Nontrivial One-loop Recursive Reduction Relation

In arXiv:2204.03190, we proposed a universal method to reduce one-loop integrals with both tensor structure and higher-power propagators. But the method is quite redundant as it does not utilize the results of lower rank cases when addressing certain tensor integrals. Recently, we found a remarkable recursion relation arXiv:2203.16881,2205.03000, where a tensor integral is reduced to lower-rank integrals and \textit{lower terms} corresponding to integrals with one or more propagators being canceled. However, the expression of the lower terms is unknown. In this paper, we derive this non-trivial recursion relation for non-degenerate and degenerate cases and provides an explicit expression for the lower terms, thus simplifying and speeding up the reduction process.

Recently, the analytical structure of one-loop integrals is studied by investigating Feynman parametrization in the projective space for its compactness and the close relation to geometry [52,53].Inspired by these papers, we find it could be convenient to do reduction for one-loop integrals in projective space.Furthermore the symmetry and compactness of reduction coefficients are illustrated clearly by this method, as shown in [1].However, with this technique, we have to expand a general one-loop integral into the combination of E n,k [V i ] first, then reduce every E n,k [V i ] to the basis, after that, we sum over all contributions to obtain the final reduction result.Here E n,k [V i ] denotes integrals defined in projective space.
where T is a general rank-k tensor and ∆ is a simplex in n-dimensional space defined by X I > 0, ∀I = 1, 2, . . ., n.The homogeneous coordinates X I are denoted by a square bracket X = [x 1 : x 2 : . . .: x n ], and two coordinates are equivalent to each other up to a scaling, i.e., [x 1 : x 2 : . . .: x n ] ∼ [kx 1 : kx 2 : . . .: kx n ] for any k = 0.The measure in the projective space is given by the differential form The matrix Q appearing in the denominator XQX = Q IJ X I X J has component Q IJ = Q IJ = m 2 I + m 2 J − (q I − q J ) 2 /2, I, J = 1, . . ., n.For simplicity, we denote E n,k [V i ] ≡ E n,k [T = ⊗V i ⊗ L k−i ] for arbitrary vector V and the constant vector L ≡ [1 : 1 : (1. 3) The order of V and L does not matter, as both are contracted with X.While the compact recursion relation makes the reduction faster, we found that the running time increases sharply with the number of propagators n and the rank r, as shown in Table 1.However, using the recursion relation of E n,k [V i ] to investigate the analytical structure of reduction coefficients is not as convenient, as it provides only parts of the entire expression.Recently, we discovered a remarkable recursion relation for one-loop reduction, where a tensor integral is reduced to lower-rank integrals.The resulting lower terms correspond to integrals where one or more propagators are canceled n + Lower Terms (1.4) where |G| is the Gram determinant of I n and the tensor integrals are defined as Finally, we reach a summation of scalar E n,n−D 's with the "wrong" dimension D = D + 2s, where s ∈ Z.Therefore, for every term, we need to shift dimension D to D. Finally, we sum over all contributions to obtain the reduction coefficients.We ran our code in Mathematica and found that the time cost grows sharply.For more details, please see the examples in Section 3 of [1].
The formula (1.4) is proved by considering acting on it with two differential operators . The expressions for A r , B r are known while the lower terms is still a mystery.In this paper, we derive the explicit form of the lower terms by applying the universal recursion relation of We obtain two forms of the lower terms, the first one involves integrals in (D − 2)-dimensional space while the second one involves integrals in the same D-dimensional space 1 .The recursion relation not only makes the reduction process extremely simple and effective, but also serves as a tool to study the analytical structures of reduction coefficients like singularities.The paper is structured as follows: In Section 2, we revisit the reduction technique in projective space.In Section 3, we give the recursion relation for non-degenerate Q and then illustrate it with some examples.The proof is listed in the Appendix A. In Section 4, we discuss how to deal with the degenerate case and show our method with some examples.Finally, we provide some discussion in Section 5.

