Color Dressed Unitarity and Recursion for Yang-Mills Two-Loop All-Plus Amplitudes
David C. Dunbar, John H. Godwin, Warren B. Perkins, Joseph M.W. Strong
aa r X i v : . [ h e p - ph ] N ov November 19, 2019
Color Dressed Unitarity and Recursion for Yang-Mills Two-LoopAll-Plus Amplitudes
David C. Dunbar, John H. Godwin, Warren B. Perkins and Joseph M.W. Strong
College of Science,Swansea University,Swansea, SA2 8PP, UKNovember 19, 2019
Abstract
We present a direct computation of the full color two-loop five-point all-plus Yang-Mills am-plitude using four dimensional unitarity and recursion. We present the SU ( N c ) amplitudes incompact analytic forms. Our results match the explicit expressions previously computed but donot require full two-loop integral methods. PACS numbers: 04.65.+e . INTRODUCTION Computing perturbative scattering amplitudes is not only a key tool in confronting the-ories of particle physics with experimental results but is also a gateway for exploring thesymmetries and properties of theories which are not always manifest in a Lagrangian ap-proach. Since the standard model of particle physics and many of its potential extensionsare gauge theories, gauge theory amplitudes are of particular interest. Within a Yang-Millsgauge theory a n -gluon amplitude in may be expanded in the gauge coupling constant, A n = g n − X ℓ ≥ a ℓ A ( ℓ ) n (1.1)where a = g e − γ E ǫ / (4 π ) − ǫ . Each loop amplitude can be further expanded in terms of colorstructures, C λ , A ( ℓ ) n = X λ A ( ℓ ) n : λ C λ , (1.2)separating the color and kinematics of the amplitude. The color structures C λ may beorganised in terms of powers of N c .A great deal of progress has been made in computing A ( ℓ ) n for tree amplitudes ( ℓ = 0)and one-loop amplitudes ( ℓ = 1) in SU ( N c ) gauge theory. However progress in two-loopamplitudes has been more modest: the four gluon amplitude has been computed [1, 2]for the full color and helicity structure and there is currently tremendous progress in thecomputation of the five-point amplitude. The first amplitude to be computed at five pointwas the leading in color part of the amplitude with all positive helicity external gluons (theall-plus amplitude) which was computed using d -dimensional unitarity methods [3, 4] andwas subsequently presented in a very elegant and compact form [5]. In [6], it was shown howfour-dimensional unitarity techniques could be used to regenerate the five-point leading incolor amplitude and in [7, 8] the leading in color all-plus amplitudes were obtained for six-and seven-points, these being the first six- and seven-point amplitudes to be obtained attwo-loops. The leading in color five-point amplitudes have been computed for the remaininghelicities [9, 10]. Full color amplitudes are significantly more complicated requiring a largerclass of master integrals incorporating non-planar integrals [11, 12]. In [13] the first fullcolor five-point amplitude was presented in QCD.In this article we will examine the one and two-loop partial amplitudes using a U ( N c )color trace basis where the fundamental objects are traces of color matrices T a rather thancontractions of the structure constants f abc . We examine the particular scattering amplitudein pure gauge theory where the external gluons have identical helicity, A n (1 + , · · · n + ). Thisamplitude is fully crossing symmetric which makes computation relatively more tractable butnonetheless is a valuable laboratory for studying the properties of gluon scattering. The all-plus amplitude has a singular structure which is known from general theorems together witha finite remainder part. We present a form for the finite part which is a simple combinationof dilogarithms together with rational terms. Specifically we compute directly all the colortrace structures for the five-point all-plus two-loop amplitude. Our results are in completeagreement with the results recently computed by Badger et. al. [13] and are consistent withconstraints imposed by group theoretical arguments [14, 15].Our methodology involves computing the polylogarithmic and rational parts of the finiteremainder by a combination of techniques. The polylogarithms are computed using four2imensional unitarity cuts and the rational parts are determined by recursion. We useaugmented recursion [16] to overcome the issues associated with the presence of doublepoles. II. ONE-LOOP SUB-LEADING AMPLITUDES An n -point tree amplitude can be expanded in a color trace basis as A (0) n (1 , , , · · · , n ) = X S n /Z n Tr[ T a · · · T a n ] A (0) n :1 ( a , a , · · · a n ) . (2.1)This separates the color and kinematic structures. The partial amplitudes A (0) n :1 ( a , a , · · · a n )are cyclically symmetric but not fully crossing symmetric. The sum over permutations is overpermutations of (1 , , · · · n ) up to this cyclic symmetry. This is not the only expansion andin fact other expansions exist [17] which may be more efficient for some purposes. This colordecomposition is valid for both U ( N c ) and SU ( N c ) gauge theories. If any of the externalparticles in the U ( N c ) case are U (1) particles then the amplitude must vanish. This imposes decoupling identities amongst the partial amplitudes [18]. For example