Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers

Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers

Hjalte Frellesvig, Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, 35131 Padova, ItalyINFN, Sezione di Padova, Via Marzolo 8, 35131 Padova, ItalyPerimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, CanadaDepartment of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada     Federico Gasparotto, Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, 35131 Padova, ItalyINFN, Sezione di Padova, Via Marzolo 8, 35131 Padova, ItalyPerimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, CanadaDepartment of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada     Stefano Laporta, Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, 35131 Padova, ItalyINFN, Sezione di Padova, Via Marzolo 8, 35131 Padova, ItalyPerimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, CanadaDepartment of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada     Manoj K. Mandal, Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, 35131 Padova, ItalyINFN, Sezione di Padova, Via Marzolo 8, 35131 Padova, ItalyPerimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, CanadaDepartment of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada     Pierpaolo Mastrolia, Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, 35131 Padova, ItalyINFN, Sezione di Padova, Via Marzolo 8, 35131 Padova, ItalyPerimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, CanadaDepartment of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada     Luca Mattiazzi, Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, 35131 Padova, ItalyINFN, Sezione di Padova, Via Marzolo 8, 35131 Padova, ItalyPerimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, CanadaDepartment of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada     Sebastian Mizera hjalte.frellesvig@pd.infn.it federico.gasparotto@pd.infn.it stefano.laporta@pd.infn.it manojkumar.mandal@pd.infn.it pierpaolo.mastrolia@pd.infn.it luca.mattiazzi@pd.infn.it smizera@pitp.ca Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, 35131 Padova, ItalyINFN, Sezione di Padova, Via Marzolo 8, 35131 Padova, ItalyPerimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, CanadaDepartment of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada
Abstract

We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss hypergeometric function, and the Appell function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to -form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.

1 Introduction

The highly interactive nature of elementary-particle dynamics is an extremely complex problem to describe. Feynman diagrams play a fundamental role by encoding how, within the perturbative approach to quantum field theory, complex scattering reactions may arise from the multiple interactions among simpler network components, representing either external or internal particles. For each scattering process, the sum of all diagrams gives the so called scattering amplitude, a complex function of external kinematics (and internal quantum numbers) whose absolute squared value determines the quantum-mechanical density of probability for that reaction to happen.

The shape of the diagrams informs directly about the complexity level of the process they describe, which increases with the number of interacting external particles, the number of interaction vertices, and with the number of loops. In general, Feynman diagrams represent functions of kinematic invariants formed by the external momenta and by the masses of the particles. While scattering amplitudes associated to tree-level graphs can be written in terms of rational functions, those coming from multi-loop graphs are usually decomposed into special functions admitting multifold integral representations.

Feynman integrals in dimensional regularization are known to be not mutually independent. A novel way to access their algebraic structure has been recently unveiled by some of us, who showed that relations for multi-loop Feynman integrals are controlled by intersection numbers Mastrolia:2018uzb ().

The by-now standard evaluation techniques of Feynman integrals exploit the loop-momentum shift invariance to establish integration-by-parts (IBP) relations Chetyrkin:1981qh () among integrals whose integrands are built out of products of the same set of denominators (and scalar products), but raised to different powers. IBP identities have been playing a crucial role in the calculation of multi-loop integrals, because they yield the identification of a minimal set of elements, dubbed master integrals (MIs), which can be used as a basis for the decomposition of multi-loop amplitudes. At the same time, IBP-decomposition algorithms can be applied to special integrands, built by acting on the master integrand with differential operators (w.r.t. kinematic invariants), or by multiplying their numerators by polynomials which modify their dimensions, or by considering arbitrary denominator powers, respectively turning the decomposition formulas into differential equations Barucchi:1973zm (); KOTIKOV1991158 (); KOTIKOV1991123 (); Bern:1993kr (); Remiddi:1997ny (); Gehrmann:1999as (); Henn:2013pwa (); Henn:2014qga (), dimensional recurrence relations Tarasov:1996br (); Lee:2009dh (), and finite difference equations Laporta:2001dd (); Laporta:2003jz () obeyed by MIs. Solving them amounts to the actual determination of the MIs themselves, as an alternative to the use of direct integration techniques.

