Jacob Bourjaily

Adventures in Perturbation Theory Jacob Bourjaily Penn State University & The Niels Bohr Institute based on work in collaboration with Herrmann, Langer, Trnka; McLeod, von Hippel, Vergu, Volk, Wilhelm; … Organization & Outline

✦ Spiritus Movens: the surprising simplicity of QFT ✦ Loop Integrands ‣ generalized unitarity (generally speaking) ‣ building bases big-enough (for e.g. the Standard Model) ‣ non-planar power-counting (a modest proposal) ✦ Loop Integration: what makes an integral easy? ‣ integration polemics (what constitutes being integrated?) ‣ direct integration (made easy) ✦ Loop Integrals: their generic analytic structure ‣ a bestiary of Feynman integral Calabi-Yau geometries

Spiritus Movens the Surprising Simplicity of Scattering Amplitudes Traditional Description of QFT ✦ Quantum Field Theory: the marriage of (special) relativity with quantum mechanics ✦ Theories (can be) speciﬁed by Lagrangians—or equivalently, by Feynman rules for virtual particles

✦ Predicted probability (amplitudes) from path integrals (over virtual ‘histories’):

4 Perturbation Theory and Loops ✦ Predictions (often) made perturbatively, according to the loop expansion:

[Dirac (1933)] [Feynman; Schwinger; Tomanaga (1947)] [Petermann (1957)] [Kinoshita (1990)]

✦ the most precisely tested idea in all of science! 5 derground rider is usually better o! taking a fairly simple route. lations—like an Underground rider who is willing to take very cir- Two decades later physicists extended Feynman’s technique to cuitous routes, making it hard to predict what he or she will do. Explosions of Complexitythe strong subnuclear force. By analogy with QED, the theory of Fortunately, at very short distances, including the distances rele- the strong force is known as quantum chromodynamics (QCD). vant for collisions at the LHC, the coupling diminishes in value QCD is also governed by a coupling, but as the word “strong” sug- and, for the very simplest collisions, theorists can again get away gests, its value is higher than that of the electromagnetic cou- with considering only uncomplicated Feynman diagrams. pling. On the face of it, a larger coupling increases the number of For messy collisions, though, the full complexity of the ✦ complicated diagrams that theorists must include in their calcu- Feynman technique comes rushing in. Feynman diagrams are While ultimately correct, the Feynman expansion classified by the number of external lines and WHY FEYNMAN DIAGRAMS DRIVE PHYSICISTS MAD the number of closed loops they have [see box at left]. Loops represent one of the quintessen- tial features of quantum theory: virtual parti- renders all but the most trivial predictions—Too Many to Keep Track Of cles. Though not directly observable, virtual particles have a measurable e!ect on the Each Feynman diagram strength of forces. They obey all the usual laws of nature, such as the conservation of energy A quark-quark interaction might produce more than one gluon or involve more than involving the fewest particles, at the and of momentum, with one caveat: their one virtual-particle loop, or both. The calculations quickly become unmanageable. mass can di!er from that of the corresponding “real” (that is, directly observed) particles. Zero loops One loop Loops represent their ephemeral life cycle: lowest orders of perturbation— they pop into existence, move a short distance, then vanish again. Their mass determines their life expectancy: the heavier they are, the shorter they live. The simplest Feynman diagrams ignore vir- computationally intractable One tual particles; they have no closed loops and are gluon called tree diagrams. In quantum electrody- namics, the simplest diagram of all shows two electrons repelling each other by exchanging a or theoretically inscrutable photon. Progressively more complicated diagrams add loops one by one. Physicists refer to QUANTUM PHYSICS this additive procedure as “perturbative,” mean- ing that we start with some approximate esti- mate (represented by the tree diagrams) and gradually perturb it by adding refinements (the LOOPS, loops). For instance, as the photon travels between the two electrons, it can spontaneously Two split into a virtual electron and virtual antielec- TREES gluons tron, which live a short while before annihilat- ing each other, producing a photon. The photon resumes the journey the original photon had AND THE been taking. In the next level of complexity, the electron and antielectron might themselves split temporarily. With increasing numbers of virtual particles, the diagrams describe quan- SEARCH tum e!ects with increasing precision. Even tree diagrams can be challenging. In the case of QCD, if you were brave enough to FOR NEW consider a collision involving two incoming and Three eight outgoing gluons, you would need to write gluons down 10 million tree diagrams and calculate a probability for each. An approach called recur- PHYSICS sion, pioneered in the 1980s by Frits Berends of Maybe unifying the forces of nature isn’t quite Leiden University in the Netherlands and Wal- as hard as physicists thought it would be ter Giele, now at Fermilab, tamed the problem By Zvi Bern, Lance J. Dixon and David A. Kosower for tree diagrams but had no obvious extension to loops. Worse, closed loops make the workload 34 Scientific American, May 2012 Illustration by Kenn Brown and Chris Wren, Mondolithic Studios overwhelming.6 Even a single loop causes an ex- [Bern, Dixon, Kosower,© 2012 Scientific American Scientific American (2012)] plosion in both the number of diagrams and the

[Rev.Mod.Phys. 56 (1984)]

✦ Once considered computationally intractable

7 Needs (Once) Beyond Our Reach ✦ Background amplitudes crucial for e.g. colliders

220 diagrams—thousands of terms

✦ Once considered computationally intractable

7 Needs (Once) Beyond Our Reach ✦ Background amplitudes crucial for e.g. colliders

[Nucl.Phys. B269 (1985)] ✦ Once considered computationally intractable

7 Needs (Once) Beyond Our Reach ✦ Background amplitudes crucial for e.g. colliders

7 Discovery of Shocking Simplicity ✦ Within six months, Parke-Taylor stumbled on a simple guess—unquestionably a theorist’s delight:

VoLUME 56, NUMsER 23 PHYSICAL REVIEW LETTERS 9 JUNE 1986

Amplitude for nGluon Scattering [PRL 56 (1986)]

Stephen J. Parke and T. R. Taylor Fermi Xationa/Accelerator Laboratory, Batavia, Illinois 60510 (Received 17 March 1986) A nontrivial squared helicity amplitude is given for the scattering of an arbitrary number of gluons to lowest order in the coupling constant and to leading order in the number of colors.

PACS QUmbcrs: 12.38.8x [van der Waerden (1929)]

Computations of the scattering amplitudes for the used to improve the existing numerical programs for vector gauge bosons of non-Abelian gauge theories, the QCD jet production, and in particular for the stud- 8 besides being interesting from a purely quantum- ies of the four-jet production for which no analytic field-theoretical point of view (determination of the S results have been available so far. Before presenting matrix), have a wide range of important applications. the helicity amplitude, let us make it clear that this In particular, within the framework of quantum chro- result is an educated guess which we have compared to modynamics (QCD), the scattering of vector gauge the existing computations and verified by a series of bosons (gluons) gives rise to experimentally observ- highly nontrivial and nonlinear consistency checks. able multijet production at high-energy hadron collid- For the n-gluon scattering amplitude, there are ers. The knowledge of cross sections for the gluon (n+2)/2 independent helicity amplitudes. At the scattering is crucial for any reliable phenomenology of tree level, the two helicity amplitudes which most jet physics, which holds great promise for testing QCD violate the conservation of helicity are zero. This is as well as for the discovery of new physics at present easily seen by the embedding of the Yang-Mills theory (CERN Spp S and Fermilab Tevatron) and future (Su- in a supersymmetric theory. 2 3 Here we give an ex- perconducting Super Collider) hadron colliders. ' pression for the next helicity amplitude, also at tree In this short Letter, we give a nontrivial squared level, to leading order in the number of colors in helicity amplitude for the scattering of an arbitrary SU(N) Yang-Mills theory. number of gluons to lowest order in the coupling con- If the helicity amplitude for gluons 1, . . . , n, of mo- stant and to leading order in the number of colors. To menta pi, . . . , p„and helicities Xi, . . . , A, „, is our knowledge this is the first time in a non-Abelian A'„(&i, . . . , &„), where the momenta and helicities gauge theory that a nontrivial, on-mass-shell, squared are labeled as though all particles are outgoing, then Green's function has been written down for an arbi- the three helicity amplitudes of interest, squared and trary number of external points. Our result can be summed over color, are

I2 = I P'„(+ + + + + ) c„(g,N) [0+O(g ) ], — I' = I M„( + + + + ) c„(g,N) [0+ O(g~) ], I~„(——+++ )I'= c„(q,N)[(pi p2)' x Xp[(pi p2)(p2 p3)(p3. p4) . (p„pi)] '+O(N )+O(g )], (3) where c„(g,N) =g2" 4N" 2(N2 —1)/2" 4n. The sum is over all permutations I'of 1, . . . , Equation (3) has the correct dimensions and symmetry properties for this n-particle scattering amplitude squared. Also it agrees with the known results4 for n =4, 5, and 6. The agreement for n =6 is numerical. More importantly, this set of amplitudes is consistent with the Altarelli and Parisi7 relationship for all n, when two of the gluons are made parallel. This is trivial for the first two helicity amplitudes but is a highly nontrivial state- ment for the last amplitude, as shown here: IM„(——+++ )I'-0, 1[(2 ' — +++ ) I'-2g'N I (- -++ z(1 —z) s ~„, IW„(- —+++ ) I'- 2g'N — ——++ ) I', 3 II 4 z(1 —z) s I~„(

1986 The American Physical Society 2459 Perturbations of Parke-Taylor ✦ What about beyond the leading order?

9 Perturbations of Parke-Taylor ✦ What about beyond the leading order?

[Bern, Dixon, Dunbar, Kosower (1994)]

9 Perturbations of Parke-Taylor ✦ What about beyond the leading order?

[Vergu (2009)]

complexity of the computations. It has also been useful to use the results for the cuts A. Double box topologies 1 a a +1 a +2 already computed when computing the coefficients of integrals detected by new cuts. In this 1 2 2 2 2 2 −   (42) xa−4,a−1xa−3,a − 2xa−4,axa−3,a−1 xa−2,a (15) 0 (23) 0 (32) Inthecaseofthedoubleboxtopologiesthemassivelegsattachedtotheverticesincident 4 4 a +3 a +4 a − 2 way, one can insure the consistency of results from different cuts and reduce the number of ! "   with the common edge have to be a sum of at least three massless momenta. The cases unknowns at the same time. where these massive legs are the sum of two massless momenta are treated separately in the 3. Two massless legs attached 1 2 2 2 2 2 Let us make a further comment about our computation procedure. The conformal inte- 0 (24) 0 (33) xa−3,a+3xa−2,a+1 − xa−3,a+1xa−2,a+3 xa,a+2 (43) 4 subsection. IV A 7. This distinction only arises for the double box topologies. % & grals with pentagon loops have numerators containing the loop momenta in combinations k l 2 l k 1 like ( + ) ,where is the loop momentum and is an external on-shell momentum. If x2 x2 x2 − x2 x2 x2 + 1 a − 1 a a +1 4 a,b+2 a+1,b b−1,b+1 a,b+1 a+1,b b−1,b+2 5.Onemasslesslegandonemassivelegattached −   (44) the propagator with momentum l is cut then, on that cut, one cannot distinguish between 1. No legs attached # 0 (34) 4 2 2 2 a +3 a − 4 a − 3 + xa,b+1xa+1,b−1xb,b+2 (16)   (k + l)2 and 2k · l.However,itiseasytoseethatonecanchoosetocutanotherpropagator $ 1 1 2 2 2 2 and in that case this ambiguity does not arise and the numerator factor is uniquely defined. 2 2 − xa−2,axa−1,a+2xa+1,a+3 (25) xa,a+2 xa−1,a+1 (8) 1 2 2 2 2 2 2 4 7.Extradoubleboxes 2 −xa−1,b−1xa,b+1xa+1,b + xa−1,b−1xabxa+1,b+1− 0 (45) ! " 4# 2 2 2 IV. RESULTS − xa−1,a+1xabxb−1,b+1 (17) $ 1 1 2 2 2 2 2 2 2 2 2 0 (26) −xa−2,bxa−1,a+2xa+1,b + xa−2,a+2xa−1,bxa+1,b+ xab xa−1,a+1 (9) 4 4 ! 0 (46) We use dual variable notation (see Ref. [48]) for the integrals. The external dual variables ! " 1 2 2 2 2 2 2 2 2 2 2 2 2 xa−2,a+3xa−1,a+1xa,a+2 − 2xa−2,a+2xa−1,a+1xa,a+3+ + xa−2,bxa−1,a+1xa+2,b − xa−2,a+1xa−1,bxa+2,b (35) 4# " are listed in clockwise direction. To the left loop we associate the dual variable xp and to x2 x2 x2 x2 x2 x2 1 2 2 2 2 2 1 2 2 2 2 2 2 + a−2,a+1 a−1,a+2 a,a+3 − a−2,a a−1,a+2 a+1,a+3 (18) x x x − x x (27) a + 1 b − 1 b 2 34 the right loop we associate the dual variable xq.Weusethenotationxij ≡ xi − xj. − x x x − x x (10) $ a−2,a a−1,b a+1,b−1 a−1,b−1 a+1,b 1 1 ab a,b−1 a−1,b ab a−1,b−1 4 ! " −   (36) −   (47) 4 ! " 4 2 We introduce the following notation which will be useful in the following bb+1 a − 1 678 4. One massive leg attached     abc··· 1 2 2 2 2 2 2 2 2 2 ! ! ! 2. One massless leg attached −xacxa+1,bxb+1,c−1 + xabxa+1,cxb+1,c−1+   = xaa! xbb! xcc! ···±(permutations of {a ,b,c,...}). (6) 4# a! b! c! ··· 0 (37) 0 (48) 1 2 2 2 2 2 2 2 2 2   x x x (19) + xa,c−1xa+1,bxb+1,c − xabxa+1,c−1xb+1,c (28) 4 a−2,a a−1,a+1 a,a+2 $ ! ! ! 1 The sign ± above takes into account the signature of the permutation of {a ,b,c,...}.It x2 x2 − x2 x2 x2 (11) 4 a,b+1 a+1,b ab a+1,b+1 a+2,b 1 a a +1 a +2 1 a − 2 a − 1 a is easy to show that ! " −   (38) −   (49) 0 (29) 4 4 1 2 2 2 2 2 2 a +2 a +3 a − 2 a + 2 b − 1 b abc··· 2 xa−1,a+1xa,b−1xa+1,b − xa−1,a+1xabxa+1,b−1 (20)       =detxij. (7) 4 ! ! ! i∈{a,b,c,··· } 1 ! " a b c ··· ! ! ! 2 2 2 2 2 2 j∈{a ,b ,c ,··· } −xa−1,bxa,a+2xa+1,b + xa−1,a+2xabxa+1,b−   4# 1 1 a − 3 a − 2 a − 1 x2 x2 x2 x2 x2 −   (50) 2 2 2 0 (30) a−3,a+1 a−2,a+2 − a−3,a+2 a−2,a+1 a,a+2 (39) 4 For some topologies, the expansion of the symbol yields terms that would cancel − xa−1,a+1xabxa+2,b (12) 4 a + 1 a +2 a +3 %& $ 0 (21) ' (   propagators. For those cases we make the convention that all the terms that would cancel propagators are absent. In fact, as we will see, terms that would cancel propagators of the 1 2 2 2 1 a + 1 b − 1 b − xabxa+1,bxb−1,b+1 (13) 6.Twomassivelegsattached −   (40) 4 4 0 (51) double pentagon topologies naturally yield coefficients for some of the topologies with a 1 2 2 2 2 2 2 b +1 b +2 a − 1 xacxa+1,bxb,c−1 − xabxa+1,cxb,c−1−   4# smaller number of propagators. 2 2 2 2 2 2 1 − xa,c−1xa+1,bxbc + xabxa+1,c−1xbc (22) − x2 x2 x2 (14) $ 0 (31) 4 a−3,a a−2,a a−1,a+1 0 (41) 0 (52)

