Four-Loop Nonplanar Cusp Anomalous Dimension in N=4 Supersymmetric Yang-Mills Theory
aa r X i v : . [ h e p - t h ] D ec Four-Loop Nonplanar Cusp Anomalous Dimension in N = 4 SupersymmetricYang-Mills Theory
Rutger H. Boels ∗ II. Institut f¨ur Theoretische Physik, Universit¨at Hamburg,Luruper Chaussee 149, D-22761 Hamburg, Germany
Tobias Huber † Naturwissenschaftlich-Technische Fakult¨at, Universit¨at Siegen,Walter-Flex-Str. 3, 57068 Siegen, Germany
Gang Yang ‡ CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China (Dated: December 17, 2018)The light-like cusp anomalous dimension is a universal function that controls infrared divergencesin quite general gauge theories. In the maximally supersymmetric Yang-Mills theory this functionis fixed fully by integrability to the three-loop order. At four loops a non-planar correction appearswhich we obtain for the first time from a numerical computation of the Sudakov form factor. Keyingredients are widely applicable methods to control the number-theoretic aspects of the appearingintegrals. Our result shows explicitly that quadratic Casimir scaling breaks down at four loops.
PACS numbers: 11.25.Db, 12.38.Bx
INTRODUCTION
Quantum field theory is used throughout physics dueto its unrivalled power to perform systematic compu-tations. Connecting the calculated quantities to ob-served experimental variables, however, can be compli-cated. Perturbation theory for the theory of the strongnuclear force, quantum chromodynamics (QCD), for in-stance can famously only be trusted on very small lengthscales [1, 2]. The physical detectors, however, involvevastly larger length scales where the perturbation the-ory is guaranteed to break down. A central role in theanalysis is played by universal functions, a prime exam-ple of which is known as the light-like cusp anomalousdimension (CAD).The universality of the CAD makes it appear in manycontexts, e.g. (1) By definition, it refers to the anomalousdimension of a Wilson loop with a light-like cusp [3, 4].(2) It provides the leading infrared (IR) behavior of on-shell amplitudes, see e.g. [5]. (3) It determines the log-arithmic growth of anomalous dimensions of high-spinWilson operators that naturally appear in the operatorproduct expansion description of deep inelastic scatter-ing processes in QCD , see e.g. [6]. (4) It is related tothe gluon Regge trajectory, and can be obtained fromthe so-called BFKL equation, see e.g. [7]. (5) Via theAdS/CFT correspondence [8], cusped Wilson operatorsin the planar limit admit a dual description in terms ofspinning strings in a curved background [9]. There istherefore ample direct physical motivation to computethis function.In N = 4 super Yang-Mills (SYM) theory, the planar cusp anomalous dimension is known non-perturbativelyvia the so-called BES equation [10] based on ideas of theAdS/CFT correspondence and integrability [11]. Con-sistency with field-theory computations through to fourloops has been obtained at the planar level [12–14], andalso confirmed by the strong coupling two-loop stringcomputations [15]. There is an intriguing connection be-tween N = 4 SYM and QCD results, via the maximaltranscendentality property: the N = 4 anomalous di-mensions correspond to the “leading-transcendentality”contribution in QCD [16, 17]. Inspired by this, the three-loop N = 4 results were first obtained in [17] using thethree-loop QCD results [18].In gauge theories with matter fields in the adjoint rep-resentation the first correction beyond the planar limitenters into the CAD at four loops. This Letter containsto our knowledge the first computation of this quan-tity. In perturbation theory four-loop non-planar formfactor integrals have to date proven to be largely pro-hibitively complicated due to the presence of IR diver-gences. Non-planar corrections are also hard to anal-yse directly within the AdS/CFT correspondence as theycorrespond to string loop corrections, and the role of in-tegrability beyond the planar limit is unclear. Moreover,a quite general conjecture has been posed by extrapo-lating three-loop results that the non-planar part of theCAD vanishes [19] in any theory. This is usually calledquadratic Casimir scaling of the CAD, and plays an im-portant role in the IR factorization in gauge theories [19–24]. This scaling is known to break down in planar N = 4SYM at strong coupling [25], as well as through instan-ton effects [26]; see also [27] for a prediction of violationat four loops. In this Letter the approach will be strictlyperturbative.The minimal scattering-like observable that containsthe cusp anomalous dimension is the Sudakov form fac-tor. In maximally supersymmetric Yang-Mills theory onecan use a correlator of a member of the stress-tensor mul-tiplet with two on-shell massless states. The first com-putation of the two-loop correction to the Sudakov formfactor in N = 4 SYM appeared in [28]. The three-loopcorrection to the QCD result was studied in a series ofpapers [29–33]. In [34] these results were fine-tuned forthe form factor in N = 4 SYM to the three-loop order.The integrand for the four-loop Sudakov form factor in N = 4 SYM was derived in [35] based on the dualitybetween color and kinematics, and its reduction to mas-ter integrals was presented in [36]. Various other cal-culations of four-loop corrections in QCD were recentlyreported [37–42]. For the five-loop integrand in N = 4SYM see [43]. REVIEWForm factor and cusp anomalous dimension
