Automation of the Dipole Subtraction Method in MadGraph/MadEvent
aa r X i v : . [ h e p - ph ] A ug Preprint typeset in JHEP style - HYPER VERSION
CP3-08-39ZU-TH 13/08
MadDipole: Automation of the Dipole SubtractionMethod in MadGraph/MadEvent
R. Frederix
Center for Particle Physics and Phenomenology (CP3),Universit´e Catholique de Louvain,Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
T. Gehrmann, N. Greiner
Institut f¨ur Theoretische Physik,Universit¨at Z¨urich,Winterthurerstrasse 190, 8057 Z¨urich, Switzerland
Abstract:
We present the implementation of the dipole subtraction formalism for thereal radiation contributions to any next-to-leading order QCD process in the Mad-Graph/MadEvent framework. Both massless and massive dipoles are considered. Startingfrom a specific ( n + 1)-particle process the package provides a Fortran code for all possibledipoles to all Born processes that constitute the subtraction term to the ( n +1)-particle pro-cess. The output files are given in the usual “MadGraph StandAlone” style using helicityamplitudes. ontents
1. Introduction 12. Construction of dipole terms 3
3. Checks 54. How to use the package 95. Conclusions 10
1. Introduction
Physics studies at the upcoming CERN LHC collider will frequently involve multi-particlefinal states. Especially searches for physics beyond the standard model rely on the recon-struction of new particles from their decay products, often through decay chains. Equally,requiring accompanying particles in the final state may serve to improve the ratio of signalto background processes, as done for example in the Higgs boson search through the vectorboson fusion channel. Meaningful searches for these signals require not only a very goodanticipation of the expected signal, but also of all standard model backgrounds yieldingidentical final state signatures. From the theoretical point of view, high precision impliesthat one has to go beyond the leading order in perturbation theory to be able to keep upwith the precision of the measurements.For leading order processes there have been many developments concerning event gener-ation and simulation tools in the last two decades such as MadGraph/MadEvent [1–3]CompHEP/CalcHEP [4]/ [5], SHERPA [6] and WHIZARD [7] and also programs usingdifferent approaches such as ALPGEN [8] and HELAC [9]. All these programs are multi-purpose event generator tools, which are able to compute any process (up to technicalrestrictions in the multiplicity) within the standard model, or within alternative theoriesspecified by their interaction Lagrangian or Feynman rules. They usually provide eventinformation which can be interfaced into parton shower, hadronization and/or detectorsimulation.Next-to-leading order (NLO) calculations are at present performed on a process-by-processbasis. The widely-used programs MCFM [10,11], NLOJET++ [12], MC@NLO [13,14] andprograms based on the POWHEG method [15–20] collect a variety of different processes– 1 –n a standardized framework, the latter two methods also match the NLO calculation ontoa parton shower.The NLO QCD corrections to a given process with a n -parton final state receive two typesof contributions: the one-loop virtual correction to the (2 → n )-parton scattering process,and the real emission correction from all possible (2 → n + 1)-parton scattering processes.For the numerical evaluation, one has to be able to compute both types of contributionsseparately.The computation of one-loop corrections to multi-particle scattering amplitudes was per-formed on a case-by-case basis up-to-now, the calculational complexity increased consider-ably with increasing number of external partons. Since only a limited number of one-loopintegrals can appear [21, 22] in the final result, the calculation of one-loop corrections canbe reformulated as determination of the coefficients of these basis integrals, plus potentialrational terms. Enormous progress [23–42] has been made in the recent past in the sys-tematic determination of the one-loop integral coefficients and rational terms, and stepstowards fully automated programs for the calculation