GKZ-hypergeometric systems for Feynman integrals

Tai-Fu Feng, Chao-Hsi Chang, Jian-Bin Chen , Hai-Bin Zhang Department of Physics, Hebei University, Baoding, 071002, China Hebei Key Laboratory of High-precision Computation and Application of Quantum Field Theory, Baoding, 071002, China Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Science, Beijing, 100190, China CCAST (World Laboratory), P.O.Box 8730, Beijing, 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Department of Physics, Taiyuan University of Technology, Taiyuan, 030024, China and Department of Physics, Chongqing University, Chongqing, 401331, China Abstract Basing on the systems of linear partial differential equations derived from Mellin-Barnes representations and Miller’s transformation, we obtain GKZ-hypergeometric systems of one-loop self energy, one-loop triangle, two-loop vacuum, and two-loop sunset diagrams, respectively. The codimension of derived GKZ-hypergeometric system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. Taking GKZ-hypergeometric systems of one-loop self energy, massless one-loop triangle, and two-loop vacuum diagrams as examples, we present in detail how to perform triangulation and how to construct canonical series solutions in the corresponding convergent regions. The series solutions constructed for these hypergeometric systems recover the well known results in literature.


I. INTRODUCTION
A central target for particle physics now is to test the standard model (SM) and to search for new physics (NP) beyond the SM [1][2][3] after the discovery of the Higgs boson [4,5].
In order to predict the electroweak observables precisely with dimensional regularization [6,7], one should evaluate the Feynman integrals exactly in the time-space dimension D = 4−2ε at first. Nevertheless each method presented in Ref. [8] has its blemishes since it can only be applied to the Feynman diagrams with special topologies and kinematic invariants.
It was proposed long ago to consider Feynman integrals as the generalized hypergeometric functions [9]. Certainly Feynman integrals satisfy indeed the systems of holonomic linear partial differential equations (PDEs) [10] whose singularities are determined by the Landau singularities. Recently the author of Ref. [11] shows that the D−module of a Feynman diagram is isomorphic to Gel'fand-Kapranov-Zelevinsky (GKZ) D−module [12][13][14][15][16].
Some Feynman integrals are already expressed as the hypergeometric series in the corresponding parameter space. In Ref. [17] the massless C 0 function is presented as the linear combination of the fourth kind of Appell function F 4 whose arguments are the dimensionless ratios among the external momentum squared, and is simplified further as the linear combination of the Gauss function 2 F 1 through the quadratic transformation [18] in the literature [19]. With some special assumptions on the virtual masses, the analytic expressions of the scalar integral C 0 are given by the multiple hypergeometric functions in Ref. [20] through the Mellin-Barnes representations. Taking the massless C 0 function as an example, the author of Ref [21] presents an algorithm to evaluate the scalar integrals of one-loop vertex-type Feynman diagrams. Certainly, some analytic results of the C 0 function can also be extracted from the expressions for the scalar integrals of one-loop massive N−point Feynman diagrams [22,23]. The Feynman parametrization and Mellin-Barnes contour integrals can be applied to evaluate Feynman integrals of ladder diagrams with three or four external lines [24]. In addition, the literature [25] also provides a geometrical interpretation of the analytic expressions of the scalar integrals from one-loop N−point Feynman diagrams.
Using the recurrence relations respecting the time-space dimension, the papers [26,27] formulate one-loop two-point function B 0 as the linear combination of the Gauss function 2 F 1 , one-loop three-point function C 0 with arbitrary external momenta and virtual masses as the linear combination of the Appell function F 1 with two arguments, and one-loop four-point function D 0 with arbitrary external momenta and virtual masses as the linear combination of the Lauricella-Saran function F s with three arguments, respectively. The expression for the scalar integral C 0 is convenient for analytic continuation and numerical evaluation because continuation of the Appell functions has been analyzed thoroughly. Nevertheless, how to perform continuation of the Lauricella-Saran function F s outside its convergent domain is still a challenge. Expressing the relevant Feynman integral as a linear combination of generalized hypergeometric functions in dimension regularization, the authors of Ref. [28] analyze the Laurent expansion of these hypergeometric functions around D = 4. The differential-reduction algorithm to evaluate those hypergeometric functions can be found in Refs. [29][30][31][32][33]. A hypergeometric system of linear PDEs is given through the corresponding Mellin-Barnes representation [34], where the system of linear PDEs is satisfied by the Feynman integral in the whole parameter space of the independent variables. Some irreducible master integrals for sunset and bubble Feynman diagrams with generic values of masses and external momenta are explicitly evaluated via the Mellin-Barnes representation in Ref. [35].
Taking some special assumptions on the virtual masses and external momenta, Ref. [36] presents some GKZ-hypergeometric systems of Feynman integrals with codimension= 0 or codimension= 1 through the Lee-Pomeransky parametric representations [37]. Using the triangulations of the Newton polytope of the Lee-Pomeransky polynomial, the authors of Ref. [38] present GKZ-hypergeometric system of the sunset diagram of codimension= 6. They also construct canonical series solutions which contain three redundant variables besides three independent dimensionless ratios among the external momentum squared p 2 and three virtual mass squared m 2 Actually it is a common defect of GKZ-hypergeometric systems originating from the Lee-Pomeransky polynomial of the corresponding Feynman diagrams that codimension is far larger than the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. To construct canonical series solutions with suitable independent variables, one should compute the restricted D-module of GKZ-hypergeometric system originating from the Lee-Pomeransky representations on corresponding hyperplane in the parameter space [39][40][41].
Some holonomic systems of linear PDEs are given through Mellin-Barnes representations of concerned Feynman integrals in Refs. [34,42,43]. Performing the Miller's transfor-mation [44,45], one derives GKZ-hypergeometric system of concerned Feynman integrals, whose codimension equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. Using those holonomic systems given in Refs. [34,42,43], we present here relevant GKZ-hypergeometric systems for Feynman integrals of one-loop self-energy, two-loop vacuum, two-loop sunset, and one-loop triangle diagrams. Taking Feynman integrals of one-loop self-energy, two-loop vacuum, and massless one-loop triangle diagrams as examples, we illuminate how to construct canonical series solutions from those relevant GKZ-hypergeometric systems [46], and how to derive the convergent regions of those series with Horn's study [47]. To shorten the length of text, we don't state those mathematical concepts and theorems that have been used in our analyses here, because they can be found in some well-known mathematical textbooks [46,[48][49][50][51][52][53][54][55].
Basing on Mellin-Barnes representations of one-loop Feynman diagrams or those multiloop diagrams with two vertices, we can derive GKZ-hypergeometric systems through Miller's transformation, whose codimension of GKZ-hypergeometric system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. Nevertheless for generic multiloop Feynman diagrams such as that presented in Refs [56,57], the corresponding codimension of GKZ-hypergeometric system derived is far larger than the number of independent dimensionless ratios, whether using Mellin-Barnes or Lee-Pomeransky representations. In order to construct canonical series solutions properly, we constrain corresponding GKZ-hypergeometric system on restricting hyperplane in the parameter space.
The generally strategy for analyzing Feynman integral includes three steps here. First we obtain the holonomic system of linear PDEs satisfied by corresponding Feynman integral through its Mellin-Barnes representation, next find GKZ-hypergeometric system via Miller's transformation, and finally construct canonical series solutions. The integration constants, i.e. the combination coefficients, are determined from the Feynman integral with some special kinematic parameters. To make the analytic continuation of those canonical series solutions from their convergent regions to the whole parameter space, one can perform some linear fractional transformations among the complex variables.
Our presentation is organized as following. Through Miller's transformation, we derive GKZ-hypergeometric systems of Feynman integrals of one-loop self energy, massless one-loop triangle, and two-loop vacuum diagrams by using the holonomic systems of linear PDEs in Refs. [34,42,43] in section II. Then we present in detail how to perform triangulation and how to construct canonical series solutions from those GKZ-hypergeometric systems in section III. Actually some well-known results are recovered with the approach presented here. In section IV, we present GKZ-hypergeometric systems for the sunset diagram with three differential masses, C 0 function with one nonzero virtual mass, and C 0 function with three differential virtual masses, respectively. The conclusions are summarized in section V. Adopting the notation in Ref. [42,43], we write the scalar integrals of one-loop self energy, massless one-loop triangle, and two-loop vacuum diagrams respectively as Generally those holonomic systems presented above originate from the Mellin-Barnes representations of corresponding Feynman integrals [34,42,43]. Using the systems of linear PDEs in Eq.
(2), one derives the following relations between f i (i = B, C, V ) and their contiguous functions (θ x i +θ y i + a i )f i the equations in Eq. (6) are changed as where Correspondingly the universal Gröbner basis of the toric ideal associated with A is The operators from Eq. (10) and Eq. (12) compose the generators of a left ideal [51] in the Weyl algebra D = C z i,1 , · · · , z i,6 , ∂ z i,1 , · · · , ∂ z i, 6 where C denotes the field of complex numbers [55]. Defining the isomorphism between the commutative polynomial ring and the Weyl algebra [46] Ψ : C[z i,1 , · · · , z i,6 , ξ i,1 , · · · , ξ i,6 ] → D, one obtains the state polytope [53] of the preimage of the universal Gröbner basis in Eq. (12) as on the hyperplane In Eq. (13) we take multi-index notation for abbreviation, i.e.
where α, β ∈ N 6 , and N = {0, 1, 2, · · ·} denotes the set of non-negative integers. The normal fan of the state polytope in Eq. (14) is the Gröbner fan of the left ideal generated by the operators in Eq. (10) and Eq. (12). Because codimension= 2 for all GKZ-hypergeometric systems, the Gröbner fan equals the hypergeometric fan. These two fans are indispensable in the construction of canonical series solutions of corresponding GKZ-hypergeometric systems.

A. Triangulation
The following matrix G A is a Gale transform of A in Eq. (11): whose column vectors compose the secondary fan Σ A of GKZ-hypergeometric system in Eq. (10). Actually the state polytope in Eq. (14) of universal Gröbner basis indicates that the hypergeometric fan H A and the Gröbner fan G A all equal the secondary fan of GKZhypergeometric system here: with e 1 = (1, 0) T , e 2 = (0, 1) T . The cones are defined as [52] Cone({e 1 , e 2 }) = {λ 1 e 1 + λ 2 e 2 | λ 1 , λ 2 ∈ R + } , where R + denotes the non-negative reals.
For a generic weight vector ω ∈ Cone({e 1 , e 2 }), the corresponding triangulation [53] △ ω = {σ a 1 , σ a 2 , σ a 3 , σ a 4 } is unimodular, and supports the toric ideal which corresponds to the initial monomial ideal in Certainly four standard pairs [46] (1, σ a j ), (j = 1, 2, 3, 4) produce the following exponent vectors of initial monomials of series solutions For a generic weight vector ω ∈ Cone({e 2 , −e 1 − e 2 }), the corresponding triangulation is also unimodular, and supports similarly the toric ideal which corresponds to the initial monomial ideal in Correspondingly four standard pairs (1, σ b j ), (j = 1, 2, 3, 4) induce the following exponent vectors of initial monomials for series solutions Finally for a generic weight vector ω ∈ Cone({e 1 , −e 1 − e 2 }), the corresponding triangu- is unimodular, and supports the toric ideal which corresponds to the initial monomial ideal in Correspondingly four standard pairs (1, σ c j ), (j = 1, 2, 3, 4) give the following exponent vectors of initial monomials for series solutions The secondary fan of GKZ-hypergeometric system in Eq. (10), ω in each cone is a representative weight vector.

B. Construction of canonical series solutions
The integer kernel ker Z (A) of the matrix A is defined as The vector u ∈ ker Z (A) can be decomposed into positive and negative part, where u + and u − are non-negative vectors with disjoint supports. In order to construct canonical series solutions of GKZ-hypergeometric system, we define the negative support of Furthermore we introduce the following subset of ker Z (A) With an exponent vector p of the initial monomial, the corresponding canonical series solution of the hypergeometric system Eq. (10) is well-defined: where the abbreviations For one-loop self energy and two-loop vacuum, p σ a Using Eq. (32), one derives Then where F 4 denotes the fourth Appell function [49]. Similarly we have For the fourth Appell function (a) n 1 +n 2 (b) n 1 +n 2 the adjacent ratios of the coefficients are To investigate the absolutely and uniformly convergent region of the double series in Eq.(38), one takes with t = n 2 /n 1 , r x = |x|, r y = |y|, respectively. The generator of the principal ideal where C[t x , t y ] denotes the polynomial ring of t x , t y on the field C. Since t x , t y ≥ 0 the equation g(t x , t y ) = 0 gives the Cartesian curve of the double power series in Eq.(38) as • For |x i | ≤ 1, |y i | ≤ 1, the Feynman integral is which is convergent in the region |x i | + |y i | < 1.
• For |x i | ≥ 1, |y i | ≤ 1, the Feynman integral is which is convergent in the region 1 + |y i | < |x i |.
• For |x i | ≤ 1, |y i | ≥ 1, the Feynman integral is which is convergent in the region 1 + |x i | < |y i |.
In order to determine those integration constants, i.e. the combination coefficients A σ,i , B σ,i , C σ,i , D σ,i with σ = a, b, c and i = B, V , we utilize expressions of the Feynman integrals at some special points of the parameter space. For the Feynman integral of one-loop self energy diagram B 0 (p 2 , m 2 1 , m 2 2 ), we employ the following expressions Using above expressions, one derives the combination coefficients as In a similar way, the combination coefficients involved in the Feynman integral of the twoloop vacuum diagram are written as For the triangulation △ ω = {σ a 1 , σ a 2 , σ a 3 , σ a 4 } of the massless one-loop triangle diagram, the canonical series solutions are constructed as Similarly the canonical series solutions corresponding to the triangulation △ ω = which is convergent in the region |x C | + |y C | < 1.
Using the expressions of Feynman integral of the massless triangle diagram at some special kinematic points (Λ 2 , Λ 2 , 0), (Λ 2 , 0, 0) etc, one obtains those integration constants as Actually the Feynman integrals presented here can be written in terms of the Gauss function 2 F 1 [19] by the well-known reduction of the Appell function of the fourth kind [18], then the analytic continuation of those Feynman integrals is made to the whole parameter space through the transformations of the Gauss functions.

IV. GKZ-HYPERGEOMETRIC SYSTEMS OF OTHER FEYNMAN INTEGRALS
A. Sunset diagram with three differential masses In order to make the notation less cluttered, we adopt the multi-index convention [55], and write the Feynman integral of the two-loop sunset diagram as with the multi-index notations a = (a 1 , Certainly the dimensionless function T p 123 complies with the third Lauricella's system of linear PDEs [42,43] x i + a 2 ) T p 123 = 0 , (k = 1, 2, 3) . (57) Using the system of linear PDES presented in Eq. (57), one derives the following relations between T p 123 and its contiguous functions as Where n 2,j ∈ R 2 , (j = 1, 2) denotes the row vector whose entry is zero except that the j−th entry is 1, and the row vector n 2 = (1, 1). In addition n 3,k ∈ R 3 , (k = 1, 2, 3) denotes the row vector whose entries are zero except that the j−th entry is 1 and n 3 = (1, 1, 1). To proceed with our analysis, we define the auxiliary function as Where the row vectors u = (u 1 , u 2 ) ∈ R 2 , v = (v 1 , v 2 , v 3 ) ∈ R 3 , and the multi-index Obviously there are the relationŝ In addition, the contiguous relations of the function defined in Eq. (59) are given aŝ where the operatorsÔ i n (n = 1, · · · , 8) arê Those operators above together withθ u j ,θ v k define the Lie algebra of the hypergeometric system [44,45] in Eq. (57). Through the transformation of indeterminates the equations in Eq. (60) are changed as where , ϑ T = (ϑ z 1 , ϑ z 2 , ϑ z 3 , ϑ z 4 , ϑ z 5 , ϑ z 6 , ϑ z 7 , ϑ z 8 ) , Correspondingly the universal Gröbner basis of the toric ideal associated with A ⊖ is The operators from Eq. (64) and Eq. (66) compose the generators of a left ideal in the Weyl algebra D = C z 1 , · · · , z 8 , ∂ z 1 , · · · , ∂ z 8 . Defining an isomorphism between the commutative polynomial ring and the Weyl algebra [46] Ψ : one obtains the state polytope [53] of the preimage of the universal Gröbner basis in Eq. (66) on the hyperplane ξ 1 = ξ 2 , ξ 3 = ξ 6 , ξ 4 = ξ 7 , The  [43,58,59].
In order to make analytic continuation of Lauricella functions from their convergent regions to the whole parameter space, we should perform some linear fractional transformations among the complex variables z 1 , · · · , z 8 . We will release our calculation results further elsewhere.

B. C 0 function with one nonzero mass
In this case, the scalar integral where the row vectors a = (a 1 , a 2 , respectively. Additionally the parameters a 1 = 4 − D, a 2 = 3 − D/2, a 3 = 1, b 1 = b 2 = 3 − D/2, p 2 3 = (p 1 + p 2 ) 2 , and the dimensionless ratios ξ 33 = −m 2 /p 2 3 , x ij = p 2 i /p 2 j , (i, j = 1, 2, 3). The dimensionless function F a,p 3 satisfies the holonomic hypergeometric system of linear PDEs Defining the auxiliary function where k = 1, 2, 3 and j = 1, 2, respectively. In addition, the contiguous relations of the auxiliary function are given aŝ where the operatorsÔ i n (n = 1, · · · , 8) are defined aŝ Certainly one can calculate the state polytope [53] corresponding to the universal Gröbner basis U Aa whose normal fan coincides with the Gröbner fan, then construct canonical series solutions in the convergent regions which are presented in Ref. [43]. In order to perform the analytic continuation of canonical series solutions from the convergent regions to the whole parameter space, one utilizes some linear fractional transformations among the complex variables z a 1 , · · · , z a 8 , then chooses u k = 1, v j = 1 (k = 1, 2, 3, j = 1, 2) finally.
V. SUMMARY

Using the system of linear PDEs satisfied by the corresponding Feynman integral in
Refs [42,43], we present GKZ-hypergeometric systems of one-loop self energy, one-loop triangle, two-loop vacuum, and the two-loop sunset diagrams, respectively. In those GKZhypergeometric systems the codimension equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared.
Actually one can derive GKZ-hypergeometric systems from Mellin-Barnes representations for the one-loop Feynman diagrams and those multiloop diagrams with two vertices, whose codimension equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. Nevertheless for the generic multiloop Feynman diagrams, the corresponding codimension of GKZ-hypergeometric system is far larger than the number of independent dimensionless ratios, whether using Mellin-Barnes or Lee-Pomeransky representations. In order to construct canonical series solutions properly, one should constrain GKZ-hypergeometric system on restricting hyperplane in the parameter space.
Taking GKZ-hypergeometric systems of one-loop self energy, one-loop massless triangle, and two-loop vacuum diagrams as examples, we present in detail how to perform triangulation and how to construct canonical series solutions in the corresponding convergent regions.
The analytic continuation of those series solutions is performed through some well known reduction of the Appell function of the fourth kind. In order to make analytic continuation of those series solutions of GKZ-hypergeometric systems of the massive sunset and massive one-loop triangle diagrams etc., one can perform the linear fractional transformations among the complex variables.
One of the techniques not involved here is how to project GKZ-hypergeometric system to the restricting hyperplane. Another calculation not contained here is how to make analytic continuation of those canonical series solutions from their convergent regions to the whole parameter space through linear fractional transformation among the complex variables. Algorithm for the first problem has been presented in literature already, and the second problem is attributed to a problem of integer programming [60] in principle. We will release our results relating to those topics in near future elsewhere.