Duals of Feynman Integrals, 2: Generalized Unitarity

The first paper of this series introduced objects (elements of twisted relative cohomology) that are Poincar\'e dual to Feynman integrals. We show how to use the pairing between these spaces -- an algebraic invariant called the intersection number -- to express a scattering amplitude over a minimal basis of integrals, bypassing the generation of integration-by-parts identities. The initial information is the integrand on cuts of various topologies, computable as products of on-shell trees, providing a systematic approach to generalized unitarity. We give two algorithms for computing the multi-variate intersection number. As a first example, we compute 4- and 5-point gluon amplitudes in generic spacetime dimension. We also examine the 4-dimensional limit of our formalism and provide prescriptions for extracting rational terms.

4 Compact support map for higher degree forms (p > 1) 23

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Recently, an option for extracting the c i that bypasses the generation of IBP identities has been suggested [38][39][40][41][42][43]. Working with Feynman integrals modulo IBP identities means that we are actually interested in the cohomology class of a Feynman integrand. These classes come with a nondegenerate pairing called the intersection number, which pair integrands with suitably defined dual forms. Then, a Feynman integral can be projected onto a chosen basis via the intersection number where f ∨ i is dual to the integrand of F i and I is the integrand for the process at hand. Thus, provided that the dual space is known, the coefficients c i can be extracted.
In [44], we identified the space of dual forms as being a certain relative twisted cohomology group. Unlike normal loop integrands, dual forms are supported on a subset of generalized unitarity cuts and contain δ-functions that force some propagators on-shell. All other factors are essentially polynomials. That is, any propagator is either cut or absent. For maximal cuts, the pairing becomes a series of residues as in generalized unitarity, yet, no information is lost even for non-maximal cuts. We then discussed the differential equations which simultaneously characterize dual forms and Feynman integrals. This provides a mathematical basis for unitarity methods, and an alternative explanation for why scattering amplitudes A are determined by on-shell subprocesses.
The (co)homological study of Feynman integrals dates back to the 1960's (a nice selection of reprints can be found in [45]). However, missing the tools of dimensional regularization [46] progress was quickly hampered. Recently, there has been a resurgence of interest in (co)homological studies of Feynman integrals. The appearance of interesting transcendental numbers (multiple zeta values and generalizations) established geometric and number theoretic connections to Feynman integrals (see [47][48][49][50][51][52][53][54] for some examples) where these "interesting" numbers arise as periods associated to the geometry of a given Feynman integral. More recently, [38][39][40][41][42][43] have shown how to make sense of the geometry (or (co)homology) of dimensionally regulated integrals.
In order to exploit the factorization of cuts into products of on-shell trees, we find it compelling to work directly in the physical momentum space, rather than in auxiliary spaces such as Feynman parameters. Since many mathematical theorems only apply to "generic" situations where propagators are raised to non-integer exponents, earlier works along this line [38][39][40][41][42][43] added infinitesimal exponents to all propagators, which promotes poles to branch cuts but unfortunately seems to obscure the connection with generalized unitarity. The main result of [44] is that the physical case (propagators with integer powers) is intrinsically simpler, as noted above: propagators become geometric boundaries (hence "relative" cohomology) and never appear in denominators.
The present paper focuses on the extraction of integral coefficients, c i , via the intersection with dual forms. The key idea is to start with a canonical algebraic (meromorphic) representative of a dual form and apply a compact support isomorphism: the c-map, to bring it to a compactly supported version, which can be used in an integral. This allows to compute intersection integrals in a sequence of algebraic steps. The c-map is unique JHEP04(2022)078 modulo exact forms, and different choices can produce various equivalent formulas for the same coefficient. Section 2 reviews the main features of dual forms. Section 3 provides an example driven introduction to intersection theory as well as the compactly supported versions of dual pentagons, boxes and triangles. The c-map for (p > 2)-dual forms is presented in section 4. In section 5 we provide a detailed example of the formalism of 4 by computing the compactly supported bubble dual form at 4-points. As an application, we compute the one-loop 4-and 5-point gluon helicity amplitudes starting from cut representations of the integrands in section 6. We also examine the 4-dimensional limit of our formalism and provide an algorithm to extract the rational terms of 4-point amplitudes (higher multiplicity is left for future work). We conclude in section 7.

Review of dual forms
In this section, we review the basics of one-loop dual forms. The first step is to define the differential form d d when d is continuous. Then, we explain how Feynman integrals are naturally elements of a twisted cohomology group and elucidate the associated dualcohomology group.

Feynman integrals and twisted cohomology
To define the measure d d for continuous d, we set d = d int −2ε where d int ∈ N is a nonnegative integer, and split momenta into a physical (integer dimensional) subspace and an extra (−2ε)-dimensional subspace. All external momentum p i live (by definition) in the physical d int -dimensional subspace, while the loop momenta i can have a non-trivial ⊥ component The measure then follows from the product rule for dimensional regularization [46], and from the fact that Feynman integrals depend only on the combination 2 ⊥ : . (2.2) Note that, viewed as a function of complex momenta, the measure is multivalued: the "twist" u has a non-integer exponent whenever ε = 0. We thus treat a one-loop integrand in dimensional regularization as a (n = d int +1)-form.
The generalization to L loops includes L(L + 1)/2 dot products i⊥ · j⊥ [44]. The standard integration contour for a Feynman integral is the R n subspace consisting of real Minkowski momenta (with the usual Feynman i0 prescription), times the region over which the Gram matrix i⊥ · j⊥ is positive definite. However, this contour plays no role in the integral reduction problem that is the focus of this paper: the intersection pairing to be defined shortly involves a (2n)-dimensional integral over all of C n .

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To more easily keep track of the multivaluedness of a Feynman form, it is convenient to factor out the twist u and work only with single-valued forms ϕ: We can completely "forget" about the multivaluedness by further introducing a covariant derivative d(u ϕ) = u∇ ω ϕ (2.4) where ∇ ω = d + ω∧ with ω = d log(u) = −εd log( 2 ⊥ ). The connection is curvature-free and keeps track of the multi-valuedness of the original integrand, somewhat analogously to the gauge potential in the Aharonov-Bohm effect.
One property of the physical integration contour will be important to us: total derivatives integrate to zero. This means that Feynman integrands are only defined modulo total derivatives or equivalently up to integration-by-parts identities: uϕ uϕ + d(uψ) where ψ is an (n−1)-form if ϕ is an n-form. Note that we could equivalently shift the single-valued form ϕ by a covariant derivative: ϕ ϕ + ∇ ω ψ. This allows us to talk about integration by parts in a single-valued framework.
The modding out by total (covariant) derivatives is precisely the idea behind de Rham cohomology groups, whose elements are equivalence classes of closed differential forms (those with vanishing total derivative) modulo exact differential forms (those that are a total (covariant) derivative). Intuitively, the integral of a closed form is unchanged under small contour deformations, while exact forms are integration-by-parts identities. Thus, Feynman integrands are elements of twisted de Rham cohomology groups: ker∇ ω : Ω n dR (X) → Ω n+1 dR (X) Im∇ ω : Ω n−1 dR (X) → Ω n dR (X) (2.5) where Ω p dR (X) is the set of all smooth p-forms on the manifold X. For Feynman integrands, the manifold is simply complex space with all singularities excised: X = C n \ { 2 ⊥ = 0} ∪ {D = 0}. Points on the zero locus of the twist { 2 ⊥ = 0} and infinity { 2 ⊥ = ∞} are called twisted singularities or twisted boundaries [38][39][40][41][42][43][44]. These singularities are regulated by ε in the sense that the covariant derivative is locally invertible. On the other hand, unregulated singularities or poles appear whenever a propagator vanishes {D = 0} [42,44,55]. Near a pole the covariant derivative can only be inverted in the absence of a residue, and is only invertible up to an integration constant.

The intersection pairing and dual forms
We review the intersection with dual forms following [44]. By Poincaré duality in (2n)dimensional space, p-forms are dual to (2n − p)-forms where we obtain a number by integrating their wedge product over the full n-dimensional complex space. For this product to make sense, the pairing must be single-valued. Thus, the dual forms come with the opposite twisting: u ∨ = u −1 . The intersection pairing is then [56,57]:

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where T denotes the transpose of the wedge product. 1 Since X is non-compact, ϕ ∨ c must have compact support so that the integral is well-defined for any representative ϕ (ie. converge near poles and branch points of the Feynman integral). 2 Demanding that (2.6) is invariant under changes of representative ϕ implies that ϕ ∨ is closed, and since ϕ is closed it is easy to see that ϕ ∨ is defined modulo exact forms. Thus the duals of Feynman integrals are also elements of a cohomology. However, it is a different cohomology of compactly supported forms with the opposite twist: While smooth compact forms make it easy to see that the above pairing is well-defined and non-degenerate, they are cumbersome to work with. The main idea of [44] is to populate the space using algebraic representatives. Intuitively, we can account for the compact support property by adding small boundaries around the singularities. Integration by parts on a manifold with boundaries produces surface terms, and relative cohomology offers a simple bookkeeping scheme to track those: relative twisted de Rham cohomology is isomorphic to compactly supported twisted de Rham cohomology. As all our boundaries satisfy simple algebraic equations, the smooth relative space is further isomorphic to an algebraic version: The first space is where dual forms can be most simply written down; in [44] the differential equations satisfied by dual forms were verified to match with the (transpose of the) ones for integrals, thus verifying the isomorphism. The third space is where the intersection 2.6 is calculated in practice. Describing the isomorphisms will be the main topic of this paper. Elements of relative cohomology are simply formal sums where each term has support on one of the boundaries, which we denote as: where the first term is a bulk form while the remaining terms are supported on boundaries indicated by the symbol δ I . Here I is a set indexing a given boundary: δ 1 is the D 1 = 0 boundary, δ 12 is the D 1 = D 2 = 0 boundary and so on. It is also important to note that the δ I are totally antisymmetric in the index I: δ ij = −δ ji . Furthermore, each form is multiplied by the symbol θ which reminds us to keep track of boundary terms when taking a derivative

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The rules are easy to remember if one views θ as a literal product of step functions, each of which vanish in the neighbourhood of one boundary (see above eq. (3.1) below): θ = i θ i . Then, the notation naturally suggests: In the absence of twist, the isomorphism between smooth relative and smooth compact cohomologies would essentially replace the combinatorial symbols θ and δ by literal Heaviside and delta functions [44], which satisfy equivalent rules.
Since generic smooth forms are cumbersome to manipulate, we work primarily with elements of algebraic cohomology, which is naturally embedded as a subgroup. To compute intersection numbers we thus use a series of isomorphisms 3 Algebraic forms are, by definition, holomorphic and the c-map produces representatives in the compactly supported cohomology whose anti-holomorphic dependence enters exclusively in a very simple manner: through δ-functions supported on products of small circles. Thus, intersection numbers are computed algebraically via residues!

