Locally Finite Observables in sYM
LLocal ly -Finite Observables in sYM
Jacob L. Bourjaily, ∗ , † Cameron Langer, ∗ Kokkimidis Patatoukos ∗∗ Institute for Gravitation and the Cosmos, Department of Physics,Pennsylvania State University, University Park, PA 16802, USA † 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] A locally-finite observable is one for which there is no region of divergence anywherein the space of real loop momenta; it can therefore be computed (in principle) withoutregularization. In this work, we prove that all two-loop ratio functions in planar,maximally supersymmetric Yang-Mills theory are locally-finite. a r X i v : . [ h e p - t h ] F e b ontents Actual (‘Local’) Finiteness vs. ‘
Spurious ’ Finiteness 3 × MHV Contribution . . . . . . . . . 123.3 Cancellation of All Local Divergences in the Ratio Function . . . . . . 133.3.1 Soft-Collinear log ( δ )-Divergent Regions . . . . . . . . . . . . . 133.3.2 Soft-Collinear log( δ )-Divergent Regions . . . . . . . . . . . . . . 143.3.3 non-Soft-but-Collinear log( δ )-Divergent Regions . . . . . . . . . 16 Many of the recent advances in our understanding of and ability to compute scat-tering amplitudes in quantum field theory have relied on the simple—yet surprisinglypowerful—idea of separating the two problems of constructing the loop integrand fromcarrying out loop integration. Indeed, in the case of the planar limit of maximallysupersymmetric ( N = 4) Yang-Mills theory (sYM), the loop integrand is a perfectlywell-defined rational function which can be constructed from knowledge of its residues.More broadly, the systematic development of computational tools to determine the all-loop integrand has been a continuous source of insight into the unanticipated simplicityof an increasingly large class of field theories. In particular, investigating the structureof the loop integrand in planar sYM led to the discoveries of dual-conformal symme-try [1–3], tree- and loop-level recursion relations [4–7], connections to Grassmanniangeometry [8–11], and the development of generalized [12–22] and prescriptive [23–30]unitarity.Despite the significant advances made in representing loop integrands, the problemof loop integration remains an exceedingly difficult one. In particular, the issues ofregularization and renormalization seem unavoidable intermediate requirements of thecomputation of most observable quantities. Particularly relevant to the case of planarsYM is the fact that the scattering amplitudes themselves (as for any theory withmassless particles) are infrared (IR) divergent and require regularization. While it is not– 1 – priori clear how much of the simplicity of loop integrands remains post-integration,there exists an impressive body of evidence that certain infrared-safe quantities doin fact preserve many integrand-level properties (such as dual-conformal invariance).These include the remainder function [20, 31–34], defined as the ratio of the maximally-helicity-violating (MHV) amplitude and the Bern-Dixon-Smirnov (BDS) ansatz, as wellas the ratio function [3], which is the ratio of the N k MHV amplitude to the MHVamplitude.In practice, computing finite observables such as the ratio function has requiredthe explicit cancellation of infrared divergences among different loop orders. However,the difficulties of direct integration of multi-loop integrands has thus far prevented thisapproach from seriously competing with bootstrap methods [35–46], for example, whichsidestep both the loop integrand and integration issues entirely and have been used todetermine the remainder and ratio functions to impressively high loop orders. Despitethese remarkable achievements, these methods are limited in their applicability; inparticular, they have little to say regarding transcendental functions which are known[47–50] to be required at two loops and beyond (for large enough multiplicities).Although the ratio function is defined at a given loop order as a combination ofdivergent amplitudes, one may ask if it is possible to do better at the integrand-level.Namely, can the ratio function be represented in terms of individually, infrared-finiteintegrals, eliminating the need for any regularization? At one loop, at least, the answeris a positive one: the basis described in [23] and the work of [24] demonstrate that allinfrared finite observables in sYM can be represented directly in terms of individually,locally finite integrals. In this work, we provide a non-constructive affirmative answer tothis question at two loops by demonstrating the cancellation of all infrared divergentregions at the integrand-level . Along the way, we use the correspondence betweencollinear and soft regions of loop momentum space and infrared divergences to introducethe notion of the ‘local finiteness’ of a two-loop integral, which can be diagnosed bysimple tests at the integrand-level. This classification of infrared behavior should proveuseful in the construction of manifestly finite integrals, which have already proven tobe invaluable in a variety of contexts [8, 23, 25, 28, 29, 51–57].
