Feynman Integrals and Intersection Theory

Pierpaolo Mastrolia∗ Dipartimento di Fisica e Astronomia, Universit`adi Padova, Via Marzolo 8, 35131 Padova, Italy and INFN, Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy

Sebastian Mizera† Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada and Department of Physics & Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada

We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider the Baikov representation of maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of differential forms with logarithmic singularities on the boundaries of the corresponding integration cycles. We give an algorithm for computing a basis decomposition of an arbitrary maximal cut using so-called intersection numbers and describe two alternative ways of computing them. Furthermore, we show how to obtain Pfaffian systems of differential equations for the basis integrals using the same technique. All the steps are illustrated on the example of a two-loop non-planar triangle diagram with a massive loop.

1. INTRODUCTION Feynman integrals that differ by the powers of denominators and/or scalar products in the numerator, see, e.g., Evaluating Feynman integrals is crucial for the investi- [14–16]. The explicit knowledge of the integration do- gation of physical problems that admit a field-theoretic main is not needed for generating IBP identities: the perturbative approach: revealing weak deviations from requirement of vanishing surface terms provides a suffi- the Standard Model predictions in the behaviour of parti- cient, qualitative information to establish IBP relations. cle collision at high accuracy; the study of properties of Nevertheless, the identification of a basis of MIs, and the newly-discovered particles; understanding formal proper- integral-decomposition formulas require the solution of ties of quantum theories that are not directly deducible large-size linear systems of equations [10]. For processes from the basic structure of their Lagrangians; exposing involving multiple kinematic scales, this may represent similarities among theories which are supposed to describe an insurmountable task. In the recent years, important interactions among particles of different species; as well as technical advances have been made by refining the com- the study of the dynamics of coalescing black-hole binary monly adopted system-solving strategy for IBP identities systems whose merger gives rise to gravitational waves. [10], either due to novel algorithms [17–21] or to the de- These are just a few examples for which the computation velopment of improved software [22–26], and calculations of multi-loop Feynman integrals cannot be considered as of multi-loop multi-particle amplitudes, which were con- optional. sidered inaccessible a few years ago, have become feasible Feynman integrals, within the dimensional regular- [27–30]. ization scheme, obey contiguity relations known as At the same time, one may want to search for alter- integration-by-parts (IBP) identities [1], which play a native methods in order to perform the decomposition crucial role in the evaluation of scattering amplitudes in terms of MIs, eventually looking for mathematical beyond the tree-level approximation. IBP identities yield methods that allow for a direct integral reduction, which the identification of an elementary set of integrals, the so- bypass the need of solving a system of linear equations. called master integrals (MIs), which can be used as a basis In this work, we explore the latter idea, and we elab- arXiv:1810.03818v2 [hep-th] 5 Mar 2019 for the decomposition of multi-loop amplitudes. At the orate on a new method for establishing relations among same time, IBP relations can be used to derive differential Feynman integrals in arbitrary space-time dimensions, equations [2–9], finite difference equations [10, 11], and and for projecting them onto a basis. An archetype of dimensional recurrence relations [12, 13] obeyed by MIs. such a basis reduction is the Gauss’s contiguous relation, The solutions of those equations are valuable methods e.g., for the evaluation of MIs, as alternatives to their direct c 2F1(a, b; c; z) a 2F1(a+1, b; c+1; z) integration. 2F1(a, b, c+1; z) = + . IBP identities can be generated by considering inte- c − a a − c (1.1) grals of total derivatives that vanish on the integration This relation can be regarded as a basis reduction of the boundary. They form a system of linear relations between hypergeometric function 2F1(a, b; c+1; z) on the left-hand side in terms of two MIs on the right-hand side. Of course, such an identity can be derived by considering linear re- ∗ [email protected] lations between 2F1’s with different parameters, derived † [email protected] either from their series representation or from their in- 2 tegral representation by means of integration-by-parts. associated to a given graph (characterized by a given However, it is possible to adopt a modern mathematical number of denominators) and integrals corresponding to approach, which offers a direct solution to the decompo- subdiagrams (with fewer denominators). The homoge- sition problem [31]. Accordingly, consider the integral neous terms of IBPs can be detected by maximal cuts, representation, since the simultaneous on-shell conditions annihilate the terms corresponding to subdiagrams. For recent studies Z 1 b c−b −a of maximal cuts in the Baikov representation, see [19, 37– 2F1(a, b, c+1; z) := x (1−x) (1−zx) ϕ. B(b, c−b) C 42]. In this work, we focus on maximal cuts in order to (1.2) present our novel algorithm for the basis reduction in the −−−→ The integration domain C := (0, 1) together with the simplest possible setting. information about the branch of the integrand is called The number of independent MIs can be derived from the twisted cycle, while the single-valued differential form the properties of Baikov polynomial, therefore, after de- c 1 termining the size of the basis, which we denote by |χ|, ϕ := c−b dlog x is called the twisted cocycle. The above integral is understood as a pairing of these two objects we construct bases of twisted cycles Ci and cocycles ϕj, [31]. B(a, b) is the Euler beta function. Similarly, we can whose pairings give rise to |χ|2 integrals: consider two other logarithmic forms, Z D−D∗ x c x B(z) 2 ϕj, i, j = 1, 2,..., |χ|. (1.4) ϕ := dlog , ϕ := dlog , (1.3) Ci 1 1−x 2 c−b 1−zx They form a minimal, linearly-independent, basis in terms which upon integration give rise to 2F1(a, b; c; z) and of which any other integral of the same type can be 2F1(a+1, b; c+1; z) respectively. The decomposition prob- decomposed. Here B(z) is obtained by evaluating B on lem reduces to projecting ϕ onto a basis of ϕ and ϕ . 1 2 the maximal cut surface. Twisted cycles Ci are chosen The solution can be found by computing certain topo- as certain regions with boundaries on {B(z) = 0}, while logical invariants called intersection numbers of pairs of twisted cocycles ϕj are differential forms with logarithmic cocycles, which are rational functions in the coefficients singularities along all the boundaries of the corresponding a, b, c. They become the coefficients of the basis expansion Cj. We describe them in Section3. The choice of the on the right-hand side of the Gauss’ contiguous relation. independent integrals is rather general, and they might Inspired by this approach, we here propose to apply not correspond to the cuts of conventional MIs (although the computational techniques of intersection theory to the they can be related to them, if needed). study of Feynman integrals. In Section4 we apply the tools of intersection theory Among different representations of Feynman integrals, of the appropriate homology and cohomology groups [43, the one most closely resembling Aomoto–Gel’fand hyper- 44] to the problem of basis reduction. It can be done geometric functions is the so-called Baikov representation separately in the space of twisted cycles and cocycles. We [32]. It uses independent scalar products between exter- focus on the reduction in the space of twisted cocycles and nal and internal momenta as the integration variables, show how to apply two different techniques of evaluating instead of the components of the loop momenta. This intersection numbers: the special case of logarithmic forms change of variables introduces a Jacobian equal to the has been treated by one of us in [45], while the case of Gram determinant of the scalar products formed by both general one-forms was discussed by Cho and Matsumoto in types of momenta, referred to as the Baikov polynomial [44]. To further support the connection between Feynman D−D∗ B, raised to a power 2 for an integer D∗. Within this calculus and intersection theory, we provide references to representation, it is possible to identify a critical space- the relevant literature for the interested reader.3 time dimension D∗, in which the integral presents nice Maximal cuts of MIs obey homogeneous difference and mathematical properties. We review the Baikov represen- differential equations [47–50]. From several studies focus- tation in Section2 and give more details of its derivation ing on the solution of differential equations for Feynman in AppendixA. integrals [39, 41, 42, 48–53], MIs have been identified with Baikov polynomial fully characterizes the space on the independent components of the integration domain. which the integrals are defined. Early signs of this fact Owing to the complete characterization of the integrand were found by Lee and Pomeransky, who applied Morse and of the integration domain, explicit solutions for the theory to relate the number of MIs to counting of the maximal cuts can be found in the Baikov representation. D−D∗ 2 critical points of B 2 [33]. IBP identities are non- In general, MIs obey a system of first-order differential homogeneous relations which, in general, involve integrals equations, whose corresponding matrix has entries which are rational functions of the kinematic variables and of