A basis of one-loop dual forms
The geometry of one-loop integrals is particularly simple due to the fact that the cohomology of the bulk space is trivial and all non-trivial elements are supported on at least one boundary (for multiloop integrals, at least one propagator must be cut in every loop). Cutting a propagator sets 2 ⊥ to some quadric in the physical loop momentum variables while linearizing all other propagators. This means that the twist locus is a sphere while the propagator/boundaries are hyperplanes. Moreover, further cuts do not change the geometry because the intersection of a sphere and a hyperplane is just a lower-dimensional sphere.
Near 4-dimensions (d int = 4), one-loop dual forms with six or more cut legs vanish since the hyperplanes have vanishing intersection; this reflects the well known reducibility of integrals with six or more legs. The basis thus consists of a pentagon, boxes, triangles, 3 In a situation where the definitions of relative algebraic and relative de Rham cohomology would become ambiguous, for example if twisted and untwisted boundaries intersect, the isomorphism with H dR,c should be taken as the primary definition of the dual space.

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bubbles and tadpoles. A uniform transcendental (UT) dual-basis (for d int = 4) is [44] Tadpoles: (2.10) Bubbles: Triangles: Boxes: where the radii r • are ratios of kinematic determinants and the normalizations c i are such that the associated differential equation is pure. 4 Here, the k µ I is the loop momentum in the directions transverse to the cut I. Note that the integration measure d is a sign factor that preserves the orientation of the original measure and keeps the dual pentagon independent of the order in which the cuts are taken. Similarly, the anti-symmetry in the d • k I compensates for the anti-symmetry of the δ I so that the basis forms are independent of the order in which cuts are taken.
For example, for a planar 5-point amplitude with generic masses, the basis contains a unique pentagon, 5 boxes, 10 triangles, 10 bubbles and 5 tadpoles for a total basis size of 31. Taking various kinematic degenerations can cause the cohomology on various cuts to become trivial decreasing the size of the basis. In the total massless degeneration, all tadpole-and triangle-cut cohomologies as well as half of the bubble-cut cohomologies become trivial decreasing the basis size to 11.
Each basis dual form has singularities on the zero locus of a (cut dependent) sphere. In the case of the tadpole and bubble, the basis forms have double poles at the zero locus of the sphere. These higher order poles act as dimension shifts so that the tadpole and bubble dual forms extract the coefficient of the 2-dimensional tadpole and bubble coefficients. The dimension shifted tadpoles and bubbles are naturally pure functions in 2-dimensions.
The purity of the integrated expressions can be seen from the ε-form of the dual differential equation [44] JHEP04(2022)078 and where the brackets (a J b J ) are dot products of embedding space vectors (see [44] for the details). We will always choose a basis of Feynman forms that is dual to the basis (2.10)-(2.14): ϕ ∨ J |ϕ J ∝ δ JJ . This ensures that the kinematic connection for the basis of Feynman forms is the minus transpose of the dual kinematic connection: Ω JJ = −Ω ∨ J J . For generic masses, the basis of Feynman forms is simply the standard UT basis of scalar one-loop integrals. However, for degenerate kinematics such as in the complete massless limit, the basis of Feynman forms changes discontinuously and, in this case, are reorganized into linear combinations of the old basis such that the box (pentagon) integrals are finite (vanishing) at d = 4. In the following, we will use the standard G-notation to denote the (near 4-dimensional) Feynman integrals where C 4 and u are given by (2.2). Below we review the massless 4-and 5-point degenerate limits from [44]. At 4-points, a standard basis of UT Feynman forms is [58] Integrating these forms yields the basis of UT Feynman integrals where While this basis is not orthonormal to our dual basis, the following linear combinations are: Consequently, they satisfy the following differential equation
For massless five point scattering, the basis of Feynman integrals consists of 5 bubbles (2.26), 5 boxes with a massive corner and a single pentagon. The boxes with one massive corner have a distinct expression from (2.27) (2.29) 5 For Feynman integrals, the canonical contour R 4 × R + is closed since the twist ( 2 ⊥ ) −ε regulates singularities at the boundary 2 ⊥ = 0, ∞. Note that the integrals for generic ε are defined by analytic continuation from a region of convergence. Therefore, there is no boundary term at 2 ⊥ = 0, ∞ regardless of the sign of ε. This contour is an element of the twisted homology group H5(C 5 \ { 2 ⊥ = 0, ∞} ∪ {D = 0}, Lω).

Introduction to intersection theory, and pentagons, boxes and triangles
In this section we will illustrate how to use equation (2.6) and provide an explicit realization of the map from holomorphic (algebraic) forms to their compactly supported versions. Section 3.1 introduces the required distributions for the compactifying procedure. We then work through examples on both compact and non-compact manifolds in sections 3.2 and 3.3. The compact supported versions of the dual pentagon, box and triangle are constructed in sections 3.4, 3.5 and 3.6.

The θ and dθ distributions
In practice, we will need the following two basic distributions: • θ(z) ≡ θ(|z|−η): a 0-form that vanishes inside a disc of radius η around z = 0, and is unity outside.
The later satisfies the following useful property (see figure 1): or more simply, dθ(z), dz z = 1. One could take θ to be a smooth function but in practice the Heaviside distribution works just as well.
Boundary-supported forms in relative cohomology, with the boundary labeled by the combinatorial symbol δ I , are mapped to ambient forms in H dR,c via the Leray coboundary Figure 1. The regulating step function θ: equal to unity outside a small disc, zero inside. Its derivative dθ is essentially a δ function supported on the red contour and is dual to a residue contour -that is, the integral of dθ(z − z i ) ∧ f (z)dz takes the residue of f at z i .
The formal symbol θ becomes a (minimal) product of Heaviside step functions enforcing compact support. The restriction | I is technically the pullback of a projection map from a product of small circles to the codimension-|I| boundary. The formal rules of the algebraic dual forms (2.9) follow from the definition (3.2) and the chain rule. Using (3.1), the intersection with a dual form supported on a cut simply becomes The above equation demonstrates the power of relative cohomology: intersections localize to cut surfaces simplifying the calculation! Equations (3.1) and (3.3) form the basis of all calculations in this paper, and we now proceed to explain the general theory by means of examples.

Compact examples: Riemann sphere and elliptic curves
As a trivial example, consider the Riemann sphere CP 1 (without any twisting). The Betti numbers (dimensions of homology groups) are {b 0 , b 1 , b 2 } = {1, 0, 1}: the nonzero entries represent a point and the whole sphere. Algebraic representatives could be defined using Čech cohomology; we omit these and directly define algebraic-like objects that play the same role in the smooth compact world (see appendix D of [44] for its connection with the standard H 2 alg (CP 1 )). Algebraic-like generators of H dR,c (CP 1 ) are where a is an arbitrary point on the sphere. It is obvious that these are respectively closed 0-and 2-forms: dg i = 0. Geometrically, a two-form represents a density on the sphere, and from eq. (3.1) the second form can be regarded as a unit mass (times 2πi) concentrated near the point z = a. It is defined modulo total derivatives of smooth compact 1-form, and we restrict to algebraic like forms: meromorphic one-form multiplied by θ or dθ to mask their singularities. Since a meromorphic one-form on a sphere has at least two singular JHEP04(2022)078 points (including infinity), the class g 2 does not depend on the choice of point a: This is effectively the familiar holomorphic anomaly equation, which technically corresponds to omitting the θ factors on the left-hand-side, which would replace the right-handside by an equivalent distribution. In this compact example, the cohomology H n c (CP 1 ) is its own dual, and integration yields a non-degenerate intersection pairing between 0 and 2 forms: As a second compact example, consider an elliptic curve: , topologically a torus. It is well-known that H 1 is now two-dimensional (a torus has two basic 1-cycles), and we again have one-dimensional H 0 and H 2 with representatives similar to eq. (3.4). A standard basis of algebraic 1-forms spanning H 1 is: While it is easy to see that both forms are naively closed (they depend only on holomorphic variables), the second one is not a well-defined smooth-compact form due to its singularity near infinity. However, it is readily patched up to define a well-defined closed form. To see this, a good local coordinate there is w = The fact that the residue vanishes is significant: otherwise the second form couldn't be closed due to a holomorphic anomaly. However, the form is still not closed since d(dw/w 2 ) = 0 is a derivative of the holomorphic anomaly. The following general procedure will be used throughout the paper to fix this. We excise a small neighborhood of the singularity by slapping on a step function θ(w), and then we add a dθ term to "patch it up" and make the result closed: This is closed provided ψ is a local 0-form primitive: dψ = x y dx. Because dθ(w) is concentrated on a small circle around w = 0, in practice, ψ need only be computed as a Laurent series in w. From eq. (3.8), where C is an arbitrary integration constant, which has no effect on the cohomology class of g 1 (the form dθ(w) is cohomologous to zero). Note that a primitive ψ only exists when the residue vanishes (log w would not be a valid single-valued primitive). Thus we defined a valid de Rham class g 1 canonically associated to x dx/y: it "looks" like it everywhere JHEP04(2022)078 except near w = 0, where its singularity is regulated in a unique way. The upshot of this method is that it is easy to compute intersections, because regularization has brought in an antiholomorphic differential dz (through dθ): 6 There are two important lessons from these examples. First, using the "building block" distributions θ and dθ it is straightforward to write algebraic-like representatives of de Rham cohomology classes, where all the "antihomolorphic" dependence is hidden inside δ-functions. Second, thanks to these δ-functions, intersections of such forms can always be computed algebraically (by residues), even on topologically nontrivial spaces such as an elliptic curve.