In this work we examine the infrared behavior of the two-loop ratio function, at theintegrand-level in planar, maximally supersymmetric ( N =4) Yang-Mills theory (sYM)and show that it (the ratio function) is locally finite. Starting from the local represen-tations of all planar N =4 two-loop integrands as introduced in [25], we demonstratehow the divergences associated with particular collinear and soft-collinear regions ofloop momentum space cancel at the integrand-level between the N k MHV × MHV and– 2 –HV × MHV terms in the ratio function. The two kinds of cancellations which arisecan be understood either as grouping of terms into BCFW recursions of the sets, or asresidue theorems on particular sub-leading singularities.This work is organized as follows. In section 2 we introduce the notion of the ‘local’finiteness of a loop integrand, and contrast this with the weaker notion of being merely (or ‘spuriously’) finite in some particular regularization scheme. We call integrands‘spuriously’ finite if their finiteness is regularization-scheme-dependent. In section 2 weclassify all possible locally-divergent regions of loop momentum space responsible forinfrared divergences at two loops, and enumerate every potentially-divergent merge-generated integrand relevant for the ratio function in planar sYM. In section 3, afterreviewing the less-divergent representation of the two-loop integrand in [25] we use theclassification of divergences to demonstrate the cancellation of every local divergencein the two-loop ratio function, thus demonstrating its local finiteness. Finally, wesummarize the main implications of this work and provide some potentially interestingavenues for future research. Actual (‘Local’) Finiteness vs. ‘
Spurious ’ Finiteness
The infrared singularities of loop amplitudes involving massless particles can bediagnosed by examining the behavior of integrands in the collinear region: where aloop momentum becomes proportional to a massless external particle’s momentum, (cid:96) ∝ p a , with p a =0. This co-dimension three configuration corresponds to a compositeresidue and is associated with the following on-shell function: . (2.1)The collinear region has a simple description in the dual momentum co-ordinates x a ,which trivialize momentum conservation through the definition p a =: x a +1 − x a . Theinverse propagators are given in dual co-ordinates as( a | b ) := ( x a − x b ) = ( p a + . . . + p b − ) , ( (cid:96) i | a ) := ( x (cid:96) i − x a ) , ( (cid:96) i | (cid:96) j ) := ( x (cid:96) i − x (cid:96) j ) , (2.2)where the x (cid:96) i are the dual points associated to the loop momenta (cid:96) i . In these variables,the collinear region of loop momentum space (at one loop) associated with the on-shell function (2.1) can be approached as follows. First, one cuts either ( (cid:96) | a ) = 0 or( (cid:96) | a +1) = 0, upon which the other propagator factorizes into two pieces; cutting both ofthese factors (at co-dimension two) corresponds to the collinear residue. To be slightlymore explicit, if we identify the loop momentum flowing into the leg a as (cid:96) , then on thesupport of (cid:96) = 0 we can write (cid:96) =: λ (cid:96) (cid:101) λ (cid:96) in terms of spinor helicity variables; in terms of– 3 –hese, the second inverse propagator becomes ( (cid:96) + a ) → (cid:104) (cid:96) a (cid:105) [ (cid:96) a ]. The collinear regionis where both of these factors vanish, which sets (cid:96) ∝ p a . In this work, it will be useful toexpress this condition in terms of dual momenta, where the aforementioned collinearconfiguration associated with leg a can be parametrized as (cid:96) = α x a + (1 − α ) x a +1 , (2.3)where α is the single remaining degree of freedom in the loop momentum on thistriple-cut. Any one-loop integral whose integrand has support on such a co-dimensionthree residue inescapably requires regularization to evaluate, and will generically yield at least /(cid:15) , log( m ) [58] or log( δ ) divergences in the dimensional-regularization (seee.g. [59–61]), mass-(or ‘Higgs’-)regularization (see e.g. [58]) and conformal-regularization[24] schemes, respectively.The so-called ‘soft-collinear’ regions correspond to co-dimension four residues wherethe loop momentum is collinear to consecutive massless momenta (or, equivalently, thedual loop momentum is coincident with a dual momentum coordinate), and correspondsto the following on-shell function: . (2.4)In the parametrization of (2.3), the soft-collinear singularities where (cid:96) = x a or (cid:96) = x a +1 correspond to poles at the locations α → α →
