Feynman integral relations from parametric annihilators

We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space integration by parts relations, which are well known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee–Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works.


Introduction
At higher orders in perturbative quantum field theory, the computation of observables via Feynman diagrams involves a rapidly growing number of Feynman integrals. Fortunately, the number of integrals which need to be computed explicitly can be reduced drastically by use of linear relations i c i I i = 0 ( * ) between different Feynman integrals I i , with coefficients c i that are rational functions of the space-time dimension d and the kinematic invariants characterizing the physical process (masses and momenta of elementary particles).
The most commonly used method to derive such identities is the integration by parts (IBP) method introduced in [22,92]. In this approach, the relations ( * ) are obtained as integrals of total derivatives in the momentum space representation of Feynman integrals. Combining these relations, any integral of interest can be expressed as a linear combination of some finite, preferred set of master integrals. 1 Laporta's algorithm [50] provides a popular approach to obtain such reductions, and various implementations of it are available [2,62,75,76,80,86,98]. However, the increase of complexity of todays computations has recently motivated considerable theoretical effort to improve our understanding of the IBP approach and the efficiency of automated reductions [35,40,52,55,57,70,78,97,99]. This includes a method by Baikov [5,7,8,82], which is based on a parametric representation of Feynman integrals.
It is interesting to ask for the number of master integrals which remain after such reductions. This number provides an estimate for the complexity of the computation and informs the problem of constructing a basis of master integrals by an Ansatz. In the recent literature, algorithms to count master integrals were proposed and implemented in the computer programs Mint [58] and Azurite [32].
We propose an unambiguous definition for the number of master integrals as the dimension of an appropriate vector space. This definition and our entire discussion are independent of the method of the reduction. The main result of this article shows that this number is a well-understood topological invariant: the Euler characteristic of the complement of a hypersurface {G = 0} associated to the Feynman graph. Therefore, many powerful tools are available for its computation.
To arrive at our result, we follow Lee and Pomeransky [58] and view Feynman integrals as a Mellin transform of G −d/2 , where G is a certain polynomial in the Schwinger parameters. Each of these parameters corresponds to a denominator (inverse propagators or irreducible scalar products) of the momentum space integrand. The classical IBP relations relate Feynman integrals which differ from each other by integer shifts of the exponents of these denominators. As Lee [56] and Baikov [7] pointed out, such shift relations correspond to annihilation operators of the integrands of parametric representations. 2 In our set-up, these are differential operators P satisfying PG −d/2 = 0.
We recall that such parametric annihilators provide all shift relations between Feynman integrals, in particular the ones known from the classical IBP method in momentum space. The obvious question, whether the latter suffice to obtain all shift relations, seems to remain open. As a positive indication in this direction, we show that the momentum space relations contain the inverse dimension shift.
Ideals of parametric annihilators are examples of D-modules. Loeser and Sabbah studied the algebraic Mellin transform [60] of holonomic D-modules and proved a dimension formula in [59,61], which, applied to our case, identifies the number of master integrals as an Euler characteristic. The key property here is holonomicity, which was studied in the context of Feynman integrals already in [46] and of course is crucial in the proof [77] that there are only finitely many master integrals.
It is furthermore worthwhile to notice that algorithms [65,67] have been developed to compute generators for the ideal of all annihilators of G −d/2 , see also [72, section 5.3]. Today, efficient implementations of these algorithms via Gröbner bases are available in specialized computer algebra systems such as Singular [4,25]. We hope that these improvements may stimulate further progress in the application of D-module theory to Feynman integrals [30,78,79,90].
We begin our article with a review of the momentum space and parametric representations of scalar Feynman integrals and recall how the Mellin transform translates shift relations to differential operators that annihilate the integrand. In Sect. 2.4 we illustrate how the classical IBP identities obtained in momentum space supply special examples of such annihilators. The relations between integrals in different dimensions are addressed in Sect. 2.5, where we relate them to the Bernstein-Sato operators and show that these can be obtained from momentum space IBPs. Our main result is presented in Sect. 3, where we apply the theory of Loeser and Sabbah to count the master integrals in terms of the Euler characteristic. Practical applications of this formula are presented in Sect. 4, which includes a comparison to other approaches and results in the literature. Finally, we discuss some open questions and future directions.
In "Appendix A", we give an example to illustrate our definitions in momentum space, present proofs of the parametric representations and demonstrate algebraically that momentum space IBPs are parametric annihilators. The theory of Loeser and Sabbah is reviewed in "Appendix B", which includes complete, simplified proofs of those theorems that we invoke in Sect. 3. Finally, "Appendix C" discusses the parametric annihilators of a two-loop example in detail.

Annihilators and integral relations
In this section, we elaborate a method to obtain relations between Feynman integrals from differential operators with respect to the Feynman parameters and show that these relations include the well-known IBP relations from momentum space.

Feynman integrals and Schwinger parameters
At first we fix conventions and notation for Feynman integrals in momentum space and recall their representations using Schwinger parameters. While the former is the setting for most traditional approaches to study IBP identities, it is the latter (in particular in its form with a single polynomial) which provides the direct link to the theory of D-modules that our subsequent discussion will be based on.
We consider integrals (also called integral families [98]) that are defined by an d L-fold integral (L is the loop number), over so-called loop momenta 1  The denominators are (at most) quadratic forms in the L loop momenta and some number E of linearly independent external momenta p 1 , . . . , p E . In most applications, the denominators are inverse Feynman propagators associated with the momentum flow through a Feynman graph that arises from imposing momentum conservation at each vertex, see Example 3. However, we will only restrict ourselves to graphs from Sect. 3.2 onwards, and keep our discussion completely general until then. An integral (2.1) is a function of the indices ν = (ν 1 , . . . , ν N ) (denominator exponents), the dimension d of spacetime and kinematical invariants (masses and scalar products of external momenta). However, we suppress the dependence on kinematics in the notation and treat kinematical invariants as complex numbers throughout. The dimension and indices are understood as free variables; that is, we consider Feynman integrals as meromorphic functions of (d, ν) ∈ C 1+N in the sense of Speer [83]. Then, the Feynman integral (2.1) can be written as

5)
Formulas (2.5) and (2.6) are known since the sixties, and we refer to [64,81] for detailed discussions and for the original references. The trivial consequence (2.7) was popularized only much more recently by Lee and Pomeransky [58], and it is this representation that we will use in the following. In Appendix A.1, we include proofs for these equations and provide further technical details. Fig. 1 with massless Feynman propagators, D 1 = − 2 and D 2 = −( + p) 2 . We find = x 1 + x 2 , Q = −x 2 p and J = −x 2 p 2 according to (2.2). Hence, the graph polynomials (2.3) become U = x 1 + x 2 and F = (− p 2 )x 1 x 2 such that

Example 3 Consider the graph in
(2.  4 From there, analytic continuation defines a unique, meromorphic extension of every Feynman integral to the whole parameter space C 1+N . The poles are simple and located on affine hyperplanes defined by linear equations with integer coefficients. For these foundations of analytic regularization, we refer to [83,84].
For a number of reasons, in particular the preservation of fundamental symmetries, the most widely used scheme in quantum field theory is dimensional regularization [23,87]. It consists of specializing the ν ∈ Z N to integers and only keeps the dimension d as a regulator. In this case, the poles are not necessarily simple anymore.
The uniqueness of the analytic continuation of a Feynman integral (as a function of d and ν) is very important; in particular, it means that an identity between Feynman integrals is already proven once it has been established locally in the non-empty domain of convergence of the involved integral representations. In other words, in any calculation with analytically regularized Feynman integrals, we may simply assume, without loss of generality, that the parameters are such that the integrals converge. The resulting relation then necessarily remains true everywhere by analytic continuation.

Example 5
The one-loop propagator in Example 3 can be computed in terms offunctions: Integral (2.8) converges only in a certain domain of (ν, d), but there it evaluates to (2.9), which has a unique meromorphic continuation. Its poles lie on the infinite family of hyperplanes defined by {d/2 − ν 1 = k}, {d/2 − ν 2 = k} and

Integral relations and the Mellin transform
In this section, we summarize how relations between ν-shifted Feynman integrals can be identified with differential operators that annihilate G −d/2 . This method was suggested in [56] (and in [7] for the Baikov representation). The parametric representation (2.7) can be interpreted as a multi-dimensional Mellin transform. For our purposes, we slightly deviate from the standard definition, as for example given in [18] and include the factors (ν i ) that occur in (2.7).
whenever this integral exists. As a special case, we define Recall that, as mentioned in Remark 4, we do not have to worry about the actual domain of convergence of (2.10) in the algebraic derivations below. The key features of the Mellin transform for us are the following elementary relations; see [18] for their form without the 's in (2.10).
because the boundary terms vanish inside the convergence domain of both integrals. This just says that if lim f ) = 0 vanishes and the analogous argument applies to the upper bound x i → ∞.

Operator algebras and annihilators
The Mellin transform relates differential operators acting on G −d/2 with operators that shift the indices ν of the Feynman integrals I(ν). In this section, we formalize this connection algebraically in the language of D-modules. For the most part, we only need basic notions which we will introduce below, and point out [24] as a particularly accessible introduction to the subject.

