An Explicit Expression of Generating Function for One-Loop Tensor Reduction

This work introduces an explicit expression for the generation function for the reduction of an $n$-gon to an $(n-k)$-gon. A novel recursive relation of generation function is formulated based on Feynman Parametrization in projective space, involving a single ordinary differential equation. The explicit formulation of generation functions provides crucial insights into the complex analytic structure inherent in loop amplitudes.


Background and Motivation
Scattering amplitudes are fundamental concepts in modern quantum field theory.They serve as not only a source of precise predictions and explanations for experiments conducted at the LHC [1] but also make a significant contribution to the analysis of the analytical structure within perturbative quantum field theory.Moreover, they play a pivotal role in our exploration of potential new phenomena in physics.Amplitudes, which result from perturbative expansions, can be classified into two primary categories: tree diagrams and loop diagrams, based on their respective orders.
At the level of loop diagrams, in the context of constructing Feynman integral expressions, one of the most significant methods over the past few decades has been the utilization of on-shell techniques [7,8,[15][16][17][18][19][20]22].On-shell methods are grounded in a fundamental physical concept: when a propagator becomes on-shell, a physical process factorizes into the product of two physical processes.Building upon this concept, the unitarity cut method [17,18,20,22,23] has been developed, allowing the calculation of a one-loop diagram to be reduced to a product of several tree-level diagrams.This approach significantly simplifies the computations compared to the Feynman rules.In addition to this, within the CHY framework, there have been relevant developments in constructing Feynman integrands for loop diagrams.These efforts primarily involve taking the forward limit of two external legs in the (n + 2)-point tree-level amplitudes.This approach establishes a relationship between n-point one-loop diagrams and (n + 2)-point tree-level amplitudes [24][25][26][27][28][29][30][31][32][33].
This article will primarily focus on the tensor reduction problem at one-loop level.This problem has been under investigation for an extended period; however, a recent breakthrough reveals that the introduction of a virtual auxiliary vector R, can streamline the conventional Passarino-Veltman reduction process [86][87][88][89][90].When an auxiliary vector R is incorporated into the IBP method, the improvement of efficiency of reduction has been shown in [91][92][93].
Another notable advancement is the introduction of the "generating function" [94][95][96][97][98].In fact, the concept of a generating function is well-established in both physics and mathematics.The application of generating functions in higher-order QFT calculations was introduced earlier in [94] and has been utilized up to the level of 3-and 4-loop calculations.In [95], Kosower employed integration-by-parts (IBP) generating vectors to derive appropriate IBP reduction relations in certain special 2-loop case.More recently, these functions have also been discussed in [96], detailing how to directly proceed from these representations to the physical result.
In the reduction process, the reduction coefficients for high tensor rank still present a considerable challenge.However, when we sum up the reduction coefficients of different tensor ranks, we might arrive at a simpler solution.For instance, in our method, the numerator of the integral is (2l • R) r with rank r.We can sum them up in two typical ways as shown in the equations below: In recent research [97,98], Bo Feng proposed a recursive method to calculate the generating function, establishing several partial differential equations based on the auxiliary R method.Subsequent work [98] has not only improved upon this at the one-loop level but also extended it to the 2-loop level by setting up and solving differential equations of these generating functions, utilizing the Integration-by-Parts (IBP) method.Both pieces of work underscore the considerable potential of generating functions.However, in both studies, authors provide only an iterative approach to compute the generating functions, and it remains strenuous to directly write generating functions explicitly, even at the one-loop level.Fortunately, we discovered a new recursive relation of generating functions based on investigations into Feynman parametrization in the projective space [90,99,100].This new relation consists of a single ordinary differential equation on t instead of a complex set of partial differential equations, allowing us to directly write an explicit expression of the generating function for the reduction of n-gon to (n − k)-gon for universal k without recursion.
The organization of this paper is as follows.Section 2 introduces essential notations used throughout the paper and establishes our new recursive relation.In Section 3, we compute the generating function of n-gon to n-gon as a warm-up.Section 4 and Section 5 detail the derivation of the generating function of n-gon to (n − 1)-gon and n-gon to (n − 2)gon, respectively, as two non-trivial examples.After summarizing the previous results in Section 6, we provide an explicit expression of the generating function for n-gon to (n − k)gon of universal k.Section 7 offers an inductive proof of our results.Finally, Section 8 provides a brief summary and discussion.Beside, we present the solution of the typical differential equation in Appendix A. We also provide the numerical verification in Appendix B.

Preparation
In this paper, our goal is to provide an explicit expression for the generating functions of one-loop tensor reduction.As we know, by introducing an auxiliary vector R, the general one loop integral in D dimensions can be written as In this formula, n represents the number of propagators, and r denotes the tensor rank number.