of one-loop multi-parton amplitudeswere made with the packages CutTools [43], BlackHat [44], Rocket [45] and GOLEM [46].The real emission corrections contain soft and collinear singularities, which become ex-plicit only after integration over the appropriate real radiation phase space yielding a hard n -parton final state. They are canceled by the singularities from the virtual one-loop contri-butions, thus yielding a finite NLO correction. To systematically extract the real radiationsingularities from arbitrary processes, a variety of methods, based either on phase-spaceslicing [47] or on the introduction of process-independent subtraction terms [48] have beenproposed. Several different algorithms to derive subtraction terms are available: residuesubtraction [49], dipole subtraction [50, 51] and antenna subtraction [52–55].Especially the dipole subtraction formalism, which provides local subtraction terms for allpossible initial and final state configurations [50] and allows to account for radiation offmassive partons [51], is used very widely in NLO calculations. It has recently also been au-tomated in the SHERPA framework [56] and the TeVJet framework [57], and most recentlyin the form of independent libraries [58] interfaced to MadGraph. The dipole subtractionwithin the SHERPA framework is not available as a stand-alone tool, while within theTeVJet framework, the user needs to provide all the necessary process dependent infor-mation. Moreover both approaches have only included massless particles for the dipoles.There is still no general tool available which is able to produce the dipole terms for anarbitrary process and which can also deal with massive partons.In this paper we present MadDipole, an automatic generation of the dipole subtractionterms using the MadGraph/MadEvent framework. The results are Fortran subroutineswhich return the squared amplitude for all possible dipole configurations in the usualMadGraph style. We describe the construction of the dipole subtraction terms and theirimplementation in Section 2. Results from various checks of the implementation are pro-vided Section 3, and instructions on the practical usage of the package are contained inSection 4. – 2 – . Construction of dipole terms
The fundamental building blocks of the subtraction terms in the dipole formalism [50, 51]are dipole splitting functions V ij,k , which involve only three partons: emitter i , unresolvedparton j , spectator k . A dipole splitting function accounts for the collinear limit of j with i , and for part of the soft limit of j in between i and k . The dipole factors, which constitutethe subtraction terms, are obtained by multiplication with reduced matrix elements, wherepartons i , j and k are replaced by recombined pseudo-partons e ij , e k . The full soft behavioris recovered after summing all dipole factors.Throughout the whole paper we are using the notation introduced in Refs. [50] and [51].Independent on whether we have initial or final state particles we can write an arbitrarydipole in the form D ij,k ∼ m h , ... ˜ ij, ..., ˜ k, ..., m + 1 | T k · T ij T ij V ij,k | , ... ˜ ij, ..., ˜ k, ..., m + 1 i m . (2.1)The amplitude factors h . . . | (‘bra’) and | . . . i (‘ket’) on the right hand side are tensors incolor space. The helicities of the external particles in them are a priori fixed (but can besummed over for unpolarized processes), while the helicities of the pseudo-partons have tobe summed over after contraction with the dipole splitting function.These Born-level amplitude factors are provided by the usual MadGraph code. The twoelements that combine the ket with the bra are the additional color structure T k · T ij T ij andthe dipole splitting function V ij,k . For the calculation of the color factors there already exist routines in the MadGraph pro-gram. Our intension was to use exactly these routines because this code is very well-confirmed and efficient. We have included the additional color operator, T k · T ij , byrewriting the internal MadGraph color labelling for the ket-side only. After insertion ofthis color operator the color