From multiple unitarity cuts to the coproduct of Feynman integrals

We develop techniques for computing and analyzing multiple unitarity cuts of Feynman integrals, and reconstructing the integral from these cuts. We study the relations among unitarity cuts of a Feynman integral computed via diagrammatic cutting rules, the discontinuity across the corresponding branch cut, and the coproduct of the integral. For single unitarity cuts, these relations are familiar. Here we show that they can be generalized to sequences of unitarity cuts in different channels. Using concrete one- and two-loop scalar integral examples we demonstrate that it is possible to reconstruct a Feynman integral from either single or double unitarity cuts. Our results offer insight into the analytic structure of Feynman integrals as well as a new approach to computing them.

1 Introduction The precise determination of physical observables in quantum field theory involves computing multiloop Feynman integrals. The difficulty of these integrals has led to their extensive study and the development of various specialized integration techniques.
One approach to computing Feynman integrals has been to analyze the discontinuities across their branch cuts. Like the integrals themselves, their discontinuities can be computed by diagrammatic rules [1][2][3] in which diagrams are separated into two parts, with the intermediate particles at the interface of the two components restricted to their mass shells, resulting in the so-called cut integrals. This on-shell restriction can simplify the integration, and its result, considerably. Traditionally, the integral might then be reconstructed directly from one of its discontinuities by a dispersion relation [1][2][3][4][5]. Alternatively, modern unitarity methods [6][7][8][9][10][11][12][13] make use of discontinuities to constrain an integral through its expansion in a basis of Feynman integrals.
A large class of Feynman integrals can be expressed in terms of transcendental functions called multiple polylogarithms, which are defined by certain iterated integrals and include classical polylogarithms as a special case. Multiple polylogarithms, and iterated integrals in general, carry a lot of unexpected algebraic structure. In particular, they form a Hopf algebra [14,15], which is a natural tool to capture discontinuities. By now, there is considerable evidence that the coproduct of a Feynman integral of transcendental weight n, with massless propagators, satisfies a condition known as the first entry condition [16]: the terms in the coproduct of transcendental weight (1, n − 1) can be written in the form where s i ranges over all Mandelstam invariants, and f s i is the discontinuity of the integral as a function of the variable s i . One might wonder whether the deeper structure of the Hopf algebra contains useful information about a fuller range of discontinuities and perhaps even points to techniques for reconstructing a full integral from its discontinuities. In this paper we present evidence that it does. We develop techniques to evaluate the cut integral explicitly, and we verify in several examples that the functions f s i in eq. (1.1) are indeed given by sums of cut integrals.
We emphasize that we work in real kinematics, which allows us to use explicit real-phase space parametrizations. Furthermore, we see that even if the original Feynman integral is finite in D = 4 dimensions, it is sometimes necessary to regularize the corresponding cut integrals. Indeed, although individual cut diagrams can be infrared divergent, their sum is finite, through a mechanism similar to the cancellation of infrared divergences in a total cross section. We use dimensional regularization.
While it might not seem very surprising that the functions f s i are related to cut integrals, the question of whether the coproduct of the cuts themselves allows for a similar interpretation in terms of generalized cuts is more intriguing. We analyze this question in several examples at one loop, and the three-point ladder at two loops. In order to do so, we first extend the diagrammatic cutting rules of ref. [2,3], which have only been formulated for single unitarity cuts so far, to allow for sequential unitarity cuts in multiple channels. We observe that several new features arise that were not present in the case of single unitarity cuts, and that we can obtain consistent results even in this case by restricting the computation to real kinematics, which implies in particular that diagrams with onshell massless three-point vertices must vanish in dimensional regularization. Furthermore, we see that beyond single unitarity cuts, the results depend crucially on the phase-space boundaries imposed by the kinematic region where each cut diagram is computed, and not only on the set of cut propagators. Equipped with this new set of rules, we show that we can correctly reproduce the relevant components of the iterated coproduct of these specific integrals, thus strengthening our hope of a deeper connection between a Feynman integral and its cuts and coproduct.
The paper is organized as follows. In section 2, we give a brief review of multiple polylogarithms and their Hopf algebra, and we discuss the class of pure transcendental functions that we expect to be able to analyze. In section 3, we present definitions of the three types of discontinuities that we consider: Disc is the difference in value as a function crosses its branch cut; Cut is the value obtained by cutting diagrams into parts; and δ is a function identified algebraically inside the coproduct. Each of these discontinuities is defined not just for a single cut, but for sequences of unitarity cuts in different Mandelstam invariants or related variables. We close this section with statements of our two conjectured relations, one between Cut and Disc, and one between Disc and δ. By combining the two relations, we claim that diagrammatic cuts correspond to functions within the coproduct. In section 4, we give examples of our relations at one-loop, including a presentation of our technique for evaluating cut integrals. Our main example is the three-mass triangle, but we include the four-mass box and the two-mass-hard box as well and discuss their different properties. Sections 5 and 6 contain the main example of this paper, namely the two-loop three-point ladder integral with massless propagators. In section 5, we compute unitarity cuts and verify our relations. In section 6, we compute sequences of two unitarity cuts, explain how to make our relations concrete, and verify them; we then consider sequences of three unitarity cuts and explain why they vanish. In section 7, we review dispersion relations and we argue that the information they contain is the same as the information contained in specific entries of the coproduct: we show that the symbols of the ladder (and the one-loop triangle) can be reconstructed from even limited knowledge of its cuts, using the integrability condition. In section 8, we close with discussion of outstanding issues and suggestions for future study. Appendix A summarizes our key conventions for evaluating Feynman diagrams and cut diagrams. Appendix B collects results of one-loop diagrams, cut and uncut, that are used throughout the paper. In appendix C we give explicit results for single unitarity cuts of the two-loop ladder. Finally, in appendix D we summarize the calculation for two sets of double cuts of the two-loop ladder, and give explicit expressions for their result.

The Hopf algebra of multiple polylogarithms
Feynman integrals in dimensional regularization usually evaluate to transcendental functions whose branch cut structures reflect the branch cuts of the loop integral. Although it is known that generic Feynman integrals can involve elliptic functions [17][18][19][20][21][22], large classes of Feynman integrals can be expressed through the classical logarithm and polylogarithm functions, log z = and generalizations thereof (see, e.g., ref. [23][24][25][26][27][28][29], and references therein). In the following we will concentrate exclusively on integrals that can be expressed entirely through polylogarithmic functions. Of special interest in this context are the so-called multiple polylogarithms, and in the rest of this section we will review some of their mathematical properties.

Multiple polylogarithms
Multiple polylogarithms are defined by the iterated integral [15,30] G(a 1 , . . . , a n ; z) = z 0 dt t − a 1 G(a 2 , . . . , a n ; t) , (2.2) with a i , z ∈ C. In the special case where all the a i 's are zero, we define, using the obvious vector notation a n = (a, . . . , a n ), 3) The number n of integrations in eq. (2.2), or equivalently the number of a i 's, is called the weight of the multiple polylogarithm. In the following we denote by A the Q-vector space spanned by all multiple polylogarithms. In addition, A can be turned into an algebra. Indeed, iterated integrals form a shuffle algebra, G( a 1 ; z) G( a 2 ; z) = a ∈ a 1 a 2 G( a; z) , (2.4) where a 1 a 2 denotes the set of all shuffles of a 1 and a 2 , i.e., the set of all permutations of their union that preserve the relative orderings inside a 1 and a 2 . It is obvious that the shuffle product preserves the weight, and hence the product of two multiple polylogarithms of weight n 1 and n 2 is a linear combination of multiple polylogarithms of weight n 1 + n 2 . We can formalize this statement by saying that the algebra of multiple polylogarithms is graded by the weight, A n with A n 1 · A n 2 ⊂ A n 1 +n 2 , (2.5) where A n is the Q-vector space spanned by all multiple polylogarithms of weight n, and we define A 0 = Q. Multiple polylogarithms can be endowed with more algebraic structures. If we look at the quotient space H = A/(π A) (the algebra A modulo π), then H is a Hopf algebra [14,15]. In particular, H can be equipped with a coproduct ∆ : H → H ⊗ H, which is coassociative, Li n−k (z) ⊗ log k z k! . (2.9) For the definition of the coproduct of general multiple polylogarithms we refer to refs. [14,15]. The coassiciativity of the coproduct implies that it can be iterated in a unique way. If (n 1 , . . . , n k ) is a partition of n, we define ∆ n 1 ,...,n k : H n → H n 1 ⊗ . . . ⊗ H n k . (2.10) Note that the maximal iteration of the coproduct, corresponding to the partition (1, . where ∧ denotes the usual wedge product on differential forms. While H is a Hopf algebra, we are practically interested in the full algebra A where we have kept all factors of π. Based on similar ideas in the context of motivic multiple zeta values [36], it was argued in ref. [37] that we can reintroduce π into the construction by considering the trivial comodule A = Q[iπ] ⊗ H. The coproduct is then lifted to a comodule map ∆ : A → A ⊗ H which acts on iπ according to ∆(iπ) = iπ ⊗ 1. In the following we will, by abuse of language, refer to the comodule as the Hopf algebra A of multiple polylogarithms.
Let us conclude this review of multiple polylogarithms and their Hopf algebra structure by discussing how differentiation and taking discontinuities (see section 3 for precise definition of discontinuity in this work) interact with the coproduct. In ref. [37] it was argued that the following identities hold: In other words, differentiation only acts in the last entry of the coproduct, while taking discontinuities only acts in the first entry.

Pure Feynman integrals
In the rest of this paper we will be concerned with connected Feynman integrals in dimensional regularization. Close to D = 4 − 2 dimensions, an L-loop Feynman integral F (L) then defines a Laurent series, In the following we will concentrate on situations where the coefficients of the Laurent series can be written exclusively in terms of multiple polylogarithms and rational functions, and a well-known conjecture states that the weight of the transcendental functions (and numbers) that enter the coefficient F of an L loop integral is less than or equal to 2L − k. If all the polylogarithms in F (L) k have the same weight, the integral is said to have uniform (transcendental) weight. In addition, we say that an integral is pure if the coefficients F (L) k do not contain rational or algebraic functions of the external kinematical variables.
It is clear that pure integrals are the natural objects to study when trying to link Hopf algebraic ideas for multiple polylogarithms to Feynman integrals. For this reason we will only be concerned with pure integrals in the rest of this paper. However, the question naturally arises of how restrictive this assumption is. In ref. [38] it was noted that if a Feynman integral has unit leading singularity [39], i.e., if all the residues of the integrand, obtained by integrating over compact complex contours around the poles of the integrand, are equal to one, then the corresponding integral is pure. Furthermore, it is well known that Feynman integrals satisfy integration-by-parts identities [40], which, loosely speaking, allow one to express a loop integral with a given propagator structure in terms of a minimal set of so-called master integrals. In ref. [41] it was conjectured that it is always possible to choose the master integrals to be pure integrals, and the conjecture was shown to hold in several nontrivial cases [42][43][44]. Hence, if this conjecture is true, it should always be possible to restrict the computation of the master integrals to pure integrals, which justifies the restriction to this particular class of integrals.
Another restriction on the class of Feynman integrals considered in this paper is that we consider all propagators to be massless. In this case, it is known that the branch points of the integral, seen as a function of the invariants s ij = 2p i · p j , where p i are the external momenta (which can be massive or massless), are the points where one of the invariants is zero or infinite [4]. It follows then from eq. (2.14) that the first entry of the coproduct of a Feynman integral can only have discontinuities in these precise locations. In particular, this implies the so-called first entry condition, i.e., the statement that the first entries of the symbol of a Feynman integral with massless propagators can only be (logarithms of) Mandelstam invariants [16]. This observation, combined with the fact that Feynman integrals can be given a dispersive representation, provides the motivation for the rest of this paper, namely the study of the discontinuities of a pure Feynman integral with massless propagators through the lens of the Hopf algebraic language reviewed at the beginning of this section.