11 12 13 14 15 16

C. Box-Pentagon topologies 4. One massless, one massive leg attached 2. One massless leg attached In the formula above we drop terms that would cancel propagators (in this case, the 1 a a +1 b − 1 2 −   (53) terms containing xpq). This expression has 78 terms when expanded. 4 b +1 c − 1 c 1 a − 2 a − 1 a a +1 1 a − 1 a a +1 a +2 1.Nolegsattached  −   =   4 4 0 (68) a +2 a +3 a − 3 q a +3 a − 3 a − 2 q 1 a + 1 a +2 b − 1 bp     −   (74) 6. Two massive legs attached 1 4 B. Kissing double-box topologies 1 −x2 x2 x2 x2 + x2 x2 x2 x2 − bb+1 a − 1 aq x2 x2 x2 x2 − x2 x2 (60) 4 a−3,a+1 a−2,a+3 a−1,q a,a+2 a−3,a−1 a−2,a+3 a+1,q a,a+2   4 ab a+1,q a,b+1 a−1,b ab a−1,b+1 1 a a +1 bb+1 % ! " −   . (69) − x2 x2 x2 x2 +2x2 x2 x2 x2 + 4 b +2 c − 1 cq a−3,a+2 a−2,a+1 a−1,q a,a+3 a−3,a+1 a−2,a+2 a−1,q a,a+3 In the formula above we drop terms that would cancel propagators (in this case, the 1 a a +1 b − 1 bp   2 2 2 2 2 2 2 2 −   (78) 1 2 + xa−3,a+1xa−2,a+3xa−1,a+2xaq + xa−3,a+2xa−2,a+1xa−1,a+3xaq− x2 x2 x2 2 2 2 2 2 2 Note that in the previous formula we suppress the terms containing xb+1,q which would terms are bp, bq and pq). This expression has 15 terms when expanded. 4 xa,a+2xa+1,q xa−1,a+2xa,a+3 − xa−1,a+3xa,a+2 (61) cc+1 d − 1 dq 1 a a +1 b − 1 b 1 a a +1 bb+1 2 2 2 2 2 2 2 2 2   −   +     = ! " otherwise cancel a propagator of the underlying topology. When expanded out, the expres- − 2xa−3,a+1xa−2,a+2xa−1,a+3xaq + xa−3,a+2xa−2,axa−1,qxa+1,a+3− 4 bb+1 a − 1 a 4 b − 1 b a − 1 a       sion above has 12 terms. − 2x2 x2 x2 x2 +2x2 x2 x2 x2 − 3. One massive leg attached In the formula above we drop terms that would cancel propagators (in this case, the terms 1 a−3,a a−2,a+2 a−1,q a+1,a+3 a−3,a−1 a−2,a+2 aq a+1,a+3 2 2 2 2 2 2 2 2 2. One massless leg attached xa−1,b+1xa+1,b−1 xab − xa−1,b−1xa+1,b+1 xab + 2 2 2 2 2 2 2 2 2 4 1 a − 2 a − 1 a a +1 − x x x x − x x x x + containing xpq). When expanded, the above expression contains 96 terms. The number of % & ' & ' −   . (70) a−3,a a−2,a+3 a−1,a+2 a+1,q a−3,a+2 a−2,a a−1,a+3 a+1,q 2 2 2 2 2 2 2 2 x x x x x x x 4 a +2 b − 1 bq 2 2 2 2 2 2 2 2 conformal dressings is 160 (the number of coefficients unrelated by symmetries is lower). + a−1,a+1 b−1,b+1 ab − a−1,b a,b+1 a+1,b−1 ab− +2x x x x −2x x x x . & '   a−3,a a−2,a+2 a−1,a+3 a+1,q a−3,a−1 a−2,a+2 a,a+3 a+1,q 1 a a +1 b − 1 bp 2 2 2 2 2 2 2 2 2 & x x x x x x x x 1 2 2 2 2 2 2 2 2 In the previous formula we suppress the terms containing xa+1,q which would otherwise −   (75) − a−1,b+1 a,b−1 a+1,b ab + a−1,b−1 a,b+1 a+1,b ab+ x x − x x x x − x x (62) (72) 4 4 a−1,b+1 ab a−1,b a,b+1 a+1,q a+2,b a+1,b a+2,q bb+1 c − 1 cq 2 2 2 2 ! "! " cancel a propagator of the underlying topology. We have written down this formula to emphasize how nontrivial it is. We suppress   E. Assembly of the result + xa−1,bxa,b−1xa+1,b+1xab (54) ( the terms containing x2 and x2 ,respectively.Thesetermswouldotherwisecancela 1 2 2 2 2 2 2 2 2 2 2 a−2,q a+2,q In the formula above we drop terms that would cancel propagators (in this case, the x x x x + x x x − x x x As explained in Sec. II, for the MHV amplitudes the ratio between the !-loop amplitude a a b b a a a a 4 a−1,b ab a+1,q b−1,b+1 a,b+1 a+1,b b−1,q ab a+1,b+1 b−1,q 1 + 1 +2 − 1 1 − 1 +1 +2 ! " 5. Two massless legs attached propagator of the underlying topology. We will see below that the box-pentagon topologies terms containing x2 , x2 or x2 ). This expression has 16 terms when expanded. −   +     (63) bp bq pq and the tree-level amplitude can be written as a sum between parity even and parity odd 4 bb+1 a − 1 a 4 b b +1 b − 1 b with massless legs attached to the vertices of the edge common to both loops can in fact be       contributions (55) 1 1 a a +1 b − 1 b seen to originate in double-pentagon topologies, by cancelling some propagators. 2 2 2 2 2 2 2 2   (71) 4. Two massless legs attached (!) (!),even (!),odd xa−4,axa−3,axa−2,qxa−1,a+1 − xa−4,a+1xa−3,axa−2,axa−1,q+ M = M + M . (79) 4# 4 b +1 b +2 a − 1 q n n n 1 a + 1 a +2 b − 1 b 1 a + 1 a +2 b +1 b +2 2 2 2 2 2 2 2 2   −   +     +2xa−4,axa−3,a+1xa−2,axa−1,q − xa−4,axa−3,axa−2,a+1xa−1,q In the previous formula we suppress the terms containing x2 which would otherwise Then, the even part can be written 4 b +1 b +2 a − 1 a 4 b − 1 b a − 1 a $ a+1,q D. Double pentagon topologies       (64) cancel a propagator of the underlying topology. 1 a + 1 a +2 b − 1 bp M (2),even = −π−De2γ# dDx dDx s c I , (80) (56)   n p q i i i − (76) % & & 1. No legs attached 4 b +1 b +2 a − 1 aq σ i∈Topologies   1 a a +1 b − 1 b 1 a a +1 bb+1 3. One massive leg attached where the first sum runs over cyclic and anti-cyclic permutations of the external legs, the −   +     (57) 4 bb+1 c − 1 c 4 b − 1 b c − 1 c In the formula we drop terms that would cancel propagators (in this case, the terms second sum runs over all the topologies, si is a symmetry factor associated to topology i,       1 a a +1 b − 1 bp 2 c is the numerator of the topology i,aslistedinSec.IVandI is the denominator or the −   (73) containing xpq). This expression has 64 terms when expanded. i i 0 (65) 4 1 a a +1 b − 1 b 1 a a +1 b +1 b +2 bb+1 a − 1 aq i −   +       product of propagators in the topology . 4 b +1 b +2 c − 1 c 4 b − 1 b c − 1 c       5. One massless, one massive leg attached Apart from the parity odd part which we have not computed, there is also a contribution (58) In the expansion of the above formula we drop terms that would cancel propagators (in 1 M (2),µ x2 x2 − x2 x2 x2 x2 − x2 x2 (66) which is not detectable from four-dimensional cuts, denoted by .Thispartofthe aq a+1,b ab a+1,q bc b+1,c−1 b,c−1 b+1,c this case, the terms containing x2 , x2 , x2 , x2 ,orx2 ). This expression has 6 terms when 4 ! "! " ap aq bp bq pq 1 a a +1 b − 1 b 1 a a +1 cc+1 result is such that its integrand vanishes in four dimensions, but the integral itself can give −  +     (59) expanded. 1 a a +1 b − 1 bp 4 cc+1 d − 1 d 4 b − 1 b d − 1 d 1 −   (77) contributions to the divergent and finite parts. In Ref. [32], for n =6case,thispartofthe       x2 x2 x2 x2 − x2 x2 (67) 4 4 a−1,a+1 aq a+1,b−1 a+2,b a+1,b a+2,b−1 b +1 b +2 c − 1 cq # M (1),µ ! "   result was found to be closely related to O( )contributionsatoneloop, . Based on previous computations we expect that the odd part and the µ integrals will not be needed in order to compare with the Wilson loop results. The odd parts could be

17 18 19 20 21 22

9 Perturbations of Parke-Taylor ✦ What about beyond the leading order?

[Arkani-Hamed, JB, Cachazo, Trnka (2010)]

9 Perturbations of Parke-Taylor ✦ What about beyond the leading order?

[Arkani-Hamed, JB, Cachazo, Trnka (2011)]

9 What About After Integration? ✦ Integrate the Parke-Taylor 2-to-4 amplitude in sYM ‣ divergences exponentiate, leaving a ﬁnite remainder ✦ Heroically computed by Del Duca, Duhr, Smirnov in 2010, in terms of ‘Goncharov’ polylogarithms

Preprint typeset in JHEP style - PAPER VERSION IPPP/10/21, DCPT/10/42 CERN-PH-TH/2010-059

The Two-Loop Hexagon Wilson Loop in =4SYM N

Vittorio Del Duca PHDepartment,THUnit,CERNCH-1211,Geneva23,Switzerland INFN, Laboratori Nazionali Frascati, 00044 Frascati (Roma), Italy E-mail: [email protected]

Claude Duhr Institute for Particle Physics Phenomenology, University of Durham Durham, DH1 3LE, U.K. E-mail: [email protected]

Vladimir A. Smirnov Nuclear Physics Institute of Moscow State University Moscow 119992, Russia E-mail: [email protected]

Abstract: In the planar =4supersymmetricYang-Millstheory,theconformalsym- [Del Duca,N Duhr, Smirnov (2010)] 10 metry constrains multi-loop n-edged Wilson loops to be given in terms of the one-loop n-edged Wilson loop, augmented, for n 6, by a function of conformally invariant cross ≥ ratios. That function is termed the remainder function. In a recent paper, we have dis- played the ﬁrst analytic computation of the two-loop six-edged Wilson loop, and thus of the corresponding remainder function. Although the calculation was performed in the quasi- multi-Regge kinematics of a pair along the ladder, the Regge exactness of the six-edged

arXiv:1003.1702v2 [hep-th] 10 Mar 2010 Wilson loop in those kinematics entails that the result is the same as in general kinematics. We show in detail how the most diﬃcult of the integrals is computed, which contribute to the six-edged Wilson loop. Finally, the remainder function is given as a function of uniform transcendental weight four in terms of Goncharov polylogarithms. We consider also some asymptotic values of the remainder function, and the value when all the cross ratios are equal.

Keywords: QCD, MSYM, small x. What About After Integration? ✦ Integrate the Parke-Taylor 2-to-4 amplitude in sYM ‣ divergences exponentiate, leaving a ﬁnite remainder