Definition 8
The Weyl algebra A N in N variables x 1 , . . . , x N is the non-commutative algebra of polynomial differential operators Note that with the multi-index notations x α = x α 1 1 · · · x α N N and ∂ β = ∂ β 1 1 · · · ∂ β N N , every operator P ∈ A N can be written uniquely in the form by commuting all derivatives to the right (only finitely many of the coefficients c αβ ∈ C are nonzero). Extending the coefficients c αβ from C to polynomials C[s] in a further, commuting variable s, we obtain the algebra Later, we will also consider the case A N k := A N ⊗ C k of coefficients that are rational functions k := C(s). The integrands of the Mellin transform (2.10) naturally form a A N [s]module A N [s] f s , which we will introduce now.
Example 11 Given f ∈ C[x 1 , . . . , x N ], we always have the trivial annihilators (2.14) Note that an annihilator ideal is a module over A N [s], that is, whenever P f s = 0, also (Q P) f s = Q(P f s ) = 0 for any operator Q ∈ A N [s]. These ideals are studied in D-module theory [24,72] and in principle annihilators can be computed algorithmically with computer algebra systems such as Singular [3,4,25].

Definition 12
The algebra S N of shift operators in N variables is defined by This algebra is clearly isomorphic to the Weyl algebra A N , since under the identificationsî + ↔ ∂ i and j − ↔ x j the commutation relations are identical. In fact, a different isomorphism is given byî + ↔ x i and j − ↔ −∂ j , and it is this identification that corresponds to the Mellin transform (see Lemma 7). We therefore denote it by 5 . (2.16) The conceptual difference between S N and A N is that we think of A N as acting on functions f (x) by differentiation, whereas S N acts on functions F(ν) of a different set ν = (ν 1 , . . . , ν N ) of variables (the indices of Feynman integrals) by shifts of the argument and, in case ofî + , a multiplication with ν i : (2.17) These operators are used very frequently in the literature on IBP relations, as for example in [35,52,78,81]. An important role is played by the operators

Example 15
For the bubble graph in Fig. 1 with G = x 1 +x 2 − p 2 x 1 x 2 from Example 3, is easily checked to annihilate G s . We therefore get the shift relation (s + n 1 + n 2 ) I = p 21 + (s + n 1 ) I = p 2 (s + n 1 + 1)1 + I.
According to (2.18), this relation can also be written as We prefer to work with the modified Feynman integral I from (2.11), because it is directly related to the Mellin transform. However, it is straightforward to translate relations between I into relations for the actual Feynman integral I.

Example 16
Substituting which also follows from expression (2.9) of I(ν) in terms of -functions.
Then, its inverse is given by

On the correspondence to momentum space
In this section, we first recall the integration by parts (IBP) relations for Feynman integrals that are derived in momentum space, following [35]. We then note that these provide a special set of parametric annihilators and discuss some open questions in regard of this comparison of IBPs in parametric and momentum space.
Since the denominators D a from (2.1) are quadratic forms in the In order to express the IBP relations coming from momentum space in terms of integrals (2.1), we need to assume, for this and the following section, that we consider N = | | denominators such that the N × N square matrix A defined by (2.26) is invertible. 6 We think of A We are interested in relations of the Feynman integral I from (2.1), that is,

Definition 19
The momentum space IBP relations of I(ν 1 , . . . , ν N ) are those relations between scalar Feynman integrals that are obtained from Stokes' theorem where the operators o i j are defined in terms of the momenta 7 as The following, explicit form of these relations as difference equations is essentially due to Baikov [6,7]; see also Grozin [35]. For completeness, we include the proof in appendix A.1.

Proposition 20 Given a set of N = | | denominators D such that the matrix
(2.34) The coefficients C bi aj are defined as (2.35) 7 Recall that q 1 , . . . , q L denote the loop momenta, whereas q L+1 , . . . , q M are the external momenta.
Proof First recall rescaling (2.11) between the Feynman integral I and the Mellin transform I of G s . As we saw in (2.22), this means that and so if we substitute (2.11) into O i j I = 0 for the operators from (2.34), then apart from the substitutionâ + b − → x a (−∂ b ) which does not change ω, the remaining terms with shiftsâ + do increment ω by one and thus acquire an additional factor of Since the O i j are first-order differential operators, we have the inclusions To test this conjecture, we should take all known shift relations for Feynman integrals, and check if they can be realized as elements of Mom (after localizing at C(s, θ)).
In the remainder of this section, we will address such a relation, namely the one originating from the well-known dimension shifts.

Dimension shifts
The representation I = M e −F /U · U s from (2.5) shows, through Corollary 13, that by substituting x i →î + . This raising dimension shift was pointed out by Tarasov [89], 9 and had been observed before in special cases [26]. For   [10,73]. Given a solution of (2.44) for f = G, we get a lowering dimension shift relation: In general, computing a Bernstein operator is not at all trivial. But in the case of a complete set of irreducible scalar products (N = | | and A is invertible), an explicit formula for the lowering dimension shift follows from Baikov's representation [7] of Feynman integrals. We use form (2.48) given by Lee in [53,54]: 10 In fact, H (s)G + sU = e x e (G∂ e − (s − 1)(∂ e G)) follows from the trivial annihilators (2.14) of G s−1 . 11 Tkachov proposed in [93] to use a generalization of (2.44) to several polynomials (the individual Symanzik polynomials U and F , instead of G = U + F ), which in the physics literature is referred to as Bernstein-Tkachov theorem. However, this result is in fact due to Sabbah [71] (see also [36]).
Recall that (q 1 , . . . , q M ) = ( 1 , . . . , L , p 1 , . . . , p E ) denotes the combined loopand external momenta (M = L + E). We introduce the Gram determinants (2.47) and remark that Gr := Gr L+1 (s) = det( p i · p j ) 1≤i, j≤E depends only on the external momenta. Furthermore, note that Gr 1 (s) is a polynomial in the scalar products s {i, j} = q i · q j . By (2.29), we can think of it also as a polynomial in the denominators D. Theorem 29 (Baikov representation [53]) The Feynman integral (2.1) can be written as where c ∈ Q is a rational constant and the Baikov polynomial P(y) has degree at most M = L + E. The contour of integration in (2.48) is such that P vanishes on its boundary.
We include the proof in "Appendix A.3". For us, the interesting feature of this alternative formula is that the dimension appears with a positive sign in the exponent of the integrand. We can therefore directly read off Proof According to (2.48), I(d + 2) is obtained by multiplying the integrand of I(d) with (−1) L P(y)/Gr and adjusting the -factors in the prefactor as and so on. Multiplying the integrand of the Baikov representation with y e is equivalent to decrementing ν e ; hence, the multiplication of the integrand by P(y) can be written as the action of P(1 − , . . . , N − ) on the integral.
We ask if this annihilator is contained in Mom; in other words, whether the lowering dimension shift relation (2.49) is a consequence of the momentum space IBP identities. This is what we will establish in the following Proposition 31. First let us write the Baikov polynomial P(y) explicitly: The block decomposition ( is an L × L matrix whose entries are quadratic in the denominators. By (2.29), where we suppress the explicit summation signs over a = 1, . . . , N in the first summand and over a, b = 1, . . . , N and r , s = L + 1, . . . , M in the second summand.
Contracting with the matrix A {k,l} c (for k, l ≤ L) by summing over c, we conclude that Note that the indices r and s take values > L, whereas i and j are ≤ L. Hence, δ { j,s},{k,l} = δ {i,r },{k,l} = 0. So, recalling (A.2), we finally arrive at We can now invoke an identity of Turnbull [94], see also [29] for a combinatorial and [20] for an algebraic proof, which relates this product of determinants to a determinant of the productQ ·˜ . This is non-trivial, because the elements of these two matrices do not commute, according to (2.53). Turnbull's identity, as stated in [20,Proposition 1.4], applies precisely to this kind of very mild non-commutativity (2.53) and states that is a simple diagonal matrix and col-det denotes the column-ordered determinant So let us now compute the entries of the product ofQ from (2.51) with˜ . Firstly, Note that q s · q m = G s,m such that contraction of the second summand with G −1 r ,s produces δ r ,m . So the sum over m collapses, and up to the term with O k s ,Q i, j˜ j,k is Putting our results together, we arrive atQ So if we ignore all terms that lie in the (left) ideal generated by the momentum space (shift) operators O i j , the column determinant (2.55) of the matrixQ ·˜ + Q col from (2.54) can be replaced by an ordinary determinant det B of the diagonal matrix such that indeed we conclude with the result that (2.57) To phrase this in terms of Recall from (2.22) that the left-hand side of (2.58) can be written, in terms of the homogeneous components Q r (with degree r ) of det Q(−∂) = r Q r , as If r ≤ L +1, this is a polynomial differential operator, and we thus obtained an explicit Bernstein-Sato operator as in Definition 26.

Corollary 32
If the degree of the Baikov polynomial P(y) is not more than L + 1, then the Bernstein-Sato polynomial b(s) of the Lee-Pomeransky polynomial G is a divisor of (s + 1)b(s). In particular, all roots of b(s)/(s + 1) are simple and at half-integers.
Note that deg P(y) ≤ min {2L, M} = L +min {L, E} by Definition 28 and Eq. (2.51), so in particular, the corollary applies to all propagator graphs (E = 1) and to all graphs with one loop (L = 1).

