# Feynman integral relations from parametric annihilators

Thomas Bitoun University of Oxford Christian Bogner Humboldt-Universität zu
Berlin René Pascal Klausen††footnotemark:  Erik Panzer All Souls College,
University of Oxford

###### 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.

##  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

$\sum_{i}c_{i}\mathcal{I}_{i}=0$ ($\ast$)

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, 90]. In this approach, the relations
$(\ast)$ 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_.111For different applications, various criteria for
choosing the master integrals have been suggested. These include uniform
transcendentality [38], finiteness [94] or finiteness of coefficients [21].
Laporta’s algorithm [49] provides a popular approach to obtain such
reductions, and various implementations of it are available [2, 61, 96, 74,
75, 79, 85]. 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
[34, 51, 54, 56, 95, 69, 77, 97, 39]. This includes a method by Baikov [5, 7,
81, 8], 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 [57] and
Azurite [31].

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 [57] 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 [55] and Baikov [7] pointed out, such
shift relations correspond to annihilation operators of the integrands of
parametric representations.222Tkachov’s idea [91] to insert Bernstein-Sato
operators in the integrand with two Symanzik polynomials was used for
numerical computations of one- and two-loop integrals [26, 27, 67, 68, 9].  In
our set-up, these are differential operators $P$ satisfying

$P\mathcal{G}^{-d/2}=0.$

We recall that such parametric annihilators provide all shift relations
between Feynman integrals, in particular the ones known from the classical IBP
method in momentum space. The obvious question, whether the latter suffice to
obtain all shift relations, seems to remain open. As a positive indication in
this direction, we show that the momentum space relations contain the inverse
dimension shift.

Ideals of parametric annihilators are examples of $D$-modules. Loeser and
Sabbah studied the _algebraic Mellin transform_ [59] of holonomic $D$-modules
and proved a dimension formula in [58, 60], 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 [45] and of course is crucial in the proof [76] that
there are only finitely many master integrals.

It is furthermore worthwhile to notice that algorithms [64, 66] have been
developed to compute generators for the ideal of all annihilators of
$\mathcal{G}^{-d/2}$, see also [71, section 5.3]. Today, efficient
implementations of these algorithms via Gröbner bases are available in
specialized computer algebra systems such as Singular [24, 4]. We hope that
these improvements may stimulate further progress in the application of
$D$-module theory to Feynman integrals [88, 78, 77, 29].

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 section 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 section
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
section 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 the following section 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 section 3. Finally,
appendix C discusses the parametric annihilators of a two-loop example in
detail.

#### Acknowledgments

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 section 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 [58, 60,
59] 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.

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 [93])
and FeynArts [36].

##  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_ [96]) 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_
$\mathsf{D}=(\mathsf{D}_{1},\ldots,\mathsf{D}_{N})$:

$\mathcal{I}(\nu_{1},\ldots,\nu_{N})=\left(\prod_{j=1}^{L}\int\frac{\mathrm{d}^{d}\ell_{j}}{\mathrm{{i}}\pi^{d/2}}\right)\prod_{a=1}^{N}\mathsf{D}_{a}^{-\nu_{a}}.$
(2.1)

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 section 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 $(d,\nu)\in\mathbbm{C}^{1+N}$ in the
sense of Speer [82].

######  Definition 1.

To each denominator $\mathsf{D}_{a}$ of a list
$\mathsf{D}=(\mathsf{D}_{1},\ldots,\mathsf{D}_{N})$, we associate a variable
$x_{a}$ ($1\leq a\leq N$) called _Schwinger parameter_. The linear combination

$\sum_{a=1}^{N}x_{a}\mathsf{D}_{a}=-\sum_{i,j=1}^{L}\Lambda_{ij}(\ell_{i}\cdot\ell_{j})+\sum_{i=1}^{L}2(Q_{i}\cdot\ell_{i})+J$
(2.2)

decomposes into quadratic, linear and constant terms in the loop
momenta.333The 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.  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

$\mathcal{U}\mathrel{\mathop{:}}=\det\Lambda,\quad\mathcal{F}\mathrel{\mathop{:}}=\mathcal{U}\left(Q^{\intercal}\Lambda^{-1}Q+J\right)\quad\text{and}\quad\mathcal{G}\mathrel{\mathop{:}}=\mathcal{U}+\mathcal{F}.$
(2.3)

Schwinger parameters yield useful representations of the Feynman integrals
(2.1):

######  Proposition 2.

Let us denote the _superficial degree of convergence_ by

$\omega\mathrel{\mathop{:}}=\left\lvert\nu\right\rvert-L\frac{d}{2}\quad\text{where}\quad\left\lvert\nu\right\rvert\mathrel{\mathop{:}}=\sum_{i=1}^{N}\nu_{i}.$
(2.4)

Then the Feynman integral (2.1) can be written as

$\displaystyle\mathcal{I}(\nu_{1},\ldots,\nu_{N})$
$\displaystyle=\left(\prod_{i=1}^{N}\int_{0}^{\infty}\frac{x_{i}^{\nu_{i}-1}\mathrm{d}x_{i}}{\Gamma\left(\nu_{i}\right)}\right)\frac{e^{-\mathcal{F}/\mathcal{U}}}{\mathcal{U}^{d/2}},$
(2.5) $\displaystyle\mathcal{I}(\nu_{1},\ldots,\nu_{N})$
$\displaystyle=\Gamma(\omega)\left(\prod_{i=1}^{N}\int_{0}^{\infty}\frac{x_{i}^{\nu_{i}-1}\mathrm{d}x_{i}}{\Gamma\left(\nu_{i}\right)}\right)\frac{\delta\left(1-\sum_{j=1}^{N}x_{j}\right)}{\mathcal{U}^{d/2-\omega}\mathcal{F}^{\omega}}\quad\text{and}$
(2.6) $\displaystyle\mathcal{I}(\nu_{1},\ldots,\nu_{N})$
$\displaystyle=\frac{\Gamma\left(\frac{d}{2}\right)}{\Gamma\left(\frac{d}{2}-\omega\right)}\left(\prod_{i=1}^{N}\int_{0}^{\infty}\frac{x_{i}^{\nu_{i}-1}\mathrm{d}x_{i}}{\Gamma(\nu_{i})}\right)\mathcal{G}^{-d/2}.$
(2.7)

The formulas (2.5) and (2.6) are known since the sixties and we refer to [63,
80] for detailed discussions and for the original references. The trivial
consequence (2.7) was popularized only much more recently by Lee and
Pomeransky [57] 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.

Figure 2.1: The one-loop bubble graph with momentum flow.

######  Example 3.

Consider the graph in figure 2.1 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

$\mathcal{I}(\nu_{1},\nu_{2})=\frac{\Gamma(\frac{d}{2})}{\Gamma(d-\nu_{1}-\nu_{2})}\int_{0}^{\infty}\frac{x_{1}^{\nu_{1}-1}\mathrm{d}x_{1}}{\Gamma(\nu_{1})}\int_{0}^{\infty}\frac{x_{2}^{\nu_{2}-1}\mathrm{d}x_{2}}{\Gamma(\nu_{2})}\left(x_{1}+x_{2}-p^{2}x_{1}x_{2}\right)^{-d/2}.$
(2.8)

######  Remark 4 (meromorphicity).

Depending on the values of $d$ and $\nu$, the integrals (2.1) and (2.5)–(2.7)
can be divergent. However, there exists a non-empty, open domain in
$\mathbbm{C}^{1+N}\ni(d,\nu)$ where all of them are convergent and agree with
each other.444The only exception are cases of zero-scale subintegrals, like
massless tadpoles. Such integrals are zero in dimensional regularization and
therefore irrelevant for our considerations.  From there, analytic
continuation defines a unique, meromorphic extension of every Feynman integral
to the whole parameter space $\mathbbm{C}^{1+N}$. The poles are simple and
located on affine hyperplanes defined by linear equations with integer
coefficients. For these foundations of analytic regularization555The widely
used _dimensional regularization_ is the special case when
$\nu\in\mathbbm{Z}^{N}$ are all integers and only the dimension $d$ remains as
a regulator. In this case, the poles are not necessarily simple anymore.  we
refer to [83, 82].

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:

$\mathcal{I}(\nu_{1},\nu_{2})=(-p^{2})^{d/2-\nu_{1}-\nu_{2}}\frac{\Gamma(d/2-\nu_{1})\Gamma(d/2-\nu_{2})\Gamma(\nu_{1}+\nu_{2}-d/2)}{\Gamma(\nu_{1})\Gamma(\nu_{2})\Gamma(d-\nu_{1}-\nu_{2})}.$
(2.9)

The 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 $k\in\mathbbm{Z}_{\leq
0}$.

###  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 [55] (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 $\nu=(\nu_{1},\ldots,\nu_{N})\in\mathbbm{C}^{N}$. The twisted (multi-
dimensional) Mellin transform of a function
$f\colon\mathbbm{R}_{+}^{N}\longrightarrow\mathbbm{C}$ is defined as

$\mathcal{M}\\!\left\\{f\right\\}(\nu)\mathrel{\mathop{:}}=\left(\prod_{i=1}^{N}\int_{0}^{\infty}\frac{x_{i}^{\nu_{i}-1}\mathrm{d}x_{i}}{\Gamma(\nu_{i})}\right)f(x_{1},\ldots,x_{N}),$
(2.10)

whenever this integral exists. As a special case we define

$\widetilde{\mathcal{I}}(\nu)\mathrel{\mathop{:}}=\mathcal{M}\\!\left\\{\mathcal{G}^{-d/2}\right\\}(\nu)\quad\text{such
that}\quad\mathcal{I}(\nu)=\frac{\Gamma(d/2)}{\Gamma(d/2-\omega)}\widetilde{\mathcal{I}}(\nu).$
(2.11)

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 $\alpha,\beta\in\mathbbm{C}$, $\nu\in\mathbbm{C}^{N}$, $1\leq i\leq N$ and
$f,g\colon\mathbbm{R}_{+}^{N}\longrightarrow\mathbbm{C}$. Writing
$\mathsf{e}_{i}$ for the $i$-th unit vector, the (twisted) Mellin transform
has the following properties:

  1. 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. 2.

Multiplication:
$\mathcal{M}\\!\left\\{x_{i}f\right\\}(\nu)=\nu_{i}\mathcal{M}\\!\left\\{f\right\\}(\nu+\mathsf{e}_{i})$
and

  3. 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,

$\int_{0}^{\infty}\frac{x_{i}^{\nu_{i}-1}\mathrm{d}x_{i}}{\Gamma(\nu_{i})}\partial_{i}f=\left[\frac{x_{i}^{\nu_{i}-1}}{\Gamma(\nu_{i})}f\right]_{x_{i}=0}^{\infty}-\int_{0}^{\infty}\frac{x_{i}^{\nu_{i}-2}\mathrm{d}x_{i}}{\Gamma(\nu_{i}-1)}f,$

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 $x_{i}\rightarrow\infty$. ∎

###  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 [23] 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

${A}^{N}\mathrel{\mathop{:}}=\mathbbm{C}\left\langle
x_{1},\ldots,x_{N},\partial_{1},\ldots,\partial_{N}\ \ |\ \
\partial_{i}x_{j}=x_{j}\partial_{i}+\delta_{ij}\ \text{for all}\ 1\leq i,j\leq
N\right\rangle,$ (2.12)

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

$P=\sum_{\alpha,\beta}c_{\alpha\beta}x^{\alpha}\partial^{\beta}\quad\text{with}\quad\alpha,\beta\in\mathbbm{N}_{0}^{N},$

by commuting all derivatives to the right (only finitely many of the
coefficients $c_{\alpha\beta}\in\mathbbm{C}$ are non-zero). Extending the
coefficients $c_{\alpha\beta}$ from $\mathbbm{C}$ to polynomials
$\mathbbm{C}[s]$ in a further, commuting variable $s$, we obtain the algebra
${A}^{N}[s]\mathrel{\mathop{:}}={A}^{N}\otimes_{\mathbbm{C}}\mathbbm{C}[s]$.
Later, we will also consider the case
${A}^{N}_{k}\mathrel{\mathop{:}}={A}^{N}\otimes_{\mathbbm{C}}k$ of
coefficients that are rational functions
$k\mathrel{\mathop{:}}=\mathbbm{C}(s)$.

######  Definition 9.

Given a polynomial $f\in\mathbbm{C}[x]$, the ${A}^{N}[s]$-module
$\mathbbm{C}[s,x,1/f]\cdot f^{s}$ consists of elements of the form
$(p/f^{k})\cdot f^{s}$ (where $p\in\mathbbm{C}[s,x]$, $k\in\mathbbm{N}_{0}$)
with the ${A}^{N}[s]$-action

$q\left(\frac{p}{f^{k}}\cdot
f^{s}\right)\mathrel{\mathop{:}}=\frac{qp}{f^{k}}\cdot
f^{s},\quad\partial_{i}\left(\frac{p}{f^{k}}\cdot
f^{s}\right)\mathrel{\mathop{:}}=\frac{f(\partial_{i}p)+(s-k)p(\partial_{i}f)}{f^{k+1}}\cdot
f^{s}$ (2.13)

for any polynomial $q\in\mathbbm{C}[s,x]$. This is just the natural action by
multiplication and differentiation, $\partial_{i}\mapsto\partial/(\partial
x_{i})$. We abbreviate $f^{s+k}\mathrel{\mathop{:}}=f^{k}\cdot f^{s}$ for
$k\in\mathbbm{Z}$. The cyclic submodule generated by $f^{s}$ is denoted as
${A}^{N}[s]f^{s}$.

######  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:

$\operatorname{Ann}_{{A}^{N}[s]}(f^{s})\mathrel{\mathop{:}}=\left\\{P\in{A}^{N}[s]\colon
Pf^{s}=0\right\\}.$

######  Example 11.

Given $f\in\mathbbm{C}[x_{1},\ldots,x_{N}]$, we always have the trivial
annihilators

$f\partial_{i}-s(\partial_{i}f)\in\operatorname{Ann}\left(f^{s}\right)\quad\text{for}\quad
1\leq i\leq N.$ (2.14)

Note that an annihilator ideal is a module over ${A}^{N}[s]$, that is,
whenever $Pf^{s}=0$, also $(QP)f^{s}=Q(Pf^{s})=0$ for any operator
$Q\in{A}^{N}[s]$. These ideals are studied in $D$-module theory [71, 23] and
in principle annihilators can be computed algorithmically with computer
algebra systems such as Singular [24, 4, 3]. 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
$P\in\operatorname{Ann}_{{A}^{N}[s]}\left(\mathcal{G}^{s}\right)$ 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

${S}^{N}\mathrel{\mathop{:}}=\mathbbm{C}\left\langle\hat{\mathbf{1}}^{+},\ldots,\hat{\mathbf{N}}^{+},\mathbf{1}^{-},\ldots,\mathbf{N}^{-}\
\ |\ \
[\hat{\mathbf{i}}^{+},\hat{\mathbf{j}}^{+}]=[\mathbf{i}^{-},\mathbf{j}^{-}]=0\
\text{and}\ [\hat{\mathbf{i}}^{+},\mathbf{j}^{-}]=\delta_{i,j}\right\rangle.$
(2.15)

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
by666 The use of the symbol $\mathcal{M}\\!\left\\{\cdot\right\\}$ for both
the analytic Mellin transform (2.10) and the isomorphism (2.16) should not
lead to any confusion. It is suggestive of, and justified by, corollary 13.

$\mathcal{M}\\!\left\\{\cdot\right\\}\colon{A}^{N}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{S}^{N},\quad
P\mapsto\mathcal{M}\\!\left\\{P\right\\}\mathrel{\mathop{:}}=\left.P\right|_{x_{i}\mapsto\hat{\mathbf{i}}^{+},\partial_{i}\mapsto-\mathbf{i}^{-}\
\text{for all $1\leq i\leq N$}}.$ (2.16)

The conceptual difference between ${S}^{N}$ and ${A}^{N}$ is that we think of
${A}^{N}$ as acting on functions $f(x)$ by differentiation, whereas ${S}^{N}$
acts on functions $F(\nu)$ 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 $\nu_{i}$:

$(\mathbf{i}^{-}F)(\nu)\mathrel{\mathop{:}}=F(\nu-\mathsf{e}_{i})\quad\text{and}\quad(\hat{\mathbf{i}}^{+}F)(\nu)\mathrel{\mathop{:}}=\nu_{i}F(\nu+\mathsf{e}_{i}).$
(2.17)

These operators are used very frequently in the literature on IBP relations,
as for example in [77, 80, 51, 34]. An important role is played by the
operators

$\mathbf{n}_{i}\mathrel{\mathop{:}}=\hat{\mathbf{i}}^{+}\mathbf{i}^{-},\quad\text{which
act by simple multiplication:}\quad(\mathbf{n}_{i}F)(\nu)=\nu_{i}F(\nu).$
(2.18)

Their commutation relations are

$[\hat{\mathbf{i}}^{+},\mathbf{n}_{j}]=\hat{\mathbf{i}}^{+}\delta_{i,j}\quad\text{and}\quad[\mathbf{i}^{-},\mathbf{n}_{j}]=-\mathbf{i}^{-}\delta_{i,j}.$
(2.19)

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,

$\mathcal{M}\\!\left\\{Pf^{s}\right\\}=\mathcal{M}\\!\left\\{P\right\\}\left[\mathcal{M}\\!\left\\{f^{s}\right\\}\right]\quad\text{for
all operators}\quad P\in{A}^{N}[s].$ (2.20)

######  Corollary 14.

Every annihilator
$P\in\operatorname{Ann}_{{A}^{N}[d]}\left(\mathcal{G}^{s}\right)$ of
$\mathcal{G}^{s}=\mathcal{G}^{-d/2}$ yields a shift relation
$\mathcal{M}\\!\left\\{P\right\\}\in{S}^{N}[s]\mathrel{\mathop{:}}={S}^{N}\otimes\mathbbm{C}[s]$
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 figure 2.1 with
$\mathcal{G}=x_{1}+x_{2}-p^{2}x_{1}x_{2}$ from example 3,

$(-p^{2})x_{1}(s-x_{1}\partial_{1})+(s-x_{1}\partial_{1}-x_{2}\partial_{2})\in\operatorname{Ann}(\mathcal{G}^{s})$

is easily checked to annihilate $\mathcal{G}^{s}$. We therefore get the shift
relation

$(s+\mathbf{n}_{1}+\mathbf{n}_{2})\widetilde{\mathcal{I}}=p^{2}\hat{\mathbf{1}}^{+}(s+\mathbf{n}_{1})\widetilde{\mathcal{I}}=p^{2}(s+\mathbf{n}_{1}+1)\hat{\mathbf{1}}^{+}\widetilde{\mathcal{I}}.$

According to (2.18), this relation can also be written as

$(-p^{2})\nu_{1}\widetilde{\mathcal{I}}(\nu_{1}+1,\nu_{2})=-\frac{s+\nu_{1}+\nu_{2}}{s+\nu_{1}+1}\widetilde{\mathcal{I}}(\nu_{1},\nu_{2}).$
(2.21)

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
$P=\sum_{\alpha,\beta}c_{\alpha,\beta}x^{\alpha}\partial^{\beta}\in\operatorname{Ann}_{{A}^{N}[s]}\left(\mathcal{G}^{s}\right)$,
we substitute (2.11) to see

$0=\mathcal{M}\\!\left\\{P\right\\}\widetilde{\mathcal{I}}=\sum_{\alpha,\beta}c_{\alpha,\beta}\left(\prod_{i,j=1}^{N}(\hat{\mathbf{i}}^{+})^{\alpha_{i}}(-\mathbf{j}^{-})^{\beta_{j}}\right)\frac{\Gamma(-s-\omega)}{\Gamma(-s)}\mathcal{I}$

and then recall from (2.4) that $\omega=\sum_{i}\nu_{i}+Ls$ to conclude

$0=\frac{\Gamma(-s)\mathcal{M}\\!\left\\{P\right\\}\widetilde{\mathcal{I}}}{\Gamma(-s-\omega)}=\sum_{\alpha,\beta}c_{\alpha,\beta}\frac{\Gamma(-s-\omega-\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert)}{\Gamma(-s-\omega)}\left(\prod_{i,j=1}^{N}(\hat{\mathbf{i}}^{+})^{\alpha_{i}}(-\mathbf{j}^{-})^{\beta_{j}}\right)\mathcal{I}.$
(2.22)

Notice that the fraction
$\Gamma(-s-\omega-\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert)/\Gamma(-s-\omega)$
is a rational function in $\omega$ (and thus in $\nu$ and $s$), due to the
functional equation for $\Gamma$, since $\left\lvert\alpha\right\rvert$ and
$\left\lvert\beta\right\rvert$ 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

$(-p^{2})\nu_{1}\mathcal{I}(\nu_{1}+1,\nu_{2})=\frac{(s+\nu_{1}+\nu_{2})(2s+\nu_{1}+\nu_{2}+1)}{s+\nu_{1}+1}\mathcal{I}(\nu_{1},\nu_{2}),$

which also follows from the 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 $P\in\operatorname{Ann}_{{A}^{N}[s]}(\mathcal{G}^{s})$ 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
$P=\mathcal{M}^{-1}\\!\left\\{R\right\\}\in{A}^{N}[s]$ 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,
$P\in\operatorname{Ann}_{{A}^{N}[s]}(\mathcal{G}^{s})$. This follows from

######  Theorem 17 (Inverse Mellin transform, e.g. [18, Theorem 3.5]).

Suppose that the (twisted) Mellin transform
$f^{\star}(\nu)\mathrel{\mathop{:}}=\mathcal{M}\\!\left\\{f\right\\}(\nu)$ of
$f(x)$ from (2.10) converges in a domain of the form
$a_{i}\leq\operatorname{Re}(\nu_{i})\leq b_{i}$ for all $1\leq i\leq N$, where
$a,b\in\mathbbm{R}^{N}$. Then its inverse is given by

$f(x)=\mathcal{M}^{-1}\\!\left\\{f^{\star}\right\\}(x)=\left(\prod_{i=1}^{N}\int_{\sigma_{i}+\mathrm{{i}}\mathbbm{R}}\mathrm{d}\nu_{i}\
\frac{\Gamma(\nu_{i})}{x_{i}^{\nu_{i}}\cdot
2\pi\mathrm{{i}}}\right)f^{\star}(\nu),\quad\text{where}\quad
x\in\mathbbm{R}_{+}^{N}.$ (2.23)

This multiple integral along lines parallel to the imaginary axis converges
for $a_{i}\leq\sigma_{i}\leq b_{i}$ ($1\leq i\leq N$) 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
$P\widetilde{\mathcal{I}}=0$ for a polynomial shift operator $P\in{S}^{N}[s]$
does arise in this way.

######  Corollary 18.

Let
$\operatorname{Ann}_{{S}^{N}[s]}(\widetilde{\mathcal{I}})\subseteq{S}^{N}[s]$
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:

$\mathcal{M}\\!\left\\{\cdot\right\\}\colon\operatorname{Ann}_{{A}^{N}[s]}\left(\mathcal{G}^{s}\right)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\operatorname{Ann}_{{S}^{N}[s]}\left(\widetilde{\mathcal{I}}\right).$
(2.24)

Instead of focussing on the annihilators themselves, we can also look at the
${A}^{N}[s]$-module
${A}^{N}[s]\cdot\mathcal{G}^{s}\cong{A}^{N}[s]/\operatorname{Ann}_{{A}^{N}[s]}(\mathcal{G}^{s})$
of the integrands and the ${S}^{N}[s]$-module
${S}^{N}[s]\cdot\widetilde{\mathcal{I}}\cong{S}^{N}[s]/\operatorname{Ann}_{{S}^{N}[s]}(\widetilde{\mathcal{I}})$
of all (shifted) Feynman integrals. The Mellin transform gives an isomorphism

$\mathcal{M}\\!\left\\{\cdot\right\\}\colon{A}^{N}[s]\cdot\mathcal{G}^{s}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{S}^{N}[s]\cdot\widetilde{\mathcal{I}}.$
(2.25)

###  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 [34]. 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
$M=L+E$ momenta
$q=(q_{1},\ldots,q_{M})\mathrel{\mathop{:}}=(\ell_{1},\ldots,\ell_{L},p_{1},\ldots,p_{E})$,
we can write them in the form

$\mathsf{D}_{a}=\sum_{\left\\{i,j\right\\}\in\Theta}\mathcal{A}_{a}^{\left\\{i,j\right\\}}s_{\left\\{i,j\right\\}}+\lambda_{a}$
(2.26)

such that the coefficients $\mathcal{A}_{a}^{\left\\{i,j\right\\}}$ and
$\lambda_{a}$ are independent of loop momenta and the pairs

$\Theta\mathrel{\mathop{:}}=\left\\{\left\\{i,j\right\\}\colon 1\leq i\leq L\
\text{and}\ 1\leq j\leq M\right\\}$ (2.27)

label the $\left\lvert\Theta\right\rvert=\frac{L(L+1)}{2}+LE$ loop-momentum
dependent scalar products

$s_{\left\\{i,j\right\\}}\mathrel{\mathop{:}}=q_{i}q_{j}=q_{j}q_{i}.$ (2.28)

In order to express the IBP relations coming from momentum space in terms of
the integrals (2.1), we need to assume, for this and the following section,
that we consider $N=\left\lvert\Theta\right\rvert$ denominators such that the
$N\times N$ square matrix $\mathcal{A}$ defined by (2.26) is invertible.777For
integrals associated to Feynman graphs, the number of edges is often less than
$\left\lvert\Theta\right\rvert$. In this case, one augments the list of
inverse propogators by an appropriate choice of additional quadratic forms in
the loop momenta, called _irreducible scalar products_ (ISPs), to achieve
$N=\left\lvert\Theta\right\rvert$. See example 62.  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

$s_{\left\\{i,j\right\\}}=\sum_{a=1}^{N}\mathcal{A}_{\left\\{i,j\right\\}}^{a}\left(\mathsf{D}_{a}-\lambda_{a}\right)\quad\text{for
all}\quad\left\\{i,j\right\\}\in\Theta.$ (2.29)

We are interested in relations of the Feynman integral $\mathcal{I}$ from
(2.1), that is,

$\mathcal{I}(\nu_{1},\ldots,\nu_{N})=\left(\prod_{j=1}^{L}\int\frac{\mathrm{d}^{d}\ell_{j}}{\mathrm{{i}}\pi^{d/2}}\right)f\quad\text{with
the integrand}\quad f=\prod_{i=1}^{N}\mathsf{D}_{i}^{-\nu_{i}}.$ (2.30)

######  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

$\left(\prod_{n=1}^{L}\int\mathrm{d}^{d}\ell_{n}\right)\mathbf{o}_{j}^{i}f=0,$
(2.31)

where the operators $\mathbf{o}_{j}^{i}$ are defined in terms of the
momenta888Recall that $q_{1},\ldots,q_{L}$ denote the loop momenta whereas
$q_{L+1},\ldots,q_{M}$ are the external momenta. as

$\mathbf{o}_{j}^{i}\mathrel{\mathop{:}}=\frac{\partial}{\partial q_{i}}\cdot
q_{j}=\sum_{\mu=1}^{d}\frac{\partial}{\partial
q_{i}^{\mu}}q_{j}^{\mu}\quad\text{for}\quad
i\in\left\\{1,\ldots,L\right\\},j\in\left\\{1,\ldots,M\right\\}.$ (2.32)

The following, explicit form of these relations as difference equations is
essentially due to Baikov [6, 7]; see also Grozin [34]. For completeness, we
include the proof in appendix A.1.

######  Proposition 20.

Given a set of $N=\left\lvert\Theta\right\rvert$ 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

$\mathbf{O}_{j}^{i}\mathcal{I}\left(\nu_{1},\ldots,\nu_{N}\right)=0$ (2.33)

where $\mathbf{O}_{j}^{i}$ denotes shift operators, indexed by $1\leq i\leq L$
and $1\leq j\leq M$, that are given by

$\mathbf{O}_{j}^{i}\mathrel{\mathop{:}}=\begin{cases}\displaystyle
d\delta_{ij}-\sum_{a,b=1}^{N}C_{aj}^{bi}\hat{\mathbf{a}}^{+}\left(\mathbf{b}^{-}-\lambda_{b}\right)&\text{for
$j\leq L$ and}\\\
\displaystyle-\sum_{a,b=1}^{N}C_{aj}^{bi}\hat{\mathbf{a}}^{+}\left(\mathbf{b}^{-}-\lambda_{b}\right)-\sum_{a=1}^{N}\sum_{m=L+1}^{M}A_{a}^{\left\\{i,m\right\\}}q_{j}q_{m}\hat{\mathbf{a}}^{+}&\text{for
$j>L$.}\end{cases}$ (2.34)

The coefficients $C^{bi}_{aj}$ are defined as

$C_{aj}^{bi}\mathrel{\mathop{:}}=\begin{cases}\sum_{m=1}^{M}\mathcal{A}_{a}^{\\{i,m\\}}\mathcal{A}_{\left\\{m,j\right\\}}^{b}\left(1+\delta_{mi}\right)&\text{if
$j\leq L$ and}\\\
\sum_{m=1}^{L}\mathcal{A}_{a}^{\\{i,m\\}}\mathcal{A}_{\left\\{m,j\right\\}}^{b}\left(1+\delta_{mi}\right)&\text{if
$j>L$.}\end{cases}$ (2.35)

######  Corollary 21.

To every difference equation $\mathbf{O}_{j}^{i}\mathcal{I}=0$ from momentum
space IBP, there corresponds a parametric annihilator
$\widetilde{\mathbf{O}}_{j}^{i}\in\operatorname{Ann}_{{A}^{N}[s]}\left(\mathcal{G}^{s}\right)$
of the form

$\displaystyle\widetilde{\mathbf{O}}_{j}^{i}$
$\displaystyle=d\delta_{ij}+\sum_{a,b=1}^{N}C_{aj}^{bi}x_{a}\left(\partial_{b}+\lambda_{b}H\right)\quad\text{for}\quad
i,j\leq L\quad\text{and}$ (2.36) $\displaystyle\widetilde{\mathbf{O}}_{j}^{i}$
$\displaystyle=\sum_{a,b=1}^{N}C_{aj}^{bi}x_{a}\left(\partial_{b}+\lambda_{b}H\right)-\sum_{a=1}^{N}\sum_{m=L+1}^{M}\mathcal{A}_{a}^{\left\\{i,m\right\\}}q_{j}q_{m}x_{a}H\quad\text{for}\quad
i\leq L<j,$ (2.37)

where
$H\mathrel{\mathop{:}}=\frac{(L+1)d}{2}+\sum_{c=1}^{N}x_{c}\partial_{c}$.

###### Proof.

First recall the 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

$\frac{\Gamma(-s-\omega)}{\Gamma(-s)}\hat{\mathbf{a}}^{+}\mathcal{I}=\Gamma(-s-\omega)\hat{\mathbf{a}}^{+}\frac{\widetilde{\mathcal{I}}}{\Gamma(-s-\omega)}=\left(-s-\omega-1\right)\hat{\mathbf{a}}^{+}\widetilde{\mathcal{I}}=\hat{\mathbf{a}}^{+}(-s-\omega)\widetilde{\mathcal{I}},$

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

$-s-\omega=(L+1)(-s)-\sum_{i}\mathbf{n}_{i}\mapsto(L+1)(-s)+\sum_{c=1}^{N}x_{c}\partial_{c}=H.$

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. ∎

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.2.999For an algebraic proof of the analogous statement in
the Baikov representation, see [34, section 9].

######  Definition 22.

By $\operatorname{Mom}$ 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:

$\operatorname{Mom}\mathrel{\mathop{:}}=\sum_{i,j}{A}^{N}[s]\cdot\widetilde{\mathbf{O}}_{j}^{i}\quad\subseteq\quad\operatorname{Ann}_{{A}^{N}[s]}\left(\mathcal{G}^{s}\right).$
(2.38)

Since the $\widetilde{\mathbf{O}}_{j}^{i}$ are first order differential
operators, we have the inclusions

$\operatorname{Mom}\subseteq\operatorname{Ann}^{1}_{{A}^{N}[s]}(\mathcal{G}^{s})\subseteq\operatorname{Ann}_{{A}^{N}[s]}(\mathcal{G}^{s}),$
(2.39)

where $\operatorname{Ann}^{1}$ 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
($\operatorname{Ann}^{1}\subsetneq\operatorname{Ann}$). It is therefore
interesting that we observe the equality
$\operatorname{Ann}^{1}=\operatorname{Ann}$ 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 $\operatorname{Mom}$ 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
$\theta_{e}\mathrel{\mathop{:}}=x_{e}\partial_{e}$ such that
$\nu_{e}=\mathcal{M}\\!\left\\{-\theta_{e}\right\\}$.

######  Question 24.

Given any annihilator $P\in\operatorname{Ann}_{{A}^{N}[s]}(\mathcal{G}^{s})$,
does there exist a polynomial $q\in\mathbbm{C}[s,\theta]$ such that
$qP\in\operatorname{Mom}$? In other words, does

$\mathbbm{C}(s,\theta)\otimes_{\mathbbm{C}[s,\theta]}\operatorname{Ann}_{{A}^{N}[s]}(\mathcal{G}^{s})=\mathbbm{C}(s,\theta)\otimes_{\mathbbm{C}[s,\theta]}\operatorname{Mom}\quad\text{hold?}$
(2.40)

To test this conjecture, we should take all known shift relations for Feynman
integrals, and check if they can be realized as elements of
$\operatorname{Mom}$ (after localizing at $\mathbbm{C}(s,\theta)$). 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

$\mathcal{I}(d)=\mathcal{M}\\!\left\\{\mathcal{U}\right\\}\mathcal{I}(d+2)$
(2.41)

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 [87],101010Tarasov 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).  and had been observed before in
special cases [25]. For
$\widetilde{\mathcal{I}}(d)=[\Gamma(d/2-\omega)/\Gamma(d/2)]\,\mathcal{I}(d)$,
the relation takes the form

$\widetilde{\mathcal{I}}(d)=\frac{s}{s+\omega}\mathcal{U}(\hat{\mathbf{1}}^{+},\ldots,\hat{\mathbf{N}}^{+})\widetilde{\mathcal{I}}(d+2).$
(2.42)

At the same time, the representation
$\widetilde{\mathcal{I}}=\mathcal{M}\\!\left\\{\mathcal{G}^{s}\right\\}$
implies also that

$\widetilde{\mathcal{I}}(d)=\mathcal{G}(\hat{\mathbf{1}}^{+},\ldots,\hat{\mathbf{N}}^{+})\widetilde{\mathcal{I}}(d+2)=\mathcal{M}\\!\left\\{\mathcal{G}\right\\}\widetilde{\mathcal{I}}(d+2).$
(2.43)

######  Remark 25.

The equality of (2.42) and (2.43) implies, via the Mellin transform, that

$H(s)\mathcal{G}+s\mathcal{U}=-(s-\sum_{a}x_{a}\partial_{a}+Ls)\mathcal{G}+s\mathcal{U}\in\operatorname{Ann}(\mathcal{G}^{s-1}).$

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).111111In 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 $\mathcal{G}^{s-1}$.

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
$\widetilde{\mathcal{I}}(d)=\mathcal{M}\\!\left\\{\mathcal{G}^{-d/2}\right\\}$:121212Tkachov
proposed in [91] 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 [70] (see also [35]).

######  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
$b(s)\in\mathbbm{C}[s]$ with

$P(s)f^{s+1}=b(s)f^{s}.$ (2.44)

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 non-zero
solution [10, 72]. Given a solution of (2.44) for $f=\mathcal{G}$, we get a
lowering dimension shift relation:

$\widetilde{\mathcal{I}}(d+2)=\frac{1}{b(s-1)}\mathcal{M}\\!\left\\{P(s-1)\right\\}\widetilde{\mathcal{I}}(d).$
(2.45)

######  Corollary 27.

If we allow the coefficients to be rational functions $k=\mathbbm{C}(s)$,
every integral in $d+2k$ 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

${A}^{N}_{k}\cdot f^{s}=k[x,1/f]\cdot f^{s}.$ (2.46)

###### 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 $n\in\mathbbm{N}$. ∎

In general, computing a Bernstein operator is not at all trivial. But in the
case of a complete set of irreducible scalar products
($N=\left\lvert\Theta\right\rvert$ and $\mathcal{A}$ is invertible), an
explicit formula for the lowering dimension shift follows from Baikov’s
representation [7] of Feynman integrals. We use the form (2.48) given by Lee
in [52, 53]:

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

$\mathsf{Gr}_{n}(s)\mathrel{\mathop{:}}=\det\begin{pmatrix}q_{n}\cdot
q_{n}&\cdots&q_{n}\cdot q_{M}\\\ \vdots&\ddots&\vdots\\\ q_{M}\cdot
q_{n}&\cdots&q_{M}\cdot q_{M}\\\
\end{pmatrix}=\det\left(s_{\left\\{i,j\right\\}}\right)_{n\leq i,j\leq M}$
(2.47)

and remark that
$\mathsf{Gr}\mathrel{\mathop{:}}=\mathsf{Gr}_{L+1}(s)=\det(p_{i}\cdot
p_{j})_{1\leq i,j\leq E}$ 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_ $\mathcal{P}(y)\in\mathbbm{C}[y_{1},\ldots,y_{N}]$ is
the polynomial defined by
$\mathcal{P}(\mathsf{D}_{1},\ldots,\mathsf{D}_{N})=\mathsf{Gr}_{1}(s)$.

######  Theorem 29 (Baikov representation [52]).

The Feynman integral (2.1) can be written as

$\mathcal{I}(d)=\frac{c\cdot\pi^{-LE/2-L(L-1)/4}}{\Gamma\left(\frac{d-E-L+1}{2}\right)\cdots\Gamma\left(\frac{d-E}{2}\right)}\cdot\frac{(-1)^{Ld/2}}{\mathsf{Gr}^{(d-E-1)/2}}\left(\prod_{e=1}^{N}\int\frac{\mathrm{d}y_{e}}{y_{e}^{\nu_{e}}}\right)\cdot\left\\{\mathcal{P}(y)\right\\}^{(d-N-1)/2}$
(2.48)

where $c\in\mathbbm{Q}$ 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 [53]).

A Feynman integral in $d+2$ dimensions can be expressed as an integral in $d$
dimensions by131313 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) $a_{L}=a(a+1)\cdots(a+L-1)$.

$\mathcal{I}(d+2)=\frac{(-1)^{L}}{\left(\frac{d-L-E+1}{2}\right)\cdots\left(\frac{d-E}{2}\right)}\frac{\mathcal{P}(\mathbf{1}^{-},\ldots,\mathbf{N}^{-})}{\mathsf{Gr}}\mathcal{I}(d).$
(2.49)

###### 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. ∎

The 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 (2.41), we find that

$\left\\{\frac{\mathcal{P}(\mathbf{1}^{-},\ldots,\mathbf{N}^{-})}{\mathsf{Gr}}\mathcal{U}(\hat{\mathbf{1}}^{+},\ldots,\hat{\mathbf{N}}^{+})-\frac{E-d+2}{2}\cdots\frac{E-d+(L+1)}{2}\right\\}\mathcal{I}(d)=0.$
(2.50)

We ask if this annihilator is contained in $\operatorname{Mom}$; 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 $(s_{i,j})_{1\leq i,j\leq
M}=\left(\begin{smallmatrix}V&B\\\ B^{\intercal}&G\\\
\end{smallmatrix}\right)$ with $V=(\ell_{i}\cdot\ell_{j})_{i,j\leq L}$,
$B=(\ell_{i}\cdot p_{j})_{i\leq L,j\leq E}$ and $G=(p_{i}\cdot p_{j})_{i,j\leq
E}$ shows that

$\frac{\mathcal{P}(y)}{\mathsf{Gr}}=\det Q(y)\quad\text{where}\quad
Q(\mathsf{D})\mathrel{\mathop{:}}=V-BG^{-1}B^{\intercal}$

is an $L\times L$ matrix whose entries are quadratic in the denominators. By
(2.29),

$Q_{i,j}=\mathcal{A}^{a}_{\left\\{i,j\right\\}}(\mathsf{D}_{a}-\lambda_{a})-\mathcal{A}^{a}_{\left\\{i,r\right\\}}(\mathsf{D}_{a}-\lambda_{a})G^{-1}_{r,s}\mathcal{A}^{b}_{\left\\{j,s\right\\}}(\mathsf{D}_{b}-\lambda_{b})$
(2.51)

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.

The 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:

$\det
Q(\mathbf{1}^{-},\ldots,\mathbf{N}^{-})\cdot\mathcal{U}(\hat{\mathbf{1}}^{+},\ldots,\hat{\mathbf{N}}^{+})-\prod_{j=2}^{L+1}\frac{\left(E+j-d\right)}{2}\in\sum_{i,j}{S}^{N}[d]\cdot\mathbf{O}_{j}^{i}.$
(2.52)

###### Proof.

Let us abbreviate
$\tilde{\Lambda}\mathrel{\mathop{:}}=\Lambda(\hat{\mathbf{1}}^{+},\ldots,\hat{\mathbf{N}}^{+})$
from (A.2) such that $\mathcal{U}=\det\Lambda$ and similarly
$\tilde{Q}\mathrel{\mathop{:}}=Q(\mathbf{1}^{-},\ldots,\mathbf{N}^{-})$ for
the matrix (2.51). Using (2.51) and (2.15), we compute

$\displaystyle\left[\tilde{Q}_{i,j},\hat{\mathbf{c}}^{+}\right]$
$\displaystyle=\mathcal{A}^{a}_{\left\\{i,j\right\\}}\left[\mathbf{a}^{-},\hat{\mathbf{c}}^{+}\right]-G^{-1}_{r,s}\mathcal{A}^{a}_{\left\\{i,r\right\\}}\mathcal{A}^{b}_{\left\\{j,s\right\\}}\left((\mathbf{a}^{-}-\lambda_{a})\left[\mathbf{b}^{-}-\lambda_{b},\hat{\mathbf{c}}^{+}\right]+\left[\mathbf{a}^{-}-\lambda_{a},\hat{\mathbf{c}}^{+}\right](\mathbf{b}^{-}-\lambda_{b})\right)$
$\displaystyle=-\mathcal{A}^{c}_{\left\\{i,j\right\\}}+G^{-1}_{r,s}\left(\mathcal{A}^{a}_{\left\\{i,r\right\\}}\mathcal{A}^{c}_{\left\\{j,s\right\\}}(\mathbf{a}^{-}-\lambda_{a})+\mathcal{A}^{c}_{\left\\{i,r\right\\}}\mathcal{A}^{b}_{\left\\{j,s\right\\}}(\mathbf{b}^{-}-\lambda_{b})\right).$

Contracting with the matrix $\mathcal{A}_{c}^{\left\\{k,l\right\\}}$ (for
$k,l\leq L$) by summing over $c$, we conclude that

$\left[\tilde{Q}_{i,j},\mathcal{A}_{c}^{\left\\{k,l\right\\}}\hat{\mathbf{c}}^{+}\right]=-\delta_{\left\\{i,j\right\\},\left\\{k,l\right\\}}+G^{-1}_{r,s}\left(\mathcal{A}^{a}_{\left\\{i,r\right\\}}\delta_{\left\\{j,s\right\\},\left\\{k,l\right\\}}(\mathbf{a}^{-}-\lambda_{a})+\delta_{\left\\{i,r\right\\},\left\\{k,l\right\\}}\mathcal{A}^{b}_{\left\\{j,s\right\\}}(\mathbf{b}^{-}-\lambda_{b})\right).$

Note that the indices $r$ and $s$ take values $>L$, whereas $i$ and $j$ are
$\leq L$. 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

$\left[\tilde{Q}_{i,j},\tilde{\Lambda}_{k,l}\right]=\frac{1+\delta_{k,l}}{2}\delta_{\left\\{i,j\right\\},\left\\{k,l\right\\}}=\frac{\delta_{i,k}\delta_{j,l}+\delta_{i,l}\delta_{j,k}}{2}.$
(2.53)

Recall (2.3), that $\mathcal{U}(x)=\det\Lambda(x)$, such that

$\det
Q(\mathbf{1}^{-},\ldots,\mathbf{N}^{-})\cdot\mathcal{U}(\hat{\mathbf{1}}^{+},\ldots,\hat{\mathbf{N}}^{+})=\det\tilde{Q}\cdot\det\tilde{\Lambda}.$

We can now invoke an identity of Turnbull [92], see also [28] 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

$\det\tilde{Q}\cdot\det\tilde{\Lambda}=\operatorname{col-
det}\left(\tilde{Q}\cdot\tilde{\Lambda}+Q_{col}\right),\quad\text{where}\quad(Q_{col})_{i,j}\mathrel{\mathop{:}}=-\frac{L-i}{2}\delta_{i,j}$
(2.54)

is a simple diagonal matrix and $\operatorname{col-det}$ denotes the column-
ordered determinant

$\operatorname{col-det}A\mathrel{\mathop{:}}=\sum_{\sigma\in
S_{N}}\operatorname{sgn}(\sigma)A_{\sigma(1),1}\cdots A_{\sigma(N),N}.$ (2.55)

So let us now compute the entries of the product of $\tilde{Q}$ from (2.51)
with $\tilde{\Lambda}$. Firstly,

$\mathcal{A}^{b}_{\left\\{j,s\right\\}}(\mathbf{b}^{-}-\lambda_{b})\tilde{\Lambda}_{j,k}=-\frac{1}{2}\mathcal{A}^{b}_{\left\\{j,s\right\\}}(1+\delta_{j,k})\mathcal{A}^{\left\\{j,k\right\\}}_{c}(\mathbf{b}^{-}-\lambda_{b})\hat{\mathbf{c}}^{+}=-\frac{1}{2}C_{cs}^{bk}\left\\{\hat{\mathbf{c}}^{+}(\mathbf{b}^{-}-\lambda_{b})-\delta_{b,c}\right\\}$

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 $s>L\geq k$. So we can rewrite, due to (2.34),

$\displaystyle\mathcal{A}^{b}_{\left\\{j,s\right\\}}(\mathbf{b}^{-}-\lambda_{b})\tilde{\Lambda}_{j,k}=\frac{1}{2}\mathbf{O}_{s}^{k}+\frac{1}{2}\sum_{m>L}\mathcal{A}^{\left\\{k,m\right\\}}_{b}\hat{\mathbf{b}}^{+}(q_{s}\cdot
q_{m}).$

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

$-\frac{1+\delta_{j,k}}{2}\mathcal{A}^{a}_{\left\\{i,j\right\\}}(\mathbf{a}^{-}-\lambda_{a})\mathcal{A}_{c}^{\left\\{j,k\right\\}}\hat{\mathbf{c}}^{+}-\frac{1}{2}\mathcal{A}^{a}_{\left\\{i,r\right\\}}(\mathbf{a}^{-}-\lambda_{a})\mathcal{A}^{\left\\{k,r\right\\}}_{b}\hat{\mathbf{b}}^{+}=-\frac{1}{2}C^{ak}_{bi}(\mathbf{a}^{-}-\lambda_{a})\hat{\mathbf{b}}^{+}\\\
=\frac{1}{2}\left\\{\mathbf{O}_{i}^{k}-d\delta^{k}_{i}+C^{ak}_{ai}\right\\}=\frac{1}{2}\left\\{\mathbf{O}_{i}^{k}-d\delta^{k}_{i}+(M+1)\delta^{k}_{i}\right\\},$

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

$\tilde{Q}_{i,j}\tilde{\Lambda}_{j,k}=\frac{1}{2}\left\\{(M+1-d)\delta^{k}_{i}+\mathbf{O}_{i}^{k}-\mathcal{A}^{a}_{\left\\{i,r\right\\}}(\mathbf{a}^{-}-\lambda_{a})G^{-1}_{r,s}\mathbf{O}_{s}^{k}\right\\}.$
(2.56)

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

$B_{i,j}=\frac{\delta_{i,j}}{2}\left(M+1-d-L+i\right)=\frac{\delta_{i,j}}{2}\left(E+1-d+i\right),$

such that indeed we conclude with the result that

$\det\tilde{Q}\cdot\det\tilde{\Lambda}\equiv\prod_{i=1}^{L}\frac{E+1-d+i}{2}\mod\sum_{i,j}{S}^{N}[d]\cdot\mathbf{O}_{j}^{i}.\qed$

The Mellin transform
$\mathcal{I}(d)=\mathcal{M}\\!\left\\{\mathcal{U}^{s}e^{-\mathcal{F}/\mathcal{U}}\right\\}$
identifies (2.50) with the annihilator

$\left\\{\det
Q(-\partial)\,\cdot\,\mathcal{U}-\tilde{b}(s)\right\\}\bullet\mathcal{U}^{s}e^{-\mathcal{F}/\mathcal{U}}=0,\quad\text{where}\quad\tilde{b}(s)\mathrel{\mathop{:}}=\prod_{j=2}^{L+1}\left(s+\frac{E+j}{2}\right).$
(2.57)

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

$\Gamma(H(s))\cdot\det
Q(-\partial)\cdot\frac{1}{\Gamma(H(s)-L-1)}\cdot\mathcal{G}^{s+1}=-(s+1)\tilde{b}(s)\mathcal{G}^{s},$
(2.58)

where
$H(s)=\mathcal{M}^{-1}\\!\left\\{-s-\omega\right\\}=-s(L+1)+\sum_{i=1}^{N}\theta_{i}$.
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

$\sum_{r}\frac{\Gamma(H(s))}{\Gamma(H(s)-L-1+r)}\,Q_{r}=\sum_{r\leq
L+1}\left[\prod_{i=1}^{L+1-r}(H(s)-i)\right]Q_{r}+\sum_{r>L+1}\left[\prod_{i=0}^{r-L-1}\frac{1}{H(s)+i}\right]Q_{r}.$

If $r\leq L+1$, 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
$\deg\mathcal{P}(y)\leq\min\left\\{2L,M\right\\}=L+\min\left\\{L,E\right\\}$
by definition 28 and equation (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 [58], that the number
of master integrals equals the Euler characteristic of the complement of the
hypersurface defined by $\mathcal{G}=0$ inside the torus $\mathbbm{G}_{m}^{N}$
(we write $\mathbbm{G}_{m}=\mathbbm{A}\setminus\left\\{0\right\\}$ for the
multiplicative group and $\mathbbm{A}$ 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 [23].

######  Definition 33.

By $V_{\mathcal{G}}$ we denote the vector space of all Feynman integrals
associated to $\mathcal{G}$, over the field
$\mathbbm{C}(s,\nu)\mathrel{\mathop{:}}=\mathbbm{C}(s,\nu_{1},\ldots,\nu_{N})$
of rational functions (in the dimension and indices). More precisely, with
$\widetilde{\mathcal{I}}_{\mathcal{G}}\mathrel{\mathop{:}}=\mathcal{M}\\!\left\\{\mathcal{G}^{s}\right\\}$,

$V_{\mathcal{G}}\mathrel{\mathop{:}}=\sum_{n\in\mathbbm{Z}^{N}}\mathbbm{C}(s,\nu)\cdot\widetilde{\mathcal{I}}_{\mathcal{G}}(\nu+n)=\mathbbm{C}(s,\nu)\otimes_{\mathbbm{C}[s,\nu]}\left({S}^{N}[s]\cdot\widetilde{\mathcal{I}}_{\mathcal{G}}\right).$
(3.1)

The _number of master integrals_ is the dimension of this vector space:

$\mathfrak{C}\left(\mathcal{G}\right)\mathrel{\mathop{:}}=\dim_{\mathbbm{C}(s,\nu)}V_{\mathcal{G}}.$
(3.2)

Note that this is the same as the dimension of the space
$\sum_{n}\mathbbm{C}(s,\nu)\mathcal{I}_{\mathcal{G}}(\nu+n)$ of Feynman
integrals (2.1), because the ratios
$\mathcal{I}_{\mathcal{G}}(\nu+n)/\widetilde{\mathcal{I}}_{\mathcal{G}}(\nu+n)=\Gamma(-s)/\Gamma(-s-\omega-\left\lvert
n\right\rvert)$ with $\left\lvert n\right\rvert=n_{1}+\ldots+n_{N}$ are all
related by a rational function in $\mathbbm{C}(s,\nu)$, see (2.22).

######  Remark 34 (Warning).

The phrase “master integrals” is used in the existing physics literature to
denote various different quantities, and none of those notions coincides
exactly with ours. The main sources for discrepancies are:

  1. 1.

Almost always the integrals are considered only for integer indices
$\nu\in\mathbbm{Z}^{N}$, 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. 2.

We only discuss relations of integrals that are expressible as linear shift
operators acting on a single integral. This setup cannot account for relations
of integrals of different graphs (with some fixed values of the indices), as
for example discussed in [47]. It also excludes symmetry relations, which are
represented by permutations of the indices $\nu_{e}$.

  3. 3.

Some authors do not count integrals if they can be expressed in terms of
$\Gamma$-functions or products of simpler integrals, for example [47, 40].

Taking care of these subtleties, we will demonstrate in section 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 [76] for the case of
integer indices $\nu\in\mathbbm{Z}^{N}$, 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

$\mathfrak{C}\left(\mathcal{G}\right)=\dim_{\mathbbm{C}(s,\theta)}\left(\mathbbm{C}(s,\theta)\otimes_{\mathbbm{C}[s,\theta]}{A}^{N}[s]\cdot\mathcal{G}^{s}\right),$

where $\mathbbm{C}[s,\theta]=\mathbbm{C}[s,\theta_{1},\ldots,\theta_{N}]$
denotes the polynomials in the dimension $s=-d/2$ and the operators
$\theta_{e}\mathrel{\mathop{:}}=x_{e}\partial_{e}=\mathcal{M}^{-1}\\!\left\\{-\nu_{e}\right\\}$,
and $F\mathrel{\mathop{:}}=\mathbbm{C}(s,\theta)$ stands for their fraction
field (the rational functions in these variables). Since $F$ contains
$k\mathrel{\mathop{:}}=\mathbbm{C}(s)$, we can equivalently work over this
base field throughout and write

$\mathfrak{C}\left(\mathcal{G}\right)=\dim_{F}(F\otimes_{R}\mathscr{M})$ (3.3)

in terms of $R\mathrel{\mathop{:}}=k[\theta]$ and the module
$\mathscr{M}={A}^{N}_{k}\\!\cdot\\!\mathcal{G}^{s}$ over the Weyl algebra
${A}^{N}_{k}\mathrel{\mathop{:}}={A}^{N}\otimes_{\mathbbm{C}}k={A}^{N}[s]\otimes_{\mathbbm{C}[s]}k$
over the field $k=\mathbbm{C}(s)$. 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 [43, 44], and holonomic modules in zero variables are precisely the
finite-dimensional vector spaces. The holonomicity of the parametric integrand
was already exploited in [45] to show that Feynman integrals fulfill a
holonomic system of differential equations, and it is also a key ingredient in
the proof in [76].

The number defined in (3.3) has been studied by Loeser and Sabbah [58] in a
slightly different setting, namely for holonomic modules over the algebra

$\mathcal{D}^{N}_{k}\mathrel{\mathop{:}}=k[x_{1}^{\pm 1},\ldots,x_{N}^{\pm
1}]\langle\partial_{1},\ldots,\partial_{N}\rangle=k[x^{\pm
1}]\otimes_{k[x]}{A}^{N}_{k}={A}^{N}_{k}[x^{-1}]$ (3.4)

of linear differential operators on the torus $\mathbbm{G}_{m,k}^{N}$. 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 $\mathcal{O}(\mathbbm{A}^{N}_{k})=k[x]$ to
rational functions $\mathcal{O}(\mathbbm{G}_{m,k}^{N})=k[x^{\pm 1}]$ whose
denominator is a monomial in the coordinates $x_{i}$. Equivalently, we can
also view $\mathcal{D}^{N}_{k}=\iota^{\ast}{A}^{N}_{k}$ as the pull-back under
the (open) inclusion

$\iota\colon\mathbbm{G}_{m,k}^{N}\lhook\joinrel\longrightarrow\mathbbm{A}^{N}_{k}.$

The pull-back along $\iota$ turns every ${A}^{N}_{k}$-module $\mathscr{M}$
into a $\mathcal{D}^{N}_{k}$-module $\iota^{\ast}\mathscr{M}$, namely the
localization $\iota^{\ast}\mathscr{M}=k[x^{\pm
1}]\otimes_{k[x]}\mathscr{M}=\mathscr{M}[x^{-1}]$. Importantly, if
$\mathscr{M}$ is holonomic, so is its pull-back $\iota^{\ast}\mathscr{M}$. The
starting point for this section is

######  Theorem 35 (Loeser & Sabbah [58, 60]).

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
$\dim_{F}(F\otimes_{R}\mathscr{M})=\chi\left(\iota^{\ast}\mathscr{M}\right)$.

In appendix B, we provide a self-contained proof of this crucial theorem,
simpler and more explicit than in [60]. For now, let us content ourselves with
reducing it to the known situation on the torus.

###### Proof.

We can invoke
$\dim_{F}(F\otimes_{R}\iota^{\ast}\mathscr{M})=\chi(\iota^{\ast}\mathscr{M})<\infty$
from [60, Théorème 2]. To conclude, we just need to note that
$F\otimes_{R}\mathscr{M}$ and
$F\otimes_{R}\mathscr{M}[x^{-1}]=F\otimes_{R}\iota^{\ast}\mathscr{M}$ 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 $1\otimes x_{i}$. ∎

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

$\chi\left(\mathscr{M}^{\prime}\right)\mathrel{\mathop{:}}=\chi(\operatorname{DR}(\mathscr{M}^{\prime}))=\sum_{i}(-1)^{i}\dim_{k}H^{i}\big{(}\operatorname{DR}(\mathscr{M}^{\prime})\big{)},$
(3.5)

of the algebraic de Rham complex of
$\mathscr{M}^{\prime}\mathrel{\mathop{:}}=\iota^{\ast}\mathscr{M}$. This is
the complex

$\operatorname{DR}(\mathscr{M}^{\prime})\mathrel{\mathop{:}}=\left(\Omega^{\bullet}_{\mathbbm{G}_{m,k}^{N}}\otimes_{\mathcal{O}(\mathbbm{G}_{m,k}^{N})}\mathscr{M}^{\prime}[N],\mathrm{d}\right)$
(3.6)

of $\mathscr{M}^{\prime}$-valued differential forms on the torus
$\mathbbm{G}_{m,k}^{N}$, 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 $0\leq i\leq N$. The extremal cohomolgy groups are easily
identified as

$H^{-N}\left(\operatorname{DR}(\mathscr{M}^{\prime})\right)=\bigcap_{i=1}^{N}\ker\partial_{i}\quad\text{and}\quad
H^{0}\left(\operatorname{DR}(\mathscr{M}^{\prime})\right)\cong\left.\mathscr{M}^{\prime}\middle/\sum_{i=1}^{N}\partial_{i}\mathscr{M}^{\prime}\right.=\pi_{\ast}\mathscr{M}^{\prime},$
(3.7)

with the latter also known as push-forward of $\mathscr{M}^{\prime}$ under the
projection
$\pi\colon\mathbbm{G}_{m,k}^{N}\relbar\joinrel\twoheadrightarrow\mathbbm{A}_{k}^{0}$
to the point. Since holonomicity is preserved under direct images, we conclude
that $\dim_{k}H^{0}\left(\operatorname{DR}(\mathscr{M}^{\prime})\right)$ is
finite.141414Recall that a holonomic module over the point
$\mathbbm{A}^{0}_{k}$ is the same as a finite-dimensional $k$-vector space.
In fact, the same is true for the other de Rham cohomology groups, which shows
that (3.5) is indeed well-defined.151515The de Rham complex
$\operatorname{DR}({A}^{N}_{k})$ is a resolution of $k[x]$ by free
${A}^{N}_{k}$-modules, such that
$H^{\bullet}(\operatorname{DR}(\mathscr{M}^{\prime}))$ are the (left) derived
functors of
$\pi_{\ast}\mathscr{M}^{\prime}=H^{0}(\operatorname{DR}(\mathscr{M}^{\prime}))$.
In the language of derived categories, saying that
$\pi_{\ast}\mathscr{M}^{\prime}$ is holonomic actually means precisely that
$\operatorname{DR}(\mathscr{M}^{\prime})$ is a complex with cohomology groups
that are holonomic modules over the point—that is, finite-dimensional vector
spaces over $k$.

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 $\mathcal{G}\in\mathbbm{C}[x_{1},\ldots,x_{N}]$ be a polynomial and set
$k=\mathbbm{C}(s)$. Then the Euler characteristics of the algebraic de Rham
complexes of the holonomic ${A}^{N}_{k}$-module
$\iota^{\ast}{A}^{N}_{k}\mathcal{G}^{s}$ and the holonomic
${A}^{N}_{\mathbbm{C}}$-module $\mathbbm{C}[x^{\pm
1},\mathcal{G}^{-1}]=\mathcal{O}(\mathbbm{G}_{m,k}^{N}\setminus\mathbbm{V}(\mathcal{G}))$
coincide:

$\chi(\iota^{\ast}{A}^{N}_{k}\mathcal{G}^{s})=\chi\left(\mathbbm{C}[x^{\pm
1},\mathcal{G}^{-1}]\right).$ (3.8)

In particular, we can dispose of the parameter $s$ completely and compute with
the algebraic de Rham complex of $\mathbbm{C}[x^{\pm 1},\mathcal{G}^{-1}]$,
which is the ring of regular functions of the complement of the hypersurface
$\mathbbm{V}(\mathcal{G})=\left\\{x\colon\mathcal{G}(x)=0\right\\}$ in the
torus $\mathbbm{G}_{m}^{N}$. 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

$\mathfrak{C}\left(\mathcal{G}\right)=\chi\left(\mathbbm{C}[x^{\pm
1},\mathcal{G}^{-1}]\right),$ (3.9)

the Euler characteristic of the algebraic de Rham complex of the complement of
the hypersurface $x_{1}\cdots x_{N}\cdot\mathcal{G}=0$ inside the affine plane
$\mathbbm{A}^{N}$. Via Grothendieck’s comparison isomorphism, this is the same
as the topological Euler characteristic, up to a sign:161616 This sign arises
from the shift by $N$ in the definition (3.6) of the de Rham complex
$\operatorname{DR}$.

$\mathfrak{C}\left(\mathcal{G}\right)=(-1)^{N}\chi\left(\mathbbm{C}^{N}\setminus\left\\{x_{1}\cdots
x_{N}\cdot\mathcal{G}=0\right\\}\right)=(-1)^{N}\chi\left(\mathbbm{G}_{m}^{N}\setminus\left\\{\mathcal{G}=0\right\\}\right).$
(3.10)

######  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.171717It fails precisely
if, for some $r\in\mathbbm{N}$, $s-r$ is a zero of the Bernstein-Sato
polynomial of $\mathcal{G}$.  It can thus happen that
${A}^{N}\cdot\mathcal{G}^{s}\subsetneq\mathbbm{C}[x,\mathcal{G}^{-1}]$ is a
proper subalgebra (note $k=\mathbbm{C}(s)=\mathbbm{C}$). 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 [86].

###### Proof of proposition 36.

Given an ${A}^{N}_{\mathbbm{C}}$-module $\mathscr{M}$ and a polynomial
$f\in\mathbbm{C}[x]$, set
$\mathscr{M}^{\prime}\mathrel{\mathop{:}}=\mathscr{M}[f^{-1}]$ and consider
the ${A}^{N}_{\mathbbm{C}}$-module $\mathscr{M}^{\prime}f^{s}$ formed by
products of $f^{s}$ with elements
$m\in\mathscr{M}^{\prime}[s]\mathrel{\mathop{:}}=\mathscr{M}^{\prime}\otimes_{\mathbbm{C}}\mathbbm{C}[s]$.
As vector spaces, $\mathscr{M}^{\prime}f^{s}\cong\mathscr{M}^{\prime}[s]$ via
$mf^{s}\mapsto m$, but the ${A}^{N}_{\mathbbm{C}}$ action on
$\mathscr{M}^{\prime}f^{s}$ has twisted derivatives to take into account the
factor $f^{s}$:

$x_{i}\bullet
mf^{s}\mathrel{\mathop{:}}=x_{i}mf^{s}\quad\text{and}\quad\partial_{i}\bullet
mf^{s}\mathrel{\mathop{:}}=\left\\{(\partial_{i}m)+sm\frac{\partial_{i}f}{f}\right\\}f^{s}.$
(3.11)

Following Malgrange [62], we introduce the action of a further variable $t$ by
setting

$t\bullet
m(s)f^{s}\mathrel{\mathop{:}}=m(s+1)f^{s+1}\quad\text{and}\quad\partial_{t}\bullet
m(s)f^{s}\mathrel{\mathop{:}}=-sm(s-1)f^{s-1},$ (3.12)

where we use the intuitive abbreviation
$f^{s+r}\mathrel{\mathop{:}}=f^{r}\cdot f^{s}$ for $r\in\mathbbm{Z}$. 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 $\mathscr{M}^{\prime}f^{s}$ becomes an
${A}^{N+1}_{\mathbbm{C}}$-module in the $N+1$ variables
$(x_{1},\ldots,x_{N},t)$. Note that $\partial_{t}t=-s$, so

$\frac{\mathscr{M}^{\prime}f^{s}}{\partial_{t}\mathscr{M}^{\prime}f^{s}}=\frac{\mathscr{M}^{\prime}f^{s}}{st^{-1}\mathscr{M}^{\prime}f^{s}}=\frac{\mathscr{M}^{\prime}f^{s}}{s\mathscr{M}^{\prime}f^{s}}\cong\mathscr{M}^{\prime}$

is an isomorphism of ${A}^{N}_{\mathbbm{C}}$-modules.181818This is also clear
from the fact that
$\mathscr{M}^{\prime}f^{s}/(\partial_{t}\mathscr{M}^{\prime}f^{s})=\pi_{\ast}(\mathscr{M}^{\prime}f^{s})$
is the push-forward of $\mathscr{M}^{\prime}f^{s}$ under the projection
$\pi\colon\mathbbm{A}^{N+1}\longrightarrow\mathbbm{A}^{N}$ forgetting the last
coordinate. Namely, since
$\mathscr{M}^{\prime}f^{s}=F_{\ast}\mathscr{M}^{\prime}$ as we discuss below,
$\pi_{\ast}(F_{\ast}\mathscr{M}^{\prime})=(\pi\circ
F)_{\ast}\mathscr{M}^{\prime}=\operatorname{id}_{\ast}\mathscr{M}^{\prime}=\mathscr{M}^{\prime}$.
Since $\partial_{t}$ is injective on $\mathscr{M}^{\prime}f^{s}$ (it raises
the degree in $s$), the de Rham complex
$\operatorname{DR}(\mathscr{M}^{\prime}f^{s})$ is quasi-isomorphic to
$\operatorname{DR}(\mathscr{M}^{\prime}f^{s}/\partial_{t}\mathscr{M}^{\prime}f^{s})=\operatorname{DR}(\mathscr{M}^{\prime})$,
see corollary 70. So we can conclude the equality

$\chi(\mathscr{M}^{\prime}f^{s})=\chi(\mathscr{M}^{\prime}),$ (3.13)

once we assume that $\mathscr{M}$ is holonomic to ensure that these Euler
characteristics are well defined. Indeed, the holonomicity of
$\mathscr{M}^{\prime}$ and $\mathscr{M}^{\prime}f^{s}$ holds because

  * •

$\mathscr{M}^{\prime}=j^{\ast}\mathscr{M}$ is the pull-back of $\mathscr{M}$
under the inclusion
$j\colon\mathbbm{A}^{N}_{\mathbbm{C}}\setminus\left\\{f=0\right\\}\lhook\joinrel\longrightarrow\mathbbm{A}^{N}_{\mathbbm{C}}$,

  * •

$\mathscr{M}^{\prime}f^{s}=F_{\ast}\mathscr{M}^{\prime}$ is the push-forward
of $\mathscr{M}^{\prime}$ under the closed embedding
$F\colon\mathbbm{A}^{N}_{\mathbbm{C}}\lhook\joinrel\longrightarrow\mathbbm{A}^{N+1}_{\mathbbm{C}}$
which sends $x$ to $(x,f(x))$.191919It follows from (3.12) that
$\mathscr{M}^{\prime}f^{s}=\bigoplus_{n\geq
0}\partial_{t}^{n}\mathscr{M}^{\prime}\cong F_{\ast}\mathscr{M}$ as
$\mathbbm{C}[x]$-modules, since
$\partial^{n}_{t}\mathscr{M}^{\prime}=\partial^{n}_{t}t^{n}\mathscr{M}^{\prime}\equiv
s^{n}\mathscr{M}^{\prime}\mod s^{<n}\mathscr{M}^{\prime}$. Furthermore, the
derivatives act on $F_{\ast}\mathscr{M}$ 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).

Alternatively, the filtration
$\Gamma_{j}\mathscr{M}^{\prime}f^{s}\mathrel{\mathop{:}}=f^{-j}\sum_{i\leq
j}s^{i}\Gamma_{2j(\deg f)-i}\mathscr{M}$ induced by any good filtration
$\Gamma_{\bullet}$ on $\mathscr{M}$ directly shows the holonomicity of
$\mathscr{M}^{\prime}f^{s}$, since its dimension grows like $j^{N+1}$ for
large $j$. We now invoke the theory of Loeser-Sabbah to deduce that

$\chi(\mathscr{M}^{\prime}f^{s})=\dim_{{\mathbbm{C}}(\theta,t\partial_{t})}\mathscr{M}^{\prime}f^{s}(\theta,t\partial_{t})=\dim_{k(\theta)}\mathscr{N}(\theta)=\chi(\mathscr{N})$

where
$\mathscr{N}\mathrel{\mathop{:}}=\mathscr{M}^{\prime}f^{s}(t\partial_{t})$
denotes the algebraic Mellin transform (B.1) of $\mathscr{M}^{\prime}f^{s}$
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
$k=\mathbbm{C}(s)$. So
$\mathscr{N}=\mathscr{M}^{\prime}\otimes_{\mathbbm{C}}\mathbbm{C}(s)f^{s}=\mathscr{M}\otimes_{\mathbbm{C}}kf^{s}$
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
$\chi(\mathscr{M}\otimes_{\mathbbm{C}}kf^{s})=\chi(\mathscr{M}[f^{-1}])$, and
the special case of $\mathscr{M}=\mathbbm{C}[x^{\pm 1}]$ with $f=\mathcal{G}$
proves the claim. ∎

######  Remark 39.

More abstractly, Proposition 36 can also be seen as an application of the
theory of characteristic cycles [32]: 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 [50,
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 [32, 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
$\chi(\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(f))\mathrel{\mathop{:}}=\chi(\mathbbm{C}[x^{\pm
1},f^{-1}])=\chi((\mathbbm{C}^{\ast})^{N}\setminus\mathbbm{V}(f))$ vanishes
for $f=\mathcal{G}$. For example, this happens if $f$ is homogeneous in a
generalized sense: Suppose we can find
$\lambda_{0},\ldots,\lambda_{N}\in\mathbbm{Z}$, not all zero, such that

$f\left(x_{1}t^{\lambda_{1}},\ldots,x_{N}t^{\lambda_{N}}\right)=t^{\lambda_{0}}f(x_{1},\ldots,x_{N})\quad\text{in}\quad\mathbbm{C}[x,t^{\pm
1}];$ (3.14)

which is equivalent (apply $\partial_{t}$ and set $t=1$) to the existence of a
linear annihilator,

$P^{\lambda}_{s}\bullet f^{s}=0,\quad\text{of the form}\quad
P^{\lambda}_{s}\mathrel{\mathop{:}}=\sum_{i=1}^{N}\lambda_{i}\theta_{i}-s\lambda_{0}\in\mathbbm{Z}[s,\theta]\setminus\left\\{0\right\\}.$
(3.15)

######  Lemma 40.

Given $f\in\mathbbm{C}[x_{1},\ldots,x_{N}]$, the Mellin transform
$\mathfrak{M}\mathrel{\mathop{:}}=\mathscr{M}\otimes_{\mathbbm{C}(s)[\theta]}\mathbbm{C}(s,\theta)$
of $\mathscr{M}\mathrel{\mathop{:}}=\mathcal{D}^{N}_{k}\\!\cdot\\!f^{s}$ is
zero if, and only if, $f^{s}$ is annihilated by a polynomial in the Euler
operators $\theta$:

$\operatorname{Ann}_{{A}^{N}_{k}}(f^{s})\cap\mathbbm{C}[s,\theta]\neq\left\\{0\right\\}.$
(3.16)

###### Proof.

Clearly, $\mathfrak{M}=\left\\{0\right\\}$ requires $f^{s}$ to be mapped to
zero in the localization $\mathfrak{M}$ of $\mathscr{M}$ at
$\mathbbm{C}[s,\theta]\setminus\left\\{0\right\\}$, and therefore the
existence of a non-zero polynomial $P(\theta,s)\in\mathbbm{C}[s,\theta]$ with
$P(\theta,s)\bullet f^{s}=0$. Conversely, given such an operator, its shifts
$P(\theta-\alpha,s+r)$ by $(r,\alpha)\in\mathbbm{Z}^{1+N}$ 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 $gf^{s+r}\otimes h$
for some $r\in\mathbbm{Z}$,
$h\in\mathbbm{C}[s,\theta]\setminus\left\\{0\right\\}$ and a Laurent
polynomial $g\in k[x^{\pm 1}]$. ∎

In particular, the presence of a linear annihilator (3.15) implies
$\mathfrak{M}=\left\\{0\right\\}$ and hence
$\chi(\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(f))=0$ via corollary 37. Note
that we could equally phrase this in terms of the hypersurface
$\mathbbm{V}(f)\subset\mathbbm{G}_{m}^{N}$ itself as $\chi(\mathbbm{V}(f))=0$,
because the Euler characteristics are related through
$\chi(\mathbbm{V}(f))=-\chi(\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(f))$ (see
section 3.3).

When $f=\mathcal{G}$ comes from Feynman graph $G$ (as in the next section), it
is not difficult to see that the homogeneity (3.14) occurs precisely when $G$
has a _tadpole_.202020A 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
$\lambda_{0}\mathrel{\mathop{:}}=L_{\gamma}$. Thus we obtain (3.14) by setting
$\lambda_{e}=1$ if $e\in\gamma$ and $\lambda_{e}=0$ otherwise.  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. This reasoning
explains a common practice in Feynman integral calculations, namely that
Feynman integrals associated to graphs with tadpoles are declared to vanish.

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 $f\in\mathbbm{C}[x_{1},\ldots,x_{N}]$ denote a polynomial. Then the
hypersurface $\left\\{f=0\right\\}$ inside the torus $\mathbbm{G}_{m}^{N}$ has
vanishing Euler characteristic precisely when there are
$\lambda_{0},\ldots,\lambda_{N}\in\mathbbm{Z}$, not all zero, such that (3.14)
holds.

Geometrically, the 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 $\mathbbm{Z}^{N}$
and hence construct a matrix $A\in\mathsf{GL}_{N}(\mathbbm{Z})$ with first row
$A_{1i}=\lambda_{i}$. In the associated coordinates $y$, defined by

$x_{i}=\prod_{j=1}^{N}y_{j}^{A_{ji}}\quad\text{and}\quad
y_{i}=\prod_{j=1}^{N}x_{j}^{A^{-1}_{ji}}\quad\text{where}\quad
A^{-1}_{ji}\mathrel{\mathop{:}}=\left(A^{-1}\right)_{ji},$

the polynomial $f$ takes the form $f(x)=y_{1}^{\lambda_{0}}g(\bar{y})$ for
some Laurent polynomial $g\in\mathbbm{C}[\bar{y}^{\pm 1}]$ 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 $f\in\mathbbm{C}[x_{1},\ldots,x_{N}]$ denote a polynomial. Then
$\mathbbm{V}(f)\subset\mathbbm{G}_{m}^{N}$ has Euler characteristic zero if
and only if it is isomorphic to a product of $\mathbbm{G}_{m}$ times a
hypersurface $\left\\{g=0\right\\}\subset\mathbbm{G}_{m}^{N-1}$.

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 $f$:

$\mathsf{NP}\left(\sum_{\alpha\in\mathbbm{Z}^{N}}c_{\alpha}x^{\alpha}\right)\mathrel{\mathop{:}}=\operatorname{conv}\left\\{\alpha\in\mathbbm{Z}^{N}\colon
c_{\alpha}\neq 0\right\\}\subset\mathbbm{R}^{N}.$ (3.17)

Since every monomial $x^{\alpha}$ is an eigenvector of the operators (3.15),
$P^{\lambda}_{s}(\theta)\bullet x^{\alpha}=P^{\lambda}_{s}(\alpha)x^{\alpha}$,
it is annihilated by $P^{\lambda}_{1}$ exactly when $\alpha$ belongs to
$F_{\lambda}\mathrel{\mathop{:}}=\left\\{P^{\lambda}_{1}(\alpha)=0\right\\}$,
the hyperplane
$F_{\lambda}=\left\\{\alpha\colon\alpha_{1}\lambda_{1}+\ldots+\alpha_{N}\lambda_{N}=\lambda_{0}\right\\}$.
In particular, $0=P^{\lambda}_{s}\bullet f^{s}=sf^{s-1}P^{\lambda}_{1}\bullet
f$ 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

$\dim\mathsf{NP}\left(f\right)<N.$ (3.18)

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
$f\in\mathbbm{C}$). In the case $N=1$, the variety
$\mathbbm{V}(f)\subset\mathbbm{C}^{\ast}$ is a finite set and hence its Euler
characteristic coincides with its cardinality. Therefore,
$\chi(\mathbbm{V}(f))=0$ 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
$\chi(\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(f))=\chi(\mathbbm{V}(f))=0$.
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 $0\neq
P(\theta,s)\in\mathbbm{C}[s,\theta]$ such that $P(\theta,s)\bullet f^{s}=0$,
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 $\mathsf{NP}\left(f\right)\subseteq\left\\{\alpha\colon
P^{\lambda}_{1}(\alpha)\leq 0\right\\}$. Under the 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 $t$:

$f(x_{1}t^{\lambda_{1}},\ldots,x_{N}t^{\lambda_{N}})=t^{\lambda_{0}}f_{\sigma}(x)\left(1+\mathcal{O}\left(t^{-1}\right)\right),\quad\text{where}\quad
f_{\sigma}(x)\mathrel{\mathop{:}}=\sum_{\alpha\in\mathbbm{Z}^{N}\cap\sigma}c_{\alpha}x^{\alpha}$
(3.19)

and $\mathcal{O}\left(t^{-1}\right)$ denotes a rational function in
$t^{-1}\mathbbm{C}(x)[t^{-1}]$. Note that $P(\theta,s)\bullet
f^{s}(\left\\{x_{i}t^{\lambda_{i}}\right\\})=0$ is still zero, because the
rescaling of $x$ commutes with the Euler operators $\theta_{i}\bullet
h(x_{i}t^{\lambda_{i}})=(\theta_{i}\bullet h(x_{i}))|_{x_{i}\mapsto
x_{i}t^{\lambda_{i}}}$. 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

$0=P(\theta,s)\bullet
f_{\sigma}^{s}(x)\left(1+\mathcal{O}\left(t^{-1}\right)\right)^{s}=P(\theta,s)\bullet
f_{\sigma}^{s}+\mathcal{O}\left(t^{-1}\right)f_{\sigma}^{s},$

where $\mathcal{O}\left(t^{-1}\right)$ on the right-hand side denotes a formal
series in $t^{-1}\mathbbm{C}(x,s)[[t^{-1}]]$. In particular, the coefficient
of $t^{0}$ must vanish, and we conclude that $P(\theta,s)\bullet
f_{\sigma}^{s}=0$. Label the variables such that $\lambda_{N}\neq 0$, 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

$P(\theta,s)=P(\theta^{\prime},0,s)+P^{\lambda}_{s}(\theta,s)\cdot
Q(\theta,s)$

for some polynomial $Q(\theta,s)\in\mathbbm{C}[\theta,s]$, such that the first
summand depends only on
$\theta^{\prime}\mathrel{\mathop{:}}=(\theta_{1},\ldots,\theta_{N-1})$ and
$s$. Since $\mathsf{NP}\left(f_{\sigma}\right)=\sigma\subset F_{\lambda}$ is
contained in the hyperplane $F_{\lambda}=\left\\{\alpha\colon
P^{\lambda}_{1}(\alpha)=0\right\\}$, we see $P^{\lambda}_{s}\bullet
f^{s}_{\sigma}=f^{s-1}P^{\lambda}_{1}\bullet f_{\sigma}=0$ and thus
$Q(\theta,s)$ drops out in

$0=P(\theta,s)\bullet f^{s}_{\sigma}=P(\theta^{\prime},0,s)\bullet
f^{s}_{\sigma}=P(\theta^{\prime},0,s)\bullet g^{s},$

where $g\mathrel{\mathop{:}}=f|_{x_{N}=1}\in\mathbbm{C}[x_{1},\ldots,x_{N-1}]$
is a polynomial in less than $N$ variables. If $P(\theta^{\prime},0,s)$ were
non-zero, lemma 40 would show
$\chi(\mathbbm{G}_{m}^{N-1}\setminus\mathbbm{V}(g))=0$, 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 $N-1$.212121Observe 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 $\lambda_{N}\neq 0$), and therefore
$\dim\mathsf{NP}\left(g\right)=\dim\mathsf{NP}\left(f_{\sigma}\right)$.

Therefore, $P(\theta^{\prime},0,s)$ must be zero and we conclude that
$P(\theta,s)=P^{\lambda}_{s}\cdot Q$ has a linear factor
$P^{\lambda}_{s}(\theta,s)$.222222The existence of a linear annihilator could
also be deduced from [30, Théorème 9.2].  Now set
$m\mathrel{\mathop{:}}=Q(\theta,s)\bullet f^{s}$, which is non-zero, 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
$r\in\mathbbm{Z}$ and a Laurent polynomial $a\in\mathbbm{C}(s)[x^{\pm}]$.
After multiplying with with a polynomial in $\mathbbm{C}[s]$, we may even
assume $0\neq a\in\mathbbm{C}[s,x^{\pm 1}]$ with $P^{\lambda}_{s}\bullet
af^{s+r}=0$. Applying the Leibniz rule and dividing by $af^{s+r}$, we find

$0=\frac{P^{\lambda}_{0}\bullet
a}{a}-s\lambda_{0}+(s+r)\frac{P^{\lambda}_{0}\bullet f}{f}.$

Since the degree of $P^{\lambda}_{0}\bullet a$ 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
$s$, $P^{\lambda}_{0}\bullet f=\lambda_{0}f$, which yields the sought-after
$P^{\lambda}_{s}\bullet f^{s}=0$. ∎

######  Remark 43.

In summary, we showed that the following six conditions on a polynomial
$f\in\mathbbm{C}[x_{1},\ldots,x_{N}]$ 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
$\chi(\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(f))=\chi(\mathbbm{V}(f))=0$ and
(6) divisibility of $\mathbbm{V}(f)$ by $\mathbbm{G}_{m}$ as stated in
corollary 42.

To conclude, let us interpret our observation in the light of the well-known
result, due to Kouchnirenko [48, Théorème IV] and Khovanskii [46, Theorem 2 in
section 3], that relates the Euler characteristic to the volume of the Newton
polytope:

######  Theorem 44.

For almost all polynomials $f\in\mathbbm{C}[x_{1},\ldots,x_{N}]$ with a fixed
Newton polytope, we have
$\chi(\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(f))=(-1)^{N}\cdot
N!\cdot\operatorname{Vol}\mathsf{NP}\left(f\right)$.

For polynomials $f$ whose non-zero coefficients are sufficiently generic,
proposition 41 follows from theorem 44. Our proof shows that when
$\operatorname{Vol}\mathsf{NP}\left(f\right)=0$, the theorem applies without
any constraints on the non-zero coefficients of $f$. In fact, this statement
extends to the case when $N!\operatorname{Vol}\mathsf{NP}\left(f\right)=1$,
because it is known that
$(-1)^{N}\cdot\chi(\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(f))$ is always
bounded from above by $N!\operatorname{Vol}\mathsf{NP}\left(f\right)$, for all
$f$, leaving only the possibilities $\left\\{0,1\right\\}$ for the signed
Euler characteristic
$(-1)^{N}\cdot\chi(\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(f))$.

###  3.2 Graph polynomials

Our discussion so far applies to all integrals of the type (2.1)—the defining
data is thus the set $\mathsf{D}=(\mathsf{D}_{1},\ldots,\mathsf{D}_{N})$ of
denominators, which is sometimes also called an _integral family_ [96]. The
denominators can be arbitrary quadratic forms in the loop momenta; the
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.232323Momentum 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 _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,242424The infinitesimal imaginary part $\mathrm{{i}}\epsilon$ is
irrelevant for our purpose of counting integrals and will be henceforth
ignored.

$\frac{1}{\mathsf{D}_{e}}=\frac{1}{-k_{e}^{2}+m_{e}^{2}-\mathrm{{i}}\epsilon}\quad(1\leq
e\leq N),$ (3.20)

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

$\mathfrak{C}\left(G\right)\mathrel{\mathop{:}}=\mathfrak{C}\left(\mathcal{G}_{G}\right)\quad\text{where}\quad\mathcal{G}_{G}\mathrel{\mathop{:}}=\mathcal{U}_{G}+\mathcal{F}_{G}.$
(3.21)

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 the 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 [87] that all such integrals with ISPs are in
fact linear combinations of scalar integrals in higher dimensions $d+2k$, for
some $k\in\mathbbm{N}$. 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, 80], which go back at least to [63].

######  Proposition 46.

The Symanzik polynomials of a graph $G$ can be written as

$\mathcal{U}_{G}=\sum_{T}\prod_{e\notin
T}x_{e}\quad\text{and}\quad\mathcal{F}_{G}=\mathcal{U}_{G}\sum_{e=1}^{N}x_{e}m_{e}^{2}-\sum_{F}p_{F}^{2}\prod_{e\notin
T}x_{e},$ (3.22)

where $T$ runs over the spanning trees of $G$ and $F$ enumerates the spanning
two-forests of $G$ ($p_{F}$ denotes the sum of all external momenta flowing
into one of the components of $F$).

In section 4.2, we will use these formulas to count the sunrise integrals.

######  Example 47.

The graph polynomials of the bubble graph (figure 2.1) are, in general
kinematics,

$\mathcal{U}=x_{1}+x_{2}\quad\text{and}\quad\mathcal{F}=(x_{1}+x_{2})(x_{1}m_{1}^{2}+x_{2}m_{2}^{2})-p^{2}x_{1}x_{2}.$
(3.23)

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
$\mathbbm{V}(\mathcal{G})\mathrel{\mathop{:}}=\left\\{\mathcal{G}=0\right\\}$
to its class $[\mathcal{G}]$ in the Grothendieck ring
$K_{0}(\text{Var}_{\mathbbm{C}})$. This ring is the free Abelian group
generated by isomorphism classes $[X]$ of varieties over $\mathbbm{C}$, 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 $1=[\mathbbm{A}^{0}]$ 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 $\mathbbm{L}=[\mathbbm{A}^{1}]$ of the
affine line is called _Lefschetz motive_ and fulfils $\chi(\mathbbm{L})=1$.
For several polynomials $P_{1},\ldots,P_{n}$, we write
$\mathbbm{V}(P_{1},\ldots,P_{n})\mathrel{\mathop{:}}=\left\\{P_{1}=\cdots=P_{n}=0\right\\}$.

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.252525Such _linear reductions_ were first investigated by Stembridge
in [84] and have led, via the $c_{2}$-invariant [73] of Schnetz, to the
discovery of graph hypersurfaces that are not of mixed Tate type [17].  Let us
state such a relation explicitly, since our setting is slightly different than
usual: For us, the natural ambient space is the torus $\mathbbm{G}_{m}^{N}$
and not the affine plane $\mathbbm{A}^{N}$.

######  Lemma 48.

Let $A,B\in\mathbbm{C}[x_{1},\ldots,x_{N-1}]$ and consider the linear
polynomial $A+x_{N}B$. Then

$[\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}\left(A+x_{N}B\right)]=\mathbbm{L}\cdot\left[\mathbbm{G}_{m}^{N-1}\setminus\mathbbm{V}\left(A,B\right)\right]-[\mathbbm{G}_{m}^{N-1}\setminus\mathbbm{V}(A)]-[\mathbbm{G}_{m}^{N-1}\setminus\mathbbm{V}(B)]$
(3.24)

holds in the Grothendieck ring. In particular, the Euler characteristic is

$\chi\left(\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}\left(A+x_{N}B\right)\right)=-\chi\left(\mathbbm{G}_{m}^{N-1}\setminus\mathbbm{V}\left(A\cdot
B\right)\right).$ (3.25)

###### Proof.

Consider the hypersurface
$\mathbbm{V}(\mathcal{G})\subset\mathbbm{G}_{m}^{N}$, defined by
$\mathcal{G}\mathrel{\mathop{:}}=A+x_{N}B$, under the projection
$\pi\colon\mathbbm{G}_{m}^{N}\longrightarrow\mathbbm{G}_{m}^{N-1}$ that
forgets the last coordinate $x_{N}$.

As long as $AB\neq 0$, 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 $\mathbbm{G}_{m}$ 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 $\mathbbm{G}_{m}$. This fibration proves (3.24);
equivalently, we can write it via $[\mathbbm{V}(A\cdot
B)]=[\mathbbm{V}(A)]+[\mathbbm{V}(B)]-[\mathbbm{V}(A,B)]$ and
$[\mathbbm{G}_{m}]=\mathbbm{L}-1$ as

$[\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}\left(A+x_{N}B\right)]=[\mathbbm{G}_{m}]\left([\mathbbm{G}_{m}^{N-1}\setminus\mathbbm{V}(A)]+[\mathbbm{G}_{m}^{N-1}\setminus\mathbbm{V}(B)]\right)-\mathbbm{L}\cdot[\mathbbm{G}_{m}^{N-1}\setminus\mathbbm{V}(A\cdot
B)].$

Applying the Euler characteristic proves (3.25) due to
$\chi(\mathbbm{A}^{1})=1$ and $\chi(\mathbbm{G}_{m})=0$. ∎

######  Corollary 49.

Set
$\widetilde{\mathcal{U}}\mathrel{\mathop{:}}=\left.\mathcal{U}\right|_{x_{N}=1}$
and
$\widetilde{\mathcal{F}}\mathrel{\mathop{:}}=\left.\mathcal{F}\right|_{x_{N}=1}$.
Then

$\displaystyle\chi\left(\mathbbm{G}_{m}^{N}\setminus\\!\\!\mathbbm{V}(\mathcal{G})\right)$
$\displaystyle=\chi\left(\mathbbm{G}_{m}^{N-1}\setminus\\!\\!\mathbbm{V}(\widetilde{\mathcal{U}},\widetilde{\mathcal{F}})\right)-\chi\left(\mathbbm{G}_{m}^{N-1}\setminus\\!\\!\mathbbm{V}(\widetilde{\mathcal{U}})\right)-\chi\left(\mathbbm{G}_{m}^{N-1}\setminus\\!\\!\mathbbm{V}(\widetilde{\mathcal{F}})\right)$
(3.26)
$\displaystyle=-\chi\left(\mathbbm{G}_{m}^{N-1}\setminus\\!\\!\mathbbm{V}(\widetilde{\mathcal{U}}\cdot\widetilde{\mathcal{F}})\right).$
(3.27)

###### Proof.

Recall that $\mathcal{U}$ and $\mathcal{F}$ are homogeneous of degrees $L$ and
$L+1$, respectively (corollary 63). Since multiplication with
$x_{N}\in\mathbbm{G}_{m}$ 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
$x_{N}\neq 0$, this shows that
$[\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(\mathcal{G})]=[\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(\widetilde{\mathcal{U}}+x_{N}\widetilde{\mathcal{F}})]$
such that the claim is just a special case of lemma 48. ∎

######  Example 50.

Both graphs consisting of a pair of massless edges,

$G_{\mathrm{series}}=\vbox{\hbox{\includegraphics[scale={0.5}]{Graphs/serieslegs}}}\quad\text{and}\quad
G_{\mathrm{parallel}}=\vbox{\hbox{\includegraphics[scale={0.5}]{Graphs/parallellegs}}},$
(3.28)

with the external momentum $p$ such that $p^{2}\neq 0$, 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
$\mathcal{U}_{\mathrm{series}}=1$,
$\mathcal{F}_{\mathrm{series}}=-p^{2}(x_{1}+x_{2})$,
$\mathcal{U}_{\mathrm{parallel}}=x_{1}+x_{2}$,
$\mathcal{F}_{\mathrm{parallel}}=-p^{2}x_{1}x_{2}$ such that

$\mathcal{G}_{\mathrm{series}}=1-p^{2}(x_{1}+x_{2})\quad\text{and}\quad\mathcal{G}_{\mathrm{parallel}}=x_{1}+x_{2}-p^{2}x_{1}x_{2}.$

In both cases, the number of master integrals (3.10) is
$\mathfrak{C}\left(G\right)=-\chi(\mathbbm{G}_{m}\setminus\mathbbm{V}(\widetilde{\mathcal{U}}\widetilde{\mathcal{F}}))=\chi(\mathbbm{G}_{m}\cap\mathbbm{V}(\widetilde{\mathcal{U}}\widetilde{\mathcal{F}}))=\chi(\mathbbm{G}_{m}\cap\mathbbm{V}(1+x_{1}))=\chi(\left\\{-1\right\\})=1$
according to (3.27). ∎

Much more on the Grothendieck ring calculus of graph hypersurfaces
$\mathbbm{V}(\mathcal{U})$ can be found, for example, in [1] and [17]. These
techniques can be used to prove some general statements about the counts of
master integrals. Let us give just one example:

$G=\vbox{\hbox{\includegraphics[scale={0.6}]{Graphs/1scalesub}}}\qquad\mapsto\qquad
G^{\prime}=\vbox{\hbox{\includegraphics[scale={0.6}]{Graphs/edge}}}$

Figure 3.1: A 1-scale subgraph $\gamma$ of $G$ is replaced by a single edge in
$G^{\prime}$.

######  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 $G\setminus\gamma$.262626Such a graph
$\gamma$ is called _massless propagator_ or _$p$ -integral_.  Write
$G^{\prime}$ for the graph obtained from $G$ by replacing $\gamma$ with a
single edge (see figure 3.1), then

$\mathfrak{C}\left(G\right)=\mathfrak{C}\left(\gamma\right)\cdot\mathfrak{C}\left(G^{\prime}\right).$
(3.29)

###### 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
$\mathcal{U}_{G}=\mathcal{U}_{\gamma}\cdot\mathcal{U}_{G/\gamma}+\mathcal{F}_{\gamma}^{\prime}\cdot\mathcal{U}_{G\setminus\gamma}$
where we set
$\mathcal{F}_{\gamma}^{\prime}\mathrel{\mathop{:}}=\left.\mathcal{F}_{\gamma}\right|_{p^{2}=-1}$.
Going through the same considerations for $\mathcal{F}_{G}$ shows that

$\mathcal{G}_{G}=\mathcal{U}_{\gamma}\cdot\mathcal{G}_{G/\gamma}+\mathcal{F}_{\gamma}^{\prime}\cdot\mathcal{G}_{G\setminus\gamma}.$

Now label the edges in $\gamma$ as $1,\ldots,N_{\gamma}$ and rescale all
Schwinger parameters $x_{e}$ with $2\leq e\leq N_{\gamma}$ by $x_{1}$. Due to
the homogeneity of $\mathcal{U}_{\gamma}$ and $\mathcal{F}_{\gamma}$ from
corollary 63, we see that
$[\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(\mathcal{G}_{G})]=[\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(A+x_{1}B)]$
where $A=\widetilde{U}_{\gamma}\mathcal{G}_{G/\gamma}$ and
$B=\widetilde{\mathcal{F}}_{\gamma}^{\prime}\mathcal{G}_{G\setminus\gamma}$ in
terms of
$\widetilde{U}_{\gamma}\mathrel{\mathop{:}}=\left.\mathcal{U}_{\gamma}\right|_{x_{1}=1}$
and
$\widetilde{\mathcal{F}}_{\gamma}^{\prime}\mathrel{\mathop{:}}=\left.\mathcal{F}_{\gamma}^{\prime}\right|_{x_{1}=1}$.
Applying (3.25), we obtain a separation of variables:

$\displaystyle\chi\left(\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(\mathcal{G}_{G})\right)$
$\displaystyle=-\chi\left(\mathbbm{G}_{m}^{N-1}\setminus\mathbbm{V}(\widetilde{\mathcal{U}}_{\gamma}\widetilde{\mathcal{F}}_{\gamma}^{\prime}\mathcal{G}_{G/\gamma}\mathcal{G}_{G\setminus\gamma})\right)$
$\displaystyle=-\chi\left(\mathbbm{G}_{m}^{N_{\gamma}-1}\setminus\mathbbm{V}(\widetilde{\mathcal{U}}_{\gamma}\widetilde{\mathcal{F}}_{\gamma}^{\prime})\right)\cdot\chi\left(\mathbbm{G}_{m}^{N-N_{\gamma}}\setminus\mathbbm{V}(\mathcal{G}_{G/\gamma}\mathcal{G}_{G\setminus\gamma})\right)$
$\displaystyle=-\chi\left(\mathbbm{G}_{m}^{N_{\gamma}}\setminus\mathbbm{V}(\mathcal{G}_{\gamma})\right)\cdot\chi\left(\mathbbm{G}_{m}^{N-N_{\gamma}+1}\setminus\mathbbm{V}(\mathcal{G}_{G^{\prime}})\right).$

In the last line we used (3.27), upon noting the contraction-deletion formula
$\mathcal{G}_{G^{\prime}}=\mathcal{G}_{G/\gamma}+x_{0}\mathcal{G}_{G\setminus\gamma}$
in terms of the additional Schwinger parameter $x_{0}$ for the (massless) edge
that replaces $\gamma$ in $G^{\prime}$ (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
$\mathbbm{V}(\widetilde{\mathcal{F}}_{\gamma})=\mathbbm{V}(\widetilde{\mathcal{F}}_{\gamma}^{\prime})$,
as long as $p^{2}\neq 0$. Finally, recall (3.10). ∎

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

$\mathcal{I}_{G}(\nu)=\left.\mathcal{I}_{\gamma}(\nu_{\gamma})\right|_{p^{2}=-1}\cdot\mathcal{I}_{G^{\prime}}(\omega_{\gamma},\nu^{\prime})$
(3.30)

of the integrals of $\gamma$ and $G^{\prime}$. Here, we denote by
$\nu_{\gamma}$ and $\nu^{\prime}$ the indices corresponding to the edges in
$\gamma$ and outside $\gamma$, respectively, such that
$\nu=(\nu_{\gamma},\nu^{\prime})$. Note that the edge replacing $\gamma$ in
$G^{\prime}$ (see figure 3.1) 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 $\gamma$).

$\vbox{\hbox{\includegraphics[scale={0.6}]{Graphs/series}}}\xmapsto{(S)}\vbox{\hbox{\includegraphics[scale={0.6}]{Graphs/edge}}}\reflectbox{$\xmapsto{\reflectbox{\scriptsize$(P)$}}$}\vbox{\hbox{\includegraphics[scale={0.6}]{Graphs/parallel}}}$

Figure 3.2: The series (S) and parallel (P) operations consist of replacing a
sequential or parallel pair of massless edges with a single edge.

######  Corollary 52.

Let $G$ be a graph with a pair $\left\\{e,f\right\\}$ of massless edges in
series or in parallel. Then
$\mathfrak{C}\left(G\right)=\mathfrak{C}\left(G^{\prime}\right)$ where
$G^{\prime}$ is the graph obtained by replacing the pair with a single edge.
In other words, repeated application of the series-parallel operations from
figure 3.2 does not change the number of master integrals.

###### Proof.

Combine lemma 51 with example 50. ∎

##  4 Tools and examples

The Euler characteristic of a singular hypersurface can be computed
algorithmically via several methods.272727We 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.  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 section 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

$\chi(E)=\chi(B)\cdot\chi(F)$ (4.1)

is true in general and not restricted to the linear case. In section 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 section 4.3 we
discuss some available implementations of these algorithms in computer algebra
systems.

In the final section 4.4, we comment on the relation of our result to other
approaches in the physics literature.

###  4.1 Linearly reducible graphs

$\displaystyle P_{1}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.95}]{Graphs/propj}}}$
$\displaystyle P_{2}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.95}]{Graphs/propl}}}$
$\displaystyle P_{3}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.95}]{Graphs/propm}}}$
$\displaystyle P_{4}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.95}]{Graphs/propn}}}$
$\displaystyle P_{5}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.95}]{Graphs/propo}}}$
$\displaystyle P_{6}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.95}]{Graphs/propp}}}$
$\displaystyle P_{7}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.95}]{Graphs/propq}}}$

$\displaystyle F_{1}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.85}]{Graphs/ffu}}}$
$\displaystyle F_{2}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.85}]{Graphs/ffv}}}$
$\displaystyle F_{3}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.85}]{Graphs/ffw}}}$
$\displaystyle F_{4}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.85}]{Graphs/ffx}}}$
$\displaystyle F_{5}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.85}]{Graphs/ffy}}}$
$\displaystyle F_{6}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.85}]{Graphs/ffz}}}$
$\displaystyle F_{7}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.85}]{Graphs/fftheta}}}$
$\displaystyle F_{8}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.45}]{Graphs/KaspersGraph}}}$
$\displaystyle F_{9}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.45}]{Graphs/KleinerBoels}}}$

Figure 4.1: Some linearly reducible propagators ($P_{i}$) and form factors
($F_{i}$). All internal edges are massless, and the form factors have two
massless external legs ($p_{1}^{2}=p_{2}^{2}=0$) and one massive leg
$p_{3}^{2}\neq 0$, indicated by the label $m$.

If the polynomial $\mathbbm{V}(f)=a+x_{N}b$ 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 $\mathbbm{V}(f)$ (or its complement). Analogous
formulas also exist in the case of a variety $\mathbbm{V}(f_{1},\ldots,f_{n})$
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 [84, 73].

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 figure 4.1.

At the end of a full linear reduction, the class of
$\mathbbm{G}_{m}^{N}\setminus\mathbbm{V}(\mathcal{G})$ in the Grothendieck
ring is expressed as a polynomial in the Lefschetz motive $\mathbbm{L}$. To
get the Euler characteristic, one then merely needs to substitute
$\mathbbm{L}=[\mathbbm{A}^{1}]\mapsto\chi(\mathbbm{A}^{1})=1$.

######  Example 53.

Consider the $2$-loop propagator $\mathrm{WS}_{3}^{\prime}$ from figure 4.2.
Linear reductions give

$[\mathbbm{G}_{m}^{5}\setminus\mathbbm{V}(\mathcal{G}_{\mathrm{WS}_{3}^{\prime}})]=-\mathbbm{L}^{4}+5\mathbbm{L}^{3}-13\mathbbm{L}^{2}+21\mathbbm{L}-15$
(4.2)

in the Grothendieck ring. Substituting $\mathbbm{L}\mapsto 1$ shows that
$\mathfrak{C}\left(\mathrm{WS}_{3}^{\prime}\right)=3$ via (3.10).

In this way, we calculated the Euler characteristics for the graphs in figure
4.1, using an implementation of the linear reductions similar to the method of
Stembridge [84]. Our results are listed in table 1.

$G$ | $P_{1}$ | $P_{2}$ | $P_{3}$ | $P_{4}$ | $P_{5}$ | $P_{6}$ | $P_{7}$ | $F_{1}$ | $F_{2}$ | $F_{3}$ | $F_{4}$ | $F_{5}$ | $F_{6}$ | $F_{7}$ | $F_{8}$ | $F_{9}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathfrak{C}\left(G\right)$ | 16 | 10 | 10 | 10 | 10 | 15 | 22 | 1 | 4 | 5 | 4 | 5 | 20 | 24 | 12 | 13
Table 1: The number $\mathfrak{C}\left(G\right)$ of master integrals, computed
as the Euler characteristic 3.10, for the graphs in figure 4.1.

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
$\mathbbm{V}(\mathcal{U})$ was computed for all wheel graphs in [17]. It is
possible to adapt such calculations to our setting (where the ambient space is
$\mathbbm{G}_{m}^{N}$ instead of $\mathbbm{A}^{N}$). 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
figure 4.2), is

$\mathfrak{C}\left(\mathrm{WS}_{L}^{\prime}\right)=\mathfrak{C}\left(\mathrm{WS}_{L}^{\prime\prime}\right)=\frac{L(L-1)}{2}.$
(4.3)

$\displaystyle\mathrm{WS}_{3}^{\prime}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.5}]{Graphs/Wheel2}}}$
$\displaystyle\mathrm{WS}_{4}^{\prime}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.5}]{Graphs/Wheel3}}}$
$\displaystyle\mathrm{WS}_{5}^{\prime}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.5}]{Graphs/Wheel4}}}$
$\displaystyle\mathrm{WS}_{L}^{\prime}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.5}]{Graphs/WheelL}}}$
$\displaystyle\mathrm{WS}_{4}^{\prime\prime}$
$\displaystyle=\reflectbox{$\vbox{\hbox{\includegraphics[scale={1}]{Graphs/Wheel3sp}}}$}$
$\displaystyle\mathrm{WS}_{5}^{\prime\prime}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={1}]{Graphs/Wheel4sp}}}$
$\displaystyle\mathrm{WS}_{L}^{\prime\prime}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.5}]{Graphs/WheelLsp}}}$
Figure 4.2: The propagator graphs $\mathrm{WS}_{L}^{\prime}$
($\mathrm{WS}_{L}^{\prime\prime}$) with $L-1$ loops are obtained from cutting
a rim (spoke) of the wheel $\mathrm{WS}_{L}$ with $L$ loops.

The proof of this and related results will be presented elsewhere.

###  4.2 Sunrise graphs

In [42], the number of master integrals was computed for all sunrise
integrals; based on a Mellin-Barnes representation and the _differential
reduction_ [41, 19] of an explicit solution in terms of Lauricella
hypergeometric functions. To our knowledge, this has hitherto been the only
non-trivial282828We consider families that arise simply by duplication of
massless propagators, like those shown in [19, figure 2], as trivial (due to
corollary 52).  infinite family of Feynman integrals with explicitly known
master integral counts. Their first result can be phrased as292929Beware that
the number $2^{L+1}-L-2$ given in [42, 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.

$\mathrm{S}_{1}=\vbox{\hbox{\includegraphics[scale={0.6}]{Graphs/Sunrise1}}}$
$\mathrm{S}_{2}=\vbox{\hbox{\includegraphics[scale={0.6}]{Graphs/Sunrise2}}}$
$\mathrm{S}_{3}=\vbox{\hbox{\includegraphics[scale={0.6}]{Graphs/Sunrise3}}}$
$\mathrm{S}_{L}=\vbox{\hbox{\includegraphics[scale={0.6}]{Graphs/SunriseL}}}$

Figure 4.3: The sunrise graphs $\mathrm{S}_{L}$ with $L$ loops.

######  Proposition 55.

The $L$-loop sunrise graph $\mathrm{S}_{L}$ from figure 4.3 with $L+1$ non-
zero masses (and non-exceptional external momentum) has
$\mathfrak{C}\left(\mathrm{S}_{L}\right)=2^{L+1}-1$ master integrals.

We will now demonstrate that this result can be obtained from a
straightforward computation of the Euler characteristic, according to
corollary 37.

###### Proof.

The graph polynomials (3.22) for the sunrise graph are

$\mathcal{U}=\sum_{i=1}^{L+1}\prod_{j\neq
i}x_{j}=\left(\prod_{i=1}^{L+1}x_{i}\right)\left(\sum_{i=1}^{L+1}\frac{1}{x_{i}}\right)\quad\text{and}\quad\mathcal{F}=(-p)^{2}\prod_{i=1}^{L+1}x_{i}+\mathcal{U}\sum_{i=1}^{L+1}x_{i}m_{i}^{2}.$
(4.4)

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 $\chi(\mathbbm{G}_{m}^{L})=0$. We
thus obtain

$(-1)^{L}\mathfrak{C}\left(\mathrm{S}_{L}\right)=\chi\left(\mathbbm{G}_{m}^{L}\setminus\mathbbm{V}\left(1+\sum_{i=1}^{L}x_{i}^{-1}\right)\right)+\chi\left(\mathbbm{G}_{m}^{L}\setminus
X^{L}_{p^{2}}\right),$ (4.5)

where we introduced the notation

$X^{L}_{p^{2}}\mathrel{\mathop{:}}=\mathbbm{V}\left(-p^{2}+\Big{[}m_{L+1}^{2}+\sum_{i=1}^{L}m_{i}^{2}x_{i}\Big{]}\cdot\Big{[}1+\sum_{i=1}^{L}x_{i}^{-1}\Big{]}\right)\subset\mathbbm{G}_{m}^{L}.$
(4.6)

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

$\mathfrak{C}\left(\mathrm{S}_{L}\right)=1+(-1)^{L}\cdot\chi\left(\mathbbm{G}_{m}^{L}\setminus
X^{L}_{p^{2}}\right).$ (4.7)

Now let us consider the projection
$\pi\colon\mathbbm{G}_{m}^{L}\longrightarrow\mathbbm{G}_{m}^{L-1}$ that
forgets $x_{L}$. Set
$y\mathrel{\mathop{:}}=m_{L+1}^{2}+\sum_{i<L}m_{i}^{2}x_{i}$ and
$z\mathrel{\mathop{:}}=1+\sum_{i<L}x_{i}^{-1}$, such that
$X_{p^{2}}^{L}=\left\\{x_{L}p^{2}=(1+zx_{L})(y+m_{L}^{2}x_{L})\right\\}\subset\mathbbm{G}_{m}^{L}$.
We note that the discriminant $D$ of this quadric in $x_{L}$ factorizes into

$D=(m_{L}^{2}-p^{2}+yz)^{2}-4m_{L}^{2}yz=\left(yz-[p+m_{L}]^{2}\right)\cdot\left(yz-[p-m_{L}]^{2}\right),$
(4.8)

such that
$\mathbbm{G}_{m}^{L-1}\subset\mathbbm{V}(D)=X^{L-1}_{(p+m_{L})^{2}}\mathbin{\dot{\cup}}X^{L-1}_{(p-m_{L})^{2}}$
is the disjoint union of two hypersurfaces.303030We assume $p^{2}\neq 0$ and
$m_{L}^{2}\neq 0$, which guarantees that $(p+m_{L})^{2}\neq(p-m_{L})^{2}$.
Since the factors are related to the $(L-1)$-loop sunrise by (4.7), we find

$\chi(\mathbbm{V}(D))=2\cdot(-1)^{L}\cdot(\mathfrak{C}\left(\mathrm{S}_{L-1}\right)-1).$
(4.9)

Over a point $x^{\prime}\in X^{L-1}_{p\pm m_{L}}\subset\mathbbm{V}(D)$ in the
discriminant, the fibre of $\pi^{-1}(x^{\prime})$ has precisely one solution
$(x^{\prime},x_{L})$ in $X^{L}_{p^{2}}$, determined by
$x_{L}=-y/[m_{L}(m_{L}\pm p)]$:

$\left[(\mathbbm{G}_{m}^{L}\setminus
X^{L}_{p^{2}})\cap\mathbbm{V}(D)\right]=([\mathbbm{G}_{m}]-1)\cdot[\mathbbm{V}(D)].$
(4.10)

If $D(x^{\prime})\neq 0$ is non-zero and also $yz\neq 0$, then the fibre
$\pi^{-1}(x^{\prime})$ has precisely two distinct solutions $x_{L}$ in the
quadric $X_{p^{2}}^{L}$. Hence,
$\chi(\pi^{-1}(x^{\prime}))=\chi(\mathbbm{G}_{m})-2=-2$ and thus

$\chi\left((\mathbbm{G}_{m}^{L}\setminus
X_{p^{2}}^{L})\setminus\\!\\!\mathbbm{V}(yzD)\right)=-2\chi\left(\mathbbm{G}_{m}^{L-1}\setminus\\!\\!\mathbbm{V}(yzD)\right)=2\chi(\mathbbm{V}(D))+2\chi(\mathbbm{V}(yz)),$
(4.11)

where used that $\mathbbm{V}(D)\cap\mathbbm{V}(yz)=\emptyset$ for non-
exceptional values of $p^{2}$, such that $(p\pm m_{L})^{2}\neq 0$ 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
$X_{p^{2}}^{L}=\left\\{x_{L}p^{2}=(1+zx_{L})m_{L}^{2}x_{L}\right\\}$ is
$x_{L}=0\notin\mathbbm{G}_{m}$; 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 $p^{2}\neq m_{L}^{2}$):

$[(\mathbbm{G}_{m}^{L}\setminus
X_{p^{2}}^{L})\cap\mathbbm{V}(yz)]=([\mathbbm{G}_{m}]-2)\cdot[\mathbbm{V}(yz)]+[\mathbbm{V}(y)]+[\mathbbm{V}(z)].$
(4.12)

We can now combine (4.10)–(4.12) via
$[Y]=[Y\cap\mathbbm{V}(D)]+[Y\cap\mathbbm{V}(yz)]+[Y\setminus\mathbbm{V}(D\cdot
yz)]$ for $Y=\mathbbm{G}_{m}^{L}\setminus X_{p^{2}}^{L}$ into the reduction
formula

$\chi(\mathbbm{G}_{m}^{L}\setminus
X^{L}_{p^{2}})=\chi(\mathbbm{V}(D))+\chi(\mathbbm{V}(y))+\chi(\mathbbm{V}(z))=2\cdot(-1)^{L}\cdot\mathfrak{C}\left(\mathrm{S}_{L-1}\right).$
(4.13)

Here, we inserted (4.9) and used
$\chi(\mathbbm{V}(y))=\chi(\mathbbm{V}(z))=-\chi(\mathbbm{G}_{m}^{L-1}\setminus\mathbbm{V}(z))=(-1)^{L}$,
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

$\mathfrak{C}\left(\mathrm{S}_{L}\right)=2\mathfrak{C}\left(\mathrm{S}_{L-1}\right)+1,$

which is obviously solved by the claimed
$\mathfrak{C}\left(\mathrm{S}_{L}\right)=2^{L+1}-1$. It merely remains to
verify the base case $L=1$, and indeed,
$\mathfrak{C}\left(\mathrm{S}_{1}\right)=1+\chi(X_{p^{2}}^{1})=3$ follows
easily from (4.7) since
$X_{p^{2}}^{1}=\left\\{x_{1}p^{2}=(1+x_{1})(m_{2}^{2}+m_{1}^{2}x_{1})\right\\}\subset\mathbbm{G}_{m}$
consists of precisely two points. ∎

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
[42]:313131The additional term $-\delta_{0,L-R}$ in [42, equation (4.13)]
subtracts a _reducible_ integral that can be attributed to a subtopology. Our
counting, however, accounts for all master integrals.

######  Proposition 56.

The $L$-loop sunrise graph with $R\leq L$ non-zero masses, $L+1-R$ vanishing
masses and non-exceptional external momentum, has
$\mathfrak{C}\left(\mathrm{S}_{L}\right)=2^{R}$ master integrals.

###### Proof.

By corollary 52, we may replace all massless edges by a single (massless) edge
without changing the number of master integrals; hence we can assume $L=R\geq
1$ (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, $\chi(\mathbbm{V}(y))=0$ vanishes
because $y=\sum_{i<L}m_{i}^{2}x_{i}$ has become homogeneous in $x$ such that
$[\mathbbm{V}(y)]=[\mathbbm{G}_{m}]\cdot[\mathbbm{V}(y)\cap\left\\{x_{1}=1\right\\}]$.
Therefore, (4.13) takes the form $\chi(\mathbbm{G}_{m}^{L}\setminus
X_{p^{2}}^{L})=(-1)^{L}\cdot\left(2\mathfrak{C}\left(\mathrm{S}_{L-1}\right)-1\right)$
and yields, via (4.7), the recursion

$\mathfrak{C}\left(\mathrm{S}_{L}\right)=2\mathfrak{C}\left(\mathrm{S}_{L-1}\right).$

We are done after verifying the base case: Indeed,
$\mathfrak{C}\left(\mathrm{S}_{1}\right)=1+\chi(X_{p^{2}}^{1})=2$ 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 $\mathbbm{G}_{m}$. ∎

###  4.3 General algorithms

The computer algebra system Macaulay2 [33] provides the function Euler in the
package CharacteristicClasses. It implements the algorithm of [37] 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

$\displaystyle[\mathbbm{A}^{N}\setminus\mathbbm{V}(x_{1}\cdots
x_{N}\mathcal{G})]$
$\displaystyle=[\mathbbm{P}^{N}\setminus\mathbbm{V}(x_{1}\cdots
x_{N}\tilde{\mathcal{G}})]-[\mathbbm{P}^{N-1}\setminus\mathbbm{V}(x_{1}\cdots
x_{N}\mathcal{F})]$ (4.14)
$\displaystyle=[\mathbbm{P}^{N}\setminus\mathbbm{V}(x_{0}x_{1}\cdots
x_{N}\tilde{\mathcal{G}})]$ (4.15)