at tree-level settingleg 1 to be U (1) and extracting the coefficient of Tr[ T T · · · T n ] implies that A (0) n :1 (1 , , , · · · n ) + A (0) n :1 (2 , , , · · · n ) + · · · A (0) n :1 (2 , · · · , , n ) = 0 . (2.2)This provides a consistency check on the partial amplitudes. At loop level these decouplingidentities provide powerful relationships between the different pieces of the amplitude.In a U ( N c ) gauge theory the one-loop n -point amplitude can be expanded as [18] A (1) n (1 , , , · · · , n ) = X S n /Z n N c Tr[ T a · · · T a n ] A (1) n :1 ( a , a , · · · a n )+ [ n/ X r =2 X S n / ( Z r − × Z n +1 − r ) Tr[ T a · · · T a r − ]Tr[ T b r · · · T b n ] A (1) n : r ( a · · · a r − ; b r · · · b n ) . (2.3)The A (1) n :2 are absent (or zero) in the SU ( N c ) case. For n even and r = n/ Z in the summation to ensure each color structure only appears once. The partialamplitudes A (1) n : r ( a · · · a r − ; b r · · · b n ) are cyclically symmetric in the sets { a · · · a r − } and { b r · · · b n } and obey a “flip” symmetry, A (1) n : r (1 , , · · · ( r − r, · · · n ) = ( − n A (1) n : r ( r − , · · · , n, · · · r ) . (2.4)Amplitudes involving the scattering of gauge bosons also occur in string theories. From astring theory viewpoint the A (1) n : r with r > U (1) and extracting the coefficient of Tr[ T T · · · T n ] implies A (1) n :2 (1; 2 , , · · · n ) + A (1) n :1 (1 , , , · · · n ) + A (1) n :1 (2 , , , · · · n ) + · · · A (1) n :1 (2 , · · · , , n ) = 0 (2.5)3nd consequently the A (1) n :2 can be expressed as a sum of ( n −
1) of the A (1) n :1 . By repeatedapplication of the decoupling identities all the A (1) n : r can be expressed as sums over the A (1) n :1 [18], A (1) n ; r (1 , , . . . , r − r, r + 1 , . . . , n ) = ( − r − X σ ∈ COP { α }{ β } A (1) n ;1 ( σ ) (2.6)where α i ∈ { α } ≡ { r − , r − , . . . , , } and β i ∈ { β } ≡ { r, r +1 , . . . , n − , n } [Note that theordering of the first set of indices is reversed with respect to the second]. COP { α }{ β } is theset of all permutations of { , , . . . , n } with n held fixed that preserve the cyclic ordering ofthe α i within { α } and of the β i within { β } , while allowing for all possible relative orderingsof the α i with respect to the β i . For example if { α } = { , } and { β } = { , , } , then COP { α }{ β } contains the twelve elements(2 , , , , , (2 , , , , , (2 , , , , , (3 , , , , , (3 , , , , , (3 , , , , , (1 , , , , , (1 , , , , , (1 , , , , , (3 , , , , , (3 , , , , , (3 , , , , . The simplest one-loop QCD n -gluon helicity amplitude is the all-plus amplitude with allexternal helicities positive. The tree amplitude vanishes for this particular amplitude andconsequently, the one-loop amplitude is rational (to order ǫ ). The leading in color one-looppartial amplitude has an all- n expression [19] A (1) n :1 (1 + , + , . . . , n + ) = − i h i h i · · · h n i X ≤ i 1] (2.8)and for r ≥ A (1) n : r (1 + , + , · · · , r − + ; r + · · · n + ) = − i ( K ··· r − ) ( h i h i · · · h ( r − 1) 1 i )( h r ( r + 1) i · · · h n r i ) . (2.9) Here a null momentum is represented as a pair of two component spinors p µ = σ µα ˙ α λ α ¯ λ ˙ α . We are usinga spinor helicity formalism with the usual spinor products h a b i = ǫ αβ λ αa λ βb and [ a b ] = − ǫ ˙ α ˙ β ¯ λ ˙ αa ¯ λ ˙ βb . Also s ab = ( k a + k b ) = h a b i [ b a ] = h a | b | a ], tr − [ ijkl ] ≡ tr( (1 − γ )2 /k i /k j /k k /k l ) = h i j i [ j k ] h k l i [ l i ],tr + [ ijkl ] ≡ tr( (1+ γ )2 /k i /k j /k k /k l ) = [ i j ] h j k i [ k l ] h l i i and ǫ ( i, j, k, l ) = tr + [ ijkl ] − tr − [ ijkl ]. A (1) n :1 grows as124 n ( n − n − n − 3) (2.10)while the summation over COP terms grows with n as ∼ ( n − r − n − r )! . (2.11)A further complication arises for one-loop amplitudes where the external helicities arenot identical, the simplest case being the single-minus amplitude with one negative helicitygluon and the rest positive helicity. Double poles arise in these amplitudes for complexmomenta where factorisations as in fig. 1 occur. FIG. 1: The origin of the double pole. The double pole corresponds to the coincidence of thesingularity arising in the 3-pt integral with the factorisation corresponding to K = s ab → The factorisation takes the form V (1) ( a + , b + , K + ) s ab × s ab × A (0) n − ( K − , · · · ) ∼ [ a b ] h a b i × A (0) n − ( K − , · · · ) (2.12)where V (1) ( a + , b + , K + ) s ab = − i a b ] [ b K ] [ K a ] s ab (2.13)is the one-loop three-point vertex [21]. For n > 4, the all-plus one-loop amplitude does notcontain double poles since the tree amplitude on the RHS of fig. 1 vanishes. The doublepoles in the single-minus amplitudes can be seen explicitly in the five-point case [22], A (1)5:1 (1 − , + , + , + , + ) = i h i (cid:20) − [2 5] [1 2] [5 1] + h i [4 5] h ih i h i h i − h i [3 2] h ih i h i h i (cid:21) (2.14)where there are h a b i − singularities for h a b i = h i , h i and h i .Again the sub-leading in color partial amplitudes can be obtained in terms of the lead-ing in color partial amplitudes using decoupling identities. The naive application of (2.6)5bscures the simplicity of the sub-leading terms. In particular, there are no double poles inthe one-loop sub-leading partial amplitudes