Three definitions of discontinuities
In this section we present our definitions and conventions for the discontinuities of Feynman integrals with respect to external momentum invariants, also called cut channels. There are three operations giving systematically related results: a discontinuity across a branch cut of the Feynman integral, which we denote by Disc and define in section 3.1 below; unitarity cuts computed via Cutkosky rules and the diagrammatic rules of ref. [2,3], which we extend here to multiple cuts and denote by Cut (section 3.2); and a corresponding δ operation on the coproduct of the integral (section 3.3). Discontinuities taken with respect to one invariant are familiar, but sequential discontinuities must be constructed with care in order to derive equivalent results from the three operations. In this section, we present these concepts in general terms. Concrete illustrations appear in the following sections.
Let F be a pure Feynman integral, and let s and s i denote Mandelstam invariants (squared sums of external momenta), labeled by i in the case where we consider several of them. These invariants come with an iε prescription inherited from the Feynman rules for propagators. In the case of planar integrals, such as the examples we will consider in the following sections, the integral is originally calculated in the Euclidean region, where all Mandelstam invariants of consecutive legs are negative, so that branch cuts are avoided. It may then be analytically continued to any other kinematic region by the prescription The most natural kinematic variables for a given integral might be more complicated functions of the momentum invariants. We denote these general kinematic variables by x or x i . Indeed, it is known that the Laurent expansion coefficients in eq. (2.15) are periods (defined, loosely speaking, as integrals of rational functions), which implies that the arguments of the polylogarithmic functions are expected to be algebraic functions of the external scales [45].

Disc: Discontinuity across branch cuts
The operator Disc x F gives the direct value of the discontinuity of F as the variable x crosses the real axis. If there is no branch cut in the kinematic region being considered, then the value is zero. Concretely, where the iε prescription must be inserted correctly in order to obtain the appropriate sign of the discontinuity. For example, Disc x log(x + i0) = 2πi θ(−x). We will discuss the sign in more detail at the end of this section, when we relate Disc to the other definitions of discontinuities.
The sequential discontinuity operator Disc x 1 ,...,x k is defined recursively: Note that Disc may be computed in any kinematic region after careful analytic continuation, but if it is to be related to the value of Cut, it should be computed in the region where only the cut invariants are positive, and the rest are negative. In particular, sequential Disc can be computed in different regions at each step.

Cut: Cut integral
The operator Cut s gives the sum of cut Feynman integrals, in which some propagators in the integrand of F are replaced by Dirac delta functions. These propagators themselves are called cut propagators. The sum is taken over all combinations of cut propagators that separate the diagram into two parts, in which the momentum flowing through the cut propagators from one part to the other corresponds to the Mandelstam invariant s. Furthermore, each cut is associated with a consistent direction of energy flow between the two parts of the diagram, in each of the cut propagators. In this work, we follow the conventions for cutting rules established in ref. [2,3], and extend them for sequential cuts.
First cut. Let us first review the cutting rules of ref. [2,3]. We start by enumerating all possible partitions of the vertices of a Feynman diagram into two sets, colored black (b) and white (w). Each such colored diagram is then evaluated according to the following rules: • Black vertices, and propagators joining two black vertices, are computed according to the usual Feynman rules.
• White vertices, and propagators joining two white vertices, are complex-conjugated with respect to the usual Feynman rules.
• Propagators joining a black and a white vertex are cut with an on-shell delta function, a factor of 2π to capture the complex residue correctly, and a theta function restricting energy to flow in the direction b → w.
For a massless scalar theory, the rules for the first cut may be depicted as: The dashed line indicating a cut propagator is given for reference and does not add any further information. We write Cut s to denote the sum of all diagrams belonging to the same momentum channel, i.e., in each of these diagrams, if p is the sum of all momenta through cut propagators flowing in the direction from black to white, then p 2 = s. Note that cut diagrams in a given momentum channel will appear in pairs that are black/white color reversals -but of each pair, only one of the two can be consistent with the energies of the fixed external momenta, giving a potentially nonzero result.
Sequential cuts. The diagrammatic cutting rules of ref. [2,3] reviewed so far allow us to consistently define cut integrals corresponding to a single unitarity cut. The aim of this paper is however the study of sequences of unitarity cuts. The cutting rules of ref. [2,3] are insufficient in that case, as they only allow us to partition a diagram in two parts, corresponding to connected areas of black and white vertices. We now present an extension of the cutting rules to sequences of unitarity cuts on different channels. At this stage, we only state the rules, whose consistency is then backed up by the results we find in the remainder of this paper. In a sequence of diagrammatic cuts, energy-flow conditions are overlaid, and complex conjugation of vertices and propagators is applied sequentially. We continue to use black and white vertex coloring to show complex conjugation. Colors are reversed as cuts are crossed. We illustrate an example in fig. 1, which will be discussed below.
Consider a multiple-channel cut, Cut s 1 ,...,s k I. It is represented by the sum of all diagrams with a color-partition of vertices for each of the cut invariants s i = p 2 i . Assign a sequence of colors {c 1 (v), . . . , c k (v)} to each vertex v of the diagram, where each c i takes the value 0 or 1. For a given i, the colors c i partition the vertices into two sets, such that the total momentum flowing from vertices labeled 0 to vertices labeled 1 is equal to p i . A vertex v is finally colored according to c(v) ≡ k i=1 c i (v) modulo 2, with black for c(v) = 0 and white for c(v) = 1. The rules for evaluating a diagram are as follows.  • A propagator joining vertices u and v is cut if c i (u) = c i (v) for any i. There is a theta function restricting the direction of energy flow from 0 to 1 for each i for which c i (u) = c i (v). If different cuts impose conflicting energy flows, then the product of the theta functions is zero and the diagram gives no contribution.
• We exclude crossed cuts, as they do not correspond to the types of discontinuities captured by Disc and δ. 1 In other words, each new cut must be located within a region of identically-colored vertices with respect to the previous cuts. In terms of the color labels, we require that for any two values of i, j, exactly three of the four possible distinct color sequences {c i (v), c j (v)} are present in the diagram.
• Likewise, we exclude sequential cuts in which the channels are not all distinct. This restriction is made only because we have not found a general relation between such cuts and Disc or δ. In principle, there is no obstacle to computing these cut diagrams.
For massless scalar theory, the rules for sequential cut diagrams may then be depicted thus: Let us make some comments about the diagrammatic cutting rules for multiple cuts we just introduced. First, we note that these rules are consistent with the corresponding rules for single unitarity cuts presented at the beginning of this section. Second, using these rules, it is clear that sequential cuts are independent of the order of cuts. Indeed, none of our rules depends on the order in which the cuts are listed. Finally, the dashed line is an incomplete shorthand merely indicating the location of the delta functions, but not specifying the direction of energy flow, for which one needs to refer to the color indices. Our diagrams might also include multiple cut lines on individual propagators, such as p . (3.9) We also introduce notation allowing us to consider individual diagrams contributing to a particular cut, and possibly restricted to a particular kinematic region. When no region is specified, for the planar examples given in this paper, it is assumed that the cut invariants are taken to be positive while all other consecutive Mandelstam invariants are negative. We write Cut s,[e 1 ···ew],R D (3.10) to denote a diagram D cut in the channel s, in which exactly the propagators e 1 · · · e w are cut, and computed in the kinematic region R. Rules of complex conjugation and energy flow will be apparent in the context of such a diagram.
Examples of sequential cuts. We briefly illustrate the diagrammatics of sequential cuts. Consider taking two cuts of a triangle integral. At one-loop order, a cut in a given channel is associated to a unique pair of propagators. We list the four possible color partitions {c 1 (v), . . . , c k (v)} in fig. 1. The first graph is evaluated according to the rules above, giving first cut second cut Figure 2: An example of crossed cuts, which we do not allow.
The second and third graphs evaluate to zero, since the color partitions give conflicting restrictions for the energy flow on the propagator labeled p. The fourth graph is similar to the first, but with energy flow located on the support of θ(−p 0 )θ(−q 0 )θ(−r 0 ). Just as for a single unitarity cut, in which only one of the two colorings is compatible with a given assigment of external momenta, there can be at most one nonzero diagram for a given topology of sequential cuts.
In the examples calculated in the following sections of this paper, we will thus omit writing the sequences of colors {c 1 (v), . . . , c k (v)}. We may also omit writing the theta functions for energy flow in the cut integrals.
We include an example of crossed cuts, which we do not allow, in fig. 2. Notice that there are four distinct color sequences in the diagram, while we only allow three for any given pair of cuts.

δ: Entries of the coproduct
We denote by δ x 1 ,...,x k F the cofactor of the first entries log x 1 ⊗ · · · ⊗ log x k in the coproduct ∆ 1,...,1,n−k F , where we must be careful to account for relations between log x and log(−x), for example, or more generally, log(f (x)) for any function f (x). Stated more precisely, if F is of transcendental weight n, and ∆ 1,1,...,1 k times ,n−k F = {a 1 ,...,a k } log a 1 ⊗ · · · ⊗ log a k ⊗ g a 1 ,...,a k , (3.11) where the a i are functions of some (combination of) variables x i , then where 13) and the congruence symbol indicates that δ x 1 ,...,x k F can be defined only modulo 2πi. If the integral contains overall numerical factors of π, they should be factored out before performing this operation.
The definition of δ x 1 ,...,x k F is motivated by the relation eq. (2.14) between discontinuities and coproducts. In particular, if δ x F ∼ = g x , then Disc x F ∼ = (Disc x ⊗ id)(log x ⊗ g x ) = ±2πi g x . The sign is determined by the iε prescription for x in F and will be discussed in more detail in the following subsection.
The first entry condition [16] mentioned at the close of Section 2 implies that this operation can be performed in a physical momentum channel for the first cut. But we will see in our main examples that the later arguments a 2 , a 3 , . . . of the coproduct are not necessarily momentum invariants, so we must formulate a clear prescription for matching δ x 1 ,...,x k F to physical discontinuities.

