All-gluon amplitudes with off-shell recursion in multiplet bases

Malin Sjodahl; Oskar Bolinder; Rikkert Frederix

REVIEW 3 major objections 3 minor 1 cited by

All-gluon amplitudes with off-shell recursion in multiplet bases

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims all-gluon color-summed squared amplitudes can be computed in $O(17^n)$ operations with off-shell recursion in orthogonal multiplet bases, beating factorial scaling.

desk verdict A genuinely new multiplet-basis recursion formalism, but the headline O(17^n) complexity depends on an unproven combinatorial count that may actually be Catalan-sized. read the letter →

arxiv 2507.22636 v1 pith:4EDT74HU submitted 2025-07-30 hep-ph

Oskar Bolinder , Rikkert Frederix , Malin Sjodahl This is my paper

classification hep-ph

keywords QCDamplitudesmultipletbasesoff-shellrecursionWigner6jsymbolscolorsummationcomputationalcomplexityall-gluonscatteringtree-level

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that color-summed squared amplitudes for tree-level processes with $n$ gluons can be computed in $O(17^n)$ operations by using off-shell recursion directly in orthogonal color multiplet bases. The authors construct multiplet-projected currents from lower-point currents, handling the color algebra with Wigner $6j$ coefficients and the birdtrack graphical calculus. Their optimized recursion reuses intermediate color configurations, reducing a naive $O(129^n)$ cost to $O(17^n)$. If the claim is right, multiplet bases would outperform trace, color-flow, and adjoint bases, whose cost scales as $O(((n-1)!)^2)$, already at moderate gluon multiplicity. This matters for precision high-multiplicity QCD predictions at the LHC.

What carries the argument

The central object is the multiplet-projected off-shell current, built recursively by coupling two lower-point currents through a three-gluon vertex. The color coefficients are obtained with Wigner $6j$ coefficients through a swap rule that moves a gluon past its neighbor and introduces a sum over intermediate representations. The optimized algorithm reuses identical intermediate birdtrack diagrams across different partitions, so the number of operations per gluon order becomes $8^{k-2}u_k$ with $u_k$ the number of unique orderings encountered, argued to scale like $O(2^k)$; this is what turns the naive $O(129^n)$ into $O(17^n)$.

What would settle it

Instrument the swapping procedure: for $k = 8, 10, 12, 14$, count the number of unique intermediate orderings $u_k$ actually produced. If $u_k$ grows faster than roughly a constant times $2^k$, the base 17 in eq. (24) is too small; equivalently, an implementation's total operation count for $n$ up to about 14 can be compared with the predicted $17^n$ growth.

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Extended reading notes

Core claim

The central claim is that tree-level $n$-gluon color-summed squared amplitudes can be computed directly in an orthogonal color multiplet basis by building off-shell currents from lower-point currents, and that after caching and partial summation the computational cost is $O(17^n)$, as stated in eq. (24). The recursion projects every current onto definite color representations using Wigner $6j$ coefficients, and because the basis is orthogonal the final color sum costs only $O(8^{n-3})$. This scaling improves over the naive $O(129^n)$ bound and over the $O(((n-1)!)^2)$ cost of trace, color-flow, and adjoint bases, making multiplet bases competitive for moderate gluon multiplicity.

Load-bearing premise

The claim that the optimized algorithm costs $O(17^n)$ rests on the heuristic that the number of distinct intermediate gluon orderings $u_k$ grows like $O(2^k)$; this is argued by counting partitions, not proved, and the caching that achieves it is described only in prose.

