A complete reduction of one-loop tensor 5- and 6-point integrals
Th. Diakonidis, J. Fleischer, J. Gluza, K. Kajda, T. Riemann, J.B. Tausk
aa r X i v : . [ h e p - ph ] D ec Preprint typeset in JHEP style - HYPER VERSION
DESY 08-174BI-TP 2008/39SFB-CPP-08-97HEPTOOLS 08-046
A complete reduction of one-loop tensor5- and 6-point integrals
Th. Diakonidis a , J. Fleischer a,b , J. Gluza c , K. Kajda c , T. Riemann a , J. B. Tausk a a Deutsches Elektronen-Synchrotron, DESY, Platanenallee 6, 15738 Zeuthen, Germany b Fakult¨at f¨ur Physik, Universit¨at Bielefeld, Universit¨atsstr. 25, 33615 Bielefeld, Germany c Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland A BSTRACT : We perform a complete analytical reduction of general one-loop Feynman integralswith five and six external legs for tensors up to rank R = 3 and , respectively. An elegantformalism with extensive use of signed minors is developed for the cancellation of inverse Gramdeterminants. The 6-point tensor functions of rank R are expressed in terms of 5-point tensorfunctions of rank R − , and the latter are reduced to scalar four-, three-, and two-point functions.The resulting compact formulae allow both for a study of analytical properties and for efficientnumerical programming. They are implemented in Fortran and Mathematica.K EYWORDS : NLO Computations, QCD, QED, Feynman Integrals. ontents
1. Introduction 22. Representing tensor integrals by scalar integrals in shifted space-time dimensions 33. Pentagons 5 R = 2 tensor integrals 63.3 R = 3 tensor integrals 7
4. Hexagons 14 R = 2 tensor integrals 154.3 R = 3 tensor integrals 164.4 R = 4 tensor integrals 18
5. Numerical results and discussion 22A. Gram determinants and algebra of signed minors 32B. Reduction of dimensionally shifted five- and four-point integrals 35References 37 – 1 – . Introduction
At the proton-proton collider LHC and the planned e + e − collider ILC, a large number of particlesper event may be produced. The hope is to discover one or several Higgs bosons or supersymmetricparticles, which are typically expected to be quite heavy. The interest is also directed to the studyof known massive particles like the W and Z bosons or the top quark. Since the production ratesare large, a proper description of the cross-sections will typically include one-loop corrections to n -particle reactions, where some of the final state particles may be massive.The Feynman integrals for reactions with up to four external particles have been systematicallystudied and evaluated in numerous studies. We just want to mention here the seminal papers [1]and [2] and the Fortran packages FF [3] and LoopTools [4], which represent the state of the artuntil now. The treatment of Feynman integrals with a higher multiplicity than four becomes quiteinvolved if questions of efficiency and stability become vital, as it happens with the calculationalproblems related to high-dimensional phase space integrals over sums of thousands of Feynmandiagrams with internal loops.In this article, we will concentrate on the evaluation of massive one-loop Feynman integralswith n external legs and some tensor structure, I µ ··· µ R n = Z d d kiπ d/ Q Rr =1 k µ r Q nj =1 c ν j j , (1.1)where the denominators c j have indices ν j and chords q j , c j = ( k − q j ) − m j + iε. (1.2)We will study in the following the cases n = 5 with R ≤ and n = 6 with R ≤ , and we willconventionally assume q n = 0 . The space-time dimension is d = 4 − ǫ .There are several strategies one might follow. One is the reduction of higher-point tensorintegrals to tensor integrals with less external lines and/or lower tensor rank [5, 6, 7, 8]; a secondapproach is essentially numerical [9, 10] or semi-numerical [11, 12, 13]. A third one rests on theunitarity cut method [14, 15, 16, 17]. In this case, a one-loop amplitude is evaluated as a whole, byusing Cutkosky rules, instead of computing loop integrals from each of the Feynman diagrams. Itis impossible to give here a comprehensive survey of recent activities, and we would like to referto e.g. [18, 19, 20, 21] for recent overviews on the subject.Here, we will advocate yet another approach and reduce the tensor integrals algebraicallyto sums over a small set of scalar two-, three- and four-point functions, which we assume to beknown. Whether such a complete reduction is competitive with the other approaches might bedisputed. Evidently, this depends on the specific problem under investigation. For a study of gaugeinvariance and of the ultraviolet (UV) and infrared (IR) singularity structure of a set of Feynmandiagrams, it is evident that a complete reduction is advantageous, and it may also be quite usefulfor a tuned, analytical study of certain regions of potential numerical instabilities.We have chosen a strictly algebraic approach and will rely heavily on the algebra of signedminors which was worked out in detail by Melrose in [22]. One of the basic observations ofMelrose was that in four dimensions all the scalar integrals can be reduced to scalar 4-point func-tions and simpler ones. In [23], a representation of arbitrary one-loop tensor integrals in terms– 2 –f scalar integrals was derived. The representation includes, however, scalar integrals with higherindices ν j and higher space-time dimensions d + 2 l . The subsequent reduction to scalar integralswith only the original indices and the generic space-time dimension d is possible with the use ofintegration-by-parts identities [24] and generalizations of them with dimensional shifts. The latterhave been derived in [25], and a systematical application to one-loop integrals may be found in[26]. Basically, the reduction problem has been solved this way for n -point functions. There wasone attempt to use the Davydychev-Tarasov reduction for the description of one-loop contributionsto the process e + e − → Hν ¯ ν [27], and the numerical problems due to the five-point functions werediscussed in some detail. To a large extent they root in the appearance of inverse powers of Gramdeterminants. This feature of the Davydychev-Tarasov reduction was identified as disadvantageoussoon after its derivation, e.g. in [28], where a strategy for avoiding these problems was developed.Besides the problem of inverse powers of the Gram determinant of the corresponding Feynmandiagram, there are additional kinematical singularities related to sub-diagrams. This will not bediscussed here; we refer to e.g. [28, 29, 5, 6, 7, 8, 13, 17] and references therein.In this article, we investigate the reduction of tensor integrals with five and six external legswhich are of immediate importance in applications at the LHC. In Section 2 we represent tensorintegrals by scalar integrals in shifted space-time dimensions with shifted indices. Section 3 andSection 4 contain our main result. In Section 3 we go one step further in the reduction of five-pointtensors compared to [26] and demonstrate how to cancel all inverse powers of the Gram determinantappearing in the Davydychev-Tarasov reduction. Earlier results for tensors of rank two may befound in [30]. Section 4 contains the reduction of tensorial six-point functions to tensorial 5-pointfunctions. The corresponding Gram determinant is identically zero [26, 6, 8], and the reductionbecomes quite compact. Some numerical results and a short discussion are given in Section 5.The numerics is obtained with two independent implementations, one made in Mathematica, andanother one in Fortran. The Mathematica program hexagon.m with the reduction formulae ismade publicly available [31], see also [32] for a short description. For numerical applications, onehas to link the package with a program for the evaluation of scalar one- to four-point functions,e.g. with LoopTools [4, 33, 3],
CutTools [34, 12],
QCDLoop [35]. Appendices are devotedto some known, but necessary details on Gram determinants and the algebra of signed minors andto a short summary about the reduction of dimensionally shifted four- and five-point integrals.