Relations among Disc, Cut, and δ
Cut diagrams and discontinuities. The rules for evaluating cut diagrams are designed to compute their discontinuities. The fact that such a relation exists at all follows from the largest time equation. For the first cut, the derivation may be found in ref. [2,3]. The original relation is where the sum runs over all momentum channels. In terms of diagrams with colored vertices, the left-hand side is the all-black diagram plus the all-white diagram. The righthand side is -1 times the sum of all diagrams with mixed colors. We can isolate a single momentum channel s by analytic continuation into a kinematic region where among all the invariants, only s is on its branch cut. Specifically, for planar integrals such as the examples given in this paper, we take s > 0 while all other invariants of consecutive momenta are negative. There, the left-hand side of eq. (3.14) can be recast 2 as Disc s F , while the right-hand side collapses to a single term: For sequential cuts, we claim that Cut s 1 ,...,s k F captures discontinuities in variables x 1 , . . . , x k which are related to arguments of the multiple polylogarithms, in a relation of the form We recall that no two of the invariants s 1 , . . . , s k should be identical, nor may any pair of them cross each other in the sense given in the cutting rules above. We now make this relation precise by explaining how to obtain the variables x 1 , . . . , x k from the Mandelstam invariants s 1 , . . . , s k . The procedure is the following.
• We assume prior knowledge of the set of variables from which the x i are drawn.
• Let R[s 1 , . . . , s j ] denote the kinematic region in which the invariants inside the brackets are positive while all other invariants are negative. The left-hand side of eq. (3.16) is evaluated in the region R[s 1 , . . . , s k ]. 3 On the right-hand side, we proceed step by step according to the definition eq. (3.2), and each Disc x i is evaluated in the region R[s 1 , . . . , s i ].
• By the traditional cutting rules cited above, we can take the first variable to be a Mandelstam invariant, x 1 = s 1 . For each subsequent i ∈ {2, . . . , k}, x i runs over all values for which log(x i ) has branch points in common with log(s i ), and for which the variable x i can approach the branch point independently of all the other x j within the region R[s 1 , . . . , s i ].
• The iε prescription for x i is inherited naturally from the iε prescription of s i in the region R[s 1 , . . . , s i ].
While sequential cuts are independent of the order in which the channels are listed, the correspondences to Disc are derived in sequence, so that the right-hand side of eq. (3.16) takes a different form when channels on the left-hand side are permuted. Thus, eq. (3.16) implies relations among the Disc x 1 ,...,x k F .
The right-hand side of eq. (3.16) may sometimes coincide with Disc s 1 ,...,s k F . We will find an instructive counterexample with k = 3 in Section 6.4, where the correspondence breaks down because there are only two independent variables to take the positions x 1 , x 2 , and thus there is no possibility for any x 3 to approach a branch point independently. The relation (3.16) is therefore a statement that cutting rules contain information about the nature of the variables x i which are the natural arguments of the function F .
Coproduct and discontinuities. As a consequence of eq. (2.14), the first discontinuity of F is captured by the operation δ. We claim that sequential discontinuities of F are captured by δ as well, in a relation of the form The congruence symbol indicates that the relation is valid modulo (2πi) k+1 , consistent with the definition of δ x 1 ,...,x k . Since the coproduct is the same in all kinematic regions, we have inserted the schematic factor Θ to express the restriction to the region where the left-hand side is to be compared with Cut. For k ≥ 2, the relation eq. (3.17) is not at all obvious, because later entries in the coproduct do not distinguish between log x i and log(−x i ), for example, and so we cannot tell whether the argument is on its branch cut, in general. Our claim is that the arguments are always on their branch cuts, so that the relation is valid, in the case of pure Feynman integrals, and where the left-hand side is related to cuts on invariants s 1 , . . . , s k through a relation of the type eq. (3.16), i.e. matching the branch points of their logarithms and allowing the x i to approach their branch points independently. Again, the left-hand side must be computed step by step in the corresponding kinematic regions, namely R[s 1 , . . . , s i ] for Disc x i . The operator δ x 1 ,...,x k can likewise be expressed sequentially as δ x 1 (δ x 2 (· · · (δ x k ))), and the factor Θ encodes a corresponding product of theta functions relating Disc x i to δ x i at each step.
To make the relation eq. (3.17) completely precise, we must specify how to fix the sign of each term. The branch cut of log x i is taken conventionally, along the negative real axis. Between the functions log x i and log(−x i ), we select the one on the branch cut in the region R[s 1 , . . . , s i ], i.e. where the argument is negative, which can be written in either case as log(x i (1 − 2θ(x i ))). The kinematic restriction allows a clear iε prescription to be inherited by x i from s i , in the region R[s 1 , . . . , s i ]. Thus we follow the iε prescription to see whether x i (1 − 2θ(x i )) is above or below the branch cut, and attach a factor of +2πi if above and −2πi if below.
For example, let us take a look at the first entries. The coproduct of F can be written so that each term has its first entry of the form log(−s 1 ), where s 1 is a Mandelstam invariant. As stated below eq. (3.16), we simply take x 1 = s 1 . Since it is a cut invariant, we work in the region where x 1 > 0. But our claim is that the coproduct sees the discontinuity coming from log(−x 1 ), rather than the function log(x 1 ). We must follow the iε to determine its sign. The original iε prescription for propagators leads to the prescription s i + iε for invariants. Thus we have −(x 1 + iε) = −x 1 − iε, and so we pick up a factor of −2πi from the first entry, giving In this paper, we give evidence for the validity of eq. (3.16) and eq. (3.17) by matching cut diagrams and coproduct entries directly, as well as by computing discontinuities in some cases.

One-loop examples
In this section, we present three simple examples of discontinuities of one-loop integrals to demonstrate the relations discussed in the previous section. We first consider the threemass triangle in some depth, which is an illuminating introduction to the two-loop ladder example in the following section, as their kinematic analyses have many common features. The second, brief, example is the four-mass box, whose functional form is closely related to the triangle although the cut diagrams are quite different. Finally, we discuss the infrareddivergent "two-mass-hard" box, which will be used as a building block for cuts of the two-loop ladder and also demonstrates the validity of consistent dimensional regularization.

Three-mass triangle
The triangle in D = 4 dimensions. We begin by analyzing the three-mass triangle integral with massless propagators. According to our conventions, which are summarized in appendix A, the three mass triangle integral in D = 4 − 2 dimensions is defined by Figure 3: The triangle integral, with loop momentum defined as in the text; and with cuts in the p 2 2 and p 2 3 channels.
where γ E = −Γ (1) denotes the Euler-Mascheroni constant. As the focus of the paper will be the computation of cut diagrams, it is of utmost importance to keep track of all imaginary parts. We follow the conventions for massless scalar theory listed in the preceding section. In particular, until cuts are introduced, all vertices (denoted by a black dot, see fig. 3) are proportional to i, and all propagators have an explicit factor of i in the numerator and follow the usual Feynman +iε prescription. These factors lead to the explicit minus sign in eq. (4.1). Note that we do not include a factor of i −1 per loop into the definition of the integration measure. Many different expressions are known for the three-mass triangle integral, both in arbitrary dimensions [49,50] as well as an expansion around four space-time dimensions in dimensional regularization [51][52][53]. Note that the three-mass triangle integral is finite in four dimensions, and we therefore put = 0 and only analyze the structure of the integral in exactly four dimensions. We start by giving a short review of this function. It is clear that, up to an overall factor carrying the dimension of the integral, the three-mass triangle can only depend on the dimensionless ratios of momentum invariants, Furthermore, it is convenient to introduce variables z,z, satisfying the relations An explicit solution to the above relations is given by where we define with the Källén function λ(a, b, c) defined by We note that, for positive values of λ, we always have z >z. Since eq. (4.3) is symmetric in z andz, there is a second solution in whichz > z, which could be interpreted as taking the negative branch of the square root in eq. (4.4). In most of our calculations, we will indeed restrict ourselves to the region where z >z, for concreteness. In the regions where all p i have the same sign, there is a portion of kinematic phase space in which λ is negative, so that (z,z) take complex values.
In terms of the variables (4.4), the triangle integral takes the form where Some comments are in order: we see that the three-mass triangle is of homogeneous transcendental weight two, i.e., it is only a function of dilogarithms and products of ordinary logarithms. It is, however, not a pure function in the sense of the definition in section 2, but it is multiplied by an algebraic function of the three external scales p 2 i (or equivalently, a rational function of z,z and p 2 1 ), which is the leading singularity. In the following we are only interested in the transcendental contribution, and we therefore define, for arbitrary values of the dimensional regulator , ) is a pure function at every order in the expansion. Let us now consider the discontinuities of the triangle integral. It is well known that the branch points of a Feynman integral with massless propagators are the points where the Mandelstam invariants approach 0 or ∞. It is easy to see that in the (z,z) plane these branch points correspond to z orz taking values among 0, 1, ∞. The correspondence is given explicitly in Table 1. The first-entry condition for Feynman integrals discussed in section 2 implies that the symbol of the three-mass triangle can only have u 2 = zz and u 3 = (1 − z)(1 −z) as its leftmost entry. The coproduct of the one-loop three mass triangle can be computed explicitly from eq. (4.9), with the result where in the second equality we made the first entry condition explicit. Our aim is to interpret the coproduct of the one-loop three-mass triangle in terms of cut diagrams, through the relations of Section 3. In the rest of this section we present, as a warm-up, the explicit computation of the unitarity cut of the one-loop three-mass triangle.

Branch point
Limit value Table 1: Branch points of the triangle, in terms of Mandelstam variables or the z,z of equation (4.3). The Mandelstam invariants can approach the branch point at ∞ from either positive or negative values. We will let z andz vary independently, and therefore we are sensitive only to the first set of branch points, where Mandelstam invariants approach 0. Table 2: Some kinematic regions of 3-point integrals, classified according to the signs of the Mandelstam invariants and the sign of λ, as defined in eq. (4.5). In the first six rows, λ > 0, so that z andz are real-valued, and we take z >z without loss of generality.
Unitarity cuts of the one-loop three-mass triangle. It is well known [1,2,4] that the discontinuity in a physical channel is given by replacing propagators in the Feynman integral by delta functions, as depicted in fig. 3b. As already discussed, the branch points of the three-mass triangle are wherever one of the external masses approaches zero or infinity, or equivalently where z orz approaches one of the points {0, 1, ∞}. The restriction of kinematic region will make clear which of these various branch points are accessible. The correspondence between signs of Mandelstam invariants and values of z,z is given in Table  2.
In the following we review the cut integral calculation. Although it is not necessary in this example, we now work in D = 4 − 2 dimensions, as a warmup to the two-loop integral where the D-dimensional formalism will be important at the level of cuts. We will work in the region which we denote by R * , where all the invariants are positive and λ < 0 (and thusz = z * ), because having z andz complex simplifies the calculation. The cut integral we want to compute reads . Without loss of generality we can select our frame and parametrize the loop momentum as follows: where θ ∈ [0, π] and |k| > 0, and 1 D−2 ranges over unit vectors in the dimensions transverse to p 2 and p 3 . Momentum conservation fixes the value of α in terms of the momentum invariants to be With this frame and parametrization, the cut integration measure becomes The D-dimensional cut triangle integral, with energy flow conditions suited for the p 2 channel, is .
Performing the change of variables, and turning to the dimensionless variables (4.2) and (4.4), the cut integral becomes The results for the cuts on different channels can be obtained in a similar way and are collected in appendix B.
Let us now consider a sequence of cuts on the p 2 2 and p 2 3 channels, consistent with energy flow from leg three to leg two (see fig. 3c). We must work in a region where p 2 2 , p 2 3 > 0; we choose R 2,3 . The cut integral is (4.17) Using the parametrization (4.12), we find (4.18) Summary and discussion. We now interpret the results for the cuts of the triangle integral we just computed in terms of the coproduct. It is trivial to analytically continue to the region R 2 in which p 2 2 > 0 and p 2 1 , p 2 3 < 0. In keeping with the familiar cutting rules in a single momentum channel, we recover the discontinuity of the function with a minus sign, (4.19) and similarly for the cuts on the other channels, in agreement with eq. (3.16). This result is in agreement with computing the discontinuity from the coproduct of the triangle integral, eq. (4.10), according to the relation eq. (3.17), Proceeding to a sequence of two discontinuities, 4 let us relate Cut p 2 2 ,p 2 3 to Disc and then to δ. The first step is to identify the variable in which the discontinuity is taken. For the triangle, we see that the natural variables appearing in the multiple polylogarithms are taken from four possible values, We must work in the region R 2,3 . In terms of z,z, we see from Table 2 thatz < 0, z > 1. Table 1 shows that the only branch point for p 2 3 within this region is z → 1. Therefore, the discontinuity in p 2 3 can be understood entirely as the discontinuity in the only variable of eq. (4.21) whose logarithm shares this branch point, namely (1 − z). Finally, to get the correct sign of the discontinuity, we observe in fig. 3b (after the p 2 cut and before the p 3 cut) that the p 3 vertex is in the white complex-conjugated region of the diagram. Therefore, we take the discontinuity from the conjugated iε prescription, namely p 2 3 − iε, which implies (1 − z) + iε inside this kinematic region. Thus we compute as a consequence of eq. (4. 19), in full agreement with eq. (4.18) and eq. (3.16): To compare this same discontinuity to the coproduct, we take the same variables as in Disc and read from eq. (4.10) that δ p 2 2 ,1−z T = −1/2. To attach the factors of 2πi with the correct signs into the relation eq. (3.17), we follow the iε at each step. As explained below eq. (3.17), the first entry always gives a factor of (−2πi). For the second factor, since 1 − z is negative in our kinematic region, we deduce that we are picking up the discontinuity of log(1 − z) rather than log(−(1 − z)). As above, the prescription is (1 − z) + iε, giving a factor of (2πi). In total, the relation eq. (3.17) between Disc and δ is which agrees with eq. (4.22) after accounting for the factor relating T to T .

