Baikov-Lee Representations Of Cut Feynman Integrals
aa r X i v : . [ h e p - ph ] J u l TCDMATH-17-12
Prepared for submission to JHEP
Baikov-Lee Representations Of Cut Feynman Integrals
Mark Harley, Francesco Moriello, Robert M. Schabinger
Hamilton Mathematics Institute, School of Mathematics, Trinity College Dublin,College Green, Dublin 2, Ireland
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We develop a general framework for the evaluation of d -dimensional cut Feyn-man integrals based on the Baikov-Lee representation of purely-virtual Feynman integrals.We implement the generalized Cutkosky cutting rule using Cauchy’s residue theorem andidentify a set of constraints which determine the integration domain. The method appliesequally well to Feynman integrals with a unitarity cut in a single kinematic channel andto maximally-cut Feynman integrals. Our cut Baikov-Lee representation reproduces theexpected relation between cuts and discontinuities in a given kinematic channel and fur-thermore makes the dependence on the kinematic variables manifest from the beginning.By combining the Baikov-Lee representation of maximally-cut Feynman integrals and theproperties of periods of algebraic curves, we are able to obtain complete solution sets forthe homogeneous differential equations satisfied by Feynman integrals which go beyondmultiple polylogarithms. We apply our formalism to the direct evaluation of a number ofinteresting cut Feynman integrals. ontents ǫ results for selected purely-virtual Feynman integrals 29 B.1 The one-external-mass one-loop triangle 30B.2 The massless one-loop box 30B.3 The one-external-mass six-line two-loop double triangle 30
C Baikov-Lee for the purely-virtual one-external-mass one-loop bubble 31
As has long been known, the evaluation of multi-loop Feynman integrals is an importantcomponent of many high-precision collider physics calculations. Even at the two-loop level,both purely-virtual and cut Feynman integrals often provide a remarkable computationalchallenge. This is particularly true if one proceeds analytically, and, as a consequence,a number of specialized techniques have been developed to aid in the evaluation of suchintegrals. Most direct analytic integration techniques have traditionally employed somevariant of the well-known Feynman (or Schwinger) parametric representation [1] as a start-ing point in the purely-virtual case, due to the fact that it makes the dependence of theintegral on Lorentz-invariant quantities manifest. The aim of this paper is to supply ananalogous framework for cuts which is suitable not only for the direct calculation of real– 1 –adiative master integrals, but also for the maximally-cut integrals relevant to the study ofpurely-virtual Feynman integrals in the method of differential equations [2–8]. As we shallsee, our work is a natural out-growth of earlier work on the subject of integral reductionwhich we apply in a novel way to evaluate various types of cut Feynman integrals.It is unfortunately the case that phenomenologically-relevant multi-loop integration byparts reductions [9, 10] often require dedicated effort and substantial resources to compute.Although it is probably fair to say that most recent higher-order perturbative calculationsrely on some variant of Laporta’s algorithm [11–14], several other interesting and usefulalgorithms have been worked out and implemented over the years (see e.g. [15–24]). How-ever, even very different approaches, such as the one advocated by Baikov [16], sometimesturn out to have ramifications for Laporta’s method as well. Some time ago, the derivationof the Baikov formula for purely-virtual Feynman integrals was clarified by Lee and thenused to great effect as a generating function for integration by parts relations [25]. Theproduct of his analysis, what we shall hereafter refer to as the Baikov-Lee representation,will be of great interest to us in this work. Larsen and Zhang argued in [21] that evaluatingthe integration by parts relations generated by the Baikov representation on the support ofvarious cuts dramatically improves the approach to integral reduction originally advocatedby Gluza, Kajda, and Kosower [17]. In fact, from their work and a similar study by Ita [20],one can readily guess that the Baikov-Lee representation ought to offer a useful startingpoint for the evaluation of cut Feynman integrals.Even though cut Feynman integrals play every bit as important a role as purely-virtualFeynman integrals in the perturbative computation of collider observables, methods fortheir direct evaluation have not been as thoroughly developed. It is entirely possible that,at least in part, this state of affairs has persisted due to the fact that it is not completelytrivial to write down a Baikov representation for Feynman integrals directly in Minkowskispace. The issue is that the standard derivation of the Baikov-Lee representation (see e.g. [28] for a detailed exposition) relies heavily upon Euclidean geometric intuition whichdoes not immediately generalize. In this paper, we write down a simple recipe for theanalytical continuation of the Euclidean Baikov-Lee formula by drawing an analogy to themore familiar situation that one encounters in the derivation of the Feynman parametricrepresentation. To pass from uncut to cut propagators, we use sequential applicationsof Cauchy’s residue theorem to implement a natural generalization of Cutkosky’s cuttingrule [29, 30]. Finally, the integration region is determined by analyzing the analyticalstructure of the integrand and applying the available constraints from physics. At eachstep of the calculation, one integrates between branch points of the integrand with respectto the current variable of integration. This procedure allows one to write down Baikov-Lee Certainly, we do not wish to suggest that such techniques do not exist. For instance, reference [26]describes some pertinent traditional and state-of-the-art direct integration techniques in great detail. Infact, a setup which bears at least some rudimentary resemblance to the one discussed in this paper wasdeveloped for tree-level cross section calculations in d = 4 long ago [27]. Due to the fact that the generalized Cutkosky rule of reference [30] is actually written in terms of deltadistributions and their derivatives, certain subtleties apply. To be consistent, one must first integrate out acomplete set of scalar products localized by the distributions before attempting any of the more non-trivialintegrations over scalar products. We would like to thank Ruth Britto for emphasizing this point to us. – 2 –epresentations for a wide class of cut Feynman integrals. Crucially, this approach makesthe dependence of the cut integral on the kinematic invariants of the problem manifest andeliminates the need to set up a convenient reference frame to carry out integrations overthe components of the cut loop momenta. As mentioned above, the Baikov-Lee representation for cut Feynman integrals alsohas important applications to the more indirect method of differential equations. Writingdifferential equations with respect to the available kinematic parameters allows for thecomplete determination of a large class of Feynman integrals in terms of an appropriateset of iterated integrals, order-by-order in the parameter of dimensional regularization, ǫ . For simple families of Feynman integrals with relatively few ratios of scales, the func-tion space associated to Feynman integrals is spanned by iterated integrals with rationalintegrating factors. This set of iterated integrals, comprised of the well-known multiplepolylogarithms [33], has been studied and popularized by many authors ([34–36] to namea few). In the polylogarithmic case, it is always possible, by suitably choosing the basisof Feynman integrals [37–39], to reduce the problem to one which admits an elementaryformal solution in terms of Chen iterated integrals [40]. Using the symbol-coproduct calcu-lus [41–44], performance-optimized solutions which can be readily interfaced with a MonteCarlo integration program may be constructed for the purposes of phenomenology (see e.g. [45–48]). Even through to weight four, this is not always straightforward to do in practice,despite the fact that the weight four function space has been studied extensively and is inprinciple well-understood [49].For general Feynman integrals, far more complicated analytic structures may appear;even simple-looking two-loop integrals which depend on sufficiently many kinematic vari-ables may already involve elliptic polylogarithms and related functions [50–55]. In thiscontext, maximally-cut Feynman integrals play an important role because they satisfy thehomogeneous differential equations for the associated uncut Feynman integrals [30, 31]. This property is particularly useful when Feynman integrals cannot be expressed in termsof multiple polylogarithms, and iterated integrals over special functions need to be con-sidered [59–65]. In fact, higher-order differential equations appear in non-polylogarithmiccases, and, at the present time, no general solution algorithm is known. Nevertheless, itwas observed by Primo and Tancredi in [31] that, upon setting ǫ to zero, maximally-cutFeynman integrals can often be computed in closed form, allowing one to find at least asingle homogeneous solution to the higher-order differential equations under considerationby direct integration. Provided that a complete, linearly independent set of homogeneoussolutions can be found, the full solution can finally be determined using the variation ofparameters technique. In our opinion, it is of great importance to supply an algorithmwhich comes up with not just one, but rather a complete set of homogeneous solutions. We To appreciate this point, we invite the reader to compare and contrast the maximally-cut Feynmanintegral calculation which appears in both references [31] and [32]. Although, we will leave a detailed discussion for future work, there is nothing stopping us from applyingthis technique also to the differential equations satisfied by multi-scale cut Feynman integrals in the reverse-unitarity method [56–58]. – 3 –rovide a general prescription and show that it allows for the straightforward constructionof complete sets of homogeneous solutions for the non-polylogarithmic examples of Section4. The plan of this paper is as follows. In Section 2, we explain in detail how to writedown Baikov-Lee representations for Feynman integrals cut in a single kinematic channeland how to work out an explicit description of the relevant integration domain. Our focuswill be on the physical case relevant to cross section calculations because the procedurefor maximally-cut Feynman integrals is closely analogous and has already been discussedin reference [32]. Although we do not have a computer program which finds the integra-tion limits for arbitrarily complicated cut Feynman integrals, we have a solid conceptualunderstanding which could in the future lead us to an explicit algorithm. In Section 3, wego through a number of well-studied one- and two-loop examples of Feynman integrals cutin a single kinematic channel in order to give the reader a feeling for how explicit com-putations typically proceed when one adopts a Baikov-Lee representation as the startingpoint. To the best of our knowledge, our treatment of classical cut Feynman integrals inthe Baikov-Lee representation is new and effectively extends the work of Frellesvig and Pa-padopoulos [32] beyond the maximally-cut case. Although we work with generic values ofthe spacetime dimension for pedagogical purposes, our calculations strongly suggest that,in practice, an expansion in ǫ under the integral sign must be the way to go for all but thevery simplest of cut Feynman integrals.In Section 4, we move on to maximally-cut Feynman integrals which evaluate to com-plete elliptic integrals. We discuss a general solution strategy applicable to many problemsof practical interest and then demonstrate the general procedure by focusing on exampleswhich are suitable for exposition. We emphasize in particular the utility of integrating outone loop at a time, as this generically leads to simpler Baikov-Lee representations. Finally,we conclude in Section 5 and outline our plans for future research. We also include a num-ber of appendices for pedagogical purposes and cross-checks. To streamline the expositionin the body of the paper, we summarize a number of purely mathematical results fromthe theory of hypergeometric-like functions in Appendix A. In Appendix B, we reproducephysical-region results from the literature for the uncut versions of the integrals consideredin Section 3. This allows the reader to easily verify our results using the classical relationbetween discontinuities and cuts in a given kinematic channel (see e.g. [68–70]). Finally,in Appendix C, we evaluate a simple uncut Feynman integral using the Baikov-Lee setupto help the less familiar reader understand the relation between our prescriptions for cutFeynman integrals and the usual prescriptions for Baikov’s method in the purely-virtualcase. A few days prior to the appearance of this paper, we became aware of a recent preprint, [66], which dis-cusses many of the same technical issues for non-polylogarithmic Feynman integrals. In fact, the maximally-cut case was also discussed recently in the Baikov approach by yet another group [67], with, however, adifferent set of physical problems in mind. – 4 –
General formalism
In this section, we define our notation, recall some results from the literature, and explainhow we generalize the Baikov-Lee representation to the case of cut Feynman integrals.
