Numerical Evaluation of Multi-loop Integrals

Transcription

1 Numerical Evaluation of Multi-loop Integrals Sophia Borowka MPI for Physics, Munich In collaboration with: J. Carter and G. Heinrich Based on arxiv: [hep-ph] DESY-HU Theory Seminar, Berlin May 24th, 212 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 1

2 The LHC Era has begun We are probing energies which have never been reached at colliders before High experimental precision is possible due to high luminosities Precise theoretical predictions become necessary S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 2

3 Searching for the Higgs ATLAS 12 CMS 12 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 3

4 Higgs Production in gluon fusion Multi-dimensional parameter integrals need to be evaluated which can contain UV, soft and collinear singularities LO (NLO in other processes) NLO NNLO S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 4

5 Higher Order Corrections to the Higgs Production 1 2 σ(pp H+X) [pb] s = 14 TeV 1 LO NLO NNLO M H [GeV] Harlander & Kilgore 2 Anastasiou, Melnikov, Petriello 5 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 5

6 Making Predictions in the LHC Era A lot of progress has been achieved towards the goal of describing hadron collider processes consistently at NLO Calculations beyond NLO are also progressing well, but automation is difficult, and analytic methods to calculate e.g. two-loop integrals involving massive particles reach their limit Numerical methods are in general easier to automate, problems mainly are Extraction of IR and UV singularities Numerical convergence in the presence of integrable singularities (e.g. thresholds) Speed/accuracy Many people are/have been working on purely numerical methods, e.g. Soper/Nagy et al., Binoth/Heinrich et al., Kurihara et al., Passarino et al., Lazopoulos et al., Anastasiou et al., Denner/Pozzorini, Freitas et al., Weinzierl et al., Czakon/Mitov et al.,... S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 6

7 Making Predictions in the LHC Era A lot of progress has been achieved towards the goal of describing hadron collider processes consistently at NLO Calculations beyond NLO are also progressing well, but automation is difficult, and analytic methods to calculate e.g. two-loop integrals involving massive particles reach their limit Numerical methods are in general easier to automate, problems mainly are Extraction of IR and UV singularities (solved with SecDec 1.) Numerical convergence in the presence of integrable singularities (e.g. thresholds) Speed/accuracy Many people are/have been working on purely numerical methods, e.g. Soper/Nagy et al., Binoth/Heinrich et al., Kurihara et al., Passarino et al., Lazopoulos et al., Anastasiou et al., Denner/Pozzorini, Freitas et al., Weinzierl et al., Czakon/Mitov et al.,... S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 7

8 Making Predictions in the LHC Era A lot of progress has been achieved towards the goal of describing hadron collider processes consistently at NLO Calculations beyond NLO are also progressing well, but automation is difficult, and analytic methods to calculate e.g. two-loop integrals involving massive particles reach their limit Numerical methods are in general easier to automate, problems mainly are Extraction of IR and UV singularities (solved with SecDec 1.) Numerical convergence in the presence of integrable singularities (e.g. thresholds) (solved with SecDec 2.) Speed/accuracy Many people are/have been working on purely numerical methods, e.g. Soper/Nagy et al., Binoth/Heinrich et al., Kurihara et al., Passarino et al., Lazopoulos et al., Anastasiou et al., Denner/Pozzorini, Freitas et al., Weinzierl et al., Czakon/Mitov et al.,... S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 8

9 Public Implementations of the Sector Decomposition Method on the Market sector decomposition (uses GiNaC) C. Bogner & S. Weinzierl 7 FIESTA (uses Mathematica, C) A. Smirnov, V. Smirnov & M. Tentyukov 8 9 SecDec (uses Mathematica, Perl, Fortran/C++) J. Carter & G. Heinrich 1 Limitation until recently: Multi-scale integrals were limited to the Euclidean region (i.e., no thresholds) NOW: Extension of SecDec to general kinematics! SB, J. Carter & G. Heinrich 12 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 9

10 SecDec 2. Computes... Feynman graphs for arbitrary kinematics, and more general parametric functions with no poles within the integration region Feynman graph or parametric function S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 1