Euler characteristic as number of master integrals
Here, we will show, using the theory of Loeser and Sabbah [59], that the number of master integrals equals the Euler characteristic of the complement of the hypersurface defined by G = 0 inside the torus G N m . (We write G m = A\ {0} for the multiplicative group and A for the affine line.) For a full understanding of this section, some knowledge of basic D-module theory is indispensable, but we tried to include sufficient detail for the main ideas to become clear to non-experts as well. In particular, we will give self-contained proofs that only use D-module theory at the level of [24].

Definition 33
By V G we denote the vector space of all Feynman integrals associated to G, over the field C(s, ν) := C(s, ν 1 , . . . , ν N ) of rational functions (in the dimension and indices). More precisely, with I G := M {G s }, (3.1) The number of master integrals is the dimension of this vector space: Note that this is the same as the dimension of the space n C(s, ν)I G (ν + n) of Feynman integrals (2.1), because the ratios

Remark 34
The phrase "master integrals" is used with different meanings in the physics literature. The main sources for discrepancies are: 1. Almost always the integrals are considered only for integer indices ν ∈ Z N , instead of as functions of arbitrary indices. In this setting, integrals with at least one ν e = 0 can be identified with quotient graphs ("subtopologies") and are often discarded from the counting of master integrals. 2. We only discuss relations of integrals that are expressible as linear shift operators acting on a single integral. This set-up cannot account for relations of integrals of different graphs (with some fixed values of the indices), as for example discussed in [48]. It also excludes symmetry relations, which are represented by permutations of the indices ν e . 3. Some authors do not count integrals if they can be expressed in terms of -functions or products of simpler integrals, for example [41,48].
Taking care of these subtleties, we will demonstrate in Sect. 4 that our definition gives results that do match the counting of master integrals obtained by other methods.
A fundamental result for methods of integration by parts reduction is that the number of master integrals is finite. This was proven in [77] for the case of integer indices ν ∈ Z N , using the momentum space representation. Below we will show that this result holds much more generally, for unconstrained ν, and that it becomes a very natural statement once it is viewed through the parametric representation. Notably, it remains true for Mellin transforms M {G s } of arbitrary polynomials G-the fact that G comes from a (Feynman) graph is completely irrelevant for this section.
Recall that, by the Mellin transform, we can rephrase statements about integrals in terms of the parametric integrands. In line with (2.25) and (3.1), we can rewrite (3.2) as denotes the polynomials in the dimension s = −d/2 and the operators θ e := x e ∂ e = M −1 {−ν e }, and F := C(s, θ) stands for their fraction field (the rational functions in these variables). Since F contains k := C(s), we can equivalently work over this base field throughout and write which is a fundamental result due to Bernstein [10]. Holonomic modules are, in a precise sense, the most constrained and behave in many ways like finite-dimensional vector spaces. For example, sub-and quotient modules, direct and inverse images of holonomic modules are again holonomic [44,45], and holonomic modules in zero variables are precisely the finite-dimensional vector spaces. The holonomicity of the parametric integrand was already exploited in [46] to show that Feynman integrals fulfil a holonomic system of differential equations, and it is also a key ingredient in the proof in [77].
The number defined in (3.3) has been studied by Loeser and Sabbah [59] in a slightly different setting, namely for holonomic modules over the algebra of linear differential operators on the torus G N m . Note that D N k is just the localization of A N k at the coordinate hyperplanes x i = 0; that is, the coefficients of the derivations are extended from polynomials O( in the coordinates x i . Equivalently, we can also view D N k = ι * A N k as the pull-back under the (open) inclusion The pull-back along ι turns every The starting point for this section is Theorem 35 (Loeser and Sabbah [59,61] In "Appendix B", we provide a self-contained proof of this crucial theorem, simpler and more explicit than in [61]. For now, let us content ourselves with reducing it to the known situation on the torus.
This result not only implies the mere finiteness of the number of master integrals, but in addition gives a formula for this number-it is the Euler characteristic, given by of the algebraic de Rham complex of M := ι * M . This is the complex Note that the r -forms ω are shifted to sit in degree r − N of the complex, which is thus supported in degrees between −N and 0; hence (3.5) is a finite sum over −N ≤ i ≤ 0. The extremal cohomology groups are easily identified as with the latter also known as push-forward of M under the projection π : G N m − A 0 k to the point. Since holonomicity is preserved under direct images, we conclude that dim k H 0 DR(M ) is finite. 13 In fact, the same is true for the other de Rham cohomology groups, which shows that (3.5) is indeed well defined. 14 Since we are interested in Feynman integrals, we consider the special case where the A N k -module M is simply M = A N k · G s from Definition 9. Its elements can be written uniquely in the form The action (2.13) of the derivatives, however, is twisted by a term proportional to s: Despite this twist, we find that the Euler characteristic stays the same: be a polynomial and set k = C(s). Then, the Euler characteristics of the algebraic de Rham complexes of the holonomic A N k -module In particular, we can dispose of the parameter s completely and compute with the

Corollary 37 The number of master integrals of an integral family with N denominators is
Remark 38 We stress that this geometric interpretation of the number of master integrals is valid for dimensionally regulated Feynman integrals, that is, we consider them as meromorphic functions in d (and ν). This is reflected in our treatment of s = −d/2 as a symbolic parameter. If, instead, one specializes to a fixed dimension like d = 2 (s = −1) or d = 4 (s = −2), then (2.46) is no longer true in general. 16 It can then happen that A N · G s 13 Recall that a holonomic module over the point A 0 k is the same as a finite-dimensional k-vector space. 14 In the language of derived categories, saying that π * M is holonomic actually means precisely that DR(M ) is a complex with cohomology groups that are holonomic modules over the point-that is, finite-dimensional vector spaces over k. 15 This sign arises from the shift by N in the Definition (3.6) of the de Rham complex DR.
C[x, G −1 ] is a proper submodule (note k = C(s) = C). While Theorem 35 still applies and relates the number of master integrals in a fixed dimension to the Euler characteristic of D N ·G s , this is not always equal to the topological Euler characteristic (3.10). This is expected, since the number of master integrals is known to be different in fixed dimensions [88]. 17

Proof of Proposition 36 Given an
C action on M f s has twisted derivatives to take into account the factor f s : Following Malgrange [63], we introduce the action of a further variable t by setting where we use the intuitive abbreviation f s+r := f r · f s for r ∈ Z. One easily verifies Note that the operator t acts invertibly on M , such that the identity ∂ t t = −s gives ∂ t = −st −1 and thus once we assume that M is holonomic to ensure that these Euler characteristics are well defined. Indeed, the holonomicity of M and M f s holds because 17 In these references, the indices are restricted to integers. However, our calculations in Sect. 4 show agreement of this way of counting with our set-up where the indices are treated as free parameters. 18 It would be more consistent with Definition 9 to write M [s] f s instead of M f s , but we prefer the shorter form to avoid clutter and to stress that we view it as an A N C -module, not a A N C[s] -module. 19 This is also clear from the 20 Alternatively, the filtration j M f s := f − j i≤ j s i 2 j(deg f )−i M induced by any good filtration • on M directly shows the holonomicity of M f s , since its dimension grows like j N +1 for large j. We now invoke the theory of Loeser-Sabbah to deduce that where N := M f s (t∂ t ) denotes the algebraic Mellin transform (B.1) of M f s with respect to the coordinate t. But note that, according to (3.12), localizing at t∂ t = −s −1 just extends the coefficients to k = C(s).

Remark 39
More abstractly, Proposition 36 can also be seen as an application of the theory of characteristic cycles [33]: It is known that the Euler characteristic only depends on the characteristic cycle of a A N k -module, which follows from the Dubson-Kashiwara formula [51, equation (6.6.4)]. Therefore, it is sufficient to show that the ]G s have the same characteristic cycles, via [33,Theorem 3.2]. This follows from the fact that these modules are identical up to the twist by the isomorphism