The Sudakov form factor involves only a single scale q which is the Lorentzian norm of the sum of the twomassless momenta, i.e. q = ( p + p ) with p = p =0. Dimensional analysis and maximal supersymmetry fixthe form factor F ( l ) at l loops to be given by F ( l ) = F tree g l ( − q ) − lǫ F ( l ) , (1)where the coupling constant is normalised as g = g C A (4 π ) (4 πe − γ E ) ǫ . For a classical Lie-group with Lie-algebra [ T a , T b ] = if abc T c and structure constants f abc ,gauge invariance dictates the color structure to be givenby Casimir invariants. Up to three-loop order, only pow-ers ( C A ) l of the quadratic Casimir appear, for which f acd f bcd = C A δ ab holds. At four loops the quartic invari-ant d = d abcdA d abcdA /N A appears in addition to ( C A ) ,with N A the number of generators of the group and d abcdA = 16 [ f αaβ f βbγ f γcδ f δdα + perms.( b, c, d )] . (2)Starting from six loops, even higher group invariants ap-pear, see e.g. [35]. In SU ( N c ), N A = N c − C A = N c and d = N c /
24 ( N c + 36) hold. Without loss of gener-ality, we will focus on SU ( N c ) group below.The form factor has no ultraviolet (UV) divergencessince the operator is protected, leaving only IR diver-gences. If dimensional regularization with D = 4 − ǫ is used to regulate the latter, F ( l ) is a purely numericalfunction of gauge group invariants and ǫ . This function isrelated to the cusp anomalous dimension γ ( l )cusp at l loops by [5, 44–47],(log F ) ( l ) = − (cid:20) γ ( l )cusp (2 lǫ ) + G ( l )coll lǫ + Fin ( l ) (cid:21) + O ( ǫ ) . (3)At l loops the planar part ∝ N lc of F ( l ) has leading di-vergence ∝ /ǫ l . This function needs to be expandeddown to ǫ − to extract the l -loop CAD, and also higherterms in the Laurent expansion in ǫ from lower-loop con-tributions are required. As mentioned, the first occur-rence of non-planar (i.e. subleading-color) corrections tothe CAD is at four loops, due to the appearance of thequartic Casimir invariant d . This invariant thereforebreaks quadratic Casimir scaling explicitly. The relationbetween form factor and cusp anomalous dimension forthe non-planar part at four loops is h F (4) i NP = − γ (4)cusp, NP (8 ǫ ) + O (cid:0) ǫ − (cid:1) , (4)i.e. [ F (4) ] NP has only a double pole in ǫ . Individual inte-grals that contribute to [ F (4) ] NP will, however, typicallyshow the full 1 /ǫ divergence. The general CAD is be-lieved to be expressible as a rational-coefficient polyno-mial of Riemann Zeta values ζ n , and their multi-indexgeneralizations, such as multiple zeta values (MZVs) andEuler sums, see e.g. [48]. MZVs are denoted by ζ n ,n ,... and have a transcendentality degree which is the sumof their indices, P i n i . At l loops, the planar CAD in N = 4 SYM has uniform transcendentality degree 2 l − N = 4SYM has been computed [12–14] to be γ (4)cusp , P = − ζ − ζ . (5)We will provide strong evidence that also the non-planarCAD is of uniform transcendentality six at four loops.In QCD, the known CAD has the same maximal tran-scendentality degree as in N = 4, but also contains lowertranscendentality degree constants. The maximal tran-scendentality coefficients match between planar N = 4and QCD, an observation known as the maximal tran-scendentality principle [16, 17]. Integrands, integrals, integral relations
The non-planar part of the Sudakov form factor in N = 4 SYM was obtained as a linear combinationof a number of four-loop integrals in [35] using color-kinematics duality [49, 50]. The integrals take the genericform I = ( q ) Z d D l . . . d D l N ( l i , p j ) Q k =1 D k , (6)where D i are propagators and the numerators N ( l i , p j )are quadratic polynomials of Lorentz products of the fourindependent loop and two independent external on-shellmomenta. The explicit expressions of these integrals canbe found in [35]. There are 14 distinct integral topologiesthat contribute to the non-planar CAD, labelled (21) –(34) in [35], each with 12 internal lines. We will see be-low that only 10 of them, (21) – (30) as shown in Fig. 1,contribute to the non-planar form factor if a basis of uni-formly transcendental integrals is used.Integrands are only identified up to terms that inte-grate to zero. Infinitesimal linear reparametrizations ofthe loop momenta generate such terms, which are knownas integration-by-parts (IBP) identities [51, 52]. Withthese identities the form factor was simplified in [36] us-ing the Reduze code [53]. A particular subset of theserelations, dubbed ‘rational IBP’ relations and obtainedin [54], will play an important role for the problem athand. Note that integral relations due to graph sym-metries are a particular subset of the rational IBP rela-tions. Although simpler integrals emerged in [36] com-pared to [35], these have largely evaded integration so fardue to their overwhelming complexity. The obstacle tocomputing the CAD is therefore to find a complete set ofintegrals which are simple enough to integrate. UNIFORMLY TRANSCENDENTAL INTEGRALS
