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UTAS-PHYS-93-40 Tensor Integrals for Two Loop Standard Model Calculations 4 Dirk Kreimer∗ 9 9 Dept.of Physics 1 Univ.of Tasmania n G.P.O.Box 252C a J Hobart, 7001 8 Australia 6 December 1993 v 3 2 2 2 Abstract 1 Wegive a newmethod for the reduction of tensor integrals to finiteintegral repre- 3 sentationsandUVdivergentanalyticexpressions. Thisincludesanewmethodfor the 9 handlingof theγ-algebra. / h p In calculating two loop corrections to the Standard Model one is confronted with two main - problems. One is the analytical difficulty of integrals involving different masses. Often one p is restricted to approximations as for example zero mass assumptions for light particles [1]. e h A further problem stems from the difficulty in the algebraic sector, where the increasing : number of terms for the tensor structure has to be handled. For the case of two-point v i functions anelegantmethodisavailabletoreducealltensorintegralsto abasicsetofscalar X integrals[2]. Neverthelessitisclearthatthesemethodswillrunintodifficultieswhenapplied r to three- or four-point functions. Also, for arbitrary mass cases, one is restricted to either a numericalevaluationsorasymptoticexpansions[3]. Thisisduetothefactthatforarbitrary mass cases it is not possible to express the generic scalar integrals through known special functions [4]. In this paper we will present a general method to express two-loop integrals in terms of finite integrals suitable to numerical evaluations plus a set of products of one-loop integrals containingtheUVsingularpart. Themethodpresentedhereisaswellapplicabletoarbitrary tensorintegralsbut willproveespecially powerfulwhenapplied to two-loopgraphsdirectly, as it will bypass the whole tensor structure problem by constructing what can be called the characteristic polynomial of the graph. The final result is appropriate for the use of integral representation generated from the ones in [5] but the method can also be applied with other choices for the generic integrals, as the way how we handle the tensor structure is independent from the analyticalapproach one uses. The methods presented in the following apply to two-loop n-point Green’s function for arbitrary n, but in examples we will restrict ourself to two-loop two- and three-point problems. We will demonstrate the use of our method on the topology of the following figure. ∗email: [email protected] 1 1 5 '$ 3 2 4 Fig.1: The master two-loop two-point function; the labels denote the propagators. &% A general two-loop integral has the following representation T(nl,nk) I = dDldDk , (1) N MN l k Z where (n ,n ) denotes the rank of its tensor structure in l,k loop momenta. Here N is the l k l part of the denominator containing only propagators depending on l, N the same for loop k momentum k and M contains the propagatorsinvolving both loop momenta. We omit in the following the cases where terms in the numerator cancel the M part of the denominator completely out as these terms give rise to products of one-loop integrals only. Let us assume I has an overall degree of divergence ω and degrees of divergence ω ,ω l k for the subdivergences. To make this integral convergent we have to subtract its l resp. k subdivergent behaviour as well as its overalldivergence. To this end let us define the following N˜ = N | , l l mi=0,qi=0 N˜ = N | , k k mi=0,qi=0 M˜ = M | , (2) mi=0,qi=0 Ljl = (N˜lM˜ −NlM)jl, (N˜lM˜)jl Kjk = (N˜kM˜ −NkM)jk, (N˜kM˜)jk where| meanstoevaluatethecorrespondingexpressionwithallmassesandexternal mi=0,qi=0 momenta set to zero. Now consider the replacement 1 1 → Ljl, N