General One-loop Reduction in Generalized Feynman Parametrization Form

Recently there is an alternative reduction method proposed by Chen in [1,2]. In this paper, using the one-loop scalar integrals with propagators having higher power, we show the power of the improved version of Chen's new method in which we used some tricks to cancel the dimension shift and the terms we do not want. We present the explicit examples of bubble, triangle, box and pentagon with one propagators doubled. With these results, we have completed our previous computations in \cite{wang} with the missed tadpole coefficients.


Introduction
To give more precise theoretical prediction of scatting amplitude of a given process, calculation of high loops integrals becomes important. For these calculations, the PV-reduction method [4] is one of the most used ideas. One way to implement the reduction method is to use Integrating-by-Parts (IBP) relation [5,6,7]. As one of the most powerful techniques for loop integrals reduction, IBP gives a large number of recurrence relations, and one could get the reduction of the simpler integrals directly by Gauss elimination. However, as the number and power of propagators become higher and higher, the IBP method becomes hard and inefficient. Finding more efficient reduction methods becomes an important direction.
Unitarity cut method is one alternative reduction method and has been proved to be very useful for one-loop integrals [8,9,10,11,12,13,14,15,16,17,18,19]. For physical one-loop process, the power of propagator is just one, but if the method is a complete method, it should be able to give the reduction of integrals with higher power of propagators. Such a situation is not just a theoretical curiosity. In fact, it appears in the higher loop diagrams as a sub-diagram. Furthermore, although for one-loop integrals the scalar basis is natural, in general the choice of basis can be different, depending on the physical input. For example, for one-loop bubble, the basis with one propagator having power two could be useful as part of UT-basis [20,21].
In our previous work [3], by combining the trick of differential operators and unitarity cut, we successfully got the analytical reduction result of one-loop integrals with high power propagators and gave the coefficients to all the basis except the tadpoles' coefficients. Since the tadpole have only one propagator, the unitarity method could not be used to get the tadpole part. To complete our investigation, we want to find the missing tadpole coefficients by some efficient methods.
Except the unitarity cut method, there are other proposals to overcome the difficulty in IBP, by using some tricks and other representations of integrals in recent years, such as Baikov representation [22,23] and Feynman parametrization representation [24,25] for loop integrals. In recent years, Chen has proposed a new representation for loop integrals [1,2]. His method is based on the generalized Feynman parametrization representation, i.e., an extra parameter x n+1 has been introduced to combine the U , F in the standard Feynman parametrization representation. Such a generalization will bring some benefits in deriving the IBP recurrence relation, as will shown in this paper.
As a common feature, the IBP recurrence relation derived using the generalized Feynman parametrization representation will naturally have terms in different spacetime dimension. Since we always concern the reduction in a certain dimension D, which is usually set to be 4 − 2ǫ for the reason of renormalization, we want to cancel these terms in different dimension. This is usually not an easy work. In [26] Gluza, Kajda and Kosower have shown how to avoid the change of power of propagators in the standard momentum space. Larsen and Zhang have considered the Baikov representation and showed how to eliminate both dimension shifting and the change of power of propagators [27,28,29,30,31,32]. These methods require the solution of syzygy equations, which is not easy to figure out in general. In Chen's second paper [2], he proposed a new technique to simplifying the recurrence relation based on the non-commutative algebra.
Motivated by above discussion and preparing Chen's method for the high-loop computations, in this paper, we will use the Chen's method to find the missing tadpole coefficients in our previous work. Furthermore, we will use the idea to remove terms with dimensional shifting in the derived IBP relation to give a simpler reduction method with the analytic results written by the elements of the coefficients matrix Â .
The plan of the paper is following. In section 2, we have reviewed the Chen's new method and illustrated with a simple example in the section 2.1. In the example, the integrals in different dimension will naturally emerge. We discussed the physical meaning of the boundary terms, which contributes to the sub-topologies. To cancel the dimension in the parametrization form and simplify the IBP relation, in the section 2.2 we proposed a new trick by adding free auxiliary parameters based on the fact that the F in the integrand is a homogeneous function of x i with degree L + 1. By our trick, we successfully canceled the dimension shift and dropped the terms that we do not concern to a certain extent, and give a simplified IBP relation in which all the integrals are in the certain dimension D and integrals except the target have lower total power of propagators. We gave our analytic result by the determinant of the cofactor of the matrixÂ, which is completely determined by the graph. In section 3, combined with our trick, we calculated the triangle I 3 (1, 1, 2), box I 4 (1, 1, 1, 2), and pentagon I 5 (1, 1, 1, 1, 2) in the parametric form proposed by Chen , and gave the analytic result of all the coefficients to the master basis, especially the tadpole parts as the complement of our previous work.

