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Numerical approach to multi-loop integrals. K. Kato (Kogakuin University) with E. de Doncker, N.Hamaguchi, T.Ishikawa, T.Koike, Y. Kurihara, Y.Shimizu, F. Yuasa The XXth International Workshop High Energy Physics and Quantum Field Theory September 25, 2011 Sochi, Russia. motivation. - PowerPoint PPT Presentation
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Numerical approach to multi-loop integrals
K. Kato (Kogakuin University)with
E. de Doncker, N.Hamaguchi, T.Ishikawa, T.Koike, Y. Kurihara, Y.Shimizu, F. Yuasa
The XXth International WorkshopHigh Energy Physics and Quantum Field Theory
September 25, 2011 Sochi, Russia
motivation
• Theoretical prediction for High Energy Physics can be done by the perturbative calculation in Quantum Field Theory . (QFTHEP)
• Sometimes important information comes from multi-body final states. Experimentalists presents (after hard work) high-statistic data. This requires higher order calculation.
Large scale calculation is inevitable.
How to handle large scale computation?
• It is beyond man-power.• Automated systems to perform perturbative
calculation in QFT have been developed.• Many systems are successfully working in tree
and 1-loop level.GRACE, CompHEP, CalcHEP, FeynArt/Calc, FDC,…
• Next generation of systems should manage 2-loop and higher orders.
• One of the essential components is the general multi-loop calculation library.
• Formulae for 2-loop integrals are given for many cases: However, it seems to be difficult to write ‘general solution’ .
W
b Z
W
te
te
t
t
multi-loop integrals(scalar)
2220
1
4
21
/// )(
)(
)(
)/()(
nLNn
rrnL
N
iVU
xdx
nLNI
N
rr
rr
N
r r Dx
xdxN
D )(
)()!(
11
1
1
imqD rrr 22
N
r r
L
jn
jn
Di
dI
110
1
2 )(
Introduce Feynman parameters to combine denominators
Integrate by loop momenta
24 n
VVU, Polynomials of x’s: depends on masses and momenta
TARGET
Integration of singular function
0 1
0 10 1
Analytic
DCMContour deformation
1
0 2 1 ixsxm
dx
)(
jc
j A0
ji
)(ziziz
1P
1
0
Numerical
would be hard for multi-x case
example
Direct Computation Method(DCM)
22
11
// )(
)()(
nLNn
rr
N
iVU
xdxI
Target
Simple example
)( JJ 0lim
1
1
yx iyxDdxdyJ
),(0
Analytic
If D has zero in the integration domain, we keep finite.
If D has no zero in the integration domain, take and perform numerical integration .
0
122
yx D
iDdxdyJ
)(
Denominator is positive: Numerical evaluation is possible.
DCM(cont.), extrapolation
Wynn’s algorithm ( Math. magic)
jc
j A0 )( jJJ lim
),(),(),(),(
kjakjakjakja
1
1111
010 ),(),(),( jaJja jInput
Even k terms give good estimation
DCM= regularized integration + series extrapolation
Examples: 2-loop box (Yuasa)
Following loop diagrams are successfully calculated by DCM. Mostly scalar integrals, but inclusion of numerator will be straightforward since DCM is based on numerical integration.
1-loop : 3, 4, 5, 6 – point functions2-loop: 2, 3, 4 – point functions
Following slides are the results for 2-loop box.
Numerical results of Two-loop planar box with masses
fs s /m2
m=50 GeV, M = 90 GeV, t = -1002 GeV2
ACAT2011 5-9 September 2011F.Yuasa/KEK
x 10-12
Numerical results of Two-loop non-planar box with masses
fs s /m2
x 10-12
p12 p2
2 p32 p4
2 m2,
m1 m2 m4 m6 m,
m3 m5 m7 M
ACAT2011 5-9 September 2011F.Yuasa/KEK
Re. fs CPU time
6.0 16 hours
7.0 2 days
10.0 1 week
Intel(R) Xeon(R) CPU X5460 @ 3.16GHz
m=50 GeV, M = 90 GeV, t = -1002 GeV2
extrapolation control (Koike)
GeV1050
GeV90GeV5003
22
.m
Ms
jc
jA
0
211.2700 .
cm A
Example
Prepare integral values for m=0 and j=0 ,.. ,140.The first term is of m=0,..,120.The 21 terms starting from j=m are the target of extrapolation.
0
The choice of epsilons M
m2qs
22 mp
m
22 mp
13
5105.3 2104.2 1106.1 4101.1 0
Re
Im
m
Best region
Analytical value
Analytical value
Values afterextrapolation
Error in extra-polation
JPS 17Sept. 2011 T.Koike
Real part, M-dependencem
0102.6 5105.3
3101.9
2104.2 1106.1 4101.1 3102.4
0
m
Re
Re
0102.6 5105.3
3101.9
2104.2 1106.1 4101.1 3102.4
07103.1
Re
Re
JPS 17Sept. 2011 T.Koike
2
2
2
2
separation of singularity (de Donker)
22
11
// )(
)()(
nLNn
rr
N
iVU
xdxI
24 nThis integral might have IR divergence and/of UV divergence as pole(s) of .
24 n
We need double extrapolation for bothwhen V has zero in the integral region.
,
Separation of IR poles is successful even for double-pole cases.
)(
OCC
I 0
1
)(
OCCC
I 0
122
ACAT2011 5-9 September E. de Doncker
analytic
1-loop vertex with IR
Each term is obtained after extrapolation extrapolation
(linear)
summary
• Direct computation method(DCM) is a unique numerical method to calculate loop integrals for general masses and momenta.
• Some items remain before it will become an important component in an automated system for higher order radiative corrections.- Study the validity of the method for wider class of mass configuration- Numerical handling of UV/IR divergence- Improve parameter selection technique for iterated computation- Accelerate computation using modern IT technology