The derivation of the IBP-decomposition formulas requires the solution of a large system of linear relations, generated by imposing that integrals of total derivatives vanish on the integration boundary Laporta:2001dd (), see also smirnov2005evaluating (); Grozin:2011mt (); Zhang:2016kfo (); Kotikov:2018wxe (). For multi-loop multi-scale scattering amplitudes, solving the system of IBP relations may however represent a formidable task, whose accomplishment has been motivating important refinements of the system-solving strategies vonManteuffel:2014ixa (); Peraro:2016wsq (); Boehm:2018fpv (); Kosower:2018obg (); Chawdhry:2018awn (); vonManteuffel:2012np (); Lee:2012cn (); Smirnov:2008iw (); Maierhoefer:2017hyi (); Georgoudis:2016wff (); Maierhofer:2018gpa (); Smirnov:2019qkx (). Together with novel algorithms for simplifying the solution of systems of differential equations Henn:2013pwa (), which triggered further studies Argeri:2014qva (); Lee:2014ioa (); Gehrmann:2014bfa (); Gituliar:2017vzm (); Lee:2017oca (); Adams:2018yfj (), the calculations of several multi-scale, multi-loop, multi-particle amplitudes became feasible Laporta:2017okg (); Anastasiou:2015ema (); Bonciani:2015eua (); Gehrmann:2015dua (); Bonetti:2017ovy (); Borowka:2016ypz (); Baglio:2018lrj (); Lindert:2017pky (); Jones:2018hbb (); Maltoni:2018zvp (); Gehrmann:2015bfy (); Badger:2017jhb (); Abreu:2018jgq (); Abreu:2018zmy (); Chicherin:2018yne (); Abreu:2018aqd (); Chicherin:2019xeg (); Abreu:2019rpt ().

The most recent developments in the research of mathematical methods for the evaluation of Feynman integrals have been benefiting from a special representation known as Baikov representation Baikov:1996iu (), where instead of the components of the loop momenta, the propagators themselves supplemented by independent scalar products between external and internal momenta, are the integration variables. This change of variables introduces a Jacobian equal to the Gram determinant of the scalar products formed by both types of momenta, referred to as the Baikov polynomial. The Baikov polynomial fully characterizes the space on which the integrals are defined, and in particular the number of MIs can be inferred from the number of its critical points Lee:2013hzt (), see also Marcolli:2008vr (); marcolli2010feynman (); Bitoun:2017nre (). IBP identities may relate integrals corresponding to a given graph to integrals that correspond to its sub-graphs. We may consider these two sets of integrals, respectively, as to the homogeneous and the non-homogenous terms of the IBP relations. The homogeneous terms of IBP identies can be detected by maximal cuts, since the multiple cut-conditions annihilate the terms corresponding to subdiagrams Larsen:2015ped (); Bosma:2017hrk (); Frellesvig:2017aai (); Zeng:2017ipr (); Bosma:2017ens (); Harley:2017qut (); Boehm:2018fpv (). By the same arguments, maximal cuts of MIs correspond to the homogeneous solutions of dimensional recurrence relations and of the differential equations, which, in general, are non-homogeneous equations Lee:2012te (); Remiddi:2016gno (); Primo:2016ebd (); Primo:2017ipr (). Similar ideas were introduced Anastasiou:2002yz () and lead to relation between multi-loop integrals and phase-space integrals, known as reverse unitarity. The homogeneous solutions play an important role in the construction of canonical systems of differential equation for MIs Henn:2013pwa (), as it was observed in Remiddi:2016gno (); Primo:2016ebd (); Primo:2017ipr (), generalizing the role of Magnus exponential matrix Argeri:2014qva () to the case of elliptic equations.

The number of MIs for a given integral family, the order of the differential equations they obey, and the classification of the homogeneous solutions according to the independent components of the integration domain, revealed a natural correspondence between the MIs and the geometric properties of the integration domain Laporta:2004rb (); Remiddi:2016gno (); Primo:2016ebd (); Primo:2017ipr (); Bosma:2017ens (); vonManteuffel:2017hms (); Frellesvig:2017aai (); Harley:2017qut (); Adams:2018kez (), easily accessed within Baikov representation.

