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.
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1 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
between different Feynman integrals \(\mathcal {I}_{i}\), 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.Footnote 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 \(\left\{ \mathcal {G}=0 \right\} \) 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 \(\mathcal {G}^{-d/2}\), where \(\mathcal {G}\) 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.Footnote 2 In our set-up, these are differential operators P satisfying
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 \(\mathcal {G}^{-d/2}\), 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 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.
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 well-known 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 D-modules that our subsequent discussion will be based on.
We consider integrals (also called integral families [98]) that are defined by an \(dL\)-fold integral (\(L\) is the loop number), over so-called loop momenta\(\ell _1,\ldots ,\ell _{L}\) in \(d\)-dimensional Minkowski space, of a product of powers of denominators\(\mathsf {D}=(\mathsf {D}_1,\ldots ,\mathsf {D}_N)\):
The denominators are (at most) quadratic forms in the \(L\) loop momenta and some number \(E\) of linearly independent external momenta\(p_1,\ldots ,p_E\). 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\(\nu =(\nu _1,\ldots ,\nu _N)\) (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
To each denominator \(\mathsf {D}_a\) of a list \(\mathsf {D}=(\mathsf {D}_1,\ldots ,\mathsf {D}_N)\), we associate a variable \(x_a\) (\(1\le a \le N\)) called Schwinger parameter. The linear combination
decomposes into quadratic, linear and constant terms in the loop momenta.Footnote 3 This defines a symmetric \(L\times L\) matrix \(\Lambda \), a vector Q of \(L\) linear combinations of external momenta and a scalar \(J\). We define furthermore the polynomials
Schwinger parameters yield useful representations of the Feynman integrals (2.1):
Proposition 2
Let us denote the superficial degree of convergence by
Then, the Feynman integral (2.1) can be written as
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.
Example 3
Consider the graph in Fig. 1 with massless Feynman propagators, \(\mathsf {D}_1=-\ell ^2\) and \(\mathsf {D}_2=-(\ell +p)^2\). We find \(\Lambda = x_1 + x_2\), \(Q= -x_2 p\) and \(J= -x_2 p^2\) according to (2.2). Hence, the graph polynomials (2.3) become \(\mathcal {U}= x_1+x_2\) and \(\mathcal {F}= (-p^2)x_1 x_2\) such that
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 non-empty, open domain in \(\mathbb {C}^{1+N} \ni (d,\nu )\) where all of them are convergent and agree with each other.Footnote 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 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 of \(\Gamma \)-functions:
Integral (2.8) converges only in a certain domain of \((\nu ,d)\), but there it evaluates to (2.9), which has a unique meromorphic continuation. Its poles lie on the infinite family of hyperplanes defined by \(\left\{ d/2-\nu _1=k \right\} \), \(\left\{ d/2-\nu _2=k \right\} \) and \(\left\{ \nu _1+\nu _2-d/2=k \right\} \), indexed by \(k \in \mathbb {Z}_{\le 0}\).
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 multi-dimensional 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
Let \(\nu =(\nu _{1},\ldots ,\nu _{N})\in \mathbb {C}^{N}\). The twisted (multi-dimensional) Mellin transform of a function \(f:\mathbb {R}_{+}^{N}\longrightarrow \mathbb {C}\) is defined as
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 \(\Gamma \)’s in (2.10).
Lemma 7
Let \(\alpha ,\beta \in \mathbb {C}\), \(\nu \in \mathbb {C}^{N}\), \(1\le i \le N\) and \(f,g:\mathbb {R}_+^{N} \longrightarrow \mathbb {C}\). Writing \(\mathsf {e}_{i}\) for the i-th unit vector, the (twisted) Mellin transform has the following properties:
-
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
The linearity is immediate from (2.10), and the functional equation \(\Gamma (\nu _i+1) = \nu _i\Gamma (\nu _i)\) provides the multiplication rule \( \left[ x_i^{\nu _i-1}/\Gamma (\nu _i) \right] x_i = \nu _i x_i^{\nu _i}/\Gamma (\nu _i+1)\). The differentiation rule is a consequence of integration by parts,
because the boundary terms vanish inside the convergence domain of both integrals. This just says that if \(\lim _{x_i\rightarrow 0} (x_i^{\nu _i-\epsilon -1} f)\) is finite for some \(\epsilon >0\), then \(\lim _{x_i\rightarrow 0} (x_i^{\nu _i-1} f) = 0\) vanishes and the analogous argument applies to the upper bound \(x_i\rightarrow \infty \). \(\square \)
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 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 \(x_{1},\ldots ,x_{N}\) is the non-commutative algebra of polynomial differential operators
such that the commutators are \([x_i,x_j]=[\partial _i,\partial _j]=0\) and \([\partial _i, x_j] = \delta _{i,j}\) (Kronecker delta).
Note that with the multi-index notations \( x^{\alpha } = x_{1}^{\alpha _{1}}\cdots x_{N}^{\alpha _{N}} \) and \( \partial ^{\beta } = \partial _{1}^{\beta _{1}}\cdots \partial _{N}^{\beta _{N}} \), every operator \(P\in {A}^{N}\) can be written uniquely in the form
by commuting all derivatives to the right (only finitely many of the coefficients \(c_{\alpha \beta }\in \mathbb {C}\) are nonzero). Extending the coefficients \(c_{\alpha \beta }\) from \(\mathbb {C}\) to polynomials \(\mathbb {C}[s]\) in a further, commuting variable s, we obtain the algebra \({A}^{N}[s] :={A}^{N} \otimes _{\mathbb {C}} \mathbb {C}[s]\). Later, we will also consider the case \({A}^{N}_k :={A}^{N} \otimes _{\mathbb {C}} k\) of coefficients that are rational functions \(k:=\mathbb {C}(s)\). 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.
Definition 9
Given a polynomial \(f \in \mathbb {C}[x]\), the \({A}^{N}[s]\)-module \(\mathbb {C}[s,x,1/f]\cdot f^s\) consists of elements of the form \((p/f^k) \cdot f^s\) (where \(p\in \mathbb {C}[s,x]\), \(k\in \mathbb {N}_0\)) with the \({A}^{N}[s]\)-action
for any polynomial \(q\in \mathbb {C}[s,x]\). This is just the natural action by multiplication and differentiation, \(\partial _i \mapsto \partial /(\partial x_i)\). We abbreviate \(f^{s+k} :=f^k \cdot f^s\) for \(k \in \mathbb {Z}\). The cyclic submodule generated by \(f^s\) is denoted as \({A}^{N}[s] f^s\). If we extend the coefficients to the rational functions in s, we write \({A}^{N}_k f^s = {A}^{N}[s] f^s \otimes _{\mathbb {C}[s]} \mathbb {C}(s)\).
Definition 10
The s-parametric annihilators of a polynomial f in \(x_1,\ldots ,x_N\) are the elements of the (left) ideal of operators in \({A}^{N}[s]\) whose action on \(f^{s}\) is zero:
Example 11
Given \(f\in \mathbb {C}[x_1,\ldots ,x_N]\), we always have the trivial annihilators
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 D-module 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
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 identifications \(\hat{\mathbf{i }}^{+} \leftrightarrow \partial _i\) and \(\mathbf{j }^{-} \leftrightarrow x_j\) the commutation relations are identical. In fact, a different isomorphism is given by \(\hat{\mathbf{i }}^{+} \leftrightarrow x_i\) and \(\mathbf{j }^{-} \leftrightarrow -\partial _j\), and it is this identification that corresponds to the Mellin transform (see Lemma 7). We therefore denote it byFootnote 5
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 \(F(\nu )\) of a different set \(\nu =(\nu _1,\ldots ,\nu _N)\) of variables (the indices of Feynman integrals) by shifts of the argument and, in case of \(\hat{\mathbf{i }}^{+}\), a multiplication with \(\nu _i\):
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
Their commutation relations are
In terms of the \({S}^{N}\) action (2.17), we can rephrase the essence of Lemma 7 as
Corollary 13
The (twisted) Mellin transform (2.10) is compatible with the actions of the algebras \({A}^{N}\) and \({S}^{N}\) under their identification (2.16). In other words,
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
For the bubble graph in Fig. 1 with \(\mathcal {G}=x_1+x_2-p^2 x_1 x_2\) from Example 3,
is easily checked to annihilate \(\mathcal {G}^s\). We therefore get the shift relation
According to (2.18), this relation can also be written as
We prefer to work with the modified Feynman integral \(\widetilde{\mathcal {I}}\) from (2.11), because it is directly related to the Mellin transform. However, it is straightforward to translate relations between \(\widetilde{\mathcal {I}}\) into relations for the actual Feynman integral \(\mathcal {I}\). Namely, if \( P =\sum _{\alpha ,\beta } c_{\alpha ,\beta } x^{\alpha } \partial ^{\beta } \in {{\mathrm{Ann}}}_{{A}^{N}[s]}\left( \mathcal {G}^s \right) \), we substitute (2.11) to see
and then recall from (2.4) that \( \omega = \sum _i \nu _i + Ls\) to conclude
Notice that the fraction \(\Gamma (-s-\omega -\left|\alpha \right|+\left|\beta \right|)/\Gamma (-s-\omega )\) is a rational function in \(\omega \) (and thus in \(\nu \) and s), due to the functional equation for \(\Gamma \), since \(\left|\alpha \right|\) and \(\left|\beta \right|\) are integers.
Example 16
Substituting \(\widetilde{\mathcal {I}}(\nu _1,\nu _2)=\mathcal {I}(\nu _1,\nu _2) \Gamma (-2s-\nu _1-\nu _2)/\Gamma (-s)\) and \(\widetilde{\mathcal {I}}(\nu _1+1,\nu _2) =\mathcal {I}(\nu _1+1,\nu _2) \Gamma (-2s-\nu _1-1-\nu _2)/\Gamma (-s)\) from (2.11) into (2.21) results in
which also follows from expression (2.9) of \(\mathcal {I}(\nu )\) in terms of \(\Gamma \)-functions.
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
(Inverse Mellin transform, e.g. [18, Theorem 3.5]) Suppose that the (twisted) Mellin transform \(f^{\star }(\nu ) :=\mathcal {M}\left\{ f \right\} (\nu )\) of f(x) from (2.10) converges in a domain of the form \(a_i\le {{\mathrm{Re}}}(\nu _i) \le b_i\) for all \(1\le i \le N\), where \(a,b \in \mathbb {R}^N\). Then, its inverse is given by
This multiple integral along lines parallel to the imaginary axis converges for \(a_i\le \sigma _i \le b_i\) (\(1\le i \le N\)) and does not depend on the concrete choice of \(\sigma _i\).
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
Let \({{\mathrm{Ann}}}_{{S}^{N}[s]}(\widetilde{\mathcal {I}}) \subseteq {S}^{N}[s]\) denote the \({S}^{N}[s]\)-module of polynomial shift operators that annihilate a (rescaled) Feynman integral (2.11). Then the Mellin transform (2.16) restricts to a bijection between these relations and parametric annihilators:
Instead of focussing on the annihilators themselves, we can also look at the \({A}^{N}[s]\)-module \({A}^{N}[s] \cdot \mathcal {G}^s \cong {A}^{N}[s]/{{\mathrm{Ann}}}_{{A}^{N}[s]}(\mathcal {G}^s)\) of the integrands and the \({S}^{N}[s]\)-module \({S}^{N}[s] \cdot \widetilde{\mathcal {I}}\cong {S}^{N}[s]/{{\mathrm{Ann}}}_{{S}^{N}[s]}(\widetilde{\mathcal {I}})\) of all (shifted) Feynman integrals. The Mellin transform gives an isomorphism
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.
Since the denominators \(\mathsf {D}_a\) from (2.1) are quadratic forms in the \(M=L+vE\) momenta \(q=(q_1,\ldots ,q_{M}) :=(\ell _1,\ldots ,\ell _{L},p_1,\ldots ,p_{E})\), we can write them in the form
such that the coefficients \(\mathcal {A}_a^{\left\{ i,j \right\} }\) and \(\lambda _a\) are independent of loop momenta and the pairs
label the \(\left|\Theta \right| = \frac{L(L+1)}{2}+LE\) loop-momentum-dependent scalar products
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 \(N=\left|\Theta \right|\) denominators such that the \(N\times N\) square matrix \(\mathcal {A}\) defined by (2.26) is invertible.Footnote 6 We think of \(\mathcal {A}_{a}^{\left\{ i,j \right\} }\) as the element of \(\mathcal {A}\) in row a and column \(\left\{ i,j \right\} \) and write \(\mathcal {A}_{\left\{ i,j \right\} }^{a}\) for the entry in row \(\left\{ i,j \right\} \) and column a of \(\mathcal {A}^{-1}\), such that the inverse of (2.26) can be written as
We are interested in relations of the Feynman integral \(\mathcal {I}\) from (2.1), that is,
Definition 19
The momentum space IBP relations of \(\mathcal {I}(\nu _{1},\ldots ,\nu _{N})\) are those relations between scalar Feynman integrals that are obtained from Stokes’ theorem
where the operators \(\mathbf{o }_{j}^{i}\) are defined in terms of the momentaFootnote 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 \(N=\left|\Theta \right|\) denominators \(\mathsf {D}\) such that the matrix \(\mathcal {A}\) defined by (2.26) is invertible, every momentum space IBP relation can be written explicitly as
where \(\mathbf O _{j}^{i}\) denotes shift operators, indexed by \(1\le i \le L\) and \(1\le j \le M\), that are given by
The coefficients \(C^{bi}_{aj}\) are defined as
Corollary 21
To every difference equation \(\mathbf O _{j}^{i} \mathcal {I}=0\) from momentum space IBP, there corresponds a parametric annihilator \(\widetilde{\mathbf{O }}_{j}^{i} \in {{\mathrm{Ann}}}_{{A}^{N}[s]}\left( \mathcal {G}^s \right) \) of the form
where \( H :=\frac{(L+1)d}{2} + \sum _{c=1}^{N}x_{c}\partial _{c} \).
