numerical evaluation of multi-loop integrals...s. borowka (mpi for physics) numerical evaluation of...

Numerical Evaluation of Multi-loopIntegrals

Sophia Borowka

MPI for Physics, Munich

In collaboration with: J. Carter and G. Heinrich

Based on arXiv:1204.4152 [hep-ph]

DESY-HU Theory Seminar, BerlinMay 24th, 2012

http://secdec.hepforge.org

S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 1

The LHC Era has begun

I We are probing energies which have never been reached atcolliders before

I High experimental precision is possible due to high luminosities

I Precise theoretical predictions become necessary


Searching for the Higgs

ATLAS ’12 CMS ’12


Higgs Production in gluon fusion

Multi-dimensional parameter integrals need to be evaluated whichcan contain UV, soft and collinear singularities

I LO (NLO in other processes)

I NLO

I NNLO


Higher Order Corrections to the Higgs Production

1

10

102

100 120 140 160 180 200 220 240 260 280 300

σ(pp → H+X) [pb]

MH

[GeV]

LONLONNLO

√s = 14 TeV

Harlander & Kilgore ’02 Anastasiou, Melnikov, Petriello ’05


Making Predictions in the LHC Era

I A lot of progress has been achieved towards the goal ofdescribing hadron collider processes consistently at NLO

I Calculations beyond NLO are also progressing well, butautomation is difficult, and analytic methods to calculate e.g.two-loop integrals involving massive particles reach their limit

I Numerical methods are in general easier to automate,problems mainly are

I Extraction of IR and UV singularitiesI Numerical convergence in the presence of integrable

singularities (e.g. thresholds)I Speed/accuracy

Many people are/have been working on purely numericalmethods, 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., ...






I Extraction of IR and UV singularities (solved with SecDec 1.0)I Numerical convergence in the presence of integrable

singularities (e.g. thresholds)I Speed/accuracy









I Extraction of IR and UV singularities (solved with SecDec 1.0)I Numerical convergence in the presence of integrable

singularities (e.g. thresholds) (solved with SecDec 2.0)I Speed/accuracy





Public Implementations of the SectorDecomposition Method on the Market

I sector decomposition (uses GiNaC) C. Bogner & S. Weinzierl ’07

I FIESTA (uses Mathematica, C) A. Smirnov, V. Smirnov & M.

Tentyukov ’08 ’09

I SecDec (uses Mathematica, Perl, Fortran/C++) J. Carter & G.

Heinrich ’10

Limitation until recently:Multi-scale integrals were limited to the Euclidean region (i.e., nothresholds)

NOW:Extension of SecDec to general kinematics! SB, J. Carter & G. Heinrich

’12


SecDec 2.0 Computes ...

I Feynman graphs for arbitrary kinematics, and more generalparametric functions with no poles within the integrationregion

orgraphFeynman

functionparametric


Parametric Functions

A general parametric function can be

I a phase space integral where IR divergences are regulateddimensionally

I polynomial functions, e.g. hypergeometric functions

pFp−1(a1, ..., ap; b1, ..., bp−1;β)


Operational Sequence of the SecDec Program

graph info Feynmanintegral

iterated sectordecomposition

contourdeformation

subtractionof poles

expansionin ǫ

numericalintegration

resultn∑

Cmǫm

1 2 3

456

7 8

multiscale?

yesno

m=−2L


General Feynman Integral

I Graph infos are converted into tensorial Feynman integralGµ1...µR in D dimensions at L loops with N propagators topower νj of rank R

I After loop momentum integration, a generic scalar Feynmanintegral

G =(−1)Nν

∏Nj=1 Γ(νj)

Γ(Nν − LD/2)

∞∫

0

N∏

j=1

dxj xνj−1j δ(1−

N∑

l=1

xl)UNν−(L+1)D/2(~x)