The key idea is to find a representation of the four-loopform factor such that all integrals have uniform transcen-dentality if the dimensional regularization parameter ǫ isassigned transcendentality −
1. Such integrals will be re-ferred to as UT integrals. Such a representation of theform factor makes manifest the expected maximal tran-scendentality property of N = 4 SYM, and has beenachieved at three loops in [34]. Crucially for our pur-poses, the UT integrals turn out to be much simpler tointegrate than generic integrals.Beyond explicit computation, there are various waysto prove a general integral is a UT integral: • A UT integral can be expressed in the so-calleddLog form [55, 56]. • A set of integral basis that are all UT integrals canlead to certain simple differential equations [57]. • The leading singularities, or equivalently, theresidues at all poles of a UT integral should be aconstant [37, 56, 58], which is expected to be a suf-ficient and necessary condition.We have obtained a dLog form, for instance, in topolo-gies (21) and (23) for the integrals with numerator (cid:2) ( l − p ) (cid:3) (7)each (see Fig. 1 for the labelling of momenta) and intopology (28) with numerator( l − l − p ) × ( l − p ) . (8) The dLog form of the topology-(21) integral as well asits analytic result were also obtained in [37].Finding a dLog form is in general a difficult problem.Moreover, the differential equations method is not di-rectly applicable for single-scale problems such as theSudakov form factor (see [37, 59] for a work-around).The last criterion in the above bullet-point list, however,allows an algorithm. For this the four-dimensional loopmomenta are parametrised as l = α p + α p + α q + α q , (9)where p and p are the external on-shell momenta, and q , q can be chosen as the two complex solutions to q i = q i · p j = 0 ∀ i, j and q · q = − p · p . (10)With this, the integration volume changes to Z d l ( . . . ) = ( p · p ) Z Y i =1 dα i ( . . . ) . (11)In this so-called parametric form, the leading singular-ity is studied by computing subsequent residues for alloccurring scalar parameters (16 at four loops). For this,one needs to choose an order of taking residues in thescalar parameters. If, after taking a residue in a partic-ular parameter, one encounters other than a simple polein a remaining parameter, the integral is not UT. In prin-ciple, there are 16! ∼ × different orders in whichthe residues can be taken. We will explain below how wedeal with this in practice.Besides checking UT properties of single integrals, thisprocedure can be used to constrain the space of potentialUT integrals of a given topology. Start with an Ansatzof a linear combination of momentum products of massdimension four. Given a set of integrals for a given topol-ogy one can see if one or more members fail the UT testfor a particular randomly chosen order of residues. If aquadratic or higher pole is found, one can derive a lin-ear constraint on the set of coefficients in the Ansatz toevade the pole. Solving these constraints gives a smallerset of integrals. In the case at hand about a thousandsuch random checks typically yields a list of candidateUT integrals that do not easily yield further constraints.Through this strategy a set of candidate integrals was ob-tained for each topology. We note that a related strategywas used in [37, 56, 58].For ease of integration it is advantageous to have aform of the numerator as a product of two factors, bothquadratic in momenta. To find linear combinations ofUT candidate integrals with this property one writesa product-type Ansatz and derives a set of quadratic-constraint equations. In some cases a solution may notexist or may not be useful. For the non-planar four-loopform factor, this happens in topologies (26) and (25), q p p ℓ ℓ ℓ ℓ q p p ℓ ℓ ℓ ℓ q p p ℓ ℓ ℓ ℓ q p p ℓ ℓ ℓ ℓ q p p ℓ ℓ ℓ ℓ q p p ℓ ℓ ℓ ℓ q p p ℓ ℓ ℓ ℓ q p p ℓ ℓ ℓ ℓ q p p ℓ ℓ ℓ ℓ q p p ℓ ℓ ℓ ℓ (21)(26) (22)(27) (23)(28) (24)(29) (25)(30) FIG. 1. Integral topologies that contribute to the non-planar form factor at four loops in a basis of UT integrals. respectively. Here the Ansatz is widened by adding inte-grals with two fewer propagators, for instance in topology(25) with numerator (cid:2) ( p − l ) + 2( l − l ) + ( l − l ) − ( l − l ) − ( p − l ) (cid:3) − l − l ) ( p − l + l − l ) . (12)Beyond integrals for which a dLog form was obtained,all UT candidates have passed a significant number ofresidue checks, e.g. the two most complicated ones are intopology (26) and have passed 10 such checks. More-over, as discussed in the next section, the precision ofthe explicit numerical computation permits a conversionof all