(2) 1 1 u 1 1 1 u 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 u 1 1 1 1 1 1 1 R (u1,u2,u3)= (H.1) 2 2 2 6,W L G , − , 0, 1;1 G , − , 0, ; 1 + G , 0, 0, ; 1 + G , 0, 0, ; 1 + G , 0, 0, ; 1 0, u231, , 0;1 + 0, u231, , 1;1 0, u231, , ; 1 , u123, , ; 1 + , u123, − , 1;1 , v213, , 0;1 , v213, , 1;1 + 1 1 u 1 1 1 1 1 1 1 4 1 u1 u1 + u2 1 − 4 1 u1 u1 + u2 1 1 u1 u3 u2 2 u3 u1 + u3 2 u3 u2 + u3 − 4G 1 u2 4G 1 u2 − 4G 1 u2 1 u2 − 4G 1 u1 1 u1 1 u1 4G 1 u1 u1 + u2 1 − 4G 1 u2 1 u2 − 2G 1 u2 1 u2 π2G , 2 − ; 1 + π2G , ; 1 + π2G , ; 1 + ! − − " ! − − − " ! " ! " ! " ! − " ! − " ! − − " ! − − − " ! − − " ! − − " ! − − " 24 1 u u + u 1 24 u u + u 24 u u + u 1 1 u2 1 1 1 u2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 u3 1 1 u3 1 1 1 1 u2 1 1 1 1 1 1 1 1 1 1 1 ! 1 1 2 " ! 1 1 2 " ! 1 1 3 " G , − , 1, 0;1 G , − , , 0;1 + G , 0, , ; 1 G , 0, , ; 1 G , 0, , ; 1 0, u231, − , 1;1 + 0, u231, − , ; 1 , u123, − , ; 1 + , u123, , 0;1 + , v213, , ; 1 + , v231, 0, 0; 1 1 −1 u 1− 1 1 1 1 1 1 4 1 u1 u1 + u2 1 − 4 1 u1 u1 + u2 1 1 u1 4 u3 u1 u1 + u3 − 4 u3 u2 u2 + u3 − 4 u3 u3 u1 + u3 − 4G u2 + u3 1 4G u2 + u3 1 1 u2 − 4G 1 u1 u1 + u2 1 1 u1 4G 1 u1 u3 4G 1 u2 1 u2 1 u2 4G 1 u2 − π2G , 3 − ; 1 + π2G , ; 1 + π2G , ; 1 + ! − − " ! − − − " ! " ! " ! " ! − " ! − − " ! − − − " ! − " ! − − − " ! − " 24 1 u u + u 1 24 u u + u 24 u u + u 1 1 u2 1 1 1 1 u2 1 1 1 1 1 1 1 1 2 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! 2 2 3 " ! 2 1 2 " ! 2 2 3 " G , − , , 1;1 G , − , , ; 1 G , 0, , ; 1 π , u123;1 + π , v123;1 + 0, u312, 0, ; 1 0, u312, , 0;1 0, u312, , 0;1 + , v123, 0, 0; 1 , v123, 0, 1; 1 + , v123, 0, ; 1 , v231, 0, 1; 1 + , v231, 0, ; 1 , v231, 1, 0; 1 1 −1 u 1− 1 1 1 1 1 1 4 1 u1 u1 + u2 1 1 u1 − 4 1 u1 u1 + u2 1 1 u1 1 u1 − 4 u3 u3 u2 + u3 − 24 G 1 u1 8 G 1 u1 4G 1 u3 − 4G u2 − 4G 1 u3 4G 1 u1 − 4G 1 u1 4G 1 u1 1 u1 − 4G 1 u2 4G 1 u2 1 u2 − 4G 1 u2 − π2G , 1 − ; 1 + π2G , ; 1 + π2G , ; 1 + ! − − − " ! − − − − " ! " ! − " ! − " ! − " ! " ! − " ! − " ! − " ! − − " ! − " ! − − " ! − " 24 1 u u + u 1 24 u u + u 24 u u + u 1 1 u2 1 u2 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! 3 1 3 " ! 3 1 3 " ! 3 2 3 " G , − , − , 1;1 + π , v132;1 π , u231;1 + π , v213;1 + 0, u312, , 1;1 0, u312, , ; 1 , v123, 1, 0; 1 , v123, 1, ; 1 + , v231, 1, ; 1 + , v231, , 0;1 3 1− 1 − 3 1 1 3 1 1 4 1 u1 u1 + u2 1 u1 + u2 1 8 G 1 u1 − 24 G 1 u2 8 G 1 u2 4G 1 u3 − 4G 1 u3 1 u3 − 4G 1 u1 − 2G 1 u1 1 u1 2G 1 u2 1 u2 4G 1 u2 1 u2 − ✦ G 0, 0, , ; 1 + G 0, 0, , ; 1 + G 0, 0, , ; 1 + ! − − − " ! − " ! − " ! − " ! − " ! − − " ! − " ! − − " ! − − " ! − − " 2 u u + u 2 u u + u 2 u u + u 1 1 u2 1 u2 1 1 1 1 1 2 1 1 2 1 1 2 1 1 u1 1 1 u1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! 1 1 2 " ! 1 1 3 " ! 2 1 2 " G , − , − , ; 1 G , 0, 0, ; 1 + π , v231;1 π , u312;1 + π , v312;1 + 0, u312, − , 1;1 + 0, u312, − , ; 1 + , v123, , 0;1 , v123, , 1;1 + , v231, , 1;1 + , v231, , ; 1 3 1 1 3 1 1 3 1 1 4 1 u1 u1 + u2 1 u1 + u2 1 1 u1 − u1 u2 8 G 1 u2 − 24 G 1 u3 8 G 1 u3 4G u1 + u3 1 4G u1 + u3 1 1 u3 4G 1 u1 1 u1 − 2G 1 u1 1 u1 2G 1 u2 1 u2 4G 1 u2 1 u2 1 u2 − G 0, 0, , ; 1 + G 0, 0, , ; 1 + G 0, 0, , ; 1 ! − − − − " ! " ! − " ! − " ! − " ! − " ! − − " ! − − " ! − − " ! − − " ! − − − " 2 u u + u 2 u u + u 2 u u + u − 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Heroically2 2 3 3 1 3 3 2 3 computedG , 0, 0, ; 1 G , 0, 0, ; 1 + G , 0, 0, ; 1 π , v321;1 by0, 0, , v123;1 0, 0, , v132;1Del0, v123, 0, ; 1 Duca,0, v123, 1, ; 1 + 0, v123, , 0;1 , v123, Duhr,, ; 1 + , v132, 0, 0; 1 Smirnov, 0, 0, v312;1 , 0, 0, v321;1 , 0, , v312;1 ! " ! " ! " 1 1 1 1 1 1 1 1 2 u1 u1 + u2 − u1 u3 2 u1 u1 + u3 − 8 G 1 u3 − 4G 1 u1 − 4G 1 u1 − 4G 1 u1 − 2G 1 u1 4G 1 u1 − 4G 1 u1 1 u1 1 u1 4G 1 u1 − 4G 1 u3 − 4G 1 u3 − 2G 1 u3 1 u3 − G 0, , 0, ; 1 + G 0, , 0, ; 1 G 0, , 0, ; 1 + ! " ! " ! " ! − " ! − " ! − " ! − " ! − " ! − " ! − − − " ! − " ! − " ! − " ! − − " 2 u u u u + u − 2 u u 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! 1 2 " ! 1 1 2 " ! 1 3 " G , 0, , ; 1 G , 0, , ; 1 G , 0, , ; 1 0, 0, , v213;1 0, 0, , v231;1 0, 0, , v312;1 0, v123, , 1;1 + 0, v123, , ; 1 0, v132, 0, ; 1 , v132, 0, 1; 1 + , v132, 0, ; 1 , v132, 1, 0; 1 , 0, , v321;1 , 0, v312, 1; 1 1 1 1 1 1 1 1 1 1 1 4 u1 u1 u1 + u2 − 4 u1 u1 u1 + u3 − 4 u1 u2 u1 + u2 − 4G 1 u2 − 4G 1 u2 − 4G 1 u3 − 2G 1 u1 4G 1 u1 1 u1 − 4G 1 u1 − 4G 1 u1 4G 1 u1 1 u1 − 4G 1 u1 − 2G 1 u3 1 u3 − 4G 1 u3 − G 0, , 0, ; 1 G 0, , , ; 1 G 0, , , ; 1 ! " ! " ! " ! − " ! − " ! − " ! − " ! − − " ! − " ! − " ! − − " ! − " ! − − " ! − " u u + u − 2 u u u + u − 2 u u u + u − 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! 1 1 3 " ! 1 1 1 2 " ! 1 1 1 3 " G , 0, , ; 1 G , 1, , 0;1 + 0, 0, , v321;1 0, 0, v123, ; 1 + (0, 0,v132, 0; 1) 0, v132, , 0;1 0, v132, , ; 1 0, v213, 0, ; 1 , v132, 1, ; 1 + , v132, , 0;1 , 0, v312, ; 1 , 0, v321, 1; 1 1 1 1 1 1 1 1 1 1 1 1 4 u1 u3 u1 + u3 − 4 1 u2 u1 4G 1 u3 − 4G 1 u1 G − 4G 1 u1 − 4G 1 u1 1 u1 − 4G 1 u2 − 2G 1 u1 1 u1 4G 1 u1 1 u1 − 4G 1 u3 1 u3 − 4G 1 u3 − G 0, , , ; 1 G 0, , , ; 1 G 0, , 0, ; 1 + ! " ! − " ! − " ! − " ! − " ! − − " ! − " ! − − " ! − − " ! − − " ! − " 2 u u u + u − 2 u u u + u − 2 u u 1 1 1 1 1 1 u3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! 1 2 1 2 " ! 1 3 1 3 " ! 2 1 " G , , 1, ; 1 + G , − , 0, 1;1 0, 0, v132, ; 1 + (0, 0,v213, 0; 1) 0, 0, v213, ; 1 0, v213, , 0;1 0, v213, , ; 1 + 0, v231, 0, ; 1 , v132, , 1;1 + , v132, , ; 1 , 0, v321, ; 1 , , 0, v312;1 1 1 1 1 1 1 1 2 1 u2 1 u2 1 u2 4 1 u2 u2 + u3 1 − 4G 1 u1 G − 4G 1 u2 − 4G 1 u2 − 4G 1 u2 1 u2 4G 1 u2 − 2G 1 u1 1 u1 4G 1 u1 1 u1 1 u1 − 4G 1 u3 1 u3 − 2G 1 u3 1 u3 − G 0, , 0, ; 1 G 0, , 0, ; 1 + G 0, , 0, ; 1 ! − − − " ! − − " ! − " ! − " ! − " ! − − " ! − " ! − − " ! − − − " ! − − " ! − − " u u + u − 2 u u u u + u − 1 1 u3 1 1 1 1 u3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 ! 2 1 2 " ! 2 3 " ! 2 2 3 " G , − , 0, ; 1 + G , − , 1, 0;1 0, 0, v231, ; 1 0, 0, v312, ; 1 + (0, 0,v321, 0; 1) 0, v231, 1, ; 1 + 0, v231, , 0;1 0, v231, , 1;1 + , 0, 0, v213;1 , 0, 0, v231;1 , 0, , v213;1 , , 0, v321;1 , , , v312;1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 u2 u2 + u3 1 1 u2 4 1 u2 u2 + u3 1 − 4G 1 u2 − 4G 1 u3 G − 2G 1 u2 4G 1 u2 − 2G 1 u2 4G 1 u2 − 4G 1 u2 − 2G 1 u2 1 u2 − 2G 1 u3 1 u3 − 4G 1 u3 1 u3 1 u3 − G 0, , , ; 1 G 0, , , ; 1 G 0, , , ; 1 ! − − − " ! − − " ! − " ! − " ! − " ! − " ! − " ! − " ! − " ! − − " ! − − " ! − − − " 2 u u u + u − 2 u u u + u − 2 u u u + u − 1 1 u3 1 1 1 1 u3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 ! 2 1 1 2 " ! 2 2 1 2 " ! 2 2 2 3 " G , − , , 0;1 + G , − , , 1;1 0, 0, v321, ; 1 0, , 0, v123;1 0, , 0, v132;1 0, v231, , ; 1 + 0, v312, 0, ; 1 0, v312, 1, ; 1 + , 0, , v231;1 , 0, v213, 1; 1 , , , v321;1 , , v312, 1; 1 1 1 1 1 1 u 1 1 4 1 u2 u2 + u3 1 1 u2 4 1 u2 u2 + u3 1 1 u2 − 4G 1 u3 − 4G 1 u1 − 4G 1 u1 − 4G 1 u2 1 u2 4G 1 u3 − 2G 1 u3 2G 1 u2 1 u2 − 4G 1 u2 − 4G 1 u3 1 u3 1 u3 − 2G 1 u3 1 u3 − 2 ! − − − " ! − − − " ! − " ! − " ! − " ! − − " ! − " ! − " ! − − " ! − " ! − − − " ! − − " G 0, , , ; 1 + G 0, − , 0, ; 1 + 1 1 u3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 u2 u3 u2 + u3 4 u1 + u2 1 1 u1 G , − , , ; 1 0, , , v ;1 0, , , v ;1 0, v , , 0;1 0, v , , 1;1 + 0, v , , ; 1 , 0, v , ; 1 , 0, v , 1; 1 , , v , ; 1 , , v , 1; 1 ! " ! − − " 4 1 u u + u 1 1 u 1 u − 2G 1 u 1 u 123 − 2G 1 u 1 u 132 − 4G 312 1 u − 2G 312 1 u 4G 312 1 u 1 u − 4G 1 u 213 1 u − 4G 1 u 231 − 4G 1 u 1 u 312 1 u − 2G 1 u 1 u 321 − 1 u2 1 1 1 u2 1 1 ! 2 2 3 2 2 " ! 1 1 " ! 1 1 " ! 3 " ! 3 " ! 3 3 " ! 2 2 " ! 2 " ! 3 3 3 " ! 3 3 " in 2010,G 0, − , , 0;1 G 0, − in, , 1;1 + terms1 −1 u 1− −u 1 − of1 −‘Goncharov’1 − 1 1 − −1 1 1 1 − 1 1 −1 1 −1 −1 polylogarithms1 −1 −1 1 −1 1 1 −1 −1 −1 1 −1 − 4 u + u 1 1 u − 4 u + u 1 1 u 3 3 ! 1 2 − − 1 " ! 1 2 − − 1 " G , − , − , 1;1 + 0, , v123, 1; 1 0, , v123, ; 1 0, , v132, 1; 1 0, v321, 0, ; 1 0, v321, , 0;1 0, v321, , ; 1 , 0, v231, ; 1 , , 0, v213;1 , , v321, ; 1 , u312, 0, 1; 1 + 1 u 1 1 1 1 u 1 u 1 1 4 1 u2 u2 + u3 1 u2 + u3 1 4G 1 u1 − 4G 1 u1 1 u1 − 4G 1 u1 − 4G 1 u3 − 4G 1 u3 − 4G 1 u3 1 u3 − 4G 1 u2 1 u2 − 2G 1 u2 1 u2 − 4G 1 u3 1 u3 1 u3 − 4G 1 u3 G 0, 2 − , , ; 1 G 0, 2 − , 2 − , ; 1 ! − − − " ! − " ! − − " ! − " ! − " ! − " ! − − " ! − − " ! − − " ! − − − " ! − " 4 u + u 1 1 u 1 u − 4 u + u 1 u + u 1 1 u − 1 1 u3 1 u3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 ! 1 2 − − 1 − 1 " ! 1 2 − 1 2 − − 1 " G , − , − , ; 1 G , 0, 0, ; 1 + 0, , v132, ; 1 0, , 0, v213;1 0, , 0, v231;1 , 0, 0, v123;1 , 0, 0, v132;1 , 0, , v123;1 , , 0, v231;1 , , , v213;1 , u312, 0, ; 1 , u312, 1, 0; 1 + 1 1 1 1 1 1 1 1 4 1 u2 u2 + u3 1 u2 + u3 1 1 u2 − u2 u1 4G 1 u1 1 u1 − 4G 1 u2 − 4G 1 u2 − 4G 1 u1 − 4G 1 u1 − 2G 1 u1 1 u1 − 2G 1 u2 1 u2 − 4G 1 u2 1 u2 1 u2 − 4G 1 u3 1 u3 − 4G 1 u3 G 0, , 0, ; 1 G 0, , 0, ; 1 + G 0, , 0, ; 1 + ! − − − − " ! " ! − − " ! − " ! − " ! − " ! − " ! − − " ! − − " ! − − − " ! − − " ! − " 2 u u − 2 u u u u + u 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 ! 3 1 " ! 3 2 " ! 3 1 3 " G , 0, 0, ; 1 G , 0, 0, ; 1 + G , 0, 0, ; 1 0, , , v213;1 0, , , v231;1 , 0, , v132;1 , 0, v123, 1; 1 , , , v231;1 , , v213, 1; 1 , u312, , 0;1 + , u312, , 0;1 1 1 1 1 1 1 1 1 1 1 2 u2 u1 + u2 − u2 u3 2 u2 u2 + u3 − 2G 1 u2 1 u2 − 2G 1 u2 1 u2 − 2G 1 u1 1 u1 − 4G 1 u1 − 4G 1 u2 1 u2 1 u2 − 2G 1 u2 1 u2 − 4G 1 u3 u2 4G 1 u3 1 u3 − G 0, , 0, ; 1 G 0, , , ; 1 G 0, , , ; 1 ! " ! " ! " ! − − " ! − − " ! − − " ! − " ! − − − " ! − − " ! − " ! − − " u u + u − 2 u u u + u − 2 u u u + u − 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! 3 2 3 " ! 3 1 1 3 " ! 3 2 2 3 " G , 0, , ; 1 G , 0, , ; 1 G , 0, , ; 1 0, , v213, 1; 1 0, , v213, ; 1 0, , v231, 1; 1 , 0, v123, ; 1 , 0, v132, 1; 1 , , v213, ; 1 , , v231, 1; 1 , u312, , 1;1 + , u312, , ; 1 + 1 1 1 1 1 1 1 1 4 u2 u1 u1 + u2 − 4 u2 u2 u1 + u2 − 4 u2 u2 u2 + u3 − 4G 1 u2 − 4G 1 u2 1 u2 − 4G 1 u2 − 4G 1 u1 1 u1 − 4G 1 u1 − 4G 1 u2 1 u2 1 u2 − 2G 1 u2 1 u2 − 4G 1 u3 1 u3 4G 1 u3 1 u3 1 u3 G 0, , , ; 1 G 0, , , ; 1 + ! " ! " ! " ! − " ! − − " ! − " ! − − " ! − " ! − − − " ! − − " ! − − " ! − − − " 2 u u u + u − 2 u u u + u 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 u1 1 1 1 u1 1 1 ! 3 3 1 3 " ! 3 3 2 3 " G , 0, , ; 1 G , 1, , 0;1 + 0, , v231, ; 1 0, , 0, v312;1 0, , 0, v321;1 , 0, v132, ; 1 , , 0, v123;1 , , v231, ; 1 , u231, 0, 1; 1 + , u312, − , 1;1 , u312, − , ; 1 + 1 u 1 1 1 u 1 1 4 u2 u3 u2 + u3 − 4 1 u3 u2 4G 1 u2 1 u2 − 4G 1 u3 − 4G 1 u3 − 4G 1 u1 1 u1 − 2G 1 u1 1 u1 − 4G 1 u2 1 u2 1 u2 − 4G 1 u2 4G 1 u3 u1 + u3 1 − 4G 1 u3 u1 + u3 1 1 u3 G 0, 1 − , 0, ; 1 + G 0, 1 − , , 0;1 ! " ! − " ! − − " ! − " ! − " ! − − " ! − − " ! − − − " ! − " ! − − " ! − − − " 4 u + u 1 1 u 4 u + u 1 1 u − 1 1 1 1 1 1 u1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! 1 3 − − 3 " ! 1 3 − − 3 " G , , 1, ; 1 + G , − , 0, 1;1 0, , , v312;1 0, , , v321;1 , , 0, v132;1 , , , v123;1 , u231, 0, ; 1 , u231, 1, 0; 1 + , v312, 0, 0; 1 , v312, 0, 1; 1 + , v312, 0, ; 1 1 u 1 1 1 u 1 1 1 2 1 u3 1 u3 1 u3 4 1 u3 u1 + u3 1 − 2G 1 u3 1 u3 − 2G 1 u3 1 u3 − 2G 1 u1 1 u1 − 4G 1 u1 1 u1 1 u1 − 4G 1 u2 1 u2 − 4G 1 u2 4G 1 u3 − 4G 1 u3 4G 1 u3 1 u3 − G 0, 1 − , , 1;1 + G 0, 1 − , , ; 1 ! − − − " ! − − " ! − − " ! − − " ! − − " ! − − − " ! − − " ! − " ! − " ! − " ! − − " 4 u + u 1 1 u 4 u + u 1 1 u 1 u − 1 1 u1 1 1 1 1 u1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! 1 3 − − 3 " ! 1 3 − − 3 − 3 " G , − , 0, ; 1 + G , − , 1, 0;1 0, , v312, 1; 1 0, , v312, ; 1 0, , v321, 1; 1 , , , v132;1 , , v123, 1; 1 , u231, , 0;1 + , u231, , 0;1 , v312, 1, 0; 1 , v312, 1, ; 1 + 1 u 1 u 1 1 1 u 1 1 4 1 u3 u1 + u3 1 1 u3 4 1 u3 u1 + u3 1 − 4G 1 u3 − 4G 1 u3 1 u3 − 4G 1 u3 − 4G 1 u1 1 u1 1 u1 − 2G 1 u1 1 u1 − 4G 1 u2 u1 4G 1 u2 1 u2 − 4G 1 u3 − 2G 1 u3 1 u3 G 0, 1 − , 1 − , ; 1 + G 0, 3 − , 0, ; 1 + ! − − − " ! − − " ! − " ! − − " ! − " ! − − − " ! − − " ! − " ! − − " ! − " ! − − " 4 u + u 1 u + u 1 1 u 4 u + u 1 1 u 1 1 u1 1 1 1 1 u1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! 1 3 − 1 3 − − 3 " ! 2 3 − − 2 " G , − , , 0;1 + G , − , , 1;1 0, , v321, ; 1 0, u123, 0, ; 1 0, u123, , 0;1 + , , v123, ; 1 , , v132, 1; 1 , u231, , 1;1 + , u231, , ; 1 + , v312, , 0;1 , v312, , 1;1 + 1 u 1 1 1 u 1 1 4 1 u3 u1 + u3 1 1 u3 4 1 u3 u1 + u3 1 1 u3 − 4G 1 u3 1 u3 − 4G 1 u1 − 4G 1 u1 4G 1 u1 1 u1 1 u1 − 2G 1 u1 1 u1 − 4G 1 u2 1 u2 4G 1 u2 1 u2 1 u2 4G 1 u3 1 u3 − 2G 1 u3 1 u3 G 0, 3 − , , 0;1 G 0, 3 − , , 1;1 + ! − − − " ! − − − " ! − − " ! − " ! − " ! − − − " ! − − " ! − − " ! − − − " ! − − " ! − − " 4 u + u 1 1 u − 4 u + u 1 1 u 1 1 u1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 u3 1 1 1 u3 1 1 1 1 1 1 1 1 ! 2 3 − − 2 " ! 2 3 − − 2 " G , − , , ; 1 0, u123, , 1;1 0, u123, , ; 1 , , v132, ; 1 , u123, 0, 1; 1 + , u231, − , 1;1 , u231, − , ; 1 + , v312, , ; 1 + , v321, 0, 0; 1 1 u3 1 1 1 1 u3 1 u3 1 1 4 1 u3 u1 + u3 1 1 u3 1 u3 − 4G 1 u1 − 4G 1 u1 1 u1 − 4G 1 u1 1 u1 1 u1 − 4G 1 u1 4G 1 u2 u2 + u3 1 − 4G 1 u2 u2 + u3 1 1 u2 4G 1 u3 1 u3 1 u3 4G 1 u3 − G 0, − , , ; 1 G 0, − , − , ; 1 ! − − − − " 4 ! − " ! − − " ! − − − " ! − " ! − − " ! − − − " ! − − − " ! − " 1 Preprint1 u 1 typesetu 1 79π in JHEP style - PAPER1 VERSIONu2 1 1 u2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 u2 + u3 1 1 u2 1 u2 − 4 u2 + u3 1 u2 + u3 1 1 u2 − 1 1 IPPP/10/21, DCPT/10/42 ! − − − " ! − − − " G , − , − , 1;1 + 0, u123, − , 1;1 + 0, u123, − , ; 1 , u123, 0, ; 1 , u123, 1, 0; 1 + , v213, 0, 0; 1 , v213, 0, 1; 1 + , v213, 0, ; 1 , v321, 0, 1; 1 + , v321, 0, ; 1 , v321, 1, 0; 1 1 1 1 1 1 1 1 4 1 u u + u 1 u + u 1 − 360 4G u1 + u2 1 4G u1 + u2 1 1 u1 − 4G 1 u1 1 u1 − 4G 1 u1 4G 1 u2 − 4G 1 u2 4G 1 u2 1 u2 − 4G 1 u3 4G 1 u3 1 u3 − 4G 1 u3 − G , 1, , 0;1 + G , , 1, ; 1 + ! − 3 1 3 − 1 3 − " ! − " ! − − " ! − − " ! − " ! − " ! − " ! − − " ! − " ! − − " ! − " 4 1 u u 2 1 u 1 u 1 u 1 1 u1 1 u1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ! − 1 3 " ! − 1 − 1 − 1 " G , − , − , ; 1 G , 0, 0, ; 1 0, u123, , 0;1 0, u231, 0, ; 1 0, u231, , 0;1 , u123, , 0;1 , u123, , 1;1 + , v213, 1, 0; 1 , v213, 1, ; 1 + , v321, 1, ; 1 + , v321, , 0;1 4 1 u u + u 1 u + u 1 1 u − u u − 4G u3 − 4G 1 u2 − 4G u1 − 4G 1 u1 1 u1 − 4G 1 u1 1 u1 4G 1 u2 − 2G 1 u2 1 u2 2G 1 u3 1 u3 4G 1 u3 1 u3 − ! − 3 1 3 − 1 3 − − 3 " ! 3 1 " ! " ! − " ! " ! − − " ! C− ERN-PH-TH/2010-059− " ! − " ! − − " ! − − " ! − − "

