Feynman integral relations from parametric annihilators
Abstract
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.
Keywords
Feynman integrals Master integrals Integration by parts IBP Dmodule Euler characteristic1 Introduction
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 wellunderstood topological invariant: the Euler characteristic of the complement of a hypersurface \(\left\{ \mathcal {G}=0 \right\} \) associated to the Feynman graph. Therefore, many powerful tools are available for its computation.
Ideals of parametric annihilators are examples of Dmodules. Loeser and Sabbah studied the algebraic Mellin transform [60] of holonomic Dmodules 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 \(\mathcal {G}^{d/2}\), 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 Dmodule 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 twoloop example in detail.
2 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 wellknown IBP relations from momentum space.
2.1 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 Dmodules that our subsequent discussion will be based on.
An integral (2.1) is a function of the indices\(\nu =(\nu _1,\ldots ,\nu _N)\) (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 \((d,\nu )\in \mathbb {C}^{1+N}\) in the sense of Speer [83].
Definition 1
Schwinger parameters yield useful representations of the Feynman integrals (2.1):
Proposition 2
Example 3
Remark 4
(meromorphicity) Depending on the values of \(d\) and \(\nu \), integrals (2.1) and (2.5)–(2.7) can be divergent. However, there exists a nonempty, open domain in \(\mathbb {C}^{1+N} \ni (d,\nu )\) where all of them are convergent and agree with each other.^{4} From there, analytic continuation defines a unique, meromorphic extension of every Feynman integral to the whole parameter space \(\mathbb {C}^{1+N}\). 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 \(\nu \in \mathbb {Z}^N\) 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 \(\nu \)) is very important; in particular, it means that an identity between Feynman integrals is already proven once it has been established locally in the nonempty 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
2.2 Integral relations and the Mellin transform
In this section, we summarize how relations between \(\nu \)shifted Feynman integrals can be identified with differential operators that annihilate \(\mathcal {G}^{d/2}\). This method was suggested in [56] (and in [7] for the Baikov representation).
The parametric representation (2.7) can be interpreted as a multidimensional Mellin transform. For our purposes, we slightly deviate from the standard definition, as for example given in [18] and include the factors \(\Gamma (\nu _i)\) that occur in (2.7).
Definition 6
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 \(\Gamma \)’s in (2.10).
Lemma 7
 1.
Linearity: \( \mathcal {M}\left\{ \alpha f+\beta g \right\} (\nu ) =\alpha \mathcal {M}\left\{ f \right\} (\nu ) + \beta \mathcal {M}\left\{ g \right\} (\nu ) \),
 2.
Multiplication: \( \mathcal {M}\left\{ x_i f \right\} (\nu ) =\nu _i \mathcal {M}\left\{ f \right\} (\nu +\mathsf {e}_{i}) \) and
 3.
Differentiation: \( \mathcal {M}\left\{ \partial _i f \right\} (\nu ) =\mathcal {M}\left\{ f \right\} (\nu \mathsf {e}_{i}) \).
Proof
2.3 Operator algebras and annihilators
The Mellin transform relates differential operators acting on \(\mathcal {G}^{d/2}\) with operators that shift the indices \(\nu \) of the Feynman integrals \(\mathcal {I}(\nu )\). In this section, we formalize this connection algebraically in the language of Dmodules. 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
Definition 9
Definition 10
Example 11
Note that an annihilator ideal is a module over \({A}^{N}[s]\), that is, whenever \(P f^s=0\), also \((QP)f^s = Q(P f^s) = 0\) for any operator \(Q\in {A}^{N}[s]\). These ideals are studied in Dmodule theory [24, 72] and in principle annihilators can be computed algorithmically with computer algebra systems such as Singular [3, 4, 25]. For the study of the Feynman integrals (2.7), we set \(s=d/2\) and \(f=\mathcal {G}=\mathcal {U}+\mathcal {F}\) is the polynomial from (2.3). Via the Mellin transform, the elements of \({A}^{N}[s] \mathcal {G}^s\) are the integrands of shifts of the Feynman integral \(\mathcal {M}\left\{ \mathcal {G}^s \right\} \). Due to Lemma 7, every annihilator \( P \in {{\mathrm{Ann}}}_{{A}^{N}[s]}\left( \mathcal {G}^s\right) \) corresponds to an identity of Feynman integrals with shifted indices \(\nu \).
Definition 12
Corollary 13
Corollary 14
Every annihilator \(P \in {{\mathrm{Ann}}}_{{A}^{N}[s]}\left( \mathcal {G}^{s} \right) \) of \(\mathcal {G}^{s}=\mathcal {G}^{d/2}\) yields a shift relation \(\mathcal {M}\left\{ P \right\} \in {S}^{N}[s] :={S}^{N} \otimes \mathbb {C}[s]\) of the Feynman integral \(\widetilde{\mathcal {I}}\) from (2.11): \(\mathcal {M}\left\{ P \right\} \widetilde{\mathcal {I}}= 0\).
Example 15
Example 16
Let us recapitulate these observations: By the Mellin transform (2.16), every annihilator \(P \in {{\mathrm{Ann}}}_{{A}^{N}[s]}(\mathcal {G}^s)\) gives rise to a linear relation \(\mathcal {M}\left\{ P \right\} \widetilde{\mathcal {I}}=0\) of the rescaled Feynman integral \(\widetilde{\mathcal {I}}\) via a shift operator \(\mathcal {M}\left\{ P \right\} \in {S}^{N}[s]\). Also we noted that according to \(\widetilde{\mathcal {I}}= \mathcal {I}\, \Gamma (s\omega )/\Gamma (s) \) from (2.11), such a relation is equivalent to a relation of the original Feynman integral \(\mathcal {I}\) as in (2.22).
