Universal Treatment of Reduction for One-Loop Integrals in Projective Space

Recently a nice work about the understanding of one-loop integrals has been done in [1] using the tricks of the projective space language associated to their Feynman parametrization. We find this language is also very suitable to deal with the reduction problem of one-loop integrals with general tensor structures as well as propagators with arbitrary higher powers. In this paper, we show that how to combine Feynman parametrization and embedding formalism to give a universal treatment of reductions for general one-loop integrals, even including the degenerated cases, such as the vanishing Gram determinant. Results from this method can be written in a compact and symmetric form.

Although in practice, we will not meet many situations where propagators have higher power, a complete reduction method should be able to deal with it. From this point of view, IBP method is a complete method, because it treats these complicated cases with the same framework as the one without higher poles. Recently, combining unitarity cut and derivation over mass, the reduction coefficients for higher pole cases can be calculated [31], except for the tadpole coefficients.
Recently, we have proposed an improved PV-reduction method for one-loop integrals [32,33]. The reduction coefficients can be expressed with the cofactors of the Gram matrices and have some symmetry. Thus it is useful to understand these symmetries appearing in our middle recursion relations and the final results. Notably, 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 [1]. Inspired by the geometric angle, we find it could be convenient to do reduction for one-loop integrals in projective space. By our study in this paper, one can see that the symmetry and simplicity of reduction coefficients are illustrated clearly with the denotations in [1].
Motivated by the work [1], we will develop an alternative method for determining the reduction coefficients of one-loop integrals in D = 4 − 2ǫ dimension. The general tensor integrals with higher poles are related to integrals E n,k [T ] in projective space where ∆ is a simplex in n-dimensional space X I , which is defined by H I X = X I > 0, ∀I = 1, 2, . . . , n. The T is a general tensor, which is contracted with k X's. 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 forms Xd n−1 X = ǫ I 1 ,I 2 ,...,In (n − 1)!
As pointed out in [1], the integral E n,k satisfies a nice recursion relation, which will be recalled in Appendix A. It is exactly the property we will use to do the reduction in this paper. This paper is structured as follows. In section 2, we discuss how to write a general one-loop integral as the sum of integrals E n,k in projective space. In section 3, we derive recursion relations for E n,k [V i ⊗ L k−i ] and dimension recursion relations for non-degenerate Q, and then apply them to the reduction of one-loop integrals. In section 4, we discuss the reduction framework for degenerate Q. In section 5, we show how to obtain the general expression of reduction coefficient from n-gon tensor integrals to n-gon scalar integrals while general expressions of reduction coefficients are given in Appendix C. More reduction results are listed in Appendix B.