for n > A (1) n :2 ( a ; b , b , b , · · · b n ). Thiscan be expressed as a sum over the A (1) n :1 , A (1) n :2 ( a ; b , b , , · · · b n ) = − A (1) n :1 ( a , b , b , · · · b n ) − A (1) n :1 ( b , a , b , · · · b n ) · · ·− A (1) n :1 ( b , b , · · · a , b n )(2.15)where the sum is over the n − a may be inserted within b , b · · · b n .If we consider the double pole in h a b i this will only occur in the first two terms and willbe of the form − V (1) ( a +1 , b +2 , K + ) s a b × A (0) n − ( K − , b , · · · b n ) − V (1) ( b +2 , a +1 , K + ) s a b × A (0) n − ( K − , b , · · · b n ) , (2.16)which vanishes since V (1) ( a + , b + , K + ) is antisymmetric. The double pole in h b b i alsovanishes, but via a different route. Only the second term in (2.15) does not contribute andwe obtain − V (1) ( b +2 , b +3 , K + ) s b b × (cid:18) A (0) n − ( a , K − , b , · · · b n )+ A (0) n − ( K − , a , b , · · · b n )+ · · · + A (0) n − ( K − , b , · · · a , b n ) (cid:19) . (2.17)This vanishes due to the decoupling identity for the tree amplitude A (0) n − (2.2). Similararguments show the vanishing of double poles for all A (1) n : r with r > n -point expressions. Explicitly, we can find all- n formulae for A (1) n :2 (1 − ; 2 + , · · · n + ) and A (1) n :3 (1 − , + ; 3 + , · · · n + ): A (1) n :2 (1 − ; 2 + , + , · · · , n + ) = − i P ≤ i A general two-loop amplitude may be expanded in a color trace basis as A (2) n (1 , , · · · , n ) = N c X S n /Z n Tr( T a T a · · · T a n ) A (2) n :1 ( a , a , · · · , a n )+ N c [ n/ X r =2 X S n / ( Z r − × Z n +1 − r ) Tr( T a T a · · · T a r − ) Tr( T b r · · · T b n ) A (2) n : r ( a , a , · · · , a r − ; b r · · · b n )+ [ n/ X s =1 [( n − s ) / X t = s X S n / ( Z s × Z t × Z n − s − t ) Tr( T a · · · T a s ) Tr( T b s +1 · · · T b s + t ) Tr( T c s + t +1 · · · T c n ) × A (2) n : s,t ( a , · · · , a s ; b s +1 · · · b s + t ; c s + t +1 · · · c n )+ X S n /Z n Tr( T a T a · · · T a n ) A (2) n :1 B ( a , a , · · · , a n ) . (3.1)Again, for n even and r = n/ Z in the summation to ensure eachcolor structure only appears once. In the s, t summations there is an extra Z when exactlytwo of s , t and n − s − t are equal and an extra S when all three are equal.For five-point amplitudes this reduces to A (2)5 (1 , , , , 5) = N c X S /Z Tr( T a T a T a T a T a ) A (2)5:1 ( a , a , a , a , a )+ N c X S /Z Tr( T a ) Tr( T b T b T b T b ) A (2)5:2 ( a ; b , b , b , b )+ N c X S / ( Z × Z ) Tr( T a T a ) Tr( T b T b T b ) A (2)5:3 ( a , a ; b , b , b )+ X S / ( Z × Z ) Tr( T a ) Tr( T b ) Tr( T c T c T c ) A (2)5:1 , ( a ; b ; c , c , c )+ X S / ( Z × Z × Z ) Tr( T a ) Tr( T b T b ) Tr( T c T c ) A (2)5:1 , ( a ; b , b ; c , c )+ X S /Z Tr( T a T a T a T a T a ) A (2)5:1 B ( a , a , a , a , a ) (3.2)which for an SU ( N c ) gauge group simplifies to A (2)5 (1 , , , , 5) = N c X S /Z Tr( T a T a T a T a T a ) A (2)5:1 ( a , a , a , a , a )+ N c X S / ( Z × Z ) Tr( T a T a ) Tr( T b T b T b ) A (2)5:3 ( a , a ; b , b , b )+ X S /Z Tr( T a T a T a T a T a ) A (2)5:1 B ( a , a , a , a , a ) . (3.3)Thus there are three independent functions to be determined: A (2)5:1 , A (2)5:3 and A (2)5:1 B . Bythemselves the U (1) decoupling identities do not determine any of the three, however they7an be used to obtain the specifically U ( N c ) functions A (2)5:2 , A (2)5:1 , and A (2)5:1 , : A (2)5:2 (1; 2 , , , 5) = − A (2)5:1 (1 , , , , − A (2)5:1 (2 , , , , − A (2)5:1 (2 , , , , − A (2)5:1 (2 , , , , ,A (2)5:1 , (4; 5; 1 , , 3) = − A (2)5:2 (5; 1 , , , − A (2)5:2 (5; 1 , , , − A (2)5:2 (5; 1 , , , − A (2)5:3 (4 , 5; 1 , , − A (2)5:3 (4 , 5; 1 , , 3) + X COP { , }{ , , } A (2)5:1 (1 , , , , A (2)5:1 , (1; 2 , 3; 4 , 5) = − A (2)5:3 (2 , 3; 1 , , − A (2)5:3 (2 , 3; 1 , , − A (2)5:3 (4 , 5; 1 , , − A (2)5:3 (4 , 5; 1 , , . (3.4)Decoupling identities do not relate the A (2) n :1 B to the other terms but do impose a tree-likeidentity, A (2) n :1 B (1 , , , · · · n ) + A (2) n :1 B (2 , , , · · · n ) + · · · A (2) n :1 B (2 , · · · , , n ) = 0 , (3.5)which in itself does not specify A (2) n :1 B completely. There are however further color restrictionsbeyond the decoupling identities [14, 15] which may be obtained by recursive methods.These, together with eq. (3.5) determine the A (2)5:1 B in terms of the A and A A (2)5:1 B (1 , , , , 5) = − A (2)5:1 (1 , , , , 5) + 2 A (2)5:1 (1 , , , , 4) + A (2)5:1 (1 , , , , − A (2)5:1 (1 , , , , 5) + 2 A (2)5:1 (1 , , , , − A (2)5:1 (1 , , , , − A (2)5:1 (1 , , , , 2) + 2 A (2)5:1 (1 , , , , 5) + A (2)5:1 (1 , , , , A (2)5:1 (1 , , , , 3) + A (2)5:1 (1 , , , , − X Z (1 , , , , (cid:18) A (2)5:3 (1 , 2; 3 , , − A (2)5:3 (1 , 3; 2 , , (cid:19) . (3.6)Our calculation we determine A (2)5:1 B directly and we use (3.6) as a consistency check. IV. SINGULARITY STRUCTURE OF THE ALL-PLUS TWO-LOOP AMPLI-TUDES The IR singular structure of a color partial amplitude is determined by general theo-rems [23]. Consequently we can split the amplitude into singular terms U (2) n : λ and finite terms F (2) n : λ , A (2) n : λ = U (2) n : λ + F (2) n : λ + O ( ǫ ) . (4.1)As the all-plus tree amplitude vanishes, U (2) n : λ simplifies considerably and is at worst 1 /ǫ . Ingeneral an amplitude has UV divergences, collinear IR divergences