Calculation of multi-loop integrals with SecDec-3 · 2015-06-19 · Calculation of multi-loop integrals with SecDec-3.0 JohannesSchlenk Max Planck Institute for Physics [email protected]

Calculation of multi-loop integrals with SecDec-3.0

Johannes Schlenk

Max Planck Institute for Physics

[email protected]

SecDec collaboration: S. Borowka, G. Heinrich, S. Jones, M. Kerner, JS, T. Zirke

RADCOR-Loopfest 2015Los Angeles, USA

June 19, 2015

Overview

1. The sector decomposition method2. New features of SecDec-3.03. Application to the process gg → HH

4. Conclusions

J. Schlenk (MPP) Secdec-3.0 June 19, 2015 2 / 19

Motivation

I Run II of the LHC at 13TeV started recentlyI In the absence of new physics signals, precision measurements become

more importantI Need for precise predictions at higher loop orders involving multiple

scalesI Master integrals are an important building block for these calculations

OPEN-PHO-ACCEL-2015-007-11, CERN


Sector Decomposition

I ∼∫ ∞

0dNx δ(1− xN)

UN−(L+1)D/2

FN−LD/2 =c−2L

ε2L+

c−2L+1

ε2L−1 + . . .

I Decompose integration region into sectors with simple singularitystructure

I Subtract divergences and expand in the regularization parameter εI Integrate the finite coefficients (numerically)I Applicable to loop integrals and more general parametric integrals

Hepp (1966); Denner, Roth (1996); Binoth, Heinrich (2000)


Decomposition algorithms

Authors # ofsectors

infiniterecursion

Description

Binoth, Heinrich small possible heuristic algorithmBogner, Weinzierl large no based on Hironaka’s polyhe-

dra gameSmirnov, Smirnov,Tentyukov

smallish no hybrid of the first two

Kaneko, Ueda small no Geometric strategy


SecDecI Public implementation of the sector decomposition method Carter, Heinrich

(2010); Borowka, Heinrich (2011)

SecDec-3.0Borowka, Heinrich, Jones, Kerner, JS, Zirke

(2015)

I Improved user interfaceI New decomposition

strategies

Other codesI sector_decomposition

Bogner, Weinzierl (2007)

I Fiesta Smirnov, Smirnov,

Tentyukov (’08,’09,’13)

http://secdec.hepforge.org/

Feynman loop integral

(generated automatically)

any integral matchingloop integral structure

more generalparametric function

(inserted by user) (inserted by user)

primary sectordecomposition

loop directory general directory

factorization

contourdeformation

multiscale?multiscale? yes yesnono

subtraction of poles

expansion in numerical integration result:

iterated sectordecomposition



Laurent series in


SecDec - Installation

Installation:tar -xvf SecDec-3.0.7.tar.gzcd SecDec-3.0.7make(make check)

Included programs:Cuba Hahn, Bases Kawabata, CQuad Gonnet (2010)

Dependencies:Mathematica version 7 or higher, Perl, C/Fortran compiler, (Normaliz Bruns,

Ichim, Roemer, Soeger for geometric decomposition)


Improved user interface

secdec -p param.input -m math.m -k kinem.inputparam.input

graph=2Lboxepsord=0contourdef=Trueintegrator=3epsrel=1.e-3,1.e-2,1.e-2,2.e-2epsabs=1.e-5,1.e-4,1.e-4,1.e-4

General options andparameters for numericalintegration

kinem.input

p1 3 -2p2 2 -1

List of kinematic points

math.mmomlist={k1,k2};proplist={k1^2,(k1+p2)^2,(k1-p1)^2,(k1-k2)^2,

(k2+p2)^2,(k2-p1)^2,(k2+p2+p3)^2,(k1+p3)^2};numerator={1};powerlist={0,1,1,1,1,1,1,-1};ExternalMomenta={p1,p2,p3,p4};externallegs=4;prefactor=Gamma[1+eps]^2;KinematicInvariants = {s,t};Masses={};ScalarProductRules = {