One-loop integrals in projective space
In this paper, we will discuss the reduction of the most general one-loop integrals where q j = j−1 i=1 p i .As pointed out in [32], we can recover the tensor structure by multiplying each index with an auxiliary vector R i,µ i . Furthermore, we can combine these R i to R = r i=1 x i R i to simplify the expression (2.1) in a Lorentzian invariant form to To recover the result of (2.1), one can expand R and extract the coefficient of r i=1 x i from the auxiliary formulas (2.2). Thus, we will transform the general forms (2.2) into projective space suggested in [1]. First, to make our formulas elegant, we denote y µ ≡ k µ , y i ≡ q i . Then (2.2) becomes Then we put the whole formula into the embedding space with two higher dimensions with where we use the light-cone coordinates, i.e. the metric η +− = η −+ = − 1 2 , η µν = diag(+, − − −−) while all other entries vanish. For clarity, we will use the capital letters M, N to denote the components of vectors in the embedding space, Greek letters µ, ν to denote the components of Lorentzian vectors, and lower-case letters i, j for the external legs. We will also use capital letters Y, X simultaneously to denote vectors in the embedding space and projective space without ambiguity. Therefore, we can simplify the denominator of (2.3) into the inner product of two vectors in the embedding space, and the quadratic expression has been somehow linearized, for example, After defining it is easy to check that (2.3) becomes (1) (GL(1) acts as an overall scaling of the Y coordinates) and the factor (Y · Y ∞ ) v−D−r is necessary for the last expression to be genuinely an integral over the projective light cone. Using the most general Feynman parametrization and putting the Feynman parameters into the projective space, (2.7) becomes Then the Feynman parametrization integral has been written into the compact form ∆ Xd n−1 X (see [1] for more details). We can further simplify (2.8) using the common trick as (2.9) Up to now, the last integral in (2.9) can be done easily. One way to solve it is to translate it back to the form (2.3), which is 1 and we have ∂W M can be done easily after using the fact Y ∞ · Y ∞ = 0 and we get where we have written (−2Y ∞ · W ) = i x i ≡ L · X with L = [1 : 1 : ... : 1] 2 . The action of R M ∂ ∂W M is more complicated. By power counting, we have the general expansion where k can be an arbitrary number and i has the same parity as r due to the power of R 2 must be an integer. The expansion coefficients C k r,i are determined by initial conditions C k 0,i = δ i,0 , C k 1,i = 2kδ i,1 and the recursion relation (2.14) From the recursion relation, one can solve C k r,i for general r, i as Plugging (2.13) into (2.12) we get Since the remaining integrals are over the projective space of Feynman parameters, we should rewrite the formula by (2.17) Then we get For general one loop integrals (2.1), one can calculate Q as Now the expression is written as the integration over the X-projective space. Using the result of [1] The reason that we can ignore the action of where ∆ is a simplex in n-dimensional space defined by H i X = X i > 0, ∀i = 1, 2, . . . , n and T is a k-th tensor contracted with k X's, the general one loop integral in the projective space (2.18) can be written as where for simplicity, we write E n,k [V a ⊗ L k−a ] ≡ E n,k [V a ] by neglecting the power of L.
For the later use, we need to do symmetrization for the tensor To do so, we can use the same trick as used in (2.1) and (2.2), i.e., Z ≡ v−n i=1 z i H i and S = tV + Z. Thus (2.21) can be obtained from after taking the coefficients of . Taking care of numerical factors, the final expression is A special case of (2.23) is that for v n = 1 n and r = 0, we have where I n;D is the scalar integral of n propagators in D dimension. Before ending this section, we want to point out in our discussion, for example in (2.8), that the power v − D − r could be positive or negative with the arbitrary choice of r. Since we have kept the dimension D arbitrary, we can take D properly (even a negative number) to make v − D − r a positive integer to make later discussion legitimate. At the end of reduction, we can analytically continue D to the proper dimension. We have checked with several examples that such a continuation is allowed.
In this paper, we mainly discuss Feynman integrals in D = 4 − 2ǫ-dimension space. At the one-loop level, the master integrals are related to E n,n−D [L n−D ], n = 1, 2, . . . , 5. So the main task of one-loop integral reduction is to reduce general integral E n,k [S a ⊗ L k−a ] to the basis E n≤5,n−D [L n−D ].

Reduction for non-degenerate Q
Having transformed our problem (2.1) to the form (2.23), in this section, we will show how to use the tricks of integrals in projective space (see [1]) to generate recursion relations of E n,k [V a ⊗ L k−a ] . By applying these recursion relations iteratively, one can reduce a general one-loop integral to the basis with coefficients written by elegant expressions. In other words, the PV-reduction can be done universally in the new projective space form. As we will point out, the reduction coefficients will have an interesting pattern other than the obvious permutation symmetry 3 . Moreover, the reduction process can be carried out in Mathematica automatically.

