Numerical evaluation of multi-scale integrals with SecDec 3

Transcription

1 Numerical evaluation of multi-scale integrals with SecDec 3 Sophia Borowka University of Zurich Project in collaboration with G. Heinrich, S. Jones, M. Kerner, J. Schlenk, T. Zirke [hep-ph] (CPC, in press) Amplitudes 215, Zurich, July 6 th, 215 S. Borowka (University of Zurich) SecDec-3. 1

2 Precise theoretical predictions in the LHC era A lot of progress has been achieved towards the goal of describing hadron collider processes consistently at NLO S. Borowka (University of Zurich) SecDec-3. 2

3 Precise theoretical 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 multiple mass scales still need to be improved S. Borowka (University of Zurich) SecDec-3. 2

4 Precise theoretical 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 multiple mass scales still need to be improved Numerical methods are in general easier to automate, multiple scales do not pose a problem S. Borowka (University of Zurich) SecDec-3. 2

5 Precise theoretical 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 multiple mass scales still need to be improved Numerical methods are in general easier to automate, multiple scales do not pose a problem highly interesting in light of the current need for predictions involving massive particles! S. Borowka (University of Zurich) SecDec-3. 2

6 Numerical evaluation of Feynman integrals Many people are/have been working on purely numerical methods, e.g. Anastasiou/Beerli/Kunszt et al., Becker/Reuschle/Weinzierl et al., Binoth/Heinrich et al., Boughezal/Melnikov/Petriello et al., Czakon et al., Freitas et al., Kurihara et al., Nagy/Soper et al., Passarino et al.,... Problems beyond the one-loop level mainly are Extraction of IR and UV singularities Numerical convergence in the presence of integrable singularities (e.g. thresholds) Speed / accuracy S. Borowka (University of Zurich) SecDec-3. 3

7 Numerical evaluation of Feynman integrals Many people are/have been working on purely numerical methods, e.g. Anastasiou/Beerli/Kunszt et al., Becker/Reuschle/Weinzierl et al., Binoth/Heinrich et al., Boughezal/Melnikov/Petriello et al., Czakon et al., Freitas et al., Kurihara et al., Nagy/Soper et al., Passarino et al.,... Problems beyond the one-loop level 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 (University of Zurich) SecDec-3. 4

8 Numerical evaluation of Feynman integrals Many people are/have been working on purely numerical methods, e.g. Anastasiou/Beerli/Kunszt et al., Becker/Reuschle/Weinzierl et al., Binoth/Heinrich et al., Boughezal/Melnikov/Petriello et al., Czakon et al., Freitas et al., Kurihara et al., Nagy/Soper et al., Passarino et al.,... Problems beyond the one-loop level 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 (University of Zurich) SecDec-3. 5

9 Numerical evaluation of Feynman integrals Many people are/have been working on purely numerical methods, e.g. Anastasiou/Beerli/Kunszt et al., Becker/Reuschle/Weinzierl et al., Binoth/Heinrich et al., Boughezal/Melnikov/Petriello et al., Czakon et al., Freitas et al., Kurihara et al., Nagy/Soper et al., Passarino et al.,... Problems beyond the one-loop level 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 (further improved in SecDec 3) S. Borowka (University of Zurich) SecDec-3. 6

10 IR and UV singularity extraction beyond 1-loop Diverse methods have been worked out R*-operation Chetyrkin, Tkachov, V.A.Smirnov 7s, 8s Polynomial exponent raising Tkachov 96, Passarino Sector decomposition Binoth & Heinrich Computation of residues within Mellin-Barnes representation Anastasiou, Daleo 6; Czakon 6 Subtraction terms Freitas 12; Becker, Weinzierl 12 Quasi-finite basis Panzer 14; Manteuffel, Schabinger, Panzer 14 + important other works on UV renormalization Bogoliubov, Parasiuk, Hepp, Zimmermann, Broadhurst, Kreimer, Connes,... S. Borowka (University of Zurich) SecDec-3. 7