Four-mass box
The four-mass box is also finite in four dimensions, and may in fact be expressed by the same function as the three-mass triangle [51]. If we label the momenta at the four corners by p 1 , p 2 , p 3 , p 4 , as in fig. 4a, and define s = (p 1 + p 2 ) 2 , t = (p 2 + p 3 ) 2 , then the box in the Euclidean region is given by where we have introduced variables Z,Z defined as follows: Since the functional form is the same as for the three-mass triangle, most of the multiple cuts can be analyzed exactly the same way. Because the transcendental weight is two, we are limited to a sequence of two cuts in computing δ. This limitation is consistent with Cut, as any real-valued cut of all four propagators of the diagram vanishes; and with Disc as related to the other discontinuities by the rules of section 3, as there are only the two variables Z,Z in which to take discontinuities.
Ordinary single-channel cuts are consistent when calculated by each of the three methods listed in the previous subsection. In view of the permutation symmetry, we can say without loss of generality that the first cut is in the channel p 2 2 . For a second cut channel, we only need to distinguish two types: p 2 4 , or any of the others. Suppose we choose p 2 3 . Then, the analysis of discontinuities from direct analytic continuation and from the coproduct is exactly the same as in the triangle example. The corresponding cut integral, with three delta functions and one of the original propagators, is shown in fig. 4b and produces the leading singularity.
The truly new kind of multiple cut to consider is the discontinuity of Disc p 2 2 B 4m in the p 2 4 channel, shown in fig. 4c. In a region where p 2 2 , p 2 4 > 0, all other invariants are negative, and λ is real-valued, we must have (1 − Z)/(1 −Z) > 0. So, either by considering the discontinuity directly, or from the coproduct, we find 5 Recalling the similarity of the functional form of this box to the triangle example, this calculation is analogous to trying to cut the triangle twice in the same channel. For the box, however, we can actually set up a cut integral to capture this sequential discontinuity. It would have all four of its propagators replaced by delta functions. This is the familiar "quadruple cut" [8], which is evaluated at its complex-valued solutions. Here, in our correspondence between cut integrals and discontinuities, we insist on real parametrization of the loop momentum. Thus there is no solution to the four delta functions, and we conclude that the cut integral vanishes, in agreement with eq. (4.26).

Two-mass-hard box
We close this section with the example of the two-mass-hard box, since some of its discontinuities are needed for our two-loop calculations. This example illustrates several features different from the previous examples, even apart from the presence of external massless legs: we must work in dimensional regularization consistently, and we can use Mandelstam invariants directly rather than new variables.
Because of the infrared divergences of this integral, we employ dimensional regularization. The coproduct structure requires that we work order by order in the regularization parameter. We take the result from ref. [49], with an additional factor of ie γ E inserted to match our conventions. In the Euclidean region, the box is given by In the following equations, we drop the O( ) terms. The coproduct is evaluated order by order in the Laurent expansion in . At order 1/ 2 , it is trivial and there is clearly no discontinuity. At order 1/ , the coproduct is simply the function itself, At order 0 , we are interested in the ∆ 1,1 term of the coproduct, which is given by Discontinuity in the t-channel. Using the analytic continuation of the dilogarithm for x > 0, we find that the discontinuity of B 2mh in the t-channel, with all other invariants negative, is given by From the point of view of the terms of the coproduct in eq. (4.29), we find and thus Disc t B 2mh ∼ = −2πi Θ δ t B 2mh , as expected.
Sequential discontinuities. Since the two-mass-hard box has four momentum channels, there are six pairs to consider as generalized cut integrals, or sequential discontinuities. Cutting any of the channel pairs (s, p 2 3 ), (s, p 2 4 ), or (p 2 3 , p 2 4 ) cuts the same set of three propagators, as shown in fig. 5a, and gives the leading singularity. The result of the integral (in the respective kinematic regions) is −4π 2 i/(st), which matches the value computed from the coproduct, eq. (4.32), or the direct evaluation of discontinuities.
Cutting the channel pair (t, p 2 3 ) or (t, p 2 4 ) corresponds to a cut integral in which a massless three-point vertex has been isolated, as shown in fig. 5, diagrams (b) and (c). It is well known that a three-point on-shell vertex in real Minkowski space requires collinear momenta. Let us see how this property figures in the cut integral. Parametrize the loop The delta functions set x = w = 0, so that = yp 2 , which is the familiar collinearity condition. If D > 4, then the integral over w vanishes. For D = 4 exactly, one can find a finite result for the integral. (It would again give the leading singularity, −4π 2 i/(st).) Looking at the coproduct, eq. (4.32), or the p 2 i -channel discontinuity of eq. (4.31), noting the appearance of p 2 i in the denominator of (1 − t/p 2 i ), we see that the sequential discontinuity for either of the channel pairs (t, p 2 3 ) and (t, p 2 4 ) is zero. Thus we see that it is correct to insist on D > 4, keeping the dimensional regularization parameter nonzero, even though the cut itself is finite in four dimensions.
Finally, the channel pair (s, t) is excluded because the cuts cross, in the sense given in the cutting rules of the previous section. Note that in the coproduct, eq. (4.29), there are terms proportional to log(−s) ⊗ log(−t) and log(−t) ⊗ log(−s). If we were to compute the cut integral, it would be zero, not only because of the on-shell three-point vertices, but also because there is no real-valued momentum solution for any box with all four propagators on shell, even in D = 4. The relations between Cut s,t and δ s,t break down at the level of Disc s,t : because the cuts cross, there is no clear iε prescription for the second cut invariant. This is the reason we exclude the possibility of crossed cuts.
We have seen again that sequential discontinuities, cut integrals, and entries of coproducts agree-provided that we take < 0 for infrared-divergent integrals, with the consequence that on-shell three-point vertices force cut integrals to vanish.

Unitarity cuts at two loops: the three-point ladder diagram
The two-loop, three-point, three-mass ladder diagram with massless internal lines, fig. 6, is finite in four dimensions [51]. In terms of the variables z,z defined in eq. (4.4), it is given by a remarkably simple expression: where we have defined the pure function Because the two-loop three-point ladder in four dimensions is given by weight four functions, its coproduct structure is much richer than the one-loop cases of the preceding section. Since one of our goals is to match the entries in the coproduct to the cuts of the integral, we list below for later reference all the relevant components of the coproduct, of the form ∆ 1, . . . , 1 k times ,n−k . We have Notice that the first entry of ∆ 1,1,1,1 is (the logarithm of) a Mandelstam invariant, in agreement with the first entry condition.
In the rest of this section we evaluate the standard unitarity cuts of the ladder graph of fig. 6, which give the discontinuities across branch cuts of Mandelstam invariants in the time-like region. Our goal is, first, to relate these cuts to specific terms of ∆ 1,3 of T L (p 2 1 , p 2 2 , p 2 3 ), and, in the following section, to take cuts of these cuts and relate them to ∆ 1,1,2 .
In contrast to the one-loop case, individual cut diagrams are infrared divergent. Again, we choose to use dimensional regularization. Even though T L (p 2 1 , p 2 2 , p 2 3 ) is finite in D = 4 dimensions, its unitarity cuts need to be computed in D = 4−2 dimensions. The finiteness of T L (p 2 1 , p 2 2 , p 2 3 ) for = 0 imposes cancellations between -poles of individual cut diagrams. These cancellations can be understood in the same way as the cancellation of infrared singularities between real and virtual corrections in scattering cross sections.
The cut diagrams will be computed in the region R * , wherez = z * and all the Mandelstam invariants are timelike. This restriction is consistent with the physical picture of amplitudes having branch cuts in the timelike region of their invariants. When comparing the results of cuts with δ, but particularly with Disc, we will be careful to analytically continue our result to the region where only the cut invariant is positive, as this is where Disc is evaluated.
Before we start computing the cut integrals, we briefly outline our approach to these calculations. We will compute the cuts of this two-loop diagram by integrating first over a carefully chosen one-loop subdiagram, with a carefully chosen parametrization of the internal propagators. We make our choices according to the following rules, which were designed to simplify the calculations as much as possible: • Always work in the center of mass frame of the cut channel p 2 i . The momentum p i is taken to have positive energy.
• The routing of the loop momentum k 1 is such that k 1 is the momentum of a propagator, and there is either a propagator with momentum (p i − k 1 ) or a subdiagram with (p i − k 1 ) 2 as one of its Mandelstam invariants.
• The propagator with momentum k 1 is always cut.
• Whenever possible, the propagator with momentum (p i − k 1 ) is cut.
• Subdiagrams are chosen so to avoid the square root of the Källén function as their leading singularity. This is always possible for this ladder diagram.
These rules, together with the parametrization of the momenta where θ ∈ [0, π], |k 1 | > 0, and 1 D−2 ranges over unit vectors in the dimensions transverse to p i and p j , make the calculation of these cuts particularly simple. It is easy to show that [12] Figure 7: Two-particle cuts in the p 2 3 -channel.
The changes of variables are also useful (the y variable is useful mainly when (p i − k 1 ) is not cut).