On the other hand, if we are given a shift relation \(R\widetilde{\mathcal {I}}= 0\) where \(R\in {S}^{N}[s]\), then \(R=\mathcal {M}\left\{ P \right\} \) corresponds to the differential operator \(P=\mathcal {M}^{1}\{R\}\in {A}^{N}[s]\) under the Mellin transform (2.16). From the vanishing \(\mathcal {M}\left\{ P \mathcal {G}^s \right\} =0\) we can conclude that this operator must be an annihilator, \(P \in {{\mathrm{Ann}}}_{{A}^{N}[s]}(\mathcal {G}^s)\). This follows from
Theorem 17
So not only do we get relations for Feynman integrals from parametric annihilators of \(\mathcal {G}^s\), but in fact every relation of the form \(R\widetilde{\mathcal {I}}=0\) for a polynomial shift operator \(R\in {S}^{N}[s]\) does arise in this way.
Corollary 18
2.4 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.
Definition 19
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
Corollary 21
Proof
Note that the proof of the identity \(\widetilde{\mathbf{O }}_{j}^{i} \mathcal {G}^s = 0\) given in Corollary 21 rests on the inverse Mellin transform. An alternative, direct algebraic proof is given in Appendix A.2.^{8}
Definition 22
Question 23
For a Feynman integral with a complete set of irreducible scalar products, is the second inclusion in (2.39) an equality? In other words, are the sparametric annihilators of Lee–Pomeransky polynomials \(\mathcal {G}\) linearly generated?
Regarding the first inclusion in (2.39), we do know that it is strict (see the example in Appendix C.4). However, it seems that \({{\mathrm{Mom}}}\) does provide all identities once we enlarge the coefficients to rational functions in the dimension \(s=d/2\) and the indices \(\nu _e\). Let us write \(\theta =(\theta _1,\ldots ,\theta _N)\) and \(\theta _e :=x_e \partial _e\) such that \(\nu _e=\mathcal {M}\left\{ \theta _e \right\} \).
Question 24
To test this conjecture, we should take all known shift relations for Feynman integrals, and check if they can be realized as elements of \({{\mathrm{Mom}}}\) (after localizing at \(\mathbb {C}(s,\theta )\)). In the remainder of this section, we will address such a relation, namely the one originating from the wellknown dimension shifts.
2.5 Dimension shifts
Remark 25
A lowering dimension shift, expressing \(\mathcal {I}(d+2)\) in terms of \(\mathcal {I}(d)\), corresponds to a Bernstein–Sato operator of \(\mathcal {G}\) under the Mellin transform \(\widetilde{\mathcal {I}}(d)=\mathcal {M}\left\{ \mathcal {G}^{d/2} \right\} \)^{11}:
Definition 26
Corollary 27
Proof
By (2.44), \(f^{sn} = \frac{P(sn)}{b(sn)} \cdots \frac{P(s1)}{b(s1)} f^s \in {A}^{N}_k \cdot f^s\) for all \(n\in \mathbb {N}\). \(\square \)
In general, computing a Bernstein operator is not at all trivial. But in the case of a complete set of irreducible scalar products (\(N=\left\Theta \right\) and \(\mathcal {A}\) 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]:
Definition 28
The Baikov polynomial\(\mathcal {P}(y) \in \mathbb {C}[y_1,\ldots ,y_N]\) is the polynomial defined by \(\mathcal {P}(\mathsf {D}_1,\ldots ,\mathsf {D}_N) = \mathsf {Gr}_1(s)\).
Theorem 29
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
Corollary 30
Proof
According to (2.48), \(\mathcal {I}(d+2)\) is obtained by multiplying the integrand of \(\mathcal {I}(d)\) with \((1)^{L} \mathcal {P}(y)/\mathsf {Gr}\) and adjusting the \(\Gamma \)factors in the prefactor as \(\Gamma \left( \frac{d+2E}{2}\right) = \frac{dE}{2} \Gamma \left( \frac{dE}{2} \right) \) and so on. Multiplying the integrand of the Baikov representation with \(y_e\) is equivalent to decrementing \(\nu _e\); hence, the multiplication of the integrand by \(\mathcal {P}(y)\) can be written as the action of \(\mathcal {P}(\mathbf{1 }^{},\ldots ,\mathbf{N }^{})\) on the integral. \(\square \)
Proposition 31
Proof
Corollary 32
If the degree of the Baikov polynomial \(\mathcal {P}(y)\) is not more than \(L+1\), then the Bernstein–Sato polynomial b(s) of the Lee–Pomeransky polynomial \(\mathcal {G}\) is a divisor of \((s+1){\tilde{b}}(s)\). In particular, all roots of \(b(s)/(s+1)\) are simple and at halfintegers.
Note that \(\deg \mathcal {P}(y) \le \min \left\{ 2L,M \right\} = L+ \min \left\{ L,E \right\} \) 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 (\(L=1\)).
3 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 \(\mathcal {G}=0\) inside the torus \(\mathbb {G}_{m}^N\). (We write \(\mathbb {G}_{m}= \mathbb {A}{\setminus }\left\{ 0 \right\} \) for the multiplicative group and \(\mathbb {A}\) for the affine line.) For a full understanding of this section, some knowledge of basic Dmodule theory is indispensable, but we tried to include sufficient detail for the main ideas to become clear to nonexperts as well. In particular, we will give selfcontained proofs that only use Dmodule theory at the level of [24].