SP[p1,p1]->0,SP[p2,p2]->0,SP[p3,p3]->0,SP[p4,p4]->0,SP[p1,p2]->s/2,SP[p2,p3]->t/2,SP[p1,p3]->-s/2-t/2};

Dim=4-2*eps;

Graph data and kinematics


New features

I Support for linear propagators including contour deformation (assumes+iδ prescription)

1q·n+i δ , ��1

q·n+i δ sign(q·n∗)I Propagators with nonpositive exponents allowedI Simplified scans over parameter rangesI Option to run compilation and numerical integration on a cluster

(Condor, PBS, LSF)I Numerical integration with CQuad (1D) and Mathematica possibleI General parametric integrals: input can contain ε-dependent symbolic

functions (masked during the decomposition stage)


Decomposition algorithms in SecDec-3.0

Implemented strategies:

I X: Heuristic strategy Binoth, Heinrich

I G1: Original geometric strategyKaneko, Ueda

I G2: Improved geometricdecomposition

Flag strategy in param.input toswitch between strategies:

strategy=G2

Normaliz Bruns, Ichim, Roemer, Soeger is usedfor the calculation of convexpolytopes

Feynman loop integral

(generated automatically)

any integral matchingloop integral structure

more generalparametric function

(inserted by user) (inserted by user)

primary sectordecomposition

loop directory general directory

factorization

contourdeformation

multiscale?multiscale? yes yesnono

subtraction of poles

expansion in numerical integration result:




Laurent series in


Geometric Method - General concept

I Eliminate xN using the Cheng-Wu theorem:

I ∼∫ ∞

0dNx δ(1− xN)N U

N−(L+1)D/2

FN−LD/2

I Calculate Newton polytope of U · F · N

∆ =

{ConvHull({vi})⋂

F

{m ∈ RN−1 | 〈m,nF 〉+ aF ≥ 0

} e1

e2

v1

v2

v3

v4 v5

n1 n2

n3n4

x11 x

02 + x2

1 x12 + x1

1 x22 + x0

1 x12 + x1

1 x12

I Change of variables in sector j : xi =∏

F∈Sj y〈ei ,nF 〉F

I For each vertex Sj contains facets incident to it (triangulation)


Geometric Method - Vacuum integral

m

∼∫ ∞

0dx1dx2

1P(x1, x2)2−ε

P(x1, x2) = x1 + x1x2 + x2


Geometric Method - Vacuum integral

Vertices:

P(x1, x2) = x11x

02 + x1

1x12 + x0

1x12

va =

(10

), vb =

(11

), vc =

(01

)

Facet normals:

n1 =

(−10

)n2 =

(0−1

)n3 =

(11

)a1 = 1 a2 = 1 a3 = −1

x1

x2

1

01a

bc

n3n1

n2


Change of variables

xi =3∏

F=1

y〈ei ,nF 〉F → x1 = y−1

1 y3x2 = y−1

2 y3

P(x1, x2) = x1 + x1x2 + x2= y−1

1 y−12 y1

3 (y1 + y2 + y3) x1

x2

1

0 1a

bc

n3n1

n2

I Include Jacobi determinantI Result for all three sectors (Sa = {3, 1},Sb = {1, 2},Sc = {2, 3}):∫ 1

0dy1dy2dy3

y ε1yε2y−1−ε3

(y1 + y2 + y3)2−ε

[δ(1− y1) + δ(1− y2) + δ(1− y3)

]


Diagram X G1 G2

2821 s

2668 s

1664 s

3681 s

3609 s

2355 s

1

2 3

4

5

6

7

5483 s

50615 s

3044 s

infiniterecursion

725 s

761 s

273365510 s

3206311856 s

27137443 s

Number of produced sectors and timings for the decomposition

Application: gg → HH

I Process gg → HH with full top mass dependence known to LO(loop-induced) Glover, van der Bij (1988)

I NLO in mt →∞ limit Plehn, Spira, Zerwas (1996); Dawson, Dittmaier, Spira (HPAIR 1998)

I + 1mt

expansion Grigo, Hoff, Melnikov, Steinhauser (2013)