structure is no longer multiplied by its own complex conju-gate and therefore the routine that squares the color needed to be altered, to multiplythe modified ket by its original complex conjugate. We emphasize that due to the facto-rial growth of the color factors MadGraph can not handle more than seven colored particles.For the insertion of the splitting function V ij,k several changes with respect to the originalcode are required. One has to keep in mind that in general the splitting function is a tensorin helicity space, i.e. , V ij,k ≡ h µ | V ij,k | ν i = V µνij,k . As MadGraph deals with helicity amplitudes, we have to write the dipole in a slightlydifferent way to be able to include the calculation of the splitting function in the code.Neglecting the color for a moment we start from the definition of the dipole in (2.1) and– 3 –y inserting a full set of helicity states − g µν = P λ ǫ ∗ µ ( λ ) ǫ ν ( λ ) we get D ij,k ∼ m h , ... ˜ ij, ..., ˜ k, ..., m + 1 | µ V µνij,k ν | , ... ˜ ij, ..., ˜ k, ..., m + 1 i m (2.2)= m h , ... ˜ ij, ..., ˜ k, ..., m + 1 | µ ′ (cid:16) − g µ ′ µ (cid:17) V µνij,k (cid:16) − g ν ′ ν (cid:17) ν ′ | , ... ˜ ij, ..., ˜ k, ..., m + 1 i m = X λ a ,λ b m h . . . | µ ′ ǫ ∗ µ ′ ( λ b ) ǫ µ ( λ b ) V µνij,k ǫ ∗ ν ( λ a ) ǫ ν ′ ( λ a ) ν ′ | . . . i m = X λ a ,λ b m h . . . | λ b V ( λ b , λ a ) λ a | . . . i m with V ( λ b , λ a ) = ǫ µ ( λ b ) V µνij,k ǫ ∗ ν ( λ a ) and ǫ µ ( λ ) µ | . . . i m = λ | . . . i m .The polarization vectors are calculated using the HELAS routines [59] already available inthe MadGraph code. The parts that are diagonal in helicity space are trivial to calculate inthat sense that one only has to multiply the MadGraph output for the squared amplitudefor a given helicity combination with the splitting function. To calculate the off-diagonalhelicity terms, the amplitude for each helicity combination is stored and then combinedwith the according amplitude with opposite helicity.For the calculation of the splitting functions and for the remaping of the momenta we usemodified versions of the routines used in MCFM [10, 11]. If some of the masses of the external particles are non-zero, in particular for processesinvolving top and/or bottom quarks, there are dipoles for which the unresolved parton ismassive. In this case the collinear singularities are regulated by the mass of the unresolvedparton and the unsubtracted matrix element does therefore no longer diverge in thesecollinear limits, but only develops potentially large logarithms. Our code still calculatesall possible dipoles, also in which the unresolved parton is massive, but puts them in aseparate subroutine, dipolsumfinite(...) , that is not evaluated by default. In the limitof large center of momentum energy or, similarly, small external masses, the user can easilyinclude the non-divergent dipoles to subtract the associated large logarithms, which canthen be included analytically through the integrated subtraction terms. In the limit of zeroexternal masses we have checked that the results obtained after summing all dipoles arethe same as obtained by generating the code with massless particles from the start.
The calculation of the subtraction terms is only necessary in the vicinity of a soft and/orcollinear limit. Away from these limits the amplitude is finite and there is in principle noneed to calculate the computationally heavy subtraction terms. The distinction betweenregions near to a singularity from regions without need for a subtraction can be parame-terized by a parameter usually labelled α with α ∈ [0 , i.e. , with partons in the initial state, is described in Ref. [12].