Editorial extensions

If this is right

For moderate $n$, the multiplet-basis recursion becomes cheaper than trace, color-flow, and adjoint bases, which scale as $O(((n-1)!)^2)$.
The final color sum is cheap because the basis is orthogonal, costing only $O(8^{n-3})$ operations.
Replacing gluons by quarks in the recursion is expected to reduce intermediate sums and improve scaling, so the method extends beyond all-gluon processes.
An implementation inside a Monte Carlo event generator would allow the exact complexity to be measured for moderate $n$ and the framework to be used for high-multiplicity predictions.
Leading-color flow assignment needed for parton showering can be handled by two-step event generation methods, as noted in the paper's outlook.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: comparing $17^n$ with $((n-1)!)^2$ under raw operation counts suggests the crossover with the trace basis occurs around $n=12$; exact constants could shift this either way.
Editorial extension: if the $O(17^n)$ scaling holds, stochastic color sampling may become unnecessary for all-gluon tree-level processes at moderate $n$, avoiding the Monte Carlo variance it introduces.
Editorial extension: a direct test is to instrument an implementation and count the number $u_k$ of unique intermediate orderings for $k$ up to about 15; if $u_k$ grows faster than $2^k$, the base 17 in eq. (24) would need to be revised upward.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Punchline: this is a worthwhile methods paper for multiplet-basis color recursion, but the advertised O(17^n) complexity is not yet proven. The u_k ~ 2^k assumption in eq. (24) does not survive contact with the stress test: it counts the initial two-block partitions, not the intermediate orderings created by swaps. The set of orderings the swapping procedure can pass through looks like it could be the 321-avoiding permutations, of which there are Catalan (~4^k), not 2^k. If so the base becomes ~33, which still beats squared-factorial for large n but moves the crossover from around n=12 to around n=16-17.

What is genuinely new: the construction of off-shell recursion directly in orthogonal multiplet bases for general helicity, using Wigner 6j and 3j symbols. The naive O(129^n) counting is clear and checkable. The birdtrack manipulations are careful, and the paper is honest that the optimized scaling is an analysis, not an implementation. The auxiliary antisymmetric tensor for splitting the four-gluon vertex is a standard trick from color-dressed recursion, so no concern there.

Soft spots, in proportion: (1) the u_k count is the load-bearing one; no proof, no numerical check, and the footnote with the reverse-bubble-sort alternative is a sketch, not a derivation. (2) No code or small-n test, so even the 2^k variant is unverified. Neither is fatal to the framework; the recursion and the 129^n bound stand on their own, and the paper frames the optimization as a proposal.

Who it's for: amplitude practitioners working on color summation and high-multiplicity automation, and those tracking the multiplet-basis program of Sjodahl et al. A serious referee should ask for an explicit algorithm that achieves u_k = k 2^{k-1} - 1, or a numerical count of u_k for k up to ~10 to settle 2^k vs 4^k. I would send it to peer review with that demand; despite my skepticism about the headline base, the paper deserves referee time.

Referee Report

3 major / 3 minor

Summary. The paper develops an off-shell recursion for computing tree-level all-gluon amplitudes directly in orthogonal SU(N_c) multiplet bases, using Wigner 3j/6j symbols and birdtrack manipulations. It defines k-gluon currents with fixed canonical orderings and constructs the color coefficients by swapping adjacent gluons with 6j factors. A naive counting of currents and partitions gives O(129^n); the authors then claim that an optimized algorithm with caching and partial summation reduces the cost to O(17^n), assuming that the number u_k of distinct orderings encountered during the swapping procedure grows like 2^k. The paper contains no implementation and defers numerical verification to future work.

Significance. If the O(17^n) bound is correct, the paper offers a novel and potentially important route to color-summed multi-gluon amplitudes that could beat the factorial scaling of trace, color-flow, and adjoint bases at moderate n. The use of orthogonal multiplet bases is conceptually clean, the naive counting bound is explicit, and the W coefficients are obtained from previously derived 6j symbols rather than from fitted parameters. These are real strengths. The central weakness is that the optimized bound rests on an unproven combinatorial assertion about u_k, and no numerical or machine-checked evidence is provided; the paper itself identifies implementation as future work.

major comments (3)