0, respectively. An integrandwith support on a soft-collinear residue will generically yield 1 /(cid:15) , log ( m ) or log ( δ )divergences upon integration in the dimensional-regularization, mass-regularization anddual-conformal-regularization schemes, respectively. To be clear, throughout this work,the statement that an integral is ‘log k ( δ )-divergent’ should be understood to refer tothe highest degree of divergence present in the integrated result. Of course, genericallylog k ( δ )-divergent integrals will also have sub-leading divergences.Because the one-loop collinear region is parity-invariant, this yields a simple cri-terion to test the convergence of a one-loop integral: multiply the integrand by twoadjacent propagators, and check if the result vanishes when evaluated in the collinearregion (2.3). In accordance with [24], we define a truly -convergent or ‘locally finite’ one-loop integral to be one whose integrand vanishes in every collinear region. Constructinglocally finite one-loop integrands is especially simple using the ‘chiral’ numerators de-scribed at length in e.g. [24, 28]. In fact, the ‘chiral octagons’ introduced in [23] canbe used to make the infrared structure of any one-loop object completely manifest,– 4 –s they naturally separate into manifestly locally finite and divergent terms—and no(non-vanishing) combination of divergent integrands is convergent.Our definition of local finiteness is to be contrasted with the weaker requirementof being merely (i.e. ‘spuriously’) finite in a particular regularization scheme. The exis-tence of spuriously-finite integrals at one-loop have been well-studied in the literature,where the existence of dual-conformal symmetry was used as a proxy for local (ac-tual, scheme-independent) finiteness (see e.g. [52, 62, 63] for examples of combinationsof one-loop box integrals that are spuriously finite in dimensional regularization). Aspuriously finite combination of one loop integrals is easier to find for the conformalregulator; for example, the following sum of four box integrals: I spurious := − +
234 5 6 1 −
345 6 7 2 . (2.5)With the conformal regulator, this combination of divergent boxes happens to be finite: I δ spurious = 12 log (cid:18) (1 | | | | (cid:19) log (cid:18) (3 | | | | | | (cid:19) + O ( δ ) ; (2.6)The fact that this combination is not truly finite is clear from its dimensionally-regulated expression: I dim-regspurious = 1 (cid:15) (cid:20) log (cid:18) (1 | | | | | | | | | | (cid:19) + 12 log (cid:18) (1 | | | | | | (cid:19)(cid:21) + O ( (cid:15) ) . (2.7)This reflects the fact that this combination of boxes does not satisfy the local finitenesscriterion given above, as can be seen by noting that e.g. only the first box in (2.5) hassupport in the collinear region associated with leg 1. The observation that a particu-lar combination of integrals is finite in some regularization scheme is not particularlymeaningful.The extension to two loops is relatively straightforward: infrared divergences areassociated with regions of loop momentum space where one or both loop momenta arecollinear to massless external momenta, with the most divergent regions correspondingto additional soft-collinear singularities. A locally finite two-loop integrand must vanishin every collinear region of the form (cid:96) = α x a + (1 − α ) x a +1 , (cid:96) = β x b + (1 − β ) x b +1 . (2.8)We will use the shorthand that a loop momentum in the collinear region associated withleg a satisfies (cid:96) i ∈ coll( a ). The maximal degrees of divergence associated to particular– 5 –ollinear and soft-collinear residues follow almost trivially from the above one-loopdiscussion. When the two loops are sent to different collinear regions, i.e. a (cid:54) = b , non-zero residues associated to double-soft-collinear singularities lead to log ( δ ) or ‘one-loopsquare’ divergences in the integrated result. An example of an integrand (which is notrelevant for the representation of the ratio function we consider below) with a log ( δ )divergence associated to the double-soft-collinear singularity where (cid:96) = x , (cid:96) = x isthe scalar double-box