2. Representing tensor integrals by scalar integrals in shifted space-time dimensions
At first we give the reduction of tensor integrals to a set of scalar integrals for arbitrary n -pointfunctions. Following [23, 26], assuming here the indices of propagators to be equal to one, ν r = 1 ,one has: I µn = Z d k µ n Y r =1 c − r = − n − X i =1 q µi I [ d +] n,i , (2.1) We will extensively quote from article [26], so we introduce here the notation (I.num) for a reference to equation(num) there. – 3 – µ νn = Z d k µ k ν n Y r =1 c − r = n − X i,j =1 q µi q νj n ij I [ d +] n,ij − g µν I [ d +] n , (2.2) I µ ν λn = Z d k µ k ν k λ n Y r =1 c − r = − n − X i,j,k =1 q µi q νj q λk n ijk I [ d +] n,ijk + 12 n − X i =1 ( g µν q λi + g µλ q νi + g νλ q µi ) I [ d +] n,i , (2.3) I µνλρn = Z d k µ k ν k λ k ρ n Y r =1 c − r = n − X i,j,k,l =1 q µi q νj q λk q ρl n ijkl I [ d +] n,ijkl − n − X i,j =1 ( g µν q λi q ρj + g µλ q νi q ρj + g νλ q µi q ρj + g µρ q νi q λj + g νρ q µi q λj + g λρ q µi q νj ) n ij I [ d +] n,ij + 14 (cid:16) g µν g λρ + g µλ g νρ + g µρ g νλ (cid:17) I [ d +] n , (2.4)where [ d +] is an operator shifting the space-time dimension by two units and I [ d +] l ,stu ··· p, i j k ··· = Z [ d +] l n Y r =1 c δ ri + δ rj + δ rk + ···− δ rs − δ rt − δ ru −··· r , Z d ≡ Z d d kiπ d/ , (2.5)where [ d +] l = 4 + 2 l − ǫ (observe that p is the number of scalar propagators of the “ p -pointfunction” and that equal lower and upper indices cancel, p ≤ n ). In (2.2–2.4), the coefficients n ij , n ijk and n ijkl were introduced. These stand for the product of factorials of the number ofequal indices: e.g. n iiii = 4! , n ijii = 3! , n iijj = 2!2! , n ijkk = 2! , n ijkl = 1! (indices i, j, k, l alldifferent from each other). Of particular relevance are the following relations for the successiveapplication of recurrence relations to reduce higher dimensional integrals: n ij = ν ij ,n ijk = ν ij ν ijk ,n ijkl = ν ij ν ijk ν ijkl , (2.6)where ν ij = 1 + δ ij ,ν ijk = 1 + δ ik + δ jk ,ν ijkl = 1 + δ il + δ jl + δ kl . (2.7)– 4 –n the next step the integrals in higher dimension have to be reduced to integrals in generic dimen-sion. Here particular attention has to be paid to I [ d +]5 . Reducing the tensor integrals, this term dropsout in general [36, 7].
3. Pentagons
We start with the reduction of the pentagons. This will also provide the basis for calculating thehexagons as we shall see.
For the scalar n = 5( d − ! I [ d +]5 = ! I − X s =1 s ! I s (3.1)With I [ d +]5 finite for d = 4 , we have in this limit E ≡ I = 1 ! X s =1 s ! I s , (3.2)i.e. the scalar five-point function is expressed in the limit d → in terms of scalar four-pointfunctions, which are obtained by scratching in the five terms of the sum the s th scalar propagator,respectively. This was already derived in [22], see eq. (6.1) there.Similarly, for the tensor integral of rank R = 1 (vector) in (2.1) we obtain: I µ = X i =1 q µi I ,i , (3.3)with I ,i ≡ E i = − I [ d +]5 ,i = ( d − i ! ! I [ d +]5 − ! X s =1 i s ! I s , (3.4)where again in the limit d → the I [ d +]5 disappears. These two cases are simple and lead to adirect reduction to scalar integrals, without the Gram determinant () occurring anyway. In thefollowing we want to reduce tensor integrals of higher rank and show, like in [5, 8], that also inthese cases the Gram determinant can be cancelled. The I ,i should not be confused with quantities introduced in Equation (2.5). – 5 – .2 R = 2 tensor integrals The tensor integral of rank 2 can be written without a g µν -term: I µ ν = X i,j =1 q µi q νj I ,ij , (3.5)which is obtained by replacing g µν by g µ ν = 2 X i,j =1 ij ! ! q µi q νj (3.6)(assuming q · · · q I [ d +]5 . It is known [36, 7] that it always cancelsin the end. This provides a very useful check on our calculations, which we have performed inevery particular case under consideration. Anticipating this cancellation, we will, for the ease ofour discussion, drop terms proportional to I [ d +]5 wherever they appear in the following derivation.With this in mind we can write for I ,ij in (2.2) with (B.2): I ,ij = ν ij I [ d +] ,ij = − j ! ! I [ d +]5 ,i + X s =1 ,s = i sj ! ! I [ d +] ,s ,i = j ! ! I ,i + X s =1 ,s = i sj ! ! I [ d +] ,s ,i , (3.7)and by means of (B.3) we obtain: I ,ij = 1 ! ! X s =1 ,s = i ss ! ( − j ! s i ! ss ! − sj ! si s ! ! + s ! s s ! ij ! ) I s − ij ! ! ! X s =1 ,s = i ss ! s ! s s ! I s – 6 – ! ! X s,t =1 ,s = i,t ss ! ( − j ! t s i ! ss ! − sj ! t si s ! ! + s ! t s s ! ij ! ) I st + ij ! ! ! X s,t =1 ,s = i,t ss ! s ! t s s ! I st . (3.8)Using (3.6) again, we find I µ ν = X i,j =1 q µi q νj E ij + g µν E , (3.9) E ij = X s =1 S ,sij I s + X s,t =1 S ,stij I st , (3.10)where S ,sij = 1 ! X s =1 ss ! X s ij , (3.11) S ,stij = − ! X s,t =1 ss ! X stij (3.12)and X s ij and X stij are defined in (A.22). Finally, E = −
12 1 ! X s =1 s ! ss ! " s s ! I s − X t =1 t s s ! I st . (3.13)In this way we have cancelled the Gram determinant for the tensor of rank 2. For later reference,we note that, by taking into account (B.4), we can also write E = −
12 1 ! X s =1 s ! I [ d +] ,s . (3.14) R = 3 tensor integrals The tensor integral of rank 3 can be written as: I µ ν λ = X i,j,k =1 q µi q νj q λk I ,ijk . (3.15)– 7 –e will now rewrite this into another representation, thereby avoiding Gram determinants () inthe denominators of the new tensor coefficients E ijk , E k : I µ ν λ = X i,j,k =1 q µi q νj q λk E ijk + X k =1 g [ µν q λ ] k E k , (3.16) E ijk = X s =1 S ,sijk I s + X s,t =1 S ,stijk I st + X s,t,u =1 S ,stuijk I stu . (3.17)According to (2.3) we have with (3.6): I ,ijk = − ν ij ν ijk I [ d +] ,ijk + jk ! ! I [ d +] ,i + ik ! ! I [ d +] ,j + ij ! ! I [ d +] ,k . (3.18)By means of recursion (B.1), taking into account (3.7) and keeping in mind to drop I [ d +]5 , we have: I ,ijk = k ! ! ν ij I [ d +] ,ij − X s =1 ,s = i,j sk ! ! ν ij I [ d +] ,s ,ij + ij ! ! X s =1 sk ! ! I [ d +] ,s = k ! ! I ,ij + ij ! ! X s =1 sk ! ! I [ d +] ,s − X s =1 ,s = i,j sk ! ! ν ij I [ d +] ,s ,ij . (3.19)Collecting the terms proportional to () − we have with I ,ij = · · · + 2 ij
0@ 1A E and (3.14): ij ! ! !