Let us begin by discussing our notation for purely-virtual, L -loop Feynman integrals andrecalling some useful facts about them. For the direct integration of purely-virtual Feyn-man integrals, a very common starting point is the Feynman (or Schwinger) parametricrepresentation (see e.g. [71] for a detailed exposition). In many cases, it is convenient towrite down the Feynman parametric representation in Euclidean space, treating all n ex-ternal momenta, { p i } , on an equal footing by taking them all to be outgoing. In the mostgeneral case [72], it suffices to consider Feynman integrals of the form I E = Z d d k · · · Z d d k L N Y ℓ =1 (cid:0) Q ℓ ( k i , p j ) + m ℓ (cid:1) − ν ℓ , (2.1)where Q ℓ ( k i , p j ) denotes the momentum of the ℓ -th propagator and the N propagators in(2.1) are linearly independent. It is often the case that one can profitably work with theFeynman representation for the all-plus metric and ultimately obtain results which differfrom the Minkowski space results in an appropriate Euclidean kinematic region only bytrivial phases. This approach has been used to great effect in recent years by Brown, Panzer, andothers [73–80], culminating recently in an impressive calculation of the six-loop β functionin φ theory [81]. Working through the details of the straightforward derivation (see e.g. [71]), one finds the all-plus Feynman parametrization I E = π Ld Γ (cid:0) ν − Ld (cid:1)Q Ni =1 Γ( ν i ) " N Y j =1 Z ∞ d x j δ (1 − x N ) U ν − ( L +1) d/ F Ld/ − ν E N Y k =1 x ν k − k , (2.2)where U E and F E are respectively the first and second Symanzik polynomials [82] in Eu-clidean space and ν = P Ni =1 ν i . However, for the evaluation of Feynman integrals relevantto the computation of collider observables, it is arguably more natural to work in Minkowskispace from the very beginning, considering Feynman integrals of the form I M = Z d d k · · · Z d d k L N Y ℓ =1 (cid:0) Q ℓ ( k i , p j ) − m ℓ + i (cid:1) − ν ℓ (2.3)with a momentum flow suitable for the description of a scattering experiment.As is well-known, the derivation of Eq. (2.2) goes through with minor modifications ifone works directly in Minkowski space. The mostly-minus Feynman parameter represen-tation has the form Of course, certain assumptions must be satisfied. For a more in-depth discussion, see reference [73]. – 5 – M = i L π Ld e − iπν Γ (cid:0) ν − Ld (cid:1)Q Ni =1 Γ( ν i ) " N Y j =1 Z ∞ d x j δ (1 − x N ) U ν − ( L +1) d/ F Ld/ − ν M N Y k =1 x ν k − k , (2.4)where U M and F M are the first and second Symanzik polynomials in Minkowski space. Forour subsequent analysis of the Baikov-Lee representation, it is important to note that thefunctional dependence of I E and I M on the external kinematics is nearly identical. Givensome spanning set of external kinematic invariants, { ω , . . . , ω n ( n − / } , constructed alongthe lines described in [5], we can straightforwardly obtain one from the other, I M = i L e − iπν I E (cid:12)(cid:12)(cid:12)(cid:12) ω i →− ω i , { p ∗ j }→−{ p ∗ j } , (2.5)provided that we remember the + i { p ∗ j } to be the set of external momenta which happen to be incoming in the physicalkinematics of interest. To understand the above relation, recall that U M = U E and thatone can generate the Minkowski space function F M from F E by flipping the signs of allgeneralized Mandelstam variables and external masses which appear and, subsequently,appropriately adjusting the signs of the external momenta which must now be regarded asincoming. For what concerns the explicit examples discussed in the following sections, we essen-tially adopt the conventions of reference [79]. That is to say, for our actual calculations,we consider Minkowskian purely-virtual Feynman integrals of the form I = Z d d k iπ d/ · · · Z d d k L iπ d/ N Y ℓ =1 (cid:0) Q ℓ ( k i , p j ) − m ℓ + i (cid:1) − ν ℓ (2.6)in physical kinematics. Note that we do not include factors in the measure to preventthe Euler-Mascheroni constant from appearing in ǫ -expanded expressions because, in thispaper, we either study cut Feynman integrals at O ( ǫ ) or to all orders in ǫ . To simplify ourdiscussion later on, it is also convenient to introduce complex-conjugated purely-virtualFeynman integrals, I † , where the + i − i iπ d/ factors in (2.6) above are replaced by factors of − iπ d/ . The maximally-cut examplesof Section 4 are far less sensitive to such details because overall phases make no differenceat all if the only goal is to produce a valid solution to a given homogeneous differentialequation. For the sake of definiteness, we will use the same normalization conventions inboth Sections 3 and 4. At this juncture, it is of critical importance to clarify that, strictly speaking, this statement is not true.Obviously, a vector in Euclidean space which squares to zero is identically zero, whereas this is not the casein Minkowski space. However, one may simply write a formal expression for a Euclidean Feynman integralwith the squares of certain momenta set to zero, remembering that, to be rigorous, one would have to workout the connection between Euclidean and Minkowski space representations with fake external masses andthen set them to zero after the fact. Naturally, we have assumed that Euclidean and Minkowskian generalized Mandelstam invariants aredefined in the usual way. For instance, one would have t = ( p + p ) in Euclidean space but t = ( p − p ) in the usual physical kinematics for 2 → – 6 – .2 The Euclidean Baikov-Lee representation and its analytical continuation To write the Euclidean Baikov-Lee formula succinctly, let us first recall that the Gramdeterminant on the K linearly independent vectors { q i } is given by G ( q , . . . , q K ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q · · · q · q K ... . . . ... q · q K . . . q K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.7)If we let q i be an element of the set { k , . . . , k L , P , . . . , P n − } , the Baikov-Lee represen-tation of the purely-virtual Feynman integral I E defined above in Eq. (2.1) is then I E = π L (3+2 d − n − L ) / Q L − r =0 Γ (cid:0) d − n − r +12 (cid:1) [ G ( P , . . . , P n − )] ( d − n ) / Z · · · Z D L Y i =1 n + L − Y j = i d( q i · q j ) ×× [ G ( q , . . . , q n + L − )] ( d − n − L ) / N Y ℓ =1 (cid:0) Q ℓ ( q i · q j ) + m ℓ (cid:1) − ν ℓ , (2.8)where D is the domain of integration.Even if one works in Euclidean space, finding an explicit description of the integrationdomain is in general a non-trivial task. To understand how this works in practice, let usconsider the evaluation of the one-loop bubble with no internal masses in the Baikov-Leeapproach. This example will both illustrate a general strategy for the determination of theintegration region (briefly discussed in [32]) and give the reader a sense as to why it is moreconvenient to integrate purely-virtual Feynman integrals using Feynman parameters as astarting point. The arguments advanced in this section are generally applicable, but it maybe quite challenging to work out the details in examples with many integration variablesand/or rich analytic structures. For future applications, we expect tools for the explicitsolution of systems of inequalities such as the Reduce routine of
Mathematica to play animportant role.For the one-loop bubble, a possible routing of the propagator momenta is Q = k Q = k − p . In this case, (2.8) becomes p = Z D Z d( q )d( q · q ) π − ǫ (cid:0) p q − ( q · q ) (cid:1) − ǫ Γ (cid:0) − ǫ (cid:1) ( p ) − ǫ q (cid:0) q − q · q + p (cid:1) , (2.9)where we have set d = 4 − ǫ . At this stage, it is important to note that the form ofEq. (2.2) and the definitions of U E and F E (see e.g. [82]) together imply that Euclidean Here, { P i } is nothing but a convenient permutation of the set of independent external momenta, { p i } .This formulation is convenient because it is often desirable to eliminate a momentum other than the n -th.Our treatment of the s -cut of the massless one-loop box integral in Section 3.2 clearly illustrates this point. – 7 –eynman integrals are positive definite if all input kinematic variables are positive definite.This is a powerful analytic constraint from the Baikov-Lee point of view and effectivelydetermines the shape of the integration region. The Baikov polynomial p q − ( q · q ) inside the integrand above depends on the variables of integration and is raised to a non-integer power. This is a generic feature of Baikov-Lee calculations. The point is that, inorder to keep the solution real-valued, one must consider e.g. q · q to lie between thebranching points ± p p q .Apart from the obvious positivity of q , there are no further constraints on the variablesof integration in this case, and we arrive at p = Z ∞ d( q ) Z √ p q − √ p q d( q · q ) π − ǫ (cid:0) p q − ( q · q ) (cid:1) − ǫ Γ (cid:0) − ǫ (cid:1) ( p ) − ǫ q (cid:0) q − q · q + p (cid:1) . (2.10)Although, the Baikov polynomial is no more than quadratic in the scalar products involvingthe loop momenta, the situation may become more complicated once the first scalar productis integrated out. In favorable cases, it is possible to find an analogous integration variableat each step of the calculation. However, it is not guaranteed that all polynomial structuresremaining in the integrand after some number of integration steps have at most quadraticdependence on the remaining variables of integration. Although it is probably clear already,let us emphasize that the two-fold integral above is far, far more complicated than thetrivial one-fold integral which one finds in the Feynman parametric approach to this simpleproblem. The evaluation of (2.10) involves non-trivial hypergeometric function identities,and we refer the interested reader to Appendix C for a detailed discussion.As we shall see, the procedure described above for the limits of integration is concep-tually even simpler for Feynman integrals cut in a single kinematic channel. In such cases,one can also make use of the fact that the region of integration is bounded; cut Feynmanintegrals of this type will be closely related to the real radiative master integrals for somephysical decay or scattering process which has a finite amount of energy and momentumin the initial state [69]. The story is otherwise analogous to what was described above forpurely-virtual Euclidean Feynman integrals because, up to phase, one again has a naturalpositivity condition. That is to say, in favorable cases, one can integrate each variablebetween branching points of the current integrand and then assign a positive orientationto the integration contour for the current variable ( i.e. one must integrate q · q from − p p q to p p q in the above example, not vice versa). We have applied these ideas toexplicitly evaluate a variety of Feynman integrals cut in a single kinematic channel at one,two, and three loops.Of course, before defining the cut Baikov-Lee representation which we will studythroughout the rest of this paper, we first need