On the status of expansion by regions
Abstract
We discuss the status of expansion by regions, i.e. a wellknown strategy to obtain an expansion of a given multiloop Feynman integral in a given limit where some kinematic invariants and/or masses have certain scaling measured in powers of a given small parameter. Using the Lee–Pomeransky parametric representation, we formulate the corresponding prescriptions in a simple geometrical language and make a conjecture that they hold even in a much more general case. We prove this conjecture in some partial cases.
1 Introduction
If a given Feynman integral depends on kinematic invariants and masses which essentially differ in scale, a very natural and often used idea is to expand it in powers of a given small parameter. As a result, the integral can be written as a series of factorized quantities which are simpler than the original integral itself and it can be substituted by a sufficiently large number of terms of such an expansion. The strategy of expansion by regions [1] (see also [2] and Chapter 9 of [3]) introduced and applied in the case of threshold expansion [1] is a strategy to obtain an expansion of a given multiloop Feynman integral in a given limit specified by scalings of kinematic invariants and/or masses characterized by powers of a given small parameter of expansion. For example, for a limit with two variables, \(q^2\) and \(m^2\), where \(m^2\ll q^2\) and the parameter of expansion is \(m^2/q^2\), one analyzes various regions in a given integral over loop momenta and, in every region, expands the integrand, i.e. a product of propagators, in parameters which are there small. Then the integration in the integral with so expanded propagators is extended to the whole domain of the loop momenta and, finally, one obtains an expansion of the given integral as the corresponding sum over the regions.
Although this strategy certainly looks suspicious for mathematicians it was successfully applied in numerous calculations. It has the status of experimental mathematics and should be applied with care, starting, first, from oneloop examples, by checking results by independent methods. Using the analysis in a toy example of an expansion of a onedimensional integral [4] presented in [2], Jantzen [5] provided detailed explanations of how this strategy works by starting from regions determined by some inequalities and covering the whole integration space of the loop momenta, then expanding the integrand and then extending integration and analyzing all the pieces which are obtained, with the hope that ‘readers would be convinced that the expansion by regions is a wellfounded method’. However, this interesting and instructive analysis can hardly be considered as a base of mathematical proofs. Let us realize that we are dealing with dimensionally regularized Feynman integrals, i.e. integrals over loop momenta of spacetime dimension \(d=42\varepsilon \) which is considered as a complex regularization parameter. Therefore it is not clear in which sense inequalities and limits for these integrals are understood because the integrands and the integrals are functions of ddimensional loop and/or external momenta so that they should be treated like some algebraic objects rather than usual functions in integer numbers of dimensions. In practice, one usually does not bother about such problems and performs calculations implicitly applying some axioms for the integration procedure, and a consistency of the whole calculation checked in some way looks quite sufficient.
The parametric representation in the case where propagators enter with general powers \(a_i\) can be obtained from (1) by including the overall factor \(\varGamma (ah d/2)/(\prod _i \varGamma (a_i))\), with \(a=\sum a_i\), and the product \(\prod _i x_l^{a_i1}\) in the integrand. The representation with negative integer indices \(a_i= \ n_i\) can be obtained from this one by taking the limit \(a_i \rightarrow  \ n_i\) where the pole at \(a_i =  \ n_i\) arising from \(x_i^{a_i1}\) is cancelled by the pole of \(\varGamma (a_i)\) in the denominator. However, to make the presentation simpler, we will consider only the case of all the indices equal to one.
The expansion by regions was also formulated in the language of the corresponding parametric integrals [6] (see also [2] and Chapter 9 of [3]). One can consider quite general limits for a Feynman integral which depends on external momenta \(q_i\) and masses and is a scalar function of kinematic invariants \(q_i \cdot q_j\) and squares of masses and assume that each kinematic invariant and a mass squared has certain scaling \(\rho ^{\kappa _i}\) where \(\rho \) is a small parameter. A nontrivial point when applying the strategy of expansion by regions, either in momentum space or in parametric representation, is to understand which regions are relevant to a given limit. For example, for the threshold expansion, these are hard, potential, soft and ultrasoft regions, as it was claimed in [1] and further confirmed in practice in multiple calculations.