–99– –100– –101– –102– –103– –104– 1 1 1 1 1 1 1 1 1 u 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , v , , 1;1 + , v , , ; 1 + G , 3 − , 1;1 H (0; u ) , 0, v ;1 H (0; u )+ , , v ;1 H (0; u ) 0, , v ;1 H (0; u ) 0, , v ;1 H (0; u )+ v , 1, ; 1 H (0; u ) v , , 1;1 H (0; u )+ 0, , v ;1 H (0; u )+ 0, , v ;1 H (0; u ) 2G 1 u 321 1 u 4G 1 u 321 1 u 1 u 4 1 u u + u 1 1 − 4G 1 u 321 1 2G 1 u 1 u 312 1 − 4G 1 u 123 2 − 4G 1 u 132 2 4G 132 1 u 2 − 4G 132 1 u 2 4G 1 u 312 3 4G 1 u 321 3 − ! 3 3 " ! 3 3 3 " ! 2 2 3 " ! 3 " ! 3 3 " ! 1 " ! 1 " ! 1 " ! 1 " ! 3 " ! 3 " 1 − 1 − 1 − 1 − 1 − 1 1 −1 u 1− 1 1 1 1 1 −1 1 1− 1− 1 −1 1 −1 1 −1 1 −1 1 − 1 1 − 1 v , 0, 1, ; 1 + v , 0, , 1;1 + v , 1, 0, ; 1 G , 3 − , ; 1 H (0; u )+ G , 0, ; 1 H (0; u ) , , v ;1 H (0; u ) , u , 0; 1 H (0; u ) 0, , v ;1 H (0; u )+ 0, , v ;1 H (0; u ) v , 1, ; 1 H (0; u )+ v , , 1;1 H (0; u )+ 0, u , ; 1 H (0; u ) 0, u , ; 1 H (0; u )+ 2G 123 1 u 2G 123 1 u 2G 123 1 u − 4 1 u u + u 1 1 u 1 2 u u 1 − 2G 1 u 1 u 321 1 − 4G 1 u 312 1 − 4G 1 u 213 2 4G 1 u 231 2 − 4G 213 1 u 2 4G 213 1 u 2 4G 123 1 u 3 − 4G 123 u 3 ! 1 " ! 1 " ! 1 " ! 2 2 3 2 " ! 2 1 " ! 3 3 " ! 3 " ! 2 " ! 2 " ! 2 " ! 2 " ! 1 " ! 3 " 5 −1 1 −1 5 1− 1 1− 1 − − 1 1 1 1 1 −1 − 1 1 −1 u 1 1 −1 1 −1 1 −1 1 −1 1 −1 1 u 1 v , 1, 1, ; 1 + v , 1, , 0;1 v , 1, , 1;1 + G , 0, ; 1 H (0; u )+ G , , ; 1 H (0; u )+ , u , ; 1 H (0; u )+ , u , 1 − ; 1 H (0; u )+ 0, , v ;1 H (0; u )+ 0, , v ;1 H (0; u )+ v , 1, ; 1 H (0; u )+ v , , 1;1 H (0; u ) 0, u , ; 1 H (0; u ) 0, u , 3 − ; 1 H (0; u ) 4G 123 1 u 2G 123 1 u − 4G 123 1 u 4 u u + u 1 4 u u u + u 1 4G 1 u 312 1 u 1 4G 1 u 312 u + u 1 1 4G 1 u 312 2 4G 1 u 321 2 4G 231 1 u 2 4G 231 1 u 2 − 4G 231 1 u 3 − 4G 231 u + u 1 3 − ! 1 " ! 1 " ! 1 " ! 2 1 2 " ! 2 1 1 2 " ! 3 3 " ! 3 1 3 " ! 3 " ! 3 " ! 2 " ! 2 " ! 2 " ! 2 3 " 1 1− 1 1 − 1 1 − 1 1 1 u 1 1 1 u 1 1 1 −1 − 1 1 − 1 − 1 − 1 1 − u 1 3 −1 3 −1 1 −1 1 1 − v , 1, , ; 1 + v , , 0, 1;1 + v , , 1, 0;1 G , 1 − , 0;1 H (0; u )+ G , 1 − , ; 1 H (0; u ) , v , 0; 1 H (0; u )+ , v , ; 1 H (0; u ) 0, u , ; 1 H (0; u ) 0, u , 2 − ; 1 H (0; u ) v , 1, ; 1 H (0; u ) v , , 1;1 H (0; u )+ 0, v , ; 1 H (0; u )+ 0, v , ; 1 H (0; u )+ 2G 123 1 u 1 u 2G 123 1 u 2G 123 1 u − 4 1 u u + u 1 1 4 1 u u + u 1 1 u 1 − 4G 1 u 312 1 2G 1 u 312 1 u 1 − 4G 123 1 u 2 − 4G 123 u + u 1 2 − 4G 312 1 u 2 − 4G 312 1 u 2 2G 123 1 u 3 2G 231 1 u 3 ! 1 1 " ! 1 " ! 1 " ! 3 1 3 " ! 3 1 3 3 " ! 3 " ! 3 3 " ! 1 " ! 1 2 " ! 3 " ! 3 " ! 1 " ! 2 " 5 1− − 1 1 − 1 − 1 −1 u 1− u 1 − 1 1 − 1 − 1 −1 1 −1 −1 1 1− 1 1 − 1 −1 1 −1 1 −1 1 −1 v , , 1, 1;1 + v , , 1, ; 1 + G , 1 − , 1 − ; 1 H (0; u )+ G , 0, ; 1 H (0; u ) , v , 0; 1 H (0; u ) , v , ; 1 H (0; u )+ 0, u , ; 1 H (0; u ) 0, u , ; 1 H (0; u )+ v , 1, ; 1 H (0; u )+ v , , 1;1 H (0; u )+ 0, v , ; 1 H (0; u )+ 0, v , ; 1 H (0; u ) 4G 123 1 u 2G 123 1 u 1 u 4 1 u u + u 1 u + u 1 1 2 u u 1 − 4G 1 u 321 1 − 2G 1 u 321 1 u 1 4G 312 u 2 − 4G 312 1 u 2 4G 321 1 u 2 4G 321 1 u 2 4G 312 1 u 3 4G 321 1 u 3 − ! 1 " ! 1 1 " ! 3 1 3 1 3 " ! 3 1 " ! 3 " ! 3 3 " ! 2 " ! 3 " ! 3 " ! 3 " ! 3 " ! 3 " 1 −1 1 1 − −1 1 1 1 1− 1 − −1 1 1 1 1 − 1 1 − 1 − 1 1 1 − 1 1 1 1 − 1 −1 1 1 1 − 1 1 − v123, , , 1;1 v132, 1, 1, ; 1 v132, 1, , 1;1 G , 0, ; 1 H (0; u1)+ G , , ; 1 H (0; u1)+ v123, 1, ; 1 H (0; u1)+ v123, , 1;1 H (0; u1)+ 0, v123, ; 1 H (0; u2)+ 0, v213, ; 1 H (0; u2)+ G , ; 1 H (0; u1) H (0; u2)+ G , ; 1 H (0; u1) H (0; u2)+ , 0, v123;1 H (0; u3)+ , 0, v132;1 H (0; u3) 2G 1 u1 1 u1 − 4G 1 u1 − 4G 1 u1 − 4 u3 u1 + u3 4 u3 u1 u1 + u3 4G 1 u1 4G 1 u1 2G 1 u1 4G 1 u2 =4 4 u1 u1 + u2 4 u2 u1 + u2 4G 1 u1 4G 1 u1 − ! − − " ! − " ! − " ! " ! " ! − " ! − " ! − " ! − " ! " ! " ! − " ! − " 1 1 1 1 1 1 1 The1 1 Two-Loop1 Hexagon1 1 1 1 Wilson1 Loop1 1 in1 SYM1 1 u1 1 1 1 1 1 1 1 v , , 1, 1;1 v , 1, 1, ; 1 v , 1, , 1;1 0, , v ;1 H (0; u )+ 0, , v ;1 H (0; u )+ v , 1, ; 1 H (0; u )+ v , , 1;1 H (0; u )+ 0, v , ; 1 H (0; u ) 0, v , ; 1 H (0; u )+ G , − ; 1 H (0; u ) H (0; u ) , , v ;1 H (0; u )+ , , v ;1 H (0; u ) 4G 132 1 u − 4G 213 1 u − 4G 213 1 u − 4G 1 u 123 1 4G 1 u 132 1 4G 132 1 u 1 4G 132 1 u 1 4G 231 1 u 2 − 2G 312 1 u 2 4 1 u u + u 1 1 2 − 2G 1 u 1 u 123 3 2G 1 u 1 u 132 3 − ! 1 " ! 2 " ! 2 " ! 1 " ! 1 " ! 1 " ! 1 " ! 2 " ! 3 " ! 3 1 3 " ! 1 1 " ! 1 1 " 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 1 − 1 1 − 1 −1 − 1 1 1 −1 − 1 1 − −1 v , , 1, 1;1 + v , 0, 1, ; 1 + v , 0, , 1;1 + 0, , v ;1 H (0; u ) 0, , v ;1 H (0; u )+ v , 1, ; 1 H (0; u )+ v , , 1;1 H (0; u ) , 0, v ;1 H (0; u ) , 0, v ;1 H (0; u )+ , u ;1 H (0; u ) H (0; u ) , v ;1 H (0; u ) H (0; u ) , u , 1; 1 H (0; u )+ , u , ; 1 H (0; u )+ 4G 213 1 u 2G 231 1 u 2G 231 1 u 4G 1 u 213 1 − 4G 1 u 231 1 4G 213 1 u 1 4G 213 1 u 1 − 4G 1 u 123 2 − 4G 1 u 132 2 4G 1 u 312 1 2 − 4G 1 u 312 1 2 − 4G 1 u 123 3 4G 1 u 123 1 u 3 ! 2 " ! 2 " ! 2 " ! 2 " ! 2 " ! 2 " ! 2 " ! 1 " ! 1 " ! 3 " ! 3 " ! 1 " ! 1 1 " 1 − 1 5 −1 1 −1 1 −1 1 −1 3 −1 3 −1 1 −1 1 1− 1 N1 1 −1 5 − 1 −1 1 1 − 1 − v , 1, 0, ; 1 v , 1, 1, ; 1 + v , 1, , 0;1 0, , v ;1 H (0; u ) 0, , v ;1 H (0; u ) v , 1, ; 1 H (0; u ) v , , 1;1 H (0; u )+ , , v ;1 H (0; u ) , , v ;1 H (0; u ) , v ;1 H (0; u ) H (0; u )+ π2H (0; u ) H (0; u ) , u , ; 1 H (0; u ) , v , 0; 1 H (0; u ) 2G 231 1 u − 4G 231 1 u 2G 231 1 u − 4G 1 u 312 1 − 4G 1 u 321 1 − 4G 231 1 u 1 − 4G 231 1 u 1 2G 1 u 1 u 123 2 − 2G 1 u 1 u 132 2 − 4G 1 u 321 1 2 24 1 2 − 4G 1 u 123 u 3 − 4G 1 u 123 3 − ! 2 " ! 2 " ! 2 " ! 3 " ! 3 " ! 2 " ! 2 " ! 1 1 " ! 1 1 " ! 3 " ! 1 3 " ! 1 " 5 1− 1 1− 1 1 − 1 1 − 1 1 − 1 3 −1 3 −1 1 −1 − 1 1 − −1 1 −1 1 1 1 1 1 −1 1 1 − 1 v , 1, , 1;1 + v , 1, , ; 1 + v , , 0, 1;1 + 0, u , ; 1 H (0; u ) 0, u , ; 1 H (0; u )+ v , 1, ; 1 H (0; u )+ v , , 1;1 H (0; u ) , u , 0; 1 H (0; u ) , u , ; 1 H (0; u )+ G 0, , ; 1 H (0; u ) G 0, , ; 1 H (0; u )+ , v , ; 1 H (0; u )+ , v , 0; 1 H (0; u )+ 4G 231 1 u 2G 231 1 u 1 u 2G 231 1 u 4G 231 u 1 − 4G 231 1 u 1 4G 312 1 u 1 4G 312 1 u 1 − 4G 1 u 123 2 − 4G 1 u 123 1 u 2 4 u u + u 3 − 4 u u + u 3 2G 1 u 123 1 u 3 4G 1 u 132 3 ! 2 " ! 2 2 " ! 2 " ! 1 " ! 2 " ! 3 " ! 3 " ! 1 " ! 1 1 " ! 1 1 3 " ! 2 2 3 " ! 1 1 " ! 1 " 1 1− 5 1− − 1 1 − 1 1 1 1 − u 1 1 −1 1 −1 1 −1 u 1 − 1 1− 1 u 1 1 3 1 1 1 −1 −1 1 −1 v , , 1, 0;1 v , , 1, 1;1 + v , , 1, ; 1 + 0, u , ; 1 H (0; u ) 0, u , 1 − ; 1 H (0; u )+ v , 1, ; 1 H (0; u ) v , , 1;1 H (0; u ) , u , 2 − ; 1 H (0; u )+ , v , 0; 1 H (0; u )+ G 0, 2 − , ; 1 H (0; u ) G 0, , ; 1 H (0; u ) , v , ; 1 H (0; u ) , 0, v ;1 H (0; u )+ 2G 231 1 u − 4G 231 1 u 2G 231 1 u 1 u 4G 312 1 u 1 − 4G 312 u + u 1 1 4G 321 1 u 1 − 4G 321 1 u 1 − 4G 1 u 123 u + u 1 2 4G 1 u 123 2 4 u + u 1 1 u 3 − 4 u u + u 3 − 2G 1 u 132 1 u 3 − 4G 1 u 213 3 ! 2 " ! 2 " ! 2 2 " ! 3 " ! 1 3 " ! 3 " ! 3 " ! 1 1 2 " ! 1 " ! 1 2 1 " ! 3 1 3 " ! 1 1 " ! 2 " 1 −1 1 1 − 1 1 − 1− 1 −1 1 1 − 1 1 −1 3 1 − 1 1 −1 1 − 1 1 − 3 1 1− − 1 u 1 1 1 −1 − 1 1 − 1 v , , , 1;1 + v , 0, 1, ; 1 + v , 0, , 1;1 + 0, v , ; 1 H (0; u )+ 0, v , ; 1 H (0; u ) G 0, , ; 1 H (0; u ) G 0, , ; 1 H (0; u ) , v , ; 1 H (0; u ) , v , 0; 1 H (0; u ) G 0, , ; 1 H (0; u ) G 0, 3 − , ; 1 H (0; u ) , 0, v ;1 H (0; u ) , , v ;1 H (0; u )+ 2G 231 1 u 1 u 2G 312 1 u 2G 312 1 u 4G 123 1 u 1 4G 132 1 u 1 − 4 u u + u 2 − 4 u u + u 2 − 2G 1 u 123 1 u 2 − 4G 1 u 132 2 − 4 u u + u 3 − 4 u + u 1 1 u 3 − 4G 1 u 231 3 − 2G 1 u 1 u 213 3 ! 2 2 " ! 3 " ! 3 " ! 1 " ! 1 " ! 1 1 2 " ! 2 1 2 " ! 1 1 " ! 1 " ! 3 2 3 " ! 2 3 2 " ! 2 " ! 