– 4 –sing the notation of Ref. [12], the contribution from the subtraction term to the differen-tial cross section can be written as dσ Aab = X { n +1 } d Γ ( n +1) ( p a , p b , p , ..., p n + 1) 1 S { n +1 } × ( X pairs i,j X k = i,j D ij,k ( p a , p b , p , ..., p n +1 ) F ( n ) J ( p a , p b , p , .., ˜ p ij , ˜ p k , .., p n +1 )Θ( y ij,k < α )+ X pairs i,j (cid:20) D aij ( p a , p b , p , ..., p n +1 ) F ( n ) J (˜ p a , p b , p , .., ˜ p ij , .., p n +1 )Θ(1 − x ij,a < α )+( a ↔ b ) (cid:21) + X i = k h D aik ( p a , p b , p , ..., p n +1 ) F ( n ) J (˜ p a , p b , p , .., ˜ p k , .., p n +1 )Θ( u i < α ) + ( a ↔ b ) i + X i h D ai,b ( p a , p b , p , ..., p n +1 ) F ( n ) J (˜ p a , p b , ˜ p , ..., ˜ p n +1 )Θ(˜ v i < α ) + ( a ↔ b ) i ) . (2.3)The functions D ij,k , D aij , D aik and D ai,b are the dipole terms for the various combinationsfor emitter and spectator. P { n +1 } denotes the summation over all possible configurationsfor this ( n + 1)-particle phase space which is labelled as d Γ ( n +1) and the factor S { n +1 } is thesymmetry factor for identical particles. We have introduced four different α -parameters,one for each type of dipoles. In our code they are called alpha ff , alpha fi , alpha if and alpha ii for the final-final, finial-initial, initial-final and initial-initial dipoles, respectively.The actual values for these parameters are by default set to unity, corresponding to theoriginal formulation of the dipole subtraction method [50, 51], but can be changed by theuser in the file dipolsum.f .It has to be kept in mind that the integrated dipole factors, which are to be added withthe virtual n -parton contribution, will also depend on α . For case of massless partons, the α -dependence of the integrated terms is stated in [12, 60] while for massive partons resultsfor most cases can be found in [61, 62].
3. Checks
The MadDipole package provides a code, check dip.f , which allows the user to test thelimits of the ( n + 1)-particle matrix element and the dipole subtraction terms. This codebuilds up a trajectory of randomly selected phase space point approaching a given soft orcollinear limit of the ( n + 1)-parton matrix element and yields the values of matrix element,sum of all the dipoles, and their ratio along this trajectory. The result is printed to thescreen in a small table for which each successive row is closer to the singularity. The ratiobetween matrix element and the sum of the dipoles should go to unity. We have testedour code in all possible limits, both for massless as well as massive dipoles and found no– 5 – a) (b) Figure 1:
Matrix element squared | M R | (upper plots, solid line) and the subtraction terms D (upper plots, dashed/dotted/dot-dashed lines) for the process e + ( p ) e − ( p ) → Z → q ( p )¯ q ( p ) g ( p )as a function of 1 − x ¯ q = ( p .p ) / ( p .p ) and x g = 1 − ( p .p ) / ( p .p ) in figures (a) and (b),respectively. Also plotted are the ratio D/ | M R | , the difference | M R | − D (averaged over 100random points per bin) and the maximal difference max( || M R | − D | ) per bin. The dashed linesinclude the dipoles for each point in phase space, α ff = 1, while for the dotted α ff = 0 . α ff = 0 .
01 the phase space for the dipoles has been restricted to the collinear/softregions. inconsistencies. Choosing small values for α -parameters, e.g. , α = 0 .
1, improves the com-putation time and the convergence of the subtraction procedure.To show that the subtraction terms are implemented correctly we provide a couple of ex-amples in the form of plots and argue that the cancellation between the matrix elementsquared and the subtraction term is as expected. In the figures 1 and 2 we show the matrixelement squared and the subtraction term as a function of a variable that represents a softand/or collinear limit of the process specified. For these figures we have binned the x -axis(equally sized bins for the logarithmic scale) and generated random points in phase spaceto fill each of the bins with exactly 100 events. In the upper plot, | M R | and D are the perbin averages of the matrix element squared and the subtraction term, respectively. Thesecond to upper plot shows the per bin average of the ratio of the matrix element squaredand the subtraction term, while the third plot from the top shows the per bin average ofthe difference. The lowest plot shows the absolute value of the maximal difference amongthe 100 points in a bin. To show the effects of the phase space restriction for the dipoles,see section 2.3, all the plots are given for α = 1 (dashed lines), α = 0 . α = 0 .