results up to O ( ǫ − ) to analytic expressions whichexhibit uniform transcendentality. Taken together, thesefeatures provide strong evidence that all candidates areindeed UT integrals.Having obtained a set of UT candidate integrals, oneneeds to express the form factor as a linear combina-tion of them. Since the form factor is expected to beuniformly transcendental itself, this linear combinationmust involve only rational numbers – not ǫ . The afore-mentioned ‘rational IBP’ relations obtained in [54] proveextremely useful for this task.The first result of this Letter is that the non-planarpart of the four-loop form factor obtained in [36] can beexpressed in terms of 23 UT candidate integrals, provid-ing strong evidence for maximal transcendentality in thenon-planar sector. The full list of these integrals and theappropriate linear combination is given in the appendixbelow, see also [60]. Interestingly, for topologies (31) –(34), no UT candidate integrals could be found. NUMERICAL INTEGRATION
A variety of techniques exist to compute loop integralsboth analytically and numerically, see e.g. [61] for anoverview. Here mainly sector decomposition and Mellin-Barnes (MB) integration have been used, as well as anexact integral result of topology (21) [37]. Two computer implementations of sector decomposi-tion [62] have been used: mostly FIESTA [63–66], withcross-checks for simpler integrals using SecDec [67–69].An important empirical observation within the sector de-composition approach is the feature that UT integralsusually generate considerably fewer integration termscompared to non-UT siblings of comparable complex-ity, even though for instance single-sector contributionsare not UT separately. The numerical integration withFIESTA is performed using the VEGAS algorithm [70]as implemented in the CUBA library [71]. When usingSecDec, the CUHRE and DIVONNE algorithms are ap-plied.A second integration technique applied here is MB in-tegration. Several automated tools exist, such as MBtools [72, 73] and AMBRE [74–76]. Given an integral ofthe form (6), the main challenge is to derive valid MBrepresentations for crossed four-loop topologies. Havingachieved the latter, the obtained representations are inmany cases too high-dimensional to be integrated. Still,efficient MB representations for a subset of (planar andcrossed) integrals could be derived by means of a hy-brid of the loop-by-loop approach and using the F and U graph polynomials (see [60] for details). If an efficientMB representation is available, the numerical precision ofthe integral is typically 3 to 4 orders of magnitude bettercompared to sector decomposition.Before presenting our results, a thorough investiga-tion of the numerical uncertainties is in order. In sec-tor decomposition, several runs with different integra-tion settings were performed. We observe that errorbars reported by FIESTA scale with 1 / √ N s , with N s the number of sampling points (typically in the range ofa few up to several 100 millions). Moreover, we observethat fluctuations upon increasing N s are well within thereported error bars. For the leading non-trivial poles, ǫ −{ , , , } , the numerical precision in each UT integralis high enough to write the number as a small rationalmultiple of { , ζ , ζ , ζ } , respectively. The reliability ofthe error bars reported by FIESTA was checked by com-paring to MB and (semi-)analytic results where available. TABLE I. Non-planar form factor result and errors. The ǫ − entry is afflicted with a rounding error. ǫ order − − − − − − − − . × − +4 . × − − . × − − . × − +3 . × − + 0.0007 +1.56uncertainty ± . × − ± . × − ± . × − ± . × − ± . × − ± ± Altogether, this comparison involves more than seventydata points. The fluctuations for all integrals used arewithin the reported FIESTA uncertainties, in many casesby over an order of magnitude.The uncertainties will be conservatively interpreted asthe standard deviation of a Gaussian distribution, anderrors are added in quadrature. Our results show no evi-dence of systematic effects in the FIESTA errors: devia-tions to other results include both positive and negativesigns. A hypothetical systematic error will be modelledby adding uncertainties from individual UT integrals lin-early. In conclusion, there is no need to manually inflatethe reported uncertainties. Further details will be pro-vided in [60].
Results
Our results for the non-planar four-loop form factor upto an overall factor are summarised in table I. The firstsix orders in the ǫ -expansion must vanish by equation(4).The ǫ − coefficient should vanish for each integral, andindeed does well within error bars. The coefficients of ǫ −{ , , , , } can be non-zero in individual integrals butmust cancel in the linear combination of the form factor,which is indeed the case, both using direct numerics (seetable I) as well as the obtained analytic expressions.The non-trivial coefficient for the non-planar form fac-tor at order ǫ − yields the sought-after non-planar four-loop CAD: γ (4)cusp, NP = − × (1 . ± .