FNν−LD/2(~x)

where Nν =∑N

j=1 νj and where U and F can be constructedvia topological cuts


Construction of the Functions U and FExample: 1-loop box

x2 x4

p2

p1

p3

p4

2

1

3

4

U(~x) =x1 + x2 + x3 + x4 (L-line cut),

F0(~x) =s12x1x3 + s23x2x4 + p21x1x2 + p22x2x3 + p23x3x4 + p24x1x4(L+1-line cut)

and

F(~x) =−F0(~x) + U(~x)N∑

j=1

xjm2j − iδ





contourdeformation

subtractionof poles

expansionin ǫ


resultn∑

Cmǫm

1 2 3

456

7 8

multiscale?

yesno

m=−2L


Sector Decomposition

I Overlapping divergences are factorized

y

x

−→ + −→(2)

(1)

+

y

x

t

t

∫ 1

0

dx

∫ 1

0

dy1

(x + y)2+ε=

∫ 1

0

dx

∫ 1

0

dt1

x1+ε(1 + t)2+ε+

∫ 1

0

dt

∫ 1

0

dy1

y1+ε(1 + t)2+ε

I Iterated sector decomposition is done, where dimensionallyregulated soft, collinear and UV singularities are factored outHepp ’66, Binoth & Heinrich ’00





contourdeformation

subtractionof poles

expansionin ǫ


resultn∑

Cmǫm

1 2 3

456

7 8

multiscale?

yesno

m=−2L


Contour Deformation I

I For kinematics in the physical region, F can still vanish

FBubble = m2(1 + t1)2 − s t1 − iδ

but a deformation of the integration contour

Re(z)

Im(z)

10

and Cauchy’s theorem can help∮

cf (t)dt =

∫ 1

0f (t)dt +

∫ 0

1

∂z(t)

∂tf (z(t))dt = 0


Contour Deformation II

Re(z)

Im(z)

10

I The integration contour is deformed by

~t → ~z = ~t + i~y ,

yj(~t)= −λtj(1− tj)∂F(~t)

∂tjSoper ’99

I Integrand is analytically continued into the complex plane

F(~t)→ F(~t + i~y(~t)) = F(~t) + i∑

j

yj(~t)∂F(~t)

∂tj+O(y(~t)2)

Soper, Nagy, Binoth; Kurihara et al., Anastasiou et al., Freitas et al., Weinzierl et al.


Find the Optimal Deformation Parameter λ I

I Robust method: check the maximally allowed λ for F andmaximize the modulus at critical points

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

Re

(F(t

1))

2+

Im(F

(t1))

2

t1

lambda = 1.2lambda = 1.0lambda = 0.5

example is for1-loop bubble,m2 = 1.0,s = 4.5with Feynmanparameter t1


Find the Optimal Deformation Parameter λ II

I Faster convergence: minimize the complex argument of F

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6

Im(F

(t1))

/Re

(F(t

1))

t1

lambda = 1.2lambda = 1.0lambda = 0.5

example is for1-loop bubble,m2 = 1.0,s = 4.5with Feynmanparameter t1





contourdeformation

subtractionof poles

expansionin ǫ


resultn∑

Cmǫm

1 2 3

456

7 8

multiscale?

yesno

m=−2L


Subtraction, Expansion, Numerical Integration

SubtractionI The factorized poles in a subsector integrand I ∝ U ,F are

extracted by subtraction (e.g. logarithmic divergence)

∫ 1

0

dtj t−1−bjεj I(tj , ε) = −I(0, ε)

bjε+

∫ 1

0

dtj t−1−bjεj (I(tj , ε)− I(0, ε))

Expansion

I After the extraction of poles, an expansion in the regulator εis done

Numerical Integration

I Monte Carlo integrator programs containted in CUBA libraryor BASES can be used for numerical integrationHahn et al. ’04 ’11, Kawabata ’95





contourdeformation

subtractionof poles

expansionin ǫ


resultn∑

Cmǫm

1 2 3

456

7 8

multiscale?