01 (dot-dashed lines).In figure 1(a) we show the matrix element squared and the subtraction term as a function of1 − x ¯ q , where x ¯ q is the fraction of the energy carried by the anti-quark, x ¯ q = s + s s = 1 − s s ,with s ij = p i .p j . For this process, e + ( p ) e − ( p ) → Z → q ( p )¯ q ( p ) g ( p ), there are onlyfinal-final state dipoles contributing to the subtraction term. The center of mass energyis set equal to the Z boson mass √ s = m Z . To restrict the discussion to the collinear– 6 –ivergence only, points close to the soft divergence ( x ¯ q = x ¯ q = 1) have been removed byforcing x q + x ¯ q < . x ¯ q →
1, as 1 /x ¯ q . The ratio D/ | M R | goes to 1 and theaverage values of the differences fluctuate close to 0 as can be seen in the second and thirdplots from the top. The numerical fluctuations for small 1 − x ¯ q can be completely explainedby statistical fluctuations. They are of the order of 1% of the maximal difference givenin the lower plot. As can be expected, the cancellations are not exact, which is shownby the lower plot. The maximal difference between | M R | and D rises like 1 / p − x ¯ q ,which does not lead to a divergent phase space integral, because the integration measure isproportional to x ¯ q . The small peaks/fluctuations in the region for small x ¯ q are due to thefact that we are approaching the other collinear limit, i.e. , for which the gluon is collinearto the anti-quark x q →
1, where the matrix elements squared and the subtraction termalso diverge.In figure 1(b) the same matrix elements and subtraction terms are presented, but as afunction of the fraction of the energy carried away by the gluon x g = 2 − x q − x ¯ q . Thelimit for which x g goes to zero represents the soft divergence of this process, while thecollinear divergences for this process are removed by excluding phase space points forwhich x ¯ q > (1 + x q ) / x q > (1 + x ¯ q ) /
2. The same conclusion as for figure 1(a) canbe drawn here: the matrix element squared and the subtraction term diverge in the softlimit, their ratio goes to one and the average difference to zero, while the absolute value ofthe maximal difference still rises when approaching the soft limit, but does not lead to adivergent phase space integral.An example for a collinear limit between final state and initial state particles is givenin figure 2(a). In this plot the matrix element squared and the subtraction term for theprocess e − ( p ) q ( p ) → e − ( p ) q ( p ) g ( p ) are given as a function of the invariant mass of theinitial state quark and the final state gluon s /s . As this invariant mass goes to zero,the matrix element squared and the subtraction term diverge like 1 /s , and their ratiogoes to one. To remove the other possible divergences a cut on the momentum transfered s /s > . s /s > . | M R | − D depends on the number of dipoles included for the phase space point.If all the dipoles are included for all points the difference goes to a smaller constant than ifwe restrict the phase space of the subtraction term to be close to the singularities by setting α <
1. Due to this restriction only the dipoles to cancel that divergence are included in thesubtraction term and therefore give a smaller constant contribution, hence the difference | M R | − D is larger. Also here the maximal left-over difference, the lowest plot, increasesfor small invariant masses but does not lead to a divergent phase space integral.In figure 2(b) an example with massive final state particles is shown. The process is t ¯ t production at a linear collider, e + ( p ) e − ( p ) → t ( p )¯ t ( p ) g ( p ) at 1 TeV center of massenergy. The plot shows a behavior very similar to the massless case, fig. 1, and the conclu-– 7 – a) (b) Figure 2:
Matrix element squared | M R | (upper plots, solid line) and the subtraction term D (upper plots, dashed/dotted/dot-dashed lines) for (a) the process e − ( p ) q ( p ) → e − ( p ) q ( p ) g ( p )as a function of s /s = ( p .p ) / ( p .p ) and (b) the process e + ( p ) e − ( p ) → Z → t ( p )¯ t ( p ) g ( p )as a function of x g = 1 − ( p .p ) / ( p .p ). Also plotted are the ratio D/ | M R | , the difference | M R | − D (averaged over 100 random points per bin) and the maximal difference max( || M R | − D | )per bin. The dashed lines include the dipoles for each point in phase space, α = 1, while for thedotted α = 0 . α = 0 .
