[PPT] - Recursive equations for arbitrary scattering processes Costas G. PowerPoint Presentation

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Recursive equations for arbitrary scattering processes

Costas G. Papadopoulos HEP2006, April 13-16, 2006, Ioannina

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High Energy and high Luminosity make necessary

precision calculations → Loops and Legs

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10 10 2 10 3 10 4 10 5 20 40 60 80 100 120 140 160 180 200 220 Centre-of-mass energy (GeV) Cross-section (pb)

CESR DORIS PEP PETRA TRISTAN KEKB PEP-II

SLC LEP I LEP II

Z W+W-

e+e−→hadrons

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e+ γ e− γ/Z f

−

f e+ e− γ/Z f

−

γ f e+ e− γ γ/Z f

−

f e+ e− γ/Z f

−

f γ

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e+ γ e− γ/Z f

−

f e+ e− γ/Z f

−

γ f

γ,Z/W f f

−/f’

γ,Z/W γ,Z/W W W/γ,Z γ,Z/W Z/W H Z/W Z/W Z/W Z/W H

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ALEPH DELPHI L3 OPAL 1990-1992 data 1993-1995 data typical syst. exp. luminosity error theoretical errors:

QED luminosity

peak

30 30.2 30.4 30.6 30.8 91.2 91.25 91.3

Ecm [GeV]

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ALEPH DELPHI L3 OPAL LEP 91.1893±0.0031 91.1863±0.0028 91.1894±0.0030 91.1853±0.0029 91.1875±0.0021 common: 0.0017 χ2/DoF = 2.2/3 mZ [GeV]

91.18 91.19 91.2

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2 4 6 100 20 400

mH [GeV] ∆χ2

Excluded

Preliminary

∆αhad = ∆α(5)

0.02761±0.00036 0.02747±0.00012 Without NuTeV

theory uncertainty

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The structure of an event

Warning: schematic only, everything simplified, nothing to scale, . . . p p/p Incoming beams: parton densities

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p p/p u g W+ d Hard subprocess: described by matrix elements

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p p/p u g W+ d c s Resonance decays: correlated with hard subprocess

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p p/p u g W+ d c s Initial-state radiation: spacelike parton showers

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p p/p u g W+ d c s Final-state radiation: timelike parton showers

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p p/p u g W+ d c s Multiple parton–parton interactions . . .

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p p/p u g W+ d c s . . . with its initial- and final-state radiation

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Beam remnants and other outgoing partons

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Everything is connected by colour confinement strings Recall! Not to scale: strings are of hadronic widths

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The strings fragment to produce primary hadrons

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Many hadrons are unstable and decay further

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High Energy and high Luminosity make necessary

precision calculations → Loops and Legs

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Gauge boson production at the LHC

Precision Phenomenology and Collider Physics – p.14

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Gauge boson production at the LHC

Gold-plated process

Anastasiou, Dixon, Melnikov, Petriello

NNLO perturbative accuracy better than 1% ⇒ use to determine parton-parton luminosities at the LHC

Precision Phenomenology and Collider Physics – p.15

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High Energy and high Luminosity make necessary

precision calculations → Loops and Legs

PYTHIA

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HEP - NCSR Democritos Over the last years we have seen several improvements

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HEP - NCSR Democritos Over the last years we have seen several improvements

Dyson-Schwinger recursive equations successfully

implemented in HELAC - Progress Report

C. G. Papadopoulos and M. Worek, arXiv:hep-ph/0512150.

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HEP - NCSR Democritos Over the last years we have seen several improvements

Dyson-Schwinger recursive equations successfully

implemented in HELAC - Progress Report

C. G. Papadopoulos and M. Worek, arXiv:hep-ph/0512150.
New insights in multi-gluon tree amplitude

calculations

F. Cachazo, P. Svrcek and E. Witten, JHEP 0409 (2004) 006
R. Britto, F. Cachazo and B. Feng, Nucl. Phys. B 715 (2005) 499

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HEP - NCSR Democritos Over the last years we have seen several improvements

Dyson-Schwinger recursive equations successfully

implemented in HELAC - Progress Report

C. G. Papadopoulos and M. Worek, arXiv:hep-ph/0512150.
New insights in multi-gluon tree amplitude

calculations

F. Cachazo, P. Svrcek and E. Witten, JHEP 0409 (2004) 006
R. Britto, F. Cachazo and B. Feng, Nucl. Phys. B 715 (2005) 499
Progress in multi-loop calculations
C. Anastasiou, Z. Bern, L. J. Dixon and D. A. Kosower, Phys. Rev. Lett. 91 (2003) 251602
Z. Bern, L. J. Dixon and V. A. Smirnov, Phys. Rev. D 72 (2005) 085001

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Reliable cross section computation and event

generation for multiparticle processes, with ∼ 10-12 particles in the final state.

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Reliable cross section computation and event

generation for multiparticle processes, with ∼ 10-12 particles in the final state.

– HELAC:

A.Kanaki and C.G.Papadopoulos, CPC 132 (2000) 306, hep-ph/0002082.

Matrix element computation algorithm, based on Dyson-Schwinger equations, including: EWK, QCD, fermion masses, reliable arithmetic, running couplings and masses . . ..

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Reliable cross section computation and event

generation for multiparticle processes, with ∼ 10-12 particles in the final state.

– HELAC:

A.Kanaki and C.G.Papadopoulos, CPC 132 (2000) 306, hep-ph/0002082.

Matrix element computation algorithm, based on Dyson-Schwinger equations, including: EWK, QCD, fermion masses, reliable arithmetic, running couplings and masses . . .. PHEGAS:

C.G.Papadopoulos, CPC 137 (2001) 247, hep-ph/0007335

Monte-Carlo phase space integration/generation based

n optimized multichannel approach.

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Reliable cross section computation and event

generation for multiparticle processes, with ∼ 10-12 particles in the final state.

– HELAC:

A.Kanaki and C.G.Papadopoulos, CPC 132 (2000) 306, hep-ph/0002082.

Matrix element computation algorithm, based on Dyson-Schwinger equations, including: EWK, QCD, fermion masses, reliable arithmetic, running couplings and masses . . .. PHEGAS:

C.G.Papadopoulos, CPC 137 (2001) 247, hep-ph/0007335

Monte-Carlo phase space integration/generation based

n optimized multichannel approach.

hep-ph/0012004 and Tokyo 2001,(CPP2001) Computational particle physics, p. 20-25

T. Gleisberg, et al. Eur. Phys. J. C 34 (2004) 173

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–

Old Feynman graphs → computational cost ∼ n!

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–

Old Feynman graphs → computational cost ∼ n! New Dyson-Schwinger → computational cost ∼ 3n

P.Draggiotis, R.H.Kleiss and C.G.Papadopoulos, Phys. Lett. B439 (1998) 157

F. Caravaglios and M. Moretti, Phys. Lett. B358 (1995) 332
F. A. Berends and W. T. Giele, Nucl. Phys. B 306 (1988) 759.

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–

Old Feynman graphs → computational cost ∼ n! New Dyson-Schwinger → computational cost ∼ 3n

P.Draggiotis, R.H.Kleiss and C.G.Papadopoulos, Phys. Lett. B439 (1998) 157

F. Caravaglios and M. Moretti, Phys. Lett. B358 (1995) 332
F. A. Berends and W. T. Giele, Nucl. Phys. B 306 (1988) 759.
Example: e−e+ → e−e+e−e+ in QED:

e e γ e γ e e e γ e e e e γ e e

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The Dyson-Schwinger recursion

Imagine a theory with 3- and 4- point vertices and just one field.