yesno

m=−2L


Results

I Successful application of the public SecDec 1.0 program tomassless multi-loop diagrams up to 5-loop 2-point functionsand 4-loop 3-point functions for Euclidean kinematics

I Successful application of SecDec 2.0 to various multi-scaleexamples, e.g., the massive 2-loop vertex graph, planar andnon-planar 6- and 7-propagator massive 2-loop box diagrams

I Timings for the 2-loop vertex diagram and a relative accuracyof 1% using the CUBA 3.0 library on an Intel(R) Core i7 CPUat 2.67GHz

p3

p1

p2

1

2

3

4

5

s/m2 timing (finite part)

3.9 13.6 secs14.0 12.1 secs


Results II: Massive Two-loop Vertex Graph G

-6

-5

-4

-3

-2

-1

0

1

2

3

4

0 2 4 6 8 10 12 14

Fin

ite

re

al a

nd

im

ag

ina

ry p

art

of

P1

26

s

Real - AnalyticReal - SecDecImag - AnalyticImag - SecDec

p3

p1

p2

1

2

3

4

5

Kotikov et al. ’97, Davydychev & Kalmykov ’04, Ferroglia et al. ’04, Bonciani et al.

’04


Results III: Massive Non-planar 6-propagator Graph

-35

-30

-25

-20

-15

-10

-5

0

5

-1.5 -1 -0.5 0 0.5 1

Massiv

e B

np6, finite p

art

s12

Real (SecDec)

Imag (SecDec)

m 6= 0

massless case: Tausk ’99


Results IV: Planar Massive Two-loop Box

-1.5e-12

-1e-12

-5e-13

0

5e-13

1e-12

1.5e-12

2e-12

2.5e-12

3e-12

2 4 6 8 10 12 14 16

Fin

ite

pa

rt o

f p

lan

ar

2-lo

op

bo

x

fs

m=50,M=90,s23=-10000

Real (SecDec)Imag (SecDec)

Fujimoto et al. ’11


Results V: Non-planar Massive Two-loop Box

-4e-13

-2e-13

0

2e-13

4e-13

6e-13

8e-13

1e-12

1.2e-12

0 2 4 6 8 10 12 14 16

Fin

ite

pa

rt o

f m

assiv

e n

on

-pla

na

r 2

-lo

op

bo

x

s/m2

m=50,M=90,s23=-10000

Real (SecDec)Imag (SecDec)

-0.2

0

0.2

0.4

0.6

0.8

-20 -15 -10 -5 0 5 10 15 20

I(s,

t)

fs

Re

Im

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-100 -90 -80 -70 -60 -50 -40 -30 -20

I(s,

t)

fs

Im

Fujimoto et al. ’11


Install SecDec 2.0

I Download:http://secdec.hepforge.org

I Install:tar xzvf SecDec.tar.gzcd SecDec-2.0./install

I Prerequisites:Mathematica (version 6 or above), Perl, Fortran and/or C++compiler


User Input I

I param.input: parameters for integrand specification andnumerical integration


User Input III template.m: definition of the integrand

(Mathematica syntax)

p3

p1

p2

1

2

3

4

5


Program Test Run

I ./launch -p param.input -t template.m


Get the Result

I resultfile P126 full.res


Summary

I With SecDec the numerical evaluation of multi-loop integralsis possible for arbitrary kinematics

I SecDec can also be used for more general parametricfunctions (e.g. phase space integrals)

I Useful to check analytic results for multi-loop masterintegrals, e.g. 2-loop boxes, 3-loop form factors, ...

I Download the public SecDec program:http://secdec.hepforge.org


Outlook

I Can be used to calculate 2-loop corrections involving severalmass scales, e.g. QCD/EW/MSSM corrections

I Implement contour deformation for more general parametricfunctions

I Include option to use Mathematica’s numerical integrator

I Implement further variable transformation to tacklesingularities very close to pinch singularities


numerical evaluation of multi-loop integrals...s. borowka (mpi for physics) numerical evaluation of...

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