561 2 34 . (2.9)Indeed, it may be readily verified that this integral is log ( δ )-divergent in the conformal-regularization scheme [64]. If one of the corners of this topology were to be massive asin , (2.10)then the most divergent configuration would send just one loop soft-collinear, (cid:96) = x ,with (cid:96) taken in the collinear region associated with leg 1. This integral would thereforebe log ( δ )-divergent; adding another mass to one of the massless corners on the rightloop would reduce the maximal degree of divergence to log ( δ ), and so-on.Sub-leading log ( δ ) and log( δ ) divergences are also associated with a special ‘over-lapping’ collinear region where in (2.8) we set a = b —where both loops are collinearwith the same massless external leg. Clearly, such divergent regions can be accessiblefor non-planar integrals only. In fact, in the ‘almost finite’ representation of the two-loop ratio function of [25], which we review in more detail in section 3.1, it turns outthat such overlapping collinear regions directly correspond to the only remaining localdivergences.The overlapping region is qualitatively distinct from the generic case because, onthe support of the composite cut ( (cid:96) | a ) = ( (cid:96) | a +1) = ( (cid:96) | a ) = ( (cid:96) | a +1) = 0, the inter-nal propagator vanishes as well, as ( (cid:96) | (cid:96) ) ∼ ( a | a +1) = 0. Because of this, double- andsingle-soft-collinear singularities are associated with log ( δ ) and log( δ ) divergences, re-spectively. – 6 – .2 Classification of Local Divergences of ‘Merger’ Integrands The representation of the two-loop ratio function given in [25]—reviewed belownear (3.10)—is written in terms of ‘mergers’ of specific one-loop integrands involvingthe point at infinity, denoted ‘ X ’, in dual momentum space. The merge operation wasdefined in [25] for two X -dependent chiral boxes as I L ( (cid:96) | X ) ⊗ I R ( (cid:96) | X ) := (cid:20) I (cid:48) L ( (cid:96) ) ( N L | X )( (cid:96) | X ) (cid:21) ⊗ (cid:20) I (cid:48) R ( (cid:96) ) ( N R | X )( (cid:96) | X ) (cid:21) := I (cid:48) L ( (cid:96) ) ( N L | N R )( (cid:96) | (cid:96) ) I (cid:48) R ( (cid:96) ) , (2.11)where (implicit) symmetrization with respect to loop-momentum labels ensures themerge operation is itself symmetric. To organize the cancellation of infrared singular-ities in the two-loop ratio function, it suffices to identify the location of all possibledivergences in the merger of two (arbitrary) chiral boxes, I ia,b,c,d ( (cid:96) , X ) ⊗ I je,f,g,h ( (cid:96) , X ).By construction, every chiral box integrand is ‘one-loop finite’ in the sense of sec-tion 2.1. Thus, every merger of two such integrands will trivially vanish when the twoloops go to distinct collinear regions, e.g. (cid:96) ∈ coll( a ) and (cid:96) ∈ coll( b ), for a (cid:54) = b . Theremaining divergent regions are those when both loops go to the same collinear region, (cid:96) , (cid:96) ∈ coll( a ). Clearly, in order for an integrand to have support on this co-dimension-six residue, it must have (at least) one massless corner where leg a sits, for both loops.We shall find a graphical representation of the merger integrands, where the chiralityof each box can be labeled by coloring its three-point vertices either white (for MHV)or blue (for MHV), to be particularly useful for keeping track of potential sources ofdivergence. Recalling that the merger of two boxes of the same chirality vanishes, wemay write the potentially divergent mergers as all integrands of the form, a ⊗ a (2.12)where, throughout this work, we use a ‘dashed’ wedge to indicate an arbitrary legrange, that is, either massless or massive (but not empty), to include all necessarydegenerations. Strictly massive legs, on the other hand, will be drawn with a solidwedge, while strictly massless legs will be drawn with a single leg, as has been donealready in e.g. (2.5) and (2.10) above. Let us consider each possible leg distribution in(2.12) and locate all sources of infrared divergences.While the generic merger of two three-mass boxes which share a massless leg a doesnot arise in the ratio function, as a simple application of the above discussion we notethat it is indeed log( δ )-divergent, with non-vanishing support on the co-dimension sixresidue (cid:96) , (cid:96) ∈ coll( a ), i.e. – 7 –2.13)Of course, this integrand has support on the additional cuts where we localize (cid:96) , (cid:96) completely by cutting additional propagators. However, these do not correspond tosoft singularities, and as such do not lead to enhanced divergences in the integratedresult. In general, we refer to such co-dimension eight residues accessible from thedoubly-collinear region as collinear-but-not-soft . In fact, mergers involving three andtwo-mass-easy