to analytically continue Eq. (2.8). The keyidea is to recognize that (2.2) and (2.8) are nothing but two different integral representa-tions of the same function, I E . Since our recipe to pass from I E to I M , Eq. (2.5), doesnot depend at all on the details of the Feynman representation, it is natural to apply it to– 8 –q. (2.8) as well, thereby obtaining a putative physical, Minkowski space version of theBaikov-Lee representation, I M = i L π L (3+2 d − n − L ) / e − iπν Q L − r =0 Γ (cid:0) d − n − r +12 (cid:1) [ G ( P , . . . , P n − )] ( d − n ) / Z · · · Z D L Y i =1 n + L − Y j = i d( q i · q j ) ×× [ G ( q , . . . , q n + L − )] ( d − n − L ) / N Y ℓ =1 (cid:0) Q ℓ ( q i · q j ) + m ℓ (cid:1) − ν ℓ (cid:12)(cid:12)(cid:12)(cid:12) ω i →− ω i , { P ∗ j }→−{ P ∗ j } = i L π L (3+2 d − n − L ) / e − iπν Q L − r =0 Γ (cid:0) d − n − r +12 (cid:1) (cid:2) ¯ G ( P , . . . , P n − ) (cid:3) ( d − n ) / Z · · · Z ¯ D L Y i =1 n + L − Y j = i d( q i · q j ) ×× (cid:2) ¯ G ( q , . . . , q n + L − ) (cid:3) ( d − n − L ) / N Y ℓ =1 (cid:0) ¯ Q ℓ ( q i · q j ) + m ℓ (cid:1) − ν ℓ , (2.11)where ¯ G ( P , . . . , P n − ), ¯ D , ¯ G ( q , . . . , q n + L − ), and ¯ Q ℓ ( q i · q j ) denote the various objectswhich appear in Eq. (2.11) after the replacements prescribed by (2.5) have been imple-mented. The key ingredient missing from the discussion so far is the generalized Cutkosky cuttingrule written down by Lee and Smirnov in reference [30]. In a nutshell, they suggest that onecan treat cut Feynman integrals with propagator denominators raised to powers greaterthan one by simply differentiating both sides of Cutkosky’s relation,1 k + i − k − i − πiθ ( k ) δ (cid:0) k (cid:1) , (2.12)an appropriate number of times with respect to k . To avoid digressing into a lengthydiscussion of distributional calculus, it is convenient to actually define our Baikov-Leerepresentation for Feynman integrals cut in a single kinematic channel using the familiarlanguage of residue calculus. The idea is that, up to a possible overall sign, the process ofputting some number of propagators on the mass shell is completely equivalent to perform-ing sequential residue computations which localize a subset of the scalar product integrationvariables. Our logic is similar to that of reference [85], except that, for our purposes, wefind it more natural to repeatedly apply Cauchy’s residue theorem to Eq. (2.11). Due tothe fact that we will use the main result of this section for the explicit examples discussed Due to the fact that the Baikov-Lee representation utilizes scalar product integration variables whichhave non-trivial dependence on the external momenta, it is not obvious that one can work in this way.However, we have found experimentally that this prescription does in fact make sense provided that oneallows { P ∗ j } → −{ P ∗ j } to act on the relevant scalar product integration variables as well. We thank Gil Paz for pointing out that the distributional identity behind the generalized cutting rulewas available in textbooks on the subject ( e.g. [83]) long before the appearance of reference [30]. The direct integration of delta distributions and their derivatives is straightforward [84], but it seemssomewhat less convenient from the perspective of implementation in a computer algebra system. – 9 –n the following sections, we find it natural to work with the absolute normalization of Eq.(2.6) in what follows.For the sake of discussion, suppose that a particular unitarity cut of I in the ω i chan-nel, say the j -th out of M , puts n j propagators on shell. By assumption, these propagatorsare linearly independent and there must therefore exist a subset of the scalar productsdepending on the loop momenta which one can sequentially integrate out using n j applica-tions of the residue theorem. To simplify our notation, let { ¯ s i } be the subset of the scalarproduct integration variables to be localized by the cut propagators, { s k } be the set of L ( L − / nL − n j scalar product integration variables left over, and { ¯ Q ℓ } ( j ) ω i − cut bethe momenta of the cut propagators. Note that, at this stage, the order of both { ¯ s i } and { ¯ Q ℓ } ( j ) ω i − cut should be fixed to match the order in which the localization of the propagatorsand associated scalar products will be implemented by the residue theorem.Finally, we define the Baikov-Lee representation of the j -th Feynman integral cut inthe ω i channel to be I ( j ) ω i − cut = ( − πi ) n j π L (3 − n − L ) / e − iπν Q L − r =0 Γ (cid:0) d − n − r +12 (cid:1) (cid:2) ¯ G ( P , . . . , P n − ) (cid:3) ( d − n ) / Z · · · Z ¯ D ( j ) ωi − cut L ( L − / nL − n j Y k =1 d s k ×× sgn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ { ¯ Q ℓ } ( j ) ω i − cut ∂ { ¯ s i } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! Res { ¯ s i } ((cid:2) ¯ G ( q , . . . , q n + L − ) (cid:3) ( d − n − L ) / N Y ℓ =1 (cid:0) ¯ Q ℓ (¯ s i , s k ) + m ℓ (cid:1) − ν ℓ ) , (2.13)where the sgn function returns the sign of its argument and Res denotes the sequence ofresidue computations which localizes the { ¯ s i } . The presence of the sgn factor is neces-sary because sequential residue computations do differ from sequential localizations imple-mented with delta distributions and their derivatives in one important aspect. The issueis that Res { a } (cid:26) f ( z ) a − z (cid:27) = − f ( a ) , (2.14)but Z ∞−∞ d z δ ( a − z ) f ( z ) = Z ∞−∞ d z δ ( z − a ) f ( z ) = f ( a ) (2.15)for arbitrary test functions f ( z ) regular at z = a . In fact, one must include the sign of theJacobian factor in (2.13) above, or the definition yields nonsensical results which may varydepending upon precisely what momentum routing is chosen for the cut Feynman integralunder consideration.We explicitly evaluate a number of cut Feynman integrals in Section 3 using Eq. (2.13)as a starting point, and we find that our definition is consistent in all cases. In particular,we find the expected unitarity relation between sums of cut Feynman integrals and the In some cases, such as that of the one-loop double-cut bubble integral, the set { s k } is actually empty. – 10 –irect discontinuities of their purely-virtual counterparts [68],Disc ω i ( I ) = − M X j =1 I ( j ) ω i − cut . (2.16)In fact, we have successfully used our framework to study a large number of other examplesof comparable complexity at one, two, and three loops. However, since we have no proofthat our formulation is equivalent to the usual one where one considers all Feynman inte-grals to be embedded in an ambient generalized scalar field theory, it is important to writedown a cut integral definition and associated unitarity relation along the lines of [69, 70].If ˆ I = i L I , thenˆ I ( j ) ω i − cut = i L (2 π ) n j π L (3 − n − L ) / e − iπν Q L − r =0 Γ (cid:0) d − n − r +12 (cid:1) (cid:2) ¯ G ( P , . . . , P n − ) (cid:3) ( d − n ) / Z · · · Z ¯ D ( j ) ωi − cut L ( L − / nL − n j Y k =1 d s k ×× sgn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ { ¯ Q ℓ } ( j ) ω i − cut ∂ { ¯ s i } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! Res { ¯ s i } ((cid:2) ¯ G ( q , . . . , q n + L − ) (cid:3) ( d − n − L ) / N Y ℓ =1 (cid:0) ¯ Q ℓ (¯ s i , s k ) + m ℓ (cid:1) − ν ℓ ) (2.17)and Disc ω i (cid:16) P n ˆ I o ˆ I (cid:17) = M X j =1 P n ˆ I ( j ) ω i − cut o ˆ I ( j ) ω i − cut , (2.18)where P n ˆ I o and P n ˆ I ( j ) ω i − cut o are scalar field theory phase factors defined in e.g. [70].For the purposes of this paper, we employ Eq. (2.13) with the attitude that it stream-lines the exposition in Section 3 and makes it easier for the reader to check our analysis usingthe results from the literature collected in Appendix B ( i.e. it allows us to forget about theannoying Feynman graph-dependent phase factors on both sides of (2.18)). Before leavingthis section, let us emphasize that one can also employ the formalism discussed above totreat maximally-cut Feynman integrals in an analogous fashion. Interesting examples ofmaximally-cut Feynman integrals will be discussed in Section 4. To get a feeling for the ideas put forward in Section 2, we now consider a number of illustra-tive one- and two-loop examples. In the spirit of reference [70], we compute the s -channelcuts of selected Feynman integrals using our cut Baikov-Lee representation, Eq. (2.13),and demonstrate that, in all cases, our results match the predictions of the optical theorem( i.e. the predictions obtained by using Appendix B to compute the direct discontinuitieson the left-hand side of Eq. (2.16)). We define the direct discontinuity of a Feynman integral in Appendix B. – 11 – .1 The one-external-mass one-loop triangle
In this section, we consider the s -channel cut of the one-external-mass one-loop trianglewith positive integer propagator exponents, ν ν ν p p = Z d d k iπ d/ p − k ) ] ν [( p + k ) ] ν [ k ] ν . (3.1)This example clearly demonstrates the applicability of our formalism to propagators ofhigher multiplicity, and its elementary nature should give the reader ample opportunity toadjust to our notation. Clearly, there is only one cut Feynman integral which needs to beevaluated. We have ν ν ν p p = ( − πi ) π − ( − ν Γ(1 − ǫ ) ( − s / / − ǫ sgn (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) ∂ { s − s , s + 2¯ s } ∂ { ¯ s , ¯ s } (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) ×× Z ¯ D s − cut d s Res { ¯ s , ¯ s } ( (cid:0) − ¯ s ¯ s s − s s / (cid:1) − ǫ ( s − s ) ν ( s + 2¯ s ) ν s ν ) (3.2)in the cut Baikov-Lee representation of Eq. (2.13), where s = k , ¯ s = k · p , ¯ s = k · p ,and s = ( p + p ) .Eq. (3.2) can be conveniently rewritten as ν ν ν p p = 2 − ν − ν π ( − ν + ν is − ǫ Γ(1 − ǫ ) ×× Z ¯ D s − cut d s s ν Res { ¯ s , ¯ s } (cid:26) (4¯ s ¯ s + s s ) − ǫ (¯ s − s / ν (¯ s + s / ν (cid:27) (3.3)to make manifest the fact that its right-hand side is purely imaginary for s > ν ν ν p p = − ν − ν π Γ( ν − ǫ )( − ν is − ǫ Γ( ǫ )Γ(1 − ǫ )Γ( ν ) ×× Z ¯ D s − cut d s s ν + ν − ǫ Res { ¯ s } ( ¯ s ν − ( s + 2¯ s ) − ν − ǫ (¯ s + s / ν ) = − π Γ( ν − ǫ )( − ν is − ν − ǫ Γ( ǫ )Γ(1 − ǫ )Γ( ν )Γ(1 + ν − ν ) Z ¯ D s − cut d s s ν + ν − ǫ ×× ( s − s ) − ν − ν − ǫ F (cid:16) − ν , − ν − ǫ ; 1 + ν − ν ; s s (cid:17) . (3.4)– 12 –t this stage of the calculation, we have to analyze the integrand to determine the s integration domain. By studying the form of Eq. (3.4), we see immediately that theintegrand has precisely two branching points, at s = 0 and at s = s . From the generaldiscussion in Section 2.2, it therefore follows that ν ν ν p p = − π Γ( ν − ǫ )( − ν is − ν − ǫ Γ( ǫ )Γ(1 − ǫ )Γ( ν )Γ(1 + ν − ν ) Z s d s s ν + ν − ǫ ×× ( s − s ) − ν − ν − ǫ F (cid:16) − ν , − ν − ǫ ; 1 + ν − ν ; s s (cid:17) . (3.5)For ǫ such that Re ( ν + ν + ǫ ) < Re ( ν + ν + ǫ ) <
2, the above integral convergesand we find that ν ν ν p p = − πs − ν − ǫ Γ( ν − ǫ )Γ(3 − ν − ν − ǫ )Γ(2 − ν − ν − ǫ )( − ν i Γ( ǫ )Γ(1 − ǫ )Γ( ν )Γ(1 + ν − ν )Γ(5 − ν − ν − ν − ǫ ) ×× F (1 − ν , − ν − ǫ, − ν − ν − ǫ ; 1 + ν − ν , − ν − ν − ν − ǫ ; 1) . (3.6)Under an additional assumption, Eq. (3.6) can be simplified using the Saalsch¨utzsummation formula. For ν >