21) 1 N c , (13)where 3072 = 2 × ×
64 is the normalization factor aris-ing from the permutational sum, the color factor [35], andthe denominator of (4), respectively. The significance ofa deviation from zero is 7 . σ . This is the second majorresult of this Letter. Adding individual uncertainties lin-early would yield 1 . ± .
62; still significantly non-zero.Compared to the planar result (5): γ (4)cusp , P ∼ − N c = 3 is used, its central value becomes γ (4)cusp, NP ∼− DISCUSSION
In this article we present the first computation ofthe non-planar correction to the cusp anomalous dimen-sion in the maximally supersymmetric Yang-Mills theory,which starts at four loops.While integrating a generic set of four-loop form fac-tor master integral remains quite challenging, a key idealeading to the present result is to express the form factorin a basis of uniformly transcendental integrals, whichcan be constructed algorithmically. While a generic inte-gral contains mixed transcendentality degrees, the UT in-tegrals contain only numbers of fixed (maximal for N = 4SYM) transcendentality in each order of the ǫ expansion.What is interesting and deserves further study is the em-pirical observation that such simplicity is inherited insector decomposition, where each sector is no longer UTseparately. Once a good set of (candidate) UT basis inte-grals is determined, the full form factor can be expressedin this basis by using a simple subset of the IBP relationsor, in principle, directly by unitarity cuts. In the case athand, our results provide strong evidence for the max-imal transcendentality of the four-loop non-planar formfactor in N = 4 SYM. In particular this implies that thefour-loop CAD can be written as a rational linear sum ofweight-six transcendental numbers.The UT basis finding algorithm is widely applicable.An immediate interesting application of already obtainedresults would be to four-loop propagator integrals inQCD, see e.g. [77]. A UT basis in QCD would alwaysinvolve pre-factors with non-negative powers of ǫ , sim-plifying computations potentially drastically, see [78] forsimilar ideas.Based on the UT basis, a numerical computation hasalso yielded the first information on the value of the non-planar CAD in N = 4, which is statistically significantlynon-zero. In particular, our result shows that quadraticCasimir scaling breaks down at four loops in this the-ory and is therefore not expected to hold in any othertheory such as QCD. Direct further research directionsinclude improving precision at order 1 /ǫ and comput-ing further orders in the ǫ -expansion which contain theso-called collinear anomalous dimension. Computing thefour-loop non-planar CAD analytically, also in other the-ories such as QCD, is a prime further goal. ACKNOWLEDGEMENTS
The authors would like to thank Dirk Seidel for col-laboration in the early stages of this project as well asBernd Kniehl and Sven-Olaf Moch for discussions and en-couragement. This work was supported by the GermanScience Foundation (DFG) within the Collaborative Re-search Center 676 “Particles, Strings and the Early Uni-verse”. GY is supported in part by the Chinese Academyof Sciences (CAS) Hundred-Talent Program, by the KeyResearch Program of Frontier Sciences of CAS, and byProject 11647601 supported by National Natural ScienceFoundation of China (NSFC).
Note added:
During the review of this Letter twopreprints [79] and [80] have appeared which report viola-tion of quadratic Casimir scaling at four loops in QCD.