Non-compact examples: complex plane and degenerate 2 F 1
The simplest non-compact complex space is perhaps the complex plane C = CP 1 \ {∞}. Because of non-compactness, its de Rham cohomology H n dR (C) is not isomorphic to its dual H n dR,c (C). Indeed, H 2 dR (C) = 0 (any two-form density can be pushed out to infinity), while H 0 dR,c (C) = 0 (there exists no compactly-supported constant function). However, the remaining CP 1 generators in eq. (3.4) survive, and the cohomology and its dual admit the following generators: Although these are now clearly non-isomorphic, the non-degenerate pairing between them has survived. This is the most general form of Poincaré duality that we will use in this paper: that integration gives a nondegenerate pairing between H 2n−k dR,c and H k dR . We finally turn to a realistic model for Feynman integrals: a family of 2 F 1 hypergeometric integral which contains both poles and branch points (arguably the two essential features of a Feynman integral). Consider the family, for (possibly negative) integers m, n, p: 14) The exponents α i ∈ C are assumed noninteger, and we will write this as u ϕ where u = z α 0 (1 − z) α 1 denotes the multi-valed part; x ∈ C\[0, 1] and B is the Euler beta function. 6 Upon writing the volume 2-cycle on the torus as the product of geometric A and B 1-cycles, this essentially gives Legendre's famous period relation:

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For any closed contour, integration-by-parts in z produces valid identities between these functions since surface terms vanish (due to the non-integer exponents at the boundaries 0, 1 and ∞). 7 The space of independent integrals is thus the cohomology where we introduced the covariant derivative ∇ ω = d + ω with connection We will be interested in the problem of integral reduction: what is a basis of the cohomology (3.15), and how do we express a given integral I m,n,p in this basis? For hypergeometric functions with generic indices, this problem has been considered in many references [56,[60][61][62][63][64][65][66]; in the generic setup, ω contains an extra term α 2 d log(z − x) which turns that pole into a branch point. The cohomology is two-dimensional (reflecting the fact that 2 F 1 satisfies a second-order differential equation) and is dual to essentially the same space with connection −ω. Their non-degenerate pairing allows for efficient extraction of coefficients.
We now detail the analogous story in the degenerate case (3.14). The key is to use the appropriate space of dual forms, which is now spanned by compactly supported forms that vanish near the boundary z = x: For concreteness, we choose the following convenient representatives of H 1 dR [55]: It is easy to prove directly that any form in (3.15) can be reduced to this basis (establishing that the cohomology is at most two-dimensional), using the following observations. 8 First, forms with singularities stronger than a single pole at 0, 1, z, ∞ can be eliminated, by adding ∇ ω ψ of a suitable 0-form ψ that has a singularity at only one of these points. This leaves the three forms d log(z), d log(1 − z) and d log(z − x) as a potential basis, however one combination is cohomologous to zero, ∇ ω 1 = α 0 d log z + α 1 d log(1 − z), which removes one. That the two classes in eq. (3.18) are indeed linearly independent will be confirmed by the non-degenerate intersection pairing computed shortly. We may depict graphically the two forms in eq. (3.18) as paths from 0 to infinity, and x to infinity, respectively, as shown in figure 2(a). One might then guess, geometrically, "orthonormal" dual cycles in figure 2(b) which intersect the cycles of figure 2(a) only once. We will now find explicit dual forms which precisely reflect this picture. 7 In this example, closed contours can have the endpoints 0, 1 and ∞. The endpoint x is special and its treatment depends on which space we are considering: X or X ∨ (see figure 2.). On X, the endpoint x can only be encircled since there is no non-integer twisting at x and the forms may be singular at x. On the other hand, x is a valid endpoint on X ∨ since the forms are non-singular there. However, taking x as an endpoint generates boundary terms when integrating-by-parts. 8 There is no need to make either of these forms compact support, because their singularities are outside the space = CP 1 \{0, 1, x, ∞}. For generic twist αi, one can also show that H 0 dR (X; ∇ω) = 0 = H 2 dR (X; ∇ω).

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n A a 5 H I N E 3 l n M Q p U 2 + o W t X S D c 2 E h I e w a k O k H u U U k I t i 3 o P U V W m U a p J l t v U j / r k + 4 u M P d 8 S t D 3 L G r f 9 1 y M
. This will be a very convenient choice for our second dual basis element: The factor u(z)/u(x) is required to make the second form closed; it is single-valued along the small circle on which dθ is supported. Let us now elaborate on the first form, which is a "compactly supported version" of the algebraic form d log z 1−z . It is constructed by a procedure analogous to that used in eq. (3.10): we slap a step functions then add patch-up terms to make the result closed. In general: where z i ∈ {0, 1, x, ∞} and ψ i = ∇ −1 −ω ϕ ∨ is a local primitive of ϕ ∨ around z i . This primitive is unique around each point with a twist, and can be obtained as a Laurent series. For example, near the origin, we have which implies that ∇ −ω can be uniquely inverted order by order in a Laurent series. The procedure generates denominators of the form 1/(n − α 0 ) where n is an integer. For The untwisted point is no different:

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but notice that the first term is now an integration constant, not determined by the differential equation. We set to zero the constants around untwisted boundaries as a canonical definition of the c-map. However, note that representatives of the relative cohomology cannot have poles at untwisted points: if ϕ ∨ had a reside, one would not be able to invert the covariant derivative. Instead of series solutions, one could formally write the primitive ψ z i as the integral where {s a } is a set of "kinematic" variables (in the present case {s a } = {x}). While evaluation of this integral can be challenging (it is essentially a twisted period), it does provide primitives valid to all orders in (z − z i ), which is sometimes useful. Given a holomorphic form ϕ, its intersection with the dual basis (3.19), defined by integration as usual (see eq. (2.6)), can now be explicitly computed: Notice that if ϕ only has logarithmic singularities, only the leading term of the primitives (3.22) is required. Similarly, for logarithmic forms, the factor u(z) u(x) is irrelevant. Then the intersections simplify to The full primitives ψ i (and u(z)/u(x)) are however very important if ϕ has higher-order poles. The number of required terms is dictated by the order of the poles in ϕ. Eq. (3.26) immediately yields the intersection with the basis |ϕ j = {d log z, d log(z − w)} defined in eq. (3.18): Notice that it is diagonal, as anticipated from the geometric picture in figure 2. As a simple application, consider the reduction of the integral I 1,2,2 (defined in eq. (3.14)), corresponding to the form ϕ 1,2,2 = dz z(1−z) 2 (z−x) 2 . Since it has double poles at 1 and x, all the terms in two of the primitives (3.22) are needed, and we find which implies the following identity between hypergeometric functions: which may be tested directly via numerical integration or series expansion in 1/x. Note also that here I 1,0,0 = B(α 0 , 1 + α 1 ) is a constant independent of x. The −α 0 factor originates from inverting the intersection (3.27).

Differential equation
As a further application, let us obtain the differential equation satisfied by the integral I 0,0,1 . At the level of cohomology, we would like the Picard-Fuchs (or Gauss-Manin) connection: where d full ≡ dx∂ x + dz∂ z includes both internal variables and external parameters, ∇ z ω only z-derivatives, and where Ω ij does not depend on z. The calculation is particularly straightforward if we keep the basis elements in their d log form, interpreting the total derivative d as d full -we now drop the "full" superscript. (The form d log(z − x) then includes a term dx x−z , which is a 0-form with respect to z that does not contribute to the calculation below [67].) Then The form on the right is logarithmic and we can use the simplified residue formula (3.26). For example, (3.32) Note that the large parenthesis is a 2-form proportional to dz ∧ dx; its residue at z = 0 simply extracts the d log z term and puts z = 0 in what it multiplies. (Our sign convention is that Res z=0 of a two-form plucks a factor dz/z from the left, which is compatible with the way the residue appears in our formula: Res z=0 ϕ ≡ 1 2πi dθ(z) ∧ ϕ.) It is easy to see that Ω i1 = 0 (there is no dx to be found in ω ∧ d log(z)), and the remaining component Ω 22 is a single residue at z = x of the above parenthesis. Thus altogether , (3.33) which may be compared (after integrating with respect to z) with the hypergeometric equation. To be fully explicit, the differential equation is: .
The fact that I 1,0,0 (x) ≡ C is constant was observed already.
To complete the analogy with the physical problem of Feynman integrals, one would consider the twists α i as small parameters: α i ∝ ε, and specialize to the ε → 0 limit. The differential equation (3.34) then has the canonical form of ref. [68]: proportional to ε. It can be readily integrated order by order in terms of harmonic polylogarithms, using the boundary condition that I 0,0,1 (∞) = 0 (clear from the definition (3.14)): JHEP04(2022)078

Zero-dimensional: pentagon dual
For the remainder of this section, we take d int = 4. Then, the maximal topology is the pentagon with propagators Any additional propagators will be linearly dependent on the D i≤5 since vector space of external momentum is 4-dimensional.
Compactifying the pentagon dual form is trivial since ( 3.37) where u = ( ⊥ ) −ε , u 12345 = u| 12345 = (r 2 12345 ) −ε and we have used equation (3.2) for the compactly supported Leray map. It is now straightforward to extract the pentagon coefficient of a general amplitude ϕ It is simply the max cut residue of ϕ weighted by u/u 12345 . Note that u/u 12345 is only needed when the propagators in ϕ have higher order poles.