Unitarity cut in the p 2 3 channel
We present the computation of the cuts in the p 2 3 channel in some detail, in order to illustrate our techniques for the evaluation of cut diagrams outlined above. We follow the conventions of appendix A. We then collect the different contributions and check the cancellation of divergent pieces and the agreement with the term δ u 3 F (z,z) in eq. (5.3).
There are four cuts contributing to this channel, and our aim is to show that δ u 3 F (z,z). (5.10) Two-particle cuts. There are two two-particle cut diagrams contributing to the p 2 3channel unitarity cut, Cut p 2 3 , [45] T L (p 2 1 , p 2 2 , p 2 3 ) and Cut p 2 3 , [12] T L (p 2 1 , p 2 2 , p 2 3 ), shown in fig. 7. We start by considering the diagram in fig. 7a, which is very simple to compute because the cut completely factorizes the two loop momentum integrations into a one-mass triangle and the cut of a three-mass triangle: We substitute the following expressions for the one-loop integrals, which we have compiled in appendix B, where we have used p 2 1 = p 2 1 + iε to correctly identify the minus sign associated with p 2 1 in this region where p 2 1 > 0. As expected, the result is divergent for → 0: the origin of the divergent terms is the one-loop one-mass triangle subdiagram. Expanding up to O( ), we get Expressions for the coefficients f (i) [45] (z,z) are given in appendix C. We now go on to fig. 7b. We can see diagrammatically that the integration over k 2 is the (complex-conjugated) two-mass-hard box we have already studied in Section 4.3, with masses p 2 1 and p 2 2 . More precisely, we have To proceed, we parametrize the momenta as in eq. (5.6), with (i, j) = (3, 1). Then, we rewrite the momentum integration as The two delta functions allow us to trivially perform the k 0 and |k| integrations. For the remaining integral, it is useful to change variables to cos θ = 2x − 1, as in eq. (5.8), and we get, (5.14) The factor e −iπ was determined according to the iε prescription of the invariants. After expansion in , all the integrals above are simple to evaluate in terms of multiple polylogarithms. We write this expression as: and give the expressions for the coefficients f (i) [12] (z,z) in appendix C.
Three-particle cuts. There are two three-particle cut diagrams that contribute to the p 2 3 -channel unitarity cut, Cut p 2 3 ,[234] T L (p 2 1 , p 2 2 , p 2 3 ) and Cut p 2 3 ,[135] T L (p 2 1 , p 2 2 , p 2 3 ), shown in fig. 8. As these two cuts are very similar, we only present the details for the computation of the cut in fig. 8a, and simply quote the result for fig. 8b. In both cases, we note that the integration over k 2 is the cut in the (p 3 − k 1 ) 2 -channel of a two-mass one-loop triangle, with masses p 2 3 and (p 3 − k 1 ) 2 . More precisely, for the cut in fig. 8a we have We take the result for the cut of the two mass triangle given in appendix B and insert it into eq. (5.16), where we have used the δ-function to set k 2 1 = 0, and we have dropped the ±iε. We have included the θ-functions because the cut of the two-mass triangle is only nonzero when the (p 3 − k 1 ) 2 -channel is positive. It is also important to recall that the positive energy flow across the cut requires k 1,0 > 0, so we have included this θ-function explicitly. We use the parametrization of eq. (5.6), with (i, j) = (3, 2) and both changes of variables in eq. (5.8), since the propagator with momentum (p 3 − k 1 ) is not cut. The two conditions imposed by the θ-functions imply that 0 ≤ y ≤ 1 .

(5.18)
We then get We can now expand the hypergeometric function into a Laurent series in using standard techniques [59], and we then perform the remaining integration order by order. As usual, we write the result in the form [234] (z,z) + f The diagram of fig. 8b can be calculated following exactly the same steps, the only difference being that when using the parametrization of eq. (5.6) we have (i, j) = (3, 1). The result is Explicit expressions for the f (i) [234] (z,z) and f (i) [135] (z,z) are given in appendix C.
Summary and discussion. Let us now combine the results for each p 2 3 -channel cut diagram and compare the total with Disc and the relevant terms in the coproduct. We observe the sum is very simple, compared to the expressions for each of the cuts.
Note that, as imposed by the fact that the two-loop ladder is finite in four dimensions, the sum of the divergent terms of each diagram vanishes. In fact, this cancellation happens in a very specific way: the sum of the two-particle cuts cancels with the sum of the threeparticle cuts. If we write We call the divergent contribution of two particle cuts a virtual contribution because it is associated with divergences of loop diagrams, whereas the divergent contribution of three particle cuts, the real contribution, comes from integrating over a three-particle phase space. This cancellation is similar to the cancellation of infrared divergences for inclusive cross sections, although in this case we are not directly dealing with a cross section, but merely with the unitarity cuts of a single finite Feynman integral. A better understanding of these cancellations might prove useful for the general study of the infrared properties of amplitudes, and it would thus be interesting to understand how it generalizes to other cases. As expected, the sum of the finite terms does not cancel. We get Since all divergences have cancelled, we can set = 0 and write the cut-derived discontinuity of the integral as For comparison with Disc, we now analytically continue this result to the region R 3 where only the cut invariant is positive: p 2 3 > 0 and p 2 1 , p 2 2 < 0. In terms of the z andz variables, the region is: z > 1 >z > 0. None of the functions in eq. (5.26) has a branch cut in this region, and thus there is nothing to do for the analytic continuation and the result is valid in this region as it is given above, This is consistent with the expectation that the discontinuity function would be real in the region where only the cut invariant is positive [2,3]. The relations with Disc and δ are now easy to find. As expected, we find, We recall that this is not an unexpected result: it is just the relation between discontinuities and cuts of Feynman diagrams, see, e.g., ref. [1][2][3][4][5]60], related in turn to the language of the coproduct. The computation of the two cuts diagrams follows the same strategy as before, i.e., we compute the cut of the two-loop diagram by integrating over a carefully chosen one-loop subdiagram.

Unitarity cut in the
Computation of the cut diagrams. We start by computing Cut p 2 2 , [46] T L (p 2 1 , p 2 2 , p 2 3 ). As suggested by the momentum routing in fig. 9a, we identify the result of the k 2 integration with the complex conjugate of an uncut two-mass triangle, with masses (p 3 + k 1 ) 2 and p 2 3 :

(5.29)
Using the result for the triangle given in appendix B and proceeding in the same way as with the p 2 3 -channel cuts, we get (setting (i, j) = (2, 3) in eq. (5.6)) Cut p 2 2 , [46],R * T L (p 2 1 , p 2 2 , p 2 3 ) = 2π The cut integral Cut p 2 2 ,[136] T L (p 2 1 , p 2 2 , p 2 3 ) is slightly more complicated. Using the routing of loop momenta of fig. 9b, we look at it as the k 1 -integration over the cut of a three-mass box, where Cut t B 3m (l 2 2 , l 2 3 , l 2 4 ; s, t) is the t-channel cut of the three-mass box with masses l 2 i , for i ∈ {2, 3, 4}, l 2 1 = 0, s = (l 1 + l 2 ) 2 and t = (l 2 + l 3 ) 2 . In our case: The result for the t-channel cut of the three-mass box is given in appendix B in the region where the uncut invariants are negative, and t is positive. Since we work in the region where all the p 2 i are positive, some terms in the expression (B.11) need to be analytically continued using the ±iε prescriptions given above. Using eq. (5.6) with (i, j) = (2, 3) and introducing the variables x and y according to eq. (5.8), we have: 6 log(−s) = log p 2 1 + log u 3 + iπ , (5.32) 6 Strictly speaking, this analytic continuation is valid forz = z * , with Re(z) < 1. For the case of Re(z) > 1, the factors of iπ are distributed in other ways among the different terms, but the combination of all terms is still the same.
Summary and discussion. Similarly to the p 2 3 -channel cuts, we first analyze the cancellation of the singularities in the sum of the two cuts contributing to the p 2 2 -channel, and check the agreement with δ u 2 T L (p 2 1 , p 2 2 , p 2 3 ) given in eq. (5.3). In this case we only have single poles, and we see that the poles cancel, as expected: This cancellation can again be understood as the cancellation between virtual (from cut [46]) and real contributions (from cut [136]). Adding the finite contributions, we find f (0) [46] (z,z) + f Hence, the cut of the two-loop ladder in the p 2 2 channel is

(5.36)
Since this result was computed in the region where all invariants are positive, we now analytically continue to the region R 2 where p 2 2 > 0 and p 2 1 , p 2 3 < 0. For the z andz variables, this corresponds to 1 > z > 0 >z. The analytic continuation of the Li 2 and Li 3 functions is trivial, because their branch cuts lie in the [1, ∞) region of their arguments. However, the continuation of log u 2 needs to be done with some care, since u 2 becomes negative. We can determine the sign of the iε associated with u 2 by noticing that where we associate a −iε to p 2 1 because it is in the complex-conjugated region of the cut diagrams. We thus see that the −iπ term in eq. (5.36) is what we get from the analytic continuation of log (−u 2 − iε) to positive u 2 . In region R 2 , we thus have This agrees with the expectation that the discontinuity function should be real in the region where only the cut invariant is positive [2,3]. Furthermore, we again observe the expected relations with Disc and δ, Diagrammatically, the relation can be written as follows: 5.3 Unitarity cut in the p 2 1 channel Given the symmetry of the three-point ladder, the cut in the p 2 1 channel shown in fig. 10 can be done in exactly the same way as the p 2 2 channel, so we will be brief in listing the results for completeness.
For the sum of the two cut integrals, the reflection symmetry can be implemented by exchanging p 1 and p 2 in eq. (5.36), along with transforming z → 1/z andz → 1/z. The total cut integral is then We now analytically continue p 2 2 and p 2 3 to the region R 1 where we should match Disc. In this region, we havez < 0 and z > 1. Similarly to the previous case, we take p 2 2 − iε to find that log(u 2 − iε) → log(−u 2 ) − iπ, and thus In the last line, we have confirmed that the cut result agrees with a direct evaluation of the discontinuity of T L (p 2 1 , p 2 2 , p 2 3 ) in the region R 1 . The δ discontinuity evaluated from the coproduct is simply related to the discontinuities in the p 2 2 and p 2 3 channels. Indeed, we can rewrite eq. (5.3) as which agrees with Disc p 2 1 T L from eq. (5.41) modulo π 2 .

Sequence of unitarity cuts
In the previous section we gave a diagrammatic interpretation of the δ u 2 F (z,z) and δ u 3 F (z,z) terms of eq. (5.3) as unitarity cuts in p 2 2 and p 2 3 respectively. In this section we will take sequences of two unitarity cuts as defined in section 3.2 and match the result to entries of the coproduct.
Unlike the single unitarity cuts, which could be computed in the kinematic region R * where √ λ is imaginary and thusz = z * , and then analytically continued back to the region in which Disc is evaluated, the calculation of double unitarity cuts (in real kinematics) has to be done in the region where z,z and √ λ are real in order to get a nonzero result. Moreover, we must work in the specific region in terms of z andz corresponding to positive cut invariants and negative uncut invariant.
We start by reviewing and applying the general procedure to relate the sequential application of the Disc operator to cut integrals and to specific terms in the coproduct, as in eq. (3.16) and (3.17). It is hoped that in the context of a specific example, the procedure will become clearer and more intuitive. Next, as an example, we focus on the cases of Cut p 2 3 ,p 2 1 and Cut p 2 2 ,p 2 1 , comparing the results to the terms δ u 3 ,z F (z,z) and (δ u 2 ,z + δ u 2 ,1−z )F (z,z) of the ∆ 1,1,2 F (z,z) component of the coproduct in eq. (5.4).
Then, we present our method to evaluate the necessary cuts. We check that we indeed reproduce the expected terms of the coproduct and satisfy the relations we expect, and that the relations (3.16) and (3.17) among Disc, Cut, and the coproduct components hold. We stress that the fact that we reproduce the expected relations between Disc, Cut and the coproduct components is a highly nontrivial check on the consistency of the extended cutting rules of section 3.2. In particular, we see that our assumption that we can restrict ourselves to real kinematics is justified. Finally, we observe that, unlike the case of single unitarity cuts, it is insufficient to define cut diagrams only through the set of propagators that go on shell, but the results for the integrals strongly depend on the phase space boundaries, which are specified by the correct choice of kinematic region.