No master integrals
Corollary 37 shows in particular that there are no master integrals, C (G) = 0, precisely when the Euler characteristic For example, this happens if f is homogeneous in a generalized sense: Suppose we can find λ 0 , . . . , λ N ∈ Z, not all zero, such that (3.14) which is equivalent (apply ∂ t and set t = 1) to the existence of a linear annihilator, with P(θ, s) · f s = 0. Conversely, given such an operator, its shifts P(θ − α, s + r ) by (r , α) ∈ Z 1+N annihilate the elements x α · f s+r , which are therefore all mapped to zero in M. By linearity, this proves M = {0}, because every element of M can be written as

is zero if, and only if, f s is annihilated by a polynomial in the Euler operators θ :
In particular, the presence of a linear annihilator (3.15) implies M = {0} and hence χ(G N m \V( f )) = 0 via Corollary 37. Note that we could equally phrase this in terms of ). When f = G comes from Feynman graph G (as in the next section), it is not difficult to see that homogeneity (3.14) occurs precisely when G has a tadpole. 21 If this is the case, the integrals from Proposition 2 do not converge for any values of s and ν. In fact, M = {0} dictates that the only value one can assign to M { f s } (ν) which is consistent with integration by parts relations is zero.
The purpose of this section is to show that the simple homogeneity condition (3.14) is not only sufficient for a vanishing Mellin transform, but it is also necessary: Geometrically, homogeneity (3.14) can be interpreted as follows: Dividing by the greatest common divisor, we may assume that λ 0 , . . . , λ N are relatively prime. Thus, we may extend (λ 1 , . . . , λ N ) to a basis of the lattice Z N and hence construct a matrix A ∈ GL N (Z) with first row A 1i = λ i . In the associated coordinates y, defined by , the polynomial f takes the form f (x) = y λ 0 1 g(ȳ) for some Laurent polynomial g ∈ C[ȳ ±1 ] in the remaining variablesȳ = (y 2 , . . . , y N ). In particular, the hypersurface { f = 0} = {g = 0} can be defined by an equation independent of the coordinate y 1 . 21 A tadpole here means a proper subgraph γ G which shares only a single vertex with the rest of G and does not depend on masses or external momenta. In this case, G G = U γ F G/γ factorizes such that the variables x i with i ∈ γ only appear in the homogeneous polynomial U γ of degree λ 0 := L γ . Thus, we obtain (3.14) by setting λ e = 1 if e ∈ γ and λ e = 0 otherwise.

has Euler characteristic zero if and only if it is isomorphic to a product of G m times a
To prove Proposition 41, we will look at the Newton polytope NP ( f ) of f , which is defined as the convex hull of the exponents of monomials that appear in f : Since every monomial x α is an eigenvector of operators (3.15), P λ We can therefore reformulate the equivalent conditions (3.14) and (3.15) as Such polytopes have zero N -dimensional volume, and we call them degenerate.

Proof of Proposition 41
We proceed by induction over the dimension N , and we will assume f to be non-constant. (The proposition holds trivially for any constant f ∈ C.) In the case N = 1, the variety V( f ) ⊂ C * is a finite set, and hence, its Euler characteristic coincides with its cardinality. Therefore, χ(V( f )) = 0 if and only if f has no zero inside the torus. This is only possible if f is proportional to a monomial x r 1 ; in particular, f must be homogeneous and we are done. Now consider N > 1 and assume that χ(G N m \V( f )) = χ(V( f )) = 0. Recall that (3.14) is equivalent to degeneracy (3.18) of NP ( f ), so we only need to rule out the non-degenerate case. We achieve this by exploiting the hypothesis dim NP ( f ) = N to construct a linear annihilator P λ s of f s , which implies (3.18) in contradiction to the non-degeneracy of NP ( f ).
To start, we use Lemma 40 to find a polynomial 0 = P(θ, s) ∈ C[s, θ] such that P(θ, s) · f s = 0, and we choose one with minimal total degree in s and θ . Then pick an (N − 1) dimensional face σ = NP ( f ) ∩ F λ , which we can write as the intersection of NP ( f ) with a hyperplane F λ for some integers λ 0 , . . . , λ N such that NP ( f ) ⊆ α : P λ 1 (α) ≤ 0 . Under rescaling (3.14), all monomials of f = α c α x α with α ∈ σ ⊂ F λ acquire a factor of t λ 0 , while the remaining monomials with and O t −1 denotes a rational function in t −1 C(x)[t −1 ]. Note that P(θ, s) · f s ( x i t λ i ) = 0 is still zero, because the rescaling of x commutes with the Euler Therefore, applying P(θ, s) to the s-th power of the right-hand side of (3.19) and dividing by t sλ 0 yields where O t −1 on the right-hand side denotes a formal series in t −1 C(x, s)[[t −1 ]]. In particular, the coefficient of t 0 must vanish, and we conclude that P(θ, s) · f s σ = 0. Label the variables such that λ N = 0, then we can divide P(θ, s) by the linear form P λ s (θ, s) from (3.15), as a polynomial in θ N , to obtain a decomposition for some polynomial Q(θ, s) ∈ C[θ, s], such that the first summand depends only on θ := (θ 1 , . . . , θ N −1 ) and s. Since is a polynomial in less than N variables. If P(θ , 0, s) were nonzero, Lemma 40 would show χ(G N −1 m \V(g)) = 0, such that we could apply our induction hypothesis to g and conclude that g is homogeneous in our generalized sense. We saw that this is equivalent to the degeneracy of NP (g), which contradicts that NP (g) ∼ = NP ( f σ ) = σ is of dimension N − 1. 22 Therefore, P(θ , 0, s) must be zero and we conclude that P(θ, s) = P λ s · Q has a linear factor P λ s (θ, s). 23 Now set m := Q(θ, s)· f s , which is nonzero, because P(θ, s) was chosen as an annihilator of f s of minimal degree. We may write this element in the form m = a · f s+r for some r ∈ Z and a Laurent polynomial a ∈ C(s)[x ± ]. After multiplying with a polynomial in C[s], we may even assume 0 = a ∈ C[s, x ±1 ] with P λ s · a f s+r = 0. Applying the Leibniz rule and dividing by a f s+r , we find Since the degree of P λ 0 · a in s is at most the degree (in s) of a itself, this first summand on the right has a finite limit as s → 0. We therefore must have a cancellation of the terms linear in s, P λ 0 · f = λ 0 f , which yields the sought-after P λ s · f s = 0. Remark 43 In summary, we showed that the following six conditions on a polynomial f ∈ C[x 1 , . . . , x N ] are equivalent: (1) homogeneity (3.14) of f , (2) existence (3.15) of an annihilator of f s linear in θ , (3) existence (3.16) of an annihilator of f s polynomial 22 Observe that NP (g) is the orthogonal projection of NP ( f σ ) onto the coordinate hyperplane {α N = 0}. This projection restricts to an isomorphism between {α N = 0} and F λ (because λ N = 0), and therefore dim NP (g) = dim NP ( f σ ). 23 The existence of a linear annihilator could also be deduced from [

Graph polynomials
Our discussion so far applies to all integrals of type (2.1)-the defining data are thus the set D = (D 1 , . . . , D N ) of denominators, which is sometimes also called an integral family [98]. The denominators can be arbitrary quadratic forms in the loop momenta; decomposition (2.2) then defines the associated polynomials U, F and G through (2.3). In particular, the denominators do not have to be related to the momentum flow through a (Feynman) graph in any way. However, we will from now on consider the most common case in applications: integrals associated to a Feynman graph with Feynman propagators.

Definition 45
Given a connected Feynman graph G with N internal edges, E + 1 external legs and L loops, imposing momentum conservation at each vertex determines the momenta k e flowing through each edge e in terms of the E external and L loop momenta. 24 The Symanzik polynomials U G and F G of the graph G are the polynomials U and F from (2.3) for the set D = (D 1 , . . . , D N ) of inverse Feynman propagators, 25 1 where m e is the mass associated to the particle propagating along edge e. The number C (G) of master integrals of the Feynman graph G is defined in terms of (3.2) as (3.21) 24 Momentum conservation implies that the sum p 1 + · · · + p E+1 = 0 of the incoming momenta on all external legs vanishes; hence only E of them are independent. 25 The infinitesimal imaginary part i is irrelevant for our purpose of counting integrals and will be henceforth ignored.
This class of integrals (using only the propagators in the graph) is sometimes referred to as scalar integrals and might appear to be insufficient for applications, since in general one needs to augment the inverse propagators by additional denominators, called irreducible scalar products (ISPs), in order to be able to express arbitrary numerators of the momentum space integrand in terms of integrals (2.1); see Example 62. Therefore, one might expect that, in order to count all these integrals, one ought to replace G G in (3.21) by the polynomial associated to the full set of denominators, including the ISPs. However, it is well known since [89] that all such integrals with ISPs are in fact linear combinations of scalar integrals in higher dimensions d + 2k, for some k ∈ N. Furthermore, those can be written as scalar integrals in the original dimension d by Corollary 27. Therefore, (3.21) is the correct definition to count the number of master integrals of (any integral family determined by) a Feynman graph.
We can therefore invoke the following, well-known combinatorial formulas for the Symanzik polynomials [14,81], which go back at least to [64].

22)
where T runs over the spanning trees of G and F enumerates the spanning two-forests of G. ( p F denotes the sum of all external momenta flowing into one of the components of F.) In Sect. 4.2, we will use these formulas to count the sunrise integrals.

Example 47
The graph polynomials of the bubble graph ( Fig. 1) are, in general kinematics, It is important to keep in mind that, even with a fixed graph, the number of master integrals will vary depending on the kinematical configuration-e.g. whether a propagator is massive or massless, or whether an external momentum is non-exceptional or sits on a specific value (like zero or various thresholds). We will always explicitly state any assumptions on the kinematics, and hence stick with the simple notation (3.21). the Euler characteristic factors through the Grothendieck ring, since it is compatible with these relations: χ(X ) = χ(X \Z ) + χ(Z ) and χ(X × Y ) = χ(X ) · χ(Y ). The class L = [A 1 ] of the affine line is called Lefschetz motive and fulfils χ(L) = 1. For several polynomials P 1 , . . . , P n , we write V(P 1 , . . . , P n ) := {P 1 = · · · = P n = 0}.

