Prescriptive Unitarity with Elliptic Leading Singularities
Jacob L. Bourjaily, Nikhil Kalyanapuram, Cameron Langer, Kokkimidis Patatoukos
PPrescriptive Unitarity with
El liptic
Leading Singularities
Jacob L. Bourjaily, a,b
Nikhil Kalyanapuram, a Cameron Langer, a Kokkimidis Patatoukos aa Institute for Gravitation and the Cosmos, Department of Physics,Pennsylvania State University, University Park, PA 16802, USA b Niels Bohr International Academy and Discovery Center, Niels Bohr Institute,University of Copenhagen, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark [email protected] , [email protected] , [email protected] , [email protected] We investigate the consequences of elliptic leading singularities for the unitarity-based representations of two-loop amplitudes in planar, maximally supersymmetricYang-Mills theory. We show that diagonalizing with respect to these leading singu-larities ensures that the integrand basis is term-wise pure (suitably generalized, tothe elliptic multiple polylogarithms, as necessary). We also investigate an alternativestrategy based on diagonalizing a basis of integrands on differential forms; this strat-egy, while neither term-wise Yangian-invariant nor pure, offers several advantages interms of complexity. a r X i v : . [ h e p - t h ] F e b ontents Homological
Diagonalization—with Respect to
Contours . . . . . . . 123.2
Cohomological
Diagonalization—with Respect to
Forms . . . . . . . . 143.3 Consistency Checks for Amplitude Integrands . . . . . . . . . . . . . 153.4 Smooth Degenerations . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Generalized unitarity has proven an extremely powerful framework for the repre-sentation of scattering amplitudes at large multiplicity and/or loop order. The basicidea is that any loop integrand —a rational differential form on the space of internalloop momenta—can be viewed as an element of a basis of standardized Feynmanloop integrands. Provided the basis of integrands is large enough, it can be used torepresent all the scattering amplitudes of any theory and spacetime dimension. Thisidea has a long history (see e.g. [1, 2]); it was formalized and used to famous effectin e.g. [3–9], and has been recently refined, generalized, and put to use for manyimpressive applications (see e.g. [10–15]).Among the many advantages of this approach is that the basis of integrands, solong as it is large enough, is sufficient to represent literally all amplitudes (arbitrarymultiplicity and states) in a wide class of theories at any loop order. Thus, theintegrands in the basis need only be integrated once and for all—reusable for anyprocess of interest. As loop integration has been (and remains) among the hardestproblems in perturbative quantum field theory, this is a very important feature. Thismakes clear the importance of choosing ‘good’ integrands for a basis—the precisemeasures of which have evolved greatly with time. (Roughly speaking, a good basiswould consist of integrands which can be integrated ‘most easily’ or which result inthe ‘simplest’ expressions.)Another advantage to unitarity is that coefficients of particular amplitudes withrespect to a basis can be computed in terms of mostly (and often wholly) on-shell– 1 –ata—specifically, on-shell functions [8, 9, 16–21]. When these on-shell functions areleading singularities, they have no internal degrees of freedom. Historically, leadingsingularities have been defined as maximal co-dimension residues (of polylogarithmicdifferential forms); more recently, this definition has been modified and generalizedto include any full-dimensional compact contour integral of a scattering amplitudeintegrand [22]. Leading singularities have played a key role in the development of ourmodern understanding of quantum field theory (see e.g. [5, 15, 23–30]), and manyof the remarkable aspects of scattering amplitudes (their simplicity, and wide rangeof symmetries) were discovered in this context. For example, the BCFW recursionrelations for tree-amplitudes were first discovered in this setting [3, 31, 32], as was theinfinite-dimensional Yangian symmetry of planar maximally supersymmetric Yang-Mills theory (sYM) [33–35], and the correspondence between on-shell functions insYM and residues of the positroid volume-form in Grassmannian manifolds [36, 37].When leading singularities are used to determine the coefficients of loop ampli-tudes with respect to some integrand basis, generalized unitarity becomes a relativelysimple problem of linear algebra—matching the ‘cuts’ of field theory against the cor-responding cuts of the integrand basis. Until recently, however, it was unclear if leading singularities represented complete information about perturbative scatteringamplitudes even in the simplest theories. The reason for this uncertainty lies in thefact that, for sufficiently large multiplicity and/or loop order, scattering amplitudeintegrands in most theories are not ‘ d log’ differential forms [38–46] and cannot becharacterized by (maximal co-dimension) residues alone. For such cases, the tra-ditional definition of leading singularity becomes incomplete; and the most typicalstrategy to deal with non-polylogarithmic contributions has been to use the highestco-dimension residues that exist (sub n -leading singularities), and then use sufficientnumbers of off-shell evaluations to match a loop integrand functionally on the re-maining degrees of freedom. (Examples of such strategies being used can be foundin [15, 16].) The result of this approach, however, has many obvious disadvantages;in particular, it results in representations of amplitudes that (at least term-by-term)involve references to arbitrary choices (for the off-shell evaluation) which can breakmany of the niceties that scattering amplitudes are known to posses.Before we discuss any concrete examples, it is worth highlighting a conventionaldifference between this work relative to virtually all existing literature: we have cho-sen to write all loop integration measures in terms of - d L (cid:126)(cid:96) where - d := d/ (2 π ) (byanalogy to ‘ (cid:126) ’). As such, many of our results differ by powers of (2 πi ) relative tothose found elsewhere. This choice is motivated by the fact that an integrand nor-malized to have unit residues with respect to the measure d L (cid:126)(cid:96) will have unit contour integrals with respect to - d L (cid:126)(cid:96) . As such, most formulae appear identical to other lit-erature; a notable exception, however, is the case of sub-leading singularities, forwhich our convention requires relative factors of i .– 2 –o illustrate how prescriptive unitarity can work when there are elliptic contri-butions, consider the elliptic double-box integrand for massless, scalar ϕ -theory infour dimensions: ⇔ - d (cid:96) - d (cid:96) ( (cid:96) | (cid:96) | (cid:96) | (cid:96) | (cid:96) )( (cid:96) | (cid:96) | (cid:96) |
1) =: I ϕ db . (1.1)Above, ( (cid:96) i | a ) represents an ordinary, scalar inverse propagator expressed in dual-momentum coordinates (the details of which we review below) and - d := d π . At anyrate, (1.1) is an 8-dimensional (rational) differential form on the space of loop mo-menta. Taking a contour integral which puts all seven propagators on-shell, however,results in an elliptic differential form on the remaining variable : (cid:73) {| ( (cid:96) i | a ) | = (cid:15) }| ( (cid:96) | (cid:96) ) | = (cid:15) I ϕ db = − ic y (2 | | | - d αy ( α ) (1.2)where we have used α to represent the final loop momentum variable, y ( α ) is anirreducible quartic (with coefficients that depend on the momenta of the particlesinvolved), and c y is a factor introduced to render y ( α ) monic . The precise detailsare not important to us now; but it is easy to see that (1.2) represents an ellipticdifferential form, without any further ‘residues’ on which we may define a traditionalleading singularity.Recently [22], a broader definition of leading singularity has been introducedto include any full-dimensional compact contour-integral of a scattering amplitudeintegrand. With this new definition, we can in fact define a leading singularity forthe double-box integrand by integrating (1.2) over, for example, the a -cycle, Ω a , ofthe elliptic curve: (cid:73) Ω a (cid:73) {| ( (cid:96) i | a ) | = (cid:15) }| ( (cid:96) | (cid:96) ) | = (cid:15) I ϕ db = 2 π c y (2 | | |
9) 1 (cid:112) ( r − r )( r − r ) K [ ϕ ] =: e ϕ a ; (1.3)here, ϕ is a cross-ratio in the roots r i of the quartic (the details of which are notimportant to us now). If we wanted to chose a basis of integrands which wouldbe normalized to ‘match’ this leading singularity prescriptively, we would need tonormalize the double-box integrand accordingly: I ϕ db (cid:55)→ I db := I ϕ db / e ϕ a = - d (cid:96) - d (cid:96) π (2 | | | (cid:112) ( r − r )( r − r ) /K [ ϕ ]2 c y ( (cid:96) | (cid:96) | (cid:96) | (cid:96) | (cid:96) )( (cid:96) | (cid:96) | (cid:96) | . (1.4) There are two solutions to these seven (quadratic) equations; here we write the contour on oneof them. – 3 –hus, a prescriptive representation of the 10-particle scattering amplitude in thistheory at two loops would involve a term I db × e ϕ a = (cid:16) I ϕ db / e ϕ a (cid:17) × e ϕ a = I ϕ db . (1.5)While this example may seem overly trivial (especially considering that the orig-inal scalar integrand in (1.1) is literally a term in the Feynman expansion!), there-writing of it according to prescriptive unitarity according to (1.5) has a remark-able feature: the now-normalized basis integrand I db is in fact dual-conformallyinvariant and pure in the sense defined by the authors of [47]—as such, it is arguablythe simplest possible form of the integral (and, presumably, the easiest to integrate).(For a broader discussion of integrand ‘purity’, we suggest the reader consult refs.[47, 48]; for the present, we merely mention this to emphasize that such integrands,and the differential equations that they satisfy (see e.g. [49]) have been defined so asto manifest many remarkable properties.) In this work, we generalize and expand upon the discussion above to the case oftwo-loop amplitudes in planar, maximally supersymmetric ( N = 4) Yang-Mills theory(sYM). A closed formula for all such amplitude integrands was first derived in ref. [15](representing an early application of what became known as ‘prescriptive’ unitarity[16]), which succeeded despite the presence of elliptic integrals due to carefully-made(but arbitrary) choices for off-shell evaluations in combinations of leading and sub-leading singularities. The resulting representations given in [15, 50, 51] involvedterms that were neither Yangian-invariant, nor integrands that were pure.In section 2 we review the salient elements of two-loop prescriptive unitarity, aswell as the novel generalization of elliptic leading singularities introduced in [22]. Insection 3, the main result of this paper, we revisit the prescriptive unitarity story inthe light of our recent work. In particular, we derive two novel representations ofamplitudes in planar sYM at two loops, both defined completely prescriptively andunambiguously. The first, described in section 3.1, involves a prescriptive integrandbasis chosen by diagonalization on leading singularities (in the new, broader sense); itis intrinsically homological, and results in a representation of amplitudes that, term-by-term, involves Yangian-invariant coefficients and pure integrals. In section 3.2,we describe an alternative representation based instead on a cohomological diagonal-ization of the integrand basis. The resulting form is simpler in many ways (especiallyalgebraically), but involves coefficients that are not Yangian-invariant and a basis ofloop integrands that are not generally pure.– 4 – Review: Prescriptive Integrand Bases for 2-Loop sYM
In this section, we briefly review the ingredients of the representation of two-loop integrands in planar sYM as described in ref. [15]. More complete details canbe found in [15, 16]. For what we need in the following sections, the details of hownumerators are chosen for the double-pentagons and pentaboxes will not be criticalto us—except for the role played by the double-box integrands as ‘contact terms’ ofthese basis elements.
A very useful (and arguably accidental) feature of planar integrands at two loopsis that a complete and not over-complete basis of dual-conformal integrands exists.This is in contrast to one loop or three or more loops, for which dual-conformalityapparently requires over-completeness (see e.g. [52–54]).At two loops, a dual-conformal basis can be chosen that consists of three classesof integrands—the double-boxes, pentaboxes, and double-pentagons: , i , ji . (2.1)To be clear, these pictures represent the corresponding set of scalar, massless Feyn-man propagators and the indices { i, j } ∈ { , } indicate particular choices of loop-dependent numerators.All the integrands in our basis can be normalized so-as to be dual-conformal(and when possible, pure). For those integrands with exclusively residues of maxi-mal co-dimension, they are normalized to have unit leading singularities on a choiceof such a contour, and made to vanish on all such defining contours for all other in-tegrands in the basis. Most of the integrands, however, have support on double-boxsub-topologies which have elliptic structures and therefore cannot be realized as d logdifferential forms.It is useful to bear in mind that the loop-independent numerators and the overallnormalization of the integrands in (2.1) have considerable flexibility. In particular,even after imposing dual-conformality, the space of possible numerators is relativelylarge. For the pentabox integrands, dual-conformal, loop independent numeratorsspan a six-dimensional space, which may be decomposed into four contact-term nu-merators (proportional to one of the four inverse-propagators associated with theedges of the pentagon side of the integrand), and two complementary ‘top-level’degrees of freedom (schematically indexed by i ∈ { , } ).Graphically, this ambiguity reflects the fact that the double-box topology can beobtained by contracting one of the edges { a, b, c, d } ,– 5 – bcdef g i ⊃ b cdef g , a cdef g , a bdef g , a bcef g Algebraically, this can be understood by decomposing the vector space of numeratorsinto the following basis,[ (cid:96) ]:= span (cid:8) ( (cid:96) |Y ) , ( (cid:96) |Y ) , ( (cid:96) | a ) , ( (cid:96) | b ) , ( (cid:96) | c ) , ( (cid:96) | d ) (cid:9) , (2.2)where Y , are the solutions to the quadruple cut ( (cid:96) | a ) = ( (cid:96) | b ) = ( (cid:96) | c ) = ( (cid:96) | d ) = 0,whose precise form does not concern us here. Thus, each pentabox integrand, a bcdef g i ⇔ - d (cid:96) - d (cid:96) n i ( (cid:96) |Y i )+ n ia ( (cid:96) | a )+ n ib ( (cid:96) | b )+ n ic ( (cid:96) | c )+ n id ( (cid:96) | d )( (cid:96) | a )( (cid:96) | b )( (cid:96) | c )( (cid:96) | d )( (cid:96) | (cid:96) )( (cid:96) | e )( (cid:96) | f )( (cid:96) | g ) (2.3)is not fully-specified until the four (loop-independent, but kinematic-dependent) ‘con-stants’ n i have been specified. The top-level normalization, n i is determined byrequiring that (some combination of) the polylogarithmic leading-singularity whichencircles the eight Feynman propagators of the pentabox is unity. In prescriptive unitarity, the basis of integrands is chosen to be diagonal withrespect to some choice of leading singularities or ‘cuts’ (or, as we will explore later,with respect to forms). To see how this works, consider the double-pentagon inte-grands. Among all the integrands in the basis (2.1), only the double-pentagons havesupport on the so-called ‘kissing-box’ leading singularities of field theory: ji . (2.4)Here, the indices { i, j } ∈ { , } label the two solutions (for each loop separately) tothe cut-equations which put these eight propagators on-shell. Thus, we choose the2 × regardless of the particular choices for the– 6 –ontact terms of the double pentagon. Thus, there remains a (6 − two non-contact-term degreesof freedom in their numerators, they cannot be used to match all four pentaboxleading-singularities: ji (2.5)where, as before { i, j } ∈ { , } label the particular solutions to the cut-equationswhich put the eight propagators on-shell. Relatedly, it is not possible to make thedouble-pentagon integrands vanish on all pentabox contours. This apparent tensionis in fact fairly trivial and easy to resolve: we merely need to make some choice oftwo of the pentabox leading singularities (or two independent combinations thereof),on which to normalize the pentabox integrands, and then require that the double-pentagons vanish on each of these contours. Once this is done, all four of the pentaboxleading singularities will be matched in the basis: two, manifestly by the pentaboxes,and the complementary pair matched indirectly by residue theorems involving thedouble-pentagon integrands. This may seem magical, but follows necessarily fromthe completeness of the integrand basis.What remains open, however, are the questions of the double-boxes: how shouldthey be normalized, and how should the double-box contact terms of the pentaboxesand double-pentagons be defined? For all the double-box integrands which do sup-port residues of maximal co-dimension (that is, any which involves at least onethree-particle vertex), the strategy above may be iterated once more without furthersubtlety, using a ‘composite’ leading singularity. (For amplitudes with fewer than tenparticles, all double-boxes have support on such additional ‘cuts’; and a completebasis can be made polylogarithmic and ‘pure’ in this way.)For double-box integrands without (traditional) leading singularities, however,the description above falls short (or at least, is incomplete). We will not review howthis question was resolved by the authors of ref. [15, 16, 55], in part because we willfind more elegant solutions here. For the sake of our current investigation, however,let us assume that a spanning set of maximal co-dimension, polylogarithmic contours– 7 –ave been chosen to define the top-level degrees of freedom of the pentabox anddouble-pentagon integrands; as such, we may take for granted that all the pentaboxand kissing-box leading singularities of field theory are matched by these integrandsin the basis—leaving only the question of matching the sub-leading singularitiesassociated with non-polylogarithmic double-boxes. That is, we need only to addressthe double-box integrands—to determine their normalizations, their coefficients infield theory, and how to ‘diagonalize’ the pentaboxes and double-pentagons withrespect to these choices. Let us therefore consider these integrands in some detail. Up to a normalizing factor denoted n db , the double-box integrands in the DCI-power-counting basis may be defined to be I db ⇔ a bcef g (cid:96) (cid:96) ⇔ - d (cid:96) - d (cid:96) n db ( a | c )( d | f )( b | e )( (cid:96) | a )( (cid:96) | b )( (cid:96) | c )( (cid:96) | (cid:96) )( (cid:96) | d )( (cid:96) | e )( (cid:96) | f ) . (2.6)Here, we have included a conventional factor in the numerator which ensures dual-conformal invariance, and we have used dual-momentum coordinates for which themassless external momenta are given by p a =: x a +1 − x a (with cyclic labeling under-stood), and with the association of (cid:96) i ⇔ x (cid:96) i . In terms of dual-momentum coordinates,the Lorentz invariants are defined by ( a | b ):= ( x a − x b ) .What should the normalization of this integral be so as to make the representa-tion of field theory amplitudes maximally transparent? One answer comes from thefact that the full-dimensional compact contour integral of the amplitude in planarsYM—which ‘encircles’ the seven poles corresponding to the propagators of (2.6)and then uses one of the fundamental cycles of the elliptic curve—is Yangian invari-ant [22]. Let us briefly review this story here, but in the general case—where themomenta flowing into the corners of the box are arbitrary. Let us start with the sub-leading singularity associated with a contour in fieldtheory with the topology of a double-box integral as shown in (2.6). That is, we’dlike to define the sub-leading singularity associated with db ± ( α ):= , (2.7) As described in [15], it suffices for us to consider only MHV amplitudes at the vertices—as gen-eral amplitudes can then be generated by multiplication by the corresponding on-shell amplitudes. – 8 –here α represents the single on-shell degree of freedom of the sub-leading singularity,and ± denotes which of the two branches of the seven-cut equations is chosen. Thiscould easily be represented in the Grassmannian according to [56, 57], but we preferto disambiguate its structure by writing it explicitly. To do this, we express db ± ( α )as simple-pole-enclosing contour integral over the co-dimension-six ‘kissing triangle’function, for which the non-vanishing term involves three R -invariants, - d α - d βα β R (cid:2) , (cid:98) a, (cid:98) c, (cid:98) d, (cid:98) f (cid:3) R (cid:2)(cid:98) a, B, b, C, c (cid:3) R (cid:2) (cid:98) d, E, e, F , f (cid:3) , (2.8)which is a special case of the general expression given in Appendix A and B of [15],written here in terms of R -invariants with (shifted) momentum super-twistor [58]arguments defined as (cid:98) a ( α ) := Z a + α Z A (cid:98) d ( β ) := Z d + β Z D (cid:98) c ( α ) := (cid:0) C c (cid:1) (cid:84) (cid:0)(cid:98) a B b (cid:1) (cid:98) f ( β ) := (cid:0) F f (cid:1) (cid:84) (cid:0) (cid:98) d E e (cid:1) . (2.9)To compute the double-box sub-leading singularity, we consider a ‘residue’ contourencircling the pole at (cid:104) (cid:98) a (cid:98) c (cid:98) d (cid:98) f (cid:105) = 0, which is a quadratic condition involving both α and β . Without any loss of generality, we are free to solve this condition for β ; the(two) branches of solutions to (cid:104) (cid:98) a (cid:98) c (cid:98) d (cid:98) f (cid:105) = 0 are then given by β ∗± := (cid:104) (cid:98) a (cid:98) c d (cid:0) D E e (cid:1) (cid:84) (cid:0)
F f (cid:1) (cid:105) + (cid:104) (cid:98) a (cid:98) c D (cid:0) d E e (cid:1) (cid:84) (cid:0) F f (cid:1) (cid:105) ± y ( α ) /c y (cid:104) (cid:98) a (cid:98) c (cid:0) D E e (cid:1) (cid:84) (cid:0)
F f (cid:1) D (cid:105) , (2.10)where y ( α ) is a quartic polynomial defined as1 c y y ( α ):= (cid:16) (cid:104) (cid:98) a (cid:98) c d (cid:0) D E e (cid:1) (cid:84) (cid:0)
F f (cid:1) (cid:105) + (cid:104) (cid:98) a (cid:98) c D (cid:0) d E e (cid:1) (cid:84) (cid:0) F f (cid:1) (cid:105) (cid:17) − (cid:104) (cid:98) a (cid:98) c D (cid:0) d E e (cid:1) (cid:84) (cid:0) F f (cid:1) (cid:105)(cid:104) (cid:98) a (cid:98) c d (cid:0) D E e (cid:1) (cid:84) (cid:0)
F f (cid:1) (cid:105) , (2.11)and the factor of 1 /c y is included to render the quartic y ( α ) monic by construction .The residue of the kissing-triangle hexa-cut function at β (cid:55)→ β ∗± follows by notingthat (cid:73) (cid:104) (cid:98) a (cid:98) c (cid:98) d (cid:98) f (cid:105) =0 (cid:32) - d β (cid:104) (cid:98) a (cid:98) c (cid:98) d (cid:98) f (cid:105) (cid:33) = ± ic y y ( α ) . (2.12)Notice that the ± sign on the left hand side reflects the choice of the branch of thecut equations, and y ( α ) is a square root of the quartic (2.11).Since there are two different solutions to (cid:104) (cid:98) a (cid:98) c (cid:98) d (cid:98) f (cid:105) = 0, it makes sense to define‘the’ double-box sub-leading singularity db ( α ) as the difference of the kissing-triangleon the two contours. We therefore define (cid:104) db + ( α ) − db − ( α ) (cid:105) =: db ( α ) =: - d αy ( α ) (cid:99) db ( α ) , (2.13)– 9 –here in the second equality we have defined a useful ‘hatted’ double-box functionwhere the inverse of the square root of the quartic has been factored out.Let us now return to (2.7). The double-box on-shell function has various fac-torization channels corresponding to the six amplitudes appearing in its definition.Each of these channels corresponds to a simple pole of db ( α ) at, say, α = a i ∈ C onwhich we may define an additional contour about a d log pole. According to prescrip-tive unitarity, each such residue is topologically equivalent to a ‘pentabox’ leadingsingularity, which we denote by pb i . We can, therefore, expand our next-to-leadingsingularity in a basis of those factorizations into pentaboxes according to db ( α ) =: - d αy ( α ) db + - d αy ( α ) (cid:88) i y ( a i )( α − a i ) pb i , (2.14)where the α -independent coefficient of the basis element without any extra simplepoles, denoted by db , is defined as: db := (cid:99) db ( α ) − (cid:88) i y ( a i )( α − a i ) pb i . (2.15)While it is not manifest in (2.15), the coefficient db is, by construction, independentof α . This trivial—yet important—point follows directly from Liouville’s theoremof elementary complex analysis: as we have removed every pole, including the poleat α = ∞ , the ‘function’ db is entire on the Riemann sphere, that is, it must be a constant . At the risk of being overly illustrative, let us emphasize that this impliesthat we may equivalently write db := (cid:99) db ( α ∗ ) − (cid:88) i y ( a i )( α ∗ − a i ) pb i , (2.16)where α ∗ is an arbitrarily chosen point.The recent work [22] defines an ‘elliptic’ leading singularity by integrating thedouble-box seven-cut differential form db ( α ) over either the a or b ‘cycle’ of theelliptic curve defined by y ( α ), e a,b := (cid:73) Ω a,b db ( α ) = (cid:73) Ω a,b - d αy ( α ) (cid:99) db ( α ) = ± (cid:73) Ω a,b db ± ( α ) . (2.17)To perform the integral over the a -cycle, (or any cycle for that matter) it is quiteuseful to factorize the monic quartic polynomial defined in (2.11) in terms of itsroots, y ( α ) =:( α − r )( α − r )( α − r )( α − r ) , (2.18)where, for positive kinematics [59], the roots form two complex conjugate pairs { r , r } and { r , r } , and are ordered so that Re ( r ) > Re ( r ) and Im ( r , ) > a, b -cycle integrals (2.17) can be writtenin terms of elliptic integrals involving the dual-conformal invariant cross-ratio ϕ := ( r − r )( r − r )( r − r )( r − r ) =: r r r r , (2.19)which, in our conventions, is always within the interval [0 ,
1] for positive kinematics.Choosing the branch cuts to connect each complex conjugate pair of roots, we nowdefine the a cycle contour, Ω a , to enclose the cut between r , . To compute the full a -cycle integral we use the two formulae (cid:73) Ω a - d αy ( α ) = 2 iπ √ r r K [ ϕ ] , (2.20)and (cid:73) Ω a - d α y ( a i )( α − a i ) y ( α ) = 2 iπ √ r r y ( a i )( r − a i ) (cid:32) K [ ϕ ]+ r ( r − a i ) Π (cid:20) ( r − a i ) r ( r − a i ) r , ϕ (cid:21)(cid:33) , (2.21)where our definitions of the complete elliptic integrals of the first and third kinds, K [ ϕ ] and Π[ q, ϕ ], respectively, are in agreement with Mathematica ’s. We haveprovided a different—and perhaps more easily generalizable—representation of theseelliptic period integrals in terms of Lauricella functions in appendix A. The attentivereader will notice that these two integral formulae differ from the results quoted in[22]; this discrepancy is a consequence of our use of - d = d/ (2 π ) throughout thiswork. Following the discussion in [22], observe that in the definition (2.14) of thedouble-box one-form, the coefficient of the pentabox pb i is the same as in (2.16), butfor the appearance of α ∗ in place of α . Upon performing the integration, if we choose α → r the terms involving K [ ϕ ] pb i thus cancel, and we are left with e a = 2 iπ √ r r (cid:32) K [ ϕ ] (cid:99) db ( α ∗ → r ) + (cid:88) i y ( a i ) r ( r − a i )( r − a i ) Π (cid:20) ( r − a i ) r ( r − a i ) r , ϕ (cid:21) pb i (cid:33) . (2.22)If instead we consider the b -cycle, say the one encircling a branch cut whichconnect two roots with different real part (e.g. r and r ), we have e b = 2 iπ √ r r (cid:32) K [1 − ϕ ] (cid:99) db ( α ∗ → r ) + (cid:88) i y ( a i ) r ( r − a i )( r − a i ) Π (cid:20) ( r − a i ) r ( r − a i ) r , − ϕ (cid:21) pb i (cid:33) , (2.23)which is the same with the first expression after the exchange r ↔ r . As discussedat greater length in [22], both expressions (2.22) and (2.23) are non-trivially Yangianinvariant, as may be verified by direct computation using, for example, the level onegenerators written in momentum super-twistor space.– 11 – New Prescriptive Representations: Two Approaches
We shall now argue that the results of the previous section suggest two naturalprescriptions for the normalization of the double-box integrand which avoid entirelythe arbitrary choices of the original prescriptive unitarity program.
Homological
Diagonalization—with Respect to
Contours
One natural choice for the normalization of the double-box integrand n db wouldbe analogous to the choice made in the introduction for scalar ϕ theory. That is,we may choose to normalize the integrand so that it integrates to 1 on the contourassociated with, say, the a -cycle of the elliptic curve; since the loop-momentum-dependent part of the integrand is proportional to 1 /y ( α ), this fixes its normalization,using (2.20), to be n db := π (cid:112) ( r − r )( r − r ) c y K [ ϕ ] . (3.1)This choice of normalization ensures that we match the elliptic leading singularityin field theory, e a , manifestly with the double-box integrand directly (cid:73) Ω a (cid:73) {| ( (cid:96) i | a ) | = (cid:15) }| ( (cid:96) | (cid:96) ) | = (cid:15) A = e a (cid:73) Ω a (cid:73) {| ( (cid:96) i | a ) | = (cid:15) }| ( (cid:96) | (cid:96) ) | = (cid:15) I db = e a . (3.2)Notice that this representation, however, does not match the b -cycle leadingsingularity of the amplitude manifestly at all! Although we have normalized thedouble-boxes so that the a -cycle contours of field theory are manifest, the result onthe b -cycle is, rather, e a (cid:73) Ω b (cid:73) {| ( (cid:96) i | a ) | = (cid:15) }| ( (cid:96) | (cid:96) ) | = (cid:15) I db = − i e a K [1 − ϕ ] K [ ϕ ] (3.3)= 2 π √ r r (cid:32) K [1 − ϕ ] (cid:99) db ( α ∗ → r )+ (cid:88) i y ( a i ) r ( r − a i )( r − a i ) K [1 − ϕ ] K [ ϕ ] Π (cid:20) ( r − a i ) r ( r − a i ) r , ϕ (cid:21) pb i (cid:33) which is not at all equal to (cid:73) Ω a (cid:73) {| ( (cid:96) i | a ) | = (cid:15) }| ( (cid:96) | (cid:96) ) | = (cid:15) A = e b (3.4)which was given in (2.23). This is not in fact a problem: as we will see, the pentaboxes(and double-pentagon integrands) do have support on the b -cycle integrals (even aftertheir contact terms have been fixed), and exactly give the necessary contributionsto reproduce the correct b -cycle leading singularity in (2.23). (With hindsight, this‘magic’ can be seen to follow directly from the completeness of the integrand basis.)Let us now discuss the implications of prescriptive unitarity for the contact-termrules of the pentabox and double-pentagon integrands. As always, prescriptivity– 12 –equires that our integrands be diagonal in a choice of contours; therefore, in the ho-mological scheme, the contact terms are determined by the requirement that all otherintegrands in our basis vanish identically on all elliptic Ω a -cycle contours associatedwith double-box sub-topologies.To see how this works in practice, consider a pentabox integrand which containsa double-box contact-term: a bcef g (cid:96) (cid:96) ⊂ a ρbcef g n iρ ( (cid:96) ) (cid:96) ⇔ - d (cid:96) - d (cid:96) (cid:2) n i ( (cid:96) |Y i )+ n iρ ( (cid:96) | ρ )+ . . . (cid:3) ( (cid:96) | a )( (cid:96) | b )( (cid:96) | c )( (cid:96) | ρ )( (cid:96) | (cid:96) )( (cid:96) | e )( (cid:96) | f )( (cid:96) | g )(3.5)where n iρ is the coefficient of the term in the numerator proportional to ( (cid:96) | ρ ).Schematically, the full contribution of the pentabox on the a -cycle contour can bewritten as (cid:73) Ω a (cid:73) {| ( (cid:96) i | a ) | = (cid:15) }| ( (cid:96) | (cid:96) ) | = (cid:15) a ρbcef g n iρ ( (cid:96) ) (cid:96) = (cid:73) Ω a - d α (cid:20) y ( a ρ )( α − a ρ ) y ( α ) + n iρ y ( α ) (cid:21) , (3.6)where a ρ corresponds to the pole where ( (cid:96) | ρ ) = 0. Notice that the leading term followsdirectly from the fact that the integral is normalized to have unit leading singularityon some pentabox contour. Diagonalization of the basis according to homology—leading singularities—fixes this contact-term coefficient, n iρ , of the pentabox by thecriterion that (3.6) vanishes. Namely, we must choose n iρ := − y ( a ρ )( r − a ρ ) (cid:34) r ( r − a ρ ) Π (cid:20) ( r − a i ) r ( r − a i ) r , ϕ (cid:21) /K [ ϕ ] (cid:35) . (3.7)One important thing to note is that the pentabox integrands with these contact-terms chosen do not vanish on the b -cycle contours. This is a good thing!—as thenormalization we have chosen for the double-boxes makes the correctness of the am-plitude integrand on the b -cycle extremely non-manifest in this representation. It isnot hard to see, however, that when these contact terms are used, they generate onthe b -cycle exactly the terms needed to cancel the ‘wrong’ elliptic-Π terms involvingthe pentabox leading singularities that arise from the double-box integrands in (3.3).Furthermore, the contact terms of the pentabox integrands also contribute the cor-rect pieces involving the pentabox leading singularities appearing in e b (2.23) oncethe corresponding contributions from the double-pentagons are included (so as tomatch all pentabox leading singularities).This diagonalization strategy has several obvious advantages. For one thing, itis morally the direct realization of ‘prescriptive unitarity’ according to a choice of– 13 –eading singularities. Moreover, as emphasized in the introduction, it should havethe property that all integrands defined in this way are pure , and all coefficients areYangian-invariant.Nevertheless, there are several reasons to be dissatisfied with this basis of in-tegrands. For example, it deeply obscures the fact that the integrand is a rational differential form in loop momenta. Choosing basis integrands whose normalizationdepends on the roots of quartics makes this representation fairly unwieldy in prac-tice (at least for most computer algebra packages). Therefore, we are motivated toconsider a slightly different strategy, with huge advantages in terms of (algebraic)complexity, but which abandons the desire for a pure integrand basis—and requiresthe use of non-Yangian-invariant coefficients. Cohomological
Diagonalization—with Respect to
Forms
The attentive reader may already have guessed an alternative strategy for match-ing amplitudes in sYM—namely, according to the various differential forms thatappear in the double-box sub-leading singularity db ( α ) in (2.14): db ( α ) =: - d αy ( α ) db + - d αy ( α ) (cid:88) i y ( a i )( α − a i ) pb i . (3.8)Considering the fact that the co-dimension seven contour of the scalar double-boxintegrand (2.6) which encircles its seven propagators results in (cid:73) {| ( (cid:96) i | a ) | = (cid:15) }| ( (cid:96) | (cid:96) ) | = (cid:15) - d (cid:96) - d (cid:96) n db ( a | c )( d | f )( b | e )( (cid:96) | a )( (cid:96) | b )( (cid:96) | c )( (cid:96) | (cid:96) )( (cid:96) | d )( (cid:96) | e )( (cid:96) | f ) = n db ( − i c y ) - d αy ( α ) , (3.9)it would be natural to choose n db to be + i , and to match its coefficient in therepresentation of field theory amplitudes to be c y db . (With this factor of c y —implicit in the definition (3.8)—this on-shell function is little-group neutral (if notYangian-invariant).) Moreover, it is easy to see that this combination of integrandand coefficient automatically matches the leading term in e b in (2.23). The only termsmissing from both leading singularities are those involving the pentabox leadingsingularities pb i .These not-yet-matched pieces of e a,b can easily be seen to arise from the pentaboxand double-pentagon contributions. Consider again how the contact terms appearin the pentabox integrands’ contributions to the a -cycle, say: (cid:73) Ω a (cid:73) {| ( (cid:96) i | a ) | = (cid:15) }| ( (cid:96) | (cid:96) ) | = (cid:15) a ρbcef g n iρ ( (cid:96) ) (cid:96) = (cid:73) Ω a - d α (cid:20) y ( a ρ )( α − a ρ ) y ( α ) + n iρ y ( α ) (cid:21) . (3.10)Taking the contact-term numerator n iρ (cid:55)→ K and Π)in (2.21) is exactly that which appears in the pentabox integrands already—and thecoefficients of these integrands are precisely the pb i needed to reproduce the ‘missing’pieces in the elliptic leading singularities e a,b .This definition of the double-box and the corresponding rule for the contactterms of the pentabox and double-pentagon integrands results in an extremely sim-ple prescription for the integrand. Moreover, it is morally equivalent to a choiceof diagonalization with respect to the various (local) differential-forms in loop mo-mentum space. As such, we call such a prescription a cohomological choice for ourbasis.Despite the obvious advantages, we have checked that the coefficient of thedouble-box normalized in this way (namely, c y db ) is not Yangian-invariant . More-over, the double-box integrands are not pure. We strongly suspect that the pentaboxand double-pentagon integrands (that contain elliptic contact-term components) aresimilarly non-pure. Thus, despite the algebraic and conceptual simplicity of the co-homological approach just described, we suspect that the homological prescriptiverepresentation will ultimately prove the superior one for integration. As already mentioned in section 2, the pentabox integrands lack the requisitenumber of degrees of freedom to match all pentabox cuts of field theory term-by-term. However, the expressions for the elliptic leading singularities e a,b involve themseparately—and arising at poles in different locations in the α -plane. Thus, ourdescription above regarding the pentaboxes’ roles in these leading singularities doesnot alone ensure that we have matched everything in field theory. The missing in-gredient, of course, are the kissing-box leading singularities times double-pentagons.These terms when combined with those of the pentaboxes ensure that all the pentaboxleading singularities do get matched correctly, and individually. Thus, we have beensomewhat schematic in our analysis above, relying on the fact that these parts of theamplitude integrands are guaranteed to work correctly once taken in combination.While ensured to work, we have checked this completely in the case of the 10-particle N MHV amplitude. Specifically, we have checked that both diagonalizationprocedures described above—the homological and the cohomological—result in inte-grand representations that exactly match the results of BCFW recursion, say. Thus,we are confident this procedure is free of any over-looked subtleties.
The reader may have considered our preference for normalizing integrands withrespect to the a -cycle Ω a rather arbitrary. It was not entirely so. As discussedin [22], the a -cycle integral of the elliptic curve smoothly degenerates to 1 in allrelevant cases (or kinematic limits). Thus, our analysis above, if applied to the caseof double-box integrands that do support polylogarithmic contour integrals, reduces– 15 –aturally to ordinary polylogarithmic ones. (Recall that a unit-residue integrand d L (cid:126)(cid:96) is equivalent to a unit- contour integrand for - d L (cid:126)(cid:96) .) Thus, we can apply the results ofthis work to truly general double-boxes, and thereby construct pure-integrand basesthat happen to be ‘ d log’ whenever the double-boxes’ elliptic curves turn out to bedegenerate. Beyond two loops (and for non-planar theories at two loops), non-polylogarithmicstructures beyond elliptic integrals abound [39, 44–46, 60]. Examples of scalar inte-grals with such structures include the three-loop traintrack and wheel integrals, , , (4.1)the maximal cuts (sub-leading singularities encircling all propagators) of which areknown to involve Calabi-Yau 2- and 3-folds, respectively.To see how the corresponding non-polylogarithmic leading singularities can beincorporated analogously to what we have described for the elliptic case, let us brieflyoutline the structure expected for the traintrack contribution.On the maximal cut surface encircling all 10 propagators of the traintrack, thesub -leading singularity should take the form: tr ( α, β ) =: - d α - d βy ( α, β ) (cid:98) tr ( α, β ) , (4.2)where y ( α, β ) is an irreducible quartic in both variables { α, β } simultaneously [44].As with the double-box sub-leading singularity (2.7), the traintrack in sYM will havemultiple co-dimension one, simple poles around which there are elliptic sub-leadingsingularities. We may express this by decomposing (4.2) according to (cid:98) tr ( α, β ) =: (cid:98) tr + (cid:88) a i y ( a i , β )( α − a i ) y ( a i , β ) el a i ( β ) + (cid:88) b i y ( α, b i )( β − b i ) y ( α, b i ) el b i ( α ) , (4.3)where the terms in the sum el a i and el b i are elliptic sub-leading singularities arisingfrom single-pole factorizations of the original traintrack. These in turn can be furtherdecomposed analogously to (2.14), resulting in an expression of the form (cid:98) tr ( α, β ) =: (cid:98) tr + (cid:88) a i y ( a i , β )( α − a i ) y ( a i , β ) (cid:98) el a i + (cid:88) b i y ( α, b i )( β − b i ) y ( α, b i ) (cid:98) el b i + (cid:88) a i ,b j y ( a i , b j )( α − a i )( β − b j ) pl a i ,b j , (4.4)where pl a i ,b j are the ‘penta-ladder’ polylogarithmic leading singularities associatedwith simultaneous factorizations in two different loops.– 16 –he number and detailed form of terms appearing in this decomposition willdepend on the number of factorization channels of the traintrack (which depends onmultiplicity), but the basic structure is clear: (4.4) is nothing but a decompositionof the sub -leading singularity (4.2) into a basis of differential forms involving oneor two simple poles, respectively—with superfunction coefficients. (For the scalartraintrack contribution to the three-loop 12-particle amplitude in scalar ϕ theory,only the leading term (cid:98) tr is required—which happens to be ( − ϕ tree-amplitudes.)The generalization of this analysis to the three-loop wheel integral—which in-volves a Calabi-Yau three-fold surface—is relatively straightforward, resulting in adecomposition of the sub -leading singularity into a top-level, irreducible volume-form times some ‘CY ’ leading-singularity, three separate sums of K K
3’ leading singularities; three, double-nested sumsof elliptic integrals with two simple poles times ‘elliptic’ leading singularities; and,finally, a triple-sum of terms with simple poles times polylogarithmic leading singu-larities.It is worth mentioning that, unlike the three-loop traintrack which is known to have support in sYM (as argued in [44]), the three-loop wheel is not any singlecomponent of an amplitude in planar sYM; as such, it is possible that the CY leadingsingularity vanishes. Thus, this makes its evaluation an important open challenge—left for future work. (While we are unaware of closed analytic formulae for the periodintegrals that would be required for such a check—analogous to those in (2.20) and(2.21) for the elliptic case—we are relatively optimistic that numerical integrationwill work for these low-dimensional cases.)The implications of these higher-dimensional Calabi-Yau leading singularities forprescriptive unitarity should be clear. In particular, we suspect that if the three-loopbasis of planar integrands outlined in [16] were diagonalized homologically , the re-sult would be a complete representation of amplitudes involving term-wise ‘pure’integrals times Yangian-invariants; and the cohomological diagonalization of this ba-sis into separate forms should be extremely straightforward to implement from thedecompositions as in (4.4). – 17 – Conclusions and Future Directions
In this work we have made use of the new, broadened definition of leading singu-larities (beyond the polylogarithmic case) to derive two new prescriptive representa-tions of two-loop scattering amplitude integrands in planar sYM. This analysis wasillustrative of a more general strategy, with applications well beyond the planar limitand to theories with less or no supersymmetry. In many ways, our results directlyreflect the primary goals of prescriptive unitarity: constructing loop-integrand basesthat are diagonal in a spanning set of contours . For scattering amplitudes free ofnon-polylogarithmic structures, this strategy directly reproduces the d log differentialforms of the traditional approach; but as we have seen, the generalization beyondpolylogarithms is extremely natural. We have also motivated a different strategy:how to construct a prescriptive integrand basis diagonal with respect to cohomol-ogy ; the resulting basis may not be pure and the coefficients required may not beYangian-invariant, but the ultimate representation of loop integrands is dramaticallysimpler from an algebraic point of view.We have described how this story illustrates a broader one—with applicationswell beyond the case of elliptic leading singularities in planar theories at two loops.It would be extremely interesting to apply these lessons more widely to generate(purportedly) ‘pure’ master integrals for applications beyond the planar limit, andto theories without supersymmetry. Although we have not proven the ‘purity’ ofthis broader class of integrals, and although the strategies and techniques requiredto efficiently exploit the differential structure of pure integrals are still being devel-oped (even in the elliptic case—but see e.g. [61–63]), we strongly suspect that theprescriptive bases we have constructed will prove computationally valuable as masterintegrals for diverse applications. For example, constructing such bases for massivetheories now appears straightforward, and undoubtedly has more immediate appli-cations for real physical applications (see e.g. [62–66]). But we leave such analysesto future work. Acknowledgements
The authors gratefully acknowledge fruitful contributions from Marcus Spradlin dur-ing the early stages of this work, and for fruitful conversations with Nima Arkani-Hamed, Song He, Enrico Herrmann, Jaroslav Trnka, and Cristian Vergu. This workwas performed in part at the Aspen Center for Physics, which is supported by Na-tional Science Foundation grant PHY-1607611, and the Harvard Center of Math-ematical Sciences and Applications. This project has been supported by an ERCStarting Grant (No. 757978), a grant from the Villum Fonden (No. 15369), by agrant from the Simons Foundation (341344, LA) (JLB).– 18 –
Hypergeometric Representations of the Elliptic Integrals
Although the representations for the complete elliptic integrals given in (2.20)and (2.21) above are fairly standard ones (with relatively efficient implementationsin
Mathematica , for example), it is worthwhile to outline an alternative form forthese integrals—which we hope has some promise to generalize beyond the ellipticcase. In this appendix, we outline how these elliptic periods can be expressed interms of Lauricella hypergeometric functions. (We refer the reader to e.g. [67–69] forsome discussions on these functions in the context of Feynman integrals.)Consider first the elliptic period integral given in (2.20): I (1) a := (cid:73) Ω a - d αy ( α ) = 2 iπ √ r r K [ ϕ ] , (A.1)where y ( α ):= ( α − r )( α − r )( α − r )( α − r ), r ij := ( r i − r j ), and ϕ := ( r − r )( r − r )( r − r )( r − r ) =: r r r r . (A.2)(We refer the reader to section 2.4 for our conventions regarding the ordering of theroots r i .) This integral can be re-cast slightly by the change of variables x := α − r r − r , (A.3)upon which (A.1) becomes I (1) a := (cid:73) Ω a - d αy ( α ) = iπ √ r r (cid:90) dx (cid:112) x (1 − x )(1 − (21;31) x )(1 − (21;41) x ) , (A.4)where we have introduced the shorthand( ab ; cd ):= r a − r b r c − r d . (A.5)From the definition of the Lauricella hypergeometric function F ( n ) (cid:16) a, b , . . . , b n , c (cid:12)(cid:12)(cid:12) x , . . . , x n (cid:17) :=Γ( c )Γ( a )Γ( c − a ) (cid:90) dx x a − (1 − x ) c − a − (1 − x x ) − b . . . (1 − x n x ) − b n (A.6)we have the identification I a = i Γ (cid:0) (cid:1) π √ r r F (2) (cid:18) , , , (cid:12)(cid:12)(cid:12) (21;31) , (21;41) (cid:19) . (A.7)Applying the same transformation to the second fundamental period integral(2.21), I (2) a := (cid:73) Ω a - d α y ( p )( α − p ) y ( α ) , (A.8)results in a representation I (2) a = − i y ( p )Γ (cid:0) (cid:1) π ( p − r ) √ r r F (3) (cid:18) , , , , (cid:12)(cid:12)(cid:12) (21;31) , (21;41) , (21; p (cid:19) . (A.9)– 19 – eferences [1] Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, “One-Loop n -Point GaugeTheory Amplitudes, Unitarity and Collinear Limits,” Nucl. Phys.
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