[§3.2, Eq. (24)] The O(17^n) claim depends entirely on the assertion u_k ~ 2^k. The stated justification counts the number of partitions of k objects into two unsorted sets, which is the number of starting configurations, not the number of distinct orderings generated by the swapping procedure. After an adjacent swap of an out-of-order pair in a two-block start, the configuration can have multiple descents and need not be a concatenation of two increasing blocks; for example, for k=5 the ordering [1,3,2,5,4] can arise as an intermediate but is not of the initial two-block form. The reachable set is therefore not bounded by the provided argument and may grow as the Catalan number ~4^k/k^{3/2}; if so, the base in Eq. (24) becomes ~33 rather than 17. Since the abstract and the conclusion rest on this exponent, please provide a proof of the 2^k bound or a direct enumeration of u_k for small k.
[§3.2, Eqs. (22)-(23)] The optimized algorithm is specified only in prose; there is no pseudocode, cache-key definition, or formal recurrence showing that the number of operations is 8^{k-2} u_k. The description of merging contributions once a gluon reaches its final position does not by itself control the number of distinct intermediate orderings, because different merging sequences can produce the same or different canonical states in ways that are not quantified. This is not merely a presentation issue, because Eq. (23) is the load-bearing step that converts the naive O(129^n) bound into O(17^n). A precise statement of the caching invariant and a bound on the size of the cache are needed.
[§3, footnote 2 and §3.1] The four-gluon interaction is removed by introducing an auxiliary non-propagating anti-symmetric tensor-like particle, but the color representation of this particle and the corresponding W coefficients in the multiplet basis are not defined. Since the recursion is built entirely from three-gluon vertices and the W coefficients are computed from gluon-line swaps, the reader cannot verify that the vertex-splitting preserves the color projection for four-gluon amplitudes. Please spell out the representation content and the modified swapping rule, or cite a derivation that covers this case in multiplet bases.

minor comments (3)

[Eq. (14), footnote 4] The factor 8^{i-2} gives fractional operation counts for i=1 (and i=k-1); the max(...,1) prescription noted for k=1 should also be applied to the subcurrent factors, e.g. m_2 becomes 1 rather than 1/128. The asymptotic base is unaffected, but the displayed formula is not literally an upper bound for small k.
[§3.1] The statement that the recursion is 'of at most exponential complexity' is trivially true for a finite sum; the substance is the value of the base, which should be stated more carefully to avoid conflating the naive and optimized bounds.
[§3.2] The paper does not discuss memory requirements of the caching scheme; storing intermediate states for all 8^{k-2} u_k orderings may be a practical bottleneck, and a brief comment on this trade-off would strengthen the complexity analysis.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the O(17^n) scaling is a combinatorial counting estimate, and the cited multiplet-basis and 6j results are independent prior derivations rather than self-referential inputs.

full rationale

The central complexity claim, eq. (24), is obtained by counting the number of k-gluon currents (C(n-1,k) 8^{k-2}) and estimating the number u_k of unique gluon orderings encountered in the bubble-sort swapping procedure as O(2^k). This is an arithmetic estimate, not a fit: no parameter is tuned to a target output, and the 'prediction' is not defined in terms of the quantity it predicts. The Wigner 6j machinery is imported from refs. [16,17] and multiplet bases from refs. [11,12]; these are separate published constructions and calculations with their own derivations, and they do not assume the paper's O(17^n) result, so the self-citations are legitimate evidence rather than a load-bearing circular chain. The swapping rule, eq. (8), is derived here from completeness and vertex-correction identities. The acknowledged caveat, that u_k ~ 2^k is a heuristic and the conclusion states that an implementation is needed to 'study the exact complexity of the algorithm for moderately large values of n', is a matter of rigor in a combinatorial estimate, not circularity. Accordingly, no step in the derivation reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central claim is a complexity estimate, so no parameters are fitted to data. It depends on five stated assumptions: the 8^{n-3} bound on basis size, the auxiliary-field replacement of the four-gluon vertex, the validity of the swap rule for color structures, the heuristic u_k ~ 2^k scaling, and the availability of efficient precomputed Wigner 6j symbols from prior work. The u_k assumption is the most fragile because it directly produces the 17 in the claimed asymptotic base.

assumptions (5)

domain assumption At most 8^{n-3} multiplet basis vectors span the n-gluon color space (each new gluon adds at most 8 allowed representations in the product 8⊗α).
Used in section 2.1 and throughout the counting in eqs. (13), (14), and (23); taken from ref [11].
domain assumption The four-gluon interaction can be replaced by effective three-gluon vertices involving a non-propagating anti-symmetric tensor-like auxiliary particle.
Stated in section 3 before eq. (11); follows ref [25]; essential for the recursion to treat only triple-gluon vertices.
standard math The swap rule in eq. (8) correctly moves a gluon past a chain of general representations, apart from possible minus signs noted only in passing.
Introduced in section 2.2 and used in section 3.2 to derive all color coefficients; the sign subtleties for vertices read in reverse order are mentioned but not worked out completely.
ad hoc to paper The number of unique orderings u_k encountered during the swapping procedure scales like O(2^k).
Introduced in section 3.2 after eq. (23); justified by the observation that the number of partitions of k objects into two unsorted sets is O(2^k). This heuristic is load-bearing for the claimed O(17^n) scaling.
domain assumption All required Wigner 3j and 6j symbols can be computed efficiently once and for all, so their precomputation does not affect the asymptotic complexity.
Assumed in section 2.2 and section 3; relies on refs [16,17] by the same group. If precomputation had non-negligible cost, the 17^n bound would change.