Let us imagine, for a moment, that the objective of a calculation is simply the decomposition in terms of master integrals of just one multi-loop Feynman integral. The IBP reduction algorithm can be seen as a collective integral decomposition. The computational machinery does not act on individual integrals, one at a time, but it is based on the solution of systems of equations where the wanted integral appears related, within linear relations, to many additional integrals. Its decomposition is then achieved together with the decomposition of other integrals – even if the latter might not be of interests, for instance. The IBP decomposition, although very effective, is computationally expensive.

The new computational strategy proposed in ref. Mastrolia:2018uzb () offers a change of perspective: it targets the direct decomposition of individual integrals in terms of master integrals, bypassing the system solving procedure characterizing the integration-by-parts reduction.

This task can be achieved by applying to Feynman integrals concepts and computational tools borrowed from the intersection theory of differential forms cho1995 (); matsumoto1998 (); Mizera:2017rqa (). It is a recent branch of algebraic geometry and topology, which was developed to study the Aomoto–Gel’fand hypergeometric functions aomoto2011theory (). This class of functions has two important properties: their integrands are multivalued, and vanish at the boundaries of the integration domain - exactly like Feynman integrals in dimensional regularization. Baikov representation makes these properties manifest, and allows to establish an explicit correspondence between Feynman integrals and the functions which can be studied via intersection theory. In particular, any Feynman integral is cast as a -form integral, characterized by three basic elements: the integration contour, part of its integrand given by a multivalued function (associated to the Baikov polynomial), and a differential form (corresponding to the genuine product of denominators times the integration measure).

In general, two integrals can give the same result if: they have the same integration domain, but their integrands differ by a term whose primitive vanishes on the integration boundaries; and/or they have the same integrand, but their integration domain differs by a contour on which the primitive vanish anyhow. Therefore each integral is actually a pairing of representatives of two equivalence classes: characterized by the integration variety (homology class) or by the integrand (cohomology class). Intersection theory allows for a derivation of relations among integrals belonging to those equivalence classes, which in the case of hypergeometric functions correspond to Gauss’ contiguity relations. In ref. Mastrolia:2018uzb (), the concepts of intersection theory were applied to Feynman integrals in order to show that it is possible: i) to identify a basis of master integrals; ii) to decompose any individual integral in the chosen basis simply by a projection technique; iii) to derive differential equations for master integrals. The so-called intersection number of differential forms cho1995 (); matsumoto1998 () constitutes the crucial novel operation that allows to implement the notion of scalar products between differential forms, which ultimately determine the coefficients of the integral decomposition. In this way, the problem of reducing a given Feynman integral in terms of master integrals can be solved by projections: any integral can be decomposed just like an arbitrary vector can be projected onto a chosen basis of a vector space. This analysis was performed by considering integrals on maximal cuts admitting 1-form representations Mastrolia:2018uzb ().

In this work, we elaborate on the decomposition-by-intersection of Feynman integrals onto a basis of master integrals, and we systematically apply it to an extensive list of cases, in order to show its advantages.

We begin by recalling basics of intersection theory for hypergeometric functions, and show their correspondence to the Baikov representation, both in the standard formulation Baikov:1996iu () and in the Loop-by-Loop version Frellesvig:2017aai (). We then define how intersection theory allows to determine the dimension of the integral space, and discuss different options for the choice of the integral bases. Afterwards, we introduce the intersections numbers and give the master decomposition formula for the direct evaluation of the coefficients of the reduction in terms of basis integrals. This formula can be also applied to derive differential equations and dimensional recurrence relations for generic basis integrals.

Before addressing Feynman calculus, we consider the derivation of contiguity relations for special functions, such as the Euler function, the Gauss hypergeometric function, and the Appell function, which belong to the more general class of Lauricella functions. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. The 1-form integral representations accounts for multi-loop integrals (on maximal cuts) which have either one irreducible scalar product (ISP), or that have multiple ISPs but can be expressed as a one-fold integral using the Loop-by-Loop approach,

In a few instructive cases, we show the direct constructions of differential equations and dimensional recurrence relations for master integrals, and discuss how the different choice of the basis may impact of the form of the result. Special emphasis is given to basis of monomial forms and to basis of dlog forms, in particular showing how, the latter obey a canonical systems of differential equations.

As stressed, the main part of this work deals with the application of intersection theory to 1-forms. The complete decomposition of multi-loop Feynman integrals in terms of master integrals (not just the ones belonging to maximal-cut diagrams, but to the complete chain of sub-diagrams, which would correspond to a smaller number of cuts) requires the application of the intersection theory for -forms. In the literature, the case of intersection numbers of dlog -forms has been understood matsumoto1998 (); Mizera:2017rqa (), but Feynman integrals belong to the wider class of generic rational -forms.

