi m2 i+1 i0 ηp i+1 p N i=1 p 3 p i+2

8 Loop-Tree Duality at one loop 2 i + (q 2 i m 2 i ) 1 q 2 j m 2 j i0 (q j q i ) L (1) (p 1,p 2,...,p N )= X Z`1 NY (qi ) G D (q i ; q j ) j=1 j6=i future-like vector corresponds to 2 0, 0 > 0 p 1 δ(q p i i ) p i+1 q q i p 2 q = N 1 (q qi m2 i+1 i0 ηp i+1 p N i=1 p 3 p i+2

9 Loop-Tree Duality at one loop The Loop-Tree Duality contains only single cuts while introducing a modified i0-prescription, the dual prescription.

10 Loop-Tree Duality at one loop The Loop-Tree Duality contains only single cuts while introducing a modified i0-prescription, the dual prescription. Number of single cut dual contributions equals the number of legs.

11 Loop-Tree Duality at one loop The Loop-Tree Duality contains only single cuts while introducing a modified i0-prescription, the dual prescription. Number of single cut dual contributions equals the number of legs. The singularities of the loop diagram appear as singularities of the dual integrals.

12 Loop-Tree Duality at one loop The Loop-Tree Duality contains only single cuts while introducing a modified i0-prescription, the dual prescription. Number of single cut dual contributions equals the number of legs. The singularities of the loop diagram appear as singularities of the dual integrals. Tensor loop integrals and physical scattering amplitudes are treated in the same way since the Loop-Tree Duality works only on propagators.

13 Loop-Tree Duality at one loop The Loop-Tree Duality contains only single cuts while introducing a modified i0-prescription, the dual prescription. Number of single cut dual contributions equals the number of legs. The singularities of the loop diagram appear as singularities of the dual integrals. Tensor loop integrals and physical scattering amplitudes are treated in the same way since the Loop-Tree Duality works only on propagators. Virtual corrections are recast in a form, that closely parallels the contribution of real corrections.

14 Singular behavior of the loop integrand [Buchta, Chachamis, Draggiotis, Malamos, Rodrigo 14] The loop integrand becomes singular at hyper- boloids with and q (+) i,0 = qq 2 i + m2 i i0 q ( ) i,0 = qq 2 i + m2 i i0 (solid lines) (dashed lines) and origin in k i,µ LT-Duality is equivalent to integrating along the! forward hyperboloids!

15 Singular behavior of the loop integrand [Buchta, Chachamis, Draggiotis, Malamos, Rodrigo 14] The loop integrand becomes singular at hyper- boloids with and q (+) i,0 = qq 2 i + m2 i i0 q ( ) i,0 = qq 2 i + m2 i i0 (solid lines) (dashed lines) and origin in k i,µ LT-Duality is equivalent to integrating along the! forward hyperboloids! Forward-forward intersection! These singularities cancel among! dual integrals

16 Singular behavior of the loop integrand [Buchta, Chachamis, Draggiotis, Malamos, Rodrigo 14] The loop integrand becomes singular at hyper- boloids with and q (+) i,0 = qq 2 i + m2 i i0 q ( ) i,0 = qq 2 i + m2 i i0 (solid lines) (dashed lines) and origin in k i,µ LT-Duality is equivalent to integrating along the! forward hyperboloids! Forward-forward intersection! These singularities cancel among! dual integrals Forward-backward intersection! These singularities remain and require! contour deformation

17 Massless case: G. Sborlini s talk! Singular behavior of the loop integrand [Buchta, Chachamis, Draggiotis, Malamos, Rodrigo 14] The loop integrand becomes singular at hyper- boloids with and q (+) i,0 = qq 2 i + m2 i i0 q ( ) i,0 = qq 2 i + m2 i i0 (solid lines) (dashed lines) and origin in k i,µ LT-Duality is equivalent to integrating along the! forward hyperboloids! Forward-forward intersection! These singularities cancel among! dual integrals Forward-backward intersection! These singularities remain and require! contour deformation

18 Numerical Implementation: Basics L (1) (p 1,p 2,...,p N )= X Z`1 (qi ) NY G D (q i ; q j ) j=1 j6=i

19 Numerical Implementation: Basics L (1) (p 1,p 2,...,p N )= X Z`1 (qi ) NY G D (q i ; q j ) j=1 j6=i

20 Numerical Implementation: Basics L (1) (p 1,p 2,...,p N )= X Z`1 (qi ) NY G D (q i ; q j ) j=1 j6=i Rewrite dual propagator like this (qi )G D (q i ; q j )=2 i (q i,0 q (+) i,0 ) 2q (+) i,0 1 (q (+) i,0 + k ji,0) 2 (q (+) j,0 )2 qq with k ij = q i q j and q (+) i,0 = 2 i + m2 i0 i The resulting N contributions have to be integrated over the loop three-momenta.

21 Numerical Implementation: Basics q (+) i,0 + q(+) j,0 + k ji,0 =0 q (+) i,0 q (+) j,0 + k ji,0 =0 forward-backward forward-forward!!!!!!! 1. The first equation describes an ellipsoid in the loop three-!!! momentum and demands k ji,0 < 0.!!!!!!!! An ellipsoid is the result of the intersection of a forward with a!!!!!!!! backward hyperboloid.! The origins of the hyperboloids are separated in a time-like (light-like) fashion, expressed by the condition:! k 2 ji (m j + m i ) 2 0, k ji,0 < 0 2. The second equation describes a hyperboloid as a result of the intersection of two forward light-cones of space-like (light-l.) separation.! kji,0 2 (m j m i ) 2 apple 0 k ji,0 may be positive or negative:

22 Singularities in Loop-Momentum Space `z `z `z `y `y `y `x `x `x Triangle with one ellipsoid and two hyperboloid singularities