1 Strictly speaking, cycles and cocycles are equivalence classes, and here we consider their representatives. 3 2 Recent applications of the theory of homology and cohomology Similar connections can be made in other representations, such classes to the coaction of one-loop (cut) Feynman integrals can as the parametric one [33] (see also [34, 35]). A correspondence be found in [46]. between the Euler characteristic of the sum of Symanzik polynomials and the number of MIs was also recently studied in [36]. 3

D−E−L−1 the dimensional regularization parameter D. The num- γ := , ber of MIs depends on the kinematics of process under 2 consideration and on the number of loops, but there is a with a constant Jacobian c, which we will drop from now certain freedom in choosing them. In particular, one may on. The integrand involves a rescaled Baikov polynomial: choose a set of MIs whose system of differential equations is linear in D [54], and use the solutions of the homogeneous equations to write a transformation matrix, known det G{`1,...,`L,p1,...,pE } as resolvent matrix of the homogeneous system (around B := , (2.4) det G{p1,...,pE } D = D0, for any chosen value of D0, which can be dic- tated by the physical dimensions of the problem under which is a ratio of two Gram determinants. Recall that a consideration) [48, 49]. The resolvent matrix is employed Gram matrix of Lorentz vectors {qi} is defined as G{qi} := to change the basis of MIs, and to define a special set of [qi ·qj]. The integration domain Γ is given by imposing L basic integrals that obey canonical systems of differential conditions: equations [8, 55], for which the differential equation ma- det G trix has a simple D-dependent term, factored out of the {ì,...,`L,p1,...,pE } Γi := > 0 , (2.5) kinematics. In Section5, we discuss Pfaffian systems of det G{ì+1,...,`L,p1,...,pE } differential equations satisfied by the basis integrals, and show that they can be derived using our basis reduction so that Γ := Γ1∩Γ2∩· · ·∩ΓL. Notice that this implies B > algorithm, without employing the IBP identities. 0 everywhere on the integration domain. For convenience, Even though the techniques described here are generally we review a derivation of the Baikov representation in applicable, throughout the paper we consider functions AppendixA. admitting integral representations over one variable and The representation (2.3) is particularly friendly towards leave applications to multi-variate cases until future stud- computing cuts. By a linear transformation we can make ies. For illustration purposes, we apply our novel method a further change of variables into za := Da for all M to a two-loop non-planar triangle diagram with a mas- inverse denominators. A single cut corresponds to taking sive loop, showing the decomposition algorithm in D a circular integration contour {|za| = ε}, which sets Da dimensions, as well as discussing features of the system on-shell. Repeating this procedure N times we obtain a of differential equations for the corresponding MIs, both maximal cut, which takes the general form [37]: in D and in 4 dimensions. Z γ Mν1ν2···νM := B(z) ϕ(z), (2.6) C 2. BAIKOV REPRESENTATION where we ignored an overall constant Jacobian. The integrand depends on the ISPs Consider scalar Feynman integrals with L loops, E+1 external momenta, and N propagators in a generic di- z := (zN+1, zN+2, . . . , zM ). (2.7) mension D: The Baikov polynomial on the maximal cut B(z) is given Z L D N Y d `j Y 1 I := . (2.1) by setting z1 = ··· = zN = 0 in eq. (2.4). Similarly, ν1ν2···νN D/2 νa D π Da ϕ(z) is a differential (M−N)-form obtained as a result R i=1 a=1 of the residue computation. For example, if the original We focus on Euclidean space in all-plus signature for integral had all propagators undoubled, we have ϕ(z) = QM νa simplicity of discussion. Baikov considered a change of a=N+1 dza/za . Finally, the integration domain C is an integration variables into all independent scalar products intersection of Γ with the cut surface {z1 = ··· = zN = 0}. between loop and external momenta [32], Whenever this intersection is empty, the maximal cut vanishes and the diagram is reducible. qi ∈ {`1, `2, . . . , `L, p1, p2, . . . , pE}, (2.2) For one-loop diagrams the maximal cut fully localizes the integral and hence the size of the basis is one, and (here we assume that D ≥ E, so that external momenta the coefficients of the decomposition can be obtained do not satisfy additional relations [56, 57]). There are 1 simply by means of residue theorem. Let us consider an M := LE + 2 L(L + 1) such kinematic invariants, ì · qj. interesting example of a two-loop non-planar diagram In order to perform the change of variables, one needs to 2 2 2 with internal mass m and p1 = s, p2 = p3 = 0: introduce M−N extra inverse denominators Da known as irreducible scalar products (ISPs) with exponents νa ≤ 0. `2 p2 The original integral (2.1) is recovered when νN+1 = ··· = `1 νM = 0. After the dust settles, one finds [13, 58]: p1 (2.8) p Z QL QE+L 3 i=1 j=i d(ì ·qj) γ 2 2 2 Iν1ν2···νM = c B , (2.3) QM νa D1 = `1, D4 = (p3 − `1 + `2) − m , Γ Da a=1 2 2 2 2 D2 = `2 − m , D5 = (`1 − `2) − m , 2 2 2 D3 = (p1 − `1) , D6 = (p2 − `2) − m . 4