Definition 33
Note that this is the same as the dimension of the space \(\sum _n \mathbb {C}(s,\nu ) \mathcal {I}_{\mathcal {G}}(\nu +n)\) of Feynman integrals (2.1), because the ratios \(\mathcal {I}_{\mathcal {G}}(\nu +n)/\widetilde{\mathcal {I}}_{\mathcal {G}}(\nu +n) = \Gamma (s)/\Gamma (s\omega \leftn \right)\) with \(\leftn \right=n_1+\cdots +n_N\) are all related by a rational function in \(\mathbb {C}(s,\nu )\), see (2.22).
Remark 34
 1.
Almost always the integrals are considered only for integer indices \(\nu \in \mathbb {Z}^N\), instead of as functions of arbitrary indices. In this setting, integrals with at least one \(\nu _e=0\) 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 setup 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 \(\nu _e\).
 3.
Some authors do not count integrals if they can be expressed in terms of \(\Gamma \)functions or products of simpler integrals, for example [41, 48].
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 \(\nu \in \mathbb {Z}^N\), using the momentum space representation. Below we will show that this result holds much more generally, for unconstrained \(\nu \), and that it becomes a very natural statement once it is viewed through the parametric representation. Notably, it remains true for Mellin transforms \(\mathcal {M}\left\{ \mathcal {G}^s \right\} \) of arbitrary polynomials \(\mathcal {G}\)—the fact that \(\mathcal {G}\) comes from a (Feynman) graph is completely irrelevant for this section.
Holonomic modules are, in a precise sense, the most constrained and behave in many ways like finitedimensional 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 finitedimensional 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].
Theorem 35
(Loeser and Sabbah [59, 61]) Let \(\mathscr {M}\) denote a holonomic \({A}^{N}_k\)module. Then, \(F \otimes _R \mathscr {M}\) is a finitedimensional vector space over F. Moreover, its dimension is given by the Euler characteristic \(\dim _F (F \otimes _R \mathscr {M}) = \chi \left( \iota ^{*} \mathscr {M}\right) \).
In “Appendix B”, we provide a selfcontained 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.
Proof
We can invoke \(\dim _F(F \otimes _R \iota ^{*} \mathscr {M}) = \chi (\iota ^{*}\mathscr {M}) < \infty \) from [61, Théorème 2]. To conclude, we just need to note that \(F \otimes _R \mathscr {M}\) and \(F \otimes _R \mathscr {M}[x^{1}] = F \otimes _R \iota ^{*}\mathscr {M}\) are isomorphic vector spaces (over F). This is clear since each coordinate \(x_i\) is invertible after localizing at F: Due to \(\partial _i x_i = 1+x_i \partial _i\), we find that \((1+x_i \partial _i)^{1} \otimes \partial _i \in F\otimes _R {A}^{N}_k\) is an inverse to \(1\otimes x_i\). \(\square \)
Since we are interested in Feynman integrals, we consider the special case where the \({A}^{N}_k\)module \(\mathscr {M}\) is simply \(\mathscr {M}={A}^{N}_k \cdot \mathcal {G}^s\) from Definition 9. Its elements can be written uniquely in the form \(h\cdot \mathcal {G}^s\), where \(h \in k[x,\mathcal {G}^{1}]\), such that \({A}^{N}_k \cdot \mathcal {G}^s \cong k[x,\mathcal {G}^{1}]\) by (2.46) are isomorphic as k[x]modules: \(x_i (h \mathcal {G}^s) = (x_i h) \mathcal {G}^s\). The action (2.13) of the derivatives, however, is twisted by a term proportional to s: \(\partial _i (h \mathcal {G}^s) = \mathcal {G}^s (\partial _i h + sh(\partial _i \mathcal {G})/\mathcal {G})\). Despite this twist, we find that the Euler characteristic stays the same:
Proposition 36
In particular, we can dispose of the parameter s completely and compute with the algebraic de Rham complex of \(\mathbb {C}[x^{\pm 1},\mathcal {G}^{1}]\), which is the ring of regular functions of the complement of the hypersurface \(\mathbb {V}(\mathcal {G})=\left\{ x : \mathcal {G}(x)=0 \right\} \) in the torus \(\mathbb {G}_{m}^N\). Combining Theorem 35 with Proposition 36, we thus obtain our main result:
Corollary 37
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 \(\nu \)). 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 \({A}^{N} \cdot \mathcal {G}^s \subsetneq \mathbb {C}[x,\mathcal {G}^{1}]\) is a proper submodule (note \(k=\mathbb {C}(s)=\mathbb {C}\)). While Theorem 35 still applies and relates the number of master integrals in a fixed dimension to the Euler characteristic of \(\mathcal {D}^{N} \cdot \mathcal {G}^s\), 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

\(\mathscr {M}'=j^{*} \mathscr {M}\) is the pullback of \(\mathscr {M}\) under the inclusion Open image in new window ,

\(\mathscr {M}' f^s = F_{*} \mathscr {M}'\) is the pushforward of \(\mathscr {M}'\) under the closed embedding Open image in new window which sends x to (x, f(x)).^{20}
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 \({A}^{N}_k\)modules \(k[x^{\pm 1},\mathcal {G}^{1}]\) and \(k[x^{\pm 1}] \mathcal {G}^s\) 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 \(\partial _i \mapsto \partial _i + s (\partial _i \mathcal {G})/\mathcal {G}\) of \(\mathcal {D}^{N}_k[\mathcal {G}^{1}]\).