11 Parametric Functions A general parametric function can be a phase space integral where IR divergences are regulated dimensionally polynomial functions, e.g. hypergeometric functions pf p 1 (a 1,..., a p ; b 1,..., b p 1 ; β) S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 11

12 Operational Sequence of the SecDec Program graph info Feynman integral iterated sector decomposition no multiscale? yes 6 expansion in ǫ 5 subtraction of poles 4 contour deformation 7 8 numerical integration result n Cm ǫ m m= 2L S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 12

13 General Feynman Integral Graph infos are converted into tensorial Feynman integral G µ 1...µ R in D dimensions at L loops with N propagators to power ν j of rank R After loop momentum integration, a generic scalar Feynman integral G = ( 1)Nν N j=1 Γ(ν j) Γ(N ν LD/2) N j=1 dx j x ν j 1 j δ(1 N l=1 x l ) U Nν (L+1)D/2 ( x) F Nν LD/2 ( x) where N ν = N j=1 ν j and where U and F can be constructed via topological cuts S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 13

14 Construction of the Functions U and F Example: 1-loop box p p 4 x 2 x 4 p p 3 U( x) =x 1 + x 2 + x 3 + x 4 (L-line cut), F ( x) =s 12 x 1 x 3 + s 23 x 2 x 4 + p 2 1x 1 x 2 + p 2 2x 2 x 3 + p 2 3x 3 x 4 + p 2 4x 1 x 4 (L+1-line cut) and F( x) = F ( x) + U( x) N x j mj 2 iδ j=1 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 14

15 Operational Sequence of the SecDec Program graph info Feynman integral iterated sector decomposition no multiscale? yes 6 expansion in ǫ 5 subtraction of poles 4 contour deformation 7 8 numerical integration result n Cm ǫ m m= 2L S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 15

16 Sector Decomposition Overlapping divergences are factorized 1 dx x 1 dy y 1 (x + y) 2+ɛ = (1) + (2) 1 dx 1 dt 1 x 1+ɛ (1 + t) 2+ɛ + x t dt dy t y 1 y 1+ɛ (1 + t) 2+ɛ Iterated sector decomposition is done, where dimensionally regulated soft, collinear and UV singularities are factored out Hepp 66, Binoth & Heinrich S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 16

17 Operational Sequence of the SecDec Program graph info Feynman integral iterated sector decomposition no multiscale? yes 6 expansion in ǫ 5 subtraction of poles 4 contour deformation 7 8 numerical integration result n Cm ǫ m m= 2L S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 17

18 Contour Deformation I For kinematics in the physical region, F can still vanish F Bubble = m 2 (1 + t 1 ) 2 s t 1 iδ but a deformation of the integration contour Im(z) 1 Re(z) and Cauchy s theorem can help 1 f (t)dt = f (t)dt + c 1 z(t) f (z(t))dt = t S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 18

19 Contour Deformation II Im(z) The integration contour is deformed by 1 Re(z) t z = t + i y, y j ( t)= λt j (1 t j ) F( t) t j Soper 99 Integrand is analytically continued into the complex plane F( t) F( t + i y( t)) = F( t) + i j y j ( t) F( t) t j + O(y( t) 2 ) Soper, Nagy, Binoth; Kurihara et al., Anastasiou et al., Freitas et al., Weinzierl et al. S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 19

20 Find the Optimal Deformation Parameter λ I Robust method: check the maximally allowed λ for F and maximize the modulus at critical points 3 lambda = 1.2 lambda = 1. lambda =.5 Re(F(t 1 )) 2 +Im(F(t 1 )) example is for 1-loop bubble, m 2 = 1., s = 4.5 with Feynman parameter t t 1 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 2

21 Find the Optimal Deformation Parameter λ II Faster convergence: minimize the complex argument of F.8.7 lambda = 1.2 lambda = 1. lambda =.5 Im(F(t 1 ))/Re(F(t 1 )) example is for 1-loop bubble, m 2 = 1., s = 4.5 with Feynman parameter t t 1 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 21

