Numerical Evaluation of Loop Integrals

Transcription

1 Numerical Evaluation of Loop Integrals Institut für Theoretische Physik Universität Zürich Tsukuba, April 22 nd 2006 In collaboration with Babis Anastasiou

2 Rationale (Why do we need complicated loop amplitudes?) Perturbative calculations play a key role in understanding and predicting phenomena in particle physics Precise determination of parameters New phenomena New phenomena parameters omplicated backgrounds parameters Perturbative calculations for processes with higher particle multiplicities, number of loops and kinematical scales

3 Difficulties and challenges omplicated analytic continuation of results when many scales are present Analytic structure (kinematics) Infrared singularities Tensor numerators Integrals must be regularized and singularities etracted Proliferate the number of terms

4 Difficulties and challenges omplicated analytic continuation of results when many scales are present Analytic structure (kinematics) Infrared singularities Tensor numerators Integrals must be regularized and singularities etracted Proliferate the number of terms...and we would like automatization

5 Lots of inspired work - Reduction to master integrals Laporta, Anastasiou, Lazopoulos, A.V. Smirnov, V.A. Smirnov, Tarasov, Baikov, Steinhauser,... - Sophisticated methods to reduce one-loop integrals and handling eceptional kinematics Giele, Glover, R.K. Ellis, Zanderighi, Denner, Dittmaier, Binoth, Heinrich,Pilon, Schubert, Kauer, Hameren, Vollinga, Weinzerl, del Aguila, Pittau, Sopper, Nagy,... - Sector decomposition: numerical evaluation in the euclidean region Binoth, Heinrich,Pilon, Schubert, Kauer, Anastasiou, Melnikov, Petriello - Twistor developments Britto, Buchbinder, achazo, Feng, Witten, Bern, Dion, Kosower,... - Mellin-Barnes representations zakon, V.A. Smirnov, Tausk, Veretin, Anastasiou, Tejeda-Yeomans, Heinrich, Davydychev, Ussyukina,... Thursday s talk by G. Salam

6 New method for the numerical evaluation of loop integrals using Mellin-Barnes representations

7 Loop integrals and Mellin-Barnes representations I 4 = Γ ( 4 d ) 2 d 1 d 2 d 3 d 4 ( s 1 3 t 2 4 i0) 4 d 2

8 Loop integrals and Mellin-Barnes representations I 4 = Γ ( 4 d ) 2 d 1 d 2 d 3 d 4 ( s 1 3 t 2 4 i0) 4 d 2 1 (a + b) α = b α 1 2πi dw ( a b ) w Γ( w) Γ(α + w) Γ(α) separates green and red series of poles Sum of residues reproduces the series epansion arg a arg b < π All regions in s-t plane w

9 Loop integrals and Mellin-Barnes representations I 4 = Γ ( 4 d ) 2 d 1 d 2 d 3 d 4 ( s 1 3 t 2 4 i0) 4 d 2 1 (a + b) α = b α 1 2πi dw ( a b ) w Γ( w) Γ(α + w) Γ(α) separates green and red series of poles Sum of residues reproduces the series epansion arg a arg b < π All regions in s-t plane w

10 Loop integrals and Mellin-Barnes representations I 4 = Γ ( 4 d ) 2 d 1 d 2 d 3 d 4 ( s 1 3 t 2 4 i0) 4 d 2 1 (a + b) α = b α 1 2πi dw ( a b ) w Γ( w) Γ(α + w) Γ(α) separates green and red series of poles Sum of residues reproduces the series epansion arg a arg b < π All regions in s-t plane w

11 Loop integrals and Mellin-Barnes representations I 4 = Γ ( 4 d ) 2 d 1 d 2 d 3 d 4 ( s 1 3 t 2 4 i0) 4 d 2 I 4 = ( ) 4 d l l=1 1 2πi dwγ( w)γ(2 + ɛ + w) ( 2 4 t) w ( 1 3 s) 2 ɛ w w - contour separates poles - Feynman parameters factorize and can be integrated out

