Integrated dipoles with MadDipole in the MadGraph framework
aa r X i v : . [ h e p - ph ] J un Preprint typeset in JHEP style - HYPER VERSION
ZU-TH 06/10
Integrated dipoles with MadDipole in the MadGraphframework
R. Frederix a T. Gehrmann a , N. Greiner a,b a Institut f¨ur Theoretische Physik, Universit¨at Z¨urich,Winterthurerstrasse 190, 8057 Z¨urich, Switzerland b Department of Physics, University of Illinois at Urbana-Champaign,1110 West Green Street, Urbana, IL 61801, USA
Abstract:
Heading towards a full automation of next-to-leading order (NLO) QCD cor-rections, one important ingredient is the analytical integration over the one-particle phasespace of the unresolved particle that is necessary when adding the subtraction terms tothe virtual corrections. We present the implementation of these integrated dipoles in theMadGraph framework. The result is a package that allows an automated calculation forthe NLO real emission parts of an arbitrary process. ontents
1. Introduction 12. Structure of NLO dipole subtraction terms 3 α -parameter 42.2 Regularization scheme dependence 5
3. Expansion of integrated dipoles in ǫ
4. Implementation and how to use the integrated dipoles 85. Checks 116. Conclusions 13A. α -dependence of the integrated dipoles 13 A.1 Final-final 14A.1.1 Case (g) 14A.1.2 Case (h) 15A.2 Final-initial 16A.2.1 Case (c) 16A.2.2 Case (d) 17A.2.3 Case (e) 17A.3 Initial-final 18A.4 Initial-initial 18
1. Introduction
Multi-particle final states are the basis of many physics studies at the CERN LHC. Insearching for physics beyond the standard model, one is aiming to identify new particlesfrom their decay products, which could often result from decay chains. Likewise, accompa-nying final state particles may help to improve the ratio of signal to background processes,as done for example in the Higgs boson search through the vector boson fusion channel.Meaningful searches for these signatures require not only a very good anticipation of theexpected signal, but also of all standard model background processes which could result inidentical final state signatures. From the theoretical point of view, high precision implies– 1 –hat 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 gen-eration and simulation tools in the last two decades such as MadGraph/MadEvent [1–3]CompHEP/CalcHEP [4]/ [5], SHERPA [6, 7] and WHIZARD [8] and also programs usingdifferent approaches such as ALPGEN [9] and HELAC [10]. 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 passed into parton shower, hadronization and/or detector simu-lation through standard interfaces.Next-to-leading order (NLO) calculations are at present performed on a process-by-process basis. The widely-used programs MCFM [11, 12], NLOJET++ [13], MC@NLO[14–17] and programs based on the POWHEG method [18–23] collect a variety of differ-ent processes in a standardized framework, the latter two methods also match the NLOcalculation onto a parton shower. The POWHEG box [24] provides a toolkit for adaptingfurther NLO calculations to match onto parton showers.The NLO QCD corrections to a given process with a n -parton final state receive twotypes of contributions: the one-loop virtual correction to the (2 → n )-parton scatteringprocess, and the real emission correction from all possible (2 → n + 1)-parton scatteringprocesses. For the numerical evaluation, one has to be able to compute both types ofcontributions separately.The computation of one-loop corrections to multi-particle scattering amplitudes wasperformed on a case-by-case basis up to now, the calculational complexity increased consid-erably with increasing number of external partons. Since only a limited number of one-loopintegrals can appear in the final result [25–27], the calculation of one-loop corrections can bereformulated as the determination of the coefficients of these basis integrals, plus potentialrational terms. Based on this observation, a variety of methods for the systematic deter-mination of the one-loop integral coefficients and the rational terms have been formulated,and first fully automated programs for the calculation of one-loop multi-parton amplitudesare becoming available with the packages CutTools [28, 29], BlackHat [30], Rocket [31] andGOLEM [32], as well as independent libraries [33].These recent technical advances have allowed the calculation of NLO corrections toseveral 2 → → W +3 jet production [35–38], Z + 3 jet production [39], pp → t ¯ tb ¯ b [40–42] and pp → t ¯ tjj [43], as well as for the quark-antiquark contribution to pp → b ¯ bb ¯ b [44].The real emission corrections contain soft and collinear singularities, which becomeexplicit only after integration over the appropriate real radiation phase space yielding ahard n -parton final state. They are canceled by the singularities from the virtual one-loop contributions, thus yielding a finite NLO correction. To systematically extract the– 2 –eal radiation singularities from arbitrary processes, a variety of methods, based either onphase-space slicing [45] or on the introduction of process-independent subtraction terms [46]have been proposed. Several different algorithms to derive subtraction terms are available:residue (or FKS) subtraction [47] and variants thereof [48], dipole subtraction [49, 50] andantenna subtraction [51–54].Especially the dipole subtraction formalism, which provides local subtraction termsfor all possible initial and final state configurations [49] and allows to account for radiationoff massive partons [50], is used very widely in NLO calculations. The generation ofdipole terms for subtracting the singular behaviour from the real radiation subprocesses hasbeen automated in various event generators: in the SHERPA framework [55], the TeVJetframework [56], the HELAC framework [57] and in the form of independent libraries [58]interfaced to MadGraph. The MadDipole package [59] provides an implementation withinMadGraph. An implementation of the residue subtraction method is also available withinMadGraph [60].For a full NLO calculation, the dipole terms have to be integrated over the dipolephase space, and added with the virtual corrections to obtain the cancellation of infraredsingularities. So far, only one of the implementations [57] provides these integrated dipoleterms including all masses and possible phase-space restrictions, and constructs the corre-sponding integrated subtraction terms. It is