Recursion relation
In this subsection, we derive the recursion relations of E n,k [V a ⊗ L k−a ]. We first consider the case Q is non-degenerate. The key equation is following (see Eq.(4.2) in [1]) The proof of the formula can be found in the Appendix A. By integrating (3.1) we get where summing over b is implicitly and to simplify our denotations, we have defined Let us give a little explanation for the first term on the right-hand side of (3.2). When integrating a total derivative term, we should choose a patch. For simplicity, we assume X i = 1. Then we get the contribution from the boundary X b = 0 and X b = +∞ for b = i. By the dimensional regularization, the term with X b = +∞ gives zero. For the term Xd n−2 X , only when the first index of Q −1 takes the value b, the contribution is nonzero, which is equivalent to write as (H b Q −1 T ).
When we repeatedly do the recursion relation, there will be a set of X b setting to zero. Writing the index set as to be the new vector in lower-dimensional projective space obtained by removing these components belonging to the set b j from the original vector X. With this understanding, the meaning of is clear. The equation (3.4) represents the integral got by removing propagators belonging to b j . Now we consider the reduction of E n,k [T ] with T = V i ⊗ L k−i as given in (2.23). Since tensor T is contracted with k X's, we can symmetrize its last k − 1 indices as below where the σ is the permutation acting on the tensor V i−1 ⊗ L k−i . By applying (3.1), one has where for simplicity the L tensor part has been omitted. To make our formula more compact, here we have defined (AB) ≡ AQ −1 B. Here we want to remark a subtle point. In (3.6), the Q is n × n matrix as defined in E n,k and appears in front of E ) −1 is the inverse of the matrix obtained by removing the rows and columns of the index set a j from the original matrix Q.
As the main result of the whole paper, recursion relation (3.6) plays a crucial role in the reduction of one-loop integrals. By comparing the power of V , one sees that it has been reduced from the LHS to RHS. Furthermore, from (2.23) one sees that the power is given by v + i − n, where v contains the contribution of higher power of propagators and i contains the contribution of the tensor numerator, thus (3.6) provides the universal reduction of both cases. As shown in the Fig(1), after iteratively using (3.6), we get a linear combination of E (a n−n ′ ) n ′ ≤n,k ′ <k , i.e., the scalar integral I n ′ ;D ′ in dimension D ′ = n ′ − k ′ . So starting with a general one-loop integral, one can always reduce it to the scalar integrals in different dimensions with coefficients being rational functions of external momenta. If we prefer the scalar basis in a given D-dimensional space, we need to find the formula to shift the dimension of the scalar basis to a fixed D.

Dimension recursion
As we have seen in (2.24), the integral E n,k corresponds to the scalar n-gon diagram in (n−k)dimensional space. To find dimension recursion relations, we set V = L in (3.6) and get (3.7) To reduce E n,k , where n − k = D + 2s, s ∈ Z, s = 0, we can iteratively use (3.7), which is established for the scalar integrals already. Noticing that in the RHS of (3.7), the first term has the same dimension as the LHS with one propagator removed, while the second term has two higher dimensions with the same number of propagators. Depending on the sign of s, we can take different manipulations.
• s > 0: For this case, we need to reduce an integral in a higher dimension to D-dimension, so we solve the second term in the RHS of (3.7) and get It is obviously that such a rewriting (3.8) is legitimate when and only when (LL) = 0.
Both sides have the same dimension, but the RHS of (3.9) has one less propagator. One well-known example of (LL) = 0 is that the bubble with null external momentum is not a basis anymore, and it is reduced to two tadpoles. Having established (3.8) and (3.9), we can reduce (D + 2s)-dimensional integrals to D-dimensional iteratively using either (3.8) or (3.9) depending on if (LL) is zero or not at that step 4 .
As pointed out already, the first term on the RHS corresponds to the scalar integrals in the same dimension but with the j-th propagator removed and the second term corresponds to scalar integrals in D ′ = n − k + 2 dimension. For the boundary situation, i.e., n = 1, the first term vanishes and the second term gives a higher dimensional scalar basis. Repeating it, we can reduce E n,k to scalar integrals in D-dimensional space.