and soft IR divergences.As the tree amplitude vanishes, both the UV divergences and collinear IR divergences are8roportional to n and cancel leaving only the soft IR singular terms [24]. The leading case, U (2) n :1 , is proportional to the one-loop amplitude, U (2) n :1 = A (1) n :1 × I (2) n (4.2)where I (2) n = " − n X i =1 ǫ (cid:18) µ − s i,i +1 (cid:19) ǫ . (4.3)In appendix A the form of the two-loop IR divergences for the other un-renormalised partialamplitudes are presented in a color trace basis.Given the general expressions for U (2) n : λ , the challenge is to compute the finite parts of theamplitude: F (2) n : λ . This finite remainder function F (2) n : λ can be further split into polylogarithmicand rational pieces, F (2) n : λ = P (2) n : λ + R (2) n : λ . (4.4)We calculate the former piece using four-dimensional unitarity and the latter using recursion. V. UNITARITY D -dimensional unitarity techniques can be used to generate the integrands [3] for thefive-point amplitude which can then be integrated to obtain the amplitude [5]. Howeverthe organisation of the amplitude in the previous section allows us to obtain the finitepolylogarithms using four-dimensional unitarity [25, 26] where the cuts are evaluated infour dimension with the corresponding simplifications. With this simplification the all-plusone-loop amplitude effectively becomes an additional on-shell vertex and the two-loop cutseffectively become one-loop cuts with a single insertion of this vertex. The non-vanishingfour dimensional cuts are shown in fig. 2. a ) ++ ++ K k k K 4+ + − + − + − + − + I m e b ) ++ ++ ++ − + −− + I m c ) ++ + − + −− ++ − I m d ) ++ ++ −− ++ I FIG. 2: Four dimensional cuts of the two-loop all-plus amplitude involving an all-plus one-loopvertex (indicated by • ). In the boxes K may be null but K must contain at least two externallegs. The cuts allow us to determine the coefficients, α i , of box, triangle and bubble functionsin the amplitude. The integral functions are I ( K ) = ( − K ) − ǫ ǫ (1 − ǫ ) , (5.1)9 ( K ) = 1 ǫ ( − K ) − − ǫ , I (cid:0) K , K (cid:1) = 1 ǫ ( − K ) − ǫ − ( − K ) − ǫ ( − K ) − ( − K ) , (5.2)and I ( S, T, K , K ) = − ST − K K " − ǫ h ( − S ) − ǫ + ( − T ) − ǫ − ( − K ) − ǫ − ( − K ) − ǫ i + Li (cid:18) − K S (cid:19) + Li (cid:18) − K T (cid:19) + Li (cid:18) − K S (cid:19) + Li (cid:18) − K T (cid:19) − Li (cid:18) − K K ST (cid:19) + 12 ln (cid:18) ST (cid:19) (5.3)where S = ( k + K ) and T = ( k + K ) .The bubbles in principle would determine the ( − s ) ǫ /ǫ infinities. However, explicit calcula-tion using, for example, a canonical basis approach [27] shows that they have zero coefficient.This is a property of this particular helicity configuration and is due to the vanishing of thetree amplitude. The triangles only contribute to U (2) n : λ , while the box functions contribute toboth U (2) n : λ and the finite polylogarithms. Separating these pieces we have I me ( S, T, K , K ) = I me (cid:12)(cid:12)(cid:12)(cid:12) IR − ST − K K F m [ S, T, K , K ] (5.4)where F m is a dimensionless combination of polylogarithms.The IR terms combine to give the correct IR singularities [28], (cid:18)P α i I ,i (cid:12)(cid:12)(cid:12)(cid:12) IR + P α i I ,i + P α i I ,i (cid:19) λ = U (1) ,ǫ n : λ (1 + , + , · · · , n + ) (5.5)where U (2) ,ǫ n (1 + , + , · · · , n + ) is the order ǫ truncation. We have checked the relation of(5.5) by using four dimensional unitarity techniques to compute the coefficients and thencomparing to the expected form of U (2) n given in appendix A for n up to 9 points.The remaining parts of the box integral functions generate the finite polylogarithms. Theexpression for P (2) n is [28] P (2) n = X i c i F mi (5.6)whereF [ S, T, K , K ] = Li (cid:18) − K S (cid:19) + Li (cid:18) − K T (cid:19) + Li (cid:18) − K S (cid:19) + Li (cid:18) − K T (cid:19) − Li (cid:18) − K K ST (cid:19) + 12 ln (cid:18) ST (cid:19) (5.7)and, in the specific case where K = 0,F [ S, T, K ] ≡ F [ S, T, , K ]= Li (cid:18) − K S (cid:19) + Li (cid:18) − K T (cid:19) + 12 ln (cid:18) ST (cid:19) + π . (5.8)10et us now consider the specific five-point case where only the K = 0 case occurs. b + c + a + e + d + • ℓ , p ℓ , mℓ , nℓ , q + − + − + − + − FIG. 3: The labelling and internal helicities of the quadruple cut. In this case the one-loop corner is a four-point amplitude and the color partial amplitudessimplify since A (1)4:1 (1 , , , 4) = A (1)4:1 (1 , , , 3) = A (1)4:1 (1 , , , , (5.9)which implies that A (1)4:2 (1; 2 , , 4) = − A (1)4:1 (1 , , , 4) and A (1)4:3 (1 , 2; 3 , 4) = 6 A (1)4:1 (1 , , , 4) (5.10)so that the full color amplitude factorises into color and kinematic terms A (1)4 ( ℓ , ℓ , d, e ) = C × A (1)4:1 ( ℓ , d, e, ℓ ) . (5.11)Since the three-point tree amplitudes also factorise, the quadruple cut of this box functionfactorises as C ′ × A (0)3 ( a, ℓ , ℓ ) A (0)3 ( b, ℓ , ℓ ) A (0)3 ( c, ℓ , ℓ ) A (1)4:1 ( ℓ , d, e, ℓ ) . (5.12)The solution to the quadruple cuts in this case is ℓ = − h b a ih c a i ¯ λ b λ c , ℓ = − h b c ih a c i ¯ λ b λ a ,ℓ = h a c i ¯ λ a + h b c i ¯ λ b h a c i λ a , ℓ = h c a i ¯ λ c + h b a i ¯ λ b h c a i λ c . (5.13)So that A (0)3 ( a, ℓ , ℓ ) A (0)3 ( b, ℓ , ℓ ) A (0)3 ( c, ℓ , ℓ ) A (1)4:1 ( ℓ , d, e, ℓ ) = 2 i s ab s bc × [ d e ] h a b i h b c i h c a i = 13 s ab s bc × A (1)5:3 ( d, e ; a, b, c ) . (5.14)Consequently, P (2)5: λ ∼ X A (1)5:3 ( d, e ; a, b, c ) × F mabc ; de (5.15)11here F mabc ; de ≡ F m [ s ab , s bc , s de ]. We can determine the terms in the summation by expanding C ′ using U ( N c ) identities: C ′ ( de ; abc ) = X m,n,p,q (cid:16) (Tr[ amn ] − Tr[ man ])(Tr[ bpn ] − Tr[ pbn ])(Tr[ pcq ] − Tr[ pqc ]) (cid:17) × (cid:16) N c Tr[ mqed ] + N c Tr[ meqd ] / N c Tr[ qemd ] / N c Tr[ qmed ] − m ] Tr[ qde ] − q ] Tr[ mde ] − d ] Tr[ emq ] − d ] Tr[ eqm ]+ 3 Tr[ de ] Tr[ mq ] + 3 Tr[ dm ] Tr[ eq ] + 3 Tr[ dq ] Tr[ em ] + { d ↔ e } (cid:17) = N c (cid:16) Tr[ deabc ] + Tr[ edabc ] − Tr[ badec ] − Tr[ baedc ] (cid:17) + N c (cid:16) − d ](Tr[ eabc ] − Tr[ baec ]) − e ](Tr[ dabc ] − Tr[ badc ]) − Tr[ a ](Tr[ debc ] + Tr[ edbc ] − Tr[ bdec ] − Tr[ bedc ]) − Tr[ b ](Tr[ deac ] + Tr[ edac ] − Tr[ aedc ] − Tr[ adec ]) − Tr[ c ](Tr[ deab ] + Tr[ edab ] − Tr[ adeb ] − Tr[ aedb ])+8 Tr[ de ](Tr[ abc ] − Tr[ bac ]) + Tr[ da ](Tr[ bec ] − Tr[ ebc ])+ Tr[ db ](Tr[ aec ] − Tr[ eac ]) + Tr[ dc ](Tr[ aeb ] − Tr[ eab ]) − Tr[ ea ](Tr[ dbc ] − Tr[ bdc ]) − Tr[ eb ](Tr[ dac ] − Tr[ adc ]) − Tr[ ec ](Tr[ dab ] − Tr[ adb ]) (cid:17) +3 (cid:16) − d ] Tr[ e ](Tr[ abc ] − Tr[ bac ]) + Tr[ d ] Tr[ a ](Tr[ ebc ] − Tr[ bec ])+ Tr[ d ] Tr[ b ](Tr[ eac ] − Tr[ aec ]) + Tr[ d ] Tr[ c ](Tr[ eab ] − Tr[ aeb ])+ Tr[ e ] Tr[ a ](Tr[ dbc ] − Tr[ bdc ]) + Tr[ e ] Tr[ b ](Tr[ dac ] − Tr[ adc ])+ Tr[ e ] Tr[ c ](Tr[ dab ] − Tr[ adb ]) (cid:17) +6 (cid:16) Tr[ deabc ] − Tr[ dcbae ] + Tr[ dcbea ] − Tr[ daebc ] + Tr[ dceba ] − Tr[ dabec ] + Tr[ dcaeb ] − Tr[ dbeac ]+ Tr[ dbaec ] − Tr[ dceab ] + Tr[ dabce ] − Tr[ decba ] + Tr[ daecb ] − Tr[ dbcea ] + Tr[ dbeca ] − Tr[ daceb ] (cid:17) . (5.16)This is an expansion of the form C ′ ( de ; abc ) = X λ a λ ( de ; abc ) C λ (5.17)where the C λ are the different color structures. Consequently the polylogarithmic part ofthe partial amplitudes is P (2)5: λ = X ( de ; abc ) a λ ( de ; abc ) A ( d, e ; a, b, c ) × F mabc ; de . (5.18)Specifically we recover the previous results of [5] and [13]. Defining S = Z ( a, b, c, d, e ), S = Z ( b, c, d, e ) and S = Z ( a, b ) × Z ( c, d, e ) we have12 (2)5:1 ( a, b, c, d, e ) = X S − A (1)5:3 ( d, e ; a, b, c )F mabc ; de ,P (2)5:3 ( a, b : c, d, e ) = 43 X S (cid:16) A (1)5:3 ( a, b ; c, d, e ) F mcde,ab + 14 A (1)5:3 ( a, c ; b, e, d ) (F mbed ; ac + F mbde ; ac − F mdbe ; ac ) (cid:17) . (5.19)We also determine directly the remaining SU ( N c ) partial amplitude, P (2)5:1 B ( a, b, c, d, e ) = 2 X S (cid:16) A (1)5:3 ( a, b ; c, d, e ) F mcde ; ab + A (1)5:3 ( a, c ; b, e, d ) (F mbed ; ac + F mbde ; ac − F mdbe ; ac ) (cid:17) . (5.20)This expression matches that obtained by using the results of (5.19) in (3.6).The specifically U ( N c ) partial amplitudes may also be extracted directly: P (2)5:2 ( a ; b, c, d, e ) = − X S (cid:16) A (1)5:3 ( a, b ; c, d, e ) F mcde ; ab + 12 A (1)5:3 ( b, c ; a, e, d ) (F made ; bc + F mdea ; bc − F mdae ; bc ) (cid:17) (5.21)and P (2)5:1 , ( a ; b ; c, d, e ) = − X S (cid:16) A (1)5:3 ( a, b ; c, d, e ) F mcde ; ab + A (1)5:3 ( a, c ; b, e, d ) (F mbed ; ac + F mbde ; ac − F mdbe ; ac ) (cid:17) . (5.22)As a check we have confirmed that these satisfy the decoupling identities (3.4). VI. RECURSION The remaining part of the amplitude is the rational function R (2) n : λ . In [8] we described atechnique for evaluating this for the leading in color partial amplitude. We review this hereand describe the extensions necessary to determine the full-color amplitude.As R (2) n : λ is a rational function we can obtain it recursively given sufficient informationabout its singularities. Britto-Cachazo-Feng-Witten (BCFW) [29] exploited the analyticproperties of n -point tree amplitudes under a complex shift of their external momenta tocompute these amplitudes recursively. Explicitly the BCFW shift acting on two momenta,say p a and p b , is ¯ λ a → ¯ λ ˆ a = ¯ λ a − z ¯ λ b , λ b → λ ˆ b = λ b + zλ a . (6.1)This introduces a complex parameter, z , whilst preserving overall momentum conservationand keeping all external momenta null. Alternative shifts can also be employed, for examplethe Risager shift [30] which acts on three momenta, say p a , p b and p c , to give λ a → λ ˆ a = λ a + z [ b c ] λ η ,λ b → λ ˆ b = λ b + z [ c a ] λ η ,λ c → λ ˆ c = λ c + z [ a b ] λ η , (6.2)13here λ η must satisfy h a η i 6 = 0 etc., but is otherwise unconstrained.After applying either of these shifts, the rational quantity of interest is a complex functionparametrized by z i.e. R ( z ). If R ( z ) vanishes at large | z | , then Cauchy’s theorem appliedto R ( z ) /z over a contour at infinity implies R = R (0) = − X z j =0 Res h R ( z ) z i(cid:12)(cid:12)(cid:12) z j . (6.3)If the function only contains simple poles, Res (cid:2) R ( z ) /z (cid:3)(cid:12)(cid:12) z j = Res[ R ( z )] (cid:12)(cid:12) z j /z j and we canuse factorisation theorems to determine the residues. Higher order poles do not present aproblem mathematically, for example, given a function with a double pole at z = z j and itsLaurent expansion, R ( z ) = c − ( z − z j ) + c − ( z − z j ) + O (( z − z j ) ) , (6.4)the residue is simply Res h R ( z ) z i(cid:12)(cid:12)(cid:12) z j = − c − z j + c − z j . (6.5)To determine this we need to know both the leading and sub-leading poles. As discussedabove, loop amplitudes can contain double poles and, at this point, there are no generaltheorems determining the sub-leading pole.Both the BCFW and Risager shifts break cyclic symmetry of the amplitude by actingon specific legs and the Risager shift further introduces the arbitrary spinor η . While it ishard to determine a priori the large z behaviour of an unknown amplitude, recovering cyclicsymmetry (and η independence) are powerful checks. For the two-loop all-plus amplitudethis symmetry recovery does not occur for the BCFW shift (the one-loop all-plus amplitudeshave the same feature). However, symmetry is recovered if we employ the Risager shift (6.2).The Risager shift excites poles corresponding to tree:two-loop and one-loop:one-loop fac-torisations. The former involve only single poles and their contributions are readily obtainedfrom the rational parts of the four-point two-loop amplitude [2]: R (2)4:1 ( K + , c + , d + , e + ) = 13 A (1)4:1 ( K + , c + , d + , e + ) (cid:18) s ce s cd s de + 8 (cid:19) ,R (2)4:3 ( K + , c + ; d + , e + ) = 19 A (1)4:3 ( K + , c + ; d + , e + ) (cid:18) s cd s ce s de + s ce s cd s de + s de s cd s ce + 24 (cid:19) . The one-loop:one-loop factorisations involve double poles and we need to determine thesub-leading pieces. By considering a diagram of the form fig. 4 using an axial gauge for-malism, we can determine the full pole structure of the rational piece, including the non-factorising simple poles. We have used this approach previously to compute one-loop [31–33]and two-loop amplitudes [6–8], we labelled this process augmented recursion . In axial gaugeformalism helicity labels can be assigned to internal off-shell legs and vertices expressed interms of nullified momenta [34, 35], K ♭ = K − K K.q q (6.6)14 IG. 4: Diagram containing the leading and sub-leading poles as s ab → 0. The axial gaugeconstruction permits the off-shell continuation of the internal legs. where q is a reference momentum. The two off-shell legs are, α = α ( ℓ ) = ℓ + a and β = β ( l ) = b − ℓ , (6.7)where we also define the sum of these legs, P αβ = α + β = a + b = P ab , which is independentof the loop momentum, ℓ .The principal helicity assignment in fig. 4, gives Z d Λ c ( α + , a + , b + , β − ) τ (1) ,cn ( α − , β + , c + , ..., n + ) (6.8)where Z d Λ c ( α + , a + , b + , β − ) ≡ ic Γ (2 π ) D Z d D ℓℓ α β V ( α, a, ℓ ) V ( ℓ, b, β ) , (6.9)the vertices are in axial gauge and τ (1) ,cn is a doubly off-shell current.As we are only interested in the residue on the s ab → s αβ → α , β = 0,(C2) The current is the one-loop, single-minus amplitude in the on-shell limit α , β → s αβ = 0.This process is detailed in [8].We now apply the method to the full color amplitude. The U ( N c ) color decompositionof d Λ c contains a common kinematic factor so we have the color decompositions τ (1) ,cn = X λ C λ τ (1) n : λ and Z d Λ c = C Λ Z d Λ (6.10)where Z d Λ ( α + , a + , b + , β − ) = ic Γ (2 π ) D Z d D ℓℓ α β [ a | ℓ | q i [ b | ℓ | q ih a q i h b q i h β q i h α q i . (6.11)Hence the full color contribution is X λ C Λ C λ Z d Λ ( α + , a + , b + , β − ) τ (1) n : λ ( α − , β + , c + , · · · n + ) . (6.12)15he various τ (1) n : λ can be expressed as sums of the leading amplitudes τ (1) n :1 via a series of U (1) decoupling identities.We now focus on the five-point case, where there are two distinct forms of the leadingcurrent, τ (1)5:1 ( α − , β + , c + , d + , e + ) and τ (1)5:1 ( α − , c + , β + , d + , e + ) , (6.13)which we call the ’adjacent’ and ’non-adjacent’ leading currents respectively. τ (1)5:1 ( α − , β + , c + , d + , e + ) has been calculated previously for a specific choice of the axialgauge spinor λ q = λ d [6]. Since we require currents for which all the legs have been per-muted it is necessary to derive this current for arbitrary λ q . The non-adjacent case has notpreviously been considered. The derivation of the adjacent current is given in Appendix B.This current is given by τ (1)5:1 ( α − , β + , c + , d + , e + ) = F cdedp (cid:20) s αβ (cid:18) [ q e ][ c e ] [ q | P αβ | c i + [ c | q | d i [ q | P αβ | q i [ c | e | d i + [ e | q | d i [ q | P αβ | q i [ e | c | d i (cid:19)(cid:21) + i h c d i h α q i h β q i (cid:20) h α c i [ c | β | d ih d e i h e α i + h c e i [ d e ] h d e i (cid:18) [ q | P αβ | d i [ q | P αβ | q i h q c i h q α i [ q | α | q ih α c i [ q | P αβ | c i − h q d i [ q | P αβ | d i [ q | β | q i [ q | P αβ | q i [ q | P αβ | c i (cid:19)(cid:21) + F cdesb + i h c d i − [ β e ] [ q e ][ e α ] [ α q ] + [ e | q | α i ([ e β ] [ β q ] [ q | P αβ | q i − [ β q ] [ e | P αβ | q i )[ α q ] [ q | P αβ | q i ! + O ( h α β i ) + O ( α ) + O ( β ) (6.14)where F cdedp = i h α q i h β q i h q | αβ | q i s αβ h e c i [ c e ] h c d i h d e i [ e | P αβ | q i [ c | P αβ | q i (6.15)and F cdesb = − i e | P αβ | α i [ β q ] [ α q ] [ q | P αβ | q i s αβ [ e | P αβ | q ih c d i . (6.16)Setting λ q = λ d in (6.14) reproduces the current presented in [6].The non-adjacent leading current is τ (1)5:1 ( α − , c + , β + , d + , e + ) = i