1, Eq. (A.7) implies that ν ν ν p p = − i sin( πǫ ) s − ν − ǫ Γ(2 − ν − ν − ǫ )Γ(2 − ν − ν − ǫ )Γ( ν − ǫ )Γ( ν )Γ( ν )Γ(4 − ν − ǫ ) . (3.7)Actually, the principle of analytical continuation allows us to conclude that Eq. (3.7) isnot only valid for arbitrary positive integer propagator exponents as desired, but that iteven furnishes a definition of the cut Feynman integral for generic complex values of thepropagator exponents. In fact, this analytical continuation of the above result will proveuseful later on in Section 3.3. Finally, one can readily check using (3.7) and the s > s ν ν ν p p = − ν ν ν p p . (3.8) For our next example, we consider the s -channel cut of the massless one-loop box integral, p p p p = Z d d k iπ d/ k ( k + p ) ( k + p + p ) ( k + p ) . (3.9)Although it is again the case that just one cut Feynman integral needs to be evaluated,it is useful to study this example because it illustrates the applicability of our frameworkto multi-scale problems. In fact, it is not obvious to us that this cut can be computedto all orders in ǫ using traditional cut parameterizations. We shall see that it is also notentirely straightforward in the Baikov-Lee approach; the cut Feynman integral of interest– 13 –ere contains structures which bear a remarkable resemblance to those which appearedduring the evaluation of certain three-loop, single-scale cut Feynman integrals [86].From Eq. (2.13), we obtain p p p p = − π / ( − su ( s + u )) ǫ i Γ(1 / − ǫ ) Z Z ¯ D s − cut d s d s × (3.10) × Res { ¯ s , ¯ s } (cid:16) − ¯ s su ( s + u ) − s ( s + u ) − (¯ s u + s s ) − s ( s + u )(¯ s u − s s ) (cid:17) − / − ǫ ¯ s (¯ s + 2 s ) (¯ s + s − s/ s /
2) (¯ s + 2 s ) for our cut Baikov-Lee representation after performing some trivial manipulations to mas-sage the expressions into a more convenient form. In Eq. (3.10), we have made thedefinitions s = k · p , s = k · p , ¯ s = k , ¯ s = k · p , s = ( p + p ) , and u = ( p − p ) .Carrying out the residue computations, we find p p p p = − π / ( − su ( s + u )) ǫ i Γ(1 / − ǫ ) Z Z ¯ D s − cut d s s d s s ×× Res { ¯ s } (cid:16) − s ( s + u ) − (¯ s u + s s ) − s ( s + u )(¯ s u − s s ) (cid:17) − / − ǫ ¯ s + s − s/ = − ǫ π / ( − u ( s + u )) ǫ is / Γ(1 / − ǫ ) Z Z ¯ D s − cut d s s d s s ×× (cid:16) s s u − s (cid:0) u + 4( s + s ) u + 4( s − s ) (cid:1) (cid:17) − / − ǫ . (3.11)Next, we integrate out the variable s . As explained in Section 2.2, the integrationruns between the real zeros of the polynomial from the last line of Eq. (3.11), s ± = s ( s + 2 u ) − su/ ± p − u ( s + u ) p s ( s − s ) s . (3.12)In fact, by observing that the s integration is completely analogous to the integration withrespect to q · q carried out in Appendix C, we can already anticipate from the branch cutstructure of the s ± (with respect to s ) that the s integration will run between 0 and s/ p p p p = − ǫ π / ( − u ( s + u )) ǫ is / Γ(1 / − ǫ ) Z s/ d s s Z s +2 s − d s s ×× (cid:16) s s u − s (cid:0) u + 4( s + s ) u + 4( s − s ) (cid:1) (cid:17) − / − ǫ = 4 πs − − ǫ iu Γ(1 − ǫ ) Z s/ d s (cid:0) s s (cid:1) − − ǫ (cid:0) − s s (cid:1) − ǫ (cid:18)q s + u − u s s + q − s s (cid:19) × – 14 – F , − ǫ ; 1 − ǫ ; 4 q s + u − u s s (cid:0) − s s (cid:1)(cid:18)q s + u − u s s + q − s s (cid:19) = 2 πs − − ǫ iu Γ(1 − ǫ ) Z d x x − − ǫ (1 − x ) − ǫ (cid:16)q s + u − u x + √ − x (cid:17) ×× F , − ǫ ; 1 − ǫ ; 4 q s + u − u x (1 − x ) (cid:16)q s + u − u x + √ − x (cid:17) , (3.13)where we have made the change of variables s = s/ x in the last line.At this point, the above result may be rewritten to exhibit a F of argument z (1+ z ) ,where z can be chosen to be either q − x s + u − u x or its reciprocal. From this observation, wesee that a simple strategy to eliminate the square roots appearing in Eq. (3.13) is to splitthe integral at the point where z = 1 and then apply a quadratic hypergeometric functiontransformation which is valid for | z | < i.e. Eq. (A.5)) to both terms. Carrying outthese steps, we find p p p p = 2 πs − − ǫ iu Γ(1 − ǫ ) Z − u/s d x x − − ǫ (1 − x ) − − ǫ r s + u − u x − x ! ×× F , − ǫ ; 1 − ǫ ; 4 r s + u − u x − x r s + u − u x − x ! − πs − − ǫ i ( s + u )Γ(1 − ǫ ) Z − u/s d x x − − ǫ (1 − x ) − ǫ (cid:18) q − x s + u − u x (cid:19) ×× F , − ǫ ; 1 − ǫ ; 4 q − x s + u − u x (cid:18) q − x s + u − u x (cid:19) (3.14)= 2 πs − − ǫ iu Γ(1 − ǫ ) Z − u/s d x x − − ǫ (1 − x ) − − ǫ F , ǫ ; 1 − ǫ ; s + u − u x − x ! − πs − − ǫ i ( s + u )Γ(1 − ǫ ) Z − u/s d x x − − ǫ (1 − x ) − ǫ F , ǫ ; 1 − ǫ ; 1 − x s + u − u x ! . Now that the square root structures have been eliminated, we can deal with the twoterms in (3.14) above by mapping them to linear combinations of known generalized Euler– 15 –ntegrals. The first step is to make the change of variables x = y s + u − u + y in the first integraland the change of variables x = s + u − u y in the second integral to bring them into thegeneralized Euler form. In fact, after making these transformations, one can immediatelyevaluate the second integral by applying integration formula (A.10): p p p p = − πs − − ǫ ( − u ) − − ǫ i ( s + u ) − ǫ Γ(1 − ǫ ) Z d y y − − ǫ (cid:18) − us + u y (cid:19) ǫ F (1 , ǫ ; 1 − ǫ ; y ) − πs − − ǫ ( − u ) − ǫ i ( s + u ) ǫ Γ(1 − ǫ ) Z d y y − ǫ (cid:18) − s + uu y (cid:19) ǫ F (1 , ǫ ; 1 − ǫ ; y )= − πs − − ǫ ( − u ) − − ǫ i ( s + u ) − ǫ Γ(1 − ǫ ) Z d y y − − ǫ (cid:18) − us + u y (cid:19) ǫ F (1 , ǫ ; 1 − ǫ ; y ) − π Γ( − ǫ ) s − ǫ ( − u ) − − ǫ i ( s + u ) ǫ Γ(1 − ǫ )Γ(1 − ǫ ) F (cid:16) , − ǫ, − ǫ ; 1 − ǫ, − ǫ ; 1 + us (cid:17) . (3.15)In our opinion, it is most convenient to deal with the remaining integral by replacing F (1 , ǫ ; 1 − ǫ ; y ) with its integral representation, Eq. (A.1), and then applying Eq.(A.2) to integrate out the variable y with an Appell series. This yields p p p p = − πs − − ǫ ( − u ) − − ǫ i ( s + u ) − ǫ Γ(1 − ǫ ) Z d t (1 − t ) − − ǫ F (cid:18) − ǫ ; − ǫ, ǫ ; 1 − ǫ ; us + u , t (cid:19) − π Γ( − ǫ ) s − ǫ ( − u ) − − ǫ i ( s + u ) ǫ Γ(1 − ǫ )Γ(1 − ǫ ) F (cid:16) , − ǫ, − ǫ ; 1 − ǫ, − ǫ ; 1 + us (cid:17) . (3.16)In this case, the sum of the second and third parameters of the F is equal to the fourthparameter and reduction formula (A.9) therefore immediately leads to p p p p = − πs − − ǫ ( − u ) − − ǫ i ( s + u ) − ǫ Γ(1 − ǫ ) Z d t − t F (cid:18) − ǫ, − ǫ ; 1 − ǫ ; us + u − t − t (cid:19) − π Γ( − ǫ ) s − ǫ ( − u ) − − ǫ i ( s + u ) ǫ Γ(1 − ǫ )Γ(1 − ǫ ) F (cid:16) , − ǫ, − ǫ ; 1 − ǫ, − ǫ ; 1 + us (cid:17) . (3.17)Finally, we can map the remaining integral onto a linear combination of standard Eulerintegrals via connection formula (A.4), p p p p = 2 π Γ( − ǫ ) s − ǫ isu Γ( − ǫ ) Z d t (1 − t ) − − ǫ (cid:16) − (cid:16) su (cid:17) t (cid:17) ǫ + 2 πs − ǫ ( − u ) − − ǫ i ( s + u ) ǫ Γ(1 − ǫ ) Z d t (1 − t ) − − ǫ F (cid:16) , − ǫ ; 1 − ǫ ; (1 − t ) (cid:16) us (cid:17)(cid:17) − π Γ( − ǫ ) s − ǫ ( − u ) − − ǫ i ( s + u ) ǫ Γ(1 − ǫ )Γ(1 − ǫ ) F (cid:16) , − ǫ, − ǫ ; 1 − ǫ, − ǫ ; 1 + us (cid:17) – 16 – 2 i sin( πǫ ) s − ǫ Γ ( − ǫ )Γ( ǫ ) su Γ( − ǫ ) F (cid:16) , − ǫ ; 1 − ǫ ; 1 + su (cid:17) + 2 πs − ǫ ( − u ) − − ǫ i ( s + u ) ǫ Γ(1 − ǫ ) Z d r r − − ǫ F (cid:16) , − ǫ ; 1 − ǫ ; r (cid:16) us (cid:17)(cid:17) − π Γ( − ǫ ) s − ǫ ( − u ) − − ǫ i ( s + u ) ǫ Γ(1 − ǫ )Γ(1 − ǫ ) F (cid:16) , − ǫ, − ǫ ; 1 − ǫ, − ǫ ; 1 + us (cid:17) , (3.18)where we have obtained the second equality by evaluating the integral on the first line ofEq. (3.18) with the help of (A.1). Applying Eq. (A.3) to the final integral remaining onthe right-hand side of (3.18), we see that the cut Feynman integral evaluates to p p p p = 2 i sin( πǫ ) s − ǫ Γ ( − ǫ )Γ( ǫ ) su Γ( − ǫ ) F (cid:16) , − ǫ ; 1 − ǫ ; 1 + su (cid:17) . (3.19)Note that the final integral on the right-hand side of (3.18) exactly cancels the F term,thereby removing all dependence on the generalized hypergeometric series from the result.One can check using (3.19) and the physical region ( s > − s < u <
0) evaluation of(3.9) given in Eq. (B.3) thatDisc s p p p p = − p p p p . (3.20) Our final example will be the s -channel cut of the one-external-mass six-line two-loopdouble triangle, p p = Z d d k iπ d/ Z d d k iπ d/ p + k ) ( p + k ) ( k − k ) k k ( p − k ) . (3.21)The s -channel cut of this integral was chosen because three cut Feynman integrals con-tribute to it and one of these integrals has an integration domain which is non-trivial todetermine. This example will show the reader what Baikov-Lee computations look like be-yond one loop, where irreducible scalar products and real-virtual contributions come intoplay for the first time. As before, we begin with Eq. (2.13). This time, however, we mustenumerate the distinguishable cut Feynman integrals which contribute. Let us consider thetriple cut first and the double cut second. The conjugate of the double-cut contributioncan be obtained from the double cut without any additional calculation. After performing some trivial manipulations, we obtain a cut Baikov-Lee representationof the form p p = − − ǫ s − ǫ i Γ(1 − ǫ ) Z Z Z Z ¯ D (1) s − cut d s d s d s d s Res { ¯ s , ¯ s , ¯ s } (3.22) By virtue of the + i – 17 – − s (¯ s − ¯ s s ) / s (¯ s ¯ s s − ¯ s (¯ s s + s s ) + s s s ) − (¯ s s − s s ) (cid:17) − / − ǫ ( s + 2 s ) (¯ s + 2 s ) (¯ s − s / − ¯ s /
2) ¯ s s (¯ s − s / for the triple-cut contribution. In Eq. (3.22), we have made the definitions s = k · p , s = k , s = k · p , s = k · p , ¯ s = k , ¯ s = k · k , ¯ s = k · p , and s = ( p + p ) .Carrying out the residue computations, we find p p = 2 − ǫ s − ǫ i Γ(1 − ǫ ) Z Z Z Z ¯ D (1) s − cut d s d s d s d s s s ( s + 2 s ) Res { ¯ s , ¯ s } (3.23) (cid:16) − s (¯ s + 2 s s ) / s ( s s s − ¯ s (¯ s s + s s ) − s s s ) − (¯ s s − s s ) (cid:17) − / − ǫ (¯ s − s / s ) (¯ s − s / = 2 − ǫ s − ǫ i Γ(1 − ǫ ) Z Z Z Z ¯ D (1) s − cut d s d s d s d s s s ( s + 2 s ) Res { ¯ s } (cid:26) s − s / (cid:16) − (¯ s s − s s ) + s ( s s s − ( s − s )(¯ s s + s s ) / − s s s ) − s ( s + 2 s ) / (cid:17) − / − ǫ (cid:27) = 2 − ǫ s − ǫ i Γ(1 − ǫ ) Z Z Z Z ¯ D (1) s − cut d s d s d s d s s s ( s + 2 s ) (cid:16) − ( s s − s s ) / s ( s s s − ( s − s )( s s + 2 s s ) / − s s s ) − s ( s + 2 s ) / (cid:17) − / − ǫ . We begin our treatment of the unbarred integration variables with s . Following thediscussion of Section 2.2, we obtain our integration domain by studying the zeros of poly-nomial structures inside the integrand. Due to the fact that the polynomial − ( s s − s s ) / s ( s s s − ( s − s )( s s + 2 s s ) / − s s s ) − s ( s + 2 s ) / − / − ǫ , the integrand obtained above in (3.23) has the branchpoints s ± = ( s s + s ( s + s )) s − s s s/ ± √ s p s (2 s + s ) s ( s − s )(2 s + s )2 s (3.24)in s . As in the previous example, we can actually deduce the limits of integration onthe remaining variables by simply looking at the branch cut structure of s ± . First, thepolynomial structures s (2 s + s ) and s ( s − s ) under the radical tell us that the limitsof integration for s are either [0 , − s ] or [ − s ,