Relation between Cut and the coproduct, for sequential cuts of the ladder
We start by deriving the exact form of the expected relations between Cut p 2 i ,p 2 j F and truncated entries of the coproduct, δ x,y F , via Disc x,y F , according to (3.16) and (3.17). This is a generalization of what was done for the one-loop triangle in Section 4.1 to the case of the ladder diagram. It is possible to write the coproduct such that x = p 2 i , an exact Mandelstam invariant in accordance with the first-entry condition, but here we take x ∈ {u 2 , u 3 } and y ∈ {z,z, 1 − z, 1 −z}, for a direct correspondence with ∆ 1,1,2 F as written in eq. (5.4).
We present one example in detail and then list the results for all sequences of two cuts below, with some details of the derivation listed in Table 3. Let us look at the case i = 1, j = 2. The first discontinuity is taken in the p 2 1 channel, which is captured by −[− Disc u 2 − Disc u 3 ]; one minus sign appears because p 2 1 is in the denominator of u 2 and u 3 , and the other is inherent in the relation eq. (3.16). For the second discontinuity, we must work in the kinematic region R 1,2 where p 2 1 , p 2 2 > 0 and p 2 3 < 0, or equivalently 0 <z < 1 < z. 7 Approaching the branch point p 2 2 → 0 can be done either by z → 0 or z → 0, according to Table 1. The former limit is not contained in the region R 1,2 , so we have only y =z and not y = z. Thus we have already arrived at the relation where the iε prescription on the right-hand side follows from the rules of the cut diagram, which for us is p 2 1 + iε, p 2 2 − iε. There were two minus signs from a double discontinuity in eq. (3.16), and one from exchanging p 2 1 for u 2 and u 3 . To derive the correct sign in the relation between Disc and δ, we must probe the branch cuts of log(u 2 ), log(u 3 ), and log(z) on the negative real axis. We have shown once and for all in eq. (3.18) that the first entry introduces a minus sign in the relation eq. (3.17). For the second entry, we are again in the region R 1,2 , wherez > 0, so we must take log(−z) i j region x y from p 2 j − iε log approaching branch cut F and δ x,y F , via Disc x,y F , as described in section 6.1. It is necessary that Disc be given the same iε prescription as the cut diagram. Here it is always p 2 i + iε and p 2 j − iε.
rather than log(z). This argument of the logarithm inherits a positive imaginary part, −z + iε, from the imaginary part of p 2 2 − iε, so it is above its branch cut. Therefore the second discontinuity does not introduce a minus sign. We have only the single minus sign from the first entry, and a factor of 2πi for each of the two cuts, giving the final relation The other five cases are analyzed similarly. Some information for the steps in the derivation is listed in Table 3. The resulting relations are summarized as follows:

Double unitarity cuts
In this section we describe the computation of the sequences of two unitarity cuts corresponding to Cut p 2 1 • Cut p 2 3 and Cut p 2 1 • Cut p 2 2 ; see fig. 11 and fig. 12. All the cut integrals can be computed following similar techniques as the ones outlined in Section 5, so we will be brief and only comment on some special features of the computation. Details on how to compute the integrals can be found in appendix D.1, and the explicit results for all the cuts in fig. 11 and fig. 12 are given in appendices D.2 and D.3 respectively.
First, we note that, since we are dealing with sequences of unitarity cuts, the cut diagrams correspond to the extended cutting rules introduced in section 3.2. In particular, in section 3.2 we argued that cut diagrams with crossed cuts should be discarded, and such diagrams are therefore not taken into account in our computation. (In this example, all possible crossed cut diagrams would vanish anyway, for the reason given next.) Figure 11: Cut diagrams contributing to the Cut p 2 1 • Cut p 2 3 sequence of unitarity cuts.
Second, some of the cut integrals vanish because of energy-momentum constraints. Indeed the cut in fig. 11e vanishes in real kinematics because it contains a three-point vertex where all the connected legs are massless and on shell. Hence, the cut diagram cannot satisfy energy momentum conservation in real kinematics with D > 4 (recall the example of the two-mass-hard box). We will set this diagram to zero, and we observe a posteriori that this is consistent with the other results, thus supporting our approach of working in real kinematics.
Let us now focus on the cuts that do not vanish. As we mentioned previously, the cuts are computed by integrating over carefully chosen one-loop subdiagrams. In particular, for simplicity we avoid integrating over three-mass triangles, cut or uncut, because the leading singularity of this diagram is the square root of the Källén function, which leads to integrands that are not directly integrable using the tools developed for multiple polylogarithms. In Tables 4 and 5 we summarize the preferred choices of subdiagrams for the first loop integration. We observe that it is insufficient to define a cut integral by the subset of propagators that are cut. Indeed, some cut integrals in the two tables have the same cut propagators, but are computed in different kinematic regions due to the rules of  Section 3, leading to very different results 8 . Finally, depending on the cut integral and the kinematic region where the cut is computed, the integrands might become divergent at specific points, and we need to make sense of these divergences to perform the integrals. In the case where the integral develops an end-point singularity, we explicitly subtract the divergence before expanding in , using the technique known as the plus prescription. For example, if g(y, ) is regular for all y ∈ [0, 1], Cut two-mass triangle, masses p 2 3 and (p 1 + k 1 ) 2 , in (p 1 + k 1 ) 2 channel, fig. 12b Cut two-mass triangle, masses p 2 3 and (p 2 + k 1 ) 2 , in (p 2 + k 1 ) 2 channel, fig. 12c and p 2 2 , in t = (p 1 − k 1 ) 2 channel, fig. 12d. Table 5: Cuts contributing to the Cut p 2 1 • Cut p 2 2 sequence of unitarity cuts.
then, for < 0, we have: The remaining integral is manifestly finite, and we can thus expand in under the integration sign. However, we also encounter integrands which, at first glance, develop simple poles inside the integration region. A careful analysis however reveals that the singularities are shifted into the complex plane due to the Feynman iε prescription for the propagators. As a consequence, the integral develops an imaginary part, which can be extracted by the usual principal value prescription, where PV denotes the Cauchy principal value, defined by where g(y) is regular on [0, 1] and y 0 ∈ [0, 1]. Note that the consistency throughout the calculation of the signs of the iε of uncut propagators and subdiagram invariants, as derived from the conventions of the extended cutting rules of section 3.2 (see also appendix A), is a nontrivial consistency check of these cutting rules.

Summary and discussion
As expected from the relations eq. (3.16) and eq. (3.17) among Cut, Disc and δ, and in particular from eq. (6.3), we observe that and Cut p 2 , and, Based on the results presented above, it is natural to ask whether double unitarity cuts reproduce the discontinuity of single unitarity cuts on a diagram by diagram basis. For instance, if we consider (p 2 1 , p 2 2 ) sequences of cuts, is it true that: 10) The answer to this question is not simple. Indeed, while eq. (6.9) is true, eq. (6.10) is not. This is because these kinds of diagram by diagram relations are very sensitive to the branch cut structure of single cut diagrams. Interestingly, all these subtleties are washed out when considering full sets of double unitarity cuts, and the results given in eq. (6.7) and (6.8) are valid despite them. We verified that for the case of the (p 2 1 , p 2 3 ) cuts of the ladder, diagram by diagram relations hold for all single cut diagrams.
Because this falls outside of the subject of this paper, which is to relate sequences of unitarity cuts to iterated discontinuities and to the coproduct of uncut Feynman diagrams, we will not comment further on these relations. However, we believe this is an interesting subject for further study.

More than two unitarity cuts
Having considered a sequence of two unitarity cuts, it is natural to wonder about a sequence of three unitarity cuts in the three distinct channels of the ladder. Since the ladder is of transcendental weight four, we might expect the result of three cuts to give a function of weight one. It turns out, however, that the sequential cut in all three channels Figure 13: Cut [12456] on the three channels p 2 1 , p 2 2 , p 2 3 .
simply gives zero. In this section, we explain briefly how this result is understood from the points of view of Disc, the coproduct and Cut.
In the list of diagrams with cuts in the three channels p 2 1 , p 2 2 , p 2 3 , all but one have the property that one of the internal vertices has all three of its incident propagators cut, giving zero. The one remaining diagram is Cut [12456] , shown in fig. 13. This diagram turns out to vanish as well. In the figure, we have oriented the internal arrows according to positive energy flow. (We have assumed positive energy of p 3 and p 1 , and negative energy of p 2 , but all the other cases are similar or trivial.) To see the vanishing of this cut diagram, recall that we must evaluate this cut in a region where all three invariants p 2 i are positive. Use the momentum parametrization The two cut conditions on propagators 4 and 5, namely k 2 1 = 0 and (p 3 + k 1 ) 2 = 0 together, imply that k 10 = − p 2 3 /2, which violates the restriction on energy flow, k 10 > 0. The coproduct itself certainly allows truncations equalling the transcendental weight of the function, so there is no problem in writing nonvanishing expressions of the form δ x 1 ,x 2 ,x 3 ,x 4 F (z,z) for the ladder, and similarly for Disc x 1 ,x 2 ,x 3 ,x 4 . However, in relating these truncations to physical discontinuities, we must establish the correspondence between the x i and the invariants p 2 j , according to the rules stated in Section 3. Notably, the rules of Section 3 state that each variable x i must be able to approach its branch points independently of all the other x j . Since F (z,z) is merely a function of two variables, we are then limited to two iterations of the truncation of the coproduct. Any third truncation, δ x 1 ,x 2 ,x 3 F (z,z), does not correspond to a triple discontinuity of the integral.