N l l which improves ω and ω at least by j and leaves ω unchanged, and a similar replacement l l k for k. Consequently 1 1 → KjkLjl, (3) N MN N MN l k l k results (at least) in an improvement ω →ω+j +j =:ω′, l k ω →ω +j =:ω′, l l l l ω →ω +j =:ω′. k k k k Let us introduce the notation ω′ >0 ⇔ {ω′ >0,ω′ >0,ω′ >0}. Let us further define the l k notation C :=C−1 for arbitrary expressions C so that 0 1 jl j Lj0l = (N˜lM˜)jl "l=1(cid:18)ll(cid:19)(−NlM)l(N˜lM˜)jl−l#, X 2 1 jk j K0jk = (N˜kM˜)jk "k=1(cid:18)kk(cid:19)(−NkM)k(N˜kM˜)jk−l#, X 1 1 jl jk LjlKjk = (cid:2) (cid:3)0 (N˜lM˜)jl (N˜kM˜)jk Xl=0l+Xkk=>00  j j l k (−NlM)l(N˜lM˜)jl−l(−NkM)k(N˜kM˜)jk−k (4) l k (cid:18) (cid:19)(cid:18) (cid:19) (cid:21) where we emphasize that the sums start with l=1 resp. k =1. We then have the algebraic identity T(nl,nk) T(nl,nk) T(nl,nk) N MN = N MN LjlKjk − N MN LjlKjk 0 ∀jl,jk. (5) l k l k l k (cid:2) (cid:3) (cid:2) (cid:3) Now there clearly exists for a given T and given N ,M,N powers j,j such that l k l k ω′,ω′,ω′ are all positive, ω′ > 0, so that the first expression on the right hand side of l k equation (5) is finite. So it can be calculated in D = 4 dimensions. But by inspecting the sums 1 jl j Ljl = (N˜lM˜)jl "l=0(cid:18)ll(cid:19)(−NlM)l(N˜lM˜)jl−l#, X 1 jk j Kjk = (N˜kM˜)jk "k=0(cid:18)kk(cid:19)(−NkM)k(N˜kM˜)jk−l#, X we can identify all terms in the expansion which fulfil ω > 0. As these terms can be calculated in four dimensions they cancel against the corresponding terms of the second expressionin the right hand side of equation (5). Note that to identify the divergentpart a simplecountingofdimensionsofmassesandexteriormomentainthenumeratorofequation (5) is sufficient. So we define the following expressions 1 jl j Lˆjl = (N˜lM˜)jl ωXl=6>00(cid:18)ll(cid:19)(−NlM)l(N˜lM˜)jl−l,     1 jk j Kˆjk = (N˜kM˜)jk ωXk6>=00(cid:18)kk(cid:19)(−NkM)k(N˜kM˜)jk−l,     Lˆj0l = (N˜lM1˜)jl ωXlj=6>l10(cid:18)jll(cid:19)(−NlM)l(N˜lM˜)jl−l,     Kˆ0jk = (N˜kM1˜)jk ωXkj6>=k10(cid:18)jkk(cid:19)(−NkM)k(N˜kM˜)jk−l, (6)     where means summation only over divergent contributions. We obtain ω6>0 P T(nl,nk) = T(nl,nk) LˆjlKˆjk − T(nl,nk) LˆjlKˆjk . (7) NlMNk NlMNk NlMNk 0 h i h i 3 Still the first of the two terms on the right hand side of equation (7) can be calculated in D =4 dimensions. We willnow consideranexample which willbe useful to demonstrate the method. Consider Fig.(1) with propagators P = l2−m2, 1 1 P = (l+q)2−m2, 2 2 P = (l+k)2−m2, 3 3 P = (k−q)2−m2, 4 4 P = k2−m2, 5 5 and choose T(2,1) = l l k , µ ν σ N = P P , l 1 2 M = P , 3 N = P P , k 4 5 N˜ = l4, l M˜ = (l+k)2, N˜ = k4, k →ω =−1, ω =0, ω =1, j =2, j =0, l k l k so that we have T(2,1) T(2,1) T(2,1) T(2,1)N M L2 = −2 + l , NlMNk NlMNk N˜lMÑk (N˜lM˜)2Nk T(2,1) T(2,1) T(2,1)N M L2 = −2 + l , NlMNk 0 N˜lMÑk (N˜lM˜)2Nk T(2,1) T(2,1) T(2,1) T(2,1)(2l·q) Lˆ2 = − + , NlMNk NlMNk N˜lMÑk N˜lM˜l2Nk T(2,1) T(2,1) T(2,1)(2l·q) Lˆ2 = − + , (8) NlMNk 0 N˜lMÑk N˜lM˜l2Nk where weused thatP =P˜ −m2,P =P˜ +2l·q+q2−m2,P =P˜ −m2. P˜ meansagain 1 1 1 2 2 2 3 3 3 i the suppression of all masses and exterior momenta in the corresponding propagator. We will continue later with this example and discuss now how to calculate the UV divergentpart of equation (7), which is contained in the second term of the right hand side of this equation. These integrals are of the form T(nl,nk) , (9) (l2)i(l+k)jN k T(nl,nk) , (10) (k2)i(l+k)jN l T(nl,nk) or ≡0. (11) (l2)i(l+k)j(k2)r The second equation, (10), is just a relabeling of (9). The third equation, (11), vanishes identically in dimensional regularization. The integer powers i,j in the above equations are determined by