invented entities (1)

Auxiliary non-propagating anti-symmetric tensor-like particle
purpose: To split the four-gluon vertex into effective three-gluon vertices so the recursion only needs triple-gluon vertices.
Borrowed from ref [25]; purely computational device, not a physical particle, no falsifiable prediction.

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Cite this review

Pith. "Pith review of All-gluon amplitudes with off-shell recursion in multiplet bases." pith.science (2026). https://pith.science/paper/4EDT74HU

@misc{pith2026250722636,
  author       = {Pith},
  title        = {Pith review of: All-gluon amplitudes with off-shell recursion in multiplet bases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4EDT74HU}},
  note         = {Machine review of arXiv:2507.22636}
}

abstract

The efficient computation of color-summed QCD amplitudes at high parton multiplicities remains a central challenge for precision collider predictions. Existing approaches using trace, color-flow, or adjoint bases suffer from non-orthogonality, which complicates the color algebra and scales poorly with multiplicity. In this work, we present an off-shell recursive framework for computing all-gluon tree-level amplitudes directly in orthogonal multiplet bases. Utilizing Wigner $6j$ coefficients, we construct an algorithm that builds multiplet-projected off-shell currents from lower-point currents. By optimizing the recursion through partial summation and caching, we find that the computational complexity of calculating $n$-gluon color-summed squared amplitudes scales as $\mathcal{O}(17^n)$. This demonstrates the potential competitiveness of multiplet bases for high-multiplicity processes.

Figures

Figures reproduced from arXiv: 2507.22636 by the authors.

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Works this paper leans on

35 extracted references · 14 canonical work pages · cited by 1 Pith paper

[1]

Paton and H.-M

J.E. Paton and H.-M. Chan,Generalized veneziano model with isospin, Nucl. Phys. B10 (1969) 516. one can easily find optimizations. For example, by considering currents with both canonical and reverse ordering (at a price of needing to compute twice as many currents) and performing a reverse bubble sort to order the gluons in the birdtrack diagrams, the va...

work page 1969
[2]

Mangano, S.J

M.L. Mangano, S.J. Parke and Z. Xu,Duality and Multi - Gluon Scattering, Nucl. Phys. B298 (1988) 653

work page 1988
[3]

Sjodahl,Color structure for soft gluon resummation: A General recipe, JHEP 09 (2009) 087 [0906.1121]

M. Sjodahl,Color structure for soft gluon resummation: A General recipe, JHEP 09 (2009) 087 [0906.1121]

arXiv 2009
[4]

Del Duca, L.J

V. Del Duca, L.J. Dixon and F. Maltoni,New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B571 (2000) 51 [hep-ph/9910563]

arXiv 2000
[5]

Maltoni, K

F. Maltoni, K. Paul, T. Stelzer and S. Willenbrock,Color Flow Decomposition of QCD Amplitudes, Phys. Rev. D67 (2003) 014026 [hep-ph/0209271]

arXiv 2003
[6]

PHEGAS: a phase-space generator for automatic cross-section computation

C.G. Papadopoulos,PHEGAS: A Phase space generator for automatic cross-section computation, Comput. Phys. Commun.137 (2001) 247 [hep-ph/0007335]

work page Pith review arXiv 2001
[7]

Mangano, M

M.L. Mangano, M. Moretti, F. Piccinini, R. Pittau and A.D. Polosa,ALPGEN, a generator for hard multiparton processes in hadronic collisions, JHEP 07 (2003) 001 [hep-ph/0206293]

arXiv 2003
[8]

Gleisberg and S

T. Gleisberg and S. Hoeche,Comix, a new matrix element generator, JHEP 12 (2008) 039 [0808.3674]

arXiv 2008

Show all 35 references

[9]

