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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)
Th.Diakonidis - ACAT2010 Jaipur, India
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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
Th.Diakonidis - ACAT2010 Jaipur, India
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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)
Th.Diakonidis - ACAT2010 Jaipur, India
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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/
Th.Diakonidis - ACAT2010 Jaipur, India
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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
Th.Diakonidis - ACAT2010 Jaipur, India
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Starting from a pentagon
For the randomly chosen phase space point:
A mixed case of massless and massive particles
Th.Diakonidis - ACAT2010 Jaipur, India
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Pentagon
Th.Diakonidis - ACAT2010 Jaipur, India
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Box case
Th.Diakonidis - ACAT2010 Jaipur, India
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Triangle
Th.Diakonidis - ACAT2010 Jaipur, India
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Bubble
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Some sample results for hexagons
For the randomly chosen phase space point given by:
Th.Diakonidis - ACAT2010 Jaipur, India
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Results for scalar, vector and 2nd rank six point functions:
Th.Diakonidis - ACAT2010 Jaipur, India
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3rd rank 6 point functions
Th.Diakonidis - ACAT2010 Jaipur, India
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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]
Th.Diakonidis - ACAT2010 Jaipur, India
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Comparisons with golem95 (for 5th rank)
A good agreement (8 digits)
(QCDLoop was used for the scalar master integrals)
Th.Diakonidis - ACAT2010 Jaipur, India
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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
Th.Diakonidis - ACAT2010 Jaipur, India
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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
Th.Diakonidis - ACAT2010 Jaipur, India
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DIANA
Th.Diakonidis - ACAT2010 Jaipur, India
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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
Th.Diakonidis - ACAT2010 Jaipur, India
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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
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