j,j . l k Let us comment on the vanishing of the integralin (11) here. It might happen that this cancellationisduetoamutualcancellationofaninfraredandanultravioletdivergence,asit appears in the integral dDl /l4. Now the UV singular partof the integralin (11)will have R 4 its counterpart in the final integral representation in (7), where it is needed to cancel some otherUVsingularparts. Soitseemsthatwehaveinstalledaspuriousinfraredsingularityin our integral representation. This should then be regularized by some appropriate IR cutoff andthiswouldinstallacorrespondingscaleintheintegralin(11)whichwouldmakeitnon- vanishing. But these infrared singularities appear in the integral representation derived in [5]alwaysas endpointsingularities. As this problemwith spuriousIR singularitiescanonly appearwhen simultaneouslyn >0 and n >0 it will (inthe Feynman gauge)be restricted l k to the two-loop boson self-energies. Nevertheless it is true in general that IR singularities appear as endpoint singularities in the method of [5] which avoidsparametrizations. Let us study these endpoint singularities a little bit more. Investigating the case N = P , l 1 M = P , 3 N = P , k 5 we find the following expression for the subtraction of UV divergences 1 1 1 M˜ −m2 − − + 3 . (12) NlMNk N˜lMÑk NlMÑ˜k N˜lM˜2N˜k Thelastterm∼m2 istheproblematicone. Theintegralrepresentationforitcanbewritten 3 as ∞ ∞ 1 1 dx dy , (13) |x||x|+|y |+|x−y | Z−∞ Z−∞ where we used x2+iη =|x| in the limit η →0. Equation (13) has an apparent endpoint singularity at x =0. But these endpoint singularities are a well defined distribution in the p η-limitwheninterpretedasHadamard’sFinitePartHFP [6]. Itsrelationtothepropagator isknownforlong[7]. Infact,thedistribution ∞ dx/ x2+iη equalsHFP[1/|x|]. Using −∞ this,wehandletheseendpointsingularitieswiththehelpofHFP andfindthecorresponding R p integral representation ∞ ∞ 1 1 dx dy( +x↔−x)+.... |x||x|+|y |+|x−y | Z1 Z−∞ Herethedotscorrespondtofinitetermsforwhichtheendpointsingularityinxhascanceled outafter the y integrationandso HFP reduces to anordinaryRiemannintegral. As HFP is a local operation which modifies test functions only at the location of the singularity our UV behaviour remains unchanged and the sum in equation (12) is still UV finite. This can also be explicitly checked by working out all the different cases for the modulus function in equation(13)indetail. Sothe integralrepresentationforequation(12)isfreeofIRandUV singularities. One can also use HFP to separate nonspurious IR singularities when they appear. As for the integral (11) to appear we need UV singular behaviour in both loop variables, it is clear that the above subtlety can only arise when there is a logarithmic UV divergence in the second loop integration, which is exactly the case for the above example. In this sense the above example is generic and serves as a proof in general. We can still maintain the vanishing of equation (11) by using HFP in our integral representations. This can also be interpreted as a consistency between HFP and DR. Let us continue now our general discussion. We can do one integration in equation (9) leading to remaining integrals of the form T˜ C dDk , (14) (k2)αN k Z where α is an integer only when D = 4 and T˜ is some tensor in k. The coefficient C will contain a pole in (D−4) in general. This and possible UV divergences of the k integration itself forbid to calculate these integrals in four dimensions. 