A systematical procedure to find relevant regions was developed in Ref. [7] using Feynman parametric representation (1) and geometry of polytopes connected with the basic functions U and F. This procedure was implemented as a public computer code asy.m [7] which is now included in the code FIESTA [8]. Using this code one can not only find relevant regions but also obtain the corresponding terms of expansion and evaluate numerically coefficients at powers and logarithms of the given expansion parameter. Although there is no mathematical justification of this procedure, numerous applications have shown that the code asy.m works consistently at least in the case where all the terms in the function F are positive. An attempt to extend this procedure and the corresponding code asy.m to some cases where some terms of the function F are negative was made in Ref. [9] where it was explained how potential and Glauber regions can be revealed.^{1}
We find it very natural to use Feynman parametric representations and the geometrical description of expansion introduced in Ref. [7] to mathematically prove expansion by regions. In fact, for the moment, only an indirect proof of expansion by regions, for limits typical of Euclidean space (where one has two different regions which can be called large and small) exists, – see the proof for the offshell largemomentum limit in [11] and Appendix B.2 of [2]. The point is that, for limits typical of Euclidean space (for example, the offshell largemomentum limit or the largemass limit), one can write down the corresponding expansion in terms of a sum over certain subgraphs of a given graph [12, 13, 14], and there is a correspondence between these subgraphs and their loop momenta which are considered large while the other loop momenta are considered small.
We believe that the prescriptions of expansion by regions hold also for integrals (5) with a general polynomial P with positive coefficients and not only for polynomials of the form (6) where the two terms are basic functions for some graph. The goal of our paper is, at least, to formulate prescriptions of expansion by regions for general polynomials with positive coefficients in an unambiguous mathematical language, to justify how terms of the leading order of expansion are constructed and to draw attention of both physicists and mathematicians who might find it interesting to prove it in a general order of expansion.
In the next section we use the geometrical description of expansion by regions on which the code asy.m [7] was based. In this paper, we consider limits with two scales where one introduces a small parameter as their ratio. Let us emphasize that this can be various important limits which are typical of Minkowski space, for example, the Sudakov limit or the Regge limit (with \(t \ll s\) where s and t are Mandelstam variables.) In this description, regions correspond to special facets of the Newton polytope associated with the product of UF of the two basic polynomials in (1). We immediately switch here to prescriptions based on the LP [15] parametric representation (5) and formulate prescriptions for a general polynomial with positive coefficients, rather than polynomial (6). Therefore, these prescriptions will be based on facets of the corresponding Newton polytope. Of course, prescriptions based on representation (5) are algorithmically preferable because the degree of the sum of the two basic polynomials is smaller than the degree of their product UF (used in asy.m) so that looking for facets of the corresponding Newton polytope becomes a simpler procedure.^{2} Therefore, the current version of the code asy.m included in FIESTA [8] is now based on this more effective procedure.
Since we are oriented at mathematical proofs we want to be mathematically correct. Let us realize that up to now we did not discuss whether integral (1) or (5) can be understood as a convergent integral at some values of d. Let us keep in mind a situation where a Feynman integral is both ultravioletly and infrared divergent so that increasing Re\((\varepsilon )\) regulates ultraviolet divergences and decreasing Re\((\varepsilon )\) regulates infrared divergences. Such situations are not exotic at all. However, in practical calculations of Feynman integrals one usually does not bother about the existence of such a convergence domain and/or tries to define the given integral in some other way if such a domain does not exist. Well, after calculation are made, one has a result which is a function of \(d=4=2\varepsilon \) usually presented by first terms of a Laurent expansion near \(\varepsilon =0\), and such a result is well defined!
In Sect. 3, we refer to some papers where attempts to define Feynman integrals before calculations are made and comment on how Feynman integrals are understood when they are evaluated. Then we turn to parametric representation (5) in Sect. 4 and explain how we can define this representation in terms of convergent integrals. In Sect. 5, we explicitly show that, in the case of Feynman integrals, i.e. where the polynomial is given by (6), with two basic functions constructed for a given graph, the two kinds of prescriptions based either on the Feynman parametric representation or on the LP parametric representation are equivalent.
Equipped with our definition based on analytic regularization, we then turn in Sect. 6 to the main conjecture, prove it it in the leading order in a special situation and analyze it in the leading order of expansion in the general situation. In Sect. 7, we prove the main conjecture in the simple case, where only one facet contributes. In Sect. 8, we summarize our results and discuss perspectives.
2 The main conjecture
The symbol \(\sim \) in (8) is the standard symbol of an asymptotic expansion. As it will be explained shortly, every term in the righthand side of (8) is homogeneous with respect to the expansion parameter, t, so that one can sort out various terms of the expansion according to their order in t and construct the sum of first terms, up to order \(t^N\). Then, according to the definition of the asymptotic expansion, the corresponding remainder defined as the difference between the initial integral and these first terms, is of order \(o(t^n)\).
The contribution of a given essential facet is defined by the change of variables \(x_i \rightarrow t^{r^{\gamma }_i} x_i\) in the integral (5) and expanding the resulting integrand in powers of t. This leads to the following definitions.

