Elsevier

Physics Letters B

Volume 701, Issue 5, 27 July 2011, Pages 646-653
Physics Letters B

Calculating contracted tensor Feynman integrals

https://doi.org/10.1016/j.physletb.2011.06.033Get rights and content

Abstract

A recently derived approach to the tensor reduction of 5-point one-loop Feynman integrals expresses the tensor coefficients by scalar 1-point to 4-point Feynman integrals completely algebraically. In this Letter we derive extremely compact algebraic expressions for the contractions of the tensor integrals with external momenta. This is based on sums over signed minors weighted with scalar products of the external momenta. With these contractions one can construct the invariant amplitudes of the matrix elements under consideration, and the evaluation of one-loop contributions to massless and massive multi-particle production at high energy colliders like LHC and ILC is expected to be performed very efficiently.

Introduction

In a recent article [1] (hereafter quoted as Ref. I), we have worked out an algebraic method to present one-loop tensor integrals in terms of scalar one-loop 1-point to 4-point functions. The tensor integrals are defined asInμ1μR=ddkiπd/2r=1Rkμrj=1ncj, with denominators cj, having chords qj,cj=(kqj)2mj2+iε. Here, we use the generic dimension d=42ε. The central problem are the 5-point tensor functions. We derived algebraic expressions for them in terms of higher-dimensional scalar 4-point functions with raised indices (powers of the scalar propagators). There are several ways to go. One option is to avoid the appearance of inverse Gram determinants 1/()5. For rank R=5, e.g.,I5μνλρσ=s=15[i,j,k,l,m=15qiμqjνqkλqlρqmσEijklms+i,j,k=15g[μνqiλqjρqkσ]E00ijks+i=15g[μνgλρqiσ]E0000is], see equations (I.4.60), (I.4.61). The tensor coefficients are expressed in terms of integrals I4,i[d+]l,s, e.g. according to (I.4.62):Eijklms=1(00)5{[(0lsm)5nijkI4,ijk[d+]4,s+(il)+(jl)+(kl)]+(0s0m)5nijklI4,ijkl[d+]4,s}. The scalar integrals areIp,ijk[d+]l,stu=d[d+]lkiπ[d+]l/2r=1n1cr1+δri+δrj+δrk+δrsδrtδru, where p is the number of internal lines and [d+]l=42ε+2l. Further, we use the notations of signed minors (I.2.14). At this stage, the higher-dimensional 4-point integrals still depend on tensor indices, namely through the indices i,j etc. The most complicated explicit example I4,ijkl[d+]4,s appears in (1.4). Now, in a next step, one may avoid the appearance of inverse sub-Gram determinants ()4. Indeed, after tedious manipulations, one arrives at representations in terms of scalar integrals I4[d+]l plus simpler 3-point and 2-point functions, and the complete dependence on the indices i of the tensor coefficients is contained now in the pre-factors with signed minors. One can say that the indices decouple from the integrals. As an example, we reproduce the 4-point part of (I.5.21),nijklI4,ijkl[d+]4=(0i)(00)(0j)(00)(0k)(00)(0l)(00)d(d+1)(d+2)(d+3)I4[d+]4+(0i0j)(0k)(0l)+(0i0k)(0j)(0l)+(0j0k)(0i)(0l)+(0i0l)(0j)(0k)+(0j0l)(0i)(0k)+(0k0l)(0i)(0j)(00)3d(d+1)I4[d+]3+(0i0l)(0j0k)+(0j0l)(0i0k)+(0k0l)(0i0j)(00)2I4[d+]2+ In (1.6), one has to understand the 4-point integrals to carry the corresponding index s and the signed minors are (0k)(0sks)5 etc. This type of relations may be called “recursion relations for small Gram determinants”.

In an alternative treatment, tensor reduction formulas free of gμν terms were derived in Ref. I. In that case, inverse powers of ()5 are tolerated. The most involved object studied was (I.3.20):I5μνλρσ=I5μνλρQ0σs=15I4μνλρ,sQsσ, with the 4-point tensor functions (I.3.29) and (I.3.18)I4μνλρ,s=Q0s,μQ0s,νQ0s,λQ0s,ρI4s+3()52(ss)52QsμQsνQsλQsρI4[d+]2,s+()5(ss)5Qs[μQsνJ4λ,sQ0s,ρ]+,J4μ,s=Q0s,μI4[d+]+t=15Qts,μI3[d+],st. The dots in (1.8) indicate 3-point functions, andQsμ=i=15qiμ(si)5()5,s=0,,5,Qst,μ=i=15qiμ(stit)5(tt)5,s,t=1,,5. Also here, the tensor coefficients have been represented by scalar functions free of tensor indices.

We remark that all the above-mentioned results are due to a systematic application of methods described and developed in [2]. For the present Letter the “algebra of signed minors” [3] plays a particularly important role. This method was also used in [2] and further developed to its full power in [4]. In the latter article also 6-point functions have been treated on this basis.

In the next section we will develop a very efficient method to evaluate realistic matrix elements with tensor integral representations of the above kind.

Section snippets

Contracting the tensor integrals

To apply the approach most efficiently one should construct projection operators for the invariant amplitudes of the matrix elements under consideration. These projectors, of course, depend on the tensor basis and have to be constructed for each process specifically. If done like that, the tensor indices of the loop integrals are saturated by contractions with external momenta pr. The chords in (1.2) are given in terms of the external momenta as qi=(p1+p2++pi), with qn=0, and inversely pr=qr1

Conclusions

The contracted rank R=1,,3 tensors of the 5-point function have been expressed by scalar integrals, accompanied by compact expressions for sums over products of chords and signed minors. It is evident how the general case has to be treated, once a table of sums as given in the appendix is available. The scalar integrals may be defined in higher dimensions or in the generic dimension, depending on the preferred algorithm of the final numerical evaluations and on questions related to a treatment

Acknowledgements

We acknowledge useful discussions with S. Moch and P. Uwer. J.F. thanks DESY for kind hospitality. Work is supported in part by Sonderforschungsbereich/Transregio SFB/TRR 9 of DFG “Computergestützte Theoretische Teilchenphysik” and European Initial Training Network LHCPHENOnet PITN-GA-2010-264564.

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