From cuts to dispersion relations and coproducts
In previous sections we introduced computational tools to compute cut integrals, and we showed that extended cutting rules in real kinematics lead to consistent results. Furthermore, we argued that the entries in the coproduct of a Feynman integral can be related to its discontinuities and cut integrals. While these results are interesting in their own right, we present in this section a short application of how to use the knowledge of (sequences of) cut integrals, namely how to reconstruct some information about the original Feynman integral based on the knowledge of its cuts.
It is obvious from the first entry condition that if all cuts are known, we can immediately write down the component (1, n − 1) of a pure integral of weight n. In particular, for the one-and two-loop triangle integrals investigated in previous sections, we immediately obtain ∆ 1,1 (T (z,z)) = log u 2 ⊗ δ u 2 T (z,z) + log u 3 ⊗ δ u 3 T (z,z) , and the quantities δ u i T (z,z) and δ u i F (z,z) are directly related to the discontinuities of the integral through eqs. (4.20), (5.27) and (5.38). Note that eq. (7.1) determines the functions T (z,z) and F (z,z) uniquely up to terms proportional to π. Similarly, in eq. (6.3) we have shown how the double discontinuities of the two-loop ladder triangle are related to the entries in the coproduct. We can then immediately write 2) and the values of δ u i ,α F (z,z) can be read off from eq. (6.3). 9 Thus, we see that the knowledge of all double discontinuities enables us to immediately write down the answer for the (1,1,2) component of the two-loop ladder triangle. Note that the knowledge of eq. (7.2) uniquely determines the function F up to terms proportional to zeta values. While the previous application is trivial and follows immediately from the first entry condition and the knowledge of the set of variables that can enter the symbol in these particular examples, it is less obvious that we should be able to reconstruct information about the full function by looking at a single unitarity cut, or at a specific sequence of two unitarity cuts. In the rest of this section we give evidence that this is true nevertheless.
The main tool for determining a Feynman integral from its cuts is the dispersion relation, which expresses a given Feynman integral as the integral of its discontinuity across a certain branch cut. Traditionally used in the context of the study of strongly interacting theories, dispersion relations appear more generally as a consequence of the unitarity of the S-matrix, and of the analytic structure of amplitudes [60]. These relations are valid in perturbation theory, order by order in an expansion of the coupling constant. It was shown in refs. [1][2][3][4][5] that individual Feynman integrals can also be written as dispersive integrals. The fundamental ingredient in the proof of the existence of this representation is the largest time equation [2], which is also the basis of the cutting rules. In the first part of this section we briefly review dispersion relations for Feynman integrals, illustrating them with the example of the one-loop three-mass triangle integral discussed in section 4.1. In the second part we show that, at least in the case of the integrals considered in this paper, we can use the modern Hopf algebraic language to determine the symbol of the integrals from a single sequence of unitarity cuts. We note however that this reconstructibility works for the full integral, and not for individual terms in the Laurent expansion in . We therefore focus on examples which are finite in four dimensions, so that we can set = 0. 10

Dispersion relations
Dispersion relations are a prescription for computing an integral from its discontinuity across a branch cut, taking the form where ρ(p 2 1 , s, . . .) = Disc p 2 2 F (p 2 1 , p 2 2 , . . .) p 2 2 =s , as computed with eq. (3.1), and the integration contour C goes along that same branch cut. The above relation can be checked using eqs. (3.1) and (6.5).
In order to illustrate the use of dispersion relations, we briefly look at the case of the scalar three-mass triangle. Its p 2 2 -channel discontinuity was computed in eq. (4.19), and we recall it here expressed in terms of Mandelstam invariants, This leads to a dispersive representation for the three-mass triangle of the form .
(7.5) Note that the integration contour runs along the real positive axis: it corresponds to the branch cut for timelike invariants of Feynman integrals with massless internal legs. Already for this not too complicated diagram we see that the dispersive representation involves a rather complicated integration.
The main difficulty in performing the integral above comes from the square root of the Källén function, whose arguments depend on the integration variable. However, defining x = s/p 2 1 , and introducing variables w andw similar to eq. (4.3) by 6) or equivalently, 10 A counterexample to the reconstructibility of individual terms in the Laurent expansion is given by the two-mass-hard box: it is clear from eq. (4.31) that a cut in a single channel can fail to capture all terms of the symbol.
we can rewrite the dispersive integral as, where the integration region for w andw is deduced from the region where the discontinuity is computed (see, e.g., table 2). Written in this form, the remaining integration is trivial to perform in terms of polylogarithms, and we indeed recover the result of the three-mass triangle, eq. (4.7). For the three-mass triangle, we can in fact take a second discontinuity and reconstruct the result through a double dispersion relation because the discontinuity function, eq. (7.4), has a dispersive representation itself [1,54]. Note that this representation falls outside of what is discussed in ref. [5], and we are not aware of a proof of its existence from first principles. The double discontinuity is simply given, up to overall numerical and scale factors, by the inverse of the square root of the Källén function, see eq. (4.22). We obtain The integral is trivial to perform and leads to the correct result. 11 We see that we can obtain the result for the one-loop three-mass triangle from the knowledge of its single and double cuts. Note that an important ingredient in order to perform the dispersive integral was the choice of variables in which to write the dispersive integral. While the choice of the variables z andz is not obvious a priori when looking at the corresponding Feynman integral, these variables appear naturally when parametrizing the phase space integrals corresponding to the cut integrals. This gives hope that for more complicated Feynman integrals, computing their cuts could be a good way to identify the most suitable variables in which to express the uncut integral (the equivalent of the z and z variables that appeared naturally in this example), as they are simpler functions that already have basic characteristics of the full Feynman integral. We will not explore this point further in this paper, and we leave it for future work.

Reconstructing the coproduct from a single unitarity cut
As discussed above, Feynman diagrams can be fully recovered from unitarity cuts on a given channel through dispersion relations. These relations rely on two ingredients: 11 We have redefined w andw by replacing u3 by y in eq. (7.7). Just as for the single dispersion integral, the integration region is deduced from the region where the double discontinuity is computed, R2,3 in this case. Changing variables to β = 1 w and γ = 1 1−w makes the integral particularly simple to evaluate.
the discontinuity of a function across a specific branch cut and the position of that particular branch cut. Given the relations between the (1, n − 1) entries of the coproduct, discontinuities and single unitarity cuts established in previous sections, it is clear that the full information about the Feynman integral is encoded in any one of these entries of the coproduct, since it contains the same information about the function as a dispersive representation. We should thus be able to reconstruct information about the full function by looking at a single cut in a given channel. For simplicity, we only work at the level of the symbol in the rest of this section, keeping in mind that we lose information about zeta values in doing so. In a nutshell, we observe that if we combine the first entry condition and the results for (the symbols of) the discontinuities with the integrability condition (2.13), we immediately obtain the symbol of the full function. In the following we illustrate this procedure on the examples of the one-loop triangle and two-loop ladder triangle. Starting from the result for the unitarity cut on a single channel, the procedure to obtain the symbol of the full function can be formulated in terms of a simple algorithm, which involves two steps: (i) check if the tensor satisfies the integrability condition, and if not, add the relevant terms required to make the tensor integrable.
(ii) check if the symbol obtained from the previous step satisfies the first entry condition, and if not, add the relevant terms. Then return to step (i).
We start by illustrating this procedure on the rather simple example of the three-mass triangle of Section 4.1. From eq. (4.19), the symbol of the cut on the u 2 channel is where we emphasize that the rational function is to be interpreted as the symbol of a logarithm. Since we considered a cut on the p 2 2 channel,the first entry condition implies that we need to prepend u 2 = zz to the symbol of the discontinuity. Thus we begin with the tensor We then proceed as follows.
• Step (i): This tensor is not the symbol of a function, as it violates the integrability condition. To satisfy the integrability condition, we need to add the two terms The full tensor is not the symbol of a Feynman diagram, since the two new terms do not satisfy the first entry condition.
• Step (ii): To satisfy the first entry condition, we add two new terms: At this stage, the sum of terms obeys the first entry condition and the symbol obeys the integrability condition, so we stop our process.
Putting all the terms together, we obtain S(T (z,z)) = 1 2 which agrees with the symbol of the one-loop three mass triangle in D = 4 dimensions, eq. (4.10). While the previous example might seem too simple to be representative, we show next that the same conclusion still holds for the two-loop ladder. In the following we use our knowledge of the cut in the p 2 3 channel, eq. (5.26), and show that we can again reconstruct the symbol of the full integral F (z,z). Combining eq. (5.26) with the first entry condition, we conclude that S(F (z,z)) must contain the following terms: If we follow the same steps as in the one-loop case, we can again reconstruct the symbol of the full function from the knowledge of the symbol of the cut in the p 2 3 channel alone. More precisely, we perform the following operations: • Step (i): To obey the integrability condition, we must add to the expression above the following eight terms: • Step (ii): The terms we just added violate the first entry condition. To restore it we must add eight more terms that combine with the ones above to have Mandelstam invariants in the first entry, • Step (i): The newly added terms violate the integrability condition. To correct it, we must add two new terms, • Step (ii): We again need to add terms that combine with the two above to have invariants in the first entry, At this point the symbol satisfies both the first entry and integrability conditions, and we obtain a tensor which agrees with the symbol for F (z,z) (5.5). We note that for both examples considered above, the same exercise could have been done using the results for cuts in other channels.

Reconstructing the coproduct from double unitarity cuts
While the possibility of reconstructing the function from a single cut in a given channel might not be too surprising due to the fact that Feynman integrals can be written as dispersive integrals over the discontinuity in a given channel, we show in this section that in this particular case we are able to reconstruct the full answer for ∆ 1,1,2 F from the knowledge of just one sequential double cut. Note that ∆ 1,1,2 F is completely equivalent to the symbol S(F ). Indeed, the weight two part of ∆ 1,1,2 F is defined only modulo π, which is precisely the amount of information contained in the symbol.
To be more concrete, let us assume that we know the value of Cut p 2 3 ,p 2 2 F , and thus we have determined that δ u 3 ,z F = − log z logz + 1 2 log 2 z . (7.11) Since the symbols of log u i and δ u 3 ,z F have all their entries drawn from the set {z,z, 1 − z, 1 −z}, we make the assumption that ∆ 1,1,2 F can be written in the following general form: where f u 3 ,z = δ u 3 ,z F and the remaining f u i ,α denote some a priori unknown functions of weight two (defined only modulo π). Imposing the integrability condition in the first two entries of the coproduct gives the following constraints among the f u i ,α : If we require in addition thatF = −F , where the tilde denotes exchange of z andz (because its leading singularity is likewise odd under this exchange), we find in additioñ Thus, we can write (7.14) Notice that up to this stage all the steps are generic: we have not used our knowledge of the functional form of the double cut f u 3 ,z , but only the knowledge of the set of variables entering its symbol and the antisymmetry of the leading singularity under the exchange of z andz.
Next we have to require that eq. (7.14) be integrable in the second and third component. Assuming again that we only consider symbols with entries drawn from the set {z, 1 − z,z, 1 −z}, we use eq. (7.11) and impose the integrability condition eq. (2.13), and we see that the symbols of the two unknown functions in eq. (7.14) are uniquely fixed, in agreement with eq. (5.4).
We stress that the fact that we can reconstruct ∆ 1,1,2 F from a single sequence of cuts is not related to the specific sequence we chose. For example, if we had computed only Cut p 2 1 ,p 2 2 F and thus determined that −f u 2 ,z −f u 3 ,z = −Li 2 (z)+Li 2 (z)+log z logz − 1 2 log 2 z, the integrability condition would fix the remaining two free coefficients in a similar way. Finally, we could consider Cut p 2 3 ,p 2 1 F , but since this cut is obtained by a simple change of variables from Cut p 2 3 ,p 2 2 F through the reflection symmetry of the ladder, it is clear that integrability fixes the full symbol once again.
Let us briefly consider the analogous construction for the one-loop triangle, where the f u i ,α are simply constant functions. A double cut, without loss of generality say Cut p 2 2 ,p 2 3 , gives a constant value for f p 2 2 ,1−z , as in eq. (4.23) and eq. (4.24). We would conclude in the analog of eq. (7.14) above that we have a consistent solution with f u 3 ,z =f u 3 ,z and f u 2 ,z = f u 3 ,1−z = 0, which is indeed the ∆ 1,1 of the triangle, obtained by a consistent completion algorithm as in the previous subsection.
While it is quite clear that the reason why the algorithm of section 7.2 converged was the existence of a dispersive representation of Feynman integrals, it is not clear to us at this stage whether the existence of a double dispersive representation is a necessary condition for the reconstruction based on the knowledge of ∆ 1,1,2 done in this section to work, although it does seem reasonable that it would be the case.
In closing, we notice that in this example, the integrability condition eq. (7.12) implies that Cut p 2 , through the relations listed in eq. (6.3). It would be interesting to see whether there is a general link between the integrability of the symbol and the permutation invariance of a sequence of cuts.