An operator \(M_{\gamma }\) can equivalently be defined by introducing a parameter \(\rho _{\gamma }\), replacing \(x_i\) by \(\rho ^{r^{\gamma }_{i}} x_i\) , pulling an overall power of \(\rho _{\gamma }\), expanding in \(\rho _{\gamma }\) and setting \(\rho _{\gamma }=1\) in the end. It is reasonable to use this variant when one needs to deal with products of several operators \(M_{\gamma }\).
 The leading order term of a given facet \(\gamma \) corresponds to the leading order of the operator \(M^0_{\gamma } \):$$\begin{aligned}&\int _0^\infty \ldots \int _0^\infty [ M^0_{\gamma } \left( P(x_1,\ldots ,x_n,t)\right) ^{\delta } ] \mathrm{d}x_1 \ldots \mathrm{d}x_{n} \nonumber \\&\quad =t^{L(\gamma )\delta +\sum _{i=1}^n r^{\gamma }_i}\nonumber \\&\qquad \times \int _0^\infty \ldots \int _0^\infty \, (P^{\gamma }_0(x_1,\ldots ,x_n) )^{\delta } \mathrm{d}x_1 \ldots \mathrm{d}x_n . \nonumber \\ \end{aligned}$$(13)
 In fact, with the above definitions, we can write down the equation of the hyperplane generated by a given facet \(\gamma \) as follows$$\begin{aligned} w_{n+1}=\sum _{i=1}^n r^{\gamma }_i w_i+L(\gamma ) . \end{aligned}$$(14)
 Let us agree that the action of an operator \(M_\gamma \) on an integral reduces to the action of \(M_\gamma \) on the integrand described above. Then we can write down the expansion in a shorter way,$$\begin{aligned} G(t,\varepsilon )\sim \sum _{\gamma } M_{\gamma } G(t,\varepsilon ) \end{aligned}$$(15)

In the usual Feynman parametrization (1), the expansion by regions in terms of operators \(M_{\gamma }\) is formulated in a similar way, and this is exactly how it is implemented in the code asy.m [7]. The expansion can be written in the same form (15) but the operators \(M_{\gamma }\) act on the product of the two basic polynomials U and F raised to certain powers present in (1). Now, each of the two polynomials is decomposed in the form (10) and so on.

It is well known that dimensional regularization might be not sufficient to regularize individual contributions to the asymptotic expansion. A natural way to overcome this problem is to introduce an auxiliary analytic regularization, i.e. to introduce additional exponents \(\lambda _i\) to power of the propagators. This possibility exists in the code asy.m [7] included in FIESTA [8]. One can choose these additional parameters in some way and obtain a result in terms of an expansion in \(\lambda _i\) followed by an expansion in \(\varepsilon \). If an initial integral can be well defined as a function of \(\varepsilon \) then the cancellation of poles in \(\lambda _i\) serves as a good check of the calculational procedure, so that in the end one obtains a result in terms of a Laurent expansion in \(\varepsilon \) up to a desired order. We will systematically exploit analytic regularization below for various reasons.

We consider the case of two kinematic parameters for simplicity. In the general case, with several kinematic invariants \(q_i \cdot q_j\) and squares of masses, where each of these variables, \(s_i\), has certain scaling, i.e. \(s_i \rightarrow \rho ^{\kappa _i}s_i\), with \(\rho \) a small parameter, one can formulate similar prescriptions. Then, the expansion is given by a similar sum over facets of a Newton polynomial which is determined for each choice of the variables \(s_i\). (This is how the code asy.m [7] works in this case.)
3 Convergence and sector decompositions
When formulating the main conjecture in the previous section we did not discuss conditions under which integral (5) is convergent. As is well known, dimensional regularization is introduced for Feynman integrals, i.e. when polynomial is given by (6), in order that various divergences become regularized so that the integral becomes a meromorphic function of the regularization parameter \(\varepsilon \). Then one can deal with the regularized quantity where divergences manifest themselves as various poles at \(\varepsilon =0\). However, a given Feynman integral can have both ultraviolet divergences which can be regularized by increasing Re\((\varepsilon )\) and (offshell or onshell) infrared^{3} as well as collinear divergences which can be regularized by decreasing Re\((\varepsilon )\). Then, typically, there is no domain of \(\varepsilon \) where the integral is convergent. In numerous calculations, one does not bother about this problem. Rather, various methods of evaluating Feynman integrals are applied and in the end of a calculation, one arrives at a result which looks like several terms of a Laurent expansion in \(\varepsilon \).