01 the phase space for the dipoles has been restricted to thecollinear/soft regions. sions drawn there apply to this plot as well.As a further check we have tested the code extensively against MCFM [10, 11]. We havegenerated random points in phase space and compared the subtraction terms calculatedby MCFM with the subtraction terms calculated by our code. See table 1 for a list ofprocesses that have been checked. We observed differences only in the case where dipoleswere introduced entirely to cancel collinear limits, which can be made independently of thespectator particle. In our code all possible dipoles are calculated, which implies a sum overall spectator particles. However, if there is only a collinear divergence, i.e. , the unresolvedparton cannot go soft, this sum is redundant and one dipole with the appropriate coefficientis enough to cancel the singularity. In MCFM, these special limits are implemented usinga single spectator momentum, while MadDipole sums over all spectator momenta, therebyyielding a different subtraction term. We have checked in the relevant cases that close tothe singularities the MCFM subtraction terms behave identical to the subtraction termscalculated by our code.We also tested the CPU time which is needed to produce the squared matrix element andthe dipoles for a given phase space point. These checks were performed with an IntelPentium 4 processor with 3.20Ghz. As an example we picked out three different processes:1) gg → gggg : |M| : 26ms, P dipoles : 68ms2) u ¯ u → d ¯ dggg : |M| : 10ms, P dipoles : 45ms3) u ¯ u → u ¯ uggg : |M| : 34ms, P dipoles : 0.15s– 8 – rocess subprocessesDrell-Yan ( W ) q ¯ q ′ → W + ( → e + ν e ) gqg → W + ( → e + ν e ) q ′ Drell-Yan ( Z ) q ¯ q → Z ( → e + e − ) gqg → Z ( → e + e − ) q Drell-Yan ( Z +jet) q ¯ q → Z ( → e + e − ) q ′ ¯ q ′ q ¯ q → Z ( → e + e − ) q ¯ qq ¯ q → Z ( → e + e − ) ggq ¯ g → Z ( → e + e − ) qgg ¯ g → Z ( → e + e − ) q ¯ q top quark pair ( t ¯ t ) q ¯ q → t ( → bl + ν l )¯ t ( → ¯ bl − ¯ ν l ) gqg → t ( → bl + ν l )¯ t ( → ¯ bl − ¯ ν l ) qgg → t ( → bl + ν l )¯ t ( → ¯ bl − ¯ ν l ) gt -channel single top gg → t ¯ bq ¯ q ′ with massive b -quark [63] qq ′ → t ¯ bq ′ q ′′ qq ′ → t ¯ bq ′ q ′′ qg → t ¯ bq ′ g Table 1:
Set of processes for which the MadDipole code has been tested against MCFM for randompoints in phase space. All the possible initial-initial, initial-final and final-initial, dipoles for masslessand massive final state particles have been checked with this set of subprocesses. No inconsistencieswere found.
The time which is needed to produce the
Fortran code is strongly dependent on the processand ranges from a few seconds to at most a few minutes. The process gg → g is currentlynot yet feasible within MadGraph because of the size of the color factors. Once MadGraphhas been adjusted to handle this process, it will equally become accessible for MadDipole.