Reduction method in parametric form by Chen
In this section, we will introduce a new reduction method proposed by Chen in [1]. The general form of loop integral is given by where for simplicity, we have denoted l = (l 1 , l 2 , l 3 , · · · , l L ) and k = (k 1 , k 2 , k 3 , · · · , k n ). Since in this paper, we consider only the scalar's integrals with N (l) = 1, let us label I(L; λ 1 + 1, · · · , λ n + 1) = d D l 1 · · · d D l L 1 By the procedure of Feynman parametrization, thus the loop integrals can be done as is a homogeneous function of α i with degree L, while the V (α) is a homogeneous function of α i with degree L + 1, and the 1 The relation has been verified in many places based on the method in graph theory loop integral becomes to To derive the parametric form suggested by Chen, we do the following. Using the α-representation of general propagators, where the "iǫ" has been neglected, we get To go further, we change the integral variables as α i = ηx i . Since there are totally n independent variables, we must put another constraint condition. In general, we could let i∈S(1,2,3,···n) where S is an arbitrary non-trivial subset of {1, 2, 3, · · · n}. After carrying out the integration over η, the second line of eq.(2.5) becomes to Finally we by Mellin transformation 2 we could write the (2.9) as Putting all together, we now finally get the parametric form of scalar loop integrals (2.5),

The IBP identity in parametric represent
The parametric form of (2.14) is the starting point of Chen's proposal. The IBP relations in this form is given by 34 where i = 1, ..., n + 1 and the dΠ (n) in the second term is The second term in (2.15) contributes to a boundary term which leads to the sub-topologies to the former term.
To illustrate the IBP relation (2.15), we present the reduction of I 2 (1, 2) as an example. The general form of one-loop bubble integrals is given by (2.17) 3 In some sense, the parametric form can be considered as the generalized Feynman parametrization form. Thus the IBP relation (2.15) could be called the IBP relation in the generalized Feynman parametrization form. 4 The IBP relation requires the term in the bracket of the first term to be degree (−n), which can be obtained by multiplying any monomial of degree one. Here in (2.15) we have multiplied xn+1 by our experiences from later examples, but one can make other choices. and the corresponding parametric form is (in this article we ignore the former factor π LD 2 ) with λ 0 = − D 2 and λ 3 = −3−m−n−2λ 0 . Using the eq.(2.15), we could get three IBP recurrence relations. Taking ∂ ∂x 1 first, the first term in (2.15) gives Here we need to explain the notation i λ 0 ;−1,n . From the middle expression of (2.22), we see that it is the parametric form of tadpole d D l (l 2 −m 2 λ 0 i λ 0 −1;m,n + 2m 2 1 λ 0 i λ 0 −1;m+1,n + ∆λ 0 i λ 0 −1;m,n+1 + δ m,0 i λ 0 ;−1,n = 0 (2.23) When we set m = n = 0 in (2.23), it reads Similarly, we could take the differential ∂ ∂x 2 and get the second IBP relation Naively, we should solve i λ 0 ;0,1 by i λ 0 ;0,0 from (2.24) and (2.25). However, for bubble part, we have λ 0 − 1 instead of λ 0 . This one could be fixed by rewriting λ 0 → λ 0 + 1 since λ 0 is a free parameter. However, the boundary tadpole part i λ 0 ;0,−1 will become i λ 0 +1;0,−1 , i.e., having the dimensional shifting, which is a common feature in the parametric IBP relation.
To deal with it, using the parametric form of tadpoles and taking the ∂ ∂x 1 and ∂ ∂x 3 , we could get two IBP relations from which we solve Putting (2.28) to (2.24) and (2.25), we can solve the i λ 0 −1;0,1 . After shifting λ 0 → λ 0 + 1, we finally get Translating back to scalar basis, we get the reduction of I 2 (1, 2) as with the coefficients The result is confirmed with the FIRE6 [33,34].