the purpose of this paper to implement thegeneration of the integrated subtraction terms into MadDipole, which will consequentlyallow to carry out the full dipole subtraction within the MadGraph/MadEvent framework.The output results are Fortran subroutines which return the squared amplitude for allpossible unintegrated and integrated dipole configurations in the usual MadGraph style.With a complete treatment of the real NLO radiation, MadDipole is a crucial buildingblock of automated NLO calculations. This automation is a very high priority for LHCpredictions [61], and can likely be accomplished only in a collaborative effort, with differentgroups supplying different pieces, linked through standard interfaces [62].The paper is structured as follows: in Section 2, we briefly review the structure of un-integrated and integrated dipole terms, Section 3 discusses the expansion of the integrateddipoles, and normalization conventions used in this. The MadDipole implementation ofthe integrated dipoles is described in Section 4, with functionality checks documented inSection 5. Finally, we conclude with Section 6.
2. Structure of NLO dipole subtraction terms
The fundamental building blocks of the subtraction terms in the dipole formalism [49, 50]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.– 3 –hroughout the whole paper we are using the notation introduced in Refs. [49] and [50].The dipole factors are subtracted from the real radiation contribution at NLO. They sub-tract the singular contributions where one parton from the NLO real radiation contributionbecomes soft and/or collinear, such that the phase-space integral of this contribution can bepreformed numerically, including arbitrary kinematic restrictions on the final-state phasespace.The dipole factors are integrated analytically over the dipole phase space (which fullyincludes the infrared singular regions), such that the integrated dipole factors have thesame kinematic structure as the virtual one-loop NLO corrections and the collinear coun-terterms from mass factorization. These integrated dipole contributions can then be addedto the virtual and mass-factorization corrections, thereby accomplishing the cancellationof infrared singularities.Several algorithms to automatically generate the unintegrated dipole terms for arbi-trary processes have been devised and implemented in various matrix element generatorframeworks. They are available for SHERPA [55], HELAC [57], for MadGraph [59] as wellas a implementation of stand-alone routines [56, 58]. Also implementations of integrateddipoles are available in the same codes. However, only the implementation based on theHELAC framework has the full set of integrated dipoles for arbitrarly masses and phase-space restrictions [57]. It was used in the context of the calculations of NLO correctionsto pp → t ¯ tb ¯ b [42] and pp → t ¯ tjj [43]. In this work, we document our implementationof the integrated dipoles in MadDipole, which is a package in the MadGraph framework.With this extension, MadDipole computes the full NLO dipole subtraction of real radia-tion and performs the infrared cancellations in a fully automated manner for color- andhelicity-summed matrix elements squared. Our implementation was already used in thecomputation of the NLO corrections to pp → b ¯ bb ¯ b in the quark-initiated channel [44]. α -parameter 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 and regions without need for a subtraction can be parametrizedby a parameter usually labelled α with α ∈ [0 , i.e. ,with partons in the initial state, is described in Ref. [13].Using the notation of Ref. [13], the contribution from the subtraction term to the– 4 –ifferential cross section in the real radiation channel 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 ) . (2.1)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. In MadDipole, we have introduced four different α -parameters, one for each type of dipoles [59]. 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 tounity, corresponding to the original formulation of the dipole subtraction method [49, 50],but can be changed by the user.The integrated dipole factors, which are to be added with the virtual n -parton contri-bution, also depend on α . For case of massless partons, the α -dependence of the integratedterms is stated in [13, 63] while for massive partons results for most cases can be foundin [57, 64, 65]. The remaining cases, i.e. , the (finite) massless-to-massive splittings, can befound in the appendix. Calculating objects that contain divergences requires a systematic prescription of how todeal with these divergences, i.e. , a scheme for their regularization. In NLO calculations, thesame regularization scheme has to be applied in the real emission part and in the virtualcorrections. Both these contributions will differ between different regularization schemes,while their sum ( i.e. , the full NLO result) is scheme-independent. Therefore it is necessaryto clearly specify which regularization scheme one is using.In QCD calculations, there are mainly two types of regularization schemes used, namelydimensional regularization [66–69] and dimensional reduction [70–72]. Both extend the– 5 –imensionality of space-time to d = 4 − ǫ , resulting in divergences becoming explicit aspoles in 1 /ǫ . A discussion about their subtypes and their differences can be found in [73].In the real radiation contribution, the dependence on the regularization scheme doesnot yet appear explicitly at the level of the unintegrated dipole terms, and we consequentlydid not address this issue in the previous release of MadDipole [59].The helicity subroutines on which MadGraph and MadDipole are build evaluate matrixelements in four dimensions. Therefore we can compute the subtraction terms only inregularization schemes in which the external particles are defined in four dimensions. Thetwo most widely used are the ’t Hooft-Veltman scheme (tHV) in dimensional regularization,which is the default used in our implementation, and the four-dimensional helicity scheme(FDH). Both methods differ only by a finite shift [49, 74]: V I ( ǫ ) tHV → V FDH I ( ǫ ) = V I ( ǫ ) tHV − ˜ γ I + O ( ǫ ) , (2.2)˜ γ q = ˜ γ ¯ q = 12 C F , ˜ γ g = 16 C A . (2.3)In the massive case there is no dependence on the regularization scheme [75]. There is asimple flag in our code that allows to change between these two schemes.