The Grothendieck ring of varieties
If a variety is described by polynomials that are linear in one of the variables, we can eliminate this variable to reduce the ambient dimension. 26 Let us state such a relation explicitly, since our setting is slightly different than usual: For us, the natural ambient space is the torus G N m and not the affine plane A N . Lemma 48 Let A, B ∈ C[x 1 , . . . , x N −1 ] and consider the linear polynomial A+x N B. Then holds in the Grothendieck ring. In particular, the Euler characteristic is Proof Recall that U and F are homogeneous of degrees L and L + 1, respectively (Corollary 63). Since multiplication with x N ∈ G m is invertible, we can rescale such that the claim is just a special case of Lemma 48.

Example 50
Both graphs consisting of a pair of massless edges, (3.28) with the external momentum p such that p 2 = 0, have a single master integral C (G) = 1.
Proof According to (3.22), the graph polynomials of the graphs in (3.28) In both cases, the number of master integrals (3.10) is (3.27).
Much more on the Grothendieck ring calculus of graph hypersurfaces V(U) can be found, for example, in [1,17]. These techniques can be used to prove some general statements about the counts of master integrals. Let us give just one example: Lemma 51 Let G be a Feynman graph with a subgraph γ such that all propagators in γ are massless and γ has only two vertices which are connected to external legs or edges in G\γ . 27 Write G for the graph obtained from G by replacing γ with a single edge (see Fig. 2), then Proof Every spanning tree T of G restricts on γ either to a spanning tree or to a spanning two-forest. In the first case, T \γ is a spanning tree of G/γ (the graph where γ is contracted to a single vertex); in the second case, T \γ is a spanning tree of G\γ . Note that the two-forests T ∩ γ in the second case determine F γ from (3.22), since all propagators in γ are massless. Therefore, we find U G = U γ · U G/γ + F γ · U G\γ where we set F γ := F γ p 2 =−1 . Going through the same considerations for F G shows that 28 Now label the edges in γ as 1, . . . , N γ and rescale all Schwinger parameters x e with 2 ≤ e ≤ N γ by x 1 . Due to the homogeneity of U γ and F γ from Corollary 63, we see . Applying (3.25), we obtain a separation of variables: In the last line we used (3.27), upon noting the contraction-deletion formula G G = G G/γ + x 0 G G\γ in terms of the additional Schwinger parameter x 0 for the (massless) edge that replaces γ in G . (This formula is easily checked by considering which spanning trees and forests contain this edge or not.) Note that for the subgraph γ , the value of p 2 does not matter for V( F γ ) = V( F γ ), as long as p 2 = 0. Finally, recall (3.10).
Of course, this result is well known on the function level: If G has a 1-scale subgraph γ , then the Feynman integral of G factorizes into the product of the integrals of γ and G . Here, we denote by ν γ and ν the indices corresponding to the edges in γ and outside γ , respectively, such that ν = (ν γ , ν ). Note that the edge replacing γ in G (see Fig. 2) gets the index ω γ = e∈γ ν e − L γ · (d/2) from (2.4), which depends on the indices of γ (L γ denotes the loop number of γ ). Fig. 3 does not change the number of master integrals.

Corollary 52 Let G be a graph with a pair {e, f } of massless edges in series or in parallel. Then, C (G) = C G where G is the graph obtained by replacing the pair with a single edge. In other words, repeated application of the series-parallel operations from
Proof Combine Lemma 51 with Example 50. 28 Note that the intersection of a two-forest F of G with γ has at most two trees connected to external vertices of γ . Any further tree in F ∩ γ is thus necessarily a full component of F and forces its contribution to F G to vanish by p 2 F = 0 (due to the absence of external legs).

Fig. 3
The series (S) and parallel (P) operations consist of replacing a sequential or parallel pair of massless edges with a single edge

Tools and examples
The Euler characteristic of a singular hypersurface can be computed algorithmically via several methods. 29 In this section, we demonstrate how some of these techniques can be used to compute the number of master integrals in various examples.
We begin with methods based on fibrations. In particular, the Euler characteristic can be computed very easily for the class of linearly reducible graphs, see Sect. 4.1. However, the decomposition of the Euler characteristic of the total space E of a fibration E −→ B with fibre F into the product is true in general and not restricted to the linear case. In Sect. 4.2, we use a quadratic fibration in order to count the master integrals of all sunrise graphs. Apart from these geometric approaches, which seem to work very well for Feynman graphs, there are general algorithms for the computation of de Rham cohomology and the Euler characteristic of hypersurfaces. In Sect. 4.3 we discuss some available implementations of these algorithms in computer algebra systems.
In final Sect. 4.4, we comment on the relation of our result to other approaches in the physics literature.

Linearly reducible graphs
If the polynomial V( f ) = a + x N b is linear in a variable x N , we saw in Lemma 48 that we can easily eliminate this variable x N in the computation of the Euler characteristic (or the class in the Grothendieck ring) of the hypersurface V( f ) (or its complement). Analogous formulas also exist in the case of a variety V( f 1 , . . . , f n ) of higher codimension, given that all of the defining polynomials f i = a i + x N b i are linear in x N . Such linear reductions have been used heavily in the study of graph hypersurfaces and are straightforward to implement on a computer [74,85].
If such linear reductions can be applied repeatedly until all Schwinger parameters have been eliminated, the graph is called linearly reducible [15]. Linear reducibility is particularly common among graphs with massless propagators; we give some examples in Fig. 4. 29 We will not discuss Kouchnirenko's Theorem 44 here, because in most examples we found that it does not apply. It seems that coefficients of graph polynomials are often not sufficiently generic.  Table 1 The number C (G) of master integrals, computed as the Euler characteristic 3.10, for the graphs in  16  in the Grothendieck ring. Substituting L → 1 shows that C WS 3 = 3 via (3.10).
In this way, we calculated the Euler characteristics for the graphs in Fig. 4, using an implementation of the linear reductions similar to the method of Stembridge [85]. Our results are listed in Table 1.
Beyond the computation of such results for individual graphs, it is possible to obtain results for some infinite families of linearly reducible graphs. In particular, efficient computations are possible for graphs of vertex width three [16]. For example, the class in the Grothendieck ring of V(U) was computed for all wheel graphs in [17]. It is possible to adapt such calculations to our setting (where the ambient space is G N m instead of A N ). For example, we could prove Proposition 54 The number of master integrals of the massless propagators obtained by cutting a wheel WS L with L loops, either at a rim or a spoke (see Fig. 5), is The proof of this and related results will be presented elsewhere.

Sunrise graphs
In [43], the number of master integrals was computed for all sunrise integrals; based on a Mellin-Barnes representation and the differential reduction [19,42] of an explicit solution in terms of Lauricella hypergeometric functions. To our knowledge, this has hitherto been the only non-trivial 30 infinite family of Feynman integrals with explicitly known master integral counts. Their first result can be phrased as 31

Proposition 55
The L-loop sunrise graph S L from Fig. 6 with L + 1 nonzero masses (and non-exceptional external momentum) has C (S L ) = 2 L+1 − 1 master integrals.
We will now demonstrate that this result can be obtained from a straightforward computation of the Euler characteristic, according to Corollary 37.
Proof The graph polynomials (3.22) for the sunrise graph are We note that for the first term in (3.26), we find that U = F = 0 imply i x i = 0, which has no solutions in the torus-hence, this term contributes χ(G L m ) = 0. We thus obtain 30 We consider families that arise simply by duplication of massless propagators, like those shown in [19, Figure 2], as trivial (due to Corollary 52). 31 Beware that the number 2 L+1 − L − 2 given in [43, equation (4.5)] counts only irreducible master integrals, which means that it discards the L + 1 integrals associated to the subtopologies obtained by contracting any of the edges. Our conventions, however, do take these integrals into account. where we introduced the notation The first Euler characteristic in (4.5) is readily evaluated to (−1) L by applying (3.25) repeatedly (being on the torus, we may replace x −1 i by x i ), so we conclude that Now let us consider the projection π : We note that the discriminant D of this quadric in x L factorizes into 2 is the disjoint union of two hypersurfaces. 32 Since the factors are related to the (L − 1)-loop sunrise by (4.7), we find Over a point x ∈ X L−1 p±m L ⊂ V(D) in the discriminant, the fibre of π −1 (x ) has precisely one solution (x , x L ) in X L p 2 , determined by x L = −y/[m L (m L ± p)]: (4.10) If D(x ) = 0 is nonzero and also yz = 0, then the fibre π −1 (x ) has precisely two distinct solutions x L in the quadric X L p 2 . Hence, χ(π −1 (x )) = χ(G m ) − 2 = −2 and thus (4.11) where used that V(D) ∩ V(yz) = ∅ for non-exceptional values of p 2 , such that ( p ± m L ) 2 = 0 in (4.8). The reason that we need to exclude the case when yz = 0 in (4.11) is that for y = 0, one of the solutions of X L ∈ G m ; whereas for z = 0 the equation for X L p 2 = x L p 2 = y + m 2 L x L 32 We assume p 2 = 0 and m 2 L = 0, which guarantees that ( p + m L ) 2 = ( p − m L ) 2 .
becomes linear. In both cases, there is only one solution in the fibre, and there is none if both y = z = 0 vanish. (We assume p 2 = m 2 L ):