One-dimensional: box dual
The algebraic box dual (2.13) is a one-form supported on the box cut. Since the Leray coboundary preserves compact support, it suffices to find the compactly supported version of this one-form before we apply the coboundary to it.
Specializing to the 1234-box, we have Here, 1234 = · e 1234 where e 1234 is a unit basis vector of the external kinematic space perpendicular to the momenta at each corner of the box.
On the 1234-cut, the twist is u ∨ 1234 = u ∨ | 1234 = r 2 1234 + 2 1234 ε and there is also a boundary corresponding to the pentagon cut

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Thus, we will need primitives for φ ∨ 1234 near the twisted singular points {±r 1234 , ∞} as well as the boundary point 0 ≡ r 2 12345 − r 2 1234 . These 1-form primitives are easily computed: Moreover, at one-loop, closed form expressions for the primitives exist (appendix A). Then, the 1234-box coefficient for any Feynman integral ϕ is given by the intersection where the index α runs over the twisted singular points and boundary. Note that if there are no D 5 propagators, the contribution from the 0 residue will always vanish. It is enlightening to take the ε → 0 limit of (3.45) and compare with the BCF formula for the box coefficient [4]. Only the two twisted singularities contribute: There are powers of ε on the left (c 0 = 4 ε ) because the box forms in our basis of Feynman integrals contain a factor of ε 2 . The two singular points of the twist 1234 = ±r 1234 correspond to the two on-shell solutions to the quad-cut in 4-dimensions, reproducing the BCF formula. In general, eq. (3.45) contains additional ε corrections; the residue at D 5 = 0 can contribute if some propagator is squared, and the residue at infinity is needed for ϕ that have numerators.

Two-dimensional null coordinates: triangle dual
In this section, we provide an efficient algorithm for compactifying certain 2-forms bypassing the need to introduce the machinery of fibration (section 4).
Coordinate choices greatly influence the compactification procedure. For example, one can often find coordinates where the cohomology on a fibre vanishes (or is smaller than it ought to be). Rather than being problematic, such singular coordinate choices can actually be used to simplify the compactification process. As an example, we will compactify the triangle dual form in null coordinates and obtain remarkably simple expressions for triangle dual intersection numbers. (Related but distinct coordinates could also be considered, such JHEP04(2022)078 as those used in [9][10][11], where the 2 ⊥ integral is kept for the last stage. Here we simply aim to experiment with rotations of Cartesian coordinates to see what kind of formulas we can obtain. ) We start by considering a massive triangle assuming no boundaries in order to demonstrate the procedure simply (boundaries will be added after). Specializing to the 123triangle, equation (2.12) yields, where the twist on the 123-cut is Note that we have rescaled the coordinates of (2.12) for ease of reading (effectively setting r 123 = 1).
While triangle dual can straightforwardly be compactified in these coordinates using the fibration technology of section 4 to deal with z 1 and z 2 one at a time, we can try to be more clever and linearize the twist. Changing to null coordinates z 1 → 1 2 (x + y), z 2 → i 2 (x − y) the twist and triangle dual form become Looking as a function of y one sees that the H 1 of the fibre vanishes for generic base point x; in the language of section 4 such a coordinate choice leads to singular fibres. To see what is happening, consider trying to make primitives for φ ∨ 123 : (the subscripts will become clear later). Taking the covariant derivative of the first reveals that this primitive is almost global x is a holomorphic anomaly (3.5) that is caused by the fact that the y-primitive becomes singular at x = 0. Thus, the light-cone coordinate transformation has concentrated the entire cohomology near the singular fibre x = 0. A similar formula exists for the other almost global primitive that concentrates the cohomology on the y = 0 line instead.
One shortcut for computing the particular intersection number φ ∨ 123 |φ would be to use the fact that the (non-dual) form φ also has almost global primitives. Then, one can show that φ ∨ 123 is cohomologous to #δ 2 (x) and φ is cohomologous to δ 2 (y). Since the wedge product of these delta function forms has compact support the intersection number is well defined without further compactification and localizes on x = y = 0.
However, as before, we prefer to fully compactify the dual forms and leave Feynman integrands untouched. The trick to constructing the compactly supported version of φ ∨ 123 will be to insist that ψ x-small is used when x < and ψ x-big is used when x > (see figure 3); this will yield a remarkably simple formula for the triangle coefficients in d-dimensions! JHEP04(2022)078 As the starting point, consider the following ansatz for c has compact support but is not closed. We can remove all the terms with a single dθ in ∇ c by adding some patch-up terms The terms highlighted in red showcase how the primitive in the neighbourhood of 1−xy = 0 is split between the regions where x > and x < . This splitting introduces an additional θ function at x = 0 and means that the intersection number can potentially localize on x = 0 even though it is not a singular point of the twist! Testing c for closure, we find The double dθ terms can be removed by constructing primitives for the difference ψ ∨ x-big − ψ ∨ x-small in the regions where the dθ's overlap (1) c , we find the closed and compactly supported form This result is remarkably simple -only the double primitives ψ ∨ 0,∞ and ψ ∨ ∞,∞ are needed! The choice to break up the neighbourhood about 1 − xy = 0 into the regions x > and x < is not unique and is somewhat analogous to choosing a fibration ordering. Especially when dealing with spherical twists, choices such as these are needed and break the spherical symmetry -the codimension-2 points where the intersection number localizes cannot be deduced from the twist alone.
Our work is not done yet. Equation (3.57) does not extract the triangle coefficients of boxes or pentagons since it is not compactly supported on the additional boundaries present in a box or pentagon integral. Fortunately, it is much simpler to enforce compact support at the boundaries. The trick is to consider the restriction of [φ ∨ 123 ] c to the boundary (say, D 4 for concreteness). Then, if we can find a primitive for the restriction of [φ ∨ 123 ] c , the Leray construction (see, (3.2) or [44]) effectively solves our problem. For example, let ψ 4 be a (compactly-supported) primitive for [φ ∨ 123 ] c | D 4 =0 . Then, the following is cohomologous to [φ ∨ 123 ] c above:

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where α denotes the singular points on the boundary D 4 (generically, there will be one or two finite singularities in addition to the singularity at infinity). In the case where we do not care about Feynman forms with higher order poles, the primitives ψ 4,α are easily computed since this boundary is simply a C 1 . We have tested this formula extensively on various examples with massive triangles as well as dimension shifting identities, as detailed below; it will also be used in section 6.2 for analyzing 4-dimensional limits of bubble coefficients. This procedure can then be repeated one boundary at a time. For each iteration, we obtain additional twist-boundary contributions. When there are two or more additional boundaries (like for the pentagon), we also find boundary-boundary contributions to (3.59). Proceeding in this way, one can derive residue formulas for triangle coefficients with an arbitrary number of boundaries.

Compact support map for higher degree forms (p > 1)
In this section we present an algorithm for constructing the compactly supported versions for multi-variate (p > 1) dual forms.
In its most basic form the compactly supported version of ϕ ∨ is obtained by adding exact terms which remove support in tubular neighbourhoods of { 2 ⊥ = 0} and {D i = 0}. For 1-forms, this is captured by equation (3.20). While we arrived at (3.20) by slapping step functions in front of ϕ ∨ and patching up to ensure that the resulting 1-form is closed, one can check that the difference [ϕ ∨ ] c − ϕ ∨ is exact. That is, our construction of the compactly supported 1-forms coincides with the construction of [69]. Explicitly, where Θ = α θ α = 1 − αθ α ,θ α = 1 − θ α and the θ α are step functions with support outside the neighbourhood of the singular points indexed by α. Here, the ψ α are local primitives of ϕ ∨ with respect to some connection ω ∨ . Analogously, it is possible to construct compactly supported versions of (p > 1)-forms by multiplying the original form by step functions with support outside tubular neighbourhoods about { 2 ⊥ = 0} ∪ {D i = 0} and patching up with terms ∧ j∈J dθ i ψ J such that the resulting p-form is closed at each nested boundary J [69]. While this method works well for low p, the multi-variate problem rapidly becomes computationally expensive since one needs a ψ J for each way of approaching each boundary. To avoid this combinatorial build up, it is often more efficient to break the problem into a series of p = 1 problems. This is achieved by a procedure called fibration [39,40,70]. While fibration plays an important role in practical intersection computations, we stress that it is just one particular way to compute the compact isomorphism [ϕ ∨ ] c .
Before moving on to the details, we summarize the fibration based multi-variate compactifying procedure here (see also figure 4): • Suppose that ϕ ∨ is an algebraic (p>1)-form in {z i } p i=1 and we wish to know its compactly supported version. • We choose an order of fibration, such that z i is fibred over z j>i : z 1 is the first variable to be integrated out and the last base is z p .

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• On the i th fibre, we choose a basis of 1-forms {f Here, the f (i) a are a 1-forms in dz i that are independent of z j<i and are generically vector valued (we will use bold typeface to denote vector-valued forms: f (i) a ). The original p-form can be decomposed in terms of the fibre basis elements ϕ ∨ f (1) · f (2) · · · f (p) · n (4.2) where, generically, is a vector-valued 1-form and the f (i) = a f (i) a are matrix-valued 1-forms. Here, n is a constant vector that picks out the correct combination of components with coefficients such that the above equality holds. 9 Note that there is an implicit wedge product in the vector/matrix multiplication above.
• Commuting the covariant derivative across the basis elements f (i) a , modulo total derivatives, determines the connection on the (i + 1) th fibre ( ω (i+1) ). We define a new form F (i) that differs from f (i) by an IBP form V (i) so as to remove the total derivatives and get a strict equality: (4.3) 9 As the astute reader may have guessed, the elements of n can be computed using intersection numbers.
We will however choose the {f (i) a } for a given ϕ ∨ such that the components of n are obvious.