Then it is straightforward to write an equation that gives the amplitude for 1 → n = + + + + +

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The Dyson-Schwinger recursion = + + a(n) = δn,1 +

n!

n1!n2!a(n1)a(n2)δn1+n2,n + n! n1!n2!n3!

a(n1)a(n2)a(n3)δn1+n2+n3,n

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⇒ Systematic approach: bµ(P) = ψ(P) = ¯ ψ(P) =

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⇒ Systematic approach: bµ(P) = ψ(P) = ¯ ψ(P) = = + bµ(P) =

n

i=1

δP =pibµ(pi) +

P =P1+P2

(ig)Πµ

ν

¯ ψ(P2)γν ψ(P1)ǫ(P1, P2)

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= + ψ(P) =

n

i=1

δP =piψ(pi) +

P =P1+P2

(ig)b /(P2) (P /1 + m) P 2

1 − m2 ψ(P1)ǫ(P1, P2)

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Let n external particles with momenta pµ

i , i = 1 . . . , n, and define the

momentum P µ P µ =

i∈I

pµ

i ,

I ⊂ {1, . . . , n}

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Let n external particles with momenta pµ

i , i = 1 . . . , n, and define the

momentum P µ P µ =

i∈I

pµ

i ,

I ⊂ {1, . . . , n} the binary vector m = (m1, . . . , mn), where its components take the values 0 or 1 : P µ =

n

i=1

mi pµ

i .

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Let n external particles with momenta pµ

i , i = 1 . . . , n, and define the

momentum P µ P µ =

i∈I

pµ

i ,

I ⊂ {1, . . . , n} the binary vector m = (m1, . . . , mn), where its components take the values 0 or 1 : P µ =

n

i=1

mi pµ

i .

Moreover this binary vector can be uniquely represented by the integer m =

n

i=1

2i−1mi , 0 ≤ m ≤ 2n − 1

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Let n external particles with momenta pµ

i , i = 1 . . . , n, and define the

momentum P µ P µ =

i∈I

pµ

i ,

I ⊂ {1, . . . , n} the binary vector m = (m1, . . . , mn), where its components take the values 0 or 1 : P µ =

n

i=1

mi pµ

i .

Moreover this binary vector can be uniquely represented by the integer m =

n

i=1

2i−1mi , 0 ≤ m ≤ 2n − 1 Replace bµ(P) → bµ(m) .

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♣ Convenient ordering of integers in binary representation ⇒ level l, defined by l =

n

i=1

mi .

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♣ Convenient ordering of integers in binary representation ⇒ level l, defined by l =

n

i=1

mi . ♣ external momenta are of level 1

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♣ Convenient ordering of integers in binary representation ⇒ level l, defined by l =

n

i=1

mi . ♣ external momenta are of level 1 ♣ the total amplitude corresponds to the unique level n integer 2n − 1 A = b(1) · b(2n − 2)

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♣ Convenient ordering of integers in binary representation ⇒ level l, defined by l =

n

i=1

mi . ♣ external momenta are of level 1 ♣ the total amplitude corresponds to the unique level n integer 2n − 1 A = b(1) · b(2n − 2) This ordering dictates the natural path of the computation : start- ing with level-1 sub-amplitudes, we compute the level-2 ones using the Dyson-Schwinger equations and so on up to level n − 1

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The solution e−(1) e+(2) → e−(4) ¯ νe(8) u(16) ¯ d(32) 1 10 33 2

2

8 1 1 12 33 4

2

8 1 1 48 34 16

3

32 4 2 26

4

10 33 16

3

. . . 2 62

2

10 33 52

1

2 62

2

12 33 50

1

2 62

2

58 31 4

2

2 62

2

58 32 4

2

2 62

2

60 31 2

2

2 62

2

60 32 2

2

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W e ν _

+ +

2 8 10

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Dirac algebra simplification: 2-dim vs 4-dim and chiral

representation, including mf = 0.

The sign factor:

ǫ(P1, P2) → ǫ(m1, m2) we define ǫ(m1, m2) = (−1)χ(m1,m2) χ(m1, m2) =

2

i=n

ˆ m1i  

i−1

j=1

ˆ m2j   where hatted components are set to 0 if the corresponding external particle is a boson.

Full EWK theory, both Unitary and Feynman gauges.
A. Denner, Fortsch. Phys. 41, 307 (1993).

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HELAC

Construction of the skeleton solution of the Dyson-Schwinger
equations. At this stage only integer arithmetic is performed. This is

part of the initialization phase.

Dressing-up the skeleton with momenta, provided by PHEGAS and wave

functions, propagators, n-point functions in general.

Unitary and Feynman gauges implemented. Due to multi-precision

arithmetic,tests of gauge invariance can be extended to arbitrary precision.

All fermions masses can be non-zero.
All Electroweak and QCD vertices are implemented, including Higgs

and would-be Goldstone bosons.

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Colour Configuration - EWK⊕QCD

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Colour Configuration - EWK⊕QCD

Ordinary approach SU(N)-type

Aa1...an =

Tr(T aσ1 . . . T aσn )

A(σ1 . . . σn)

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Colour Configuration - EWK⊕QCD

Ordinary approach SU(N)-type

Aa1...an =

Tr(T aσ1 . . . T aσn )

A(σ1 . . . σn) Cij =

Tr(T aσ1 . . . T aσn )Tr(T

aσ′

1 . . . T

aσ′

n )

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Colour Configuration - EWK⊕QCD

Ordinary approach SU(N)-type

Aa1...an =

Tr(T aσ1 . . . T aσn )

A(σ1 . . . σn) Cij =

Tr(T aσ1 . . . T aσn )Tr(T

aσ′

1 . . . T

aσ′

n )

Quarks and gluons treated differently

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Colour Configuration - EWK⊕QCD

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Colour Configuration - EWK⊕QCD

New approach U(N)-type

Each color-configuration amplitude is proportional to Di = δ1,σi(1)δ2,σi(2) . . . δn,σi(n) where σi represents the i-th permutation of the set 1, 2, . . . , n.

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Colour Configuration - EWK⊕QCD

New approach U(N)-type

Each color-configuration amplitude is proportional to Di = δ1,σi(1)δ2,σi(2) . . . δn,σi(n) where σi represents the i-th permutation of the set 1, 2, . . . , n. ⋆ quarks 1 . . . n ⋆ antiquarks σi(1 . . . n) and ⋆ gluons = q¯ q

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Colour Configuration - EWK⊕QCD

New approach U(N)-type

Each color-configuration amplitude is proportional to Di = δ1,σi(1)δ2,σi(2) . . . δn,σi(n) where σi represents the i-th permutation of the set 1, 2, . . . , n. ⋆ quarks 1 . . . n ⋆ antiquarks σi(1 . . . n) and ⋆ gluons = q¯ q Cij =

DiDj = Nα

c ,

α = σ1, σ2

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Colour Configuration - EWK⊕QCD

New approach U(N)-type

Each color-configuration amplitude is proportional to Di = δ1,σi(1)δ2,σi(2) . . . δn,σi(n) where σi represents the i-th permutation of the set 1, 2, . . . , n. ⋆ quarks 1 . . . n ⋆ antiquarks σi(1 . . . n) and ⋆ gluons = q¯ q Cij =

DiDj = Nα

c ,

α = σ1, σ2 ♠ exact color treatment ⇒ low color charge

Problem: number of colour connection configurations: ∼ n! where n is the number of gluons or q ¯ q pairs. ⇒ Monte-Carlo over continuous colour-space.

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σi) (j, (j,σj) g (i,σi) g g δ σ

i j

fabcta

AB tb CDtc EF = − i

4(δADδCF δEB − δAF δCB δED) δ1σ2δ2σ3δ3σ1

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(0, σi) σi) (i, q _ q (i,0) g

ta

ABtb CD = 1

2(δADδCB − 1 Nc δAB δAC ) 1 √ 2

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(i,0) (0,σi) q _ g δi σi q (0,0)

ta

ABtb CD = 1

2(δADδCB − 1 Nc δAB δAC ) 1 √2Nc

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(1,σ )

1

(4,σ ) (3,σ ) (2,σ )

2 4 3

δ1σ3δ3σ2δ2σ4δ4σ1 2g12g34 − g13g24 − g14g23

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The

1 Nc expansion

Cij =

DiDj = Nα

c ,

α = σ1, σ2 The leading term σ1 = σ2, Nn

c

The subleading terms: how many δ’s survive after contraction, which in Combinatorial Analysis are known to be the Stirling numbers (−)n−mS(m)

n

where n is the number of ‘objects’ and m the number of ‘surviving’ δ’s, called cycles. For instance the

1 Nc -term is related to n(n−1) 2

permutations !