Review of one-loop reduction in projective space
In recent work [1], we study general one-loop integrals in D-dimensional spacetime with both tensor structure and higher poles 2 1 While writing this paper, we became aware of similar results obtained by Chen et al. [54], who used the IBP method in Baikov representation.Our work complements theirs by providing an alternative approach that emphasizes geometric interpretations and explicit constructions. 2 Here to manifest permutation symmetry of propagators we use the notations differing from (1.5).
-3 -where the loop momentum , auxiliary vector R and external momentum q j live in D = (d − 2 )-dimensional spacetime.We denote v n = {v 1 , v 2 , . . ., v n } the power list of the n propagators.In the formula, we introduce an auxiliary vector R to make the expression compact 3 .one can see any general tensor structure can be recovered by applying differential operators of R on the standard expression, for example After some algebra [1].we find (2.1) can be written as a compact form in terms of E n,k . We have defined the vectors V , H i as and the expansion coefficient in (2.3) is where we require i to have the same parity as r.We denote I v;D as the one-loop integrals before taking coefficient of t i z vn−1 , whose reduction result is where b j ≡ {b 1 , b 2 , . . ., b j } is the length-j label list of the propagators being canceled.It is found that the reduction of general one-loop integrals is solved by a simple recursion relation for where α n,k = 1 n+k−2 and Q is an arbitrary symmetric matrix.In the formula we need to sum the (k − 1) ways to contract indices between Q and T n−1,k−1 means the integral obtained from the n-dimensional E n,k by deleting the b th component of X.The reduction process is achieved by following steps: • Step 1: Expanding a general one-loop integral according to (2.3).
• Step 2: Depending on whether Q is degenerate, take reduce every term in the expansion (2.3) to scalar basis.
In this paper, we mainly discuss pure tensor reduction which is involved in the real scattering process.Consider non-degenerate Q, we can choose Q = Q −1 , and then (2.7) becomes (2.9) where we suppress the summation of b = 1, 2, . . ., n and have defined the compact notation where {j} is the sub-list of the label list of removed propagators and Q ({j}) denotes of Q matrix of corresponding one-loop integrals.Although we have used compact notations in every iteration to make all steps of the reduction process purely algebraic, it can be timeconsuming for large n and r.To estimate the running time, we can consider an even rank of r = 2m for simplicity.First, there are r/2 = m terms E n,n−D−2m+i [V i ] to deal with, where i = 0, 2, 4, . . ., 2m.We can regard D = n − k as the effective dimension for and in the expansion (2.3), every term has a "wrong" dimension of D = D + 2m − i > D.
We also notice that the last two terms in (2.9) increase the dimension from D to D + 2. Each iteration of (2.9) extracts one V or two V 's at the cost of higher dimensions.Roughly speaking, the time to reduce E n,n−D−δD [V i ] satisfies the recursion relation By applying (2.9) iteratively, we can finally reach the scalar integrals with wrong dimension.The last step is to shift higher dimension back to D β n,k+2 (LL) . (2.12) So the time for dimension shifting satisfies The total time to reducing an one-loop tensor integral is For simplicity, we ignore the time cost of summing over all terms in each step, and let T n [δD = 0, V 0 ] = t 0 .We have listed the theoretical running times in when we compare it with Table 1, we observe that for n = 3, 4, the actual running time grows faster than what we have analyzed.This is because we have not accounted for the time cost of summing terms for every iteration in Mathematica.These summation processes can be very cumbersome for larger n and r.
After observing the significant increase in running time, one may wonder if there is a more efficient recursive method for evaluating the integrals I (r) n;D directly, rather than using their expansion pieces E n,k .This question can be addressed by combining IBP and sazagy methods, as done in [59].However, instead of this approach, we have derived general recursion formulas for arbitrary n and r directly at the integral level with the techniques in projective space.

Recursion relation for non-degenerate Q
In this section, we give the explicit expression for the recursion relation.The proof is given in Appendix A. Then we list some examples to illustrate reduction process using the recursion relations.