5 Calculating the above integrals (14) for arbitrary D and α is possible but a difficult analyticaltask. Butthereisanalgebraicapproachtotheseintegrals. Wewillfirsttransform the UV divergent integrals in equation (7) via a partial integration in such a way that the first l integrationis UV finite and its UV divergencesare shifted to the k integration. Then weuseamethodsimilartothemethodabovetosplitthefinalkintegrationoncemoreintoa finite integration,analytically computable in an easy manner and UV divergentintegralsof massivetadpolestructure. As aconsequenceeverythingisexpressibleby algebraicmethods in one-loop functions. Defining l ...l I(i,n,m) := dDl µ1 µi , we have (l2)n((l+k)2)m Z ∂ l l ...l 0 = dDl ρ µ1 µi (15) ∂l (l2)n((l+k)2)m ρ Z = (D+i−2n−m)I(i,n,m)−mI(i,n−1,m+1)+mk2I(i,n,m+1), I(i,0,m) =I(i,m,0) =0 ∀i,m. (16) As D+i≥2(n+m) for UV divergent integrals, we have (D+i−2n−m)6=0. Therefore we can always solve equation (16) for I(i,n,m) in the following. We can repeat this procedure until we have expressed I(i,n,m) through integrals of the form I(i,n′,m′) with powers m′,n′ such that they are all finite in the l integration. We will end up with integrals of the form k ...k (k2)β dDk µ1 µr , N k Z where againβ is anintegeronly ifD =4 but allcoefficients ofthese integralsarenow finite in four dimensions. But the k integration is still UV singular, and to find an expression for these integrals we do a last replacement 1 = 1 (Kî−Kî) where (17) N N 0 k k (N˙ −N )i Ki := k k and N˙ i k N˙ :=N | , k k qi=0 where it is understood that when expanding the numerators the sums run only over the divergent parts, as indicated by theôn the various K. Note that we have chosen propa- gatorswith vanishing exteriormomenta but non-vanishingmasses to profitfromsymmetric integration properties and we remind the reader that we cannot set the masses to zero be- cause this would result in vanishing tadpole integrals. So we end up with finite integrals and separated UV divergent integrals of the form k ...k (k2)β µ1 µr , N˙ k which are, as they are massive tadpole integrals, easy to evaluate in D dimensions. The finiteintegralsin(17)arerelatedtostandardmassiveone-loopintegralswithintegerpowers of propagators. Let us continue our example now. By inspecting equation (8) we see that we have to consider I(2,2,1) and I(3,3,1). The last one is already finite in the l integration. For the first one we have −1 2 I(2,2,1) = k2I(2,1,3)−k2I(2,2,2) . (D−3) (D−2) (cid:20) (cid:21) 6 The remaining integrals are of the form k ...k dDk µ1 µi . Z P4P5(k2)j+4−2D By using P = P˙ ,P = P˙ −2k·q+q2 we separate this according to the above equations 5 5 4 4 (17). Notethatasaconsequenceofoursubtractionoperationonlytermswithanevenpower of k in the numerator give contributions to UV divergent integrals. The finite integrals are of the form k ...k (−2k·q+q2)r dDk µ1 µi , Z P˙4mP4P5(k2)j+4−2D where only terms of even power in k contribute in the numerator. Via partial fraction decompositions in k2,P ,P˙ this can be reduced to standard one- 5 4 loopintegralswithintegerpowersofpropagatorswhicharealgebraicallyrelatedtoone-loop integrals with unit powers of propagators via mass derivatives. All these reductions and separations can be handled by symbolic calculation routines. So it is possible to reduce an arbitrary two-loop tensor integral into finite integral representations and a standard set of one-loop integrations. For the examples discussed in this paperthiswasalwayspossibleusingREDUCE[8]ona486notebookwithinlessthan5min- utes. Note that even for the two-loop four-point function N is not worse than a one-loop k three-point function (apart from the trivial topology arising from self-energy insertions), so thattheresultingone-loopintegralsarerelativelysimple. Theknowledgeofarbitrarytensor integrals of one-loop