2 2 We also choose an ISP D7 = 2(p2 + `1) − p1, for later |C] encode all IBP identities and contour deformations. convenience. In this case, E = L = 2, M−N = 1 and the Their pairing, denoted by hϕ|C], is deﬁned to be equal to maximal cut in Baikov representation (2.6) becomes: the maximal cut (2.6),

Z D−5 Z 2 ν γ M111111|−ν = B(z) z dz, (2.9) Mν1ν2···νM = B(z) ϕ(z) =: hϕ|C] . (3.4) C C where we relabelled z7 → z, ν7 → ν for clarity. The In general, Baikov polynomial on the maximal cut B(z) rescaled Baikov polynomial reads: admits a decomposition into irreducible components:

1 K B(z) = − (z2 − s2)(z2 − ρ2), (2.10) Y 64s2 B(z) = bi(z). (3.5) i=1 where ρ := ps(s + 16m2). In the kinematic regime s, m2 > 0 its roots are ordered as −ρ < −s < s < ρ. From now on we will consider only the cases in which each The constraints (2.5) imply that the integration region is: bi(z) has degree at most M−N, so that the corresponding variety {bi(z) = 0} does not have self-intersections. We −−−−−→ −−−→ also assume that the dimension D in the exponent of the C = (−ρ, −s) ∪ (s, ρ). (2.11) Baikov polynomial is generic, and in particular not an integer. −−−→ γ Here (a, b) denotes an oriented interval between a and b. The logarithm of B(z) is called a Morse function whenever its critical points (also called saddle points) are non- degenerate. From now on we will assume that this is the 3. MINIMAL BASIS FOR MAXIMAL CUTS case. It means that we can use Morse theory to analyze the properties of the integrals (2.6) and, in particular, to Integrals of the type (2.6) admit a beautiful interpreta- read-off the size of the basis of twisted cycles and cocycles, tion in terms of Aomoto–Gel’fand hypergeometric func- see, e.g.,[31, 61]. tions [59, 60], where they are understood as pairings of Then the number of independent twisted cycles and PM−N δ−M+N twisted cycles C and cocycles ϕ. In order to see this, let cocycles, |χ|, is given by |χ| = δ=0 (−1) Cδ, us consider the following one-form: where Cδ is the number of critical points with a Morse index δ. The Morse index is the number of independent ω := d log B(z)γ . (3.1) directions along which the Morse function decreases away from the critical point [61]. Under the assumptions given It defines a flat connection ∇ω := d + ω∧, related to the in [31], all critical points have the maximal index δ = integration-by-parts (IBP) relations, M−N and hence: Z Z γ γ |χ| = number of solutions of ω = 0 . (3.6) B(z) ∇ωξ(z) = d (B(z) ξ(z)) = 0 (3.2) C C Here we used the fact that ω = 0 determines the critical 5 for any (M−N−1)-form ξ and (M−N)-dimensional cycle points. When all irreducible components bi(z) are linear, C. This means that we can define equivalence classes hϕ| |χ| is also equal to the number of bounded regions in M−N of (M−N)-forms ϕ up to terms ∇ωξ that integrate to R \ {B(z) = 0} [62]. zero: Let us discuss how to construct bases of twisted cycles Ci and cocycles ϕi for i = 1, 2,..., |χ|. The real section of M−N hϕ| : ϕ ∼ ϕ + ∇ωξ. (3.3) the integration domain, R \ {B(z) = 0}, decomposes into multiple disconnected regions, called chambers. Each Similarly, we have equivalence classes |C] of cycles C up to chamber is a valid choice of a basis element Ci. For each terms integrating to zero. More precisely, if the integral Ci we can construct the corresponding twisted cocycle ϕi (2.6) over the cycle Ce vanishes (for example, when Ce is a with logarithmic singularities along the boundaries of Ci. loop contractible to a point), then C and C + Ce belong to For the algorithmic constructions, see, e.g.,[31, 63–66] the same equivalence class |C].4 The two classes hϕ| and and the more recent study [67]. It is currently known how to do it in the cases when Ci is bounded by hyperplanes

