Th.Diakonidis - ACAT2010 Jaipur, India1 Calculating one loop multileg processes A program for the case of In collaboration with B.Tausk ( T.Riemann & J

Th.Diakonidis - ACAT2010 Jaipur, India

1

Calculating one loop multileg processes A program for the case of

gg tt gg

( , )

Theodoros Diakonid

DESY ZE

s

UTHEN

i

In collaboration with B.Tausk (T.Riemann & J. Fleischer)


2

Motivation and goals Recent years have seen the emergence of first results

for 2 → 4 scattering processes

To that scope one of the challenges posed is the need to compute one-loop tensor integrals with up to 6 legs (Tord Riemann’s talk)

Public code produced for this reduction

(hexagon.m by K. Kajda) Fortran code (used for the process)

Program for NLO corrections of the process gg tt gg


3

Fortran Package For Tensor integrals, we have a Fortran implementation

package (Th. Diakonidis & B. Tausk)

The present implementation includes: Six point functions up to rank five (Hexagon.F) Five point functions (all 5 ranks) (Pentagon.F) Boxes (all 4 ranks) (Box.F) Triangles (all 3 ranks) (Triangle.F) Bubbles (all 2 ranks) (Bubble.F)


4

The code so far uses:

QCDLoop (R.K. Ellis and G. Zanderighi) (Finite part and and terms)

To calculate the scalar master integrals after the reduction

It can be adapted to any Fortran package for 1,2,3,4 point functions

A lot of cross checks have been done so far (shown after) and we also cross checked the results with an independent code by Peter Uwer

1/ 21/


5

The triangle

5I ,

3stI

,3

stI

5I

5I

5I

,4

sI 5I

,4

sI

,4

sI

,2

stuI

,4sI ,

3stI ,

2stuI ,

1stuvI

Here we have to add some extra terms in the cases of boxes, triangles and bubbles with the exception of 1st rank

0E 0D 0C 0B 0A


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Starting from a pentagon

For the randomly chosen phase space point:

A mixed case of massless and massive particles


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Pentagon


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Box case


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Triangle


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Bubble


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Some sample results for hexagons

For the randomly chosen phase space point given by:


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Results for scalar, vector and 2nd rank six point functions:


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3rd rank 6 point functions


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4th rank 6-point


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More results (massless case)

For the phase space point given by:

p1 =(0.5, 0, 0, 0.5)p2 =(0.5, 0, 0, -0.5)p3 =(-0.19178191,-0.12741180,-0.08262477,-0.11713105)p4 =(-0.33662712, 0.06648281, 0.31893785, 0.08471424)p5 =(-0.21604814, 0.20363139,-0.04415762,-0.05710657)p6 =-(p1+p2+p3+p4+p5)

M1=0, M2=0, M3=0, M4=0, M5=0, M6=0

Golem95: T.Binoth, J.-Ph.Guillet, G. Heinrich, E.Pilon, T.Reiter [arXiv:hep-ph/0810.0992]


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Comparisons with golem95 (for 5th rank)

A good agreement (8 digits)

(QCDLoop was used for the scalar master integrals)


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Study of the process

Part of the processLes Houches wishlist

C.Buttar et al. Les Houches Physics at TeV Colliders 2005, Standard Model and Higgs working group: Summary report,

hep-ph/0604120Z. Bern et al., The NLO multileg working group: Summary report, hep-ph/0803.0494

To the same direction:Bredenstein et al. NLO QCD corrections to production at the LHC: 1. quark-antiquark annihilation JHEP08 108 (2008) hep-ph/ 0807.1248 2. gluon-gluon annihilation hep-ph/ 0905.0110 (1001.4727,1001.4006) G. Bevilaqua et al. Assault on the NLO Wishlist: hep-ph/0907.4723, 1002.4009

gg tt gg

2pp tt jets

ttbb

pp t t bb


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One Loop Diagrams for the process

The diagram construction was done using DIANA.

A Feynman Diagram Analyser DIANAM. Tentyukov, J. Fleischer hep-ph/ 9904258

After applying all the Feynman rules the final output of Diana gives: Hexagons Pentagons Boxes Triangles Bubbles

} 4510 loop diagrams

in total


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DIANA


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Color Manipulation

Total number of C.S = 50

Divided in 4 categories:

1. 24

2. 12

3. 8

4. 6

31 2 4

1 2{ }( )aa a a

f fTr T T T T

31 2 4

1 2( )aa a a

f fT T T T

31 2 4

1 2{ }( )aa a a

f fTr T T T T

31 2 4

1 2{ }aa a a

f fTr T T T T g1

g2

g3

g4

g2

g1

g3

g4

g1

g3

g2

g4


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DIANADiagram

construction Output (form) hex_m.frm

:bub_m.frm

50 different Structures

color.F

Color2fortran.frmSUn.prc

MAPLE INPUT cRank0.m(1…4)

:cRank5.m(1…4)

OPTIMIZATIONggttgg.m

FORTRAN OPTcFi_rtSum3(1…4)

:cS_rtSum23(1…4)

hex_mf.frm:

bub_mf.frm

Passrt_hex.F:

Passrt_bub.F

Hex(Sum_6(4)):

Bubble(Sum_2(4))

Main fortran program

gm(line,n1,…,n9)Spinor structures

MADGRAPHmomenta.dat


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Conclusions

Results from the analytical recursive reduction presented by T.Riemann

Reduction formulas have been implemented in a Mathematica and a Fortran program

Fortran program for the calculation of the NLO contribution of the process shown explicitlygg tt gg

Documents

Th.Diakonidis - ACAT2010 Jaipur, India1 Calculating one loop multileg processes A program for the case of In collaboration with B.Tausk ( T.Riemann & J