Determine Arbitrary Feynman Integrals by Vacuum Integrals

By introducing an auxiliary parameter, we find a new representation for Feynman integrals, which defines a Feynman integral by analytical continuation of a series containing only vacuum integrals. The new representation therefore conceptually translates the problem of computing Feynman integrals to the problem of performing analytical continuations. As an application of the new representation, we use it to construct a novel reduction method for multi-loop Feynman integrals, which is expected to be more efficient than known integration-by-parts reduction method. Using the new method, we successfully reduced all complicated two-loop integrals in $gg\to HH$ process and $gg\to ggg$ process.

Introduction. -Computation of Feynman loop integrals is in the heart of modern physics, which is important both for testing the particle physics Standard Model and for discovering new physics. A good method to compute one-loop integrals was proposed as early as in 1970s, the strategy of which is to first express scattering amplitudes in terms of linear combinations of master integrals (MIs) and then compute these MIs [1][2][3]. Based on this method, one can compute one-loop scattering amplitudes systematically and efficiently if the number of external legs is no more than 4. With further improvement of the traditional tensor reduction [4] and the development of unitarity-based reduction [5][6][7], computation of multi-leg one-loop scattering amplitudes is also a solved problem right now.
Yet, about 40 years later, it is still a challenge to compute multi-loop integrals, even for two-loop integrals with 4 external legs. The mainstream approach to calculate multi-loop integrals in literature is similar to that at oneloop level, by first reducing Feynman integrals to MIs and then calculating these MIs. However, both of the two steps are much harder to achieve than one-loop case.
Although compact and explicit expressions for oneloop MIs can be easily obtained [2,3], the computation of multi-loop MIs is very challenging. There are many methods in literature to compute multi-loop MIs, such as the sector decomposition [8], Mellin-Barnes representation [9,10], and the differential equation method [11][12][13][14], but none of them provides a satisfactory solution. In Ref. [15], we proposed a systematic and efficient method to calculate multi-loop MIs by constructing and numerically solving a system of ordinary differential equations (ODEs). The differential variable, say η, is an auxiliary parameter introduced to all Feynman propagators. With the ODEs, physical results at η = 0 + are fully determined by boundary conditions chosen at η = ∞, which can be obtained almost trivially. Therefore, MIs can be treated as special functions as far as there is a good reduction method to set up ODEs.
Reduction of multi-loop integrals is an even harder problem. Significantly different from one-loop case, propagators in a multi-loop integral are usually not enough to form a complete set to expand all independent scalar products, either between a loop momentum and an external momentum or between two loop momenta. As a consequence, the unitarity-based multi-loop reduction [16][17][18][19][20][21][22][23][24][25][26] has difficulty to fully reduce scattering amplitudes. Although the integration-by-parts (IBP) reduction [27][28][29][30][31] is general enough to reduce any scattering amplitude to MIs, the incompleteness of multi-loop propagators makes it hard to generate efficient reduction relations. Currently, IBP reduction is mainly based on Laporta's algorithm [28], which is a brute force algorithm that becomes extremely inefficient for slightly complicated problems. E.g., it cannot give a complete reduction for Higgs pair hadroproduction at two-loop order [32]. Improvements for IBP reduction method can be found in [33,34] and references therein.
To get a satisfactory solution for multi-loop calculation, new ideas seem to be indispensable. Inspired by our previous work [15], in this Letter we construct a novel method to compute Feynman loop integrals. The most important observation is that, with the introduction of the auxiliary parameter η, any Feynman integral can be defined as the analytical continuation of a calculable asymptotic series, which we call series representation of the Feynman integral. As the series contains only vacuum integrals, it can be easily computed order by order. This new representation therefore translates the problem of computing Feynman integrals to the problem of performing analytical continuations. The series representation can also be used to set up reduction relation between any Feynman integral and MIs, as while as to set up ODEs for the MIs. Setting up reduction relations using series representation has an advantage that the incompleteness of multi-loop propagators does not introduce any difficulty. With the reduction relations and ODEs, the desired analytical continuations can be achieved eas-ily. With a two-loop example, we show that our method can be much more efficient than existed ones.
A series representation for Feynman integral.
-Following Ref. [15], we introduce a dimensionally regularized L-loop Feynman integral with an auxiliary parameter η, are usual inverse Feynman propagators with m α being corresponding masses and q α being linear combinations of loop momenta i and external momenta p i , ν α are powers of propagators which can be any integer, and s = (s 1 , . . . , s r ) are independent kinematic variables in the problem. The physical integral that we want to get is with 0 + defining the causality of Feynman amplitudes.