De Angelis, J.R

M. De Angelis, J.R. Forshaw and S. Plätzer,Resummation and Simulation of Soft Gluon Effects beyond Leading Color, Phys. Rev. Lett.126 (2021) 112001 [2007.09648]

2021 arXiv
[10]

Isaacson and S

J. Isaacson and S. Prestel,Stochastically sampling color configurations, Phys. Rev. D 99 (2019) 014021 [1806.10102]

2019 arXiv
[11]

Keppeler and M

S. Keppeler and M. Sjodahl,Orthogonal multiplet bases in SU(Nc) color space, JHEP 09 (2012) 124 [1207.0609]

2012 arXiv
[12]

Sjodahl and J

M. Sjodahl and J. Thorén,QCD multiplet bases with arbitrary parton ordering, JHEP 11 (2018) 198 [1809.05002]

2018 arXiv
[13]

Alcock-Zeilinger and H

J. Alcock-Zeilinger and H. Weigert,Transition Operators, J. Math. Phys.58 (2017) 051703 [1610.08802]

2017 arXiv
[14]

Cvitanovic,Group Theory: Birdtracks, Lie’s, and Exceptional Groups, Princeton University Press (5, 2020)

P. Cvitanovic,Group Theory: Birdtracks, Lie’s, and Exceptional Groups, Princeton University Press (5, 2020)

2020
[15]

Sjodahl and J

M. Sjodahl and J. Thorén,Decomposing color structure into multiplet bases, JHEP 09 (2015) 055 [1507.03814]

2015 arXiv
[16]

Alcock-Zeilinger, S

J. Alcock-Zeilinger, S. Keppeler, S. Plätzer and M. Sjodahl,Wigner 6j symbols for SU(N): Symbols with at least two quark-lines, J. Math. Phys.64 (2023) 023504 [2209.15013]

2023 arXiv
[17]

Keppeler, S

S. Keppeler, S. Plätzer and M. Sjodahl,Wigner 6j symbols with gluon lines: completing the set of 6j symbols required for color decomposition, JHEP 05 (2024) 051 [2312.16688]. 13

2024 arXiv
[18]

Berends and W.T

F.A. Berends and W.T. Giele,Recursive Calculations for Processes with n Gluons, Nucl. Phys. B306 (1988) 759

1988
[19]

Caravaglios and M

F. Caravaglios and M. Moretti,An algorithm to compute Born scattering amplitudes without Feynman graphs, Phys. Lett. B358 (1995) 332 [hep-ph/9507237]

1995 arXiv
[20]

Caravaglios, M.L

F. Caravaglios, M.L. Mangano, M. Moretti and R. Pittau,A New approach to multijet calculations in hadron collisions, Nucl. Phys. B539 (1999) 215 [hep-ph/9807570]

1999 arXiv
[21]

Draggiotis, R.H.P

P. Draggiotis, R.H.P. Kleiss and C.G. Papadopoulos,On the computation of multigluon amplitudes, Phys. Lett. B439 (1998) 157 [hep-ph/9807207]

1998 arXiv
[22]

Britto, F

R. Britto, F. Cachazo and B. Feng,New recursion relations for tree amplitudes of gluons, Nucl. Phys. B715 (2005) 499 [hep-th/0412308]

2005 arXiv
[23]

Britto, F

R. Britto, F. Cachazo, B. Feng and E. Witten,Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett.94 (2005) 181602 [hep-th/0501052]

2005 arXiv
[24]

Cachazo, P

F. Cachazo, P. Svrcek and E. Witten,MHV vertices and tree amplitudes in gauge theory, JHEP 09 (2004) 006 [hep-th/0403047]

2004 arXiv
[25]

C. Duhr, S. Hoeche and F. Maltoni,Color-dressed recursive relations for multi-parton amplitudes, JHEP 08 (2006) 062 [hep-ph/0607057]

2006 arXiv
[26]

Parke and T.R

S.J. Parke and T.R. Taylor,An Amplitude forn Gluon Scattering, Phys. Rev. Lett.56 (1986) 2459

1986
[27]

Y.-J. Du, M. Sjödahl and J. Thorén,Recursion in multiplet bases for tree-level MHV gluon amplitudes, JHEP 05 (2015) 119 [1503.00530]