Proof
First recall rescaling (2.11) between the Feynman integral \(\mathcal {I}\) and the Mellin transform \(\widetilde{\mathcal {I}}\) of \(\mathcal {G}^s\). As we saw in (2.22), this means that
and so if we substitute (2.11) into \(\mathbf O _{j}^{i} \mathcal {I}=0\) for the operators from (2.34), then apart from the substitution \(\hat{\mathbf{a }}^{+}\mathbf{b }^{-} \mapsto x_a(-\partial _b)\) which does not change \(\omega \), the remaining terms with shifts \(\hat{\mathbf{a }}^{+}\) do increment \(\omega \) by one and thus acquire an additional factor of
This proves that \(\mathcal {M}\left\{ \widetilde{\mathbf{O }}_{j}^{i} \right\} \widetilde{\mathcal {I}}= 0\) for the operators in (2.36) and (2.37) and Theorem 17 concludes the proof. \(\square \)
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.Footnote 8
Definition 22
By \({{\mathrm{Mom}}}\) we denote the left \({A}^{N}[s]\)-module generated by the annihilators \(\widetilde{\mathbf{O }}_{j}^{i}\) from Corollary 21, corresponding to the momentum space IBP identities:
Since the \(\widetilde{\mathbf{O }}_{j}^{i}\) are first-order differential operators, we have the inclusions
where \({{\mathrm{Ann}}}^1\) denotes the \({A}^{N}[s]\)-module generated by all first-order annihilators. Note that for a generic polynomial \(\mathcal {G}\), one would not expect that all of its annihilators can be obtained from linear ones (\({{\mathrm{Ann}}}^1\subsetneq {{\mathrm{Ann}}}\)). It is therefore interesting that we observe the equality \({{\mathrm{Ann}}}^1={{\mathrm{Ann}}}\) in all cases of Feynman integrals that we checked.
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 s-parametric 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
Given any annihilator \(P\in {{\mathrm{Ann}}}_{{A}^{N}[s]}(\mathcal {G}^s)\), does there exist a polynomial \(q \in \mathbb {C}[s,\theta ]\) such that \(q P \in {{\mathrm{Mom}}}\)? In other words, does
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 well-known dimension shifts.
2.5 Dimension shifts
The representation \(\mathcal {I}= \mathcal {M}\left\{ e^{-\mathcal {F}/\mathcal {U}}\cdot \mathcal {U}^s \right\} \) from (2.5) shows, through Corollary 13, that
where \(\mathcal {M}\left\{ \mathcal {U} \right\} = \mathcal {U}(\hat{\mathbf{1 }}^{+},\ldots ,\hat{\mathbf{N }}^{+})\) is obtained from the polynomial \(\mathcal {U}(x_1,\ldots ,x_N)\) by substituting \(x_i \mapsto \hat{\mathbf{i }}^{+}\). This raising dimension shift was pointed out by Tarasov [89],Footnote 9 and had been observed before in special cases [26]. For \(\widetilde{\mathcal {I}}(d)=\mathcal {I}(d)\cdot \Gamma (d/2-\omega )/\Gamma (d/2)\), the relation takes the form
At the same time, the representation \(\widetilde{\mathcal {I}}=\mathcal {M}\left\{ \mathcal {G}^s \right\} \) implies also that
Remark 25
The equality of (2.42) and (2.43) implies, via the Mellin transform, that
Indeed, \( H(s) \mathcal {G}^s = s(L\mathcal {U}+ (L+1)\mathcal {F}) \mathcal {G}^{s-1}-s(L+1)\mathcal {G}^s = -s\mathcal {U}\mathcal {G}^{s-1} \) follows from the homogeneity of \(\mathcal {U}\) and \(\mathcal {F}\), see (A.5).Footnote 10
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 \(\widetilde{\mathcal {I}}(d)=\mathcal {M}\left\{ \mathcal {G}^{-d/2} \right\} \)Footnote 11:
Definition 26
A Bernstein–Sato operator \(P(s)\in {A}^{N}[s]\) for a non-constant polynomial f is a polynomial differential operator such that there exists a polynomial \(b(s) \in \mathbb {C}[s]\) with
Such operators always exist, and the Bernstein–Sato polynomial is the unique monic polynomial b(s) of smallest degree for which (2.44) has a nonzero solution [10, 73]. Given a solution of (2.44) for \(f=\mathcal {G}\), we get a lowering dimension shift relation:
Corollary 27
If we allow the coefficients to be rational functions \(k=\mathbb {C}(s)\), every integral in \(d+2n\) dimensions can be written as an integral in \(d\) dimensions. In other words, the multiplication with f is invertible on \({A}^{N}_k f^s\). Put still differently, \(f^s\) generates the full module
Proof
By (2.44), \(f^{s-n} = \frac{P(s-n)}{b(s-n)} \cdots \frac{P(s-1)}{b(s-1)} f^s \in {A}^{N}_k \cdot f^s\) 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]:
Recall that \((q_1,\ldots , q_{M})=(\ell _{1},\ldots ,\ell _{L},p_{1},\ldots ,p_{E})\) denotes the combined loop- and external momenta (\(M= L+E\)). We introduce the Gram determinants
and remark that \(\mathsf {Gr}:=\mathsf {Gr}_{L+1}(s) = \det (p_i \cdot p_j)_{1\le i,j\le E}\) depends only on the external momenta. Furthermore, note that \(\mathsf {Gr}_1(s)\) is a polynomial in the scalar products \(s_{\left\{ i,j \right\} }=q_i \cdot q_j\). By (2.29), we can think of it also as a polynomial in the denominators \(\mathsf {D}\).
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
(Baikov representation [53]) The Feynman integral (2.1) can be written as
where \(c\in \mathbb {Q}\) is a rational constant and the Baikov polynomial \(\mathcal {P}(y)\) has degree at most \(M=L+E\). The contour of integration in (2.48) is such that \(\mathcal {P}\) 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
Corollary 30
(lowering dimension shift [54]) A Feynman integral in \(d+2\) dimensions can be expressed as an integral in \(d\) dimensions byFootnote 12
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 \(\Gamma \left( \frac{d+2-E}{2}\right) = \frac{d-E}{2} \Gamma \left( \frac{d-E}{2} \right) \) 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. \(\square \)
Equation (2.49) can be thought of as a Bernstein equation for \(\mathcal {I}(d)\), or, equivalently, as a special type of integral relation: Combining (2.49) with the raising dimension shift (2.49), we find that
We ask if this annihilator is contained in \({{\mathrm{Mom}}}\); 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 \(\mathcal {P}(y)\) explicitly: The block decomposition \((s_{i,j})_{1\le i,j\le M} = \left( {\begin{matrix} V &{} B \\ B^{\intercal } &{} G \\ \end{matrix}} \right) \) with \(V=(\ell _{i}\cdot \ell _{j})_{i,j \le L}\), \(B=(\ell _{i} \cdot p_{j})_{i\le L,j\le E}\) and \(G=(p_{i}\cdot p_{j})_{i,j\le E}\) shows that
is an \(L\times L\) matrix whose entries are quadratic in the denominators. By (2.29),
where we suppress the explicit summation signs over \(a=1,\ldots ,N\) in the first summand and over \(a,b=1,\ldots ,N\) and \(r,s=L+1,\ldots ,M\) in the second summand.
Proposition 31
Annihilator (2.50), corresponding to the lowering dimension shift (2.49), is contained in the ideal of shift operators generated by the momentum space IBP’s from Proposition 20:
Proof
Let us abbreviate \({\tilde{\Lambda }} :=\Lambda (\hat{\mathbf{1 }}^{+},\ldots ,\hat{\mathbf{N }}^{+})\) from (A.2) such that \(\mathcal {U}=\det \Lambda \) and similarly \({\tilde{Q}} :=Q(\mathbf{1 }^{-},\ldots ,\mathbf{N }^{-})\) for matrix (2.51). Using (2.51) and (2.15), we compute
Contracting with the matrix \(\mathcal {A}_c^{\left\{ k,l \right\} }\) (for \(k,l\le L\)) by summing over c, we conclude that
Note that the indices r and s take values \(>L\), whereas i and j are \(\le L\). Hence, \(\delta _{\left\{ j,s \right\} ,\left\{ k,l \right\} }=\delta _{\left\{ i,r \right\} ,\left\{ k,l \right\} } = 0\). So, recalling (A.2), we finally arrive at
Recall (2.3), that \(\mathcal {U}(x) = \det \Lambda (x)\), such that
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 product \({\tilde{Q}} \cdot {\tilde{\Lambda }}\). 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 \({{\mathrm{col-det}}}\) denotes the column-ordered determinant
So let us now compute the entries of the product of \({\tilde{Q}}\) from (2.51) with \({\tilde{\Lambda }}\). Firstly,
according to (2.35). Note that \(C_{bs}^{bk} = \sum _{b,j} (1+\delta _{j,k}) \mathcal {A}_b^{\left\{ j,k \right\} } \mathcal {A}^b_{\left\{ j,s \right\} } = \sum _j (1+\delta _{j,k}) \delta _{\left\{ j,k \right\} ,\left\{ j,s \right\} } = 0\) due to \(s>L\ge k\). So we can rewrite, due to (2.34),
Note that \(q_s \cdot q_m = G_{s,m}\) such that contraction of the second summand with \(G^{-1}_{r,s}\) produces \(\delta _{r,m}\). So the sum over m collapses, and up to the term with \(\mathbf O _{s}^{k}\), \({\tilde{Q}}_{i,j} {\tilde{\Lambda }}_{j,k}\) is
where the term \(C_{ai}^{ak} = \sum _{a,j} (1+\delta _{j,k}) \mathcal {A}_a^{\left\{ j,k \right\} } \mathcal {A}^a_{\left\{ j,i \right\} } = \sum _j (1+\delta _{j,k}) \delta _{\left\{ j,k \right\} ,\left\{ j,i \right\} } =\delta _{i}^{k} (M+1)\) comes from commuting \(\mathbf{a }^{-}\) with \(\hat{\mathbf{b }}^{+}\). Putting our results together, we arrive at
So if we ignore all terms that lie in the (left) ideal generated by the momentum space (shift) operators \(\mathbf O _{j}^{i}\), the column determinant (2.55) of the matrix \({\tilde{Q}} \cdot {\tilde{\Lambda }} + Q_{col}\) 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
\(\square \)
The Mellin transform \(\mathcal {I}(d)=\mathcal {M}\left\{ \mathcal {U}^s e^{-\mathcal {F}/\mathcal {U}} \right\} \) identifies (2.50) with the annihilator
To phrase this in terms of \(\widetilde{\mathcal {I}}(d) = \mathcal {M}\left\{ \mathcal {G}^s \right\} = \mathcal {I}(d) \cdot \Gamma (-s-\omega )/\Gamma (-s)\), we can use \(\mathcal {U}\mathcal {G}^s = -H(s+1) \mathcal {G}^{s+1}/(s+1)\) from Remark 25 to conclude that
where \(H(s)=\mathcal {M}^{-1}\{-s-\omega \} = -s(L+1)+\sum _{i=1}^N \theta _i\). 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 \(\det Q(-\partial ) = \sum _r Q_r\), as
If \(r \le L+ 1\), 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 \(\mathcal {P}(y)\) 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 half-integers.
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 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 \(V_{\mathcal {G}}\) we denote the vector space of all Feynman integrals associated to \(\mathcal {G}\), over the field \(\mathbb {C}(s,\nu ) :=\mathbb {C}(s,\nu _1,\ldots ,\nu _N)\) of rational functions (in the dimension and indices). More precisely, with \(\widetilde{\mathcal {I}}_{\mathcal {G}} :=\mathcal {M}\left\{ \mathcal {G}^s \right\} \),
The number of master integrals is the dimension of this vector space:
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 \(\mathcal {I}_{\mathcal {G}}(\nu +n)/\widetilde{\mathcal {I}}_{\mathcal {G}}(\nu +n) = \Gamma (-s)/\Gamma (-s-\omega -\left|n \right|)\) with \(\left|n \right|=n_1+\cdots +n_N\) are all related by a rational function in \(\mathbb {C}(s,\nu )\), see (2.22).
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 \(\nu \in \mathbb {Z}^N\), 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 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 \(\nu _e\).
-
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].
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 \(\nu \in \mathbb {Z}^N\), 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.
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
where \(\mathbb {C}[s,\theta ] = \mathbb {C}[s,\theta _1,\ldots ,\theta _N]\) denotes the polynomials in the dimension \(s=-d/2\) and the operators \(\theta _e :=x_e \partial _e = \mathcal {M}^{-1}\{-\nu _e\}\), and \(F :=\mathbb {C}(s,\theta )\) stands for their fraction field (the rational functions in these variables). Since F contains \(k :=\mathbb {C}(s)\), we can equivalently work over this base field throughout and write
in terms of \(R :=k[\theta ]\) and the module \(\mathscr {M}= {A}^{N}_k \cdot \mathcal {G}^s\) over the Weyl algebra \({A}^{N}_k :={A}^{N} \otimes _{\mathbb {C}} k = {A}^{N}[s] \otimes _{\mathbb {C}[s]} k\) over the field \(k=\mathbb {C}(s)\). Crucially, \({A}^{N}_k \cdot \mathcal {G}^s\) is a holonomic\({A}^{N}_k\)-module, 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 \(\mathbb {G}_{m}^N\). Note that \(\mathcal {D}^{N}_k\) 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 \(\mathcal {O}(\mathbb {A}^N_k)=k[x]\) to Laurent polynomials \(\mathcal {O}(\mathbb {G}_{m}^N)=k[x^{\pm 1}]\) in the coordinates \(x_i\). Equivalently, we can also view \(\mathcal {D}^{N}_k= \iota ^{*} {A}^{N}_k\) as the pull-back under the (open) inclusion
The pull-back along \(\iota \) turns every \({A}^{N}_k\)-module \(\mathscr {M}\) into a \(\mathcal {D}^{N}_k\)-module \(\iota ^{*} \mathscr {M}\), namely the localization \(\iota ^{*} \mathscr {M}= k[x^{\pm 1}] \otimes _{k[x]} \mathscr {M}= \mathscr {M}[x^{-1}]\). Importantly, if \(\mathscr {M}\) is holonomic, so is its pull-back \(\iota ^{*} \mathscr {M}\). The starting point for this section is
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 finite-dimensional 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 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.