11 The method of sector decomposition Idea and method of sector decomposition pioneered by Hepp 66, Denner & Roth 96, Binoth & Heinrich 1 x 1 1 dx 1 dx 2 (x 1 + x 2 ) 2+ɛ 1 = = = 1 1 y (1) + (2) 1 1 dx 1 dx 2 (x 1 + x 2 ) 2+ɛ (θ(x 1 x 2 ) + θ(x 2 x 1 )) x1 1 dx 1 dx 2 (x 1 + x 2 ) 2+ɛ + 1 dx 1 dt x 1 (x 1 + x 1 t) 2+ɛ x t x2 1 dx 2 dx 1 (x 1 + x 2 ) 2+ɛ 1 dx 2 d t + t 1 x2 1+ɛ ( t + 1) 2+ɛ iterative sector decomposition is highly automatable S. Borowka (University of Zurich) SecDec-3. 8 y

12 Public codes using the sector decomposition method Public codes: sector decomposition (uses GiNaC) Bogner & Weinzierl 7 supplemented with CSectors Gluza, Kajda, Riemann, Yundin 1 for construction of integrand in terms of Feynman parameters FIESTA* (uses Mathematica, C) A.V. Smirnov, V.A. Smirnov, Tentyukov 8 9, A.V. Smirnov 13 SecDec* (uses Mathematica, Fortran/C++) Carter & Heinrich 1; SB, Carter, Heinrich 12; SB & Heinrich 13; SB, Heinrich, Jones, Kerner, Schlenk, Zirke 15 *Multi-scale integrals not limited to the Euclidean region SB, J. Carter & G. Heinrich 12; A.V. Smirnov 13 S. Borowka (University of Zurich) SecDec-3. 9

13 SecDec 3 can tackle... Feynman graph or parametric function SecDec is a tool to numerically compute General Feynman integrals for arbitrary kinematics and with numerators Integrals matching a Feynman integral structure More general parametric functions S. Borowka (University of Zurich) SecDec-3. 1

14 Feynman loop integrals Scalar multi-loop integral in Feynman parametrization 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, s ij ) with N ν = N j=1 ν j in D dimensions with L loops, N propagators to power ν j Feynman integrals with (contracted) numerators of rank R U and F can be constructed via topological cuts or by specifying the individual propagators in momentum space S. Borowka (University of Zurich) SecDec-3. 11

15 Modified Feynman loop integrals More general user-defined polynomial integrals matching the Feynman loop integral structure G user = P(ε) 1 N a dx j x j (ε) j N ( x, s ij, ε) U expou(ε) ( x, s ij ) F expof(ε) ( x, s ij ) j=1 with a prefactor P and a numerator function N and exponents a j U and F can have negative exponents, also a j < allowed integrals without δ-constraint F is used for construction of deformation of the integration contour in the physical region user has more responsibility when using deformation of integration contour S. Borowka (University of Zurich) SecDec-3. 12

16 Multi-dimensional parameter integrals A general parametric function can be a phase space integral where IR divergences are regulated dimensionally, e.g. dφ D ME 2 ds 13 ds 23 s13 1 ε F(s 13, s 23 ) s 13 + s ε F(x, y) dx dy x x + y functions of the type of hypergeometric functions, e.g. 3F 2 (a 1,..., a 3 ; b 1, b 2 ; β) 1 dxdy x a1 1 (1 x) b1 a1 1 y a2 1 (1 y) b2 a2 1 (1 βxy) a3 NEW in SecDec 3: additional ε-dependent functions g(ε, x) can be included (no iterated sector decomposition applied) S. Borowka (University of Zurich) SecDec-3. 13

17 The program - Outline 1a 2 graph info Feynman integral 1b user defined function 3 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 (University of Zurich) SecDec-3. 14

18 Operational sequence of the SecDec 3 program 1a 2 graph info Feynman integral 1b user defined function no 3 iterated sector decomposition 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 (University of Zurich) SecDec-3. 15

19 Improved user interface in SecDec 3 SecDec needs 3 input files: param.input: minimal info needed to run SecDec math.m: graph definition, enhanced flexibility kinem.input: contains point name and numerical values for kinematics S. Borowka (University of Zurich) SecDec-3. 16

20 Operational sequence of the SecDec 3 program 1a 2 graph info Feynman integral 1b user defined function no 3 iterated sector decomposition 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 (University of Zurich) SecDec-3. 17