3.1 No master integrals
Lemma 40
Proof
Clearly, \(\mathfrak {M}=\left\{ 0 \right\} \) requires \(f^s\) to be mapped to zero in the localization \(\mathfrak {M}\) of \(\mathscr {M}\) at \(\mathbb {C}[s,\theta ]{\setminus }\left\{ 0 \right\} \), and therefore the existence of a nonzero polynomial \(P(\theta ,s)\in \mathbb {C}[s,\theta ]\) with \(P(\theta ,s)\cdot f^s =0\). Conversely, given such an operator, its shifts \(P(\theta \alpha ,s+r)\) by \((r,\alpha )\in \mathbb {Z}^{1+N}\) annihilate the elements \(x^{\alpha } \cdot f^{s+r}\), which are therefore all mapped to zero in \(\mathfrak {M}\). By linearity, this proves \(\mathfrak {M}=\left\{ 0 \right\} \), because every element of \(\mathfrak {M}\) can be written as \(g f^{s+r} \otimes h\) for some \(r \in \mathbb {Z}\), \(h \in \mathbb {C}[s,\theta ]{\setminus }\left\{ 0 \right\} \) and a Laurent polynomial \(g \in k[x^{\pm 1}]\). \(\square \)
In particular, the presence of a linear annihilator (3.15) implies \(\mathfrak {M}=\left\{ 0 \right\} \) and hence \(\chi (\mathbb {G}_{m}^N{\setminus }\mathbb {V}(f))=0\) via Corollary 37. Note that we could equally phrase this in terms of the hypersurface \(\mathbb {V}(f) \subset \mathbb {G}_{m}^N\) itself as \(\chi (\mathbb {V}(f))=0\), because the Euler characteristics are related through \(\chi (\mathbb {V}(f))=\chi (\mathbb {G}_{m}^N{\setminus }\mathbb {V}(f))\) (see Sect. 3.3).
When \(f=\mathcal {G}\) 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 \(\nu \). In fact, \(\mathfrak {M}=\left\{ 0 \right\} \) dictates that the only value one can assign to \(\mathcal {M}\left\{ f^s \right\} (\nu )\) 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:
Proposition 41
Let \(f \in \mathbb {C}[x_1,\ldots ,x_N]\) denote a polynomial. Then, the hypersurface \(\left\{ f=0 \right\} \) inside the torus \(\mathbb {G}_{m}^N\) has vanishing Euler characteristic precisely when there are \(\lambda _0,\ldots ,\lambda _N \in \mathbb {Z}\), not all zero, such that (3.14) holds.
Corollary 42
Let \(f \in \mathbb {C}[x_1,\ldots ,x_N]\) denote a polynomial. Then, \(\mathbb {V}(f) \subset \mathbb {G}_{m}^N\) has Euler characteristic zero if and only if it is isomorphic to a product of \(\mathbb {G}_{m}\) times a hypersurface \(\left\{ g=0 \right\} \subset \mathbb {G}_{m}^{N1}\).
Proof of Proposition 41
We proceed by induction over the dimension N, and we will assume f to be nonconstant. (The proposition holds trivially for any constant \(f\in \mathbb {C}\).) In the case \(N=1\), the variety \(\mathbb {V}(f) \subset \mathbb {C}^{*}\) is a finite set, and hence, its Euler characteristic coincides with its cardinality. Therefore, \(\chi (\mathbb {V}(f))=0\) if and only if f has no zero inside the torus. This is only possible if f is proportional to a monomial \(x_1^r\); in particular, f must be homogeneous and we are done.
Now consider \(N>1\) and assume that \(\chi (\mathbb {G}_{m}^N{\setminus }\mathbb {V}(f))=\chi (\mathbb {V}(f))=0\). Recall that (3.14) is equivalent to degeneracy (3.18) of \(\mathsf {NP}\left( f\right) \), so we only need to rule out the nondegenerate case. We achieve this by exploiting the hypothesis \(\dim \mathsf {NP}\left( f\right) =N\) to construct a linear annihilator \(P^{\lambda }_s\) of \(f^s\), which implies (3.18) in contradiction to the nondegeneracy of \(\mathsf {NP}\left( f\right) \).
Remark 43
In summary, we showed that the following six conditions on a polynomial \(f\in \mathbb {C}[x_1,\ldots ,x_N]\) are equivalent: (1) homogeneity (3.14) of f, (2) existence (3.15) of an annihilator of \(f^s\) linear in \(\theta \), (3) existence (3.16) of an annihilator of \(f^s\) polynomial in \(\theta \), (4) degeneracy (3.18) of the Newton polytope \(\mathsf {NP}\left( f\right) \), (5) vanishing of the Euler characteristic \(\chi (\mathbb {G}_{m}^N{\setminus }\mathbb {V}(f))=\chi (\mathbb {V}(f))=0\) and (6) divisibility of \(\mathbb {V}(f)\) by \(\mathbb {G}_{m}\) as stated in Corollary 42.
To conclude, let us interpret our observation in the light of the wellknown result, due to Kouchnirenko [49, Théorème IV] and Khovanskii [47, Theorem 2 in section 3], that relates the Euler characteristic to the volume of the Newton polytope:
Theorem 44
For almost all polynomials \(f \in \mathbb {C}[x_1,\ldots ,x_N]\) with a fixed Newton polytope, we have \(\chi (\mathbb {G}_{m}^N{\setminus }\mathbb {V}(f))=(1)^N \cdot N! \cdot {{\mathrm{Vol}}}\mathsf {NP}\left( f\right) \).