2015 arXiv
[28]

Keppeler,Birdtracks for SU(N), SciPost Phys

S. Keppeler,Birdtracks for SU(N), SciPost Phys. Lect. Notes3 (2018) 1 [1707.07280]

2018 arXiv
[29]

Peigné,Introduction to color in QCD: Initiation to the birdtrack pictorial technique, in6th Chilean School of High Energy Physics, 2, 2023 [2302.07574]

S. Peigné,Introduction to color in QCD: Initiation to the birdtrack pictorial technique, in6th Chilean School of High Energy Physics, 2, 2023 [2302.07574]

2023 arXiv
[30]

MacFarlane, A

A.J. MacFarlane, A. Sudbery and P.H. Weisz,On Gell-Mann’s lambda-matrices, d- and f-tensors, octets, and parametrizations of SU(3), Commun. Math. Phys.11 (1968) 77

1968
[31]

Butera, G.M

P. Butera, G.M. Cicuta and M. Enriotti,Group Weight and Vanishing Graphs, Phys. Rev. D21 (1980) 972

1980
[32]

Sjodahl,Color evolution of 2→ 3 processes, JHEP 12 (2008) 083 [0807.0555]

M. Sjodahl,Color evolution of 2→ 3 processes, JHEP 12 (2008) 083 [0807.0555]

2008 arXiv
[33]

A. Alex, M. Kalus, A. Huckleberry and J. von Delft,A numerical algorithm for the explicit calculation ofSU (n) and SL(n,c ) Clebsch-Gordan coefficients, Journal of Mathematical Physics52 (2011) 023507. 14

2011
[34]

Dytrych, D

T. Dytrych, D. Langr, J.P. Draayer, K.D. Launey and D. Gazda,SU3lib: A C++ library for accurate computation of Wigner and Racah coefficients of SU(3), Comput. Phys. Commun.269 (2021) 108137

2021
[35]

Frederix and T

R. Frederix and T. Vitos,Leading-colour-based unweighted event generation for multi-parton tree-level processes, JHEP 12 (2024) 201 [2409.12128]. 15

2024 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Paton and H.-M

J.E. Paton and H.-M. Chan,Generalized veneziano model with isospin, Nucl. Phys. B10 (1969) 516. one can easily find optimizations. For example, by considering currents with both canonical and reverse ordering (at a price of needing to compute twice as many currents) and performing a reverse bubble sort to order the gluons in the birdtrack diagrams, the va...

work page 1969

[2] [2]

Mangano, S.J

M.L. Mangano, S.J. Parke and Z. Xu,Duality and Multi - Gluon Scattering, Nucl. Phys. B298 (1988) 653

work page 1988

[3] [3]

Sjodahl,Color structure for soft gluon resummation: A General recipe, JHEP 09 (2009) 087 [0906.1121]

M. Sjodahl,Color structure for soft gluon resummation: A General recipe, JHEP 09 (2009) 087 [0906.1121]

arXiv 2009

[4] [4]

Del Duca, L.J

V. Del Duca, L.J. Dixon and F. Maltoni,New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B571 (2000) 51 [hep-ph/9910563]

arXiv 2000

[5] [5]

Maltoni, K

F. Maltoni, K. Paul, T. Stelzer and S. Willenbrock,Color Flow Decomposition of QCD Amplitudes, Phys. Rev. D67 (2003) 014026 [hep-ph/0209271]

arXiv 2003

[6] [6]

PHEGAS: a phase-space generator for automatic cross-section computation

C.G. Papadopoulos,PHEGAS: A Phase space generator for automatic cross-section computation, Comput. Phys. Commun.137 (2001) 247 [hep-ph/0007335]

work page Pith review arXiv 2001

[7] [7]

Mangano, M

M.L. Mangano, M. Moretti, F. Piccinini, R. Pittau and A.D. Polosa,ALPGEN, a generator for hard multiparton processes in hadronic collisions, JHEP 07 (2003) 001 [hep-ph/0206293]

arXiv 2003

[8] [8]

Gleisberg and S

T. Gleisberg and S. Hoeche,Comix, a new matrix element generator, JHEP 12 (2008) 039 [0808.3674]

arXiv 2008