12 Loop integrals and Mellin-Barnes representations I 4 = Γ ( 4 d ) 2 d 1 d 2 d 3 d 4 ( s 1 3 t 2 4 i0) 4 d 2 I 4 = 1 2πi Γ( 2ɛ) The integral only picks up the right residues if the contour separates red and green poles: 1 < w < 0 2 w < ɛ < 1 w < 0 the representation is only valid for ɛ < 0 but we need a Laurent series around ɛ = 0... I 4 = 2 ɛ ɛ dwγ( w)γ(2 + ɛ + w)γ 2 (1 + w)γ 2 ( 1 ɛ w) ( t) w ( s) 2 ɛ w w 1 2

13 Loop integrals and Mellin-Barnes representations I 4 = Γ ( 4 d ) 2 d 1 d 2 d 3 d 4 ( s 1 3 t 2 4 i0) 4 d 2 I 4 = 1 2πi Γ( 2ɛ) The integral only picks up the right residues if the contour separates red and green poles: 1 < w < 0 2 w < ɛ < 1 w < 0 the representation is only valid for ɛ < 0 but we need a Laurent series around ɛ = 0... I 4 = 2 ɛ ɛ dwγ( w)γ(2 + ɛ + w)γ 2 (1 + w)γ 2 ( 1 ɛ w) ( t) w ( s) 2 ɛ w w 1 2

14 Loop integrals and Mellin-Barnes representations I 4 = Γ ( 4 d ) 2 d 1 d 2 d 3 d 4 ( s 1 3 t 2 4 i0) 4 d 2 I 4 = 1 2πi Γ( 2ɛ) The integral only picks up the right residues if the contour separates red and green poles: 1 < w < 0 2 w < ɛ < 1 w < 0 the representation is only valid for ɛ < 0 but we need a Laurent series around ɛ = 0... I 4 = 2 ɛ ɛ dwγ( w)γ(2 + ɛ + w)γ 2 (1 + w)γ 2 ( 1 ɛ w) ( t) w ( s) 2 ɛ w w 1 2

15 Loop integrals and Mellin-Barnes representations w w w 2 1 = Original representation 3/2 < ɛ < 1/2 Similar to the original 1/2 < ɛ < 1/2 Single residue 1/2 < ɛ < 1/2 Simpler than original integral

16 Loop integrals and Mellin-Barnes representations w w w 2 1 = Allows series epansion in ɛ Original representation 3/2 < ɛ < 1/2 I 4 = ɛ πi Similar to the original 1/2 < ɛ < 1/2 Single residue 1/2 < ɛ < 1/2 Simpler than original integral dwγ( w)γ(2 + w)γ 2 (1 + w)γ 2 ( 1 w) ( t) w ( s) 2 w + O(ɛ 2 ) + ( t) ɛ s t Γ( ɛ) 2 Γ(1 + ɛ) Γ( 2ɛ) {γ E log( s) + log( t) + 2ψ( e) ψ(1 + e)}

17 Loop integrals and Mellin-Barnes representations - Iterative procedure, until reaching ɛ = 0. - Poles in ɛ appear eplicitly when taking residues. - Remaining integrals can be epanded in Laurent series. - Straightforward to etend to more complicated loop integrals. - alculation of remaining integrals: analytically, series resummation very-well suited for numerical integration - Whole procedure implemented in MATHEMATIA and MAPLE I 4 = ɛ πi dwγ( w)γ(2 + w)γ 2 (1 + w)γ 2 ( 1 w) ( t) w ( s) 2 w + O(ɛ 2 ) + ( t) ɛ s t Γ( ɛ) 2 Γ(1 + ɛ) Γ( 2ɛ) {γ E log( s) + log( t) + 2ψ( e) ψ(1 + e)}

18 Evaluation of integrals - Analytic evaluation implies the infinite sum over residues etremely difficult for multidimensional MB integrals involved analytic continuation in kinematical variables - Numerical integration on the other hand: rapid damping of integrand along the contour when Im(w) ± thanks to etra gamma functions coming from Feynman par. integration trivial analytic continuation in kinematical variables: only powers and logs I 4 = ɛ πi dwγ( w)γ(2 + w)γ 2 (1 + w)γ 2 ( 1 w) ( t) w ( s) 2 w + O(ɛ 2 ) + ( t) ɛ s t Γ( ɛ) 2 Γ(1 + ɛ) Γ( 2ɛ) {γ E log( s) + log( t) + 2ψ( e) ψ(1 + e)}