25 5 X s =1 " ! sk ! − k ! s ! I [ d +] ,s = ij ! ! ! X s =1 s k ! I [ d +] ,s , (3.20)i.e. we have already cancelled one Gram determinant. We multiply (3.19) by ! such that wecan make use of ! sk ! = s k ! ! + s ! k ! , (3.21)– 8 –hich will give us another factor () . Adding all contributions, we obtain ! I ,ijk = X ′ ! ss ! ( k ! ss ! (cid:2) X sij I s − X stij I st (cid:3) + ij ! ss ! s k ! " s s ! I s − t s s ! I st − ! sk ! " si s ! sj s ! + i sj s ! s s ! I s − " sj s ! t si s ! + i sj s ! t s s ! I st !) + X ′ ! ! sk ! ss ! s ts t ! t sj s ! " s ti s t ! I st − u s ti s t ! I stu + ( i ↔ k ) + ( j ↔ k ) ≡ A + X ′ ( ! ss ! ij ! ss ! s k ! " s s ! I s − t s s ! I st + 1 ! ! sk ! ss ! s ts t ! t sj s ! " s ti s t ! I st − u s ti s t ! I stu + ( i ↔ k ) + ( j ↔ k ) (3.22)The symbol P ′ in these equations denotes a sum P s,t,u =1 in terms proportional to I stu , P s,t =1 interms proportional to I st , and P s =1 in terms proportional to I s . Concerning the symmetrizationin (3.22), we point out that the original expression (3.18) is obviously symmetric under ( i ↔ j ) ,while this is not explicitly seen in (3.22) anymore. Later on, however, this symmetry will becomeapparent again.All terms with factors of the type ij ! can be considered, due to (3.6), as belonging to some g µν term. For other terms we have to use (3.21), which yields terms with () to be cancelled. Theseare explicitly given in the coefficients of I s , I st and I stu , i.e. (3.30, 3.32, 3.33). Apart from theterms in the last line of (3.22) and the ij ! term, the remaining contributions to the coefficients– 9 –f I s and I st , inserting X sij and X stij , can be written as A = − X s =1 ! ss ! k ! ( ss ! " s i ! sj s ! − js i ! s s ! + s ! " si s ! sj s ! + i sj s ! s s ! I s + X s,t =1 ! ss ! k ! ( ss ! " s j ! t si s ! − is j ! t s s ! + s ! " sj s ! t si s ! + j si s ! t s s ! I st . (3.23)Here the following “master formula” ( Equation (A.13) of [22] ) is of great help: si ! s τ s ! = s ! s τi s ! + ss ! s τ i ! , τ = 0 , , . . . , (3.24)which yields explicitly: ss ! s i ! + s ! si s ! = si ! s s ! , (3.25)and: ss ! js i ! − s ! i sj s ! = − sj ! si s ! , (3.26)so that (3.23) reads: A = − X s =1 ! ss ! k ! · (" si ! sj s ! + sj ! si s ! s s ! I s − X t =1 " sj ! s s ! t si s ! + si ! sj s ! t s s ! I st ) . (3.27)Next we will use : k ! si ! = − is k ! ! + ik ! s ! . (3.28)As trivial as this relation may look, it plays the crucial role of splitting off ik ! in order to produce g µν terms. It might also have been written as: k ! si ! = sk i ! ! + i ! sk ! , (3.29)– 10 –ut then it would not fulfill its purpose.The first term at the rhs. of (3.28) cancels a () , while the second term enters the g µν -terms,all of which are collected in (3.36). The complete coefficient of I s in (3.16) is thus given by: S ,sijk = 13 ! ss ! ( − s k ! " si s ! sj s ! + i sj s ! s s ! + " is k ! sj s ! + js k ! si s ! s s ! + ( i ↔ k ) + ( j ↔ k ) ) . (3.30)Finally we have to investigate the last line of (3.22), being left with the factor k ! s ! asbefore in (3.21). The master formula (3.24) then yields: s ! t sj s ! = sj ! t s s ! − ss ! t s j ! . (3.31)The ss ! in the second term of (3.31) cancels and the remaining factor is antisymmetric in s and t , i.e. this term drops out after summation over s, t . Using again (3.28) and dropping for the timebeing the contribution to g µν terms, we finally write the coefficients of I st and I stu in the followingway, taking care of the original ( i ↔ j ) symmetry in (3.22): S ,stijk = 13 ! ss ! ( s k ! " t si s ! sj s ! + i sj s ! t s s ! + ss ! s ti s t ! s ts t ! t sj s ! − " is k ! sj s ! + js k ! si s ! t s s ! − " is k ! t sj s ! + js k ! t si s ! ss ! s t s t ! s ts t ! +( i ↔ k ) + ( j ↔ k ) ) , (3.32)and S ,stuijk = − ! ss ! s ts t ! ( s k ! t sj s ! u s ti s t ! − – 11 – " js k ! u s ti s t ! + is k ! u s tj s t ! t s s ! + ( i ↔ k ) + ( j ↔ k ) ) . (3.33)At the end we can determine the g µν terms from the above by collecting all terms containing factorsof the type ij ! : X j =1 g [ µν q λ ] j E j = 2 ! X ijk =1 " jk ! E i + ik ! E j + ij ! E k q µi q νj q λk , (3.34)where the square bracket means symmetrization of the included indices, g [ µν q λ ] k = g µν q λk + g µλ q νk + g νλ q µk , (3.35)and use has been made of (3.6). Collecting all terms of type ij ! in (3.22) we have: ! E j = − X s =1 ss ! " s ! sj s ! − ss ! s j ! s s ! I s + 12 X s,t =1 ( ss ! " s ! sj s ! − ss ! s j ! t s s ! + 1 ss ! s ! t sj s ! s s ! + 1 ss ! s ts t ! s ! t s s ! s tj s t ! ) I st − X s,t,u =1 ss ! s ts t ! s ! t s s ! u s tj s t ! I stu . (3.36)The following relation can be proven by multiplication with ss ! , transforming it into the relationfor an extensional of Equation (A.8) of [22]: s ! µ s tj s t ! = sj ! µ s t s t ! − µ s j ! s ts t ! + t s j ! t sµ s ! , µ = 0 , , · · · , . – 12 –3.37)It turns out to be useful for the simplification of the coefficients of I st and I stu in (3.36). For thecoefficient of I st , we apply relation (3.37) with µ = 0 . The last term on the r.h.s. of (3.37) iscombined with the term on the third line of (3.36) using (3.31): ss ! ss ! s ts t ! t s j ! t s s ! + s ! t sj s ! s s ! = 1 ss ! sj ! t s s ! s s ! − t s j ! s t s t ! s ts t ! . , (3.38)After summation over s and t , the last term on the r.h.s. will vanish. Furthermore we apply (3.24)taking τ = 0 .For the coefficient of I stu in (3.36) we apply relation (3.37) with µ = u . Since I stu is sym-metric in s, t and u , we consider the sum over all permutations of any fixed set of values of s, t and u . We find that X permutations ss ! s ts t ! t s s ! " u s j ! s ts t ! − t s j ! t su s ! = 0 , (3.39)so that the two last terms on the r.h.s. of (3.37) can be dropped in this case. Thus we have: ! E j = − X s =1 ss ! " s ! sj s ! − sj ! s s ! s s ! I s + 12 X s,t =1 ss ! s ! sj s ! − sj ! t s s ! s ts t ! t s s ! I st − X s,t,u =1 ss ! sj ! ss ! u s t s t ! s ts t ! t s s ! I stu . (3.40)Collecting all contributions, our final result for the tensor of rank 3 can be written as: I µ ν λ = X i,j,k =1 q µi q νj q λk E ijk + X k =1 g [ µν q λ ] k E k , (3.41)– 13 – ijk = X s =1 S ,sijk I s + X s,t =1 S ,stijk I st + X s,t,u =1 S ,stuijk I stu , (3.42)and the coefficients S ,sijk , S ,stijk , S ,stuijk are given in (3.30), (3.32) and (3.33) and E k in (3.40).