22 Operational Sequence of the SecDec Program graph info Feynman integral iterated sector decomposition no multiscale? yes 6 expansion in ǫ 5 subtraction of poles 4 contour deformation 7 8 numerical integration result n Cm ǫ m m= 2L S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 22

23 Subtraction, Expansion, Numerical Integration Subtraction The factorized poles in a subsector integrand I U, F are extracted by subtraction (e.g. logarithmic divergence) 1 Expansion dt j t 1 b j ɛ I(, ɛ) j I(t j, ɛ) = b j ɛ + 1 dt j t 1 b j ɛ j (I(t j, ɛ) I(, ɛ)) After the extraction of poles, an expansion in the regulator ɛ is done Numerical Integration Monte Carlo integrator programs containted in CUBA library or BASES can be used for numerical integration Hahn et al. 4 11, Kawabata 95 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 23

24 Operational Sequence of the SecDec Program graph info Feynman integral iterated sector decomposition no multiscale? yes 6 expansion in ǫ 5 subtraction of poles 4 contour deformation 7 8 numerical integration result n Cm ǫ m m= 2L S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 24

25 Results Successful application of the public SecDec 1. program to massless multi-loop diagrams up to 5-loop 2-point functions and 4-loop 3-point functions for Euclidean kinematics Successful application of SecDec 2. to various multi-scale examples, e.g., the massive 2-loop vertex graph, planar and non-planar 6- and 7-propagator massive 2-loop box diagrams Timings for the 2-loop vertex diagram and a relative accuracy of 1% using the CUBA 3. library on an Intel(R) Core i7 CPU at 2.67GHz p p1 p2 s/m 2 timing (finite part) secs secs S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 25

26 Results II: Massive Two-loop Vertex Graph G 4 1 p1 4 p p2 2 Finite real and imaginary part of P Real - Analytic -5 Real - SecDec Imag - Analytic Imag - SecDec s Kotikov et al. 97, Davydychev & Kalmykov 4, Ferroglia et al. 4, Bonciani et al. 4 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 26

27 Results III: Massive Non-planar 6-propagator Graph 5-5 Massive Bnp6, finite part m Real (SecDec) Imag (SecDec) s 12 massless case: Tausk 99 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 27

28 Results IV: Planar Massive Two-loop Box 3e e-12 m=5,m=9,s 23 =-1 Real (SecDec) Imag (SecDec) 2e-12 Finite part of planar 2-loop box 1.5e-12 1e-12 5e-13-5e-13-1e e fs Fujimoto et al. 11 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 28

29 fs Im Results V: Non-planar Massive Two-loop Box 1.2e-12 1e-12 m=5,m=9,s 23 =-1 Real (SecDec) Imag (SecDec) Finite part of massive non-planar 2-loop box 8e-13 6e-13 4e-13 2e-13-2e I(s,t) Re Im -4e s/m fs I(s,t) Fujimoto et al. 11 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 29

30 Install SecDec 2. Download: Install: tar xzvf SecDec.tar.gz cd SecDec-2../install Prerequisites: Mathematica (version 6 or above), Perl, Fortran and/or C++ compiler S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 3

31 User Input I param.input: parameters for integrand specification and numerical integration S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 31

32 User Input II template.m: definition of the integrand (Mathematica syntax) p p1 p2 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 32

33 Program Test Run./launch -p param.input -t template.m S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 33

34 Get the Result resultfile P126 full.res S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 34

35 Summary With SecDec the numerical evaluation of multi-loop integrals is possible for arbitrary kinematics SecDec can also be used for more general parametric functions (e.g. phase space integrals) Useful to check analytic results for multi-loop master integrals, e.g. 2-loop boxes, 3-loop form factors,... Download the public SecDec program: S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 35

36 Outlook Can be used to calculate 2-loop corrections involving several mass scales, e.g. QCD/EW/MSSM corrections Implement contour deformation for more general parametric functions Include option to use Mathematica s numerical integrator Implement further variable transformation to tackle singularities very close to pinch singularities S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 36