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 \(1\otimes x_i\). \(\square \)
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 \(\mathscr {M}':=\iota ^{*} \mathscr {M}\). This is the complex
of \(\mathscr {M}'\)-valued differential forms on the torus \(\mathbb {G}_{m}^N\), with the connection \(\mathrm{d}^{}(\omega \otimes m) = \mathrm{d}^{}\omega \otimes m + \sum _{i=1}^N (\mathrm{d}^{}x_i \wedge \omega ) \otimes \partial _i m\). Note that the r-forms \(\omega \) 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 \(-N \le i \le 0\). The extremal cohomology groups are easily identified as
with the latter also known as push-forward of \(\mathscr {M}'\) under the projection to the point. Since holonomicity is preserved under direct images, we conclude that \(\dim _k H^0\left( {{\mathrm{DR}}}(\mathscr {M}') \right) \) is finite.Footnote 13 In fact, the same is true for the other de Rham cohomology groups, which shows that (3.5) is indeed well defined.Footnote 14
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
Let \(\mathcal {G}\in \mathbb {C}[x_1,\ldots ,x_N]\) be a polynomial and set \(k=\mathbb {C}(s)\). Then, the Euler characteristics of the algebraic de Rham complexes of the holonomic \({A}^{N}_k\)-module \(\iota ^{*} {A}^{N}_k \mathcal {G}^s\) and the holonomic \({A}^{N}_{\mathbb {C}}\)-module \(\mathbb {C}[x^{\pm 1},\mathcal {G}^{-1}]=\mathcal {O}(\mathbb {G}_{m}^N {\setminus } \mathbb {V}(\mathcal {G}))\) coincide:
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
The number of master integrals of an integral family with N denominators is
the Euler characteristic of the algebraic de Rham complex of the complement of the hypersurface \(x_1\ldots x_N \cdot \mathcal {G}=0\) inside the affine plane \(\mathbb {A}^N\). Via Grothendieck’s comparison isomorphism, this is the same as the topological Euler characteristic, up to a sign:Footnote 15
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.Footnote 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].Footnote 17
Proof of Proposition 36
Given an \({A}^{N}_{\mathbb {C}}\)-module \(\mathscr {M}\) and a polynomial \(f \in \mathbb {C}[x]\), set \(\mathscr {M}' :=\mathscr {M}[f^{-1}]\) and consider the \({A}^{N}_{\mathbb {C}}\)-module \(\mathscr {M}'f^s\) formed by products of \(f^s\) with elements \(m\in \mathscr {M}'[s] :=\mathscr {M}' \otimes _{\mathbb {C}} \mathbb {C}[s]\).Footnote 18 As vector spaces, \(\mathscr {M}' f^s \cong \mathscr {M}'[s]\) via \(mf^s \mapsto m\), but the \({A}^{N}_{\mathbb {C}}\) action on \(\mathscr {M}'f^s\) 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 \(f^{s+r}:=f^r \cdot f^s\) for \(r\in \mathbb {Z}\). One easily verifies \([\partial _t,t] = 1\) and \([\partial _i,\partial _t]=[\partial _i,t]=[x_i,\partial _t]=[x_i,t] = 0\), such that \(\mathscr {M}' f^s\) becomes an \({A}^{N+1}_{\mathbb {C}}\)-module in the \(N+1\) variables \((x_1,\ldots ,x_N,t)\). Note that the operator t acts invertibly on \(\mathscr {M}'\), such that the identity \(\partial _t t = -s\) gives \(\partial _t = -st^{-1}\) and thus
is an isomorphism of \({A}^{N}_{\mathbb {C}}\)-modules.Footnote 19 Since \(\partial _t\) is injective on \(\mathscr {M}' f^s\) (it raises the degree in s), the de Rham complex \({{\mathrm{DR}}}(\mathscr {M}'f^s)\) is quasi-isomorphic to \({{\mathrm{DR}}}(\mathscr {M}' f^s/\partial _t \mathscr {M}' f^s) = {{\mathrm{DR}}}(\mathscr {M}')\), see Corollary 70. So we can conclude the equality
once we assume that \(\mathscr {M}\) is holonomic to ensure that these Euler characteristics are well defined. Indeed, the holonomicity of \(\mathscr {M}'\) and \(\mathscr {M}' f^s\) holds because
-
\(\mathscr {M}'=j^{*} \mathscr {M}\) is the pull-back of \(\mathscr {M}\) under the inclusion ,
-
\(\mathscr {M}' f^s = F_{*} \mathscr {M}'\) is the push-forward of \(\mathscr {M}'\) under the closed embedding which sends x to (x, f(x)).Footnote 20
Alternatively, the filtration \(\Gamma _j \mathscr {M}' f^s :=f^{-j} \sum _{i\le j} s^i \Gamma _{2j(\deg f)-i} \mathscr {M}\) induced by any good filtration \(\Gamma _{\bullet }\) on \(\mathscr {M}\) directly shows the holonomicity of \(\mathscr {M}' f^s\), since its dimension grows like \(j^{N+1}\) for large j. We now invoke the theory of Loeser–Sabbah to deduce that
where \(\mathscr {N}:=\mathscr {M}'f^s (t\partial _t)\) denotes the algebraic Mellin transform (B.1) of \(\mathscr {M}'f^s\) with respect to the coordinate t. But note that, according to (3.12), localizing at \(t\partial _t = -s-1\) just extends the coefficients to \(k=\mathbb {C}(s)\). So \(\mathscr {N}= \mathscr {M}' \otimes _{\mathbb {C}} \mathbb {C}(s)f^s = \mathscr {M}\otimes _{\mathbb {C}} k f^s\) is just the holonomic \({A}^{N}_{k}\)-module on the left-hand side of (3.8), because, over k, f is invertible by Corollary 27. We have proven \(\chi (\mathscr {M}\otimes _{\mathbb {C}} k f^s) = \chi (\mathscr {M}[f^{-1}])\), and the special case of \(\mathscr {M}=\mathbb {C}[x^{\pm 1}]\) with \(f = \mathcal {G}\) proves the claim. \(\square \)
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
Corollary 37 shows in particular that there are no master integrals, \(\mathfrak {C}\left( \mathcal {G}\right) =0\), precisely when the Euler characteristic \(\chi (\mathbb {G}_{m}^{N}{\setminus } \mathbb {V}(f)) :=\chi (\mathbb {C}[x^{\pm 1},f^{-1}]) = \chi ( (\mathbb {C}^{*})^N {\setminus } \mathbb {V}(f))\) vanishes for \(f=\mathcal {G}\). For example, this happens if f is homogeneous in a generalized sense: Suppose we can find \(\lambda _0,\ldots ,\lambda _N \in \mathbb {Z}\), not all zero, such that
which is equivalent (apply \(\partial _t\) and set \(t=1\)) to the existence of a linear annihilator,
Lemma 40
Given \(f\in \mathbb {C}[x_1,\ldots ,x_N]\), the algebraic Mellin transform \(\mathfrak {M}:=\mathscr {M}\otimes _{\mathbb {C}(s)[\theta ]} \mathbb {C}(s,\theta )\) of \(\mathscr {M}:=\mathcal {D}^{N}_k\cdot f^s\) is zero if, and only if, \(f^s\) is annihilated by a polynomial in the Euler operators \(\theta \):
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.Footnote 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.
Geometrically, homogeneity (3.14) can be interpreted as follows: Dividing by the greatest common divisor, we may assume that \(\lambda _0,\ldots ,\lambda _N\) are relatively prime. Thus, we may extend \((\lambda _1,\ldots ,\lambda _N)\) to a basis of the lattice \(\mathbb {Z}^N\) and hence construct a matrix \(A \in \mathsf {GL}_{N}( \mathbb {Z} )\) with first row \(A_{1i} = \lambda _i\). In the associated coordinates y, defined by
the polynomial f takes the form \(f(x)=y_1^{\lambda _0} g({\bar{y}})\) for some Laurent polynomial \(g \in \mathbb {C}[{\bar{y}}^{\pm 1}]\) in the remaining variables \({\bar{y}}=(y_2,\ldots ,y_N)\). In particular, the hypersurface \(\left\{ f=0 \right\} =\left\{ g=0 \right\} \) can be defined by an equation independent of the coordinate \(y_1\).
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 \(\left\{ g=0 \right\} \subset \mathbb {G}_{m}^{N-1}\).
To prove Proposition 41, we will look at the Newton polytope\(\mathsf {NP}\left( f\right) \) of f, which is defined as the convex hull of the exponents of monomials that appear in f:
Since every monomial \(x^{\alpha }\) is an eigenvector of operators (3.15), \(P^{\lambda }_s(\theta )\cdot x^{\alpha } = P^{\lambda }_s(\alpha ) x^{\alpha }\), it is annihilated by \(P^{\lambda }_1\) exactly when \(\alpha \) belongs to \(F_{\lambda } :=\left\{ P^{\lambda }_1(\alpha )=0 \right\} \), the hyperplane \(F_{\lambda }= \left\{ \alpha : \alpha _1\lambda _1+\cdots +\alpha _N \lambda _N=\lambda _0 \right\} \). In particular, \(0=P^{\lambda }_s \cdot f^s = s f^{s-1} P^{\lambda }_1 \cdot f\) is equivalent to the Newton polytope \(\mathsf {NP}\left( f\right) \subset F_{\lambda }\) being contained in that hyperplane. 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 \(f\in \mathbb {C}\).) 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 non-degenerate 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 non-degeneracy of \(\mathsf {NP}\left( f\right) \).
To start, we use Lemma 40 to find a polynomial \(0\ne P(\theta ,s) \in \mathbb {C}[s,\theta ]\) such that \(P(\theta ,s)\cdot f^s = 0\), and we choose one with minimal total degree in s and \(\theta \). Then pick an \((N-1)\) dimensional face \(\sigma = \mathsf {NP}\left( f\right) \cap F_{\lambda }\), which we can write as the intersection of \(\mathsf {NP}\left( f\right) \) with a hyperplane \(F_{\lambda }\) for some integers \(\lambda _0,\ldots ,\lambda _N\) such that \(\mathsf {NP}\left( f\right) \subseteq \left\{ \alpha : P^{\lambda }_1(\alpha ) \le 0 \right\} \). Under rescaling (3.14), all monomials of \(f=\sum _{\alpha } c_{\alpha } x^{\alpha }\) with \(\alpha \in \sigma \subset F_{\lambda }\) acquire a factor of \(t^{\lambda _0}\), while the remaining monomials with \(\alpha \in \mathsf {NP}\left( f\right) {\setminus } \sigma \) come with a smaller exponent \(\sum _{i=1}^N \alpha _i \lambda _i<\lambda _0\) of t:
and \(\mathcal {O}\left( t^{-1} \right) \) denotes a rational function in \(t^{-1} \mathbb {C}(x)[t^{-1}]\). Note that \(P(\theta ,s)\cdot f^s(\left\{ x_i t^{\lambda _i} \right\} ) = 0\) is still zero, because the rescaling of x commutes with the Euler operators \(\theta _i\cdot h(x_i t^{\lambda _i}) = (\theta _i \cdot h(x_i))|_{x_i\mapsto x_i t^{\lambda _i}}\). Therefore, applying \(P(\theta ,s)\) to the s-th power of the right-hand side of (3.19) and dividing by \(t^{s\lambda _0}\) yields
where \(\mathcal {O}\left( t^{-1} \right) \) on the right-hand side denotes a formal series in \(t^{-1} \mathbb {C}(x,s) [[t^{-1}]]\). In particular, the coefficient of \(t^0\) must vanish, and we conclude that \(P(\theta ,s) \cdot f_{\sigma }^s=0\). Label the variables such that \(\lambda _N \ne 0\), then we can divide \(P(\theta ,s)\) by the linear form \(P^{\lambda }_s(\theta ,s)\) from (3.15), as a polynomial in \(\theta _N\), to obtain a decomposition
for some polynomial \(Q(\theta ,s)\in \mathbb {C}[\theta ,s]\), such that the first summand depends only on \(\theta ':=(\theta _1,\ldots ,\theta _{N-1})\) and s. Since \(\mathsf {NP}\left( f_{\sigma }\right) =\sigma \subset F_{\lambda }\) is contained in the hyperplane \(F_{\lambda }=\left\{ \alpha : P^{\lambda }_1(\alpha )=0 \right\} \), we see \(P^{\lambda }_s \cdot f^s_{\sigma } = f^{s-1} P^{\lambda }_1 \cdot f_{\sigma } = 0\) and thus \(Q(\theta ,s)\) drops out in
where \(g :=f|_{x_N=1} \in \mathbb {C}[x_1,\ldots ,x_{N-1}]\) is a polynomial in less than N variables. If \(P(\theta ',0,s)\) were nonzero, Lemma 40 would show \(\chi (\mathbb {G}_{m}^{N-1}{\setminus } \mathbb {V}(g))=0\), 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 \(\mathsf {NP}\left( g\right) \), which contradicts that \(\mathsf {NP}\left( g\right) \cong \mathsf {NP}\left( f_{\sigma }\right) =\sigma \) is of dimension \(N-1\).Footnote 22
Therefore, \(P(\theta ',0,s)\) must be zero and we conclude that \(P(\theta ,s)=P^{\lambda }_s \cdot Q\) has a linear factor \(P^{\lambda }_s(\theta ,s)\).Footnote 23 Now set \(m:=Q(\theta ,s)\cdot f^s\), which is nonzero, because \(P(\theta ,s)\) was chosen as an annihilator of \(f^s\) of minimal degree. We may write this element in the form \(m=a\cdot f^{s+r}\) for some \(r\in \mathbb {Z}\) and a Laurent polynomial \(a \in \mathbb {C}(s)[x^{\pm }]\). After multiplying with a polynomial in \(\mathbb {C}[s]\), we may even assume \(0 \ne a \in \mathbb {C}[s,x^{\pm 1}]\) with \(P^{\lambda }_s \cdot af^{s+r} = 0\). Applying the Leibniz rule and dividing by \(af^{s+r}\), we find
Since the degree of \(P^{\lambda }_0 \cdot a\) in s is at most the degree (in s) of a itself, this first summand on the right has a finite limit as \(s\rightarrow 0\). We therefore must have a cancellation of the terms linear in s, \(P^{\lambda }_0 \cdot f = \lambda _0 f\), which yields the sought-after \(P^{\lambda }_s \cdot f^s = 0\). \(\square \)
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 well-known 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
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.Footnote 24 The Symanzik polynomials\(\mathcal {U}_G\) and \(\mathcal {F}_G\) of the graph G are the polynomials \(\mathcal {U}\) and \(\mathcal {F}\) from (2.3) for the set \(\mathsf {D}=(\mathsf {D}_1,\ldots ,\mathsf {D}_N)\) of inverse Feynman propagators,Footnote 25
where \(m_e\) is the mass associated to the particle propagating along edge e. The number \(\mathfrak {C}\left( G\right) \) of master integrals of the Feynman graph G is defined in terms of (3.2) as
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, well-known combinatorial formulas for the Symanzik polynomials [14, 81], which go back at least to [64].