Discussion
In this paper we studied cut Feynman diagrams with two objectives. The first was to develop techniques for analytic evaluation of such integrals, and the second to formulate precise relations between cut integrals and uncut ones, providing an interpretation of the coproduct and the symbol of the latter.
Techniques for direct computation of cut integrals in D spacetime dimensions are far less developed than those for ordinary (uncut) loop integrals. A well established technique for the calculation of multi-loop diagrams is the integration over an off-shell subdiagram. The ultimate advantage of cut integrals is that multi-loop cut diagrams reduce to integrals over products of simpler lower-loop integrals with extra on-shell external legs. This was illustrated here at the two-loop level, where different cuts where computed using one-loop triangle and box integrals with massless or a limited number of massive external legs. This method has the potential to be applied to more complicated multi-loop and multi-leg cut integrals.
Throughout this paper we took D = 4 − 2 -dimensional cuts. This is a necessity when dealing with infrared-divergent cut integrals: notably, individual cuts of (multiloop) integrals that are themselves finite in four dimensions may be divergent when the internal propagators that are put on shell are massless. The sum of all cuts on a given channel corresponds, according to the largest time equation [2,3], to the discontinuity of the uncut integral; given that the latter is finite, one expects complete cancellation of the singularities among the different cuts. This situation was encountered here upon taking unitarity cuts of the two-loop ladder graph, where we have seen that the pattern of cancellation is similar to the familiar real-virtual cancellation mechanism in cross sections, although this example does not correspond to a cross section. Understanding this pattern of cancellation is useful for the general program of developing efficient subtraction procedures for infrared singularities, and it would be interesting to explore how this generalizes for other multi-loop integrals.
Taking a step beyond the familiar case of a single unitarity cut, we developed here the concept of a sequence of unitarity cuts. To consistently define this notion, we extended the cutting rules of refs. [2,3] to accommodate multiple cuts on different channels in an appropriately chosen kinematic region. The cutting rules specify a unique prescription for complex conjugation of certain vertices and propagators, which is dictated by the channels on which cuts are taken. Importantly, the result does not depend on the order in which the cuts are applied. The kinematic region is chosen such that the Mandelstam invariants corresponding to the cut channels are positive, corresponding to timelike kinematics. In its center-of-mass Lorentz frame, this invariant defines the energy flowing through the set of on-shell propagators. The energy flow through all these propagators has a consistent direction that is dictated by the external kinematics; for any given propagator this direction must be consistent with the direction of energy flow assigned to it by any other cut in the sequence. We further exclude crossed cuts, as well as iterated cuts in the same channel since they are not related to discontinuities as computed in this paper. Finally, we restrict ourselves to real kinematics. These cutting rules pass numerous consistency checks and they form a central result of the present paper. Understanding what information is contained in crossed cuts and in iterated cuts in the same channel as well as what can be obtained by allowing for complex kinematics are of course interesting questions for further study.
Having specified the definition of a sequence of unitarity cuts, we find the following correspondence, which we conjecture to be general, among (a) the sum of all cut diagrams in the channels s 1 , . . . s k , which we denote by Cut s 1 ,...,s k ; (b) a sequence of discontinuity operations, which we denote by Disc x 1 ,...,x k , where the x i are algebraic functions of the Mandelstam invariants; (c) and the weight n − k cofactors of the terms in the coproduct of the form ∆ 1,1,...,1,n−k , where each of the k weight one entries of a specific term in ∆ 1,1,...,1,n−k is associated with one of the x i in a well defined manner, which we call δ x 1 ,...,x k .
The correspondence is formulated in eqs. (3.16) and (3.17). We illustrated it using the two-loop ladder triangle example where one may take up to two sequential cuts with any combination of channels, obtaining nontrivial results; the relations are summarized by eqs. (6.3). In examples with more loops and legs, we expect that a deeper sequence of unitarity cuts may be attainable.
We find that while the leftmost entry of the symbol (or equivalently of the ∆ 1,...,1,n−k terms in the coproduct) is always one invariant out of the subset of the Mandelstam invariants in which the function has a branch cut (the first entry condition [16]), all other entries may not necessarily be such a variable, but may instead be drawn from a longer list, {x i }, sometimes called the symbol alphabet. These are also the natural variables appearing as arguments of logarithms and polylogarithms in both cut diagrams and the original uncut one. For example, in the two-loop ladder triangle considered through O( 0 ), the alphabet consists of four letters {z,z, 1 − z, 1 −z} defined in eq. (4.4). In general, letters in the symbol alphabet x i are algebraic functions of the Mandelstam invariants: they are the solutions of quadratic equations which emerge upon solving the simultaneous on-shell conditions imposed by cuts. Consequently, there is hope that cuts can identify the relevant variables in terms of which the uncut integral can be most naturally expressed.
Because the arguments of polylogarithms, and equivalently the second and subsequent entries of the coproduct ∆ 1,1,...,1,n−k terms, are not the Mandelstam invariants themselves, while any unitarity cut is defined by a channel that does correspond to a Mandelstam invariant s i , the relation between cuts and discontinuities in eq. (3.16) is more complicated starting from the second cut. Nevertheless, we have seen how these variables are related. The rule is that the relevant branch points are common to s i and x i , and these branch points can be approached by x i independently of the other variables x j . Also, the iε prescription of x i is inherited from that of s i , so that the relation of eq. (3.16) can be made precise.
We verified that the expected relations between sequences of cuts, sequences of discontinuities and the relevant terms of the coproduct hold in the cases of the double cut of the one-loop triangle, the four-mass box and the two-mass-hard box. We then explored in detail the much less trivial two-loop three-mass ladder diagram, for which we also observed agreement with the expected relations.
Given that cut diagrams are simpler to compute (owing to the fact that they reduce to integrals over products of simpler lower-loop amplitudes) and may identify the most convenient variables, it is natural to ponder whether the result of a cut diagram can be uplifted to obtain the uncut function. In the case of a single unitarity cut, this can always be done through a dispersion integral [1][2][3][4][5]. In the case of a sequence of unitarity cuts, this requires a multiple dispersion relation, and the general conditions for these to exist are not known.
In section 7 of the present paper we made some progress in developing methods for the reconstruction of a Feynman integral from its cuts. Our first observation, considering the reconstruction of the one-loop three-mass triangle from either its single or double cut, was that while dispersion relations may appear as complicated integrals, they become simple when expressed in terms of the natural variables x i . In these variables the dispersion integral in the case considered falls into the class of iterated integrals amenable to the Hopf algebra techniques. This is of course consistent with the fact that each dispersion integral is expected to raise the transcendental weight of the function by one: it is the opposite operation to taking the discontinuity of the function across its branch cut. It is clearly important to study this connection between dispersion integrals and iterated polylogarithmic integrals for other examples.
We next presented ways to reconstruct information about the full function from the knowledge of a single set of cuts, along with the symbol alphabet. This was achieved by using two main constraints: the integrability of the symbol and the first entry condition. More precisely, we showed how to reconstruct the symbol of the full integral from the knowledge of (the symbol of) a single unitarity cut in one of the channels. We believe that our approach to reconstruction is valid generally, provided the existence of a dispersive representation of Feynman integrals. We also showed that in the case of the two-loop ladder (and the much simpler one-loop triangle) it is possible to reconstruct all the terms of the ∆ 1,1,2 component of the coproduct of the uncut integral from the knowledge of a single sequence of double cuts. How general this procedure is is less obvious to us, and it is certainly worth investigating.
Another very intriguing observation based on the examples at hand concerns the connection between the integrability condition of the symbol and the equality of sequences of unitarity cuts between which the order is permuted. As mentioned above, the result of a sequence of unitarity cuts does not depend on the order in which the cuts are applied. Therefore the double cut relations summarized in eqs. (6.3) must satisfy Cut p 2 i ,p 2 j = Cut p 2 j ,p 2 i . This in turn implies highly nontrivial relations between different ∆ 1,1,2 components; for example the r.h.s. of eq. (6.3a) must be the same as the r.h.s. of eq. (6.3b), and similarly for the other pairs. The crucial observation is that these relations indeed hold owing to the integrability constraints as summarized in eq. (7.12). Note that the latter are based solely on the symbol alphabet and the integrability condition of eq. (2.13). We leave it for future study to determine how general the connection is between integrability and permutation invariance of a sequence of cuts.
In conclusion, we developed new techniques to evaluate cut Feynman integrals and relate these to the original uncut ones. In dealing with complicated multi-loop and multileg Feynman integrals there is a marked advantage to computing cuts, where lower-loop information can be systematically put to use. While cut integrals are simpler than uncut ones, they depend on the kinematics through the same variables, {x i }, which characterize the analytic structure of the integral. Identifying this alphabet is crucial in relating cuts to terms in the coproduct, and then either integrating the dispersion relation or reconstructing the symbol of the uncut integral algebraically. We have demonstrated that the language of the Hopf algebra of polylogarithms is highly suited for understanding the analytic structure of Feynman integrals and their cuts. Finally, we have shown that there is a great potential for computing Feynman integrals by using multiple unitarity cuts, and further work in this direction is in progress. e a Tecnologia, Portugal, through a doctoral degree fellowship (SFRH/BD/69342/2010). R.B. was supported in part by the Agence Nationale de la Recherche under grant ANR-09-CEXC-009-01 and is grateful to the BCTP of Bonn University for extensive hospitality in the course of this project. R.B. and C.D. thank the Higgs Centre for Theoretical Physics at the University of Edinburgh for its hospitality. E.G. was supported in part by the STFC grant "Particle Physics at the Tait Institute" and thanks the IPhT of CEA-Saclay for its hospitality.

A Notation and conventions
Feynman rules. Here we summarize the Feynman rules for cut diagrams in massless scalar theory. For a discussion of their origin, as well as the rules for determining whether a propagator is cut or uncut, see section 3. There can be multiple dashed lines, indicating cuts, on the same propagator, without changing its value. There is a theta function restricting the direction of energy flow on a cut propagator, whose origin is detailed in Section 3.
In the examples, we omit writing the theta function, as there is always at most one nonvanishing configuration.
uncut one-loop triangle with one mass (p 2 3 ) and the double cut of a three-mass triangle, with masses p 2 1 , p 2 2 and p 2 3 , in the channels p 2 1 and p 2 3 .