For polynomials f whose nonzero coefficients are sufficiently generic, Proposition 41 follows from Theorem 44. Our proof shows that when \({{\mathrm{Vol}}}\mathsf {NP}\left( f\right) =0\), the theorem applies without any constraints on the nonzero coefficients of f. In fact, this statement extends to the case when \(N!{{\mathrm{Vol}}}\mathsf {NP}\left( f\right) =1\), because it is known that \((1)^N\cdot \chi (\mathbb {G}_{m}^N{\setminus }\mathbb {V}(f))\) is always bounded from above by \(N!{{\mathrm{Vol}}}\mathsf {NP}\left( f\right) \), for all f, leaving only the possibilities \(\left\{ 0,1 \right\} \) for the signed Euler characteristic \((1)^N\cdot \chi (\mathbb {G}_{m}^N{\setminus }\mathbb {V}(f))\).
3.2 Graph polynomials
Our discussion so far applies to all integrals of type (2.1)—the defining data are thus the set \(\mathsf {D}=(\mathsf {D}_1,\ldots ,\mathsf {D}_N)\) 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 \(\mathcal {U}\), \(\mathcal {F}\) and \(\mathcal {G}\) 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
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 \(\mathcal {G}_G\) 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 \(k\in \mathbb {N}\). 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, wellknown combinatorial formulas for the Symanzik polynomials [14, 81], which go back at least to [64].
Proposition 46
In Sect. 4.2, we will use these formulas to count the sunrise integrals.
Example 47
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 nonexceptional 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).
3.3 The Grothendieck ring of varieties
Since we are from now on only interested in the Euler characteristic, we can simplify calculations by abstracting from the concrete variety \(\mathbb {V}(\mathcal {G}) :=\left\{ \mathcal {G}=0 \right\} \) to its class \([\mathcal {G}]\) in the Grothendieck ring \(K_0(\text {Var}_{\mathbb {C}})\). This ring is the free Abelian group generated by isomorphism classes [X] of varieties over \(\mathbb {C}\), modulo the inclusion–exclusion relation \([X]=[X{\setminus } Z] + [Z]\) for closed subvarieties \(Z \subset X\). It is a unital ring for the product \([X]\cdot [Y] = [X \times Y]\) with unit \(1=[\mathbb {A}^0]\) given by the class of the point. Crucially, the Euler characteristic factors through the Grothendieck ring, since it is compatible with these relations: \(\chi (X) = \chi (X{\setminus } Z) + \chi (Z)\) and \(\chi (X\times Y) = \chi (X)\cdot \chi (Y)\). The class \(\mathbb {L}= [\mathbb {A}^1]\) of the affine line is called Lefschetz motive and fulfils \(\chi (\mathbb {L})=1\). For several polynomials \(P_1,\ldots ,P_n\), we write \(\mathbb {V}(P_1,\ldots ,P_n) :=\left\{ P_1=\cdots =P_n=0 \right\} \).
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 \(\mathbb {G}_{m}^N\) and not the affine plane \(\mathbb {A}^N\).
Lemma 48
Proof
Consider the hypersurface \(\mathbb {V}(\mathcal {G})\subset \mathbb {G}_{m}^{N}\), defined by \(\mathcal {G}:=A+x_N B\), under the projection \(\pi :\mathbb {G}_{m}^N \longrightarrow \mathbb {G}_{m}^{N1}\) that forgets the last coordinate \(x_N\).
Corollary 49
Proof
Recall that \(\mathcal {U}\) and \(\mathcal {F}\) are homogeneous of degrees \(L\) and \(L+1\), respectively (Corollary 63). Since multiplication with \(x_N \in \mathbb {G}_{m}\) is invertible, we can rescale all variables \(x_i\) with \(i<N\) by \(x_N\). This change of coordinates transforms \(\mathcal {G}\) into \(x_N^{L} ({\widetilde{\mathcal {U}}}+x_N {\widetilde{\mathcal {F}}})\). Since \(x_N \ne 0\), this shows that \( [\mathbb {G}_{m}^N {\setminus } \mathbb {V}(\mathcal {G})] = [\mathbb {G}_{m}^N {\setminus } \mathbb {V}({\widetilde{\mathcal {U}}} + x_N {\widetilde{\mathcal {F}}})] \) such that the claim is just a special case of Lemma 48. \(\square \)
Example 50
Proof
Lemma 51
Proof
Corollary 52
Let G be a graph with a pair \(\left\{ e,f \right\} \) of massless edges in series or in parallel. Then, \(\mathfrak {C}\left( G\right) = \mathfrak {C}\left( G'\right) \) where \(G'\) is the graph obtained by replacing the pair with a single edge. In other words, repeated application of the seriesparallel operations from Fig. 3 does not change the number of master integrals.
4 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.
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.
4.1 Linearly reducible graphs
If the polynomial \(\mathbb {V}(f)=a+x_N b\) 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 \(\mathbb {V}(f)\) (or its complement). Analogous formulas also exist in the case of a variety \(\mathbb {V}(f_1,\ldots ,f_n)\) 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.
At the end of a full linear reduction, the class of \(\mathbb {G}_{m}^N{\setminus } \mathbb {V}(\mathcal {G})\) in the Grothendieck ring is expressed as a polynomial in the Lefschetz motive \(\mathbb {L}\). To get the Euler characteristic, one then merely needs to substitute \(\mathbb {L}=[\mathbb {A}^1] \mapsto \chi (\mathbb {A}^1)=1\).
Example 53
G  \(P_1\)  \(P_2\)  \(P_3\)  \(P_4\)  \(P_5\)  \(P_6\)  \(P_7\)  \(F_1\)  \(F_2\)  \(F_3\)  \(F_4\)  \(F_5\)  \(F_6\)  \(F_7\)  \(F_8\)  \(F_9\) 
\(\mathfrak {C}\left( G\right) \)  16  10  10  10  10  15  22  1  4  5  4  5  20  24  12  13 
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 \(\mathbb {V}(\mathcal {U})\) was computed for all wheel graphs in [17]. It is possible to adapt such calculations to our setting (where the ambient space is \(\mathbb {G}_{m}^N\) instead of \(\mathbb {A}^N\)). For example, we could prove
Proposition 54
The proof of this and related results will be presented elsewhere.