boxes—in which there are no consecutive massless legs—are also free ofany soft singularities, and fit into the same classification according to infrared structure.As such, we may write this class of mergers succinctly ascollinear-but-not-soft log( δ ): a ⊗ a. (2.14)The next class of mergers involves a three- and two-mass-hard box (as well asdegenerations). Any merger with (at least) one box with two consecutive masslesslegs allows access to the soft-collinear singularity in the corresponding loop variable.Single-soft-collinear singularities generate log( δ ) divergences upon integration, and allrelevant cases can be grouped as the following (including degenerations):soft-collinear log( δ ): a ⊗ a , a ⊗ a. (2.15)Finally, there are those mergers with consecutive massless legs in both boxes. Thesemergers have support on double-soft-collinear singularities, and lead to log ( δ ) diver-gences upon integration:double-soft-collinear log ( δ ): a ⊗ a. (2.16)In section 3.3 we will make use of this classification to organize contributions to andconfirm the cancellation of all local divergences in the two-loop ratio function of planarsYM. – 8 – Local-Finiteness of the Two-Loop Ratio Function in sYM
To examine the local finiteness of the two-loop ratio function in planar sYM, thelocal integrand-level representations for all one and two-loop amplitudes introduced in[25]—and implemented in the attached
Mathematica package ‘ two loop amplitudes ’(see also [24, 65, 66])—are a particularly good starting point as they make the infraredstructure of the amplitude building blocks entirely manifest. By tailoring a basis ofintegrands to match a minimal set of independent on-shell data sufficient to fix the fullamplitude integrand, the one- and two-loop amplitudes admit convenient separationsinto manifestly finite and divergent parts. At one loop, the chiral box expansion reads[24]: A ( k ) , n = =: A ( k ) , n, fin (cid:122) (cid:125)(cid:124) (cid:123)(cid:88) a,b,c,d (cid:16) f a,b,c,d I a,b,c,d + f a,b,c,d I a,b,c,d (cid:17) + =: A ( k ) , n, div (cid:122) (cid:125)(cid:124) (cid:123) A ( k ) , n I div , (3.1)where: A ( k ) ,Ln denotes the L -loop n -point N k MHV amplitude; f ia,b,c,d denote the familiar‘quadruple-cut’ on-shell functions associated with the two co-dimension four residuesputting four propagators on-shell,( (cid:96) | a ) = ( (cid:96) | b ) = ( (cid:96) | c ) = ( (cid:96) | d ) = 0 ⇔ f ia,b,c,d := ia d cb ; (3.2)and the integrands I ia,b,c,d are chiral boxes, given schematically as I ia,b,c,d := ia d cb ⇔ ( (cid:96) | N i )( Y i | X )( (cid:96) | a )( (cid:96) | b )( (cid:96) | c )( (cid:96) | d )( (cid:96) | X ) , (3.3)which are designed to have support on one of the associated quadruple cuts and vanishon the other, and where X := x ∞ is the point at infinity in (dual) loop momentumspace. Explicit expressions for these integrands given in terms of momentum twistorsmay be found in [25], though their explicit form will not be needed here. Finally,the divergent part of one loop amplitudes is obtained by matching term-by-term allcomposite leading singularities with ‘chiral triangle’ integrands,– 9 – ( k ) , n = ⇔ I a ( X ) := ( X | a )( a − | a +1)( (cid:96) | a − (cid:96) | a )( (cid:96) | a +1)( (cid:96) | X ) , and I div := (cid:88) a I a ( X ) . (3.4)In the expansion (3.1), each chiral box integrand is dressed with the appropriateN k MHV on-shell function. Because the colors and labels of an integrand and its associ-ated on-shell function coefficient are identical, we shall find convenient a diagrammaticnotation which combines both the on-shell function and the chiral box integrand mul-tiplying it in (3.1) into a single figure using spherical (colored—to indicate N k MHVdegree) vertices: ia d cb := ia d cb × ia d cb . (3.5)As usual, we shall use blue (white) vertices to indicate both the respective MHV (MHV)amplitudes in the on-shell function and the corresponding quad-cut solution on whichthe integrand has support. As an example, this notation allows us to write the MHVone-loop amplitude integrand as a sum of two-mass-easy boxes (with one-mass degen-erations included in the sum): A (0) , n = (cid:88) a