h α q i h β q i h α e i [ e c ] h c α i h d e i − [ e c ] [ e | α | d i [ c | α | d i ! + O ( h α β i ) . (6.17)These currents must be integrated before extracting the rational pole. The non-adjacentcase integrates to the simple form, Z d D ℓℓ α β i a | ℓ | q i [ b | ℓ | q ih a q i h b q i h a e i [ e c ] h c a i h d e i = i e c ] h a e i [ a b ] h d e i h c a i h a b i . (6.18)where the second term in eq. (6.17) has been dropped since it is a quadratic pentagon anddoes not contain any rational terms. The integrated adjacent case is a generalisation of the16revious result [6]. Summing over all the channels excited by the Risager shift we recover thefull two-loop color decomposition. We present compact forms of the SU ( N c ) rational piecesbelow, including the first compact form for the rational piece of the maximally non-planaramplitude obtained via a direct computation. We find complete agreement with previouscalculations [13] and R (2)5:1 B satisfies the constraint (3.6). R (2)5:1 ( a + , b + , c + , d + , e + ) = i h a b i h b c i h c d i h d e i h e a i X S (cid:16) tr [ deab ] s de s ab + 5 s ab s bc + s ab s cd (cid:17) , (6.19) R (2)5:3 ( a + , b + ; c + , d + , e + ) = 2 i h a b i h b a i h c d i h d e i h e c i X S (cid:18) tr − [ acde ]tr − [ ecba ] s ae s cd + 32 s ab (cid:19) (6.20)and R (2)5:1 B ( a + , b + , c + , d + , e + ) = 2 iǫ ( a, b, c, d ) (cid:16) C PT ( a, b, e, c, d ) + C PT ( a, d, b, c, e )+ C PT ( b, c, a, d, e ) + C PT ( a, b, d, e, c ) + C PT ( a, c, d, b, e ) (cid:17) (6.21)where C PT ( a, b, c, d, e ) = 1 h a b i h b c i h c d i h d e i h e a i . (6.22)These expressions are valid for both U ( N c ) and SU ( N c ) gauge groups and are remarkablycompact.We note that there are double poles at leading and sub-leading in color, but not at sub-sub-leading. As R (2)4:1 B vanishes [2] the poles in R (2)5:1 B do not correspond to tree:two-loopfactorisations, instead they arise from contributions of the type shown in fig.4 where thecorresponding current has no pole in s ab . VII. CONCLUSIONS Computing perturbative gauge theory amplitudes to high orders is an important butdifficult task. In this article, we have demonstrated how the full color all-plus five-pointamplitude may be computed in simple forms. We have computed all the color componentsdirectly and only used color relations between them as checks. In passing, we have givensimple all- n expressions for the one-loop subleading in color amplitudes and presented the n -point IR divergences in a color basis approach. Our methodology obtains these resultswithout the need to determine two-loop non-planar integrals. VIII. ACKNOWLEDGEMENTS DCD was supported by STFC grant ST/L000369/1. JMWS was supported by STFCgrant ST/S505778/1. JHG was supported by the College of Science (CoS) Doctoral TrainingCentre (DTC) at Swansea University. 17 ppendix A: Infra-Red Divergences The singular behaviour of two-loop gluon scattering amplitudes is known from a generalanalysis [23]. The leading IR singularity for the n -point two-loop amplitude is [36] − s − ǫab ǫ f aij f bik × A (1) n ( j, k, · · · , n ) (A1)where A (1) n is the full-color one-loop amplitude. We wish to disentangle this simple equationinto the color-ordered partial amplitudes. It will be convenient to use a more list basednotation for the partial amplitudes where we use A ( l ) n ( S ) = A ( l ) n ( { a , a , · · · a n } ) ≡ A ( l ) n :1 ( a , a , · · · a n ) , (A2) A ( l ) n ( S ; S ) for A ( l ) n : r and A ( l ) n ( S ; S ; S ) for A ( l ) n : s,t .First we define I i,j ≡ − ( s ij ) − ǫ ǫ (A3)and we have for a list S = { a , a , a , · · · , a r } , I r [ S ] = r X i =1 I a i ,a i +1 (A4)where the term I a r ,a r +1 ≡ I a r ,a is included in the sum. We also define I j [ S , S ] and I k [ S , S ], I j [ S , S ] = I j [ { a , a · · · a r } , { b , b , · · · b s } ] ≡ ( I a ,a r + I b ,b s − I a ,b − I a r ,b s ) ,I k [ S , S ] = I k [ { a , a · · · a r } , { b , b , · · · b s } ] ≡ ( I a ,b s + I b ,a r − I a ,b − I a r ,b s ) (A5)giving I r [ S ⊕ S ] = I r [ S ] + I r [ S ] + I k [ S , S ] − I j [ S , S ] (A6)where { a · · · a r } ⊕ { b · · · b s } = { a · · · a r , b · · · b s } . In this language the leading and sub-leading IR singularities at one-loop are A (1) n ( S ) = A (0) n ( S ) × I r [ S ] ,A (1) n ( S ; S ) = X S ′ ∈ C ( S ) X S ′ ∈ C ( S ) A (0) n ( S ′ ⊕ S ′ ) × I j [ S ′ , S ′ ] . (A7)The set C ( S ) is the set of cyclic permutations of S .18t two-loops, we have A (2) n ( S ) = A (1) n ( S ) × I r [ S ] ,A (2) n ( S ; S ) = A (1) n ( S ; S ) × ( I r [ S ] + I r [ S ])+ X S ′ ∈ C ( S ) X S ′ ∈ C ( S ) A (1) n ( S ′ ⊕ S ′ ) × I j [ S ′ , S ′ ] ,A (2) n ( S ; S ; S ) = X S ′ ∈ C ( S ) X S ′ ∈ C ( S ) A (1) n ( S ; S ′ ⊕ S ′ ) × I j [ S ′ , S ′ ]+ X S ′ ∈ C ( S ) X S ′ ∈ C ( S ) A (1) n ( S ; S ′ ⊕ S ′ ) × I j [ S ′ , S ′ ]+ X S ′ ∈ C ( S ) X S ′ ∈ C ( S ) A (1) n ( S ; S ′ ⊕ S ′ ) × I j [ S ′ , S ′ ] ,A (2) n,B ( S ) = X U ( S ) A (1) n ( S ′ ; S ′ ) × I k [ S ′ , S ′ ] , (A8)where U ( S ) is the set of all distinct pairs of lists satisfying S ′ ⊕ S ′ ∈ C ( S ) where the sizeof S ′ i is greater than one. For example U ( { , , , , } ) = (cid:26) ( { , } , { , , } ) , ( { , } , { , , } ) , ( { , } , { , , } ) , ( { , } , { , , } ) , ( { , } , { , , } ) (cid:27) . (A9) Appendix