4. Hexagons
The 6-point function has the nice property that the tensors of rank R can be reduced to a sum of six5-point tensors of rank R − . This property has also been derived in [5]; an earlier demonstrationof this property, however, has been given already in [26]. The simplification in this case is due tothe fact that () ≡ , which has extensively been discussed in [26]. Beyond that, in our approach,the above results for the 5-point tensors can be directly used, thus reducing the 6-point tensors ofup to rank R = 4 to scalar 4- and 3- and 2-point integrals. Particularly simple results are thusobtained for the 6-point tensors using the results of Appendix A and Sections 3.1 and 3.2. Whatwas missing in [26] is exactly this simplification, which comes with the cancellation of the Gramdeterminant () ; see Appendix A of that paper. According to (I.33) we write (see [22] and also (I.55)): I = X r =1 r ! ! E r = X r =1 rk ! k ! E r , k = 1 , . . . , , (4.1)and (3.2) now reads: E r ≡ I r = 1 r r ! X s =1 ,s = r rs r ! I rs . (4.2)Here we see already the general scheme of reducing 6-point functions to 5-point functions: Ingeneral, in any signed minor ( · · · ) a further column rr ! is scratched, resulting in a ( · · · ) andin the scalar functions a further propagator is scratched.As in (3.3) and (3.4), with the use of (I.57), we obtain: I µ = X i =1 q µi I ,i , (4.3)– 14 – ,i = − I [ d +]6 ,i = ( d − i ! ! I [ d +]6 − ! X r =1 i r ! I r . (4.4)While in (3.4) the first part vanishes in the limit d → , here its disappearance is due to (I.61): X i =1 q µi i ! = 0 . (4.5)Indeed (4.5) will play a crucial role for the higher tensor reduction. The resulting form in (4.4) isalready the generic form for the higher tensors too! Therefore it appears useful to introduce thevector, applying further (A.15) of [22] and (I.61): v µr = − ! X i =1 i r ! q µi = − k ! X i =1 rk i ! q µi , k = 0 , . . . , , (4.6)summing over all 5 (dependent) vectors. v r projected on these vectors reads: v r · q i = − δ ir − ( Y i − Y ) r ! ! = − δ ir + ( q i + m − m i ) r ! ! . (4.7)With this definition we can write in a compact way: I µ = X r =1 v µr E r . (4.8) R = 2 tensor integrals The equation (2.2) reads in this case: I µ ν = X i,j =1 q µi q νj ν ij I [ d +] ,ij − g µν I [ d +]6 , (4.9)– 15 –nd by using (I.59) we have: ν ij I [ d +] ,ij = − ( d − i ! ! I [ d +] ,i + i j ! ! I [ d +]6 + 1 ! X r =1 ,r = i j r ! I [ d +] ,r ,i . (4.10)We consider the limit d → and use (I.67): g µν = 2 ! X i,j =1 i j ! q µi q νj . (4.11)Writing it like in (3.5), I µ ν = X i,j =1 q µi q νj I ,ij , (4.12)we obtain by using (3.4): I ,ij = − ! X r =1 ,r = i j r ! E ri , (4.13)to be compared with (4.4). For completeness we specify E ri , which we read off from (3.4) to be: E ri = − r r ! X s =1 i r s r ! I rs , (4.14)and finally: I µ ν = X i =1 q µi X r =1 ,r = i v νr E ri . (4.15)We remark that due to (4.14), E ri = 0 for r = i and correspondingly this will be the case forall higher tensors such that limitations like r = i could be dropped but are convenient to keep innumerical programs. R = 3 tensor integrals Equation (2.3) reads in this case: I µ ν λ = − X i,j,k =1 q µi q νj q λk ν ij ν ijk I [ d +] ,ijk + 12 X i =1 ( g µν q λi + g µλ q νi + g νλ q µi ) I [ d +] ,i , (4.16)– 16 –nd with (I.60) we have: ν ij ν ijk I [ d +] ,ijk = − ( d − k ! ! I [ d +] ,ij + k i ! ! I [ d +] ,j + k j ! ! I [ d +] ,i + 1 ! X r =1 ,r = i,j k r ! ν ij I [ d +] ,r ,ij . (4.17)The first term on the r.h.s. is eliminated due to (4.5) and the next two terms cancel due to (4.11).Taking into account I [ d +]5 , relation (3.7) now reads: I r ,ij = ν ij I [ d +] ,r ,ij − i rj r ! rr ! I [ d +] ,r . (4.18)As a further representation of g µν we have (see (I.75)): g µν = 2 rr ! X i,j =1 i rj r ! q µi q νj , r = 1 . . . . (4.19)Using again (I.57) and the definition I µ νλ = X i,j,k =1 q µi q νj q λk I ,ijk , (4.20)we obtain: I ,ijk = − ! X r =1 ,r = i,j k r ! I r ,ij . (4.21)From (3.10) and (4.19) I r ,ij reads: I r ,ij = E rij + 2 i rj r ! rr ! E r , (4.22)– 17 –o that we get: I µ ν λ = X i,j =1 q µi q νj X r =1 ,r = i,j v λr E rij + X i,j =1 q µi q νj X r =1 i rj r ! rr ! v λr E r , (4.23)where in the second term we can drop the limitation r = i, j since it is automatically fulfilled dueto the numerator i rj r ! , vanishing for r = i and r = j . Thus summation over i and j is possible,using (4.19), with a result: I µ ν λ = X i,j =1 q µi q νj X r =1 ,r = i,j v λr E rij + g µν X r =1 v λr E r , (4.24)or: I µ ν λ = X r =1 v λr I µ ν ,r , (4.25)with I µ ν ,r = X i,j =1 ,i,j = r q µi q νj E rij + g µν E r . (4.26) R = 4 tensor integrals The tensor integral in (2.4) contains three different integrals in higher dimension, which have to bereduced or to be eliminated. We begin with I [ d +] n,ijkl using (I.26). For convenience we use x insteadof : ! n ν ijkl l + I [ d +] x n,ijk ≡ ! n ν ijkl I [ d +] x n,ijkl = n X r =1 l r ! n [ d + 2 x − ( n + 3)] I [ d +] x n,ijk − n X s =1 l s ! n ν ijks I [ d +] x n,ijk − n X r,s =1; r = s l r ! n ν ijks r − s + I [ d +] x n,ijk = ( [ n + 4 − ( d + 2 x )] l ! n − l i ! n − l j ! n − l k ! n ) I [ d +] x n,ijk − n X r =1 l r ! n n X s =1; s = r ν ijks I [ d +] x ,rn,ijks . (4.27)The last double sum in (4.27), assuming all indices i, j, k to be different, reads: − l i ! n n X s =1; s = i ν jks I [ d +] x n,jks − l j ! n n X s =1; s = j ν iks I [ d +] x n,iks − l k ! n n X s =1; s = k ν ijs I [ d +] x n,ijs – 18 – n X r =1; r = i,j,k l r ! n n X s =1; s = r ν ijks I [ d +] x ,rn − ,ijks . (4.28)Now adding corresponding terms in (4.27) and (4.28), e.g. for r = i , we get: − l i ! n n X s =1; s = i ν jks I [ d +] x n,jks − l i ! n I [ d +] x n,ijk = − l i ! n n X s =1 ν jks I [ d +] x n,jks = l i ! n I [ d +] ( x − n,jk , (4.29)due to (I.29). In case two indices are equal, e.g. i = j = k , we have: − l i ! n n X s =1; s = i (1 + 2 δ is + δ ks ) I [ d +] x n,iks − l i ! n I [ d +] x n,iis = − l i ! n n X s =1 (1 + δ is + δ ks ) I [ d +] x n,iks ≡ − l i ! n n X s =1 ν iks I [ d +] x n,iks , (4.30)like (4.29), i.e. if two indices agree, this integral occurs only once. As final result we have: ! n ν ijkl I [ d +] x n,ijkl = [ n + 4 − ( d + 2 x )] l ! n I [ d +] x n,ijk + [ ijk ] ( l )r ed + n X r =1; r = i,j,k l r ! n I [ d +] ( x − ,rn − ,ijk , (4.31)where according to (4.29) and the discussion thereafter: [ ijk ] ( l ) = l i ! n I [ d +] ( x − n,jk + l j ! n I [ d +] ( x − n,ik + l k ! n I [ d +] ( x − n,ij (4.32)and [ ijk ] ( l )r ed = [ ijk ] ( l ) without repetition, e.g. [ iii ] ( l )r ed = l i ! n I [ d +] ( x − n,ii .Now, making use of n ijkl = ν ij ν ijk ν ijkl , we see that due to (4.5) the first part in (4.31) dropsout after