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The decomposition of the original top form is unchanged: • The compactly supported version of φ ∨ is given by replacing the F (i) with their compactly supported versions (with respect to z i ) via the 1-form c-map Thanks to the commutation relation (4.3), showing that [ϕ ∨ ] c is cohomologous to ϕ ∨ is relatively straightforward.
While the additional vector and matrix structure may look complicated, our algorithm is still efficient since all connections are (block) lower triangular due to that fact that we are using relative cohomology rather than the parametric deformation as in [40,41,70]. Note that only the top anti-holomorphic term in [ϕ ∨ ] c will contribute to the intersection with a Feynman integrand since Feynman integrands are a top holomorphic forms. In section 4.1, we describe the fibration process that allows one to turn the multi-variate problem into a series of single-variate problems. Then, we illustrate how to compactly the fibration of ϕ ∨ one variable at a time in section 4.3. This compactification scheme leads to a simple recursive formula for the computation of any intersection number.

Fibration
We would like to write ϕ ∨ schematically as where f (i) is a 1-form in dz i that is independent of z j<i . Since only f (1) depends on z 1 , it is useful to split the action of the covariant derivative into a piece that acts on z 1 and a piece that acts on z i>1 : Here, the first term is the covariant derivative on the 1 st fibre (F 1 = z 1 ∈ C \ poles(ω ∨ )) while the second term is the covariant derivative on the 1 st base B 1 = (z 2 , . . . , z p ) ∈ C p \ poles(ω (1) ) where ω (1) is some undetermined connection resulting from the splitting into fibre and base. To determine the connection on the first base, we need to specify a basis on the fibre cohomology 1 . Then, the covariant derivative of any basis element can be expressed in terms of the same basis  where the coefficients ω (1) ab are 1-forms in the z i>1 . Note that since the base connection is matrix-valued, the elements of the base cohomology are vector-valued differential forms.
It would be preferable if (4.9) was an exact equality on the level of differential forms rather than an equivalence of cohomology classes. Therefore, we define F

a is any vector-valued form on the base,
Repeating this process, we define the 1-dimensional fibres and the accompanying (p−i)dimensional bases where ω (0) ≡ ω ∨ is the connection on the total space, B 0 . Then, for each fibre, we choose a basis, {f (i) a }, for the fibre cohomology H 1 a is a vector-valued form if the cohomology H 1 B i−1 is more than one-dimensional. As a result, on the subsequent bases we get vector-valued version of the above: where F (i) a is a completion of f (i) a by adding IPB vectors proportional to dz j≥i . The original top form ϕ ∨ can then be expressed as a general linear combination ϕ ∨ = F (1) · F (2) · · · F (p) · n . (4.15)

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The vector of forms, F (1) · F (2) · · · F (p) , span the un-fibred cohomology H p (C p \ { 2 ⊥ = 0}, ∪ i {D i = 0}; ω ∨ ). 10 In general, the coefficients n a can be computed via the intersection pairing or IBP relations. However, one can choose fibration bases such that ϕ ∨ is easily related to the f (i) a . With this notation, it is easy to see how ∇ ∨ commutes across the F 's and acts on an element of the base cohomology φ This relation allows us to apply the 1-form compact support isomorphism to each F (i) on F i independently of the other F j .

Boundaries
The next piece of fibration technology we need is the twist on B i . Since the i th cohomology is vector-valued, the associated twist u (i) is a matrix valued 0-form. The twist satisfies the following differential equations Formally, these are solved by the path ordered exponential Fortunately, explicit expressions are not required, only the abstract properties of u (i) are needed. Lastly, we need to define the Leray coboundary for vector-valued forms. One way is to use the matrix-valued twist: Then, the columns of the Leray δ D i can be used as basis forms for the boundaries. However, in practice, we find that it is easier to take a more abstract approach that does not involve the matrix-valued twist. It is easiest to illustrate the idea with an example. Suppose that there are two boundaries D a and D b on the first fibre. The basis of form on F 1 will contain a bulk form f and two boundary forms f that when wedged with F (1) produces a form on the total space

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Each component φ ∨ a belongs to the twisted cohomology produced by the corresponding diagonal component of the connection ω (1) aa . By applying the total space derivative to (4.21), the action of the B 1 covariant derivative can be determined to avoid over counting. Choosing to put all double boundary contributions in the φ ∨ 3 position, equation (4.22) becomes where the B 1 covariant derivative is defined to be Here, encodes the boundary information reproducing the double boundary term in (4.22) and When constructing a basis for the second fibre, the columns of L will be used in place of the columns of (4.19).

Compactifying
Most of the heavy lifting has already been done by fibration. Once ϕ ∨ has been written in terms of the F (i) , the compact version of ϕ ∨ is obtained by the replacement F (i) a → [F (i) a ] c , following eq. (4.5).
In the remainder of this section, we define [F (i) ] c , show that the above replacement is cohomologous to the original form ϕ ∨ and describe how to compute the intersection number.
Basis forms associated to boundaries are conveniently represented by Leray forms, which already have compact support. Explicitly, if f (i) a is a Leray basis form (a column of (4.19)), then F (i) a = [F (i) a ] c . Therefore, the remaining task is to compactify the basis elements that do not already have compact support (bulk forms or non-Leray forms).
Suppose that f (i) a bk is a bulk form. It's compactly supported version, [f (i) a bk ] c , is defined via the 1-form c-map (4.1) on the i th fibre α , and, the z i,α are the twisted singular and untwisted boundary points on F i . Here, The vector primitives (4.27) appearing in (4.29) are computed in the same manner as their 1-dimensional counterparts. The only new complication is that matrix-valued fibre connections ω ∨ F i need to be inverted. For one-loop integrals, the connections are lowertriangular and this inversion is efficient. 11 Knowing how ∇ B i commutes past [F (i) a bk ] c , is essential to proving that (4.5) is cohomologous to ϕ ∨ . Crucially, it is possible to define compact versions of F (i) a bk that also satisfy equation (4.14) However, to prove (4.30), we need to know how ∇ (i) acts on the primitives ψ (i) which can be found in appendix B.
We can now show that (4.5) is cohomologous to ϕ ∨ . Assume that we are in the middle of the compactifying procedure (4.31) To convert F (q) into [F (q) ] c , we subtract the total derivative (4.32) Repeating this procedure for each fibre, one finds that ϕ ∨ is indeed cohomologous to [ϕ ∨ ] c . Inserting (4.5) into the formula for the intersection number one finds where and ϕ is a generic Feynman integral. After fibration, the intersection number is realized recursively as a vector product of 1-dimensional intersection numbers. The vector-valued Feynman forms ϕ (i) are forms on the base B i not F i and are obtained by projecting ϕ (i−1) 11 Boundaries help keep the fibration connection simple. If all the boundaries were instead twisted singularities, the connection would be some generic matrix.

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onto the basis of H 1 F i . This formula is analogous to those derived in [40,70] -the difference being that we have chosen a different fibration order.
More concretely, the intersection number can be expressed as a series of consecutive residues. The top anti-holomorphic part of the dual form contains p-dθ functions each of which take a residue when integrated over Res zp=zp,α p · · · Res z 1 =z 1,α 1 ψ (1) whereφ is the differential stripped version of ϕ.

d-dimensional bubble coefficients
In this section we discuss the dual forms which extract bubble coefficients for planar 4-point massless amplitudes, testing out simple examples of dimension shifting identities. The 5point case is a strightforward generalization of the 4-point case detailed here. Application to gluon amplitudes are discussed in the next section. Section 5.1 fixes our conventions for 4-point integrals. A fibration for the massless 4-point problem is worked out in detail in section 5.2. Once the relation between the fibration basis and the UT dual form basis is know, the compactly supported version of the UT dual form basis is straightforwardly obtained. In section 5.3, we intersect the UT dual form basis with a basis of Feynman forms and discover that the UT dual boxes are dual to combinations of Feynman integrals, which are finite. As the fist application of the formalism of section 4, we extract the generalized unitarity coefficients of dimension shifted integrals in section 5.4.

Conventions for massless 4-particle kinematics
The propagators are defined to be 3) We use the 4-point rational parameterization of appendix C.1 for the external momenta and cartesian coordinates for the loop momentum Here, the z i are dimensionless variables and the strange numerical labeling is connected to the fibration of section 5.2. Moreover, defining 3 to be purely imaginary ensures that the propagators do not contain explicit factors of i but introduces a factor of i into the integration measure of both dual and non-dual forms (this will cancel in the end).

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Specializing to the 13-bubble cut, sets z 2 ⊥ = 1 4 − z 2 2 − z 2 3 + z 2 4 and z 1 = 1/2. Then, on this cut, the remaining boundaries become where x = t/s (recall that s = −(p 1 + p 2 ) 2 and t = −(p 2 + p 3 ) 2 are the 4-point Mandelstam invariants). In these coordinates, the twist is hyperboloidal instead of spherical: In order to avoid unnecessary square roots and factors of i, we keep the hyperboloidal twist instead of the spherical twist of (2.10)-(2.14) and [44]. With these conventions 13-bubble dual form becomes To complete the basis on the 13-cut, the 1234-box dual form must be included. On the 1234-cut, the twist factorizes into a product of hyperplanes Up to an overall kinematic factor, the box dual form is simply the volume form on the 1234-cut divided by some power of the twist. The exact kinematic factor can be obtained from (2.13) by a change of coordinates and we define as the 1234-box dual form.