Examples
To illustrate our method and avoid complicated computation in general cases, we first consider the reduction of tensor bubbles where We set q 1 = 0 for simplicity, then Here we give some results for different choices of the rank and the powers to illustrate our idea.
where the black points represent zero terms. The red, orange and cyan arrows represent the first, second and third terms respectively in (3.6).
• Tensor bubble with primary propagator We first consider reducing a rank-1 bubble I {1,1} . Since there are no higher poles, we just choose S = V, r = 1, v = n = 2 in (2.23) and we have (3.14) First, we use (3.6) to reduce E 2,2−D [V ] and get where the first term E (i) 1,1−D corresponds to D-dimensional scalar tadpoles generated by removing the i-th propagator of bubble I 2 while the second term E 2,−D corresponds to a (D + 2)-dimensional bubble. We need to lower the dimension of the second term further. Here we assume (LL) = 0, by using (3.8), we get where the two terms in the numerator correspond to the D-dimensional bubble and two tadpoles. Plugging (3.16) and (3.15) to (3.14) and recognizing them as master integral according to (2.24) we have So the reduction coefficients are (3.18) • Tensor bubble with massless legs One can notice there is a pole of q 2 2 in the reduction coefficients of I 2 , which comes from (LL) For q 2 2 = 0, we have (LL) = 0, so we have Here we need to reduce E 2,−D where we have used (3.9) and Using (2.24), we finally get Explicitly, we have • Scalar bubble with higher poles Then we consider reducing scalar bubbles Due to there is no tensor structure in the numerator, we just set S = Z = z 1 H 1 + z 2 H 2 , r = 0 in (2.23) where and get where the first term E (i) 1,3−D corresponds to (D−2)-dimensional scalar tadpoles generated by removing the i-th propagator of bubble I 2 while the second term E 2,2−D corresponds to wanted D-dimensional bubble. We need to lift the dimension of the first term further. By using (3.10), we get where we have used E (3.28) There are two configurations Explicitly, using (3.13), we find the reduction coefficients are (3.31) • Tensor bubble with higher poles At last, we consider a combined case, I v 2 ;v=3 . Here we need to set S = tV + Z, Z = z 1 H 1 + z 2 H 2 . Setting v = 3, r = 1, n = 2 in (2.23), we have First, we use (3.6) iteratively to pull out all S's in the numerator Among the four terms above, we only need to deal with the last term E 2,−D since it corresponds to a (D+2)-dimensional bubble, which has been discussed in (3.21). Finally, we have (3.34) One can get the reduction results for I {2,1} , I {1,2} . For example, where for simplicity, we will not present explicit expressions for these coefficients.
Note that the reduction coefficients in these examples are rational functions. For some special masses and momenta configurations, denominators can become zero, which leads to several kinds of divergences. Since only (LL) ≡ LQ −1 L appears in the denominators, all divergences come from Q-matrix and its all sub-matrices, which have det Q = 0 or (LL) = 0. For example, the pole of q 2 2 in (3.18) comes from LQ −1 L (see (3.19)). The divergence of C {2,1}→2; 2 is given by m 2 1 (m 1 − m 2 ) 2 − q 2 2 (m 1 + m 2 ) 2 − q 2 2 = 0, which is corresponds to det Q (2) = 0 or det Q = 0. One can find the pole (LL) = 0 comes from the dimension shifting process (3.8), which can be addressed by employing (3.7) to reduce E n,k to lower topology. To deal with the divergences coming from det Q = 0, we need to consider the reduction method for degenerate Q elaborated in the next section.

Reduction for degenerate Q
In this section, we generalize our reduction method to degenerate Q. The basic idea is to generalize the recursion relation (3.1) to the formula (A.9). When Q is degenerate, the characteristic equation Qξ = 0 always has solutions, and we denote the N Q as the null space spanned by linearly independent ξ's.

QL = 0
When Q is degenerate, the recursion relation (3.1) in the last sections breaks down for det Q = 0. Our idea is to consider the tensor structure with one L in the first place and make other (k − 1) indices completely symmetric by summing over all permutations between i V 's and (4.1) For a degenerate Q, we can always find a matrix Q = [ξ 1 , ξ 2 , . . . , ξ n ],ξ i ∈ N Q so that Q Q = 0. Then the LHS of (4.1) vanishes. If Q * L = 0, where Q * is the adjugate matrix of Q, we can take Q = Q * . With the denotatioṅ (4.1) becomes Depending on the value of(LL), we have following two cases: • (1) When(LL) = 0, (4.3) can be rewritten as where the first term in RHS corresponds to the lower topologies and the second term has tensor rank reduced by one.
• (2) When(LL) = 0, (4.3) can be rewritten as where although the tensor rank increased by one on the RHS, it belongs to the lower topologies. One particular thing of (4.5) is that since V depends on the auxiliary R, we will always have(V L) = 0. Another tricky point is that although R appears in the denominator in (4.5) in the middle steps, it will be canceled in the final reduction coefficients.