Expression of lower term
As pointed in [3], there excites a non-trivial recursion relation for one-loop tensor integrals4 The two coefficients are Notice that due to B r=1 = 0, the formula (3.1) works for r ≥ 1.For r = 0, the integral I n for general Q is a master basis, so it cannot be reduced further.The lower term Lt (r) n corresponds to integrals with one or more propagators being canceled.In [3], we check the consistency of (3.1) by applying two differential operators of R (i.e., Although with the differential operators D i , T , one can establish a reduction framework for one-loop integrals, it is quite difficult to work out Lt (r) n with this technique for two types of complex recursion relations are involved.On the other hand, one-loop reduction in projective space only relies on one simple and symmetric recursion relation (2.7).As shown in Appendix A, one can figure out the expression of Lt where we have omitted the summation over b from 1 to n on the RHS.Notably, these RHS terms represent integrals in (D − 2)-dimensional spacetime.A reduction in (D − 2)dimensional spacetime is straightforward to implement: where , the reduction coefficients for D-dimensional integrals, are known.In the case of non-degenerate Q (b) , we can further reduce the lower topology term with integrals in the same dimension by reducing the right-hand side of (3.3).After some algebra, we obtain The details of this reduction are presented in Appendix A, where the necessary algebraic manipulations are carried out.The presented formula offers superior feasibility, as a result of being able to utilize the reduction results of lower topologies directly.As such, the resultant formula achieved herein is more symmetrical in nature: The present formula contains tensor integrals with lower ranks and integrals whereby one propagator is canceled, consequently, these integrals are already known to undergo an iterative reduction process.Utilizing the recursion formula, the reduction results for any tensor integral n can be obtained.The recursion diverges at (LL) = 0, which corresponds to the singularity of the degenerate Gram matrix det G = 0 5 .The singularity is addressed simply by multiplying both sides of (3.6) with (LL) and subsequently taking the limit (LL) → 0, so we obtain the recursion relation for vanishing Gram determinant It is noteworthy that R should only appear in the numerator in the final expression of reduction coefficients.Therefore, it is anticipated that the (V L) in A r+1 can be eliminated.Actually, There exist two distinct identities when |G| = 0.There are two identities for |G| = 0: which simplify the recursion relation (3.8) to It is easy to check the spurious singularity (V L) disappears in the final result for r = 0, 1: • r = 0 • r = 1 For higher rank case, one can iteratively check (V L) disappears in the denominator.

Running time and tensor structure of reduction coefficients
As stated in the introduction, utilizing recursion relations at the integrals level is a preferable approach to dealing directly with E n,k [V i ].Firstly, the new method exhibits notably enhanced efficiency over the previous approach.When comparing Table 1 with Table 3, it becomes evident that the equation given in (3.6) drastically reduces the required computation time in comparison to the direct expansion method.For instance, the time required for n = 5, r = 3 in Table 3 is approximately 1/330000 of the time listed in Table 1.The reason behind this significant difference can be explained by investigating T n,r , which denotes the amount of time required to reduce an r-rank n-gon integral.From (3.6), we have the following relation: which means the computational complexity of the proposed method exhibits a desirable linear relationship with respect to lower values of n and r.For example The formula presented in (3.n .As a result, we are able to derive results for tensor ranks of considerable magnitude within a notably brief interval of time. significant difference between the two: (3.13) involves the total time as opposed to its pieces in (2.11).Setting T n,0 = t 0 , we can calculate T n,r in Table 4.The reduction process is extremely fast since there is no need for an E n,k expansion or a dimension shifting process.Moreover, we can quickly obtain a numeric result by replacing the coefficients in the recursions with certain numbers.
Additionally, the recursion relation in (3.6) makes it apparent that the general structure of the reduction coefficients for arbitrary n and r can be studied.By observing that the singularity of reduction coefficients of a particular sector is (LL) ({j}) ∝ |G ({j}) |, where j is the sub-list of the label list of removed propagators, it is clear that the coefficient of the top-sector C (r) n→n only has the singularity of |G|.On the other hand, the coefficients of the next sector C (r) n→n; i can have the singularity of (LL) (i) ∝ |G (i) |.It is interesting to note that the degree of singularity grows linearly with rank r since every iteration of ( The structure F i 1 i 2 ,...,im;k 0 ,k 1 ,...,km is utilized to collect contributions from A r , B r , A r; b , B r; b of every iteration.Hence, the function where {j} ⊂ {i}.It is important to note that F exhibits symmetry about i 1 , i 2 , . . ., i m , as demonstrated in (3.48).In this paper, we present several results for rank-6 triangle to illustrate this concept.
The equivalence of the divergence degrees between C 3→3 and C 3→3; 3 for (LL) is due to the fact that, for the sub-sector in which one propagator is removed, setting k 0 = 6 and k 1 = 0 in (3.15) is permissible.However, for the sector with two propagators removed, i.e., C (6) 3→3; 23 , the largest divergence degree of (LL) is 5, since k 1 > 0 and k 0 +k 1 +k 2 ≤ r = 6.The leading divergence terms correspond to k 0 = 5, k 1 = 1, and k 0 = 0.The lower degree of divergence terms are more complex, as they involve several additional contributions throughout the iteration process.Notably, the divergence of (LL) (j) in the sub-sector coefficients is also observed, and all of these characteristics are codified in (3.15).Furthermore, the symmetry of the coefficient C The permutation symmetry appears in higher-point case, which makes the results extremely simple, as shown with more reduction results in Appendix B. In Section 4, we will discuss modified recursion relations that enable us to approach degenerate cases in a manner similar to that which we employ for non-degenerate cases.So we will not provide detailed discussion of these degenerate cases in this paper.