Proposition 46
The Symanzik polynomials of a graph G can be written as
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).
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.Footnote 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
Let \(A,B \in \mathbb {C}[x_1,\ldots ,x_{N-1}]\) and consider the linear polynomial \(A+x_N B\). Then
holds in the Grothendieck ring. In particular, the Euler characteristic is
Proof
Consider the hypersurface \(\mathbb {V}(\mathcal {G})\subset \mathbb {G}_{m}^{N}\), defined by \(\mathcal {G}:=A+x_N B\), under the projection \(\pi :\mathbb {G}_{m}^N \longrightarrow \mathbb {G}_{m}^{N-1}\) that forgets the last coordinate \(x_N\).
As long as \(AB\ne 0\), the unique solution of \(\mathcal {G}=0\) in the fibre is \(x_N=-A/B\). If \(A=0\), the solution \(x_N=0\) is not in \(\mathbb {G}_{m}\) and the fibre is empty, and it is also empty whenever \(B=0\). The only exception to this emptiness is over the intersection \(A=B=0\), where \(x_N\) is arbitrary and the fibre is the full \(\mathbb {G}_{m}\). This fibration proves (3.24); equivalently, we can write it via \([\mathbb {V}(A\cdot B)] = [\mathbb {V}(A)] + [\mathbb {V}(B)] - [\mathbb {V}(A,B)]\) and \([\mathbb {G}_{m}]=\mathbb {L}- 1\) as
Applying the Euler characteristic proves (3.25) due to \(\chi (\mathbb {A}^1)=1\) and \(\chi (\mathbb {G}_{m})=0\). \(\square \)
Corollary 49
Set \({\widetilde{\mathcal {U}}} :=\mathcal {U} \big |_{x_N=1}\) and \({\widetilde{\mathcal {F}}} :=\mathcal {F} \big |_{x_N=1}\). Then
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
Both graphs consisting of a pair of massless edges,
with the external momentum p such that \(p^2 \ne 0\), have a single master integral \(\mathfrak {C}\left( G\right) =1\).
Proof
According to (3.22), the graph polynomials of the graphs in (3.28) are \( \mathcal {U}_{\text {series}} = 1\), \( \mathcal {F}_{\text {series}} = - p^2 (x_1 + x_2)\), \( \mathcal {U}_{\text {parallel}} = x_1 + x_2\), \( \mathcal {F}_{\text {parallel}} = -p^2 x_1 x_2\) such that
In both cases, the number of master integrals (3.10) is \( \mathfrak {C}\left( G\right) =-\chi (\mathbb {G}_{m}{\setminus } \mathbb {V}({\widetilde{\mathcal {U}}}{\widetilde{\mathcal {F}}})) = \chi (\mathbb {G}_{m}\cap \mathbb {V}({\widetilde{\mathcal {U}}} {\widetilde{\mathcal {F}}})) = \chi (\mathbb {G}_{m}\cap \mathbb {V}(1+x_1)) = \chi (\left\{ -1 \right\} ) =1 \) according to (3.27). \(\square \)
Much more on the Grothendieck ring calculus of graph hypersurfaces \(\mathbb {V}(\mathcal {U})\) 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 \(\gamma \) such that all propagators in \(\gamma \) are massless and \(\gamma \) has only two vertices which are connected to external legs or edges in \(G{\setminus }\gamma \).Footnote 27 Write \(G'\) for the graph obtained from G by replacing \(\gamma \) with a single edge (see Fig. 2), then
Proof
Every spanning tree T of G restricts on \(\gamma \) either to a spanning tree or to a spanning two-forest. In the first case, \(T{\setminus } \gamma \) is a spanning tree of \(G/\gamma \) (the graph where \(\gamma \) is contracted to a single vertex); in the second case, \(T{\setminus } \gamma \) is a spanning tree of \(G{\setminus } \gamma \). Note that the two-forests \(T\cap \gamma \) in the second case determine \(\mathcal {F}_{\gamma }\) from (3.22), since all propagators in \(\gamma \) are massless. Therefore, we find \( \mathcal {U}_G = \mathcal {U}_{\gamma } \cdot \mathcal {U}_{G/\gamma } + \mathcal {F}_{\gamma }' \cdot \mathcal {U}_{G{\setminus } \gamma } \) where we set \(\mathcal {F}_{\gamma }':=\mathcal {F}_{\gamma } \big |_{p^2=-1}\). Going through the same considerations for \(\mathcal {F}_G\) shows thatFootnote 28
Now label the edges in \(\gamma \) as \(1,\ldots ,N_{\gamma }\) and rescale all Schwinger parameters \(x_e\) with \(2\le e \le N_{\gamma }\) by \(x_1\). Due to the homogeneity of \(\mathcal {U}_{\gamma }\) and \(\mathcal {F}_{\gamma }\) from Corollary 63, we see that \([\mathbb {G}_{m}^N {\setminus } \mathbb {V}(\mathcal {G}_G)] = [\mathbb {G}_{m}^N {\setminus } \mathbb {V}(A+x_1 B)]\) where \(A={\widetilde{U}}_{\gamma } \mathcal {G}_{G/\gamma }\) and \(B={\widetilde{\mathcal {F}}}_{\gamma }' \mathcal {G}_{G{\setminus } \gamma }\) in terms of \({\widetilde{U}}_{\gamma } :=\mathcal {U}_{\gamma } \big |_{x_1=1}\) and \({\widetilde{\mathcal {F}}}_{\gamma }' :=\mathcal {F}_{\gamma }' \big |_{x_1=1}\). Applying (3.25), we obtain a separation of variables:
In the last line we used (3.27), upon noting the contraction-deletion formula \(\mathcal {G}_{G'} = \mathcal {G}_{G/\gamma } + x_0 \mathcal {G}_{G{\setminus } \gamma }\) in terms of the additional Schwinger parameter \(x_0\) for the (massless) edge that replaces \(\gamma \) in \(G'\). (This formula is easily checked by considering which spanning trees and forests contain this edge or not.) Note that for the subgraph \(\gamma \), the value of \(p^2\) does not matter for \(\mathbb {V}({\widetilde{\mathcal {F}}}_{\gamma })=\mathbb {V}({\widetilde{\mathcal {F}}}_{\gamma }')\), as long as \(p^2\ne 0\). Finally, recall (3.10). \(\square \)
Of course, this result is well known on the function level: If G has a 1-scale subgraph \(\gamma \), then the Feynman integral of G factorizes into the product
of the integrals of \(\gamma \) and \(G'\). Here, we denote by \(\nu _{\gamma }\) and \(\nu '\) the indices corresponding to the edges in \(\gamma \) and outside \(\gamma \), respectively, such that \(\nu =(\nu _{\gamma },\nu ')\). Note that the edge replacing \(\gamma \) in \(G'\) (see Fig. 2) gets the index \(\omega _{\gamma } = \sum _{e \in \gamma } \nu _e - L_{\gamma }\cdot (d/2)\) from (2.4), which depends on the indices of \(\gamma \) (\(L_{\gamma }\) denotes the loop number of \(\gamma \)).
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 series-parallel operations from Fig. 3 does not change the number of master integrals.
Proof
4 Tools and examples
The Euler characteristic of a singular hypersurface can be computed algorithmically via several methods.Footnote 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 \(E\longrightarrow B\) 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.
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
Consider the 2-loop propagator \({\mathrm{WS}}_{3}'\) from Fig. 5. Linear reductions give
in the Grothendieck ring. Substituting \(\mathbb {L}\mapsto 1\) shows that \(\mathfrak {C}\left( {\mathrm{WS}}_{3}'\right) =3\) 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 \(\mathbb {V}(\mathcal {U})\) 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 number of master integrals of the massless propagators obtained by cutting a wheel \({\mathrm{WS}}_{L}\) 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.
4.2 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-trivialFootnote 30 infinite family of Feynman integrals with explicitly known master integral counts. Their first result can be phrased asFootnote 31
Proposition 55
The \(L\)-loop sunrise graph from Fig. 6 with \(L+1\) nonzero masses (and non-exceptional external momentum) has 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 \(\mathcal {U}=\mathcal {F}=0\) imply \(\prod _{i} x_i = 0\), which has no solutions in the torus—hence, this term contributes \(\chi (\mathbb {G}_{m}^{L}) = 0\). We thus obtain
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 \(x_i^{-1}\) by \(x_i\)), so we conclude that
Now let us consider the projection \(\pi :\mathbb {G}_{m}^{L} \longrightarrow \mathbb {G}_{m}^{L-1}\) that forgets \(x_{L}\). Set \(y :=m_{L+1}^2+\sum _{i<L} m_i^2 x_i\) and \(z :=1+\sum _{i<L} x_i^{-1}\), such that \(X_{p^2}^{L} = \left\{ x_{L} p^2 = (1+z x_{L})(y+ m_{L}^2 x_{L}) \right\} \subset \mathbb {G}_{m}^{L}\). We note that the discriminant D of this quadric in \(x_{L}\) factorizes into
such that \(\mathbb {G}_{m}^{L-1} \subset \mathbb {V}(D) = X^{L-1}_{(p+m_{L})^2} \mathbin {\dot{\cup }}X^{L-1}_{(p-m_{L})^2}\) is the disjoint union of two hypersurfaces.Footnote 32 Since the factors are related to the \((L-1)\)-loop sunrise by (4.7), we find
Over a point \(x'\in X^{L-1}_{p\pm m_{L}} \subset \mathbb {V}(D)\) in the discriminant, the fibre of \(\pi ^{-1}(x')\) has precisely one solution \((x',x_{L})\) in \(X^L_{p^2}\), determined by \(x_{L}=-y/[m_{L} (m_{L}\pm p)]\):
If \(D(x')\ne 0\) is nonzero and also \(yz \ne 0\), then the fibre \(\pi ^{-1}(x')\) has precisely two distinct solutions \(x_L\) in the quadric \(X_{p^2}^{L}\). Hence, \(\chi (\pi ^{-1}(x'))=\chi (\mathbb {G}_{m})-2=-2\) and thus
where used that \(\mathbb {V}(D) \cap \mathbb {V}(yz)=\emptyset \) for non-exceptional values of \(p^2\), such that \((p \pm m_{L})^2 \ne 0\) 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 \(X_{p^2}^{L} = \left\{ x_{L}p^2=(1+z x_{L})m_{L}^2 x_{L} \right\} \) is \(x_{L}=0 \notin \mathbb {G}_{m}\); whereas for \(z=0\) the equation for \(X_{p^2}^{L}=\left\{ x_{L}p^2=y+m_{L}^2 x_{L} \right\} \) 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 \(p^2 \ne m_{L}^2\)):
We can now combine (4.10)–(4.12) via \([Y]=[Y\cap \mathbb {V}(D)]+[Y\cap \mathbb {V}(yz)]+[Y{\setminus } \mathbb {V}(D\cdot yz)]\) for \(Y=\mathbb {G}_{m}^{L}{\setminus } X_{p^2}^{L}\) into the reduction formula
Here, we inserted (4.9) and used \(\chi (\mathbb {V}(y))=\chi (\mathbb {V}(z))=-\chi (\mathbb {G}_{m}^{L-1}{\setminus }\mathbb {V}(z))=(-1)^{L}\), which follows from repeated application of (3.25)—just as above, when we computed the first term in (4.5). According to (4.7), we can write the reduction as the recursion
which is obviously solved by the claimed . It merely remains to verify the base case \(L=1\), and indeed, follows easily from (4.7) since \(X_{p^2}^1 = \left\{ x_1 p^2 = (1+x_1)(m_2^2+m_1^2 x_1) \right\} \subset \mathbb {G}_{m}\) consists of precisely two points. \(\square \)
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]Footnote 33:
Proposition 56
The \(L\)-loop sunrise graph with \(R \le L\) nonzero masses, \(L+1-R\) vanishing masses and non-exceptional external momentum, has 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 \(m_{L+1}=0\).
We can apply the exact same recursion as in the proof of Proposition 55; the only difference to (4.13) is that now, \(\chi (\mathbb {V}(y)) = 0\) vanishes because \(y=\sum _{i<L} m_i^2 x_i\) has become homogeneous in x such that \([\mathbb {V}(y)]=[\mathbb {G}_{m}]\cdot [\mathbb {V}(y)\cap \left\{ x_1=1 \right\} ]\). Therefore, (4.13) takes the form and yields, via (4.7), the recursion
We are done after verifying the base case: Indeed, from (4.7) is clear since \(X_{p^2}^1 = \left\{ p^2 x_1 = (1+x_1) m_1^2 x_1 \right\} \) is the single point \(x_1=p^2/m_1^2-1\) in \(\mathbb {G}_{m}\). \(\square \)
4.3 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 \(\mathcal {G}\) to \({\tilde{\mathcal {G}}} = x_0 \mathcal {U}+ \mathcal {F}\), 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 using Euler for up to six loops.
Example 57
Consider the one-loop sunrise graph with \(m_1^2=m_2^2=-p^2=1\), which is a non-degenerate kinematic configuration. According to Example 47, its Lee–Pomeransky polynomial is \(\mathcal {G}= (x_1+x_2)(x_1+x_2+1)+x_1 x_2\). The Macaulay2 script
computes the output 0 for \(\chi (\mathbb {V}(x_0 x_1 x_2 {\tilde{\mathcal {G}}}) \cap \mathbb {P}^2)\). Using \(\chi (\mathbb {P}^2)=3\) and (4.15), we conclude in agreement with Proposition 55.
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 \({\mathrm{WS}}_{3}'\) 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 D-modules and Gröbner bases and tends to demand more resources than the method discussed above.
Example 58
Consider again the massive one-loop sunrise from Example 47. The program
computes the following cohomology groups of \(X=\mathbb {G}_{m}^2 {\setminus } \mathbb {V}(\mathcal {G})\): \(H^0(X) \cong \mathbb {Q}\), \(H^1(X) \cong \mathbb {Q}^3\) and \(H^2(X)\cong \mathbb {Q}^5\). Hence, as in Example 57.