QL = 0
Since QL = 0, every term in (4.3) vanishes. Now we can put V in the first place of tensor structure and using (A.9) to reach Using Q Q = QL = 0, (4.6) becomes and we have Again, although R appears in denominator through(V V ) in middle steps, it will be canceled in the final reduction coefficients.

Dimension recursion
Having reduced to scalar integrals with different dimensions, we want to shift the dimension to a given D. Depending on various situations, we have: • For QL = 0, we can always choose Q Q = 0,(LL) = 0. Then using (4.4) for the case i = 0, the second term vanishes, and we have n−1,k+1 . (4.9) • For QL = 0 we use (4.8) with i = 1. Then we shift V → L + ǫK with a reference K such that K ⊥ N Q and get (4.10) Comparing both sides, especially the ǫ term, we have We lower the topology in the RHS, so we can reduce it further by using the equation recursively. The dependence of the choice of K in the middle steps will vanish in the final reduction coefficients as shown in later examples.

Examples
In this section, we illustrate our method for degenerate Q. To avoid unnecessary and complicated calculations and compare with the reduction procedures for non-degenerate Q discussed in section (3.3), we focus on bubbles with some special masses and momenta configurations.
• QL = 0: Massless scalar bubble with equal 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 5 .
We find Q defined in (3.13) degenerates to a corank-1 matrix Since Q Q = 0 and Q * L = 0, using (2.24) we have (4.14) One can see the RHS is not irreducible anymore by employing (4.11) where the reference vector K satisfies Q * K = 0 and the RHS corresponds to integrals of tadpole topology acquired by removing one propagator from I 2 . We then use (3.6) to pull out the K Note that the Q-matrix in (4.16) is just a number Combining (4.15),(4.16) and (4.14), we finally find One can check that the result above does not depend on the choice of K = (a, b) as long as a = b.
• QL = 0: Massless tensor bubble with equal internal masses Here we consider the reduction of the tensor bubble I where the RHS corresponds to integrals of tadpoles with non-degenerate Q-matrix Q (b) = m 2 . We can use (3.6) reduce it iteratively There are two terms in the last line, we need to reduce the second one E Using( we find the reduction relation where I 2; 1 is just the tadpole I 1 [m] with mass m.
• QL = 0: Scalar bubble We then consider the scalar bubble I (1) 2 with degenerate Q but m 1 = m 2 . Here we will show that it can be reduced to two tadpoles using our method. The equation det Q = 0 gives two solutions (4.26) One can check that QL = 0 for m 1 = m 2 . Due to(LL) = 0, we can use (4.4) to reduce E 2,2−D in the expansion and we get Plugging (4.28) into (4.27), we have One can find the explicit expression easily. For q 2 2 = (m 1 + m 2 ) 2 , the explicit expression is (4.30) • QL = 0: Tensor bubble Here we discuss the reduction of the rank-1 bubble with q 2 2 = (m 1 ± m 2 ) 2 . First, we expand it using (2.23) (4.31) Choosing Q = Q * , due to(LL) = LQ * L = 0, we can use (4.4) to reduce the RHS to where the second term has been discussed in the last example. We just need to reduce the first term E , which corresponds to a tadpole. Using (3.6), we find where the RHS is a D-dimensional scalar tadpole. Plugging (4.33),(4.32) into (4.31) we get (4.34) Then we refer to the result (4.29), and find For q 2 2 = (m 1 + m 2 ) 2 , the explicit expression is (4.36)

General expression of C (r) n→n
It seems hard to solve (3.6), but if we only care the reduction coefficients to the same topology, we can ignore the first term for it contributes only to lower topologies. Then by iteratively using (3.6), keeping only the second and the third term, one find that Although the first term has the same topology, with different choices of i, j, the dimension is different, thus we need to reduce it further.
To simplify our denotation, we define Here we give the computation for tensor reduction with all propagators power one 7 . Thus, we need to reduce all E n,n−D−(r−i) [V i ] to E n,n−D . After using (3.8) to repeatedly lift k we have Since the second term in (5.4) belongs to lower topologies, which can be ignored, finally we have From it, we read out reduction coefficient where we require r, i, j having the same parity.