3. Expansion of integrated dipoles in ǫ Integration of the dipoles makes their infrared singularities explicit as poles in the di-mensional regularization parameter ǫ . The formal structure of the integrated dipoles isindependent of the configuration we are considering. For definiteness, we discuss only thefinal-final case, the same structure also holds in all other cases. For initial state hadrons, aadditional collinear contribution is present, which is rendered finite by mass factorization,which we will describe here as well. In the final-final case, the integrated dipole function is written as Z [d p i ( e p ij , e p k )] 1( p i + p j ) − m ij h V ij,k i ≡ α s π − ǫ ) (cid:18) πµ e p ij · e p k (cid:19) ǫ V ij ( ǫ ) . (3.1)Depending on the configuration, the splitting function and the propagator on the left handside of (3.1) change their form; the structure of the result on the right hand side remainsthe same.The factor V ij ( ǫ ) is determined by the specific configuration. It is singular in the limit ǫ →
0, and is expanded as a Laurent series in ǫ . Prefactors taken out from the expansionmust be consistent between the dipoles and the virtual one-loop corrections to the processunder consideration. – 6 –n the MS-scheme, only universal factors are taken out, and this expansion can bewritten symbolically as Z [d p i ( e p ij , e p k )] 1( p i + p j ) − m ij h V ij,k i ≡ α s π − ǫ ) (cid:18) πµ e p ij · e p k (cid:19) ǫ V ij ( ǫ )= e γǫ (4 π ) ǫ (cid:16) y ij ;1 ǫ + y ij ;2 ǫ + y ij ;3 (cid:17) , (3.2)where γ is the Euler-Mascheroni constant, γ = 0 . . . . . This structure is the basis forour implementation.The specific values of y ij ;1 , y ij ;2 and y ij ;3 depend on the splitting one is considering. Forinstance if one takes the massless final-final quark-gluon splitting, i.e. , where the emitteris a massless quark int the final state, the unresolved particle is a gluon, and the spectatoris also a massless final state particle, then the integrated splitting function is given by V qg ( ǫ ) = C F (cid:18) ǫ + 32 ǫ + 5 − π (cid:19) . (3.3)Consequently, the expansion coefficients in (3.2) become y qg ;1 = α s π C F ,y qg ;2 = α s π C F (cid:18)
32 + log (cid:18) µ e p ij · e p k (cid:19)(cid:19) ,y qg ;3 = α s π C F (cid:18)
12 log (cid:18) µ e p ij · e p k (cid:19) + 32 log (cid:18) µ e p ij · e p k (cid:19) − π
12 + 5 (cid:19) . (3.4) The cases with initial state radiation, i.e. , initial-initial and initial-final, are slightly moreinvolved. Not all singularities that occur in the real emission process are cancelled by thevirtual corrections. The integrated initial-final dipole functions take the form: Z [d p i ( e p k ; p a , z )] 1( p i + p j ) − m ij n s ( e ai ) n s ( a ) h V aik i≡ α s π − ǫ ) (cid:18) πµ p a · e p k (cid:19) ǫ V a,ai ( z ; ǫ )= e γǫ (4 π ) ǫ (cid:18) y i,j ;1 ( z ) ǫ + y i,j ;2 ( z ) ǫ + y i,j ;3 ( z ) (cid:19) . (3.5)The left-over singularities arise from collinear splitting off the initial emitter particle. Forexample, for the initial-final dipole describing the gluon radiation off an incoming quark,one has V q,q ( z ; ǫ ) = − ǫ P qq ( z ) + δ (1 − z ) (cid:20) V qg ( ǫ ) + (cid:18) π − (cid:19) C F (cid:21) + B q,q ( z ) + O ( ǫ ) , (3.6)where B q,q ( z ) contains regular functions and plus-distributions in z .– 7 –he collinear singularity remains and is absorbed into the parton distribution function.This is done by introducing a collinear counterterm and its contribution to the cross sectionis given by (6.6) of [49]: dσ Ca ( p ; µ F ) = − α s π − ǫ ) X b Z dz (cid:20) − ǫ (cid:18) πµ µ F (cid:19) ǫ P ab ( z ) + K ab F . S . ( z ) (cid:21) dσ Bb ( zp ) , (3.7)where in MS scheme, K ab F . S . ( z ) = 0.Neglecting the sum over the partonic subprocesses, we have a counterterm contributionof the form I abc ( ǫ ) = − α s π − ǫ ) (cid:18) πµ µ F (cid:19) ǫ (cid:20) − ǫ P ab ( z ) (cid:21) (3.8)which is added to the integrated dipole.In the MS scheme the expansion of (3.8) is given by I abc ( ǫ ) = e γǫ (4 π ) ǫ (cid:18) y ca,b ;2 ǫ + y ca,b ;3 (cid:19) . (3.9)The coefficients y ca,b ;2 and y ca,b ;3 are given by y ca,b ;2 = α s π · P ab ( z ) y ca,b ;3 = α s π · P ab ( z ) log (cid:18) µ µ F (cid:19) . (3.10) The final output the program provided to the user is then given by all