The same functionality is provided by Singular’s deRham.lib library via the command deRhamCohomology. The Singular analogue of Example 58 is
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 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.
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 \(\mathfrak {C}\left( G\right) = 15\), 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 two-loop sunrise with one massless line has 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 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 \(H^N(\mathbb {C}^N{\setminus } \mathbb {V}(\mathcal {G}))\).Footnote 34 In most cases, the program Mint computes this number correctly, which then agrees with the other mentioned methods.Footnote 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 \(\nu \). 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
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 \(\mathfrak {C}\left( G\right) \) the integrals associated to subsectors (and corrects for double counting). Note that if \(\gamma \) 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 \(\mathcal {G}_{G/\gamma }=\mathcal {G} \big |_{x_e=0\forall e \in \gamma }\), we find that
is the Euler characteristic of the hypersurface complement inside affine space (as compared to the torus \(\mathbb {G}_{m}^N\) 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, \(\widehat{\mathfrak {C}}\left( G/\left\{ 1,2 \right\} \right) =-1\) 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 \(G/\left\{ 1,2,3 \right\} \) and \(G/\left\{ 1,2,4 \right\} \) have one master integral each, if we consider them individually. However, they are embedded into the graph \(G/\left\{ 1,2 \right\} \), which has only a single master integral, \(\mathfrak {C}\left( G/\left\{ 1,2 \right\} \right) =1\). This is sometimes referred to as a “relation between subtoplogies”, and the negative value of \(\widehat{\mathfrak {C}}\left( G/\left\{ 1,2 \right\} \right) =-1\) 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.Footnote 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 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 \({{\mathrm{Ann}}}(\mathcal {G}^s)={{\mathrm{Ann}}}^1(\mathcal {G}^s)\) is linearly generated, remains to be settled. A positive answer to either of these would imply that the labour-intensive 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 set-up 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 \(\left\{ \mathcal {G}=0 \right\} \) 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 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 \(\nu _a\) 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 one-scale subgraphs (both situations introduce non-integer indices). For a recent step into this direction, see [91].
Notes
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.
The only exception are cases of zero-scale subintegrals, like massless tadpoles. Such integrals are zero in dimensional regularization and therefore irrelevant for our considerations.
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.
Recall that \(q_{1},\ldots , q_{L}\) denote the loop momenta, whereas \(q_{L+1},\ldots , q_{M}\) are the external momenta.
For an algebraic proof of the analogous statement in the Baikov representation, see [35, section 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).
In fact, \( H(s)\mathcal {G}+ s\mathcal {U}= \sum _e x_e \left( \mathcal {G}\partial _e - (s-1) (\partial _e \mathcal {G}) \right) \) follows from the trivial annihilators (2.14) of \(\mathcal {G}^{s-1}\).
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]).
The first fraction can also be written as \(2^{L}/\left( d-L-E+1 \right) _{L}\) in terms of the Pochhammer symbol (raising factorial) \(a_{L} = a (a+1) \cdots (a+L-1)\).
Recall that a holonomic module over the point \(\mathbb {A}^0_k\) is the same as a finite-dimensional k-vector space.
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, finite-dimensional vector spaces over k.
This sign arises from the shift by N in the Definition (3.6) of the de Rham complex \({{\mathrm{DR}}}\).
It fails precisely if, for some \(r \in \mathbb {N}\), \(s-r\) is a zero of the Bernstein–Sato polynomial of \(\mathcal {G}\).
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.
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.
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 push-forward 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, \(\pi _{*} (F_{*} \mathscr {M}') = (\pi \circ F)_{*} \mathscr {M}' = {{\mathrm{id}}}_{*} \mathscr {M}' = \mathscr {M}'\).
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 \(\partial _i \bullet m(s) = (\partial _i - (\partial _i f) \partial _t)m(s) = (\partial _i +s (\partial _i f) t^{-1})m(s) = (\partial _i +s (\partial _i f)/f)m(s-1) \) in accordance with (3.11).
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.
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 \(\dim \mathsf {NP}\left( g\right) =\dim \mathsf {NP}\left( f_{\sigma }\right) \).
The existence of a linear annihilator could also be deduced from [31, Théorème 9.2].
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.
The infinitesimal imaginary part \(\text {{i}}\epsilon \) is irrelevant for our purpose of counting integrals and will be henceforth ignored.
Such a graph \(\gamma \) is called massless propagator or p-integral.
Note that the intersection of a two-forest 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).
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.
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.
We assume \(p^2 \ne 0\) and \(m_{L}^2 \ne 0\), which guarantees that \((p+m_{L})^2 \ne (p-m_{L})^2\).
The additional term \(-\delta _{0,L-R}\) in [43, Equation (4.13)] subtracts a reducible integral that can be attributed to a subtopology. Our counting, however, accounts for all master integrals.
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.
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.)
As a referee kindly pointed out, this relation also follows from momentum space IBPs.
We use the signature \((1,-\,1,\ldots ,-\,1)\) for the Minkowski metric.
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.
Recall that our metric has signature \((1,-1,\ldots ,-1)\), so the integrations over the \(d-1\) 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.
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 \(\mathbb {R}_+^N\).
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 \(\mathscr {M}x_N^{\nu }\).
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 \(c<\infty \).
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.
Let \(\pi ^{N}:\mathbb {G}_{m}^N\longrightarrow \left\{ \text {pt} \right\} \) denote the projection to a point, such that \(\pi ^N = \pi ^{N-1} \circ \pi \). The identity \(\pi ^{N}_+ = \pi ^{N-1}_+ \circ \pi _{+}\) of the corresponding push-forwards 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 \(H^0(\pi _+ \mathscr {M}) = \mathscr {M}/(\partial _N \mathscr {M})\).
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 \(\partial _N^k m =0\).
The first statement is clear since \(\ker \partial _N \subseteq \mathscr {N}\). The second claim follows from the identity \(x^k\partial ^k = (\partial x -k)x^{k-1}\partial ^{k-1} = \cdots = \prod _{i=1}^k (\partial x -i)\), which implies \(0=x_N^k \partial _N^k m = (-1)^k (k!) m \mod \partial _N \mathscr {N}\) whenever \(\partial _N^k m = 0\).
References
Aluffi, P., Marcolli, M.: Feynman motives and deletion–contraction relations. In: Topology of Algebraic Varieties and Singularities, vol. 538 of Contemporary Mathematics, pp. 21–64. American Mathematical Society. arXiv:0907.3225 [math-ph] (2011)
Anastasiou, C., Lazopoulos, A.: Automatic integral reduction for higher order perturbative calculations. J. High Energy Phys. 7, 046 (2004). arXiv:hep-ph/0404258
Andres, D.: Algorithms for the computation of Sato’s b-functions in algebraic D-module theory. Diploma thesis, Rheinisch-Westfälische Technische Hochschule Aachen (2010)
Andres, D., Brickenstein, M., Levandovskyy, V., Martín, J.: Constructive \(D\)-module theory with singular. Math. Comput. Sci. 4(2), 359–383 (2010)
Baikov, P.A.: Explicit solutions of the 3-loop vacuum integral recurrence relations. Phys. Lett. B 385, 404–410 (1996). arXiv:hep-ph/9603267
Baikov, P.A.: Explicit solutions of the \(n\)-loop vacuum integral recurrence relations. Preprint arXiv:hep-ph/9604254 (1996)
Baikov, P.A.: Explicit solutions of the multi-loop integral recurrence relations and its application. Nucl. Instrum. Methods A 389(1), 347–349 (1997). arXiv:hep-ph/9611449
Baikov, P.A.: A practical criterion of irreducibility of multi-loop Feynman integrals. Phys. Lett. B 634, 325–329 (2006). arXiv:hep-ph/0507053
Bardin, D.Y., Kalinovskaya, L.V., Tkachov, F.V.: New algebraic numeric methods for loop integrals: some 1-loop experience. In: High Energy Physics and Quantum Field Theory. Proceedings, 15th International Workshop, QFTHEP 2000, Tver, Russia, 14–20, 2000, pp. 230–232. arXiv:hep-ph/0012209 (2000)
Bernshtein, I.N.: The analytic continuation of generalized functions with respect to a parameter. Funct. Anal. Appl. 6(4), 273–285 (1972)
Binosi, D., Theußl, L.: JaxoDraw: a graphical user interface for drawing Feynman diagrams. Comput. Phys. Commun. 161, 76–86 (2004). arXiv:hep-ph/0309015
Björk, J.-E.: Rings of Differential Operators, vol. 21. North-Holland Mathematical Library, North-Holland, Amsterdam (1979)
Boels, R.H., Kniehl, B.A., Yang, G.: Master integrals for the four-loop Sudakov form factor. Nucl. Phys. B 902, 387–414 (2016). arXiv:1508.03717 [hep-th]
Bogner, C., Weinzierl, S.: Feynman graph polynomials. Int. J. Mod. Phys. A 25, 2585–2618 (2010). arXiv:1002.3458 [hep-ph]
Brown, F.C.S.: The massless higher-loop two-point function. Commun. Math. Phys. 287, 925–958 (2009). arXiv:0804.1660 [math.AG]
Brown, F.C.S.: On the periods of some Feynman integrals. Preprint arXiv:0910.0114 [math.AG] (2009)
Brown, F.C.S., Schnetz, O.: A K3 in \(\phi ^{4}\). Duke Math. J. 161, 1817–1862 (2012). arXiv:1006.4064 [math.AG]
Brychkov, Y.A., Glaeske, H.J., Prudnikov, A.P., Tuan, V.K.: Multidimensional Integral Transformations. Gordon and Breach Science Publishers, Philadelphia (1992)
Bytev, V.V., Kalmykov, M.Y., Kniehl, B.A.: Differential reduction of generalized hypergeometric functions from Feynman diagrams: one-variable case. Nucl. Phys. B 836, 129–170 (2010). arXiv:0904.0214 [hep-th]
Caracciolo, S., Sportiello, A., Sokal, A.D.: Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type identities. I: generalizations of the Capelli and Turnbull identities. Electron. J. Comb. 16, 103 (2009). arXiv:0809.3516 [math.CO]
Chetyrkin, K.G., Faisst, M., Sturm, C., Tentyukov, M.: \(\epsilon \)-finite basis of master integrals for the integration-by-parts method. Nucl. Phys. B 742, 208–229 (2006). arXiv:hep-ph/0601165
Chetyrkin, K.G., Tkachov, F.V.: Integration by parts: the algorithm to calculate \(\beta \) functions in 4 loops. Nucl. Phys. B 192, 159–204 (1981)
Collins, J.C.: Renormalization. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1984)
Coutinho, S.C.: A Primer of Algebraic \(D\)-Modules, Vol. 33 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1995)
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 4-0-2—a computer algebra system for polynomial computations (2015). http://www.singular.uni-kl.de. Accessed 1 Jan 2018
Derkachov, S.E., Honkonen, J., Pis’mak, Y.M.: Three-loop calculation of the random walk problem: an application of dimensional transformation and the uniqueness method. J. Phys. A Math. Gen. 23, 5563–5576 (1990)
Ferroglia, A., Passarino, G., Passera, M., Uccirati, S.: A frontier in multi-scale multi-loop integrals: the algebraic-numerical method, nuclear instruments and methods in physics research section a: accelerators, spectrometers, detectors and associated equipment 502 (2003). In: Proceedings of the VIII International Workshop on Advanced Computing and Analysis Techniques in Physics Research (ACAT), Moscow, June 24–28, pp. 391–395 (2002)
Ferroglia, A., Passera, M., Passarino, G., Uccirati, S.: All-purpose numerical evaluation of one-loop multi-leg Feynman diagrams. Nucl. Phys. B 650, 162–228 (2003). arXiv:hep-ph/0209219
Foata, D., Zeilberger, D.: Combinatorial proofs of Capelli’s and Turnbull’s identities from classical invariant theory. Electron. J. Comb. 1, 1 (1994). arXiv:math/9309212
Fujimoto, J., Kaneko, T.: GRACE and loop integrals. PoS LL2012, 047 (2012). PoS(LL2012)047
Gabber, O., Loeser, F.: Faisceaux pervers \(\ell \)-adiques sur un tore. Duke Math. J. 83(3), 501–606 (1996)
Georgoudis, A., Larsen, K.J., Zhang, Y.: Azurite: an algebraic geometry based package for finding bases of loop integrals. Comput. Phys. Commun. (2017). arXiv:1612.04252 [hep-th]. https://bitbucket.org/yzhphy/azurite
Ginsburg, V.: Characteristic varieties and vanishing cycles. Invent. Math. 84(2), 327–402 (1986)
Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/. Accessed 1 Jan 2018
Grozin, A.G.: Integration by parts: an introduction. Int. J. Mod. Phys. A 26(17), 2807–2854 (2011). arXiv:1104.3993 [hep-ph]. Extended Version of the Lectures at the School Computer Algebra and Particle Physics at DESY Zeuthen, Germany, March 21–25 (2011)