Supplemental material:UT integrals used in the non-planar form factor
Here we list explicitly the 23 UT integrals I ( n )1 − thatbuild the non-planar form factor. The superscript ( n ) in-dicates the 12 propagators from topology ( n ) in Fig. 1 ofthe main article, where also the labelling of the momentais taken from. In this way, we only have to list the numer-ator of each integral. Moreover, each integral I ( n ) i getssupplemented by a rational pre-factor c i according to ~c = { / , / , / , − , / , − / , − / , , , , , , − / , , , , , , , , − , / , / } . The non-planar form factor isthen obtained via P i =1 ,..., c i I ( n ) i . Integrals I − , I − , and I − , are 12-, 11-, and 10-line integrals,respectively. I (21)1 = [( ℓ − p ) ] I (22)2 = − ( ℓ − p ) [ ℓ + ℓ − ℓ + ( ℓ − ℓ + p ) + ( ℓ − ℓ − p ) ] I (23)3 = [( ℓ − p ) ] I (24)4 = ( ℓ − p ) [( q − ℓ − ℓ ) + ( ℓ + p ) ] I (25)5 = (cid:2) ( p − ℓ ) + 2( ℓ − ℓ ) + ( ℓ − ℓ ) − ( ℓ − ℓ ) − ( p − ℓ ) (cid:3) − ℓ − ℓ ) ( p − ℓ + ℓ − ℓ ) I (26)6 = [( ℓ − ℓ − ℓ ) − ( ℓ − ℓ − p ) − ( ℓ − p ) − ℓ ] × [ ℓ − ℓ − ℓ + ( ℓ − ℓ ) ] + 4 ℓ ( ℓ − p ) + ( ℓ − ℓ ) ( ℓ − ℓ + ℓ − p ) I (26)7 = 4 [( ℓ − ℓ )( ℓ − ℓ + ℓ − p )] × [( ℓ − ℓ )( ℓ − ℓ + ℓ − p )] − ( ℓ − ℓ ) ( ℓ + ℓ − ℓ ) − ℓ ( ℓ − p ) − ℓ − ℓ ) ( ℓ − ℓ + ℓ − p ) − ℓ ( ℓ − p ) − ℓ ( ℓ − ℓ + ℓ + ℓ − q ) I (27)8 = 12 (cid:2) ℓ − ℓ − ( ℓ − ℓ − p ) (cid:3) × (cid:2) ( ℓ − ℓ − ℓ ) + ( ℓ + p ) (cid:3) I (28)9 = ( ℓ − ℓ − p ) (cid:2) ( ℓ − ℓ ) − ( ℓ − p ) (cid:3) I (29)10 = 12 (cid:2) ℓ − ℓ − ( ℓ − ℓ − p ) (cid:3) [ ℓ · ( ℓ − ℓ + ℓ − p )] I (30)11 = ( ℓ − ℓ − p ) [( p − ℓ ) + ( ℓ − ℓ ) − ( ℓ − p ) ] I (27)12 = 12 ( ℓ − ℓ ) (cid:2) ℓ − p ) + ( ℓ − p ) − ℓ + ℓ − ( ℓ − ℓ ) + 2 ( p + p ) (cid:3) I (28)13 = 12 ( ℓ − ℓ ) (cid:2) ℓ − ℓ − p ) + ( ℓ − p ) + ℓ − ( ℓ − ℓ ) (cid:3) I (29)14 = ( ℓ − p ) (cid:2) ( ℓ − ℓ + ℓ ) + ( ℓ − p ) − ℓ (cid:3) I (29)15 = 12 ( ℓ − p − p ) (cid:2) ( ℓ − ℓ ) − ( ℓ − p ) − ( ℓ − p ) − ( p + p ) (cid:3) I (30)16 = ( ℓ − p − p ) ( ℓ + p ) I (30)17 = 12 ( ℓ − p ) (cid:2) ℓ + p ) − ( ℓ + p + ℓ − ℓ ) (cid:3) I (30)18 = 12 ( ℓ − ℓ ) (cid:2) ℓ − ℓ + p ) − ℓ (cid:3) I (22)19 = ( ℓ − ℓ ) ( p − ℓ + ℓ ) I (22)20 = ℓ ( p − ℓ ) I (24)21 = ( p − ℓ − ℓ ) ( ℓ − p − p ) I (24)22 = ℓ ( ℓ − p − p ) I (28)23 = ( ℓ − p ) ( ℓ − ℓ + ℓ − p ) . ∗ [email protected] † [email protected] ‡ [email protected][1] D. J. Gross and F. Wilczek,Phys. Rev. Lett. , 1343 (1973).[2] H. D. Politzer, Phys. Rev. Lett. , 1346 (1973).[3] A. M. Polyakov, Nucl. Phys. B164 , 171 (1980).[4] G. P. Korchemsky and A. V. Radyushkin,
InternationalSeminar: Quarks 86 Tbilisi, USSR, April 15-17, 1986 ,Phys. Lett.
B171 , 459 (1986).[5] Z. Bern, L. J. Dixon, and V. A.Smirnov, Phys. Rev.
D72 , 085001 (2005),arXiv:hep-th/0505205 [hep-th].[6] G. P. Korchemsky, Mod. Phys. Lett. A4 , 1257 (1989).[7] A. V. Kotikov and L. N. Li-patov, Nucl. Phys. B582 , 19 (2000), arXiv:hep-ph/0004008 [hep-ph].[8] J. M. Maldacena, Int. J. Theor. Phys. , 1113 (1999),[Adv. Theor. Math. Phys.2,231(1998)],arXiv:hep-th/9711200 [hep-th].[9] S. S. Gubser, I. R. Klebanov, andA. M. Polyakov, Nucl. Phys. B636 , 99 (2002),arXiv:hep-th/0204051 [hep-th].[10] N. Beisert, B. Eden, and M. Stau-dacher, J. Stat. Mech. , P01021 (2007),arXiv:hep-th/0610251 [hep-th].[11] N. Beisert et al. , Lett. Math. Phys. , 3 (2012),arXiv:1012.3982 [hep-th].[12] Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower,and V. A. Smirnov, Phys. Rev. D75 , 085010 (2007),arXiv:hep-th/0610248 [hep-th].[13] F. Cachazo, M. Spradlin, andA. Volovich, Phys. Rev.