The study in Ref. [15] shows that, as η → ∞, there is only one integration region for M(D, s , η), where all components of loop momenta are at the order of |η| 1/2 . Therefore, all propagators can be expanded like where is a linear combination of loop momenta i , and p is a linear combination of external momenta p i . After the expansion, all external momenta and masses do not present in denominators anymore, thus each term of the expansion can be interpreted as vaccum integrals with equal internal squared masses −iη. Inserting Eq. (3) into Eq. (1) and rescaling all loop momenta by η 1/2 , we get an asymptotic expansion around η = ∞, where M bub µ0 (D, s ) consist of vacuum integrals with equal internal squared masses −i, which can be easily expressed as linear combinations of vacuum MIs, Here B L is the total number of L-loop equal-mass vacuum MIs, with B 1 = 1, B 2 = 2, B 3 = 6 and so on. Thus, we have the decomposition where µ is a r-dimensional vector in Ω r µ0 ≡ { µ ∈ N r | µ 1 + · · · + µ r = µ 0 }, C µ0...µr k (D) are fractional polynomials of D.
As I bub L,k (D) can be easily calculated [35][36][37][38][39][40], the series (4) defines an analytical function around η = ∞, which therefore determines M(D, s , η) for any value of η based on analytical continuation. Especially, the desired physical value at η = 0 + is fully determined. As a result, the expression (4) can be thought as a series representation of M(D, s , 0). We note that the series representation can be applied not only for individual scalar Feynman integrals, but also for tensor integrals and scattering amplitudes. As far as we know, this is the only series representation of Feynman integral in literature. All other representations, such as Feynman (or Schwinger) parametric representation [41], the Baikov representation [42] and Mellin-Barnes representation [9], which have played important roles in the study of Feynman integrals, are integral representations.
Comparing with integral representations, there is a nice feature of the series representation. As values of the series representation in the η → ∞ region can be easily computed order by order in 1/η expansion, by analytical continuation one can obtain physical value at η = 0 + . Thus the problem of computing Feynman integrals is translated to the problem of performing analytical continuations. This conceptual change of interpretation of Feynman integrals may both deepen our understanding of scattering amplitudes and result in powerful methods to compute scattering amplitudes.
However, evaluation of M(D, s , 0) using this series representation is a highly nontrivial problem. The reason is that, in practice, the Eq. (4) must be truncated to some orders in 1/η expansion, which makes the analytic continuations to η = 0 + be very hard both numerically and analytically. Fortunately, the analytical continuations can be achieved thanks to the fact that any given family of Feynman integrals can always form a finite-dimensional linear space, as we will explain in the following.
Reduction relations from series representation.
-An important property of Feynman loop integrals is that the number of MIs is finite [43]. More precisely, for loop integrals constructed from any given set of propagators, there exists a finite set of loop integrals (called MIs, which can be found easily [44]) so that all other loop integrals can be expressed as linear combinations of them, with coefficients being fractional polynomials of kinematic variables and spacetime dimension. In other words, loop integrals with given set of propagators form a finite-dimensional linear space. In the following discussion, we will suppress the dependence on D, s and η in loop integrals whenever it does not introduce any confusion. For a given problem, let us assume that the dimension of the linear space is n. Then any given set of n + 1 integrals {M 1 , . . . , M n+1 } must be linearly dependent, which means that there exists n + 1 polynomials We denote the mass dimension of M i by Dim(M i ) and the mass dimension of Q i by 2d i , then they are constrained by Therefore, for each choice of d 1 , all other d i would be fixed. For fixed d i , we can expand Q i (D, s , η) as where Q λ0...λr i (D) are fractional polynomials of D to be determined.
By calculating the series representation to sufficiently high order in 1/η, we can generate enough linear equations to pin down all unknown Q λ0···λr i (D). It is worth mentioning that, if the value of d 1 is chosen too small, there will be no solution for Q λ0···λr i (D); while if d 1 is chosen too large, there will be more than one solution, which is easy to understand because the Eq. (6) is unchanged if we multiplied it by any polynomial. Therefore, to find out the minimal value of d 1 so that Eq. (6) holds, we generate and solve the linear equations (10) for each choice of d 1 , running from a small enough value to larger values. We stop once solutions for Q λ0···λr i (D) are found.
In practice, we can choose a few special values of D and determine Q λ0···λr i (D) for each special value. As It is needed to emphasize that, as Q λ0···λr i (D) are independent of η, their values determined by series representation in the η → ∞ region are the same as their values in other regions. Therefore, we get an analytic relation (6) valid for any value of η. Especially, by taking η → 0 + limit in Eq. (6), we get a correct reduction relation between M i (D, s , 0).
Analytical continuation. -As series representation can relate any loop integral to MIs, the problem of performing analytical continuations for arbitrary loop integrals is reduced to the problem of performing analytical continuations for MIs.
Let us denote I(D, s, η) as the vector of a complete set of n MIs. As ∂ ∂η I(D, s, η) are special loop integrals, the reduction method described above can also express them in terms of linear combinations of I (D, s, η). Therefore, we obtain a system of ODEs, where A(D, s, η) is the calculable n × n coefficient matrix. With the ODEs, analytical continuation of MIs from η = ∞ to η = 0 + can be obtained straightforwardly by numerically solving the ODEs with BCs chosen at η = ∞. The process is well-studied mathematically, and final results can be obtained efficiently to high precision [15]. We eventually find that all MIs, and thus arbitrary loop integrals, can be determined unambiguously by the series representation, which basically involves only vacuum integrals.