4 If required, a more rigorous, mathematical definition can be given M−N by following Ref. [31]: on the space X = C \{B(z) = 0} 5 The idea of determining the size of the basis using the Euler k we define twisted cohomology groups H (X, ∇ω) := ker(∇ω : characteristic χ was first considered in Feynman integral literature k k+1 k−1 k Ω (∗D) → Ω (∗D))/∇ωΩ (∗D) 3 hϕ|, where Ω (∗D) is the by Lee and Pomeransky [33], see also [36]. In the present work, sheaf of smooth holomorphic k-forms on X with poles along the however, we do not make any claims beyond maximal cuts in singular divisor D of B(z). Locally finite twisted homology groups generic dimension D. lf Hk (X, Lω) 3 |C] with coefficients in a rank-1 local system Lω, k lf are isomorphic by H (X, ∇ω) ' HomC(Hk (X, Lω), C), which induces the pairing hϕ|C] := Mν1ν2···νM in (2.6). In this case, k lf M−N dim H (X, ∇ω) = dim Hk (X, Lω) = δk,M−N (−1) χ. 5 and at most one hypersurface {bi(z) = 0} with degree as Feynman integrals on their own (but they may corre- up to M−N [67]. This is enough for considering many spond to non-trivial combination of them). For example, interesting examples of maximal cuts [41]. evaluating (3.1) for the diagram (2.8) we have: For instance, whenever a given chamber Ci is simplex- 2 2 2 like, i.e., bounded by exactly M−N+1 hyperplanes z 2z − s − ρ ω = (D−5) dz. (3.11) {bj(z) = 0}, j = 1, 2,...,M−N+1, we have: (z2 − s2)(z2 − ρ2) b b b ϕ = d log 1 ∧ d log 2 ∧ · · · ∧ d log M−N . (3.7) We find |χ| = 3 solutions of the condition ω = 0: i b b b 2 3 M−N+1 r s2 + ρ2 z = 0, ± . (3.12) In situations when {B(z) = 0} is non-normally crossing, ∗ 2 i.e., more than M−N hypersurfaces intersect at a single point, one needs to consider a blowup in the neighbour- Their positions are irrelevant for the counting, but they hood of this point. will be used later on for other purposes (the size of the ba- The choice of logarithmic basis comes with multiple sis is alternatively determined to be 3 from the dimension advantages. For instance, its singularity structure in of the pure braid group associated to (3.10).) There is a the variable γ is manifest: neighbourhood of each co- certain freedom is choosing the bases of twisted cycles, as dimension k ≤ M−N boundary of Cj gives contributions it is only required that they have endpoints on the points at order γ−k, e.g., hypersurfaces contribute at order γ−1, {−ρ, −s, s, ρ, ∞}. A natural choice is as follows: while their intersections at order γ−2: −−−−−→ −−−−→ −−−→ C1 = (−ρ, −s), C2 = (−s, s), C3 = (s, ρ). (3.13) 2 CP \{B(z)=0} {b1(z) = 0} The choice of twisted cocycles is also arbitrary, as long C γ−2 (3.8) as they have poles only at the points {−ρ, −s, s, ρ, ∞}. {b2(z) = 0} j γ−1 Given the above basis of cycles, a natural counterpart is: {b3(z) = 0} z+ρ z+s ϕ = d log , ϕ = d log , Hence the leading divergence comes from the highest co- 1 z+s 2 z−s dimension boundaries (points), around which the integral z−s ϕ = d log , (3.14) behaves as γ−(M−N). To be precise, a given co-dimension 3 z−ρ k boundary of the form which are designed to have residues ±1 on the two end- {b1(z) = 0} ∩ {b2(z) = 0} ∩ · · · ∩ {bk(z) = 0} (3.9) points of the corresponding Ci. They are special cases of (3.7). The matrix P consists of nine linearly independent −k contributes to a given integral hϕi|Cj] at order γ if and Appell functions F1 that form a basis. only if it belongs to the boundary of a given twisted cycle Note a symmetry for the two independent kinematic ∂Cj and at the same time the twisted cocycle ϕi has a invariants, s → −s, ρ → −ρ, under which non-vanishing residue at the boundary (3.9). At this stage let us remark that it is always possible to C1 ↔ −C3, C2 → −C2, shift D → D +2n for n ∈ in the exponent of the Baikov Z ϕ1 ↔ −ϕ3, ϕ2 → −ϕ2, (3.15) polynomial [12], at a cost at redefining ϕ → B(z)−nϕ (the requirement of single-valuedness of ϕ imposes n ∈ while B(z) is invariant. Therefore the entries of the matrix Z). This changes γ → γ + n and hence the singularity P are related as Pij → P4−i,4−j and we end up with a structure, but does not affect the choice of the basis itself. five-dimensional functionally-independent basis. We can organize all possible pairings of twisted cycles In the Feynman integral literature it is customary to and cocycles into the twisted period matrix P with entries: talk about bases of MIs, which have to consist of Feynman integrals, and hence have a fixed integration domain C and Z different powers of ISPs. In this language, the diagram γ Pij := hϕi|Cj] = B(z) ϕi. (3.10) (2.8) in generic dimension D has 3 linearly independent C j MIs (hϕi|C] for i = 1, 2, 3), and 2 after imposing the above symmetry, in agreement with [49, 52]. Its |χ|2 components provide a minimal linearly- independent basis for any integral of the form (2.6), as was first observed in [49], and later also [39–41] in the 4. BASIS REDUCTION WITH INTERSECTION Baikov representation. In other words, by choosing ϕi NUMBERS as basic integrands for the maximal cuts of the master integrals, the entries of the P matrix correspond to the The goal of a basis reduction is expressing an arbitrary independent solutions of the homogeneous system of dif- integral of the form (2.6) in terms of the |χ|2 basis func- ferential equations the latter obey. Let us stress that tions in P. This can be done separately in the space of elements of the basis might not be necessarily recognized 6 cycles and cocycles. In order to do so, we introduce the original Baikov integration domain C from (2.11) already notion of a metric on these spaces. Assuming existence of decomposes as: dual spaces |ϕi and [C|, let us consider pairings between their basis elements: |C] = |C1] + |C3]. (4.6)

