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