I + full mt dependence in real radiation and parton shower Frederix, Hirschi,

Mattelaer, Maltoni, Torrielli, Vryonidou, Zaro (2014); Maltoni, Vryonidou, Zaro (2014)

I NNLO in mt →∞ limit De Florian, Mazzitelli (2013); Grigo, Melnikov, Steinhauser (2014).I Approximation is poor above production thresholdI Need for NLO calculation with full top mass dependenceI Requires computation of unknown two-loop integrals



Borowka, Heinrich, Greiner, Jones,

Kerner, Luisoni, Mastrolia, JS,

Schubert, Stoyanov, Di Vita, Zirke

2

1

3

4

C0 +C−1

ε+

C−2

ε2

m2t = 1

m2H = 0.75

3 4 5 6 7 8 9 10s12-1

-0.5

0

0.5

1

cos(θ)

-4

-2

0

2

ggHHp2 eps0 (Re)

3 4 5 6 7 8 9 10s12-1

-0.5

0

0.5

1

cos(θ)

-0.5-0.4-0.3-0.2-0.1

0

ggHHp2 ε-1 (Re)

3 4 5 6 7 8 9 10s12-1

-0.5

0

0.5

1

cos(θ)

-0.1-0.05

0 0.05

0.1 0.15

ggHHp2 ε-2 (Re)

3 4 5 6 7 8 9 10s12-1

-0.5

0

0.5

1

cos(θ)

-4

-2

0

2

ggHHp2 eps0 (Im)

3 4 5 6 7 8 9 10s12-1

-0.5

0

0.5

1

cos(θ)

-0.1 0

0.1 0.2 0.3 0.4 0.5

ggHHp2 ε-1 (Im)

3 4 5 6 7 8 9 10s12-1

-0.5

0

0.5

1

cos(θ)

-0.1-0.05

0 0.05

0.1 0.15

ggHHp2 ε-2 (Im)

Plots by Anton StoyanovJ. Schlenk (MPP) Secdec-3.0 June 19, 2015 17 / 19


3

1

42

C0 +C−1

ε

m2t = 1

m2H = 0.75

3 4 5 6 7 8 9 10s12-1

-0.5

0

0.5

1

cos(θ)

0 0.1 0.2 0.3 0.4 0.5

ggHHnp2a eps0 (Re)

3 4 5 6 7 8 9 10s12-1

-0.5

0

0.5

1

cos(θ)

0

0.1

0.2

ggHHnp2a eps-1 (Re)

3 4 5 6 7 8 9 10s12-1

-0.5

0

0.5

1

cos(θ)

0 0.1 0.2 0.3

ggHHnp2a eps0 (Im)

3 4 5 6 7 8 9 10s12-1

-0.5

0

0.5

1

cos(θ)

0

0.1

ggHHnp2a eps-1 (Im)

Plots by Anton Stoyanov


Conclusions and OutlookI SecDec: Public implementation of the sector decomposition algorithm

http://secdec.hepforge.org/I New version SecDec-3.0:

I Improved user interfaceI New geometric decomposition strategiesI Linear propagatorsI Cluster mode

OutlookI Improvement of geometric decomposition algorithmI Interface to Gosam-2LoopI Application to phenomenologically relevant processes

Thank you for your attentionJ. Schlenk (MPP) Secdec-3.0 June 19, 2015 19 / 19

Conclusions and OutlookI SecDec: Public implementation of the sector decomposition algorithm

http://secdec.hepforge.org/I New version SecDec-3.0:

I Improved user interfaceI New geometric decomposition strategiesI Linear propagatorsI Cluster mode

OutlookI Improvement of geometric decomposition algorithmI Interface to Gosam-2LoopI Application to phenomenologically relevant processes

Thank you for your attentionJ. Schlenk (MPP) Secdec-3.0 June 19, 2015 19 / 19

Documents

Calculation of multi-loop integrals with SecDec-3 · 2015-06-19 · Calculation of multi-loop integrals with SecDec-3.0 JohannesSchlenk Max Planck Institute for Physics [email protected]