21 Sector decomposition algorithms Iteration of the sector decomposition leads to extraction of IR and UV divergences Heuristic algorithm (Binoth & Heinrich, strategy X) so far most efficient one, included since SecDec-1. Infinite recursion may appear Other strategies avoid infinite recursion Bogner, Weinzierl 7 8, A. Smirnov, Tentyukov 8, Kaneko, Ueda 9 1 For complicated examples: lead to more decomposed sectors or functions of higher complexity NEW in SecDec-3 Geometric strategy by Kaneko and Ueda (G1) Geometric strategy Kaneko and Ueda combined with Cheng-Wu theorem (G2) S. Borowka (University of Zurich) SecDec-3. 18

22 Sector decomposition strategy G2 Exploits Cheng-Wu theorem Cheng, Wu 87: resolve δ-constraint of Feynman integral for one variable, integrate all other Feynman parameters to infinity Then perform the sector decomposition using computational geometry Kaneko, Ueda 9 1 Calculate Newton polytope of F U N If vertex lies in more than N 1 facets of the polytope, a triangulation is performed (with Normaliz Bruns, Ichim, Roemer, Soeger) Perform a change of variables to map upper bound of N 1 Feynman parameters to 1 S. Borowka (University of Zurich) SecDec-3. 19

23 Comparison of decomposition strategies Diagram Strategy X Strategy G1 Strategy G2 282 sectors 1 s 368 sectors 1 s 266 sectors 8 s 36 sectors 9 s 166 sectors 4 s 235 sectors 5 s sectors 3 s 56 sectors 15 s 34 sectors 4 s infinite recursion 72 sectors 5 s 76 sectors 1 s sectrs 551 s 3263 sectrs s sectrs 443 s S. Borowka (University of Zurich) SecDec-3. 2

24 Operational sequence of the SecDec 3 program 1a 2 graph info Feynman integral 1b user defined function no 3 iterated sector decomposition 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 (University of Zurich) SecDec-3. 21

25 Extension to physical kinematics For kinematics in the physical region, F can still vanish after sector decomposition 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 (University of Zurich) SecDec-3. 22

26 Deformation of the integration contour to integrate Im(z) mass thresholds Integrand is analytically continued into the complex plane 1 Re(z) F( t) F( t + i y( t)) = F( t) + i j y j ( t) F( t) t j + O(y( t) 2 ) The integration contour is deformed by t z = t + i y, y j ( t)= λt j (1 t j ) F( t) t j Soper 99 Soper, Nagy; Binoth; Anastasiou/Beerli/Kunszt et al., Kurihara et al., Freitas et al., Becker/Reuschle/Weinzierl et al. S. Borowka (University of Zurich) SecDec-3. 23

27 Operational sequence of the SecDec 3 program 1a 2 graph info Feynman integral 1b user defined function no 3 iterated sector decomposition 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 (University of Zurich) SecDec-3. 24

28 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 with Integrators in Cuba-4 library Hahn et al BASES Kawabata 95 NEW in SecDec 3: CQuad Gonnet 1 (fastest for 1-dim), NIntegrate Wolfram Research S. Borowka (University of Zurich) SecDec-3. 25

29 Summary of new features in SecDec version 3 SB, Heinrich, Jones, Kerner, Schlenk, Zirke 15 Two additional decomposition algorithms based on computational geometry (avoid infinite recursion) Numerators can be given in terms of inverse propagators Linear propagators can be treated ε-dependent symbolic functions allowed in parametric integrals Restructured user input helps interfacing with reduction programs 2 new integrators included CQuad, NIntegrate Usage of batch systems facilitated, scans over parameter ranges accelerated Internal structure largely rewritten SecDec is ready for large-scale applications! S. Borowka (University of Zurich) SecDec-3. 26

30 Download SecDec 3 S. Borowka (University of Zurich) SecDec-3. 27

31 Install SecDec 3 Install: tar xzvf SecDec-3..7.tar.gz cd SecDec-3..7 make (make check) Prerequisites: Mathematica (version 7 or above), Perl, Fortran and/or C++ compiler, Normaliz Bruns, Ichim, Roemer, Soeger for usage of geometric decomposition strategies S. Borowka (University of Zurich) SecDec-3. 28

32 Selection of applications - Outline 1) Master integrals 2) Large(r)-scale applications 3) Miscellaneous S. Borowka (University of Zurich) SecDec-3. 29