Notations
Let us start with the introduction of some notations that we'll be using subsequently.
• Some n-dimension vectors: where 1 is in the b-th position.
• Q is an n × n matrix defined as • The notations (AB) and (AB) b , with a label list b = {b 1 , b 2 , ..., b k } for two vectors A and B, are defined as follows: • If Ω is an analytic expression composed of (AB) or (AB) b , then [Ω] a with a label set a outside the square brackets represents the analytic expression obtained by appending subscript a on each term of the form (AB) or (AB) b present in Ω.For example then • I (2.9)

Recursive relation
After introducing the above notations, we begin to derive the recursive relations for the generating function of the reduction coefficients.As pointed out in [100] and [90], there exists a non-trivial recursion relation for one-loop tensor integrals for r ≥ 11 ), (2.10) where the coefficients are We focus on the summation of the tensor integrals with form (2.12)Then, we multiply both sides of equation (2.10) by t r and sum over r from 1 to ∞.The following relations can be used: (2.13) We obtain a differential function for the generating function φ n (t) : (2.14) We know that any integral can be written as the linear combination of certain irreducible scalar integrals (which are the master integrals in arbitrary spacetime dimension) with coefficients as the rational functions of external momenta, masses and space-time dimension D.
Thus, equation (2.14) is transformed into a recursive formula for the generating functions of the reduction coefficients. where is the reduction coefficient of scalar integral I 3 n-gon to n-gon For the case of reducing an n-gon to an n-gon, that is, b = ∅, we have, C n→n = 1 and the terms inside the curly braces on the right-hand side of equation (2.16) vanish.Consequently, the differential equation for the generating function GF n→n (t) simplifies to: The general solution to this differential equation is2 : where The Generalized Hypergeometric Function is defined as: where we have used the Pochhammer symbol At first glance, one might expect the undetermined constant to be resolved by the initial condition GF n→n (0) = C (0) n→n = 1.However, this doesn't work in our case.So, how should we determine C 1 ?The definition of GF n→n (t) specifies that the generating functions should be represented as a Taylor series of t.Nevertheless, the term cannot be expanded into a Taylor series3 of t.Thus, the constant C 1 = 0. Ultimately, we obtain the generating function of the reduction from an n-gon to an n-gon as follows:

Analytical Results
Upon deriving the formula for the generating function pertaining to the reduction from an n-gon to an n-gon, as delineated in equation (3.7), we subsequently elucidate the corresponding analytical formalism.Substituting (3.3) into (3.7),we can obtain: where (3.10)By expanding the generating function as a series in terms of t, we obtain: where n→n are the reduction coefficients of tensor integral to the irreducible scalar integral (Master Integral) I n .Furthermore, using the Taylor series expansion, )! , we can obtain the coefficients for each order in the series expansion: 4 n-gon to (n − 1)-gon Next, let us consider the case of the (n − 1)-gon.Without loss of generality, we select the Master Integral as follows: In this case, C n→n; a 1 = 0. Then equation (2.16) transforms into: where GF n; a 1 →n; a 1 (t) can be obtained from (3.7).All we need to do is replace all instances of n with (n − 1) in expression (3.7), and replace all instances of (LL), (V L), where the notation as defined in Section 2. The general solution to equation (4.2) is 4 Clearly, the initial condition GF n→n, a 1 (0) = 0 is insufficient to determine the undetermined coefficient C 1 .However, since we require the generating function to be a Taylor series of t, we must have C 1 = 0. 5  The integral in (4.5) cannot be evaluated directly.An intuitive idea would be to first expand the integrand into a series in terms of u, and then integrate.However, fully expanding a hypergeometric function into a series would render the expression overly complex.Therefore, we decide to select a suitable integral formula and perform a series expansion only on part of the integrand.The integral formula we select is (4.6) Then in order to evaluate the integral explicitly, we need to change the integration variable Note that in (4.5) there are terms involving the derivative of the generating function GF n; a 1 →n; a 1 , which requires the use of the following derivative formula for the generalized hypergeometric function Substituting (4.3) in (4.5), after altering the integration variable, the integral part becomes where (4.10) and the terms in the large square brackets can be expanded as a Taylor series of u.The factor u −1+D−n in the integrand will cancel the factor t 1−D+n outside the integrand.This makes the result a Taylor series of t after integrating.
We expand as a Taylor series of x.By formula we have the expansion coefficients as follows: Then the integral part (excluding the constant) becomes: By using integral formula of generalized hypergeometric function(4.6),we can explicitly solve for the generating function of the reduction from an n-gon to an (n − 1)-gon, 5 n-gon to (n − 2)-gon Now, let us discuss the reduction from an n-gon to an (n−2)-gon.Without loss of generality, we select the Master Integral: In this case, C (0) n→n; a 1 ,a 2 = 0. Then recursive relation (2.16) transforms into The general solution of (5.2) is where GF n; a 1 →n; a 1 ,a 2 (t) and GF n; a 2 →n; a 1 ,a 2 (t) can be obtained from (4.15) by (5.4) We select the undetermined constant C 1 = 0 since the generating function should be a Taylor series of t.We can see that in the integrand of expression (5.3), a 1 and a 2 are completely symmetric.We can calculate only one half and obtain the other half by exploiting this symmetry.Moreover, since we are going to use integral formula (4.6) to compute the integral, we can divide GF n; a 1 →n; a 1 ,a 2 (t) and GF n; a 2 →n; a 1 ,a 2 (t) according to the generalized hypergeometric function as follows: where a 1 (n − 1; m 1 )] a 2 can be factored out of the integral sign.By this separation, the integral in equation ( 5.3) can be divided into several parts.We can calculate each part separately and add it up.Next we will use the following part as an example to illustrate the process: (5.7) The main difference from what we did in Section 4 is that we use another derivative formula of the generalized hypergeometric function (5.8) After changing the integration variable from u to the integral (5.7) becomes (5.10) where coefficients are (5.11)Note that labels a 1 , a 2 are no longer symmetric in the above coefficients.By equation (4.12), we can expand the integral (5.10) as follows: (5.12) where the expansion coefficients are m ′ (a 1 ; a 2 ) come from the Taylor expansion of terms like (1 By formula (4.12), the Taylor expansion coefficients are (5.15) Again, we need to emphasize label a 1 , a 2 are not symmetric in N (1) . By using integral formula(4.6),we can calculate the integral (5.7).Eventually, by summing everything up, we obtain the final generating function of n-gon to (n − 2)-gon as 6 Generating function of n-gon to (n − k)-gon From(4.15) and(5.16),we found GF n→n; a 1 ,a 2 (t) has an analogous functional form with GF n→n; a 1 (t).This suggests that the generating function GF n→n; I k (t) (where the label list I k = {a 1 , a 2 , ..., a k }) should also have the same functional form.In this section, we will provide the form of GF n→n; I k (t) directly.A proof by induction will be given in the next section to show it does satisfy the recursion relation (2.10).
(6.1) Now we can construct two types of generalized hypergeometric functions based on the array as follows: where For example, HG 1 (n, 3; {0, 0, 1}; z) (6.4) Then generating function GF n→n; I k is where The first summation in the penultimate line of (6.5) runs over all the permutations of I k = {a 1 , a 2 , ..., a k }.For example for k = 3, this summation should run over 7) It's important to note that the second summation on that line does not include b k .For example for k = 4, this summation runs over without b 4 but For k = 4, {b 1 , b 2 , b 3 } = {0, 1, 0} as an example, the C (6.10)There is a slight difference between the definitions of 0 and 1 when compared to the definitions in formal section (5.13).Here, it includes one more variable k ′ , defined as 11) The variables n ′ , k ′ , m ′ , and sum must be non-negative integers.The variable a can be fixed either a single label or a list of labels, while b solely represents a single label.Upon setting m ′ = m 2 , n ′ = n, k = 1, a = a 1 , b = a 2 , and sum = m 1 , we revert to the form (5.13) as outlined in the previous section.The other coefficients P 1 (a; b), P 2 (a; b), Q 0 (a; b), Q 1 (a; b), N m ′ (a; b) are defined by merely replacing all instances of the labels a 1 and a 2 in (5.11), and (5.15) with variables a and b respectively.We have thus completed the explanation of every term in GF n→n; I k (t) (6.5).Furthermore, if we additionally define C a 1 ∅ ≡ 1 for k = 1, we find that the function (6.5) aligns with GF n→n; a 1 (t)(4.15)and GF n→n; a 1 ,a 2 (t)(5.16).

Triangle to Tadpole
Now we present the reduction of an tensor triangle to tadpole GF 3→3: 2,3 as an example to illustrate the analyticity.The generating function can be obtain from (6.5) by selecting n = 3, k = 2, a 1 = 2, a 2 = 3. Expanding the generating function as a series of t, we can obtain: where are the reduction coefficients of tensor triangle to the master integral (6.14) Moreover, we list the first two non-zero orders in the expansion: (6.15) In addition, for the general reduction of an n-gon to an (n − k)-gon for k ≥ 1, we expand The reduction coefficients are (6.17) The sources of those coefficients are shown as follows: • N (l 1 ) 1 comes from • N 2 (m 1 , ..., m k ) comes from • By (3.13), M where the factor S i ({b 1 , ..., b k−1 }) are defined as

Proof
In this section, we will provide an inductive proof of parameter k to verify the correctness of (6.5).The main methodology echoes the computation of GF n→n; a 1 ,a 2 (t).Suppose that expression (6.5) holds for k.We can evaluate the generating function for reduction of an n-gon to (n−(k +1))-gon by solving the differential equation (2.16).After writing down the general solution, we divided the integration into several parts according to the generalized hypergeometric function.During the steps of integration, we firstly alter the integration variable.Subsequently, we conduct the Taylor expansion on a part of the integrand function.Lastly, by selecting the suitable integration formula(4.6),we affirm that the statement (6.5) holds true for k + 1. Readers who are already familiar with this process may choose to skip this section.The derivation and integration rules that we employ during our proof are outlined in (5.8) and (4.6).In the context of (6.2), these two rules transform into:

) and
Supposing generating function of n-gon to (n−k)-gon GF n→n;I k (t) satisfies (6.5), then recursive relation of GF n→n; I k+1 (t) with I k+1 = {a 1 , a 2 , ..., a k+1 } is The general solution is (7.4) The undetermined constant C 1 is determined to be zero, as the generating function must conform to the requirement of being a Taylor series in terms of t.Where GF n; a i →n; I k+1 (t) = [GF n−1→n−1; I k+1 /a i (t)] a i .We split it into where k ,a i in (7.5) are independent of t, they can be pulled out of the integral.Then solution (7.4) can be separated into Next we focus on the part inside the curly braces.After changing the integration variable u to x I k+1 (u) : and applying derivative formula (7.1), terms inside the big curly braces of (7.7) become (7.9) The coefficients P 1 (a; b), , and B 2 (a; b) are defined by simply replacing all instances of labels a 1 and a 2 in (5.11), (5.15) with the label list {a ′ 1 , ..., a ′ k } and the single label a i , respectively.Then, upon carrying out the series expansion, it transforms into (7.10) According to integration rule (7.2), above equation becomes When everything is added together, we obtain the generating function of an n-gon to an (n − (k + 1))-gon as follows: From the definition of C And obviously, Then, we can see that (7.12) precisely aligns with our result (6.5) in the k + 1 case.Therefore, we have successfully validated the correctness of our result.

Conclusion
In this paper, we present an explicit expression for the generating function for the reduction of an n-gon to an (n − k)-gon, where k is a general value.We formulate a novel recursive relation of generating functions, which is based on our investigation into Feynman Parametrization in projective space.This newly established relation comprises a single ordinary differential equation in terms of variable t.To solve this equation, it is required to carry out the integral part by the following steps: (1) Changing the integration variable.( 2) Implementing the Taylor expansion on the appropriate part of the integrand.(3) Selecting the suitable integration formula.In the end, we unearthed the rule of the general term formula of generating functions.
In addition, there are several comments that we need to supplement.Firstly, it is understood that the reduction coefficients should be rational functions of some Lorentz invariants such as q i • q j , M 2 i and spacetime dimension D. However, in the expression (3.7) and (6.5), there exists irrational terms such as the square root term In fact, by the following quadratic transformation equation (3.7) can be changed to a rational form We can observe that all terms in the above expression are rational terms.Therefore, even though the generating function we obtain contains irrational terms, it does not alter the fact that the final reduced coefficients remain rational. 7xpressing rational numbers with irrational numbers is not unusual in mathematics, with the general term formula of the famous Fibonacci sequence being an example This may suggest that in the existing work, the study of the analytic structure of the master integrals or reduction coefficients is perhaps not yet deep enough.
In numerical computations on computers, maintaining a function in rational form allows us to preserve the accuracy of the entire numerical calculation through methods such as Finite Field Reconstruction, ensuring complete precision.When irrational terms are present, we are forced to use floating-point numbers, which can lead to increased computational time and a reduction in precision.However, based on the definition of coefficient {b 1 ,...,b k−1 } (n) (6.9) and the formula of (2.11)(4.10)(4.13)(5.11), it can be seen that the only irrational term appearing in the generating function (6.5) from an n-gon to an (n−k)-gon is (8.1).Therefore, by extending the rational number field Q to the field of form Q + √ X • Q, we can similarly employ finite field reconstruction methods to maintain computational accuracy and save computational time.Nonetheless, we maintain our belief that there may exist a rationalized analytical representation for the generating function of one-loop tensor integral.Investigating methods to derive such a rationalized expression is a meaningful avenue for future research.
Secondly, the generating function we discuss in this paper is in any spacetime dimension D where all scalar integrals with single pole are irreducible.When the spacetime dimension D takes a finite value (such as 4 − 2ǫ), some scalar integrals will become reducible scalar integrals, then what would the form of the generating functions be in this case?Additionally, this paper focuses on one-loop tensor integrals that solely encompass single poles.As for one-loop integrals involving higher poles, what would be the corresponding generating functions for reduction coefficients?We have already computed some results, which will be presented in future articles.
Thirdly, the simplicity of the method in this paper lies in the fact that the recursion relation established is an ordinary differential equation rather than a set of partial differential equations.The foundation of this relation is the study of the analytic structure of Feynman Parameterization in projective space.Could this logic be extrapolated to two loops, thus obtaining the general form of the two-loop generating function, or at least, obtain some simpler recursion relations?In [95], Kosower focus on two-loop integrals, where he only deriving generating functions for the reduction coefficients associated with the most intricate master integral.His approach is applied to only a limited set of concrete examples.In contrast, our goal is to offer a comprehensive and systematic treatment encompassing all generating function for general two-loop integrals.For the case of two-loop diagrams, we have undertaken some initial attempts and obtained preliminary results, which will be presented in future work.Nevertheless, it remains a significant challenge.
We split the integral of p(x) into Then the general solution of the homogeneous part is By (A.2), the solution of the original equation is In the case of the reduction of an n-gon to an n-gon, the non-homogeneous part is a constant g(x) = (a − 1).By following integral formula: (A.9) we can solve y(x) = e − p(x) dx q(x)e p(x) dx dx where x − , x + are two roots of the quadratic equation

Appendix B Numerical check
In this section we will verify our generating function numerically by comparing to the result produced by the cpp version of FIRE6 [48].The tensor integral we reduce is defined as follows:

B.1 Tadpole
Let us first begin with a trivial Tadpole case, where we have only one master integral.We choose the numeric value of the kinetic variable as follows: • Tadpole to Tadpole The numerical Taylor expansion of the generating function is given as follows: which match the result of FIRE6 perfectly. 8We can also choose x+ = . In this way, we will have another form of solution.Two forms are equivalent due to the properties of hypergeometric functions.

B.2 Bubble
We choose the numeric value of the kinetic variable as follows: The numerical Taylor expansion of the generating function is given as follows:

B.3 Triangle
We choose the numeric value of the kinetic variable as follows: , q 1 • q 2 → 0, q 1 • q 3 → 0, The numerical Taylor expansion of the generating function is given as follows:

B.4 Box
We choose the numeric value of the kinetic variable as follows:

B.5 Pentagon
We choose the numeric value of the kinetic variable as follows:

. 4 )
In this context, A b and B b are two vectors derived by removing all the b i -th components from vectors A and B. Q b b is a matrix obtained by eliminating all the b i −th rows and b i −th columns from matrix Q.For example, with n = 4 and b = {2, 3}, we have -loop integral where all propagators of I (r) n with index b i ∈ b are removed

n
is an irreducible scalar integral, the reduction coefficient C (0) n→n; b is 1 if b is the null set, i.e, b = ∅ and C (0) n→n; b = 0 otherwise.GF n; b i →n; b (t) in the right hand side of (2.16) is the generating function for the reduction coefficient of φ n; b i (t) to Master Integral I (0) n; b .The equation (2.16) serves as the cornerstone of this paper, drawing a connection between the generating functions in the reduction of an n-gon to an (n − k)-gon and an (n − 1)-gon to an ((n − 1) − (k − 1))-gon.At first, if we set b = ∅, the terms inside the curly braces on the right-hand side of (2.16) vanish.The remaining part is a first-order non-homogeneous ordinary differential equation with respect to the generating function GF n→n (t).By solving this differential equation, we obtain analytic expression of the generating function for an ngon to an n-gon.Once we have obtained the analytic expression for the generating function for the reduction of an n-gon to (n − (k − 1))-gon, due to the arbitrariness of n, we can also know the analytic expression for an (n − 1)-gon to ((n − 1) − (k − 1))-gon (which is what appears on the right side of the equation).Then, the equation (2.16) becomes a first-order non-homogeneous differential equation for the generating function of an n-gon to (n − k)-gon.Continuously repeating this process, we can calculate all the generating functions.

Table 2 .
Reduction Result of Bubble to Bubble by FIRE6

Table 4 .
Reduction Result of Bubble to D 1 Tadpole by FIRE6

Table 6 .
Reduction Result of Triangle to D 2 D 3 Bubble by FIRE6