Massless 4-point fibration
In order to use the fibration c-map of section 4, we need to first specify a fibration. The dimension of the fibre cohomology greatly depends on the choice of coordinates and ordering of fibres. For example, dimH 1 F 1 = 3 if z 2 is chosen as the first fibre (one bulk and two boundary basis forms) while dimH 1 F 1 = 2 if either z 3 or z 4 are chosen (one bulk and one boundary basis form). Alternatively, we could change coordinates such that z 3 no longer appears in the propagators. Then, dimH 1 F 1 = 1 if z 3 is chosen as the first fibre. Since such choices are specific to the problem at hand, we will proceed naively in our chosen coordinates and choose z 2 as the first fibre, z 3 as the second and z 4 as the third and last fibre.

Fibre 1
The first fibre is F 1 = z 2 ∈ C \ {u (0) = 0} and we choose the d log basis 12 3 = δ 4 (1), (5.11) 12 While the Leray forms are not technically d log's, we will still call them logarithmic since they are dual to d log's.

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for the fibre cohomology H 1 F 1 . Next, we determine how the covariant derivative of the total space acts on the fibre cohomology in order to obtain the connection on B 1 . The total space covariant derivative acts simply on the Leray forms: where the Q (1) aa are the quadrics appearing in the boundary twists (u aa ) ε , which are the diagonal components of the matrix twist u (1) ). Since our boundaries are non-generic (massless external and internal kinematics), the boundary twist factorizes into a product of hyperplanes (5.14) The covariant derivative of the bulk-from, f 1 , is slightly more complicated since we must add the covariant derivative of an IBP-form in order to project ∇ ∨ f (1) While the off-diagonal connections ω (1) 1a may not be d log, they share the same singularities as the corresponding diagonal component ω (1) aa . 13 Alternatively, one can compute ω (1) ab using single variable intersection numbers on F 1 . However, we will not show this here as it involves introducing a space dual to the (dual form) fibre cohomology.
To understand how the basis bulk forms (f in this case) are chosen, suppose that the twist u (0) = Q ε is a function some generic quadric Q (0) . Assuming that z 2 is still the fibre variable, we can construct a d log form from the z 2 -roots of Q (0) :

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where Disc z 2 Q (0) is the discriminant of Q (0) with respect to the variable z 2 and r i are the z 2 -roots of Q (0) . Despite the apparent square root, equation (5.21) is always algebraic since the square root cancels once the d log is expanded. When chosen as a basis element, the bulk form (5.21) produces (ε + 1 2 ) d log Q (1) 11 as its diagonal component in the base connection. The connection ω 11 introduces singularities at the roots of Q (1) 11 on the base B 1 . However, on the full manifold, these singularities are not allowed (except at z 2 = 0). Indeed, the wedge product f (1) 1 ∧ ω (1) 11 is free of spurious singularities since (5.21) contains a factor of Disc z 2 Q (0) in the numerator. For this reason, the normalization of (5.21) is particularly natural. 14 While convenient, normalizing (5.21) by the discriminant is not necessary. For example, dz 2 /Q (0) could be used as a basis form on F 1 instead. However, the off-diagonal connections would then contain singularities that do not appear in the some of the diagonal connections. This makes it impossible to find primitives for a certain class of base forms. Therefore, base forms must contain enough zeros at the bulk singularities introduced by the off-diagonal connections such that a local primitive always exists.
The last step on F 1 is to construct the improved basis forms by combining them with their corresponding IBP form: The remaining basis forms f (1) 2,3 are already in the desired form: F 2,3 . Summarizing, the improved basis satisfies where (5.24)

Fibre 2
Now, we can specify the B 1 cohomology. Following the discussion at the end of section 4.1, a generic form on B 1 is a 3-vector where each component belongs to the twisted cohomology of the corresponding diagonal of the connection such that the wedge product with the vector valued form F (1) produces a valid element of the full un-fibred cohomology. The connection on B 1 is given by where (4.26) becomes (5.26) 14 This normalization is also natural from the perspective of single variable intersection numbers. Since the intersection of identical d log forms is proportional to the inverse discriminant, f 1 |f (1) 1 11 , normalizing the dual-basis brings a factor of the discriminant into the numerator of f (1) 1 .

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Having understood the B 1 covariant derivative, we can define the second fibre F 2 = z 3 ∈ C \ poles(ω (1) ), which has covariant derivative ∇ F 2 = ∇ (1) | dz 2 =0 . The F 2 -cohomology is 4-dimensional: one boundary form generated by L (1) and three bulk forms generated by the diagonal elements of ω (1) . For the basis of the boundary cohomology, we choose the Leray form f (2) Acting with ∇ (1) , we can read off the component ω as a basis for the bulk cohomology. Using the IBP-forms it is fairly easy to check that where Note that the d log basis is somewhat special since the V (2) 1,2,3 are unit vectors multiplied by a 1-form. Generically, the other components of V (2) 1,2,3 are complicated if f (2) 1,2,3 is not a d log form. The last step on F 2 is to construct the improved basis forms ba .

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The matrix structure of ω (2) describes 4 different scenarios. The first row represents localization on the twisted singularities of F 2 after localizing on the twisted singularities of F 1 while the second and third row represents localization on the twisted singularities of F 2 after localizing on the boundaries D 2 | 13 or D 4 | 13 on F 1 . The last row represents localization on the double boundary D 2 | 13 = D 4 | 13 = 0.

Fibre 3
With ω (2) in hand, we define the second base which is also the third and last fibre B 2 = F 3 = C \ {poles(ω (2) )}. Since ω (2) is a 4 × 4 matrix, the elements of B 2 cohomology are 4vectors whose a th -component belongs to the twisted cohomology generated by ω (2) aa . Given a generic element of the base cohomology φ ∨ B 2 , the covariant acts as since there are no boundaries on B 2 .
Having understood the action of the covariant derivative on the last fibre, we can choose a basis for the fibre cohomology H 1 F 3 . Since there are no boundaries, the basis forms are bulk forms generated by the diagonal components of ω (2) . However, the diagonal components ω (2) aa for a = 2, 3 are degenerate since Disc z 4 Q (2) aa = 0. Since ω (2) aa ∝ d log(linear polynomial of z 4 ) 2 for a = 2, 3 there is not enough singularities to generate a non-trivial twisted cohomology class. Physically, this corresponds to the vanishing of triangle integrals in the massless limit. The triangle-dual components (a = 2, 3) of 2 .
2 is defined below. Thus, dimH 1 F 3 = 2 and we choose the basis

Basis for full cohomology
A basis for the full cohomology on the 13-bubble cut is given by the columns of the vector While the second column is directly proportional to the box dual form, the logarithmic basis form in the first column still needs to be related to the 13-bubble dual form. Using integration-by-parts, we find that The conversion N is non-diagonal since we are projecting the non-logarithmic uniform transcendental bubble (5.8) onto the logarithmic fibration basis (5.44). While choosing a non-d log fibration basis could help diagonalize N it also makes the fibration more complicated.
The intersection numbers of a given Feynman integrand ϕ with the basis of uniform transcendental dual forms ((5.8) and (5.10)) is simply Moreover, the compactly supported versions of our basis on the 13-cut are

Intersection matrix on 13-cut
To extract the coefficients of the bubble and box Feynman integrals, the intersection matrix of the fibration basis (5.44) with a basis of Feynman forms is needed. We choose our initial basis of Feynman forms to be the UT basis of [58] ϕ naive = ϕ bubs ϕ box , ϕ bubs = ε(2ε−1) 49) and will discover that the UT dual forms ((5.8) and (5.10)) extract the coefficients corresponding to linear combinations of (5.49) such that the new box integral is finite! 15 We begin by computing the intersection matrix for the dual form fibration basis (5.44) and Feynman integrand basis (5.49)

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These intersection numbers are computed recursively starting from (4.34). Projecting ϕ naive onto the F 1 -basis, we find ϕ (1) Then, following (4.35), the columns of ϕ (1) naive are projected onto the F 2 -basis Then the intersection matrix is obtained by projecting the columns of ϕ (2) naive onto the F 3 -basis With C naive in hand, the UT bubble/box (5.49) coefficients can be computed for any Feynman integrand ϕ: naive . (5.54) Note that it is surprising that the intersection matrix above is diagonal since the double pole in the bubble dual form ((2.11) or equivalently (5.8)) was needed to establish duality with the standard UT basis of Feynman forms [44]. However, in the massless degeneration, the bubble dual forms are dual to a linear combination of Feynman forms instead.
To find the basis of Feynman forms dual to (5.8) and (5.10), we look for a basis of integrands, which we refer to as ϕ, such that It follows that the integrands dual to (5.8) and (5.10) are up to forms with vanishing 13-residue. Performing the analogous calculation on the 24-cut completely fixes ϕ 1234 (see (2.22)).  Interestingly, the UT basis of dual forms has picked a particularly nice basis of Feynman integrals where ϕ 1234 integrates to something finite (by "finite", we mean that the integrated expression is not more singular than the integrand). Even without knowing the part of ϕ 1234 that vanishes on the 13-cut, we can check that the discontinuity of I [ϕ 1234 ] across the 13-cut is finite

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and Disc 13 is 1 2πi times the discontinuity across the 13-channel. This can also be verified by taking the 13-channel discontinuity of (2.27) directly.
The corresponding ϕ coefficients are simply for any given FI ϕ. Since we only know the compactly supported versions of the fibration dual basis, ϕ ∨ has to be expressed in terms of ϕ ∨ f .

Simple check: dimension shifting identities
As a simple application of the newly obtained compactly supported bubble and box dual forms, we consider the projection of higher-dimensional bubbles and boxes onto the 4dimensional bubble and box.