insertion into (2.4). The second contribution of (4.31) yields: ! n n − X l =1 q ρl n − X i,j,k =1 ν ij ν ijk [ ijk ] ( l )r ed q µi q νj q λk . (4.33)We have: ν ij ν ijk [ ijk ] ( l )r ed = [ ijk ] ( l ) + l i ! n δ jk I [ d +] ( x − n,jk – 19 – l j ! n δ ik I [ d +] ( x − n,ik + l k ! n δ ij I [ d +] ( x − n,ij , (4.34)with the help of which (4.33) reads: ! n n − X l =1 q ρl n − X i,j,k =1 " q µi l i ! n (1 + δ jk ) I [ d +] ( x − n,jk q νj q λk + q νj l j ! n (1 + δ ik ) I [ d +] ( x − n,ik q µi q λk + q λk l k ! n (1 + δ ij ) I [ d +] ( x − n,ij q µi q νj . (4.35)Using (I.67) we have for d = 4 : n g µρ n jk I [ d +] ( x − n,jk q νj q λk + g νρ n ik I [ d +] ( x − n,ik q µi q λk + g λρ n ij I [ d +] ( x − n,ij q µi q νj o , (4.36)and we see that this contribution is canceled by the last three terms of the type I [ d +] ( x − n,jk in (2.4).The first three terms of this type are evaluated by means of (I.59) to yield: n ij I [ d +] ( x − n,jk = 1 ! n ( [ n + 2(2 − x ) − d ] j ! n I [ d +] ( x − n,i + i j ! n I [ d +] ( x − n + n X r =1; r = i j r ! n I [ d +] ( x − ,rn − ,i . (4.37)Inserting this into (2.4), the first part yields a vanishing contribution due to (4.5) . The second termyields, again due to (4.5): − ! n n − X i,j =1 n g µν q λi q ρj + g µλ q νi q ρj + g νλ q µi q ρj o i j ! n I [ d +] ( x − n = − (cid:16) g µν g λρ + g µλ g νρ + g µρ g νλ (cid:17) I [ d +] ( x − n , (4.38)which cancels the last term in (2.4) and the total contribution thus reads: ! n n − X i,j,k,l =1 ν ij ν ijk q µi q νj q λk q ρl n X r =1; r = i,j,k l r ! n I [ d +] ( x − ,rn − ,ijk − n − X i,j =1 (cid:16) g µν q λi q ρj + g µλ q νi q ρj + g νλ q µi q ρj (cid:17) n X r =1; r = i j r ! n I [ d +] ( x − ,rn − ,i , (4.39)– 20 –educing the 6-point tensor to 5-point tensors in lower dimensions. For further reduction we putexplicitly n = 6 and x = 4 and write (3.18) in the form ν ij ν ijk I [ d +] ,r ,ijk = − I r ,ijk + j rk r ! rr ! I [ d +] ,r ,i + i rk r ! rr ! I [ d +] ,r ,j + i rj r ! rr ! I [ d +] ,r ,k . (4.40)With (4.19) it is now easy to see that the square bracket in (4.40) cancels out the second part in(4.39) and using the definition: I µ ν λ ρ = X i,j,k,l =1 q µi q νj q λk q ρl I ,ijkl , (4.41)we obtain: I ,ijkl = − ! X r =1 ,r = i,j,k l r ! I r ,ijk . (4.42)Again, with (3.17) and (4.19) I r ,ijk reads: I r ,ijk = E rijk + 2 i rj r ! rr ! E r k + 2 i rk r ! rr ! E r j + 2 j rk r ! rr ! E r i , (4.43)so: I µ ν λρ = X i,j,k =1 q µi q µj q λk X r =1 ,r = i,j,k v ρr E rijk + X i,j,k =1 q µi q µj q λk X r =1 ,r = i,j,k v ρr i rj r ! rr ! E r k + 2 i rk r ! rr ! E r j + 2 j rk r ! rr ! E r i , (4.44)and with the same argument like the one used after (4.23) we obtain the final result: I µ ν λρ = X i,j,k =1 q µi q µj q λk X r =1 ,r = i,j,k v ρr E rijk + g µν X k =1 q λk X r =1 ,r = k v ρr E r k + g µλ X j =1 q νj X r =1 ,r = j v ρr E r j + g νλ X i =1 q µi X r =1 ,r = i v ρr E r i , (4.45)– 21 –r: I µ ν λρ = X r =1 v ρr I µ ν λ,r , (4.46)with: I µ ν λ,r = X i,j,k =1; i,j,k = r q µi q νj q λk E rijk + X k =1 ,k = r g [ µν q λ ] k E r k . (4.47)
5. Numerical results and discussion
In order to illustrate the numerical results which can be obtained with the described approach, wewill evaluate a representative collection of tensor coefficients. We rely on two implementations ofthe formalism, one has been established in Fortran, and the other one in the Mathematica package hexagon.m .In the following, we denote the scalar five-point function by E and the scalar six-point func-tion by F . The tensor decompositions of pentagons E and hexagons F read: E µ = X i =1 q µi E i , (5.1) E µν = X i,j =1 q µi q νj E ij + g µν E , (5.2) E µνλ = X i,j,k =1 q µi q νj q λk E ijk + X i =1 g [ µν q λ ] i E i , (5.3) F µ = X i =1 q µi F i , (5.4) F µν = X i,j =1 q µi q νj F ij , (5.5) F µνλ = X i,j,k =1 q µi q νj q λk F ijk + X i =1 g µν q λi F i , (5.6) F µνλρ = X i,j,k,l =1 q µi q νj q λk q ρl F ijkl + X i,j =1 q µi q [ νj g λρ ] F ij . (5.7)Please observe the difference of E , F and E , F in the following. The kinematics is visualizedin Figure 5.1. Deviating from the first sections, we have chosen here q = 0 in order to stay closeto common conventions of other numerical packages.For the evaluation of the scalar two-, three- and four-point functions, which appear after thecomplete reduction, we have implemented two numerical libraries: • For massive internal particles:
Looptools 2.2 [4, 33];– 22 – igure 5.1:
Momenta flow used in the numerical examples for six- and five-point integrals. • If there are also massless internal particles:
QCDLoop-1.4 [35].We observed that
Looptools may become unstable in the presence of massless internal particles,while
QCDLoop seems to be generally slower. Our Mathematica package has an implementationof only
Looptools .For completeness, we would like to mention also other publicly available Fortran packages fortensor functions, which we found useful for comparisons: • Six-point tensors with massive internal particles: none; • Five-point tensors with massive internal particles:
Looptools [4, 33] ; • Five-point tensors with both massive and massless particles: none; • Five- and six-point tensors with only massless internal particles: golem95 [37].The two independent numerical implementations have been checked in several ways: • By internal comparisons of the two codes, relying on the formulae presented in this article;With alternative, direct representations of the tensor integrals with sector decomposition [38] and Mellin-Barnes representations [39, 40]; • By simplifying the numerator structures algebraically and subsequent evaluation of the re-sulting integrals of lower rank; • By direct comparison with other tensor integral packages [33, 37].Some of the comparisons were documented in [32]. We used a Mathematica interface to the
GINAC package sector decomposition in order to have a convenientway to evaluate tensor Feynman integrals. – 23 –e restrict ourselves to a few phase-space points, see Tables 5.1 to 5.3. The first configurationcorresponds to the reaction gg → t ¯ tq ¯ q , with external momenta generated by Madgraph [41, 42].The second configuration comes from [37], while the third is a slight modification of the first one.The kinematical input is completed by adding the masses of internal particles.We begin with massive six-point tensors. For the kinematics introduced above, we determinethe tensor components with our Fortran pacakge as shown in Tables 5.4 to 5.6. They are complex,finite numbers. Only independent components of the tensors are shown, all the remaining ones areobtained by permutations of indices.Selected tensor coefficients of five-point tensors for the case of massive internal particles areshown in Table 5.7. The coefficients have been compared with