2 2 " 1 − 1 − 5 1 − 1 1 − 1 −1 1 −1 3 1 1 1 u 1 1 1 −1 −1 1 −1 1 1 1 1 1 u− 1 − 1 −1 1 1− 1− v , 1, 0, ; 1 v , 1, 1, ; 1 + v , 1, , 0;1 0, v , ; 1 H (0; u )+ 0, v , ; 1 H (0; u )+ G 0, , ; 1 H (0; u ) G 0, 2 − , ; 1 H (0; u ) , v , ; 1 H (0; u )+ , 0, v ;1 H (0; u )+ G , 1, ; 1 H (0; u )+ G , 2 − , 1;1 H (0; u ) , , v ;1 H (0; u ) , u , 0; 1 H (0; u ) 2G 312 1 u − 4G 312 1 u 2G 312 1 u − 2G 231 1 u 1 2G 312 1 u 1 4 u u + u 2 − 4 u + u 1 1 u 2 − 2G 1 u 132 1 u 2 4G 1 u 213 2 4 1 u u 3 4 1 u u + u 1 3 − 2G 1 u 1 u 231 3 − 4G 1 u 231 3 − ! 3 " ! 3 " ! 3 " ! 2 " ! 3 " ! 2 2 3 " ! 1 2 1 " ! 1 1 " ! 2 " ! 1 3 " ! 1 1 2 " ! 2 2 " ! 2 " 5 1− 1 1− 1 1 − 1 1 1 − 1 1 − 1 1 1 1 u 1− −1 1 −1 − 1 1 − 1 1 −1 u 1 1 − 1 1− 1 1 −1 − 1 1 −1 u 1 v , 1, , 1;1 + v , 1, , ; 1 + v , , 0, 1;1 + , 0, v ;1 H (0; u )+ , 0, v ;1 H (0; u )+ G 0, , ; 1 H (0; u )+ G 0, 1 − , ; 1 H (0; u )+ , 0, v ;1 H (0; u )+ , , v ;1 H (0; u )+ G , 2 − , ; 1 H (0; u )+ G , 0, ; 1 H (0; u ) , u , ; 1 H (0; u )+ , u , 3 − ; 1 H (0; u ) 4G 312 1 u 2G 312 1 u 1 u 2G 312 1 u 4G 1 u 123 1 4G 1 u 132 1 4 u u + u 2 4 u + u 1 1 u 2 4G 1 u 231 2 2G 1 u 1 u 213 2 4 1 u u + u 1 1 u 3 2 u u 3 − 4G 1 u 231 1 u 3 4G 1 u 231 u + u 1 3 − ! 3 " ! 3 3 " ! 3 " ! 1 " ! 1 " ! 3 2 3 " ! 1 3 3 " ! 2 " ! 2 2 " ! 1 1 2 1 " ! 1 3 " ! 2 2 " ! 2 2 3 " 1 1− 5 1− − 1 1 − 1 1 −1 1 1− 1 1 1 1 u 1 1 1− −u 1 1 1 −1 1 1− 1− 1 1− 1 − − 1 1 1 1 1 −1 − 1 1 − 1 − v , , 1, 0;1 v , , 1, 1;1 + v , , 1, ; 1 + , , v ;1 H (0; u )+ , , v ;1 H (0; u )+ G , 2 − , 0;1 H (0; u )+ G , 2 − , ; 1 H (0; u ) , , v ;1 H (0; u )+ , v , 1; 1 H (0; u )+ G , 0, ; 1 H (0; u )+ G , , ; 1 H (0; u )+ , v , 0; 1 H (0; u ) , v , ; 1 H (0; u )+ 2G 312 1 u − 4G 312 1 u 2G 312 1 u 1 u 2G 1 u 1 u 123 1 2G 1 u 1 u 132 1 4 1 u u + u 1 2 4 1 u u + u 1 1 u 2 − 2G 1 u 1 u 231 2 4G 1 u 213 2 4 u u + u 3 4 u u u + u 3 4G 1 u 213 3 − 2G 1 u 213 1 u 3 ! 3 " ! 3 " ! 3 3 " ! 1 1 " ! 1 1 " ! 1 1 2 " ! 1 1 2 1 " ! 2 2 " ! 2 " ! 1 1 3 " ! 1 3 1 3 " ! 2 " ! 2 2 " 1 −1 1 1 − 1 1 − 1− 1 −1 − 1 1 − −1 1 −1 u 1− u 1 − 1 1 − 1 − 1 −1 − 1 1 −1 1 1 u 1 1 1 u 1 1 1 −1 1 −1 −1 v , , , 1;1 v , 1, 1, ; 1 v , 1, , 1;1 Vittorio, v , 1; 1 H (0; u )+ Del, v , Duca; 1 H (0; u )+ G , 2 − , 2 − ; 1 H (0; u )+ G , 0, ; 1 H (0; u ) , v , ; 1 H (0; u )+ , v , 1; 1 H (0; u )+ G , 3 − , 0;1 H (0; u )+ G , 3 − , ; 1 H (0; u ) , v , 0; 1 H (0; u )+ , v , ; 1 H (0; u )+ 2G 312 1 u 1 u − 4G 321 1 u − 4G 321 1 u − 4G 1 u 123 1 4G 1 u 123 1 u 1 4 1 u u + u 1 u + u 1 2 2 u u 2 − 4G 1 u 213 1 u 2 4G 1 u 231 2 4 1 u u + u 1 3 4 1 u u + u 1 1 u 3 − 4G 1 u 231 3 2G 1 u 231 1 u 3 ! 3 3 " ! 3 " ! 3 " ! 1 " ! 1 1 " ! 1 1 2 1 2 " ! 1 2 " ! 2 2 " ! 2 " ! 2 2 3 " ! 2 2 3 2 " ! 2 " ! 2 2 " 1 −1 − 3 1 1 − − 1 −1 1 −1 −1 1 1− 1 − −1 1 1 1 1 −1 −1 1 −1 1 −1 u 1− u 1 − 1 1 − 1 − 1 −1 1 −1 − v , , 1, 1;1 G 0, , ; 1 H (0; u ) , v , 1; 1 H (0; u )+ , v , ; 1 H (0; u )+ G , 0, ; 1 H (0; u )+ G , , ; 1 H (0; u ) , v , ; 1 H (0; u ) , 0, v ;1 H (0; u )+ G , 3 − , 3 − ; 1 H (0; u )+ G , 0, ; 1 H (0; u ) , 0, v ;1 H (0; u )+ , 0, v ;1 H (0; u )+ 4G 321 1 u − 4 u u + u 1 − 4G 1 u 132 1 4G 1 u 132 1 u 1 4 u u + u 2 4 u u u + u 2 − 4G 1 u 231 1 u 2 − 4G 1 u 312 2 4 1 u u + u 1 u + u 1 3 2 u u 3 − 4G 1 u 312 3 4G 1 u 321 3 ! 3 " ! 1 1 2 " ! 1 " ! 1 1 " ! 1 1 2 " ! 1 2 1 2 " ! 2 2 " ! 3 " ! 2 2 3 2 3 " ! 2 3 " ! 3 " ! 3 " 3 1 − 1 1 1 1 1 −1 1 −1 − 3 1 1 3 1 1 1 −1 − 1 1 − 1 1 1− 1 − −1 1 1 1 1 −1 1 1− 1 1 G 0, , ; 1 H (0; u1) G 0, , ; 1 H (0; u1) , 0, v213;1 H (0; u1) , 0, v231;1 H (0; u1)+ G , 0, ; 1 H (0; u2) G , 0, ; 1 H (0; u2)+ , 0, v321;1 H (0; u2) , , v312;1 H (0; u2)+ G , 0, ; 1 H (0; u3)+ G , , ; 1 H (0; u3) , , v312;1 H (0; u3)+ , , v321;1 H (0; u3)+ 4 u1 u1 + u3 − 4 u2 u1 + u2 − 4G 1 u2 PHDepartment,THUnit,CERNCH-1211,Geneva23,Switzerland− 4G 1 u2 4 u2 u1 + u2 − 4 u2 u2 + u3 4G 1 u3 − 2G 1 u3 1 u3 4 u2 u2 + u3 4 u2 u3 u2 + u3 − 2G 1 u3 1 u3 2G 1 u3 1 u3 ! " ! " ! " ! " ! " ! " ! " ! " ! " ! " ! " ! " 1 1 1 1 u 1 1 1 −1 1 1− 1 1 1 1 1 1 1 1 1 1 1 −1 1 1− 1− 3 1 1 3 1 1 1 −1 − 1 1 − −1 G 0, , ; 1 H (0; u ) G 0, 1 − , ; 1 H (0; u )+ , , v ;1 H (0; u ) , , v ;1 H (0; u ) G , , ; 1 H (0; u )+ G , , ; 1 H (0; u )+ , , v ;1 H (0; u ) , u , 1; 1 H (0; u )+ G , 0, ; 1 H (0; u ) G , 0, ; 1 H (0; u )+ , v , 1; 1 H (0; u )+ , v , ; 1 H (0; u )+ 4 u u + u 1 − 4 u + u 1 1 u 1 2G 1 u 1 u 213 1 − 2G 1 u 1 u 231 1 − 4 u u u + u 2 2 u u u + u 2 2G 1 u 1 u 321 2 − 4G 1 u 312 2 4 u u + u 3 − 4 u u + u 3 4G 1 u 312 3 4G 1 u 312 1 u 3 ! 3 1 3 " ! 1 3 − − 3 " ! − 2 − 2 " ! − 2 − 2 " ! 2 1 1 2 " ! 2 2 1 2 " ! − 3 − 3 " ! − 3 " ! 3 1 3 " ! 3 2 3 " ! − 3 " ! − 3 − 3 " 1 u3 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 G 0, − , ; 1 H (0; u1) G , 0, ; 1 H (0; u1) , u231, 1; 1 H (0; u1)+ , u231, ; 1 H (0; u1)+ G , , ; 1 H (0; u2)+ G , , ; 1 H (0; u2) , u312, ; 1 H (0; u2)+ , u312, ; 1 H (0; u2) G , , ; 1 H (0; u3)+ G , , ; 1 H (0; u3)+ , v321, 1; 1 H (0; u3)+ , v321, ; 1 H (0; u3) 4 u2 + u3 1 1 u2 − 4 u1 u1 + u2 − 4G 1 u2 INFN,4G 1 Laboratoriu2 u1 Nazionali2 u2 Frascati,u2 u2 + u3 4 u 000442 u3 u2 + u3 Frascati− (Roma),4G 1 u3 u2 Italy4G 1 u3 1 u3 − 4 u3 u1 u1 + u3 4 u3 u2 u2 + u3 4G 1 u3 4G 1 u3 1 u3 − ! " ! " ! " ! " ! " ! " ! " ! " ! " ! " ! " ! " 3 1 1− − 1 1 1 1 1 −1 1 1− 1 1 1 1 1 1 u 1 1 −1 1 1− 1− 1 1 1 1 1 1 1 1 3 − 1 3 − 1 − G , 0, ; 1 H (0; u )+ G , , ; 1 H (0; u )+ , u , ; 1 H (0; u )+ , v , 0; 1 H (0; u )+ G , 1, ; 1 H (0; u )+ G , 1 − , 1;1 H (0; u ) , v , 0; 1 H (0; u ) , v , ; 1 H (0; u )+ G , , ; 1 H (0; u )+ G , , ; 1 H (0; u ) v , 1, ; 1 H (0; u ) v , , 1;1 H (0; u )+ 4 u u + u 1 2 u u u + u 1 4G 1 u 231 1 u 1 4G 1 u 213 1 4 1 u u 2 4 1 u u + u 1 2 − 4G 1 u 312 2 − 2G 1 u 312 1 u 2 2 u u u + u 3 2 u u u + u 3 − 4G 123 1 u 3 − 4G 123 1 u 3 ! 1 1 3 " ! 1 1 1 2 " ! 2 2 " ! 2 " ! 3 2 " ! 3 1 3 " ! 3 " ! 3 3 " ! 3 3 1 3 " ! 3 3 2 3 " ! 1 " ! 1 " 1 1 1 1 1 1 1 1 1 −1 −1 1 −1 1 −1 u 1 1 − 1 1− 1 1 −1 1 −1 −1 1 1 1 1 1 −1 1 −1 G , , ; 1 H (0; u )+ G , , ; 1 H (0; u )+ , v , ; 1 H (0; u ) , v , 0; 1 H (0; u ) G , 1 − , ; 1 H (0; u )+ G , 0, ; 1 H (0; u ) , v , 0; 1 H (0; u )+ , v , ; 1 H (0; u )+ 0, , v ;1 H (0; u )+ 0, , v ;1 H (0; u ) v , 1, ; 1 H (0; u )+ v , , 1;1 H (0; u ) 2 u u u + u 1 4 u u u + u 1 2G 1 u E-mail:213 1 u 1 − 4G [email protected] u 231 1 − 4 1 u u + u 1 1 u 2 2 u u 2 − 4G 1 u 321 2 2G 1 u 321 1 u 2 4G 1 u 123 3 4G 1 u 132 3 − 4G 132 1 u 3 4G 132 1 u 3 − ! 1 1 1 3 " ! 1 2 1 2 " ! 2 2 " ! 2 " ! 3 1 3 3 " ! 3 2 " ! 3 " ! 3 3 " ! 1 " ! 1 " ! 1 " ! 1 " 1 1 1 1 1 1 1 1 −1 −1 1 −1 1 1− 1 − − 1 1 1 1 3 − 1 3 − 1 − 1 −1 1 −1 1 −1 1 −1 G , , ; 1 H (0; u ) G , 1, ; 1 H (0; u )+ , v , ; 1 H (0; u )+ , 0, v ;1 H (0; u ) G , 0, ; 1 H (0; u )+ G , , ; 1 H (0; u )+ v , 1, ; 1 H (0; u )+ v , , 1;1 H (0; u ) 0, , v ;1 H (0; u )+ 0, , v ;1 H (0; u )+ v , 1, ; 1 H (0; u ) v , , 1;1 H (0; u )+ 4 u u u + u 1 − 4 1 u u 1 2G 1 u 231 1 u 1 4G 1 u 312 1 − 4 u u + u 2 4 u u u + u 2 4G 123 1 u 2 4G 123 1 u 2 − 4G 1 u 213 3 4G 1 u 231 3 4G 213 1 u 3 − 4G 213 1 u 3 ! 1 3 1 3 " ! − 2 1 " ! − 2 − 2 " ! − 3 " ! 3 2 3 " ! 3 2 2 3 " ! − 1 " ! − 1 " ! − 2 " ! − 2 " ! − 2 " ! − 2 " Claude Duhr –105– –106– –107– –108– –109– –110– 1 1 1 1 1 1 3 1 3 1 1 2 1 2 1 u1 + u3 1 1 u1 + u3 1 1 2 1 1 2 1 1 2 1 H (0; u ) 0, 1, 1; + H (0; u ) 0, 1, 1; H (0; u ) 0, 1, 1; 1 1 3 1 3 1 3 1 v231, 1, ; 1 H (0; u3)+ v231, , 1;1 H (0; u3)+ π H (0, 1; (u1 + u2)) + π H (0, 1; u3)+ H (0; u1) H (0; u2) H 0, 1; − H (0; u2) H 0, 0, 1; − H (0; u1) H (0, 0, 1; (u1 + u3)) π H (0; u3) 1; π H (0; u1) 1; + π H (0; u3) 1; 4 2 H u 4 2 H v − 4 3 H v − 1, 1, 0, 1; + 1, 1, 1, 1; + 1, 1, 1, 1; + 1, 1, 1, 1; 4G 1 u2 4G 1 u2 12 12 4 u1 1 − 2 u1 1 − − 24 H v132 − 24 H v213 24 H v213 − ! 312 " ! 123 " ! 