contributions ofthe coefficients y i and y ci respectively multiplied with a born level matrix element that ismodified by its color structure. This explicitly means ǫ : y · m h · · · m | T i · T k T i | · · · m i m , ǫ : ( y + y c ) · m h · · · m | T i · T k T i | · · · m i m ,finite: ( y + y c ) · m h · · · m | T i · T k T i | · · · m i m .The contributions from the collinear counterterms are of course only present if there areinitial state QCD particles involved. By adding the collinear counterterm, only endpointsingularities, which occur at z = 1, remain. Those then completely cancel with the virtualcorrections. The finite pieces can contain regular functions of z as well as δ -functions andplus-distributions.
4. Implementation and how to use the integrated dipoles
The installation and running of the new package is very similar to the already existingMadDipole package: – 8 –. Download the MadDipole package (version 4.4.35 or later),
MG ME DIP V4.4.??.tar.gz ,from one of the MadGraph websites, e.g. , http://madgraph.phys.ucl.ac.be/ .2. Extract and 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/newprocess generates the code for the ( n + 1)-particle matrix el-ement and for all dipole terms and their integrated versions. After running thisyou will find a newly generated directory ./SubProcesses/P0 yourprocess ( e.g. , ./SubProcesses/P0 e+e- uuxg ) which contains all required files.In the ./SubProcesses/P0 yourprocess directory all the files relevant to that par-ticular subprocesses are generated. In particular this includes the ( n + 1) particle matrixelements in the file matrix.f and the dipoles in the files dipol???.f , where ??? standsfor a number starting from . Furthermore the directory has two files, dipolsum.f and intdipoles.f , where the sum of the dipoles and their integrated versions are calculated,respectively.Here we discuss in more detail the syntax and implementation of the integrated dipolesin the intdipoles.f file. In this file there are two subroutines of the form intdipoles(P, X, Z, PSWGT EPSSQ, EPS, FIN) and intdipolesfinite(P, X, Z, PSWGT EPSSQ, EPS, FIN) ,where the input parameters are the phase space point P(0:3,nexternal) , the Bjorken x values of both incoming parton distributions, X(1:2) , and the momentum fraction ofthe incoming parton that goes into the hard process after an initial state radiation or ifthe spectator is an initial state particle, Z . It is the latter quantity which is denoted with x ij,a , x ik,a and x i,ab respectively in [49], but we shall refer to it here simply as z . In thenumerical integration, it must be taken uniformly over the interval z ∈ [0; 1] to ensurecorrect representation of the distributions. Furthermore, the phase space weight shouldbe passed as well. These first four arguments should be provided by the user. For thegiven phase space point, the routines evaluate the sum of all integrated dipole subtractionterms (integrated dipole factors multiplied with the appropriate reduced matrix elements)after mass factorization of the collinear singularities. The routines call external subroutines(explained below) from dipolesum.f which supply the parton distributions appearing withthe reduced matrix elements, apply event rejection cuts and pass the event information intohistograms.The integrated dipoles in the two subroutines correspond to the unintegrated dipolesin dipolsum(..) and dipolsumfinite(..) . The dipoles in the dipolsumfinite(..) and intdipolesfinite(..) subroutines are not needed to cancel singularities because theycorrespond to gluon splittings into massive particles. However, they can be useful for– 9 –hecks when taking the limit of vanishing quark masses, or to cancel some large logarithmsin the real emission matrix elements [59].The output parameters are the coefficients of the divergent and finite terms: EPSSQ is the coefficient of 1 /ǫ , EPS is the coefficient of 1 /ǫ . After inclusion of the collinearcounterterms they contribute with a factor δ (1 − z ), and are real numbers. FIN is thefinite coefficient. The calculation of the
FIN coefficient requires some explanation, sinceits coming from distributions in z . In the intdipoles(..) and intdipolesfinite(..) subroutines a three-dimensional array FINITE is filled with the contributions from thevarious dipoles:
FINITE(1) : regular function in z , FINITE(2) : coefficient of δ (1 − z ), FINITE(3) : coefficient of δ ( z + − z ).The last entry appears only for massive dipoles, with z + = 1 − µ Q , (4.1)where µ Q is the rescaled fermion mass occurring in the splitting.In this decomposition, the δ -functions and (+)-distributions are carried out by takinginto account the convolution with the product of a reduced matrix element g ( z ) (generatedby MadDipole out of MadGraph) and a parton distribution function h ( z ) (supplied by theuser through the subroutine dipolepdf ) Z d z δ (1 − z ) g ( z ) h ( z ) = Z d z g (1) h (1) | {z } in FINITE(2) , (4.2) Z d z ( f ( z )) + g ( z ) h ( z ) = Z d z h f ( z ) g ( z ) h ( z ) | {z } in FINITE(1) − f ( z ) g (1) h (1) | {z } in FINITE(2) i . (4.3)In the massive case the point z = 1 can not be reached in all cases but we may have areduced endpoint z + , such that instead of a δ (1 − z ) we then have a δ -distribution of theform δ ( z + − z ) which is the third entry of the array of the finite terms. If we have sucha reduced endpoint then also the (+)-distribution is generalized into a z + -distribution asdefined in (A.13). As before, we have the following implementation: Z d z δ (1 − z + ) g ( z ) h ( z ) = Z d z g ( z + ) h ( z + ) | {z } in FINITE(3) , (4.4) Z d z ( f ( z )) z + g ( z ) h ( z ) = Z d z Θ( z + − z ) h f ( z ) g ( z ) h ( z ) | {z } in FINITE(1) − f ( z ) g ( z + ) h ( z + ) | {z } in FINITE(3) i . (4.5)By making a transformation of variables and computing the parton distribution func-tion not at x (Bjorken’s x ) but at x/z , the matrix element itself becomes independentof z as the initial state particle, that radiates the unresolved particle, then comes withthe momentum fraction of x/z · z = x . Therefore the set of momenta which are used to– 10 –alculate the matrix element do not depend on z . This means that g ( z ) = g ( z + ) = g (1) inEqs. (4.2–4.5).The final result FIN is the sum of the three contributions:
FIN = (cid:0) FINITE(1) /z + FINITE(2) + FINITE(3) /z + (cid:1) × PSWGT . (4.6)Since FINITE(3) appears only in massless-to-massive splittings, it is always zero for the intdipoles(..) subroutine. Conversely, the
EPSSQ and
EPS should be zero when comingfrom the subroutine intdipolesfinite(..) .For ease of use, we have provided three dummy subroutines/functions that the usermight want to fill when using the code. In the code there are consistent calls to thesesubroutines. These subroutines/functions can be found in the file dipolsum.f and are passcutsdip(P) : In this
LOGICAL FUNCTION the user should provide a set of cuts that hewants to be applied to the phase space points. Note that in general the phase-spacemapping for each unintegrated dipole is different; there needs to be a call to thisfunction for each dipole. It should return
FALSE if the point fails the cuts. By defaultevery point passes the cuts. dipolepdf(P,leg1,leg2,WGT) : In this
SUBROUTINE the user should provide the value forhis/her favourite PDF set for the two incoming particles with PDG codes passedby leg1 and leg2 . The factorization scale should be defined in the include file dipole.inc . The weight from the PDF should be returned in the argument
WGT ,which is set to by default. writehist(P,WGT) : In this
SUBROUTINE the phase-space point (for each dipole) are pro-vided together with its weight. The user could use these to fill histograms or saventuples.Besides these three subroutines/functions, the more technical parameters, like the α -parameter (see Sec. 2.1), the number of flavors, the renormalization scheme and the scalescan be set in the include file dipole.inc . When changing any of the parameters in this filethe code should be recompiled (after removing the object files) for these changes to takeeffect.Besides the already existing check checking program, that checks the limits of the realemission matrix element minus the subtraction terms, we provide the user with anothersample program, checkint , to calculate the value of the integrated subtraction terms fora given (or random) phase space point.More details and latest news, updates, bug fixes, etc. can be found at http://cp3wks05.fynu.ucl.ac.be/twiki/bin/view/Software/MadDipole .