19 Tensor integrals d d k k µ 1 I n,m =... kµ m i π d 2 (k + q 1 ) 2 (k + q 2 ) 2 (k + q n ) 2 = ( ) dw i Γ (scalar) d d+r ( w) h(m,r) ( w) r m i Γ (scalar) d d+r ( w): analogous to the representation of the scalar case with shifted dimension d d + r h (m,r) ( w): polynomial in the MB variables with tensor coefficients (eternal momenta). Does not affect the analytic continuation in ɛ -This decomposition allows to perform a unique analytic continuation in ɛ for each topology, using a general polynomial. -Whole diagrams can be evaluated at the same time.

20 1 loop massless heagon - Many important processes with si eternal legs at the LH - Recent calculation of 6 gluons amplitude [Ellis, Giele, Zanderighi] - Our calculation: 8 dimensional MB representation analytic continuation for arbitrary tensors sampled several points in the physical region 2 4 check in the euclidean region with sector decomposition check some phase space points with eplicit reductions using IBP identities - Evaluation of tensors up to rank 6 - Ready to evaluate whole diagrams

21 A rank 6 tensor in the physical region q 2 k q 2 k q 3 k q 4 k q 5 k q 6 k Point c 2 c 1 c 0 P (0) + i (0) P (0) + i (0) P (0) + i (0) P (0) + i (0) (6) i (4) (2) i (2) (2) i (2) (4) i (7) 79. (2.) i 485. (2.) (6.) i (5.) 214. (2.) i 663. (1.) 26. (2.) i 531. (2.)

22 2 loop boes - planar double bo with one off-shell leg test of analytic continuation in ɛ test of numerical integration feasibility effective 3-fold integral V.A. Smirnov (MB), Gehrmann and Remiddi (diff. eq.) non-trivial check of analytic continuation in kin. var. - planar double bo with two adjacent off-shell legs first calculation in the physical region relevant for heavy boson pair production at colliders euclidean points by Binoth and Heinrich with sec. dec. effective 3-fold integral good numerical convergence

23 2-Bo: 1 off-shell leg numeric/analytic numeric/analytic s 23 M 2 = 1 s 13 = 3/10 M 2 = 1/10 s = t

24 2-Bo: 2 off-shell legs Re(c 0 ) Im(c 0 ) t s = 1 M 2 1 = 1/20 M 2 2 = 1/ t

25 3 loop boes - on-shell triple bo analytic result by V.A. Smirnov with MB technique all physical regions poles up to e 6 effective 5-fold integral - triple bo with one off-shell leg first evaluation of a 3 loop bo with 3 scales evaluation in different physical regions production of a heavy boson in association with a jet effective 6-fold integral

26 3-Bo: on-shell Re( c 0 ) numeric/analytic t t

27 3-Bo: 1 off-shell leg Re(c 0 ) s 23 Re(c 0 ) t 10 2 Im(c 0 ) Im(c 0 ) s M 2 = 1 s 13 = 3/10 M 2 = 1/10 s = 1 t

28 Summary - Framework for the numerical evaluation of loop integrals using Mellin-Barnes representations algorithmic etraction of infrared singularities very well suited for multi-loop multi-scale problems all phase space regions without cumbersome analytic continuations generalization for efficient evaluation of tensor integrals direct numeric integration of contour integrals - Full automatization of the whole procedure - Numerical integration: possible to evaluate numerically with good accuracy fast and compact code

29 Summary - Framework for the numerical evaluation of loop integrals using Mellin-Barnes representations algorithmic etraction of infrared singularities very well suited for multi-loop multi-scale problems all phase space regions without cumbersome analytic continuations generalization for efficient evaluation of tensor integrals direct numeric integration of contour integrals - Full automatization of the whole procedure - Numerical integration: possible to evaluate numerically with good accuracy fast and compact code - Novel results: evaluation of high rank tensors for the 1 loop heagon first evaluation of two loop bo with two adjacent massive legs first evaluation of three loop bo with one massive leg