123 " 4H v321 2H v123 2H v231 2H v312 ! − " ! − " ! − " ! − " ! " ! " ! " 1 1 1 1 1 1 ! " ! " ! " ! " 1 1 1 1 1 2 Instituteu1 + u3 1 1 2 for1 2 Particleu2 + u3 1 Physics Phenomenology,1 u2 + u3 1 University1 2 of Durham1 1 2 1 1 2 1 H (0; u ) 0, 1, 1; + H (0; u ) 0, 1, 1; + H (0; u ) 0, 1, 1; v312, 1, ; 1 H (0; u3)+ v312, , 1;1 H (0; u3)+ π H 0, 1; − + π H (0, 1; (u1 + u3)) π H 0, 1; − + H (0; u3) H (0, 0, 1; (u1 + u3)) H (0; u1) H 0, 0, 1; − π H (0; u1) 1; + π H (0; u3) 1; + π H (0; u1) 1; 4 2 H v 4 3 H v 4 1 H v − 4G 1 u3 4G 1 u3 24 u1 1 12 − 24 u3 1 − 2 u3 1 − 8 H v231 8 H v231 8 H v312 − ! 132 " ! 132 " ! 213 " References ! − " ! − " ! − " ! − " ! − " ! " ! " ! " 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 u2 + u3 1 1 2 1 1 2 1 1 2 1 H (0; u ) 0, 1, 1; H (0; u ) 0, 1, 1; + H (0; u ) 0, 1, 1; + v321, 1, ; 1 H (0; u3)+ v321, , 1;1 H (0; u3)+ π H (0, 1; (u2 + u3)) G 0, ; 1 H (1, 0; u1) H (0; u3) H 0, 0, 1; − H (0; u2) H (0, 0, 1; (u2 + u3)) π H (0; u2) 1; + π H (0; u1) 1; π H (0; u2) 1; 3 1 3 4G 1 u 4G 1 u 12 − 2 u + u − 2 u 1 − − 8 H v 24 H v − 24 H v − 4 H v213 − 4 H v231 4 H v231 [1] C. Anastasiou, Z. Bern, L. J. Dixon and D. A. Kosower, “Planar amplitudes in maximally ! 3 " ! 3 " ! 1 2 " ! 3 " ! 312 " ! 321 " ! 321 " ! " ! " ! " 1 1 1 − − 1 1 Durham,1 1 1 DH1 3LE, U.K. − 1 1 1 1 1 1 1 1 1 1 1 1 1 1 supersymmetric Yang-Mills theory,” Phys. Rev. Lett. 91 (2003) 251602 2 H (0; u ) 0, 1, 1; H (0; u ) 0, 1, 1; H (0; u ) 0, 1, 1; + G , ; 1 H (0; u1) H (0; u3)+ G 0, ; 1 H (1, 0; u1)+ G , ; 1 H (1, 0; u1)+ H (0; u3) H (0, 0, 1; (u2 + u3)) H (0; u2) H (0, 1, 0; u1) H (0; u3) H (0, 1, 0; u2) H (0; u2) H (0; u3) 0, 1; H (1, 0; u2) 0, 1; + π 0, 1; + 1 2 1 [arXiv:hep-th/0309040]. 4 u u + u 2 u + u 4 u u + u − 2 − 2 − 4 H u − 4 H u 24 H u 4 H v312 − 4 H v312 − 4 H v321 ! 1 1 3 " ! 1 3 " ! 1 1 2 " ! 123 " ! 123 " ! 123 " ! " ! " ! " 1 1 u 1 1 1 1 1 1 1 1 1 u1 + u2 1 1 1 1 1 1 1 1 1 1 1 1 1 3 H (0; u1) H (0, 1, 0; u3)+ H (0; u2) H 0, 1, 1; − 2 H (0; u ) 0, 1, 1; + H (0; u ) 1, 0, 1; + H (0; u ) 1, 0, 1; + [2] Z. Bern, J. S. Rozowsky and B. Yan, “Two-loop four-gluon amplitudes in N = 4 G , − ; 1 H (0; u1) H (0; u3)+ G , ; 1 H (1, 0; u1)+ G , ; 1 H (1, 0; u1)+ π 0, 1; H (0; u1) H (0; u3) 0, 1; H (1, 0; u3) 0, 1; 2 3 1 4 1 u u + u 1 4 u u + u 4 u u + u 2 4 u2 1 − 24 H u − 4 H u − 4 H u − 4 H v321 4 H u123 4 H u231 super-Yang-Mills,” Phys. Lett. B 401 (1997) 273 [arXiv:hep-ph/9702424]. ! 2 2 3 " ! 1 1 3 " ! 2 1 2 " ! − " ! 231 " ! 231 " ! 231 " ! " ! " ! " 1 1− 1 − 1 1 1 1 E-mail:u 1 1 [email protected] 1 1 u1 + u2 1 1 u1 + u3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 H (0; u3) H 0, 1, 1; − + H (0; u1) H 0, 1, 1; − 2 H (0; u ) 1, 0, 1; + H (0; u ) 1, 0, 1; H (0; u ) 1, 0, 1; G , ; 1 H (0; u1) H (0; u3) , u231;1 H (0; u1) H (0; u3) G , − ; 1 H (1, 0; u1)+ G , ; 1 H (1, 0; u1) H (0; u1) H (0; u2) 0, 1; H (1, 0; u1) 0, 1; + π 0, 1; 2 2 3 [3] Z. Bern, M. Czakon, D. A. Kosower, R. Roiban and V. A. Smirnov, “Two-loop iteration of 4 u u + u − 4G 1 u − 4 1 u u + u 1 4 u u + u − 4 u2 1 4 u1 1 − 4 H u − 4 H u 24 H u − 4 H u312 4 H v123 − 4 H v123 − ! 3 1 3 " ! 2 " ! 3 1 3 " ! 3 1 3 " ! − " ! − " ! 312 " ! 312 " ! 312 " ! " ! " ! " five-point N = 4 super-Yang-Mills amplitudes,” Phys. Rev. Lett. 97 (2006) 181601 1 1 1 −1 1 −1 − 3 3 1 u1 + u3 1 1 u2 + u3 1 1 1 1 1 1 1 1 1 1 1 , v ;1 H (0; u ) H (0; u ) , v ;1 H (0; u ) H (0; u )+ , u ;1 H (1, 0; u ) H (0, 0; u ) H (1, 0; u ) H (0, 0; u ) H (1, 0; u )+ H (0; u2) H 0, 1, 1; − H (0; u1) H 0, 1, 1; − + H (0; u ) H (0; u ) 0, 1; + H (0, 0; u ) 0, 1; + H (0; u2) 1, 0, 1; + H (0; u3) 1, 0, 1; + H (0; u1) 1, 0, 1; [arXiv:hep-th/0604074]. 213 1 3 231 1 3 312 1 2 1 3 1 4 u1 1 − 4 u3 1 2 3 2 4 H v 4 H v 4 H v − 4G 1 u2 − 4G 1 u2 4G 1 u3 − 4 − 4 ! − " ! − " 4 H v123 4 H v123 ! 132 " ! 132 " ! 213 " ! − " ! − " ! − " 1 u2 + u3 1 1 1 ! " ! " 1 1 1 1 1 1 [4] F. Cachazo, M. Spradlin and A. Volovich, “Iterative structure within the five-particle 5 1 1 u2 1 1 u1 + u3 1 1 1 1 1 1 1 1 1 1 π2H (0; u ) H (0; u )+ G , ; 1 H (0; u ) H (0; u )+ H 0, 1; H (1, 0; u ) π2H (1, 0; u ) G 0, ; 1 H (1, 0; u ) H (0; u3) H 0, 1, 1; − + H (0; u2) H (1, 0, 0; u1) H (0; u3) H (1, 0, 0; u1) H (0, 0; u ) 0, 1; + π2 0, 1; H (0; u ) H (0; u ) 0, 1; + H (0; u3) 1, 0, 1; H (0; u1) 1, 0, 1; + H (0; u3) 1, 0, 1; + 1 3 − 2 3 − 1 1 2 4 u3 1 2 − 2 − 3 2 3 4 H v − 4 H v 4 H v two-loop amplitude,” Phys. Rev. D 74, 045020 (2006) [arXiv:hep-th/0602228]. 24 4 1 u1 u1 + u2 1 4 u1 1 − 3 − 2 u1 + u2 − ! − " 4 H v123 6 H v123 − 4 H v132 ! 213 " ! 231 " ! 231 " ! − − " ! − " ! " 1 1 1 ! " ! " ! " 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 u2 1 H (0; u1) H (1, 0, 0; u2)+ H (0; u3) H (1, 0, 0; u2)+ H (0; u1) H (1, 0, 0; u3) 1 1 1 1 1 2 1 H (0; u ) 1, 0, 1; H (0; u ) 1, 0, 1; H (0; u ) 1, 0, 1; + [5] V. Del Duca, C. Duhr, E. W. Nigel Glover and V. A. Smirnov, “The one-loop pentagon to G , ; 1 H (0; u2) H (0; u3)+ G , ; 1 H (0; u2) H (0; u3) GVladimir0, ; 1 H (1, 0; u2)+ G A., − Smirnov; 1 H (1, 0; u2)+ 2 2 2 − H (0, 0; u2) 0, 1; + H (0, 0; u3) 0, 1; + π 0, 1; 1 2 1 4 H v312 − 4 H v312 − 4 H v321 1001 4 u2 u2 + u3 4 u3 u2 + u3 − 2 u2 + u3 4 1 u1 u1 + u2 1 4 H v132 4 H v132 6 H v132 − ! " ! " ! " higher orders in epsilon,” JHEP (2010) 042 [arXiv:0905.0097 [hep-th]]. ! " ! " ! " ! − − " 1 1 u1 + u2 1 ! " ! " ! " 1 1 1 1 1 1 1 1 1 1 1 1 1 1 H (0; u2) H (1, 0, 0; u3) H (0; u3) H 1, 0, 1; − 1 1 1 1 H (0; u ) 1, 0, 1; + H (0; u ) 1, 1, 1; H (0; u ) 1, 1, 1; , u123;1 H (0; u2) H (0; u3) , v123;1 H (0; u2) H (0; u3) G , ; 1 H (1, 0; u2)+ G , ; 1 H (1, 0; u2)+ 2 − 4 u2 1 − H (0; u1) H (0; u3) 0, 1; + H (0, 0; u1) 0, 1; + 2 2 3 [6] V. Del Duca, C. Duhr and E. W. Nigel Glover, “The five-gluon amplitude in the ! " 4 H v321 H v123 − H v123 − 4G 1 u1 − 4G 1 u1 − 4 u1 u1 + u2 4 u2 u1 + u2 1 u + u 1 1 − u + u 1 4 H v213 4 H v213 ! " ! " ! " high-energy limit,” JHEP 0912 (2009) 023 [arXiv:0905.0100 [hep-th]]. ! − " ! − " ! " ! " H (0; u ) H 1, 0, 1; 1 3 H (0; u ) H 1, 0, 1; 2 3 ! " ! " 1 1 1 1 1 5 2 1 1 1 1 1 1 2 − 1 − 1 1 1 2 1 1 1 H (0; u ) 1, 1, 1; + H (0; u ) 1, 1, 1; + H (0; u ) 1, 1, 1; , v132;1 H (0; u2) H (0; u3)+ π H (0; u2) H (0; u3)+ G , ; 1 H (1, 0; u2)+ G , ; 1 H (1, 0; u2) 4 u1 1 − 4 u3 1 − H (0, 0; u3) 0, 1; + π 0, 1; H (0; u1) H (0; u3) 0, 1; + 1 3 1 4G 1 u 24 4 u u + u 4 u u + u − ! − " ! − " 4 H v 6 H v − 4 H v H v231 H v231 H v312 − [7] L. F. Alday, J. M. Henn, J. Plefka and T. Schuster, “Scattering into the fifth dimension of 1 2 2 Nuclear3 3 Physics2 3 Institute of Moscow State3 Universityu1 + u2 1 213 213 231 ! " ! " ! " ! − " ! " ! " 7H (0, 0, 0, 0; u ) 7H (0, 0, 0, 0; u ) 7H (0, 0, 0, 0; u )+ H 0, 0, 0, 1; + ! " ! " ! " 1 3 1 3 1 N=4 super Yang-Mills,” JHEP 1001 (2010) 077 [arXiv:0908.0684 [hep-th]]. 