5. Checks
To verify the implementation we have performed two different kinds of checks: indepen-dence on the phase space restriction parameter α and comparison with the implementationof dipoles in the MCFM program [11, 12]. – 11 – -3 -2 -1
10 1 PSfrag replacements α ii σ u ¯ u → e + e − gα if σu ¯ u → t ¯ t g (a) -4 -3 -2 -1
10 1 PSfrag replacements α ii σu ¯ u → e + e − g α if σ u ¯ u → t ¯ t g (b) Figure 1:
Two examples for the cancellation of the α -dependence between unintegrated andintegrated dipoles on specific contributions. Since these do not correspond to full physical processes,we use an arbitrary normalization. The plot on the left shows the α dependence of the initial-initialdipole where a gluon is radiated of the initial quark lines. On the right the α dependence of aninitial-final dipole where the spectator is massive is shown. For both figures only the α parametershown is varied, all others are kept fixed. The upper dashed line is the first contribution of (5.1), i.e. , the sum of matrix element and unintegrated dipoles. The lower dashed line is the secondcontribution, i.e. , the finite terms of the integrated dipoles. The (red) solid line is the sum of both.For the sum we also included the Monte-Carlo error to show that the results for different values areconsistent with each other. Both the non-integrated and the integrated dipoles depend on the α -parameter, thedependence on this parameter should cancel in the sum. To validate this, we require: Z n +1 (cid:0) dσ R − dσ A (cid:1) + Z n (finite parts of int. dip.) = const , (5.1)which must be a constant in that sense that it should not depend on α . We have validatedthis for all 27 different cases (emitter/spectator in initial/final state, and mass assign-ments) listed in the appendix. Figure 1 shows the dependence on the α -parameter for twoexamples: in the left plot is the dependence shown on the α ii -parameter that governs theinitial-initial dipole phase-space restriction, and in the right plot for a processes with mas-sive spectators the dependence on the α if -parameter (that restricts the initial-final dipolephase space) is shown. The solid (red) lines display the quantity defined in (5.1), which isobserved to be independent on α over several orders of magnitude.This is a very powerful check because it includes several aspects of the implementation.It verifies not only the correct implementation of the θ -functions for the unintegratedsubtraction terms and the α -dependent correction terms for the integrated dipoles, butindependence on this parameter can only be achieved if the correct parton distributionfunctions are called with the right arguments, and if the various terms contributing tothe integrated dipoles are summed correctly. Moreover, an inconsistent infrared-unsafeimplementation of the cuts (by the user) will lead to a dependence on the α parameter.The α -independence does, however, not probe important features related to the polestructure of the subtraction terms, namely the scheme dependence and the factorization– 12 –cale dependence, neither does it show other finite contributions. Therefore our secondcheck was a direct comparison of our implementation against MCFM [11, 12]. We havecompared the results for single phase space points, which allowed us to directly probethe implementation of the integrated terms. The various terms (regular functions, δ -and plus-distributions) could be checked separately. We note that not all possibilities(massless/massive, initial/final emitter/spectator) are present in MCFM and therefore notall could be checked. This includes in particular the finite dipoles: massless-to-massivesplittings have no divergences, but do have a potentionally large logarithm which might beuseful to subtract.
6. Conclusions
The MadDipole package [59] provides dipole subtraction terms for the evaluation of real ra-diation corrections in NLO calculations within the framework of the MadGraph/MadEventmatrix element and event generator [1–3]. In this work, we described the extension of theMadDipole package to include the integrated dipole terms, which are required to combinethe real radiation corrections to a given process with the virtual corrections and collinearcounterterms of the parton distributions. With the newly developed subroutines, Mad-Dipole provides the unintegrated and integrated dipole subtraction terms.The integrated subtraction terms are convoluted with the user-supplied parton distri-butions and are combined with the collinear counterterms from mass factorization. Con-sequently, infrared singularities appear only in the kinematic endpoints, which correspondto the kinematics of the virtual corrections. The MadDipole output can thus be readilycombined with the results from one-loop matrix element generators, which are currentlyunder rapid development [28–32].A first application of the MadDipole package, in combination with the GOLEM one-loop amplitude generator [32], was in the calculation of NLO corrections to the quark-antiquark contribution to pp → b ¯ bb ¯ b [44]. Many more applications are likely to follow. Acknowledgements