To demonstrate the cancellation of all overlapping collinear divergences appearingterm-by-term in the representation of the two-loop ratio function (3.10), it is useful toexamine the divergent part of the MHV × MHV contribution separately first. The one-loop, finite part of the MHV amplitude integrand A (0) , n is a sum of two-mass easy boxes(and one-mass degenerations) as shown in (3.6). Consider the local divergence associ-ated with, say, leg a , and the corresponding collinear region where the loop momentaare parametrized as (cid:96) = α x a + (1 − α ) x a +1 , (cid:96) = β x a + (1 − β ) x a +1 . (3.11)The only mergers appearing in A (0) , n, fin ⊗ A (0) , n, fin with support in this region involve amassless leg a in both the left and right boxes; that is, both must be one of thefollowing chiral box integrands: a or a. (3.12)As the merger of two boxes of the same chirality always vanishes, this implies that thefull MHV × MHV divergence arising from the collinear region associated with leg a canbe written as A (0) , n, fin ⊗ A (0) , n, fin ⊃ (cid:88) b a ⊗ b a (3.13)In fact, the local divergence is even simpler than the representation (3.13) suggests. Ifwe consider the residue associated with setting ( (cid:96) i | b ) →
0, where b is generic, there aretwo contributing terms in (3.13) which cancel pairwise:Res (cid:96) ,(cid:96) ∈ coll( a )( (cid:96) i | b )=0 A (0) , n, fin ⊗ A (0) , n, fin = 2 ⊗ b a + b a = 0 . (3.14)The same pairwise cancellation works out for every collinear-but-not-soft pole. As aconsequence, the residue (3.13) in the collinear region is purely double-soft-collinear,and in terms of the parametrization (3.11) can be written as a two-form in α, β ,Res (cid:96) ,(cid:96) ∈ coll( a ) A (0) , n, fin ⊗ A (0) , n, fin = 2 dα dβα ( α − β ( β − . (3.15)This implies that in the two-loop ratio function, all local divergences that are not of thedouble-soft-collinear type must cancel amongst the N k MHV × MHV terms themselves.– 12 – .3 Cancellation of All Local Divergences in the Ratio Function
The remaining divergent terms in the representation of the ratio function given in(3.10) are written in terms of the mergers of the finite parts of N k MHV and MHV one-loop integrands. Every potentially divergent merger was classified in subsection 2.2, sowe may organize the divergent contributions which must cancel according to the threeclasses of divergent integrands introduced there. From the result of the previous sub-section, it follows that the double-soft-collinear log ( δ ) divergences in N k MHV × MHVmust combine to give twice the tree amplitude, while all other local divergences—thoseassociated with singly-soft-collinear and collinear-but-not-soft singularities—must can-cel separately. log ( δ ) -Divergent Regions Let us consider the double-soft-collinear singularity where (cid:96) = (cid:96) = x a . From the generalclassification of subsection 2.2, every non-zero merger with two consecutive masslesslegs a − , a in both left and right boxes appearing in the ratio function have thesedivergences. For the MHV × MHV part of the ratio function, there is a single suchmerger (which contributes with multiplicity two),Res (cid:96) ,(cid:96) = x a A ( k ) , n A (0) , n, fin ⊗ A (0) , n, fin = 2 A ( k ) , n × a ⊗ a = 2 A ( k ) , n . (3.16)Alternatively, this follows from the residue of the MHV × MHV mergers on the fullhexa-cut surface (3.15). For the N k MHV × MHV terms, there are six merger topologiesof the general form (2.16). Adopting the convention that every N k MHV box is drawnon the left-hand-side of the merger symbol ⊗ , the integrands with this local divergencemay be written as a ⊗ a , a ⊗ a. (3.17)Conveniently, we may group the terms in the N k MHV × MHV part of the ratio functionwith support on the double-soft-collinear residue (cid:96) = (cid:96) = x a into two ‘BCFW’ sets [4]according to: – 13 –es (cid:96) = x a (cid:96) = x a A ( k ) , n, fin ⊗ A (0) , n, fin = (cid:32) a + a + (cid:88) a (cid:33) ⊗ a + (cid:32) a + a + (cid:88) a (cid:33) ⊗ a . (3.18)As each integrand merger in this expression is unit on the double-soft-collinear singu-larity (cid:96) = (cid:96) = x a , each factor in parentheses in (3.18) becomes a particular BCFWrecursion of the N k MHV tree amplitude, combining to give exactly a total contribu-tion of − A ( k ) , n to the ratio function; this precisely cancels against the ‘MHV × MHV’contribution (3.16).The above argument was for the double-soft-collinear singularity where (cid:96) = (cid:96) = x a .The local divergence associated with the double-soft-collinear singularity where theloops approach different dual points, e.g. (cid:96) = x a and (cid:96) = x a +1 , cancels by an analogousgrouping of the relevant integrands into BFCW sets. log( δ ) -Divergent Regions The cancellation of the log( δ ) divergences associated with a single-soft-collinear singu-larity is, by the general arguments above, among the N k MHV × MHV terms themselves .The set of mergers with single-soft-collinear divergences was given in (2.15); however, toorganize their cancellation, it is useful to consider the subset of these mergers which havesupport on a co-dimension eight residue where one loop is soft-collinear, e.g. (cid:96) i → x a ,while the other loop is collinear and cuts a non-adjacent propagator, (cid:96) j ∈ coll( a ) and( (cid:96) j | b ) →