33 Master Integrals S. Borowka (University of Zurich) SecDec-3. 3

34 Non-planar 2L box with 2 mass scales s = (p 1 + p 2 ) 2 m1 2 = 1, m2 2 =.522, p2 3 = p2 4 = m2 2, p2 1 = p2 2 =, t = m1 p3 p4 p2 p1 m1=1,m2=mh/mt=125/173, t= m1=1,m2=mh/mt=125/173, t= Re[HHNP1].2 Im[HHNP1] Real part s s Imaginary part timings: secs (CPU time), rel. accuracy: 1 3, abs. accuracy: 1 5 SB & Heinrich Nov 14 S. Borowka (University of Zurich) SecDec-3. 31

35 All-massive planar 7-propagator 2L box 13 independent mass scales, full numerical approach Many scales are not a bottleneck Finite part Finite part of planar 2-loop box Real (SecDec) Imag (SecDec) s 12 s 12 = (p 1 + p 2 ) 2 p1 p2 m1 m3 m2 m4 m7 m5 m 2 1 = 2, m2 2 = 6, m 2 3 = 7, m2 4 = 8, m 2 5 = 9, m2 6 = 1, m 2 7 = 12, p2 1 = 1, p 2 2 = 3, p2 3 = 4, p 2 4 = 5, s 23 =.25 timings: 1-8 secs (SecDec 2), rel. accuracy 1 3, abs. accuracy: 1 8 s 12 + s 23 + s 13 = ( 4 i=1 p i ) 2 SB Jun 14 S. Borowka (University of Zurich) SecDec m6 p4 p3

36 Top-quark pair NNLO Two of most complicated 2-loop diagrams: p 4 p 4 m 2 p 2 p 1 m 1 p 1 p 2 m 1 p 3 p 3 (a) ggtt1 (b) ggtt2 ggtt1: enters heavy fermionic corrections: no analytical result available fully numerical approach easy ggtt2: enters light fermionic corrections: more complicated infrared singularity structure, spurious divergences, numerical cancellations pure numerical approach difficult mixed approach: analytical preparation SB & Heinrich Mar 13 S. Borowka (University of Zurich) SecDec-3. 33

37 Results for the non-planar massive 2L-diagram ggtt1 s 12 = (p 1 + p 2 ) 2 p1 m2 p2 m1 p4 m 2 1 = m2 2 = 1, p2 3 = p2 4 = m2 1, p2 1 = p2 2 =, s 23 = 1.25 p3 5 Non-planar scalar massive 2L box diagram ggtt1 2 Non-planar rank 2 massive 2L box diagram ggtt1 1 Finite real and imaginary part Finite real and imaginary part Real (SecDec) Imag (SecDec) s 12 Scalar integral -3 Real (SecDec) Imag (SecDec) s 12 Rank 2 integral timings (SecDec 2): secs (scalar), 5-7 secs (rank 2), rel. accuracy: 1 3, abs. accuracy: 1 5 SB & Heinrich Mar 13 S. Borowka (University of Zurich) SecDec-3. 34

38 Results for the non-planar massive 2L-diagram ggtt2 mixed analytical & numerical approach 1 Real (SecDec) Imag (SecDec) p4 Finite part Finite part of non-planar 2L box diagram ggtt s s = (p 1 + p 2 ) 2 p1 m 2 1 = 1, p2 p 2 1 = p2 2 =, p 2 3 = p2 4 = m2 1, s 23 = 1.25 timings (SecDec 2): 25-4 secs, rel. accuracy 5 1 3, abs. accuracy: 1 5 analytic results: Manteuffel & Studerus Sep 13 SB & Heinrich Mar 13 m1 p3 S. Borowka (University of Zurich) SecDec-3. 35

39 Massive 2-loop master integrals S. Borowka (University of Zurich) SecDec-3. 36

40 Off-shell 2-loop box master integrals S. Borowka (University of Zurich) SecDec-3. 37

41 Large(r)-scale projects S. Borowka (University of Zurich) SecDec-3. 38

42 Higgs-boson self-energy diagrams for O(α s α t ) p 2 = result: Heinemeyer, Hollik, G. Weiglein 98 p 2 result: SB, Hahn, Heinemeyer, Heinrich, Hollik Apr 14; Degrassi, Di Vita, Slavich Oct 14 Tensor reduction with TwoCalc Weiglein et al. 93 & FormCalc Hahn et al Numerical evaluation of momentumdependent integrals with a preliminary version of SecDec 3 φ = h, H, A S. Borowka (University of Zurich) SecDec-3. 39