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Summation/Integration over color

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Summation/Integration over color M({pi}n

1 , {εi}n 1 , {ai}n 1 ) ∼

P (2,...,n)

Tr(ta1 . . . tan)A({pi}n

1 , {εi}n 1 )

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Summation/Integration over color M({pi}n

1 , {εi}n 1 , {ai}n 1 ) ∼

P (2,...,n)

Tr(ta1 . . . tan)A({pi}n

1 , {εi}n 1 )

M({pi}n

1 , {εi}n 1 , {Ii, Ji}n 1 ) ∼

P (2,...,n)

δI1,P (J1) . . . δIn,P (Jn)A({pi}n

1 , {εi}n 1 )

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Summation/Integration over color M({pi}n

1 , {εi}n 1 , {ai}n 1 ) ∼

P (2,...,n)

Tr(ta1 . . . tan)A({pi}n

1 , {εi}n 1 )

M({pi}n

1 , {εi}n 1 , {Ii, Ji}n 1 ) ∼

P (2,...,n)

δI1,P (J1) . . . δIn,P (Jn)A({pi}n

1 , {εi}n 1 )

{ai}n

1 {εi}n 1

|M({pi}n

1 , {εi}n 1 , {ai}n 1 )|2 = g2n−4 ε

ij

AiCijA∗

j

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Summation/Integration over color M({pi}n

1 , {εi}n 1 , {ai}n 1 ) ∼

P (2,...,n)

Tr(ta1 . . . tan)A({pi}n

1 , {εi}n 1 )

M({pi}n

1 , {εi}n 1 , {Ii, Ji}n 1 ) ∼

P (2,...,n)

δI1,P (J1) . . . δIn,P (Jn)A({pi}n

1 , {εi}n 1 )

{ai}n

1 {εi}n 1

|M({pi}n

1 , {εi}n 1 , {ai}n 1 )|2 = g2n−4 ε

ij

AiCijA∗

j

P (2,...,n)

∼ n!

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Summation/Integration over color M({pi}n

1 , {εi}n 1 , {ai}n 1 ) ∼

P (2,...,n)

Tr(ta1 . . . tan)A({pi}n

1 , {εi}n 1 )

M({pi}n

1 , {εi}n 1 , {Ii, Ji}n 1 ) ∼

P (2,...,n)

δI1,P (J1) . . . δIn,P (Jn)A({pi}n

1 , {εi}n 1 )

{ai}n

1 {εi}n 1

|M({pi}n

1 , {εi}n 1 , {ai}n 1 )|2 = g2n−4 ε

ij

AiCijA∗

j

P (2,...,n)

∼ n!

{Ii,Ji}n

1

∼ 3n × 3n

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The Dyson-Schwinger recursion equation for gluon in a general way can be written as follows: [Aµ(P); (A, B)] =

n

i=1

[ δ(P − pi) Aµ(pi); (A, B)i]+

[ (ig) Πµ

ρ V ρνλ(P, p1, p2)Aν (p1)Aλ(p2)σ(p1, p2); (A, B) = (C, D)1⊗(E, F)2]

−

[ (g2) Πµ

σ Gσνλρ(P, p1, p2, p3)Aν (p1)Aλ(p2)Aρ(p3)σ(p1, p2 + p3);

(A, B) = (C, D)1 ⊗ (E, F)2 ⊗ (G, H)3] +

P =p1+p2

[ (ig) Πµ

ν

¯ ψ(p1)γν ψ(p2)σ(p1, p2); (A, B) = (0, D)1 ⊗ (C, 0)2] where A, B, C, D, E, F, G, H = 1, 2, 3.

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L = −1 4F a

µν F µνa,

F a

µν = ∂µAa ν − ∂ν Aa µ + gfabcAb µAc ν

L = −1 2Ha

µν Hµνa + 1

4Ha

µν F µνa.

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[Aµ(P); (A, B)] =

n

i=1

[ δ(P − pi) Aµ(pi); (A, B)i]+ [ (ig) Πµ

ρ V ρνλ(P, p1, p2)Aν (p1)Aλ(p2)σ(p1, p2); (A, B) = (C, D)1 ⊗(E, F)2]

+ [ (ig) Πµ

σ (gσλgνρ−gνλgσρ) Aν (p1)Hλρ(p2)σ(p1, p2); (A, B) = (C, D)1⊗(E, F)2]

+ [ (ig) Πµ

ν

¯ ψ(p1)γν ψ(p2)σ(p1, p2); (A, B) = (0, D)1 ⊗ (C, 0)2] and [Hµν (P); (A, B)] =

P =p1+p2

[ (ig) (gµλgνρ−gνλgµρ) Aλ(p1)Aρ(p2)σ(p1, p2); (A, B) = (C, D)1 ⊗ (E, F)2].

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(A, B) = (C, 0) ⊗ (0, D) = (C, D)w=1, if C = D.

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(A, B) = (C, 0) ⊗ (0, D) = (C, D)w=1, if C = D. (A, B) = (C, 0) ⊗ (0, D) = (1, 1)w1 ⊕ (2, 2)w2 ⊕ (3, 3)w3, if C = D.

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(A, B) = (C, 0) ⊗ (0, D) = (C, D)w=1, if C = D. (A, B) = (C, 0) ⊗ (0, D) = (1, 1)w1 ⊕ (2, 2)w2 ⊕ (3, 3)w3, if C = D. (1, 0) ⊗ (0, 1) = (1, 1)2/3 ⊕ (2, 2)−1/3 ⊕ (3, 3)−1/3

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NCC =

nq

A=0

nq−1

B=0

nq−A−B

C=0
nq!

A!B!C! 2 δ(nq = A + B + C)

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NCC =

nq

A=0

nq−1

B=0

nq−A−B

C=0
nq!

A!B!C! 2 δ(nq = A + B + C)

Process NALL CC NCC NF CC (%) gg → 2g 6561 639 59.1 gg → 3g 59049 4653 68.4 gg → 4g 531441 35169 77.4 gg → 5g 4782969 272835 85.0 gg → 6g 43046721 2157759 90.4 gg → 7g 387420489 17319837 94.0 gg → 8g 3486784401 140668065 96.4

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Process NALL CC NCC NF CC (%) gg → u ¯ u 729 93 93.5 gg → gu ¯ u 6561 639 91.6 gg → 2gu ¯ u 59049 4653 92.6 gg → 3gu ¯ u 531441 35169 94.6 gg → 4gu ¯ u 4782969 272835 96.4 gg → 5gu ¯ u 43046721 2157759 97.8 gg → 6gu ¯ u 387420489 17319837 98.6 gg → c¯ cc¯ c 6561 639 99.1 gg → gc¯ cc¯ c 59049 4653 98.8 gg → 2gc¯ cc¯ c 531441 35169 99.0 gg → 3gc¯ cc¯ c 4782969 272835 99.3 gg → 4gc¯ cc¯ c 43046721 2157759 99.6

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Process σMC ± ε (nb) ε (%) gg → 7g (0.53185 ± 0.01149)×10−2 2.1 gg → 8g (0.33330 ± 0.00804)×10−3 2.4 gg → 9g (0.17325 ± 0.00838)×10−4 4.8 gg → 5gu ¯ u (0.38044 ± 0.01096)×10−3 2.8 gg → 3gc¯ cc¯ c (0.95109 ± 0.02456)×10−5 2.6 gg → 4gc¯ cc¯ c (0.81400 ± 0.02583)×10−6 3.2

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Process σMC ± ε (nb) ε (%) gg → Zu ¯ ugg (0.18948 ± 0.00344)×10−3 1.8 gg → W + ¯ udgg (0.62704 ± 0.01458)×10−3 2.3 gg → ZZu ¯ ugg (0.16217 ± 0.00420)×10−6 2.6 gg → W +W −u ¯ ugg (0.27526 ± 0.00752)×10−5 2.7 d ¯ d → Zu ¯ ugg (0.38811 ± 0.00569)×10−5 1.5 d ¯ d → W +¯ csgg (0.18765 ± 0.00453)×10−5 2.4 d ¯ d → ZZgggg (0.99763 ± 0.02976)×10−7 2.9 d ¯ d → W +W −gggg (0.52355 ± 0.01509)×10−6 2.9

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 100 200 300 400 500 600 700 800

dσ/dMjj Mjj

Figure 1: Invariant mass distribution of 2 gluons in the gg → 5g

process. Solid line crosses denote SPHEL case whereas dashed, the

Monte Carlo one.