Gyoja, A.: Bernstein–Sato’s polynomial for several analytic functions. J. Math. Kyoto Univ. 33(2), 399–411 (1993)
Hahn, T.: Generating Feynman diagrams and amplitudes with FeynArts 3. Comput. Phys. Commun. 140, 418–431 (2001). arXiv:hep-ph/0012260
Helmer, M.: Algorithms to compute the topological Euler characteristic. Chern–Schwartz–MacPherson class and Segre class of projective varieties. J. Symb. Comput. 73, 120–138 (2016)
Henn, J.M.: Multiloop integrals in dimensional regularization made simple. Phys. Rev. Lett. 110, 251601 (2013). arXiv:1304.1806 [hep-th]
Ita, H.: Two-loop integrand decomposition into master integrals and surface terms. Phys. Rev. D 94, 116015 (2016). arXiv:1510.05626 [hep-th]
Kalmykov, M.Y., Kniehl, B.A.: Counting master integrals: integration by parts vs. differential reduction. Phys. Lett. B 702(4), 268–271 (2011). arXiv:1105.5319 [math-ph]
Kalmykov, M.Y., Kniehl, B.A.: Mellin–Barnes representations of Feynman diagrams, linear systems of differential equations, and polynomial solutions. Phys. Lett. B 714, 103–109 (2012). arXiv:1205.1697 [hep-th]
Kalmykov, M.Y., Kniehl, B.A.: Counting the number of master integrals for sunrise diagrams via the Mellin–Barnes representation. JHEP 2017, 031 (2017). arXiv:1612.06637 [hep-th]
Kashiwara, M.: \(B\)-functions and holonomic systems. Rationality of roots of \(B\)-functions. Invent. Math. 38, 33–53 (1976)
Kashiwara, M.: On the holonomic systems of linear differential equations, II. Invent. Math. 49, 121–135 (1978)
Kashiwara, M., Kawai, T.: Holonomic systems of linear differential equations and Feynman integrals. Publ. Res. Inst. Math. Sci. Kyoto 12, 131–140 (1977)
Khovanskii, A.G.: Newton polyhedra and the genus of complete intersections. Funct. Anal. Appl. 12(1), 38–46 (1978)
Kniehl, B.A., Kotikov, A.V.: Counting master integrals: integration-by-parts procedure with effective mass. Phys. Lett. B 712, 233–234 (2012). arXiv:1202.2242 [hep-ph]
Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32, 1–31 (1976)
Laporta, S.: High-precision calculation of multiloop Feynman integrals by difference equations. Int. J. Mod. Phys. A 15(32), 5087–5159 (2000). arXiv:hep-ph/0102033
Laumon, G.: Sur la catégorie dérivée des \({\cal{D}}\)-modules filtrés, vol. 1016 of Lecture Notes in Mathematics, pp. 151–237. Springer, Berlin (1983)
Lee, R.N.: Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals. J. High Energy Phys. 2008, 31 (2008). arXiv:0804.3008 [hep-ph]
Lee, R.N.: Calculating multiloop integrals using dimensional recurrence relation and \(\cal{D}\)-analyticity. Nucl. Phys. B Proc. Suppl. 205, 135–140. arXiv:1007.2256 [hep-ph]. 10th DESY Workshop on Elementary Particle Theory: Loops and Legs in Quantum Field Theory, Woerlitz, Germany, April 25–30 (2010)
Lee, R.N.: Space-time dimensionality \(\cal{D}\) as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to \(\cal{D}\). Nucl. Phys. B 830(3), 474–492 (2010). arXiv:0911.0252 [hep-ph]
Lee, R.N.: Presenting LiteRed: a tool for the Loop InTEgrals REDuction. Preprint arXiv:1212.2685 [hep-ph] (2012)
Lee, R.N.: Modern techniques of multiloop calculations. In: Proceedings, 49th Rencontres de Moriond on QCD and High Energy Interactions, pp. 297–300. arXiv:1405.5616 [hep-ph] (2014)
Lee, R.N.: LiteRed 1.4: a powerful tool for reduction of multiloop integrals. J. Phys. Conf. Ser. 523, 012059 (2014). arXiv:1310.1145 [hep-ph]
Lee, R.N., Pomeransky, A.A.: Critical points and number of master integrals. JHEP 2013, 165 (2013). arXiv:1308.6676 [hep-ph]
Loeser, F., Sabbah, C.: Caractérisation des \(\cal{D}\) -modules hypergéométriques irréductibles sur le tore. C. R. Acad. Sci. Paris Sér. I Math. 312(10), 735–738 (1991)
Loeser, F., Sabbah, C.: Equations aux différences finies et déterminants d’intégrales de fonctions multiformes. Comment. Math. Helv. 66(3), 458–503 (1991)
Loeser, F., Sabbah, C.: Caractérisation des \(\cal{D}\)-modules hypergéométriques irréductibles sur le tore, II. C. R. Acad. Sci. Paris Sér. I Math. 315, 1263–1264 (1992)
Maierhöfer, P., Usovitsch, J., Uwer, P.: Kira—a Feynman integral reduction program. arXiv:1705.05610 [hep-ph] (2017)
Malgrange, B.: Le polynôme de Bernstein d’une singularité isolée, vol. 459 of Lecture Notes in Mathematics, pp. 98–119. Springer, Berlin (1975)
Nakanishi, N.: Graph theory and Feynman integrals. Mathematics and Its Applications, vol. 11. Gordon and Breach, New York (1971)
Oaku, T.: Algorithms for the \(b\)-function and \(D\) -modules associated with a polynomial. J. Pure Appl. Algebra 117—-118(Supplement C), 495–518 (1997)
Oaku, T., Takayama, N.: An algorithm for de Rham cohomology groups of the complement of an affine variety via \(D\)-module computation. J. Pure Appl. Algebra 139(1–3), 201–233 (1999). arXiv:math/9801114
Oaku, T., Takayama, N.: Algorithms for \(D\)-modules—restriction, tensor product, localization, and local cohomology groups. J. Pure Appl. Algebra 156(2), 267–308 (2001). arXiv:math/9805006
Passarino, G.: An approach toward the numerical evaluation of multi-loop Feynman diagrams. Nucl. Phys. B 619(1), 257–312 (2001). arXiv:hep-ph/0108252
Passarino, G., Uccirati, S.: Algebraic-numerical evaluation of Feynman diagrams: two-loop self-energies. Nucl. Phys. B 629(1–3), 97–187 (2002). arXiv:hep-ph/0112004
Ruijl, B., Ueda, T., Vermaseren, J.A.M.: The diamond rule for multi-loop Feynman diagrams. Phys. Lett. B 746, 347–350 (2015). arXiv:1504.08258 [hep-ph]
Sabbah, C.: Proximité évanescente. II. Équations fonctionnelles pour plusieurs fonctions analytiques. Compos. Math. 64(2), 213–241 (1987)
Saito, M., Sturmfels, B., Takayama, N.: Gröbner Deformations of Hypergeometric Differential Equations, Vol. 6 of Algorithms and Computation in Mathematics. Springer, Berlin (2000)
Sato, M., Shintani, T., Muro, M.: Theory of prehomogeneous vector spaces (algebraic part)—the English translation of Sato’s lecture from Shintani’s note. Nagoya Math. J. 120, 1–34 (1990)
Schnetz, O.: Quantum field theory over \(\mathbb{F}_{q}\). Electron. J. Combin. 18, P102 (2011). arXiv:0909.0905 [math.CO]
Smirnov, A.V.: Algorithm FIRE—Feynman Integral REduction. J. High Energy Phys. 10, 107 (2008). arXiv:0807.3243 [hep-ph]
Smirnov, A.V.: FIRE5: a C++ implementation of Feynman Integral REduction. Comput. Phys. Commun. 189, 182–191 (2015). arXiv:1408.2372 [hep-ph]
Smirnov, A.V., Petukhov, A.V.: The number of master integrals is finite. Lett. Math. Phys. 97(1), 37–44 (2011). arXiv:1004.4199 [hep-th]
Smirnov, A.V., Smirnov, V.A.: Applying Gröbner bases to solve reduction problems for Feynman integrals. J. High Energy Phys. 1, 001 (2006). arXiv:hep-lat/0509187
Smirnov, A.V., Smirnov, V.A.: On the reduction of Feynman integrals to master integrals. In: Proceedings, 11th International Workshop on Advanced computing and analysis techniques in physics research (ACAT 2007), Vol. ACAT2007, p. 085. arXiv:0707.3993 [hep-ph] (2007)
Smirnov, A.V., Smirnov, V.A.: FIRE4, LiteRed and accompanying tools to solve integration by parts relations. Comput. Phys. Commun. 184, 2820–2827 (2013). arXiv:1302.5885 [hep-ph]
Smirnov, V.A.: Analytic Tools for Feynman integrals. Springer Tracts in Modern Physics, vol. 250. Springer, Berlin (2012)
Smirnov, V.A., Steinhauser, M.: Solving recurrence relations for multi-loop Feynman integrals. Nucl. Phys. B 672, 199–221 (2003). arXiv:hep-ph/0307088
Speer, E.R.: Generalized Feynman Amplitudes. Annals of Mathematics Studies, vol. 62. Princeton University Press, Princeton (1969)
Speer, E.R.: Ultraviolet and infrared singularity structure of generic Feynman amplitudes. Ann. Inst. H. Poincaré Sect. A 23(1), 1–21 (1975)
Stembridge, J.R.: Counting points on varieties over finite fields related to a conjecture of Kontsevich. Ann. Comb. 2(4), 365–385 (1998)
Studerus, C.: Reduze—Feynman integral reduction in C++. Comput. Phys. Commun. 181, 1293–1300 (2010). arXiv:0912.2546 [physics.comp-ph]
’t Hooft, G., Veltman, M.: Regularization and renormalization of gauge fields. Nucl. Phys. B 44, 189–213 (1972)
Tancredi, L.: Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations. Nucl. Phys. B 901, 282–317 (2015). arXiv:1509.03330 [hep-ph]
Tarasov, O.V.: Connection between Feynman integrals having different values of the space–time dimension. Phys. Rev. D 54, 6479–6490 (1996). arXiv:hep-th/9606018
Tarasov, O.V.: Reduction of Feynman graph amplitudes to a minimal set of basic integrals. In: Loops and Legs in Gauge Theories. Proceedings, Zeuthen Workshop on Elementary Particle Theory, Rheinsberg, Germany, April 19–24, 1998, vol. 29, p. 2655. arXiv:hep-ph/9812250 (1998)
Tarasov, O.V.: Massless on-shell box integral with arbitrary powers of propagators. Preprint arXiv:1709.07526 [hep-ph] (2017)
Tkachov, F.V.: A theorem on analytical calculability of 4-loop renormalization group functions. Phys. Lett. B 100, 65–68 (1981)
Tkachov, F.V.: Algebraic algorithms for multiloop calculations The first 15 years. What’s next? Nucl. Instrum. Methods Phys. Res. Sect. A 389, 309–313 (1997). arXiv:hep-ph/9609429. New Computing Techniques in Physics Research V. Proceedings, 5th International Workshop AIHENP, Lausanne, September 2–6 (1996)
Turnbull, H.W.: Symmetric determinants and the Cayley and Capelli operator. Proc. Edinb. Math. Soc. 8, 76–86 (1948)
Vermaseren, J.A.M.: Axodraw. Comput. Phys. Commun. 83, 45–58 (1994)
von Manteuffel, A., Panzer, E., Schabinger, R.M.: A quasi-finite basis for multi-loop Feynman integrals. J. High Energy Phys. 2015, 120 (2015). arXiv:1411.7392 [hep-ph]
von Manteuffel, A., Schabinger, R.M.: A novel approach to integration by parts reduction. Phys. Lett. B 744, 101–104 (2015). arXiv:1406.4513 [hep-ph]
von Manteuffel, A., Studerus, C.: Reduze 2—distributed Feynman integral reduction. Preprint arXiv:1201.4330 [hep-ph] (2012)
Zhang, Y.: Integration-by-parts identities from the viewpoint of differential geometry. In: 19th Itzykson Meeting on Amplitudes 2014 (Itzykson2014) Gif-sur-Yvette, France, June 10–13, 2014. arXiv:1408.4004 [hep-th] (2014)
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 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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Appendices
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.
1.1 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,Footnote 37
which depends on the mass \(m_e\) of the particle e and the \(d\)-dimensional momentum \(k_e \in \mathbb {R}^{d}\) flowing through this edge. Enforcing momentum conservation at each vertex fixes all \(k_e\) in terms of \(E\) independent external momenta \(p_1,\ldots ,p_E\) and \(L\) free loop momenta \(\ell _1,\ldots ,\ell _{L}\). Note that the actual number of external legs of G is \(E+1\), since overall momentum conservation \(\sum _{i=1}^{E+1} p_i=0\) imposes one relation among the external momenta. Taking only the inverse Feynman propagators \(\mathsf {D}_e\) 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, this yields
Typically, the number \(\left|\Theta \right|=L(L+1)/2+LE\) of independent scalar products \(s_{\left\{ i,j \right\} }\) 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 \(\left|\Theta \right|\) denominators with the property that the matrix \(\mathcal {A}\) defined by
becomes invertible. This means that all loop-momentum-dependent scalar products can be written as linear combinations of the denominators, see (2.29). The additional denominators introduced in this way are called irreducible scalar products.
Example 62
We again consider the graph from Fig. 8 with 6 internal edges labelled as in Example 61. In the massless case, the inverse propagators are just \(\mathsf {D}_e = -k_e^2\) and their explicit decomposition into the \(\left|\Theta \right|=7\) scalar products takes the form
The matrix \(\mathcal {A}\) has rank 6 and annihilates \((0,0,0,1,0,0,0)^\intercal \). Thus, we can choose \(\mathsf {D}_7 = \ell _1 p_1\) as an irreducible scalar product to complete the basis of quadratic forms in the loop momenta. The matrix \(\mathcal {A}\) 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 \(N \times \left|\Theta \right|\) matrix \(\mathcal {A}\) of the inverse propagators has full rank N) in order to be able to extend them to a basis of quadratic forms by choosing \(\left|\Theta \right|-N\) irreducible scalar products.Footnote 38
For each denominator \(\mathsf {D}_a\) we introduce a scalar \(x_{a}\), which is known as Schwinger-, Feynman- or \(\alpha \)-parameter. In Definition 1 we have introduced the decomposition
which determines a symmetric \(L\times L\) matrix \(\Lambda \), a vector Q and a scalar \(J\). With their help we defined the polynomials \(\mathcal {U}\), \(\mathcal {F}\) and \(\mathcal {G}=\mathcal {U}+\mathcal {F}\) in (2.3). Explicitly, from Definition 1 and (2.26) we can read off that
Since \(\Lambda _{ij}\) is an \(L\times L\) matrix with entries that are linear in the Schwinger parameters, the polynomial \(\mathcal {U}\) is homogeneous of degree \(L\). By Cramer’s rule, \((\det \Lambda )\Lambda ^{-1}_{ij}\) is homogeneous of degree \(L-1\) and the linearity of \(Q\) and \(J\) in the \(x_a\) implies
Corollary 63
\(\mathcal {U}\) and \(\mathcal {F}\) are homogeneous polynomials in the variables \(x_{1},\ldots , x_{N}\) with the degrees \(\deg (\mathcal {U})=L\) and \(\deg (\mathcal {F})=L+1\). Hence, for \(\mathcal {G}=\mathcal {U}+\mathcal {F}\), we have
Let us now come to the proof of Proposition 2 following [64] and [58].