Let us now remember that there is a mathematical definition of a dimensionally regularized Feynman integral in the case where both ultraviolet and offshell infrared divergences are present. Speer defines [16] such an integral^{4} as an analytic continuation of the corresponding dimensionally and analytically regularized integral, i.e. with all propagators \(1/(p_l^2+m^2i 0)\) replaced by \(1/(p_l^2+m^2i 0)^{1+\lambda _l}\), from a domain of analytic regularization parameters \(\lambda _l\) where the integral is absolutely convergent. Moreover, Speer proves explicitly that such a domain of parameters \(\lambda _l\) is nonempty.
Both Hepp [17] and Speer [16] sectors are introduced globally, i.e. once and forever. In fact, the Speer sectors^{5} correspond to oneparticleirreducible subgraphs and their infrared analogues. These sector decompositions were successfully applied for proving various results on regularized and renormalized Feynman integrals.
Global sector decompositions for Feynman integrals with onshell infrared divergences and/or collinear divergences are unknown. Binoth and Heinrich were first to construct recursive sector decompositions [22, 23, 24]. The first step in their procedure was to introduce the set of primary sectors corresponding to the set of the lines of a given graph, \(\varDelta _l = \left\{ (x_1,\ldots ,x_n) \left. \right x_i\le x_l,\; i\ne l\right\} \), the sector variables are introduced by \(x_i = y_i x_l, \; i\ne l\). The integration over \(x_l\) is then taken due to the delta function in the integrand and one arrives at an integral over unit hypercube over \(y_i\).
There are several public codes where various strategies of recursive sector decompositions are implemented [8, 25, 26, 27, 28]. In the case, where the basic polynomial F is positive, Bogner and Weinzierl [28] presented first examples of strategies which terminate, i.e. provide, after a finite number of steps, a desired factorization (18) of the integrand in each final sector.
When recursive sector decompositions are applied in practice, using a code for numerical evaluation, one does not care that, generally, there is no domain of parameter \(\varepsilon \) where initial integral (1) is convergent. However, in the case of Euclidean external momenta, one could remember about the Speer’s definition [16] and use it to prove that this naive way is right. Indeed, starting from the analytically regularized parametric representation (16) and using some terminating strategy one can arrive at a factorization in final sector of the form (17), where the exponents of the final sector variables \(a_i+b_i\varepsilon \) obtain an additional linear combination of parameters \(\lambda _i\). Then one can use the same procedure of making explicit poles in the regularization parameters by a generalization of (18). As a result one can observe that starting from the Speer’s domain of parameters \(\lambda _i\) where the given parametric integral is convergent one can continue analytically all the terms resulting from the sector decomposition and the procedure of extracting poles just by setting all the \(\lambda _i\) to zero.
In the next section, we will first derive conditions of convergence of integral (5) and then conditions of convergence of integral (19). We will prove that there exists a nonempty domain of \(\lambda _i\) where the integral is convergent. Then, similarly to how this was done by Speer for Feynman integrals at Euclidean external momenta [16], we will formulate a definition of integrals (5) at general \(\delta =2\varepsilon \) which, in particular, gives a definition of dimensionally regularized Feynman integrals with possible onshell infrared and collinear divergences.
4 Convergence of the LP representation
Let \(\pi (S)\) be the projection of the set S on the hyperplane \(w_{n+1}=0\), let \(\pi ({\mathcal {N}}_P)\) be the projection of \({\mathcal {N}}_P\) on the same hyperplane, and \(\pi (\gamma )\) be the corresponding projections of essential facets. It turns out that it is reasonable to turn to a more general family of integrals (5) by assuming that P is given by (7) where the set S is a finite set of rational numbers. The following proposition holds.
Proposition 1
The integral (5) is convergent if and only if \(A=\left( \frac{1}{\delta },\ldots ,\frac{1}{\delta }\right) \in {\mathbb R}^{n}\) is inside \(\pi ({\mathcal {N}}_P)\).
Proof
Let us assume that the statement is not true, i.e. that the integral G(1) is convergent but the point A is outside the interior of the polytope \(\pi ({\mathcal {N}}_P)\). Let us, first, consider the case where A is outside \(\pi ({\mathcal {N}}_P)\). Since \(\pi ({\mathcal {N}}_P)\) is a convex set, there exist a plane \(p_1w_1+\cdots +p_n w_n+p_0=0\) such that \(\pi ({\mathcal {N}}_P)\) and A are on its opposite sides. One can choose a plane such that all \(p_i\ne 0\). Let \(p_1w_1+\cdots +p_n w_n+p_0<0\) for all the points w of the polytope and let \(p_1\frac{1}{\delta }+\cdots +p_n \frac{1}{\delta }+p_0>0\), or \(p_1+\cdots +p_n>\delta p_0\).