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SPHEL approximation based on MHV amplitudes
a,ε

|M({pi}n

1 , {εi}n 1 , {ai}n 1 )|2 = 2g2n−4Nn−2 c

(N2

c − 1)

× 2n − 2(n + 1) n(n − 1)

1≤i≤j≤n

(pi · pj)4

P (2,...,n)

1 (p1 · p2)(p2 · p3) . . . (pn · p1),

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Process tSPHEL tMC tMC/tSPHEL gg → 2g 0.372 × 10−3 0.519 × 10−1 139.52 gg → 3g 0.776 × 10−3 0.135 × 100 173.97 gg → 4g 0.252 × 10−2 0.364 × 100 144.44 gg → 5g 0.122 × 10−1 0.143 × 101 117.21 gg → 6g 0.806 × 10−1 0.497 × 101 61.66 gg → 7g 0.639 × 100 0.133 × 102 20.81 gg → 8g 0.569 × 101 0.334 × 102 5.87 gg → 9g 0.567 × 102 0.923 × 102 1.63 gg → 10g 0.620 × 103 0.267 × 103 0.43

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The high-colour processes The idea is to replace colour summation with integration and then follow a MC approach Gµ

AB (Pi) = 8

a=1

Ga(Pi)ηa(z) = √ 6

ziAz∗

iB − 1

3δAB

ǫµ

λ(Pi)

ψA(Pi) = √ 3 u(Pi) ziA ¯ ψA(Pi) = √ 3 ¯ u(Pi) z∗

iA

In such a representation the amplitude can be seen M(z1, z2, . . .) =

z1 · zσ(i)1z2 · zσ(i)2 . . . Ai

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and the MC is over

[dz] ≡

3

i=1

dzidz∗

i

δ(

3

i=1

ziz∗

i − 1)

where

[dz]GAB GCD =
[dz]

√ 6

zAz∗

B − 1

3δAB √ 6

zC z∗

D − 1

3δAB

= 1

2

δADδCB − 1

3δAB δCD

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Multi-jet processes

Beyond any colour treatment a summation over different flavours is also needed. Up to now the most straightforward way was to count the distinct processes and then multiply with a multiplicity factor, i.e. process Flavour gg → ggg 1 q ¯ q → ggg 8 qg → qgg 8 qg → qgg 8 gg → q ¯ qg 5 q ¯ q → q ¯ qg 8 q ¯ q → r ¯ rg 32 qq → qqg 8 q ¯ r → q ¯ rg 24 qr → qrg 24 qg → qq ¯ q 8 qg → qr ¯ r 32 gq → qq ¯ q 8 gq → qr ¯ r 32

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initial-state type distinct processes multiplicity factor A (gg) C1(n) χ(n0, n1, . . . , nf ; f ) B (q ¯ q) C2(n) χ(n0, n2, . . . , nf ; f − 1) C (gq and qg) C2(n − 1) χ(n0, n2, . . . , nf ; f − 1) D (qq) C2(n − 2) χ(n0, n2, . . . , nf ; f − 1) E (qq′ and q ¯ q′) C3(n − 2) χ(n0, n3, . . . , nf ; f − 2) In order to clarify what we mean we consider the example of the type A initial state. Each distinct process is defined by an array (n0, n1, . . . , nf ). For instance, in the case

f four-jet production we have

(4,0,0,0,0,0) gg → gggg (2,1,0,0,0,0) gg → ggq ¯ q (0,2,0,0,0,0) gg → q ¯ qq ¯ q (0,1,1,0,0,0) gg → q ¯ qr ¯ r

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C1(n) =

n0+2n1+...+2nf =n

Θ(n1 ≥ n2 ≥ . . . ≥ nf ) C2(n) =

n0+2n1+...+2nf =n

Θ(n2 ≥ n3 ≥ . . . ≥ nf ) and C3(n) =

n0+2n1+...+2nf =n

Θ(n3 ≥ n4 ≥ . . . ≥ nf ) A distinct process, given by the array (n0, n1, . . . , nf ) has a multiplicity factor : χ(n0, n1, . . . , nf ; f ) = nf (nf − 1)...(nf − j + 1)/j! j = f if

f

i=1

ni = 0 j = f − 1 if

f−1

i=1

ni = 0 . . . j = 1 if n1 = 0 j = 0

therwise

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Now we can think of a flavour-MC, so the wave function is multiplied by an Nf -dimensional array representing flavour , f = Nf (f1, f2, ...) such that

fifj
= δij

with a weight proportional to the relevant pdf for initial state flavours In that case a process like gg → ggq ¯ qq ¯ q will actually represent a plethora of processes. The number of distinct processes is now given by 9k + 3 if n = 2k and 9k + 7 if n = 2k + 1 # of jets 2 3 4 5 6 7 8 9 10 # of D-processes 12 16 21 24 30 34 39 43 48 # of dist.processes 10 14 28 36 64 78 130 154 241 total # of processes 126 206 621 861 1862 2326 4342 5142 8641

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Multi-jet rates pT i > 60 GeV, θij > 30o |ηi| < 3 # jets 3 4 5 6 7 8 σ(nb) 91.41 6.54 0.458 2.97 ×10−2 2.21 ×10−3 2.12 ×10−4 % Gluon 45.7 39.2 35.7 35.1 33.8 26.6

A new code ⇒ JetI

anybody to tell us how many Feynman graphs in gg → 8g ?
or gg → 2g3u3¯

u ?

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Feynman graphs in gg → 8g

10,525,900 !!

or gg → 2g3u3¯

u 946,050!

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PHEGAS

Phase space

dΦn = (2π)4−3n

n

i=1

d3pi 2Ei δ

Ei − w
δ3
pi
RAMBO, VEGAS-based nice but completely inefficient!

dσn = FLUX × |M2→n|2dΦn need appropriate mappings of peaking structures, plus optimization!

Efficiency ⇒ to a large number of generators, each
ne for a specific class of processes.

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HEP - NCSR Democritos Multichannel approach I =

f(

x)dµ( x) = f( x) p( x)p( x)dµ( x) p( x) =

Mch

i=1

αi pi( x)

Mch

i=1

αi = 1 I → f( x) p( x)

E2N →

f( x) p( x) 2 − I2

⋆ Optimize αi ⇒ Minimize E ⋆

R.Kleiss and R.Pittau, Comput. Phys. Commun. 83, 141 (1994).

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HEP - NCSR Democritos New Dyson-Schwinger equations: subamplitude is a combination of several peaking structures! problem unsolved? QCD antennas

P.D.Draggiotis, A.van Hameren and R.Kleiss, hep-ph/0004047.

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HEP - NCSR Democritos New Dyson-Schwinger equations: subamplitude is a combination of several peaking structures! problem unsolved? QCD antennas

P.D.Draggiotis, A.van Hameren and R.Kleiss, hep-ph/0004047.

Old Feynman graphs: exhibit single peaking structure! problem solved

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Back to Feynman graphs: e e e W γ ν u d 1 2 4 16 32 8 48 56 58 The corresponding intrinsic representation looks like 62

2

4

2

58 31 58 31 2

2

56 2 56 2 48 33 8 1 48 33 16

3

32 4

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HEP - NCSR Democritos Time-like momenta q2 ≥ 0 Q Q

1 2

Q

dΦn = . . . dQ2

1

2π dQ2

2

2π dΦ2(Q → Q1, Q2) . . . = . . . dQ2

1

2π dQ2

2

2π d cos θ dφ λ1/2(Q2, Q2

1, Q2 2)

32π2 Q2 . . .