Proof of Proposition 2
We consider the Feynman integral defined in (2.1),
Using the Schwinger trick to exponentiate each denominator,Footnote 39
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 \(\ell ' :=\ell - \Lambda ^{-1} Q\) asFootnote 40
In summary, we therefore arrive at the integral representation (2.5):
We now multiply with \(1=\int _0^{\infty } \delta (\rho -\sum _{j=1}^N x_j) \mathrm{d}^{}\rho \) and substitute \(x_a \rightarrow \rho x_a\).Footnote 41 The Jacobian \(\rho ^N\), the monomials \(x_i^{\nu _i-1}\) and \(\delta (\rho -\sum _j x_j) \rightarrow \delta (1-\sum _j x_j)/\rho \) contribute the power \(\rho ^{\left|\nu \right|-1}\), whereas the homogeneity of \(\mathcal {F}\) and \(\mathcal {U}\) from Corollary 63 implies that \(\mathcal {U}\rightarrow \rho ^{L}\mathcal {U}\) and \(\mathcal {F}/\mathcal {U}\rightarrow \rho \mathcal {F}/\mathcal {U}\). Overall, by realizing that the integral over \(\rho \) is
we arrive at the first parametric formula (2.6). Similarly, we multiply the integrand of (2.7) with \(1=\int _0^{\infty } \delta (\rho -\sum _{i} x_i) \mathrm{d}^{}\rho \) and substitute \(x_i \rightarrow \rho x_i\). Using \(\mathcal {U}\rightarrow \rho ^{L} \mathcal {U}\) and \(\mathcal {F}\rightarrow \rho ^{L+1}\mathcal {F}\) from Corollary 63, the integral over \(\rho \) becomes
and combines with the prefactors in (2.7) to reproduce (2.7). \(\square \)
We conclude the section with the proof of Proposition 20 following Grozin [35]:
Proof of Proposition 20
The action of \(\mathbf{o }_{j}^{i}\) 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 \(\left\{ j,m \right\} \in \Theta \) 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 \(\nu _a/\mathsf {D}_a\) is equivalent to the action of the operator \(\hat{\mathbf{a }}^{+}\) defined in (2.17), whereas multiplication with \(\mathsf {D}_b\) lowers the index \(\nu _b\) and corresponds to \(\mathbf{b }^{-}\). \(\square \)
1.2 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 \({{\mathrm{Ann}}}\left( \mathcal {G}^{-d/2} \right) \). 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.
Lemma 64
The operators \(\widetilde{\mathbf{O }}_{j}^{i}\) from (2.36) and (2.37) corresponding to the momentum space IBP relations annihilate the parametric integrand \(\mathcal {G}^{-d/2}\):
Proof
Let us first consider the case \(j\le L\). After acting with \(\widetilde{\mathbf{O }}_{j}^{i}\) from (2.36) on \(\mathcal {G}^{-d/2}\) and dividing by \((d/2) \mathcal {G}^{-d/2-1}\), we are left to prove the vanishing of
where we exploited the homogeneity from (A.5). Using (2.3), we note that
because \(\partial _b J= \lambda _b\) according to (A.4). In order to evaluate \(\partial _b \mathcal {U}\) with Jacobi’s formula \((\partial _b \mathcal {U})/\mathcal {U}= \sum _{r,s=1}^{L} \Lambda ^{-1}_{r,s} \partial _b \Lambda _{s,r}\), we use (A.2) to compute
which restricts m to either r or s. So in particular, \(m\le L\) and we can use (A.2) in
which proves that for arbitrary j (independent of whether \(j\le L\) 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 \(\partial _b Q\) can be read off from (A.3) and the sum over b yields
because \(j\le L< r\) excludes the possibility that \(m=s\) and \(r=j\). Thus with (A.3),
cancels the contribution from (A.13) in (A.12) and finishes the proof in the case \(j\le L\). If instead we have \(j>L\), then we must replace \(\delta _{\left\{ m,j \right\} ,\left\{ s,r \right\} } = \delta _{j,r} \delta _{m,s}\) in (A.14) such that
where we used (A.2) once more. Now recall that (A.11) remains true and becomes zero for \(j>L\) because \(\delta _{i,j}=0\) since \(i\le L\). For the same reason, \(\delta _{j,s}=0\) in (A.13) and therefore, using (A.9),
This is precisely cancelled by the additional contribution to (A.8) coming from \(\widetilde{\mathbf{O }}_{j}^{i}\) in (2.37) in the case \(j>L\): The additional term acts on \(\mathcal {G}^{-d/2}\) as
after dividing by \((d/2) \mathcal {G}^{-d/2-1}\). Note that here we used (A.5) and (A.3). \(\square \)
1.3 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 \(q_1, \ldots , q_{M}\) are vectors in a Euclidean vector space and write
for their Gram matrices and determinants. Note that
and thus, by adding \(-(p_n \cdot p_{\bullet }) V_{n+1}^{-1}\) times the lower \(M-n\) rows to the first row,
Indeed, formula \({\text {pr}}_{{{\mathrm{lin}}}\left\{ q_{n+1},\ldots ,q_{M} \right\} }^{\perp } (v) = \sum _{i,j=n+1}^{M} q_i \left( V^{-1}_{n+1} \right) _{i,j} (q_j \cdot v)\) for the orthogonal projection of v onto the space spanned by \(q_{n+1},\ldots ,q_{M}\) shows that
Now assume our integrand f only depends on the scalar products \(s_{i,j} = q_i \cdot q_j\), and we want to integrate out the first loop momentum \(q_1\). Let us decompose \(q_1 = q_{\perp } + q_{\parallel }\) into the component \(q_{\parallel } \in {{\mathrm{lin}}}\left\{ q_2,\ldots ,q_{M} \right\} \) that lies in the space spanned by the other momenta, and the component \(q_{\perp }\) in its orthogonal complement. According to (A.17), \(G_n^{1/2}\) is the volume of the parallelotope spanned by \(q_n,\ldots ,q_{M}\). Hence, changing coordinates from \(q_{\parallel }\) to \((s_{1,2},\ldots ,s_{1,M})\) yields
The integral over the orthogonal component is, due to \(s_{1,1}=q_1^2=q_{\perp }^2 + q_{\parallel }^2\), given by
Note that the lower boundary \(s_{1,1}=q_{\parallel }^2\) corresponds to \(0=q_{\perp }^2=G_1/G_2\). Altogether,
Transforming the remaining loop integrations analogously, all but two of the Gram determinants cancel, and we conclude that
where \(M=L+E\) is the sum of the number \(L\) of loops and the number \(E\) of linearly independent external momenta. Note that \(G_{L+1} = \det (p_i \cdot p_j)_{1\le i,j\le E}\) is the Gram determinant of the external momenta and independent of the integration variables.
Proof of Theorem 29
Since every denominator \(\mathsf {D}_e\) is a linear combination of \(s_{i,j}\) (and some loop momentum independent constant \(\lambda _e\)) according to (2.26), an affine change of variables allows us to integrate over the values of \(\mathsf {D}_e\)’s instead of \(s_{i,j}\)’s. This transformation only introduces a constant Jacobian \(c' = \det (\mathcal {A}^a_{\left\{ i,j \right\} })\). We set \(f(s)=\prod _e \mathsf {D}_e^{-\nu _e}\) in formula (A.19) above. The (Euclidean) integration domain is determined, according to (A.17), by \(0 < \Vert {\text {pr}}_{{{\mathrm{lin}}}\left\{ q_{n+1},\ldots ,q_{M} \right\} ^{\perp }}^{\perp }(q_n)\Vert ^2 = G_n/G_{n+1}\) for \(1\le n \le L\). Therefore, a point on the boundary of the integration domain is determined by \(G_n=0\) for some \(1\le n \le L\), which is equivalent to a linear dependence \(q_n \in {{\mathrm{lin}}}\left\{ q_{n+1},\ldots ,q_{M} \right\} \) 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 \(\int \mathrm{d}^{d} \ell _k/(\text {{i}}\pi ^{d/2})\) exactly into the measure \(\int \mathrm{d}^{d} \ell _k/\pi ^{d/2}\) on the left-hand side of (A.19), we only have to remember that, due to our mostly-minus signature \((1,-\,1,\ldots ,-\,1)\) of the Minkowski metric, the Euclidean scalar products on the right-hand side of (A.19) receive a factor \((-1)\). For example, the \(M\times M\) determinant \(G_1\) turns into \((-1)^{M} \mathsf {Gr}_1\); similarly, \(G_{L+1}\) becomes \((-1)^{E}\mathsf {Gr}\). 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). \(\square \)
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 \(\mathscr {M}\) over the algebra \(\mathcal {D}^{N}_k={A}^{N}_k[x^{-1}]\) 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 \(\theta _i :=x_i \partial _i\) and set
for the algebraic Mellin transform [60, Section 1.2]. We begin with the finite-dimensionality, which was proven in [60, Lemma 1.2.2]:
Lemma 65
Let \(\mathscr {M}\) denote a holonomic \(\mathcal {D}^{N}_k\)-module. Then, for any \(1\le i \le N\), its algebraic Mellin transform \(\mathscr {M}(\theta _i,\ldots ,\theta _N)\) is a holonomic \(\mathcal {D}^{i-1}_{k(\theta _i,\ldots ,\theta _N)}\)-module.
Corollary 66
The full Mellin transform \(\mathscr {M}(\theta _1,\ldots ,\theta _N)\) is a finite-dimensional vector space over the field \(k(\theta _1,\ldots ,\theta _N)\).
Proof of Lemma 65
Since \(\mathscr {M}(\theta _i,\ldots ,\theta _N) = [\mathscr {M}(\theta _{i+1},\ldots ,\theta _N)](\theta _i)\), it suffices (by induction over i) to consider the case \(i=N\). Introducing a new indeterminate \(\nu \), we extend the scalars from k to \(k[\nu ]\) to obtain a \(\mathcal {D}^{N}_{k[\nu ]}\)-module \(\mathscr {M}[\nu ] :=\mathscr {M}\otimes _k k[\nu ]\). It sits in an exact sequence
of \(\mathcal {D}^{N-1}_{k[\nu ]}\)-modules, where \(\mathfrak {M} = \mathscr {M}\) denotes the initial module \(\mathscr {M}\) with the action of \(\nu \) defined as \(\nu m :=-x_N \partial _N m\). Since \(k(\nu )\) is flat, this sequence remains exact after tensoring with \(k(\nu )\) over \(k[\nu ]\). Through identification of \(\nu \) with \(-\theta _N-1\), we conclude that
are isomorphic as \(\mathcal {D}^{N-1}_{k(\nu )}\)-modules. The left-hand side is the quotient \(\mathscr {M}x_N^{\nu }/\partial _N \mathscr {M}x_N^{\nu }\) of the \(\mathcal {D}^{N}_{k(\nu )}\)-module \(\mathscr {M}x_N^{\nu } :=\mathscr {M}\otimes _k k(\nu )\) defined by the original action of \(\mathcal {D}^{N-1}_{k}\) and \(x_N^{\pm 1}\) on \(\mathscr {M}\), but twisting the operator \(\partial _N\) to act like \(\partial _N+ \nu /x_N\).Footnote 42 The holonomicity of \(\mathscr {M}\) implies that \(\mathscr {M}x_N^{\nu }\) is also holonomic,Footnote 43 and hence its push-forward \(\pi _{*} (\mathscr {M}x_N^{\nu }) = \mathscr {M}x_N^{\nu } / \partial _N (\mathscr {M}x_N^{\nu }) \cong \mathscr {M}(\theta _N)\), with respect the projection that forgets the last coordinate, is also holonomic. \(\square \)
Now we want to relate this dimension to the de Rham complex \( {{\mathrm{DR}}}(\mathscr {M}) = K^{}(\mathscr {M};\partial ) \), which is a special case of the Koszul complex:
Definition 67
For commuting k-linear endomorphisms \(s=(s_1,\ldots ,s_N)\) of a k-vector space \(\mathscr {M}\), let
denote the Koszul complex with r-forms sitting in degree \(r-N\). The \(r-N\) cochains
have natural coordinates with respect to the basis \(\mathsf {e}_{I} :=\mathsf {e}_{i_1} \wedge \cdots \wedge \mathsf {e}_{i_r} \in \Lambda ^{r} k\) indexed by r-sets \(I=\left\{ i_1<\cdots <i_r \right\} \). The cochain map reads \(\mathrm{d}^{}(\mathsf {e}_{I} \otimes m) = \sum _{i \notin I} (\mathsf {e}_{i} \wedge \mathsf {e}_{I}) \otimes s_i m\).
Remark 68
Since \(\theta _j \left( \prod _{i\in I} x_i \right) m = \left( \prod _{i \in I\cup \left\{ j \right\} } x_i\right) \partial _j m\) (for \(j \notin I\)), the rule
defines a cochain map. It has an inverse, defined by \(\mathsf {e}_{I} \otimes m \mapsto \mathsf {e}_{I} \otimes x^{-I}m\). We thus conclude that \({{\mathrm{DR}}}(\mathscr {M})=K^{}(\mathscr {M};\partial )\) and \(K^{}(\mathscr {M};\theta )\) are quasi-isomorphic and therefore share the same Euler characteristic.