Proposition 56
The L-loop sunrise graph with R ≤ L nonzero masses, L + 1 − R vanishing masses and non-exceptional external momentum, has C (S L ) = 2 R master integrals.
Proof By Corollary 52, we may replace all massless edges by a single (massless) edge without changing the number of master integrals; hence, we can assume L = R ≥ 1.
(The totally massless case R = 0 reduces to the trivial case of a single edge.) Label the edges such that the massless edge is m L+1 = 0.
We can apply the exact same recursion as in the proof of Proposition 55; the only difference to (4.13) is that now, χ(V(y)) = 0 vanishes because y = i<L m 2 i x i has become homogeneous in x such that We are done after verifying the base case: Indeed, C (

General algorithms
The computer algebra system Macaulay2 [34] provides the function Euler in the package CharacteristicClasses. It implements the algorithm of [38] for the computation of the Euler characteristic. This program requires projective varieties as input, so we need to homogenize G toG = x 0 U + F, and can then use one of to express the sought-after number of master integrals as the Euler characteristic of a projective hypersurface complement. We found that this algorithm performs well for small numbers of variables (edges): The examples in Table 2 require not more than a couple of minutes of runtime. For more variables, however, the computations tend to rapidly become much more time consuming and often impracticable. Apart from the results in Table 2, we also verified Proposition 55 for the sunrise graphs S L using Euler for up to six loops.
Recall that the number of master integrals depends on the kinematical configuration; in Table 2 we give the results both for massless and for massive internal propagators. In particular, note how the massless 2-loop propagator WS 3 from Example 53 with only C WS 3 = 3 master integrals grows to carry C WS 3 = 30 master integrals in the fully massive case. Furthermore, Macaulay2 also provides an implementation (the command deRham) of algorithm [66] of Oaku and Takayama for the computation of the individual de Rham cohomology groups. This uses D-modules and Gröbner bases and tends to demand more resources than the method discussed above.

Comparison to other approaches
We successfully reproduced all of our results above (the wheels WS L with L ≤ 6, the sunrises S L with L ≤ 4 loops and the graphs from Fig. 4 and Table 2) with the program Azurite [32], which provides an implementation of Laporta's approach [50]. While it employs novel techniques to boost performance, in the end it solves linear systems of equations between integrals obtained from annihilators of the integrand of Baikov's representation (2.48) in order to count the number of master integrals.
The observed agreement with our results is to be expected, since the identification of integral relations with parametric annihilators that we elaborated on in Sect. 2.3 works equally for the Baikov representation, which can also be interpreted as a Mellin transform. Note, however, that we must use the options Symmetry -> False and GlobalSymmetry -> False for Azurite in order to switch off the identification of integrals that differ by a permutation of the edges. The reason being that, in our approach, all edges e carry their own index ν e and no relation between these indices for different edges is assumed.
Unfortunately, due to the way Azurite treats subsectors, this can occasionally lead to an apparent mismatch. However, this is rather a technical nuisance than an actual disagreement.

Example 59
For the graph G in Fig. 7, the Euler characteristic gives C (G) = 15, whereas both Reduze [98] and Azurite produce 16 master integrals. The problem arises from the subsector where the edges 1 and 2 are contracted: As shown in Fig. 7, it does have a remaining external momentum p 4 , such that the momenta running through edges 3 and 4 are different-however, since p 2 4 = 0, the graph polynomials (and hence the Feynman integrals) are identical to those of the vacuum graph G in Fig. 7. Since Since the only momentum p 4 running through the graph after contracting 1 and 2 is null ( p 2 4 = 0), the associated integral is the same as for G edges 3 and 4 in G have the same mass, they can be combined and thus G clearly has only a single master integral: the product of two tadpoles. But Azurite and Reduze instead consider the subsectors of G/ {1, 2} obtained by contracting a further edge (3 or 4), and obtain the two tadpoles (see Fig. 7) consisting only of edges {4, 5} and {3, 5}, respectively, as master integrals. Of course, these would be recognized as identical if symmetries were allowed, but the point is that even without using symmetries, there is only a single master integral for G (as computed by the Euler characteristic).
Our results are also consistent with the conclusions obtained within the differential reduction approach [42]; indeed, we demonstrated in Sect. 4.2 how the master integral counts of [43] for the sunrise graphs emerge directly from the computation of the Euler characteristic. Let us point out again, however, that some care is required for these comparisons, since those works refer to irreducible master integrals, which excludes integrals that can be expressed with gamma functions. In particular, the fact that the two-loop sunrise S 2 with one massless line has C (S 2 ) = 4 master integrals (see Proposition 56) is consistent with [41]. We are counting all master integrals and are not concerned here with the much more subtle question addressed by the observation that two of these integrals may be expressed with gamma functions.
Finally, let us note that also the work of Lee and Pomeransky [58] addresses a different problem: Considering only integer indices ν ∈ Z N , how many top-level master integrals are there for a graph G? This means that integrals obtained from subsectors (graphs G/e with at least one edge e contracted) are discarded. Geometrically, the number of the remaining master integrals is identified with the dimension of the cohomology group H N (C N \V(G)). 34 In most cases, the program Mint computes this number correctly, which then agrees with the other mentioned methods. 35 We refer to [13, section 4] and [43, section 6] for detailed discussions of this comparison. Note that the dimension (and a basis) of the top cohomology group can also be computed with the command deRhamCohom from the Singular library dmodapp.lib. The concept of top-level integrals does not literally make sense in our setting of arbitrary, non-integer indices ν. Here, there is no relation at all between integrals of a quotient graph G/e and integrals of G. (The former do not depend on ν e at all; the latter do.) However, discarding integrals from quotient graphs suggests a definition of a number as follows.

Remark 60
Using the inclusion-exclusion principle, one might be tempted to define as the number of top-level master integrals, since it subtracts from all master integrals C (G) the integrals associated to subsectors (and corrects for double counting). Note that if γ contains a loop, the corresponding term in the sum should be set to zero. (We only consider contractions with the same loop number as G.) The reverse relation, is consistent with the intuition that the total set of master integrals is obtained as the union of all top-level masters. By G G/γ = G| x e =0∀e∈γ , we find that is the Euler characteristic of the hypersurface complement inside affine space (as compared to the torus G N m as ambient space). We find that this number behaves exactly as expected and is consistent with our calculations in Sect. 4. However, we point out that this number can take negative values. For example, C (G/ {1, 2}) = −1 is negative for the graph from Fig. 7. In fact, this is necessary for the consistency of sum (4.17) of all master integrals: Namely, as we discussed in Example 59, the two subtopologies G/ {1, 2, 3} and G/ {1, 2, 4} have one master integral each, if we consider them individually. However, they are embedded into the graph G/ {1, 2}, which has only a single master integral, C (G/ {1, 2}) = 1. This is sometimes referred to as a "relation between subtoplogies", and the negative value of C (G/ {1, 2}) = −1 is precisely correcting the total counting.
Note that symmetries play no role in this discussion -the "extra" relation is detected by the Euler characteristic and thus corresponds to a parametric annihilator. 36

Outlook
We have studied linear relations between Feynman integrals that arise from parametric annihilators of the integrand G s in the Lee-Pomeransky representation. Seen as a multivariate (twisted) Mellin transform, the integration bijects these special partial differential operators with relations of various shifts (in the indices) of a Feynman integral. In particular, every classical IBP relation (derived in momentum space) is of this type.
The question whether all shift relations of Feynman integrals (equivalently, all parametric annihilators of G s ) follow from momentum space relations remains open (see Question 24). We showed that the well-known lowering and raising operators with respect to the dimension are consequences of the classical IBPs. A next step would be to clarify if the same applies to the relations implied by the trivial annihilators (2.14). Similarly, Question 23 asking whether the annihilator Ann(G s ) = Ann 1 (G s ) is linearly generated, remains to be settled. A positive answer to either of these would imply that the labour-intensive computation of the parametric annihilators could be simplified considerably (Mom is known explicitly, and Ann 1 can be calculated through syzygies).
The main insight of this article is a statement on the number of master integrals, which we define as the dimension of the vector space of the corresponding family of Feynman integrals over the field of rational functions in the dimension and the indices. Since we treat all indices ν a as independent variables, this definition does not account for symmetries (automorphisms) of the underlying graph. An important next step, in particular for practical applications, is to incorporate these symmetries into our setup by studying the action of the corresponding permutation group. The widely used partition of master integrals into top-level and subsector integrals can be mimicked in our framework, as discussed in Remark 60.
Our result shows that the number of master integrals is not only finite, but identical to the Euler characteristic of the complement of the hyperspace {G = 0} determined by the Lee-Pomeransky polynomial G. This statement follows from a theorem of Loeser and Sabbah. We exemplified several methods to compute this number and found agreement with other established methods. We expect that, combining the available tools for the computation of the Euler characteristic, it should be possible to compile a program for the efficient calculation of the number of master integrals for a wide range of Feynman graphs.
Let us conclude by emphasizing, once again, that the main objects of the approach elaborated here-the s-parametric annihilators generating the integral relations, and the Euler characteristic giving the number of master integrals-are well-studied objects in the theory of D-modules and furthermore algorithms for their automated computation are available in principle.
In particular, we hope that this parametric, D-module theoretical and geometrical approach can also shed light on the problems most relevant for perturbative calculations in QFT: the construction of a basis of master integrals and the actual reduction in arbitrary integrals to such a basis. For this perspective, we would like to point out that our approach of treating the indices ν a as free variables, in particular not tied to take integer values, is desirable in order to deal with dimensionally regulated integrals in position space, and for the ability to integrate out one-scale subgraphs (both situations introduce non-integer indices). For a recent step into this direction, see [91]. integrals, Jørgen Rennemo for suggesting literature relevant for Sect. 3.1 and Viktor Levandovskyy for communication and explanations around D-modules and in particular their implementation in Singular. Thomas Bitoun thanks Claude Sabbah for feedback and bringing the work [59][60][61] to our attention. Thomas acknowledges funding through EPSRC grant EP/L005190/1. Christian Bogner thanks Deutsche Forschungsgemeinschaft for support under the project BO 4500/1-1. We are grateful to the referees for their constructive comments. This research was supported by the Munich Institute for Astro-and Particle Physics (MIAPP) of the DFG cluster of excellence "Origin and Structure of the Universe". Furthermore we are grateful for support from the Kolleg Mathematik Physik Berlin (KMPB) and hospitality at Humboldt-Universität Berlin. Images in this paper were created with JaxoDraw [11] (based on Axodraw [95]) and FeynArts [37].
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A. Integral representations
In this appendix, we add technical details on the material of Sect. 2. We summarize the various well-known parametric representations, including their proofs, and the explicit relation to momentum space via Propositions 2 and 20. Furthermore, we give an alternative, algebraic proof for Corollary 63.