Example. -We take the sunrise diagram in Fig.1 as a simple but nontrivial example to illustrate how our method works. Let us consider a family of Feynman in-tegralsÎ with inverse propagators (13) This family forms a 2-dimensional linear space, with basis can be chosen as {Î 211 ,Î 111 }. To computeÎ ν1ν2ν3 , we introduce similar Feynman integrals I ν1ν2ν3 by changing D i → D i + iη. Note that, I ν1ν2ν3 form a 5-dimensional linear space, with basis can be chosen as {I 211 , I 121 , I 111 , I 110 , I 011 }. By taking η → 0 + limit to the 5 basis, the obtainedÎ 110 andÎ 011 vanish in dimensional regularization, andÎ 121 can be further reduced toÎ 211 andÎ 111 . Suppose that we are now interested in the computation ofÎ ν11 with large ν.
We should first expand I ν11 and the corresponding 5 basis in large η region to obtain series representations. E.g., we have where I bub 2,1 and I bub 2,2 are respectively the factorizable and non-facorizable vacuum MIs shown in Fig.2. The next step is to set up a reduction relation between I ν11 and the 5 basis using the series representations. However, setting up the relation directly through Eq.(6) is almost impossible for large ν because the difference between the mass dimension of I ν11 and that of the 5 basis is very large, which results in too many Q λ0λ1λ2 which can be set up by solving a system of 30 linear equations with 17 unknown variables. For the integrals I ν20 and I ν10 introduced in the above two reduction relations, we can reduce them even easily because they are essentially one-loop integrals. In this way, we can eventually get the desired reduction relation. Before continuing, it is interesting to compare our reduction of I ν11 to the corresponding 5 basis with the IBP reduction ofÎ ν11 to the corresponding 2 basis. We list the time consumed by our reduction and that by the IBP reduction using FIRE5 [31] in Tab. I for different values of ν. Although that our method is realized in Mathematica while FIRE5 is written in C++, and that the reduction of I ν11 seems to be much harder than the reduction of I ν11 , the time consumed by our method is significantly shorter than that by FIRE5, especially for large ν. There are mainly two reasons why our method is more efficient. The first reason is that we can generate reduction relation for any set of integrals, as far as they are linearly dependent. So we always generate relations to reduce an integral to some "simpler" integrals. While in IBP method, one does not know how to generate these good relations, and thus one needs to generate plenty of relations to eventually get the desired reduction, as shown in Tab. I for the number of relations generated by the two methods. The second reason is that, although reduction relations we obtain are analytical, to get them we do not need to manipulate analytical expressions, but only rational numbers. We further note that the reduction of I ν11 can be used not only for sunrise integrals with one massive propagators, but also for two or three massive propagators depending on the choice of the value of η.
By taking η → 0 + limit for the reduction relation between I ν11 and the 5 basis, we obtain a relation between I ν11 and {Î 211 ,Î 121 ,Î 111 }. Thus the value of the former can be got once values of the later are known. To compute the later integrals, we use series representations to set up ODEs for the 5 basis w.r.t. η, the procedure of which is similar to the reduction of I ν11 described above. By solving the ODEs [15], we get the values ofÎ 211 ,Î 121 , andÎ 111 , and thus the value ofÎ ν11 can be obtained. For example, with m 2 = 1 and p 2 = 3.3 we get 10 4Î (100)11 = 1.0307153 −1 + (4.9596399 + 3.2380877i) + (8.0586259 + 15.581168i) + O( 2 ) .
As far as the best knowledge of us, there is no other method that can computeÎ (100) 11 to the same precision that we quoted. FIESTA4 [45] can only computeÎ ν11 with ν < 7, in which cases the obtained results agree with ours. Summary and outlook. -In this Letter, we find a novel representation for Feynman integrals, which is defined as analytical continuation of a calculable asymptotic series. Distinguished from all existed representations, it is a series representation but not integral representation. The new representation translates the problem of computing Feynman integrals to the problem of performing analytical continuations. This new perspective of Feynman integrals may be helpful to deepen our understanding of Feynman integrals and scattering amplitudes.
To realize analytical continuations, we first use the series representation to set up reduction relations between Feynman integrals and MIs, as while as to set up ODEs for the MIs. Then the desired analytical continuations can be achieved easily by solving the ODEs. As series representation can generate reduction relations freely, we can always choose to generate more efficient relations comparing with IBP reduction. With a two-loop example, we show that our method to compute Feynman integrals can be much more efficient than all existed methods.
Our method can be further improved from many aspects. One possible improvement is to take advantage of finite fields technique (see Ref. [46] for an introduction) to avoid large numbers in the middle of calculation process. Another possible improvement is to find out simpler reduction relations, which may only involve Feynman integrals in the subspace of the entire linear space. For example, we indeed find that, if ν 2 > 2, there is a very simple relation between I ν1ν2ν3 and the following three integrals: {I (ν1+2)(ν2−2)ν3 , I (ν1+1)(ν2−2)ν3 , I ν1(ν2−1)ν3 } .
Besides, the number of Feynman integrals involved in dimensional recurrence relations is usually also less than the dimension of the entire linear space.