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2

− q Q Q q q Q

2 2 2 1

−

dΦn = . . . dQ2

1

2π dQ2

2

2π dΦ2(Q → Q1, Q2) . . . = . . . dQ2

1

2π dQ2

2

2π dt dφ 1 32π2 Q | q2| . . . t = (Q1 − q2)2 = m2

2 + Q2 1 − E2

Q (Q2 + Q2

1 − Q2 2) + λ1/2

Q | q2| cos θ

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Find limits of t(Q2

1, cos θ):

t± = m2

2 + Q2 1 − E2

Q (Q2 + Q2

1 − Q2 2) ± λ1/2

Q | q2| In order to find the maximum of t+ we study the function ∂t+/∂Q2

1

in the region Q2

1,min < Q2 1 < (Q − Q2)2. Since

∂2t+ ∂(Q2

1)2 = −4Q2Q2 2λ−3/2 |

q2| Q ≤ 0 and ∂t+/∂Q2

1|Q2

1=(Q−Q2)2 → − ∞

we just consider two cases (| q2| = 0):

1. ∂t+/∂Q2

1|Q2

1=Q2 1,min < 0 in which case

tmax = t+,max = t+(Q2

1 = Q2 1,min), and

2. ∂t+/∂Q2

1|Q2

1=Q2 1,min > 0 in which case one can easily derive

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tmax = t+(Q2

1 = x−) with

x− = Q2 +Q2

2−2 Q Q2

1 − E2/Q √α , α =

1 − E2

Q 2 − | q2| Q 2 > 0 ♠ Q2

1-limits:

The limits for the Q2

1-integration for given t can now be fixed by the

condition | cos θ| ≤ 1 or equivalently Π(Q2

1) ≤ 0

with Π(Q2

1) =

t − Q2

1 − m2 2 + E2

Q (Q2 + Q2

1 − Q2 2)

2 − | q2| Q 2 λ If y1 ≤ y2 are the two roots of the polynomial Π(Q2

1) then we have

1. For a > 0 , y− < Q2

1 < y+, with y− = max(y1, Q2 1,min) and

y+ = min(y2, Q2

1,max)

2. For a < 0 we have to satisfy two conditions Q2

1 < y1 or y2 < Q2 1

and Q2

1,min < Q2 1 < Q2 1,max

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At the end we get:

dΦn →

dsi pi(si)
dtj pj(tj)
dφk
d cos θl
p(x) are chosen so that ’singularities’ are smoothed out!
(s − m2)2 + m2Γ2

for massive unstable particles, W±, Z.

sν

for time-like massless propagators, e.g. γ, gluons, fermions.

|t|ν

for space-like massless propagators.

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Final states number of FG √s (GeV) Cross section (fb) u ¯ d s ¯ c γ 90(74) 200 199.75 (16) e− ¯ νe µ+ νµ γ 108(100) 200 29.309 (25) µ− ¯ νµ u ¯ d γ γ 587(210) 500 1.730 (58) µ− ¯ νµ u ¯ d c ¯ c 209(102) 500 0.1783 (20) µ− ¯ νµ u ¯ d c ¯ c γ 2142(339) 500 0.02451 (65)

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6 (and more) fermion production

Input parameters and cuts as defined in hep-ph/0206070 and the WG
DS (Dyson-Schwinger) elements, HC (helicity configurations) and CC

(colour connections) determine the matrix element computational cost.

FG (Feynman graphs) determine the phase-space generation cost.

This is drastically reduced by using ’removing channels’ techniques.

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Top-quark channels final state QCD AMEGIC++ [fb] HELAC [fb] b¯ bu ¯ dd ¯ u yes 32.90(15) 33.05(14) yes 49.74(21) 50.20(13) no 32.22(34) 32.12(19) no 49.42(44) 50.55(26) b¯ bu ¯ ugg – 11.23(10) 11.136(41) – 9.11(13) 8.832(43) b¯ bgggg – 18.82(13) 18.79(11) – 24.09(18) 23.80(17) b¯ bu ¯ de− ¯ νe yes 11.460(36) 11.488(15) yes 17.486(66) 17.492(41) no 11.312(37) 11.394(18) no 17.366(68) 17.353(31) b¯ be+νee− ¯ νe – 3.902(31) 3.885(7) – 5.954(55) 5.963(11) b¯ be+νeµ− ¯ νµ – 3.847(15) 3.848(7) – 5.865(24) 5.868(10) b¯ bµ+νµµ− ¯ νµ – 3.808(16) 3.861(19) – 5.840(30) 5.839(12)

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Vector fusion with Higgs exchange final state QCD AMEGIC++ [fb] HELAC [fb] e−e+u ¯ ud ¯ d yes 0.6842(85) 0.6858(31) yes 1.237(15) 1.265(5) no 0.6453(62) 0.6527(35) no 1.206(14) 1.2394(75) e−e+u ¯ ue−e+ – 6.06(36)e-03 6.113(87)e-03 – 6.58(23)e-03 6.614(80)e-03 e−e+u ¯ uµ−µ+ – 9.24(12)e-03 9.04(11)e-03 – 9.25(17)e-03 9.145(74)e-03 νe ¯ νeu ¯ dd ¯ u yes 1.15(3) 1.176(6) yes 2.36(7) 2.432(12) no 1.14(3) 1.134(5) no 2.35(7) 2.429(13) νe ¯ νeu ¯ de− ¯ νe – 0.426(11) 0.4309(48) – 0.916(30) 0.9121(48) νe ¯ νeu ¯ dµ− ¯ νµ – 0.425(12) 0.4221(30) – 0.878(27) 0.8888(47)

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Vector fusion without Higgs exchange final state QCD AMEGIC++ [fb] HELAC [fb] e−e+u ¯ ud ¯ d yes 0.4838(50) 0.4842(25) yes 1.0514(97) 1.0445(51) no 0.4502(31) 0.4524(23) no 1.0239(79) 1.0227(43) e−e+u ¯ ue−e+ – 3.757(98)e-03 3.577(43)e-03 – 4.082(56)e-03 4.214(46)e-03 e−e+u ¯ uµ−µ+ – 5.201(61)e-03 5.119(70)e-03 – 5.805(67)e-03 5.828(49)e-03 νe ¯ νeu ¯ dd ¯ u yes 0.15007(53) 0.15070(64) yes 0.4755(21) 0.4711(24) no 0.12828(42) 0.12793(55) no 0.4417(19) 0.4398(21) νe ¯ νeu ¯ de− ¯ νe – 0.04546(13) 0.04564(19) – 0.16033(63) 0.16011(78) νe ¯ νeu ¯ dµ− ¯ νµ – 0.0423(12) 0.04180(16) – 0.14383(53) 0.14439(65)

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Higgs production through Higgsstrahlung final state QCD AMEGIC++ [fb] HELAC [fb] µ−µ+µ− ¯ νµe− ¯ νe – 0.03244(27) 0.03210(15) – 0.03747(29) 0.03749(32) µ−µ+u ¯ de− ¯ νe – 0.0924(8) 0.09306(46) – 0.1106(22) 0.10901(66) µ−µ+µ−µ+e−e+ – 2.828(67)e-03 2.923(52)e-03 – 2.731(65)e-03 2.691(42)e-03 µ−µ+u ¯ ud ¯ d yes 0.2534(24) 0.2540(16) yes 0.2634(22) 0.2642(15) no 0.2441(23) 0.2471(15) no 0.2593(22) 0.2589(14) µ−µ+u ¯ uu ¯ u yes 1.125(8)e-02 1.135(22)e-02 yes 8.767(65)e-03 8.978(58)e-03 no 7.929(57)e-03 8.078(92)e-03 no 6.098(35)e-03 6.013(26)e-03