We would like to thank Fabio Maltoni for many useful discussions and John Campbell forsome help on the checks against MCFM. NG wants to thank the University of Zurich forthe kind hospitality during final work on the project. This research was supported in partby the Swiss National Science Foundation (SNF) under contract 200020-126691 and in partby the U. S. Department of Energy under contract No. DE-FG02-91ER40677. A. α -dependence of the integrated dipoles In this appendix we give a list of the references, where the α -dependent terms can be foundin the literature. For the few cases which were not known previously, we give the resultbelow. – 13 – .1 Final-final Configuration Reference Remarks(a) Q → Q g , m k > α (b) Q → Q g , m k = 0 [64] (A.22,A.23)(c) q → q g , m k > q → q g , m k = 0 [63] (22)(e) g → g g , m k > α (f) g → g g , m k = 0 [63] (24)(g) g → Q ¯ Q , m k > g → Q ¯ Q , m k = 0 this work, (A.9)(i) g → q ¯ q , m k > α (j) g → q ¯ q , m k = 0 [63] (23)In the massive case there is an ambiguity how one defines α . The upper limit of theintegration over y is given by the variable y + . One can now define α in such a way thatthe maximal value for α is given by y + . In that case the integration range is given bythe interval y ∈ [ α, y + ]. Another possibility is to have α = 1 as the maximal value whichimplies that the allowed integration range is given by αy + < y ij,k < y + . (A.1)The first definition has been used in [42] whereas we use the latter.The two cases (g) and (h) are in principle not needed as the result is still finite dueto the masses of the quarks. For the sake of completeness we also included this cases andtherefore give the result. A.1.1 Case (g)
Case (g) is the splitting of a gluon into two massive quarks ( g → Q ¯ Q ) with a massivespectator. The splitting function for that process is given in Eq.(5.18) of [50], h V Q ¯ Q,k i = 8 πµ ǫ α s T R v ij,k ( − − ǫ " ˜ z i (1 − ˜ z i ) − (1 − κ ) z + z − − κµ Q µ Q + (1 − µ Q − µ k ) y ij,k , (A.2)where we neglect the last term because of κ = 0 in our implementation. The integrationranges are 2 µ j − µ j − µ k < y ij,k < y + = 1 − µ k (1 − µ k )1 − µ j − µ k , (A.3)1 − v ij,i v ij,k < ˜ z i < v ij,i v ij,k . (A.4)– 14 –ere, the integration over y ij,k does not start at y ij,k = 0 but at a value larger thanzero. But introducing the α parameter and calculating the correction terms implies newintegration boundaries for the y ij,k integration according to (A.1):2 µ j − µ j − µ k < αy + , (A.5)which implies that α must not be chosen to be too small for this splitting. With thisrestriction, we find I (g) ij,k ( ǫ, α ) = I (g) ij,k ( ǫ ) − T R (cid:16)(cid:16) bd (cid:16) − aµ j log (cid:16) αc y + − d q c − µ j − µ j (cid:17) + 2 ac log (cid:16) αc y + − d q c − µ j − µ j (cid:17) +2 ac log (cid:16) αc y + − d q c − µ j − µ j (cid:17) + 4 aµ j log (cid:16) αc y + − d q c − µ j − µ j (cid:17) − a (cid:0) c + c − µ j + 2 µ j (cid:1) log (cid:16) (1 − αy + ) (cid:0) − c − µ j (cid:1) / (cid:0) c − µ j + 1 (cid:1)(cid:17) +8 aµ j log (cid:16) − b q c − µ j + c y + − µ j (cid:17) − ac log (cid:16) − b q c − µ j + c y + − µ j (cid:17) − ac log (cid:16) − b q c − µ j + c y + − µ j (cid:17) − aµ j log (cid:16) − b q c − µ j + c y + − µ j (cid:17) +2 a (cid:0) c + c − µ j + 2 µ j (cid:1) log (cid:16) ( y + − (cid:16) − (cid:0) − c − µ j (cid:1) / (cid:17) (cid:0) c − µ j + 1 (cid:1)(cid:17) − c q µ j − c log( − αcy + + d )) + 4 cµ j q µ j − c log( − αcy + + d )) − c q µ j − c log( − αcy + + d )) + 3 c q µ j − c log( − b + cy + )) − cµ j q µ j − c log( − b + cy + )) + 2 c q µ j − c log( − b + cy + )) (cid:17) +2 bd q µ j − c (cid:0) c − c + 1) µ j + 4 µ j (cid:1) tan − µ j q α c y − µ j + c q µ j − c (cid:0) c y + (cid:0) α by + − αb − d ( y + − (cid:1) + 4 cµ j ( b ( αy + −
1) + d ( − y + ) + d ) + 4 µ j ( b − d ) (cid:1) − bd q µ j − c (cid:0) c − c + 1) µ j + 4 µ j (cid:1) tan − µ j q c y − µ j / (cid:16) c (cid:0) µ j − c (cid:1) / q c y − µ j q α c y − µ j (cid:17)(cid:17) (A.6) where we used the following abbreviations: a = q − µ k b = q c y − µ j c = − µ j + µ k d = q α c y − µ j . A.1.2 Case (h)