As additional main results of this manuscript, we present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation. They are important new development both for physical and mathematical research areas, as they represent the first step towards the extension of the formalism to generic -form representations. Owing to the results of the research presented in this work, we are confident that this objective is within reach.

The decomposition formulae computed through the use of intersection numbers for 1- and 2-forms are directly verified to agree with the ones obtained using integration-by-parts identities on the maximal cut. As reported in the examples discussed later, we employed several codes for checking our results, and when available, we compared them with the literature. Let us finally observe that, in a few cases, the number of master integrals (on the maximal cuts) found by means of intersection theory is smaller than the one found by applying the IBP-decomposition to the interested integral families: the mismatch has been mitigated by finding the additional, missing relations at the cost of applying the IBP-reduction to integrals families with a larger number of denominators.

The results presented in this paper demonstrate that a fascinating property of Feynman integrals has been found, which opens a completely new path to access their algebraic structures. Together with the idea of dimensional regularization, its main application to streamlining reductions unto master integrals, is expected to yield significant computational advantages for the evaluation of high-multiplicity scattering amplitudes at higher orders in perturbation theory.

The paper is organized as follows. In Sec. 2 we introduce the basics of hypergeometric integrals, and their description in terms of (co)homology and intersection theory. Significantly this section introduces the master decomposition formula eq. (14). Then follows Sec. 3 in which we discuss Feynman integrals, the Baikov representations, unitarity cuts, and the connection to intersection theory. The section also discusses relations between Feynman integrals such as reductions unto master integrals, differential equations, and dimensional recurrence.

In Sec. 4 we discuss certain specific mathematical functions to which our theory is applicable, which serve as our first examples. These are the -function, the , the Appell , and the Lauricella . The next two sections, Secs. 5 and 6, contain our first examples of the use of the theory to Feynman integrals: a four-loop vacuum integral, and a three-loop two-point function respectively. These two integrals are put in a one-form representation by the standard Baikov parametrization. In Sec. 7 we discuss the massless sunrise integral. This integral can be put into a one-form representation by the Loop-by-Loop version of Baikov parametrization, where the standard version of Baikov representation would have left it as a two-fold integral. In Sec. 8 we return to the non-planar triangle of ref. Mastrolia:2018uzb (), and show how to use our method to deal with doubled propagators. Then in Sec. 9 follows an example of a planar triangle diagram with only one master integral, showing the intersection approach alongside a traditional cut-based extraction. Sec. 10 discusses a certain planar diagram, that is of interest due to peculiar properties of the Baikov polynomial, something that in the past has led to ambiguities in the counting of master integrals in that sector. Secs. 11 and 12 discuss two double-box integrals, with and without an internal mass. The massive case is of interest as this is our first example in which the intersection theory on the maximal cut, detects a relation that is not usually found by IBP identities. The next two sections applies the theory to some cases of physical interest, namely Bhabha scattering in Sec. 13, and associated Higgs production ( and ) in Sec. 14. These two cases are at the edge of what is possible to fully reduce with traditional IBP methods. Then follows Sec. 15 on the celebrated penta-box in a planar and a non-planar version and Sec. 16 about its generalizations to cases with massive legs, and significantly to cases with more legs, including an -leg example. In Sec. 17 we look some high-loop integrals (planar and non-planar) that contribute to production, and we show several -loop generalizations thereof (that we denote “rocket diagrams”), including their reductions unto master integrals, and their differential equations.

The following two sections contain discussions on the extension of the one-form algorithm described in this paper, to higher forms. Sec. 18 discusses one approach that consists of iterating the one-form algorithm combined with direct integration. Sec. 19 discusses another genuinely multivariate approach in which the intersection numbers are computed using intersecting hyperplanes. Finally Sec. 20 contains our conclusions and discussion. The paper ends with an App. A in which we discuss the relation between the number of critical points and the number of master integrals. In particular we calculate the number of critical points with both the standard and the Loop-by-Loop approaches to Baikov parametrization on the maximal cut, and we discuss in detail the cases in which the two numbers thereby obtained are in disagreement.