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Backgrounds to Higgsstrahlung final state QCD AMEGIC++ [fb] HELAC [fb] µ−µ+µ− ¯ νµe− ¯ νe – 0.01845(14) 0.01843(13) – 0.03054(23) 0.03092(19) µ−µ+u ¯ de− ¯ νe – 0.05284(57) 0.05209(33) – 0.08911(53) 0.08925(48) µ−µ+µ−µ+e−e+ – 2.204(52)e-03 2.346(49)e-03 – 2.280(66)e-03 2.277(62)e-03 µ−µ+u ¯ ud ¯ d yes 0.1412(10) 0.1404(11) yes 0.2092(12) 0.2075(13) no 0.1358(20) 0.1341(12) no 0.2040(12) 0.2015(11) µ−µ+u ¯ uu ¯ u yes 5.937(24)e-03 5.937(25)e-03 yes 6.134(29)e-03 6.108(27)e-03 no 2.722(10)e-03 2.710(11)e-03 no 3.290(12)e-03 3.303(12)e-03

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Triple Higgs coupling final state QCD AMEGIC++ [fb] HELAC [fb] µ−µ+b¯ bb¯ b yes 2.560(26)e-02 2.583(26)e-02 yes 3.096(60)e-02 3.019(43)e-02 no 1.711(55)e-02 1.666(28)e-02 no 2.34(12)e-02 2.36(10)e-02 Backgrounds to triple Higgs coupling final state QCD AMEGIC++ [fb] HELAC [fb] µ−µ+b¯ bb¯ b yes 7.002(32)e-03 7.044(22)e-03 yes 6.308(24)e-03 6.364(21)e-03 no 2.955(11)e-03 2.972(12)e-03 no 3.704(15)e-03 3.695(13)e-03

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FS CC DS FG HC σ(fb) T µ−µ+b¯ bb¯ b 2 256+256 1158 24 0.00827(18) b¯ bb¯ bu ¯ dµ−νµ 2 938+938 23116 18 0.001576(17) u ¯ de− ¯ νeγ 1 119 108 6 10.87(18) Input parameters: Working Group

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g g → b¯ b b¯ b W−W+

challenging process, from a computational point of view
a nice example to demonstrate the ability of PHEGAS/HELAC to deal with

QCD processes.

background of t¯

tH production MC points result error efficiency efficiency w > 0 (fb) (fb) (%) w > 0 (%) 99442 4.716 0.024 3.3 33

energy √s = 500 GeV and to 1 × 106 MC points.
Feynman graphs for this process is 960, with 4! colour configurations,

without taking into account electroweak contributions from Z and γ intermediate states.

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Parameters used are gQCD = 1, mtop = 175 GeV and Γtop = 1.5 GeV. Moreover the following set of cuts has been applied: Mq,q′ > 20GeV, Eq > 20GeV, | cos θ(q, beam)| < 0.9,

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p p → t ¯ t b ¯ b b ¯ b

Another challenging process, from a computational point of view
A nice example to demonstrate the ability of PHEGAS/HELAC to deal

with QCD processes in a realistic setup.

A background of t¯

tHH production, which seems interesting in a high-luminosity LHC version for HHH coupling.

Feynman graphs for this process is 1454 (gg), with 5! colour

configurations.

A two-phase implementation has been set up.
Srtucture functions and αs from PDFLIB, CTEQ-4L (LO).
Kinematical decays of t → bW+ has been implemented
Cuts: pb

T > 20GeV , |ηb| < 2.5, ∆R > 0.5

The result is 1.053 ± 0.073 (fb) @ LHC

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e− e+ → e− e+ µ− µ+

σtot (in nb) √s(GeV) BDK NEXTCALIBUR 20 98.9 ± 0.6 99.20 ± 0.98 35 131.4 ± 2.2 131.03 ± 0.88 50 154.4 ± 0.9 152.33 ± 0.83 100 205.9 ± 1.2 204.17 ± 1.73 200 — 263.50 ± 1.31 200 (all) — 265.58 ± 1.44

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e− e+ → e− e+ e− e+

σtot (in nb ×107) √s(GeV) BDK NEXTCALIBUR 20 0.920 ± .011 0.905 ± .011 35 1.070 ± .015 1.079 ± .014 50 1.233 ± .018 1.214 ± .016 100 1.459 ± .025 1.485 ± .020 200 — 1.776 ± .019 200 (all) — 1.787 ± .030

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Higher-order corrections

Fermion-loop corrections have been implemented and studied for up

to 3-point vertices.

We have started the computation of 4-point contributions.

– FORM has been used to reduce the expressions to Passarino-Veltman coefficient functions – FF has been updated to include level-4 tensor coefficient functions for 4-point integrals. – Implementation and checking in HELAC is in progress.

This implementation will allow to study 4 fermion+γ and 6 fermion

production including the running of electroweak couplings.

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Fermion-loop corrections to six-fermion production process

e−e+ → µ− ¯ νµu ¯ d τ−τ+

Number of Feynman Graphs: 208
Number of DS vertices: 140
Cuts: El, Eq > 5GeV and mll, mqq > 10GeV
Results: E = 500GeV
σ0/ab = 54, 96(26) σ1/ab = 57, 31(28) K/100 = 4.28(2)
MC data: generated: 1M(961792) used: 404842 time:6 1/2 h

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Latest On-shell recursive equations

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Latest On-shell recursive equations A(1 . . . n) =

n−2

j=2

n−2

j=2

P 2

1...j

R. Britto, F. Cachazo and B. Feng, Nucl. Phys. B 715 (2005) 499
R. Britto, F. Cachazo, B. Feng and E. Witten, Phys. Rev. Lett. 94 (2005) 181602

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A(p1, λ1, . . . , pn, λn) =

λX =±1

n−1

k=1

A(ˆ p1, λ1, . . . , pk, λk, −ˆ pX , −λX ) 1 P 2

1...k

A(ˆ pX , λX , pk+1, λk+1, . . . , ˆ pn, λn)

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A(p1, λ1, . . . , pn, λn) =

λX =±1

n−1

k=1

A(ˆ p1, λ1, . . . , pk, λk, −ˆ pX , −λX ) 1 P 2

1...k

A(ˆ pX , λX , pk+1, λk+1, . . . , ˆ pn, λn)

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A(p1, λ1, . . . , pn, λn) =

λX =±1

n−1

k=1

A(ˆ p1, λ1, . . . , pk, λk, −ˆ pX , −λX ) 1 P 2

1...k

A(ˆ pX , λX , pk+1, λk+1, . . . , ˆ pn, λn) ABCF 1→n =

n−1

k=1

A1→k+1A1→n−k+1 (1)

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The equation ABCF 1→n = A1→n cannot be true because of the following reasons:

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The equation ABCF 1→n = A1→n cannot be true because of the following reasons:

Firstly, contributions where no propagator line exists between the 1st

and the nth particle, are not included in the BCFW equation

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The equation ABCF 1→n = A1→n cannot be true because of the following reasons:

Firstly, contributions where no propagator line exists between the 1st

and the nth particle, are not included in the BCFW equation These contributions are given by A0

1→n = A1→n−1 +

n1+n2=n−1

A1→n1A1→n2 (2)

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Secondly, BCFW is multiple-counting contributions of the form

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Secondly, BCFW is multiple-counting contributions of the form

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Secondly, BCFW is multiple-counting contributions of the form

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In fact we suggest that this over counting is exactly equal to the multiplicity of propagator lines connecting particles 1 and n. To make our arguments more quantitative we start with the Berends-Giele (or Dyson-Schwinger for ordered graphs) recursive equation for a generic theory with 3− and 4−vertices.

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In fact we suggest that this over counting is exactly equal to the multiplicity of propagator lines connecting particles 1 and n. To make our arguments more quantitative we start with the Berends-Giele (or Dyson-Schwinger for ordered graphs) recursive equation for a generic theory with 3− and 4−vertices. A1→n =

n1+n2=n

A1→n1A1→n2 +

n1+n2+n3=n

A1→n1A1→n2A1→n3

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A1→n = ABCF 1→n + A0

1→n − D

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A1→n = ABCF 1→n + A0

1→n − D

D =

n−1

M =3

(M − 2)

M

k=0

  M k   Dn

M +k

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A1→n = ABCF 1→n + A0

1→n − D

D =

n−1

M =3

(M − 2)

M

k=0

  M k   Dn

M +k

where M − 1 is the number of propagators of the particular overcounted class of diagrams and Dn

M =

n1+...+nM =n

A1→n1 . . . A1→nM counts the number of diagrams within that class.