3H (0; u2) H (0, 0; u1) H (0; u3)+3H (0; u1) H (0, 0; u2) H (0; u3)+ 1 1 3 3 1 2 3 − 1 1 1 1 1 2 1 H (0; u ) 1, 1, 1; 0, 0, 0, 1; 0, 0, 0, 1; , u123;1 H (1, 0; u2) H (0, 0; u1) H (1, 0; u2) H (0, 0; u3) H (1, 0; u2)+ − − 2 u2 1 H (0, 0; u1) 0, 1; + H (0, 0; u3) 0, 1; + π 0, 1; 2 1 u1 + u2 1 1 4G 1 u − 4 − 4 ! − " 4 H v 4 H v 6 H v − H v312 − 2H u123 − 2H u231 − [8] J. M. Henn, S. G. Naculich, H. J. Schnitzer and M. Spradlin, “Higgs-regularized three-loop H (0; u ) H 0, 1; − H (0; u )+ H (0; u ) H (0, 1; (u + u )) H (0; u )+ 1 3 u1 + u3 1 231 231 231 ! " ! " ! " 2 3 1 1 3 3 ! − " 3H (0, 0, 0, 1; (u + u )) + H 0, 0, 0, 1; + 3H (0, 0, 0, 1; (u + u )) + ! " ! " ! " 3 1 1 1 1 4 u2 1 2 1 u1 + u2 1 1 1 2 1 2 − 1 3 1 1 1 1 0, 0, 0, 1; 3 0, 0, 0, 1; 3 0, 0, 0, 1; 3 0, 0, 0, 1; four-gluon amplitude in N=4 SYM: exponentiation and Regge limits,” arXiv:1001.1358 ! − " H 0, 1; − H (1, 0; u2) H (1, 0; u1) H (1, 0; u2) π H (1, 0; u2) 2 u1 1 H (0; u1) H (0; u2) 0, 1; + H (0, 0; u1) 0, 1; + 1 u2 + u3 1 1 4 u 1 − 4 − 3 − ! − " 4 H v 4 H v 2H u312 − H v132 − H v213 − H v321 − [hep-th]. H (0; u ) H 0, 1; − H (0; u )+ H (0; u ) H (0, 1; (u + u )) H (0; u )+ Moscow2 119992, Russia 3 u2 + u3 1 9 312 312 ! " ! " ! " ! " 1 3 2 2 3 3 ! − " H 0, 0, 0, 1; + 3H (0, 0, 0, 1; (u + u )) + H (0, 0, 1, 0; u )+ ! " ! " 1 1 1 1 1 1 4 u3 1 2 1 1 1 1 − 2 3 1 1 1 1 2 1 1 1 0, 0, 1, 1; 0, 0, 1, 1; 0, 0, 1, 1; ! − " G 0, ; 1 H (1, 0; u3) G 0, ; 1 H (1, 0; u3)+ 2 u3 1 4 H (0, 0; u2) 0, 1; + π 0, 1; H (0; u1) H (0; u2) 0, 1; + [9] Z. Bern, L. J. Dixon and V. A. Smirnov, “Iteration of planar amplitudes in maximally 3 3 ! " 2H u123 − 2H u231 − 2H u312 − H (0; u ) H (1, 0; u ) H (0; u )+ H (0; u ) H (1, 0; u ) H (0; u )+ 2 u1 + u3 − 2 u2 + u3 9 9− 1 1 4 H v312 6 H v312 − 4 H v321 ! " ! " ! " supersymmetric Yang-Mills theory at three loops and beyond,” Phys. Rev. D 72 (2005) 4 2 1 3 4 1 2 3 ! " ! " H (0, 0, 1, 0; u )+ H (0, 0, 1, 0; u ) H (0, 1, 0, 0; u ) H (0, 1, 0, 0; u ) ! " ! " ! " 1 1 1 1 1 1 1 1 1 1 1 u3 1 4 2 4 3 − 2 1 − 2 2 − 1 1 1 1 1 2 1 0, 1, 0, 1; 0, 1, 0, 1; 0, 1, 0, 1; + 085001 [arXiv:hep-th/0505205]. 1 1 1 1 G , ; 1 H (1, 0; u3)+ G , − ; 1 H (1, 0; u3)+ H (0, 0; u1) 0, 1; + H (0, 0; u2) 0, 1; + π 0, 1; 2H u − 2H u − 2H u , v213;1 H (0, 0; u1)+ , v231;1 H (0, 0; u1)+ 4 u1 u1 +E-mail:u3 4 1 [email protected] u2 + u3 1 1 1 u1 + u2 1 1 u1 + u3 1 4 H v321 4 H v321 6 H v321 − ! 123 " ! 231 " ! 312 " ! " ! " ! " ! " ! " 4G 1 u2 4G 1 u2 1 1 1 1 − 1 − 1 1 H (0, 1, 0, 0; u3)+ H 0, 1, 0, 1; − + H 0, 1, 0, 1; − + 1 1 1 1 1 1 1 1 [10] Z. Bern, L. J. Dixon, D. A. Kosower, R. Roiban, M. Spradlin, C. Vergu and A. Volovich, ! − " ! − " 2 2 2 u2 1 2 u1 1 0, 1, 1, 1; + 0, 1, 1, 1; + ζ3H (0; u1)+ζ3H (0; u2)+ζ3H (0; u3)+ 1 1 1 1 23 2 G , ; 1 H (1, 0; u3) π H (1, 0; u3)+ G , ; 1 H (1, 0; u3)+ ! − " ! − " H (0; u2) H (0; u3) 1, 1; + H (0, 0; u2) 1, 1; + 4H v 4H v “The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills , v312;1 H (0, 0; u1)+ , v321;1 H (0, 0; u1) π H (0, 0; u1)+ 4 u2 u2 + u3 − 3 4 u3 u1 + u3 1 u2 + u3 1 2 H v123 2 H v123 ! 123 " ! 132 " 4G 1 u3 4G 1 u3 − 24 ! " ! " H 0, 1, 0, 1; − + H (0, 1, 1, 0; u1)+H (0, 1, 1, 0; u2)+H (0, 1, 1, 0; u3) ! " ! " 5 5 5 1 1 1 1 Theory,” Phys. Rev. D 78 (2008) 045007 [arXiv:0803.1465 [hep-th]]. ! − " ! − " 1 1 1 1 1 2 u3 1 − 1 1 11 2 1 1 2 1 ζ H (1; u )+ ζ H (1; u )+ ζ H (1; u )+ ζ 1; + ζ 1; + 1 1 1 1 G , ; 1 H (1, 0; u3) , u231;1 H (1, 0; u3)+ ! − " H (0, 0; u3) 1, 1; + π 1, 1; π 1, 1; 3 1 3 2 3 3 3 3 , v ;1 H (0, 0; u )+ , v ;1 H (0, 0; u )+ 4 u u + u − 4G 1 u 1 u + u 1 1 u + u 1 2 H v 24 H v − 24 H v − 2 2 2 2 H u123 2 H u231 [11] L. F. Alday and J. Maldacena, “Comments on gluon scattering amplitudes via AdS/CFT,” 123 2 132 2 ! 3 2 3 " ! 2 " H 0, 1, 1, 1; 1 2 H 0, 1, 1, 1; 1 3 ! 123 " ! 123 " ! 132 " ! " ! " 4G 1 u1 4G 1 u1 3 3 − 3 − − 1 1 1 1 1 1 1 1 1 1 1 1 1 1 JHEP 0711 (2007) 068 [arXiv:0710.1060 [hep-th]]. ! − " ! − " H (0; u ) H (0; u ) H (1, 0; u ) H (0, 0; u ) H (1, 0; u ) H (0, 0; u ) H (1, 0; u )+ 4 u2 1 − 4 u1 1 − 2 ζ3 1; 1, 0, 0, 1; 1, 0, 0, 1; 1, 0, 0, 1; + 1 1 1 1 1 2 3 1 3 2 3 ! − " ! − " π 1, 1; H (0; u1) H (0; u3) 1, 1; + H (0, 0; u1) 1, 1; + 2 H u − 2H u − 2H u − 2H u , v312;1 H (0, 0; u2)+ , v321;1 H (0, 0; u2) 4 − 4 − 4 1 u2 + u3 1 u1 + u2 1 u1 + u3 1 24 H v213 − 2 H v231 2 H v231 312 123 231 312 4G 1 u 4G 1 u − 1 u + u 1 1 1 H 0, 1, 1, 1; − + H 1, 0, 0, 1; − + H 1, 0, 0, 1; − + ! " ! " ! " ! " ! " ! " ! " [12] J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev, “The hexagon Wilson loop ! 3 " ! 3 " 2 3 4 u 1 u 1 u 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 25 − 23 − 1 1 H 0, 1; − H (1, 0; u3) H (1, 0; u1) H (1, 0; u3) H (1, 0; u2) H (1, 0; u3)+ 3 2 1 H (0, 0; u ) 1, 1; + π2 1, 1; H (0; u ) H (0; u ) 1, 1; + ζ3 1; + ζ3 1; + ζ3 1; + ζ3 1; + ζ3 1; + and the BDS ansatz for the six-gluon amplitude,” Phys. Lett. B 662 (2008) 456 2 4 u3 1 − 4 − 4 ! − " ! − " ! − " 3 1 2 4 H v 4 H v 4 H v 4 H v 4 H v H (0, 0; u1) H (0, 0; u2) π H (0, 0; u2)+ , v123;1 H (0, 0; u3)+ ! − " u2 + u3 1 2 H v231 24 H v231 − 2 H v312 ! 123 " ! 132 " ! 213 " ! 231 " ! 312 " [arXiv:0712.4138 [hep-th]]. 4 − 24 4G 1 u1 1 2 1 2 1 2 1 H 1, 0, 0, 1; − + 2H (1, 0, 1, 0; u1)+2H (1, 0, 1, 0; u2)+2H (1, 0, 1, 0; u3)+ ! " ! " ! " 1 1 1 1 1 1 1 1 ! − " π H (1, 1; u1)+ π H (1, 1; u2)+ π H (1, 1; u3)+ H (0; u2) H (0, 0, 0; u1)+ u3 1 1 1 1 1 11 2 1 ζ 1; + 0, 1, 1, 1; + 0, 1, 1, 1; + 0, 1, 1, 1; + 1 1 1 1 24 24 24 2 ! − " H (0, 0; u1) 1, 1; + H (0, 0; u2) 1, 1; + π 1, 1; 4 3H v 4H v 4H v 4H v [13] J. Bartels, L. N. Lipatov and A. Sabio Vera, “BFKL Pomeron, Reggeized gluons and , v132;1 H (0, 0; u3)+ , v213;1 H (0, 0; u3)+ Abstract: 1 u1 + u2 1 1 u1 + u3 1 2 H v312 2 H v312 24 H v312 − 321 213 231 312 1 1 1 u + u 1 ! " ! " ! " ! " 4G 1 u1 4G 1 u2 H (0; u ) H (0, 0, 0; u )+ H (0; u ) H (0, 0, 0; u ) InH (0; u ) H the0, 0, 1; 1 2 − planar H 1, 1, 0, 1=4supersymmetricYang-Millstheory,theconformalsym-; − + H 1, 1, 0, 1; − + ! " ! " ! " 1 1 1 1 1 1 1 1 Bern-Dixon-Smirnov amplitudes,” Phys. Rev. D 80 (2009) 045002 [arXiv:0802.2065 ! − " ! − " 3 2 1 3 2 4 u2 1 4 u1 1 1 2 1 1 1 1 1 0, 1, 1, 1; + 1, 0, 1, 1; + 1, 0, 1, 1; + 1, 0, 1, 1; + 1 1 2 2 − 2 u2 1 − ! − " ! − " π 1, 1; + H (0; u2) 0, 0, 1; + H (0; u3) 0, 0, 1; + [hep-th]]. , v ;1 H (0, 0; u )+3H (0; u ) H (0; u ) H (0, 0; u ) ! − " 1 u2 + u3 1 1 1 1 24 H v 2 H u 2 H u 4H v321 4H v123 4H v132 4H v213 231 3 1 2 3 1 u1 + u2 1 321 123 123 ! " ! " ! " ! " 10 4G 1 u2 − Del Duca,H 1, 1, 0, 1; − + H (1, 1 , 1, 0;Duhr,u1)+ H (1, 1, 1, 0; u2)+ H (1, 1, 1, 0; u3) Smirnov! " ! " (2010)! " 1 1 1 1 1 1 1 1 H (0; u ) H 0, 0, 1; H (0; u ) H (0, 0, 1; (u + u )) ! " 3 − 1 1[2 4 u3 1 2 2 2 − 1 1 1 1 1 1 ] [14] J. Bartels, L. N. Lipatov and A. Sabio Vera, “N=4 supersymmetric Yang Mills scattering − N 1, 0, 1, 1; + 1, 0, 1, 1; + 1, 0, 1, 1; + 1, 1, 0, 1; + 25 25 23 2 1 2 2 u2 1 − − ! " H (0; u1) 0, 0, 1; + H (0; u3) 0, 0, 1; + H (0; u1) 0, 0, 1; + ! " − 4H v231 4H v312 4H v321 4H v123 amplitudes at high energies: the Regge cut contribution,” arXiv:0807.0894 [hep-th]. H (0, 0; u1) H (0, 0; u3) H (0, 0; u2) H (0, 0; u3) π H (0, 0; u3)+ π H (0, 1; u1)+ − 1 u + u 1 1 2 1 1 2 1 1 2 1 2 H u231 2 H u231 2 H u312 ! " ! " ! " ! " 4 − 4 − 24 12 1 3 π H (0; u3) 1; π H (0; u1) 1; π H (0; u2) 1; + ! " ! " ! " H (0; u2) H (0, 0, 1; (u1 + u2)) H (0; u1) H 0, 0, 1; − 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 u1 + u2 1 1 metry constrains multi-loop24 nH -edgedu123 − 24 H Wilsonu231 − 24 H loopsu312 to be given in terms of the one-loop1, 1, 0, 1; + 1, 1, 0, 1; + 1, 1, 0, 1; + 1, 1, 0, 1; + [15] R. M. Schabinger, “The Imaginary Part of the N = 4 Super-Yang-Mills Two-Loop Six-Point − 2 u1 1 − ! " ! " ! " H (0; u2) 0, 0, 1; + H (0; u3) 0, 1, 1; + H (0; u1) 0, 1, 1; + π H (0, 1; u2) π H 0, 1; − + H (0; u1) H (0; u2) H (0, 1; (u1 + u2)) + ! − " 1 1 1 1 1 1 2 H u 4 H u 4 H u 4H v132 4H v213 4H v231 4H v312 12 − 24 u2 1 2 π2H (0; u ) 1; π2H (0; u ) 1; + π2H (0; u ) 1; ! 312 " ! 123 " ! 231 " ! " ! " ! " ! " MHV Amplitude in Multi-Regge Kinematics,” JHEP 0911 (2009) 108 [arXiv:0910.3933 ! − " 8 2 H v − 8 3 H v 24 2 H v − n-edged Wilson loop, augmented,! 123 " for! n123 " 6,! by132 " a function of conformally invariant cross [hep-th]]. –111– – 112 – –113– ≥ –114– –115– –116– ratios. That function is termed the remainder function. In a recent paper, we have dis- played the first analytic computation of the two-loop six-edged Wilson loop, and thus of the corresponding remainder function. Although the calculation was performed in the quasi- multi-Regge kinematics of a pair along the ladder, the Regge exactness of the six-edged