2 Basics of Hypergeometric Integrals

In this section we review a few concepts from the theory of hypergeometric functions and Feynman integrals that serve as a basis for the remainder of the paper.

Consider an integral over the variables of the general form:

(1)

where is a multi-valued function and is a differential -form. We assume that vanishes on the boundaries of , , so that, upon integration no surface-term is leftover. For example, choosing

(2)

gives the Euler beta function for . More generally, integrals of the type (1) are called Aomoto–Gel’fand hypergeometric functions aomoto1977structure (); Gelfand (), or simply hypergeometric functions.

As with any integral, there could exist many forms that integrate to give the same result . Let us consider the total derivative of times any -differential form :

(3)

By Stokes’ theorem, the result is zero due to our choice of the integration domain . Let us manipulate the above integral so that it is of the form (1):

(4)

In the final equality we defined a connection , which differs from the usual derivative by the one-form :

(5)

Since the above expression integrates to zero, we have

(6)

Hence and carry the same information and we can talk about equivalence (cohomology) classes of forms that integrate to the same result:

(7)

In other words, whenever two forms are equal to each other up to integration-by-parts identities, they belong to the same equivalence class. This class is called a twisted cocycle. The word twisted refers to the fact that the usual derivative operator is replaced by the covariant derivative given in (5), as a consequence of the presence of the multi-valued function in the hypergeometric integral. We often refer to any representative of the class (7) as twisted cocycle, as well as drop the subscript when it is clear from the context.111For completeness, let us mention that, similarly, there are equivalence (homology) classes of integration domains that give the same result for the integral (1), called twisted cycles , though we do not make use of this fact in the current manuscript. A remarkable observation is that we can pair up and to obtain the integral from (1), which we denote by

(8)

This integral representation, as a bilinear in and , is suitable for establishing linear relations between hypergeometric functions. In fact, let us assume that the number of linearly-independent twisted cocycles is , and indicate an arbitrary basis of forms,

(9)

A basis decomposition is achieved by expressing an arbitrary twisted cocycle, say , as a linear combination of the above ones. This goal be achieved as follows. Introduce a dual (and auxiliary) space of twisted cocycles, whose basis is denoted by for , and consider the matrix , whose entries are the pairing ,

(10)

This pairing is called the intersection number of and . We then construct the matrix , defined as,

(11)

The columns of the matrix are labelled by for an arbitrary , while the rows are labelled by . Each entry is given by a pairing (bilinear) of the corresponding row and column. In the second equality, we expose the structure of as a submatrix , a column vector and a row vector , respectively with elements and (for ).

The fact that the cocycles labelling the rows and columns are necessarily linearly dependent (since the basis is -dimensional) and that each entry of is a bilinear, implies that the determinant of this matrix vanishes. Using the well-known identity for the determinant of a block matrix, we find:

(12)

In addition, cannot be zero (by definition), since it is formed from bilinears between two bases. Therefore we conclude that:

(13)

Given the arbitrariness of , we obtain the master decomposition formula

(14)

which provides an explicit way of projecting onto a basis of . Following Mastrolia:2018uzb (), in this paper we use (14) to perform the decomposition of Feynman integrals in terms of master integrals, on the maximal cut. For example, by contracting both sides with the twisted cycle (which boils down to multiplying by and integrating over ), we have a linear identity between integrals:

(15)

Similarly, the same idea can be used to derive linear system of differential equations satisfied by the basis integrals in some external variable . It is enough to notice that

(16)

where . Let us remark that even if depends on , the differential operator commutes with the integral sign, due to the vanishing of on the boundary of . Therefore, the problem reduces to projecting on the right-hand side back onto a basis using (14).

One should think of and as parameterizing a vector space of inequivalent integrands of a hypergeometric function. In this sense provides a metric on this space. Naturally, the prescription (14) is only useful if computing invariants of the type is efficient. We argue that this is the case. It turns out that the dual space of twisted cocycles has a straightforward interpretation as the equivalence classes:

(17)

where the only difference to (7) is the use of the connection instead of . The resulting bilinear:

(18)

is called the intersection number of and . This term is conventionally used in the literature on hypergeometric functions, but it does not mean that is an integer. In general, it can be a rational function of external parameters. The characteristic property of the intersection number is that it is a bilinear in the two equivalence classes. We give multiple ways of computing it throughout the text.