A.1. Momentum space and Schwinger parameters
As in Sect. 3.2, we consider a connected Feynman graph G with N internal edges, E +1 external legs and loop number L, which is related by Euler's formula L = N − V + 1 to the number V of vertices of G. Let us consider the case where each edge e of G is associated with a Feynman propagator, 37 1 which depends on the mass m e of the particle e and the d-dimensional momentum k e ∈ R d flowing through this edge. Enforcing momentum conservation at each vertex fixes all k e in terms of E independent external momenta p 1 , . . . , p E and L free loop momenta 1 , . . . , L . Note that the actual number of external legs of G is E + 1, since overall momentum conservation E+1 i=1 p i = 0 imposes one relation among the external momenta. Taking only the inverse Feynman propagators D e as denominators, Eq. (2.1) defines the Feynman integral associated to G.

Example 61
The graph in Fig. 8 has V = 5 vertices, N = 6 internal edges and L = 2 loops. It depends on two independent external momenta p 1 and p 2 . A choice of loop momenta and the resulting momentum flow is depicted in Fig. 8. With all masses zero, Typically, the number | | = L(L + 1)/2 + L E of independent scalar products s {i, j} in (2.28) is larger than the number of edges in a graph G. We can then extend the initial set of denominators (given as the inverse propagators of the graph) by a suitable choice of additional quadratic (or linear) forms in the loop momenta, such that we reach a set of | | denominators with the property that the matrix A defined by The matrix A has rank 6 and annihilates (0, 0, 0, 1, 0, 0, 0) . Thus, we can choose D 7 = 1 p 1 as an irreducible scalar product to complete the basis of quadratic forms in the loop momenta. The matrix A then acquires an additional row (0, 0, 0, 1, 0, 0, 0) and becomes invertible.
Note that we assume that the inverse propagators of G (the initial set of denominators) are linearly independent (that is, the N × | | matrix A of the inverse propagators has full rank N ) in order to be able to extend them to a basis of quadratic forms by choosing | | − N irreducible scalar products. 38 For each denominator D a we introduce a scalar x a , which is known as Schwinger-, Feynman-or α-parameter. In Definition 1 we have introduced the decomposition N a=1 which determines a symmetric L × L matrix , a vector Q and a scalar J . With their help we defined the polynomials U, F and G = U + F in (2.3). Explicitly, from Definition 1 and (2.26) we can read off that Let us now come to the proof of Proposition 2 following [64] and [58]. 38 If there are linear dependencies between the inverse propagators, these relations imply that the Feynman integral can be expressed in terms of contracted graphs with linearly independent inverse propagators. For example, if αD 1 + βD 2 = 1, then iterated use of 1/(D 1 D 2 ) = α/D 2 + β/D 1 allows one to ultimately eliminate one of D 1 or D 2 . Therefore, requesting linear independence is no restriction.

Proof of Proposition 2
We consider the Feynman integral defined in (2.1), Using the Schwinger trick to exponentiate each denominator, 39 1 the integral in (A.6) turns into According to (A.1) and (2.3), we can complete the square in the exponent to perform the Gaußian integrals over the shifted loop momenta := − −1 Q as 40 In summary, we therefore arrive at the integral representation (2.5): We now multiply with 1 = ∞ 0 δ(ρ − N j=1 x j )dρ and substitute x a → ρx a . 41 The Jacobian ρ N , the monomials 39 Integral (A.7) converges only for Re(ν a ) > 0 and therefore restricts the domain of convergence for the parametric integral. However, this has no consequences for algebraic relations, see Remark 4. 40 Recall that our metric has signature (1, −1, . . . , −1), so the integrations over the d − 1 spacelike components are Euclidean and give √ π L U each. The timelike integrations are understood as contour integrals and yield the same factor after rotating the integration contour to the imaginary axis, according to the Feynman i -prescription. 41 Much more generally, we could replace N j=1 x j in the δ-constraint with any other function as long as it is homogeneous of degree 1 and positive on R N + .
the power ρ |ν|−1 , whereas the homogeneity of F and U from Corollary 63 implies that U → ρ L U and F/U → ρF/U. Overall, by realizing that the integral over ρ is we arrive at the first parametric formula (2.6). Similarly, we multiply the integrand of (2.7) with 1 = ∞ 0 δ(ρ − i x i )dρ and substitute x i → ρx i . Using U → ρ L U and F → ρ L+1 F from Corollary 63, the integral over ρ becomes and combines with the prefactors in (2.7) to reproduce (2.7).
We conclude the section with the proof of Proposition 20 following Grozin [35]:

Proof of Proposition 20
The action of o i j on the integrand from (2.30) is According to (2.26), the chain rule gives and we can express the scalar products q j q m with { j, m} ∈ in terms of denominators using (2.29). The remaining terms with j, m > L are products of external momenta, so We conclude by noticing that multiplying the integrand f with ν a /D a is equivalent to the action of the operatorâ + defined in (2.17), whereas multiplication with D b lowers the index ν b and corresponds to b − .

A.2. Algebraic proof for Corollary 21
With the proof of Corollary 21 we have shown that, for every momentum space IBP relation, there is a corresponding annihilator in Ann G −d/2 . The proof rests on the inverse Mellin transform, which may be seen as a convenient but rather abstract argument. As a more direct alternative, we prove the statement in a purely algebraical way by use of properties of the graph polynomials.
where we exploited the homogeneity from (A.5). Using (2.3), we note that because ∂ b J = λ b according to (A.4). In order to evaluate ∂ b U with Jacobi's formula which proves that for arbitrary j (independent of whether j ≤ L or j > L) Via (A.9), this identity reduces the proof of (A.8) to showing that vanishes. The last term is easily evaluated with (A.10) and gives whereas the derivative ∂ b Q can be read off from (A.3) and the sum over b yields where we used (A.2) once more. Now recall that (A.11) remains true and becomes zero for j > L because δ i, j = 0 since i ≤ L. For the same reason, δ j,s = 0 in (A.13) and therefore, using (A.9),

A.3. The Baikov representation
In this section, we discuss the representation of Feynman integrals suggested by Baikov in [7], whose complete form (2.48) was given by Lee in [53,54]. We will give some details on the derivation of this formula (see also [35, section 9]), which was presented in [53] and applied in our discussion of the lowering dimension shift in Sect. 2.5.
Assume that q 1 , . . . , q M are vectors in a Euclidean vector space and write q M · q n · · · q M · q M ⎞ ⎟ ⎠ = q i · q j n≤i, j≤M and G n := det V n (A. 16) for their Gram matrices and determinants. Note that where q • · q n := ⎛ ⎜ ⎝ q n+1 · q n . . .
and thus, by adding −( p n · p • )V −1 n+1 times the lower M − n rows to the first row, Now assume our integrand f only depends on the scalar products s i, j = q i · q j , and we want to integrate out the first loop momentum q 1 . Let us decompose q 1 = q ⊥ + q into the component q ∈ lin {q 2 , . . . , q M } that lies in the space spanned by the other momenta, and the component q ⊥ in its orthogonal complement. According to (A.17), G 1/2 n is the volume of the parallelotope spanned by q n , . . . , q M . Hence, changing coordinates from q to (s 1,2 , . . . , The integral over the orthogonal component is, due to s 1,1 = q 2 1 = q 2 ⊥ + q 2 , given by of variables allows us to integrate over the values of D e 's instead of s i, j 's. This transformation only introduces a constant Jacobian c = det(A a {i, j} ). We set f (s) = e D −ν e e in formula (A. 19) above. The (Euclidean) integration domain is determined, according to (A.17), by 0 < pr ⊥ lin{q n+1 ,...,q M } ⊥ (q n ) 2 = G n /G n+1 for 1 ≤ n ≤ L. Therefore, a point on the boundary of the integration domain is determined by G n = 0 for some 1 ≤ n ≤ L, which is equivalent to a linear dependence q n ∈ lin {q n+1 , . . . , q M } and hence implies G 1 = 0.
Note that we have to analytically continue (A.19) from Euclidean to Minkowski space in order to obtain the Feynman integral (2.1). As Wick rotation turns d d k /(iπ d/2 ) exactly into the measure d d k /π d/2 on the left-hand side of (A. 19), we only have to remember that, due to our mostly-minus signature (1, − 1, . . . , − 1) of the Minkowski metric, the Euclidean scalar products on the right-hand side of (A.19) receive a factor (−1). For example, the M × M determinant G 1 turns into (−1) M Gr 1 ; similarly, G L+1 becomes (−1) E Gr. Overall, analytic continuation gives an additional factor of We absorb the last factor, together with the Jacobian c , into the constant prefactor c, and have thus finally arrived at (2.48).