23 Contour deformation One-dimensional example! Shape of the contour deformation f(`x) = `2x 1 E 2 + i0 10 Corresponding deformation: 5 `x! `0x =`x `2 x E 2 Im(lx') 0 +i `x exp 2E 2 E: location of the singularity contour deformation Integration path before defo poles D: `! ` + i ` exp G 2 D width Re(lx')

24 Numerical Implementation: Results Implementation in C++. The code runs on an Intel i7 (3.4GHz) desktop machine.! Finite Triangle, Box, Pentagon with no deformation needed: 4 digits in 0.5s! Pentagon with deformation: 4 digits in 25s Real part Imaginary part Analytic value E E-10 LT Duality (234)E (234)E-14 Numerical result produced with Cuhre (Cuba Library) [Hahn 05],! analytic values by LoopTools [Hahn 99].

25 Numerical Implementation: Results Triangle (all internal masses equal), varying the mass around the threshold:! Red curve is LoopTools, blue points is LTD Re Im Imaginary part Real part m 2m s s 1.2 Red and blue are one top of each other! Good precision close to threshold! 1.4

26 Tensor integrals L (1) (p 1,p 2,...,p N ) = X Z e (qi ) NY G D (q i ; q j )N (`, {p i }) i `1 j=1 j6=i The presence of numerators does not spoil the cancellation of hyperboloid singularities among dual integrals!! q Delta function fixes `0-component to `0 = k i,0 + q 2.! i + m2 i As a direct consequence, the numerator takes a different form for every dual contribution.! Analytic values for hexagons from SecDec 3.0. [Borowka, Heinrich, Jones,! Kerner, Schlenk, Zirke 15] Rank 3! Hexagon Real Part Imaginary Part SecDec (1913)E (560)E-7 LTD (8)E (8)E-7

27 Numerical Implementation: Results Pentagon, varying the Mandelstam variable s (p 1 ), all internal masses different:! Red curve is LoopTools, blue points is LTD Im Re Real part Imaginary part s s Numerator ` p 1 ` p 2 ` p 3 is also varied!

28 Conclusions and outlook The Tree-Loop Duality lets us rewrite loop integrals (scattering amplitudes) as linear combinations of tree-level objects.

29 Conclusions and outlook The Tree-Loop Duality lets us rewrite loop integrals (scattering amplitudes) as linear combinations of tree-level objects. It aims for a holistic approach, treating real and virtual corrections simultaneously in a Monte Carlo event generator.

30 Conclusions and outlook The Tree-Loop Duality lets us rewrite loop integrals (scattering amplitudes) as linear combinations of tree-level objects. It aims for a holistic approach, treating real and virtual corrections simultaneously in a Monte Carlo event generator. Partial cancellation of singularities among Dual Integrals.

31 Conclusions and outlook The Tree-Loop Duality lets us rewrite loop integrals (scattering amplitudes) as linear combinations of tree-level objects. It aims for a holistic approach, treating real and virtual corrections simultaneously in a Monte Carlo event generator. Partial cancellation of singularities among Dual Integrals. Deals well with multi-leg scalar and tensor integrals!

32 Deformation: 1+3 dim 0 1 For each individual ellipsoid include: def = q i p q 2 i + q j q q 2 j A exp G 2 D (q j; q i ) A ij ij A ij scaling factor width of the deformation } Can be chosen differently for each! individual ellipsoid

33 Deformation: 1+3 dim 0 1 For each individual ellipsoid include: def = q i p q 2 i + q j q q 2 j A exp G 2 D (q j; q i ) A ij ij A ij scaling factor width of the deformation } Can be chosen differently for each! individual ellipsoid Sum over the entire group: i~apple = X def j j2group Final deformation: ~`! ~`0 = ~` + i~apple

34 p l p l 1 q l p 1 Extension to two loops q 0 p l 2 q l 1 q 1 q l+1 p l+1 p 2 [Bierenbaum, Catani, Draggiotis, Rodrigo 10] q l 2 q l+2 q 2 8 p l+2 Notation: q i = >< `1 + p 1,i i 2 1 `2 + p >: i,l 1 i 2 2 `1 + `2 + p i,l 1 i 2 3 l 2 q N l 1 p N 1 1 = {0, 1,...,r}, 3 = {l +1,l+2,...,N} G F ( k )= Y 2 = {r +1,r+2,...,l} i2 k G F (q i ), G D ( k )= X i2 k (qi ) Y j2 k j6=i p r+1 G D (q i ; q j ) q r+1 p N q r p r For a set of looplines belonging to the same loop (of a multi loop diagram): Z Z `i G F ( 1 [ 2 [ [ n )= `i G D ( 1 [ 2 [ [ n ) Subsequently apply the LTD to the other loops of the diagram.

35 p l p l 1 q l p 1 q 0 p l 2 q l 1 q 1 q l+1 p 2 Extension to two loops q l 2 q l+2 p l+1 q 2 p l+2 l 2 l 1 p N 1 q N Each application to a loop introduces an extra single cut.! p r+1 q r+1 p r Apply it as many times as there are loop: Opening loops to trees.! p N q r Every application converts Feynman into dual propagators. Since the LTD can only be applied to Feynman P.s, the dual p.s of the unification of several subsets must be reexpressed in terms of dual and Feynman P.s. before going to the next loop.! One loop line might take part in more that one loop, see middle line in graph. Z Z L (2) (p 1,p 2,...,p N )= { G D ( 1 )G F ( 2 )G D ( 3 )+G D ( 1 )G D ( 2 [ 3 )+G D ( 3 )G D ( 1 [ 2 )} `1 `2 Reiterate the procedure for higher order loop integrals.