Keywords: QCD, MSYM, small x. What About After Integration? ✦ Integrate the Parke-Taylor 2-to-4 amplitude in sYM ‣ divergences exponentiate, leaving a ﬁnite remainder ✦ Heroically computed by Del Duca, Duhr, Smirnov in 2010, in terms of ‘Goncharov’ polylogarithms

Classical Polylogarithms for Amplitudes and Wilson Loops

A. B. Goncharov,1 M. Spradlin,2 C. Vergu,2 and A. Volovich2 1Department of Mathematics, Brown University, Box 1917, Providence, Rhode Island 02912, USA 2Department of Physics, Brown University, Box 1843, Providence, Rhode Island 02912, USA We present a compact analytic formula for the two-loop six-particle maximally helicity violating remainder function (equivalently, the two-loop lightlike hexagon Wilson loop) in N =4supersym- metric Yang-Mills theory in terms of the classical polylogarithm functions Lik with cross-ratios of momentum twistor invariants as their arguments. In deriving our formula we rely on results from the theory of motives. [Goncharov, Spradlin, Vergu, Volovich (2010)]

INTRODUCTION Wilson loop diagrams to obtain an analytic expression (2) for R6 as a 17-page linear combination of generalized The past few years have witnessed revolutionary ad- polylogarithm functions [16, 17] (see also [18]). vances in our understanding of the structure of scattering The motivation for the present work is the belief that amplitudes, especially in = 4 supersymmetric Yang- if SYM theory is really as beautiful and rich as recent Mills theory (SYM). It isN easy to argue that the seeds developments indicate, then there must exist a more en- (2) of modern progress were sown already in the 1980s with lightening way of expressing the remainder function R6 . the discovery of the Parke-Taylor formula for the sim- Ideally, like the Parke-Taylor formula at tree level, the ex- plest nontrivial amplitudes: tree-level maximally helicity pression should provide encouragement and guidance as violating (MHV) gluon scattering. The mere existence we seek deeper understanding of SYM at loop level. 10 (2) of such a simple formula for a quantity which otherwise We present our new formula for R6 in the next sec- would have been prohibitively difficult to calculate us- tion and then describe the algorithm by which it was ing traditional Feynman diagram methods signalled the obtained. tantalizing possibility that a great vista of unanticipated structure in scattering amplitudes awaited exploration. THE REMAINDER FUNCTION R(2) In contrast to the situation at tree level, it is fair to 6 say that recent progress at loop level has mostly been (2) evolutionary rather than revolutionary, driven primarily The remainder function R6 is usually presented as a by faster computers, improved algorithms (both analytic function of the three dual conformal cross-ratios and numeric), and software for multiloop calculations s12s45 s23s56 s34s61 u1 = ,u2 = ,u3 = , (1) which has been made publicly available. Yet we hope s123s345 s234s123 s345s234 that a great new vista of unexplored structure awaits us 2 also at loop level in SYM theory. of the momentum invariants si j =(ki + + kj ) , though we will see shortly that··· cross-ratios of··· momen- This paper is concerned with the planar two-loop six- tum twistor invariants are more natural variables. In particle MHV amplitude [1, 2], which in a sense is the terms of simplest nontrivial SYM loop amplitude. The known infrared and collinear behavior of general amplitudes, con- u1 + u2 + u3 1 √∆ xi± = uix±,x± = − ± , (2) arXiv:1006.5703v2 [hep-th] 7 Oct 2010 veniently encapsulated in the ABDK/BDS ansatz [3, 4], 2u1u2u3 determines the n-particle MHV amplitude at each loop 2 order L 2 up to an additive finite function of kinematic where ∆ =(u1 + u2 + u3 1) 4u1u2u3,wefind ≥ (L) − − invariants called the remainder function Rn . Given the 3 presumption of dual conformal invariance [5, 6] for SYM (2) + 1 R (u ,u ,u )= L (x ,x−) Li (1 1/ui) amplitudes (not yet proven, but supported by all avail- 6 1 2 3 4 i i − 2 4 − (L) i=1 " # able evidence [1, 3, 4, 7, 8]), Rn can depend on confor- ! 2 3 2 4 mal cross-ratios only. Since there are no cross-ratios for 1 1 4 π 2 π Li (1 1/ui) + J + J + . (3) (2) − 8 2 − 24 12 72 n =4, 5, the first nontrivial remainder function is R6 . $i=1 % (2) ! The same function R6 is also believed [9–12] to arise Here we use the functions as the expectation value of the two-loop lightlike hexagon Wilson loop in SYM theory [13, 14] (after appropriate + 1 + 4 L (x ,x−)= log(x x−) subtraction of ultraviolet divergences, e.g. [15]). Numer- 4 8!! ical agreement between the two remainder functions was 3 m ( 1) + m + established in [1, 14]. In a heroic effort, Del Duca, Duhr, + − log(x x−) ("4 m(x )+"4 m(x−)) (4) (2m)!! − − m=0 and Smirnov (DDS) explicitly evaluated the appropriate ! Amplitudes: a Virtuous Cycle

Compute Something beyond the reach of recent imagination

Exploit Simplicity Discover Simplicity to build more powerful computational technology beyond expectations

Understand Why study, understand, explain it, & explore its consequences

11 Constructing Integrands for Loop Amplitudes (constructively) Novel Representations of Integrands ✦ Powerful new tools now exist for understanding and computing integrands in perturbation theory Recursion Relations Q-cuts and Forward Limits

Prescriptive Unitarity Correlator Bootstraps

13 Novel Representations of Integrands ✦ Powerful new tools now exist for understanding and computing integrands in perturbation theory

Prescriptive Unitarity

13 The Cuts of Loop Amplitudes ✦ On-Shell Functions: scattering amplitudes, and functions built thereof—as networks of amplitudes

✦ Locality: amplitudes independent, so multiplied ✦ Unitarity: internal particles unseen, so summed

14 The Cuts of Loop Amplitudes ✦ On-Shell Functions: scattering amplitudes, and functions built thereof—as networks of amplitudes

✦ deﬁned for all all quantum ﬁeld theories— exclusively in terms of physical (observable) states ✦ can be used to reconstruct all loop amplitudes

14 General, Generalized Unitarity [Bern, Dixon, Kosower; Dunbar; …] ✦ Integrands are rational functions—so may be expanded into an arbitrary (but complete) basis:

L=2 n = f (2.27) A L XL where the ‘ladder’ integrands are chosen from our basis (2.19), constructed in the way described above, and each coecient f is a a single on-shell function: L

, , 2 8 9 L=2 <> => n = f (2.27) A L (2.28) :> ;> 8 9 L f > , , > X L 2 > > where the ‘ladder’ integrands are chosen from our basis (2.19), constructed<> in the => way described above, and each coecient f is a a single on-shell function: > > L > > Readers familiar with:> the earlier work in ref. [119] will notice that;> the representation of two loop amplitudes described here is considerably more compact15 (and , , arguably more straightforward). There are several reasons for this. 2 8 9 > The primary distinction> between the representation of two loop amplitudes in < (2.27)andthatdescribedinref.[= 119] is that here we have made no use of composite residues such as (2.16(2.28)). Because these residues are entirely responsible for :> ;> the infrared divergences of scattering amplitudes which are known to exponentiate 8 9 f > , , according to the BDS> ansatz described in ref. [152], it is well-motivated to make this L 2 > > <> exponentiation manifest=> in an integrand-level representation. This was achieved in the formulation described in ref. [119], but at the cost of distinguishing the terms > in (2.27)accordingtothepossiblemasslessnessoftheexternallegrangesofthe> > > Readers familiar with:> the earlier work in ref. [119]integrals, will notice and that using;> thedi↵erent repre- cuts/coecients for the di↵erent cases—namely, using sentation of two loop amplitudes described here is considerablycomposite more residues compact for the (and massless cases, and those similar to what we described arguably more straightforward). There are several reasonsabove for whenever this. composite residues would not exist. Our choice here to not make such distinctions is primarily pedagogical: breaking The primary distinction between the representation of two loop amplitudes in the basis of integrals into more cases requires a degree of unessential complication (2.27)andthatdescribedinref.[119] is that here we haveand a made longer discussion. no use of At com- three loops, choosing not to exploit composite residues posite residues such as (2.16). Because these residues areleads entirely to a considerably responsible more for compact formulation, but it is worth mentioning that the infrared divergences of scattering amplitudes which arewe have known been to unable exponentiate to make the exponentiation of infrared divergences manifest according to the BDS ansatz described in ref. [152], it is well-motivatedat three loops even to if make these this distinctions had been made. As such, it is not merely the interest of brevity that motivates our choice to ignore any possible composite exponentiation manifest in an integrand-level representation. This was achieved in residues that may exist. Reﬁning our representation of three loops to make the the formulation described in ref. [119], but at the cost ofexponentiation distinguishing of infrared the terms divergences manifest would be an interesting exercise, but in (2.27)accordingtothepossiblemasslessnessoftheexternallegrangesofthemust be left for future research. integrals, and using di↵erent cuts/coecients for the di↵erent cases—namely, using composite residues for the massless cases, and those similar to what we described above whenever composite residues would not exist. Our choice here to not make such distinctions is primarily pedagogical: breaking –20– the basis of integrals into more cases requires a degree of unessential complication and a longer discussion. At three loops, choosing not to exploit composite residues leads to a considerably more compact formulation, but it is worth mentioning that we have been unable to make the exponentiation of infrared divergences manifest at three loops even if these distinctions had been made. As such, it is not merely the interest of brevity that motivates our choice to ignore any possible composite residues that may exist. Reﬁning our representation of three loops to make the exponentiation of infrared divergences manifest would be an interesting exercise, but must be left for future research.

–20– General, Generalized Unitarity [Bern, Dixon, Kosower; Dunbar; …] ✦ Integrands are rational functions—so may be expanded into an arbitrary (but complete) basis:

3 Prescriptive Representation of All Three Loop Amplitudes Let us now apply the prescriptive approach to construct a closed-form representation ✦ Onceof all n an-point independent NkMHV amplitudes basis in planar is SYM chosen, at three loops. coefﬁcients Until now, the only arecases determined known (for arbitrary by multiplicity) (evaluations were the three/cuts loop on) MHV cuts integrands found in [9, 119]. In this section, we construct representations valid for all amplitudes,

L=3 n = f f , (3.1) A W + L XW XL where the integrals span all contact-term topologies of those drawn above, and the non-contact terms of each are fully determined to match specific field theory cuts f W and f . As indicated in (3.1), the possible integrands come in two principle topologies15 L which we will call ‘wheels’ and ‘ladders’, respectively. In the next subsection, we will demonstrate that this corresponds to a complete basis for three loop integrands, and we give a complete enumeration of the contact-term topologies (relative to what is drawn in (3.1)) that appear in our basis. In the following subsection we illustrate the cuts which define our basis (and the field theory coecients); complete details are provided in Appendix A. General aspects of (3.1)arediscussedinsection3.3. 3.1 Constructing a Diagonal Integrand Basis for Three Loop Integrals As outlined in section 2.4, the first step in applying prescriptive unitarity is to construct a complete basis of integrals (relevant to a particular quantum field theory). At two loops, we saw that all integrals (with the correct power counting for SYM) could be expanded into those involving at most four external propagators—generally, double-pentagon integrals and contact terms thereof. At three loops, the same rule applies: any integral involving more than four external propagators is expandable into those with fewer. For (single-loop-momentum) factors of integrands involving a single internal propagator, the argument is the same at two loops. New at three loops is the possibility that one loop involves two internal propagators. The fact that a heptagon involving five external and two internal propagators (with numerators spanning a 50-dimensional space according to (2.11)) can be decomposed into those involving at most four external propagators is similarly obvious (in terms of counting), and easy to verify by counting. See Table 2 for more general counting. This fact demonstrates that general integrands with the wheel topology (the first terms in (3.1)) can involve at most four external propagators per loop, and that the ladder integrals drawn in (3.1)areactuallyreducibleinto:

8 9 > , >. (3.2) > > ⇢<> =>

> > > > : –23– ; General, Generalized Unitarity [Bern, Dixon, Kosower; Dunbar; …] ✦ Integrands are rational functions—so may be expanded into an arbitrary (but complete) basis: