Polytope symmetries of Feynman integrals

Feynman integrals appropriately generalized are $\mathsf A$-hypergeometric functions. Among the properties of $\mathsf A$-hypergeometric functions are symmetries associated with the Newton polytope. In ordinary hypergeometric functions these symmetries lead to linear transformations. Combining tools of $\mathsf A$-hypergeometric systems and the computation of symmetries of polytopes, we consider the associated symmetries of Feynman integrals in the Lee-Pomeransky representation. We compute the symmetries of $\mathtt n$-gon integrals up to $\mathtt n=8$, massive banana integrals up to 5-loop, and on-shell ladders up to 3-loop. We apply these symmetries to study finite on-shell ladder integrals up to 3-loop.


I. INTRODUCTION
The evaluation of Feynman integrals is a fundamental problem in perturbative quantum field theory that is deeply connected with modern methods in mathematics [1].Among those is the recognition that they can be evaluated in terms of generalized hypergeometric series known as A-hypergeometric functions, which were proposed by Gel'fand-Kapranov-Zelevinsky(GKZ) as a unified approach to hypergeometric functions [2].The evaluation of Feynman integrals using GKZ is done by restricting the general solution of the system of differential equations that a Feynman integral satisfies when the coefficients of the Symanzik polynomials are promoted to be undetermined [3,4] .The evaluation can be done using two equivalent approaches, namely by constructing triangulations of the Newton polytope of the sum of the Symanzik polynomials or by the Gröbner deformation approach [5].These methods have been recently automated in a package [6,7].
The matrix of exponents of the sum of Symanzik polynomials, encoded in a matrix A, is key in various aspects of the approach because its convex hull defines the Newton polytope.Its (regular) triangulations give valid solutions to the system of differential equations and, in the case where the parameters are generic, its volume determines the rank of the system of differential equations.Moreover, the Newton polytope determines the convergence domain of a generic Feynman integral [8].It is also useful in numerical evaluations of Feynman integrals [9][10][11] and Landau analysis [12][13][14].
In this paper, we will apply the GKZ approach to study permutation symmetries of the Newton polytope of Feynman integrals encoded in A. Permutation symmetries and the conditions in which they hold were studied for A-hypergeometric functions by Forsgård-Matusevich-Sobieska in Ref. [15].They correspond to linear transformations of hypergeometric functions [16].As we will see, permutation symmetries may in general shift the kinematic dependence from the second Symanzik polynomial to the first and vice versa, and may also permute the kinematic dependence of it between the monomials.This * leonardo.delacruz@ipht.frchange is compensated by a modification of the powers of the propagators.
The remainder of this paper is organized as follows.In Section II, we review generalized Feynman integrals.In Section III, we present the mathematical methods and give a simple example of their applications.In Section IV, we present our results and apply them in Section V to the case of on-shell ladder integrals.Our conclusions are presented in Section VI.

II. GENERALIZED FEYNMAN INTEGRALS
We will consider a Feynman integral in Euclidean space in dimensional regularization.A L-loop integral with N propagators and E independent external momenta may be written as where the denominators have the form The matrices, M i , Q i , and J i have dimensions L × L, L × E, and 1 × 1, respectively.The exponents have been collected into the vector α = (α 1 , . . ., α N ).The parametric representations of Eq.( 1) are expressed in terms of the so-called Symanzik polynomials U and F. These can be computed from the matrices so the Symanzik polynomials are The polynomial U is a homogeneous polynomial in z of degree L, while F is homogeneous of degree L + 1 [17].
In Euclidean kinematics, U, F are positive semi-definite functions of the Feynman parameters.These polynomials can also be obtained from the topology of the graphs.
Their properties are summarized in Refs.[18,19].The Lee-Pomeransky representation is given by [20] where we have used the multi-index notation and R + = (0, ∞).The overall factor and the polynomial G(z) are where x ij... := x i + x j + . . .From now on, we will focus on the integral stripped from the kinematic-independent gamma factors, namely I G (α, β) := I LP (α)/ξ Γα with β := d/2.The polynomial G(z) may be written as follows where the sum runs over all monomials in G and a i ∈ N N are the exponent vectors of each monomial.The kinematic content of the integrals is in the coefficients c i .
Let us denote by n the total number of monomials and consider the (N + 1) × n matrix of exponents of G, which we write as The corresponding coefficients can be grouped in a vector c = (c 1 , . . ., c n ).The Newton polytope of Newton(A) = conv(A) is defined as the convex hull of the columns of the matrix A. Promoting the coefficients in the polynomial G(z) to be indeterminate leads to the generalized Feynman integral [3,4] where κ := −(d/2, α) and Ω is some cycle of integration.The coefficients c in Eq.( 8) are now variables so we have indicated explicitly it writing G(c, z).The first row in ( 9) represents an overall factor z 0 in G, whose integral is taken over S 1 and set to unity.The integral ( 10) is a solution of the A-hypergeometric system of differential equations: where θ j := c j ∂/∂c j .This system is usually denoted by H A (κ).For generic parameters κ, the rank of the system satisfies the inequality [5] rank where Vol(Newton(A)) is the normalized volume of the Newton polytope [21].The Euler integral ( 10) is a special case of the integrals studied by GKZ [2].More generally, Feynman integrals in parametric representation are special instances of the so called Euler-Mellin integrals and their transition to A hypergeometric functions was discussed in Ref. [8], where non-compact cycles of integration were constructed.These are the appropriate cycles to study Feynman integrals.On the other hand, compact cycles are useful to study maximal cuts [22].The evaluation of these integrals is done using series solutions of the system of PDEs upon restriction [3,4], see also [23].

III. MATHEMATICAL METHODS
We are interested in symmetries of a Feynman integrals, which encode symmetries of the lattice polytope Newton(A).Determining the symmetries of polytopes is a fundamental problem in polyhedral computations, see e.g, the review [24].

A. Normal form of a polytope
The information about the monomials present in G is recorded in A so the order of the columns is irrelevant as it defines the same system of PDEs.However, for computations a convenient way of deducing the symmetries of a polytope is to choose a special ordering known as the normal form of the polytope.We say that A is determined only up to a S n ×GL(N, Z) symmetry, where S n is the symmetry group of permutations of the vertices and GL(N, Z) is the group of coordinate transformations of a N -dimensional lattice [25].An algorithm to obtain the normal form was used in Refs.[25,26] to classify reflexive polyhedra in 3 and 4 dimensions and later implemented in PALP [27].A detailed description of the (PALP) normal form that we will use in this work can be found in Appendix B of Ref. [28].

B. Two results by Forsgård-Matusevich-Sobieska (FMS)
Let ϕ be a monomial automorphism associated with A of the form ϕ(z i ) = z ti for i = 1, . . ., N + 1, where t i ∈ Z N +1 .Let T denote the matrix whose columns are the exponent vectors t i , T = (t 1 , . . ., t N +1 ).Applied on the monomials of G(c, z), ϕ induces a permutation on A so that TA = AP, which encodes a polytope symmetry of Newton(A).The following statements are proved in Ref. [15]: where cP and Tκ denote the standard matrix multiplication.
Theorem 4.4: Let F (κ, c) be an A-hypergeometric function for which there exists a transformation valid for generic parameters κ; where P is a permutation matrix and R(κ) is a constant with respect to c. Then TA = AP.That is, P encodes a polytope symmetry of A A few comments are in order.The second statement is only a partial converse of the first one since not all polytope symmetries induce an automorphism of A. For Feynman integrals the factors in Eq.( 14)-( 15) are unity.
The cycle of integration of the Euler-type integral and they are invariant under ϕ up to homotopy.
The calculation of the full set of symmetries of a general Feynman integral is of course a daunting task.Permutation symmetries grow factorially with the number of monomials in the Symanzik polynomials.The most general application of FMS is to answer the question of whether certain permutation is a symmetry or not.A direct search can be done by choosing some permutation matrix, say P try , and solve a linear system for T, namely If the system has a solution the corresponding pair is a symmetry of the polytope.The symmetry thus computed is a symmetry of the generalized Feynman integral and it is a property of the Newton polytope.In the examples below, we will restrict our direct searches to cases with up to 11 monomials.When the number of monomials is higher we use PALP [27,29] by first computing the normal form of A and identifying the permutations that leave the polytope invariant.One then solves Eq.( 16) for T.
Here the empty cycle () represents the permutation where all are fixed points.Their corresponding T matrices are   where 1 i the i × i identity matrix.The set of matrices form a group under matrix multiplication and thus we have included the identity matrix in the counting as we will do in the rest of the paper.The monomial automorphisms of A associated, say, with T 7 are The pair (T 7 , P 7 ) encodes an identity for the generalized Feynman integral, specifically where ).On the restricted integral this corresponds to the identity Notice that the transformation has moved the mass dependence to one of the monomials of U and compensated it by shifting the powers of z 1 , z 2 in the integrand.It is easy to verify that the second integral evaluates to which agrees with the result of the first integral upon using an identity of the Gauß hypergeometric function 2 F 1 [3].The matrices (19) and the permutations matrices of (18) imply that there are six additional representations of the Feynman integral that can be obtained using Eq.( 14).
We have checked numerically that they indeed match.
IV. RESULTS