Case (h) describes also the splitting of a gluon into a massive quark pair ( g → Q ¯ Q ) howeverwith a massless spectator. While the splitting function is the same (A.2), the integration– 15 –oundaries are different, namely 2 µ i − µ i < y ij,k < − v ij,i < ˜ z i < v ij,i . (A.8)The phase space integral gets multiplied by Θ( α − y ij,k ) and integration leads to I (h) ij,k ( ǫ, α ) = I (h) ij,k ( ǫ ) − T R q α (cid:0) − µ j (cid:1) − µ j α − µ j − α + q α (cid:0) − µ j (cid:1) − µ j + (cid:0) µ j − (cid:1) − log (cid:18) − (cid:18)q α (cid:0) − µ j (cid:1) − µ j + α (cid:0) µ j − (cid:1)(cid:19)(cid:19) + 2 tan − µ j q α (cid:0) − µ j (cid:1) − µ j + log (cid:16) − (cid:16) µ j + q − µ j − (cid:17)(cid:17) − − µ j q − µ j + q − µ j . (A.9) A.2 Final-initial
Configuration Reference Remarks(a) Q → Q g [64] (A.13,A.14)(b) q → q g [64](A.17) / [13] (11-16) Different approaches(c) g → Q ¯ Q this work, (A.12,A.14,A.16)(d) g → q ¯ q [13] (11-16) Different approach in MadDipole(e) g → g g [13] (11-16) Different approach in MadDipoleThe cases (b), (d), and (e) can be found in [13]. Their result however contains alreadythe sum of different contributions making use of the I - and K -flavor kernels. As it mixesdifferent contributions this is not suitable for the MadDipole implementation. Thereforewe use [64] for (b) and derive results for (d) and (e) following the approach in [64]. Ofcourse, both approaches are equivalent. Again, for the sake of completeness we also addthe finite case (c). A.2.1 Case (c)
The one particle phase space for final-initial dipoles is given in Eq.(5.48) of [50]: Z [d p i ( e p ij ; p a , x )] = 14 (2 π ) − ǫ (2 e p ij p a ) − ǫ Z x + d x ij,a δ ( x − x ij,a ) (1 − x + µ ij ) − ǫ × Z d d − Ω Z z + ( x ) z − ( x ) d˜ z i [ z + ( x ) − ˜ z i ] − ǫ [˜ z i − z − ( x )] − ǫ , (A.10)and the integrated splitting function is given by Eq.(5.53) of [50] as: Z [d p i ( e p ij ; p a , x )] 1( p i + p j ) − m ij h V aij i ≡ α s π − ǫ ) (cid:18) πµ e p ij p a (cid:19) ǫ I aij ( x ; ǫ ) . (A.11)– 16 –or case (c), the splitting of a gluon into massive quarks ( g → Q ¯ Q ), the integrated spittingfunction is given in Eq.(5.57) of [50]: I aQ ¯ Q ( x ; ǫ ) = T R n [ J aQ ¯ Q ( x, µ Q )] x + + δ ( x + − x ) h J a ;S Q ¯ Q ( µ Q ; ǫ ) + J a ;NS Q ¯ Q ( µ Q ) io + O( ǫ ) , (A.12)where the x + -distribution is defined as: Z d x (cid:16) f ( x ) (cid:17) x + g ( x ) ≡ Z d x f ( x )Θ( x + − x ) [ g ( x ) − g ( x + )] . (A.13)Imposing the cut on the α -parameter implies that the phase space in (A.10) is multipliedwith Θ( α − x ija ).This leads to a modification of the x + -distribution terms and we get in analogy toEq.(5.62) of [50] [ J aQ ¯ Q ( x, µ Q , α )] x + = 23 − x + 2 µ Q (1 − x ) s − µ Q − x − α , (A.14)where we define: Z dx f ( x ) ( g ( x )) − α = Z − α dx g ( x ) ( f ( x ) − f (1)) (A.15)The non-singular terms J a ;NS Q ¯ Q ( µ Q ) receive: J a ;NS Q ¯ Q ( µ Q , α ) = J a ;NS Q ¯ Q ( µ Q ) + 29 − µ Q s (cid:0) − µ Q (cid:1) (cid:0) α − µ Q (cid:1) α + 4 − s (cid:0) − µ Q (cid:1) (cid:0) α − µ Q (cid:1) α − µ Q + 5 (cid:1) / (cid:16)q − µ Q (cid:17) + 6 log (cid:16)q α − µ Q + √ α (cid:17) − (cid:16)q − µ Q + 1 (cid:17)(cid:17) . (A.16) A.2.2 Case (d)
Case (d) is just the limit µ Q → i.e. ,[ J aq ¯ q ( x, , α )] + = 23 (cid:18) − x (cid:19) − α , (A.17)which leads to the following additional non-singular terms: J a ;NS q ¯ q (0 , α ) = J a ;NS q ¯ q (0) + 23 log α. (A.18) A.2.3 Case (e)
In the case of the splitting ( g → g g ) the general structure of the integrated splittingfunction is given by Eq.(5.66) of [50]: I agg ( x ; ǫ ) = 2 C A (cid:8) [ J agg ( x )] + + δ (1 − x ) J a ;S gg ( ǫ ) (cid:9) + O( ǫ ) . (A.19)– 17 –he first term [ J agg ( x )] + contains all +-distributions but is not a +-distribution itself. Inthe presence of the α -parameter we find[ J agg ( x, α )] + = (cid:18) − x ln 11 − x −
116 11 − x (cid:19) − α + 21 − x ln(2 − x )Θ( α − x ) , (A.20)which leads to a modification of the terms proportional to δ (1 − x ) of Eq.(5.68) of [50] ofthe following form: J a ;S gg ( x, α ) = J a ;S gg ( x ) − log α −
116 log α. (A.21) A.3 Initial-final
Configuration Reference Remarks(a) ˜ ij : q, emitter : q [64] (A.9, A.11)/ [13] (11-16) [13] only massless(b) ˜ ij : g, emitter : q [65] (A.8)(c) ˜ ij : q, emitter : g [65] (A.10)(d) ˜ ij : g, emitter : g [65] (A.11)From the analytical point of view the limit of a vanishing spectator mass can be performedwithout any problems. However taking the result for a massive spectator and setting themass to zero in the numerical implementation causes problems for the cases (b) and (d).For these two cases we calculated the limit analytically and implemented a massive and amassless version. A.4 Initial-initial
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