LoopTools 2.2 and indeed weagree. For the massive six-point functions, there is no alternative package publicly available.In presence of massless internal particles, we face potential infrared singularities. Then, theloop functions are Laurent series in ǫ , starting with a term proportional to ǫ , and one has to careabout re-normalizations compared to our basic definition 1.1. A popular measure is [35, 37]: M = ( µ ) − d Γ(1 − ǫ )Γ(1 + ǫ )Γ (1 − ǫ ) Z d d kiπ d/ . (5.8)When discussing Feynman integrals with a dependence on inverse powers of ǫ there appears adependence of their constant terms on these conventions. For convenience of the reader, the tablesare produced with a normalization as introduced in Equation 5.8, with the choice µ = 1 .For the case of six-point and five-point functions with only massless internal particles, we showonly a few sample coefficients in Table 5.9 and Table 5.8, which are produced with our Fortranpackage. The phase space point chosen here is defined in Table 5.2. We checked that, withindouble precision, we completely agree with corresponding numbers produced with golem95 .Finally, to complete the list of relevant results, we show also sample tensor coefficients for thecase of both massive and massless internal particles, for five-point tensors in Table 5.10 and forsix-point tensors in Table 5.11. For this case with mixed internal masses, there is no other publiclyreleased code available. Please notice that we show here five-point tensor coefficients , while in the case of six-point tensors we have showntensor components . The tensor components are representation independent and should be preferred as numerical output.For the five-point tensors with massive internal particles, however, we have arranged for a one-to-one correspondencewith output of
LoopTools 2.2 , so it might be interesting to have, in this case, the tensor coefficients instead. p p p – 0.20369415 E+03 – 0.47579512 E+02 0.42126823 E+02 0.84097181 E+02 p – 0.20907237 E+03 0.55215961 E+02 – 0.46692034 E+02 – 0.90010087 E+02 p – 0.68463308 E+01 0.53063195 E+01 0.29698267 E+01 – 0.31456871 E+01 p – 0.15878244 E+02 – 0.12942769 E+02 0.15953850 E+01 0.90585932 E+01 m = 110 . , m = 120 . , m = 130 . , m = 140 . , m = 150 . , m = 160 . Table 5.1:
The components of external four-momenta for the six-point numerics; all internal particles aremassive. For five-point functions, we shrink line 2 and fix p + p → p in order to retain momentumconservation. – 24 – p p – 0.19178191 – 0.12741180 – 0.08262477 – 0.11713105 p – 0.33662712 0.06648281 0.31893785 0.08471424 p – 0.21604814 0.20363139 – 0.04415762 – 0.05710657 p = − ( p + p + p + p + p ) , m = · · · = m = 0 . Table 5.2:
The external four-momenta for the six-point numerics; all internal particles are massless. Thisset of momenta comes from [37]. For five-point functions, we shrink line 2 and fix p + p → p in order toretain momentum conservation. To summarize, we have presented in this article tensor integrals of rank R ≤ for five-pointfunctions and of rank R ≤ for six-point functions. This is sufficient for the calculation of e.g.four fermion production at the LHC with NLO QCD corrections.There are further reactions of interest which will need higher-point functions and higher ranksof five- and six-point functions. The details of their reductions have been left for a later investiga-tion. p p p – 0.20369415 E+01 – 0.47579512 E+00 0.42126823 E+00 0.84097181 E+00 p – 0.20907237 E+01 0.55215961 E+00 – 0.46692034 E+00 – 0.90010087 E+00 p – 0.68463308 E–01 0.53063195 E–01 0.29698267 E–01 – 0.31456871 E–01 p – 0.15878244 E+00 – 0.12942769 E+00 0.15953850 E–01 0.90585932 E–01 m = 0 . , m = 0 . , m = 0 . , m = 1 . , m = 0 . , m = 0 . Table 5.3:
The external four-momenta for the six-point numerics; one internal mass is finite. For five-pointfunctions, we shrink line 2 and fix p + p → p in order to retain momentum conservation. – 25 – – 0.223393 E–18 – i 0.396728 E–19 µ F µ µ ν F µν Table 5.4:
Tensor components for scalar, vector, and rank R = 2 six-point functions; kinematics defined inTable 5.1 and Figure 5.1. – 26 – ν λ F µνλ Table 5.5:
Tensor components for a massive rank R = 3 six-point function; kinematics defined in Table 5.1and Figure 5.1. – 27 – ν λ ρ F µνλρ Table 5.6:
Tensor components for a massive rank R = 4 six-point function; kinematics defined in Table5.1 and Figure 5.1. – 28 – E E – 7.86411 E-16 + i 1.03994 E-15 E – 8.18587 E-11 + i 1.80354 E-11 E E Table 5.7:
Selected tensor coefficients of five-point tensor functions with massive internal particles; kine-matics defined in Table 5.1. ǫ /ǫ /ǫ F – 57.8724994 – i 9248.84583 – 3167.69411 – i 2981.57728 – 1003.89197 F – 867.761166 + i 859.212722 273.495904 + i 483.076108 153.767901 F F – 185.635891 + i 1465.754753 487.259427 + i 525.6914058 174.2745041 F – 2.64116950 – i 4.28827971 – 0.8480346995 – i 0.4557274228 – 0.1450625441 Table 5.8:
Tensor coefficients F , F , F and tensor components F , F of six-point functions; allinternal particles are massless, kinematics of Table 5.2. – 29 – /ǫ /ǫ E E E E E – 1035.29689 – i 1422.01085 – 452.640112 – i 254.226520 – 80.9228146 + i 0.0 E – 0.81227772 – i 5.68434189 E–14 – 2.04281037 E–14 – i 2.84217094 E–14 – 7.10542736 E–15 + i 0.0 Table 5.9:
Selected tensor coefficients of five-point tensor functions with massless internal particles; kinematics defined in Table 5.2. ǫ /ǫ /ǫ E – 0.289852933 E+04 + i 0.228935552 E+03 – 0.945038648 E+02 + i 0.454178453 E+02 0.7112330546 E+01 + i 0.0 E E – 0.79409571852 E+01 + i 0.5445326927 E+00 – 0.3008645503 E+00 + i 0.9457613783 E–01 0.1027869989 E–01 + i 0.0 E E Table 5.10:
Selected tensor components of five-point tensor functions with both massive and massless internal particles; kinematics defined in Table 5.3. ǫ /ǫ /ǫ F F F – 0.1014018623 E+02 + i 0.1797332619 E+01 – 0.5914958485 E–01 + i 0.3275539398 E+00 0.7678550480 E–01 + i 0.0 F – 0.5007216712 E+00 + i 0.4194342396 E–01 – 0.1642316924 E–01 + i 0.7789453935 E–02 0.1225024390 E–02 + i 0.0 F Table 5.11:
Selected tensor components of six-point tensor functions with both massive and massless internal particles; kinematics defined in Table 5.3. –30– cknowledgements
Work supported by Sonderforschungsbereich/Transregio SFB/TRR 9 of DFG “Computergest¨utz-te Theoretische Teilchenphysik” and by the European Community’s Marie-Curie Research Trai-ning Networks MRTN-CT-2006-035505 “HEPTOOLS” and MRTN-CT-2006-035482 “FLAVIA-net”. K.K. acknowledges a scholarship from the UPGOW project co-financed by the EuropeanSocial Fund. J.F. likes to thank DESY for the kind hospitality. We thank Th. Binoth, A. Denner,S. Dittmaier, Th. Hahn, C. Papadopoulos and P. Uwer for useful discussions.– 31 – . Gram determinants and algebra of signed minors