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1 → n A ABCF A0 ABCF + A0 − A D 3 3 1 2 4 10 6 5 1 1 5 38 29 17 8 8 6 154 136 64 46 46 7 654 636 259 241 241 8 2871 2992 1098 1219 1219 9 12925 14190 4815 6080 6080 10 59345 67860 21659 30174 30174 11 276835 327080 99385 149630 149630

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Kinematical identities

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Kinematical identities ˆ f(p; z) =

j
ˆ

f(p; z)(z − zj)

z=zj

1 z − zj

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Kinematical identities ˆ f(p; z) =

j
ˆ

f(p; z)(z − zj)

z=zj

1 z − zj 1 ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

k

=

j=1..k
1

ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

j−1 ˆ

pj+1 . . . ˆ p2

k

z=zj

1 ˆ p2

j

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Kinematical identities ˆ f(p; z) =

j
ˆ

f(p; z)(z − zj)

z=zj

1 z − zj 1 ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

k

=

j=1..k
1

ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

j−1 ˆ

pj+1 . . . ˆ p2

k

z=zj

1 ˆ p2

j

with zj such that ˆ pj(zj)2 = (pj + zjǫ)2 = p2

j + 2zjpj ·ǫ = 0

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Kinematical identities ˆ f(p; z) =

j
ˆ

f(p; z)(z − zj)

z=zj

1 z − zj 1 ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

k

=

j=1..k
1

ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

j−1 ˆ

pj+1 . . . ˆ p2

k

z=zj

1 ˆ p2

j

with zj such that ˆ pj(zj)2 = (pj + zjǫ)2 = p2

j + 2zjpj ·ǫ = 0

1 p2

1p2 2 . . . p2 k

=

j=1..k
1

ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

j−1 ˆ

p2

j+1 . . . ˆ

p2

k

z=zj

1 p2

j

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zρ ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

k

=

j=1..k
zρ

ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

j−1 ˆ

p2

j+1 . . . ˆ

p2

k

z=zj

1 ˆ p2

j

which gives the very useful set of identities, valid for every ρ < k

j=1..k
zρ

ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

j−1 ˆ

pj+1 . . . ˆ p2

k

z=zj

1 p2

j

= 0

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Finally, if z = k, the function in the left hand side is no longer vanishing at z → ∞. Subtracting its limit at infinity, however, we have a new function that does, so

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Finally, if z = k, the function in the left hand side is no longer vanishing at z → ∞. Subtracting its limit at infinity, however, we have a new function that does, so zk ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

k

− lim

z→∞

zk ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

k

=

j=1..k
zk

ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

j−1 ˆ

pj+1 . . . ˆ p2

k

z=zj

1 ˆ p2

j

and taking the limit z → 0 we get

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Finally, if z = k, the function in the left hand side is no longer vanishing at z → ∞. Subtracting its limit at infinity, however, we have a new function that does, so zk ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

k

− lim

z→∞

zk ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

k

=

j=1..k
zk

ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

j−1 ˆ

pj+1 . . . ˆ p2

k

z=zj

1 ˆ p2

j

and taking the limit z → 0 we get − lim

z→∞

zk ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

k

=

j=1..k
zk

ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

j−1 ˆ

pj+1 . . . ˆ p2

k

z=zj

1 p2

j

r

−1 k

j=1 2ǫ·pj

=

j=1..k
zk

ˆ p2

1 ˆ

p2

2 . . . ˆ

p2

j−1 ˆ

pj+1 . . . ˆ p2

k

z=zj

1 p2

j

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Gauge choice: we choose to use p1 = pn and pn = p1

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Gauge choice: we choose to use p1 = pn and pn = p1

ǫµ

+1 =

˜ λ˙

a(p1)σµ, ˙ aaλa(pn)

√ 2n1 ǫµ

−n = −

˜ λ˙

a(p1)σµ, ˙ aaλa(pn)

√ 2 [n1]

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Gauge choice: we choose to use p1 = pn and pn = p1

ǫµ

+1 =

˜ λ˙

a(p1)σµ, ˙ aaλa(pn)

√ 2n1 ǫµ

−n = −

˜ λ˙

a(p1)σµ, ˙ aaλa(pn)

√ 2 [n1] ǫ+1·pn = 0 = ǫ−n ·p1. ǫµ ≡ 1 2 ˜ λ˙

a(p1)σ ˙ aa µ λa(pn)

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As a consequence, any diagram in which the first and the last leg meet in a three-vertex vanishes.

1 n = 0

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Momentum shift

p1 → p1 + zǫ pn → pn − zǫ λa(p1) → λa(p1) + zλa(pn) ˜ λ˙

a(pn) → ˜

λ˙

a(pn) − z˜

λ˙

a(p1)

(3) As a result, the denominators of ǫ+1 and ǫ−n become n1 → n1 + znn = n1 and [n1] → [n1] − z[11] = [n1]

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Let us call ‘hatted’ diagrams graphs of the form

A B C

ˆ 1 ˆ n F(m1, . . . , ˜ mj, ˜ mj+1, . . . , mk+1) =

. . . . . . m1 ˜ mj ˜ mj+1 mn 1 n

H(m1, . . . , mk+1; j) =

ˆ

F(m1, . . . , mk+1)ˆ p2

1j

z=zi

1 p2

1j

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= ˆ Vµνρ = gµν (ˆ p1 − p2)ρ + gνρ(p2 − ˆ p3)µ + gρµ(ˆ p3 − ˆ p1)ν = gµν (p1 − p2)ρ + gνρ(p2 − p3)µ + gρµ(p3 − p1)ν −zǫσ(2gµρgνσ − gµσgνρ − gµν gσρ) = Vµνρ − zǫσ(2gµρgνσ − gµσgνρ − gµνgσρ) = +

ǫσ

(4) where

ǫσ

= −zVµνρσ (5) with Vµνρσ the QCD four-vertex.

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If we write Jµ

Q for the current coming from the blob Q, we have

H(m1, m2; 1) ≡

A B

=

ǫ1µJAν ˆ

V µνρ ˆ VρκλJκ

Bǫλ n

z0

1 p2

1A

=

ǫ1µJAν (V µνρ − zǫσV σµνρ)(Vρκλ − zǫτ Vτ ρκλ)Jκ

Bǫλ n

z0

1 p2

1A

=

ǫ1µJAν V µνρVρκλJκ

Bǫλ n − zǫ1µJ1ν ǫσV σµνρVρκλJκ B ǫλ n

−zǫ1µJAν V µνρǫτ Vτ ρκλJκ

B ǫλ n

+z2ǫ1µJAν ǫσV σµνρǫτ Vτ ρκλJκ

B ǫλ n

z0

1 p2

1A

(6)

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H(m1, m2; 1) ≡

A B

= =

A B

− [2(ǫ·p1A)(ǫ1·JA)(ǫn ·JB )]z0 1 p2

1A

with z0 = − p2

1A

2(p1Aǫ) (7)

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H(m1, m2; 1) ≡

A B

=

A B

+ (8)

r

H(m1, m2; 1) = F(m1, m2) + F( ˜ m1, ˜ m2) (9)

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A B C

+

A B C

(10) +

A C B

+

A C B

=

A B C

+

A C B

+

A C B

(11)

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Progress in MultiLoops

MHV rules to calculate one-loop amplitudes
A. Brandhuber, B. J. Spence and G. Travaglini, Nucl. Phys. B 706 (2005) 150

L 1 M H V M H V L 2 m 2 m 1 m 1 ✠ 1 m 2 + 1

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BCF recursive equations to calculate one-loop amplitudes
Z. Bern, L. J. Dixon and D. A. Kosower, Phys. Rev. D 73 (2006) 065013

An(0) = cΓ   Cn(0) −

poles α

Res

z=zα

Rn(z)

z   −

poles α

Res

z=zα

Rn(z) z ≡ RD

n (k1, . . . , kn)

