Direct reduction of multiloop multiscale scattering amplitudes

We propose an alternative approach based on series representation to directly reduce multi-loop multi-scale scattering amplitude into set of freely chosen master integrals. And this approach avoid complicated calculations of inverse matrix and dimension shift for tensor reduction calculation. During this procedure we further utilize the Feynman parameterization to calculate the coefficients of series representation and obtain the form factors. Conventional methodologies are used only for scalar vacuum bubble integrals to finalize the result in series representation form. Finally, we elaborate our approach by presenting the reduction of a typical two-loop amplitude for W boson production.

Introduction. The CERN Large Hadron Collider (LHC) is the most accurate experiment on the elementary particle physics at present, and the next generation lepton colliders have been proposed aiming at higher accuracy. They all demand the high precision theoretical predictions to include higher orders of either electroweak or QCD corrections [1]. However, the higher order corrections may become seriously challenging due to the evaluation of the multi-loop Feynman diagrams, which usually can be decomposed into several steps of calculations. And practically one of the most difficult calculations is to reduce the loop amplitude into linear combination of master integrals.
For the one-loop amplitude many different reduction algorithms have been developed after decades of effort. The Passarino-Veltman reduction algorithm [2][3][4][5] has been widely used in enormous number of investigations on the next-to-leading order (NLO) effects for the Standard Model (SM) processes and some new physics processes. Later the implementation of unitarity algorithm [6][7][8][9][10][11][12][13] on the one-loop amplitude provided very interesting and inspiring prospect on the amplitude structure. Meanwhile the algorithm based on unitarity also presents excellent numerical efficiency. Consequently, by implementing these modern reduction algorithms the SM NLO calculations have been automated [14][15][16][17][18][19][20]. Other methods can be found from [21][22][23][24][25][26]. At the multi-loop level in the consideration of efficiency the amplitude has to be reduced into linear combination of finite number of master integrals [27], which can be further calculated analytically or numerically. In contrast to the one-loop case, the achievement of multi-loop reduction conventionally includes two separate steps, i.e. the tensor reduction and the scalar integral reduction using integration by part (IBP) identities [28].
First the tensor reduction is used to isolate the loop momenta from fermion chains, polarization vectors or product of them, which will be factorized out of the loop integral to construct the form factors. Specifically one of conventional approaches is the projection method [29,30] that has been commonly used in the calculations of high order QCD corrections to the Higgs production [31][32][33] and the vector boson productions [34,35]. The key to projection method is the projector basis relying on the analytic inversion of projection matrix. However, for some complicated scattering processes, e.g. the full next-to-next-to-leading order QCD correction to singletop production [36], the project matrix could become so big that its inversion may seriously challenge the computation resource. Another approach for tensor reduction is Tarasov's method [37] based on Schwinger parameterization [38,39]. It can avoid irreducible numerator but shift the space-time dimensions of obtained scalar integrals. Thus it is inevitable to shift the dimensions of scalar integrals back to the conventional D-dimension or the same dimension at least. And this commonly needs to resolve the dimension recurrence relations, which however is as difficult as the matrix inversion in projection method. Besides another popular approach is using IBP identities [40,41], which however also confronts serious difficulties in the multi-scale processes. During this modern age of evaluation, computational algebraic based algorithms [42][43][44] successfully implemented on N=4 Yang-Mills theory and numerical unitarity method for multigluon amplitudes [45][46][47].
Then after the successful tensor reduction the loop amplitude becomes linear combination of scalar integrals, whose number could be order O(10 4 ) for complicated processes. Consequently as the second step usually the IBP reduction is introduced to reduce the scalar integrals into a much smaller number of master integrals. The most popular method for IBP reduction is Laporta algorithm [28], which has been implemented by several codes [41,[48][49][50][51]. Another interesting method [50,52] for IBP reduction recently has been developed based on algebraic geometry. Due to the fact that IBP reduction relies heavily on the IBP relations the choice of master integral set can not be arbitrary, so the resulting reduction expressions may confront unacceptable inflation [53,54]. Therefore, to efficiently evaluate the multi-scale multiloop amplitude one better keep the freedom to choose master integrals. And this can be achieved by series representation [55], which in fact can also be used to solve the tensor reduction as we will show in the following.
In this letter, based on the series representation [55,56] we propose an alternative reduction approach that can directly reduce loop amplitude to master integrals so that the complexity of tensor reduction and IBP reduction can be relieved. In next section the main idea will be explained in detail. Then its application on one typical two-loop diagram of W boson production as an example will be shown. Finally the conclusion is made.