In this brief review, we only scratched the surface of the fascinating theory of hypergeometric functions. We refer the interested reader to aomoto2011theory (); yoshida2013hypergeometric () for review of twisted (co)homologies and their intersection theory, as well as Mizera:2016jhj (); Mizera:2017rqa (); Mizera:2017cqs (); Mastrolia:2018uzb () for some recent applications of these ideas to physics.

In the following, we focus on Feynman integrals. In order to translate them into the form (1) we make use of the Baikov representation in the standard form Baikov:1996iu () and the Loop-by-Loop approach developed in Frellesvig:2017aai ().

3 Feynman Integral Decomposition

Consider scalar Feynman integrals with loops, external momenta, and (generalised) denominators222 amounts to the total number of scalar products which can be built with the loop momenta and the independent external momenta , and corresponds to the sum of the so called reducible and irreducible scalar products. The former can be expressed in terms of the denominators of graph propagators, while the latter are independent of them. Nevertheless, they also can be interpreted as auxiliary denominators, not related to any internal line of the graph. in a generic dimension :

(19)

where stands for either a genuine denominator or an irreducible scalar product (ISP).

In Baikov representation, one changes the integration variables, from the loop momenta to the denominators , at the cost of introducing a Jacobian, see, e.g., Lee:2009dh (); Lee:2010wea () or Appendix A of Mastrolia:2018uzb (). Here we summarize the final forms of the standard and Loop-by-Loop Baikov representations.

  1. Standard Baikov Representation. In this case, after the change of variables, the Feynman integral may be written as,

    (20)

    where

    (21)

    and

    (22)

    and where is the Baikov polynomial computed as a determinant of the Gram matrix of scalar products, depending on loop momenta, and is a constant pre-factor (independent of the integration variables), which may depend on the external kinematic invariants and on the dimensional regulator . The integration contour is defined such that vanishes on its boundaries.

    We can re-express it, in the language of intersection theory, as a bilinear pairing,

    (23)

    with

    (24)
  2. Loop-by-Loop (LBL) Baikov Representation. In this case, after the change of variables, the the number of integration variables can be smaller than the (because ISPs have been integrated out). For this case, the integral have the form

    (25)

    with

    (26)

    and where

    (27)

    where is a rational function of the (that is if all with are 0).

  3. Cut Integrals. Within the Baikov representation, the on-shell cut-conditions are most naturally expressed as a contour integration. Any multiple -cut integral, with , becomes

    (28)

    where the deformed contour is defined as

    (29)

    with the -contours denoting a small loop in the complex plane around the pole at . Accordingly, the integration domain of the cut-integral is given by the geometric intersection of with the planes identifying the on-shell conditions,

    (30)

    In general, the domain may admit a decomposition into subregions,

    (31)

    though only of them can be independent. After integrating over the cut variables, the left over (phase-space) integral reads as,

    (32)

    with

    (33)
    (34)
    (35)
    (36)

    where vanishes on the boundary , and is a rational function (see eqs. (27)). Therefore, also the -cut integral keeps admitting a bilinear pairing representation,

    (37)
    Notation.

    In the following examples, for ease of notation, we drop the prime symbol , and use directly , , , and to express the various quantities on the cut. Moreover, in the univariate case where after the maximal cut the integrals are characterized by a single ISP, we use the notation , where is the power of the remaining irreducible scalar product.

3.1 Intersection Numbers of One-Forms

In this section we specialize to the case when are -forms. Consider,

(38)

and define as the set of poles of ,

(39)

Note that can also include the pole at infinity if .333 The number of master integrals is equal, up to a sign, to the Euler characteristic of the space , on which the forms are defined, where the number of poles in is exactly , provided that all are not non-negeative integers. See also Lee:2013hzt (); Bitoun:2017nre () for discussion of Euler characteristic in the context of Feynman integrals. Earlier considerations on possible relations between the number of MIs and geometric properties of differential manifolds can be found in Kosower:2011ty (); CaronHuot:2012ab ()

Given two (univariate) 1-forms and , we define the intersection number as cho1995 (); matsumoto1998 ()

(40)

where, is a function (0-form), solution to the differential equation , around , i.e.,

(41)

where was defined in eq. (5) (the notation indicates the Laurent expansion of around ). The above equation can be also solved globally, however only a handful of terms in the Laurent expansion around are needed to evaluate the residue in (40). In particular, after defining , and the ansatz,

(42)
(43)

the differential equation in eq. (41) freezes all unknown coefficients . In other words, the Laurent expansion of around each , is determined by the Laurent expansion of and of . A given point contributes only if the condition is satisfied, and the above expansion exists only if is not a non-positive integer.