4.2 Sunrise graphs
Proposition 55
The \(L\)loop sunrise graph Open image in new window from Fig. 6 with \(L+1\) nonzero masses (and nonexceptional external momentum) has Open image in new window 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
It should be clear that our calculation can be adapted to the situation when some masses are zero. Let us demonstrate how to obtain another result of [43]^{33}:
Proposition 56
The \(L\)loop sunrise graph with \(R \le L\) nonzero masses, \(L+1R\) vanishing masses and nonexceptional external momentum, has Open image in new window 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 \(L=R \ge 1\). (The totally massless case \(R=0\) reduces to the trivial case of a single edge.) Label the edges such that the massless edge is \(m_{L+1}=0\).
4.3 General algorithms
Counts of master integrals according to (3.10) computed with Macaulay2’s Euler for some graphs for massless and massive internal propagators. (As no symmetries are regarded, all masses can be assumed to be different from each other.) All external momenta are assumed to be nondegenerate (nonzero and not on any internal mass shell) in both cases
Graph G  

\(\mathfrak {C}\left( G\right) \) massless  4  11  3  4  20 
\(\mathfrak {C}\left( G\right) \) massive  7  15  30  19  55 
Example 57
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 2loop propagator \({\mathrm{WS}}_{3}'\) from Example 53 with only \(\mathfrak {C}\left( {\mathrm{WS}}_{3}'\right) =3\) master integrals grows to carry \(\mathfrak {C}\left( {\mathrm{WS}}_{3}'\right) =30\) 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 Dmodules and Gröbner bases and tends to demand more resources than the method discussed above.
Example 58
4.4 Comparison to other approaches
We successfully reproduced all of our results above (the wheels \({\mathrm{WS}}_{L}'\) with \(L\le 6\), the sunrises Open image in new window with \(L\le 4\) 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 \(\nu _e\) and no relation between these indices for different edges is assumed.
Example 59
For the graph G in Fig. 7, the Euler characteristic gives \(\mathfrak {C}\left( G\right) = 15\), 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 \(p_4^2=0\), the graph polynomials (and hence the Feynman integrals) are identical to those of the vacuum graph \(G'\) in Fig. 7. Since 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 \(G/\left\{ 1,2 \right\} \) obtained by contracting a further edge (3 or 4), and obtain the two tadpoles (see Fig. 7) consisting only of edges \(\left\{ 4,5 \right\} \) and \(\left\{ 3,5 \right\} \), 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 twoloop sunrise Open image in new window with one massless line has Open image in new window 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 \(\nu \in \mathbb {Z}^N\), how many toplevel 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 \(H^N(\mathbb {C}^N{\setminus } \mathbb {V}(\mathcal {G}))\).^{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 toplevel integrals does not literally make sense in our setting of arbitrary, noninteger indices \(\nu \). 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 \(\nu _e\) at all; the latter do.) However, discarding integrals from quotient graphs suggests a definition of a number as follows.
Remark 60
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}
5 Outlook
We have studied linear relations between Feynman integrals that arise from parametric annihilators of the integrand \(\mathcal {G}^s\) 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 \(\mathcal {G}^s\)) follow from momentum space relations remains open (see Question 24). We showed that the wellknown 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 \({{\mathrm{Ann}}}(\mathcal {G}^s)={{\mathrm{Ann}}}^1(\mathcal {G}^s)\) is linearly generated, remains to be settled. A positive answer to either of these would imply that the labourintensive computation of the parametric annihilators could be simplified considerably (\({{\mathrm{Mom}}}\) is known explicitly, and \({{\mathrm{Ann}}}^1\) 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 \(\nu _a\) 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 toplevel 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 \(\left\{ \mathcal {G}=0 \right\} \) determined by the Lee–Pomeransky polynomial \(\mathcal {G}\). 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 sparametric annihilators generating the integral relations, and the Euler characteristic giving the number of master integrals—are wellstudied objects in the theory of Dmodules and furthermore algorithms for their automated computation are available in principle.
In particular, we hope that this parametric, Dmodule 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 \(\nu _a\) 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 onescale subgraphs (both situations introduce noninteger indices). For a recent step into this direction, see [91].
Footnotes
 1.
 2.
 3.
The scalar products \(\ell _{i} \cdot \ell _{j} = \ell _i^1 \ell _j^1  \sum _{\mu =2}^{d} \ell _{i}^{\mu } \ell _{j}^{\mu }\) are understood with respect to the Minkowski metric.
 4.
The only exception are cases of zeroscale subintegrals, like massless tadpoles. Such integrals are zero in dimensional regularization and therefore irrelevant for our considerations.
 5.
 6.
For integrals associated to Feynman graphs, the number of edges is often less than \(\left\Theta \right\). In this case, one augments the list of inverse propagators by an appropriate choice of additional quadratic forms in the loop momenta, called irreducible scalar products (ISPs), to achieve \(N=\left\Theta \right\). See Example 62.
 7.
Recall that \(q_{1},\ldots , q_{L}\) denote the loop momenta, whereas \(q_{L+1},\ldots , q_{M}\) are the external momenta.
 8.
For an algebraic proof of the analogous statement in the Baikov representation, see [35, section 9].
 9.
Tarasov considered the special case where all inverse propagators are of the form \(\mathsf {D}_e = k_e^2  m_e^2\), and hence, \(\mathcal {U}\) is just the graph (first Symanzik) polynomial from (3.22).