Let us now consider the case, where the point A is at the boarder of the set \(\pi ({\mathcal {N}}_P)\). Since G(1) is a continuous function of \(\delta \) then the convergence of the integral as some \(\delta \) leads to the convergence in a sufficiently small vicinity, i.e. once can find an external point \((\frac{1}{\mu },\ldots ,\frac{1}{\mu })\) of the polytope \(\pi ({\mathcal {N}}_P)\), where the integral \(\int _0^\infty \ldots \int _0^\infty ({\widetilde{P}}(x))^{\mu } \mathrm{d}x_1\ldots \mathrm{d}x_{n}\) is convergent so that we come to a contradiction. \(\square \)
 (a)Let \(Q_1\) and \(Q_2\) be polynomials with positive coefficients. If the integralis divergent then the integrals$$\begin{aligned} \int _0^\infty \ldots \int _0^\infty \left( Q_1(x)+Q_2(x)\right) ^{\delta } \mathrm{d}x_1\ldots \mathrm{d}x_{n} \end{aligned}$$are also divergent.$$\begin{aligned} \int _0^\infty \ldots \int _0^\infty \left( Q_i(x)\right) ^{\delta } \mathrm{d}x_1\ldots \mathrm{d}x_{n} \end{aligned}$$
 (b)If a polynomial Q(x) with positive coefficients contains terms \(x_1^{w_1}\ldots x_n^{w_n}\) and \(x_1^{u_1} \ldots x_n^{u_n}\), then the convergence of the following two integrals is equivalent:and$$\begin{aligned} \int _0^\infty \ldots \int _0^\infty \left( Q(x)\right) ^{\delta } \mathrm{d}x_1\ldots \mathrm{d}x_{n} \end{aligned}$$where \(\beta _i = w_i+z(u_iw_i)\), \(z\in [0,1]\),$$\begin{aligned} \int _0^\infty \ldots \int _0^\infty \left( Q(x)+x_1^{\beta _1}\ldots x_n^{\beta _n}\right) ^{\delta } \mathrm{d}x_1\ldots \mathrm{d}x_{n} , \end{aligned}$$
Suppose now that the condition of Proposition 1 does not hold, i.e. the point \(A=\left( \frac{1}{\delta },\ldots ,\frac{1}{\delta }\right) \) is not inside \(\pi ({\mathcal {N}}_P)\). Then we introduce a general analytic regularization and turn to integral (19). We have
Proposition 2
The integral (19) is convergent if the point \(\left( \frac{1+\lambda _1}{\delta },\ldots ,\frac{1+\lambda _n}{\delta }\right) \in {{\mathbb {R}}}^{n}\) is inside \(\pi ({\mathcal {N}}_P)\).
Proof
\(\left. i=1,\ldots ,n; v_{n+1}=w_{n+1} \right\} \). Using the convex property of the polytopes \({\mathcal {N}}_P\) and \({\mathcal {N}}_{{\bar{P}}}\) we arrive at the desired statement.
The function \({\bar{P}}\) is no longer a polynomial but we assume this possibility in Proposition 1. Now, it is clear that we can adjust parameters \(\lambda _i\) using a blowingdown or blowing up (with \(1<\lambda _i<0\) or \(\lambda _i>1\)) to provide convergence by putting \(\frac{1+\lambda _i}{\delta }\) between the left and the right values of the ith coordinates of \(\pi ({\mathcal {N}}_P)\). \(\square \)
Let us formulate this statement as an analogue of the Speer’s theorem [16].
Corollary 1
The integral (19) is an analytic function of parameters \(\lambda _i\) in a nonempty domain.
This domain exists for any given \(\delta \equiv 2\varepsilon \). Now we define the integral (5) as a function of \(\varepsilon \) as the analytic continuation of the integral (19) from the convergence domain of parameters \(\lambda _i\) to the point where all \(\lambda _i=0\) by referring to sector decompositions in the same way as it was outlined in the previous section.
It suffices then to explain how sector decompositions can be introduced for the LP integrals. If we are dealing with a Feynman integral, with Eq. (6), we turn to (1) so that we can apply standard terminating strategies. If this is a more general integral, with a positive polynomial P one can reduce it to integrals over unit hypercubes, for example, by the following straightforward procedure. Make the variable change \(x_i=y_i/(1y_i)\) to arrive at an integral over a unit hypercube. In order to avoid singularities near \(y_i=1\), decompose each integration over \(y_i\) in two parts: from 0 to 1 / 2 and 1 / 2 to 1 and change variables again in order to have integrations over unit hypercubes. As a result, one arrives at integrals to which terminating strategies [28] can be applied.