We prove Theorem 35 by an induction over the number of variables. The base case is
Theorem 69
([59, Théorème 1]) If \(\mathscr {M}\) denotes a holonomic \(\mathcal {D}^{1}_k\)-module, then
Proof
We can pick a generator of \(\mathscr {M}\) (by holonomicity, \(\mathscr {M}\) is cyclic as a \(\mathcal {D}^{1}_k\)-module) and extend it to a (finite) basis of \(\mathscr {M}(\theta _1)\) as a vector space over \(k(\theta _1)\), due to Corollary 66. Let \(\mathscr {N}\subset \mathscr {M}\) denote the \(k[\theta _1]\)-module generated by such a basis, hence \(\mathscr {N}(\theta _1)=\mathscr {M}(\theta _1)\). By construction, \(\mathscr {M}= {A}^{1}_k \mathscr {N}= \sum _{j\in \mathbb {Z}} x_1^j \mathscr {N}\) is exhaustively filtered by the finitely generated \(k[\theta _1]\)-modules \(\mathscr {N}_j :=\sum _{i=-j}^j x_1^i \mathscr {N}\).Footnote 44
Since \(\mathscr {M}(\theta _1)=\mathscr {N}(\theta _1) = \mathscr {N}_1(\theta _1)\) is finitely generated, there is a nonzero polynomial \(b(\theta _1)\in k[\theta _1]\) such that \(b(\theta _1) \mathscr {N}_1 \subseteq \mathscr {N}\). Therefore, using \((\theta _1 -1)x_1 = x_1 \theta _1\),
shows that the polynomials \(b_{j+1}(\theta _1) :=b(\theta _1+j)b(\theta _1-j) \in k[\theta _1]\) have the property \(b_{j+1}(\theta _1)\mathscr {N}_{j+1} \subseteq \mathscr {N}_j\). Let \(Z=b^{-1}(0)\) denote the zeroes of \(b_1=b^2\), then note that the zeroes of \(b_{j+1}\) are \((Z+j) \cup (Z-j)\) and get pushed away from zero for increasing j. In particular, there exists some \(j_0 \in \mathbb {N}\) such that \(b_{j}(0) \ne 0\) for all \(j > j_0\). For each such value of j, we can find \(u_j,v_j \in k[\theta _1]\) such that \(1 = u_j(\theta _1) b_j(\theta _1) + v_j(\theta _1) \theta _1\); then
holds for every \(m \in \mathscr {N}_j\). This proves \(\ker (\theta _1) \cap \mathscr {N}_j \subseteq \mathscr {N}_{j-1}\) for all \(j>j_0\), and therefore \(\ker (\theta _1) \subseteq \mathscr {N}_{j_0}\). Similarly, we conclude \(\mathscr {M}/ (\theta _1 \mathscr {M}) \cong \mathscr {N}_{j_0} / (\mathscr {N}_{j_0} \cap \theta _1 \mathscr {M}\)). But given some \(m =\theta _1 x \in \mathscr {N}_{j_0}\) with \(x \in \mathscr {N}_j\), \( x = u_j(\theta _1) b_j(\theta _1) x + v_j(\theta _1) m \in \mathscr {N}_{j-1} + \mathscr {N}_{j_0} \) proves that \(\mathscr {N}_{j_0} \cap \theta _1(\mathscr {N}_j) = \mathscr {N}_{j_0} \cap \theta _1(\mathscr {N}_{j-1})\) for all \(j>j_0\). In consequence, we have proven that
in other words, the Koszul complexes \({{\mathrm{DR}}}(\mathscr {M})=K^{}(\mathscr {M};\partial _1)\) and \(K^{}(\mathscr {N}_{j_0};\theta _1)\) are quasi-isomorphic (see Remark 68). The statement of the theorem thus reduces to the identity
for a finitely generated \(k[\theta _1]\)-module \(\mathscr {N}_{j_0}\). Since both sides are additive under short exact sequences, this claim reduces (via a finite free resolution) to the case of a free rank one \(k[\theta _1]\)-module, i.e. \(k[\theta _1]\), which is clear: \(k[\theta _1](\theta _1) = k(\theta _1)\) is of dimension one over \(k(\theta _1)\), while \(\ker (\theta _1) = \left\{ 0 \right\} \) is trivial and \(k[\theta _1]/(\theta _1 k[\theta _1]) = k\) is one-dimensional. \(\square \)
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 higher-dimensional lattices.
Proof of Theorem 35
Let \(\mathscr {M}\) denote a holonomic \(\mathcal {D}^{N}_k\)-module, and suppose we have proven Theorem 35 for all holonomic modules in less than N variables. In particular, we may invoke the claim for the \(\mathcal {D}^{N-1}_k\)-modules \(\ker \partial _N\) and \(\mathscr {M}/ \partial _N \mathscr {M}\), as these are holonomic because they are the cohomologies of the complex
which computes the push-forward of \(\mathscr {M}\) along the projection \(\pi :\mathbb {G}_{m}^N \longrightarrow \mathbb {G}_{m}^{N-1}\) that forgets the last coordinate. So we already know that
where \(k':=k(\theta ')\) and \(\mathscr {M}' :=\mathscr {M}(\theta ')\) with \(\theta ':=(\theta _1,\ldots ,\theta _{N-1})\). Analogously, \(\chi (\ker \partial _N) = \dim _{k'} \ker (\partial _N')\), where \(\partial _N'\) denotes the action of \(\partial _N\) on \(\mathscr {M}'\). In conclusion, we know that
where we recognized the first line as the Euler characteristic of the de Rham complex of the \(\mathcal {D}^{1}_{k'}\)-module \(\mathscr {M}'\) and applied Theorem 69 to get to the last line (\(\mathscr {M}'=\mathscr {M}(\theta ')\) is holonomic by Lemma 65). So we only need to show that the left-hand side is equal to \(\chi (\mathscr {M})\).
This is well known and follows from the Grothendieck spectral sequence.Footnote 45 Alternatively, an elementary way to obtain the identity \(\chi (\mathscr {M}/(\partial _N\mathscr {M}))-\chi (\ker \partial _N)=\chi (\mathscr {M})\) is given by the long exact sequence
in de Rham cohomology [12, Ch. 2, Proposition 4.13]. \(\square \)
For completeness, let us demonstrate (B.5), by following the standard construction in the proof of Kashiwara’s theorem: First note that
defines a \(\mathcal {D}^{N}_k\)-submodule of \(\mathscr {M}\).Footnote 46 The exact sequence \( 0 \rightarrow \mathscr {N}\rightarrow \mathscr {M}\rightarrow \mathscr {M}/\mathscr {N}\rightarrow 0 \) of holonomic \(\mathcal {D}^{N}_{k}\)-modules induces a sequence of de Rham complexes, \( 0 \rightarrow {{\mathrm{DR}}}(\mathscr {N}) \rightarrow {{\mathrm{DR}}}(\mathscr {M}) \rightarrow {{\mathrm{DR}}}(\mathscr {M}/\mathscr {N}) \rightarrow 0 \), which is also exact (\(\mathscr {M}\mapsto {{\mathrm{DR}}}(\mathscr {M})=\Lambda ^{\bullet } k \otimes _k \mathscr {M}\) is exact by flatness of \(\Lambda ^{\bullet }k\)). Hence, we get a long exact sequence in cohomology:
The key observation now is that \(\partial _N\) is injective on \(\mathscr {M}/\mathscr {N}\) and surjective on \(\mathscr {N}\).Footnote 47 We thus get short exact sequences
of \(\mathcal {D}^{N-1}_k\)-modules. The induced short exact sequences of de Rham complexes provide quasi-isomorphisms \({{\mathrm{DR}}}(\ker \partial _N) \simeq {{\mathrm{DR}}}(\mathscr {N})[1]\) and \({{\mathrm{DR}}}(\mathscr {M}/\partial _N\mathscr {M}) \simeq {{\mathrm{DR}}}(\mathscr {M}/\mathscr {N})\), because the de Rham complex of a \(\mathcal {D}^{N}_k\)-module \(\mathscr {N}\) is the mapping cone of the map \(\partial _N:K^{}(\mathscr {N};\partial ')\longrightarrow K^{}(\mathscr {N};\partial ')\). Hence we obtain (B.5) from (B.6) due to
To clarify this final step, first note that separating \(\partial _N\) from \(\partial ':=(\partial _1,\ldots ,\partial _{N-1})\) yields an isomorphism of k-vector spaces
In this representation, the differential is given by
which is known as the mapping cone of \(\partial _N:K^{}(\mathscr {N};\partial ') \longrightarrow K^{}(\mathscr {N};\partial ')\). Here, we denote by \({\mathrm{d}^{}}'\) the differential of \(K^{}(\mathscr {N};\partial ')\).
If \(\partial _N\) is surjective, we can find x with \(\partial _N x = y\) and hence \(\mathrm{d}^{}\Phi (x\oplus 0)=\Phi ({\mathrm{d}^{}}'x \oplus y)\) for every y. Therefore, every element of \(K^{}(\mathscr {N};\partial )\) has a representative of the form \(\Phi (x \oplus 0)\), modulo exact forms. But such a form is closed, \(\mathrm{d}^{}\Phi (x\oplus 0)=0\), if and only if \(x \in \ker \partial _N \cap \ker {\mathrm{d}^{}}'\).
Corollary 70
If \(\partial _N\) is surjective, then \(K^{}(\mathscr {N};\partial )\) and \(K^{}(\ker \partial _N;\partial ')\) [1] are quasi-isomorphic. If \(\partial _N\) is injective, then \(K^{}(\mathscr {N};\partial )\) and \(K^{}(\mathscr {N}/\partial _N \mathscr {N};\partial ')\) are quasi-isomorphic.
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 \({\mathrm{WS}}_{3}'\) 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 \(\mathcal {G}=\mathcal {U}+\mathcal {F}\) is given by the Symanzik polynomials
By Definition 6 the modified Feynman integral
is related to the Feynman integral by
with \(\omega = \nu _1+\nu _2+\nu _3+\nu _4+\nu _5+2s\) and \(s=-d/2\).
1.1 C.1. From annihilators to integral relations
A set of generators of the annihilator ideal \({{\mathrm{Ann}}}_{A_5[s]}\left( \mathcal {G}^s\right) \) 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 \({{{\mathrm{Ann}}}_{A_5[s]}\left( \mathcal {G}^s\right) }\) gives rise to an integral relation. According to Lemma 7, we just need to replace each \(x_i\) by \(\hat{\mathbf{i }}^{+}\) and each \(\partial _i\) by \(-\mathbf{i }^{-}\) to obtain a shift relation between modified Feynman integrals. For example, for the generator \(P_1\), we obtain the shift operator
satisfying
For the (unmodified) Feynman integral \(\mathcal {I}(\nu _1,\ldots ,\nu _5)\) we obtain the corresponding shift relation via (2.22). From \(P_1\) we obtain the operator
which satisfies
1.2 C.2. Linear annihilators
It is useful to consider the linear annihilators \({{\mathrm{Ann}}}^1_{{A}^{N}[s]}(\mathcal {G}^s) \subseteq {{\mathrm{Ann}}}_{{A}^{N}[s]}(\mathcal {G}^s).\) Recall that they are the annihilators which are linear differential operators, being of the form \(P=q + \sum _{i=1}^N p_i \partial _i\) with \(p,q_1,\ldots ,q_N \in \mathbb {Q}[s,x_1,\ldots ,x_N]\). We compute the generators of \({{\mathrm{Ann}}}^1(\mathcal {G}^s)\) as generators of the Syzygy-module of \((\mathcal {G},\partial _1 \mathcal {G},\ldots ,\partial _N \mathcal {G})\), using the Singular command syz. For our example we obtain the following 8 generators:
Using Singular we find that for our two-loop example every generator \({P_i}\) can be expressed as a linear combination of the linear generators \({L_i}\) over \({{\mathrm{Ann}}}_{{A}^{5}[s]}(\mathcal {G}^s)\). For instance, the first five annihilators satisfy
We emphasize that such a relation exists for all the \(P_i\). In particular such a relation also exists for the quadratic \(P_{13}\), which however is too long to be shown here. As a consequence we can view \({{\mathrm{Ann}}}_{{A}^{N}[s]}(\mathcal {G}^s)\) as generated by the linear \(L_i\), which will simplify the discussion in Sect. C.4.
1.3 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
we obtain six IBP relations \(\mathbf O _{j}^{i} \mathcal {I}=0\) with the shift operators
Following the steps in the proof of Corollary 21, we derive for each shift operator \(\mathbf O _{j}^{i}\) a parametric annihilator \(\widetilde{\mathbf{O }}_{j}^{i}\). We obtain
These operators are useful to compare both approaches as discussed next.
1.4 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 \(\mathcal {G}\) 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 \({P_1,\ldots ,P_{13}}\) 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 \(\nu _1,\ldots ,\nu _5\) and do not answer the question for arbitrary values of the \(\nu _i\). We therefore approach the problem on the level of parametric differential operators instead.
We find that actually not all parametric annihilators are contained in \({{\mathrm{Mom}}}\); however, they turn out to still be consequences of the momentum space IBP relations in the following sense: While we checked that \(P_1 \notin {{\mathrm{Mom}}}\), we can find a polynomial \(q_1\in \mathbb {Q}[s,x_1\partial _1,\ldots ,x_5\partial _5]\) such that \(q_1 P_1\in {{\mathrm{Mom}}}\). Recall that, under the Mellin transform, such a \(q_1\) corresponds to a polynomial in the dimension and in the \(\nu _e\). The interesting question then is if we can find a polynomial \(q\in \mathbb {Q}[s,x_1\partial _1,\ldots ,x_N\partial _N]\) for every \(P\in {{\mathrm{Ann}}}_{{A}^{N}[s]}(\mathcal {G}^s)\) such that \(q P \in {{\mathrm{Mom}}}\). If we can find such a \(q_i\) for every generator \(P_i\), we can express every annihilator in terms of the \(\widetilde{\mathbf{O }}_{j}^{i}\). The \(q_i\) are the denominators of the coefficients in such a linear combination.
In Sect. 1 we have seen for our example that \({{\mathrm{Ann}}}_{{A}^{N}[s]}(\mathcal {G}^s)\) is generated by the linear annihilators \(L_i\). As a consequence, it is sufficient to show that for each \(L_i\) there is a \({\tilde{q}}_i\) such that \({\tilde{q}}_i L_i \in {{\mathrm{Mom}}}\) for \(i=1,\ldots ,8\). Indeed we can construct such \({q_1,\ldots ,q_8}\) by an explicit Ansatz. For example we obtain the identity
with
Using these results and the expressions for the generators of \({{\mathrm{Ann}}}_{{A}^{5}[s]}(\mathcal {G}^s)\) in terms of the \(L_i\), we can derive every annihilator from the IBP operators. We have done this computation with the same conclusion for several further graphs of low loop-order. These computations support our conjectures phrased in Questions 23 and 24.
1.5 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
and obtain the same number of master integrals with Azurite. The three master integrals suggested by Azurite are
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
and compute with one of the sets \(\left\{ I_1,I_2 \right\} \), \(\left\{ I_1,I_3 \right\} \).
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Bitoun, T., Bogner, C., Klausen, R. et al. Feynman integral relations from parametric annihilators. Lett Math Phys 109, 497–564 (2019). https://doi.org/10.1007/s11005-018-1114-8
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DOI: https://doi.org/10.1007/s11005-018-1114-8