 10.
In fact, \( H(s)\mathcal {G}+ s\mathcal {U}= \sum _e x_e \left( \mathcal {G}\partial _e  (s1) (\partial _e \mathcal {G}) \right) \) follows from the trivial annihilators (2.14) of \(\mathcal {G}^{s1}\).
 11.
Tkachov proposed in [93] to use a generalization of (2.44) to several polynomials (the individual Symanzik polynomials \(\mathcal {U}\) and \(\mathcal {F}\), instead of \(\mathcal {G}=\mathcal {U}+\mathcal {F}\)), 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]).
 12.
The first fraction can also be written as \(2^{L}/\left( dLE+1 \right) _{L}\) in terms of the Pochhammer symbol (raising factorial) \(a_{L} = a (a+1) \cdots (a+L1)\).
 13.
Recall that a holonomic module over the point \(\mathbb {A}^0_k\) is the same as a finitedimensional kvector space.
 14.
The de Rham complex \({{\mathrm{DR}}}({A}^{N}_k)\) is a resolution of k[x] by free \({A}^{N}_k\)modules, such that \(H^{\bullet } ({{\mathrm{DR}}}(\mathscr {M}'))\) are the (left) derived functors of \(\pi _{*} \mathscr {M}'=H^0({{\mathrm{DR}}}(\mathscr {M}'))\). In the language of derived categories, saying that \(\pi _{*} \mathscr {M}'\) is holonomic actually means precisely that \({{\mathrm{DR}}}(\mathscr {M}')\) is a complex with cohomology groups that are holonomic modules over the point—that is, finitedimensional vector spaces over k.
 15.
This sign arises from the shift by N in the Definition (3.6) of the de Rham complex \({{\mathrm{DR}}}\).
 16.
It fails precisely if, for some \(r \in \mathbb {N}\), \(sr\) is a zero of the Bernstein–Sato polynomial of \(\mathcal {G}\).
 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 setup where the indices are treated as free parameters.
 18.
It would be more consistent with Definition 9 to write \(\mathscr {M}'[s]f^s\) instead of \(\mathscr {M}' f^s\), but we prefer the shorter form to avoid clutter and to stress that we view it as an \({A}^{N}_{\mathbb {C}}\)module, not a \({A}^{N}_{\mathbb {C}[s]}\)module.
 19.
This is also clear from the fact that \(\mathscr {M}' f^s/(\partial _t \mathscr {M}' f^s) = \pi _{*} (\mathscr {M}' f^s)\) is the pushforward of \(\mathscr {M}' f^s\) under the projection \(\pi :\mathbb {A}^{N+1} \longrightarrow \mathbb {A}^N\) forgetting the last coordinate. Namely, since \(\mathscr {M}' f^s = F_{*} \mathscr {M}'\) as we discuss below, \(\pi _{*} (F_{*} \mathscr {M}') = (\pi \circ F)_{*} \mathscr {M}' = {{\mathrm{id}}}_{*} \mathscr {M}' = \mathscr {M}'\).
 20.
It follows from (3.12) that \(\mathscr {M}'f^s=\bigoplus _{n \ge 0} \partial _t^n \mathscr {M}' \cong F_{*} \mathscr {M}\) as \(\mathbb {C}[x]\)modules, since \(\partial ^n_t \mathscr {M}'=\partial ^n_t t^n \mathscr {M}' \equiv s^n \mathscr {M}' \mod s^{<n} \mathscr {M}'\). Furthermore, the derivatives act on \(F_{*} \mathscr {M}\) by \(\partial _i \bullet m(s) = (\partial _i  (\partial _i f) \partial _t)m(s) = (\partial _i +s (\partial _i f) t^{1})m(s) = (\partial _i +s (\partial _i f)/f)m(s1) \) in accordance with (3.11).
 21.
A tadpole here means a proper subgraph \(\gamma \subsetneq G\) which shares only a single vertex with the rest of G and does not depend on masses or external momenta. In this case, \(\mathcal {G}_G=\mathcal {U}_{\gamma } \mathcal {F}_{G/\gamma }\) factorizes such that the variables \(x_i\) with \(i\in \gamma \) only appear in the homogeneous polynomial \(\mathcal {U}_{\gamma }\) of degree \(\lambda _0:=L_{\gamma }\). Thus, we obtain (3.14) by setting \(\lambda _e = 1\) if \(e\in \gamma \) and \(\lambda _e=0\) otherwise.
 22.
Observe that \(\mathsf {NP}\left( g\right) \) is the orthogonal projection of \(\mathsf {NP}\left( f_{\sigma }\right) \) onto the coordinate hyperplane \(\left\{ \alpha _N=0 \right\} \). This projection restricts to an isomorphism between \(\left\{ \alpha _N=0 \right\} \) and \(F_{\lambda }\) (because \(\lambda _N \ne 0\)), and therefore \(\dim \mathsf {NP}\left( g\right) =\dim \mathsf {NP}\left( f_{\sigma }\right) \).
 23.
The existence of a linear annihilator could also be deduced from [31, Théorème 9.2].
 24.
Momentum conservation implies that the sum \(p_1+\cdots +p_{E+1}=0\) of the incoming momenta on all external legs vanishes; hence only \(E\) of them are independent.
 25.
The infinitesimal imaginary part \(\text {{i}}\epsilon \) is irrelevant for our purpose of counting integrals and will be henceforth ignored.
 26.
 27.
Such a graph \(\gamma \) is called massless propagator or pintegral.
 28.