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The relationship between FI's of different dimensions was first noted in [71]. After writing a FI in parametric form, it is clear that the spacetime dimension only enters through the exponent of a homogeneous polynomial P (α) −d/2 in the Schwinger parameters {α i }. All propagators are given formal masses that can be set to zero later and P (α) is independent of the masses. Then, one constructs a differential operator out of ∂ m 2 i such that its action on a FI gives the same integral except with an extra power of P in the integrand effectively changing d → d − 2. The corresponding differential operator is simply obtained by the replacement α i → ∂ m 2 i in P . Using standard IBP techniques, the action of P (∂ 2 m i ) on any FI can be decomposed into a know basis of d-dimensional integrals. For one-loop integrals, [71] gives compact expressions relating (d + 2)-dimensional integrals to d-dimensional integrals. Suppose that we are considering integrals with ≤ n points. For the maximum topology, we have the propagators D i=1,...,n−1 = ( + i j=1 p j ) 2 + m 2 i and D n = 2 + m 2 n . We label all other m ≤ n-point integrals using the convections set by the n-point integral. Then, for some subset J ⊂ 1, . . . , n, is the d-dimensional FI corresponding to the subset J and J | νa→νa−1 is the same integral with a reduced power of the a th propagator. Here, the (•) · (•) corresponds to minors of the Gram determinant defined in section 2.5 of [44].
We can derive analogous dimension shifting identities using intersections numbers. First, recall that the (d int − 2ε)-dimensional FI associated to the form ϕ is Then, note that the FI associated to the form ( 2 ⊥ ) m ϕ corresponds to the same integral but in (d int + 2m − 2ε)-dimensions where {ϕ i } is a basis of (d int − 2ε)-dimensional Feynman integrands, {ϕ ∨ i } is a basis for the corresponding dual space and C ij = ϕ ∨ i |ϕ j is the associated intersection matrix. The projection of higher-dimensional bubbles and boxes onto 4-dimensional bubbles and boxes is collected in tables 1 and 2, which agrees with equation (5.61).

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< l a t e x i t s h a 1 _ b a s e 6 4 = " F Y O g g R 3 7 X e M t H i p e Z X o x 2 3 b I J J o = " > A A A I U n i c t V X N b 9 s 2 F F f q b X G 8 b k 3 X 4 y 7 E l A D J o L i S 7 G Q F 2 g z F t s M u B T Z g a Q v E g U F R l E 2 U o g i S s u s R + q / 2 p + y y 7 b i / Y Z e d 9 k h / x F + n A R V A i / q 9 r x / f e 3 z O J G f a x P E f B w 9 a H 3 3 8 y W H 7 q P P p w 8 8 + f 3 T 8 + I v X u q o V o T e k 4 p V 6 m 2 F N O R P 0 x j D D 6 V u p K C 4 z T t 9 k 7 7 5 3 8 j c T q j S r x C 9 m J u l d i U e C F Y x g A 9 D w c e v V I K M j J m x R F g X j t L E F n Y m R w n L c H B 0 d d

Some gluon amplitudes in generic dimension
In this section we compute one-loop 4-point and 5-point gluon amplitudes using the method detailed in section 5 to extract bubble and box coefficients. Our main goal is to see how the d-dimensional coefficients contrast with their four-dimensional limit, and how intermediate steps are affected by coordinate choices.
Since our dual forms all localize to at least a bubble cut, we only need the helicity integrands on cuts. These cut integrands are obtained by gluing massive tree-level helicity amplitudes. To reproduce the rules of dimensional regularization, these are supplemented by d g − 5 scalars running in the loop (see appendix D.2 for details). As is conventional, we record the spacetime dimension d g used in numerator algebra separately from that spanned by the loop momenta.
In section 6.1, we compute the generalized unitarity coefficients of the one-loop 4point helicity amplitudes and construct the integrated expressions. To make contact with traditional generalized unitarity, we also compactify the bubble form in the strict d = 4 limit avoiding fibration all together in section 6.2. The generalized unitarity coefficients of 5-point gluon amplitudes are extracted in section 6.3.

4-gluon scattering
Projecting the integrands defined by (D.9) onto the dual form basis yields the generalized unitarity coefficients for the one-loop helicity amplitudes. Since not many kinematic variables are involved, the calculation can be done exactly so we record the exact dependence for each of the independent helicity choices: (+ + ++), (− + ++), (− − ++), (− + −+). For each choice, we divide the amplitude by a dimensionless factor L • that contains the little group weight (see equation (D.10)).
We find the finite amplitudes 13) and the divergent amplitudes

14)
A one-loop Note that all we have also divided these amplitudes by a one-loop factor g 2 N c (µ 2 ) ε e −εγ E /(4π) 2−ε . The divergences (infrared and ultraviolet) are as expected from general results and going to the SCET hard function simply cancels out the first line [72,73].
Comparing with known results for the one-loop 4-point helicity amplitudes [74], we find agreement. 16 One can also check that there are no spurious u-channel singularities. Here, d g = 4 − 2εδ R where δ R is 0 in the FDH scheme and 1 in the 't Hooft-Veltman scheme.

4-dimensional limit of bubble coefficients (using null variables)
It is clear from the above that d-dimensional coefficients package a lot of information that cancels in the four-dimensional limit (compare eqs. (6.1)-(6.10) with the subsequent amplitudes). Thus, it is instructive to try to extract the relevant terms in the d → 4 limit directly at the level of dual forms. By examining this limit, we will discover how to construct primitives that extract 4-dimensional information such as rational terms.

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where y = √ x + √ 1 + x rationalizes the square roots that appear in the rotation above. While there are many rotations that make the boundaries free of z 4 , (6.18) simplifies the expression for the primitives. Applying (6.18), the boundaries become free of z 4 (6.19) and the bubble dual form (5.8) becomes We now integrate out z 4 first since propagators are free of it (ie. we treat it as the first fibre as in eq. (5.21)). The intersection of a Feynman form ϕ with the bubble dual form becomes where φ = Res 13 [ϕ], (ϕ ∨ 13 ) B is the projection of the bubble dual form onto the base and φ B is the projection of the Feynman form onto the base. Here, we have chosen the following basis on the fibre where This choice of fibre basis yields the following base connection: (1) . (6.24) Projecting the bubble dual onto the z 2 , z 3 base, for example Now that we have a 2-form, we can use the methods of section 3.6 to compactify (φ ∨ 13 ) B . The double primitives of (φ ∨ 13 ) B need to be computed to O(ε) since the leading term of φ B scales as 1/ε. The 1/ε terms of φ B are present if and only if φ has a simple pole at 2 ⊥ | 13 = Q (0) = 0. The higher order terms in φ B come from poles at z 4 = ∞ (we assume that φ does not have higher order poles in Q (0) ). Since the leading term of the double primitive also scale as 1/ε (at the collinear singularity of a massless triangle), we also need to expand φ B up to O(ε). However, as we will see later, the O(ε) term of φ B is orthogonal to the 1/ε term of the double primitives.

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Changing to light cone variables, the connection and bubble dual form become In these coordinates, there exists almost global primitives for the bubble dual form . (6.28) The difference of these almost global primitives depends only on the combination . (6.29) Thus, the bulk double primitive can be computed in terms of w about the twisted points w = −1, ∞ ((z + , z − ) → (0, ∞), (∞, ∞)). Computing this double primitive order by order in ε, yields To make the bubble dual form compactly supported about the boundaries D 2 and D 4 we also need primitives for the restriction of ψ ∨ 1 to the boundaries . (6.33) We will specialize to the D 2 boundary since the D 4 is obtained by simply inverting y. On the D 2 boundary, there are twisted singularities at z − = y, ∞. In these neighbourhoods, the primitives are

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To make the bubble dual form truly with compact support a primitive for ψ 1 | D 2 near the intersection D 2 ∩ D 4 and O(D 2 ) corrections to (6.34) and (6.35) are needed. However, such corrections are only needed when the Feynman forms have propagators raised to higher powers. The D 4 primitives are obtained from the D 2 primitives by the replacement y → 1/y. The boundary primitive restricted to the 123-triangle cut contains poles at ε = 0 and at z − = y. Mathematically, the z − = y pole simply comes from the twist on the triangle cut. Physically, the point z − = y corresponds to the collinear singularity in the massless triangle. From this point of view, it is natural that the z − = y and ε = 0 poles are coupled.
Putting everything together, we obtain the following residue formula for divergent and finite terms of 13-bubble coefficient The first thing to note is that while the O(ε) terms of ψ ∨ • are required, the O(ε) terms of φ ∨ B are not needed. The O(ε) term of φ B simply has too many powers of Q (1) in the numerator to have a residue at the collinear singularity of massless triangles. After restricting to D 2 , the twist Q (1) | D 2 ∝ (z − −y) 2 has a double zero at the collinear singularity canceling the pole in ψ ∨ D 2 ,y . A similar argument reveals that the O(ε 0 ) term of φ B is orthogonal to the O(ε −1 ) term of ψ ∨ • . Moreover, the residues in c (bub) ij and c (rat) ij are only non-zero when a subset of the loop momentum components go to infinity. More specifically, c (bub) ij receives contributions from residues at 2 ⊥ = 0 and i = ∞ while c (rat) ij receives contributions from residues at 2 ⊥ = ∞ and i = ∞. These formulas highlight two important physical facts. First, the rational term only receives contributions from the infinite momentum limit. Secondly, the bubble term does not receive contributions form the collinear singularity of the massless triangle.
While an algorithm for extracting the rational terms of one-loop amplitudes already exists [10,11], an interesting aspect of our method is its systematic nature and thus expect it to be useful for higher loop generalizations. We see some technical differences: in [10,11], the rational terms are generated exclusively by the 2 ⊥ = ∞ limit, while here we see contributions from both 2 ⊥ = 0 and 2 ⊥ = ∞. It would be interesting to better understand the relation between the formulas. The differences may be due to their use of betteradapted coordinate choices, or to taking residues in 2 ⊥ last rather than first. In any case, these differences suggest that the above formulas can be improved. Table 3. Residues that contribute to (6.36) at each order in ε. Since Q (0) is 2 ⊥ restricted to the 13-cut, the Q (0) = 0 residues pick up the leading 4-dimensional terms while residues at Q (0) = ∞, corresponding to integrals with a 2 ⊥ in the numerator and contribute to the rational term only.
In this section, we have presented a compact formula for extracting the rational term of 4-point integrals (6.36). While some work is needed to extend our algorithm to any multiplicity at one-loop, our method is systematic and we expect that it will be useful for higher loop generalizations.