Symmetry Properties.

Intersection numbers of one-forms have the following symmetry property under the exchange of and ,

(44)

Notice that on the r.h.s. the intersection number is evaluated with respect to the form (instead of ).

Logarithmic Forms.

When both and are logarithmic, meaning that for all points , then the formula (40) simplifies to

(45)

Note that in this case the intersection number becomes symmetric in and , i.e.,

(46)

while (44) still holds.

Vector Space Metric, Integral Decomposition and Master Integrals.

Following the discussion in Sec. 2, consider an -dimensional vector space, and its dual space, whose basis are respectively represented as, and with . We use intersection numbers to define a metric on this space

(47)

which gives rise to matrix . According to the master decomposition formula eq. (14), any element of the space can be decomposed in terms of , as

(48)

Therefore, the pairing of on the l.h.s. and on the r.h.s. with the integration cycle , univocally gives rise to the decomposition (on the cut) of the Feynman integral in terms of master integrals , by means of projections built with intersection numbers, i.e.

(49)

where

(50)

and

(51)

The main goal of this work is to show that the decomposition formulas for Feynman integrals obtained by intersection numbers are equivalent to the one derived by the standard integration-by-parts identities (IBPs). Very interestingly, using intersection numbers, the system-solving strategy inherent to the IBP-decomposition is completely bypassed Mastrolia:2018uzb ().

Reducible Integrals and Maximal Cuts.

As shown, the number of independent basis forms, and hence MIs, is given by . Therefore, for any given integral family the existence of MIs is due to the existence of the solutions for . It is possible to identify a few special cases:

  • Reducibility. Absence of master integrals, amounting to , can happen either when Baikov polynomial on the maximal cut is vanishing, , or when is linear in the integration variable, : in the former case, does not exist; in the latter case, , therefore , and has no solutions. In these cases, the integral family is reducible, namely the corresponding integrals can be expressed as a combination of the master integrals of the subtopologies.

  • Maximal Cuts. Baikov polynomial is a non-zero constant on the maximal cut. This means that no ISP is left over to parametrize the cut integral. In other words, the integral is fully localized by the cut-conditions. In this case, the condition is always satisfied, and there is master integral.
    This situation may occur, for instance, at one-loop, where maximal cuts are indeed maximum cuts.

Choices of Bases.

The bases and can be different from each other, but is a possible choice too. We decompose 1-form employing either a monomial basis

(52)

or a dlog-basis, of the type,

(53)

where are poles of .

Alternatively, orthonormal bases for twisted cocycles can be chosen as follows. Out of the set of poles pick two special ones, say and . Then construct bases of one-forms using:

(54)

for . With this choice, the intersection matrix becomes the identity matrix,

(55)

as can be shown directly using the residue prescription (40), and therefore the basis decomposition formula simplifies to

(56)

3.2 System of Differential Equations

Let us give more details about deriving systems of differential equations using intersection numbers.

Consider the system of differential equations in for the basis ,

(57)

in general depending on the space-time dimension and external variables . Let us consider the l.h.s. of eq. (57), after taking the derivative in ,

(58)

where . Here can be decomposed in terms of , by means of intersection numbers,

(59)
(60)
(61)

where summation over indices is implied and we introduced the intersection matrix

(62)

as well as defined the matrix as,

(63)

appearing in the r.h.s. of eq. (57).

In Mastrolia:2018uzb (), it was observed that in the case of dlog-basis defined for integrals within the standard Baikov representation (for which ), the matrix is -factorized, and so it is the matrix. Therefore the system of differential equations for the dlog-basis is canonical Henn:2013pwa () by construction, around the critical dimension .

Master Integrals in dimensions correspond to integrals of the form

(64)

where may depend on as well. Therefore, if,

(65)

then the system of differential equations for reads,

(66)
(67)
Solutions.