43 Numerical evaluation of momentum-dependent integrals 34 mass configurations run with SecDec, e.g. Scalar 2L bubble with m 1 =m 2 =173.2, m 2 =1173.2, m 3 =15. Scalar 2L bubble with m 1 = m 2 =173.2, m 3 =826.8, m 4 =15. 5 Real (SecDec) Imag (SecDec).5 Real (SecDec) Imag (SecDec) Real and imaginary finite part of T Real and imaginary finite part of T e+6 p 2 T 1234, finite part e+6 p 2 T 11234, finite part differences of kinematic invariants of up to 14 orders of magnitude rel. accuracy better than 1 5, timings range from.1 1 secs S. Borowka (University of Zurich) SecDec-3. 4

44 Full 2-loop process: gg HH Higgs-boson pair production in gluon fusion interesting for measurement of Higgs-boson self-coupling LO (1-loop) known Glover, van der Bij 88 NLO in m t limit Plehn, Spira, Zerwas 96; Dawson, Dittmaier, Spira 98 NLO with m t but supplemented with 1/m t expansion Grigo, Hoff, Melnikov, Steinhauser 13 NNLO in m t limit De Florian, Mazzitelli 13 NNLO m t with all matching coefficients Grigo, Melnikov, Steinhauser 14 NNLO m t + NNLL threshold resummation De Florian, Mazzitelli 15 Full mass dependence in real radiation part + matching to parton shower Frederix, Hirschi, Mattelaer, Maltoni, Torrielli, Vryonidou, Zaro 14; Maltoni, Vryonidou, Zaro 14 Full top-mass dependence at NLO missing so far! S. Borowka (University of Zurich) SecDec-3. 41

45 gg HH - Two-loop integrals SB, Heinrich, Greiner, Jones, Kerner, Luisoni, Mastrolia, Schlenk, Schubert, Stoyanov, Di Vita, Zirke Requires computation of unknown two-loop integrals 4 independent scales: s 12, s 23, m H, m t numerical evaluation with SecDec m H = 125 GeV m t = 173 GeV Plot by Gudrun Heinrich S. Borowka (University of Zurich) SecDec-3. 42

46 gg HH - Two-loop integral examples I = P 1 ε + P m H = 125 GeV m t = 173 GeV Plot by Gudrun Heinrich S. Borowka (University of Zurich) SecDec-3. 43

47 gg HH - Two-loop integral examples HHNP2b, Re(P) I = P 1 ε + P m H = 125 GeV m t = 173 GeV s23/mt s12/mt Plot by Gudrun Heinrich S. Borowka (University of Zurich) SecDec-3. 44

48 from Jim Talbert s talk at RadcorLoopfest 215 Can make use of the new ε dependent dummy functions feature in SecDec 3 We work together to implement further new features needed for their project S. Borowka (University of Zurich) SecDec-3. 45

49 Miscellaneous S. Borowka (University of Zurich) SecDec-3. 46

50 WIMP el.-magnetic form factors, 2-loop S. Borowka (University of Zurich) SecDec-3. 47

51 Model for neutrino mass generation, 3-loop S. Borowka (University of Zurich) SecDec-3. 48

52 Summary and Outlook Summary SecDec allows for computation of diverse integrals contributing to scattering amplitudes SecDec 3: new decomposition strategies, improved user interface, negative propagator powers and linear propagators allowed, integrators added, efficiency increased New features in SecDec 3 prepare for large(r) scale applications Outlook Push limits of the method further (minimize spurious singularities) Interface to other programs, e.g. FIRE, LiteRed, Reduze Application to phenomenologically relevant processes S. Borowka (University of Zurich) SecDec-3. 49

53 Backup S. Borowka (University of Zurich) SecDec-3. 5

54 Change of variables in decomposition strategy G2 x i = y ei, nf F F S j F : facet S j : Set of facets a vertex lies in y F : new coordinate for each facet F e i : unit vector in R N 1 n F : primitive normal vector of the facet F Kaneko, Ueda 9 1 S. Borowka (University of Zurich) SecDec-3. 51