D75 , 105011 (2007),arXiv:hep-th/0612309 [hep-th].[14] J. M. Henn and T. Huber, JHEP , 147 (2013),arXiv:1304.6418 [hep-th].[15] R. Roiban and A. A. Tseytlin, JHEP , 016 (2007),arXiv:0709.0681 [hep-th].[16] A. V. Kotikov and L. N. Lipatov,Nucl. Phys. B661 , 19 (2003), [Erratum: Nucl.Phys.B685,405(2004)], arXiv:hep-ph/0208220 [hep-ph].[17] A. Kotikov, L. Lipatov, A. Onishchenko, andV. Velizhanin, Phys.Lett.
B595 , 521 (2004),arXiv:hep-th/0404092 [hep-th].[18] S. Moch, J. A. M. Vermaseren, andA. Vogt, Nucl. Phys.
B688 , 101 (2004),arXiv:hep-ph/0403192 [hep-ph].[19] T. Becher and M. Neubert, JHEP , 081 (2009), [Erra-tum: JHEP11,024(2013)], arXiv:0903.1126 [hep-ph].[20] E. Gardi and L. Magnea, JHEP , 079 (2009),arXiv:0901.1091 [hep-ph].[21] L. J. Dixon, Phys. Rev. D79 , 091501 (2009),arXiv:0901.3414 [hep-ph].[22] T. Becher and M. Neubert,Phys. Rev.
D79 , 125004 (2009), [Erratum: Phys.Rev.D80,109901(2009)], arXiv:0904.1021 [hep-ph].[23] L. J. Dixon, E. Gardi, and L. Magnea,JHEP , 081 (2010), arXiv:0910.3653 [hep-ph].[24] V. Ahrens, M. Neubert, and L. Vernazza,JHEP , 138 (2012), arXiv:1208.4847 [hep-ph].[25] A. Armoni, JHEP , 009 (2006),arXiv:hep-th/0608026 [hep-th].[26] G. P. Korchemsky, (2017), arXiv:1704.00448 [hep-th].[27] L. F. Alday and J. M. Maldacena, JHEP , 019 (2007),arXiv:0708.0672 [hep-th].[28] W. van Neerven, Z.Phys. C30 , 595 (1986).[29] P. Baikov, K. Chetyrkin, A. Smirnov, V. Smirnov,and M. Steinhauser, Phys.Rev.Lett. , 212002 (2009),arXiv:0902.3519 [hep-ph].[30] R. Lee, A. Smirnov, and V. Smirnov,JHEP , 020 (2010), arXiv:1001.2887 [hep-ph].[31] T. Gehrmann, E. Glover, T. Huber, N. Iki-zlerli, and C. Studerus, JHEP , 094 (2010),arXiv:1004.3653 [hep-ph].[32] T. Gehrmann, E. W. N. Glover, T. Huber, N. Ik-izlerli, and C. Studerus, JHEP , 102 (2010),arXiv:1010.4478 [hep-ph].[33] A. von Manteuffel, E. Panzer, and R. M.Schabinger, Phys. Rev. D93 , 125014 (2016),arXiv:1510.06758 [hep-ph]. [34] T. Gehrmann, J. M. Henn, and T. Huber, JHEP ,101 (2012), arXiv:1112.4524 [hep-th].[35] R. H. Boels, B. A. Kniehl, O. V. Tarasov, and G. Yang,JHEP , 063 (2013), arXiv:1211.7028 [hep-th].[36] R. H. Boels, B. A. Kniehl, andG. Yang, Nucl. Phys. B902 , 387 (2016),arXiv:1508.03717 [hep-th].[37] J. M. Henn, A. V. Smirnov, V. A. Smirnov,and M. Steinhauser, JHEP , 066 (2016),arXiv:1604.03126 [hep-ph].[38] J. Henn, A. V. Smirnov, V. A. Smirnov, M. Steinhauser,and R. N. Lee, (2016), 10.1007/JHEP03(2017)139,arXiv:1612.04389 [hep-ph].[39] T. Ahmed, J. M. Henn, and M. Steinhauser, (2017),arXiv:1704.07846 [hep-ph].[40] A. von Manteuffel and R. M. Sch-abinger, Phys. Rev. D95 , 034030 (2017),arXiv:1611.00795 [hep-ph].[41] J. Davies, A. Vogt, B. Ruijl, T. Ueda, andJ. A. M. Vermaseren, Nucl. Phys.
B915 , 335 (2017),arXiv:1610.07477 [hep-ph].[42] R. N. Lee, A. V. Smirnov, V. A. Smirnov, and M. Stein-hauser, (2017), arXiv:1705.06862 [hep-ph].[43] G. Yang, Phys. Rev. Lett. , 271602 (2016),arXiv:1610.02394 [hep-th].[44] A. H. Mueller, Phys.Rev.
D20 , 2037 (1979).[45] J. C. Collins, Phys.Rev.