0. Concretely, we may parametrize the on-shell loop momenta in this case as (cid:96) i = x a , (cid:96) j = β (cid:63) x a + (1 − β (cid:63) ) x a +1 , (3.19)where β (cid:63) is the (unique) solution to ( (cid:96) | b ) = 0, whose explicit form is not important forour discussion. The cancellation of single-soft-collinear divergences in the ratio functionis equivalent to the statement that all such co-dimension eight residues vanish.There are two kinds of cancellations we must consider, depending on which loopgoes soft-collinear. If we consider the divergence associated with sending the N k MHVloop to the soft-collinear point x a +1 and ‘collecting’ on the pole associated with thedual momentum x b , there are exactly two mergers involving the same N k MHV chiralbox (and associated on-shell function) which cancel pairwise:Res (cid:96) = x a (cid:96) ∈ coll( a )( (cid:96) | b )=0 A ( k ) , n, fin ⊗ A (0) , n, fin ⊃ a ⊗ b a + b a = 0 . (3.20)– 14 –n the other hand, if we consider going soft-collinear in the MHV loop and collecton poles in the N k MHV boxes, it is easy to see the cancellation cannot (generically) bepairwise for the simple reason that there are no two-term identities between distinctN k MHV one-loop on-shell functions. Instead, the cancellation of local divergencesarises as a particular instance of Cauchy’s residue theorem [27, 67]. To illustrate thisexplicitly, let us first consider the terms in the ratio function involving N k MHV boxeswhose on-shell coefficients include a white massless vertex with leg a . Schematically,the contributing terms readRes (cid:96) = x a (cid:96) ∈ coll( a )( (cid:96) | b )=0 A ( k ) , n, fin ⊗ A (0) , n, fin ⊃ (cid:32) b a + b a + b a (3.21)+ b a + ab + b a (cid:33) ⊗ a . This expression should be understood as being summed over all the possible leg dis-tributions on the four- or higher-point vertices. It may be immediately verified usinge.g. the tools found in [24] that, for any given choice of helicity degree k , the particularsum of on-shell functions appearing in the N k MHV side of (3.21) vanishes when theappropriate signs from a residues of the cut integrands are taken into account. In fact,the necessary identity can be directly understood as the residue theorem associatedwith the boundaries of the chiral two-mass triangle sub-leading singularity in sYM.Namely, letting ‘ ∂ [ · ]’ denote the set of all maximal co-dimension residues obtained byfactorization of a diagram’s vertices, then the resulting collection of leading singulari-ties: ∂ ab = ab , ab , ab , ab , ab , ab (3.22)which satisfies a residue theorem with precisely the relative signs obtained by evaluatingthe merger integrands on the soft-collinear residue of (3.21). The additional contribut-ing terms on this residue, all of which involve N k MHV boxes with an MHV masslessvertex involving leg a , cancel as a consequence of the residue theorem involving thetwo-mass triangle of the opposite chirality, namely,– 15 – ab = ab , ab , ab , ab , ab , ab (3.23)Of course, for a given choice of k , not all of the on-shell functions appearing in the gen-eral residue theorems (3.22) and (3.23) will be non-zero. For example, for N ( k =1) MHVamplitudes the latter identity (3.23) will involve only two terms. In fact, even thecancellation in (3.20) can be regarded as another instance of a two-term residue the-orem for MHV amplitude integrands. The remaining non-soft-but-collinear divergentregions can also be dealt with using almost the exact same argument, as we shall nowdemonstrate. log( δ ) -Divergent Regions The final class of divergences which must cancel to prove the local finiteness of the ratiofunction are those associated to collinear-but-not-soft singularities in both loops. Wemust demonstrate the cancellation of all locally divergent regions in loop momentumspace parametrized as (cid:110) (cid:96) = α (cid:63) x a + (1 − α (cid:63) ) x a +1 , (cid:96) = β (cid:63) x a + (1 − β (cid:63) ) x a +1 , ( (cid:96) ( α (cid:63) ) | b ) = ( (cid:96) ( β (cid:63) ) | c ) = 0 (cid:111) . (3.24)Here, α (cid:63) , β (cid:63) are the solutions to cutting two additional (non-adjacent) propagators( (cid:96) | b ) and ( (cid:96) | c ) in the two loops, respectively. In this case, neither loop can access thesoft-collinear region, and the relevant mergers were given in (2.14). It is once againuseful to organize the cancellations according to whether we ‘collect’ on the MHV loopor the N k MHV loop.First, consider the mergers which contribute in the divergent region where bothloops are collinear to leg a , and the MHV loop is fully localized by setting ( (cid:96) | c ) = 0.For each N k MHV box there are two MHV boxes with support on this residue, and thecancellation is pairwise,Res (cid:96) ,(cid:96) ∈ coll( a )( (cid:96) | c )=0 A ( k ) , n, fin ⊗ A (0) , n, fin ⊃ a ⊗ + = 0 . (3.25)Here, we can see these two residues cancel as functions of the remaining loop degree offreedom, even prior to localizing e.g. ( (cid:96) | b ) = 0. If, on the other hand, we collect on (acollinear-but-not-soft pole of) the N k MHV loop, the contributing on-shell functions are:– 16 –es (cid:96) ,(cid:96) ∈ coll( a )( (cid:96) | b )=( (cid:96) | c )=0 A ( k ) , n, fin ⊗ A (0) , n, fin ⊃ (cid:32) b a + b a + b a (3.26)+ b a + b a + ab (cid:33) ⊗ c a = 0(where we are implicitly summing over all possible factorizations with the requisitepole structure). Once again, these combine to give a residue theorem generated by(3.23), cancelling all of these local divergences.We have therefore exhausted our classification of local divergences in two loopamplitude integrands, showing that each possibly divergent region of loop momentumspace is cancelled between terms appearing in the expression for the ratio functionintegrand given in (3.10). Thus, we have proven—for all multiplicity and arbitraryN k MHV degree, that two-loop ratio functions in planar sYM are locally finite . In this paper, we have investigated the infrared behavior of the two-loop ratiofunction in planar, maximally supersymmetric ( N = 4) Yang-Mills theory (sYM). Byintroducing integrand-level tests to locally diagnose two-loop infrared-finiteness, wehave identified the correspondence between particular soft and collinear residues atthe integrand-level, and the maximal degree of divergence which arise upon integra-tion in some regularization scheme. We have shown that the particular combinationof infrared-divergent amplitudes in the two-loop ratio function is locally finite by ex-plicitly demonstrating the cancellation of all divergences present in the local integrandrepresentation of [25].Having demonstrated the finiteness of the ratio function, there is an immediatefollow-up to this work: namely, finding a new representation of ratio functions whichmakes their finiteness manifest at the integrand-level. With such, the ratio functionwould be possible to compute without regularization—perhaps making the preservationof the simplicities of the integrand (such as dual conformal invariance) more manifestafter integration.Because local finiteness is a homogeneous constraint on any loop integrand, it isrelatively straightforward to construct the space of locally-finite box-power-countingtwo loop integrands for any multiplicity (as a subspace of all integrands with boxpower-counting). However, it is by no means guaranteed that a basis for this space of– 17 –ntegrands exists which is compatible with dual-conformality, purity, ease of integration,or even stability in form with increasing multiplicity. We suspect that a basis of pure,dual-conformal, locally finite integrands exist for all multiplicity at two loops analogousto the one described at one-loop in [23]; however, we must leave this search to futurework.Although our proof of local finiteness of the ratio function at two loops relied heavilyon the representation (3.10) derived in [29], and involved checking a number of specificcase, there is reason to suspect that a more powerful, all-loop argument may followfrom a different line of reasoning. Indeed, in [68] it was shown that the divergences (atthe integrand-level) of the logarithm of the amplitude in the Wilson-loop picture wereentirely captured by the contributions from cusps; such an understanding of the local(in loop-momentum-space) structure responsible for infrared divergences is somethingthat would be fruitful to explore for a broader class of infrared-(or ultraviolet-)finiteobservables. Acknowledgements
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