The system of differential equations in eq. (65) can be used to deduce a single homogeneous differential equation of order for each separately (). For each , the independent solutions of such an equation can be found by building the pairing

(68)

where are the independent sub-regions considered in eq. (31), see, e.g., Primo:2016ebd (); Bosma:2017ens (); Primo:2017ipr (). The matrix is the resolvent matrix of the system of differential equations. For instance, by choosing a -dimensional basis formed by and its derivatives up the -order, becomes the Wronski matrix, whose determinant is the Wronskian of the differential equation obeyed by .

The matrix plays an important role in the construction of canonical systems of differential equation Henn:2013pwa (), as it was observed in Remiddi:2016gno (); Primo:2016ebd (); Primo:2017ipr (), generalizing the role of Magnus exponential matrix Argeri:2014qva () to the case of elliptic equations. More generally, in the theory of hypergeometric functions, is known as twisted period matrix. It can be used, for instance, to build the so called twisted Riemann period relations cho1995 (), a fundamental identity giving quadratic relations between hypergeometric functions. A proper study of twisted Riemann period relations to Feynman integrals goes beyond the scope of the current manuscript, and it is left to future investigations.

3.3 Dimensional Recurrence Relation

Within the standard Baikov representation, the dependence of Feynman integrals is carried solely by the prefactor and by the exponent of the Baikov polynomial . Let us write the MIs in dimensions as,

(69)

with

(70)

and consider the decomposition of the in terms of the basis ,

(71)

This equation can be interpreted as a change of basis, from with to with . We can, therefore, decompose in terms of the new basis , as

(72)

which can be written in the suggestive fashion,

(73)

with . Upon the pairing with , it yields the recursion formula for the integral ,

(74)

where the coefficients , computed by means of the master decomposition formula eq. (48), may depend on and on the kinematics. Finally, by a simple redefinition of the coefficients, the dimensional recurrence relation for the MIs arises,

(75)

with

4 Special Functions

One-variable integrals of the hypergeometric type considered in this paper, may always444If the integrand is just a product of linear terms with the integration path being between two of the , a Möbius transform can bring it into the form discussed in the text. be expressed in the form

(76)

For , this integral (up to pre-factors) corresponds to the Euler beta-function, the Gauss hypergeometric function , and the Appell function repectively, and the general case is known as the Lauricella functions.

In this section, we apply the ideas of intersection theory to these paradigmatic cases with their increasing level of complexity, in order to derive contiguity relations, which for hypergeometric functions play the same role that IBP identities play for Feynman integrals555Recent applications of the theory of hypergeometric functions to the coaction of one-loop (cut)Feynman integrals can be found in Abreu:2017enx (); Abreu:2018nzy ()..

4.1 Euler Beta Integrals

We start by discussing integral relations associated to a simple class of integrals such as the Euler beta function, defined as

(77)

4.1.1 Direct Integration

Let us consider integrals of the type

(78)

These integrals admit a closed-form expression in terms of functions,

(79)

from which it is possible to derive a relation between and ,

(80)

For instance, when , it reads

(81)

4.1.2 Integration-by-Parts Identities

Let us recover the same relation from integration by parts identities. With the choice of as above, the following integration-by-parts identity holds

(82)

The action of the differential operator under the integral sign yields the following equation,

(83)

Therefore we obtain the recurrence relation

(84)

which, for , gives

(85)

4.1.3 Intersections

We are going to (re)derive, once more, the relations between Euler beta integrals using intersection numbers. We consider integrals defined as,

(86)

with

(87)
(88)
Monomial Basis.

implies the existence of 1 master integral, which we choose as . The goal of this calculation is to derive the relation between and ,

(89)

which can be derived by decomposing in terms of ,

(90)

Notice that since , the intersection matrix has just one element .

We need to evaluate the intersection numbers , and .

For each pole , we identify (the series expansion of around ), and determine the associated function (the series expansion of around ), by solving the following differential equation,

(91)

After inserting the series expansion of and an ansatz for in the above equation, we get an equation at each order on , which together determines the coefficients in the ansatz for . In practice, we define , and take the Laurent expansions of

(92)

and the ansatz,

(93)

to solve the following differential equation,

(94)

In our case we have,

  • For , :

    min max

    with

    (95)

    Therefore

    (96)
  • For , :

    min max

    with

    (97)
    (98)

    Therefore

    (99)

Notice that in the above formulas only the