Examples
In this section, we provide examples to demonstrate the use of one-loop recursion with non-degenerate Q.These examples serve to illustrate how the method can be applied in practice, thereby providing a deeper insight into its efficacy and versatility.

Tadpole
First, we consider the simplest case, i.e, the tensor tadpole Due to the tadpole has one propagator, there is only one master integral I 1 and Lt (r) 1 = 0. We set q 1 = 0 throughout the paper, then we find The recursion relation (3.1) for tensor tadpole is quite simple It is easy to figure out the general result

Bubble
To avoid unnecessary complicate calculations and to show the advantages of reduction in projective space, we consider tensor bubbles as a non-trivial example where we have set q 1 = 0.The recursion relation reads The recursion gives the reduction coefficients written in a compact form (3.27) To simplify notation, we use R 2 = s 00 , R • q 2 = s 01 , and q 2 2 = s 11 for the bubble.By performing direct calculations, we obtain the matrix And the sub-matrices of Q are With the expressions for Q and its sub-matrices, it becomes effortless to compute these concise cells using the recursive relationship (3.26), as illustrated below: Using the reduction results for tensor tadpoles, it becomes possible to iteratively apply the recursion relation to reduce a tensor bubble up to any desired rank.This can be illustrated by the following example: • r = 1 Then use the results of rank-1 bubble reduction, we finally find These results agree with that provided by PV reduction.But comparing (3.31) with its compact form (3.27), one can see analytical structure and permutation symmetries of propagators manifest in projective space language.Here, consider the permutation exchanging the two propagators of bubble: σ(1) = 2, σ(2) = 1.From (3.27), we find . (3.35) Same things happen for higher n, r.Moreover, reduction recursion relies on n trivially for n-gon tensor integrals.for example, one can easily obtain the reduction coefficients of a rank-1 triangle It has the same form as rank-1 bubble.One can notice that there is a singularity s 11 = q 2 2 = 0 in the recursion relation (3.31), which comes from (LL) = 0 in (3.26).We address the singularity by (3.8) we finally find (3.39) So we have We can utilize this recursion to iteratively reduce a tensor bubble, as illustrated below: • r = 0 (3.41) • r = 1 Then using the result we have got 2; 1 = 2s 01 I 2; 1 , I 2; 2 = 0 , (3.43) we finally find Moreover, there exists another singularity, specifically when m 1 = m 2 , within the recursion relation.In such an instance, the associated matrix Q becomes degenerate, and it will be analyzed in the next section.

Triangle
Next we consider a nontrivial case, the triangle with We denote R 2 = s 00 , R • q i = s 0(i−1) , q i • q j = s (i−1)(j−1) and write the reduction results as By simple iteration, we can find the reduction result for rank r = 1, 2 By direct calculation, one finds the Q matrix are given by (3.50) It is obvious that there is a singularity at det G = 0 in these expressions.To make our expression simple and the singularity appear obviously, we can take {m Then we use (3.10) to address the singularity at t = 0 where You can find for rank r = 0, the scalar triangle degenerates to three bubbles For rank r = 1, 2, 3, we list the necessary results while other coefficients can be obtained by permutations of propagators.All results are checked with Fire6.

Hexagon
We consider = 4 − 2 and all external momenta are in 4−dimensional spacetime.One can prove that the matrix Q for a hexagon isn't degenerate for general moment.But here we have LQ −1 L = 0.So using (3.10), we find • r = 0 • r = 1 For n > 6, the matrix Q is degenerate, we discuss it in the next section.

Recursion of degenerate Q
In the preceding section, we presented a reduction for non-degenerate Q based on the formula presented in equation (3.1).However, there are certain cases where the recursion in equation (3.1) breaks down.Specifically, there are two cases: (A) when the matrix Q degenerates, rendering expressions such as (W Z) ≡ W.Q −1 .Z ill-defined, and (B) when Q is non-degenerate but (LL) = 0, which has been addressed in equation (3.8).
In this section, our focus primarily lies on adjusting the formula in equation (3.1) to accommodate for degenerate Q.Then we give some examples to demonstrate how the aforementioned modification can be applied in practical scenarios, thereby providing a deeper understanding of the efficacy and versatility of the modified formula.