Discussion
Although the main target of the paper [1] is to understand general one-loop integrals from a geometric point of view, it does contain many important and valuable results. In this paper, we have elaborated on the method to compute reduction coefficients for general one-loop integrals.
The essential idea of the method is to put the whole one-loop Feynman integrals in the projective space, in which integrals have compact forms and geometry properties. Using a vital recursion relation of E n,k [T ], one achieves the wanted reduction. The advantage and the most promising point of this method is that we can solve any one-loop integrals with higher poles and tensor structures at the same time, which is demonstrated by some examples in section 3 and 4. The language of projective space simplifies the reduction process a lot by keeping the elegant contractions like (HL), (LL) in these recursion relations without expansion, thus making the whole reduction process a symbolic calculation. For some programs based on the traditional IBP method, like FIRE, LiteRed,Kira, etc [35][36][37][38][39][40][41], they do reduction by solving linear equations where the determinant of the Gram matrix appears with the full-expanded form, which makes the final results too complicated to read. The appearance of (reduced) Gram matrix has also been observed in our recent work [32,33].
One obvious idea is to generalize the above method to the reduction of two-loops and higher-loops. Recently, using the improved PV-reduction method with auxiliary vectors, we have shown how to do the general tensor reduction for two-loop sunset integrals [42]. From the results in [32,33] and results in this paper, we see that these two methods have some correspondences. In other words, they treat the same thing from different but related angles. Since the improved PV-reduction method has some results for two-loop reductions, the projective space method should be generalized to two-loop too.

A Recursion relations of E n,k [T ]
In this section, we recall the proof of the relation (3.1) given in [1]. Let us calculate directly where we require T is completely symmetric of the last (k−1) indices i 2 i 3 . . . i k . For simplicity, we denote Using the fact which is nothing but the wanted (3.1). The derivation above can be generalized to the case detQ = 0, where the Q −1 does not exist. In this case, we can replace Q −1 by arbitrary matrix Q in (A.1) and repeat the same derivations to reach the similar expression likes (A.8), except Q −1 replaced by Q. Rearranging (A.8) we get where (Q QT ) has the same rank as T In (A.9), the LHS is the first term in RHS of (A.8), while the first term on the RHS is the boundary contribution of LHS of (A.8).

B.1 Bubbles
Here list the results for rank from r = 1 to r = 4, where we set q 1 = 0.

B.2 Triangles
For triangle topology, we have presented results for scalar triangles with higher poles. Here we present some examples, including the tensor triangles without higher poles and with higher poles.
For the tensor triangles without higher poles, we have: • r = 1: The expression is the same as (B.2) for bubble topology. This phenomenon is not an accident, and will be persistent to other topologies.
• r = 2: Again, the last two coefficients are very similar to these given in (B.3).
For the tensor with higher poles, we have: • v = 4, r = 1: Here we have suppressed the process to take the coefficient of tz v 3 −1 ,i.e, | tz v 3 −1 . The similarity of the above expression with the one given in (3.25) is obvious.
• v = 4, r = 2: Here, when calculating the reduction coefficients, we take the coefficient of z v 3 −1 for terms contains one R 2 and one S while we take the coefficient of t 2 z v 3 −1 for terms contains three S's.

B.3 Boxes
For the box topology, we present three cases : As pointed out in the triangle topology, the reduction coefficients have some similarity between different topology. In fact, the similarity is classified by the pair (r, v − n) as one can check by using the results listed in the Appendix and the main body of the paper.

B.4 Pentagons
For pentagon topology, the reduction coefficients are similar, so we just present one example, i.e., scalar pentagon with higher poles.
• v = 6 (B.14) One thing we want to point out is that these coefficients have a manifest permutation symmetry. Using these observations, the expression can be very compact, as shown above.