=

partitions P
h=±

  R(kP1, . . . , ˆ kj, . . . , kP−1, − ˆ P h) × i P 2 × Atree(kP 1, . . . , ˆ kl, . . . , kP −1, ˆ P −h) + Atree(kP1, . . . , ˆ kj, . . . , kP−1, − ˆ P h) × i P 2 × R(kP 1, . . . , ˆ kl, . . . , kP −1, ˆ P −h)    ,

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T L

^

(a) + − 2 1 4 5 + − L T −

^

5+

^

+ 4 1− − + (d) − +

^ ^

L T 2 4 5 1 + − T L

^

2 3 5 + − (b) (c) 5+ − + T L 4 −

^

2 3 − (e) L T 5 3 − + − (f) 3 + 3 2 3 + − + 4 4 +

^

+ + + − + − +

^

+ + 1

^

+ − 2

^ ^

1 + − 1

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A(1−, 2−, 3+, . . . , n+) QCD one-loop amplitudes completed
D. Forde and D. A. Kosower, Phys. Rev. D 73 (2006) 061701

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All order results for N = 4 SYM
Z. Bern, L. J. Dixon and V. A. Smirnov, Phys. Rev. D 72 (2005) 085001

Mn ≡ 1 +

∞

L=1

aLM(L)

n

(ǫ) = exp  

∞

l=1

al

f(l)(ǫ)M(1)

n

(lǫ) + C(l) + E(l)

n (ǫ)

  . a ≡ Ncαs 2π (4πe−γ )ǫ , f(l)(ǫ) = f(l) + ǫf(l)

1

+ ǫ2f(l)

2

.

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M(2)

4

(ǫ) = 1 2

M(1)

4

(ǫ) 2 + f(2)(ǫ) M(1)

4

(2ǫ) + C(2) + O(ǫ) ,

C. Anastasiou, Z. Bern, L. J. Dixon and D. A. Kosower, Phys. Rev. Lett. 91, 251602 (2003)

M(3)

4

(ǫ) = −1 3

M(1)

4

(ǫ) 3 + M(1)

4

(ǫ) M(2)

4

(ǫ) + f(3)(ǫ) M(1)

4

(3 ǫ) + C(3) +O(ǫ) , Extended to 5-point amplitudes by

Z. Bern, M. Czakon, D. A. Kosower, R. Roiban and V. A. Smirnov, arXiv:hep-th/0604074.

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Outlook

PHEGAS / HELAC: a framework for high-energy phenomenology
Standard Model fully included

⋆ High color charge processes: multijet production

P.Draggiotis, R.H.Kleiss and C.G.Papadopoulos, Phys. Lett. B439 (1998) 157;

Eur. Phys. J. C 24 (2002) 447 hep-ph/0202201
C. G. Papadopoulos and M. Worek, arXiv:hep-ph/0512150

⋆ Higher order corrections – Direct approach. Ongoing work to better understand Dyson-Schwinger equations and loop calculations: stepping equations, recursive actions, etc. – Running couplings and masses: 4-point FL contributions and BBC non-local approch to go beyond 4-fermion final states.

SUSY and new particles

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The years to come we will live a very exciting period for particle

physics phenomenology

We will learn lots of new things by confronting precision calculations

and developed relevant tools with the experimental data

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For an 1 → n color ordered amplitude the number of 3-vertex

n−1

k=1

(n − k)k

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For an 1 → n color ordered amplitude the number of 3-vertex

n−1

k=1

(n − k)k and for the 4-vertex their number is

n−1

k=2

(n − k) k(k − 1) 2

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For an 1 → n color ordered amplitude the number of 3-vertex

n−1

k=1

(n − k)k and for the 4-vertex their number is

n−1

k=2

(n − k) k(k − 1) 2 where n − k is the number of k-words in the length n object (1, 2, 3, 4) → (1, 2)(2, 3)(3, 4) (1, 2, 3, 4) → (1, 2, 3)(2, 3, 4)

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numberofterms =   n + 2 3   +   n + 2 4  

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numberofterms =   n + 2 3   +   n + 2 4  

gg → ng 2 3 4 5 6 7 8 9 # 5 15 35 70 126 210 330 495 FG-O 3 10 38 154 654 2871 12,925 59,345 FG-U 4 25 220 2,485 34,300 559,405 10,525,900 224,449,225

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numberofterms =   n + 2 3   +   n + 2 4  

gg → ng 2 3 4 5 6 7 8 9 # 5 15 35 70 126 210 330 495 FG-O 3 10 38 154 654 2871 12,925 59,345 FG-U 4 25 220 2,485 34,300 559,405 10,525,900 224,449,225

a(n) = 1 2!

n!

n1!n2! a(n1)a(n2)δn1+n2,n + 1 3!

n!

n1!n2!n3!a(n1)a(n2)a(n3)δn1+n2+n3,n a(1) = 1

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V A = (V 0 + Vz, V 0 − Vz, Vx + iVy, Vx − iVy) , A = 1, . . . , 4 . Polarization state-vectors are given by ǫA

−

=

−pT

√ 2| p| , pT √ 2| p| , (px + ipy)(| p| + pz) √ 2| p|pT , (px − ipy)(−| p| + pz) √ 2| p|pT

ǫA

+

=

pT

√ 2| p| , −pT √ 2| p| , (px + ipy)(| p| − pz) √ 2| p|pT , (px − ipy)(−| p| − pz) √ 2| p|pT

ǫA

=

|

p|

p2 +

pzp0 | p|

p2 ,

| p|

p2 −

pzp0 | p|

p2 ,

(px + ipy)p0 | p|

p2

, (px − ipy)p0 | p|

p2
As for the Dirac matrices we are using the chiral representation. The wave functions

which describe massive spinors are given by: u+(p) =       r/c a(px + ipy)/r −mb/r −m(px + ipy)/r       ¯ u+(p) =       mb/r m(px − ipy)/r −r/c −a(px − ipy)/r       u−(p) =       m(px − ipy)/r −mb/r −a(px − ipy)/r r/c       ¯ u−(p) =       a(px + ipy)/r −r/c −m(px + ipy)/r mb/r      

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v+(p) =       −m(px − ipy)/r mb/r −a(px − ipy)/r r/c       ¯ v+(p) =       a(px + ipy)/r −r/c m(px + ipy)/r −mb/r       v−(p) =       r/c a(px + ipy)/r mb/r m(px + ipy)/r       ¯ v−(p) =       −mb/r −m(px − ipy)/r −r/c −a(px − ipy)/r       where: a = p0 + | p|, b = pz + | p|, c = 2| p|, r = √ abc For a massless particle the spinors are uR(p) =      

p0 + pz

(px + ipy)/

p0 + pz

      ¯ uR(p) =       −

p0 + pz

−(px − ipy)/

p0 + pz

      uL(p) =       −(px − ipy)/

p0 + pz
p0 + pz

      ¯ uL(p) =       (px + ipy)/

p0 + pz

−

p0 + pz

     

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Vµ = ¯ ψ(P1)γµ (gRωR + gLωL) ψ(P2) turns out to be V A =       −gRψ1 ¯ ψ3 − gLψ4 ¯ ψ2 −gRψ2 ¯ ψ4 − gLψ3 ¯ ψ1 −gRψ2 ¯ ψ3 + gLψ4 ¯ ψ1 −gRψ1 ¯ ψ4 + gLψ3 ¯ ψ2       where ψi( ¯ ψi), i = 1, . . . , 4 are the components of the spinor ψ(P2) ¯ ψ(P1)

and

ωL = 1 2 (1 − γ5) , ωR = 1 2 (1 + γ5) . On the other hand, the spinor u = (P / + m)b /(P1)ωRψ(P2) can be reduced to u =       (−b2p1 + b3p4)ψ1 + (b4p1 − b1p4)ψ2 (b3p2 − b2p3)ψ1 + (−b1p2 + b4p3)ψ2 m(b2ψ1 − b4ψ2) m(−b3ψ1 + b1ψ2)       where ψi, bi, pi, i = 1, . . . , 4 are the components of P , b(P1) and ψ(P2) respectively.