0] and that the limits of integration for s are either [0 , s ] or [ s , s , what we shall It is worth pointing out that our logic here is similar to that used to formulate the compatibility graphsmethod for purely-virtual Feynman integrals [75]. Namely, we do not expect new singularity structures toarise beyond those which are already encoded in the analytical structure of the initial integrand. – 18 –onsider to be the final variable of integration, is positive or negative. In the physicalregion, the polynomial 2 s + s under the radical implies that the variable s is a negativenumber which runs between − s/ s integration runsbetween 0 and − s and that the s integration runs between s and 0. The upshot is that p p = 2 − ǫ s − ǫ i Γ(1 − ǫ ) Z − s/ d s Z − s d s Z s d s Z s +4 s − d s ×× s s ( s + 2 s ) (cid:16) − ( s s − s s ) / − s ( s + 2 s ) / s ( s s s − ( s − s )( s s + 2 s s ) / − s s s ) (cid:17) − / − ǫ . (3.25)In this case, all of the remaining integrations are elementary. The s integration isclosely analogous to the integration with respect to q · q carried out in Appendix C andthe other integrations may be carried out with the help of a computer algebra system suchas Mathematica . We find p p = 2 πs − ǫ i Γ (1 − ǫ ) Z − s/ d s ( − s ) − ǫ (2 s + s ) − ǫ Z − s d s s − − ǫ ×× Z s d s (cid:16) s − s (cid:17) − ǫ (cid:16) s (2 s + s ) (cid:17) − − ǫ = 2 πs − ǫ iǫ Γ(1 − ǫ ) Z − s/ d s s (2 s + s ) − ǫ Z − s d s s − − ǫ ( − s − s ) − − ǫ = 2 − − ǫ π Γ ( − ǫ ) s − ǫ iǫ Γ ( − ǫ ) Z − s/ d s ( − s ) − − ǫ (2 s + s ) − ǫ = − i sin(2 πǫ ) s − − ǫ Γ( − − ǫ )Γ(1 + 2 ǫ )Γ ( − ǫ )Γ(1 − ǫ )Γ( − ǫ ) (3.26)for the triple-cut contribution.Our next task is to calculate the double-cut contribution. In fact, we can write downthe answer immediately by recycling calculations that we have already carried out. Werequire only the result obtained in Section 3.1 for the s -channel cut of the one-external-mass one-loop triangle integral together with the well-known result for its purely-virtualcounterpart. The key observation is that, on the support of the double cut, the virtual partof the cut one-external-mass six-line two-loop double triangle is precisely a purely-virtualone-loop triangle with external mass k . From Eq. (B.2), we see that k = e iπǫ Γ(1 + ǫ )Γ ( − ǫ )Γ(1 − ǫ ) (cid:0) k (cid:1) ǫ . (3.27) All possible limits of integration for s and s force one of the s integration limits to be 0. Here, the + i – 19 –t therefore follows that we can treat the integration over k using Eq. (3.7) with propagatorexponents ν = 1, ν = 1, and ν = 2 + ǫ . The desired result is p p = e iπǫ Γ(1 + ǫ )Γ ( − ǫ )Γ(1 − ǫ ) ǫp p = − ie iπǫ sin( πǫ ) s − − ǫ Γ ( − − ǫ )Γ(1 + ǫ )Γ(2 + 2 ǫ )Γ ( − ǫ )Γ( − ǫ )Γ(1 − ǫ ) . (3.28)The final cut Feynman integral of interest is the conjugate double-cut contribution.The result may be simply obtained by taking the complex conjugate of the virtual part ofthe double-cut contribution. We have p p = e − iπǫ Γ(1 + ǫ )Γ ( − ǫ )Γ(1 − ǫ ) ǫp p = − ie − iπǫ sin( πǫ ) s − − ǫ Γ ( − − ǫ )Γ(1 + ǫ )Γ(2 + 2 ǫ )Γ ( − ǫ )Γ( − ǫ )Γ(1 − ǫ ) . (3.29)Finally, the sum of the three contributions to the s -channel cut, Eqs. (3.26), (3.28), and(3.29), may be rewritten as p p + p p + p p = (3.30) − i sin(2 πǫ ) s − − ǫ Γ( − − ǫ )Γ(1 + 2 ǫ )Γ ( − ǫ )Γ(1 − ǫ )Γ( − ǫ ) (cid:16) Γ( − ǫ ) − Γ(1 + ǫ )Γ( − ǫ ) (cid:17) . Using (3.30) and the s > s p p = − p p + p p + p p . (3.31) As mentioned in the introduction, it was observed in [31] that maximally-cut Feynman in-tegrals [87, 88] in the ǫ → ǫ → y ( x, t ) = r(cid:16) x − a ( t ) (cid:17)(cid:16) x − a ( t ) (cid:17)(cid:16) x − a ( t ) (cid:17)(cid:16) x − a ( t ) (cid:17) , (4.1)where t is a parameter and { a i ( t ) } is the set of branch points of y ( x, t ), with a i ( t ) = a j ( t )for i = j . Now, for some Feynman integral, I , let us suppose that we are able to obtain aone-fold integral representation for its maximal cut, ¯ I , of the form¯ I = Z ¯ D { ¯ si } d x R ( x, y ( x, t ) , t ) , (4.2)where R ( x, y ( x, t ) , t ) is a rational function of its arguments. If R ( x, y ( x, t ) , t ) has no polesin x , it is then the case that a complete set of solutions to the homogeneous part of thedifferential equations satisfied by I may be obtained by considering¯ I ( a i ( t ) , a j ( t )) = Z a j ( t ) a i ( t ) d x R ( x, y ( x, t ) , t ) (4.3)for i = j . If poles are present, one must also consider closed contour integrals around eachpole, ¯ I ( γ k ) = I γ k d x R ( x, y ( x, t ) , t ) , (4.4)to find all possible solutions. In (4.4), γ k denotes a closed contour which encircles the k -thpole of R ( x, y ( x, t ) , t ), but no other pole or branch point of the integrand. Note that, quitegenerically, the set of solutions obtained in this manner will actually be overcomplete.By using properties of elliptic curves we can argue that (4.3) and (4.4) representa complete set of solutions. Integrals ¯ I ( a i ( t ) , a j ( t )) for i = j and ¯ I ( γ k ) are periods ofthe elliptic curve y ( x, t ) [90] and the given homogeneous differential equation for I withrespect to t is nothing but the associated Picard-Fuchs equation. By construction, thesedifferential equations are the same for every period, and a complete set of periods provides It has been clear for a long time that Picard-Fuchs equations play a very important role in the theoryof Feynman integrals (see e.g. [91, 92]). – 21 – complete set of solutions to the Picard-Fuchs equation (for a comprehensive review ofthe subject of periods see [90] and the references therein).The prescription described above represents a substantial generalization of that de-scribed in Section 2.2. When considering a cut Feynman integral associated with theunitarity cut in some physical kinematic channel, one naturally expects to obtain a resultwhich is real-valued up to an overall phase; in this context, however, no such constraintapplies and the integration domain is no longer uniquely determined. Indeed, Eq. (4.3)implies that, in the elliptic case, we can integrate the maximal cut of I over six distinctdomains and this obviously leads to some solutions which possess both real and imaginaryparts. If the integrand has poles one must also include solutions of the form (4.4). Thiswould be relevant, for example, when considering a complete elliptic integral of the thirdkind.In the absence of poles, complete elliptic integrals admit a simple description as closedcontour integrals which wrap the torus [89]. As claimed above, it is clear from this pointof view that our solutions cannot form a linearly independent set. To see this, recall thatthe fundamental group of the torus, Z × Z , is isomorphic to the first homology group(Hurewicz’s theorem [93]). This means that, in the absence of poles, the torus admits justtwo independent cycles for us to integrate along. We can therefore conclude that four ofthe six functions generated by applying the prescription given in (4.3) above are actuallyspurious and may be disposed of. In general, one must also check whether contour integralsof the form (4.4) around different poles of the integrand yield linearly dependent results.The above discussion generalizes and systematizes the analysis of, e.g. , [94] to genericcomplete elliptic integrals of the form (4.2). Moreover, it is possible to generalize it tocurves of higher genus, i.e. when the square of (4.1) is a polynomial of degree greater thanfour, by considering the set of periods over the relevant higher-genus Riemann surface. Theapplication of these techniques to curves of genus greater than one goes beyond the scopeof the present paper but will likely play a role in future calculations. In the following, weconsider illustrative examples taken from the virtual corrections to Higgs + jet with exacttop mass dependence. As a first example, we consider the maximal cut of the two-loop crossed form factor, p p = Z d d k iπ d/ d d k iπ d/ Y i =1 D − i , (4.5)where we have made the definitions, D = ( k + p ) − m D = ( k − k − p − p ) D = k − m D = ( k − p ) − m D = ( k − k ) D = k − m . (4.6)The evaluation of this Feynman integral is relevant to the calculation of the non-planarpart of the two-loop virtual corrections to Higgs + jet with exact top mass dependence.– 22 –ts maximal cut was considered in [31], where it was evaluated with a traditional cutparametrization in an effort to obtain a solution to the homogeneous part of the associatedsystem of differential equations. Using Eq. (2.13), we arrive at the following one-foldintegral representation of the maximally-cut Feynman integral, p p = 2 − ǫ π ( s − p ) − ǫ Γ (1 − ǫ ) ×× Z ¯ D { ¯ si } d s (cid:0) s (cid:0) s − p + 2 s (cid:1) (cid:0) m s − s (cid:0) s − p + 2 s (cid:1)(cid:1)(cid:1) − / − ǫ , (4.7)where we have made the definitions s = k · p and s = ( p + p ) , and we have chosen towork in the physical region above threshold where s > p > m > d = 4.We have, lim ǫ → p p = 16 π s − p Z ¯ D { ¯ si } d s y (cid:0) s , m , p , s (cid:1) , (4.8)where y (cid:0) s , m , p , s (cid:1) = q s (cid:0) s − p + 2 s (cid:1) (cid:0) m s − s (cid:0) s − p + 2 s (cid:1)(cid:1) . (4.9)Following our general prescription, concrete results are obtained by integrating betweenbranch points of the integrand. In other words, we obtain six possible solutions by pickingdistinct pairs of elements from the set of branch points of the integrand, (cid:26) p − s − ρ , p − s , , p − s + ρ (cid:27) , (4.10)where we have introduced the convenient shorthand ρ = q m s + (cid:0) s − p (cid:1) (4.11)in (4.10). Note that, in the kinematic region that we are working in, the elements of (4.10)are real-valued and ordered from smallest to largest.As discussed above, the solutions to our homogeneous differential equations will becomplete elliptic integrals and these may be thought of as periods of the torus. The torusadmits two linearly independent cycles and we therefore expect to find just two linearlyindependent periods. As we shall see, it is convenient to take f ≡ π s − p Z p − s ) / d s y (cid:0) s , m , p , s (cid:1) (4.12) f ≡ π s − p Z ( p − s + ρ ) / d s y (cid:0) s , m , p , s (cid:1) (4.13)– 23 –o be our independent basis elements.That f and f are actually independent periods may be seen by writing them ina standard form. For f , this is easily achieved by making the appropriate analyticalcontinuation of Eq. (4.9) and then changing variables according to [31] s = p − s t − ( s − p ) s − p − ρ (1 − t ) . The result is f = − π i (cid:0) s − p (cid:1) (cid:0) s − p + ρ (cid:1) K ρ (cid:0) s − p (cid:1)(cid:0) s − p + ρ (cid:1) ! (4.14)in the physical kinematic region of interest. Similar considerations lead to f = 32 π (cid:0) s − p (cid:1) q ρ (cid:0) s − p (cid:1) K − (cid:0) s − p − ρ (cid:1) ρ (cid:0) s − p (cid:1) ! , (4.15)again in the kinematic region of interest. The other possible solutions may be written aslinear combinations of f and f and it is clear that some of them will have both real andimaginary parts. For example, we have from (4.10) and the explicit expressions for f and f that 16 π s − p Z ( p − s + ρ ) / p − s ) / d s y (cid:0) s , m , p , s (cid:1) = f + f , (4.16)where f is purely imaginary and f is purely real in the region where s > p > m > We consider the following two-loop Higgs + jet integral, denoted in [64] as f A , p p p p = Z d d k iπ d/ d d k iπ d/ Y i =1 D − i , (4.17) In Eqs. (4.14) and (4.15), K ( z ) is the complete elliptic integral of the first kind, K ( z ) = Z d t p (1 − t )(1 − z t ) . – 24 –ith propagators, D = ( k + p + p ) − m D = ( k + p ) − m D = k − m D = ( k + k + p + p ) − m D = ( k + k + p ) − m D = k . (4.18)As usual, the kinematics is s = ( p + p ) t = ( p − p ) u = ( p − p ) p = s + t + u. (4.19)The maximal cut of this Feynman integral was also considered more recently in bothreferences [31] and [32]. For the purposes of our analysis in this section, it is convenient towork in the kinematic region where s > p > s > m >