two- and three-point functions with integer powers of propagators is indeed sufficient for the calculation of two-loop problems by this method. Having demonstrated the handling of the UV divergent part let us now discuss the integral representations for the finite integrals. In [5] an integral representation is given which transforms the integral to a twofold integral over the parallel space variables. This canbe generalizedto arbitrarytensor integralswhichappearas polynomialsin paralleland orthogonal space variables in the numerator. So we have to integrate out the orthogonal space variables. This can be easily handled by use of partial integrations and applying the residuetheoremafterwards. Butalittlemodificationarisesforthetermswhichsubtractthe UVdivergence. HerewecannotuseapartialfractiondecompositionintheP˜,P˜ propagator 1 2 as these propagatorsare equal. Doing the z and k -integration as in [5] we are left with ⊥ l ⊥ dl [log terms]; i=2,3. ⊥(l2−l2)i Z 0 ⊥ But we can integrate these terms most easily by rewriting 1 1 ∂i 1 = | . (18) (l2−l2)i i!∂µi(l2−l2 −µ) µ=0 0 ⊥ 0 ⊥ A similar remark applies to the three-point two-loop functions. The case of four-point functions will be discussed elsewhere, but will give no modifications in general [9]. Never- theless we include the principal reduction of two-loopbox integrals in the discussionbelow. Uptonowwehavestudiedparticulartensorintegralsonly. Letusnowdiscussamethod how to calculate tensor integrals more directly. We again use the notation of [5] for the parallel and orthogonal space variables and will discuss as a specific example the following graph, which appears in the flavour changing fermion self-energy. The two-loop radiative corrections to order O(α )×g2 are currently being investigated for these self-energies [10]. s 1,qj 5,W '$ 3,q i 2,g 4,q Fig.2: Again themaster topology; thelabels specify propagators and parti- i cles. &% 7 Let us decompose the Clifford algebra /l = l γ||−l γi , || ⊥i ⊥ k/ = k γ||−k γi , || ⊥i ⊥ γ ,γ = 0, || ⊥ wherewesimply use the factth(cid:8)atwe c(cid:9)ansplitaloopmomentuminDR initscomponentin the direction of the exterior momentum q = q and its orthogonal complement. The same || splitting can be done for higher dimensional parallel spaces (the linear span of the exterior momenta). ThisgivesrisetoacorrespondingsplittingintheCliffordalgebrawhichcanalso be written down in a covariant manner l·q l = q , ||µ q2 µ l·q l = l − q , ⊥µ µ q2 µ q/ γ = γ − q , ⊥µ µ q2 µ q/ γ = q , ||µ q2 µ q q µ ν g = g − , ⊥µν µν q2 q/ q/ γ ,γ = q ,γ − q =0. ||ν ⊥µ q2 ν µ q2 µ (cid:26) (cid:27) (cid:8) (cid:9) The generalizationto higher dimensional parallel spaces is clear. We can now use this decomposition for a reordering where we bring all parallel space γ matrices and the anticommuting γ [11] to the left so that we have 5 γ n5γn1...γniγ ...γ , 5 ||1 ||i ⊥µ1 ⊥µk for an i dimensional parallel space and a rank k tensor in the orthogonalspace. The n are i the number of appearances of γ and parallel space γ-matrices. Using γ 2 = 1,γ2 = 1 we 5 5 ||i can simplify the above expession. For the case that we do not have free Lorentz indices in our Green function we can also simplify the orthogonal space part of this expression. In this case we are allowedto replace the orthogonalspace γ matrices identically by metrical ⊥ tensors g ,(gµ = (D−i)), even when we do not apply a trace to the string. Note that ⊥ ⊥µ this means that all the lengthy trace and γ algebra calculations of perturbation theory will not appear for the case of no free Lorentz indices. It also means that we only have to consider contributions with an even