Improvement of parametric IBP
As we have seen from the previous subsection, the IBP relation given in (2.15) will contain the integrals with dimension shift, which makes the reduction program a bit troublesome. We would like a recurrence relation without dimension shift. As we reviewed in the introduction there are several references dealt with this or related problems. Based on these work, an improved version of IBP relation has been given in [2] (see Eq.(2.12), (2.13) ). All these methods require the solution of syzygy equations, which is not an easy task in general. However, for our one-loop integrals, the function F (x) is a homogeneous function of x i with degree two 6 . This good property makes the related syzygy equations simple, which can be solved straightly 7 . In this paper, we will develop a direct algorithm to write down IBP relations without the dimension shift and the terms having unwanted higher power of propagators.
In the generalized parametric representation, our improved IBP relation is to multiply a degree zero coefficient z i , for example, , in (2.15). Since the degree of the new integrand does not change, the IBP identity still holds. Summing them together we get 8 Since the second boundary term involve integrals with sub-topologies, we focus on the first term. Expanding it, we got From (2.13), one can see the power λ 0 of F is related to dimension. To cancel the dimension shift, we need to choose the proper coefficients Since coefficients z i are not polynomials, (2.34) is not the "normal sygyzy equation" and one can not directly use the technique developed for polynomial ring. In [2], Chen developed a method based on the lift and down operators. Here for the one loop integrals, we can solve it directly with some free auxiliary parameters, as we will show shortly. When putting back solutions to the IBP recurrence relation, we could choose these free parameters to cancel both the dimension shift and unwanted terms with higher power of propagators, which leads to a simpler recurrence relation. Now let us explain the idea in details. Note that in one loop case, the homogeneous function F is a degree two function of x i , so we can write where A is the symmetric matrix 9 . Thus we have 8 Note the summation of i is form 1 to n + 1, where we have included the auxiliary parameter xn+1, which is an apparent different from the tradition Feynman parametrization. 9 In general it is not necessary to makeÂ be symmetry matrix, and this is just one choice. But for the simplification of the following calculation, since we will later set an antisymmetric matrixKA, it is convenient to make the convention to setÂ be symmetry matrix.
where the coefficients' matrixK is a real symmetry matrix. In fact we can do more. Using the trick that with any antisymmetric matrix K A , we could add (2.37) to (2.36) to get a more general form Noticing that because the arbitrary matrixK A of rank n + 1, there are n(n+1) 2 free independent parameters, a 1 , · · · , a n(n+1) 2 in the matrixQ in (2.38).
Now putting (2.38) back to (2.34), we could solveẑ aŝ Noticing that since z is degree zero, we should have B homogenous function of degree −1. In our article, we choose B = 1 x n+1 . The choice of z given by (2.39) will guarantee to remove the dimension shift in the IBP relation. Furthermore, by choosing particular value of these free parameters ofQ, we could cancel some unwanted terms. In the later computations, we will give some examples to illustrate this trick.

Reduction of one-loop integrals
As we have mentioned in the introduction, one motivation of the paper is to complete the reduction of scalar basis with general powers. Using the unitarity cut method in [3], we are able to find reduction coefficients of all basis, except the tadpole. In this section, we will use the improved IBP relation (2.32) to find the tadpole coefficients as well as other coefficients.

The bubble's case
Let us start from the bubble topology. Although we have done it already in (2.30), here we will redo it using the improved IBP relation (2.32). The parametric form of bubble is given by (2.18), (2.19) and (2.20). Using our label, we havef Adding the antisymmetric matrix K A , we havê

Deriving the recurrence relation
Expanding the (2.32), we got the IBP recurrence relation where the δ 2 is the boundary term, which we will compute later. Other coefficients are Since we want to get the reduction of I 2 (1, 2), starting from m = n = 0, we want to eliminate terms with indices (m + 1, n) and (m + 1, n − 1), while keeping the term with index (m, n + 1). Thus we impose c m+1,n = 0 and c m+1,n−1 = 0, which can be satisfied by choosing the free parameters 10 After this choice, the matrixQ becomes tô For this example, one can check that we can not add another constraint to fix a1.
and it left us five terms with non-zero coefficients 11 .
The boundary δ 2 term: The δ 2 term is given by where the λ i represents the power of x i . It is worth to emphasize that since z i contains x i , the total power λ i of x i is not equal to m, n, λ 3 in general. Expanding it, we get 12 Remembering our extended notation explained under (2.22), we have and the δ 2 term could be written as where the subscript r in δ 2;r and Q ij;r means that the a 2 and a 3 should be replaced by (3.6).
Since the m and n could not be −1, the first and fifth terms are actually zero.
Now we could use (3.4) and (3.11) to get our result directly. Setting m = 0 and n = 0, all other terms in (3.4) are equal to zero, and we are left with 13 with the coefficients From it we could directly write down the answer Translating back to scalar integrals, it is I 2 (1, 2) = c 12→11 I 2 (1, 1) + c 12→10 I 2 (1, 0) + c 12→20 I 2 (2, 0) + c 12→01 I 2 (0, 1) + c 12→02 I 2 (0, 2) (3. 16) with c 12→20 = 0 and 13 When setting m = n = 0, except the boundary term δ2, among other seven terms in (3.4), the coefficients of the second and the third terms have been chosen to be zero. For the other five terms, one can show that cm−1,n+1, cm,n−1, cm−1,n are zero by using the last line of (3.5). There is another technical point. When m = n = 0, the seventh term will contain i λ 0 ;−1,0 , which looks like the one defined in (3.10). But they are, in fact, different. The one appeared in (3.4) with the measure dΠ (3) while the one appeared in (3.10) with measure dΠ (2) . 14 The reduction of tadpole with higher power is simple. Noticing that I2(1, 0) ∝ (m 2 1 ) by dimensional analysis, one can take the derivative over m 2 1 to get the wanted reduction coefficients.
with the coefficients which is given in (2.30).