4. How to use the package
The installation and running of the MadDipole package is very similar to the usual Stand-Alone version of the MadGraph code. In this section, we only provide a brief descriptionfor MadDipole, while more information can also be found on the MadGraph wiki page, http://cp3wks05.fynu.ucl.ac.be/twiki/bin/view/Software/MadDipole .1. Download and extract the MadDipole package,
MG ME DIP V4.4.3.tar.gz , from oneof the MadGraph websites, e.g. , http://madgraph.hep.uiuc.edu/ .2. Run make in the MadGraphII directory.3. Copy the
Template directory into a new directory, e.g. , MyProcDir to ensure thatyou always have a clean copy of the Template directory.4. Go to the new
MyProcDir directory and specify your process in the file ./Cards/proc card.dat .This is the ( n + 1)-particle process you require the subtraction term for.5. Running ./bin/new process generates the code for the ( n + 1)-particle matrix el-ement and for all dipole terms. After running this you will find a newly generated– 9 –irectory ./Subprocesses/P0 yourprocess ( e.g. , ./Subprocesses/P0 e+e- uuxg )which contains all required files.6. For running the check program change to this directory and run make and ./check .The directory ./Subprocesses/P0 yourprocess contains all necessary files needed for fur-ther calculations. As in the usual MadGraph code the ( n + 1)-particle matrix element isincluded in the file matrix.f . For the dipoles there exist several files called dipol???.f where ??? stands for a number starting from . Each file contains only one dipole. Notethat the syntax for calling the dipoles is exactly the same as calling the ( n + 1)-particlematrix element. In particular, also the dipoles have to be called with ( n + 1) momentarather than with only n momenta. The remapping of the momenta going from the ( n + 1)-particle phase space to a n -parton phase space is done within the code for the dipoles.The file dipolsum.f calculates the sum over all dipoles for a given ( n + 1)-particle phasespace point. It contains two subroutines called DIPOLSUM and
DIPOLSUMFINITE . As dis-cussed in Section 2.2 above, the subroutine
DIPOLSUMFINITE contains the dipoles whichonly contribute potentially large logarithms but not a real singularity.In both subroutines the contribution of a certain dipole is only added to the sum if thephase space restriction specified by the α parameter is fulfilled. The value of the four α parameters can be changed in these two subroutines. They are all set to unity by default.
5. Conclusions
In this paper we have presented MadDipole, an implementation to fully automatize thecalculation of the dipole subtraction formalism for massless and massive partons in theMadGraph/MadEvent framework. The implementation is done in such a way that theuser only needs to specify the desired ( n + 1)-particle process and our code returns a For-tran code for all dipoles combined with possible Born processes which can lead to the( n + 1) process specified by the user.For the calculation of the new color factors we have used as far as possible the routinesalready provided by the original MadGraph code. We inserted the two additional operatorsfor emitter and spectator and modified the evaluation of the squared color factors.For the contributions that are not diagonal in helicity space we again used the alreadyavailable routines for calculating amplitudes for a given helicity combination and combinedamplitudes with different helicity combinations to yield the off-diagonal helicity contribu-tions to the subtraction terms.We have validated the code on numerous different processes with massive and masslesspartons using two checking procedures. The ratio of M n +1 / P ( dipoles ) was confirmed toapproach unity as one approaches any soft or collinear limit. The package includes a file check dip.f which allows the user to reproduce this check for any process under consid-eration.Secondly, we compared our code against the results of MCFM [10,11], where subtraction isperformed using the dipole formalism, finding point-wise agreement wherever anticipated.– 10 –ifferences with MCFM are understood to be due to different details in the implementa-tion.The MadDipole package allows the automated computation of real radiation dipole subtrac-tion terms required for NLO calculations. Together with the fastly developing automationof one-loop calculations of multi-leg processes, it could lead to a full automation of NLOcalculations for collider processes. In view of the large number of potentially relevantmulti-particle production processes at LHC, such automation will be crucial for precisionphenomenology, in order to establish and interpret potential deviations from standardmodel expectations. Acknowledgments
We would like to thank Fabio Maltoni and Keith Hamilton for useful comments and dis-cussions. NG wants to thank Gabor Somogyi for useful discussions and comparing results.He also would like to thank the University of Louvain for kind hospitality where part ofthis work was done. This work was partially supported by the Belgian Federal SciencePolicy (IAP 6/11) and by the Swiss National Science Foundation (SNF) under contract200020-117602.
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