0, and p − s > t .As we are considering a four-point function at two loops, there are nine scalar prod-uct integration variables, and it would therefore seem that we must consider a three-foldintegral representation of the maximal cut. Fortunately, one can obtain a one-fold inte-gral representation of the maximal cut by proceeding recursively loop-by-loop. First, weintegrate out the one-loop triangle subintegral defined by the propagators D , D , and D , p k p = Z d d k iπ d/ k (( k + k + p ) − m ) (( k + k + p + p ) − m ) , (4.20)by localizing k , k · p , and k · p . To do so, we evaluate the maximal cut of (4.20) usingEq. (2.13), our cut Baikov-Lee representation. Of course, the maximal cut of a one-looptriangle involves no non-trivial integrations and one immediately finds p k p = 4 π Γ(1 − ǫ ) (cid:16)(cid:0) s − p − k · p (cid:1) + 4 (cid:0) k + 2 k · p (cid:1) p (cid:17) − / ǫ × (4.21) × (cid:16) − m ( s − k · p ) − p (cid:0) k + 2 k · p + m (cid:1) (cid:0) k + 2 k · p + 2 k · p − s + m (cid:1)(cid:17) − ǫ after carrying out the residue computations.We now integrate out the remaining loop by localizing k , k · p , and k · p withthe remaining cut conditions. A moment’s thought reveals that Eq. (2.13) may still bestraightforwardly applied to loop-by-loop Baikov-Lee calculations; the only difference isthat the results of the previous loop integrations appear in the current integrand. In otherwords, if we make the definitions s = k · p , ¯ s = k , ¯ s = k · p , and ¯ s = k · p , wehave p p p p = − π / i Γ (1 / − ǫ ) (cid:2) ¯ G ( p , p , p ) (cid:3) − ǫ Z ¯ D { ¯ si } d s × (4.22) × Res { ¯ s , ¯ s , ¯ s } ( (cid:2) ¯ G ( k , p , p , p ) (cid:3) − / − ǫ (¯ s + m ) (¯ s + 2¯ s + m ) (¯ s + 2¯ s + 2 s − s + m ) p k p !) – 25 –nd can immediately writelim ǫ → p p p p = 32 π Z ¯ D { ¯ si } d s y (cid:0) s , m , p , s, t (cid:1) , (4.23)where y (cid:0) s , m , p , s, t (cid:1) = q(cid:0) p + 2 s (cid:1) − m p q s (cid:0) m t (cid:0) s + t − p (cid:1) − s ( t + 2 s ) (cid:1) (4.24)in the region of interest.We now turn to the problem of finding a complete set of homogeneous solutions to thedifferential equations for the uncut integral, proceeding as described at the beginning ofSection 4. Possible solutions are obtained by integrating between the branch points of theintegrand, (cid:26) − p − σ, − p σ, − t − τs , − t τs (cid:27) , (4.25)where we have set σ = q m p and τ = q m st (cid:0) s + t − p (cid:1) . (4.26)In the kinematic region that we have chosen to work in, the elements of (4.25) are real-valued and ordered from smallest to largest.As guaranteed by the form of Eqs. (4.23) and (4.24), we again find just two linearlyindependent solutions, g ≡ π Z − t − τs − p + σ d s y (cid:0) s , m , p , s, t (cid:1) (4.27)and g ≡ π Z − t + τs − t − τs d s y (cid:0) s , m , p , s, t (cid:1) . (4.28)That these two solutions are actually linearly independent follows from the explicit formulaswritten in terms of complete elliptic integrals of the first kind, g = − π iK (cid:18) − στs ( p − t ) +8 στ − m ( s p + t ( s + t − p )) (cid:19)r s (cid:16) s (cid:0) p − t (cid:1) + 8 στ − m (cid:0) s p + t (cid:0) s + t − p (cid:1)(cid:1)(cid:17) (4.29)and g = 32 π K (cid:18) στs ( p − t ) +8 στ − m ( s p + t ( s + t − p )) (cid:19)r s (cid:16) s (cid:0) p − t (cid:1) + 8 στ − m (cid:0) s p + t (cid:0) s + t − p (cid:1)(cid:1)(cid:17) , (4.30)which may be derived by making appropriate changes of variables in Eqs. (4.27) and(4.28). We have explicitly checked that g and g satisfy the appropriate homogeneoussecond-order differential equations. To derive (4.29), one must analytically continue the second square root structure in Eq. (4.24) above. – 26 –
Conclusions
In this paper, we formulated and studied cut Baikov-Lee representations, both for Feynmanintegrals cut in a single kinematic channel and for maximally-cut Feynman integrals. Fora wide class of interesting problems, our framework provides a convenient setup for theexplicit computation of cut Feynman integrals. It makes the dependence on the Lorentz-invariant kinematic variables manifest and may be used directly or in conjunction with othermethods such as sector decomposition [95, 96]. Although some elements of our analysisin Section 2 relied upon physically-motivated plausibility arguments and experimentation,we subsequently presented a substantial amount of evidence in Sections 3 and 4 that ourmaster formula, Eq. (2.13), is correct. It would be very interesting in future work to consider still more non-trivial examples such as the s -channel cut of the massless two-loop non-planar double box; as discussedin [79], examples for which one cannot avoid imaginary parts on the mass shell requirespecial care and may be instructive. In an effort to remove as many superfluous assumptionsas possible from the formulation given in Section 2, it would of course also be desirable toput the theoretical foundations of the cut Baikov-Lee representation on a firmer footing.Although we have employed the familiar language of classical complex analysis throughoutthis work, it might be interesting to reformulate our findings in more modern languagealong the lines of [85, 98]. It is unclear to us, however, that such a reformulation willimmediately lead to clarifications.In fact, there exist several interesting classes of cut Feynman integrals which were notdiscussed in this work at all. First of all, it would be interesting to study representativesequential cuts of the type discussed in reference [70]. One should also check whether acut-discontinuity relation of the type discussed in [70] also exists for “crossed” sequentialcuts such as the s -channel + t -channel cut of the massless one-loop box of Section 3.2.Although success is less certain, it might be interesting to use the Baikov-Lee formalismdeveloped in this work to study iterated cuts in a single channel. As a start, one couldconsider the Feynman integral analog of the double two-particle cuts at two loops discussedin [99]. For such iterated cuts, it is not obvious that a cut-discontinuity relation exists atall, and it would therefore be interesting to take a fresh look at the problem using ourBaikov-Lee machinery. Finally, it goes almost without saying that we would very muchlike to apply our techniques to the evaluation of the master integrals relevant to the currentgeneration of phenomenologically-important unsolved problems in perturbative quantumfield theory. Although we have employed Eq. (2.13) throughout this paper, Eq. (2.17) is actually on more solidground from the theoretical point of view. The difference is that we have included all of the phases in Eq.(2.17) which one would find by considering the relevant Feynman integrals to be Feynman graphs inside ofan appropriate generalized scalar field theory. It is not clear to us why Eq. (2.13) works as well as it does. When a reference evaluation is not available, it is important to check analytical results numerically. Wetherefore note that the recently-released program pySecDec [97] should allow for the evaluation of a wideclass of cut Feynman integrals numerically (up to some fixed order in ǫ ) with moderate user input. – 27 – cknowledgments The authors would especially like to thank Ruth Britto for many interesting discussions,support, and comments on the manuscript. We also gratefully acknowledge an illumi-nating discussion with Stefan M¨uller-Stach and thank him for reading excerpts from ourmanuscript. RMS would like to thank Sven-Olaf Moch for an interesting discussion which,at least in part, inspired the author to pursue this line of research. This project has receivedfunding from the European Research Council (ERC) under the European Union’s Horizon2020 research and innovation programme under grant agreement No 647356 (CutLoops).Our figures were generated using
Jaxodraw [100], based on
AxoDraw [101].
A Mathematical relations for hypergeometric-like functions
In this appendix, we review some well-known facts used in this paper about hypergeometricfunctions and their generalizations. First of all, let us recall the usual integral representa-tion of the hypergeometric function which provides the analytical continuation of the hy-pergeometric series in non-exceptional cases. For | arg(1 − z ) | < π and Re ( c ) > Re ( a ) > F ( a, b ; c ; z ) = Γ( c )Γ( a )Γ( c − a ) Z d t t a − (1 − t ) c − a − (1 − tz ) − b . (A.1)In fact, for | arg(1 − z ) | < π , | arg(1 − z ) | < π , and Re ( c ) > Re ( a ) >
0, a completelyanalogous formula holds for the Appell F function [103]: F ( a ; b , b ; c ; z , z ) = Γ( c )Γ( a )Γ( c − a ) Z d t t a − (1 − t ) c − a − (1 − tz ) − b (1 − tz ) − b . (A.2)We also encounter the generalized hypergeometric function F , which, for the purposes ofthis paper, may be defined via the integral representation F ( a , a , a ; b , b ; z ) = Γ( b )Γ( a )Γ( b − a ) Z d t t a − (1 − t ) b − a − F ( a , a ; b ; tz ) , (A.3)which is valid for | arg(1 − z ) | < π and Re ( b ) > Re ( a ) > F . Of particular interest to us are, F ( a, b ; c ; z ) = Γ( c )Γ( b − a )Γ( b )Γ( c − a ) (1 − z ) − a F (cid:18) a, c − b ; a − b + 1; 11 − z (cid:19) + Γ( c )Γ( a − b )Γ( a )Γ( c − b ) (1 − z ) − b F (cid:18) b, c − a ; b − a + 1; 11 − z (cid:19) , (A.4) F (cid:18) a, b ; 2 b ; 4 z (1 + z ) (cid:19) = (1 + z ) a F (cid:18) a, a − b + 12 ; b + 12 ; z (cid:19) , (A.5)– 28 –nd F (cid:18) a, a + 12 ; c ; z (cid:19) = (cid:18) √ − z (cid:19) − a F (cid:18) a, a − c + 1; c ; 1 − √ − z √ − z (cid:19) . (A.6)Eq. (A.4) is valid for non-integral a − b , | arg(1 − z ) | < π , and | arg( − z ) | < π , whereas Eq.(A.5) is valid for 2 b = − , − , − , . . . and, crucially, | z | <
1. Eq. (A.6) is valid so long asthe condition | arg(1 − z ) | < π is satisfied.We also require some reduction identities, two for the generalized hypergeometric func-tion F and one for the Appell F function. The Saalsch¨utz summation formula [104], F ( a , a , a ; b , b ; 1) = Γ( b )Γ(1 + a − b )Γ(1 + a − b )Γ(1 + a − b )Γ(1 − b )Γ( b − a )Γ( b − a )Γ( b − a ) , (A.7)applies if b + b − a − a − a = 1 and one element of { a , a , a } is a negative integer.A more non-trivial summation formula involving two F functions on the left-hand sideis [105] − Γ(1 − a )Γ(1 + a )Γ( a − a )Γ( b )Γ( a )Γ(1 + a − a )Γ(1 − a + a )Γ( b − a ) ×× F ( a , a − a , a − b ; 1 + a − a , a − a ; 1) + F ( a , a , a ; b , a ; 1)= Γ( b )Γ(1 − a )Γ(1 + a )Γ( a − a )Γ( a )Γ(1 − a + a )Γ( b − a ) . (A.8)Eq. (A.8) is valid for Re (1 + b − a − a ) > F function collapses to a F if c = b + b [103], F ( a ; b , b ; b + b ; z , z ) = (1 − z ) − a F (cid:18) a, b ; b + b ; z − z − z (cid:19) . (A.9)The standard integral representation given above for the generalized hypergeometricfunction F is only one of a number of Euler integrals involving F which may be evaluatedusing the F series. Many evaluations of such generalized Euler integrals are given inreference [107]. The main result of interest to us is Z dt t γ − (1 − t ) ρ − (1 − tz ) − σ F ( α, β ; γ ; t ) = Γ( γ )Γ( ρ )Γ( γ + ρ − α − β )Γ( γ + ρ − α )Γ( γ + ρ − β ) ×× (1 − z ) − σ F (cid:18) ρ, σ, γ + ρ − α − β ; γ + ρ − α, γ + ρ − β ; zz − (cid:19) , (A.10)which is valid for | arg(1 − z ) | < π , Re ( γ ) > Re ( ρ ) >
0, and Re ( γ + ρ − α − β ) > B All-order-in- ǫ results for selected purely-virtual Feynman integrals In this appendix, we collect some useful results from the Feynman integral literature. Allresults which follow are presented in the normalization of Eq. (2.6) and are valid to allorders in the parameter of dimensional regularization, ǫ , for generic phase-space points inphysical kinematics. In order to compare with the cut calculations performed in Section– 29 –, we must explain how to parse the Disc operation introduced in Section 2. The directdiscontinuity of Feynman integral I in the s -channel is simplyDisc s ( I ) = I ( s + i { v j } \ s ) − I ( s − i { v j } \ s ) , (B.1)where { v j } denotes the set of variables (the parameter of dimensional regularization, gen-eralized Mandelstam variables, and, in general, internal masses) that I is a function of. B.1 The one-external-mass one-loop triangle
The one-external-mass one-loop triangle with generic propagator exponents is given by [71] ν ν ν p p = e iπǫ s − ν − ǫ Γ(2 − ν − ν − ǫ )Γ(2 − ν − ν − ǫ )Γ( ν − ǫ )Γ( ν )Γ( ν )Γ(4 − ν − ǫ ) , (B.2)to all orders in ǫ , where s = ( p + p ) > ν = P i =1 ν i . B.2 The massless one-loop box
The massless one-loop box is given by [71, 109] p p p p = − Γ ( − ǫ )Γ( ǫ ) su Γ( − ǫ ) (cid:20) ( − u ) − ǫ F (cid:16) , − ǫ ; 1 − ǫ ; 1 + us (cid:17) + e iπǫ s − ǫ F (cid:16) , − ǫ ; 1 − ǫ ; 1 + su (cid:17) (cid:21) (B.3)to all orders in ǫ , where s = ( p + p ) and u = ( p − p ) . In the physical region, s > − s < u < B.3 The one-external-mass six-line two-loop double triangle
The purely-virtual counterpart of the one-external-mass six-line two-loop double triangleintegral studied in Section 3 is given by p p = e iπǫ s − − ǫ Γ( − − ǫ )Γ(1 + 2 ǫ )Γ ( − ǫ )Γ(1 − ǫ )Γ( − ǫ ) (cid:16) Γ( − ǫ ) − Γ(1 + ǫ )Γ( − ǫ ) (cid:17) (B.4)to all orders in ǫ , where s = ( p + p ) >
0. To our knowledge, Eq. (B.4) was first derivedby van Neerven [109]. His idea was to first calculate all s -channel cuts and then deducethe associated purely-virtual result using unitarity. Consequently, to obtain (B.4) withoutreferring to cuts, it was necessary for us to evaluate the integral ourselves using Feynmanparameters. This exercise is elementary and may be carried out using the loop-by-loopintegration strategy suggested in [110]. – 30 – Baikov-Lee for the purely-virtual one-external-mass one-loop bubble
In this appendix, we complete the calculation of the Euclidean purely-virtual one-loopbubble integral with no internal masses which was initiated in Section 2. Our point ofdeparture will be Eq. (2.10), p = Z ∞ d( q ) Z √ p q − √ p q d( q · q ) π − ǫ (cid:0) p q − ( q · q ) (cid:1) − ǫ Γ (cid:0) − ǫ (cid:1) ( p ) − ǫ q (cid:0) q − q · q + p (cid:1) . (C.1)The first step is to map the domain of the first integration variable, q · q , onto theunit interval. This can be achieved straightforwardly by making the change of variables q · q = 2 p p q z − p p q . We arrive at p = Z ∞ d( q ) Z d z − ǫ π − ǫ (cid:0) q (cid:1) − ǫ ( z (1 − z )) − ǫ Γ (cid:0) − ǫ (cid:1) (cid:16) q + 2(1 − z ) p p q + p (cid:17) . (C.2)By comparing Eq. (C.2) to the form of Eq. (A.1), it is now obvious that the z integralmay be evaluated in terms of the F series.The one-fold integral that remains, p = Z ∞ d( q ) π − ǫ (cid:0) q (cid:1) − ǫ Γ(2 − ǫ )( q + p ) F (cid:18) ,
12 ; 2 − ǫ ; 4 q p ( q + p ) (cid:19) , (C.3)is most naturally evaluated by splitting the integral at the point q = p and then mappingboth the integral from 0 to p and the integral from p to ∞ onto the unit interval. This willallow for the simultaneous application of quadratic transformation (A.6) to both integrals.Making the change of variables q = p x in the first integral and the change of variables q = p /x in the second integral, we find p = Z d x π − ǫ (cid:0) p (cid:1) − ǫ Γ(2 − ǫ )(1 + x ) (cid:16) x − ǫ + x − ǫ (cid:17) F (cid:18) ,
12 ; 2 − ǫ ; 4 x (1 + x ) (cid:19) = Z d x π − ǫ (cid:0) p (cid:1) − ǫ Γ(2 − ǫ ) (cid:16) x − ǫ + x − ǫ (cid:17) F (1 , ǫ ; 2 − ǫ ; x ) . (C.4)At this stage, we can straightforwardly evaluate both integrals using Eq. (A.3). Theresult obtained in this manner, p = π − ǫ (cid:0) p (cid:1) − ǫ Γ(2 − ǫ ) (cid:18) ǫ F (1 , ǫ, ǫ ; 2 − ǫ, ǫ ; 1)+ 11 − ǫ F (1 , ǫ, − ǫ ; 2 − ǫ, − ǫ ; 1) (cid:19) , (C.5) A variable change of this form often allows one to recognize the definite integrals which arise fromall-orders-in- ǫ Feynman integral calculations as Euler integrals of hypergeometric type (see e.g. [69]). – 31 –s correct but far more complicated than it needs to be. In this case, we can simplify theresult by applying summation formula (A.8) to eliminate the second F series in Eq. (C.5)above: p = π − ǫ (cid:0) p (cid:1) − ǫ Γ (1 − ǫ )Γ( ǫ )Γ(2 − ǫ ) . (C.6)Needless to say, the above result agrees with what one obtains (far more easily) using theFeynman representation. References [1] R. P. Feynman,
Spacetime approach to quantum electrodynamics , Phys. Rev. (1949)769–789.[2] A. Kotikov, Differential equations method: New technique for massive Feynman diagramscalculation , Phys. Lett.
B254 (1991) 158–164.[3] A. Kotikov,
Differential equations method: The Calculation of vertex type Feynmandiagrams , Phys. Lett.
B259 (1991) 314–322.[4] A. Kotikov,
Differential equation method: The Calculation of N -point Feynman diagrams , Phys. Lett.
B267 (1991) 123–127.[5] Z. Bern, L. J. Dixon, and D. A. Kosower,
Dimensionally regulated one-loop integrals , Phys.Lett.
B302 (1993) 299–308, [ hep-ph/9212308 ].[6] Z. Bern, L. J. Dixon, and D. A. Kosower,
Dimensionally regulated pentagon integrals , Nucl.Phys.
B412 (1994) 751–816, [ hep-ph/9306240 ].[7] E. Remiddi,
Differential equations for Feynman graph amplitudes , Nuovo Cim.
A110 (1997) 1435–1452, [ hep-th/9711188 ].[8] T. Gehrmann and E. Remiddi,
Differential equations for two-loop four-point functions , Nucl. Phys.
B580 (2000) 485–518, [ hep-ph/9912329 ].[9] F. Tkachov,
A Theorem on Analytical Calculability of Four-Loop Renormalization GroupFunctions , Phys. Lett.
B100 (1981) 65–68.[10] K. Chetyrkin and F. Tkachov,
Integration by Parts: The Algorithm to Calculate betaFunctions in 4 Loops , Nucl. Phys.
B192 (1981) 159–204.[11] S. Laporta,
High-precision calculation of multi-loop Feynman integrals by differenceequations , Int. J. Mod. Phys.
A15 (2000) 5087–5159, [ hep-ph/0102033 ].[12] A. von Manteuffel and C. Studerus,
Reduze 2 - Distributed Feynman Integral Reduction , .[13] A. von Manteuffel and R. M. Schabinger, A novel approach to integration by partsreduction , Phys. Lett.
B744 (2015) 101–104, [ ].[14] A. V. Smirnov,
FIRE5 : a
C++ implementation of Feynman Integral REduction , Comput.Phys. Commun. (2015) 182–191, [ ].[15] S. G. Gorishnii, S. A. Larin, L. R. Surguladze, and F. V. Tkachov,
Mincer: Program forMulti-loop Calculations in Quantum Field Theory for the Schoonschip System , Comput.Phys. Commun. (1989) 381–408.[16] P. A. Baikov, Explicit solutions of the multi-loop integral recurrence relations and itsapplication , Nucl. Instrum. Meth.
A389 (1997) 347–349, [ hep-ph/9611449 ].[17] J. Gluza, K. Kajda, and D. A. Kosower,
Towards a Basis for Planar Two-Loop Integrals , Phys. Rev.
D83 (2011) 045012, [ ]. One can see that this is the case by consulting a standard text such as Smirnov [71]. – 32 –
18] R. N. Lee,
Presenting
LiteRed : a tool for the Loop InTEgrals REDuction , .[19] B. Ruijl, T. Ueda, and J. Vermaseren, The diamond rule for multi-loop Feynman diagrams , Phys. Lett.
B746 (2015) 347–350, [ ].[20] H. Ita,
Two-loop Integrand Decomposition into Master Integrals and Surface Terms , Phys.Rev.
D94 (2016), no. 11 116015, [ ].[21] K. J. Larsen and Y. Zhang,
Integration-by-parts reductions from unitarity cuts and algebraicgeometry , Phys. Rev.
D93 (2016), no. 4 041701, [ ].[22] T. Ueda, B. Ruijl, and J. A. M. Vermaseren,
Calculating four-loop massless propagatorswith Forcer , in
ACAT 2016 conference proceedings , [ ].[23] S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, and B. Page,
Sub-leading Poles in theNumerical Unitarity Method at Two Loops , .[24] S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, B. Page, and M. Zeng, Two-LoopFour-Gluon Amplitudes with the Numerical Unitarity Method , .[25] R. N. Lee, Calculating multi-loop integrals using dimensional recurrence relation and D -analyticity , Nucl. Phys. Proc. Suppl. (2010) 135–140, [ ].[26] C. Anastasiou, C. Duhr, F. Dulat, and B. Mistlberger,
Soft triple-real radiation for Higgsproduction at N LO , JHEP (2013) 003, [ ].[27] R. Kumar, Covariant phase-space calculations of n -body decay and production processes , Phys. Rev. (1969) 1865–1875.[28] A. G. Grozin,
Integration by parts: An Introduction , Int. J. Mod. Phys.
A26 (2011)2807–2854, [ ].[29] R. E. Cutkosky,
Singularities and discontinuities of Feynman amplitudes , J. Math. Phys. (1960) 429–433.[30] R. N. Lee and V. A. Smirnov, The Dimensional Recurrence and Analyticity Method forMulti-component Master Integrals: Using Unitarity Cuts to Construct HomogeneousSolutions , JHEP (2012) 104, [ ].[31] A. Primo and L. Tancredi, On the maximal cut of Feynman integrals and the solution oftheir differential equations , Nucl. Phys.
B916 (2017) 94–116, [ ].[32] H. Frellesvig and C. G. Papadopoulos,
Cuts of Feynman Integrals in Baikov representation , JHEP (2017) 083, [ ].[33] J. A. Lappo-Danilevsky, Th´eorie algorithmique des corps de Riemann , Rec. Math. Moscou (1927) 113.[34] A. B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes , Math. Res.Lett. (1998) 497–516, [ ].[35] E. Remiddi and J. Vermaseren, Harmonic polylogarithms , Int. J. Mod. Phys.
A15 (2000)725–754, [ hep-ph/9905237 ].[36] T. Gehrmann and E. Remiddi,
Two-loop master integrals for γ ∗ → jets: The Planartopologies , Nucl. Phys.