number of γ-matrices in the orthogonal space. To prove this statement note that as long as we have no free indices (after contracting out all double indices) γ -matrices can only be generated by the loop momenta/l,k/. As the only ⊥ covariant available for the orthogonalspace integration is the total symmetric combination of metric tensors g (the Levi Civita tensor does not appear in symmetric integration) we ⊥ only need the symmetric part of the Clifford algebra in orthogonalspace. This corresponds to a replacement of γ matrices by g tensors, as all indices have to be contracted by a ⊥ ⊥ total symmetric tensor. This method can be generalized. In the case that we cannot avoid free indices we still canuse the aboveargumentforthe γ matricesgeneratedby orthogonalspace components ⊥ of loop momenta. But note that in most cases it is possible to contract free indices with metricaltensorsorexteriormomentawithoutlosinginformation. Forexample,todetermine the formfactorsappearingin radiativecorrectionsof the fermionic vertex,one cancontract with exterior momenta p,p′ say. Then one can use all advantages of the case of no free indices. 8 Our example was calculatedusing this method by installing the proposedorderingalgo- rithm via simple LET rules in REDUCE. Comparing this with the same calculation using the high energy packet in REDUCE, it was found that the method we suggest here was orders of magnitude quicker. Even for the one-loop case we can report on a calculation of two- and three-point functions in the unitary gauge in the SM where significant advantages were obtained for the γ algebra [12]. This method fails for the anomaly. The interesting covariant is ∼ ǫ pρpσ and there µνρσ 1 2 exists no tensor constructed from symmetric metrical tensors and exterior momenta p ,p 1 2 to give a nonvanishing contraction with this covariant. One nevertheless can calculate the resulting trace by this method. But then one has to consider the case of γ with six γ 5 matrices, two in the linear span of p ,p , two in the orthogonal complement but still in 1 2 four dimensional Minkowski space and two with arbitrary indices, which gives rise to the non-cyclic trace of [11]. Continuing with our example we are left with the tensors 1,γ ,q/,γ q/. Each of these 5 5 tensors is multiplied by an integral representation for the finite part, which is of the form (using (7),(18) and the results in [5]) 1 [c L −c L +c L −c L ] 25 325 15 315 14 314 24 324 (P −P )(P −P ) 1 2 4 5 ∂ ∂2 − [(c(3,0)+c(2,1)+c(2,0)+c(1,1))L ] −l·q [(c(3,0)+c(2,1))L ] ∂µ sub |µ=0 ∂µ2 sub |µ=0 ∂ ∂2 − [(c(0,3)+c(1,2)+c(0,2))K ] +k·q [(c(0,3)+c(1,2))K ] . ∂µ sub |µ=0 ∂µ2 sub |µ=0 Welisttheexplicitcoefficientsc ,c ,c ,c ,c(i,j)inanappendix. ForthecaseofFig.(2) 25 15 24 14 it took only about 20 seconds in REDUCE to generate them for the complete graph. Let us summarize the situation for the most interesting two-loop graphs. We assume that the corresponding calculations are done in a renormalizable gauge, though it is not necessary to restrict to this case. Two-point functions: We have two topologies. For the trivial self-energy insertions we can always arrange that power counting improves by 2n (assuming l is the subloop momentum), as we can always l arrange exterior momenta not to flow through the subloop. For this topology we then have n =0always,whilen =1orn =2correspondingtoω =1orω =2,thatiscorresponding k l l to boson or fermion self-energies. For the other (master) topology of our example we have (n =0,n =2) or vice versa if k l ω =1 and (n =2,n =1) or vice versa if ω =2. l k Three-point functions: We have three topologies, the ladder