B: Obtaining the adjacent current We build the rational part of the full color five-point amplitude recursively, using aug-mented recursion to determine the sub-leading poles arising in the one-loop:one-loop factori-sations. For this we need an approximation to the doubly massive current τ (1)5 ( α, β, c, d, e )shown in fig.4. As we are only interested in the residue on the s ab → s αβ → α , β = 0,(C2) The current is the one-loop, single-minus amplitude in the on-shell limit α , β → s αβ = 0.Condition (C2) requires the current τ (1)5: λ to reproduce the full partial amplitude A (1)5: λ inthe α → β → τ (1)5:1 . The cyclic and flip symmetries inherited from A (1)5:1 mean that any of the τ (1)5: λ can be related to τ (1)5:1 ( α, β, c, d, e ) and τ (1)5:1 ( α, c, β, d, e ) up topermutations of the legs { c, d, e } .To build the current we start with the one-loop, five-point, single-minus partial amplitude A (1)5:1 ( α − , β + , c + , d + , e + ) = X j = i,ii,iii A (1)5:1 j (cid:0) α − , β + , c + , d + , e + (cid:1) (B1)19here A (1)5:1 i (cid:0) α − , β + , c + , d + , e + (cid:1) = i h c d i h c e i h α d i [ d e ] h α β i h d e i h β c i ,A (1)5:1 ii (cid:0) α − , β + , c + , d + , e + (cid:1) = − i h c d i [ β e ] [ α β ] [ e α ]and A (1)5:1 iii (cid:0) α − , β + , c + , d + , e + (cid:1) = i h c d i h α c i h β d i [ β c ] h d e i h α e i h β c i . + − − + c + d + e + α − β + A (1)4:1 FIG. 5: Factorisations of the current on the s αβ → Condition (C1) requires our approximation to the current to reproduce the correct leadingsingularities as s αβ → 0, the sources of these are depicted in fig.5 [6]. We determine thesewithin the axial gauge formalism. The two channels give F cdedp ≡ [ β k ] h α q i h β q i h k q i s αβ A (1)4:1 ( k − , c + , d + , e + ) = i h α q i h β q i h q | αβ | q i s αβ h e c i [ c e ] h c d i h d e i [ e | P αβ | q i [ c | P αβ | q i and F cdesb ≡ h α k i [ β q ] [ α q ] [ k q ] 1 s αβ A (1)4:1 ( k + , c + , d + , e + ) = − i h α k i [ β q ] [ α q ] [ k q ] 1 s αβ [ e k ] h c d i (B2)where k = α + β = − c − d − e which is null on the pole.Using the identity1 h α β i h β c i = 1 h α q i h β q i (cid:18) h q | αβ | q i [ q | P αβ | q i s αβ [ q | P αβ | c i + h q β i h q c i [ q | α | q ih β c i [ q | P αβ | c i (cid:19) (B3)20nd the expansion[ β | P ♭αβ | d i [ β | P αβ | q i = [ q | P αβ | d i [ q | P αβ | q i + s αβ h q d i [ β q ][ β | P αβ | q i [ q | P αβ | q i + O ( s αβ ) (B4)we find A (1)5:1 i = F cdedp (cid:20) s αβ (cid:18) [ q e ][ c e ] [ q | P αβ | c i + [ c | q | d i [ q | P αβ | q i [ c | e | d i + [ e | q | d i [ q | P αβ | q i [ e | c | d i (cid:19) + O ( s αβ ) (cid:21) . (B5)We see that A (1)5:1 i generates the correct singularity as h α β i → 0. This terms generatesthe double pole when integrated and the form in (B5) explicitly exposes the subleadingcontribution.The F cdesb factorisation arises when [ α β ] → 0. This we obtain from A (1)5:1 ii . Using, k ♭ = k − k k.q q = α ♭ + β ♭ + δq, (B6)where δ = α α.q + β β.q − s αβ k.q , (B7)we have F cdesb = i s αβ " [ e β ] [ β q ] h β α i [ α q ] h c d i + δ [ e | q | α i ([ e β ] [ β q ] [ k q ] + [ β q ] [ e k ])[ α q ] [ k q ] h c d i . (B8)Now A (1)5:1 ii can be rewritten as A (1)5:1 ii = − i h c d i [ β e ] [ q e ][ e α ] [ α q ] − i h c d i [ β e ] [ q β ][ α β ] [ α q ] (B9)and noting that h β α i s αβ − α β ] = h β α i [ α β ] − s αβ s αβ [ α β ] = ( α ♭ + β ♭ ) − s αβ s αβ [ α β ] = − (cid:18) α α.q + β β.q (cid:19) k.qs αβ [ α β ] , (B10)we see that A (1)5:1 ii has the form F cdesb +∆ α +∆ β +∆ s αβ as [ α β ] → 0, where ∆ α is proportionalto α /s αβ etc.. As ∆ s αβ does not contribute on the pole, we don’t have to replicate it in thecurrent and therefore include a contribution to the current of the form A (1)5:1 ii − ∆ α − ∆ β tosatisfy condition (C1). This does not compromise condition (C2). Upon integration the α and β factors in ∆ α and ∆ β generate s ab factors which cancel the pole. We therefore donot require these forms explicitly. For the purposes of integration it is convenient to expressthe term with the [ α β ] pole in terms of F cdesb . To maintain condition (C2) we must retain∆ s αβ .The remaining piece of the one-loop amplitude, A (1)5:1 iii , contains no poles as h α β i → α β ] → X α ih Y α i = h X α ih Y α i h Y a ih Y a i = h X a ih Y a i + O ( h α a i ) (B11)as terms of O ( h α a i ) do not ultimately contribute to the residue.The adjacent leading current is then τ (1)5:1 ( α − , β + , c + , d + , e + ) = F cdedp (cid:20) s αβ (cid:18) [ q e ][ c e ] [ q | P αβ | c i + [ c | q | d i [ q | P αβ | q i [ c | e | d i + [ e | q | d i [ q | P αβ | q i [ e | c | d i (cid:19)(cid:21) + i h c d i h α q i h β q i (cid:20) h a c i [ c | β | d ih d e i h e a i + h c e i [ d e ] h d e i (cid:18) [ q | P αβ | d i [ q | P αβ | q i h q c i h q a i [ q | α | q ih a c i [ q | P αβ | c i − h q d i [ q | P αβ | d i [ q | β | q i [ q | P αβ | q i [ q | P αβ | c i (cid:19)(cid:21) + F cdesb + i h c d i − [ β e ] [ q e ][ e α ] [ α q ] + [ e | q | α i ([ e β ] [ β q ] [ k q ] + [ β q ] [ e k ])[ α q ] [ k q ] 2 k.q ! + O ( h α β i ) + O ( α ) + O ( β ) . 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