In this section relations are derived, which will turn out to be indispensable in our tensor reductions.We begin with some notational remarks on Gram determinants G n − , G n − = | q j q k | , j, k = 1 , · · · , n − . (A.1)The modified Cayley determinant of a diagram with n internal lines with chords q j is: () n = | C jk | , j, k = 0 , · · · , n, (A.2) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . . . Y Y . . . Y n Y Y . . . Y n ... ... ... . . . ... Y n Y n . . . Y nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , with Y jk = − ( q j − q k ) + m j + m k . (A.3)From our choice q n = 0 , it follows that both determinants are related: () n = − G n − , (A.4)and we will usually call () n the Gram determinant of the Feynman integral. Signed minors [22] are determinants (with a sign convention) which are obtained by excludingrows and columns from the modified Cayley determinant () n . They are denoted by the symbol j j · · · j m k k · · · k m ! n , (A.5)labelling the rows j , j , · · · , j m and columns k , k , · · · , k m which have been excluded from () n .The sign of a signed minor is defined by ( − j + j + ··· + j m + k + k + ··· + k m × Signature [ j , j , · · · j m ] × Signature [ k , k , · · · k m ] , (A.6)where Signature gives the sign of permutations to place the indices in increasing order. Thisagrees e.g. with the definition of the operator
Signature [ List ] in Mathematica. As an examplemay serve the quantity ∆ n : ∆ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y Y . . . Y n Y Y . . . Y n ... ... . . . ... Y n Y n . . . Y nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ! n . (A.7)We now will derive two relations between signed minors. Let us introduce A sij ≡ − j ! s i ! ss ! − sj ! si s ! ! + s ! s s ! ij ! . (A.8)– 32 –e are going to show that this expression can be factorized as A sij = ! X sij , (A.9)and provide an explicit expression for X sij . To begin with, we show that A sij is symmetric in theindices i and j for fixed s . Obviously the third term on the right hand side of (A.8) is symmetricsince we consider a symmetric determinant. The symmetry of the first two terms means ss ! " i ! j s ! − j ! i s ! + ! " si ! sj s ! − sj ! si s ! = 0 . (A.10)The first square bracket of (A.10) can be evaluated using (A.13) of [22], i.e. j ! i s ! = − ! si j ! + i ! j s ! (A.11)and (A.10) then results in si ! sj s ! + sj ! ss i ! + ss ! si j ! = 0 . (A.12)This is proved by multiplication with () and using Eqn. (A.8) of [22] with r = 2 , i.e. i lj k ! ! = ij ! lk ! − ik ! lj ! ; i, j, k, l = 0 , . . . , . (A.13)Inserting this, products of three factors of the form ik ! cancel by pairs, q.e.d. .For the following, relations (A.11) and (A.12) of [22] become important, i.e. n X i =1 i ! = () (A.14)and n X i =1 ji ! = 0 , ( j = 0) . (A.15)Further, “extensionals“ are needed, i.e. relations valid for () can be extended to any minor of () ;an extensional of (A.14) e.g. is n X i =1 j k i ! = jk ! . (A.16) Assuming here () = 0 means no limitation since we are just looking for an algebraic relation. – 33 –s the simplest case we now immediately obtain from (A.8) A sss = 0 , i.e. X sss = 0 . (A.17)Applying (A.14) and (A.15) to (A.8), we see X j =1 A sij = − ! s i ! ss ! (A.18)and due to the symmetry in i and j we also have X i =1 A sij = − ! s j ! ss ! , (A.19)which gives us a hint of how X sij might look, namely due to (A.18) it should contain a term − s i ! sj s ! . A further contribution must vanish after summing over i . Due to (A.17) itmust contain a factor js i ! . The second factor of this contribution can only be depend on s and has been determined by explicit calculation to be s s ! . Thus we conclude: X sji = X sij = − s i ! sj s ! + js i ! s s ! . (A.20)We come now to the second relation between signed minors. While (A.20) will be needed forthe reduction of -point tensors to scalars I s , for the reduction of -point tensors to scalars I st wealso need − j ! t s i ! ss ! − sj ! t si s ! ! + s ! t s s ! ij ! = ! X stij , (A.21)where again we have to show that indeed () factorizes and we have to give an explicit expressionfor X stij . The left-most term on the left hand side is an auxiliary term. It is antisymmetric in s and t after the cancellation of ss ! and vanishes after summation over s and t because I st is symmetricin s and t . The cancellation of ss ! has to be checked explicitly in every case where (A.21) isapplied. Observe that P j =1 js i ! = 0 but P i =1 js i ! = − P i =1 ji s ! = − js ! . – 34 –e observe that the expressions for X sij (A.8) and X stij (A.21) differ only by replacing one by t . Therefore the following ansatz is implied for X stij . X stij = − s j ! t si s ! + is j ! t s s ! . (A.22)Now we directly evaluate X stij () using (A.13): X stij () = − " ! sj ! − s ! j ! t si s ! + " s ! ij ! − is ! j ! t s s ! (A.23)and the remaining equation to be verified is s ! j ! t si s ! − is ! j ! t s s ! = − j ! t s i ! ss ! , (A.24)which is done by multiplying again with () and again using (A.13). This gives us at the same timealso a more general proof for X sij (A.20), putting t = 0 . B. Reduction of dimensionally shifted five- and four-point integrals
In this appendix we provide explicitly the needed recursion relations for the reduction of the five-and four-point functions. In spite of the fact that here, essentially, only two different relationsof [26] are applied for different indices and dimension, namely (I.30) and (I.31), we consider ithelpful and sometimes even necessary, to provide them in detail. A special case of (I.31) is (B.4).The others are special cases of (I.30). For the six-point function relation (I.26) plays a major roleand will be quoted when applied. ν ijk I [ d +] ,ijk = − k ! ! I [ d +] ,ij + X s =1 ,s = i,j sk ! ! I [ d +] ,s ,ij + ik ! ! I [ d +] ,j + jk ! ! I [ d +] ,i , (B.1) ν ij I [ d +] ,ij = − j ! ! I [ d +]5 ,i + X s =1 ,s = i sj ! ! I [ d +] ,s ,i + ij ! ! I [ d +]5 . (B.2)The four-point function’s shift is (I.44): I [ d +] ,s ,i = − si s ! ss ! I s + X t =1 ,t = s t si s ! ss ! I st , (B.3)– 35 –nd the four-point integrals occurring in the reduction are (I.50): I [ d +] ,s = s s ! ss ! I s − X t =1 ,t = s t s s ! ss ! I st d − (B.4)In applications we can put d = 4 since I [ d +]4 is UV- and IR-finite. Beyond that, as it is donefrequently [8], I [ d +]4 can be used as well as a “master integral” (see e.g. (3.14)) without reductionto the generic dimension. – 36 – eferences [1] G. ’t Hooft and M. Veltman, Scalar one loop integrals , Nucl. Phys.