Cij := hϕi|ϕji, Hkl := [Ck|Cl]. (4.1) and hence no detailed computation is necessary. Examples for other maximal cuts will appear elsewhere. These pairings are called intersection numbers. Using Let us stress that intersection numbers entering the simple linear algebra, we can decompose an arbitrary expression (4.2) can be computed for any basis, which does twisted cocycle hϕ| into a basis of hϕi| as follows: not necessarily have to be the logarithmic one introduced in Section3. For example, one could construct a basis |χ| of maximal cuts with diﬀerent powers of ISPs, e.g., (2.9) X −1 hϕ| = hϕ|ϕji (C )ji hϕi|, (4.2) with ν = 0, 1, 2. i,j=1

Concerning the decomposition of Feynman integrals in A. Intersection Numbers of Logarithmic Forms terms of basic integrals, (4.2) constitutes the first main result of this work, hence we define to be the master Similarly, intersection numbers hϕ |ϕ i can be evalu- decomposition formula. i j ated in multiple different ways, see, e.g.,[31, 44, 45, 69, 71– Similarly for a twisted cycle |C] in a basis of |C ]: l 80, 82–84]. They are rational functions in kinematic in- |χ| variants and the dimension D. It was recently found that X |C] = |C ](H−1) [C |C]. (4.3) for logarithmic forms ϕi and ϕj there exists a formula l lk k localizing on the critical points given by ω = 0 [45]: k,l=1 M −1 −1 Z Y Here hϕ|ϕji (C )ji and (H )lk [Ck|C] are coefficients hϕ |ϕ i = (−1)M−N dz δ(ω ) ϕ ϕ . (4.7) of the expansions. Putting these two decompositions to- i j a a bi bj a=N+1 gether, we find that the original integral hϕ|C] is expressed in terms of basis functions in P as follows: PM Here ωa are components of ω = a=N+1 ωadza, and ϕb de- |χ| QM notes a differential-stripped cocycle ϕ =: ϕ a=N+1 dzi. X −1 −1 b hϕ|C] = hϕ|ϕji (C )ji Pil (H )lk [Ck|C]. (4.4) Let use apply it to the two-loop example (2.8). For i,j,k,l=1 simplicity, we are going to choose the same representatives (3) for cocycle bases of both hϕi| and |ϕji. The above In fact, this statement is completely general and holds formula (4.7) becomes: for any Feynman integral in arbitrary parametrization, as long as one is able to identify |ϕi and [C| and their X 1 pairings. The advantage of the Baikov representation of hϕi|ϕji = − ϕbi ϕbj , (4.8) ∂ω/∂z z=z maximal cuts is that such identifications can be made, z∗ b ∗ which allows for explicit computations. where the sum goes over the three critical points z from For completeness, we define the dual space as equiv- ∗ (3.12) and we have a Jacobian ∂ω/∂zb coming from evalu- alence classes |ϕi : ϕ ∼ ϕ + ∇−ωξ with the connection 6 ating the delta function. Performing this computation for ∇−ω and similarly for dual twisted cycles [C|. With every combination of hϕi| and |ϕji gives us the matrix C this choice, intersection numbers [Ci|Cj] are trigonometric from (4.1): functions of the dimension D [43]. They can be computed straightforwardly by considering all the places where Ci  2 −1 0  and C intersect geometrically (additional care needs to 2 j C = −1 2 −1 . (4.9) be taken when boundaries of Ci and Cj are non-normally D−5 0 −1 2 crossing). In the current manuscript, we will not make use of intersection numbers for cycles: there exist nu- It is always possible to choose the dual basis |ϕji to be merous ways of evaluating them, and we refer the reader orthonormal, i.e., such that C = 1, which simplifies the to, e.g.,[31, 43, 68–82]. In the example at hand, the decomposition (4.2).7

6 The latter is an equivalence class of cycles [C| : C ∼ C + Ce such 7 For instance, an orthonormal basis to (3) is given by: that Z Z B(z)−γ ϕ(z) = B(z)−γ ϕ(z) (4.5) |ϕ1i = γ d log(z+ρ), |ϕ2i = −γ d log(z−s)(z−ρ), C C+Ce |ϕ3i = −γ d log(z−ρ), (4.10) for any ϕ(z). Notice the negative sign in the exponent of the Baikov polynomial compared to (2.6). with γ = (d − 5)/2, though we will not make use of it in the text. 7

B. Intersection Numbers of Non-Logarithmic ρ − s hϕ|ϕ3i = . (4.17) One-Forms 2D−9 This completes a decomposition of hϕ| into a basis, which In order to complete the decomposition according after evaluating (4.2) reads: to (4.2), we ought to compute the remaining intersection numbers hϕ|ϕji. Let us consider the maximal cut D−5 hϕ| = ρ hϕ |+hϕ |+hϕ | + s hϕ | . (4.18) (2.9) with no numerators, ν = 0, for which we have ϕ = dz. 2(2D−9) 1 2 3 2 This form has a double pole at infinity, which means that we cannot use (4.7). In this case we employ an alternative The relation in (4.18) is an IBP identity. Finally, using formula for general one-forms due to Cho and Matsumoto the cycle decomposition (4.6), we find: [44] (see also [83, 85]): D−5 X h i hϕ|C]= ρ P11+P21+P31 + s P21 (4.19) hϕ|ϕji = Resp ψp ϕj , (4.11) 2(2D−9) p∈{B(z)=0} + (Pi1→Pi3) . −1 where for each p one needs to compute ψp := ∇ω ϕ around z=p, i.e., one needs to find a unique holomorphic Recall that the terms Pi3 = hϕi|C3] are related to Pi1 = function ψp solving hϕi|C1] by the symmetry s → −s, ρ → −ρ, therefore only three elemetns of the basis need to be computed. The ∇ ψ = ϕ locally near p. (4.12) ω p prefactor of γ = (D−5)/2 guarantees that the integral For a review of the derivation of (4.11), see, e.g., Ap- hϕ|C] is finite when γ → 0. pendix A of [45]. For our example, it involves a sum over Let us stress that the choice of logarithmic forms as all boundary components {B(z) = 0} = {−ρ, −s, s, ρ, ∞}. a basis for the maximal cut is not a limitation. It was inspired by the mathematical literature, but other choices Since the connection ∇ω decreases the order of the pole by one, it is enough to consider an ansatz are possible, such as monomial powers, or, more generally, rational functions of the integration variables. The com- ∞ putational load and the resulting formulas may depend X (α) α ψp = ψp (z − p) , (4.13) on the choice of the basis, but the physical results do not. α=1+ordpϕ Also, we remark that the formula for the intersection numbers (4.11) is specific to one-forms, though, in general, where ordpϕ denotes the order of the zero of ϕ around maximal cuts of Feynman integrals may admit m-forms p, e.g., ordp dz/(z − p) = −1. (Analogous expansion in representations, where the integration variables corre- 1/z is done when p = ∞.) Plugging it into (4.12) one spond to the ISPs. Intersection numbers for multivariate (α) can solve for the first few coefficients ψp order-by-order logarithmic forms were presented in [83], and its extension in (z−p). Notice that ordpψp = ordpϕ + 1. The residues to non-logarithmic forms, and application to Feynman in (4.11) are non-zero only if ψp ϕj has at least a simple integrals will be discussed in a future publication. pole, or in other words:

ordpϕ + ordpϕj ≤ −2. (4.14) 5. PFAFFIAN SYSTEMS

In our case, the basis elements ϕj have at most simple The derivation of Pfaffian systems of differential equa- poles around each p, while ϕ = dz has only a double pole tions follows the same decomposition algorithm. Let us at infinity. Using the condition (4.14) we conclude that denote with d0 a differential on the appropriately chosen p = ∞ is the only contributing point in (4.11). Hence we space of kinematic invariants K. Acting with it on the compute: basis elements, we have: 1 ψ = z + O(1) (4.15) d0P = d0hϕ |C ] = hΦ |C ] , (5.1) ∞ 2D−9 ij i j i j using the procedure above, but expanding around infin- with ity. Plugging the result into the formula for intersection hΦ |C ] := h (d0 + d0 log Bγ ) ϕ |C ] . (5.2) numbers (4.11) we find:8 i j i j 9 ρ − s 2s We want to bring it into the Pfaffian form: hϕ|ϕ1i = , hϕ|ϕ2i = , 2D−9 2D−9 0 d Pij = Ωik ∧ Pkj for all j. (5.3)

8 For logarithmic forms ϕi and ϕj , the formula (4.11) reduces to 9 Recall that in the deﬁnition of Pij we stripped away Jacobians, X Resp ϕi Resp ϕj i.e., set c c0 = 1 in (A.9). They can be easily restored and will hϕi|ϕj i = , (4.16) Resp ω modify the Pfaﬃan system. p∈{B(z)=0} which is an alternative way of obtaining the entries of C in (4.9). 8

Here Ω is a one-form on K. In order to find it, it is enough The linear system (5.3) can be converted into a higher- to project each Φi on the right-rand side of (5.1) in the order differential equations for each of the i’s separately. basis P, again using (4.2). Hence the entries of Ω are For fixed i, the solutions of such an equation can be computed with intersection numbers: expressed in terms of the basis of Pij for different j’s, i.e., different solutions correspond to distinct choices of the |χ| integration cycles. X −1 Ωik = hΦi|ϕli (C )lk. (5.4) l=1 6. FOUR DIMENSIONS We expect that for the logarithmic bases described in this work the matrix Ω should be proportional to γ. In addition, it can be shown that the leading order in γ of So far we have been working in a generic dimension P coincides with the intersection numbers, i.e., D/∈ Z. In this section we illustrate how to apply the same techniques directly in the strict D → 4 limit for the −(M−N)+1 maximal cut of the diagram (2.8). Setting D=4 we have: Pij = Cij + O(γ ), (5.5) if the bases of cocycles are logarithmic. −1/2 Let us derive differential equations in the example at −1/2 1 2 2 2 2 B(z) = − 2 z − s z − ρ , (6.1) hand. We choose K = {x ∈ C | x = ρ/s}, which gives: 64s s 2s2x(x − 1) which behaves as z−2 when z → ∞ (as opposed to z2(D−5) Φ = −1 + γ dz ∧ d0x, 1 (z + sx)2 (z + s)(z − sx) in the generic case). Hence the integral has a trivial monodromy at infinity, which changes its topological proper- 4s3x 0 ties. We will see that this means the size of the basis |χ| Φ2 = γ 2 2 2 2 2 dz ∧ dx, (5.6) (z − s )(z − s x ) drops from 3 to 2. s 2s2x(x − 1) In order to be able to consistently use the techniques Φ = −1 + γ dz ∧ d0x, 3 (z − sx)2 (z − s)(z + sx) of the previous sections and make the connection to elliptic curves known from the literature [49, 52, 88], let us where γ = (D−5)/2. Using the definition (4.11), we redefine evaluate: 2 2 −1/2 −1/2 1 z −s ϕ ˆ 0 Be(z) := − , ϕ := . (6.2) hΦi|ϕli =: Cil dx, (5.7) 64s2 z2−ρ2 e z2 − ρ2 with −1/2 −1/2 This leaves the combination B(z) ϕ = Be(z) ϕe in-  7x2+2x−1 2 x−1  variant, but makes the integral defined on (x−1)x(x+1) − x−1 − x(x+1)  2  1 Cˆ =  − 2 4x −  . (5.8) X := CP \ {−ρ, −s, s, ρ}, (6.3)  x−1 (x−1)(x+1) x−1  x−1 2 7x2+2x−1 − x(x+1) − x−1 (x−1)x(x+1) with the point at infinity included (hence we require that ϕe does not have poles at z = ∞). Indeed, the double- Notice that singularities in x can only occur when two cover of X is an elliptic curve branching at the four points punctures out of {−sx, −s, s, sx, ∞} collide, i.e., where {−ρ, −s, s, ρ}. For simplicity, we continue to work directly x is −1, 0, 1 or ∞. The matrix C was already computed on the base space X. in (4.9), which after plugging in (5.4) gives the one-form That the size of the basis is 2 can be counted by solving Ω: 1 ωe := − 2 d log Be(z) = 0, which has two solutions, z∗ = X (p) 0 0, ∞. Alternatively, it is easily seen that |C1] and |C2] Ω = γ Ω d log(x−p), (5.9) from (3.13) define two independent cycles on the elliptic p∈{−1,0,1} curve, while |C3] is homologous to |C1]. Finally, let us see the same fact from intersection numbers, by choosing the with bases of twisted cocycles hϕi| and |ϕji as in (3). (This  1 0 −1 1 1 1 might not be the optimal choice for D=4, but we use it (−1) (0) Ω =  1 2 1 , Ω = 0 0 0 , for consistency.) Using the definition (4.11) with ωe, but −1 0 1 1 1 1 summing over p ∈ {−ρ, −s, s, ρ} we find:   2 0 0  1 3ρ 1  Ω(1) = −1 0 −1 . (5.10) 2 s−ρ    3ρ−2s − 6ρ 3ρ−2s  Ce = − 3 s−ρ s−ρ s−ρ , (6.4) 0 0 2 3ρ (s+ρ)  3ρ  1 s−ρ 1 Note the symmetry Ωij = Ω4−i,4−j, which descents from symmetries of P. This is a Fuchsian system, which can where be solved using standard techniques [8,9, 54, 86, 87]. Ce ij := hϕei|ϕeji . (6.5) 9