5 Equivalence of the new and the old prescriptions
Up to now, the code asy.m [7] included in FIESTA [8] was based on prescriptions formulated in Sect. 2 but with the use of the representation (1) and the corresponding product UF of the two basic functions, rather than with the use of (5). Let us prove that the two prescriptions are equivalent.
Let us keep in mind that the functions U and F are homogeneous in the variables \(x_i\), with different homogeneity degrees.
Proposition 3
Let U and F be two homogeneous functions of the variables \(x_i\) with different homogeneity degrees such that the Newton polytope \({\mathcal {N}}_{U+F}\) for \(U+F\) has dimension \(n+1\). Equivalently, \({\mathcal {N}}_{U F}\) has dimension n. Then there is a onetoone correspondence between essential facets of \({\mathcal {N}}_{U+F}\) and essential facets of \({\mathcal {N}}_{U F}\). This correspondence is obtained by the projection on the hyperplane orthogonal to the vector \(\{1,\ldots ,1,0\}\) which we will denote by \(v_0\).
Proof
Let \(\varGamma \) be an essential facet of \({\mathcal {N}}_{U+F}\). It has dimension n. Since \({\mathcal {N}}_{U}\) and \({\mathcal {N}}_{F}\) have dimension not greater than n this means that if \(\varGamma \) does not intersect with one of them it should contain the other Newton polytope whose dimension is n. Then, due to homogeneity, its normal vector is proportional to \(v_0\) but this cannot be the case for an essential facet. Hence \(\varGamma \) has a nonempty intersection with both Newton polytopes.
Let us analyze intersections \(\varGamma _U\) and \(\varGamma _F\) of the facet \(\varGamma \) with \({\mathcal {N}}_{U}\) and \({\mathcal {N}}_{F}\), correspondingly. The hyperplane generated by \(\varGamma \) has dimension n and can be defined as a vector sum of the hyperplane generated by \(\varGamma _{U}\), the hyperplane generated by \(\varGamma _{F}\) and some vector which connects a point of \(\varGamma _{U}\) and a point of \(\varGamma _{F}\). Therefore, the vector sum of the hyperplane generated by \(\varGamma _{U}\) and the hyperplane generated by \(\varGamma _{F}\) has dimension \(n1\).
Furthermore, both hyperplanes are orthogonal to the vector \(r^{\varGamma }\) and to the vector \(v_0\), therefore they are also orthogonal to \(r^{\varGamma }_0\), the projection of the vector \(r^{\varGamma }\) on the hyperplane orthogonal to \(v_0\).
Now it suffices to show that \(r^{\varGamma }_0\) corresponds to a facet of \({\mathcal {N}}_{UF}\). Indeed, the minimal values of scalar products of points of this polytope with the vector \(r^{\varGamma }_0\) is achieved from the pairwise sums of the points of the facets of \(\varGamma _U\) and \(\varGamma _F\). The linear space spanned by these points can be generated by the vector sum of the hyperplanes spanned over the sets \(\varGamma _{U}\) and \(\varGamma _{F}\) but we have just shown that this space has dimension \(n1\), i.e is a facet.
Now let us turn to the inverse statement. Let \(\varGamma \) be a facet of \({\mathcal {N}}_{UF}\). Its normal vector \(r^{\varGamma }\) is orthogonal to \(v_0\). Let us consider the sets \(\nu _U\) and \(\nu _F\) consisting of points with the minimal scalar product with \(r^{\varGamma }_0\) of \({\mathcal {N}}_{U}\) and \({\mathcal {N}}_{F}\), respectively.
The sum of the hyperplanes spanned on \(\nu _U\) and \(\nu _F\) coincides with the hyperplane spanned on \(\varGamma \). Therefore, it has dimension \(n1\).
Let the scalar product of \(r^{\varGamma }_0\) and the points of \(\nu _U\) be \(u_0\) and the scalar product of \(r^{\varGamma }_0\) and the points of \(\nu _F\) be \(f_0\). Furthermore, let the scalar product of \(v_0\) and points of \(\nu _U\) be \(u_1\) and the scalar product of \(v_0\) and points of \(\nu _F\) be \(f_1\) The fact that the scalar product is fixed follows from the homogeneity, and we know than \(u_1 \ne f_1\).
Let us find such a vector \(r = r^{\varGamma }_0 + \{\alpha ,\ldots ,\alpha ,0\}\) that its scalar product with points of \(\nu _U\) and \(\nu _F\) is the same. To do this we solve the equation \(u_0 + x u_1 = f_0 + \alpha f_1\), so that \(\alpha = (f_0u_0) / (u_1  f_1)\).
Let \(\varGamma \) be the face of \({\mathcal {N}}_{U F}\) spanned over points having the minimal product with r. The hyperplane spanned over \(\varGamma \) is the sum of hyperplanes spanned over \(\nu _U\) and \(\nu _F\) and some vector connecting a point of \(\nu _U\) and a point of \(\nu _F\). The dimension of the sum of first two hyperplanes is \(n1\) however the connecting vector does not belong to this sum since it is not orthogonal to \(v_0\). Therefore \(\varGamma \) is a facet of \({\mathcal {N}}_{U F}\) and \(r = r^{\varGamma }\). \(\square \)
6 The leading order
Let us, first, assume that the conditions of Proposition 1 hold. We have
Proposition 4
Proof
If the condition of Proposition 1 does not hold we can use Proposition 2 and adjust an analytic regularization to provide convergence. Let \(\varGamma \) be an essential facet. Then, like in the proof of Proposition 2, we can adjust parameters \(\lambda _i\) by putting \(\frac{1+\lambda _i}{\delta }\) between the left and the right values of the ith coordinates of \(\pi (\varGamma )\). After this, we can follow the same arguments as in the proof of Proposition 3 and obtain the following generalized version.
Proposition 5
Let us emphasize that the projection \(\pi ({\mathcal {N}}_P)\) of the Newton polytope can be covered by the corresponding projections \(\pi (\gamma )\) of essential facets. The intersection of any pair \(\pi (\gamma _1)\) and \(\pi (\gamma _2)\) of projections of the facets has dimension less than n. Therefore, the contribution of one of the facets can be made leading by adjusting analytic regularization parameters. We can refer again to sector decompositions in order to prove that the contribution of each facet is a meromorphic function of parameters \(\varepsilon \) and \(\lambda _i\) so that then we can expand a result for this contribution in the limit of small \(\lambda _i\) up to a finite part in \(\lambda _i\) keeping possible singular terms in \(\lambda _i\), and then expand in \(\varepsilon \) at \(\varepsilon \rightarrow 0\).
We can use this procedure as an unambiguous definition of the leading contribution of a given facet, i.e. we can clarify the prescriptions in Sect. 2 and define it as the expanded analytic continuation of the contribution described in Proposition 5 in the two successive limits, \(\lambda _i\rightarrow 0\) and \(\varepsilon \rightarrow 0\). However, it is necessary to specify how the limit \(\lambda _i\rightarrow 0\) is taken. At least two practical variants were in use: (1) take the limits \(\lambda _i\rightarrow 0\) for \(i=1,2,\ldots \), or in some other fixed order, keeping expansion up to \(\lambda _i^0\); (2) choose, \(\lambda _i=p_i \lambda _1\), \(i=2,3,\ldots \), where \(p_i\) is the ith prime number and then take the limit \(\lambda _1\rightarrow 0\). The second variant was systematically used, in particular, in Refs. [29, 30]. In both cases, the definitions depends on the order of parameters \(\lambda _i\) but final results for the whole expansion should be independent of this choice if the initial integral is convergent at \(\lambda _i=0\).
It is clear that one has to choose the same way of taking the limit \(\lambda _i\rightarrow 0\) for all the facets. Possible individual singularities in \(\lambda _i\) should cancel in the sum of contributions of different facets. Then \(\lambda _i\rightarrow 0\) and we are left with expansion in \(\varepsilon \). Of course, the order of contributions to the expansion is measured in powers of t when the limit \(\lambda _i\rightarrow 0\) is already taken. The true leading order of the expansion is given by a sum of contributions of some essential facets which can be called leading.
7 General order for one essential facet
Let us consider a simple situation with one essential facet. For Feynman integrals, this can be, for example, an expansion in the small momentum limit, where a given Feynman graph has no massless thresholds. Then one can refer to general analytic properties of Feynman amplitudes and claim that the Feynman integral is analytic up to the first threshold so that if can be expanded in a Taylor series at zero external momenta. Of course, there is only one essential facet in the corresponding Newton polytope associated with the polynomial P in (5) and the limit looks trivial. However, our goal is an integral with an arbitrary polynomials with positive coefficients, so that the situation with one essential facet should not be qualified as trivial. We have the following
Proposition 6
Proof
Let us start from Eq. (29). The second term in the brackets tends to zero at \(t\rightarrow +0\), so that one can obtain a series in powers of t by expanding this expression with respect to the second term, according to the prescriptions formulated in Sect. 2. This is, generally, not a Taylor expansion. Rather, this is an expansion in powers of \(t^{1/q}\) where q is the least common multiple of the rationals \(\kappa _{w,\varGamma }\).
Let us define \({\widetilde{u}}_i= \sum \nolimits _{j=1}^{m} u_i^j k_j\). Taking into account the convex property of the set \(\pi ({\mathcal {N}}_P)\), the property of \(k_j\) and the fact that there is only one essential facet \(\varGamma \), we can conclude that the point \(\frac{1}{m}({\widetilde{u}}_1,\ldots , {\widetilde{u}}_n)\) is an internal point of \(\pi ({\mathcal {N}}_P)\). Let us prove, using Proposition 2, that the convergence property of the integral (35) of \(E(y_1,\ldots ,y_n)\) is equivalent to the condition that the point \(A=\frac{1}{m+\delta }({\widetilde{u}}_1+1,\ldots , {\widetilde{u}}_n+1)\) is inside \(\pi ({\mathcal {N}}_P)\). Let us assume that this is not true, i.e. A is not an internal point of \(\pi ({\mathcal {N}}_P)\). Then there should exist a hyperplane \(\sum \nolimits _{i=1}^n p_iw_i+p_0=0\) such that \(\pi ({\mathcal {N}}_P)\) and A belong to the different sides from this hyperplane, or on this hyperplane.
 1.
The inequality \(\sum \nolimits _{i=1}^n p_iw_i+p_0\le 0\) holds for \(w\in \pi ({\mathcal {N}}_P)\).
 2.
The relation \(\frac{1}{\delta +m}\sum \nolimits _{i=1}^n p_i({\widetilde{u}}_i+1)+p_0\ge 0\) holds for the point A.
 3.
The condition of convergence of the initial integral is \(\frac{1}{\delta }\sum \nolimits _{i=1}^n c_i+c_0<0\).
 4.
The relation \(\sum \nolimits _{i=1}^n p_i(\frac{1}{m}{\widetilde{u}}_i{\widetilde{w}}_i)<0\) holds for some point \({\widetilde{w}}\in \pi ({\mathcal {N}}_P)\) because \(\frac{1}{m}({\widetilde{u}}_1,\ldots , {\widetilde{u}}_n)\) is an internal point of the convex set \(\pi ({\mathcal {N}}_P)\).
8 Summary

We performed an analysis of convergence of the LP representation, proved a generalization of the Speer’s theorem for integrals (5) and presented a general definition of dimensionally regularized integrals (5).

We presented a direct proof of equivalence of expansion by regions for Feynman integrals based on the standard Feynman parametric representation (1) and the LP representation (5). This change is now implemented in FIESTA [8] so that revealing regions is now performed in a much more effective way just because the degree of polynomial \(P=U+F\) in (5) is less than the degree of the product of the polynomials UF.

We proved our prescriptions for the contribution of the leading order for each essential facet.

We proved our prescriptions in the general order in the simple situation with one essential facet.
We believe that the commutativity of the expansion procedure with the operation of analytic continuation with respect to the regularization parameter can be proven so that this will give a justification of the prescriptions at least in the leading order of expansion. Another possible scenario would be to prove the prescriptions in a general order of expansion by constructing a remainder with the help of the operator \(\prod _i(1M_i^{n_i})\) with appropriately adjusted subtraction degrees \(n_i\). The problem would be divided into two parts: justifying the necessary asymptotic estimate of the remainder where an auxiliary analytic regularization is not needed and obtaining terms of the corresponding expansion where, generally, an analytic regularization is necessary.
Footnotes
 1.
After our paper has been sent to the archive, a new approach (based on Landau equations) to reveal regions corresponding to a given limit has appeared [10]. Its authors show on examples that the potential and Glauber regions can be revealed within their prescriptions.
 2.
In fact, this step is performed within asy.m with the help of another code qhull. It is most timeconsuming and can become problematic in higherloop calculations.
 3.
We follow the terminology introduced in the sixties and seventies. Ultraviolet (infrared) divergences arise from integration over large (small) loop momenta. By offshell infrared divergences we mean divergences at small loop momenta in situations where external momenta are not put on a mass shell. In particular, external momenta can be considered Euclidean (any partial sum of external momenta is spacelike) (see for example Ref. [16]), or a Feynman integral can be considered as a tempered distribution with respect to external momenta (for example, in very wellknown papers on renormalization [17, 18, 19, 20]). Onshell infrared divergences appear when an external momentum is considered on a mass shell, \(p^2=m^2\), in particular a massless mass shell. In the latter case, collinear divergences can appear due to integration near lightlike lines.
 4.
Without massless detachable subgraphs; this means that there are no onevertexirreducible subgraphs with zero incoming momenta. The corresponding integrals would be scaleless integrals.
 5.
Notes
Acknowledgements
The work was supported by RFBR, grant 170200175A. We are grateful to Roman Lee and Alexey Pak for stimulating discussions.
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