Amplitude Reduction. Recently, series representation of Feynman integral has been proposed to reduce the scalar integrals into master integrals [55] and to numerically evaluate the master integrals [56]. It is very promising since it can be applied to multi-scale multi-leg integrals and has freedom to choose master integrals. Intriguingly we find that the series representation can also be directly implemented on the loop amplitude, which in general can be written as where are the denominators of loop propagators. Numerator ) may contain fermion chains, polarization vectors or both.
In order to obtain the expression of loop amplitude in series representation, we first modify all the denominators, where P i ≡ Q i +K i is the momentum of the i-th propagator. Q i and K i are defined as linear combinations of loop momenta and external momenta, respectively. Therefore, we obtain the modified amplitude M(η), which depends on parameter η. Then the loop amplitude can be decomposed as linear combination of tensor integrals The summation is over all tensor structures in the given amplitude. By using Feynman parameterization [57] for tensor integrals, we can express the tensor integral as where N ν ≡ n i=1 ν i and N (m) ν ≡ N ν − m − LD/2. U and F are the first and second Symanzik polynomials, respectively. And the definitions of the symbols in square bracket can be found in Ref. [57]. Here the parameter x i is corresponding to one of the loop propagators, D(x i ) ≡ D i . By using Taylor series for η → ∞ one can obtain Now it can be seen that all the tensor structures are only related to the external momenta. Consequently the fermion chains or the polarization vectors can be attached to the external momenta to generate the form factors. And the coefficients of the form factors become integrals on Feynman parameters {x i }, for instance where D can be different from the space-time dimension D. Then we can map the parameters {x i } into L(L + 1) categories For each category, we can insert unit integral, e.g., dy i δ(y i − x i1 − · · · − x i k ) = 1. Meanwhile because U can be constructed from the 1-tree cuts on the Feynman loop diagram, it can be found that U only depends on {y i }. Then the parameters x i1 , . . . , x i k can be integrated along with the δ-function as = dy i y ni 1 +···+ni k +k−1 i Γ(n i1 + 1) · · · Γ(n i k + 1) Γ(n i1 + · · · + n i k + k) .
And finally the integrals on {y i } can be reconstructed as vacuum bubble integrals, for instance at two-loop level For the remaining scalar vacuum bubble integrals we can further implement the IBP reduction [41] and the dimension shift operation. Finally the modified loop amplitude can be expressed as the series representation in terms of vacuum bubble master integrals. It is necessary to emphasize that the IBP reduction and the dimension shift operation are implemented only on the vacuum bubble integrals, which are process independent and can be easily prepared once for all.
Obviously now we have successfully achieved the tensor reduction for loop amplitude. Finally we can rewrite the modified loop amplitude as and where F i is the form factor and C i is the relevant coefficient. r max i is the maximum rank of loop momenta in the form factor coefficient C i . And I (vac),D L,j represents the j-th L-loop vacuum bubble master integral. The series coefficient A ijp depends only on scalar products of external momenta and space-time dimension D in dimensional regularization. If dim(A i10 ) is defined as the mass dimension of A i10 , the mass dimension of Then for the given amplitude we can choose a proper set of master integrals { I i (η)} whose mass dimensions are close to dim(C i ). And the reduction can be achieved by matching the form factor coefficient C i with the series representations of master integrals { I i (η)} as explained in Ref. [55]. Formally where C i (η) is the reduction coefficient of relevant I i (η) for modified loop amplitude. In the conventional approach the reduction coefficients could be obtained by using tensor reduction and IBP reduction, which could be very difficult as been reviewd in previous section. However, as we have shown above by directly implementing series representation on modified loop amplitude, the difficulties in both tensor reduction and IBP reduction can be relieved. And the final reduction relation for the original loop amplitude can be obtained by taking the limit η → 0 + , Although in order to achieve loop amplitude reduction this set of master integrals themselves may not be convenient to evaluate analytically or numerically, one may make further apply reduction increasingly to the final set of master integrals that can satisfy the requirement of evaluation.
Example. In this section we take one typical twoloop diagram of W boson production shown in Fig.1 as an example to demonstrate our approach. The diagram is plotted by using Jaxodraw [58] based on Axodraw [59]. Its relevant modified amplitude can be written as where the denominators from loop propagators are and And to make complete integral family for two-loop onefinal-state amplitude we need additional one denominator For reader's convenience we also explicitly show the nu- merator of the amplitude N (q 1 , q 2 ,k 1 , k 2 , k 3 ) = 16 9v (k 2 )γ α ( k 2 + q 1 )γ β ( k 3 + q 2 ) × ε(k 3 )P L q 2 γ β ( k 1 − q 1 )γ α u 1 (k 1 ).
By implementing the approach mentioned in previous section, we can directly extract the only form factor And it is known that at two-loop level there are two vacuum bubble master integrals, and I (vac),D 2,2 In series representation the modified loop amplitude can be expressed as Finally for the matching procedure we choose 25 master integrals,