B601 (2001) 248–286, [ hep-ph/0008287 ].[37] A. V. Kotikov,
The Property of maximal transcendentality in the N = 4 SupersymmetricYang-Mills , in
Diakonov, D. (ed.): Subtleties in quantum field theory , pp. 150–174, (2010),[ ].[38] J. M. Henn,
Multi-loop integrals in dimensional regularization made simple , Phys. Rev. Lett. (2013), no. 25 251601, [ ].[39] J. M. Henn, A. V. Smirnov, and V. A. Smirnov,
Analytic results for planar three-loopfour-point integrals from a Knizhnik-Zamolodchikov equation , JHEP (2013) 128,[ ].[40] K.-T. Chen,
Iterated path integrals , Bull. Am. Math. Soc. (1977) 831–879. – 33 –
41] A. B. Goncharov,
Galois symmetries of fundamental groupoids and noncommutativegeometry , Duke Math. J. (2005) 209, [ math/0208144 ].[42] F. Brown,
On the decomposition of motivic multiple zeta values , .[43] C. Duhr, H. Gangl, and J. R. Rhodes, From polygons and symbols to polylogarithmicfunctions , JHEP (2012) 075, [ ].[44] C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes , JHEP (2012) 043, [ ].[45] T. Gehrmann, A. von Manteuffel, and L. Tancredi, The two-loop helicity amplitudes for qq ′ → V V → leptons , JHEP (2015) 128, [ ].[46] A. von Manteuffel and L. Tancredi, The two-loop helicity amplitudes for gg → V V → leptons , JHEP (2015) 197, [ ].[47] R. Bonciani, V. Del Duca, H. Frellesvig, J. M. Henn, F. Moriello, and V. A. Smirnov, Next-to-leading order QCD corrections to the decay width H → Zγ , JHEP (2015) 108,[ ].[48] A. von Manteuffel and R. M. Schabinger, Numerical Multi-Loop Calculations via FiniteIntegrals and One-Mass EW-QCD Drell-Yan Master Integrals , JHEP (2017) 129,[ ].[49] H. Frellesvig, D. Tommasini, and C. Wever, On the reduction of generalized polylogarithmsto Li n and Li , and on the evaluation thereof , JHEP (2016) 189, [ ].[50] F. Brown and A. Levin, Multiple Elliptic Polylogarithms , .[51] S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph , J. Number Theor. (2015) 328–364, [ ].[52] S. Bloch, M. Kerr, and P. Vanhove,
A Feynman integral via higher normal functions , Compos. Math. (2015) 2329–2375, [ ].[53] L. Adams, C. Bogner, and S. Weinzierl,
The two-loop sunrise graph in two spacetimedimensions with arbitrary masses in terms of elliptic dilogarithms , J. Math. Phys. (2014), no. 10 102301, [ ].[54] L. Adams, C. Bogner, and S. Weinzierl, The two-loop sunrise integral around four spacetimedimensions and generalisations of the Clausen and Glaisher functions towards the ellipticcase , J. Math. Phys. (2015), no. 7 072303, [ ].[55] L. Adams, C. Bogner, A. Schweitzer, and S. Weinzierl, The kite integral to all orders interms of elliptic polylogarithms , J. Math. Phys. (2016) 122302, [ ].[56] C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD , Nucl. Phys.
B646 (2002) 220–256, [ hep-ph/0207004 ].[57] C. Anastasiou, L. J. Dixon, K. Melnikov, and F. Petriello,
Dilepton rapidity distribution inthe Drell-Yan process at NNLO in QCD , Phys. Rev. Lett. (2003) 182002,[ hep-ph/0306192 ].[58] C. Anastasiou, L. J. Dixon, K. Melnikov, and F. Petriello, High precision QCD at hadroncolliders: Electroweak gauge boson rapidity distributions at NNLO , Phys. Rev.
D69 (2004)094008, [ hep-ph/0312266 ].[59] M. Caffo, H. Czyz, S. Laporta, and E. Remiddi,
The Master differential equations for thetwo-loop sunrise selfmass amplitudes , Nuovo Cim.
A111 (1998) 365–389,[ hep-th/9805118 ].[60] S. Laporta and E. Remiddi,
Analytic treatment of the two-loop equal mass sunrise graph , Nucl. Phys.
B704 (2005) 349–386, [ hep-ph/0406160 ].[61] L. Adams, C. Bogner, and S. Weinzierl,
The iterated structure of the all-order result for thetwo-loop sunrise integral , J. Math. Phys. (2016), no. 3 032304, [ ]. – 34 –
62] S. Bloch, M. Kerr, and P. Vanhove,
Local mirror symmetry and the sunset Feynmanintegral , .[63] E. Remiddi and L. Tancredi, Differential equations and dispersion relations for Feynmanamplitudes. The two-loop massive sunrise and the kite integral , Nucl. Phys.
B907 (2016)400–444, [ ].[64] R. Bonciani, V. Del Duca, H. Frellesvig, J. M. Henn, F. Moriello, and V. A. Smirnov,
Two-loop planar master integrals for Higgs → partons with full heavy-quark massdependence , JHEP (2016) 096, [ ].[65] A. von Manteuffel and L. Tancredi, A non-planar two-loop three-point function beyondmultiple polylogarithms , .[66] A. Primo and L. Tancredi, Maximal cuts and differential equations for Feynman integrals.An application to the three-loop massive banana graph , .[67] J. Bosma, M. Sogaard, and Y. Zhang, Maximal Cuts in Arbitrary Dimension , .[68] G. ’t Hooft and M. J. G. Veltman, Diagrammar , NATO Sci. Ser. B (1974) 177–322.[69] A. Gehrmann-De Ridder, T. Gehrmann, and G. Heinrich, Four particle phase-spaceintegrals in massless QCD , Nucl. Phys.
B682 (2004) 265–288, [ hep-ph/0311276 ].[70] S. Abreu, R. Britto, C. Duhr, and E. Gardi,
From multiple unitarity cuts to the coproductof Feynman integrals , JHEP (2014) 125, [ ].[71] V. A. Smirnov, Evaluating Feynman integrals , Springer Tracts Mod. Phys. (2004)1–244.[72] A. V. Smirnov and A. V. Petukhov,
The Number of Master Integrals is Finite , Lett. Math.Phys. (2011) 37–44, [ ].[73] E. Panzer, Feynman integrals and hyperlogarithms . Ph.D. thesis, Humboldt U., Berlin, Inst.Math., (2015), .[74] F. Brown,
The massless higher-loop two-point function , Commun. Math. Phys. (2009)925–958, [ ].[75] F. Brown,
On the periods of some Feynman integrals , .[76] E. Panzer, On the analytic computation of massless propagators in dimensionalregularization , Nucl. Phys.
B874 (2013) 567–593, [ ].[77] E. Panzer,
On hyperlogarithms and Feynman integrals with divergences and many scales , JHEP (2014) 071, [ ].[78] E. Panzer, Algorithms for the symbolic integration of hyperlogarithms with applications toFeynman integrals , Comput. Phys. Commun. (2015) 148–166, [ ].[79] A. von Manteuffel, E. Panzer, and R. M. Schabinger,
A quasi-finite basis for multi-loopFeynman integrals , JHEP (2015) 120, [ ].[80] A. von Manteuffel, E. Panzer, and R. M. Schabinger, On the Computation of Form Factorsin Massless QCD with Finite Master Integrals , Phys. Rev.
D93 (2016), no. 12 125014,[ ].[81] M. Kompaniets and E. Panzer,
Renormalization group functions of φ theory in theMS-scheme to six loops , PoS
LL2016 (2016) 038, [ ].[82] C. Bogner and S. Weinzierl,
Feynman graph polynomials , Int. J. Mod. Phys.
A25 (2010)2585–2618, [ ].[83] I. M. Gel´fand and G. E. Shilov,
Generalized Functions, Volume I: Properties andOperations , AMS Chelsea Publishing, Volume: 377 (1964) 1–437.[84] G. B. Folland,
Fourier analysis and its applications , Wadsworth & Brooks/ColeMathematics Series (1992) 1–428.[85] S. Abreu, R. Britto, C. Duhr, and E. Gardi,
Cuts from residues: the one-loop case , . – 35 –
86] Y. Li, A. von Manteuffel, R. M. Schabinger, and H. X. Zhu, N LO Higgs boson andDrell-Yan production at threshold: The one-loop two-emission contribution , Phys. Rev.
D90 (2014), no. 5 053006, [ ].[87] Z. Bern, V. Del Duca, L. J. Dixon, and D. A. Kosower,
All non-maximally-helicity-violatingone-loop seven-gluon amplitudes in N = 4 super Yang-Mills theory , Phys. Rev.
D71 (2005)045006, [ hep-th/0410224 ].[88] R. Britto, F. Cachazo, and B. Feng,
Generalized unitarity and one-loop amplitudes in N = 4 super Yang-Mills , Nucl. Phys.
B725 (2005) 275–305, [ hep-th/0412103 ].[89] J. Carlson, S. M¨uller-Stach, and C. Peters,
Period mappings and Period Domains , Cambridge University Press: Cambridge Studies in Advanced Mathematics (2003) 1–425.[90] M. Kontsevich and D. Zagier, Periods , Springer (2001) 771–808.[91] S. M¨uller-Stach, S. Weinzierl, and R. Zayadeh,
A Second-Order Differential Equation forthe Two-Loop Sunrise Graph with Arbitrary Masses , Commun. Num. Theor. Phys. (2012)203–222, [ ].[92] S. M¨uller-Stach, S. Weinzierl, and R. Zayadeh, Picard-Fuchs equations for Feynmanintegrals , Commun. Math. Phys. (2014) 237–249, [ ].[93] A. Hatcher,
Algebraic Topology , Cambridge University Press (2002) 1–538.[94] L. Adams, C. Bogner, and S. Weinzierl,
The two-loop sunrise graph with arbitrary masses , J. Math. Phys. (2013) 052303, [ ].[95] T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergentmulti-loop integrals , Nucl. Phys.
B585 (2000) 741–759, [ hep-ph/0004013 ].[96] C. Bogner and S. Weinzierl,
Resolution of singularities for multi-loop integrals , Comput.Phys. Commun. (2008) 596–610, [ ].[97] S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, J. Schlenk, and T. Zirke, pySecDec : a toolbox for the numerical evaluation of multi-scale integrals , .[98] S. Abreu, R. Britto, C. Duhr, and E. Gardi, The algebraic structure of cut Feynmanintegrals and the diagrammatic coaction , .[99] Z. Bern, L. J. Dixon, and D. A. Kosower, A Two-loop four-gluon helicity amplitude inQCD , JHEP (2000) 027, [ hep-ph/0001001 ].[100] D. Binosi and L. Theussl, JaxoDraw : A Graphical user interface for drawing Feynmandiagrams , Comput. Phys. Commun. (2004) 76–86, [ hep-ph/0309015 ].[101] J. A. M. Vermaseren,
Axodraw , Comput. Phys. Commun. (1994) 45–58.[102] N. N. Lebedev, Special Functions And Their Applications , in
Silverman, Richard A. (ed.):Selected Russian Publications In The Mathematical Sciences , pp. 1–303, (1965).[103] M. J. Schlosser,
Multiple Hypergeometric Series: Appell Series and Beyond , in
Schneider,Carsten and Bl¨umlein, Johannes (ed.): Computer Algebra in Quantum Field Theory -Integration, Summation, and Special Functions , pp. 305–324, (2013), [ ].[104] L. J. Slater,
Generalized Hypergeometric Functions , Cambridge University Press (1966)1–268.[105] http://functions.wolfram.com/ . The Wolfram Functions Site maintained by WolframResearch, Inc.[106] T. H. Koornwinder,
Identities of non-terminating series by Zeilberger’s algorithm , J.Comput. Appl. Math. (1998) 449–461, [ math/9805010 ].[107] J. Letessier and G. Valent, Some Integral Relations Involving Hypergeometric Functions , SIAM J. Appl. Math. (1988), no. 1 214–221.[108] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products , Jeffrey, Alanand Zwillinger, Daniel (ed.): Edition Seven (2007) 1–1150. – 36 –
Dimensional Regularization of Mass and Infrared Singularities inTwo-Loop On-shell Vertex Functions , Nucl. Phys.
B268 (1986) 453–488.[110] R. J. Gonsalves,
Dimensionally regularized two-loop on-shell quark form factor , Phys. Rev.
D28 (1983) 1542.(1983) 1542.