The rank of this matrix is 2, which signals that there are amplitudes in terms of a basis of master integrals, and the only two linearly-independent twisted cocycles. From here recent development of ideas for the evaluation of the lat- it is also seen that hϕ1| and hϕ3| are linearly dependent, ter seem to offer a new perspective on Feynman calculus. and hence we choose the basis to consist of hϕ1|, hϕ2| and The exploitation of existing relations between multi-loop similarly for the dual cocycles. This means we can no integrals is of fundamental importance to minimize the longer exploit the symmetry properties under s → −s, computational load required for the evaluation of scatter- ρ → −ρ. ing amplitudes that, according to the number of involved As in the previous sections, we decompose hϕe| = particles and to their masses, depend on several kinematic dz/(z2−ρ2) into the basis by computing (4.2): scales. Integration-by-parts identities have been playing a fundamental role since their discovery, about forty years 2 ago. X −1 s+ρ hϕ| = hϕ|ϕji (Ce )jih ϕi| = ρ hϕ1|+ hϕ2|, (6.6) e e e e e 2 e In this work, we introduced the tools of intersection i,j=1 theory, borrowed from the theory of Aomoto–Gel’fand hypergeometric functions, to Feynman integrals. We iden- where we used: tified a correspondence between the forms associated to ρ2−2s2+3ρs 2s generalized hypergeometric functions and Feynman inte- hϕ|ϕ i = , hϕ|ϕ i = − , (6.7) e e1 3ρ3(ρ2−s2) e e2 ρ2(ρ2−s2) grals in the Baikov representation, and demonstrated the applicability of intersection theory, by discussing choices of bases of twisted cycles and cocycles, basis reduction and we used the corresponding 2×2 minor of Ce from (6.4). in both spaces, as well as derivation of the differential Alternatively, we could have computed that hϕ3| = hϕ1| and obtained the same result (6.6) from (4.18). equations. We applied it to the special case of a 2-loop Let us derive differential equations for this basis. The non-planar 3-point function with internal massive propagators, both in arbitrary D dimensions and in the four forms Φ from (5.1) are related to those in (5.6) by Φ = ei ei dimensional case, which, as studied in the recent litera- Φ /(z2−ρ2) and γ = −1/2. Hence we have: i ture, is known to involve elliptic integrals. In particular, we analyzed maximal cuts of Feynman integrals, which hΦ |ϕ i =: C d0x, (6.8) ei el b il have the property that the twisted cocycles have singu- with larities only along the vanishing of Baikov polynomials, that defines the cut surface. For non-maximal cuts, by " 1 2 # definition, there exist singularities of twisted cocycles that 1 3x2 x(1−x) Cb = 4 2 2 4 , (6.9) do not coincide with the cut surfaces. In these cases, one s x (x −1) − 2 x(1+x) x −1 needs to consider relative twisted homology and cohomology groups. The definition of intersection numbers in Plugging into the expression for Ω in (5.4) (upon replacing such cases was discussed in the mathematics literature ϕ → ϕe), we find: only very recently [89]. It is expected that it can also help  1−2x 1  with lifting the assumption of genericity of D, so that − one can study integer dimensions directly in more general (x−1)x 2x  0 Ω =  2 1  d x. (6.10) cases. We leave these questions for future research. − Even in the realm of maximal cuts, the variety defined x2−1 x+1 by the vanishing of the Baikov polynomial, could become This provides a linear system of two coupled differential more involved at higher loops or with more kinematic 0 equations, d Pij = Ωik ∧ Pkj. We can solve them to scales, which complicates the determination of relevant obtain second-order decoupled equations for P1j and P2j bases of twisted cocycles. It would be interesting to separately: study whether there exist consistent changes of variables that simplify the decomposition (3.5) into linear factors, 2 2 x(x+1)(x−1) ∂xP1j perhaps at the cost of introducing different exponents, as +2(x−1)(2x2−1) ∂ P + (2x2−3x−1) P = 0, (6.11) is the case for, e.g., the Appell function F4 [31]. x 1j 1j Our analysis can be considered as a preliminary explo- 2 2 2 x x −1 ∂xP2j + 5x −1 ∂xP2j + 3x P2j = 0. (6.12) ration of the possibility of developing a method for the direct decomposition of Feynman amplitudes in terms of In each case, a basis for the two solutions is provided by a basis of independent integrals by means of projections, different choices of twisted cycles. where the elements of intersection theory allow to define a metric within the space of Feynman integrals. 7. DISCUSSION