B. The theory of Loeser-Sabbah
This section is devoted to Theorem 35, which was first stated in [59]. Beware that the original argument is flawed; a correct (but terse) proof was given in [61]. Our aim here is to provide a simplified and more detailed derivation. Throughout we will consider modules M over the algebra D N k = A N k [x −1 ] of differential operators (3.4) on the torus in some number N of variables x i , over some field k of characteristic zero. To lighten the notation, let us abbreviate θ i := x i ∂ i and set -modules, where M = M denotes the initial module M with the action of ν defined as νm := −x N ∂ N m. Since k(ν) is flat, this sequence remains exact after tensoring with k(ν) over k [ν]. Through identification of ν with −θ N − 1, we conclude that x ±1 N on M , but twisting the operator ∂ N to act like ∂ N + ν/x N . 42 The holonomicity of M implies that M x ν N is also holonomic, 43 and hence its push-forward Proof We can pick a generator of M (by holonomicity, M is cyclic as a D 1 k -module) and extend it to a (finite) basis of M (θ 1 ) as a vector space over k(θ 1 ), due to Corollary 66. Let N ⊂ M denote the k[θ 1 ]-module generated by such a basis, hence 2 , then note that the zeroes of b j+1 are (Z + j) ∪ (Z − j) and get pushed away from zero for increasing j. In particular, there exists some j 0 ∈ N such that b j (0) = 0 for all j > j 0 . For each such value of j, we can find u j , v j ∈ k[θ 1 ] such that 1 = u j (θ 1 )b j (θ 1 ) + v j (θ 1 )θ 1 ; then holds for every m ∈ N j . This proves ker(θ 1 ) ∩ N j ⊆ N j−1 for all j > j 0 , and therefore ker( In consequence, we have proven that in other words, the Koszul complexes DR(M ) = K (M ; ∂ 1 ) and K (N j 0 ; θ 1 ) are quasi-isomorphic (see Remark 68). The statement of the theorem thus reduces to the identity With this starting point, we can now prove Theorem 35 by induction. In fact, the higher-dimensional case can be seen as a straightforward corollary of the univariate case above. In contrast to [61], our demonstration avoids any reference to higherdimensional lattices.

Proof of Theorem 35
where k := k(θ ) and M := M (θ ) with θ := (θ 1 , . . . , θ N −1 ). Analogously, χ(ker ∂ N ) = dim k ker(∂ N ), where ∂ N denotes the action of ∂ N on M . In conclusion, we know that where we recognized the first line as the Euler characteristic of the de Rham complex of the D 1 k -module M and applied Theorem 69 to get to the last line (M = M (θ ) is holonomic by Lemma 65). So we only need to show that the left-hand side is equal to χ(M ). This is well known and follows from the Grothendieck spectral sequence. 45 Alternatively, an elementary way to obtain the identity χ(M /(∂ N M ))−χ(ker ∂ N ) = χ(M ) is given by the long exact sequence → H i+2 (DR(ker ∂ N )) → · · · (B.5) 45 Let π N : G N m −→ {pt} denote the projection to a point, such that π N = π N −1 • π . The identity π N + = π N −1 + • π + of the corresponding push-forwards in the derived category of D N k -modules implies that The key observation now is that ∂ N is injective on M /N and surjective on N . 47 We thus get short exact sequences To clarify this final step, first note that separating ∂ N from ∂ := (∂ 1 , . . . , ∂ N −1 ) yields an isomorphism of k-vector spaces y). 46 One only needs to check that x ±1 47 The first statement is clear since ker ∂ N ⊆ N . The second claim follows from the identity In this representation, the differential is given by which is known as the mapping cone of ∂ N : K (N ; ∂ ) −→ K (N ; ∂ ). Here, we denote by d the differential of K (N ; ∂ ).
If ∂ N is surjective, we can find x with ∂ N x = y and hence d (x ⊕0) = (d x ⊕ y) for every y. Therefore, every element of K (N ; ∂) has a representative of the form (x ⊕ 0), modulo exact forms. But such a form is closed, d (x ⊕ 0) = 0, if and only if x ∈ ker ∂ N ∩ ker d .
The proof of the second statement is very similar to the surjective case and left as a straightforward exercise.

C. A two-loop example
We demonstrate some main points of this article by a pedagogical example. The complete results of this calculation can be obtained from https://doi.org/10.5287/ bodleian:2RkGjPNG0, "Annihilators of the two-loop master integral". Consider the massless two-loop two-point graph with five propagators, graph WS 3 in Fig. 5. To this graph, we associate the family of integrals with two-loop momenta q 1 = l 1 , q 2 = l 2 and one external momentum q 3 = p. We normalize to − p 2 = 1. The graph polynomial G = U + F is given by the Symanzik polynomials with ω = ν 1 + ν 2 + ν 3 + ν 4 + ν 5 + 2s and s = −d/2.

C.1. From annihilators to integral relations
A set of generators of the annihilator ideal Ann A 5 [s] (G s ) can be derived in Singular [25] with algorithms introduced in [4]. Using the command SannfsBM, we obtain a set of 13 generators. To give an impression, the first five of them read We notice that this set includes one generator which is quadratic in the differential operators, reading Every operator in Ann A 5 [s] (G s ) gives rise to an integral relation. According to Lemma 7, we just need to replace each x i byî + and each ∂ i by −i − to obtain a shift relation between modified Feynman integrals. For example, for the generator P 1 , we obtain the shift operator

C.3. From IBP relations to annihilators
Going in the other direction, we can derive annihilators from momentum space IBP relations. In the usual way, inserting the differential operators Following the steps in the proof of Corollary 21, we derive for each shift operator O i j a parametric annihilator O i j . We obtain These operators are useful to compare both approaches as discussed next.

C.4. Comparing annihilators and IBP operators
According to Corollary 21, every momentum space IBP relation corresponds to a parametric annihilator. For our two-loop example, this is given by the fact that We may furthermore ask if the reverse is true: Can every annihilator of G be derived from IBP relations? If the answer would be no, the approach via parametric annihilators would provide new integral identities. While this question remains open for the general case, we can test it for simple Feynman graphs such as the present two-loop example.
In a first attempt, we could consider the shift relations obtained from the generators P 1 , . . . , P 13 and try to confirm that they are combined IBP relations. If we use one of the well-known implementations of Laporta's algorithm to reproduce e.g. Eq. (C.5), we have to fix the values of ν 1 , . . . , ν 5 and do not answer the question for arbitrary values of the ν i . We therefore approach the problem on the level of parametric differential operators instead.
We find that actually not all parametric annihilators are contained in Mom; however, they turn out to still be consequences of the momentum space IBP relations in the following sense: While we checked that P 1 / ∈ Mom, we can find a polynomial q 1 ∈ Q[s, x 1 ∂ 1 , . . . , x 5 ∂ 5 ] such that q 1 P 1 ∈ Mom. Recall that, under the Mellin transform, such a q 1 corresponds to a polynomial in the dimension and in the ν e . The interesting question then is if we can find a polynomial q ∈ Q[s, x 1 ∂ 1 , . . . , x N ∂ N ] for every P ∈ Ann A N [s] (G s ) such that q P ∈ Mom. If we can find such a q i for every generator P i , we can express every annihilator in terms of the O i j . The q i are the denominators of the coefficients in such a linear combination.

C.5. The number of master integrals
The result of the previous subsection implies that the parametric approach and momentum space IBP lead to the same number of master integrals for this example. Indeed, as mentioned in Sect. 4.1, we compute the Euler characteristic Notice that symmetries of the graph were not taken into account here, which in Azurite is assured by setting Symmetry -> False and GlobalSymmetry -> False. For an integral reduction in practice, one would of course make use of the symmetry I(ν 1 , ν 2 , ν 3 , ν 4 , ν 5 ) = I(ν 2 , ν 1 , ν 4 , ν 3 , ν 5 ) and compute with one of the sets {I 1 , I 2 }, {I 1 , I 3 }.