5-particle examples
We now illustrate how the above calculations, using the fibration method, generalize to higher points, focusing on one-loop 5-gluon amplitudes.
The corresponding integrals were described in (2.26) and (2.28). Recall that in our basis, all infrared and ultraviolet divergences are pushed into the bubble coefficients; the box contribution is a finite weight-two transcendental function, and the pentagon contributes at O(ε). Due to the nature of the 5-point problem, the full ε-dependence of the generalized unitarity coefficients is too complicated to present here. Expressions for the 5-point generalized unitarity coefficients to all orders in ε can be found in the ancillary file.
Since the (+ + + + +) and (− + + + +) helicity amplitudes have no four-dimensional cuts (see appendix D), they are finite and at order O(ε 0 ) come entirely from the bubble coefficients. In fact, the finite part of the (+ + + + +) and (− + + + +) amplitudes are proportional to the sum of all the bubble coefficients (in the ε → 0 limit) reproducing the standard result [75] A

JHEP04(2022)078 7 Conclusions
In [44], we identified the space Poincaré dual to the vector space of Feynman integrals (twisted cohomology) as a certain twisted relative cohomology. Its elements, called dual forms, appear simpler than their Feynman counterparts since they are localized to generalized unitarity cuts. Intuitively, dual forms are integrals in (d int + 2ε)-dimensions that are forced to vanish near the zero locus of the propagators ({D i = 0}) instead of having poles. The wedge product of a dual form and Feynman form can thus be meaningfully integrated, yielding an algebraic invariant called the intersection number.
Since the intersection number is invariant under adding integration-by-part identities, it allows to reduce complicated integrals to a minimal basis without generating such identities [38][39][40][41][42][43]. Our work differs from previous intersection-based approaches in two ways. First, we insist on working directly in loop momentum space rather than an auxiliary parameter space in order to exploit the factorization of amplitudes on their cuts. Second, we do not modify the non-generic (integer) propagator exponents that arise naturally in Feynman integrals. The second condition allows dual forms to localize on maximal generalized unitarity cut, thus providing a systematic method for computing integral coefficients for arbitrary sub-topologies and away from the 4-dimensional limit.
Dual forms are best characterized by algebraic representatives, which are holomorphic top forms supported on various cuts. The main ingredient to compute intersection numbers is the c-isomorphism, which maps the algebraic version of a form to a compactly supported version by adding certain "dθ" anti-holomorphic terms (see section (3.1)). We provide two methods for computing the c-map of multi-variate dual forms: fibrations in Cartesian and light-cone variables. Since the c-map is only an isomorphism in cohomology, different coordinate choices generally yield distinct residue-like formulas that compute the same integral coefficients. While fibration is perhaps the most systematic algorithm at the moment and applicable to p-forms of any degree, it requires one to compute the c-map for the entire basis of dual cohomology at once and is currently hard to apply to a single dual form on is own. For 2-forms, light-cone variables provide a direct and efficient solution.
As an example of our formalism, one-loop 4-and 5-point gluon amplitudes were extracted in different ways from d-dimensional cuts. While these amplitudes have long been known, we focused here on validating the method and also to determine how coordinate choices affect intermediate steps. It would be interesting to see if these would further simplify using other choices or integration orders, for example those used in [10]. An irreducible issue is that ε-dependant generalized unitarity coefficients at higher points admit intrinsically bulky analytic expressions. With this in mind, we took first steps in section 6.2 to analyze the physically relevant d → 4 limit, focusing on 4-gluon scattering, and exploiting the fact that the Laurent series around d = 4 of (the c-image of) dual forms can be computed directly under the integration sign. Amusingly, the duals of lower-transcendental integrals turn out to be pure higher transcendental forms: the dual of a rational term contains a dilogarithm primitive, etc. Since the method is systematic, we expect that it could prove useful at higher loop orders. Other fibration strategies, such as a loop-by-loop approach, could streamline the calculation of multi-variate intersection numbers. Such considerations are left for future work.
Similarly, the vector valued analogue is It follows that where the primitive of the boundary/Leray forms on F i are the corresponding column of [u (i−1) ] −1 · [u (i−1) · F (i) a bk ]| α=b bd . One subtlety here is that the rows of f (i) a bk have to contain different θ's in order to construct compactly supported forms satisfying (4.30). While it is not correct to compactify each column with respect to the location of all twisted singularities, it does not hurt in situations where it can be done. The extra terms produced by this over compactification do not contribute to the intersection simply because the projection of Feynman forms onto the B i ((4.34) and (4.35)) never develop the singularities needed survive the corresponding residues. In such cases, each row is uniformly compactified with respect to all the twisted singularities of the connection.

C Rational parameterizations
Analytic calculations of scatting amplitudes usually involves complicated functions of scalar products p i · p j and spinor products ij , [ij]. For example, the region momenta {x 2 i,i+1 , x i,i+2 } forms a basis of Mandelstam-like variables for kinematic space in the case of planar scattering. However, due to momentum conservation and algebraic constraints (like the Schouten identity) these scalar products are not mutually independent. Instead, we work with momentum twistor variables, which automatically satisfy all identities (Schouten identity, momentum conservation, etc.). Moreover, momentum twistor variables are rational functions of the x ij (and vice versa) making conversation between these variables trivial.
Momentum twistors are functions of the holomorphic spinor variables λ i and antiholomorphic spinors µ i The usual anti-holomorphic spinor is defined in terms of the µ i [76] Thus, parametrizing the components of Z = (Z 1 Z 2 · · · ) fixes our representation of the external kinematics. We can use the symmetries of Z (Poincaré and a U(1) little group JHEP04(2022)078 scaling) to reduce the number of unfixed twistor components. For example, consider nparticle scattering. The momentum twistor Z has 4n components but symmetries fix n+10 components leaving 3n − 10 free components. While the momentum twistor parameterization may not seem very useful for 4-point scattering, it considerably simplifies 5-point kinematics. As a warm up, we the momentum twistor parametrization for 4-point scattering in section C.1. Then the 5-point parameterization is given in section C.2.

C.1 4pt parametrization
We choose the following representation for the 4-point momentum twistor 18 In our parameterization, the twistor variables are just the ordinary Mandelstam variables: 12 [21] = s and 23 [32] = t. Twistor variables can be used to calculate any physical expression that is does not contain an overall helicity factor. We will use the following cartesian coordinates for the loop momentum

C.2 5pt parametrization
The following momentum twistor parameterization for the 5-point kinematics was used to simplify the intermediate results of the intersection calculations.
The generalized unitarity coefficients of the one-loop 5-point helicity amplitudes in the attached ancillary file are expressed in terms of these parameters.

D.1 Some massive tree amplitudes
We follow the conventions of [78]. In brief: metric is mostly-plus and spinors satisfy ij [ji] = −2p i ·p j = s ij when the p i , p j are massless and for a general set I, s I ≡ −( i∈I p i ) 2 . Note that this implies a minus sign in the Clifford algebra: p / 1 p / 1 = −p 2 1 = +m 2 1 . For massive vectors polarizations (boldface), left-and right-spinors are related as |1 = p / 1 m 1 |1], normalized to 1 † 1 = 2m [79]. Note that a minus sign is required for bras by consistency: 1| = −[1| p / 1 m 1 . To construct the one-loop cut integrands we employ factorization: where for a particle of spin j the polarization sum satisfies: λ aλ 2j λ b 2j = m 2j ab 2j . Since the internal loop momentum can be massive, we need 3-,4-and 5-point tree amplitudes with two massive legs to construct all one-loop 4-and 5-point helicity amplitudes. We find concise formulas for amplitudes where the massless gluons all have the same helicity: (D. 2) The first one is the "minimal coupling" from ref. [79]; note that the ratio is independent of the reference µ thanks to the on-shell conditions. The other amplitudes were obtained by applying BCFW recursion (shifting |3] and |4 ). Same-helicity amplitudes involving a massive scalar (instead of vector) are obtained by the simple replacement: 12 2 → m 2 . General formulas for massive scalar and (n − 2) positive-helicity gluons were given in ref. [80] and we expect that by multiplying these formulas by 12 2 gives the corresponding amplitude for massive vectors. Amplitudes with non-identical helicity are slightly more complicated. For example, at 4-points [79]:
(D.8) Since the momentum twistor parameterization is only unambiguous for quantities that are little group invariant, we multiply each cut amplitude by a factor that absorbs any little group weight. That is, we actually compute the generalized unitarity coefficients of the little group invariant integrals When it exists, we factor our the tree level amplitude as done in the second line above. Also note that all but L −+++ are symmetric in s and t. This allows us to predict all but the (− + ++) t-channel coefficients from symmetry. For planar five-gluon scattering, up to dihedral and parity symmetries, there are four inequivalent helicity configurations: (+ + + + +), (− + + + +), (− − + + +), (− + − + +). (D.11) Each has five distinct bubble cuts. The corresponding integrands are obtained by multiplying the massive 4-and 5-point trees from section D.1. While the trees are relatively concise, and the polarization sums would be straightforward in d = 4 dimensions, the result