A. Bananas
We will consider now the L-loop fully massive banana graphs (a.k.a.sunset graphs) shown in Fig. 1.We parametrize their denominators as . . ., and label the integrals by their denominators masses.In the above order a massless propagators is recorded as 0 and 1 otherwise.A L-loop banana integral is then labeled by a binary word w of length |w| = L + 1.For example, w = 10101 labels a 4-loop integral where D 2 and D 4 are massless.The results of the calculations are summarized in Table I.
Comparing the results at one loop we notice that there is only one symmetry for w = 11.This corresponds to the relabeling z 1 ↔ z 2 , α 1 ↔ α 2 .We could also interpret this relabeling as the exchange m 1 ↔ m 2 , which leaves the polynomial invariant and it is also a symmetry of the result when expressed as a linear combination of two 2 F 1 [30].Thus there are less symmetries than in the case w = 10.These are inherited from the symmetries of the Gauß hypergeometric function whose group of matrices is also generated by eight matrices.
At higher loops we find that the only symmetries present for the massive bananas are due to relabeling.We have performed direct searches up to 2-loop for the fully massive case and up to 4-loop in the 1-mass case.A direct search is of course prohibitive already at 3-loop for w = 1111 where there are 17! permutation-matrices to test.Instead, we have computed the number of symmetries with PALP and explicitly checked through Eq.( 16) that the permutation matrices that lead to symmetries are relabelings.We have checked this up to 5-loop.

V. THE EVALUATION OF TRANSFORMED FEYNMAN INTEGRALS
As an application we consider the evaluation of the transformed ladder integrals in d = 6 where they are finite.Let us consider first the generalized box integral In the restricted case c = (1, 1, 1, 1, s, t) and Ω = R 4 + .We find that in this case there is a group of 72 polytope symmetries.Let us consider for example the pair ) which leads to the identity where κ ′ = −(β, α 2 , −α 12 + β, −α 34 + β, α 3 ).Restricting the integral and setting d = 6, we have κ = −(3, 1, 1, 1) and I G = I G ((t, 1, 1, s, 1, 1), κ).The calculation of the remaining cases leads to integrals of the form I G (cP, Tκ) = I G (cP, κ).Since the product Tκ is invariant, out of the 72 symmetries there are only 4 distinct polynomials G produced by cP.
The double box can be treated along the same lines.There are 8 polytope symmetries whose permutation cycles are given in Eq. (28).The integral converges for In tune with the one loop case, we find for all cases.
We have checked numerically that the resulting integrals for the box and double box agree.At 3-loop the symmetries leave κ invariant and thus the above property is satisfied trivially.

VI. CONCLUSIONS
In this paper we have computed the symmetries of the Newton polytope of a Feynman integrals.On the restricted integral these symmetries represent new equivalent representations of Feynman integrals obtained by permuting coefficients in the Symanzik polynomials.We have verified up to 5-loop that the massive sunset with all masses different possesses no other polytope symmetries than those associated with relabelings.For ladder integrals we have used our methods in d = 6, where they are finite.In these cases we have found that the symmetries leave the powers of the propagators invariant Tκ = κ.It would be interesting to prove this statement in general and study Feynman integrals with numerators [31].

TABLE I :
Symmetry counting for the banana graph: the numbers include the identity matrix.Starred numbers were computed through PALP.