Numerical Evaluation of Multi-loop Integrals Sophia Borowka MPI for Physics, Munich In collaboration with G. Heinrich Based on arxiv:124.4152 [hep-ph] HP 8 :Workshop on High Precision for Hard Processes, Munich September 5th, 212 http://secdec.hepforge.org S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 1
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 Highly precise theoretical predictions are necessary S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 2
New particle consistent with the Higgs found Announcement of a new particle finding on July 4th 212 Local p-value 1-1 1-2 1 CMS s -1 = 7 TeV, L = 5.1 fb s -1 = 8 TeV, L = 5.3 fb 1σ 2σ -3 1 Observed 3σ ATLAS 12 CMS 12 Exp. for SM Higgs Boson -4 1 7 TeV Observed 4σ 8 TeV Observed -5 1 11 115 12 125 13 135 14 145 15 m H (GeV) S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 3
Making Predictions in the LHC Era At NLO: lot of progress (multi-leg, automation,...) S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 4
Making Predictions in the LHC Era At NLO: lot of progress (multi-leg, automation,...) Beyond NLO: calculations progressing well, but automation is difficult S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 4
Making Predictions in the LHC Era At NLO: lot of progress (multi-leg, automation,...) Beyond NLO: calculations progressing well, but automation is difficult Analytic methods to calculate e.g. two-loop integrals involving massive particles reach their limit S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 4
Making Predictions in the LHC Era At NLO: lot of progress (multi-leg, automation,...) Beyond NLO: calculations progressing well, but automation is difficult 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 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 4
Making Predictions in the LHC Era At NLO: lot of progress (multi-leg, automation,...) Beyond NLO: calculations progressing well, but automation is difficult 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 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 5
Making Predictions in the LHC Era At NLO: lot of progress (multi-leg, automation,...) Beyond NLO: calculations progressing well, but automation is difficult 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 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 6
Making Predictions in the LHC Era At NLO: lot of progress (multi-leg, automation,...) Beyond NLO: calculations progressing well, but automation is difficult 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., Freitas et al., Weinzierl et al.,... S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 6
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) S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 7
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 7
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 8
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 9
Operational Sequence of the SecDec Program 1 2 3 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 1
General Feynman Integral Graph infos are converted into (scalar or contracted tensor) Feynman integral in D dimensions at L loops with N propagators to power ν j of rank R After loop momentum integration, generic scalar Feynman integral reads 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 11
Operational Sequence of the SecDec Program 1 2 3 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
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 + 1 1 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, Denner & Roth 96, Binoth & Heinrich S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 13
Operational Sequence of the SecDec Program 1 2 3 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 14
Contour Deformation I For kinematics in the physical region, F can still vanish F Bubble = s t 1 (1 t 1 ) + m 2 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 15
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., Becker et al. S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 16
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 2.5 Re(F(t 1 )) 2 +Im(F(t 1 )) 2 2 1.5 1.5 example is for 1-loop bubble, m 2 = 1., s = 4.5 with Feynman parameter t 1.2.4.6.8 1 t 1 robust method default: smalldefs=, largedefs= S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 17
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 )).6.5.4.3.2 example is for 1-loop bubble, m 2 = 1., s = 4.5 with Feynman parameter t 1.1.4.42.44.46.48.5.52.54.56.58.6 t 1 Singular points lie far from endpoints ( and 1) of integration region, use smalldefs=1 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 18
Operational Sequence of the SecDec Program 1 2 3 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 19
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 2
Operational Sequence of the SecDec Program 1 2 3 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 21
Results Successful application of SecDec 1. to massless multi-loop diagrams up to 5-loop 2-point functions and 4-loop 3-point functions for Euclidean kinematics Successful application of the public 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 p3 3 4 5 1 2 p1 p2 s/m 2 timing (finite part) 3.9 9.5 secs 14. 3.6 secs S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 22
Results II: Massive Two-loop Vertex Graph G 4 1 p1 4 p3 3 3 5 2 p2 2 Finite real and imaginary part of P126 1-1 -2-3 -4 Real - Analytic -5 Real - SecDec Imag - Analytic Imag - SecDec -6 2 4 6 8 1 12 14 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 23
Results III: Massive Non-planar 6-propagator Graph 5-5 Massive Bnp6, finite part -1-15 -2 m =.5-25 -3 Real (SecDec) Imag (SecDec) -35-1.5-1 -.5.5 1 s 12 massless case: Tausk 99 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 24
.4.3.2.1 -.1 -.2-1 -9-8 -7-6 -5-4 -3-2 fs Im Results IV: 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-13.8.6.4 I(s,t) Re Im -4e-13.2 2 4 6 8 1 12 14 16 s/m 2 -.2-2 -15-1 -5 5 1 15 2 fs I(s,t) Fujimoto et al. 11 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 25
Results V: Non-planar ggtt Contribution p3 p1 m2 p2 m1 3 2 p4 Finite part of non-planar 2L box diagram ggtt1 1-1 -2-3 -4 m 1, m 2 =.5 s 23 =.4-5 Real (SecDec) Imag (SecDec) -6-1 -.5.5 1 1.5 2 2.5 3 s 12 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 26
Install SecDec 2. Download: http://secdec.hepforge.org 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 27
User Input I param.input: parameters for integrand specification and numerical integration S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 28
User Input II template.m: definition of the integrand (Mathematica syntax) p3 3 4 5 1 2 p1 p2 S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 29
Program Test Run./launch -p param.input -t template.m S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 3
Get the Result resultfile P126 full.res S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 31
Conclusion 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,... Outlook Implement contour deformation for more general parametric functions Implement further variable transformation to tackle singularities very close to pinch singularities Application to 2-loop processes involving several mass scales, e.g. QCD/EW/MSSM corrections S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 32