Modified recursion relations
In the context of practical calculations of scattering amplitudes, it is apparent that there may be instances where the matrix Q becomes degenerate.This typically occurs when working in finite dimensions while considering special kinetic configurations.Under such circumstances, our recursion (3.1) may break down.For instance, in the dimension regularization scheme, where D = d − 2 , and the external momenta exist in d-dimensional spacetime, the matrix Q must be degenerate if n exceeds d + 2. Furthermore, certain momenta or mass configurations can also lead to a degenerate matrix Q.For example, in a scattering process involving photons or gluons, some external momenta or inner propagators may be massless, leading to a degenerate Q.We provide a derivation of (3.1) in Appendix A, which can also be adapted to cover degenerate Q.The only modification required is to replace Q −1 with an arbitrary symmetric matrix Q in the intermediate steps.
Ultimately, this leads to a general recursion relation with every term containing a Q.
where we have defined where we split the coefficient B r into two parts B (RR) r and B and the first one contains no Q.To keep the homogeneous condition for Q, we need to add a Q into I (r)  Similar to the non-degenerate Q, depending on whether (LL) is zero or not, we have two cases to deal with: We fist consider the simple case (LL) = 0, then we get a recursion relation similar to (3.6) but without the coefficient B (RR) r Then there are two subcases: -QL = 0: The coefficient A r is nonzero, so -QL = 0: The coefficient A r vanishes, so finally we get Both two recursion relations work for r ≥ 0.
At last, we consider a very special case where the sub-matrices Q (b) degenerate too, which makes some coefficients in (4.1) diverge.one need to use the (D − 2)-dimensional lower terms then reduce these terms by (3.4).Another method is to expand reduction coefficients and some degenerate basis according to Q (b) , then taking the limit Q (b) → 0 (see [3]).

Examples
Here we give some examples to illustrate how to apply the recursion relations in this section to reduction for degenerate cases.To compare with the non-degenerate case that we have discussed in Section 3.3, here we mainly deal with bubbles and triangles.

Bubble
• L QL = 0, QL = 0: Massless scalar bubble with the same internal masses To check the validity of our method, we first consider a scalar bubble with m 1 = m 2 = m and q 2 2 = 0, defined as which can be reduced to two tadpoles 6 .
We find Q degenerates to a corank-1 matrix and Notice that (LL) = 0, (V L) = 0, from (4.10), we have where I 1 [m] means the scalar tadpole with mass m and we have used 2; 1 = 0 .(4.16) Then we consider the rank-1 case where we have used 6 One can check this by using FIRE or direct calculation.
-21 - • L QL = 0: Massless scalar bubble with different internal masses We then consider the bubbles with degenerate Q and m 1 = m 2 .Here we will show that it can be reduced to two tadpoles.The equation det Q = 0 gives two solutions Here we take the solution q 2 2 = (m 1 + m 2 ) 2 as an example, and we set (4.20) Notice (LL) ≡ L QL = 0, we use the formula (4.6) which is valid for r ≥ 0. The reduction for r = 0 is the same as what we do in [1], which gives The reduction for r ≥ 1 is With the expressions below it is easy to figure out the reduction result (4.24) One can calculate reduction coefficients up to any rank by applying (4.22) iteratively.

Triangle
We have discussed triangles in the last section, similarly, here to make our expression simple, we can take {m  We first solve the reduction of r = 0 by direct expansion, then applying the recursion relation to obtain One can check it is right for higher ranks, for example, C (4.28)