D22 , 1478 (1980).[46] A. Sen, Phys.Rev.
D24 , 3281 (1981).[47] L. Magnea and G. F. Sterman,Phys.Rev.
D42 , 4222 (1990).[48] J. Bl¨umlein, D. J. Broadhurst, and J. A. M.Vermaseren, Comput. Phys. Commun. , 582 (2010),arXiv:0907.2557 [math-ph].[49] Z. Bern, J. J. M. Carrasco, and H. Jo-hansson, Phys. Rev.
D78 , 085011 (2008),arXiv:0805.3993 [hep-ph].[50] Z. Bern, J. J. M. Carrasco, and H. Jo-hansson, Phys. Rev. Lett. , 061602 (2010),arXiv:1004.0476 [hep-th].[51] K. Chetyrkin and F. Tkachov,Nucl.Phys.
B192 , 159 (1981).[52] F. Tkachov, Phys.Lett.
B100 , 65 (1981).[53] A. von Manteuffel and C. Studerus, (2012),arXiv:1201.4330 [hep-ph].[54] R. H. Boels, B. A. Kniehl, and G. Yang,
Proceedings,13th DESY Workshop on Elementary Particle Physics:Loops and Legs in Quantum Field Theory (LL2016):Leipzig, Germany, April 24-29, 2016 , PoS
LL2016 , 039(2016), arXiv:1607.00172 [hep-th].[55] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo,and J. Trnka, Phys. Rev. Lett. , 261603 (2014),arXiv:1410.0354 [hep-th].[56] Z. Bern, E. Herrmann, S. Litsey, J. Stankow-icz, and J. Trnka, JHEP , 202 (2015),arXiv:1412.8584 [hep-th].[57] J. M. Henn, Phys. Rev. Lett. , 251601 (2013),arXiv:1304.1806 [hep-th].[58] Z. Bern, E. Herrmann, S. Litsey, J. Stankow-icz, and J. Trnka, JHEP , 098 (2016),arXiv:1512.08591 [hep-th].[59] J. M. Henn, A. V. Smirnov, and V. A. Smirnov,JHEP , 088 (2014), arXiv:1312.2588 [hep-th].[60] R. H. Boels, T. Huber, and G. Yang, To appear (2017).[61] V. A. Smirnov, Springer Tracts Mod.Phys. , 1 (2012). [62] T. Binoth and G. Hein-rich, Nucl.Phys. B585 , 741 (2000),arXiv:hep-ph/0004013 [hep-ph].[63] A. Smirnov and M. Tentyukov,Comput.Phys.Commun. , 735 (2009),arXiv:0807.4129 [hep-ph].[64] A. Smirnov, V. Smirnov, and M. Ten-tyukov, Comput.Phys.Commun. , 790 (2011),arXiv:0912.0158 [hep-ph].[65] A. V. Smirnov, Comput.Phys.Commun. , 2090 (2014),arXiv:1312.3186 [hep-ph].[66] A. V. Smirnov, Comput. Phys. Commun. , 189 (2016),arXiv:1511.03614 [hep-ph].[67] J. Carter and G. Heinrich,Comput.Phys.Commun. , 1566 (2011),arXiv:1011.5493 [hep-ph].[68] S. Borowka, J. Carter, and G. Hein-rich, Comput.Phys.Commun. , 396 (2013),arXiv:1204.4152 [hep-ph].[69] S. Borowka, G. Heinrich, S. Jones, M. Kerner, J. Schlenk, et al. , (2015), arXiv:1502.06595 [hep-ph].[70] G. P. Lepage, CLNS-80/447 (1980). [71] T. Hahn, Comput. Phys. Commun. , 78 (2005),arXiv:hep-ph/0404043 [hep-ph].[72] M. Czakon, Comput.Phys.Commun. , 559 (2006),arXiv:hep-ph/0511200 [hep-ph].[73] A. V. Smirnov and V. A.Smirnov, Eur. Phys. J.
C62 , 445 (2009),arXiv:0901.0386 [hep-ph].[74] J. Gluza, K. Kajda, and T. Riemann,Comput.Phys.Commun. , 879 (2007),arXiv:0704.2423 [hep-ph].[75] J. Gluza, K. Kajda, T. Riemann, andV. Yundin, Eur.Phys.J.
C71 , 1516 (2011),arXiv:1010.1667 [hep-ph].[76] J. Bl¨umlein, I. Dubovyk, J. Gluza, M. Ochman,C. G. Raab, et al. , PoS
LL2014 , 052 (2014),arXiv:1407.7832 [hep-ph].[77] B. Ruijl, T. Ueda, and J. A. M. Vermaseren, (2017),arXiv:1704.06650 [hep-ph].[78] A. von Manteuffel, E. Panzer, and R. M. Schabinger,JHEP02