Note that the intersection of a twoforest F of G with \(\gamma \) has at most two trees connected to external vertices of \(\gamma \). Any further tree in \(F\cap \gamma \) is thus necessarily a full component of F and forces its contribution to \(\mathcal {F}_G\) to vanish by \(p_F^2=0\) (due to the absence of external legs).
 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.
 30.
 31.
Beware that the number \(2^{L+1}L2\) 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.
 32.
We assume \(p^2 \ne 0\) and \(m_{L}^2 \ne 0\), which guarantees that \((p+m_{L})^2 \ne (pm_{L})^2\).
 33.
The additional term \(\delta _{0,LR}\) in [43, Equation (4.13)] subtracts a reducible integral that can be attributed to a subtopology. Our counting, however, accounts for all master integrals.
 34.
Actually, they initially refer to a different, relative cohomology group; but in the description of their implementation in Mint they seem to work with this total cohomology group.
 35.
Occasional mismatches are known, like for the graph \(F_9\) from Fig. 4 that was addressed in [13, section 4.1]. These discrepancies are due to an error in the implementation of Mint that misses contributions from critical points at infinity. (We thank Yang Zhang for bringing this to our attention.)
 36.
As a referee kindly pointed out, this relation also follows from momentum space IBPs.
 37.
We use the signature \((1,\,1,\ldots ,\,1)\) for the Minkowski metric.
 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 \(\alpha \mathsf {D}_1+\beta \mathsf {D}_2 = 1\), then iterated use of \(1/(\mathsf {D}_1 \mathsf {D}_2)=\alpha /\mathsf {D}_2 + \beta /\mathsf {D}_1\) allows one to ultimately eliminate one of \(\mathsf {D}_1\) or \(\mathsf {D}_2\). Therefore, requesting linear independence is no restriction.
 39.
 40.
Recall that our metric has signature \((1,1,\ldots ,1)\), so the integrations over the \(d1\) spacelike components are Euclidean and give \(\sqrt{\pi ^{L} \mathcal {U}}\) 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 \(\text {{i}}\epsilon \)prescription.
 41.
Much more generally, we could replace \(\sum _{j=1}^N x_j\) in the \(\delta \)constraint with any other function as long as it is homogeneous of degree 1 and positive on \(\mathbb {R}_+^N\).
 42.
This just encodes the natural action \(\partial _N x_N^{\nu } m = x_N^{\nu }(\partial _N + \nu /x_N) m\) on products of elements m of \(\mathscr {M}\) with the function \(x_N^{\nu }\)—hence the suggestive notation \(\mathscr {M}x_N^{\nu }\).
 43.
Given a good filtration \(\Gamma _{\bullet }\) of \(\mathscr {M}\), \(\Gamma _{j}' (\mathscr {M}x_N^{\nu }) :=(x_N^{j}\Gamma _{2j} \mathscr {M}) \otimes _k k(\nu )\) defines a filtration of \(\mathscr {M}x_N^{\nu }\) with \(\dim _{k(\nu )} \Gamma _j' (\mathscr {M}x_N^{\nu }) \le \dim _k \Gamma _{2j} \mathscr {M}\le c \cdot (2j)^N\) for some \(c<\infty \).
 44.
Due to \(\theta _1 x_1^{i} \mathscr {N}= x_1^i (\theta _1+i) \mathscr {N}\subseteq x_1^i\mathscr {N}\subseteq \mathscr {N}_j\), indeed \(\mathscr {N}_j\) is a \(k[\theta _1]\)module.
 45.
Let \(\pi ^{N}:\mathbb {G}_{m}^N\longrightarrow \left\{ \text {pt} \right\} \) denote the projection to a point, such that \(\pi ^N = \pi ^{N1} \circ \pi \). The identity \(\pi ^{N}_+ = \pi ^{N1}_+ \circ \pi _{+}\) of the corresponding pushforwards in the derived category of \(\mathcal {D}^{N}_k\)modules implies that \(\chi ({{\mathrm{DR}}}(\mathscr {M}))=\chi (\pi ^N_+(\mathscr {M})) = \sum _i (1)^i \chi ( H^i (\pi _+ \mathscr {M}))\), where \(H^{1}(\pi _+ \mathscr {M}) = \ker \partial _N\) and \(H^0(\pi _+ \mathscr {M}) = \mathscr {M}/(\partial _N \mathscr {M})\).
 46.
One only needs to check that \(x_N^{\pm 1} \mathscr {N}\subseteq \mathscr {N}\), which follows from \(\partial _N^{k+1} x_N^{\pm 1} m = \big [(k+1)(\partial _N x_N^{\pm 1}) + x_N^{\pm 1} \partial _N) \partial _N^k m = 0\) whenever \(\partial _N^k m =0\).
 47.
The first statement is clear since \(\ker \partial _N \subseteq \mathscr {N}\). The second claim follows from the identity \(x^k\partial ^k = (\partial x k)x^{k1}\partial ^{k1} = \cdots = \prod _{i=1}^k (\partial x i)\), which implies \(0=x_N^k \partial _N^k m = (1)^k (k!) m \mod \partial _N \mathscr {N}\) whenever \(\partial _N^k m = 0\).
Notes
Acknowledgements
We like to thank Mikhail Kalmykov, Roman Lee, Robert Schabinger, Lorenzo Tancredi and Yang Zhang for interesting discussions and helpful comments on integration by parts for Feynman integrals, Jørgen Rennemo for suggesting literature relevant for Sect. 3.1 and Viktor Levandovskyy for communication and explanations around Dmodules 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 4500/11. 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 HumboldtUniversität Berlin. Images in this paper were created with JaxoDraw [11] (based on Axodraw [95]) and FeynArts [37].
Supplementary material
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