Numerical Evaluation of Multi-loop Integrals...S. Borowka (MPI for Physics) Numerical evaluation of...
Transcript of 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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 2
Searching for the Higgs
ATLAS ’12 CMS ’12
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 3
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 4
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 5
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., ...
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 6
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 singularities (solved with SecDec 1.0)I 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., ...
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 7
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 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
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., ...
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 8
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 9
SecDec 2.0 Computes ...
I Feynman graphs for arbitrary kinematics, and more generalparametric functions with no poles within the integrationregion
orgraphFeynman
functionparametric
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 10
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)
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 11
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 12
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 13
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 14
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 15
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 16
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 17
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 18
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.
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 19
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 20
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 21
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 22
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 23
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 24
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 25
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 26
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 27
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 28
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 29
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 30
User Input I
I param.input: parameters for integrand specification andnumerical integration
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 31
User Input III template.m: definition of the integrand
(Mathematica syntax)
p3
p1
p2
1
2
3
4
5
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 32
Program Test Run
I ./launch -p param.input -t template.m
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 33
Get the Result
I resultfile P126 full.res
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 34
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 35
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
S. Borowka (MPI for Physics) Numerical evaluation of multi-loop integrals 36