The general case of bubbles
Now let us consider the more complicated examples, i.e., the bubble with general higher power of propagators. By the choice (3.6) we got an IBP recurrence realtion (3.7) and use it we could reduce the bubbles i λ 0 ,m,n+1 to the simpler bubbles having less total power of propagators and no higher power in D 2 . Similarly, by choosing the different values of a 2 and a 3 , we could get another IBP recurrence realtion to reduce the integral to those having no higher power in D 1 . The choice is (3.20) and the corresponding IBP recurrence is  In the example I 2 (1, 3), we just need to reduce D 2 from power 3 to 1. The strategy is to use (3.7) two times. In the first step, by setting m = 0 and n = 1 in (3.7) we got For the first term in (3.24), setting m = 0 and n = 0 in (3.7) again we have Putting (3.25) into (3.24) and using the reduction of tadpole 15 we get with the coefficients The result is confirmed with FIRE6. In this example, we just need to solve 2 equations in reducing bubbles' topology.
With the same idea going down, we just need to solve 14 equation to complete reduce the I 2 (3, 5). The analytic expression by these 14 equations have also been confirmed by FIRE6. 15 In general, we could repeat the similar procedure to give the tadpoles' IBP recurrence relation, and calculate them step by step. Here, for simplicity, we could just use the trick, I2(1, 0) ∝ (m

The triangle's case
The triangle I 3 (m + 1, n + 1, q + 1) is given by The parametric form of it is Using the expression (2.10), we have Thus we can read out matriceŝ
c 0,0,0 i λ 0 ,0,0,0 + c 0,0,1 i λ 0 ,0,0,1 + δ 3;000 = 0 (3.42) with the coefficients One can see that in (3.42), only two terms of triangle topologies are left: one is the scalar basis and one is the target we want to reduce. Other five terms in (3.39) disappear by the expression in (3.40). Thus there 16 Since the boundary term having only one xi = 0, it reduces to the sub-topologies with only one propagator pinched.

Summary and further discussion
In this paper, we consider the one-loop scalar integrals in the parametric representation given by Chen. However, in the recurrence relation, there are usually some terms that we do not want, as well as some terms with dimensional shifting in general, which makes our calculation not easy and efficient. In Chen's later paper [2], he used a method based on non-commutative algebra to cancel the dimension shift. Different from others methods, in the one loop case, we have used a straight method by solving the linear equation systems to simplify the IBP recurrence relation in the parametric representation. Benefited from the fact that the F is a homogeneous function of x i with degree two in one-loop's situation, we could solve the x i by ∂F ∂x i with some free parameters. Then combining all the IBP identities with particular coefficients z i , and then choose particular values of the free parameters, we succeed to cancel the dimension shift and the terms with higher total power. As the complement of the tadpole coefficients in the reduction to our previous paper, we calculated several examples and gave the analytic result of the reduction.
For further research, there are some questions needed to be considered. In the previous calculation, we can see that the coefficients we constructed z i is not polynomial since it has the denominator with the form x γ n+1 , so we could not directly use the technic of syzygy. Also, the application of Chen's method to higher loop is definitely another future direction. For this case the homogeneous function F (x) is of degree L + 1, where L is the number of loops. For high loop's case, we should consider how to construct the coefficients z i efficiently, and find a relation similar to (2.37) to cancel the terms we do not need. Thirdly, the sub-topologies are totally decided by the boundary term in the parametric representation, and this may lead to some simplification of calculation.