The mathematical structure of scattering amplitudes is richer than what is currently known. The study of geometric aspects related to the decomposition of multi-loop 10

ACKNOWLEDGMENTS Out of the independent momenta (for simplicity we assume that D ≥ E, though it is not necessary) We thank the organizers and participants of the UCLA Bhaumik Institute’s “QCD Meets Gravity 2017” work- qi ∈ {`1, `2, . . . , `L, p1, p2, . . . , pE}. (A.2) shop, where this work was initiated. S.M. thanks Nima involved in the scattering process, we can construct Arkani-Hamed, Ruth Britto, Freddy Cachazo, Alfredo the Gram matrix G := [q · q ], which has M := Guevara, Keiji Matsumoto, Oliver Schlotterer, Ellis Ye {qi} i j LE + 1 L(L + 1) independent entries depending on loop Yuan, and Yang Zhang for useful comments and discus- 2 sions. P.M. wishes to acknowledge stimulating discussions momenta, ì · qj. We use them to express each of the N with Amedeo Primo, Ulrich Schubert and Lorenzo Tan- inverse propagators as: credi on contours and starters, and with Hjalte Frellesvig, X ij 2 Federico Gasparotto, Stefano Laporta, Manoj K. Mandal, Da = Aa ì ·qj + ma, (A.3) Luca Mattiazzi, Giovanni Ossola, Ettore Remiddi, Ray i,j Sameshima, William Torres, and Roberto Volpato. This where the sum goes over M kinematic invariants ` · q research was supported in part by Perimeter Institute i j and m denotes the mass of an internal particle. Here for Theoretical Physics, and by the Supporting TAlent a A defines an N × M matrix, whose rows are labelled in ReSearch at Padova University (UniPD STARS Grant by propagators and columns by Lorentz products ` · 2017). Research at Perimeter Institute is supported by i q . For later convenience, we extend A into an M × M the Government of Canada through the Department of j invertible matrix by introducing additional propagators Innovation, Science and Economic Development Canada D , D ,..., D called irreducible scalar products. and by the Province of Ontario through the Ministry of N+1 N+2 M We choose their powers to be ν ≤ 0, so that they only Research, Innovation and Science. a appear in numerators. The Baikov representation makes manifest the prop- agator structure of a given Feynman integral. In order Appendix A: Derivation of the Baikov to do so, we first change the integration variables from Representation µ ì to ì · qj. This is done by decomposing each loop k ⊥ In this appendix we review the derivation of the Baikov momentum ì = ì + ì , into the (E+L−i)-dimensional representation [13, 32, 58] for a scalar Feynman integral space spanned by {`1, `2, . . . , ì−1, p1, p2, . . . , pE} and the with L loops, E+1 external momenta, and N propagators: orthogonal complement [58]. (The order in which loop momenta are labelled does not matter for the final result.) The integration measure becomes: Z L D N Y d `j Y 1 I := . (A.1) ν1ν2···νN D/2 νa D π Da R i=1 a=1

Z dD` Z dE+L−i`k dD−E−L+i`⊥ i = i i D/2 D/2 D π D π R R E+L D−E−L−2+i −E−L+i 1Z det G 2 π − 2 Y {ì,...,`L,p1,...,pE } = det G d(ì ·qj) . (A.4) Γ D−E−L+i {ì+1,...,`L,p1,...,pE } det G 2 Γi j=i {ì+1,...,`L,p1,...,pE }

Gram determinants arise as volumes of parallelotopes we obtain: spanned by the relevant momenta and enter as Jacobians Z QL QE+L for the change of variables. (When i = L, the Gram D−E−L−1 i=1 j=i d(ì ·qj) 2 Iν1ν2···νM = c B , (A.6) determinant in the denominator is that of only external QM νa Γ Da momenta.) In the second line the orthogonal directions a=1 ⊥ ì were integrated out directly in spherical coordinates, where which results in the constraint on the integration domain: L−M L − 2 π 2 det G c = {p1,...,pE } , (A.7) QL Γ D−E−L+i ⊥ 2 det G{ì,...,`L,p1,...,pE } i=1 2 Γi := (ì ) = > 0 . (A.5) det G{ì+1,...,`L,p1,...,pE } is a constant, the integration domain is Γ := Γ1 ∩ Γ2 ∩ Applying this decomposition to all L loop momenta, · · · ∩ ΓL, and the rescaled Baikov polynomial reads det G B := {`1,...,`L,p1,...,pE } , (A.8) det G{p1,...,pE } 11

1 The constraints on Γ imply that B > 0 everywhere on the where a := 2πi {|za| = ε} and C is the intersection integration domain. Since the inverse propagators Da are related to `i·qj by C := Γ ∩ {z1 = 0} ∩ {z2 = 0} ∩ · · · ∩ {zN = 0}. (A.11) a linear transformation (A.3), we can make an additional change of variables to express the Feynman integrals in Evaluating (A.9) on this contour, we obtain the Baikov representation of maximal cuts: terms of M variables za := Da: Z M D−E−L−1 Z 2 D−E−L−1 Y dza Mν1ν2···νM := B(z) ϕ(z), (A.12) 0 2 Iν1ν2···νM = c c B νa , (A.9) C za Γ a=1 where z = (zN+1, zN+2, . . . , zM ) denotes the collective where c0 = (det A)−1 is a constant Jacobian. We set integration variable and c c0 = 1 for convenience. B and Γ are the same quantities as before, but expressed in terms of the new variables z . B(z) = B . (A.13) a z1=···=zN =0 The expression (A.9) is called the Baikov representation. Maximal cut corresponds to choosing the integration The (M−N)-form ϕ(z) is a result of the residue compu- contour: tation, i.e.,

I M 1 ∧ 2 ∧ · · · ∧ N ∧ C, (A.10) D−E−L−1 D−E−L−1 Y dza − 2 2 ϕ(z) := B(z) B νa . (A.14) za 1∧···∧ N a=1

For example, when ν1 = ··· = νN = 1, we simply have QM νa ϕ(z) = a=N+1 dza/za .

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