topology, the crossed topology and the self-energy in- sertion topology. For the last one we always have (n = 1,n = 0) while the other ones l k reduce to (n =0,n =1) or (n =0,n =2), depending what kind of vertex one considers k l k l (the fermionic one has only logarithmic overalldivergence,while e.g. triple boson couplings arelineardivergent). Notethatforthecrossedtopology,asM containstwopropagators,the UVdivergentone-loopintegralsreducetotwo-pointfunctions andsothis morecomplicated topology is simpler in its divergent structure. This corresponds to the fact that it has no proper subdivergence, while the ladder topology has a vertex type subdivergence and thus involves three-point one-loop functions. Four-point functions: We need four exteriorbosons to have a UV divergence of logarithmic degree. We then have ω = 0 and the case (n = 1,n = 0) is always sufficient. Again the complicated topologies l k involve simpler one-loop integrals. 9 We see that only for boson self-energies we have the case (n 6= 0,n 6= 0) so that only l k for them do we have to use HFP explicitly. Note also that all(n =1,n =0)cases reduce l k to a very simple subtraction which simply sets the exterior momenta and masses to zero in the subtracted term, and this is sufficient especially for the two-loop fermionic vertex corrections. Weconcludethataneasyseparationofdivergencesfortwo-loopfunctionscanbeachieved. We sawthatforthe subtractionofthe UVdivergencesofthe realtwo-looppartverysimple one-loopfunctions weresufficient. Especiallyonly termsoforder1/(D−4)wereused. This wasnotunexpectedas the order1/(D−4)2 polesofa two-loopgraphcanbe obtainedfrom productsofone-loopfunctions. Thisis trueingeneralfortheleadingdivergenceinan-loop calculation. So we will find the leading divergences in the terms where the M part of the denominator is cancelled, giving rise to decoupled one-loop integrals. For our example in Fig.(2) we list the corresponding expressions in the appendix. As the individual terms of equation(7) correspondto terms subtractingl divergences,k divergencesand overalldiver- gences, one might wonder if it is possible to implement the whole renormalization program (that is the addition of one-loop counterterm graphs) at this level to obtain even simpler expressions. We intend to investigate this question in the future. Acknowledgements It is a pleasure to thank D. Broadhurst and K. Schilcher for stimulating discussions on the subject. This work was supported in part by the Deutsche Forschungsgemeinschaft and under grant number A69231484from the Australian Research Council. Appendix In this appendix we list the relevant polynomials in l ,k for the final integral represen- 0 0 tation of the graph of Fig.(2) explicitly. Our notation follows [5]. We list these results to give an idea of the complexity resp. simplicity of the integral representation for a complete graph. We see that the integral representation, compared with the pure integral representation for the scalar master function, is similar in its qualitative and quantitative difficulty. Let us first list the propagators of Fig.(2): P :=l2−m2 P :=(l−q)2, 1 j 2 P :=(l+k)2−m2, 3 i P :=(k+q)2−m2 P :=k2−m2. 4 i 5 w We define q := q2 and ˆq/:=q//q. c :=8·γpˆq/· − 2·q·l2 15 5 0 −2·q·l0·k0(cid:0)+m(cid:0)2w·l0+(cid:1)m2w·k0−m2i ·k0+m2j ·l0+m2j ·k0 +4·γ5·mj · −(2·q·l0)−2·q·k0−m2w+3·m2i +m2j (cid:1) +8·ˆq/· − 2(cid:0)·q·l02 −2·q·l0·k0+m2w·l0+m2w·k0−m(cid:1) 2i ·k0 +m2j ·l(cid:0)0+(cid:0)m2j ·k0 (cid:1)+4·mj · 2·q·l0+2·q·k0+m2w−3·m2i −m2j (cid:1) (cid:0) (cid:1) c :=8·γ ˆq/· − q2·l 14 5 0 −q2·k0−2·q(cid:0)·l(cid:0)02−4·(cid:1)q·l0·k0−2·q·k02+m2i ·l0+m2j ·l0 10