Conclusion and Outlook
In this paper, we derive non-trivial recursive relations for reducing one-loop tensor integrals and extend it for degenerate Q.As in [1], we first express one-loop integrals as a sum of integrals E n,k [V i ] in projective space.However, unlike the formal approach where each E n,k [V i ] is naively reduced to a scalar basis, we use the basic recursive relation of E n,k [V i ] to derive a single recursion relation for one-loop Feynman integrals.We start by discussing the non-degenerated case and demonstrate that a r-rank tensor integral can be reduced to (r − 1) and (r − 2)-rank tensor integrals of the same sector and the sub-sector (with one propagator removed).This recursive relation at the integral level avoids the need for an E n,k [V i ] expansion and term-by-term reduction, Furthermore, the previous computation results can be reused in the next iteration, which significantly speeds up the calculation process.We present the results of the time taken for both methods in Table 1   Then, we present an explicit derivation for the recursion relations for degenerate cases by modifying the original recursions for non-degenerate Q.Although it may appear to be an ad-hoc patch, this approach actually provides a coherent framework with an explicit derivation given in Appendix A, where one can replace Q −1 with a general symmetric matrix Q.The reduction framework is shown in Figure 1.Depending on the determinant of Q and the choice of Q, we utilize one of five different recursions.The recursion relations are expressed in lower rank or lower topology integrals with the same dimensions, except in the very specific cases where both Q and Q (b) are degenerate.The reduction coefficient tensor structure and runtime analysis for degenerate cases are similar to those for nondegenerate cases.Therefore, this article will not delve into further detail on this topic.
There are some things to be explored in future, which we briefly comment below.
• The occurrence in which both Q and Q (b) are degenerate can be resolved by introducing additional Q b in the reduction of E n−1,k .This process is similar to the approach taken in the present context, albeit more intricate.
• As demonstrated in [3], degenerate integrals can also be reduced by means of taking limits based on the expressions of reduction coefficients pertaining to non-degenerate Q.In this context, an anzatz is proposed whereby one basis splits into a combination of other bases.The unknown coefficients within the anzatz are then evaluated by requiring that all singularities cancel each other out.
• In the case of one-loop integrals, E n,k is defined by only two tensor structures, namely V I X I and L I X I , enabling the extraction of all V elements in order to reduce it.However, in attempting to extend this same methodology to two-loop integrals, the tensor integrals must be written in projective space, revealing more complex tensor structures such as W IJ X I J, V I X I , (H b ) I X I , and so forth, coupled with a cubic polynomial, Q IJK X I X J X K , appearing in the denominator.Due to the difficulty in calculating the inverse of Q and extracting the unwanted V elements, a different approach must be taken.I believe this is related to the appearance of irreducible scalar product in the case of two loops, but further research is needed to confirm this.
denominator (LL).Two different recursion relations can be considered for different tensor orders.
Rearrange the second relation we get The above formula applies for i > 0, but it can also be extended to the case where i = 0 by replacing V with L: then it becomes safe to take i = 0: .
a vector.Notice that in the derivation of (4.1) we assume Q (b) , i.e., the sub-matrix of Q, is non-degenerate.If not, we need to start with (3.3) and introduce another (n − 1)× (n − 1) general matrices Q (b) to reduce Lt (r) n .It is obvious that (4.1) can return to the non-degenerate case by choose Q 1) vanishes, and (4.1) becomes (LL)I (r) n = A r I (r−1)

6 )
This formula is valid for r ≥ 1.For r = 0, from the discussion in[1], I n doesn't belong to the basis anymore.One can first writeI n = (−1) n Γ(n − D/2)E n,n−D , then applying the recursion relation for E n,n−D E n,n−D = − (H b L) (n − D + 1) (LL) n−D+1can be reduced further depending on whether Q (b) is degenerate or not.

7 )
k+2 /α n,k+2 − (H b L)E For the sake of simplicity, we introduce the notation k 0 = n − D − r, k(i) = k 0 + i, and s = D/2 + r − n.Using equations (A.5) and (A.7), we can reduce the first summation on the right-hand side of equation (A.1

Table 1 .
Actual running time of tensor reduction by direct expansion.First, we expand a one-loop tensor integral according to (2.3), where we set v = n and S = V .Next, we choose

Table 2 .
Theoretical running time T n,r for analytical tensor reduction by direct expansion.We set T n [δD = 0, V 0 ] = t 0 then calculate T n,r through (2.11),(2.13)and(2.14).One can see T n,r suffers from sharp growth as n, r increase, which is consistent with practice.

Table 3 .
Actual 13)is similar to the one in(2.11).However, there is a running time of analytical tensor reduction by recursion formula.In comparison with Table1, we restrict our focus to the non-degenerate scenario.In this context, we rely solely on the employ of the recursion formula provided by(3.6)toiteratively reduce I (r)

Table 4 .
Theoretical running time of analytical tensor reduction by recursion relation.To streamline the process, we have set T n,0 to equal t 0 .Subsequently, we compute T n,r by means of (3.13).Upon comparison with Table2, it becomes apparent that the rate of increase of running time is considerably slower.

Table 3 .
and Additionally, the recursive relation enables us to study the general structure, such as singularity, of reduction coefficients, as shown by our explicit examples in the main text and the results listed in Appendix B.