B153 (1979) 365–401.[2] G. Passarino and M. Veltman,
One loop corrections for e + e − annihilation into µ + µ − in the Weinbergmodel , Nucl. Phys.
B160 (1979) 151.[3] G. van Oldenborgh,
FF: A package to evaluate one loop Feynman diagrams , Comput. Phys. Commun. (1991) 1–15.[4] T. Hahn and M. Perez-Victoria, Automatized one-loop calculations in four and d dimensions , Comput.Phys. Commun. (1999) 153, [ hep-ph/9807565 ].[5] A. Denner and S. Dittmaier,
Reduction of one-loop tensor 5-point integrals , Nucl. Phys.
B658 (2003)175–202, [ hep-ph/0212259 ].[6] A. Denner and S. Dittmaier,
Reduction schemes for one-loop tensor integrals , Nucl. Phys.
B734 (2006) 62–115, [ hep-ph/0509141 ].[7] T. Binoth, J. Guillet, and G. Heinrich,
Reduction formalism for dimensionally regulated one-loop n-point integrals , Nucl. Phys.
B572 (2000) 361–386, [ hep-ph/9911342 ].[8] T. Binoth, J. Guillet, G. Heinrich, E. Pilon, and C. Schubert,
An algebraic/numerical formalism forone-loop multi-leg amplitudes , JHEP (2005) 015, [ hep-ph/0504267 ].[9] A. Ferroglia, M. Passera, G. Passarino, and S. Uccirati, All-purpose numerical evaluation of one-loopmulti-leg Feynman diagrams , Nucl. Phys.
B650 (2003) 162–228, [ hep-ph/0209219 ].[10] Y. Kurihara and T. Kaneko,
Numerical contour integration for loop integrals , Comput. Phys.Commun. (2006) 530–539, [ hep-ph/0503003 ].[11] G. Ossola, C. Papadopoulos, and R. Pittau,
Reducing full one-loop amplitudes to scalar integrals atthe integrand level , Nucl. Phys.
B763 (2007) 147–169, [ hep-ph/0609007 ].[12] G. Ossola, C. Papadopoulos, and R. Pittau,
CutTools: a program implementing the OPP reductionmethod to compute one-loop amplitudes , JHEP (2008) 042, [ arXiv:0711.3596 ].[13] R. K. Ellis, W. Giele, and G. Zanderighi, Semi-numerical evaluation of one-loop corrections , Phys.Rev.
D73 (2006) 014027, [ hep-ph/0508308 ].[14] Z. Bern, L. Dixon, and D. Kosower,
Progress in one-loop QCD computations , Ann. Rev. Nucl. Part.Sci. (1996) 109–148, [ hep-ph/9602280 ].[15] L. Dixon, Calculating scattering amplitudes efficiently , hep-ph/9601359 .[16] Z. Bern, L. Dixon, and D. Kosower, On-Shell Methods in Perturbative QCD , Annals Phys. (2007)1587–1634, [ arXiv:0704.2798 ].[17] W. T. Giele and G. Zanderighi,
On the Numerical Evaluation of One-Loop Amplitudes: the GluonicCase , hep-ph/0805.2152 .[18] S. Weinzierl, Automated calculations for multi-leg processes , PoS
ACAT (2007) 005,[ arXiv:0707.3342 ].[19] Z. Bern et al. , The NLO multileg working group: Summary report , hep-ph/0803.0494 .[20] J. Bl¨umlein, S. Moch and T. Riemann (eds.), Loops and Legs in Quantum Field Theory, Proceedingsof the 9th DESY Workshop on Elementary Particle Theory, 20-25 April 2008, Sondershausen,Germany , Nucl. Phys. Proc. Suppl. (2008). – 37 –
21] T. Binoth,
LHC phenomenology at next-to-leading order QCD: theoretical progress and new results , PoS
ACAT (2008). To appear.[22] D. B. Melrose,
Reduction of Feynman diagrams , Nuovo Cim. (1965) 181–213.[23] A. Davydychev, A simple formula for reducing Feynman diagrams to scalar integrals , Phys. Lett.
B263 (1991) 107–111.[24] K. Chetyrkin and F. Tkachov,
Integration by parts: The algorithm to calculate β functions in fourloops , Nucl. Phys.
B192 (1981) 159–204.[25] O. Tarasov,
Connection between Feynman integrals having different values of the space-timedimension , Phys. Rev.
D54 (1996) 6479–6490, [ hep-th/9606018 ].[26] J. Fleischer, F. Jegerlehner, and O. Tarasov,
Algebraic reduction of one-loop Feynman graphamplitudes , Nucl. Phys.
B566 (2000) 423–440, [ hep-ph/9907327 ].[27] F. Jegerlehner and O. Tarasov,
FIRCLA, one-loop correction to e + e − → ν ¯ νH and basis of Feynmanintegrals in higher dimensions , Nucl. Phys. Proc. Suppl. (2003) 83–87, [ hep-ph/0212004 ].[28] J. Campbell, E. W. N. Glover, and D. Miller,
One-loop tensor integrals in dimensional regularisation , Nucl. Phys.
B498 (1997) 397–442, [ hep-ph/9612413 ].[29] G. Devaraj and R. G. Stuart,
Reduction of one-loop tensor form-factors to scalar integrals: A generalscheme , Nucl. Phys.
B519 (1998) 483–513, [ hep-ph/9704308 ].[30] J. Fleischer, J. Gluza, K. Kajda, and T. Riemann,
Pentagon diagrams of Bhabha scattering , Acta Phys.Polon.
B38 (2007) 3529–3536, [ arXiv:0710.5100 ∼ On the tensor reductionof one-loop pentagons and hexagons , Nucl. Phys. Proc. Suppl. (2008) 109–115,[ arXiv:0807.2984 ].[33] T. Hahn and M. Rauch,
News from FormCalc and LoopTools , Nucl. Phys. Proc. Suppl. (2006)236–240, [ hep-ph/0601248 ].[34] A. van Hameren, J. Vollinga, and S. Weinzierl,
Automated computation of one-loop integrals inmassless theories , Eur. Phys. J.
C41 (2005) 361–375, [ hep-ph/0502165 ].[35] R. K. Ellis and G. Zanderighi,
Scalar one-loop integrals for QCD , JHEP (2008) 002,[ arXiv:0712.1851 ].[36] Z. Bern, L. Dixon, and D. Kosower, Dimensionally regulated pentagon integrals , Nucl. Phys.
B412 (1994) 751–816, [ hep-ph/9306240 ].[37] T. Binoth, J. P. Guillet, G. Heinrich, E. Pilon, and T. Reiter,
Golem95: a numerical program tocalculate one-loop tensor integrals with up to six external legs , arXiv:0810.0992 .[38] C. Bogner and S. Weinzierl, Resolution of singularities for multi-loop integrals , Comput. Phys.Commun. (2008) 596–610, [ arXiv:0709.4092 ].[39] J. Gluza, K. Kajda, and T. Riemann,
AMBRE - a Mathematica package for the construction ofMellin-Barnes representations for Feynman integrals , Comput. Phys. Commun. (2007) 879–893,[ arXiv:0704.2423 ]. – 38 –
40] M. Czakon,
Automatized analytic continuation of Mellin-Barnes integrals , Comput. Phys. Commun. (2006) 559–571, [ hep-ph/0511200 ].[41] T. Stelzer and W. Long,
Automatic generation of tree level helicity amplitudes , Comput. Phys.Commun. (1994) 357–371, [ hep-ph/9401258 ].[42] F. Maltoni and T. Stelzer, MadEvent: Automatic event generation with MadGraph , JHEP (2003)027, [ hep-ph/0208156 ].].